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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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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
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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
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Ou, J. W., & Cheng, K. C. (1973). Viscous dissipation effects in the entrance region
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289–301.
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with Dirichlet Wall Boundary Conditions. Applied Scientific Research, 36, 13-34.
Papoutsakis, E., Ramkrishna, D., & Lim, H. C. (1980). The Extended Graetz Problem
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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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Shah, R. K., & London, A. L. (1978). Laminar Flow Forced Convection in Ducts.
Retrieved from https://www.elsevier.com
Slater, L. J. (1960). Confluent Hypergeometric Functions. Cambridge University Press.
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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]
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
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
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
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
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
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
d e m o d e m o d e m o d e m o
Fig.4.
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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]
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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