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Az eacuteszreveacuteteleket hibaacutekat keacuterjuumlk a csabahoshdsbmehu ciacutemre kuumlldeni

Feladatgyűjtemeacuteny az

Aacuteramlaacutestechnikai geacutepek

taacutergyhoz

BME Hidrodinamikai Rendszerek Tanszeacutek

copy Dr Kullmann Laacuteszloacute Dr Hős Csaba

Dr Vaacuteradi Saacutendor Varga Roxaacutena Csizmadia Peacuteter Szaboacute Viktor eacutes Hajgatoacute Gergely koumlzreműkoumldeacuteseacutevel

wwwhdsbmehu

Budapest 2011 januaacuter 31

Utolsoacute moacutedosiacutetaacutes 2016 juacutenius 13

Tartalom

1 Gaacutezok sűriacuteteacutese kompresszorok munkafolyamatai 2 2 Euler-turbinaegyenlet sebesseacutegi haacuteromszoumlg 7 3 Teljesiacutetmeacutenyek veszteseacutegek hataacutesfokok 8 4 Dimenzioacutetlan uumlzemi mennyiseacutegek jellemző fordulatszaacutem affinitaacutes 11 5 Csővezeteacutek jelleggoumlrbe munkapont az uumlzem stabilitaacutesa 15 6 Soros paacuterhuzamos uumlzem 21 7 Szivattyuacute vezeacuterleacutesi moacutedok 25 8 Allievi-elmeacutelet hirtelen zaacuteraacutes 28 9 Axiaacutelis radiaacutelis iraacutenyuacute hidraulikai eredetű erő 29 10 Kavitaacutecioacute sziacutevoacutekeacutepesseacuteg NPSH 31 11 Ventilaacutetorok uumlzeme 33 12 Volumetrikus elven műkoumldő geacutepek hidraulikus hajtaacutesok 39 13 Meacutereacutesekhez kapcsoloacutedoacute feladatok 42 14 Elmeacuteleti mintakeacuterdeacutesek 45

Aacuteramlaacutestechnikai Geacutepek feladatgyűjtemeacuteny BME Hidrodinamikai Rendszerek Tanszeacutek wwwhdsbmehu

2

1 Gaacutezok sűriacuteteacutese kompresszorok munkafolyamatai

11 Gaacutezt 1 bar abszoluacutet nyomaacutesroacutel 4 bar tuacutelnyomaacutesra sűriacutetuumlnk a gaacutezaacutellandoacute 288 JkgK az aacutellandoacute nyomaacuteson meacutert fajhő cp = 1000 JkgK Mekkora a κ kitevő A sűriacuteteacutesi folyamatot leiacuteroacute politropa kitevője n = 15 Hataacuterozza meg az ideaacutelis (izentroacutepikus) eacutes valoacutes (politropikus) kompresszioacute veacuteghőmeacuterseacutekleteacutet eacutes a politropikus fajlagos munkaacutet Hataacuterozza meg az izotermikus hasznos munkaacutet eacutes az izotermikus hataacutesfokot A besziacutevott gaacutez sűrűseacutege 116 kgm3 A houml asymp h koumlzeliacuteteacutes elfogadhatoacute

Megoldaacutes

2 4 1 5p bar= + =

R eacutes κ ismereteacuteben szaacutemolhatoacute =

Ebből

= minus minus = minus 1000

288 minus 1000= 1405

hőmeacuterseacuteklet az ideaacutelis gaacuteztoumlrveacutenyből szaacutemolhatoacute

= =

10

288 lowast 116= 2993

Az izentropikus kompresszioacute veacuteghőmeacuterseacuteklete

=

= 2993 51

= 47598

A politropikus kompresszioacute veacuteghőmeacuterseacuteklete

=

= 2993 51

= 5118

Az izotermikus hasznos munka

13 = 13 =

10

11613 5

1 = 1387

A bevezetett munka eacutes az izotermikus hataacutesfok

= minus = 1000 lowast 5118 minus 2993 = 2125

= 13 =

1387

2125= 653

A politropikus fajlagos munka

= 1313 minus 1

minus = 15

15 minus 12885118 minus 2993 = 1832

12 Ideaacutelis gaacutez izentroacutepikus aacutellapotvaacuteltozaacutesaacutet felteacutetelezve szaacutemiacutetsa ki egy mamp = 2 kgs

toumlmegaacuteramot szaacutelliacutetoacute kompresszor sziacutevoacutecsonkjaacutenak keresztmetszeteacutet (00099 m2) ha a besziacutevott levegőt a sziacutevoacutecsonkban c = 200 ms sebesseacutegre kell felgyorsiacutetani A kompresszor nyugvoacute T0 = 290 K hőmeacuterseacutekletű p0 = 1 bar abszoluacutet nyomaacutesuacute teacuterből sziacutevja be a levegőt A fajhő cp = 1000 JkgK a fajhőviszony κ = 14 (Abszoluacutet rendszerről leacuteveacuten szoacute az oumlsszentalpia aacutellandoacute )

13 Hataacuterozza meg ndash izentroacutepikus hasznos munkaveacutegzeacutest felteacutetelezve ndash az adiabatikus

aacutellapotvaacuteltozaacutes T2 veacuteghőmeacuterseacutekleteacutet (490 K) eacutes h2 entalpiaacutejaacutet (2181 kJkg) t0 = 0 degC-hoz keacutepest ha R = 0288 kJkgsdotK t1 = 10 degC p1 = 1 bar p2 = 4 bar (abszoluacutet nyomaacutes) κ = 14 az adiabatikus kompresszioacute veszteseacutege Yrsquo = 70 kJkg Szaacutemiacutetsa ki a cp fajhőt


3

(1008 kJkg K) Mekkora a veszteseacuteg miatti entroacutepia noumlvekedeacutes (154 kJkg) A h(s p=aacutell) diagram koumlzeliacutetőleg leiacuterhatoacute a

ln

273 (e 1)p

s R p

c

ph c+

= minus

keacuteplettel amelyben h s R cp meacuterteacutekegyseacutege [kJkg] illetve [kJkgsdotK] eacutes p [bar]-ban helyettesiacutetendő

14 27 degC hőmeacuterseacutekletű 1 bar nyomaacutesuacute (ideaacutelis gaacuteznak tekinthető R = 288 JkgK κ =

14) levegőt 3 bar nyomaacutesuacute teacuterbe juttatunk be kompresszor segiacutetseacutegeacutevel A valoacutedi aacutellapotvaacuteltozaacutest koumlzeliacutető politroacutepa kitevője n = 15 Szaacutemiacutetsa ki a kompresszorboacutel kileacutepő levegő abszoluacutet hőmeacuterseacutekleteacutet (4327 K) eacutes sűrűseacutegeacutet (2407 kgm3) Szaacutemiacutetsa ki az izentroacutepikus veacuteghőmeacuterseacutekletet (4106 K) az izentroacutepikus hataacutesfokot (833 ) az izentroacutepikus hasznos munkaacutet (11148 kJkg) Szaacutemiacutetsa ki a veszteseacutegek legyőzeacuteseacutere fordiacutetandoacute teljesiacutetmeacutenyt (668 kW) ha a szaacutelliacutetandoacute gaacutez toumlmegaacuterama 3 kgs

15 A leacutegszaacutelliacutetoacute geacutepeket aszerint szoktaacutek osztaacutelyozni hogy a levegő oumlsszenyomoacutedaacutesaacutet figyelembe kell-e venni vagy nem illetve a geacutepet hűteni kell-e vagy nem Eszerint a geacutep ventilaacutetor ha 1 le p2p1 lt 13 fuacutevoacute ha 13 le p2p1 lt 3 kompresszor ha 3 le p2p1 Tegyuumlk fel hogy a leacutegszaacutelliacutetoacute geacutep p1 = 1 bar nyomaacutesuacute t1 = 20 degC hőmeacuterseacutekletű levegőt sziacutev be Izentroacutepikus aacutellapotvaacuteltozaacutest (κ = 14) felteacutetelezve mekkora a ( ) 112 ρρρ minus relatiacutev sűrűseacutegvaacuteltozaacutes a ventilaacutetor-fuacutevoacute hataacuteron (0206 = 206 ) eacutes

mekkora a t2 veacuteghőmeacuterseacuteklet a fuacutevoacute - kompresszor hataacuteron (128 deg C) 16 Egy D = 600mm belső aacutetmeacuterőjű csővezeteacuteken foumlldgaacutezt szaacutelliacutetanak Nyomaacutesfokozoacute

kompresszorok L = 75km-enkeacutent vannak telepiacutetve a csővezeteacutek menteacuten A kompresszor telep nyomoacutecsonkjaacuten a p1 = 80bar abszoluacutet nyomaacutesuacute ρ1 = 85kgm3 sűrűseacutegű foumlldgaacutez v1 = 64 ms sebesseacuteggel leacutep be a csővezeteacutekbe A csősuacuterloacutedaacutesi teacutenyező λ = 0018 A csővezeteacutekben az aacutellapotvaacuteltozaacutes izotermikus a

nyomaacutesvaacuteltozaacutesra eacuterveacutenyes az Aacuteramlaacutestanban tanult 21

11

22

21

22v

D

Lp

pp ρλ=minus

oumlsszefuumlggeacutes (Lajos Tamaacutes Az aacuteramlaacutestan alapjai) A nyomaacutes iacutegy a csővezeteacutek veacutegeacuten szaacutemiacutethatoacute a kompresszortelep ezt a nyomaacuteseseacutest fedezi Mekkora a nyomaacutes sűrűseacuteg eacutes a sebesseacuteg a csővezeteacutek szakasz veacutegeacuten Mekkora a szaacutelliacutetott foumlldgaacutez toumlmegaacuterama Mekkora a szuumlkseacuteges politropikus kompresszor teljesiacutetmeacuteny n = 145 politropikus kitevő eseteacuten eacutes ez haacuteny ezreleacuteke a H = 43 MJkg fűtőeacuterteacutekű foumlldgaacutezban tovaacutebbiacutetott teljesiacutetmeacutenynek

Megoldaacutes

A nyomaacutes eacutes sűrűseacuteg a koumlvetkező kompresszortelep sziacutevoacutecsonkjaacuten

( ) barvD

Lppp 541146

2

85

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22 25252

11

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2

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p ==ρ=ρ

A szaacutelliacutetott toumlmegaacuteram

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vAm 7553815346854

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11

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11 ==π=ρπ=ρ=amp


4

A sebesseacuteg a csővezeteacutek veacutegeacuten

2211 vAvA ρ=ρ 2211 vpvp = smvp

pv 444469361

2

12 ===

Fajlagos munka teljesiacutetmeacuteny veszteseacuteg

kg

MJ

p

pp

n

nY

n

n

pol 25015411

80

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1

451

14515

1

2

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2

2 =

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minus=

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ρminus=

minusminus

MWYmP polpol 42382508153 === amp

MWkgMJskgHmQbelső 6613438153 === amp

384300058 058

6613pol

belső

P MW

Q MW= = = energiaveszteseacuteg aacutellomaacutesonkeacutent

(Oumlsszehasonliacutetaacuteskeacuteppen a Paksi atomerőmű 4 blokkja egyenkeacutent 440 MW teljesiacutetmeacutenyű)

17 Egy kompresszor uumlzeme soraacuten a nyomoacutecsonkban 25 bar abszoluacutet nyomaacutest eacutes 187 degC-os hőmeacuterseacutekletet meacutertuumlnk a besziacutevott levegő hőmeacuterseacuteklete 27 degC a leacutegkoumlri nyomaacutes 1020 hPa (1 hPa = 100 Pa) Hataacuterozza meg a politropikus kitevőt (n = 191 R eacuterteacuteke nem szuumlkseacuteges) eacutes a politropikus hataacutesfokot κ = 14 eseteacuten (ηpol = 06)

18 Az aacutebraacuten laacutethatoacute kompresszor levegőt

szaacutelliacutet a koumlrnyezeteacuteből egy zaacutert tartaacutelyba A besziacutevott levegő (R = 286 JkgK κ = 14) nyomaacutesa 1 bar hőmeacuterseacuteklete 27 degC A kompresszioacute veacutegnyomaacutesa 4 bar a sziacutevoacutecsonkbeli teacuterfogataacuteram 25 m3s a sűriacuteteacutesre jellemző politropikus kitevő n = 154 A sziacutevoacute- eacutes nyomoacutecsonk azonosan 125 mm aacutetmeacuterőjű suacuterloacutedaacutesi ellenaacutellaacutesuktoacutel eltekintuumlnk

bull Ellenőrizze hogy a sziacutevoacutecsőbeli Mach-szaacutem kisebb-e 07-neacutel

o Sebesseacuteg a sziacutevoacutecsőben

35

0

00 1661

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p=

sdot=

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smA

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o Hőmeacuterseacuteklet a sziacutevoacutecsőben

Kkg

JRc p sdot

=minus

sdot=

minussdot= 1001

141

28641

1κκ

Kc

cTT

p

327910012

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o Mach-szaacutem ellenőrzeacutese


5

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=sdotsdot

=TR

cMa

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bull Szaacutemiacutetsa ki a gaacutezba bevezetett munkaacutet eacutes teljesiacutetmeacutenyt valamint a politropikus hasznos munkaacutet (a kinetikus energia megvaacuteltozaacutesaacutet vegye figyelembe)

o A sziacutevoacutecsőbeli nyomaacutes

3

141

11

1

0

101 9750

300

32791661

m

kg

T

T=

sdot=

sdot=

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PaTRp 7788232792869750111 =sdotsdot=sdotsdot= ρ o A sűriacuteteacutes veacuteghőmeacuterseacuteklete

Kp

pTT

n

n

749577882

1043279

541

15415

1

1

212 =

sdotsdot=

sdot=

minusminus

o A levegő sebesseacutege a nyomoacutecsonkban

3

1541

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1

1

212 8212

3279

74959750

m

kg

T

T n

=

sdot=

sdot=

minusminusρρ

s

kgqm 44297505211 =sdot=sdot= ρamp

s

m

A

mc 470

12508212

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22 =

sdotsdot

sdot=

sdot=

πρamp

o A bevezetett munka eacutes ndashteljesiacutetmeacuteny

( ) ( )kg

kJ

kg

JTTcY pbe 21721661632797495100112 asymp=minussdot=minussdot=

kWW

mcc

YP bep

484483967

4422

7203470216616

2

2221

22

asymp=

=sdot

minus+=sdot

minus+= amp

o A politropikus hasznos munka

( ) ( )

kg

kJ

kg

J

TTRn

nYpol

4176176421

327964952861541

541

1 12

asymp=

=minussdotsdotminus

=minussdotsdotminus

=

bull A zaacutert tartaacutelyba aacuteramloacute levegő nyugalomba keruumll majd a tartaacutelyfalon keresztuumll ndash aacutellandoacute teacuterfogaton ndash visszahűl a koumlrnyezeti hőmeacuterseacutekletre Szaacutemiacutetsa ki a visszahűleacutes veacutegeacuten a tartaacutelyban uralkodoacute nyomaacutest

barPaT

pTp 4210422

7495

104300 5

5

2

203 asympsdot=sdotsdot=sdot=


6

bull Mennyi lenne a fajlagos hasznos munka a sűriacuteteacutest izentroacutepikusnak illetve izotermikusnak felteacutetelezve A veacutegnyomaacutest mindkeacutet esetben tekintse azonosnak a politropikus folyamateacuteval

Kp

pTT s 9445

77882

1044279

41

1415

1

1

212 =

sdotsdot=

sdot=

minusminusκ

κ

( ) ( )kg

kJ

kg

JTTcY spizentroacutep 516716746742797446100112 asymp=minussdot=minussdot=

kg

kJ

kg

J

p

pTRYizoterm 131130752

77882

104ln4279286ln

5

1

21 asymp=

sdotsdotsdot=

sdotsdot=

A szaacuteraz levegő alaacutebbi h-s diagrammjaacuten berajzoltuk a p[bar] = aacutell eacutes a v[m3kg] = 1ρ = aacutell vonalakat Ugyancsak megrajzoltuk a 0 ndash 1 ndash 2 ndash 3 aacutellapotvaacuteltozaacutes vonalaacutet

250

300

350

400

450

500

65 66 67 68 69 7 71 72 73 74 75

s [kJkgK]

h[kJkg]

11090705v = 03

p=051234


7

2 Euler-turbinaegyenlet sebesseacutegi haacuteromszoumlg 21 Egy 240 mm aacutetmeacuterőjű 20 mm kileacutepő szeacutelesseacutegű jaacuteroacutekereacutek 40 ls vizet szaacutelliacutet 1440min

fordulatszaacutemon A relatiacutev aacuteramlaacutes aacutetlagos szoumlge a kileacutepeacutesneacutel β 2aacutetl = 25deg A beleacutepeacutes perduumlletmentes Szaacutemiacutetsa ki az elmeacuteleti szaacutelliacutetoacutemagassaacutegot (229 m) eacutes rajzolja meg leacutepteacutekhelyesen az aacutetlagos kileacutepő sebesseacutegi haacuteromszoumlget

22 Axiaacutelis szivattyuacute agyaacutetmeacuterője 250 mm a szaacuternylapaacutetos (propeller) kereacutek kuumllső

aacutetmeacuterője 400 mm A kereacutek fordulatszaacutema 1470min A sziacutevoacutecsonkon perduumlletmentesen leacutep be a viacutez A szivattyuacute hidraulikai hataacutesfoka 85 szaacutelliacutetoacutemagassaacutega 6 m teacuterfogataacuterama 036 m3s A lapaacutetok aacuteltal a folyadeacuteknak aacutetadott fajlagos munka a lapaacutet menteacuten az agytoacutel a kuumllső aacutetmeacuterőig veacutegig aacutellandoacute Mekkora β1 ill β2 szoumlget zaacuter be a beleacutepő illetve a kileacutepő relatiacutev sebesseacuteg a keruumlleti sebesseacuteggel az agynaacutel (137deg167deg) Mekkoraacutek ezek a szoumlgek a kereacutek kuumllső aacutetmeacuterőjeacuten (87deg 94deg)

23 400 mm aacutetmeacuterőjű jaacuteroacutekereacutek fordulatszaacutema 1440min a kileacutepeacutesi keresztmetszetben az

aacutetlagos meridiaacutensebesseacuteg-komponens 25 ms A relatiacutev eacutes a keruumlleti sebesseacuteg aacutetlagos szoumlge 25deg Ezt a szoumlget a lapaacutetveacuteg kihegyezeacuteseacutevel 28deg-ra noumlveljuumlk A lapaacutetveacuteg elveacutekonyodaacutesa miatt a meridiaacuten sebesseacuteg az eredeti eacuterteacutek 90 -aacutera csoumlkken A beleacutepeacutes perduumlletmentes Haacuteny szaacutezaleacutekkal vaacuteltozik meg az elmeacuteleti szaacutelliacutetoacutemagassaacuteg (46 )

24 Perduumlletmentes beleacutepeacutest ηh = 86 -os hidraulikai eacutes ηv = 95 -os volumetrikus

hataacutesfokot felteacutetelezve iacuterja fel annak a centrifugaacutelis aacutetoumlmleacutesű haacutetrahajloacute lapaacutetozaacutesuacute szivattyuacute jaacuteroacutekereacuteknek a dimenzioacutetlan elmeacuteleti ψ(φ) jelleggoumlrbe egyenleteacutet amelyiknek 9 veacutekony lapaacutetja van (iacutegy a lapaacutetozaacutes joacute koumlzeliacuteteacutessel veacutegtelen sűrűnek tekinthető) (ψ = 172(1- 3485φ) = 172 ndash 5994φ) A jaacuteroacutekereacutek lapaacutetok kileacutepő szoumlge β2 = 40ordm a jaacuteroacutekereacutek szeacutelesseacutege a kereacutekaacutetmeacuterő 9 -a Szaacutemiacutetsa ki az 1450min fordulatszaacutemmal forgatott 200 mm jaacuteroacutekereacutek aacutetmeacuterőjű vizet szaacutelliacutetoacute szivattyuacute jellemző fordulatszaacutemaacutet (ψopt = 1001 nq = 546) Az optimaacutelis pontban φopt = 012


8

3 Teljesiacutetmeacutenyek veszteseacutegek hataacutesfokok

31 Egy szivattyuacute hajtaacutesaacutehoz szuumlkseacuteges teljesiacutetmeacuteny az optimaacutelis uumlzemaacutellapotban 25 kW a fordulatszaacutem 1440min a szaacutelliacutetott teacuterfogataacuteram 006 m3s A szivattyuacute volumetrikus hataacutesfoka 092 hidraulikai hataacutesfoka 085 taacutercsasuacuterloacutedaacutesi veszteseacutege 09 kW mechanikai veszteseacutege 13 kW A szivattyuacute vizet szaacutelliacutet Mennyi a szivattyuacute szaacutelliacutetoacutemagassaacutega (303 m) eacutes jellemző fordulatszaacutema (273) Vaacutezlaton mutassa be a jaacuteroacutekereacutek meridiaacuten-metszeteacutenek alakjaacutet

32 Egyfokozatuacute 1470min fordulatszaacutemuacute radiaacutelis aacutetoumlmleacutesű szivattyuacute 0055 m3s vizet

szaacutelliacutet szaacutelliacutetoacutemagassaacutega ekkor 45 m A szivattyuacute hidraulikai teljesiacutetmeacutenyveszteseacutege 25 kW mechanikai teljesiacutetmeacutenyveszteseacutege 13 kW taacutercsasuacuterloacutedaacutesi veszteseacutegteacutenyezője 0065 A szivattyuacute hajtaacutesaacutehoz ebben az uumlzemaacutellapotban 32 kW teljesiacutetmeacutenyre van szuumlkseacuteg Szaacutemiacutetsa ki a belső eacutes a hasznos teljesiacutetmeacutenyt (307 kW 2428 kW) a volumetrikus mechanikai hidraulikai eacutes oumlsszhataacutesfokot (932 959 907 759) Hataacuterozza meg az elmeacuteleti eacutes a reacuteseken keringő teacuterfogataacuteramot (0059 m3s 0004 m3s) az elmeacuteleti szaacutelliacutetoacutemagassaacutegot (4963 m) eacutes a szivattyuacute hidraulikus veszteseacutegmagassaacutegaacutet (463 m) A nemzetkoumlzi hataacutesfokszint

2

320max 44

lg2900480940

sdotminussdotminus= minus q

opt

nqη

Mennyivel marad el ennek a szivattyuacutenak a hataacutesfoka a nemzetkoumlzi szinttől (25)

33 Tiszta vizet szaacutelliacutetoacute szivattyuacute teacuterfogataacuterama 0055 m3s szaacutelliacutetoacutemagassaacutega 45 m A szivattyuacute hidraulikai teljesiacutetmeacutenyveszteseacutege 25 kW mechanikai veszteseacutege 13 kW A szivattyuacute hajtaacutesaacutehoz 34 kW teljesiacutetmeacutenyre van szuumlkseacuteg A taacutercsasuacuterloacutedaacutes veszteseacutegteacutenyezője 0065 Szaacutemiacutetsa ki a belső az elmeacuteleti eacutes a hasznos teljesiacutetmeacutenyt (327 kW 3057 kW 2428 kW) az oumlssz- a hidraulikus mechanikus volumetrikus hataacutesfokot (714 907 962 876 ) az elmeacuteleti eacutes a reacutes-teacuterfogataacuteramot (00628 m3s 00078 m3s) az elmeacuteleti szaacutelliacutetoacutemagassaacutegot (4963 m )eacutes a hidraulikai veszteseacutegmagassaacutegot (463 m)

34 A tiszta hideg vizet szaacutelliacutetoacute szivattyuacute megadott adatai alapjaacuten szaacutemiacutetsa ki Helm Qelm

hidrη eacutes volη eacuterteacutekeacutet (212 m 001954 m3s 849 938 ) eacutes a radiaacutelis aacutetoumlmleacutesű

szivattyuacute metszetrajzaacuten jeloumllje be a veszteseacutegek keletkezeacutesi helyeacutet Pbevezetett=435 kW Q=1100 dm3min H=180 m Prsquomech=16 kW taacutercsaν =3 hrsquo=32m


ηhidr eacutes ηvol eacuterteacutekeacutet eacutes a radiaacutelis aacutetoumlmleacutesű szivattyuacute metszetrajzaacuten jeloumllje be a veszteseacutegek keletkezeacutesi helyeacutet Pbevezetett = 30 kW Q = 2100 dm3min H = 60 m Prsquomech = 11 kW tsν = 3 eacutes hrsquo = 12 m

Megoldaacutes

A belső teljesiacutetmeacuteny belső bev mechP P P 30 11 289kWprime= minus = minus =

Az elmeacuteleti teljesiacutetmeacuteny eebelsőtse HgQkWPP sdotsdotsdot==sdotminus=sdotminus= ρν 03328928)0301()1(


9

Az elmeacuteleti szaacutelliacutetoacutemagassaacuteg eacutes teacuterfogataacuteram pedig

3 2

e

3e

e kg me m s

H H h 60 12 72m

P 28033kW mQ 00397

gH 1000 981 72m s

prime= + = + =

= = =ρ times times

A hidraulikai eacutes volumetrikus hataacutesfok

hidr vole e

H 60 Q 0035833 eacutes 882

H 72 Q 00397η = = = η = = =

36 Egy tiszta vizet szaacutelliacutetoacute oumlrveacutenyszivattyuacute műkoumldeacutesi tartomaacutenya 0 lt Q lt Qmax = 024

m3s A hajtoacute motor leadott teljesiacutetmeacutenye 30 kW A csapaacutegyakban toumlmszelenceacuteben keletkező suacuterloacutedaacutesi nyomateacutek Ms = 20 Nm A haacutetragoumlrbiacutetett lapaacutetozaacutesuacute radiaacutelis jaacuteroacutekereacutek fordulatszaacutema n = 1440 min Mekkora a belső teljesiacutetmeacuteny A reacutesveszteseacuteg elhanyagolhatoacute A folyadeacuteksuacuterloacutedaacutesboacutel szaacutermazoacute veszteseacutegmagassaacuteg eacuterteacuteke a maximaacutelis teacuterfogataacuteramnaacutel hrsquo

suacuterl(Qmax)=56 m a levaacutelaacutesi veszteseacuteg a Qmax2 tervezeacutesi pont koumlruumll szimmetrikus maacutesodfokuacute fuumlggveacuteny eacuterteacutekei a műkoumldeacutesi tartomaacuteny szeacutelein hrsquo

lev(0)= hrsquolev(Qmax)=7 m A [m3s]-ban megadott teacuterfogataacuteram fuumlggveacutenyekeacutent a

szivattyuacute oumlsszhataacutesfokaacutet leiacuteroacute keacuteplet ( )QQQouml minussdotsdot=minus max60][η A szivattyuacute a

legkisebb oumlsszes hidraulikai veszteseacuteghez tartozoacute teacuterfogataacuteramnaacutel műkoumldik Mekkora az oumlsszhataacutesfoka hasznos teljesiacutetmeacutenye eacutes szaacutelliacutetoacutemagassaacutega Szaacutemiacutetsa ki a szivattyuacute taacutercsasuacuterloacutedaacutesi veszteseacutegteacutenyezőjeacutet Rajzolja meg leacutepteacutekhelyesen ebben a munkapontban a teljesiacutetmeacutenyszalagot

Megoldaacutes

kWPP szivbemoth 30 ==

kWWnMMP ssmech 0163301660

14402202 ==sdotsdotsdot=== ππω

kWPPP mechszivbebelső 98426016330 =minus=minus=

2

2

max

2bQ

QQahhh suacuterllev +

minus=+=

mQ

aQ

Qa QQQ 742

2max

max0

2

max ==

minus == 522

111486240

74msa =sdot=

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52

222297

240

65msb ==

022

2 max

=+

minus= bQQ

QadQ

dh sm

Q

ba

aQopt 10

23max =

+=

( ) mbQQ

QaQh optoptopt 16612

2

2

max =+

minus=

kWghQPhidr 1411661819100010 =sdotsdotsdot== ρ

( ) ( ) 84060][ max =minussdotsdot=minus optoptoptouml QQQQη

gHQkWPP optszivbeoumlszivh ρη ==sdot== 22530840


10

mgQ

PH

opt

szivh 68825819100010

25200 =sdotsdot

==ρ

mhHHe 85426166168825 =+=+=

kWgHQP eee 3442685426819100010 =sdotsdotsdot== ρ

kWPPP ebelsőt 6403442698426 =minus=minus=

0237098426

640

===belső

tsts P

Pν


11

4 Dimenzioacutetlan uumlzemi mennyiseacutegek jellemz ő fordulatszaacutem affinitaacutes

41 Becsuumllje meg egy 220 mm jaacuteroacutekereacutek aacutetmeacuterőjű 2880min fordulatszaacutemmal hajtott szivattyuacute teacuterfogataacuteramaacutet szaacutelliacutetoacutemagassaacutegaacutet eacutes oumlsszhataacutesfokaacutet(0087 m3s 5026 m 835 ) A szivattyuacute jellemző fordulatszaacutema 45 Az optimaacutelis uumlzemi pont becsuumllt hataacutesfokmaximuma eacutes a szivattyuacute alapeacutertelmezeacutes szerinti nyomaacutesszaacutema az alaacutebbi keacutepletekkel szaacutemiacutethatoacute

η max = 094 ndash 0048 Qopt-032 ndash 029(lg [nq44] )2 ψ = [ 300 (270+nq) ]

94

42 Egy szivattyuacute optimaacutelis teacuterfogataacuterama 16 dm3s fordulatszaacutema 960 1min jellemző

fordulatszaacutema 255 Mekkora a szivattyuacute szaacutelliacutetoacutemagassaacutega (8 m) Mekkora a σfordulatszaacutem-teacutenyező (0162) A Cordier-diagram joacute koumlzeliacuteteacutessel a

δ = (21 (141 + lgσ ))134 alakuacute aacutetmeacuterőteacutenyező-fordulatszaacutemteacutenyező keacuteplettel iacuterhatoacute le Mekkora a Cordier-

diagram előiacuteraacutesaacutenak megfelelő jaacuteroacutekereacutek aacutetmeacuterő [ 75050 ψϕσ = eacutes 50250 ϕψδ = ] (207 mm)

43 Mekkora a megadott H = 45 ndash 2781[s2m5] Q2 η [] = 72 - 14694 (Q -

007)2 jelleggoumlrbeacutejű 1470min fordulatszaacutemuacute 320 mm jaacuteroacutekereacutek aacutetmeacuterőjű egyfokozatuacute szivattyuacute jellemző fordulatszaacutema (293) Ceacutelszerűen vaacutelasztott leacutepteacutekekkel rajzolja meg a ( )DnDn ϕψ eacutes az ( )Dnϕη dimenzioacutetlan jelleggoumlrbeacutek

grafikonjaacutet goumlrbeacutenkeacutent 4-4 szaacutemiacutetott pont alapjaacuten

44 Egy CR 8-60 tiacutepusuacute 2850min fordulatszaacutemuacute hat fokozatuacute taacutepszivattyuacute szaacutelliacutetoacutemagassaacutega H = 68 ndash 02 Q2 oumlsszhataacutesfoka η = 066 ndash 000731 (Q-95)2 A teacuterfogataacuteram meacuterteacutekegyseacutege a keacutepletekben [m3h] Hataacuterozza meg a szivattyuacuteba eacutepiacutetett jaacuteroacutekereacutek jellemző fordulatszaacutemaacutet A jellemző fordulatszaacutem eacuterteacuteke alapjaacuten aacutellapiacutetsa meg a jaacuteroacutekereacutek tiacutepusaacutet Hataacuterozza meg a vizet szaacutelliacutetoacute szivattyuacute teljesiacutetmeacuteny-felveacuteteleacutet az uumlresjaacuterati 0 teacuterfogataacuteram eseteacuten extrapolaacutelva a Q = 15 1 05 m3h teacuterfogataacuteramuacute pontok teljesiacutetmeacuteny adatait valamint a LrsquoHospital szabaacutelyt alkalmazva

Megoldaacutes

Az qn jellemző fordulatszaacutemot az optimaacutelis uumlzemi pont (hataacutesfok-maximum) eacutes egy

jaacuteroacutekereacutek adataival kell szaacutemolni

opt

d2 000731 (Q 95) 0

dQ

η = minus times times minus = iacutegy 3 3

opt

m mQ 95 0002639

h s= =

Az egy fokozat aacuteltal leacutetrehozott szaacutelliacutetoacutemagassaacuteg (az optimaacutelis uumlzemi pontban)

2

6 opt1fokozatopt

H (Q ) 68 02 95H 8325m

6 6

minus times= = =

iacutegy a jellemző fordulatszaacutem

12

34

optq

1opt

Qn n 299

H= = azaz a jaacuteroacutekereacutek radiaacutelis aacutetoumlmleacutesű


12

A bevezetett teljesiacutetmeacuteny a teacuterfogataacuteram fuumlggveacutenyeacuteben

444444 3444444 21

4444 84444 76

η

ηρ

η 23

233

59007310660

20689810

)(

)(

)(

)(

minus

sdotminus

sdotminussdot

sdot

=sdotsdotsdot==

h

mQ

h

mQ

s

mQ

Q

QHgQ

Q

QPP

H

hbev

A fenti keacutepletet tekintve belaacutethatoacute hogy zaacutert toloacutezaacuteraacutellaacutesnaacutel mind a hasznos teljesiacutetmeacuteny mind a hataacutesfok eacuterteacuteke zeacuterus Ez egyben azt is jelenti hogy a fenti oumlsszefuumlggeacutessel a bevezetett teljesiacutetmeacuteny a zaacuteraacutesi pontban nem szaacutemolhatoacute

Veacutegezzuumlnk előszoumlr lineaacuteris extrapolaacutecioacutet a zaacuteraacutesi teljesiacutetmeacutenyfelveacutetel becsleacuteseacutere Ehhez elsőkeacutent kiszaacutemiacutetjuk egy 0 teacuterfogataacuteram koumlzeli pontban a teljesiacutetmeacutenyfelveacutetelt mint referencia eacuterteacuteket majd keacutet tovaacutebbi pontban (egyenkoumlzűen Q = 0 m^3h feleacute tartva) is meghataacuterozzuk a teljesiacutetmeacutenyfelveacutetelt Iacutegy szaacutemiacutethatoacute keacutet teljesiacutetmeacutenyfelveacutetel-differencia melyekből lineaacuterisan extrapolaacutelva szaacutemiacutethatoacute a harmadik melyből a zaacuteraacutesi teljesiacutetmeacutenyfelveacutetel meghataacuterozhatoacute Az eacuterteacutekek taacuteblaacutezatos formaacuteban

Q [m3h] 15 1 05 0 Hb(Q) [m] 6755 678 6795 68 Q [m3s] 417e-4 278e-4 139e-4 0 Ph [W] 276 185 93 0 η(Q) [-] 0192 0132 0068 0 Pbe [W] 1437 1401 1364 1324 ∆Pbe [W] - -36 -38 -40

A pontos eredmeacutenyt a LrsquoHospital-szabaacutely alkalmazaacutesaacuteval kapjuk

)0(

)0(

)(

)(lim

)(

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)(lim

0 ηηhh

Qtx

P

Q

QP

tg

tf

xg

xf =rarr=rarrrarr

A derivaacutelaacutes elveacutegzeacutese előtt vegyuumlk eacuteszre hogy a teacuterfogataacuteram eacutes a hozzaacute tartozoacute konstansok nem SI meacuterteacutekegyseacutegben szerepelnek a szaacutelliacutetoacutemagassaacuteg eacutes a hataacutesfok keacutepleteacuteben Ezeacutert első leacutepeacutesben mindent SI meacuterteacutekegyseacutegre kell aacutetiacuterni

[ ]

[ ]

22

32

2

332

6

2

232

5

23

22

3600

593600007310660

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3600

1593600007310660

360020689810

minussdotsdotminus

sdotsdotsdotsdotminussdotsdotsdot=

=

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sdot

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QgQg

h

sh

m

s

mQ

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s

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h

s

mQ

h

s

m

hm

s

mQ

ms

kg

P

H

bev

ρρη

4444444444444 34444444444444 21

44444444 844444444 76


13

A derivaacutelaacutest elveacutegezve

WggP

P h

Qbev 1334

3600

59023600007310

036006068

)0(

)0(

2

22

0

=

minussdotsdotsdotminus

sdotsdotsdotsdotminussdotsdot===

ρρη

45 Az n1 = 1450min fordulatszaacutemuacute szivattyuacute jelleggoumlrbeacuteje H1 = 40 m ndash 40000 s2m5 Q2

Szaacutemiacutetsa ki az n2 = 2900min fordulatszaacutemuacute motorral hajtott szivattyuacute jelleggoumlrbeacutejeacutenek 5 pontjaacutet a Q2 = 001 m3s ndash 005 m3s teacuterfogataacuteram intervallumban 001 m3s-onkeacutent mivel a meacutereacutesek szerint az affinitaacutes ebben a tartomaacutenyban eacuterveacutenyes Iacuterja fel az n2 fordulatszaacutemuacute szivattyuacute H2(Q2) jelleggoumlrbeacutejeacutenek egyenleteacutet (H2 = 160 ndash 40000 Q2

2)

46 Az nq = 25 jellemző fordulatszaacutemuacute centrifugaacutel szivattyuacutek ventilaacutetorok optimaacutelis nyomaacutesszaacutema ψ = 1 (nem ψn D ) Mekkora a szuumlkseacuteges D2 jaacuteroacutekereacutek aacutetmeacuterő a direkt hajtaacutesuacute n = 1470min fordulatszaacutemuacute 20 m szaacutelliacutetoacutemagassaacuteguacute vizet szaacutelliacutetoacute szivattyuacute illetve a 1000 Pa oumlssznyomaacutes-noumlvekedeacutest leacutetrehozoacute 12 kgm3 sűrűseacutegű levegőt szaacutelliacutetoacute ventilaacutetor eseteacuten (0257 m illetve 053 m) Mekkora a keacutet geacutep teacuterfogataacuterama (00259m3s illetve 0226m3s) Mekkora a keacutet geacutep teljesiacutetmeacutenyfelveacutetele 70 oumlsszhataacutesfok eseteacuten (7259 kW illetve 0323 kW) eacutes mekkora hajtoacutenyomateacutekra kell meacuteretezni a tengelyt 15-szeres biztonsaacuteg eseteacuten (707 Nm illetve 315 Nm)

47 Vizet szaacutelliacutetoacute oumlrveacutenyszivattyuacute fordulatszaacutema 1450min szaacutelliacutetoacutemagassaacutega 17 m

teacuterfogataacuterama 003 m3s a tervezeacutesi uumlzemaacutellapotban Mekkora a jellemző fordulatszaacutem (30) Mekkora aacutetmeacuterőjű jaacuteroacutekereacutek eseteacuten lesz a szivattyuacute nyomaacutesszaacutema az ilyen tiacutepusuacute szivattyuacutek eseteacuten szokaacutesos ψ = 1 eacuterteacutekű (240 mm) Szaacutemiacutetsa ki a ϕ mennyiseacutegi szaacutemot (0036) Mekkora teacuterfogataacuteramot szaacutelliacutet milyen szaacutelliacutetoacutemagassaacuteg elleneacuteben ez a szivattyuacute optimaacutelis uumlzemben 970min motorfordulatszaacutem eseteacuten (002 m3s 761 m)

48 Az n = 2880min fordulatszaacutemuacute szivattyuacute-jaacuteroacutekereacutek idealizaacutelt jelleggoumlrbeacutejeacutenek

egyenlete Heinfin = 145 m ndash 1570 sm2 sdotQe ahol infinsdot= ee HH λ eacutes a perduumllet apadaacutesi

teacutenyező eacuterteacuteke 085 A jaacuteroacutekereacutek kuumllső aacutetmeacuterője 250 mm szeacutelesseacutege ezen az aacutetmeacuterőn 14 mm Perduumllet mentes beleacutepeacutest felteacutetelezve mekkora a lapaacutetszoumlg a kileacutepeacutesneacutel (1255deg ) A lapaacutetveacutegek kieacutelezettek (Ψ2 = 1) Szaacutemiacutetsa ki a valoacutedi jelleggoumlrbe Q = 003 m3s teacuterfogataacuteramuacute pontjaacuteban a ϕ mennyiseacutegi szaacutem (00162) eacutes a ψ nyomaacutesszaacutem eacuterteacutekeacutet (0895) A volumetrikus hataacutesfok becsuumllt eacuterteacuteke e pontban 95 a hidraulikai hataacutesfok becsuumllt eacuterteacuteke ugyanitt 80

49 Az n1 = 1460min fordulatszaacutemuacute szivattyuacute jelleggoumlrbeacuteje H1 = 40 m ndash 40000 s2m5middotQ1

2 Szaacutemiacutetsa ki az n2 = 2920min fordulatszaacutemuacute motorral hajtott szivattyuacute jelleggoumlrbeacutejeacutenek oumlt pontjaacutet 001 m3s-onkeacutent a Q2 = 001 m3s divide 005 m3s teacuterfogataacuteram intervallumban tudva hogy a meacutereacutesek szerint az affinitaacutes ebben a tartomaacutenyban eacuterveacutenyes Iacuterja fel az n2 fordulatszaacutemuacute szivattyuacute H2(Q2) jelleggoumlrbeacutejeacutenek

egyenleteacutet (megoldaacutes 225

2

2 40000160 Qm

smH sdotminus= )

410 Egy ventilaacutetor aacutetmeacuterő-teacutenyezője 175 fordulatszaacutem-teacutenyezője 045 Milyen jaacuteroacutekereacutek

aacutetmeacuterő eseteacuten szaacutelliacutet a geacutep 27000 m3h 12 kgm3 sűrűseacutegű levegőt 1440min fordulatszaacutemon Mekkora ekkor a ventilaacutetor aacuteltal leacutetesiacutetett nyomaacutesnoumlvekedeacutes eacuterteacuteke


14

Megoldaacutes

ϕψδ 2141751 minus== ϕψσ 2143450 minus==

1 2045175 07875 0773σδ ψψ minus= = = rarr =

21 4

0287ψϕ

δ

= =

2

3322 2 22

2

4 4 27000 3600761

0287 1440 604

Q QmmD

nD nD

u

ϕπ ϕπ ππ

sdot= rarr = = =sdot sdot

123

22 0761 1440 60 574 n m su D π π= = sdot sdot =

222

120773 574 1528

2 2p Pau

ρψ∆ = = sdot sdot =


15

5 Csővezeteacutek jelleggoumlrbe munkapont az uumlzem stabilitaacutesa 51 Az aacutebraacuten laacutethatoacute buacutevaacuterszivattyuacute Q = 30 ls vizet nyom a D aacutetmeacuterőjű gyűjtővezeteacutekbe

amelyik oumlt azonos kivitelű eacutes teacuterfogataacuteramuacute buacutevaacuterszivattyuacute vizeacutet szaacutelliacutetja a medenceacutebe Hataacuterozza meg a berendezeacutes szaacutelliacutetoacutemagassaacutegaacutet (1107 m) eacutes a szuumlkseacuteges szivattyuacute szaacutelliacutetoacutemagassaacutegot (1280 m) 503502503 2 ==== kdkdvcssz ζζζζ d

= 160 mm 220=kDζ D = 400 mm A csősuacuterloacutedaacutesi teacutenyező a d aacutetmeacuterőjű csőben

0021 a D aacutetmeacuterőjű vezeteacutekben 0018

52 Szaacutemiacutetsa ki hogy milyen szaacutelliacutetoacutemagassaacuteguacute (3553 m) szivattyuacutet kell beeacutepiacuteteni egy

berendezeacutesbe hogy azon 1200 dm3min viacutezaacuteramot szaacutelliacutetson joacute hataacutesfokkal A sziacutevoacutecső aacutetmeacuterője Ds = 120 mm veszteseacutegteacutenyezője sζ = 36 a nyomoacutecső aacutetmeacuterője

Dn = 100 mm veszteseacutegteacutenyezője (kileacutepeacutesi veszteseacuteg neacutelkuumll) nζ = 14 Az aknaacuteba

eacutepiacutetett szivattyuacute nyitott kuacutetboacutel sziacutev amelynek viacutezszintje 25 m meacutelyen van az alapszint alatt eacutes a nyomoacutecső 5 m magasan veacutegződik az alapszint felett a szabad levegőn onnan a viacutez belezuhog egy uszoda medenceacutejeacutebe Rajzolja meg az elrendezeacutes vaacutezlataacutet

53 Az Erzseacutebet-hiacuted budai oldalaacuten a Szent Gelleacutert szobor alatti mű-viacutezeseacutes alsoacute gyűjtő

medenceacutejeacutet a felső 22 m-rel magasabban levő szabad kifolyaacutesuacute csőveacutegekkel 2 darab paacuterhuzamosan futoacute 30 m hosszuacute cső koumlti oumlssze Az egyik cső 70 mm melyben 15 ms gazdasaacutegos aacuteramlaacutesi sebesseacuteget felteacutetelezuumlnk A maacutesik cső 100 mm belső aacutetmeacuterőjű Az egyenes csőszakaszok suacuterloacutedaacutesi teacutenyezője λ = 0019 A szikla iacuteveit koumlvető csoumlvek kis iraacutenytoumlreacuteseinek eredő veszteseacutegteacutenyezője ζ = 05 Szaacutemiacutetsa ki a maacutesik csőben az aacuteramlaacutesi sebesseacuteget majd hataacuterozza meg a beeacutepiacutetett ndash mindkeacutet csoumlvet taacuteplaacuteloacute ndash szivattyuacute teacuterfogataacuteramaacutet Szaacutemiacutetsa ki mindkeacutet csőre a szaacutelliacutetoacutemagassaacuteg igeacutenyt 65-os szivattyuacute+hajtoacutemotor hataacutesfokot felteacutetelezve szaacutemolja ki a 100 napos turista szezonra jutoacute villamos energia koumlltseacuteget Az energiadiacutej 40 FtkWh Megoldaacutes

ζkD

ζkD ζkD

ζkd

ζ2kd

4 Q

Q

210 m2 m

3 m

26m11 m

7 m

emptyD

emptyd

9 m

Buacutevaacuterszivattyuacute

Sziacutevoacutekosaacuter eacutes visszacsapoacuteszelep ζsz ζvcs


16

Nyomaacutes a berendezeacutes elejeacuten-veacutegeacuten leacutegkoumlri ezeacutert mzH st 22=∆=

mD

L

g

vHH stc 1123150

070

300190

8192

51221

2

2

1

21

1 =

++sdotsdotsdot

+=

+++= ζλ

A nyomaacuteskuumlloumlnbseacutegnek a 2-es cső keacutet veacutegeacuteben meg kell egyeznie az 1-es csőeacutevel vagyis a szaacutelliacutetoacutemagassaacutegok megegyeznek mHH cc 112312 ==

mv

D

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g

vHH stc 1123150

10

300190

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++sdotsdotsdot

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Ebből

s

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s

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ππππ

kWgHQPh 441123819100001940 =sdotsdotsdot== ρ

kWP

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44 ===+η

kWhnapnaphkWTPW villvill 351623910024776 =sdotsdot==

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54 Neacutegy fokozatuacute 68 m szaacutelliacutetoacutemagassaacuteguacute 1450min fordulatszaacutemuacute szivattyuacute 120 mm

belső aacutetmeacuterőjű viacutezszintesen fektetett csoumlvoumln szaacutelliacutet 003 m3s vizet A csősuacuterloacutedaacutesi teacutenyező 0025 Haacuteny meacuteter hosszuacute csőszakaszonkeacutent kell uacutejabb szivattyuacutet beeacutepiacuteteni ha a szivoacutecsonkbeli nyomaacutes alaacute nem csoumlkkenhet a nyomaacutes a csővezeteacutekben A cső folyoacutemeacuteter-aacutera az aacutetmeacuterő neacutegyzeteacutevel araacutenyos A csőaacutetmeacuterőt 160 mm-re bőviacutetve milyen hosszonkeacutent kell az előző felteacutetel mellett uacutejabb szivattyuacutet beeacutepiacuteteni Ha azonos tiacutepusuacute szivattyuacuteboacutel kevesebbet kell beeacutepiacuteteni a szaacutelliacutetoacutecsőbe akkor a szivattyuacute koumlltseacuteghaacutenyad csoumlkken milyen fuumlggveacutenye a hosszra jutoacute szivattyuacutekoumlltseacuteg a csőaacutetmeacuterőnek Rajzolja meg alakra helyesen az oumlsszes beruhaacutezaacutesi koumlltseacuteget a csőaacutetmeacuterő fuumlggveacutenyeacuteben

Megoldaacutes


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A szivattyuacutek darabszaacutema iacutegy az aacutera is a csővezeteacutek L meacutereteacutevel fordiacutetottan araacutenyos

55 D

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A cső aacutera a cső keruumlleteacutenek eacutes falvastagsaacutegaacutenak szorzataacuteval araacutenyos A bdquokazaacutenformulardquo miatt a falvastagsaacuteg araacutenyos a csőaacutetmeacuterővel iacutegy a cső aacutera 2DKAacute ccső sdot= ahol Kc

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Ezzel a teljes aacuter 25 DK

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csőaacutetmeacuterő


18

55 Hataacuterozza meg egy toacuteba eacutepiacutetendő szoumlkőkutat műkoumldtető szivattyuacute munkapontjaacutet A 30 mm aacutetmeacuterőjű fuacutevoacutekaacuteboacutel a toacute viacutezfelsziacuteneacuten kileacutepő fuumlggőleges viacutezsugaacuternak 20 m magasra kell emelkednie A cseppekre szakadoacute viacutezsugaacuter a fuumlggőleges hajiacutetaacuteskeacutent szaacutemiacutethatoacute elmeacuteleti magassaacuteg 80 -aacutera emelkedik A szivattyuacute sziacutevoacuteoldalaacuten ζsz = 07 veszteseacutegteacutenyezőjű szűrőkosaacuter van a nyomoacuteoldalon 1 m hosszuacute λ = 002 suacuterloacutedaacutesi teacutenyezőjű cső valamint keacutet darab egyenkeacutent ζiacutev = 02 veszteseacutegteacutenyezőjű csőiacutev van A szivattyuacute csonkjai eacutes a nyomoacutecső egyaraacutent 80 mm belső aacutetmeacuterőjű Keacutesziacutetsen vaacutezlatot a berendezeacutesről Mekkora az 1470min fordulatszaacutemuacute keacutetfokozatuacute szivattyuacute jellemző fordulatszaacutema Milyen magasra emelkedik a viacutezsugaacuter ha a fordulatszaacutem 970min-re csoumlkken

Megoldaacutes

bull A szuumlkseacuteges teacuterfogataacuteram

o Elmeacuteleti magassaacuteg (amely a bdquokileacutepeacutesi veszteseacutegesrdquo tagboacutel adoacutedik majd)

mh

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o Viacutezsugaacuter sebesseacutege a fuacutevoacutekaacutenaacutel a mozgaacutesi- eacutes a helyzeti energiaacuteboacutel

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o A szuumlkseacuteges teacuterfogataacuteram

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bull A szuumlkseacuteges szaacutelliacutetoacutemagassaacuteg

o Sebesseacutegmagassaacuteg a nyomoacutecsőben

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o A sziacutevoacute- eacutes nyomoacutecső egyuumlttes veszteseacutegmagassaacutega

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o A szivattyuacute szaacutelliacutetoacutemagassaacutega

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bull A szivattyuacute jellemző fordulatszaacutema

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bull Viacutezsugaacuter magassaacutega a megvaacuteltozott fordulatszaacutem mellett o A megvaacuteltozott teacuterfogataacuteram

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A keacutet szaacutemiacutetaacutes koumlzoumltti elteacutereacutes a megvaacuteltozott teacuterfogataacuteram kerekiacuteteacuteseacuteből adoacutedik

56 Egy Dk = 2800 mm aacutetmeacuterőjű vasbeton (λk = 003) csővezeteacutekbe egy Db = 1000 mm aacutetmeacuterőjű aceacutelcsoumlvet (λb = 002) helyeznek (l aacutebra) Mindkeacutet csőszakasz hossza 540 m a csoumlvek veacutegein 3 m a viacutezszint kuumlloumlnbseacutege Hataacuterozza meg a teacuterfogataacuteramot

a) a belső cső neacutelkuumlli eacutes b) a belső cső beeacutepiacuteteacutese utaacuteni esetben

A suacuterloacutedaacutesi teacutenyezőt a feluumlletek nagysaacutegaacuteval suacutelyozza Megoldaacutes a) A sebesseacuteg majd abboacutel a teacuterfogataacuteram szaacutemiacutetaacutesa

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b) Ebben az esetben a sebesseacuteg az egyeneacuterteacutekű csősuacuterloacutedaacutesi teacutenyezővel eacutes a hidraulikai aacutetmeacuterővel szaacutemiacutethatoacute

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Azaz a belső csővezeteacutek beeacutepiacuteteacuteseacutevel a teacuterfogataacuteram az eredeti ~73-aacutera csoumlkken


21

6 Soros paacuterhuzamos uumlzem

61 Egy HI = 70 m - 50000 s2m5 Q2 jelleggoumlrbeacutejű szivattyuacute a Hc = 20 m + 10000 s2m5 Q2 jelleggoumlrbeacutejű csővezeteacutekben vizet szaacutelliacutet Mekkora a teacuterfogataacuteram (00285 m3s) A teacuterfogataacuteramot 0032 m3s ra kiacutevaacutenjaacutek noumlvelni Rendelkezeacutesre aacutell egy HII = 80 m ndash 50000 s2m5 Q2 jelleggoumlrbeacutejű maacutesodik szivattyuacute A kiacutevaacutent munkapontot a koumlzoumls nyomoacutevezeteacutekbe beeacutepiacutetett fojtaacutes segiacutetseacutegeacutevel kiacutevaacutenjuk beaacutelliacutetani A szivattyuacutekat sorosan vagy paacuterhuzamosan kapcsolva oldhatoacute meg a feladat kisebb fojtaacutesi veszteseacuteggel Vaacutelaszaacutet diagram segiacutetseacutegeacutevel grafikusan igazolja (Soros kapcsolaacutes Prsquof=545 kW paacuterhuzamos kapcsolaacutes Prsquof=988kW)

62 Keacutet szivattyuacute paacuterhuzamos uumlzemben dolgozik A szivattyuacutek koumlzoumltt hosszuacute emelkedő

oumlsszekoumltő cső van ennek jelleggoumlrbe egyenlete Hc1 = 20 m + 10000 s2m5 Q2 Az alacsonyabban leacutevő szivattyuacute jelleggoumlrbeacuteje HI = 80 m - 50000 s2m5 Q2 a maacutesik szivattyuacute jelleggoumlrbeacuteje HII = 45 m ndash 50000 s2m5 Q2 (mindkeacutet esetben a sziacutevoacutecsoumlvet is beleeacutertve) A koumlzoumls nyomoacutecső a II jelű szivattyuacutetoacutel indul jelleggoumlrbeacuteje Hc2 = 15 m + 10000 s2m5 Q2 Rajzolja meg a kapcsolaacutest eacutes a jelleggoumlrbeacuteket leacutepteacutekhelyesen fekvő A4 lapon (Az aacutebraacutezolaacuteshoz a jelleggoumlrbe pontokat elegendő 001 m3sndashonkeacutent kiszaacutemiacutetani) Hataacuterozza meg az egyes szivattyuacutek sajaacutet munkapontjaacutet (HI QI illetve HII QII)a fenti kapcsolaacutes eseteacuten

Megoldaacutes

bull Meghataacuterozzuk az I szivattyuacute eacutes a koumlzoumls csővezeteacutek eredő jelleggoumlrbeacutejeacutet HI-Hc1 bull Ezt az eredő jelleggoumlrbeacutet (paacuterhuzamosan) oumlsszeadjuk a II szivattyuacuteval iacutegy kapjuk a (HI

ndash Hc1) + HII jelleggoumlrbeacutet bull Ezt a jelleggoumlrbeacutet messuumlk el a Hc2 csővezeteacutek jelleggoumlrbeacutevel iacutegy megkapjuk a rendszer

munkapontjaacutet bull Veacuteguumll a munkapontot (viacutezszintesen) vetiacutetsuumlk vissza a HII ill a HI-Hc1 jelleggoumlrbeacutere eacutes

onnan fuumlggőlegesen a HI jelleggoumlrbeacutere

Tehaacutet a keacutet szivattyuacute munkapontja (0022 m3s 565 m illetve 00167 m3s 31 m)


22

63 Az aacutebra szerinti elrendezeacutesben keacutet szivattyuacute egy viacuteztorony toumllteacuteseacutet eacutes az SZ jelű szerelveacutenyaacutegban leacutevő fogyasztoacutek ellaacutetaacutesaacutet veacutegzi Az elemek jelleggoumlrbeacutei (H [m]-ben Q [litmin] ndashben helyettesiacutetendő )

HI = 50 ndash 02 Q2 HII = 35 ndash 03 Q2 HL1 = 10 + 03 Q2 HL2 = 25 + 04 Q2 HL3 = 5 + 1 Q2

Rajzolja meg a jelleggoumlrbeacuteket neacutehaacuteny pont alapjaacuten eacutes grafikusan hataacuterozza meg az Li aacutegakban kialakuloacute teacuterfogataacuteramokat (Q1 Q2 Q3) (57 73 44 litmin) az eredő munkapont adatait (Q H) (117 litmin 242 m) eacutes a szivattyuacutek sajaacutet munkapontjaacutet (QI HI) (QII HII) (57 litmin 436 m 6 litmin 242 m)

T

L1

L2

L3 SZ

I

II

64 Egy kis meacuteretű telepuumlleacutest a Hs = 70 ndash 330 Q2 jelleggoumlrbeacutejű szivattyuacute laacutetja el viacutezzel Nappali uumlzemben a Hce1 = 25 + 30 Q2 eacutejszakai uumlzemben a Hce2 = 25 + 750 Q2 eredő jelleggoumlrbe helyettesiacuteti a telepuumlleacutes viacutezhaacuteloacutezataacutet A szivattyuacute nyomoacutecsonkjaacutehoz kapcsoloacutedoacute magas taacuterozoacute jelleggoumlrbeacuteje HT = 40 + 55 |Q| Q A szaacutelliacutetoacutemagassaacutegokat megadoacute oumlsszefuumlggeacutesekben [Q] = m3s [H] = m Keacutesziacutetsen vaacutezlatot a telepuumlleacutes viacutezműveacuteről Hataacuterozza meg szerkeszteacutessel a szivattyuacute (033 m3s 029 m3s) a telepuumlleacutes (06 m3s 015 m3s) eacutes a taacuterozoacute teacuterfogataacuteramaacutet (0275 m3s -014 m3s) eacutes a szivattyuacute szaacutelliacutetoacutemagassaacutegaacutet (36 m 41 m) mindkeacutet uumlzemaacutellapotban (A szerkeszteacuteshez hasznaacuteljon millimeacuteter papiacutert)

65 Egy 1440min fordulatszaacutemmal jaacuteroacute ventilaacutetor jelleggoumlrbeacuteje 216 0331oumlp q∆ = minus

hataacutesfoka ( )qq minus=η 2258 A csővezeteacutek jelleggoumlrbe 205 0124oumlp q∆ = + Az

oumlsszefuumlggeacutesekben hasznaacutelt meacuterteacutekegyseacutegek [ ] kPapouml =∆ [ ] smq 3= [ ] =η

Meghataacuterozandoacutek a munkapont adatai (qM1=1556m3s ∆poumlM1=08kPa ηM1=581 PbeM1=2143kW) Meghataacuterozandoacutek keacutet paacuterhuzamosan kapcsolt ventilaacutetor eseteacuten a munkapont (M2) adatai eacutes a hajtaacutes teljesiacutetmeacutenyszuumlkseacuteglete (qM2=2307m3s ∆poumlM2=116kPa ηM2=70 PbeM1=2x1912kW) Az M2 munkapontot egy ventilaacutetorral milyen fordulatszaacutemon lehet megvaloacutesiacutetani (nuacutej = 1945min) (Az affinitaacutes toumlrveacutenye eacuterveacutenyes)

66 A 2159037 qH s sdotminus= jelleggoumlrbeacutejű szivattyuacuteval egy viacutezszintes terep

oumlntoumlzőrendszereacutenek egymaacutessal paacuterhuzamosan koumltoumltt aacutegait taacuteplaacuteljuk Meacuterteacutekegyseacutegek q [m3h] H [m] Minden aacuteg egy csővezeteacutekből eacutes egy a csővezeteacutek veacutegeacuten leacutevő szoacuteroacutefejből aacutell Az egyes aacutegak csővezeteacutek hossza 20 m (belső) aacutetmeacuterője 25 mm a becsuumllt csősuacuterloacutedaacutesi teacutenyező 003 Az oumlntoumlzőfejek 2 bar tuacutelnyomaacutes eseteacuten 4 m3h teacuterfogataacuteramot engednek aacutet jelleggoumlrbeacutejuumlk 2qKH fejfej sdot= alakban iacuterhatoacute

a) Keacutesziacutetse el a berendezeacutes kapcsolaacutesi vaacutezlataacutet 3 aacuteg eseteacuten


23

b) Mekkora a teacuterfogataacuteram egyetlen műkoumldő aacuteg eseteacuten c) Haacuteny paacuterhuzamos aacutegat tud a szivattyuacute kiszolgaacutelni uacutegy hogy a szoacuteroacutefejeken

legalaacutebb 2 bar tuacutelnyomaacutest biztosiacutetunk a szuumlkseacuteges szoacuteraacutesi taacutevolsaacuteg megtartaacutesaacutehoz

Megoldaacutes bull Keacutesziacutetse el a berendezeacutes kapcsolaacutesi vaacutezlataacutet 3 aacuteg eseteacuten

bull Mekkora a teacuterfogataacuteram egyetlen műkoumldő aacuteg eseteacuten

Ehhez előszoumlr hataacuterozzuk meg 1 cső eacutes 1 fej ezekből pedig 1 aacuteg jelleggoumlrbeacutejeacutet

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Ha a szivattyuacute csak 1 aacutegra dolgozik a teacuterfogataacuteram eacutes a szaacutelliacutetoacutemagassaacuteg 22 66591159037 QQ sdot=sdotminus

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bull Haacuteny paacuterhuzamos aacutegat tud a szivattyuacute kiszolgaacutelni uacutegy hogy a szoacuteroacutefejeken legalaacutebb 2 bar tuacutelnyomaacutest biztosiacutetunk a szuumlkseacuteges szoacuteraacutesi taacutevolsaacuteg megtartaacutesaacutehoz A feladat szoumlvegezeacutese egyeacutertelműveacute teszi hogy a szoacuteroacutefejek előtti 2 bar tuacutelnyomaacutes biztosiacutetaacutesaacutehoz aacutegankeacutent legalaacutebb 4 m3h teacuterfogataacuteramot kell szaacutelliacutetani Legyen az aacutegak szaacutema n a teacuterfogataacuteram pedig n4 m3h Ekkor a munkapont-egyenlet


24

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A szivattyuacute keacutet aacutegat tud a megadott peremfelteacutetelekkel kiszolgaacutelni

Ellenőrzeacutes A szivattyuacute aacuteltal szaacutelliacutetott teacuterfogataacuteram

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Tuacutelnyomaacutes a szoacuteroacutefej előtt

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25

7 Szivattyuacute vezeacuterleacutesi moacutedok

71 H [m]= 100 ndash 00025 Q2 eacutes az η []= 15 Q - 00075 Q2 jelleggoumlrbeacutekkel rendelkező szivattyuacute a Hc [m]= 324 + 00015 Q2 jelleggoumlrbeacutejű csővezeteacuteken keresztuumll szaacutelliacutet vizet (Q meacuterteacutekegyseacutege [dm3min]) Mekkora a szaacutelliacutetott teacuterfogataacuteram (130 dm3min) mekkora a szivattyuacute hajtaacutesaacutehoz szuumlkseacuteges teljesiacutetmeacuteny (1799 kW) A teacuterfogataacuteramot fojtaacutessal kiacutevaacutenjuk csoumlkkenteni 80 dm3min-re Hogyan szaacutemiacutethatoacute ebben az aacutellapotban a fojtaacutesos uumlzem f fajlagos energiakoumlltseacutege (Qrsquo= 80 dm3min = 1333 dm3s H(Qrsquo) = 84 m Pbe(Qrsquo) = 1526 kW iacutegy f = Pbe(Qrsquo)Qrsquo = 1144 kJdm3) Szaacutemiacutetsa ki eacutes rajzolja meg a fojtaacutessal vezeacuterelt rendszer η ber(Q) (berendezeacutes hataacutesfok-teacuterfogataacuteram) jelleggoumlrbeacutejeacutet a 0 40 80 120 130 dm3min teacuterfogataacuteramuacute pontokban

72 A szivattyuacute amelynek jelleggoumlrbeacutejeacutet a Hs = 70 m - 90000 s2m5sdotQ2 keacuteplet adja meg a

Hc = 30 m + 100000 s2m5sdotQ2 jelleggoumlrbeacutejű fővezeteacutekre dolgozik Iacuterja fel annak a megcsapoloacute-vezeteacuteknek a jelleggoumlrbe egyenleteacutet amelynek megnyitaacutesaacuteval a fővezeteacuteken a teacuterfogataacuteram 480 dm3minndashra csoumlkken A megcsapoloacute-vezeteacutek statikus terhelőmagassaacutega 25 m A feladat megoldaacutesaacutehoz rajzolja fel az elrendezeacutes vaacutezlataacutet eacutes a jelleggoumlrbeacutek grafikonjaacutet koumlzeliacutető pontossaacuteggal H(Q) koordinaacutetarendszerben de a megoldaacutest analitikus uacuteton szaacutemiacutetsa ki

Megoldaacutes

Az uacutej teacuterfogataacuteram 3 3

480 0008uacutej

dm mQ

min s= =

Tudjuk hogy a csővezeteacutek jelleggoumlrbeacutejeacuten rajta lesz az uacutej munkapont iacutegy ki tudjuk szaacutemiacutetani a szaacutelliacutetoacutemagassaacutegot a munkapontban

5 2

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s s m m

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364m H (Q ) H (Q )

= + times =

= =

Mivel a csővezeteacutek eacutes a megkeruumllő vezeteacutek

paacuterhuzamosan van kapcsolva a szaacutelliacutetoacutemagassaacuteg koumlzoumls iacutegy a szivattyuacute aacuteltal szaacutelliacutetott teacuterfogataacuteram

3

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90000 s

minus= rarr = =

A szivattyuacute aacuteltal szaacutelliacutetott eacutes a rsquocrsquo csővezeteacuteken aacutetaacuteramloacute (kiacutevaacutent) teacuterfogataacuteram kuumlloumlnbseacutege

az rsquomrsquo megcsapoloacutevezeteacuteken taacutevozik 3

m s uacutej

mQ Q Q 001132

s= minus =

A megcsapoloacutevezeteacutek jelleggoumlrbeacuteje tehaacutet egy olyan parabola mely 2m stm mH H BQ= +

alakuacute eacutes aacutethalad a 3

m c

m(Q H ) (001132 364m)

s= ponton Mivel tudjuk hogy

st mH 25m= koumlnnyű B eacuterteacutekeacutet meghataacuterozni

2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26

azaz a megcsapoloacute vezeteacutek jelleggoumlrbeacuteje 2mH 25 88963Q= +

73 1470 min fordulatszaacutemuacute szivattyuacute eacutes a raacutekapcsolt csővezeteacutek jelleggoumlrbeacuteit keacutepletekkel adjuk meg Hsz = 45 [m] ndash 2781 [s2m5] Q2 η [] = 72 - 14694 (Q - 007)2 Hcs = 20 [m] + 1125[s2m5] Q2 Mekkora a szivattyuacute teacuterfogataacuterama eacutes szaacutelliacutetoacutemagassaacutega (Q = 008 m3s H = 272 m) Milyen fordulatszaacutemmal kell jaacuteratni a szivattyuacutet hogy az adott csővezeteacutekben 005 m3s teacuterfogataacuteramot szaacutelliacutetson (Az affinitaacutes toumlrveacutenye hasznaacutelhatoacute) (1195 min) Mennyi lesz a berendezeacutes hataacutesfoka ha a kiacutevaacutent teacuterfogataacuteramot nyomoacuteoldali fojtaacutessal aacutelliacutetjuk be (396 ) Mekkora ekkor fojtaacutes teljesiacutetmeacutenyveszteseacutege (748 kW) eacutes a berendezeacutes fajlagos energiafogyasztaacutesa (565 kJkg)

74 A H [m]= 100 ndash 00025 Q2 eacutes az η []= 15 Q - 00075 Q2 jelleggoumlrbeacutekkel rendelkező

szivattyuacute a Hc [m]= 324 + 00015 Q2 jelleggoumlrbeacutejű csővezeteacuteken keresztuumll szaacutelliacutet vizet (Q meacuterteacutekegyseacutege dm3min) Mekkora a szaacutelliacutetott teacuterfogataacuteram (130 dm3min) mekkora a szivattyuacute hajtaacutesaacutehoz szuumlkseacuteges teljesiacutetmeacuteny (1799 kW) A teacuterfogataacuteramot megkeruumllő vezeteacutekkel kiacutevaacutenjuk csoumlkkenteni Rajzolja meg a megkeruumllő vezeteacutek jelleggoumlrbeacutejeacutet amely eseteacuten 80 dm3min teacuterfogataacuteram folyik a megadott jelleggoumlrbeacutejű csővezeteacuteken Hogyan szaacutemiacutethatoacute ebben az aacutellapotban a megkeruumllő vezeteacutekes uumlzem fajlagos f energiakoumlltseacutege ndash nem kell kiszaacutemiacutetani az eacuterteacutekeacutet (a megkeruumllő vezeteacutekes vezeacuterleacutes eseteacuten a szivattyuacute 1523 dm3min teacuterfogataacuteramot szaacutelliacutet szaacutelliacutetoacutemagassaacutega 42 m Ehhez ndash mivel ekkor a szivattyuacute hataacutesfoka 545 - 1919 kW teljesiacutetmeacutenyre van szuumlkseacuteg Iacutegy f = 1919 kW 1333 dm3s = 1439 kJ dm3)

75 Zaacutert csővezeteacutek hurkot alkot melynek egy szakaszaacuten keacutet szivattyuacute van paacuterhuzamosan

beeacutepiacutetve Az S1 szivattyuacute jelleggoumlrbeacuteje Hs1 = 10 + 10 Q ndash 1000 Q2 az S2 szivattyuacute ami uumlzemzavari tartaleacutek-geacutep jelleggoumlrbeacuteje Hs2 = 10 ndash 10000 Q2 A csővezeteacutek jelleggoumlrbeacuteje mivel a statikus szaacutelliacutetoacutemagassaacuteg zeacuterus Hc = 4000 Q2 parabola A szivattyuacutek koumlzoumls sziacutevoacute eacutes koumlzoumls nyomoacutecsonkjaacutet egy proacutebavezeteacutek is

c

p

S1

S2


27

oumlsszekoumlti amelyben uumlzemszerűen a toloacutezaacuter zaacuterva van Ha teacutevedeacutesből nem zaacuterjaacutek el a toloacutezaacuterat teljesen akkor ennek a proacutebavezeteacuteknek a jelleggoumlrbeacuteje Hp = 40000 Q2 Ez esetben mekkora a teacuterfogataacuteram a c illetve a p csővezeteacutekben ha a) csak az S1 szivattyuacute jaacuter (Q = 0057 m3s) b) csak az S2 szivattyuacute jaacuter (Q = 0028 m3s) c) mindkeacutet szivattyuacute jaacuter (Q = 0061 m3s) Keacutesziacutetsen pontos eacutes leacutepteacutekhelyes vaacutezlatot a jelleggoumlrbeacutekről eacutes a vaacutelaszt ennek a grafikonnak alapjaacuten adja meg Q meacuterteacutekegyseacutege az oumlsszes jelleggoumlrbe eseteacuten [m3s]

76 1470min fordulatszaacutemuacute szivattyuacute eacutes a raacutekapcsolt csővezeteacutek jelleggoumlrbeacuteit keacutepletekkel

adjuk meg Hsz = 45 ndash 2781[s2m5] Q2 η [] = 72 - 14694 (Q - 007)2 Hcs = 20 + 1125[s2m5] Q2 Mekkora a szivattyuacute teacuterfogataacuterama eacutes szaacutelliacutetoacutemagassaacutega (Q = 008 m3s H = 272 m) Milyen fordulatszaacutemmal kell jaacuteratni a szivattyuacutet hogy az adott csővezeteacutekben 005 m3s teacuterfogataacuteramot szaacutelliacutetson (Az affinitaacutes toumlrveacutenye hasznaacutelhatoacute) (n = 1205 1min) Mennyi lesz a fojtaacutessal vezeacuterelt berendezeacutes hataacutesfoka ha a kiacutevaacutent teacuterfogataacuteramot nyomoacuteoldali fojtaacutessal aacutelliacutetjuk be (ηber = 397)

77 Mennyi vizet szaacutelliacutet a Hs = 70 ndash 45000 Q2 jelleggoumlrbeacutejű szivattyuacute a Hc = 20 + 20000

Q2 jelleggoumlrbeacutejű csővezeteacuteken (00277m3s) A teacuterfogataacuteramot 0015 m3s-ra kell csoumlkkenteni Ez megtehető fojtaacutessal vagy megkeruumllő (by-pass) vezeteacutekes vezeacuterleacutessel Rajzolja meg mindkeacutet megoldaacutes szivattyuacute ndash cső - szelep elrendezeacuteseacutet Mekkora a fojtoacuteszelepen eacutebredő hidraulikai teljesiacutetmeacuteny veszteseacuteg fojtaacutes eseteacuten (52kW) mekkora ez a veszteseacuteg a megkeruumllő vezeteacutekes vezeacuterleacutes eseteacuten (40kW) A szivattyuacute teljesiacutetmeacutenyfelveacutetele Pbe = 94 + 240 Q ndash 50000 Q3 Mekkora az f fajlagos energia felhasznaacutelaacutes a keacutet esetben (0237 ill 0285 kWhm3) A keacutepletekben a meacuterteacutekegyseacutegek [m] [m3s] [kW]


28

8 Allievi-elmeacutelet hirtelen zaacuteraacutes

81 Egy NA150-es vezeteacutekben viacutez aacuteramlik Q = 44 m3h teacuterfogataacuterammal a csővezeteacutekben az alapnyomaacutes 5 bar (relatiacutev) a hullaacutemsebesseacuteg 1200 ms Szaacutemiacutetsa ki a csővezeteacutek elejeacuten talaacutelhatoacute szivattyuacute kieseacutese eseteacuten (az aacuteramlaacutesi sebesseacuteg főidőn beluumll zeacuterusra csoumlkken) kialakuloacute nyomaacuteshullaacutem amplituacutedoacutejaacutet Fennaacutell a kavitaacutecioacute veszeacutelye Szaacutemiacutetsa ki hogy ugyanezen teacuterfogataacuteram mellett milyen csőaacutetmeacuterőre van szuumlkseacuteg ahhoz hogy a szivattyuacute kieseacutese eseteacuten se essen a csőbeli nyomaacutes a leacutegkoumlri nyomaacutes alaacute

Megoldaacutes Az Allievi-elmeacutelet szerint

cap sdotsdot=∆ ρ Az aacuteramlaacutesi sebesseacuteg a csőben

s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ

A nyomaacuteshullaacutem amplituacutedoacute barPap 3882800069012001000 ==sdotsdot=∆

Ebből a legkisebb relatiacutev nyomaacutes a csővezeteacutekben 5-83 = -33 bar lenne de ne felejtsuumlk el hogy ez fizikailag lehetetlen a csővezeteacutekben -1 bar relatiacutev nyomaacutes joumln leacutetre 0 bar relatiacutev nyomaacutes alatt a viacutez kavitaacutel A szuumlkseacuteges csőaacutetmeacuterő a kavitaacutecioacute elkeruumlleacuteseacutehez

m19303600105

44412001000

p

Q4aD

D

Q4acap

52=

sdotπsdotsdotsdotsdotsdot=

πsdot∆sdotsdotsdotρ=rarr

πsdotsdotsdotsdotρ=sdotsdotρ=∆

NA200-as cső beeacutepiacuteteacuteseacutevel a kavitaacutecioacute főidőn beluumlli zaacuteraacutes eseteacuten is elkeruumllhető

82 Egy ivoacuteviacutezvezeteacutek rekonstrukcioacuteja soraacuten a reacutegi azbesztcement (AC) vezeteacuteket egy szakaszon aceacutelcsőre csereacutelik Az AC vezeteacutekben a hullaacutemsebesseacuteg 920 ms miacuteg az aceacutelcsőben 1200 ms Szaacutemiacutetsa ki hogy 07 ms sebesseacutegű aacuteramlaacutes 7 bar alapnyomaacutes eseteacuten a csővezeteacutek veacutegeacutet hirtelen (főidőn beluumll) lezaacuterva mekkora a legnagyobb nyomaacutescsuacutecs (AC dp=644 bar pmax=1344 bar aceacutelcső dp=84 bar pmax=154 bar)

83 8 km hosszuacute viacutezszintes NA 200 taacutevvezeteacutek nyiacutelt felsziacutenű medence foumlleacute szaacutelliacutet 3600

lmin vizet A csősuacuterloacutedaacutesi teacutenyező 0018 Mekkora nyomaacutest kell leacutetrehozni a szivattyuacuteval a cső elejeacuten a stacionaacuterius uumlzem fenntartaacutesaacutehoz (1313 bar) Időben lineaacuteris sebesseacutegcsoumlkkeneacutest felteacutetelezve a főidő haacutenyszorosa alatt szabad elzaacuterni a szivattyuacute nyomoacutecsonkjaacutehoz kapcsoloacutedoacute toloacutezaacuterat ahhoz hogy a nyomaacutes ott ne csoumlkkenjen a leacutegkoumlri nyomaacutes alaacute (175Tf) A nyomaacuteshullaacutem 1200 ms sebesseacuteggel terjed Mekkora a főidő (Tf = 1333s) Rajzolja meg a sebesseacuteg-idő fuumlggveacuteny grafikonjaacutet


29

9 Axiaacutelis radiaacutelis iraacutenyuacute hidraulikai eredet ű erő

91 Szaacutemiacutetsa ki hogy egy 300 mm aacutetmeacuterőjű jaacuteroacutekereacutek bordaacutezott haacutetlapjaacutenak kuumllső feluumlleteacutere mekkora axiaacutelis erő hat ha a jaacuteroacutekereacutek keruumllete menteacuten a nyomaacutes 23 bar a tengelytoumlmiacuteteacutes joacute (nem alakul ki reacutesaacuteramlaacutes a jaacuteroacutekereacutek haacutetlapja menteacuten) az agyaacutetmeacuterő 50 mm A jaacuteroacutekereacutek fordulatszaacutema 1470 min A bordaacutezott haacutetlap menteacuten a viacutez aacutetlagos szoumlgsebesseacutege a jaacuteroacutekereacutek szoumlgsebesseacutegeacutenek 85 -a Iacuterja fel a haacutetlap menteacuten a nyomaacuteseloszlaacutest a sugaacuter fuumlggveacutenyeacuteben Rajzolja meg a jaacuteroacutekereacutek meridiaacuten metszeteacutet eacutes rajzolja be a kiszaacutemiacutetott erő iraacutenyaacutet

Megoldaacutes

sradjkf 1316014702

850850 =sdotsdotsdot== πωω

Krp fhaacutetlap += 22

2ωρ

K meghataacuterozhatoacute abboacutel hogy ( ) Krrp fhaacutetlap += 2222 2ωρ

Iacutegy a nyomaacuteseloszlaacutes ( ) ( )222

22 2

rrprp fhaacutetlap minusminus= ωρ

A keresett erő pedig

( ) =sdot+sdot

minus=sdot= intintint drrrdrrrpdrrrpF f

r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot

minus= intint drrdrrrpr

r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ

=minussdotsdotsdot+minussdot

sdotsdotminussdotsdot=4

0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ

kNN 36959356681852538 ==+= a sziacutevoacuteoldal feleacute hat

92 Szaacutemiacutetsa ki hogy mekkora eacutes milyen iraacutenyuacute radiaacutelis erő hat egy D2=260 mm aacutetmeacuterőjű

b3=60 mm kuumllső szeacutelesseacutegű iker-jaacuteroacutekereacutek re melynek csigahaacutezaacuteban a sarkantyuacutetoacutel szaacutemiacutetott szoumlg fuumlggveacutenyeacuteben a keruumllet menti nyomaacuteseloszlaacutes linearizaacutelt koumlzeliacuteteacutese

( ) ( )πϕ=ϕ 270000p [Pa] midőn a szivattyuacute teacuterfogataacuterama 100 dm3s szemben az optimaacutelis 160 dm3s teacuterfogataacuterammal Enneacutel a reacuteszterheleacutesneacutel a vizet szaacutelliacutetoacute szivattyuacute szaacutelliacutetoacutemagassaacutega 16 m az optimaacutelis pontban 14 m a fordulatszaacutem 970 min Mekkora a jellemző fordulatszaacutem Joacute becsleacutes-e hogy az FR radiaacutelis erő

gHbDq

qF

optR ρ

minus= 32

2

1360 keacuteplettel koumlzeliacutethető

Bronstein Matematikai zsebkoumlnyv int ϕϕ+ϕ=ϕϕϕ sincoscos d


30

int ϕϕminusϕ=ϕϕϕ cossinsin d

Megoldaacutes A szivattyuacute jellemző fordulatszaacutema

65314160

970 43

21

43

21

=sdot==opt

optq H

qnn

Az elemi erő komponensei

( ) ( ) ( ) =ϕϕϕπ

minus=ϕϕϕminus=ϕϕϕminus= intintintπππ 2

0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ

minus=ϕϕϕminus= intintππ 2

0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2

=sdotsdotsdotsdotsdot

minussdot=

minus= ρ

Az elteacutereacutes kevesebb mint 2 azaz a becsleacutes elfogadhatoacute

93 Egy 300 mm aacutetmeacuterőjű vizet szaacutelliacutetoacute jaacuteroacutekereacutek keruumllete menteacuten a nyomaacutes 23 bar a tengelytoumlmiacuteteacutes joacute (nem alakul ki reacutesaacuteramlaacutes a jaacuteroacutekereacutek haacutetlapja menteacuten) az agyaacutetmeacuterő 50 mm A jaacuteroacutekereacutek fordulatszaacutema 1470 min Iacuterja fel a haacutetlap melletti teacuterben a nyomaacutes sugaacuter menti vaacuteltozaacutesaacutet az alaacutebbi keacutet esetben

bull a jaacuteroacutekereacutek haacutetlapjaacuten radiaacutelis bordaacutek vannak iacutegy a haacutetlap menteacuten a viacutez aacutetlagos szoumlgsebesseacutege a jaacuteroacutekereacutek szoumlgsebesseacutegeacutenek 85 -a (936kN)

bull a jaacuteroacutekereacutek haacutetlapja sima iacutegy a haacutetlap menteacuten a viacutez aacutetlagos szoumlgsebesseacutege a jaacuteroacutekereacutek szoumlgsebesseacutegeacutenek 50 -a (1358kN)

bull Szaacutemiacutetsa ki hogy mennyivel eacutes milyen iraacutenyban vaacuteltozik az axiaacutelis erő a maacutesodik esetben az elsőhoumlz keacutepest (42 kN-nal nő)


31

10 Kavitaacutecioacute sziacutevoacutekeacutepesseacuteg NPSH

101 Szaacutemiacutetsa ki a 180 m3h teacuterfogataacuteramuacute 23 degC hőmeacuterseacutekletű vizet szaacutelliacutetoacute egyfokozatuacute oumlrveacutenyszivattyuacute megengedhető sziacutevoacutemagassaacutegaacutet ha a szaacutelliacutetoacutemagassaacuteg enneacutel a teacuterfogataacuteramnaacutel 30 m a szivattyuacute NPSHr eacuterteacuteke 463 m A szivattyuacute nyiacutelt medenceacuteből sziacutev a leacutegnyomaacutes 1023 mbar A sziacutevoacutecsonk 04 m-rel van a jaacuteroacutekereacutek tengelye alatt a sziacutevoacutecső veszteseacutege az adott teacuterfogataacuteramnaacutel a hsrsquo = 652 [s2m5] Q2 keacutepletből szaacutemiacutethatoacute A teliacutetett gőznyomaacutes pg = 1704 + 0107 (t-15) + 0004 (t-15)2 kPa a

oo 3015 lele t tartomaacutenyban Mekkora a Thoma-feacutele kavitaacutecioacutes szaacutem

Megoldaacutes

A szivattyuacute megengedett beeacutepiacuteteacutesi magassaacutega

t gsmeg s r s

p p (t)H e NPSH h (Q)

g

minusprime= minus minus minus

ρ

A sziacutevoacutecső aacuteramlaacutesi veszteseacutege 2

sh (Q) 652 005 163mprime = times =

A teliacutetett gőznyomaacutes ot 23 C= -on

2gp 1704 0107 (23 15) 0004 (23 15) 2816kPa= + times minus + times minus =

Iacutegy smeg

1023 2816H 04 463 163 3481 m

981

minus= minus minus minus = a Thoma-feacutele kavitaacutecioacutes

szaacutem pedig rNPSH (Q) 463m0154

H(Q) 30mσ = = =

102 Kisnyomaacutesuacute gőzkazaacutenba (pkazaacuten = p0) jelleggoumlrbeacuteivel adott szivattyuacute szaacutelliacutetja a taacutepvizet

a kondenzaacutetorboacutel (pkond = pg) A keringetett toumlmegaacuteram 27 kgs A viacutezszintes egyenes csőszakaszok hossza 10 m aacutetmeacuterője 100 mm csősuacuterloacutedaacutesi teacutenyezője 003 A sziacutevoacutecsőbe beeacutepiacutetett szerelveacutenyek egyuumlttes veszteseacutegteacutenyezője 5 A meleg taacutepviacutez sűrűseacutege 983 kgm3 Milyen magasra kell elhelyezni a kondenzaacutetort a szivattyuacute sziacutevoacutecsonkja foumlleacute hogy a geodetikus hozzaacutefolyaacutes a kavitaacutecioacute mentes uumlzemet biztosiacutetsa (937 m negatiacutev sziacutevoacutemagassaacuteg) Vegye figyelembe a fuumlggőleges ndash a viacutezszinteseacutevel azonos aacutetmeacuterőjű eacutes csősuacuterloacutedaacutesi teacutenyezőjű ndash csőszakasz veszteseacutegeit is Mekkora a Thoma-feacutele σ kavitaacutecioacutes szaacutem (00334)

H [m] = 82 ndash 4800 Q [m3s]2 NPSH [m] = 16 + 1360 Q [m3s]2 103 002 m3s aacutellandoacute teacuterfogataacuteramuacute vizet szaacutelliacutetoacute viacutezszintes tengelyű oumlrveacutenyszivattyuacute

laboratoacuteriumi meacutereacutese soraacuten a 125 mm aacutetmeacuterőjű viacutezszintes tengelyű sziacutevoacutecsonkra koumltoumltt egyik veacutegeacuten nyitott U-csoumlves higanyos manomeacuteter kiteacutereacutese ∆ h A manomeacutetert a sziacutevoacutecsővel oumlsszekoumltő műanyag csőben levegő van a higany sűrűseacutege 13600 kgm3 A leacutegkoumlri nyomaacutes 1 bar a hideg viacutez teliacuteteacutesi gőznyomaacutesa a koumlrnyezeti hőmeacuterseacutekleten 2500 Pa Rajzolja le a meacutereacutesi elrendezeacutest A H [m] szaacutelliacutetoacutemagassaacuteg eacutes a ∆h [mm] manomeacuteter-kiteacutereacutes kapcsolataacutet a

minus=

∆minusminus

4

400

1 18h

eH


32

oumlsszefuumlggeacutessel iacuterhatjuk le Rajzolja meg a H( h∆ ) grafikonjaacutet h∆ = 20 300 380 385 390 mm kiteacutereacutesekneacutel szaacutemiacutetott pontokboacutel A szaacutelliacutetoacutemagassaacuteg 3 -os csoumlkkeneacuteseacutehez tartozoacute meacutereacutesi pontot tekintve kritikusnak szaacutemiacutetsa ki az NPSHr eacuterteacutekeacutet az adott teacuterfogataacuteramnaacutel (482 m)

104 Hataacuterozza meg a szuumlkseacuteges csőaacutetmeacuterőt (0073 m rarr NA 80 mm) ha a szivattyuacute olyan

zaacutert tartaacutelyboacutel sziacutev 30 dm3s vizet amelyben a nyomaacutes a viacutezfelsziacuten felett 40 kPa A sziacutevoacutecső egyeneacuterteacutekű hossza 5 m az adott hőmeacuterseacutekleten a viacutezgőz nyomaacutesa a hataacutergoumlrbeacuten 28 kPa a csősuacuterloacutedaacutesi teacutenyező 002 a szivattyuacute szuumlkseacuteges NPSH eacuterteacuteke a fenti teacuterfogataacuteramnaacutel NPSHr = 32 m eacutes a sziacutevoacutecsonk 3 m meacutelyen van a sziacutevoacuteoldali viacutezszint alatt

105 Mekkora a szuumlkseacuteges geodetikus hozzaacutefolyaacutes ha a szivattyuacute olyan tartaacutelyboacutel sziacutev

amelyben a folyadeacutek felsziacutene foumlloumltt a folyadeacutek hőmeacuterseacutekleteacutehez tartozoacute teliacuteteacutesi gőznyomaacutes uralkodik (5252 m) NPSHr = 5 m A sziacutevoacutecső egyeneacuterteacutekű hossza 6 m aacutetmeacuterője 150 mm a csősuacuterloacutedaacutesi teacutenyező 002 A szivattyuacute 20 dm3s 920 kgm3 sűrűseacutegű folyadeacutekmennyiseacuteget szaacutelliacutet A sziacutevoacutecsonk koumlzepe 200 mm-rel meacutelyebben van mint a viacutezszintes helyzetű szivattyuacute tengely Mekkora a Thoma-feacutele σ kavitaacutecioacutes szaacutem eacuterteacuteke ha a szaacutelliacutetoacutemagassaacuteg 50 m (01)


zaacutert tartaacutelyboacutel sziacutev 30 dm3s vizet amelyben a nyomaacutes a viacutezfelsziacuten felett 20 kPa A sziacutevoacutecső egyeneacuterteacutekű hossza 6 m az adott hőmeacuterseacutekleten a viacutezgőz nyomaacutesa a hataacutergoumlrbeacuten 25 kPa a csősuacuterloacutedaacutesi teacutenyező 0023 a szivattyuacute szuumlkseacuteges NPSHr eacuterteacuteke a fenti teacuterfogataacuteramnaacutel NPSHr = 37 m eacutes a sziacutevoacutecsonk 2 m meacutelyen van a sziacutevoacuteoldali viacutezfelsziacuten alatt


33

11 Ventilaacutetorok uumlzeme 111 1470min fordulatszaacutemuacute axiaacutelis ventilaacutetor 08 m3s 12 kgm3 sűrűseacutegű levegőt szaacutelliacutet

A jaacuteroacutekereacutek kuumllső aacutetmeacuterője 460 mm az agyaacutetmeacuterő 250 mm A levegő perduumllet mentesen leacutep a jaacuteroacutekereacutekbe Az abszoluacutet sebesseacuteg keruumlleti komponense a jaacuteroacutekereacutek utaacuten 6 ms Rajzolja meg leacutepteacutekhelyesen a beleacutepő eacutes kileacutepő sebesseacutegi haacuteromszoumlget a jaacuteroacutekereacutek lapaacutetok legnagyobb sugaraacuten Mekkora az ideaacutelis oumlssznyomaacutesnoumlvekedeacutes a jaacuteroacutekereacutekben Ebből mennyi a statikus nyomaacutesnoumlvekedeacutes eacutes utoacuteterelő raacutecs alkalmazaacutesaacuteval ez az eacuterteacutek mennyivel noumlvelhető Mennyivel noumlveli tovaacutebb a statikus nyomaacutest a gyűrű keresztmetszetet keacutetszereseacutere noumlvelő diffuacutezor 84-os hidraulikai hataacutesfok eseteacuten mekkora a teljes ventilaacutetor (jaacuteroacutekereacutek + vezetőkereacutek + diffuacutezor) nyomaacutesszaacutema

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34

112 Terelő neacutelkuumlli axiaacutelis aacutetoumlmleacutesű ventilaacutetor 2 m3s 12 kgm3 sűrűseacutegű levegőt szaacutelliacutet 120 Pa statikus nyomaacutesnoumlvekedeacutes eseteacuten Mekkora a 450 mm aacutetmeacuterőjű csőbe beeacutepiacutetett ventilaacutetor aacuteltal leacutetesiacutetendő oumlssznyomaacutes noumlveleacutes (215 Pa) Mekkora a ventilaacutetor hasznos hidraulikai teljesiacutetmeacutenye (430 W) Mekkora a ventilaacutetor hataacutesfoka ha a hajtoacutemotor a fenti uumlzemaacutellapotban 500 W teljesiacutetmeacutenyt ad le (86 ) A ventilaacutetor fordulatszaacutema 960min Mekkora a ventilaacutetor hangteljesiacutetmeacuteny szintje (778 dB) LW = 97+10lg[Q∆pouml(1η-1)]+32lg u2a [dB] A hangsebesseacuteg mint ismeretes 340 ms Rajzolja meg a ventilaacutetorlapaacutet legnagyobb sugaruacute pontjaacutehoz tartozoacute valoacutes sebesseacutegi haacuteromszoumlgeket (uk = 2262 ms cax = 1258 ms ∆cu = 792 ms) A kis aacutetmeacuterőjű jaacuteroacutekereacutekagy axiaacutelis sebesseacuteg noumlvelő hataacutesaacutetoacutel eacutes a fenti adatokboacutel ki nem szaacutemiacutethatoacute forgaacutesi- eacutes reacutesveszteseacutegektől eltekinthet

113 Takarmaacutenyszaacuteriacutetoacute szellőztető padozat egymaacutestoacutel elvaacutelasztott egyseacutegekre van osztva A

szaacuteriacutetoacute levegő ellaacutetaacutesaacutera keacutet azonos ∆pouml = 1200 ndash 300 [Pasdots2m6] Q2 jelleggoumlrbeacutejű centrifugaacutel ventilaacutetort eacutepiacutetettek be A szellőzőpadozatok ∆poumlpad = 900 [Pasdots2m6] Q2 bdquojelleggoumlrbeacutevelrdquo jellemezhetők Mekkora a szaacutelliacutetott levegő teacuterfogataacuterama ha egy padozatot egy ventilaacutetor laacutet el (1 m3s) Keacutet paacuterhuzamosan kapcsolt padozat eseteacuten mindkeacutet ventilaacutetorra szuumlkseacuteg van Mekkora lesz a leacutegaacuteram ha a keacutet ventilaacutetort sorba kapcsoljaacutek (1706 m3s) mekkora ha paacuterhuzamosan kapcsoljaacutek (2 m3s)

114 Siacutepaacutelya hoacuteaacutegyuacute axiaacutelventilaacutetora leacutegkoumlri nyomaacutesuacute 132 kgm3 sűrűseacutegű nyugvoacute hideg

levegőt gyorsiacutet 30 ms sebesseacutegre A jaacuteroacutekereacutek utaacuteni keverőteacuterben 4 kgs viacutezpermetet gyorsiacutetunk fel nyugalomboacutel a levegővel azonos sebesseacutegre Mennyi a viacutez impulzusvaacuteltozaacutesa (120 kgms2) mekkora nyomaacuteskuumlloumlnbseacuteg szuumlkseacuteges ehhez a 02 m2 keresztmetszetű keverőteacuter keacutet veacutege koumlzoumltt (600 Pa) Mekkora a ventilaacutetor statikus- eacutes oumlssznyomaacutes noumlveleacutese ha a hoacuteaacutegyuacuteboacutel a levegő-viacutezpermet kevereacutek koumlzvetlenuumll a leacutegkoumlrbe jut ki Rajzolja meg a statikus eacutes az oumlssznyomaacutes aacuteramvonal menti vaacuteltozaacutesaacutet (A kevereacutek toumlmegaacuterama a viacutezpermet eacutes a levegő toumlmegaacuteramaacutenak oumlsszege 1192 kgs A kevereacutek sebesseacutege 30 ms a 02 m2 keresztmetszetben iacutegy a kevereacutek teacuterfogataacuterama 6 m3s aacutetlagos sűrűseacutege 1987kg m3 iacutegy a kevereacutek dinamikus nyomaacutesa 894 Pa

115 Mekkora tuacutelnyomaacutes szuumlkseacuteges egy 22x40 m2 alapteruumlletű teniszsaacutetor kifesziacuteteacuteseacutehez ha

a saacutetorponyva toumlmege 3000 kg (334 Pa) A saacutetor felaacutelliacutetaacutesa soraacuten a oumlp∆ = 70 Pa- 42

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594 Pa

594 Pa

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35

saacutetorba fuacutejja az 13 kgm3 sűrűseacutegű levegőt A ventilaacutetor nyomoacutecsonkjaacutenak keresztmetszete 02 m2 A maacuter felaacutelliacutetott foumlldhoumlz leroumlgziacutetett feszes teniszsaacutetorban mekkora leacutegveszteseacuteget tud a ventilaacutetor poacutetolni aacutellandoacutesult aacutellapotban ha a szellőzőnyiacutelaacutesok szabadba nyiacuteloacute teljes feluumllete 005 m2 (0469 m3s) Mekkora ekkor a tuacutelnyomaacutes a saacutetorban (57 Pa)

116 Az aacutebraacuten laacutethatoacute PC processzor hűtő

axiaacutelventilaacutetor kerekeacutenek kuumllső aacutetmeacuterője 47 mm agyaacutetmeacuterője 215 mm fordulatszaacutema 2740min A gondos aacuteramlaacutestechnikai tervezeacutes alapjaacuten 85 -ra tehető a hidraulikai hataacutesfok de a kereacutek eacutes a haacutez koumlzoumltti taacuteg reacutesek miatt a volumetrikus hataacutesfok csupaacuten 75 A lapaacutetok be- eacutes kileacutepő szoumlge a lapaacutetok koumlzeacutepsugaraacuten β1 = 20ordm ill β2 = 40ordm Rajzolja meg ezen adatok alapjaacuten a be- eacutes kileacutepő sebesseacutegi haacuteromszoumlget szaacutemiacutetsa ki a ventilaacutetor aacuteltal szaacutelliacutetott 125 kgm3 sűrűseacutegű leacutegaacuteramot (1104 lmin) eacutes az aacuteltala leacutetesiacutetett oumlssznyomaacutes-noumlvekedeacutest a tervezeacutesi uumlzemaacutellapotban (1453 Pa)

117 Autoacutehűtő neacutevleges feluumllete 012 m2 aacuteramlaacutesi veszteseacutegteacutenyezője ζ = 12 A hűtőventilaacutetor jelleggoumlrbe uumlzemi tartomaacutenya a ∆pouml = 147[Pa] ndash 300[Pamiddots2m6](Q-03)2 fuumlggveacutennyel koumlzeliacutethető Az 11 kgm3 sűrűseacutegű meleg levegőt a hűtőn keresztuumll sziacutevja a ventilaacutetor melynek jaacuteroacutekereacutek aacutetmeacuterője 310 mm agyaacutetmeacuterője 140 mm A ventilaacutetorboacutel a levegő szabadon a motorteacuterbe aacuteramlik eacutes ott lefeacutekeződik ekoumlzben nyomaacutesa ismeacutet eleacuteri a p0 leacutegkoumlri nyomaacutest Iacuterja fel az aacutelloacute ndash koumlzlekedeacutesi laacutempaacutenaacutel jaacuteroacute motorral vaacuterakozoacute ndash geacutepkocsi eseteacuten a Bernoulli-egyenletet a szabadboacutel a hűtőn keresztuumll a ventilaacutetor előtti pontig eacutes a maacutesik Bernoulli-egyenletet a ventilaacutetor utaacuteni ponttoacutel a motorteacuterig Ezek eacutes a ventilaacutetor jelleggoumlrbeacuteje alapjaacuten hataacuterozza meg a leacutegaacuteramot (06828 m3s) Rajzolja meg leacutepteacutekhelyesen a statikus dinamikus eacutes oumlssznyomaacutes vaacuteltozaacutesaacutet a geacutepkocsi előtti ponttoacutel a hűtőn ventilaacutetoron aacutet a motorteacuterig tartoacute aacuteramvonal menteacuten

118 Ventilaacutetor meacuterőaacutellomaacuteson a sziacutevoacutecső aacutetmeacuterője 200 mm a nyomoacutecső aacutetmeacuterője 150 mm a sziacutevoacutecsőbe eacutepiacutetett meacuterőperem szűkiacutetőelem aacutetmeacuterője 140 mm aacutetfolyaacutesi szaacutema 058 Az expanzioacutes szaacutem 1-nek vehető Egy uumlzemaacutellapotban a meacuterőperemre kapcsolt viacuteztoumllteacutesű manomeacuteter kiteacutereacutese 65 mm a nyomoacute eacutes sziacutevoacutecsonk koumlzeacute kapcsolt viacuteztoumllteacutesű manomeacuteter kiteacutereacutese 110 mm a szaacutelliacutetott levegő sűrűseacutege 118 kgm3 A motor 2 Nm nyomateacutekkal hajtja a ventilaacutetort fordulatszaacutema 2880min Mennyi a ventilaacutetor statikus eacutes oumlssznyomaacutes noumlveleacutese statikus eacutes oumlsszhataacutesfoka ebben az uumlzemaacutellapotban Megoldaacutes A teacuterfogataacuteram ill a sebesseacutegek a sziacutevoacute- eacutes nyomoacutecsonkban


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119 Az aacutebraacuten laacutethatoacute irodai ventilaacutetor 3 lapaacutetos axiaacutelis

jaacuteroacutekerekeacutenek fordulatszaacutema a legerősebb fokozatban n3 = 1160min A jaacuteroacutekereacutek kuumllső aacutetmeacuterője 360 mm a kereacutekagyat eltakaroacute lemez aacutetmeacuterője 125 mm A jaacuteroacutekereket mindkeacutet oldalon veacutedő droacutethaacuteloacute droacutet- vastagsaacutega δ = 09 mm a droacutethaacuteloacute osztaacutesa a = 109 mm a rombusz alakuacute haacuteloacuteszemek hegyes szoumlge α = 62ordm Mekkora a haacuteloacute f szabadteljes keresztmetszet viszonya

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koumlzvetlenuumll a haacuteloacute utaacuten ndash ahol a leacutegsugaacuter meacuteg nem teruumll szeacutet ndash v =34 ms Rajzolja meg a nyomaacutes vaacuteltozaacutesaacutet a ventilaacutetor forgaacutestengelyeacutevel paacuterhuzamos aacuteramvonal menteacuten Mekkora a ventilaacutetor aacuteltal leacutetesiacutetett ∆pouml oumlssznyomaacutes noumlvekedeacutes eacutes a ventilaacutetor hasznos teljesiacutetmeacutenye A motor neacutevleges teljesiacutetmeacutenye 50W Mieacutert van szuumlkseacuteg erre a tuacutelmeacuteretezett teljesiacutetmeacutenyre Mekkora az oumlssznyomaacutes noumlvekedeacutes az 1 fokozatban melynek fordulatszaacutema n1 = 813min Rajzolja meg a beleacutepő eacutes kileacutepő sebesseacutegi haacuteromszoumlget a 120 mm-es koumlzbuumllső sugaacuteron a hidraulikus hataacutesfok becsuumllt eacuterteacuteke 85

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12 Volumetrikus elven m űkoumldő geacutepek hidraulikus hajtaacutesok 121 Egy fogaskereacutek-szivattyuacute jelleggoumlrbeacuteje ][004309311min][ 3 barpdmQ ∆sdotminus= 35

bar nyomaacuteskuumlloumlnbseacutegneacutel a volumetrikus hataacutesfok 9874 Mekkora az elmeacuteleti teacuterfogataacuteram eacutes a geometriai teacuterfogat ha a tengely fordulatszaacutema 80 fordmin A szivattyuacute 85-os oumlsszhataacutesfoka eseteacuten mekkora a hajtoacute nyomateacutek

Megoldaacutes A teacuterfogataacuteram 35 bar ellennyomaacutesnaacutel

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fuumlggő 800 kg toumlmegű terhet 12 mmin sebesseacuteggel kell emelni Mekkora legyen a 92 volumetrikus hataacutesfokuacute 960min fordulatszaacutemuacute fogaskereacutek-szivattyuacute geometriai teacuterfogataacuterama Mekkora teacuterfogatot szaacutelliacutet a szivattyuacute egy fordulat alatt Mekkora a szivattyuacute aacuteltal leacutetesiacutetendő tuacutelnyomaacutes eacutes mekkora a szivattyuacutet hajtoacute motor teljesiacutetmeacutenye eacutes tengely nyomateacuteka A fogaskereacutek-szivattyuacute hataacutesfoka 74 Keacutesziacutetsen vaacutezlatot a fogaskereacutek-szivattyuacuteroacutel a forgaacutesiraacuteny a sziacutevoacute- eacutes a nyomoacutecsonk bejeloumlleacuteseacutevel Mekkora lesz a szivattyuacute folyadeacutekszaacutelliacutetaacutesa teljesiacutetmeacutenyfelveacutetele eacutes nyomateacutekigeacutenye ha ugyanezt a terhet emeljuumlk fel de 1440min fordulatszaacutemuacute motorral hajtjuk

Megoldaacutes A fogaskereacutek szivattyuacute aacutebraacutejaacutet laacutesd az előadaacutesjegyzetben A fogaskereacutek-szivattyuacute geometriai teacuterfogataacuterama eacutes geometriai teacuterfogata

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123 Fuumlggőleges tengelyű munkahenger dugattyuacuteja 700 kg toumlmegű terhet tart amelyet

legfeljebb 64 mms sebesseacuteggel szabad suumlllyeszteni A henger belső aacutetmeacuterője 50 mm a dugattyuacuteruacutedeacute - ezen taacutemaszkodik a suacutely ndash 28 mm A hengert taacuteplaacuteloacute fogaskereacutek-szivattyuacute jelleggoumlrbeacuteje a meacutereacutesi diagram szerint Q [litermin] = 86 ndash 00467 ∆ p[bar] A dugattyuacute aloacutel a 970 kgm3 sűrűseacutegű hidraulika-olaj fojtoacuteszelepen keresztuumll taacutevozik amelynek aacutetfolyaacutesi teacutenyezője 07 Mekkora legyen a fojtoacuteszelep maximaacutelisan nyithatoacute keresztmetszete Mekkora a fogaskereacutek-szivattyuacute legnagyobb hasznos teljesiacutetmeacutenye

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124 Rajzolja meg egy fogaskereacutek-szivattyuacutet tartalmazoacute hidraulikus taacutepegyseacuteggel hajtott

forgoacute mozgaacutest előaacutelliacutetoacute fogaskereacutek-motoros hidraulikus hajtaacutes kapcsolaacutesi vaacutezlataacutet paacuterhuzamos fojtaacutessal A vaacutezlaton az elemek szimboacutelumaacutehoz iacuterjon megnevezeacuteseket Mi a kapcsolat a hidromotor tengelynyomateacuteka eacutes a hasznosiacutetott nyomaacuteseseacutes koumlzoumltt

125 Ismert Q(∆p) jelleggoumlrbeacutejű hidraulikus taacutepegyseacuteggel hidromotort hajtunk Adott a

hidromotor ηvm volumetrikus hataacutesfoka a hidromotor Vgm geometriai folyadeacuteknyeleacutese a hidromotor ηoumlm oumlsszhataacutesfoka A hidromotor neacutevleges Mm terhelő nyomateacuteka ismereteacuteben hataacuterozza meg a hidromotor tengelyeacutenek nm fordulatszaacutemaacutet Vaacutezolja fel az elrendezeacutest olajtartaacutely taacutepegyseacuteg csővezeteacutekek hidromotor ndash tengelyeacuten teherrel

126 Rajzolja meg egy hidraulikus taacutepegyseacuteg leacutenyeges elemeinek szimbolikus kapcsolaacutesi

vaacutezlataacutet eacutes iacuterja fel a taacutepegyseacuteg Q(p∆ ) jelleggoumlrbeacutejeacutenek meghataacuterozaacutesaacutehoz szuumlkseacuteges egyenleteket (erőegyensuacutely kontinuitaacutes Bernoulli-egyenlet) eacutes az egyenletekben szereplő mennyiseacutegeket a nyomaacuteshataacuteroloacute szelep metszet-rajzaacuten jeloumllje be


42

13 Meacutereacutesekhez kapcsoloacutedoacute feladatok 131 Szaacutemiacutetsa ki az aacutebraacuten laacutethatoacute elrendezeacutes eseteacuten a teacuterfogataacuteramot

Adatok mmd 50=

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45

14 Elmeacuteleti mintakeacuterdeacutesek 141 Milyen moacutedszereket ismer radiaacutelis aacutetoumlmleacutesű oumlntoumltt jaacuteroacutekerekek neacutevleges uumlzemi

pontjaacutenak forgaacutecsoloacute megmunkaacutelaacutessal toumlrteacutenő beaacutelliacutetaacutesaacutera Vaacutezlattal eacutes roumlvid szoumlveges magyaraacutezattal ismertesse ezen moacutedszerek leacutenyegeacutet (l előadaacutesjegyzet 11 fejezet)

142 Rajzoljon stabilis eacutes labilis jelleggoumlrbe aacuteggal rendelkező szivattyuacute jelleggoumlrbeacutet

Rajzoljon olyan csővezeteacutek jelleggoumlrbeacutet amely a szivattyuacuteval labilis uumlzemet eredmeacutenyez eacutes magyaraacutezza meg az uumlzem labilitaacutesaacutenak okaacutet Rajzoljon olyan csővezeteacutek jelleggoumlrbeacutet amelyik a szivattyuacute jelleggoumlrbe bdquolabilisrdquo aacutegaacuten stabilis munkapontot eredmeacutenyez Mi az uumlzem stabilitaacutesaacutenak felteacutetele (l előadaacutesjegyzet 91 fejezet)

143 Hasonliacutetsa oumlssze kis koumlzepes eacutes nagy jellemző fordulatszaacutemuacute szivattyuacutek szaacutelliacutetoacutemagassaacuteg teljesiacutetmeacutenyfelveacutetel eacutes hataacutesfok diagramjainak alakjaacutet az optimaacutelis pont adataival dimenzioacutetlaniacutetott fajlagos mennyiseacutegek koordinaacutetarendszereacuteben Milyen koumlvetkezteteacutesek vonhatoacutek le a diagramokboacutel

144 Definiaacutelja ventilaacutetorok statikus eacutes oumlssznyomaacutes noumlvekedeacuteseacutet Vaacutezlaton mutassa meg az oumlssznyomaacutes komponenseinek vaacuteltozaacutesaacutet egy egyfokozatuacute terelőraacutecs neacutelkuumlli a kileacutepeacutesneacutel a jaacuteroacutekereacutek utaacuten diffuacutezorral ellaacutetott axiaacutelis aacutetoumlmleacutesű ventilaacutetor eseteacuten egy koumlzepes aacuteramvonalon

145 Ismertesse az oumlrveacutenygeacutepek elmeacuteleti eacutes valoacutes jelleggoumlrbeacutejeacutet haacutetrahajloacute (β2lt90o) lapaacutetozaacutes eseteacuten H-Q diagram segiacutetseacutegeacutevel Indokolja az elteacutereacutesek fizikai okait azaz a veszteseacutegforraacutesokat eacutes azok vaacuteltozaacutesaacutet a teacuterfogataacuteram fuumlggveacutenyeacuteben

146 Ismertesse a leacutegszaacutelliacutetoacute geacutepek osztaacutelyozaacutesaacutet a kompresszioacute-viszony fuumlggveacutenyeacuteben Melyek a fő kuumlloumlnbseacutegek az egyes geacuteptiacutepusok koumlzoumltt

147 Rajzoljon előrehajloacute normaacutelis eacutes haacutetrahajloacute lapaacutetozaacutesuacute centrifugaacutelis jaacuteroacutekereket eacutes aacutebra segiacutetseacutegeacutevel ismertesse az elmeacuteleti jelleggoumlrbeacutejuumlket (levezeteacutes nem szuumlkseacuteges)

148 Ismertesse az oumlrveacutenyelven műkoumldő eacutes a volumetrikus geacutepek koumlzoumltti fő kuumlloumlnbseacutegeket (Jellemző fordulatszaacutem Q-∆p kapcsolat) Ismertessen neacutehaacuteny (legalaacutebb 2) volumetrikus geacuteptiacutepust

149 Rajzolja fel egy centrifugaacutelszivattyuacute forgoacute jaacuteroacutekerekeacutenek haacutetlapjaacuten kialakuloacute nyomaacuteseloszlaacutest jellegre helyesen Adja meg a nyomaacuteseloszlaacutes egyenleteacutet eacutes a nyomaacuteseloszlaacutesboacutel szaacutermazoacute axiaacutelis erőkomponens kiszaacutemiacutetaacutesaacutenak alapoumlsszefuumlggeacuteseacutet (a veacutegkeacuteplet nem kell) Definiaacutelja a keacutepletben szereplő mennyiseacutegeket

1410 Milyen veszteseacutegtiacutepusok jelentkeznek egy szivattyuacuteban Ismertesse az egyes veszteseacutegek fizikai okaacutet keletkezeacutesi helyeacutet Vaacutelaszaacutet teljesiacutetmeacutenyszalag segiacutetseacutegeacutevel szemleacuteltesse

1411 Egy oumlrveacutenyszivattyuacute szaacutelliacutetaacutesa az utoacutebbi időben drasztikusan lecsoumlkkent (az optimaacutelis ndash tervezeacutesi - teacuterfogataacuteram alig 20-aacutera) eacutes a geacutep zaj- eacutes rezgeacutesszintje ezzel


46

egyidejűleg jelentősen megnőtt (kuumlloumlnoumlsen a csapaacutegyak koumlzeleacuteben) A szivattyuacute egy kondenzaacutecioacutes tartaacutelyboacutel szaacutelliacutet a geacutep maga 3 m-rel a tartaacutelybeli viacutezszint felett helyezkedik el A szivattyuacutezott koumlzeg meleg (70oC-os) viacutez Mi lehet a meghibaacutesodaacutes oka A vaacutelaszaacutet indokolja

1412 Mutassa meg a T-s diagramban bejeloumllt teruumlletek alapjaacuten egy politroacutep kompresszioacute hataacutesfokaacutenak szaacutemiacutetaacutesaacutet

1413 Rajzolja meg a toumlmegegyseacutegre feliacutert Bernoulli entalpia eacutes a teacuterfogataacuteram vaacuteltozaacutesaacutet egy szivattyuacute sziacutevoacute eacutes nyomoacutecsonkja koumlzoumltt (sziacutevoacuteteacuter ndash jaacuteroacutekereacutek ndash nyomoacuteteacuter jeloumlleacuteseacutevel)

1414 Egy szivattyuacutezaacutesi feladat tervezeacutesi alapadatai szaacutelliacutetoacutemagassaacuteg 10 m a szaacutelliacutetandoacute viacutez teacuterfogataacuterama 24 m3h Adja meg a feladat ellaacutetaacutesaacutera alkalmas 1440 fordmin fordulatszaacutemuacute szivattyuacute tiacutepusaacutet

1415 Oumlrveacutenyszivattyuacute 45m3h viacutezmennyiseacuteget szaacutelliacutet egy 200mm-es 53 km hosszuacute csővezeteacutekben melyben az alapnyomaacutes 4bar tuacutelnyomaacutes A szivattyuacute aacuteramszuumlnet miatt 2 s alatt leaacutell Adjon becsleacutest a kialakuloacute nyomaacutesdepresszioacute eacuterteacutekeacutere abszoluacutet nyomaacutesban A nyomaacuteshullaacutem terjedeacutesi sebesseacutege 1100ms Javasoljon keacutet kuumlloumlnboumlző veacutedekezeacutesi moacutedszert

1416 Egy vizet szaacutelliacutetoacute viacutezszintes 200m hosszuacute 300mm aacutetmeacuterőjű eacutes l=0018-as csősuacuterloacutedaacutesi teacutenyezőjű csővezeteacutek feleacuteneacutel (100m) egy olyan elveacuteteli pont talaacutelhatoacute mely a (mindenkori) teacuterfogataacuteram feleacutet fogyasztja Iacuterja fel a csővezeteacutek jelleggoumlrbeacutejeacutet H[m]=B (Q[m3h])2 alakban

1417 Egy vizet szaacutelliacutetoacute szivattyuacute H=40m Q=20m3h munkapontban dolgozik egy keringető csővezeteacutekrendszerre (melynek statikus szaacutelliacutetoacutemagassaacutega zeacuterus) A geacutepet hajtoacute villamos motor fordulatszaacutemaacutet megkeacutetszerezik Adjon becsleacutest az uacutej motor szuumlkseacuteges teljesiacutetmeacutenyeacutere

1418 Egy meacutelytengeri olajkitermeleacutesben alkalmazott viacutez visszataacuteplaacutelaacutesi feladat soraacuten 130liters teacuterfogataacuteramot kell szaacutelliacutetanunk 400 bar nyomaacuteskuumlloumlnbseacuteg mellett A szivattyuacutet hajtoacute villanymotor fordulatszaacutema 1500min a visszanyomott viacutez sűrűseacutege 1016 kgm3 Javasoljon szivattyuacutetiacutepust eacutes indokolja a vaacutelasztaacutesaacutet

1419 Egy 38 km hosszuacute ivoacuteviacutez taacutevvezeteacutekben a hullaacutemsebesseacuteg 800 ms az aacuteramlaacutesi sebesseacuteg 12 ms az aacutellandoacutesult aacutellapotbeli alapnyomaacutes 5 bar tuacutelnyomaacutes Sebesseacutegben lineaacuteris zaacuteraacutest felteacutetelezve mennyi idő alatt zaacuterhatoacute el a vezeteacutek elejeacuten talaacutelhatoacute toloacutezaacuter hogy a nyomaacutes ne csoumlkkenjen leacutegkoumlri nyomaacutes alaacute

1420 Iacuterja fel egy radiaacutelis aacutetoumlmleacutesű szivattyuacute jaacuteroacutekereacutek haacutetlapjaacutera hatoacute axiaacutelis erő szaacutemiacutetaacutesaacutehoz a haacutetlap menti nyomaacutesvaacuteltozaacutes egyenleteacutet felteacutetelezve hogy a haacutetlap eacutes a haacutez koumlzoumltti teacuterben a folyadeacutek aacutetlagos szoumlgsebesseacutegeacutet ωf-et tovaacutebbaacute a geometriai meacutereteket eacutes a jaacuteroacutekereacutek aacuteltal leacutetesiacutetett nyomaacutest ismeri

1421 Sorolja fel a radiaacutelis aacutetoumlmleacutesű egy vagy toumlbbfokozatuacute oumlrveacutenyszivattyuacutek axiaacutelis erő csoumlkkenteacutesi tehermentesiacuteteacutesi moacutedjait eacutes roumlviden magyaraacutezza el hogy milyen elven alapulnak ezek a moacutedszerek


2

1 Gaacutezok sűriacuteteacutese kompresszorok munkafolyamatai

11 Gaacutezt 1 bar abszoluacutet nyomaacutesroacutel 4 bar tuacutelnyomaacutesra sűriacutetuumlnk a gaacutezaacutellandoacute 288 JkgK az aacutellandoacute nyomaacuteson meacutert fajhő cp = 1000 JkgK Mekkora a κ kitevő A sűriacuteteacutesi folyamatot leiacuteroacute politropa kitevője n = 15 Hataacuterozza meg az ideaacutelis (izentroacutepikus) eacutes valoacutes (politropikus) kompresszioacute veacuteghőmeacuterseacutekleteacutet eacutes a politropikus fajlagos munkaacutet Hataacuterozza meg az izotermikus hasznos munkaacutet eacutes az izotermikus hataacutesfokot A besziacutevott gaacutez sűrűseacutege 116 kgm3 A houml asymp h koumlzeliacuteteacutes elfogadhatoacute

Megoldaacutes

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Ebből

= minus minus = minus 1000

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= =

10

288 lowast 116= 2993

Az izentropikus kompresszioacute veacuteghőmeacuterseacuteklete

=

= 2993 51

= 47598

A politropikus kompresszioacute veacuteghőmeacuterseacuteklete

=

= 2993 51

= 5118

Az izotermikus hasznos munka

13 = 13 =

10

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1 = 1387

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= minus = 1000 lowast 5118 minus 2993 = 2125

= 13 =

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= 1313 minus 1

minus = 15

15 minus 12885118 minus 2993 = 1832

12 Ideaacutelis gaacutez izentroacutepikus aacutellapotvaacuteltozaacutesaacutet felteacutetelezve szaacutemiacutetsa ki egy mamp = 2 kgs

toumlmegaacuteramot szaacutelliacutetoacute kompresszor sziacutevoacutecsonkjaacutenak keresztmetszeteacutet (00099 m2) ha a besziacutevott levegőt a sziacutevoacutecsonkban c = 200 ms sebesseacutegre kell felgyorsiacutetani A kompresszor nyugvoacute T0 = 290 K hőmeacuterseacutekletű p0 = 1 bar abszoluacutet nyomaacutesuacute teacuterből sziacutevja be a levegőt A fajhő cp = 1000 JkgK a fajhőviszony κ = 14 (Abszoluacutet rendszerről leacuteveacuten szoacute az oumlsszentalpia aacutellandoacute )

13 Hataacuterozza meg ndash izentroacutepikus hasznos munkaveacutegzeacutest felteacutetelezve ndash az adiabatikus

aacutellapotvaacuteltozaacutes T2 veacuteghőmeacuterseacutekleteacutet (490 K) eacutes h2 entalpiaacutejaacutet (2181 kJkg) t0 = 0 degC-hoz keacutepest ha R = 0288 kJkgsdotK t1 = 10 degC p1 = 1 bar p2 = 4 bar (abszoluacutet nyomaacutes) κ = 14 az adiabatikus kompresszioacute veszteseacutege Yrsquo = 70 kJkg Szaacutemiacutetsa ki a cp fajhőt


3


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Lp

pp ρλ=minus


Megoldaacutes


( ) barvD

Lppp 541146

2

85

60

750000180108021080

22 25252

11


2

2

1

1

ρ=

ρpp

3

1

212 2612

80

541185 mkg

p

p ==ρ=ρ


htskgvD

vAm 7553815346854

60

4

2

11

2

11 ==π=ρπ=ρ=amp


4



pv 444469361

2

12 ===


kg

MJ

p

pp

n

nY

n

n

pol 25015411

80

2612

105411

1451

4511

1

451

14515

1

2

1

2

2 =

minus

minus=

minus

ρminus=

minusminus



384300058 058

6613pol

belső

P MW








35

0

00 1661

300286

10mkg

TR

p=

sdot=

sdot=ρ

smA

qc 7203

1250

5242

11 =

sdotsdot==

π


Kkg

JRc p sdot

=minus

sdot=

minussdot= 1001

141

28641

1κκ

Kc

cTT

p

327910012

7203300

2

221

01 =sdot

minus=sdot

minus=



5

70610327928641

7203

1

1 lt=sdotsdot

=sdotsdot

=TR

cMa

κ



3

141

11

1

0

101 9750

300

32791661

m

kg

T

T=

sdot=

sdot=

minusminusκρρ


Kp

pTT

n

n

749577882

1043279

541

15415

1

1

212 =

sdotsdot=

sdot=

minusminus


3

1541

11

1

1

212 8212

3279

74959750

m

kg

T

T n

=

sdot=

sdot=

minusminusρρ

s


s

m

A

mc 470

12508212

44242

22 =

sdotsdot

sdot=

sdot=

πρamp


( ) ( )kg

kJ

kg


kWW

mcc

YP bep

484483967

4422

7203470216616

2

2221

22

asymp=

=sdot

minus+=sdot

minus+= amp


( ) ( )

kg

kJ

kg

J

TTRn

nYpol

4176176421

327964952861541

541

1 12

asymp=

=minussdotsdotminus

=minussdotsdotminus

=


barPaT

pTp 4210422

7495

104300 5

5

2



6


Kp

pTT s 9445

77882

1044279

41

1415

1

1

212 =

sdotsdot=

sdot=

minusminusκ

κ

( ) ( )kg

kJ

kg


kg

kJ

kg

J

p


77882

104ln4279286ln

5

1

21 asymp=

sdotsdotsdot=

sdotsdot=


250

300

350

400

450

500

65 66 67 68 69 7 71 72 73 74 75

s [kJkgK]

h[kJkg]

11090705v = 03

p=051234


7










8





2

320max 44

lg2900480940


opt

nqη








Megoldaacutes




9


3 2

e

3e

e kg me m s

H H h 60 12 72m

P 28033kW mQ 00397

gH 1000 981 72m s

prime= + = + =

= = =ρ times times


hidr vole e

H 60 Q 0035833 eacutes 882

H 72 Q 00397η = = = η = = =







Megoldaacutes





2

2

max

2bQ


minus=+=

mQ

aQ

Qa QQQ 742

2max

max0

2

max ==

minus == 522

111486240

74msa =sdot=

mbQ QQ 65max2 ==

52

222297

240

65msb ==

022

2 max

=+

minus= bQQ

QadQ

dh sm

Q

ba

aQopt 10

23max =

+=

( ) mbQQ


2

2

max =+

minus=





10

mgQ

PH

opt

szivh 68825819100010

25200 =sdotsdot

==ρ

mhHHe 85426166168825 =+=+=



0237098426

640

===belső

tsts P

Pν


11




94









Megoldaacutes



opt

d2 000731 (Q 95) 0

dQ


opt

m mQ 95 0002639

h s= =


2

6 opt1fokozatopt

H (Q ) 68 02 95H 8325m

6 6

minus times= = =


12

34

optq

1opt

Qn n 299



12


444444 3444444 21

4444 84444 76

η

ηρ

η 23

233

59007310660

20689810

)(

)(

)(

)(

minus

sdotminus

sdotminussdot

sdot

=sdotsdotsdot==

h

mQ

h

mQ

s

mQ

Q

QHgQ

Q

QPP

H

hbev





)0(

)0(

)(

)(lim

)(

)(

)(

)(lim

0 ηηhh

Qtx

P

Q

QP

tg

tf

xg

xf =rarr=rarrrarr


[ ]

[ ]

22

32

2

332

6

2

232

5

23

22

3600

593600007310660

36002068

3600

1593600007310660

360020689810

minussdotsdotminus


=

sdot

minus

sdot

sdot

minusminus

sdot

sdot

minussdot

sdot

sdot=

Q

QgQg

h

sh

m

s

mQ

h

s

m

h

s

mQ

h

s

m

hm

s

mQ

ms

kg

P

H

bev

ρρη

4444444444444 34444444444444 21

44444444 844444444 76


13


WggP

P h

Qbev 1334

3600

59023600007310

036006068

)0(

)0(

2

22

0

=



ρρη



2)










2

2 40000160 Qm

smH sdotminus= )




14

Megoldaacutes


1 2045175 07875 0773σδ ψψ minus= = = rarr =

21 4

0287ψϕ

δ

= =

2

3322 2 22

2

4 4 27000 3600761

0287 1440 604

Q QmmD

nD nD

u

ϕπ ϕπ ππ


123


222

120773 574 1528

2 2p Pau



15











ζkD

ζkD ζkD

ζkd

ζ2kd

4 Q

Q

210 m2 m

3 m

26m11 m

7 m

emptyD

emptyd

9 m




16


mD

L

g

vHH stc 1123150

070

300190

8192

51221

2

2

1

21

1 =

++sdotsdotsdot

+=

+++= ζλ


mv

D

L

g

vHH stc 1123150

10

300190

8192221

2

22

2

22

2 =

++sdotsdotsdot

+=

+++= ζλ

Ebből

s

mv 7418192

15010

300190

2211232 =sdotsdot

++sdot

minus=

s

dm

s

mv

Dv

DQ

3322

2

22

1

21 41901940741

4

1051

4

070

44==+=+=

ππππ


kWP

Pmsz

hvill 776

650

44 ===+η





Megoldaacutes


17

smD

Qv 6522

4120

030

4 22=

sdot==

ππ

++minus++=minus= 1

211

2

222

12 22z

g

v

g

pz

g

v

g

peeHber ρρ



spp =1 spp =2

022 1

211

2

222

12 =

++minus++=minus= z

g

v

g

pz

g

v

g

peeHber ρρ

Hg

v

D

LhHH berc =+=+=

20


Innen mv

gHDL 5910

65220250

120688192222

=sdot

sdotsdotsdot==λ

42πD


52

24

162

DCQ

DDHgL sdot=


λπ

5 aacutellD

LC ==

mmD 160 = mD

DLD

D

LDCL 3835

120160

91055

55

5 =

sdot=

sdot=sdot=sdot=


55 D

k

DC

K

L

KAacutesziv =



konstans


D




18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













3


ln

273 (e 1)p

s R p

c

ph c+

= minus








11

22

21

22v

D

Lp

pp ρλ=minus


Megoldaacutes


( ) barvD

Lppp 541146

2

85

60

750000180108021080

22 25252

11


2

2

1

1

ρ=

ρpp

3

1

212 2612

80

541185 mkg

p

p ==ρ=ρ


htskgvD

vAm 7553815346854

60

4

2

11

2

11 ==π=ρπ=ρ=amp


4



pv 444469361

2

12 ===


kg

MJ

p

pp

n

nY

n

n

pol 25015411

80

2612

105411

1451

4511

1

451

14515

1

2

1

2

2 =

minus

minus=

minus

ρminus=

minusminus



384300058 058

6613pol

belső

P MW








35

0

00 1661

300286

10mkg

TR

p=

sdot=

sdot=ρ

smA

qc 7203

1250

5242

11 =

sdotsdot==

π


Kkg

JRc p sdot

=minus

sdot=

minussdot= 1001

141

28641

1κκ

Kc

cTT

p

327910012

7203300

2

221

01 =sdot

minus=sdot

minus=



5

70610327928641

7203

1

1 lt=sdotsdot

=sdotsdot

=TR

cMa

κ



3

141

11

1

0

101 9750

300

32791661

m

kg

T

T=

sdot=

sdot=

minusminusκρρ


Kp

pTT

n

n

749577882

1043279

541

15415

1

1

212 =

sdotsdot=

sdot=

minusminus


3

1541

11

1

1

212 8212

3279

74959750

m

kg

T

T n

=

sdot=

sdot=

minusminusρρ

s


s

m

A

mc 470

12508212

44242

22 =

sdotsdot

sdot=

sdot=

πρamp


( ) ( )kg

kJ

kg


kWW

mcc

YP bep

484483967

4422

7203470216616

2

2221

22

asymp=

=sdot

minus+=sdot

minus+= amp


( ) ( )

kg

kJ

kg

J

TTRn

nYpol

4176176421

327964952861541

541

1 12

asymp=

=minussdotsdotminus

=minussdotsdotminus

=


barPaT

pTp 4210422

7495

104300 5

5

2



6


Kp

pTT s 9445

77882

1044279

41

1415

1

1

212 =

sdotsdot=

sdot=

minusminusκ

κ

( ) ( )kg

kJ

kg


kg

kJ

kg

J

p


77882

104ln4279286ln

5

1

21 asymp=

sdotsdotsdot=

sdotsdot=


250

300

350

400

450

500

65 66 67 68 69 7 71 72 73 74 75

s [kJkgK]

h[kJkg]

11090705v = 03

p=051234


7










8





2

320max 44

lg2900480940


opt

nqη








Megoldaacutes




9


3 2

e

3e

e kg me m s

H H h 60 12 72m

P 28033kW mQ 00397

gH 1000 981 72m s

prime= + = + =

= = =ρ times times


hidr vole e

H 60 Q 0035833 eacutes 882

H 72 Q 00397η = = = η = = =







Megoldaacutes





2

2

max

2bQ


minus=+=

mQ

aQ

Qa QQQ 742

2max

max0

2

max ==

minus == 522

111486240

74msa =sdot=

mbQ QQ 65max2 ==

52

222297

240

65msb ==

022

2 max

=+

minus= bQQ

QadQ

dh sm

Q

ba

aQopt 10

23max =

+=

( ) mbQQ


2

2

max =+

minus=





10

mgQ

PH

opt

szivh 68825819100010

25200 =sdotsdot

==ρ

mhHHe 85426166168825 =+=+=



0237098426

640

===belső

tsts P

Pν


11




94









Megoldaacutes



opt

d2 000731 (Q 95) 0

dQ


opt

m mQ 95 0002639

h s= =


2

6 opt1fokozatopt

H (Q ) 68 02 95H 8325m

6 6

minus times= = =


12

34

optq

1opt

Qn n 299



12


444444 3444444 21

4444 84444 76

η

ηρ

η 23

233

59007310660

20689810

)(

)(

)(

)(

minus

sdotminus

sdotminussdot

sdot

=sdotsdotsdot==

h

mQ

h

mQ

s

mQ

Q

QHgQ

Q

QPP

H

hbev





)0(

)0(

)(

)(lim

)(

)(

)(

)(lim

0 ηηhh

Qtx

P

Q

QP

tg

tf

xg

xf =rarr=rarrrarr


[ ]

[ ]

22

32

2

332

6

2

232

5

23

22

3600

593600007310660

36002068

3600

1593600007310660

360020689810

minussdotsdotminus


=

sdot

minus

sdot

sdot

minusminus

sdot

sdot

minussdot

sdot

sdot=

Q

QgQg

h

sh

m

s

mQ

h

s

m

h

s

mQ

h

s

m

hm

s

mQ

ms

kg

P

H

bev

ρρη

4444444444444 34444444444444 21

44444444 844444444 76


13


WggP

P h

Qbev 1334

3600

59023600007310

036006068

)0(

)0(

2

22

0

=



ρρη



2)










2

2 40000160 Qm

smH sdotminus= )




14

Megoldaacutes


1 2045175 07875 0773σδ ψψ minus= = = rarr =

21 4

0287ψϕ

δ

= =

2

3322 2 22

2

4 4 27000 3600761

0287 1440 604

Q QmmD

nD nD

u

ϕπ ϕπ ππ


123


222

120773 574 1528

2 2p Pau



15











ζkD

ζkD ζkD

ζkd

ζ2kd

4 Q

Q

210 m2 m

3 m

26m11 m

7 m

emptyD

emptyd

9 m




16


mD

L

g

vHH stc 1123150

070

300190

8192

51221

2

2

1

21

1 =

++sdotsdotsdot

+=

+++= ζλ


mv

D

L

g

vHH stc 1123150

10

300190

8192221

2

22

2

22

2 =

++sdotsdotsdot

+=

+++= ζλ

Ebből

s

mv 7418192

15010

300190

2211232 =sdotsdot

++sdot

minus=

s

dm

s

mv

Dv

DQ

3322

2

22

1

21 41901940741

4

1051

4

070

44==+=+=

ππππ


kWP

Pmsz

hvill 776

650

44 ===+η





Megoldaacutes


17

smD

Qv 6522

4120

030

4 22=

sdot==

ππ

++minus++=minus= 1

211

2

222

12 22z

g

v

g

pz

g

v

g

peeHber ρρ



spp =1 spp =2

022 1

211

2

222

12 =

++minus++=minus= z

g

v

g

pz

g

v

g

peeHber ρρ

Hg

v

D

LhHH berc =+=+=

20


Innen mv

gHDL 5910

65220250

120688192222

=sdot

sdotsdotsdot==λ

42πD


52

24

162

DCQ

DDHgL sdot=


λπ

5 aacutellD

LC ==

mmD 160 = mD

DLD

D

LDCL 3835

120160

91055

55

5 =

sdot=

sdot=sdot=sdot=


55 D

k

DC

K

L

KAacutesziv =



konstans


D




18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













4



pv 444469361

2

12 ===


kg

MJ

p

pp

n

nY

n

n

pol 25015411

80

2612

105411

1451

4511

1

451

14515

1

2

1

2

2 =

minus

minus=

minus

ρminus=

minusminus



384300058 058

6613pol

belső

P MW








35

0

00 1661

300286

10mkg

TR

p=

sdot=

sdot=ρ

smA

qc 7203

1250

5242

11 =

sdotsdot==

π


Kkg

JRc p sdot

=minus

sdot=

minussdot= 1001

141

28641

1κκ

Kc

cTT

p

327910012

7203300

2

221

01 =sdot

minus=sdot

minus=



5

70610327928641

7203

1

1 lt=sdotsdot

=sdotsdot

=TR

cMa

κ



3

141

11

1

0

101 9750

300

32791661

m

kg

T

T=

sdot=

sdot=

minusminusκρρ


Kp

pTT

n

n

749577882

1043279

541

15415

1

1

212 =

sdotsdot=

sdot=

minusminus


3

1541

11

1

1

212 8212

3279

74959750

m

kg

T

T n

=

sdot=

sdot=

minusminusρρ

s


s

m

A

mc 470

12508212

44242

22 =

sdotsdot

sdot=

sdot=

πρamp


( ) ( )kg

kJ

kg


kWW

mcc

YP bep

484483967

4422

7203470216616

2

2221

22

asymp=

=sdot

minus+=sdot

minus+= amp


( ) ( )

kg

kJ

kg

J

TTRn

nYpol

4176176421

327964952861541

541

1 12

asymp=

=minussdotsdotminus

=minussdotsdotminus

=


barPaT

pTp 4210422

7495

104300 5

5

2



6


Kp

pTT s 9445

77882

1044279

41

1415

1

1

212 =

sdotsdot=

sdot=

minusminusκ

κ

( ) ( )kg

kJ

kg


kg

kJ

kg

J

p


77882

104ln4279286ln

5

1

21 asymp=

sdotsdotsdot=

sdotsdot=


250

300

350

400

450

500

65 66 67 68 69 7 71 72 73 74 75

s [kJkgK]

h[kJkg]

11090705v = 03

p=051234


7










8





2

320max 44

lg2900480940


opt

nqη








Megoldaacutes




9


3 2

e

3e

e kg me m s

H H h 60 12 72m

P 28033kW mQ 00397

gH 1000 981 72m s

prime= + = + =

= = =ρ times times


hidr vole e

H 60 Q 0035833 eacutes 882

H 72 Q 00397η = = = η = = =







Megoldaacutes





2

2

max

2bQ


minus=+=

mQ

aQ

Qa QQQ 742

2max

max0

2

max ==

minus == 522

111486240

74msa =sdot=

mbQ QQ 65max2 ==

52

222297

240

65msb ==

022

2 max

=+

minus= bQQ

QadQ

dh sm

Q

ba

aQopt 10

23max =

+=

( ) mbQQ


2

2

max =+

minus=





10

mgQ

PH

opt

szivh 68825819100010

25200 =sdotsdot

==ρ

mhHHe 85426166168825 =+=+=



0237098426

640

===belső

tsts P

Pν


11




94









Megoldaacutes



opt

d2 000731 (Q 95) 0

dQ


opt

m mQ 95 0002639

h s= =


2

6 opt1fokozatopt

H (Q ) 68 02 95H 8325m

6 6

minus times= = =


12

34

optq

1opt

Qn n 299



12


444444 3444444 21

4444 84444 76

η

ηρ

η 23

233

59007310660

20689810

)(

)(

)(

)(

minus

sdotminus

sdotminussdot

sdot

=sdotsdotsdot==

h

mQ

h

mQ

s

mQ

Q

QHgQ

Q

QPP

H

hbev





)0(

)0(

)(

)(lim

)(

)(

)(

)(lim

0 ηηhh

Qtx

P

Q

QP

tg

tf

xg

xf =rarr=rarrrarr


[ ]

[ ]

22

32

2

332

6

2

232

5

23

22

3600

593600007310660

36002068

3600

1593600007310660

360020689810

minussdotsdotminus


=

sdot

minus

sdot

sdot

minusminus

sdot

sdot

minussdot

sdot

sdot=

Q

QgQg

h

sh

m

s

mQ

h

s

m

h

s

mQ

h

s

m

hm

s

mQ

ms

kg

P

H

bev

ρρη

4444444444444 34444444444444 21

44444444 844444444 76


13


WggP

P h

Qbev 1334

3600

59023600007310

036006068

)0(

)0(

2

22

0

=



ρρη



2)










2

2 40000160 Qm

smH sdotminus= )




14

Megoldaacutes


1 2045175 07875 0773σδ ψψ minus= = = rarr =

21 4

0287ψϕ

δ

= =

2

3322 2 22

2

4 4 27000 3600761

0287 1440 604

Q QmmD

nD nD

u

ϕπ ϕπ ππ


123


222

120773 574 1528

2 2p Pau



15











ζkD

ζkD ζkD

ζkd

ζ2kd

4 Q

Q

210 m2 m

3 m

26m11 m

7 m

emptyD

emptyd

9 m




16


mD

L

g

vHH stc 1123150

070

300190

8192

51221

2

2

1

21

1 =

++sdotsdotsdot

+=

+++= ζλ


mv

D

L

g

vHH stc 1123150

10

300190

8192221

2

22

2

22

2 =

++sdotsdotsdot

+=

+++= ζλ

Ebből

s

mv 7418192

15010

300190

2211232 =sdotsdot

++sdot

minus=

s

dm

s

mv

Dv

DQ

3322

2

22

1

21 41901940741

4

1051

4

070

44==+=+=

ππππ


kWP

Pmsz

hvill 776

650

44 ===+η





Megoldaacutes


17

smD

Qv 6522

4120

030

4 22=

sdot==

ππ

++minus++=minus= 1

211

2

222

12 22z

g

v

g

pz

g

v

g

peeHber ρρ



spp =1 spp =2

022 1

211

2

222

12 =

++minus++=minus= z

g

v

g

pz

g

v

g

peeHber ρρ

Hg

v

D

LhHH berc =+=+=

20


Innen mv

gHDL 5910

65220250

120688192222

=sdot

sdotsdotsdot==λ

42πD


52

24

162

DCQ

DDHgL sdot=


λπ

5 aacutellD

LC ==

mmD 160 = mD

DLD

D

LDCL 3835

120160

91055

55

5 =

sdot=

sdot=sdot=sdot=


55 D

k

DC

K

L

KAacutesziv =



konstans


D




18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













5

70610327928641

7203

1

1 lt=sdotsdot

=sdotsdot

=TR

cMa

κ



3

141

11

1

0

101 9750

300

32791661

m

kg

T

T=

sdot=

sdot=

minusminusκρρ


Kp

pTT

n

n

749577882

1043279

541

15415

1

1

212 =

sdotsdot=

sdot=

minusminus


3

1541

11

1

1

212 8212

3279

74959750

m

kg

T

T n

=

sdot=

sdot=

minusminusρρ

s


s

m

A

mc 470

12508212

44242

22 =

sdotsdot

sdot=

sdot=

πρamp


( ) ( )kg

kJ

kg


kWW

mcc

YP bep

484483967

4422

7203470216616

2

2221

22

asymp=

=sdot

minus+=sdot

minus+= amp


( ) ( )

kg

kJ

kg

J

TTRn

nYpol

4176176421

327964952861541

541

1 12

asymp=

=minussdotsdotminus

=minussdotsdotminus

=


barPaT

pTp 4210422

7495

104300 5

5

2



6


Kp

pTT s 9445

77882

1044279

41

1415

1

1

212 =

sdotsdot=

sdot=

minusminusκ

κ

( ) ( )kg

kJ

kg


kg

kJ

kg

J

p


77882

104ln4279286ln

5

1

21 asymp=

sdotsdotsdot=

sdotsdot=


250

300

350

400

450

500

65 66 67 68 69 7 71 72 73 74 75

s [kJkgK]

h[kJkg]

11090705v = 03

p=051234


7










8





2

320max 44

lg2900480940


opt

nqη








Megoldaacutes




9


3 2

e

3e

e kg me m s

H H h 60 12 72m

P 28033kW mQ 00397

gH 1000 981 72m s

prime= + = + =

= = =ρ times times


hidr vole e

H 60 Q 0035833 eacutes 882

H 72 Q 00397η = = = η = = =







Megoldaacutes





2

2

max

2bQ


minus=+=

mQ

aQ

Qa QQQ 742

2max

max0

2

max ==

minus == 522

111486240

74msa =sdot=

mbQ QQ 65max2 ==

52

222297

240

65msb ==

022

2 max

=+

minus= bQQ

QadQ

dh sm

Q

ba

aQopt 10

23max =

+=

( ) mbQQ


2

2

max =+

minus=





10

mgQ

PH

opt

szivh 68825819100010

25200 =sdotsdot

==ρ

mhHHe 85426166168825 =+=+=



0237098426

640

===belső

tsts P

Pν


11




94









Megoldaacutes



opt

d2 000731 (Q 95) 0

dQ


opt

m mQ 95 0002639

h s= =


2

6 opt1fokozatopt

H (Q ) 68 02 95H 8325m

6 6

minus times= = =


12

34

optq

1opt

Qn n 299



12


444444 3444444 21

4444 84444 76

η

ηρ

η 23

233

59007310660

20689810

)(

)(

)(

)(

minus

sdotminus

sdotminussdot

sdot

=sdotsdotsdot==

h

mQ

h

mQ

s

mQ

Q

QHgQ

Q

QPP

H

hbev





)0(

)0(

)(

)(lim

)(

)(

)(

)(lim

0 ηηhh

Qtx

P

Q

QP

tg

tf

xg

xf =rarr=rarrrarr


[ ]

[ ]

22

32

2

332

6

2

232

5

23

22

3600

593600007310660

36002068

3600

1593600007310660

360020689810

minussdotsdotminus


=

sdot

minus

sdot

sdot

minusminus

sdot

sdot

minussdot

sdot

sdot=

Q

QgQg

h

sh

m

s

mQ

h

s

m

h

s

mQ

h

s

m

hm

s

mQ

ms

kg

P

H

bev

ρρη

4444444444444 34444444444444 21

44444444 844444444 76


13


WggP

P h

Qbev 1334

3600

59023600007310

036006068

)0(

)0(

2

22

0

=



ρρη



2)










2

2 40000160 Qm

smH sdotminus= )




14

Megoldaacutes


1 2045175 07875 0773σδ ψψ minus= = = rarr =

21 4

0287ψϕ

δ

= =

2

3322 2 22

2

4 4 27000 3600761

0287 1440 604

Q QmmD

nD nD

u

ϕπ ϕπ ππ


123


222

120773 574 1528

2 2p Pau



15











ζkD

ζkD ζkD

ζkd

ζ2kd

4 Q

Q

210 m2 m

3 m

26m11 m

7 m

emptyD

emptyd

9 m




16


mD

L

g

vHH stc 1123150

070

300190

8192

51221

2

2

1

21

1 =

++sdotsdotsdot

+=

+++= ζλ


mv

D

L

g

vHH stc 1123150

10

300190

8192221

2

22

2

22

2 =

++sdotsdotsdot

+=

+++= ζλ

Ebből

s

mv 7418192

15010

300190

2211232 =sdotsdot

++sdot

minus=

s

dm

s

mv

Dv

DQ

3322

2

22

1

21 41901940741

4

1051

4

070

44==+=+=

ππππ


kWP

Pmsz

hvill 776

650

44 ===+η





Megoldaacutes


17

smD

Qv 6522

4120

030

4 22=

sdot==

ππ

++minus++=minus= 1

211

2

222

12 22z

g

v

g

pz

g

v

g

peeHber ρρ



spp =1 spp =2

022 1

211

2

222

12 =

++minus++=minus= z

g

v

g

pz

g

v

g

peeHber ρρ

Hg

v

D

LhHH berc =+=+=

20


Innen mv

gHDL 5910

65220250

120688192222

=sdot

sdotsdotsdot==λ

42πD


52

24

162

DCQ

DDHgL sdot=


λπ

5 aacutellD

LC ==

mmD 160 = mD

DLD

D

LDCL 3835

120160

91055

55

5 =

sdot=

sdot=sdot=sdot=


55 D

k

DC

K

L

KAacutesziv =



konstans


D




18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













6


Kp

pTT s 9445

77882

1044279

41

1415

1

1

212 =

sdotsdot=

sdot=

minusminusκ

κ

( ) ( )kg

kJ

kg


kg

kJ

kg

J

p


77882

104ln4279286ln

5

1

21 asymp=

sdotsdotsdot=

sdotsdot=


250

300

350

400

450

500

65 66 67 68 69 7 71 72 73 74 75

s [kJkgK]

h[kJkg]

11090705v = 03

p=051234


7










8





2

320max 44

lg2900480940


opt

nqη








Megoldaacutes




9


3 2

e

3e

e kg me m s

H H h 60 12 72m

P 28033kW mQ 00397

gH 1000 981 72m s

prime= + = + =

= = =ρ times times


hidr vole e

H 60 Q 0035833 eacutes 882

H 72 Q 00397η = = = η = = =







Megoldaacutes





2

2

max

2bQ


minus=+=

mQ

aQ

Qa QQQ 742

2max

max0

2

max ==

minus == 522

111486240

74msa =sdot=

mbQ QQ 65max2 ==

52

222297

240

65msb ==

022

2 max

=+

minus= bQQ

QadQ

dh sm

Q

ba

aQopt 10

23max =

+=

( ) mbQQ


2

2

max =+

minus=





10

mgQ

PH

opt

szivh 68825819100010

25200 =sdotsdot

==ρ

mhHHe 85426166168825 =+=+=



0237098426

640

===belső

tsts P

Pν


11




94









Megoldaacutes



opt

d2 000731 (Q 95) 0

dQ


opt

m mQ 95 0002639

h s= =


2

6 opt1fokozatopt

H (Q ) 68 02 95H 8325m

6 6

minus times= = =


12

34

optq

1opt

Qn n 299



12


444444 3444444 21

4444 84444 76

η

ηρ

η 23

233

59007310660

20689810

)(

)(

)(

)(

minus

sdotminus

sdotminussdot

sdot

=sdotsdotsdot==

h

mQ

h

mQ

s

mQ

Q

QHgQ

Q

QPP

H

hbev





)0(

)0(

)(

)(lim

)(

)(

)(

)(lim

0 ηηhh

Qtx

P

Q

QP

tg

tf

xg

xf =rarr=rarrrarr


[ ]

[ ]

22

32

2

332

6

2

232

5

23

22

3600

593600007310660

36002068

3600

1593600007310660

360020689810

minussdotsdotminus


=

sdot

minus

sdot

sdot

minusminus

sdot

sdot

minussdot

sdot

sdot=

Q

QgQg

h

sh

m

s

mQ

h

s

m

h

s

mQ

h

s

m

hm

s

mQ

ms

kg

P

H

bev

ρρη

4444444444444 34444444444444 21

44444444 844444444 76


13


WggP

P h

Qbev 1334

3600

59023600007310

036006068

)0(

)0(

2

22

0

=



ρρη



2)










2

2 40000160 Qm

smH sdotminus= )




14

Megoldaacutes


1 2045175 07875 0773σδ ψψ minus= = = rarr =

21 4

0287ψϕ

δ

= =

2

3322 2 22

2

4 4 27000 3600761

0287 1440 604

Q QmmD

nD nD

u

ϕπ ϕπ ππ


123


222

120773 574 1528

2 2p Pau



15











ζkD

ζkD ζkD

ζkd

ζ2kd

4 Q

Q

210 m2 m

3 m

26m11 m

7 m

emptyD

emptyd

9 m




16


mD

L

g

vHH stc 1123150

070

300190

8192

51221

2

2

1

21

1 =

++sdotsdotsdot

+=

+++= ζλ


mv

D

L

g

vHH stc 1123150

10

300190

8192221

2

22

2

22

2 =

++sdotsdotsdot

+=

+++= ζλ

Ebből

s

mv 7418192

15010

300190

2211232 =sdotsdot

++sdot

minus=

s

dm

s

mv

Dv

DQ

3322

2

22

1

21 41901940741

4

1051

4

070

44==+=+=

ππππ


kWP

Pmsz

hvill 776

650

44 ===+η





Megoldaacutes


17

smD

Qv 6522

4120

030

4 22=

sdot==

ππ

++minus++=minus= 1

211

2

222

12 22z

g

v

g

pz

g

v

g

peeHber ρρ



spp =1 spp =2

022 1

211

2

222

12 =

++minus++=minus= z

g

v

g

pz

g

v

g

peeHber ρρ

Hg

v

D

LhHH berc =+=+=

20


Innen mv

gHDL 5910

65220250

120688192222

=sdot

sdotsdotsdot==λ

42πD


52

24

162

DCQ

DDHgL sdot=


λπ

5 aacutellD

LC ==

mmD 160 = mD

DLD

D

LDCL 3835

120160

91055

55

5 =

sdot=

sdot=sdot=sdot=


55 D

k

DC

K

L

KAacutesziv =



konstans


D




18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













7










8





2

320max 44

lg2900480940


opt

nqη








Megoldaacutes




9


3 2

e

3e

e kg me m s

H H h 60 12 72m

P 28033kW mQ 00397

gH 1000 981 72m s

prime= + = + =

= = =ρ times times


hidr vole e

H 60 Q 0035833 eacutes 882

H 72 Q 00397η = = = η = = =







Megoldaacutes





2

2

max

2bQ


minus=+=

mQ

aQ

Qa QQQ 742

2max

max0

2

max ==

minus == 522

111486240

74msa =sdot=

mbQ QQ 65max2 ==

52

222297

240

65msb ==

022

2 max

=+

minus= bQQ

QadQ

dh sm

Q

ba

aQopt 10

23max =

+=

( ) mbQQ


2

2

max =+

minus=





10

mgQ

PH

opt

szivh 68825819100010

25200 =sdotsdot

==ρ

mhHHe 85426166168825 =+=+=



0237098426

640

===belső

tsts P

Pν


11




94









Megoldaacutes



opt

d2 000731 (Q 95) 0

dQ


opt

m mQ 95 0002639

h s= =


2

6 opt1fokozatopt

H (Q ) 68 02 95H 8325m

6 6

minus times= = =


12

34

optq

1opt

Qn n 299



12


444444 3444444 21

4444 84444 76

η

ηρ

η 23

233

59007310660

20689810

)(

)(

)(

)(

minus

sdotminus

sdotminussdot

sdot

=sdotsdotsdot==

h

mQ

h

mQ

s

mQ

Q

QHgQ

Q

QPP

H

hbev





)0(

)0(

)(

)(lim

)(

)(

)(

)(lim

0 ηηhh

Qtx

P

Q

QP

tg

tf

xg

xf =rarr=rarrrarr


[ ]

[ ]

22

32

2

332

6

2

232

5

23

22

3600

593600007310660

36002068

3600

1593600007310660

360020689810

minussdotsdotminus


=

sdot

minus

sdot

sdot

minusminus

sdot

sdot

minussdot

sdot

sdot=

Q

QgQg

h

sh

m

s

mQ

h

s

m

h

s

mQ

h

s

m

hm

s

mQ

ms

kg

P

H

bev

ρρη

4444444444444 34444444444444 21

44444444 844444444 76


13


WggP

P h

Qbev 1334

3600

59023600007310

036006068

)0(

)0(

2

22

0

=



ρρη



2)










2

2 40000160 Qm

smH sdotminus= )




14

Megoldaacutes


1 2045175 07875 0773σδ ψψ minus= = = rarr =

21 4

0287ψϕ

δ

= =

2

3322 2 22

2

4 4 27000 3600761

0287 1440 604

Q QmmD

nD nD

u

ϕπ ϕπ ππ


123


222

120773 574 1528

2 2p Pau



15











ζkD

ζkD ζkD

ζkd

ζ2kd

4 Q

Q

210 m2 m

3 m

26m11 m

7 m

emptyD

emptyd

9 m




16


mD

L

g

vHH stc 1123150

070

300190

8192

51221

2

2

1

21

1 =

++sdotsdotsdot

+=

+++= ζλ


mv

D

L

g

vHH stc 1123150

10

300190

8192221

2

22

2

22

2 =

++sdotsdotsdot

+=

+++= ζλ

Ebből

s

mv 7418192

15010

300190

2211232 =sdotsdot

++sdot

minus=

s

dm

s

mv

Dv

DQ

3322

2

22

1

21 41901940741

4

1051

4

070

44==+=+=

ππππ


kWP

Pmsz

hvill 776

650

44 ===+η





Megoldaacutes


17

smD

Qv 6522

4120

030

4 22=

sdot==

ππ

++minus++=minus= 1

211

2

222

12 22z

g

v

g

pz

g

v

g

peeHber ρρ



spp =1 spp =2

022 1

211

2

222

12 =

++minus++=minus= z

g

v

g

pz

g

v

g

peeHber ρρ

Hg

v

D

LhHH berc =+=+=

20


Innen mv

gHDL 5910

65220250

120688192222

=sdot

sdotsdotsdot==λ

42πD


52

24

162

DCQ

DDHgL sdot=


λπ

5 aacutellD

LC ==

mmD 160 = mD

DLD

D

LDCL 3835

120160

91055

55

5 =

sdot=

sdot=sdot=sdot=


55 D

k

DC

K

L

KAacutesziv =



konstans


D




18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













8





2

320max 44

lg2900480940


opt

nqη








Megoldaacutes




9


3 2

e

3e

e kg me m s

H H h 60 12 72m

P 28033kW mQ 00397

gH 1000 981 72m s

prime= + = + =

= = =ρ times times


hidr vole e

H 60 Q 0035833 eacutes 882

H 72 Q 00397η = = = η = = =







Megoldaacutes





2

2

max

2bQ


minus=+=

mQ

aQ

Qa QQQ 742

2max

max0

2

max ==

minus == 522

111486240

74msa =sdot=

mbQ QQ 65max2 ==

52

222297

240

65msb ==

022

2 max

=+

minus= bQQ

QadQ

dh sm

Q

ba

aQopt 10

23max =

+=

( ) mbQQ


2

2

max =+

minus=





10

mgQ

PH

opt

szivh 68825819100010

25200 =sdotsdot

==ρ

mhHHe 85426166168825 =+=+=



0237098426

640

===belső

tsts P

Pν


11




94









Megoldaacutes



opt

d2 000731 (Q 95) 0

dQ


opt

m mQ 95 0002639

h s= =


2

6 opt1fokozatopt

H (Q ) 68 02 95H 8325m

6 6

minus times= = =


12

34

optq

1opt

Qn n 299



12


444444 3444444 21

4444 84444 76

η

ηρ

η 23

233

59007310660

20689810

)(

)(

)(

)(

minus

sdotminus

sdotminussdot

sdot

=sdotsdotsdot==

h

mQ

h

mQ

s

mQ

Q

QHgQ

Q

QPP

H

hbev





)0(

)0(

)(

)(lim

)(

)(

)(

)(lim

0 ηηhh

Qtx

P

Q

QP

tg

tf

xg

xf =rarr=rarrrarr


[ ]

[ ]

22

32

2

332

6

2

232

5

23

22

3600

593600007310660

36002068

3600

1593600007310660

360020689810

minussdotsdotminus


=

sdot

minus

sdot

sdot

minusminus

sdot

sdot

minussdot

sdot

sdot=

Q

QgQg

h

sh

m

s

mQ

h

s

m

h

s

mQ

h

s

m

hm

s

mQ

ms

kg

P

H

bev

ρρη

4444444444444 34444444444444 21

44444444 844444444 76


13


WggP

P h

Qbev 1334

3600

59023600007310

036006068

)0(

)0(

2

22

0

=



ρρη



2)










2

2 40000160 Qm

smH sdotminus= )




14

Megoldaacutes


1 2045175 07875 0773σδ ψψ minus= = = rarr =

21 4

0287ψϕ

δ

= =

2

3322 2 22

2

4 4 27000 3600761

0287 1440 604

Q QmmD

nD nD

u

ϕπ ϕπ ππ


123


222

120773 574 1528

2 2p Pau



15











ζkD

ζkD ζkD

ζkd

ζ2kd

4 Q

Q

210 m2 m

3 m

26m11 m

7 m

emptyD

emptyd

9 m




16


mD

L

g

vHH stc 1123150

070

300190

8192

51221

2

2

1

21

1 =

++sdotsdotsdot

+=

+++= ζλ


mv

D

L

g

vHH stc 1123150

10

300190

8192221

2

22

2

22

2 =

++sdotsdotsdot

+=

+++= ζλ

Ebből

s

mv 7418192

15010

300190

2211232 =sdotsdot

++sdot

minus=

s

dm

s

mv

Dv

DQ

3322

2

22

1

21 41901940741

4

1051

4

070

44==+=+=

ππππ


kWP

Pmsz

hvill 776

650

44 ===+η





Megoldaacutes


17

smD

Qv 6522

4120

030

4 22=

sdot==

ππ

++minus++=minus= 1

211

2

222

12 22z

g

v

g

pz

g

v

g

peeHber ρρ



spp =1 spp =2

022 1

211

2

222

12 =

++minus++=minus= z

g

v

g

pz

g

v

g

peeHber ρρ

Hg

v

D

LhHH berc =+=+=

20


Innen mv

gHDL 5910

65220250

120688192222

=sdot

sdotsdotsdot==λ

42πD


52

24

162

DCQ

DDHgL sdot=


λπ

5 aacutellD

LC ==

mmD 160 = mD

DLD

D

LDCL 3835

120160

91055

55

5 =

sdot=

sdot=sdot=sdot=


55 D

k

DC

K

L

KAacutesziv =



konstans


D




18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













9


3 2

e

3e

e kg me m s

H H h 60 12 72m

P 28033kW mQ 00397

gH 1000 981 72m s

prime= + = + =

= = =ρ times times


hidr vole e

H 60 Q 0035833 eacutes 882

H 72 Q 00397η = = = η = = =







Megoldaacutes





2

2

max

2bQ


minus=+=

mQ

aQ

Qa QQQ 742

2max

max0

2

max ==

minus == 522

111486240

74msa =sdot=

mbQ QQ 65max2 ==

52

222297

240

65msb ==

022

2 max

=+

minus= bQQ

QadQ

dh sm

Q

ba

aQopt 10

23max =

+=

( ) mbQQ


2

2

max =+

minus=





10

mgQ

PH

opt

szivh 68825819100010

25200 =sdotsdot

==ρ

mhHHe 85426166168825 =+=+=



0237098426

640

===belső

tsts P

Pν


11




94









Megoldaacutes



opt

d2 000731 (Q 95) 0

dQ


opt

m mQ 95 0002639

h s= =


2

6 opt1fokozatopt

H (Q ) 68 02 95H 8325m

6 6

minus times= = =


12

34

optq

1opt

Qn n 299



12


444444 3444444 21

4444 84444 76

η

ηρ

η 23

233

59007310660

20689810

)(

)(

)(

)(

minus

sdotminus

sdotminussdot

sdot

=sdotsdotsdot==

h

mQ

h

mQ

s

mQ

Q

QHgQ

Q

QPP

H

hbev





)0(

)0(

)(

)(lim

)(

)(

)(

)(lim

0 ηηhh

Qtx

P

Q

QP

tg

tf

xg

xf =rarr=rarrrarr


[ ]

[ ]

22

32

2

332

6

2

232

5

23

22

3600

593600007310660

36002068

3600

1593600007310660

360020689810

minussdotsdotminus


=

sdot

minus

sdot

sdot

minusminus

sdot

sdot

minussdot

sdot

sdot=

Q

QgQg

h

sh

m

s

mQ

h

s

m

h

s

mQ

h

s

m

hm

s

mQ

ms

kg

P

H

bev

ρρη

4444444444444 34444444444444 21

44444444 844444444 76


13


WggP

P h

Qbev 1334

3600

59023600007310

036006068

)0(

)0(

2

22

0

=



ρρη



2)










2

2 40000160 Qm

smH sdotminus= )




14

Megoldaacutes


1 2045175 07875 0773σδ ψψ minus= = = rarr =

21 4

0287ψϕ

δ

= =

2

3322 2 22

2

4 4 27000 3600761

0287 1440 604

Q QmmD

nD nD

u

ϕπ ϕπ ππ


123


222

120773 574 1528

2 2p Pau



15











ζkD

ζkD ζkD

ζkd

ζ2kd

4 Q

Q

210 m2 m

3 m

26m11 m

7 m

emptyD

emptyd

9 m




16


mD

L

g

vHH stc 1123150

070

300190

8192

51221

2

2

1

21

1 =

++sdotsdotsdot

+=

+++= ζλ


mv

D

L

g

vHH stc 1123150

10

300190

8192221

2

22

2

22

2 =

++sdotsdotsdot

+=

+++= ζλ

Ebből

s

mv 7418192

15010

300190

2211232 =sdotsdot

++sdot

minus=

s

dm

s

mv

Dv

DQ

3322

2

22

1

21 41901940741

4

1051

4

070

44==+=+=

ππππ


kWP

Pmsz

hvill 776

650

44 ===+η





Megoldaacutes


17

smD

Qv 6522

4120

030

4 22=

sdot==

ππ

++minus++=minus= 1

211

2

222

12 22z

g

v

g

pz

g

v

g

peeHber ρρ



spp =1 spp =2

022 1

211

2

222

12 =

++minus++=minus= z

g

v

g

pz

g

v

g

peeHber ρρ

Hg

v

D

LhHH berc =+=+=

20


Innen mv

gHDL 5910

65220250

120688192222

=sdot

sdotsdotsdot==λ

42πD


52

24

162

DCQ

DDHgL sdot=


λπ

5 aacutellD

LC ==

mmD 160 = mD

DLD

D

LDCL 3835

120160

91055

55

5 =

sdot=

sdot=sdot=sdot=


55 D

k

DC

K

L

KAacutesziv =



konstans


D




18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













10

mgQ

PH

opt

szivh 68825819100010

25200 =sdotsdot

==ρ

mhHHe 85426166168825 =+=+=



0237098426

640

===belső

tsts P

Pν


11




94









Megoldaacutes



opt

d2 000731 (Q 95) 0

dQ


opt

m mQ 95 0002639

h s= =


2

6 opt1fokozatopt

H (Q ) 68 02 95H 8325m

6 6

minus times= = =


12

34

optq

1opt

Qn n 299



12


444444 3444444 21

4444 84444 76

η

ηρ

η 23

233

59007310660

20689810

)(

)(

)(

)(

minus

sdotminus

sdotminussdot

sdot

=sdotsdotsdot==

h

mQ

h

mQ

s

mQ

Q

QHgQ

Q

QPP

H

hbev





)0(

)0(

)(

)(lim

)(

)(

)(

)(lim

0 ηηhh

Qtx

P

Q

QP

tg

tf

xg

xf =rarr=rarrrarr


[ ]

[ ]

22

32

2

332

6

2

232

5

23

22

3600

593600007310660

36002068

3600

1593600007310660

360020689810

minussdotsdotminus


=

sdot

minus

sdot

sdot

minusminus

sdot

sdot

minussdot

sdot

sdot=

Q

QgQg

h

sh

m

s

mQ

h

s

m

h

s

mQ

h

s

m

hm

s

mQ

ms

kg

P

H

bev

ρρη

4444444444444 34444444444444 21

44444444 844444444 76


13


WggP

P h

Qbev 1334

3600

59023600007310

036006068

)0(

)0(

2

22

0

=



ρρη



2)










2

2 40000160 Qm

smH sdotminus= )




14

Megoldaacutes


1 2045175 07875 0773σδ ψψ minus= = = rarr =

21 4

0287ψϕ

δ

= =

2

3322 2 22

2

4 4 27000 3600761

0287 1440 604

Q QmmD

nD nD

u

ϕπ ϕπ ππ


123


222

120773 574 1528

2 2p Pau



15











ζkD

ζkD ζkD

ζkd

ζ2kd

4 Q

Q

210 m2 m

3 m

26m11 m

7 m

emptyD

emptyd

9 m




16


mD

L

g

vHH stc 1123150

070

300190

8192

51221

2

2

1

21

1 =

++sdotsdotsdot

+=

+++= ζλ


mv

D

L

g

vHH stc 1123150

10

300190

8192221

2

22

2

22

2 =

++sdotsdotsdot

+=

+++= ζλ

Ebből

s

mv 7418192

15010

300190

2211232 =sdotsdot

++sdot

minus=

s

dm

s

mv

Dv

DQ

3322

2

22

1

21 41901940741

4

1051

4

070

44==+=+=

ππππ


kWP

Pmsz

hvill 776

650

44 ===+η





Megoldaacutes


17

smD

Qv 6522

4120

030

4 22=

sdot==

ππ

++minus++=minus= 1

211

2

222

12 22z

g

v

g

pz

g

v

g

peeHber ρρ



spp =1 spp =2

022 1

211

2

222

12 =

++minus++=minus= z

g

v

g

pz

g

v

g

peeHber ρρ

Hg

v

D

LhHH berc =+=+=

20


Innen mv

gHDL 5910

65220250

120688192222

=sdot

sdotsdotsdot==λ

42πD


52

24

162

DCQ

DDHgL sdot=


λπ

5 aacutellD

LC ==

mmD 160 = mD

DLD

D

LDCL 3835

120160

91055

55

5 =

sdot=

sdot=sdot=sdot=


55 D

k

DC

K

L

KAacutesziv =



konstans


D




18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













11




94









Megoldaacutes



opt

d2 000731 (Q 95) 0

dQ


opt

m mQ 95 0002639

h s= =


2

6 opt1fokozatopt

H (Q ) 68 02 95H 8325m

6 6

minus times= = =


12

34

optq

1opt

Qn n 299



12


444444 3444444 21

4444 84444 76

η

ηρ

η 23

233

59007310660

20689810

)(

)(

)(

)(

minus

sdotminus

sdotminussdot

sdot

=sdotsdotsdot==

h

mQ

h

mQ

s

mQ

Q

QHgQ

Q

QPP

H

hbev





)0(

)0(

)(

)(lim

)(

)(

)(

)(lim

0 ηηhh

Qtx

P

Q

QP

tg

tf

xg

xf =rarr=rarrrarr


[ ]

[ ]

22

32

2

332

6

2

232

5

23

22

3600

593600007310660

36002068

3600

1593600007310660

360020689810

minussdotsdotminus


=

sdot

minus

sdot

sdot

minusminus

sdot

sdot

minussdot

sdot

sdot=

Q

QgQg

h

sh

m

s

mQ

h

s

m

h

s

mQ

h

s

m

hm

s

mQ

ms

kg

P

H

bev

ρρη

4444444444444 34444444444444 21

44444444 844444444 76


13


WggP

P h

Qbev 1334

3600

59023600007310

036006068

)0(

)0(

2

22

0

=



ρρη



2)










2

2 40000160 Qm

smH sdotminus= )




14

Megoldaacutes


1 2045175 07875 0773σδ ψψ minus= = = rarr =

21 4

0287ψϕ

δ

= =

2

3322 2 22

2

4 4 27000 3600761

0287 1440 604

Q QmmD

nD nD

u

ϕπ ϕπ ππ


123


222

120773 574 1528

2 2p Pau



15











ζkD

ζkD ζkD

ζkd

ζ2kd

4 Q

Q

210 m2 m

3 m

26m11 m

7 m

emptyD

emptyd

9 m




16


mD

L

g

vHH stc 1123150

070

300190

8192

51221

2

2

1

21

1 =

++sdotsdotsdot

+=

+++= ζλ


mv

D

L

g

vHH stc 1123150

10

300190

8192221

2

22

2

22

2 =

++sdotsdotsdot

+=

+++= ζλ

Ebből

s

mv 7418192

15010

300190

2211232 =sdotsdot

++sdot

minus=

s

dm

s

mv

Dv

DQ

3322

2

22

1

21 41901940741

4

1051

4

070

44==+=+=

ππππ


kWP

Pmsz

hvill 776

650

44 ===+η





Megoldaacutes


17

smD

Qv 6522

4120

030

4 22=

sdot==

ππ

++minus++=minus= 1

211

2

222

12 22z

g

v

g

pz

g

v

g

peeHber ρρ



spp =1 spp =2

022 1

211

2

222

12 =

++minus++=minus= z

g

v

g

pz

g

v

g

peeHber ρρ

Hg

v

D

LhHH berc =+=+=

20


Innen mv

gHDL 5910

65220250

120688192222

=sdot

sdotsdotsdot==λ

42πD


52

24

162

DCQ

DDHgL sdot=


λπ

5 aacutellD

LC ==

mmD 160 = mD

DLD

D

LDCL 3835

120160

91055

55

5 =

sdot=

sdot=sdot=sdot=


55 D

k

DC

K

L

KAacutesziv =



konstans


D




18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













12


444444 3444444 21

4444 84444 76

η

ηρ

η 23

233

59007310660

20689810

)(

)(

)(

)(

minus

sdotminus

sdotminussdot

sdot

=sdotsdotsdot==

h

mQ

h

mQ

s

mQ

Q

QHgQ

Q

QPP

H

hbev





)0(

)0(

)(

)(lim

)(

)(

)(

)(lim

0 ηηhh

Qtx

P

Q

QP

tg

tf

xg

xf =rarr=rarrrarr


[ ]

[ ]

22

32

2

332

6

2

232

5

23

22

3600

593600007310660

36002068

3600

1593600007310660

360020689810

minussdotsdotminus


=

sdot

minus

sdot

sdot

minusminus

sdot

sdot

minussdot

sdot

sdot=

Q

QgQg

h

sh

m

s

mQ

h

s

m

h

s

mQ

h

s

m

hm

s

mQ

ms

kg

P

H

bev

ρρη

4444444444444 34444444444444 21

44444444 844444444 76


13


WggP

P h

Qbev 1334

3600

59023600007310

036006068

)0(

)0(

2

22

0

=



ρρη



2)










2

2 40000160 Qm

smH sdotminus= )




14

Megoldaacutes


1 2045175 07875 0773σδ ψψ minus= = = rarr =

21 4

0287ψϕ

δ

= =

2

3322 2 22

2

4 4 27000 3600761

0287 1440 604

Q QmmD

nD nD

u

ϕπ ϕπ ππ


123


222

120773 574 1528

2 2p Pau



15











ζkD

ζkD ζkD

ζkd

ζ2kd

4 Q

Q

210 m2 m

3 m

26m11 m

7 m

emptyD

emptyd

9 m




16


mD

L

g

vHH stc 1123150

070

300190

8192

51221

2

2

1

21

1 =

++sdotsdotsdot

+=

+++= ζλ


mv

D

L

g

vHH stc 1123150

10

300190

8192221

2

22

2

22

2 =

++sdotsdotsdot

+=

+++= ζλ

Ebből

s

mv 7418192

15010

300190

2211232 =sdotsdot

++sdot

minus=

s

dm

s

mv

Dv

DQ

3322

2

22

1

21 41901940741

4

1051

4

070

44==+=+=

ππππ


kWP

Pmsz

hvill 776

650

44 ===+η





Megoldaacutes


17

smD

Qv 6522

4120

030

4 22=

sdot==

ππ

++minus++=minus= 1

211

2

222

12 22z

g

v

g

pz

g

v

g

peeHber ρρ



spp =1 spp =2

022 1

211

2

222

12 =

++minus++=minus= z

g

v

g

pz

g

v

g

peeHber ρρ

Hg

v

D

LhHH berc =+=+=

20


Innen mv

gHDL 5910

65220250

120688192222

=sdot

sdotsdotsdot==λ

42πD


52

24

162

DCQ

DDHgL sdot=


λπ

5 aacutellD

LC ==

mmD 160 = mD

DLD

D

LDCL 3835

120160

91055

55

5 =

sdot=

sdot=sdot=sdot=


55 D

k

DC

K

L

KAacutesziv =



konstans


D




18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













13


WggP

P h

Qbev 1334

3600

59023600007310

036006068

)0(

)0(

2

22

0

=



ρρη



2)










2

2 40000160 Qm

smH sdotminus= )




14

Megoldaacutes


1 2045175 07875 0773σδ ψψ minus= = = rarr =

21 4

0287ψϕ

δ

= =

2

3322 2 22

2

4 4 27000 3600761

0287 1440 604

Q QmmD

nD nD

u

ϕπ ϕπ ππ


123


222

120773 574 1528

2 2p Pau



15











ζkD

ζkD ζkD

ζkd

ζ2kd

4 Q

Q

210 m2 m

3 m

26m11 m

7 m

emptyD

emptyd

9 m




16


mD

L

g

vHH stc 1123150

070

300190

8192

51221

2

2

1

21

1 =

++sdotsdotsdot

+=

+++= ζλ


mv

D

L

g

vHH stc 1123150

10

300190

8192221

2

22

2

22

2 =

++sdotsdotsdot

+=

+++= ζλ

Ebből

s

mv 7418192

15010

300190

2211232 =sdotsdot

++sdot

minus=

s

dm

s

mv

Dv

DQ

3322

2

22

1

21 41901940741

4

1051

4

070

44==+=+=

ππππ


kWP

Pmsz

hvill 776

650

44 ===+η





Megoldaacutes


17

smD

Qv 6522

4120

030

4 22=

sdot==

ππ

++minus++=minus= 1

211

2

222

12 22z

g

v

g

pz

g

v

g

peeHber ρρ



spp =1 spp =2

022 1

211

2

222

12 =

++minus++=minus= z

g

v

g

pz

g

v

g

peeHber ρρ

Hg

v

D

LhHH berc =+=+=

20


Innen mv

gHDL 5910

65220250

120688192222

=sdot

sdotsdotsdot==λ

42πD


52

24

162

DCQ

DDHgL sdot=


λπ

5 aacutellD

LC ==

mmD 160 = mD

DLD

D

LDCL 3835

120160

91055

55

5 =

sdot=

sdot=sdot=sdot=


55 D

k

DC

K

L

KAacutesziv =



konstans


D




18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













14

Megoldaacutes


1 2045175 07875 0773σδ ψψ minus= = = rarr =

21 4

0287ψϕ

δ

= =

2

3322 2 22

2

4 4 27000 3600761

0287 1440 604

Q QmmD

nD nD

u

ϕπ ϕπ ππ


123


222

120773 574 1528

2 2p Pau



15











ζkD

ζkD ζkD

ζkd

ζ2kd

4 Q

Q

210 m2 m

3 m

26m11 m

7 m

emptyD

emptyd

9 m




16


mD

L

g

vHH stc 1123150

070

300190

8192

51221

2

2

1

21

1 =

++sdotsdotsdot

+=

+++= ζλ


mv

D

L

g

vHH stc 1123150

10

300190

8192221

2

22

2

22

2 =

++sdotsdotsdot

+=

+++= ζλ

Ebből

s

mv 7418192

15010

300190

2211232 =sdotsdot

++sdot

minus=

s

dm

s

mv

Dv

DQ

3322

2

22

1

21 41901940741

4

1051

4

070

44==+=+=

ππππ


kWP

Pmsz

hvill 776

650

44 ===+η





Megoldaacutes


17

smD

Qv 6522

4120

030

4 22=

sdot==

ππ

++minus++=minus= 1

211

2

222

12 22z

g

v

g

pz

g

v

g

peeHber ρρ



spp =1 spp =2

022 1

211

2

222

12 =

++minus++=minus= z

g

v

g

pz

g

v

g

peeHber ρρ

Hg

v

D

LhHH berc =+=+=

20


Innen mv

gHDL 5910

65220250

120688192222

=sdot

sdotsdotsdot==λ

42πD


52

24

162

DCQ

DDHgL sdot=


λπ

5 aacutellD

LC ==

mmD 160 = mD

DLD

D

LDCL 3835

120160

91055

55

5 =

sdot=

sdot=sdot=sdot=


55 D

k

DC

K

L

KAacutesziv =



konstans


D




18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













15











ζkD

ζkD ζkD

ζkd

ζ2kd

4 Q

Q

210 m2 m

3 m

26m11 m

7 m

emptyD

emptyd

9 m




16


mD

L

g

vHH stc 1123150

070

300190

8192

51221

2

2

1

21

1 =

++sdotsdotsdot

+=

+++= ζλ


mv

D

L

g

vHH stc 1123150

10

300190

8192221

2

22

2

22

2 =

++sdotsdotsdot

+=

+++= ζλ

Ebből

s

mv 7418192

15010

300190

2211232 =sdotsdot

++sdot

minus=

s

dm

s

mv

Dv

DQ

3322

2

22

1

21 41901940741

4

1051

4

070

44==+=+=

ππππ


kWP

Pmsz

hvill 776

650

44 ===+η





Megoldaacutes


17

smD

Qv 6522

4120

030

4 22=

sdot==

ππ

++minus++=minus= 1

211

2

222

12 22z

g

v

g

pz

g

v

g

peeHber ρρ



spp =1 spp =2

022 1

211

2

222

12 =

++minus++=minus= z

g

v

g

pz

g

v

g

peeHber ρρ

Hg

v

D

LhHH berc =+=+=

20


Innen mv

gHDL 5910

65220250

120688192222

=sdot

sdotsdotsdot==λ

42πD


52

24

162

DCQ

DDHgL sdot=


λπ

5 aacutellD

LC ==

mmD 160 = mD

DLD

D

LDCL 3835

120160

91055

55

5 =

sdot=

sdot=sdot=sdot=


55 D

k

DC

K

L

KAacutesziv =



konstans


D




18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













16


mD

L

g

vHH stc 1123150

070

300190

8192

51221

2

2

1

21

1 =

++sdotsdotsdot

+=

+++= ζλ


mv

D

L

g

vHH stc 1123150

10

300190

8192221

2

22

2

22

2 =

++sdotsdotsdot

+=

+++= ζλ

Ebből

s

mv 7418192

15010

300190

2211232 =sdotsdot

++sdot

minus=

s

dm

s

mv

Dv

DQ

3322

2

22

1

21 41901940741

4

1051

4

070

44==+=+=

ππππ


kWP

Pmsz

hvill 776

650

44 ===+η





Megoldaacutes


17

smD

Qv 6522

4120

030

4 22=

sdot==

ππ

++minus++=minus= 1

211

2

222

12 22z

g

v

g

pz

g

v

g

peeHber ρρ



spp =1 spp =2

022 1

211

2

222

12 =

++minus++=minus= z

g

v

g

pz

g

v

g

peeHber ρρ

Hg

v

D

LhHH berc =+=+=

20


Innen mv

gHDL 5910

65220250

120688192222

=sdot

sdotsdotsdot==λ

42πD


52

24

162

DCQ

DDHgL sdot=


λπ

5 aacutellD

LC ==

mmD 160 = mD

DLD

D

LDCL 3835

120160

91055

55

5 =

sdot=

sdot=sdot=sdot=


55 D

k

DC

K

L

KAacutesziv =



konstans


D




18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













17

smD

Qv 6522

4120

030

4 22=

sdot==

ππ

++minus++=minus= 1

211

2

222

12 22z

g

v

g

pz

g

v

g

peeHber ρρ



spp =1 spp =2

022 1

211

2

222

12 =

++minus++=minus= z

g

v

g

pz

g

v

g

peeHber ρρ

Hg

v

D

LhHH berc =+=+=

20


Innen mv

gHDL 5910

65220250

120688192222

=sdot

sdotsdotsdot==λ

42πD


52

24

162

DCQ

DDHgL sdot=


λπ

5 aacutellD

LC ==

mmD 160 = mD

DLD

D

LDCL 3835

120160

91055

55

5 =

sdot=

sdot=sdot=sdot=


55 D

k

DC

K

L

KAacutesziv =



konstans


D




18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













18


Megoldaacutes



mh

hhh

val 2580

20 ====

η


mghmvd =

2

2

s



smd

vQ d 0156504

0301522

43

22

=sdotsdot== ππ



mg

vsmv

D

dv D

dD 49508192

1153

211531522

80

30 2222

=sdot

=rarr=sdot

=

=


mD

L

g

vhhh iacutevsz

Dnyszcső 6680

080

10202027049502

2

2 =

sdot+sdot+sdot=

++sdot=+= λζζ


mgg

pphh

g

vvHH oo

csősn

st 668252566802

11531153

2

22

22



mH

H fokozat 83122

66825

21 ===

13278312

0156501470

43

21

431

21

=sdot==opt

optq H

Qnn


19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













19


min970 =n

s

m

n

nQQ

3 01030

1470

970015650 =sdot=sdot=


smA

Qv

dd 5714

030

0103042

=

sdotsdot==

π


mg

vh h 67880

8192

5714

2

22 =sdot

sdot== η

o Ellenőrzeacutes

smvn

nv dd 62141522

1470

970 =sdot==

mhn

nh val 71820

1470

97022

=sdot

=

=





s

m

L

hDgv

g

v

D

Lh k

k

193540030

38281922

2

2

=sdot

sdotsdotsdot=sdot

sdotsdotsdot=rarr

sdotsdotsdot=

λλ

h

m

s

mvAQ

332

7071364191934



02740182

102082030 =+

sdot+sdot=sdot+sdot

sdotsdot+sdotsdot=

+sdot+sdot

=ππ

πλπλλλλbk

bbkk

bk

bbkke

DD

DD

KK

KK

( ) mDDDD

DD

K

AD bk

bk

bk

h 81182444

22

=minus=minus=sdot+

sdotminus

sdot=sdot=

π

π


20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













20

s

m

L

hDgv

e

h 68254002740

38181922=

sdotsdotsdotsdot=

sdotsdotsdotsdot

=λ

( )h

m

s

mvAQ

3322

5183039146824




21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













21





Megoldaacutes







22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













22




T

L1

L2

L3 SZ

I

II










23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













23






23

23

2

24

23

2

24

1

39170

36008190250

8

0250

20030

3600

8

sdot=

sdot

sdotsdotsdot

sdotsdot

=

sdot

sdotsdotsdot

sdotsdot=

h

mQ

h

mQ

h

s

h

mQ

h

sgD

D

LH

aacutegaacuteg

aacutegcső

π

πλ

mg

pH barfej 38720

81910

1023

51

2 =sdot

sdot=sdot

=ρ

[ ] 23

5

22

3

23

2

21

21 27421

4

38720

sdot

=

sdot

=sdot=

h

mQ

m

h

h

mQ

h

m

mQ

Q


barfejfej

23

23

111 66591)2742139170(

sdot=

sdot+=+=

h

mQ

h



mHh

mQ 733546659154

159066591

37 23

=sdot=rarr=+

=



24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













24

22

2 1666591

16159037 nn


2022161590

166659137 rarr=sdot

sdotminus=n



h

mQQQ

3

2

22

2 028

2

665911590

37

2

66591159037 =



barmh

mQ

m

hH aacutegfej 05249200142742127421

23

5

2

1 ==sdot=

sdot

=


25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













25





Megoldaacutes


480 0008uacutej

dm mQ

min s= =


5 2

c uacutej

s s m m

H (Q ) 30 10 0008 364m

364m H (Q ) H (Q )

= + times =

= =



3

s s s

70 364 m364 H (Q ) Q 001932

90000 s

minus= rarr = =



m s uacutej

mQ Q Q 001132

s= minus =



m c

m(Q H ) (001132 364m)



2

c st m2 2 5

m

H H 364 25 sB 88963

001132 mQ

minus minus= = =


26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













26







c

p

S1

S2


27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













27







28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













28





s

m

D

Q

A

Qc 690

3600150

444422

=sdotsdot

sdot=sdot

sdot==ππ



m19303600105

44412001000

p

Q4aD

D

Q4acap

52=









29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













29



Megoldaacutes

sradjkf 1316014702



2ωρ



22 2



( ) =sdot+sdot


r

r

r

r

f

r

r

haacutetlaph

agyagyagy

πωρπωρπ 22

22

2 2222

22

222

=sdot+sdot


r

f

r

r

f

agyagy

22

3222

22 2

22

2πωρπωρ

=minus

sdot+minus

sdot

minus=42

222

244

2222

222

22

agyf

agyf

rrrrrp ωρπωρπ



0250150131

2

10002

2

0250150150131

2

100010322

442

22225 ππ





gHbDq

qF

optR ρ

minus= 32

2




30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













30



65314160

970 43

21

43

21

=sdot==opt

optq H

qnn


( ) ( ) ( ) =ϕϕϕπ


0

23

23

2

0

2

0

cos2

70000

22 cos cos d

Dbd

DbpdApFx

[ ] 0sincos2

70000

220

23 =ϕϕ+ϕ

πminus= πDb

( ) ( ) =ϕϕϕπ


0

23

2

0

sin2

70000

2 sin d

DbdApFy

[ ] Nm

Nmm

Db

Db 54670000

2

2600602

2

70000

2cossin

2

70000

2 22

320

23 ==π

π=ϕϕminusϕ

πminus= π

A becsleacutes

NgHbDq

qF

optR 5371681910060260

160

10013601360 3

2

32

2


minussdot=

minus= ρ







31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













31




Megoldaacutes


t gsmeg s r s


g


ρ





Iacutegy smeg

1023 2816H 04 463 163 3481 m

981



H(Q) 30mσ = = =





minus=

∆minusminus

4

400

1 18h

eH


32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













32









33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













33



Megoldaacutes



4 ∙

∙ 4 ∙ 08

046 025 ∙ 683


∙ ∙ 046 ∙ ∙ 147060 3541

683

3541 0193 rarr 109deg

6833541 6 0232 rarr 131deg



∆ ∆ ouml 2$ ∙ 255 12 2$ ∙ 683 6 2054


∆ 2 ∙

2 ∙

122 ∙ 6 216

∆ 1313 2 ∙ amp

2 (

) 2 ∙

34 ∙

122 ∙ 34 ∙ 683

21


∙ 1313 mert 1313 2 ∙ 1313


∆ ouml ∆ ∆ ∆ 1313

∙

( 2054 216 21

∙

( 255


∆ ouml + ∙ ∆ ouml 084 ∙ 255 214

∆ ouml 2$ ∙ 25512 2$ ∙ 3541

0339



34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













34









594 Pa

594 Pa

894 Pa

600 Pa


oumlssznyomaacutes



35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













35







36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













36

smhgd

Q mpv 2940181

0650819100024

1401580

2

43

2

1

2


= πρ

ρπαε

222

0314204

204

mD

A ss =sdot== ππ

smA

Qc

ss 3589

0314202940 ===

Pacs 75135892181

222 =sdot=ρ

222

176704

1504

mD

A nn =sdot== ππ

smA

Qc

nn 6416

176702940 ===

Pacn 316364162181

222 =sdot=ρ


Pacppp ssnst 4102775111081910002



2 =+=+∆=∆ ρ

A hataacutesfokok

5060288022


πωη

M

pQ

P

Pst

moth

sthst

58060288022


πωη

M

pQ

P

Pouml

moth

oumlhouml



eacutes mieacutert


sina

)a(a1f

2


1f1

f1371

minus+minussdot=ζ





a

δ α


37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













37

Megoldaacutes

82062sin910

90)10(909101

sina

)a(a1f

22=sdot+sdotminus=


o


( ) 29501820

11803711

f

1f1371

22

minus+sdot=

minus+minussdot=ζ


( ) Pa011590143605901v2



2t

2

m089504

DDA =πminus=


s

m30404308950AvQ

3

=sdot==


p0 ∆pouml p0



813p

n

np ouml

2

ouml

2

3

11ouml =∆

=∆

=∆


s

m57714

60

1160240

60

nDu 3 =sdotπsdot=

sdotπsdot=

s

m43vcax ==

o113u

carctg ax

1 ==β


38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













38


m7400

u

pc

h

oumlu2 =

sdotρη∆

= innen

s


o81383713

43arctg

w

carctg

u2


cax

w1

w2

cax

u

u

c2

β1

β2

c2u


39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













39





min931198740

7811 3dmQ

Qv

e ===η

31491080

9311dm

n

QV e

g ===


WpQPh 68710356010


=∆=

WP

Pouml

houml 808

850687 ===

η

Nmn

PPM oumlouml 596

60802

808

2=

sdotsdot===

ππω




( ) ( ) 232222

1064914

020050

4m


min521105853920

20106491 3343

dmsmAvQ

Qvv

sg ===

η=

η= minus

minus

333

42202240min960

min521cmdm

dm

n

QV g

g ====



40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













40

barPam

smkg

A

mgp 6474759248

0016490

8198002

2

====∆


kWWPasmpQ


740

4759248920105853 34

===∆

==minus

ηη

NmW

n

PPM motorhmotorh

t 121

60

9602

2121

2 =

sdotsdot=

sdotsdot==

ππω


min2532

960

1440521

3

dm

n

nQQ

n

n

Q

Q =sdot=sdot=rarr=

kW183960

1440122

n

nPP

n

n

P

P

bebe

be

be =sdot=sdot=rarr=

ttt

t MMp

p

M

M=rarr

∆∆asymp



Megoldaacutes


( )2 2 2gyA D d 0001348m

4

π= minus =

2 2A D 0001964m4

π= =

Erőegyensuacutely

( )( )

0 szelep gy

gy 0 d 0

A p p A p mg

A p p mg A p

+ ∆ = + =

= + ∆ + +


min

dm1765

s

m0640m0013480vAQ

32

dgysziv =sdot==


sziv86 Q 86 5176p 733bar

00467 00467


p0

p0

p+p0

Qsziv

A

Agy p

mg

szelep


41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













41


dm547

s

m0640m0019640AvQ

32

dszelep =sdot==


2Q A p= micro ∆

ρ


6

gyszelep

A p mg 0001348 733 10 700 981p 8527bar

A 0001964



2

5szelep

olaj

szelepszelep mm3541

102785970

270

00012570

p2

QA =

sdotsdot=

∆ρ

micro=



h szivopt opt

dP86 2 00467 p 0 p 921bar


∆ Ebben a

pontban opt

lQ 43









42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













42


Adatok mmd 50=

mmD 80= 20050270 mA = mmh 300=∆

Megoldaacutes


D

dβ




1

1

144

=minus

=minus

=β

E


52812

Re

100029018400312059590


D

C βββ ahol νDv

D1Re =





1454551011

0802Re

61 =

sdotsdot== minusν

DvD

3727012 =β 023308 =β 3088052 =β


43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













43

=


750652812

Re

100029018400312059590

D

C βββ

=


7506

145455

103088000290023301840372700312059590

60700145455

3192860320

750=+=

659400863160700 =sdot== CEα


smpd

q 011150016910659401000

370822

4

050165940

2

43

2

1

2


παε


smA

qvszaacutem 2182

0050270

0111501 ===


8490983902182

21822

1

1 ==minus=minus

=szaacutem

szaacutembecsv v




1613091011

0802182Re

61 =

sdotsdot== minusν

DvD

60670161309

3192860320

750=+=C

659100863160670 =sdot== CEα

smq 011145001691065910 3=sdot=

smA

qvszaacutem 2172

0050270

01114502 ===

04500004502172

21722182

2

21 ==minus=minus

=szaacutem







22

22

2p

hq

hP p

P

q

Ph ∆

∆partpart+


q


p

Ph =∆part

part iacutegy

( ) 22222pqP qp


qqE σ2le


44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













44

q

Ee q


smqEq

q37

4

1062632

1041720030

2

0030

2minus


minussdot=σ

p

Ee p

p ∆== ∆


PapE p

p4

5

1052

1050020

2

020





222 3414905146293 WWW =+= W

hP 2212=σ



4424 asymp==W

W

P

E

h

Ph


( )00010225001000150

222222

222222

2

2222

2

2

=+=

+

=

∆+

=

∆+

=

∆


h

P ee

p

E

q

E

pqpq

qp

Ph

σσσσσ

010110=h

P

Ph

σ

WhP 22512=σ

WEhP 4524=


45















46













45















46













46












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