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Conjugate mixed convection laminar non-Darcy ®lm condensation along a vertical plate in a porous medium Ming-I. Char a,*, Jin-Dain Lin a, Han-Taw Chen b a b

Department of Applied Mathematics, National Chung Hsing University, Taichung, 402, China Department of Mechanical Engineering, National Cheng Kung University, Tainan, 701, China Received 23 February 2000; accepted 10 May 2000 _ (Communicated by E.S. S ß UHUBI)

Abstract This study numerically investigates the coupling of the wall conduction with laminar mixed-convection ®lm condensation along a vertical plate within a saturated vapor porous medium. The Darcy±Brinkman± Forchheimer model is utilized to treat the ¯ow ®eld and the eect of heat conduction across the wall is taken into account. The governing system of equations with their corresponding boundary conditions are ®rst transformed into a dimensionless form by a non-similar transformation and the resulting equations are then solved by the cubic spline collocation method. Of interest are the eects of the conjugate heat transfer parameter A, the Jakob number Ja, the Peclet number Pe and the inertia parameter C on the ¯uid±solid interfacial temperature distribution and the local heat transfer rate. The results indicate that the eect of wall conduction has great in¯uences on the ®lmwise condensation heat transfer, and is to reduce the local heat transfer rate as well as the dimensionless interfacial temperature in comparison with the isothermal plate case. It is found that the local heat transfer rate increases with a decrease in the Jakob number, the Peclet number, and the inertial parameter or an increase in the conjugate heat transfer parameter. In addition, the overall surface heat transfer rate for dierent values of A and Ja are also obtained. Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction The problem of mixed convection condensation phenomena in a porous medium occurs in various ®elds of science and engineering. Common examples are heat and ¯uid ¯ow for some

*

Corresponding author. Tel.: +886-4-2861030; fax: +886-4-2873028. E-mail address: [email protected] (M.-I. Char).

0020-7225/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 0 ) 0 0 0 7 4 - 4

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Nomenclature A conjugate heat transfer parameter C inertial coecient Da Darcy number g gravitational acceleration Gr Grashof number h heat transfer coecient hfg latent heat of condensation Ja Jakob number (subcooling parameter) k thermal conductivity K permeability L length of the plate Nu average Nusselt number Nux local Nusselt number p pressure Pe Peclet number Pr Prandtl number t thickness of the plate T temperature constant temperature of cooled Tb side of plate u,v velocity components in x- and y-directions, respectively x,y rectangular coordinates

Greeks a eective thermal diusivity C inertial parameter d ®lm thickness e porosity of the medium g pseudo-similarity variable h dimensionless temperature k density ratio, q/q K thermal diusiviy ratio, a/a l dynamic viscosity t kinematic viscosity n dimensionless streamwise coordinate q density of the ¯uid w stream function x viscosity ratio, l/l Superscripts n false time level of n n + 1 false time level of n + 1 vapor phase Subscripts i, j grid point location s saturated condition w condition in the platel

industrial drying and cooling processes, enhanced recovery of petroleum resources, packed-bed heat exchangers, solidi®cation of castings, geothermal reservoirs, and many others. Cheng [1] ®rst investigated the problem of ®lm condensation along an inclined cooled plate embedded in a Darcian porous medium by means of similarity transformation. The analysis was later extended by Kumari et al. [2] to a frustum of a cone without transverse curvature eect. Wang and Tu [3] studied the eect of non-condensable gas on forced convection condensation along a horizontal plate embedded in a porous medium. Also, an analytical approach to solve the laminar ®lm condensation outside a horizontal elliptical tube embedded in a Darcian porous medium was performed by Chiou et al. [4]. All previous studies [1±4], regarding ®lm condensation along the solid surface in a porous medium, deal primarily with the mathematical simpli®cation based on the Darcy ¯ow model. However, the non-Darcian eects become signi®cant in the porous media with high porosities such as foam metals and ®brous media. Using the similarity transformation, Nakayama [5] studied the Darcy±Forchheimer (DF model) ®lm condensation within a porous medium in the presence of gravity and forced ¯ow. Renken et al. [6] analyzed, using a ®nite-dierence method, the eect of vapor velocity on ®lm condensation heat transfer along an isothermal surface

M.-I. Char et al. / International Journal of Engineering Science 39 (2001) 897±912

899

embedded in a non-Darcy porous medium. Lucas [7] used the integral and ®nite-dierence treatment to attack the problem of combined body force and forced convection ®lm condensation of mixed vapors in open space. Shu and Wilks [8] carried out a detailed asymptotic and numerical study of the mixed-convection laminar ®lm condensation along a semi-in®nite vertical plate in a clean and open space. The foregoing studies [1±8], the thermal boundary condition at the solid surface was assumed to be isothermal and, thus, the interaction between the solid surface and its adjacent boundary layer was neglected. Consideration of the thermal interaction between the solid body and the ¯uid ¯ow, the conjugate heat transfer problem, more closely approximates the physical situation. Along this direction of research, Vynnycky and Kimura [9] analyzed, making use of a ®nite-dierence procedure, the conjugate free convection along a vertical plate in a porous medium. Pop and Nakayama [10] employed the integral method to study the conjugate free convection from long vertical plate ®ns in a non-Newtonian ¯uid-saturated porous medium. Numerical and asymptotic studies have been carried out by Pop et al. [11] who investigated the conjugate mixed convection on a vertical surface in a porous medium. Char et al. [12] performed an analysis of conjugate heat transfer occurring in the laminar boundary layer on a continuous, moving plate, employing the cubic spline collocation numerical method. Recently, Char and Cheng [13] used the previous numerical method [12] to examine the eect of wall conduction on the heat transfer characteristics of the natural convection ¯ow of micropolar ¯uids along a ¯at plate. The aforementioned investigations dealing with conjugate heat transfer problem [9±13] are not involving changes of phase. The problem of conjugate mixed-convection laminar ®lm condensation in a porous medium, which, to the authors' knowledge, is rarely addressed. This has motivated the present investigation. The purpose of this article is to investigate the conjugate mixed-convection laminar ®lm condensation along a conducting vertical plate in a porous medium ®lled with dry saturated vapor. The Darcy±Brinkman±Forchheimer (DBF) model is employed to describe the ¯uid ¯ow in the porous medium. Assuming a thin cooled plate is embedded in the porous medium so that heat conduction within the plate is one-dimensional. A cubic spline collocation method has been employed to solve this conjugate heat transfer problem. Results of interest, such as the ¯uid±solid interfacial temperature distribution and the local heat transfer rate are presented to highlight the in¯uence of the wall conduction. The overall heat transfer rate for the ®lm condensation is also examined at dierent values of the controlling parameters. 2. Mathematical formulation Consider a vertical ¯at plate of length L and ®nite thickness t, which is embedded in a ¯uidsaturated porous medium ®lled with a dry saturated vapor. The physical model and coordinate system are shown in Fig. 1, where the streamwise coordinate is denoted by x and that normal to it is denoted by y. At free stream, a dry saturated vapor continuously ¯ows downward in the positive x-direction at the uniform velocity U1 . The free stream temperature is the saturation temperature of the vapor Ts , whereas the temperature of the outside surface of the cooled plate is maintained at a constant temperature Tb and below the saturation temperature. Due to the gravity, a laminar ®lm of condensate runs downward along the plate. To describe the vapor phase,

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M.-I. Char et al. / International Journal of Engineering Science 39 (2001) 897±912

Fig. 1. Physical model and coordinate system.

the intrinsic coordinates x ; y are aligned so that x measures the distance along the interface from the leading edge of the plate and y measures the distance normal to the interface. In the analysis, the following conventional assumptions are made: the ¯ow is steady, laminar, incompressible and two-dimensional; the Boussinesq approximation is applicable; capillarity is negligible; the ¯uid-saturated porous media is isotropic and homogeneous; and the ¯uids is in local thermal equilibrium with the porous matrix. It is also assumed that the liquid ®lm thickness is relatively small as compared to a typical dimension of the plate and thus x x . Under these assumptions, the set of governing equations (namely, the continuity equation, the momentum equation based on the Darcy±Brinkman±Forchheimer model, and the energy equation) and their corresponding boundary and matching conditions, are given for each phase as follows: For the condensate ®lm x P 0, 0 6 y 6 d x ou ov 0; ox oy l o2 u e oy 2 u

l u qg K

1 dp dx

oT oT o2 T v a 2: ox oy oy

qCu2 0;

2

3

M.-I. Char et al. / International Journal of Engineering Science 39 (2001) 897±912

For the vapor x P 0, y y

d x P 0

ou ov 0; ox oy l o2 u e oy 2

901

4

l u q g K

dp dx

q Cu2 0;

T Ts

5 6

which obviously satis®es the energy equation. The boundary and matching conditions under which Eqs. (1)±(6) are to be solved are At the plate surface (y 0): T Tw x

u v 0;

7

At the vapor±liquid interface (y d x): T Ts ;

8

u u ;

9

dd q u dx l

v

q

dd u dx

v

;

ou ou l ; oy oy

10

11

and dd q u dx

v hfg k

oT : oy

12

At the vapor boundary layer edge (y ! 1): u U1 ;

13

where u and v are the Darcian velocity components in the x- and y-directions, respectively, T, p, and g the temperature, pressure, and gravitational acceleration, q and l the density and dynamic viscosity of the condensate, a and k the eective thermal diusivity and eective thermal conductivity of the ¯uid saturated porous medium, K and C the permeability and inertial coecient of the porous medium, e the porosity of the packed bed, hfg the latent heat of condensation, and d

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M.-I. Char et al. / International Journal of Engineering Science 39 (2001) 897±912

is the liquid ®lm thickness which is to be determined. Furthermore, Tw (x) is the surface temperature of the plate, which is not known a priori. The superscript denotes the vapor quantities, while no superscript is assigned for the liquid phase. Moreover, the pressure gradient term dp=dx appearing in Eqs. (2) and (5) may be eliminated using the boundary condition given by Eq. (13). It is noted that in the foregoing Eq. (7) Tw (x) depends on x; one objective of this work is to estimate Tw (x) and one further governing equation is, therefore, required. If the thickness of the plate t is relatively small compared with the length of the plate L, heat conduction within the plate can be considered to be one-dimensional. Under this condition, the axial conduction term in the heat conduction equation of the plate can be omitted and the governing equation for temperature distribution within the plate can be expressed as o2 Tw 0; oy 2

0 6 x 6 L;

t 6 y 6 0:

14

Its corresponding boundary conditions are kw

oTw oy

Tw Tb

k

oT x; 0 oy

at y

and Tw T x; 0 at y 0

15

t;

16

where Tw is the temperature, kw the thermal conductivity, and t is the thickness of the plate. By using Eqs. (14)±(16), the couplings can be obtained as Tw x T x; 0

kt oT x; 0 Tb : kw oy

17

To facilitate the analysis, the following transformations are introduced to non-dimensionalize the preceding equations p x y p Ts T Pex ; w a Pex f n; g; h ; n ; g L x Ts Tb p y p U1 x g Pex ; w a Pex f n; g; Pex ; a x

Pex

U1 x ; a 18

where w and w are the stream functions de®ned in the usual way, f and f the reduced stream functions for the ¯ows, and Pex and Pex are the local Peclet numbers, respectively. Introducing Eq. (18) into Eqs. (2), (3) and (5) gives 2 ! 1 o3 f n 1 of C 1 of PrGr n 0; 19 n 3 e og DaPe x og Pr k og Pe2 o2 h 1 oh of oh f n 2 og 2 og on og

of oh og on

0;

20

M.-I. Char et al. / International Journal of Engineering Science 39 (2001) 897±912

1 o3 f n 1 e og3 DaPeK

of og

Cx n 1 PrkK

! 2

of og

0:

903

21

The dimensionless forms of the corresponding boundary conditions are f n; 0 2n

of ; on

of n; 0 0; og

h n; 0

1

oh=ogjg0 n1=2 A

p 1 of n; d 1 of n; 0 f n; d n ; k K f n; 0 n 2 on 2 on of n; d of n; 0 ; og og

Ja

;

22

x

o2 f n; d p o2 f n; 0 K ; og2 og2

oh n; d of n; d 1 n f n; d 0; og on 2

of n; 1 1: og

23

24

In the foregoing equation, Pe U1 L=a is the Peclet number; Da K 2 =L is the Darcy number; C LC is the inertial parameter; Gr g q q L3 =qm2 is the Grashof number; Pr m=a is the Prandtl number, and the dimensionless parameters k, x, and K are the density ratio, viscosity ratio, and thermal diusivity ratio, respectively, which are de®ned as k q=q ; x l=l ; K a=a

25

p A kw L=kt Pe is the conjugate heat transfer parameter and can be regarded as a measure of deviation from the constant wall temperature solution. It may be noted that for the limit case A ! 1, the thermal boundary condition (22) on the plate becomes isothermal. Ja Cp Ts Tb =hfg is the Jakob number, which is a measure of the relative degree of subcooling in the ®lm. Finally, the heat transfer result in terms of the local Nusselt number can be expressed as hx oh p Pex ; 26 Nux k og g0 where h is the local heat transfer coecient. The average Nusselt number Nu is de®ned as Nu hL=k, where h is the average heat transfer coecient over the length L of the plate. In terms of non-dimensional variables, we have Nu=Pe

1=2

Z

0

1

oh n og g0

1=2

dn:

27

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M.-I. Char et al. / International Journal of Engineering Science 39 (2001) 897±912

3. Numerical method The transformed governing equations (19)±(21) subject to the boundary conditions (22)±(24) have been solved by using the cubic spline collocation method [13±16]. The steady-state solution of the system can be obtained by using a pseudotransient formulation in which a false transient term is added to each equation. Eqs. (19)±(21) using the false transient technique are discretized by un1 uni;j 1 n1 C 1 i;j Lui;j ni e Pr k Ds

uni;j 2

hn1 hni;j 1 n n1 i;j Ln1 hi;j fi;j lhi;j 2 Ds n1

un i;j

ui;j

Ds

fi;jn

1 Cx ni 1 Ln1 ui;j e PrKk

1 1 ni DaPe x

fin 1;j ni ln1 hi;j Dni

2 un i;j

uni;j

GrPr n; Pe2 i

uni;j

hni Dni

hni;j

1 ni 1 DaPek

1;j

28

ni ;

un i;j

29

30

where u of =og ;

u of =og; 2

2

Lh o h=og ;

lu ou=og;

lu ou =og ;

2

Lu o2 u=og2 ; 2

Lu o u =og ;

Dni ni

lh oh=og; ni 1 :

Ds sn1 sn denotes the false time step and the superscript n refers to the iteration number. After some rearrangement, Eqs. (28)±(30) can be expressed in the following spline approximation form: n1 Hi;jn1 Fi;j Gi;j ln1 Hi;j Si;j LHi;j ;

31

where H denotes the functions of u, h and u . The quantities Fi;j ; Gi;j and Si;j are known coecients calculated at previous steps (Table 1). Eq. (31) combined with cubic spline relations described by Rubin and Khosla [15], at n 1th iteration may be written in the tridiagonal form as n1 n1 ai;j /n1 i;j 1 bi;j /i;j ci;j /i;j1 di;j ;

32

where / represents the function h; u; u and its ®rst- and second-order derivatives. Therefore, Eq. (32) can be easily solved for the quantity /n1 i;j by use of the Thomas algorithm. The computational procedure followed is ®rst to solve the energy equation (20), which provides the temperature ®eld necessary for the solution of the reduced ®lm stream function equation (19). Solution of the reduced vapor stream function equation (21) then completes the procedure. This cycle of computation is repeated until convergence is achieved. The criterion for the convergence of the solutions is that the maximum relative change in all the dependent variables satisfy

M.-I. Char et al. / International Journal of Engineering Science 39 (2001) 897±912

905

Table 1 The coecients of Eq. (31) H h

F

G

hni;j

hni 1;j

hni;j

Dsni

u

uni;j

DsGrPr ni Pe2

u

un i;j 1

1 Dsfi;jn Dsni 2

Dni

2 un i;j

1 k

uni;j

2 DsC

DsCx n 1 PrKk i

j/n1 /ni;j jmax i;j < 5 10 7 : j/ni;j jmax

Pr un i;j

ni

1 x

Ds n DaPek i

uni;j

Ds n DaPe i

S

fi;jn

fin 1;j Dni

Ds

0

Ds=e

0

Ds=e

33

4. Results and discussion Inspection of the foregoing analysis reveals that the dimensionless parameters include e, Gr, Da, Pe, Pr, C, A, Ja, k, x and K. For the convenience of numerical investigation, computations have been carried out for various values of the parameters Ja, Pe, C and A with ®xed values of e 0:8, Gr 1010 , Da 10 6 , Pr 0:2, k 320, x 2:8, K 0:024. It may be noted that the present analysis includes the study of Darcy±Forchheimer (DF model) steady ®lm condensation along an isothermal vertical ¯at plate, corresponding to the case studied by Nakayama [5] for A ! 1 and C 0. In order to assess the accuracy of the present numerical method, we have compared the local heat transfer rates (as a function of Ja) with the corresponding approximate solutions of Nakayama [5]. The comparison is shown in Fig. 2 and found in good agreement. Results for the dimensionless interfacial temperature distributions as a function of n are plotted in Fig. 3 for various values of Ja with A 10, Pe 5 103 and C 5. It is seen from the ®gure that for a given Ja, the dimensionless interfacial temperature hw (Ts Tw =Ts Tb ) increases monotonically with increasing n. This ®gure also reveals that the higher values of Ja, the higher is the dimensionless interfacial temperature hw for ®xed values of A, C and Pe. This is because of the decrease in the liquid ®lm thickness as Ja increases which gives rise to a decrease in the temperature of the ¯uid on the surface Tw (x). Fig. 4 depicts the representative distributions of the dimensionless interfacial temperature hw along the plate for various values of A at Ja 0:02, C 5 and Pe 5 103 . To demonstrate the eect of wall conduction, it is remarked that in the limit when A becomes in®nity, the thermal boundary condition on the plate becomes that of a constant wall temperature Tb . Eq. (22) for h under this circumstance reduces to h n; 0 1. This ®gure shows that the lower values of A lead to larger deviations of the temperature of the ¯uid on the surface from A ! 1. The basic cause of this behavior is that lower values of A correspond to lower wall conductance kw L and higher convective cooling k or Pe, which promote greater surface temperature Tw , namely, smaller

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M.-I. Char et al. / International Journal of Engineering Science 39 (2001) 897±912

Fig. 2. Local heat transfer rate for constant wall temperature case A ! 1.

Fig. 3. Variation of interfacial temperature pro®les with n at dierent values of Ja for A 10, C 5 and Pe 5 103 .

M.-I. Char et al. / International Journal of Engineering Science 39 (2001) 897±912

907

Fig. 4. Variation of interfacial temperature pro®les with n at dierent values of A for Ja 0:02, C 5 and Pe 5 103 .

dimensionless surface temperature hw . Also, the dimensionless interfacial temperature increases as n increases. Fig. 5 shows the variation of the dimensionless interfacial temperature pro®les with n for various values of Pe. The results indicate a decrease of Pe predicts greater surface temperature variations. The reason for this behavior is that the smaller the Pe, the higher is the heat transfer rates. The streamwise variation of the dimensionless interfacial temperature hw is shown in Fig. 6 for dierent values of C. It is found that as C increases for ®xed values of A, Ja and Pe, the dimensionless interfacial temperature increases. This can be explained from the fact that inertial eect reduces the velocity within the boundary layer, and therefore leads to a decrease in the surface temperature Tw . Representative distributions of the heat ¯ow rate expressed by h0 n; 0 along the streamwise coordinate n are shown in Fig. 7 for dierent values of Ja. It is observed that the local heat transfer rate decreases as Ja increases. This is due to the increase in the liquid ®lm thickness as Ja increases which results in a reduction in the local heat transfer rate. This ®gure also reveals that the lower values of Ja lead to larger local heat transfer rate variations. Fig. 8 gives the plot of the local heat transfer rate for dierent values of A. It is observed that as A increases, the local heat transfer rate h0 n; 0 increases. This eect is attributed to the fact that a higher values of A, representing a lower thermal resistance of the solid wall, yields a higher heat transfer rate. Fig. 8 also shows that the in¯uence of the conjugate heat transfer parameter A is to

908

M.-I. Char et al. / International Journal of Engineering Science 39 (2001) 897±912

Fig. 5. Variation of interfacial temperature pro®les with n at dierent values of Pe for A 10, Ja 0:2 and C 5.

Fig. 6. Variation of interfacial temperature pro®les with n at dierent values of C for A 10, Ja 0:01 and Pe 5 103 .

M.-I. Char et al. / International Journal of Engineering Science 39 (2001) 897±912

Fig. 7. Eect of Ja on the local heat transfer rate for A 10, C 5 and Pe 5 103 .

Fig. 8. Eect of A on the local heat transfer rate for Ja 0:02, C 5 and Pe 5 103 .

909

910

M.-I. Char et al. / International Journal of Engineering Science 39 (2001) 897±912

Fig. 9. Eect of Pe on the local heat transfer rate for A 10, Ja 0:2 and C 5.

Fig. 10. Eect of C on the local heat transfer rate for A 10, Ja 0:01 and Pe 5 103 .

M.-I. Char et al. / International Journal of Engineering Science 39 (2001) 897±912

911

Fig. 11. Eect of A and Ja on overall heat transfer rate for C 5 and Pe 5 103 .

reduce the heat transfer rate as compared to the isothermal plate case (A ! 1). This reduction increases with decreasing A. Fig. 9 shows the variation of the local heat transfer rate h0 n; 0 with n for various values of Pe. The result indicates that larger values of Pe predict smaller local heat transfer rate. Also, the local heat transfer rate increases as n increases. Fig. 10 gives the eect of C on the local heat transfer rate for A 10, Ja 0:01 and Pe 5 103 . It is seen from Fig. 10 that the local heat transfer rate h0 n; 0 decreases as the inertial parameter C increases. An increase in the parameter C implies a higher inertial eect which results in a reduction in the momentum transport in the boundary layer, and hence the local heat transfer rate. Representative distributions of the overall heat transfer rate expressed by Nu=Pe1=2 are presented in Fig. 11 for dierent values of A and Ja. It may be seen that increasing the parameter A can enhance the overall heat transfer rate. However, increasing the parameter Ja can lower the overall heat transfer rate. As shown in Fig. 11, we conclude that under the in¯uences of A and Ja, the behavior of the overall heat transfer rate is similar to that of the local heat transfer rate.

5. Conclusions An analysis is made on the mixed-convection laminar non-Darcy ®lm condensation from a vertical plate embedded in a porous media, taking into consideration wall conduction eect.

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M.-I. Char et al. / International Journal of Engineering Science 39 (2001) 897±912

Numerical results to the transformed governing equations have been obtained by using the cubic spline collocation method. Conclusions are summarized as: 1. The eect of wall conduction has great in¯uences on the ®lmwise condensation heat transfer. The results indicate that the eect of wall conduction is to reduce the local surface heat transfer rate h0 n; 0 as well as the dimensionless interfacial temperature hw when compared to the isothermal plate case (without the eect of wall conduction). 2. As the parameter A increases, the interfacial temperature hw and local surface heat transfer rate h0 n; 0 increase. 3. The interfacial temperature distribution hw increases, while the local surface heat transfer rate h0 n; 0 decreases, with an increase in the Jakob number Ja, the Peclet number Pe, and the inertial parameter C. Acknowledgements This research was supported by the National Science Council of the R.O.C. through grant NSC 89-2212-E-005-010. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

P. Cheng, Int. J. Heat Mass Transfer 24 (1981) 983. M. Kumari, I. Pop, G. Nath, Int. J. Heat Mass Transfer 27 (1984) 2155. C. Wang, C. Tu, Int. J. Heat Mass Transfer 32 (1989) 1847. J.S. Chiou, S.A. Yang, C.K. Chen, Chem. Eng. Commun. 127 (1994) 125. A. Nakayama, Warme-und Sto ubertragung 27 (1992) 119. K.J. Renken, M. Aboye, M. Carneiro, K. Meechan, Int. Commun. Heat Mass Transfer 20 (1993) 1. K. Lucas, Int. J. Heat Mass Transfer 19 (1975) 1273. J.J. Shu, G. Wilks, J. Fluid Mech. 300 (1995) 207. M. Vynnycky, S. Kimura, Int. J. Heat Mass Transfer 38 (1995) 219. I. Pop, A. Nakayama, Int. Commun. Heat Mass Transfer 21 (1994) 297. I. Pop, D. Lesnic, D.B. Ingham, Int. J. Heat Mass Transfer 58 (1994) 1517. M.I. Char, C.K. Chen, J.W. Cleaver, Int. J. Heat Fluid Flow 11 (1990) 257. M.I. Char, C.L. Chang, Int. J. Heat Mass Transfer 40 (1997) 3641. S.G. Rubin, R.A. Graves, Comput. Fluids 3 (1975) 1. S.G. Rubin, P.K. Khosla, AIAA J. 14 (1976) 851. P. Wang, R. Kahawita, Int. J. Comput. Math. 13 (1983) 271.

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