ORIGINAL
A semianalytical approach for temperaturedistribution in Dean ﬂow
Charbel Habchi
•
Mahmoud Khaled
•
Thierry Lemenand
•
Dominique Della Valle
•
Ahmed Elmarakbi
•
Hassan Peerhossaini
Received: 16 December 2012/Accepted: 17 August 2013/Published online: 28 August 2013
SpringerVerlag Berlin Heidelberg 2013
Abstract
Numerical simulations of the ﬂow ﬁeld andheat transfer require the conjugate solution of the Navier–Stokes and energy equations, a highly computeintensiveprocess. Here a semianalytical approach is proposed tosolve the energy equation in curved pipes. It requires theﬂow velocity ﬁeld, the wall temperature, and the temperature at only one point of the ﬂow crosssection to providethe entire temperature ﬁeld.
List of symbols
a
Duct radius, m
d
Pipe diameter, m
D
Thermal diffusivity, m
2
s
r
Radial coordinate in the pipe cross section
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
x
2
þ
y
2
p
R
Bend curvature radius, m
T
Temperature, K
u, v
Radial velocities in the pipe cross section, m s

1
W
Mean axial velocity, m s

1
x, y
Cartesian coordinates
z
Curvilinear coordinate
Greek symbols
e
Relative error between exact and approximate solution
h
Curvature angle in the bend plane, rad
l
Dynamic viscosity, Pa s
t
Kinematic viscosity, m
2
s

1
q
Density, kg m

3
Subscripts
app
Solution obtained from the present approach
ext
Exact solution
c
Mixer centerline
i, j, k
Iteration number
w
Wall0 Constant values
Notations
0
Firstdegree partial derivative
00
Seconddegree partial derivative
Dimensionless numbers
De
Dean number
=
(
W
2
a
3
)/(
R
t
2
)
Re
Reynolds number
=
Wd
/
t
1 Introduction
The secondary ﬂow arising in curved tubes by the effect of centrifugal forces (usually called Dean cells) takes theshape of a pair of counterrotating rollcells. These Deanrollcells enhance radial mass and heat transfer comparedto a straight pipe [1, 2]. This property is useful in numerous
applications involving the mixing of viscous ﬂuids: in
C. Habchi
M. KhaledEnergy and ThermoFluids Group ETF, School of Engineering,Lebanese International University LIU, P.O. Box 146404,Mazraa, Beirut, LebanonT. Lemenand
D. Della Valle
H. PeerhossainiThermoﬂuid Complex Flows and Energy Research Group,Laboratoire de Thermocine´tique de Nantes, CNRS UMR 6607,LUNAM Universite´, 44306 Nantes, FranceD. Della ValleONIRIS, 44322 Nantes, FranceA. ElmarakbiSchool of Computing and Technology, University of Sunderland, Sunderland SR6 0DD, UK H. Peerhossaini (
&
)Institut des Energies de Demain (IED), Univ Paris Diderot,Sorbonne Paris Cite´, 75013 Paris, Franceemail: hassan.peerhossaini@univparisdiderot.fr
1 3
Heat Mass Transfer (2014) 50:23–30DOI 10.1007/s002310131222z
chemical reactions, multifunctional heat exchangers andfoodprocessing applications, especially when the curvedpipe segments are mounted in a twistedpipe conﬁgurationto create chaotic advection [3, 4].
The local heat transfer process in a helically coiled heatexchanger has been widely studied experimentally, particular by Acharya et al. [3] and Mokrani et al. [5]. It has
been shown that the presence of the Dean instability has asigniﬁcant effect on heat transfer and extensively modiﬁesthe temperature proﬁles. Dean instability, appearing abovea critical Dean number, forms two additional vortices,called Dean vortices, that appear on the concave wall of thecurved tube and rotate in the direction opposite to the Deanrollcells.Lemenand and Peerhossaini [6] simpliﬁed the Navier–Stokes and energy equations and obtained a thermal modelto simulate heat transfer in helically coiled and chaotictwisted pipe. The large database obtained from this simpliﬁed heat transfer model provides correlations to theNusselt number
Nu
.Naphon and Wongwises [7] reviewed the heat transferand ﬂow characteristics of singlephase and twophaseﬂows in curved pipes. They reported numerous relevantcorrelations for heat transfer coefﬁcients and friction factors in different curvedtube conﬁgurations. This reviewpoints to the lack of analytical solutions for the energyequation in curved ducts and two main reasons can be putforward.On the one hand, the experimental study of heat transferin this type of geometry requires special temperaturemeasurement techniques that, in addition to being difﬁcultto realize [7] and generally intrusive, interact with the ﬂowand affect the measurement results. On the other hand,numerical simulations of the ﬂow ﬁeld and the heat transferrequire the conjugate solution of the Navier–Stokes andenergy equations, which is highly demanding in terms of computer memory and capacity.Analytical solutions could be an appropriate alternativeto the above methods and should allow accessible solutionsto this difﬁcult problem. The availability of analyticalsolutions for the velocity proﬁle in a curved pipe (Dean’ssolution [8, 9]) led to the ‘‘discovery’’ of chaotic advection
in twistedpipe ﬂow by Jones et al. [10]. Meanwhile, anequivalent analytical solution for the thermal ﬁeld of theDean ﬂow has not appeared in the open literature. Therefore, the ‘‘thermal chaotic advection’’ has so far beendeduced from the hydrodynamic ﬁeld.Here we propose an analytical resolution of the energyequation in curved pipes that allows computation of thetemperature proﬁle in the tube cross section from theknowledge of the wall temperature and the temperature atone point in the ﬁeld. The main practical interest of thismethod is in heat transfer experiments, since it permitsreduction of the number of sensors to only one in the ﬂowsection. While this property can be extended to any type of internal ﬂow, the results presented here concern the curvedpipe ﬂow, i.e. regular Dean ﬂow.This method is based on the resolution of the partialdifferential equations (PDEs) for temperature derived fromthe nonlinear energy equation. The prerequisites are theknown velocity components at each point in the ﬂow crosssection, which can generally be measured by nonintrusiveoptical techniques (LDA, PIV) or can be computed.The paper is organized as follows. In Sect. 2 we formulate the problem. Section 3 describes the mathematicalapproach to solving the energy equation. Results and discussion are presented in Sect. 4, and in Sect. 5 we draw
some conclusions.
2 Problem formulation
In fully developed laminar pipe ﬂow, the temperature ﬁeldcan be reduced to a twodimensional planar ﬁeld due to theaxial symmetry. The energy differential equation in Eulerian coordinates is then independent of the streamwisecoordinate
z
(Fig. 1) and can be written as
u x
;
y
ð Þ
o
T
o
x
þ
v x
;
y
ð Þ
o
T
o
y
¼
D
o
2
T
o
x
2
þ
o
2
T
o
y
2
:
ð
1
Þ
The velocity ﬁeld is provided by the analytical solutiongiven in Habchi et al. [11]. The solution was calculatedfrom the streamfunction
w
developed by Jones et al. [10]for the Dean [8, 9] asymptotic solution, given by
w
¼
t
Re
2
288
W a
4
r
2
1
r
2
2
y
ð
2
Þ
The Dean perturbation analysis [9] is valid for smallDean numbers De
\
100 and small curvature ratios (
a
/
R
).Although some assumptions are made in deriving thevelocity ﬁeld in curved pipe ﬂow, Jones et al. [10] showedthat the qualitative trends of experimental results in pipeﬂow are captured by this velocity ﬁeld. Previous studies[11, 12] showed that the predicted deformation rates
derived from Dean’s solution are in fair agreement with
Fig. 1
Eulerian frame of reference and secondary ﬂow in a curvedpipe24 Heat Mass Transfer (2014) 50:23–30
1 3
experimental values because authors exhibit a relativedifference of 10 % maximum, proving that Dean’s analytical approach captures the basic physics underlying theﬂow.In the local coordinate system (
x
,
y
,
z
) in Fig. 1, thesecondary velocity components
u
and
v
can be written as
u
¼
o
w
o
yv
¼
o
w
o
x
ð
3
Þ
Hence, the velocity ﬁeld follows:
u
¼
a
1
r
2
6
y
2
3
r
2
4
r
2
1
r
2
v
¼
6
a
xy
1
r
2
3
r
2
;
ð
4
Þ
where
a
=
Re
/144.Equation (1) in its general form is a nonlinear differential equation of second order. In the next section wedescribe a semianalytical approach to solving these typesof equations.
3 A semianalytical approach
The nonlinear energy Eq. (1) can be linearized by considering that each point
P
k
in the tube cross section belongsto a virtual disk
D
k
on which the velocity components areuniformly distributed and are equal to that of the point(
x
k
,
y
k
) in the temperature computation domain, i.e. of point
P
k
, with
k
=
1
?
n
. Thus, considering
n
points in thetube cross section, we have
n
superposed virtual disks of radius 1, on each of which the velocity components
u
k
and
v
k
are uniform, obtained from the hydrodynamic of theproblem (Eq. 4) as shown in Fig. 2. With this assumption,
the solution of the nonlinear energy Eq. (1) on the tubecross section is reduced to the solutions of
n
linear PDEs,one on each virtual disk
D
k
. This approach is used to obtainthe temperature distribution in Dean ﬂow for a givenconstant wall temperature.Hence, we assume that the temperature ﬁeld on a givencomputation domain of
n
points can be obtained from thesolutions of a system of
n
PDEs, each written as:
u
k
o
T x
;
y
ð Þ
o
x
þ
v
k
o
T x
;
y
ð Þ
o
y
¼
D
o
2
T x
;
y
ð Þ
o
x
2
þ
o
2
T x
;
y
ð Þ
o
y
2
with
k
¼
1
!
n
for
x
2
þ
y
2
¼
1
o
T x
;
y
ð Þ
o
x
¼
o
T x
;
y
ð Þ
o
y
¼
0
T x
;
y
ð Þ ¼
T
w
:
8><>:
ð
5
Þ
The velocity values
u
k
and
v
k
in Eq. (5) are coefﬁcientsfor each equation
k
referring to a virtual disk
D
k
, and varyonly among the different
n
equations, i.e. among the
n
virtual disks. Hence, the solution of each PDE of thesystem of Eq. (5) is the same and only the values of
u
k
and
v
k
vary from one solution to another. Thus, the system of Eq. (5) can be solved by adopting a parametric procedureand solving the following equation:
u
0
o
T x
;
y
ð Þ
o
x
þ
v
0
o
T x
;
y
ð Þ
o
y
¼
D
o
2
T x
;
y
ð Þ
o
x
2
þ
o
2
T x
;
y
ð Þ
o
y
2
i
ð Þ
for
x
2
þ
y
2
¼
1
o
T x
;
y
ð Þ
o
x
¼
o
T x
;
y
ð Þ
o
y
¼
0
ii
ð Þ
T x
;
y
ð Þ ¼
T
w
iii
ð Þ
8><>:
ð
6
Þ
where
u
0
and
v
0
are respectively equal to
u
k
and
v
k
at thepoint
P
k
.Once this differential equation is solved, the solution fora given point
P
k
in the tube cross section can be obtainedby replacing
u
0
and
v
0
by their corresponding values
u
k
and
v
k
known from the Dean’s velocity ﬁeld solution (Eq. 4) at
P
k
.The solution of the Eq. (6) is obtained as follows:
•
ﬁrst, the homogenous system and boundary conditionsgiven, respectively in Eqs. (6i) and (6ii) are solved by
Fourier’s method (see Cain and Meyer [13] for moredetails) for separation of variables (i.e. Eq. 7) whichconsists in solving the system of Eq. (6i) and (6ii) as
orthogonal function series,
•
then, we satisfy condition (6iii) on the sum of the
k
nodal functions obtained from the transcendentalequation (i.e. Eq. 12) arising from the solution of thesystem (6i) and (6ii).
If one isolates Eqs. 6i and 6ii from Eq. 6iii, the dif
ferential system becomes homogeneous since its partialdifferential equation as well as the boundary equation arehomogeneous. Thus, the Fourier method can be applied tothe present analytical approach (with Eq. 6iii satisﬁedlater after solving for the separate functions and summingthe Fourier functions):
T x
;
y
ð Þ ¼
f x
ð Þ
g y
ð Þ
:
ð
7
Þ
Hence, the PDE is reduced to two ordinary differentialequations
u
0
f
0
x
ð Þ
f x
ð Þ
D f
00
x
ð Þ
f x
ð Þ ¼
v
0
g
0
y
ð Þ
g y
ð Þ þ
Dg
00
y
ð Þ
g y
ð Þ ¼
b
2
;
ð
8
Þ
where
±
b
2
is a constant.Here we solve Eq. (8) on the
y
axis and compare the analytical results to those obtained experimentally by Mokraniet al. [5]. On the
y
axis, by substituting
x
=
0 in Eq. (4), thevelocity components of the Dean ﬂow are
u
0
;
y
ð Þ ¼
a
7
y
4
23
y
2
þ
4
y
2
1
v
0
;
y
ð Þ ¼
0
:
ð
9
Þ
Heat Mass Transfer (2014) 50:23–30 25
1 3
Equation (8) can then be written:
u
0
D f
0
0
ð Þ
f
0
ð Þ
f
00
0
ð Þ
f
0
ð Þ ¼
g
00
y
ð Þ
g y
ð Þ ¼
c
2
i
ð Þ
for
y
¼
1
o
T
0
;
y
ð Þ
o
y
¼
0
ii
ð Þ
T
0
;
y
ð Þ ¼
T
w
iii
ð Þ
8><>:
ð
10
Þ
To satisfy the symmetry conditions, the expression
g
00
(
y
)/
g
(
y
)
= ±
c
2
has a solution only for

c
2
, and thus itfollows that:
g y
ð Þ ¼
A
cos
cy
ð Þ þ
B
sin
cy
ð Þ
;
ð
11
Þ
where
A
and
B
are constants.Applying the boundary condition (10ii) to Eq. (11)
leads to the following transcendental equation
T
0
;
y
ð Þ ¼
X
1
i
¼
0
C
i
f
0
ð Þ
cos
i
p
y
ð Þ½ ð
12
Þ
where
C
i
is a constant.Now solving the lefthand side of Eq. (10i) andcombining its solution
f
(0) with Eq. (12) into
T
(0,
y
)
=
f
(0)
g
(
y
) leads to:
f
0
ð Þ ¼
f
00
0
ð Þ
C
2
i
f
0
0
ð Þ
C
2
i
u
0
ð
i
Þ
T
0
;
y
ð Þ ¼
X
1
i
¼
0
f
00
0
ð Þ
C
i
f
0
0
ð Þ
C
i
u
0
cos
i
p
y
ð Þ¼
X
1
i
¼
0
E
i
1
þ
F
i
u
0
ð Þ
cos
i
p
y
ð Þ½ ð
ii
Þð
13
Þ
As the PDE is of order 2, an additional boundarycondition is needed to obtain the two constants
E
i
and
F
i
;the temperature in the tube center can be used for thispurpose: this choice is arbitrary; another temperature in theﬂow cross section could be used instead. Then theadditional boundary condition is:
y
¼
0
!
T
¼
T
0
;
c
ð
14
Þ
Solving the temperature distribution on the
y
axis yieldsa parametric expression that depends on the velocity ﬁeld,the ﬂuid temperature at the tube center
T
0,
c
, and the walltemperature
T
w
:
T y
ð Þ ¼
T
w
X
1
j
¼
0
1
ð Þ
j
þ
u
0
u
0
;
c
T
0
;
c
T
w
þ
1
ð Þ
j
þ
1
cos
j
p
y
ð Þ
j
þ
1
ð Þ
j
þ
2
ð Þ
:
ð
15
Þ
In Eq. (15),
u
0
takes the value of
u
(0,
y
) obtained fromEq. (9) at the
y
position where the temperature
T
(
y
) iscalculated. By extending the resolution along the
y
axis, i.e.by considering the solution of the
n
PDEs, one can write:
T y
ð Þ ¼
T
w
X
1
j
¼
0
1
ð Þ
j
þ
u y
ð Þ
u
0
;
c
T
0
;
c
T
w
þ
1
ð Þ
j
þ
1
cos
j
p
y
ð Þ
j
þ
1
ð Þ
j
þ
2
ð Þ
:
ð
16
Þ
The analytical solution developed here provides thetemperature distribution on the crosssection diameter fromonly one known temperature at the pipe center (the other ison the wall), thus reducing the instrumentation required andminimizing the ﬂow disturbance in practical applications.The temperature distribution depends implicitly on the ﬂowpattern and the thermophysical properties of the ﬂuid.
4 Validity domain
To examine the validity domain of the semianalyticalapproach proposed here, we compare the exact analytical
Fig. 2
Schematic description of the semianalytical approach26 Heat Mass Transfer (2014) 50:23–30
1 3
solution with the approximate solution obtained by thisapproach for a second order linear differential equationgiven by Eq. (17).
Dd
2
T dx
2
i
¼
u
j
x
i
ð Þ
dT dx
i
;
ð
17
Þ
where
u
j
(
x
i
)isavariablevelocitycomponent,with
i
=
1to3and
j
=
1 to 3.The classical exact solution of this differential equationis obtained from the following relation:
T x
i
ð Þ ¼
mK
ð
x
i
Þ þ
p
;
ð
18
Þ
where
m
and
p
are constants, and
K x
i
ð Þ ¼
R
x
i
0
exp
R
x
0
u x
i
ð Þ
=
D
½
dx
dx
.The boundary conditions are given by:
x
i
¼
x
1
!
T
¼
T
1
x
i
¼
x
2
!
T
¼
T
2
:
(
ð
19
Þ
Hence, the exact solution takes the form:
T
ext
x
i
ð Þ ¼
T
1
T
2
K x
1
ð Þ
K x
2
ð Þ
K x
i
ð Þþ
T
2
K x
1
ð Þ
T
1
K x
2
ð Þ
K x
1
ð Þ
K x
2
ð Þ
:
ð
20
Þ
For the semianalytical approach and each virtual disk,we suppose that
u
(
x
) is locally constant and hence:
Dd
2
T dx
2
i
¼
u
0
dT dx
i
:
ð
21
Þ
where
u
0
is now a parameter which will be varied in theﬁnal solution.Hence, from separation of variables one can obtain:
T x
i
ð Þ ¼
b Du
0
exp
u
0
D x
i
þ
e
;
ð
22
Þ
with
b
and
e
as constants.By applying the same boundary conditions as before,and replacing
u
0
by
u
(
x
), the ﬁnal solution of the semianalytical approach is written as:
T
app
x
i
ð Þ ¼
T
1
T
2
ð Þ
exp
u x
i
ð Þ
D
x
i
exp
u x
i
ð Þ
D
x
1
h i
exp
u x
i
ð Þ
D
x
1
exp
u x
ð Þ
D
x
2
h i
þ
T
1
:
ð
23
Þ
In fact the semianalytical solution for temperature is afunction of a characteristic Pe´clet number deﬁned as
xu
(
x
)/
D
and a temperature gradient deﬁned as
D
T
/
L
, where
L
=
x
1

x
2
and
D
T
=
T
1

T
2
, and hence it takes intoaccount these different key parameters.Finally, the relative error between the exact solution andthe semianalytical solution can be given by the followingexpression:
e
¼
1
n
X
n j
¼
1
T
app
;
j
x
i
ð Þ
T
ext
;
j
x
i
ð Þ
T
ext
;
j
x
i
ð Þ
:
ð
24
Þ
Therefore, the validity of the semianalytical solutiondepends on the value of
e
(in %) considered as acceptableby the user.In the validation procedure presented below, theparameter
e
has been calculated in the
x
and
y
directionbased on the analytical expressions for
u
(
x
) and
v
(
x
) givenby Eq. (4). It has been shown that
e
values are lower than5 % in the two directions. In the following section, wepresent an experimental validation of the semianalyticalapproach as well as a parametric study based on thissolution given in Eq. (16).
5 Results and discussion
5.1 Experimental validationAnalytical results for Reynolds number 97 are comparedwith the experimental results of Mokrani et al. [5] in Fig. 3
(to the best of our knowledge the only available experimental proﬁle in the open literature). It should be noted thatthe experimental temperature distribution of Mokrani et al.[5] serves only as a qualitative reference, since it wasaimed not at verifying theoretical temperature proﬁles butrather at comparing the global heat transfer efﬁciency of helical and chaotic heat exchangers.From Fig. 3, two observations can be made: ﬁrst, at thecenter of the Dean rollcells the measured temperatures aremore uniform and higher than the analytical ones. This canbe attributed, to a certain extent, to the intrusive mixingcaused by the thermocouple probe (as shown on Fig. 4),which intensiﬁes the mixing and then the convective heat
Fig. 3
Comparison of experimental [5] and analytical temperatureproﬁles for
Re
=
97Heat Mass Transfer (2014) 50:23–30 27
1 3