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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 0≠h, if 0

1→−

+ii fvm

wall

fvm

wall TT also 0→− ii fvm

wall

fem

wall qq and 0→− ii 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 0≠h, if 0

1→−

+ii fvm

wall

fvm

wall qq also 0→− ii fvm

wall

fem

wall TT and 0→− ii 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 Ωs (13)

(

)

fluidwallwall TThq

−

=

on Ωf (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

α

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

α

:

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

α

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

α

, 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 1≤Bi . Note that for problems with 1>Bi the hFFB method is stable regardless the value of h

~

.

For problems with 1≤Bi 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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