NUMERICAL SIMULATION OF AXIAL FLOW COMPRESSORS

Transcription

1 Thesis presented to the Faculty of the Division of Graduate Studies of The Technological Institute of Aeronautics in partial fulfillment of the requirements for the Degree of Master in Science of Course of Aeronautical and Mechanical Engineering, Area Aerodynamics, Propulsion and Energy. Jesuíno Takachi Tomita NUMERICAL SIMULATION OF AXIAL FLOW COMPRESSORS Prof. Dr. João Roberto Barbosa Advisor Prof. Dr. Homero Santiago Maciel Head of the Division of Graduate Studies Campo Montenegro São José dos Campos, SP - Brazil 003

2 NUMERICAL SIMULATION OF AXIAL FLOW COMPRESSORS Jesuíno Takachi Tomita Graduate Committee: Amílcar Porto Pimenta João Roberto Barbosa Helder Fernando de França Mendes Carneiro Márcio J. Prado Schimidt Nelson Manzares Filho Chairmain - ITA Advisor (IEME ITA) (ITA/IAE) (ITA) (UNIFEI) ITA

3 LIST OF CONTENTS CHAPTER I. Introduction... 1 I.1. Theme Identification... 1 I.. Existing Knowledge... 1 I.3. Bibliographical Review... 4 I.4. Objectives... 7 I.5. Achievements... 8 CHAPTER II. The Axial Compressor... 9 II.1. Introduction... 9 II.. Important Parameters for Analysis II..1. Tip Speed II... Axial velocity, Axial velocity ratio and Flow Coefficient II..3. Hub Tip Ratio II..4. Inlet Mach Number II..5. Exit Mach Number II..6. Compressor Axial Channel II.3. Design Point II.4. Off-Design Point II.5. Prediction of Off-Design Performance II.5.1. Performance Characteristics... 0 II Blade-Element Method... II Stage-Stacking Method... 3 II Simplified Method... 3 i

4 II.5.. Inlet Guide Vanes... 4 II.5.3. Variable Geometry... 5 II.5.4. Compressor Air Bleed... 5 II.6. Numerical Simulation... 6 II.6.1. Current Methods... 6 II The Mean-line Method... 6 II The Radial Equilibrium Method... 6 II The Streamline Curvature Method... 7 II The Finite Differences Method... 7 II The Finite Element Method... 7 II The Finite Volume Method... 7 II.6.. Method Used in this Work... 8 II Description... 8 II.6... Developed Algorithm... 8 CHAPTER III. Numerical Implementation III.1. Compressor Design (Design-Point)... 3 III.1.1. Channel Design... 3 III.1.. Stage Blading III.1.3. Minimum Loss Parameters III.. Compressor Performance Prediction (Off-Design Performance) CHAPTER IV. Computational Implementation IV.1. Developed Algorithm IV.. Iterative Processes ii

5 IV.3. Interpolations CHAPTER V. Results V.1. A Study for the Computer Program (AFCC) Validation V.. Axial Compressor Design V.3. Performance Analysis of an Existing Axial Compressor V.4. Variable Geometry CHAPTER VI. Conclusions VI.1. Analysis of Obtained Results VI.. The Objectives of This Work VI.3. Suggestions for Research Continuation CHAPTER VII. Bibliographical References Appendix A: Axial Flow Compressor Design Appendix B: Axial Flow Compressor Performance iii

6 LIST OF FIGURES Figure II-1: Examples or rotor and stator blade rows of a 9-stage axial flow compressor Figure II-: Aero gas turbine illustration compressor(s) location Figure II-3: Compressor with Variable Geometry Figure II-4: Axial compressor blading and thermodynamics... 1 Figure II-5: Axial flow compressor in turbojet engine Figure II-6: Axial compressor channel Figure II-7: Axial compressor map Figure II-8: Rotating stall in the compressor annulus... 1 Figure II-9: Propagating stall in a cascade... 1 Figure II-10: IGV influence in the velocity triangles... 4 Figure II-11: Meanline representation... 8 Figure III-1: Blade and flow angles Figure III-: Slope factor for minimum loss incidence angle (n), as function of inlet air angle á 1 and solidity ó = c/s Figure III-3: Minimum loss incidence angle for zero camber (i 0) 10, as function of inlet air angle á 1 and solidity ó = c/s Figure III-4: Thickness correction for zero-camber (k i ), as function of thickness chord ratio and airfoil specification Figure III-5: Deduced variation of average rotor reference incidence angle minus low speed two dimensional cascade rule reference incidence angle with relative Mach number (i -i ) D D iv

7 Figure III-6: Deduced variation of slope factor m, in the deviation angle rule (n) as function of inlet air angle á 1 and solidity ó = c/s Figure III-7: Zero-camber deviation angle ( δ ) 0 10, at reference minimum loss incidence angle deduced from low speed cascade data for 10 percent thick Figure III-8: Value of solidity exponent b in deviation angle rule Figure III-9: Thickness correction for zero camber deviation angle ( K δ ), as function of thichness chord ratio and airfoil specification... 6 Figure III-10: Deviation angle slope dδ di D at reference incidence angle Figure III-11: Deduced variation of average rotor deviation angle minus low speed two dimensional cascade rule deviation angle at compressor reference incidence angle with relative inlet Mach number ( ) δ δ c D Figure IV-1: Scheme of the AFCC in the flowchart representation Figure IV-: Flowchart showing the iterative processes implemented in AFCC Figure V-1: Input file for the 8-stage axial flow compressor design Figure V-: 8-stage axial flow compressor performance maps (actual and prediction) Figura 49: Figure V-4: Input data of the AFCC for off-design case (dimensions in mm) Figure V-5: Performance maps for the reference [18] and calculated compressor Figure V-6: Efficiency curves of the 8-stage axial flow compressor Figure V-7: Variation of the loss with the incidence for first rotor and N=19600 rpm. Refer 90 Figure V-8: Variation of the loss with the incidence for first stator and N=19600 rpm Figure V-9: Comparison of the results for the model of reference [7] v

8 Figure V-10: Efficiency curves calculated using the model of reference [7] Figure V-11: Variable Geometry influence on the compressor efficiency Figure V-1: Variable Geometry on the compressor pressure ratio Figure V-13: Variable IGV angles and its influence in the pressure ratio Figure V-14: Variable IGV angles and its influence in the efficiency Figure V-15: Effect of variable IGV on pressure ratio at N = rpm Figure V-16: Effect of variable IGV on efficiency at N = rpm Figure V-17: Effect of variable IGV on pressure ratio at N = rpm Figure V-18: Effect of variable IGV on efficiency at N = rpm Figure V-19: Effect of variable IGV on pressure ratio in different rotations Figure V-0: Effect of variable IGV on efficiency in different rotations Figure V-1: Effect of several IGV angles on pressure ratio in different rotations Figure V-: Effect of several IGV angles on efficiency in different rotations Figure V-3: Influence of variable stators in compressor pressure ratio Figure V-4: Influence of variable stators in compressor efficiency vi

9 LIST OF TABLES Table II-1: Axial compressor evolution Table III-1: Equations for V and * n d V * n u Table III-: Equations for ( is -iopt ) and ( s opt ) u i -i l Table III-3: Equations for C l or v lv d u C Table V-1: Comparison of AFCC/Reference of the compressor external radii Table V-: Comparison of AFCC/Reference of the compressor hub radii Table V-3: Comparison of AFCC/Reference of the compressor axial dimensions Table V-4: Comparison of AFCC/Reference of the compressor inlet blade angles Table V-5: Comparison of AFCC/Reference of the compressor outlet blade angles Table V-6: Comparison of AFCC/Reference of the compressor camber blade angles Table V-7: Comparison of AFCC/Reference of the compressor stagger blade angles Table V-8: Comparison of AFCC/Reference of the parameter de Haller Table V-9: Comparison of AFCC/Reference of the compressor blade deflections Table V-10 Map for the 8 stage compressor Table V-11: Variation of the loss with the incidence angle Table V-1: Variable IGV and the respective speed vii

10 LIST OF SYMBOLS LATIN LETTERING A b C Clv Área Solidity exponent Chord Lift coeficient C d D z DH DF h i h t Drag coefient Axial distance de Haller number Diffusion factor Hub (blade position) Hub-tip ratio Incidence i* Nominal incidence ( i0 ) 10 ( i i ) c D Minimum loss incidence angle for zero camber Rotor incidence angle minus low speed two dimensional cascade reference incidence angle with relative Mach number K b K i K ä Blockage coeficient Thickness correction for zero-camber incidence angle Thickness correction for zero-camber deviation angle viii

11 . m Mn m N n Pt Mass flow Mach number Slope factor in the deviation angle rule Rotational speed Slope factor in the incidence angle rule; number of stages Total or stagnation pressure PR Ps Pressure ratio Static Pressure R r r c s s c Tt Gas constant radius Casing radius Space (picht) Space chord ratio Total or stagnation temperature t Ts Tip (blade position) Static temperature U V Tangential velocity Velocity ix

12 GREEK LETTERING á á 0 á 1 á á 3 â â 1 â â 3 â 4 ã ä ( δ0 ) 10 Air angle Inlet air angle at the rotor Relative flow angle at the rotor inlet Absolute flow angle at the rotor outlet Flow angle at the stator inlet Blade angle Inlet rotor blade angle Outlet rotor blade angle Inlet stator blade angle Outlet stator blade angle Ratio of specifc heats Deviation angle Zero camber deviation angle at reference minimum loss incidence angle deduced from low speed cascade data for 10 percent thick ( δ δ ) c D Rotor deviation angle minus low speed two dimensional cascade deviation angle at compressor reference incidence angle with relative inlet Mach number dδ di ÄT t å æ D Deviation angle slope at reference incidence angle Total temperature increase Deflection Stagger blade angle x

13 ç p ç a ç c è ó Polytropic efficiency Adiabatic efficiency Compressor efficiency Blade camber angle Solidity xi

14 SUBSCRIPTS 0 Rotor inlet; ambient conditions 1 Rotor outlet Stator inlet 3 Stator outlet 4 Rotor inlet a d des e h l m ml opt r s t u W Axial Down Design Compressor inlet Hub Lower Mean Minimum loss Optimum Rotor Compressor outlet; stator Tip Upper Swirl velocity direction Infinite number of blades xii

15 ABSTRACT This work deals with the numerical simulation of axial flow compressors, from design to performance prediction. The stage performance prediction uses the meanline flow properties. Stage-stacking is used to analyse a multi-stage compressor. A computer program, written in FORTRAN, was developed and is able to design an axial flow compressor given air mass flow, total pressure ratio, overall efficiency and design speed. All geometrical data relevant to the compressor performance prediction is calculated after it has been bladed. From the geometrical data and for given compressor speed, the performance is calculated: pressure ratio and efficiency are determined from stall to choke. A complete compressor performance map can be produced. Validation was carried out with data obtained from the literature. Good agreement has been achieved. xiii

16 ACKNOWLEGMENTS: Firstly, I would like to thank my advisor and friend Professor João Roberto Barbosa, expert in the Gas Turbine area, for his support and teaching during these past 4 months. To researcher Noel Seyb, thanks for his scientific information which made possible the validation of the present work results. Thanks to my friends turbineiros, Ramiro, Marco, Cleverson, Ricardo, Genival, Franco and André, for the encouragement during the days and nights working and for the friendship of Professors Helder, Zaparoli and Ézio. Thanks also to the friends of energy area: Rocamora and Saito for the help during the work edition. To Duane, Gustavo Rivas, Ligia, Luzia, Viviane, Renato, Firmino and Hudson for the friendship during the whole period. I would like everyone to know that I will always be ready to lend a helping hand in whatever is within my reach. Thank FAPESP Fundação de Amparo à Pesquisa do Estado de São Paulo for the support to this research. xiv

17 Dedicated to My family, specially for my super mother Izabel Tiyoka Tomita, my grandmother Tomiko Dano Tomita, my grandfather Yoshihiko Tomita (in memoriam) and to my darling fiancée Maira Dias Marin. Thank God for this opportunity. Jesuino Takachi Tomita March 003 xv

18 CHAPTER I. Introduction CHAPTER I. Introduction I.1. Theme Identification A numerical simulator of axial flow compressors is a tool that supplies the user with results of preliminary calculations for multistage axial compressors analysis in the steady state condition. The main results for operation evaluation in axial compressors are the values of the parameters found in their performance maps (or operation maps) such as overall pressure ratio and overall efficiency for each mass flow and rotational speed values. However, to make a numerical solution useful it is necessary to use many theoretical and practical concepts, and specific methodologies so as to obtain results close to the test bench ones. I.. Existing Knowledge Development plans for gas turbines in Brazil were already part of the Creation Project of CTA, in However, the research only took a large impulse in the 70 s, with the creation of a Research Programat CTA. To have an idea of the importance given to the research in gas turbines, one can mention the plan of the Turbines Laboratory, composed of several test benchs, e.g. compressors (800 kw), combustion chambers (up to 1300 K), turbines (100 kw), and complete engines (turboprops and turbojets) in the range of 1000 kw. Some of these test facilities were built e.g. the compressors test bench; others like the test facilities for turbojets. The installations were rather appropriate. In the 70 s, CTA decided to invest heavily in gas turbines. Several partnerships options with traditional industries as Rolls-Royce (England), Garrett (United States), Pratt & Whitney 1

19 CHAPTER I. Introduction (United States and Canada), Lucas Aerospace (England), Kongsberg (Norway) were considered. Major obstacle was human resources, a fact that the Turbines Program seriously. As an example, the joint-project with Rolls-Royce seeking to design, manufacture and develop a turboprop in the power range of 300 kw, with immediate application to aircrafts in the Bandeirante airplane class, could not be carried out with the active participation of CTA, due to lack of human resources, although, parallel to the negotiations of the mentioned project, an ambitious program for human resources training had begun. Cranfield Institute of Technology (nowadays Cranfield University), was the main training site for CTA engineers and ITA professors in the important areas to gas turbines. Such a program began in 1967, contemplating undergraduate, specialization and graduate courses. Several researchers underwent graduated studies in gas turbines (turbomachines, compressors, turbines, performance and materials), aiming at creating and diffusing a highly specialized turbomachine group in Brazil. Most of the knowledge developed at that time is still latent at CTA. In some cases even progress has being achieved e.g. the gas turbine simulation. Unfortunately, lack of government al actions to consolidate the gas turbine tecnology, therefore creating the manufactur ing capability caused the Turbine Program to almost halt. With the advent of the aircraft industry, thousands of gas turbines have been imported. A large application market has been created but the opportunity to create a local industry has not been considered, even as an off-set requirement, either for the production of parts and components or for the assembly of the whole engine. It is worth pointing that our aircraft industry paid the costs of design and development of new engines to fulfil its needs. The opportunity to initiate a gas turbine industry in Brazil, is being missing since we are not organized enough to foresee the opportunities on time. Although we do not have an indigenous gas turbine industry, there is still room and need

20 CHAPTER I. Introduction for specialists in gas turbines, mainly in the areas of performance analysis and engine applications. The match of an appropriate turbine to a specific application is a must since it would impart a large economy for the user, either as operation costs (fuel consumption), or as maintenance costs. CTA has been attempting to build a modern Turbine Laboratory [1], with a Compressors Test Facility (1500 kw), a Turbines Test Bench (000 kw of braking power, rpm) and a Combustion Chamber Test Cell (1500 K, 10 x 10 5 Pa). Its reserch and development program contemplated the design and development of small gas turbines []. The Turbine Laboratory was conceived to meet the reserch needs of CTA, and of the scientific community. All the laboratory facilities, yet nonexistent in the country, require large resources investment which are sought among the government agencies. On the other hand, the aeronautical industry has had difficulty in hiring trained and competent personel. It is not uncommom foreign people being hired at the international wage and working under a Brasilian professional whose salary is a small fraction of theirs. It is not different the case of the thermal power industry that uses gas turbine. ITA and CTA have struggled to graduate engineers and specialists in the areas related to, gas turbines. However, the effort has not been enough. Besides CTA, partner since a long time ago in several human resources programs, such as UNESP - campus of Guaratinguetá - and UNIFEI - of Itajubá, have been supporting our program, and offering courses and dedicating to reserch in areas of gas turbine applications. The University of Campinas (UNICAMP) and the University of Ijuí (UNIJUÍ) also have modest gas turbine activities. Although desirable, there is no teachers and researchers integration. International co-operation has been sought like the one covered by the Alpha-Geophiles Project between ITA and Cranfield University, conceived during post-doctoral work of this thesis supervisor at Cranfield in It aimed at human resources training of Latin America 3

21 CHAPTER I. Introduction engineers on biomass use as fuel for gas turbines, for electricity generation. This project brought together the universities of Cranfield, Genoa (Italy), Tessalônic (Greece), and the industry Guascor (Spain). Funded by the European Community. Ten professionals were trained (CTA, UNIFEI and University of Montevidéu). Courses were presented at ITA by european specialists, and followed by a 6 months training program at the european partners installations and visits to many industries [3, 4]. I.3. Bibliographical Review In this country there is little engagement in gas turbine reserch and development. ITA contributes with large part to this activity; UNIFEI has been increasing R&D applications. Other universities are at the very beginning of their activities on gas turbines, mostly on aplications. Therefore, almost all available literature is generated in foreign countries, being the U.K. and the U.S.A. major contributors. Strong correlation exists between these two countries and the sources of gas turbine literature because the major gas turbine industries are from these countries. The difficulty to get hold of the papers, reports, thesis, and others informations, still is present since ITA s library has little to offer. Most of the information used in this thesis was kindly supplied either the supervisor s by his colleagues at Cranfield University. Reference [5], contains an analysis of axial compressor performance with identification of the used methodology. The method of streamline curvature is used for the flow calculation along a multistage axial compressor of known geometry. Starting from the calculated flow properties, compressor performance is calculated and performance maps can be plotted.. Reference [6] deals with the axial compressor losses. The author discusses the most important losses due to the flow in axial compressor, producing curves for loss analysis. A - D analysis of each loss is made aiming the simulation of the actual conditions (three 4

22 CHAPTER I. Introduction dimensions). A brief summary of the scattered information pertinent to the evaluation of the aerodynamic losses in axial turbomachines is presented. Reference [7] relates a program created by Rolls-Royce, starting from mathematical models that have a good correlation with experimental data. That work is based on the use of experimental data to calculate the lift coefficient, minimum loss conditions (incidence and deviation) and losses parameters at the blade sections. Reference [8] correlates some high speed cascade test results for compressor blades, on a more fundamental, although empirically based, so that interpolation, and a certain amount of extrapolation can be carried out with a reasonable degree of accuracy. The authors present, the aerodynamic justifications for the methods of analysis adopted. Many curves for the calculation of critical and maximum Mach numbers are presented. Reference [9] shows the analysis necessary to define the optimum incidence, deviation and deflection angles. Several rules have been proposed which ensure a satisfactory cascade design to fulfill given requirements and these have been of considerable practical importance. An example of cascade characteristics calculation is presented, where the optimum incidence is analyzed. The method applied in that work is associated with reference [7]. Reference [10] describes a mean-line stage-stacking method used for axial compressor prediction and presents a computer program, in FORTRAN language, for the assembly of a compressor map. The program uses a methodology for analysis of losses based on the shapes of known axial compressors maps. Reference [11] presents a computational program to calculate the efficiency of a singlestage axial compressor, analyzed with the one-dimensional condition, with the pressure ratio equal to 1.. The author describes the importance of analyzing the incidence, deviation, profile losses, secondary losses and boundary-layer limits, from the hub to tip, related with the tip clearance, in order to predict the axial compressor performance. The deviation angle is 5

23 CHAPTER I. Introduction obtained through Carter's rule and Lieblen s model [1] is used to calculate the profile losses. The effect of the relative Reynolds number corrections to the friction losses is calculated by Koch s model and the Mach number correction is done by Jansen and Moffatt procedure [14]. The program needs as input data the following parameters: inlet hub radius, rotor and stator chords, rotor and stator blades pitchs, rotor and stator maximum thickness, tip clearance, absolute and relative inlet flow angles and the inlet and outlet blade angles. Reference [13] describes the main components of compressor blade losses, and suggests ways to evaluate them at off-design. A compressor performance prediction program was developed, based on blade row stacking at mid radius. The results of this program are presented in the graphic form including a compressor map. The loss model used is not well documented. Reference [14] describes in some detail the methods used for generation of a computer program for the off-design performance calculation of axial compressors. The main subjects presented are associated with the losses, axial velocity ratios, correction for the effects of blade thickness on turning and Mach number. An elaborate study is included to find the flow angle at the blade trailing edge and the blade row pressure loss. Reference [15] refers to the complete aerodynamic design of a multi-stage axial flow compressor using the streamline curvature approach, aiming at a configuration of the compressor which would be appropriate for a complete aero and mechanical design. The author presented 1-D and -D models, as well as the loss model he used. A concise analysis about off-design point is also commented upon. Reference [16] explains the similarity of complexity of the business process for multistage compressors to that of complete aero-engines or propulsion systems. The details about the elements of the design process are commented. This publication is very interesting for better understanding the theory and research applied in industry. This work relates the 6

24 CHAPTER I. Introduction importance of the multidisciplinary design process in the propulsion systems industries, showing the simultaneous action of the engineering team in the compressor design, and the phases of the project of axial compressors, from the simplest to the more complex analysis models and calculations in the construction of those equipments. Reference [17] relates the computer program designed for the analysis of single- stage axial compressor. The streamline curvature method is used and described. This work is limited to a single-stage compressor and inlet guide vanes may be included. Reference [18] is a program (spreadsheet) for design and prediction of an 8-stage, constant outer diameter, axial compressors. The model has been tuned for comparison with available experimental data. References [19 to 5], were used during this work in a complementary basis. It is worth nothing that References [19] and [3] are very similar since they were closely supervised by my supervisor in an effort to integrate high technology advances in the design of axial compressors and providing the institutions, where the work were being done, with an efficient computer program. I.4. Objectives The objectives of this work were set out as: 1. To define a methodology for the design and analysis of multistage axial flow compressor;. To apply the methodology to generate usable compressor maps; 3. To incorporate stators variable geometry as a means to improve compressor performance; 4. To handle both existing and new compressors. 7

25 CHAPTER I. Introduction I.5. Achievements A modular computational program, written in FORTRAN language, named AFCC (Axial Flow Compressor Code), has been developed that is capable to design and do the off-design performance calculation of a compressor with variable stators. It was validated after comparison of its output with available data of an 8-stage axial compressor supplied privately by one of the supervisor s colleagues. 8

26 CHAPTER II. The Axial Compressor CHAPTER II. The Axial Compressor II.1. Introduction An axial compressor is usually made of many stages each one composed by a rotor and a stator (rotating cascade and fixed cascade, respectively Figure II-11). Knowing the air properties at the compressor inlet, the air properties at outlet would be calculated from cascade data [1], or from other correlational data [1], and from other literature sources like [1, 4, ] (fixed geometry), and [6, 7] (variable geometry). Rotor Stator Rotor + Stator Figure II-1: Examples or rotor and stator blade rows of a 9-stage axial flow compressor. Flow properties at the stage outlet are calculated combining a rotor with a stator. The whole compressor characteristics are calculated stacking the compressor stages: the inlet conditions of a representative stage is obtained from the outlet conditions of the previous stage. Friction, shock waves, secondary flow and levels of velocities levels (Mach numbers), influence the stage performance most. 9

27 CHAPTER II. The Axial Compressor The compressor is the component that mo st strongly influences the gas turbine performance, either for the peculiar characteristic of operation instability or for the high consumption of energy during the air compression. Figure II- shows the location of the axial compressor in a gas turbine. Intake Compressor Combustion Chamber Turbine After-Burner Figure II-: Aero gas turbine illustration compressor(s) location. The high performance compressors, those that promote a high compression per stage, require a flow with high velocity in the blade channels. The flow velocity is associated with energy losses when the flow direction does not match the blade channels direction. Thus, at off-design operation the compressor efficiency can vary significantly, due to a high losses and deterioration of the gas turbine performance as result. To decrease the losses in the compressor operation, use is made of variable geometry. The position of the stators (angles) is altered during operation in order to align the flow with the blades (Figure II-3). 10

28 CHAPTER II. The Axial Compressor Setting stator blades to a new position is equivalent to having a new compressor. Therefore, the same methodology developed for the compressor design and analysis would be enough for the evaluation of the compressor with varying geometry. Stators with Variable Geometry Figure II-3: Compressor with Variable Geometry. Figure II-4 gives indication of the relationships among blade rows and the thermodynamic processes involved. 11

29 CHAPTER II. The Axial Compressor Figure II-4: Axial compressor blading and thermodynamics. A schematic drawing of an axial flow compressor as installed in a turbojet engine is shown in Figure II-5. In the general configuration, the first row of blades, the IGV (Inlet Guide Vanes), imparts a rotation to the air to establish a specified velocity distribution ahead of the first rotor. The pre-rotation of the air is then changed in the first rotor, and energy is thereby added in accordance with Euler s equation. This energy is manifested as increases in total temperature and total pressure of the air leaving the rotor. Usually accompanying these increases are increases in static pressure and in absolute velocity of the air. A part, or all, of the rotation is then removed in the next stator, thus converting velocity head to static pressure and this stator also sets up the distribution of air flow for the subsequent rotor row. The air passes successively through rotors and stators in such a manner to increase the total pressure of the air to the degree required in the gas turbine. As the air is compressed, its density is increased and the annular flow area is reduced to correspond to the decreasing volume. This 1

30 CHAPTER II. The Axial Compressor change in area may be accomplished by means of varying tip or hub diameter or both. In this compression process there are losses that result in an increase in the air entropy. Thus, in passing through a compressor, the velocity, the pressure, the temperature, the density, the entropy, and the radius of a given particle of air are changed across each of the blade rows. Figure II-5: Axial flow compressor in turbojet engine. The axial flow compressor is the principal type of compressor used in aircraft gas turbine engines. Although some of the early turbojet engines incorporated the centrifugal compressor, currently the axial compressor has a broader range of applications. This dominance is a result of the ability of the axial flow compressor to better satisfy the basic requirements of the aircraft gas turbine. In general, the axial compressor has a high efficiency, high air flow capacity per unit frontal area, and high pressure ratio per stage. Because of the demand for rapid engine acceleration and for operation over a wide range of flight conditions, this high level of 13

31 CHAPTER II. The Axial Compressor aerodynamic performance must be maintained over a wide range of speeds and flows. In Table II-1, some examples of axial compressor evolution in the last decades are shown. Table II-1: Axial compressor evolution. Engine Year Thrust (kn) Pressure Ratio Number of stages Pressure Ratio per stage Avon ,145 Spey ,196 RB ,7 Trent ,81 Note that with the development of new technologies, it is possible to build a compressor with higher pressure ratio and smaller number of stages. II.. Important Parameters for Analysis A compressor is a complex equipment, required to work at a wide range of flow pumping capacity. Therefore, constraints on both mechanical and aerodynamical matters must be taken into account. Some of the corresponding parameters are listed below. II..1. Tip Speed Tip speed is the blade tip peripheral velocity. At design, attention should be given to that value due to the maximum strength carried out by the materials used for the blades and disks (or drums). For tip speeds of around 350 m/s, stress problems are not critical for the actual available materials. In the case of fans, this that velocity may be over 450 m/s. 14

32 CHAPTER II. The Axial Compressor II... Axial velocity, Axial velocity ratio and Flow Coefficient A high axial velocity is required to provide a high mass flow rate per unit of frontal area, which is important for high performance gas turbine, but the axial velocity in the axial compressor is limited by aerodynamic reasons. Axial velocities for an industrial gas turbine will usually be of the order of 150 m/s and for advanced aero engines they could be up to 00 m/s. Axial velocity ratio, the outlet to inlet axial velocity ratio is kept around 1.0 in order to reduce axial loading. The flow coefficient, the ratio of the inlet axial velocity to the tangencial velocity (of blade), would be selected in the range of 0.5 to The higher the value the higher the relative Mach number and the possibility of higher shock losses. II..3. Hub Tip Ratio This is the hub and tip radii ratio. At high values of hub tip ratio, tip clearance effects will be comparatively higher due to the lower blade height. At low hub-tip ratios blade stresses may become prohibitive and secondary flows become powerful. Due to these two effects hub-tip ratio should be greater than 0.60 for the first stage and higher for the others stages, although fans of large bypass engine have hub-tip ratio as low as II..4. Inlet Mach Number For aero gas turbines, the use of high inlet Mach number to minimize frontal area leads to high relative velocities at the first stage blade tip, and hence inefficiency. Values between 0.4 and 0.6 are common, the highest level being for aero engines in supersonic applications. 15

33 CHAPTER II. The Axial Compressor II..5. Exit Mach Number To minimize and prevent excessive pressure loss, the exit Mach number can not be high. This value should not be higher than 0.35 (ideally 0.5). II..6. Compressor Axial Channel The flow passages in the axial direction form the axial channel. This channel is of almost influence on the compressor performance, so that it must be designed carefully. Well designed axial channel would result in efficient compressor, usually after little development. On the other hand, lack of careful design would result in low efficience compressor, where development will certainly require redesign of the channel (Figure II-6). Compressor Constant outer diameter Combustion chamber Turbine Constant mean diameter Figure II-6: Axial compressor channel. In the present work, constant outer diameter (COD), constant mean diameter (CMD), the variable mean diameter (VMD) and constant inner diameter (CID), were implemented for the 16

34 CHAPTER II. The Axial Compressor design, giving the user a range of options from which it is possible to chose the best one to fit his needs. The use of a constant inner diameter is often found in industrial units, allowing the use of rotor discs of the same diameter, which lowers of manufacturing costs. In aero engines the frontal area is critical due to the drag forces. In turbines frontal area is not so important and with reverse flow combustion chambers large differences in the compressor and turbine diameters can be easily accommodated. For the aircraft engines, in order to decrease the weight and the size of the compressor, consequently of the gas turbine, constant outer diameter channels are commonly found since it would result in minimum number of stages. II.3. Design Point Design point is the condition at which the cycle parameters are specified for the machine development. Generally at the design point the efficienc y of the components is maximum and the efficiency of engine is maximized. Thus, the design point is chosen as the condition that the engine will work for a longer time. The problem of designing axial compressors becomes the problem of accurately calculating the flow through the compressor blade rows. In order to be accurate and have the greatest range of applicability, these calculations should be based on the fundamental laws of motion and thermodynamics as close as possible. At the same time, they should be of such a nature that they can be made readily with available computing techniques and equipment. II.4. Off-Design Point It is a condition of engine operation at which some of its parameters are different from the adopted for the design. 17

35 CHAPTER II. The Axial Compressor Thus, varying the rotational speed or the air mass flow from the design point, the equipment will not have the same efficiency. If analyzed in a global way in the gas turbine, the turbine would transmit the same power to the compressor shaft, but this won't realize the same work. That is the reason why the engine efficiency drops, due to the reduction of each component efficiency. Thus, off-design performance is defined as the compressor performance at flow conditions and speeds other than those for which the compressor was specifically designed. The off-design analysis differs from the design case in that the compressor geometry is given and the objective is to find the compressor outlet conditions for a range of speeds and mass flows. If the compressor has variable geometry, the work range at off-design is larger and its efficiency and pressure ratio can be increased for each mass flow required, chosing appropriate stator blade settings. The off-design point performance calculation is denominated direct compressor problem, whereas the original design is called indirect problem. The off-design analysis is one of the most difficult tasks for the compressor designers [1]. II.5. Prediction of Off-Design Performance A typical axial compressor performance map is shown in Figure II-7. 18

36 CHAPTER II. The Axial Compressor Figure II-7: Axial compressor map. The regions of the performance map discussed herein are those to the right of the surge/ stall limit line along lines of constant rotational speed, in other words, the complete compressor map, in the section II

37 CHAPTER II. The Axial Compressor II.5.1. Performance Characteristics In the compressor map, the rotation and mass flow values are relative to design values, denominated corrected speed and corrected mass flow, where T 01 and P 01 represent the total temperature and the total pressure at ambient conditions. Each curve represents one different rotation. At the design point the rotation is 100%, represented in the Figure II-7 by 1.0. In a well designed compressor this point would be the most efficienty. For other constant rotations (corrected), the compressor will be operating at off-design points. Thus, for each rotation curve, at the design or off-design, one can identify the value of the pressure ratio and efficiency for each mass flow value (corrected). The constant rotation curves are limited by the minimum mass flow where there is stall (or surge) and by the maximum mass flow, at wich choke occurs. The stall limit (or surge line), represents the locus of minimum flow points just before the performance of the compressor deteriorates abruptly. Experimental data and analyses give credibility to the existence of at least three distinct phenomena during stall operation, the first two being aerodynamic phenomena and the last an aero elastic phenomenon: 1. Rotating stall consists of large stall zones covering several blade passages and propagating at some fraction of rotor speed in the absolute flow direction of rotor rotation (Figure II-8). Steady state operation in rotating stall is undesirable due to deteriorated compressor performance and, hence, engine performance, and the possibility of including destructive high cycle blade vibration. The propagation mechanism can be described by considering the blade row to be a cascade of blades similar to that shown in Figure II-9. The flow perturbation caused in the blade number two reachs a stalled condition before the other blades in the cascade, then this stalled 0

38 CHAPTER II. The Axial Compressor blade does not produce sufficie nt pressure rise to maintain the flow around it, consequently effective flow blockage or a zone of reduced flow develops. Compressor investigations have shown rotating stall to be the most prevalent type of stall phenomenon. Figure II-8: Rotating stall in the compressor annulus. Figure II-9: Propagating stall in a cascade. 1

39 CHAPTER II. The Axial Compressor. Individual blade stall is characterized by the development of large separated flow regions or zones of low flow in the wake of each blade. 3. Stall flutter is a blade oscillation and must be distinguished from the more familiar classical flutter. The stall flutter takes place at high angles of attack and is associated with individual blade stall. The choking point represents the maximum mass flow that can pass in the compressor channel for a given rotation. Thus, in order to increase the mass flow it is necessary to increase the operational rotation. Three techniques for predicting design and off-design compressor performance are summarized below. In this work the technique used is described in section II II Blade-Element Method In this method, the compressor performance is obtained by first integrating radially the compressor blade element data to obtain blade row performance, in other words, compressor blades are evolved by a radial stacking of a series of blade sections to form the complete blade. Therefore, the blade row characteristics can be determined if the performance of each blade element is known and overall compressor performance can then be obtained by an integration of the performance of each blade row. In this method some allowances must be made for boundary-layer growth, blade row interaction effects, and radial mixing of blade wakes. The approach is currently used, but much information required to carry out this calculation is not usually available. This method has the greatest potentiality for providing a picture of the internal flow mechanism through a compressor at off-design operation conditions. Thus, the performance

40 CHAPTER II. The Axial Compressor of successive stages can be determined by utilizing the computed outlet conditions of one stage as the inlet conditions to the following stage. II Stage-Stacking Method In this method [5, 1], overall compressor performance for a range of speeds and mass flows may be estimated. The performance of each stage of the multistage compressor is obtained and presented so that its performance is a function only of its inlet equivalent mass flow and rotational speed. For adopted values of compressor mass flow and speed, the first stage performance yields the inlet equivalent flow and rotational speed to the second stage. A stage-by-stage calculation through the compressor gives the individual stage pressure and temperature ratios, and efficiency can be calculated for the adopted values of compressor mass flow and rotational speed. The stage-stacking method may be used as a research tool to investigate compressor offdesign problems or to estimate the performance of an untested compressor. This method is also useful for determining the effects of interstage bleed and variable geometry on compressor performance. It has been chosen as the method for the research set out in the beginning of this work. II Simplified Method This method is based on data obtained from overall performance maps of previously designed compressors. It is considerably more simplified than the first two methods, because there are no integration procedures or stage-stacking techniques involved in the computation of performance maps. Correlation curves are provided to facilitate the calculation. There are three phases of the calculation procedure. First, points of maximum efficiency at each speed are calculated. Second, the stall limit line is determined, and values along the 3

41 CHAPTER II. The Axial Compressor stall limit line are referred to as stall limit values. Finally, points along lines of constant speed are calculated from the stall limit to maximum flow (in this case is a choke mass flow). The integration of these phases results in a complete compressor performance map. II.5.. Inlet Guide Vanes The Inlet Guide Vane (IGV) is a stator row located in front of the first rotor row. The use of the IGV is related to the inlet flow in the first rotor, because the IGV will determine the flow incidence angle at the leading edge of the following blade. IGVs are used to impart a pré-swirl to the incoming rotor flow, reducing the relative Mach number and mass flow. There are IGVs with variable geometry used to extend the compressor operation range in the off-design condition. Figure II-10 shows the velocity triangles and the influence of IGV in the first rotor inlet. V u V u V um V u1 V um W V V 1 W V V 1 W 1 W 1 U U (a) without IGV (b) with IGV Figure II-10: IGV influence in the velocity triangles. 4

42 CHAPTER II. The Axial Compressor II.5.3. Variable Geometry When the compressor is submitted to operation at off-design, the efficiency decreases due to rotation and mass flow variations, consequently decreasing the pressure ratio. In many cases this occurs due to the flow separation in the blades cascade, thus an alternative is to vary the stators stagger angles (re-stagger) for aligning the flow with the blades. Hence, the compressor map operation is different for each variation of the stators stagger angles. Theoretically, for each re-stagger distinct compressors are obtained. Variable inlet guide vanes have, however, been effectively used as means of improving engine acceleration characteristics, and adjustable guide vanes and stator provide a potential technique for improving the flexibility of high pressure ratio compressor operation without sacrificing design point performance. Then, adjustable inlet guide vanes and stator blades have also been proposed as variable geometry features for alleviation of part-speed performance problems and improvement of flexibility of engine operation. The added weight and complexity of the engine and its control system can, however, add an extra cost. II.5.4. Compressor Air Bleed Compressor discharge bleed has been used effectively to alter the matching of the compressor and turbine at part speed so as to avoid serious intermediate speed and to avoid complete compressor stall problems. This technique, however, does not eliminate the potential of blade vibrations instigated by rotating stall. Interstage air bleed in the multistage compressor has the potential of effectively altering the matching of the inlet and exit stages and thus appreciably reducing the speed at which rotating stall is encountered, as well as to avoid the complete compressor stall or increasing surge margin at intermediate speeds. 5

43 CHAPTER II. The Axial Compressor The use of interstage bleed will add some weight and complexity to the engine, but, in general, offers improvement with regard to intermediate speed compressor stall or surge margin and to blade vibration problems resulting from rotating stall. II.6. Numerical Simulation II.6.1. Current Methods There are several methods developed for the design and flow analysis of axial compressors. AGARD created the group [8] to study the existent methods of compressors analysis. They were classified as: II The Mean-line Method It consists of calculations based on the mean streamline of the axial compressor channel. The flow properties as temperature, pressure, velocity and the dimensions of the equipment, including the blading angles, are determinated at half blade height. The fact of just using one streamline makes necessary the use of empiric correlations for loss correlations originating from experimental data. The method, when simulated in computer, presents fast numeric convergence and sufficiently accurate results for a first analysis of the compressors performance, allowing to obtain the compressor operating characteristics through the construction of its map. II The Radial Equilibrium Method The analysis of the compressor is based on the meridian plane that contains the axis of rotation of the equipment, thus this plane is divided in control surfaces from hub to tip of the blades. The flow properties are determined using conservation laws and empiric al 6

44 CHAPTER II. The Axial Compressor correlations. This method allows obtaining the geometry of the blades in several points from hub to tip of each row according to the division done in the meridional plane. II The Streamline Curvature Method In this method the calculation is based on the resolution of the Euler s equations aided by empirical correlations of experimental data. Many streamlines are adopted from hub to tip of the blades and divided in sections applied in the inlet and outlet of each row. Hence, all flow properties can be determined at each point of the intersection among the sections of the cascade and the streamlines forming control surfaces. II The Finite Differences Method The flow as well as its properties are resolved through the finite differences equations obtained from the conservation laws. However, it is necessary to define the mesh type to be used since it determines the detailing level of the solution. II The Finite Element Method The flow domain is divided into a group of finite elements and the properties are calculated in the resulting nodal points. This method is generally easier to use with complex geometries due to the fact that it employs non-structured grids. II The Finite Volume Method Knowing the geometry of the compressor, the channel where the flow passes is divided in to several elementary volumes. The conservation equations are integrated in each control volume and the resulting algebric set of equations is solved by mathematical algorithms until the results are obtained, as it is also in the other methods. 7

45 CHAPTER II. The Axial Compressor II.6.. Method Used in this Work II Description The method used in this work is a mean-line method, described in section II All flow properties (temperature, pressure and velocity) are calculated for the mean height of each row (rotor and stator). In this method it is possible to calculate all the dimensions of the compressor. However the inlet and outlet angles of blades are calculated only for the medium height of each row as shown in the Figure II-11. Rotor Stator Flow direction Meanline Figure II-11: Meanline representation. The mean-line method provides accurate results and fast mathematical convergence, allowing to obtain the necessary data for the construction of the compressor map. II.6... Developed Algorithm The developed algorithm for the design point and off-design point is the following: A) Design-Point: 1. Read the input parameters; 8

46 CHAPTER II. The Axial Compressor. Calculate the channel geometry and dimensions at inlet and outlet; 3. Calculate the pressure or temperature distribution for each stage or blade row; 4. Calculate the compressor channel; 5. Calculate the blade and flow angles; 6. Calculate the rotor bladings such as the parameters of analysis as: de Haller number, diffusion factor, lift coefficient, optimum conditions, and others; 7. Calculate the stator bladings parameters such as the parameters of analysis as: de Haller number, diffusion factor, lift coefficient, optimum conditions, and others; 8. Calculate the minimum loss conditions; 9. Calculate the performance data; 10. Calculate losses; 11. Calculate choke and stall points; 1. Calculate the overall conditions (pressure ratio, efficiency, temperature increase). B) Off-design points and performance maps: 1. Read in the input parameters of the compressor (dimensions and geometrical data);. Calculate the optimum conditions (flow angle, incidence, deviation, lift coefficient and others); 3. Calculate the flow proper (total and static temperature, total and static pressures, velocities) at each blade row; 4. Calculate the inlet and outlet Mach numbers at each blade row; 5. Calculate the actual (predicted) conditions of the flow angles at inlet and outlet of the each blade; 6. Loss calculation; 7. Calculation of choke and stall mass flows for each rotational speed; 9

47 CHAPTER II. The Axial Compressor 8. Calculate the map points for each rotational speed; 9. Calculate the overall conditions (pressure ratio, efficiency, temperature increase) for each mass flow of the equivalent rotational speed; If the axial compressor, at the off-design point, has a variable geometry, use this algorithm to repeat all calculations above mentioned, for each blade re-stagger. 30

48 CHAPTER III. Numerical Implementation CHAPTER III. Numerical Implementation For the creation of the AFCC program, it was first necessary to develop a model for the calculations at the mean-line for the stage-stacking method. The calculation of the axial compressor here developed is divided in two parts: 1. The indirect problem: It is the axial compressor design. To accomplish the calculations it is necessary to specify the input parameters for the compressor to be designed, such as air mass flow, pressure ratio, ambient conditions and efficiency.. The direct problem: It is the performance calculation of an existent compressor, in this case, the axial compressor is known, through its dimensions and geometry. Figure III-1 shows the nomenclature adopted in this work. 31

49 CHAPTER III. Numerical Implementation Figure III-1: Blade and flow angles. III.1. Compressor Design (Design-Point) III.1.1. Channel Design The design of axial compressor requires the following input parameters:. h P, T, m,pr,, U, η,n, γ, α, T, K,M,M t t0 t0 t p 0 t b n0 ns Design starts with calculation of channel inlet and exit areas to accommodate the desired mass flow and pressure ratio (items 1 to 11). A pressure rise distribution (or temperature distribution) is defined, followed by the calculations of total pressure and temperature at each stage inlet (and outlet). A distribution of axial velocities and stators exit angles across the compressor is given and the areas at each blade row are calculated. Geometrical relationships among chord, blade 3

50 CHAPTER III. Numerical Implementation height, axial spacing are given so that the blade axial lengths are calculated together with the compressor length. Velocities triangles are then calculated. Follow the calculations of incidence, losses and deflectios, from which the entire compressor geometry is available. From the endwall axial velocity distribution it is calculated the blockage factor. Iteration is then needed to correct the initial guess for these blockage factors. The appropriate 1-D formulae used in this work are indicated below. Some were derived by the author; most of then came from reference [7]. All are displayed without any previous considerations because it was felt that they are either straight forward or incorporate empirical constants that better and fully would be explained at the appropriate references. 1. The inlet area of axial compressor: A e = PM. m T t0 t0 n0 γ+ 1 γ 1 γ γ 1 R 1+ ( Mn ) 0 ( III-1 ). The external radii at the compressor inlet, considering a compressor with constant outer diameter. r = A h ð1- t e c ( III- ) 3. The hub radii at the compressor inlet. r he h = r t c ( III-3 ) 33

51 CHAPTER III. Numerical Implementation 4. The total pressure at the compressor outlet. P = P PR ts t0 ( III-4 ) 5. The compressor efficiency. γ 1 γ PR -1 ç c = γ -1 p PR γη 1 ( III-5 ) 6. The total temperature at the compressor outlet. γ 1 γ T T = T PR -1 ts t + 0 η t0 c ( III-6 ) 7. The total temperature increase at the compressor. T = T -T t t t total s 0 ( III-7 ) 8. The outlet area of the compressor. A s = PM. m T ts ts ns γ+ 1 γ 1 γ γ 1 R 1+ ( Mn ) s ( III-8 ) 9. The hub radii at the compressor outlet. r hs = A r - ð s c ( III-9 ) 10. The hub-tip ratio h t s r = r h s c h t, at the compressor outlet. s ( III-10 ) 34

52 CHAPTER III. Numerical Implementation 11. The mean pressure ratio between each stage. PR m = PR 1 n ( III-11 ) 1. The efficiency in each stage. γ 1 γ PR m -1 η stage = γ 1 γηp PR m -1 ( III-1 ) 13. The rotor efficiency (taken from experimental data for tested compressors). η = η + rotor stage ( 1-ηstage ) 4 ( III-13 ) 14. The rotational speed at the compressor. 30U N = πr c t ( III-14 ) 15. The axial velocity at the compressor inlet. Adopted the initial axial velocity ( V a = 00 m/s). V = Q R T Q ae t0 A ( III-15 ) where,. m T t 0 Q= P t π k r -r A 0 Q = 1- ( e ) b c h V a γr T t 0 γ-1 1 γ-1 35 ( III-16 ) ( III-17 )

53 CHAPTER III. Numerical Implementation P= V -V a ae ( III-18 ) Thus, the axial velocity can be iteratively calculated, starting from an arbitrary value of 00 m/s, until converge nce is achieved at P < The axial velocity at the compressor outlet. Adopted the initial axial velocity V a = 00 m/s. V as Q = RT Q A ts ( III-19 ) where,. m Tt s Q = P t π s k b r c -r h ( s ) ( III-0 ) Q Va A = 1- γr Tt s γ-1 1 γ-1 ( III-1 ) P= Va-Va s ( III- ) Likewise, the axial velocity can be iteratively calculated until P < The hub rotor radii at the first stage. A r = r - π r1 h1 c r ( III-3 ) where, 36

54 CHAPTER III. Numerical Implementation A =. mrtt 1 0 r1 1 PVk t 0 ae b γ -1 V a e cos ( 0 ) 1- α γr T γ -1 t0 ( III-4 ) 18. The swirl velocity at the mean height, for all rotors. ( r + r c h ) 1 U r m = πn r 30 ( III-5 ) 19. Total pressure at the rotor inlet, for all rotors. ç T estagio t1 P t =P 4 t +1 0 Tt 0 ã ã-1 ( III-6 ) 0. Total pressure at the stator outlet, for all stators. γ ηrotor T γ-1 t1 Pt = P 3 t T t0 ( III-7 ) For the second to the last stators use is made of the respective total temperature increase at the previous rotors. 1. The absolute velocity at the rotor inlet and in the swirl velocity direction, for all rotors. V ( ) = V tan α ( III-8 ) w0 ae 0. Static temperature at the rotor, for all rotors. 37

55 CHAPTER III. Numerical Implementation T = T - s0 t0 ( Va + Vw ) e 0 γ R γ -1 ( III-9 ) 3. The relative flow angle at the rotor inlet, for all rotors. α = U ( α ) mr 1 atan -tan 0 V ae ( III-30 ) 4. Static pressure at the first rotor. P γ T γ -1 s0 s = P 0 t 0 T t0 ( III-31 ) 5. Mach number at the first rotor outlet. M n1 = V cos a e ( α ) 1 γrt s 0 ( III-3 ) 6. Total pressure at the second rotor outlet. ( γ -1) P = P 1+ M t1 s0 n1 γ γ -1 ( III-33 ) 7. The hub radii of the second rotor. A r h = r r c - π r 1 ( III-34 ) where, 38

56 CHAPTER III. Numerical Implementation A. mrt 1 t4 r = 1 Pt V 4 a k e b γ-1 V a e cos( 4 ) 1- α γr T t4-1 γ ( III-35 ) 8. The hub radii of first stator. r h1s = ( r + r h1 h ) r r ( III-36 ) 9. The swirl velocity at the first stator. U ms ( c rh ) π N r + = 60 1 s ( III-37 ) 30. The relative velocity at the stator exit and in the swirl velocity direction. V w3 γr T + U V tan α γ -1 = U ( ) t1 mr ae 0 3 ( III-38 ) 31. The axial velocity at the first stator outlet. Adopted the initial axial velocity V a = 00 m/s. V = Q R T a3 t4 QA ( III-39 ) where,. m Tt 4 Q = P t π 3 k b r c -r h1 s ( ) ( III-40 ) 39

57 CHAPTER III. Numerical Implementation Q V1 A = 1- γ R Tt 4 γ-1 1 γ-1 ( III-41 ) V = V + V 1 w 3 a ( III-4 ) P= Va-Va 3 ( III-43 ) As previously stated the axial velocity is iteratively calculated, until P < Static temperature at the first stator outlet. T = T - s3 t4 ( Va + Vw ) 3 3 γ R γ -1 ( III-44 ) 33. Static pressure at the first stator outlet. P γ T -1 s γ 3 s = P 3 t 3 T t4 ( III-45 ) 34. The flow angle at the first stator inlet. V w3 α 3 = atan V a 3 ( III-46 ) 35. The absolute flow angle at the first rotor outlet. U α = atan -tan ( α ) 3 3 V a 3 ( III-47 ) 36. Mach number at the first stator inlet. 40

58 CHAPTER III. Numerical Implementation M n Va3 cos = γrts ( α ) 3 ( III-48 ) 37. Total pressure at the first stator inlet. γ 1 P = P 1+ ( γ-1) M t s3 n γ γ -1 ( III-49 ) 38. Mach number at the first stator outlet. Va 3 cos ( α3 ) M n = 3 γrt s3 ( III-50 ) 39. The s c rotor from the Zweifel method, for all rotors. cos ( αm ) ( ) ( ) ( ) s = c tan α -tan α cos α rotor 1 ( III-51 ) where, 1 tan( α M) = tan( α 1) + tan ( α) ( III-5 ) ( ) ( ) sec α = 1+ tan α ( III-53 ) M M cos 1 α M = sec ( ) ( α ) M ( III-54 ) 40. The s c stator from the Zweifel method for all stators. cos ( αm ) ( ) ( ) ( ) s = c tan α -tan α cos α stator ( III-55 ) 41

59 CHAPTER III. Numerical Implementation where, 1 tan ( α M) = tan ( α 3) + tan ( α0) ( III-56 ) ( ) ( ) sec α = 1+ tan α ( III-57 ) M M cos 1 α M = sec ( ) ( α ) M ( III-58 ) 41. The s c limitations. For rotors and stators, s c is limited in the range 0.5 to The inlet blade angle of the first rotor. α -6.5-α B r 1 Ar β 1 = r s c rotor 1-Br - BrMr A r ( III-59 ) where, M r = α ( III-60 ) s = Ar 1-Mr c rotor ( III-61 ) B r s = 0.19 c rotor 4 ( III-6 )

60 CHAPTER III. Numerical Implementation 43. The inlet blade angle of first stator. α -3.5-α r 3 0 Ar β 1 = s 1-B - B M s s s B s c A s stator ( III-63 ) where, Ms = α 0 (III-64 ) s = As 1-Ms c stator ( III-65 ) B s s = 0.19 c stator ( III-66 ) 44. Incidence at the first rotor. ir =α1-β1 r ( III-67 ) 45. Incidence at the first stator. is =α3-β1 s ( III-68 ) 46. Deflection at the first rotor. ε = α -α r 1 ( III-69 ) 47. Deflection at the first stator. ε = α -α s 3 4 ( III-70 ) 48. Axial distance at the first rotor (channel length). 43

61 CHAPTER III. Numerical Implementation Dz 1r = 0 ( III-71 ) 49. Axial distance at stators (channel length). D = D + z1 z s 1r ( r-r c h ) s ( III-7 ) 50. Axial distance at rotors (channel length). ( r-r c h ) r Dz = D z + r 1s 1.5 ( III-73 ) 51. De Haller number at the first rotor. DH r V cos = V cos ( α ) ( α ) a3 1 ae ( III-74 ) 5. De Haller number at the first stator. DH s V cos = V cos ( α ) ( α ) ae 1 a3 ( III-75 ) 53. Mach number at the compressor outlet. ( ) M = M K cos α ( III-76 ) ns n0 b Adiabatic efficiency. ç = a ã ( PR ) ã-1-1 Tt 0 T -T t4 t0 ( III-77 ) 55. Polytropic efficiency 44

62 CHAPTER III. Numerical Implementation ã-1 ln PR ã ç p = ( PR -1) ln + 1 ça ( ) ã ã-1 ( III-78 ) III.1.. Stage Blading Following are the indications for a rotor blading. Stator blading follows the same procedure so that the formulae given below would incorporate the values for the stator. Where there is different approach, both rotor and stator will be separately analysed. 1. Inlet blade angle. β =α -i 1r 1 r ( III-79 ). Outlet blade angle. α β = r s -β M c 1r r A r rotor ( III-80 ) where, M r = α ( III-81 ) s = Ar 1-Mr c rotor ( III-8 ) 3. Camber angle. 45

63 CHAPTER III. Numerical Implementation θ=β -β 1r r ( III-83 ) 4. Stagger angle. θ ζ= -β 1 r ( III-84 ) 5. Deviation angle. δ=α -β s ( III-85 ) 6. Diffusion factor. s DF 1-DH 0.5 = + r c rotor V tan ( α )- V tan ( α ) ae 1 a3 3 V cos ae ( α ) 1 ( III-86 ) 7. Optimum incidence. iopt = iopt + i ( III-87 ) where iopt, is given by [18] = θ 10 + θ 10 + θ θ+ ( III-88 ) iopt where, V C * = n i s c lv1 rotor ( III-89 ) Hence, 46

64 ( ) ( ) ( ) CHAPTER III. Numerical Implementation Clv = s tan α 1 1 -tan α cos αmclv1 c ( III-90 ) V * * s * * * * ( Vn -Vn ) 10+ ( Vd Vn -Vu Vn ) c d u u d * rotor n = * * Vd -Vu ( III-91 ) And * V n is calculated using data from Table III-1 for blade stagger. 1 ( α M ) = ( α 1) + ( α) tan tan tan Clv1 ( III-9 ) cos ( αm ) C lv1 = 1 ( M ) tan 1 α + Clv1 ( III-93 ) α 1 =β 1 +i r r ( III-94 ) For * V n calculation: * s Vd = 10 c rotor ( III-95 ) where * V d is the lower integer value contained in the brackets. V = V + 1 * * u d ( III-96 ) - ζ= ( β 1 +β ) r r ( III-97 ) * * V n and d nu V are calculate with the information from Table III-1, whose curves are curve-fits to experimental data, arranged at integer 47

65 CHAPTER III. Numerical Implementation values of. * VU or * V D. Table III-1: Equations for * V and V (rotors). * n d n u * V u or V * d * * * V = f(ζ; V ) and V = f(ζ; V ) * n u ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ u n d d 8. Optimum lift coefficient (limited to the value of.4) for individual blade section. ( α ) s cos Cl = tan( ) ( ) v 1 i r opt -tan opt β + α c rotor cos α M Clvopt 1 tan α = tan β + i + tan α ( ) ( ) M Cl 1r opt vopt ( III-98 ) ( III-99 ) 1 cos α M Clvopt = tan α M Clvopt + 1 ( III-100 ) 9. Optimum lift coefficient for an ideal blade row (infinite number of blades). C = lv s Clv 6 opt c s 6-1 c rotor rotor ( III-101 ) 10. Calculation of ( s opt ) i - i to calculate the stall incidence. calc 48

66 CHAPTER III. Numerical Implementation ( is - i opt ) = ( is -iopt ) - ( is - iopt ) ( Cl - ) -10C lv + ( is - iopt ) calc 1 l u u l ( III-10 ) Considering the maximum value of ( i -i s opt ) max = ( s opt ) i -i and, Considering the minimum value of ( i -i s opt ) min = ( is -i opt ) l u, this max and min values is limited to the value.5, where Table III-, gives the expressions for the calculation ( is -iopt ) i -i provided C l is in the most probable range of occurrence, 6 to 8. Where: u and ( s opt ) l u i -i => To ( s opt ) u C lu = 10C (greatest integer contained in the expression in brakets) lv opt i -i => and to ( s opt ) brakets). l C lu = 10C - (greatest integer contained in the expression in lv opt 49

67 CHAPTER III. Numerical Implementation Table III-: Equations for ( is -iopt ) and ( s opt ) C ( s opt ) l u u i -i. u l i -i and( is -i opt ) = f(α ; C ) l u α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α +.5 l 11. Stall incidence. i stall = (i s - i opt ) calc + iopt ( III-103 ) 1. Nominal deflection. ε=β + i -α ( III-104 ) 1r opt ( ) ε =β + i - α i - i ( III-105 ) stall 1r opt s opt ε = 0.8ε nominal stall ( III-106 ) 13. Nominal incidence. ( ) * i = x is -iopt + i calc opt ( III-107 ) The nominal incidence is obtained from the curve å versus i, regarding the chosen aerodynamic profile. For the DCA blades and using a Newton-Raphson scheme given in reference [18], x is taken from: 50

68 CHAPTER III. Numerical Implementation ( ε - ε) E = = 0.73x x x x E ( III-108 ) nominal ( i -i s opt ) ε 14. Relation ε. * * ε i +β1 -α r * = * ε ε ( III-109 ) where, * * ε = i +β1 -α r ( III-110 ) 15. Optimum Mach number at the rotor outlet. This value is limited to 0.99 if ' i opt >9º or C >.4, where lv opt ' iopt is the integer value of i opt, according to [18]. M = M nc n 1opt c1 ( III-111 ) where, nc n 1 c ( ) cos α M = M cos i ( opt +β1 ) r ( III-11 ) 0.45 s 0.07 c M n = M c n -1 c 0.07 c rotor 0.45 s rotor ( III-113 ) ' ( )( ) M = Y -Y C -C + Y ( III-114 ) nc d u lvd lv d where, 51

69 CHAPTER III. Numerical Implementation ' s cos Clv = tan ( iopt + 1 )-tan ( ) r c β α rotor cos α ( α ) ( M ' ) Clv ( III-115 ) and, 1 ( α ) = ( ) ( ) M ' opt + β + α 1 r tan tan i tan Clv ( III-116 ) cos ( M ' ) α = Clv 1 ( M ' ) tan α + 1 Clv ( III-117 ) C is for the range of incidence of: -10 to 15. lv opt ' Y D and Y U are calculated accordind Table III-3 where Y D = f( i opt ; C l ) and Y v U = d ' f( i opt +1; C lv u ). Hence: s cos ' Cl = tan( β 1 + iopt )-tan ( α ) vd c rotor cos α ( α ) ( M Cl vd ) ( III-118 ) for which, 1 ' ( α M ) = ( β 1 + opt ) + ( α ) tan tan i tan Cl v d cos ( M C lvd ) α = 1 ( M C lvd ) tan α + 1 ( III-119 ) ( III-10 ) and 5

70 CHAPTER III. Numerical Implementation s ' cos Cl = tan( ) ( ) v 1 iopt 1 -tan u c β + + α rotor cos α ( α ) ( M C lvu ) ( III-11 ) for which, 1 ' ( α M ) = ( β 1 + opt + ) + ( α ) Cl vu tan tan i 1 tan ( III-1 ) cos ( M C lvu ) α = 1 ( M C lvu ) tan α + 1 ( III-13 ) 53

71 CHAPTER III. Numerical Implementation Table III-3: Equations for Cl or v lv d u C. i opt For Y D :X = C lv and for Y d U : X = C lv u X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Design Mach number at the rotor outlet. This value is limited to 0.99 if ' i >9º or C >.4, where i ' is the integer value of i, lv opt according to [18]. M = M nc n 1 c des 1 ( III-14 ) where, nc nc 1des ( ) cos α M = M cos i ( opt +β1 ) r ( III-15 ) 54

72 CHAPTER III. Numerical Implementation 0.45 s 0.07 c M n = M c n -1 c 0.07 c rotor 0.45 s rotor ( III-16 ) ' ( )( ) M = Y -Y C -C + Y nc d u lv l d v d ( III-17 ) where, ' s cos Cl = tan( i + ) ( ) v 1 -tan r c β α rotor cos α ( α ) ( M ' ) C lv ( III-18 ) and, 1 tan α tan M ( i ) tan ( ) C ' = + β 1 + α r lv ( III-19 ) cos α M = ' Cl v 1 tan α + 1 M C ' lv ( III-130 ) C is for the range of incidence of: -10 to 15. lv opt Y D and Y U are calculated accordind Table III-3 (this table is presented in the preceding item), where Y D = f( i ' ; C lv d ) and Y U = f( i ' +1; C lv u ). Hence: s ' cos Cl = tan( ) ( ) v 1 i -tan d β + α c rotor cos α ( α ) ( M ) C lv d ( III-131 ) for which, 1 ' ( α M ) = ( β 1+ ) + ( α) Cl vd tan tan i tan ( III-13 ) 55

73 CHAPTER III. Numerical Implementation cos ( M C lvd ) α = 1 ( MC lvd ) tan α + 1 ( III-133 ) and ( α ) s ' cos Cl = tan( β 1+ i + 1) -tan ( α) vu c rotor cos αm C lvu ( III-134 ) for which, 1 ' tan α M = tan ( β 1+ i + 1) + tan ( α) C lvu ( III-135 ) 1 cos α M C lvu = tan α M + 1 C lvu ( III-136 ) Thus, to calculate the C l vd and Cl v u to obtain M n c1des, only follow the M n c1opt calculation o f the preceding item. III.1.3. Minimum Loss Parameters 1. Minimum loss incidence calculations [1]. The model is the NASA SP-36 model: i ml =K(i i 0 ) 10 +n θ +(id -i D ) ( III-137 ) 56

74 CHAPTER III. Numerical Implementation The quantities involved in this exression are interpolated from the curves given in Figure III- to Figure III-5 [1]. Figure III-: Slope factor for minimum loss incidence angle (n), as function of inlet air angle á 1 and solidity ó = c/s. 57

75 CHAPTER III. Numerical Implementation Figure III-3: Minimum loss incidence angle for zero camber (i 0) 10, as function of inlet air angle á 1 and solidity ó = c/s. Figure III-4: Thickness correction for zero-camber (k i ), as function of thickness chord ratio and airfoil specification. 58

76 CHAPTER III. Numerical Implementation Figure III-5: Deduced variation of average rotor reference incidence angle minus low speed two dimensional cascade rule reference incidence angle with relative Mach number (id -i D). To account for high speed flows, if the relative inlet Mach number is greater than unity, the minimum loss incidence is corrected [5] as indicated in equation ( III-138 ). t 4 * c Mn M relative n i = i +0.5 atan θ π sen 1 sin ml ml + 1 M θ nrelative ( III-139 ). Minimum loss deviatiom. Likewise, the minimum loss deviation is given by equation ( III-140 ), from the NASA SP-36 model: dδ δ =K δ + mσθ+ i i + δ δ di b ( ) ( ) ( ) ml δ 0 10 c D c D D ( III-141 ) 59

77 CHAPTER III. Numerical Implementation The quantities involved are taken from Figure III-6 to Figure III-11 Figure III-6: Deduced variation of slope factor m, in the deviation angle rule (n) as function of inlet air angle á 1 and solidity ó = c/s.. 60

78 CHAPTER III. Numerical Implementation Figure III-7: Zero-camber deviation angle ( δ ) 0 10 deduced from low speed cascade data for 10 percent thick., at reference minimum loss incidence angle Figure III-8: Value of solidity exponent b in deviation angle rule. 61

79 CHAPTER III. Numerical Implementation Figure III-9: Thickness correction for zero camber deviation angle ( K δ ), as function of thichness chord ratio and airfoil specification. Figure III-10: Deviation angle slope dδ di D at reference incidence angle. 6

80 CHAPTER III. Numerical Implementation Figure III-11: Deduced variation of average rotor deviation angle minus low speed two dimensional cascade rule deviation angle at compressor reference incidence angle with δ δ. relative inlet Mach number ( ) c D 3. Critical Mach number. α = i +β 1ml ml 1 ( rotors) ( III-14 ) α = i +β ml ml 3 ( stators ) s cos C = tan( α ) tan ( α ) c α ( α ) ml ( 1 ) lv 1 ml ml ml cos ml ( III-143 ) ( III-144 ) 63

81 CHAPTER III. Numerical Implementation ( ) ( ) ( ) ml c = l ml v l ml v l ml v + ml M C C C ( lv ) ( lv ) ( lv ) 3 iml iml iml C C C + ml ml ml iml iml i ( III-145 ) M = M 1crit c ml ml 0.45 s 1 t c c cos α 0.45 s cos α t c c ( ) ml ( 1 ) ml ( III-146 ) 4. Diffusion factor and loss coefficient at the minimum loss condition. ( α1 ) ml ( α ) cos 1s DF = 1 + tan α tan α cos α cos c ( ) ( ) ( ) ml 1ml ml 1ml ml ( III-147 ) ( ) ( ) ( ) DF DF DF ω ml = 1s cos ( α ml ) c ml ml ml ( III-148 ) III.. Compressor Performance Prediction (Off-Design Performance) For the analysis of the axial compressor at off-design, the calculation procedure is indicated below: 1. Optimum deviation, may be calculated either by Carter s rule [5]. δ = mθ opt s c ( III-149 ) 64

82 CHAPTER III. Numerical Implementation where m = ζ+ ζ ( III-150 ) Or by curve- fitting experimental results [7] 4 3 ζ ζ ζ δ opt = ζ ζ a ζ a a s θ c 100 c c c ( III-151 ) Where ζ is the stagger.. The optimum relative outlet angle. α =β +δ opt r opt ( III-15 ) 3. Optimum incidence. The optimum incidence may be calculated either from the same method used at the design point calculation in section III.1 or by curve-fitting of experimental results [7]. 3 θ θ θ i opt = θ a a c c ( III-153 ) 3 * ζ ζ ζ V n = ζ s ζ s s s c c c c 3 4 s s c c ( III-154 ) 65

83 CHAPTER III. Numerical Implementation 4. Maximum and minimum values of the ( s opt ) i -i. This value may be calculated at design-point section III.1 or by curve- fitting of experimental results [7]. ( is -i opt ) = opt ( opt ) ( opt ) calc 1.84 ( α opt ) Cl C v lv 3 α α α + ( III-155 ) 5. Stall incidence. i stall = (i s - i opt ) calc + iopt ( III-156 ) 6. Nominal deflection. ε=β+ 1 iopt -α opt ( III-157 ) ( ) å = â +i -á i -i stall opt opt s opt ( III-158 ) ε = 0.8ε nominal stall ( III-159 ) 7. Axial velocity at the rotor inlet. The initial velocity is iteratively calculated and the first guesses for velocity is: V a = 00m/s. V Q = RT QA a0 t0 ( III-160 ) V = V 1 a 66

84 CHAPTER III. Numerical Implementation V V1 a = QA = 1- γ R Tt 0 γ-1 1 γ-1 ( III-161 ) P= Va-Va 0 ( III-16 ) The axial velocity is be iteratively calculated until P < The tangential speed at the mean radii. N Um = ( rc + rh) π ( III-163 ) Relative flow angle at the rotor inlet. U α = atan -tan ( α ) m 1 0 V a0 ( III-164 ) 10. Incidence angle. i =α1 β1 ( III-165 ) 11. Flow deflection. * 4 * 3 i i i i ε ε ε= * ε * i i i i * + + * ε ε e e * * e e * ( III-166 ) where, * * ε = i +β1 α ( III-167 ) opt 67

85 CHAPTER III. Numerical Implementation and * i i * ε e is limited to the values -1.1 i i * * ε 0.8. e 1. Relative flow angle at the rotor outlet. α =α1 ε ( III-168 ) 13. Axial velocity at the stator inlet. The axial velocity is iteratively calculated as indicated below. Initial guesses are: V ' a3 = 00m/s and Pt 0 ρ= RT t 0 ( c h ) A =π r -r s ( III-169 ) ( III-170 ) then, Um tan( α 3) = -tan ( α ) ( III-171 ) V ' a 3 cos ( ) α = 3 1 ( ) 1+ tan α 3 ( III-17 ) a. m V = ( III-173 ) ñak b ( ) V = V tan α ( III-174 ) ' w3 a3 3 V ( ) = V tan α ( III-175 ) w0 a0 0 68

86 CHAPTER III. Numerical Implementation ( ) ' 3 = a + 3 w3 V V V ( III-176 ) P = V V < ( III-177 ) ' a3 3 The axial velocity at the stator inlet is be iteratively calculated until P < Total temperature increase. T= t V U -V U w3 m w0 m ã R ã-1 ( III-178 ) 18. Total temperature at the stator inlet. T =T + T t3 t0 t ( III-179 ) 19. Static temperature at the stator inlet. V3 Ts = T 3 t - 3 γ R γ -1 ( III-180 ) 0. Relative Mach number at the rotor inlet. Tt = T s ( γ -1) 3 3 Mn -1 3 ( III-181 ) 1. Static temperature at the rotor inlet. T 1 V γ 1 a 0 s = T 0 t 0 cos( α0 ) γr ( III-18 ) 69

87 CHAPTER III. Numerical Implementation. Absolute Mach number at the rotor inlet. M n 0 = V 0 ( γrts ) 0 ( III-183 ) Relative Mach number at the rotor inlet is calculated the same way as for the design point calculation. 3. Static pressure at the rotor inlet. P P s0 s = 0 t 0 T t0 γ 1 T γ ( III-184 ) 4. Total pressure at the rotor outlet. ( γ 1) P = P 1+ M t1 s0 n1 γ γ 1 ( III-185 ) 5. Losses calculation. Conditions at row outlet are calculated after the losses are obtained. Losses are calculated according to [18] by equation ( III-186 ). loss highspeed cos( α1) = C dhighspeed s 3 ( cos ( αm )) c ( III-187 ) where, cos ( M ) α = tan 1 ( α ) + tan ( α ) ( III-188 )

88 CHAPTER III. Numerical Implementation and C d highspeed is given by equation ( III-189 ). Mn M 1 n M c n M opt 1 nc opt Cd = C highspeed d lowspeed 1 Mn 1 M c n opt c opt ( III-190 ) 4 3 Cd = EQUIV EQUIV + lowspeed EQUIV EQUIV EQUIV EQUIV ( III-191 ) where * * i i i i * + * ε equiv ε EQUIV = * i i ε equiv e ( III-19 ) is limited in the range of: -1.1 to Mn Mn Mn M 1 c 1 n opt copt M + n 1M + c n opt copt 4 3 * Mn 1 Mnc Mn1 M i i opt n c opt = * 1M nc 1M ε equiv opt nc opt Mn1 Mn M M copt n n copt Mncopt Mncopt ( III-193 ) 6. Total pressure at the stator outlet. ( ) P = P loss P P ( III-194 ) t t1 highspeed t1 s0 71

89 CHAPTER III. Numerical Implementation 7. Mach number at the stator outlet. M M n n3 cos ( α3 ) ( α ) cos = ( III-195 ) 8. Static pressure at the stator outlet. P s Pt = ( γ 1) 1+ M n γ γ 1 ( III-196 ) 9. Inlet flow angle at the stator. U α = atan tan α ( ) m 3 V a 3 ( III-197 ) where, α =α1 ε ( III-198 ) 30. Total pressure at the stator outlet. ( ) P = P loss P P ( III-199 ) t t1 highspeed t1 s0 31. Static pressure in the stator outlet. P s3 Pt = ( γ 1) 1+ M n γ γ 1 ( III-00 ) 3. Total pressure in the stator inlet. ( γ 1) P = P 1+ M t3 s3 n3 γ γ 1 ( III-01 ) 7

90 CHAPTER III. Numerical Implementation 33. Total pressure in the rotor inlet. ( ) P = P loss P P ( III-0 ) t0 t3 highspeed t3 s3 34. Overall parameters, pressure ratio, total temperature increase in the compressor, efficiency. PR overall P = P ts t 0 ( III-03 ) T Tt = overall T ts t0 ( III-04 ) γ η overall = overall T t0 ( PR ) γ -1-1 T toverall ( III-05 ) 73

91 CHAPTER IV. Computational Implementation CHAPTER IV. Computational Implementation IV.1. Developed Algorithm Figure IV-1 shows the block diagram for the computer program developed for the compressor design and analysis. It comprises: a) a branch for the design, whose output are all the compressor geometrical data needed for the analysis module; b) a branch for the performance calculation of an existing or a calculated compressor like the one produced through branch a). Branch a) deals with the compressor design, starting with the axial channel, then the blading, ending with estimation of design point performance. Branch b) starts with the existing/calculated compressor data reading, followed by the determination of the optimal operating conditions. Compressor stall and compressor choke mass flows are determined, so that the range of operating mass flows is determinated: minimum mass flow coincides with the stall mass flow and maximum mass flow with the choke mass flow. For a set of mass flows in this interval the losses are evaluated, followed by the compressor non-dimensional parameters. After all mass flows are dealed with, the compressor maps are generated. 74

92 CHAPTER IV. Computational Implementation AFCC DP Design Point or Off-Design Point ODP Input Data Input Data Channel Calculation Optimum Conditions Calculations Rotor Blading Choke Analysis Stator Blading Stall Analysis Minumum Loss Conditions Interpolation of the Mass Flow Points for Compressor Map Exit Data Loss Calculations End Calculation of the Compressor Map Parameters Exit Data Compressor Map Data End Figure IV-1: Scheme of the AFCC in the flowchart representation. 75

93 CHAPTER IV. Computational Implementation IV.. Iterative Processes Iterative processes are used to calculate the unknown temperature, velocity, density, and other parameters. Initial guesses are taken from probable close values. Equations are set out in an order such that it is possible to check whether or not initial guess was correct. By trial-anderror a converged value is calculated. For instance, the calculation of the axial velocity at any section is carried out like the procedure shown in block diagram below. 76

94 CHAPTER IV. Computational Implementation Begin Inlet area calculation Air density (ñ), calculated with the inlet total temperature (T t ) Axial velocity (Va), and meridional velocity (V), calculated with the ñ value Static temperature (T s ), and static pressure (P s ) calculations ñ = ñ1 New density value (ñ1), calculated with the static temperature (T s ) New axial velocity (V a1 ) calculated with ñ1. Va1 Va < (precision) N Y Axial velocity = Va1 Static temperature = T s End Figure IV-: Flowchart showing the iterative processes implemented in AFCC. Since density is unknown, total density is taken as initial guess. 77

95 CHAPTER IV. Computational Implementation A relaxation coefficient is sometimes desirable to speed up or even to allow convergence. Convergence is accepted when guess and calculated values are less than a convenient given error. IV.3. Interpolations Interpolation from the figures shown in the text is linear. The curves are translated into tables where entries are the variables indicated on their axis and the ones shown on the curves. Interpolation details and procedures are from [5]. 78

96 CHAPTER V. Results CHAPTER V. Results V.1. A Study for the Computer Program (AFCC) Validation The design module was validated using reference [5]. The compressor under study is a 8-stage axial compressor, for kg/s, of air at ambient conditions of KPa and 88 K, pressure ratio of and polytropic efficiency of 88.8 %. The off-design module was validated in two ways: a) using the same loss model as reference [18]. In this case, as expected, the results are exactly what was predicted by the reference; b) using the proposed loss model [7]. For this case, Figure V-9 and Figure V-10 showns the compressor map obtained from reference [7]. Good approximation was achieved. V.. Axial Compressor Design Figure V-1 shows a view of the 8-stage axial flow compressor design input file. 79

97 CHAPTER V. Results Figure V-1: Input file for the 8-stage axial flow compressor design. Table V-1 to Table V-9 shows the calculated values and the relative differences. It is possible to verify that there is agreement as far as dimensions are concerned but only reasonable agreement if gas parameters are compared. This may be explained due to different possible design assumptions for the compressors, like the stage loss split between rotors and stators. Needless to say that further investigation in the design process is needed, like the use of more complex methodology like the streamline curvature. 80

98 CHAPTER V. Results Table V-1: Comparison of AFCC/Reference of the compressor external radii. Compressor tip radius (mm) Stage Row AFCC Reference Difference (mm) 1 R1 166, 166,1 0,10 S1 166, 166,1 0,10 R 166, 166,1 0,10 S 166, 166,1 0,10 3 R3 166, 166,1 0,10 S3 166, 166,1 0,10 4 R4 166, 166,1 0,10 S4 166, 166,1 0,10 5 R5 166, 166,1 0,10 S5 166, 166,1 0,10 6 R6 166, 166,1 0,10 S6 166, 166,1 0,10 7 R7 166, 166,1 0,10 S7 166, 166,1 0,10 8 R8 166, 166,1 0,10 S8 166, 166,1 0,10 Outlet 166, 166,1 0,10 Table V-: Comparison of AFCC/Reference of the compressor hub radii. Compressor hub radius (mm) Stage Row AFCC Reference Difference (mm) 1 R1 10,1 10, -0,10 S1 111,7 109,3,40 R 11, 116,5 4,70 S 17,1 13,9 3,0 3 R ,4 1,60 S3 136,7 135,6 1,10 4 R4 140,4 139,9 0,50 S4 14,9 14,7 0,0 5 R5 145,4 145,6-0,0 S5 147, 147,6-0,40 6 R6 148,9 149,6-0,70 S6 150,3 150,7-0,40 7 R7 151,6 151,9-0,30 S7 151,8 15-0,0 8 R8 15,1 15,1 0,00 S8 15,1 15, -0,10 81

99 CHAPTER V. Results Table V-3: Comparison of AFCC/Reference of the compressor axial dimensions. Compressor axial dimensions (mm) Stage Row AFCC Reference Difference (mm) 1 R S1 36,3 0,40 15,90 R 66,3 4,60 3,70 S 9,3 60,00 3,30 3 R3 114,4 77,00 37,40 S ,50 43,50 4 R4 151,1 104,0 46,90 S4 166,6 115,70 50,90 5 R5 180,4 17,0 53,0 S5 193,1 140,00 53,10 6 R6 04,5 15,60 51,90 S6 15,1 164,10 51,00 7 R7 4,7 175,10 49,60 S7 34, 185,60 48,60 8 R8 43,6 195,90 47,70 S8 5,9 05,80 47,10 Outlet 6, ,300 Table V-4: Comparison of AFCC/Reference of the compressor inlet blade angles. Inlet blade angles Stage Row AFCC Reference Difference 1 R1 54,6 53,83 0,7700 S1 33,96 4,76-8,8000 R 54,84 56,7-1,4300 S 4,36 46,14-3, R3 54, ,3900 S3 43,08 45,55 -, R4 53,93 54,91-0,9800 S4 44, 45,98-1, R5 5,98 53,79-0,8100 S5 45,53 46,98-1, R6 51,79 5,04-0,500 S6 46,95 48,68-1, R7 50,51 51,89-1,3800 S7 48,68 51,4 -,700 8 R8 53,56 55,44-1,8800 S8 51,55 55,47-3,900 8

100 CHAPTER V. Results Table V-5: Comparison of AFCC/Reference of the compressor outlet blade angles. Outlet blade angles Stage Row AFCC Reference Difference 1 R1 3,57 38,64-6,070 S1-3,61 4,9-8,510 R 31,58 39,14-7,560 S 0,83 0,0471 0,783 3 R3 30,8 38,83-8,010 S3 5,8 5,98-0,700 4 R4 8,94 37,38-8,440 S4 9,63 11,59-1,960 5 R5 6,47 35,0-8,550 S5 13,96 17,09-3,130 6 R6 3,5 3,38-8,880 S6 18,4,44-4,00 7 R7 3,41 34,71-11,300 S7 3 7,51-4,510 8 R8 9,39 41,16-11,770 S8 7, 3,7-5,070 Table V-6: Comparison of AFCC/Reference of the compressor camber blade angles. Camber blade angles Stage Row AFCC Reference Difference (degree) R1,03 14,95 7,08 1 S1 43,57 46,53 -,96 R 3,7 16,76 6,51 S 41,53 44, -,67 R3 3,8 17,11 6,69 3 S3 37,81 38,0-0,1 R4 4,99 17,58 7,41 4 S4 34,59 3,91 1,68 R5 6,5 19,01 7,49 5 S5 31,58 8,8 3,30 R6 8,3 19,95 8,35 6 S6 8,53 4,5 4,03 R7 7,11 17,6 9,49 7 S7 5,68,31 3,37 R8 4,17 10,13 14,04 8 S8 4,35 7,41-3,06 83

101 CHAPTER V. Results Table V-7: Comparison of AFCC/Reference of the compressor stagger blade angles. Stagger blade angles Stage Row AFCC Reference Difference (degree) R1-43,6-46,,7 1 S1-18, -18,9 0,8 R -43, -47,7 4,5 S -1,5-3,1 1,6 R3-4,7-47,4 4,7 3 S3-4, -5,8 1,6 R4-41,4-46,1 4,7 4 S4-6,9-8,8 1,9 R5-39,7-44,8 5,1 5 S5-9,7-3,0,3 R6-37,6-3,0-5,6 6 S6-3,7-4, 9,5 R7-37,0-35,6-1,4 7 S7-35,8-43,3 7,5 R8-41,5-39,5 -,0 8 S8-39,4-48,3 8,9 Table V-8: Comparison of AFCC/Reference of the parameter de Haller. de Haller parameter Stage Row AFCC Reference Difference 1 R1 0,65 0,61 0,04 S1 0,87 0,81 0,06 R 0,65 0,6 0,05 S 0,8 0,76 0,06 3 R3 0,66 0,63 0,03 S3 0,8 0,76 0,04 4 R4 0,68 0,63 0,05 S4 0,78 0,73 0,05 5 R5 0,69 0,64 0,05 S5 0,76 0,7 0,04 6 R6 0,69 0,63 0,06 S6 0,75 0,68 0,07 7 R7 0,69 0,6 0,07 S7 0,68 0,63 0,05 8 R8 0,67 0,56 0,11 S8 0,65 0,49 0,16 84

102 CHAPTER V. Results Table V-9: Comparison of AFCC/Reference of the compressor blade deflections. Blade deflections Stage Row AFCC Reference Difference 1 R1 15,7 10,0 5,7 S1 36,3 35,9 0,4 R 17, 11,6 5,6 S 34,9 34, 0,7 3 R3 17,7 1,4 5,3 S3 31,5 9,4,0 4 R4 18,7 13,1 5,6 S4 8,6 5,6 3,0 5 R5 0,0 14,7 5,4 S5 6,0,1 3,9 6 R6 1,5 15,7 5,8 S6 3,3 19, 4,1 7 R7 19,8 14,0 5,8 S7 1,0 17, 3,7 8 R8 17,7 5,3 1,4 S8 0,5,4-1,9 The results calculated at the AFCC, are presented in the Appendix A. V.3. Performance Analysis of an Existing Axial Compressor For the validation of AFCC at off-design conditions, an eight stage compressor was chosen from reference [18] and the results compared both the same loss model and the loss model from reference [7]. Results from reference [18] has been validated with experimental data. Figure V- shows the calculated from reference [18] and actual performance of 8-stage compressor. 85

103 CHAPTER V. Results 8-stage axial flow compressor pressure ratio Pressure ratio x Mass flow Pressure ratio Experimental data Reference results Mass Flow (kg/s) Figure V-: 8-stage axial flow compressor performance maps (actual and prediction). 8-stage axial flow compressor efficiency Efficiency x Mass flow 1,0 Efficiency 0,9 0,8 0,7 Experimental data Reference results 0, Mass Flow (kg/s) Figure V-3: 8-stage axial flow compressor efficiency. 86

104 CHAPTER V. Results Based on the quality of the calculated parameters from [18], it is assumed that loss model and performance calculation methodology are adequate for initial compressor design and assessment. For the AFCC validation at off-design, the same 8-stage axial compressor requirement at design point has been used. The corresponding input data file is shown on Figure V-4. Note that all compressor dimensions (radius) are presented in millimetre (mm). Figure V-4: Input data of the AFCC for off-design case (dimensions in mm). 87