The effect of wall conduction on the performance of regenerative heat exchangers

0360-5442/92 $5.00+ 0.00 Copyright0 1992PergamonPressLtd

EnergyVol. 17, No. 12, pp. 1199-1213,1992 Printedin GreatBritain.All rightsreserved

THE EFFECT OF WALL CONDUCTION PERFORMANCE OF REGENERATIVE EXCHANGERS

ON THE HEAT

CHIEN-MING SHEN and W. M. WOREK? Department of Mechanical Engineering (m/c 251), University of Illinois at Chicago, P.O. Box 4348, Chicago, IL 60680, U.S.A. (Received 10 February 1992)

Abstract-The effect of heat conduction in the wall on the effectiveness of rotary regenerators is described. In order to document the effect of wall conduction, it is assumed that the thermal conductance of the solid is finite, both parallel and perpendicular to the gas flow. Two parameters introduced for modeling conduction effects are the Biot numbers in the y(Bi,) and x(BiX) directions. Our results show that for certain values of Bi, and Bi,, the effectiveness including wall conduction can deviate substantially from results obtained when wall conduction is neglected.

INTRODUCTION

Regenerators may be generally classified as dynamic or static. For dynamic regenerators, also called rotary or Ljunstrom regenerators, there is a single matrix arranged in the form of a flat disk or hollow drum. The solid wall or matrix is exposed to the hot and cold streams by rotating the matrix. The static (also called fixed-bed) regenerators operate continuously only when two identical regenerators are used. The heating and cooling processes in the fixed-bed regenerator are reversed by the closing and opening of valves that direct the fluid stream to the regenerators. Although heat regenerators have been used in industrial applications for nearly 6 decades, the mathematical representation of fixed-bed regenerators have remained virtually the same since the initial publications of Nusselt’ and Hausen.’ The mathematical modeling of rotary regenerators has also remained the same since the original publication of Coppage and London.9 Most workers investigating the performance of a regenerator use models which are based on assumptions initially presented by Nusselt. Nusselt made the important assumptions that the thermal conductance of the solid is zero parallel to the gas flow and infinitely large normal to the gas flow. Baclic and Heggs’ investigated the case of negligible thermal resistance in the wall with their new method of a strong and weak approach, and Skiepkoz6*” investigated the effect of latitudinal conduction (i.e. conduction perpendicular to the direction of flow) and presented the effect of latitudinal conduction on regenerator performance in terms of an inverse Peclet number (Pe-‘). The performance of rotary regenerators is affected by the following factors: (i) Rotormuss and speed-Karlsson and Helm’ have documented the effect of rotary mass and speed on the performance of a rotary regenerator. Coppage and London’ showed that C: should be 5 or more for good performance.4 Kays and London3 showed that when the rotational speed of the rotor is greater than approximately 15 ‘pm, the effectiveness of a regenerator is constant. More recently, San, Worek, and LavanU showed that when the parameter C: is greater than about 10, the effectiveness of a regenerator is essentially constant. (ii) Carryover-Due to carryover from one side of the regenerator to the other, regenerators are limited to gas-gas tTo whom all correspondence should be addressed. 1199

CHIEN-MINGSHEN and W. M. WOREK

1200

heat-exchange applications. The degradation of the regenerator performance resulting from this factor is directly proportional to the ratio of the rotor speed to the fluid-flow rate. If C& and CE, are greater than 10, the effect of carryover will be less than 1%.4 Banks” has suggested that this contribution may be predicted by treating the fluid carryover as a fluid flow bypassing the regenerator matrix. (iii) Leakage-The leakage problem becomes serious if a rotary regenerator is used where the pressure ratio between two fluid streams is of the order of 2-6.4 (iv) Conductivity of thermal storage material-From previous work, it is known that large longitudinal conduction (i.e. in the direction of flow) degrades regenerator performance and large latitudinal conduction is advantageous, because more mass is available for heat transfer without increasing carryover losses. Therefore, the ideal matrix material should have zero conductance in the direction of fluid flow and infinite heat conductance in the direction perpendicular to the fluid flow. A great deal of work has been done on regenerators using such an ideal matrix materia17~‘2’1~18~U)~21~24~~ and also on matrix materials that have finite conductance perpendicular to the flow direction. *5~Z~23~26*27 In this study, we concentrate on the effect on component performance of matrix material conductance both parallel and perpendicular to the flow direction. (v) Matrix geometry-For near-ambient or low-temperature applications, a corrugated sheet metal construction may be used.475 (vi) Matrix material-A set of desirable characteristics for the thermal storage material has been suggested by Schmidt and Willmott.6 Their specifications for the matrix are the following: high specific heat, high thermal diffusivity, high density, reversible heating and cooling performance, chemical and geometrical stability, noncombustible, noncorrosive and nontoxic, low vapor pressure to reduce the cost of containment, low material cost and low storage-unit fabrication costs, and sufficient mechanical strength to support compression loads resulting from the stacking of the storage core.

MATHEMATICAL

MODELS

Two models are used in this study. In the first or non-conduction model, heat conduction in the fluid is negligible and the thermal conductance of the thermal storage wall is zero parallel to and infinite perpendicular to the fluid flow. In the second or finite-conduction model, heat conduction in the fluid is also assumed to be negligible and the thermal conductance of the thermal storage wall is finite, both parallel and perpendicular to the direction of fluid flow. For simplicity, the following assumptions have been made in both mathematical models: constant fluid and thermal storage-material properties; a fully-developed flow in the fluid channel; a fully-developed thermal boundary layer in the fluid channel; the regenerator is thermally insulated from the surroundings; negligible thermal radiation as compared to convection and conduction heat transfer; no carryover and mixing of the fluid while switching from one period to another. A schematic of a regenerator and the nomenclature used are shown in Fig. 1. If the fluid channels and the thermal storage wall are identical and if the fluid flows are the same in each of the channels, there are surfaces of symmetry in the regenerator. The analysis may be simplified by considering only one-half of the channel width for both fluid channels and for the thermal storage walls, as is shown in Fig. 2. Nonconduction

model

Since we assume for this model no conduction in the storage wall, the control volumes for the fluid and thermal storage wall mass will be chosen as in Figs. 3(a) and (b), respectively. From conservation of energy, we obtain for the hot fluid

!%+LdTh+- hAh h

td,h

3.X

Chfd,h

(G

-

Tw,h) = 0.

Wall conduction effect on rotary regenerator performance

1201

Flow chofinel Thermal storogc

WON

Fig. 1. Schematic and major nomenclature used for a counterflow regenerator,

For the thermal storage wall during the hot period,

(2) The governing equation for the cold fluid is

(3)

CHIEN-MINGSHEN and W. M. WOREK

1202

r-l ////lllllll.

1

C, cl

ChTh

--

ch

di

FIT,,

L

aTh ax

dx 1

=h

%%*x

h

(Th +

-CT,-T I_

Y.h ) *x

(a)

Control

h-

An “.

volume

(T,

of fluid

-TV,,)

L \

8

dx

(b) Control

volume

of wall

Fig. 3. Control volume for the non-conduction model.

and for thermal storage wall during the cold period,

(4) The negative sign in Eq. (3) represents a counterflow regenerator. The boundary and initial conditions for the counterllow regenerator Th(O,tb) = Th,iy 0 s th s Pi,

(5)

T,(L) tc) = Tc,iy 0 S tc 6 PC,

(6)

Tw&, t,, = Pi) = T,&,

t, = 0),

0 6 x s L,

(7)

T,,Jx, tc = I’._)= Tw&, th = 0),

0 sx s L.

(8)

Equations (l)-(4) are the dimensional order to obtain the nondimensional parameters are introduced:

x=;,

are the following:

Y=$,

governing equations for the non-conduction model. In governing equations, the following nondimensional

r,, =i

,

(th-+$

1203

Wall conduction effect on rotary regenerator performance

Q

_

h-

Th

-

c,i

Th,i- T,,i ’

Q,=

T,- Tc,i

@w,h=

G,i - T,,i ’

Using these relations, Eqs. (l)-(4)

Tw,h- Tc,i

%c

G,i - T,,i ’

=

Tw,h- Tc,i G,i - Z,i ’

can be rewritten as follows. For the hot fluid,

for the thermal storage wall,

We may similarly write the governing equations of the cold period. For the cold fluid,

2 =NTU,(O,

(11)

- Q,,J;

for the thermal storage wall,

(12) The dimensionless boundary and initial conditions are 8,,(o,rh)=1,

06r,sl;

(13)

&(l,r,)=O,

OGt,al;

(14)

@,,h(.& t,, = 1) = &,C(x,

t, = o),

0 d x d 1;

(15)

tB,,&,

t,, = o),

0 d x s 1.

(16)

rC = 1) = 0,.,(x,

Finiteconduction model According to assumptions stated previously, conduction only occurs in the thermal storage wall. The equations governing heat transfer in the fluid are the same as in the nonconduction model [Eqs. (1) and (3)]. An element of the thermal storage wall is shown in Fig. 4. Applying conservation of energy, we obtain for the hot period 3% dr,=

k,Ak.hL &h

$T,

h

&Ah

6 a2Tw

%-Fay”

h

(17)

Similarly, for the cold period,

We now introduce the two Biot numbers Bi, = hLl K,,

(19)

Bi,, = h 6/K,.

(20)

and

Where Bi, is essentially Bi,,(L/G). The Stanton number is St,

=

h/@-h&,

(21)

and the porosity u = Af/Afr = A,/(A, + Ak).

(22)

1204

CHIEN-MINGSHEN and W. M. WOREK

-K,

-K,+dy

+

dx

t JT**h ay

+

“;$’

dy)

a7w.h ar I-K,

dy

aT*,h ( ar

+dy

++dr)

*

Y A

a%,h

AhdX -Km-

I

L

ay

dx

Fig. 4. Control volume for the finite-conduction model.

Using these definitions, we obtain the dimensionless during the hot period in the form

2=

equation for heat conduction

3+ (&@W($--)3;

(&--(ah)(F)&)

in the wall

(23)

similarly, for the cold period,

(24) The dimensionless boundary and initial conditions are &(O,rrJ=l,

osr,e1,

(25)

Q,(O,r,)=l,

O~Z,~l,

(26)

sx

6,&f,

r,, = 1) = f3,,#,

tc = o),

0

%,C(x,

rc = 1)=&.,(x,

r,,=o),

t,dxsl.

4 1,

(27) (28)

There are two types of boundary conditions that can be used to solve the equation governing heat transfer in the thermal storage wall. For the first, we assume that the end of the wall is insulated and for the second that the end of the wall loses heat by convection to the fluid. We have verified that both boundary conditions give essentially the same results for the parameters used in our study. For convenience, the insulated boundary conditions were used throughout

Wall conduction effect on rotary regenerator

performance

1205

this study, i.e. for the hot period, dO,,JdX

= 0,

at X = 0,

(29)

dO,,JdX

= 0,

at X = 1,

(30)

%&JdY

= 0,

at Y = 0,

(31)

dO,,JdY

= Bi,(O, - 6&J,

at Y = 1;

(32)

for the cold period, %,,JaX

= 0,

at X = 0,

(33)

%D,,J?JX = 0,

at X = 1,

(34)

aO,,JdY

at Y = 0,

(35)

&,JaY

= 0,

= Bi,,(O, - O,,J,

at Y = 1.

(36)

Cyclic equilibrium and thermal effectiveness

A regenerator has reached cyclic equilibrium (cyclic steady state) when the heat transferred to the matrix during the flow of the hot fluid stream is equal to the heat released from the matrix during the flow of the cold fluid stream.’ Hence, for cyclic equilibrium, the performance of the regenerator may be evaluated in terms of the effectiveness E. To correlate the number of transfer units on the hot and cold sides, the modified number of transfer units3 is introduced as WU,

= (l/C,i”)[(l/hAiJ

NUMERICAL

+ (l/h&)]-‘*

(37)

FORMULATION

In order to simulate the performance of a rotary regenerator, Eqs. (9)-(12) for the non-conduction mode1 and Eqs. (23) and (24) for the finite-conduction mode1 were solved numerically with the appropriate boundary and initial conditions. The performance of a regenerator without conduction in the wall has been given previously by Lamberson.” In this case, the numerical scheme is simple and the computational time is short compared to that for the finite-conduction model. When the conduction effect in the regenerator wall is considered, the number of nodes that must be used increases dramatically. Bahnke and Howard” studied the effect of longitudinal conduction on the performance of a regenerator. In this study, we are interested in the effects of both longitudinal and latitudinal conduction on performance. A convenient approach for solving these governing equations is an explicit finite-difference method. Since we are interested in cyclic equilibrium of the regenerator, the initial temperature field of the thermal storage wall is not given. In order to solve the governing equations, we need to guess the initial temperature of the thermal storage wall. The procedure for calculation is iterative until the temperature field of the thermal storage wall has converged and becomes independent of the initial guess. The last and most important step is an examination of the heat balance. The heat-balance error can be determined by computing the difference in enthalpy change for the fluid on the hot side from that on the cold side, as compared to the enthalpy change on the hot side. This error is expressed as R

_ err

-

Ch(l -

%,O)- cc&,0

Ch(l - al,o)

I= Il

cc Qc.0 -c,<1

-&,J

*

(3%

To ensure that the program is stable and convergent throughout the calculating process, we need an increased number of subdivisions in space and time. In order to determine the trade-off between CPU time and heat-balance error, some information has been tabulated on

CHIEN-MING SHENand

1206

W. M. WOREK

Table 1. Criteria to end computations for both numerical models. convergence

non-conduction CPU:=, time

criteria

lo-’

1 0.5843 I 0.07075 / I

1O-4

model? iteration number

RvL %

1

I

1.0

I

1 0.05592 1

RJ,

time

%

41

model* iteration number

1 14.25 1 1.589 1 I

I

I

I

17

I

1 25.17 1 1.978 1 I

I

I

30

I

I

0.0604 /

1.222

1o-B

24

finite-conduction

CPU&,

II

II

I

50

/ 32.76 11.982 / I

I

39

I

t/Based on 50 nodes in X, 1000 nodes in time, C’ = 1.0, C; = 5.0,NTU, = 4.0,initial wall temperature=l.O. * Based on 20 nodes in X, 20 nodes in Y, 2500 nodes in time, C’ = 1.0, C; = 5.0, NTU. = 4.0, Bi. = 50.0, Bi, = 2.5, initial wall temperature= 1.0. § Using CPUb.,. = 2.466 set, see Eq. (39). ill Relative

heat-balance

error, see Eq. (38).

reaching thermal equilibrium and is shown in Tables l-3. Table 1 shows the dependence of the heat-balance error and CPU time on the stop criterion when it is defined as the maximum amount that the temperature field can change from one time step to the next. Table 1 shows that for a stop criterion of lo-*, the heat-balance error is acceptable for both models. Table 2 refers to the non-conduction model for C: = 5.0, NTU, = 4.0, and a convergence criterion of 10m4. Table 3 holds for the finite-conduction model for C: = 5.0, NTU,= 4.0, Bi, = 50.0, Bi, = 2.5, and the stop criterion of 10e4. The third and sixth columns of Table 2 and the fourth and seventh columns of Table 3 give the relative CPU time, which is defined as CPU,,, = CPU(sec)/CPU,,,(sec), where the CPUbase is the base CPU time in set for 50 subdivisions in space and 1000 subdivisions in time for the non-conduction model (i.e. CPUbase = 2466sec). It should be emphasized that for the unbalanced case (C* # l), an increased number of nodes in the Table 2. CPU time and heat balance error vs grid size for the non-conduction model.

‘I Based

on C; = 5.0. NTU,, = 4.0convergence criterion =lO-‘, initial wall temperature=l.O. *Based OIL C; = 5.0, NTLro = 4.0. convergence criterion =lO-‘, initial wall temperature= 1.0. 5 Using CPUb.,.= 2.466 set, see Eq. (39). ll Relative heat-balance error, see Eq. (38).

Wall conduction effect on rotary regenerator Table 3. CPU time and heat-balance

/ subdiv-

I subdiv-

isions in X

isions in Y

2500

10

2500

subdivisions in time

2500

10000

I 1 I I 1

1

C’ = $I_, %

10

7.863

2.10

20

10

13.99

20

20

25.17

20

1 210.7

/

1207

error vs grid size for the finite-conduction

CPU& time

40

performance

I

l.Ot

iteration

CP&

number

model.

C’ = 0.5* RB, ’ iteration number

time

%

33

4.984

10.64

21

2.02

32

9.153

2.99

21

1.98

30

16.77

2.92

20

33

I 140.2

13.33 1

/ 0.642 1

22

t Based on C’; = 5.0, NTU, = 4.0, Bi. = 50.0, Bi, = ‘2.5, convergence criterion=lO-‘, initial wall temperature= 1.0. * Based on C; = 5.0, NZV. = 4.0, Bi, = 50.0, Bi, = 2.5, convergence criterion=lO-’ initial wall temperature= 1.0. § Using CPU*,_. = 2.466 set, see Eq. (39). ll Relative heat-balance error, see Eq. (38).

longitudinal direction are needed to achieve the desired heat-balance error as compared to the balanced case (C* = 1). For all the results presented in the next section, the heat-balance error will be within 0.8% for the non-conduction model and within 2.0% for the finite-conduction model.

RESULTS

Validation of the numerical method

In order to validate the numerical method used in this study, we have compared our results for the case of negligible and finite wall conduction with the results of Kays and London.3 Table 4 holds for the non-conduction model based on C* = 1.0 and 0.5 for C: = 10.0. The comparison in Table 4 shows that the differences between our results and those presented by Table 4. Validation of the non-conduction

NTU,

C’ = 1.0 Effectiveness Present Kays & London( 1)* Study(2)

2.0

0.664

I 0.666

4.0

0.796

I ;

10.0

0.904

Error A§ % 0.30

’ 0.66 1

i’ Based on C’; = 10.0. * Taken from Kays & Loudow Kays & Londow3

C’=O.5 Effectiveness Present i Kays 6i London( l)q Study(2)

0.799

0.910

model.

I

0.X

( Error 45 %

0.773

0.39

0.922

/

0.923

j 0.11

0.995

1

0.986

~ 0.90

5 Calculated from A = l(1) - (2)1/(l).

7 Taken from

CHIEN-MINGSHEN and W. M. WOREK

1208

Table 5. Validation of the finite-conduction model.

NTU,

C’ = 1.0 Effectiveness / Kays k ~ Present London(l)* j Study(2)

c-=0.95 Effectiveness I Error I Present ( A§ i Kays k / % 1London(l)( Study(a)

I 10.0

15.0

I

0.901

0.930

/

0.898

0.33

0.921

0.919

0.928

0.22

0.948

0.951

Error

/ A§ / % I 1 0.22 I ! 032

’

‘b Based on C; = 10.0, Bi, = 0.1. ,I = 0.01. *Taken from Kays & London.3 5 Calculated from A = l(1) - (Z)//(l). ll Taken from Kays & Londom3

Kays and London3 are less than 0.9%. Table 5 refers to the finite-conduction model with the longitudinal conduction parameter rZ defined by Kays and London3 for correlating the results. The conduction parameter is now defined by A = (l/Bi,)&thlC,i,).

(40)

The Bi,, used in Table 5 is 0.1 while C: = 10.0, whereas Kays and London3 used BiY = 0.0 and C: > 5.0. Table 5 shows that the differences between these two studies are within 0.7%. Regenerator/effectiveness As described in the previous discussion, the ideal thermal storage material is one that has zero conductivity in the direction of fluid flow and infinite conductivity in the direction perpendicular to the fluid flow. For such an ideal material, Bi, is infinite and Bi,, is zero. Szego and Schmidt14 have proposed a limiting Bi,, (i.e. Bi,, G 0.1) for which the non-conduction model can be used and effects introduced by finite-conduction can be neglected. Our results confirm this limiting Biy. In order to compare the differences obtained between the non-conduction and finiteconduction models, the results for these two models are shown in Figs. 5 and 6. Figure 5 is based on C* = 1.0, C: = 5.3, and Bi, = 50.0; and Fig. 6 is based on C* = 1.0, CT = 5.3 and Bi,, = 0.1. From these results, we see that if Bi,, d 0.1 and Bi, 3 9, ideal performance is attained. Also, similar results are obtained when 0.5 s C* < 1.0. Bradshaw, Johnson, and McLachlan13 have suggested a formula which can be used to correlate the non-conduction and finite-conduction models. It is called an equivalent conductivity model. The formula given by Jeffreson15 and by Lai, Dudukovic, and Ramachandranz3 for a symmetrical design (i.e. NTU,,/NTC, = 1.0) and balanced fluid flow (C* = 1.0) is NTUA = [ 1 + (Bi,/S)]NTU,*,

(41)

where NTUA is a modified number of transfer units considering latitudinal conduction in the solid for the equivalent conductivity model, and NTU,* is the modified number of transfer units for the non-conduction model. Equation (41) indicates that for the same performance (effectiveness E) of a regenerator, the ratio of the modified number of transfer unit of the equivalent conductivity model to that of non-conduction model is [l + (B&/S)]. It should be pointed out that Eq. (41) has been proposed for spherical particles in a packed bed. Using this equation and comparing the data presented in Fig. 5, we find that Eq. (41) is not a good approximation to account for latitudinal conduction in the solid.

Wall conduction effect on rotary regenerator performance

1209

100 I-

-

-

- -

Non-conduction

modal

Finitr-conduction

modal

90

60

a ?

70

Y

I t60 .-p! t; 0 z W60

40

30

20 0

I

I

I

1

2

3

1 4

Modified

number

I

I

I

I

I

5

6

7

6

9

of transfer

unit

10

1

(NTU,

Fig. 5. Effectiveness vs NTU,, for C* = 1.0, C: = 5.3, Bi, = 50.0.

1oc ----_ -

Non-conduction mod.1 Flnlto -conduction mod.1

60

40

30

20 0

1

2 Modified

3

4 number

of

5 transfer

6 unit

7

1NTU,)

Fig. 6. Effectiveness vs NTU,, for C* = 1.0, C: = 5.3, Biy = 0.1.

6

9

1210

SHENand W. M. WOREK

CHIEN-MING

so

260

” t2

10.0

1

70

-

60

-

50

-

5.0

3 Y z W

0

NTU,

0.13

0.26

0.39

0.62

- 1.0

0.65

0.76

0.91

1.17

1.04

1.30

Fig. 7. Effectiveness vs St,,(l + V,/V,)(l - ~)/a, for C* = 1.0, Cf = 5.3, Bi, = 0.1, Bi, = 50.0.

Two other parameters that affect the performance of a regenerator are St,(l - a)/a and S&(1 - a)/o. To examine the effect of these parameters on system performance, we define

(42) 7 shows the dependence of the effectiveness on St,(l + V&)(1 - a)/~, for C* = 1.0, C: = 5.3, Bi,, = 0.1, and Bi, = 50.0. These results indicate that this parameter does not affect the performance of a regenerator significantly when NTU, < 5.0. Figure

The conduction parameters To investigate the effects of longitudinal and latitudinal conduction, conduction parameter y and longitudinal conduction parameter 6 as

y-l E(non-conduction)

- s(latitudinal .s(non-conduction)

we define the latitudinal

conduction)1 (43)

and

p-1 E(non-conduction)

- s(longitudina1 conduction)1

c(non-conduction)

,

(44)

In Fig. 8 the latitudinal conduction parameter is plotted vs NTU, for C* equal to 1.0, for C: = 5.3 and Bi, = 50.0. From Fig. 8, we see that when the NTU, is increased, the effect of latitudinal conduction decreases; the values of y are maxima around NTU, = 1.5. Similar results are obtained when 0.5 d C* < 1.0. In Fig. 9, the longitudinal conduction parameter /I is presented as a function of NTU, for C* = 1.0, and also for C: = 5.3, Bi,, = 0.1. From Fig. 9, we see when NTU, increases, the effect of longitudinal conduction increases for low Bi, while for high Bi,, the effect decreases with NTU,. Furthermore, no maximum deviation exists. Also, similar results are obtained when 0.5 s C* < 1.0. respectively.

Wall conduction effect on rotary regenerator

performance

1211

27.0

24.3

0

I

I

I

I

I

I

I

I

1

2

3

4

5

6

7

8

Modified

number of transfer

unit

0.1 9

10

(NTU,)

Fig. 8. Latitudinal conduction parameter vs NTU,, for C* = 1.0, CT = 5.3, Bi, = 50.0.

0

I

2

Modified

3

4

5

number of transfer

6

unit

7

6

9

I NTU, )

Fig. 9. Longitudinal conduction parameter vs NTU,, for C* = 1.0, C: = 5.3, Bi,, = 0.1.

EGY 17:12-H

CHIEN-MING SHEN and W. M. WOREK

1212

CONCLUSION

A non-conduction model and a finite-conduction model have been presented for a counterflow regenerator. The governing equations for regenerator performance are solved using an explicit finite-difference technique. All of the results refer to NTUJNTU, = 1.0 while the values of C: are kept in a range that is normally encountered in regenerators (C: > 5.0). The results show that for design purposes, if BiY d 0.1 and Bi, 3 9.0, the non-conduction model can be used to calculate the regenerator performance. For Bi,, b 0.1 and Bi, d 9.0, the equivalent conductivity model given by previous investigators23 is not a good approximation for which the non-conduction results account for latitudinal conduction in the thermal storage wall. When NTU, increases, the effect of latitudinal conduction decreases and the values of y are maxima around NTU, = 1.5. Also, the effect of the factor C* on latitudinal conduction is not significant. When NTU, is increased, the effect of longitudinal conduction increases for low Bi, while it decreases with NTU, for high Bi,.

REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13.

W. Nusselt, 2. VDZ 71, 85 (1927). H. Hausen, Tech. Mech. Thermo-Dynam. 1,219 (1930). W. M. Kays and A. L. London, Compact Heat Exchangers, McGraw-Hill, New York, NY (1964). A. P. Fraas and M. N. Ozisik, Heat Exchanger Design, Wiley, New York, NY (1965). H. Hausen, in Heat Exchangers-Design and Theory Sourcebook, pp. 207-222, N. Afgan and E. U. Schlunder eds., McGraw-Hill, New York, NY (1974). F. W. Schmidt and A. J. Willmott, Thermal Energy Storage and Regeneration, McGraw-Hill, New York, NY (1981). B. S. Baclic and P. J. Heggs, in Advances in Heat Transfer, J. P. Hartnett and T. F. Irvine Jr. eds., Vol. 20, Academic Press, New York, NY (1990). H. Karlsson and S. Holm, Trans. ASME 65, 64 (1943). J. E. Coppage and A. L. London, Trans. ASME 75,779 (1953). T. J. Lamberson, Trans. ASME 80, 586 (1958). G. D. Bahnke and C. P. Howard, Trans. ASME 86, 105 (1964). A. J. Willmott, Znt. J. Heat Mass Transfer 12, 997 (1969). A. V. Bradshaw, A. Johnson, N. H. McLachlan, and Y. T. Chiu, Trans. Inst. Chem. Engrs 48, T77

(1970). 14. J. Szego and F. W. Schmidt, J. Heat Transfer 100,740 (1978). 15. C. P. Jeffreson, AZChE J. 18,409(1972). 16. A. J. Willmott and R. C. Duggan, Znt. J. Heat Mass Transfer 23, 655 (1980). 17. C. H. Li, J. Heat Transfer 105,611 (1983). 18. A. E. Saez and B. J. McCoy, Znt. J. Heat Mass Transfer 26, 49 (1983). 19. P. J. Banks, J. Heat Transfer 105,216 (1984). 20. B. S. Baclic, J. Heat Transfer 107,214 (1985). 21. Z. Wozniak, Numer. Heat Transfer 8, 751 (1985). 22. J. Y. San, W. M. Worek, and Z. Lavan, Energy-The International Journal 12,485 (1987). 23. S. Lai, M. P. Dudukovic, and P. A. Ramachandran, Numer. Heat Transfer 11,125(1987). 24.F. E. Romie, J. Heat Transfer 109,563(1987). 25.A. Hill and A. J. Willmott, Znt. J. Heat Mass Transfer 30, 241 (1987). 26. T. Skiepko, Znt. J. Heat Mass Transfer 31, 2227 (1988). 27. T. Skiepko, Znt. J. Heat Mass Transfer 32, 1443 (1989). NOMENCLATURE A = Heat transfer area (m’) Ak = Total cross-sectional area

gitudinal conduction

(m*)

At = Total free flow area \m’) Afr = Total frontal area (m )

Bi, = Biot number in the x direction

for

longi-

Bi,, = C= C* = C, =

Biot Heat Heat Heat rage C:= Heat

number in the y direction capacity rate (W/K) capacity ratio of Cmin and C,,,, capacity rate of the thermal stowall (W/K) capacity ratio of C, and C

Wall

C:= Cmin= C,, = h=

conduction effect on rotary regenerator performance

Heat capacity ratio of C, and Cmin Minimum of C, or C, (W/K) Maximum of C, or C,, (W/K) Heat transfer coefficient (W/m’ K) k = Conductivity of the fluid (W/mK) K, = Conductivity of the thermal storage material (W/mK) L = Length of the fluid channel or the thermal storage wall (m) NTU = Number of transfer units (nondimensional) P = Period (set) St = Stanton number T, = Temperature of thermal storage wall (“C) T = Fluid temperature (“C) t = Time during the hot or cold period (set) td = Dwell time of the fluid (set) V = Average velocity of the fluid (m/set) x = Longitudinal position (m) X = Dimensionless longitudinal position y = Latitudinal position (m) Y = Dimensionless latitudinal position

1213

Greek letters 8, = Dimensionless

wall temperature 6 = Dimensionless time-averaged temperature of the fluids 8 = Dimensionless fluid temperature 6 = Half thickness of the thermal storage wall (m) E = First-law effectiveness u = Porosity of the thermal storage matrix y = Latitudinal conduction parameter /3 = Longitudinal conduction parameter t = Dimensionless periods

Subscripts

c= h= i= 0= t=

Cooling period Heating period Inlet Overall, outlet Total