Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus: an exact solution

Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus An exact solution Engineering Science and Technology, an International Journal xxx (2017)[.]

Full Length Article

Combined effect of heat source, porosity and thermal radiation on mixed

convection flow in a vertical annulus: An exact solution

Michael O Oni

Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria

a r t i c l e i n f o

Article history:

Received 18 September 2016

Revised 5 December 2016

Accepted 25 December 2016

Available online xxxx

Keywords:

Heat source

Thermal radiation

Mixed convection

Vertical annulus

Porous material

Exact solution

a b s t r a c t This paper examines the effect of heat source, thermal radiation and porosity on mixed convection flow in

a vertical annulus filled with porous material The inner surface of outer cylinder is assumed to be the heated surface Closed-form expression for temperature, velocity, Nusselt number, skin-friction and mass flow rate are obtained in terms of Bessel’s function and modified Bessel’s function of first and second kind Based on depicted graphs, fluid temperature and Nusselt number increase with increase in radiation parameter and heat source parameter while velocity as well as skin-friction decreases with increase in radiation parameter and heat source parameter at the surfaces of the cylinder

Ó 2017 Karabuk University Publishing services by Elsevier B.V This is an open access article under the CC

BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1 Introduction

Over the years, owing to the fact that cylinders have been used

in nuclear waste disposal, energy extortion in catalytic beds and

undergrounds, convective heat transfer about cylindrical

geome-tries has begun to attract the attention of many researchers since

some fluids are good emitter and absorber of thermal radiation,

it is of interest to study the effect of heat source on temperature

distributions and heat transfer when the fluid is capable of

emit-ting and absorbing thermal radiation This can be attributed to

the fact that heat transfer by thermal radiation is becoming of

greater importance when we are concerned with space

applica-tions (space journey) and higher operating temperatures

Several investigations have been carried out on problem of heat

transfer by radiation as an important application of space and

tem-perature related problems Greif et al.[1]obtained an exact

solu-tion for the problem of laminar convective flow in a vertical

heated channel in the optically thin limit They concluded that in

the optically thin limit, the fluid does not absorb its own emitted

radiation which means that there is no self-absorption but the fluid

does absorb radiation emitted by the boundaries Viskanta [2]

investigated the forced convective flow in a horizontal channel

permeated by uniform vertical magnetic field taking radiation into

account In his work, he studied the effects of magnetic field and radiation on the temperature distribution and the rate of heat transfer in the flow and found that the effect of magnetic field is

to decrease fluid velocity Later Gupta and Gupta[3]studied the effect of radiation on the combined free and forced convection of

an electrically conducting fluid flowing inside an open-ended ver-tical channel in the presence of a uniform transverse magnetic field for the case of optically thin limit They found that radiation tends

to increase the rate of heat transfer of the fluid there by reducing the effect of natural convection

Later, Hossain and Takhar[4]analyzed the effect of radiation using the Rosseland diffusion approximation which leads to non-similar solution for the forced and free convection of an optically dense viscous incompressible fluid past a heated vertical plate with uniform free stream and uniform surface temperature, while Hos-sain et al.[5]studied the effect of radiation on free convection from

a porous vertical plates

The role of thermal radiation is of major importance in the design of many advanced energy convection systems operating at high temperature and due to increase in science and technology, radiative heat transfer becomes very important in nuclear power plants, gas turbines and various propulsion devices for aircraft, missiles and space vehicles[6–10]

In cylindrical geometry, the studies of heat generation and ther-mal radiation have been studied by several authors Chamkha[11]

analyzed the heat and mass transfer of a MHD flow over a moving

http://dx.doi.org/10.1016/j.jestch.2016.12.009

2215-0986/Ó 2017 Karabuk University Publishing services by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

E-mail address: Michaeloni29@yahoo.com

Contents lists available atScienceDirect Engineering Science and Technology,

an International Journal

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / j e s t c h

Please cite this article in press as: M.O Oni, Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus:

permeable cylinder with heat generation or absorption and

chem-ical reaction He concluded that the role of Hartmann number is to

decrease fluid velocity Trujillo et al.[12]studied the heat and mass

transfer process during the evaporation of water from a circular

cylinder through CFD modeling In other related work, Mujtaba

and Chamkha[13] discussed the heat and mass transfer from a

permeable cylinder in a porous medium with magnetic field and

heat generation/absorption Ganesan and Loganathan [14,15]

investigated an unsteady natural convective flow past

semi-infinite vertical cylinder with heat and mass transfer under

differ-ent physical situations Hossain et al.[16]reported the radiation

conduction interaction on mixed convection from a horizontal

cir-cular cylinder using an implicit finite-difference scheme

Also, Ganesan and Loganathan[17]studied the radiation and

mass transfer effects on flow of an incompressible viscous fluid

past a moving vertical cylinder Gnaneswar and Reddy[18,19]

ana-lyzed the radiation and mass transfer effects on an unsteady MHD

free convection flow of an incompressible viscous fluid past a

mov-ing vertical cylinder Radiation effects on hydromagnetic free

con-vective and mass transfer flow of a gas past a circular cylinder with

uniform heat and mass flux was studied by Hakiem[20] He also

found that magnetic field parameter retards fluid velocity Yih

[21]analyzed the radiation effect on natural convection over a

ver-tical cylinder embedded in a porous media Suneetha and Bhaskar

[22]analyzed the radiation and mass transfer effects on MHD free

Convection flow past a moving vertical cylinder embedded in a

porous medium Other related articles on radiation effect on heat

transfer of mixed convection flow for different fluid can be seen

in[23–25]

The purpose of this paper is to examine theoretically the effects

of thermal radiation and porosity on viscous, incompressible and

heat generating fluid in a vertical annulus filled with porous

mate-rial Exact solution for temperature, velocity, skin-friction and

Nus-selt number are obtained and the effects of governing parameters

are discussed with the aid of line graphs

2 Mathematical analysis Consider the steady laminar mixed convection flow of a viscous incompressible heat generating fluid The axis of cylinder is taken along the z-axis, while r-axis is taken in the radial direction The inner surface of the outer cylinder is assumed to be heated to a temperature Tw greater than that of surrounding fluid and outer surface of the inner cylinder having temperature T0 The radius

of the inner and outer cylinder walls are a and b respectively as

Nomenclature

a radius of inner cylinder

b radius of outer cylinder

g gravitational acceleration

h convective heat transfer coefficient

H dimensionless heat generating (source) parameter

In modified Bessel’s function of first kind on order n

Where n¼ 0; 1; 2; 3;

Jn Bessel’s function of first kind on order n Where

n¼ 0; 1; 2; 3;

Kn modified Bessel’s function of second kind on order n

Where n¼ 0; 1; 2; 3;

Nu0 rate of heat transfer at the outer surface of the inner

cylinder

Nu1 rate of heat transfer at the inner surface of the outer

cylinder

p dimensional pressure difference

P dimensionless pressure difference

Q0 dimensional heat generating (source) parameter

R dimensionless coordinate

T dimensional temperature of the fluid

T0 initial temperature

Tw final temperature

u dimensional velocity of fluid

U dimensionless velocity of fluid

Yn Bessel’s function of second kind on order n Where

n¼ 0; 1; 2; 3;

Z dimensionless coordinate Greek alphabets

a thermal diffusivity

b coefficient of thermal expansion

j mean absorption coefficient

k aspect ratioðb=aÞ

h dimensionless temperature of fluid

Subscript

1 outer surface of inner cylinder

k inner surface of outer cylinder

Fig 1 Schematic diagram.

Please cite this article in press as: M.O Oni, Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus:

seen in Fig 1 The following assumptions are made in order to

obtain an exact solution:

i The fluid motion is fully developed both thermally and

hydrodynmaically

ii All physical quantities except pressure gradient are

indepen-dent on z-axis

iii Viscous dissipation and displacement currents terms are

neglected in the energy equation

iv The fluid is considered to be a gray, absorbing emitting

radi-ation but non-scattering medium

Under the above assumptions, the governing equations of the

flow are respectively the continuity, momentum and energy

equations:

dðv0Þ

meff r

d

dr r

du0 dr

þ bgðT  T0Þ mu0

K 1

k

qCp

1 r

d

dr r

dT0 dr

q1Cp

1 r

d

drðrqrÞ þQ0ðT  T0Þ

u0¼ 0 T ¼ T0 at r¼ a

By using the Rosseland approximation (Brewster [26]), the radiative heat flux is given by qr

qr¼ 4r

3k

@T4

Following[27], the function T4in Eq.(5)can be expressed as a linear function by expanding it in a Taylor series about Tw and neglecting higher powers of T as follows:

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

R

H = 0.5

H = 2.5

N = 0.5, 1.5, 2.5, 3.5

Fig 2 Temperature profile versus R varying H and N at k = 2.0.

0 0.2 0.4 0.6 0.8 1

R

N = 0.5

N = 3.5

λ = 2.2, 2.0, 1.8

Fig 3 Temperature profile versus R varying k and N at H = 1.5.

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T4¼ 4T2

wT 3T4

Introducing the following dimensionless parameters:

U¼u0

u0; Re ¼ul0a; k ¼ba; c¼meff

m ; P ¼upa0l; Da¼aK2;

h ¼ T T0

Tw T0; R ¼ra; Z ¼az; N ¼ jk

4rT3; H2

¼Q0b2

k ;

Gr¼gbðT2 T0Þa3

Eqs.(2)–(6)in dimensionless form become:

c

R

d

dR R

dU

dR

þGrReh DaU dPdZ¼ 0 ð8Þ

1 R

d

dR R

dh dR

þ 3H 2 N

U¼ 0 h ¼ 0 at R ¼ 1

The solution of Eqs.(8)–(10)is obtained as:

hðRÞ ¼ C1J0ðm1RÞ þ C2Y0ðm1RÞ ð11Þ UðRÞ ¼ C3I0

R ffiffiffiffiffiffiffiffiffi

Dac

p

!

þ C4K0

R ffiffiffiffiffiffiffiffiffi

Dac

p

!

þGr Re

Da

cm2Daþ 1

  C½ 1J0ðm1RÞ

þC2Y0ðm1RÞ  DadP

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

R

H = 0.5

H = 2.5

γ = 0.5, 1.0, 1.5

Fig 4 Velocity profile versus R varying H andcat N = 1.5, k = 2.0, Gr/Re = 50, Da = 0.1.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

R

N = 0.5

N = 3.5

Da = 0.001, 0.01, 0.1

Fig 5 Velocity profile versus R varying Da and N at H = 2.5,c= 1.5, k = 2.0, Gr/Re = 50.

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The rate of heat transfer between the surfaces of the cylinders

and the fluid is obtained by differentiating the temperature and

is given by:

Nu1¼dh

dR

R ¼1

¼ m1½C1J1ðm1Þ þ C2Y1ðm1Þ ð13Þ

Nuk¼ dh

dR

R ¼k

¼ m1½C1J0ðkm1Þ þ C2Y0ðkm1Þ ð14Þ

The skin friction at the outer surface of inner cylinder and inner

surface of outer cylinder is respectively given by:

s1¼dU

dR

R ¼1

¼ ffiffiffiffiffiffiffiffiffi1

Dac

p C3I1

1 ffiffiffiffiffiffiffiffiffi

Dac

p

!

 C4K1

1 ffiffiffiffiffiffiffiffiffi

Dac

p

!

GrRe Dam1

ðcm2Daþ 1Þ½C1J1ðm1Þ þ C2Y1ðm1Þ ð15Þ

sk¼ dUdR

R¼k¼ ffiffiffiffiffiffiffiffiffi1

Dac

p C4K1 ffiffiffiffiffiffiffiffiffik

Dac

p

!

 C3I1 ffiffiffiffiffiffiffiffiffik

Dac

p

!

þGr Re

Dam1

ðcm2Daþ 1Þ½C1J1ðkm1Þ þ C2Y1ðkm1Þ ð16Þ

The amount of fluid passing through the annulus is given by the volume flow rate and defined as:

V¼

Zk 1

where n1; n2; n3; n4are constants defined in theAppendix The pressure gradient is obtained using the conservation law as:

Z k 1 RUðRÞdR ¼

Z k 1

0 0.5 1 1.5 2

R Gr/Re = 200, 150, 100, 50, 0, -50, -100, -150, -200

Fig 6 Velocity profile versus R varying Gr/Re at H = 1.5, N = 1.5,c= 1.5, k = 2.0.

1

1

1.5

1.5

2

2

2.5

2.5

3

3

3

3

3.5

3.5

3.5

3.5

4

4

4

4

4.5

4.5

4.5

5

5

5

5 5

H

2 3 4 5 6 7 8

Fig 7 Nusselt number for different values of H and N at k = 2.0, R = 1.

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dZ¼p1

p2

ð20Þ

where p1and p2are constants defined inAppendix

The critical Gr=Re is obtained by:

dU

dR

Using Eq.(21), the reverse flow occurrence at the outer surface

of inner cylinder and inner surface of outer cylinder is respectively

given by:

Gr

Re

R¼1¼t23

Gr Re

R¼1¼t25

there tifor i¼ 1; 2; 3; are constant defined inAppendix

3 Results and discussion The solution obtained from Eqs.(11)–(23)are seen to be govern

by heat source parameterðHÞ, radiation parameter ðNÞ, Darcy num-berðDaÞ, aspect ratio ðkÞ, viscosity ratio ðcÞ and mixed convection parameterðGr=ReÞ In order to see the effect of these parameters, lines graphs are plotted to capture the physical situation Through-out this work, the heat source parameter is taken over the range

0:5 6 H 6 3:0 with reference value of 2.5, thermal radiation over

1

1

1.5

1.5

2

2

2.5

2.5

2.5

2.5

3

3

3

3

3.5

3.5

3.5

3.5

4

4

4

4

4.5

4.5

4.5

4.5

5

5

5

5 5

H

0 1 2 3 4 5

Fig 8 Nusselt number for different values of H and N at k = 2.0, R = k.

0 5 10 15 20 25 30

H

τ1

N = 0.5

N = 3.5

Da = 0.1, 0.01, 0.001

Fig 9 Skin friction for different values of H, Da and N atc= 1.5, k = 2.0, Gr/Re = 50, (R = 1).

Please cite this article in press as: M.O Oni, Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus:

0:5 6 N 6 3:5, 0:001 6 Da 6 0:1 and 0:5 6c6 1:5 to capture

different physical problems Mixed convection parameter

200 6 ðGr=ReÞ 6 200 has been chosen to capture cases when

the natural convection dominates, ðGr=ReÞ P 0 and otherwise

ðGr=ReÞ 6 0

Fig 2presents the effect of heat source parameterðHÞ and

ther-mal radiation parameterðNÞ on temperature distribution at different

points in the annulus, the effect of heat generation and radiation

parameter is to increase the fluid temperature It is observed that

for small heat generation parameter the effect of thermal radiation

is insignificant on fluid temperature in the annulus

Fig 3on the other hand depicts effect of aspect ratioðkÞ on

tem-perature distribution in the annulus It is found that ask increases,

fluid temperature decreases In addition, magnitude of

tempera-ture is seen increase with increase in thermal radiation parameter

Fig 4 shows combined effect of viscosity ratio ðcÞ and heat

sourceðHÞ on fluid velocity in the annulus It is observed that fluid

velocity is an increasing function of heat source parameterðHÞ and

viscosity ratio ðcÞ at the center of the annulus but the reverse is observed at the region close to the surfaces of the annulus Two points of inflexion are noticed in this figure, at these points, fluid velocity is independent on heat source parameter

In similar manner,Fig 5gives the combined effect of radiation ðNÞ and porosity of porous material ðDaÞ on fluid velocity As expected, as ðDaÞ increases, velocity also increases This can be attributed to the fact that Darcy numberðDaÞ is directly propor-tional to the permeability which widens the pores of the porous material and as such enhances fluid motion The reverse situation

is noticed at the region close to the surfaces of the cylinders In addition, the maximum velocity is reached at the center of the annulus

Fig 6presents the effect of mixed convection parameterðGr=ReÞ

on fluid velocity For positive values ofðGr=ReÞ i.e when natural convection dominates over forced convection, fluid velocity is seen

to be higher at the inner surface of the outer cylinder than outer surface of the inner cylinder On the other hand, when

0 5 10 15 20 25 30

H

τ λ

N = 0.5

N = 3.5

Da = 0.1, 0.01, 0.001

Fig 10 Skin friction for different values of H, Da and N atc= 2.0, k = 1.5, Gr/Re = 50 (R = k).

-30 -25 -20 -15 -10 -5 0 5 10

H

τ1

γ =0.5

γ = 1.0

γ = 1.5

Gr/Re = 150, 100, 50, 0

Fig 11 Skin friction for different values of H,cand Gr/Re at k = 1.5, N = 1.5, Da = 0.1.

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ðGr=ReÞ < 0, the reverse result is obtained When ðGr=ReÞ ¼ 0, i.e.

no natural convection (purely forced convection), the velocity is

seen to be symmetric Also, a point of inflexion is noticed at the

center of the annulus At this point, fluid velocity is independent

whether forced or natural convection

Figs 7 and 8display the rate of heat transfer at the outer surface

of the inner cylinder and inner surface of outer cylinder

respec-tively for different values of radiation and heat source parameter

It is observed that radiationðNÞ and heat source ðHÞ enhance rate

of heat transfer at the outer surface of inner cylinder This can be

attributed to the fact that increase heat source and radiation

parameter, enhances fluid temperature which in turn increase

the rate at which heat is transfers between the fluid and the

sur-faces of the cylinders In addition, the rate of heat transfer is seen

to be higher at the outer surface of inner cylinder than inner

sur-face of outer cylinder

Figs 9 and 10illustrate combined effect of radiationðNÞ

poros-ityðDaÞ and heat source ðHÞ on the skin-friction at the outer

sur-face of inner cylinder and inner sursur-face of outer cylinder At both

surfaces of the cylinders, the skin-friction is seen to decrease with

increase in Darcy numberðDaÞ, radiation parameter ðNÞ and heat

source parameterðHÞ It is interesting to note that skin-friction is

independent on radiation or porosity for small value of heat source

parameter This can be explained form the solution obtained, that

for small heat source parameter, temperature profile is

indepen-dent of H and hence the velocity/skin friction Further, skin friction

is higher at the inner surface of outer cylinder than outer surface of

inner cylinder

Figs 11 and 12depict combined effect of viscosity ratio and

mixed convection parameter on the skin-friction at the surfaces

of the cylinders The skin-friction at the outer surface of inner

cylinder inFig 11is seen to decrease with increase in heat source

parameter and mixed convection parameter but increase with

increase in viscosity ratio The reversed result is noticed for

skin-friction at the inner surface of outer cylinder inFig 12 Also, the

maximum skin friction is observed at the inner surface of outer

cylinder

Table 1presents the critical values of mixed convection

param-eter varying radiation paramparam-eter, Darcy number and viscosity

ratio It is found that the critical Gr=Re increase with increase in

viscosity ratio but decreases with increase in radiation parameter

as well as Darcy number

4 Conclusions

In this work, an exact solution of combined effects of heat source and thermal radiation on mixed convection flow in a verti-cal annulus filled with porous material is obtained Closed-form expression for temperature distributions, velocity profiles, Nusselt number (representing the rate of heat transfer), skin-friction and mass flow rate are obtained in term of Bessel’s function and mod-ified Bessel’s function of first and second kinds respectively The effects of governing parameters such as heat source parameter ðHÞ, radiation parameter ðNÞ, mixed convection parameter ðGr=ReÞ, Darcy number ðDaÞ, aspect ratio ðkÞ and viscosity ratio

ðcÞ on temperature, velocity, Nusselt number and skin-friction are illustrated with the use of line graphs Based on the figures depicted, the following conclusions can be drawn:

6 8 10 12 14 16 18 20 22 24

H

τ λ

γ = 0.5

γ = 1.0

γ = 1.5

Gr/Re = 150, 100, 50, 0

Fig 12 Skin friction for different values of H,cand Gr/Re at k = 2.0, N = 1.5 (R = k).

Table 1 Numerical values for critical   Gr

for different values of N; Da andcat H ¼ 1:5; k ¼ 2:0.

R¼k

Please cite this article in press as: M.O Oni, Combined effect of heat source, porosity and thermal radiation on mixed convection flow in a vertical annulus:

1 Thermal radiation as well as heat source parameter increase

fluid temperature

2 Fluid velocity decreases with increase in heat source, viscosity,

radiation and mixed convection parameter but increases with

increase in Darcy number at the outer surface of inner cylinder

while the reverse trend is occurs at the inner surface of outer

cylinder

3 The rate of heat transfer at the surfaces of the cylinders

increases with increase in radiation and heat source

4 The skin-friction at the surfaces of the cylinders is reduced by

porosity (Darcy number), heat source parameter and radiation

parameter

5 Reverse flow occurrence at the surfaces of the cylinder

decreases with increase in porosity and radiation parameter,

but increases with increase in viscosity ratio

6 Reverse flow in the annulus can be avoided by choosing

appro-priate values for heat source parameter, radiation parameter,

Darcy number and viscosity ratio

When Gr=Re ¼ 0, this work reduces to pressure driven flow and

the effect of N and H on velocity profile and skin friction is

suppressed

Acknowledgements

The author is thankful to his supervisors Prof B.K Jha and Prof

A.O Ajibade for their support throughout the compilation of this

article and their fatherly impartations

Appendix

m1¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3NH2

3Nþ 4

s

; m2¼

ffiffiffiffiffiffiffiffiffi 1

cDa

s

; m3¼ 1

m2

C1¼ Y0ðm1Þ

½Y0ðkm1ÞJ0ðm1Þ  J0ðkm1ÞY0ðm1Þ;

C2¼ J0ðm1Þ

½Y0ðkm1ÞJ0ðm1Þ  J0ðkm1ÞY0ðm1Þ

C3¼ t4

dP

dZþ t5

Gr

Re; C4¼ t8

dP

dZþ t9

Gr

Re;

n1¼ C3m3½kI1ðkm2Þ  I1ðm2Þ

n2¼ C4m3½K1ðm2Þ  kK1ðkm2Þ;

n3¼ C1DaGr

Reðcm2Daþ 1Þm1½kJ1ðkm1Þ  J1ðm1Þ

n4¼ C2DaGr

Reðcm2Daþ 1Þm1

½kY1ðkm1Þ  Y1ðm1Þ;

n5¼ DadPdZðk

2 1Þ

2

t1¼ Da½K0ðkm2Þ  K0ðm2Þ; t2¼ DaK0ðm2Þ

ðcm2Daþ 1Þ

t3¼ ½K0ðkm2ÞI0ðm2Þ  I0ðkm2ÞK0ðm2Þ; t4¼t1

t3; t5¼t2

t3

t6¼ Da½I0ðkm2Þ  I0ðm2Þ; t7¼  DaI0ðm2Þ

ðcm2Daþ 1Þ; t8¼t6

t3;

t9¼t7

t3

t10¼Daðk

2 1Þ

2 ; t11¼ðk

2 1Þ 2

t12¼ Da

ðcm2Daþ1Þ

C1

m1ðkJ1ðkm1Þ  J1ðm1ÞÞ þC2

m1ðkY1ðkm1Þ  Y1ðm1ÞÞ

t13¼ kI1ðkm2Þ  I1ðm2Þ; t14¼ kK1ðkm2Þ  K1ðm2Þ

t15¼ m3ðt9t14 t5t13Þ  t12; p1¼ t11þGr

Re½t15;

p2¼ m3ðt4t13 t8t14Þ  t10

t16¼ Dam1

ðcm2Daþ 1Þ; t17¼ t8m2K1ðm2Þ  t4m2I1ðm2Þ;

t18¼ t5m2I1ðm2Þ  t9m2K1ðm2Þ  t16ðC1J1ðm1Þ þ C2Y1ðm1ÞÞ

t19¼ t8m2K1ðkm2Þ  t4m2I1ðkm2Þ; t22¼p2

p1

t20¼ t5m2I1ðkm2Þ  t9m2K1ðkm2Þ  t16ðC1J1ðkm1Þ

þ C2Y1ðkm1ÞÞ; t21¼t11

p1

t23¼ t17t21; t24¼ t18 t17t22; t25¼ t19t21; t24¼ t20 t19t22:

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