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Chemical Engineering and Processing 47 (2008) 893905
Numerical study of turbulent forced convectionin coiled flow inverter
Monisha Mridha, K.D.P. Nigam
Department of Chemical Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi 110016, India
Received 21 July 2006; received in revised form 8 February 2007; accepted 9 February 2007
Available online 3 March 2007
Abstract
A numerical study is done to investigate turbulent forced convection in a new device of coiled flow inverter. The proposed device works onthe technique based on flow inversion by changing the direction of centrifugal force in helically coiled tubes thus enabling rotation of the plane
of vortex. The objective of the present study is to characterize the flow development and temperature fields in coiled flow inverter (CFI) under
turbulent flow for the range of 10,000
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Table 1
Published studies for heat transfer for turbulent flow in curved tubes
Author Method Range of parameters Remarks
1/ NRe NPr
Jeschke [8] Experimental 6.1, 18.2 150,000 0.7 (NNu)(NPr)0.4 = 0.045(1 + (3.54/))(NRe)0.76. The work
was for a limited range of parameters
White [9] Experimental 15.15, 50, 2050 15,000100,000 7 fc = 0.08N1/4Re + 0.012/
. Heat transfer coefficient may be
predicted from fluid friction data
Kirpikov [10] Experimental 10, 13,18 10,00045,000 7 (NNu)(NPr)0.4 = 0.0456(NRe)0.85(1/)0.21. Heat transfer
coefficient was obtained using the wall to bulk temperature
difference
Ito [21] Experimental 16.4, 40,100, 250,
648
2,000400,000 7 Proposed an empirical equation for the critical Reynolds
number, NRe cr = 2 104()0.32
Seban and
McLaughlin [11]
Experimental 17, 104 6,00065,000 2.9657 (NNu)(NPr)0.4 = 0.023(NRe)0.85(1/)0.1. For turbulent flow,
the results for heat transfer coefficient were simplified and
average heat transfer coefficients for the periphery was
predicted more accurately using friction factors for curved tubes
Rogers and Mayhew
[12]
Experimental 10.8, 13.3, 20.12 3,00050,000 7 (NNu) = 0.021(NRe)0.85(NPr )0.4()0.1. Non-isothermal frictionfactors and heat transfer coefficients were estimated and were
recommended for design purposes
Mori and Nakayama
[13,14]
Theoretical
and
experimental
18.7, 40 10,000200,000 1 For NPr 1, NNu = (NPr/26.2(N2/3Pr 0.074))N
4/5Re ()
1/10 [1 + (0.098/{NRe()2}1/5)]; for NPr >1, NNuN
0.4Pr =
(1/41.0)N5/6Re ()
1/12 [1 + (0.061/{NRe()2.5}1/6)]. In thefirst order approximation, heat transfer in a curved pipe doesnt
differ for uniform wall temperature or uniform heat flux, in both
laminar and turbulent regions
Schmidt [15] Experimental The empirical formula presented is as following: NNufd /NNus
=1.0 + 3.6(1 )0.8, where NNus = 0.023N0.8Re N0.4Pr
Shchukin [16] Experimental 6.2104 NRec
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Table 1 ( Continued)
Author Method Range of parameters Remarks
1/ NRe NPr
Zheng et al. [17] Numerical 20 10,000100,000 0.7 An interaction phenomenon between turbulent forced
convection and thermal radiation of an absorbing-emitting gas
in a curved pipe at different temperature ratio, optical thickness,
and wall emissivity was studied. There was no influence ofthermal radiation, optical thickness, wall emissivity, and
temperature ratio on velocity fields but slightly affected the
temperature fields when only radiation-participating medium
was considered. The Nusselt number was affected by the wall
emissivity
Cioncolini and Santini
[18]
Experimental 6.9369 2,90016,000 7 Coil curvature was found effective in smoothing the emergence
of turbulence. Criterion for predicting turbulence emergence in
coiled pipes was proposed for different curvature ratios
Table 2
Published studies for enhanced heat transfer by chaotic advection
Author Method Range of parameters Remarks
D/d Re Pr
Acharya et al. [24] Numerical and experimental 5 3,00010,000 7 hi = 512.04N0.138Re . There was an enhancement of68% of heat transfer coefficient in helical coil with
alternating axis as compared to conventional helical
coil with constant axis. The pressure drop for the
chaotic configuration was 1.52.5% more than the
regular helical coil
Mokrani et al. [25] Experimental 5.5 60200 130 The effect of chaotic advection in the chaotic heat
exchanger on temperature uniformity and overallefficiency was studied. Flatter temperature
distribution was found in the chaotic coil as
compared to the regular helical coil. The relative
enhancement of the chaotic heat exchanger was
1328%
Chagny et al. [26] Experimental 11 3030,000 6.5820 Comparison for heat transfer between chaotic type
heat exchanger and a helical coil type heat exchanger
was made. At low Reynolds numbers, heating was
more homogeneous and heat transfer was intensified
in the chaotic advection regime without any increase
in energy expenditure. There was no influence on
heat transfer for Prandtl numbers higher than 225
Acharya et al. [27] Numerical 5, 10 501,200 0.110 NNu = 0.7()0.18
N0.5Re N
0.375Pr for Pr 1, NNu =
0.7()0.18N0.5Re N0.3Pr for Pr 1. The modified coil
geometry had 720% more heat transfer with
respect to regular coil with little change in pressure
drop. There was little enhancement with Reynolds
number for Pe 60
Lemenand and Peerhossaini [28] Numerical 11 100300 30100 NNu = 1.045N0.303Re N0.287Pr N0.033bends . A simplifiedthermal model was implemented to simulate heat
transfer in a helically coiled and chaotic
configuration
Kumar and Nigam [29] Numerical 10 251,200 0.74150 The bent coil configuration displays a 2030% heat
transfer enhancement as compared to the straight
helical coil with 56% increase in pressure drop
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results reported by Saxena and Nigam [30] shows that the devise
was behaving like a plug flow reactor.
Recently, Kumar and Nigam [29] numerically studied the
hydrodynamics and heat transfer in helical coiled tubes and
coiled flow inverter with curvature ratio, = 0.1 for a working
range of 25NRe 1200. They reported that the coiled flowinverter displays a 2030% heat transfer enhancement as com-
pared to the straight helical coils under laminar flow conditions
with the small increase in pressure drop.
In the present study, an attempt has been made to predict
the hydrodynamics and heat transfer in the coiled flow inverter
for turbulent flow. The flow development and temperature fields
in coiled flow inverter with = 0.1 and pitch of 0 and 0.02m
have been investigated for the range of Reynolds number from
10,000 to 30,000. Simulations were done with fluids (air, water,
kerosene and ethylene glycol) of Prandtl number ranging from
0.7 to 150.
2. Mathematical formulation
2.1. Governing equations
The geometry and system of coordinates considered are
shown in Fig. 1. The circular pipe studied, which has a diameter
of dt, is coiled at radius of Rc (=dc/2). The distance between
the two turns (the pitch) is reported by H. The bends introduced
in between the helical coils are of 90 and each helical tubehas same length before and after the bend. At the inlet ( = 0),fluid enters at a temperature T0 with a velocity ofu0. The wall
of the pipe is heated under constant temperature, Tw. The flow
was considered to be steady, and constant thermal properties
were assumed, except for the density of air. The values of T0,Tw, and physical and thermal properties of different fluids have
been reported in Table 3. The ideal gas law assumption was used
to the flowing fluid air.
TheRNG k modelproposedby Yakhot and Orszag [31] was
used to model the turbulent flow and heat transfer in the helical
Fig. 1. (a) System of coordinates, (b) geometry of coiled flow inverter and (c)
unstructured grid on one cross-section of the coiled flow inverter.
Table 3
Properties used in the numerical simulation T0 =300K, Tw = 350KProperties Air Water Kerosene Ethylene glycol
(kg/m3) Ideal gas law 998.2 780 1111.4
Cp (J/kg K) 1006.43 4182 2090 2415
(W/m K) 2.4E2 6E1 1.49E1 2.52E1 (kg/m s) 1.7894E
5 1.003E
3 2.4E
3 1.57E
2
Molecular weight 2 8.96 18.0152 157.30 62.0482
coil andcoiled flow inverter because the RNGmodel included an
additional term in its equation that significantly improved the
accuracy forrapidly strained flows,such as those in curved pipes.
The effect of swirl on turbulence is included in the RNG model,
enhancing accuracy for swirling flows. Researchers [1,17,32]
have used k model to simulate the turbulent flow and heat
transfer in curved tube for the same range of flow rate as done in
the present work. The maximum difference between the present
study and the past numerical and experimental data [14,15] was
less than 5% for the same parameter range. The time-averaged,
fully elliptic three-dimensional differential governing equations
can be written in tensor form in the Cartesian system as follows:
state : p = RT (for air) (1)
mass :ui
xi= 0 (2)
momentum :(uiuj)
xj
= xj
eff
ui
xj+ uj
xi
2
3eff
uk
xk
p
xi(3)
energy :(uiCpT)
xi
= xi
T
eff
T
xi
+ duidxj
eff
ui
xj+ uj
xi
2
3eff
uk
xkij
(4)
turbulent kinetic energy :(uik)
xi
= xi
keff k
xi
+ tS2 + Gb (5)
dissipation rate of turbulent kinetic energy :(ui)
xi
= xi
eff
xi
+ C1
ktS
2 C22
k R (6)
The effective viscosity, eff can be defined as
eff= mol
1 +
C
mol
k
2(7)
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where mol is the molecular viscosity. The coefficients T, kand in Eqs. (4)(6) are the inverse effective Prandtl numbers
for T, k, and , respectively.
The inverse effective Prandtl numbers, T, kand arecom-
puted using the following formula derived analytically by the
RNG theory:
1.39290 1.39290.6321
+ 2.39290 + 2.39290.3679 = moleff (8)
where 0 is equal to 1/Pr, 1.0, and 1.0, for the computation of
T, k, , respectively.
When a non-zero gravity field and temperature gradient are
present simultaneously, the k model account for the genera-
tion of k (kinetic energy) due to buoyancy [Gb in Eq. (5)] and
the corresponding contribution to the production of (energy
dissipation) in Eq. (6). In FLUENT, the effects of buoyancy are
always included despite the fact that the effect of buoyancy is
not so significant at very high Reynolds number. The generation
of turbulence due to buoyancy is given by
Gb = git
Prt
T
xi(9)
where Prt is the turbulent Prandtl number for energy and gi is
the component of the gravitational vector in the ith direction. In
the caseof the RNG k model, Prt = 1/T, and , the coefficient
of thermal expansion, is defined as =1/(p/T)p.In the Eq. (6), R is given by
R = C3(1 /0)
1 + 32
k(10)
where = S. k/, 0 4.38, = 0.012. The model constants C,C1, and C2 are equal to 0.085, 1.42 and 1.68, respectively [17].The term S in Eqs. (5) and (6) is the modulus of the mean
rate-of-strain tensor, defined as S2SijSij , whereSij =
1
2
ui
xj+ uj
xi
(11)
The two-layerbased, non-equilibriumwallfunction was usedfor
the near-wall treatment of flow in the given geometry. The non-
equilibrium wall functions are recommended for use in complex
flows because of thecapability to partly account for the effects of
pressure gradients and departure from equilibrium. The numer-
ical results for turbulent flow tend to be more susceptible togrid dependency than those for laminar flow due to the strong
interaction of the mean flow and turbulence. The distance from
the wall at the wall-adjacent cells must be determined by con-
sidering the range over which the log-law is valid. The size of
wall adjacent cells can be estimated from yp( y+p /u), whereu (w/)0.5 = u(cf/2)0.5. In the present study, the y+p wastaken in the range of 3060.
2.2. Boundary conditions
No-slip boundary condition, ui = 0, andconstant temperature,
Tw, were imposed at the wall. At the inlet, uniform profiles for
all the dependent variables were employed:
u = u0, T = T0, k = k0, = 0 (12)The turbulent kinetic energy at the inlet, k0, and the dissipation
rate of turbulent kinetic energy at the inlet, 0, are estimated by
k0 =3
2 (u0I)2, 0 = C3/4
k03/2
L (13)
The turbulence intensity level, I, is defined as u/u 100%,where u is root-mean-square turbulent velocity fluctuation.
At the outlet, the diffusion fluxes for all variables in exit
direction are set to be zero:
n(ui,T ,k,) = 0 (14)
where n is used to represent the normal coordinate direction
perpendicular to the outlet plane.
2.3. Parameter definitions
The following non-dimensional parameters and variables
were used in order to characterize the heat transfer in chaotic
configuration:
NRe =u0dt
, NDe = NRe
, = dt
dc,
= Tb TwTin Tw
, Tb =
1
uA
A0
u TDA
,
f =w
(1/2)u20, fm =
1
2 2
0f d,
NNu, =qwdh
(Tw Tb), NNu,m =
1
A
A0
NNu, d (15)
where is the curvature ratio, Tb the bulk temperature, f and
NNu,, local friction factor and Nusselt number along the circum-
ference of the pipe,respectively,fm andNNu,m, the circumference
average friction factor and Nusselt number and u0 denotes the
velocity at the inlet of the tube.
3. Numerical computation
3.1. Numerical method
The governing equations for mass, momentum and heat
transfer in the helical pipe were solved in the master Carte-
sian coordinate system with a control-volume finite difference
method (CVFDM) similar to that introduced by Patankar [33].
Fluent 6.2 [36] program was used as a numerical solver for the
present three-dimensional simulation. An unstructured (block-
structured) non-uniform grid system was used to discretize the
governing equations. Fig. 1(c) illustrates the grid topology used
on one cross-section. The convection term in the governing
equations was modeled with the bounded second-order upwind
scheme and the diffusion term was computed using the mul-
tilinear interpolating polynomials nodes Ni(X, Y, Z). The final
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Table 4
Grid independent test (NDe = 6325, NPr= 0.7, and = 0.1)
Total grids (cross-
sectional axial)NNu,max NNu,m Mori and
Nakayama [14]
1110 440 121.91 82.701280 440 125.75 82.92 84.761710
440 127.15 83.01
1860 440 127.15 83.011710 510 127.15 83.01
discrete algebraic equation for variable at each node is a set
of nominally linear equations that can be written as
app =
nb
anbnb + C (16)
where the subscript nb denotes neighbor value. The coefficients
ap and anb, contain convection and diffusion coefficients. C is
the source of in the control volume surrounding point p. The
SIMPLEC algorithm introduced by Van Doormaal and Raithby
[34] was used to resolve the coupling between velocity and pres-
sure. To accelerate convergence, the under-relaxation technique
was applied to all dependent variables. In the present study, the
under-relaxation factor for the pressure,p, was 0.3; that for tem-
perature, T, was 0.9; that for the velocity component in the i
direction, ui, was 0.5; that for body force was 0.8; that for kand
, was 0.7.
3.2. Convergence criteria
The numerical computation is considered converged when
the residual summed over all the computational nodes at the nth
iteration, Rn, satisfies the following criterion:
Rn
Rm 106 (17)
where Rm refers to the maximum residual value of variable
summed over all the computation cells after the mth iteration
and Rn, the value at the nth iteration.
A grid refinement study was conducted to determine an ade-
quate distribution. Boundary layer has been considered while
meshing the geometry in GAMBIT. Table 4 presents a com-parison of the predicted results at different grid distributions
(cross-sectional axial) for a fully developed turbulent flowin coiled tube. The sectional number refers to the total num-
ber of elements on one cross-section ( = constant) of the pipe.
Table 4 indicates that for the computation domain (0 <
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Fig. 3. Computed velocity contours in coiled flow inverter (one bend) with
= 0.1,NDe
= 6325,NPr
= 0.7at differentcrosssections: (a) = 15
, (b) = 30
,
(c) = 60, (d) = 90, (e) =120, (f) =180, (g) =270, (h) =360 and(i) =720.
ent workers [1920,2223]. No such rotation of velocity fields
in the laminar fully developed flow was observed in the study
of Kumar and Nigam [29]. This suggests that effect of torsion is
less significant for laminar flow as compared to turbulent flow
conditions. A fully developed flow was obtained at =270 asit can be seen clearly that the velocity fields does not have much
change after =270.The velocity field from the outlet of coiled tube was intro-
duced at the inlet of coiled flow inverter with one bend. It can be
seen from Fig. 3 that the orientation of flow field gets changeddue to the 90 bend. The above result reveals that the velocityfields, which had maximum velocity in coiled tube, would have
minimum velocity after 90 bend. Similarly, the velocity vectorsthat were at minimum velocity would have maximum velocity
Fig. 4. Computed velocity contours in coiled flow inverter (two bends) with
= 0.1,NDe = 6325,NPr= 0.7at differentcrosssections: (a) = 15, (b) = 30,
(c) = 60, (d) = 90, (e) =120, (f) =180, (g) =270, (h) =360 and
(i) =720.
Fig. 5. Computed velocity contours for fluids of various Prandtl number in: (i)
torus (H= 0) and (ii) helical coil (H= 0.02 m) for NDe = 6325 and =360.
after inversion. The initial inversion in case of laminar flow [29]
was observed at 120 while in the present case it was 180. Thisobservation is expected due to intense turbulence. The flow then
again reassembled back to almost 90. It can be also seen thatthe velocity fields does not have much change after =270;indicating a fully developed flow. The oscillations of velocity
fields were observed due to torsion.
The velocity fields from the outlet of first bend were intro-
ducedattheinletofsecondbend. Fig.4 showscomputedvelocity
fields at different cross sections in coiled flow inverter havingtwo bends. The same phenomenon was observed after the sec-
ond bend as observed after the first bend. The orientation of flow
fields again changed almost to 90.Computations were also carried out in torus and helical coil
with definite pitch in order to see the effect of torsion. Fig. 5
shows the computed velocity fields of fluids with different
Prandtl number at the outlet of torus and helical coil. The fig-
ure shows that the velocity contours in both the geometry were
gradually shifted towards outer wall as was increased. The
contours in torus were symmetric to the centerline between the
outer most point to the innermost point. The velocity distribu-
tion at the outlet was found to be similar for fluids of higher
Prandtl number. It was also observed that the velocity contoursin helical coil were asymmetric in nature because of rotational
flow, which is generated due to torsion. There was no change
in behavior of velocity contours even at higher Prandtl number.
This suggests that buoyancy has a little role to play in developed
turbulent flow as compared to developed laminar flow [35].
The complete axial velocity profiles for helical pipe and
coiled flow inverter at different angular planes are shown in
Figs. 68. The axial velocity profiles on horizontal centerline
at different axial planes in coiled tube are shown in Fig. 6(i).
This figure shows that at low values of , the velocity profiles
werealmost symmetrical on the horizontal centerline. This result
agrees with the axial velocity field. With the increase of , the
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Fig.6. Developmentof axial velocityprofile on: (i) horizontal centerlineand (ii)
vertical centerline in coiled tube with = 0.1, NDe = 6325, NPr = 0.7 at different
axial planes: (a) = 15, (b) = 30, (c) = 60, (d) = 90, (e) =120, (f)
=180, (g) =270 and (h) =360.
axial velocity became asymmetrical. At the horizontal center-
line, the maximum velocity shifted to the outside of the pipe
because of the unbalanced centrifugal force on the main flow.
The velocity profiles in turbulent flow were shifted more toward
theouterwall as compared to thevelocityprofiles in laminar flow
presented by Kumar and Nigam [29]. It was also observed that
the velocity profiles on horizontal centerline were flatter than
in the case of laminar flow. The axial velocity profiles at the
vertical centerline are shown in Fig. 6(ii). The velocity became
slightly asymmetrical. The asymmetrical nature may be due to
the torsion effect acting on the fluid. The profiles on the ver-
tical centerline were also flatter in turbulent flow as compared
to laminar flow. The axial velocity profiles at different axial
planes on horizontal centerline in coiled tube with one bend are
shown in Fig. 7(i). It can be seen that the velocity profiles were
Fig.7. Developmentof axial velocityprofile on: (i) horizontal centerlineand (ii)
vertical centerline, in coiled flow inverter (one bend) with = 0.1, NDe = 6325,
NPr= 0.7at differentaxialplanes:(a) = 0,(b) = 15, (c) = 30,(d) = 60,
(e) = 90, (f) =120, (g) =180, (h) =270 and (i) =360.
Fig. 8. Development of axial velocity profile on: (i) horizontal centerline and
(ii) vertical line in coiled flow inverter (two bends) with = 0.1, NDe = 6325,
NPr = 0.7at differentaxialplanes:(a) = 0,(b) = 15,(c) = 30,(d) = 60,
(e) = 90, (f) =120, (g) =180, (h) =270 and (i) = 360.
almost asymmetrical at initial stage but as increased, the pro-
files became more and more symmetrical in nature. Fig. 7(ii)
shows the development of axial velocity profile on vertical cen-
terline at different axial planes in coiled tube with one bend.
With the increase of , the axial velocity shifted to the outside
of the pipe because of the unbalanced centrifugal force. Fig. 7
shows that the velocity profile at vertical and horizontal center-
line was interchanged from that of velocity profile in straight
coiled tube. This is because of reorientation of flow fields due
to 90 bend. Fig. 8 shows the axial velocity profiles at differentaxial planes in coiled tube with two bends. Same phenomenon
is obtained after second bend. The velocity profiles at vertical
and horizontal centerline were again interchanged.
4.2. Description of temperature fields
Figs. 911 represent the development of temperature field
at different axial positions in the coiled tube and coiled flow
inverter with one and two bends, respectively. Fig. 9 shows
the variation of computed temperature contours at various
cross-sectional planes in coiled tube. The developments of the
temperature fields agree with that of the axial velocity fields.
There was negligible effect of secondary flow near the tube
inlet. It can be seen that as increased, the secondary flow wasenhanced. Due to the secondary flow, the temperature fields with
lower values were pushed towards outer wall region. A com-
parison of dimensionless temperature contours at =270 and =360 reveals that there was not much change in temperaturedistribution. This indicates that the temperature boundary layer
has become fully developed, hence heat transfer is fully devel-
oped at = 270. When is small, the temperature fields were
symmetric to the centerline between the outer most point to the
innermost point. But at the later stage, rotation of temperature
contours was found, similar to the velocity fields. The tempera-
ture contours in the present study were found to be more uniform
as compared to the laminar flow [29]. In the laminar flow, two
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Fig. 9. Computed temperature contours in coiled tube with = 0.1, NDe = 6325,
NPr= 0.7 at different cross sections: (a) = 15, (b) = 30, (c) = 60, (d)
= 90, (e) =120, (f) =180, (g) =270, (h) =360 and (i) =720.
Dean roll cells were observed which were absent in this study.
Similar distinction between laminar and turbulent temperature
profiles was also made by Yang and Ebadian [1]. The absence
of Dean roll cells in turbulent flow may be due to the reason
that the secondary flow is not as prominent as in case of laminar
flow. Also the thermal diffusivity is high in case of turbulent
flow. Fluids at high thermal diffusivity rapidly adjust their tem-
perature to that of their surroundings, because they conduct heat
quickly. Hence there was no incursion or penetration in fluid as
was found in laminar flow. As a result the Dean Roll cells were
absent in fluid with turbulent flow. Fig. 10 shows the devel-opment of temperature contours at different axial positions in
coiled flow inverter with one bend. The direction of temperature
Fig. 10. Computed temperature contours in coiled flow inverter (one bend) with
= 0.1,NDe = 6325,NPr= 0.7at differentcrosssections: (a) = 15, (b) = 30,
(c) = 60, (d) = 90, (e) =120, (f) =180, (g) =270, (h) =360 and
(i) =720.
Fig. 11. Computed temperature contours in bent helix (two bends) with = 0.1,
NDe = 6325, NPr= 0.7 at different cross sections: (a) = 15, (b) = 30, (c) = 60, (d) = 90, (e) =120, (f) =180, (g) =270, (h) =360 and (i) =720.
contours gradually changed as was increased. It can be seen
that there was no appreciable change in distribution of tempera-
ture contours after =270. This further confirms that the flowwas fully developed. The maximum temperature was obtained
at the inner wall and minimum temperature at the outer wall
of the tube. Fig. 11 shows computed temperature fields at dif-
ferent cross sections in coiled flow inverter having two bends.
The same phenomenon was observed after the second bend as
observed after thefirst bend. It was also observed that thetemper-ature became more and more uniform as increased; indicating
a good mixing of fluids due to flow inversion. Further work is
being done considering the affect of variable properties on fluid
flowing through coiled flow inverter. The work will be included
in the part II of the paper.
Fig. 12 shows the temperature contours for fluids of vari-
ous Prandtl number at the outlet of torus and helical coil. It
was found that due to secondary flow, the contours were pushed
towards the outer wall of both the geometries. Slight distortion
of temperature contours were found in the helical coil similar
to the velocity contours. It was also observed that as the Prandtl
number was increased, the temperature fields became more and
more uniform. Reason for this can be because of the fact thatPrandtl number is the ratio of momentum diffusivity (kinematic
viscosity) to thermal diffusivity. Thermal diffusivity is the ratio
of heat conducted through the material to the heat stored per unit
volume. When Prandtl number is small, it means that the ther-
mal diffusivity is high and heat diffuses very quickly compared
to the velocity (momentum). If Prandtl number is high then the
thermal diffusivity is small. This means that a big part of the heat
is absorbed by the fluid and only a small portion is conducted
through it.
The development of computed temperature profile on hor-
izontal and vertical centerline at different axial planes in
coiled tube and coiled flow inverter at one and two bends, are
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Fig. 16. Fully developed axial velocity profile on: (i) horizontal centerline and
(ii) vertical centerline in coiled tube and different coiled flow inverter with
= 0.1, NDe = 6325, NPr= 0.7.
Fig.17. Fullydevelopedtemperature profile on: (i) horizontal centerline and (ii)
vertical centreline in coiled tube and different coiled flow inverter with = 0.1,
NDe = 6325, NPr= 0.7.
Fig. 18. Comparison of computed fullydeveloped Nusselt number forNPr=0.7
and = 0.1, with the data of Mori and Nakayama [14] and Schmidt [15].
The empirical formula presented by Schmidts experimental
work[15] is as following:
NNufd
NNus= 1.0 + 3.6(1 )0.8 (19)
where NNus = 0.023N0.8Re N0.4Pr .It can be observed from Fig. 18 that the present predictions
of Nusselt number were in good agreement with the available
results. The maximum deviationbetween the present predictions
and the empirical correlation is less than 5%.
The results of the numerical computations for enhancement
of heat transfer with Dean number (NDe) in coils with two bends
is shown in Fig. 19. From the Fig. 19, it can beseen thatthe Nus-selt number increased with the increase in Dean number. It can
also be observed that there was 413% heat transfer enhance-
ment in coiled flow inverter as compared to the straight helical
coil. The figure also shows 3545% increase of heat transfer
in coiled flow inverter as compared to straight tube. The heat
gain for straight helical coil over straight tube was found to be
Fig. 19. Nusselt number variation with Dean number at NPr= 0.7 and = 0.1;
in straight tube, coiled tube and coiled flow inverter with two bends.
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Fig. 20. Nusselt number variation with Prandtl number at NDe = 6325 and
= 0.1; in straight tube, coiled tube and coiled flow inverter with two bends.
3235%. Computations were also carried out to study the effect
of Prandtl number. Fig. 20 shows 412% enhancement of heat
transfer in coiled flow inverter as compared to straight coiled
tube as Prandtl number was increased. It can also be observed
that the heat transfer in coiled flow inverter was 2443%
more as compared to straight tube with increase in Prandtl
number.
Itcanbeseenfrom Fig.21 thatthe increase of friction factorin
bent coil tube to that of straight coiled tube at low values of Dean
number was about 9%. The percentage increase of friction fac-
tor in coiled flow inverter to straight coiled tube decreased with
increase in value of Dean Number and it reduces to about 2% at
NDe = 9487. This may be due to the fact that the hydrodynamic
effect of both helical coil and coiled flow inverter becomes com-
parable at higher Dean number. So both configurations becomemore and more equivalent in performance. Similar observations
were found by Chagny et al. [26] who studied the chaotic flow
obtained by alternately turning the axis of curved tubes (half cir-
cles) by 90. The Fig. 21 also shows 2930% increase in frictionfactor in coiled flow inverter as compared to straight tube. The
increase in friction factor of straight helical coil over straight
tube is 2328%.
Fig. 21. Friction factor variation with Dean number at NPr = 0.7 and = 0.1; in
straight tube, coiled tube and coiled flow inverter with two bends.
6. Conclusion
In the present study, hydrodynamics and heat transfer of tur-
bulent forced convection in an innovative heat exchanger having
coils with one and two bends with circular cross section have
been investigated. The developments of velocity fields at differ-
ent axial positions in straight helical coil and coiled flow inverter
with one and two bends have been reported under turbulent flow
conditions. A slight rotation of velocity contours was observed
in the coiled tubes. This may be due to the torsion caused by
turbulent flow. It was found that buoyancy doesnt have major
role in the turbulent forced convection of fluid in coiled tubes.
The velocity fields were increasingly uniform as the numbers of
bends in the coiled tube were increased. This may be because
of increase in radial mixing. Similar results were found with the
temperature fields in coiled tube and coiled flow inverter. The
cold regions present in the straight coil are modified in bent coil
due to radial mixing. Eventually, the heat transfer was enhanced.
It was found that the enhancement of heat transfer in coiled flow
inverter as compared to straight coil and straight tube is morethan the increase in friction factor. The effect of heat transfer
in fluid of higher Prandtl number was also studied. The study
shows that the heat transfer increases with increase in Prandtl
number.
Acknowledgements
The authors gratefullyacknowledge the Ministry of Chemical
and Fertilizers, GOI, India for funding the project.
Appendix A. Nomenclature
A area (m2)
cf skin friction coefficient
C1, C2, C turbulent model constant
Cp specific heat (kJ/kg K)
dc diameter of the coil (m)
dh hydraulic diameter of the helical pipe (m)
dt diameter of the helical pipe (m)
D dispersion coefficient (m2/s)
H pitch (m)
k turbulent kinetic energy (m2/s2)
L length of reactor (m)
n coordinate direction perpendicular to a surfaceNDe Dean number [NDe = NRe
]
NNu Nusselt number [NNu, = qwdh/(Tw Tb)]NPr molecular Prandtl number [NPr = Cp/]NRe Reynolds number [NRe = u0dt/]p pressure (N/m2)
q heat flux (W/m2)
T temperature (K)
Tb fluid bulk temperature on one cross-section (K)
u velocity component in flow direction (m/s)
u0 inlet velocity (m/s)
ui velocity component in i-direction (i = 1, 2 and 3) (m/s)
u root-mean-square turbulent velocity fluctuation (m/s)
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M. Mridha, K.D.P. Nigam / Chemical Engineering and Processing 47 (2008) 893905 905
U axial velocity component (m/s)
Uavg average velocity over a cross-section (m/s)
x spatial position (m)
xi master Cartesian coordinate in i-direction (i = 1, 2 and
3) (m)
Greek symbols
T inverse effect Prandtl number for T
inverse effect Prandtl number for
inverse effect Prandtl number for
thermal conductivity (W/m K)
ij dirac delta function
dissipation ratio of turbulent kinetic energy (m2/s2)
axial angle () curvature ratio (= dt/dc) viscosity (kg/m s)
eff effective viscosity (kg/ms)
non-dimensional temperature ((Tb Tw)/(Tin Tw)) circumferential quantity
density of fluid (kg/m3) shear stress (N/m2)
Subscripts
0 inlet conditions
2nd secondary flow
b bulk quantity
m circumferential average quantity
t turbulent quantity
w wall condition
References
[1] G. Yang, M.A. Ebadian, Turbulent forced convection in a helicoidal pipe
with substantial pitch, Int. J. Heat Mass Transfer 39 (1996) 20152022.
[2] C.X. Lin, M.A. Ebadian, Developing turbulent convective heat transfer in
helical pipes, Int. J. Heat Mass Transfer 40 (1997) 38613873.
[3] W.R. Dean, Note on the motion of fluid in a curved pipe, Philos. Mag. 4
(1927) 208223.
[4] W.R. Dean, The streamline motion of fluid in a curved pipe, Philos. Mag.
7 (1928) 673695.
[5] V. Kubair, N.R. Kuloor, Heat transfer to Newtonian fluids in coiled pipes
in laminar flow, Int. J. Heat Mass Transfer 9 (1966) 6375.
[6] S.A. Berger, L. Talbot, L.S. Yao, Flow in curved pipes, Annu. Rev. Fluid
Mech. 15 (1983) 461512.
[7] R.K. Shah, S.D. Joshi, Convective heat transfer in curved ducts, in: Hand-
book of Single-Phase Convective Heat Transfer, Wiley, New York, 1987.
[8] Jeschke, Warmeuberggang ud Druckverlust in Rohrschlangen, Z. Ver. dt.Ing. 69 (1925) 1526.
[9] C.M. White, Fluid Friction and its relation to heat transfer, Trans. Instn.
Chem. Eng. 10 (1932) 6686.
[10] A.V. Kirpikov, Heat transfer in helically coiled pipes, Trudi. Moscov. Inst.
Khim. Mashinojtrojenija 12 (1957) 4356.
[11] R.A. Seban, E.F. McLaughlin, Heat transfer in tube coils with laminar and
turbulent flow, Int. J. Heat Mass Transfer 6 (1963) 387395.
[12] G.F.C. Rogers, Y.R. Mayhew, Heat transfer and pressure loss in heli-
cally coiled tubes with turbulent flow, Int. J. Heat Mass transfer 7 (1964)
12071216.
[13] Y. Mori, W. Nakayama, Study on forced convective heat transfer in curved
pipes (2nd report, turbulent region), Int. J. Heat Mass Transfer 10 (1967)
3759.
[14] Y. Mori, W. Nakayama, Study on forced convective heat transfer in curved
pipes (3rd report, theoretical analysis under the condition of uniform wall
temperature and practical formulae), Int. J. Heat Mass Transfer 10 (1967)681695.
[15] E.F. Schmidt,Warmeubergang und Druckverlustin Rohrschlangen, Chem.-
Ing.-Technol. 36 (1967) 781789.
[16] V.K. Shchukin, Correlation of experimental data on heat transfer in curved
pipes, Therm. Eng. 16 (1969) 7276.
[17] B. Zheng, C.X. Lin, M.A. Ebadian, Combined turbulent forced convection
and thermal radiation in a curved pipe with uniform wall temperature,
Numer. Heat Transfer Part A 44 (2003) 149167.
[18] A. Cioncolini, L. Santini, An experimental investigationregarding the lam-
inar to turbulent flow transition in helicallycoiled pipes,Exp. Therm.Fluid
Sci. 30 (2006) 367380.
[19] M. Germano, On the effect of torsion in a helical pipe flow, J. Fluid Mech.
125 (1982) 18.
[20] H.C. Kao, Torsion effect on fully developed flow in a helical pipe, J. Fluid
Mech. 184 (1987) 335356.[21] H. Ito, Flow in curved pipes, JSME Int. J. 30 (1987) 543552.
[22] K. Yamamoto,T. Akita, H. Ikeuchi, Y. Kita, Experimentalstudy of theflow
in a helical circular tube, Fluid Dyn. Res. 16 (1995) 237249.
[23] T.J. Huttl, R. Friedrich, Influence of curvature and torsion on turbulent
flow in helically coiled pipes, Int. J. Heat Fluid Flow 21 (2000) 345
353.
[24] N. Acharya, M. Sen, H.C. Chang, Heat transfer enhancement in coiled
tubes by chaotic mixing, Int. J. Heat Mass Transfer 35 (1992) 24752489.
[25] A. Mokrani,C. Castelain, H. Peerhossaini, The effects of chaotic advection
on heat transfer, Int. J. Heat Mass Transfer 40 (1997) 30893104.
[26] C. Chagny, C. Castelain, H. Peerhossaini, Chaotic heat transfer for heat
exchanger design and comparison with a regular regime for a large range
of Reynolds numbers, Appl. Therm. Eng. 20 (2000) 16151648.
[27] N. Acharya, M. Sen, H.C. Chang, Analysis of heat transfer enhance-
ment in coiled-tube heat exchangers, Int. J. Heat Mass Transfer 44 (2001)
31893199.
[28] T. Lemenand, H. Peerhossaini, A thermal model for prediction of the Nus-
selt number in a pipe with chaotic flow, Appl. Therm. Eng. 22 (2002)
17171730.
[29] V. Kumar, K.D.P. Nigam, Numerical simulation of steady flow fields in
coiled flow inverter, Int. J. Heat Mass Transfer 48 (2005) 48114828.
[30] K.D.P. Nigam, A.K. Saxena, Coiled configuration for flow inversion and
its effect on residence time distribution, AlChE J. 30 (1984) 363368.
[31] V. Yakhot, S.A. Orszag, Renormalization group analysis of turbulence. I.
Basic theory, J. Sci. Comput. 1 (1986) 151.
[32] O. Kaya, I. Teke, Turbulent forced convection in helically coiled square
duct with one uniform temperature and three adiabatic walls, Heat Mass
Transfer 42 (2005) 129137.
[33] S.V. Patankar, NumericalHeat Transferand FluidFlow, Hemisphere, Wash-
ington, DC, 1980.
[34] J.P. VanDoormaal, G.D.Raithby, Enhancementsof theSIMPLEmethodfor
predicting incompressible flow problems, Numer. Heat Transfer 7 (1984)
147158.
[35] J. Lee, H.A. Simon, J.C.F. Chow, Buoyancy in developed laminar curved
tube flows, Int. J. Heat Mass Transfer 28 (1985) 631640.
[36] Fluent Users Guide, Version 6.2, Fluent Inc., Lebanon, NH, 2005.