Calculation of mean temperature difference of multi-tubepass crossflow type heat exchanger

Heal Recovery Systems & ClIP Vol. 8, No. 2, pp. 173-182, 1988 Printed in Great Britain

0890-4332/88 $3.00 + 0.00 Pergamon Press plc

C A L C U L A T I O N OF M E A N T E M P E R A T U R E D I F F E R E N C E OF MULTI-TUBEPASS CROSSFLOW TYPE HEAT E X C H A N G E R JuN DAR CHEN and SING TSU TSAI China Steel Corporation, 1 Lin Hal Rd, Hsiao-Kang, Kaohsiung, Taiwan, Republic of China (Recewed 22 June 1987) Abstract--For multi-tubepass crossflow type heat exchangers with one fluid mixed and the other unmixed, a computation model, based on analytical derivation, has been developed to find the mean temperature difference. Various commonly used configurations have been investigated. Some calculated results, in terms of temperature correction factor, are offered as practical references. Effects of parameters, such as tubepass number, dimensionless temperature and flow arrangement, etc., on the mean temperature difference are presented and discussed.

NOMENCLATURE

A~

coefficients in equation (6) Pascal triangular coefficients, see equation (9) coefficients, see equation (8) c? F, temperature correction factor dimemiunless temperature parameter (--R x S) K N number of tubepass R dimensionless temperature parameter (ffi(T1 - T2)/(t2 - h ) ) ¥ dimensionless mean temperature difference ( f f i A t / ( T i - h)) dimensionless temperature parameter (=(t2 - t,)/(Ti - t,)) S T temperature for hot fluid [°C] t temperature for cold fluid [°C] At mean temperature difference [°C] mean temperature difference for pure counter-flow [°C] At,. X dimensionless parameter, see equation (10) Subscripts i ith tubepass counter pure counter-flow 1 inlet 2 outlet

s,

INTRODUCTION

Heat exchangers are employed to transfer heat from a hot fluid to a cold fluid which has a lower initial temperature. The numerous applications of heat exchangers in industry are based on the recognition that re-utilizing the heat of a process increases operation economy. With the recent rise in energy cost, the demands to recover the waste heat of a process have increased accordingly. Based on their simplicity and directness, heat exchangers have been widely employed in recent heat recovery systems. Consequently, many efforts have been made to improve their efficiency and compactness. Since temperature difference is the driving force for heat transfer, precise calculations of temperature difference will cause performance evaluations to be more accurate. To achieve this aim and simplify the calculation, a concept of temperature correction factor had been devised [1]*. Taking account of the difficulty in calculating temperature difference analytically for complex configurations, it is therefore necessary to develop a computation model. The temperature

*See

also [9]. 173

174

JUN DAR CHEN and SING TSU TSAI

correction factors for parallel flow and counter-flow, which are widely employed in many industries, have been developed in detail and are diagrammatically shown in [2]. However, those for crossflow are not so well developed due to fewer applications in the past. Nusselt [3]* derived analytically the expression of the mean temperature difference for crossflow with both fluids unmixed. This initiates the subsequent investigations for further types. For types with one fluid mixed and the other unmixed, Smith [4]* derived the expression of temperature difference for single-tube pass and introduced a series concept to derive that for counter-crossflow with two tubepass. Takahashi [5] follows the results of Smith and derives some general expressions for both parallel crossflow and counter-crossflow with any number of tube passes. Owing to increasing use of this type in heat recovery systems, the present work develops a computation model to calculate the mean temperature difference for various configurations of this type. To serve as references for manual design and efficiency calculation, results will be diagrammatically presented in terms of temperature correction factor. T H E O R E T I C A L ANALYSIS I. A n a l y t i c a l derivation

Usually, the mean temperature difference, At, for any configuration can be expressed by the formula At = F, x Afin,

(1)

where F, = temperature correction factor = - -

r

(2)

(r)counter

and Aft, is the temperature difference of pure counter-flow. In general, the value of (r)a,.,t~ is known when the inlet and outlet temperatures of both fluids are given. Thus, if the r value for a given configuration is available, the corresponding temperature correction factor can be found. In deriving the computation model, the following assumptions are made [4]: (1) one fluid is completely mixed and the other unmixed; (2) flow capacities of each tubepass are identical; (3) there is constancy of the product of heat transfer coefficient and are in each tubepass. In cases studied here, K will be used with reference to the mixed fluid and S to the unmixed fluid. For single tubepass configuration, Smith derived the r expression explicitly as: -S r = ln[l + ( S / K ) I n ( I

- K)]"

(3)

For each configuration with more tubepasses, the following relations can be obtained by following the derivation of Smith:

l(S)

ri

N

K,

K

s,

s

--=--

i

1, N

(4)

i=l,N

(5)

A.K7 = 0

(6)

and N rim0

where K,, Se, r,~ are values of K, S, r corresponding to the ith tubepass, and K, is the value of K corresponding to the first tubepass. By employing relations (4)-(5) and equation (3), the r expression for that configuration can be expressed as: -S r = N x In[1 + ( S / K ) I n ( 1 - g0]" (7) *See also [91.

Multi-tubepass crossflow type heat exchanger

~

IIII "', IIII

175

"-:~

(a)

Fig. 1. Schematic diagram for crosdow type with B configuration: (a) parallel cro~flow; Co) counter-cronflow

Thus, if the appropriate Km is given from equation (6), the r value for that configuration can be obtained immediately.

2. Expressionsfor Ao's Three cases are investigated in the present work. The schematic diagrams are shown in Figs 1 and 2. Since the expressions for the values of A, depend strongly upon the configuration, those

t re

Y (a) Type 1.

Y

(b) Type 2.

11

u_..- ~

t~ --.f

If' 1 (c) Type 3.

u_..- r

(d) Type 4.

Fig. 2. Schematic diagram for crossflow type with miscellaneous configuration.

176

JUN DAR CHEN and SlrqG Tsu TsAI

for each case will be presented separately. Before this is done, some commonly used coefficients are defined; these are c," =

N!

(8)

(N--i)! x i!

Bi = ( - 1)~-i

x

C~

(9)

and S X=~.

(lO)

Case A: parallel crossflow. In this case, the configuration is arranged as a series combination of single-tubepasses. Inlets of both fluids are arranged so as to enter the heat exchanger in the first tubepass [see Fig. l(a)]. General expressions of A,'s are derived and expressed as: Ao= Bo x K

(11)

At =BI

(12)

A ~ = B , x ( I + X ) "-l

for

n>l.

(13)

Case B: counter-cross.flow. If the flow direction of one fluid in Case A is reversed and the other unchanged (see Fig. 1(b)), this forms the configuration studied by this case. The general expressions of A,'s for this case can also be derived and expressed as: A0 = B0 x K

(14)

At = BI

(15)

A ~ = B ~ x [ ( 1 - S ) ( I ~ I X J ) + X ~ - ' 1 for n > l .

(16)

Case C: miscellaneous crossflow. To meet space restrictions, miscellaneous configurations are usually employed in practical applications. Four types of this case are investigated in the present work (see Fig. 2). The reason for this choice will be given later. The expressions for the A,'s are summarized in Table 1. 3. Computation procedure It can be seen that equation (6) must be solved first to provide the appropriate K~ required, In case of N larger than three, it is difficult to solve equation (6) analytically. Thus numerical methods are adopted. Since only real roots are required, Newton's classical method for polynomials [6] is sufficient and is employed in this work. The flow chart of computation procedure used is shown in Fig. 3. The criterion of relative error used to terminate the root-finding calculation is l0 -4 to ensure the accuracy of found roots. RESULTS

AND

DISCUSSIONS

Since the main object of this work is to meet the demands of practical reference, calculated results will be presented in terms of temperature correction factor. By the results of calculation, effects of parameters are also discussed. Table

I.

Expressions of A.'s for the miscellaneousconfif~'ation epse

Type I and type 2 A3=-4(1+~.S / 2 - 3 S ( 1 + S / 1+ S -3S

Type 3 and type 4 S 2 A 3 = ~ -4I I + ~ S, + ( ~ )S] +2S ( I + S ) ~6(1+~t-3S

A~~ - 4

A~~ - 4

Ao ffiK

Ao ffiffi K

Multi.tubepass crossflow type heat exchanger

~.

177

Start

1 J

Input clara

I.

.I CaLcul.ateroot of equation (6) Reset coefficients J New guess

1

No

N'N-I No

]

~ Yes Detete a~se root

I

J CotcuLater

I

1

CaLculate F~ I

I I

Print Ft

I Stop

)

Fig. 3. Flow chart for calculation procedure.

To verify the feasibility of the present computational model, flow configurations for which analytical expressions have been derived are first calculated and compared. Except within a limit region, the maximum relative error is within 0.01% which shows excellent agreement for calculated results. The limit occurs when K + S equals one for parallel crossflow and K equals one for counter-crossflow [4]. This implies that the root of equation (6) equals the K value of the heat exchanger. The agreement between calculated and analytical values is downgraded when this limit is approached. There are two major causes for this discrepancy. Firstly, the relative error will increase as the exact value decreases if the absolute error is approximately constant and this is the case in the limit region. Secondly, this limit suggests that only the first tubepass takes care of the task of heat exchange and no contribution is made by other tubepasses. This is physically unrealistic. Moreover, this unreality also causes a numerical singularity which is computationally intractable by this model. However, since this unreality is unphysical in practical applications, the neglect of those values in that region is acceptable. Therefore, only values of Ft that are larger than 0.6 will be given in this presentation for the sake of accuracy and practicality. With regard to the effect of tubepasses, relations between F, and tubepass number for parallel and counter-crossflow are presented in Figs 4 and 5, respectively. As the number of tubepasses increases beyond four, there is little variation in F,. As a result, for each case F, will finally approach

_

k

i

2

1

-

! 2

i

! 1

\

I 4

I 5

I S

m

4

-

:

:

¢

I 7

5

I

,.~ S

I

.;. I

7

•

• M.4, s-0.3 • R=O.O , S-0.3 • It-2.0 , S.'0.3

¢

(b) R varies, S fixed.

3

!

:.

(a) R fixed, S varies.

I 3

• n - o . s , s-o.t • R-O.6 , S',0.3

O

J.

J.

I 8

PASS

PASS

Fig. 4. Relation between F t and tubepass. (Parallel crossflow.)

O. 70

O. 75

0.80

0.05

0.90

0.95

Ft 1.00

0.80

0.05

0.90

0.95

Ft

k

I 1

I 2

I

2

I

1

I

I

4

R,,O.6

5

I

R,,O.6 R=0.6

m

6

I

, S=0.3 , S=0.5

, S,,0.1

i

I

5

R-O.8 R-2.0

R=0.4

6

I

, S=0.3 , S-0.3

, S..0.3

(b) R varies, S fixed.

I

4

!

3

• •

•

(a) R fixed, S varies.

3

• • •

-u

I

m

7

I

7

i

I

8

I

8

r

PASS

PASS

Fig. 5. Relation between F, and tubepass. (Counter-crossflow-)

O. 7 0

0.75

0.80

0.85

0.90

0.95

1.00

Ft

0.80

0.95

0.90

0.95

1.00

Ft

c~

¢3.

Z

Z

Multi-tubepass creasflow type heat exchanger

179

its asymptotic value for given values of R and S. However, as Figs 4 and 5 show, the approaches to the asymptotic values are different for parallel and counter-flow. It can also be seen that the asymptotic value decreases when one parameter increases with the other held constant. While the rate of decrease for parallel crossflow becomes faster with the parameter increase, that for counter-crossflow becomes slower. Consequently, the asymptotic value of each set of R and S for counter-crossflow will approach the value of one [7]. This implies that, for counter-crossflow with more than four tubepasses, it is usually sufficient to determine the mean temperature difference as that of pure counter-flow. Because of little variation in performance for more than four tubepasses, only those miscellaneous configurations with four tubepasses need to be studied in practice. It can be found that, although flows are arranged in a reverse way, type l and type 2 (termed pair A hereinafter) have the same set of An's, which implies that both types have the same performance. Similar results are also found for type 3 and type 4 (termed pair B hereinafter). In scrutinizing the constructions of these types, one finds that both type 1 and type 2 have the counter-flow sense which means neither the inlets nor the outlets of both fluids are in the same tubepass. The equality of these two types is not known exactly unless a detailed investigation of each tubepass is performed. However, this investigation is beyond the scope of the present work. Similarly, a parallel flow sense, which means either the inlets or the outlets of both fluids are in the same tubepass, is found in type 3 and type 4. Consequently, the limit mentioned above will also exist in these types. The equalities found in miscellaneous cases thus verify that the symmetry of flow reversibility proposed by Pignotti [8] is also available in these cases. As indicated in [1], F, is a function of R and S if flow configuration is given. Since the discrepancy due to tubepass increase is too small to be distinguished in a figure, a common figure can be used for tubepasses numbering more than four. To provide a reference for practical application, the relations between F,, and R and S for two-four or more passes are presentred in Fig. 6 for parallel crossflow and in Fig. 7 for counter-crossflow. Figure 8 gives results for miscellaneous cases. These figures are similar to those of other relevant studies. That is if one parameter is kept constant, F, will decrease as the other increases. Moreover, rapid decrease occurs when parameters approach the limiting region mentioned above. To investigate the effect of configuration in global heat transfer performance, comparisons between these configurations are made in Fig. 9. Since the heat transfer performance of counter-flow is superior to that of parallel flow, it is not surprising to find that the performance for counter-crossflow is also superior to that for parallel crossflow. This finding also validates the combination of a crossflow heat exchanger with a counter-flow operation, so commonly used in practical applications. Since the configurations of miscellaneous cases can be regarded as a combination of parallel crossflow and counter-crossflow, the performances of pair A and pair B are intermediate between those of counter-crossflow and parallel crossflow, as one may anticipate. Moreover, it is found that the performance of pair A is superior to that of pair B. The explanation for this superiority still requires a detailed investigation and is not given here. However, the counter-flow sense of pair A and the parallel flow sense of pair B will be helpful in explaining this superiority. By this comparison, it can be concluded that designing crossflow type exchangers on the basis of counter-crossflow will result in a conservative design.

CONCLUSIONS

The following conclusions can be drawn from the analyses of this work. I. A computation model to calculate the mean temperature difference of crossflow type heat exchangers with multi-tubepasses has been successfully developed based on a semi-analytical approach. 2. The flow arrangement plays an important role in heat transfer performance. By changing the number of tubepasses under certain constraints, the performance can be enhanced for countercrossflow. However, the performance becomes worse for parallel crossflow. 3. The asymptotic tendency of temperature correction factors F, shows that increasing the number of tubepasses indefinitelyis of no significance in varying the mean temperature difference.

0.1

I

,

I

0.1

/

..

0.3 0.4 0.5 0.6 0.7 S= ( t 2 - t l ) / ( T l - t l )

/i\l\

\,\~,~\

(b) Three-tubepass.

0 . 3 0 . 4 O.B 0.6 0 . 7 S= ( t 2 - t l ) / ( T l - t l )

///1\~\

(c) Four- and more tubepass.

0.2

0.2

1.0

0.8

0.9

1.0

Fig. 6. Temperature correction factor for crossflow with one fluid mixed and the other unmixed. (Parallel crossflow.)

0.0

o,

0.7

o.,

1.0

0.0

o,

0.6

0 , 3 0 , 4 0 , 5 0,6 0.7 S= ( t 2 - t l ) / (T1 -El )

I

0.3 0,4 0.5 0,6 0.7 S- ( t 2 - t l l / ( T l - t l ) (b) Thrcc-tubcpass.

0.8

0.8

0,9

0.9

0.8

0.9

1.0

1.0

1.0

l il ! ! I ~//1!

(a) Two-tubepass.

0.7

(c) Four- and more tubcpass.

0,2

0.5

S= (t2-t I I / (T1 - t l )

0.30~

ill

I

1

0,2

0.~

~!~

Fig. 7. Temperature correction factor for crossflow with one fluid mixed and the other unmixed (Counter-crossflow.)

O.O O~

0.6

0.7

Ft 0 . 8

0.9

1.0

D.6 0.00.

0.7

0.6 0.0 O.

0,7

0.9

1.0

Ft 0.8

~

0.9

Ft 0,8

0.8

0.8

0.9

(a) Two-tubepass.

0 . 3 0 . 4 0 . 5 O.G 0 . 7 S= ( t 2 - t l ) / ( T l - t l l

0.9

0.2

1.0

0.1

0.7

Ft 0.8

1.0

0.0

0.7

,1~/~o~,',-~~~ ~,~

0.9

0.9

Ft 0.8

1.0

1.0

C

C~

c~

Z

\

0.1

0.2

0.3

(a) Pair A.

0.4 0.5 0.6 S- (t .2 -t l) / (Tl-t.1)

(b) Pair B.

0.4 0,5 0.6 S= ( t 2 - t l ) / ( T I - L I )

0.3

0.7

0.7

\

0.8

'\

0.8

0.9

0.9

1.0

1.0

Fig. 8. Temperature correction factor for crossflow with one fluid mixed and the other unmixed. (Miscellaneous type.)

0.1

0.2

.+ +'I +\'+\

-!+

0.6 0.0

0.7

Ft, 0 . 8

0.9

1.0

0.8 0.0

0.7

Ft. 0 . 8

0.9

1.0

,

I

o.,

o.~

' \ ~, '~'.. .

|

o.~

o.~ (b) R -- 2.0.

S" ( t 2 - t l I / I T 1 - t l

o.~

\

(a) R = 0.6.

o.,

'ii

\

i\"

\

I

, \

)

0.4 0.5 0.6 S= ( t 2 - t l ) / ( T I - L 1 )

;

0.3

I

I

0.2

I

0.1

~

x

\ ~ \

!

"'

..... ~11eI .......... Pair A . . . . . . Pmlr B

,

CROSSFLON TYPE

II l

O.B

,'. !

-. ,

I

0.9

1.0

!

,I

\

, 1 , ! , ! ~ _ 0.? 0.8 0.9 1.0

CROSSFLOW TYPE --~Counter ..... PsPalle1 .......... Pslr A . . . . . . Psir B

0.7

,. I

\

\

Fig. 9. Comparison of temperature difference correction factor in various cases.

0.6 0.0

0.7

Ft 0.8

0.s

1.0

0.6 0.0

-

-

Ft 0,8

0.7

-

0.9

1.0

oo

W

=.

=b 0

0

G

182

JUN DAR CHEN and SING TSU TSAI

4. A c c o m p a n y i n g an increase in one o f the p a r a m e t e r s R o r S, the a s y m p t o t i c value o f t e m p e r a t u r e c o r r e c t i o n f a c t o r F, decreases r a p i d l y for p a r a l l e l crossflow a n d slowly for c o u n t e r crossflow. F i n a l l y , the t e m p e r a t u r e c o r r e c t i o n f a c t o r for c o u n t e r - c r o s s f i o w will a p p r o a c h a value o f one for m o r e t h a n f o u r tubepasses. 5. T h e h e a t t r a n s f e r p e r f o r m a n c e o f c o u n t e r crossflow is s h o w n to be s u p e r i o r to t h a t o f parallel crossflow as one w o u l d expect. T h e p e r f o r m a n c e s o f m i s c e l l a n e o u s c o n f i g u r a t i o n s , however, are i n t e r m e d i a t e between these t w o cases for the s a m e n u m b e r o f tubepasses. 6. T h e s y m m e t r y flow reversibility is s h o w n to h o l d in m i s c e l l a n e o u s c o n f i g u r a t i o n s o f crossflow t y p e h e a t exchangers. Acknowledgements--The authors would like to express much appreciation to Dr S. S. Hsieh and Mr P. S. Wang for their constructive criticism in the manuscript. Thanks are also due to Mr C. P. Hou for his help in data arrangement and figure plotting. The authors are also grateful to the China Steel Corporation for its support in collecting the references.

REFERENCES 1. R. A. Bowman, A. C. MueUer and W. M. Nagle, Mean temperature difference in design, Trans. Am. Sac. mech. Engrs 62, 283-294 (1940). 2. Tubular Exchanger Manufacturers Association, Standard of Tubular Exchanger Manufactures Association, 6th Edn. Tubular Exchanger Manufacturers Association, New York (1978). 3. W. Nusselt, Eine neue Formel fiir den Wirmedurchgang im Kreuz.strom, Tech. Mech. U. Thermodynam. 1, 417-422 (1930). 4. D. M. Smith, Mean temperature difference in crossflow, Engineering 138, 474-481 and 606-607 (1934). 5. Y. Takahashi, Mean temperature difference in multi-pass crossflow heat exchangers, Trans. Japan Soc. mech. Engrs 8, 1-9 (1942). 6. S. D. Conte and Carl de Boor, Elementary Numerical Analysis, 2nd Edn. McGraw-Hill, Maidenhead (1972), 7. J. E. Mott, J. T. Person and W. R. Brock, Computerized design of a minimum cost heat exchanger, ASMEpaper no. 72-1"17"-26(1972). 8. A. Pignotti, Flow Reversibility of Heat Exchangers, J. Heat Transfer 106, 361-368 (1984). 9. D. Q. Kern, Process Heat Transfer. McGraw-Hill, Maidenhead 0965).