amo advanced modeling and optimization, volume 15 number 3, … · 2016-10-28 · amo – advanced...

AMO – Advanced Modeling and Optimization, Volume 15 Number 3, 2013

Chemically reacting MHD free convection flow on a vertical porous plate in a porous

medium with Dufour effect.

A.J. Omowaye

Department of Mathematical Sciences

Federal University of Technology,

P.M.B 7 04,Akure,Ondo-State, Nigeria.

Abstract.

Unsteady flow of an incompressible electrically conducting viscous fluid past a semi-infinite

accelerating vertical porous medium is considered. The diffusion- thermo effect was taken

into consideration. The coupled partial differential equations describing the conservation of

mass, momentum and energy are obtained and solved analytically. The present analytical

results were compared with those available in the literature excellent agreement was

observed. The effects of flow parameters and thermo physical properties on the flow,

velocity, temperature and concentration fields across the boundary layer are investigated.

The forms of wall shear stress, Nusselt number and Sherwood number are derived. The fluid

velocity decreases as either Schmidt number, porosity parameter, Hartmann number

increased while temperature increased as Dufour number increased. The concentration

decreased as Schmidt number increased.

Keywords: Diffusion-thermo, MHD, free convection, chemically reacting, boundary layer.

Corresponding Author Email [email protected]

AMO - Advanced Modeling and Optimization. ISSN: 1841-4311

A.J.Omowaye

746

1. INTRODUCTION

The problem of heat and mass transfer to unsteady magneto hydrodynamics (MHD) flow is

encountered in a variety of applications such as MHD power generator and pumps,

accelerators, aerodynamics heating, petroleum industry, electrostatic precipitation,

purification of crude and molten metals [[Schlicting,1986], [Sutton and Sharman,1965]].

Fluid convection at vertical plates resulting from buoyancy forces find applications in

several industrial and technological field such as nuclear reactors, heat exchangers,

electronic cooling equipments and aeronautics among others [Sutton and Sharman,1965].

Unsteady natural convection heat and mass transfer is of immense importance in the design

of control systems for modern free convection heat exchange devices. Furthermore, heat

transfer processes that required the evaluation of the performance of thermal equipment in the

unsteady free convection regime include start-up, shut-down, pump failure and so on. I must

remark here that ,free convection induced by temperature gradients has been studied

extensively by researchers with the assumption that other influences are so small that they are

neglected. Mean while, the study of transport phenomena in a porous media has attracted the

attension of theorists and experimentalists in recent years, due to its scope in various fields

of engineering and environmental sciences. In recent times porous media models are being

simulating more general situations such as flow through packed and fluidized beds. The

majority of the studies on convection heat transfer in porous media are based on Darcy’s

law [[Darcy,1856],[ Ingham and Pop,2002]].

[Chamkha,2002] discussed the problem of unsteady laminar combined forced-free

convection flow in a square cavity in the presence of internal heat generation/absorption and

Chemically reacting MHD free convection flow on a vertical porous plate in a porous…

747

a magnetic field was formulated. Both the top and bottom horizontal walls of the cavity were

insulated while the left and right vertical walls were kept constant at different temperatures.

The left vertical wall was moving in its own plane at a constant speed while all other walls

fixed. A uniform magnetic field was applied in the horizontal direction normal to the

moving wall. A temperature dependent heat source or sink was assumed to exist within the

cavity .The governing equations were solved numerically by the finite-volume approach

along with alternating direct implicit (ADI) procedure. Parametric study was conducted and

the results was in agreement with the existing work in the literature. [Kham,2006] reported

unsteady boundary layer free convection flow of an incompressible electrically conduction

viscoelactic second order fluid over a vertical permeable flat plate, where temperature and

concentration were responsible for the convective buoyancy current. The flow was affected

by a constant suction of the fluid through the permeable wall in the presence of a

temperature-dependent heat source\sink and applied transverse magnetic field. This intricate

mathematical problem was solved analytically. The effect of various non-dimensional

physical parameters such as the viscoelastic parameter, Grashof number, modified Grashof

number, source/sink parameter, frequency parameter, time dependency parameter, Prandtl

number, Schmidt number, permeability parameter and magnetic parameter. Some of the

several findings of the results were the combined effects of increasing the values of

viscoelastic parameter, modified Grashof number and permeability parameter was to

enhance the horizontal velocity profile largely in the boundary layer. [Mansour et al,2007]

studied the effects of combined heat and mass diffusion effects in the presence of viscous

dissipation and chemical reaction in MHD natural convection flow saturated in porous media

A.J.Omowaye

748

with suction or injection flows. An approximate numerical solution for the flow problem was

obtained by solving the governing equations using shooting technique with fourth order

Runge-Kutta integration scheme. The results obtained show that the flow field was

influenced appreciably by the presence of chemical reaction, viscous dissipation and suction

or injection flow. [Soundalgekar et al,1995] presented an exact solution to the flow of a

viscous incompressible fluid past an infinite vertical oscillating plate ,in the presence of a

foreign mass by Laplace transform technique when the plate temperature was linearly varying

as time .The velocity profiles were shown on graphs and numerical values of the skin-

friction were listed in a table. It was observed that the skin-friction increases with increasing

Sc, t ,or Gr but decrease with increasing Gm [Mansour etal,2007]studied MHD flow of a

micro polar fluid due to heat and mass transfer through a porous medium bounded by an

infinite vertical porous plate in the presence of a transverse magnetic field in slip-flow

regime .The effects of flow parameters and thermo physical properties on the flow,

temperature and concentration field across the boundary layer were discussed The form of

wall shear stress ,wall couple stress, Nusselt number and Sherwood number were derived .It

was shown that skin-friction coupled stress, heat transfer and mass transfer with various

values i showed complete oscillating nature.

In many transport processes in nature flow is driven by density differences caused by

temperature gradients, chemical composition gradients and material composition as

highlighted by [Gebhart and Pera,1971]. It is therefore important to study flow induced by

concentration differences independently of simultaneously with temperature differences when

heat and mass transfer occurs simultaneously in a moving fluid, the relationship between the

fluxes and the driving potentials are of a more intricate nature [Kafoussians and


749

Williams,1995]. It has been found that an energy flux can be generated not only by

temperature gradients but by composition gradients as well. The energy flux caused by the

composition gradient is called the Dufour effect. If on the other hand, mass fluxes are created

by the temperature gradients, it is called the soret effect.. These effects are generally of a

small order of magnitude and are often neglected in heat and mass transfer processes [Jha and

Ajibade,2010]. However, soret and the Dufour effects have been found to be of importance as

the soret effect is utilized for isotope separation and in a mixture of gases of light and

medium molecular weight, the Dufour effects was found to be a considerable order of

magnitude such that it cannot be neglected [Eckert and Drake,1972].

Recently, [Beg et al,2007]obtained numerical solutions of chemically reacting mixed

convective heat and mass transfer along inclined and vertical plates with the soret and the

Dufour effects and concluded that skin friction increases with positive increase in

concentration-to-thermal-buoyancy ratio parameter (N). In the present analysis, it is proposed

to study unsteady free convective flow of an electrically conducting fluid with chemical

reaction and mass transfer past an accelerating vertical porous plate embedded in a porous

medium in the presence of Dufour and magnetic field. A novel feature of this work is to

present analytical solutions for velocity, temperature and concentration distributions and

these solutions are shown graphically. In addition ,the analytical solutions obtained in this

present work ,are very important as they serve as accuracy checks for experimental and

asymptotic method.

2. MATHEMATICAL ANALYSIS

The flow considered is an unsteady flow of an incompressible electrically conducting viscous

fluid past a semi-infinite accelerating vertical porous plate in a porous medium. A constant

A.J.Omowaye

750

fluid suction or blowing is imposed at the plate surface with a uniform transverse magnetic

field. By assuming a very small magnetic Reynolds number the induced magnetic is

neglected. It is assumed that the plate surface temperature and concentration are varying

exponentially with time and a chemical species diffused into the ambient fluid, initiating a

first order irreversible chemical reaction. The x-axis is taking along the plate and y-axis is

taken normal to it. In the governing equations the temperature is governed by concentration,

leading to the diffusion – thermo(Dufour) effect. The flow configuration and co-ordinate

system is shown in fig 1

Figure 1: Flow configuration and coordinate system

Also, the Boussinesq approximation is invoked, thereby confining the density variation to the

buoyancy term. Based on these simplifying assumptions and discussion, the model equations

in dimensionless form after neglecting the bar system for clarity and assumed constant

suction the governing equations can be written as

2

2

2( )a

u u uM H u GrT GcC

t y y

(2.1)

2 2

2

2 2

1

Pr Pr

DT T T C

t y y y

(2.2)

0

mtUe g

,T C

x

y

u

v


751

2

2

1C C CC

t y Sc y

(2.3)

with

, 0mt mt mtu he T e C e at y

0 , 0 0u T C as y (2.4)

The non-dimensional quantity introduced in the above equations are defined

as0

''

''

''

'''

0

''

0 ,,Pr,,,u

uh

CC

CCC

k

C

TT

TTT

utt

yuy

w

p

w

( 2.5)

22 0 1

22 2 2

( ) ( ) ( ), , , ,

( )

T w c w wa

w

g T T g C C D C CGr Sc Gc H D

v D v v T T

Here Pr

is the Prandtl number,D2 is the Dufour number, which is the coefficient of the

concentration-energy diffusion, Sc is the Schmidt number, Ha is the Hartmann number, Gr is

the temperature Grashof number and Gc is the mass Grashof number. The physical quantities

used in equation (2.5) are defined in the nomenclature. Without loss of generality, it has been

assumed that the fluid velocity, temperature and concentration vary exponentially in time

.Therefore, we take

( , ) ( ) ( , ) ( ) ( , ) ( )mt mt mtu y t F y e T y t y e C y t y e (2.6)

which transform equations (2.1)-(2.3) to ordinary differential equations of the form

'' ' 2( ) 0aF F m M H F Gr Gc (2.7)

'' ' ''Pr Pr 2 0m D (2.8)

'' ' ( ) 0Sc Sc m (2.9)

where m is a real number and the prime symbol denotes differentiation with respect to y.

The boundary conditions after transformation are

A.J.Omowaye

752

1 1 0F h at y (2.10)

0 0 0 0F as y

Solving the system of second order ordinary differential equations (2.7)-(2.9) with boundary

conditions (2.10) we obtain the following results

F(y) = 8 1 2

5 6 7

a y a y a ya e a e a e

2 1

4 4( ) (1 )a y a y

y a e a e

1( )a y

y e

where the constants 1 8...a a are given in the appendix.

The skin friction coefficient, Nusselt number and the Sherwood number are important

physical parameters for this problem. These can be defined as

2

0 0

(0),f

y

dF duC

u dy dy

'

0 0

(0) ,( )

ww

w y

q dTNu q k

ku T T dy

'

0

(0) ,( )

ww

w w y

J dcSh J D

C C D dy

According to the analytical solutions reported before Cf ,Nu and Sh take on the respective

forms:

Cf = 721685 aaaaaa

Nu = 4142 )1( aaaa

Sh = 1a

3 RESULTS AND DISCUSSION


753

The present article has considered chemically reacting MHD free convection flow of a

viscous incompressible fluid on a vertical porous plate in a porous medium with the

Dufour effect. The governing equations along with the boundary conditions have been

solved in the preceeding section in order to give the details of flow fields, thermal and

concentration distributions. The effects of the main controlling parameters as they appear

in the governing equations are discussed in the current section. In the entire numerical

computations we have chosen t = 0,m = 0.5,h = 0.1 while Sc, 2D ,Ha ,M, Gr, Gc and

are varied over ranges which are listed in the figure legends. Since air and water are two

very important fluids, the choice of Pr are 0.71 and 7.0 corresponding to air and water

respectively. In air, the diffusing chemical species of common interest have Schmidt

numbers in the range 0.1-10,therefore this range is considered . For water, the species

H2,H2O,CO2,salt and propylBenezene are perphas the typical species of most interest

[Gebhart and Pera,1971]. In addition, we have focus attention on positive values of the

buoyancy parameters Gr > 0 (which corresponds to the cooling problem) and Gc >0

(which indicates that the free stream concentration is less than the concentration at the

boundary surface) . The cooling problem is often encountered in engineering applications

[Phiri and Makinde,2007].It should be noted that >0, =0 and <0 represent

destructive, no and generative chemical reactions respectively. The expressions for the

concentration, temperature, velocity, Nusselt number and skin-friction are presented

graphically in figs 2-17. Fig2 depicts the species concentration for different gases

hydrogen (H2 :Sc = 0.24) water vapour (H20: Sc = 0.60)and propyl Benzene :Sc =

2.62.The values of Schmidt number (Sc) are chosen to represent the most common

diffusing chemical species which are of interest. A comparison of curves in the figure

A.J.Omowaye

754

shows a decrease in concentration distribution with an increase in Schmidt number

because the smaller values of Sc are equivalent to increasing chemical molecular

diffusivity (D).Hence, the concentration of the species is higher for small values of Sc and

lower for larger values of Sc ,this results is in agreement with [[Mansour et al,2007],[

Phiri and Makinde,2007],[Soundalgekar et al,1995]]. The concentration profiles also

decreases with increase in generative chemical reaction fig 3.Moreso,the concentration

profiles attain maximum values at the plate and decrease exponentially with y and finally

tends to zero as y tends to infinity. Figs 4-7 show variations of the temperature profiles

along the span-wise coordinate y for different values of Prandtl number, Dufour

number ,Schmidt number and generative chemical reaction. In fig 4, the temperature

decreases with increase in the Prandtl number. This is in agreement with physical fact that

the thermal boundary layer thickness decreases with increase in Prandtl number. The

reason underlying such a behavior is that the high Prandtl number fluid has a relatively

low thermal conductivity. This results in the reduction of the thermal boundary layer

thickness, this is in agreement with [[Chamkha,2003],[ Jha and Ajibade,2010],[ Mansour

et al,2007]]. Fig 5,shows an increase in temperature for different values of Dufour. It is

evident that Dufour effect assists the temperature of the fluid to increase. This means that

concentration exerts a greater influence on temperature this agrees with [Jha and

Ajibade,2010].Figs 6 and 7 show a typical variations in the temperature profiles along the

span wise coordinate y for different values of the Schmidt number and generative

chemical reaction parameter. The graphs show an increase in temperature as Schmidt

number and chemical reaction parameter increases. It is noteworthy in figs 4,6,7 that

,there exists an overshooting of the temperature for small values of Pr, Sc, (e.g. Pr=


755

0.71,Sc = 0.24, =0.1),the temperature overshoot decreases with increasing Pr, Sc, and

vanishes for higher values of Pr, Sc, ( e.g. Pr= 7.0,Sc = 2.62, =0.9).In addition, in figs

4 -7 the fluid temperature reaches its maximum value at short distance from the plate and

decrease to zero value away from the plate. Figs 8-14 depict the velocity profiles for

different important parameters. Generally speaking, the streamwise velocity profiles show

an increase, a peak value near the wall of the porous plate and then decrease gradually to

free stream zero value far away from the plate. In fig8 the velocity of the fluid decreases

as Schmidt number increases. This reduction in velocity profiles is accompanied by

simultaneous reduction in the boundary layer thickness .This observation is in agreement

with those reported in [[Chamkha,2003],[ Phiri and Makinde,2007]]. Figs 9 and 10 depict

the influence of Grashof number (Gr) and mass Grashof number (Gc) on the velocity

profiles. Obviously increasing the Grashof number and mass Grashof number aids the

flow because buoyancy force which stretch the fluid ,produces higher fluid velocities. It is

in agreement with what is reported in [ [Chamkha,2003],[ Mansour et al,2007],[Phiri and

Makinde,2007],[ Soundalgekar et al,1995]].Moreover,fig11 reveals that on increasing the

values of the permeability parameter (M) the profiles of F tends to decrease this is in

agreement with what is reported in [[Ingham and Pop,2002],[ Mansour et al,2007],[Phiri

and Makinde,2007]].Also, it is observed that, keeping other parameter fixed, as Hartmann

number increases, the velocity decreases.

The presence of a magnetic field in an electrically conducting fluid introduces a force

called Lorentz force which acts against the flow if the magnetic field is applied in the

normal direction as considered in the present problem .This type of resistive force tends

to slow down the flow field. This result agree with what is reported in [[Chamkha, 2003],[

A.J.Omowaye

756

Mansour et al,2007],[ Phiri and Makinde,2007]]. Figs 13 and 15 illustrate the influence of

chemical reaction parameter and Prandtl number on velocity profiles for various value of

these parameters. It is observed that there is decrease in velocity as these parameter

increases, this is in line with what are reported in [[Chamkha,2003],[ Mansour et al,2007],[

Phiri and Makinde,2007]]. In fig 14,we observed that the fluid velocity reaches its maximum

value at short distance from the plate and decreases to zero value away from the plate. It is

interesting to note that Dufour have effect of increasing the velocity of the fluid as reported

by [Jha and Ajibade,2010]. Fig 16 shows a reduction in skin friction at the plate surface with

increase in chemical reaction parameter and Hartmann number . Also, there is evidence that it

will converge at far distance from the plate. Figs 17 and 18 show the effect of Schmidt

number and chemical reaction parameter on Nusselt number and Sherwood number ,from

these figures we observed that both Nusselt number and Sherwood number increase with

increase in Schmidt number and chemical reaction parameter.


757

Figure 2 : Concentration Profiles taking m = 0.5,g=0.1, 0.1 for

various Sc

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4y

C(y)Sc =0.24

Sc=0.60

Sc = 2.62

Figure 3: Concentration Profiles taking m =0.5,Sc = 0.24 for various

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4y

C (y) 0.1

0.5

0.9

A.J.Omowaye

758

Figure 6:Temperature profiles taking Pr = 0.71,m = 0.5,D2 = 2.0, =

0.1 for various Sc

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3 4 5 6 7 8 9 10y

( y )

Sc = 0.24

Sc = 0.60

Sc = 2.62

Figure 7: Temperature profiles taking Pr =m 0.71,m = 0.5,D2 = 2.0,

Sc = 0.24 for various

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10y

( y ) 0.1

0.4

0.9


759

Figure 8: Velocity profiles taking D2 = 2.0,Pr = 0.71,M = 2.0,h = 0.1,Gc

= 2.0,Gr = 3.0,Sc = 0.24,Ha = 0.1, = 0.1 for various Sc

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10y

F ( y )Sc = 0.24

Sc = 0.60

Sc = 0.78

Figure 9:Velocity profiles taking Pr = 0.71,M = 2.0,D2 = 2.0,Gr =

3.0,Sc = 0.24,Ha = 0.1, = 0.1 for various Gc

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5 6 7 8 9 10

y

F ( y )

Gc = 2.0

Gc = 4.0

Gc = 6.0

A.J.Omowaye

760

Figure 10: Velocity profiles taking Pr = 0.71,h = 0.1,Gc = 2.0,D2 =

2.0,Gc = 2.0,D2 = 2.0,Ha = 0.1Sc = 0.24, = 0.1 for various Gr

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10y

F ( y )

Gr =3.0

Gr = 6.0

Gr = 9.0

Figure 11:Velocity profiles taking Pr = 0.71,h = 0.1,Gr = 3.0,Gc =

2.0,Sc = 0.24, D2 = 2.0,Ha = 0.1, = 0.1 for various M

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 10y

F ( y )M = 2.0

M = 4.0

M = 6.0


761

F ig 12:Velocity profiles taking Pr = 0.71,D2 = 2.0,M = 2.0,Sc = 0.24,Gr

= 3.0,Gc = 2.0,h = 0.1,g = 0.1 for various Ha

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

y

F ( y)Ha = 0.1

Ha = 0.3

Ha = 0.5

Figure 13:Velocity profiles taking Pr = 0.71,M = 2.0,D2 = 2.0,Gr =

3.0,Gc = 2.0,Ha = 0.1,Sc = 0.24 for various

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

y

F ( y )

0.1

0.4

0.9

A.J.Omowaye

762

Figure 15:Velocity profiles taking Sc =0.24,M = 2.0,Gr = 3.0,Gc = 2.0,h

= 0.1,Ha = 0.1,D2 = 2.0, = 0.1 for various Pr

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10y

F ( y ) Pr = 0.71

Pr = 2.0

Pr = 7.0

Figure 16: Skin-friction coefficient taking Pr = 0.71,Sc = 0.24

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1 1.2

Ha

Cf 0.1

0.4

0.9


763

Figure 17:Temperature gradient

-18.4

-16.4

-14.4

-12.4

-10.4

-8.4

-6.4

-4.4

-2.4

-0.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4

PrNu

Sc = 0.24

Sc = 0.62

Sc = 2.62

4. CONCLUDING REMARK

The general problem of chemically reacting MHD free convective flow on a vertical porous

plate in a porous medium with Dufour effect has been studied theoretically and solved

exactly without approximations. The parameters that govern the flow situation are Prandtl

number, Schmidt number, chemical reaction parameter ,Grashof number, mass Grashof

number , Dufour number, Hartmann number and porous parameter. Extensive computations

have been carried out for different values of these parameters entering in the problem and

their results are presented graphically .We must remark that Dufour effect exerts a very

A.J.Omowaye

764

significant influence on the temperature and velocity distribution. The main conclusions of

the present work are as follows :

1. The fluid velocity decreased as either Schmidt number ,porosity parameter, Hartmann

number, chemical reaction parameter, Prandtl number increased while the velocity

increased as thermal Grashof number, mass Grashof number and Dufour number

increased.

2. The fluid temperature decreased as Prandtl number increased and increased as

Dufour number ,Schmidt number and chemical reaction parameter increased.

3. The fluid concentration decreased as Schmidt and chemical reaction parameter

increased.

4. The skin-friction coefficient decreased as chemical reaction parameter increased.

5. The Nusselt number and Sherwood number increased as Schmidt and chemical

reaction parameter increased.

NOMENCLATURE

C – Concentration

Cw- Wall concentration

C∞- Ambient concentration

Cp- Specific heat at constant pressure

Cf- Skin- friction coefficient

D- Mass diffusion coefficient

D2- Dufour number

D1-Dimensional coefficient of the

diffusion-thermo effect.

F- Dimensionless velocity

g- Acceleration due to gravity

Gc-Mass Grashof number

Gr- Thermal Grashof number

Jw-Rate of transfer of species

concentration


765

K- Fluid thermal conductivity

M- Porosity parameter

Nu- Nusselt number

Pr-Prandtl number

qw- Local surface heat flux

Sc- Schmidt number

Sh- Sherwood number

T- Fluid temperature

T∞-Ambient temperature

Tw-Wall temperature

t- Time

U- Plate velocity parameter

u- Velocity of the fluid in the upward

direction

v- Velocity of the fluid in the

horizontal direction

x- Coordinate axis along the plate in

the vertically upward direction.

y-Coordinate axis normal (horizontal)

to the plate

m,h-Real numbers

Greek Symbol

α- Thermal diffusivity

βc-Coefficient of concentration

expansion

βT- Coefficient of thermal expansion

β0-Magnetic induction

ρ- Fluid density

ν- Kinematic viscosity

μ- Fluid dynamic viscosity

σ-Fluid electrical conductivity

γ-Chemical reaction parameter

θ-Dimensionless temperature

- Dimensionless concentration

- Local wall shear stress

A.J.Omowaye

766

APPENDIX

2

1

( ) 4( )

2

Sc Sc ma

2

2

(Pr) 4 Pr

2

Sc ma

3 4(1 )a a

2

14 2

1 1

2

( Pr Pr)

D aa

a a m

5 6 7( )a h a a

46 2 2

1 1

( )

( ( )a

Gc Graa

a a m M H

4

7 2 2

2 2

(1 )

( ( )a

Gr aa

a a m M H

2

8

1 1 4( )

2

am M Ha


767

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