Numerical study of conjugate heat transfer in a double-pipe with exponential fins using DGFEM

Accepted Manuscript Research Paper Numerical study of conjugate heat transfer in a double-pipe with exponential fins using DGFEM Waseem Ahmad, Khalid Saifullah Syed, Muhammad Ishaq, Ahmad Hassan, Zafar Iqbal PII: DOI: Reference:

S1359-4311(16)32046-4 http://dx.doi.org/10.1016/j.applthermaleng.2016.09.171 ATE 9196

To appear in:

Applied Thermal Engineering

Received Date: Revised Date: Accepted Date:

7 December 2015 25 August 2016 30 September 2016

Please cite this article as: W. Ahmad, K. Saifullah Syed, M. Ishaq, A. Hassan, Z. Iqbal, Numerical study of conjugate heat transfer in a double-pipe with exponential fins using DGFEM, Applied Thermal Engineering (2016), doi: http:// dx.doi.org/10.1016/j.applthermaleng.2016.09.171

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Title: Numerical study of conjugate heat transfer in a double-pipe with exponential fins using DGFEM

$XWKRUQDPHVDQGDIILOLDWLRQVWaseem Ahmada , Khalid Saifullah Syeda , Muhammad Ishaqb, Ahmad Hassanc, Zafar Iqbald a Centre for Advanced Studies in Pure and Applied Mathematics (CASPAM), Bahauddin Zakariya University, Multan 60800, Pakistan b Department of Computer Sciences, Institute of Information Technology, Mailsi road, off Multan road, Vehari 61100, Pakistan c Department of Basic Sciences and Humanities, University College of Engineering & Technology, Bahauddin Zakariya University, Multan 60800, Pakistan d Department of Mathematics, Government. Emerson College, Multan 60800, Pakistan

&RUUHVSRQGLQJDXWKRUMuhammad Ishaq Email of corresponding author: [email protected]

1

Numerical study of conjugate heat transfer in a double-pipe with exponential fins using DGFEM

ABSTRACT The present work is aimed at studying the performance of exponential fin by numerical simulation of the conjugate heat transfer problem of fully developed laminar flow through the finned annulus of a double-pipe. The exponential fin shape has been considered for the first time in literature according to the knowledge of the authors. The governing partial differential equations are numerically solved by employing the discontinuous Galerkin finite element method. A comprehensive investigation of the effects of the ratio of thermal conductivities of the solid and fluid, the number of longitudinal fins, the ratio of radii of pipes and the fin thickness on the thermal performance of the finned duct has been carried out. It is found that significant gain in the Nusselt number may be achieved by increasing the ratio of conductivities. A value of the ratio greater than 500 may not be cost-effective. The exponential fin has been found to outperform the triangular fin by 0.02% to 15.09%. The study shows that in order to have optimal performance, a large number of higher and thinner fins will have to be augmented to a double pipe with larger ratio of radii.

keywords: Laminar, Double-pipe, Exponential fin, Conjugate heat transfer, DGFEM

Nomenclature


inner radius of the inner pipe, 




outer radius of the inner pipe, 




inner radius of the outer pipe, 




radius equal to fin height from the centre of the inner pipe, 


 

cylindrical coordinates, 
  

        Rhat 

dimensionless form of
 
 
 
 and


dimensionless form of the outer radius of the inner pipe

 ƍ

uniform heat input per unit axial length, 



the pressure, 

2



angle formed by the line of symmetry between two consecutive fins and the line joining the centre of the inner pipe and the fin base,




half of the angle formed by two lines joining the centre of the inner pipe and the fin base,




axial velocity component, 



dimensionless axial velocity component



fluid dynamic viscosity (absolute),  

ɤ

thermal diffusivity,  

!

thermal conductivity of the fluid, # $

%

dimensionless fin height relative to the annular gap

&'

hydraulic diameter, 

"

&'  dimensionless hydraulic diameter of finned geometry ()

flow cross-sectional area, 

*+

 specific heat capacity, .,- $

/0

bulk mean fluid temperature, 1

/

wall or fluid temperature at the solid-fluid interface, 1

20

dimensionlessbulk mean fluid temperature

3 

dimensionless heated parameter

4+ 45

Pressure gradient in finned geometry,  6

87 888

average Nusselt number based on hydraulic diameter

98

average heat transfer coefficient

:

dimensionless Fanning friction factor




dimensionless Prandlt number

3



dimensionless Reynolds number

;

ratio of conductivity of the wall-fin material to that of the fluid

Subscripts %

hydraulic

*

cross-section

<

bulk

9

heated

Superscript 

dimensionless quantity

Introduction

The techniques employed for heat transfer enhancement in the heat exchangers may be categorized into the active and passive techniques. In the active techniques, external forces are required for example, electric field, acoustic or surface vibration and fluid additives[1], while the passive techniques require special surface geometries such as rough surfaces and extended surfaces. Extended surfaces are generally known as fins. In many engineering applications like electric power generation, chemical processing, environmental control, etc., fins are usually the dominant way to enhance the heat transfer rate from a surface to its surrounding medium. Harper and Brown [2] made the first really significant attempt to provide mathematical analysis of the interesting interplay between convection and conduction in and upon a single extended surface. Harper and Brown called this a cooling fin, which later became known merely as a fin. It is most probable that Harper and Brown were the pioneers even though Jakob [3] pointed out that published mathematical analyses of extended surfaces could be traced all the way back to 1789. He pointed out that Fourier [4] and Despretz [5-7] published mathematical analyses of the temperature variation of metal bars or rods. In fluid flow and heat transfer equipments, the circular ducts with augmented fins are widely used. In the case of Double Pipe Heat Exchanger (DPHE), fins are normally straight, non-porous and uniformly distributed around the periphery of the inner pipe. Researchers consider two assumptions regarding the conductivity of the material of heated surface, one is highly conductive material with negligible conductive resistance and the other is that with significant conductive resistance. The first

4

assumption simplifies the model and in view of the absence of temperature distribution in the wall-fin assembly does not require the energy equation to be a part of the model. However, this assumption may lead to significant overestimates in the heat transfer results. To be more realistic, we take into account the conductive resistance of the material of wall-fin assembly and couple the heat equation with the Navier Stokes's and energy equations. In this way, consideration of the heat transfer surface makes the heat transfer problem a conjugate heat transfer problem in which the modes of heat transfer in the fluid are coupled with that in the solid. A brief literature survey of the conjugate heat problems is presented in the next paragraph. Mori at el. [8] considered the conjugate problem of heat transfer in the circular tube and found that the heat transfer characteristics were significantly affected by variation in the conductivities of the fluid and wall, and the thickness of the wall. Faghri and Sparrow [9] studied heat transfer in a circular pipe and discussed the heat transfer characteristics of simultaneous axial conduction in the wall and the fluid. The radial temperature gradients became negligible due to the assumption of thin wall. Their results showed significant upstream flow of heat in the pipe wall. In internally finned tube, Soliman [10] studied the effects of conductivities of wall and fluid on the heat transfer characteristics. Uniform temperature boundary condition was employed in the angular direction and variable temperature boundary condition in the radial direction. He reported that heat transfer characteristics strongly depend on the ratio of conductivities and the dimensions of finned tube govern the intensity of this effect. He also proposed a viable correction parameter to check the effect of fin conductance parameter on the Nusselt number. Krishan [11] used Laplace transform to solve the governing equations of conjugate heat transfer in a pipe with finite thickness and investigated the wall conduction effects. For an unsteady and fully developed flow, he considered a step change in the heat flux boundary condition imposed at the outer periphery of the pipe. Barrozi and Pagliarani [12] solved the conjugate heat transfer problem in a circular pipe numerically by employing the finite element method to solve the heat equation in the solid wall. They avoided determination of the temperature field in the flow field and applied Duhamel’s theorem to compute the interfacial temperature. They imposed the uniform heat flux boundary condition and investigated that the axial heat conduction in the wall might not be overlooked. Sakakibara et al. [13] performed conjugate heat transfer investigation in the annulus region analytically. They employed Duhamel’s theorem to determine the interfacial temperature and considered three boundary conditions, namely, constant heat flux, constant wall temperature, and constant heat transfer coefficient with constant ambient fluid temperature. Their results showed inverse relation between heat conduction in the solid wall and the ratio of the conductivities of the solid to fluid. Tao [14] studied conjugate heat transfer in a finned double-pipe. He considered the fully developed flow and calculated temperature distribution in fluids both in the tube and the annulus, and in the tube wall simultaneously. His results showed the effect of heat

5

capacities of the two fluids on heat transfer enhancement. Pagliarani [15] reported that the wall conductance may be ignored for wall thickness values of less than 0.1 of the pipe diameter. It is no longer true for higher values of wall thickness. Kettner et al. [16] investigated numerically that there is no noticeable effect of the ratio of thermal conductivities of the solid to fluid when the fins of small height were considered. However, this conductivity ratio showed significant influence when the fin height relative to the pipe radius was taken more than 40%. Fiebig et al. [17-18] studied conjugate heat transfer characteristics in the finned tube element and reported that the heat flux and the heat transfer coefficient were affected significantly by the fin efficiency parameters. Nguyen et al. [19] inserted a cutting edged disc in a pipe to control fluid flow and heat transfer. They observed different heat transfer rates based on the Reynolds number ranging from 335 to 845. Cutting edged baffle with flat side showed highest heat transfer rate. Syed [20] investigated finned double pipe numerically. He considered the fins of rectangular shape which were distributed uniformly around the periphery. He employed the boundary condition of uniform rate of heat transfer per unit axial length (H1) and of constant temperature (T1) at the inner pipe surface. He used one-dimensional fin equation and concluded that heat loss increased in the fin as the fin conductivity was increased. Syed, et al. [21] presented optimization of a finned double pipe with trapezoidal fins by using genetic algorithm and trust region method. They studied the optimization results corresponding to both the hydraulic and the equivalent diameters and concluded that the optimal configurations were dependent on the choice of the characteristic length used to define the Nusselt number. Z. Iqbal, et al. [22] studied optimal shape of longitudinal fins augmented to the outer surface of the inner pipe for maximizing the Nusselt number. They showed significant improvement in the Nusselt number for both the equivalent and the hydraulic diameters. Ishaq [23] presented the results of their investigation of triangular fins of different height in finned double pipe. Fully developed laminar flow with isothermal boundary conditions were used to investigate the influence of the number of fins and variations in the fin height, on the thermal design of the finned duct. The governing momentum and energy equations were solved numerically utilizing the Discontinuous Galerkin Finite Element method (DGFEM) subject to H1 thermal boundary conditions. Mazhar and K. Syed [24] extended the work of Syed [20].They examined laminar conjugate problem of heat transfer in the thermal entrance region of the finned annulus. They simulated this problem using finite difference method for the hydrodynamically fully developed flow. The authors also discussed thermally developed and thermally developing flows and found the entry region more efficient for heat transfer than the region where the flow was fully developed. They also reported that the ratio of thermal conductivities of the solid to the fluid has significant effect near the entrance region. The present work is aimed at studying the performance of a newly introduced fin-shape generated by the exponential function through conjugate heat transfer problem in the finned annulus of a double

6

pipe. Here, we investigate the influence of the number of fins, the ratio of the radii of inner and outer pipes, the fin thickness and the ratio of conductivity of the solid material used to manufacture the inner pipe and the fin, to that of the fluid on the hydrodynamic and thermal performance of the double pipe heat exchange system. We will study the behaviour of performance indicator like the Nusselt number and give some recommendations on the design of this heat exchange system for industrial applications.

Problem Analysis and Mathematical Formulation

We consider laminar, steady and hydrodynamically fully-developed flow of a viscous, incompressible, Newtonian and constant property fluid in the annulus of the double pipe. Longitudinal fins with profile generated by the exponential function are attached to the outer surface of the inner pipe. The fins material is non-porous and uniformly distributed around the periphery. It is also assumed that the wall and fins are made up of the same material so that there is no temperature discontinuity at the interface made by the fin base and the inner pipe wall. Figure 1 and 2 show the cross-section and the computational domain of the geometry respectively. The inner radius of the inner pipe is denoted by
 ,

the outer radius of the inner pipe by
 and the inner radius of the outer-pipe by
 . All the body forces and viscous dissipation are negligible. Pressure gradient is the only driving force in the axial direction. The wall-fin assembly is taken to be of finite conductivity. At the inner surface of the inner pipe wall, the flow is subjected to the thermal boundary condition of uniform heat input per unit axial length denoted by  ƍ

with circumferentially uniform temperature at any cross-section. The axial conduction in the wall and fluid is also neglected. An adiabatic thermal condition is imposed at the inner surface of the outer-pipe wall. At the fin surface and the outer surface of the inner pipe, we assume continuity of temperature and heat flux across the interface between the wall-fin assembly and the fluid. The objective is to study thermal performance of this heat transfer system with newly introduced fin profile through numerical simulation of laminar convection in the annular region and investigate the effects of various configurations of the finned annulus and the conductivity of the wall-fin assembly relative to that of the fluid. The governing equations of the present conjugate problem comprise the momentum equation, the energy equation and the heat equation.

Momentum Equation:

For the fully-developed laminar flow, the governing momentum equation according to the above stated assumptions, can be written as,

7

=> ? =@ >

 => ?

 =?

 4+

A @ =@ A @> =B>  C 45

(1)

whereD is the axial velocity component,  is the pressure and E
  F represent cylindrical coordinate system. The boundary conditions due to the viscosity of the fluid and the symmetry in the domain are: (a)

No slip conditions at the solid boundaries:   G at

 and G H  H  (inner pipe surface)   G at IJ
K <
LMN   IJ whereD<  @

T VWXDEYZ[F SRDE U F T\ VWX Y

U ]^PE_`aFb@\ ]^P _

O PQR B

and  H  H  A  (fin surface)

(2b)

and 
  b0@\ ]^P _ NcJ 

  G at

 and G H  H  A  (outer pipe surface) (b)

(2a)

(2c)

Symmetry conditions =? =B =? =B

 G at   G and
 H
H
 (radial symmetry line between two consecutive fins) (2d)  G at    A  and
 H
H
 (radial symmetry line dividing the fin into two halves) (2e)

To transform the above model into dimensionless form, the following dimensionless variables are defined:  

? , ?def

@ @ @

 ,   \ ,   U @g

@g

(3)

@g

where #Oh is the maximum velocity in the annulus region without fins and defined as #Oh   4+

K iC 45
 Ej K # A k # IJ # F in which # is the corresponding radial position of the point of maximum velocity and defined as # 

@d @g

l

bm> . SRDE6mF

Using transformations defined in Eq. 3, the

governing momentum equation 1, can be reduced into the following dimensionless form,  =?  m =m

A

=> ?  =m>

A

 => ?  m> =B>



bi n

(4)

where 1  Ej K # A k # IJ # F. The corresponding dimensionless boundary conditions are: a) No-slip conditions at the solid boundaries

   G at   and G H  H  (inner pipe surface) O

   G at IJ K < LMN   IJ PQRB and  H  H  A  (fin surface) o VWXDEYZ[F SRDE U p F oVWXY  ]^PE_`aFbm ]^P_ U

where <  m

(5a) (5b)

and   NcJ b0m]^P_

   Gat  j and G H  H  A  (outer pipe surface)

(5c)

b) Symmetry conditions at the lines of symmetry =?  =B

 G at   G and  H H j (radial symmetry line between two consecutive fins)

8

(5d)

=?  =B

 G at   D A  and  H H j (radial symmetry line dividing the fin into two halves) (5e)

where  is the dimensionless radial coordinate of the tip of the fin and     K  is the dimensionless fin height in Figure 2. It may be noted that the normalization of the velocity field carried out in Eq. 2 under the assumption of fully developed flow makes the governing momentum Eq. 1 independent of the 4+

Reynolds number. Therefore the present model is valid for 45 being constant and too small to make the flow turbulent.

Energy and Heat Equations:

The energy and heat equations with corresponding boundary conditions are described below. The energy equation with the temperature distribution / q in the fluid is  = =s t  => s t r
u A @ =@ =@ @ > =B>



 ɤ

=s t =5

(6)

The appropriate boundary conditions are: =s t =@

=s t =B =s t =B

 G at

 and G H  H  A  (adiabatic condition at the outer pipe surface)

(7a)

 G at   G and
 H
H
 (radial symmetry line between two consecutive fins)

(7b)

 G at   D A  and
 H
H
 (radial symmetry line dividing the fin into two halves)

(7c)

The 2D heat equation with the corresponding boundary conditions is => s v =@ >

 =s v =@

A@

 => s v =B>

A @>

 G,

(8)

where / w represents temperature distribution in the wall-fin assembly. / w  / at

 and G H  H  A 

(9a)

=s v

 G at   G and
 H
H


(9b)

=s v =B

 G at    A  and
 H
H


(9c)

=B

Continuity of temperature at the wall-fin assembly: / w  / q at

 and G H  H 

/ w  / q at IJ
K <
LMN  

O IJ PQR B

(9d) and  H  H  A  (fin surface)

(9e)

Continuity of flux at the wall-fin assembly: !w !w

=s v =@ =s v =B

 !q  !q

=s t =@ =s t =B

at

 and G H  H  at IJ
K <
LMN   IJ

O PQR B

(9f) and  H  H  A  (fin surface)

9

(9g)

The energy and heat equation with their associated boundary conditions may be converted into dimensionless form by employing the same dimensionless transformations (Eq. 3) and the dimensionless temperature 2E
F 

sE@5Fbsx E5F y ƍ .zt

into Eqs. 6 and 8 together with corresponding boundary conditions.

The dimensionless form of energy equation is: = > {t =m>

 ={t =m

Am

 = > {t =B>

A m>

?

 | 8?888~

(10)

}

The dimensionless form of heat equation is: = > {v =m>

A

 ={v m =m

A

 = > {v m> =B>

G

(11)

The boundary conditions in dimensionless form are given below. 2 w  G at   and G H  H  A  ={t =m

={v =m

={t =B ={v =B

={t =B

(12a)

 G at  j and G H  H  A 

(12b)

 G at   G and  H H 

(12c)

 G at   G and  H H j

(12d)

 G at    A  and  H H 

(12e)

 G at   D A  and  H H j

(12f)

The interface conditions of continuity of temperature and heat flux may be expressed mathematically as given below.

2 w  2 q at   and G H  H 

(12g) O

2 w  2 q at IJ K < LMN   IJ PQR B and  H  H  A  (fin surface) ={v =m ={v =B

 ={t =m

;

at   and G H  H 

 ={t

(12h) (12i)

O

 ; =B at IJ K < LMN   IJ PQR B and  H  H  A  (fin surface)

(12j)

where ;  !w !q represents the thermal conductivity ratio of the wall-fin assembly and the fluid. Numerical Solution Procedure

We are to solve elliptic partial differential equations. Here we formulate a 2D elliptic model problem using Discontinuous Galerkin Finite Element Method (DGFEM). Ciarlet [25] has shown the uniqueness and existence of solutions of Poisson problem. Consider a model elliptic partial differential Equation

10

K€ E*€F A   :

(13)

and reduce it to two-dimensional Poisson equation with and * equal to 0 and 1 respectively. This choice of values of coefficients depends on the models used in the present work. Therefore, Eq.13 becomes, K€   :EF, ‚ƒŸ4 .

(14)

The above equation may be represented in terms of auxiliary variable „ by splitting a second

order equation into two coupled first order equations by Bassi and Rebay [26] as € „  :E‚F, €  „.

(15)

Now writing „ in terms of its components „  E„ h  „ … F and using this into Eq. 15, we get first

order system as =†

=†

„ h  =h ,„ …  , =… =‡f =h

A

=‡ˆ =…

(16)

 :EF.

(17)

In the present work, we employ the following strong form of Equation 13 applying integration twice as described in [27]. We choose the grid points ‚‰  Š  j k ‹ 7+ , known as Legendre-GaussLobatto (LGL) nodes discussed in [28]. ŒŸ „3h‰ ‰ ‚  ŒŸ ‰ E‚Fh 3‰ ‚ K Œ=Ÿ ‘’h E3‰ K E3‰ F F“‰ E‚F ‚

(18)

ŒŸ „3 ‰ ‚  ŒŸ ‰ E‚F… 3‰ ‚ K Œ=Ÿ‘’… E3‰ K E3‰ F F“‰ E‚F ‚

(19)

Ž

…‰

Ž

Ž

Ž

Ž

Ž

…‰



…‰

…‰ 

ŒŸ h „3h‰ A … „3 “ ‰ E‚F‚ K ŒŸ r‘’h „3h‰ K „3h‰ “ “u A ”‘’… r„3 K „3 “ u• ‰ E‚F‚  Ž

Ž

Œ=Ÿ :3‰ “‰ E‚F‚

(20)

Ž

These Equations (18)-(20) leads to the following system of differential algebraic equations. 

(21)



(22)

–‰ „3h‰  — h 3‰ K ŒŸ r‘’h 3‰ K 3‰ “ “u ˜‰3 E‚F‚ Ž

…‰

–‰ „3  — … 3‰ K ŒŸ r‘’… 3‰ K 3‰ “ “u ˜‰3 E‚F‚ …‰

Ž



…‰ 

…‰

— h „3h‰ A — … „3 K ŒŸ r‘’h „3h‰ K „3h‰ “ “u A ”‘’… r„3 K „3 “ u• ˜‰3 E‚F‚  –‰ :3‰ Ž

…

(23)

‰ where –™  ŒŸ ‰ E‚F™‰ E‚F‚ is mass matrix, —™h  ŒŸ ‰ E‚F™‰ E‚F‚ and —™  ŒŸ ‰ E‚F… ™‰ E‚F‚ Ž

Ž

Ž

are stiffness matrices. The choice of numerical flux is very important for the stability of the DGFEM. Two elements having common face/edge share information between each other using these numerical fluxes. The domain is discretized into triangular elements in the present work, therefore the communication is exchanged between the elements having common edge. Also it may be noted that consistent fluxes are chosen.

11

In the present investigation, the stabilized internal penalty fluxes are employed and these are defined as 3  š›3 œ,

„3  š›€3 œ K žŸ3  , 

where š›€3 œ  E€3 b A €3 ` F, Ÿ3    3 ` ‘` K 3 b ‘b and ‘b and ‘` are the vectors normal to

the internal (‘-’) and adjacent interfaces (‘+’). For ž  G, the above equation gives central flux. The role of added term is to penalize the solution so that large jumps are disallowed in  and the parameter ž controls this jump. For the present conjugate problem in finned annulus, the working algorithm is described in the following steps: generation of initial mesh in the computation domain (Figure 2); extracting mesh in this region and solving the momentum equation to get velocity field in the flow region and finally solving the energy and heat equations simultaneously in the numerical domain. As the solution is assumed to be discontinuous across the element faces in the DGFEM, the continuity of temperature at the solid-fluid interfaces cannot be directly imposed. However, it is automatically weakly implemented by approximating the Riemann’s problem using the stabilized internal penalty flux. The flux continuity is imposed in the normal direction to the interface. The boundary conditions equations (12i) and (12j) can be 

q

represented by a single equation „¡w  Ÿ „¡ where „¡ is the temperature gradient along the normal. Since we are using DGFEM formulation in Cartesian coordinate system, the normal flux may be decomposed 

q

into its x- and y- components. The flux condition, then, takes the form E„h P ¢£ A „… P ¤£F  Ÿ E„h ¢£ A „… q ¤£F and gives the following two scalar equations 

q

(24)



q

(25)

„hw  Ÿ „h  „…w  Ÿ „…  q

where „h 

=s t q ,„ =h …



=s t w ,„ =… h



=s v , =h

„…w 

=s v . =…

On the fluid side, when we solve the energy equation, we employ fluid side fluxes as interior fluxes and the solid side fluxes as exterior fluxes and when we solve the heat equation in the solid medium, the case is vice versa. Here ‘-’ and ‘+’superscripts represent interior and exterior fluxes respectively. In this way, we implement the boundary conditions equations (12i) and (12j) by the following choice of interior and exterior fluxes. 

q`

„hwb  „hw , „hw`  Ÿ „h , qb

„h

q

q`

 „h , „h

 Ÿ„hw` ,

12

q

q

„h  „hb K „h`  „h K Ÿ„hw , 

q

„hw  „hb K „h`  „hw K „h , Ÿ and 

q`

„…wb  „…w , „…w`  Ÿ „… , qb

„…

q

q

q`

 „… , „…

 Ÿ„…w` , q

„…  „…b K „…`  „… K Ÿ„…w ,  Ÿ

q

„…w  „…b K „…`  „…w K „… . Mesh Refinement

DG-FEM is known to render solutions of higher order. We have performed several numerical experiments in order to choose appropriate triangular mesh size. Initial and finer meshes have been 888 and :  for different values of the ratio of radii . Finer mesh is obtained after employed to compute 87

regular refinement of initial mesh where all the triangles are divided into four triangles of the same shape. The results have been computed for different values of  keeping the fin height, the fin thickness and the conductivity constant, i.e., %   G ¥,   ¦ and Ÿ  §GG respectively. The number of elements in the 8888 initial and finer mesh depend on the value of . Table 1 gives the values of average Nusselt number 7

and : at initial and finer meshes along with their percentage differences keeping the half fin angle   ¦ , %   G ¥ and the conductivity of fin material Ÿ  §GG. The number of fins ¨ considered for

this verification are 9, 18 and 27 while ratios of radii  are 0.2, 0.5 and 0.7. For different values of  and

¨, the minimum and maximum percentage differences of Nu at initial and at finer mesh are 0.0044% and 0.2340% and that of :  are Ϭ͘ϬϮϰϮ% and Ϭ͘Ϯϵϰϵ% respectively. These differences are not significant

which justifies the use of initial mesh for the current investigation. Therefore, we have computed the present results on initial mesh with second order local polynomial approximation.

Validity of Results The said numerical technique is validated for some cases i.e. a laminar flow in an annulus with zero fin height as well as for fin height equal to 1. We then compared the results with [20, 23, 29] and found that the percentage error relative to the analytical solution in :  is less than 0.1220% and that of

8888 7 is less than 0.1510%. Thus the results excellently demonstrate the accuracy and validity of the numerical technique.

13

Results and Discussions

Our objective in the present work is to study the hydrodynamic and thermal characteristics of the newly considered fin profile generated by the exponential function, in a physically realistic situation of conjugate heat transfer in the annulus of a double pipe. Inner pipe wall thickness is considered to be 5% of the inner radius of outer pipe. The ratio of conductivities of the wall-fin material to that of the fluid is chosen to have the following values: 1, 2, 5, 10, 20, 50, 100, 500, 1000, 5000 and 500,000,000. The last value of the conductivity ratio, i.e., 500,000,000 may be regarded as if the wall-fin material has infinite conductivity. The present study is more realistic because of taking into account the thermal resistance of the solid material than that carried out in [23-24, 30] in which it was neglected. Our major focus is to investigate the performance of an exponential fin. We will study the effects of different geometry parameters, namely, the ratio of radii , the half fin angle , the fin height relative to the annular gap % ,

8888 to be defined later. the number of fins –, and the conductivity ratio ; on the average Nusselt number 7

The following values of the above parameters are used for the computation of the present work.

  ›G k G ¦ G © G § G ¥ G ªœ

–  ›¥ ¬ jk j§ j« kj k© kª ¦Gœ

,

  ›j k ¦ © §œ

,

%   ›G k G © G ¥ G «œ

,

­

whereD%   bm and   is the actual fin height. Nusselt Number

The Nusselt number is a dimensionless measure of the heat transfer coefficient. The average

8888, is defined as Nusselt number based on the hydraulic diameter &' denoted by 7 °

8888  ®¯ 3, 7 z

(26)

where ! is the thermal conductivity of the fluid and 98 is the average heat transfer coefficient defined as y ± . Es ² x bs³ F

98  +

Here /0 is the bulk mean fluid temperature, / K /0 is the driving temperature difference,

 ´ is the heat transfer rate per unit length of the pipe and 3 is the heated perimeter.

In terms of dimensionless quantities &' , 3 , and 20 , 8888 7may be expressed as 

® 8888 7  K + ¯ { , ²

(27)

³

where 3  

+² @g

and 20 

µ ?s . µ ?

First of all, we present velocity and temperature contours. These may be regarded as the primary sources for understanding the behavior of the model and validating it physically. Figures 3(a)-3(d) present

14

velocity contours for two different values of the number of fins and those of fin height for specified values of the other parameters.

p  G ¦ ȕ  ¦^  ¨  ›¥ jkœ and ·   ›G © G «œ. The figures show The values of the parameters are ¶ that the velocity of the fluid at the solid surfaces always remains zero and it starts increasing as we move away from the solid boundaries. This fact may be observed by the contours running along the inner pipe, fin and outer pipe surfaces. This behavior validates the mathematical model of the problem and the numerical solution procedure. Our interest lies in identifying the regions of large velocity gradients and high velocity zone together with their response to parametric variations so that we may anticipate where large heat convection rates would be and which parameters would be more influential in this regard. We note from Figure 3(a) that for smaller number of fins, ¨  ¥, and fin height, ·   G ©, the high velocity regime lies between the tips of consecutive fins shown by closed contours in Figure 3(a) and extends circumferentially towards the clearance between the fin tip and the outer pipe. This high velocity zone can be confined within the inter-fin spacing by increasing the fin height or can almost be uniformly distributed along the circumferential direction by increasing the number of fins with small fin height as may be seen in Figures 3(b) and 3(c) respectively. Figure 3(c) also demonstrates one dimensional radial distribution of the axial velocity encompassing a larger region in the clearance for larger number of shorter fins unaffected by the presence of fins. We can anticipate that this region may not have significant contribution in the heat loss. Increasing the number of fins for smaller fin height has another negative impact of the development of almost dead velocity zone with very small velocity gradients at the inner pipe as may be seen in Figure 3(c). The remedy to these drawbacks is to increase the fin height by which 1D flow is narrowed down and pushed towards the outer pipe and its major portion is trapped in the inner most closed contours between the tips of the fins and the velocity gradients at the inner pipe surface also appear to improve as may be observed in Figure 3(d). In this way the velocity gradients at the fin surface, particularly, its upper part also show high rise reflecting the major role of the fin played in the enhancement of convective heat transfer. On the other hand, large velocity gradients at the outer pipe surface are significant contributors in the pressure loss without playing any role in the improvement of heat transfer rate and thus are mere over heads. Figures

4(a)-4(e)

show

isotherms

p  G ¦ , ¸  ¦^ , ¨  ¥ , ·   G « and for ¶

;  j § jGG §GG jGGG. Figure 4(a) for ;  j is important in validating the physical behavior of the

model and the numerical solution procedure. Circular contours within the inner pipe wall and outside it in its vicinity passing through the fluid region and the fins indicate that heat is transferred at equal rate in the radial direction from the inner surface of the inner pipe wall and this is what it should be, for ;  j, the conductivity of the wall-fin assembly is equal to that of the fluid. These contours also show successful implementation of the boundary conditions of continuity of temperature and that of flux at the solid-fluid

15

interfaces and the success of discontinuous Galerkin finite element method in capturing the continuity behavior. As we move away from the inner pipe wall in the radial direction, the circular shape of the contours starts changing in the way that circular arcs between the consecutive fins tend to straighten and become chords of the arcs. Its implication is that the temperature tends to decrease in the radial direction within the fluid region. This is physically quite natural as the convection of heat increases by moving away from the almost dead velocity zone near the inner pipe wall to the large velocity zone between the tips of the fins. As Ÿ is increased from 1 to 5, i.e., the conductivity of the wall-fin assembly is increased relative to that of the fluid, the contours no longer remain circular even in the inner pipe wall and we may witness relatively larger temperature gradient in the pipe wall indicated by the contour entering the pipe wall near the fin base. This variation in the radial temperature gradient within the pipe wall owes to larger rate of heat loss at the fin-fluid interface than that at the wall-fluid interface. Consequently, peripheral temperature variation develops everywhere, which gets stronger by moving towards the high velocity zone and for larger fin heights it remains significant up to the vicinity of the outer pipe wall. This behavior is more clear and intensified in Figures 4(c) and 4(d) for larger values of Ÿ. Only one contour passing through the fin for Ÿ  §GG and no contour passing through the fin for Ÿ  jGGG in Figure 4(d)

and 4(e) respectively indicate that fin temperature is nearly equal to the wall temperature because of which the dimensionless temperature is almost zero in the whole fin for Ÿ  jGGG and in most of the fin region for Ÿ  §GG. This shows that depending on the number of fins and fin height a value of ; in the

interval ¹§GG jGGGº may be chosen for which the assumption of no conductive resistance within the wall-fin assembly will be valid and consequently the conjugate problem will be reduced to the problem comprising only the momentum and energy equations within the fluid.

Now we investigate the thermal performance of our heat exchange system. This will include results of average Nusselt number, assessment of validity of the common assumption of infinite conductivity of highly conductive material of wall-fin assembly and comparison of the performance of exponential fin with the triangular fin. 8888 increases with Ÿ for all the values of the number of fins The figures 5(a)-5(h) show that, in general, 7

and the fin height, and demonstrates asymptotic behavior as Ÿ » ’. Since increase in Ÿ means increase in the rate at which heat is supplied to the solid-fluid interface through conduction mode of heat transfer,

the increasing behaviour of 8888 7-curves shows that the rate at which heat is convected by the fluid also increases indicating that for smaller values of Ÿ, the heat carrying capacity of the flow was not being fully utilized. Therefore, as the supply of heat at the solid-fluid interface increased, the Nusselt number also increased. However, the rate of increase of the Nusselt number is not uniform with increase in Ÿ. For

16

8888-curves increase in a concave down fashion for all values of the other the shortest fin, %   G k, the 7 parameters showing that the rate of increase of 8888 7 is decreasing. In view of the velocity contours

presented earlier, this is because the fin surface is away from the high velocity zone. However, for larger

values of the fin height, as Ÿ is increased, the 8888 7-curves first increase in concave up fashion and then change their concavity at a certain value of Ÿ in the range §G H Ÿ H §GGDdepending on the values of the other parameters, and become concave down with ultimate asymptotic behaviour. This behaviour

8888 first increases more rapidly withŸ but, in most of the cases, indicates that for larger fin height the7

around Ÿ  §GG an approximate balance is established between the rate at which heat is being conducted to the solid-fluid interface and that at which heat is being convected away by the fluid from the heated surface so that for Ÿ ¼ §GG there is no significant improvement in the heat transfer coefficient with further increase in Ÿ. Since the pressure gradient is constant in this study, therefore, for smaller number of shorter fins, the fluid will have more velocity as well as capacity to carry heat from the wall-fin assembly. The fin height and the number of fins may be increased to balance the capacity of fluid of carrying heat from the assembly. We note from the figure that the effect of – is more significant for

larger values ofDŸ. Moreover, for   G ¦ the curves get wider apart with increase in Ÿ for all the values of the fin height. For   G §, the curves do so only for %   G k, 0.4 while for %   G ¥ the curve for –  jk gets closer to the curve for –  ¥ when Ÿ is large and for %   G «, the curves do so for

–  jk and 18 and override that for –  ¥ at around Ÿ  §GG and 1000 respectively. The intersecting curves in Figure 5(h) show complex behavior of the heat transfer co-efficient for larger values of the parameters , %  and Ÿ. Obviously, the heat transfer co-efficient depends on how much fin surface area

is washed by the high velocity fluid zone and how quickly heat is conducted to the solid-fluid interface which, in turn, depends on the values of the parameter –, , % , , ½ and Ÿ We note from all the above

mentioned Figures 5(a)-5(h), that the 8888 7 curves become almost horizontal for all values of the other parameters when Ÿ ¼ §GG. This behavior shows that for the materials of wall-fin assembly having

conductivity ratio greater than 500, the assumption of infinite conductivity or negligible thermal resistance in the wall-fin assembly may be employed. The quantitative analysis of this assumption will be 8888 is 4.82 for –  ¥,   G §,   ¦ ,D%   G k and Ÿ  ’ made later. The overall maximum value of 7 p  G ¦,   ¦ ,D%   G « and Ÿ  j. and the minimum value is 0.0742 for –  ¦G,D In

Figures

6(a)-6(d),

we

have

drawn 8888 7 against – at   G § ,   ¦ and

Ÿ ¾ ›jGG §GG jGGG §GGGœ for %   ›G k G © G ¥ G «œ. We note from these figures thatfor shorter fins

8888 decreases monotonically without changing its concavity with an increase in the %  ¾ ›G k G ©œ7

number of fins, with its maximum value being 4.82 at –  ¥,   G §,   ¦ ,%   G k and Ÿ = 5000.

17

8888 first increases to attain its maximum value 4.18 at –  jk for   G §, In Figure 6(d), for %   G «, 7   ¦ and Ÿ  §GGG and then starts decreasing. The reason for monotonic decrease for %   G k G ©, is that as the number of fins is increased the inter fin spacing becomes narrow because of which the flow speed decreases resulting into reduction in heat convection through the lateral surface of the fin. Because of shorter fins, high velocity zone finds its place in the clearance between the fin tip and the outer pipe and the rapidly moving fluid only washes the top surface of the fin. On the other hand, when fin height is larger, %   G «, the clearance region becomes narrow and as the number of fins is increased, the highvelocity zone shifts upward but significantly washes a larger fin surface area that results into increase in the Nusslet number. However, further increase in the number of fins reduces the free flow area and consequently flow speed decreases to the extent that the rate of heat transfer starts deteriorating that is

8888 -curve shows non-monotonic behaviour. The 7 8888 curve for %   G ¥ shows why for %   G «, the 7 transition between the monotonically rapidly decreasing curve with its concavity upward for the shorter fins and non-monotonic curve with a peak for the highest fins. Although it is decreasing but its concavity at the beginning has changed from upward to downward so that it decreases steadily with the number of fins. The curve for %   G « is perhaps of our major interest. First point to be noted is that there is no

8888 for –  jk and –  j§ when Ÿ is high. Therefore 15 number significant difference in the values of 7

of fins will also give almost optimal values of the heat transfer coefficient for %   G «. The second point worth noting is that although heat transfer coefficient is maximum for –  ¥ and %   G k and there are

other combinations of – and %  as well that render larger heat transfer coefficient than that for –  j§ and %   G «, yet they provide too small heat transfer surface area to meet the heat duty requirement of

industrial applications. The third point to be mentioned is this that this is the most greatly influenced curve by variation in Ÿ and its peak gets higher as Ÿ is increased. Therefore, the present results

recommend the use of 15 fins with 80% height of the annular gap when   G §,   ¦ and Ÿ is as much high as possible.

The effects of  on 8888 7 may be studied through Figures 7(a)-7(h) in which 8888 7 has been plotted

against  at   ¦ for %   ›G © G «œ , Ÿ  ›jGG §GG jGGG §GGGœ and –  ›¥ jk j« k© ¦Gœ

Figures 7(a)-7(d) show that when %   G ©, 8888 7 increases with  for all values of –.This is because

increase in  broadens the inter-fin spacing near the inner pipe and it starts contributing in convection which it was not doing earlier. This, in turn, improves the convection rates in this region. Note that

although increase in  reduces free flow area in the vicinity of the heated surface and apparently this should reduce the heat transfer rate by convection, but, this does not happen. In fact, this area had no 8888. That is why the significant contribution in the convection and was lowering the average value of 7

18

8888 increases when  is increased.Moreover, there is no significant influence of average value of 7

8888 curves for these shorter fins. This is because main bulk flow is in the variation in Ÿ on the behavior of 7 clearance between fin-tip and outer-pipe, therefore increase in Ÿ does not bear significant outcomes. It

should also be noted that smaller number of fins have larger 8888 7 for all values of . However, when %  is increased to 0.8, Figures 7(e)-7(h) show that smaller number of fins start losing their performance at

certain values of . That is why the 8888 7 curves for –  ¥ jk and j« attain peaks at   G ¦ G § and G ¥ respectively and then start falling when Ÿ  ›§GG jGGG §GGGœ. Whereas Figures 7(a)-7(d) show that for

8888 could be further increased by increasing  for all values of – Figures 7(e)shorter fins, %   G ©,D7

7(h) show thatfor higher fins, %   G «, for –  ¥ jk and j« , a balance between conduction and convection has been achieved at   G ¦ G § and G ¥ respectively and a further increase in  deteriorates

8888 for –  ¥ jk and j«. However, for larger number of fins, –  k© ¦G such a balance is not reached. 7

consequently, larger number of fins start outperforming smaller number of fins at higher values of . This

feature is of major interest because in industrial applications large heat duty requirement necessitates the use of larger number of fins for providing larger heat transfer surface area for which optimal values of other parameters are required. Therefore, Figure 7(h) provides very important information for industrial applications. For smaller values of  when – is increased, the inter-fin spacing near the inner pipe

surface becomes very small and since this region is surrounded by the stationary solid surfaces from three sides, the no slip conditions make the flow almost stagnant thus making the role of this region in convection almost negligible. That is why when  is smaller, 8888 7 decreases with – . However, for %   G « and Ÿ  ›§GG jGGG §GGGœ when  is increased to 0.7, the inter-fin spacing near the inner pipe

is not too narrow to retard the rate of convection. Therefore,for   G ª, 8888 7 increases as – increases to 24. For –  ¦G, the value of 8888 7 remains in between the values for –  ¥ jk and –  j« k©.

Figures 8(a)-8(h) present 8888 7 plotted against  for Ÿ  ›jGG §GG jGGG §GGGœ , – 

›¥ jk j« k© ¦Gœ and %   G ¥ when   ›G §G ªœ. These figures help us in choosing the fin base thickness for optimal convection. We note from Figures 8(a)-8(d) that when   G §, for – H jk, 8888 7

increases with  for all values of Ÿ; for –  j«, 8888 7 has a peak at   ¦ for ŸƒD›jGG §GG jGGGœ and at

  j for Ÿ  §GGG; for –ҹj«, 8888 7 attains insignificant peaks for ŸƒD›jGG §GGœ while it decreases with  for ŸƒD›jGGG §GGGœ. These results show that for smaller number of fins, – H jk, 8888 7 is monotonically

increasing with  for all values of Ÿ, where j H  H §. This is because for smaller number of fins we have larger free flow area in the inter-fin spacing and increasing the fin thickness gives more supply of heat at the fin-fluid interface which is quickly convected by the fluid in the inter fin spacing resulting into increase in the Nusselt number. Note that increasing the fin thickness decreases the free flow area but this does not have any effect on the convection rate because due to smaller number of fins the fins are too far

19

away to deteriorate the convection rate due to increase in the fin thickness. This happens for all values of 888 increases up to a certain value of  Ÿ. However, when the number of fins is larger i.e., – ¼ j«, 87

depending on the number of fins and fin conductivity ratio Ÿ and then starts decreasing. Since increase in

 means more supply of heat but reduction in free flow area that may result into decrease in convection rate, therefore depending on the number of fins and conductivity ratio, a balance between conduction and convection rates is achieved at a certain value of  which may be called optimal value of . Below this value conduction rate is slow in comparison to the convecting capacity of the flow and above this value, convection rate becomes slower than the conduction rate. It is to be noted that the decreasing 8888 7 curves

show that this balance is achieved at  H j. When the ratio of radii is increased from 0.5 to 0.7, we note 8888 increases with  for all values of Ÿ; for –  k©, 7 8888 has a from Figures 8(e)-8(h) that for – H j«, 7 8888 attains peaks at peak at   © for Ÿ  DjGG and at   ¦ for Ÿƒ›D§GG jGGG §GGGœ; for –  ¦G, 7

  ¦ and k for Ÿ  DjGG and §GG respectively while it decreases with  for Ÿƒ›DjGGG §GGGœ. These

results show that the optimal value of  has increased by increasing the ratio of radii. This is because for larger ratio of radii, inter fin spacing increases which gives more room for increasing the fin thickness. The value of  at which this balance is achieved, goes down if the number of fins or the conductivity ratio is increased for specified values of other parameters and it goes up if the ratio of radii is increased. Physically this behavior is quite realistic as for larger number of fins, fin thickness is to be decreased for providing sufficient free flow area; by increasing the conductivity ratio, thinner fin can conduct heat to the fin-fluid interface at a sufficiently faster rate to balance the convection rate; by increasing the ratio of radii, more inter-fin spacing and thus free flow area becomes available at the fin base which enhances the convection rate as well as gives room to increase the fin thickness for achieving balance in conduction and convection rates. Thus we conclude that as the number of fins or fin conductivity is increased, the thickness of the fin should be decreased to attain higher Nusselt number.

We now investigate the validity of a well-known assumption of ignoring the conductive resistance in a wall-fin assembly composed of highly conductive material for the heat exchange system under consideration. Table 2 gives % overestimates in 8888 7 due to this assumption relative to the finite values of the conductivity parameter Ÿ when   G §,   ¦ and ½  G G§. If the tolerable overestimates

are considered upto 5% rounded to the nearest integer then this table shows that, for ŸҸD§G, this is not a valid assumption for any values of % and –. However, for §G H Ÿ H jGG, this assumption is valid only

for shorter fins of height %   G k with –  ¥ jk j« k© and ¦G. For Ÿ ¼ §GG, the assumption is valid for all values of fin height and number of fins considered here with the following exceptions: (i) Ÿ  §GG, %   G « and jk H – H ¦G, (ii) Ÿ  jGGG, %   G « and –  ¦G. These results show that for

very highly conductive materials having Ÿ ¼ jGGG, the conductive resistance may be safely ignored with

20

overestimates upto 6% for a large number of very high fins. However, for Ÿ  §GG, this overestimate may go upto 12%. 8888 achieved by the present exponential fin over the triangular fins Now we present % gain in 7 studied in [27]. The comparison has been carried out in Table 3 for the values of the parameters shown in the table. We observe that for all the cases considered in Table 3, the exponential fin renders higher values of the heat transfer coefficient and that the percent gain in 8888 7 ranges from 0.0183 to 15.0852. For smaller values of % , %  H G ©, there is no considerable benefit of employing exponential fin over the

triangular fins. However, for larger values of % , particularly for %   G «, exponential fin significantly

outperforms the triangular fins in most of the cases of – and Ÿ. It is interesting to note that the maximum 8888 depends significantly on the values of Ÿ. For kG H Ÿ H jGG and Ÿ  §GGG, it ranges from gain in 7 12% to 15% while for Ÿ  §GG it is 3.77% and for Ÿ  jGGG, it is more than 8%.

We now present pressure drop analysis of the heat exchange system under consideration. This analysis is usually performed by studying the behaviour of the product of the Fanning friction factor and the Reynolds number which is regarded as a dimensionless measure of the frictional loss. This product may be defined as[31]. :   K

 4+ ®¯ > , ° C 45 ?

where &'  © ¿

q@ÀÀDq­D)@wwbwÀ)Á¡O­DO@ÀO . ÀÁÁÀ4D+À@#ÀÁÀ@

Transformations defined in Equation3 transform this product into the following dimensionless form :  

> ®¯ , ° n?

where 1 is already defined in equation 4.

Figures 9(a)–9(d) present :  plotted against the number of fins for specified values of the other

parameters. Figure 9(a) shows that for very short fins, %   G k, :  decreases with the number of fins

for all of values of . However, the rate of decrease is larger for smaller  and it tends to become steady

as  is increased. It may also be noted that for any number of fins :  increases with . Physically this

behavior may be understood in view of the fact that for smaller  the inter-fin spacing near the inner-pipe

surface is narrower than that for larger . As the number of fins is increased, it gets more and more narrow for smaller  so that the fluid in the inter-fin spacing gets highly retarded resulting into very small

velocity gradients at the inner-pipe and fin’s lateral surfaces which, in turn, reduces :  On the other

hand for larger , the clearance between the fin-tip and outer pipe is smaller while inter-fin spacing is

larger than in the case of smaller . Therefore, as the number of fins is increased, because of the narrower

21

clearance and wider inter-fin spacing, bulk of the flow remains in the inter-fin spacing which does not let :  decrease rapidly. As the fin height is increased to %   G ©, Figure 9(b) shows no significant change

in the behaviour of :  . However, for higher fins, i.e. %   G ¥ G « , non-monotonic behaviour of : with  as well as –may be witnessed in Figures 9(c)-9(d). Figure 9(c) shows that when %   G ¥,

:  is smaller for   G ª than that for   G § in the range ¥ H – H jk. However, for –  jk,: 

increases with . Another point to be noted from this figure is this that :  does not show significant

change with –for   G ª. This behaviour gives us the choice of employing any number of fins with

%   G ¥ within the specified range without any significant increase in the frictional loss. Figure 9(d)

shows more interesting behaviour of : for very high fins i.e. %   G «. First of all, we note from the

figure that for – H ¬, :  is minimum when   G §; for ¬ à – H j§, it is minimum for   G ªand

for j§ à – H ¦G, the minimum value is rendered by   G ¦. As the choice of   G ¦ is of no practical interest because of poor heat transfer performance as we will see later in this section, neglecting its curve

we note that :  remains smaller in the range ¬ à – H k© for   G ª than that for   G §. However,

for very large number of fins k© à – H ¦G, frictional loss for   G ª overrides that for   G § .

Although in view of the frictional loss   G § may be recommended for either small number of fins, i.e. – H ¬, or very large number of fins, i.e. –  k©, a comparison of heat transfer enhancement relative to

888, :  increase in the frictional loss would be more interesting and useful. Table 4 gives the values of 87

and 8888 7:  for %   G «,   ¦, Ÿ  §GG,  ¾ ›G ¦ G § G ªœ and – ¾ ›jk j« k© ¦Gœ. We note that

the heat transfer enhancement per unit increase in frictional loss represented by the ratio 8888 7:  is

minimum for all values of – when   G ¦. Therefore,   G ¦ may be of no practical interest. The values of this ratio for   G § and   G ª are comparable when –  jk. Therefore in view of other requirements either value of  may be employed. For other values of –, i.e. – ¾ ›j« k© ¦Gœ,   G ª

outperforms   G §with regard to the values of 8888 7:  by 24%, 51% and 75% respectively. In this

way maximum percent increase in 8888 7:  is achieved for the case of maximum number of fins, – 

¦G, considered in this work, which is highly remarkable. If we look for the maximum value of the ratio 8888: , it is obtained for –  j« and   G ª. Therefore for optimal heat transfer performance while 7

taking the frictional loss into account a double pipe heat exchange system with   G ª may be recommended for any number of fins when %   G « and   ¦. Conclusions Conjugate problem of heat transfer in the annulus with longitudinal exponential fins has been studied in the present work. This fin profile generated by the exponential function has been considered for the first time in the literature as per information of the authors. The velocity and temperature contours

22

have been discussed to gain fundamental insight into the physics of the problem under consideration which, in addition, validate the mathematical model of the problem and the numerical solution procedure. The behavior of the average Nusselt number has been investigated against different geometry parameters like the conductivity ratio Ÿ, the number of fins –, the ratio of radii  and the half fin angle . We may draw the following conclusions from our present study. 1. Fin height and the number of fins may effectively be employed to control the extents and location of the high velocity zone and the velocity gradients at the wetted surfaces which, in turn, give control on the heat convection rate. 2. Increasing the conductivity of wall-fin assembly to get Ÿ  §GG may not be cost-effective, as for almost all choices of the geometric parameters, there is no significant gain in the Nusselt number by raising Ÿ above §GG.Therefore, Ÿ  §GG is an appropriate choice. 3. The ratio of radii of the inner and outer pipes has significant impact on the number of fins that gives optimal Nusselt number. The optimal number of fins increases by increasing this ratio so that for highly conductive fins with 80% height of the annular gap and thickness

ȕ  ¦^ , this optimal number is 12 when   G 5 and it becomes 24 when  is increased to 0.7. 4. For larger number of fins, the thinner fins outperform the thicker fins so that the optimal value of ȕ will decrease by increasing the number of fins for specified values of the other parameters. 5. For optimal heat transfer performance while taking the frictional loss into account a double

pipe heat exchange system with   G ª may be recommended for any number of fins when %   G « and   ¦.

6. The assumption of negligible conductive resistance may incur an overestimate in the average value of the Nusselt number up to 11.6% for Ÿ  §GG, 5.4% for Ÿ  jGGG and 1.0% for Ÿ  §GGG when a large number of higher fins E¨  ¦G and ·   G «F is employed.

7. The exponential fin profile is more efficient than the triangular profile from 0.02% to 15.09% for ¨  jk k©, G k H · H G « and j H Ÿ H §GGG. Acknowledgement The authors of this paper acknowledge the financial support provided by the Higher Education Commission of Pakistan (HEC) under Indigenous Fellowship Scheme.

23

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28. 29. 30.

31.

K. Syed, Simulation of fluid flow through a double-pipe heat exchanger, Ph. D. Thesis, Department of Mathematics University of Bradford, 1997. K.S. Syed, Z. Iqbal and M. Ishaq, Optimal configuration of finned annulus in a double pipe with fully developed laminar flow, Applied Thermal Engineering, 31(2011)14351446. Z. Iqbal,K.S. Syed and M. Ishaq, Optimal fin shape in finned double pipe with fully developed laminar flow, Applied Thermal Engineering, 51(2013)1202-1223. Muhammad Ishaq, Khalid Saifullah Syed, Zafar Iqbal, Ahmad Hassan and Amjad Ali, DG-FEM based simulation of laminar convection in an annulus with triangular fins of different heights, International Journal of Thermal Sciences, 72(2013) 125-146. I. Mazhar and K. Syed, Analysis of Thermally Developing Laminar Convection in the Finned Double-Pipe Heat Exchanger, Heat Transfer Research, 45(2014). P. G. Ciarlet,The finite element method for elliptic problems, vol. 40 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations," Journal of Computational Physics, 131(1997)267-279. M. Ishaq, FEM based numerical solutions of incompressible flows in a finned doublepipe, Ph. D. Thesis, Centre for Advanced Studies in Pure and Applied Mathematics (CASPAM), BahauddinZakariya University, Multan, Pakistan, 2012. J. S. Hesthaven and T. Warburton, Nodal discontinuous Galerkin methods: algorithms, analysis, and applications: Springer, 2007. Khalid S. Syed, Muhammad Ishaq, Muhammad Bakhsh, Laminar convection in the annulus of a double-pipe with triangular fins, Computers & Fluids, 44(2011)43-55. I. Mazhar, Numerical study of laminar heat transfer through a finned double-pipe heat exchanger, Ph. D. Thesis, Centre for Advanced Studies in Pure and Applied Mathematics (CASPAM), BahauddinZakariya University Multan, Pakistan, 2006. A. P. Fraas, Heat exchanger design: John Wiley & Sons, 1989.



25

List of Figures Figure 1. View of the cross-section of the geometry. Figure 2. Zoomed portion of computational interest.

p  G ¦ ¸  ¦^  ¨  ¥Dand ·   G © Figure 3(a). Velocity contours for ¶

p  G ¦ ¸  ¦^  ¨  ¥Dand ·   G « Figure 3(b). Velocity contours for ¶

p  G ¦ ¸  ¦^  ¨  jkDand ·   G © Figure 3(c). Velocity contours for ¶

p  G ¦ ¸  ¦^  ¨  jkDand ·   G « Figure 3(d). Velocity contours for ¶ p  G ¦, ¸  ¦^ , ¨  ¥, ·   G «and ;  j. Figure 4(a). Isotherms for ¶

p  G ¦, ¸  ¦^ , ¨  ¥, ·   G «and ;  D§ Figure 4(b). Isotherms for ¶

p  G ¦, ¸  ¦^ , ¨  ¥, ·   G «and ;  jGG Figure 4(c). Isotherms for ¶

p  G ¦, ¸  ¦^ , ¨  ¥, ·   G «and ;  D§GG Figure 4(d). Isotherms for ¶

p  G ¦, ¸  ¦^ , ¨  ¥, ·   G «and ;  DjGGG Figure 4(e). Isotherms for ¶

888 at   G ¦   ¦ , %   G k and –  ›¥ jk j« k© ¦Gœ Figure 5(a). Effect of conductivity on 87

Figure 5(b). Effect of conductivity on 8888 7 at   G ¦,   ¦ , %   G © and –  ›¥ jk j« k© ¦Gœ 8888 at   G ¦,   ¦ , %   G ¥ and –  ›¥ jk j« k© ¦Gœ Figure 5(c). Effect of conductivity on 7

Figure 5(d). Effect of conductivity on 8888 7 at   G ¦,   ¦ , %   G « and –  ›¥ jk j« k© ¦Gœ 8888 at   G §,   ¦ , %   G k and –  ›¥ jk j« k© ¦Gœ Figure 5(e). Effect of conductivity on 7

Figure 5(f). Effect of conductivity on 8888 7 at   G §,   ¦ , %   DG © and –  ›¥ jk j« k© ¦Gœ 8888 at   G §,   ¦ , %   G ¥ and –  ›¥ jk j« k© ¦Gœ Figure 5(g). Effect of conductivity on 7

Figure 5(h). Effect of conductivity on 8888 7 at   G §,   ¦ , %   G «, and –  ›¥ jk j« k© ¦Gœ 8888 at   G §,   ¦  %   ›G k G ©  G ¥ G «œ and ;  jGG Figure 6(a). Effect of – on 7

Figure 6(b). Effect of – on 8888 7 at   G §,   ¦  %   ›G k G ©  G ¥ G «œ and ;  §GG

8888 at   G §,   ¦  %   ›G k G ©  G ¥ G «œ and ;  jGGG Figure 6(c). Effect of – on 7

Figure 6(d). Effect of – on 8888 7 at   G §,   ¦  %   ›G k G ©  G ¥ G «œ and ;  §GGG

8888 plotted against  at   ¦ , %   G © –  ›¥ jk j« k© ¦Gœ and ;  jGG. Figure 7(a). 7

8888 plotted against  at   ¦ , %   G © –  ›¥ jk j« k© ¦Gœ and ;  §GG. Figure 7(b). 7

8888 plotted against  at   ¦ , %   G © –  ›¥ jk j« k© ¦Gœ and ;  jGGG. Figure 7(c). 7

8888 plotted against  at   ¦ , %   G © –  ›¥ jk j« k© ¦Gœ and ;  §GGG. Figure 7(d).D7 8888 plotted against  at   ¦ , %   G « –  ›¥ jk j« k© ¦Gœ and ;  jGG. Figure 7(e). 7 8888 plotted against  at   ¦ , %   G « –  ›¥ jk j« k© ¦Gœ and ;  §GG. Figure 7(f). 7

8888 plotted against  at   ¦ , %   G « –  ›¥ jk j« k© ¦Gœ and ;  jGGG. Figure 7(g). 7

26

8888 plotted against  at   ¦ , %   G « –  ›¥ jk j« k© ¦Gœ and ;  §GGG. Figure 7(h). 7 8888 plotted against  at   G § %   G ¥ and ;  jGG. Figure 8(a). 7

8888 plotted against  at   G § %   G ¥ and ;  §GG. Figure 8(b). 7

8888 plotted against  at   G § %   G ¥ and ;  jGGG. Figure 8(c). 7

8888 plotted against  at   G § %   G ¥ and ;  §GGG. Figure 8(d). 7 8888 plotted against  at   G ª %   G ¥ and ;  jGG. Figure 8(e).D7 8888 plotted against  at   G ª %   G ¥ and ;  §GG. Figure 8(f). 7

8888 plotted against  at   G ª %   G ¥ and ;  jGGG. Figure 8(g). 7 8888 plotted against  at   G ª %   G ¥ and ;  §GGG. Figure 8(h).D7

Figure 9(a). :  plotted against –at   ¦ , %   G k and   ›G ¦ G § G ªœ

Figure 9(b). : Dplotted against –at   ¦ , %   G © and   ›G ¦ G § G ªœ Figure 9(c). :  plotted against –at   ¦ , %   G ¥ and   ›G ¦ G § G ªœ

Figure 9(d). :  plotted against –at   ¦ , %   G « and   ›G ¦ G § G ªœ

27

List of Tables

Table 1. Percentage difference of Nu and :  at initial and finer mesh for   ¦  %   G ¥ ;  §GG 888 for   G §,   ¦ and ½  G G§ Table 2. Percentage overestimates Nusselt number 87

8888 by exponential fins relative to a Triangular fin for   G §,   ¦ and Table 3. Percentage gain in 7 ½  G j

Table 4. Comparison of Heat Transfer Enhancement relative to Frictional Loss for %   G «,   ¦ and ;  §GG

28

Table(s)

Table 1. Percentage difference of Nu and ݂ܴ݁ at initial and finer mesh for ߚ ൌ ͵௢ ǡ ‫ כ ܪ‬ൌ ͲǤ͸ǡ π ൌ ͷͲͲǤ ෡ ࡾ

0.2

0.5

0.7

തതതത ࡺ࢛

ࡹ

(finer)

തതതത ࡺ࢛(initial)

9

2.29

18

തതതത percentage ࡺ࢛

ࢌࡾࢋ

ࢌࡾࢋ

ࢌࡾࢋ percentage

difference

(finer)

(initial)

2.2899

0.0044

17.973

17.926

0.2622

0.8183

0.8191

0.0977

11.051

11.047

0.0362

27

0.4014

0.4015

0.0249

6.6163

6.6179

0.0242

9

3.9354

3.9393

0.099

19.654

19.648

0.0305

18

2.5394

2.5413

0.0748

18.785

18.748

0.1974

27

1.387

1.3886

0.115

14.964

14.920

0.2949

9

4.1299

4.1396

0.234

19.998

19.990

0.0400

18

4.0699

4.0768

0.169

20.001

19.985

0.0800

27

3.2866

3.2914

0.146

19.771

19.743

0.1418

difference

തതതതfor ܴ෠ ൌ ͲǤͷ, ߚ ൌ ͵ and ߜ ൌ ͲǤͲͷ. Table 2. Percentage overestimates Nusselt number ܰ‫ݑ‬ ࡴ‫כ‬

0.2

0.4

0.6

0.8

ࡹ

Ω

1

10

20

50

100

500

1000

5000

6

53.0

10.8

6.2

2.8

1.5

0.4

0.2

0.0

12

73.0

17.1

9.8

4.5

2.4

0.5

0.3

0.0

18

88.2

20.0

11.6

5.0

2.6

0.6

0.3

0.3

24

97.0

21.5

12.1

5.3

2.8

0.4

0.4

0.0

30

102.8

20.8

11.4

4.9

2.9

0.9

0.5

0.5

6

101.9

33.5

21.5

10.8

5.9

1.4

0.7

0.2

12

183.8

58.9

36.1

16.9

9.2

1.8

0.9

0.0

18

226.5

67.6

40.8

19.6

10.7

2.6

1.3

0.4

24

244.8

66.0

40.2

18.8

10.3

2.4

1.2

0.6

30

251.0

59.3

34.0

15.6

7.7

1.6

0.8

0.0

6

167.1

61.7

41.3

21.4

12.2

3.0

1.5

0.5

12

415.8

138.9

86.6

42.0

23.4

5.0

2.7

0.5

18

559.2

159.8

93.4

41.0

21.6

4.3

2.3

0.4

24

600.0

154.4

87.6

37.8

19.9

4.2

1.7

0.0

30

614.3

147.4

85.5

39.2

21.0

4.3

2.6

0.8

6

212.7

78.3

51.2

25.9

14.6

3.4

1.7

0.5

12

744.0

232.3

139.8

64.8

34.4

7.4

3.7

0.7

18

1401.9

376.9

213.6

94.1

48.5

10.1

5.1

1.0

24

1913.1

454.0

252.8

109.5

55.6

11.2

5.5

1.0

30

2169.9

467.8

256.9

109.4

56.3

11.6

5.7

0.9

തതതത by exponential fins relative to a Triangular fin for ܴ෠ ൌ ͲǤͷ, ߚ ൌ ͵and Table 3. Percentage gain in ܰ‫ݑ‬ ߜ ൌ ͲǤͳ.

ࡴ‫כ‬

1

10

0.2

ࡹ

12

0.30

0.2

24

0.4

തതതത for different values of Ω Percentage gain in ܰ‫ݑ‬ 20

50

100

500

1000

5000

0.22

0.08

0.035

0.08

0.08

0.10

0.07

0.71

0.41

0.33

0.02

0.16

0.28

0.36

0.31

12

1.52

1.93

1.78

0.84

0.20

0.35

0.51

0.48

0.4

24

2.41

1.90

1.02

0.66

1.45

2.26

2.31

2.45

0.6

12

3.58

5.74

6.31

3.88

1.55

1.15

1.53

1.84

0.6

24

5.78

6.13

6.67

3.16

0.68

6.44

7.41

8.27

0.8

12

5.79

10.00

13.13

11.49

7.65

0.56

0.81

2.04

0.8

24

10.16

8.25

12.84

15.09

12.06

3.78

8.65

13.57

Table 4. Comparison of Heat Transfer Enhancement relative to Frictional Loss for‫ כ ܪ‬ൌ ͲǤͺ, ߚ ൌ ͵ and ࡹ

π ൌ ͷͲͲ.

෡ ൌ ૙Ǥ ૜ ࡾ ࢌࡾࢋ

തതതത ࡺ࢛Τࢌࡾࢋ 0.15847

3.93

12

തതതത ࡺ࢛

18

1.85

16.5

0.112121

24

1.17

13.8

30

0.75

11.1

2.9

18.3

തതതത ࡺ࢛

෡ ൌ ૙Ǥ ૞ ࡾ ࢌࡾࢋ

തതതത ࡺ࢛Τࢌࡾࢋ

തതതത ࡺ࢛

෡ ൌ ૙Ǥ ૠ ࡾ ࢌࡾࢋ

തതതത ࡺ࢛Τࢌࡾࢋ

18.2

0.215934

3.7529

17.3232 0.216637

3.56

19.2

0.185417

4.0984

17.8315 0.229839

0.084783

2.77

18.8

0.14734

4.1988

18.7772

0.067568

1.99

17.3

0.115029

3.9054

19.3154 0.202191

0.22361

Figure 1&2

Figure 1. View of the cross-section of the geometry.

Figure 2. Zoomed portion of computational interest.

Fig 3(a) velocity_rhat0.3_M6_beta3_H04

o

*

R^ = 0.3, β = 3 , M = 6 and H = 0.4 0.6

0.5

0.4

0.3

0.2

0.1

0

Fig 3(b) velocity_rhat0.3_M6_beta3_H08

o

*

R^ = 0.3, β = 3 , M = 6 and H = 0.8 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Fig 3(c) velocity_rhat0.3_M12_beta3_H0.4

R^ = 0.3, β = 3o, M = 12 and H* = 0.4 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Fig 3(d) velocity_rhat0.3_M12_beta3_H08

o

*

R^ = 0.3, β = 3 , M = 12 and H = 0.8 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

Fig 4(a) temp_rhat0.3_M6_beta3_conduct1_H0.8

o

*

R^ = 0.3, β = 3 , M = 6, H = 0.8 and Ω = 1 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

Fig 4(b) temp_rhat0.3_M6_beta3_conduct5_H08

o

*

R^ = 0.3, β = 3 , M = 6, H = 0.8 and Ω = 5 1.2

1

0.8

0.6

0.4

0.2

0

Fig 4(c) temp_rhat0.3_M6_beta3_conduct100_H0.8

R^ = 0.3, β = 3o, M = 6, H* = 0.8 and Ω = 100 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Fig 4(d) temp_rhat0.3_M6_beta3_conduct500_H08

R^ = 0.3, β = 3o, M = 6, H* = 0.8 and Ω = 500 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Fig 4(e) temp_rhat0.3_M6_beta3_conduct1000_H0.8

o

*

R^ = 0.3, β = 3 , M = 6, H = 0.8 and Ω = 1000 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Figure 5(a)

4.5 o

*

Rhat=0.3,β=3 ,H =0.2

4 3.5

Nu

3 2.5 2 1.5 M= 6 M= 12 M= 18 M= 24 M= 30

1 0.5 0

1

2

5

10

20

50 Ω

100

500

1000

5000

∞

Figure 5(b)

4 o

*

Rhat=0.3,β=3 ,H =0.4

3.5 M= 6 M= 12 M= 18 M= 24 M= 30

3

Nu

2.5 2 1.5 1 0.5 0

1

2

5

10

20

50 Ω

100

500

1000

5000

∞

Figure 5(c)

4.5 o

*

Rhat=0.3,β=3 ,H =0.6

4 3.5

Nu

3 2.5 2 1.5 M= 6 M= 12 M= 18 M= 24 M= 30

1 0.5 0

1

2

5

10

20

50 Ω

100

500

1000

5000

∞

Figure 5(d)

4 o

*

Rhat=0.3,β=3 ,H =0.8

3.5 3

Nu

2.5 2 1.5 1

M= 6 M= 12 M= 18 M= 24 M= 30

0.5 0

1

2

5

10

20

50 Ω

100

500

1000

5000

∞

Figure 5(e)

5 o

*

Rhat=0.5,β=3 ,H =0.2

4.5 4

Nu

3.5 3 2.5 2

M= 6 M= 12 M= 18 M= 24 M= 30

1.5 1

1

2

5

10

20

50 Ω

100

500

1000

5000

∞

Figure 5(f)

4.5 o

*

Rhat=0.5,β=3 ,H =0.4

4 3.5

Nu

3 2.5 2 1.5 M= 6 M= 12 M= 18 M= 24 M= 30

1 0.5 0

1

2

5

10

20

50 Ω

100

500

1000

5000

∞

Figure 5(g)

4.5 o

*

Rhat=0.5,β=3 ,H =0.6

4 3.5

Nu

3 2.5 2 1.5 M= 6 M= 12 M= 18 M= 24 M= 30

1 0.5 0

1

2

5

10

20

50 Ω

100

500

1000

5000

∞

Figure 5(h)

4.5 o

*

Rhat=0.5,β=3 ,H =0.8

4 3.5

Nu

3 2.5 2 1.5 M= 6 M= 12 M= 18 M= 24 M= 30

1 0.5 0

1

2

5

10

20

50 Ω

100

500

1000

5000

∞

Figure 6

(a)

(c) Figure 6(a-d). Effect of ‫ ܯ‬on തതതത ܰ‫ ݑ‬at ܴ෠ ൌ ͲǤͷ, ߚ ൌ ͵௢ and π ൌ ሼͳͲͲǡ ͷͲͲǡ ͳͲͲͲǡ ͷͲͲͲሽ.

(b)

(d)

Figure 7(a)

5 4.5 4 3.5

Nu

3 2.5 2 M= 6

1.5

M= 12

1

M= 18 o

0.5 0 0.2

M= 24

*

β=3 ,H =0.4,Conductivity=100

0.3

0.4

0.5

ˆ R

M= 30

0.6

0.7

Figure 7(b)

5 4.5 4 3.5

Nu

3 2.5 2 M= 6

1.5

M= 12

1

M= 18

0.5 0 0.2

o

M= 24

*

β=3 ,H =0.4,Conductivity=500

0.3

0.4

0.5

ˆ R

M= 30

0.6

0.7

Figure 7(c)

5 4.5 4 3.5

Nu

3 2.5 2 M= 6

1.5

M= 12

1

M= 18

0.5 0 0.2

o

M= 24

*

β=3 ,H =0.4,Conductivity=1000

0.3

0.4

0.5

ˆ R

M= 30

0.6

0.7

Figure 7(d)

5 4.5 4 3.5

Nu

3 2.5 2 M= 6

1.5

M= 12

1

M= 18

0.5 0 0.2

o

M= 24

*

β=3 ,H =0.4,Conductivity=5000

0.3

0.4

0.5

ˆ R

M= 30

0.6

0.7

Figure 7(e)

4 3.5 3

Nu

2.5 2 1.5 M= 6 M= 12

1

M= 18

0.5

o

M= 24

*

β=3 ,H =0.8,Conductivity=100

0 0.2

0.3

0.4

0.5

ˆ R

M= 30

0.6

0.7

Figure 7(f)

4.5 4 3.5

Nu

3 2.5 2 1.5

M= 6 M= 12

1

M= 18

0.5 0 0.2

o

M= 24

*

β=3 ,H =0.8,Conductivity=500

0.3

0.4

0.5

ˆ R

M= 30

0.6

0.7

Figure 7(g)

4.5 4 3.5

Nu

3 2.5 2 1.5

M= 6 M= 12

1

M= 18

0.5 0 0.2

o

M= 24

*

β=3 ,H =0.8,Conductivity=1000

0.3

0.4

0.5

ˆ R

M= 30

0.6

0.7

Figure 7(h)

4.5 4 3.5

Nu

3 2.5 2 1.5

M= 6 M= 12

1

M= 18

0.5 0 0.2

o

M= 24

*

β=3 ,H =0.8,Conductivity=5000

0.3

0.4

0.5

ˆ R

M= 30

0.6

0.7

Figure 8

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figures 8(a-h). തതതത ܰ‫ ݑ‬plotted against ߚ at ܴ෠ ൌ ሼͲǤͷǡͲǤ͹ሽ, ‫ כ ܪ‬ൌ ͲǤ͸ and π ൌ ሼͳͲͲǡ ͷͲͲǡ ͳͲͲͲǡ ͷͲͲͲሽ.

Figure 9(a)

24 β=3o,H*=0.2

22 20

fRe

18 16 14 Rhat= 0.3

12

Rhat= 0.5

10 Rhat= 0.7

8

6

9

12

15

18 M

21

24

27

30

Figure 9(b)

24 o

*

β=3 ,H =0.4

22 20

fRe

18 16 14 12 Rhat= 0.3

10 Rhat= 0.5

8 6

Rhat= 0.7

6

9

12

15

18 M

21

24

27

30

Figure 9(c)

22 o

*

β=3 ,H =0.6

20 18

fRe

16 14 12 Rhat= 0.3

10

Rhat= 0.5

8 Rhat= 0.7

6

6

9

12

15

18 M

21

24

27

30

Figure 9(d)

20 o

*

β=3 ,H =0.8

19 18

fRe

17 16 15 14 Rhat= 0.3

13 Rhat= 0.5

12 11

Rhat= 0.7

6

9

12

15

18 M

21

24

27

30