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Persamaan BernoulliPersamaan BernoulliPersamaan BernoulliPersamaan BernoulliDinamikaDinamika zatzat caircair yang yang dipengaruhidipengaruhigayagaya –– gayagaya yangyang bekerjabekerjagayagaya –– gayagaya yang yang bekerjabekerja..
AsumsiAsumsi ::AsumsiAsumsi ::1.1. ZatZat caircair idealideal22 ZatZat caircair homogenhomogen dandan taktak kompresibelkompresibel2.2. ZatZat caircair homogenhomogen dandan taktak kompresibelkompresibel3.3. AliranAliran kontinuekontinue dandan sepanjangsepanjang garisgaris
arusarusarusarus4.4. SatuSatu dimensidimensi : : kecepatankecepatan meratamerata padapada
seluruhseluruh tampangtampangseluruhseluruh tampangtampang5.5. Gaya : Gaya : gayagaya beratberat dandan tekanantekanan
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a∂ dAdspp ⎟⎠⎞
⎜⎝⎛
∂+
ds s ⎠⎝ ∂
dzdA α
pdAds dA
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Gaya Gaya –– gaya yang bekerja :gaya yang bekerja :
1.1. Gaya tekanan :Gaya tekanan : dAdsspp;dAp ⎟
⎠⎞
⎜⎝⎛
∂∂
+
2.2. Gaya berat :Gaya berat : dAdsγ
Hukum Newton II : Hukum Newton II : ΣΣ F = F = m .m . aa
dsdAadAdsgdAdssppdAp ραρ
∂
=−⎟⎠⎞
⎜⎝⎛
∂∂
+− cos
agsp ραρ =−∂∂
− cos
dtdVa = ds
sVdt
tVdV
∂∂
+∂∂
=sVV
tVa
∂∂
+∂∂
=; V = V(t, s) ;; V = V(t, s) ;
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z∂⎟⎞
⎜⎛ ∂∂∂∂ VVVzp
scos
∂=α ⎟
⎠⎞
⎜⎝⎛
∂+
∂ρ=
∂ρ−
∂−
sV
tsg
sp
01 ∂∂∂∂ VVVzp 0=∂
+∂
+∂
+∂ρ ts
Vs
gsp
01 ∂∂∂ VVpzAli St d / 0=∂∂
+∂∂
ρ+
∂∂
sVV
sp
sg
01++
dVVdpdzg
Aliran Steady /Permanen :
0=++ds
Vdsds
gρ
01=++ VdVdpgdz
Hanya fungsi s :
0=+ρ
+ VdVdpgdz
vp1 2
Persamaan EulerPersamaan Euler
cg
vpzg
x =+γ
+→∫→2
1Persamaan BernoulliPersamaan Bernoulli × γ
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PERSAMAAN BERNAULLIPERSAMAAN BERNAULLIPERSAMAAN BERNAULLIPERSAMAAN BERNAULLI
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Persamaan Bernoulli pada zat cair riil :Persamaan Bernoulli pada zat cair riil :Persamaan Bernoulli pada zat cair riil :Persamaan Bernoulli pada zat cair riil :Terjadi kehilangan tenagaTerjadi kehilangan tenaga
Akibat gesekanAkibat gesekanAkibat gesekanAkibat gesekanPerubahan tampangPerubahan tampang
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garis tekanan
hhf
1p
he
e
gv2
2
garis tenagaγ1p
γ2p
3p
h
garis tenaga
e
z
γ
1z2z
3z
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Persamaan Bernoulli
∑∑ ++++=++2
1
2
1
222
2
211
1 22 fe hhg
vpzg
vpzγγ
∑∑ ++++=332
333 2 fe hhvpz ∑∑
113 2 fegγ
vkh2 2
11 ⎟⎞
⎜⎛ Ak
gkhe 2
=2
11 ⎟⎠
⎞⎜⎝
⎛−=
Ak
gv
DLfhf 2
2=
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Aplikasi Persamaan Bernaulli:Aplikasi Persamaan Bernaulli:Tekanan HidrostatisTekanan HidrostatisTekanan HidrostatisTekanan Hidrostatis
vpzvpz22
222
2
211
1 ++=++gg 22 21 γγ
021 == vv
( )γγ
121
2 pzzp+−= 1
aphp +γ=2
2
P
h
P
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Tekanan StagnasiTekanan Stagnasi22
gvpz
gvpz ss
s 22
2200
0 +γ
+=+γ
+S
Po
Vo
0=sv
2 2
( )g
vpzzps
s
200
0 +γ
+−=γ
1
gv2
0
Sp2
00 21 vpps ρ+=
SO
γ0p
γSp
Szz0
0z0z
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Alat ukur kecepatan :Alat ukur kecepatan :
gvpph s
2
2=
γ−
=
p
gv2
2
= h
gvp
gvp ss
22
22+
γ=+
γ
γp
v γSp
hg
vpps ==γ−
2
2
hgγ 2
γ−
==ppgghv s22
v
γSp
γ
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APLIKASI PERSM.BERNOULLI :TABUNG PITOT /TABUNG PITOT /
PRANDTL
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Alat ukur debit :Alat ukur debit :Po
Dc
czz =0
2
20
2
2 QpQpc +=+
Do
Pc2
02 22 gAgAc γγ
−=⎟⎟
⎞⎜⎜⎛
− 02 11 ppQ c
hm
γ=⎟⎟
⎠⎜⎜⎝
220 22 gAgA
Qc
− cc ppgAQ 02
2 2γ
⎟⎠⎞⎜
⎝⎛−
= c
c
c pp
AAgAQ 0
2
0
2
1
2
21
0
21
4
242
⎟⎠⎞
⎜⎝⎛
γ−
⎤⎡ ⎞⎛
π= cc pp
D
DgQ
01
⎠⎝
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛− c
DD
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• Sebuah Venturimeter dilengkapi dengan manometer air raksaseperti terlihat pada gambar. Kemiringan sudut venturimeterterhadap horisontal adalah, φ = 30°. Bila diketahui beda tinggimuka air raksa pada manometer adalah 0.35 m, tentukan besardebit aliran (air, s = 1) yang melalui venturi meter tersebut.(Catatan: nilai koefisien debit Cd = 0 96; g = 9 81 m/s2)(Catatan: nilai koefisien debit, Cd = 0.96; g = 9.81 m/s ).