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Latin American applied research

Print version ISSN 0327-0793

Lat. Am. appl. res. vol.42 no.2 Bahía Blanca Apr. 2012

 

ARTICLES

Advances in nonlinear stress analysis of a steam cooled gas turbine blade

M. M. Jafari, G. A. Atefi† and J. Khalesi

Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, 16844, Iran.
mmjafari110@iust.ac.ir, atefi@iust.ac.ir, javadkhalesi@gmail.com

† Corresponding author


Abstract - In this paper to study the thermoelastic behavior of a gas turbine blade, a computer code based on Conjugate Heat Transfer method was developed to solve the coupled external flow field and the internal temperature field inside a gas turbine blade. An interpolation method based on the 3D shape functions was used to calculate the correct temperature values at the boundaries to determine the stress field. For the materials with temperature dependent mechanical and physical properties, the thermal stresses exhibit a nonlinear behavior. In these cases, an analytic solution for the energy and equilibrium equations to obtain the temperature distribution and the stress field, knowing K(T), a(T), E(T) would be impossible and the numerical schemes should be employed instead. The results show that the stress for the materials with temperature dependent properties is higher than that for the cases where the blade material properties are constant with temperature. Further, the temperature field and the flow field obtained by the present method were compared to the available experimental findings. The results show a good agreement.

Keywords - Nonlinear Stress; Cooling; Temperature Dependent; CHT Method.


 

I. INTRODUCTION

In recent years, interest in design and analysis of sensitive elements of gas turbines as a power product is increased. One of the main negative aspects of gas turbine units can be erosion of its hot parts which increases the maintenance cost. The main element of a gas turbine is its blades which from design technology point of view, production and metallurgy are very important (Mallet et al., 1995). Among turbine stages, the first row blades are very sensitive and vulnerable. These blades are prone to erosion due to the thermal and structural stresses and stresses associated with the centrifugal forces (Amr et al., 2006; Tofighi et al., 2008). For an accurate prediction of the rate of erosion and the harms come to the blades, obtaining stress and temperature distribution and determination of critical points is of great importance.

In the past decades, the maximum working temperature for gas turbine due to temperature limits of the blade materials and lack of cool-treating technology has been limited to about 800°C which is the main reason for low efficiency of these turbines. Various attempts have been made so far to improve the working conditions of the blades and reduce the effect of external flow on the blade and enhance the gas turbines efficiency.

With advent of superalloys which in general are Nickel-base superalloys, increase the temperature limits of the blade up to 300°C. These alloys show high strength at high temperatures and have good strength against erosion. However, erosion problem of these blades have not been solved, and for decreasing the erosion one uses single-crystal coating and directionally-solidification coating methods (Amr et al., 2006).

Conventionally cooled gas turbines are using compressor air for cooling purposes, this method in known as Closed Circuit Steam Cooling. There are several advantages by using compressed air from compressor (CCSC) comparing to the air cooling (Krüger et al., 2001). In several references (Bohn et al., 1995, 1997, 2001, 2002, 2003; Bohn and Schnenborn, 1996; Amaral et al., 2010; Atefi et al., 2010a-d; Montenay et al., 2000; Heidmann et al., 2003; York and Leylek, 2003; Lassaux et al., 2004; Verstraete et al., 2007; Jafari et al., 2011; Bayerlein and Sockel, 1991; Han et al., 2000; Verdicchio et al., 2001) two viewpoints are presented for solving CHT on the basis of meeting continuity of temperature and heat flux over the common boundary of fluid and solid: In the first one, all flow part equations inside the body consist of a unit simultaneous equation system, and continuity of temperature and heat flux is considered implicitly. This method which is known as the conjugate method is computationally efficient and requires only one solver. In this method, the connection between fluid and solid surface is ignored if the continuity of nodes exist (Bohn et al., 2003). This viewpoint is used when external and internal temperature field are coupled (Kassab et al., 2003). The second one which is known as a coupled method, heat and flow field is calculated separately. Heat field inside the solid body and the fluid flow are coupled over solid surface by a proper boundary condition. In this viewpoint, field of inside the blade is considered steady state and is analyzed by a finite element method and flow field over the blade by a finite volume method. Coupling is ignored in numerical and analytical problems, for example, in a turbo machine problem, flow solution and heat transfer is solved separately. A separate heat transfer coefficient for determination of blade temperature field is obtained by solving external and internal flows (Kassab et al., 2003). The advantage of this method is using a standard mesh for every part of domain (Bohn et al., 2002).

The second method was used in this paper. To determine the critical points, the temperature was first obtained by solving the flow field over the blade and considering the convection effects. The temperature gradient was then determined on the surface and the heat conduction in the laminar sub layer was calculated. This value was transferred inside the blade to calculate the heat conduction and temperature inside it. This procedure yields a new value for temperature on the blade surface. Finally, the strain and stress inside the blade were calculated and the critical points on the surface where either the temperature or the stress has maximum value were determined.

Finite volume, finite element or a combination of both is the usual approaches for solving the coupling problems. A few example of using the CHT method in turbo machine problems can be found in the works of Bohnl et al. (1997), Hahn et al. (2000) and Kassab et al. (2003).

The studies carried out so far need some experimental initial data to solve the problem and the investigations were limited to the materials with temperature independent properties. In the present paper, temperature distribution and stress field inside a blade with a steam cooled channel is obtain using CHT method which is an introduction for prediction of life time of the blade that is done on the subsequent works. A new CHT method, considering all material properties as temperature dependent, has been developed to study a turbine rotor blade with cooling. With this method, there will be no need to any experimental information as initial data. Further, the results obtained by this method give a better approximation of the reality than those of the previous classic approaches.

II. MATERIAL PEROPERTIES

Nickel-based super alloys are the highest temperature resistant metals used in industries (Stephen, 2006). Among the various applications of these alloys, the turbine blades working in high temperature environments are the most critical ones. The necessity for increasing output and efficiency in industrial gas turbines results in the need for increased capability materials for both creep and fatigue. It also puts increased demand on thermal and environmental coatings to provide protection for long times in higher temperature environments. The blade material was Inconel 738LC and its nominal composition is listed in Table 1.

Table 1: composition of Inconel 738LC nickel base superalloy.

Generally, all thermal parameters, such as mass density, specific heat and thermal conductivity depend on temperature, while the latter is more temperature dependent than the others. To consider the effect of temperature dependent parameters on temperature fields, Taylor and Kirchhoff transformation (Zhou and Hashida, 2001) or the numerical methods can be applied. Density and thermal conductivity of the Inconel 738LC at the mean temperature are 8110 (Kg/m3) and 16.2 (W/m.K), respectively. The temperature dependent thermal conductivity of the Inconel 738LC is shown in Fig.1.

Fig. 1: Material properties of Inconel-738 LC

The Young's modulus, coefficient of thermal expansion and Poisson's ratio of the Inconel 738LC at mean temperature are 205 (Gpa), 13.3E-6 (/K) and 0.33, respectively. The Young's modulus and the coefficient of thermal expansion of Inconel 738LC alloy decreases and increases respectively, as increasing the temperature. The temperature dependent Young's modulus and coefficient of thermal expansion of the Inconel 738LC alloy are shown in Fig. 1.

III. FORMULATION

A. Flow field

To simulate the flow field, the 3D compressible N-S equations over a hybrid mesh using implicit finite volume method is solved. The governing equations for the conservative variable in arbitrary, body-fitted coordinate's ξ, η and ζ with the fluxes in normal direction to ξ, η and ζ = constant are as below:

(1)
(2)
(3)

where

(4)

Where Spalart-Allmaras model has been used in above equation for turbulence modeling and the non dimensional wall distance, y+, was chosen to be less than one. The resultant linear equation system is solved by an iterative method. The temperature distribution and heat flux over the blade surface provide the boundary condition for the finite element part (Atefi et al., 2010b,d) Also, the conservative equations in implicit form are obtained by using Newton method. An upwind scheme is used to calculate inviscid fluxes and a central difference method is used for viscous fluxes. For accurate modeling of the boundary layer, structured mesh was used for the regions close to the external blade surface and an unstructured mesh for the far field as shown in Fig.2.

Fig. 2: Numerical grid for boundary layer and channel

B. Thermal and stress analysis

Using the CHT approach mentioned in this paper, it is also possible to calculate the thermal stresses produced in cooled blades due to the temperature gradients. The maximum thermal stresses occur around the hole, where high temperature gradients exist. Taking in to account the dependency of blade material properties on temperature, the stress field calculation will be a nonlinear problem and numerical methods must be used to solve it. It should be pointed out that the thermal stresses only exist in the 3D cases and the plain stress assumptions are not feasible in a 3D problem.

Fourier heat equation

The heat conduction equation for a solid is obtained from the energy equation. Using the Fourier heat conduction law without heat source:

(5)

Equation (5) is linear as long as k depends on position. For the cases when k depends on temperature, the equation would be nonlinear and numerical schemes should be employed. In the present paper, the finite element method was used to solve it. Boundary conditions for the heat conduction equation typically consist of initial temperature distribution in the entire domain, and temperature and/or heat transfer rate prescription on the boundaries (Bunker, 2006; Bonifaz and Richards, 2009). If thermal conductivity depended to temperature, the energy equation becomes:

(6)

In steady state Eq. (6) becomes:

(7)

Constitutive equations

The stresses in a rotor coolant blade are mainly due to thermal expansion, rotational speed and structural loads. To determine these stresses, the displacement thermoelastic equation is expressed in terms of the displacement vectors based on the temperature dependent parameters including the elastic modulus E and the coefficient of thermal expansion a. Given the elastic modulus and coefficient of thermal expansion in this case, the resulting system of differential equations can be solved. The results are valid as long as the Poisson's ratio is constant and the elastic modulus E and the coefficient of thermal expansion a depend on temperature (Atefi et al., 2010a,c).

The Hooke's law including thermal tensions is in the form of Eq. (8) (Trostel, 1958):

(8)

Note that the strain tensor is expressed as , Substituting the stress tensor from (8) in to the equilibrium equation :

(9)

and

Also

(10)
(11)

and:

(12)
(13)
(14)

From Eq. (11):

Thermal stresses and heat conduction equations both are non-linear and hence an iterative method should be used for convergence. In this paper, a 16-node parallel processing system is used to decrease the computation time.

IV. DATA TRANSFORMATION

In this paper, a processing routine was developed to transfer the data between the finite volume and the finite element solvers to determine the values of the stress. In this routine, an interpolation scheme based on the 3D shape functions was employed to determine the correct values of the transferred parameters (Liu and Quek, 2003). Once the points in 3 dimensions were identified by the search algorithm, they are mapped from the physical to the computational domain (Fig. 3) and the unknown variables are then calculated by the method of interpolating the 3D shape functions using the Newton method for nonlinear system of equations (Liu and Quek, 2003). This mapping is one by one since there is one and only one point in the computational domain corresponding to each point in the physical domain. To interpolate the data for the unknown variables, the values of x, h and z are first evaluated using Eq. (15):

(15)


Fig. 3. An eight-nodal hexahedron element and the coordinate systems.

where i indicates the coordinate system of a cell in the old computational domain, Ni(x,h,z) is defined as:

(16)

As shown in Fig. 3, the values of x, h and z are known and between -1 to 1. The unknown are x, h and z. After substituting the values and numerical solution using the Newton method with enough accuracy, these unknowns are determined. Since this mapping as one by one, the calculated values for x, h and z are necessarily in the range of [-1, 1]. The flow variables are determined using:

(17)

where 's are the flow variables for the points 1 though 8 in Fig. 3 and the values determined for are the flow variables such as velocity, temperature, pressure, for the identified points (Liu and Quek, 2003).

V. PROBLEM GEOMETRY AND BOUNDARY CONDITIONS

The general geometric configuration under consideration is shown on Fig. 4(a). This configuration contains three turbine blades which turbine number 2 has steam-cooled channels. Cooling the blade is done by passing high pressure steam through the channel. For this propose, 22 circular channels are improvised in this blade which contain two set of holes with different diameters. The diameters of the small holes are 1mm and the larger ones are 1.5 mm that is shown in Fig. 4(b).


Fig. 4: (a) Geometric configuration of solution domain. (b) Cooling channel arrangement of the cooled blade.

The height of the blades is 76.7 mm and the blades spacing is 55.2 mm. The field solution is considered as a rectangular channel of 446 mm length which the dimensions of the field solution entry are 191×76.7 mm and at exit is 111×76.7 mm. Boundary condition for external flow and steam-cooled channel is shown in table 2 and 3. To validate the present results the geometry parameters and the boundary conditions were chosen according to Krüger et al. (2001).

Table 2: external flow boundary conditions and characteristic numbers.

Table 3: cooling channel boundary conditions

VI. NUMERICAL RESULTS

To check the accuracy of the present method, the results are compared with the experimental data. The surface pressure and temperature, temperature distribution inside the blade, the temperature distribution along a section line starting upstream the blade near the leading edge, extending to the inside region ending at the center of leading edge hole have been compared with those of Krüger et al. (2001).

Figure 5 show pressure distribution over the surface of blade number 2 at 50% height of blade. Good agreement is observed between the present results and those of reference (Krüger et al., 2001). As expected, the maximum pressure value occurs at the leading edge, and the minimum value at the suction side of the blade.


Fig. 5: Surface pressure distribution at 50% height of coolant blade and compare with ref. (Krüger et al., 2001)

In Fig. 6 the temperature distribution over blade surface at different heights are compared with Krüger et al. (2001). As can be seen the present results are in a good agreement with the experimental data. Figure 7(a) shows temperature contour at section of 50% height of blade number 2. As can be seen, the maximum temperature occurs at trailing edge of the blade and the minimum is in the cooling channels, which grows radially. Figure 7(b) shows the temperature contour over 3-D blade. It can be seen that in absence of the cooling channel, the maximum temperature occurs at the trailing edge. To show the temperature continuity and validity of the calculations over a line at leading edge region of the blade shown in Fig. 4(b), temperature values are present in Fig. 8. Temperature variation from leading edge to S1 channel decreases linearly from 1050° K to 950° K. The linear variation of the temperature indicates that the thermal conductivity variation in steady state does not affect the blade temperature distribution (Atefi et al., 2010c).


Fig. 6: Surface temperature at different blade height and compare with ref. (Krüger et al., 2001)


Fig. 7: (a) Temperature contour at 50% height of coolant blade. (b) Temperature contour on 3D blade


Fig. 8: Temperature along section line at the leading edge (Krüger et al., 2001)

Figures 9(a) and 9(b) show the pressure and Mach number contours at 50% height of cascade respectively. According to Fig. 9(a), the pressure at the suction side is lower and at the pressure side is higher than the reference pressure, as is expected. The Mach contours in Fig. 9(b) shows the stagnation point at the leading edge. The flow reaches to its maximum velocity at the upper surface and then decelerating to the rear stagnation at the trailing edge. Note that the Mach number growth on the upper surface was not more than 16% inlet Mach number due to the high temperature on the blade. To calculate the thermal stresses over the blade, both temperature dependent and temperature independent material properties are considered. Cubic elements, shown in Fig. 10, are used to calculate these stresses.


Fig. 9: (a) Pressure coefficient contour at 50% height of coolant blade(b) Mach number contour at 50% height of coolant blade.


Fig. 10: Numerical grid for stress analysis (FE solver).

Figure 11(a) shows the Von Misses stresses due to the thermal loads, when the material properties are considered constant. In this case the maximum value of stress, that is 399 Mpa and located at trailing edge on 5% section, occurs near the holes at the end of the blade as expected. The minimum value is 5.89 Mpa and located at S9 on 95% section. For variable material properties, the aforementioned stresses are shown in Fig. 11(b). The maximum and the minimum values of stress are located at the same locations as those for the constant properties material. Note that for the variable properties material, the maximum and minimum values of stress due to thermal loads were 409 and 6.104 Mpa respectively, while for the constant properties case, these values were 399 and 5.89 Mpa. Generally, the maximum and the minimum values of the stress due to the thermal loads for the blades with temperature dependent material properties are higher than those with temperature independent properties. This result has been previously observed by Atefi et al. (2010a,c) and Allen (1982).


Fig. 11: (a) Stress contour in coolant blade with constant properties due to thermal loads (b) Stress contour in coolant blade with temperature dependent properties due to thermal loads.

For a better view of the stress variation due to the thermal loads, the sections at height 5%, 50% and 95% are cut. In the subsequent figures, the stress variation due to the thermal loads along the direction normal to the blade height will be presented. These variations are considerable and play an important role in crack growth.

Figures 12(a) and 12(b) show the Von Misses stresses due to the structural loads at the blade pressure and suction sides respectively. According to these figures, the leading edge experiences the maximum stress which is about 9.5 Kpa. The minimum stress occurs at the trailing edge and is about 0.1 Kpa.


Fig. 12: Stress contour in coolant blade due to structural loads

Figures 13 to 15 show the Von Misses stress contours at the sections 5%, 50% and 95% for both constant material properties and temperature dependent cases. At the section 95% of the blade height, according to Fig. 4(b), a path between two channels (section 'A-A') is considered and the variation of stress on this path is calculated and is shown in Fig. 16. The normal axis on this figure is Misses stresses and the horizontal in dimensionless distance S/L. According to this figure, the maximum and the minimum Misses stress values in the case of temperature dependent properties are more than the constant properties ones.


Fig. 13: (a) Stress contour at 5% height of coolant blade with constant properties (b) Stress contour at 5% height of coolant blade with temperature dependent properties.


Fig. 14: (a) Stress contour at 50% height of coolant blade with constant properties (b) Stress contour at 50% height of coolant blade with temperature dependent properties.


Fig. 15: (a) Stress contour at 95% height of coolant blade with constant properties (b) Stress contour at 95% height of coolant blade with temperature dependent properties.


Fig. 16: Stress field at A-A section at 95% height of coolant blade.

VII. CONCLUSION

In this paper, a new CHT method is used based on FEM/FVM for solving flow field and conductive heat transfer and thermal stresses on a turbine blade. In this new method, the thermal stress field has been calculated by obtaining the necessary boundary conditions from simultaneous solution of the flow field and the heat transfer equations. In addition to the conventional constant properties material for the turbine blade, in this new method, all of the material parameters were considered as temperature dependent. Since this temperature dependent behavior was completely modeled in the present calculations, the critical values of temp and stress on the blade, obtained by the present method, can be used to estimate the blade life time with a high accuracy. The results have been compared with the available experimental data and remarkable agreement was achieved, approving the accuracy of the present method. The results show that the maximum and minimum values of the thermal stress for temperature dependent materials are higher than the constant properties ones. Since all blade parameters vary with temperature in actual cases, to attain higher estimation accuracy, this temperature dependency should be considered in the associated calculations.

According to the results, among the stresses developed on gas turbine blades, the thermal stresses are the dominant ones and in the presence of high temperature gradients and in absence of the centrifugal forces, the stresses associated with the structural loads can be neglected.

NOMENCLATURE

A (mm2) area
cp (J/kg K) specific heat
D strain tensor
etot specific total energy
ex , ey , ez unit vectors
δ unit tensor
E, E0 (N/mm2) young's modulus
E, F, G flux vectors (body fitted coordinate)
f ( N/m3) body forces
flux vectors (Cartesian coordinate)
J cell volume
K (W/m K) thermal conductivity
mass flow
q heat flux vector
qx ( W/m2) x- component of heat transfer rate per area
R, S, T fluxes
Re Reynolds number
t (s) time
T ( K) temperature
U vector of conservative variable
u (m) displacement vector
u, v, w ( m/s) velocity in Cartesian coordinate
$ ( N/mm2) stress tensor
x, y, z Cartesian coordinate
y (mm) distance from wall
y+ non dimensional wall coordinate
a (1/K) coefficient of thermal expansion
ε temperature dependent parameter
ρ ( kg/m3) density
μ ( kg/m s) dynamic viscosity
v Poisson' ratio
τij ( N/mm2) components of shear stress tensor
ξ , η body-fitted coordinates
Ni(x,h,z) Shape function
xp, yp, zp Coordinate in physical domain
xi, yi, zi Coordinate in comp. domain
(Val)p Value of variable in physical domain
(Val)i Value of variable in comp. domain

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Received: February 20, 2011.
Accepted: August 15, 2011.
Recommended by Subject Editor Eduardo Dvorkin.

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