- Email: [email protected]
Optimum design of a subsonic axial-flow compressor stage
APPLIED ENERGY
Applied Energy 80 (2005) 187–195
www.elsevier.com/locate/apenergy
Optimum design of a subsonic axial-flow compressor stage Lingen Chen
a,*
, Fengrui Sun a, Chih Wu
b
a
b
Faculty 306, Naval University of Engineering, Wuhan 430033, PR China Department of Mechanical Engineering, US Naval Academy, Annapolis, Md 21402, USA Available online 6 May 2004
Abstract The design of an axial-flow compressor stage for subcritical Mach numbers has been formulated as a non-linear multi-objective mathematical programming problem with the objective of minimizing the aerodynamic losses and the weight of the stage, while maximizing the compressor’s stall margin. Aerodynamic as well as mechanical constraints are considered in the optimization solution. The prediction model for estimating the performance characteristics, such as efficiency, weight and stall margin, of the compressor stage is presented. The present design optimization procedure can be applied to a multi-stage compressor. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Axial flow compressor; Optimum design; Efficiency; Weight; Stall margin
1. Introduction The main design requirements of the compressor stage involve an acceptable level of thermodynamic efficiency, adequate surge margin, and a reduced weight, etc. However, the weight reduction inevitably causes loss of efficiency, and the higher efficiency will lead to decreases of the surge margin. As a result, the efficiency and surge-margin improvement, and the weight reduction must be compromised according to a certain design criterion. This is especially important in the design of a multi-stage axial-flow compressor. It is highly desirable that optimization techniques be applied in the design of compressor stages in order to obtain improvements. *
Corresponding author. Tel.: +86-27-83615046; fax: +86-27-83638709. E-mail addresses: [email protected], [email protected] (L. Chen).
0306-2619/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2004.03.008
188
L. Chen et al. / Applied Energy 80 (2005) 187–195
Much work has been reported on the application of optimization techniques to axial-flow compressor design, see [1–15]. In this paper, weight and stall margin are considered together with efficiency as the optimization objectives: an optimum design method for a subsonic axial-flow compressor stage is provided. 2. Performance evaluation of an axial compressor stage Figs. 1 and 2 define the important geometrical data, the velocity triangles and reference stations within an axial compressor stage at the subcritical Mach number under consideration. The performance evaluation method provided by Casey [16] is adopted in this paper. This method is a relatively simple mean-line prediction method, in which a one-dimensional (1D) analysis of the velocity vectors at the root 0:5 mean square (rms) radius, r ¼ ½ðrt2 þ rh2 Þ=2 , where rt and rh are the tip and hub radii, gives the aerodynamic loading of stage, and well-established empirical correlations are used to determine the losses and the operating range. The correlations used in the method have been selected on the basis of simplicity and appropriateness and have been modified where necessary to give better agreement with the many test cases analyzed. 2.1. Incidence, deviation and operating range The reference incidence angle iref are calculated from the equations and diagrams given by Lieblein [17]. The deviation angle at the reference incidence angle is calculated using a modified form of Carter’s rule [18]. The variation of deviation angle ðdbi Þ with incidence is calculated in the manner suggested by Lieblein [17]. The operating range ðdbÞ of a cascade is calculated as follows: db ¼ dbi ðdb=dbi ÞMa , where dbi is a function of solidity ðrÞ, blade camber angle ðU0 Þ and inlet blade angle ðb01 Þ of the rotor blade: ðdb=dbi ÞMa ¼ 1 if Ma < 0.2 and ðdb=dbi ÞMa ¼ 10A if Ma > 0.2, where A ¼ 2:5(Ma)0.2)4:4 . 2.2. Profile losses The total pressure profile-loss coefficients of rotor and the stator are calculated as follows: x ¼ xi ðx=xi ÞRe ðx=xi Þinc ðx=xi ÞMa ; ð1Þ where xi is the incompressible total-pressure-loss coefficient of the blade profile at the rms radius and at the reference angle [14], ðx=xi ÞRe Þ is the correction for Reynolds number and surface roughness [19], ðx=xi Þinc Þ is the correction for incidence [20] and ðx=xi ÞMa Þ is the correction for Mach number [20]. 2.3. Stage efficiency The efficiency that would exist if there were no end-wall losses is gpro ¼ 1 ðxr W12 þ xs C22 Þ=½2uðC2u C1u Þ;
ð2Þ
L. Chen et al. / Applied Energy 80 (2005) 187–195
189
Fig. 1. Stage geometry.
where xr and xs are the rotor and stator blade profile-loss coefficients, W is the relative velocity, C is the absolute velocity, and Cu is the circumferential velocity. The stage efficiency that accounts for end-wall losses is [19] g ¼ gpro ð1 2d =hÞ=ð1 2s =hÞ;
ð3Þ
where h is the annulus height, d is the mean displacement thickness and s is the mean tangential-force thickness of the end-wall boundary layers. d and s are functions of X, the ratio of the pressure rise coefficient to the maximum pressure rise
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Fig. 2. Stage velocity triangles.
coefficient, X ¼ CP =CPmax , where CP is the effective stage static-pressure rise coefficient defined by Koch [21] and CPmax is the value of CP at the stall point. 2.4. Stall margin A static pressure rise coefficient is defined as [21] CP ¼ DP =½qðW12 þ C22 Þ=2:
ð4Þ
The maximum value of the effective static-pressure rise coefficient is calculated as Cpmax ¼ CPD ðCP =CPD ÞRe ðCP =CPD Þe ðCP =CPD ÞDz ;
ð5Þ
where CPD is the maximum static-pressure rise coefficient from the diffuser correlation, ðCP =CPD ÞRe is the Reynolds number correction, ðCp =CPD Þe is a correction for tip-clearance correction, and ðCP =CPD ÞDZ is the correction for axial spacing. These corrections are given by Koch [21]. In the calculation, the rotor and stator stall margin values ðCPr ; CPs Þ are combined into a weighted averaged stage in which the blade-row inlet dynamic heat was used as the weighting factor. 2.5. Stage weight Total weight of the stage is estimated by a simplified method. The rotor and stator blade weights are calculated by integrating the blade cross-sectional area along the radius and multiplying the blade number. The blade cross-sectional area at an arbitrary radius is proportional to the square of a chord length. The shape of the annular walls is assumed to be a constant-thickness cylinder. The thickness of each wall is calculated from the equilibrium equation between the flow pressure and permissible stress of the material, including the safety factor. For the constant-stress distribution along the radius with the root stress, the thickness of the disk varies
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191
along the radius according to an exponential function. Integrating thickness from the hub to the centerline gives the disk weight. The total weight ðGÞ of the stage is the sum of the blade weights, the annular wall weights and the disk weight.
3. Statement of optimization problem Any optimization problem involves the identification of the optimum design variables, objective functions and constraints of the problem. 3.1. Optimum-design variables The parameters related to the compressor-stage performance involve the followings: mass-flow rate (m), total-pressure ratio (P03 =P01 ), rotational speed (N ), total pressure (P0 ) and total temperature (T0 ) at the stage inlet, densities (qa and qm ) of air and metal material, hub diameter (Dh ), blade height (h), average roughness (Ra ), rotor and stator blade chord-lengths (Cr and Cs ), rotor and stator blade circumferential spacing (Sr and Ss ), rotor and stator blade maximum thicknesses (tr and ts ), axial blade spacing (dzr and dzs ), tip clearances (er and es ), blade geometric angles (a01 , a02 , b01 , and b02 ), blade stagger angles (h0r and h0s ), and blade camber angles (/0r and /0s ). The former five parameters are the original data of the design. Rs , tr , ts , Dzr , Dzs , er , es , h0r , h0s , /0r and /0s may be taken as fixed input parameters. The velocity angles and the blade angles are related at follows: a1 ¼ a01 þ ds ;
ð6Þ
b2 ¼ b02 þ dr ;
ð7Þ
b1 ¼ tg1 ðu=Cx tga1 Þ;
ð8Þ
a1 ¼ tg1 ðu=Cx tgb2 Þ:
ð9Þ
Using the iteration calculation of deviation and incidence, the blade angles may be obtained from the velocity angles. There are only two independent velocity angles because of the correction of the velocity triangle. Therefore, the following parameters are taken as the design variables: hub diameter ðDh Þ, blade height ðhÞ, chord lengths (Cr and Cs ), circumferential spacing (Sr and Ss ), absolute inlet velocity angles ða1 Þ of the blade and relative outlet-velocity angle ðb1 Þ of the blade, i.e. T
X ¼ ½Dh ; h; Cr ; Cs ; Sr ; Ss ; a1 ; b2 :
ð10Þ
3.2. Objective functions The maximization of stage efficiency ðgÞ and stall margin ðCP Þ as well as the minimization of stage weight ðGÞ are taken as the objective functions, that is, find ½X that minimizes
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fF1 ¼ 1 gðX Þ;
F2 ¼ GðX Þ;
F3 ¼ 1 CP ðX Þg
ð11Þ
and F ¼ k1 ½1 gðX ÞG0 =ð1 g0 Þ þ k2 GðX Þ þ k3 ½1 CP ðX ÞG0 =ð1 CP0 Þ;
ð12Þ
where F is a linear combination of weight, efficiency and stall margin; g0 ; G0 , and CP0 are the initial guesses for the stage efficiency, weight and stall margin; k1 ; k2 and k3 are weighting factors, and they satisfy the following equations k1 þ k2 þ k3 ¼ 1; ð13Þ ð14Þ k1 2 ½0; 1; k2 2 ½0; 1; k3 2 ½0; 1: The coefficients G0 =ð1 g0 Þ and G0 =ð1 CP0 Þ are balancing factors to make three terms of comparable magnitude. The weighting factors may be selected according to a different method. The two usual ways are equal-balancing weighting ðk1 ¼ k2 ¼ k3 ¼ 1=3Þ and binomial-coefficient weighting ðk1 ¼ 0:5; k2 ¼ k3 ¼ 0:25, according to the order of importance, g > G > CP ). Hence, two objective functions are constructed by the different weighting factors F4 ¼ ½F1 G0 =ð1 g0 Þ þ F2 þ F3 G0 =ð1 CPD Þ=3;
ð15Þ
F5 ¼ 0:5F1 G0 =ð1 g0 Þ þ 0:25F2 þ 0:25F3 G0 =ð1 CPD Þ:
ð16Þ
3.3. Design constraints The following requirements must be met in considering the aerodynamic, the strength of materials and the suitability of the model. These requirements represent 18 terms of constraints and form 46 unequal constraint functions. (1) The rotational velocity of the rotor-blade tip should be within certain bounds. (2) The height-chord ratios of the rotor and stator blades should be within certain bounds. (3) The circumferential spacing-chord ratios of the rotor and stator blades should be within certain bounds. (4) The hub diameter should be within certain bounds. (5) The diameter–height ratio should lie within the bounds of the one-dimensional flow assumption. (6) The axial velocity of flow should be within certain bounds. (7) The degree of reaction at the rms radius should be within certain bounds. (8) The absolute velocity angle at the stator’s blade outlet should be within certain bounds. (9) The relative and absolute velocity angles at the inlet and outlet of the rotor blade should be within certain bounds. (10) The degree of reaction at the root of the blade should be larger that the specified value (estimated according to the blade-twist criteria of a1 ¼ const.). (11) The equivalent diffusion ratios of the rotor and stator should be less than the specified value. (12) The stage static pressure rise coefficient should be larger that the specified value.
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(13) The relative velocity Mach numbers at the inlet and outlet of the rotor blade should be within certain bounds. (14) The number of blades of the rotor and stator should be within certain bounds. (15) The inlet-blade angle of rotor should be within certain bounds. (16) The inlet-blade angles of the rotor and stator should be larger than the outletblade angles. (17) The Reynolds numbers of the rotor and stator blade should be within certain bounds. (18) The stresses developed at the root of the rotor and stator blade should be less than the permissible value. Constraint (5) is due to the one-dimension design. Constraints (11) and (12) are the dual limitations to prevent the stage from stalling. Constraint (13) is due to the subsonic analysis. Constraints (15) and (17) represent the valid ranges of the loss correlation equations by Casey [16]. 4. Mckenzie stage as a numerical example The model presented above represents a non-linear programming problem of three, five objective functions with constrained functions and several variables. There are many procedures available for solving the above problem. In this paper, the single objective-optimization problem is solved by using the exterior SUMT (in which the Powell unconstrained minimization technique is improved by Sargent) along with the parabolic interpolation method of one-dimensional minimization. This procedure has been used successfully in solving many problems for the design optimization of turbomachinary [22–25]. The Mckenzie stage is taken as a numerical example for the optimum design of subsonic axial compressor stages. Mckenzie [26] gave details of measurements on a four-stage compressor with Dt =Dh ¼ 1:25, in which, identical stages were installed. The blades were of C5 profile with t=c ¼ 0:1; /0r ¼ /0s ¼ 40° ; h0r ¼ h0s ¼ 50° . The stage could have various numbers of blades. The stage with 60 blades (S=C ¼ 0:941) is taken as the initial plan of the optimization calculation. Table 1 summaries the design constraints for the numerical example. Table 2 represents the results of the five-optimization problems. The results show that the method presented is valid. Table 1 Design constraints 400 P Ut P 40 9 P h=Cr P 2 9 P h=Cs P 2 1:0 P Sr =Cs P 0:4 1:0 P Ss =Cs P 0:4 1 P Dt P 0:1 10 P Dh =h P 4 400 P Cx P 50 0:9 P Rm P 0:1
80 P a1 P 5 80 P b1 P 5 Rh P 0 2:0 P Deqr 2:02 P Deqs CP P 0:45 0:7 P Mw1 P 0:1 0:7 P Mw2 P 0:1 90 P Zr P 15
90 P Zs P 15 70 P b01 P 30 a02 P a01 b01 ¼ b02 1:0 107 P Rer P 1:0 104 1:0 107 P Res P 1:0 104 80 P a2 P 5 6:9 108 P rtr 6:9 108 P rts
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Table 2 Optimum results Performance
Initial plan
F1
F2
F3
F4
F5
g (%) CP G=G0 (%)
87.24 0.516 100.0
88.47 0.464 94.3
85.43 0.520 43.7
82.02 0.577 95.8
87.56 0.552 51.5
88.02 0.545 64.6
5. Conclusion A design optimization program for a subsonic axial-flow compressor stage has been developed by applying the numerical optimization techniques to a simulation algorithm which consists of thermodynamic compression relations, cascade geometric variables, empirical loss-correlations, and simple stress relations. Using this program, example optimization problems for maximum efficiency, minimum weight, maximum stall margin, a balanced weighting optimum between the three indexes, as well as a binomial coefficients weighting optimum among the three indexes have been deduced. The results show the program is valid and effective. If we take account the matching of stages, the method presented may be extended to the design optimization of multi-stage subsonic axial-flow compressor.
Acknowledgements This paper is supported by The Foundation for the Authors of National Excellent Doctoral Dissertations of the PR China (Project No. 200136).
References [1] Wall RA. Axial-flow compressor performance prediction. AGARD-LS-83 1976;(June):4.1–4.34. [2] Gu C, Miao Y. Blade design of axial-flow compressors by the method of optimal control theory. Trans ASME J Turbomachinary 1987;109(1):99–107. [3] Tuccille R. A proposal for optimized design of multi-stage compressors. ASME Paper No. 89-GT-34; 1989. [4] Hearsey RM. Numerical optimization of axial compressor design. ASME Paper No. 89-GT-14; 1989. [5] Egorov IN, Fomin VN. Numerical method of optimization of a multi-stage axial compressor: experimental and computational aerothermodynamics of internal flows. New York: World Publishing Corporation; 1990. p. 495–503. [6] Tuccillo R. Optimum design of an axial-flow compressor. ASME IGTI 1990;5:27–233. [7] Geoge H, Stuart B. Preliminary design of axial compressors using artificial-intelligence and numericaloptimization techniques. ASME Paper No. 91-GT-334; 1991. [8] Chen L, Zhang J. Efficiency optimization of an axial-flow compressor stage. Exp Invest Gas-Turbine, 1992(1):16–20. [9] Chen L. A brief introduction of multi-objective optimization for an axial-flow compressor stage. GasTurbine Tech 1992;5(1):11–3.
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[10] Egorov IN, Krekinin GV. Multi-criterion stochastic optimization of an axial compressor. ASME IGTI 1992;7:563–70. [11] Egorov IN. Deterministic and stochastic optimization of a variable axial compressor. ASME Paper No. 93-GT-397; 1993. [12] Chen L. Some new developments on the optimum design of turbomachinary during the past decade. J Eng Thermal Energy Pow 1992;7(4):214–21. [13] Sun J, Elder R. Numerical optimization of a stator vane setting in multi-stage axial-flow compressors. Proc Instn Mech Eng 1998;212(A4):247–59. [14] Gallimore SJ. Axial flow compressor design. Proc Instn Mech Eng 1999;213(C6):437–49. [15] Calvert WJ, Ginder RB. Transonic fan and compressor design. Proc Instn Mech Eng 1999;213(C6):419–36. [16] Casey MV. A mean-line prediction method for estimation the performance characteristics of an axialcompressor stage. In: Proceedings of the IMechE 1987, Turbomachinary Efficiency Prediction and Improvement; 1987. p. 145–55. [17] Lieblein S. Incidence and deviation angle correlations for compressor cascades. Trans ASME J Basic Eng 1960;82(Sept):575–87. [18] Davis WR. A computer program for the analysis and design of the flow in turbomanchinery. Carleton University Report ME/A 70-1, Ottowa; 1970. [19] Koch CC, Smith LH. Loss sources and magnitudes in an axial-flow compressor. Trans ASME J Eng Pow 1976;98(3):411–24. [20] Jansen W, Moffatt WC. The off-design analysis of an axial-flow compressor. Trans ASME J Eng Pow 1967;89(4):443–62. [21] Koch CC. Stalling pressure rise capability of an axial-flow compressor stage. Trans ASME J Eng Pow 1981;103(4):645–55. [22] Chen L, Wu C, Blank D, Sun F. Preliminary design optimization of a marine dual tandem gear. Int J Pow Energy Sys 1997;17(3):218–22. [23] Chen L, Wu C, Ni N, Cao Y, Sun F. Optimum design of centrifugal-compressor stages. Int J Pow Energy Sys 1998;18(1):12–5. [24] Chen L, Wu C, Blank D, Sun F. The multi-objective optimum design method for a radial-flow turbine with the criteria of optimum twist at the outlet of blade. Int J Pow Energy Sys 1998;18(1):16–20. [25] Chen L, Zhang J, Wu C, Blank D, Sun F. Analysis of multi-objetive decision-making for a marine steam turbine. Int J Pow Energy Sys 1998;18(2):96–101. [26] Mckenzie AB. The design of axial compressor blading based on tests of a low speed compressor. Proc IMechE 1980;194(6):103–11.
Applied Energy 80 (2005) 187–195
www.elsevier.com/locate/apenergy
Optimum design of a subsonic axial-flow compressor stage Lingen Chen
a,*
, Fengrui Sun a, Chih Wu
b
a
b
Faculty 306, Naval University of Engineering, Wuhan 430033, PR China Department of Mechanical Engineering, US Naval Academy, Annapolis, Md 21402, USA Available online 6 May 2004
Abstract The design of an axial-flow compressor stage for subcritical Mach numbers has been formulated as a non-linear multi-objective mathematical programming problem with the objective of minimizing the aerodynamic losses and the weight of the stage, while maximizing the compressor’s stall margin. Aerodynamic as well as mechanical constraints are considered in the optimization solution. The prediction model for estimating the performance characteristics, such as efficiency, weight and stall margin, of the compressor stage is presented. The present design optimization procedure can be applied to a multi-stage compressor. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Axial flow compressor; Optimum design; Efficiency; Weight; Stall margin
1. Introduction The main design requirements of the compressor stage involve an acceptable level of thermodynamic efficiency, adequate surge margin, and a reduced weight, etc. However, the weight reduction inevitably causes loss of efficiency, and the higher efficiency will lead to decreases of the surge margin. As a result, the efficiency and surge-margin improvement, and the weight reduction must be compromised according to a certain design criterion. This is especially important in the design of a multi-stage axial-flow compressor. It is highly desirable that optimization techniques be applied in the design of compressor stages in order to obtain improvements. *
Corresponding author. Tel.: +86-27-83615046; fax: +86-27-83638709. E-mail addresses: [email protected], [email protected] (L. Chen).
0306-2619/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2004.03.008
188
L. Chen et al. / Applied Energy 80 (2005) 187–195
Much work has been reported on the application of optimization techniques to axial-flow compressor design, see [1–15]. In this paper, weight and stall margin are considered together with efficiency as the optimization objectives: an optimum design method for a subsonic axial-flow compressor stage is provided. 2. Performance evaluation of an axial compressor stage Figs. 1 and 2 define the important geometrical data, the velocity triangles and reference stations within an axial compressor stage at the subcritical Mach number under consideration. The performance evaluation method provided by Casey [16] is adopted in this paper. This method is a relatively simple mean-line prediction method, in which a one-dimensional (1D) analysis of the velocity vectors at the root 0:5 mean square (rms) radius, r ¼ ½ðrt2 þ rh2 Þ=2 , where rt and rh are the tip and hub radii, gives the aerodynamic loading of stage, and well-established empirical correlations are used to determine the losses and the operating range. The correlations used in the method have been selected on the basis of simplicity and appropriateness and have been modified where necessary to give better agreement with the many test cases analyzed. 2.1. Incidence, deviation and operating range The reference incidence angle iref are calculated from the equations and diagrams given by Lieblein [17]. The deviation angle at the reference incidence angle is calculated using a modified form of Carter’s rule [18]. The variation of deviation angle ðdbi Þ with incidence is calculated in the manner suggested by Lieblein [17]. The operating range ðdbÞ of a cascade is calculated as follows: db ¼ dbi ðdb=dbi ÞMa , where dbi is a function of solidity ðrÞ, blade camber angle ðU0 Þ and inlet blade angle ðb01 Þ of the rotor blade: ðdb=dbi ÞMa ¼ 1 if Ma < 0.2 and ðdb=dbi ÞMa ¼ 10A if Ma > 0.2, where A ¼ 2:5(Ma)0.2)4:4 . 2.2. Profile losses The total pressure profile-loss coefficients of rotor and the stator are calculated as follows: x ¼ xi ðx=xi ÞRe ðx=xi Þinc ðx=xi ÞMa ; ð1Þ where xi is the incompressible total-pressure-loss coefficient of the blade profile at the rms radius and at the reference angle [14], ðx=xi ÞRe Þ is the correction for Reynolds number and surface roughness [19], ðx=xi Þinc Þ is the correction for incidence [20] and ðx=xi ÞMa Þ is the correction for Mach number [20]. 2.3. Stage efficiency The efficiency that would exist if there were no end-wall losses is gpro ¼ 1 ðxr W12 þ xs C22 Þ=½2uðC2u C1u Þ;
ð2Þ
L. Chen et al. / Applied Energy 80 (2005) 187–195
189
Fig. 1. Stage geometry.
where xr and xs are the rotor and stator blade profile-loss coefficients, W is the relative velocity, C is the absolute velocity, and Cu is the circumferential velocity. The stage efficiency that accounts for end-wall losses is [19] g ¼ gpro ð1 2d =hÞ=ð1 2s =hÞ;
ð3Þ
where h is the annulus height, d is the mean displacement thickness and s is the mean tangential-force thickness of the end-wall boundary layers. d and s are functions of X, the ratio of the pressure rise coefficient to the maximum pressure rise
190
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Fig. 2. Stage velocity triangles.
coefficient, X ¼ CP =CPmax , where CP is the effective stage static-pressure rise coefficient defined by Koch [21] and CPmax is the value of CP at the stall point. 2.4. Stall margin A static pressure rise coefficient is defined as [21] CP ¼ DP =½qðW12 þ C22 Þ=2:
ð4Þ
The maximum value of the effective static-pressure rise coefficient is calculated as Cpmax ¼ CPD ðCP =CPD ÞRe ðCP =CPD Þe ðCP =CPD ÞDz ;
ð5Þ
where CPD is the maximum static-pressure rise coefficient from the diffuser correlation, ðCP =CPD ÞRe is the Reynolds number correction, ðCp =CPD Þe is a correction for tip-clearance correction, and ðCP =CPD ÞDZ is the correction for axial spacing. These corrections are given by Koch [21]. In the calculation, the rotor and stator stall margin values ðCPr ; CPs Þ are combined into a weighted averaged stage in which the blade-row inlet dynamic heat was used as the weighting factor. 2.5. Stage weight Total weight of the stage is estimated by a simplified method. The rotor and stator blade weights are calculated by integrating the blade cross-sectional area along the radius and multiplying the blade number. The blade cross-sectional area at an arbitrary radius is proportional to the square of a chord length. The shape of the annular walls is assumed to be a constant-thickness cylinder. The thickness of each wall is calculated from the equilibrium equation between the flow pressure and permissible stress of the material, including the safety factor. For the constant-stress distribution along the radius with the root stress, the thickness of the disk varies
L. Chen et al. / Applied Energy 80 (2005) 187–195
191
along the radius according to an exponential function. Integrating thickness from the hub to the centerline gives the disk weight. The total weight ðGÞ of the stage is the sum of the blade weights, the annular wall weights and the disk weight.
3. Statement of optimization problem Any optimization problem involves the identification of the optimum design variables, objective functions and constraints of the problem. 3.1. Optimum-design variables The parameters related to the compressor-stage performance involve the followings: mass-flow rate (m), total-pressure ratio (P03 =P01 ), rotational speed (N ), total pressure (P0 ) and total temperature (T0 ) at the stage inlet, densities (qa and qm ) of air and metal material, hub diameter (Dh ), blade height (h), average roughness (Ra ), rotor and stator blade chord-lengths (Cr and Cs ), rotor and stator blade circumferential spacing (Sr and Ss ), rotor and stator blade maximum thicknesses (tr and ts ), axial blade spacing (dzr and dzs ), tip clearances (er and es ), blade geometric angles (a01 , a02 , b01 , and b02 ), blade stagger angles (h0r and h0s ), and blade camber angles (/0r and /0s ). The former five parameters are the original data of the design. Rs , tr , ts , Dzr , Dzs , er , es , h0r , h0s , /0r and /0s may be taken as fixed input parameters. The velocity angles and the blade angles are related at follows: a1 ¼ a01 þ ds ;
ð6Þ
b2 ¼ b02 þ dr ;
ð7Þ
b1 ¼ tg1 ðu=Cx tga1 Þ;
ð8Þ
a1 ¼ tg1 ðu=Cx tgb2 Þ:
ð9Þ
Using the iteration calculation of deviation and incidence, the blade angles may be obtained from the velocity angles. There are only two independent velocity angles because of the correction of the velocity triangle. Therefore, the following parameters are taken as the design variables: hub diameter ðDh Þ, blade height ðhÞ, chord lengths (Cr and Cs ), circumferential spacing (Sr and Ss ), absolute inlet velocity angles ða1 Þ of the blade and relative outlet-velocity angle ðb1 Þ of the blade, i.e. T
X ¼ ½Dh ; h; Cr ; Cs ; Sr ; Ss ; a1 ; b2 :
ð10Þ
3.2. Objective functions The maximization of stage efficiency ðgÞ and stall margin ðCP Þ as well as the minimization of stage weight ðGÞ are taken as the objective functions, that is, find ½X that minimizes
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fF1 ¼ 1 gðX Þ;
F2 ¼ GðX Þ;
F3 ¼ 1 CP ðX Þg
ð11Þ
and F ¼ k1 ½1 gðX ÞG0 =ð1 g0 Þ þ k2 GðX Þ þ k3 ½1 CP ðX ÞG0 =ð1 CP0 Þ;
ð12Þ
where F is a linear combination of weight, efficiency and stall margin; g0 ; G0 , and CP0 are the initial guesses for the stage efficiency, weight and stall margin; k1 ; k2 and k3 are weighting factors, and they satisfy the following equations k1 þ k2 þ k3 ¼ 1; ð13Þ ð14Þ k1 2 ½0; 1; k2 2 ½0; 1; k3 2 ½0; 1: The coefficients G0 =ð1 g0 Þ and G0 =ð1 CP0 Þ are balancing factors to make three terms of comparable magnitude. The weighting factors may be selected according to a different method. The two usual ways are equal-balancing weighting ðk1 ¼ k2 ¼ k3 ¼ 1=3Þ and binomial-coefficient weighting ðk1 ¼ 0:5; k2 ¼ k3 ¼ 0:25, according to the order of importance, g > G > CP ). Hence, two objective functions are constructed by the different weighting factors F4 ¼ ½F1 G0 =ð1 g0 Þ þ F2 þ F3 G0 =ð1 CPD Þ=3;
ð15Þ
F5 ¼ 0:5F1 G0 =ð1 g0 Þ þ 0:25F2 þ 0:25F3 G0 =ð1 CPD Þ:
ð16Þ
3.3. Design constraints The following requirements must be met in considering the aerodynamic, the strength of materials and the suitability of the model. These requirements represent 18 terms of constraints and form 46 unequal constraint functions. (1) The rotational velocity of the rotor-blade tip should be within certain bounds. (2) The height-chord ratios of the rotor and stator blades should be within certain bounds. (3) The circumferential spacing-chord ratios of the rotor and stator blades should be within certain bounds. (4) The hub diameter should be within certain bounds. (5) The diameter–height ratio should lie within the bounds of the one-dimensional flow assumption. (6) The axial velocity of flow should be within certain bounds. (7) The degree of reaction at the rms radius should be within certain bounds. (8) The absolute velocity angle at the stator’s blade outlet should be within certain bounds. (9) The relative and absolute velocity angles at the inlet and outlet of the rotor blade should be within certain bounds. (10) The degree of reaction at the root of the blade should be larger that the specified value (estimated according to the blade-twist criteria of a1 ¼ const.). (11) The equivalent diffusion ratios of the rotor and stator should be less than the specified value. (12) The stage static pressure rise coefficient should be larger that the specified value.
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(13) The relative velocity Mach numbers at the inlet and outlet of the rotor blade should be within certain bounds. (14) The number of blades of the rotor and stator should be within certain bounds. (15) The inlet-blade angle of rotor should be within certain bounds. (16) The inlet-blade angles of the rotor and stator should be larger than the outletblade angles. (17) The Reynolds numbers of the rotor and stator blade should be within certain bounds. (18) The stresses developed at the root of the rotor and stator blade should be less than the permissible value. Constraint (5) is due to the one-dimension design. Constraints (11) and (12) are the dual limitations to prevent the stage from stalling. Constraint (13) is due to the subsonic analysis. Constraints (15) and (17) represent the valid ranges of the loss correlation equations by Casey [16]. 4. Mckenzie stage as a numerical example The model presented above represents a non-linear programming problem of three, five objective functions with constrained functions and several variables. There are many procedures available for solving the above problem. In this paper, the single objective-optimization problem is solved by using the exterior SUMT (in which the Powell unconstrained minimization technique is improved by Sargent) along with the parabolic interpolation method of one-dimensional minimization. This procedure has been used successfully in solving many problems for the design optimization of turbomachinary [22–25]. The Mckenzie stage is taken as a numerical example for the optimum design of subsonic axial compressor stages. Mckenzie [26] gave details of measurements on a four-stage compressor with Dt =Dh ¼ 1:25, in which, identical stages were installed. The blades were of C5 profile with t=c ¼ 0:1; /0r ¼ /0s ¼ 40° ; h0r ¼ h0s ¼ 50° . The stage could have various numbers of blades. The stage with 60 blades (S=C ¼ 0:941) is taken as the initial plan of the optimization calculation. Table 1 summaries the design constraints for the numerical example. Table 2 represents the results of the five-optimization problems. The results show that the method presented is valid. Table 1 Design constraints 400 P Ut P 40 9 P h=Cr P 2 9 P h=Cs P 2 1:0 P Sr =Cs P 0:4 1:0 P Ss =Cs P 0:4 1 P Dt P 0:1 10 P Dh =h P 4 400 P Cx P 50 0:9 P Rm P 0:1
80 P a1 P 5 80 P b1 P 5 Rh P 0 2:0 P Deqr 2:02 P Deqs CP P 0:45 0:7 P Mw1 P 0:1 0:7 P Mw2 P 0:1 90 P Zr P 15
90 P Zs P 15 70 P b01 P 30 a02 P a01 b01 ¼ b02 1:0 107 P Rer P 1:0 104 1:0 107 P Res P 1:0 104 80 P a2 P 5 6:9 108 P rtr 6:9 108 P rts
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Table 2 Optimum results Performance
Initial plan
F1
F2
F3
F4
F5
g (%) CP G=G0 (%)
87.24 0.516 100.0
88.47 0.464 94.3
85.43 0.520 43.7
82.02 0.577 95.8
87.56 0.552 51.5
88.02 0.545 64.6
5. Conclusion A design optimization program for a subsonic axial-flow compressor stage has been developed by applying the numerical optimization techniques to a simulation algorithm which consists of thermodynamic compression relations, cascade geometric variables, empirical loss-correlations, and simple stress relations. Using this program, example optimization problems for maximum efficiency, minimum weight, maximum stall margin, a balanced weighting optimum between the three indexes, as well as a binomial coefficients weighting optimum among the three indexes have been deduced. The results show the program is valid and effective. If we take account the matching of stages, the method presented may be extended to the design optimization of multi-stage subsonic axial-flow compressor.
Acknowledgements This paper is supported by The Foundation for the Authors of National Excellent Doctoral Dissertations of the PR China (Project No. 200136).
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