Transient Three-dimensional Numerical Analysis of Forced Convection Flow and Heat Transfer in a Curved Pipe

IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE)
e-ISSN: 2278-1684,p-ISSN: 2320-334X, Volume 9, Issue 5 (Nov. - Dec. 2013), PP 47-57
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Transient Three-dimensional Numerical Analysis of Forced
Convection Flow and Heat Transfer in a Curved Pipe
Lect. Dr. Wajeeh Kamal Hasan
Refrigeration and Air Conditioning Engineering Department, Al-Rafidain University Collage, Iraq
Abstract: A three-dimensional transient numerical study of a constant property Newtonian fluid in curved pipe
under laminar flow conditions is presented for a uniform wall temperature boundary condition. Numerical
solutions were obtained using the control volume method described by Patankar for the range
of . The working fluid was water. The transient flow pattern and the temperature distribution
on the tube section were derived for different values of the Reynolds number. Graphical results for velocity and
temperature are presented and analyzed. Results have shown that the maximum velocity in center of velocity
profile increase with increasing of Reynolds number. In curved pipes, time averaged results exhibited Dean
circulation and a strong velocity and temperature stratification in the radial direction. Flow and heat transfer
were strongly asymmetric, with higher values near the outer pipe bend.
Keywords: curved pipe, forced convection, internal flow, laminar flow, transient heat transfer
Nomenclature
cp Specific heat capacity (J. kg-1
. k-1
)
De Dean number
k thermal conductivity (W. m−1
. k−1
)
p pressure (N . m-2
)
r radial coordinate
R curvature radius of pipe (m)
Re Reynolds number
T temperature (K)
t time (sec.)
v velocity (m .s-1
)
z axial coordinate
Greek letters
θ angular coordinate
ρ density (kg.m-3
)
Subscripts
i internal
r radial direction
z axial direction
θ tangential direction
I. Introduction
Interest in the problem of heat transfer in fully developed pipe flows dates back over a hundred years.
The forced convection problems in curved pipes are frequently encountered in various heat exchanges, cooling
or heating systems, chemical reactors, heat engines and other apparatus, equipment and devices. It is well
known that the mode of fluid in a curved pipe is characterized by a secondary flow field, which is superimposed
upon the axial-velocity flow field. The nature of the viscous flow in a curved pipe, as compared with simple
straight-pipe parabolic flow, causes relatively high average heat- and mass-transfer rates per unit axial pressure
drop, especially for fluid with a high Prandtl number and high Schmidt number. Of fundamental interest to the
development of a complete understanding of viscous-flow phenomena in toroidal tubes is the nature of the
velocity and temperature distributions in the fully-developed flow region.
As regards Flow field and pressure drop, in curved pipes a characteristic secondary motion develops in
the cross section due to the imbalance between pressure and inertial (centrifugal) forces [1]. The fluid moves
towards the outer bend side near the equatorial midplane, returns towards the inner side along two near-wall
boundary layers, and then forms two symmetric secondary cells (Dean vortices) having a characteristic velocity
scale .The most popular friction correlations for curved pipes are those by Ito [4]:

Transient Three-dimensional Numerical Analysis of Forced Convection Flow and Heat Transfer in a
(laminar flow) (1)
(turbulent flow) (2)
in which f is the Darcy friction coefficient (four times the Fanning friction coefficient) and Re is the bulk
Reynolds number. Although its correlations date back to the 1950s, they have been confirmed to a very high
degree of approximation by recent computational [2] and experimental [5] studies. Several authors attempted to
characterize the transition to turbulence in curved pipes on the basis of friction coefficient measurements [5],
flow visualization [6] or local flow/temperature measurements [7]. The main result of these studies is that
curvature delays transition to turbulence with respect to straight pipes. For example, Srinivasan et al. [8]
proposed the following correlation for the critical Reynolds number:
(3)
This exhibits the correct asymptotic behavior for d = 0 and predicts, for example, Recr = 10068 for δ = 0.1 and
Recr = 15902 for δ = 0.3, values considerably higher than that (Recr = 2100) applicable to straight pipes. Among
computational studies, Di Piazza and Ciofalo [2] presented detailed results on the breakdown of steady laminar
flow and the transition to turbulence in closed toroidal pipes having curvatures δ of 0.1 and 0.3. They found
periodic and quasi-periodic flow regimes, characterized by traveling waves, in a relatively narrow interval of
Reynolds numbers and a rich scenario of transitions and bifurcations, including super- and sub-critical Hopf
bifurcations and double Hopf bifurcations. Chaotic flow was obtained only for Re = 8000. Friedrich and co-
workers [9, 10] presented numerical simulation results for toroidal and helical pipes at Re = 5600 (Reτ = 230)
and δ = 0.1. The authors performed a statistical processing of the solution (e.g. by computing Reynolds
stresses), but the case they studied was a time-dependent laminar flow rather than a truly turbulent flow [2].
Moreover, only a tract of pipe 7.5 diameters long was modeled, using periodic boundary conditions at the ends;
such conditions are inadequate for a truly predictive simulation of traveling waves since they force the wave
length to be an exact sub-multiple of the arbitrary domain length. Similar remarks hold for the work of Webster
and Humphrey [11], who presented flow visualizations and numerical simulations for curved pipes. The length
of the computational domain was chosen to be equal to the wavelength that was experimentally measured. When
the experiments evidenced traveling wave instability, the corresponding computational results showed
oscillating velocities with maximum rms values in the Dean Vortex region. A quantitative comparison with
experiments was not possible since these latter were limited to flow visualization. As regards heat transfer,
many experimental studies were performed in the 1960s and the 1970s on the average heat transfer rate in
curved and helical pipes, e.g. by Seban and McLaughlin [12] and Mori and Nakayama [13]; only some of these
investigations explored the influence of the Prandtl number on heat transfer, and very few investigated the local
heat transfer rate distribution. The results of most of such studies will not be reported here and will not be used
for validation purposes because they are either insufficiently confirmed, or limited to a range of parameters too
far from that of interest here; the reader is referred to the recent reviews presented by Naphon and Wongwises
[14] and by Vashist et al. [15]. Rogers and Mayhew [16] proposed for the Nusselt number in helical pipes the
following power-law correlation:
(4)
This formula basically is a curvature correction to the Dittus–Bolter correlation; since it does not contain the
pipe torsion, it does not distinguish between planar and helical curved pipes. Unfortunately, Eq. (4) does not
exhibit the correct asymptotic behavior for small δ, predicting Nu = 0 for straight pipes.
Xin and Ebadian [17] presented an experimental study on heat transfer in helical pipes; the authors
explored two values of curvature, i.e., δ = 0.027 and 0.08, Re from 5 x 103
to 1.1 x 105
, and three different
fluids, i.e., air (Pr = 0.7), water (Pr = 5), and ethylene glycol (Pr = 175), covering a broad range of Prandtl
numbers. Results for air and water (0.7 < Pr < 5) were approximated by the following correlation:
(5)
with an RMS deviation of 18% for 0.7 < Pr < 5.5 x 103
< Re < 105
, 0.0267 < δ < 0.0884. Eq. (5), like Eq. (4),
does not distinguish between planar and helical pipes; unlike Eq. (4), it offers a reasonable asymptotic behavior
for straight pipes, but is strictly applicable only to the narrow curvature range of the experiments. As regards
computational studies on heat transfer in curved pipes, they are relatively numerous, but most of them regard
laminar flow or geometrical configurations much different from those discussed in the present paper. Some
authors concentrated on the simulation of the shell-side heat transfer in heat exchangers with helically coiled
tubes [18,19] or considered helical coils just as components of the larger system represented by the heat
exchanger or steam generator [20,21]. Litster et al. [22] studied laminar mass transfer in helical pipes using the
CFX4.3 finite volume code. Bolinder and Sundèn [23] computed laminar heat transfer in helical coils of square
cross section. Centrifugal and gravitational buoyancy in laminar flow was the focus of several studies [24–27].
Yang et al. [28] studied laminar flows in serpentine pipes, in which the pipe axis lies in a plane but left- and

right-turn bends alternate periodically. Other studies are dedicated to the possible mixing effects of chaotic
advection even in the presence of steady laminar flow, especially when coils with three-dimensionally
alternating axes are used [29,30].
The purpose of this work is to investigate numerically the transient three-dimensional laminar flow heat
transfer in a 135°
curved circular pipe under the thermal boundary conditions of constant wall temperature. At
the inlet of the curved pipe, a parabolic velocity profile is imposed. Water is used as the working fluid. The
effects of governing non-dimensional parameters, such as Reynolds number, Re, on the flow characteristics
involving axial flow, secondary flow pattern and temperature profiles are investigated in details.
II. Description of the Physical Problem
The geometry and characteristics of the curved pipe used in the present study are shown in Fig.1. The
inner diameter of the pipe is 0.01 m and the outer 0.012 m, respectively. It is bending to a 135°
angle with a
radius of 0.02 m. beyond the bend, 0.03 m of the direct part is also accounted for.
III. Mathematical Model
The transient laminar flow (20 < De < 1200) in the curved pipe is considered. All the thermo- physical
properties are assumed to be constant. Water is considered to be an incompressible Newtonian fluid. Viscous
dissipation and body force are considered to be negligibly small and dropped from the energy and momentum
equations respectively. The extra body force due to curvature is considered in the momentum equation.
Coordinate system is shown in Figure 1. In this system, the radius of curvature of the pipe axis is shown by R.
The distance along the pipe axis is shown by z, the distance in the radial direction measured from the pipe axis
by r. The velocity components in the directions of increasing r, θ, z are vr, vθ, vz, respectively. On the basis of
the simplifying assumptions, the transient three-dimensional governing equations are summarized as follows:
The equation of continuity:
The equations of Motion (momentum equation):
(7-a)
(7-b)
(7-c)
Where:
The energy equation:
(9)
3.1 Boundary and Initial Conditions:
The governing equations of the fluid flow are non-linear and coupled partial differential equations,
subjected to the following boundary conditions. At the inlet of the curved pipe, a parabolic velocity profile is
imposed. This eliminates the hydrodynamic entrance length required to obtain a fully developed velocity profile
before heating begins. The temperature of the water is set equal to 350 K at the inlet of the computational
domain. Standard no slip boundary condition was used on the walls of the curved pipe. A constant wall
temperature boundary condition was imposed on the external walls of curved pipe. The walls of the curved pipe
are kept at 300 K. Outflow boundary conditions were enforced for the outlet section. This boundary condition
implies zero normal gradients for all flow variables except pressure. At the initial state, the water velocity, vr =
vθ = vz = 0, everywhere, water temperature, T= 0, instantly, they are changed from the initial state to the inlet
conditions.
R
r
q
z
fig. (1): Geometry and coordinate system

3.2 Numerical Procedure
A mathematical model, i.e. differential equations, with initial and boundary conditions, has been
discretised using the control volume method described by Patankar [31]. The velocities and pressure of heat
transfer fluid are solved by using the SIMPLER scheme [31]. The resulting discretisation equations have been
solved simultaneously using Biconjugate Gradient Stabilized iterative procedure. Due to nonlinearity of the
problem, iterations are needed during each time step. The first order time derivatives are discretizated using the
second order Backward Differences Formulae time-stepping method. A convergence criterion is set at 1.0e-10
for all variables of the system. In order to improve convergence, under relaxation factors were used both for the
velocity components and temperature. Typical values used from 0.2 - 1.0.
IV. Results and Discussion
The numerical computations were carried out for the curved pipe. The forced convection heat transfer
in curved pipe was investigated under laminar flow conditions. The Reynolds number Re was varied in the
range of 100 to 200. The parametric study conducted produced a large number of results. In order to present
some characteristic velocity and isotherm contours, some samples of results are chosen as shown below. Figs.
(3a, 5a, and 7a) show the typical velocity magnitude contours on a section along (r-z) plane while Figs. (3b, 5b,
and 7b) show the contours of velocity at the outlet plane of the pipe for different Reynolds numbers. These
velocity contours provide information on the effect of Reynolds numbers on the flow field. The typical
temperature contours predicted by the present numerical computations for flow in the pipe on a section along (r-
z) plane are shown in Figs. (2a, 4a, and 6a) and Figs. (2b, 4b, and 6b) show these contours at the outlet plane of
the pipe for different Reynolds numbers. An inspection of these figures reveals the effect of Reynolds numbers
on the heat transfer. A close study of these figures indicates that the centrifugal body forces induced by the wall
curvature have a strong influence on the heat transfer. At low Reynolds numbers the centrifugal body forces are
weaker and the convective currents have strong influence on the flow. The velocity profile presents the
characteristic parabolic shape. At higher Reynolds numbers the centrifugal body forces are now much stronger
and the secondary flow motion begins to show its effect on flow and heat transfer. The distortion of the velocity
profile is not negligible. As a consequence, also the temperature profile is distorted showing an increase of the
gradients at the wall. There are stronger temperature gradients on the convex surface of the pipe. In other
words, the convex surface of the pipe influences heat transfer more than the concave surface of the pipe. Figs.
(3b, 5b and 7b) reveal that the flow development is symmetric about the horizontal pipe centerline.
At the low Reynolds numbers depending on the curvature, the maximum axial velocity stays on the
semi-inner portion of the curved pipe where the viscous forces dominate to the centrifugal forces generated due
to the curvature of the curved pipe. When the centrifugal forces dominate to viscous forces, typically at higher
Reynolds numbers, the maximum axial velocity covers the semi-outer portion of the curved. As the Reynolds
number increases, the Dean vortices generated due to centrifugal forces at the outer and inner curved walls of
the pipe become stronger and depending on the Reynolds number they break into two or more small vortices.
Transient temperature and velocity distributions obtained by numerical calculation are also shown in these
figures for different time periods. Contours of temperatures for different time periods are shown in Figs. (2, 4
and 6). These figures indicate that heat transfer during the initial transient period is primarily governed by the
radial conduction mechanism, and heat transfer by axial convection and conduction effects increases with time.
As the time increase, temperature fluctuations are small, interest a thin near-wall region, and are azimuthally
non-uniform, being more intense in the outer bend region.
the velocity's contours for different time periods are also shown in Figs. (3, 5 and 7).with a parabolic
inlet velocity, the fluid particles close to the center of curvature have larger centrifugal forces than the others
near the inlet, and they tend to push the fluid far away from the center of curvature and the secondary velocities
are much less intense and tend to concentrate near the outer bend region as the time increase. also, from these
figures, we can observes that ,with increasing time, one pair of vortices appeared to grow up due to the high
radial pressure gradient and substantial tendency of the main flow toward outer wall which lead the flow to be
changed.

(a) (b)
t= 2 sec t= 2 sec
t= 6 sec t= 10 sec
t= 16 sect= 14 sec
fig.(2): contours of temperature of the pipe's fluid for different time periods, Re=100, on a
section along: (a) r-z plane , (b) r-θ plane
(a) (b)

t= 2 sec t= 2 sec
t= 6 sec t= 6 sec
t= 12 sec t= 14 sec
fig.(3): velocity contours of the pipe's fluid for different time periods, Re=100, on a section
along: (a) r-z plane , (b) r-θ plane
(a)
(b)
t= 2 sec t= 2 sec

t= 10 sec t= 10 sec
t= 16 sec t=16 sec
(a) (b)
t= 2 sec t= 2 sec
t= 6 sec t= 6 sec

t= 12 sec t= 14 sec
(a)
(b)
t= 2 sec t= 2 sec
t= 10 sec t= 10 sec
t= 16 sec t= 16 sec

(a) (b)

t= 2 sect= 2 sec
t= 6 sec t= 6 sec
t= 8 sec t= 14 sec
V. Conclusions
A numerical model was developed to predict the transient flow and heat transfer behavior of water
throughout a curved pipe. Laminar forced convection heat transfer with constant wall temperature has been
investigated. Simultaneous effects of the centrifugal forces have been shown. The effects of the Reynolds
number on velocity and temperature are presented. The results of numerical analysis show that when the inlet
temperature is suddenly changed from the initial temperature, conduction in the radial direction is the governing
heat transfer mechanism at the initial transient period, but as time passes, axial convection and conduction
effects are relatively increased and become an important heat transfer after a certain time. In curved pipes,
results showed the presence of secondary (Dean) circulation and of a strong stratification of stream wise
velocity and temperature along the radius of curvature. At the low Reynolds numbers, the maximum axial
velocity stays on the semi-inner portion of the curved pipe. When the centrifugal forces dominate to viscous
forces, typically at higher Reynolds numbers, the maximum axial velocity covers the semi-outer portion of the
curved and the Dean vortices generated due to centrifugal forces at the outer and inner curved walls of the pipe
become stronger and they break into two or more small vortices.
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Transient Three-dimensional Numerical Analysis of Forced Convection Flow and Heat Transfer in a Curved Pipe

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