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A NOVEL METHOD FOR THE COMPUTATION OF CONJUGATE HEAT TRANSFER WITH COUPLED SOLVERS

Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
A NOVEL METHOD FOR THE COMPUTATION OF CONJUGATE HEAT TRANSFER
WITH COUPLED SOLVERS
Verstraete Tom* and Rene Van den Braembussche
Turbomachinery Department
Von Karman Institute for Fluid Dynamics
Waterloosesteenweg 72, 1640 Sint-Genesius-Rode, Belgium
(* Corresponding author: verstraete@vki.ac.be)
ABSTRACT. This paper provides an overview of three commonly used methods for solving
conjugate heat transfer problems with different solvers for the fluid and solid domains. A fourth new
method with improved stability properties is introduced. A novel stability criterion is presented for all
coupled methods which allows selecting a suitable method for each application.
The results of the conjugate heat transfer in a flat plate are compared for the four different methods and
are validated with analytical solutions. Finally, the conjugate heat transfer is computed in a first stage
turbine blade with 5 internal cooling channels. The convergence history between two different methods
is compared.
NOMENCLATURE
BEM Boundary Element Method
CFD Computational Fluid Dynamics
CHT Conjugate Heat Transfer
FEM Finite Element Method
FFTB Flux Forward Temperature Back
FVM Finite Volume Method
hFFB heat transfer coefficient Forward Flux Back
hFTB heat transfer coefficient Forward Temperature Back
TFFB Temperature Forward Flux Back
INTRODUCTION
Two main strategies exist in solving the Conjugate Heat transfer (CHT) problem numerically,
depending on how the continuity of temperature and heat flux are imposed on the common walls
between the fluid and solid.
One approach integrates the entire set of equations in the fluid and solid as a single system and treats
the continuity of temperature and heat flux implicitly. The full coupled system of equations is solved
together. This approach, in literature referred to as the conjugate method, is computationally efficient,
but requires that both the fluid and solid are handled by a similar numerical approach and put together
into a unified framework.
A second approach calculates separately the flow and the thermal fields with a coupling provided by the
boundary conditions at the interface. This approach allows different stand-alone flow and solid
platforms to be used within an iterative procedure to obtain the continuity of temperature and heat flux.
The drawback of this approach, known as the coupled method, is the need for sequential iterations
between the two platforms and interpolation of the boundary conditions from one grid to the other.
Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
This paper provides an overview of three commonly used methods for the second approach. A novel
fourth approach is introduced. Due to the common problem of instabilities that occur during the
coupled CHT method [e.g. Divo 2003], a new stability criterion is presented that allows to understand
and predict the instabilities. Finally, some applications are given.
COUPLED METHODS
The characteristic time constant of the heat transfer in a fluid system is one order of magnitude smaller
than for the solid [e.g., Montenay et al. 2000]. This could lead to a slow convergence for conjugate
methods where many more iterations are needed in the solid domain to obtain convergence. A
decoupling of both computations, where the solid is solved by a steady state approach and the boundary
conditions at the fluid domain are updated only after several time steps, could lead to a faster
convergence of the entire system.
This method is mostly referred to as the coupled approach and uses different solvers for both domains.
In this method, the solid domain is solved by a steady state Finite Element Method (FEM) or Boundary
Element Method (BEM), which is more appropriate than a Finite Volume Method (FVM) with time
stepping to obtain the steady state solution. The fluid domain is solved by the FVM with explicit or
implicit schemes, and is interrupted after a number of time steps for an update of the boundary
conditions at the interface. For the flow solver, the solid seems to be calculated with an infinitely large
time step.
The main advantage of the coupled approach is that one can use standard solvers and grid generators
for each domain. Those codes have been extensively verified and their limitations and capabilities are
well known. In case the solid conduction is computed by a FEM, the same mesh can also be used for
the stress and vibration analysis which can make use of the CHT temperature results. Thermal stresses
can be computed straightaway and temperature dependent material laws can be used to compute the
stress resulting from centrifugal forces and pressure forces on the blade surface.
Next sections will discuss several coupling algorithms for the conjugate heat transfer analysis. The
name of the different methods refers to the transfer of quantities relative to the FVM.
The Flux Forward Temperature Back Method A first method of the coupled approach is the Flux
Forward Temperature Back (FFTB) method, in which the wall temperature distribution is imposed to
the fluid solver and the resulting heat flux distribution is imposed as a boundary condition to the solid
conduction solver. The latter predicts an updated temperature distribution at the fluid solver solid
boundaries. This loop is repeated until the temperature and heat flux are continuous between both
domains. A schematic overview of the method is given in Fig. 1.
Only few authors are found in the literature that use the FFTB method. Verdicchio et al. [2001] uses it
in an axisymmetric model for the prediction of the heat transfer in the internal cavities of turbine discs.
A relaxation coefficient of 0.3 is required for convergence. Illingworth et al. [2005] reports on the use
of the FFTB method between a commercial CFD code (FLUENT) and an in house FEM code for the
pre-swirl system of an aero engine.
The Temperature Forward Flux Back Method As an alternative to the FFTB method, one can also
impose the heat flux distribution as a boundary condition for the fluid computation and the resulting
wall temperature to the solid conduction solver. The updated heat flux is then returned as a boundary
condition to the fluid solver. This method is mostly referred to as the Temperature Forward Flux Back
(TFFB) method. A schematic overview is given in Fig. 2.
Divo et al. [2002, 2003] and Heidmann et al. [2000, 2003] report on the use of the TFFB method for the
computation of the conjugate heat transfer effects on a realistic filmcooled turbine blade. The method
uses an explicit FVM code and a Boundary Element Method (BEM) for the computation of the heat
transfer in the solid. The method uses a relaxation coefficient of 0.8.
Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
He et al. [1995] also uses a BEM method for the solid domain. However, their method differs from the
TFFB method as both FVM and BEM computations use the same temperature distribution at the
interface as a boundary condition. An update of the wall temperature is based on a weighted average of
the heat fluxes.
Figure 1. Flow chart of the FFTB method.
Figure 2. Flow chart of the TFFB method.
The Heat Transfer Coefficient Forward Temperature Back Method A third method uses the
convective heat transfer equation (Eqn. (1)) to update the boundary conditions at the FEM side. The
resulting wall temperature is returned to the FVM domain. The method therefore is called the heat
transfer coefficient forward temperature back method, or abbreviated the hFTB method. A flow chart of
the method is shown in Fig. 3.
(
)
fluidwallwall TThq
=
(1)
The method starts with an initial temperature distribution wall
T at the boundary of the flow solver. The
results of the NS computation are used to estimate the heat transfer coefficient h and the ambient fluid
temperature fluid
T. Substituting them in Eqn. (1) provides an implicit relation between wall
T and wall
q
that can be used as a boundary condition for the solid conduction computation. The advantage of using
Eqn. (1) as boundary condition is an automatic adjustment of wall
q as a function of the new wall
T. The
latter one is then returned to the fluid solver and the loop is repeated until convergence.
Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
Figure 3. Flow chart of the hFTB method.
Figure 4. Flow chart of the hFFB method.
The remaining problem is the definition of h and fluid
T from the NS solution. They also need to satisfy
Eqn. (1) in which wall
T is the imposed boundary condition and wall
q is the solution of the fluid solver.
However, there is only one equation for two unknowns. One possibility is to make an extra fluid flow
calculation with a different wall temperature [e.g. Verdicchio et al. 2001] or even an adiabatic one
(wall
q = 0, e.g. Montenay [2000]). Substituting the two solutions of wall
q in Eqn. (1) and assuming that
h and fluid
T remain unchanged provides Eqn. (2) defining h.
12
12
wallwall
wallwall
TT
qq
h
= (2)
fluid
T can then be calculated by Eqn. (1) as function of the imposed wall
T and corresponding wall
q. The
difficulty is to ensure a positive value of h on the entire solid wall because negative values of h would
make the conduction problem ill-posed (see Montenay [2000]). The latter are likely to occur in regions
where the heat flux is changing sign.
A simpler and more stable approach is by imposing a constant positive value of h. Following shows that
the value of h only influences the convergence rate and does not affect the final result. The boundary
conditions at the ith iteration are:
(
)
i
fluid
fvm
wall
fvm
wall TThq ii = (3)
(
)
i
fluid
fem
wall
fem
wall TThq ii = (4)
Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
ii fem
wall
fvm
wall TT =
+1 (5)
The first equation is used to compute, for a fixed value of h, the value of fluid
T as a function of the
values of fvm
wall
T and fvm
wall
q defined by the fluid computation. The second equation is the boundary
condition for the conduction calculation in the solid. This results in a new fem
wall
T and fem
wall
q on the solid
wall. The third equation defines the boundary condition for the next iteration of the fluid computation.
Subtracting Eqn. (3) from Eqn. (4) gives:
(
)
iiii fvm
wall
fem
wall
fvm
wall
fem
wall TThqq = (6)
and substituting Eqn. (5) results in:
(
)
iiii fvm
wall
fvm
wall
fvm
wall
fem
wall TThqq =+1 (7)
This means that, for 0h, if 0
1
+ii fvm
wall
fvm
wall TT also 0ii fvm
wall
fem
wall qq and 0ii fvm
wall
fem
wall TT .
Hence the value of h has no effect on the solution once the continuity of temperature and heat flux
between both domains is satisfied. It affects only the convergence history as can be seen from Eqn. (7).
A smaller value of h results in a larger change of the wall temperature between two successive
iterations, for a given flux difference. This leads to a faster convergence but may also lead to
divergence of the method, as will be explained in the section on stability. The choice of h is a trade off
between computational time and stability.
Amano et al. [1994], Montenay et al. [2000] and Verdicchio et al. [2001] use the hFTB method for the
computation of the conjugate heat transfer in an engine internal cavity with an axisymmetric model.
Lassaux et al. [2004] and Heselhaus et al. [1992] report on the use of the hFTB method for the 3D
conjugate heat transfer analysis of typical blades. In a later study, Heselhaus [1998] presents the results
of the hFTB method for an axial turbine guide vane convectively cooled by a multi-pass cooling
channel. The number of NS time steps per coupling step is not kept constant but an inventive scheme is
used to determine it. The NS computation is interrupted each time the average temperature change in
the cell next to the wall exceeds a threshold value or when a fixed number of time steps is performed
(250). This results in a small number of NS time steps at the start of the calculation, and a gradual
increase as the wall temperature converges. However, as will be shown in the next section, a reduction
of number of time steps tends to destabilize the hFTB method when the wall temperature boundary
condition changes rapidly. Consequently, it is better to use a large number of time steps at the
beginning of the coupling process.
The Heat Transfer Coefficient Forward Flux Back Method As an alternative to the hFTB method,
the quantity returned to the FVM domain can be a heat flux. This results in a novel method with
different stability properties and is called the hFFB method. The flow chart is given in Fig. 4.
Similar to the hFTB, the value of h does not affect the final result, but influences only the convergence.
The boundary conditions at the ith iteration are:
(
)
i
fluid
fvm
wall
fvm
wall TThq ii = (8)
(
)
i
fluid
fem
wall
fem
wall TThq ii = (9)
ii fem
wall
fvm
wall qq =
+1 (10)
Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
The first equation is used to compute, for a fixed value of h, the value of fluid
T in function of the value
fvm
wall
T and fvm
wall
q defined by the fluid computation. The second equation is the boundary condition for the
conduction calculation in the solid.
This results in a new fem
wall
T and fem
wall
q on the solid wall. The third equation defines the boundary
condition for the next iteration of the fluid computation. Subtracting Eqn. (8) from Eqn. (9) gives:
(
)
iiii fvm
wall
fem
wall
fvm
wall
fem
wall TThqq = (11)
and substituting Eqn. (10) results in:
(
)
iiii fvm
wall
fvm
wall
fvm
wall
fem
wall qq
h
TT =+1
1 (12)
This means that, for 0h, if 0
1
+ii fvm
wall
fvm
wall qq also 0ii fvm
wall
fem
wall TT and 0ii fvm
wall
fem
wall qq .
Hence the value of h has no effect on the solution once the continuity of temperature and heat flux
between both domains is satisfied. It affects only the convergence history.
STABILITY OF THE COUPLED METHODS
The stability of all four methods will be discussed in this section. Giles [1997] provides a stability
analysis of the solid-fluid coupling. The stability of a 1D model is analyzed by applying the stability
theory of Godunov and Ryabenkii [1964] on the discretized set of equations. Several simplifications are
made, such as a uniform grid on both sides of the interface and the omission of the convection terms in
the fluid domain. The latter one simplifies the fluid equations to the ones governing in the solid domain,
however with a much lower conductivity. Giles concludes that the key point for achieving numerical
stability is the use of Neuman boundary conditions (heat flux) for the structural calculation and
Dirichlet boundary conditions (temperature) for the fluid calculations. However, this does not
correspond to the stability behavior found in practice (Verstraete [2008]). The main reason is that Giles
uses a time marching technique (both explicit and implicit) in solid and fluid domains, and updates the
boundary conditions at every iteration. In practice a FEM or BEM method is used for the solid which
provides a steady state response to a given boundary condition and not a transient one as assumed by
Giles. Similarly, in the fluid domain the boundary condition is only updated after a given number of
time steps and not after each time step.
Heselhaus [1998] extended the theory of Giles by implementing convective boundary conditions for the
solid domain and investigated the stability behavior if boundary conditions are exchanged after 2 time
steps instead of one. It was shown that the method gains stability with increasing number of time steps.
In this paper, a new simplistic convergence criterion is derived, which is based on the physics of the
problem rather than on the discretized equations. However, it reveals the true nature of the divergence
problems. The main assumptions are based on the relation between temperature and heat flux at the
boundary for both solid and fluid domains.
Consider the 1D conjugate heat transfer problem sketched in Fig. 5. A temperature Ts is specified at one
boundary of a solid, while at the other wall a fluid flows, and thus heat is transferred by convection.
Suppose the fluid temperature fluid
T and the heat transfer coefficient h are known. The problem consists
in finding the wall heat flux wall
q and temperature wall
T at the interface.
Following equations define the 1D conjugate heat transfer problem:
()
walls
s
wall TT
L
q=
λ
on s (13)
(
)
fluidwallwall TThq
=
on f (14)
The solution of this simple problem is given by Eqn. (15):
Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
Bi
TBiT
Tfluids
wall
+
+
=1 (15)
with
s
hL
Bi
λ
= (16)
the Biot number.
Figure 5. The one-dimensional conjugate heat transfer problem.
Stability of the FFTB method In the FFTB method a first guess 0
wall
T of the wall temperature is used
to solve the fluid domain. Suppose this initial guess differs by a value 0
from the correct wall
temperature:
0
0
α
+= wallwall TT (17)
The heat flux according to this wall temperature is then:
(
)
()
0
0
00
α
α
+=
+=
=
hq
hTTh
TThq
wall
fluidwall
fluidwallwall
(18)
This heat flux is then imposed to the solid heat transfer equation and results in an update of the wall
temperature:
321
1
0
0
01
α
α
λ
α
λ
λ
BiT
hL
q
L
T
q
L
TT
wall
S
wall
S
S
wall
S
Swall
=
=
=
(19)
At the ith iteration the temperature equals to:
(
)
i
wall
i
wall BiTT += 0
α
(20)
and the heat flux equals to:
(
)
hBiqq i
wall
i
wall += 0
α
(21)
The wall temperature converges to wall
T if and only if
1<Bi (22)
Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
thus for cases with a higher thermal gradient in the fluid than in the solid. As can be seen from Eqn.
(20), the convergence to the correct temperature will be faster for smaller Biot number.
Subtracting the
()
th
i1 wall temperature from the ith temperature results in:
(
)
(
)
BiBiTT i
i
wall
i
wall +=
1
1
0
1
α
(23)
and for the wall heat fluxes this results in:
(
)
11 =i
wall
i
wall
i
wall
i
wall TThqq (24)
The wall heat flux converges together with the wall temperature, but at a rate defined by h. If the FFTB
method is stopped with a difference T
Δ
in temperature, the difference in heat flux will be equal to
Th Δ.
Stability of the TFFB method The TFFB method starts with a wall temperature guess to solve the
solid domain. Let 0
wall
T be that guess and suppose it differs from the correct wall temperature by a value
0
:
0
0
α
+= wallwall TT (25)
The heat flux according to this wall temperature is then:
()
()
0
0
00
α
λ
α
λλ
λ
+=
+=
=
L
q
L
TT
L
TT
L
q
S
wall
S
wallS
S
wallS
S
wall
(26)
This heat flux is then imposed to the fluid domain and results in an update of the wall
temperature:
{
1
0
0
0
1
α
α
λ
α
Bi
T
hLh
q
T
h
q
TT
wall
Swall
fluid
wall
fluidwall
=
+=
+=
(27)
At the ith iteration the temperature equals to:
i
wall
i
wall Bi
TT
+= 1
0
α
(28)
and the heat flux equals to:
h
Bi
qq
i
wall
i
wall
+= 1
0
α
(29)
The wall temperature converges to wall
T if and only if
1>Bi (30)
The convergence criterion is the opposite to the one of the FFTB method and converges for cases
with a higher thermal gradient in the solid than in the fluid. The convergence to the correct
temperature will be faster for larger Biot numbers.
Subtracting the
()
th
i1 wall temperature from the ith temperature results in:
Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
+
=
BiBi
TT
i
i
wall
i
wall
1
1
11
0
1
α
(31)
and for the wall heat fluxes this results in:
()
11 =i
wall
i
wall
S
i
wall
i
wall TT
L
qq
λ
(32)
Opposed to the FFTB method, the thermal resistance in the solid L
S/
λ
determines the heat flux
difference for a given temperature difference at convergence.
Stability of the hFTB method The hFTB method differs from both previous methods by the
convective boundary conditions used for the solid domain. In order to avoid confusion, h
~will be used
as the heat transfer coefficient for the method (not to be confused with the real heat transfer coefficient
h) and fluid
T
~
will be used for the ambient fluid temperature for the solid boundary condition (not to be
confused with the real ambient fluid temperature fluid
T).
Suppose the initial guess of the wall temperature differs from the correct wall temperature by a value
0
α
:
0
0
α
+= wallwall TT (33)
The heat flux for the fluid domain will be:
(
)
0
00
α
+=
=
hq
TThq
wall
fluidwallwall (34)
As explained in the section on coupled methods, the fluid temperature 0
~
fluid
T given to the solid boundary
equation for a fixed h
~ coefficient is computed as:
+=
++=
+=
iB
Bi
T
T
h
h
h
q
T
h
q
T
fluid
wall
wall
wall
wall
fluid
~
1
~
~~
~
~
0
00
0
0
0
α
αα
(35)
with
h
q
TT wall
wallfluid ~
~= (36)
and
S
Lh
iB
λ
~
~= (37)
respectively the fluid temperature and Biot number if the heat transfer coefficient is changed to h
~.
The computed 0
~
fluid
T fluid is given to the solid as a boundary condition. Solving the solid domain yields:
()
11
wallS
S
wall TT
L
q=
λ
(38)
(
)
011
~
~
fluidwallwall TThq = (39)
with Eqn. (39) the convective boundary condition applied to the solid wall. The solution is given by
Eqn. (15) and for present application:
Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
43421
1
0
0
0
1
1
~
~
~
1
~
1
~
~
1
~~
~
1
~
~
α
α
α
+
+=
+
+
+
+
=
+
+
=
iB
BiiB
T
iB
Bi
iB
iB
iB
TiBT
iB
TiBT
T
wall
fluidS
fluidS
wall
(40)
Similar to previous methods, the wall temperature at the ith iteration can thus be written as:
0
1
~
~
α
+
+=
i
wall
i
wall iB
BiiB
TT (41)
As explained for both TFFB and FFTB methods, this series converges if and only if
1
1
~
~
<
+
iB
BiiB (42)
This can be rewritten as
1
~
~
+<iBBiiB (43)
If iBBi ~
the expression simplifies to
1
1
~
~
>
+<
B
i
iBBiiB (44)
which is always satisfied. However, if iBBi ~
>
1
~
2
1
~
~
+
<
+<
i
B
B
i
iBiBBi (45)
Thus, the hFTB method converges for each Biot number as long as the artificial Biot number iB
~
is
chosen high enough, this means
2
1
~
>Bi
iB (46)
Note that for problems with Bi < 1 the hFTB is stable regardless the value of h
~
. For
problems with Bi > 1 an appropriate choice for h
~
leads to convergence.
In case the Biot number cannot be determined but the heat transfer coefficient is known, we can state
that a sufficient but not necessary criterion to converge is:
2
~h
h> (47)
Equation (41) illustrates that the convergence to wall
T will be faster the closer
(
)
(
)
1
~
/
~
+iBBiiB is to 0.
If BiiB =
~
, the first wall temperature will already be equal to the final result and remains constant
during further iterations. For values BiiB <
~, the convergence will be slower with smaller iB
~. This
eventually leads to divergence if
(
)
2/1
~
< BiiB . On the other hand, when BiiB >
~
, the convergence
will also be slower with increasing iB
~, however without risk of divergence.
For this simple 1D problem the heat transfer coefficient h is constant over the wall surface but in a real
3D problem this is not the case. This remark suggests to introduce not a constant h
~
, but a variable one
that is as close as possible to the real h distribution. However, a guess for h is not very simple. The
Nusselt number can be expressed as a function of Reynolds number and Prandtl number for only very
simple cases (e.g. Schlichting [1979]). The best practice is thus to guess a constant value of h
~ for the
Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
entire wall larger than the maximum value of h, so that convergence is guaranteed. However, a too high
value of h
~ will result in a larger wall heat flux difference between solid and fluid at convergence (see
Eqn. (7)).
Stability of the hFFB method The hFFB method differs from the previous method by the type of
boundary conditions returned to the fluid domain. The discussion on the stability of the method is
similar to the on in previous section, however, the heat flux is the quantity transferred between both
domains and will therefore be investigated now.
Suppose the initial guess of the heat flux differs by a value 0
from the correct wall heat flux:
0
0
α
+= wallwall qq (48)
This initial heat flux would be a result of the fluid computation if following initial temperature is given
as boundary condition to the fluid domain:
h
TT wallwall
1
0
0
α
+= (49)
The analysis made in previous case for this wall temperature is still valid. By replacing 0
α
in Eqn. (33)
by h/
0
, one will obtain the wall temperature result for the solid domain (Eqn. (40)):
hBi
BiiB
TT wallwall
1
1
~
0
1
α
+
+= (50)
However, it is not this solid wall temperature that is returned to the fluid domain, but the solid heat flux
corresponding to that wall temperature:
(
)
4434421
1
0
11
1
~
~
α
α
λ
λ
+
=
=
iB
BiiB
Lh
q
TT
L
q
S
wall
wallS
S
wall
(51)
The wall heat flux at the ith iteration can thus be written as:
0
1
~
~
1
α
+
+=
i
wall
i
wall iB
BiiB
Bi
qq (52)
this series converges if and only if
1
1
~
~
1<
+
iB
BiiB
Bi (53)
This can be rewritten as
(
)
1
~
~
+<iBBiBiiB (54)
If iBBi
~
the expression simplifies to
(
)
()
BiiB
iBBiiBBi
+<
+<
1
~
0
1
~~
(55)
which is always satisfied. However, if iBBi
~
<
(
)
()
2
~
~
2
~~
1
~~
+
>
+<
+<
i
B
iB
Bi
iBBiiB
iBBiBiiB
(56)
Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
Thus, the hFTB method converges for each Biot number as long as the artificial Biot number iB
~ is
chosen low enough, this means
Bi
Bi
iB
<1
2
~ (57)
for 1Bi . Note that for problems with 1>Bi the hFFB method is stable regardless the value of h
~
.
For problems with 1Bi an appropriate choice for h
~ leads toconvergence.
In case the Biot number cannot be determined but the heat transfer coefficient is known, we can state
that a sufficient but not necessary criterion to converge is:
hh < 2
~
(58)
Equation (52) illustrates that the convergence to wall
q will be faster the closer 1
~
~
1
+
iB
BiiB
Bi is to 0. if
BiiB =
~, the first wall heat flux will already equal to the final result. For values BiiB >
~, the
convergence will be slower the higher iB
~ is chosen, which eventually leads to divergence if
Bi
Bi
iB
>1
2
~. On the other hand, when BiiB <
~, the convergence will also be slower the lower iB
~ is
chosen but with no risk for divergence.
The difference in temperature between the solid and fluid domain is proportional to
(
)
hqq i
wall
i
wall /
1
,
(Eqn. (12)). In order to have the temperature difference minimal, a high value of h is a proper choice.
However, this demand is in conflict with the convergence criterion Eqn. (28)). Therefore, the highest
possible value of h allowing convergence is the best choice.
Discussion The above mentioned convergence criteria are derived with simplistic assumptions on the
response of both domains on a change of heat flux or temperature at the interface. The solid response is
based on the thermal resistance defined by L/λs, while the fluid response is based on the heat transfer
coefficient h. It is concluded that if the solid response is more sensitive to a change in interface
temperature than the fluid, i.e. if it results in a larger change of the boundary heat flux, the temperature
boundary condition should be given to the fluid domain to obtain convergence. Although in a 3D CHT
problems the response of both solid and fluid computations are not as simple as in the 1D model, the
mechanism of the instability is the same.
It is also assumed that the temperature or heat flux response of each domain is a steady state response,
which is true for the solid domain but depends on the number of FVM iterations (implicit or explicit) of
the fluid domain. In most cases, the FVM simulation is not run till full convergence before going back
to the solid domain. Instead, the FVM is only run for few iterations and gradually converges throughout
the entire CHT process, with exchange of boundary conditions with the solid domain. Due to the
unconverged solution the stability criterion is not strictly respected, as an incorrect response is returned
to the solid domain. The effect of this depends on the quantity imposed at the fluid domain.
If a heat flux boundary condition is applied to a FVM solver, such as in the TFFB or hFFB methods, a
change of the wall heat flux boundary condition will result in a gradual convergence towards a new
wall temperature, without abrupt change. If the FVM solver is stopped before convergence, a smaller
temperature change will be predicted with respect to a full converged solution, which means that
reducing the number of FVM iterations artificially increases h and the Biot number (see Eqn 2 and 16).
Therefore, the TFFB method will also behave stable for Biot values lower than 1 when few FVM
iterations are performed. Similarly, the hFFB method will behave stable even when the stability
criterion Eqn (53) or Eqn (58) is not met, as long as the number of FVM iterations is kept low.
On the other hand, if a temperature boundary condition is used for the fluid solver, as in the FFTB or
hFTB method, the change in wall temperature will cause abrupt changes in the heat flux which will
Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
gradually converge towards the new solution. If the FVM method is stopped before convergence, the
heat flux response of the fluid domain will again be as if a larger value of h is used (see Eqn 2) and thus
as if the Biot number of the CHT problem is larger. Especially if the FVM computation is stopped after
few iterations, the heat flux results at the wall are meaningless as they correspond to the response of a
heavily perturbed thermal boundary layer that still needs to converge to its steady state. With an
artificially increased Biot number, the FFTB and hFTB methods will behave less stable than predicted
by the stability criterion. For the FFTB method it is almost impossible to converge the CHT problem
without the use of a relaxation coefficient, as show in [Verstraete 2008]. The hFTB method has the
advantage that the convergence can be controlled by the value of h but requires a much higher value
than predicted by the stability criterion.
Taking into account that the FFTB and hFTB methods both require a large number of FVM iterations
per solid-fluid interaction, the TFFB and hFFB methods are preferred above the former. The heat flux
boundary condition for the fluid domain guarantees a smooth change of the wall temperature and
increases the stability of these two methods to problems with low Biot number. Especially for the hFFB
method, the stability can be guaranteed for very low Biot numbers by choosing an appropriate value of
h.
RESULTS
CHT in a Flat Plate The four different methods are compared to each other for the two-dimensional
flat plate test case. All results are compared with the analytical solutions of Luikov by using the
differential heat transfer equation and the boundary layer equation (Luikov [1974]). The dimensions of
the flat plate, the boundary conditions and the flow properties can be found in Table 1. The conduction
in the solid was determined such that the average Biot number of the flat plate equals one. In the
leading edge of the flat plate the high value of h results in Biot numbers above unity, while downstream
the Biot number becomes below one. This allows to test the stability criterion.
Figure 6 shows the variation of the wall temperature results of all 4 methods over the length of the
plate. Note that for the FFTB method only a converged solution could be obtained with a relaxation
coefficient of 0.9, as can be expected by the stability criterion. The h coefficient for the hFTB method is
800 W/m2K and a value of 50 W/m2K is used for the hFFB method. A total of 30 solid-fluid iterations
are performed for each computation, with 100 FVM explicit iterations in the fluid per solid-fluid
iterations. The TFFB method behaves stable without relaxation coefficient, even in the downstream part
where the local Biot number is below unity. This is due to the heat flux boundary condition applied to
the FVM solver, which artificially increases the Biot number.
Table 1
Dimensions of the flat plate, boundary conditions and flow properties.
Flat Plate length 0.2 m
Flat Plate thickness 0.01 m
Inlet flow total temperature 1000 K
Bottom temperature flat plate 600 K
Solid conductivity 0.3 W/mK
Reynolds number 21000
Prandtl number 0.6629
Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
As can be seen from Fig. 6 all results are similar except for the hFTB method. This is due to a
remaining difference in heat flux between the solid and fluid solver. The CHT solutions agree well on
the first part of the flat plate with the most accurate analytic Luikov solution which uses the differential
heat transfer equations. The difference between analytical predictions and numerical computations is
the largest near the trailing edge.
x[m]
Twall [K]
00.05 0.1 0.15 0.2
750
800
850
900
950
Luikov diff. h.t.e.
Luikov bound. lay. eqn.
hFTB
TFFB
hFFB
FFTB relax 0.90
Figure 6. Comparison of the results for the different methods with the analytical solutions (Luikov
[1974]).
CHT in a cooled turbine blade A second application of the coupled methods for CHT analysis
consists of the heat transfer computation of a cooled turbine blade. The prediction of the solid
temperature is essential for the assessment of the structural integrity of the blade. It allows to compute
the thermal stresses and together with the centrifugal stresses the lifetime of the blade can be computed
(Amaral et al. [2008]).
The test case in the present study is a first stage turbine blade cooled by 5 internal cooling channels.
The blade characteristics are summarized in Table 2 and the boundary conditions are listed in Table 3.
For a complete description of the CHT analysis the reader is referred to Amaral et al. [2008]. Present
study aims to investigate the difference in convergence between the hFTB and hFFB methods. Both
methods are preferred above the FFTB and TFFB method as the stability can be controlled by the value
of h.
Table 2.
Blade geometry specifications.
blade height 0.2m
number of blades 90
Hub axial length 0.13m
Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
hub pitch to chord ratio 0.667
Table 3.
Stage operating conditions.
Inlet total pressure 7.0E5 Pa
Inlet total temperature 1400K
Inlet Mach number 0.3
Outlet static pressure 3.5E5 Pa
Outlet Mach number 0.8
Mass flow rate 400kg/s
Figure 7 shows the temperature result for the hFFB method with h = 500 W/m2K. The position of the
cooling channels can be recognized by the chordwise variation of the temperature.
The convergence history is plot in Fig. 8. It shows the evolution of the maximum solid temperature
difference between two successive CHT iterations. The hFTB method uses a h = 3000 W/m2K while
the hFFB method uses a h = 500 W/m2K, both chosen for stability reasons. The hFFB method shows a
faster convergence with respect to the hFTB method and does not level off at a maximum temperature
difference around 1K as does the hFTB method. Hence it is the preferred method.
CONCLUSIONS
This paper introduces a novel method for solving CHT problems by a coupled approach. A new
stability criterion has been derived that allows a deeper insight into the instabilities often encountered
when solving CHT problems. The newly developed method turns out to be the most appropriate choice
for CHT computations according to this criterion as it has the flexibility to converge in a large range of
Biot numbers by using an appropriate value of h. In addition, similar to the TFFB method its stability is
enhanced by the use of a heat flux boundary condition for the fluid domain, which results in smooth
wall temperature modifications.
All methods have been validated on the flat plate test case, and both the hFTB and hFFB method have
been compared on a real problem involving the computation of the heat transfer in a internally cooled
gas turbine blade.
Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
Figure 7. Temperature distribution of a turbine blade with internal cooling channels.
Iteration
Lmax norm
0 2 4 6 8 10 12 14 16
10
-1
10
0
10
1
10
2
hFFB
hFTB
Figure 8. Convergence history of the hFTB and hFFB method.
Int. Symp. on Heat Transfer in Gas Turbine Systems
9*14 August 2009, Antalya, Turkey
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Conference Paper
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A conjugate heat transfer solver has been developed and applied to a realistic film-cooled turbine vane for a variety of blade materials. The solver used for the fluid convection part of the problem is the Glenn-HT general multiblock heat transfer code. The solid conduction module is based on the Boundary Element Method (BEM), and is coupled directly to the flow solver. A chief advantage of the BEM method is that no volumetric grid is required inside the solid — only the surface grid is needed. Since a surface grid is readily available from the fluid side of the problem, no additional gridding is required. This eliminates one of the most time consuming elements of the computation for complex geometries. Two conjugate solution examples are presented — a high thermal conductivity Inconel nickel-based alloy vane case and a low thermal conductivity silicon nitride ceramic vane case. The solutions from the conjugate analyses are compared with an adiabatic wall convection solution. It is found that the conjugate heat transfer cases generally have a lower outer wall temperature due to thermal conduction from the outer wall to the plenum. However, some locations of increased temperature are seen in the higher thermal conductivity Inconel vane case. This is a result of the fact that film cooling is a two-temperature problem, which causes the direction of heat flux at the wall to change over the outer surface. Three-dimensional heat conduction in the solid allows for conduction heat transfer along the vane wall in addition to conduction from outer to inner wall. These effects indicate that the conjugate heat transfer in a complicated geometry such as a film-cooled vane is not governed by simple one-dimensional conduction from the vane surface to the plenum surface, especially when the effects of coolant injection are included.
Conference Paper
This paper considers the coupling of a finite element thermal conduction solver with a steady, finite volume fluid flow solver. Two methods were considered for passing boundary conditions between the two codes — transfer of metal temperatures and either convective heat fluxes or heat transfer coefficients and air temperatures. These methods have been tested on two simple rotating cavity test cases and also on a more complex real engine example. Convergence rates of the two coupling methods were compared. Passing heat transfer coefficients and air temperatures was found to give the quickest convergence. The coupled method gave agreement with the analytic solution and a conjugate solution of the simple free disc problem. The predicted heat transfer results for the real engine example showed some encouraging agreement, although some modelling issues are identified.
Conference Paper
This first paper describes the Conjugate Heat Transfer (CHT) method and its application to the performance and lifetime prediction of a high pressure turbine blade operating at a very high inlet temperature. It is the analysis tool for the aerothermal optimization described in a second paper. The CHT method uses three separate solvers: a Navier-Stokes (NS) solver to predict the non-adiabatic external flow and heat flux, a Finite Element Analysis (FEA) to compute the heat conduction and stress within the solid, and a 1D aero-thermal model based on friction and heat transfer correlations for smooth and rib-roughened cooling channels. Special attention is given to the boundary conditions linking these solvers and to the stability of the complete CHT calculation procedure. The Larson-Miller parameter model is used to determine the creep-to-rupture failure lifetime of the blade. This model requires both the temperature and thermal stress inside the blade, calculated by the CHT and FEA. The CHT method is validated on two test cases: a gas turbine rotor blade without cooling and one with 5 cooling channels evenly distributed along the camber line. The metal temperature and thermal stress distribution in both blades are presented and the impact of the cooling channel geometry on lifetime is discussed.
Conference Paper
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Conference Paper
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An interaction model between the fluid and solid media is generally required in turbine configurations but remains a difficult issue. A coupling procedure between a Navier-Stokes code and a conduction solver is therefore the only way to achieve heat transfer prediction in all flow situation. The objective of this work is to present such a procedure , which has been developed by Snecma and based on a Finite Volume Navier-Stokes code and a commercial Finite Element solver. To demonstrate the quality of the procedure, a conjugate heat transfer computation in a turbine blade internal cavity is described in detail. Nomenclature s C solid calorific capacity p C fluid specific heat L geometric reference scale w q wall heat flux T wall temperature λ µ p C = Pr Prandtl number µ ρ L U = Re Reynolds number ρ density λ conductivity µ viscosity Superscripts s and f refer to solid and fluid quantities respectively. Subscript w refers to wall quantities.
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