Heat and mass transfer effect of chemically reactive fluid ... · PDF fileHeat and mass transfer effect of chemically reactive ... polymer production, ... Heat and mass transfer effect

Heat and mass transfer effect of chemically reactive fluid on flow over

an accelerated vertical surface in presence of radiation with constant

heat flux

DR. ARPITA JAIN

Head, Department of Mathematics,

JECRC UDML College of Engineering,

Kukus, Jaipur, Rajasthan, India

E-mail*: [email protected]

ABSTRACT. This paper aims to investigate the influence of chemical reaction and the

combined effects of heat and mass transfer on laminar boundary layer flow over a

moving vertical plate in presence of radiation with heat is supplied to the plate at

constant rate. The governing equations for this investigations are solved analytically by

Laplace-transform technique. Graph results are presented for temperature,

concentration, velocity, skin friction, Nusselt number and Sherwood number. The

effects of various parameters on flow variables are illustrated graphically and the

physical aspects of the problem are discussed.

KEY WORDS: Free convection, heat and mass transfer, chemical reaction, radiation,

accelerated plate.

1. INTRODUCTION:

The analysis of free convection flow near a vertical plate has been carried out as an

important application in many industries. Numerous investigations are performed by

using both analytical and numerical methods. The first exact solution of the Navier-

Stokes equation was given by Stokes [1] and explains the motion of a viscous in-

International Journal of Advancements in Research & Technology, Volume 2, Issue 11, November-2013 ISSN 2278-7763

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compressible fluid past an impulsively started infinite horizontal plate in its own

plane. This is known as Stokes’s first problem in the literature. If the plate is in a

vertical direction and gives an impulsive motion in its own plane in a stationary

fluid, then the resulting effect of buoyancy force was first studied by Sundalgekar

[2] by Laplace transformation technique and the effects of heating or cooling of the

plate by free convection cur- rents were discussed. Natural convection has been

analyzed extensively by many investigators. Some of them are Revankar [3], Li et.al.

[4].

Furthermore, the free convection flows together with heat and mass transfer are of

great importance in geophysics, aeronautics, and engineering. In several process

such as drying, evaporation of water at body surface, energy transfer in a wet

cooling tower, and flow in a desert cooler, heat and mass transfer occurs

simultaneously A number of investigations have already been carried out with

combined heat and mass transfer under the assumption of different physical

situations. The illustrative examples of mass transfer can be found in the book of

Cussler [5] .Combined heat and mass transfer flow past a surface analyzed by

Chaudhary et. al. [6], Muthucumaraswamy et. al. [7,8] and Rajput et.al.[9 ]with

different physical conditions. Chaudhary et. al. [10] pioneered unsteady heat and mass

transfer flow past a surface by Laplace Transform method.

Combined heat and mass transfer problems with chemical reaction are of

importance in many processes and have, therefore, received a considerable amount of

attention in recent years. Chemical reaction can be codified as either homogeneous or

heterogeneous processes. A homogeneous reaction is one that occurs uniformly through

a given phase. In contrast, a heterogeneous reaction takes place in a restricted region or


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within the boundary of a phase. A reaction is said to be first order, if the rate of

reaction is directly proportional to the concentration itself which has many applications

in different chemical engineering processes and other industrial applications such as

polymer production, manufacturing of ceramics or glassware and food processing. Das

et al. [11] considered the effects of first order chemical reaction on the flow past an

impulsively started infinite vertical plate with constant heat flux and mass transfer.

Muthucumarswamy et. al. [12] studied study of chemical reaction effects on vertical

plate with variable temperature.

In the above mentioned studies the effects of radiation on flow has not been

considered. Actually, many processes in new engineering areas occur at high

temperature and knowledge of radiation heat transfer becomes imperative for the

design of the pertinent equipment. Nuclear power plants, gas turbines and the various

propulsion devices for aircraft, missiles, satellites, and space vehicles are examples of

such engineering areas. The effects of radiation on free convection on the accelerated

flow of a viscous incompressible fluid past an infinite vertical plate has many

important technological applications in the astrophysical, geophysical and engineering

problem. Unsteady Free Convection flow past a Vertical plate with chemical reaction

under different temperature condition on the plate is elucidated by Bhaben Ch. Neog

et.al.[13] and Rajesh V. [14] et.al. Thermal radiation effect on flow past a vertical plate

with mass transfer is examined by Muralidharan M. et. al. [15] and Rajput et.al. [16].

Natural convective flow past a plate with constant mass flux in the presence of

radiation is studied by Chaudhary et. al. [17]

However, it seems less attention was paid on free convection flows near a vertical

plate subjected to a constant heat flux boundary condition even though this situation


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involves in many engineering applications. In many problems particularly those

involving the cooling of electrical nuclear components, the wall heat flux is specified.

In such problems, over heating burns out and melt down are very important issues.

From practical stand point, an important wall model is considered with constant

heat flux. Ogulu et. al. [18] and Narahari, M. et. al. [19] examined the flow past a

surface with constant heat flux. Free convection effects on flow past an infinite vertical

accelerated with constant heat flux is pioneered by Chaudhary et. al. [20 ].

Hence, the objective of this paper is to study the influence of chemical reaction and

the combined effects of heat and mass transfer on flow over a moving vertical plate in

presence of radiation with heat is supplied to the plate at constant rate.

2. MATHEMATICAL ANALYSIS: We consider a two-dimensional flow of an

incompressible and electrically conducting viscous fluid along an infinite vertical plate.

The x'-axis is taken on the infinite plate and parallel to the free stream velocity and y'-

axis normal to it. Initially, the plate and the fluid are at same temperature T '

with

concentration level 'C at all points. At time t' > 0, the plate concentration is changed

to 'wC with heat supplied at a Constant rate to the plate and it accelerates with a

velocity3 '

RU t

in its own plane. It is assumed that there exist a homogeneous

chemical reaction of first order with constant rate K l between the diffusing species

and the fluid. Since the plate is infinite in extent therefore the flow variables are the

functions of y' and t' only. The fluid is considered to be gray absorbing-emitting

radiation but non scattering medium . The radiative heat flux in the x'-direction is

considered negligible in comparison with of y'-direction. Then neglecting viscous

dissipation and assuming variation of density in the body force term (Boussinesq’s


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approximation), the problem can be governed by the following set of equations:

' 2 'r

' 2 'p p

qT T 1C Ct y' y

…(1)

' 2 ''

l' 2

C CD k Ct y'

…(2)

2' ' ' '

c2

u' u' g (T T ) g (C C ')t' y'

…(3)

with following initial and boundary conditions

0 t',y'allforCC,TT,0u' '''' …(4)

3 ' ''Rw'

' ' ' ' ' ' '

U t T qu' , - , C' C at y' 0, t' 0y

u 0, T T ,C C as y , t 0

The radiative heat flux term, by using the Rosseland’s approximation is given by

4

r4 T'q

y'3

…(5)

where UR is reference velocity, g is gravitational acceleration, Cp is specific

heat at constant pressure, D is mass diffusivity, is thermal expansion coefficient,

C is concentration expansion coefficient, is density, is thermal conductivity

of fluid, is mean absorption coefficient, is electrical conductivity of fluid,

is kinematic viscosity and , qr is radiative heat flux, σ is Stefan-Boltzmann

constant.

We assume that the temperature differences within the flow are such that T' 4

may be expressed as a linear function of the temperature T'. This is accomplished by

expanding T' 4 in a Taylor series about T '

and neglecting higher-order terms


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4 3 4T' 4 T' T' 3T' …(6)

By using equations (5) and (6), equation (1) gives

32 2

p 2 2

16 T'T' T' T'C t' y' 3 y'

…(7)

Introducing the following dimensionless quantities2' ' 'R l

R R R

U kt y ut , y , u , k ,t L U

2

' ' 2p

4R

R

C T T g qPr , Sc , ,GqD UU

' '' 'C w l

' ' 3 2w R R

g (C C )C C kC , Gm , kC C U U

, '3R

4 T

' ' 1/3w RT T T ,U ( g t) ,

3132R

31

2R T)gtTgL

. …(8)

where LR is reference length, tR is reference time, Gm is modified

Grashof number, Pr is Prandtl number, Sc is Schmidt number and

u is dimensionless velocity component, is dimensionless temperature,

C is dimensionless concentration, is viscosity of fluid, t is time in

dimensionless coordinate, R is radiation parameter and k is chemical reaction

parameter.

The governing Equations (1) to (3) reduce to the following non-dimensional form


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2Pr ( ) t R y

2413

...(9)

2

2

C 1 C kC t Sc y

…(10)

2

2

u u G Gm C t y

…(11)

with the following initial and boundary conditions

u 0 , 0, C 0 for all y, t 0 …(12)

u t, 1, C 1 at y 0, t 0y

u 0, 0, C 0 as y , t 0

…(13)

On taking Laplace-transform of Equations (9) to (11) and Boundary conditions

(12, 13), we get

( )R

41

30Prp

dyd

2

2

…(14)

2

2

d k+p Sc C 0dy

…(15)

2

2

d u p u G ( y,p) -G m Cdy

…(16)

2

1 d 1 1u , , C a t y 0 , t 0p d y p p

u 0 , 0 , C 0 a s y , t 0

…(17)

Where p is the Laplace -transform parameter.

Solving Equations (14) to (16) with the help of Boundary condition (17), we get

/

pPr exp ( y )H H(y,p)Pr p

3 2 ...(18)


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exp ( y (kSc+p Sc))C(y,p)

p

…(19)

2

exp ( y p)u (y,p)

p

/

G exp ( y p) Gm exp ( y p)HPr kScPr p ( -1) p (Sc-1)(p+ )H Sc

5 2

1

Gm exp ( y pSc kSc )kScp (Sc 1) p +

Sc 1

/

p pr G exp ( y )H HPr p (Pr/ H )

5 2 1…(20)

On taking inverse Laplace-transform of Equations (18) to (20), we get

t H - Pr Prerfc ( ) t erfcPr H H

2

…(21)

1C {exp(2 kSc t )erfc ( Sc kt ) exp( 2 kSc t )erfc ( Sc kt )}2

…(22)

For Pr =Sc 1

exp ( )u t (1+2 )erfc( ) -

22 2

3/2

2 2 2H G t 4 (1 )exp ( ) 6 4 erfc( )PrPr 3 ( -1)H


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3/2H G t Pr Pr Pr Pr Pr Gm( )exp( ) ( ) erfc( ) erfc( )PrPr H H H H H k Sc( )H

2 2 24 1 6 43 1

Gm kSc t kSc t kSc t kSc t kSc t{exp( )((exp(2 )erfc( ) exp( 2 )erfc( )2kSc 1 Sc 1 Sc 1 Sc 1 Sc 1 Sc

Gm kSc t kSc t k t kSct k texp( )(exp(2 )erfc( Sc ) exp( 2 )erfc( Sc )2kSc 1 Sc 1 Sc 1 Sc 1 Sc 1 Sc

Gm exp(2 kSct erfc( Sc k t ) exp( 2 kSct erfc( Sc k t )2kSc

…(23)

Where HR

41

3

In expressions, erfc (x1+i y1) is complementary error function of complex argument

which can be calculated in terms of tabulated functions in Abramowitz et al. [21].The

tables given in Abramowitz et al. [21] do not give erfc (x1 + i y1) directly but an

auxiliary function W1(x1 + iy1) which is defined as

21111111 iyx{exp xiyWiy(xerfc

Some properties of W1 (x1 + iy1) are

112111 yixWiyxW

1122

11111 yixWiy{-(xexp2yixW

where 112 iyxW is complex conjugate of 111 iyxW


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3. SKIN-FRICTION:

From velocity field , skin-friction at the plate in non dimensional form is expressed as:

y

uy

t H G t PrPrPr Ht ( -1)H

0

2 1

Gm kSc t kSc t kScexp( ) erf ( )kSc 1 Sc 1 Sc 1 Sc

Gm kSc erf ktkSc

Gm kSc t k t kScexp( ) erf ( )kSc 1 Sc 1 Sc 1 Sc

…(24)

4. NUSSELT NUMBER

From temperature field, the rate of heat transfer in non-dimensional form is

expressed as

y 0

Nu( ) y

10

( )

10

Prt H

…(25)

5. SHERWOOD NUMBER

From the concentration field, the rate of concentration transfer, which when

expressed in non-dimensional form, is given by

y 0

Shy


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kSc Sc( exp( k t))t

12

2 2…(26)

6. DISCUSSION: In order to get physical insight into the problem, the values of

Schmidt number are chosen to represent the presence of species by Hydrogen (0.22),

Water vapor (0.60) and Carbon dioxide (0.96) at 250C temperature and 1 atmospheric

pressure, the values of Pr are chosen 0.71, 7 which represent air and water respectively

at 200C temperature and 1 atmospheric pressure. The values of other parameters are

chosen arbitrary.

Figure 1 reveals temperature profiles against (distance from the plate). It is

obvious from the figure that the magnitude of temperature is maximum at the plate and

then decays to zero asymptotically. Further, the magnitude of the temperature for air is

greater than that of water. This is due to the fact that thermal conductivity of fluid

decreases with increasing Pr, resulting a decrease in thermal boundary layer thickness.

It is also seen that it decreases steeply for Pr = 7 than that of Pr = 0.71. Furthermore, it

is noticed that an increase in radiation parameter to 4, 15 the temperature decreases

but it increases with increasing time.

The species concentration profiles verses is plotted in F igure 2. It is

observed that the concentration at all points in the flow field decreases with and tends

to zero as . Furthermore, an increase in the value of Sc leads to a fall in the

concentration. Physically, it is true since increase of Sc means decrease of molecular

diffusivity which results in decrease of concentration boundary layer. Hence, the

concentration of species is higher for small values of Sc. It is also observed that

concentration decreases with an increase in time.


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Figure 3 illustrates the influences of Sc, t, and Pr on the velocity against η. It is

noticed that at the plate, fluid velocity is equal to value of time then it increases and

attains maximum velocity in the vicinity of the plate(η<0.6) after that it decreases and

vanish far away from the plate for both Pr = 7 and 0.71 whereas with an increase in Sc it

decreases continuously to asymptotic value. Further, it also increases with an increase

in time at each point in the flow field for Hydrogen gas. The effect of time on velocity

boundary layer is more dominant than Sc for both water and air. Finally, we observed

that the velocity for Hydrogen(Sc=0.22) is higher than that of carbon di oxide (Sc=0.96)

for both air and water. Physically, it is possible since an increase in Sc means increase

in kinematic viscosity or viscosity of fluid due to which the velocity of fluid decreases.

It is also obvious from figure that increase or decrease in thickness of velocity boundary

layer with an increase in parameters value is greater near the plate then away from the

plate and on moving away from the plate this difference is negligible Figure 4

illustrates the influences of Gm, G and Pr on the velocity. This Figure reveals that the

maximum velocity attains near the plate then decreases and faded away from the plate.

Further velocity for Pr=7 is less than that of Pr=0.71 since increasing Pr means

increasing viscosity which in turn reduces the velocity of flow. Moreover, it is observed

that an increase in G, Gm leads to an increase in velocity for both Pr=7 and Pr =0.71

when Hydrogen gas is present in the flow field. The reason is that the values of Grashof

number and modified Grashof number has the tendency to increase the mass buoyancy

effect. The increase in velocity due to increase in G and Gm is more near the plate than

away from the plate. Figure 5 elucidates the effects of k, R and Pr on velocity profile. It

is found that the velocity decreases with an increase in chemical reaction parameter k

from 0.2 to 10 for both Pr=7 and 0.71. The same phenomenon is observed with an


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increase in radiation parameter from 4 to 20. It is obvious since the radiation parameter

defines the relative contribution of conduction heat transfer to the thermal radiation

transfer. Moreover, the variation of the velocity across the boundary layer with R is

found to be negligibly small.

Figure 6 elucidates the effects of Sc, k, Pr, G, Gm on skin friction which is plotted

against time t. The maximum value of skin friction occurs for smaller values of t and

then it decreases rapidly with increasing t. It is clear from the Figure that skin-friction

increases with an increase in Sc, Pr and k but reverse effect is observed with an increase

in G and Gm. The Figure depicts that skin friction is higher for Sc=0.96 in comparison

to for Sc=0.22. Physically, it is correct since an increase in Sc serves to increase

momentum boundary layer thickness. Further, increasing Pr means increasing viscosity

of fluid which increases the magnitude of skin friction. Moreover increasing value of

chemical reaction parameter reflects decrease in kinematic viscosity or viscosity of fluid

which results increase in the value of skin friction The boundary layer separation

occurs in the fluid flow. The variation in the values of skin friction is more dominant

with parameters Pr and Gm in comparison with other parameters.

Figure 7 exhibits the Nusselt number against time. It is concluded from the

Figure that there is a increase in it with an increase in the value of radiation parameter.

Further, the value of Nusselt number for water is greater than air. It is consistent with

the fact that smaller values of Pr are equivalent to increasing thermal conductivities and

therefore heat is able to diffuse away from the plate more rapidly than higher values of

Pr, hence the rate of heat transfer is reduced.

Figure 8 depicts the effect of chemical reaction parameter k and Schmidt

number Sc on Sherwood number. It is observed that Sherwood number increase with an


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increase in k and Sc. Since increase in Sc means decrease in molecular diffusivity which

in turn gives rise to increase in Sherwood number as Sherwood number is the ratio of

convective and diffusive mass transfer coefficient. Chemical reaction parameter is the

interfacial mass transfer so Sherwood number increases with an increase in k.

REFERENCES:

[1] Stokes GG, On the Effect of Internal Friction of Fluids on the Motion of

Pendulums, Mathematical and Physical Papers 1851; 3; 8-106.

[2] Soundalgekar VM, Free Convection Effects on Stokes’s Problem for a

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oscillating infinite vertical plate, Mechanics Research Comm. 2000; 27; 241-246.

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MHD free convection flow past an oscillating plate embedded in porous

medium, Rom.Journ.Phys.(Romania) 2007; 52;505-524.

[7] Muthucumaraswamy R and Vijayalakshmi A, Effects of heat and mass

transfer on flow past an oscillating vertical plate with variable temperature., Int.

J. of Appl. Math and Mech. 2008; 4; 59 – 65.


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[8] Muthucumaraswamy R, Sundar RM and Subramanian VSA, Unsteady flow past

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[15] Muralidharan M and Muthucumaraswamy R, Thermal radiation on

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[16] Rajput US and Kumar S, Radiation Effects on MHD flow past an

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[17] Chaudhary RC and Jain A, Unsteady free convection flow past an

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[21] Abramowitz BM and Stegun IA, Handbook of Mathematical Functions, New

York: Dover Publications; 1970.


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Figure1: Temperature profile


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Figure 2: Concentration profile for k=0.2


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Figure 3 : Velocity profile for R=4, k=0.2, G=5, Gm=2


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Figure 4: Velocity profile R=4, k=0.2, SC=0.22, t=0.2


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Figure 5: Velocity profile for sc=0.22, t=0.2, G=5, Gm=2


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Figure 6: Skin-friction for R=4.


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Figure 7: Nusselt number


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Figure 8 : Sherwood number


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Heat and mass transfer effect of chemically reactive fluid ... · PDF fileHeat and mass transfer effect of chemically reactive ... polymer production, ... Heat and mass transfer effect