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10Статья поступила в редакцию 01.08.12. Ред. рег. № 1381
The article has entered in publishing office 01.08.12. Ed. reg. No. 1381
УДК 669.791
ЧИСЛЕННЫЕ ИССЛЕДОВАНИЯ ЭФФЕКТА СОРЕ, КОМБИНИРОВАННОГО С ДВОЙНОЙ ДИФФУЗИЕЙ В НАПОЛНЕННОЙ ЖИДКОСТЬЮ КУБИЧЕСКОЙ ПОЛОСТИ ПОД ВОЗДЕЙСТВИЕМ ПЕРЕСЕКАЮЩИХСЯ ТЕПЛОВЫХ И РАСТВОРЕННЫХ ГРАДИЕНТОВ
1 2 2 2 3
М. Ламсаади , М. Наими , Х. Эль Харфи , А. Раджи , М. Хаснауи
Университет султана Мулэя Слимана, полидисциплинарный факультет, междисциплинарная научно-исследовательская лаборатория (LIRST), В. Р. 592, Бени-Меллал, Марокко Tel.: (212) 5 23 48 51 12/22/82; Fax: (212) 5 23 48 52 01; E-mail: lamsaadima@yahoo.fr 2Университет султана Мулэя Слимана, научно-технический факультет, кафедра физики, лаборатория моделирования потоков и перемещений (LAMET), В. Р. 592, Бени-Меллал, Марокко
3Университет Кади Айада, Научный факультет Семлалия, кафедра физики, лаборатория механики жидкости и энергетики (LMFE),
В.Р. 2390, Маракеш, Марокко
Заключение совета рецензентов: 15.08.12 Заключение совета экспертов: 20.08.12 Принято к публикации: 25.08.12
В работе описывается численное исследование эффекта Соре на сложных устоявшихся растворах, индуцированных двойной диффузной естественной конвекцией в кубической полости, заполненной жидкостью, под воздействием пересекающихся тепловых и растворенных градиентов. Расчеты, ограниченные водными растворами, проводились для определяющих факторов, таких как критерий Льюиса, Le, коэффициент Соре, M, коэффициент плавучести, N, число Прандтля, Pr, и тепловое число Рэлея, Ra7, а именно Le = 10, -111 < M < 122, -0,1 < N < 0,1, Pr = 7 и 5-104 < Ra7 < 5-105. Влияние M на структуру, нагрев и массовый перенос анализируется для данных значений N и Rar, а значительные изменения наблюдались в разных растворах в зависимости от M.
Ключевые слова: двойная диффузия, перенос тепла, массовый перенос, сложные растворы, численное исследование, кубическая полость, эффект Соре.
NUMERICAL STUDY OF SORET EFFECT COMBINED WITH DOUBLE DIFFUSION IN A FLUID-FILLED SQUARE CAVITY SUBJECTED TO CROSS THERMAL AND
SOLUTE GRADIENTS
M. Lamsaadi1, M. Naïmi2, H. Elharfi2, A. Raji2, M. Hasnaoui3
'Sultan Moulay Slimane University, Polydisciplinary Faculty, Interdisciplinary Laboratory of Research in Sciences and Technologies (LIRST),
B.P. 592, Beni-Mellal, Morocco Tel.: (212) 5 23 48 51 12/22/82; Fax: (212) 5 23 48 52 01; E-mail: lamsaadima@yahoo.fr 2Sultan Moulay Slimane University, Faculty of Sciences and Technologies, Physics Department, Laboratory of Flows and Transfers Modeling
(LAMET), B.P. 523, Beni-Mellal, Morocco 3Cadi Ayyad University, Faculty of Sciences Semlalia, Physics Department, Laboratory of Fluid Mechanics and Energetics (LMFE),
B.P. 2390, Marrakech, Morocco
Referred: 15.08.12 Expertise: 20.08.12 Accepted: 25.08.12
Numerical study of Soret effect on multiple steady state solutions inducted by double diffusive natural convection in a fluid-filled square cavity, submitted to cross thermal and solute gradients, has been reported in this paper. The computations, which have been limited to water-based solutions, have been carried out for governing parameters, namely the Lewis number, Le, the Soret parameter, M, the buoyancy ratio, N, the Prandtl number, Pr, and the thermal Rayleigh number, Rar, such that Le = 10, -111 <M< 122, -0,1 < N< 0,1, Pr = 7 and 5-104< Ra7 < 5105. The effect ofMon the flow structure and heat and mass transfers has been analyzed for given values of N and Rar, and significant changes have been observed from one solution to another, depending on M.
Keywords: double diffusion; heat transfer; mass transfer; multiple solutions; numerical study; square cavity, Soret effect.
Nomenclature
BF - bicellular flow
D - mass diffusivity (m2/s)
D* - thermodiffusion coefficient
g - acceleration due to gravity (m/s2)
HBF - horizontal bicellular flow
HBAF - horizontal bicellular antinatural flow
HBNF - horizontal bicellular natural flow
H' - height of the cavity (m)
Le - Lewis number, Eq. (8)
M - Soret parameter, Eq. (8)
MF - monocellular flow
MCF - monocellular clockwise flow
MCCF - monocellular counter-clockwise flow
N - buoyancy ratio, Eq. (8)
Nu - mean Nusselt number, Eq. (9)
Pr - Prandtl number, Eq. (8)
RaT - thermal Rayleigh number, Eq. (8)
S - dimensionless concentration, [= (S' - S'L )/AS']
S'r - reference concentration (kg/m3), [= (SR + S'L)/2]
S'L - concentration of the left wall (kg/m3)
SR - concentration of the right wall (kg/m3)
Sh - mean Sherwood number, Eqs. (9, 10)
T - dimensionless temperature, [= (T' - TU )/AT' ]
TU - temperature of the upper wall (K)
T'L - temperature of the lower wall (K)
T' - reference temperature (K), [= (T[ + TU)/2]
(u, v) - dimensionless axial and transverse velocities,
[=(u', v ')l(al H')]
VBF - vertical bicellular flow
VBAF - vertical bicellular antinatural flow
VBNF - vertical bicellular natural flow
(x, y) - dimensionless axial and transverse co-ordinates
[=( x', y 'V H']
Greek symbols
a - thermal diffusivity of fluid at the reference temperature (m2/s)
PT - thermal expansion coefficient of fluid at the reference temperature (1/K)
Ps - solutal expansion coefficient of fluid at the reference
concentration (m3/kg)
v - kinematic viscosity of fluid (m2/s)
AT' - temperature difference between the upper and
lower walls (K), [= (T[ - TU)]
AS - concentration difference between the right and left walls (kg/m3), [=(SR - SL)]
Q - dimensionless vorticity, [= Q'/(a/H'2)]
^ - dimensionless stream function, [=
Superscript
- dimensional variable
Subscripts
L - left or lower max - maximum value min - minimal value r - reference value R - right S - solutal
T - thermal or threshold U - upper
1. Introduction
Thermodiffusion phenomenon, or Soret effect, is a solute transfer that may happen in a mixture under the influence of a temperature gradient even with an initial uniform concentration. The solute migrates toward the hot or cold side depending on the mixture composition and, sometimes, on the temperature range [1]. Soret effect may become important in some applications such as the solidification of binary alloys, groundwater pollutant migration, chemical reactors, and geosciences. Such an effect is not well understood as it, sometimes, exhibits specific phenomena. Multiple steady/oscillatory states, subcritical flows, hysteresis behaviors, standing/traveling waves, and Hopf bifurcations are some examples of these phenomena that were observed experimentally as well as numerically.
Most of the past studies on heat and mass transfers, where the investigations were mainly conducted
numerically or/and analytically including different boundary conditions and geometries [2-4], were concerned with double diffusive convection without Soret effect. Comparatively, few investigations considered the double diffusive convection with Soret effect, such as the work by Chavepeyer and platen [5], who studied numerically thermogravitational separation induced within a differentially heated cavity with the presence of obstacles. They conclded that the Soret coefficient is not affected by the presence of obstacles when considering a simple thermal diffusive regime. On the other hand, while examining thermal diffusion in a binary mixture within a porous enclosure subject to a horizontal thermal gradient, Benano et al [6] showed that, depending on the Soret number, multiple convection-roll flow patterns may take place when solutal and thermal buoyancy forces act in the opposing directions. In the case of permeable boundaries, Alex and Patil [7] studied the effect of variable gravity field
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on the onset of Soret driven convection in a horizontal porous layer where the upper and lower boundaries are maintained at constant but different temperatures and concentrations. They found that only large magnitude of Soret parameter affects the convection pattern. This effect was also considered by Bennacer et al. [8] in the case of convection in a horizontal porous cavity submitted to cross gradients of temperature and concentration. Their results show that, when the vertical concentration gradient is stabilizing, multiple steady-state solutions become possible over a range of buoyancy ratios, which depends strongly on Soret parameter. Er-Raki et al. [9] studied Soret effect on boundary layer flows induced in a vertical porous layer subjected to horizontal heat and mass fluxes. It was found that, depending on the sign of the buoyancy ratio, the boundary layer thickness increases or decreases with Soret parameter. Considering a horizontal porous layer subject to uniform heat flux, Bourich el al. [10] investigated Soret effect on natural convection and determined analytically, in the limit of a shallow layer, the critical conditions for the onset of subcritical and stationary convection. For the same configuration, Bahloul et al. [11] determined the thresholds for finite-amplitude, oscillatory and monotonic convection instabilities in terms of the governing parameters using linear and non linear stability analyses. The existence of sub-critical convection was predicted for negative values of Soret parameter. A similar problem was studied by Bourich et al. [12] by performing a comparative study for the limiting cases, which considered Darcy porous and clear fluid media. A closed form analytical solution for a shallow enclosure was derived from the parallel flow concept and the onset of over-stabilities was predicted using a linear stability analysis. An appropriate normalization for Rayleigh number was used to demonstrate that the flow behavior is similar to that predicted by the parallel flow approximation. The same authors [13-14] investigated analytically and numerically Soret effect on thermosolutal convection, induced in a horizontal Darcy porous layer subjected to constant heat and mass fluxes, and on thermal natural convection within a horizontal porous enclosure uniformly heated from below by a constant heat flux using the Brinkman extended Darcy model. In their first study, the thresholds for the onset of supercritical and sub-critical convection were predicted explicitly as functions of the governing parameters. They demonstrated that there exist combinations of the governing parameters for which Soret effect imposes a vertical non-linear stratification of the concentration field, even for a convective regime, and that a reversal horizontal concentration gradient is also possible. In the second study, they found that the separation parameter has a strong effect on the thresholds of instabilities and on the heat and mass transfer characteristics.
Most of the previous studies dealt with shallow horizontal or tall vertical porous layers to allow a closed form of analytical solution. The square enclosure flow
configuration could reveal some new flow features, as was demonstrated by Mansour et al [15-16], when studying Soret effect on double diffusive natural convection in a porous square cavity subjected to cross thermal and solute gradients. The flow patterns are very different from those observed previously in the case of a rectangular porous layer. In these studies, the effect of the buoyancy ratio and the Soret parameter on the maintenance and disappearance of the multiple steady-state solutions obtained in the case of purely thermal convection was analyzed. It was found that only one monocellular flow mode persists at large values of N (or -N), both in the presence and in the absence of Soret effect. Some flow modes are destroyed by the solute buoyancy forces in the absence of Soret effect (M = 0), but reappear in some range of M. Soret effect may affect considerably the heat and mass transfers in the medium, since it leads to an enhancement or a reduction of the mass transfer, depending on the flow structure and the sign of M. There are situations where a solution transfers the solute toward the wall of the highest concentration. Such behavior was observed when the temperature gradient and the magnitude of the Soret parameter M are such that the thermodiffusion flux opposes and overcomes the convection flux, resulting in a net mass flux directed from the least concentrated wall toward the most concentrated one. Note that in the study of Mansour et al. [16], a three cellular flow was observed.
To the best of our knowledge, there are no investigations dealing with Soret effect in fluid-filled enclosures subjected to cross temperature and concentration gradients. Therefore, it would be interesting to conduct a study in this sense, in order to bring some insight to the effects of such geometries. Hence, the aim of the present work is to study Soret effect on the multiplicity of solutions in a square cavity filled with a Newtonian fluid mixture and subjected to cross gradients of temperature and concentration. The results presented here are limited to water-based solutions, for which the Prandtl number Pr = 7.
2. Mathematical formulation
The geometry considered in this study is sketched in Fig. 1. It consists of a square cavity of size H'xH' filled with a Newtonian fluid mixture (water-based solution). The vertical left and right walls of the cavity are adiabatic and submitted, respectively, to different but uniform concentrations S' and SR (S'L < SR), while its upper and lower faces are impermeable and maintained, respectively, at constant temperatures TU and T' (TU < T'). In order to simplify the problem, the following hypotheses have been made:
• all the physical properties of the fluid mixture are supposed constant and evaluated at reference temperature T' and concentration S'r except the density in the buoyancy term where it is assumed to vary linearly with the temperature and concentration (Boussinesq approximation);
• the viscous dissipation is negligible;
• the flow is laminar. In fact, in most buoyancy driven motions, the fluid circulation is slow due to moderate temperature gradients [17];
• the fluid is incompressible. For pressures close to that of atmosphere, liquids are a rigorous approximation of incompressible fluids;
• the third dimension of the cavity is such that the problem can be considered as two-dimensional. This is generally relatively well satisfied and provides insight into the more complicated three-dimensional flows [17].
У ,v
SL
T '
1 U
J Л H k г' ь
1 r
SR > SL
TL > TU
x ,u
Рис. 1. Схема геометрии и системы координат Fig. 1. Sketch of the geometry and co-ordinates system
On the basis of what precedes and taking into account Soret effect, the dimensionless governing equations, written in terms of vorticity, Q, temperature, T, concentration, S, and stream function, are:
3Q duQ dvQ -+-+-=
dt dx dy
„ . d2Q d2Q _
~àXF + эУ2"l+PrRa'
dT ArdS — + N— dx dx
dT + ЭК) + ЭК) = V2T ;
dt dx dy
(1)
(2)
dS d(uS) d(vS) 1 , 2 2X
— + + = — (V2S +MV2T) (3) dt dx dy LeV '
and
V2 у = -Q,
To close the problem, the following appropriate dimensionless boundary conditions have been used:
dT
u = v = ^ = — = S = 0 for x = 0; (6)
dx
u = v = y = — = S -1 = 0 for x = 1; (7)
dx
u = v = y = — + M — = T -1 = 0 for y = 0; (8) dy dy
u = v = V = —+M — = T = 0 for y = 1. (9)
dy dy
As for the vorticity, which is unknown at the boundaries, the relation of woods [18] has been adopted, for its accuracy and stability.
The above equations reveal five governing dimensionless parameters, namely, the Lewis number, Le, the Soret parameter, M, the buoyancy ratio, N, the Prandtl number, Pr, and the thermal Rayleigh number, RaT, whose expressions are:
D'S'AT ' P S AS ' Le = a/D, M =-N = •
DAS '
PT AT '
Pr = v/a and R^ =
gpT AT 'H '
av
(10)
Moreover, to characterize heat and mass transfers in the vertical and horizontal directions, the Nusselt and Sherwood numbers, respectively defined by
1 dT
Nu = f
J0 dy
dx ;
(11)
У=1
1
Sh = J[3S/3x|x=0 + M dT/3x|x=0 ] dy (12)
0
have been used. The first and second terms in the expression of Sh characterize the Fich diffusion and
Soret fluxes, respectively. In our case dT/ 3x[ and Soret flux is zero, which yields:
(4)
Sh = jdS/dx|x 0dy .
= 0
(13)
where
dw dw , „ dv du
u , v =--and U = -——.
dx
dy
dx dy
(5)
The dimensionless variables have been obtained by
a, a/ H',
AS' = (SR - SL) and a time, velocity, vorticity, characteristic temperature, characteristic concentration, and stream function, respectively.
using the characteristic scales H', H'2/i a/H'2, AT' = (TL - TU ) corresponding to length,
3. Numerics
The two-dimensional governing equations have been discretized by using the well-known second order central finite difference method with a regular mesh size. The integration of Eqs. (1)-(3) has been performed with the Alternating Direction Implicit method (ADI). To satisfy the mass conservation, Eq. (4) has been solved by a Point Successive Over Relaxation method (PSOR) with an optimum relaxation factor calculated by the Frankel formula woods [18]. A grid of 81*81 has been considered
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0 and 1
sufficient to obtain accurate results. At each time step, 8t, which has been varied between 10-7 and 10-4, depending on the values of the governing parameters, the convergence criterion ^lyf*1 f
¥ ТЛ < 10-4
has been satisfied for у, — being the value of the
stream function at the node (i, j) for the kth iteration level.
Furthermore, the present code has been validated against the results obtained, in the case of a square cavity differentially heated, by Vola et al. [19] and De Vahl Davis [20]. Comparative results are summarized in Table 1 where it can be seen that the agreement is very good since the maximum difference does not exceed 2%.
Таблица 1
Проверка численного кода при сравнении с предыдущими исследованиями (Pr = 0,71, Rar = 104)
Table 1
Validation of the numerical code against previous studies (Pr = 0.71, Ra7 = 104)
Present work [19] [20]
l¥max| Nu l¥max| Nu l¥max| Nu
5.072 2.248 5.032 2.240 5.071 2.238
Notice that VBAF/HBAF can be obtained from VBNF/HBNF by a simple rotation of n around the cavity center, but both the solutions induce the same heat and mass transfer for any values of Rar, M and N; situation that does not hold for MF when N ^ 0. Then, only BF natural solutions have been considered in this work.
4.1. Streamlines, isotherms and isoconcentrations
4.1.1 Monocellular flow The Soret effect on the flow, temperature and concentration fields corresponding to MF, is illustrated in Figs. 2 and 3 for N = 0.1, Rar = 105 and two ranges of values of M. These figures show that an increase of |M| affects strongly the isoconcentrations, due to Soret effect which tends to promote the solute gradient as explained before. Indeed, in the absence of Soret effect (M = 0), the isoconcentrations appear very crowded by double diffusion near the most and least salted vertical walls.
4. Results and discussion
At first, it is to remind that the numerical results presented in this study are limited to water-based solutions, for which Pr = 7, although previous studies [2122] showed that this parameter remains without effect beyond Pr = 1. In addition to that, the computations have been carried out with Le = 10, -111 < M < 122, -0,1 < N < 0,1, and 5-104 < Rar < 5-105. The choice of these ranges of the values of M, N and Rar has been encouraged by the fact that it is possible to find out a wide variety of steady solutions in such situations. These correspond to a monocellular (clockwise or counterclockwise) flow or a vertical or horizontal bicellular (natural or antinatural) flow, denoted MF (MCF or MCCF) and BF (VBNF, VBAF, HBNF or HBAF), respectively. Recall that the existence of the monocellular and vertical bicellular flows was proved numerically by Robillard et al. [23], for thermally driven convection (N = 0), by Mansour et al. [6] and Bourich et al. [24] for thermo-solute convection (N ^ 0) with and without Soret effect, respectively, in a square porous cavity. As all these solutions may exist for the same set of the governing parameters, the choice of the initial conditions has been crucial in this study. On the other hand, to obtain a flow pattern with m vertical (n = 1) or n horizontal (m = 1) cells for M = N = 0, a stream function field of form ^(x, y) = = Asin(mnx)sin(nny), where the constant A is such that 1 < A < 50, and a conduction temperature field have been used for an adequate value of Rar.
Рис. 2. Направления течения, изотермы и изоконцентрации моноклеточного потока, направленного против часовой стрелки, для N = 0.1, RaT = 105 и различные значения M: (a) M = -37, (b) M = -10, (с) M = 0, (d) M = 10 и (е) M = 65 Fig. 2. Streamlines, isotherms and isoconcentrations of MCCF
for N = 0.1, RaT = 105 and various values of M: (a) M = -37, (b) M = -10, (c) M = 0, (d) M = 10 and (e) M = 65
4.1.2 Bicellular flow
Figs. 4 and 5 display two examples of streamlines, isotherms and isoconcentrations, corresponding to VBNF and HBNF, obtained for (N = 0.1, RaT = 105) and (N = 0.05, RaT = 5-105), respectively, and various values of M. Notice that, the existence of such regimes depends generally on the value the above mentioned governing parameters as shown in Table 3.
First of all, it is easy to see the asymmetrical pattern of the flow, which consists of two cells with different sizes depending on M. Besides, the effect of M is such that the isoconcentrations undergo an important change with Ml, expressed by a gradual disappearance of the vertical solute boundary layers (three layers for VBNF and two ones for HBNF) to the benefit of the horizontal ones, as explained in the previous subsection (cf. 4.1.1), but unlike MF, a slight modification is observed for the streamlines and the isotherms for M < 0, whereas the direction of the rotation of the cells remains unchanged, for each solution (VBNF or HBNF) whatever the sign and the magnitude of M.
Рис. 3. Направления течения, изотермы и изоконцентрации моноклеточного потока, направленного по часовой стрелке, для N = 0.1, RaT = 105 и различные значения M: (a) M = -4,6, (b) M = -2, (с) M = 0, (d) M = 10 и (e) M = 48 Fig. 3. Streamlines, isotherms and isoconcentrations of MCF for N = 0.1, RaT = 105 and various values of M: (a) (a) M = -4.6, (b) M = -2, (с) M = 0, (d) M = 10 and (e) M = 48
As |M increases the distribution of the concentrations changes so that the related isolines become closely spaced next to the horizontal walls due to Soret effect initiated by the temperature gradient, which is important in the vertical direction. In contrast, a slight modification is observed for the streamlines for M > 0, while the direction of the rotation of the cell and the isotherms remain unaffected whatever the sign of M.
Note that the two figures correspond to two solutions which differ in their structure and direction of rotation, although the range of the values of M related to Fig. 3 (MCF) is included in the one related to Fig. 2 (MCCF), which proves the multiplicity of solutions for MF. In addition to that, all the solutions observed are symmetric with respect to the cavity center, owing to the unicellular and symmetrical natures of the flow and boundary conditions, respectively.
Рис. 4. Направления течения, изотермы и изоконцентрации вертикального двухклеточного естественного потока для N = 0.1, RaT = 105 и различные значения M: (a) M = -50,2, (b) M = -10, (с) M = 0, (d) M = 5 и (e) M = 14,1 Fig. 4. Streamlines, isotherms and isoconcentrations of VBNF
for N = 0.1, RaT = 105 and various values of M: (a) M = -50.2, (b) M = -10, (с) M = 0, (d) M = 5 and (e) M = 14.1
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Таблица 2
Значения интенсивности потока и различные
значения M для различных растворов Интервалы существования различных растворов
Table 2
Values of the flow intensity corresponding and various values of M for different solutions
Suprax, i^mini)
(RaT, N)
(lO5, O.l)
(5-10, 0.05)
M
-lO
lO
-lO
lO
MCCF
24.041
24.б42
25.125
44.155
44.70б
45.O94
VBF
19.501
l9.l99
1S.516
HBF
27.б22
27.34б
27.372
4.3 Heat and mass transfer The Soret effect on heat and mass transfer inducted by MCCF, MCF, VBF and HBF is presented in Figs. 6 and 7,
in terms of the evolutions of the quantities Nu and Sh with M for (N = 0.1, RaT = 105) and (N = 0.05, RaT = 5-105).
MCCF
■MCF
VBF
Рис. 5. Направления течения, изотермы и изоконцентрации горизонтального двухклеточного естественного потока
для N = 0.05, RaT = 5105 и различные значения M: (a) M = -23,8, (b) M = -5, (с) M = 0, (d) M = 5 и (e) M = 19,8 Fig. 5. Streamlines, isotherms and isoconcentrations of HBNF
for N = 0.05, RaT = 5-105 and various values of M: (a) M = -23.8, (b) M = -5, (с) M = 0, (d) M = 5 and (e) M = 19.8
4.2. Flow intensity
In Table 2 are displayed the values of the flow intensity corresponding to (N = 0.1, RaT = 105) and (N = 0.05, RaT = 5-105) and various values of M, i.e. the situations discussed above. It appears that MF is the most intense flow for all the explored values of M, and that for each solution a very small variation of the quantity Sup(^max, |^mm |) around its value related to the case M = 0 (absence of Soret effect) is observed, which means that Soret effect does not influence significantly the flow intensity at least for the flow regimes obtained with the above values of the couple (N, RaT).
-60 -40
60 70
b
Рис. 6. Критерий Нуссельта (a) и число Шервуда (b) по сравнению с M для различных растворов, полученные с RaT = 105 и N = 0,1 Fig. 6. Nusselt number (a) and Sherwood number (b) versus M for different solutions obtained with Ra^ = 105 and N = 0.1
O
O
a
Nu 6.0-1
5.6
4.8
4.0
3.2
■MCCF -MCF . -HBF
-60 -40
40
80
M
120
b
80 m 120
Рис. 7. Критерий Нуссельта (а) и число Шервуда (b) по сравнению с M для различных растворов, полученные с RaT = 5105 и N = 0,05 Fig. 7. Nusselt number (а) and Sherwood number (b) versus M for different solutions obtained with RaT = 5105 and N = 0.05
As can be seen, Nu reaches a maximum which seems important in the case of VBF/MCCF for (N = 0.1, Rar = 105)/(N = 0.05, Rar = 5-105) indicating that the best heat transfer is induced by this kind of flow in almost all its range of existence, which is wider for MCCF. However, in the absence of VBF, which disappears by increasing Rar, the weakest value of Nu (expressing the worst heat transfer) is ensured by HBF. Indeed, since the heat transfer is in the upward direction, the presence of horizontal counter-rotating cells prevents the fluid circulation from the lower hot wall to the upper cold one. Besides, the interface between the flow cells (where the heat transfer is dominated by conduction, as can be seen from the isotherms of Fig. 5) resists the upward heat transfer.
As for mass transfer, it is seen that Sh increases/deceases with M in the case of the MCCF/MCF since the flow circulation is such that the right/left side of the cavity is hotter than the left/right one. In fact, according to Eq. 3, Soret effect imposes the solute migration from the hot region to the cold one at
M > 0. For these solutions, there are ranges of M where Sh< 0. This happens when the temperature gradient and the sign of M are such that Soret flux is oriented from the left wall (less salted) towards the right one (most salted), with Ml large enough to compensate the ordinary convective mass flux induced by the concentration difference between the vertical walls (and directed from right to left). According to Eq. 13, Sh< 0 means that Soret effect results in an accumulation of solute in the vicinity of the right wall in such a way that the dimensionless concentration there becomes greater than unity (i.e. the dimensional concentration greater than that of the most salted wall). Remember that negative values of Sh was obtained by Mansour et al. [15-16] in the case of porous square cavity. For BF, Sh remains always positive and varies slightly with M. Figs. 6b and 7b, show also that in the range of M where more than one solution exist simultaneously, Sh changes considerably from one solution to another. Hence, with three solutions, Sh> 0 for MCCF and BF (VBF or HBF) and Sh< or > 0 for MCF, whereas with two solutions the sign of Sh depends mainly on that of M.
All these figures show arrows indicating the transitions made by the obtained solutions, whose limits of existence are clearly specified in Table 3, for various values of N and Rar. To determine these limits, the procedure used has consisted of starting the calculations from M = 0 and increasing (decreasing) M gradually (the solution obtained with a given value of M has used as initial guess for the next value of this parameter). With such procedure, it has been found that a given solution persists until M crosses a threshold value, then a transition occurs towards MCCF/ MCF for (N > 0)/(N < 0) or to an oscillatory behavior that does not depend on Rar. In such a situation (N > 0)/(N < 0), other transitions from MCCF/MCF towards BF may occur at (M < 0)/(M > 0). Moreover, for relatively high values of Ml, when (N > 0)/(N < 0), the number of solutions reduces to two and only (MCCF and MCF) or (MCCF/MCF and BF) subsist for (M > 0)/(M < 0) and (M < 0)/(M > 0), respectively. For Rar = 5-105, MCF/MCCF has been obtained only for (N > 0)/(N < 0) and (M > 0)/(M < 0), i.e. that these flow regimes disappear with Soret effect, which implies that their reappearance is mainly due to this one. Notice that for such value of Rar, it has not been possible to generate VBF whatever M and N values, whereas HBF remains possible to be found for particular negative or positive values of M provided that |N| < 0.1, as shown in Table 3.
Last, it is to underline that it has been possible to obtain the same rate of heat and mass transfers with MCCF and MCF depending on whether that M < 0 or M > 0 and N < 0 or N > 0. For example, with M > 0 and
N > 0, Nu and Sh values related to MCCF are identical to those corresponding to MCF for M < 0 and N < 0.
а
International Scientific Journal for Alternative Energy and Ecology № 10 (114) 2012
© Scientific Technical Centre «TATA», 2012
Таблица 3
Области существования различных решений
Table 3
Ranges of existence of different solutions
RaT N MCCF MCF VBF VBH
5404 0.05 -93 < M < 122 —43 < M < 107 —105 < M < 51.8
0.1 —4б < M < б3 —15 < M < 52 —51 < M < 13
-0.05 -107 < M < 43 —122 < M < 93 —51.8 < M < 105
-0.1 —52 < M < 15 —б3 < M < 4б —13 < M < 51
105 0.05 —23.5 <M< 10б 9.б2 < M< 82 —23.8 < M < 19.8
0.1 —19.8 < M < 59.1 12.8б < M< 33
-0.05 —82 < M < —9.б2 —10б < M< 23.5 —19.8 < M < 23.8
-0.1 —33 < M < —12.8б —59.1 < M < 19.8
5105 0.05 —54.2 < M < —122 —22.4 < M < 101 —111 < M < 42.7
0.1 —37 < M < б5 —4.б <M< 48 —50.2 < M < 14.1
-0.05 —101 < M < 22.4 —122 < M < 54.2 —42.7 < M < 111
-0.1 —48 < M < 4.б —б5 < M < 37 —14.1 < M < 50.2
6. Conclusion
Multiplicity of solutions and heat and mass transfers for combined Soret effect and double diffusive convection within a square cavity, filled with a Newtonian liquid and submitted to cross temperature and concentration gradients, have been studied numerically. The computations have been performed with Le = 10, -111 < M < 122, -0.1 < N < 0.1, Pr = 7 and 5-104 < RaT < 5-105 under the assumptions commonly used in thermo-solute natural convection problems. The main results of this study are as follows:
- Soret parameter may have a strong effect on the multiplicity of solutions and heat and mass transfer;
- Depending on the values of M, one, two, or three solutions are possible for very small values of |N|;
- There are situations where Sh > 0 for one solution and Sh < 0 for another;
- For large values of RaT, monocellular flows, which appear for N = 0, and disappears for N ^ 0 with Soret effect, reappears when this latter exists but only for a certain range of M > 0 or M < 0 depending on the sign of N;
- Bicellular flows existence depends on RaT and N, whatever the sign of M;
- Unlike the porous square cavity, where three vertical cells were found, it has not been possible in our case to obtain more than two.
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International Scientific Journal for Alternative Energy and Ecology № 10 (114) 2012
© Scientific Technical Centre «TATA», 2012
