# Effects of Exothermic/Endothermic Chemical Reactions with Arrhenius Activation Energy on MHD Free Convection and Mass Transfer Flow in Presence of Thermal Radiation

Authors: Kh. Abdul Maleque
Journal: Journal of Thermodynamics
 (2013)

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## Abstract

A local similarity solution of unsteady MHD natural convection heat and mass transfer boundary layer flow past a flat porous plate within the presence of thermal radiation is investigated. The effects of exothermic and endothermic chemical reactions with 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 locally similarity transformation (Maleque (2010)). Numerical solutions to the reduced nonlinear 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 obtained for both steady and unsteady cases then presented graphically in the form of velocity, temperature, and concentration profiles. Comparison has been made for steady flow (A=0) and shows excellent agreement with Bestman (1990), hence encouragement for the use of the present computations.


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## Body

## 1. Introduction

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 modified Arrhenius law (IUPAC Goldbook definition of modified Arrhenius equation) is usually of the form (Tencer et al. [1])
(1)K=B(TT∞)nexp[-EakT],
whereKis the rate constant of chemical reaction and B that is the preexponential factor simply prefactor (constant) is based on the fact that increasing the temperature frequently causes a marked increase in the rate of reactions. Ea is the activation energy, and k=8.61×10-5 eV/K is the Boltzmann constant which is the physical constant relating energy at the individual particle level with temperature observed at the collective or bulk level.In areas such as geothermal or oil reservoir engineering, the prvious 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 the 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 becomes 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 [2]. Thereafter Mills [3] and Beevers and Craine [4] 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. [5] and Wang et al. [6].A simple model involving binary reaction was studied by Bestman [7]. 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 Dikshit [8]. Bestman [9] and Alabraba et al. [10] took into account the effect of the Arrhenius activation energy under the different physical conditions. Recently Kandasamy et al. [11] 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. [12, 13] 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 Abdul Maleque [14, 15] investigated the similarity solution on unsteady incompressible fluid flow with binary chemical reactions and activation energy. In the present paper, we investigate a numerical solution of unsteady Mhd natural convection heat and mass transfer boundary layer flow past a flat porous plate taking into account the effect of Arrhenius activation energy with exothermic/endothermic chemical reactions and thermal radiation. This problem is an extension work studied by Abdul Maleque [16].


## 2. Governing Equations

We consider the boundary wall to be of infinite extend, so that all quantities are homogeneous inx, and hence all derivatives with respect to x are neglected. The x-axis is taken along the plate, and y-axis is perpendicular to the plate. Assume that a uniform magnetic field B0 is applied perpendicular to the plate and the plate is moving with uniform velocity Uo in its own plane. In presence of exothermic/endothermic binary chemical reaction with Arrhenius activation energy, a uniform magnetic field and thermal radiation   thus the governing equations are
(2)∂v∂y=0,(3)∂u∂t+v∂u∂y=υ∂2u∂y2+gβ1(T-T∞)+gβ2(C-C∞)-σB02ρu,(4)ρcp(∂T∂t+v∂T∂y)=κ∂2T∂y2+βkr2(TT∞)n×exp[-EakT](C-C∞)-∂qr∂y,(5)∂C∂t+v∂C∂y=Dm∂2C∂y2-kr2(TT∞)nexp[-EakT](C-C∞).The boundary conditions of previous system are(6)u=U0,v=v(t),T=Tw,C=Cwaty=0,t>0,u=0,T⟶T∞,C⟶C∞aty=∞,t>0,
where(u,v)is the velocity vector, Tis the temperature, Cis the concentration of the fluid, μ is the fluid viscosity, σ is the electrical conductivity, υ is the kinematic coefficient of viscosity, β1 and β2 are the coefficients of volume expansions for temperature and concentration, respectively, κ is the heat diffusivity coefficient, cp is the specific heat at constant pressure, Dm is the coefficient of mass diffusivity, kr2 is the chemical reaction rate constant, β(=±1) is exothermic/endothermic parameter, and (T/T∞)nexp[-Ea/kT] is the Arrhenius function wherenis a unit less constant exponent fitted rate constants typically lie in the range -1<n<1.The radiative heat fluxqr is described by Roseland approximation such that
(7)qr=-4σ13k1∂T4∂y,
where σ1 and k1 are the Stefan Boltzmann constant and mean absorption coefficient, respectively. Following Makinde and Olanrewaju [13], we assume that the temperature differences within the flow are sufficiently small, so that the Tn can be expressed as a linear function. By using Taylor’s series, we expand Tn about the free stream temperature T∞ and neglecting higher order terms. This result of the following approximation:
(8)Tn≈(1-n)T∞n+nT∞n-1T.
Using (9), we have
(9)∂qr∂y=-4σ13k1∂2T4∂y2≈-16T∞3σ13k1∂2T∂y2,(10)(TT∞)n≈(1-n)+nTT∞.


## 3. Mathematical Formulations

In order to solve the governing equations (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 nondimensional variables are now introduced:
(11)η=yδ(t),uU0=f(η),T-T∞Tw-T∞=θ(η),C-C∞Cw-C∞=ϕ(η).
From the equation of continuity (3), we have
(12)v(t)=-v0υδ(t),
where v0 is the dimensionless suction/injection velocity at the plate, v0>0 corresponds to suction, and v0<0 corresponds to injection.Introducing (9), (10), the dimensionless quantities from (11) and v from (12) in (4), (5), and (6), we finally obtain the nonlinear ordinary differential equations as
(13)f′′+(ηδδ′υ+v0)f′-Mf+G0γθ+Gmϕ=0,(14)θ′′+(Pr1+N)[(ηδδ′υ+v0)θ′+βλ2(1+nγθ)exp(-E1+γθ)ϕ(ηδδ′υ+v0)]=0,(15)Sc-1ϕ′′+(ηδδ′υ+v0)ϕ′-λ2(1+nγθ)exp(-E1+γθ)ϕ=0.
Here, Grashof number Gr=G0γ=(gβT∞δ2/υU0)γ; the temperature relative parameter γ=(Tw-T∞)/T∞.Modified (Solutal) Grashof numberGm=gβ*(Cw-C∞)δ2/υU0; Prandtl number Pr=ρυcp/κ; magnetic interaction parameter M=σδ2B02/μ; schmidt number Sc=υ/DM; the nondimensional activation energy E=Ea/k(Tw-T∞); the dimensionless chemical reaction rate constant λ2=kr2δ2/υ; radiation parameter N=16σ1T∞3/3κk1.Equations from (13) to (15) 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 Abdul Maleque [17], one can try a class of solutions of (13) to (15) by assuming that
(16)δδ′υ=A(constant).
From (16), we have
(17)δ(t)=2Aυt+L2,
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. Since δ is a scaling factor as well as similarity parameter, any other values of A in (13) would not change the nature of the solution except that the scale would be different. Finally introducing (16) in (13) to (15), respectively, we have the following dimensionless nonlinear ordinary differential equations:
(18)f′′+(Aη+v0)f′-Mf+G0γθ+Gmϕ=0(19)θ′′+(Pr1+N)[(-E1+γθ)(Aη+v0)θ′+βλ2(1+nγθ)exp(-E1+γθ)ϕ]=0,(20)Sc-1ϕ′′+(Aη+v0)ϕ′-λ2(1+nγθ)exp(-E1+γθ)ϕ=0.
The boundary conditions equation (4) becomes
(21)f(0)=1,θ(0)=1,ϕ(0)=1,f(∞)=0,θ(∞)=0,ϕ(∞)=0.In all over equations primes denote the differentiation with respect toη. Equations from (18)–(20) are solved numerically under the boundary conditions (21) using Nachtsheim-Swigert iteration technique.


## 4. Numerical Solutions

Equations (18)–(20) are solved numerically under the boundary conditions (21) using Nachtsheim-Swigert [18] iteration technique. In (21), there are three asymptotic boundary conditions and hence follow three unknown surface conditions f′(0), θ′(0), and ϕ′(0). Within the context of the initial-value method and the Nachtsheim-Swigert iteration technique the outer boundary conditions may be functionally represented by the first-order Taylor’s series as
(22)f(ηmax)=f(X,Y,Z)=fc(ηmax)+ΔXfX+ΔYfY+ΔZfZ=δ1,(23)θ(ηmax)=θ(X,Y,Z)=θc(ηmax)+ΔXθX+ΔYθY+ΔZθZ=δ2,(24)ϕ(ηmax)=ϕ(X,Y,Z)=ϕc(ηmax)+ΔXϕX+ΔYϕY+ΔZϕZ=δ3,
with the asymptotic convergence criteria given by
(25)f′(ηmax)=f′(X,Y,Z)=fc′(ηmax)+ΔXfX′+ΔYfY′+ΔZfZ′=δ4,(26)θ′(ηmax)=θ′(X,Y,Z)=θc′(ηmax)+ΔXθX′+ΔYθY′+ΔZθZ′=δ5,(27)ϕ′(ηmax)=ϕ′(X,Y,Z)=ϕc′(ηmax)+ΔXϕX′+ΔYϕY′+ΔZϕZ′=δ6,
where X=f′(0), Y=θ′(0), Z=ϕ′(0), and X,Y,Z subscripts indicate partial differentiation, for example, fY′=∂f′(ηmax)/∂θ′(0). The subscriptcindicates the value of the function at ηmax to be determined from the trial integration.Solutions of these equations in a least square sense require determining the minimum value ofE=δ12+δ22+δ32+δ42+δ52+δ62 with respect to X, Y, and Z. To solve ΔX, ΔY, and ΔZ, we require to differentiate E with respect to X, Y and Z, respectively. Thus adopting this numerical technique, a computer program was set up for the solutions of the basic nonlinear differential equations of our problem where the integration technique was adopted as the six-ordered Runge-Kutta method of integration. The results of this integration are then displayed graphically in the form of velocity, temperature, and concentration profiles in Figures 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 equation defining the wall skin friction is
(28)τ=-μ(∂u∂y)y=0=-μU0δf′(0).
Hence the skin-friction coefficient is given by
(29)12Cf=τρU02⟹12ReCf=-f′(0),
here, the Reynolds number Re=ρU0δ/μ.Figure 1
Effects ofβ and E on temperature profiles.Figure 2
Effects ofβ and E on concentration profiles.Figure 3
Effects ofβ and E on velocity profiles.Figure 4
Effects ofβ andλ on velocity profiles.Figure 5
Effects ofβ and λ on temperature profiles.Figure 6
Effects ofβ and λ on concentration profiles.Figure 7
Effects ofN on temperature profiles.Figure 8
Effects ofN on velocity profiles.Figure 9
Effects ofγ on velocity profiles.Figure 10
Effects ofγ on temperature profiles.Figure 11
Effects ofγ on concentration profiles.Figure 12
Effects ofv0 on velocity profiles.Figure 13
Effects ofv0 on temperature profiles.Figure 14
Effects ofGm on velocity profiles.Figure 15
Effects ofM on velocity profiles.The heat flux(qw) and the mass flux (Mw) at the wall are given by
(30)qw=-κ(∂T∂y)y=0=-κΔTδθ′(0),Mw=-DM(∂C∂y)y=0=-DMΔCδϕ′(0).
Hence the Nusselt number (Nu) and the Sherwood number (Sh) are obtained as
(31)Nu=qwδκΔT=-θ′(0),Sh=MwδDMΔC=-ϕ′(0).
These previous coefficients are then obtained from the procedure of the numerical computations and are sorted in Table 1.Table 1
Effects on skin friction, heat transfer, and mass transfer coefficients. ForG0=10, Gm=1, N=0.5, Pr=0.71,M=0.5,Sc=0.6,λ=5, n=0.5, v0=3, γ=0.5, and E=1.
-
f
′
(
0
)
-
θ
′
(
0
)
-
ϕ
′
(
0
)
E for β=1
0.0
−1.65163
−0.91802
5.78634
0.5
−1.60009
−0.88980
5.75523
1.0
−1.54562
−0.85343
5.70174
E for β=-1
0.0
1.30241
4.49966
5.55159
0.5
0.98702
4.19573
5.21093
1.0
0.63543
3.87806
4.81246
λ for β=1
1.0
−0.43158
1.73298
2.42627
2.0
−0.63434
1.34699
2.91446
5.0
−1.54562
−0.85343
5.70174
λ for β=-1
1.0
−0.27903
1.98971
2.42088
2.0
−0.07554
2.32400
2.84424
5.0
0.63543
3.87806
4.81246


## 5. Results and Discussions

The parameters entering into the fluid flow are Grashof numberGr=G0γ, Solutal (modified) Grashof number Gm, suction parameter v0, Prandtl number Pr, the nondimensional chemical reaction rate constant λ2, Schmidt number Sc, the magnetic parameter M, the nondimensional activation energy E, the radiation parameter N, the exothermic/endothermic parameter β, and the temperature relative parameter γ.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 Figures1–14 for the different values of Gr,Gm,λ,M,E,v0,β, and γ. The value of γ is taken to be both positive and negative, since these values represent, respectively, cooling and heating of the plate.The values ofGr is taken to be large (Gr=10, G0=10, and γ=1), since the value corresponds to a cooling problem that is generally encountered in nuclear engineering in connection with the cooling of reactors. In air (Pr=0.71) the diluting chemical species of most common interest have Schmidt number in the range from 0.6 to 0.75. Therefore, the Schmidt number Sc=0.6 is considered. In particular, 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 figures and tables, the representative velocity, temperature, and concentration profiles and the values of the physically important parameters, that is, the local shear stress, the local rates of heat and mass transfer, are illustrated for uniform wall temperature and species concentration in Figures 1–17 and in Tables 1-2.Table 2
Effects on skin friction, heat transfer, and mass transfer coefficients. ForG0=10, Gm=1, N=0.5, Pr=0.71,M=0.5,Sc=0.6,λ=5, n=0.5,v0=3, γ=0.5, and E=1.
-
f
′
(
0
)
-
θ
′
(
0
)
-
ϕ
′
(
0
)
N
0.0
−0.87260
−1.44125
5.68996
2.0
−3.05961
−0.30027
5.71624
5.0
−4.93466
−0.04756
5.72770
M
0.0
−1.85010
−0.85309
5.70164
2.0
−0.75246
−0.85443
5.70203
5.0
0.46744
−0.85628
5.70257
γ
−2.00000
7.32171
3.65294
0.07835
−1.00000
5.66728
3.48636
0.26098
0.00000
3.31180
1.88820
2.28076
1.00000
−1.54562
−0.85343
5.70174
2.00000
−7.85616
−3.67839
9.25776
A
0.00000
−1.27274
−0.26974
3.97243
2.00000
1.20419
0.56519
3.91549Figure 16
Effects ofA on velocity profiles.Figure 17
Effects ofA on temperature profiles.


### 5.1. Effects of Activation Energy (E) on the Concentration, the Temperature, and the Velocity Profiles for Exothermic/Endothermic Chemical Reaction

In chemistry,activation energy is defined as the energy that must be overcome in order for a chemical reaction to occur. Activation energy may also 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.An exothermic reaction is a chemical reaction that releases energy in the form of light and heat. It is the opposite of an endothermic reaction. It gives out energy to its surroundings. The energy needed for the reaction to occur is less or greater than the total energy released for exothermic or endothermic reaction, respectively. Energy is obtained from chemical bonds. When bonds are broken, energy is required. When bonds are formed, energy is released. Each type of bond has specific bond energy. It can be predicted whether a chemical reaction will release or require heat by using bond energies. When there is more energy used to form the bonds than to break the bonds, heat is given off. This is known as an exothermic reaction. When a reaction requires an input or output of energy, it is known as an exothermic or endothermic reaction.Effects of activation energy (E) and exothermic parameter (β) on the temperature, the concentration and velocity profiles are shown in Figure 1 to Figure 3, respectively. β=1 and β=-1 represent exothermic and endothermic chemical reactions, respectively. Figure 1 shows that a small decreasing effect of temperature profile is found for increasing values of E for exothermic reaction (β=1), but mark opposite effects are found for endothermic reaction in Figure 1. That is the temperature profile increases for increasing values of E for endothermic reaction β=-1. From (1) we observe that chemical reaction rate (K) decreases with the increasing values of activation energy (Ea). We also observe from (20) that increase in activation energy (E) leads to decrease λ2exp(-E/θ) as well as to increase in the concentration profiles shown in Figure 2. Small and reported effects are found for β=1 and β=-1, respectively. The parameter E does not enter directly into the momentum equation, but its influence comes through the mass and energy equations. Figure 3 shows the variation of the velocity profiles for different values of E. From this figure it has been observed that the velocity profile mark increases with the increasing values of E for endothermic chemical reaction. But negligible effects are found for exothermic reaction.


### 5.2. Effects ofλ on the Concentration, the Temperature, and the Velocity Profiles for Exothermic/Endothermic Reaction

Considering chemical reaction rate constantλ2 is always positive. Figures 4–6 represent the effect of chemical rate constant λ on the velocity, the temperature, and the concentration profiles, respectively.We observe from Figures4 and 5 that velocity and temperature profiles increase with the increasing values of λ for exothermic reaction, but opposite effects are found in these figures for endothermic reaction. It is observed from (1) that increasing temperature frequently that causes a marked increase in the rate of reactions is shown in Figure 4. As the temperature of the system increases, the number of molecules that carry enough energy to react when they collide also increases. Therefore, the rate of reaction increases with temperature. As a rule, the rate of a reaction doubles for every 10°C increase in the temperature of the system.Last part of (20) shows that λ2exp(-E/θ) increases with the increasing values of λ. We also observe from this equation that increase in λ2exp(-E/θ) means that increase inλleads to the decrease in the concentration profiles. This is in great agreement with Figure 6 for both cases β=±1. It is also observed from this figure that for exothermic reaction the mass boundary layer is close to the plate other than endothermic reaction.


### 5.3. Effects of Radiation ParameterN on Temperature and Velocity Profiles

The effects of radiation parameterN on the temperature and the velocity profiles are shown in Figures 7 and 8. From Figure 7, it can be seen that an increase in the values of N leads to a decrease in the values of temperature profiles within the thermal boundary layers η<0.22, while outside η>0.22, the temperature profile gradually increases with the increase of the radiation parameter N. It is appear from Figure 8 that when N=0, that is, without the radiation, the velocity profile shows its usual trend of gradually decay. As radiation becomes larger the profiles overshoot the uniform velocity close to the boundary.


### 5.4. Effects ofγ on the Velocity, Temperature, and Concentration Profiles

The value ofγ=ΔT/T∞ is taken to be both positive and negative, since these values represent, respectively, cooling and heating of the plate. The effects of temperature relative parameter on velocity, temperature, and concentration profiles are shown in Figures 9–11.The velocity profiles generated due to impulsive motion of the plate is plotted in Figure9 for both cooling (γ>0) and heating (γ<0) of the plate keeping other parameters fixed (G0=10, Gm=1.0, M=Ec=0.5, Pr=0.71, Sc=0.6, λ=5, v0=3.0, E=1.0, and n=1). In Figure 9, velocity profiles are shown for different values of γ. We observe that velocity increases with increasing values of γ for the cooling of the plate. From this figure it is also observed that the negative increase in the temperature relative parameter leads to the decrease in the velocity field. That is, for heating of the plate (Figure 9), the effects of the γ on the velocity field have also opposite effects, as compared to the cooling of the plate. Figure 10 shows the effects of temperature relative parameter γ on the temperature profiles. It has been observed from this figure that for heating plate the temperature profile shows its usual trend of gradually decay, and the thermal boundary layer is close to the plate. But for cooling plate that is temperature relative parameter γ becomes larger, the profiles overshoot the uniform temperature close to the thermal boundary. Opposite effects of temperature profile are found for concentration profiles shown in Figure 11.


### 5.5. Effects of Suction/Injection (v0) on the Velocity and the Temperature Profiles

The effects of suction and injection (v0) for G0=10, Gm=1.0, M=Ec=0.5, Pr=0.71, Sc=0.6, λ=5, γ=0.5, E=1.0, and n=1on the velocity profiles and temperature profiles are shown, respectively, in Figures 12 and 13. For strong suction (v0>0), the velocity and the temperature 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 Figures 12 and 13 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.


### 5.6. Effects ofGm and M on the Velocity Profiles

The effects of Solutal Grashof numberGm and magnetic interaction parameter M on velocity profiles are shown in Figures 14 and 15, respectively. Solutal Grashof number Gm>0 corresponds that the chemical species concentration in the free stream region is less than the concentration at the boundary surface. Figure 14 presents the effects of Solutal Grashof number Gm on the velocity profiles. It is observed that the velocity profile increases with the increasing values of Solutal Grashof number Gm.Imposition of a magnetic field to an electrically conducting fluid creates a drag like force called Lorentz force. The force has the tendency to slow down the flow around the plate at the expense of increasing its temperature. This is depicted by decreases in velocity profiles as magnetic parameterM increases as shown in Figure 15.


### 5.7. Effects ofA on the Velocity and the Temperature Profiles

HereA=0 and A≠0 represent steady and unsteady flows, respectively. The effects ofAon velocity and temperature profiles are shown in Figures 16 and 17, respectively. For unsteady flow (A≠0) both figures show that the profiles rapidly decay and close to the plate. But for steady flow (A=0) the profiles overshoot the uniform velocity/temperature close to the boundary.


### 5.8. The Skin-Friction, the Heat Transfer, and the Mass Transfer Coefficients

The skin-friction, the heat transfer, and the mass transfer coefficients are tabulated in Tables1 and 2 for different values of E, β, λ, N, M, γ, and A. We observe from Table 1 that the skin-friction coefficient (-f′(0)) and the Nusselt number Nu(=-θ′(0)) increase for increasing values of dimensionless activation parameter E for exothermic chemical reaction, but opposite actions are found for endothermic reaction. That is for endothermic reaction, it is found that both shearing stress and heat transfer coefficients decrease for increasing values of E. For both cases exothermic/endothermic reactions, the mass transfer coefficient (=-ϕ′(0)) decreases for increasing values of E. We also observe from Table 1 that the skin-friction coefficient (-f′(0)) and the Nusselt number Nu(=-θ′(0)) decrease for increasing values of dimensionless reaction rate constant λ for exothermic chemical reaction, but opposite effects are found for endothermic reaction. That is for endothermic reaction, it is found that both shearing stress and heat transfer coefficients increase for increasing values of λ. For both cases exothermic/endothermic reactions, the mass transfer coefficient (=-ϕ′(0)) remarkably increases for increasing values of λ. From Table 2 it has been observed that the skin-friction coefficient remarkably decreases and Nusselt number remarkable increases for increasing values of radiation parameter N. The skin-friction coefficient increases for increasing values of magnetic parameter M. From Table 2 we also found that skin friction and heat transfer coefficients decrease, but mass transfer coefficient increases for increasing values of temperature relative parameter γ.


### 5.9. Comparison between Our Numerical Result and the Analytical Result of Bestman [7]

Bestman studied steady natural convective boundary layer flow with large suction. He solved his problem analytically by employing the perturbation technique proposed by Singh and Dikshit [8]. In our present work, take A=0 in (16) for considering steady flow. The values of the suction parameter v02=10 is taken to see the effects of large suction. Gr=Gm=1.0, γ=1, M=N=0, λ=5, E=5, and n=1 are also chosen with a view to compare our numerical results with the analytical results of Bestman [7]. The comparison of velocity profiles as seen in Figure 18 highlights the validity of the numerical computations adapted in the present investigation.Figure 18
Comparison of our calculated velocity profile and the velocity profile of Bestman [7].


## 5.1. Effects of Activation Energy (E) on the Concentration, the Temperature, and the Velocity Profiles for Exothermic/Endothermic Chemical Reaction

In chemistry,activation energy is defined as the energy that must be overcome in order for a chemical reaction to occur. Activation energy may also 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.An exothermic reaction is a chemical reaction that releases energy in the form of light and heat. It is the opposite of an endothermic reaction. It gives out energy to its surroundings. The energy needed for the reaction to occur is less or greater than the total energy released for exothermic or endothermic reaction, respectively. Energy is obtained from chemical bonds. When bonds are broken, energy is required. When bonds are formed, energy is released. Each type of bond has specific bond energy. It can be predicted whether a chemical reaction will release or require heat by using bond energies. When there is more energy used to form the bonds than to break the bonds, heat is given off. This is known as an exothermic reaction. When a reaction requires an input or output of energy, it is known as an exothermic or endothermic reaction.Effects of activation energy (E) and exothermic parameter (β) on the temperature, the concentration and velocity profiles are shown in Figure 1 to Figure 3, respectively. β=1 and β=-1 represent exothermic and endothermic chemical reactions, respectively. Figure 1 shows that a small decreasing effect of temperature profile is found for increasing values of E for exothermic reaction (β=1), but mark opposite effects are found for endothermic reaction in Figure 1. That is the temperature profile increases for increasing values of E for endothermic reaction β=-1. From (1) we observe that chemical reaction rate (K) decreases with the increasing values of activation energy (Ea). We also observe from (20) that increase in activation energy (E) leads to decrease λ2exp(-E/θ) as well as to increase in the concentration profiles shown in Figure 2. Small and reported effects are found for β=1 and β=-1, respectively. The parameter E does not enter directly into the momentum equation, but its influence comes through the mass and energy equations. Figure 3 shows the variation of the velocity profiles for different values of E. From this figure it has been observed that the velocity profile mark increases with the increasing values of E for endothermic chemical reaction. But negligible effects are found for exothermic reaction.


## 5.2. Effects ofλ on the Concentration, the Temperature, and the Velocity Profiles for Exothermic/Endothermic Reaction

Considering chemical reaction rate constantλ2 is always positive. Figures 4–6 represent the effect of chemical rate constant λ on the velocity, the temperature, and the concentration profiles, respectively.We observe from Figures4 and 5 that velocity and temperature profiles increase with the increasing values of λ for exothermic reaction, but opposite effects are found in these figures for endothermic reaction. It is observed from (1) that increasing temperature frequently that causes a marked increase in the rate of reactions is shown in Figure 4. As the temperature of the system increases, the number of molecules that carry enough energy to react when they collide also increases. Therefore, the rate of reaction increases with temperature. As a rule, the rate of a reaction doubles for every 10°C increase in the temperature of the system.Last part of (20) shows that λ2exp(-E/θ) increases with the increasing values of λ. We also observe from this equation that increase in λ2exp(-E/θ) means that increase inλleads to the decrease in the concentration profiles. This is in great agreement with Figure 6 for both cases β=±1. It is also observed from this figure that for exothermic reaction the mass boundary layer is close to the plate other than endothermic reaction.


## 5.3. Effects of Radiation ParameterN on Temperature and Velocity Profiles

The effects of radiation parameterN on the temperature and the velocity profiles are shown in Figures 7 and 8. From Figure 7, it can be seen that an increase in the values of N leads to a decrease in the values of temperature profiles within the thermal boundary layers η<0.22, while outside η>0.22, the temperature profile gradually increases with the increase of the radiation parameter N. It is appear from Figure 8 that when N=0, that is, without the radiation, the velocity profile shows its usual trend of gradually decay. As radiation becomes larger the profiles overshoot the uniform velocity close to the boundary.


## 5.4. Effects ofγ on the Velocity, Temperature, and Concentration Profiles

The value ofγ=ΔT/T∞ is taken to be both positive and negative, since these values represent, respectively, cooling and heating of the plate. The effects of temperature relative parameter on velocity, temperature, and concentration profiles are shown in Figures 9–11.The velocity profiles generated due to impulsive motion of the plate is plotted in Figure9 for both cooling (γ>0) and heating (γ<0) of the plate keeping other parameters fixed (G0=10, Gm=1.0, M=Ec=0.5, Pr=0.71, Sc=0.6, λ=5, v0=3.0, E=1.0, and n=1). In Figure 9, velocity profiles are shown for different values of γ. We observe that velocity increases with increasing values of γ for the cooling of the plate. From this figure it is also observed that the negative increase in the temperature relative parameter leads to the decrease in the velocity field. That is, for heating of the plate (Figure 9), the effects of the γ on the velocity field have also opposite effects, as compared to the cooling of the plate. Figure 10 shows the effects of temperature relative parameter γ on the temperature profiles. It has been observed from this figure that for heating plate the temperature profile shows its usual trend of gradually decay, and the thermal boundary layer is close to the plate. But for cooling plate that is temperature relative parameter γ becomes larger, the profiles overshoot the uniform temperature close to the thermal boundary. Opposite effects of temperature profile are found for concentration profiles shown in Figure 11.


## 5.5. Effects of Suction/Injection (v0) on the Velocity and the Temperature Profiles

The effects of suction and injection (v0) for G0=10, Gm=1.0, M=Ec=0.5, Pr=0.71, Sc=0.6, λ=5, γ=0.5, E=1.0, and n=1on the velocity profiles and temperature profiles are shown, respectively, in Figures 12 and 13. For strong suction (v0>0), the velocity and the temperature 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 Figures 12 and 13 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.


## 5.6. Effects ofGm and M on the Velocity Profiles

The effects of Solutal Grashof numberGm and magnetic interaction parameter M on velocity profiles are shown in Figures 14 and 15, respectively. Solutal Grashof number Gm>0 corresponds that the chemical species concentration in the free stream region is less than the concentration at the boundary surface. Figure 14 presents the effects of Solutal Grashof number Gm on the velocity profiles. It is observed that the velocity profile increases with the increasing values of Solutal Grashof number Gm.Imposition of a magnetic field to an electrically conducting fluid creates a drag like force called Lorentz force. The force has the tendency to slow down the flow around the plate at the expense of increasing its temperature. This is depicted by decreases in velocity profiles as magnetic parameterM increases as shown in Figure 15.


## 5.7. Effects ofA on the Velocity and the Temperature Profiles

HereA=0 and A≠0 represent steady and unsteady flows, respectively. The effects ofAon velocity and temperature profiles are shown in Figures 16 and 17, respectively. For unsteady flow (A≠0) both figures show that the profiles rapidly decay and close to the plate. But for steady flow (A=0) the profiles overshoot the uniform velocity/temperature close to the boundary.


## 5.8. The Skin-Friction, the Heat Transfer, and the Mass Transfer Coefficients

The skin-friction, the heat transfer, and the mass transfer coefficients are tabulated in Tables1 and 2 for different values of E, β, λ, N, M, γ, and A. We observe from Table 1 that the skin-friction coefficient (-f′(0)) and the Nusselt number Nu(=-θ′(0)) increase for increasing values of dimensionless activation parameter E for exothermic chemical reaction, but opposite actions are found for endothermic reaction. That is for endothermic reaction, it is found that both shearing stress and heat transfer coefficients decrease for increasing values of E. For both cases exothermic/endothermic reactions, the mass transfer coefficient (=-ϕ′(0)) decreases for increasing values of E. We also observe from Table 1 that the skin-friction coefficient (-f′(0)) and the Nusselt number Nu(=-θ′(0)) decrease for increasing values of dimensionless reaction rate constant λ for exothermic chemical reaction, but opposite effects are found for endothermic reaction. That is for endothermic reaction, it is found that both shearing stress and heat transfer coefficients increase for increasing values of λ. For both cases exothermic/endothermic reactions, the mass transfer coefficient (=-ϕ′(0)) remarkably increases for increasing values of λ. From Table 2 it has been observed that the skin-friction coefficient remarkably decreases and Nusselt number remarkable increases for increasing values of radiation parameter N. The skin-friction coefficient increases for increasing values of magnetic parameter M. From Table 2 we also found that skin friction and heat transfer coefficients decrease, but mass transfer coefficient increases for increasing values of temperature relative parameter γ.


## 5.9. Comparison between Our Numerical Result and the Analytical Result of Bestman [7]

Bestman studied steady natural convective boundary layer flow with large suction. He solved his problem analytically by employing the perturbation technique proposed by Singh and Dikshit [8]. In our present work, take A=0 in (16) for considering steady flow. The values of the suction parameter v02=10 is taken to see the effects of large suction. Gr=Gm=1.0, γ=1, M=N=0, λ=5, E=5, and n=1 are also chosen with a view to compare our numerical results with the analytical results of Bestman [7]. The comparison of velocity profiles as seen in Figure 18 highlights the validity of the numerical computations adapted in the present investigation.Figure 18
Comparison of our calculated velocity profile and the velocity profile of Bestman [7].


## 6. Conclusions

In this paper, we investigate the effects of chemical reaction rate and Arrhenius activation energy on an unsteady MHD natural convection heat and mass transfer boundary layer flow past a flat porous plate in presence of thermal radiation.The Nachtsheim and Swigert [18] iteration technique based on sixth-order Runge-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 ofγ for the cooling of the plate, and the negative increase in the temperature relative parameter γ leads to the decrease in the velocity field. That is, for heating of the plate the effects of the temperature relative parameter γ on the velocity field have also opposite effects, as compared to the cooling of the plate.(2)
Solutal Grashof numberGm>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 Gm.(3)
Increase inλ leads to increase in the velocity, and temperature profiles for exothermic reaction but opposite effects are found for endothermic reaction. For β=±1 increase in λ leads to the decrease in the concentration profiles, but for exothermic reaction, the mass boundary layer is close to the plate other than endothermic reaction.(4)
The velocity profile mark increases with the increasing values ofE for endothermic chemical reaction. But negligible effect is found for exothermic reaction. Temperature decreases for increasing values of E for exothermic reaction, but opposite effects are found for endothermic reaction. Activation energy (E) leads to increase in the concentration profiles, but small and reported effects are found for β=1 and β=-1, respectively.(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)
Imposition of a magnetic field to an electrically conducting fluid creates a drag like force called Lorentz force. The force has the tendency to slow down the flow around the plate at the expense of increasing its temperature. This is depicted by decreases in velocity profiles as magnetic parameterM increases(7)
An increase in the values ofN leads to a decrease in the values of temperature profiles within the thermal boundary layers η<0.22, while outside η>0.22 the temperature profile gradually increases with the increase of the radiation parameter N.(8)
For unsteady flow (A≠0) the velocity and the temperature profiles rapidly decay and close to the plate. But for steady flow (A=0) the profiles overshoot the uniform velocity/temperature close to the boundary.



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