EFFECTS OF MASS TRANSFER ON FLOW PASTAN IMPULSIVELY STARTED INFINITE VERTICALPLATE WITH NEWTONIAN HEATING AND CHEMICALREACTION

V. Rajesh UDC 536.25

An exact solution to the problem of flow due to the impulsive motion of an infinite vertical plate in its ownplane in the presence of i) species concentration, ii) Newtonian heating at the plate, and iii) first-orderchemical reaction has been derived by the Laplace transform technique. The influence of various parametersentering into the problem on the velocity field and skin friction for both air and water in the cases of bothcooling and heating of the plate is discussed.

Keywords: natural convection, incompressible fluid, heat and mass transfer, Newtonian heating, chemicalreaction.

Introduction. In many transport processes in both nature and industrial applications, heat and mass transfer isa consequence of buoyancy effects caused by diffusion of heat and chemical species. The study of such processes isuseful for improving a number of chemical technologies, such as polymer production and food processing. In nature,the existence of pure air or water is impossible, and some foreign mass may be present either naturally or mixed withthem. A large number of research works concerning transfer processes with chemical reactions have been reported. Inparticular, the study of chemical reaction and heat and mass transfer is of considerable importance in chemical and hy-drometallurgical industries. Chambre and Young [1] analyzed a first-order chemical reaction in the neighborhood of ahorizontal plate. Dass, Deka, and Soundalgekar [2] studied the effect of a homogeneous first-order chemical reactionon flow past an impulsively started infinite vertical plate with a uniform heat flux and mass transfer. The mass transfereffects on a moving isothermal vertical plate in the presence of chemical reaction was considered in [3] where the di-mensionless governing equations were solved by the usual Laplace transform technique. Muthucumaraswamy [4] stud-ied the effects of chemical reaction on a moving isothermal vertical plate with variable mass diffusion. A theoreticalstudy of the chemical reaction effects on a vertical oscillating plate with variable temperature was given in [5, 6]. Ra-jesh and Varma [7] considered the chemical reaction effects on a free convection flow past an exponentially acceler-ated vertical plate. Rajesh [8] studied the MHD and chemical reaction effects on a free convection flow with variabletemperature and mass diffusion. Moreover, the radiation and chemical reaction effects on flow past a vertical platewith ramped wall temperature was presented in [9].

In all the studies cited above the flow is driven by either a prescribed surface temperature or a prescribed sur-face heat flux. Here, a somewhat different driving mechanism for unsteady free convection along a vertical surface isconsidered where it is assumed that the flow is caused by Newtonian heating from the surface. Heat transfer charac-teristics are dependent on the thermal boundary conditions. In general, there are four types of boundary conditions forheating processes specifying the wall-to-ambient temperature distributions, namely, prescribed wall temperature distribu-tion (power law distribution along the surface is usually used); prescribed surface heat flux distribution; conjugate con-ditions where heat is supplied through a bounding surface of finite thickness and finite heat capacity and the interfacetemperature is not known a priori but depends on the intrinsic properties of the system, namely, the thermal conduc-tivities of the fluid and solid; Newtonian heating where the heat transfer rate from the bounding surface with a finiteheat capacity is proportional to the local surface temperature. The last-named case occurs in many important engineer-ing devices, for example,

Journal of Engineering Physics and Thermophysics, Vol. 85, No. 1, January, 2012

1062-0125/12/8501-0221�2012 Springer Science+Business Media, Inc. 221

Department of Engineering Mathematics, GITAM University, Hyderabad Campus, Hyderabad-502 329, A.P.,India; email: v.rajesh.30@gmail.com. Published in Inzhenerno-Fizicheskii Zhurnal, Vol. 85, No. 1, pp. 205–211, Janu-ary–February, 2012. Original article submitted January 3, 2011; revision submitted October 17, 2011.

(i) in heat exchangers where conduction in a solid tube wall is greatly influenced by the convection in thefluid flowing over it;

(ii) in conjugate heat transfer around fins where conduction within the fin and convection in the fluid sur-rounding it must be simultaneously analyzed in order to obtain vital design information;

(iii) in convective flow setup where the bounding surfaces absorb heat of solar radiation.Therefore, we conclude that the conventional assumption on the absence of interaction of conduction-convec-

tion coupled effects is not always realistic, and this interaction must be considered in evaluating the conjugate heattransfer processes in many practical engineering applications. Merkin [10] was the first to consider the free convectionboundary layer over a vertical flat plate immersed in a viscous fluid, and then the authors of the works [11–13] con-sidered the cases of vertical and horizontal surfaces embedded in a porous medium. The studies [10–13] deal withsteady free convection. In the present paper, the effects of mass transfer on the unsteady free convection boundary-layer flow past a vertical plate with Newtonian heating and chemical reaction are considered. The dimensionless gov-erning equations are solved using the Laplace transform technique [14]. The literature concerning this subject can befound in [15, 16].

Mathematical Analysis. The unsteady free convection flow of a viscous incompressible fluid past an impul-sively started infinite vertical plate with Newtonian heating and mass transfer is considered. The x′ axis is taken alongthe plate in the vertical upward direction and the y′ axis is chosen to be normal to the plate. Initially, for time t′ ≤ 0,the plate and fluid are at the same temperature T∞′ in a stationary state with the concentration C∞′ at all points. At timet′ > 0, the plate is set into impulsive motion in the vertical upward direction against the gravitational field with a char-acteristic velocity u0. It is assumed that the rate of heat transfer from the surface is proportional to the local surfacetemperature T ′, and the concentration near the plate is raised to Cw′ . It is also assumed that the effect of viscous dis-sipation in the energy equation is negligible and there is a first-order chemical reaction between the diffusing speciesand the fluid. Since the plate is considered infinite in the x′ direction, all physical variables are independent of x′ andare functions of y′ and t′ only. In the Boussinesq approximation, the flow is governed by the following equations:

∂u′∂t′

= gβ (T ′ − T∞′) + gβ∗

(C ′ − C∞′) + ν

∂2u′

∂y′2 , (1)

ρCp ∂T ′∂t′

= κ ∂2

T ′

∂y′2 , (2)

∂C ′∂t′

= D ∂2

C ′

∂y′2 − Kr (C ′ − C∞

′) (3)

with initial and boundary conditions

t′ ≤ 0 : u′ =0 , T ′ = T∞′ , C ′ = C∞

′ for all y′ ;

t′ > 0 : u′ = u0 , ∂T ′∂y′

= − hκ

T ′ , C ′ = Cw′ at y′ = 0 ,

u′ = 0 , T ′ → T∞′ , C ′ → C∞

′ as y′ → ∞ .

(4)

We introduce the following dimensionless quantities:

u = u′

u0

, t = t′u0

2

ν , y =

y′u0

ν , Pr =

μCp

κ , Gr =

gβνT∞′

u03 , θ =

T ′ − T∞′

T∞′

, Sc = ν

D ,

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Gm = gβ∗ν (Cw

′ − C∞′)

u03 , C =

C ′ − C∞′

Cw′ − C∞

′ , K =

Krν

u02 .

(5)

Substituting Eqs. (5) into Eqs. (1)–(4), we obtain the following dimensionless equations:

∂u

∂t = Gr θ + Gm C +

∂2u

∂y2 , (6)

∂θ∂t

= 1

Pr ∂2θ∂y

2 , (7)

∂C

∂t =

1

Sc ∂2

C

∂y2 − KC . (8)

The initial and boundary conditions in dimensionless form are

t ≤ 0 : u = 0 , θ = 0 , C = 0 for all y ;

t > 0 : u = 1 , ∂θ∂y

= − (1 + θ) , C = 1 at y = 0 ,

u = 0 , θ → 0 , C → 0 as y → ∞ .

(9)

The dimensionless governing equations (6)–(8) subject to the boundary conditions (9) are solved by the usualLaplace transform technique, and the solutions take the form

u (y, t) = B1 + Gr Pr

Pr − 1 ⎛⎜⎝B2 − B3 −

B4

√⎯⎯⎯Pr − B7 + B8 +

B9

√⎯⎯⎯Pr

⎞⎟⎠ +

Gr

Pr − 1 (B10 − B5) +

Gm

KSc (B3 − B6 − B11 + B12) , (10)

θ (y, t) = B7 − B8 , (11)

C (y, t) = B11 , (12)

where

B1 = erfc ⎛⎜⎝

y2√⎯t

⎞⎟⎠ , B2 = exp ⎛⎜

⎝−

y√⎯⎯⎯Pr

+ t

Pr⎞⎟⎠ erfc

⎛⎜⎝

⎜⎜

y2√⎯t

− √⎯⎯ tPr

⎞⎟⎠

⎟⎟ , B3 = erfc ⎛⎜

⎝

y2√⎯t

⎞⎟⎠ ,

B4 = 2√⎯ tπ

exp ⎛⎜⎝−

y2

4t

⎞⎟⎠ − y erfc ⎛⎜

⎝

y2√⎯t

⎞⎟⎠ , B5 =

⎛⎜⎝t +

y2

2

⎞⎟⎠ erfc ⎛⎜

⎝

y2√⎯t

⎞⎟⎠ − y√⎯ t

π exp

⎛⎜⎝−

y2

4t

⎞⎟⎠ ,

B6 = exp (− bt)

2 ⎡⎢⎣exp (− y√⎯⎯⎯− b) erfc ⎛⎜

⎝

y2√⎯t

− √⎯⎯⎯⎯− tb ⎞⎟⎠ + exp (y√⎯⎯⎯− b) erfc ⎛⎜

⎝

y2√⎯t

+ √⎯⎯⎯⎯− tb ⎞⎟⎠⎤⎥⎦ ,

B7 = exp ⎛⎜⎝− y +

tPr

⎞⎟⎠ erfc

⎛⎜⎝

⎜⎜y√⎯⎯⎯Pr2√⎯t

− √⎯⎯ tPr

⎞⎟⎠

⎟⎟ , B8 = erfc

⎛⎜⎝

y√⎯⎯⎯Pr2√⎯t

⎞⎟⎠ ,

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B9 = 2√⎯ tπ

exp ⎛⎜⎝−

y2Pr

4t

⎞⎟⎠ − y√⎯⎯⎯Pr erfc

⎛⎜⎝y√⎯⎯⎯Pr2√⎯t

⎞⎟⎠ , B10 =

⎛⎜⎝t +

y2Pr2

⎞⎟⎠ erfc

⎛⎜⎝

y√⎯⎯⎯Pr2√⎯t

⎞⎟⎠ − y√⎯⎯tPr

π exp

⎛⎜⎝−

y2Pr

4t

⎞⎟⎠ ,

B11 = 12

⎡⎢⎣exp (y√⎯⎯⎯⎯KSc) erfc

⎛⎜⎝y√⎯⎯⎯Sc2√⎯t

+ √⎯⎯⎯Kt⎞⎟⎠ + exp (− y√⎯⎯⎯⎯KSc) erfc

⎛⎜⎝

y√⎯⎯⎯Sc2√⎯t

− √⎯⎯⎯Kt⎞⎟⎠

⎤⎥⎦ ,

B12 = exp (− bt)

2 ⎡⎢⎣exp (− y√⎯⎯⎯⎯⎯⎯⎯⎯⎯(K − b) Sc) erfc

⎛⎜⎝

y√⎯⎯⎯Sc2√⎯t

− √⎯⎯⎯⎯⎯⎯⎯⎯(K − b) t ⎞⎟⎠

+ exp (y√⎯⎯⎯⎯⎯⎯⎯⎯⎯(K − b) Sc) erfc ⎛⎜⎝

y√⎯⎯⎯Sc2√⎯t

+ √⎯⎯⎯⎯⎯⎯⎯⎯(K − b) t ⎞⎟⎠

⎤⎥⎦ , b =

KScSc − 1

.

Skin Friction. The skin friction in the dimensionless form is

τ = − dudy

⎪⎪⎪y=0

. (13)

We have from Eqs. (10)–(13)

τ = 1

√⎯⎯πt −

Gr√⎯⎯⎯Pr√⎯⎯⎯Pr + 1

⎡⎢⎣

⎢⎢exp ⎛⎜

⎝

tPr

⎞⎟⎠

⎛⎜⎝1 + erf

⎛⎜⎝√⎯⎯ t

Pr

⎞⎟⎠

⎞⎟⎠

⎤⎥⎦

⎥⎥ +

Gr√⎯⎯⎯Pr√⎯⎯⎯Pr + 1

+ 2√⎯ tπ

Gr

√⎯⎯⎯Pr + 1

− GmKSc

⎡⎣√⎯⎯⎯− b exp (− bt) erf (√⎯⎯⎯⎯− bt) + √⎯⎯⎯⎯KSc erf (√⎯⎯⎯Kt ) − √⎯⎯⎯⎯⎯⎯⎯⎯⎯(K − b) Sc exp (− bt) erf (√⎯⎯⎯⎯⎯⎯⎯⎯(K − b) t )⎤⎦ . (14)

Fig. 1. Velocity profiles at Sc = 0.22, Pr = 0.71 (a) and 7 (b) for differentvalues of Gr, Gm, K, and t.

224

Fig. 3. Velocity profiles at K = 0.2, Sc = 0.22, Pr = 0.71 (a) and 7 (b) fordifferent values of Gr, Gm, and t.

Fig. 2. Velocity profiles at K = 0.2, Pr = 0.71 (a) and 7 (b) for different val-ues of Gr, Gm, Sc, and t.

225

Results and Discussion. In order to get a physical insight into the problem, we present the velocity profilesfor different values of the parameters K, Sc, Gr, and Gm for the cases of heating (Gr < 0) and cooling (Gr > 0) ofthe plate*. Heating and cooling take place by virtue of the free convection current due to the temperature and concen-tration gradients. The values of the Prandtl number Pr are chosen to be 0.71 and 7, which corresponds to air andwater, respectively.

Figure 1 presents the velocity profiles at Sc = 0.22 and different values of the chemical reaction parameter Kin the cases of cooling and heating of the plate for both air (Fig. 1a) and water (Fig. 1b). As expected, the presence

Fig. 4. Velocity profiles at Sc = 0.22 with various K (a), at K = 0.2 with variousSc (b), and at K = 0.2, Sc = 0.22 (c) for different values of Pr, Gr, and Gm.

*From the EditorsDimensionless criteria are usually positive quantities. Thus, the Editors consider that it could be logical to somehowdifferently label the parameters Gr and Gm, introduced by the author, for example, as Gr′, so that we could have Gr = |Gr′|.

226

of a chemical reaction significantly affects the velocity profiles. It should be mentioned that the studied case refers toa destructive chemical reaction. In fact, as K increases, a considerable reduction in the velocity is observed for bothair and water in the case of cooling of the plate. In this case, it is seen that the velocity increases with y near theplate, reaches a maximum, then decreases away from the plate, and finally takes an asymptotic value for all the valuesof K. However, a reverse effect is seen in the case of heating of the plate. It is found that at t = 0.4 and K increasingfrom 0.2 to 2 and 5, respectively, the maximum velocity of the fluid decreases by 4.02% and 9.05% for air and by4.58% and 9.19% for water in the case of cooling of the plate (Gr = 2, Gm = 5), whereas the minimum velocity ofthe fluid increases by 17.06% and 37.56% for air and by 20.91% and 45.62% for water when the plate is heated (Gr= –2, Gm = –5). It is also seen from the figure that the velocity increases with time for both air and water in thecase of cooling of the plate, and the opposite tendency takes place in the case of heating.

Figure 2 shows the velocity profiles for different values of the Schmidt number. It is seen that the velocitydecreases with increase in Sc for both air and water in the case of cooling of the plate. An increasing Schmidt numberimplies that viscous forces dominate over the diffusional effects. Therefore, an increase in Sc will counteract momen-tum diffusion since viscosity effects will increase and molecular diffusivity will be reduced. Thus, the flow will be de-celerated with a rise in Sc. However, the reverse effect is observed with plate heating. It is found that at K = 0.2,t = 0.2, y = 0.4, and Sc increasing from 0.22 to 0.60 and 0.78, respectively, the velocity decreases by 11.28% and14.24% for air and by 11.95% and 15.09% for water in the case of cooling of the plate, whereas it increases by94.91% and 119.91% for air and by 64.36% and 81.69% for water in the case with heating.

Figure 3 illustrates the influences of the thermal Grashof number Gr and mass Grashof number Gm on thevelocity field. It is seen from Fig. 3a that the velocity increases with Gr for air in the case of cooling of the plate.The thermal Grashof number represents the effect of the thermal buoyancy force relative to the viscous hydrodynamicforce. The flow is accelerated due to the enhancement of the buoyancy force corresponding to the increase in Gr. Posi-tive values of Gr correspond to cooling of the plate by natural convection. Here, heat is conducted away from the ver-tical plate into the fluid, which increases the temperature and thereby enhances the buoyancy force. However, theopposite phenomenon is observed in the case of heating of the plate. Figure 3b shows that the effect of Gr on thevelocity is negligible for water in the cases of both cooling and heating of the plate. The mass Grashof number Gmdefines the ratio of the species buoyancy force to the viscous hydrodynamic force. It is seen from Fig. 3 that the ve-locity increases with Gm for both air and water in the case of cooling of the plate, and the reverse effect is observedwith heating.

The skin friction against time for different values of the parameters is presented in Fig. 4. It is observed thatthe skin friction increases with K or Sc and decreases with increase in Gr or Gm for both air and water in the caseof cooling of the plate, and the reverse effect takes place for heating. It is also seen that the skin friction decreaseswith time for cooling and increases for heating of the plate. It is found that at Sc = 0.22 and t = 0.6, when K in-creases from 0.2 to 2 and 5, respectively, the skin friction increases by 6.68% and 14.32% for air and by 11.84% and25.36% for water in the case of cooling of the plate, whereas it decreases by 4.96% and 10.63% for air and by7.33% and 15.70% for water in the case of heating of the plate. At K = 0.2 the skin friction increases by 12.33%and 15.73% for air and by 21.84% and 27.86% for water in the case of cooling of the plate, whereas it decreases by9.15% and 11.67% for air and by 13.52% and 17.25% for water in the case of heating when Sc increases from 0.22to 0.60 and 0.78, respectively.

NOTATION

C, dimensionless species concentration in the fluid; C′, concentration; C∞′ , concentration far away from theplate; Cw′ , concentration near the plate; Cp, specific heat at constant pressure; D, diffusivity; Gm, mass Grashof num-ber; Gr, thermal Grashof number; g, acceleration due to gravity; h, heat transfer coefficient; Kr, chemical reaction pa-rameter; K, dimensionless chemical reaction parameter; Pr, Prandtl number; Sc, Schmidt number; T ′, temperature; T∞′ ,ambient temperature; t, dimensionless time; t′, time; u, dimensionless fluid velocity in the x′ direction; u′, fluid veloc-ity; u0, velocity of the plate; x′ and y′, Cartesian coordinates along the plate and normal to it; y, dimensionless coor-dinate; β, volumetric coefficient of thermal expansion; β∗, volumetric coefficient of expansion with concentration; θ,dimensionless temperature; κ, thermal conductivity; μ, viscosity; ν, kinematic viscosity; ρ, density; τ, dimensionlessskin friction. Indices: w, wall; ∞, free stream conditions.

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