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1 Proceedings of ASME Turbo Expo 2008 Power of Land, Sea, and Air June 9-13, 2008, Berlin, Germnay Draft Paper-GT Optimum design and sensitivity analysis of axial flow compressor with combination of analytical method, qualitative and quantitative rules and genetic algorithm Mohsen Reza Soltani Graduate Student School of Mechanical Engineering Sharif University of Technology Tehran, , Iran Hiwa Khaledi Research Engineer Power & Propulsion Group Middle East Petrogas (MPG) Company Tehran, Iran Mohammad Bagher Ghofrani Associate Professor School of Mechanical Engineering Sharif University of Technology Tehran, , Iran sharif.edu ABSTRACT Simulation and prediction of gas turbine performance is a very important issue in design process or in actual behavior analysis. In these models physical behavior of components such as compressor, combustion chamber and turbine are simulated related to each other. Compressor is the most important part of simulation. The paper presents a model for simulating compressor using stage stacking procedure with aid of genetic algorithm. The most important feature of the proposed method is that qualitative and quantitative rules based on turbo-machinery knowledge of compressor are used as constraints to the genetic algorithm to find the corrected situations of design. The model is based on an analytical solution and provides an insight into the effects of choices made during the compressor design process on performance and off-design stage matching. The results of the model highlight the capability of the method which accurately reproduces the available data. In addition to obtaine design conditions, this model can find and calculate stages that are highly loaded and this information is vital to control compressor. Nomenclature C Air velocity (m / sec) H Enthalpy per unit mass ( Kj / Kg) U Blade speed (m / sec) W Work per unit mass ( Kj / Kg) NomenclatureSubscripts 0 Stagnation value 1 Value at rotor inlet 2 Value at rotor outlet i Number of any stages of the compressor a Axial Greek symbols Flow coefficient φ ψ η α β Δ ζ INTRODUCTION Pressure rise coefficient Efficiency Air absolute angle Air relative angle Changes of a variable Temperature coefficient COMPRESSOR MODELING The axial flow compressor consists of a series of stages, each stage comprising a row of rotor blades followed by a row of stator blades. The working fluid is accelerated by the rotor blades and then decelerated in the stator blade passage where in 1 Copyright 2008 by ASME

2 the kinetic energy transferred in the rotor is converted to static pressure. Applying the steady flow energy equation to the rotor and considering the change in angular momentum of the air in passing through the rotor, the following expression for power input to the stage can be deduced.(fig (1)) T 02 W = m c T 01 P ( T ) dt = mu ( C W 2 C W1 = muca(tanα2 tanα1) Or: H02 H01 = UCa(tanα2 tanα1) (2) = UCa(tan β1 tan β2) By considering velocity diagram, equation (2) can be rewritten H 02 H01 Ca = 1 (tanα 1 + tan β 2) (3) 2 U U C The term a is known as the flow coefficient (φ ) and U H H 02 U 2 01 as the temperature coefficient (ζ ). ) (1) Fig (2) Stage characteristic STAGE MATCHING The overall performance of a gas turbine engine is governed by the performance characteristics of its constituent components and by the laws of compatibility of continuity, energy and momentum (axial and angular), which determine the interaction or matching between these conditions. It is essential that all individual stages of a compressor operate in the region of high efficiency without encountering either stall or choke at normal operating condition. For achieving correct matching of the stages the operation of several identical stages must be considered in series. The procedure is known as stage stacking. The stage stacking method is a simple one-dimensional calculation. Details of the stage stacking procedure can be found in the works of Robbins and Dugan [5], Stone [6], and Doyle and Dixon [7]. In Fig (3) schematic of this method is shown briefly. Fig (1) Velocity diagram of stage The overall performance of a stage can be expressed in terms of flow coefficient, temperature coefficient and pressure coefficient. Pressure coefficient is the product of the stage efficiency and temperature coefficient. ψ η = (4) ζ The ψ -φ curve is drawn for the case where the efficiency is maximum and it was named stage characteristic. Fig (3) schematic of Stage-Stacking method There are various types of procedure available for stage matching. Two procedures are described in this paper. One of the common procedures sets all the stages to operate at the peak stage efficiency at the design speed. Such a procedure is used in designing most commercial and industrial compressors because engine performance must be guaranteed at the design point and manufacturers have generally been reluctant to sacrifice design point performance for improved off-design performance. In this procedure with changing in inlet condition and reducing speed, the pressure ratio of stages and total pressure ratio decrease in comparison with design condition. The drop off in efficiency with reduced speed is explained in Fig (3) where the 2 Copyright 2008 by ASME

3 stage pressure ratio is plotted against the flow parameters for constant percentage of inlet stage design equivalent speed. The brokelines indicate the operating range of the first, eighth, twelfth, and sixteenth stages. Stage pressure ratio Flow coefficient 50% 70% 80% 100% stage 1 stage 8 stage 12 stage 14 Fig (3) Stage operation at first procedure at 50 and 100 percent of design speed It can be seen from Fig (3) that first stage operate at peak efficiency at design speed but with reducing speed the flow coefficient decrease and angle of attack increase. Stage twelve was found to operate over the minimum range of angle of attack. This suggests that in such a compressor the twelfth stage could be highly loaded. The sixteenth stage operates at approximately peak efficiency at design speed but as speed decrease the flow coefficient increase and the angle of attack decrease. It can be understood that this procedure of matching is good for design point and has a high efficiency but at off-design point total efficiency and pressure ratio drop off. The first compressor that was modeled in this paper was assumed to design with this procedure. Because all of the stages operate at peak efficiency ( η d ), it can be assumed that the flow coefficients are equal for all stages and the pressure coefficient are equal too. In the second procedure that described in this paper, each stage was set to take full advantage of its efficient range of angle of attack by operating at a different value of flow coefficient or angle of attack at the design speed as indicated in Fig (4). The first stage was set to operate to the right side of the peak efficiency, or high flow coefficient and low angle of attack and the exit stage was set to operate at the low flow coefficient or high angle of attack. The intermediate stages were set between these limits such that the middle stage operated at the peak efficiency point. Fig (4) Design stage operating point for second procedure This procedure improves off-design performance. For example the axial velocity of LM2500 compressor is decreased as it is shown in Fig (5). Axial Velocity (ft/s) Fig (5) Axial velocity distribution of LM2500 compressor [ ] GENETIC ALGORITHM Genetic algorithm is a subset of evolutionary algorithms that model biological processes to optimize highly complex cost functions. A genetic algorithm allows a population composed of many individuals to evolve under specified selection rules to a state that maximizes the fitness (minimize the cost function). Genetic algorithm simultaneously searches from a wide sampling of the cost surface and deals with a large number of parameters. FINDING AND APPLYING QUANTITATIVE AND QUALITATIVE RULE FOR MODELING COMPRESSOR Two types of compressor are modeled in this paper. The first one is industrial gas turbine and first procedure is assumed to simulate it. The general characteristic of first compressor is as follow: Number of stages 12 Total pressure ratio 7.75 Weight flow per sq ft area 33.5 Tip speed 950 ft / s Absolute inlet air angle In this compressor the tip diameter is constant. According to the previous section, the flow coefficient and pressure coefficient and efficiency are equal for all stage. In the computer program that is developed for modeling compressor some parameters that are necessary must be guessed. These parameters are stage pressure ratio, flow coefficient and efficiency and they set as algorithm genetic chromosomes. After guessing these parameters, they enter the model as inlet condition and other parameters calculated and compressor is simulated stage by stage. For modeling compressor some criteria must be considered. It can be seen from equation (1) that to obtain a high temperature rise in a stage the designer combine : High blade speed High axial velocity High fluid deflection ( β1 β2 ) in the rotor blade. Blade stress limits the blade speed, so the tip speed of blade must be considered in modeling. Increasing the axial velocity, increase the Mach number and consequently increase losses in the compressor, so it must be checked. 3 Copyright 2008 by ASME

4 High fluid deflection implies a high rate of diffusion, so it must be assessed for allowable diffusion. One of the earliest V criteria used is the de haller number, defined as 2 > V1 For the second compressor, second procedure was assumed. In this compressor, each stage has specific flow coefficient. So the number of variables causes some problem such as taking long time to converge or not converging and finding wrong answers. The general characteristic of first compressor is as follow: Number of stages 12 Total pressure ratio 7.73 Weight flow per sq ft area Tip speed 950 Absolute inlet air angle As it said before in this section some qualitative and quantitative rules are developed to find accurate answers. In the stage matching section it is shown that flow coefficient in the early is more than and flow coefficient in the later stage is less than optimum point. So this variation can be estimated as a linear function: y = ax + b (5) By estimating like this, the number of variables reduces to two instead of 12 ( N stage ) but in reality flow coefficient of stages may not be follow this function and have some fluctuations. Thus, a tolerance is assumed for each stage and function change: y = ax + b + e( i) i = 1,2,..., N stage (6) The term e(i) is a tolerance for each stage. Now the number of variables is 2 + N stage instead of N stage, but it helps model to find answers accurately. Another rule is estimating pressure ratio for each stage. As it said in previous sections the intermediate stages operate over the minimum range of flow coefficient in various speed, as a result, they could be highly loaded and stage pressure ratio for middle stage is higher than early and last stages. Stage pressure ratio can be estimated with two linear functions with having tolerance in each stage. (Fig (6)) y = ax + b + e( i) x x m i = 1,..., N stage (7) y = cx + d + e( i) x x m The term x m is a stage that pressure ratio is higher than other stages. Term e(i) is a tolerance for each stage. The number of variables increases to 4 + N stage but it helps problem to converge easier. Pressure ratio Fig (6) Estimated function for stage pressure ratio Efficiency can be simulated like stage pressure ratio and flow coefficient. It can be seen from Fig (4) that how efficiency vary and it can be simulated with polynomial function, The range of variation of parameters is as important as the type of variation in the first compressor flow coefficient is constant for every stage and it vary between 0.6 and In the second compressor flow coefficient isn t constant and they vary more widely between 0.5 and 0.8. The range of variation of stage pressure ratio depends on gas turbine type and its application. Stage pressure ratio of air derivative gas turbine is higher than industrial gas turbine; anyway, they vary between 1.13 and In addition, the stage that has a highest pressure ratio ( x m ) must be guessed. The variation of this stage depends on the number of stages and of course it must be in the middle stages. For this case it assumed to be between stage 3 and 8. Efficiency varies between 0.85 and For the first compressor, efficiency is constant and for the second compressor, it varies like Fig (4). In addition to indicate the type of variation and range of variation of variables, that they are the chromosome of Genetic Algorithm, Selecting cost function has important effect on problem convergence. Cost function can be defined like this: n X m m Qc Qm min Fob w = i Q i = 1 m i The term Q c and Q m are calculated parameter and measured parameter. n m is the number of measured data and w i is factor for indicating the importance of variables. General characteristic of compressor like air mass flow, inlet pressure, inlet temperature, total pressure ratio and sometimes a cross sectional layout of compressor can be found in literature. Outlet temperature and compressor work can be calculated with zero dimensional model (thermodynamic model).[khaledi]. In this computer program mass flow, inlet pressure and inlet temperature are as inlet variables to program. Hub-tip ratio can be measured from cross sectional layout and they enter program as cost function variable ( Q m ). Also the total pressure ratio and temperature outlet is cost function variable too. These parameters don t have effect on calculation and they are suitable for comparing. In Fig (7) and Fig(8) hub-tip ratio for first and second compressor are shown, it is clear that computer program converg and calculated data are fully adopted on the measured data. 2 N stage 4 Copyright 2008 by ASME

5 hub-tip ratio hub-tip ratio Fig (7) Hub-tip ratio for first compressor Fig (8) Hub-tip ratio for second compressor RESULTS AND DISCUSSIONS The stage pressure ratio and pressure coefficient for the first compressor are shown in Fig (9) and Fig (10). It can be seen that calculated data has a little difference with available data and adopted with assumptions (rules). In this compressor because efficiency and flow coefficient are constant for each stage the pressure coefficient remains constant. Pressure ratio In the second compressor limitation and complexity is more than first compressor. Stage pressure ratio, pressure coefficient and flow coefficient are shown in Fig (11), Fig (12) and Fig (13). As it can be seen from these results, calculated data are approximately adopted with available data with good precision and it shows the correctness of assumptions (rules). As it is shown in Fig (11), it has a peak in stage 7 and it represents that this stage is highly loaded. In addition, flow coefficient decrease linearly and assuming tolerance for each stage helps program to simulate compressor accurately. Pressure Coefficient Flow Coefficient Pressure ratio Fig (11) Stage pressure ratio for second compressor Fig (12) Pressure coefficient for second compressor Pressure coefficient Fig (9) Stage pressure ratio for first compressor Fig (13) Flow coefficient for second compressor In this paper the rotor absolute inlet angle is assumed 22.5 for each stage (This assumption isn t bad and for nearly most of the compressors this angle is close to this number). By assuming this the rotor relative angle enter to rotor is shown in Fig (14). as it can be seen, this angle increase and it calculated for meanline Stage number Fig (10) Pressure coefficient for first compressor 5 Copyright 2008 by ASME

6 Relative Angle (Deg) Rotor Fig (14) Rotor Relative Inlet Angle for second compressor In CF6-6 compressor the rotor relative inlet angle increase too. (Fig (15)) Rotor Relative Inlet Angle Tip Hub As a result, relative angle enter to rotor can be simulated by linear function with small tolerance (like flow coefficient function). CONCLUSION In the present work the compressor is modeled and the unknown parameters such as flow coefficient, pressure coefficient, stage pressure ratio and efficiency are guessed. The inlet condition is limited to general characteristic such as mass flow, total pressure ratio and cross sectional layout. The characteristics of the proposed method is that the unknown parameters are determined by combining a stage stacking method and by using genetic algorithm as minimization tool. REFRENCES [1] J.Kurzke, 2003, "Model Based Gas Turbine Parameter Corrections ", ASME Turbo Expo [2] J.Kurzke, 1995, "Advanced User Friendly Gas Tuebine Performance Calculations on a Personal Computer ", International Gas Turbine and Aerospace Congress [3] J.Kurzke, 2002, "Performance Modeling Methodology : Efficiency Definitions for Cooled Single and Multistage Turbines", ASME Turbo Expo. [4] Philip P.Walsh, 2004, Gas Turbine Performance, Blackwell Scinence [5] Stone, A., 1958, "Effects of Stage Characteristics and Matching on Axial Flow Compressor Performance", Transactions of the ASME, Vol. 80, pp [6] Robbins, W. H., Dugan, J. F., 1965, "Prediction of Off- Design Performance of Multi-Stage Compressors", NASA SP-36. [7] Doyle, M. D., Dixon, S. l., 1962, "The Stacking of Compressor Stage Characteristics to Give an Overall Compressor Performance Map", The Aeronautical Quarterly, pp [8] Novak, R. A., and Hearsey, R. M. 1977, A Nearly Three Dimensional Intrablade Computing System for Turbomachinery, J. Fluid Eng., Vol.99, p [9] Denton, J. D., 1975, A Time Marching Method for Twoand Three-Dimensional Blade-to-Blade Flow, Aeronautical Research Council (U.K.) R & M [10] H.Cohen, G.F.C. Rogers, H.I.H. Saravanamuttoo, 1996, Gas Turbine Theory [11] J. H. Holland, Adaptation in natural and artificial systems, Ann Arbor, MI: University of Michigan Press, Michigan, 1975 [12] Benser, W. A. Aerodynamic Design of Axial Flow Compressors. Chapter ХШ. N.A.C.A. RM 56 BO3b, [13] J.H.Kim, T.S.Kim, J.L.Sohn and S.T.Ro, 2003," Comparative Analysis of Off-Design Performance Characteristic of Single and Two-Shaft Industrial Gas Turbine ", Transaction of the ASME [14] Ricardo Procacci,Franco Rispoli, "Off Design Performance Evaluation of Deteriorated Variable Geometry Axial Flow Compressors",ASME Cogen-Turbo Power Conference,1995. [15] Muir, D. E., Saravanamuttoo, H.I.H., Marshall, D.J., 1989, Health monitoring of variable geometry gas turbine for the Canadian navy, ASME Journal Of Engineering for Gas Turbine and Power, vol.111, PP [16] Tabakoff, W., Lakshminasimha, A. N., and Pasin M., 1990, Simulation of Compressor Performance Deterioration Due to Erosion, Transactions of the ASME, Vol. 112, PP, [17] Finger, H. B.Dugan, J. F., Analysis of Stage Matching and Off-Design Performance of Multistage Axial Flow Compressor. NACA TN 2059,1950. [18] Benvenuti E., Casper R., Development of High- Speed,High Efficiency Power Turbine for the LM2500+,ASME Turbo Expo 1995 Technical paper # 95- GT-410. [19] J. F. Klapproth, M. L. Miller, D. E. Parker, Aerodynamic development and performance of the CF6-6/LM2500 compressor, AIAA [20] R. L. Haupt, S. E. Haupt, Practical genetic algorithms, John Wiley and Sons Inc., New York [21] Tabari, A., Khaledi, H. and Banisi, A. H., 2006, Comparative Evaluation of Advanced Gas Turbine Cycles with Modified Blade Cooling Models, ASME Turbo Expo. 6 Copyright 2008 by ASME