Transport of Aerosols in the Presence of Electric Field with Interphase Mass Transfer

Transport of Aerosols in the Presence of Electric Field with Interphase Mass Transfer

Meena Priya. P* and Nirmala P.Ratchagar#

Mathematics Section, Faculty of Engineering and Technology, Annamalai University, Annamalai Nagar- 608 002, India

Abstract

We have developed a mathematical model for evaluating the unsteady convective dispersion of aerosols with inter phase mass transfer by poorly conducting couple stress fluid in a channel in the presence of a transverse electric field. The validity of time dependent dispersion coefficient is widened by using a generalized dispersion coefficient. The exact expression is obtained for the dispersion coefficient (K2) which shows it decreases with increase in couple stress parameter a and reaction rate . The anlaysis leads to the novel result for convection coefficient (K1) and dispersion coefficient (K2) (which is a measure of the longitudinal dispersion coefficient of the aerosol). It is found that the value of K2 depends on the value of reaction rate ( ) and electric field (we) whereas the values of K1 is constant in both the cases. Finally, the effect of and the electric field (we) on the axial distribution of the mean concentration m is investigated in detail.

keywords: electrodes, generalized dispersion, aerosols, chemical reaction.

1. Introduction

   The problem caused by the air pollution are complex and they affect natural processes, strongly influencing the ecological balance [2]. For this reason, it is important to study and understand the dispersion process of pollutants in the atmosphere in order to predict the possible impact of the pollution on the diverse ecosystems involved. Pollutants released from various sources into the atmosphere in the form of suspended ultrafine particles are aerosols [1] affect the environment directly or indirectly. The couple stress fluid theory developed by stokes

   [9] represents the simplest generalization of the classical viscous fluid theory that sustains couple stresses and the body couples.

   The presence of earth's surface influences the concentration of ions, aerosols and radioactive particles, through its control over the wind, temperature and water vapour distributions [4], [8]. Atmospheric electricity abounds in the environment [7]. Recently it has become evident that lightening is a form of electricity associated with thunderstorms [10]. The resulting ground level concentration patterns have to be estimated for a wide variety of air quality analysis for social planning and industrial growth [3].

   It is not always feasible to measure/monitor concentration of species at various vulnerable points of a city. However man insight in this regard could be achieved with the help of suitable mathematical models. In this paper, we developed a mathematical model to study the unsteady convective diffusion of atmospheric aerosols with interphase mass transfer in a couple stress fluid flow through a channel in the presence of electric field. The atmospheric flow and concentration change of aerosols are commonly described by a set of partial differential equations, which are mathematical formulation of one or more of the conservation law of physics. These include the equations of mass, momentum species along with Maxwell's equation which involve advection and diffusion terms as a main constituent. They have been solved using generalised dispersion model [5] with appropriate boundary conditions and the results have been depicted graphically.

2. Mathematical Formulation

   The physical configuration shown in the figure 1 consists of an infinite horizontal couple stress fluid layer bounded on both sides by electro- conducting impermeable right plates embedded with electrodes located at y = h and electric

   potentials The equation (7) is zero because in a poorly

   V x at y

   h

   -h and

   V

   h x x0

   at y h

   conducting fluid, induced magnetic field is negligible and there is no applied magnetic field.

   are maintained on theseboundarieswhere V is

   potential. y

   J E(Ohm' s

   law)

   (8)

   V (x h

   x0 )

   electrodes

   y=+h

   The above equations are solved using the following boundary conditions on velocity and potential are,

   No slip condition,

   Couple stress fluid

   -Xs/2

   x

   slug

   Xs/2

   u 0 at

   y h, h

   (9)

   V x y=-h

   h

   The couple stress condition,

   d 2u

   Fig. 1. Physical Configuration

   In this paper, we make the assumption that the

   dy2

   0 at y h

   (10)

   electrical conductivity ( ) is negligibly small and hence the magnetic field is negligible. This assumption makes the electric field , to be conservative.

   V x h

   V (x h

   at y x0 )

   h

   at y h

   (11)

   i.e. =- (1)

   In Cartesian form, using the above approximation equation (3) becomes

   The basic equations are p

   0

   Conservation of mass for an incompressible flow x

   2u 4u

   2

   2

   e Ex , y2

   . =0 (2)

   Conservation of momentum

   In a poorly conducting fluid, the electrical conductivity is assumed to vary linearly with temperature in the form

   q (q.

   t

   )q

   p 2 q

   4 q

   e E(3)

   [1 (T

   h

   0 b

   T )] 0

   (12)

   Where is a couple stress parameter. Conservation of species

   where h is the coefficient of volumetric expansion.

   We assume the flow is fully developed and

   C (q. )C

   t

   D 2C

   (4) Conserva

   unidirectional in the x direction. This means the velocity is independent of time and all physical

   tion of charges

   quantities except pressure and concentration are independent of x, so that the velocity and

   e (q. )

   .J 0

   (5)

   temperature will be functions of y only. Using

   t e the following dimensionless quantities,

   Maxwells equation

   y* y ; u* u ; E * Ex ; * e ;

   E

   . e (Gauss law)

   (6)

   h x V e 0V h h p

   0

   E

   0 (Faraday ' s

   law)

   (7)

   P* P

   h

   x

   2 ; x* h

   Where V is electric potential, we get electric potential through electrodes.

   Tb 1at y 1

   Tb 1at y 1

   (19)

   V

   2

   Equations (3) to (11) becomes,

   is Tb=y (20)

   d 4u dy4

   p d 2u p

   l 2 dy2 l 2

   e Ex 0

   vp p

   l 2 x

   Therefore equation (12) becomes

   = 0[1+ h Ty] = 0 (1+ y) 0 e y

   We assume that the fluid with pollutants is isotropic and homogenous so that molecular diffusivity D, viscosity are all constants [6].

   e y (Where = h T) (21) From equations (14) and (21) we get,

   4u 2 d 2u 2 2

   d 2 d

   a

   y4 dy2

   a wePe Ex a P

   (13) 0

   (22)

   dy2 dy

   where

   V 2 P h

   Its solution satisfying the boundary condition

   (17) is

   we 0 ; P , l ,a

   x l

   x x0 [e

   e y ]

   (23)

   is the couple stress parameter

   Equation (5) becomes, =0 using equation (1) we get,

   2sin h

   using the dimensionless quantities and equation (23), equation (6), (7) and (8) reduce to

   2 = . = – 2 = – x2

   2e y

   ( ) + . =0 (14) e

   The boundary conditions on velocity, couple stress and electric potentials after dimensionless are

   eEx=

   0 ; 2

   x e

   2 2 y

   0

   2 sin h

   (24)

   u = 0 at y= 1 (15)

   d 2u

   The solution of equation (13) satisfying the condition (15), (16) is

   0

   dy2

   at y 1

   (16)

   u b0 y b

   P y2

   2

   b2 sinh ay

   x at y 1

   (17)

   b coshay

   b4e y

   (25)

   x x0 at y 1

   3

   Where b0

   The solution for , according to (14)depends on

   a sinh( ) 1 1 1 1

   which in turn depends on the temperature Tb as

   0

   = 2 a2

   2 a2

   ; b1

   P 2 a2

   in (12). In a poorly conducting fluid, <<1 and hence any perturbation on it is negligible and hence it depends on the conduction temperature

   a0 cosh( ) 1 1 ;

   2 a2 a2 2

   Tb namely,

   d 2T

   b sin h

   2 sinh a a2

   a0

   2 a2 ;

   b 0

   (18)

   1. a cosh

      a

      dy2

      b P 0 ;

      Hence the solution satisfying the boundary conditions

      3 cosha

      2

      b a0

      4 2 2 a2

   2. a2

   W 2 x

   Where U= u ,u is the average velocity of the

   and a0

   e 0

   2sin h

   u

   flow, Peis the Peclet number and is the

   The average velocity is given by,

   dimensionless reaction rate constant, and the initial conditions reduce to

   1 1

   P b sinh a

   u udy b1 3

   2 0 6 a

   (0,X, ) = (X)Y( ), (30a)

   b4 sinh

   (26)

   at 1,

   (30b)

   Into this flow, introduced a slug input of concentration C(0,x,y) and the local concentration C(t,x,y) of the solute which satisfies the convective diffusion equation

   at 1,

   , , , , 0

   X

   (30c)

   C u( y, t) C t x

   2C 2C

   D x2 y2

   (27)

   In the generalized dispersion model [5], we let

   k

   along with the initial condition

   , X ,

   f , m

   (31)

   C(0, x, y) C (x)Y ( y)

   0 1 1

   X

   (28a)

   Where

   k k '

   k

   Where C0 is a reference concentration. The corresponding boundary condition at the channel

   1 , X , ) d is

   m

   1

   2 1

   the dimensionless

   walls are

   D C

   y

   Ks C at

   y h and

   average concentration. (31a)

   Equation (29) is multiplied throughout by

   1 and

   2

   C

   D y Ks C at y h

   (28b)

   Wh

   integrated with respect to y from -1 to 1. Introducing the definition of m , we get

   ere Ks is the reaction rate constant catalysed by the walls.

   2 1

   m 1 1

   m

   C t, , y

   C t, , y 0

   x

   (28c) 1

   2 2

   P

   X

   2

   e 1

   1

   U , d

   (32)

   On introducing the following non-dimensional

   2 x 1

   quantities,

   Dt ;

   p

   C ; X

   C

   Dx ; y ;

   pu h

   Using equation (31) in (32), the dispersion model for m is obtained as

   0

   u hu K H

   m K ( ) m

   (33)

   i

   U ( ,

   ) ; P

   ; s

   i X i

   u e D D i 0

   Equation (27) and (28) become

   Where Kis are given by

   1 2 2

   U

   (29)

   Ki( )

   i 2

   P2

   1 fi ,1

   2

   e

   X P 2 X 2 2

   e

   1 1

   2

   fi 1 ,

   1

   U , d

   (34)

   Where f-1=0 and i2 is the Kroneckar delta defined by

   fk ,1

   fk ,1

   (37b)

   1, i j

   ij 0, i j

   f k ,0 0

   (37c)

   The exchange coefficient K ( ) accounts for the 1 1

   0 fk , d

   k0 , k

   0,1,2 (37d )

   nonzero solute flux at the channel wall and 2 1

   negative sign indicates the depletion of solute in the system with time caused by the irreversible reaction, which occurs at the channel wall. The presence of non-zero solute flux at the walls of the channel also affects the higher order Ki due

   to the explicit appearance of ( ,1) in

   The function f0 and the exchange coefficient K0 are independent of the velocity field and can be solved easily. It should be pointed out here that a simulatenous solution has to be obtained from these two quantities since K0, which can be obtained from (34) as

   equation (34). Equation (33) can be truncated K 0

   after the term involving K2 without causing

   1

   1. f0

   2. y 1

   (38)

   serious error because K3, K4, etc. become negligibly small compared to K2.The resulting

   Substituting k =0 in (36) we get the differential equation or f0 as

   model for the mean concentration is

   2

   f0 0

   f

   2

   2

   f0 K0

   (39)

   m K0 ( ) m K1

   1. K2

      X

      m (35)

      X 2

      We now derive an initial condition for f0

      using

      To solve this equation, we need the coefficients Ki( ) in addition to the appropriate initial and

      (31a) by taking = 0 in that equation to get

      1 1

      boundary conditions. For this, the corresponding function fk must be determined. So, substituting

      m 0, X

      0, X , d

      2 1

      (40)

      (31) into (29) and using equation (32), the

      following set of differential equations for fk are generated.

      Substituting =0 in (31) and setting fk( ) = 0 (k=1,2,3) gives us the initial condition for f0 as

      0, X ,

      f f 1

      f0 0,

      0, X

      (41)

      k 2 k

      2

      k

      Ufk 1

      2 fk 2

      P

      e

      m

      We note that the left hand side of (41) is function

      Ki fk 1 K

      i 0

      0,1,2,…

      (36)

      of only and the right hand side is a function of

      both X and . Thus clearly the initial concentration distribution must be a separable

      where f-1 = f-2 = 0.

      We note that to evaluate Kis we need to know the fks which are obtained by solving

      function of X and . This is the justification for the chosen form of (0,X, ) in (41). Substituting (30) into (41), we get

      (36) for fks subject to the boundary conditions.

      f0 0,

      1

      1 d

      (42)

      fk ,0

      finite

      (37a) 2 1

      The solution of the reaction diffusion equation

      (39) with these conditions may be formulated as

      f0 ,

      g0 ,

      exp

      t

      K0 ( )d

      0

      n

      n

      (43)

      ensured convergence of the series seen in the expansions for f0 and K0. Having obtained f0, we get K0 from (38) in the form

      From which it follows that g0 ( , ) has to satisfy

      g g

      2

      0 0

      9

      An n

      exp

      2 sin

      0

      n

      n

      2 (44) K

      n 0

      (50)

      along with the conditions

      9 An n 0 n

      n exp

      2 sin

      f0 0,

      g0 0, 1 1

      d

      (45a)

      K0 is independent of velocity distribution.

      2 1 As , we get the asympotic solution for K0

      0

      from (50) as

      g0 ( ,0)= finite (45b)

      K0( ) = 2

      (51)

      g0 ,1

      g0 ,1

      (45c)

      Where 0 is the first root of the equation (47). Physically this represents first order chemical reaction coefficient having obtained K0( ), we can now get K1( ), from (34) with (i=1)

      n

      The solution of (44) subject to conditions (45) is

      knowing f0 ( , ) and f1( , ). Likewise K2( ), K3( ), require the knowledge of K0, K1, f0, f1

      g0 An

      n 0

      cos n

      exp 2

      (46)

      and f2. Equation (49) in limit reduces to

      Where ns are the roots of

      f0 ,

      0 cos

      0

      sin 0

      (52)

      ntan n= , n=0,1,2,3.. (47) and Ans are given by

      We then find f1, K1,f2, and K2. For asymptotically long times, i.e., , (34) and (36) give us Kis and fks as

      1

      2 ( ) cos n d

      K i 2

      i P2

      fi ,1

      1

      Ufi 1

      , d ,

      A 1

      e 1

      (48)

      n sin 2 1

      (i 1,2,3)

      (53)

      1 n ( )d

      d 2 fk

      2 n 1

      2 fk

      U K f

      Now from (43) is follows that

      d 2 0

      1 k 1

      f0 ,

      2g0 ,

      1

      1

      P

      K

      f

      2 2 k 2 ,

      e

      (K 1,2)

      (54)

      g0

      1

      2

      9

      An exp n

      , d

      cos n

      The fks must satisfy the conditions (31a) and this permits the eigen function expansion in the form

      9

      n 0

      (49)

      9 An

      exp

      <>2 sin

      fk ,

      B j ,k cos( j ),

      j 0

      k 1,2,3,….

      (55)

      n

      n

   2. 0 n

   The first ten roots of the transcendental equation

   (47) are obtained using mathematica and are given in Table 1. We find that these ten roots

   Substituting (55) in equation (54) and multiplying the resulting equation by Cos jand integrating with respect to from -1 to 1, we get after simplification

   B 1 1 B

   K

   k B ,

   and the parameters of the problem a and . This

   P

   j,k

   2 2 2

   j 0 e

   1

   j,k 2

   i j k i

   i 1

   distribution is valid only for long time and is a gross approximation at short and moderate times.

   sin 2

   1 j

   2 j

   9

   Bj,k 1

   j 0

   I ( j,1)

   (56)

   The initial conditions for solving (33) can be obtained from (30a) by taking the cross-sectional average. Since we are making long time

   Where

   1

   k 1,2

   evaluations of the coefficients, we note that the side effect is independent of m on the initial concentration distribution. In view of this, the solution of (33) with asymptotic coefficients can

   be written as

   I j,l

   U cos

   1

   j .cos l d

   I (l, j)

   (57)

   m , X

   1

   (58)

   2Pe K 2

   Bj , 1

   0, Bj ,0 0

   for

   j 1 to 9

   exp

   K

   0

   2

   X K

   1

   4K

   (63)

   The first expansion coefficient B0,k in (55) can be expressed in terms of Bj,k (j=1 to 9) by using the conditions (37) as

   Where

   ,

   2

   0, m , 0

   (64)

   B0,k

   0

   sin

   9

   Bj,k

   sin

   j , k

   1,2,3,…

   m X

   (59)

   0 j 1 j

   Further, from (55) and (52) we find that

3. Results and Discussions

   We have modeled the solvent as a couple-stress fluid and studied dispersion of solute in a couple- stress fluid bounded by electrodes considering

   B0,0

   0

   sin 0

   (60)

   heterogenous chemical reaction, on the interphase. The walls of the channel act as catalysts to the first order reaction and the effect of wall reaction on dispersion is discussed. The

   Substituting i=1 in (53) and using (57), (58) and

   (60) in the resulting equation, we get

   problem brings into focus three important dispersion coefficients namely the exchange coefficient (-K0), the convective coefficient (-K1)

   1

   K I 0,0

   1 sin 2 0

   (61)

   and diffusive coefficient (K2). The exchange coefficient arises only due to the interphase mass transfer and hence it is unaffected by the flow

   2 0

   Substituting i=2 in (53) and using (56), (57) and

   (60) in the resulting equation, we get

   and thus is independent of the couple-stress

   parameter a. The asymptotic values of these three coefficients are plotted in figure 3 to 6 for

   K 1

   Sin 0

   9

   B .I

   (62)

   various value of a and reaction rate parameter .

   P

   1

   e

   2 2 sin 2

   2

   0

   0

   1

   0 j 0

   sin 2

   j,1

   1

   j,0

   From these figures we predict the following.

   Figure 2 is a plot of the velocity against the non- dimensional transverse coordinate for different values of electric number We. It is seen that the

   velocity profile is a parabolic curve. We note that

   j

   1

   WhereBj -1= 2

   2 0 0 .I ( j,0)

   0

   2 0 sin 0

   the velocity increases with the increase in electric number. From figure 3, it is evident that

   Using the asymptotic coefficients K0( ),K1( ) and K2( ), in (33). One can determine the mean concentration distribution as a function of X,

   K0( ) increases with increase in the value of the wall reaction parameter and is unaffected by the couple-stress parameter a.

   The convective coefficient -K1 is plotted in figure 4 versus wall reaction parameter for different couple stress parameter a with a fixed value of slip parameter = 0.1. From these figures we conclude that increase in as well as decrease in a is to increase K1. This is advantageous in maintaining the laminar of flow. Figure 5 is a plot of dispersion coefficient K2 against electric number We for different values of

   e

   . From this we conclude that increase in and We is to decrease the effective dispersion coefficient K2. The scaled dispersion coefficient K2- P 2 is plotted versus in figure 6 for

   different values of a. From this figure, it is clear that the increase in a and , the effective dispersion coefficient decreases. These are useful in the control of dispersion of a solute.

   The cross sectional average concentration m is plotted versus x in figures 7 and 8 respectively for different values of a, and for fixed values of the other parameters given in these figures. It is clear that the increase in and a increases m as expected on the physical grounds. m is also plotted in figures 9 and 10 against the dimensionless time for different values of and a for fixed values of the other parameters given in these figures. We note that the peak of m decreases with an increase in occuring at the lower interval of time . We also note that the peak increases with an increase in a but occurs at almost at the same interval of time . These informations are useful to understand the transport of solute at different times.

   The proposed model and analysis presented here also suggests that to remove the pollutants from the atmosphere, external species may be introduced in the atmosphere which can interact with the pollutant and remove it by some suitable removal processes based upon the physical and chemical properties of the pollutants as well as that of externally introduced species.

   Acknowledgment: The author's interest in this work was originally stimulated by corre- spondence with Dr. N. Rudraiah, Honorary Professor, UGC-Centre for Advanced Studies in Fluid Mechanics, Department of Mathematics, Bangalore University. The authors are thankful for his helpful suggestion and constructive criticism.

4. References

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6. Gupta, P.S., and Gupta, A.S. Effect of homogenous and heterogenous reactions on the dispersion of a solute in the laminar flow between two plates. Proc. roy. Soc. London., A 330, (1972), 89.

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Table 1: Roots of the equation n tan n =

Figure 2. Velocity profiles of aerosols or different We

Figure 3. Plots of exchange coefficient versus reaction rate parameter

Figure 5. Effect of electric number We on depersion coefficient K2 for different values of

Figure 4. Plot of convective coefficient K, with wall reaction parameter for different

values of a.

e

Figures 6. Plots of scaled dispersion coefficient K2( )- P -2 against for different values of a.

Figures 7. Mean concentration ( m) varying along axial distance x for different values of a and for fixed = 0.06, =0.01

Figures 8. Mean concentration ( m) varying along axial distance x for different values of and for fixed = 0.06, a=1

Figures 9. Mean concentration ( m) varying along dimensionless time for different values of a at x=0.6, =0.01

Figures 10. Mean concentration ( m) varying along dimensionless time for different values of

at x=0.6, a=1