Heat transfer in a reacting power law fluid with thermal radiation over a flat plate.

Abstract

We examine the temperature field of a reacting power law fluid caused by the exothermic reaction of the fluid molecules as it flows in the presence of thermal radiation over a flat plate. Of particular interest are the effects of the thermal radiation parameter and the Frank-Kamenetskii parameter on the temperature field. And the result shows that for the flow of a reacting power law fluid over a flat plate in the presence of a thermal radiation, the quenching time for the reacting flow decreases with increases in the Frank-Kamenetskii parameter. On the other hand, the quenching time for the reacting flow increases with the increase of the thermal radiation parameter.

Subject Classification: Fluid Dynamics.

Keywords: Frank-Kamenetskii parameter, thermal radiation parameter, reacting power law fluid.

Introduction

The study of non--Newtonian fluid has been of much interest to scientist because some industrial materials are non--Newtonian; in food, polymer, petrochemical, rubber, paint and biological industries, fluids with the non- Newtonian behaviours are encountered. Of particular interest is power law fluid for which the shear stress i , is given by

[[tau].sub.yx] =-m [[absolute value of [partial derivative]u/[partial derivative]y].sup.n-1] [partial derivative]u/[partial derivative]y,

where,

m is the flow index

[partial derivative]u/[partial derivative]y is the shear rate n is the power law exponent. When n is less than 1, the fluid is pseudoplastic, for n equal to 1, the fluid is Newtonian and when n greater than 1, the fluid is dilatant. Some examples of a power law fluid are commercial carboxymethylcelllulose in water, cement rock in water, napalm in kerosene, lime in water, Illinois yellow clay in water. The studies of the flow of non-Newtonian fluids have over the past years attracted the keen interest of scientist. Basov and Shelukhin [1] introduced solution for the one-dimensional equations of Bingham compressible flow. This provided the possibility of describing a joint motion of rigid and fluid zones without incorporation of free boundaries corresponding to fluid rigid interfaces. A global unique solvability is proved.

Hakin [4] examined the existence of a slow steady flow of viscoelastic fluids (Maxwell type model, Jeffey's--type model). In the equations of conservation of momentum, the extra stress tension is decomposed into a Newtonian part (a solvent) and a purely elastic part (the polymer) which is related to the velocity field by a differential constitutive equation of Mute-Metzer type.

Vander Hout [8] investigated the linearized stability problem for fibre spinning of Newtonian and power law fluids in a one-dimensional isothermal setting, which is reduced to a single volterra integral equation. A rigorous linear stability criterion is presented with computed corresponding critical stretch ratios, using method of complex analysis.

Pascal [7] investigated the spread of a gravity current consisting of a fluid of Non-Newtonian power law rheology along a rigid horizontal plane under a shallow layer with a free theory approximation to establish a two layer model which couples the dynamics of the two layers. The problem corresponding to the release of a constant volume is solved numerically and the solution compared to the exact similarity solutions in a limiting case. The effect of the various physical parameters is analyzed and comparison is made with the result of the simple layer model.

Hassanien et al [5] discussed the flow and heat transfer in a power law fluid over a non-isothermal stretching sheet. They presented a boundary layer analysis for the problem of flow and heat transfer in a power law fluid over a continuous stretching sheet with variable wall temperature. They performed parametric studies to investigate the effect on non-Newtonian flow index, generalized Prandtl number, power law surface temperature and surface mass transfer rate. Their results showed that friction factor and heat transfer depend strongly on the flow parameters. Howell et al [6] examined momentum and heat transfer on a continuously moving surface in a power law fluid. They examined the momentum and heat transfer occurring in the laminar boundary layer on a continuously moving and stretching two dimensional surface in non-Newtonian power law fluid. Their results include situations when the velocity is non linear and when the surface is stretching linearly.

In this paper, we discuss the steady temperature field of a reacting non-Newtonian power-law fluid caused by the exothermic reaction of the fluid molecules as it flows in the presence of the thermal radiation over a flat plate. The discussion showed the effects of the frank-Kamenetskii parameter (M) and the thermal radiation parameter ([T.sub.R]) on the flow rate and temperature field and the quenching time which is indicated by the rate at which the temperature tends to zero.

Mathematical Formulation

We consider the steady flow of reacting Newtonian power law fluid over a flat in the presence of a thermal radiation and the relevant governing equations for the steady flow are given below: Continuity Equation:

[partial derivative]u/[partial derivative]x + [partial derivative]v /[partial derivative]y = 0 (1)

Momentum Equation:

p (u [partial derivative]u/[partial derivative]x + v [partial derivative]u/[partial derivative]y) = [partial derivative][[tau].sub.yx]/ [partial derivative]y (2)

Energy Equation

[rho]c (u [partial derivative]T/[partial derivative]x + v [partial derivative]T/[partial derivative]y) = K [[partial derivative].sup.2] T + [partial derivative][y.sup.2] + [partial derivative][q.sub.r]/ [partial derivative]y + [AQe.sup.-E/RT] (3)

where,

[q.sub.r] =-e[sigma] [partial derivative][T.sup.4]/[partial derivative]y (4)

[[tau].sub.yx] = m[(-[partial derivative]u/[partial derivative]y).sup.n] (5)

With the boundary conditions

u = Ax, v = [V.sub.m], T = [T.sub.o], [partial derivative]T/[partial derivative]y = -[(v/[[beta].sup.2-n][x.sup.1-n]).sup.1/n+1] at y = 0

and

u [right arrow] 0, T [right arrow] [T.sub.[infinity]] as y [right arrow] [infinity]

Where, [[tau].sub.yx] is the stress, u is the x - component velocity, v is the y - component velocity, [rho] is the density, T is the temperture, E is the activation Energy, [q.sub.r] is the radiation Energy, e is the emmisivity [sigma] is the Stefan Boltzmann constant.

Method of Solution

Introducing the stream function formulation:

u = [partial derivative][psi]/ [partial derivative]y v=-[partial derivative][psi]/ [partial derivative]x (6)

the continuity equation (1) is automatically satisfied.

Define a similarity variable [eta] as;

[eta] = y [([[beta].sup.2-n] [x.sup.1-n]/v).sup.1/n+1] (7)

such that

[psi] = (v[[beta].sup.2n-1][x.sup.-2n])1/n+1 f([eta]) (8)

Using equations (5), (6), (7) and (8) in the momentum equation (2), we obtain,

n f"'[(-f").sup.n-1] + 2n/n+1 f f'-[(f,).sup.2] 0 (9)

f(0)=0, f'(0)=(1/n) 1/2, f" (0)=-1, f'([infinity])=0 (10)

The equation (9) has a simple exact analytical solution of the form

f ([eta]) = [alpha] + [Dl.sup.-[alpha][eta]] (11)

Satisfying the boundary and initial conditions (10), yields

f ([eta]) = [(1/n).sup.1/4] - [(1/n).sup.1/4] exp (-[eta]/[n.sup.1/4]) (12)

Now consider the energy equation (3)

u [partial derivative]T/[partial derivative]x + v [partial derivative]T/[partial derivative]y = k/[rho]c [[partial derivative].sup.2]T/[partial derivative][y.sup.2] + 1/ [rho]c [partial derivative]q/[partial derivative]y + [AQe.sup.-E/RT]/[rho]c (13)

Let,

[theta] = (T-[T.sub.0]) [member of]/[RT.sub.0.sup.2] (14)

Where,

[member of] = [RT.sub.0]/E (15)

Expanding [T.sup.4] in a Taylor series about [T.sub.[infinity]] and neglecting the higher order terms, we obtain,

[T.sup.4] = [4T.sub.[infinity].sup.3] T-3[T.sub.[infinity].sup.4](16)

Substituting equations (14), (15) and (16) into equation (13), we obtain,

u [partial derivative][theta]/[partial derivative]x + v [partial derivative][theta]/[partial derivative]y = k/[rho]c [[partial derivative].sup.2][theta]/ [partial derivative][y.sup.2]-k4e[sigma][T.sub.[infinity].sup.3]/[rho] ck [[partial derivative].sup.2][theta]/[partial derivative][y.sub.2] + A[theta]E/[RT.sub.0.sup.2] [rho] c [e.sup.-[theta]] (17)

as [member of] [right arrow] 0

Let

[T.sub.R] = 4e[sigma] [T.sub.[infinity].sup.3]/k, be the Thermal Radiation Parameter,

and

M = AQE/[RT.sub.0.sup.2] [rho]c, be the Frank-Kamentskii Parameter

Then, equation (17) becomes,

u [partial derivative][theta]/[partial derivative]x + v[partial derivative][theta]/ [partial derivative]y = k/[rho]c (1-[T.sub.R]) [[partial derivative].sup.2] [theta]/[partial derivative][y.sup.2] + [Me.sup.-[theta]] (18)

Define

[theta] = g([eta]) (19)

Using equations (7) and (8) in equation (6), we obtain

u=[beta]xf' ([eta]) (20)

and

v = 2n/n+1(v[[beta].sup.2n-1] [x.sup.n-1])1/ n+1 f ([eta]) + 1 - n/n + 1 [eta](v[[beta].sup.2n-1][x.sup.n-1])1/n+1 f'([eta]) (21)

Substituting equations (19), (6), (7), (8), (20), and (21) into equation (18), we obtain,

1/[Pr.sub.n] (1 - [T.sub.R])g" + 2n/n + 1 fg' + M/[beta] [e.sup.-[theta]] = 0 (22)

g(0) = 1, g' (0) = -1, g([infinity]) = 0 (23)

Remark:

[Pr.sub.n] = [[[rho].sup.n+1] [C.sup.n+1], [v.sup.2]/[k.sup.n+1][[beta].sup.3(1-n)][x.sup.2(1-n)] is the Prandtl number for power law fluid.

Substituting equation (12) into equation (22) and resolve the resulting non--linear differential equation into system of equation as follows:

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

Taking the first derivative of the system of equation (24), then the resulting non--linear differential equation obtained when equation (12) is substituted into equation (22) is resolved into system equation (25) as stated below; And the initial conditions in (23) become,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

And the initial conditions in (23) become,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](26)

We solve problem (25) together with the initial condition (26) for various values of M, the Frank Kamenetskii parameter and [T.sub.R], the thermal radiation parameter. The result is presented in terms of temperature profiles in figure 1-8.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

Discussion of Result and Conclusion

The effects of the thermal radiation and Frank- Kamenetskii parameters on the heat transfer in a reacting power law fluid over a flat plate in the presence thermal radiation were examined and the results follow:

The temperature profiles for the effect of the Frank- Kamenetskii parameter (M) on the heat transfer in a reacting power law fluid with thermal radiation were shown in figures 1-4. Tables 1-4 showed the value of [eta] for which the temperature g ([ets]) tends to zero for various value of the power law exponent n: n = 1/3,1/2, 2/3, and 5/6.

From Tables 1-4; as the values of the power law exponent increases the rate at which the temperature g([eta]) approaches zero increases. The Frank-Kamenetskii parameter increases the rate at which the temperature g ([eta]) tends to zero increases. This implies that as the Frank- Kamenetskii parameter increases the rate of flow increase and the time length for the reaction flow decreases.

The temperature profiles 5-8, show the effect of thermal radiation parameter ([T.sub.R]) on the heat transfer in a reacting non-Newtonian power law fluid over a flat plate. The value of [eta] for which the temperature g([eta])[right arrow]0 as [eta][right arrow] [infinity] are shown below in tables 5-8;

It is clear from tables 5-8 that as the value of the power law exponent increases, the rate at which the temperature g([eta]) approaches zero increases. Also, as the thermal radiation parameter increases, the rate at which the temperature approaches zero decreases, which implies more energy is emitted at higher thermal radiation parameter. The system becomes hotter with an increase in the thermal radiation parameter. The fact that the temperature g([eta]) tends to zero as [eta][right arrow][infinity], suggests a slow flow and as the temperature increases the system releases most of its heat. Thus, as the thermal radiation parameter increases the rate of flow decreases and the time length for the reacting flow increases.

In conclusion, this study shows that for the flow of a reacting power law fluid over a flat plate in the presence of a thermal radiation, the quenching time for the reacting flow decreases with an increase in the Frank- Kamenetskii parameter. On the other hand, the quenching time for the reacting flow increases with increases in the thermal radiation parameter.

Acknowledgements

The author is grateful to Professor R.O Ayeni for his encouragements and useful contributions.

Reference

[1] Basov I.V, Shelukhin V.V., 1999, Generalized solutions to the Equations of compressible Bingham flows. Zamm.Z.Angem. Math. Mech., 79(3), pp. 185-192.

[2] Crane L.J., 1970, Flow past a stretching plane, Z. Angem. Math. Phy., 21, pp. 645-647.

[3] Grubka L.J, Bobba K.M., 1985, Heat transfer characteristics of a continuous stretching surface with variable temperature. ASME Journal of Heat Transfer 107, pp. 248-250.

[4] Hakin A., 1999, Existence of slow steady flows of viscoelastic fluids of Mute Metzer type. Compu. Math. Appl, 38(11-12), pp. 239-247.

[5] Hassanien LA, Abdullah A.A, Gorla R.S.R., 1998, Heat Transfer in a power law fliud over a non-isothermal stretching sheet., Math. Compu. Modeling, 28(9), pp. 105-116.

[6] Howell T.G, Jeng DR, De Witt K.J, 1997, Momentum and Heat transfer on a continuous moving surface in a power law fluid., Int. J. Heat Mass Transfer, 40(8), pp. 1853-1861.

[7] Pascal J.P., 2000, The spread of a non-Newtonian power law fluid under a shallow ambient layer, ZAMM.Z. Angem. Math. Mech., 80(6), pp.399-409.

[8] Vander Hout. K., 2000, Draw resonance in isothermal fiber spinning of Newtonian and power-law fluid., European J. Appl. Math, 11(2), pp. 129-136.

B.I Olajuwon

Department of Mathematics, University of Agriculture,

Abeokuta, Ogun State, Nigeria

E-mail: olajuwonishola@ yahoo.com

Table 1: M = 0

n             1/3           1/2           2/3           5/6
[eta]         1.4706        1.0471        0.8824        0.7882

Table 2: M = 0.2

n             1/3           1/2           2/3           5/6
[eta]         1.3529        1.0353        0.8706        0.7765

Table 3: M = 0.5

n             1/3           1/2           2/3           5/6
[eta]         1.2588        1.0118        0.8588        0.7647

Table 4:M = 1.0

n             1/3           1/2           2/3           5/6
[eta]         0.9765        0.8824        0.7765        0.7059

Table 5: [T.sub.R] = 0

n             1/3           1/2           2/3           5/6
[eta]         0.8207        0.7655        0.7241        0.6966

Table 6: [T.sub.R] = 0.2

n             1/3           1/2           2/3           5/6
[eta]         0.8642        0.8000        0.7500        0.7000

Table 7: [T.sub.R] = 0.4

n             1/3           1/2           2/3           5/6
[eta]         0.9745        0.8468        0.7702        0.7192

Table 8: [T.sub.R] = 0.8

n             1/3           1/2           2/3           5/6
[eta]         2.4286        1.6357        1.1000        0.8214