EFFECT OF SOLUTAL BUOYANCY FORCES ON THERMAL CONVECTION IN CONFINED NON-NEWTONIAN POWER-LAW FLUIDS
T. Makayssi*, M. Naïmi*'a, M. Lamsaadi*, M. Hasnaoui**, A. Raji*, A. Bahlaoui*
*Sultan Moulay Slimane university, Faculty of Sciences and Technologies, Physics Department, UFR of Sciences and Engineering of Materials, Team of Flows and Transfers Modelling (EMET), BP 523, Béni-Mellal, Morocco
**Cadi Ayyad University, Faculty of Sciences Semlalia, Physics Department, UFR TMF, Laboratory of Fluid Mechanics and Energetics (LMFE),
BP 2390, Marrakech, Morocco
"Corresponding author. Tel.: (212) 23 48 51 12/22/82; Fax: (212) 23 48 52 01. E-mail: naimi@fstbm.ac.ma, naimima@yahoo.fr
Received: 24 Sept 2007; accepted: 5 Nov 2007
This paper reports the results of an analytical and numerical study on natural convection heat transfer with and without solutal buoyancy forces in a non-Newtonian power-law fluid contained in a horizontal rectangular shallow enclosure submitted to uniform heat and mass fluxes along its short vertical sides, while the horizontal ones are insulated and impermeable. An approximate theoretical solution is developed, on the basis of the parallel flow assumption, and validated numerically by solving the full governing equations. A comparison between results obtained in presence and in absence of solutal buoyancy forces is done. The effect of the non-Newtonian behavior on fluid flow and heat transfer characteristics is examined.
Keywords: thermodynamic analysis in renewable energy, thermal natural convection, double diffusive convection, rectangular enclosures, non-Newtonian fluids
Education: A national doctorate (INPL, Nancy, France, 1989) and a doctorate of state (University Cadi Ayyad, Marrakech, Morocco, 2001). The title of the former is: "Etudes des lois d'écoulement et de transfert de chaleur pour des fluides non-Newtoniens en espace annulaire tournant: approche réaliste de l'échangeur de chaleur à surface raclée" while the latter entitled: "Contribution à l'étude de l'effet Marangoni thermique dans les fluides non-Newtoniens en conditions de gravité et de microgravité". Experience: A teacher researcher, with the degree of higher teaching Professor, at the Faculty of Sciences and Technologies of Sultan Moulay Slimane University (Béni-Mellal, Morocco). Main range of scientific interests: natural, double diffusive and capillary convections in non-Newtonian fluids.
Publications: 8 main papers since 2005.
Nomenclature
A - aspect ratio of the cavity, Eq. (11)
CT - dimensionless temperature gradient in the x-direction
CS - dimensionless concentration gradient in the x-direction
D - mass diffusivity (m2/s)
g - gravitational acceleration (m/s2)
H - height of the enclosure (m)
j' - constant mass flux per unit area (kg/m2-s)
K - consistency index for a power-law fluid at the reference
temperature (Pa-f)
Le - Lewis number, Eq. (11)
L' - length of the rectangular enclosure (m)
N - buoyancy ratio, Eq. (11)
n - flow behavior index for a power-law fluid at the reference temperature
Nu - local Nusselt number, Eqs. (12), (13) and (33) Nu - average Nusselt number, Eqs. (14) and (33) Pr - generalised Prandtl number, Eq. (11)
q' - constant heat flux per unit area (W/m2) RaT - generalized thermal Rayleigh number, Eq. (11) S - dimensionless concentration [= ( - S'c )/AS * ] Sc - reference concentration at the geometric center of the enclosure (kg/m3)
Sh - local Sherwood number, Eqs. (12), (13) and (33) Sh - mean Sherwood number, Eqs. (14) and (33) T - dimensionless temperature [= ( - Tc')/AT* ] Tc'- reference temperature at the geometric center of the enclosure (K)
AT* - characteristic temperature [= qH'fX\ (K) AS* - characteristic concentration [= j'H'/D\ (kg/m3) (u, v) - dimensionless axial and transverse velocities [= (u',vj/(a/H j]
(x, y) - dimensionless axial and transverse co-ordinates
[=(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)
X - thermal conductivity of fluid at the reference temperature (W/m-C)
- dynamic viscosity for a Newtonian fluid at the reference temperature (Pa-s)
|i„ - dimensionless apparent viscosity of fluid, Eq. (7) p - density of fluid at the reference temperature (kg/m3) Q - dimensionless vorticity [= Q(a/H'2 )] y - dimensionless stream function [= y/a]
Superscript
' - dimensional variable
Subscripts
c - value relative to the centre of the enclosure (x, y) = (A/2, 1/2) * - characteristic variable
Introduction
Thermal or simple natural convection is a flow due to density variations with temperature in gravitational field. Double-diffusive natural convection, i.e. flows generated by buoyancy due to simultaneous temperature and concentration gradients, 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, [1]. The literature on double-diffusive natural convection shows that the majority of investigations were focused on the enclosures of rectangular form [2].
In the past, many studies concerning Newtonian fluid flows in enclosures, driven simultaneously by thermal and solutal buoyancy effects, were carried out. These can be classified under three types, according to the thermal and solutal boundary conditions adopted. In the first type, the cavity is subjected to a vertical solutal gradient and a horizontal thermal one [3]. In the second type, both the temperature and concentration gradients are imposed transversally [4]. In the third type, which is the present case, both the thermal and solutal gradients are imposed laterally [5].
To our knowledge, for non-Newtonian fluids, except the work performed by Benhadji et al. [6] in the case of a porous horizontal rectangular layer, where double-diffusive convection is generated inside a power-law fluid by application of horizontal or vertical uniform heat and mass fluxes, there is no investigations dealing with fluid-filled enclosures.Otherwise, the majority of investigations concerning non-Newtonian fluids dealt with thermal driven buoyancy convection [7, 8]. Non-Newtonian flows are of importance and very present in many industrial applications such as paper making, oil drilling, slurry transporting, food processing, polymer engineering and many others. Some of these applications are discussed in Jaluria [9]. In order to contribute to fill the gap left by the lack of studies on the field, at least partly, the present investigation focuses on the effect of solutal buoyancy forces on natural convection heat transfer inside a two-dimensional horizontal rectangular enclosure, filled with a non-Newtonian fluid. The cavity is submitted to uniform heat and mass fluxes from its short vertical sides, while its long horizontal boundaries are insulated and impermeable.
Mathematical formulation
The studied configuration, sketched in Fig. 1, is a rectangular enclosure of height H and length L' with the long horizontal rigid walls insulated and impermeable and the short vertical ones submitted to constant heat and mass densities of fluxes, q' andj', respectively.
q
y ' v
0 x , u
dT' _dS' dy dy
_ 0
Control volume-
H
q
dT' dS'
_ 0
dy' dy'
Fig. 1. Sketch of the cavity and co-ordinates system
International Scientific Journal for Alternative Energy and Ecology № 6 (62) 2008
© Scientific Technical Centre «TATA», 2008
The non-Newtonian fluids considered here are those for which the rheological behavior can be described by the power-law model, proposed by Ostwald-De Waele [10], whose expression, in term of laminar apparent viscosity, is
( f
V-'a = k
du' dX
V v
( dv' Y ^
дУ
(
du' dv dy' dX
A2
V
(1)
= Pr
^aV 2O + 2
d^a дО, дца dO,
dx dx dy dy _
+ so
(2)
where
V >=-O;
dy dx
К =
where n is the power-law index and к is an empirical coefficient known as the consistency factor, which is an indicator of the degree of fluids viscosity. Note that for n = 1 the power-law model reduces to the Newton's law by setting к = ц. Thus, the deviation of n from unity characterizes the degree of non-Newtonian behavior of the fluid. Specifically, when n is in the range 0 < n < 1 the fluid is said to be pseudo-plastic (or shear-thinning) and the viscosity is found to decrease by increasing the shear rate. On the other hand when n > 1 the fluid is said to be dilatant (or shear-thickening) and the viscosity increases by increasing the shear rate. Dilatant fluids are generally much less frequent than pseudo-plastic ones. Though the Ostwald-de Wale model does not converge to a Newtonian behavior in the limit of zero and maximum shear rates, it presents however the advantage to be simple and mathematically tractable. In addition, the rheological behavior of many substances can be adequately represented by this model for relatively large range of shear rates (or shear stresses) making it useful, at least for engineering purpose, and justifying its use in most theoretical investigations of fluids having pseudoplastic or dilatant behaviors. On the other hand, the main assumptions made here are those commonly used, i.e., the flow is laminar and two-dimensional, the viscous dissipation is negligible, the interactions between heat and mass exchanges, known under the name of Soret and Duffour effects, are negligible, the fluid is incompressible and its physical properties are considered temperature independent except the density in the buoyancy term which obeys the Boussinesq approximation. Then, the dimensionless governing equations, written in terms of vorticity, Q, temperature, T, concentration, S, and stream function, are:
dO d(
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