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## Latin American applied research

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*versión impresa* ISSN 0327-0793

### Lat. Am. appl. res. vol.41 no.4 Bahía Blanca oct. 2011

**ARTICLES**

**Soret and dufour effects on heat and mass transfer due to a stretching cylinder saturated porous medium with chemically-reactive species**

**S. M. M. El-Kabeir**

*Department of Mathematics, South Valley University, Faculty of Science, Aswan, Egypt. elkabeir@yahoo.com*

*Abstract* The diffusion-thermo and thermal-diffusion effects on heat and mass transfer by boundary layer flow over a stretching cylinder embedded in a porous medium have been studied numerically in the a presence of chemical reaction effect. The governing nonlinear partial differential equations are transformed into a set of coupled ordinary differential equations, which are solved numerically by using Runge-Kutta method with shooting techniques. Numerical results are obtained for the velocity, temperature and concentration distributions, as well as the skin friction coefficient, local Nusselt number and local Sherwood number for several values of the parameters, namely, the Reynolds number, Darcy number, chemical reaction parameter, Dufour and Soret numbers. The obtained results are presented graphically and the physical aspects of the problem are discussed.

*Keywords* Heat and Mass Transfer; Porous Medium; Stretching Cylinder; Dufour and Soret Effects; Chemical Reaction.

**I. INTRODUCTION**

Convective flow through porous media has many 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 packed 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. Comprehensive reviews on this area have been made by many researchers such as Nield and Bejan (1999), Vafai (2000), and Ingham and Pop (1998 and 2002).

The combined heat and mass transfer problems with chemical reactions are of importance in many processes, and therefore have received a considerable amount of attention in recent years. In processes, such as drying, evaporation at the surface of a water body, energy transfer in a wet cooling tower and the flow in a desert cooler, the heat and mass transfer occurs simultaneously. Chemical reactions can be codified as either homogeneous or heterogeneous processes. A homogeneous reaction is one that occurs uniformly through a given phase. In contrast, a heterogeneous reaction takes place in a restricted region or within the boundary of a phase. A reaction is said to be the first order if the rate of reaction is directly proportional to the concentration itself. In many chemical engineering processes, a chemical reaction between a foreign mass and the fluid does occur. These processes take place in numerous industrial applications, such as the polymer production, the manufacturing of ceramics or glassware, the food processing and so on. The effects of chemical reaction and mass transfer on flow past an impulsively infinite vertical plate with constant heat flux were studied by Das *et al.* (1994). Andersson *et al.* (1994) have studied the flow and mass diffusion of a chemical species with first-order and higher order reactions over a linearly stretching surface. Anjalidevi and Kandasamy (1999) have analyzed the steady laminar flow along a semi-infinite horizontal plate in the presence of a species concentration and chemical reaction. Muthucumaraswamy (2002) has studied the effect of a chemical reaction on a moving isothermal vertical infinitely long surface with suction. Analytical solutions for the overall heat and mass transfer on MHD flow of a uniformly stretched vertical permeable surface with the effects of heat generation/absorption and chemical reaction were presented by Chamkha (2003). El-Kabeir and Modather (2007) studied the effect of chemical reaction on the heat and mass transfer by MHD flow over a vertical cone surface in micropolar fluids with heat generation/absorption. Rashad and El-Kabeir (2010) have studied the effects of thermal/mass diffusions and chemical reaction on the heat and mass transfer by unsteady mixed convection boundary layer past a vertical stretching sheet embedded in a porous medium. Rashad *et al.* (2011) studied the coupled heat and mass transfer by mixed convection about solid sphere saturated porous medium in the presence of chemical reaction effect and using Brinkman-Forchheimer extended Darcy model.

On other hand, Soret and Dufour effects are important for intermediate molecular weight gases in coupled heat and mass transfer in binary systems, often encountered in chemical process engineering. When species are introduced at a surface in a fluid domain, with a different (lower) density than the surrounding fluid, both Soret (thermo-diffusion) and Dufour (diffuso-thermal) effects can become influential. It is also found that the diffusion thermo effect is much stronger for injection of hydrogen and helium (molecular weight less than air) than for injection of argon, carbon dioxide, and xenon (molecular weight greater than air). It is observed, for a light gas such as helium or hydrogen is injected into the boundary layer, the temperature induced buoyancy forces are augmented, and the heat transfer rates are increased. Therefore, the heat transfer rate (Nusselt number) increases in the presence of thermal diffusion and diffusion thermo effects with blowing rate. Li *et al.* (2006) used an implicit finite volume method to investigate thermal-diffusion (Soret) and diffusion-themo (Dufour) effects in a strongly endothermic chemically-reacting flow in a porous medium, showing that for low convectional velocity is lower or higher initial temperature of the feeding gas, Soret and Dufour effects have a strong influence on the regime. Partha *et al.* (2006) obtained similarity solutions for double dispersion effects on free convection hydromagnetic heat and mass transfer in a non-Darcy porous medium with Soret and Dufour effects, showing that in both aiding and opposing buoyancies, Dufour and Soret numbers considerably affect the wall mass transfer and heat transfer rates, both of which are also reduced with stronger magnetic field. The thermal-diffusion and diffusion-thermo effects on the heat and mass transfer characteristics of free convection past a continuously stretching permeable surface in the presence of magnetic field and radiation are studied by Abd El-Aziz (2008). Postelnicu (2007) has discussed the influence of chemical reaction on heat and mass transfer by natural convection from vertical surfaces embedded in fluid-saturated porous medium considering Soret and Dufour effects. Cheng (2009) studied the Soret and Dufour effects on natural convection heat and mass transfer from a vertical cone in a porous medium. Partha (2009) investigated the suction/injection effects on thermophoresis particle deposition in a non-Darcy porous medium under the influence of Soret and Dufour effects. The effects of Soret, Dufour, chemical reaction and thermal radiation on MHD non-Darcy unsteady mixed convective heat and mass transfer over a stretching sheet were analyzed by Pal and Mondal (2011).

The main objective of this paper is to study the chemical reaction, thermal-diffusion and diffusion-thermo effects on coupled heat and mass transfer due to a stretching cylinder embedded in a medium using the Brinkman-Darcy extended model. Similarity transformations are used to transform the partial differential equations to ordinary differential equations which then are solved numerically. The present study represents an extension of the works of Wang (1988) and Chamkha *et al.* (2010) by including mass transfer along a stretching cylinder saturated in the a presence of chemical reaction, Soret and Dufour effects.

**II. PROBLEM FORMULATION**

Consider steady, laminar, two-dimensional, heat and mass transfer boundary layers flow of a viscous incompressible fluid over a stretching tube embedded in a porous medium. The radius of a stretching tube is a in the axial direction in a fluid at rest as shown in Fig. 1, where the z-axis is measured along the axis of the tube and the r-axis is measured in the radial direction. It is assumed that the surface of the tube is maintained at a constant temperature *T _{w}* and a constant concentration

*C*, and the ambient temperature and concentration far away from the surface of the cone

_{w}*T*and

_{8}*C*are assumed to be uniform, where

_{8}*T*>

_{w}*T*and

_{8}*C*>

_{w}*C*. The physical properties of the fluid are assumed to be constant. The concentration of diffusing species thermal-diffusion and diffusion-thermal energy effects are taken into account, and a first-order homogeneous chemical reaction is assumed to take place in the flow. The governing equations for this physical situation are based on the usual balance laws of mass, linear momentum, energy and mass diffusion modified to account for the physical effects mentioned above. These equations are given by (Wang, 1988, and Chamkha et al., 2010) :

_{8}(1) | |

(2) | |

(3) | |

(4) | |

(5) |

Fig. 1. Physical model and coordinate system.

Subject to the boundary conditions

(6) |

where *W _{w}*=2

*cz*,

*c*is a positive constant, where

*u*and

*w*denote the velocity components in the

*z*- and

*r*- directions respectively,

*K*is permeability of porous medium, n is the kinematic viscosity,

*T*and

*C*are the temperature and concentration, respectively, ? is the density,

*D*is the mass diffusivity,

*C*is the specific heat capacity,

_{p}*C*is the concentration susceptibility,

_{s}*a*is the thermal diffusivity,

*T*is the mean fluid temperature,

_{m}*k*is the thermal diffusion ratio,

_{T}*T*

_{8}is the free stream temperature,

*C*

_{8}is the free stream concentration.

*K*

^{/}is the dimensional of chemical reaction. Following Wang (1988), the similarity transformation can be given as:

(7) |

Substituting (7) into Eqs. (2)-(5) , we get the following ordinary differential equations:

(8) | |

(9) | |

(10) | |

The boundary conditions (6) become

(11) |

The pressure *p* can now be determined from Eq. (3) in the form:

(12) | |

(13) |

where a prime denotes partial differentiation with respect to *?* and

(14) |

are the Reynolds number, Prandtl number, Schmidt number, dimensionless of chemical reaction parameter, Darcy number, Dufour and Soret numbers.

Physical quantities of interest are the skin friction coefficient *C _{f}*, the Nusselt number

*Nu*and Sherwood number

*Sh*, which are defined, respectively as:

(15) |

with *k* being the thermal conductivity. Further,*t _{w}*,

*q*and

_{w}*q*are the shear stress, the heat transfer and the mass transfer from the surface of the cylinder, respectively, and they are given by

_{m}(16) |

Substituting Eqs. (7) and (16) into Eq. (15) yields

(17) |

**III. NUMERICAL METHOD**

The set of Eqs. (8) to (10) under the boundary conditions (11) has been solved numerically using the Runge-Kutta integration scheme with shooting method. We let *f=x*_{1}**, ***f'=x*_{2}**, ***f"=x*_{3}**, ***?=x*_{4}**, ***?'=x*_{5}**, ***=x*_{6}**, ***=x*_{7}. Equations (8) to (10) are transformed into systems of first order differential equations as follows:

(18) | |

subject to the following initial conditions:

(19) |

In a shooting method, the unspecified initial conditions; *S*_{1}, *S*_{2} and *S*_{3} in Eq. (19) are assumed. Equation (18) is then integrated numerically as an initial valued problem to a given terminal point. The accuracy of the assumed missing initial condition is then checked by comparing the calculated value of the dependent variable at the terminal point with its given value. If a difference exists, improved values of the missing initial conditions must be obtained and the process is repeated. A step size of *?* = 0.001 was selected to be satisfactory for a convergence criterion of 10^{-7} in nearly all cases. The maximum value of ?_{8}, to each group of parameters *Da, Sc*, *D _{f}* ,

*Sr*,

*Re*, and

*Pr*are determined when the values of unknown boundary conditions at

*?*= 0 does not change to successful loop with error less than 10

^{-7}. From the process of numerical computation, the local skin friction coefficient, the local Nusselt number and the local Sherwood number, which are respectively proportional to

*f'*(1), -

*?'*(1) and worked out and their numerical values presented graphically.

**IV. RESULTS AND DISCUSSIONS**

The system of ordinary differential Eqs. (8)-(10) with the boundary conditions (11) has been solved numerically by applying the shooting iteration technique together with Runge-Kutta forth-order integration scheme. From the process of numerical computation, the skin-friction coefficient, the Nusselt and the Sherwood numbers, the pressure which are respectively proportional to *f"*(1), -*?"*(1), and , are also worked out and their numerical values are presented in a tabular form. In order to access the accuracy of the numerical results, various comparisons with previously published works for the cases of the absence of mass transfer and permeability of porous medium. These comparisons are presented in Tables 1. It is obvious from these tables that excellent agreement between the results exist.

Table 1 Comparison of results for-* f"*(1)

Numerical calculations have been carried out for different values of the Darcy number *Da*, the Reynolds number *Re*, the chemical reaction parameter , Dufour number *D _{f}* and Soret number

*S*. Throughout the study Prandtl number is kept constant at (

_{r}*Pr*= 7.0) which represents water polluted by the species Benzene (

*Sc*= 1.60). The value of the Schmidt number (

*Sc*) was chosen to represent the most common diffusing chemical species which are of interest and the values of Dufour and Soret numbers are chosen in such a way that their product is constant provided that the mean temperature

*T*is kept constant as well.

_{m}Figures 2-5 present the effect of the Darcy number *Da* (permeability of porous medium), on the velocity along the cylinder *f'*, temperature *?* and concentration , and the pressure profiles, respectively. Physically, the presence of a porous medium in the flow presents resistance to flow. Thus, the resulting resistive force tends to slow the motion of the fluid along the cylinder surface and causes increases in its temperature, concentration, and pressure profiles. This is depicted in Figures 2-5 by the increasing in the velocity profiles and decreases in the values of either of the temperature, concentration or pressure as the Darcy number *Da*increases.

Fig. 2. Effect of Darcy number *Da* on the velocity profiles.

Fig. 3. Effect of Darcy number *Da* on the temperature profiles.

Fig. 4. Effect of Darcy number *Da* on the concentration profiles.

Fig. 5. Effect of Darcy number *Da* on the pressure profiles.

Figures 6-9 show the velocity, temperature, concentration, and pressure profiles, for various values of Reynolds number *Re*, respectively. In these figures, the Reynolds number represents the relative significance of inertia effect compared to the viscous effect. Thus, as Re increases, all the velocity, temperature and concentration profiles decrease, while increasing the Reynolds number results in higher pressure distribution. These decreases in the velocity, temperature and concentration values are followed by corresponding decreases in both the hydrodynamic, thermal and solutal (concentration) boundary-layer thickness.

Fig. 6. Effect of the Reynolds number *Re* on the velocity profiles.

Fig. 7. Effect of the Reynolds number *Re* on the temperature profiles.

Fig. 8. Effect of the Reynolds number *Re* on the concentration profiles.

Fig. 9. Effect of the Reynolds number *Re* on the pressure profiles.

Figures 10-11 illustrate the behavior of the distributions on the local skin-friction coefficient *C _{f}*, local Nusselt number

*Nu*, and local Sherwood number

*Sh*due to changes in the values of

*Da*and

*Re*, respectively. As mentioned before and seen from Figures 2-5, for a given value of the Reynolds number, increasing the Darcy number

*Da*causes the temperature and concentration at the cylinder surface to decrease, while the opposite effect with the velocity profiles. This behavior results increases slightly in the local Nusselt number, whereas both the local skin-friction coefficient and Sherwood number decrease. However,

*Nu*enhances, while both

*C*,and

_{f}*Sh*reduce as the Darcy number

*Da*increases, it can be seen that all of the local skin-friction coefficient with an increase of the Reynolds number

*Re*, This observation agrees with the results shown in table 1. The magnitude of the skin friction coefficient and local Nusselt number increases with Reynolds number Re which is consistent with the behavior reported by Wang (1988) and Chamkha

*et al.*(2010). In addition, an opposite trend is observed for the local Sherwood number

*Sh*which reduces as

*Re*increases.

Fig. 10. Effects of the Reynolds number *Re* on the local the skin-friction coefficient with various values of Darcy number *Da*.

Fig. 11. Effects of the Reynolds number *Re* on the local Nusselt and local Sherwood numbers with various values of Darcy number *Da*.

The effects of the chemical reaction parameter , Dufour number *D _{f}* and Soret number

*S*on the temperature and concentration profiles are shown in Figs. 12-15. It is seen from these figures that the concentration of the fluid decreases with increase of destructive reaction (>1) of chemical reaction whereas the temperature of the fluid are significant increasing with increase of destructive reaction. Moreover, It is evident to note that the increase of chemical reaction significantly alters the concentration boundary layer thickness but not momentum and thermal boundary layers.

_{r}

Fig. 12. Effects of Dufour number *D _{f}* and Soret number

*Sr*on the temperature profiles.

Fig. 13. Effects of Dufour number *D _{f} *and Soret number

*Sr*on the concentration profiles.

Fig. 14. Effect of the chemical reaction parameter on the temperature profiles.

Fig. 15. Effect of the chemical reaction parameter on the concentration profiles.

Also, it is observed that the concentration of the fluid is increasing with increasing *Sr* (or decreasing *D _{f}*), but the temperature decrease as

*Sr*increases (or

*D*decreases). This behavior is a direct consequence of the Soret effect, which produces a mass flux from lower to higher solute concentration driven by the temperature gradient. Moreover, it is obvious that the governing equations (8)-(10) are uncoupled. Therefore, changes in the values of ,

_{f}*D*and

_{f}*S*will cause no changes in both of the distributions of velocity of fluid, and for this reason, no figures for these variables are presented herein.

_{r}Figures 16-17 show the effect of the chemical reaction parameter on the local skin-friction coefficient *C _{f}*, local Nusselt number

*Nu*, and the local Sherwood number

*Sh*for various values of the Dufour number

*D*and Soret number

_{f}*S*, respectively. It can be seen that as the Dufour number

_{r}*D*increases, the Nusselt number reduces, while the Sherwood number enhances. This is because either increase in concentration difference or decrease in temperature difference leads to an increase in the value of

_{f}*D*resulting trends similar to the above observation. Similarly, either a decrease in concentration difference or an increase in temperature difference leads to an increase in the value of the Soret number

_{f}*Sr*. Therefore, increasing the parameter

*Sr*causesincreases in the local Nusselt number while it produces decreases in the local Sherwood number. Finally, it can be observed that as increases, the local Sherwood number increases, while the opposite effect is found for the local Nusselt number. This is because as increases, the concentration difference between the cylinder surface and the fluid decreases and so the rate of mass transfer at the cylinder surface must increase, while the rate of heat transfer decrease as a result of the decrease in the fluid temperature.

Fig. 16. Effects of Dufour number *D _{f} *and Soret number

*Sr*on the local Nusselt and local Sherwood numbers with various values of Darcy number

*Da*.

Fig. 17. Effect of chemical reaction *g* on the local Nusselt and local Sherwood numbers with various values of Darcy number *Da*.

**V. CONCLUSIONS**

In the present work, the chemical reaction, thermal-diffusion and diffusion-thermo effects on coupled heat and mass transfer due to a stretching cylinder embedded in a medium using the Brinkman-Darcy extended model has been investigated. Similarity solutions are obtained for a linearly stretching cylinder with a constant surface temperature and concentration Graphical results for the velocity and temperature profiles as well as the skinfriction coefficient and local Nusselt and Sherwood numbers are presented and discussed for various parametric conditions. It was found that the skin-friction coefficient increased as the Reynolds number increased, while the opposite trend with increasing the Darcy number. It was found that the local Nusselt number decreased, whereas and Sherwood increased due to increases in either the Reynolds number or Darcy number (reflect the permeability of porous medium effect). Also, increases in the values of Soret number (reciprocal of thermal-diffusion effect) produced increases in the local Nusselt number and decreases in the local Sherwood number. However, the opposite behavior was predicted as Dufour number (reciprocal of diffusion-thermo effect) or the chemical reaction parameter was increased, where the Sherwood number was increased while the Nusselt number was decreased. It is hoped that the present work will serve as a motivation for future experimental work which seems to be lacking at the present time.

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**Received: May 31, 2010 Accepted: January 21, 2011 Recommended by subject editor: Walter Ambrosini**