Heat Transfer on Flow Past a Linearly Vertical Accelerated Plate With Constant Temperature and Variable Mass Diffusion

Heat Transfer on Flow Past a Linearly Vertical Accelerated Plate With Constant Temperature and Variable Mass Diffusion

1 M Sundar Raj, 2G Nagarajan

1,2Department of Mathematics, Panimalar Engineering College, Chennai 600 123 , Tamilnadu, India. sundarrajkani@gmail.com, sridin_naga@yahoo.co.in

Abstract-An analysis of flow past a linearly accelerated infinite vertical plate is offered in the presence of variable mass diffusion with constant temperature. The temperature of the plate is raised to Tw and species concentration level closer to the plate rises linearly with respect to the time. The non-dimensional governing equations are solved by Laplace-transform methods. The effects of concentration , temperature and velocity are calculated for different parameters like thermal Schmidt number , Prandtl number , Grashof number, mass Grashof number and time. It shows that the velocity

considered in horizondal direction . At time t 0 , the plate and fluid are at the same temperature T . The plate is linearly accelerated at time t 0 , with a velocity u u0t in its own

plane and the temperature of the plate is raised to Tw and the mass is diffused to the fluid from the plate with respect to time

1. Then by Boussinesq's approximation the governing equations

   of unsteady flow are as follows:

   x

   increases due to increasing values of thermal Grashof or mass Grashof number. It is also observed that the velocity increases with decreasing values of the Schmidt number.

   Keywords: accelerated, constant temperature, isothermal, vertical plate, heat transfer, mass diffusion.

   u0

   C'w

   Tw

   g

   U

   C

   C '

   T

   B0 y

   1. INTRODUCTION

      The effects of heat and mass transfer plays an important role are spacecraft design, solar energy collectors, filtration processes, nuclear reactors the drying of porous materials in textile industries and the saturation of porous materials by chemicals, design of chemical processing equipment and pollution of the environment. Effects of mass transfer on flow past a uniformly accelerated vertical plate was studied by Soundalgekar[1]. The above problem

      was extended to include heat and mass transfer effects subjected to variable suction or injection by Kafousias and Raptis[2]. It is proposed to study the effects of on flow past a linearly accelerated isothermal infinite vertical plate in the presence of variable mass diffusion with constant temperature. The dimensionless governing equations are solved using the Laplace-transform technique. The solutions in terms of error complementary function and also exponential form.

   2. MATHEMATICAL FORMULATION

      The flow of a incompressible fluid past a linearly vertical accelerated infinite plate with constant temperature and variable mass diffusion has been considered. The x -axis is taken in the vertical direction along the plate and the y-axis is

      Figure: Physical model of the problem

      7

      ( t = 0.2,0.4,0.6),Gr = Gc = 5 at t = 0.2 are studied and

      The initial and boundary conditions in non- dimensional form are

   3. METHOD OF SOLUTION

      The non-dimensional governing equations (4) subject to the conditions (5) are solved by Laplace method and we get the following solutions

      presented in fig 1. It is observed that the velocity increases with increasing values of t. Fig 2. demonstrates the effects of different thermal Grashof number (Gr =2,5) and mass Grashof number (Gc=2,5) on the velocity at time t = 0.2. It was observed that the velocity increases with increasing of the thermal Grashof or mass Grashof number.

      Fig 3 represents the result of concentration at time t=0.2 for varies Schmidt number (Sc=0.16, 0.3,0.5,2.01). The result of concentration is important in concentration field. The profiles have the common feature that the concentration decreases in a monotone fashion from the surface to a zero value far away in the free stream. It was observed that the concentration increases with decreasing values of the Schmidt number.

      where,

      = Y

      2 t

   4. RESULTS

      In sequence to get a physical problem, numerical computations are carried out for different parameters Gr, Gc, Sc, Pr and t upon the nature of the flow and transport. The value of the Schmidt number Sc is taken to be 0.6 which corresponds to water-vapor. Also, the values of Prandtl number Pr are chosen such that they represent air (Pr =0.71) and water (Pr =7.0). The values of the concentration ,velocity and temperature are calculated for different parameters like Prandtl number, Schmidt number

      ,thermal Grashof(Gr), mass Grashof number(Gc) and time. The velocity profiles for different Sc = 0.6, Pr = 0.71

      The result of velocity for varies values of the Schmidt number (Sc=0.16, 0.3,0.6,2.01) ,Gr=Gc=5 and time t=0.2 are shown in fig 4. The trend shows that the velocity increases with decreasing Schmidt number . It was observed that the variation of the velocity with the magnitude of the Schmidt number.

      The temperature profiles are calculated for water and air from Equation (6) and these are shown in Fig 5. at time t = 0.2. The result of the Prandtl number plays an important role in temperature field. It was observed that the temperature increases with decreasing Prandtl number. This shows that the heat transfer is more in air than in water.

   5. CONCLUSIONS

The effects of Heat and Mass transfer on flow past a linearly accelerated infinite vertical plate in the presence of variable mass diffusion have been studied. The non- dimensional governing equations are solved by the usual Laplace-method. The result of concentration ,velocity and temperature and for various parameters like thermal Grashof (Gr), mass Grashof number(Gc), Schmidt number(Sc), Prandtl number(Pr) and time (t) are studied graphically. The study concludes that the velocity increases with increasing values of thermal Grashof number(Gr), mass Grashof number(Gc) and time (t). But the velocity increases with decreasing Schmidt number (Sc). The wall concentration increases with decreasing values of the Schmidt number.