CC BY
10ТЕРМОДИНАМИЧЕСКИМ АНАЛИЗ В АЛЬТЕРНАТИВНОЙ ЭНЕРГЕТИКЕ
THERMODYNAMIC ANALYSIS IN RENEWABLE ENERGY
Статья поступила в редакцию 19.09.2011. Ред. рег. № 1102 The article has entered in publishing office 19.09.11. Ed. reg. No. 1102
УДК 669.791
КРАТНЫЕ РЕШЕНИЯ ДВОЙНОЙ ДИФФУЗИИ В КВАДРАТНОЙ ПОЛОСТИ ПРИ УСЛОВИИ ПЕРЕСЕЧЕНИЯ ГРАДИЕНТОВ ТЕМПЕРАТУРЫ И КОНЦЕНТРАЦИИ
12 2 2 3
М. Ламсаади , М. Наими , Х. Эльхарфи , А. Раджи , М. Хаснауи
Университет султана Мулэя Слимана, полидисциплинарный факультет, междисциплинарная научно-исследовательская лаборатория (LIRST), В.Р. 592, Бени-Меллал, Марокко 2Университет султана Мулэя Слимана, научно-технический факультет, кафедра физики, лаборатория моделирования потоков и перемещений (LAMET), В. Р. 592, Бени-Меллал, Марокко Тел.: (212) 5 23 48 51 12/22/82; факс: (212) 5 23 48 52 01 E-mail: naimi@fstbm.ac.ma, naimima@yahoo.fr ^Университет Кади Айада, Научный факультет Семлалия, кафедра физики, лаборатория механики жидкости и энергетики (LMFE),
В.Р. 2390, Маракеш, Марокко
Заключение совета рецензентов: 30.09.11 Заключение совета экспертов: 05.10.11 Принято к публикации: 10.10.11
Численные результаты двумерной двухдиффузионной естественной конвекции в квадратной полости, заполненной ньютоновской жидкостью и представленной для пересекающихся градиентов тепла и концентрированного раствора, показаны в этой работе. Расчеты, ограниченные водными растворами, были проведены для управляющих параметров, варьируемых в следующем диапазоне: Le = 10 (число Льюиса), -1 < N < 1 (коэффициент плавучести), Pr = 7 (число Прандтля) и 103 < Rar < 107 (число Рэлея). Проанализированы влияния управляющих параметров на структуру потока, передачу тепла и массы. Показано, что сила Архимеда, индуцированная горизонтальными концентрационными градиентами, исключает разнообразность решений, полученных в чистой тепловой конвекции, где N превышает некоторое пороговое значение, зависящее от Rar. Для N < 0 / (N > 0) правостороннее/(левостороннее) решение, рассматриваемое в одной счетной ячейке, сохраняется для всех исследуемых диапазонов управляющих параметров, показанных в данной работе.
Ключевые слова: двойная диффузия, передача тепла, передача массы, кратное решение, численное моделирование, квадратная полость.
DOUBLE DIFFUSIVE MULTIPLE SOLUTIONS IN A SQUARE CAVITY SUBJECT TO CROSS GRADIENTS OF TEMPERATURE AND CONCENTRATION
M. Lamsaadi1, M. Naimi2, 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 2Sultan Moulay Slimane University, Faculty of Sciences and Technologies, Physics Department, Laboratory of Flows and Transfers Modeling (LAMET), B.P. 523, Beni-Mellal, Morocco Tel.: (212) 5 23 48 51 12/22/82; Fax: (212) 5 23 48 52 01 E-mail: naimi@fstbm.ac.ma, naimima@yahoo.fr 3Cadi Ayyad University, Faculty of Sciences Semlalia, Physics Department, Laboratory of Fluid Mechanics and Energetics (LMFE),
B.P. 2390, Marrakech, Morocco
Referred: 30.09.11 Expertise: 05.10.11 Accepted: 10.10.11
Numerical results of two-dimensional double diffusive natural convection in a square cavity filled with Newtonian fluid and submitted to cross gradients of heat and solute concentration are reported in this paper. The computations, which have been limited to water-based solutions, have been carried out for governing parameters varying in the following range: Le = 10 (Lewis number), -1 < N < 1 (buoyancy ratio), Pr = 7 (Prandtl number) and 103 < Rar < 107 (Rayleigh number). The effects of the governing parameters on the flow structure and heat and mass transfer are analyzed. It is demonstrated that the solutal buoyancy force, induced by horizontal concentration gradients, eliminates the multiplicity of solutions obtained in pure thermal convection when N exceeds some threshold value, which depends on Rar. For N < 0 / (N > 0), the monocellular clockwise/(counterclockwise) solution is maintained for all the explored ranges of the governing parameters considered in this study.
Keywords: Double diffusion; Heat transfer; Mass transfer; Multiple solutions; Numerical study; Square cavity.
International Scientific Journal for Alternative Energy and Ecology № 10 (102) 2011
© Scientific Technical Centre «TATA», 2011
Mohamed Naïmi
Vita
National doctorate (INPL, Nancy, France, 1989)
State doctorate (Cadi Ayyad University, Marrakech, Morocco, 2001)
Professor of Mechanical Engineering (Sultan Moulay Slimane University, Faculty of Sciences and Technologies, Physics Department, Beni-Mellal, Morocco)
Director of Flows and Transfers Modelling Laboratory (LAMET)
Research Topics
- Natural convection in non-Newtonian fluids;
- Thermosolutal convection in non-Newtonian fluids.
Some Recent Publications
1. M. Lamsaadi, M. Naimi, M. Hasnaoui, A. Bahlaoui and A. Raji,Multiple steady state solutions for natural convection in a tilted rectangular slot containing non-Newtonian power law fluids and subject to a transverse thermal gradient, Numerical Heat Transfer, Part A, vol. 51, N° 3 & 4, pp.393-414, 2007.
2. M. Lamsaadi, M. Naimi, A. Bahlaoui A. Raji, M. Hasnaoui and M. Mamou, Parallel Flow Convection in a Shallow Horizontal Cavity Filled with non-Newtonian Power-law Fluids and Subject to Horizontal and Vertical Uniform Heat Fluxes, Numerical Heat Transfer, Part A, Vol. 53, N° 2, pp. 178-203, 2008.
3. T. Makayssi, M. Naimi, M. Lamsaadi, M. Hasnaoui, A. Raji and A. Bahlaoui, Effect of solutal buoyancy forces on thermal convection in confined non-Newtonian power-law fluids, International Scientific Journal for Alternative Energy and Ecology, Vol. 62, N° 6, pp. 77-86, 2008.
4. Makayssi, M. Lamsaadi, M. Naimi, M. Hasnaoui, A. Raji, and A. Bahlaoui, Natural double-diffusive convection in a shallow horizontal rectangular cavity uniformly heated and salted from the side and filled with non-Newtonian power-law fluids: the cooperating case, Energy Conversion and Management, Vol. 49, pp. 2016-2025, 2008.
Nomenclature
BF bicellular flow
D mass diffusivity (m2/s)
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)
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)
Rar thermal Rayleigh number, Eq. (8)
S dimensionless concentration, [= (S' - S'L)/AS']
S' y reference concentration (kg/m3), [= (S'R + S'L)/2]
S'L concentration of the left wall (kg/m3)
S'r concentration of the right wall (kg/m3)
Sh mean Sherwood number, Eq. (9)
T dimensionless temperature, [= (T - 7VU)/A7V
Tu temperature of the upper wall (K)
Tl temperature of the lower wall (K)
T y reference temperature (K), [= (T v + 7L)/2]
O, v) dimensionless axial and transverse velocities,
[= («', v' )/(a/H )A7
VBF vertical bicellular flow
VBAF vertical bicellular antinatural flow
VBNF vertical bicellular natural flow
(x y) dimensionless axial and transverse co-ordinates
[= (X, y )/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 L + 7V)] As concentration difference between the right
and left walls (kg/m3), [= (S'R - S'L)] Q dimensionless vorticity, [= Q'/(a/H'2)]
y dimensionless stream function, [=y' /a]
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
Introduction
Double-diffusive natural convection, which is a fluid flow generated by buoyancy due to simultaneous temperature and concentration gradients in the gravitational field, can be found in wide range of situations. In nature, such flows are encountered in the oceans, lakes, solar ponds, shallow coastal waters and the atmosphere. In industry, examples include chemical processes, crystal growth, energy storage, material and food processing, etc... For a review of the fundamental works in this area, see, for instance, Ostrach [1] and Viskanta et al. [2].
The literature related to natural double-diffusive convection shows that the majority of analytical, numerical and experimental investigations were focused on the square and rectangular porous cavities. On this subject, the books of Bejan [3] and Nield and Bejan [4] constitute basic references.
In the past, most of the existing studies on double diffusive convection were concerned with rectangular cavities where the temperature and concentration gradients were either horizontal [5-14] or vertical [1521], including different kinds of boundary conditions (constant temperature and concentration or uniform fluxes of heat and mass) and methods of solutions (the investigations were mainly conducted numerically or/and analytically). Comparatively, few investigations considered configurations where cross gradients of temperature and concentration are imposed. For this kind of boundary conditions, numerical results of thermosolutal natural convection were reported by Mohamad and Bennacer [22] in the case of a horizontal porous cavity heated horizontally and salted from the bottom. It was found that multiple solutions were possible for a modified Grashof number Grm = 1000, and a buoyancy ratio number, N, varying between 0.8 and 1. In this range of N, bifurcation from monocellular dominating flow to bicellular dominating one was observed. In addition, the concentration gradient reversal was possible in the case of thermally driven flow. The stability of the same problem was explored by Bennacer et al. [23], who predicted oscillatory behaviors within a limited range of N. Analytical and numerical study of double diffusive natural convection, developed in a horizontal porous layer with short vertical walls and long horizontal ones submitted respectively to uniform heat and mass fluxes, was conducted by Kalla et al. [24]. The existence of multiple steady state solutions was demonstrated for a given set of the governing parameters. Two and three dimensional thermosolutal convection in a horizontal enclosure filled with a saturated porous medium and submitted to cross gradients of temperature and concentration was studied numerically by Mohamad and Bennacer [25]. The enclosure was differentially heated and stably stratified species concentration was imposed vertically. It was found that the two dimensional model is generally
sufficient to model the heat and mass transfer properly for the ranges of the governing parameters.
Most of the previous studies dealt with horizontal shallow 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 Bourich al. [26] and Mansour [27], when studying two-dimensional double diffusive natural convection in a porous cavity subjected to cross thermal and solutal gradients. The flow patterns are very different from those observed previously in the case of a horizontal porous layer. In these studies the effects of the governing parameters on the flow and heat and mass transfer were analyzed. It was demonstrated that the solutal buoyancy forces induced by horizontal concentration gradients eliminate the multiplicity of solutions, obtained in thermally driven convection (N = 0), when the buoyancy ratio exceeds some critical value depending on the Lewis, Le, and Rayleigh, Ra, numbers. For N > 0 / (N < 0) the unicellular counterclockwise/ (clockwise) solution is maintained for all the explored ranges of the parameters governing this study. Note that in the Mansour study [27] a three cellular flow was observed.
To the best of our knowledge, there are no investigations dealing with fluid-filled enclosures subjected to cross temperature and concentration gradients. Therefore, a corresponding study must be taken in order to bring some insight to the effects of such geometries. Hence, the present work is concerned with double diffusive natural convection problem in a square enclosure filled with a Newtonian fluid mixture and differentially heated from the horizontal permeable walls while the vertical adiabatic ones are submitted to a concentration difference. In what follows, a numerical solution of the full governing equations is given for a wide range of the governing parameters, whose influence on the flow and heat and mass transfer is amply discussed. The results presented here are limited to water-based solutions, for which the Prandtl number Pr = 7.
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'L 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' except the density in the buoyancy term where it is assumed to vary linearly
International Scientific Journal for Alternative Energy and Ecology № 10 (102) 2011
© Scientific Technical Centre «TATA», 2011
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 [28];
• the fluid is incompressible. For pressures close to that of atmosphere, liquids are a rigorous approximation of incompressible fluids;
• there are no interactions between heat and mass exchanges known under the name of Soret and Duffour;
• 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 [28].
Fig. 1. Sketch of the geometry and co-ordinates system Рис. 1. Чертеж геометрии и системы координат
On the basis of what precedes, the dimensionless governing equations, written in terms of vorticity, Q, temperature, T, concentration, S, and stream function, are:
дО + d(uO) + Э(уО) дt дх ду
It is to note that such a formulation presents the advantage to reduce the number of equations, by eliminating the pressure, which is without interest in this study, and to be more appropriate for two-dimensional flows.
The dimensionless variables are obtained by using the characteristic scales H', H'2/a, a/H', o/H'2, AT' = (T' - TU), AS' = (SR - S'L ) and a corresponding
to length, time, velocity, vorticity, characteristic temperature, characteristic concentration, and stream function, respectively.
To close the problem, the following appropriate dimensionless boundary conditions have been used:
dT
u = v = ^ = — = S = 0 for x = 0, dx
dT
u = v = ^ = — = S -1 = 0 for x = 1; (6) dx
u = v = ^ = — = T -1 = 0 for y = 0, дУ d S
u = v = ^ =-= T = 0 for y = 1. (7)
dУ
As for the vorticity, which is unknown at the boundaries, the relation of woods [29] has been adopted, for its accuracy and stability.
On the other hand, it emerges from Eqs. (1)-(3) that the problem is governed by four dimensionless parameters, namely, the Lewis number, Le, the buoyancy ratio, N, the Prandtl number, Pr, and the thermal Rayleigh number, RaT, whose expressions are
Le = /D, N = PSAS'/PTAT', Pr = v/o
and RaT = ;
TДT'H'7аv .
(S)
„ . д2о Э2О\ „ (дт BS = Pr l —г + —г + PrRaT| —+ N—
дх ду I v Эх дх
(1)
Finally, to characterize the heat and mass transfer through the cavity, respectively in the vertical and horizontal directions, the Nusselt and Sherwood numbers, defined as:
and
where
dT+duT)+dvT) = V2T
dt дх ду
dS + duS)+dvS)=_L v 2 S ■
dt дх ду Le
V2y = -О ,
Эш Эш dv du
тг, У = and О = —— — Эу дх дх ду
(2)
(3)
(4)
(5)
Nu = j (dT/dy)y= 0dx and Sh = j (dS/dx)dy (9)
0 0
have been used.
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
ss
formula [29]. A grid of 81*81 has been considered 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 ^ [y* j - y* j | |y j < 10-4 •,j / j has been satisfied for y, y^. being the value of the
stream function at the node (i, j) for the kth iteration level.
Moreover, the present code has been validated against the results obtained, in the case of a square cavity differentially heated, by Vola et al. [30] and De Vahl Davis [31]. 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%.
Thermally driven convection (N = 0): influence of RaT Fig. 2 display an example of streamlines (left), isotherms (middle) and isoconcentrations (right), corresponding to MCF (a), MCCF (b), VBNF (c) and VBAF (d), obtained for Rar = 5-104. Note that MCF isolines are the images, through a vertical mirror, of MCCF ones while those corresponding to VBNF are the images of VBAF ones, through a horizontal mirror. Therefore, the intensities of the flow cells and the heat and mass transfer mean rates corresponding to MCF, and VBNF are identical to those related to MCCF and VBAF, respectively (this does not hold when N ± 0 for MF, as will be seen later).
Таблица 1
Сравнение численного кода с предыдущими исследованиями (Pr = 0.71, Raj- = 104)
Table 1
Validation of the numerical code against previous studies (Pr = 0,71, Rar = 104)
Present work [30] [31]
IVmaxI Nu IVmaxI Nu IVmaxI Nu
5.072 2.248 5.032 2.240 5.071 2.238
Results and discussion
First of all, 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 [32, 33] showed that this parameter remains without effect beyond Pr = 1. In addition to that, the computations have been carried out with Le = 10, -1 < N < 1 and 103 < Rar < 107.
On the other hand, the present problem is characterized by steady solutions that may correspond to a monocellular (clockwise or counter-clockwise) flow or a vertical or horizontal bicellular (natural or antinatural) flow, denoted MF (MCF or MCCF) and BF (VBNF, VBAF, HBNF or HBAF), respectively. It is to mention that the existence of the monocellular and vertical bicellular flows was proved numerically in the past by Robillard et al. [34] for thermally driven convection (N = 0) and by Bourich et al. [26] and Mansour et al. [27] for thermosolutal convection (N ^ 0) without Soret effect in confined porous media. As all these solutions may exist for the same set of the governing parameters, the choice of the initial conditions is crucial in this study. To obtain a flow pattern with m vertical cells (n = 1) or n horizontal cells (m = 1) for N = 0, a stream function field of form y(x, y) = Asin(mnx)sin(nnx), where the constant A is such that 1 < A < 50, and a conduction temperature field was used for an adequate value of Rar.
Рис. 2. Направления течений, изотермы и изоконцентрации для N = 0 и Rar = 5104: а - MCF; b - MCCF; с - VBNF; d - VBAF Fig. 2. Streamlines, isotherms and isoconcentrations for N = 0 and Rar = 5-104: a - MCF; b - MCCF; с - VBNF; d- VBAF
On the other hand, an augmentation of Rar to 10 (Fig. 3) lets appear, in addition to MF (Fig. 3(a-b)), two kinds of BF: a dissymmetric VBF (Fig. 3(c-d)) and a symmetric HBF (Fig. 3(e-f)), depending on the initial conditions. The former consists on two vertical cells of different size and intensities, which explains their dissymmetry, and the latter is characterized by two horizontal cells, symmetrical with respect to the horizontal axis of the enclosure, hence the same sizes and intensities that they have. A further increase of Rar to 5-105 (Fig. 4) modifies MF (the core of the
International Scientific Journal for Alternative Energy and Ecology № 10 (102) 2011
© Scientific Technical Centre «TATA», 2011
centrosymmetrical cell takes an elliptical form and the streamlines become crowded near the vertical sides as shown in Fig. 4 (a-b)), removes VBF but maintains HBF, that is why the intensities of MF and HBF have undergone strong augmentations compared with the previous situation.
cold vertical side (where the fluid comes from the cold wall) and one hot vertical side (where the fluid comes from the hot wall). The undulations express the fact that, in the case of VBF, the changes in the temperature gradients are more important in the fluid medium. As for the concentration field, a vertical layer of strong concentration gradient in the core of the cavity appears for VBF. This layer is located at the interface between the counter-rotating cells, where the mass transfer by diffusion is dominant. In the case of HBF, thermal and solute layers develop at the horizontal interface, located between the two cells. These layers, which are the result of strong temperature and concentration gradients, become important with an increasing RaT and convey the domination of the diffusion regime.
Рис. 3. Направления течений, изотермы и изоконцентрации для N = 0 и RaT = 105: а - MCF; b - MCCF; c- VBNF; d- VBAF; e - HBNF; f - HBAF Fig. 3. Streamlines, isotherms and isoconcentrations for N = 0 and Raj = 105: a - MCF; b - MCCF; c - VBNF; d- VBAF; e - HBNF; f- HBAF
In addition, these figures show clearly that the temperature and concentration fields, obtained for the same set of governing parameters, change significantly from one flow pattern to another. It can be seen, for instance, that the number of undulations shown by the isotherms in the horizontal direction increases with that of the flow cells. The reason is that each cell has one
Рис. 4. Направления течений, изотермы и изоконцентрации для N = 0 и Raj = 5105: а - MCF; b - MCCF; c - HBNF; d - HBAF Fig. 4. Streamlines, isotherms and isoconcentrations for N = 0 and RaT = 5-105: a - MCF; b - MCCF; c - HBNF; d - HBAF
Fig. 5 displays the flow intensity (a), Sup(ymax,min|), and heat (b), Nu , and mass (c), Sh,
transfer rates versus RaT. The effect of the flow pattern on these quantities can be easily seen. The arrows show the transition that each solution undergoes when RaT is gradually decreased starting from relatively large values of this parameter. It has been found that MF, VBF and HBF exist for RaT > 2320, RaT > 11970 and RaT > 8-104, respectively, and the effect of the flow pattern (multiplicity of solutions) on heat transfer depends
strongly on Rar. Indeed, according to Fig. 5, b, for Rar < 33500, the highest value of Nu (i.e. the best heat transfer) is achieved by MF. However, for Rar > 33500, VBF becomes the most favorable to heat transfer. Finally, for Rar > 8-104, the weakest value of Nu (i.e. the worst heat transfer) is ensured by HBF. The reason is that, as the heat transfer occurs in the vertical 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 Figs. 3 and 4) plays a resisting role against the heat transfer in the vertical direction.
Рис. 5. Интенсивность потока (а), число Нуссельта (b) и число Шервуда (с) по сравнению с Rar для различных решений, полученных с N = 0 Fig. 5. Flow intensity (a), Nusselt number (b) and Sherwood number (с) versus Ra r for different solutions obtained with N = 0
As for the mass transfer, Fig. 5, c shows that up to Rar = 8-105 MF is the most favorable to the mass transfer, which is, in such a situation, the weakest with VBF. In fact, as the mass transfer occurs in the horizontal direction, the presence of vertical cells tends to slow down the fluid circulation from the most concentrated side (right wall) to the least concentrated one (left wall). Moreover, the interface between the flow cells (where the mass transfer is mainly due to molecular diffusion, as indicated by the isoconcentrations of Figs. 2 and 3) puts up a resistance to mass transfer in the horizontal direction.
It should be noted also, from Fig. 5, that, whatever the flow mode, an increase of Rar leads to an increase of the quantities Sup (y max, |y min |), Nu and Sh, due to the
obvious contribution of the thermal buoyancy effects in promoting the convection.
Double diffusive natural convection: influence of Nfor a given RaT To get enough information about the effect of the buoyancy ratio, N, on the different solutions, numerical results in terms of streamlines, isotherms, and isoconcentrations are depicted in Figs. 6-8, for various values of N and Rar.
Fig. 6. Направления течений, изотермы и изоконцентрации MCCF и VBNF для Rar = 5104 и различные значения N (а и с - N = 0,1, b и d - N = -0,1) Fig. 6. Streamlines, isotherms and isoconcentrations of MCCF and VBNF for Rar = 5104 and various values of N (a and с - N = 0,1, b and d- N = -0,1)
International Scientific Journal for Alternative Energy and Ecology № 10 (102) 2011
© Scientific Technical Centre «TATA», 2011
to the detriment of the right one which gives rise to a symmetrical flow with respect to a vertical mirror. In such a situation, as the interface between the cells is tilted in the counterclockwise (N > 0) or clockwise (N < 0) direction, an inclination of the thermal undulations and of the solutal layers, occur. Finally, in the case of HBF, when N is increased/(decreased) above/(below) 0, the lower/(upper) cell is intensified and enlarged expressing a certain dissymmetry of the flow (the interface between the two cells is switched upward/(downward) from mid-height).
(f)
Рис. 7. Направления течений, изотермы и изоконцентрации MCCF, VBNF и HBNF для RaT = 105 и различные значения N (а и c - N = 0,1, b и d - N = -0,1, e - N = 0,2 и f - N = -0,2) Fig. 7. Streamlines, isotherms and isoconcentrations of MCCF, VBNF and HBNF for Raj = 105 and various values of N (a и c - N = 0,1, b and d - N = -0,1, e - N = 0,2 and f - N = -0,2)
Thus, as can be seen from these figures, the effect of N on the flow structure is such that a sudden variation of this parameter from a positive to a negative value (absolutely identical) does not affect MF, but changes notably VBF and HBF.
In fact, MF keeps the same direction of rotation (the counterclockwise one) and intensity, and the corresponding isolines of temperature and concentration show almost identical tendencies, which means that the buoyancy forces direction has no effect on the situation for this kind of flow, owing to the weak value of I N|. As regards VBF, the left cell decreases in size and intensity
Рис. 8. Направления течений, изотермы и изоконцентрации MCCF и HBNF для RaT = 5105 и различные значения N (а и c
- N = 0,02, b и d - N = -0,02, e - N = 0,05 и f - N = -0,05) Fig. 8. Streamlines, isotherms and isoconcentrations of MCCF
and HBNF for RaT = 5105 and various values of N (а и c -N = 0,02, b and d - N = -0,02, e - N = 0,05 and f - N = -0,05)
At the same time, the isotherms become crowded in the interface between the two cells and near the horizontal walls. Outside these zones, they present strong distortions, particularly in the region occupied by the cell of lower size. Concerning the isoconcentrations, these ones squeeze up near the vertical boundaries and at the interface level.
parameter), it is found that a given solution persists until N crosses a threshold value, NT, then a transition occurs towards MCCF (MCF) or to another solution (VBF or HBF) which, in turn, transits towards MCCF (MCF), depending on RaT. The transitions are such that, for relatively high values of |n|, the multiplicity of solutions disappears (due to the importance of solutal buoyancy forces) and only MCCF (MCF) subsists in the enclosure.
Figs. 9, 10 and 11 display the variations of Sup (vmax, mm I) (a), Nu (b) and Sh (c) with N for RaT = 5-104, 105 and 5-105, respectively.
Рис. 9. Интенсивность потока (а), число Нуссельта (b) и число Шервуда (с) по сравнению с N для различных решений, полученных с Rar = 5104 Fig. 9. Flow intensity (a), Nusselt number (b) and Sherwood number (с) versus N for different solutions obtained with Rar = 5-104
In what follows, the procedure of the determination of the threshold values of N is described. Starting the calculations from N = 0 and increasing (decreasing) N gradually (the solution obtained with a given value of N is used as initial guess for the next value of this
Рис. 10. Интенсивность потока (a), число Нуссельта (b) и число Шервуда (с) по сравнению с N для различных решений, полученных с Rar = 105 Fig. 1о. Flow intensity (a), Nusselt number (b) and Sherwood number (с) versus N for different solutions obtained with Rar = 105
International Scientific Journal for Alternative Energy and Ecology № 10 (102) 2011
© Scientific Technical Centre «TATA», 2011
Рис. 11. Интенсивность потока (а), число Нуссельта (b) и число Шервуда (c) по сравнению с N для различных решений, полученных с RaT = 5105 Fig. 11. Flow intensity (а), Nusselt number (b) and Sherwood number (c) versus N for different solutions obtained with Raj = 5-105
(that is why the presented figures have not the same scale). With regard to mass transfer, it is easy to see that this one is the best with MF for the reasons given in the previous subsection.
On the other hand, as shown in Fig. 9, the domain of the existence of VBF seems to be restricted compared to that of MF. Moreover, although Sup (y max, min |) is the
most important with MF, there exists a range of N values (|n| < 0.1) where VBF is the most favorable to heat transfer (Nu (VBF) > Nu (MF). Fig. 10, shows analogous behavior with MF and VBF as before, but the presence of HBF, whose domain of existence is narrower (| N| < 0.02) than that of VBF, leads only to better mass
transfer compared to VBF (Sh(NBF) > Sh (VBF). Finally, Fig. 11 confirms the fact that with HBF the heat transfer is the weakest over all its domain of existence (|n| < 0.058), as explained before (cf. 4.1). In all these figures, the arrows indicate the transitions made, whose threshold values of N are displayed in Table 2.
A minute inspection of this table reveals that an increase of Rar precipitates the transition from MCCF/(MCF) towards MCF/(MCCF) and from VBF towards MCF or MCCF (|Nr| decreases with Rar), but delays that from HBF towards MCF or MCCF (¡Nr| increases with Rar). Note that all the transitions from MCCF, MCF or VBF occur towards MCF or MCCF, but when Rar becomes relatively large, the transition from MCF/(MCCF) towards MCCF/(MCF) is in cascade (a transition is observed first towards HBF and then towards MCCF/(MCF)). This behavior is a consequence of the delay in the disappearance of HBF with respect to MCF/(MCCF) for a high value of Rar.
Note that Nu/Sh remains nearly constant while varying | N| in the case of the HBF/(VBF), but it decreases notably with this parameter for VBF/(HBF). This is mainly due to the arrangement of the flow, which prevents the fluid circulation from the lower/(left) wall to the upper/(right) one (i.e. from the most heated/(salted) wall to the least salted/(heated) one) for VBF/(HBF).
First of all, the symmetry of all the solutions, with respect to the vertical line passing through N = 0, is perfect (MCF and MCCF considered together as a unique curve, since they just exchange their roles when passing from N > 0 to N < 0), which explains the identical flow and heat and mass transfer characteristics induced by each kind of flow (MF or BF) for N > 0 and N < 0.
In addition, for all the selected values of RaT, the value of Sup (y max, |y min |) corresponding to MF is the
highest, which means that this flow is the strongest among all those observed in the studied configuration. Such a quantity tends to be constant with an increasing I N|, leading to an asymptotic behavior of y for MF that gets more and more precocious with an increasing RaT
Conclusion
The problem related to multiplicity of solutions of natural double diffusion convection in a square cavity, filled with water-based solutions as Newtonian fluids (Pr = 7) and submitted to vertical ascendant temperature and horizontal concentration gradients, has been studied by numerical way. The computations have been conducted for Le = 10, -1 < N < 1, and 103 < RaT < 107 under the assumption of 2-D laminar flow. It is emerges from this study the following:
- Multiple solutions, consisting of monocellular and bicellular (horizontal or vertical) flows, exist in the absence of solutal buoyancy forces (N = 0) depending on RaT. In such a situation, when all the observed solutions
exist simultaneously, the highest heat and mass transfer rates are induced by the monocellular flow.
- In the presence of solutal buoyancy forces (N ï 0), the multiplicity of solutions manifest itself for a range of N values, such that, N < 1 with its extend depending on RaT. The transition between these solutions can be direct or in cascade according to the value of RaT. The multiplicity disappears totally when N is large enough.
On the other hand, in its range of existence, the vertical bicellular flow ensure the best heat transfer, while the monocellular one give rise to the best mass transfer for any value of N.
- Unlike Mansour [27], who obtained three vertical cells in a porous square cavity, it has not been possible in our case (fluid-filled square cavity) to find more than two vertical cells.
Таблица 2
Переходы, полученные для различных решений и их соответствующие пороговые значения NT
для различных значений Raj-
Table 2
Transitions obtained for different solutions and their corresponding threshold values,
NT for various values of RaT
Ra_ t Flow structure
5 -104 105 5 -105 MTF MCF VBF
Transition towards t Transition towards т Transition towards V t
MCF -0.224 MCCF 0.224 MCCF 0.146
MCF -0.146
Flow structure
MTF MCF VBF HBF
Transition towards К t Transition towards т Transition towards AL t Transition towards t
MCF -0.132 MCCF 0.132 MCCF 0.116 MCCF 0.02
MCF -0.116 MCF -0.02
Flow structure
MCCF MCF HBF
Transition towards Transition towards nr Transition towards nr
IIBF -0.02 HBF 0.02 MCCF 0058
MCF -0.058
References
1. Ostrach S. Natural Convection with Combined Driving Forces // Phys. Chem. Hydro. 1, 1980.
2. Viskanta R., Bergman T.L., Incopera F.P. Double-Diffusive Natural Convection in Natural Convection, Fundamentals and Applications, Hemisphere, Washington DC; 1985.
3. Bejan A. The boundary layer regime in a porous layer with uniform heat flux from the side // Int. J Heat Mass Transfer. 1983. 26. P. 1339-1346.
4. Nield D. A., Bejan A. Convection in Porous Media, Springer Verlag; 1992.
5. Trevisan O.V., Bejan A. Natural convection with combined heat and mass transfer buoyancy effects in a porous medium // Int. J Heat Mass Transfer. 1985. 28. P. 1597-611.
6. Trevisan O.V., Bejan A. Mass and heat transfer by natural convection in a vertical slot filled with porous medium // Int. J Heat Mass Transfer. 1986. 29. P. 403-15.
7. Alavyoon F., Masuda Y., Kimura S., On natural convection in vertical porous enclosures due to opposing fluxes of heat and mass prescribed at the vertical walls // Int. J Heat Mass Transfer. 1994. 37. P. 195-206.
8. Mamou M., Vasseur P., Bilgen E. Multiple solutions for double diffusive convection in a vertical porous enclosure // Int. J Heat Mass Transfer. 1995. 38. P. 1787-98.
9. Nithiarasu P., Seetharamu K.N., Sundarajan T. Double-diffusive natural convection in an enclosure filled with fluidsaturated porous medium, a generalized non-Darcy approach // Num. Heat Transfer Part A. 1996. 30. P. 413-26.
International Scientific Journal for Alternative Energy and Ecology № 10 (102) 2011
© Scientific Technical Centre «TATA», 2011
10. Karimi-Fard M., Charrier-Mojtabi M.C., Vafai K. Non-Darcian effects on double-diffusive convection within a porous medium // Num. Heat Transfer Part A.
1997. 31. P. 837-52.
11. Nishimura M., Wakamatsu M., Morega A.M. Oscillatory double diffusive convection in a rectangular enclosure with combined horizontal temperature and concentration gradients // Int. J Heat Mass Transfer.
1998. 41. P. 1601-11.
12. Mamou M., Vasseur P., Bilgen E. Double diffusive convection instability in a vertical porous enclosure // J. Fluid Mech. 1998. 368. P. 263-89.
13. Amahmid A., Hasnaoui M., Mamou M, Vasseur P. Boundary layer flows in a vertical porous enclosure induced by opposing buoyancy forces // Int. J Heat Mass Transfer. 1999. 42. P. 3599-608.
14. Amahmid A., Hasnaoui M., Vasseur P. Etude analytique et numérique de la convection naturelle dans une couche poreuse de Brinkman doublement diffusive // Int. J Heat Mass Transfer. 1999. 42. P. 2991-3005.
15. Nield D.A. Onset of thermohaline convection in a porous medium // Water Resour. Res. 4 1968. P. 553560.
16. Poulikakos D. Double-diffusive convection in a horizontally sparsely packed porous layer // Int. Comm. Heat Mass Transfer. 1986. 13. P. 587-598.
17. Murray B.T., Chen C.F. Double diffusive convection in a porous medium // J Fluid Mech. 1989. 201. P. 147-166.
18. Trevisan O.V., Bejan A. Mass and heat transfer by high Rayleigh number convection in a porous medium heated from below // Int. J Heat Mass Transfer. 1987. 30. P. 2341-56.
19. Chen F., Chen C.F. Double-diffusive fingering convection in a porous medium // Int. J Heat Mass Transfer. 1993. 36. P. 793-807.
20. Amahmid A., Hasnaoui M., Mamou M., Vasseur P. Double-diffusive parallel flow induced in a horizontal Brinkman porous layer subjected to constant heat and mass fluxes, analytical and numerical studies // J Heat Mass Transfer. 1999. 35. P. 409-421.
21. Sezai I., Mohamad A.A. Three-dimensional double diffusive convection in a porous cubic enclosure due to opposing gradients of temperature and concentration // J Fluid Mech. 1999. 400. P. 333-353.
22. Mohamad A.A., Bennacer R. Natural convection in a confined saturated porous medium with horizontal temperature and vertical solutal gradients // Int. J Thermal Sc. 2001. 40. P. 82-93.
23. Bennacer R., Mohamad A.A., Akrour D. Transient natural convection in an enclosure with horizontal temperature and vertical solutal gradients // Int. J Thermal Sc. 2001. 40. P. 899-910.
24. Kalla L., Vasseur P., Benacer R., Beji H., Duval R. Double diffusive convection within a horizontal porous layer salted from the bottom and heated horizontally // Int. Comm. Heat Mass Transfer. 2001. 28. P. 1-10.
25. Mohamad A.A., Bennacer R. Double diffusion natural convection in an enclosure filled with saturated porous medium subjected to cross gradients; stably stratified fluid // Int. J Heat Mass Transfer. 2002. 45. P. 3725-40.
26. Bourich M., Amahmid A., Hasnaoui M. Double Diffusive Convection in a Porous Enclosure Submitted to Cross Gradients of Temperature and Concentration // Energy Conversion Management. 2004. 45. P. 16551670.
27. Mansour A., Amahmid A., Hasnaoui M., Bourich M. Multiplicity of Solutions Induced by Thermosolutal Convection in a Square Porous Cavity Heated from below and submitted to horizontal concentration gradient in the presence of Soret effect // Num. Heat Transfer Part A. 2006. 49. P. 69-94.
28. Siginer D.A., Valenzuela-Rendon A. On the laminar free convection and stability of grade fluids in enclosures // Int J Heat Mass Transfer. 2000. 43. P. 3391-3405.
29. Roache P.J. Computational fluid dynamics, Hermosa Publishers, Albuquerque, New Mexico, 1982.
30. Vola D, Boscardin L., Latché J.C. Laminar unsteady flows of Bingham fluids, a numerical strategy and some benchmark results // J. Comp. Physics. 2003. 187. P. 441-456.
31. De Vahl Davis G. Natural convection of air in a square cavity: a bench mark numerical solution // Int. J. Num. Method Fluids. 1983. 3. P. 249-264.
32. Mamou M., Vasseur P., Hasnaoui M. On numerical stability analysis of double-diffusive convection in confined enclosures // J Fluid Mech. 2001. 433. P. 209-250.
33. Lamsaadi M., Nai'mi M., Hasnaoui M. Natural convection of non-Newtonian power-law fluids in a shallow horizontal rectangular cavity uniformly heated from below // Heat and Mass Transfer. 2005 . 41. 3. P. 239-249.
34. Robillard L., Wang C.H., Vasseur P. Multiple Steady States in a Confined Porous Medium with Localized Heating from Below // Int. J Heat Mass Transfer. 1988. 13. P. 91-110.
