9. Appendix 1 – Illustration of some concepts in heat ...people.exeter.ac.uk/MAPatric/icam2/Notes 27-39.pdf · The “operating line” is a plot of the corresponding values of

ICAM 2 – Heat Exchangers

27

9. Appendix 1 – Illustration of some concepts in heat exchangers 9.1. Heat Exchanger “Effectiveness”

This is defined as: maxRateTransferHeatPossibleMaximum

RateTransferHeatActualq

q==ε

The maximum heat transfer rate is for an “infinite area” counter flow heat exchanger. )(max cihihh TTCmq −= & )(max cihicc TTCmq −= & i.e. in general: )()( minmax cihi TTCmq −= & In a “real” heat exchanger )()( hohihhcicocc TTCmTTCmq −=−= &&

)()(

)()()(

)(

minmin cihi

hohihh

cihi

cicoccTTCm

TTCmTTCm

TTCm−

−=

−−

=∴&

&

&

&ε

So: cccihi

cico CmCmTTTT

&& =−−

= min)(if)()(

ε and hhcihi

hohi CmCmTTTT

&& =−−

= min)(if)()(

ε

9.2. Transfer Unit Consider a counter flow heat exchanger:

ccchhh dTCmdTCmdq && ==

cchh CmCm <&Take

)(sconstant

1

o=

>=hh

cc

c

hCmCm

dTdT

&

&

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28

The “operating line” is a plot of the corresponding values of Th against Tc in the exchanger (with slope so > 1 for cchh CmCm && < ). This is compared with the “equilibrium line” which is drawn such that Th = Tc (slope = 1).

For this case: )( cchh CmCm && < and ∆

=−−

=δ

εcihi

hohiTTTT

min)( CmUANTU

&= is a non-dimensional expression of the “heat exchanger size”.

The greater the number of transfer units, the closer the operating line approaches to the equilibrium line. 9.3. Comparison between LMTD factor and NTU-effectiveness approaches Compare two exchangers: [1] Perfect counter flow (as reference) [2] “Real” exchanger Each has the same cohocihihhcc TTTTCmCmUq ,,,,,,, && but their areas are different.

Then 1actual

wcounterflo ≤=A

AFt and

actual

wcounterflo)NTU(

)NTU(=tF for the same value of ε

Also the parameter S = ε for min)( CmCm cc && =

S =

cc

hhCmCm

&

&ε for min)( CmCm hh && =

parameter R =Rmin

maxR

max

minC1

)()(

orC)()(

either ===CmCm

CmCm

CmCm

hh

cc&

&

&

&

&

&

i.e. a “one-to-one” correspondence exists between the two sets of parameters.

Slope of operating line:

cico

hohi

hh

cco TT

TTCmCm

s−−

==&

&

In this situation

Rmax

min C)()(1

==CmCm

so &

&




29

9.4. Temperature “cross” in a shell and tube heat exchanger Consider a one shell-pass, two tube-pass heat exchanger:

In the above case cchh CmCm && <

hohi

cico

hohi

cohicicocohi

hohi

cihi

hohi

TTTT

TTTTTTTT

TTTTTT

−−

+−−

=−+−

−=

−−

=∴1

ε ……..(9.4.1)

But if there is a temperature cross: hoco TT = and if min)( CmCm hh && =

then hohi

cico

cc

hhTTTT

CmCm

CmCm

−−

===&

&

&

&

max

minR )(

)(C

Substituting for CR in Equation 9.4.1: RC1

1+

=cε

This can be drawn on the appropriate NTU plot to illustrate the occurrence of the temperature cross.

A temperature cross is imminent when: coho TT =

but )()(

)(

min cihi

cicoccTTCm

TTCm−

−=

&

&ε

or )()(

)(

min cihi

hohihhTTCm

TTCm−

−=

&

&ε




30

9.5. Temperature distribution in a (1,2) shell and tube heat exchanger The detailed temperature distribution depends on the relative paths taken by the shell and tube fluids. 9.6. Graphical representation of Heat Exchangers in Series

Take hhCmCm && =min)( so slope 1>om& Overall effectiveness: From the diagram, Similarly: Thus

iεεεε === 321

o

o

cihi

hohiTTTT

∆=

−−

=δ

ε

oooo

oo

ooo

oommmy

y/1

1/1

/1)/(1

4ε

εδ

δδ

δ−

−=

∆−∆−

=−∆

−∆=

ooo myy

myy

myy

/11

/11

/11

3

3

3

4

2

2

2

3

1

1

1

2ε

εε

εε

ε−

−=

−−

=−

−=

oi oi

immy

yyy

yy

yy

/11

/113

11

2

2

3

3

4

1

4ε

εε

ε−

−=

−−

=⋅⋅= ∏=




31

In general, for “n” passes: ………(9.6.1) Inverting: Solving for ε , If all the units are identical ( pi εε = ) and we put CR = 1/mo (Capacity Ratio)

{ }

{ } R/1

R

/1R

C)1/()C1(

1)1/()C1(

−−−

−−−= n

pp

npp

εε

εεε or inverting:

{ }

{ } R/1

R

/1R

C)1/()C1(1)1/()C1(

−−−

−−−= n

n

pεε

εεε …..….(9.6.2)

Special case of CR = 1

∏=

−−

=−

− n

i oi

i

o mm 1 /11

/11

εε

εε

∏=

−−

=−

− n

i i

oio mm

1 1/1

1/1

εε

εε

∏

∏

=

=

−

−−

−

−−

=n

i i

i

n

i i

i

1 o

o

1

o

m1

1m/1

11

m/1

εε

εε

ε

From the above expression, for the special case of CR = 1, ε is undefined. The diagram on the left may be used to investigate the situation. In this case the operating line has a slope of unity. We consider “n” identical exchangers in series. The strip AXZ'C represents the first of these. As before:

o

oDFDE

∆==

δε

{ } { }

p

p

p

p

nn

nnn

nn

n

ε

εε

εεε

δδδ

ε

δδδ

εδδ

δε

)1(1:tosimplifiesThis

/1//1/

)/(/

/ Zbut Z

)ZZ(XY and strips n"" are theresince/n AXXYbut XZXY

:exchangerfirst For the

oooo

oo

ooo

ooo

oop

−+=

−−=

∆−∆−∆

=∴

−−∆=∴−=′

′−∆====




32

Heat exchangers in Series-Parallel configuration

In general for “n” passes: ∏=

−−

=−

− n

i oi

i

o mm 1 /11

/11

εε

εε

For the special case where CR = 0, then ∞→om

{ } { }∏∏==

−−=−=−∴n

ii

n

ii

1111and11 εεεε

Thus, for CR = 0, the temperature of the secondary fluid remains “constant” and so the situation applies to any type of exchanger and any type of connection for the secondary fluid (series, parallel or combinations of the two). 10. Reference Books

1. JANNA, William S, “Engineering Heat Transfer”, Van-Nostrand Reinhold International 2. HOLMAN, J P, “Heat Transfer”, McGraw-Hill 3. KREITH, F, “Transmission de la Chaleur” (Translation of “Principles of Heat Transfer”),

Masson et Cie. 4. KAYS, W M and LONDON, A L, “Compact Heat Exchangers”, McGraw-Hill 5. SACADURA, J F, “Initiation aux Transferts Thermiques”, Technique et Documentation.

6. SHAMSUNDAR, N, “A property of the log-mean temperature-difference correction

factor”, Mechanical Engineering News, 19(3), 14-15, 1982. 7. LIENHARD, J (iv) and LIENHARD, J (v), “A Heat Transfer Textbook”, 3rd edition

(August 2001). Available for download on the Internet at: http://web.mit.edu/lienhard/www/ahtt.html




33

11. Tables of Theoretical Relationships for Heat Exchangers 11.1. Heat exchanger effectiveness relations

min)(NTUN CmUA &== maxminR )()(C mCmC= =ε effectiveness

Flow geometry Relation Double pipe: Parallel-flow

[ ]{ }

)C1()C1(Nexp1

R

R+

+−−=ε

Counter-flow

[ ]

[ ])C1(NexpC1)C1(Nexp1

RR

R−−−

−−−=ε

Counter-flow ( 1CR = )

( )1NN +=ε

Cross flow: Both fluids unmixed

−−−=

nC1)nNC(exp

exp1R

Rε where 22.0Nn −=

Both fluids mixed

1

R

RN1

)NCexp(1C

)Nexp(11

−

−

−−+

−−=ε

max)( Cm& mixed, min)( Cm& unmixed

( ) [ ]{ }))Nexp(1(Cexp1C1 RR −−−−=ε

max)( Cm& unmixed, min)( Cm& mixed

( )[ ]{ })NCexp(1C1exp1 RR −−−−=ε

Shell and tube: One shell pass, 2, 4, 6 tube passes

{ }{ }

1

2/12 R

2/12 R2/12

R)C1(exp1)C1(exp1

)C1(12−

+−−

+−++++=

NNCRε

All exchangers with CR = 0:

)Nexp(1 −−=ε




34

11.2. NTU relations for heat exchangers

maxminR )()(C mCmC= =ε effectiveness min)(NTUN CmUA &==

Flow geometry Relation Double pipe: Parallel-flow

{ }

)C1()C1(1log

NR

R+

+−−=

εe

Counter-flow

−−−

=ε

ε

RR C-11log

)C1(1N e

Counter-flow (CR=1)

)-(1

Nε

ε=

Cross flow: max)( Cm& mixed, min)( Cm& unmixed

[ ])C1(log)C/1(1logN RR ε−+−= ee

max)( Cm& unmixed, min)( Cm& mixed

[ ])1(logC1log)C/1(N RR ε−+−= ee

Shell and tube: One shell pass, 2, 4, 6 tube passes

++−−

+−−−+−= −

2/12RR

2/12RR2/12

R)C1(C1)2()C1(C1)2(

log)C1(Nε

εe

All exchangers with CR=0:

)1(logN ε−−= e




35

12.1. Shell and Tube Heat Exchanger 12.2. Correction Factor Plot for (1,2) Shell and Tube Heat Exchanger

12. Charts




36

12.3. Ten Broeck plot for outlet temperature of a (1,2) Shell and Tube Heat Exchanger 12.4. & 12.5. Effectiveness-NTU plots for (a) Double pipe; (b) (1,2) Shell and Tube Exchangers




37

12.6. Correction Factor Plot for Crossflow Heat Exchanger with Both Fluids Unmixed

12.7. Effectiveness-NTU Plot for Crossflow Heat Exchanger with Both Fluids Unmixed




38

12.8. Correction Factor Plot for Crossflow Heat Exchanger with One Fluid Mixed

12.9. Effectiveness-NTU Plot for Crossflow Heat Exchanger with One Fluid Mixed




39

12.10. Correction Factor Plot for Crossflow Heat Exchanger with Both Fluids Mixed

12.11. Effectiveness-NTU Plot for Crossflow Heat Exchanger with Both Fluids Mixed



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9. Appendix 1 – Illustration of some concepts in heat ...people.exeter.ac.uk/MAPatric/icam2/Notes 27-39.pdf · The “operating line” is a plot of the corresponding values of