For Review Only - หน้าหลักrdo.psu.ac.th/sjstweb/Ar-Press/60-July/5.pdf · For Review Only Keywords: Graetz problem, Sturm-Liouville problem, Partial differential equation,

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Exact solution of boundary value problem describing the

convective heat transfer in fully- developed laminar flow through a circular conduit

Journal: Songklanakarin Journal of Science and Technology

Manuscript ID SJST-2017-0071

Manuscript Type: Original Article

Date Submitted by the Author: 02-Mar-2017

Complete List of Authors: belhocine, ali; Mechanical Engineering, , Mechanical Engineering,

Wan Omar, Wan Zaidi ; Mechanical Engineering, Mechanical Engineering

Keyword: Graetz problem, Sturm-Liouville problem, Partial differential equation, dimensionless variable

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Songklanakarin Journal of Science and Technology SJST-2017-0071 belhocine

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Research Article

Exact solution of boundary value problem describing the convective heat transfer

in fully- developed laminar flow through a circular conduit

Ali Belhocine1*

and Wan Zaidi Wan Omar2

1 Faculty of Mechanical Engineering, University of Sciences and the Technology of

Oran, L.P 1505 El - MNAOUER, USTO 31000 ORAN Algeria

2Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 81310 UTM

Skudai, Malaysia

* Corresponding author, Email address: [email protected]

Abstract

This paper proposes an exact solution of the classical Graetz problem in terms of

an infinite series represented by a nonlinear partial differential equation considering two

space variables, two boundary conditions and one initial condition. The mathematical

derivation is based on the method of separation of variables whose several stages were

illustrated to reach the solution of the Graetz problem. A MATLAB code was used to

compute the eigenvalues of the differential equation as well as the coefficient series.

However, the expression of the Nusselt number as infinite series solution and the Graetz

number was defined based on the heat transfer coefficient and the heat flux from the

wall to the fluid. In addition, the analytical solution was compared to the numerical

values obtained by the same author using FORTRAN program show that the orthogonal

collocation method giving better results. It is important to note that the analytical

solution is in good agreement with published numerical data.

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Keywords: Graetz problem, Sturm-Liouville problem, Partial differential equation,

dimensionless variable

Nomenclature

a :parameter of confluent hypergeometric function

b : parameter of confluent hypergeometric function

cp :heat capacity

Cn : coefficient of solution defined in Eq. (38)

F (a;b;x) : standard confluent hypergeometric function

K : thermal conductivity

L : length of the circular tube

NG : Graetz number

Nu : Nusselt number

Pe : Peclet number

R : radial direction of the cylindrical coordinates

r1 : radius of the circular tube

T : temperature of the fluid inside a circular tube

T0 : temperature of the fluid entering the tube

Tω : temperature of the fluid at the wall of the tube

υmax : maximum axial velocity of the fluid

Greek letters

βn : eigenvalues

ζ : dimensionless axial direction

θ : dimensionless temperature

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ξ : dimensionless radial direction

ρ : density of the fluid

µ : viscosity of the fluid

1. Introduction

The solutions of one or more partial differential equations (PDEs), which are

subjected to relatively simple limits, can be tackled either by analytical or numerical

approach. There are two common techniques available to solve PDEs analytically,

namely the variable separation and combination of variables. The Graetz problem

describes the temperature (or concentration) field in fully developed laminar flow in a

circular tube where the wall temperature (or concentration) profile is a step-function

(Shah & London, 1978). The simple version of the Graetz problem was initially

neglecting axial diffusion, considering simple wall heating conditions (isothermal and

isoflux), using simple geometric cross-section (either parallel plates or circular

channels), and also neglecting fluid flow heating effects, which can be generally

denoted as Classical Graetz Problem (Braga, de Barros, & Sphaier, 2014). Min, Yoo,

and Choi (1997) presented an exact solution for a Graetz problem with axial diffusion

and flow heating effects in a semi-infinite domain with a given inlet condition. Later,

the Graetz series solution was further improved by Brown (1960).

Hsu (1968) studied a Graetz problem with axial diffusion in a circular tube,

using a semi- infinite domain formulation with a specified inlet condition. Ou and

Cheng (1973) employed the separation of variables method to study the Graetz problem

with finite viscous dissipation. They obtained the solution in the form of a series whose

eigenvalues and eigenfunctions of which satisfy the Sturm–Liouville system. The

solution technique follows the same approach as that applicable to the classical Graetz

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problem and therefore suffers from the same weakness of poor convergence behavior

near the entrance. The same techniques have been used by other authors to derive

analytical solutions, involving the same special functions (Fehrenbach, De Gournay,

Pierre, & Plouraboué, 2012; Plouraboue & Pierre, 2007; Plouraboue & Pierre, 2009).

Basu and Roy (1985) analyzed the Graetz problem for Newtonian fluid by taking

account of viscous dissipation but neglecting the effect of axial conduction.

Papoutsakis, Damkrishna, and Lim (1980) presented an analytical solution to the

extended Graetz problem with finite and infinite energy or mass exchange sections and

prescribed wall energy or mass fluxes, with an arbitrary number of discontinuities.

Coelho, Pinho, and Oliveira (2003) studied the entrance thermal flow problem for the

case of a fluid obeying the Phan-Thienand Tanner (PTT) constitutive equation. This

appears to be the first study of the Graetz problem with a viscoelastic fluid. The solution

was obtained with the method of separation of variables and the ensuing Sturm–

Liouville system was solved for the eigenvalues by means of a freely available solver,

while the ordinary differential equations for the eigenfunctions and their derivatives

were calculated numerically with a Runge–Kutta method. In the work of Bilir (1992), a

numerical profile based on based on the finite difference method was developed by

using the exact solution of the one-dimensional problem to represent the temperature

change in the flow direction.

Ebadian and Zhang (1989) analyzed the convective heat transfer properties of a

hydrodynamically, fully developed viscous flow in a circular tube. Lahjomri and

Oubarra (1999) investigated a new method of analysis and improved solution for the

extended Graetz problem of heat transfer in a conduit. An extensive list of contributions

related to this problem may be found in the papers of Papoutsakis, Ramkrishna, Henry,

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and Lim (1980) and Liou and Wang (1990). In addition, the analytical solution

proposed efficiently resolves the singularity and this methodology allows extension to

other problems such as the Hartmann flow (Lahjomri, Oubarra, & Alemany, 2002),

conjugated problems (Fithen & Anand, 1988) and other boundary conditions.

Transient heat transfer for laminar pipe or channel flow was analyzed by many authors.

Ates , Darici, and Bilir (2010) investigated the transient conjugated heat transfer in

thick walled pipes for thermally developing laminar flow involving two-dimensional

wall and axial fluid conduction. The problem was solved numerically by a finite-

difference method for hydrodynamically developed flow in a two-regional pipe, initially

isothermal in which the upstream region is insulated and the downstream region is

subjected to a suddenly applied uniform heat flux. Darici, Bilir, and Ates (2015) in their

work solved a problem in thick walled pipes by considering axial conduction in the

wall. They handled a transient conjugated heat transfer in simultaneously developing

laminar pipe flow. The numerical strategy used in this work is based on a finite

difference method in a thick walled semi-infinite pipe which is initially isothermal, with

hydrodynamically and thermally developing flow and with a sudden change in the

ambient temperature. Darici and Sen (2017) numerically investigated a transient

conjugate heat transfer problem in micro channels with the effects of rarefaction and

viscous dissipation. They also examined the effects of other parameters on heat transfer

such as the Peclet number, the Knudsen number, the Brinkman number and the wall

thickness ratio.

Recently, Belhocine (2015) developed a mathematical model to solve the

classic problem of Graetz using two numerical approaches, the orthogonal collocation

method and the method of Crank-Nicholson.

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In this paper, the Graetz problem that consists of two differential partial

equations will be solved using separation of variables method. The Kummer equation is

employed to identify the confluent hypergeometric functions and its properties in order

to determine the eigenvalues of the infinite series which appears in the proposed

analytical solution. Also, theoretical expressions of the Nusselt number as a function of

Graetz number were obtained. In addition, the exact analytical solution presented in this

work was validated by the numerical value data previously published by the same

author represented by the orthogonal collocation method which giving better results.

2. Background of the Problem

As a good model problem, we consider steady state heat transfer to fluid in a fully

developed laminar flow through a circular pipe (Fig.1). The fluid enters at z=0 at a

temperature of T0 and the pipe walls are maintained at a constant temperature of Tω. We

will write the differential equation for the temperature distribution as a function of r and

z, and then express this in a dimensionless form and identify the important

dimensionless parameters. Heat generation in the pipe due to the viscous dissipation is

neglected, and a Newtonian fluid is assumed. Also, we neglect the changes in viscosity

in temperature variation. A sketch of the system is shown below.

Fig.1.

2.1. The heat equation in cylindrical coordinates

The general equation for heat transfer in cylindrical coordinates developed by Bird,

Stewart, and Lightfoot (1960) is as follows;

TC

kzTu

pZ

2∇=∂∂

ρ (1)

∂∂

+∂∂

+

∂∂

∂∂=

∂∂+

∂∂+

∂∂+

∂∂

2

2

2

2

2

11z

TT

rrTr

rrk

zTuT

r

u

rTu

tTC Zrp θθ

ρ θ+

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∂∂

+∂∂

+

∂∂

+∂∂

+

∂∂

+

+

∂∂

+

∂∂

22222

112

z

u

r

uu

rz

u

z

uu

u

rr

u rZZZr

r

θµ

θµ θθ +

µ2

1

∂∂+

∂∂

r

u

rr

u

r

r θ

θ (2)

Considering that the flow is steady, laminar and fully developed flow (Re ⟨ 2400), and if

the thermal equilibrium had already been established in the flow, then 0=∂∂

tT . The

dissipation of energy would also be negligible. Other physical properties would also be

constant and would not vary with temperature such as ρ, µ, Cp ,k. This assumption also

implies incompressible Newtonian flow. Axisymmetric temperature field ( 0=∂∂θT

),

where we are using the symbol θ for the polar angle.

By applying the above assumptions, Eq. (2) can be written as follows:

pZ

C

kzTu

ρ=

∂∂

∂∂

∂∂

rTr

rr1 (3)

Given that the flow is fully developed laminar flow (Poiseuille flow), then the velocity

profile would have followed the parabolic distribution across the pipe section,

represented by

( )

−=

2

12RruuZ (4)

Where u2 is the Maximum velocity existing at the centerline. By replacing the speed

term in Eq. (3), we get:

−2

12R

ru

pC

kzT

ρ=

∂∂

∂∂

∂∂

rTr

rr1 (5)

Solving the equations require the boundary conditions as set in Fig.1, thus

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• 0=z , 0TT=

• 0=r , 0=∂∂

rT (6)

• Rr = , ωTT=

It is more practical to study the problem with standardized variables from 0 to 1. To do

this, new variables without dimension (known as adimensional) are introduced, defined

as 0TT

TT

−−=

ω

ωθ , Rrx= and

Lzy= . The substitution of the adimensional variables in

Eq.(5) gives

∂∂−

+∂∂−

=∂∂−

−

2

2

2

000

2

22 )()(1)(12

xR

TT

xR

TT

xRc

kyL

TT

R

Rxu

p

θθρ

θ ωωω (7)

After making the necessary arrangements and simplifications, the following simplified

equation is obtained.

∂∂

+∂∂=

∂∂−

2

2

2

2 12

)1(xxxRuC

kL

yx

P

θθρ

θ

where the term k

CRu pρ2 is the adimensional number known as Peclet number (Pe),

which in fact is Reynolds number divided by Prandtl number. In steady state condition,

the partial differential equation resulting from this, in the adimensional form can be

written as follows:

∂∂

∂∂

=∂∂

−x

xxxRPe

L

yx

θθ 1)1( 2 (9)

This equation, if subjected to the new boundary conditions, would be transformed to the

followings:

• z=0, ⇒= 0TT y =0, 1=θ ,

• r = 0, ⇒=∂∂ 0

rT x=0, 0),0(0 ≠⇒=

∂∂ y

xθθ ,

(8)

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• r =R, ⇒= ωTT x=1, 0),1(0 =⇒= yθθ .

It is hereby proposed that the separation of variables method could be applied, to solve

Eq. (9).

3. Analytical Solution using Separation of Variables Method

In both qualitative and numerical methods, the dependence of solutions on the

parameters plays an important role, and there are always more difficulties when there

are more parameters. We describe a technique that changes variables so that the new

variables are “dimensionless”. This technique will lead to a simple form of the equation

with fewer parameters. Let the Graetz problem is given by the following governing

equation

∂∂

+∂∂

=∂∂

−2

22 1)1(

xxxPeR

L

yx

θθθ (10)

where, Pe is the Peclet number, L is the tube length and R is the tube radius. With the

following initial conditions:

IC : y = 0 , 1=θ

BC1 : x=0 , 0=∂∂

xθ

BC2 : x=1 , 0=θ

Introducing dimensionless variables as demonstrated (Huang, Matloz, Wen, & William,

1984) as follows:

Rrx ==ξ (11)

2max Rvc

kz

pρζ = (12)

Lzy= (13)

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By substituting Eq. (13) into Eq. (12) then it becomes:

2max Rvc

Lyk

pρζ = (14)

Knowing that uv 2max=

Therefore,

Rk

Rcu

Ly

Rcu

Lyk

pp.

22 2 ρρζ == (15)

Notice that the term k

Rcu pρ2 in Eq. (15) is similar to the Peclet number, P.

Thus, Eq. (15) can be written as

RPe

Ly=ζ (16)

Based on Eqs. (11)- (16), ones can write the following expressions;

ξθθ∂∂

=∂∂

x (17)

2

2

2

2

ξθθ

∂∂

=∂∂

x (18)

ζθ

ζθζθ

∂∂

=∂∂

∂∂

=∂∂

PeR

L

yy. (19)

Now, by replacing Eqs. (17)-(19) into Eq. (10), the governing equation becomes:

∂∂

+∂∂

=∂∂

−2

22 1)1(

ξθ

ξθ

ξζθ

ξPeR

L

PeR

L (20)

Eq. (20) will be reduced to:

2

22 1)1(

ξθ

ξθ

ξζθ

ξ∂∂

+∂∂

=∂∂

− (21)

The right term in Eq. (21) can be simplified as follows:

∂∂

∂∂

=∂∂

+∂∂

ξθ

ξξξξ

θξθ

ξ11

2

2

(22)

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Finally, the equation that characterizes the Graetz problem can be written in the form of:

∂∂

∂∂

=∂∂

−ξθ

ξξξζ

θξ

1)1(

2 (23)

Now, using an energy balance method in the cylindrical coordinates, Eq. (23) can be

decomposed into two ordinary differential equations. This is done by assuming constant

physical properties of a fluid and neglecting axial conduction and in steady state. The

associated boundary conditions for a constant-wall-temperature case for Eq. (23) are as

follows:

at entrance ζ = 0 , θ =1

at wall ξ = 1 , θ =0

at center ξ = 0 , 0=∂∂ξθ

and dimensionless variables are defined by:

θ = 0TT

TT

−−

ω

ω ,

1rr=ξ and

21max rvc

kz

pρζ =

while the separation of variables method is given by

)()( ξζθ RZ= (24)

Finally, Eq. (23) can be expressed as follows:

ζβ dZ

dZ 2−= (25)

and

0)1( 22

2

2

=−++ Rd

dR

d

Rdξξβ

ξξξ (26)

where 2β is a positive real number and represents the intrinsic value of the system.

The solution of Eq. (25) can be given as:

ζβ 2

1

−= ecZ (27)

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where 1c is an arbitrary constant. In order to solve Eq. (26), transformations of

dependent and independent variables need to be made by taking:

(Ι) 2βξ=v

(Π) ( ) ( )vSevR v 2−=

Thus, Eq. (26) is now given by;

042

1)1(

2

2

=

−−−+ Sd

dS

d

Sd βν

νν

ν (28)

Eq. (28) is also called as confluent hypergeometric as cited in (Slater, 1960) and it is

commonly known as Kummer equation.

3.1. Theorem of Fuchs

A homogeneous linear differential equation of the second order is given by

0)()( ''' =++ yZQyZPy (29)

If P(Z) and Q(Z) admit a pole at point Z=Z0, it is possible to find a solution developed

in the whole series provided that the limits on )()(lim 00ZPZZZZ −→ and

)()(lim 2

00ZQZZZZ −→ exist.

The method of Frobenius seeks a solution in the form of

n

n

n ZaZZy ∑∞

=

=0

)( λ (30)

where, λ is a coefficient to be determined whilst properties of the hypergeometric

functions are defined by ;

++−+++−+++

++++

++= LL

LL

!)1()2)(1(

)1()2)(1(

!2)1(

)1(1),,(

2

n

Z

n

nZZZF

n

θβββαααα

ββαα

βα

βα

),,( γβαF converge Z∀ (31)

Using derivation against Z, the function is now become

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[ ]

++

−+++−+++

++

++++

+++

++=

LL

LL

!)1()2)(1(

)1()2)(1(

!2)2)(1(

)2)(1(

)1(

)1(1),,(

2

n

Z

n

n

ZZZF

dZ

d

n

αβββαααα

ββαα

βα

βα

βα

= ),1,1( ZF ++ βαβα

(32)

From Eq. (32), ones will get ;

−

−−=

− 222 ,2,42

31

2

1)(,1,

42

1ξβ

ββ

ξβξββ

ξ nn

n

nnn FF

q

d (33)

Thus, the solution of Eq. (26) can be obtained by:

−=

− 222 ,1,

4212

βξβξβFecR (34)

0,1,42

1=

− n

nF β

β (35)

where n = 1, 2, 3 ... and eigenvalues nβ are the roots of Eq. (35). These can be readily

computed in MATLAB since it has a built-in hypergeometric function calculator. The

solutions of our equation are the eigenfunctions to the Graetz problem. It can be shown

by a series expansion that the eigenfunctins are:

−= − 22

,1,42

1)(

2

ξββ

ξ ξβn

n

n FeG n (36)

where F is the confluent hypergeometric function or the Kummer function. These

functions are power series in ξ similar to an exponential function (Abramowitz &

Stegun ,1965). The functions represented above have the symmetry property since they

are even functions. Hence the boundary condition at ξ=0 is satisfied. Since the system is

linear, the general solution can be determined using superposition approach:

∑∞

=

−−

−=

1

22,1,

42

1..

22

n

n

n

n FeeC nn ξββ

θ ξβζβ (37)

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The constants in Eq. (37) can be sought using orthogonality property of the Sturm-

Liouville systems after the initial condition is being applied as stated below;

∫

−−

−

−

=−

−

1

0

2

23

2

,1,42

1.)(

,2,42

3.121

2

ξξββξξ

βββ

ξβ

β

dFe

Fe

c

nn

nn

n

n

n

n

(38)

The integral in the denominator of Eq. (38) can be evaluated using numerical

integration.

For the Graetz problem, it is noticed that:

∂∂

∂∂

=∂∂

−ξθ

ξξζ

θξξ )(

3 (39)

where )( 3ξξ − is the function of the weight / nβ eigenvalues

B.C 0,1 == θξ

B.C 0,0 =∂∂

=ξθ

ξ

IC 1,0 == θζ

∑∑∞

=

−−∞

=

−

−==

1

22

1

,1,42

1)(

222

n

nn

n

n

nn FeeCGeC nnn ξββ

ξθ ξβζβζβ (40)

−= − 22,1,

42

1)(

2

ξββ

ξ ξβn

nn FeG n is the function of the weight

Sturm-Liouville problem.

0)1(1 22 =−−

nn

n Gd

dG

d

dβξ

ξξ

ξξ (41)

0=ξd

dGn for 0=ξ , 0=nG for 1=ξ (42)

IC 1,0 == θζ

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∑∞

=

−

−===1

22,1,

42

11)0(

2

n

nn

n FeC n ξββ

ζθ ξβ (43)

Relation of orthogonality

)(,0)()()(1

0jixYxYxW ji ≠=∫ (44)

0,1,42

1,1,

42

1)(

221

0

22322

=

−

−− −−∫ ξξββ

ξββ

ξξ ξβξβdFeFe m

mn

n mn )(, mn ≠∀ (45)

∑

∑∞

=

−−

∞

=

−−

−

−−+

−−=

∂∂

1

222

1

22

,2,42

31

2

1)(

,1,42

1)(

22

22

n

nn

n

nn

n

nn

nn

FeeC

FeeC

nn

nn

ξββ

βξβ

ξββ

ξβξθ

ξβζβ

ξβζβ

(46)

∑∞

=

−−

−−=

∂∂

1

222,1,

42

1)(

22

n

n

n

nn FeeC nn ξββ

βζθ ξβζβ

(47)

By considering

− nnF β

β,1,

42

1 , Eq. (39) and onwards can be given as;

1

0

1

0

1

0

3 )(ξθ

ξξθ

ξξ

ξζθ

ξξ∂∂

=

∂∂

∂∂

=∂∂

− ∫∫ d (48)

1

1

0

3)(=∂

∂=

∂∂

−∫ξξ

θξ

ζθ

ξξ d (49)

∑

∑

∫ ∑

∞

=

−−

∞

=

−−

∞

=

−−

−

−−+

−−=

−−−

1

22

1

22

1

0 1

2223

,2,42

31

2

1)(

,1,42

1)(

,1,42

1)()(

2

2

22

n

nn

n

nn

n

n

n

nn

n

nn

nn

FeeC

FeeC

dFeeC

nn

nn

nn

ββ

ββ

ββ

β

ξξββ

βξξ

βζβ

βζβ

ξβζβ

(50)

∑

∑ ∫∞

=

−−

∞

=

−−

−

−−=

−−−

1

22

1

22

1

0

32

,2,42

31

2

1)(

,1,42

1)()(

2

22

n

n

n

n

nn

n

n

n

nn

FeeC

dFeeC

nn

nn

ββ

ββ

ξξββ

ξξβ

βζβ

ξβζβ

(51)

By combining Eqs. (49), (50) and (51), the equation can be reduced to;

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ξξββ

ξξ ξβdFe n

nn∫

−− −1

0

223 ,1,42

1)(

2

−

−= −

nn

n

Fe n ββ

ββ

,2,42

31

2

1 2 (52)

Let’s multiply Eq. (43) by Eq. (53) and then integrate Eq. (54),

−− − 223,1,

42

1)(

2

ξββ

ξξ ξβm

mFe m (53)

ξξββ

ξββ

ξξ

ξξββ

ξξ

ξβ

ξβ

ξβ

dFe

FeC

dFe

nn

mm

n

n

mm

n

m

m

−

−−=

−−

−

−∞

=

−

∫∑

∫

22

1

0

223

1

1

0

223

,1,42

1

.,1,42

1)(

,1,42

1)(

2

2

2

(54)

The outcomes of multiplication and integration process will produce the following:

(i) If (n ≠ m) the result is equal to zero (0)

(ii) If (n = m) the result is

ξξββ

ξξ ξβdFe n

nn∫

−− −1

0

223,1,

42

1)(

2

ξξββ

ξξ ξβdFeC n

nn

n

21

0

23 ,1,42

1)(

2

∫

−−= − (55)

Substituting Eq. (52) into Eq.(55), the equation becomes;

=

−

− −

nn

n

Fe n ββ

ββ

,2,42

31

2

1 2 ξξββ

ξξ ξβdFeC n

nn

n

21

0

23 ,1,42

1)(

2

∫

−− − (56)

And the constants Cn can be obtained by;

ξξββ

ξξ

ββ

β

ξβ

β

dFe

Fe

C

nn

nn

n

n

n

n

21

0

23

2

,1,42

1)(

,2,42

31

2

1

2

∫

−−

−

−

=−

−

(57)

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4. Results and discussion

4.1. Evaluation of the first four eigenvalues and the constant Cn

A few values of the series coefficients are given in Table 1 together with the

corresponding eigenvalues. The results of calculated values of the center temperature as

a function of the axial coordinate ζ are also summarized in Table 2.

Tables 1-2.

The leading term in the solution for the center temperature is there for:

)0(9774.0)0,(704.2

Geζ

ζθ−

≈ (58)

4.2. Graphical representation of the exact solution of the Gratez problem

The center temperature profile is shown in Fig.2 using five terms to sum the series.

As seen in this figure, the value of dimensionless temperature (θ) decreases with

increasing values of dimensionless axial position (ζ). Note that the five-term series

solution is not accurate for ζ<0.05 More terms needed here for the series to converge.

Fig.2.

4.3. Comparison between the analytical model and the previous model simulation

results

In order to compare the previous numerical results carried out previously by

Belhocine (2015) with the analytical model of our heat transfer problem, we chose to

present the results of numerical distribution of temperature with the method of

orthogonal collocation which gives the best results. Fig.3 plots the comparison results. It

is clear from Fig. 3 that there is a good agreement between numerical results and center

analytical solution of the Graetz problem.

Fig. 3.

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4.4. Heat transfer coefficient correlation

The heat flux from the wall to the fluid )(zqω is a function of axial position. It can be

calculated directly by using the result:

),()( zRr

Tkzq∂∂

=ω (59)

but as we noted earlier, it is customary to define a heat transfer coefficient )(zh via

)()()( bTTzhzq −= ωω (60)

where the bulk or cup-mixing average temperature bT is introduced. The way to

experimentally determine the bulk average temperature is to collect the fluid coming out

of the system at a given axial location, mix it completely, and measure its temperature.

The mathematical definition of the bulk average temperature was given in an earlier

section.

∫

∫=

R

R

b

drrvr

drzrTrvr

T

0

0

)(2

),()(2

π

π (61)

where the velocity field )1()( 22

0 Rrvrv −= . You can see from the definition of the

heat transfer coefficient that it is related to the temperature gradient at the tube wall in a

simple manner:

)(

),(

)(bTT

zRr

Tk

zh−

∂∂

=ω

(62)

We can define a dimensionless heat transfer coefficient, which is known as the Nusselt

number.

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)(

)1,(

22

)(ζθ

ζξθ

ζbk

RhNu

∂∂

−== (63)

Where bθ is the dimensionless bulk average temperature.

By substituting from the infinite series solution for both the numerator and the

denominator, the Nusselt number can be written as follows.

∑ ∫

∑∞

=

−

∞

=

−

−

∂∂

−==

1

1

0

3

1

)()(4

)(

22

)(2

2

n

nn

n

nn

dGeC

GeC

k

RhNu

n

n

ξξξξ

ξξ

ζζβ

ζβ

(64)

The denominator can be simplified by using the governing differential equation for

)(ξnG , along with the boundary conditions, to finally yield the following result.

∑

∑∞

=

−−

∞

=

−

∂∂

∂∂

=

12

1

)1(2

)1(

2

2

2

n

n

n

n

n

nn

Ge

eC

GeC

Nu

n

n

n

ξβ

ξ

ζβζβ

ζβ

(65)

We can see that for large ζ , only the first term in the infinite series in the numerator,

and likewise the first term in the infinite series in the denominator is important.

Therefore, as; 656.32

,2

1 =→∞→β

ζ Nu . From our equation, it can be expressed:

∑

∑

∞

=

−−

∞

=

−−

−

−+

−

−

−

=

1

22

1

22

, 2

2

.,2,42

321

2

1

1.,2,42

3212

n

N

nn

n

n

n

N

nn

n

nGr

aNu

Gr

n

n

Gr

n

n

eFeC

eFeCN

Nβπ

β

βπβ

ββ

βπ

ββ

β (66)

where kL

RvcN

p

Gr2

2

maxπζ= is the Graetz number.

Fig.4 is the plot showing Nusselt number along the nondimensional length of a

tube with uniform heat flux. As expected, the graphs show that the Nusselt number is

very high at the beginning of the entrance region of the tube and thereafter decreases

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exponentially to the fully developed Nusselt number. Fig. 4 indicates that the Nusselt

number decreased in the entry region and rapidly reached a constant value in the fully

developed region

Fig.4.

4. Conclusion

In this paper, an exact solution of the Graetz problem is successfully obtained

using the method of separation of variables. The hypergeometric functions are

employed in order to determine the eigenvalues and constants, Cn and later to a find

solution for the Graetz problem. The mathematical method performed in this study can

be applied to the prediction of the temperature distribution in steady state thermally

laminar heat transfer based on the fully developed velocity for fluid flow through a

circular tube. In future work extensions, we recommend performing the Graetz solution

by separation of variables in a variety of ways of accommodating non-Newtonian flow,

turbulent flow, and other geometries besides a circular tube. It is important to note that

the present analytical solutions of the Graetz problem are in good agreement with

previously published numerical data of the author. It will be also interesting to solve the

equation of the Graetz problem using experimental data for comparison purposes with

the proposed exact solution.

References

Abramowitz, M., & Stegun, I. (1965) Handbook of Mathematical Functions, Dover,

New York.

Ates, A., Darıcı, S., & Bilir, S. (2010). Unsteady conjugated heat transfer in thick

walled pipes involving two-dimensional wall and axial fluid conduction with

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uniform heat flux boundary condition. International Journal of Heat and Mass

Transfer, 53(23), 5058–5064.

Basu, T., & Roy, D. N. (1985). Laminar heat transfer in a tube with viscous dissipation.

International Journal of Heat and Mass Transfer, 28, 699–701.

Belhocine, A. (2015). Numerical study of heat transfer in fully developed laminar flow

inside a circular tube. International Journal of Advanced Manufacturing

Technology, 1-12.

Bilir, S. (1992). Numerical solution of Graetz problem with axial conduction.

Numerical Heat Transfer Part A-Applications, 21, 493-500.

Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (1960). Transport Phenomena, John

Wiley and Sons, New York.

Braga, N. R., de Barros, L. S., & Sphaier, L. A. (2014). Generalized Integral Transform

Solution of Extended Graetz Problems with Axial Diffusion ICCM2014 28-30th

July, Cambridge, England, pp.1-14.

Brown, G. M. (1960) .Heat or mass transfer in a fluid in laminar flow in a circular or

flat conduit.AIChE Journal, 6, 179–183.

Coelho, P. M., Pinho, F. T., & Oliveira, P. J. (2003). Thermal entry flow for a

viscoelastic fluid: the Graetz problem for the PTT model. International Journal of

Heat and Mass Transfer, 46, 3865–3880.

Darıcı, S., Bilir, S., & Ates, A. (2015). Transient conjugated heat transfer for

simultaneously developing laminar flow in thick walled pipes and minipipes.

International Journal of Heat and Mass Transfer, 84, 1040–1048.

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Ebadian, M. A., & Zhang, H. Y. (1989). An exact solution of extended Graetz problem

with axial heat conduction. International Journal of Heat and Mass Transfer,

32(9), 1709-1717.

Fehrenbach, J., De Gournay, F., Pierre, C., & Plouraboué, F. (2012). The Generalized

Graetz problem in finite domains. SIAM Journal on Applied Mathematics, 72,

99–123.

Fithen, R. M„ & Anand, N. K. (1988). Finite Element Analysis of Conjugate Heat

Transfer in Axisymmetric Pipe Flows. Numerical Heat Transfer, 13, 189-203.

Graetz, L. (1882). Ueber die Wärmeleitungsfähigkeit von Flüssigkeiten. Annalen der

Physik, 254, 79. doi: 10.1002/andp.18822540106

Hsu, C. J. (1968). Exact solution to entry-region laminar heat transfer with axial

conduction and the boundary condition of the third kind. Chemical Engineering

Science, 23(5), 457–468.

Huang, C. R., Matloz, M., Wen, D. P., & William, S. (1984). Heat Transfer to a

Laminar Flow in a Circular Tube. AIChE Journal, 5, 833.

Lahjomri, J., & Oubarra, A. (1999). Analytical Solution of the Graetz Problem with

Axial Conduction. Journal of Heat Transfer, 1, 1078-1083.

Lahjomri, J., Oubarra, A., & Alemany, A. (2002). Heat transfer by laminar Hartmann

flow in thermal entrance eregion with a step change in wall temperatures: The

Graetz problem extended. International Journal of Heat and Mass Transfer, 45(5),

1127-1148.

Liou, C. T., & Wang, F. S. (1990). A Computation for the Boundary Value Problem of

a Double Tube Heat Exchanger. Numerical Heat Transfer Part A, 17, 109-125.

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Min, T., Yoo, J. Y., & Choi, H. (1997). Laminar convective heat transfer of a bingham

plastic in a circular pipei. Analytical approach—thermally fully developed flow

and thermally developing flow (the Graetz problem extended). International

Journal of Heat and Mass Transfer, 40(13), 3025–3037.

Ou, J. W., & Cheng, K. C. (1973). Viscous dissipation effects in the entrance region

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289–301.

Papoutsakis, E., Damkrishna, D., & Lim, H. C. (1980). The Extended Graetz Problem

with Dirichlet Wall Boundary Conditions. Applied Scientific Research, 36, 13-34.

Papoutsakis, E., Ramkrishna, D., & Lim, H. C. (1980). The Extended Graetz Problem

with Prescribed Wall Flux. AlChE Journal, 26(5), 779-786.

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axial cylinders. International Journal of Heat and Mass Transfer, 50(23-24), 4901–

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Pierre, C., & Plouraboué, F. (2009). Numerical analysis of a new mixed-formulation for

eigenvalue convection-diffusion problems. SIAM Journal on Applied

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Sen, S., & Darici, S. (2017).Transient conjugate heat transfer in a circular microchannel

involving rarefaction viscous dissipation and axial conduction effects. Applied

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Slater, L. J. (1960). Confluent Hypergeometric Functions. Cambridge University Press.

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Research Article



Ali Belhocine1*





Skudai, Malaysia


List of figures

Fig.1. Schematics of the classical Graetz problem and the coordinate system

Fig.2. Variation of dimensionless temperature profile (θ) with dimensionless axial

distance (ζ)

Fig. 3. A comparison between the present analyticalcal results with the numerical

Orthogonal collocation data of Belhocine (2015)

Fig.4. Nusselt number versus dimensionless axial coordinate

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Fig.1.

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

1,1

Dimensionless Temperature

θθ θθ

Dimensionless Axial Position ζζζζ

Approximation Solution θ(ζ ,0)θ(ζ ,0)θ(ζ ,0)θ(ζ ,0)

Center Analytical Solution with 5 terms θθθθ

d e m o d e m o d e m o d e m o









Fig.2.

R

r

z

T(R, z )=Tω

Fluid at

T0

vr (z)

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0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

1,1

1,2

Dimensionless Temperature

θθ θθ

Longitudinal Coordinate ζζζζ

Center Analytical Solution

Orthogonal Collocation Solution










Fig. 3.

0,0 0,2 0,4 0,6 0,8 1,0

3

4

5

6

7

8

9

10

11

Nusselt Number

Nu

Dimensionless Axial Position ζζζζ

Nu

3,656









Fig.4.

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Research Article



Ali Belhocine1*





Skudai, Malaysia


List of tables

Table 1. Eigenvalues and constants for Graetz’s problem.

Table 2. Results of the center temperature functions θ (ζ)

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Table 1.

n Eigenvalues βn Coefficient Cn )0( =ξnG

1 2.7044 0.9774 1.5106

2 6.6790 0.3858 -2.0895

3 10.6733 -0.2351 -2.5045

4 14.6710 0.1674 -2.8426

5 18.6698 -0.1292 -3.1338

Table 2.

ζ Temperature (θ) )0,(ζθ

0 1.0000000 1.0000000

0,05 0,93957337 1,02424798

0,1 0,70123412 0,71053981

0,15 0,49191377 0,49291463

0,25 0,23720134 0,2372129

0,5 0,03811139 0,03811139

0,75 0,0061231 0,0061231

0,8 0,00424771 0,00424771

0,85 0,00294671 0,00294671

0,9 0,00204419 0,00204419

0,95 0,00141809 0,00141809

0,96 0,00131808 0,00131808

0,97 0,00122512 0,00122512

0,98 0,00113871 0,00113871

0,99 0,0010584 0,0010584

1 0,00098376 0,00098376

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