Study of Heat Transfer Process in a Regenerator

0263±8762/99/$10.00+0.00 € Institution of Chemical Engineers Trans IChemE, Vol 77, Part A, March 1999

STUDY OF HEAT TRANSFER PROCESS IN A REGENERATOR S. MURALIKRISHNA Department of Mechanical Engineering, Sri Ramakrishna Engineering College, Coimbatore, India

A

simple closed form solution to the problem of heat transfer through the bed of a regenerator has not previously been available in the literature. In the present work, analytical solutions to two different cases, namely with and without the inclusion of conductive terms are reported. The solution is applicable to both heating and cooling periods. The model is validated by comparison with the better known Hausen’ s theory and Hegg’ s experimental data. The comparison shows that the present model is in good agreement with the experimental data and the predictions are more accurate than using Hausen’ s theory. Keywords: regenerator; performance; heating; cooling; conduction; similarity

INTRODUCTION

The convective term signi® cantly contributes to the heat transfer process, especially when the conductive term is negligible, for example, non-metal and steel heat exchangers. So far, closed form solutions to the model equations are not available in the literature. Thus, a de® nite theory to design a regenerator cannot be found. In the present paper, the model equations of a regenerator are formulated by taking an energy balance over a control volume and a solution is presented for both heating and cooling periods. The solid fraction is considered while writing the governing equations, in contrast to the earlier models. A set of results is reported for two cases, i.e., with and without the inclusion of axial dispersion and conduction. The results show that the analytical model predicts the performance of a regenerator with a fair degree of accuracy. The effectiveness of a regenerator increases with an increase in Fourier number in the case of a model without axial conduction and dispersion. The square of the Peclet number dominates the heat transfer process in the case of a model with axial dispersion and conduction.

Regenerators are devices used to recover waste thermal energy and are used as a heat recovery device in incinerators, air-conditioning systems, foundries and in the chemical process industries. They are classi® ed as ® xed bed regenerators and rotary regenerators. Compared to a rotary regenerator, it is easy to manufacture a ® xed bed regenerator and it requires minimal maintenance. On the other hand, a rotary regenerator which incorporates a reduction gear box, costs more comparatively. Also, leakage losses and carryover occur in a rotary regenerator. Both systems have a high value of effectiveness and are commonly employed to recover waste heat when compactness is desired. Model equations and their complete solution are required for optimal design. The heat transfer process in a ® xed bed regenerator has been extensively studied and analytical solutions to the model equations have been reported1 ,2 . In Hausen’ s model, the effect of storage in the ¯ uid has been neglected and a graphical solution is given. Schumann2 has elegantly transformed the governing differential equations by using reduced time and space co-ordinates. A control volume approach is not used to write the model equations and the solid fraction is not taken into account in either of these models. Subsequently, numerical solutions to these models have also been provided in the literature3 ,4 . The numerical schemes consume time and require the use of a computer. Investigations have been carried out to reduce the computational time5 ,6 . Wilmott and Duggan6 have reported an approximate series form of solution; the solutions are expressed as a polynomial. Design methods were summarized by Coppage and London7 , while transient studies were discussed by Romie8 , but these were not complete in all respects. The effects of axial conduction have also been studied9 ,1 0 ,1 1 . Chung Hsiung Li1 2 has provided information on temperature distribution in the matrix as well as in the air stream. Skiepko1 3 has employed the method of collocation to numerically solve Hausen’ s differential equation of Hausen. His work demonstrates that the solution procedure is complicated.

ANALYSIS OF A REGENERATOR The co-ordinate system is shown in Figure 1. The segment of a regenerator considered here, is a thin disc of thickness, dz. For the sake of generality, the segment is assumed to be at a distance z from the origin. The length and diameter of the regenerator are represented by L and D, respectively. The following assumptions were made to develop the mathematical model of a regenerator. 1. The axial velocity is assumed to remain constant during ¯ uid ¯ ow through a regenerator. 2. The ¯ uid and material phases are separately considered as continuous media and isotropic. 3. The thermal properties do not vary over the temperature range in which a regenerator operates. 4. The regenerator is supposed to be packed with spherical particles. 131

132

MURALIKRISHNA to be adopted to obtain a solution. The j-parameters are expressed as a function of the dimensionless quantities such as Fo, Re, Nu, and others with the diameter of the regenerator, D, as the characteristic length. This is deliberately introduced to isolate the aspect ratio, L/D, another parameter. This exercise provides a designer with more ¯ exibility in designing a regenerator. The initial conditions are: At h

Figure 1. Physical model and co-ordinate system.

5. There is no heat transfer between the system and surroundings.

Air stream: rf cf 1 0#

z#

h

¶tf /¶t L; 0 # t # e

nh ¶tfh /¶z

hc Ap n tfh

tr ;

tsh 1

Matrix material: rs cs e ¶tsh /¶t 0#

z#

L;

hc Ap n tfh 0#

t#

tsh

tr ;

2

The initial conditions are At t

0 and z

0

tfh

ti

3

tsh

ta

4

In order to generalize the results and obtain a solution, equations (1) and (2) are written in non-dimensional form as follows: Air stream j1 ¶Tfh /¶h j2 ¶Tfh /¶Z j3 Tfh 0 # Z # 1; 0 # h # 1;

Tsh

0 5

Matrix material: j4 ¶Tsh /¶h j5 Tfh Tsh 0 0 # Z # 1; 0 # h # 1;

1

7

Tsh

0

8

Solutions to equation (5) and equation (6) are obtained by the method of separation of variables, and the mathematical technique is dealt elsewhere1 3 and is also brie¯ y outlined in the Appendix. The temperature pro® les are given by, T

Heating period: Energy balance across the segment of a regenerator along with above assumptions yields the following set of partial differential equations for a ® xed bed regenerator:

0

Tfh

Tfh

Case (i) Model Without Axial Dispersion and Conduction

0 and Z

exp

h s

b1 b2 Z

j5 /j3 T

9

h

h f

10

Where b1

j3

j5 /j4

11

b2

j4

j1 /j2

12

Since j5 /j3 is very nearly equal to zero for air-metal heat exchangers, equation (8) satis® es equation (10). The outlet temperature of a regenerator varies with time. A time average value of the outlet temperature is a measure of the effectiveness of a regenerator. An average outlet temperature of an air stream is obtained by integrating the expression for the temperature distribution [equation (9)], at the outlet, with respect to h between the limits 0 and 1. The average outlet temperature for an air stream is given by Tfh,a

exp

b1 b2 1

exp

b1 /b1

13

Cooling period: In this period, thermal energy is transferred from the matrix material to a cool stream of air by virtue of temperature difference. In the meantime, a decrease in the internal energy of the matrix material results. Therefore, the governing equations for the cooling period are the same as those of the heating period; the difference being the time period over which this heat interaction takes place. Let this time be tp . Then, the governing equations could be written as Air stream: vc ¶tfc /¶z rf cf ¶tfc /¶t 0 # z # L; 0 # t # tp ;

hc Ap n tcf

tcs 14

6

In this model, ® ve different dimensionless groups j1 ±j5 are retained, as opposed to the reduced time and space coordinates in the previous models1 ,2 . This is because a trivial solution results if the dimensionless groups of the previous models are retained; a rigorous mathematical technique has

Matrix material: rs cs e ¶tcs /¶t 0#

z#

L;

hc Ap n tfc 0#

t#

tp

tsc 15

The cool stream of air and the matrix material are assumed to be at the same initial temperature. During the cooling Trans IChemE, Vol 77, Part A, March 1999

STUDY OF HEAT TRANSFER PROCESSES IN A REGENERATOR process, the initial conditions are given by: At t 0 and z tfc tci t

c s

Fluid phase:

L counter flow mode

thi

0#

17

In the dimensionless form, equation (14) and equation (15) could be expressed as

Tsc

0#

0 18

j4 /Y ¶Tsc /¶h1 j5 Tfc Tsc 0 # Z1 # 1; 0 # h1 # 1;

0 19

The corresponding initial conditions for a counter ¯ ow operation in dimensionless form are At h1 0 and Z1 Tf 0

0 20

1

21

Equations (18) and (19) are identical in all respects to equations (5) and (6); j1 and j2 are to be replaced by j1 /Y and j4 /Y , respectively, and j2 by j2 c . Then proceeding on the lines given in the Appendix, the solution could be written as Tfc

1

c s

1

T

exp

b 3 b 4 Z1

j5 /j3 exp

22

h1

b 3 b 4 Z1

23

h1

Where b3

b1 Y

24

b4

b2 /Y c

25

Since j5 /j3 < 0, equation (21) satis® es equation (23). The average outlet temperature of an air stream during the cooling mode is evaluated by integrating equation (22) with respect to h1 , between the limits 0 and 1. The average outlet temperature for an air stream in counter ¯ ow mode is Tfc,a

L; 0 #

1

exp

b3 b4 1

1

exp

b1 b2 /c 1

exp exp

b3

b1 Y

z#

26

The contribution of the convective term diminishes at low velocities of the air stream due to low values of the heat transfer coef® cient. The conductive terms assume importance at low velocities and govern the heat transfer process along with the convective term. The model and the assumptions are the same as given under case (i); the exception being the inclusion of axial dispersion and conductive terms in both air and matrix material. An energy balance applied across the control volume in the z-direction yields the following set of partial differential equations. Trans IChemE, Vol 77, Part A, March 1999

t#

hc A p n tf

ts 28

tr

2 2 ¶ Tf / ¶ Z

j 3 Tf

Ts 29

and j4 L /D 0#

Z#

2

2 2 ¶Ts /¶h ¶ Ts /¶Z 1; 0 # h # 1

j5 L /D

2

Tf

Ts 30

The non-dimensional co-ordinates and the groups have the same de® nition as given in the nomenclature. In dimensionless form, the initial conditions are given by equations (7) and (8). The technique of separation of variables does not yield a solution because of the complex nature of the problem. But, the solution given earlier suggests that there is a possibility of a similarity solution with a similarity variable of the form f a1 Z a2 h . Here a1 and a2 are constants to be determined from the transformed equations. The solution procedure is discussed in detail in the literature1 3 and the complete solution is expressed as Tfh T

C1 exp m1 w

h s

C2 exp m2 w

2

j5 1/D /j3 T

31

h f

32

where w

A B

Case (ii) Model With Axial Dispersion and Conduction

L; 0 #

j 1 ¶ Tf / ¶ h j2 ¶Tf /¶Z 0 # Z # 1; 0 # h # 1

m2

/b1 Y

tr

The initial and boundary conditions are the same as those given in equations (3) and (4) for the case without axial dispersion and conduction. Proceeding on the same lines as before, equations (27) and (28) are expressed in dimensionless form as follows:

m1

/b3

t#

ks e ¶2 ts /¶Z 2

rs cs e ¶ts /¶t

Matrix material:

Ts

z#

2 e ¶ tf /¶z2 27

kf 1

Solid phase:

Air stream: j3 Tfc

vz ¶tf /¶z

e ¶Tf /¶t hc A p n tf ts

rf cf 1 16

j1 /Y ¶Tfc /¶h1 j2 c ¶Tfc /¶Z1 0 # Z1 # 1; 0 # h1 # 1;

133

a1 Z

a2 h

A1 A1

33

1 1

B

1/2

B

1/2

34 35

2 2 2

j4 L /D j / 2a2 j4 L /D 4 j3

a1 /a2

j5 L /D

j4 L /D

2

2

2

j4 L /D

j1 2

2

36

j1

2

37

j1 /j2

38

The constants of integration, C1 and C2 are evaluated by considering the overall energy balance of the system. The average outlet temperature of an air stream during the heating period is given below: Tfh,a

C1 exp m1 a1 exp m1 a2

1 /m1 a2

C2 exp m2 a1 exp m2 a2

1 /m2 a2

39

For the cooling period, the average outlet temperature is Tfc,a

1

C3 exp m3 a3 exp m3 a4 C4 exp m4 a3 exp m4 a4

1 /m3 a4 1 /m4 a4

40

134

MURALIKRISHNA

Figure 2. Comparison with Hausen’ s theory and experimental data. Figure 4. Effect of Nusselt number on the performance of a regenerator.

where w1

a 3 Z1

a4 h1

m3

A1 1

1

m4 A1 B1 a3 /a4

A1 A Yc

1

41 B1 B1

1/2 1/2

2

42 43 44

2

45

a1 /a2 1/Yc

46

B/Y

RESULTS AND DISCUSSION Case (i) Model Without Conductive Transport An experimental result is an ideal choice to compare and establish the validity of the model. Heggs1 4 has presented experimental data on the performance of a medium sized regenerator. Heggs compared Hausen’ s results with his own experimental data and has observed that the most widely used Hausen’ s theory is inadequate in predicting the behaviour of regenerators; experimental points are found to lie well below the theoretical curve, even though the trend is reasonably good, as shown in Figure 2. Figure 2 shows the comparison of the present model with Heggs’ experimental data. It is observed that the present model predicts the behaviour of a regenerator with better accuracy compared to Hausen’ s theory. It could be clearly seen that the deviation of the experimental results from the predictions is pronounced in the initial periods of operation

Figure 3. Effect of Reynolds number on the performance of a regenerator.

in the case of Hausen’ s analysis. This is because of the simplifying assumptions made in formulating the problem and is due to the non-inclusion of a storage effect in the ¯ uid. It can be seen from the comparisons that the current model consistently predicts the behaviour of a regenerator, quantitatively. The ranges of parameters over which the study is conducted are as follows: Particle ratio, b Nusselt Number, Nu Reynolds Number, Re Aspect ratio, L /D Fourier number, Fo

: : : : :

5,000±10,000 1±10 20,000±180,000 0.1±10 10 ± 5 ±10 ± 1

The above ranges encompass the possible range of operation of a regenerator and are selected to study the behaviour of regenerators. The effectiveness approaches unity for L /D > 1, Nu > 10, b > 15, 000, Fo > 102 . Therefore, no appreciable variation in effectiveness results beyond the values of the parameters considered for study. Figures 3±6 show the effect of different parameters on the performance of a regenerator. It can be deduced from Figures 3 and 4 that a lower Reynolds number yields a higher value of effectiveness, as the air is in contact with the material particles for a longer period of time and results in an increased transfer of thermal energy. The performance of a regenerator for different values of particle ratio is depicted in Figure 5. The effectiveness of a

Figure 5. Effect of particle ratio on the performance of a regenerator.

Trans IChemE, Vol 77, Part A, March 1999

STUDY OF HEAT TRANSFER PROCESSES IN A REGENERATOR

135

Figure 6. Effect of aspect ratio on the performance of a regenerator.

regenerator increases with a larger value of the particle ratio. A larger value of the particle ratio signi® es a small size of particle and a large surface area per unit volume of the regenerator. At very large aspect ratios, an increase in effectiveness is negligibly small compared to the size of a regenerator. Also, lengthy regenerators are associated with a higher pressure drop which in turn demands a blower with more capacity. Figure 6 shows this aspect. In the case of a higher value of the heat transfer coef® cient, the effectiveness is expected to increase with an increase in Nusselt number. This aspect is evident from Figures 3 and 4. The ratio of process to recovery period, Y, is an important parameter and its effect on the performance of a counter¯ ow regenerator is shown in Figure 7. A small value of Y is associated with a comparatively longer time period for the recovery stream to exchange thermal energy with the material. Therefore, the effectiveness increases with a decrease in the value of Y . Here, the ratio of the velocities of the two streams is assumed to be equal to unity. Case (ii) Model With Conductive Heat Transfer The range of parameters considered for the studies is as follows: Particle ratio, b Nusselt Number, Nu Reynolds Number, Re Fourier number, Fo

100±2,000 0.1±2.0 100±300 10± 4 ±10 ± 2

Figure 8. Variation of constants of integration in the conduction-model.

These ranges span the possible range of operation of regenerators and are selected to study the response of regenerators in this case. The effectiveness assumes a value of unity for L /D > 0.01, Fo > 10 2 and Re > 500. The ranges for Nu and b are selected for representative purposes and it is found that the effect is negligible beyond the range mentioned above. Equation (31) represents the temperature pro® le for an air stream. Equation (31) reveals that the solution consists of two functions. The ® rst function decays exponentially, while the other shows an opposite trend. The temperature distribution is a combination of these two contrasting functions. Figure 8 shows the variation of constants of integration for three different values of Reynolds number. One of the constants approaches zero, while the other constant tends to unity at higher values of Fourier number. This suggests that the solution can be approximated by asymptotic solution at a higher Fourier number and is valid for slow wheels. The plots show that at a higher Reynolds number, the constants approach the limiting values more rapidly than at a lower Reynolds number. From Figure 9 it can be seen that Nusselt number and particle ratio do not exert any in¯ uence on the performance of the regenerator in this case. This is because of the lower Reynolds number. CONCLUSIONS Analytical solutions are obtained to the problem of air ¯ ow through the matrix of a ® xed bed regenerator for two cases: (i) Without axial dispersion and conduction. (ii) With axial dispersion and conduction.

Figure 7. Effect of ratio of periods on the performance of a regenerator.

Trans IChemE, Vol 77, Part A, March 1999

The analytical model satisfactorily predicts the experimental data of a regenerator and the solutions are applicable to air-metal regenerators. The solutions are valid for both heating and cooling of air.

136

MURALIKRISHNA

where j1

rf cf /kf L /D 1/Fo L /D

2

D2 /tr

rf cf Lv/kf D/D m/m

hc Ap L2 N/kf V R 2

D /D

NV S

rs cs D2 /ks tr

j4 j4

rs cs ¶ts /¶t

hc Ap n tf

hc Ap n tf

ts / 1

Tf

hc Ap L2 N/kf V R

NV s

A3

and rs cs D2 ¶ts /¶t /ks

hc Ap N/VR / NVS /VR

D2 /ks

A4 ´ tf ts In dimensionless form, equations (A3) and (A4) could be expressed as j 1 ¶ Tf / ¶ h

j2 ¶Tf /¶Z

j4 ¶Ts /¶h

j 5 Tf

j3 Tf

Ts

A5

and TS

A6

1

A9

D2 /af tr af /as

hc D/kf kf /ks 6 D/d A11

F hG Z

Ts

A12

j5 /j3 F h G Z

A13

where F h is a function of h and G Z is a function of Z alone. Differentiate equation (A12) with respect to h and Z, respectively. Also, differentiate equation (A13) with respect to h. Then, substitute the corresponding derivatives in equations (A5) and (A6) to obtain the following set of ordinary differential equations j2 F h G Z

j4 F h G Z

j3

j3

j5 F h G Z A14 A15

j5 F h G Z

The solutions to the above set of ordinary differential equations are given by f h

A1 A2

rf cf L2 v ¶tf /¶z /kf

L /D

Assume a solution of the form

exp exp

j3 j3

j5 /j4

j4

j1 /j2 Z

j5 /j4 h

A16 A17

NOMENCLATURE

2

rf cf L /D 2 D2 ¶tf /¶t /kf

NV s /

A10

hc Ap ND /NVs ks

GZ

Multiply equation (A1) by L /kf and equation (A2) by D2 /ks . Then,

3

Nu bK

e

ts /e

D2 /as tr

2

j5

APPENDIX

v¶tf /¶z

1/ 3/2 D/d

y/Fo

j1 F h G Z

rf cf ¶tf /¶t

A8

hc Ap L2 N/kf VR

1 ´ 1/N 2 Nu b L /D / b3 1/144N L /D

Figure 9. Effect of different parameters on the performance of a regenerator in the conduction-model.

The governing partial differential equations could be rearranged as

Pr Re L /D

2

hc D/kf L /D 2 6 D/d

In the case of the model without dispersion and conduction, higher values of the particle ratio and Nusselt number result in a higher value of effectiveness. A larger aspect ratio increases the effectiveness of a regenerator. Beyond a certain value of the aspect ratio, there is a negligible increase in the effectiveness of a regenerator while the size of it becomes extremely large. Further, it increases the pressure drop across a regenerator. In the case of the model with dispersion and conduction, the particle ratio and Nusselt number do not appreciably affect the performance of a regenerator. Here, an asymptotic solution will suf® ce to predict the behaviour of a regenerator. Also, the square of the Peclet number dominates the heat transfer process.

D2 /af tr A7

mcf /kf rf vD/m L /D j3

2

2

rf cf L2 n /kf L

j2

L /D

Ap b1 , b2 c d D Fo hc j1 j2 j3 j4 j5 k K L n N Nu Pr Re t

surface area of particle, pd 2 , m2 constants de® ned in equations (11) and (12), respectively. speci® c heat, kJ kg ±1K ±1 diameter of each particle, m diameter of a regenerator, m Fourier number, af tr /D2 , dimensionless ¯ uid-to-particle heat transfer coef® cient, W m ±2K ±1 (1/Fo L /D 2 , dimensionless Re Pr L /D , dimensionless Nu b 1/ b3 L /D 1/144N 1 L /D 2 , dimensionless y/Fo ), dimensionless Nu b K, dimensionless thermal conductivity, W m ±1K ±1 thermal conductivity ratio, kf /ks , dimensionless length of a regenerator, m number density, number of particles per unit volume of regenerator, m3 number of particles in a regenerator Nusselt Number, hc D/kf , dimensionless Prandtl Number, m cf /kf , dimensionless Reynolds Number, rf vh D/m, dimensionless temperature, °C

Trans IChemE, Vol 77, Part A, March 1999

STUDY OF HEAT TRANSFER PROCESSES IN A REGENERATOR ta t hi t ci Tf Ts vc vh VR Vs Y z Z Z1

initial temperature of a regenerator, °C initial temperature of a hot stream of air, °C initial temperature of a cold stream of air, °C dimensionless temperature of air stream, tf ta / thi tci dimensionless temperature of matrix material, ts ta / thi tci velocity of air stream (cooling mode), m s ±1 velocity of air stream (heating mode), m s ±1 volume of a regenerator, pD2 L /4, m3 volume of a solid spherical particle, p/6 d3 , m3 ratio of cooling to heating periods, tp /tr , dimensionless axial co-ordinate, m axial co-ordinate (heating period), z/L, dimensionless axial co-ordinate (cooling period), 1 z/L , dimensionless

Greek Symbols a thermal diffusivity, m2 s ±1 b particle ratio, 6(D/d , dimensionless c ratio of velocities, vc /vh , dimensionless e solid fraction, N Vs /VR , dimensionless h time co-ordinate, t/tr , dimensionless h1 time co-ordinate, t/tp , dimensionless m viscosity, kg m ±1s ±1 r density, kg m ±3 t time co-ordinate, s tr time allowed for heating (heating period), s tp time allowed for cooling (cooling period), s y thermal diffusivity ratio, af /as , dimensionless g effectiveness of a regenerator, 1 Tf ,a , dimensionless Subscripts f air stream s matrix material

137

5. Baclic, B. S., 1985, The application of Galerkin method of the solution of the symmetric and balanced counter¯ ow regenerator problem, Trans ASME J of Heat Transfer, 107: 214±217. 6. Wilmott, A. J. and Duggan, R. C., 1980, Re® ned closed form methods for the counter-¯ ow thermal regenerator problem, Int J Heat Mass Transfer, 23: 655±662. 7. Coppage, J. E. and London, A. L., 1953, The periodic ¯ ow regeneratorÐ a summary of design theory, Trans ASME, 80: 779±787. 8. Romie, F. E., 1993, Calculation of temperature in a single-blow regenerator having arbitrary initial and solid entering temperature, Trans ASME J of Heat Transfer, 115: 1064±1066. 9. Handley, D. and Heggs, P. J., 1968, The effect of thermal conductivity of the packing material on transient heat transfer in a ® xed bed, Int J Heat Mass Transfer, 12: 549±570. 10. Heggs, P. J. and Carpenter, K. J., 1979, A modi® cation of the thermal regenerator in® nite conduction model to predict the effects of intraconduction, Trans IChemE, 57: 228±236. 11. Bahnke, G. D. and Howard, C. P., 1964, The effect of longitudinal heat conduction on periodic ¯ ow heat exchanger performance, Trans ASME J of Engg Power, 86: 105±120. 12. Chung Hsiung Li, 1983, A numerical ® nite difference method for performance evaluation of a periodic ¯ ow heat exchanger, Trans ASME J of Heat Transfer, 105: 611±617. 13. Muralikrishna, S., 1991, Heat and Mass Transfer studies in rotary regenerators and dehumidi® ers, PhD Thesis, (IIT, Madras, India). 14. Heggs, P. J., 1983, Experimental techniques and correlation for heat exchanger surfaces: regenerators. in Low Reynolds Number Flow Heat Exchangers Ed. By S. Kakac, R. K. Shah, and A. E. Bergles, (Hemisphere Publishing Corp) pp. 369±394.

ACKNOWLEDGEMENTS

Superscripts c cooling period h heating period

The author wishes to thank his students Mr V. Aishwarya, (Graduate student (VPI)) and Mr K. Nagarajan, (Lecturer in our college) for their painstaking efforts in transforming the manuscript to the present form. Also, thanks to Dr R. Venkatesalu, Managing Trustee and the Principal for providing the necessary facilities.

REFERENCES 1. Hausen, A., 1929, Uber die theorie des warmeastauches in regeneratoren, Agnew Math Mec, 9: 173±200. 2. Lambertson, T. J., 1958, Performance factors of a periodic ¯ ow heat exchanger, Trans ASME, 75: 586±592. 3. Wilmott, A. J., 1964, Digital computer simulation of a thermal regenerator, Int J Heat Mass Transfer, 7: 1291±1302. 4. Razelos, P. and Benjamin, M. K., 1978, Computer model of thermal regenerators with variable mass ¯ ow rates, Int J Heat Mass Transfer, 21: 735±743.

Trans IChemE, Vol 77, Part A, March 1999

ADDRESS Correspondence concerning this paper should be addressed to Professor S. Muralikrishna, Department of Mechanical Engineering, Sri Ramakrishna Engineering College, Vattamalaipalayam, NGGO Colony Post, Coimbatore 641022, India. (E-mail [email protected]). The manuscript was received 3 April 1998 and accepted for publication after revision 3 December 1998.