International Communications in Heat and Mass Transfer 39 (2012) 869–876
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International Communications in Heat and Mass Transfer
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Numerical simulation of three-dimensional double diffusive free convection flow andirreversibility studies in a solar distiller☆
Kaouther Ghachem a, Lioua Kolsi a,⁎, Chamseddine Mâatki a, Ahmed Kadhim Hussein b,Mohamed Naceur Borjini a
a Unité de Métrologie en Mécanique des Fluides et Thermique, École Nationale d'Ingénieurs de Monastir, Université de Monastir, 5000, Tunisiab College of Engineering, Mechanical Engineering Department, Babylon University, Babylon City, Hilla, Iraq
☆ Communicated by W.J. Minkowycz.⁎ Corresponding author.
Numerical results of double-diffusive natural convection are presented in a three-dimensional solar distiller.The flow is considered laminar and caused by the interaction of the thermal energy and the chemical speciesdiffusions. Equations of concentration, energy and momentum are formulated using vector potential-vorticity formulations in its three-dimensional form, then solved by the finite volume method. The Rayleighnumber is fixed at Ra=105 and the effects of the buoyancy ratio are studied for opposed temperature andconcentration gradients, with a particular interest to the three-dimensional aspects and entropy generation.
Double diffusive natural convection i.e., flows generated by buoy-ancy due to simultaneous temperature and concentration gradients,occurs in different ranges of fields such as oceanography, drying pro-cesses, crystal growth processes, solar desalination, etc. This convec-tion was largely studied for aiding or opposed temperature andconcentration gradients. Studies on this subject can be found in sever-al publications [1–16]. It was noted that, for a buoyancy number nearthe unity, an oscillatory flow takes place, caused by the interactionbetween thermal and compositional cells. Nishumira et al. [13] car-ried out a careful description of the mechanism of this oscillatoryflow.
A few of studies are interested in the 3D double diffusive naturalconvection. Sezai and Mohamed [16] studied thermosolutal naturalconvection in a cubic enclosure subject to horizontal and opposinggradients of heat and solute. They indicated that the double-diffusive flow in enclosures was strictly three-dimensional for a cer-tain range of Rayleigh numbers, Lewis numbers and buoyancy ratiosrespectively. The same configuration was studied by Abidi et al. [14]but with heat and mass diffusive horizontal walls. They mentionedthat the effect of the heat and mass diffusive walls was found to re-duce the transverse velocity for the thermal buoyancy-dominated re-gime and to increase it considerably for the compositional buoyancy-
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dominated regime. The effect of radiative heat transfer on 3D doublediffusive natural convection was studied by Abidi et al. [17], and theyshowed that variations of the conduction–radiation parameter andoptical thickness affect significantly the flow structure. The combinedradiation double diffusive natural convection is also studied by Mef-tah et al. [18]. Obtained results show that radiation influences thetemperature and concentration field by creating oblique stratification.In their recent work, Bhuvaneswari et al. [19] present a numericalstudy on double diffusive mixed convection with a Soret effect in atow sided lid-driven cavity. They mentioned that heat and masstransfer are reduced if both walls are moving in the same direction,while heat and mass transfer are enhanced if the walls are movingin the opposite direction. Khadiri et al. [20] studied the double diffu-sive convection with Soret effect in a square porous cavity heated andsalted from below. They noticed that both, transition to oscillatory re-gime and heat transfer are considerably affected by the Soret param-eter. Li et al. [21] investigated the double diffusive convection flow ata high Rayleigh number, and they found that 3D modeling is neces-sary in order to have better agreement with experimental result.
The performance of a solar distiller could be governed by manyparameters such as depth of water, glass cover angle, fabrication ma-terials, temperature of water in the basin and insulation thickness,etc. In fact with the aim to maximize the distillate productivity ofthe apparatus over current solar stills, Naim et al. [23] developed asingle-stage solar desalination spirally-wound module of original de-sign that makes use of both the latent heat of condensation of theformed vapor and the sensible heat of the concentrated solution, inpreheating the incoming saline water. A recent detailed review on ac-tive solar distillation procedures and modeling could be found in thework of Sampathkumar et al. [22]. For the solar desalination of brack-ish water, the most used technology was the capillary film distiller
Subscriptsx, y, z Cartesian coordinatesdif diffusivefr frictionth thermaltot total
Superscript′ dimensional variable
870 K. Ghachem et al. / International Communications in Heat and Mass Transfer 39 (2012) 869–876
which was composed of identical cells of evaporation–condensation(Bouchekima et al. [24–26]). Ben Snoussi et al. [3] studied this config-uration by simulating numerically the natural convection in 2D rect-angular cavities. Results showed that the flow, mass and thermalfields were strongly dependent on the Rayleigh number and aspectratio.
However the three-dimensional transverse flow is primordialwhen dealing with the intensification of heat and mass transfers ina cell distiller, and it is still not extensively studied in literature. Inthis article we propose to study the double diffusive natural convec-tion with horizontal temperature and concentration gradients in athree-dimensional distiller, for a buoyancy ratio in the range of[0–10].
2. Physical problem and governing equations
The considered configuration (Fig. 1) represents a three-dimensional cavity of width W, height H and length L. In this studywe fix the aspect ratios at Ay=H/W=2 and Az=L/W=2. Differentand uniform temperatures and concentrations are imposed on twovertical walls, where the thermal buoyancy force retards the compo-sitional buoyancy force, i.e., opposing flow. All the other walls areconsidered adiabatic. The hot wall warms up by solar radiation induc-ing the evaporation of the water film; the vapor thus produced mixeswith the air then condenses when it reaches the cold wall. The fluidcontained in the cavity is assumed incompressible and the flow fol-lows the Boussinesq approximation.
The equations describing the double diffusive natural convectionare equations of continuity, momentum, energy and species diffusionrespectively:
∇:V→V¼ 0 ð1Þ
∂V→V
∂tVþ V→V: ∇
→� �
V→V¼ − 1
ρ∇→PVþ νΔV
→Vþ βt T V−T0ð Þ g→þβc CV−C0ð Þ g→ ð2Þ
∂T V∂tVþ V
→V:∇T V¼ α∇2T V ð3Þ
∂CV∂tVþ V
→V:∇C V¼ D∇2C V: ð4Þ
In order to eliminate the pressure term, which is delicate to treat,the numerical method used in this work is based on the voracity-vector potential formulation ψ
→− ω
→� �
. For this, one applies therotational to the equation of momentum. The vector potential andthe vorticity are, respectively, defined by the two following relations:
ω→V¼∇
→�V→V and V
→V¼∇
→�ψ→V: ð5Þ
In Eqs. (1), (2), (3) and (4), time t′, velocityV→′, the stream functionψ
→′,
and the vorticity ω→′, are put in their dimensional forms respectively by
W2/α, α/W, α and W2/α. The dimensionless forms of temperature andconcentration are respectively defined by: T=(T′−T′c)/(T′h−T′c)and C=(C′−C′l)/(C′h−C′l).
871K. Ghachem et al. / International Communications in Heat and Mass Transfer 39 (2012) 869–876
Based on the dimensionless variables above the governing equa-tions (stream function, vorticity and energy equations) can be writtenas:
− ω→¼ ∇2 ψ
→ ð6Þ
∂ ω→
∂t þ V→:∇
� �ω→− ω
→:∇
� �V→¼ Δ ω
→þRa:Pr:∂T∂z−N
∂C∂z ;0;−
∂T∂x þ N
∂C∂x
� �ð7Þ
∂T∂t þ V
→:∇T ¼ ∇2T ð8Þ
∂C∂t þ V
→:∇C ¼ 1
Le∇2C ð9Þ
with Pr=ν/α, Ra ¼ g:βt :W3: T Vh−T Vcð Þα:ν , N ¼ βc : C Vh−C Vð Þl
βt : T ′h−T ′ð Þc, Le ¼ α
D ¼ ScPr :
The physical boundary conditions can be defined for all the vari-ables, as follows:
• Temperature: T=1 at x=1, T=0 at x=0; ∂T∂n ¼ 0 on other walls
(adiabatic).• Concentration: C=1 at x=1, C=0 at x=0; ∂C
∂n ¼ 0 on other walls(impermeable).
• Vorticity: ωx=0, ωy ¼ −∂Vz∂x , ωz ¼ ∂Vy
∂x at x=0 and 1; ωx ¼ ∂Vz∂y ,
ωy=0, ωz ¼ −∂Vx∂y at y=0 and 1; ωx ¼ −∂Vy
∂z , ωy ¼ ∂Vx∂z , ωz=0 at
z=0 and 1.• Vector potential: ∂ψx
∂x ¼ ψy ¼ ψz ¼ 0 at x=0 and 1; ψx ¼ ∂ψy
∂y ¼ ψz ¼ 0at y=0 and 1; ψx ¼ ψy ¼ ∂ψz
∂z ¼ 0 at z=0 and 1.• Velocity: Vx=Vy=Vz=0 on all walls.
The local Nusselt and Sherwood numbers are given respectivelyby:
Sh ¼ ∂C∂x
����x¼0;1
and Sh ¼ ∂C∂x
����x¼0;1
: ð10Þ
The average values of the Nusselt and Sherwood numbers, on theisothermal walls are expressed by:
Nuav ¼ ∫1
0
∫1
0
Nu:∂y:∂z and Shav ¼ ∫1
0
∫1
0
Sh:∂y:∂z : ð11Þ
The local entropy generation rate in a three-dimensional flowwith single diffusing specie of concentration (C) can be written as[11]:
S′gen ¼ kT20
∂T V∂xV
� �2
þ ∂T V∂yV
� �2
þ ∂T V∂zV
� �2" #( )
þ μT0
(2
"∂V Vx∂xV
� �2
þ ∂V Vy∂yV
!2
þ ∂V Vz∂zV
� �2#þ ∂V Vy
∂xV þ ∂V Vx∂yV
!2
þ ∂V Vz∂yV þ ∂V Vy
∂zV
!2
þ ∂V Vx∂zV þ ∂V Vz
∂xV
� �2)
þ(R:DC0
∂CV∂xV
� �2
þ ∂C V∂yV
� �2
þ ∂CV∂zV
� �2" #
þR:DT0
∂T V∂xV
� � ∂C V∂xV
� �þ ∂T V
∂y′� � ∂C V
∂y′� �
þ ∂T V∂z′
� � ∂C V∂zV
� �� �): ð12Þ
where C0 and T0 are respectively the reference concentration andtemperature. The dimensional form of the local entropy generationis as follows:
NS ¼∂T∂x
� �2
þ ∂T∂y
� �2
þ ∂T∂z
� �2" #
þ ϕ1:
(2
"∂Vx
∂x
� �2
þ ∂Vy
∂y
!2
þ ∂Vz
∂z
� �2#þ ∂Vy
∂x þ ∂Vx
∂y
!2
þ ∂Vz
∂y þ ∂Vy
∂z
!2
þ ∂Vx
∂z þ ∂Vz
∂x
� �2" #)
þ(ϕ2
∂C∂x
� �2
þ ∂C∂y
� �2
þ ∂C∂z
� �2" #
þ ϕ3
"∂T∂x
� � ∂C∂x
� �þ ∂T
∂y
� � ∂C∂y
� �
þ ∂T∂z
� � ∂C∂z
� �#): ð13Þ
The first term of NS represents the thermal irreversibility; it isnoted as NS-th. The second term, NS-fr, represents the viscous irrevers-ibility and the third term, NS-dif., represents the diffusive irreversibili-ty. NS gives a good idea on the profile and the distribution of thegenerated local dimensionless entropy. The total dimensionless gen-erated entropy is written as follows:
Stot ¼ ∫v
Nsdv ¼ 1v∫v
Ns−th þ Ns−f r þ Ns−dif
� �dv ¼ Sth þ Sfr þ Sdif : ð14Þ
Bejan number (Be) is the ratio of heat and mass transfer irrevers-ibility to the total irreversibility due to heat transfer and fluid friction:
Be ¼ Sth þ SdifSth þ Sfr þ Sdif
: ð15Þ
Dimensionless irreversibility distribution ratios (ϕ1, ϕ2 and ϕ3),are given by:
ϕ1 ¼ μα2T0
L2kΔT2 ; ϕ2 ¼ RDT0
kC0
ΔC′
ΔT ′
" #and ϕ3 ¼ RD
kΔC′
ΔT ′
" #: ð16Þ
For N=0, there is no mass diffusion and we assume that the ther-mal and species diffusions are opposed. For 0bNb1 the flow is ther-mally dominated and for N>1 the flow is compositionally dominated.
3. Validation
The numerical technique employed to simulate the problem con-sidered in the present study, was validated by comparing the contourmaps of the streamlines, temperature and concentration for buoyancyratios of 0.8 and 1.3 with results reported earlier by Nishimura et al.[13]. Fig. 2 presents the isotherms, isoconcentration, z-vector poten-tial and velocity projection in the XY-plan for N=0.8 and N=1.3.The streamlines presented in the work of Nishimura et al. [13] arecompared with the z-component of the vector potential. It is notedthat in the 3D configurations the contours of the streamlines are notclosed. It is clear from these figures that the results obtained by thepresent model are in good agreement with those of Nishimura et al.[13].
4. Results and discussions
Numerical results are presented for Pr=0.7, Le=0.85 andSc=0.695 which cover water vapor diffusion into air. The Rayleighnumber is fixed at Ra=105, and the buoyancy ratio is varied fromN=0 to 10. The dimensionless irreversibility distribution ratios (ϕ1,ϕ2 and ϕ3), are fixed respectively at 10−4, 0.5 and 10−2.
Fig. 3 presents the iso-surfaces of concentration and tempera-ture for some buoyancy ratios. For Nb1, when the flow is thermal-ly dominated the isotherms and isoconcentrations present vertical
T C Velocity projection
Fig. 2. Plots of isotherms, isoconcentration, z-vector potential and velocity projection inthe XY-plan for N=0.8 (top) and N=1.3 (bottom).
Fig. 4. Iso-temperatures and iso-concentrations in the XY-plan.
872 K. Ghachem et al. / International Communications in Heat and Mass Transfer 39 (2012) 869–876
stratifications in the center and excessive gradients near the lowerpart of the hot wall and the upper part of the cold wall. For N>1,when the flow is compositionally dominated, the central stratifica-tion persists while the location of the stepper gradients isinversed. This result is clear in Fig. 4, which represents the iso-temperatures and iso-concentrations in the XY-plan. At N=1,where the thermal buoyancy force equals the compositional buoy-ancy force, one notices a disappearance of the vertical stratificationof isotherms and isoconcentrations, and these fields become simi-lar to purely diffusive modes. In the core of the cavity, distortionsare more marked for N=1.1 than for N=0.8. For N=5 isothermsand isoconcentrations in the core of the cavity, became horizontal-ly uniform and stably stratified in the vertical direction.
Fig. 5, presents the velocity vector projection in the XY-plan, forNb1 the flow is thermally dominated with only one clockwise rotat-ing thermal vortex. For N=1, we remark the apparition of two greatcompositional vortices and two small thermal vortices. For N slightly
N=0.8 N=1 N=1.1 N=5
Fig. 3. Iso-surfaces of concentration (top) and of temperature (bottom).
higher than 1, a re-apparition of one vortex structure is noted, butwith inversed flow rotation caused by the compositional domination.When the buoyancy ratio is higher, the flow becomes a two-inner-vortex structure for N=1.5 and 2 and four-inner-vortex structurefor N=3.
Fig. 6 presents different buoyancy ratios, and some particle trajecto-ry on half of the cavity. ForN=0.8, the internal flow (Fig. 6.a) convergesdirectly from the frontal wall toward the X–Y plan, then from this plantoward the frontal wall. The external flow (Fig. 6.b) is also convergentfrom the frontalwalls toward theX–Y plan. ForN=1, thermal and com-positional forces are of the same order, which causes complex three-dimensional particle trajectories (Fig. 6.c). For N=1.5, the internalflow (Fig. 6.d) diverges from the X–Y plan where the structure is onecell with two-inner vortices toward an intermediate plan (z=1.65)where there is coalescence of the two vortices (Fig. 6.g), then divergesfrom this plan toward the frontal wall. The external flow is from thecentral plan to the frontal plan. ForN=3, internal and external flows di-verge from the X–Y plan to the frontal plan, and the flow structurechanges from four-inner vortices at z=1 to two-inner vortices atz=1.6 then one vortex near the frontal wall at z=1.9.
Fig. 7 presents the local entropy generations in the central plan. Forthe thermal or compositional dominated flow, the generated entro-pies occur principally near the active walls. For Nb1, as predictedthe creations of thermal and compositional entropies aremainly local-ized near the bottom of the hot surface and the top of the cold surface.This figure shows also that the distribution of total entropy generationis similar to the iso-entropies due to gradients of temperature andconcentration, which indicates the predominance of these irrevers-ibilities. For N>1, thermal and compositional irreversibilities becomelocalized near the top of the hot surface and the bottomof the cold sur-face. In this case entropy generations become dominated by irrevers-ibilities due to friction. For a higher value of buoyancy ratio, localentropy generations become more close to the active walls. Whenthe thermal and compositional forces are equal the generated entro-pies are distributed in all the cavities and not localized near wallswhich implies that a suppression of boundary layer phenomenon ismet in other cases. Fig. 8 illustrates the 3D distribution of the total en-tropy generation, and it is clear that the 3D character is more pro-nounced for N=1 than in other cases. Fig. 9.a shows that thermaland compositional entropies have a similar variation as a function ofbuoyancy ratio, and except for N near the unity, they are dominatedby friction irreversibility. We remark also that all entropy generationspresent a minimum at N=1, and this minimum is due to equilibriumbetween thermal and compositional forces causing a reduction in ve-locity and a reduction of excessive temperature and solute gradients.The variation of the Bejan numbers as a function of N (Fig. 9.b)shows that only for buoyancy ratios near unity, irreversibility due tofluid friction is dominated by those ratios due to heat and mass trans-fers. For N=1 this variation presents a peak indicating a maximumdomination of the heat and mass transfer irreversibilities.
Fig. 7. Entropy generation in the XY plan, (a) N=0.8; (b) N=1; (c) N=1.3; (d) N=5.
874 K. Ghachem et al. / International Communications in Heat and Mass Transfer 39 (2012) 869–876
N=0.8 N=1 N=1.3
Fig. 8. Iso-surface of total generated entropy.
1
10
100
1000
Sdif
Sth
Sfr
Stot
S
N4 6 8 10
0 2
0 2
4 6 8 100.0
0.2
0.4
0.6
0.8
Be
N
a)
b)
Fig. 9. Total entropy generation and mean Bejan number as a function of N.
5
43
2
1
2.1
1.7
0.7
0.8
1
0.9
1.1
1.7
1.3
1
0.8
0.81
2
3
7
5
4
6
5
4
3
2
1
1.1
1.2
0.9
1
10.8
0.8
0.90
.8
0 .9
1.11.3
1.4 1.5
1.6
5
6.5
2
1
N=0.8 N=1 N=1
Fig. 10. Local Nusselt number (top) and local She
2 4 6 8 100
2
4
6
8
10
Sh av
Nuav
Nu a
v,Sh
av
N
Fig. 11. Mean Nusselt and Sherwood numbers.
875K. Ghachem et al. / International Communications in Heat and Mass Transfer 39 (2012) 869–876
Fig. 10 illustrates the local Nusselt and Sherwood numbers in thehot wall. For Nb1 the peak of heat transfer is in the bottom of thehot wall for N=0.8. For N slightly higher than unity, the peak ofheat transfer is in the top of the hot wall. For these cases, the variationof local Nusselt numbers according to the z-direction is quasi-constant in the center of the active walls and presents a slight devia-tion near the frontal walls. When N is higher, the deviation becomesmore pronounced due to the intensification of the 3D character ofthe flow. For N=1, the local Nusselt number presents a complexand non symmetric variation due to the non-organized behavior ofthe flow. The same results can be drawn for the local Sherwood num-ber, except the positions of peaks which are inversed. Mean Nusseltand Sherwood numbers (Fig. 11), present the same variation as afunction of buoyancy ratio, in fact they decrease for Nb1, increasefor N>1 and present a minimum for N=1.
5. Conclusions
A numerical analysis of double diffusive natural convection andentropy generation in a three-dimensional solar dryer is performedin this study. Some conclusions can be drawn as follows:
– The variation of the buoyancy ratio affects significantly the distri-butions of isotherms, iso-concentrations and the structure ofthe flow. Particularly for N=1, the flow is completely three-dimensional.
– All kinds of entropy generations present a minimum for N=1, thisresult is due to the competition between thermal and composi-tional forces. These entropies rise considerably when N grows.
2.5
5
15
12.5
7.5
10
20
4
7.5
21.5
18
11
14.5
25
4
3
6
5.5
10.5
3.5
16.5
13.5
7.5
1.5
8
145
23
17
11
2
.3 N=5 N=10
rwood number (bottom) as a function of N.
876 K. Ghachem et al. / International Communications in Heat and Mass Transfer 39 (2012) 869–876
– A high Bejan number is found for N=1, indicating the dominationof heat and mass irreversibilities. Outside, friction irreversibilitiesare largely dominant.
– The 3D distribution of the generated entropy is more pronouncedfor N=1.
– Distribution of local Nusselt numbers changes with changingbuoyancy ratio and takes a complex structure for N=1.
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