Abstract
The work presented here predicts centrifugal stress along blade aerofoil from root to tip at different radial sections. It also presumes a value for thermal stress based on the literature available. Based on centrifugal stress, thermal stress and life requirement it predicts the permissible blade metal temperature from L-M parameter for different radial sections. Further it estimates the external flow convective heat transfer coefficient along blade profile from leading edge to trailing edge on both suction and pressure sides of different sections of blade aerofoil from root to tip.
1. Introduction
From thermodynamic analysis it is clear that an increase in permissible turbine inlet temperature has strong effect on gas turbine performance and mainly it results in an increase in specific power output (or increased specific thrust in the case of aircraft propulsion units) and decrease in specific fuel consumption. But for a particular operating condition of the turbine components the increase in inlet temperature reduces the intended life and sometimes the components temperatures may exceed the metallurgical limits of the component materials, which may lead early failure of components. Therefore to allow higher turbine inlet temperature or to obtain increased life at a particular turbine inlet temperature or both, it is required to cool the critical components.
The technology of turbine cooling was recognised by some almost from the inception of the first turbojet engine. Cooling studies were first performed in the 1940 and many investigations were carried on in the 1950s. Around 1960, turbine cooling was first used in a commercial aircraft engine. Since that time, there has been a very rapid rise in turbine inlet temperature that has placed an even greater emphasis on turbine cooling. A continuous improvement in high-temperature materials has also helped to increase the turbine inlet temperature.
For the blade of a particular material the estimation of blade material temperature at any section, depends on the net stress on that section and life requirement. Rotor blades are subjected mainly with four types of stresses; thermal stress, centrifugal stress, centrifugal bending stress and gas bending stress. Among these stresses thermal and centrifugal stresses are dominating in nature.
Based on these stresses and life requirement, the allowable maximum material temperature at any section can be calculated from the mechanical properties of the material. To maintain the temperature of the section at or below this allowable temperature, an efficient cooling system must be designed. This calls for the prediction of heat transfer coefficients for the heat transfer from the hot gas to blade.
The present work has been carried out for a typical low-pressure axial flow gas turbine rotor blade. Centrifugal stresses have been calculated from root to tip of the blade at different sections using theoretical method. A factor of safety of ‘2' has been taken for the calculation of maximum possible stress out of centrifugal stress. This takes care of all the possible additions in the centrifugal stress due to other sources of stress and unforeseen variations in operating conditions. Larson and Miller parameter (L-M parameter) plot, for the material of the blade, has been used to predict the allowable blade metal temperatures at different sections for a particular life requirement. Different correlations have been used to estimate the external flow heat transfer coefficients based on the area of exposure of the blade material to the hot gas atmosphere and nature of flow of gas over the blade. For the turbine blade as the gas passes through it, gas properties vary from root to tip and also from inlet to exit across the blade. The present work has been dedicated for the estimation of heat transfer coefficients at three different critical sections of the blade namely Hub, Mean and Tip. The predicted results show a good compatibility with the literature available.
2. BLADE MATERIAL AND ITS STRENGTH
Gas turbine blades are exposed to a very severe thermal atmosphere. The temperature is so high that it is fairly much more than the melting points of the common high-strength materials. Besides high temperature the requirement of durability is also another factor, which makes common materials unsuitable for use. Only super alloys may be suitable for this purpose. But the current trend of continuously increasing the turbine entry temperature attracted the concentration of the designers not only towards the new materials with well-improved mechanical and thermal properties but also to restrict the temperature of the blade material by its proper cooling. So, the material should have sufficient strength to face the operating situations.
2.1. Blade properties
Blade material, MAR-M-247 (NiCo), is a Ni-based super alloy. It's a casting alloy developed for applications requiring high strength at elevated temperature up to about 1900k. The chemical composition of the material is listed below.
Thermal conductivity of the material is temperature dependent. Fig. 1.5 shows the variation of thermal conductivity with respect to temperature.
2.2. Strength of blade material
In ordinary temperature conditions the strength of the material under constant loads is estimated by tensile strength or yield strength. At high temperatures under action of constant loads in ordinary structural materials there appears the phenomenon of creep. It occurs as a result of prolonged exposure of materials to high stresses at high temperatures. This is particularly a acute problem on highly stressed rotating turbine blades and it occurs in the form of slowly and continuously developing plastic deformation. And excess of this plastic deformation causes the failure of the component. It is observed that at constant stress the higher the temperature the more quickly proceeds the process of creep i.e. the lesser the life of the component. It means that at a particular stress lesser will be the temperature higher will be the life of component. Therefore life of the component is a function of working temperature and stress. Hence to maintain the life of the component at a desire value it is required to lower the temperature of the component.
Gas turbines operate in conditions of high temperatures and therefore in highly stressed components like rotor blades there appears the phenomenon of creep. Therefore for these cases where creep is the main criterion behind component failure the ultimate tensile stress is defined as the stress at which the component fails at a certain working temperature after the expiry of a certain period of time. It means that the strength of the material subjected at high temperatures is a function of this temperature and its operational life.
2.3. L - M Parameter
There Larson-Miller parameter plot for MAR-M-247 (NiCo) has been shown fig. 1.6. This is a plot between the material rupture stress and a parameter ‘P' which is a function of working temperature, and life of the component. This parameter is called as Larson-Miller Parameter and is given by the following equation:
P = (T+460)(20+log t) x10-3
T = Test Temperature, F
t = Rupture Time, hr
The criterion of rupture of the material taken to draw the plot is creep. From this plot the allowable working temperature for a cross-section of the material can be estimated for a known value of stress on the section and life of the blade.
3. STRESSES IN ROTOR BLADE
Rotor blades of gas turbine are subjected to very high rotational speeds of the order of several thousand rpm and also are exposed to a variable thermal environment. Hence these blades are subjected to different types of stresses of different magnitudes and directions. As it is peculiar that the strength is a function of life and working temperature, The net stress at any section of the blade should not exceed the maximum allowable value. The control on the blade metal temperature is the only way to sustain the stresses for the designed life of the blade for a specific operating condition and life requirement. Therefore to know about the cooling requirement, stresses should be predicted correctly on the blades at different sections.
There are mainly four types of stresses with that rotor blades are being subjected;
3.1. Centrifugal tensile stress
Centrifugal stress in the rotor blade is due to the rotation of the blade. It is tensile in nature. This is the largest in magnitude but not necessarily the most important because it is almost a steady stress. When the rotational speed of the blade is specified, the allowable centrifugal tensile stress places a limit on the annulus area but does not affect the choice of blade chord. This stress is the basic cause of the blade failure due to the creep.
3.2. Centrifugal bending stress
If the blade design is such that the centroids of all the blade cross-sections at different radii, taken perpendicular to the radial direction, do not lie in the same radial plane, centrifugal stresses arising in the blade will try to bend the blade. This type of stress arising due to the different directions of the centrifugal stresses in different blade sections is called as centrifugal bending stress. It will produce compressive stress in one side of the blade whereas tensile stress in the opposite side. Any torsional stress arising from these centrifugal stresses is small enough to be neglected. Thus this stress is very sensitive to manufacturing errors.
3.3. Gas bending stress
The force arising from the change in angular momentum of the gas in the tangential direction, which produces the useful torque, also tries to bend the blade about the axis of rotation of the blades. The stress arising due to this bending force is called as gas bending stress. There may be change of momentum in the axial direction and in reaction turbines there will certainly be a pressure force in the axial direction. All these two will produce a bending moment in the blade about the tangential direction. The gas bending stress will be tensile in the leading and trailing edges and compressive in the back of the blade and with tapered twisted blades either the leading or trailing edge suffers with the maximum value of this stress. This is a fluctuating stress and its value becomes maximum when the rotor blade passes through the leading edge of the stator.
3.4. Gas bending stress
Turbine blade is subjected to three-dimensional temperature gradients, along the blade height, along the blade profile and along the thickness of the blade.
Due to these temperature gradients the blade fibres tend to deform unequally. This unequal deformation causes mainly two types of stresses to set up in the blade, compressive and tensile. As the blade considered is un-cooled therefore the contribution of the stress due to the temperature gradient along the thickness of the blade in net stress is not appreciable and can be neglected. Usually with the cooled blade this source of stress is main among all the sources of thermal stress.
Again the thermal stress due to the temperature gradient along the blade height would not come in picture because the blade is free to expand along the height. Only the stress due to temperature gradient along the chord of the blade will contribute in net blade stress but its magnitude would not be much because the temperature gradient along the chord is not so high.
4. HEAT TRANSFER ANALYSIS
Among three modes of heat transfer namely conduction, convection and radiation, conduction and convection heat transfers are predominant in nature while talking about heat transfer from hot gas to blade and usually referred to as convective heat transfer. At temperatures encountered in low-pressure turbine stage radiation heat transfer may not be significant and therefore it has been neglected in the present work.
Convective heat transfer from hot gas to blade aerofoil is confined to the boundary layer region very near to the surface, where large velocity and temperature gradients are present. The critical elements in the process are the boundary layer developing on the surface, the free stream total temperature. Boundary layers which act as a buffer zone between the main stream and solid offer resistance to the heat transfer. Heat transfer occurs in this viscous layer between the blade and hot gas through the both conduction and convection mechanism. Once the heat has penetrated into the flow, the energy transport occurs mainly through convection by the moving media. Therefore the buffer region or boundary layer plays a very critical role in heat transfer. The condition and property of this layer determines the rate at which the heat is transferred. Free stream turbulence and local turbulence intensity has very high influence on heat transfer from hot gas to blade. Both prevent the growth of boundary layer over blade surface. The correlation developed by Krishnamoorthy and Sukhatme predicts that 12% increase in local turbulence intensity increases the heat transfer rate by approximately 75%.
The heat flux from the hot gas to the blade outer wall can be expressed as a product of gas side heat transfer coefficient and temperature difference between hot gas and blade outer wall. Therefore estimation of heat transfer coefficient is a prerequisite for calculation of heat transfer. The external heat transfer coefficient around the blade surface is affected by mainstream turbulence, mainstream acceleration and surface curvature.
The present work is dedicated to the estimation of external convective heat transfer coefficient for only three important sections of the rotor blade e.g. Hub, Mean and Tip.
4.1. Stagnation flow region
Hot combustion gases flow over the leading edge of the blade with high turbulence level. Literatures available present that this turbulence level could be very high (10-15%) due to intense mixing in the combustion chamber and in the present work it is taken as 15% considering the estimation of heat transfer coefficient on a safer side. Because leading edge is exposed to the higher temperature and turbulence than the other portions of the blade profile hence normally it is cooled. This cooling arrangement results in a thick leading edge, leading to large acceleration and curvature effects. Such sudden acceleration can bring about considerable changes in heat transfer rates. In many cases, the flow near the leading edge may separate. Furthermore, the stagnation point flow is very sensitive to free-stream turbulence and it has been found that even low turbulence levels can cause the formation of stream wise vortices that augment heat transfer. It was found that heat transfer near the leading edge is doubled when the turbulence is increased from zero to 2.2%. Even though the cylinder case does not represent the leading edge of a turbine blade, the qualitative effects are similar, and one can expect the trend to be the same in turbine leading edge heat transfer.
Correlation has been developed by Lowery and Vachon for flow past a heated cylinder under free turbulence and the same can be used to predict heat transfer in the stagnation region at the leading edge of the blade under free stream turbulence.
In the above correlation the Nusselt and Reynolds numbers are based on the diameter of the cylinder and in the case of blade it is based on the diameter of the circular arc of leading edge section profile. This correlation is valid for Tu Red½ from 0 to 40 and it under predicts Nu after Nu reaches a maximum value. The last two terms incorporate the effect of free-stream turbulence.
Daniels and Schultz show that their leading edge (HP blade) data agrees to within 10% of the value predicted from the above correlation.
4.2. Laminar, transition and turbulent flow region
For Krishnamoorthy and Pai carried out detailed experiments using scaled up model of the high-pressure turbine vane with 2D cascade under constant heat flux boundary condition to obtain the heat transfer coefficient distribution around the Hub, Mean and Tip section profiles. They observed the different regimes of heat transfer namely laminar, transition and turbulent boundary layer. They observed that the boundary layer is predominantly turbulent on the pressure surface of the vane whereas a small laminar boundary layer exists on the suction surface followed by a transition region. The rest of the surface is covered with turbulent boundary layer.
Based on the above observations the present work assumes only the turbulent boundary layer from leading edge to trailing edge on both suction and pressure sides of different blade section profiles.
They have fitted correlations for heat transfer coefficients for the experimental data obtained from cascade test results.
The correlation for turbulent region is:
As hot gas flows from leading edge to trailing edge of the blade, the gas properties gets changed from inlet to exit of the blade aerofoil due to a continuous change in pressure and temperature of the gas. Ekert has presented a correlation for a reference temperature for high-speed laminar or turbulent flow of a fluid over a flat plate. This temperature can be used for the calculation of fluid properties at any location over the plate and along the fluid stream without any appreciable error. This correlation can also be used for the estimation of reference temperature over rotor blade because flow past rotor blade have much similarity with the high-speed laminar or turbulent flow over the flat plate.
In the above equation, velocity is the local velocity and characteristic length is the distance along the profile from leading edge stagnation point to the point of calculation, for Reynolds number calculation and the fluid properties are evaluated considering high-speed flow over flat plate at Eckert's reference temperature as given below:
Here temperatures T0 and Ta are the total temperature in relative frame and free-stream temperature at the inlet of the blade section profile respectively.
‘r' i.e. recovery factor is equal to cube root of Prandtl number.
Adiabatic wall temperature or the effective gas temperature is a temperature that the wall would reach if it were uncooled, therefore a measure of the viscous heating in the boundary layer. Prandtl number has an effect on the adiabatic wall temperature because it is the ratio of viscosity (responsible for energy dissipation) to the thermal diffusivity (mechanism allowing heat to escape from the boundary layer). This would suggest that for a given free-stream kinetic energy, a high prandtl number should lead to a high adiabatic wall temperature, and vice versa.
To account for the mainstream gas turbulence, a turbulence correction (local turbulence intensity) can be introduced as:
Where Vin and Vlocal are the circumferential velocity in relative frame and local velocity along the profile of the blade, of the gas respectively.
In the turbulent region the heat transfer coefficient in presence of turbulence in free-stream is given by:
5. DESIGN INPUTS
Conventionally, Fig. 1.1 shows a typical low-pressure gas turbine rotor blade. The upper most portion of the blade in the flange shape is blade shroud. Fig. 1.4 shows the radial location of different blade section profiles. Fig 1.2 represents the general shape of three different sections namely Hub, Mean and Tip of the blade. it is apparent from these three profiles that from root to tip the blade becomes thinner and thinner.
Fig. 1.3 details the nomenclature applicable to the turbine blade e.g. pressure side, suction side, leading edge, trailing edge, stagnation point and chamber line.
The different design input for the typical blade has been listed below:
6. CALCULATION PROCEDURE
7. RESULTS & DISCUSSIONS
Blades centrifugal tensile stresses, the maximum allowable stresses and maximum allowable blade material temperatures and heat transfer coefficients at different sections of the blade have been tabulated from table 1.1 to 1.6.
Table 1.1 represents the value of centrifugal tensile stresses at five sections of the blade e.g. Root, Hub, below Mean, Mean, above Mean and Tip. The values have a good compatibility with the data available in the literature.
Table 1.2 represents the maximum allowable stresses in the rotor blade based on the factor of safety ‘2'. And from the L-M parameter plot it also predicts the values of maximum allowable blade material temperatures at different sections. The maximum permissible value of the blade material temperature goes on increasing from root to tip of the blade height because the stress decreases with the blade height.
Table 1.3 represents the convective heat transfer coefficients at three sections e.g. Hub, Mean and Tip. The result obtained here has very good match with the results available for the similar cases in literature.
Table 1.4 represents the convective heat transfer coefficients at mid of the blade section profile at different sections on both pressure and suction sides of the blade. The results obtained have a good match with the existing data in literature.
Table 1.5 represents the heat transfer coefficient at the trailing edge of different blade section profiles on both the sides of the profile. There is not much difference in the heat transfer coefficients values on both the side for a particular blade section profile.
Fig. 1.7 represents the variation of heat transfer coefficient at Hub section profile. On pressure side the heat transfer coefficient first drops and then increases near the trailing edge while on the suction side it drops almost continuously from leading edge to trailing edge.
Fig. 1.8 and 1.9 show the variation of heat transfer coefficients at Mean and tip sections respectively. The variation follows the same pattern as on hub section. All three curves show the alike variation of heat transfer coefficient.
This is clear from the results that for a particular blade section profile, the heat transfer coefficient and hence heat transfer is highest at the leading edge whereas it is minimum at the mid profile of the pressure side and in between at the trailing edge. At mid profile the heat transfer coefficient is more at suction side than that of pressure side. But at the trailing edge it is opposite.
8. SUMMARY AND CONCLUSION
From the present analysis it is peculiar that at any blade section profile the allowable blade material temperature is fairly below than the total temperature of the gas approaching to it. Hence to sustain the designed life of the blade it must be properly cooled.
Form the results it is clear that the allowable stress at root is maximum and the maximum allowable temperature increases from the root to tip of the rotor blade but its a common practice to keep the gas temperature minimum at the root and give the maximum load to the mean section of the blade and also the temperature available of the coolant (with cooled blade) is minimum at the root of the blade, hence the failure of the blade at root can easily be avoided. Therefore in the present work the calculation of heat transfer coefficients has not been done for the root section.
From the results it is obvious that the convective heat transfer coefficient is maximum at the leading edge of the rotor blade and hence it will be exposed to maximum heat transfer. And this value is maximum at the mean profile of the blade section. Hence with the uncooled blade the leading edge will tend to attain the highest temperature very close to the temperature of hot gas approaching the leading edge. This clearly indicates that the leading edge should be cooled properly.
The trailing edge of the blade is kept very thin because it cannot be made thick due to aerodynamic limitations. It is clear from the analysis that at a particular blade section profile the heat transfer coefficient is about half of that of the leading edge. Therefore heat transfer will also be very effective at trailing edge and since the cross section at this location is very thin, it should be effectively cooled to avoid the failure of the blade. Moreover, the pressure side heat transfer coefficient is more than that of suction side hence this side should be given more attention for cooling than to suction side.
At the mid chord of the blade section profile the heat transfer coefficients are not so high and also the maximum thickness of the blade occurs somewhere near the mid chord region of the blade, which provides the more bearing area for the stresses. Hence the cooling at the mid chord region is not so important.
The analysis carried out in the present work reflects that the cooling design should be such that it can provide very effective cooling at the leading and trailing edge region of the blade aerofoil whereas the cooling of mid chord region of the blade is not so important.
9. Acknowledgments
10. Acknowledgments should be inserted at the end of the paper, before the references, not as a footnote to the title. Use an unnumbered section heading for the Acknowledgments, similar to the References heading.
11. References
References in the text should be indicated by superscript Arabic numerals in square brackets that run consecutively through the paper, and grouped together at the end of the paper in numerical order as shown in these instructions [1].
The reference list should be set in the same typeface as the body of the text. Use a hanging indent of 7.5 mm to accommodate the numbers.
The format for references is as follows:
Last name, initial, year of publication, full paper title, journal name, volume, first and last page. Use only common abbreviations in journal names.
As we saw at “Referencing and avoiding Plagiarism” here are some examples of a reference list:
Element Ni Co W Cr Al Ta Hf Ti MoC Zr B
PercentageRest10.010.08.255.53.01.51.00.70.150.050.015
For books : surname, initials (year),“title”, publisher, place of publication, e.g.
