A binary chemical reaction on unsteady free convective boundary layer heat and mass transfer flow with arrhenius activation energy and heat generation/absorption

Kh. A. Maleque

Professor of Mathematics, American International University-Bangladesh, House-23, 17, Kamal Ataturk avenue, Banani, Dhaka-1213, Bangladesh, email: maleque@aiub.edu, malequekh@gmail.com

Abstract — We investigate a local similarity solution of an unsteady natural convection heat and mass transfer boundary layer incompressible fluid flow past a moving vertical porous plate in the presence of the heat absorption and generation. The effects of chemical reaction rate which is function of temperature and Arrhenius activation energy on the velocity, temperature and concentration are also studied in this paper. The governing partial differential equations are reduced to ordinary differential equations by introducing local similarity transformation (Maleque, 2010a). Numerical solutions to the reduced non-linear similarity equations are then obtained by adopting Runge-Kutta and shooting methods using the Nachtsheim- Swigert iteration technique. The results of the numerical solution are then presented graphically in the form of velocity, temperature and concentration profiles. The corresponding skin friction coefficient, the Nusselt number and the Sherwood number are also calculated and displayed in table showing the effects of various parameters on them.

Keywords — Heat and Mass Transfer; Chemical Reaction; Activation Energy; Porous Plate; Heat Generation/Absorption.

I. INTRODUCTION

In the past many researchers have published their papers on combined heat and mass transfer laminar free convective boundary layer flows for plate with various orientations. More information on this subject can be found in the monographs by Blums et al. (1987) and Gebhart et al. (1988) and in the papers by Khair and Bejan (1985), Mongruel et al. (1996), Lin and Wu (1997), Hossain and Rees(1999), Sattar and Maleque (2000), Maleque and Sattar (2002), Chaudhary and Sharma (2006), Ezzat et al. (2011), Maleque (2009, 2010b, 2011, 2013a).

In free convection boundary layer flows with simultaneous heat mass transfer, one important criteria that is generally not encountered is the species chemical reactions with finite Arrhenius activation energy. The Arrhenius law is usually of the form (Tencer et al., 2004).

(1)

where K is the rate constant of chemical reaction and B is the pre-exponential factor simply prefactor (constant), is based on the fact that increasing the temperature frequently causes a marked increase in the rate of reactions. E_a is the activation energy and k=8.61×10^-5 eV/°K is the Boltzmann constant. In areas such as geothermal or oil reservoir engineering, the above phenomenon is usually applicable. Apart from experimental works in these areas, it is also important to make some theoretical efforts to predict the effects of the activation energy in flows mentioned above. But in this regard very few theoretical works are available in literature. The reason is that the chemical reaction processes involved in the system are quite complex and generally the mass transfer equation that is required for all the reactions involved also become complex. Theoretically, such an equation is rather impossible to tackle. Form chemical kinetic viewpoint this is a very difficult problem, but if the reaction is restricted to binary type a lot of progress can be made. The thermomechanical balance equations for a mixture of general materials were first formulated by Truesdell (1957). Thereafter Mills (1966) and Beevers and Craine (1982) have obtained some exact solutions for the boundary layer flow of a binary mixture of incompressible Newtonian fluids. Several problems relating to the mechanics of oil and water emulsions, particularly with regard to applications in lubrication practice, have been considered within the context of a binary mixture theory by Al- Sharif et al. (1993) and Wang et al. (1993). A simple model involving binary reaction was studied by Bestman (1990). He considered the motion through the plate to be large which enabled him to obtain analytical solutions (subject to same restrictions) for various values of activation energy by employing the perturbation technique proposed by Singh and Diskshit (1988). Bestman (1991) and Alabraba et al. (1992) took into account the effect of the Arrhenius activation energy under the different physical conditions. Kandasamy et al. (2005) studied the combined effects of chemical reaction, heat and mass transfer along a wedge with heat source and concentration in the presence of suction or injection. Their result shows that the flow field is influenced appreciably by chemical reaction, heat source and suction or injection at the wall of the wedge. Recently Makinde et al. (2011) and Makinde and Olanrewaju (2011) studied the problems of unsteady convection with chemical reaction and radiative heat transfer past a flat porous plate moving through a binary mixture in an optically thin environment. More recently Maleque (2013b) investigated an unsteady natural convection boundary layer flow with mass transfer and a binary chemical reaction and also studied the effects of exothermic chemical reaction. In the present paper, we investigate a numerical solution of unsteady natural convection heat and mass transfer boundary layer incompressible fluid flow past a uniform moving flat porous plate taking into account the effect of Arrhenius activation energy and heat generation /absorption. The rate of chemical reaction is also considered the function of temperature. This problem is an extension work studied by Maleque (2013c).

II. GOVERNING EQUATIOS

We consider the boundary wall to be of infinite extent so that all quantities are homogeneous in x and hence all derivatives with respect to x are neglected. Thus the governing equations are

	(2)
	(3)
	(4)
	(5)

The boundary conditions of above system are

(6)

where (u,v) is the velocity vector, T is the temperature, C is the concentration of the fluid, υ is the kinemetic coefficient of viscosity, β and β^* are the coefficients of volume expansions for temperature and concentration respectively, κ is the heat diffusivity coefficient, c_p is the specific heat at constant pressure, D_m is the coefficient of mass diffusivity, k_r² is the chemical reaction rate constant and (T-T_∞)^w exp[-E_a/k(T-T_∞)] is the Arrhenius function where w is a unit less constant exponent fitted rate constants typically lie in the range -1<w<1 (http://www.iupac.org/goldbook/mo3963.pdf).

III. MATHEMATICAL FORMULATIONS

In order to solve the governing Eqs. (2) to (5) under the boundary conditions (6), we adopt the well defined similarity technique to obtain the similarity solutions. For this purpose the following non-dimensional variables are now introduced:

(7)

From the equation of continuity (3) we have

(8)

where v₀ is the dimensionless suction/injection velocity at the plate, v₀>0 corresponds to suction and v₀<0 corresponds to injection.

Introducing the dimensionless quantities from Eq. (8) and v from Eq. (9) in Eqs. (4), (5) and (6), we finally obtain the nonlinear ordinary differential equations as

	(9)
	(10)
	(11)

Here, Grashof number G_r=gβ(T_w-T_∞)δ²/(υU₀), Modified (Solutal) Grashof number G_m=gβ^*(C_w-C_∞)δ ²/(υU₀), Prandtl Number P_r=ρυc_p/κ, Schmidt number S_c=υ/D_M, The chemical reaction rate constant λ²=k_r²δ ²(T_w-T_∞)^w/υ, dimensionless activation energy E= E_a/[k(T_w-T_∞)]. The dimensionless heat generation / absorption coefficient γ=δ ²Q₀/(υρc_p). The Eqs. (9) to (11) are similar except for the term δδ ' /υ, where time t appears explicitly. Thus the similarity condition requires that δδ'/υ must be a constant quantity. Hence following Maleque (2010a) one can try a class of solutions of the Eqs. (10) to (12) by assuming that

(12)

From Eq. (10) we have

(13)

where the constant of integration L is determined through the condition that δ=L when t=0 . Here A=0 implies that δ=L represents the length scale for steady flow and A≠0 that is, δ represents the length scale for unsteady flow. Let us now consider a class of solutions for which A=2 and hence the length scale δ from Eq. (12) becomes

(14)

which exactly corresponds to the usual scaling factor considered for various unsteady boundary layer flow (Schlichting, 1968). Since δ is a scaling factor as well as similarity parameter any other values of A in Eq. (14) would not be change the nature of the solution except that the scale would be different. Finally introducing Eq. (14) in Eqs. (9) to (11) respectively, we have the following dimensionless nonlinear ordinary differential equations

	(15)
	(16)
	(17)

The boundary conditions Eq. (6) then becomes

(18)

In all over equations primes denote the differentiation with respect to η. Equations (15) to (17) are solved numerically under the boundary conditions (18) using Nachtsheim-Swigert iteration technique.

IV. NUMERICAL SOLUTIONS

Equations (15) -(17) are solved numerically under the boundary conditions (18) using Nachtsheim and Swigert (1965) iteration technique. Thus adopting this numerical technique, a computer program was set up for the solutions of the basic non-linear differential equations of our problem where the integration technique was adopted as the six ordered Range-Kutta method of integration. The results of this integration are then displayed graphically in the form of velocity, temperature and concentration profiles in Figs. (1)-(15). In the process of integration, the local skin-friction coefficient, the local rates of heat and mass transfer to the surface, which are of chief physical interest are also calculated out. The equations defining the wall skin-friction, Nusselt number and Sherwood number are

	(19)
	(20)

and

(21)

Here, the Reynolds number R_e=ρU₀δ/μ.

Fig. 1: Effect of G_r on the velocity profiles.

Fig. 2: Effect of G_m on the velocity profiles.

Fig. 3: Effects of λ on the concentration profiles.

Fig. 4: Effects of λ on the temperature profiles.

Fig. 5: Effects of λ on the velocity profiles.

Fig. 6: Effects of E on the concentration profiles.

Fig. 7: Effects of E on the temperature profiles.

Fig. 8: Effects of E on the velocity profiles.

Fig. 9: Effects of v₀ on the velocity profiles.

Fig. 10: Effects of v₀ on the temperature profiles.

Fig. 11: Effects of v₀ on the concentration profiles.

Fig. 12: Effects of γ on the velocity profiles.

Fig. 13: Effects of γ on the temperature profiles.

Fig. 14: Effects of γ on the concentration profiles.

Fig. 15: Comparison of our calculated velocity profile and the velocity profile of Bestman (1990).

These above coefficients are then obtained from the procedure of the numerical computations and are sorted in Table 1.

Table 1: Effects of G_r, λ, w, E and v₀ on the skin friction coefficient, Nusselt number and the Sherwood number for G_r=10, G_m=1, P_r=0.7, v₀=0.6, λ=5, E=1 and γ=0.

V. RESULTS AND DISCUSSIONS

The parameters entering into the fluid flow are Grashof number G_r, Solutal (modified) Grashof number G_m, suction parameter v₀, Prandtl number P_r, the non dimensional chemical reaction rate constant λ², Schmidt number S_c and the non-dimensional activation energy E.

It is, therefore, pertinent to inquire the effects of variation of each of them when the others are kept constant. The numerical results are thus presented in the form of velocity profiles, temperature profiles and concentration profiles in Figs.1-14 for the different values of G_r, G_m , λ, E, v₀ and γ. The value of G_r is taken to be both positive and negative, since these values represent respectively cooling and heating of the plate. The values of G_r is taken to be large (G_r =10), since the value corresponds to a cooling problem that is generally encountered in nuclear engineering in connection with the cooling of reactors. In air (P_r= 0.71) the diluting chemical species of most common interest have Schmidt number in the range from 0.6 to .75. Therefore, the Schmidt number S_c=0.6 is considered. In particulars, 0.6 corresponds to water vapor that represents a diffusing chemical species of most common interest in air. The values of the suction parameter v₀ are taken to be large. Apart from the above Figures and Tables, the representative velocity, temperature and concentration profiles and the values of the physically important parameters, i.e. the local shear stress, the local rates of heat and mass transfers, are illustrated for uniform wall temperature and species concentration in Figs. 1-14 and in Table 1.

The velocity profiles generated due to impulsive motion of the plate are plotted in Fig.1 for both cooling (G_r>0) and heating (G_r<0) of the plate keeping other parameters fixed G_m=1.0, P_r=0.71, S_c=0.6, λ=5, v₀=3.0, E=1.0 and w=1. In Fig.1, velocity profiles are shown for different values of G_r. We observe that velocity increases with increasing values of G_r for the cooling of the plate. From this figure it is also observed that the negative increase in the Grashof number leads to the decrease in the velocity field. That is, for heating of the plate (Fig.1), the effects of the Grashof number G_r on the velocity field have also opposite effects, as compared to the cooling of the plate. Solutal Grashof number G_m>0 corresponds that the chemical species concentration in the free stream region is less than the concentration at the boundary surface. Figure 2 presents the effects of Solutal Grashof number G_m on the velocity profiles. It is observed that the velocity profile increases with the increasing values of Solutal Grashof number G_m.

Considering chemical reaction rate constant λ² is always positive. Figures 3-5 represent the effect of chemical rate constant λ on the concentration, the temperature and the velocity profiles respectively. We observe from the last part of Eq. (18) that λ²exp(-E/θ) increases with the increasing values of λ. We also observe from this equation that increase in λ²exp(-E/θ) means increase in λ leads to the decrease in the concentration profiles. This is in great agreement with Fig.3. It is observed from the Eq. (1) that increasing temperature frequently causes a marked increase in the rate of reactions is shown in Fig. 4. The parameter λ does not enter directly into the momentum equation but its influence comes through the mass equation. Figure5 shows the variation of the velocity profiles for different values of λ. The velocity profile increases with the increasing values of λ.

Activation energy may be defined as the minimum energy required starting a chemical reaction. The activation energy of a reaction is usually denoted by Ea, and given in units of kilojoules per mole. Effects of Activation Energy (E) on the concentration, the temperature and velocity profiles are shown in fig.6 to Fig.8 respectively. From Eq. (1) we observe that chemical reaction rate (K) decreases with the increasing values of activation energy (Ea). We also observe from Eq. (18) that increase in activation energy (E) leads to decrease λ²exp(-E/θ) as well as to increase the concentration profiles shown in Fig. 6. Equation (1) suggests that the activation energy is dependent on temperature. For fixed value of chemical reaction rate constant the temperature profile increases with the increasing values of activation energy shown in Fig.7. The parameter E does not enter directly into the momentum equation but its influence comes through the mass equation. Figure 8 shows the variation of the velocity profiles for different values of E. From this figure it has been observed that the velocity profile increases with the increasing values of E.

The effects of suction and injection (v₀) for λ= 5, E=1, G_r= 5, G_m=1, S_c=0.6 and P_r=0.71 on the velocity profiles, temperature profiles and concentration profiles are shown respectively in Fig. 9 to Fig. 11. For strong suction (v₀>0), the velocity, the temperature and the concentration profiles decay rapidly away from the surface. The fact that suction stabilizes the boundary layer is also apparent from these figures. As for the injection (v₀<0) , from Fig. 9 to Fig. 11 it is observed that the boundary layer is increasingly blown away from the plate to form an interlayer between the injection and the outer flow regions.

The positive value and the negative value of γ represent the heat generation and heat absorption respectively. The dimensionless heat generation/ absorption coefficient γ does not enter directly into the equation of continuity, Navier Stokes equations and concentration equation, its influence comes through the energy equation. In the presence of the buoyancy force, it appears that the heat generation/ absorption coefficient γ has marked effect on the boundary layer and the velocity profiles of the fluid is shown in Fig. 12. From this figure it has been observed that the effect of heat generation coefficient is to expand the boundary layer thickness and opposite effect is found for the effect of heat absorption. Thus the dimensionless heat generation/absorption coefficient has same effects on skin-friction coefficients shown in Table 1. The effects of heat absorption coefficient (γ<0) on temperature profiles and the Nusselt number are shown in Fig. 13 and Table 1. In general, the effect of heat absorption coefficient (γ<0) is to reduce the thickness of the thermal boundary layer. The temperature profiles decrease as the heat absorption coefficient γ (<0) increases as shown in Fig. 13. On the other hand, the heat generation coefficient γ (>0) leads to increase the temperature profiles. Thermal boundary layer thickness expands for increasing values of the heat generation coefficient γ (>0) as shown in Fig. 13

Finally the effects of heat generation/absorption coefficient (γ) on the dimensionless Nusselt number N_u is shown in Table 1. The Nusselt number increases with the increasing values of the heat absorption coefficient (γ<0). This is due to the fact that the heat absorption increases, the disk tends to supply more energy through its wall to maintain the constant temperature of the wall, and hence the Nusselt number increases. Opposite effects are found for the increasing values of the heat generation coefficient (γ> 0).

The dimensionless heat generation/absorption coefficient γ does not enter directly into the concentration equation, its influence comes through the energy equation. But in the presence of the activation energy, it appears that the heat generation/absorption coefficient γ has negligible effect on the concentration profiles of the fluid. The concentration profiles decrease for increasing values of γ as shown in Fig. 14. Also it has been found from Table 1 that Sherwood number S_h increases with the increasing values of γ.

The skin-friction, the heat transfer and the mass transfer coefficients are tabulated in Table 1 for different values of G_r, λ, E, w, v₀ and γ. We observe from Table 1 that the skin-friction coefficient (-f′(0)) decreases with increase in the Grashop number G_r. It is also observed from Table-1 that the rate of mass transfer coefficient increases with the increasing values of λ and the mass transfer coefficient decreases with the increasing values of E and w. It is observed from this table that this is an increasing effect of suction parameter v₀ on the skin friction coefficient, heat and mass transfer coefficients. Finally the effects of heat generation/ absorption coefficient (γ) on the dimensionless Nusselt number N_u is shown in Table 1. The Nusselt number increases with the increasing values of the heat absorption coefficient (γ<0). This is due to the fact that the heat absorption increases, the disk tends to supply more energy through its wall to maintain the constant temperature of the wall, and hence the Nusselt number increases. Opposite effects are found for the increasing values of the heat generation coefficient (γ>0).

Comparison is made between our numerical result and the analytical result of Bestman (1990). Bestman (1990) studied steady natural convective boundary layer flow with large suction. He solved his problem analytically by employing the perturbation technique proposed by Singh and Diskshit (1988). In our present work, taking A = 0 in Eq. (13) for considering steady flow. The values of the suction parameter v₀² = 10 is taken to see the effects of large suction. G_r=G_m=1.0, λ=5, E=5 and w=1 are also chosen with a view to compare our numerical results with the analytical results of Bestman (1990). The comparison of velocity profiles as seen in Fig. 15 highlight the validity of the numerical computations adapted in the present investigation.

VI. CONCLUSIONS

In this paper, we investigate the effects of chemical reaction rate and Arhenious activation energy on an unsteady natural convection heat and mass transfer boundary layer flow past a flat porous plate. The Nachtsheim and Swigert (1965) iteration technique based on sixth-order Range-Kutta and Shooting method has been employed to complete the integration of the resulting solutions.

The following conclusions can be drawn as a result of the computations:

1. Velocity increases with increasing values of G_r for the cooling of the plate and the negative increase in the Grashof number leads to the decrease in the velocity field. That is, for heating of the plate the effects of the Grashof number G_r on the velocity field have also opposite effects, as compared to the cooling of the plate.
2. Solutal Grashof number G_m>0 corresponds that the chemical species concentration in the free stream region is less than the concentration at the boundary surface. It is observed that the velocity profile increases with the increasing values of Solutal Grashof number G_m.
3. Increase in λ leads to the decrease in the concentration profiles. It is observed from the Eq. (1) that increasing temperature frequently causes a marked increase in the rate of reactions λ. The velocity profile slightly increases with the increasing values of λ.
4. Increase in activation energy (E) leads to increase the concentration, temperature and velocity profiles.
5. For strong suction (v₀>0), the velocity, the temperature and the concentration profiles decay rapidly away from the surface. As for the injection (v₀<0), it is observed that the boundary layer is increasingly blown away from the plate to form an interlayer between the injection and the outer flow regions.
6. The Nusselt number increases with the increasing values of the heat absorption coefficient (γ<0). This is due to the fact that the heat absorption increases, the disk tends to supply more energy through its wall to maintain the constant temperature of the wall, and hence the Nusselt number increases.

NOMENCLATURE

(u, v): components of velocity field.

c_p : specific heat at constant pressure

N_u : Nusselt number

P_r: Prandtl number

Q₀: Heat generation/absorption coefficient

S_h : Sherwood number

T : Temperature of the flow field

T_∞ : Temperature of the fluid at infinity

T_w: Temperature at the plate

f : Dimensionless similarity functions

E_a: The activation energy

k : The Boltzmann constant

S_c: Schmidt number

E : The non-dimensional activation energy

G_r: Grashof number

G_m : Modified (Solutal) Grashof number

GREEK SYMBOLS

θ : Dimensionless temperature

: Dimensionless concentration

η : Dimensionless similarity variable

δ : Scale factor

υ : Kinematic viscosity

ρ : Density of the fluid

κ : Thermal conductivity,

τ : Shear stress

μ : Fluid viscosity

λ² : Non-dimensional chemical reaction rate constant

β and β^* : the coefficients of volume expansions for temperature and concentration respectively,

γ : The dimensionless heat generation/ absorption coefficient

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Received: November 6, 2012
Accepted: July 4, 2013
Recommended by Subject Editor: Walter Ambrosini