Chinese Journal of Aeronautics

Chinese Journal of Aeronautics 22(2009) 243-249 www.elsevier.com/locate/cja

Application of Coolant Diffusion Equation in Numerical Simulation of Flow in Air-cooled Turbines

Wang Qiang*, Yan Peigang, Dong Ping, Feng Guotai, Wang Zhongqi

School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China

Received 10 April 2008; accepted 30 August 2008

Abstract

This article deals with time-averaged coolant diffusion equation in arbitrary, body-fitted coordinates. Schmidt number and Lewis number are used to obtain the mass diffusion coefficient. Other than the existing N-S solver, a new solver for the diffusion equation and the influences of cooling air on the thermal parameters, such as gas constant, constant-pressure specific heat, is established. Capable of simulating the flow of multi component fluid, this solver is used to simulate the flow in air-cooled two-stage turbines. The comparison of test results from the modified solver to those from the traditional N-S solver has evi-denced that a marked difference present is in thermal parameters and flow field structure.

Keywords: turbine; coolant mixing; mass diffusion; variable specific heat

1. Introduction1

The demand for improvement on thermal efficiency of turbines necessitates enhancement of their operating temperature, which imposes ever-more-increasing ther-mal stress on the blades. To protect turbine compo-nents from overheating, as an active cooling measure, the highly efficient film cooling technique has found wide application. It involves cooling air injected through orifices into the main stream. The researches on this technique are often compelled to focus on flow and heat transfer in the cooling channel[1], and the influences that cooling air injection exerts upon the flow field in the passage.

At present, several simplifications have been made when simulating flows with injected cooling air. Of them, one is to assume that the components of coolant should be the same as those of the fluid in the passage, and the other is that the specific heat should be only affected by temperature. In reality, the components of injected coolant are different from those of the gas in the passage. This entails a mixing process, which can be described by mass diffusion equation in a film-cooled turbine. The mixing process leads to component

*Corresponding author. Tel: +86-451-86412433. E-mail address: chuangwang.w@gmail. com Foundation item: National Natural Science Foundation of China

(50476028, 50576017, 50706009) 1000-9361/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi: 10.1016/S1000-9361(08)60094-1

variation. Hence, the specific heat would not be af-fected by temperature only, but the changeable com-ponents of gas too. Although in simulation of film-cooled turbines, in order to predict accurate distribution of temperature, the effects of coolant diffusion should be taken into account, and there are precious few publica-tions concerning coolant diffusion in simulation of gas turbines. This may be attributed to sheer complexity of the equation system governing the flow with consi- deration of coolant diffusion and the huge volume of computational work thus induced. In contrast, in some fields like chemical engineering, environment engi-neering[2-3], and air conditioning[4], simulations have already been carried out making allowance for the mass diffusion as well as mixing process of different fluids[5]. Nevertheless, recently there was a researcher who tried to use a commercial software, Fluent, to simulate the flow in a gas turbine associated with coolant diffusion[6].

2. Computation Scheme

2.1. Governing equation

Compared with the governing equations for the con-servative variables of invariable-component fluid, those for the conservative variables of variable-component fluid[6] contain only one additional mass diffusion equa-tion. Consequently, N-S solver for flows of invari-able-component fluid can be used for flows of vari-

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able-component fluid after some modification by add-ing component diffusion equation solver and building up the relationship between fluid components and thermal parameters.

N-S equations for dimensionless conservative vari-ables in arbitrary, body-fitted coordinates (neglecting body force and heat radiation) are as follows

1 21t r Re r

f fU E F G Q R S

(1) where Re is Reynolds number, the vector U=J [ u v

w e]T is a component-weighted parameter, J a Jacobin determinant for coordinate transformation, the com-ponent-averaged density, e the internal energy per unit mass, u, v, w are the velocity components in axial, radial, and circumferential directions. Other vectors such as f1, f2 , E, F, G, Q, R, and S could be referred in Ref.[7].

Mass diffusion equation of coolant is

( Ys) ( Ys ) ( Ys)Dt

v (2)

where Ys the concentration of cooling air. For the ‘pure’ gas without cooling air, Ys equals to 0, and for ‘pure’ cooling air, Ys to 1. By time-averaging Eq.(2) the following can be obtained:

( Ys) ( Ys ) ( Ys) ( Ys )Dt

v v (3)

where D is the coefficient of mass diffusion and v the velocity vector. For the sake of convenience, the su-perscripts of time-averaged variables in Eq.(3) could be neglected, and the fluid is assumed incompressible in derivation. Ysv is a turbulent diffusion flux. Ac-cording to the famous Boussinesq postulation, an ex-pression between the flux and concentration gradient is derived by

t(Ys)Ys ( 1,2,3)i

iv D i

x (4)

Analogous to the Fick’s law for molecular diffusion, tD is termed turbulent diffusion coefficient, which is

assumed to be characteristic of isotropic. As a dimen-sionless parameter, Schmidt number (Sc) refers to the ratio of viscosity to diffusion coefficient. The turbulent and laminar Schmidt numbers are

tt

t

ll

l

ScD

ScD

(5)

By substituting Eqs.(4)-(5) into Eq.(3), expanding the formula obtained in the cylindrical coordinates and normalizing, dimensionless mass diffusion equation in the cylindrical coordinates could be obtained as follows:

( Ys) ( Ys) ( Ys) ( Ys)u v wt z r r

t tl l

l t l t

1 Ys YsRe z Sc Sc z r Sc Sc r

t tl l

l t l t

Ys Ys 1 Ysvr Sc Sc r r r Re Sc Sc r

(6) Since equations in the arbitrary, body-fitted coordi-

nates are most usable in engineering practices, Eq.(6) should be transformed into the form of the arbitrary coordinates. Thus, the following can be obtained:

Ys Ys Ys Ys Ys1

t Ys Ys Ys Ysl

l t

1 (7)

t r

Re Sc Sc r

U E F G f

R Q R S

where

Ys YsUU J (8)

Ys Ys

Ys Ys

Ys Ys

///

z r

z r

z r

r Er Fr G

EF JG

(9)

Ys Ys

Ys Ys

Ys Ys

///

z r

z r

z r

r Qr Rr S

QR JS

(10)

Ys1 Ys1ff J (11)

Ys

Ys

Ys

Ys

Ys1

Ys

Ys

Ys

YsYsYs

YsYs

Ys

Ys

Ys

E uF vG wUf v

Qz

Rr

Sr

(12)

The parameters, such as J, z and so on, in Eqs. (7)-(11) are for coordinate transformation, and their definitions can be found in Ref.[7]. It is noted that both Schmidt numbers, tSc and lSc , are not constant but dependent upon temperature and component of fluid. However, in engineering practice, it is rather conven-ient to assume Schmidt numbers to be constant. It has been proved that the results from mass transfer ex-periments accord well with those from heat transfer experiments if Schmidt number ranges from 0.226 to 1.540[6]. Since there is no commonly recognized value of Schmidt number for simulating flows in film-cooled turbines, this article supposes the mass diffusion pro- cess in complete analogue to heat transfer process. From Sc=Le Pr, where Le is Lewis number and Pr Pramdtl number, the value of Schmidt number could

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be obtained by letting Le be 1.

2.2. Computational model

For the modified version of HIT-3D, the N-S equa-tions and the mass diffusion equation are solved sepa-rately. The detailed computational model for solving N-S equations can be referred to Ref.[7]. Here, only the model of mass diffusion is introduced. According to Crank-Nicolson scheme, the following approxima-tion of the conservative equations in the semi-implicit form could be derived

Ys

Ys Ys YsYs

[ ( )]

( ) (13)

n n nn

n

A B CI U

E F GH

where A, B, C are the Jacobians of the fluxes EYs, FYs,

and GYs; HYs is the diffusive term; time step length; and the coefficient of the scheme. The present so-

lution vector 1YsnU can be acquired by adding UYs to

the last solution YsnU , 1

Ys Ys Ysn nU U U .

The left side of Eq.(13) constitutes the implicit part, whereas the right side constitutes the explicit. To im-prove on the solving efficiency and ensure numerical stability of the implicit part, the implicit approximate factorization algorithm combined with diagonalization as well as the flux vector splitting method[8] is used. Besides, an upwind discretization is used for the split flux.

The explicit part, which includes convective term and diffusive term, is discretized with two kinds of different schemes. A third order accurate total variation diminishing (TVD) scheme is used for convective terms. The solutions with such a TVD scheme are on the cell faces leading to a Riemann problem. To reckon with the numerical stability, an approximate Riemann solver[9] is used and a central difference scheme is used for diffusive terms.

(1) Difference scheme The third order accurate TVD scheme[9] to discretize

convective terms in mass diffusion equation is as fol-lows: for the one-dimensional problem, the fluxes on the jth cell faces uj+1/2,L and uj+1/2,R could be approxi-mated to

1/ 2,L 1/ 2 1/ 2

1/ 2,R 1 3/ 2 1/ 2

1 1 14

1 1 14

j j j j

j j j j

u u u u

u u u u

(14)

where is an interpolation factor. To ensure the TVD characteristic of the flux u,

uj+1/2 should be restricted by a limiter. The interpola-tion formulas with limiters are

1/2,L 1/2 1/ 2

1/2,R 1 3/ 2 1/2

1 1 14 (15)

1 1 14

j j j j

j j j j

u u u u

u u u u

where

1 2 1 2 1 2

1 2 1 2 3 2

,

,

j j j

j j j

u u u

u u u

and satisfies the following expression: 311

(16)

By denoting ,( )d d as a flux limiter, the minimum modulus limiter is selected to be

, 1 2 3 2( ) min ,j jd d u u (17)

where is a coefficient, a difference operator, and d a flux.

(2) Turbulence model In the original version of HIT-3D, the closure of

conservative N-S equations and mass diffusion equa-tion is provided by B-L algebraic turbulence model modified with modified mixing length (MML)[10]. The two-equation low-Re turbulence model, i. e. q- model[11], is added into the modified version. The B-L model has been used in the early work[12]. q- model, which can predict not only detailed flow phenomena but also laminar-turbulence transition, will be used here.

(3) Influences of component variation on specific heat and gas constant

Since component variation affects the fuel-air ratio, it exerts influences upon specific heat and gas constant. The relationship between fuel-air ratio and specific heat and gas constant is described by[6]

g a/(1 )

pp p cC C f f (18)

wheregpC and

apC represent specific heat of gas and air

at constant pressure respectively, f fuel-air ratio, and

pc a factor pertinent to temperature.

8.314 5/ ( )R M f (19) where M( f ) is mole of gas.

(4) Boundary conditions There are five kinds of boundaries all together: inlet,

outlet, nosplitting wall, periodic interface and cooling air jet boundaries. The detailed boundary conditions imposed on N-S equations can be referred to Ref.[13], and those imposed on mass diffusion equation are as follows.

At the inlet, Ys = 0, whereas at the solid wall and the outlet, Ys is set to zero normal fluxes. At the both sides of periodic interfaces, values of Ys are identical, and at cooling air jet boundaries, Ys = 1.

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2.3. Cooling orifice locations and computing grids

In the case of a two-stage film-cooling turbine, cooling air is injected through slots and bores at upper and lower endwalls and on the blade surfaces. In Fig.1(a), HPTS denotes high-pressure turbine stator, HPTR high-pressure turbine rotor, LPTS low-pressure turbine stator, and LPTR low-pressure turbine rotor. Fig.1(b) illustrates the coolant air injection position on the blade surface.

(a) Injection orifices at upper and lower endwalls

(b) Injection orifices on blade surfaces

Fig.1 General view of cooling air injection orifice positions.

To accurately simulate flows inside the boundary, fine O-type grids are used to discretize domains around the blade. To ease computing workloads, H-type grids are used to discretize main tream domains. The computa-tional grids are shown in Fig.2.

(a) Grids on S1 section

(b) Grids on blade wall

(c) Enlarged view of grids.

Fig.2 Computational grids.

The numbers of H-type grid cells in each stage are as follows: 76×40×40 (HPTS), 76×40×40 (HPTR), 76×40×40 (LPTS), and 88×40×40 (LPTR), respec-tively. Correspondingly, those of O-type: 240×6×300 (HPTS), 220×6×300 (HPTR), 100×6×40 (LPTS), and 100×6×40 (LPTR). The dimensionless distance of the next grids to the solid wall y+, is less than five.

3. Results and Discussion

At the highest inlet temperature, the high-pressure stator of turbine needs more intense cooling which requires more cooling air than other blades. Therefore, as a representative, this article presents numerical re-sults in the high-pressure stator only. It should be noted that the parameters in following figures are dimen-sionless.

The real value of seeking for solution to diffusion equation is that the solution itself reveals the distribu-tion of cooling air in the passage and as such indicates the film cooling efficiency. Actually, cooling air in jected through holes and slots, where coolant densely concentrates, mixes up with gas in the passage making the concentration of cooling air decrease in the mixing process. Fig.3 illustrates the distribution of cooling air concentration on the high pressure stator surface. It can be seen that the concentration, after reaching its local apex, declines gradually behind the injection orifices.

Fig.4 shows the distribution of surface cooling air concentration at the midspan, and L denotes the axial chord. Denser concentration at leading edge and trail-ing edge of pressure side could be attributed to local cooling air injection. The slightly higher concentration at the pressure side than at the suction side could be ascribed to the acceleration at the latter side. Fig.5

(a) Pressure side

No.3 Wang Qiang et al. / Chinese Journal of Aeronautics 22(2009) 243-249 · 247 ·

(b) Suction side

Fig.3 Cooling air concentration on high pressure stator surface.

Fig.4 Surface cooling air concentration at midspan.

Fig.5 Surface gas constant at midspan.

shows the corresponding surface gas constant distribu-tion at midspan. It is observed that at the leading edge and the trailing edge with more cooling air concen-trated, the gas constant is lower than that in other re-gions. Eq.(19) indicates the influences of gas compo-nent variation on gas constants. It can be concluded that there exists slight variation of gas constant in the whole region ranging from 0.998 1 to 0.998 7.

For the sake of comparison, Figs.6-9 present the re-sults from the modified and the original versions of HIT-3D. Fig.6 provides the distribution of surface con-stant pressure specific heat at midspan of high-pres-

Fig.6 Surface constant pressure specific heat at midspan.

Fig.7 Surface static temperature at midspan.

Fig.8 Surface static pressure at midspan.

sure stator. Compared with results without considering component variation, the predicted Cp is lower at lead-ing and pressure side trailing edge with an about 0.1 difference, wheeas higher values exist in the other part of regions, with an around 0.2 difference. The same trend for temperature could be found in Fig.7, which shows the distribution of surface temperature. Eq.(18) indicates that Cp, as a function of fuel-air ratio, will be reduced if the fuel-air ratio or temperature is lowered. The inflow of cooling air with the fuel-air ratio equal to 0 into the main gas flow would have its fuel-air ratio decreased, which leads to decrease of constant pressure

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specific heat if considering the effects of fuel-air ratio only. However, this ratio is affected by cooling air dif-fusion process, and larger value of fuel-air ratio would soothe the cooling effects resulting in raising tempera-ture and thereby increasing constant pressure specific heat. Generally, at the leading edge and pressure side trailing edge with more cooling air concentrated and, hence, fairly good cooling effects, the specific heat and temperature considering coolant diffusion are lower than those not considering it. This is just the opposite with the other part of stator surface.

(a) Without coolant diffusion

(b) With coolant diffusion

Fig.9 Contours of Mach number at midspan.

Fig.8 shows the distribution of static surface pres-sure at midspan. As is shown, the static pressure in the case of variable components is almost the same as that in the case of invariable components at the pressure side of stator, but the former seems slightly greater than the latter at the front of suction side. Remarkable difference in pressure turns up at the rear of suction side. There is a notable rise in static pressure in regions located lengthwise at x >38 when the effects of coolant diffusion are considered, but this rise occurs at the trailing edge instead if they are not considered. Fig.9 illustrates that for the flow of variable-component gas, the Mach number gradually increases up to more than 1 in the front of suction side, followed by plunging under 1 at the position far away from leading edge, which indicates an occurance of a shock wave. In con-trast, for the flow of invariable-component gas, the Mach number remains unchanged in most parts of suc-tion side except at the trailing edge due to there being a shock. Thus the shock at the suction side moves up-stream owing to coolant diffusion, which induces a

very remarkable difference in surface static pressure.

4. Conclusions

(1) A mass diffusion equation for cooling air in the dimensionless form in arbitrary, body-fitted coordi-nates can be derived.

(2) Both Schmidt and Lewis numbers are used to build up the relationship between viscosity and mass diffusion coefficient, assuming the mass boundary layer being identical to thermal boundary layer, namely Le = 1.

(3) The solution to mass diffusion equation illus-trates the distribution of cooling air in the flow field, which is linked to the cooling efficiency of cooling air injection.

(4) Cooling air injection into main flow causes variation of gas components, thereby changing thermal parameters and temperature. Compared with the results without considering effects of mass diffusion, the gas constant varies slightly but the constant pressure spe-cific heat and temperature considerably.

(5) The variability of gas component affects behav-ior of flow in the passage. In this study, it is observed that the shock occurring in the high pressure stator moves upstream.

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Biographies:

Wang Qiang Born in 1982, he received B.S. and M.S. degrees from Harbin Institute of Technology in 2004 and 2006, respectively. He now studies as a Ph.D. candidate in the same university. His main research interest is aerother-modynamics. E-mail: chuangwang.w@gmail. com

Yan Peigang Born in 1975, he received B.S., M.S. , and Ph.D. degrees from Harbin Institute of Technology in 1999, 2001, and 2004, respectively. Now he is an associate profes-sor in the same university. His main research interest is aerothermodynamics. E-mail: Peigang_ y@sina.com Dong Ping Born in 1974, he received B.S. degree from Xi’an Jiaotong University in 1997 and M.S. degree from Harbin Institute of Technology in 2003, respectively. He now studies as a Ph.D. candidate in Harbin Institute of Technol-ogy. His main research interest is aerothermodynamics. E-mail: dps777@163.com