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Developing incompressible viscous fluid flow and heat transfer in a curved annular pipe is studied numerically. The governing equations consisting of continuity, full Navier–Stokes, and energy equations are solved using a projection method based on

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International Journal of Thermal Sciences 48 (2009) 1542–1551
Contents lists available atScienceDirect
International Journal of Thermal Sciences
www.elsevier.com/locate/ijts
A numerical analysis of developing ﬂow and heat transfer in a curvedannular pipe
M.R.H. Nobari
∗
, B.R. Ahrabi
1
, G. Akbari
1
Department of Mechanical Engineering, Amirkabir University of Technology, 424 Hafez Ave., P.O. Box 15875-4413, Tehran, Iran
a r t i c l e i n f o a b s t r a c t
Article history:
Received 12 April 2008Received in revised form 9 December 2008Accepted 10 December 2008Available online 31 December 2008
Keywords:
Annular curved pipeIncompressible ﬂowFinite differenceProjection method
Developing incompressible viscous ﬂuid ﬂow and heat transfer in a curved annular pipe is studiednumerically. The governing equations consisting of continuity, full Navier–Stokes, and energy equationsare solved using a projection method based on the second order central difference discretization.Considering the outer wall to be adiabatic, two different thermal boundary conditions involving constanttemperature and constant heat ﬂux are applied at the inner wall to analyze the heat transfer rate in thetwo different cases. The effects of governing non-dimensional parameters involving the aspect ratio, thecurvature, Reynolds number, Dean number, and Prandtl number on the ﬂow and temperature ﬁeld bothin developing and fully developed regions of the curved annular pipe are studied in detail. Two majordifferent developing patterns of the ﬂow are determined based on the location of maximum axial velocityeither in the semi-inner or in the semi-outer region of the curved annular pipe. Also the numericalresults obtained indicate that the friction factor and the Nusselt number in a curved annular pipe areboth proportional to the square root of Dean number (
κ
1
/
2
). At
κ
1
/
2
8 the friction factor for bothcurved and straight annular pipes are the same, beyond that it increases in the circular curved pipe byincreasing Dean number and decreasing aspect ratio.
©
2008 Elsevier Masson SAS. All rights reserved.
1. Introduction
Flow and heat transfer in curved pipes are used in a very largenumber of cases such as piping systems, bio-ﬂuid mechanics espe-cially blood ﬂow in catheterized artery, engineering devices suchas heat exchangers, cooling systems of rotating electrical machin-ery, chemical mixing or drying machinery, chemical reactors, chro-matography columns, and other processing equipment. Because of the wide range of applications, ﬂow and heat transfer in this con-ﬁgurations are studied extensively during the last decades.Physical aspects of the ﬂuid ﬂow inside the curved pipes arevery much complicated due to the presence of curvature generat-ing centrifugal and pressure forces in the curvature direction. Incontrast to the centrifugal forces, the pressure forces decrease inthe curvature direction as the ﬂuid particles approach centre of curvature. Mutual effects of centrifugal, pressure, inertia and vis-cous forces provide a very complex ﬂow pattern which has notphysically fully understood. A relatively detailed qualitative physi-cal description of the ﬂow in a plain curved pipe has been carriedout by Yao and Berger [1]. However, shifting from a plain curvedpipe ﬂow to the annular one makes the ﬂow pattern more com-
*
Corresponding author.
E-mail address:
mrnobari@aut.ac.ir(M.R.H. Nobari).
1
Graduate student.
plex owing to the presence of an additional internal curved pipe.In this case the secondary boundary layers start developing onthe walls of both curved pipes from their outermost point of cur-vatures, where the pressure forces are more than the centrifugalforces. On the other hand, in the core region off the two pipe walls,the reverse ﬂuid motion, i.e. from the inner to the outer radii of curvature, occurs resulting from the larger centrifugal and smallerpressure forces. This secondary core ﬂow which starts from thesymmetrical plane at the inner radii of the curvature (
φ
=
π
)
de-velops similar to a jet ﬂow and interacts with opposite ﬂowing sec-ondary boundary layers forming two pairs of vortices, a weak pairclose to the inner pipe and a strong one close to the outer pipe.This phenomenon implies a physical point that in the secondaryﬂows the diffusion of viscous forces occurs more rapidly than themain axial ﬂow owing to the presence of small inertia forces (orderof magnitude of secondary inertia forces is about 10
−
1
of the axialone). Within the entrance length up to the fully developed region,the centre location of vortices displace under the effect of the mainaxial ﬂow development. At low Dean numbers (or low Reynoldsnumbers) and curvatures this displacement is small and the centreof large vortices along with the maximum axial velocity stay in theinner half curved pipe region (region 1 in Fig. 1), but the centre of small ones appear in the very beginning of the outer half curvedregion (region 2 in Fig. 1). Otherwise, as the larger the dean num-ber and the curvature, this displacement becomes larger with thechance of breaking vortices into small ones. In these cases max-
1290-0729/$ – see front matter
©
2008 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.ijthermalsci.2008.12.004
M.R.H. Nobari et al. / International Journal of Thermal Sciences 48 (2009) 1542–1551
1543
Nomenclature
c
p
speciﬁc heat at constant pressure
D
i
inner diameter of annular pipe
D
o
outer diameter of annular pipe
D
h
hydraulic diameter,
D
h
=
D
o
−
D
i
f
frictional resistance
FR
friction factor,
f
c
/
f
s
k
thermal conductivity
N
outward normal unit vector to the boundary
Nu
average Nusselt number
Nu
local Nusselt number
p
dimensionless pressure
Pr
Prandtl number,
Pr
=
ν
/
α
q
w
heat ﬂux at the inner wall of annular pipe
r
dimensionless radial coordinate
r
i
inner radius of annular pipe
r
o
outer radius of annular pipe
R
c
curvature radius
Re
Reynolds number,
Re
=
w
m
D
h
/
ν
t
dimensionless time
T
dimensionless temperature
T
m
bulk ﬂuid temperature
u
,
v
,
w
dimensionless velocity components in
r
,φ
and
θ
di-rection, respectively
V
velocity vector
w
m
mean axial velocity
Greek symbols
α
thermal diffusivity
δ
c
curvature ratio,
δ
c
=
r
o
/
R
c
δ
r
aspect ratio,
δ
r
=
r
i
/
r
o
θ
curvature coordinate
κ
Dean number,
κ
=
2
δ
1
/
2
c
(
w
m
D
h
/
ν
)
μ
viscosity of ﬂuid
ν
kinematic viscosity
ρ
density of ﬂuid
τ
stress
φ
angular coordinate
Γ
boundary
Subscriptsc
curved annular pipefd fully developed
i
inner wallin inlet of pipe
o
outer wall
s
straight annular pipe
Superscript
dimensional parameters
Fig. 1.
Toroidal coordinates and geometry of the curved annular pipe.
imum axial ﬂow velocity stays in region 2. Considering the mainﬂow in the entrance region, at low Reynolds numbers (low Deannumbers) the growth rate of the main boundary layers (
∼
1
/
√
Re
)are larger resulting in the faster full diffusion of viscosity effects inthe main ﬂow and shorter entrance length. However, this is reversein the higher Reynolds numbers at which the maximum axial ve-locity appears in the outer half curvature region (region 2). As themain ﬂow starts developing from the entrance of the curved an-nular pipes toward the fully developed region, due to shifting thepeaks of the main ﬂow from the inner curved region toward theouter, the boundary layer thicknesses become thicker on the wallswith the lower radii of the curvature and thinner on the walls withthe higher radii of curvature. Based on the physical descriptionsoutlined above, the presence of the secondary ﬂows in the curvedpipes delays the development of the main ﬂow by deforming theevolution of the axial velocity proﬁle and increases the resistantof the ﬂuid ﬂow. Consequently, comparing to the similar straightpipes the reduction of ﬂow rate and the increase of both frictionrate and entrance length can be easily expected.Many analytical, numerical and experimental works about theﬂuid ﬂow and heat transfer in curved pipes are developed. The ﬁrstmajor study on the ﬂow in the curved pipe was made by Dean[2,3] who considered a loosely curved pipe where the ﬂow de-pends on a single non-dimensional parameter, i.e. the Dean num-ber,
K
=
2
a
/
R
(
w
max
a
/
ν
)
2
, where
a
is the radius of pipe,
R
is theradius of curvature,
w
max
is the maximum axial velocity in the cor-responding straight pipe, and
ν
is the kinematic viscosity. Dean’swork is valid for
K
576. In later works on curved pipes, a vari-ety of Dean numbers have been used by different researchers. Forexample, McConalogue and Srivastava [4] proposed the parameter
D
=
(
Ga
2
/
μ
)(
2
a
3
/
ν
2
R
)
1
/
2
, where
G
is the constant pressure gra-dient along the pipe. This parameter relates to
K
as
D
=
4
K
1
/
2
. Bythis deﬁnition of Dean number, the upper limit of Dean’s workbecomes 96. They considered intermediate range of Dean num-bers (96
D
600
)
using Fourier series method to formulate theproblem and solve the resulting equations numerically. Collins andDennis [5], and Dennis [6] used ﬁnite difference method to solvethe ﬂow equations in the range of 96
D
5000. An investigationon developing laminar ﬂow in a curved pipe was made by Soh andBerger [7] using artiﬁcial compressibility technique. They reportedtheir results for 108
.
2
κ
680
.
3, where
κ
=
2
δ
1
/
2
(
W
m
a
/
ν
)
isanother deﬁnition of Dean number. They found that curvature ratio
1544
M.R.H. Nobari et al. / International Journal of Thermal Sciences 48 (2009) 1542–1551
has great effect on the intensity of secondary ﬂow and the separa-tion which occurs near the inner wall of curved pipe. Among othersimilar works on ﬂow in a stationary curved pipe, the works of Pedley [8], Dennis and Ng [9], Ito [10] and Kao [11] can be men-tioned.Nobari and Gharali [12] have investigated the effect of internalﬁns on the ﬂuid ﬂow and heat transfer through a rotating straightpipe and a stationary curved pipe. Ishigaki [13–15] examined ﬂowand heat transfer in a rotating curved pipe and investigated the ef-fect of Coriolis force in complicating the ﬂow structure. Heat trans-fer and ﬂuid ﬂow in a curved annular pipe has been studied in thefully developed region by Karahalios [16], Petrakis and Karahalios[17]. Also, the effect of catheterization on the ﬂow characteristicsin a curved artery was studied by Karahalios [18], Ebadian [19], Jayaraman and Tiwari [20] and Dash et al. [21].In the present work, direct numerical simulation of develop-ing incompressible viscous ﬂow and heat transfer in a curvedannular pipe is studied using a projection algorithm based onthe second order ﬁnite difference discretization. For doing so, athree-dimensional staggered grid in a toroidal coordinate systemis used to discretize the governing equations including continuity,full Navier–Stokes and energy equations. To study heat transfer,two different thermal boundary conditions consisting of constanttemperature and constant heat ﬂux at the inner wall are applied.At both of these conditions, the outer wall of annular pipe is as-sumed to be adiabatic. The effects of governing non-dimensionalparameters, such as Dean number,
κ
, Reynolds number,
Re
, cur-vature ratio,
δ
c
, and aspect ratio,
δ
r
, on the ﬂow characteristicsinvolving axial ﬂow, secondary ﬂow pattern, friction factor, tem-perature proﬁles and Nusselt number are investigated in details.It should be mentioned that developing ﬂow in a curved annularpipe with a detailed physical analysis of the fully developed re-gion is carried out for the ﬁrst time in this study to the best ourknowledge.
2. Governing equations
Here, developing incompressible viscous ﬂuid ﬂow and heattransfer in a stationary curved annular pipe with circular crosssection are studied. The best coordinate system compatible withthe geometry of the physical domain is the toroidal coordinatessystem as shown in Fig. 1. Consequently, the governing equa-tions describing the ﬂow and heat transfer consisting of continuity,Navier–Stokes and energy equations are written in the toroidalcoordinates system as non-dimensional form using the followingnon-dimensional parameters
r
=
r
D
h
,
u
=
u
w
m
,
v
=
v
w
m
w
=
w
w
m
,
p
=
p
ρ
w
2
m
,
t
=
t
D
h
/
w
m
Re
=
w
m
D
h
ν
,
Pr
=
να
,
κ
=
2
δ
1
/
2
c
(
W
m
D
h
/
ν
)
T
=
T
−
T
w
T
in
−
T
w
for Case A
,
T
=
T
−
T
in
q
w
D
h
/
k
for Case B (1)where,
r
is the dimensionless radial coordinate,
D
h
the hydraulicdiameter of annular pipe (
D
o
−
D
i
)
,
u
the dimensionless veloc-ity in the radial coordinate (
r
),
w
m
the mean axial velocity,
v
thedimensionless velocity in the angular coordinate (
θ)
,
w
the di-mensionless velocity in the curvature coordinate (
φ)
,
p
the dimen-sionless pressure,
ρ
the density,
t
the dimensionless time,
Re
theReynolds number,
υ
the kinematic viscosity,
Pr
the Prandtl num-ber,
α
the thermal diffusivity,
κ
the Dean number,
δ
c
the curvatureratio (
δ
c
=
r
o
/
R
c
)
,
T
the temperature,
T
in
the inlet temperature,
T
w
the wall temperature of the inner pipe,
k
the thermal conduc-tivity of the ﬂuid, and
q
w
the constant heat ﬂux at the inner pipewall.Therefore, the corresponding non-dimensional governing equa-tions in the toroidal coordinate system can be expressed ascontinuity
∂∂
r
(
r
ξ
u
)
+
∂∂θ (ξ
v
)
+
∂∂φ(δ
rw
)
=
0 (2)
r
momentum
∂
u
∂
t
+
1
r
ξ
∂∂
r
r
ξ
u
2
+
∂∂φ(ξ
uv
)
+
∂∂θ (δ
ruw
)
−
ξ
v
2
−
δ
r
cos
φ
w
2
= −
∂
p
∂
r
+
1
Re
1
r
ξ
∂∂
r
r
ξ ∂
u
∂
r
+
∂∂φ
ξ
r
∂
u
∂φ
+
∂∂θ
δ
2
r
ξ ∂
u
∂θ
−
1
r
2
2
∂
v
∂φ
+
u
+
δ
sin
φ
vr
ξ
+
δ
2
cos
φξ
2
v
sin
φ
−
u
cos
φ
−
2
∂
w
∂θ
(3)
φ
momentum
∂
v
∂
t
+
1
r
ξ
∂∂
r
(
r
ξ
uv
)
+
∂∂φ
ξ
v
2
+
∂∂θ (δ
rvw
)
−
ξ
uv
+
δ
r
sin
φ
w
2
= −
1
r
∂
p
∂φ
+
1
Re
1
r
ξ
∂∂
r
r
ξ ∂
v
∂
r
+
∂∂φ
ξ
r
∂
v
∂φ
+
∂∂θ
δ
2
r
ξ ∂
v
∂θ
+
1
r
2
2
∂
u
∂φ
−
u
−
δ
sin
φ
ur
ξ
−
δ
2
sin
φξ
2
v
sin
φ
−
u
cos
φ
−
2
∂
w
∂θ
(4)
θ
momentum
∂
w
∂
t
+
1
r
ξ
∂∂
r
(
r
ξ
uw
)
+
∂∂φ(ξ
vw
)
+
∂∂θ
δ
c
rw
2
−
δ
rw
(
u
cos
φ
−
v
sin
φ)
= −
δξ ∂
p
∂θ
+
1
Re
1
r
ξ
∂∂
r
r
ξ ∂
w
∂
r
+
∂∂φ
ξ
r
∂
w
∂φ
+
∂∂θ
δ
2
r
ξ ∂
w
∂θ
+
2
δ
2
ξ
2
∂
u
∂θ
cos
φ
−
∂
u
∂θ
sin
φ
−
w
2
(5)energy
∂
T
∂
t
+
u
∂
T
∂
r
+
vr
∂
T
∂φ
+
δξ
w
∂
T
∂θ
=
1
Pr
·
Re
1
r
ξ
∂∂
r
r
ξ ∂
T
∂
r
+
∂∂φ
ξ
r
∂
T
∂φ
+
δ
2
∂∂θ
r
ξ ∂
T
∂θ
(6)where,
ξ
=
1
+
δ
r
cos
φ
and
δ
=
D
h
/
R
c
.As it is clear from Eqs. (3) to (6), here, the transient equationsare used to reach the steady state solution because of the sim-plicity of the parabolic equations comparing with the elliptic ones.Furthermore, the ﬂow ﬁeld is symmetric relative to the horizontalmid plane, and it is enough to take into account either upper orlower semicircle region of the curved annular pipe in the numer-ical simulation. Therefore, the following four boundary conditionscan be applied.
At the inlet
:uniform axial ﬂow
w
(
r
,φ,
0
)
=
1
,
u
(
r
,φ,
0
)
=
0
,
v
(
r
,φ,
0
)
=
0 (7)
M.R.H. Nobari et al. / International Journal of Thermal Sciences 48 (2009) 1542–1551
1545
and uniform temperature
T
(
r
,φ,
0
)
=
1 for Case A
T
(
r
,φ,
0
)
=
0 for Case B(8)
At the outlet
:hydrodynamically fully developed
∂
u
∂θ
=
∂
v
∂θ
=
∂
w
∂θ
=
0 at
θ
=
θ
fd
(9)and thermally fully developed
∂
T
∂θ
=
T T
m
∂
T
m
∂θ
for Case A
∂
T
∂θ
=
dT
w
d
θ
=
4
δ
r
RePr
δ
c
(
1
−
δ
2
r
)
for Case B at
θ
=
θ
fd
(10)where,
θ
fd
is suﬃcient enough to consider the ﬂow fully developedboth hydrodynamically and thermally. Depending on the amountof Reynolds and Dean numbers, the values of
π
and 3
π
/
2 for
θ
fd
are used in the fully developed region.
At the walls
:no slip conditions for velocity
inner wall
:
u
(
r
i
,φ,θ)
=
v
(
r
i
,φ,θ)
=
w
(
r
i
,φ,θ)
=
0
outer wall
:
u
(
r
o
,φ,θ)
=
v
(
r
o
,φ,θ)
=
w
(
r
o
,φ,θ)
=
0(11)where,
r
i
=
r
i
D
h
=
δ
r
2
(
1
−
δ
r
),
r
o
=
r
o
D
h
=
12
(
1
−
δ
r
)
(12)constant temperature at the inner wall and adiabatic outer wall(Case A)
T
=
0 at
r
=
r
i
∂
T
∂
r
=
0 at
r
=
r
o
(13)or constant heat ﬂux at the inner wall and adiabatic outer wall(Case B)
∂
T
∂
r
= −
1 at
r
=
r
i
∂
T
∂
r
=
0 at
r
=
r
o
(14)
Plane of symmetry
:
∂
u
∂φ
=
∂
w
∂φ
=
∂
T
∂φ
=
0 and
v
=
0 at
φ
=
0
,
π
(15)
3. Numerical method
In this study the projection method which is introduced byChorin [22] for the ﬁrst time is employed to solve the transientNavier–Stokes equations using forward in time and central in spaceﬁnite difference discretization. Although the unsteady solutions arephysically accurate here, the concentration is focused on the steadystate solutions where the transient terms vanish. In the projectionalgorithm, the Navier–Stokes and continuity equations are writtenas:
V
n
+
1
−
V
n
Δ
t
+
A
V
n
+∇
p
n
+
1
=
1
Re
∇
2
V
n
(16)
∇ ·
V
n
+
1
=
0 (17)
Table 1
Grid independence test for three different grid sizes (
δ
r
=
0
.
2,
δ
c
=
1
/
7,
Re
=
200and
Pr
=
5) at Case A.Grid size
FR w
max
Nu
c
/
Nu
s
Axial velocity contours30
×
20
×
30 1
.
0517 1
.
6889 1
.
306650
×
30
×
40 1
.
0672 1
.
6961 1
.
324580
×
40
×
80 1
.
0685 1
.
6968 1
.
3261
Then, deﬁning the temporary velocity,
V
∗
, Eq. (20) can be split intoEqs. (22) and (23)
V
∗
−
V
n
Δ
t
+
A
V
n
−
1
Re
∇
2
V
n
=
0 (18)
V
n
+
1
−
V
∗
Δ
t
+∇
p
n
+
1
=
0 (19)Taking the divergence of Eq. (19) and using Eq. (17) leads to thePoisson equation for the pressure:
∇
2
p
n
+
1
=
1
Δ
t
∇ ·
V
∗
(20)The Neumann condition for the pressure at the boundary is ob-tained by projecting Eq. (19) normal to the boundaries
∂
p
∂
n
n
+
1
Γ
= −
1
Δ
t
V
n
+
1
Γ
−
V
∗
Γ
·
n
(21)where, the subscript
Γ
indicates the boundary. It can be shownthat the pressure ﬁeld is independent of
V
∗
Γ
, therefore, by choosing
V
∗
Γ
=
V
n
+
1
Γ
, zero pressure gradient is obtained on the boundaries.Due to explicit discretization, the following stability conditionsmust be satisﬁed
Δ
t Re
Min
{
(
Rd
θ)
2
,(
rd
φ)
2
,(
dr
)
2
}
1
/
6Max
u
2
+
v
2
+
w
2
Re
Δ
t
2 (22)To obtain a physical pressure ﬁeld, a staggered uniform grid inwhich no pressure nodes exist on the boundaries is used.For the grid independence test of the numerical code imple-mented, three different grid sizes shown in Table 1 are consideredto compare the numerical results obtained in fully developed re-gion in the case of constant heat ﬂux at the inner wall (Case A).The comparison of the friction factors, Nusselt numbers, meantemperatures and the axial velocity contours clearly indicates theconservative property of the developed numerical code and itsgrid independent results, where the maximum error is about 1.5%.Finally, it should be declared that the maximum error of the resid-uals to reach to the steady state solution is assumed to be of theorder of 1
×
10
−
6
.
4. Results and discussion
To verify the accuracy of the code implemented here, a highcurvature case (
δ
c
=
1
/
100
)
which is too close to the straight con-centric pipe is simulated, and the results obtained are comparedwith the analytical solution in the fully developed region in Fig. 2
1546
M.R.H. Nobari et al. / International Journal of Thermal Sciences 48 (2009) 1542–1551
Fig. 2.
Comparison between axial velocity in the plane of symmetry of a curvedannular pipe with
δ
c
=
100 and analytical solution of similar straight one.
indicating a very good agreement. Now, the numerical results ob-tained for the ﬂow and heat transfer in the curved annular pipe atdifferent curvatures and aspect ratios are described in detail at thefollowing. First the ﬂow ﬁeld is analyzed and then the correspond-ing results obtained for heat transfer will be taken into account.
4.1. Axial ﬂow development
Fig. 3 shows the axial ﬂow development on the symmetry planefor a curved annular pipe with
δ
r
=
0
.
2 and
δ
c
=
1
/
7 at differentReynolds numbers. Due to curvature and no slip conditions at theinner and outer walls, the development of the axial velocity is af-fected by the interaction of the main and secondary ﬂow boundarylayers. The numerical results obtained here indicate that the ax-ial ﬂow pattern is different for low Reynolds numbers from high
(a)
Re
=
50
(
κ
=
37
.
8
)
(b)
Re
=
105
(
κ
=
79
.
4
)
(c)
Re
=
305
(
κ
=
230
.
6
)
(d)
Re
=
600
(
κ
=
453
.
6
)
Fig. 3.
Development history of axial velocity on the plane of symmetry for
δ
r
=
0
.
2,
δ
c
=
1
/
7.

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