Free Convection over a Varying wall Vertical Cylinder embedded in a Porous medium with effect of Radiation, Variable Fluid Properties and Stratification

Free Convection over a Varying wall Vertical Cylinder embedded in a Porous medium with effect of Radiation, Variable Fluid Properties and Stratification

T. Raja Rani

Military Technological College, Muscat, Oman

Abstract

The aim of the present paper is to study free convection boundary layer flows over a vertical cylinder which is embedded in a porous medium with effects of Radiation, variable viscosity, variable thermal conductivity and stratification are studied numerically. Temperature of the lateral surface of the cylinder as well as the ambient temperature is assumed to be linear functions of the axial coordinate. A similarity transformation is used to reduce partial differential equations governing the problem into ordinary differential equations and equations are solved numerically subject to boundary conditions by the use of Runge-Kutta-Gill method with shooting technique.

Under the assumption of stable stratification similarity solutions are presented for the problem. Solutions for an unstratified medium are obtained as a special case by assigning zero value to the stratification parameter

1. Qualitatively interesting graphs are discussed as functions of parameters of the problem.

   1. Introduction

      The study of free convection from the outer surface of a heated vertical cylinder embedded in a fluid saturated porous medium has important geophysical and engineering applications (refer Nield and Bejan [7]). As quoted in the reference, such studies will aid in the assessment and evaluation of geothermal resources during geophysical exploration. Further, the heat transfer coefficients of such a study will be useful in estimating the cooling rate of intrusive bodies and consequently the life span of a geothermal reservoir. It will also be useful to calculate the heat loss from underground casing and piping systems for the optimum design of geothermal power plants.

      In many practical situations the region of flow and the whole configuration will be confined to a finite region of space and so the heated boundary layer eventually hits the ceiling. This heated layer remains at the top of gradually colder layers and this leads to thermal stratification (refer Bejan[ 2]).

      Minkowycz and Cheng [6] discussed free convection at a vertical cylinder embedded in a porous medium. Cheng and his co-workers, Cheng and Minkowycz [4], Cheng and Chang [3] have obtained similarity solutions for free convection in a porous medium adjacent to vertical and horizontal plates with wall temperatures being a power function of distance. Hossain and Nakayamma [5] discussed non- Darcy free convective flow along a vertical cylinder embedded in a porous medium with surface mass flux. However, in the above studies the effect of stable ambient stratification on heat and mass transfer was not discussed. It is known that stable ambient stratification phenomenon usually occurs in cooling ponds, lakes, solar ponds and atmosphere. In view of this, several researchers have discussed the effect of stratification on different heat transfer studies. Takhar.et.al.,[10] discussed natural convection at vertical cylinder in a thermally stratified high porosity medium. Saha and Hossain [9] discussed the natural convection flow with combined buoyancy effects due to thermal and mass diffusion in thermally stratified media. Angirasa and Srinivasan [1] studied the natural convection flows due to the combined buoyancy of heat and mass diffusion in a thermally stratified medium. Raja Rani and CNB. Rao [8] studied the effects of variable fluid properties on free Convection Flows and heat transfer at an isothermal vertical plate embedded in porous medium in the presence of magnetic field and radiation.

      In the present paper the free convection flow about a vertical cylinder embedded in a saturated porous medium is discussed. Here the temperatures on the surface of the cylinder as well as the ambient temperature are assumed to vary linearly with x, where x is the distance from the leading edge of the cylinder. It is also assumed that ambient temperature is less than the temperature of the cylinder and, as a result, stable stratification occurs. However in an unstratified medium, the ambient temperature can be lower or higher than the temperature of the cylinder. In the present problem possible cases of hot cylinder and cold cylinder are considered in the case of an unstratified medium. The effects of radiation, stratification,

      variable viscosity and variable thermal conductivity on free convection at the vertical cylinder are discussed. Because of linear variation of temperatures of the cylinder and that of the ambient with the axial coordinate, similarity solutions exist for the problem.

   2. Formulation and Solution

      P Ps

      x s

      P v 0

      r K

      T T

      g u 0

      K

      1 T

      qr

      (2)

      (3)

      Let a vertical cylinder of radius a immersed in

      cp u x v r r r km r r r

      (4)

      a saturated porous medium (the physical model is shown in fig.1). Origin is chosen at the centre of cylinder at its leading edge where the boundary layer

      Boundary Conditions are:

      thickness is zero. The axial coordinate ( x ) is taken vertically along the length of the cylinder and the radial

      at r a, v 0, T Tw T0 Ax , x 0

      coordinate (r) perpendicular to it. Temperature of the

      as r , u 0, T T T0 Bx , x 0

      (5)

      surface of the cylinder Tw is assumed to vary

      where u, v are fluid velocity components, T is fluid

      linearly with the distance x from its leading edge. The

      temperature, K is Permeability, km is effective

      temperature of the ambient fluid T is also assumed to vary as a linear function of x. Stable stratification occurs under the assumption that Tw T (hot plate). Temperature of the fluid at the leading edge is taken as T0 which is the minimum temperature of the configuration. Viscosity of the fluid and effective thermal conductivity of the fluid saturated porous medium is assumed to vary as linear functions of temperature. Other fluid properties are assumed to be constant except for density which is taken to be a function of temperature only in the body force term one of the momentum equations.

      thermal conductivity of the porous medium and is

      the dynamic viscosity, qr is radiative heat flux and the Rosseland approximation used in the energy equation to describe the thermal radiative heat transfer.

      In fact the radiative heat flux qr is given

      4 T 4

      by q s , where the Stefan-

      e

      e

      r 3k y s

      Boltzmanns constant and ke is the mean absorption coefficient. Assuming temperature differences within

      qr

      the flow to be sufficiently small, the term of

      y

      equation (4) is simplified by expanding

      T 4 into the

      Taylor series about T , and neglecting higher order terms. It may be noted that because of the use of Rosseland approximation, the present analysis gets limited to optically thick fluids.

      Taking

      [1 (T T )]

      in the

      The governing equations of the Darcy model boundary

      body force term, and eliminating fluid pressure from equations (2) and (3), the governing equations are obtained as

      layer free convection flow in cylindrical coordinates are

      T

      T 1

      T q

      as follows:

      cp u

      v km r

      r (6)

      rv ru

      x r r r r r

      r x 0

      (1)

      A stream function is introduced through the relations

      u 1

      & v 1

      (7)

      becomes a stably stratified medium. However S =0

      r r

      r x

      corresponds to an unstratified medium in which case,

      Tw need not be greater than T .

      Introducing non- dimensional variables

      , f ,

      and

      The case when the parameters , k take

      parameters o through the relations

      non-zero values is referred to as the variable fluid property case, or in short VFP case. The case when the

      1 1

      K g A 2 r 2 a 2 2

      parameters

      , k take zero values is referred as the

      2 a

      constant fluid properties case, or in short CFP case.

      m

      1

      When , k take non-zero values, four cases arise (i)

      m K g A 2

      0, 0

      (ii)

      0, 0

      (iii)

      f

      ax f ()

      k k

      1

      0, k 0 (iv) 0, k 0 ( Refer [8]). As

      2 f m 2

      a result, depending on the numerical values of

      0 a K

      T T

      g A

      (8)

      and k , the results become applicable to certain fluids in specific ranges of temperatures.

      Tw T0

      3

      3

      F k f ke

      4 T

   3. SOLUTION OF THE PROBLEM

3.1.1 Parameters of the problem

s

4 T 3

The parameters of the present problem are S,

Rd s

kke

Rd, , k , 0 . The parameter S is the stratification

The governing equations in non-dimensional form are obtained as

parameter. As pointed out earlier, this parameter assumes values between 0 &1 and hence the medium will be a stably stratified medium. S=0 corresponds to an unstratified medium. Rd is the radiation parameter.

1 1 f

f

(9)

Zero value for the parameter Rd corresponds to the case

2

2

when the heat energy through radiation from the fluid is neglected. Positive numerical values of Rd correspond to intensity of thermal radiation from the

1

1 4 Rd 1

fluid. Solutions are found for the values of 0, 0.5 and

1 k

0

2

0

3

10 of the parameter Rd. Thermal radiation can causes thickening of the thermal boundary layer and hence

1 2

1

2 Rd

f

increasing values of the parameter Rd can increase

k 0 1 k

f 0

0 0

2 3

(10)

thermal boundary layer thickness. In this paper solutions are found for the values of -1, 0 and 1 of both , k . Calculations are done for the value 0 &

Boundary conditions in terms of f and are:

0.5 for S. Here 0

is the curvature parameter. The

at 0,

f 0, 1 S

value infinity corresponds to flat plate case. Calculations are done for the values 0.25, 0.5 and 1

as ,

f 0, 0

B

(11)

of 0 .

1. Numerical Solution

   where

   S 1

   A

   is the stratification parameter.

   Equations (9) and (10) are solved subject to

   Because of the assumption that Tw T , it follows that

   0 S 1 and the fluid saturated porous medium

   the conditions (11) by the use of Runge-Kutta Gill method together with a shooting technique. Appropriate results of the present analysis are

   compared with the results of Mynkowycz & Cheng [6]. Some of the qualitatively and quantitatively interesting results are presented through figures 2 to 10 and through table 1.

2. DISCUSSION OF THE RESULTS

   Variations in skin friction

   f (0)

   are

   presented in figures 2 and 3; corresponding variations

   in wall heat transfer rate ( (0)) are presented in

   figures 4 and 5 respectively. Skin friction assumes negative values for all values of the parameters when 0 0.25. As the radiation parameter takes increasing values, absolute values of the skin friction can be seen to diminish. The diminishing trend is more pronounced in an unstratified medium rather than in a stratified medium. For fluids like Methyl chloride absolute value of skin friction in the variable fluid property (VFP) case will be larger than that in the constant fluid property (CFP) case. However for fluids like Dichloro Difluro methane, opposite is the behaviour of skin friction.

   From figure 3, it may be noted that skin friction doesnt vary with 0 when radiation parameter takes a constant value. The values however change with changing values of Rd. The variation in skin friction with changing values of the stratification parameter and

   From figures 4 and 5, wall heat transfer rate can be observed to take positive values for all values of the parameters. ' (0)' assumes larger numerical

   values in the absence of stratification than in its presence. ' (0)' Diminish with increasing values of

   the radiation parameter Rd. For small values of Rd, for fluids like Dichloro Difluro methane the heat transfer coefficient in the VFP case assumes smaller values than in CFP case. Opposite is the trend for fluids like Methyl chloride.

   with changing values of

   earlier.

   , k

   is as pointed out

   From Figure 5, it may be noted that, like skin friction, the heat transfer coefficient also doesnt

   change with changing values of of Rd.

   0 for a fixed value

   Variations in fluid velocity ( f () ) are presented in figure.6. Reduced velocities can be observed in a stably stratified medium than in an unstratified medium. Velocities near the plate are relatively larger in an unstratified medium than in a stably stratified medium. Effect of variable fluid properties on fluid velocity is more significant in unstratified medium than in a stably stratified medium.

   Variations in fluid temperature are presented in figures 7 and 8. Effect of increasing values of Rd and

   medium than in an unstratified medium. Also Thermal Boundary layer thickness is larger in a stratified medium than in an unstratified medium.

   Variations in stream function are presented in figures 9 and 10. As can be expected, stream function starts from zero value at the surface of the cylinder and asymptotically approaches a constant value far away

   from it.

   increasing values of 0 on fluid temperature is to

   increase temperature in the boundary layer region near the cylinder and, effect increasing values of S is to diminish the fluid temperature. From figure 8, in a stably stratified medium, for fluids like Dichloro Difluro methane temperature in the VFP case is larger than the one in the CFP case. However for fluids like Methyl chloride temperature in the VFP case is smaller than the one in the CFP case. Temperature in the vicinity of the cylinder is smaller in a stably stratified

   From figure 9, both in an unstratified medium and in a stably stratified medium, with increasing intensity of radiation and with increasing values of 0 , stream function assumes larger numerical values. The values of stream function are smaller in a stably stratified medium than in an unstratified medium.

   From figure 10 it may be noted that, the variations in the values of the stream function between CFP and VFP cases are significant and they can be interpreted in an appropriate fashion when S=0 and when S=0.5. When S=0.5, for fluids like Dichloro Difluoro methane stream function assumes larger values in the VFP case thn in the CFP case while for fluids like Methyl chloride opposite is the behaviour between VFP and CFP case

   Conclusions:

   1. Skin friction assumes negative values for all values of the parameters when 0 0.25 . As the radiation parameter takes increasing values, absolute values of the skin friction diminish. The diminishing trend is more pronounced in an unstratified medium rather than in a stratified medium.

   2. For fluids like Methyl chloride absolute value of skin friction in the variable fluid property (VFP) case will be larger than that in the constant fluid property (CFP) case. However for fluids like Dichloro Difluro methane, opposite is the behaviour of skin friction.

   3. Velocities near the cylinder are relatively larger in an unstratified medium than in a stably stratified medium. Effect of variable fluid properties on fluid velocity is more significant in unstratified medium than in a stably stratified medium.

   4. Effect of increasing values of Rd and

      increasing values of

      0 on fluid temperature

      is to increase temperature in the boundary layer region near the cylinder and, effect increasing values of S is to diminish the fluid temperature.

   5. When S=0.5, for fluids like Dichloro Difluoro methane, the stream function assumes larger values in the VFP case than in the CFP case while for fluids like Methyl chloride opposite is the behaviour between VFP and CFP cases.

Acknowledgements

Raja Rani wishes to thank the authorities of Military College of Technology, Muscat for their encouragement and also for providing the facility for research.

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