Heatline And Isothermallines Visualization Of Forced Convection In A Porous Medium With Radiation And Varying Wall Temperature

Heatline And Isothermallines Visualization Of Forced Convection In A Porous Medium With Radiation And Varying Wall Temperature

T. Raja Rani

Higher College of Technology, Muscat, Oman

Abstract

The aim of the present paper is to study the pattern of Isothermal lines and Heat lines in a forced convection problem at a horizontal plate in a porous medium under the effect of radiation and varying wall temperature. The temperature of the plate is assumed to vary along the length of the plate as a power function of distance. A similarity transformation is used to reduce the partial differential equations governing the problem into ordinary differential equations and the equations are solved analytically subject to appropriate boundary conditions to find the temperature function, Heatfunction and hence draw the related Isothermallines and the Heatlines. Heatlines and Isothermallines are drawn for both hot wall and cold wall cases. With the effect of radiation and varying wall temperature, significant patterns of Heatlines and Isothermallines are displayed in the discussion. The lines are drawn by the use of

MATHEMATICA.

Key Words: Forced Convection, Darcy model, Isothermal lines, Heatlines and Radiation.

1. Introduction

   During the last three decades much insightful work has been done on Heatlines, Streamlines and Masslines. These tools are used to visualize the heat flow and mass transfer occurring in the boundary layers. One can as well find more complete picture of involved transport phenomenon as well as the true conditions where the boundary layer hypothesis is applied. Heatlines and Streamlines are found to be adequate to visualize and understand heat energy distribution and thermal mixing occurring inside or near a boundary layer of the given geometry.

   The Heatline concept was first introduced by Kimura and Bejan [6] and appeared for the first time in a book on convective heat transfer by Bejan [1]. Prior to that, convective heat transfer process was analyzed using mainly the isotherms. On the other hand, the adequate visualizing tools in fluid dynamics are the streamlines and not the isobars, the same occurring in the field of convective heat transfer, where the adequate visualizing tools are the Heatlines and not Isotherms as pointed out in Costa [3]. For the first time Heatline visualization of forced convection in saturated porous media was

   presented by Morega and Bejan [8]. The development of the similarity version of the Heatfunction for the laminar natural convection near a vertical wall, isothermal or under constant heat flux was presented for the first time in Costa [4]. The streamlines and Heatlines are the most effective ways to visualize the paths followed by mass and heat flowing in two-dimensional problems without source terms ( refer Costa [3]). In Dash [5] Heatline concept was also applied when dealing with heat transfer in turbulent flows through the introduction of turbulent fluxes into the Heatfunction differential equation. If the heat transfer problem under consideration occurs in the turbulent boundary layer near a wall, the Heatline concept can be applied in a simpler way, by using an effective diffusion coefficient for heat, which includes the eddy diffusivity effect of the turbulent transport (refer Bejan[2]). Heatline approach for visualization of heat flow and efficient thermal mixing with discrete heat sources was studied by Ram Satish et.al. [11]. Raja Rani & CNB Rao[10 ] studied the effect of radiation and magnetic field on mixed convection at a vertical plate embedded in a porous medium with variable fluid properties and varying wall temperature. In recent works Pawan and Supot [9] studied a natural convection flow in a rectangular enclosure filled with porous medium when the walls are insulated except the left wall which is non- uniformly heated.

   0

   The aim of the present paper is to investigate the pattern of Heatlines and Isothermallines in a forced convection problem at a horizontal plate in a porous medium under the effect of radiation and varying wall temperature. The temperature of the plate T0 is assumed to vary as a power function of distance along the plate as, T T A x , where T is the temperature of the ambient fluid, A is constant and is a real number.

   The hot wall T0 T can be obtained for A positive

   and the cold wall T0 T can be obtained for A is

   where U is free stream velocity parallel to the plate, which is taken as U B x where B is a constant. A

   negative. The physical coordinate system of the problem is presented in fig.1. The governing partial differential equations are transformed into ordinary differential equations by applying similarity transformation and the equations are solved analytically subject to appropriate boundary conditions to find the Heatlines and Isothermallines. Present work has agreed well with Morega and Bejan [8] for Heatlines of both cold wall as

   well as hot wall for isothermal plate and without

   stream function , Peclet number Pex , non-

   dimensional functions f , ; a similarity variable and radiative flux qr are introduced through the relations (2.7) given below.

   It may be noted that the flow and temperature fields in the thermal boundary layer region are expressible in

   radiation effect 0 & Rd

   0.

   closed form by introducing

   u U

   and v=0 .

2. Formulation and Solution

   Let a semi infinite plate be embedded horizontally in a porous medium saturated with an incompressible homogeneous fluid. The fluid is assumed to be a gray

   (2.6)

   Pex

   U x

   m

   medium that emits, absorbs but do not scatter thermal radiation. It is also assumed that radiation from the fluid is only taken in to consideration which is present

   in the form of unidirectional flux, transverse to the

   f

   1

   x

   m

   Pe 2

   horizontal wall. Let x-axis be taken along the wall and

   T T

   (2.7)

   y-axis perpendicular to it. The orientation of the wall is shown in fig-1. The equations governing the forced

   T0 T

   y 1

   convection boundary layer flow for the Darcy model are written as:

   Pe 2

   x

   x

   16

   T 3 T

   u v

   q s

   r

   x y 0

   (2.1)

   3 kc y

   P u 0

   x K

   (2.2)

   Boundary conditions can be easily written in terms of

   ,

   at 0, 1, f 0,

   P v 0

   (2.3)

   as , 0, f 0

   (2.8)

   y K

   T

   T

   2T q

   cp u

   x v

   y km

   y2

   • r

   y

   (2.4)

   By eliminating fluid pressure from (2.2) and (2.3) and introducing (2.7) and (2.8), the governing equation is obtained as

   where u, v are fluid velocity components, T is fluid temperature, K is Permeability, km is effective thermal

   f 0

   (2.9)

   conductivity of the porous medium, qr

   is radiative heat

   Equation (2.9) can be integrated once using the

   flux, transverse to the vertical plate. The Rosseland approximation is used in the energy equation to describe the thermal radiative heat transfer. It may be noted that by the use of Rosseland approximation, the applicability of the present analysis is limited to optically thick fluids

   only

   condition on f at to get

   f 1

   f

   1

   * x * yPeL 2

   (2.10)

   (2.11)

   Now x

   , y L

   are introduced, where

   L

   The appropriate boundary conditions are:

   U L

   . (2.12)

   0

   T T Ax , v 0, at y 0, x 0

   PeL

   T T , u U , as y , x 0

   (2.5) m

   To find the temperature function and Heatfunction for boundary layer near a hot varying wall temperature plate, equation (2.4) is rewritten as

   H * (x* , y* )

   1

   * 1

   1 2 2

   2 1

   2

   3 2

   x 2 Erfc 211.333 Rd

   exp 211.333Rd

   cp u

   T Tref

   x

   • v T Tref

   y

   km

   T Tref

   y2

   16 sT

   3kc

   T Tref

   y2

   (2.13)

   1. Boundary Layer near a hot varying wall temperature plate:

      (2.19)

   2. Boundary Layer near a cold varying wall temperature plate:

   Tref

   T

   for hot varying wall i.e., T0

   T

   . In this

   In this case the transfer of heat is from fluid to the

   wall. Here the wall is colder than the porous medium and

   case the transfer of heat is from wall to fluid and also it can be observed that the pattern of heat lines will

   introducing (2.6), (2.8) and (2.12) in (2.13) one can get

   differ from hot wall case. Here to find

   H * x* , y* ,

   1

   Erfc 1 2

   1

   (2.14)

   Tref T0 is taken and a procedure similar to that of the above is used. In this case

   21 1.333Rd 2

   * * *

   The form of equation (2.4) demands a special definition for Heatfunction, H which is valid inside the boundary

   H (x , y )

   1

   layer region and

   * 1

   1 2 2

   2 1

   H

   x 2 Erf 211.333 Rd exp 211.333Rd

   y c p u T Tref

   (2.15)

   • H

   x

   c

   vT T

   k

   T

   m y

   (2.16)

   ref

   1

   p

   1

   (2.20)

   and

   Erf 1 2

   (2.21)

   1

   where H * H

   (2.17)

   21 1.333Rd 2

   L

   km T0 T Pe 2

   H x, y can be easily obtained from (2.15) and (2.16)

   and it satisfies the energy equation (2.4), provided Tref is a constant. By using (2.6), (2.14) and (2.17) it can be seen that

   By taking = constant, Isothermallines can be drawn and by taking H * (x* , y* ) = constant, Heatlines can be drawn.

3. Solution of the Problem

   H *

   y*

   () and

   H *

   x*

   y*

   (2.18)

   1. Parameters of the Problem and Their Effect on heatlines and isothermallines

      Equation (2.18) can be integrated by using (2.6) and (2.14) to get

      1

      The flow and heat transfer depend on the parameter

      and Rd where is power of index of the wall temperature and Rd is the radiation parameter. T0 T

      H * (x* , y* ) x* 2 2

      corresponds to hot wall and T0 T corresponds to

      cold wall respectively. To determine certain important values for , the total heat convected in the flow, Q(x) at any downstream location x is considered.

      Q(x) c p T T u dy

      0

      This can be seen to be proportional to

      3 1

      x 2

      . For

      uniform heat flux surface, Q(x) should vary linearly with

      1

      x and so

      . For an adiabatic surface, Q(x) can be

      3

      1

      independent of x and so

      3

      . Zero value of

      corresponds to the isothermal case. In this study solutions are found for the values of -0.3, 0, 0.5 and 1.0 of .

      When transfer of heat energy through radiation is neglected, the parameter Rd takes zero value and for increasing intensity of the thermal radiation, the parameter takes larger values. Solutions are found for 0.5 and 5 of the parameter Rd, whereas the intensity of thermal radiation decreases in case of -0.5 of Rd. As increases the value of heatfunction increases in case of both hot wall as well as cold wall.

      The effects of simultaneous variation of the values of the parameters on the heatlines and isothermal lines are presented in the discussion.

   2. Discussion of the Results

   The visualization of both heatlines and isothermallines for both hot wall and cold wall are presented in the Figures 2-20. Heatlines near a cold wall case are presented for different values of parameters of and Rd in the Figures- 2to 8. Heat lines near a hot wall are presented for different values of parameters of and Rd in the Figures-9 to 14. Isothermallines for both cold and hot wall cases for different values of and Rd are presented in Figs- 15 to 20.

   The pattern of heatlines of Fig-2 and Fig-9 have well agreed with Morega and Bejan [8] for =0 and Rd=0. Heatlines for an isothermal wall are presented in Fig-2 to

4. For an isothermal wall as the values of Rd increase from 0 to 0.5 the values of heatfunction also increase.

Dimensionless heatfunction can be seen to increase from heat lines of figures 2, 3 and 4. The heatlines can be seen to be perpendicular to the plate near the plate while they are not so at distances away from the plate. Since

the wall is colder than the distant medium and Tref T0 ,

it follows that at the wall H

y

0 . The perpendicularity

of the heatlines at the wall is due to the fact that

Tref T0 . From Fig.3 it can be observed that as the heatfunction increases from H*=0.3 at the tip to H*=1.18 at the trailing edge. One can find same heatline pattern in Fig.2 and Fig.4.

As increases from 0 to 0.5 heatfunction increases from H*=1.13 to H*=1.22. Also it can be observed from Fig-7 that the pattern of the lines for = – 0.3 are entirely different from =0 and 0.5 as it is independent of x as already discussed in §3.1.1 and the heat function increases to 1.32. From Fig.8 it is observed that in this case denser heatlines are not found and also the value of the heatfunction nearer to x* = 0 is more as compared to that of the other figures.

The pattern of heatlines near an isothermal hot wall can be visualized in Figs-9 and 10. As radiation effect increases (as Rd increases) the heatfunction decreases. This is because the wall is warmer than the distant

porous medium ie.,

T0 T

. The pattern of the

heatlines for hot wall is just opposite to cold wall with radiation effect. From Figures 9 and 11 it is observed that as increases, heat function increases as in cold wall case. The denser heatlines near x*=0 indicate the higher fluxes near the leading edge of the heated section of the boundary. As x* increases the wall heat function H*(x*,0) decreases due to the fact that wall loses heat to the boundary layer in case of hot wall. The total heat released by wall can be seen from the heatlines at the trailing edge of y*.

The pattern of heat lines are just opposite to each other for hot wall and cold wall. The values of the heatfunction also change significantly between hot wall and cold wall cases.

The line along which temperature is a constant is known as an isothermal line. The pattern of the isothermal lines can be seen in Figures 15 to 20. Comparing Fig.15 & Fig.16 it can be observed that the temperature () values are just opposite but not negative as it satisfies the equations 2.14 and 2.21.

Conclusions:

In this paper heatlines and isothermallines for both the hot wall as well as cold wall are presented. As the effect of radiation increases, heatfunction deceases in cases of both the hot wall and cold wall. But as (that signifies varying wall temperature) increases value of the heat function also increases for both cold wall as well as hot wall. Significant pattern of heatlines and isothermallines can be visualized from the figures.

Acknowledgements:

T.Raja Rani wishes to thank Prof.C N B.Rao, for his valuable suggestions to complete this work and also wishes to thank the authorities of Higher College of Technology, Muscat for their encouragement and also for providing the facility for research.

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