Heat Transfer of Two Immiscible Fluids Flow Through a Porous Channel

Heat Transfer of Two Immiscible Fluids Flow Through a Porous Channel

Abdul Mateen

Department of UG & PG Studies in Mathematics Government Womens First Grade College, Gulbarga, Karnataka, India

Abstract – An analysis is performed to study two viscous immiscible fluid flow and heat transfer through composite porous medium between two pa. The flow is modeled by Darcy-Brinkman equation with suitable boundary and interface conditions. The partial differential equations governing the flow and heat transfer have been transformed into a system of ordinary differential equations. The obtained equations are solved analytically taking the viscous and Darcy dissipation into consideration. The effect of the variation in the porous parameter, Prandtl number, Eckert number, ratios of viscosities and thermal conductivities on the velocity and temperature fields for both the fluids is discussed.

Keywords: Heat transfer,porous channel, immiscible fluids, viscous dissipation.

1. INTRODUCTION

Investigation of the flow through porous geometries has many scientific and engineering applications, such as in the utilization of geothermal energy, underground disposal of nuclear waste material, high-performance building insulation, crude oil extraction, petroleum industries, chemical catalytic reactor, solid matrix exchangers and many others. Berman [1] investigated the flow of twodimensional steady incompressible laminar viscous fluidsthrough a porous channel where both channel plates haveequal permeability and the flow at the center line of thechannel attains maximum. Later Yuan [2] extended theproblem for different values of suction and injection Reynoldsnumbers. The combined natural and force convective flows through a horizontal porous channel connecting two reservoirs have been investigated by Haajizzadeh and Tien [3]. Steady and transient Magnetohydrodynamic flow and heat transfer in a porous medium channel has been analyzed by Chamkha [4]. Mankinde and Mohane [5] investigated the combined effect of a transverse magnetic field and radiative heat transfer to unsteady flow of a conducting optically thin fluid through a channel filled with the saturated porous medium with no-slip boundary condition, further the work is extended by Mehmoodand Ali [6] by considering the fluid slip at the lower wall. Nield and Bejan [7], Vafai[8,9], Pop and Ingham [10], Ingham et al. [11] and Bejanet al.[12] have made

comprehensive reviews of the studies of heat transfer in relation to the above applications.

In all the above-mentioned research pertain to single fluid model. Most of the problems relating to the petroleum industry, geophysics, plasma physics, and so forth involve multiphase flow situations. The multiphase flow in porous media has attracted considerable attention for many researchers. This is due to the fact that problem involving the multiphase flow, heat transfer and multi-component mass transfer in porous media arises in engineering disciplines such as geothermal energy production, multiphase trickle bed reactors, high level radioactive waste repositories and paper machines. Multiphase flow in porous media has been approached by so called Multiphase Flow Model (MFM) (Abriola [13], Bear [14])in which various phase are considered as distinct fluids with individual thermodynamic and transport properties and with different flow velocities. The transport phenomena are mathematically described by the basic principles of convection for each phase separately and by appropriate interfacial condition between various phases. Srinivasan and Vafai [15] have reported a theoretical study on immiscible fluid systems in a porous medium, taking into account the non-Darcian boundary and inertia effects. Chamkha [16] analyzed the flow of two immiscible fluids in porous and non-porous channels. Two-fluid flow and heat transfer in an inclined channel containing porous and fluid layers was studied analytically by Malashetty et al. [17]. Umavathi et al. [18, 19] analyzed unsteady flow and heat transfer of immiscible fluids in a horizontal channel.Recently, Prathap Kumar

[20] has studied the effect of homogeneous and heterogeneous reactions on the dispersion of a solute in a composite porous medium between two parallel plates. The unsteady magnetohydrodynamic flow of two immiscible fluids in a horizontal channelbounded by two parallel porous isothermal plates in the presence of an applied magnetic and electric field is investigated by LingaRaju and Nagavalli[21], unsteady oscillatory flow and heat transfer of a viscous and micropolar fluid through a horizontal channel with heated walls are analyzed by ShaikMeera and Abdul Mateen [22].

The object of the present work is to study flow and heat transfer of immiscible fluids through the porous

media in which the differential equation governing the fluid motion is based on the Darcys law which accounts for the drag exerted by the porous medium. The two plates are maintained at two different but constant temperatures. This configuration is a good approximation of some practical situations such as heat exchangers and pipes that connect system components. The cooling of these devices can be achieved by utilizing a porous surface through which a coolant, either a liquid or gas, is forced. Therefore, the results obtained here are important for the design of the wall and the cooling arrangements of these

Region-I

1 2 0

2u1 p

y x

2T u 2

K1 1 1 1 0

y2 y

Region-II

(1)

(2)

devices.

II. MATHEMATICAL FORMULATION

2u2

2 y2

p 2

x s

u2 0

(3)

2T u 2

K2 2 2 2

2 u2 0

(4)

y2

Y

y s 2

o o

o o

o o

o o o o

o o

o o

o o

o o o o

h

oo o

o o o

W1

T

Rgion-I

X

o o Rgion-II

o o

o

o

o o o o o

where u is the x-component of fluid velocity and T is the fluid temperature. The subscripts 1 and 2 correspond to region-I and region-II, respectively. The boundary conditions on velocity are the no-slip boundary conditions which required that the x-component of velocity must vanish at the wall. The boundary conditions on temperature are isothermal conditions. We also assume the

o o

o o o o o o o oo o o o o

continuity of velocity, shear stress, temperature and heat

o o

Porous

o o o o

o o

-h

o o o o

T

W2

flux at the interface between the two fluid layers at y=0.

material Figure 1. Physical configaration

The boundary and interference condition for this problem is written as

Consider the flow of an incompressible viscousfluid in a horizontal composite channel as shown in

u1 0,

u2 0,

T1 Tw1 at T2 Tw2 at

y h y h

Fig. 1

u1 u2 ,T1 T2 at

y 0

(5)

In order to derive basic equations for the problem under consideration following assumptions are made:

u1

1 y

2

u2 at

y

y 0

1. The flow is steady, laminar and fully developed with constant physical properties.

   K T1 K

   1 y

   2

   T2 at

   y

   y 0

   0

   i

2. The region-I ( 0 y h ) is occupied by a clear viscous fluid and the region-II ( h y 0 ) is

   By use of the following non-dimensional quantities:

   filled with a porousmatrix.

   ui U u*,

   y hy*,

3. All the physical dependent variables except

   Twi Tw2

   2 p

   pressure will only dependent of y.

   i

4. The plates of the channel are assumed to be finite and maintained at constant and different

   Tw1 p

   Tw2

   , ,

   s

   p

   (6)

   P U x ,

   temperatures Tw1 and Tw2 .

   1 0

5. Viscous and Darcy dissipation terms are included in this study.

   Pr

   1Cp

   K1

   U 2

   , Ec 0

   Cp (Tw1 Tw2 )

6. The flow considered in both the regions is driven

by a common pressure gradient p and

x

temperature gradients Tw1 Tw2 .

and for simplicity dropping the asterisks, equations (1) to

(5) becomes

Region-I

d 2u1

Under these assumptions and taking,

dy2

d 2

P 0

du 2

(7)

CP1 CP 2 CP

and

1 2 0

the governing

1 Ec Pr 1 0

(8)

equations of motion and energy are:

dy2

dy

Region-II The effect of the viscosities ratio on the velocity

d 2u2

dy2

2u2 P 0

(9)

and temperature distributions is shown in Figs 7 and 8, respectively. As the viscosity ratio increases, both the velocity and temperature profile decreases. This is due to

d 2

Ec Pr

du

2

the fact that as the viscous effects increase, the fluids in

2

dy2

2u2 2 0

2 dy

(10)

both regions become thicker and hence, the flow velocity in the channel is reduced causing the temperature

where 1

2

is the ratio of viscosities and

K1 is

K2

distribution in the channel to reduce as well.

the ratio of thermal conductivities.

Nomenclature

The boundary and interference condition in the CP

specific heat at constant pressure

non-dimensional form becomes

Ec Eckert number

u1 0, 1 1 at u2 0, 2 0 at u1 u2 ,1 2 at

u1 u2 , at

y y

y h y h y 0

y 0

(11)

h channel half width

K thermal conductivity

P pressure

Pr Prandtl number

1. permeability of porous matrix

   T temperature

   1 2 at y 0

   y y

   Tw wall temperature

2. time

   U0 average velocity

   Solution: The governing equations (7) to (10) are solved subject to the boundary and interface conditions (11) for the velocity and temperature distributions in both regions.

   III. RESULTS AND DISCUSSION

   In this section, a detailed parametric study has been performed for the effects of the porous parameter, Prandtl number, Eckert number, ratios of viscosities and thermal conductivities on the velocity and temperature profiles which is presented graphically in figures 2 to 8. The following parameters are fixed values P 2.0 ,

   2.0 , Pr 0.71 , Ec 0.2 , 1.0 and 1.0

3. velosity component along the plate

Greek letters

ratio of viscosities

ratio of thermal conductivities

porous parameter

fluid density

viscosity of fluid

non-dimensional temperature

Subscripts

1,2 quantities for region-I and region-II respectively.

except one the varying.

Figs. 2 and 3 display the effects of the porous medium parameter on the velocity and temperature profiles respectively. As the porous medium parameter

increases, the velocity and temperature decrease in both regions of the channel. This is expected since the porous matrix represents an obstacle to flow and therefore, reduces its velocity and temperature. This result is also similar to the case of fully developed flow through a porous medium as predicted by Rudraiahand Nagraj [23].

Figs. 4 and 5 respectively shows that an increase in Prandtl number Pr and Eckert number Ec clearly boost temperature in both the regions. Eckert number signifies the quantity of mechanical energy converted via internal friction to thermal energy. Increasing Ec values will

1.0

0.5

y 0.0

-0.5

-1.0

8.0

6.0

4.0

=2.0

Region-I

Region-II

therefore cause an increase in thermal energy contribution

to the temperature profiles.

0.0 0.4

0.8 1.2 1.6

u

Fig. 6 displays the influence of the thermal conductivity ratio on thetemperatureprofiles. Increases

in the thermal conductivity ratio have the tendency to cool down thethermal state in the channel. This is depicted in the reduction in fluid temperatures as increasesin both

the regions.

Fig. 2 Velocity profile for different values of

1.0

Region-I

1.0

Region-I

0.5 0.5

y 0.0

-0.5

Region-II

8.0

4.0

6.0

=2.0

y 0.0 Region-II

-0.5

2.0

1.0

0.5 = 0.25

-1.0

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 3 Temperature profile for different values of

-1.0

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 6 Temperature profile for different values of

1.0

Region-I

1.0

Region-I

0.5 0.5

2.0

1.0

0.5 = 0.25

y 0.0 Region-II

0.025

0.71

1.0

y 0.0

Pr=3.0

-0.5 -0.5

-1.0

0.0 0.5 1.0 1.5

Fig. 4 Temperature profile for different values of Pr

-1.0

Region-II

0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1

u

Fig. 7 Velocity profile for different values of

1.0

Region-I

1.0

Region-I

0.5

y 0.0

Region-II

Ec=0.1

0.2

0.3

0.4

0.5

y 0.0 Region-II

1.0

2.0 0.5

= 0.25

-0.5

-0.5

-1.0

0.0 0.3 0.6 0.9

Fig. 5 Temperature profile for different values of Ec

-1.0

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 8 Temperature profile for different values of

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