Effects of viscous dissipation on heat convection of viscoelastic flow inside isothermal channels and tubes

(1)

DOI: 10.1007/s13367-018-0026-6

Effects of viscous dissipation on heat convection of viscoelastic flow inside isothermal channels and tubes

Seyed Zia Daghighi and Mahmood Norouzi*

Mechanical Engineering Department, Shahrood University of Technology, Shahrood, Semnan 361 9995161, Iran (Received May 11, 2018; final revision received September 26, 2018; accepted September 27, 2018) In the present paper, analytical solutions for thermal convection of the Finitely Extensible Nonlinear Elastic model with Peterlin’s closure (FENE-P) in isothermal slits and tubes are presented by considering the viscous dissipation effects for the first time. Temperature distributions are derived in closed form, namely the HeunT function, and Frobenius series solutions for the slit and the tube flows, respectively. The effects of fluid elasticity, the Brinkman number and the extensibility parameter on the thermal convection characteristics of FENE-P fluid flows are investigated in detail. Generally, the Brinkman number causes a fall in the Nusselt number, but a rise in the centerline temperature. The results reveal that during the cooling process the Nusselt number experiences a decrease with the Brinkman number, while the centerline temperature rises. However, during the heating process the former increases and the latter decreases. As the main innovation of this study, the results show a strong relation between the Nusselt number and the Brinkman number and also between the centerline temperature and the Brinkman number.

Keywords: FENE-P model, viscoelastic fluid, forced convection, HeunT function, isothermal tubes and slits, viscous dissipation

1. Introduction

Forced convective heat transfer of non-Newtonian flows is of special interest because of the wide range of indus- trial applications in biological heat exchangers, chemical processing industries, food production, cosmetic industries, resin injection in fabrication of composite laminates, and production of plastic tools especially by polymer injection.

The thermal convection of Newtonian fluids in ducts has been studied extensively in the literature. Shah (1975) investigated the effect of cross-section shape on the heat transfer characteristics in the straight ducts and reported that rounding the corners in ducts leads to an improvement in the heat transfer characteristics. The heat transfer of fluids through elliptical, square, and rectangular ducts has been studied without considering the viscous dissipation effect (Abdel-Wahed et al., 1984; Barletta et al., 2003;

Bhatti, 1984; Chang et al., 2007; Maia et al., 2006;

Norouzi et al., 2009; Rao et al., 1969; Sakalis et al., 2002;

Sayed-Ahmed and Kishk, 2008). Brinkman (1951) presented the exact solution for fully developed flows through ducts. By considering the viscous dissipation effect, Brinkman (1951) succeeded in deriving, for the first time, the temperature distribution in the entrance region of pipes at an imposed either constant temperature or constant heat flux in the wall. Aydin (2005a) analyzed the laminar fully developed Newtonian pipe flow with viscous dissipation and derived the Nusselt number and the temperature dis-

tribution as a function of the Brinkman number analytically, and reported that the viscous dissipation affects the heat transfer strongly for a Brinkman number larger than one. The problem of fully developed heat transfer of New- tonian fluids through a concentric annuli has been worked out analytically by Coelho and Pinho (2006). They employed the viscous dissipation in their calculations and considered two types of boundary conditions for both uniform wall heat fluxes and wall temperatures. Though the heat flux and the wall temperature are uniform, temperature distribution were different on the walls resulted in asymmetric heating of the fluid. They reported that when the wall temperature is the same as the bulk temperature, the Nusselt number goes through a singularity and their results also showed that in the case of small Brinkman numbers, the bulk and fluid temperatures are constrained by the wall temperatures. Moreover, their results revealed that for the second boundary condition and when the Brinkman number is positive, the inner wall Nusselt number increases and decreases with the Brinkman number and the radius ratio, shown by κ = Ri/R_o, respectively. Bar- letta and Magyari (2007) studied analytically the thermal entrance forced convection of laminar flow through cir- cular ducts for two boundary conditions cases. They reported that the distribution of local Nusselt number is much more monotonic by the Brinkman number when the wall heat flax is uniform and also their results revealed that the thermal entrance region length increases with absolute Brinkman numbers for the case of linearly changeable wall heat flux.

The fundamental study of Finitely Extensible Nonlinear

*Corresponding author; E-mail: mnorouzi@shahroodut.ac.ir

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Elastic model with Peterlin’s closure (FENE-P) was car- ried out analytically by Oliveira (2002) for the tube and slit flows which is similar to that of Oliveira and Pinho (1999). They reported that by increasing the dimensionless Deborah number and decreasing the extensibility parameter shown as L² the velocity profile tends to become flatter, leading to higher shear rates and, in turn, higher flow rates. Oliveira and Pinho (1999) presented a hydrodynamic solution for the fully developed channel and pipe flow. Filali and Khezzar (2013) analyzed the Graetz problem for the FENE-P fluids in ducts with constant wall heat flux numerically. By neglecting the viscous dissipation effect, they reported that the fluid elasticity causes the normalized heat transfer coefficient to rise which stems from the shear thinning behavior. However, the extensibility parameter leads to a decrease in the Nus- selt number. The problem of forced convection of both linear and exponential forms of the Phan-Thien-Tanner (PTT) fluids through isothermal pipes was studied by Norouzi (2016). By neglecting the effect of viscous dissipation, Norouzi (2016) derived exact solutions for the temperature distribution as well as two correlations for the maximum temperature and the Nusselt number.

The effect of viscous dissipation on the thermal entrance region of non-laminar and non-Newtonian fluids is avail- able in the literature (Avci and Aydin, 2008; Aydin, 2005b;

Basu and Roy, 1985; Dehkordi and Memari, 2010; Ou and Cheng, 1974). Pinho and Oliveira (2000) analyzed the forced convection of fully developed linear Phan-Thien- Tanner (LPTT) fluids by taking into account the viscous dissipation within pipes and channels maintained at a constant heat flux at the wall and derived exact formulations for the temperature distribution. They reported that there is an improvement in the heat transfer by increasing either the Deborah number or the extensibility parameter inde- pendent of the sign of viscous dissipation. Coelho et al.

(2002) analyzed the problem of fully developed forced convection of Simplified Phan-Thien-Tanner (SPTT) fluid in isothermal ducts semi-analytically. Their results showed that the heat transfer would be better as the fluid elasticity is enhanced and also reported that the amount of internal heat generation at the wall region would be bigger than the similar one in the core. Oliveira et al. (2004) studied the Graetz problem with viscous dissipation for the FENE- P fluids in the pipe and channel flows semi-analytically.

Regarding the viscous dissipation and the fluid elasticity, they reported that the Nusselt number followed completely different trends; the Nusselt number decreases with the viscous dissipation, but it increases with the fluid elasticity. Pinho and Coelho (2006) studied the fully developed heat transfer of SPTT fluid flow with viscous dissipation in an annuli for imposed constant heat flux and temperature at the wall. They pointed out that at a constant wall heat flux, the Deborah number would increase the

heat transfer. Nonetheless, in the case of a constant wall temperature, the heat transfer witnessed a drop with the Deborah number. By neglecting the viscous dissipation effect, Norouzi et al. (2018) investigated the heat transfer of the viscoelastic FENE-P fluid through isothermal slits and tubes. They reported that the Nusselt number and the maximum temperature will rise if the Deborah number increases.

Polymer processing often involves non-isothermal flows.

Sometimes fluids are heated or cooled by external agen- cies. In other cases, heat is generated via viscous dissipation (Bird and Giacomin, 2016). Then it is necessary to solve the energy equation, by taking the viscous dissipation term into account which is the aim of this paper. In the present work, in continuation of the work presented by Norouzi et al. (2018), the FENE-P viscoelastic fluid through isothermal slits and tubes is examined by considering the viscous dissipation effect in the energy equation for the first time. The temperature distribution is derived in the closed form solution for the slit flow and in the Frobenius series solution for the tube case as well. The effects of fluid elasticity and the Brinkman number on the heat transfer characteristics are examined in detail.

2. Governing and Constitutive Equations

The governing equations of internal heat convection of viscoelastic flows consist of the continuity, momentum and energy equations:

(1)

(2)

(3) where is the velocity vector, is pressure, is temperature, ρ is density, is the stress tensor, cp is the spe- cific heat, and α is the thermal diffusivity. Here, the FENE-P model (Bird et al., 1980) is used as the constitutive equation to determine the stress tensor. According to the case which is examined here, the fully-developed pipe and channel flows, the simplified form of the FENE- P equation could be written as:

(4) where λ is the relaxation time, f is a dimensionless function, η is dynamic viscosity of the fluid, and a is a constant which can be expressed as:

(5) where L is the extensibility parameter of the model. In Eq.

(4), is the deformation rate and is the upper con-

. 0

∇ =V

. p .

ρ ∇ = −∇ + ∇V V τ . 2

p

T T

c α φ

∇ = ∇ +ρ

V

V˜ p˜ T˜

τ˜

2

f a

λ^∇τ+ τ= ηD

2

1 1 3 /

a= L

−

D˜ ^∇τ

(3)

vected derivate of the stress tensor defined as follows:

(6) (7) where superscript is the transpose operator.

FENE-P is one of few molecular constitutive equations that can be used in computational fluid dynamics and analytical fluid mechanics since it does not need statistical averaging at each grid point at any instance in time. This equation is able to accurately predict the shear thinning viscosity and elongational viscosity. The multi-mode form of this model is capable of being fitted to the rheological data of dilute polymeric solutions in oscillation tests. The FENE-P model is also able to predict the polymer turbu- lence drag reduction. As a disadvantage, it is unable to model the second normal stress difference.

3. Formulation

As mentioned before, the main object of this study is to present an exact solution for heat convection of the FENE- P fluid flow in tubes and slits. Oliveira (2002) presented an exact analytical solution for the velocity field of this problem by solving Eq. (1b) based on the response of Eq.

(6) for a rectilinear pressure driven flow as follows (Oliveira, 2002):

(8) where is the main flow velocity (axial velocity) and U is the mean velocity. In Eq. (8), y is the dimensionless profile direction and it is defined as for the tube case and for the slit case (Here, is the radial direction in the tube flow, R is the tube radius, is the lateral direction in the slit flow, and H is half of the distance between two parallel plates). The constants of β1 and β2

can be determined from the following relationships (Oliveira, 2002):

For slit flow:

(9) For slit flow:

(10) In Eq. (9), De is the Deborah number which is determined as De =λU/H and De = λU/R for slit and tube flow, respectively. Oliveira (2002) showed that the velocity ratio (U_N/U) can be calculated from the following formulation in terms of rheological properties:

(11)

where δ is:

(12)

In Eqs. (11) and (12), χ is a constant which is determined from the following equations (Oliveira, 2002):

(13)

By considering the rectilinear flow and by neglecting the axial conduction in comparison to the radial conduction, the energy equation (Eq. (3)) would be simplified as follows:

(14)

where j equals 0 for the slit flow and 1 for the tube flow.

The viscous dissipation term is the result of the tensor product of the velocity gradient tensor with the stress tensor which is simplified below with respect to this problem:

(15)

By calculating the viscous dissipation from the above formula, we have,

(16)

In order to non-dimensionalize the parameters used in this study, the following dimensionless groups are used:

(17)

where d_h is the hydraulic diameter (for the tube flow:

d_h= 2R and for the slit flow: d_h= 4H), Nu is the Nusselt number, h is the heat transfer coefficient, and are the mean and the wall temperature, respectively. In fully developed thermal conditions, the axial gradient of the dimensionless temperature is zero (Eckert and Drake Jr., 1987) which can thus be written as the following equation:

(18)

1 ( †)

=2 ∇ + ∇ D V V

{

^†

}

∇ .

= ∇ − ∇ + ∇

τ V τ V τ τ V

†

{ }

2 2

1 2

( ) (1 ) 1 (1 )

u y y y

U =β − +β +

u˜

y = r˜/R

y = y˜/H r˜

y˜

2 2 2

1 2 2 2 ,

3 ( / )

= / , 9 ,

2 3

N x

N N

De U U p H

U U U

β β a L

= = − η

2 2 2

,

1 2 2 2

( / )

=2 / , 16 ,

8

N x

N N

De U U p R

U U U

β β a L

= = −η

1/6 2/3 2/3

1/2 1/3

432 ( 2 )

6 UN

U

δ χ δ

= −

1/2 3/2 1/2

(4 27 ) 3

δ= + χ + χ

2 2

2 2 2 2

54 (slit) & 64 (tube)

5 3

De De

a L a L

χ= χ=

,x ,yy ,y

p

uT T jT

y c

α φ

ρ

⎛ ⎞

= ⎜ + ⎟+

⎝ ⎠

du φ τ = dr

2 2 2 2 2

, ,

2 2 2

(3 j 1) 1

x x

p y p y

sa L φ λ

η η

⎛ ⎞

= + ⎜⎜⎝ + ⎟⎟⎠

For slit flow: s = 1 2--- For tube flow: s = 2

⎩⎪

⎨⎪

⎧

⇒

2

2 2

, , , and

( )

For slit flow: , , , and

3 /

For pipe flow: , , , and

4 /

w h

m w w m

u T T U hd

u T Br Nu

U T T k T T k

x y U

x y De

H H H U H

x y U

x y De

R R R U R

η

λ φ φ

η

λ φ φ

η

= = − = =

− −

= = = =

T˜m T˜w

,

, w 0

x m w x

T T T

T T

⎛ − ⎞

=⎜⎝ − ⎟⎠ =

(4)

By expanding the above equation, it can be seen that the axial gradient of temperature is equal to:

(19) In this study, the axial mean temperature gradient is obtained by considering the balance of energy on a differential control volume and by taking into account the viscous dissipation term as follows

(20) where and are the perimeter and area of the cross- section, respectively. By applying Eq. (21) which is reported by Oliveira (2002), on Eq. (20) finally, Eq. (22) is derived for the axial gradient of the mean temperature.

(21)

(22)

The dimensionless form of the viscous dissipation given in Eq. (16) is derived by considering Eq. (17):

(23)

It is easily provable that the viscous dissipation term for De=0, the Newtonian fluid, is equal to:

(24) Regarding Eqs. (8), (16), (19), and (22), the non-dimen- sional form of the heat transfer equation of the FENE-P fluid flow can be expressed as follows:

(25)

where j = 0 and j = 1 denote the slit and tube cases, respectively. There are two boundary conditions for the above second-order non-homogeneous differential equation consisting of the constant wall temperature, Eq. (26)

and the symmetry condition at the centerline, Eq. (27).

(26)

(27) 4. Analytical Solution

The solution of Eq. (25) is the aim of the present work.

Two types of analytical solutions are derived for the isothermal slit and tube flow, the HeunT function, as a closed form mathematical function, and also the Frobenius series, respectively.

4.1. Slit flow

The solution of heat convection of the FENE-P fluid flow in an isothermal slit is obtained by solving Eq. (25) and considering j = 0.

(28) where A, B and C are:

,x ( )m x,

T =T T

2 2 2 2 2

, ,

2 2 2

2

2 2 2

( ) 1 dx

4 2

( ) 3 ( / ) (1 ( / ) ) For slit flow:

2 ( ) 8 ( / ) (1 ( / ) ) For tube flow:

V

x x

p m w m

m w m N N

p p

m w m N N

p

p r p r

AUc dT h T T Pdx dA

a L

dT h T T U U U U U

dx HUc H Uc

dT h T T U U U U U

dx RUc R

ρ λ

η η

η α

ρ ρ

η α

ρ ρ

⎛ ⎞

= − + ⎜⎜⎝ + ⎟⎟⎠

− +

= +

− +

= +

∫

²Uc_p

⎧⎪

⎪⎨

⎪⎪

⎩

P˜ A˜

(

¹⁺^α⁽^{U U}^N^{/ )}²

)

⁼^{U U}^/ ^N

, 2

( ) 3

For slit flow: ( )

2 ( ) 8

For tube flow: ( )

w m N

m x p p

w m N

m x p p

h T T U

T HUc H c

h T T U

T RUc R c

η

ρ ρ

η

ρ ρ

⎧ = − +

⎪⎪

⎨ −

⎪ = +

⎪⎩

2 2 2

(3 j U U y)( N/ ) (1 2 2y )

φ= + + β

(3 ) 2

N j y

φ = +

2 2

, ,

2 2

1

1 1 2 2

(1 ) 1 2(1 )

1 2( 1) 4 (1 2 ) 0

4 3

yy j y

T T y y

y

Nu j Br T Br y y

j

β β

β β β

⎡ ⎤

+ + − ⎣ + + ⎦

⎛ + + ⎞ − + =

⎜ − ⎟

⎝ ⎠

1 0

y= → T=

0 ,y 0

y= → T =

( )

2 2 2

1 1 2

1 2 1 4/3 1

1 1 6

1 2 1

1 2 1 2/3

2

1 1 2

1 2 1

2 1 2

( )

(8 ) (2 1) ,0,

( (8 ))

HeunT

0.7211 8 , 0.6934 (8 )

( (8 ))

1 (8 ) (2 3)

exp 12 (8 )

(8 He

0.1

unT 3

0.13 T y

Br Nu Br Nu

C Br Nu

Br Nu y B Nu

Br Nu y y

Br Nu

C

β β β

β β

β β β β β β

β β β

β β

=

⎛ + + ⎞

⎜ + ⎟

⎜ ⎟

⎜ + ⎟

⎜ − + ⎟

⎜ + ⎟

⎝ ⎠

⎛ + + ⎞ +

⎜ ⎟

⎜ + ⎟

⎝ ⎠

2 2

1 2

4/3

1 2 1

1 1 6

1 2 1

2/3

1 2 1

2

1 1 2

1 2 1

2

1 1 2 1

1 1

) (2 1) ,0,

( (8 ))

(8 )

0.7211 ,0.6934 (8 )

( (8 ))

1 (8 ) (2 3)

exp 12 (8 )

24 (

0.13

8 )

(8 HeunT

Br Nu Br Nu Br Nu

Br Nu y B Nu

Br Nu y y

Br

B

Nu

B Nu

r r

B

β β β β

β β β β β

β β β

β β β β

β β

⎛ + + ⎞

⎜ + ⎟

⎜ ⎟

⎜ + + ⎟

⎜ ⎟

⎜ + ⎟

⎝ ⎠

⎛− + + ⎞ +

⎜ ⎟

⎜ + ⎟

⎝ ⎠

+

2 2 1 1

3 1 2 1 2/3

2 1 2 1

6 1 2 1

3 2 1 2 1

2

1 1 2 2

23 1 2 1

)(2 1) ,0,0.7211 (8 ) ,

( (8 ))

(8 )

(2 3 ) (8 )

exp 1 12

(8 )(2 1) ,0,0.72 0.6934

0.1 11

(8 )

Heu T 3

n

Nu B Nu

B Nu

Ady B Nu y

y y Br Nu

B Nu

B

β β β

β β β β

β β β

β β β β

β

β β β

β β β β

⎛ + + + ⎞

⎜ + + ⎟

⎜ ⎟

⎜ + ⎟

⎝ ⎠

⎛ + + ⎞

⎜ ⎟

⎝ ⎠

+ +

−

+

∫

1 1

1 2 1 2/3

6 1 2 1

1 2 1 2 2

2

(8 ) ,

( (8 ))

(8 )

(8 ) 3

1 2

exp 6 0.6934

B Nu

B Nu y

y Br Nu

y

y C

Bd

β β

β β β

β β β β

β

⎡ ⎤

⎢ ⎥

⎢ ⎛ + ⎞⎥

⎢ ⎜ ⎟⎥

⎢ ⎜ + ⎟⎥

⎢ ⎜⎜ ⎟⎟⎥

⎢ ⎝ + ⎠⎥

⎢ ⎥

⎢ ⎛ ⎛ ⎞⎞ ⎥

⎢ ⎜ + ⎜ + ⎟ ⎟ ⎥

⎢ ⎜− ⎝ ⎠⎟ ⎥

⎢ ⎜ ⎟ ⎥

⎢ ⎝ ⎠ ⎥

⎣ ⎦

∫

(5)

(29) As can be seen, Eq. (28) presents a closed form solution containing a special mathematical function, HeunT, for the slit case. There are three unknown constant parameters in the slit temperature distribution, Eq. (28) namely, C1, C2, and Nu. By applying one of the boundary conditions, Eq.

(26), on Eq. (28), C1 would vanish. Now, another boundary condition is needed to derive the slit temperature distribution completely, which is given in Eq. (30). This is the product of the dimensionless velocity profile into the dimensionless temperature distribution throughout the area of slit. Now all that is needed is to apply the boundary conditions shown in Eqs. (27) and (30). Finally, both of the so far unknown parameters, C1= Tmax and Nu, are calculated through a system of equations, simultaneously.

(30)

4.2. Tube flow

By substitution of j = 1 in Eq. (25) and then by solving it, the solution of heat convection of the FENE-P fluid flow in an isothermal tube would be obtained in the Frobenius series solution as shown in the Eq. (31).

(31)

As can be seen from Eq. (31), the tube temperature distribution has a singularity point at r = 0 and since the temperature distribution is finite throughout the cross-section, applying the second boundary condition, Eq. (27), results in C2= 0. For greater accuracy, the tube temperature distribution up to an order of 8 is presented in Eq. (32). Sim- ilar to the slit case, the Nusselt number and the maximum temperature, (C1= Tc) are calculated by substituting Eq.

(32) into both Eq. (26) and (33).

2

1 2 1 2

4 2

2

2/3 2

1 2 1 2 1 1

2 2/3

2 1 2 1

6 1 2 1

(8 ) 3

1 2

(2 )exp

6

( (8 )) (2 1) (8 )

0.13 ,0,0.7211 ,

( (8 ))

HeunT .

0.6934 (8 )

y Br Nu y

y y

Br Nu Br Nu

B Nu

Br Nu y A

β β β β

β β

β β β β β β

β β β β

β β β

⎛ + ⎛ + ⎞⎞

⎜ ⎜⎝ ⎟ ⎟⎠

⎜ ⎟

+ −

⎜ ⎟

⎝ ⎠

⎛ + + + ⎞

⎜ + ⎟

⎜ ⎟

⎜ + ⎟

⎝ ⎠

=

1 2 2 1

1 1 2 2 1 1

3 2/3

1 2 1

2 1 2 1

6 1 2 1

2/3

1 2 1

1 1

48 1

8 2

(8 )(2 1) ,0,0.7211 (8 ) ,

( (8 ))

(8 )

HeunT

(8 )

4.16( (8 ))

(8 )(

HeunTPrime 0.13 0.6934

. 0.13

Br Nu y

B Nu B Nu

B Nu

B Nu y

Br Nu

B B

Nu

β β β

β β β β β

β β β

β β β β

β β β

β β

⎛ + ⎞⎛ + ⎞

⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

⎛ + + + ⎞

⎜ + + ⎟−

⎜ ⎟

⎜ + ⎟

⎝ ⎠

+ +

=

2

2 1 1

3 1 2 1 2/3

2 1 2 1

6 1 2 1

1 1 2 2 1 1

3 1 2

2 1 2 1

.

0.6934

2 1) ,0,0.7211 (8 ) ,

( (8 ))

(8 )

(8 )(2 1) (8 )

0.13 ,0,0.7211

( (

(8 )

HeunT

B Nu

B Nu y

Br Nu B Nu

Br Nu

β β β

β β β β

β β β

β β β β β

β β

β β β β

⎛ ⎞

⎜ ⎟

⎝

⎛ + + ⎞

⎜ + + ⎟

⎜ ⎟

⎜ + ⎟

⎝ ⎠

+ + +

+

⎠

1 2/3

6 1 2 1

1 2 1 2/3

1 1 2 2 1 1

3 2/3

1 2 1

2 1 2 1

6 1 2 1

8 )) ,

0.6934 (8 )

4.16( (8 ))

(8 )(2 1) ,0,0.7211 (8 ) ,

( (8 ))

(8 )

HeunTPrime .

(8 )

Heun

. 0.13

0.6934

T

B Nu

B Nu y

Br Nu

B Nu B Nu

B Nu

B Nu y

β β β β

β β β

β β β β β

β β β

β β β β

β α β

⎛ ⎞

⎜ + ⎟−

⎜ ⎟

⎜− + ⎟

⎝ ⎠

+

⎛ + + + ⎞

⎜ + + ⎟

⎜ ⎟

⎜ + ⎟

⎝− ⎠

1 1 2 2 1 1

3 2/3

1 2 1

2 1 2 1

6 1 2 1

3 2 1 2 1

4 2

2

1 2 1 2/3 2

(8 )(2 1) (8 )

0.13 ,0,0.7211 ,

( (8 ))

(8 )

0.6934 (8 )

(2 3 ) (8 )

(2 )exp 1 12

( (8 )) (2 1)

HeunT 0.13

Br Nu B Nu

B Nu

Br Nu

B Nu y

y y Br Nu

y y

Br C

Nu

β β β β β

β β β

β β β β

β β β

β β β β

β β

β β β β

⎛ + + + ⎞

⎜ + + ⎟

⎜ ⎟

⎜ + ⎟

⎝ ⎠

⎛ + + ⎞

+ ⎜⎜⎝ ⎟⎟⎠

+

=

+ ² ₁ ₁

2 2/3

2 1 2 1

6 1 2 1

(8 )

,0,0.7211 ,

( (8 ))

0.6934 (8 )

Br Nu

B Nu

Br Nu y

β β

β β β β

β β β

⎛ + ⎞

⎜ + ⎟

⎜ ⎟

⎜− + ⎟

⎝ ⎠

0

∫

1^{u y}^{( )T y}^{( )dy = 1}

2

1 2 1 2

2

1 2 4

1 2 2 3 2

1 2 1 1 2

1 1 1

1 1 23 2 2

3 3 2

1 2 1 2

4 3

1 2

(y)

1 1 ( (1 ) 4 (1 )Br) 4

( (1 ) 4)

1

64 8( (1 ) 2 2 (1 ) )

C 1 (4 )

2304

( (1 ) 20(1 )) 64 (1 ) 10 (1 ) 8 2 (1 )

T

Nu y

Nu Nu

Nu Br Br y

Br Nu Nu Nu

Nu Br

β β β β

β β

β β β β β β

β β

β β β β β

β β β β

β β

=

− + + + +

⎛ + + + ⎞

⎜ ⎟ −

⎜ + + + + ⎟

⎝ ⎠

+

+ + + − +

⎛ + + + +⎞

⎜⎜ +

⎝

6 7

2

1 2 1 2

1 22 4

1 2 2 3 2

1 2 1 1 2

1 1

3

1 1 2

2

(y )

1 1 ( (1 ) 4 (1 )Br) 4

( (1 ) 4)

1

64 8( (1 ) 2 2 (1 ) )

ln(y) 1 (4 )

2304

( (1 ) 20(

Br y O

Nu y

Nu Nu

Nu Br Br y

Br Nu Nu Nu

C

β β β β

β β

β β β β β β

β β

β β β

⎛ ⎞

⎜ ⎟

⎜ ⎟ +

⎜ ⎟

⎜⎛ ⎞ ⎟

⎜⎜ ⎟ ⎟

⎜⎜ ⎟ + ⎟

⎜⎜⎜ ⎟⎟ ⎟⎟ ⎟

⎜⎝ ⎠ ⎠ ⎟

⎝ ⎠

− + + + +

⎛ + + + ⎞

⎜ ⎟ −

⎜ + + + + ⎟

⎝ ⎠

+

+ + 2 2

6 7

3 3 2

1 2 1 2

4 3

1 2

1 2 1 2 2

1 1

2 2

1 2

1 2 2

1 2 1

1 )) 64 (1 ) 10 (1 ) (y ) 8 2 (1 )

1 ( (1 ) 4 (1 )Br) 4

1 ( 4 )

16

(1 ) 4 3

128 8( (1 ) 2 2

Nu y O Br Br

Nu y

Nu Br

Nu Nu

Nu

β β

β β β β

β β

β β β β

β β

β β β β

⎛ ⎞

⎜ ⎟

⎜ ⎟ +

⎜ ⎟

⎜⎛ + − +⎞ ⎟

⎜⎜ ⎟ ⎟

⎜⎜ ⎛ + + + +⎞ ⎟ + ⎟

⎜⎜⎜ ⎜⎜ ⎟⎟ ⎟⎟ ⎟

⎜⎝ ⎝ + ⎠ ⎠ ⎟

⎝ ⎠

+ + + +

+ −

+ + +

( )

4

3 2

1 2

1 1

1 2 2 12 2

1 1

2 2 3

1 2 1 2 2

3 3 2

1 2 1 2

4 3

1 2

(1 ) )

3 ( 4 )

576

(1 ) 8 4 (1 )Br

11 ( 4 )

13824

(1 ) 20 (1 ) 64 (1 ) 10 (1 ) 8 2 (1 )

y

Br Br

Nu Br

Nu

Nu Br

Nu Nu

Nu Br

Br

β β

β β β β β

β β

β β β β β

β β β β

β β

⎛ ⎞

⎜ ⎟

⎜ ⎟ +

⎜ ⎛ ⎞⎟

⎜ ⎟

⎜ ⎜ ⎟⎟

⎜ ⎝ + ⎠⎟

⎝ ⎠

⎛− +

⎜⎜

+ − + + +

+

⎛ + + + − ⎞

⎜ ⎟

⎛ + + + +⎞

⎜ ⎟

⎜ ⎜ ⎟ ⎟

⎜ ⎝ + ⎠ ⎟

⎝ ⎠

⎝

6 7

2 4 2 2 2 6 7

1 1 2 1 1 2 1 2

(y )

1 2 1 ( (1 ) 4 (1 4 )Br) (y )

4 9 144

y O

Br β y β β β βNu β β β y O

⎛ ⎞

⎜ ⎟

⎜ ⎟ +

⎜ ⎟

⎜ ⎞ ⎟

⎜ ⎟ ⎟

⎜ ⎟

⎜⎜ ⎟ + ⎟

⎜ ⎟

⎜⎜ ⎟⎠ ⎟

⎝ ⎠

⎛ +⎛ − + + + ⎞ ⎞+

⎜ ⎜⎝ ⎟⎠ ⎟

⎝ ⎠