Diffusion-thermo and thermal-diffusion effects on free convective heat and mass transfer flow in a porous medium with time dependent temperature and concentration.

Abstract

The diffusion-thermo and thermal-diffusion effects on unsteady free convection and mass transfer flow along an accelerated vertical porous plate embedded in a porous medium have been studied numerically taking the plate temperature and concentration to be functions of time. The governing nonlinear partial differential equations are transformed into a set of coupled ordinary differential equations, which are solved numerically by applying Nachtsheim-Swigert shooting iteration technique along with sixth order Runge-Kutta integration scheme. The effects of various parameters entering into the problem have been examined on the flow field for a hydrogen-air mixture as a non-chemical reacting fluid pair. The numerical results have shown that the above-mentioned effects have to be taken into consideration in the fluid, heat and mass transfer processes.

Keywords: Free convection, Porous medium, Vertical plate, Dufour effect, Soret effect.

Introduction

Convective flow through porous media has many important important applications, such as heat transfer associated with heat recovery from geothermal systems and particularly in the field of large storage systems of agricultural products, heat transfer associated with storage of nuclear waste, exothermic reaction in packedded reactors, heat removal from nuclear fuel debris, flows in soils, petroleum extraction, control of pollutant spread in groundwater, solar power collectors and porous material regenerative heat exchangers.

Coupled heat and mass transfer finds applications in a variety of engineering application, such as the migration of moisture through the air contained in fibrous insulation and grain storage installations, filtration, chemical catalytic reactors and processes, spreading of chemical pollutants in plants and diffusion of medicine in blood veins. A Comprehensive reviews on this area have been made by many researchers such as Nield and Bejan [1], Ingham and Pop [2, 3], Bejan and Khair [4] and Trevisan and Bejan [5].

Most of the above studies, however, considered constant plate temperature and concentration and have neglected the diffusion-thermo and thermal-diffusion terms from the energy and concentration equations respectively. When heat and mass transfer occur simultaneously in a moving fluid, the relations between the fluxes and the driving potentials are of more intricate nature. It has been found that an energy flux can be generated not only by temperature gradients but by composition gradients as well. The energy flux caused by a composition gradient is called the Dufour or diffusion-thermo effect. On the other hand, mass fluxes can also be created by temperature gradients and this is the Soret or thermal-diffusion effect. In general, the thermal-diffusion and diffusion-thermo effects are of a smaller order of magnitude than the effects described by Fourier's or Fick's law and are often neglected in heat and mass transfer processes. However, exceptions are observed therein. The thermal-diffusion (Soret) effect, for instance, has been utilized for isotope separation, and in mixture between gases with very light molecular weight ([H.sub.2], He) and of medium molecular weight ([N.sub.2], air) the diffusion-thermo (Dufour) effect was found to be of a considerable magnitude such that it cannot be ignored (Eckert and Drake [6]). In view of the importance of these above mentioned effects, Dursunkaya and Worek [7] studied diffusion-thermo and thermal-diffusion effects in transient and steady natural convection from a vertical surface whereas Kafoussias and Williams [8] studied the same effects on mixed free-forced convective and mass transfer boundary layer flow with temperature dependent viscosity. Recently, Anghel et al. [9] investigated the Dufour and Soret effects on free convection boundary layer over a vertical surface embedded in a porous medium. Very recently, Postelnicu [10] studied numerically the influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects.

Therefore, the objective of this work is to investigate the Diffusion-thermo and thermal-diffusion effects on unsteady free convection and mass transfer flow past an accelerated vertical porous flat plate embedded in a porous medium with time dependent temperature and concentration.

Mathematical Formulation

We consider an unsteady free convection and mass transfer flow of a viscous incompressible fluid past an infinite vertical porous plate in a porous medium. The flow is assumed to be in the x-direction, which is taken along the vertical plate in the upward direction, and the y-axis is taken to be normal to the plate. Initially the plate and the fluid are at same temperature [T.sub.[infinity]] in a stationary condition with concentration level [C.sub.[infinity]] at all points. At time t > 0 the plate is assumed to be moving in the upward direction with a velocity U(t) and the plate temperature and concentration are raised to T(t) and C(t) respectively. The physical model and co-ordinate system is shown in the following fig. A.

[FIGURE A OMITTED]

It is assumed that the plate is infinite in extent and hence all physical quantities depend on y and t only. Thus accordance with the above assumptions and Boussinesq's approximation, the basic equations relevant to the problem are:

[partial derivative]v / [partial derivative]y = 0, (1)

[partial derivative]u / [partial derivative]t + v [partial derivative]u / [partial derivative]y = [upsilon] [[partial derivative].sup.2]u / [partial derivative][y.sup.2] + g[beta](T - [T.sub.[infinity]]) + g[[beta].sup.*] (C - [C.sub.[infinity]]) - [upsilon] / K' u, (2)

[partial derivative]T / [partial derivative]t + v [partial derivative]T / [partial derivative]y = [alpha] [[partial derivative].sup.2]T / [partial derivative][y.sup.2] + [D.sub.m][k.sub.T] / [c.sub.s][c.sub.p] [[partial derivative].sup.2]C / [partial derivative][y.sup.2], (3)

[partial derivative]C / [partial derivative]t + v [partial derivative]C / [partial derivative]y = [D.sub.m] [[partial derivative].sup.2]C / [partial derivative][y.sup.2] + [D.sub.m] [k.sub.T] / [T.sub.m] [[partial derivative].sup.2]T / [partial derivative][y.sup.2], (4)

where u, v are the velocity components in the x and y directions respectively, [upsilon] is the kinematic viscosity, g is the acceleration due to gravity, [rho] is the density, [beta] is the coefficient of volume expansion, [[beta].sup.*] is the volumetric coefficient of expansion with concentration, T and [T.sub.[infinity]] are the temperature of the fluid inside the thermal boundary layer and the fluid temperature in the free stream, respectively, while C and [C.sub.[infinity]] are the corresponding concentrations. Also, K' is the permeability of porous medium, a is the thermal diffusivity, [D.sub.m] is the coefficient of mass diffusivity, [c.sub.p] is the specific heat at constant pressure, [T.sub.m] is the mean fluid temperature, [k.sub.T] is the thermal diffusion ratio and [c.sub.s] is the concentration susceptibility.

The appropriate initial and boundary conditions relevant to the problem are:

For t [less than or equal to] 0 : u = v = 0, T = [T.sub.[infinity]], C = [C.sub.[infinity]] for all y.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

In order to obtain a local similarity solution (in time) of the above problem, we introduce a similarity parameter s, which is a time dependent length scale as

[sigma] = [sigma](t). (6)

In terms of this length scale, a convenient solution of the equation (1) is considered to be in the following form

v = v(t) = -[v.sub.0] [upsilon] / [sigma], (7)

where [v.sub.0] (> 0) is the suction velocity of the fluid through the porous plate. The following dimensionless quantities are then defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where m is a non-negative integer and [U.sub.0], [T.sub.0], [C.sub.0] are respectively the free stream velocity, mean temperature and mean concentration. Here [[sigma].sub.*] = [sigma] / [[sigma].sub.0] where [[sigma].sub.0] is the value of [sigma] at t = [t.sub.0].

Then introducing the relations (6)-(8) into equations (2), (3) and (4), we have the following non-dimensional equations:

f'' + [eta] [sigma] / [upsilon] d[sigma] / dt f' + [v.sub.0]f' - (2m + 2) [sigma] / [upsilon] d[sigma] / dt - Kf + Gr[theta] + Gm[phi] = 0, (9)

[theta]'' + [eta] [sigma] / [upsilon] d[sigma] / dt Pr[theta]' + [v.sub.0] Pr[theta]' - 2m Pr [sigma] / [upsilon] d[sigma] / dt [theta] + Pr Df[phi]'' = 0, (10)

[phi]'' + [eta] [sigma] / [upsilon] d[sigma] / dt Sc[phi]' + [v.sub.0] Sc[phi]' - 2m [sigma] / [upsilon] d[sigma] / dt Sc[phi] + ScSr[theta]'' = 0, (11)

where Pr = [upsilon] / [alpha] is the Prandtl number, Sc = [upsilon] / [D.sub.m] is the Schmidt number, K = [[sigma].sup.2] / K' is

Permeability parameter, Sr = [D.sub.m][k.sub.T] ([T.sub.0] - [T.sub.[infinity]]) / [T.sub.m][upsilon]([C.sub.0] - [C.sub.[infinity]]) is the Soret number, Df = [D.sub.m][k.sub.T]([C.sub.0] - [C.sub.[infinity]]) / [c.sub.s][c.sub.p] [upsilon]([T.sub.0] - [T.sub.[infinity]]) is the Dufour number, Gr = g[beta]([T.sub.0] - [T.sub.[infinity]]) [[sigma].sup.2.sub.0] / [upsilon][U.sub.0] is the local Grashof number and Gm = g[[beta].sup.*] ([C.sub.0] - [C.sub.[infinity]]) [[sigma].sup.2.sub.0] / [upsilon][U.sub.0] is the local modified Grashof number.

The corresponding boundary conditions for t > 0 are obtained as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Now the equations (9)-(11) are locally similar except the term ([sigma] / [upsilon] d[sigma] / dt), where t appears explicitly. Thus the local similarity condition requires that ([sigma] / [upsilon] d[sigma] / dt) in the equations (9)-(11) must be a constant quantity.

Hence following the works of Hasimoto [11], Sattar and Hossain [12] and Sattar and Maleque [13] one can try a class of solutions of the equations (9)-(11) by assuming that

([sigma] / [upsilon] d[sigma] / dt) = C (a constant). (13)

Integrating (13) we have

[sigma](t) = [square root of 2C[upsilon]t] (14)

where the constant of integration is determined through the condition that [sigma] = 0 when t = 0. We have considered the problem for small time. In this case normal velocity (7) will be large i.e., suction will be large, which can be applied to increase the lift of the airfoils. From (13) choosing C = 2, the length scale [sigma](t) = 2[square root of [upsilon] t] which exactly corresponds to the usual scaling factor for various unsteady boundary layer flows (Schlichting [14]). Since sis a scaling factor as well as a similarity parameter, any value of C in (13) would not change the nature of the solutions except that the scale would be different.

Finally introducing [13] with C = 2 into equations (9)-(11) we respectively have the dimensionless equations which are locally similar in time:

f'' + 2 f[zeta] - (4 + 4m + K)f + Gr[theta] + Gm[phi] = 0, (15)

[theta]'' + 2 Pr[theta]'[zeta] - 4m Pr[theta] + Pr Df[phi]'' = 0, (16)

[phi]'' + 2Sc[phi]'[zeta] - 4mSc[phi] + ScSr[theta]'' = 0, (17)

where [zeta] = [eta] + [v.sub.0] / 2.

The equations (15)-(17) are similar together with the boundary equations (12). The above systems have been solved numerically for various values of the parameters entering into the problem. From the process of numerical computation the local Nusselt number and the local Sherwood number, which are respectively proportional to -[theta]'(0) and -[phi]'(0), are also sorted out and their numerical values are presented in tabular form.

Numerical Method

Numerical solutions to the transformed set of non-linear ordinary differential equations (15)-(17) with boundary conditions (12) were obtained, using Nachtsheim-Swigert [15] shooting iteration technique along with sixth order Runge-Kutta integration scheme. A step size of [DELTA][eta] = 0.01 was selected to be satisfactory for a convergence criterion of [10.sup.-6] in all cases. The value of [[eta].sub.[infinity]] was found to each iteration loop by the statement [[eta].sub.[infinity]] = [[eta].sub.[infinity]] + [DELTA][eta]. The maximum value of [[eta].sub.[infinity]], to each group of parameters [v.sub.0], K, m, Sr, Df , Pr, Sc, Gr and Gm determined when the value of the unknown boundary conditions at [eta] = 0 not change to successful loop with error less than [10.sup.-6]. However, different stepsizes such as [DELTA][eta] = 0.01, [DELTA][eta] = 0.006, [DELTA][eta] = 0.002 were also tried and the obtained solutions (velocity profiles) have been found to be independent of the step sizes as observed in Fig. 1. The method is validated by directly comparing its results with those of Hossain and Begum [16] for the same problem with all of K, m, Sr and Df are set to zero as shown in Fig. 2. It is seen from this figure that both results are in excellent agreement. Therefore, this lends confidence in the numerical results to be reported in the next section.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Results and Discussion

For the purpose of discussing the effects of various parameters on the flow behaviour near the plate, numerical calculations have been carried out for different values of [v.sub.0], K, m, Sr, Df, Gr, and Gm and for fixed values of Pr, Sc. The value of Prandtl number (Pr) is taken to be 0.71, which corresponds to air, and the value of Schmidt number (Sc) is chosen to represent hydrogen at 25[degrees]C and 1 atm. The values of Dufour number (Df) and Soret number (Sr) are chosen in such a way that their product is constant provided that the mean temperature [T.sub.m] is kept constant as well. However, the values of [v.sub.0] and m are chosen arbitrarily. The numerical results for the dimensionless velocity, temperature and concentration profiles are displayed in Figs.3-12. The effects of the permeability parameter (K) and free convection currents (both Gr and Gm) on the velocity field are shown in Fig. 3. From this figure it is observed that the velocity decreases with the increase of permeability parameter at a particular point of the boundary layer while it increases with the increase of both Gr and Gm (i. e. free convection currents). The effects of suction parameter ([v.sub.0]) in the velocity field are shown in Fig. 4. It is seen from this figure that the velocity profiles decrease monotonically with the increase of suction parameter indicating the usual fact that suction stabilizes the boundary layer growth. The effect of suction parameter ([v.sub.0]) on the temperature and concentration field are displayed in Figs. 5 and 6 respectively and we see that both the temperature and concentration decrease with the increases of suction parameter. Sucking decelerated fluid particles through the porous wall reduce the growth of the fluid boundary layer as well as thermal and concentration boundary layers.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

The effects of Soret and Dufour numbers on the velocity field are shown in Fig. 7. We observe that quantitatively when [eta] = 1.0 and Sr decreases from 2 to 1 (or Df increases from 0.03 to 0.06) there is 6.26% increase in the velocity value, whereas the corresponding increase is 10.45%, when Sr decreases from 0.2 to 0.08.

The effects of Soret and Dufour numbers on the temperature field are shown in Fig. 8. We observe that quantitatively when [eta] = 0.80 and Sr decreases from 2 to 1 (or Df increases from 0.03 to 0.06) there is 72.02% increase in the temperature value, whereas the corresponding increase is 24.7%, when Sr decreases from 0.2 to 0.08.

[FIGURE 8 OMITTED]

The effects of Soret and Dufour numbers on the concentration field are shown in Fig. 9. We observe that quantitatively when [eta] = 1.0 and Sr decreases from 2 to 1 (or Df increases from 0.03 to 0.06) there is 7.38% decrease in the concentration value, whereas the corresponding decrease is 8.2%, when Sr decreases from 0.2 to 0.08.

[FIGURE 9 OMITTED]

In Figs.10, 11 and 12, the effects of the non-negative integer m on the velocity, temperature and concentration profiles are shown. Because m = 0 defines the case for constant temperature and concentration, it appears from Figs. 10, 11 and 12 that as the plate temperature and concentration are changed from constant value (m = 0) to variable values (m = 1, 2 and 3), the velocity, temperature and concentration decrease significantly for all fixed parameters which indicate that time dependent temperature and concentrations has stronger decreasing effect on the velocity, temperature and concentration fields compared to constant temperature and concentration of the plate and the fluid. From Fig. 10 we also see that for m = 0, velocity profile first reaches a maximum near the leading edge of the plate then decrease to zero. Free convection effect is much clear for m = 0.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

Finally, the effects of the above-mentioned parameters on the rate of heat and mass transfer are shown in Tables 1 and 2. From table 1 we observe that for fixed Df and Sr; both the local Nusselt and Sherwood numbers increase as m increases. However, from table 2 we see that the local Nusselt number decreases, while the local Sherwood number increases as Df increases and Sr decreases.

Conclusions

In this paper the diffusion-thermo and thermal-diffusion effects on an unsteady free convection and mass transfer flow past an accelerated vertical porous plate embedded in a porous medium is studied numerically with time dependent plate temperature and concentration. A hydrogen-air mixture was selected as fluid pair used in the study due to its radically different thermodynamic properties as compared to other fluid pairs. The governing equations were developed and transformed using appropriate similarity transformations. The transformed non-linear similarity equations were then solved numerically by applying Nachtsheim-Swigert [15] shooting iteration technique along with sixth order Runge-Kutta integration scheme. The obtained results for the special cases of the present problem were compared with previously published work and found to be in excellent agreement. From the present numerical investigation we observed that velocity profiles decrease with the increase of permeability parameter while it increases with the increase of free convection currents. It was found that wall suction stabilizes the velocity, thermal as well as concentration boundary layer growth. We observed that time dependent temperature and concentrations has stronger decreasing effect on the velocity, temperature and concentration fields compared to constant temperature and concentration of the plate and the fluid. Both the local Nusselt and Sherwood numbers were found to increase as m increases. The presented analysis has also that the Dufour and Soret effects appreciably influence the flow field. Therefore we can conclude that for fluids with medium molecular weight ([H.sub.2], air), the Dufour and Soret effects should not be neglected.

References

[1] D. A. Nield and A. Bejan, Convection in Porous Media, 2nd edition, Springer, New York, 1999.

[2] D. B. Ingham and I. Pop, Transport Phenomena in Porous Media I, Pergamon, Oxford, 1998.

[3] D. B. Ingham and I. Pop, Transport Phenomena in Porous Media II, Pergamon, Oxford, 2002.

[4] A. Bejan and K. R. Khair (1985): Heat and mass transfer by natural convection in a porous medium, Int. J. Heat Mass Transfer, 28, 909-918.

[5] O. V. Trevisan and A. Bejan (1990): Combined heat and mass transfer by natural convection in a porous medium, Adv. Heat Transfer, 20, 315-352.

[6] E. R. G. Eckert and R. M. Drake, Analysis of Heat and Mass Transfer, McGraw-Hill, New York, 1972.

[7] Z. Dursunkaya and W. M. Worek (1992): Diffusion-thermo and thermal-diffusion effects in transient and steady natural convection from a vertical surface, Int. J. Heat Mass Transfer, 35, 2060-2065.

[8] N. G. Kafoussias and E. M. Williams (1995): Thermal-diffusion and diffusion-thermo effects on mixed free-forced convective and mass transfer boundary layer flow with temperature dependent viscosity, Int. J. Engng. Sci. 33, 1369-1384.

[9] M. Anghel, H. S. Takhar and I. Pop (2000): Dufour and Soret effects on free convection boundary-layer over a vertical surface embedded in a porous medium, Studia Universitatis Babes-Bolyai Mathematica, XLV, 11-21.

[10] A. Postelnicu (2004): Influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects, Int. J. Heat Mass Transfer, 47, 1467-1472.

[11] H. Hasimoto (1957): Boundary layer growth on a flat plate with suction or injection, J. Phys. Soc. Japan, 22, 68-72.

[12] M. A. Sattar and M. M. Hossain(1992): Unsteady hydromagnetic free convection flow with Hall current and mass transfer along on accelerated porous plate with time dependent temperature and concentration. Can. J. Phys., 70, 369-374.

[13] M. A. Sattar and M. A. Maleque (2000): Unsteady MHD natural convection flow along an accelerated porous plate with Hall current and mass transfer in a rotating system, J. Energy, Heat and Mass Transfer 22, 67-72.

[14] H. Schlichting, Boundary Layer Theory, 6th Edn, McGraw-Hill, New York, 1968.

[15] P. R. Nachtsheim and P. Swigert (1965): Satisfaction of the asymptotic boundary conditions in numerical solution of the system of non-linear equations of boundary layer type, NASA TND-3004.

[16] M. A. Hossain and R. A. Begum(1985): effects of mass transfer on the unsteady flow past an accelerated vertical porous plate with variable suction, Astrophys. Space Sci., 115, 145.

Nomenclature 

C = concentration

[c.sub.p] = specific heat at constant pressure

[c.sub.s]= concentration susceptibility

[D.sub.m] = mass diffusivity

Df = Dufour number

f = dimensionless velocity

g = acceleration due to gravity

Gr = local temperature Grashof number

Gm = local mass Grashof number

K = permeability parameter

K' = permeability of the porous medium

k = thermal conductivity of fluid

Nu = Nusselt number

Pr = Prandtl number

Sc = Schmidt number

Sh = Sherwood number

Sr= Soret number

[T.sub.m] = mean fluid temperature

[U.sub.0] = free stream velocity

u, v = velocity components in the x- and y- direction respectively

x, y = Cartesian coordinates along the plate and normal to it

Greek Symbols

[eta] = similarity variable

[alpha] = thermal diffusivity

[beta] = coefficient of thermal expansion 

[[beta].sup.*] = coefficient of concentration expansion

[sigma](t)=time dependent length scale

[rho] = density of the fluid

[upsilon] = kinematic viscosity

[theta] = dimensionless temperature

[phi] = dimensionless concentration

M.S. Alam (1), M.M. Rahman (2), M. Ferdows (3), Koji Kaino (3), Eunice Mureithi (4) and A. Postelnicu (5)

(1) Department of Mathematics, Dhaka University of Engineering and Technology (DUET), Gazipur-1700, Bangladesh.

(2) Department of Mathematics, University of Dhaka, Dhaka-1000, Bangladesh.

(3) Department of Advanced Science and Technology Toyota Technological Institute, Nagoya, Japan.

(4) Department of Mathematics and Applied Mathematics University of Pretoria , Pretoria 0002, South Africa.

(5) Department of Fluid Mechanics and Thermal Engineering Transsilvania University of Brasov, Romania.

Table 1: Numerical values of the local Nusselt and Sherwood numbers for
Gr = 12, Gm = 6, Pr = 0.71, [v.sub.0] = 0.5, K = 2 and Sc = 0.22.

  m      Df        Sr        Nux           [Sh.sub.x]

  0       0.15       0.4      1.225484     0.530282

  1       0.15       0.4      2.161475     1.000125

  2       0.15       0.4      2.801340     1.317094

  3       0.15       0.4      3.316499     1.571336

  0       0.075       0.8      1.319614    0.459756

  1       0.075       0.8      2.312489    0.883541

  2       0.075       0.8      2.992298    1.168513

  3       0.075       0.8      3.539834    1.396872

  0        0.05       1.2      1.420347     0.39304

  1        0.05       1.2      2.471738    0.774757

  2        0.05       1.2      3.192822    1.030367

  3        0.05       1.2      3.773872    1.234971

Table 2: Numerical values of the local Nusselt and Sherwood numbers for
Gr = 12, Gm = 6, Pr = 0.71, [v.sub.0] = 0.5, K = 2, m = 1 and
Sc = 0.22.

Df        Sr      [Nu.sub.x]       [Sh.sub.x]

0.030     2.0     2.890605         0.585832

0.037     1.6     2.659539         0.674995

0.050     1.2     2.471738         0.774757

0.075     0.8     2.161475         0.883541

0.150     0.4     2.161475         1.000125

0.600     0.1     1.937075         1.092098