- The document analyzes turbulent momentum and heat transfer in a rectangular helical duct using water and freon-12 as working fluids. - It develops a 3D model of the helical duct geometry based on prior research. Simulations are run using ANSYS Fluent and CFX to analyze flow, temperature, and turbulence fields. - The results show secondary flows developing with higher velocities near the outer wall. While some aspects agree with prior work, the simulations produce higher velocities and different vortex structures than expected. Thermal development also takes longer than velocity development.

1. Numerical Analysis of Turbulent Momentum and Heat Transfer
in a Rectangular Helical Duct using Water and Freon-12
Nathaniel H. Werner
1 Introduction
Due to their compact size and ecient heat transfer
performance, helical coiled ducts are widely employed
as heat exchangers for industrial applications such as
power generation, nuclear industry, process plants,
heat recovery systems, and refrigeration. The goal of
this analysis is to develop a better understanding of
the eect of turbulence and heat transfer on helical
rectangular ducts for heat exchange and refrigeration
systems used in both industry and by consumers.
2 Literature Review
In order to analyze turbulent ow in a rectangular
helical duct, the geometry of the duct is modeled after
what is presented in Xing et al. [1]. It has been widely
reported in the literature that heat transfer rates in
helical coiled channels are higher as than those in
straight ducts as shown by Moawed and Kurnia et.
al. [4, 5]. For instance, it has been established by
Kao [2] that the eect of torsion is signicant if the
ratio of curvature to torsion approaches unity.
Wu et al. [3] also investigated the eects of a larger
curvature ratio on the turbulent heat transfer. There
it was found that the increase in heat transfer co-
ecient is smaller for the turbulent ow than for
a laminar ow. This is expected due to the higher
mixing that is natural for turbulent ows. Kaew-On
et al. [6] studied the heat transfer of water owing
through straight and helical minichannel tubes. The
result shows that there is an enhanced heat transfer
for ducts with greater curvature ratio.
The majority of research done regarding heat
transfer and ow characteristics in helical coiled ducts
have been restricted to channels of circular cross sec-
tion. Therefore, further study of non-circular cross
sections is encouraged.
3 Methodology
The rst portion of this analysis attempts to verify
the results given in Xing et al. [1]. In order to do this
the same geometry is analyzed, with similar ow and
boundary conditions and with water as the working
uid. Obviously not all of the results in [1] can be
Figure 1: Helical Rectangular Duct
tested and veried; so this analysis will focus only on
verifying the velocity, temperature, and turbulence
intensity elds at dierent cross sections. The sec-
ond portion of this analysis will use the geometries
from the rst portion, but with a dierent working
uid, such as Freon-12. Finally the third portion of
this report will attempt to use other CFD software
packages in order to compare the results with those
found in [1].
3.1 Helical Rectangular Duct Geome-
try
The geometry for this model was built in SolidWorks
and then imported into ANSYS Workbench where
the mesh was built. Figure 1 shows the rectangu-
lar helical duct used in this analysis. The duct has
three revolutions, where each turn is specied by an
azimuthal angle θ starting from 0 degrees at the end
of the entry length and ending at 1080 degrees. The
entry length is specied by equation (5) as at least 10
times the hydraulic diameter
1. At the end of the en-
try length there is a smooth transition built into the
geometry to avoid any abrupt changes in the velocity
1This entry length is intentionally greater than what is spec-
ied by (5)
1

2. Table 1: Geometric Parameters of the Duct
Width (mm) Height (mm)
1.0 1.2
Hydraulic Diameter (mm) Entry Length (mm)
1.09 11.0
eld that could eect the downstream conditions in
the ow.
The dimensionless curvature and torsion are de-
ned by equations (2,3). These were used to match
the geometry as given by [1], for values of δ and λ of
0.192 and 0.11 respectively.
The hydraulic diameter is specied by the geomet-
ric parameters including the height and width of the
cross section by equation (4). For comparison the
helical rectangular duct used by [1] has a width of
5 mm and a height of 6 mm, while the one used in
this analysis has a width of 1 mm and a scaled height
of 6/5 or 1.2 mm by equation (1). The reason for
this scaling was to make the overall size of the ge-
ometry smaller. This scaling retains the value of the
dimensionless curvature and torsion, as well as the
hydraulic diameter as dened below. All of the geo-
metric parameters used in the design of the duct are
given in Table 1.
c =
a
b
(1)
δ =
a
R
→ a = δ · R (2)
λ =
p
2πR
→ p = λ · 2πR (3)
DH =
2ab
a + b
=
2δR
1 + c
(4)
LE ≥ 10DH =
20δR
1 + c
(5)
3.2 Numerical Method
As mentioned in section 3.1, the mesh for this model
was designed using the ANSYS Workbench meshing
tool. For each mesh, there are 15 divisions on each
edge of the inlet which gives 225 elements in each
cross section. These were set with a bias type that
was largest in the center and smallest at each end,
and then they were swept forward through the entire
geometry. A body sizing was applied for the entire
volume for an element size of 0.1 mm, and a face
sizing was applied to the inlet and the walls of the
helix with an element size of 0.08 mm. This resulted
in 379648 nodes and 328950 elements for the entire
geometry.
In Fluent the Navier Stokes and energy equations
are solved with a SST k-ω model and the SIMPLE
scheme is employed to resolve the coupling between
velocity and pressure as done by [1]. The pressure,
momentum, TKE, dissipation rate, and energy are
all discretized using second order or second order up-
wind when available. The relaxation factors are 0.1
for pressure, 1 for density and body forces, 0.7 for
momentum, 0.8 for TKE and dissipation rate, and
0.9 for energy.
In CFX there was much less control over the model
and the only parameter that could be controlled was
the turbulence, which used a traditional k-ω model.
The advection scheme was set at a high resolution,
and the turbulence numerics were set at rst order.
The residual target for all parameters was set at 1e-4.
3.3 Flow Parameters
In the ow eld the Reynolds number is dened using
the average speed of the ow, the hydraulic diameter
and the kinematic viscosity of the working uid.
Re =
V DH
ν
(6)
In order to determine if this ow is turbulent be-
fore proceeding to the analysis the critical Reynolds
number is related to the curvature ratio by equation
(2). This relationship is given by equation (7) from
[7].
Recrit = 2100(1 + 12δ1/2
) (7)
3.4 Thermal Properties of the Work-
ing Fluids
This analysis was performed using water to model a
heat exchanger, and Freon-12 to model a refrigeration
cycle. The properties
2 of the uids used in calcula-
tions are each given in Table (2) and are found at
the inlet lm temperature (8) using [8, 10, 12]. The
inlet is held constant at a temperature of 283 K (10
ºC) while the walls (after the entry length) are held
at 370 K (97 ºC). This makes the inlet lm temper-
ature equal to 327 K (53 ºC). This temperature is
higher than the maximum temperature provided for
the Freon-12 so the values were extrapolated forward
using a 4th order polynomial for the kinematic vis-
cosity and a 6th order polynomial for the thermal
conductivity. The plots for these are given in Figure
2
3
Tf =
Tinlet + Twall
2
(8)
2The density and dynamic viscosites were found in ([9, 11])
at a lm temperature of 53 ºC.
3Because the kinematic viscosity and the thermal conduc-
tivity were not given at the inlet lm temperature polynomials
needed to be generated to extrapolate the values forward.
2

3. Table 2: Properties of the Working Fluids
Water Freon-12
Density (
kg
m3 ) 988.1 1200
Kinematic Viscosity (
m2
s ) 5.146e-7 1.852e-7
Thermal Conductivity (
W
m·K ) 0.653 0.693
4 Results and Discussion
The inlet (of the entry length) is specied at a
Reynolds number of 55,000. The turbulence at the
inlet was specied using an intensity and length scale
model where the intensity was 5% and the length
scale was set as the hydraulic diameter since that
would be approximately the largest size of the largest
eddy within the turbulence. The intensity at the inlet
was set at 5% because of the straight duct being used
as the entry length where the intensities are normally
between 1-5%[12]. Because the inlet Reynolds num-
ber is much larger than the critical Reynolds num-
ber which is approximately 13,150 the ow through
the helix is clearly turbulent. Using equation (6) the
hydraulic diameter and the kinematic viscosity from
Table 2 the average velocity of the ow is calculated
for each uid. The average velocities are 25.97 m/s
and 9.35 m/s for the water ow and Freon-12 ow re-
spectively. If the inlet temperature is used the result-
ing velocity is 78.11 m/s and 10.2 m/s which exceeds
what is given by[1]. The walls are set as stationary
walls to satisfy the no slip condition and the outlet is
held at 1 atm for each of the ows.
4.1 Description of Velocity Field
The ow through the entry length will become fully
developed, and then it will experience a transition as
it travels through the helical duct. It was expected
that because of the complicated geometry of the heli-
cal duct and the turbulent nature of the ow that sig-
Figure 2: Property Extrapolation Curves of Freon-12
Figure 3: Velocity Field in Water from Fluent
nicant secondary ows would arise. The ow closer
to the outer wall will experience a higher acceleration
as it travels around the helix than the inner wall. It
is also expected that at some point the velocity eld
will become fully developed again.
4.1.1 Water Flow
Figure 3 presents the evolution of the velocity eld
in water at dierent azimuthal angles θ as given by
Fluent; Figure 4 shows the velocity elds as given
from CFX. At each angle, the left side is the outer
wall and right side is the inner wall. It can be seen
that in the results from both codes that the maximum
velocity gradually shifts to the outer region of the ow
as θ gets larger, this is in reasonable agreement with
[1].
The results from Fluent reveal that secondary ow
is enhanced with an increase in θ similar to what is
shown in [1]. This secondary ow leads to a higher
level of convective heat transfer near the outer wall
because of the higher mean velocity. This is because
the curvature of the duct causes there to be a cen-
tripetal force which tends to keep the ow in a circu-
lar motion and push the higher velocity ow towards
the outer wall. Unlike what is described by [1], the
fully developed ow takes the form of concentrated
vortices near the top and bottom corner of the outer
wall. The ow does not become fully developed until
at least 540º (1.5 revolutions) which is also in dis-
agreement with [1]. The maximum velocity is at least
3 times what is given by [1]. Although the inlet ve-
locity is not given in [1], the velocity in this analysis
likely exceeds it by looking at Figures 3 and 4.
The results from CFX reveal similar behavior of
the secondary ow as in Fluent, but there is no con-
centration of vortices in the same location. Rather,
the ow develops into regions of high velocity near
the outer, top and bottom walls. This again does
not agree with [1], but the ow does become fully
3

4. developed at 360º (1 revolution) which is in agree-
ment. CFX also overshoots the maximum velocity
by roughly 3 times what is given by [1].
In both cases the maximum velocity increases be-
tween (900º and 1080º) because of the outlet bound-
ary condition. It was observed if an exit length is
attached to the outlet of the geometry then this issue
is easily corrected but it can lead to convergence er-
rors. This is in good agreement with what is provided
by [1].
4.1.2 Freon-12 Flow
Because of the lower kinematic viscosity of Freon-
12 compared to water, the volume ow rate will in-
crease according to equation (13-10) in [9]. This is
because similar to laminar ow where the average ve-
locity varies with the inverse of kinematic viscosity,
in turbulent ow it varies with ν− 1
7 . The variation
in the velocity eld becomes minimal after 540º (1.5
revolutions) but since the ow is asymmetric about
the horizontal centerline it is not fully developed.
4.2 Description of Temperature Field
Because of the helical geometry and turbulent na-
ture of the ow, the centripetal force causes the ow
to accelerate near the outer wall. This acceleration
will tend to keep the coolest part of the ow in this
region until the ow becomes thermally fully devel-
oped. The ow becomes thermally fully developed
when equation (9) approaches 1. This is shown by
the temperature contour becoming strictly uniform.
Θ =
Tsurface − T
Tsurface − Tmean
(9)
Figure 4: Velocity Field in Water from CFX
Figure 5: Velocity Field in Freon-12 from Fluent
4.2.1 Water Flow
Figure 6 shows the evolution of the temperature eld
in water from Fluent, and Figure 7 gives the results
from CFX. Both Figures 6 and 7 show that tempera-
ture prole becomes fully developed at 720º (2 revo-
lutions). This is further than where the velocity eld
became fully developed. This is reasonable since the
walls of the helical duct after the entry length are kept
at a higher temperature than the inlet. This will ini-
tiate a thermal boundary layer after the entry length
while the momentum boundary layers have already
converged.
The shape and magnitude of the temperature are
in good agreement with [1]. It can be seen that both
of the codes produce results similar to what is docu-
mented in [1].
4.2.2 Freon-12 Flow
Because of the higher thermal conductivity of Freon-
12 compared to water, the temperature eld will be
delayed in becoming fully developed. The tempera-
ture is symmetric about the horizontal axis at 540º
Figure 6: Temperature Field in Water from Fluent
4

5. Figure 7: Temperature Field in Water from CFX
but becomes asymmetric again after 900º. This is
shown in Figure 8 where the temperature eld never
becomes fully developed which is not expected.
4.3 Description of Turbulence Intensi-
ties
Figures 9 and 10 show the evolution of the turbu-
lence intensity eld in water and Freon-12 respec-
tively. What is clearly shown is that for both u-
ids is that maximum turbulence is near the walls and
the minimum develops in the center. This agrees with
the turbulence intensity contours in [1]. However, the
shape of the contours is extremely dierent. Here, the
minimum intensity starts in the center and decreases
up until 360º (1 revolution), however it becomes fully
developed at 540º (1.5 revolutions) in the form two
pockets of low intensity near the top and bottom wall.
This agrees well with the ndings found in section
4.1.1. These are asymmetric about the vertical cen-
terline but symmetric about the horizontal centerline.
The variations in the Freon-12 turbulence intensity
Figure 8: Temperature Field in Freon-12 from Fluent
Figure 9: Turbulence Intensity Field in Water
become minimal after 540º (1.5 revolutions). This
ow is asymmetric about the vertical and horizontal
centerline which indicates that the turbulent intensity
is not fully developed for the entire ow. This is due
to the lower kinematic viscosity and higher thermal
conductivity of Freon-12 compared to water.
5 Conclusions
In this paper, three-dimensional turbulent ow and
convective heat transfer in helical rectangular ducts
have been investigated numerically. The Navier-
Stokes and energy equations are solved using a SST
k-ω turbulence model for Fluent and a traditional k-ω
model for CFX
4. In both cases the mesh was rened
near the wall to ensure accuracy of the solution and
to capture boundary layer eects.
4SST k-ω was not available in CFX only the traditional
form.
Figure 10: Turbulence Intensity Field in Freon-12
5

6. 5.1 Conclusions from the Current
Analysis
It was found that the heat transfer is enhanced on
the outer wall and reduced on the inner wall. Sig-
nicant secondary ow is shown to develop due to a
centripetal force, forcing the max velocity to the outer
wall and the minimum velocity to the inner wall.
5.2 Comparison with the Original Nu-
merical Analysis [1]
This analysis has shown that neither of the turbu-
lence models used in both Fluent and CFX do not
match the results for the velocity contours as given
by [1]. However, it was observed that for higher resid-
uals Fluent was able to match the velocity elds. At
0º for all cases (both water and Freon-12) exhibited
a dierent inlet contour than in [1] which exhibits a
concentration of maximum velocity towards the outer
wall. The temperature proles for all cases match the
results which indicates that Fluent and CFX are bet-
ter at predicting the temperature elds. The turbu-
lent intensities show no indication of matching what
is given by [1]. The results provided by [1] do not do
a good job at specifying the inlet velocity conditions,
which lead to confusion and discrepancies.
Nomenclature
Below are the symbols used to dene dierent param-
eters pertinent to the analysis
a = width of the duct cross section (mm)
b = height of the duct cross section (mm)
c = width to height ratio of the duct
DH = hydraulic diameter (mm)
k = thermal conductivity (W/m K)
LE = entry length (mm)
p = coil pitch (mm)
R = radius of the helical duct (mm)
Re =Reynolds number
V = average velocity(m/s)
δ = dimensionless curvature
λ = dimensionless torsion
ν = kinematic viscosity (m2
/s)
6

7. References
[1] Xing, Y., Fengquan Z., and Xinyu Z.. Numer-
ical Study of Turbulent Flow and Convective
Heat Transfer Characteristics in Helical Rectan-
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[2] Kao, Hsiao C. Torsion Eect on Fully Devel-
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Mechanics 184.1 (1987): 335-56.
[3] Wu, S. Y., et al. Numerical Investigation of
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Generation in a Helical Coiled Tube with Larger
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[4] Moawed, M. Experimental Study of Forced
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[5] J. C. Kurnia, A. P. Sasmito A. S. Mujumdar.
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