NUMERICALSTUDY OF MIXED CONVECTION FLOWIN ALID

ABSTRACT A numerical investigation of steady laminar mixed convection flow of a Copper-water nanofluid in a lid –driven cavity has been executed. In the present study, the left vertical and inclined walls are heated at constant temperature while the right vertical and inclined right walls are maintained at constant cold temperature. The bottom wall is adiabatic and moving with uniform velocity. The study has been carried out for Rayleigh number Ra= 10– 10, Reynolds number Re=20-100 and solid volume fraction of Cu nanoparticles Ø= 0-0.05. The effective viscosity and thermal conductivity of the nanofluid have been calculated by Brinkman and Maxwell-Garnett models, respectively. The results indicated that, the Nusselt number increases with increasing Ra, Re and Ø.


INTRODUCTION
Mixed convection has received a noticeable attraction in research due to its wide application in industry likes, solar collectors, storage of grains, disposing of waste materials, etc. The performance of these devices can be improved by using nanofluids rather than regular fluids. A nanofluid is a base liquid with suspended metallic or non-metallic nanoparticles. Because traditional fluids used for heat transfer applications such as water, mineral oils and ethylene glycol have a rather low thermal conductivity, nanofluids with relatively higher thermal conductivities have attracted enormous interest from researchers due to their potential in enhancement of heat transfer with little or no penalty in pressure drop.
Ali and Eiyad (2012) studied laminar mixed convection flow in single and doublelid driven square cavities filled with water -Al2O3 nanofluid .The left and right walls of the cavity were kept insulated while the bottom and top walls were maintained at constant temperatures with the top surface being the hot wall. It is found that small Richardson number causes reductions in the average Nusselt number. Abbasian et al. (2012) investigated numerically mixed convection laminar flow around an adiabatic body in a lid-driven enclosure filled with nanofluid (Al2O3-water)using variable thermal conductivity and viscosity. The vertical enclosure wall is maintained at constant cold temperature while the horizontal bottom wall is kept at constant hot temperature. The top wall of the enclosure is insulated and moving with uniform velocity. The ratio of body length to the enclosure length is kept constant at 1/3. The study has been carried out for Ri (0.01-100), Ø (0-0.06) and Gr (104). The results showed that, the average Nu increased by increasing Ø and reduction with Ri. Jami et al. (2013) studied numerically the mixed convection in lid-driven partially heated cavities. The results show that, for Ri>1, the average Nu is relatively low while for Ri<1 the forced convection becomes dominant and the natural convection weak, as a result of which Nu is relatively higher. The heat transfer rate increases with increasing the solid volume fraction of the nanoparticles. Sivanandam et al. (2010) investigated convection flow and heat transfer behavior of nanofluids with different nanoparticles in a square cavity. The hot left wall temperature is varied linearly with height whereas the cold right wall temperature is kept constant. It was found that, the heat transfer rate increases with increasing the volume fraction of the nanofluid for all types of nanoparticles considered. Also, the increment of Nu is strongly dependent on the nanoparticles chosen. Samehand Mansour (2012) investigated numerically the mixed convection flows in a square lid-driven cavity partially heated from below using Cu-water, Ag-water, AL2O3-water and TiO2-water nanofluid. The results showed that the increasing the Ø leads to decreasing both the activity of the fluid motion and the fluid temperature. However, it increases the corresponding Nu by adding TiO2 nanoparticles to  Eiyad and chamkha (2010) studied natural convection heat transfer characteristics in a differentially heated enclosure filled with CuO-EG-Water nanofluids for different variable thermal conductivity and variable viscosity models. The results showed that, the effect of thermal conductivity models was less significant than the viscosity models at high Rayleigh number. Mostafa and Seyed (2012) investigated natural convection fluid flow and heat transfer inside C-shaped enclosures filled with Cu-water nanofluid. The results found that the mean Nusselt number increased with the increase in Rayleigh number and volume fraction of Cu nanoparticles regardless the aspect ratio of the enclosure.
It can be concluded from the survey above, that all the studied geometries are rectangular enclosures.
Thus, the present study is dedicated to study the mixed convection in such a hooded rectangular enclosure with base is being moving either to right or left direction. This geometry can be considered as a simulation for a modified solar collector.

THEORETICAL ANALYSIS
A schematic diagram of the considered model is shown in Fig. 1. It is a twodimensional enclosure of height H and based length L, (L=H) filled with Cu-water nanofluid. The insulated lid-driven bottom wall is sliding at a constant speed [+ uo for case I and withuo for case II]. The left vertical and inclined walls are heated at constant temperature (Th) while the right vertical and inclined walls are kept at constant cold temperature (Tc). The Cu-water nanofluid is assumed Newtonian, incompressible, in thermal equilibrium, and the nanoparticles are kept uniform in shape and size. The Kufa Journal of Engineering, Vol. 9, No. 3, July 2018 131 thermo-physical properties of the base fluid and the nanoparticles, presented in Table 1, are considered constant with the exception of its density, which varies according to the Boussinesq approximation. The viscosity of the nanofluid is assumed to be a function of volume fractions of nanoparticles by using Brinkman model, Brinkman (1952). The governing equations for laminar, steady-state lid driven convection in an enclosure filled with Cu -water nanofluids are given as: Continuity equation: Momentum equation in x-direction: Where: Momentum equation in y-direction: Energy equation: The effective density of nanofluid at the reference temperature can be defined as: The heat capacitance of nanofluid can be given as: The effective thermal conductivity of the nanofluid is approximated by the Maxwell -Garnett model Equations (1) The overall average Nusselt number for the heated walls is be given by:

VALIDATION AND COMPARISON OF THE STUDY
The studied geometry in this paper is an obstructed solar collector cavity; therefore several grid size sensitivity tests together with the continuity equation ( + = 0 ) are achieved. The grid independency test is presented in Table 2. This table shows that the Nuav becomes stable beyond grid 5. Therefore, the grid 6 (nodes = 3049) is adopted in this paper. The obtained results showed an exactly validation of the velocity distribution for grid size obtained by imposing an accuracy of 10-

This accuracy is a compromised vale between the result accuracy and the time consumed in each
run. The grid domain for Ra= 104, Re =20 and Ø=0 is shown in Fig. 2-a and the distribution of ( + ) over the domain is presented in Fig. 2

RESULTS AND DISCUSSIONS
The present study is carried out for copper-water nanofluid flow at Rayleigh number Ra=104-106, Reynolds number Re=1-100 and solid volume fraction Ø=0-0.05. The discussion is established on two cases: case I and case II. In case I, the bottom wall is moving to the right while the case II, the bottom wall is moving to the left. Fig. 3 shows isotherms contour (on the left), U-velocity contour (in the middle) and V-velocity contour (on the right) for various Ra numbers, Re=20 and Ø=0.05. It is noticed that, at lower Ra number the solid concentration has more effect to increase the heat penetration; because the conduction heat transfer effect is overheated with increasing Ra number, so the solid concentration has smaller effect on thermal distribution compared buoyancy effect. When Ra number decreases, the intensity of U -velocity contour increases near the bottom wall due to its moving and buoyancy effect. In addition, V-component increases with increasing Ra number. Fig. 4 indicates the effect of Re number on these contours at Ra=104 and Ø=0.05, for isotherms as can be seen that when Re number smaller, the effect of lid-driven is insignificant and the intensity of the isotherms are concentrated close to the heated wall. However, as Re number increases the effect of lid-driven increases and hence forced convection flow. It is noticed also that with increasing Re, an isothermal zone is localized above the moving bottom wall. At Re=20, the vortex core of U-velocity contour is in the center of the cavity. When Re number increases the vortex core moves to the right side of the cavity due to increasing the forced convection. When Re number increases V-velocity increases as well.  However, as Re number increases the effect of lid-driven increases and hence forced convection flow. It is noticed also that with increasing Re, an isothermal zone is localized above the moving bottom wall. At Re=20, the vortex core of U-velocity contour is in the center of the cavity. When Re number increases the vortex core moves to the right side of the cavity due to increasing the forced convection. When Re number increases V-velocity increases as well.  All these observations are due to the strengthening of the natural convection with increasing Ra. Fig.   7 indicates the effect of the height of the hood of the cavity (D Moreover, this figure tell us that when the bottom wall is moving to the left (U= -1), the average Nusselt number manifests greater values. This is quite reasonable, because the direction of the lid coincides with the natural movement of nanofluid which is generated from the left walls towards the upper zone of the enclosure, i.e. the aiding mechanism will be obtained U= -1, and the opposing mechanism is obtained when U= +1. Fig. 8a-left shows variation of Nuav on Ra number for the second case. As a result, Nuav increases with Ra number for two cases but the values of Nuav for cases II larger than case I due to the secondary flow. Fig 8a-right indicates the effect of θav on Ra for two cases. θav decreases with increasing Ra number for two cases.
Fig. 8b-left shows the effect of Nuav on Re number. As can be seen that Nuav increases linearly approximately with Re number for two cases. Fig. 8b-right represents the variation of θav on Re for two cases. The increase in Re number leads to decreasing θav for two cases. Fig. 8c-left refers to the effect of Nuav on Ø. As can be seen that Nuav increases Ø for two cases. Also, that talk for θav at