Hindawi Publishing CorporationJournal of ThermodynamicsVolume 2013 Article ID 692516 11 pageshttpdxdoiorg1011552013692516

Research ArticleEffects of ExothermicEndothermic Chemical Reactions withArrhenius Activation Energy on MHD Free Convection and MassTransfer Flow in Presence of Thermal Radiation

Kh Abdul Maleque

Department of Mathematics American International University-Bangladesh House-23 17 Kamal Ataturk AvenueBanani Dhaka 1213 Bangladesh

Correspondence should be addressed to Kh Abdul Maleque malequeaiubedu

Received 8 March 2013 Accepted 24 May 2013

Academic Editor Felix Sharipov

Copyright copy 2013 Kh Abdul MalequeThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A local similarity solution of unsteady MHD natural convection heat and mass transfer boundary layer flow past a flat porousplate within the presence of thermal radiation is investigated The effects of exothermic and endothermic chemical reactions withArrhenius activation energy on the velocity temperature and concentration are also studied in this paper The governing partialdifferential 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 shootingmethods using the Nachtsheim-Swigert iteration technique The results of the numerical solution are obtained for both steady andunsteady cases then presented graphically in the form of velocity temperature and concentration profiles Comparison has beenmade for steady flow (119860 = 0) and shows excellent agreement with Bestman (1990) hence encouragement for the use of the presentcomputations

1 Introduction

In free convection boundary layer flows with simultaneousheat mass transfer one important criteria that is generallynot encountered is the species chemical reactions with finiteArrhenius activation energy The modified Arrhenius law(IUPAC Goldbook definition of modified Arrhenius equa-tion) is usually of the form (Tencer et al [1])

119870 = 119861(119879

119879infin

)

119899

exp [minus119864119886

119896119879] (1)

where 119870 is the rate constant of chemical reaction and 119861 thatis the preexponential factor simply prefactor (constant) isbased on the fact that increasing the temperature frequentlycauses a marked increase in the rate of reactions 119864

119886is the

activation energy and 119896 = 861 times 10minus5 eVK is the Boltzmannconstant which is the physical constant relating energy atthe individual particle level with temperature observed at thecollective or bulk level

In areas such as geothermal or oil reservoir engineeringthe prvious phenomenon is usually applicable Apart fromexperimental works in these areas it is also important tomake some theoretical efforts to predict the effects of theactivation energy in flowsmentioned above But in this regardvery few theoretical works are available in the literature Thereason is that the chemical reaction processes involved inthe system are quite complex and generally the mass transferequation that is required for all the reactions involved alsobecomes complex Theoretically such an equation is ratherimpossible to tackle Form chemical kinetic viewpoint thisis a very difficult problem but if the reaction is restricted tobinary type a lot of progress can be made The thermome-chanical balance equations for a mixture of general materialswere first formulated by Truesdell [2] Thereafter Mills [3]and Beevers and Craine [4] have obtained some exactsolutions for the boundary layer flow of a binary mixture ofincompressible Newtonian fluids Several problems relatingto the mechanics of oil and water emulsions particularlywith regard to applications in lubrication practice have been

2 Journal of Thermodynamics

considered within the context of a binary mixture theory byAl-Sharif et al [5] and Wang et al [6]

A simple model involving binary reaction was studiedby Bestman [7] He considered the motion through theplate to be large which enabled him to obtain analyticalsolutions (subject to same restrictions) for various values ofactivation energy by employing the perturbation techniqueproposed by Singh andDikshit [8] Bestman [9] andAlabrabaet al [10] took into account the effect of the Arrheniusactivation energy under the different physical conditionsRecently Kandasamy et al [11] studied the combined effects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjection Their result shows that the flow field is influencedappreciably by chemical reaction heat source and suctionor injection at the wall of the wedge Recently Makindeet al [12 13] studied the problems of unsteady convectionwith chemical reaction and radiative heat transfer past aflat porous plate moving through a binary mixture in anoptically thin environment More recently Abdul Maleque[14 15] investigated the similarity solution on unsteadyincompressible fluid flow with binary chemical reactionsand activation energy In the present paper we investigatea numerical solution of unsteady Mhd natural convectionheat and mass transfer boundary layer flow past a flat porousplate taking into account the effect of Arrhenius activationenergy with exothermicendothermic chemical reactions andthermal radiationThis problem is an extension work studiedby Abdul Maleque [16]

2 Governing Equations

Weconsider the boundarywall to be of infinite extend so thatall quantities are homogeneous in 119909 and hence all derivativeswith respect to 119909 are neglected The 119909-axis is taken alongthe plate and 119910-axis is perpendicular to the plate Assumethat a uniform magnetic field 119861

0is applied perpendicular

to the plate and the plate is moving with uniform velocity119880119900in its own plane In presence of exothermicendothermic

binary chemical reaction with Arrhenius activation energya uniform magnetic field and thermal radiation thus thegoverning equations are

120597V

120597119910= 0 (2)

120597119906

120597119905+ V120597119906

120597119910= 1205921205972

119906

1205971199102+ 1198921205731(119879 minus 119879

infin)

+ 1198921205732(119862 minus 119862

infin) minus1205901198612

0

120588119906

(3)

120588119888119901(120597119879

120597119905+ V120597119879

120597119910) = 120581

1205972

119879

1205971199102+ 1205731198962

119903(119879

119879infin

)

119899

times exp [minus119864119886

119896119879] (119862 minus 119862

infin) minus120597119902119903

120597119910

(4)

120597119862

120597119905+ V120597119862

120597119910= 119863119898

1205972

119862

1205971199102minus 1198962

119903(119879

119879infin

)

119899

exp [minus119864119886

119896119879] (119862 minus 119862

infin)

(5)

The boundary conditions of previous system are

119906 = 1198800 V = V (119905) 119879 = 119879

119908 119862 = 119862

119908

at 119910 = 0 119905 gt 0

119906 = 0 119879 997888rarr 119879infin 119862 997888rarr 119862

infinat 119910 = infin 119905 gt 0

(6)

where (119906 V) is the velocity vector 119879is the temperature 119862isthe concentration of the fluid 120583 is the fluid viscosity 120590 isthe electrical conductivity 120592 is the kinematic coefficient ofviscosity 120573

1and 120573

2are the coefficients of volume expansions

for temperature and concentration respectively 120581 is theheat diffusivity coefficient 119888

119901is the specific heat at constant

pressure 119863119898

is the coefficient of mass diffusivity 1198962119903is

the chemical reaction rate constant 120573(= plusmn1) is exother-micendothermic parameter and (119879119879

infin)119899 exp[minus119864

119886119896 119879] is

the Arrhenius function where 119899 is a unit less constant expo-nent fitted rate constants typically lie in the range minus1 lt 119899 lt 1

The radiative heat flux 119902119903is described by Roseland

approximation such that

119902119903= minus41205901

31198961

1205971198794

120597119910 (7)

where1205901and 119896

1are the StefanBoltzmann constant andmean

absorption coefficient respectively Following Makinde andOlanrewaju [13] we assume that the temperature differenceswithin the flow are sufficiently small so that the 119879119899 canbe expressed as a linear function By using Taylorrsquos serieswe expand 119879119899 about the free stream temperature 119879

infinand

neglecting higher order terms This result of the followingapproximation

119879119899

asymp (1 minus 119899) 119879119899

infin+ 119899119879119899minus1

infin119879 (8)

Using (9) we have

120597119902119903

120597119910= minus41205901

31198961

1205972

1198794

1205971199102asymp minus161198793

infin1205901

31198961

1205972

119879

1205971199102 (9)

(119879

119879infin

)

119899

asymp (1 minus 119899) + 119899119879

119879infin

(10)

3 Mathematical Formulations

In order to solve the governing equations (2) to (5) under theboundary conditions (6) we adopt thewell-defined similaritytechnique to obtain the similarity solutions For this purposethe following nondimensional variables are now introduced

120578 =119910

120575 (119905)

119906

1198800

= 119891 (120578)

119879 minus 119879infin

119879119908minus 119879infin

= 120579 (120578) 119862 minus 119862

infin

119862119908minus 119862infin

= 120601 (120578)

(11)

Journal of Thermodynamics 3

From the equation of continuity (3) we have

V (119905) = minusV0120592

120575 (119905) (12)

where V0is the dimensionless suctioninjection velocity at the

plate V0gt 0 corresponds to suction and V

0lt 0 corresponds

to injectionIntroducing (9) (10) the dimensionless quantities from

(11) and V from (12) in (4) (5) and (6) we finally obtain thenonlinear ordinary differential equations as

11989110158401015840

+ (1205781205751205751015840

120592+ V0)1198911015840

minus119872119891 + 1198660120574120579 + 119866

119898120601 = 0 (13)

12057910158401015840

+ (119875119903

1 + 119873)[(120578

1205751205751015840

120592+ V0)1205791015840

+1205731205822

(1 + 119899120574120579) exp(minus 119864

1 + 120574120579)120601] = 0

(14)

119878minus1

11988812060110158401015840

+ (1205781205751205751015840

120592+ V0)1206011015840

minus 1205822

(1 + 119899120574120579) exp(minus 119864

1 + 120574120579)120601 = 0

(15)

Here Grashof number 119866119903= 1198660120574 = (119892120573119879

infin1205752

1205921198800)120574 the

temperature relative parameter 120574 = (119879119908minus 119879infin)119879infin

Modified (Solutal) Grashof number 119866119898= 119892120573

lowast

(119862119908minus

119862infin)1205752

1205921198800 Prandtl number 119875

119903= 120588120592119888

119901120581 magnetic interac-

tion parameter119872 = 120590120575211986120120583 schmidt number 119878

119888= 120592119863

119872

the nondimensional activation energy 119864 = 119864119886119896(119879119908minus 119879infin)

the dimensionless chemical reaction rate constant 1205822 =1198962

1199031205752

120592 radiation parameter119873 = 1612059011198793

infin31205811198961

Equations from (13) to (15) are similar except for the term1205751205751015840

120592 where time 119905 appears explicitly Thus the similaritycondition requires that 1205751205751015840120592 must be a constant quantityHence following Abdul Maleque [17] one can try a class ofsolutions of (13) to (15) by assuming that

1205751205751015840

120592= 119860 (constant) (16)

From (16) we have

120575 (119905) = radic2119860120592119905 + 1198712 (17)

where the constant of integration119871 is determined through thecondition that 120575 = 119871 when 119905 = 0 Here 119860 = 0 implies that 120575 =119871 represents the length scale for steady flow and 119860 = 0 thatis 120575 represents the length scale for unsteady flow Since 120575 is ascaling factor as well as similarity parameter any other valuesof119860 in (13) would not change the nature of the solution exceptthat the scale would be different Finally introducing (16) in

(13) to (15) respectively we have the following dimensionlessnonlinear ordinary differential equations

11989110158401015840

+ (119860120578 + V0) 1198911015840

minus119872119891 + 1198660120574120579 + 119866

119898120601 = 0 (18)

12057910158401015840

+ (119875119903

1 + 119873)[ (119860120578 + V

0) 1205791015840

+1205731205822

(1 + 119899120574120579) exp(minus 119864

1 + 120574120579)120601] = 0

(19)

119878minus1

11988812060110158401015840

+ (119860120578 + V0) 1206011015840

minus 1205822

(1 + 119899120574120579) exp(minus 119864

1 + 120574120579)120601 = 0

(20)

The boundary conditions equation (4) becomes

119891 (0) = 1 120579 (0) = 1 120601 (0) = 1

119891 (infin) = 0 120579 (infin) = 0 120601 (infin) = 0

(21)

In all over equations primes denote the differentiationwith respect to 120578 Equations from (18)ndash(20) are solvednumerically under the boundary conditions (21) usingNachtsheim-Swigert iteration technique

4 Numerical Solutions

Equations (18)ndash(20) are solved numerically under the bound-ary conditions (21) using Nachtsheim-Swigert [18] iterationtechnique In (21) there are three asymptotic boundary con-ditions and hence follow three unknown surface conditions1198911015840

(0) 1205791015840(0) and 1206011015840(0) Within the context of the initial-valuemethod and the Nachtsheim-Swigert iteration technique theouter boundary conditions may be functionally representedby the first-order Taylorrsquos series as

119891 (120578max) = 119891 (119883 119884 119885)

= 119891119888(120578max) + Δ119883119891119883 + Δ119884119891119884 + Δ119885119891119885 = 1205751

(22)

120579 (120578max) = 120579 (119883 119884 119885)

= 120579119888(120578max) + Δ119883120579119883 + Δ119884120579119884 + Δ119885120579119885 = 1205752

(23)

120601 (120578max) = 120601 (119883 119884 119885)

= 120601119888(120578max) + Δ119883120601119883 + Δ119884120601119884 + Δ119885120601119885 = 1205753

(24)

with the asymptotic convergence criteria given by

1198911015840

(120578max) = 1198911015840

(119883 119884 119885)

= 1198911015840

119888(120578max) + Δ119883119891

1015840

119883+ Δ119884119891

1015840

119884+ Δ119885119891

1015840

119885= 1205754

(25)

1205791015840

(120578max) = 1205791015840

(119883 119884 119885)

= 1205791015840

119888(120578max) + Δ119883120579

1015840

119883+ Δ119884120579

1015840

119884+ Δ119885120579

1015840

119885= 1205755

(26)

1206011015840

(120578max) = 1206011015840

(119883 119884 119885)

= 1206011015840

119888(120578max) + Δ119883120601

1015840

119883+ Δ119884120601

1015840

119884+ Δ119885120601

1015840

119885= 1205756

(27)

4 Journal of Thermodynamics

For G0 = 10 Gm = 1M = 05

120574 = 1 n = 1 = 3 and 120582 = 51

05120579

120578

120573 = 1

120573 = minus1

0 1 2

E = 1

E = 0

0Sc = 06

Pr = 071 N = 1

Figure 1 Effects of 120573 and 119864 on temperature profiles

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and 120582 = 5

05 15120578

120573 = 1

120573 = minus1120601

0

1

05

01

E = 0

E = 1

0Sc = 06

Pr

Figure 2 Effects of 120573 and 119864 on concentration profiles

where 119883 = 1198911015840

(0) 119884 = 1205791015840(0) 119885 = 1206011015840(0) and 119883119884 119885subscripts indicate partial differentiation for example 1198911015840

119884=

1205971198911015840

(120578max)1205971205791015840

(0) The subscript 119888 indicates the value of thefunction at 120578max to be determined from the trial integration

Solutions of these equations in a least square senserequire determining the minimum value of 119864 = 120575

2

1+

1205752

2+ 1205752

3+ 1205752

4+ 1205752

5+ 1205752

6with respect to 119883 119884 and 119885 To

solve Δ119883 Δ119884 and Δ119885 we require to differentiate 119864 withrespect to 119883 119884 and 119885 respectively Thus adopting thisnumerical technique a computer program was set up forthe solutions of the basic nonlinear differential equations ofour problem where the integration technique was adoptedas the six-ordered Runge-Kutta method of integration Theresults of this integration are then displayed graphically in

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and 120582 = 5

120578

120573 = 1

120573 = minus1

00

f05

1

1 2

E = 0

E = 1

0Sc = 06

Pr

Figure 3 Effects of 120573 and 119864 on velocity profiles

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and E = 1

120573 = 1

120573 = minus1f

05

1

120578

00

1 2

120582 = 1

120582 = 5

0Sc = 06

Pr

Figure 4 Effects of 120573 and 120582 on velocity profiles

the form of velocity temperature and concentration profilesin Figures 1ndash15 In the process of integration the local skin-friction coefficient the local rates of heat and mass transferto the surface which are of chief physical interest are alsocalculated out The equation defining the wall skin friction is

120591 = minus120583 (120597119906

120597119910)

119910=0

= minus1205831198800

1205751198911015840

(0) (28)

Hence the skin-friction coefficient is given by

1

2119862119891=120591

1205881198802

0

997904rArr1

2119877119890119862119891= minus1198911015840

(0) (29)

here the Reynolds number 119877119890= 1205881198800120575120583

Journal of Thermodynamics 5

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and E = 1

120582 = 1

120582 = 5

120573 = 1

120573 = minus1

120578

0 1 2

1

05

0

120579

0Sc = 06

Pr

Figure 5 Effects of 120573 and 120582 on temperature profiles

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and E = 1

120582 = 1

120582 = 5

120573 = 1

120578

00

1

1

05

2

120573 = minus1

120601

0Sc = 06

Pr

Figure 6 Effects of 120573 and 120582 on concentration profiles

0 1 2 3 40

05

1

120578

120579

N = 0 2 5

For G0 = 10 Gm = 1M = 05 = 071

120582 = 5

120574 = 1

n = 1 0 = 3 and E = 1Sc = 06

Pr

Figure 7 Effects of119873 on temperature profiles

0 1 2 3 40

05

1

15

2

f

N = 0 2 5

For G0 = 10 Gm = 1M = 05 = 071

120582 = 5

120574 = 1

n = 1 0 = 3 and E = 1

120578

Sc = 06

Pr

Figure 8 Effects of119873 on velocity profiles

0 1 2

0

1

2

f

For G0 = 10 Gm = 1N = 05 = 071M= 05

120582 = 5 n = 05 = 3 and E = 10

Gr = G0120574120574

2

1

0

minus1

minus1

minus2

120578

Sc = 06

Pr

Figure 9 Effects of 120574 on velocity profiles

0 1 20

05

1For G0 = 10 Gm = 1N = 05 = 071M= 05

120582 = 5 n = 05 = 3 and E = 10

Gr = G0120574

120574

210minus1

minus2

120578

120579

Sc = 06

Pr

Figure 10 Effects of 120574 on temperature profiles

6 Journal of Thermodynamics

0 1 20

05

1

120578

120601

For G0 = 10 Gm = 1N = 05 = 071M= 05

120582 = 5 n = 05 = 3 and E = 10

Gr = G0120574

120574

minus2minus101

2

Sc = 06

Pr

Figure 11 Effects of 120574 on concentration profiles

0 1 2 3 40

1

2

3

f

120578

For G0 = 10 Gm = 1N = 05 = 071M = 05

120582 = 5 n = 05 120574 = 05 and E = 1

0 = minus3

0 = 0

0 = 3

Sc = 06

Pr

Figure 12 Effects of V0on velocity profiles

The heat flux (119902119908) and the mass flux (119872

119908) at the wall are

given by

119902119908= minus120581(

120597119879

120597119910)

119910=0

= minus120581Δ119879

1205751205791015840

(0)

119872119908= minus119863119872(120597119862

120597119910)

119910=0

= minus119863119872

Δ119862

1205751206011015840

(0)

(30)

Hence the Nusselt number (119873119906) and the Sherwood number

(119878ℎ) are obtained as

119873119906=119902119908120575

120581 Δ119879= minus 120579

1015840

(0) 119878ℎ=119872119908120575

119863119872Δ119862= minus1206011015840

(0) (31)

These previous coefficients are then obtained from the proce-dure of the numerical computations and are sorted in Table 1

5 Results and Discussions

The parameters entering into the fluid flow are Grashofnumber 119866

119903= 1198660120574 Solutal (modified) Grashof number 119866

119898

suction parameter V0 Prandtl number 119875

119903 the nondimen-

sional chemical reaction rate constant 1205822 Schmidt number

0 1 2 3 40

05

1

15

0

3

For G0 = 10 Gm = 1N = 05 = 071M = 05

120582 = 5 n = 05 120574 = 05 and E = 1

0

minus3

120578

120579

Sc = 06

Pr

Figure 13 Effects of V0on temperature profiles

0 1 20

05

1

10

5

0

f

Gm

120582 = 5 n = 05 120574 = 05 and E = 1

120578

For G0 = 10 0 = 3N = 05 = 071M = 05

Sc = 06

Pr

Figure 14 Effects of 119866119898on velocity profiles

0 1 20

05

1

M

1

0

5

f

120578

For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

120582 = 5 n = 1 0 = 3 and E = 1Sc = 06

Pr

Figure 15 Effects of119872 on velocity profiles

Journal of Thermodynamics 7

0 1 2 30

05

1

f

120578

For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

A = 0 (steady)A = 2 (unsteady)

120574 = 5 n = 1 0 = 3M = 05 and E = 1Sc = 06

Pr

Figure 16 Effects of 119860 on velocity profiles

0 1 2 30

05

1 For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

A = 0 (steady)A = 2 (unsteady)

120579

120578

120574 = 5 n = 1 0 = 3M = 05 and E = 1Sc = 06

Pr

Figure 17 Effects of 119860 on temperature profiles

119878119888 the magnetic parameter 119872 the nondimensional acti-

vation energy 119864 the radiation parameter 119873 the exother-micendothermic parameter 120573 and the temperature relativeparameter 120574

It is therefore pertinent to inquire the effects of variationof each of them when the others are kept constant Thenumerical results are thus presented in the form of velocityprofiles temperature profiles and concentration profiles inFigures 1ndash14 for the different values of 119866

119903 119866119898 120582119872 119864 V

0 120573

and 120574The value of 120574 is taken to be both positive and negativesince these values represent respectively cooling and heatingof the plate

The values of 119866119903is taken to be large (119866

119903= 10

1198660= 10 and 120574 = 1) since the value corresponds to

Table 1 Effects on skin friction heat transfer and mass transfercoefficients For 119866

0= 10 119866

119898= 1119873 = 05 119875

119903= 071119872 = 05 119878

119888=

06 120582 = 5 119899 = 05 V0= 3 120574 = 05 and 119864 = 1

minus1198911015840

(0) minus1205791015840

(0) minus1206011015840

(0)

119864 for 120573 = 100 minus165163 minus091802 57863405 minus160009 minus088980 57552310 minus154562 minus085343 570174

119864 for 120573 = minus100 130241 449966 55515905 098702 419573 52109310 063543 387806 481246

120582 for 120573 = 110 minus043158 173298 24262720 minus063434 134699 29144650 minus154562 minus085343 570174

120582 for 120573 = minus110 minus027903 198971 24208820 minus007554 232400 28442450 063543 387806 481246

a cooling problem that is generally encountered in nuclearengineering in connection with the cooling of reactors In air(119875119903= 071) the diluting chemical species of most common

interest have Schmidt number in the range from 06 to 075Therefore the Schmidt number 119878

119888= 06 is considered In

particular 06 corresponds to water vapor that represents adiffusing chemical species of most common interest in airThe values of the suction parameter V

0are taken to be large

Apart from the figures and tables the representative velocitytemperatureand concentration profiles and the values ofthe physically important parameters that is the local shearstress the local rates of heat and mass transfer are illustratedfor uniform wall temperature and species concentration inFigures 1ndash17 and in Tables 1-2

51 Effects of Activation Energy (119864) on the Concentrationthe Temperature and the Velocity Profiles for ExothermicEndothermic Chemical Reaction In chemistry activationenergy is defined as the energy that must be overcome inorder for a chemical reaction to occur Activation energymayalso be defined as the minimum energy required startinga chemical reaction The activation energy of a reaction isusually denoted by 119864

119886and given in units of kilojoules per

moleAn exothermic reaction is a chemical reaction that

releases energy in the form of light and heat It is theopposite of an endothermic reaction It gives out energy to itssurroundings The energy needed for the reaction to occur isless or greater than the total energy released for exothermicor endothermic reaction respectively Energy is obtainedfrom chemical bonds When bonds are broken energy isrequired When bonds are formed energy is released Eachtype of bond has specific bond energy It can be predictedwhether a chemical reaction will release or require heat byusing bond energiesWhen there ismore energy used to form

8 Journal of Thermodynamics

Table 2 Effects on skin friction heat transfer and mass transfercoefficients For119866

0= 10119866

119898= 1119873 = 05119875

119903= 071119872 = 05 119878

119888=

06 120582 = 5 119899 = 05 V0= 3 120574 = 05 and 119864 = 1

minus1198911015840

(0) minus1205791015840

(0) minus1206011015840

(0)

119873

00 minus087260 minus144125 56899620 minus305961 minus030027 57162450 minus493466 minus004756 572770

119872

00 minus185010 minus085309 57016420 minus075246 minus085443 57020350 046744 minus085628 570257

120574

minus200000 732171 365294 007835minus100000 566728 348636 026098000000 331180 188820 228076100000 minus154562 minus085343 570174200000 minus785616 minus367839 925776

119860

000000 minus127274 minus026974 397243200000 120419 056519 391549

the bonds than to break the bonds heat is given off This isknown as an exothermic reaction When a reaction requiresan input or output of energy it is known as an exothermic orendothermic reaction

Effects of activation energy (119864) and exothermic param-eter (120573) on the temperature the concentration and velocityprofiles are shown in Figure 1 to Figure 3 respectively 120573 = 1and 120573 = minus1 represent exothermic and endothermic chemicalreactions respectively Figure 1 shows that a small decreasingeffect of temperature profile is found for increasing valuesof 119864 for exothermic reaction (120573 = 1) but mark oppositeeffects are found for endothermic reaction in Figure 1 Thatis the temperature profile increases for increasing values of119864 for endothermic reaction 120573 = minus1 From (1) we observethat chemical reaction rate (119870) decreases with the increasingvalues of activation energy (119864

119886) We also observe from (20)

that increase in activation energy (119864) leads to decrease1205822 exp(minus119864120579) as well as to increase in the concentration

profiles shown in Figure 2 Small and reported effects arefound for 120573 = 1 and 120573 = minus1 respectively The parameter119864 does not enter directly into the momentum equation butits influence comes through the mass and energy equationsFigure 3 shows the variation of the velocity profiles fordifferent values of 119864 From this figure it has been observedthat the velocity profile mark increases with the increasingvalues of 119864 for endothermic chemical reaction But negligibleeffects are found for exothermic reaction

52 Effects of 120582 on the Concentration the Temperature and theVelocity Profiles for ExothermicEndothermic Reaction Con-sidering chemical reaction rate constant 1205822 is always positiveFigures 4ndash6 represent the effect of chemical rate constant120582 on

the velocity the temperature and the concentration profilesrespectively

We observe from Figures 4 and 5 that velocity andtemperature profiles increase with the increasing values of120582 for exothermic reaction but opposite effects are foundin these figures for endothermic reaction It is observedfrom (1) that increasing temperature frequently that causes amarked increase in the rate of reactions is shown in Figure 4As the temperature of the system increases the numberof molecules that carry enough energy to react when theycollide also increasesTherefore the rate of reaction increaseswith temperature As a rule the rate of a reaction doubles forevery 10∘C increase in the temperature of the system

Last part of (20) shows that 1205822 exp(minus119864120579) increases withthe increasing values of 120582We also observe from this equationthat increase in 1205822 exp(minus119864120579)means that increase in 120582 leadsto the decrease in the concentration profiles This is in greatagreement with Figure 6 for both cases 120573 = plusmn1 It isalso observed from this figure that for exothermic reactionthe mass boundary layer is close to the plate other thanendothermic reaction

53 Effects of Radiation Parameter 119873 on Temperature andVelocity Profiles The effects of radiation parameter119873 on thetemperature and the velocity profiles are shown in Figures 7and 8 From Figure 7 it can be seen that an increase in thevalues of 119873 leads to a decrease in the values of temperatureprofiles within the thermal boundary layers 120578 lt 022while outside 120578 gt 022 the temperature profile graduallyincreases with the increase of the radiation parameter 119873 Itis appear from Figure 8 that when 119873 = 0 that is withoutthe radiation the velocity profile shows its usual trend ofgradually decay As radiation becomes larger the profilesovershoot the uniform velocity close to the boundary

54 Effects of 120574 on the Velocity Temperature and Concen-tration Profiles The value of 120574 = Δ119879119879

infinis taken to be

both positive and negative since these values representrespectively cooling and heating of the plate The effects oftemperature relative parameter on velocity temperature andconcentration profiles are shown in Figures 9ndash11

The velocity profiles generated due to impulsive motionof the plate is plotted in Figure 9 for both cooling (120574 gt 0) andheating (120574 lt 0) of the plate keeping other parameters fixed(1198660= 10 119866

119898= 10 119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 V0= 30 119864 = 10 and 119899 = 1) In Figure 9 velocity

profiles are shown for different values of 120574 We observe thatvelocity increases with increasing values of 120574 for the coolingof the plate From this figure it is also observed that thenegative increase in the temperature relative parameter leadsto the decrease in the velocity field That is for heating ofthe plate (Figure 9) the effects of the 120574 on the velocity fieldhave also opposite effects as compared to the cooling ofthe plate Figure 10 shows the effects of temperature relativeparameter 120574 on the temperature profiles It has been observedfrom this figure that for heating plate the temperature profileshows its usual trend of gradually decay and the thermalboundary layer is close to the plate But for cooling plate

Journal of Thermodynamics 9

that is temperature relative parameter 120574 becomes largerthe profiles overshoot the uniform temperature close to thethermal boundary Opposite effects of temperature profile arefound for concentration profiles shown in Figure 11

55 Effects of SuctionInjection (V0) on the Velocity and the

Temperature Profiles The effects of suction and injection (V0)

for 1198660= 10 119866

119898= 10119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 120574 = 05 119864 = 10 and 119899 = 1 on the velocity profilesand temperature profiles are shown respectively in Figures12 and 13 For strong suction (V

0gt 0) the velocity and the

temperature decay rapidly away from the surface The factthat suction stabilizes the boundary layer is also apparentfrom these figures As for the injection (V

0lt 0) from Figures

12 and 13 it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayer between theinjection and the outer flow regions

56 Effects of 119866119898and119872 on the Velocity Profiles The effects

of Solutal Grashof number 119866119898

and magnetic interactionparameter119872 on velocity profiles are shown in Figures 14 and15 respectively Solutal Grashof number 119866

119898gt 0 corresponds

that the chemical species concentration in the free streamregion is less than the concentration at the boundary surfaceFigure 14 presents the effects of Solutal Grashof number119866119898on the velocity profiles It is observed that the velocity

profile increases with the increasing values of Solutal Grashofnumber 119866

119898

Imposition of a magnetic field to an electrically conduct-ing fluid creates a drag like force called Lorentz force Theforce has the tendency to slow down the flow around the plateat the expense of increasing its temperature This is depictedby decreases in velocity profiles as magnetic parameter 119872increases as shown in Figure 15

57 Effects of 119860 on the Velocity and the Temperature ProfilesHere 119860 = 0 and 119860 = 0 represent steady and unsteady flowsrespectively The effects of 119860 on velocity and temperatureprofiles are shown in Figures 16 and 17 respectively Forunsteady flow (119860 = 0) both figures show that the profilesrapidly decay and close to the plate But for steady flow (119860 =0) the profiles overshoot the uniform velocitytemperatureclose to the boundary

58 The Skin-Friction the Heat Transfer and the MassTransfer Coefficients The skin-friction the heat transferand the mass transfer coefficients are tabulated in Tables1 and 2 for different values of 119864 120573 120582 119873 119872 120574 and 119860We observe from Table 1 that the skin-friction coefficient(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) increasefor increasing values of dimensionless activation parameter119864 for exothermic chemical reaction but opposite actionsare found for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients decrease for increasing values of 119864 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) decreases for increasing values of 119864We also observe fromTable 1 that the skin-friction coefficient

0 1 2 3 40

05

1

Bestman (1990)Present

f

120578

Gr = Gm = 1 E = 5 20 = 10

= 071 and 120582 = 5

120573 = 0 120574 = 1 and M= N = A = 0

Sc = 1 Pr

Figure 18 Comparison of our calculated velocity profile and thevelocity profile of Bestman [7]

(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) decreasefor increasing values of dimensionless reaction rate constant120582 for exothermic chemical reaction but opposite effects arefound for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients increase for increasing values of 120582 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) remarkably increases for increasingvalues of 120582 From Table 2 it has been observed that theskin-friction coefficient remarkably decreases and Nusseltnumber remarkable increases for increasing values of radi-ation parameter119873 The skin-friction coefficient increases forincreasing values of magnetic parameter 119872 From Table 2we also found that skin friction and heat transfer coefficientsdecrease butmass transfer coefficient increases for increasingvalues of temperature relative parameter 120574

59 Comparison between Our Numerical Result and theAnalytical Result of Bestman [7] Bestman studied steadynatural convective boundary layer flow with large suctionHe solved his problem analytically by employing the per-turbation technique proposed by Singh and Dikshit [8] Inour present work take 119860 = 0 in (16) for considering steadyflow The values of the suction parameter V2

0= 10 is taken

to see the effects of large suction 119866119903= 119866119898= 10 120574 = 1

119872 = 119873 = 0 120582 = 5 119864 = 5 and 119899 = 1 are also chosen witha view to compare our numerical results with the analyticalresults of Bestman [7] The comparison of velocity profilesas seen in Figure 18 highlights the validity of the numericalcomputations adapted in the present investigation

6 Conclusions

In this paper we investigate the effects of chemical reactionrate and Arrhenius activation energy on an unsteady MHD

10 Journal of Thermodynamics

natural convection heat and mass transfer boundary layerflow past a flat porous plate in presence of thermal radiation

The Nachtsheim and Swigert [18] iteration techniquebased on sixth-order Runge-Kutta and Shooting method hasbeen employed to complete the integration of the resultingsolutions

The following conclusions can be drawn as a result of thecomputations

(1) Velocity increases with increasing values of 120574 for thecooling of the plate and the negative increase in thetemperature relative parameter 120574 leads to the decreasein the velocity fieldThat is for heating of the plate theeffects of the temperature relative parameter 120574 on thevelocity field have also opposite effects as comparedto the cooling of the plate

(2) Solutal Grashof number 119866119898gt 0 corresponds that

the chemical species concentration in the free streamregion is less than the concentration at the bound-ary surface It is observed that the velocity profileincreaseswith the increasing values of SolutalGrashofnumber 119866

119898

(3) Increase in 120582 leads to increase in the velocity andtemperature profiles for exothermic reaction butopposite effects are found for endothermic reactionFor 120573 = plusmn1 increase in 120582 leads to the decrease in theconcentration profiles but for exothermic reactionthe mass boundary layer is close to the plate otherthan endothermic reaction

(4) The velocity profile mark increases with the increas-ing values of 119864 for endothermic chemical reactionBut negligible effect is found for exothermic reactionTemperature decreases for increasing values of 119864 forexothermic reaction but opposite effects are found forendothermic reaction Activation energy (119864) leads toincrease in the concentration profiles but small andreported effects are found for 120573 = 1 and 120573 = minus1respectively

(5) For strong suction (V0gt 0) the velocity the tem-

perature and the concentration profiles decay rapidlyaway from the surface As for the injection (V

0lt 0)

it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayerbetween the injection and the outer flow regions

(6) Imposition of a magnetic field to an electrically con-ducting fluid creates a drag like force called Lorentzforce The force has the tendency to slow down theflow around the plate at the expense of increasing itstemperature This is depicted by decreases in velocityprofiles as magnetic parameter119872 increases

(7) An increase in the values of 119873 leads to a decrease inthe values of temperature profiles within the thermalboundary layers 120578 lt 022 while outside 120578 gt 022the temperature profile gradually increases with theincrease of the radiation parameter119873

(8) For unsteady flow (119860 = 0) the velocity and the tem-perature profiles rapidly decay and close to the plate

But for steady flow (119860 = 0) the profiles overshoot theuniform velocitytemperature close to the boundary

Nomenclature

(119906 V) Components of velocity field119888119901 Specific heat at constant pressure119873119906 Nusselt number

119875119903 Prandtl number1198760 Heat generationabsorption coefficient

119878ℎ Sherwood number119879 Temperature of the flow field119879infin Temperature of the fluid at infinity

119879119908 Temperature at the plate

119891 Dimensionless similarity functions119864119886 The activation energy

119896 The Boltzmann constant119878119888 Schmidt number119864 The nondimensional activation energy119866119903 Grashof number

119866119898 Modified (Solutal) Grashof number

Greek Symbols

120579 Dimensionless temperature120601 Dimensionless concentration120578 Dimensionless similarity variable120575 Scale factorV Kinematic viscosity120588 Density of the fluid120581 Thermal conductivity120591 Shear stress120583 Fluid viscosity1205822 Dimensionless chemical reaction rate

constant120573 and 120573lowast The coefficients of volume expansions for

temperature and concentrationrespectively

120574 The dimensionless heatgenerationabsorption coefficient

References

[1] M Tencer J S Moss and T Zapach ldquoArrhenius averagetemperature the effective temperature for non-fatigue wearoutand long term reliability in variable thermal conditions andclimatesrdquo IEEE Transactions on Components and PackagingTechnologies vol 27 no 3 pp 602ndash607 2004

[2] C Truesdell ldquoSulle basi della thermomeccanicardquo RendicontiLincei vol 22 no 8 pp 33ndash38 1957

[3] N Mills ldquoIncompressible mixtures of newtonian fluidsrdquo Inter-national Journal of Engineering Science vol 4 no 2 pp 97ndash1121966

[4] C E Beevers and R E Craine ldquoOn the determination ofresponse functions for a binary mixture of incompressiblenewtonian fluidsrdquo International Journal of Engineering Sciencevol 20 no 6 pp 737ndash745 1982

Journal of Thermodynamics 11

[5] A Al-Sharif K Chamniprasart K R Rajagopal andA Z SzerildquoLubrication with binary mixtures liquid-liquid emulsionrdquoJournal of Tribology vol 115 no 1 pp 46ndash55 1993

[6] S H Wang A Al-Sharif K R Rajagopal and A Z SzerildquoLubrication with binarymixtures liquid-liquid emulsion in anEHL conjunctionrdquo Journal of Tribology vol 115 no 3 pp 515ndash522 1993

[7] A R Bestman ldquoNatural convection boundary layer withsuction and mass transfer in a porous mediumrdquo InternationalJournal of Energy Research vol 14 no 4 pp 389ndash396 1990

[8] A K Singh and C K Dikshit ldquoHydromagnetic flow pasta continuously moving semi-infinite plate for large suctionrdquoAstrophysics and Space Science vol 148 no 2 pp 249ndash256 1988

[9] A R Bestman ldquoRadiative heat transfer to flow of a combustiblemixture in a vertical piperdquo International Journal of EnergyResearch vol 15 no 3 pp 179ndash184 1991

[10] M A Alabraba A R Bestman and A Ogulu ldquoLaminar con-vection in binary mixture of hydromagnetic flow with radiativeheat transfer Irdquo Astrophysics and Space Science vol 195 no 2pp 431ndash439 1992

[11] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[12] O DMakinde P O Olanrewaju andWM Charles ldquoUnsteadyconvection with chemical reaction and radiative heat transferpast a flat porous plate moving through a binary mixturerdquoAfricka Matematika vol 22 no 1 pp 65ndash78 2011

[13] O D Makinde and P O Olanrewaju ldquoUnsteady mixed convec-tion with Soret and Dufour effects past a porous plate movingthrough a binarymixture of chemically reacting fluidrdquoChemicalEngineering Communications vol 198 no 7 pp 920ndash938 2011

[14] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer flow with mass transfer and a binary chemical reactionrdquoBritish Journal of Applied Science and Technology (Sciencedo-main International) vol 3 no 1 pp 131ndash149 2013

[15] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer heat and mass transfer flow with exothermic chemicalreactionsrdquo Journal of Pure and Applied Mathematics (Advancesand Applications) vol 9 no 1 pp 17ndash41 2013

[16] Kh Abdul Maleque ldquoEffects of binary chemical reactionand activation energy on MHD boundary layer heat andmass transfer flow with viscous dissipation and heat genera-tionabsorptionrdquo ISRN Thermodynamics vol 2013 Article ID284637 9 pages 2013

[17] Kh Abdul Maleque ldquoEffects of combined temperature- anddepth-dependent viscosity and hall current on an unsteadyMHD laminar convective flow due to a rotating diskrdquo ChemicalEngineering Communications vol 197 no 4 pp 506ndash521 2010

[18] P R Nachtsheim and P Swigert ldquoSatisfaction of asymptoticboundary conditions in numerical solution of system of non-linear of boundary layer typerdquo NASA TN-D3004 1965

Submit your manuscripts athttpwwwhindawicom

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ThermodynamicsJournal of

2 Journal of Thermodynamics

considered within the context of a binary mixture theory byAl-Sharif et al [5] and Wang et al [6]

A simple model involving binary reaction was studiedby Bestman [7] He considered the motion through theplate to be large which enabled him to obtain analyticalsolutions (subject to same restrictions) for various values ofactivation energy by employing the perturbation techniqueproposed by Singh andDikshit [8] Bestman [9] andAlabrabaet al [10] took into account the effect of the Arrheniusactivation energy under the different physical conditionsRecently Kandasamy et al [11] studied the combined effects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjection Their result shows that the flow field is influencedappreciably by chemical reaction heat source and suctionor injection at the wall of the wedge Recently Makindeet al [12 13] studied the problems of unsteady convectionwith chemical reaction and radiative heat transfer past aflat porous plate moving through a binary mixture in anoptically thin environment More recently Abdul Maleque[14 15] investigated the similarity solution on unsteadyincompressible fluid flow with binary chemical reactionsand activation energy In the present paper we investigatea numerical solution of unsteady Mhd natural convectionheat and mass transfer boundary layer flow past a flat porousplate taking into account the effect of Arrhenius activationenergy with exothermicendothermic chemical reactions andthermal radiationThis problem is an extension work studiedby Abdul Maleque [16]

2 Governing Equations

Weconsider the boundarywall to be of infinite extend so thatall quantities are homogeneous in 119909 and hence all derivativeswith respect to 119909 are neglected The 119909-axis is taken alongthe plate and 119910-axis is perpendicular to the plate Assumethat a uniform magnetic field 119861

0is applied perpendicular

to the plate and the plate is moving with uniform velocity119880119900in its own plane In presence of exothermicendothermic

binary chemical reaction with Arrhenius activation energya uniform magnetic field and thermal radiation thus thegoverning equations are

120597V

120597119910= 0 (2)

120597119906

120597119905+ V120597119906

120597119910= 1205921205972

119906

1205971199102+ 1198921205731(119879 minus 119879

infin)

+ 1198921205732(119862 minus 119862

infin) minus1205901198612

0

120588119906

(3)

120588119888119901(120597119879

120597119905+ V120597119879

120597119910) = 120581

1205972

119879

1205971199102+ 1205731198962

119903(119879

119879infin

)

119899

times exp [minus119864119886

119896119879] (119862 minus 119862

infin) minus120597119902119903

120597119910

(4)

120597119862

120597119905+ V120597119862

120597119910= 119863119898

1205972

119862

1205971199102minus 1198962

119903(119879

119879infin

)

119899

exp [minus119864119886

119896119879] (119862 minus 119862

infin)

(5)

The boundary conditions of previous system are

119906 = 1198800 V = V (119905) 119879 = 119879

119908 119862 = 119862

119908

at 119910 = 0 119905 gt 0

119906 = 0 119879 997888rarr 119879infin 119862 997888rarr 119862

infinat 119910 = infin 119905 gt 0

(6)

where (119906 V) is the velocity vector 119879is the temperature 119862isthe concentration of the fluid 120583 is the fluid viscosity 120590 isthe electrical conductivity 120592 is the kinematic coefficient ofviscosity 120573

1and 120573

2are the coefficients of volume expansions

for temperature and concentration respectively 120581 is theheat diffusivity coefficient 119888

119901is the specific heat at constant

pressure 119863119898

is the coefficient of mass diffusivity 1198962119903is

the chemical reaction rate constant 120573(= plusmn1) is exother-micendothermic parameter and (119879119879

infin)119899 exp[minus119864

119886119896 119879] is

the Arrhenius function where 119899 is a unit less constant expo-nent fitted rate constants typically lie in the range minus1 lt 119899 lt 1

The radiative heat flux 119902119903is described by Roseland

approximation such that

119902119903= minus41205901

31198961

1205971198794

120597119910 (7)

where1205901and 119896

1are the StefanBoltzmann constant andmean

absorption coefficient respectively Following Makinde andOlanrewaju [13] we assume that the temperature differenceswithin the flow are sufficiently small so that the 119879119899 canbe expressed as a linear function By using Taylorrsquos serieswe expand 119879119899 about the free stream temperature 119879

infinand

neglecting higher order terms This result of the followingapproximation

119879119899

asymp (1 minus 119899) 119879119899

infin+ 119899119879119899minus1

infin119879 (8)

Using (9) we have

120597119902119903

120597119910= minus41205901

31198961

1205972

1198794

1205971199102asymp minus161198793

infin1205901

31198961

1205972

119879

1205971199102 (9)

(119879

119879infin

)

119899

asymp (1 minus 119899) + 119899119879

119879infin

(10)

3 Mathematical Formulations

In order to solve the governing equations (2) to (5) under theboundary conditions (6) we adopt thewell-defined similaritytechnique to obtain the similarity solutions For this purposethe following nondimensional variables are now introduced

120578 =119910

120575 (119905)

119906

1198800

= 119891 (120578)

119879 minus 119879infin

119879119908minus 119879infin

= 120579 (120578) 119862 minus 119862

infin

119862119908minus 119862infin

= 120601 (120578)

(11)

Journal of Thermodynamics 3

From the equation of continuity (3) we have

V (119905) = minusV0120592

120575 (119905) (12)

where V0is the dimensionless suctioninjection velocity at the

plate V0gt 0 corresponds to suction and V

0lt 0 corresponds

to injectionIntroducing (9) (10) the dimensionless quantities from

(11) and V from (12) in (4) (5) and (6) we finally obtain thenonlinear ordinary differential equations as

11989110158401015840

+ (1205781205751205751015840

120592+ V0)1198911015840

minus119872119891 + 1198660120574120579 + 119866

119898120601 = 0 (13)

12057910158401015840

+ (119875119903

1 + 119873)[(120578

1205751205751015840

120592+ V0)1205791015840

+1205731205822

(1 + 119899120574120579) exp(minus 119864

1 + 120574120579)120601] = 0

(14)

119878minus1

11988812060110158401015840

+ (1205781205751205751015840

120592+ V0)1206011015840

minus 1205822

(1 + 119899120574120579) exp(minus 119864

1 + 120574120579)120601 = 0

(15)

Here Grashof number 119866119903= 1198660120574 = (119892120573119879

infin1205752

1205921198800)120574 the

temperature relative parameter 120574 = (119879119908minus 119879infin)119879infin

Modified (Solutal) Grashof number 119866119898= 119892120573

lowast

(119862119908minus

119862infin)1205752

1205921198800 Prandtl number 119875

119903= 120588120592119888

119901120581 magnetic interac-

tion parameter119872 = 120590120575211986120120583 schmidt number 119878

119888= 120592119863

119872

the nondimensional activation energy 119864 = 119864119886119896(119879119908minus 119879infin)

the dimensionless chemical reaction rate constant 1205822 =1198962

1199031205752

120592 radiation parameter119873 = 1612059011198793

infin31205811198961

Equations from (13) to (15) are similar except for the term1205751205751015840

120592 where time 119905 appears explicitly Thus the similaritycondition requires that 1205751205751015840120592 must be a constant quantityHence following Abdul Maleque [17] one can try a class ofsolutions of (13) to (15) by assuming that

1205751205751015840

120592= 119860 (constant) (16)

From (16) we have

120575 (119905) = radic2119860120592119905 + 1198712 (17)

where the constant of integration119871 is determined through thecondition that 120575 = 119871 when 119905 = 0 Here 119860 = 0 implies that 120575 =119871 represents the length scale for steady flow and 119860 = 0 thatis 120575 represents the length scale for unsteady flow Since 120575 is ascaling factor as well as similarity parameter any other valuesof119860 in (13) would not change the nature of the solution exceptthat the scale would be different Finally introducing (16) in

(13) to (15) respectively we have the following dimensionlessnonlinear ordinary differential equations

11989110158401015840

+ (119860120578 + V0) 1198911015840

minus119872119891 + 1198660120574120579 + 119866

119898120601 = 0 (18)

12057910158401015840

+ (119875119903

1 + 119873)[ (119860120578 + V

0) 1205791015840

+1205731205822

(1 + 119899120574120579) exp(minus 119864

1 + 120574120579)120601] = 0

(19)

119878minus1

11988812060110158401015840

+ (119860120578 + V0) 1206011015840

minus 1205822

(1 + 119899120574120579) exp(minus 119864

1 + 120574120579)120601 = 0

(20)

The boundary conditions equation (4) becomes

119891 (0) = 1 120579 (0) = 1 120601 (0) = 1

119891 (infin) = 0 120579 (infin) = 0 120601 (infin) = 0

(21)

In all over equations primes denote the differentiationwith respect to 120578 Equations from (18)ndash(20) are solvednumerically under the boundary conditions (21) usingNachtsheim-Swigert iteration technique

4 Numerical Solutions

Equations (18)ndash(20) are solved numerically under the bound-ary conditions (21) using Nachtsheim-Swigert [18] iterationtechnique In (21) there are three asymptotic boundary con-ditions and hence follow three unknown surface conditions1198911015840

(0) 1205791015840(0) and 1206011015840(0) Within the context of the initial-valuemethod and the Nachtsheim-Swigert iteration technique theouter boundary conditions may be functionally representedby the first-order Taylorrsquos series as

119891 (120578max) = 119891 (119883 119884 119885)

= 119891119888(120578max) + Δ119883119891119883 + Δ119884119891119884 + Δ119885119891119885 = 1205751

(22)

120579 (120578max) = 120579 (119883 119884 119885)

= 120579119888(120578max) + Δ119883120579119883 + Δ119884120579119884 + Δ119885120579119885 = 1205752

(23)

120601 (120578max) = 120601 (119883 119884 119885)

= 120601119888(120578max) + Δ119883120601119883 + Δ119884120601119884 + Δ119885120601119885 = 1205753

(24)

with the asymptotic convergence criteria given by

1198911015840

(120578max) = 1198911015840

(119883 119884 119885)

= 1198911015840

119888(120578max) + Δ119883119891

1015840

119883+ Δ119884119891

1015840

119884+ Δ119885119891

1015840

119885= 1205754

(25)

1205791015840

(120578max) = 1205791015840

(119883 119884 119885)

= 1205791015840

119888(120578max) + Δ119883120579

1015840

119883+ Δ119884120579

1015840

119884+ Δ119885120579

1015840

119885= 1205755

(26)

1206011015840

(120578max) = 1206011015840

(119883 119884 119885)

= 1206011015840

119888(120578max) + Δ119883120601

1015840

119883+ Δ119884120601

1015840

119884+ Δ119885120601

1015840

119885= 1205756

(27)

4 Journal of Thermodynamics

For G0 = 10 Gm = 1M = 05

120574 = 1 n = 1 = 3 and 120582 = 51

05120579

120578

120573 = 1

120573 = minus1

0 1 2

E = 1

E = 0

0Sc = 06

Pr = 071 N = 1

Figure 1 Effects of 120573 and 119864 on temperature profiles

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and 120582 = 5

05 15120578

120573 = 1

120573 = minus1120601

0

1

05

01

E = 0

E = 1

0Sc = 06

Pr

Figure 2 Effects of 120573 and 119864 on concentration profiles

where 119883 = 1198911015840

(0) 119884 = 1205791015840(0) 119885 = 1206011015840(0) and 119883119884 119885subscripts indicate partial differentiation for example 1198911015840

119884=

1205971198911015840

(120578max)1205971205791015840

(0) The subscript 119888 indicates the value of thefunction at 120578max to be determined from the trial integration

Solutions of these equations in a least square senserequire determining the minimum value of 119864 = 120575

2

1+

1205752

2+ 1205752

3+ 1205752

4+ 1205752

5+ 1205752

6with respect to 119883 119884 and 119885 To

solve Δ119883 Δ119884 and Δ119885 we require to differentiate 119864 withrespect to 119883 119884 and 119885 respectively Thus adopting thisnumerical technique a computer program was set up forthe solutions of the basic nonlinear differential equations ofour problem where the integration technique was adoptedas the six-ordered Runge-Kutta method of integration Theresults of this integration are then displayed graphically in

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and 120582 = 5

120578

120573 = 1

120573 = minus1

00

f05

1

1 2

E = 0

E = 1

0Sc = 06

Pr

Figure 3 Effects of 120573 and 119864 on velocity profiles

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and E = 1

120573 = 1

120573 = minus1f

05

1

120578

00

1 2

120582 = 1

120582 = 5

0Sc = 06

Pr

Figure 4 Effects of 120573 and 120582 on velocity profiles

the form of velocity temperature and concentration profilesin Figures 1ndash15 In the process of integration the local skin-friction coefficient the local rates of heat and mass transferto the surface which are of chief physical interest are alsocalculated out The equation defining the wall skin friction is

120591 = minus120583 (120597119906

120597119910)

119910=0

= minus1205831198800

1205751198911015840

(0) (28)

Hence the skin-friction coefficient is given by

1

2119862119891=120591

1205881198802

0

997904rArr1

2119877119890119862119891= minus1198911015840

(0) (29)

here the Reynolds number 119877119890= 1205881198800120575120583

Journal of Thermodynamics 5

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and E = 1

120582 = 1

120582 = 5

120573 = 1

120573 = minus1

120578

0 1 2

1

05

0

120579

0Sc = 06

Pr

Figure 5 Effects of 120573 and 120582 on temperature profiles

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and E = 1

120582 = 1

120582 = 5

120573 = 1

120578

00

1

1

05

2

120573 = minus1

120601

0Sc = 06

Pr

Figure 6 Effects of 120573 and 120582 on concentration profiles

0 1 2 3 40

05

1

120578

120579

N = 0 2 5

For G0 = 10 Gm = 1M = 05 = 071

120582 = 5

120574 = 1

n = 1 0 = 3 and E = 1Sc = 06

Pr

Figure 7 Effects of119873 on temperature profiles

0 1 2 3 40

05

1

15

2

f

N = 0 2 5

For G0 = 10 Gm = 1M = 05 = 071

120582 = 5

120574 = 1

n = 1 0 = 3 and E = 1

120578

Sc = 06

Pr

Figure 8 Effects of119873 on velocity profiles

0 1 2

0

1

2

f

For G0 = 10 Gm = 1N = 05 = 071M= 05

120582 = 5 n = 05 = 3 and E = 10

Gr = G0120574120574

2

1

0

minus1

minus1

minus2

120578

Sc = 06

Pr

Figure 9 Effects of 120574 on velocity profiles

0 1 20

05

1For G0 = 10 Gm = 1N = 05 = 071M= 05

120582 = 5 n = 05 = 3 and E = 10

Gr = G0120574

120574

210minus1

minus2

120578

120579

Sc = 06

Pr

Figure 10 Effects of 120574 on temperature profiles

6 Journal of Thermodynamics

0 1 20

05

1

120578

120601

For G0 = 10 Gm = 1N = 05 = 071M= 05

120582 = 5 n = 05 = 3 and E = 10

Gr = G0120574

120574

minus2minus101

2

Sc = 06

Pr

Figure 11 Effects of 120574 on concentration profiles

0 1 2 3 40

1

2

3

f

120578

For G0 = 10 Gm = 1N = 05 = 071M = 05

120582 = 5 n = 05 120574 = 05 and E = 1

0 = minus3

0 = 0

0 = 3

Sc = 06

Pr

Figure 12 Effects of V0on velocity profiles

The heat flux (119902119908) and the mass flux (119872

119908) at the wall are

given by

119902119908= minus120581(

120597119879

120597119910)

119910=0

= minus120581Δ119879

1205751205791015840

(0)

119872119908= minus119863119872(120597119862

120597119910)

119910=0

= minus119863119872

Δ119862

1205751206011015840

(0)

(30)

Hence the Nusselt number (119873119906) and the Sherwood number

(119878ℎ) are obtained as

119873119906=119902119908120575

120581 Δ119879= minus 120579

1015840

(0) 119878ℎ=119872119908120575

119863119872Δ119862= minus1206011015840

(0) (31)

These previous coefficients are then obtained from the proce-dure of the numerical computations and are sorted in Table 1

5 Results and Discussions

The parameters entering into the fluid flow are Grashofnumber 119866

119903= 1198660120574 Solutal (modified) Grashof number 119866

119898

suction parameter V0 Prandtl number 119875

119903 the nondimen-

sional chemical reaction rate constant 1205822 Schmidt number

0 1 2 3 40

05

1

15

0

3

For G0 = 10 Gm = 1N = 05 = 071M = 05

120582 = 5 n = 05 120574 = 05 and E = 1

0

minus3

120578

120579

Sc = 06

Pr

Figure 13 Effects of V0on temperature profiles

0 1 20

05

1

10

5

0

f

Gm

120582 = 5 n = 05 120574 = 05 and E = 1

120578

For G0 = 10 0 = 3N = 05 = 071M = 05

Sc = 06

Pr

Figure 14 Effects of 119866119898on velocity profiles

0 1 20

05

1

M

1

0

5

f

120578

For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

120582 = 5 n = 1 0 = 3 and E = 1Sc = 06

Pr

Figure 15 Effects of119872 on velocity profiles

Journal of Thermodynamics 7

0 1 2 30

05

1

f

120578

For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

A = 0 (steady)A = 2 (unsteady)

120574 = 5 n = 1 0 = 3M = 05 and E = 1Sc = 06

Pr

Figure 16 Effects of 119860 on velocity profiles

0 1 2 30

05

1 For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

A = 0 (steady)A = 2 (unsteady)

120579

120578

120574 = 5 n = 1 0 = 3M = 05 and E = 1Sc = 06

Pr

Figure 17 Effects of 119860 on temperature profiles

119878119888 the magnetic parameter 119872 the nondimensional acti-

vation energy 119864 the radiation parameter 119873 the exother-micendothermic parameter 120573 and the temperature relativeparameter 120574

It is therefore pertinent to inquire the effects of variationof each of them when the others are kept constant Thenumerical results are thus presented in the form of velocityprofiles temperature profiles and concentration profiles inFigures 1ndash14 for the different values of 119866

119903 119866119898 120582119872 119864 V

0 120573

and 120574The value of 120574 is taken to be both positive and negativesince these values represent respectively cooling and heatingof the plate

The values of 119866119903is taken to be large (119866

119903= 10

1198660= 10 and 120574 = 1) since the value corresponds to

Table 1 Effects on skin friction heat transfer and mass transfercoefficients For 119866

0= 10 119866

119898= 1119873 = 05 119875

119903= 071119872 = 05 119878

119888=

06 120582 = 5 119899 = 05 V0= 3 120574 = 05 and 119864 = 1

minus1198911015840

(0) minus1205791015840

(0) minus1206011015840

(0)

119864 for 120573 = 100 minus165163 minus091802 57863405 minus160009 minus088980 57552310 minus154562 minus085343 570174

119864 for 120573 = minus100 130241 449966 55515905 098702 419573 52109310 063543 387806 481246

120582 for 120573 = 110 minus043158 173298 24262720 minus063434 134699 29144650 minus154562 minus085343 570174

120582 for 120573 = minus110 minus027903 198971 24208820 minus007554 232400 28442450 063543 387806 481246

a cooling problem that is generally encountered in nuclearengineering in connection with the cooling of reactors In air(119875119903= 071) the diluting chemical species of most common

interest have Schmidt number in the range from 06 to 075Therefore the Schmidt number 119878

119888= 06 is considered In

particular 06 corresponds to water vapor that represents adiffusing chemical species of most common interest in airThe values of the suction parameter V

0are taken to be large

Apart from the figures and tables the representative velocitytemperatureand concentration profiles and the values ofthe physically important parameters that is the local shearstress the local rates of heat and mass transfer are illustratedfor uniform wall temperature and species concentration inFigures 1ndash17 and in Tables 1-2

51 Effects of Activation Energy (119864) on the Concentrationthe Temperature and the Velocity Profiles for ExothermicEndothermic Chemical Reaction In chemistry activationenergy is defined as the energy that must be overcome inorder for a chemical reaction to occur Activation energymayalso be defined as the minimum energy required startinga chemical reaction The activation energy of a reaction isusually denoted by 119864

119886and given in units of kilojoules per

moleAn exothermic reaction is a chemical reaction that

releases energy in the form of light and heat It is theopposite of an endothermic reaction It gives out energy to itssurroundings The energy needed for the reaction to occur isless or greater than the total energy released for exothermicor endothermic reaction respectively Energy is obtainedfrom chemical bonds When bonds are broken energy isrequired When bonds are formed energy is released Eachtype of bond has specific bond energy It can be predictedwhether a chemical reaction will release or require heat byusing bond energiesWhen there ismore energy used to form

8 Journal of Thermodynamics

Table 2 Effects on skin friction heat transfer and mass transfercoefficients For119866

0= 10119866

119898= 1119873 = 05119875

119903= 071119872 = 05 119878

119888=

06 120582 = 5 119899 = 05 V0= 3 120574 = 05 and 119864 = 1

minus1198911015840

(0) minus1205791015840

(0) minus1206011015840

(0)

119873

00 minus087260 minus144125 56899620 minus305961 minus030027 57162450 minus493466 minus004756 572770

119872

00 minus185010 minus085309 57016420 minus075246 minus085443 57020350 046744 minus085628 570257

120574

minus200000 732171 365294 007835minus100000 566728 348636 026098000000 331180 188820 228076100000 minus154562 minus085343 570174200000 minus785616 minus367839 925776

119860

000000 minus127274 minus026974 397243200000 120419 056519 391549

the bonds than to break the bonds heat is given off This isknown as an exothermic reaction When a reaction requiresan input or output of energy it is known as an exothermic orendothermic reaction

Effects of activation energy (119864) and exothermic param-eter (120573) on the temperature the concentration and velocityprofiles are shown in Figure 1 to Figure 3 respectively 120573 = 1and 120573 = minus1 represent exothermic and endothermic chemicalreactions respectively Figure 1 shows that a small decreasingeffect of temperature profile is found for increasing valuesof 119864 for exothermic reaction (120573 = 1) but mark oppositeeffects are found for endothermic reaction in Figure 1 Thatis the temperature profile increases for increasing values of119864 for endothermic reaction 120573 = minus1 From (1) we observethat chemical reaction rate (119870) decreases with the increasingvalues of activation energy (119864

119886) We also observe from (20)

that increase in activation energy (119864) leads to decrease1205822 exp(minus119864120579) as well as to increase in the concentration

profiles shown in Figure 2 Small and reported effects arefound for 120573 = 1 and 120573 = minus1 respectively The parameter119864 does not enter directly into the momentum equation butits influence comes through the mass and energy equationsFigure 3 shows the variation of the velocity profiles fordifferent values of 119864 From this figure it has been observedthat the velocity profile mark increases with the increasingvalues of 119864 for endothermic chemical reaction But negligibleeffects are found for exothermic reaction

52 Effects of 120582 on the Concentration the Temperature and theVelocity Profiles for ExothermicEndothermic Reaction Con-sidering chemical reaction rate constant 1205822 is always positiveFigures 4ndash6 represent the effect of chemical rate constant120582 on

the velocity the temperature and the concentration profilesrespectively

We observe from Figures 4 and 5 that velocity andtemperature profiles increase with the increasing values of120582 for exothermic reaction but opposite effects are foundin these figures for endothermic reaction It is observedfrom (1) that increasing temperature frequently that causes amarked increase in the rate of reactions is shown in Figure 4As the temperature of the system increases the numberof molecules that carry enough energy to react when theycollide also increasesTherefore the rate of reaction increaseswith temperature As a rule the rate of a reaction doubles forevery 10∘C increase in the temperature of the system

Last part of (20) shows that 1205822 exp(minus119864120579) increases withthe increasing values of 120582We also observe from this equationthat increase in 1205822 exp(minus119864120579)means that increase in 120582 leadsto the decrease in the concentration profiles This is in greatagreement with Figure 6 for both cases 120573 = plusmn1 It isalso observed from this figure that for exothermic reactionthe mass boundary layer is close to the plate other thanendothermic reaction

53 Effects of Radiation Parameter 119873 on Temperature andVelocity Profiles The effects of radiation parameter119873 on thetemperature and the velocity profiles are shown in Figures 7and 8 From Figure 7 it can be seen that an increase in thevalues of 119873 leads to a decrease in the values of temperatureprofiles within the thermal boundary layers 120578 lt 022while outside 120578 gt 022 the temperature profile graduallyincreases with the increase of the radiation parameter 119873 Itis appear from Figure 8 that when 119873 = 0 that is withoutthe radiation the velocity profile shows its usual trend ofgradually decay As radiation becomes larger the profilesovershoot the uniform velocity close to the boundary

54 Effects of 120574 on the Velocity Temperature and Concen-tration Profiles The value of 120574 = Δ119879119879

infinis taken to be

both positive and negative since these values representrespectively cooling and heating of the plate The effects oftemperature relative parameter on velocity temperature andconcentration profiles are shown in Figures 9ndash11

The velocity profiles generated due to impulsive motionof the plate is plotted in Figure 9 for both cooling (120574 gt 0) andheating (120574 lt 0) of the plate keeping other parameters fixed(1198660= 10 119866

119898= 10 119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 V0= 30 119864 = 10 and 119899 = 1) In Figure 9 velocity

profiles are shown for different values of 120574 We observe thatvelocity increases with increasing values of 120574 for the coolingof the plate From this figure it is also observed that thenegative increase in the temperature relative parameter leadsto the decrease in the velocity field That is for heating ofthe plate (Figure 9) the effects of the 120574 on the velocity fieldhave also opposite effects as compared to the cooling ofthe plate Figure 10 shows the effects of temperature relativeparameter 120574 on the temperature profiles It has been observedfrom this figure that for heating plate the temperature profileshows its usual trend of gradually decay and the thermalboundary layer is close to the plate But for cooling plate

Journal of Thermodynamics 9

that is temperature relative parameter 120574 becomes largerthe profiles overshoot the uniform temperature close to thethermal boundary Opposite effects of temperature profile arefound for concentration profiles shown in Figure 11

55 Effects of SuctionInjection (V0) on the Velocity and the

Temperature Profiles The effects of suction and injection (V0)

for 1198660= 10 119866

119898= 10119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 120574 = 05 119864 = 10 and 119899 = 1 on the velocity profilesand temperature profiles are shown respectively in Figures12 and 13 For strong suction (V

0gt 0) the velocity and the

temperature decay rapidly away from the surface The factthat suction stabilizes the boundary layer is also apparentfrom these figures As for the injection (V

0lt 0) from Figures

12 and 13 it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayer between theinjection and the outer flow regions

56 Effects of 119866119898and119872 on the Velocity Profiles The effects

of Solutal Grashof number 119866119898

and magnetic interactionparameter119872 on velocity profiles are shown in Figures 14 and15 respectively Solutal Grashof number 119866

119898gt 0 corresponds

that the chemical species concentration in the free streamregion is less than the concentration at the boundary surfaceFigure 14 presents the effects of Solutal Grashof number119866119898on the velocity profiles It is observed that the velocity

profile increases with the increasing values of Solutal Grashofnumber 119866

119898

Imposition of a magnetic field to an electrically conduct-ing fluid creates a drag like force called Lorentz force Theforce has the tendency to slow down the flow around the plateat the expense of increasing its temperature This is depictedby decreases in velocity profiles as magnetic parameter 119872increases as shown in Figure 15

57 Effects of 119860 on the Velocity and the Temperature ProfilesHere 119860 = 0 and 119860 = 0 represent steady and unsteady flowsrespectively The effects of 119860 on velocity and temperatureprofiles are shown in Figures 16 and 17 respectively Forunsteady flow (119860 = 0) both figures show that the profilesrapidly decay and close to the plate But for steady flow (119860 =0) the profiles overshoot the uniform velocitytemperatureclose to the boundary

58 The Skin-Friction the Heat Transfer and the MassTransfer Coefficients The skin-friction the heat transferand the mass transfer coefficients are tabulated in Tables1 and 2 for different values of 119864 120573 120582 119873 119872 120574 and 119860We observe from Table 1 that the skin-friction coefficient(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) increasefor increasing values of dimensionless activation parameter119864 for exothermic chemical reaction but opposite actionsare found for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients decrease for increasing values of 119864 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) decreases for increasing values of 119864We also observe fromTable 1 that the skin-friction coefficient

0 1 2 3 40

05

1

Bestman (1990)Present

f

120578

Gr = Gm = 1 E = 5 20 = 10

= 071 and 120582 = 5

120573 = 0 120574 = 1 and M= N = A = 0

Sc = 1 Pr

Figure 18 Comparison of our calculated velocity profile and thevelocity profile of Bestman [7]

(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) decreasefor increasing values of dimensionless reaction rate constant120582 for exothermic chemical reaction but opposite effects arefound for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients increase for increasing values of 120582 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) remarkably increases for increasingvalues of 120582 From Table 2 it has been observed that theskin-friction coefficient remarkably decreases and Nusseltnumber remarkable increases for increasing values of radi-ation parameter119873 The skin-friction coefficient increases forincreasing values of magnetic parameter 119872 From Table 2we also found that skin friction and heat transfer coefficientsdecrease butmass transfer coefficient increases for increasingvalues of temperature relative parameter 120574

59 Comparison between Our Numerical Result and theAnalytical Result of Bestman [7] Bestman studied steadynatural convective boundary layer flow with large suctionHe solved his problem analytically by employing the per-turbation technique proposed by Singh and Dikshit [8] Inour present work take 119860 = 0 in (16) for considering steadyflow The values of the suction parameter V2

0= 10 is taken

to see the effects of large suction 119866119903= 119866119898= 10 120574 = 1

119872 = 119873 = 0 120582 = 5 119864 = 5 and 119899 = 1 are also chosen witha view to compare our numerical results with the analyticalresults of Bestman [7] The comparison of velocity profilesas seen in Figure 18 highlights the validity of the numericalcomputations adapted in the present investigation

6 Conclusions

In this paper we investigate the effects of chemical reactionrate and Arrhenius activation energy on an unsteady MHD

10 Journal of Thermodynamics

natural convection heat and mass transfer boundary layerflow past a flat porous plate in presence of thermal radiation

The Nachtsheim and Swigert [18] iteration techniquebased on sixth-order Runge-Kutta and Shooting method hasbeen employed to complete the integration of the resultingsolutions

The following conclusions can be drawn as a result of thecomputations

(1) Velocity increases with increasing values of 120574 for thecooling of the plate and the negative increase in thetemperature relative parameter 120574 leads to the decreasein the velocity fieldThat is for heating of the plate theeffects of the temperature relative parameter 120574 on thevelocity field have also opposite effects as comparedto the cooling of the plate

(2) Solutal Grashof number 119866119898gt 0 corresponds that

the chemical species concentration in the free streamregion is less than the concentration at the bound-ary surface It is observed that the velocity profileincreaseswith the increasing values of SolutalGrashofnumber 119866

119898

(3) Increase in 120582 leads to increase in the velocity andtemperature profiles for exothermic reaction butopposite effects are found for endothermic reactionFor 120573 = plusmn1 increase in 120582 leads to the decrease in theconcentration profiles but for exothermic reactionthe mass boundary layer is close to the plate otherthan endothermic reaction

(4) The velocity profile mark increases with the increas-ing values of 119864 for endothermic chemical reactionBut negligible effect is found for exothermic reactionTemperature decreases for increasing values of 119864 forexothermic reaction but opposite effects are found forendothermic reaction Activation energy (119864) leads toincrease in the concentration profiles but small andreported effects are found for 120573 = 1 and 120573 = minus1respectively

(5) For strong suction (V0gt 0) the velocity the tem-

perature and the concentration profiles decay rapidlyaway from the surface As for the injection (V

0lt 0)

it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayerbetween the injection and the outer flow regions

(6) Imposition of a magnetic field to an electrically con-ducting fluid creates a drag like force called Lorentzforce The force has the tendency to slow down theflow around the plate at the expense of increasing itstemperature This is depicted by decreases in velocityprofiles as magnetic parameter119872 increases

(7) An increase in the values of 119873 leads to a decrease inthe values of temperature profiles within the thermalboundary layers 120578 lt 022 while outside 120578 gt 022the temperature profile gradually increases with theincrease of the radiation parameter119873

(8) For unsteady flow (119860 = 0) the velocity and the tem-perature profiles rapidly decay and close to the plate

But for steady flow (119860 = 0) the profiles overshoot theuniform velocitytemperature close to the boundary

Nomenclature

(119906 V) Components of velocity field119888119901 Specific heat at constant pressure119873119906 Nusselt number

119875119903 Prandtl number1198760 Heat generationabsorption coefficient

119878ℎ Sherwood number119879 Temperature of the flow field119879infin Temperature of the fluid at infinity

119879119908 Temperature at the plate

119891 Dimensionless similarity functions119864119886 The activation energy

119896 The Boltzmann constant119878119888 Schmidt number119864 The nondimensional activation energy119866119903 Grashof number

119866119898 Modified (Solutal) Grashof number

Greek Symbols

120579 Dimensionless temperature120601 Dimensionless concentration120578 Dimensionless similarity variable120575 Scale factorV Kinematic viscosity120588 Density of the fluid120581 Thermal conductivity120591 Shear stress120583 Fluid viscosity1205822 Dimensionless chemical reaction rate

constant120573 and 120573lowast The coefficients of volume expansions for

temperature and concentrationrespectively

120574 The dimensionless heatgenerationabsorption coefficient

References

[1] M Tencer J S Moss and T Zapach ldquoArrhenius averagetemperature the effective temperature for non-fatigue wearoutand long term reliability in variable thermal conditions andclimatesrdquo IEEE Transactions on Components and PackagingTechnologies vol 27 no 3 pp 602ndash607 2004

[2] C Truesdell ldquoSulle basi della thermomeccanicardquo RendicontiLincei vol 22 no 8 pp 33ndash38 1957

[3] N Mills ldquoIncompressible mixtures of newtonian fluidsrdquo Inter-national Journal of Engineering Science vol 4 no 2 pp 97ndash1121966

[4] C E Beevers and R E Craine ldquoOn the determination ofresponse functions for a binary mixture of incompressiblenewtonian fluidsrdquo International Journal of Engineering Sciencevol 20 no 6 pp 737ndash745 1982

Journal of Thermodynamics 11

[5] A Al-Sharif K Chamniprasart K R Rajagopal andA Z SzerildquoLubrication with binary mixtures liquid-liquid emulsionrdquoJournal of Tribology vol 115 no 1 pp 46ndash55 1993

[6] S H Wang A Al-Sharif K R Rajagopal and A Z SzerildquoLubrication with binarymixtures liquid-liquid emulsion in anEHL conjunctionrdquo Journal of Tribology vol 115 no 3 pp 515ndash522 1993

[7] A R Bestman ldquoNatural convection boundary layer withsuction and mass transfer in a porous mediumrdquo InternationalJournal of Energy Research vol 14 no 4 pp 389ndash396 1990

[8] A K Singh and C K Dikshit ldquoHydromagnetic flow pasta continuously moving semi-infinite plate for large suctionrdquoAstrophysics and Space Science vol 148 no 2 pp 249ndash256 1988

[9] A R Bestman ldquoRadiative heat transfer to flow of a combustiblemixture in a vertical piperdquo International Journal of EnergyResearch vol 15 no 3 pp 179ndash184 1991

[10] M A Alabraba A R Bestman and A Ogulu ldquoLaminar con-vection in binary mixture of hydromagnetic flow with radiativeheat transfer Irdquo Astrophysics and Space Science vol 195 no 2pp 431ndash439 1992

[11] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[12] O DMakinde P O Olanrewaju andWM Charles ldquoUnsteadyconvection with chemical reaction and radiative heat transferpast a flat porous plate moving through a binary mixturerdquoAfricka Matematika vol 22 no 1 pp 65ndash78 2011

[13] O D Makinde and P O Olanrewaju ldquoUnsteady mixed convec-tion with Soret and Dufour effects past a porous plate movingthrough a binarymixture of chemically reacting fluidrdquoChemicalEngineering Communications vol 198 no 7 pp 920ndash938 2011

[14] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer flow with mass transfer and a binary chemical reactionrdquoBritish Journal of Applied Science and Technology (Sciencedo-main International) vol 3 no 1 pp 131ndash149 2013

[15] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer heat and mass transfer flow with exothermic chemicalreactionsrdquo Journal of Pure and Applied Mathematics (Advancesand Applications) vol 9 no 1 pp 17ndash41 2013

[16] Kh Abdul Maleque ldquoEffects of binary chemical reactionand activation energy on MHD boundary layer heat andmass transfer flow with viscous dissipation and heat genera-tionabsorptionrdquo ISRN Thermodynamics vol 2013 Article ID284637 9 pages 2013

[17] Kh Abdul Maleque ldquoEffects of combined temperature- anddepth-dependent viscosity and hall current on an unsteadyMHD laminar convective flow due to a rotating diskrdquo ChemicalEngineering Communications vol 197 no 4 pp 506ndash521 2010

[18] P R Nachtsheim and P Swigert ldquoSatisfaction of asymptoticboundary conditions in numerical solution of system of non-linear of boundary layer typerdquo NASA TN-D3004 1965

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ThermodynamicsJournal of

Journal of Thermodynamics 3

From the equation of continuity (3) we have

V (119905) = minusV0120592

120575 (119905) (12)

where V0is the dimensionless suctioninjection velocity at the

plate V0gt 0 corresponds to suction and V

0lt 0 corresponds

to injectionIntroducing (9) (10) the dimensionless quantities from

(11) and V from (12) in (4) (5) and (6) we finally obtain thenonlinear ordinary differential equations as

11989110158401015840

+ (1205781205751205751015840

120592+ V0)1198911015840

minus119872119891 + 1198660120574120579 + 119866

119898120601 = 0 (13)

12057910158401015840

+ (119875119903

1 + 119873)[(120578

1205751205751015840

120592+ V0)1205791015840

+1205731205822

(1 + 119899120574120579) exp(minus 119864

1 + 120574120579)120601] = 0

(14)

119878minus1

11988812060110158401015840

+ (1205781205751205751015840

120592+ V0)1206011015840

minus 1205822

(1 + 119899120574120579) exp(minus 119864

1 + 120574120579)120601 = 0

(15)

Here Grashof number 119866119903= 1198660120574 = (119892120573119879

infin1205752

1205921198800)120574 the

temperature relative parameter 120574 = (119879119908minus 119879infin)119879infin

Modified (Solutal) Grashof number 119866119898= 119892120573

lowast

(119862119908minus

119862infin)1205752

1205921198800 Prandtl number 119875

119903= 120588120592119888

119901120581 magnetic interac-

tion parameter119872 = 120590120575211986120120583 schmidt number 119878

119888= 120592119863

119872

the nondimensional activation energy 119864 = 119864119886119896(119879119908minus 119879infin)

the dimensionless chemical reaction rate constant 1205822 =1198962

1199031205752

120592 radiation parameter119873 = 1612059011198793

infin31205811198961

Equations from (13) to (15) are similar except for the term1205751205751015840

120592 where time 119905 appears explicitly Thus the similaritycondition requires that 1205751205751015840120592 must be a constant quantityHence following Abdul Maleque [17] one can try a class ofsolutions of (13) to (15) by assuming that

1205751205751015840

120592= 119860 (constant) (16)

From (16) we have

120575 (119905) = radic2119860120592119905 + 1198712 (17)

where the constant of integration119871 is determined through thecondition that 120575 = 119871 when 119905 = 0 Here 119860 = 0 implies that 120575 =119871 represents the length scale for steady flow and 119860 = 0 thatis 120575 represents the length scale for unsteady flow Since 120575 is ascaling factor as well as similarity parameter any other valuesof119860 in (13) would not change the nature of the solution exceptthat the scale would be different Finally introducing (16) in

(13) to (15) respectively we have the following dimensionlessnonlinear ordinary differential equations

11989110158401015840

+ (119860120578 + V0) 1198911015840

minus119872119891 + 1198660120574120579 + 119866

119898120601 = 0 (18)

12057910158401015840

+ (119875119903

1 + 119873)[ (119860120578 + V

0) 1205791015840

+1205731205822

(1 + 119899120574120579) exp(minus 119864

1 + 120574120579)120601] = 0

(19)

119878minus1

11988812060110158401015840

+ (119860120578 + V0) 1206011015840

minus 1205822

(1 + 119899120574120579) exp(minus 119864

1 + 120574120579)120601 = 0

(20)

The boundary conditions equation (4) becomes

119891 (0) = 1 120579 (0) = 1 120601 (0) = 1

119891 (infin) = 0 120579 (infin) = 0 120601 (infin) = 0

(21)

In all over equations primes denote the differentiationwith respect to 120578 Equations from (18)ndash(20) are solvednumerically under the boundary conditions (21) usingNachtsheim-Swigert iteration technique

4 Numerical Solutions

Equations (18)ndash(20) are solved numerically under the bound-ary conditions (21) using Nachtsheim-Swigert [18] iterationtechnique In (21) there are three asymptotic boundary con-ditions and hence follow three unknown surface conditions1198911015840

(0) 1205791015840(0) and 1206011015840(0) Within the context of the initial-valuemethod and the Nachtsheim-Swigert iteration technique theouter boundary conditions may be functionally representedby the first-order Taylorrsquos series as

119891 (120578max) = 119891 (119883 119884 119885)

= 119891119888(120578max) + Δ119883119891119883 + Δ119884119891119884 + Δ119885119891119885 = 1205751

(22)

120579 (120578max) = 120579 (119883 119884 119885)

= 120579119888(120578max) + Δ119883120579119883 + Δ119884120579119884 + Δ119885120579119885 = 1205752

(23)

120601 (120578max) = 120601 (119883 119884 119885)

= 120601119888(120578max) + Δ119883120601119883 + Δ119884120601119884 + Δ119885120601119885 = 1205753

(24)

with the asymptotic convergence criteria given by

1198911015840

(120578max) = 1198911015840

(119883 119884 119885)

= 1198911015840

119888(120578max) + Δ119883119891

1015840

119883+ Δ119884119891

1015840

119884+ Δ119885119891

1015840

119885= 1205754

(25)

1205791015840

(120578max) = 1205791015840

(119883 119884 119885)

= 1205791015840

119888(120578max) + Δ119883120579

1015840

119883+ Δ119884120579

1015840

119884+ Δ119885120579

1015840

119885= 1205755

(26)

1206011015840

(120578max) = 1206011015840

(119883 119884 119885)

= 1206011015840

119888(120578max) + Δ119883120601

1015840

119883+ Δ119884120601

1015840

119884+ Δ119885120601

1015840

119885= 1205756

(27)

4 Journal of Thermodynamics

For G0 = 10 Gm = 1M = 05

120574 = 1 n = 1 = 3 and 120582 = 51

05120579

120578

120573 = 1

120573 = minus1

0 1 2

E = 1

E = 0

0Sc = 06

Pr = 071 N = 1

Figure 1 Effects of 120573 and 119864 on temperature profiles

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and 120582 = 5

05 15120578

120573 = 1

120573 = minus1120601

0

1

05

01

E = 0

E = 1

0Sc = 06

Pr

Figure 2 Effects of 120573 and 119864 on concentration profiles

where 119883 = 1198911015840

(0) 119884 = 1205791015840(0) 119885 = 1206011015840(0) and 119883119884 119885subscripts indicate partial differentiation for example 1198911015840

119884=

1205971198911015840

(120578max)1205971205791015840

(0) The subscript 119888 indicates the value of thefunction at 120578max to be determined from the trial integration

Solutions of these equations in a least square senserequire determining the minimum value of 119864 = 120575

2

1+

1205752

2+ 1205752

3+ 1205752

4+ 1205752

5+ 1205752

6with respect to 119883 119884 and 119885 To

solve Δ119883 Δ119884 and Δ119885 we require to differentiate 119864 withrespect to 119883 119884 and 119885 respectively Thus adopting thisnumerical technique a computer program was set up forthe solutions of the basic nonlinear differential equations ofour problem where the integration technique was adoptedas the six-ordered Runge-Kutta method of integration Theresults of this integration are then displayed graphically in

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and 120582 = 5

120578

120573 = 1

120573 = minus1

00

f05

1

1 2

E = 0

E = 1

0Sc = 06

Pr

Figure 3 Effects of 120573 and 119864 on velocity profiles

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and E = 1

120573 = 1

120573 = minus1f

05

1

120578

00

1 2

120582 = 1

120582 = 5

0Sc = 06

Pr

Figure 4 Effects of 120573 and 120582 on velocity profiles

the form of velocity temperature and concentration profilesin Figures 1ndash15 In the process of integration the local skin-friction coefficient the local rates of heat and mass transferto the surface which are of chief physical interest are alsocalculated out The equation defining the wall skin friction is

120591 = minus120583 (120597119906

120597119910)

119910=0

= minus1205831198800

1205751198911015840

(0) (28)

Hence the skin-friction coefficient is given by

1

2119862119891=120591

1205881198802

0

997904rArr1

2119877119890119862119891= minus1198911015840

(0) (29)

here the Reynolds number 119877119890= 1205881198800120575120583

Journal of Thermodynamics 5

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and E = 1

120582 = 1

120582 = 5

120573 = 1

120573 = minus1

120578

0 1 2

1

05

0

120579

0Sc = 06

Pr

Figure 5 Effects of 120573 and 120582 on temperature profiles

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and E = 1

120582 = 1

120582 = 5

120573 = 1

120578

00

1

1

05

2

120573 = minus1

120601

0Sc = 06

Pr

Figure 6 Effects of 120573 and 120582 on concentration profiles

0 1 2 3 40

05

1

120578

120579

N = 0 2 5

For G0 = 10 Gm = 1M = 05 = 071

120582 = 5

120574 = 1

n = 1 0 = 3 and E = 1Sc = 06

Pr

Figure 7 Effects of119873 on temperature profiles

0 1 2 3 40

05

1

15

2

f

N = 0 2 5

For G0 = 10 Gm = 1M = 05 = 071

120582 = 5

120574 = 1

n = 1 0 = 3 and E = 1

120578

Sc = 06

Pr

Figure 8 Effects of119873 on velocity profiles

0 1 2

0

1

2

f

For G0 = 10 Gm = 1N = 05 = 071M= 05

120582 = 5 n = 05 = 3 and E = 10

Gr = G0120574120574

2

1

0

minus1

minus1

minus2

120578

Sc = 06

Pr

Figure 9 Effects of 120574 on velocity profiles

0 1 20

05

1For G0 = 10 Gm = 1N = 05 = 071M= 05

120582 = 5 n = 05 = 3 and E = 10

Gr = G0120574

120574

210minus1

minus2

120578

120579

Sc = 06

Pr

Figure 10 Effects of 120574 on temperature profiles

6 Journal of Thermodynamics

0 1 20

05

1

120578

120601

For G0 = 10 Gm = 1N = 05 = 071M= 05

120582 = 5 n = 05 = 3 and E = 10

Gr = G0120574

120574

minus2minus101

2

Sc = 06

Pr

Figure 11 Effects of 120574 on concentration profiles

0 1 2 3 40

1

2

3

f

120578

For G0 = 10 Gm = 1N = 05 = 071M = 05

120582 = 5 n = 05 120574 = 05 and E = 1

0 = minus3

0 = 0

0 = 3

Sc = 06

Pr

Figure 12 Effects of V0on velocity profiles

The heat flux (119902119908) and the mass flux (119872

119908) at the wall are

given by

119902119908= minus120581(

120597119879

120597119910)

119910=0

= minus120581Δ119879

1205751205791015840

(0)

119872119908= minus119863119872(120597119862

120597119910)

119910=0

= minus119863119872

Δ119862

1205751206011015840

(0)

(30)

Hence the Nusselt number (119873119906) and the Sherwood number

(119878ℎ) are obtained as

119873119906=119902119908120575

120581 Δ119879= minus 120579

1015840

(0) 119878ℎ=119872119908120575

119863119872Δ119862= minus1206011015840

(0) (31)

These previous coefficients are then obtained from the proce-dure of the numerical computations and are sorted in Table 1

5 Results and Discussions

The parameters entering into the fluid flow are Grashofnumber 119866

119903= 1198660120574 Solutal (modified) Grashof number 119866

119898

suction parameter V0 Prandtl number 119875

119903 the nondimen-

sional chemical reaction rate constant 1205822 Schmidt number

0 1 2 3 40

05

1

15

0

3

For G0 = 10 Gm = 1N = 05 = 071M = 05

120582 = 5 n = 05 120574 = 05 and E = 1

0

minus3

120578

120579

Sc = 06

Pr

Figure 13 Effects of V0on temperature profiles

0 1 20

05

1

10

5

0

f

Gm

120582 = 5 n = 05 120574 = 05 and E = 1

120578

For G0 = 10 0 = 3N = 05 = 071M = 05

Sc = 06

Pr

Figure 14 Effects of 119866119898on velocity profiles

0 1 20

05

1

M

1

0

5

f

120578

For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

120582 = 5 n = 1 0 = 3 and E = 1Sc = 06

Pr

Figure 15 Effects of119872 on velocity profiles

Journal of Thermodynamics 7

0 1 2 30

05

1

f

120578

For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

A = 0 (steady)A = 2 (unsteady)

120574 = 5 n = 1 0 = 3M = 05 and E = 1Sc = 06

Pr

Figure 16 Effects of 119860 on velocity profiles

0 1 2 30

05

1 For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

A = 0 (steady)A = 2 (unsteady)

120579

120578

120574 = 5 n = 1 0 = 3M = 05 and E = 1Sc = 06

Pr

Figure 17 Effects of 119860 on temperature profiles

119878119888 the magnetic parameter 119872 the nondimensional acti-

vation energy 119864 the radiation parameter 119873 the exother-micendothermic parameter 120573 and the temperature relativeparameter 120574

It is therefore pertinent to inquire the effects of variationof each of them when the others are kept constant Thenumerical results are thus presented in the form of velocityprofiles temperature profiles and concentration profiles inFigures 1ndash14 for the different values of 119866

119903 119866119898 120582119872 119864 V

0 120573

and 120574The value of 120574 is taken to be both positive and negativesince these values represent respectively cooling and heatingof the plate

The values of 119866119903is taken to be large (119866

119903= 10

1198660= 10 and 120574 = 1) since the value corresponds to

Table 1 Effects on skin friction heat transfer and mass transfercoefficients For 119866

0= 10 119866

119898= 1119873 = 05 119875

119903= 071119872 = 05 119878

119888=

06 120582 = 5 119899 = 05 V0= 3 120574 = 05 and 119864 = 1

minus1198911015840

(0) minus1205791015840

(0) minus1206011015840

(0)

119864 for 120573 = 100 minus165163 minus091802 57863405 minus160009 minus088980 57552310 minus154562 minus085343 570174

119864 for 120573 = minus100 130241 449966 55515905 098702 419573 52109310 063543 387806 481246

120582 for 120573 = 110 minus043158 173298 24262720 minus063434 134699 29144650 minus154562 minus085343 570174

120582 for 120573 = minus110 minus027903 198971 24208820 minus007554 232400 28442450 063543 387806 481246

a cooling problem that is generally encountered in nuclearengineering in connection with the cooling of reactors In air(119875119903= 071) the diluting chemical species of most common

interest have Schmidt number in the range from 06 to 075Therefore the Schmidt number 119878

119888= 06 is considered In

particular 06 corresponds to water vapor that represents adiffusing chemical species of most common interest in airThe values of the suction parameter V

0are taken to be large

Apart from the figures and tables the representative velocitytemperatureand concentration profiles and the values ofthe physically important parameters that is the local shearstress the local rates of heat and mass transfer are illustratedfor uniform wall temperature and species concentration inFigures 1ndash17 and in Tables 1-2

51 Effects of Activation Energy (119864) on the Concentrationthe Temperature and the Velocity Profiles for ExothermicEndothermic Chemical Reaction In chemistry activationenergy is defined as the energy that must be overcome inorder for a chemical reaction to occur Activation energymayalso be defined as the minimum energy required startinga chemical reaction The activation energy of a reaction isusually denoted by 119864

119886and given in units of kilojoules per

moleAn exothermic reaction is a chemical reaction that

releases energy in the form of light and heat It is theopposite of an endothermic reaction It gives out energy to itssurroundings The energy needed for the reaction to occur isless or greater than the total energy released for exothermicor endothermic reaction respectively Energy is obtainedfrom chemical bonds When bonds are broken energy isrequired When bonds are formed energy is released Eachtype of bond has specific bond energy It can be predictedwhether a chemical reaction will release or require heat byusing bond energiesWhen there ismore energy used to form

8 Journal of Thermodynamics

Table 2 Effects on skin friction heat transfer and mass transfercoefficients For119866

0= 10119866

119898= 1119873 = 05119875

119903= 071119872 = 05 119878

119888=

06 120582 = 5 119899 = 05 V0= 3 120574 = 05 and 119864 = 1

minus1198911015840

(0) minus1205791015840

(0) minus1206011015840

(0)

119873

00 minus087260 minus144125 56899620 minus305961 minus030027 57162450 minus493466 minus004756 572770

119872

00 minus185010 minus085309 57016420 minus075246 minus085443 57020350 046744 minus085628 570257

120574

minus200000 732171 365294 007835minus100000 566728 348636 026098000000 331180 188820 228076100000 minus154562 minus085343 570174200000 minus785616 minus367839 925776

119860

000000 minus127274 minus026974 397243200000 120419 056519 391549

the bonds than to break the bonds heat is given off This isknown as an exothermic reaction When a reaction requiresan input or output of energy it is known as an exothermic orendothermic reaction

Effects of activation energy (119864) and exothermic param-eter (120573) on the temperature the concentration and velocityprofiles are shown in Figure 1 to Figure 3 respectively 120573 = 1and 120573 = minus1 represent exothermic and endothermic chemicalreactions respectively Figure 1 shows that a small decreasingeffect of temperature profile is found for increasing valuesof 119864 for exothermic reaction (120573 = 1) but mark oppositeeffects are found for endothermic reaction in Figure 1 Thatis the temperature profile increases for increasing values of119864 for endothermic reaction 120573 = minus1 From (1) we observethat chemical reaction rate (119870) decreases with the increasingvalues of activation energy (119864

119886) We also observe from (20)

that increase in activation energy (119864) leads to decrease1205822 exp(minus119864120579) as well as to increase in the concentration

profiles shown in Figure 2 Small and reported effects arefound for 120573 = 1 and 120573 = minus1 respectively The parameter119864 does not enter directly into the momentum equation butits influence comes through the mass and energy equationsFigure 3 shows the variation of the velocity profiles fordifferent values of 119864 From this figure it has been observedthat the velocity profile mark increases with the increasingvalues of 119864 for endothermic chemical reaction But negligibleeffects are found for exothermic reaction

52 Effects of 120582 on the Concentration the Temperature and theVelocity Profiles for ExothermicEndothermic Reaction Con-sidering chemical reaction rate constant 1205822 is always positiveFigures 4ndash6 represent the effect of chemical rate constant120582 on

the velocity the temperature and the concentration profilesrespectively

We observe from Figures 4 and 5 that velocity andtemperature profiles increase with the increasing values of120582 for exothermic reaction but opposite effects are foundin these figures for endothermic reaction It is observedfrom (1) that increasing temperature frequently that causes amarked increase in the rate of reactions is shown in Figure 4As the temperature of the system increases the numberof molecules that carry enough energy to react when theycollide also increasesTherefore the rate of reaction increaseswith temperature As a rule the rate of a reaction doubles forevery 10∘C increase in the temperature of the system

Last part of (20) shows that 1205822 exp(minus119864120579) increases withthe increasing values of 120582We also observe from this equationthat increase in 1205822 exp(minus119864120579)means that increase in 120582 leadsto the decrease in the concentration profiles This is in greatagreement with Figure 6 for both cases 120573 = plusmn1 It isalso observed from this figure that for exothermic reactionthe mass boundary layer is close to the plate other thanendothermic reaction

53 Effects of Radiation Parameter 119873 on Temperature andVelocity Profiles The effects of radiation parameter119873 on thetemperature and the velocity profiles are shown in Figures 7and 8 From Figure 7 it can be seen that an increase in thevalues of 119873 leads to a decrease in the values of temperatureprofiles within the thermal boundary layers 120578 lt 022while outside 120578 gt 022 the temperature profile graduallyincreases with the increase of the radiation parameter 119873 Itis appear from Figure 8 that when 119873 = 0 that is withoutthe radiation the velocity profile shows its usual trend ofgradually decay As radiation becomes larger the profilesovershoot the uniform velocity close to the boundary

54 Effects of 120574 on the Velocity Temperature and Concen-tration Profiles The value of 120574 = Δ119879119879

infinis taken to be

both positive and negative since these values representrespectively cooling and heating of the plate The effects oftemperature relative parameter on velocity temperature andconcentration profiles are shown in Figures 9ndash11

The velocity profiles generated due to impulsive motionof the plate is plotted in Figure 9 for both cooling (120574 gt 0) andheating (120574 lt 0) of the plate keeping other parameters fixed(1198660= 10 119866

119898= 10 119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 V0= 30 119864 = 10 and 119899 = 1) In Figure 9 velocity

profiles are shown for different values of 120574 We observe thatvelocity increases with increasing values of 120574 for the coolingof the plate From this figure it is also observed that thenegative increase in the temperature relative parameter leadsto the decrease in the velocity field That is for heating ofthe plate (Figure 9) the effects of the 120574 on the velocity fieldhave also opposite effects as compared to the cooling ofthe plate Figure 10 shows the effects of temperature relativeparameter 120574 on the temperature profiles It has been observedfrom this figure that for heating plate the temperature profileshows its usual trend of gradually decay and the thermalboundary layer is close to the plate But for cooling plate

Journal of Thermodynamics 9

that is temperature relative parameter 120574 becomes largerthe profiles overshoot the uniform temperature close to thethermal boundary Opposite effects of temperature profile arefound for concentration profiles shown in Figure 11

55 Effects of SuctionInjection (V0) on the Velocity and the

Temperature Profiles The effects of suction and injection (V0)

for 1198660= 10 119866

119898= 10119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 120574 = 05 119864 = 10 and 119899 = 1 on the velocity profilesand temperature profiles are shown respectively in Figures12 and 13 For strong suction (V

0gt 0) the velocity and the

temperature decay rapidly away from the surface The factthat suction stabilizes the boundary layer is also apparentfrom these figures As for the injection (V

0lt 0) from Figures

12 and 13 it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayer between theinjection and the outer flow regions

56 Effects of 119866119898and119872 on the Velocity Profiles The effects

of Solutal Grashof number 119866119898

and magnetic interactionparameter119872 on velocity profiles are shown in Figures 14 and15 respectively Solutal Grashof number 119866

119898gt 0 corresponds

that the chemical species concentration in the free streamregion is less than the concentration at the boundary surfaceFigure 14 presents the effects of Solutal Grashof number119866119898on the velocity profiles It is observed that the velocity

profile increases with the increasing values of Solutal Grashofnumber 119866

119898

Imposition of a magnetic field to an electrically conduct-ing fluid creates a drag like force called Lorentz force Theforce has the tendency to slow down the flow around the plateat the expense of increasing its temperature This is depictedby decreases in velocity profiles as magnetic parameter 119872increases as shown in Figure 15

57 Effects of 119860 on the Velocity and the Temperature ProfilesHere 119860 = 0 and 119860 = 0 represent steady and unsteady flowsrespectively The effects of 119860 on velocity and temperatureprofiles are shown in Figures 16 and 17 respectively Forunsteady flow (119860 = 0) both figures show that the profilesrapidly decay and close to the plate But for steady flow (119860 =0) the profiles overshoot the uniform velocitytemperatureclose to the boundary

58 The Skin-Friction the Heat Transfer and the MassTransfer Coefficients The skin-friction the heat transferand the mass transfer coefficients are tabulated in Tables1 and 2 for different values of 119864 120573 120582 119873 119872 120574 and 119860We observe from Table 1 that the skin-friction coefficient(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) increasefor increasing values of dimensionless activation parameter119864 for exothermic chemical reaction but opposite actionsare found for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients decrease for increasing values of 119864 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) decreases for increasing values of 119864We also observe fromTable 1 that the skin-friction coefficient

0 1 2 3 40

05

1

Bestman (1990)Present

f

120578

Gr = Gm = 1 E = 5 20 = 10

= 071 and 120582 = 5

120573 = 0 120574 = 1 and M= N = A = 0

Sc = 1 Pr

Figure 18 Comparison of our calculated velocity profile and thevelocity profile of Bestman [7]

(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) decreasefor increasing values of dimensionless reaction rate constant120582 for exothermic chemical reaction but opposite effects arefound for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients increase for increasing values of 120582 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) remarkably increases for increasingvalues of 120582 From Table 2 it has been observed that theskin-friction coefficient remarkably decreases and Nusseltnumber remarkable increases for increasing values of radi-ation parameter119873 The skin-friction coefficient increases forincreasing values of magnetic parameter 119872 From Table 2we also found that skin friction and heat transfer coefficientsdecrease butmass transfer coefficient increases for increasingvalues of temperature relative parameter 120574

59 Comparison between Our Numerical Result and theAnalytical Result of Bestman [7] Bestman studied steadynatural convective boundary layer flow with large suctionHe solved his problem analytically by employing the per-turbation technique proposed by Singh and Dikshit [8] Inour present work take 119860 = 0 in (16) for considering steadyflow The values of the suction parameter V2

0= 10 is taken

to see the effects of large suction 119866119903= 119866119898= 10 120574 = 1

119872 = 119873 = 0 120582 = 5 119864 = 5 and 119899 = 1 are also chosen witha view to compare our numerical results with the analyticalresults of Bestman [7] The comparison of velocity profilesas seen in Figure 18 highlights the validity of the numericalcomputations adapted in the present investigation

6 Conclusions

In this paper we investigate the effects of chemical reactionrate and Arrhenius activation energy on an unsteady MHD

10 Journal of Thermodynamics

natural convection heat and mass transfer boundary layerflow past a flat porous plate in presence of thermal radiation

The Nachtsheim and Swigert [18] iteration techniquebased on sixth-order Runge-Kutta and Shooting method hasbeen employed to complete the integration of the resultingsolutions

The following conclusions can be drawn as a result of thecomputations

(1) Velocity increases with increasing values of 120574 for thecooling of the plate and the negative increase in thetemperature relative parameter 120574 leads to the decreasein the velocity fieldThat is for heating of the plate theeffects of the temperature relative parameter 120574 on thevelocity field have also opposite effects as comparedto the cooling of the plate

(2) Solutal Grashof number 119866119898gt 0 corresponds that

the chemical species concentration in the free streamregion is less than the concentration at the bound-ary surface It is observed that the velocity profileincreaseswith the increasing values of SolutalGrashofnumber 119866

119898

(3) Increase in 120582 leads to increase in the velocity andtemperature profiles for exothermic reaction butopposite effects are found for endothermic reactionFor 120573 = plusmn1 increase in 120582 leads to the decrease in theconcentration profiles but for exothermic reactionthe mass boundary layer is close to the plate otherthan endothermic reaction

(4) The velocity profile mark increases with the increas-ing values of 119864 for endothermic chemical reactionBut negligible effect is found for exothermic reactionTemperature decreases for increasing values of 119864 forexothermic reaction but opposite effects are found forendothermic reaction Activation energy (119864) leads toincrease in the concentration profiles but small andreported effects are found for 120573 = 1 and 120573 = minus1respectively

(5) For strong suction (V0gt 0) the velocity the tem-

perature and the concentration profiles decay rapidlyaway from the surface As for the injection (V

0lt 0)

it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayerbetween the injection and the outer flow regions

(6) Imposition of a magnetic field to an electrically con-ducting fluid creates a drag like force called Lorentzforce The force has the tendency to slow down theflow around the plate at the expense of increasing itstemperature This is depicted by decreases in velocityprofiles as magnetic parameter119872 increases

(7) An increase in the values of 119873 leads to a decrease inthe values of temperature profiles within the thermalboundary layers 120578 lt 022 while outside 120578 gt 022the temperature profile gradually increases with theincrease of the radiation parameter119873

(8) For unsteady flow (119860 = 0) the velocity and the tem-perature profiles rapidly decay and close to the plate

But for steady flow (119860 = 0) the profiles overshoot theuniform velocitytemperature close to the boundary

Nomenclature

(119906 V) Components of velocity field119888119901 Specific heat at constant pressure119873119906 Nusselt number

119875119903 Prandtl number1198760 Heat generationabsorption coefficient

119878ℎ Sherwood number119879 Temperature of the flow field119879infin Temperature of the fluid at infinity

119879119908 Temperature at the plate

119891 Dimensionless similarity functions119864119886 The activation energy

119896 The Boltzmann constant119878119888 Schmidt number119864 The nondimensional activation energy119866119903 Grashof number

119866119898 Modified (Solutal) Grashof number

Greek Symbols

120579 Dimensionless temperature120601 Dimensionless concentration120578 Dimensionless similarity variable120575 Scale factorV Kinematic viscosity120588 Density of the fluid120581 Thermal conductivity120591 Shear stress120583 Fluid viscosity1205822 Dimensionless chemical reaction rate

constant120573 and 120573lowast The coefficients of volume expansions for

temperature and concentrationrespectively

120574 The dimensionless heatgenerationabsorption coefficient

References

[1] M Tencer J S Moss and T Zapach ldquoArrhenius averagetemperature the effective temperature for non-fatigue wearoutand long term reliability in variable thermal conditions andclimatesrdquo IEEE Transactions on Components and PackagingTechnologies vol 27 no 3 pp 602ndash607 2004

[2] C Truesdell ldquoSulle basi della thermomeccanicardquo RendicontiLincei vol 22 no 8 pp 33ndash38 1957

[3] N Mills ldquoIncompressible mixtures of newtonian fluidsrdquo Inter-national Journal of Engineering Science vol 4 no 2 pp 97ndash1121966

[4] C E Beevers and R E Craine ldquoOn the determination ofresponse functions for a binary mixture of incompressiblenewtonian fluidsrdquo International Journal of Engineering Sciencevol 20 no 6 pp 737ndash745 1982

Journal of Thermodynamics 11

[5] A Al-Sharif K Chamniprasart K R Rajagopal andA Z SzerildquoLubrication with binary mixtures liquid-liquid emulsionrdquoJournal of Tribology vol 115 no 1 pp 46ndash55 1993

[6] S H Wang A Al-Sharif K R Rajagopal and A Z SzerildquoLubrication with binarymixtures liquid-liquid emulsion in anEHL conjunctionrdquo Journal of Tribology vol 115 no 3 pp 515ndash522 1993

[7] A R Bestman ldquoNatural convection boundary layer withsuction and mass transfer in a porous mediumrdquo InternationalJournal of Energy Research vol 14 no 4 pp 389ndash396 1990

[8] A K Singh and C K Dikshit ldquoHydromagnetic flow pasta continuously moving semi-infinite plate for large suctionrdquoAstrophysics and Space Science vol 148 no 2 pp 249ndash256 1988

[9] A R Bestman ldquoRadiative heat transfer to flow of a combustiblemixture in a vertical piperdquo International Journal of EnergyResearch vol 15 no 3 pp 179ndash184 1991

[10] M A Alabraba A R Bestman and A Ogulu ldquoLaminar con-vection in binary mixture of hydromagnetic flow with radiativeheat transfer Irdquo Astrophysics and Space Science vol 195 no 2pp 431ndash439 1992

[11] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[12] O DMakinde P O Olanrewaju andWM Charles ldquoUnsteadyconvection with chemical reaction and radiative heat transferpast a flat porous plate moving through a binary mixturerdquoAfricka Matematika vol 22 no 1 pp 65ndash78 2011

[13] O D Makinde and P O Olanrewaju ldquoUnsteady mixed convec-tion with Soret and Dufour effects past a porous plate movingthrough a binarymixture of chemically reacting fluidrdquoChemicalEngineering Communications vol 198 no 7 pp 920ndash938 2011

[14] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer flow with mass transfer and a binary chemical reactionrdquoBritish Journal of Applied Science and Technology (Sciencedo-main International) vol 3 no 1 pp 131ndash149 2013

[15] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer heat and mass transfer flow with exothermic chemicalreactionsrdquo Journal of Pure and Applied Mathematics (Advancesand Applications) vol 9 no 1 pp 17ndash41 2013

[16] Kh Abdul Maleque ldquoEffects of binary chemical reactionand activation energy on MHD boundary layer heat andmass transfer flow with viscous dissipation and heat genera-tionabsorptionrdquo ISRN Thermodynamics vol 2013 Article ID284637 9 pages 2013

[17] Kh Abdul Maleque ldquoEffects of combined temperature- anddepth-dependent viscosity and hall current on an unsteadyMHD laminar convective flow due to a rotating diskrdquo ChemicalEngineering Communications vol 197 no 4 pp 506ndash521 2010

[18] P R Nachtsheim and P Swigert ldquoSatisfaction of asymptoticboundary conditions in numerical solution of system of non-linear of boundary layer typerdquo NASA TN-D3004 1965

Submit your manuscripts athttpwwwhindawicom

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ThermodynamicsJournal of

4 Journal of Thermodynamics

For G0 = 10 Gm = 1M = 05

120574 = 1 n = 1 = 3 and 120582 = 51

05120579

120578

120573 = 1

120573 = minus1

0 1 2

E = 1

E = 0

0Sc = 06

Pr = 071 N = 1

Figure 1 Effects of 120573 and 119864 on temperature profiles

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and 120582 = 5

05 15120578

120573 = 1

120573 = minus1120601

0

1

05

01

E = 0

E = 1

0Sc = 06

Pr

Figure 2 Effects of 120573 and 119864 on concentration profiles

where 119883 = 1198911015840

(0) 119884 = 1205791015840(0) 119885 = 1206011015840(0) and 119883119884 119885subscripts indicate partial differentiation for example 1198911015840

119884=

1205971198911015840

(120578max)1205971205791015840

(0) The subscript 119888 indicates the value of thefunction at 120578max to be determined from the trial integration

Solutions of these equations in a least square senserequire determining the minimum value of 119864 = 120575

2

1+

1205752

2+ 1205752

3+ 1205752

4+ 1205752

5+ 1205752

6with respect to 119883 119884 and 119885 To

solve Δ119883 Δ119884 and Δ119885 we require to differentiate 119864 withrespect to 119883 119884 and 119885 respectively Thus adopting thisnumerical technique a computer program was set up forthe solutions of the basic nonlinear differential equations ofour problem where the integration technique was adoptedas the six-ordered Runge-Kutta method of integration Theresults of this integration are then displayed graphically in

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and 120582 = 5

120578

120573 = 1

120573 = minus1

00

f05

1

1 2

E = 0

E = 1

0Sc = 06

Pr

Figure 3 Effects of 120573 and 119864 on velocity profiles

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and E = 1

120573 = 1

120573 = minus1f

05

1

120578

00

1 2

120582 = 1

120582 = 5

0Sc = 06

Pr

Figure 4 Effects of 120573 and 120582 on velocity profiles

the form of velocity temperature and concentration profilesin Figures 1ndash15 In the process of integration the local skin-friction coefficient the local rates of heat and mass transferto the surface which are of chief physical interest are alsocalculated out The equation defining the wall skin friction is

120591 = minus120583 (120597119906

120597119910)

119910=0

= minus1205831198800

1205751198911015840

(0) (28)

Hence the skin-friction coefficient is given by

1

2119862119891=120591

1205881198802

0

997904rArr1

2119877119890119862119891= minus1198911015840

(0) (29)

here the Reynolds number 119877119890= 1205881198800120575120583

Journal of Thermodynamics 5

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and E = 1

120582 = 1

120582 = 5

120573 = 1

120573 = minus1

120578

0 1 2

1

05

0

120579

0Sc = 06

Pr

Figure 5 Effects of 120573 and 120582 on temperature profiles

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and E = 1

120582 = 1

120582 = 5

120573 = 1

120578

00

1

1

05

2

120573 = minus1

120601

0Sc = 06

Pr

Figure 6 Effects of 120573 and 120582 on concentration profiles

0 1 2 3 40

05

1

120578

120579

N = 0 2 5

For G0 = 10 Gm = 1M = 05 = 071

120582 = 5

120574 = 1

n = 1 0 = 3 and E = 1Sc = 06

Pr

Figure 7 Effects of119873 on temperature profiles

0 1 2 3 40

05

1

15

2

f

N = 0 2 5

For G0 = 10 Gm = 1M = 05 = 071

120582 = 5

120574 = 1

n = 1 0 = 3 and E = 1

120578

Sc = 06

Pr

Figure 8 Effects of119873 on velocity profiles

0 1 2

0

1

2

f

For G0 = 10 Gm = 1N = 05 = 071M= 05

120582 = 5 n = 05 = 3 and E = 10

Gr = G0120574120574

2

1

0

minus1

minus1

minus2

120578

Sc = 06

Pr

Figure 9 Effects of 120574 on velocity profiles

0 1 20

05

1For G0 = 10 Gm = 1N = 05 = 071M= 05

120582 = 5 n = 05 = 3 and E = 10

Gr = G0120574

120574

210minus1

minus2

120578

120579

Sc = 06

Pr

Figure 10 Effects of 120574 on temperature profiles

6 Journal of Thermodynamics

0 1 20

05

1

120578

120601

For G0 = 10 Gm = 1N = 05 = 071M= 05

120582 = 5 n = 05 = 3 and E = 10

Gr = G0120574

120574

minus2minus101

2

Sc = 06

Pr

Figure 11 Effects of 120574 on concentration profiles

0 1 2 3 40

1

2

3

f

120578

For G0 = 10 Gm = 1N = 05 = 071M = 05

120582 = 5 n = 05 120574 = 05 and E = 1

0 = minus3

0 = 0

0 = 3

Sc = 06

Pr

Figure 12 Effects of V0on velocity profiles

The heat flux (119902119908) and the mass flux (119872

119908) at the wall are

given by

119902119908= minus120581(

120597119879

120597119910)

119910=0

= minus120581Δ119879

1205751205791015840

(0)

119872119908= minus119863119872(120597119862

120597119910)

119910=0

= minus119863119872

Δ119862

1205751206011015840

(0)

(30)

Hence the Nusselt number (119873119906) and the Sherwood number

(119878ℎ) are obtained as

119873119906=119902119908120575

120581 Δ119879= minus 120579

1015840

(0) 119878ℎ=119872119908120575

119863119872Δ119862= minus1206011015840

(0) (31)

These previous coefficients are then obtained from the proce-dure of the numerical computations and are sorted in Table 1

5 Results and Discussions

The parameters entering into the fluid flow are Grashofnumber 119866

119903= 1198660120574 Solutal (modified) Grashof number 119866

119898

suction parameter V0 Prandtl number 119875

119903 the nondimen-

sional chemical reaction rate constant 1205822 Schmidt number

0 1 2 3 40

05

1

15

0

3

For G0 = 10 Gm = 1N = 05 = 071M = 05

120582 = 5 n = 05 120574 = 05 and E = 1

0

minus3

120578

120579

Sc = 06

Pr

Figure 13 Effects of V0on temperature profiles

0 1 20

05

1

10

5

0

f

Gm

120582 = 5 n = 05 120574 = 05 and E = 1

120578

For G0 = 10 0 = 3N = 05 = 071M = 05

Sc = 06

Pr

Figure 14 Effects of 119866119898on velocity profiles

0 1 20

05

1

M

1

0

5

f

120578

For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

120582 = 5 n = 1 0 = 3 and E = 1Sc = 06

Pr

Figure 15 Effects of119872 on velocity profiles

Journal of Thermodynamics 7

0 1 2 30

05

1

f

120578

For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

A = 0 (steady)A = 2 (unsteady)

120574 = 5 n = 1 0 = 3M = 05 and E = 1Sc = 06

Pr

Figure 16 Effects of 119860 on velocity profiles

0 1 2 30

05

1 For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

A = 0 (steady)A = 2 (unsteady)

120579

120578

120574 = 5 n = 1 0 = 3M = 05 and E = 1Sc = 06

Pr

Figure 17 Effects of 119860 on temperature profiles

119878119888 the magnetic parameter 119872 the nondimensional acti-

vation energy 119864 the radiation parameter 119873 the exother-micendothermic parameter 120573 and the temperature relativeparameter 120574

It is therefore pertinent to inquire the effects of variationof each of them when the others are kept constant Thenumerical results are thus presented in the form of velocityprofiles temperature profiles and concentration profiles inFigures 1ndash14 for the different values of 119866

119903 119866119898 120582119872 119864 V

0 120573

and 120574The value of 120574 is taken to be both positive and negativesince these values represent respectively cooling and heatingof the plate

The values of 119866119903is taken to be large (119866

119903= 10

1198660= 10 and 120574 = 1) since the value corresponds to

Table 1 Effects on skin friction heat transfer and mass transfercoefficients For 119866

0= 10 119866

119898= 1119873 = 05 119875

119903= 071119872 = 05 119878

119888=

06 120582 = 5 119899 = 05 V0= 3 120574 = 05 and 119864 = 1

minus1198911015840

(0) minus1205791015840

(0) minus1206011015840

(0)

119864 for 120573 = 100 minus165163 minus091802 57863405 minus160009 minus088980 57552310 minus154562 minus085343 570174

119864 for 120573 = minus100 130241 449966 55515905 098702 419573 52109310 063543 387806 481246

120582 for 120573 = 110 minus043158 173298 24262720 minus063434 134699 29144650 minus154562 minus085343 570174

120582 for 120573 = minus110 minus027903 198971 24208820 minus007554 232400 28442450 063543 387806 481246

a cooling problem that is generally encountered in nuclearengineering in connection with the cooling of reactors In air(119875119903= 071) the diluting chemical species of most common

interest have Schmidt number in the range from 06 to 075Therefore the Schmidt number 119878

119888= 06 is considered In

particular 06 corresponds to water vapor that represents adiffusing chemical species of most common interest in airThe values of the suction parameter V

0are taken to be large

Apart from the figures and tables the representative velocitytemperatureand concentration profiles and the values ofthe physically important parameters that is the local shearstress the local rates of heat and mass transfer are illustratedfor uniform wall temperature and species concentration inFigures 1ndash17 and in Tables 1-2

51 Effects of Activation Energy (119864) on the Concentrationthe Temperature and the Velocity Profiles for ExothermicEndothermic Chemical Reaction In chemistry activationenergy is defined as the energy that must be overcome inorder for a chemical reaction to occur Activation energymayalso be defined as the minimum energy required startinga chemical reaction The activation energy of a reaction isusually denoted by 119864

119886and given in units of kilojoules per

moleAn exothermic reaction is a chemical reaction that

releases energy in the form of light and heat It is theopposite of an endothermic reaction It gives out energy to itssurroundings The energy needed for the reaction to occur isless or greater than the total energy released for exothermicor endothermic reaction respectively Energy is obtainedfrom chemical bonds When bonds are broken energy isrequired When bonds are formed energy is released Eachtype of bond has specific bond energy It can be predictedwhether a chemical reaction will release or require heat byusing bond energiesWhen there ismore energy used to form

8 Journal of Thermodynamics

Table 2 Effects on skin friction heat transfer and mass transfercoefficients For119866

0= 10119866

119898= 1119873 = 05119875

119903= 071119872 = 05 119878

119888=

06 120582 = 5 119899 = 05 V0= 3 120574 = 05 and 119864 = 1

minus1198911015840

(0) minus1205791015840

(0) minus1206011015840

(0)

119873

00 minus087260 minus144125 56899620 minus305961 minus030027 57162450 minus493466 minus004756 572770

119872

00 minus185010 minus085309 57016420 minus075246 minus085443 57020350 046744 minus085628 570257

120574

minus200000 732171 365294 007835minus100000 566728 348636 026098000000 331180 188820 228076100000 minus154562 minus085343 570174200000 minus785616 minus367839 925776

119860

000000 minus127274 minus026974 397243200000 120419 056519 391549

the bonds than to break the bonds heat is given off This isknown as an exothermic reaction When a reaction requiresan input or output of energy it is known as an exothermic orendothermic reaction

Effects of activation energy (119864) and exothermic param-eter (120573) on the temperature the concentration and velocityprofiles are shown in Figure 1 to Figure 3 respectively 120573 = 1and 120573 = minus1 represent exothermic and endothermic chemicalreactions respectively Figure 1 shows that a small decreasingeffect of temperature profile is found for increasing valuesof 119864 for exothermic reaction (120573 = 1) but mark oppositeeffects are found for endothermic reaction in Figure 1 Thatis the temperature profile increases for increasing values of119864 for endothermic reaction 120573 = minus1 From (1) we observethat chemical reaction rate (119870) decreases with the increasingvalues of activation energy (119864

119886) We also observe from (20)

that increase in activation energy (119864) leads to decrease1205822 exp(minus119864120579) as well as to increase in the concentration

profiles shown in Figure 2 Small and reported effects arefound for 120573 = 1 and 120573 = minus1 respectively The parameter119864 does not enter directly into the momentum equation butits influence comes through the mass and energy equationsFigure 3 shows the variation of the velocity profiles fordifferent values of 119864 From this figure it has been observedthat the velocity profile mark increases with the increasingvalues of 119864 for endothermic chemical reaction But negligibleeffects are found for exothermic reaction

52 Effects of 120582 on the Concentration the Temperature and theVelocity Profiles for ExothermicEndothermic Reaction Con-sidering chemical reaction rate constant 1205822 is always positiveFigures 4ndash6 represent the effect of chemical rate constant120582 on

the velocity the temperature and the concentration profilesrespectively

We observe from Figures 4 and 5 that velocity andtemperature profiles increase with the increasing values of120582 for exothermic reaction but opposite effects are foundin these figures for endothermic reaction It is observedfrom (1) that increasing temperature frequently that causes amarked increase in the rate of reactions is shown in Figure 4As the temperature of the system increases the numberof molecules that carry enough energy to react when theycollide also increasesTherefore the rate of reaction increaseswith temperature As a rule the rate of a reaction doubles forevery 10∘C increase in the temperature of the system

Last part of (20) shows that 1205822 exp(minus119864120579) increases withthe increasing values of 120582We also observe from this equationthat increase in 1205822 exp(minus119864120579)means that increase in 120582 leadsto the decrease in the concentration profiles This is in greatagreement with Figure 6 for both cases 120573 = plusmn1 It isalso observed from this figure that for exothermic reactionthe mass boundary layer is close to the plate other thanendothermic reaction

53 Effects of Radiation Parameter 119873 on Temperature andVelocity Profiles The effects of radiation parameter119873 on thetemperature and the velocity profiles are shown in Figures 7and 8 From Figure 7 it can be seen that an increase in thevalues of 119873 leads to a decrease in the values of temperatureprofiles within the thermal boundary layers 120578 lt 022while outside 120578 gt 022 the temperature profile graduallyincreases with the increase of the radiation parameter 119873 Itis appear from Figure 8 that when 119873 = 0 that is withoutthe radiation the velocity profile shows its usual trend ofgradually decay As radiation becomes larger the profilesovershoot the uniform velocity close to the boundary

54 Effects of 120574 on the Velocity Temperature and Concen-tration Profiles The value of 120574 = Δ119879119879

infinis taken to be

both positive and negative since these values representrespectively cooling and heating of the plate The effects oftemperature relative parameter on velocity temperature andconcentration profiles are shown in Figures 9ndash11

The velocity profiles generated due to impulsive motionof the plate is plotted in Figure 9 for both cooling (120574 gt 0) andheating (120574 lt 0) of the plate keeping other parameters fixed(1198660= 10 119866

119898= 10 119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 V0= 30 119864 = 10 and 119899 = 1) In Figure 9 velocity

profiles are shown for different values of 120574 We observe thatvelocity increases with increasing values of 120574 for the coolingof the plate From this figure it is also observed that thenegative increase in the temperature relative parameter leadsto the decrease in the velocity field That is for heating ofthe plate (Figure 9) the effects of the 120574 on the velocity fieldhave also opposite effects as compared to the cooling ofthe plate Figure 10 shows the effects of temperature relativeparameter 120574 on the temperature profiles It has been observedfrom this figure that for heating plate the temperature profileshows its usual trend of gradually decay and the thermalboundary layer is close to the plate But for cooling plate

Journal of Thermodynamics 9

that is temperature relative parameter 120574 becomes largerthe profiles overshoot the uniform temperature close to thethermal boundary Opposite effects of temperature profile arefound for concentration profiles shown in Figure 11

55 Effects of SuctionInjection (V0) on the Velocity and the

Temperature Profiles The effects of suction and injection (V0)

for 1198660= 10 119866

119898= 10119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 120574 = 05 119864 = 10 and 119899 = 1 on the velocity profilesand temperature profiles are shown respectively in Figures12 and 13 For strong suction (V

0gt 0) the velocity and the

temperature decay rapidly away from the surface The factthat suction stabilizes the boundary layer is also apparentfrom these figures As for the injection (V

0lt 0) from Figures

12 and 13 it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayer between theinjection and the outer flow regions

56 Effects of 119866119898and119872 on the Velocity Profiles The effects

of Solutal Grashof number 119866119898

and magnetic interactionparameter119872 on velocity profiles are shown in Figures 14 and15 respectively Solutal Grashof number 119866

119898gt 0 corresponds

that the chemical species concentration in the free streamregion is less than the concentration at the boundary surfaceFigure 14 presents the effects of Solutal Grashof number119866119898on the velocity profiles It is observed that the velocity

profile increases with the increasing values of Solutal Grashofnumber 119866

119898

Imposition of a magnetic field to an electrically conduct-ing fluid creates a drag like force called Lorentz force Theforce has the tendency to slow down the flow around the plateat the expense of increasing its temperature This is depictedby decreases in velocity profiles as magnetic parameter 119872increases as shown in Figure 15

57 Effects of 119860 on the Velocity and the Temperature ProfilesHere 119860 = 0 and 119860 = 0 represent steady and unsteady flowsrespectively The effects of 119860 on velocity and temperatureprofiles are shown in Figures 16 and 17 respectively Forunsteady flow (119860 = 0) both figures show that the profilesrapidly decay and close to the plate But for steady flow (119860 =0) the profiles overshoot the uniform velocitytemperatureclose to the boundary

58 The Skin-Friction the Heat Transfer and the MassTransfer Coefficients The skin-friction the heat transferand the mass transfer coefficients are tabulated in Tables1 and 2 for different values of 119864 120573 120582 119873 119872 120574 and 119860We observe from Table 1 that the skin-friction coefficient(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) increasefor increasing values of dimensionless activation parameter119864 for exothermic chemical reaction but opposite actionsare found for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients decrease for increasing values of 119864 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) decreases for increasing values of 119864We also observe fromTable 1 that the skin-friction coefficient

0 1 2 3 40

05

1

Bestman (1990)Present

f

120578

Gr = Gm = 1 E = 5 20 = 10

= 071 and 120582 = 5

120573 = 0 120574 = 1 and M= N = A = 0

Sc = 1 Pr

Figure 18 Comparison of our calculated velocity profile and thevelocity profile of Bestman [7]

(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) decreasefor increasing values of dimensionless reaction rate constant120582 for exothermic chemical reaction but opposite effects arefound for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients increase for increasing values of 120582 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) remarkably increases for increasingvalues of 120582 From Table 2 it has been observed that theskin-friction coefficient remarkably decreases and Nusseltnumber remarkable increases for increasing values of radi-ation parameter119873 The skin-friction coefficient increases forincreasing values of magnetic parameter 119872 From Table 2we also found that skin friction and heat transfer coefficientsdecrease butmass transfer coefficient increases for increasingvalues of temperature relative parameter 120574

59 Comparison between Our Numerical Result and theAnalytical Result of Bestman [7] Bestman studied steadynatural convective boundary layer flow with large suctionHe solved his problem analytically by employing the per-turbation technique proposed by Singh and Dikshit [8] Inour present work take 119860 = 0 in (16) for considering steadyflow The values of the suction parameter V2

0= 10 is taken

to see the effects of large suction 119866119903= 119866119898= 10 120574 = 1

119872 = 119873 = 0 120582 = 5 119864 = 5 and 119899 = 1 are also chosen witha view to compare our numerical results with the analyticalresults of Bestman [7] The comparison of velocity profilesas seen in Figure 18 highlights the validity of the numericalcomputations adapted in the present investigation

6 Conclusions

In this paper we investigate the effects of chemical reactionrate and Arrhenius activation energy on an unsteady MHD

10 Journal of Thermodynamics

natural convection heat and mass transfer boundary layerflow past a flat porous plate in presence of thermal radiation

The Nachtsheim and Swigert [18] iteration techniquebased on sixth-order Runge-Kutta and Shooting method hasbeen employed to complete the integration of the resultingsolutions

The following conclusions can be drawn as a result of thecomputations

(1) Velocity increases with increasing values of 120574 for thecooling of the plate and the negative increase in thetemperature relative parameter 120574 leads to the decreasein the velocity fieldThat is for heating of the plate theeffects of the temperature relative parameter 120574 on thevelocity field have also opposite effects as comparedto the cooling of the plate

(2) Solutal Grashof number 119866119898gt 0 corresponds that

the chemical species concentration in the free streamregion is less than the concentration at the bound-ary surface It is observed that the velocity profileincreaseswith the increasing values of SolutalGrashofnumber 119866

119898

(3) Increase in 120582 leads to increase in the velocity andtemperature profiles for exothermic reaction butopposite effects are found for endothermic reactionFor 120573 = plusmn1 increase in 120582 leads to the decrease in theconcentration profiles but for exothermic reactionthe mass boundary layer is close to the plate otherthan endothermic reaction

(4) The velocity profile mark increases with the increas-ing values of 119864 for endothermic chemical reactionBut negligible effect is found for exothermic reactionTemperature decreases for increasing values of 119864 forexothermic reaction but opposite effects are found forendothermic reaction Activation energy (119864) leads toincrease in the concentration profiles but small andreported effects are found for 120573 = 1 and 120573 = minus1respectively

(5) For strong suction (V0gt 0) the velocity the tem-

perature and the concentration profiles decay rapidlyaway from the surface As for the injection (V

0lt 0)

it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayerbetween the injection and the outer flow regions

(6) Imposition of a magnetic field to an electrically con-ducting fluid creates a drag like force called Lorentzforce The force has the tendency to slow down theflow around the plate at the expense of increasing itstemperature This is depicted by decreases in velocityprofiles as magnetic parameter119872 increases

(7) An increase in the values of 119873 leads to a decrease inthe values of temperature profiles within the thermalboundary layers 120578 lt 022 while outside 120578 gt 022the temperature profile gradually increases with theincrease of the radiation parameter119873

(8) For unsteady flow (119860 = 0) the velocity and the tem-perature profiles rapidly decay and close to the plate

But for steady flow (119860 = 0) the profiles overshoot theuniform velocitytemperature close to the boundary

Nomenclature

(119906 V) Components of velocity field119888119901 Specific heat at constant pressure119873119906 Nusselt number

119875119903 Prandtl number1198760 Heat generationabsorption coefficient

119878ℎ Sherwood number119879 Temperature of the flow field119879infin Temperature of the fluid at infinity

119879119908 Temperature at the plate

119891 Dimensionless similarity functions119864119886 The activation energy

119896 The Boltzmann constant119878119888 Schmidt number119864 The nondimensional activation energy119866119903 Grashof number

119866119898 Modified (Solutal) Grashof number

Greek Symbols

120579 Dimensionless temperature120601 Dimensionless concentration120578 Dimensionless similarity variable120575 Scale factorV Kinematic viscosity120588 Density of the fluid120581 Thermal conductivity120591 Shear stress120583 Fluid viscosity1205822 Dimensionless chemical reaction rate

constant120573 and 120573lowast The coefficients of volume expansions for

temperature and concentrationrespectively

120574 The dimensionless heatgenerationabsorption coefficient

References

[1] M Tencer J S Moss and T Zapach ldquoArrhenius averagetemperature the effective temperature for non-fatigue wearoutand long term reliability in variable thermal conditions andclimatesrdquo IEEE Transactions on Components and PackagingTechnologies vol 27 no 3 pp 602ndash607 2004

[2] C Truesdell ldquoSulle basi della thermomeccanicardquo RendicontiLincei vol 22 no 8 pp 33ndash38 1957

[3] N Mills ldquoIncompressible mixtures of newtonian fluidsrdquo Inter-national Journal of Engineering Science vol 4 no 2 pp 97ndash1121966

[4] C E Beevers and R E Craine ldquoOn the determination ofresponse functions for a binary mixture of incompressiblenewtonian fluidsrdquo International Journal of Engineering Sciencevol 20 no 6 pp 737ndash745 1982

Journal of Thermodynamics 11

[5] A Al-Sharif K Chamniprasart K R Rajagopal andA Z SzerildquoLubrication with binary mixtures liquid-liquid emulsionrdquoJournal of Tribology vol 115 no 1 pp 46ndash55 1993

[6] S H Wang A Al-Sharif K R Rajagopal and A Z SzerildquoLubrication with binarymixtures liquid-liquid emulsion in anEHL conjunctionrdquo Journal of Tribology vol 115 no 3 pp 515ndash522 1993

[7] A R Bestman ldquoNatural convection boundary layer withsuction and mass transfer in a porous mediumrdquo InternationalJournal of Energy Research vol 14 no 4 pp 389ndash396 1990

[8] A K Singh and C K Dikshit ldquoHydromagnetic flow pasta continuously moving semi-infinite plate for large suctionrdquoAstrophysics and Space Science vol 148 no 2 pp 249ndash256 1988

[9] A R Bestman ldquoRadiative heat transfer to flow of a combustiblemixture in a vertical piperdquo International Journal of EnergyResearch vol 15 no 3 pp 179ndash184 1991

[10] M A Alabraba A R Bestman and A Ogulu ldquoLaminar con-vection in binary mixture of hydromagnetic flow with radiativeheat transfer Irdquo Astrophysics and Space Science vol 195 no 2pp 431ndash439 1992

[11] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[12] O DMakinde P O Olanrewaju andWM Charles ldquoUnsteadyconvection with chemical reaction and radiative heat transferpast a flat porous plate moving through a binary mixturerdquoAfricka Matematika vol 22 no 1 pp 65ndash78 2011

[13] O D Makinde and P O Olanrewaju ldquoUnsteady mixed convec-tion with Soret and Dufour effects past a porous plate movingthrough a binarymixture of chemically reacting fluidrdquoChemicalEngineering Communications vol 198 no 7 pp 920ndash938 2011

[14] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer flow with mass transfer and a binary chemical reactionrdquoBritish Journal of Applied Science and Technology (Sciencedo-main International) vol 3 no 1 pp 131ndash149 2013

[15] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer heat and mass transfer flow with exothermic chemicalreactionsrdquo Journal of Pure and Applied Mathematics (Advancesand Applications) vol 9 no 1 pp 17ndash41 2013

[16] Kh Abdul Maleque ldquoEffects of binary chemical reactionand activation energy on MHD boundary layer heat andmass transfer flow with viscous dissipation and heat genera-tionabsorptionrdquo ISRN Thermodynamics vol 2013 Article ID284637 9 pages 2013

[17] Kh Abdul Maleque ldquoEffects of combined temperature- anddepth-dependent viscosity and hall current on an unsteadyMHD laminar convective flow due to a rotating diskrdquo ChemicalEngineering Communications vol 197 no 4 pp 506ndash521 2010

[18] P R Nachtsheim and P Swigert ldquoSatisfaction of asymptoticboundary conditions in numerical solution of system of non-linear of boundary layer typerdquo NASA TN-D3004 1965

Submit your manuscripts athttpwwwhindawicom

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ThermodynamicsJournal of

Journal of Thermodynamics 5

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and E = 1

120582 = 1

120582 = 5

120573 = 1

120573 = minus1

120578

0 1 2

1

05

0

120579

0Sc = 06

Pr

Figure 5 Effects of 120573 and 120582 on temperature profiles

For G0 = 10 Gm = 1M = 05 = 071 N = 1

120574 = 1 n = 1 = 3 and E = 1

120582 = 1

120582 = 5

120573 = 1

120578

00

1

1

05

2

120573 = minus1

120601

0Sc = 06

Pr

Figure 6 Effects of 120573 and 120582 on concentration profiles

0 1 2 3 40

05

1

120578

120579

N = 0 2 5

For G0 = 10 Gm = 1M = 05 = 071

120582 = 5

120574 = 1

n = 1 0 = 3 and E = 1Sc = 06

Pr

Figure 7 Effects of119873 on temperature profiles

0 1 2 3 40

05

1

15

2

f

N = 0 2 5

For G0 = 10 Gm = 1M = 05 = 071

120582 = 5

120574 = 1

n = 1 0 = 3 and E = 1

120578

Sc = 06

Pr

Figure 8 Effects of119873 on velocity profiles

0 1 2

0

1

2

f

For G0 = 10 Gm = 1N = 05 = 071M= 05

120582 = 5 n = 05 = 3 and E = 10

Gr = G0120574120574

2

1

0

minus1

minus1

minus2

120578

Sc = 06

Pr

Figure 9 Effects of 120574 on velocity profiles

0 1 20

05

1For G0 = 10 Gm = 1N = 05 = 071M= 05

120582 = 5 n = 05 = 3 and E = 10

Gr = G0120574

120574

210minus1

minus2

120578

120579

Sc = 06

Pr

Figure 10 Effects of 120574 on temperature profiles

6 Journal of Thermodynamics

0 1 20

05

1

120578

120601

For G0 = 10 Gm = 1N = 05 = 071M= 05

120582 = 5 n = 05 = 3 and E = 10

Gr = G0120574

120574

minus2minus101

2

Sc = 06

Pr

Figure 11 Effects of 120574 on concentration profiles

0 1 2 3 40

1

2

3

f

120578

For G0 = 10 Gm = 1N = 05 = 071M = 05

120582 = 5 n = 05 120574 = 05 and E = 1

0 = minus3

0 = 0

0 = 3

Sc = 06

Pr

Figure 12 Effects of V0on velocity profiles

The heat flux (119902119908) and the mass flux (119872

119908) at the wall are

given by

119902119908= minus120581(

120597119879

120597119910)

119910=0

= minus120581Δ119879

1205751205791015840

(0)

119872119908= minus119863119872(120597119862

120597119910)

119910=0

= minus119863119872

Δ119862

1205751206011015840

(0)

(30)

Hence the Nusselt number (119873119906) and the Sherwood number

(119878ℎ) are obtained as

119873119906=119902119908120575

120581 Δ119879= minus 120579

1015840

(0) 119878ℎ=119872119908120575

119863119872Δ119862= minus1206011015840

(0) (31)

These previous coefficients are then obtained from the proce-dure of the numerical computations and are sorted in Table 1

5 Results and Discussions

The parameters entering into the fluid flow are Grashofnumber 119866

119903= 1198660120574 Solutal (modified) Grashof number 119866

119898

suction parameter V0 Prandtl number 119875

119903 the nondimen-

sional chemical reaction rate constant 1205822 Schmidt number

0 1 2 3 40

05

1

15

0

3

For G0 = 10 Gm = 1N = 05 = 071M = 05

120582 = 5 n = 05 120574 = 05 and E = 1

0

minus3

120578

120579

Sc = 06

Pr

Figure 13 Effects of V0on temperature profiles

0 1 20

05

1

10

5

0

f

Gm

120582 = 5 n = 05 120574 = 05 and E = 1

120578

For G0 = 10 0 = 3N = 05 = 071M = 05

Sc = 06

Pr

Figure 14 Effects of 119866119898on velocity profiles

0 1 20

05

1

M

1

0

5

f

120578

For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

120582 = 5 n = 1 0 = 3 and E = 1Sc = 06

Pr

Figure 15 Effects of119872 on velocity profiles

Journal of Thermodynamics 7

0 1 2 30

05

1

f

120578

For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

A = 0 (steady)A = 2 (unsteady)

120574 = 5 n = 1 0 = 3M = 05 and E = 1Sc = 06

Pr

Figure 16 Effects of 119860 on velocity profiles

0 1 2 30

05

1 For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

A = 0 (steady)A = 2 (unsteady)

120579

120578

120574 = 5 n = 1 0 = 3M = 05 and E = 1Sc = 06

Pr

Figure 17 Effects of 119860 on temperature profiles

119878119888 the magnetic parameter 119872 the nondimensional acti-

vation energy 119864 the radiation parameter 119873 the exother-micendothermic parameter 120573 and the temperature relativeparameter 120574

It is therefore pertinent to inquire the effects of variationof each of them when the others are kept constant Thenumerical results are thus presented in the form of velocityprofiles temperature profiles and concentration profiles inFigures 1ndash14 for the different values of 119866

119903 119866119898 120582119872 119864 V

0 120573

and 120574The value of 120574 is taken to be both positive and negativesince these values represent respectively cooling and heatingof the plate

The values of 119866119903is taken to be large (119866

119903= 10

1198660= 10 and 120574 = 1) since the value corresponds to

Table 1 Effects on skin friction heat transfer and mass transfercoefficients For 119866

0= 10 119866

119898= 1119873 = 05 119875

119903= 071119872 = 05 119878

119888=

06 120582 = 5 119899 = 05 V0= 3 120574 = 05 and 119864 = 1

minus1198911015840

(0) minus1205791015840

(0) minus1206011015840

(0)

119864 for 120573 = 100 minus165163 minus091802 57863405 minus160009 minus088980 57552310 minus154562 minus085343 570174

119864 for 120573 = minus100 130241 449966 55515905 098702 419573 52109310 063543 387806 481246

120582 for 120573 = 110 minus043158 173298 24262720 minus063434 134699 29144650 minus154562 minus085343 570174

120582 for 120573 = minus110 minus027903 198971 24208820 minus007554 232400 28442450 063543 387806 481246

a cooling problem that is generally encountered in nuclearengineering in connection with the cooling of reactors In air(119875119903= 071) the diluting chemical species of most common

interest have Schmidt number in the range from 06 to 075Therefore the Schmidt number 119878

119888= 06 is considered In

particular 06 corresponds to water vapor that represents adiffusing chemical species of most common interest in airThe values of the suction parameter V

0are taken to be large

Apart from the figures and tables the representative velocitytemperatureand concentration profiles and the values ofthe physically important parameters that is the local shearstress the local rates of heat and mass transfer are illustratedfor uniform wall temperature and species concentration inFigures 1ndash17 and in Tables 1-2

51 Effects of Activation Energy (119864) on the Concentrationthe Temperature and the Velocity Profiles for ExothermicEndothermic Chemical Reaction In chemistry activationenergy is defined as the energy that must be overcome inorder for a chemical reaction to occur Activation energymayalso be defined as the minimum energy required startinga chemical reaction The activation energy of a reaction isusually denoted by 119864

119886and given in units of kilojoules per

moleAn exothermic reaction is a chemical reaction that

releases energy in the form of light and heat It is theopposite of an endothermic reaction It gives out energy to itssurroundings The energy needed for the reaction to occur isless or greater than the total energy released for exothermicor endothermic reaction respectively Energy is obtainedfrom chemical bonds When bonds are broken energy isrequired When bonds are formed energy is released Eachtype of bond has specific bond energy It can be predictedwhether a chemical reaction will release or require heat byusing bond energiesWhen there ismore energy used to form

8 Journal of Thermodynamics

Table 2 Effects on skin friction heat transfer and mass transfercoefficients For119866

0= 10119866

119898= 1119873 = 05119875

119903= 071119872 = 05 119878

119888=

06 120582 = 5 119899 = 05 V0= 3 120574 = 05 and 119864 = 1

minus1198911015840

(0) minus1205791015840

(0) minus1206011015840

(0)

119873

00 minus087260 minus144125 56899620 minus305961 minus030027 57162450 minus493466 minus004756 572770

119872

00 minus185010 minus085309 57016420 minus075246 minus085443 57020350 046744 minus085628 570257

120574

minus200000 732171 365294 007835minus100000 566728 348636 026098000000 331180 188820 228076100000 minus154562 minus085343 570174200000 minus785616 minus367839 925776

119860

000000 minus127274 minus026974 397243200000 120419 056519 391549

the bonds than to break the bonds heat is given off This isknown as an exothermic reaction When a reaction requiresan input or output of energy it is known as an exothermic orendothermic reaction

Effects of activation energy (119864) and exothermic param-eter (120573) on the temperature the concentration and velocityprofiles are shown in Figure 1 to Figure 3 respectively 120573 = 1and 120573 = minus1 represent exothermic and endothermic chemicalreactions respectively Figure 1 shows that a small decreasingeffect of temperature profile is found for increasing valuesof 119864 for exothermic reaction (120573 = 1) but mark oppositeeffects are found for endothermic reaction in Figure 1 Thatis the temperature profile increases for increasing values of119864 for endothermic reaction 120573 = minus1 From (1) we observethat chemical reaction rate (119870) decreases with the increasingvalues of activation energy (119864

119886) We also observe from (20)

that increase in activation energy (119864) leads to decrease1205822 exp(minus119864120579) as well as to increase in the concentration

profiles shown in Figure 2 Small and reported effects arefound for 120573 = 1 and 120573 = minus1 respectively The parameter119864 does not enter directly into the momentum equation butits influence comes through the mass and energy equationsFigure 3 shows the variation of the velocity profiles fordifferent values of 119864 From this figure it has been observedthat the velocity profile mark increases with the increasingvalues of 119864 for endothermic chemical reaction But negligibleeffects are found for exothermic reaction

52 Effects of 120582 on the Concentration the Temperature and theVelocity Profiles for ExothermicEndothermic Reaction Con-sidering chemical reaction rate constant 1205822 is always positiveFigures 4ndash6 represent the effect of chemical rate constant120582 on

the velocity the temperature and the concentration profilesrespectively

We observe from Figures 4 and 5 that velocity andtemperature profiles increase with the increasing values of120582 for exothermic reaction but opposite effects are foundin these figures for endothermic reaction It is observedfrom (1) that increasing temperature frequently that causes amarked increase in the rate of reactions is shown in Figure 4As the temperature of the system increases the numberof molecules that carry enough energy to react when theycollide also increasesTherefore the rate of reaction increaseswith temperature As a rule the rate of a reaction doubles forevery 10∘C increase in the temperature of the system

Last part of (20) shows that 1205822 exp(minus119864120579) increases withthe increasing values of 120582We also observe from this equationthat increase in 1205822 exp(minus119864120579)means that increase in 120582 leadsto the decrease in the concentration profiles This is in greatagreement with Figure 6 for both cases 120573 = plusmn1 It isalso observed from this figure that for exothermic reactionthe mass boundary layer is close to the plate other thanendothermic reaction

53 Effects of Radiation Parameter 119873 on Temperature andVelocity Profiles The effects of radiation parameter119873 on thetemperature and the velocity profiles are shown in Figures 7and 8 From Figure 7 it can be seen that an increase in thevalues of 119873 leads to a decrease in the values of temperatureprofiles within the thermal boundary layers 120578 lt 022while outside 120578 gt 022 the temperature profile graduallyincreases with the increase of the radiation parameter 119873 Itis appear from Figure 8 that when 119873 = 0 that is withoutthe radiation the velocity profile shows its usual trend ofgradually decay As radiation becomes larger the profilesovershoot the uniform velocity close to the boundary

54 Effects of 120574 on the Velocity Temperature and Concen-tration Profiles The value of 120574 = Δ119879119879

infinis taken to be

both positive and negative since these values representrespectively cooling and heating of the plate The effects oftemperature relative parameter on velocity temperature andconcentration profiles are shown in Figures 9ndash11

The velocity profiles generated due to impulsive motionof the plate is plotted in Figure 9 for both cooling (120574 gt 0) andheating (120574 lt 0) of the plate keeping other parameters fixed(1198660= 10 119866

119898= 10 119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 V0= 30 119864 = 10 and 119899 = 1) In Figure 9 velocity

profiles are shown for different values of 120574 We observe thatvelocity increases with increasing values of 120574 for the coolingof the plate From this figure it is also observed that thenegative increase in the temperature relative parameter leadsto the decrease in the velocity field That is for heating ofthe plate (Figure 9) the effects of the 120574 on the velocity fieldhave also opposite effects as compared to the cooling ofthe plate Figure 10 shows the effects of temperature relativeparameter 120574 on the temperature profiles It has been observedfrom this figure that for heating plate the temperature profileshows its usual trend of gradually decay and the thermalboundary layer is close to the plate But for cooling plate

Journal of Thermodynamics 9

that is temperature relative parameter 120574 becomes largerthe profiles overshoot the uniform temperature close to thethermal boundary Opposite effects of temperature profile arefound for concentration profiles shown in Figure 11

55 Effects of SuctionInjection (V0) on the Velocity and the

Temperature Profiles The effects of suction and injection (V0)

for 1198660= 10 119866

119898= 10119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 120574 = 05 119864 = 10 and 119899 = 1 on the velocity profilesand temperature profiles are shown respectively in Figures12 and 13 For strong suction (V

0gt 0) the velocity and the

temperature decay rapidly away from the surface The factthat suction stabilizes the boundary layer is also apparentfrom these figures As for the injection (V

0lt 0) from Figures

12 and 13 it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayer between theinjection and the outer flow regions

56 Effects of 119866119898and119872 on the Velocity Profiles The effects

of Solutal Grashof number 119866119898

and magnetic interactionparameter119872 on velocity profiles are shown in Figures 14 and15 respectively Solutal Grashof number 119866

119898gt 0 corresponds

that the chemical species concentration in the free streamregion is less than the concentration at the boundary surfaceFigure 14 presents the effects of Solutal Grashof number119866119898on the velocity profiles It is observed that the velocity

profile increases with the increasing values of Solutal Grashofnumber 119866

119898

Imposition of a magnetic field to an electrically conduct-ing fluid creates a drag like force called Lorentz force Theforce has the tendency to slow down the flow around the plateat the expense of increasing its temperature This is depictedby decreases in velocity profiles as magnetic parameter 119872increases as shown in Figure 15

57 Effects of 119860 on the Velocity and the Temperature ProfilesHere 119860 = 0 and 119860 = 0 represent steady and unsteady flowsrespectively The effects of 119860 on velocity and temperatureprofiles are shown in Figures 16 and 17 respectively Forunsteady flow (119860 = 0) both figures show that the profilesrapidly decay and close to the plate But for steady flow (119860 =0) the profiles overshoot the uniform velocitytemperatureclose to the boundary

58 The Skin-Friction the Heat Transfer and the MassTransfer Coefficients The skin-friction the heat transferand the mass transfer coefficients are tabulated in Tables1 and 2 for different values of 119864 120573 120582 119873 119872 120574 and 119860We observe from Table 1 that the skin-friction coefficient(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) increasefor increasing values of dimensionless activation parameter119864 for exothermic chemical reaction but opposite actionsare found for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients decrease for increasing values of 119864 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) decreases for increasing values of 119864We also observe fromTable 1 that the skin-friction coefficient

0 1 2 3 40

05

1

Bestman (1990)Present

f

120578

Gr = Gm = 1 E = 5 20 = 10

= 071 and 120582 = 5

120573 = 0 120574 = 1 and M= N = A = 0

Sc = 1 Pr

Figure 18 Comparison of our calculated velocity profile and thevelocity profile of Bestman [7]

(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) decreasefor increasing values of dimensionless reaction rate constant120582 for exothermic chemical reaction but opposite effects arefound for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients increase for increasing values of 120582 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) remarkably increases for increasingvalues of 120582 From Table 2 it has been observed that theskin-friction coefficient remarkably decreases and Nusseltnumber remarkable increases for increasing values of radi-ation parameter119873 The skin-friction coefficient increases forincreasing values of magnetic parameter 119872 From Table 2we also found that skin friction and heat transfer coefficientsdecrease butmass transfer coefficient increases for increasingvalues of temperature relative parameter 120574

59 Comparison between Our Numerical Result and theAnalytical Result of Bestman [7] Bestman studied steadynatural convective boundary layer flow with large suctionHe solved his problem analytically by employing the per-turbation technique proposed by Singh and Dikshit [8] Inour present work take 119860 = 0 in (16) for considering steadyflow The values of the suction parameter V2

0= 10 is taken

to see the effects of large suction 119866119903= 119866119898= 10 120574 = 1

119872 = 119873 = 0 120582 = 5 119864 = 5 and 119899 = 1 are also chosen witha view to compare our numerical results with the analyticalresults of Bestman [7] The comparison of velocity profilesas seen in Figure 18 highlights the validity of the numericalcomputations adapted in the present investigation

6 Conclusions

In this paper we investigate the effects of chemical reactionrate and Arrhenius activation energy on an unsteady MHD

10 Journal of Thermodynamics

natural convection heat and mass transfer boundary layerflow past a flat porous plate in presence of thermal radiation

The Nachtsheim and Swigert [18] iteration techniquebased on sixth-order Runge-Kutta and Shooting method hasbeen employed to complete the integration of the resultingsolutions

The following conclusions can be drawn as a result of thecomputations

(1) Velocity increases with increasing values of 120574 for thecooling of the plate and the negative increase in thetemperature relative parameter 120574 leads to the decreasein the velocity fieldThat is for heating of the plate theeffects of the temperature relative parameter 120574 on thevelocity field have also opposite effects as comparedto the cooling of the plate

(2) Solutal Grashof number 119866119898gt 0 corresponds that

the chemical species concentration in the free streamregion is less than the concentration at the bound-ary surface It is observed that the velocity profileincreaseswith the increasing values of SolutalGrashofnumber 119866

119898

(3) Increase in 120582 leads to increase in the velocity andtemperature profiles for exothermic reaction butopposite effects are found for endothermic reactionFor 120573 = plusmn1 increase in 120582 leads to the decrease in theconcentration profiles but for exothermic reactionthe mass boundary layer is close to the plate otherthan endothermic reaction

(4) The velocity profile mark increases with the increas-ing values of 119864 for endothermic chemical reactionBut negligible effect is found for exothermic reactionTemperature decreases for increasing values of 119864 forexothermic reaction but opposite effects are found forendothermic reaction Activation energy (119864) leads toincrease in the concentration profiles but small andreported effects are found for 120573 = 1 and 120573 = minus1respectively

(5) For strong suction (V0gt 0) the velocity the tem-

perature and the concentration profiles decay rapidlyaway from the surface As for the injection (V

0lt 0)

it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayerbetween the injection and the outer flow regions

(6) Imposition of a magnetic field to an electrically con-ducting fluid creates a drag like force called Lorentzforce The force has the tendency to slow down theflow around the plate at the expense of increasing itstemperature This is depicted by decreases in velocityprofiles as magnetic parameter119872 increases

(7) An increase in the values of 119873 leads to a decrease inthe values of temperature profiles within the thermalboundary layers 120578 lt 022 while outside 120578 gt 022the temperature profile gradually increases with theincrease of the radiation parameter119873

(8) For unsteady flow (119860 = 0) the velocity and the tem-perature profiles rapidly decay and close to the plate

But for steady flow (119860 = 0) the profiles overshoot theuniform velocitytemperature close to the boundary

Nomenclature

(119906 V) Components of velocity field119888119901 Specific heat at constant pressure119873119906 Nusselt number

119875119903 Prandtl number1198760 Heat generationabsorption coefficient

119878ℎ Sherwood number119879 Temperature of the flow field119879infin Temperature of the fluid at infinity

119879119908 Temperature at the plate

119891 Dimensionless similarity functions119864119886 The activation energy

119896 The Boltzmann constant119878119888 Schmidt number119864 The nondimensional activation energy119866119903 Grashof number

119866119898 Modified (Solutal) Grashof number

Greek Symbols

120579 Dimensionless temperature120601 Dimensionless concentration120578 Dimensionless similarity variable120575 Scale factorV Kinematic viscosity120588 Density of the fluid120581 Thermal conductivity120591 Shear stress120583 Fluid viscosity1205822 Dimensionless chemical reaction rate

constant120573 and 120573lowast The coefficients of volume expansions for

temperature and concentrationrespectively

120574 The dimensionless heatgenerationabsorption coefficient

References

[1] M Tencer J S Moss and T Zapach ldquoArrhenius averagetemperature the effective temperature for non-fatigue wearoutand long term reliability in variable thermal conditions andclimatesrdquo IEEE Transactions on Components and PackagingTechnologies vol 27 no 3 pp 602ndash607 2004

[2] C Truesdell ldquoSulle basi della thermomeccanicardquo RendicontiLincei vol 22 no 8 pp 33ndash38 1957

[3] N Mills ldquoIncompressible mixtures of newtonian fluidsrdquo Inter-national Journal of Engineering Science vol 4 no 2 pp 97ndash1121966

[4] C E Beevers and R E Craine ldquoOn the determination ofresponse functions for a binary mixture of incompressiblenewtonian fluidsrdquo International Journal of Engineering Sciencevol 20 no 6 pp 737ndash745 1982

Journal of Thermodynamics 11

[5] A Al-Sharif K Chamniprasart K R Rajagopal andA Z SzerildquoLubrication with binary mixtures liquid-liquid emulsionrdquoJournal of Tribology vol 115 no 1 pp 46ndash55 1993

[6] S H Wang A Al-Sharif K R Rajagopal and A Z SzerildquoLubrication with binarymixtures liquid-liquid emulsion in anEHL conjunctionrdquo Journal of Tribology vol 115 no 3 pp 515ndash522 1993

[7] A R Bestman ldquoNatural convection boundary layer withsuction and mass transfer in a porous mediumrdquo InternationalJournal of Energy Research vol 14 no 4 pp 389ndash396 1990

[8] A K Singh and C K Dikshit ldquoHydromagnetic flow pasta continuously moving semi-infinite plate for large suctionrdquoAstrophysics and Space Science vol 148 no 2 pp 249ndash256 1988

[9] A R Bestman ldquoRadiative heat transfer to flow of a combustiblemixture in a vertical piperdquo International Journal of EnergyResearch vol 15 no 3 pp 179ndash184 1991

[10] M A Alabraba A R Bestman and A Ogulu ldquoLaminar con-vection in binary mixture of hydromagnetic flow with radiativeheat transfer Irdquo Astrophysics and Space Science vol 195 no 2pp 431ndash439 1992

[11] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[12] O DMakinde P O Olanrewaju andWM Charles ldquoUnsteadyconvection with chemical reaction and radiative heat transferpast a flat porous plate moving through a binary mixturerdquoAfricka Matematika vol 22 no 1 pp 65ndash78 2011

[13] O D Makinde and P O Olanrewaju ldquoUnsteady mixed convec-tion with Soret and Dufour effects past a porous plate movingthrough a binarymixture of chemically reacting fluidrdquoChemicalEngineering Communications vol 198 no 7 pp 920ndash938 2011

[14] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer flow with mass transfer and a binary chemical reactionrdquoBritish Journal of Applied Science and Technology (Sciencedo-main International) vol 3 no 1 pp 131ndash149 2013

[15] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer heat and mass transfer flow with exothermic chemicalreactionsrdquo Journal of Pure and Applied Mathematics (Advancesand Applications) vol 9 no 1 pp 17ndash41 2013

[16] Kh Abdul Maleque ldquoEffects of binary chemical reactionand activation energy on MHD boundary layer heat andmass transfer flow with viscous dissipation and heat genera-tionabsorptionrdquo ISRN Thermodynamics vol 2013 Article ID284637 9 pages 2013

[17] Kh Abdul Maleque ldquoEffects of combined temperature- anddepth-dependent viscosity and hall current on an unsteadyMHD laminar convective flow due to a rotating diskrdquo ChemicalEngineering Communications vol 197 no 4 pp 506ndash521 2010

[18] P R Nachtsheim and P Swigert ldquoSatisfaction of asymptoticboundary conditions in numerical solution of system of non-linear of boundary layer typerdquo NASA TN-D3004 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Superconductivity

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Soft MatterJournal of

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ThermodynamicsJournal of

6 Journal of Thermodynamics

0 1 20

05

1

120578

120601

For G0 = 10 Gm = 1N = 05 = 071M= 05

120582 = 5 n = 05 = 3 and E = 10

Gr = G0120574

120574

minus2minus101

2

Sc = 06

Pr

Figure 11 Effects of 120574 on concentration profiles

0 1 2 3 40

1

2

3

f

120578

For G0 = 10 Gm = 1N = 05 = 071M = 05

120582 = 5 n = 05 120574 = 05 and E = 1

0 = minus3

0 = 0

0 = 3

Sc = 06

Pr

Figure 12 Effects of V0on velocity profiles

The heat flux (119902119908) and the mass flux (119872

119908) at the wall are

given by

119902119908= minus120581(

120597119879

120597119910)

119910=0

= minus120581Δ119879

1205751205791015840

(0)

119872119908= minus119863119872(120597119862

120597119910)

119910=0

= minus119863119872

Δ119862

1205751206011015840

(0)

(30)

Hence the Nusselt number (119873119906) and the Sherwood number

(119878ℎ) are obtained as

119873119906=119902119908120575

120581 Δ119879= minus 120579

1015840

(0) 119878ℎ=119872119908120575

119863119872Δ119862= minus1206011015840

(0) (31)

These previous coefficients are then obtained from the proce-dure of the numerical computations and are sorted in Table 1

5 Results and Discussions

The parameters entering into the fluid flow are Grashofnumber 119866

119903= 1198660120574 Solutal (modified) Grashof number 119866

119898

suction parameter V0 Prandtl number 119875

119903 the nondimen-

sional chemical reaction rate constant 1205822 Schmidt number

0 1 2 3 40

05

1

15

0

3

For G0 = 10 Gm = 1N = 05 = 071M = 05

120582 = 5 n = 05 120574 = 05 and E = 1

0

minus3

120578

120579

Sc = 06

Pr

Figure 13 Effects of V0on temperature profiles

0 1 20

05

1

10

5

0

f

Gm

120582 = 5 n = 05 120574 = 05 and E = 1

120578

For G0 = 10 0 = 3N = 05 = 071M = 05

Sc = 06

Pr

Figure 14 Effects of 119866119898on velocity profiles

0 1 20

05

1

M

1

0

5

f

120578

For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

120582 = 5 n = 1 0 = 3 and E = 1Sc = 06

Pr

Figure 15 Effects of119872 on velocity profiles

Journal of Thermodynamics 7

0 1 2 30

05

1

f

120578

For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

A = 0 (steady)A = 2 (unsteady)

120574 = 5 n = 1 0 = 3M = 05 and E = 1Sc = 06

Pr

Figure 16 Effects of 119860 on velocity profiles

0 1 2 30

05

1 For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

A = 0 (steady)A = 2 (unsteady)

120579

120578

120574 = 5 n = 1 0 = 3M = 05 and E = 1Sc = 06

Pr

Figure 17 Effects of 119860 on temperature profiles

119878119888 the magnetic parameter 119872 the nondimensional acti-

vation energy 119864 the radiation parameter 119873 the exother-micendothermic parameter 120573 and the temperature relativeparameter 120574

It is therefore pertinent to inquire the effects of variationof each of them when the others are kept constant Thenumerical results are thus presented in the form of velocityprofiles temperature profiles and concentration profiles inFigures 1ndash14 for the different values of 119866

119903 119866119898 120582119872 119864 V

0 120573

and 120574The value of 120574 is taken to be both positive and negativesince these values represent respectively cooling and heatingof the plate

The values of 119866119903is taken to be large (119866

119903= 10

1198660= 10 and 120574 = 1) since the value corresponds to

Table 1 Effects on skin friction heat transfer and mass transfercoefficients For 119866

0= 10 119866

119898= 1119873 = 05 119875

119903= 071119872 = 05 119878

119888=

06 120582 = 5 119899 = 05 V0= 3 120574 = 05 and 119864 = 1

minus1198911015840

(0) minus1205791015840

(0) minus1206011015840

(0)

119864 for 120573 = 100 minus165163 minus091802 57863405 minus160009 minus088980 57552310 minus154562 minus085343 570174

119864 for 120573 = minus100 130241 449966 55515905 098702 419573 52109310 063543 387806 481246

120582 for 120573 = 110 minus043158 173298 24262720 minus063434 134699 29144650 minus154562 minus085343 570174

120582 for 120573 = minus110 minus027903 198971 24208820 minus007554 232400 28442450 063543 387806 481246

a cooling problem that is generally encountered in nuclearengineering in connection with the cooling of reactors In air(119875119903= 071) the diluting chemical species of most common

interest have Schmidt number in the range from 06 to 075Therefore the Schmidt number 119878

119888= 06 is considered In

particular 06 corresponds to water vapor that represents adiffusing chemical species of most common interest in airThe values of the suction parameter V

0are taken to be large

Apart from the figures and tables the representative velocitytemperatureand concentration profiles and the values ofthe physically important parameters that is the local shearstress the local rates of heat and mass transfer are illustratedfor uniform wall temperature and species concentration inFigures 1ndash17 and in Tables 1-2

51 Effects of Activation Energy (119864) on the Concentrationthe Temperature and the Velocity Profiles for ExothermicEndothermic Chemical Reaction In chemistry activationenergy is defined as the energy that must be overcome inorder for a chemical reaction to occur Activation energymayalso be defined as the minimum energy required startinga chemical reaction The activation energy of a reaction isusually denoted by 119864

119886and given in units of kilojoules per

moleAn exothermic reaction is a chemical reaction that

releases energy in the form of light and heat It is theopposite of an endothermic reaction It gives out energy to itssurroundings The energy needed for the reaction to occur isless or greater than the total energy released for exothermicor endothermic reaction respectively Energy is obtainedfrom chemical bonds When bonds are broken energy isrequired When bonds are formed energy is released Eachtype of bond has specific bond energy It can be predictedwhether a chemical reaction will release or require heat byusing bond energiesWhen there ismore energy used to form

8 Journal of Thermodynamics

Table 2 Effects on skin friction heat transfer and mass transfercoefficients For119866

0= 10119866

119898= 1119873 = 05119875

119903= 071119872 = 05 119878

119888=

06 120582 = 5 119899 = 05 V0= 3 120574 = 05 and 119864 = 1

minus1198911015840

(0) minus1205791015840

(0) minus1206011015840

(0)

119873

00 minus087260 minus144125 56899620 minus305961 minus030027 57162450 minus493466 minus004756 572770

119872

00 minus185010 minus085309 57016420 minus075246 minus085443 57020350 046744 minus085628 570257

120574

minus200000 732171 365294 007835minus100000 566728 348636 026098000000 331180 188820 228076100000 minus154562 minus085343 570174200000 minus785616 minus367839 925776

119860

000000 minus127274 minus026974 397243200000 120419 056519 391549

the bonds than to break the bonds heat is given off This isknown as an exothermic reaction When a reaction requiresan input or output of energy it is known as an exothermic orendothermic reaction

Effects of activation energy (119864) and exothermic param-eter (120573) on the temperature the concentration and velocityprofiles are shown in Figure 1 to Figure 3 respectively 120573 = 1and 120573 = minus1 represent exothermic and endothermic chemicalreactions respectively Figure 1 shows that a small decreasingeffect of temperature profile is found for increasing valuesof 119864 for exothermic reaction (120573 = 1) but mark oppositeeffects are found for endothermic reaction in Figure 1 Thatis the temperature profile increases for increasing values of119864 for endothermic reaction 120573 = minus1 From (1) we observethat chemical reaction rate (119870) decreases with the increasingvalues of activation energy (119864

119886) We also observe from (20)

that increase in activation energy (119864) leads to decrease1205822 exp(minus119864120579) as well as to increase in the concentration

profiles shown in Figure 2 Small and reported effects arefound for 120573 = 1 and 120573 = minus1 respectively The parameter119864 does not enter directly into the momentum equation butits influence comes through the mass and energy equationsFigure 3 shows the variation of the velocity profiles fordifferent values of 119864 From this figure it has been observedthat the velocity profile mark increases with the increasingvalues of 119864 for endothermic chemical reaction But negligibleeffects are found for exothermic reaction

52 Effects of 120582 on the Concentration the Temperature and theVelocity Profiles for ExothermicEndothermic Reaction Con-sidering chemical reaction rate constant 1205822 is always positiveFigures 4ndash6 represent the effect of chemical rate constant120582 on

the velocity the temperature and the concentration profilesrespectively

We observe from Figures 4 and 5 that velocity andtemperature profiles increase with the increasing values of120582 for exothermic reaction but opposite effects are foundin these figures for endothermic reaction It is observedfrom (1) that increasing temperature frequently that causes amarked increase in the rate of reactions is shown in Figure 4As the temperature of the system increases the numberof molecules that carry enough energy to react when theycollide also increasesTherefore the rate of reaction increaseswith temperature As a rule the rate of a reaction doubles forevery 10∘C increase in the temperature of the system

Last part of (20) shows that 1205822 exp(minus119864120579) increases withthe increasing values of 120582We also observe from this equationthat increase in 1205822 exp(minus119864120579)means that increase in 120582 leadsto the decrease in the concentration profiles This is in greatagreement with Figure 6 for both cases 120573 = plusmn1 It isalso observed from this figure that for exothermic reactionthe mass boundary layer is close to the plate other thanendothermic reaction

53 Effects of Radiation Parameter 119873 on Temperature andVelocity Profiles The effects of radiation parameter119873 on thetemperature and the velocity profiles are shown in Figures 7and 8 From Figure 7 it can be seen that an increase in thevalues of 119873 leads to a decrease in the values of temperatureprofiles within the thermal boundary layers 120578 lt 022while outside 120578 gt 022 the temperature profile graduallyincreases with the increase of the radiation parameter 119873 Itis appear from Figure 8 that when 119873 = 0 that is withoutthe radiation the velocity profile shows its usual trend ofgradually decay As radiation becomes larger the profilesovershoot the uniform velocity close to the boundary

54 Effects of 120574 on the Velocity Temperature and Concen-tration Profiles The value of 120574 = Δ119879119879

infinis taken to be

both positive and negative since these values representrespectively cooling and heating of the plate The effects oftemperature relative parameter on velocity temperature andconcentration profiles are shown in Figures 9ndash11

The velocity profiles generated due to impulsive motionof the plate is plotted in Figure 9 for both cooling (120574 gt 0) andheating (120574 lt 0) of the plate keeping other parameters fixed(1198660= 10 119866

119898= 10 119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 V0= 30 119864 = 10 and 119899 = 1) In Figure 9 velocity

profiles are shown for different values of 120574 We observe thatvelocity increases with increasing values of 120574 for the coolingof the plate From this figure it is also observed that thenegative increase in the temperature relative parameter leadsto the decrease in the velocity field That is for heating ofthe plate (Figure 9) the effects of the 120574 on the velocity fieldhave also opposite effects as compared to the cooling ofthe plate Figure 10 shows the effects of temperature relativeparameter 120574 on the temperature profiles It has been observedfrom this figure that for heating plate the temperature profileshows its usual trend of gradually decay and the thermalboundary layer is close to the plate But for cooling plate

Journal of Thermodynamics 9

that is temperature relative parameter 120574 becomes largerthe profiles overshoot the uniform temperature close to thethermal boundary Opposite effects of temperature profile arefound for concentration profiles shown in Figure 11

55 Effects of SuctionInjection (V0) on the Velocity and the

Temperature Profiles The effects of suction and injection (V0)

for 1198660= 10 119866

119898= 10119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 120574 = 05 119864 = 10 and 119899 = 1 on the velocity profilesand temperature profiles are shown respectively in Figures12 and 13 For strong suction (V

0gt 0) the velocity and the

temperature decay rapidly away from the surface The factthat suction stabilizes the boundary layer is also apparentfrom these figures As for the injection (V

0lt 0) from Figures

12 and 13 it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayer between theinjection and the outer flow regions

56 Effects of 119866119898and119872 on the Velocity Profiles The effects

of Solutal Grashof number 119866119898

and magnetic interactionparameter119872 on velocity profiles are shown in Figures 14 and15 respectively Solutal Grashof number 119866

119898gt 0 corresponds

that the chemical species concentration in the free streamregion is less than the concentration at the boundary surfaceFigure 14 presents the effects of Solutal Grashof number119866119898on the velocity profiles It is observed that the velocity

profile increases with the increasing values of Solutal Grashofnumber 119866

119898

Imposition of a magnetic field to an electrically conduct-ing fluid creates a drag like force called Lorentz force Theforce has the tendency to slow down the flow around the plateat the expense of increasing its temperature This is depictedby decreases in velocity profiles as magnetic parameter 119872increases as shown in Figure 15

57 Effects of 119860 on the Velocity and the Temperature ProfilesHere 119860 = 0 and 119860 = 0 represent steady and unsteady flowsrespectively The effects of 119860 on velocity and temperatureprofiles are shown in Figures 16 and 17 respectively Forunsteady flow (119860 = 0) both figures show that the profilesrapidly decay and close to the plate But for steady flow (119860 =0) the profiles overshoot the uniform velocitytemperatureclose to the boundary

58 The Skin-Friction the Heat Transfer and the MassTransfer Coefficients The skin-friction the heat transferand the mass transfer coefficients are tabulated in Tables1 and 2 for different values of 119864 120573 120582 119873 119872 120574 and 119860We observe from Table 1 that the skin-friction coefficient(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) increasefor increasing values of dimensionless activation parameter119864 for exothermic chemical reaction but opposite actionsare found for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients decrease for increasing values of 119864 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) decreases for increasing values of 119864We also observe fromTable 1 that the skin-friction coefficient

0 1 2 3 40

05

1

Bestman (1990)Present

f

120578

Gr = Gm = 1 E = 5 20 = 10

= 071 and 120582 = 5

120573 = 0 120574 = 1 and M= N = A = 0

Sc = 1 Pr

Figure 18 Comparison of our calculated velocity profile and thevelocity profile of Bestman [7]

(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) decreasefor increasing values of dimensionless reaction rate constant120582 for exothermic chemical reaction but opposite effects arefound for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients increase for increasing values of 120582 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) remarkably increases for increasingvalues of 120582 From Table 2 it has been observed that theskin-friction coefficient remarkably decreases and Nusseltnumber remarkable increases for increasing values of radi-ation parameter119873 The skin-friction coefficient increases forincreasing values of magnetic parameter 119872 From Table 2we also found that skin friction and heat transfer coefficientsdecrease butmass transfer coefficient increases for increasingvalues of temperature relative parameter 120574

59 Comparison between Our Numerical Result and theAnalytical Result of Bestman [7] Bestman studied steadynatural convective boundary layer flow with large suctionHe solved his problem analytically by employing the per-turbation technique proposed by Singh and Dikshit [8] Inour present work take 119860 = 0 in (16) for considering steadyflow The values of the suction parameter V2

0= 10 is taken

to see the effects of large suction 119866119903= 119866119898= 10 120574 = 1

119872 = 119873 = 0 120582 = 5 119864 = 5 and 119899 = 1 are also chosen witha view to compare our numerical results with the analyticalresults of Bestman [7] The comparison of velocity profilesas seen in Figure 18 highlights the validity of the numericalcomputations adapted in the present investigation

6 Conclusions

In this paper we investigate the effects of chemical reactionrate and Arrhenius activation energy on an unsteady MHD

10 Journal of Thermodynamics

natural convection heat and mass transfer boundary layerflow past a flat porous plate in presence of thermal radiation

The Nachtsheim and Swigert [18] iteration techniquebased on sixth-order Runge-Kutta and Shooting method hasbeen employed to complete the integration of the resultingsolutions

The following conclusions can be drawn as a result of thecomputations

(1) Velocity increases with increasing values of 120574 for thecooling of the plate and the negative increase in thetemperature relative parameter 120574 leads to the decreasein the velocity fieldThat is for heating of the plate theeffects of the temperature relative parameter 120574 on thevelocity field have also opposite effects as comparedto the cooling of the plate

(2) Solutal Grashof number 119866119898gt 0 corresponds that

the chemical species concentration in the free streamregion is less than the concentration at the bound-ary surface It is observed that the velocity profileincreaseswith the increasing values of SolutalGrashofnumber 119866

119898

(3) Increase in 120582 leads to increase in the velocity andtemperature profiles for exothermic reaction butopposite effects are found for endothermic reactionFor 120573 = plusmn1 increase in 120582 leads to the decrease in theconcentration profiles but for exothermic reactionthe mass boundary layer is close to the plate otherthan endothermic reaction

(4) The velocity profile mark increases with the increas-ing values of 119864 for endothermic chemical reactionBut negligible effect is found for exothermic reactionTemperature decreases for increasing values of 119864 forexothermic reaction but opposite effects are found forendothermic reaction Activation energy (119864) leads toincrease in the concentration profiles but small andreported effects are found for 120573 = 1 and 120573 = minus1respectively

(5) For strong suction (V0gt 0) the velocity the tem-

perature and the concentration profiles decay rapidlyaway from the surface As for the injection (V

0lt 0)

it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayerbetween the injection and the outer flow regions

(6) Imposition of a magnetic field to an electrically con-ducting fluid creates a drag like force called Lorentzforce The force has the tendency to slow down theflow around the plate at the expense of increasing itstemperature This is depicted by decreases in velocityprofiles as magnetic parameter119872 increases

(7) An increase in the values of 119873 leads to a decrease inthe values of temperature profiles within the thermalboundary layers 120578 lt 022 while outside 120578 gt 022the temperature profile gradually increases with theincrease of the radiation parameter119873

(8) For unsteady flow (119860 = 0) the velocity and the tem-perature profiles rapidly decay and close to the plate

But for steady flow (119860 = 0) the profiles overshoot theuniform velocitytemperature close to the boundary

Nomenclature

(119906 V) Components of velocity field119888119901 Specific heat at constant pressure119873119906 Nusselt number

119875119903 Prandtl number1198760 Heat generationabsorption coefficient

119878ℎ Sherwood number119879 Temperature of the flow field119879infin Temperature of the fluid at infinity

119879119908 Temperature at the plate

119891 Dimensionless similarity functions119864119886 The activation energy

119896 The Boltzmann constant119878119888 Schmidt number119864 The nondimensional activation energy119866119903 Grashof number

119866119898 Modified (Solutal) Grashof number

Greek Symbols

120579 Dimensionless temperature120601 Dimensionless concentration120578 Dimensionless similarity variable120575 Scale factorV Kinematic viscosity120588 Density of the fluid120581 Thermal conductivity120591 Shear stress120583 Fluid viscosity1205822 Dimensionless chemical reaction rate

constant120573 and 120573lowast The coefficients of volume expansions for

temperature and concentrationrespectively

120574 The dimensionless heatgenerationabsorption coefficient

References

[1] M Tencer J S Moss and T Zapach ldquoArrhenius averagetemperature the effective temperature for non-fatigue wearoutand long term reliability in variable thermal conditions andclimatesrdquo IEEE Transactions on Components and PackagingTechnologies vol 27 no 3 pp 602ndash607 2004

[2] C Truesdell ldquoSulle basi della thermomeccanicardquo RendicontiLincei vol 22 no 8 pp 33ndash38 1957

[3] N Mills ldquoIncompressible mixtures of newtonian fluidsrdquo Inter-national Journal of Engineering Science vol 4 no 2 pp 97ndash1121966

[4] C E Beevers and R E Craine ldquoOn the determination ofresponse functions for a binary mixture of incompressiblenewtonian fluidsrdquo International Journal of Engineering Sciencevol 20 no 6 pp 737ndash745 1982

Journal of Thermodynamics 11

[5] A Al-Sharif K Chamniprasart K R Rajagopal andA Z SzerildquoLubrication with binary mixtures liquid-liquid emulsionrdquoJournal of Tribology vol 115 no 1 pp 46ndash55 1993

[6] S H Wang A Al-Sharif K R Rajagopal and A Z SzerildquoLubrication with binarymixtures liquid-liquid emulsion in anEHL conjunctionrdquo Journal of Tribology vol 115 no 3 pp 515ndash522 1993

[7] A R Bestman ldquoNatural convection boundary layer withsuction and mass transfer in a porous mediumrdquo InternationalJournal of Energy Research vol 14 no 4 pp 389ndash396 1990

[8] A K Singh and C K Dikshit ldquoHydromagnetic flow pasta continuously moving semi-infinite plate for large suctionrdquoAstrophysics and Space Science vol 148 no 2 pp 249ndash256 1988

[9] A R Bestman ldquoRadiative heat transfer to flow of a combustiblemixture in a vertical piperdquo International Journal of EnergyResearch vol 15 no 3 pp 179ndash184 1991

[10] M A Alabraba A R Bestman and A Ogulu ldquoLaminar con-vection in binary mixture of hydromagnetic flow with radiativeheat transfer Irdquo Astrophysics and Space Science vol 195 no 2pp 431ndash439 1992

[11] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[12] O DMakinde P O Olanrewaju andWM Charles ldquoUnsteadyconvection with chemical reaction and radiative heat transferpast a flat porous plate moving through a binary mixturerdquoAfricka Matematika vol 22 no 1 pp 65ndash78 2011

[13] O D Makinde and P O Olanrewaju ldquoUnsteady mixed convec-tion with Soret and Dufour effects past a porous plate movingthrough a binarymixture of chemically reacting fluidrdquoChemicalEngineering Communications vol 198 no 7 pp 920ndash938 2011

[14] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer flow with mass transfer and a binary chemical reactionrdquoBritish Journal of Applied Science and Technology (Sciencedo-main International) vol 3 no 1 pp 131ndash149 2013

[15] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer heat and mass transfer flow with exothermic chemicalreactionsrdquo Journal of Pure and Applied Mathematics (Advancesand Applications) vol 9 no 1 pp 17ndash41 2013

[16] Kh Abdul Maleque ldquoEffects of binary chemical reactionand activation energy on MHD boundary layer heat andmass transfer flow with viscous dissipation and heat genera-tionabsorptionrdquo ISRN Thermodynamics vol 2013 Article ID284637 9 pages 2013

[17] Kh Abdul Maleque ldquoEffects of combined temperature- anddepth-dependent viscosity and hall current on an unsteadyMHD laminar convective flow due to a rotating diskrdquo ChemicalEngineering Communications vol 197 no 4 pp 506ndash521 2010

[18] P R Nachtsheim and P Swigert ldquoSatisfaction of asymptoticboundary conditions in numerical solution of system of non-linear of boundary layer typerdquo NASA TN-D3004 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Thermodynamics 7

0 1 2 30

05

1

f

120578

For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

A = 0 (steady)A = 2 (unsteady)

120574 = 5 n = 1 0 = 3M = 05 and E = 1Sc = 06

Pr

Figure 16 Effects of 119860 on velocity profiles

0 1 2 30

05

1 For G0 = 10 Gm = 1N = 05 = 071 120574 = 1

A = 0 (steady)A = 2 (unsteady)

120579

120578

120574 = 5 n = 1 0 = 3M = 05 and E = 1Sc = 06

Pr

Figure 17 Effects of 119860 on temperature profiles

119878119888 the magnetic parameter 119872 the nondimensional acti-

vation energy 119864 the radiation parameter 119873 the exother-micendothermic parameter 120573 and the temperature relativeparameter 120574

It is therefore pertinent to inquire the effects of variationof each of them when the others are kept constant Thenumerical results are thus presented in the form of velocityprofiles temperature profiles and concentration profiles inFigures 1ndash14 for the different values of 119866

119903 119866119898 120582119872 119864 V

0 120573

and 120574The value of 120574 is taken to be both positive and negativesince these values represent respectively cooling and heatingof the plate

The values of 119866119903is taken to be large (119866

119903= 10

1198660= 10 and 120574 = 1) since the value corresponds to

Table 1 Effects on skin friction heat transfer and mass transfercoefficients For 119866

0= 10 119866

119898= 1119873 = 05 119875

119903= 071119872 = 05 119878

119888=

06 120582 = 5 119899 = 05 V0= 3 120574 = 05 and 119864 = 1

minus1198911015840

(0) minus1205791015840

(0) minus1206011015840

(0)

119864 for 120573 = 100 minus165163 minus091802 57863405 minus160009 minus088980 57552310 minus154562 minus085343 570174

119864 for 120573 = minus100 130241 449966 55515905 098702 419573 52109310 063543 387806 481246

120582 for 120573 = 110 minus043158 173298 24262720 minus063434 134699 29144650 minus154562 minus085343 570174

120582 for 120573 = minus110 minus027903 198971 24208820 minus007554 232400 28442450 063543 387806 481246

a cooling problem that is generally encountered in nuclearengineering in connection with the cooling of reactors In air(119875119903= 071) the diluting chemical species of most common

interest have Schmidt number in the range from 06 to 075Therefore the Schmidt number 119878

119888= 06 is considered In

particular 06 corresponds to water vapor that represents adiffusing chemical species of most common interest in airThe values of the suction parameter V

0are taken to be large

Apart from the figures and tables the representative velocitytemperatureand concentration profiles and the values ofthe physically important parameters that is the local shearstress the local rates of heat and mass transfer are illustratedfor uniform wall temperature and species concentration inFigures 1ndash17 and in Tables 1-2

51 Effects of Activation Energy (119864) on the Concentrationthe Temperature and the Velocity Profiles for ExothermicEndothermic Chemical Reaction In chemistry activationenergy is defined as the energy that must be overcome inorder for a chemical reaction to occur Activation energymayalso be defined as the minimum energy required startinga chemical reaction The activation energy of a reaction isusually denoted by 119864

119886and given in units of kilojoules per

moleAn exothermic reaction is a chemical reaction that

releases energy in the form of light and heat It is theopposite of an endothermic reaction It gives out energy to itssurroundings The energy needed for the reaction to occur isless or greater than the total energy released for exothermicor endothermic reaction respectively Energy is obtainedfrom chemical bonds When bonds are broken energy isrequired When bonds are formed energy is released Eachtype of bond has specific bond energy It can be predictedwhether a chemical reaction will release or require heat byusing bond energiesWhen there ismore energy used to form

8 Journal of Thermodynamics

Table 2 Effects on skin friction heat transfer and mass transfercoefficients For119866

0= 10119866

119898= 1119873 = 05119875

119903= 071119872 = 05 119878

119888=

06 120582 = 5 119899 = 05 V0= 3 120574 = 05 and 119864 = 1

minus1198911015840

(0) minus1205791015840

(0) minus1206011015840

(0)

119873

00 minus087260 minus144125 56899620 minus305961 minus030027 57162450 minus493466 minus004756 572770

119872

00 minus185010 minus085309 57016420 minus075246 minus085443 57020350 046744 minus085628 570257

120574

minus200000 732171 365294 007835minus100000 566728 348636 026098000000 331180 188820 228076100000 minus154562 minus085343 570174200000 minus785616 minus367839 925776

119860

000000 minus127274 minus026974 397243200000 120419 056519 391549

the bonds than to break the bonds heat is given off This isknown as an exothermic reaction When a reaction requiresan input or output of energy it is known as an exothermic orendothermic reaction

Effects of activation energy (119864) and exothermic param-eter (120573) on the temperature the concentration and velocityprofiles are shown in Figure 1 to Figure 3 respectively 120573 = 1and 120573 = minus1 represent exothermic and endothermic chemicalreactions respectively Figure 1 shows that a small decreasingeffect of temperature profile is found for increasing valuesof 119864 for exothermic reaction (120573 = 1) but mark oppositeeffects are found for endothermic reaction in Figure 1 Thatis the temperature profile increases for increasing values of119864 for endothermic reaction 120573 = minus1 From (1) we observethat chemical reaction rate (119870) decreases with the increasingvalues of activation energy (119864

119886) We also observe from (20)

that increase in activation energy (119864) leads to decrease1205822 exp(minus119864120579) as well as to increase in the concentration

profiles shown in Figure 2 Small and reported effects arefound for 120573 = 1 and 120573 = minus1 respectively The parameter119864 does not enter directly into the momentum equation butits influence comes through the mass and energy equationsFigure 3 shows the variation of the velocity profiles fordifferent values of 119864 From this figure it has been observedthat the velocity profile mark increases with the increasingvalues of 119864 for endothermic chemical reaction But negligibleeffects are found for exothermic reaction

52 Effects of 120582 on the Concentration the Temperature and theVelocity Profiles for ExothermicEndothermic Reaction Con-sidering chemical reaction rate constant 1205822 is always positiveFigures 4ndash6 represent the effect of chemical rate constant120582 on

the velocity the temperature and the concentration profilesrespectively

We observe from Figures 4 and 5 that velocity andtemperature profiles increase with the increasing values of120582 for exothermic reaction but opposite effects are foundin these figures for endothermic reaction It is observedfrom (1) that increasing temperature frequently that causes amarked increase in the rate of reactions is shown in Figure 4As the temperature of the system increases the numberof molecules that carry enough energy to react when theycollide also increasesTherefore the rate of reaction increaseswith temperature As a rule the rate of a reaction doubles forevery 10∘C increase in the temperature of the system

Last part of (20) shows that 1205822 exp(minus119864120579) increases withthe increasing values of 120582We also observe from this equationthat increase in 1205822 exp(minus119864120579)means that increase in 120582 leadsto the decrease in the concentration profiles This is in greatagreement with Figure 6 for both cases 120573 = plusmn1 It isalso observed from this figure that for exothermic reactionthe mass boundary layer is close to the plate other thanendothermic reaction

53 Effects of Radiation Parameter 119873 on Temperature andVelocity Profiles The effects of radiation parameter119873 on thetemperature and the velocity profiles are shown in Figures 7and 8 From Figure 7 it can be seen that an increase in thevalues of 119873 leads to a decrease in the values of temperatureprofiles within the thermal boundary layers 120578 lt 022while outside 120578 gt 022 the temperature profile graduallyincreases with the increase of the radiation parameter 119873 Itis appear from Figure 8 that when 119873 = 0 that is withoutthe radiation the velocity profile shows its usual trend ofgradually decay As radiation becomes larger the profilesovershoot the uniform velocity close to the boundary

54 Effects of 120574 on the Velocity Temperature and Concen-tration Profiles The value of 120574 = Δ119879119879

infinis taken to be

both positive and negative since these values representrespectively cooling and heating of the plate The effects oftemperature relative parameter on velocity temperature andconcentration profiles are shown in Figures 9ndash11

The velocity profiles generated due to impulsive motionof the plate is plotted in Figure 9 for both cooling (120574 gt 0) andheating (120574 lt 0) of the plate keeping other parameters fixed(1198660= 10 119866

119898= 10 119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 V0= 30 119864 = 10 and 119899 = 1) In Figure 9 velocity

profiles are shown for different values of 120574 We observe thatvelocity increases with increasing values of 120574 for the coolingof the plate From this figure it is also observed that thenegative increase in the temperature relative parameter leadsto the decrease in the velocity field That is for heating ofthe plate (Figure 9) the effects of the 120574 on the velocity fieldhave also opposite effects as compared to the cooling ofthe plate Figure 10 shows the effects of temperature relativeparameter 120574 on the temperature profiles It has been observedfrom this figure that for heating plate the temperature profileshows its usual trend of gradually decay and the thermalboundary layer is close to the plate But for cooling plate

Journal of Thermodynamics 9

that is temperature relative parameter 120574 becomes largerthe profiles overshoot the uniform temperature close to thethermal boundary Opposite effects of temperature profile arefound for concentration profiles shown in Figure 11

55 Effects of SuctionInjection (V0) on the Velocity and the

Temperature Profiles The effects of suction and injection (V0)

for 1198660= 10 119866

119898= 10119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 120574 = 05 119864 = 10 and 119899 = 1 on the velocity profilesand temperature profiles are shown respectively in Figures12 and 13 For strong suction (V

0gt 0) the velocity and the

temperature decay rapidly away from the surface The factthat suction stabilizes the boundary layer is also apparentfrom these figures As for the injection (V

0lt 0) from Figures

12 and 13 it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayer between theinjection and the outer flow regions

56 Effects of 119866119898and119872 on the Velocity Profiles The effects

of Solutal Grashof number 119866119898

and magnetic interactionparameter119872 on velocity profiles are shown in Figures 14 and15 respectively Solutal Grashof number 119866

119898gt 0 corresponds

that the chemical species concentration in the free streamregion is less than the concentration at the boundary surfaceFigure 14 presents the effects of Solutal Grashof number119866119898on the velocity profiles It is observed that the velocity

profile increases with the increasing values of Solutal Grashofnumber 119866

119898

Imposition of a magnetic field to an electrically conduct-ing fluid creates a drag like force called Lorentz force Theforce has the tendency to slow down the flow around the plateat the expense of increasing its temperature This is depictedby decreases in velocity profiles as magnetic parameter 119872increases as shown in Figure 15

57 Effects of 119860 on the Velocity and the Temperature ProfilesHere 119860 = 0 and 119860 = 0 represent steady and unsteady flowsrespectively The effects of 119860 on velocity and temperatureprofiles are shown in Figures 16 and 17 respectively Forunsteady flow (119860 = 0) both figures show that the profilesrapidly decay and close to the plate But for steady flow (119860 =0) the profiles overshoot the uniform velocitytemperatureclose to the boundary

58 The Skin-Friction the Heat Transfer and the MassTransfer Coefficients The skin-friction the heat transferand the mass transfer coefficients are tabulated in Tables1 and 2 for different values of 119864 120573 120582 119873 119872 120574 and 119860We observe from Table 1 that the skin-friction coefficient(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) increasefor increasing values of dimensionless activation parameter119864 for exothermic chemical reaction but opposite actionsare found for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients decrease for increasing values of 119864 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) decreases for increasing values of 119864We also observe fromTable 1 that the skin-friction coefficient

0 1 2 3 40

05

1

Bestman (1990)Present

f

120578

Gr = Gm = 1 E = 5 20 = 10

= 071 and 120582 = 5

120573 = 0 120574 = 1 and M= N = A = 0

Sc = 1 Pr

Figure 18 Comparison of our calculated velocity profile and thevelocity profile of Bestman [7]

(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) decreasefor increasing values of dimensionless reaction rate constant120582 for exothermic chemical reaction but opposite effects arefound for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients increase for increasing values of 120582 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) remarkably increases for increasingvalues of 120582 From Table 2 it has been observed that theskin-friction coefficient remarkably decreases and Nusseltnumber remarkable increases for increasing values of radi-ation parameter119873 The skin-friction coefficient increases forincreasing values of magnetic parameter 119872 From Table 2we also found that skin friction and heat transfer coefficientsdecrease butmass transfer coefficient increases for increasingvalues of temperature relative parameter 120574

59 Comparison between Our Numerical Result and theAnalytical Result of Bestman [7] Bestman studied steadynatural convective boundary layer flow with large suctionHe solved his problem analytically by employing the per-turbation technique proposed by Singh and Dikshit [8] Inour present work take 119860 = 0 in (16) for considering steadyflow The values of the suction parameter V2

0= 10 is taken

to see the effects of large suction 119866119903= 119866119898= 10 120574 = 1

119872 = 119873 = 0 120582 = 5 119864 = 5 and 119899 = 1 are also chosen witha view to compare our numerical results with the analyticalresults of Bestman [7] The comparison of velocity profilesas seen in Figure 18 highlights the validity of the numericalcomputations adapted in the present investigation

6 Conclusions

In this paper we investigate the effects of chemical reactionrate and Arrhenius activation energy on an unsteady MHD

10 Journal of Thermodynamics

natural convection heat and mass transfer boundary layerflow past a flat porous plate in presence of thermal radiation

The Nachtsheim and Swigert [18] iteration techniquebased on sixth-order Runge-Kutta and Shooting method hasbeen employed to complete the integration of the resultingsolutions

The following conclusions can be drawn as a result of thecomputations

(1) Velocity increases with increasing values of 120574 for thecooling of the plate and the negative increase in thetemperature relative parameter 120574 leads to the decreasein the velocity fieldThat is for heating of the plate theeffects of the temperature relative parameter 120574 on thevelocity field have also opposite effects as comparedto the cooling of the plate

(2) Solutal Grashof number 119866119898gt 0 corresponds that

the chemical species concentration in the free streamregion is less than the concentration at the bound-ary surface It is observed that the velocity profileincreaseswith the increasing values of SolutalGrashofnumber 119866

119898

(3) Increase in 120582 leads to increase in the velocity andtemperature profiles for exothermic reaction butopposite effects are found for endothermic reactionFor 120573 = plusmn1 increase in 120582 leads to the decrease in theconcentration profiles but for exothermic reactionthe mass boundary layer is close to the plate otherthan endothermic reaction

(4) The velocity profile mark increases with the increas-ing values of 119864 for endothermic chemical reactionBut negligible effect is found for exothermic reactionTemperature decreases for increasing values of 119864 forexothermic reaction but opposite effects are found forendothermic reaction Activation energy (119864) leads toincrease in the concentration profiles but small andreported effects are found for 120573 = 1 and 120573 = minus1respectively

(5) For strong suction (V0gt 0) the velocity the tem-

perature and the concentration profiles decay rapidlyaway from the surface As for the injection (V

0lt 0)

it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayerbetween the injection and the outer flow regions

(6) Imposition of a magnetic field to an electrically con-ducting fluid creates a drag like force called Lorentzforce The force has the tendency to slow down theflow around the plate at the expense of increasing itstemperature This is depicted by decreases in velocityprofiles as magnetic parameter119872 increases

(7) An increase in the values of 119873 leads to a decrease inthe values of temperature profiles within the thermalboundary layers 120578 lt 022 while outside 120578 gt 022the temperature profile gradually increases with theincrease of the radiation parameter119873

(8) For unsteady flow (119860 = 0) the velocity and the tem-perature profiles rapidly decay and close to the plate

But for steady flow (119860 = 0) the profiles overshoot theuniform velocitytemperature close to the boundary

Nomenclature

(119906 V) Components of velocity field119888119901 Specific heat at constant pressure119873119906 Nusselt number

119875119903 Prandtl number1198760 Heat generationabsorption coefficient

119878ℎ Sherwood number119879 Temperature of the flow field119879infin Temperature of the fluid at infinity

119879119908 Temperature at the plate

119891 Dimensionless similarity functions119864119886 The activation energy

119896 The Boltzmann constant119878119888 Schmidt number119864 The nondimensional activation energy119866119903 Grashof number

119866119898 Modified (Solutal) Grashof number

Greek Symbols

120579 Dimensionless temperature120601 Dimensionless concentration120578 Dimensionless similarity variable120575 Scale factorV Kinematic viscosity120588 Density of the fluid120581 Thermal conductivity120591 Shear stress120583 Fluid viscosity1205822 Dimensionless chemical reaction rate

constant120573 and 120573lowast The coefficients of volume expansions for

temperature and concentrationrespectively

120574 The dimensionless heatgenerationabsorption coefficient

References

[1] M Tencer J S Moss and T Zapach ldquoArrhenius averagetemperature the effective temperature for non-fatigue wearoutand long term reliability in variable thermal conditions andclimatesrdquo IEEE Transactions on Components and PackagingTechnologies vol 27 no 3 pp 602ndash607 2004

[2] C Truesdell ldquoSulle basi della thermomeccanicardquo RendicontiLincei vol 22 no 8 pp 33ndash38 1957

[3] N Mills ldquoIncompressible mixtures of newtonian fluidsrdquo Inter-national Journal of Engineering Science vol 4 no 2 pp 97ndash1121966

[4] C E Beevers and R E Craine ldquoOn the determination ofresponse functions for a binary mixture of incompressiblenewtonian fluidsrdquo International Journal of Engineering Sciencevol 20 no 6 pp 737ndash745 1982

Journal of Thermodynamics 11

[5] A Al-Sharif K Chamniprasart K R Rajagopal andA Z SzerildquoLubrication with binary mixtures liquid-liquid emulsionrdquoJournal of Tribology vol 115 no 1 pp 46ndash55 1993

[6] S H Wang A Al-Sharif K R Rajagopal and A Z SzerildquoLubrication with binarymixtures liquid-liquid emulsion in anEHL conjunctionrdquo Journal of Tribology vol 115 no 3 pp 515ndash522 1993

[7] A R Bestman ldquoNatural convection boundary layer withsuction and mass transfer in a porous mediumrdquo InternationalJournal of Energy Research vol 14 no 4 pp 389ndash396 1990

[8] A K Singh and C K Dikshit ldquoHydromagnetic flow pasta continuously moving semi-infinite plate for large suctionrdquoAstrophysics and Space Science vol 148 no 2 pp 249ndash256 1988

[9] A R Bestman ldquoRadiative heat transfer to flow of a combustiblemixture in a vertical piperdquo International Journal of EnergyResearch vol 15 no 3 pp 179ndash184 1991

[10] M A Alabraba A R Bestman and A Ogulu ldquoLaminar con-vection in binary mixture of hydromagnetic flow with radiativeheat transfer Irdquo Astrophysics and Space Science vol 195 no 2pp 431ndash439 1992

[11] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[12] O DMakinde P O Olanrewaju andWM Charles ldquoUnsteadyconvection with chemical reaction and radiative heat transferpast a flat porous plate moving through a binary mixturerdquoAfricka Matematika vol 22 no 1 pp 65ndash78 2011

[13] O D Makinde and P O Olanrewaju ldquoUnsteady mixed convec-tion with Soret and Dufour effects past a porous plate movingthrough a binarymixture of chemically reacting fluidrdquoChemicalEngineering Communications vol 198 no 7 pp 920ndash938 2011

[14] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer flow with mass transfer and a binary chemical reactionrdquoBritish Journal of Applied Science and Technology (Sciencedo-main International) vol 3 no 1 pp 131ndash149 2013

[15] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer heat and mass transfer flow with exothermic chemicalreactionsrdquo Journal of Pure and Applied Mathematics (Advancesand Applications) vol 9 no 1 pp 17ndash41 2013

[16] Kh Abdul Maleque ldquoEffects of binary chemical reactionand activation energy on MHD boundary layer heat andmass transfer flow with viscous dissipation and heat genera-tionabsorptionrdquo ISRN Thermodynamics vol 2013 Article ID284637 9 pages 2013

[17] Kh Abdul Maleque ldquoEffects of combined temperature- anddepth-dependent viscosity and hall current on an unsteadyMHD laminar convective flow due to a rotating diskrdquo ChemicalEngineering Communications vol 197 no 4 pp 506ndash521 2010

[18] P R Nachtsheim and P Swigert ldquoSatisfaction of asymptoticboundary conditions in numerical solution of system of non-linear of boundary layer typerdquo NASA TN-D3004 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Journal of Thermodynamics

Table 2 Effects on skin friction heat transfer and mass transfercoefficients For119866

0= 10119866

119898= 1119873 = 05119875

119903= 071119872 = 05 119878

119888=

06 120582 = 5 119899 = 05 V0= 3 120574 = 05 and 119864 = 1

minus1198911015840

(0) minus1205791015840

(0) minus1206011015840

(0)

119873

00 minus087260 minus144125 56899620 minus305961 minus030027 57162450 minus493466 minus004756 572770

119872

00 minus185010 minus085309 57016420 minus075246 minus085443 57020350 046744 minus085628 570257

120574

minus200000 732171 365294 007835minus100000 566728 348636 026098000000 331180 188820 228076100000 minus154562 minus085343 570174200000 minus785616 minus367839 925776

119860

000000 minus127274 minus026974 397243200000 120419 056519 391549

the bonds than to break the bonds heat is given off This isknown as an exothermic reaction When a reaction requiresan input or output of energy it is known as an exothermic orendothermic reaction

Effects of activation energy (119864) and exothermic param-eter (120573) on the temperature the concentration and velocityprofiles are shown in Figure 1 to Figure 3 respectively 120573 = 1and 120573 = minus1 represent exothermic and endothermic chemicalreactions respectively Figure 1 shows that a small decreasingeffect of temperature profile is found for increasing valuesof 119864 for exothermic reaction (120573 = 1) but mark oppositeeffects are found for endothermic reaction in Figure 1 Thatis the temperature profile increases for increasing values of119864 for endothermic reaction 120573 = minus1 From (1) we observethat chemical reaction rate (119870) decreases with the increasingvalues of activation energy (119864

119886) We also observe from (20)

that increase in activation energy (119864) leads to decrease1205822 exp(minus119864120579) as well as to increase in the concentration

profiles shown in Figure 2 Small and reported effects arefound for 120573 = 1 and 120573 = minus1 respectively The parameter119864 does not enter directly into the momentum equation butits influence comes through the mass and energy equationsFigure 3 shows the variation of the velocity profiles fordifferent values of 119864 From this figure it has been observedthat the velocity profile mark increases with the increasingvalues of 119864 for endothermic chemical reaction But negligibleeffects are found for exothermic reaction

52 Effects of 120582 on the Concentration the Temperature and theVelocity Profiles for ExothermicEndothermic Reaction Con-sidering chemical reaction rate constant 1205822 is always positiveFigures 4ndash6 represent the effect of chemical rate constant120582 on

the velocity the temperature and the concentration profilesrespectively

We observe from Figures 4 and 5 that velocity andtemperature profiles increase with the increasing values of120582 for exothermic reaction but opposite effects are foundin these figures for endothermic reaction It is observedfrom (1) that increasing temperature frequently that causes amarked increase in the rate of reactions is shown in Figure 4As the temperature of the system increases the numberof molecules that carry enough energy to react when theycollide also increasesTherefore the rate of reaction increaseswith temperature As a rule the rate of a reaction doubles forevery 10∘C increase in the temperature of the system

Last part of (20) shows that 1205822 exp(minus119864120579) increases withthe increasing values of 120582We also observe from this equationthat increase in 1205822 exp(minus119864120579)means that increase in 120582 leadsto the decrease in the concentration profiles This is in greatagreement with Figure 6 for both cases 120573 = plusmn1 It isalso observed from this figure that for exothermic reactionthe mass boundary layer is close to the plate other thanendothermic reaction

53 Effects of Radiation Parameter 119873 on Temperature andVelocity Profiles The effects of radiation parameter119873 on thetemperature and the velocity profiles are shown in Figures 7and 8 From Figure 7 it can be seen that an increase in thevalues of 119873 leads to a decrease in the values of temperatureprofiles within the thermal boundary layers 120578 lt 022while outside 120578 gt 022 the temperature profile graduallyincreases with the increase of the radiation parameter 119873 Itis appear from Figure 8 that when 119873 = 0 that is withoutthe radiation the velocity profile shows its usual trend ofgradually decay As radiation becomes larger the profilesovershoot the uniform velocity close to the boundary

54 Effects of 120574 on the Velocity Temperature and Concen-tration Profiles The value of 120574 = Δ119879119879

infinis taken to be

both positive and negative since these values representrespectively cooling and heating of the plate The effects oftemperature relative parameter on velocity temperature andconcentration profiles are shown in Figures 9ndash11

The velocity profiles generated due to impulsive motionof the plate is plotted in Figure 9 for both cooling (120574 gt 0) andheating (120574 lt 0) of the plate keeping other parameters fixed(1198660= 10 119866

119898= 10 119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 V0= 30 119864 = 10 and 119899 = 1) In Figure 9 velocity

profiles are shown for different values of 120574 We observe thatvelocity increases with increasing values of 120574 for the coolingof the plate From this figure it is also observed that thenegative increase in the temperature relative parameter leadsto the decrease in the velocity field That is for heating ofthe plate (Figure 9) the effects of the 120574 on the velocity fieldhave also opposite effects as compared to the cooling ofthe plate Figure 10 shows the effects of temperature relativeparameter 120574 on the temperature profiles It has been observedfrom this figure that for heating plate the temperature profileshows its usual trend of gradually decay and the thermalboundary layer is close to the plate But for cooling plate

Journal of Thermodynamics 9

that is temperature relative parameter 120574 becomes largerthe profiles overshoot the uniform temperature close to thethermal boundary Opposite effects of temperature profile arefound for concentration profiles shown in Figure 11

55 Effects of SuctionInjection (V0) on the Velocity and the

Temperature Profiles The effects of suction and injection (V0)

for 1198660= 10 119866

119898= 10119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 120574 = 05 119864 = 10 and 119899 = 1 on the velocity profilesand temperature profiles are shown respectively in Figures12 and 13 For strong suction (V

0gt 0) the velocity and the

temperature decay rapidly away from the surface The factthat suction stabilizes the boundary layer is also apparentfrom these figures As for the injection (V

0lt 0) from Figures

12 and 13 it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayer between theinjection and the outer flow regions

56 Effects of 119866119898and119872 on the Velocity Profiles The effects

of Solutal Grashof number 119866119898

and magnetic interactionparameter119872 on velocity profiles are shown in Figures 14 and15 respectively Solutal Grashof number 119866

119898gt 0 corresponds

that the chemical species concentration in the free streamregion is less than the concentration at the boundary surfaceFigure 14 presents the effects of Solutal Grashof number119866119898on the velocity profiles It is observed that the velocity

profile increases with the increasing values of Solutal Grashofnumber 119866

119898

Imposition of a magnetic field to an electrically conduct-ing fluid creates a drag like force called Lorentz force Theforce has the tendency to slow down the flow around the plateat the expense of increasing its temperature This is depictedby decreases in velocity profiles as magnetic parameter 119872increases as shown in Figure 15

57 Effects of 119860 on the Velocity and the Temperature ProfilesHere 119860 = 0 and 119860 = 0 represent steady and unsteady flowsrespectively The effects of 119860 on velocity and temperatureprofiles are shown in Figures 16 and 17 respectively Forunsteady flow (119860 = 0) both figures show that the profilesrapidly decay and close to the plate But for steady flow (119860 =0) the profiles overshoot the uniform velocitytemperatureclose to the boundary

58 The Skin-Friction the Heat Transfer and the MassTransfer Coefficients The skin-friction the heat transferand the mass transfer coefficients are tabulated in Tables1 and 2 for different values of 119864 120573 120582 119873 119872 120574 and 119860We observe from Table 1 that the skin-friction coefficient(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) increasefor increasing values of dimensionless activation parameter119864 for exothermic chemical reaction but opposite actionsare found for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients decrease for increasing values of 119864 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) decreases for increasing values of 119864We also observe fromTable 1 that the skin-friction coefficient

0 1 2 3 40

05

1

Bestman (1990)Present

f

120578

Gr = Gm = 1 E = 5 20 = 10

= 071 and 120582 = 5

120573 = 0 120574 = 1 and M= N = A = 0

Sc = 1 Pr

Figure 18 Comparison of our calculated velocity profile and thevelocity profile of Bestman [7]

(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) decreasefor increasing values of dimensionless reaction rate constant120582 for exothermic chemical reaction but opposite effects arefound for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients increase for increasing values of 120582 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) remarkably increases for increasingvalues of 120582 From Table 2 it has been observed that theskin-friction coefficient remarkably decreases and Nusseltnumber remarkable increases for increasing values of radi-ation parameter119873 The skin-friction coefficient increases forincreasing values of magnetic parameter 119872 From Table 2we also found that skin friction and heat transfer coefficientsdecrease butmass transfer coefficient increases for increasingvalues of temperature relative parameter 120574

59 Comparison between Our Numerical Result and theAnalytical Result of Bestman [7] Bestman studied steadynatural convective boundary layer flow with large suctionHe solved his problem analytically by employing the per-turbation technique proposed by Singh and Dikshit [8] Inour present work take 119860 = 0 in (16) for considering steadyflow The values of the suction parameter V2

0= 10 is taken

to see the effects of large suction 119866119903= 119866119898= 10 120574 = 1

119872 = 119873 = 0 120582 = 5 119864 = 5 and 119899 = 1 are also chosen witha view to compare our numerical results with the analyticalresults of Bestman [7] The comparison of velocity profilesas seen in Figure 18 highlights the validity of the numericalcomputations adapted in the present investigation

6 Conclusions

In this paper we investigate the effects of chemical reactionrate and Arrhenius activation energy on an unsteady MHD

10 Journal of Thermodynamics

natural convection heat and mass transfer boundary layerflow past a flat porous plate in presence of thermal radiation

The Nachtsheim and Swigert [18] iteration techniquebased on sixth-order Runge-Kutta and Shooting method hasbeen employed to complete the integration of the resultingsolutions

The following conclusions can be drawn as a result of thecomputations

(1) Velocity increases with increasing values of 120574 for thecooling of the plate and the negative increase in thetemperature relative parameter 120574 leads to the decreasein the velocity fieldThat is for heating of the plate theeffects of the temperature relative parameter 120574 on thevelocity field have also opposite effects as comparedto the cooling of the plate

(2) Solutal Grashof number 119866119898gt 0 corresponds that

the chemical species concentration in the free streamregion is less than the concentration at the bound-ary surface It is observed that the velocity profileincreaseswith the increasing values of SolutalGrashofnumber 119866

119898

(3) Increase in 120582 leads to increase in the velocity andtemperature profiles for exothermic reaction butopposite effects are found for endothermic reactionFor 120573 = plusmn1 increase in 120582 leads to the decrease in theconcentration profiles but for exothermic reactionthe mass boundary layer is close to the plate otherthan endothermic reaction

(4) The velocity profile mark increases with the increas-ing values of 119864 for endothermic chemical reactionBut negligible effect is found for exothermic reactionTemperature decreases for increasing values of 119864 forexothermic reaction but opposite effects are found forendothermic reaction Activation energy (119864) leads toincrease in the concentration profiles but small andreported effects are found for 120573 = 1 and 120573 = minus1respectively

(5) For strong suction (V0gt 0) the velocity the tem-

perature and the concentration profiles decay rapidlyaway from the surface As for the injection (V

0lt 0)

it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayerbetween the injection and the outer flow regions

(6) Imposition of a magnetic field to an electrically con-ducting fluid creates a drag like force called Lorentzforce The force has the tendency to slow down theflow around the plate at the expense of increasing itstemperature This is depicted by decreases in velocityprofiles as magnetic parameter119872 increases

(7) An increase in the values of 119873 leads to a decrease inthe values of temperature profiles within the thermalboundary layers 120578 lt 022 while outside 120578 gt 022the temperature profile gradually increases with theincrease of the radiation parameter119873

(8) For unsteady flow (119860 = 0) the velocity and the tem-perature profiles rapidly decay and close to the plate

But for steady flow (119860 = 0) the profiles overshoot theuniform velocitytemperature close to the boundary

Nomenclature

(119906 V) Components of velocity field119888119901 Specific heat at constant pressure119873119906 Nusselt number

119875119903 Prandtl number1198760 Heat generationabsorption coefficient

119878ℎ Sherwood number119879 Temperature of the flow field119879infin Temperature of the fluid at infinity

119879119908 Temperature at the plate

119891 Dimensionless similarity functions119864119886 The activation energy

119896 The Boltzmann constant119878119888 Schmidt number119864 The nondimensional activation energy119866119903 Grashof number

119866119898 Modified (Solutal) Grashof number

Greek Symbols

120579 Dimensionless temperature120601 Dimensionless concentration120578 Dimensionless similarity variable120575 Scale factorV Kinematic viscosity120588 Density of the fluid120581 Thermal conductivity120591 Shear stress120583 Fluid viscosity1205822 Dimensionless chemical reaction rate

constant120573 and 120573lowast The coefficients of volume expansions for

temperature and concentrationrespectively

120574 The dimensionless heatgenerationabsorption coefficient

References

[1] M Tencer J S Moss and T Zapach ldquoArrhenius averagetemperature the effective temperature for non-fatigue wearoutand long term reliability in variable thermal conditions andclimatesrdquo IEEE Transactions on Components and PackagingTechnologies vol 27 no 3 pp 602ndash607 2004

[2] C Truesdell ldquoSulle basi della thermomeccanicardquo RendicontiLincei vol 22 no 8 pp 33ndash38 1957

[3] N Mills ldquoIncompressible mixtures of newtonian fluidsrdquo Inter-national Journal of Engineering Science vol 4 no 2 pp 97ndash1121966

[4] C E Beevers and R E Craine ldquoOn the determination ofresponse functions for a binary mixture of incompressiblenewtonian fluidsrdquo International Journal of Engineering Sciencevol 20 no 6 pp 737ndash745 1982

Journal of Thermodynamics 11

[5] A Al-Sharif K Chamniprasart K R Rajagopal andA Z SzerildquoLubrication with binary mixtures liquid-liquid emulsionrdquoJournal of Tribology vol 115 no 1 pp 46ndash55 1993

[6] S H Wang A Al-Sharif K R Rajagopal and A Z SzerildquoLubrication with binarymixtures liquid-liquid emulsion in anEHL conjunctionrdquo Journal of Tribology vol 115 no 3 pp 515ndash522 1993

[7] A R Bestman ldquoNatural convection boundary layer withsuction and mass transfer in a porous mediumrdquo InternationalJournal of Energy Research vol 14 no 4 pp 389ndash396 1990

[8] A K Singh and C K Dikshit ldquoHydromagnetic flow pasta continuously moving semi-infinite plate for large suctionrdquoAstrophysics and Space Science vol 148 no 2 pp 249ndash256 1988

[9] A R Bestman ldquoRadiative heat transfer to flow of a combustiblemixture in a vertical piperdquo International Journal of EnergyResearch vol 15 no 3 pp 179ndash184 1991

[10] M A Alabraba A R Bestman and A Ogulu ldquoLaminar con-vection in binary mixture of hydromagnetic flow with radiativeheat transfer Irdquo Astrophysics and Space Science vol 195 no 2pp 431ndash439 1992

[11] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[12] O DMakinde P O Olanrewaju andWM Charles ldquoUnsteadyconvection with chemical reaction and radiative heat transferpast a flat porous plate moving through a binary mixturerdquoAfricka Matematika vol 22 no 1 pp 65ndash78 2011

[13] O D Makinde and P O Olanrewaju ldquoUnsteady mixed convec-tion with Soret and Dufour effects past a porous plate movingthrough a binarymixture of chemically reacting fluidrdquoChemicalEngineering Communications vol 198 no 7 pp 920ndash938 2011

[14] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer flow with mass transfer and a binary chemical reactionrdquoBritish Journal of Applied Science and Technology (Sciencedo-main International) vol 3 no 1 pp 131ndash149 2013

[15] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer heat and mass transfer flow with exothermic chemicalreactionsrdquo Journal of Pure and Applied Mathematics (Advancesand Applications) vol 9 no 1 pp 17ndash41 2013

[16] Kh Abdul Maleque ldquoEffects of binary chemical reactionand activation energy on MHD boundary layer heat andmass transfer flow with viscous dissipation and heat genera-tionabsorptionrdquo ISRN Thermodynamics vol 2013 Article ID284637 9 pages 2013

[17] Kh Abdul Maleque ldquoEffects of combined temperature- anddepth-dependent viscosity and hall current on an unsteadyMHD laminar convective flow due to a rotating diskrdquo ChemicalEngineering Communications vol 197 no 4 pp 506ndash521 2010

[18] P R Nachtsheim and P Swigert ldquoSatisfaction of asymptoticboundary conditions in numerical solution of system of non-linear of boundary layer typerdquo NASA TN-D3004 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Thermodynamics 9

that is temperature relative parameter 120574 becomes largerthe profiles overshoot the uniform temperature close to thethermal boundary Opposite effects of temperature profile arefound for concentration profiles shown in Figure 11

55 Effects of SuctionInjection (V0) on the Velocity and the

Temperature Profiles The effects of suction and injection (V0)

for 1198660= 10 119866

119898= 10119872 = 119864

119888= 05 119875

119903= 071 119878

119888= 06

120582 = 5 120574 = 05 119864 = 10 and 119899 = 1 on the velocity profilesand temperature profiles are shown respectively in Figures12 and 13 For strong suction (V

0gt 0) the velocity and the

temperature decay rapidly away from the surface The factthat suction stabilizes the boundary layer is also apparentfrom these figures As for the injection (V

0lt 0) from Figures

12 and 13 it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayer between theinjection and the outer flow regions

56 Effects of 119866119898and119872 on the Velocity Profiles The effects

of Solutal Grashof number 119866119898

and magnetic interactionparameter119872 on velocity profiles are shown in Figures 14 and15 respectively Solutal Grashof number 119866

119898gt 0 corresponds

that the chemical species concentration in the free streamregion is less than the concentration at the boundary surfaceFigure 14 presents the effects of Solutal Grashof number119866119898on the velocity profiles It is observed that the velocity

profile increases with the increasing values of Solutal Grashofnumber 119866

119898

Imposition of a magnetic field to an electrically conduct-ing fluid creates a drag like force called Lorentz force Theforce has the tendency to slow down the flow around the plateat the expense of increasing its temperature This is depictedby decreases in velocity profiles as magnetic parameter 119872increases as shown in Figure 15

57 Effects of 119860 on the Velocity and the Temperature ProfilesHere 119860 = 0 and 119860 = 0 represent steady and unsteady flowsrespectively The effects of 119860 on velocity and temperatureprofiles are shown in Figures 16 and 17 respectively Forunsteady flow (119860 = 0) both figures show that the profilesrapidly decay and close to the plate But for steady flow (119860 =0) the profiles overshoot the uniform velocitytemperatureclose to the boundary

58 The Skin-Friction the Heat Transfer and the MassTransfer Coefficients The skin-friction the heat transferand the mass transfer coefficients are tabulated in Tables1 and 2 for different values of 119864 120573 120582 119873 119872 120574 and 119860We observe from Table 1 that the skin-friction coefficient(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) increasefor increasing values of dimensionless activation parameter119864 for exothermic chemical reaction but opposite actionsare found for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients decrease for increasing values of 119864 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) decreases for increasing values of 119864We also observe fromTable 1 that the skin-friction coefficient

0 1 2 3 40

05

1

Bestman (1990)Present

f

120578

Gr = Gm = 1 E = 5 20 = 10

= 071 and 120582 = 5

120573 = 0 120574 = 1 and M= N = A = 0

Sc = 1 Pr

Figure 18 Comparison of our calculated velocity profile and thevelocity profile of Bestman [7]

(minus1198911015840

(0)) and the Nusselt number 119873119906(= minus120579

1015840

(0)) decreasefor increasing values of dimensionless reaction rate constant120582 for exothermic chemical reaction but opposite effects arefound for endothermic reaction That is for endothermicreaction it is found that both shearing stress and heat transfercoefficients increase for increasing values of 120582 For bothcases exothermicendothermic reactions the mass transfercoefficient (= minus1206011015840(0)) remarkably increases for increasingvalues of 120582 From Table 2 it has been observed that theskin-friction coefficient remarkably decreases and Nusseltnumber remarkable increases for increasing values of radi-ation parameter119873 The skin-friction coefficient increases forincreasing values of magnetic parameter 119872 From Table 2we also found that skin friction and heat transfer coefficientsdecrease butmass transfer coefficient increases for increasingvalues of temperature relative parameter 120574

59 Comparison between Our Numerical Result and theAnalytical Result of Bestman [7] Bestman studied steadynatural convective boundary layer flow with large suctionHe solved his problem analytically by employing the per-turbation technique proposed by Singh and Dikshit [8] Inour present work take 119860 = 0 in (16) for considering steadyflow The values of the suction parameter V2

0= 10 is taken

to see the effects of large suction 119866119903= 119866119898= 10 120574 = 1

119872 = 119873 = 0 120582 = 5 119864 = 5 and 119899 = 1 are also chosen witha view to compare our numerical results with the analyticalresults of Bestman [7] The comparison of velocity profilesas seen in Figure 18 highlights the validity of the numericalcomputations adapted in the present investigation

6 Conclusions

In this paper we investigate the effects of chemical reactionrate and Arrhenius activation energy on an unsteady MHD

10 Journal of Thermodynamics

natural convection heat and mass transfer boundary layerflow past a flat porous plate in presence of thermal radiation

The Nachtsheim and Swigert [18] iteration techniquebased on sixth-order Runge-Kutta and Shooting method hasbeen employed to complete the integration of the resultingsolutions

The following conclusions can be drawn as a result of thecomputations

(1) Velocity increases with increasing values of 120574 for thecooling of the plate and the negative increase in thetemperature relative parameter 120574 leads to the decreasein the velocity fieldThat is for heating of the plate theeffects of the temperature relative parameter 120574 on thevelocity field have also opposite effects as comparedto the cooling of the plate

(2) Solutal Grashof number 119866119898gt 0 corresponds that

the chemical species concentration in the free streamregion is less than the concentration at the bound-ary surface It is observed that the velocity profileincreaseswith the increasing values of SolutalGrashofnumber 119866

119898

(3) Increase in 120582 leads to increase in the velocity andtemperature profiles for exothermic reaction butopposite effects are found for endothermic reactionFor 120573 = plusmn1 increase in 120582 leads to the decrease in theconcentration profiles but for exothermic reactionthe mass boundary layer is close to the plate otherthan endothermic reaction

(4) The velocity profile mark increases with the increas-ing values of 119864 for endothermic chemical reactionBut negligible effect is found for exothermic reactionTemperature decreases for increasing values of 119864 forexothermic reaction but opposite effects are found forendothermic reaction Activation energy (119864) leads toincrease in the concentration profiles but small andreported effects are found for 120573 = 1 and 120573 = minus1respectively

(5) For strong suction (V0gt 0) the velocity the tem-

perature and the concentration profiles decay rapidlyaway from the surface As for the injection (V

0lt 0)

it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayerbetween the injection and the outer flow regions

(6) Imposition of a magnetic field to an electrically con-ducting fluid creates a drag like force called Lorentzforce The force has the tendency to slow down theflow around the plate at the expense of increasing itstemperature This is depicted by decreases in velocityprofiles as magnetic parameter119872 increases

(7) An increase in the values of 119873 leads to a decrease inthe values of temperature profiles within the thermalboundary layers 120578 lt 022 while outside 120578 gt 022the temperature profile gradually increases with theincrease of the radiation parameter119873

(8) For unsteady flow (119860 = 0) the velocity and the tem-perature profiles rapidly decay and close to the plate

But for steady flow (119860 = 0) the profiles overshoot theuniform velocitytemperature close to the boundary

Nomenclature

(119906 V) Components of velocity field119888119901 Specific heat at constant pressure119873119906 Nusselt number

119875119903 Prandtl number1198760 Heat generationabsorption coefficient

119878ℎ Sherwood number119879 Temperature of the flow field119879infin Temperature of the fluid at infinity

119879119908 Temperature at the plate

119891 Dimensionless similarity functions119864119886 The activation energy

119896 The Boltzmann constant119878119888 Schmidt number119864 The nondimensional activation energy119866119903 Grashof number

119866119898 Modified (Solutal) Grashof number

Greek Symbols

120579 Dimensionless temperature120601 Dimensionless concentration120578 Dimensionless similarity variable120575 Scale factorV Kinematic viscosity120588 Density of the fluid120581 Thermal conductivity120591 Shear stress120583 Fluid viscosity1205822 Dimensionless chemical reaction rate

constant120573 and 120573lowast The coefficients of volume expansions for

temperature and concentrationrespectively

120574 The dimensionless heatgenerationabsorption coefficient

References

[1] M Tencer J S Moss and T Zapach ldquoArrhenius averagetemperature the effective temperature for non-fatigue wearoutand long term reliability in variable thermal conditions andclimatesrdquo IEEE Transactions on Components and PackagingTechnologies vol 27 no 3 pp 602ndash607 2004

[2] C Truesdell ldquoSulle basi della thermomeccanicardquo RendicontiLincei vol 22 no 8 pp 33ndash38 1957

[3] N Mills ldquoIncompressible mixtures of newtonian fluidsrdquo Inter-national Journal of Engineering Science vol 4 no 2 pp 97ndash1121966

[4] C E Beevers and R E Craine ldquoOn the determination ofresponse functions for a binary mixture of incompressiblenewtonian fluidsrdquo International Journal of Engineering Sciencevol 20 no 6 pp 737ndash745 1982

Journal of Thermodynamics 11

[5] A Al-Sharif K Chamniprasart K R Rajagopal andA Z SzerildquoLubrication with binary mixtures liquid-liquid emulsionrdquoJournal of Tribology vol 115 no 1 pp 46ndash55 1993

[6] S H Wang A Al-Sharif K R Rajagopal and A Z SzerildquoLubrication with binarymixtures liquid-liquid emulsion in anEHL conjunctionrdquo Journal of Tribology vol 115 no 3 pp 515ndash522 1993

[7] A R Bestman ldquoNatural convection boundary layer withsuction and mass transfer in a porous mediumrdquo InternationalJournal of Energy Research vol 14 no 4 pp 389ndash396 1990

[8] A K Singh and C K Dikshit ldquoHydromagnetic flow pasta continuously moving semi-infinite plate for large suctionrdquoAstrophysics and Space Science vol 148 no 2 pp 249ndash256 1988

[9] A R Bestman ldquoRadiative heat transfer to flow of a combustiblemixture in a vertical piperdquo International Journal of EnergyResearch vol 15 no 3 pp 179ndash184 1991

[10] M A Alabraba A R Bestman and A Ogulu ldquoLaminar con-vection in binary mixture of hydromagnetic flow with radiativeheat transfer Irdquo Astrophysics and Space Science vol 195 no 2pp 431ndash439 1992

[11] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[12] O DMakinde P O Olanrewaju andWM Charles ldquoUnsteadyconvection with chemical reaction and radiative heat transferpast a flat porous plate moving through a binary mixturerdquoAfricka Matematika vol 22 no 1 pp 65ndash78 2011

[13] O D Makinde and P O Olanrewaju ldquoUnsteady mixed convec-tion with Soret and Dufour effects past a porous plate movingthrough a binarymixture of chemically reacting fluidrdquoChemicalEngineering Communications vol 198 no 7 pp 920ndash938 2011

[14] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer flow with mass transfer and a binary chemical reactionrdquoBritish Journal of Applied Science and Technology (Sciencedo-main International) vol 3 no 1 pp 131ndash149 2013

[15] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer heat and mass transfer flow with exothermic chemicalreactionsrdquo Journal of Pure and Applied Mathematics (Advancesand Applications) vol 9 no 1 pp 17ndash41 2013

[16] Kh Abdul Maleque ldquoEffects of binary chemical reactionand activation energy on MHD boundary layer heat andmass transfer flow with viscous dissipation and heat genera-tionabsorptionrdquo ISRN Thermodynamics vol 2013 Article ID284637 9 pages 2013

[17] Kh Abdul Maleque ldquoEffects of combined temperature- anddepth-dependent viscosity and hall current on an unsteadyMHD laminar convective flow due to a rotating diskrdquo ChemicalEngineering Communications vol 197 no 4 pp 506ndash521 2010

[18] P R Nachtsheim and P Swigert ldquoSatisfaction of asymptoticboundary conditions in numerical solution of system of non-linear of boundary layer typerdquo NASA TN-D3004 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

10 Journal of Thermodynamics

natural convection heat and mass transfer boundary layerflow past a flat porous plate in presence of thermal radiation

The Nachtsheim and Swigert [18] iteration techniquebased on sixth-order Runge-Kutta and Shooting method hasbeen employed to complete the integration of the resultingsolutions

The following conclusions can be drawn as a result of thecomputations

(1) Velocity increases with increasing values of 120574 for thecooling of the plate and the negative increase in thetemperature relative parameter 120574 leads to the decreasein the velocity fieldThat is for heating of the plate theeffects of the temperature relative parameter 120574 on thevelocity field have also opposite effects as comparedto the cooling of the plate

(2) Solutal Grashof number 119866119898gt 0 corresponds that

the chemical species concentration in the free streamregion is less than the concentration at the bound-ary surface It is observed that the velocity profileincreaseswith the increasing values of SolutalGrashofnumber 119866

119898

(3) Increase in 120582 leads to increase in the velocity andtemperature profiles for exothermic reaction butopposite effects are found for endothermic reactionFor 120573 = plusmn1 increase in 120582 leads to the decrease in theconcentration profiles but for exothermic reactionthe mass boundary layer is close to the plate otherthan endothermic reaction

(4) The velocity profile mark increases with the increas-ing values of 119864 for endothermic chemical reactionBut negligible effect is found for exothermic reactionTemperature decreases for increasing values of 119864 forexothermic reaction but opposite effects are found forendothermic reaction Activation energy (119864) leads toincrease in the concentration profiles but small andreported effects are found for 120573 = 1 and 120573 = minus1respectively

(5) For strong suction (V0gt 0) the velocity the tem-

perature and the concentration profiles decay rapidlyaway from the surface As for the injection (V

0lt 0)

it is observed that the boundary layer is increasinglyblown away from the plate to form an interlayerbetween the injection and the outer flow regions

(6) Imposition of a magnetic field to an electrically con-ducting fluid creates a drag like force called Lorentzforce The force has the tendency to slow down theflow around the plate at the expense of increasing itstemperature This is depicted by decreases in velocityprofiles as magnetic parameter119872 increases

(7) An increase in the values of 119873 leads to a decrease inthe values of temperature profiles within the thermalboundary layers 120578 lt 022 while outside 120578 gt 022the temperature profile gradually increases with theincrease of the radiation parameter119873

(8) For unsteady flow (119860 = 0) the velocity and the tem-perature profiles rapidly decay and close to the plate

But for steady flow (119860 = 0) the profiles overshoot theuniform velocitytemperature close to the boundary

Nomenclature

(119906 V) Components of velocity field119888119901 Specific heat at constant pressure119873119906 Nusselt number

119875119903 Prandtl number1198760 Heat generationabsorption coefficient

119878ℎ Sherwood number119879 Temperature of the flow field119879infin Temperature of the fluid at infinity

119879119908 Temperature at the plate

119891 Dimensionless similarity functions119864119886 The activation energy

119896 The Boltzmann constant119878119888 Schmidt number119864 The nondimensional activation energy119866119903 Grashof number

119866119898 Modified (Solutal) Grashof number

Greek Symbols

120579 Dimensionless temperature120601 Dimensionless concentration120578 Dimensionless similarity variable120575 Scale factorV Kinematic viscosity120588 Density of the fluid120581 Thermal conductivity120591 Shear stress120583 Fluid viscosity1205822 Dimensionless chemical reaction rate

constant120573 and 120573lowast The coefficients of volume expansions for

temperature and concentrationrespectively

120574 The dimensionless heatgenerationabsorption coefficient

References

[1] M Tencer J S Moss and T Zapach ldquoArrhenius averagetemperature the effective temperature for non-fatigue wearoutand long term reliability in variable thermal conditions andclimatesrdquo IEEE Transactions on Components and PackagingTechnologies vol 27 no 3 pp 602ndash607 2004

[2] C Truesdell ldquoSulle basi della thermomeccanicardquo RendicontiLincei vol 22 no 8 pp 33ndash38 1957

[3] N Mills ldquoIncompressible mixtures of newtonian fluidsrdquo Inter-national Journal of Engineering Science vol 4 no 2 pp 97ndash1121966

[4] C E Beevers and R E Craine ldquoOn the determination ofresponse functions for a binary mixture of incompressiblenewtonian fluidsrdquo International Journal of Engineering Sciencevol 20 no 6 pp 737ndash745 1982

Journal of Thermodynamics 11

[5] A Al-Sharif K Chamniprasart K R Rajagopal andA Z SzerildquoLubrication with binary mixtures liquid-liquid emulsionrdquoJournal of Tribology vol 115 no 1 pp 46ndash55 1993

[6] S H Wang A Al-Sharif K R Rajagopal and A Z SzerildquoLubrication with binarymixtures liquid-liquid emulsion in anEHL conjunctionrdquo Journal of Tribology vol 115 no 3 pp 515ndash522 1993

[7] A R Bestman ldquoNatural convection boundary layer withsuction and mass transfer in a porous mediumrdquo InternationalJournal of Energy Research vol 14 no 4 pp 389ndash396 1990

[8] A K Singh and C K Dikshit ldquoHydromagnetic flow pasta continuously moving semi-infinite plate for large suctionrdquoAstrophysics and Space Science vol 148 no 2 pp 249ndash256 1988

[9] A R Bestman ldquoRadiative heat transfer to flow of a combustiblemixture in a vertical piperdquo International Journal of EnergyResearch vol 15 no 3 pp 179ndash184 1991

[10] M A Alabraba A R Bestman and A Ogulu ldquoLaminar con-vection in binary mixture of hydromagnetic flow with radiativeheat transfer Irdquo Astrophysics and Space Science vol 195 no 2pp 431ndash439 1992

[11] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[12] O DMakinde P O Olanrewaju andWM Charles ldquoUnsteadyconvection with chemical reaction and radiative heat transferpast a flat porous plate moving through a binary mixturerdquoAfricka Matematika vol 22 no 1 pp 65ndash78 2011

[13] O D Makinde and P O Olanrewaju ldquoUnsteady mixed convec-tion with Soret and Dufour effects past a porous plate movingthrough a binarymixture of chemically reacting fluidrdquoChemicalEngineering Communications vol 198 no 7 pp 920ndash938 2011

[14] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer flow with mass transfer and a binary chemical reactionrdquoBritish Journal of Applied Science and Technology (Sciencedo-main International) vol 3 no 1 pp 131ndash149 2013

[15] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer heat and mass transfer flow with exothermic chemicalreactionsrdquo Journal of Pure and Applied Mathematics (Advancesand Applications) vol 9 no 1 pp 17ndash41 2013

[16] Kh Abdul Maleque ldquoEffects of binary chemical reactionand activation energy on MHD boundary layer heat andmass transfer flow with viscous dissipation and heat genera-tionabsorptionrdquo ISRN Thermodynamics vol 2013 Article ID284637 9 pages 2013

[17] Kh Abdul Maleque ldquoEffects of combined temperature- anddepth-dependent viscosity and hall current on an unsteadyMHD laminar convective flow due to a rotating diskrdquo ChemicalEngineering Communications vol 197 no 4 pp 506ndash521 2010

[18] P R Nachtsheim and P Swigert ldquoSatisfaction of asymptoticboundary conditions in numerical solution of system of non-linear of boundary layer typerdquo NASA TN-D3004 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Thermodynamics 11

[5] A Al-Sharif K Chamniprasart K R Rajagopal andA Z SzerildquoLubrication with binary mixtures liquid-liquid emulsionrdquoJournal of Tribology vol 115 no 1 pp 46ndash55 1993

[6] S H Wang A Al-Sharif K R Rajagopal and A Z SzerildquoLubrication with binarymixtures liquid-liquid emulsion in anEHL conjunctionrdquo Journal of Tribology vol 115 no 3 pp 515ndash522 1993

[7] A R Bestman ldquoNatural convection boundary layer withsuction and mass transfer in a porous mediumrdquo InternationalJournal of Energy Research vol 14 no 4 pp 389ndash396 1990

[8] A K Singh and C K Dikshit ldquoHydromagnetic flow pasta continuously moving semi-infinite plate for large suctionrdquoAstrophysics and Space Science vol 148 no 2 pp 249ndash256 1988

[9] A R Bestman ldquoRadiative heat transfer to flow of a combustiblemixture in a vertical piperdquo International Journal of EnergyResearch vol 15 no 3 pp 179ndash184 1991

[10] M A Alabraba A R Bestman and A Ogulu ldquoLaminar con-vection in binary mixture of hydromagnetic flow with radiativeheat transfer Irdquo Astrophysics and Space Science vol 195 no 2pp 431ndash439 1992

[11] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[12] O DMakinde P O Olanrewaju andWM Charles ldquoUnsteadyconvection with chemical reaction and radiative heat transferpast a flat porous plate moving through a binary mixturerdquoAfricka Matematika vol 22 no 1 pp 65ndash78 2011

[13] O D Makinde and P O Olanrewaju ldquoUnsteady mixed convec-tion with Soret and Dufour effects past a porous plate movingthrough a binarymixture of chemically reacting fluidrdquoChemicalEngineering Communications vol 198 no 7 pp 920ndash938 2011

[14] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer flow with mass transfer and a binary chemical reactionrdquoBritish Journal of Applied Science and Technology (Sciencedo-main International) vol 3 no 1 pp 131ndash149 2013

[15] Kh Abdul Maleque ldquoUnsteady natural convection boundarylayer heat and mass transfer flow with exothermic chemicalreactionsrdquo Journal of Pure and Applied Mathematics (Advancesand Applications) vol 9 no 1 pp 17ndash41 2013

[16] Kh Abdul Maleque ldquoEffects of binary chemical reactionand activation energy on MHD boundary layer heat andmass transfer flow with viscous dissipation and heat genera-tionabsorptionrdquo ISRN Thermodynamics vol 2013 Article ID284637 9 pages 2013

[17] Kh Abdul Maleque ldquoEffects of combined temperature- anddepth-dependent viscosity and hall current on an unsteadyMHD laminar convective flow due to a rotating diskrdquo ChemicalEngineering Communications vol 197 no 4 pp 506ndash521 2010

[18] P R Nachtsheim and P Swigert ldquoSatisfaction of asymptoticboundary conditions in numerical solution of system of non-linear of boundary layer typerdquo NASA TN-D3004 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of