JGPP Vol. 6, No. 3, TP 2

International Journal of Gas Turbine, Propulsion and Power Systems
December 2014, Volume 6, Number 3
Conjugate Heat Transfer Analysis of Convection-cooled Turbine
Vanes Using γ-Reθ Transition Model
Gang Lin1, Karsten Kusterer1, Anis Haj Ayed1, Dieter Bohn2, Takao Sugimoto3
1
B&B-AGEMA GmbH
Jülicher Strasse 338, 52070 Aachen, Germany
E-mail: [email protected]
2
RWTH Aachen University
3
University of Hyogo
conduction, it is obvious that a coupled calculation of the fluid, the
heat transfer and the heat conductivity in the solid body can lead to
a higher accuracy in the design process.
A little bit more than 20 years ago, in the early 1990s, the
Institute of Steam and Gas Turbines at RWTH Aachen University
headed by Prof. Dieter Bohn started a long-term development of a
most sophisticated approach for the coupled calculation of fluid
flows and heat transfer with focus on the hot gas components in a
gas turbine. The numerical group of the institute developed the
homogeneous method for the conjugate calculation technique
(CCT). The method involves the direct coupling of the fluid flow
and the solid body using the same discretization and numerical
principle for both zones. This makes it possible to have an
interpolation-free crossing of the heat fluxes between the
neighboring cell faces. Thus, additional information on the
boundary conditions at the blade walls, such as the distribution of
the heat transfer coefficient, becomes redundant, and the wall
temperatures as well as the temperatures in the blade walls are a
direct result of this simulation. First results and validation cases
have been published in the 1990s for convection-cooled cases (e.g.
[3-6]) as well as for film-cooled configurations (e.g. [7-9]). The
detailed description of the conjugate calculation technique and its
validation is provided by Bohn et al. in [10].
ABSTRACT
In order to achieve high process efficiencies for the economic
operation of stationary gas turbines and aero engines, extremely
high turbine inlet temperatures at adjusted pressure ratios are
applied. The allowable hot gas temperature is limited by the
material temperature of the hot gas path components, in particular
the vanes and blades of the turbine. Thus, intensive cooling is
required to guarantee an acceptable life span of these components.
Modern computational tools as well as advanced calculation
methods support essentially on the successful design of these
thermally high-loaded components. The homogeneous, or
sometimes also mentioned as “full”, conjugate calculation
technique for the coupled calculation of fluid flows, heat transfer
and heat conduction is such an advanced numerical approach in the
design process and huge experiences on validation and application
have been collected throughout the years. This paper summarizes
the numerical approach for this method as well as provides a
collection of numerical results obtained by the authors for
validation cases for a convection-cooled turbine vane test case as
well as comparison to calculation data for this test case provided in
open literature. Furthermore, systematic studies on the impact of
calculation parameters, e.g. hot gas fluid properties, and numerical
models for turbulence calculation are performed and the numerical
results are compared to the experimental results of the test case.
NOMENCLATURE
INTRODUCTION
Due to the necessity of cooling technologies in modern gas turbines, turbulent heat transfer is of significant importance in the
thermal design process of the cooled components. Tremendous efforts have been put into the determination of empirical correlations
for the internal and external heat transfer, which are necessary for
the conventional design process. Here, analysis of turbine blade
cooling and heat transfer consists of three areas (Patankar, [1]): (a)
prediction of the heat transfer coefficients on the external surface
of the airfoil (Kays and Crawford, [2]), (b) prediction of heat
transfer in the internal cooling passages (Kays and Crawford, [2])
and (c) calculation of the temperature distribution in the blade material.
However, the accuracy of the approach based on heat transfer
coefficients is very much limited by the uncertainties of the
correlations if applied to the real gas turbine geometric
configurations and conditions. With regard to the inter-relations
between the external fluid flow, the internal fluid flow and the heat
Flength
Fonset
H
H0
L
Ma
k
Re
Rec
Ret
~
Ret
S
T
Tref
X
y+
γ
μ
μt
Manuscript Received on November 20, 2013
Review Completed on November 26, 2014
= function to control transition length
= function to control transition onset location
= heat transfer coefficient (W/m2/K)
= reference heat transfer coefficient (1135 W/m2/K )
= axial chord length (m)
= Mach number
= turbulent kinetic energy (m2/s2)
= momentum thickness Reynolds number
= momentum thickness Reynolds number, where the
intermittency starts to increase
= momentum thickness Reynolds number, where the skin
friction starts to increase
= transported variable for Ret
= streamwise distance
= temperature (K)
= reference temperature (811K)
= axiale chord position (m)
= non-dimensional wall-normal distance,
= intermittency
= dynamic viscosity (kg/m/s)
= turbulent viscosity (kg/m/s)
Copyright © 2014 Gas Turbine Society of Japan
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JGPP Vol.6, No. 3
ρ
ω
= density (kg/m3)
= specific turbulence dissipation rate (s-1)
mentioned, presented results from the CCM+ solver have been
obtained by application of this model.
TEST CASE DESCRIPTION
The famous Mark II test case for a convection-cooled vane has
been chosen for comparative calculations of the thermal load by
application of the CCT. The vane has been investigated extensively
by Hylton et al. [11] over a wide range of operating conditions in a
hot gas duct. Mark II is a high-pressure turbine nozzle guide vane,
which is convectively cooled with air by ten radial cooling channels.
Figure 1 shows the vane geometry and the arrangement of the
cooling passages. The test case no. 5411 has been chosen for the
numerical investigations. The numerical validations have been
carried out in both 2-D and 3-D cases. In 2-D case the heat transfer
coefficients and cooling air temperatures have been defined as
boundary conditions for calculation of heat transfer from internal
cooling air. In 3-D case the heat transfer from internal cooling air to
solid is calculated by given cooling air mass flow and cooling air
inlet temperature. After Hylton et al. [11] the inlet turbulence
intensity was 6.5% in the experiment, which has also be used in the
numerical investigations. Instead of turbulence length scale the
turbulent viscosity ratio has been applied as another inlet boundary
condition for turbulence conditions. The inlet turbulent viscosity
ratio has been set at 10. In the study by Mansour et al. [12] the inlet
turbulent viscosity ratio has insignificant impact on the heat
transfer coefficient distribution prediction at suction side in the case
Mark II.
Fig. 2 Mid-section of 3D polyhedral mesh for CCM+ application
The SST turbulence model [15] is a two-equation turbulence
model, which combines the advantages of both k-ω and k-ε model.
With the help of a blending function F1, the turbulence model can
be transformed between k-ε model in the freestream zone and κ-ω
model in the boundary layer zone. With this method, the advantages
of near-wall performance of k-ω model can be implemented
without any influence on sensitivity of any boundary condition in
free stream, which can lead to errors in the original k-ω model. The
equations of two transport models, one for turbulent kinetic energy
k and the other for specific turbulence dissipation rate ω, are given
as follows:

u j 
Dk
 
k  
2

   k 
    S  k


t 
Dt  x j 
 xj 
x j   eff t



 
min max 

eff


,0.1 ,1   k
The own-developed solver, CHTflow, is based on the
homogeneous method as it is described in [10]. Conjugate
calculations have been performed with structured hexahedral grids
for 2-D case. The y+ in the boundary layer is approximately y+=1.
The Baldwin-Lomax algebraic turbulence model [13] has been
applied. The model has been established during the validation
procedure of the CHTflow code in the mid 1990s and the results
have been presented in [4, 10]. These results are taken as the
reference for the present calculations with commercial CFD
software. In the recent years, this kind of homogeneous conjugate
methodology found its way also into the commercial CFD codes
and, thus, the Star CCM+ solver [14] has been applied for the new
calculation and comparison to the CHTflow results. Figure 2 shows
the mid-section of the 3-D polyhedral mesh, which has been used
for the CCM+ application. In order to establish an appropriate
calculation of the local heat transfer, the y+ value of the first cell in
the boundary layer is less than y+=1. The suitable cell number of the
grid after mesh independency study is 2.86 million including the
prism layers on the fluid-solid boundaries. Best results have been
obtained with the SST k-ω turbulence model and if not otherwise



uj 
D
 
 
2
   2 
    
     S  k


t 
t


Dt  x j 
x j  
x j 


2  1  F 1  2
Fig. 1 Geometry and boundary conditions for Mark II test case


1 k 
  xj  xj


Usually the γ-Reθ transition model is applied in the SST
turbulence model [15]. This transition model is build up by two
transport equations: one is for the intermittency γ and the other is
for the transported transition momentum thickness Reynolds
~ .
number R
et

    
 
D

 t

S 
Dt  x j 
    x j  F length ca1

 ca 2  F turb 1  ce 2  


10

F  1  c  
0.5
onset
e1

JGPP Vol.6, No. 3

~
~
D Ret
 Ret 
U 2 (
 


 t   
  ct
t 
Dt
 xj 
500 Ret
x j 



~
Ret )1  F t 



The intermittency γ determines the production of turbulent
kinetic energy in the boundary layer. It is 0 in the laminar boundary
~ is used as the criterion
layer and 1 in the fully turbulent layer. R
et
of transition onset position, which transforms non-local free stream
information into a local quantity. So that the function Fonset (shown
in eq. 3), which is used to control transition onset location, can be
expressed.
Open literature numerical data by other authors for this test
configuration can be found in [16-19], for example, as well as
numerical data [20-24] for the similar C3X test case from Hylton et
al. [11]. The same test configuration Mark II has been studied by
Yan et al. [16] using a structured mesh in the commercial solvers
CFX and Fluent with different turbulence models. The results of
SST Gamma Theta calculation in CFX solver have shown a good
agreement with the experimental data in temperature distribution.
The temperature at the reattachment point on suction side was over
predicted by about 6% to 7%. Lin et al. [17] have done the analysis
of this test case also in the CFX solver but using unstructured mesh
with the SST-γ-Reθ model. The numerical results capture the trends
of temperature along the surface with an under prediction by about
5% to 8% on the pressure side. Zeng and Qing [18] have completed
another 2D and 3D conjugate heat transfer simulation of Mark II
with unstructured mesh. The grid in the 3D calculation was
obtained by extruding of the 2D grid. The boundary conditions in
the cooling channels, for both 2D and 3D simulation, were applied
with cooling temperature and heat transfer coefficient. The SSTγ-Reθ model again showed the best accuracy among all of the used
turbulence models. The maximum difference between prediction
and test data occurred at the reattachment point with about 3% to
4% over prediction.
a) CHT flow result
RESULTS
Both applied numerical codes show reasonable results for the
Mach-number distribution in the mid-section of the vane as it is
compared in Fig. 3. There is a strong and sharp compression shock
on the front part of the vane suction side. In front of the shock the
flow is accelerated up a Mach-number of slightly over Ma=1.6.
Downstream the strong shock, the flow is accelerated again, before
a second, weaker shock can be observed shortly in front of the
trailing edge.
b) CCM+ result
Fig. 3 Mach-number distribution (mid-section)
The temperature distribution in the mid-section for the fluid
region as well as for the solid body of the vane is shown in Fig. 4.
The lowest temperature inside the vane appears near the leading
edge between the 2nd and the 3rd cooling hole. The hottest region
can be found in the trailing edge. As shown in the distribution of
isotherms in the vane, the temperature gradients are high at two
positions: One near the stagnation point at the leading edge and the
other at the suction side after the first shock. Such a high density of
the isotherms indicates locally a high heat transfer between the
fluid and solid region.
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JGPP Vol.6, No. 3
Fig. 5 Comparison of numerical and experimental results of the
vane surface temperature
a) CHT flow result
The effect of turbulence models
As shown in Fig. 6, the results obtained by application of CCM+
depend on the chosen turbulence model in the 2-D cases. The SST
k-ω model provided by the CCM+ code performs the best for the
conjugate application. Applying other turbulence models, the
calculated temperature distributions in the front part of the vane
suction side, i.e. the region with a laminar boundary layer, show a
significantly reduced accuracy. The major reason for this is the fact,
that those turbulence models provide no possibility to prescribe or
influence the turbulence onset location. The results have shown that
for all tested turbulence models the onset of turbulence production
is located to far upstream. Figure 7 illustrates this for the Realizable
k-ε model. Compared to the calculation with the SST-γ-Reθ model,
the Realizable κ-ε model starts much earlier (by X/L=0.044) to over
predict the turbulence at the suction side, which results in too much
heat flux in this laminar region. By the calculation with the
SST-γ-Reθ model, the turbulence is firstly produced at position by
X/L=0.45, which is the same as in the experiment. Although the
SST-γ-Reθ model can predict a quite well transition at the suction
side, the temperature distribution at the pressure side is even worse
predicted than the Realizable k-ε model. As shown in Fig. 7, the
boundary layer at the pressure side, near the trailing edge, is still
laminar in the calculation with SST-γ-Reθ model, while the other
calculation results are turbulent. The main reason behind this effect
is that the value of Reθt,min in the transition model, which is suitable
for the suction side, is too high for the pressure side. A value of 20
is recommended as the lower bound for Reθt,min, and this leads also
to minimum reachable Reθc. It is clear from the model equations
that decreasing Reθc results in an earlier transition onset. Figure 8
shows the surface temperature prediction with variation of Reθt,min.
The result with prescribed Reθt,min, which equals 130, agrees very
well with the test data at the suction side before the first shock.
However, a smaller Reθt,min shows better agreement for the pressure
side. The increase of Reθt,min results in a later transition at both sides
of the vane, which in turn leads to further under prediction at
suction side before the first shock and also at the pressure side.
b) CCM+ result
Fig. 4 Temperature distribution (mid-section)
Figure 5 gives a comparison of the obtained numerical surface
temperature results with the experimental results. The CHTflow
results for 2-D case show excellent agreement with the
experimental results. The CCM+ results for the suction side are also
very good in agreement with the experimental results, but on the
pressure side the surface temperatures are calculated consistently
lower than the experimental results. The 3-D CCM+ results have
improved significantly with only small deviations compared to the
experimental results along the whole surface. Here, the deviation
between 2-D and 3-D results comes from the different internal
cooling air boundary conditions and also the influence of secondary
flow.
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JGPP Vol.6, No. 3
0,9
while the measured surface temperatures have been used as
boundary conditions. Thus, the uncertainties regarding the
accuracy of the experimental heat transfer coefficients are much
higher than for the experimental surface temperatures.
Experiment
0,85
V2F
Realizable k-epsilon
SST_GammaReTheta
T/Tref [-]
0,8
CHTflow
0,9
0,75
0,85
0,7
0,6
0,55
Re_theta=100
0,75
Re_theta=200
T/Tref [-]
0,65
Experiment
0,8
Re_theta=130
0,7
0,65
-1
-0,8
-0,6
-0,4
-0,2
0
0,2
0,4
0,6
0,8
1
0,6
X/L [-]
0,55
Fig. 6 Comparison of numerical results with different turbulence
models and experimental results of the vane surface temperature
0,5
-1
-0,8
-0,6
-0,4
-0,2
0
X/L [-]
0,2
0,4
0,6
0,8
1
Fig. 8 Temperature distribution under different Reθt,min (2-D result)
The y axis in Fig. 9 shows the heat transfer coefficient normalized
with respect to H0, which is 1135 W/m2/K. Similar to the
temperature prediction, the results of Realizable κ-ε model and V2F
model also show very well agreement with the test data at the
pressure side, but much higher heat transfer coefficient at the
laminar region of suction side. In contrast, the result of SST-γ-Reθ
model agrees very well before the transition at suction side but an
under prediction at pressure side. Comparing the test data, there is
another large error of all turbulence models at the suction side after
the first shock. It is reattachment of the separated flow after the
transition, which causes a high heat flux in this reattachment point
on the blade. Although the SST-γ-Reθ model can predict the right
onset location of transition, the temperature error of calculation is
still around 20K (2.5%) at this point. Some possible reasons for this
error in the SST-γ-Reθ model are as follows: (1) at present transition
model, the Mach number effect is still not been implemented [25];
(2) in this transition model the local intermittency can exceed 1,
when the laminar boundary layer separates. This leads to a large
production of turbulent kinetic energy to get an earlier reattachment,
but at the same time also results in a locally high heat flux.
a) Realizable k-ε model
1,4
Test Data
V2F
1,2
Realizable k-epsilon
SST-GammaReTheta
H/H0 [-]
1
CHTflow
0,8
0,6
0,4
b) SST-γ-Reθ model
0,2
Fig. 7 The midspan turbulence viscosity ratio distribution under
different turbulence models (2-D result)
0
-1
-0,8
-0,6
-0,4
-0,2
0
0,2
0,4
0,6
0,8
X/L [-]
In Fig. 9, the vane surface heat transfer coefficient distributions
at midspan taken from calculation with different turbulence models
are compared to heat transfer coefficient published by Hylton et al.
[11]. The latter values have not been obtained by measured, but by
application of an FEM tool solving the energy equation in the solid,
Fig. 9 Heat transfer coefficient distribution under different
turbulence models (2-D result)
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JGPP Vol.6, No. 3
References
[1] Patankar, SV., 1994, “Advances in Numerical Prediction of
Turbine Blade Heat Transfer,” In Heat Transfer in Turbomachinery,
RJ. Goldstein, DE. Metzger and AI Leotiev (eds). Begell House,
Inc., New York and Wallingford, pp. 501-510.
[2] Kays, W. M., and Crawford, M. E., 1980, Convective Heat and
Mass Transfer, McGraw-Hill Book Company: 2nd edition.
[3] Bohn, D., Bonhoff, B., Schönenborn, H., 1995, “Combined
Aerodynamic and Thermal Analysis of a High-pressure Turbine
Nozzle Guide Vane,” IGTC-108, Yokohama, Japan.
[4] Bohn, D., and Schönenborn, H., 1996, “3-D Coupled
Aerodynamic and Thermal Numerical Analysis of a Turbine
Nozzle Guide Vane,” 4th ISHT-conference, Beijing, China.
[5] Bohn, D., Heuer, T., Kusterer, K., and Lang, G., 1997,
“Application of a Conjugate Fluid Flow and Heat Transfer Method
in the Thermal Design Process of a Convection-Cooled Turbine
Nozzle Guide Vane,” 99-AA-5, ASME Asia 1997, Singapore.
[6] Bohn, D., Becker, V., Kusterer, K., Otsuki, Y., Sugimoto, T., and
Tanaka, R., 1999, “3-D International Flow and Conjugate
Calculations of a Convective Cooled Turbine Blade with
Serpentine-Shaped and Ribbed Channels,” ASME Turbo Expo
1999, 99-GT-220, Indianapolis, USA.
[7] Bohn, D., and Bonhoff, B., 1994, “Berechnung der Kühl- und
Störwirkung eines filmgekühlten transsonisch durchströmten
Turbinengitters mit diabaten Wänden, ” VDI-Berichte 1109, S.261
– 275.
[8] Bohn, D., Bonhoff, B., Schönenborn, H., and Wilhelmi, H.,
1995, “Validation of a Numerical Model for the Coupled
Simulation of Fluid Flow and Diabatic Walls with Application to
Film-cooled Gas Turbine Blades,” VDI-Berichte-Nr. 1186, pp.
259-272.
[9] Bohn, D., and Kusterer, K., 1998, “3-D Numerical
Aerodynamic and Combined Heat Transfer Analysis of a Turbine
Guide Vane with Showerhead Ejection,” Heat Transfer
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Korea.
[10] Bohn, D., Krüger, U., and Kusterer, K., 2001, “Conjugate Heat
Transfer: An Advanced Computational Method for the Cooling
Design of Modern Gas Turbine Blades and Vanes,” In Heat
Transfer in Gas Turbines, B. Sundén and M. Faghri (eds). WIT
Press, Southampton, pp. 58-108.
[11] Hylton, L. D., Milhec, M. S., Turner, E. R., Nealy, D. A., and
York, R. E., 1983, “Analytical and Exp. Evaluation of the Heat
Transfer Distribution Over the Surfaces of Turbine Vanes”,
NASA-Report. CR 168015, NASA Lewis Research Center,
Cleveland, Ohio.
[12] Mansour, M. L., Hosseini, K. M., Liu, J. S., Goswami, S., 2006,
“Assessment of the Impact of Laminar-Turbulent Transition on the
Accuracy of Heat Transfer Coefficient Prediction in high Pressure
Turbines,” Proceedings of ASME Turbo Expo 2006,
GT2006-90273, Barcelona, Spain.
[13] Baldwin, B. S., and Lomax, H., 1978, “The Layer
Approximation and Algebraic Model for Seperated Flows,”
AIAA-paper 78-257.
[14] STAR-CCM+ Software Package, Version 6.04, CD-adapco,
Melbillr NY.
[15] Malan, P., Suluksna, K., and Juntasaro, E., 2009, “Calibrating
the γ-Reθ Transition Model for Commercial CFD,” AIAA-092407.
[16] Yan, P. G., and Wang, Z. F., 2010, “Conjuate Heat Transfer
Numerical Validation and PSE Analysis of Transonic
Internally-Cooled Turbine Cascade,” Proceedings of ASME Turbo
Expo 2010, GT2010-23251, Glasgow, UK.
[17] Lin, L., Ren, J., and Jiang, H. D., 2011, “Application of γ-Reθ
Transition Model for Internal Cooling Simulation,” Heat
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Transfer of Turbine Cascades,” In Heat Transfer - Theoretical
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The impact of gas properties
The effect of gas properties on predicted surface temperatures
has been studied with two kinds of gases: (1) ideal gas with
molecular weight of 28.9664 kg/kmol and constant specific heat of
1003.62 J/kg/K; (2) real gas with molecular weight of 28.5705
kg/kmol and specific heat based on a temperature polynomial.
Figure 10 shows the surface temperature distributions at these two
conditions. The trends of temperature along the surface are
consistent with these two calculations, and the values are almost
identical. Only the results of real gas calculation show slightly
further over prediction than ideal gas calculation at the suction side
before the transition and at the reattachment point.
Fig. 10 Temperature distribution under different gas properties (3-D
result)
CONCLUSIONS
In recent years the conjugate heat transfer method has been
implemented in several commercial CFD codes. In the present
study, the solver Star CCM+ has been applied for calculation of a
convection-cooled high pressure test vane, Mark II, and the results
have been compared to experimental results and numerical results,
which have been obtained by application of the solver CHTflow.
Three different turbulence models have been chosen for the
calculations with Star CCM+, showing significant variations in the
surface temperature prediction, especially in laminar region at the
suction side. Due to the integrated transition model, only the
SST-γ-Reθ model tends to predict a right onset location of the
transition, which results in a good agreement with the test data in
this laminar region. For SST-γ-Reθ model, the maximum
temperature error between calculation results and test data occurs in
the reattachment point after the transition (2.5%), where the heat
flux is over estimated.
Compared with the 2D calculation, 3D calculation shows overall
better agreement with test data except at the reattachment point.
Other authors also published numerical results for the same test
case, using different numerical codes. Those results show larger
deviations to the experimental temperature distributions. Most of
their studies also found that the SST-γ-Reθ model presents the best
agreement for prediction of temperature distribution in the test case
Mark II. But none of them has investigated the influence of
parameter Reθt,min on the transition at airfoil suction side. The
CHTflow results of 2D case show overall a very good agreement
with the experimental data, which is comparable with the results of
SST-γ-Reθ model. The maximal error for calculation with CHTflow
is approximately 2%.
ACKNOWLEDGMENTS
The numerical simulations presented in this paper have been
carried out with the STAR CCM+ Software of CD-adapco. Their
support is gratefully acknowledged.
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JGPP Vol.6, No. 3
[19] Wang, Z. F., Yan, P. G., Huang, H. Y., and Han, W. J., 2010,
“Conjugate Heat Transfer Analysis of a High Pressure Air-Cooled
Gas Turbine Vane,” Proceedings of ASME Turbo Expo 2010,
GT2010-23247, Glasgow, UK.
[20] Luo, J., and Razinsky, E. H., 2006, “Conjugate Heat Transfer
Analysis of a Cooled Turbine Vane Using the V2F Turbulence
Model,” ASME Turbo Expo 2006, GT2006-91109, Barcelona,
Spain.
[21] Ledezma, G. A., Laskowski, G. M., and Tolpadi, A. K., 2008,
“Turbulence Model Assessment for Conjugate Heat Transfer in a
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15

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