CALCULATION OF FINNED TUBE PRESSURE DROP AND HEAT TRANSFER USING FULLY THREE-DIMENSIONAL CFD METHODS

CALCULATION OF FINNED TUBE PRESSURE DROP AND HEAT TRANSFER
USING FULLY THREE-DIMENSIONAL CFD METHODS
Christopher J. Matice, P.E.
Brian Gao
Stress Engineering Services, Inc.
Abstract
The use of computational fluid dynamics (CFD) in the design and troubleshooting of HRSGs is
now well established. However, the large ratio of scales between the overall flow path and the
tube bundles themselves means that the bundles must be approximated using “porous media”
models, with heat transfer and flow resistance simulated using lumped parameters. The
parameters input into these models are taken from the open literature, proprietary data, and/or
approximate analytical models. In the case of plain tubes, two-dimensional CFD models have
been used to supplement this information. For the finned tubes used in HRSGs, however, 2D
models will not suffice and 3D models can quickly become unwieldy. In this work we present
the results of a parametric study of pressure drop and heat transfer for finned tube bundles
computed using fully three-dimensional CFD models. The results are compared with values
computed from correlation equations typically used in design. The advantages and limitations of
this approach to obtaining inputs for HRSG tube bundle simulations are discussed.
Introduction
This work was motivated by a desire to use computational methods to predict the onset of flow-
induced vibration (FIV) in the leading superheater tube bundles of heat recovery steam
generators (HRSG). These bundles are susceptible to vibration caused by turbulent buffeting as
the flow from the gas turbine impinges upon the leading tubes and, to a lesser extent, familiar
FIV mechanisms such as fluid-elastic whirl and vortex shedding. In the case of plain tubes
experiencing uniform flow, two-dimensional coupled flow and stiffness models, such as that
shown in Figure 1 have been used to successfully simulate FIV. In the case of a HRSG,
however, the tubes are typically finned and the approach velocity is far from uniform. Even if
we simplify the problem, considering a span over which the approach velocity can be considered
to be approximately uniform, we are still left with the problem of modeling the fins. This
requires a three-dimensional flow model.
As a first step in the development of a 3D FIV model, the flow portion of the problem is
addressed in isolation. The goal is to determine how well a reasonably-sized computational fluid
dynamics (CFD) model could predict the flow versus pressure relationship for a representative
finned tube bundle. Comparisons of the CFD results with correlation equations for flow loss in
Presented at the EPRI Boiler Tube and HRSG Tube 1/17
Failures and Inspections International Conference
April 19, 2010, Baltimore, Maryland

bundles are used to evaluate the performance of the CFD model. Numerical and turbulence
modeling issues are also addressed.
While the desire to perform fully 3D FIV simulations motivates this work, the flow model alone
is of value in predicting the steady-state performance of a bundle. This is particularly the case
for typical HRSG bundles that includes tubes of varying diameter and fins of varying height and
pitch within a single bundle. Therefore, thermal calculations are performed using the CFD
models and compared with correlation equations commonly used in the industry.
Figure 1
Two-dimensional Flow-Induced Vibration Model of Tube Bundle

Model Setup and Parameter Study Definition
The layout of the tube bundle analyzed is shown in Figure 2. Typical HRSG superheater bundles
range from 3 to 8 tube rows in depth; a six-row bundle is chosen for this analysis. The values of
several important parameters are varied, producing a set of eleven cases, as shown in Table 1.
These parameters are:
Vapproach...........Approach velocity
d......................Tube diameter
P .....................Tube pitch
hfin...................Fin height
pfin...................Fin pitch
tfin ...................Fin thickness
These parameters are non-dimensionalized, as shown in Table 2, with the Reynolds number
defined as
ν
d
Vapproach
=
Re , (1)
where ν is the kinematic viscosity of air at the approach temperature.
Figure 2
Model Definition

Table 1
Parameter List - Dimensional
Case
Vapproach d P hfin pfin tfin
(m/s) (mm) (mm) (mm) (mm) (mm)
1 15 40 100 10 5 1
2 7.5 40 100 10 5 1
3 30 40 100 10 5 1
4 15 40 80 10 5 1
5 15 40 120 10 5 1
6 15 40 100 8 5 1
7 15 40 100 12 5 1
8 15 40 100 10 4 0.8
9 15 40 100 10 6 1.2
10 15 40 100 10 5 0.8
11 15 40 100 10 5 1.2
Table 2
Parameter List – Non-Dimensional
Case Re P/d hfin/d pfin/d tfin/pfin
1 7264 2.5 0.25 0.125 0.2
2 3632 2.5 0.25 0.125 0.2
3 14528 2.5 0.25 0.125 0.2
4 7264 2 0.25 0.125 0.2
5 7264 3 0.25 0.125 0.2
6 7264 2.5 0.2 0.125 0.2
7 7264 2.5 0.3 0.125 0.2
8 7264 2.5 0.25 0.1 0.2
9 7264 2.5 0.25 0.15 0.2
10 7264 2.5 0.25 0.125 0.16
11 7264 2.5 0.25 0.125 0.24

The symmetry of the bundle is exploited by modeling only one column of tubes, as shown by the
dashed lines in Figure 2. The surfaces represented by these lines must be treated as periodic
boundaries, wherein flow leaving from one boundary reenters on the opposite boundary. The
CFD model constructed from Figure 2 is shown in Figure 3. The model mesh is shown in Figure
4. This mesh is based on the smallest length scale in the problem, the fin thickness, resulting in
an element size of approximately 1 mm and giving approximately one element for every 3
degrees of arc around each tube (126 elements around each tube). The total number of elements
is approximately 2 million.
Figure 3
Model Layout
Figure 4
Typical Mesh

The periodic symmetry imposed on the CFD model allows for sufficient detail to simulate
steady-state flow and heat transfer in the tube bundle. However, simulation of unsteady behavior
leading to FIV would require that multiple columns of tubes be modeled. The absolute minimum
number of columns modeled should be six. This allows two columns of rigid tubes to form the
boundaries of the model and four columns of mobile tubes. In the current case this would result
in a model of 12 million elements. Therefore, even a very restricted model will be quite large.
The model is analyzed using the commercial CFD program FLUENT [1]. Material parameters
for air at 800 °K (980 °F) are:
ρ = 0.4413 kg/m3
ν = 8.26e-5 m2
/s
cp = 1099 J/kg-°K
k = 0.0569 W/m-°K
The fins are modeled using a conjugate heat transfer model and are assigned properties for steel
as:
ρ = 7800 kg/m3
cp = 465 J/kg-°K
k = 40 W/m-°K
The tubes themselves are assigned a uniform temperature of 750 °K (890 °F). Simulations are
performed for both finned and plain versions of each configuration. The plain-tube version is
obtained by changing the fin material from steel to air. Therefore, the mesh is unchanged
between the finned and plain versions of each configuration.
All solutions are performed using the k-ε turbulence model with FLUENT’s enhanced wall
functions (see below for discussion). Second-order interpolation is used for the momentum
equations and a coupled solver is used to perform the iterative solution. Typical run times are on
the order of 10 hours using one processor on a quad-core PC. Significant speed-up could be
obtained through parallel processing. This would be necessary in analyzing a multi-column
bundle model of a size sufficient to obtain FIV results (i.e. at least 6 columns of tubes).

Results
Color plots of results for Case 1 are shown in Figures 5 - 7. Some “wandering” of the wake is
seen in many of the solutions, suggesting there may be a time-dependent component to the
solution. A limited number of cases were rerun in transient mode. This showed a small
variation in the solution with time but nothing appreciable. Therefore, all results presented
herein are for steady-state conditions. The results in Figures 5 – 7 indicate that the upstream and
downstream boundaries are adequately spaced to eliminate any influence of the boundary
location on the solution. In fact, the inlet boundary could be moved significantly closer to the
first tube, decreasing the mesh size by approximately 15%.
Figure 5
Velocity Results for Case 1

Figure 6
Pressure Distribution Results for Case 1
Figure 7
Heat Transfer Results for Case 1

The results of the parameter study are presented in Table 3. The pressure drop results are non-
dimensionalized using the dynamic pressure in the gap between the tubes to obtain Euler
numbers in the form given by Zukauskas and Ulinskas [2] as
N
U
p
Eu
2
2
1
ρ
∆
= , (2)
where N is the number of tubes in the bundle and U is the velocity in the gap between the tubes
defined by
1
−
=
a
a
V
U approach , (3)
with d
P
a = .
Table 3
Parameter Study Results
Case Re P/d hfin/d pfin/d tfin/pfin
Eu Nu
w/ fins
w/o
fins
w/ fins
w/o
fins
1 7264 2.5 0.25 0.125 0.2 0.44 0.25 73.0 198
2 3632 2.5 0.25 0.125 0.2 0.60 0.31 53.3 139
3 14528 2.5 0.25 0.125 0.2 0.33 0.22 95.6 287
4 7264 2 0.25 0.125 0.2 0.52 0.29 83.2 216
5 7264 3 0.25 0.125 0.2 0.38 0.22 67.0 187
6 7264 2.5 0.2 0.125 0.2 0.40 - 80.3 -
7 7264 2.5 0.3 0.125 0.2 0.49 - 65.9 -
8 7264 2.5 0.25 0.1 0.2 0.47 - 64.6 -
9 7264 2.5 0.25 0.15 0.2 0.42 - 78.5 -
10 7264 2.5 0.25 0.125 0.16 0.42 - 69.2 -
11 7264 2.5 0.25 0.125 0.24 0.46 - 75.1 -

The change in temperature computed by the CFD model is used to compute the heat gain per
meter of bundle length Q and the logarithmic mean temperature difference (LMTD) in the form
( ) ( )
[ ]
out
wall
in
wall
in
out
T
T
T
T
T
T
LMTD
−
−
−
=
ln
, (3)
where in all cases Twall = 750 °K and Tin = 800 °K. The heat transfer coefficients are then
computed as
LMTD
A
Q
h
total
= , (4)
where Atotal is the total surface area of the tubes and fins. The heat transfer coefficients are non-
dimensionalized to obtain Nusselt numbers in the form:
k
L
h
Nu = , (5)
where, following Gnielinski, et al [3], L = d for finned tubes and L = πd/2 for plain tubes.
The effect of each of the five parameters on both pressure drop and heat transfer is shown in
Figures 8 – 12. The effects rank in order with the figure numbers, with Reynolds number having
the greatest effect, followed by tube pitch and the fin parameters. Note that the effect of
acceleration through the tube bundle is already accounted for by the use of the velocity in the
narrowest gap to compute the dynamic pressure used to compute Eu. Therefore, the pitch effect
shown in Figure 9 is independent of the effect of acceleration through the narrowest gap.

Figure 8
Dependence of Pressure Drop and Heat Transfer on Reynolds Number
Figure 9
Dependence of Pressure Drop and Heat Transfer on Tube Pitch

Figure 10
Dependence of Pressure Drop and Heat Transfer on Fin Height
Figure 11
Dependence of Pressure Drop and Heat Transfer on Fin Pitch

Figure 12
Dependence of Pressure Drop and Heat Transfer on Fin Thickness
Comparison with Correlation Equations
The CFD results are compared with values of Eu and Nu computed from correlation equations
provided in references [2], [3], and [4]. The comparisons are presented in Table 4 as ratios of the
values computed from the CFD models (see Table 3) to the values computed from the correlation
equations. The results show the CFD pressure drop calculations to be on the order of 20-30%
lower than those obtained using the correlation equations. The CFD heat transfer values are on
the order of ±20% for the finned tube cases and +40% for the plain tube cases, with respect to
those given by the correlation equations. In all cases the correlation equations include
adjustments for the relatively small number of tube rows in this bundle. However, in the case of
the heat transfer calculation for the finned tubes from [3] this adjustment is rather rough (factor
of ½ on the first tube and ¾ on the second tube, as compared with tubes deeper in the bundle).
The level of agreement obtained between the CFD pressure drop results and the correlation
equations is acceptable. As described below, further mesh refinement would improve the
agreement by approximately 12%. The agreement between the correlation equations and the
underlying data is not known but deviations on the order of the deviation between the CFD
results and those from the correlation equations are typical of the point-to-point agreement
between these equations and data.

The level of agreement between the CFD heat transfer results and the correlation equations is
acceptable for the finned tube cases. In fact, the two correlations considered bracket the CFD
results. The agreement is not as good for the plain tube cases. This may be due to a combination
of turbulence modeling issues and mesh refinement (see below). Since the major interest of this
work is in the simulation of finned tubes, the results obtained here are encouraging.
Table 4
Comparison of CFD Results with Correlation Equations
Eu Ratio
(CFD/Correlation) [2]
Nu Ratio
(CFD/Correlation) [3] (CFD/Correlation) [4]
w/ fins w/o fins w/ fins w/o fins w/ fins w/o fins
0.73 0.80 1.27 1.45 0.87 1.38
0.83 0.89 1.46 1.57 1.00 1.51
0.64 0.81 1.06 1.34 0.72 1.29
0.71 0.85 1.29 1.40 0.88 1.35
0.75 0.74 1.25 1.48 0.85 1.40
0.73 - 1.36 - 0.89 -
0.73 - 1.18 - 0.83 -
0.74 - 1.17 - 0.82 -
0.73 - 1.32 - 0.88 -
0.70 - 1.21 - 0.82 -
0.76 - 1.31 - 0.89 -

Sensitivity Studies
Mesh Sensitivity
The sensitivity of the solution to mesh refinement was tested by constructing a 2D version of the
plain tube version of Case 1. The mesh was refined until the computed pressure drop stabilized.
The results are shown in Table 5. The pressure drop stabilizes when the mesh size is dropped
from 1 mm to 0.75 mm. Further decrease to 0.5 mm has only minor effect. The results with a 1
mm mesh are 9% below the stabilized value. The results for the 3D model with a 1 mm mesh
are 12% below the stabilized value. This difference between the 2D and 3D models is most
likely due the fact that while the mesh is nominally 1 mm in the 3D model, the actual mesh
ranges from 0.5 mm to 2 mm. Based on these results we can conclude that the solutions reported
herein are nearly, but not completely, mesh independent. Since element count scales with the
cube of mesh size, it is expected that the 3D model would need approximately 4.8 million
elements to achieve complete mesh independence.
Table 5
Mesh Refinement Study Results
Mesh
Size
Pressure Drop over 6 Plain Tubes
(Case 1)
2D Model 3D Model
2 mm 184.7 Pa -
1 mm 218.3 Pa 209.8
0.75 mm 241.5 Pa -
0.5 mm 239.7 Pa -
Turbulence Modeling
Due to the separated wakes that occur in flow through a tube bundle, it is expected that the
details of turbulence modeling will be important to the accuracy of the results. All of the
solutions reported herein have been obtained using the standard k-ε model. This is the default
model for most industrial problems. While this model is not preferred for calculation of flow
over a single plain tube at low Reynolds numbers, the constraint imposed by adjacent tubes and
fins makes this model an acceptable first choice for analyzing finned tube bundles. Work with
other turbulence models is ongoing. In particular, since FIV simulations are inherently time-
dependent, use of large eddy simulation (LES) techniques will be evaluated.

Of even greater importance than the turbulence model selected is the choice of wall functions.
The standard wall functions make use of the “Law of the Wall” [5], assuming a universal shape
for the boundary layer. Use of this model is valid for meshes with a near-wall spacing given by
30 < y+
< 300,
where y+
is the non-dimensional distance from the wall to the center of the first element adjacent
to the wall. In the tube bundle models y+
is on the order of 5, meaning that the assumption
behind the use of the standard wall functions is violated. For this reason, FLUENT’s enhanced
wall functions [1, pp. 12-68 – 12-71] were used in all calculations.
In order to evaluate the improvement gained by using the enhanced wall functions, Case 1-11
were run using both the standard and enhanced wall functions and the results compared. This
comparison is presented in Table 6 below. This table shows the ratio of the Eu and Nu results
computed with enhanced wall functions to those obtained with standard wall functions. The
largest change is seen in the calculation of pressure drop for the finned tubes, where an
improvement of approximately 20% is obtained using the enhanced wall functions. Since these
are the configurations that we are most interested in, the benefits of using the enhanced wall
functions are clear.
Table 6
Comparison of Enhanced and Standard Wall Functions
Ratios (Enhanced/Standard)
Eu Ratio Nu Ratio
w/ fins w/o fins w/ fins w/o fins
1.21 1.06 1.11 1.10
1.08 1.03 1.07 0.98
1.31 1.06 1.19 1.17
1.30 1.06 1.13 1.13
1.14 1.06 1.11 1.08
1.17 - 1.13 -
1.19 - 1.12 -
1.20 - 1.12 -
1.20 - 1.12 -
1.16 - 1.13 -
1.20 - 1.14 -

Conclusions
The CFD modeling of finned tube bundles shows promising results for calculation of pressure
drop over the bundle and thus for drag on individual tubes. The heat transfer results agree well
with correlation equations for finned tubes, though the comparison is not as good for plain tubes.
Application of an alternative turbulence model may improve the plain tube heat transfer results.
The goal of using models such as these for calculation of FIV in tube bundles is obtainable.
Based on the findings of the mesh refinement study, a mesh of 4.8 million elements would be
required to obtain a mesh-independent solution to a 3D model of a single column of six tubes.
This could be reduced to 4 million elements by shortening the approach mesh without impairing
the accuracy of the simulation. Judicious grading of the mesh could reduce this total further,
perhaps to 3.5 million elements. Using this number, for an FIV model encompassing 6 tube
rows and 6 columns, the minimum size needed to simulate tube motion, the total element count
would be approximately 20 million. This would be a large model that would need to be solved
for transient flow. However, given a moderately sized computing cluster, on the order of 32-64
processors, solutions could be obtained over a number of days. While analysis of finned tube
bundle FIV is not yet a turn-key engineering calculation, the tools for accurate 3D simulation are
currently available and can be applied with the use of a sufficiently large computational cluster.
Acknowledgements
We would like to thank Harbi Pordal and Anup Paul for their advice concerning the development
of the CFD models used in this work.
References
1. FLUENT 6.3 Users Manual, ANSYS, Inc., 2006.
2. Zukauskas, A. and Ulinskas, R. in Hewitt, G. F. ed., Handbook of Heat Exchanger Design,
New York, New York: Begell House, Inc., 1992, pp. 2.2.4-1 – 2.2.4-17.
3. Gnielinski, V., Zukauskas, A. and Skrinska, A., in Hewitt, G. F. ed., Handbook of Heat
Exchanger Design, New York, New York: Begell House, Inc., 1992, pp. 2.5.3-1 – 2.5.3-16.
4. Hewitt, G. F., Shires, G. L., and Bott, T. R., Process Heat Transfer, Boca Raton, Florida:
CRC Press, Inc., 1994, pp.73 – 88.
5. Hinze, J. O., Turbulence, 2nd
ed., New York, New York: McGraw-Hill, 1975, pp. 586-770.

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