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Procedia Engineering 56 (2013) 480 - 488

Procedia Engineering

www. el sevi er. com/1 ocate/procedi a

5th BSME International Conference on Thermal Engineering

Double diffusive natural convective flow characteristics in a cavity

Salma Parvin*, Rehena Nasrin, M.A. Alim, N.F. Hossain

Department of Mathematics, Bangladesh University of Engineering & Technology,Dhaka-1000, Bangladesh

Abstract

The influences of Soret and Dufour coefficients on free convection flow phenomena in a partially heated square cavity filled with water-AI2O3 nanofluid is studied numerically. The top surface has constant temperature Tc, while bottom surface is partially heated Th, with Th > Tc. The concentration in top wall is maintained higher than bottom wall (Cc < Ch). The remaining bottom wall and the two vertical walls are considered adiabatic. Water is considered as the base fluid. By Penalty Finite Element Method the governing equations are solved. The effect of the Soret and Dufour coefficients on the flow pattern and heat and mass transfer has been depicted. Comprehensive average Nusselt and Sherwood numbers, average temperature and concentration and mid-height horizontal and vertical velocities inside the cavity are presented as a function of the governing parameters. Results shows that both heat and mass transfer increased by Soret and Dufour coefficients.

© 2013 The Authors. Published by ElsevierLtd.

Selecticn and peer review under responsibility of the Bangladesh Society of Mechanical Engineers

Keywords: Soret and Dufour coefficients; double-diffusive natural convection; finite element method; water-AkO3 nanofluid.

Nomenclature

c Dimensional concentration (kg m-3)

C Non-dimensional concentration

CP Specific heat at constant pressure (kJ kg-1 K-1)

Cs Concentration susceptibility

D Solutal diffusivity (m2 s-1)

Df Dufour parameter

g Gravitational acceleration (m s-2)

h Local heat transfer coefficient (W m-2 K-1)

k Thermal conductivity (W m-1 K-1)

Kt Thermal diffusion ratio

L Lengh of the enclosure (m)

Nu Nusselt number,

Pr Prandtl number

Sc Schmidt number

Sh Sherwood number

Sr Soret parameter

Ra Rayleigh number

* Corresponding author. Tel.: 880-9665650-ext 7912. E-mail address: salpar@math.buet.ac.bd

1877-7058 © 2013 The Authors. Published by Elsevier Ltd.

Selection and peer review under responsibility of the Bangladesh Society of Mechanical Engineers doi: 10.1016/j .proeng .2013.03.150

T Dimensional temperature (°K)

u, v Dimensional x and y components of velocity (m s"1)

U, V Dimensionless velocities, U = u/L, V = v/L

X, Y Dimensionless coordinates, X = x / L, Y = y / L

x, y Dimensional coordinates (m)

Greek Symbols

a Fluid thermal diffusivity (m2 s"1)

P Thermal expansion coefficient (K"1)

(/) Nanoparticles volume fraction

6 Dimensionless temperature

H Dynamic viscosity (N s m"2)

v Kinematic viscosity (m2 s"1)

p Density (kg m"3) Subscripts

av average

c cold

f fluid

m mean

nf nanofluid

s_solid particle_

1. Introduction

The natural convection in enclosures continues to be a very active area of research during the past few decades. While a good number of works have made significant contributions for the development of the theory, an equally good number of works have been devoted to many engineering applications that include electronic or computer equipment, thermal energy storage systems and etc.

Double diffusive convection of water has been studied by Nithyadevi and Yang [1] and Sivasankaran and Kandaswamy [2, 3]. Yet, most work done considers flow inside closed enclosures, the applications included, such as pollution dispersion inlakes, chemical deposition, and melting and solidification process. Diffusion of matter caused by temperature gradients (Soret effect) and diffusion of heat caused by concentration gradients (Dufour effect) become very significant when the temperature and concentration gradients are very large. Generally these effects are considered as second order phenomenon. These effects may become important in some applications such as the solidification of binary alloys, groundwater pollutant migration, chemical reactors, and geosciences. The importance of these effects has also seen in Mansour et al. [4], Platten [5] and Patha et al. [6].

Double diffusive and Soret induced convection in a shallow horizontal enclosure is studied numerically by Mansour et al. [4]. They found that the Nusselt number has decreases in general with the Soret parameter while the Sherwood number increases or decreases with this parameter depending on the temperature gradient induced by each solution.

In the above studies convection heat transfer is due to the imposed temperature gradient between the opposing walls of the enclosure taking the entire vertical wall to be thermally active. But in many naturally occurring situations and engineering applications it is only a part of the wall which is thermally active. For example in solar energy collectors due to shading, it is only the unshaded part of the wall that is thermally active. In order to have the results to possess applications, it is essential to study heat transfer in an enclosure with partially heated active walls. Only a few studies are reported in the literature concerning heat transfer in enclosures with partially heated side walls, by Oztop [7] and Erbay et al. [8].

Natural convection in an enclosure with partially active walls is studied by Nithyadevi et al. [9] and Kandaswamy et al. [10] without Soret and Dufour effects. Present study deals with the natural convection in a square enclosure filled with water and partially heated vertical walls for three different combinations of heating location in the presence of solute concentration with Soret and Dufour effects. The hot region is located at the top, middle and bottom of the left vertical wall of the enclosure.

Oztop and Abu-Nada [11] numerically studied natural convection in partially heated rectangular enclosures filled with nanofluids. Rouboa et al. [12] analyzed convective heat transfer in nanofluid. Esfahani and Bordbar [13] studied double diffusive natural convection heat transfer enhancement in a square enclosure using nanofluids. Gorla et al. [14] analyzed mixed convective boundary layer flow over a vertical wedge embedded in a porous medium saturated with a nanofluid: Natural Convection Dominated Regime. Kuznetsov and Nield [15] performed double-diffusive natural convective

boundary-layer flow of a nanofluid past a vertical plate where similarity solution was performed in order to obtain correlation formulas giving the reduced Nusselt number as a function of the various relevant parameters. The stability boundaries for both non-oscillatory and oscillatory cases had been approximated by simple analytical expressions. For the porous medium the Darcy model is employed.

Effects of Soret Dufour, chemical reaction and thermal radiation on MHD non-Darcy unsteady mixed convective heat and mass transfer over a stretching sheet was investigated by Pal and Mondal [16]. The author used shooting algorithm with Runge-Kutta-Fehlberg integration scheme to solve the governing equations. Natural convection heat transfer of nanofluids in a vertical cavity: Effects of non-uniform particle diameter and temperature on thermal conductivity was performed by Lin and Violi [17]. Moreover, Saleh et al. [18] studied natural convection heat transfer in a nanofluid-filled trapezoidal enclosure. They found that acute sloping wall and Cu nanoparticles with high concentration were effective to enhance the rate of heat transfer.

The present work discussed the effect of Soret and Dufour parameter on double diffusive natural convection in a partially heated cavity. The results are presented in the form of streamlines, isotherms, average Nusselt number Nu and average Sherwood number Sh, average temperature of the fluid and mid height velocity in the cavity for relevant parameter.

2. Physical model

Figure 1 shows a schematic diagram of a partially heated square enclosure. The fluid in the cavity is water-based nanofluid containing AI2O3 nanoparticles with Soret and Dufour coefficients. The nanofluid is assumed incompressible and the flow is considered to be laminar. It is taken that water and nanoparticles are in thermal equilibrium and no slip occurs between them. The top horizontal wall has constant temperature Tc, while bottom wall is partially heated Th, with Th > Tc. The concentration in top wall is maintained higher than bottom wall (Cc < Ch). The remaining bottom wall and the two vertical walls are considered adiabatic. The thermophysical properties of the nanofluid are taken from Saleh et al. [18] and given in Table 1. The density of the nanofluid is approximated by the Boussinesq model.

Tc, Ch

\ U g ! 1 j

' Th, Cc

adiabatic

Fig. 1. Schematic diagram of the enclosure

Table 1. Thermo physical properties of fluid and nanoparticles [18]

Physical Properties Fluid phase (Water) M2O3

CJkgK) 4179 765

P (kg/m3) 997.1 3600

k (W/mK) 0.6 46

jMO'5 (1/K) 21 0.63

3. Governing equations

The governing equations for laminar natural convection in a cavity filled with water-alumina nanofluid in terms of the

Navier-Stokes and energy equation (non dimensional form) are given as:

ÔU д¥ n

-+-= 0

usU+yU

,SV ,,d¥

'Pn/SX

U— + V— =-f— + Pr-n/-dX ÔY —

d2U d2U

ydX2 dY \

d2V s2V"

dX2 dY2

+ RaPr(e + C)

Pnfßf

U*J_+V™= -L

dX dY Pr

U<C + V — = ±

dX dY Sc

dX2+ dY2

dX2 + dY2

dX2 + dY2

dX2 + dY2

The corresponding boundary conditions take the following form: at all solid boundaries U = V = 0 at Y = 0, 0.3 £ X< 0.7, 0 = 1, C = 0 at Y = 1, 0 = 0, C = 1

at the remaining boundaries-= 0, -= 0

the following dimensionless dependent and independent variables

X=* Y U=— V = — P = -PL-

L L Vf l dimensional.

where, pnf = (1-^)Pf+<!>PS is the density, (pC^ = pCp^+<j>{^pCpj is the heat capacitance,

Pnf = (1 -(/>) Pf+ <!>PS is the thermal expansion coefficient, anf = knf j^pC^ is the thermal diffusivity,

the dynamic viscosity of Brinkman model [19] is finf = Pf i}~'f) 2 5

and the thermal conductivity of Maxwell Garnett (MG) model [20] is knf = kf

T-T C-C Q = ——, C = —-are used to make the above equations non-

K+ 2kf- 2ф (kf-ks) ks+ 2kf+</>(kf-ks)

Prandtl number Pr = \ — \ , Schmidt number Sc = J , thermal Rayleigh number RaT =

gß,fÜ[Ch-Cc) n„fo„r coefficient n _ iD kTf(Ch~Cc)

gßrf L {rh-Tc)

solutal

Dufour coefficient Df=\ —

f 1 v)f CsCp(Th-Tc)

and Soret coefficient

are used.

Rayleigh number Rac

sJD) f^)

The average Nusselt and Sherwood numbers at the heated and concentrated surfaces of the enclosure may be expressed, respectively as

dX and Sh = -^^CdX .

Ls J dY

4. Numerical implementation

The Galerkin finite element method is used to solve the non-dimensional governing equations along with boundary conditions for the considered problem. The equation of continuity has been used as a constraint due to mass conservation

and this restriction may be used to find the pressure distribution. The penalty finite element method [21] is used to solve the Eqs. (2) - (4), where the pressure P is eliminated by a penalty constraint. The continuity equation is automatically fulfilled for large values of this penalty constraint. Then the velocity components (U, V), temperature (Q) and concentration (C) are expanded using a basis set. The Galerkin finite element technique yields the subsequent nonlinear residual equations. Three points Gaussian quadrature is used to evaluate the integrals in these equations. The non-linear residual equations are solved using Newton-Raphson method to determine the coefficients of the expansions. The convergence of solutions is assumed when the relative error for each variable between consecutive iterations is recorded below the convergence criterion s such

that n+1 - y/n\< 10"4, where n is the number of iteration and W is a function of U, V, 6 and C. 4.1. Grid independent test

An extensive mesh testing procedure is conducted to guarantee a grid-independent solution for Ra = 104, Pr = 6.2, Df = Sr = 0.5, Sc = 5, (¡) = 5% in the chamber. In the present work, we examine five different non-uniform grid systems with the following number of elements within the resolution field: 2569, 4730, 6516, 8457 and 10426. The numerical scheme is carried out for highly precise key in the average Nusselt (Nu) and Sherwood (Sh) numbers for the aforesaid elements to develop an understanding of the grid fineness as shown in Fig. 2. The scale of the average Nusselt and Sherwood numbers for 8457 elements shows a little difference with the results obtained for the other elements. Hence, considering the nonuniform grid system of 8457 elements is preferred for the computation.

Number of elements

Fig. 2. Grid test for the geometry Streamlines Isotherms Concentration

Fig. 3. Comparison between present work and Nithyadevi and Yang using Pr = 11.573, Df = Sr = 0.5, Sc = 5 and RaT = 105

4.2. Code validation

The present numerical solution is validated by comparing the current code results for streamlines, isotherms and concentration profiles using Df = Sr = 0.5, Sc = 5, Pr = 11.573 and RaT = 105 with the graphical representation of Nithyadevi and Yang [2] which was reported for double diffusive natural convection in a partially heated enclosure with Soret and Dufour effects. Fig. 3 demonstrates the above stated comparison. As shown in Fig. 3, the numerical solutions (present work and Nithyadevi and Yang [2]) are in good agreement.

5. Results and discussion

In this section, numerical results of streamlines and isotherms for various values of Soret (Sr) and Dufour (Dr) coefficients and with Al2O3 /water nanofluid in a square enclosure are displayed. Ra = RaT = Rac is assumed for the present numerical calculation. The considered values of Df and Sr are Df = Sr = (0, 0.5 and 1). But the Prandtl number Pr = 6.2, the Rayleigh number Ra = 104, the Schmidt number Sc= 5 and solid volume fraction of the nanofluid (/> = 5% are kept fixed for this study. In addition, the values of the average Nusselt and Sherwood numbers, mean temperature and concentration as well as horizontal and vertical velocities at the middle of the cavity have been calculated for different mentioned parameters.

Figure 4 (a) - (c) exposes the effect of Sr on the flow, thermal and concentration fields while Df = 0.5 and Sc = 5. At the absence of the Soret coefficient (Sr) a primary anticlockwise circulating cell occupies the bulk of the chamber. The size of the inner vortex of this cell becomes larger with the increasing of the Soret coefficient. In addition for the largest value of Sr, the streamlines form rectangular pattern whereas initially they are circular. As well as another vortex is appeared near the left wall of the chamber. The isotherms and iso-concentrations are crowded around the active location on the bottom surface of the enclosure for (Sr = 1). In addition, the temperature lines corresponding to Sr = 1 become less bended. Decreasing Soret effect leads to deformation of the thermal and concentration boundary layers at the right part of the cold upper wall and middle of the bottom surface.

(a) (b) (c)

Fig. 4. Effect of Sr on (a) streamlines, (b) Isotherms and (c) Concentration at Df = 0.5 and Sc = 5

(a) (b) (c)

Fig. 7. Effect of Df on (a) streamlines, (b) Isotherms and (c) Concentration at Sr = 0.5 and Sc = 5

The average Nusselt (Nu) and Sherwood (Sh) numbers, average temperature (6av) and concentration (Cav) along with the Soret coefficient (Sr) are depicted in Fig. 5(i)-(ii). It is seen from Fig. 5(i) that Nu enhances gradually whereas Sh remains almost invariant for mounting Sr. Consequently Fig. 5(ii) shows that (0av) devalues and (Cav) rises sequentially for all values of Soret coefficient Sr.

Figure 6(i)-(ii) shows the mid-height horizontal and vertical velocity profiles inside the chamber for different Sr effect. It is observed that the fluid particle moves with greater velocity for the absence of Soret coefficient Sr. The waviness devalues for higher values of Sr.

The effect of Df on the flow, thermal and concentration fields is presented in Fig. 7 (a) - (c) while Sr = 0.5 and Sc = 5. A primary anticlockwise recirculation cell occupying the whole cavity is found for the absence of the Dufour coefficient (Df). The fluid rises along the right wall and falls along the left wall. The size of the inner vortex of this cell becomes larger with the increasing of the Dufour coefficient. The strength of the flow circulation, the thermal current and concentration activities are much more activated with escalating Df. Increasing Df, the temperature and concentration lines at the middle part of the enclosure become vertical whereas initially they are almost horizontal. Due to rising values of Df, the temperature and concentration distributions become distorted resulting in an increase in the overall heat and mass transfer. It is worth noting that as the Dufour coefficient increases, the thickness of the thermal boundary layer near the horizontal surfaces rises which indicates a steep temperature and concentration gradients. Hence, an increase in the overall heat and mass transfer within the cavity is observed.

Figure 8(i)-(ii) displays the mean Nusselt and Sherwood numbers, average temperature (0av) and concentration (Cav) for the effect of Dufour coefficient Df. Both Nu and Sh grow up for varying Df. The rate of heat transfer is found to be more effective than the mass transfer rate. On the other hand, Qav and Cav has notable changes with different values of Df. The value of mean concentration is always higher than that of average temperature at a particular value of Dufour coefficient.

The U and V velocities at the middle of the cavity for various Df are depicted in Fig. 9 (i)-(ii). A small variation in velocity is found due to changing Df. Some perturbations are seen in the horizontal velocity graph for Df = 0 and in the vertical velocity profile for Df = 1.

6. Conclusion

The influence of nanoparticles on natural convection boundary layer flow inside a square cavity with water-Al2O3 nanofluid is accounted. Various Soret-Dufour coefficients and Schmidt number have been considered for the flow, temperature and concentration fields as well as the heat and mass transfer rate, horizontal and vertical velocities at the middle height of the enclosure while Pr, Ra and <j> are fixed at 6.2, 104 and 5% respectively. The results of the numerical analysis lead to the following conclusions:

• The structure of the fluid streamlines, isotherms and iso-concentrations within the chamber is found to significantly depend upon the Soret-Dufour coefficients..

• The Al2O3 nanoparticles with the highest Sr and Df is established to be most effective in enhancing performance of heat transfer rate than the rate of mass transfer.

• Greater variation is observed in velocities at a particular point for the changes of Sr with compared to that of Df.

• Average concentration is higher than average temperature inside the chamber for the pertinent parameters.

Overall the analysis also defines the operating range where water-Al2O3 nanofluid can be considered effectively in determining the level of heat and mass transfer augmentation.

Acknowledgements

The present work is fully supported by the department of Mathematics, Bangladesh University of Engineering & Technology.

References

[1] Nithyadevi, N., Yang, R.J., 2009. Double diffusive natural convection in a partially heated enclosure with Soret and Dufour effects, International Journal of Heat and Fluid Flow 30, p. 902.

[2] Sivasankaran, S., Kandaswamy, P., 2006. Double diffusive convection of water in a rectangular partitioned enclosure with temperature dependent species diffusivity, International Journal of Fluid Mechanics Research 33, p. 345.

[3] Sivasankaran, S., Kandaswamy, P., 2007. Double diffusive convection of water in a rectangular partitioned enclosure with concentration dependent species diffusivity, Journal of the Korean Society for Industrial and Applied Mathematics 11, p. 71.

[4] Mansour, A., Amahmid, A., Hasnaoui, M., Bourich, M., 2006. 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, Numerical Heat Transfer Part A: Applications 49, p. 69.

[5] Platten, J.K., 2006. The Soret effect: a review of recent experimental results, Journal of Applied Mechanics 73, p. 5.

[6] Patha, M.K., Murthy, P.V.S.N., Raja Sekhar, G.P., 2006. Soret and Dufour effects in a non-darcy porous medium. Journal of Heat Transfer 128, p. 605.

[7] Oztop, H.F., 2007. Natural convection in partially cooled and inclined porous rectangular enclosures. International Journal of Thermal Sciences 46, p. 149.

[8] Erbay, B., Altac, Z., Sulus, B., 2004. Entropy generation in a square enclosure with partial heating from a vertical lateral wall, Heat and Mass Transfer 40, p. 909.

[9] Nithyadevi, N., Kandaswamy, P., Lee, J., 2007. Natural convection in a rectangular cavity with partially active side walls. International Journal of Heat and Mass Transfer 50, p. 4688.

[10] Kandaswamy, P., Sivasankaran, S., Nithyadevi, N., 2007. Buoyancy-driven convection of water near its density maximum with partially active vertical walls. International Journal of Heat and Mass Transfer 50, p. 942.

[11] Oztop, H.F. Abu-Nada, E., 2008. Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, International Journal of Heat and Fluid Flow 29, p. 1326.

[12] Rouboa, A.,Silva, A., Freire, A.J., Borges, A., Ribeiro, J., Silva, P., Alexandre, J.L., 2008. "Numerical analysis of convective heat transfer in nanofluid", AIP Conference Proceedings, pp. 819-822.

[13] Javad Abolfazli Esfahani and Vahid Bordbar, 2011. Double Diffusive Natural Convection Heat Transfer Enhancement in a Square Enclosure Using Nanofluids, Journal of Nanotechnology in Engineering and Medicine 2(2), p. 021002.

[14] Rama Subba Reddy Gorla, Ali Jawad Chamkha, Ahmed Mohamed Rashad, 2011. Mixed convective boundary layer flow over a vertical wedge embedded in a porous medium saturated with a nanofluid: Natural Convection Dominated Regime, Nanoscale Research Letters 6, p 207.

[15] Kuznetsov, A.V. , Nield, D.A. , 2011. Double-diffusive natural convective boundary-layer flow of a nanofluid past a vertical plate, International Journal of Thermal Sciences 50, p. 712.

[16] Pal, D., Mondal, H., 2011. Effects of SoretDufour, chemical reaction and thermal radiation on MHD non-Darcy unsteady mixed convective heat and mass transfer over a stretching sheet, Communications in Nonlinear Science and Numerical Simulation 16, p. 1942.

[17] Kuang C. Lin, Angela Violi, 2010. Natural convection heat transfer of nanofluids in a vertical cavity: Effects of non-uniform particle diameter and temperature on thermal conductivity, International Journal of Heat and Fluid Flow 31, p. 236.

[18] Saleh, H., Roslan, R., Hashim, I., 2011. Natural convection heat transfer in a nanofluid-filled trapezoidal enclosure, International Journal of Heat and Mass Transfer 54, p. 194.

[19] Brinkman, H.C. 1952. The viscosity of concentrated suspensions and solution, Journal of Chemical Physics 20, p. 571.

[20] Maxwell-Garnett, J.C., 1904. Colours in metal glasses and in metallic films, Philosophical Transactions of the Royal Society of London A 203, p. 385.

[21] Basak, T., Roy, S., Pop, I., 2009. Heat flow analysis for natural convection within trapezoidal enclosures based on heat line concept, International Journal of Heat and Mass Transfer 52, p. 2471.