Natural Convection Heat Transfer from a Plane Wall to Thermally Stratified Environment

The effect of linear thermal stratification in stable stationary ambient fluid on free convective flow of a viscous incompressible fluid along a plane wall is numerically investigated in the present work. The governing equations of continuity, momentum and energy are solved numerically using finite difference method with Alternating Direct implicit Scheme. The velocity, temperature distributions and the Nusselt number are discussed numerically for various values of physical parameters and presented through graphs. ANSYS program also used to solve the problem. The results show that the effect of stratification parameter is marginalized with the increase in Prandtl number, and the increase in Grashof number does not practically vary the effect of stratification parameter.


Introduction
Convective heat transfer in thermally stratified ambient fluid occurs in many industrial applications and is an important aspect in the study of heat transfer. If stratification occurs, the fluid temperature is function of distance. Convection in such environment exists in lakes, oceans, nuclear reactors. The problem had been investigated by many researcher analytically and numerically, see for example (Cheesewright 1967), (Yang et al 1972), (Jaluria and Himasekhar 1983), (Kulkarni et al 1987), (Angirasa and Srinivasan 1992), (Pantokratoras 2003), (Saha and Hossain 2004), (Ahmed 2005), (Ishak et al 2008), (Deka and Neog 2009), (Singh et al 2010). Experimental works also had been reported; see (Tanny and Cohen 1998).Theoretical and experimental work had been investigated by (Chen and Eichhorn 1976). The present work investigates the problem numerically with wide range of stratification parameter for different kinds of fluids (air, water, and oil), different Grashof number and different inclination angle. To support the numerical solution the problem was solved also using ANSYS Program.

Formulation of the Problem
Consider the two dimensional thermal boundary layer flows natural convection heat transfer of an incompressible fluid along a plane wall immersed in a stable thermally stratified fluid. The coordinates system and the flow configuration are shown in figure 1. Using Boussinesq approximations, the following continuity, momentum and energy equations in nondimensional form for laminar flow adjacent to a plane wall are obtained; (1) where (Angirasa and Srinivasan 1992);   , , , The initial condition can be written in nondimensional form as follows: The boundary conditions in nondimensional form are: U = 0, V = 0, Ө = 0 at Y ∞ for X = 0 (8) The local rate of heat transfer in term of the local Nusselt number at the plate is given by; (9)

Numerical Solution
Finite Difference Method is considered as efficient technique to solve the thermal problems; therefore it has been used in the present study. The Momentum and Energy equation are solved by Alternating Direction Implicit Scheme (ADI). Numerical results were first obtained to check for grid dependency. The results showed that no considerable difference in the results of suggested grid size after (51x51) and showed that no considerable different in the results of suggested transverse distance after (0.5). Therefore in the present study the grid size of (51x51) and transverse distance of (0.5) was used. The convergence of the solution to the steady state result for large time was obtained with a convergence criterion of (1 x 10 -4 ). This criterion was chosen after varying it over a wide range so that the steady state results were essentially independent of the chosen value. The mathematical model was solved by computer program which was written by Visual basic language to solve the momentum and energy equations and to calculate Nusselt number. The Tridiagonal system of equation was used to solve the matrix of dependent variables. Heat transfer process by natural convection in stratified media was also solved by Mechanical ANSYS Parametric Design Language (APDL). The FLUID 141 element is used which can solve model of transient or steady state fluid/thermal systems that involve fluid and/or non-fluid regions.

Results and Discussion
Theoretical investigation are done for three working fluids, air (Pr = 0.7), water (Pr = 6) and oil (Pr = 6400), three Grashof numbers (1E4, 1E5 and 1E6) and three angle (-30, 0 and 30) for wide range of thermal stratification (S = 0, 0.5, 1, 1.5, 2, 3, 4). Figure 2 shows the temperature profile for the different values of the stratification level at mid high wall plane (X = 0.5). The temperature profile decreases with increasing the stratification parameter. For the higher value of stratification, the ambient temperature exceeds the wall temperatures, which lead to negative temperature profile. The figures also show that the temperature profile equal to zero at (S = 2) because of the equalization between the wall and ambient temperature. A comparative study of figures 2 to 4 indicates that the effect of stratification parameter is marginalized with the increase in Prandtl number, as the separateness among the temperature profile reduces. Also for given value of Prandtl number the velocity and thermal boundary layer thickness are almost the same while with the increase in Prandtl number the boundary layer thickness reduces. Figure 5 shows that the temperature profile decreases with the increase in Prandtl number. In addition, the reversal of temperature was found to be stronger at high Prandtl numbers and weaker at low numbers. It can be suitably remarked that the increase in Grashof number does not practically vary the effect of stratification factor on temperature profiles. Figures 6 show that with increase in Grashof number the fluid temperature deceases. This must happen because buoyancy force assists the flow by increasing fluid velocity and hence the heat is convected readily thereby reducing fluid temperature. Figure 7 illustrates the influence of the inclination angle (φ) on temperature profile for stratified media (S = 2), where observed that in addition to the influence of thermal stratification the temperature profile will be less effected by the inclination angle of the wall, this is considerably noted for high levels of thermal stratification, therefore the orientation marginalized the effect of the stratification parameter. Figure 8 shows that the velocity profile decreases with increasing the stratification parameter. It is understood, since the factor (T w -T ∞,x ) reduces with the increase in stratification factor, thus buoyancy effect very close to the plate is marginalized thereby reducing the fluid velocity. A comparative study of figures 8 to 10 indicates that the effect of stratification parameter is marginalized with the increase in Prandtl number, as the separateness among the velocity profile reduces. Figure 11 shows that the velocity profile decreases with the increase in Prandtl number. At high Prandtl numbers there is a small reversal of flow while for low Prandtl numbers the flow reversal is much stronger. It can be suitably remarked that the increase in Grashof number does not practically vary the effect of stratification factor on velocity profiles. Figure 12 shows that with increase in Grashof number the fluid velocity increases. This is because the buoyancy force assists the flow by increasing fluid velocity. Figure 13 shows that the velocity profile decreases with increasing the Grashof number. This phenomenon is clear at high Prandtl number which lowers fluid velocity. Figure 14 illustrates the influence of the inclination angle (φ) on velocity profile for stratified level (S = 2), where observed that in addition to the influence of thermal stratification the velocity profile will be less effected by the inclination angle of the wall, this is considerably noted for high levels of thermal stratification, therefore the orientation marginalized the effect of the stratification parameter. The effect of the stratification parameter is to reduce the Nusselt number. Nusselt number is equal to zero at any location of the plane wall when the wall temperature equal to ambient temperature. This equalization is result of stratification level. The figure 15 shows the deceases in Nusselt number with the stratification parameter because of the Nusselt number dependence on the temperature profile which decreased with increase in stratification parameter as mentioned above. As the Prandtl number increases the Nusselt number first deceases, then increases. An increase in Prandtl number is found to cause a decrease in thermal boundary layer thickness and an increase in the absolute value of the temperature gradient at the surface. In unstratified media the local Nusselt number for air is higher than for water and less than for oil as shown in figure 16. In stratified media, the Nusselt number has the same behavior of the unstratified environment. Nusselt number is dependent on many variables, one of these variables Grashof numbers which affects the heat transfer behavior from fluid to another. The fluids which have small Prandtl number, Nusselt number decreases with increasing the Grashof number. The reverse behavior is for fluids which have large Prandtl number where the Nusselt number increases with increasing Grashof number as shown in figure 17.The Nusselt number has the same behavior in stratified and unstratified environment.
Consider an inclined hot plate that makes an angle (φ = 30) from vertical wall plane. The difference between the buoyancy and gravity force acting on a unit volume of fluid in the boundary layer is always in the vertical direction. In the case of inclined plate, this force can be resolved into two components, the parallel force drive the flow along the plate and the normal force on the wall plane. The force that drives the motion is reduced; therefore the convection currents to be weaker and the rate of heat transfer to be lower relative to the vertical plane case. In the case (φ = -30) the opposite behavior is observed. The reason for this behavior is that the normal force component initiates upward motion in addition to the parallel motion along the wall plane, and thus the boundary layer breaks up and forms plumes. As result, the thickness of the boundary layer and thus the resistance to heat transfer decreases, and the rate of heat transfer increases relative to the vertical orientation. In the stratified media, the Nusselt number has the same behaviors of the unstratified media until temperature defect occurs, then the Nusselt number has the opposite behaviors as shown in figure 18.