vol1no3-6_hammamim

Engineering Applications of Computational Fluid Mechanics Vol. 1, No.3, pp. 216–226 (2007)

NUMERICAL STUDY OF COUPLED HEAT AND MASS TRANSFER IN A TRAPEZOIDAL CAVITY

Moez Hammami*, Mohamed Mseddi and Mounir Baccar

Laboratoire de Dynamique des Fluides Numérique et Phénomènes de Transferts Département de Génie Mécanique, Ecole Nationale d’Ingénieurs de Sfax,

Route de Sokra, B.P. W 3038 Sfax, Tunisie * E-Mail: [email protected] (Corresponding Author)

ABSTRACT: In this paper, we present a numerical three-dimensional study of coupled heat and mass transfer by natural convection occurring in a trapezoidal cavity. Our objective is to evaluate numerically the thermal and hydrodynamic behaviour of this system. It is assumed that the cavity vertical walls are thermally isolated and impermeable, and we imposed on the lower surface a constant high temperature and concentration. The upper surface is supposed to be cooled at a constant temperature and has a zero concentration.

Governing equations are solved by a finite-volume technique and provide the temperature, concentration and velocity fields in binary mixture air-water vapor system. The obtained results show that the flow configuration depends strongly on the α angle inclination of the upper wall. The influence of the cavity dimensions on heat and mass transfer rates is also examined. In particular, as the aspect ratio increases, multi-cellular flow patterns start to form.

Keywords: natural convection, trapezoidal cavity, coupled heat and mass transfers, finite-volume method, 3D simulation

1. INTRODUCTION

Due to its importance, natural convection in enclosures has been investigated by many numerical and experimental researchers, both in rectangular and, to a lesser extent, in trapezoidal geometries. As far as rectangular enclosures are concerned, the basic problem of natural convection has been quite extensively studied in both laminar and turbulent flow regimes. The study of Ostrach (1972) has made a big contribution on the fundamental aspect of natural convection in closed cavity. Let us also mention the numerical and experimental works of Nicolette, Yang and Lloyd (1985) that concern the transient bidimensional and one-phase natural convection that occurs in a cubic cavity filled with air, with one vertical partition cooled and the other partitions adiabatic. Leong, Hollands and Brunger (1998) presented results for the natural convection problem of an air-filled cubical cavity. The results are presented for a cavity with one pair of opposing hot and cold walls, and the remaining faces having a linear variation of temperature. Recently, Wu, Ewing and Ching (2006) conducted an experimental investigation into the natural

convection in an air-filled square cavity driven by a temperature difference between the vertical walls. The top wall temperature had a significant effect on the flow along the top wall. In particular, there was a flow separation on the top wall when the temperature of the top wall was increased. More recently, Sharif (2007) presented numerical study of laminar mixed convective heat transfer in two-dimensional shallow rectangular cavities with an aspect ratio of 10. The top moving lid of the cavity is at a higher temperature than the bottom wall. Computations are performed for Rayleigh numbers ranging from 105 to 107. The effects of inclination of the cavity on the flow and thermal fields are investigated for inclination angles ranging from 0° to 30°. The average Nusselt number is found to increase with cavity inclination. The rate of increase of the average Nusselt number with cavity inclination is mild for the dominating forced convection case, while it is much steeper in the dominating natural convection case. Additional works concerning the transient natural convection in closed cavities with solutal gradient has been conducted by many authors. Bennacer, Mezenner and Bouhadef (2001) investigated numerically and analytically the transient, double

Received: 3 Mar. 2006; Revised: 7 May 2007; Accepted: 12 May 2007

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Engineering Applications of Computational Fluid Mechanics Vol. 1, No. 3 (2007)

diffusive natural convection in a horizontal enclosure. The enclosure is heated and cooled along the vertical walls and solutal gradient is imposed vertically. It is found that the flow becomes unstable for finite range of solutal to thermal buoyancy ratios. It is possible to obtain different solutions on this region depending on the initial conditions. Also, the results reveal that the thermal convection may be suppressed for strongly stratified fluid. Also, Hammami, Mseddi and Baccar (2007) developed a two-dimensional numerical modeling of thermosolutal transfer in rectangular cavity for analysing the complex flow structure velocities and temperature distributions in the transient regime. The enclosure is heated and cooled along the horizontal walls and solutal gradient is imposed vertically. Resolution of coupled momentum, heat and mass transfer equations gives interesting local information concerning evolution with time of the hydrodynamic and the thermal behaviours during the storage of energy. Hammami, Mseddi and Baccar (2007) have numerically proved the importance of the salinity gradient in the accumulation of energy and in the reduction of the thermal losses by convection. They have demonstrated that concentration stratification is stable and resistant to the flow evolution. Trapezoidal geometries on the other hand have received more limited attention. Among the representative studies in trapezoidal cavity were those presented by Lam, Gani and Symons (1989) who carried out both numerical and experimental works, and Mcquain et al. (1994) who studied numerically the flow in a trapezoidal cavity. Mcquain et al. (1994) found that streamlines and vorticity distributions are sensitive to geometric changes. Indeed, the streamlines are altered as the geometry changes from a rectangle, through a series of trapezoids, to a triangle. The primary eddy becomes smaller, only partially filling the cavity. Due to its sharper and larger stagnant region, the

triangle exhibits secondary, tertiary and quaternary eddies. Thus, a triangular cavity would be comparatively less effective in the transport of mass and energy by convection. Tiwari et al. (1997) study the convective mass transfer in a trapezoidal cavity. They attempted to determine the Nusselt number for a trapezoidal cavity, which can be used for the evaluation of convective mass transfer in a solar distillation process. Boussaid, Mezenner and Bouhadef (1999) studied numerically the influence of geometry and the nature of the species in the cavity. The obtained results show that the heat and mass transfer rates vary depending on the thermo-solutals solicitations. The decrease of the Lewis number causes an increase in the heat and mass transfer rates. Recently, Reynolds et al. (2004) presented an experimental and computational study of the heat loss characteristics of a trapezoidal cavity absorber. Heat loss from the absorber occurs via a complex interaction between radiation, convection and conduction within the cavity, and then from the cavity to the surroundings. Almost all the aforementioned numerical studies have dealt with two-dimensional simulations. However, the flow structure can be quite complex and a two-dimensional approach is criticisable. The aim of the present work is to study numerically the coupled heat and mass transfer phenomena in a trapezoidal cavity with a binary mixture of air-water vapor. The study intends to improve the heat exchange coefficient between the inclined cooled upper wall and the heated lower plane. For this purpose, we suggest here a parametric study giving, at various Rayleigh numbers, the effect of the geometrical parameters of the cavity on the hydrodynamic and thermal characteristics. Integrating local information allows us to correlate the average overall heat transfer coefficient.

L

H

Cooled surface

Heated surface

l

z

x

y

α

Symmetry plane

Fig. 1 Schematic trapezoidal cavity.

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The flow inside the trapezoidal cavity, as shown in Fig. 1, can be mathematically described by the continuity, momentum, energy and mass equations. These equations are solved by finite volume method in a Cartesian three-dimensional coordinates system. Resolution is conducted in the steady laminar regime by way of the false-transient method.

2. MATHEMATICAL FORMULATION

We suppose that the fluid is incompressible and has constant physical properties except the density in the buoyancy term (the body force term), which depends linearly on both the local temperature and concentration (Boussinesq approximation). Because of the symmetry of the system, it is only necessary to conduct simulation over half of the cavity.

2.1 Basic equations

By taking into account the previous assumptions, the coupled transport equations governing heat and mass transfer in 3-D Cartesian coordinates can be written in dimensionless form as follows:

• Continuity equation 0Vdiv =

r (1)

• U–velocity component ( )

xPUgradPr-UVdiv

tU

∂∂

−=+∂∂ r (2)

• V–velocity component ( )

yPVgradPr-VVdiv

tV

∂∂

−=+∂∂ r (3)

• W–velocity component ( )=+

∂∂ WgradPr-WVdiv

tW r

( ) ( 5.0CLePr

mRa5.0TPrTRazP

−+−+∂∂

− ) (4)

• Energy equation

( 0Tgrad-TVdivt

T=+

∂∂ r ) (5)

• Concentration equation

0CgradLe1-CVdiv

tC

=⎟⎠⎞

⎜⎝⎛+

∂∂ r (6)

The parameter of practical importance in thermal system is the overall coefficient of heat transfer from the hot surface to the cold surface, which is measured by the average Nusselt number. The Nusselt and Sherwood numbers are averaged over half of the bottom of the cavity and can be expressed as:

∫ ∫=∂

∂×

−=L

0

2/l

00z

dxdyz

)z,y,x(TlL

2Nu and

( )∫ ∫

=∂∂

×−=

L

0

2/l

00z

dxdyz

z,y,xClL

2Sh (7)

2.2 Boundary and initial conditions

Initially, the binary mixture air-water vapor system is supposed to be in stagnation state: U=V=W=0. The initial temperature and concentration throughout the cavity are the same as ambient conditions: T=0 and C=0. The boundary conditions are the no-slip conditions on all the rigid wall surfaces. The vertical walls are assumed to be thermally isolated and impermeable, and the lower surface is supposed to be maintained at a constant high temperature and have a constant saturated water-vapor concentration; in dimensionless form, we have: C=1 and T=1. At the symmetrical vertical plane, the corresponding boundaries are: ∂U/∂y=0, V=0, ∂W/∂y=0, ∂T/∂y=0 and ∂C/∂y=0. The upper surface is supposed to be cooled by ambient air at a constant temperature and has a zero water-vapor concentration (T=0 and C=0).

3. NUMERICAL METHOD

To obtain numerical solution of the complete governing equations (1) to (6), finite-volume discretization method was used. The cavity domain is subdivided into a number of control volumes Ω defined in a Cartesian three-dimensional coordinates. The equation of transport to be solved is then integrated on each of these volumes, so expressing the balance of flux “JΦ”. The general conservation form of the transport parameter “Φ”, which stands for components of velocity vector, temperature and concentration scalars, can be written as follows:

dvSdvJdivdvt

⋅+−=∂Φ∂

∫∫∫∫∫∫∫∫∫Ω

ΦΩ

ΦΩ

(8)

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with: ΦΓ−Φ= ΦΦ gradVJrr

is the flux term of Φ, Γ

Φ is the corresponding conductance

coefficient, and SΦ is the corresponding source/sink

term. A staggered mesh is used in such a way—four different control volumes are defined for a given node point: one for each of the three vector components and one for the scalar variables. Then, each of the transport equations is integrated over its own control volume. The spatial discretization is obtained using a hybrid scheme interpolation. Concerning temporal discretization, the false-transient approach with an implicit scheme of alternate directions of Douglass and Gunn (1964) was used. The pressure-velocity coupling was handled by the SIMPLE algorithm of Patankar (1980). A computational domain consisting of 40×40×40 grid points with non-uniform grid spacing in the x-, y- and z-directions and a dimensionless time step 10-3 were found to be sufficient for producing accurate results at reasonable computing time. The convergence criterion required that the difference between the current and previous iterations for all of the dependent variables be 10-4.

4. PARAMETRIC STUDY

In order to ameliorate the heat transfer rate in a trapezoidal cavity, a better knowledge of the flow and thermal patterns inside this cavity is necessary. The mathematical modeling allows predicting the hydrodynamic and thermal behaviors in the cavity, by conducting a parametric study on the influence of: dimensionless length, dimensionless width, inclination angle of the upper surface and Rayleigh number. These results are given for a buoyancy ratio N=1 (Ra = RaT = Ram) and fixed values of Prandtl and Schmidt numbers (Pr=0.7, Sc=0.6), which correspond to the average binary mixture air-water vapor characteristics.

4.1 Effect of the aspect ratios

Figures 2, 3 and 4 represent velocity and temperature fields at the symmetrical vertical mid-plane of the cavity for different length ratios (L/H =1, 2 and 4). Dimensionless width is assumed to be equal to unity and the angle is fixed at 14°. These results are obtained for Ra=5x103, which corresponds to a laminar flow regime. We note that

a fundamental variation in the structure of the flow and thermal patterns occurs as the length of the cavity increases. Therefore, for a small value of dimensionless length L/H =1, Fig. 2 shows a large and powerful recirculation occupying the entire cavity. As the length of the cavity increases, multi-cellular structures are generated. Hence, for L/H =2 (Fig. 3 refers), two cells are obtained. The cell situated in the retracted side region of the cavity reveals a clear decrease of the natural convection movement which practically gives a stagnation zone. For L/H ratio equal to 4 (Fig. 4 refers), we obtain a triangular geometry cavity, and there appears a succession of vortices which become smaller and weaker until they disappear near the lower corner of the cavity. Hence, we can distinguish a stagnation zone taking place in the confined region of the system. Generation of multi-cellular structure should improve the fluid homogenisation, and subsequently convective heat transfer will occur. In fact, conjugated cells contribute to enhance the axial convection which promotes heat and mass transfer between the lower and the upper planes.

(a) Velocity distribution

(b) Temperature distribution

Fig. 2 Velocity and temperature distributions at the symmetrical vertical mid-plane for L/H =1 and Ra=5x103.

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To verify this behavior, we have reproduced in Fig. 5 the average Nusselt number as a function of the dimensionless length for different Rayleigh numbers and a constant dimensionless width (l /H =1). It appears that the Nusselt number remains unchanged when L/H is increased to 2. This is because only one cell dominates the flow field. However, when the L/H ratio is greater than 2, the heat transfer, and subsequently the mass transfer occurs, which is related to the appearance of more than one recirculation.

Ra = 5x103

Ra = 104

Ra = 2x104

Fig. 5 Effect of the dimensionless length of the cavity on the Nusselt average number for various Rayleigh numbers and l /H =1.

Fig. 6 3D temperature distribution for l /H =1 (L/H =2 and Ra=104).

Fig. 7 3D temperature distribution for l /H =5 (L/H =2 and Ra=104).

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The effect of the length ratio interacts with the width ratio. Figures 6 and 7 illustrate 3-D temperature fields induced respectively in small (l /H =1) and large (l /H =5) trapezoidal cavities; the inclination angle α is assumed to be equal to 18°. For cavites of wider width (as in Fig. 7), the temperature patterns indicate the dominance of a multi-cellular flow field giving rise to alternating downstream and upstream natural convection movement between two consecutive cells. Fig. 8 gives more information on the impact of the width of the cavity on natural convection. It shows W–velocity component profiles for Ra=104, L/H =2, and for various dimensionless width, l /H =1, 2 and 5. These profiles give the axial velocity evolution with the y coordinate, and are reproduced for x=1 and z=0.3.

Fig. 8 W-velocity component profiles for various

dimensionless widths of the cavity for x=1, z=0.3, L/H=2 and Ra=104.

Fig. 9 Effect of the dimensionless width on the

average Sherwood number for L/H =2.

For l /H =1 or 2, we notice that only one recirculation occurs between lateral surfaces of the cavity. As the width of the cavity increases (l /H =5), we obtain a sinusoidal profile indicating a succession, in parallel y-z planes, of clockwise and counterclockwise secondary cells alternately. It is noted that the width ratio has a great effect on the velocity field whereas the dimensionless width has a negligible influence on the average Nusselt and Sherwood numbers (Fig. 9 refers). This can be explained by the fact that the number of natural convective eddies formed all along the width is proportional to the cavity width.

4.2 Effect of the inclination angle

The angle of the inclined wall can be varied from zero degree (rectangular cavity) until it reaches a triangular shape. For the two inclination angles 10° and 26.5°, figures 10 and 11 show temperature and velocities distributions at the symmetrical vertical mid-plane of the cavity. These results are given for a Rayleigh number equal to 104 and aspect ratios L/H =2 and l /H =1. The obtained results show that the flow configurations depend strongly on the inclination angle α of the upper wall. For a small angle (as in Fig. 10), two cellular flow patterns have been found. The clockwise cell is comparatively weak and at the verge of disappearance. At a larger value of the angle (as in Fig. 11), the two cells mentioned for α=10° (Fig. 10 refers) were found to have merged, creating a single and stronger natural convective eddy pattern. It is noted that the configuration of the thermal and flow behaviors in this case is a type of heated triangular enclosure. Fig. 12 shows the average Nusselt number as a function of the inclination angle for L/H =1 and l /H =1. The increase of the inclination angle gives rise to thermosolutal transfer which causes the Nusselt number to increase. This is because the natural convection movement increases with the angle, as reflected by the size and the magnitude of the counterclockwise cells appeared in figures 10 and 11.

Ra = 5x103

Ra = 104

Ra = 2x104

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Fig. 10 Velocity and temperature distributions at the symmetrical vertical mid-plane for α =10° and Ra= 104.


(b) Temperature distribution Fig. 11 Velocity and temperature distributions at the

symmetrical vertical mid-plane for α =26.5° and Ra= 104

Ra = 5x103

Ra = 104

Ra = 2x104

Fig. 12 Effect of the inclination angle on the average Nusselt averaged number for various Rayleigh numbers, L/H =1 and l /H =1.

4.3 Rayleigh number effect

In general, increasing Rayleigh number Ra strengthens the buoyancy-driven cells generated by the vertical temperature gradient. With all other parameters remaining the same (L/H =2, l /H =1 and α=22°), figures 13 and 14 show the flow patterns generated for various values of Ra. Comparison of the flow patterns indicates intensification of the natural convection as Ra increases from 5x103 to 5x104. For a small Rayleigh number: Ra=5x103, the flow is characterized by a large stagnation zone in the confined zone of the cavity. Elsewhere, a big counterclockwise cell dominates the flow field. The flow pattern for Ra=5x104 is formed by succession of three obvious eddies rotating at alternate directions. The cell formed at the spacious side of the cavity is much stronger than the other two cells. These recirculations enhance the heat transferred from the heated lower surface to the cooled inclined plane.

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Fig. 13 Velocity distribution at the symmetrical vertical

mid-plane for L/H =2, α =10° and Ra= 5x103.

Fig. 14 Velocity distribution at the symmetrical vertical mid-plane for L/H =2, α=10° and Ra=5x104.

5. COMPARISON WITH ANTERIOR RESULTS

In order to check the accuracy of the findings obtained in the present study, we have reproduced the study of natural convection heat transfer in a rectangular enclosure, which was reported earlier by Corcione (2003). In his work, a numerical study of laminar natural convection in a cavity using air as the medium and having differentially heated isothermal horizontal walls and adiabatic vertical walls is reported. Simulation is conducted for aspect ratio L/H =2. Furthermore, it is assumed that the flow is incompressible and laminar, and fluid is Newtonian. Fig. 15, giving the temperature distributions in the cavity for a Rayleigh number equal to 104, shows a satisfactory qualitative agreement between our computations and those of Corcione (2003). We also compared in Fig. 16 our results with the published data of Corcione (2003) on the average Nusselt number along the hot bottom wall as a function of the Rayleigh number Ra. As well, a good agreement with the anterior results is observed.

(a) Temperature distributions (Corcione, 2003).

(b) Temperature distributions (Present study). Fig. 15 Comparison of the temperature distributions in

the cavity for a Rayleigh number equal to 104.

Fig. 16 Comparison of the average Nusselt number with

anterior results reported by Corcione (2003).

6. CONCLUSIONS

In this paper, a three-dimensional numerical modeling of heat and mass transfer in a binary mixture of air-water vapor confined in a trapezoidal cavity has been developed. The solution of the coupled momentum and heat and mass transfer equations is achieved by the control volume method. We have demonstrated that a fundamental variation in the flow and thermal structures accompanies the modification of the aspect ratios, the α inclination of the upper cooled wall and the Raleigh number. The flow changes from a predominantly mono-cellular pattern to a multi-

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cellular structure when the aspect ratios of the cavity are increased. Also, increasing Ra number favours the appearance of a succession of vortices. However, the increase in the angle of the inclined cold wall entails gradually a flow comparable to that occurs in a triangular enclosure. Appearance of multi-cellular structure improves the fluid homogenisation, and heat and mass transfer will take place. But this creates a very slow moving fluid in the retracted region of the cavity, giving a nearly stagnant region. Integrating local vertical temperature gradient at the bottom of the cavity allows us to correlate the overall heat transfer coefficient. The heat transfer profiles did not reveal a significant change in the Nusselt number—Nu values, when the width ratio increased. However, Nu was found to increase when the length ratio changed from 2 to 4. Also, we note that the Nusselt number increases as a function of the angle α and the Rayleigh number. For design purposes, an attempt has been made to correlate all of the heat transfer data obtained in this study. By cross-plotting numerical data, the Nusselt number can be correlated by using equation (9), established for length ratio L/H less than 2:

Nu = 0.17 Ra0.25 α0.16 (9)

For 2<L/H<4, the computation equation giving the heat transfer is as follows:

Nu = 0.11 Ra0.25 α0.16 (L/H)0.62 (10)

It appears that the numerical Nusselt numbers given by both equations (9) and (10) versus Ra number are approximately equal to 0.25. This result is in agreement with the general theory of natural convective flows. Finally, numerical results have been compared with literature data and a satisfactory agreement is found.

NOMENCLATURE

a = λ / (ρ Cp) thermal diffusivity C concentration Cmin minimal concentration Cmax maximal concentration Cp specific heat D diffusivity g gravitational acceleration h overall heat transfer H height of the cavity L length of the cavity

l width of the cavity P pressure T temperature T min minimal temperature T max maximal temperature U, V, W velocity components X, Y, Z spatial coordinates

Dimensionless form

C = [C – (Cmax +Cmin )/2]/(Cmax -Cmin ) L = L /H l = l /H P = P / (ρ a2 /H 2 ) T = (T – (T max + T min )/2)/( T max - T min ) U=U H / a , V=V H / a , W=W H / a x=X/ H , y=Y/ H , z=Z/ H

Greek symbols

ρ fluid density μ dynamic fluid viscosity βT thermal expansion coefficient βc compositional expansion coefficient λ thermal conductivity α inclination angle

Dimensionless numbers

Le Lewis number N buoyancy ratio Nu Nusselt number Pr Prandtl number Sc Schmidt number Sh Sherwood number Ra Rayleigh number RaT temperature Rayleigh number Ram mass Rayleigh number

REFERENCES

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