heat exchangers thermal design rudolf Žitný, Ústav procesní a zpracovatelské techniky Čvut fs...
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Heat exchangers thermal design
Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010
HEAT PROCESSESHP7
Shell and tube HE. Comparison 1-1 and 1-2 arrangements from point of view of pressure drops and heat transfer. Enthalpy balance of HE, temperature profiles, effectiveness, LMTD, sizing and rating design methods. NTU-epsilon method for parallel flows (eigenvalue problem, derived temperature profiles and eps), counter current and cross flow arrangement of streams (sheet of selected NTU-eps correlations from Rohsenow). Asymptotical properties. Zonal method. Graphical design (Roetzel Spang diagrams from VDI).
Heat exchangers
Recuperative Regenerative
Wall separating streamsWall separating streams
Direct contact
Rotating drum
HEAT EXCHANGERSHP7
HEAT EXCHANGERSHP7
Compactness
100 1000 10 000
Hydraulic diameter, mm
60 10 1 0.1
Plate
Plate and fin
Car cooler
Speciallungs
m2/m3
Shell-&-tube
HE Shell & TubeHP7
Tanguy
Shell&Tube are the most frequently used and universal heat exchangers
HE Shell & TubeHP7
ABB Lummus
)
ROD baffles
• 300 bar shell, 1400 bar pipe
• -100 up to 600oC
• For any media
• Maximal effectivness = 0.9
• Minimal T = 5 K
• 10 up to 1000 m2
Segmental baffles
Helical baffles
Terminology STHEExample cpndenser 1-2 (one pass in shell, two
passes in tubes)
HE Shell & Tube TEMAHP7
A
B
N
C
D
E 1pass
J
H
G
F 2passes
X
L(=A)M(=B)N
W
T
S
P
FRONT HEAD SHELL REAR HEAD
TEMA (Tubular Exchanger Manufact. Assoc.) specification
Floating rear heads are
necessary in case of large temperature differences (dilatation)
Front head design
depends upon
pressures, cleaning
requirements etc
HE Shell & Tube TEMAHP7
TEMA (hydraulic & thermal design based upon Delaware method)
J-faktor (Colburn)
Idea: basic correlations for the friction factor f and the heat transfer
coefficient are corrected by factors reflecting parallel streams
A,B…
A-leakage through gap tube-baffle
B-cross flow
C-bypass outside bundle
E-leakage through gap shell-baffle
Wolverine Engineering Data Book II. (Wolverine Tube Inc. 2001)
HE Shell & TubeHP7
L L
With 2 passes in tubes the mean velocity increased 2x, path increased 2x, heat transfer surface the same.
Therefore pressure drop will be increased 4x in laminar and 8x in fully developed turbulent regime (Fanning factor f is indirectly proportional to velocity in laminar, while f is constant in turbulent regime).
On the other hand increased velocity in the 2pass configuration decreases fouling rate and increases heat transfer. Nusselt number is Re1/3 and Re0.8 in laminar and turbulent regime, respectively. Therefore heat transfer coefficient increases 21/3=1.26 times in laminar, and 20.8=1.7 times in turbulent flow regime.
Example:
Comparison 1-1 and 1-2 (for the same dimensions, number of tubes, flowrate...)
HE PlateHP7
Plate & Frame heat exchanger
Chevron corrugated plate
Welded plate HE
HE Plate-Fin ALPEMA standard HP7
ALPEMA
THE STANDARDS OF THEBRAZED ALUMINIUM PLATE-FIN HEAT EXCHANGERMANUFACTURERS' ASSOCIATION
Terminology Plate-Fin HE
HE Thermal designHP7
SIZING methods (how to calculate heat transfer surface, given temperatures and flow rate)
RATING methods (how to calculate outlet temperatures and duty for given HE)
Thermal design is based upon enthalpy balances and upon calculation of temperature profiles using correlations for heat transfer coefficients. Results are frequently expressed in form of design diagrams.
Alternative approach is CFD (Computer Fluid Dynamics)
HE Thermal designHP7
W1 T1’
W1 T1’’
W2 T2’’
W2 T2’
Q=kST
Qloss
Enthalpy balance of the whole HE (steady state)
Enthalpy balance of the stream 1 (steady state)
There are three unknowns when RATING: outlet temperatures and Q (duty of HE). Heat losses Qloss are usually small and can be neglected or estimated knowing thermal resistance of insulation.
There is only one unknown when SIZING: “effective” heat transfer surface kS
.
lossQTTWTTW )'''()'''(0 222111
QTTW )'''(0 111
Capacity rate W [W/K] is calculated for one phase flow
asi i piW m c
HE Thermal designHP7
Enthalpy balances represents only 2 equations, the third one is
where k is overall heat transfer coefficient calculated from heat transfer coefficients on both streams, thermal resistance of fouling layers and thermal resistance of wall (thickness h)
TkSQ
2121
111ff
w
RRh
k
Mean temperature difference is defined as
dSTTS
T )(1
21
Mean temperature difference can be calculated only if detailed temperature profiles inside HE are known (see later).
HE Thermal designHP7
'
''ln
'''ln
TLMTD
T1’
T2’
T1’’
T2’’
T1
T2
’=T1’-T2’=T1-T2 ’’=T1’’-T2’’
QsQ
T1’
T2’
T1’’
T2’’
T1
T2
’=T1’-T2’=T1-T2 ’’=T1’’-T2’’
sS
Only in the case of two parallel flows it is possible to calculate the mean temperature difference as LMTD (logarithmic mean temperature difference)
This diagram was used in PINCH analysis. Do
you remember? (composite curves of hot
and cold streams)
HE Thermal designHP7
Q
QQ s
s )'''(')(
Q
kds
Q
dQd s
)'''()'''(
Q
kdsd)'''(
S
Q
kdsd
0
''
'
)'''(Q
kS)'''(
'
''ln
'
''ln
'''
kSQ
Proof follows from the linear relationship between temperature and enthalpy flow Qs
and this is ordinary differential equation
which can be integrated
giving
T1’
T2’
T1’’
T2’’
T1
T2
’=T1’-T2’=T1-T2 ’’=T1’’-T2’’
QsQ
Q
QQ s
s )'''(')(
'
''ln
'''
kSQ
The same holds for the counter current heat exchanger = proof is identical
etc., with the same result:T1’
T2’
T1’’
T2’’
T1
T2
’=T1’-T2’’
=T1-T2’’=T1’’-T2’
QsQ
HE Thermal designHP7
HE Thermal designHP7
)( 211
1 TTW
k
dx
dT
)( 212
2 TTW
k
dx
dT
This approach cannot be used for more complicated arrangement of streams, for example cross-flow HE (temperature distribution in streams are functions of two variables x,y). Even the HE with parallel flows with more than one pass or more than two streams must be based upon temperature profiles expressed as functions of coordinates (x,y, or heat transfes surface measured from one end of HE). In this case temperature profiles are calculated from enthalpic balances of individual streams (or individual passes).
For example the previous case of two parallel flows is solved from two balances (x-distance from one end of HE)
Enthalpy balances
(k is heat transfer coef. related to unit length of HE!! not to the unit heat transfer surface, check units)
x
HE Thermal designHP7
][]][[][ TATdx
d
2
1][T
TT
22
11
//
//]][[
WkWk
WkWkA
These differential equations can be written in matrix form
][]][[][ ZUT
]][[]][[]][[]][[ UUA
]][[
This system of coupled differential equations is transformed to uncoupled system by transformation (temperatures T(x) are transformed to new variables Z(x))
where U is eigenvector matrix associated to conductivity matrix A
where is diagonal matrix of eigenvalues 1 2. To find out eigenvector
matrix and eigenvalues given matrix A is standard operation available in scientific computer libraries, for example using single command in MATLAB [L,U]=eig(A).
HE Thermal designHP7
Substituting the transformation to diff. equations
]]][[[]][[]]][[[ ZUAZUdx
d
]]][[[]][[]][[][ 1 ZUAUZdx
d
results the uncoupled system
][]][[][ ZZdx
d
Uncoupled because [[ ]] is a diagonal matrix
xedxZ 111 )( xedxZ 2
22 )( solution
xx edUedUxT 212121111 )(
xx edUedUxT 212221212 )(
Coefficients d1, d2 are determined by boundary conditions (end temperatures).
1 1 ,...,
( )Tdd N N
x
e e
T x U e d
HE Thermal designHP7
Special case of two streams can be solved analytically and easily, because eigenvalue problem has the solution (verify)
1
2
1
1]][[
W
WU
)
11(0
00]][[
21 WWk
therefore boundary conditions at x=0
2
1
1
2
2
1
1
1
'
'
d
d
W
W
T
T giving
21
22111
''WW
TWTWd
21
212
''WWTT
d .
Temperatures at outlet (x=L, length of HE) are therefore
))''(''(1
'' 22212211
211
LeWTTTWTWWW
T
))''(''(1
'' 2
1212211
21
2LeWTTTWTW
WWT
x
'1T "
1T
"2T'
2T
L
HE Thermal design paper RoetzelHP7
Xing Luo, Meiling Li, Wilfried Roetzel: A general solution for one-dimensional multistream heat exchangers and their networks. International Journal of Heat and Mass Transfer 45 (2002) 2695–2705
A mathematical model for predicting the steady-state thermal performance of one-dimensional (cocurrent and countercurrent) multistream heat exchangers and their networks is developed and is solved analytically for constant physical properties of streams. By introducing three matching matrices, the general solution can be applied to various types of one-dimensional multistream heat exchangers such as shell-and-tube heat exchangers, plate heat exchangers and plate–fin heat exchangers as well as their networks. The general solution is applied to the calculation and design of multistream heat exchangers. Examples are given to illustrate the procedures in detail. Based on this solution the superstructure model is developed for synthesis of heat exchanger networks.
Thermal flow rate
U=kS/L
Eigenvectors of A
Eigenvalues
This paper describes the previous method of temperature profiles calculation in more details. It is always usefull to read Roetzel’s papers
Vector D is determined by boundary conditions
(temperatures)
HE Thermal design paper RoetzelHP7
Coefficients [d] follow from boundary conditions (inlet temperatures at N streams) and from the fact that outlet temperatures of M- subchannels are inlet temperatures of connected subchannels (defined by matrix [[G]]).
Inlet temperatures of subchannels are either inlet temperatures of streams or outlet temperatures of connected subchannels
''1
'2
'22
'21
'1
'12
'11
...
...
...
...
]]'[[
1
22221
11211
MMM
M
M
xMM
xM
xM
xx
xM
xx
eUeU
eUeUeU
eUeUeU
V
1[ ''] [[ '']] [[ '']] ([[ ']] [[ ]] [[ '']] ) [[ ']] [ ']N NxM MxM MxM MxM MxM MxN NT G V V G V G T
NMxNMMxMMxMMxM
MMxMx
MxMMxMNMxNMMxMx
MxM
MMxMNMxNM
TGdVGV
deUGTGdeU
xTGTGxT
]'[]]'[[][)]]''[[]][[]]'([[
][]][[]][[]][[]'[]]'[[][]][[]][[
)]''([]][[]'[]]'[[)]'(['''
[ ( )] [[ ]] [[ ]] [ ]xT x U e d
Analytical expression for temperature profiles in subchannels
This is system of M linear algebraic equations for vector [d]. Outlet temperatures can be expressed as
HE Thermal designHP7
The exponential term can be expressed in terms of two dimensionless parameters, that play important role in thermal design of HE
))1(exp(-NTU))WW
1(WkL
exp(-)WWWW
exp(-kL2
1
121
212 WeE L
The fist criterion NTU (Number of Transfer Units) is a measure of HE size
1WkLNTU (usually defined as
1WkSNTU
with the overall heat transfer coefficient k related to unit heat transfer surface)
The second parameter W is ratio of heat capacities of streams (thermal flow rates W i).
))''(''(1
'' 22212211
211
LeWTTTWTWWW
T
Please return back to the previously derived expressions for outlet temperatures
1
2
WW
W
HE Thermal designHP7
The third criterion (characteristics) of HE is effectiveness defined as an actual duty of HE to the duty of ideal HE (infinite heat transfer surface S, countercurrent flow orientation) atthe same flow rates and the same inlet temperatures, therefore
''
'''
)''(
)'''(
21
11
211
111
max TT
TT
TTW
TTW
Q
Q
The interpretation that the temperature ratio (temperature drop of the first stream)/(difference of inlet temperatures) is the ratio of actual power (duty) to the maximum power transferred by an ideal heat exchanger with infinite heat transfer surface is correct only if the stream 1 is weaker, because only than the temperature of the weaker stream can approach to the inlet temperature of the stream 2.
T1’
T1’’T2’
T2’’
W1<W2
max
Q
Q
Temperature profile for the actual HE
Temperature profile for the HE with infinite heat
transfer surface
HE Thermal designHP7
Efectiveness, NTU and W are related by some relationship - this is quite general conclusion, but the form of this relationship is different for different HE. For example for the previous parallel cocurrent HE holds
W
WNTU
W
E
TT
eWTTTWTWWW
T
TT
TTL
1
))1(exp(1
1
1
''
)))''(''(1
('
''
'''
21
221221121
1
21
11
2
remark: this relationship holds for W1<W2
Parallel flows
W
WNTU
1
))1(exp(1 W1<W2
Counter flow
))1(exp(1
))1(exp(1
WNTUW
WNTU
W1<W2
Cross flow transversally mixed weak stream ))).exp(1(
1exp(1 NTUW
W
W1<W2
Cross flow transversally mixed stronger stream )))]exp(1(exp(1[
1NTUW
W
W1<W2
HE Thermal designHP7
General approximation by Schneller
,
2cot1
2
ZNTU
ghZW where ,41 2 PWWZ
and the parameter P depends upon specific configuration (P=0.5 for cocurrent, =0.82 cross-flow, =1 counter flow).
Asymptotic behaviour for NTU<<1 NTU=
NTUTTW
TTkS
Q
Q
)''(
)''(
211
21
max
What does it mean? For small HE (small heat transfer surface related to heat capacity W) you need not worry about a specific relationship =f(NTU,W). All small HE are the same. Balthus
HE Thermal design zonal methodHP7
The fact that for small HE holds NTU= is the basis of numerical ZONAL methods (HE is substituted by a network of small HE and system of internal temperatures is solved iteratively).
T2’ W2
T1’ W1
zone NTUzone
T1’’=T1’-zone(T1’-T2’)
T2’’=T2’+W1/W2 zone(T1’-T2’)
HE Thermal designHP7
Summary of thermal design methods SIZING methods RATING methods given Temperatures and flowrates given Size (S) and inlet temperatures calculated S calculated outlet temperatures and Q
)'''( 111 TTWQ calculate from NTU and W.
TQ
kS
calculate one output temperature T1’ from
and second outlet temperature from the enthalpy balance.
HE design diagrams F-correctionHP7
Method LMTD, Bowman et al. (1940) is an example of SIZING methods.
Heat transfer surface is calculated from
T LM
QkS
F T
where TLM is logarithmic mean temperature difference (LMTD) based only upon inlet and outlet temperatures (see next). FT is a correction factor dependent upon and W.
Remark: HEDH (Heat Exchanger Design Handbook) use the symbol P for thermal effectiveness (instead of ) and the symbol R as the ratio of heat capacities (instead of W, shown in the graph)
FT<0.75 indicates unsuitable flow
arrangement (use a different configuration of
HE)
Logarithmic Mean Temperature Difference is expressed only by inlet/outlet temperatures of both streams, not taking into account specific location of inlet/outlet ports. The following definition is quite general
1 2 1 2
1 2
1 2
( ' '') ( '' ')' ''
ln'' '
LM
T T T TT
T TT T
T1’
T2’
T1’’
T2’’
T1
T2
’=T1’-T2’’
=T1-T2’’=T1’’-T2’
This LMTD corresponds to the counter-current heat exchanger. For any other (nonideal) HE it is necessary to decrease the TLM value by the F-correction
2 2 2 1 1
1 2 1 2 2
( , )
'' ' '' '
' ' ' ''
m T LMT LM
QT F P R T kS
F T
T T W T TP R
T T W T T
HP7 HE design diagrams F-correction
2 2
1 2 max
'' ' =
' '
T T QP
T T Q
2 1 1
1 2 2
'' '
' ''
W T TR
W T T
The correction TLM depends upon two parameters
As soon as the stream 2 is weaker (W1>W2, R<1) the parameter P can be called effectiveness (and denoted by symbol ). Effectiveness is the ratio of thermal power Q of the analysed exchanger to the power Qmax of an ideal (counter current) HE having infinite heat transfer surface. In that case outlet temperature of the stream 2 approaches inlet temperature of the stream 1 (T1’=T2’’)
T1’
T1’’T2’
T2’’
W1<W2
max
P Q
R Q
T1’
T1’’
T2’
T2’’
W1>W2
max
QP
Q
HP7 HE design diagrams F-correction
HP7
Examples F-correction diagrams from Kakac S. Boilers, Evaporators and condensers, Wiley 1991
HE design diagrams F-correction
T F Tm T LMF-correction of LMTD
2 2 1 1
2 1 2 2
'' ' ' '';
' ' '' '
T T T TP R
T T T T
For shell & tube heat exchangers with 2 passes in pipes T1-temperatures in shell, T2-temperatures in pipes
R
HP7 HE design diagrams F-correction12
T F Tm T LMF-correction LMTD 2 2
2 1
'' '
' '
T TP
T T
The variants A,B of the same heat exchanger differ only by reverting flow direction in shell. The variant B is IRRONEOUS with a possible temperature crossing. However effectiveness P and the FT values are the same.
T2’
T2’’
T1’
T1’’
PA
A
T2’
T2’’
T1’
T1’’ PB
B
FTA
Temperature crossing
HP7 HE design diagrams F-correction (Example)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
20
30
40
50
60
70
80
90
100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
20
30
40
50
60
70
80
90
100
HP7 HE design diagrams F-correction (Example)
l=1; length L=1 m
w1=1; heat capacity of stream 1 (shell)
w2=1; heat capacity of stream 2 (pipes)
ks=1; kS-the heat transfer surface (1 pass)
t1in=100; inlet temperature shell
t2in=10; inlet temperature tubes
n=500;
dz=l/(n-1);
t1(1:n)=t1in;
t2a(1:n)=t2in;
t2b(1:n)=t2in;
e1=exp(-2*ks*dz/w1); NTU-stream shell
e2=exp(-ks*dz/w2); NTU-stream pipes
for iter=1:30
for i=1:n-1
a=(t1(i)+t1(i+1))/2;
t2a(i+1)=a+(t2a(i)-a)*e2;
end
t2b(n)=t2a(n);
for i=n-1:-1:1
a=(t1(i)+t1(i+1))/2;
t2b(i)=a+(t2b(i+1)-a)*e2;
end
for i=n-1:-1:1
a=(t2a(i)+t2a(i+1)+t2b(i)+t2b(i+1))/4;
t1(i)=a+(t1(i+1)-a)*e1;
end
end
B
A
B
The exchanger can be calculated by MATLAB using zonal method
HP7 HE design diagrams F-correction (Example)
Exactly the same problem can be solved by the following excel program
NTU1=kS/W1- tubes, NTU2=kS/W2 - shell
You should specify inlet temperatures at tubes (blue
stream) and shell (read stream)
HP7 HE design diagrams F-correction (Example)
The previous Excel program compares the zonal method with the approximation based upon assumption that the shell stream can be divided into two streams exchanging heat with the tube streams and only at the exit these two streams are mixed.
Specify inlet temperature and heat capacity of the shell
stream
Specify kS for the first and the second
pass
Specify kS for the first and the second
pass
Specify outlet tube
temperature
HP7 HE design diagrams F-correction (Example)
When comparing the Excel and MATLAB solution you see that:
MATLAB solution is clear and readable
MATLAB solution enables to use variable number of zones easily
EXCEL programming is a hell nicely described by Hieronimus Bosch
HEAT EXCHANGERS design diagramsHP7
Unlike the Bowman’s F-correction the Kays London (1964) NTU-eps method is suitable for RATING. Given the heat transfer surface (NTU) and capacities of stream W it is possible to read effectiveness from appropriate graphs and
),'T'T(WQ min 21
HEAT EXCHANGERS design diagramsHP7
Roetzel Spang diagrams (1990) see VDI Warmeatlas
HEAT EXCHANGERS design diagramsHP7
Roetzel Spang diagrams (1990) see VDI Warmeatlas
Asymmetry of NTU curves indicates, that effectiveness depends upon exchange of
streams
R2
P2
P1
R1
NTU2
NTU1
R2
P2
P1
R1
NTU2
NTU1
R2
P2
P1
R1
NTU2
NTU1
R2
P2
P1
R1
NTU2
NTU1
Red line NTU
Not only the counter-current HE but also some kinds of cross-flow HE are capable to achieve P=1 at arbitrary R
Small HE (NTU<0.1) are really the same
HE CFD analysis selected papersHP7
Computer Fluid Dynamics analysis yields interesting data important for structural, fouling and RTD analysis, identification of flow irregularities, dead spaces, shortcuts, etc… However, accuracy of heat transfer and pressure drop prediction is seldom better than 10%. Results hold usually only to a specific design and dimensions.
Miro
HE CFD analysis papers (segmental baffles)HP7
Ender Ozden, Ilker Tari: Shell side CFD analysis of a small shell-and-tube heat exchanger. Energy Conversion and Management 51 (2010) 1004–1014
The shell side design of a shell-and-tube heat exchanger; in particular the baffle spacing, baffle cut and shell diameter dependencies of the heat transfer coefficient and the pressure drop are investigated by numerically modeling a small heat exchanger. The flow and temperature fields inside the shell are resolved using a commercial CFD package. A set of CFD simulations is performed for a single shell and single tube pass heat exchanger with a variable number of baffles and turbulent flow. The results are observed to be sensitive to the turbulence model selection. The best turbulence model among the ones considered is determined by comparing the CFD results of heat transfer coefficient, outlet temperature and pressure drop with the Bell–Delaware method results. For two baffle cut values, the effect of the baffle spacing to shell diameter ratio on the heat exchanger performance is investigated by varying flow rate.
Too small baffle cut and too long spacing
optimum
Bell–Delaware recommended segmental baffle cut values as a function of B/Ds ratio. SBC: segmental baffle cuts in no-phase-change flow; CV: baffle cuts applicable to condensing vapors
HE CFD analysis papers (segmental baffles)HP7
Ender Ozden, Ilker Tari: Shell side CFD analysis of a small shell-and-tube heat exchanger. Energy Conversion and Management 51 (2010) 1004–1014
HE CFD analysis papers (ROD baffles)HP7
Q.W. Dong, Y.Q. Wang, M.S. Liu: Numerical and experimental investigation of shellside characteristics for RODbaffle heat exchanger. Applied Thermal Engineering 28 (2008) 651–660
Geometric model of
periodic flow unit duct
modelLb
PT
Correlation used by Philips Petroleum
RODbaffle heat exchanger is a kind of shell-and-tube heat exchanger with eminent performance. Because of the characteristics of tube bundle support structure, fluid flow on shellside is longitudinal and periodical, and fluid flow is of symmetry in landscape orientation. According to the fluid flow characteristics on shellside and ignoring the impact of inlet, outlet and shell wall, a periodic flow unit duct was taken as the simplified model of the shellside to perform numerical simulation by using CFD code, FLUENT. It was found that the both errors in magnitude of the main fluid velocities and heat transfer coefficients between results of simulation and that of experiment or correlations are less than 10%, and the errors of pressure drop between simulation and correlation are no more than 20%. The detailed characteristics and relation of fluid flow and heat transfer on shellside of the RODbaffle heat exchanger were analyzed using the simulation results.
HE CFD analysis papers (helical baffles)HP7
Jian-Fei Zhang, Ya-Ling He, Wen-Quan Tao: 3D numerical simulation on shell-and-tube heat exchangers with middle-overlapped helical baffles and continuous baffles – Part II: Simulation results of periodic model and comparison between continuous and noncontinuous helical baffles. International Journal of Heat and Mass Transfer 52 (2009) 5381–5389In this paper, based on the simplified periodic model the performance predictions for heat exchanger with middle-overlapped helical baffles are carried out by 3D simulation for three different helix angles (30, 40 and 50), and the commercial codes of GAMBIT 2.3 and FLEUNT 6.3 are adopted in the simulation. It is found that the model average heat transfer coefficient per unit pressure drop of the 40 angle case is the largest, which is in qualitative agreement with the existing literature. The predicted average intersection angle of this case is the smallest, being consistent with the field synergy principle. The performance of periodic model with continuous helical baffle is also compared with that of the noncontinuous middle-overlapped helical baffles. It is found that the heat transfer coefficient per unit pressure drop of the noncontinuous middle-overlapped helical baffles is appreciably larger than that of the continuous helical baffle, indicating that the heat exchanger with noncontinuous middle-overlapped helical baffles has its advantage over the one with continuous helical baffle.
HP7
EXAMHP7
Heat exchangers
Thermal design TkSQ
2121
111ff
w
RRh
k
What is important (at least for exam)HP7
'
''ln
'''ln
TLMTDLogarithmic mean temperature difference is
applicable only for co- and counter-current HEDerive it from the temperature-enthalpy diagram
Q
kds
Q
dQd s
)'''()'''(
T1’
T2’
T1’’
T2’’
T1
T2
’=T1’-T2’=T1-T2 ’’=T1’’-T2’’
QsQ
''
'''
)''(
)'''(
21
11
211
111
max TT
TT
TTW
TTW
Q
Q
NTU(Number of Transfer Units)-(effectiveness) relationships =f(NTU,W)
1WkSNTU 1
2
WW
W
NTU method is suitable for RATING: given size of HE and flowrates (therefore NTU and W) it is possible to calculate effectiveness (and therefore outlet temperatures).
For general arrangement of parallel streams and for variable heat transfer coefficients it is possible to divide streams into subchannels and solve system of ODE for temperatures [T] in subchannels by eigenvalue method
1.
2.
3.
][]][[][ TATdx
d ][]][[][ ZUT ]][[]][[]][[]][[ UUAeigenvectors U, eigenvalues conductivity matrix A
T LM
QkS
F T
What is important (at least for exam)HP7
Bowman F-correction method suitable for SIZING (given power Q and temperature TLM heat transfer surfaces S is calculated using
diagrams))
4.
1 2 1 2
1 2
1 2
( ' '') ( '' ')' ''
ln'' '
LM
T T T TT
T TT T
This definition of TLM is general for any heat exchanger (unlike LMTD that is defined only for co- and counter-current heat exchangers)
Kays -method is suitable for RATING (given S the power Q
and temperatures are calculated using diagrams)) 5.
),'T'T(WQ min 21
Roetzel Spang-method suitable for RATING and SIZING6.