Axial Flow Compressor Mean Line Design

Transcription

1 Axial Flow Compressor Mean Line Design Niclas Falck February 2008 Master Thesis Division of Thermal Power Engineering Department of Energy Sciences Lund University, Sweden

2 Niclas Falck 2008 ISSN ISRN LUTMDN/TMHP 08/5140 SE Printed in Sweden Lund 2008

3 Preface This master thesis has been conducted at the division of Thermal Power Engineering, department of Energy Science, Lund University, Sweden. This experience has been very educational in terms of modeling and computing thermal energy devices. This thesis has been about axial flow compressors, but the approach and methodic that I have implemented in this thesis will also be useful in my future career as an Engineer regardless of branch. I want to thank my supervisor Magnus Genrup for his support and expertise in the field of turbomachinery. I also want to thank the rest of the department of Energy Science, especially my fellow master thesis workers, for an enjoyable time here in Lund.

4 Abstract The main objective in this thesis is creating a method on how one can model an axial flow compressor. The calculation used in this thesis is based on common thermodynamics and aerodynamics principles in a mean stream line analyses. Calculations based on one stream line i.e. one dimension, is a good first start to model a compressor. Most of the correlations and thermodynamics are based on one stream line, or they can be modified to work on one stream line. By just a handful of design specifications an accurate model can be generated. These specifications can be mass flow, rotational speed, number of stages, pressure ratio etc. The pressure ratio is also the one parameter that the calculation aims to satisfy. If the calculation results in a pressure ratio that is not what was specified in the beginning an adjustment must be made on one parameter. In this case the stage load coefficient is selected. By changing the stage loading coefficient and keeping the other parameters constant the pressure ratio will vary. This is done in an iterative process until the pressure ratio is converged. The purposes of modeling compressors based on correlations and thermodynamics and not model them in a CFD (Computational Fluid Dynamics) simulation program at once is that it takes a long time for a calculation to converge in a CFD program. Finding better correlations and methods on how one can model a compressor will result in fewer hours fine tuning them in advanced fluid dynamic programs and hence same time and not to mention money.

5 Content Nomenclature... 4 Introduction Background Gas turbine Compressor Stagnation property Compressor Fundamentals Compressor operation Blade to Blade Flow path Rothalpy Compressor Losses Profile-loss Endwall-loss Blade geometry Dimensionless Parameters Stage load coefficient Stage flow coefficient Stage reaction de Haller number Pressure rise coefficient Efficiency Isentropic efficiency Polytropic efficiency Operating Limits Methods of Calculation State properties Incidence and Deviation Incidence Angle

6 Content Axial Flow Compressor Mean Line Design Deviation Angle Diffusion Factor and Diffusion Ratio Losses Profile loss model Endwall loss model Total loss Pitch Chord ratio Diffusion Factor Method Hearsey Method McKenzie Method Stall/Surge Calculation procedure Input parameters Main specification Detailed specification Inlet specification Parameter variations throughout the compressor Calculation limitations Mean stream line analyses Convergence criteria s Structure of the calculation Module Module Module Module Newton-Rhapson Method Calculation process Module 0, Inlet geometry Module 1, Rotor-inlet Module 2, Rotor-outlet/stator-inlet Module 2.1 start Module Module 2.2 start

7 Axial Flow Compressor Mean Line Design Content Module Module 3, Stator-outlet Module 3.1 start Module Module 3.2 start Module Outlet Guide Vane, OGV Blade angles calculation Result LUAX-C Structure of the program User Guide to LUAX-C References Appix A, polynomial coefficients for the graphs B, MATLAB script for the calculations B.1 Main Calculation B.2 Inlet geometry calculation B.3 Pitch chord ratio B.4 Diffusion Factor and Diffusion Ratio B.5 Compressor losses B.6 Blade angles Appix C

8 Nomenclature Symbol Unit Description a [m/s] Speed of sound A [m2] Area c p [kj/kgk] Specific heat at constant pressure c v [kj/kgk] Specific heat at constant volume c [m] Chord C [m/s] Absolute velocity C [m/s] Tangential absolute velocity C m [m/s] Meridional velocity C p [-] Static pressure rise coefficient DF [-] Diffusion factor D eq [-] Equivalent diffusion ratio h [kj/kg] Static enthalpy h 0 [kj/kg] Stagnation enthalpy H [m] Blade height i [ ] Incidence I [-] Rothalpy m [kg/s] Massflow Ma [-] Mach-number N [rev/s] Rotational speed p [bar] Static pressure p 0 [bar] Stagnation pressure r [m] Radius R [J/kgK] Gas constant s [kj/kg] entropy S [m] Staggered spacing t [m] Maximum blade thickness T [K] Static temperature T 0 [K] Stagnation temperature U [m/s] Blade velocity W [m/s] Relative velocity W [m/s] Tangential relative velocity 4

9 Axial Flow Compressor Mean Line Design Nomenclature Symbol Unit Description [ ] Angle between absolute velocity and axial direction [ ] Angle between relative velocity and axial direction [ ] Stagger angle [ ] Deviation [m] Endwall clearance [%] Efficiency [-] Heat capacity ratio, Isentropic exponent Heat conductivity [Kg/m2] Density [m2/s] Kinematic viscosity Pressure loss coefficient [ ] Camber angle [-] Stage load coefficient [-] Stage flow coefficient 5

10 Introduction The development of gas turbines has in the recent years come a long way. Serious development began during the Second World War with the key interest of shaft power, but attention was shortly transferred to the turbojet engine for aircraft propulsion. The gas turbine began to compete successfully in other fields in the mid 1950s, since then it has made a successful impact in an increasing variety of applications. When combining a gas turbine with a heat recovery steam generator the heat, that otherwise would be wasted from the gas turbine outlet, can be extracted. Together with a conventional steam generator this will form a combined cycle. The efficiency of a combined cycle power plant is far better than regular gas turbine power plants. The question is than, how could we improve the efficiency of a gas turbine? One can either focus on the compressor, the combustion chamber or the turbine. In this thesis the compressor, especially the axial flow compressor, will be investigated. When designing a new compressor, a good start is to create a base design for the compressor. By just a handful of design specifications an accurate model can be generated. The modelling techniques used are based on combinations of thermodynamic and aerodynamic correlations. This base design will make up for about % of the finished design. In this first stage in designing a new compressor, designs that would not work or have pore efficiency can be avoided. Further on in the process powerful CFD (Computational Fluid Dynamics) simulation programs are being used. A CFD calculation takes a long time and hence cost a lot of money. The solution to cutting down the number of simulations is then to make the base design more accurate. 6

11 1 Background 1.1 Gas turbine A gas turbine consists mainly by three components, the compressor, the combustion chamber and the turbine, see Figure 1.1. The compressor is one a part of the entire gas turbine, but never the less, an important and probably the most complicated component to design in an aerodynamic point of view. The working fluid enters an inlet duct and continues to the compressor. The compressor pressurises the fluid and will also lead to an increase in temperature. Deping on the application it can either have a radial or an axial design deping on mass flow and pressure ratio. After the compressor, the pressure of the working fluid will have increased to bar, even above 40 in aero engines, and will have a temperature of about 500 C. By combustion of fuel in the combustion chamber, energy is added to the working fluid. A gas turbine is very flexible in terms of what sort of fuels can be used. The working fluid which now has a temperature of about C enters the last stage in the process, the turbine. Here the fluid expands and thus transferring its energy to the turbine blade in form of mechanical work. The turbine is connected to the compressor by a shaft and this lead the mechanical work from the turbine to the compressor. If the gas turbine is to be used in a multi-shaft configuration, the work provided by the turbine will just be enough to drive the compressor otherwise a load can be connected like a pump, a propeller or a generator. Combustion chamber Load Compressor Turbine Figure 1.1, Schematic figure over the main components in a gas turbine 7

12 1 Background Axial Flow Compressor Mean Line Design 1.2 Compressor There are two types of compressor designs, radial and axial flow compressors, see Figure 1.3 and Axial flow compressors are divided in a series of stages, each stage consistss of a rotating rotor and a stationary one called stator. It is difficult to get a high pressuree rise in a single stage. Unlike axial flow compressor rs, the radial compressor often consists of a single stage. It is possible to obtain a higher pressure rise over one stage in a radial compressor. An axial flow compressor can handle a much larger mass flow compared to a radial flow compressor. If one would like to have a small compact compressor a radial design is the best choice. But if high power is required, for an examplee in a jet engine for a big airliner, an axial flow is not just the best but probably the only choice. An example of a radial compressor in an aircraft is the Swedish aircraft SAAB J29 also known as Tunnan (in eng. The Barrel ) ). This has a very wide fuselagee because of the large radial compressor design, see Figure 1.2. Figure 1.2, SAAB J29 A deeper insight of the axial flow compressors construction and its design will follow and be discussed in this thesis. Figure 1.3, Axial flow compressor 8

13 Axial Flow Compressor Mean Line Design 1 Background Figure 1.4, Radial flow compressor 1.3 Stagnation property When the kinetic and potential energies of a given fluid are negligible, as is often the case, the enthalpy represents the total energy of the fluid. For high speed flows, M>0.4, the kinetic energy is highly noticeable, but the potential energy is still negligible. It is the convenient to combine the kinetic energy with the enthalpy of the fluid into a single term called stagnation (or total) enthalpy h 0, which is defined as. (1.1) If the kinetic energy is negligible the enthalpy is the referred as the static enthalpy, h. Consider a duct such as a nozzle or a diffuser where a fluid is flowing through, see Figure 1.5. The flow takes place under an adiabatic process where there is no work input or output. Assuming there is no potential energy difference through the duct for the fluid, the energy balance can then be reduced to. or 9

14 1 Background Axial Flow Compressor Mean Line Design 2 1 Figure 1.5, Steady flow of a fluid through an adiabatic duct The stagnation enthalpy will not change through a duct if there is no heat or work done to the system. Flows through nozzles or diffusers usually satisfy these conditions, and any changes in the fluid velocity will create a change in the static enthalpy of the fluid. Substituting the enthalpy with temperature instead results in the following expression or (1.2) C p represents the specific heat value for the fluid for an ideal gas. T 0 is called stagnation (or total) temperature. The term V 2 /2C p is called the dynamic temperature and corresponds to the temperature rise during an adiabatic process. The pressure a fluid obtains when brought to rest is called stagnation pressure, P 0. For ideal gases with constant specific heats, P 0 is related to the static pressure of the fluid by, represents the specific heat ratio, C p /C v. 10

15 2 Compressor Fundamentals 2.1 Compressor operation A typical axial flow compressor consists of a series of stages; each stage has a row of moving rotor blades followed by a row of stator blades which is stationary, see Figure 2.1. The rotor blades accelerates the working fluid thus gaining energy, this kinetic energy is then converted into static pressure by decelerating the fluid in the stator blades. The process is then repeated as many times as necessary to get the required pressure ratio. The number of stages in a compressor is important especially when the engine will be used in an aircraft. The main reason is that too many stages will result in an increase in weight and a large core engine length. For land based gas turbines the main reason is the cost, which will increase when adding more stages. Some different compressors used in aircrafts are shown in Table 2.1, and here one can see how compressor improvement has come along over the years. Figure 2.1, Cross-section view over a compressor flow path Engine Date Thrust Pressure Stages [kn] ratio Avon Spey RB Trent Table 2.1, Compressor evolution, aircraft engine 11

16 2 Compressor Fundamentals Axial Flow Compressor Mean Line Design As discussed earlier all the power is absorbed in the rotor and the stator transforms the kinetic energy which has been absorbed by the rotor into an increase in static pressure. The stagnation temperature remains constant throughout the stator since there is no work feed into the fluid. Figure 2.2 shows a sketch of a typical compressor stage. T 02, T 03 Temperature, T T 03 p 02 p 3 p 03 p 2 T p 01 T 01 T 1 p 1 R S Entropy, s Figure 2.2, Compressor stage and T-s diagram The stagnation pressure rise occurs wholly in the rotor, but in practice, there will be some losses in the stator due to fluid friction which will result in a decrease in stagnation pressure. There are also some losses in the rotor and the stagnation pressure rise will be less than of an isentropic compression. 2.2 Blade to Blade Flow path To get a clear picture in how a compressor works, blade to blade flow path analysis is the most fundamental part. The velocity components of the working fluid can be expressed in two velocity vectors, absolute and relative velocity. The fluid enters the rotor with an absolute velocity, C 1, and has an angle, 1, from the axial direction. Combining the absolute velocity with the blade speed, U, gives the relative velocity, W 1, with its angle 1. The mechanical energy from the rotating rotors will be transferred to the working fluid. This energy absorption will increase the absolute velocity of the fluid. After leaving the rotor the fluid will have a relative velocity, W 2, with an angle, 2, determined by the blade outlet angle. The fluid leaving the rotor is consequently the air entering the stator where a similar change in velocity will occur. Here the relative 12

17 Axial Flow Compressor Mean Line Design 2 Compressor Fundamentals velocity, W 2, will be diffused and leaving the stator with a velocity, C 3, at an angle, 3. Typically the velocity leaving the stator will be the same as the velocity entering the rotor in the next row, C 3 = C 1 and 3 = 1. By creating so called velocity triangles, see Figure 2.3, will make it easier to visualize the change of velocities and angles in a compressor stage [1]. 1 C 1 W C 1 C a1 C 1 U 2 C 2 W 2 C C a2 C 2 C Rothalpy Figure 2.3, Velocity triangles for one stage The work, W, is expressed as the enthalpy change. For adiabatic machines the heat flux, Q, is zero. Introducing the Euler equation and expanding the stagnation enthalpy gives after rearrangement. Consider the left-hand side, expanding C 2 2 as C C x2 2 + C R2 2 and then expressing the absolute tangential velocity in terms of that in the moving frame of reference C 2 = W 2 + U 2. After some manipulation to the left-hand side of the equation one obtains. This can then be used for obtaining the difference between the inlet and outlet. 13

18 2 Compressor Fundamentals Axial Flow Compressor Mean Line Design Or alternatively The term (h 02 ) rel is the stagnation enthalpy in the relative frame of reference. The rothalpy is defined as the quantity (2.1) In rotating blade rows rothalpy has properties analogous to stagnationn enthalpy in stationary passages. If the same concept of rothalpy is applied to a stationary blade row the equation reverts to conservation of stagnation enthalpy [2]. 2.4 Compressor Losses The flow in a compressor is complicated 3-D, unsteady and dominated by viscous effects, see Figure 2.4. This dissipative nature increases the entropy and a loss in pressuree occurs due to the flow effects. Figure 2.4, Flow fields in a cascade 14

19 Axial Flow Compressor Mean Line Design 2 Compressor Fundamentals The individual losses are lumped into profile- and wall-losses. These pressure losses are depent on a numerous parameters which include tip clearance, blade aspect ratio, pitch chord ratio, thickness chord ratio, Mach number and Reynolds number. The different loss models are based on mid radius and will be modelled individually for the rotor/stator Profile-loss Profile-losses are based on the effect of blade boundary layer growth (including separated flow) and wakes through turbulent and viscous dissipation. The effect of these losses is an increase of entropy due to the heat developed by the mechanical energy within the boundary layers. This results in a stagnation pressure loss [3] Endwall-loss In addition to the losses which arise from the blade surfaces, i.e. profile losses, additional losses generated on the walls. These are often called secondary losses which arises from -wall boundary layer build up, secondary flow and tip clearance. When a flow that is parallel but non-uniform in velocity and density is made to follow a curved path, the result is a three-dimensional motion with velocity normal to the overall flow direction. Cross-flow of this type is referred as secondary flow. A good analogy of this is a simple teacup. When stirring the tea in a teacup, the tea leafs will move towards the center of the cup driven by the secondary flow. The formation, development, diffusion and dissipation of these vortices as well as the kinetic energy in secondary velocities generate secondary flow losses. Somewhere between 50-70% of the losses may come from wall losses, deping of the type of turbo machinery [3]. 15

20 2 Compressor Fundamentals Axial Flow Compressor Mean Line Design 2.5 Blade geometry 1 W 1 t b1 C Profile Camber line W 2 2 b2 S Figure 2.5, Cascade notation 1 b1 2 b2 i c S t Relative air inlet angle Blade inlet angle Relative air outlet angle Blade outlet angle Stagger angle Camber angle Incidence angle, 1 - b1 Deviation angle, 2 - b2 Chord length Staggered spacing Maximum thickness Solidity, c/s Table 2.2, Cascade notation 2.6 Dimensionless Parameters Introducing a set of dimensionless parameters will give a useful guidance in designing a compressor stage. These dimensionless performance parameters define the performance of a single stage in a compressor. 16

21 Axial Flow Compressor Mean Line Design 2 Compressor Fundamentals Stage load coefficient The total enthalpy rise through a rotor blade row is expressed by the well-known Euler turbine equation, i.e. (2.2) where H is the total enthalpy rise through the rotor. It is often useful to introduce dimensionless stage performance parameters for a repeating stage, i.e. the rotor-inlet (station 1) and the stator-outlet (station 3) from the previous stage has identical velocity diagrams. Then, the stage load coefficient,, can be defined as (2.3) Stage flow coefficient The stage flow coefficient,, is defined as followed. (2.4) This expresses the ratio between the meridional velocity and the blade velocity. A high stage flow coefficient indicated a high flow through the stage relative to the blade velocity. A low whirl velocity change in a stage would also indicate a high stage flow coefficient and vice versa [1] Stage reaction The stage reaction, R, is defined as the fraction of the rise in static enthalpy in rotor compared to the rise in stagnation enthalpy throughout the entire stage. (2.5) If a compressor stage would have a stage reaction of 1.0 or 100%, the rotor would do all of the diffusion in the stage. Similar if the stage reaction is 0 than the stator will do all of the diffusion of the working fluid. It is never good to have either a stage reaction of 1.0 or 0. The literature, reference 1, suggest that a stage reaction about 0.5 i.e. the diffusion is equally divided between the two blade rows. But in practice a higher stage reaction is preferred. Increasing the stage reaction results in a decrease in whirl prior to the rotor. A smaller whirl will create a larger relative inlet velocity to the rotor row, at a constant C p, and hence make it easier for the rotor to increase the static pressure de Haller number In most compressor stages both the rotors and the stators are designed to diffuse the fluid, and hence transform its kinetic energy into an increase in static enthalpy and static pressure. The more the fluid is decelerated, the bigger pressure rise, but boundary layer growth and wall stall is limiting the process. To avoid this, de Haller proposed that the 17

22 2 Compressor Fundamentals Axial Flow Compressor Mean Line Design overall deceleration ratio, i.e. W 2 /W 1 and C 2 /C 3 in a rotor and stator respectively, should not be less than 0.72 (historic limit) in any row [1] Pressure rise coefficient Another parameter is the pressure rise coefficient. (2.6) (2.7) If axial velocity is assumed constant and the working fluid is assumed to be incompressible, then the pressure rise coefficient can also be expressed as a function of the dehaller number. This is done by applying Bernoulli s principle. (2.8) 2.7 Efficiency The term efficiency finds very wide application in turbo machinery. For all machines or stages, efficiency is defined as. There are several different ways of evaluating efficiency and these reveal different information. Two of the most widely used efficiencies are the isentropic efficiency and the polytropic efficiency Isentropic efficiency The isentropic efficiency can be expressed as the ratio between enthalpy change in an ideal compressor and the actual enthalpy change. An ideal compressor which is both adiabatic and reversible cannot alter the entropy of the gas flowing through it. These types of compressors are usually referred to as isentropic. Since there will be some 18

23 Axial Flow Compressor Mean Line Design 2 Compressor Fundamentals losses which generates an entropy rise, the actual work into the compressor will differ from an ideal one. The efficiency can then be described as, (2.9) The subscript s denotes entropy held constant. Figure 2.6 shows a typical schematic diagram over a reversible adiabatic compression. Temperature T p 02 2s 2 p 01 1 Entropy S Figure 2.6, isentropic compression The constant pressure lines in the T-S diagram, Figure 2.6, have a slope proportional to the temperature and diverge as the temperature increases. For a given pressure rise the work input needed is greater for the later stages in a compressor, this because the temperature is higher and also that the work input required by the later stages is raised because of the previous stages. The isentropic efficiency therefore gets lower as the overall pressure ratio is increased. To cope with this problem, another efficiency the socalled polytropic or small-stage efficiency may be used instead [2] Polytropic efficiency The definition of polytropic efficiency is as follows. By applying Gibbs law and the relationship between temperature and enthalpy it can be rewritten so it deps on temperatures and pressures instead. 19

24 2 Compressor Fundamentals Axial Flow Compressor Mean Line Design Integrating the expression on pressure, p leads to the following equation. (2.10) One can also assume that the specific heat capacity is constant, which is not the case in this thesis. If this is assumed, the following expression can be found [2]. (2.11) 2.7 Operating Limits There are mainly two phenomena that can cause a compressor to break down, rotating stall and surge. Gas turbines, for example, may encounter severe performance and durability problems if the compressor is not able to avoid stall and surge. In preliminary designs there is a need for reliable methods for computing the compressors stall margin capability. This because it is difficult to correct and change the compressor stall margin after its basic design has been chose. In a typical compressor it is normal that if the mass flow is reduced the pressure rise increases. At a certain point in an operating range the pressure rise is at its maximum, in a further reduction in mass flow will lead to an abrupt and definite change in flow pattern in the compressor. This change in flow pattern is known as surge and can cause the flow to start oscillating backwards and forwards, and after a while the compressor will break down. A mild version of surge causes the operating point to orbit around the point of maximum pressure rise. An audible burble is a clear indicator when the compressor is on the limit of the more severe surge [2]. The other phenomena that one should be looking for is stall. If the mass flow is reduced the axial velocity will, according to the continuity equation, also decrease. This will increase the air inlet angle and, due to the difference in air inlet angle and blade inlet angle, create incidence. With an increasing incidence angle the flow will eventually separate from the surface at the trailing edge. The separation will grow with a further 20

25 Axial Flow Compressor Mean Line Design 2 Compressor Fundamentals increase of incidence angle, and finally cover the whole upper blade. This phenomenon is called stall, and will change the performance of a compressor drastically. Rotating stall means that the stall is moved from one blade to another and an uninformed pattern will occur, see Figure 2.7. The annulus then contains regions of stalled flow, usually referred as cells, and regions of unstalled flow. Rotating stall is a mechanism which allows the compressor to adapt to a mass flow which is too small. Instead of trying to share the limited flow over the whole annulus the flow is shared unequally, so that some areas have a larger mass flow than other. The overall mass flow remains constant but the local mass flow varies as the rotating cell passes the point of observation. The cells always rotate in the direction of the rotor. Part-span cells very often rotate at close to 50 percent of the rotor speed, full-span cells usually rotate more slowly in the range of percent. Full-span cells ext axially through the whole compressor while part-span cells can exist in a single blade row [2]. Cell Unstalled flow Full-span stall Part-span stall Figure 2.7, Different types of rotating stall 21

26 3 Methods of Calculation 3.1 State properties In order to calculate the state of a fluid, an approach according to the Gibbs-Dalton is used. The model used is the NASA SP-273, and by integration, the enthalpy and entropy are known. The specific heat is expressed as fifth order polynomial. The reference values are set to zero at kpa and K. as seen in the equations above. As seen in the equation above, the temperature and the pressure must be known if the entropy and enthalpy are to be found. If let say that the enthalpy and the temperature are known instead, an iterative process is needed since the specific heat value is expressed as a fifth order polynomial. This iterative procedure uses the standard Newton method, see chapter 4.5 Newton Rhapson Method. Introduction of other property libraries are straightforward, as long as they are semi-perfect (specific heat only a function of temperature) [11]. 3.2 Incidence and Deviation There are several different methods on how to get the blade angles in a cascade. In this thesis, one method is used to calculate the angles based on a number of input variables. Howard, see reference 4, has put together a number of correlations and equations based on Johnsen and Bullock (1965), which commonly is referred to as NASA SP-36 correlations. These correlations are largely based on low speed cascade test; he also introduces some correlations for advanced transonic compressor blades by Köning, et al (1996), but this will not be taken in consideration in this thesis. 22

27 Axial Flow Compressor Mean Line Design 3 Methods of Calculation Incidence Angle Incidence is the difference between the inlet blade angel and the inlet flow angle. As the fluid flows towards the leading edge it will experience induced incidence. There is one pressure surface and one suction surface at a given blade. This different of pressure will change the ingoing flow angle as it approaches the leading edge, see Figure 3.1. _ + Figure 3.1, induced incidence By performing experimental tests on a given cascade, the incidence can be established. This incidence angle is referred as reference incidence. When testing a given cascade at different inlet flow angles, the loss coefficient,, varies with incidence. There will be an increase in both positive and negative incidence angles with a range of low values for. The pressure loss at twice the minimum loss will be the range in which the reference incidence will be located. Outside this range stall blade stall occurs. If this range of incidence is split in the middle, the point of reference incident angle will be found, see Figure 3.2. Pressure loss, Reference incidence angle i/2 i/2 Min. loss 2 x Min. loss Incidence angle, i Figure 3.2, Definition of reference incidence angle 23

28 3 Methods of Calculation Axial Flow Compressor Mean Line Design The correlations for reference incidence angle are presented below: (3.1) K sh and K it are correction factors for blade shape and thickness respectively. K sh differs whether the blade is a DCA, 65-Serie or a C-Series, see Table 3.1. K sh Blade type 0.7 DCA Series 1.1 C-Series Table 3.1, Shape factor, K sh, for the calculation of incidence angle (3.2) i 010 is the incidence angle based on 10% thick blades, see Figure 3.3. The variable n represents the incidence slope factor, see Figure 3.4. (3.3) (3.4) In the polynomials above, 1 denotes the relative inlet flow angle for the rotor, and replaced by 2 for the stator. 24

29 Axial Flow Compressor Mean Line Design 3 Methods of Calculation incidence, i flow inlet angle [degree] Figure 3.3, reference incidence angle for profiles with zero camber with variations in solidity slope factor, n flow inlet angle [degree] Figure 3.4, slope factor, n, with variations in solidity 25

30 3 Methods of Calculation Axial Flow Compressor Mean Line Design Deviation Angle Deviation angle is the difference between the blade angle from the trailing edge and the exit flow angle. It arises from a combination of two effects. Firstly the flow is decelerating on the suction surface and accelerating on the pressure surface as it approaches the trailing edge. A result from this is that the streamlines are diverging from the suction surface and converging towards the pressure surface so that the mean flow angle is less than the blade angle. This is an inviscid effect which increases in magnitude with the rate of diffusion and acceleration towards the trailing edge. Secondly the rapid boundary layer growth on the suction surface towards the trailing edge pushes the streamline away from the surface, contributing to the deviation [6]. A method to calculate the deviation angel is to use the classical Carter s rule. = deviation angle m c = an empirical function of stagger angle, = camber angle = solidity x = experimental factor (typical 2) The deviation angle is depent on both camber and stagger angle. This makes it quite difficult to calculate the deviation angle, therefore another approach is needed. The deviation angle used in the calculation is based on the reference incidence angle and the structure looks very similar to that of the reference incidence angle. (3.5) K sh is the same as for reference incidence but the correction factor for blade thickness is different. (3.6) 010 is the deviation angle based on 10% thick blades, see Figure 3.5. The variable m represents the deviation slope factor, see Figure 3.6. (3.7) 26

31 Axial Flow Compressor Mean Line Design 3 Methods of Calculation (3.8) (3.9) The modified slope factor, m, is different whether DCA, 65-series or C-Series is used. 65-series DCA and C-series (3.10) (3.11) deviation, flow inlet angle [degree] Figure 3.5, zero camber deviation angle at i ref with variations in solidity 27

32 3 Methods of Calculation Axial Flow Compressor Mean Line Design slope factor, m flow inlet angle [degree] Figure 3.6, slope factor, m, with variations in solidity 3.3 Diffusion Factor and Diffusion Ratio The blade loading is usually assessed by the diffusion factor. This relates the peak velocity on the suction surface of the blade to the velocity at the trailing edge. Values of DF in excess of 0.6 are thought to indicate blade stall and values of 0.45 might be taken as a typical design choice. The ratio between maximum velocity and the outlet velocity is called the diffusion ratio [2]. There are several methods on how to calculate the diffusion ratio as well for the diffusion factor. Some of these correlations are presented below [4]. A simplified method on how to establish the diffusion ratio and the diffusion factor is. (3.12) (3.13) Lieblein developed useful approximations for both diffusion ratio and diffusion factor. DF denotes the diffusion factor and D eq * the equivalent diffusion ratio. The derivation of the diffusion factor is based on the establishment of the velocity gradient on the 28

33 Axial Flow Compressor Mean Line Design 3 Methods of Calculation suction surface in terms of W 1, W 2 and W max in conjunction with results from cascade tests. From cascade tests it was deduced that the maximum velocity is (3.14) (3.15) Koch and Smith modified the Lieblein approach and developed a more advanced correlation for the losses which allows for the factors maximum thickness to chord (t/c) and the Axial Velocity Density Ratio, i.e. AVDR. (3.16) where (3.17) (3.18) and the density ratio between the passage throat to entry is (3.19) 29

34 3 Methods of Calculation Axial Flow Compressor Mean Line Design (3.20) where M x1 is the axial component of entry Mach number. 3.4 Losses Profile loss model The profile-loss model used is a modified version of the two dimensional low speed correlation of Lieblein, see Figure 3.7. This correlation has been established for DCA aerofoils but is used for conventional circular arc aerofoils [8] Profile loss parameter Equivalent diffusion ratio Figure 3.7, profile loss parameter with variation in Mach number The profile-loss parameter is expressed as (3.21) where is the profile-loss coefficient. 30

35 Axial Flow Compressor Mean Line Design 3 Methods of Calculation Endwall loss model Again a correlation is used to determine the losses in the wall. Based on a numerous sets of compressor data where the parameters, tip clearance, aspect ratio and mean line loading where systematically varied, Freeman was able to correlate these parameters, se Figure 3.8. To determine the loading on the blade, the diffusion factor where used [8] Endwall loss parameter Diffusion factor Figure 3.8, wall loss parameter with variation in tip clearance As seen in the loss model created by Wright and Miller, the losses starts to increase rapidly as the diffusion factor approaches A diffusion factor about 0.45 is a typical value for designing compressors. The wall-loss parameter is expressed as where is the wall-loss coefficient. (3.22) Total loss Summarizing the wall-losses and profile-losses will give the total loss in the blade row. From this the loss in stagnation pressure with respect to inlet dynamic pressure, p 01 - p 1, can be calculated [2]. 31

36 3 Methods of Calculation Axial Flow Compressor Mean Line Design Pressure losses in the blade row can be calculated, but it is preferred to express the losses in an entropy change instead. By applying Gibbs 2 nd law, the pressure change can be expressed as a change in entropy instead. By definition there is no enthalpy change, will be zero. With help from the ideal gas law the following modification can be made. After integration on the right hand side, the following expression can be found where 1 and 2 denoted the inlet and outlet conditions. The final result will be as follow. (3.23) 3.5 Pitch Chord ratio The pitch chord ratio is also known as the inverse of the solidity,, where the latter is used in the U.S. When determining the aerodynamic loading in a blade row the pitch chord ratio is an important parameter. A blade row with a low pitch chord ratio, high solidity, will probably have fewer blades in a blade row than one with a high stagger chord ratio. The aerodynamic loading will then be shared by less number of blades and thus the blade loading will increase. A higher aerodynamic loading will probably be nearer the surge/stall point than one with a lower aerodynamic load. As for the Diffusion Factor, there are several methods on how to determine the pitch chord ratio. 32

37 Axial Flow Compressor Mean Line Design 3 Methods of Calculation Diffusion Factor Method The first method on how to calculate the pitch chord ratio is the same equation as for the diffusion factor, but instead of expressing it for the diffusion factor it will express the pitch chord ratio as a function of diffusion ratio instead Hearsey Method Hearsey uses an explicit loss correlation: (3.24) to produce the following relationship for minimum (3.25) where D opt is the Diffusion Factor calculated using = opt. This correlation is based on profile losses only and underestimates the required solidity when shocks are expected McKenzie Method Another method is for determining the pitch chord ratio is taken from McKenzie, see reference 5. Here the pitch chord ratio is a function of the static pressure rise, C p. (3.26) 3.6 Stall/Surge When calculating on how near the each stage is to surge/stall a relationship is used created by Koch, see reference 9. By calculating the static pressure rise coefficient, C p, based on pitchline dynamic head, and comparing it to the maximum static pressure rise, will give a good indication of how close the stage is towards stall. The maximum static pressure rise, C p,max, is based on diffuser correlation modified by the influence of Reynolds number, tip clearance and axial spacing between blade rows. The correction factors for the static pressure rise coefficient are shown in Figures The static pressure rise coefficient and the maximum pressure rise coefficient are as followed. 33

38 3 Methods of Calculation Axial Flow Compressor Mean Line Design (3.27) Since there is a change in radius throughout the rotor, the factor (U 2 2 -U 1 2 )/2 will decrease static pressure rise. This decrease in static pressure rise will not put any more stress on the boundary layers because there will not be any further diffusion caused by the change in radius. (3.28) The effective dynamic pressure factor ef is described by Koch in the paragraph below. A parameter giving a quantitative measure of this flow coefficient/stagger angle effect, termed the effective dynamic pressure factor ef, was defined as the effective dynamic head divided by the pitchline free stream dynamic head. As indicated by the equation 3.29, the effective dynamic head was represented by a weighted average of the free stream dynamic head, the minimum possible dynamic head and the dynamic head at zero axial velocity. By trial and error, the best fit of the data was found to occur if the minimum dynamic head was weighted 2.5 times as heavily as the free stream head, and the head at zero axial velocity was weighted one half as heavily as the free stream head. The minimum dynamic head was set equal to the free stream value for vector triangles where ( + ) was greater than 90 deg., because in such a case the minimum possible dynamic head could only occur at axial velocities higher than the free stream value. Also, for vector triangles in which the upstream blade row turned the flow past the axial direction, the minimum dynamic head was not allowed to become less than the zero through-flow value, because in this case the mathematical minimum dynamic head could only occur at negative axial velocities. [9] 34

39 Axial Flow Compressor Mean Line Design 3 Methods of Calculation g 1 Rotor L g 2 U W 90 min C min C U Figure 3.9, diagram giving definition of the effective dynamic pressure factor, ef [9] (3.29) For the influences of Reynolds number, tip clearance and axial spacing between blade rows a set o graphs is used. These graphs were created by a numerous sets of tests in which different parameters were varied. The tests were performed with a General Electric low-speed multistage compressor and the blade geometry and clearance was systematically varied. These tests, plus some additional low speed experimental configurations, also provided data for the correction for Reynolds number and showed the effect of extreme values of stagger angle, flow coefficient and reaction [9]. Note that the value for the stage is determined by a weighted average value for the rotor and the stator. 35

40 3 Methods of Calculation Axial Flow Compressor Mean Line Design The equations that will be presented are for the rotor but are applicable for the stator as well [4]. (3.30) In the equation above Z denotes the number of blades in one row. (3.31) To determine the axial spacing between the rows, a guide rule can be used [10]. (3.32) C pd Diffusion length/exit passage width, L/g 2 Figure 3.10, Correlation of stalling pressure rise data for baseline testing [9] 36

41 Axial Flow Compressor Mean Line Design 3 Methods of Calculation (C p /C pd ) Tip clearence/average Pitchline Gap, /g Figure 3.11, Effect of tip clearance on stalling pressure rise coefficient [9] (C p /C pd ) Z Normalized Axial Spacing, Z/s Figure 3.12, Effect of axial spacing on stalling pressure rise coefficient [9] 37

42 3 Methods of Calculation Axial Flow Compressor Mean Line Design (C p /C pd ) Re Reynolds Number x10-5, Re Figure 3.13, Effect of Reynolds number on stalling pressure rise coefficient [9] 38

43 4 Calculation procedures 4.1 Input parameters When designing a new compressor, a number of parameters must be chosen to specify the geometry and operating conditions for the compressor. There are numerous different combinations of parameters that could specify the compressor. The different input parameters, used in this thesis, are shown in table 4.1. Main specification Type of compressor Mass flow Number of stages Pressure ratio Rotational speed Stage reaction Detailed specification Tip clearance, /c Aspect ratio, h/c Thickness chord ratio, t/c Axial velocity ratio, AVR Blockage factor, BLK Diffusion factor, DF Stage Loading distribution Inlet specification Inlet flow angle, Stage flow coefficient, Hub tip ratio, r hub /r tip Table 4.1, Input parameters for the calculations 39

44 4 Calculation procedures Axial Flow Compressor Mean Line Design Main specification The geometry of a compressor can be categorised into 3 main designs types, a Constant Outer Diameter (COD), a Constant Mean Diameter (CMD) or a Constant Hub Diameter (CID), see Figure 4.1. Figure 4.1, different compressor geometries There are several different parameters that can specify a particular compressor. A number of these parameters will be presented and used for the calculation that will follow later on. The first set of input parameters are based on the running conditions for the machine. These involve mass flow, rotational speed, pressure ratio and the number of stages. For controlling the distribution of the load between the rotor and the stator the stage reaction can be set. If this is not of importance, the outlet flow angle for the each stage must be set instead, more on this in chapter 4.3 Calculation limitations. 40

45 Axial Flow Compressor Mean Line Design 4 Calculation procedures Detailed specification There are three main geometry specifications that will be used in this thesis, tip clearance chord ratio, thickness chord ratio and blade height chord ratio also known as aspect ratio. These three parameters specify the main geometry for the blades in the compressor. If the axial velocity is set to be constant throughout the compressor, the blades at the of the compressor will be very short and thus have higher losses and more susceptible to mechanical stresses. Setting the Axial Velocity Ratio will take this in consideration. As the fluid is working itself towards the of the compressor, boundary layer growth starts to appear on the compressor housing. This will result in a narrower path for the fluid to flowing through. Introducing a Blockage Factor will account for this phenomenon. The stage load distribution is another parameter that can be set Inlet specification To be able to start the calculations for the compressor, certain dimensions and properties must be calculated first. By setting up different parameters that is only valid for the first row these dimensions and properties can be calculated. These inlet specifications consist of three parameters, inlet flow angle, stage flow coefficient and the hub radius tip radius ratio. 4.2 Parameter variations throughout the compressor Certain parameters in the compressor will vary in the compressor, namely: Tip clearance, /c Aspect ratio, h/c Thickness chord ratio, t/c Axial velocity ratio, AVR Blockage factor, BLK Diffusion factor, DF Stage Loading distribution A simple linear distribution for the parameters may, for simplicity, be used except for the stage loading. By setting the front and rear value for the different parameters a linear distribution is than created. The parameters can also be set for the rotor and the stator separately. 41

46 4 Calculation procedures Axial Flow Compressor Mean Line Design As for the blockage factor one can say that after a certain stage in the compressor the boundary layer growth will have settled. By studying several different calculations that have been made a conclusion can be made that at the 5 th stage the boundary layer growth has been stabilized. The stage load distribution throughout the compressor can be determined by setting the inlet value, the mean value and the outlet value. This will create a type of ramp function, see Figure 4.2 for an example how it can look. Using this type of ramp function gives more control over how the stage load is distributed. More advanced functions can be used or a customized distribution can be made as well, but for most cases this ramp function will be adequate. Distribution, % stage Figure 4.2, stage load distribution over a compressor 4.3 Calculation limitations Mean stream line analyses The calculations in this thesis are based on mean line stream analysis i.e. one dimension. The mean radius is used in the calculations to determine the blade speed. Normally when calculating with the mean line stream method, the mean radius will not change. But by changing the mean radius throughout one stage will give a more accurate design, see Figure 4.3. The mean radius will be kept constant in the space between rotor and stator as well for the space between each row. A change in radius in the space between each blade row won t make a big difference in the result. It is more crucial to have a change in radius in the blade them self since this will have a more noticeable effect. 42

47 Axial Flow Compressor Mean Line Design 4 Calculation procedures Rotor Stator r mean Figure 4.3, variation in radius through a single stage Convergence criteria s There are two main input parameters that the calculation must satisfy. These are the total pressure ratio,, and the stage reaction, R. To be able to have the desirable pressure ratio and stage reaction after the calculation is done, it is necessary to adjust some parameters. There are several parameters that can be chosen but two parameters have been selected, one for the pressure ratio and one for the stage reaction. The pressure ratio is adjusted with help from the stage load coefficient. Another possibility is to adjust the dehaller number. It is easier to control the load on the compressor when designing it; this is why the stage load coefficient is selected as the adjustment parameter for the pressure ratio. The stage reaction is strongly depent on the flow angles in and out of each stage. This since the stage reaction is the ratio between the work done in the rotor and the work done in the entire stage. The work done is depent on the enthalpy change and thus is depent on the absolute velocity for the inlet and outlet at each stage. The absolute angle,, decides the absolute velocity. Changing the outlet angle, 3, will change the stage reaction on the next stage. This because the inlet angle of a stage is the same as for the outlet angle from the previous stage. The question Why can not the inlet angle be adjusted? seems relevant at this point. The inlet angle cannot be adjusted since this angle is the same as for the outlet angle from the previous stage. If this would be changed then it would not have the same value as for the outlet angle from the previous stage. The calculation is divided into separate modules. This makes it easier to understand how it is all related. There are three main modules and these makes up for the calculation for one stage, see Figure 4.4. Each of these modules has sub modules in their self. In chapter 5 Calculation process, an entire stage is described on how this calculation is done. 43

48 4 Calculation procedures Axial Flow Compressor Mean Line Design 4.4 Structure of the calculation Module 0 Before the calculation can begin the inlet geometry must be determined. In this module the axial velocity, tip radius, mean radius and hub radius are being calculated. This will make up for the overall design for the compressor, if it has a constant mean radius or some of the other designs. Some of the input criterions will have variations throughout the compressor. In module 0 these variations are accomplished Module 1 The calculations at rotor-inlet are performed in module 1. In this module there is no need for approximations for certain values and hence no need for iteration procedures. Module 0 = f() R = f( 3(i-1) ) Module 1 i=1 Module 2 Stage i Module 3 i +1 R Figure 4.4, structure over the iteration procedure for the whole compressor 44

49 Axial Flow Compressor Mean Line Design 4 Calculation procedures Module 2 In this module the calculations at rotor-outlet/stator-inlet are performed. There are two parameters that must be approximated. The first is the entropy rise in the rotor. This must be known to be able to calculate the entropy at rotor-outlet. With this identified, the temperature and pressure can then be determined. The second approximation is the mean radius at rotor-outlet. There will be a change in radius throughout the rotor and to be able to determine the velocities, a radius must be known. In Figure 4.5 a more detailed view over this procedure is shown. Module 1 guess s 2-1 guess r rms,2 Module 2 Module 2.1 r rms,2 Module 2.2 s 21 Module 3 Figure 4.5, structure over the iteration procedure for a rotor (module 2) 45

50 4 Calculation procedures Axial Flow Compressor Mean Line Design Module 3 In the last of the three main modules, the stator-outlet is computed. The same method as in module 2 is used, namely two parameters must be approximated. The two parameters are somewhat the same as for the rotor, the entropy rise throughout the stator and the mean radius at stator-outlet, see Figure 4.6. Module 2 guess s 3-2 guess r rms,3 Module 3 Module 3.1 r rms,3 Module 3.2 s 32 Figure 4.6, structure over the iteration procedure for a stator (module 3) 46

51 Axial Flow Compressor Mean Line Design 4 Calculation procedures 4.5 Newton-Rhapson Method Newton s method (also called Newton-Rhapson method) for solving nonlinear equations is one of the most well-known and powerful procedures in all of numerical analysis methods. It always converges if the initial approximation is sufficiently close to the root, and it converges quadratically. Its only disadvantage is that the derivative f (x) of the nonlinear function f(x) must be evaluated. Newton s method is illustrated graphically in Figure 4.7. Lets locally approximate f(x) by the linear function g(x), which is tangent to f(x), and find the solution g(x) = 0. That solution is then taken as the next approximation to the solution of f(x) = 0. The procedure is applied iteratively to convergence. Thus, Solving the equation above for x n+1 with f (x n+1 ) = 0 yields. This equation is applied repetitively until either one or both of the following convergence criteria are satisfied: f(x) g(x) f(x) x n+1 x n x Figure 4.7, Newton s method [7] 47

52 5 Calculation process 5.1 Module 0, Inlet geometry To be able to solve the inlet geometry the inlet flow velocity, C m, must be known. Since this velocity is unknown an iterative process must be used. By approximating the value of C m, the density can be found. With help of mass continuity a new inlet flow velocity can be calculated. This value is then used to start over the calculation until converged, see Figure 5.1. The first step is to get hold off the thermodynamic properties in the inlet of the compressor. The ambient pressure and temperature is known and from these the enthalpy and entropy can be found. Now that the inlet properties are known, the iteration procedure can begin. 48

53 Axial Flow Compressor Mean Line Design 5 Calculation process Start value C m Figure 5.1, structure over the iteration procedure for the inlet geometry (module 0) 49

54 5 Calculation process Axial Flow Compressor Mean Line Design 5.2 Module 1, Rotor-inlet When starting the calculation, the geometry from the inlet calculations is used. Since the calculation for the entire stage will be repeated, the rotor-inlet conditions, i.e. station 1, will have the same velocity and radius as the stator-outlet, i.e. station 3, for the previous stage. Now that the preferences for the rotor-inlet at the first stage are known the calculations can begin. The first thing is to find out the entropy and the enthalpy for this station. From the ambient temperature and pressure the enthalpy and entropy can be found. If the calculations is not preformed on the first stage than the stagnation properties of the working fluid is taken from the previous stage. Flow angles and velocities To be able to find out the static properties of the working fluid the absolute velocity, C 1, is necessary. With the help of the velocity diagrams, see Figure 5.2, the relative and the absolute velocity can be found, as well as the relative angle, 1. 50

55 Axial Flow Compressor Mean Line Design 5 Calculation process W C 1 C m1 C 1 Figure 5.2, velocity triangle for the rotor inlet Static properties Now that the velocity is known, the static enthalpy can be calculated. And with help from the entropy other fluid dynamic properties like pressure, temperature, density etc. can be found. To be able to move from the rotor-inlet towards the outlet of the rotor a relationship between these must be used. Since the rothalpy is constant throughout the rotor, makes the rothalpy useful when calculating the outlet of the rotor. Further in to the calculations the relative Mach number and the axial Mach number will be used and since that the speed of sound, a 1, is known these two Mach numbers can be calculated. 51

56 5 Calculation process Axial Flow Compressor Mean Line Design Relative properties The dominating velocity that is acting on a rotor blade is the relative velocity, W 1 ; therefore relative stagnation properties must be calculated. Instead of calculations based on the absolute velocity, C 1, the relative velocity is used. The relative pressure and temperature are found out by the relative enthalpy and entropy. Geometry The blockage factor is here denoted as, BLK. The geometry is the same for the rotorinlet as for the stator-outlet in the previous stage. A result of this is that the blockage factor should be the same for the rotor-inlet and the stator-outlet at the previous stage. From the definition of the cross section area and the mean radius, the hub radius, the mean radius or the tip radius can be calculated deping if the compressor is of the type CID, CMD or COD. Constant Mean Diameter, CMD 52

57 Axial Flow Compressor Mean Line Design 5 Calculation process Constant Outer Diameter, COD Constant Inner Diameter, CID 5.3 Module 2, Rotor-outlet/stator-inlet There are two separate modules in module 2. The first, 2.1, is for the calculation of the entropy rise in the rotor. The second, 2.2, calculates the mean radius at rotor-outlet. Both of these are iteration processes where an approximated value is first guessed and then a new value is calculated to adjust the approximated first value Module 2.1 start Flow angles and velocities The mean radius at rotor-outlet in unknown so a value for this must be approximates to be able to find out the blade speed. A new value for this will be calculated further on in the calculation. Since a change in radius throughout the rotor is occurring a modification to the definition of the stage load coefficient must be made. A modification is made based on the blade velocity at the rotor-outlet. The coefficient omega is the rotational speed in radians per second. By substituting this with the blade speed, U=r, following relation can be found. 53

58 5 Calculation process Axial Flow Compressor Mean Line Design The stage load coefficient is known and from this the tangential part of the absolute velocity, C 2, can be calculated. Now that the tangential part of the absolute velocity is know the tangential part of the relative velocity, W 2, can be calculated. From this the absolute and relative velocities can be calculated and there angle. W 2 C C a2 C 2 Figure 5.3, velocity triangle for the stator inlet Static properties From the rothalpy calculation done previously on the rotor blade inlet, the static enthalpy at rotor blade outlet can now be calculated. Due to the losses throughout the rotor passage, an increase in entropy will occur. This increase in entropy is used to calculate the entropy at the rotor-outlet, s 2. The value of 54

59 Axial Flow Compressor Mean Line Design 5 Calculation process this entropy rise is just an approximation. A new value for this will be calculated further in to the calculations. Stagnation properties Geometry Like for the inlet of the rotor the different radiuses is calculated with help from the area and the definition of the mean radius. This value is a new value for the one that was approximated in the beginning of module Module 2.1 After the module 2.1 is completed a mean radius for the rotor-outlet has been calculated Module 2.2 start The calculation proceeds with the known rotor-outlet radius. 55

60 5 Calculation process Axial Flow Compressor Mean Line Design Rotor properties An average value for the tip radius, mean radius and hub radius is used for calculating the height of the rotor blade. Since one of the input design parameters is the aspect ratio, i.e. H/c, the chord of the blade can also be calculated. Reynolds number and AVDR are also calculated for the calculations that will follow. Reynolds number is based on the inlet velocity and viscosity at the inlet of the rotor. AVDR for the rotor is calculated because it is a definition that will be used when the equivalent diffusion ratio, D eq, will be determined. Pitch chord ratio The calculations for the pitch chord ratio, also known as pitch chord ratio, can be done based on either on the diffusion factor, the McKenzie method or on the Hearsey method. These different methods needs some input parameters and these are the same whether the method based on the diffusion factor is used or some of the other methods. The input parameters consist of the relative inlet and outlet flow angles, the different axial velocities and radiuses and also the diffusion factor. To get a deeper insight in how this calculation is performed see chapter 3.4 Pitch chord ratio. 56

61 Axial Flow Compressor Mean Line Design 5 Calculation process Equivalent diffusion ratio When calculating the diffusion ratio a similar approach as the pitch chord ratio is used. The input parameters include apart from them in the pitch chord ratio calculation also the pitch chord ratio, thickness chord ratio, axial velocity density ratio and the Mach number based on the axial velocity at rotor-inlet, see chapter 3.2 Diffusion Factor and Diffusion Ratio. Rotor losses The last step in the rotor is to calculate entropy rise due to the losses throughout the rotor. In the beginning of the rotor calculation a good approximation was made for the entropy rise. This must be corrected by calculating a true entropy rise. To see more in detail on how this calculation is made see chapter 3 Losses Module 2 At this point the geometry of the rotor is fully defined and the calculation can now proceed to the stator. 57

62 5 Calculation process Axial Flow Compressor Mean Line Design 5.4 Module 3, Stator-outlet As for the rotor, the stator calculation is divided into two sub modules, 3.1 and Module 3.1 start Flow angles and velocities The same method is applied for the stator-outlet as for the stator-inlet when calculating the different velocities. Here the absolute velocity is the only one that is interesting since there is no relative velocity. The dehaller number is calculated based on the inlet and outlet velocity of the rotor. Static properties There is no work done to the working fluid in the stator. This results in a constant stagnation enthalpy throughout the stator passage. From this the static enthalpy can be calculated based on the absolute velocity, C 3. As for the rotor, an approximation for the entropy rise must be made for the stator. This will give the entropy at the stator-outlet. Later on in the calculation an accurate entropy rise will be calculated. From the entropy and the enthalpy, the fluid dynamic properties of the working fluid can be found for the stator-outlet. 58

63 Axial Flow Compressor Mean Line Design 5 Calculation process Stagnation properties Geometry Like for the rotor-inlet and the stator-inlet the different radiuses is calculated with help from the area and the definition of the mean radius. This value of the radius at statoroutlet is then used to update the approximated value at the beginning of the calculation of the stator-outlet Module 3.1 The calculation can now proceed now that the radius at stator-outlet is known Module 3.2 start The calculation can now proceed, now that the radius at stator-outlet in known. Stator properties As for the rotor an average value for the tip radius, mean radius and hub radius is used for calculating the height of the rotor blade. 59

64 5 Calculation process Axial Flow Compressor Mean Line Design The Reynolds number is based on the inlet velocity, C 2, and viscosity, 2, for the stator. AVDR for the stator is yet again based on the density and axial velocity for the statorinlet and stator-outlet respectively. Pitch chord ratio Equivalent diffusion ratio Stator losses 60

65 Axial Flow Compressor Mean Line Design 5 Calculation process Module 3 The entire stage is now fully defined. This process is then repeated for the remaining stages. The next step in the calculation is to calculate the outlet guide vane. 5.5 Outlet Guide Vane, OGV An outlet guide vane, OGV, is an extra stator after the compressor. Its main purpose is to turn the flow so the whirl component decreases or gets eliminated. When entering the combustion chamber one wants a smooth and controlled flow with no disturbances. The calculation for the outlet guide vane is similar to the stator in the compressor. Flow angles and velocities If zero whirl is desirable then the axial velocity at the last stator-outlet is equal to the absolute velocity at the OGV outlet. C OGV 3 C 3 Figure 5.4, velocity triangle for the outlet guide vane Static properties As before the stagnation enthalpy at the OGV outlet is equal to the stagnation enthalpy at the outlet of the last stator. And again an entropy rise must me approximated since this is unknown and is necessary for further calculations. 61

66 5 Calculation process Axial Flow Compressor Mean Line Design Stagnation properties Geometry An approximation is used for the OGV radiuses. This approximation is that the radiuses at the last stator-outlet are the same as for the OGV radiuses. 62

67 Axial Flow Compressor Mean Line Design 5 Calculation process Pitch chord ratio Equivalent diffusion ratio Outlet guide vane losses The entire compressor is now fully computed and the pressure ratio and the stage reaction have converged. The next step is to determine the blade angles for the rotor row and stator row. 63

68 5 Calculation process Axial Flow Compressor Mean Line Design 5.6 Blade angles calculation To be able to calculate the blade angles an iteration procedure is needed, like the ones used in previous calculations. In this case the camber angle is estimated. With this the incidence and deviation angles can then be found. From the incidence and deviation angle a new camber angle is calculated. This is then repeated until converged, see Figure 5.5. guess Figure 5.5, structure over the iteration procedure for the blade angles One can argue whether this iteration is necessary since it is possible to explicit solve the camber angle. This approach was chosen for having flexibility for other types of correlations, like Carter s rule, see chapter 3.12 Deviation angle. This method requires the stagger angle to be known, hence the iteration is necessary. 64

69 6 Results In this chapter a compressor is calculated and the results analyzed. In the following tables, the input parameters are specified. The compressor that will be calculated has a constant mean line design with a mass flow of 122 kg/s. The desirable pressure ratio is 20 and it has 15 stages. A stage reaction of 0.55 is also set as one of the input variables. Main specification Type of compressor CMD Mass flow 122 Number of stages 15 Pressure ratio 20 Rotational speed 6600 Stage reaction 0.55 Table 6.1, input parameter example for the main specifications The detailed specifications are listed below in table 6.2. Here the diffusion factor is set constant throughout the compressor, both for the rotors and the stators. The thickness chord ratio and tip clearance are also set constant. Detailed specification First stage Last stage Tip clearance, /c Aspect ratio, h/c Thickness chord ratio, t/c Axial velocity ratio, AVR Diffusion factor, DF First stage 5 th stage Blockage factor, BLK Table 6.2, input parameter example for the detailed specifications As for the loading, the first blade rows have the highest stress. Further in to the compressor the stress on the blades will decrease. 65

70 6 Results Axial Flow Compressor Mean Line Design First stage Middle stage Last stage Stage Loading distribution Table 6.3, input parameter example for the stage loading distribution The inlet air angle is set to 15 degrees and it has a hub tip ratio of The stage flow coefficient for the inlet is set to Inlet specification Inlet flow angle, 15 Stage flow coefficient, 0.65 Hub tip ratio, r hub /r tip 0.50 Table 6.4, input parameter example for the inlet specifications After the calculations have finished a graph over the blade geometry can be constructed, see Figure 6.1. In this graph the total length of the compressor can be estimated and also the variation of the radii. Here one can clearly see that it is a constant mean radius design as it was designed to be. The red colour indicates that it is a rotor and for the stators a blue colour Figure 6.1, example of the compressor geometry for a given set of input parameters There are two efficiencies that can be calculated, polytropic and isentropic efficiency. These efficiencies and other interesting results are shown in table

71 Axial Flow Compressor Mean Line Design 6 Results Polytropic efficiency % Isentropic efficiency % Temperature rise K Inlet tip 1.08 Mass kg/s Compressor power MW Table 6.5, Results from the calculations A graph can be drawn of the surge limit of each stage, see Figure 6.2. Studying the graph one can see that in stage number 2 there is a possibility of surge. Further in to the compressor the margin to surge increases. The limit to surge is strongly depent to the diffusion factor. By decreasing the diffusion factor will result in a lower static pressure rise and hence increase the margin to surge. The diffusion factor is not something one can change by itself; it is depent on several other factors. Pitch chord ratio, s/c, is the factor that has the most profound effect on the diffusion factor. If the pitch chord ratio is decreased, then the diffusion factor will also decrease, resulting in a lower static pressure rise, see Figure C h,max C h Static pressure rise coefficient Stage Figure 6.2, Koch surge limit for this compressor example 67

72 6 Results Axial Flow Compressor Mean Line Design 0.55 C h,max C h Static pressure rise coefficient Stage Figure 6.3, Koch surge limit for this compressor example, with a change in pitch chord ratio 68

73 7 LUAX-C 7.1 Structure of the program In order to use these calculations for designing a preliminary design of a compressor a GUI, Graphic User Interface, was created. This interface was made in MATLAB and is called LUAX-C, Lund University Axial Flow Compressor. The main calculation which calculates the whole compressor is divided into separate sub programs, each program takes care of one part of the calculation, for example the losses generated in the rotor/stator. As discussed in chapter 4.1 Input parameters, a set of input parameters that specify the running conditions and geometry is needed for the calculations. These input parameters are placed in a text file. This because it is easier to make small and more specific changes in the input parameters this way and the text file is compatible with other computer programs as well. As for the input file, the result, output file, is also a text file. In this all the results from the calculations that could be of use for further study are displayed. The communication between the GUI and the calculation and other post processing programs are done with a type of databases. There are three different databases that are generated by the main calculation. STG CompInfo OGV In the first one, STG, the information for each stage is saved. Some of this information can be the different velocities, angles, geometry etc. The second one, CompInfo, consists of all the information about the compressor, for example efficiency, power, mass flow, pressure ratio etc. The last one, OGV, consists of the information about the outlet guide vane. After the calculation is finished and the database is created the data is transferred to the post processing. The post processing will create a text file which consists of all the results from the database. It will also s information to the GUI so different graphs and other demonstrations can be made to visualize the result. 69

74 7 LUAX-C Axial Flow Compressor Mean Line Design Input file Main Sub 1 GUI Sub 2 Post Process STG CompInfo OGV Result file Figure 7.1, Structure of LUAX-C 7.2 User Guide to LUAX-C In this chapter a quick walkthrough of the program is shown. This program uses MATLAB as a platform so it is necessary to have MATLAB installed on the computer where the program will run from. The first step is to change the current directory to the directory where LUAX-C is saved. This will help MATLAB to find the files and to have a place to save the output files. The main window consists of all the input parameters necessary to calculate an entire compressor, see Figure

75 Axial Flow Compressor Mean Line Design 7 LUAX-C Figure 7.2, Main window in LUAX-C Based on the filename chosen by the user, the program will create input and output files with the same file name. Next step is to define what type of compressor one wants to design and the specifications of this compressor. Here the user can chose the number of stages, mass flow, pressure ratio and rotational speed. If the stage reaction is of importance, this can also be set. If not, the whirl angle of the stage needs to be set instead. These settings are the main specifications and the detailed specifications are as follows. Tip clearance, /c Aspect ratio, h/c Thickness chord ratio, t/c Axial velocity ratio, AVR Blockage factor, BLK Diffusion factor, DF Stage Loading distribution 71

76 7 LUAX-C Axial Flow Compressor Mean Line Design All of these specifications are varied linearly throughout the compressor based on the inlet and outlet values. After all the input parameters are specified, an input file must be created. By pressing Create Data File, the variations will be created and an input data file will be created. If one wish to specify these parameters for each blade row separately, this is possible by pressing Open/Edit, see Figure 7.2. This will open the input text file where it is possible to make the changes that one wants to do. Table 7.1 shows which row in the text file that represents which input parameter and the choices that can be made for each parameter. After this, the text file must be saved since this is the input file for the calculations. Figure 7.3, input text file for LUAX-C 72

77 Axial Flow Compressor Mean Line Design 7 LUAX-C Input parameter choice File name - Compressor Type CMD, COD, CID Ambient pressure bar Ambient temperature Celsius Number of stages - Mass flow kg/s Pressure ratio - Rotational speed rpm Stage reaction select stage reaction = 1, whirl angle = 0 Stage reaction whirl angle in degree s/c calculation method DF, McKenzie, Hearsay Alpha in degree PHI in - h/t in - Epsilon/c rotor - Epsilon/c stator - Aspect ratio rotor - Aspect ratio rotor - t/c rotor - t/c stator - Diffusion factor rotor - Diffusion factor stator - Axial velocity ratio rotor from 1.0 to 0.97 Axial velocity ratio rotor from 1.0 to 0.97 Blockage factor - PSI variation from 1.0 to 0.0 Figure 7.1, List of the different parameters in the input text file Now it is time to run the program. By pressing RUN will initiate the calculations and it is important that the input file has been created. For each run the user must create the input file by pressing Create Data File. When the program has finished calculating, which can take a while deping of the computer power and type of compressor, two graphs are shown, see Figure 7.4. The one above is a graph over how the geometry varies over the length of the compressor. The red blades represents the rotors and the blue ones the stators. The second graph shows the velocity diagrams over a particular stage. Here the user can chose which stage one wants to look at. 73

78 7 LUAX-C Axial Flow Compressor Mean Line Design Figure 7.4, after the calculations has been finished By pressing Open Result File will open the result file created by the program, see Figure 7.5. This file consists of all the data that the user may need when transferring it to another program like a CFD simulation program. In this result file, the performance of the compressor are shown and several other important parameters. Figure 7.5, Result text file 74

79 Axial Flow Compressor Mean Line Design 7 LUAX-C If the user wishes to study the surge limits, this is done by pressing Surge Graph, see Figure 7.6. This opens a graph in which the user can see if the compressor is on the limit of surge on a particular stage. This surge graph is based on Koch which is discussed in this thesis. Figure 7.6, Koch surge limit for this compressor example 75

80 References [1] Saravanamutto, HIH, Rogers, GFC och Cohen, H. Gas Turbine Theory Fifth Edition, Pearson Prentice Hall, [2] Cumpsty, N.A. Compressor Aerodynamics, Krieger Publishing Company, [3] Lakshminarayana, Budugur. Fluid Dynamics and Heat Transfer of Turbomachinery, Wiley-Interscience, John Wiley & Sons, [4] Howard, J.H.G. Axial Fan and Compressor Modeling [5] McKenzie, A.B. Axial Flow Fans and Compressors, Ashgate Publishing Limited, [6] Denton, J D. Cambridge Turbomachinery Course, Whittle Laboratory, Deparment of Engineering, University of Cambridge, [7] Hoffman, Joe D. Numerical Methods for Engineers and Scientists, Marcel Dekker Inc. [8] Wright, P I och Miller, D C. An Improved Compressor Performance Prediction Model, ACGI,DIC,Rolls-Royce, Derby, [9] Koch, C.C. Stalling Preuusre Rise Capability of Axial Flow Compressor Stages, Aircraft Engine Group, General Electric Co., [10] Walsh, P.P och Fletcher, P. Gas Turbine Performance, Blackwell Science, [11] Genrup, Magnus. Degradation and Monitoring Tools for Gas and Steam Turbines, Doctoral Thesis, Department of Heat and Power Engineering, Lund Institute of Technology, Lund University, Sweden,

81 Appix A, polynomial coefficients for the graphs A polynomial fitting method has been used to interpret the graphs that are used in this thesis for the calculations. The different coefficients for each polynomial for each figure are listed in the following tables. a 4 a 3 a 2 a 1 a e e e e e e e e e e e e e e e-01 Table A.1, Polynomial coefficients for Figure 3.7 a 4 a 3 a 2 a 1 a e e e e e e e e e e e e e e e e e e e e e e e e e-01 Table A.2, Polynomial coefficients for Figure 3.8 a 5 a 4 a 3 a 2 a 1 a e e e e e e-02 Table A.3, Polynomial coefficients for Figure 3.10 a 5 a 4 a 3 a 2 a 1 a e e e e e e00 Table A.4, Polynomial coefficients for Figure 3.11 a 5 a 4 a 3 a 2 a 1 a e e e e e e00 Table A.5, Polynomial coefficients for Figure

82 Appix Axial Flow Compressor Mean Line Design For the interpretation of figure 3.13 a normal polynomial fitting was not feasible, instead a function of the type was used. a b c Table A.5, coefficients for the function in Figure 3.11 B, MATLAB script for the calculations B.1 Main Calculation % Main Calculation for the compressor %##################################################################### %## ## %## Main Calculation ## %## ## %## (c) Niclas Falck and Magnus Genrup 2008 ## %## ## %## Lund University/Dept of Energy Sciences ## %## ## %##################################################################### function [STG,CompInfo,OGV] = compressorcalculation(filename); %################# Converts textfile to a matrix ########################## CompInputdata = createinputfile(filename); %###################### Numerical settings ############################## RLX_REACT = 0.8; RLX_PR = 0.95; %################### Import data from the input matrix #################### N_stg = str2num(compinputdata{5,1}); % number of stages in the compressor PR_comp = str2num(compinputdata{7,1}); % sets the desirable compressor ratio P0_in = str2num(compinputdata{3,1}); % absolute inlet pressure T0_in = str2num(compinputdata{4,1}) ; % absolute inlet temperature RPM = str2num(compinputdata{8,1}); % revolutions per minute for k=1:n_stg % massflow variation flow(k) = str2num(compinputdata{6,k}); Alpha_in = str2num(compinputdata{12,1}); PHI_in = str2num(compinputdata{13,1}); HONT_in = str2num(compinputdata{14,1}); % hub/tip comp_type = CompInputdata{2,1}; SONC_type = CompInputdata{11,1}; set_react = str2num(compinputdata{9,1}); if set_react == 1 for i=1:n_stg 78

83 Axial Flow Compressor Mean Line Design Appix REACT_stg(i) = str2num(compinputdata{10,i}); % sets a fixed value of the degree of reation for i=1:n_stg EPSONC_rtr(i) = str2num(compinputdata{15,i}); %(blande clearence)/chord EPSONC_str(i) = str2num(compinputdata{16,i}); %(blande clearence)/chord AR_rtr(i) = str2num(compinputdata{17,i}); AR_str(i) = str2num(compinputdata{18,i}); %Aspect ratio (H/C) TONC_rtr(i) = str2num(compinputdata{19,i}); % (T/C) TONC_str(i) = str2num(compinputdata{20,i}); if strcmp(sonc_type,'custom') == 0 % SONC is not set but is calculated instead DF_rtr(i) = str2num(compinputdata{21,i}); DF_str(i) = str2num(compinputdata{22,i}); else SONC_rtr(i) = str2num(compinputdata{21,i}); SONC_str(i) = str2num(compinputdata{22,i}); AVR_rtr(i) = str2num(compinputdata{23,i}); AVR_str(i) = str2num(compinputdata{24,i}); BLK_stg(i) = str2num(compinputdata{25,i}); % Axial Velocity Ratio % Blockage Factor PSI_variation(i) = str2num(compinputdata{26,i}); %loading distribution %########################## Compressor inlet ############################## [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('pt',p0_in,t0_in,0,1); Cp_in = Cp; rho_in = rho; Visc_in = Visc; kappa_in = kappa; a_in = a; [r_rms_in, r_m_in, r_hub_in, r_tip_in] = Inletgeom(P0_in,T0_in,flow(1),RPM,HONT_in,PHI_in,Alpha_in,BLK_stg(1)); U_in = r_rms_in*pi*rpm/30; %blade speed based om RMS radius Cm_in = PHI_in*U_in; %meridional velocity C_in = Cm_in/cosd(Alpha_in); M_in = C_in/a_in; % inlet relative Mach # U_tip_in = r_tip_in*pi*rpm/30; M_tip_in = ((Cm_in^2+(U_tip_in-Cm_in*tand(Alpha_in))^2)^0.5)/a_in; area_in = flow(1)/(cm_in*rho_in); %########################################################################## %########################## Pressure ratio iteration ###################### %########################################################################## if set_react == 1 for k=1:n_stg Alpha3(k) = Alpha_in; % Init values for reaction iteration else for k=1:n_stg % Alpha3(k) = str2num(compinputdata{10,i}); % sets the whirl angles if the REACT is not set %########################## Start values for PSI ########################## factor = 1.01; 79

84 Appix Axial Flow Compressor Mean Line Design PSI = 0.4*PSI_variation; % Creates a variation of PSI PSI(2,:) = PSI*factor; % inorder to calculated the derative a second is needed %########################## Start of iteration ########################## n_pr = 0; PR_rel_error_level = 1; conv_pr = 0; while conv_pr == 0 % Solves for correct pressure ratio if abs(pr_rel_error_level) < 10^(-4) conv_pr = 1; for j=1:2 %Second run for obtaining derivative %################################################################## %###################### Reaction iteration ######################## %################################################################## REACT_rel_RMS_error = 1; n_react = 0; conv_react = 0; while conv_react == 0 %############### Compressor meridional flowpath ############### for i=1:n_stg compressor compressor if comp_type == 'CMD' r_rms(i) = r_rms_in(1); elseif comp_type == 'CID' %Sets constant HUB radius throughout the r_hub(i) = r_hub_in(1); elseif comp_type == 'COD' %Sets constant TIP radius throughout the r_tip(i) = r_tip_in(1); %########################################################## %################## Station 1 Rotor inlet ################# if i==1 r_rms_1(i) = r_rms_in; Cm1(i) = Cm_in; Alpha1(i) = Alpha_in; else r_rms_1(i) = r_rms_3(i-1); Cm1(i) = Cm3(i-1); Alpha1(i) = Alpha3(i-1); U1(i) = r_rms_1(i)*pi*rpm/30; %################ Station 1 total properties ############## if i==1 % The first stage P01(i) = P0_in; T01(i) = T0_in; [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('pt',p01(i),t01(i),0,1); H01(i) = H; S01(i) = S; S1(i) = S; else P01(i) = P03(i-1); T01(i) = T03(i-1); H01(i) = H03(i-1); S01(i) = S3(i-1); S1(i) = S3(i-1); C_theta1(i) = Cm1(i)*tand(Alpha1(i)); W_theta1(i) = U1(i)-C_theta1(i); Beta1(i) = atand(w_theta1(i)/cm1(i)); C1(i) = Cm1(i)/cosd(Alpha1(i)); W1(i) = Cm1(i)/cosd(Beta1(i)); 80

85 Axial Flow Compressor Mean Line Design Appix %################# Station 1 static properties ############ H1(i) = H01(i)-(C1(i)^2)/2; % Static enthalpy at rotor inlet [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('hs',h1(i),s1(i),0,1); P1(i) = P; T1(i) = T; Cp1(i) = Cp; rho1(i) = rho; Visc1(i) = Visc; kappa1(i) = kappa; a1(i) = a; MW1(i) = W1(i)/a1(i); % Station 1 relative Mach # MCm1(i) = Cm1(i)/a1(i); % Relative inlet meridional Mach # %############# Station 1 relative properties ############## H01_rel(i) = H1(i)+ (W1(i)^2)/2; % Relative total enthalpy [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('hs',h01_rel(i),s1(i),0,1); P01_rel(i) = P; T01_rel(i) = T; I1(i) = H1(i)+(W1(i)^2)/2-(U1(i)^2)/2; % Station 1 rothalpy %################### Station 1 geometry ################### if i==1 area1(i) = flow(i)/(cm1(i)*rho1(i)*blk_stg(i)); else area1(i) = flow(i)/(cm1(i)*rho1(i)*blk_stg(i-1)); if comp_type == 'CMD' r_hub_1(i) = ((r_rms_1(i)^2)-(area1(i)/(2*pi)))^0.5; r_tip_1(i) =((r_rms_1(i)^2)+(area1(i)/(2*pi)))^0.5; elseif comp_type == 'CID' r_hub_1(i) = r_hub(i); r_rms_1(i) = ((r_hub_1(i)^2)+(area1(i)/(2*pi)))^0.5; r_tip_1(i) =((r_rms_1(i)^2)+(area1(i)/(2*pi)))^0.5; elseif comp_type == 'COD' r_tip_1(i) = r_tip(i); r_rms_1(i) = ((r_tip_1(i)^2)-(area1(i)/(2*pi)))^0.5; r_hub_1(i) = ((r_rms_1(i)^2)-(area1(i)/(2*pi)))^0.5; height1(i) = r_tip_1(i)-r_hub_1(i); chord1(i) = height1(i)/ar_rtr(i); %########################################################## %########### Station 2 Rotor Outlet/Stator inlet ########## r_rms_2(i) = r_rms_1(i); % init value for r_rms_2 if i==1 ds21(i) = 5; % initial ds21 else ds21(i) = ds21(i-1); % initial ds21 value from previous rotor %########## Entropy rise iteration for the rotor ########## ds21_rel_error = 1; n_ds = 0; % Noff iteration steps entropy rise while abs(ds21_rel_error) > 10^(-4) % Outer loop solves for correct n_rms = 0; % Noff iteration steps RMS_rel_error = 1; exit radius while abs(rms_rel_error) > 10^(-4) % Inner loop solves for correct 81

86 Appix Axial Flow Compressor Mean Line Design Cm2(i) = Cm1(i)*AVR_rtr(i); U2(i) = r_rms_2(i)*pi*rpm/30; C_theta2(i) = PSI(j,i)*U2(i)+(r_rms_1(i)/r_rms_2(i))*C_theta1(i); % N.B. Based on delh/u2^2 W_theta2(i) = U2(i)-C_theta2(i); C2(i) = (Cm2(i)^2+C_theta2(i)^2)^0.5; W2(i) = (Cm2(i)^2+W_theta2(i)^2)^0.5; Alpha2(i) = atand(c_theta2(i)/cm2(i)); Beta2(i) = atand(w_theta2(i)/cm2(i)); dh_rtr(i) = W2(i)/W1(i); % Rotor dehaller # %########## Station 2 static properties ########### S2(i) = S1(i)+dS21(i); I2(i) = I1(i); %I.e. constant rothalpy through a rotor H2(i) = I2(i)-(W2(i)^2)/2+(U2(i)^2)/2; [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he] = state('hs',h2(i),s2(i),0,1); P2(i) = P; T2(i) = T; Cp2(i) = Cp; rho2(i) = rho; Visc2(i) = Visc; kappa2(i) = kappa; a2(i) = a; MC2(i) = C2(i)/a2(i); % Absolute inlet Mach # MCm2(i) = Cm2(i)/a2(i); % Relative inlet meridional Mach # %############# Station 2 relative properties ############## H02_rel(i) = H2(i)+ (W2(i)^2)/2; % Relative total enthalpy [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('hs',h02_rel(i),s2(i),0,1); P02_rel(i) = P; T02_rel(i) = T; %########### Station 2 total properties ########### H02(i) = H2(i)+(C2(i)^2)/2; % Exit total enthalpy [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he] = state('hs',h02(i),s2(i),0,1); P02(i) = P; T02(i) = T; %################ Station 2 geometry ############## area2(i) = flow(i)/(cm2(i)*rho2(i)*blk_stg(i)); if comp_type == 'CMD' r_hub_2(i) = ((r_rms_2(i)^2)-(area2(i)/(2*pi)))^0.5; r_tip_2(i) =((r_rms_2(i)^2)+(area2(i)/(2*pi)))^0.5; r_rms_2_new(i) = r_rms_2(i); elseif comp_type == 'CID' r_hub_2(i) = r_hub(i); r_rms_2_new(i) = ((r_hub_2(i)^2)+(area2(i)/(2*pi)))^0.5; r_tip_2(i) =((r_rms_2_new(i)^2)+(area2(i)/(2*pi)))^0.5; elseif comp_type == 'COD' r_tip_2(i) = r_tip(i); r_rms_2_new(i) = ((r_tip_2(i)^2)-(area2(i)/(2*pi)))^0.5; r_hub_2(i) = ((r_rms_2_new(i)^2)-(area2(i)/(2*pi)))^0.5; height2(i) = r_tip_2(i)-r_hub_2(i); chord2(i) = height2(i)/ar_rtr(i); RMS_rel_error = r_rms_2_new(i)/r_rms_2(i) - 1; r_rms_2(i)= r_rms_2_new(i); n_rms = n_rms+1; if n_rms > 50 %Emergency break RMS_rel_error = 0; warning('rms radius Convergence Error'); 82

87 Axial Flow Compressor Mean Line Design Appix % End of RMS loop %################## Rotor geometry #################### r_rms_rtr(i) = (r_rms_1(i)+r_rms_2(i))/2; r_hub_rtr(i) = (r_hub_1(i)+r_hub_2(i))/2; r_tip_rtr(i) =(r_tip_1(i)+r_tip_2(i))/2; HONT_rtr(i) = r_hub_rtr(i)/r_tip_rtr(i); height_rtr(i) = r_tip_rtr(i)-r_hub_rtr(i); chord_rtr(i) = height_rtr(i)/ar_rtr(i); Re_rtr(i) = W1(i)*chord_rtr(i)/Visc1(i); % Reynolds number %## Rotor Pitch-to-chord ratios and diffusion factors # rel_ang_in = Beta1(i); rel_ang_out = Beta2(i); AVDR_rtr(i) = rho2(i)*cm2(i)/(rho1(i)*cm1(i)); % Axial Velocity Density Ratio for the rotor if strcmp(sonc_type,'custom')==0 [SONC,SONC_Hearsey,SONC_McKenzie] = SONC1(rel_ang_in,rel_ang_out,Cm1(i),Cm2(i),r_rms_1(i),r_rms_2(i),DF_rtr(i)); if strcmp(sonc_type,'df') SONC_rtr(i) = SONC; elseif strcmp(sonc_type,'hearsey') SONC_rtr(i) = SONC_Hearsey; elseif strcmp(sonc_type,'mckenzie') SONC_rtr(i) = SONC_McKenzie; [DF_lbl,Deq_star_lbl,Deq,Deq_star_ks] = Deq_star1(rel_ang_in,rel_ang_out,Cm1(i),Cm2(i),r_rms_1(i),r_rms_2(i),SONC_rtr(i),TONC_rt r(i),avdr_rtr(i),mcm1(i)); DF_lbl_rtr(i) = DF_lbl; Deq_star_lbl_rtr(i) = Deq_star_lbl; Deq_rtr(i) = Deq; Deq_star_ks_rtr(i) = Deq_star_ks; [Mcrit_Hearsey,Mcrit_Sch] = MCRIT1(rel_ang_in,rel_ang_out,Cm1(i),SONC_rtr(i),TONC_rtr(i),kappa1(i)); Mcrit_Hearsey_rtr(i) = Mcrit_Hearsey; Mcrit_Sch_rtr(i) = Mcrit_Sch; %################# Rotor entropy rise ################# [OMEGA_p,OMEGA_ew, K_Re] = WRTMLR(DF_lbl_rtr(i),Deq_rtr(i),dH_rtr(i),rel_ang_out,MW1(i),AR_rtr(i),EPSONC_rtr(i),Re_ rtr(i)); K_Re_rtr(i) = K_Re; % Correction factor for Reynolds number OMEGA_p_rtr(i) = OMEGA_p; OMEGA_ew_rtr(i) = OMEGA_ew; OMEGA_rtr(i) = OMEGA_p+OMEGA_ew; OMEGA_rtr(i) = OMEGA_rtr(i)*K_Re; S_1 dp21(i) = OMEGA_rtr(i)*(P01_rel(i)-P1(i)); % Total pressure drop ds21_new(i) = -R*log(1-(dP21(i))/P01(i)); % updated value of S_2 - ds21_rel_error = ds21_new(i)/ds21(i) - 1; ds21(i) = ds21_new(i); n_ds = n_ds+1; if n_ds > 20 %Emergency break ds21_rel_error = 0; warning('entropy rise Convergence Error on rotor'); % End of entropy loop %###################### Station 3 Stator Outlet ############### r_rms_3(i) = r_rms_2(i); ds32(i) = ds21(i); % initial guess value ds32_rel_error = 1; n_ds = 0; % Noff iteration steps while abs(ds32_rel_error) > 10^(-4) % Solves for correct entropy rise 83

88 Appix Axial Flow Compressor Mean Line Design n_rms = 0; % Noff iteration steps RMS_rel_error = 1; while abs(rms_rel_error) > 10^(-4) Cm3(i) = Cm2(i)*AVR_str(i); if i == N_stg Alpha3(i) = Alpha3(i-1); C_theta3(i) = Cm3(i)*tand(Alpha3(i)); C3(i) = Cm3(i)/cosd(Alpha3(i)); dh_str(i) = C3(i)/C2(i); % Stator dehaller # %########### Station 3 static properties ########## H03(i) = H02(i); % Constant total enthalpy through the stator H3(i) = H03(i)-(C3(i)^2)/2; S3(i) = S2(i)+dS32(i); [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('hs',h3(i),s3(i),0,1); P3(i) = P; T3(i) = T; Cp3(i) = Cp; rho3(i) = rho; Visc3(i) = Visc; kappa3(i) = kappa; a3(i) = a; MC3(i) = C3(i)/a3(i);% Absolute inlet Mach # MCm3(i) = Cm3(i)/a3(i);% Relative inlet meridional Mach # %############ Station 3 total properties ########## [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('hs',h03(i),s3(i),0,1); P03(i) = P; T03(i) = T; %################ Station 3 geometry ############## area3(i) = flow(i)/(cm3(i)*rho3(i)*blk_stg(i)); if comp_type == 'CMD' r_hub_3(i) = ((r_rms_3(i)^2)-(area3(i)/(2*pi)))^0.5; r_tip_3(i) =((r_rms_3(i)^2)+(area3(i)/(2*pi)))^0.5; r_rms_3_new(i) = r_rms_3(i); elseif comp_type == 'CID' r_hub_3(i) = r_hub(i); r_rms_3_new(i) = ((r_hub_3(i)^2)+(area3(i)/(2*pi)))^0.5; r_tip_3(i) =((r_rms_3_new(i)^2)+(area3(i)/(2*pi)))^0.5; elseif comp_type == 'COD' r_tip_3(i) = r_tip(i); r_rms_3_new(i) = ((r_tip_3(i)^2)-(area3(i)/(2*pi)))^0.5; r_hub_3(i) = ((r_rms_3_new(i)^2)-(area3(i)/(2*pi)))^0.5; height3(i) = r_tip_3(i)-r_hub_3(i); chord3(i) = height3(i)/ar_str(i); RMS_rel_error = r_rms_3_new(i)/r_rms_3(i) - 1; r_rms_3(i) = r_rms_3_new(i); n_rms = n_rms+1; if n_rms > 50 %Emergency break RMS_rel_error = 0; warning('rms radius Convergence Error'); % End of RMS loop %################### Stator geometry ################## area_str(i) = (area2(i)+area3(i))/2; r_rms_str(i) = (r_rms_2(i)+r_rms_3(i))/2; 84

89 Axial Flow Compressor Mean Line Design Appix r_hub_str(i) = ((r_rms_str(i)^2)-(area_str(i)/(2*pi)))^0.5; r_tip_str(i) =((r_rms_str(i)^2)+(area_str(i)/(2*pi)))^0.5; HONT_str(i) = r_hub_str(i)/r_tip_str(i); height_str(i) = r_tip_str(i)-r_hub_str(i); chord_str(i) = height_str(i)/ar_str(i); Re_str(i) = C2(i)*chord_str(i)/Visc2(i); % Reynolds number %# Stator Pitch-to-chord ratios and diffusion factors # rel_ang_in = Alpha2(i); rel_ang_out = Alpha3(i); AVDR_str(i) = rho3(i)*cm3(i)/(rho2(i)*cm2(i));% Axial Velocity Density Ratio for the stator if strcmp(sonc_type,'custom')==0 [SONC,SONC_Hearsey,SONC_McKenzie] = SONC1(rel_ang_in,rel_ang_out,Cm2(i),Cm3(i),r_rms_2(i),r_rms_3(i),DF_str(i)); if strcmp(sonc_type,'df') SONC_str(i) = SONC; elseif strcmp(sonc_type,'hearsey') SONC_str(i) = SONC_Hearsey; elseif strcmp(sonc_type,'mckenzie') SONC_str(i) = SONC_McKenzie; [DF_lbl,Deq_star_lbl,Deq,Deq_star_ks] = Deq_star1(rel_ang_in,rel_ang_out,Cm2(i),Cm3(i),r_rms_2(i),r_rms_3(i),SONC_rtr(i),TONC_st r(i),avdr_str(i),mcm2(i)); DF_lbl_str(i) = DF_lbl; Deq_star_lbl_str(i) = Deq_star_lbl; Deq_str(i) = Deq; Deq_star_ks_str(i) = Deq_star_ks; [Mcrit_Hearsey,Mcrit_Sch] = MCRIT1(rel_ang_in,rel_ang_out,Cm2(i),SONC_str(i),TONC_str(i),kappa2(i)); Mcrit_Hearsey_str(i) = Mcrit_Hearsey; Mcrit_Sch_str(i) = Mcrit_Sch; %################## Stator entropy rise ############### [OMEGA_p, OMEGA_ew, K_Re] = WRTMLR(DF_lbl_str(i),Deq_str(i),dH_str(i),rel_ang_out,MC2(i),AR_str(i),EPSONC_str(i),Re_ str(i)); K_Re_str(i) = K_Re; % Correction factor for Reynolds number OMEGA_p_str(i) = OMEGA_p; OMEGA_ew_str(i) = OMEGA_ew; OMEGA_str(i) = OMEGA_p+OMEGA_ew; OMEGA_str(i) = OMEGA_str(i)*K_Re; dp32(i) = OMEGA_str(i)*(P02(i)-P2(i)); % P0_3 - P0_2 ds32_new(i) = -R*log(1-(dP32(i))/P02(i)); %S_3 - S_2 ds32_rel_error = ds32_new(i)/ds32(i) - 1; ds32(i) = ds32_new(i); n_ds = n_ds+1; if n_ds > 20 %Emergency break ds21_rel_error = 0; warning('entropy rise Convergence Error on stator'); % End of entropy loop % for all stages %############################################################## %############### outlet guide vane, OGV ####################### ds_ogv = ds32(n_stg); % initial guess value ds_ogv_rel_error = 1; n_ds = 0; % Noff iteration steps while abs(ds_ogv_rel_error) > 10^(-4) % Solves for correct entropy rise 85

90 Appix Axial Flow Compressor Mean Line Design AVR_OGV = 1.0; Cm_OGV = Cm3(N_stg)*AVR_OGV; C_OGV = Cm_OGV; %########### OGV static properties ########## H0_OGV = H03(N_stg); % Constant total enthalpy through the OGV H_OGV = H0_OGV-(C_OGV^2)/2; S_OGV = S3(N_stg)+dS_OGV; [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('hs',h_ogv,s_ogv,0,1); P_OGV = P; T_OGV = T; Cp_OGV = Cp; rho_ogv = rho; Visc_OGV = Visc; kappa_ogv = kappa; a_ogv = a; MCm_OGV = Cm_OGV/a_OGV; MC_OGV = C_OGV/a_OGV;% Absolute inlet Mach # dh_ogv = C_OGV/C3(N_stg); %############ OGV total properties ########## [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('hs',h0_ogv,s_ogv,0,1); P0_OGV = P; T0_OGV = T; %############ OGV geometry ########## r_rms_ogv = r_rms_3(n_stg); r_tip_ogv = r_tip_3(n_stg); r_hub_ogv = r_hub_3(n_stg); HONT_OGV = r_hub_ogv/r_tip_ogv; height_ogv = r_tip_ogv-r_hub_ogv; chord_ogv = height_ogv/ar_str(i); Re_OGV = C_OGV*chord_OGV/Visc_OGV; % Reynolds number rel_ang_in = Alpha3(N_stg); rel_ang_out = 0; AVDR_OGV = rho_ogv*cm_ogv/(rho3(n_stg)*cm3(n_stg));% Axial Velocity Density Ratio for the stator if strcmp(sonc_type,'custom')==0 [SONC_OGV,SONC_OGV_Hearsey,SONC_OGV_McKenzie] = SONC1(rel_ang_in,rel_ang_out,Cm3(N_stg),Cm_OGV,r_rms_3(N_stg),r_rms_OGV,DF_str(N_stg)); if strcmp(sonc_type,'df') SONC_OGV = SONC_OGV; elseif strcmp(sonc_type,'hearsey') SONC_OGV = SONC_OGV_Hearsey; elseif strcmp(sonc_type,'mckenzie') SONC_OGV = SONC_OGV_McKenzie; else SONC_OGV = SONC_str(N_stg); [DF_lbl,Deq_star_lbl,Deq,Deq_star_ks] = Deq_star1(rel_ang_in,rel_ang_out,Cm3(N_stg),Cm_OGV,r_rms_3(N_stg),r_rms_OGV,SONC_OGV,TON C_str(N_stg),AVDR_OGV,MCm_OGV); DF_lbl_OGV = DF_lbl; Deq_star_lbl_OGV = Deq_star_lbl; Deq_OGV = Deq; Deq_star_ks_OGV = Deq_star_ks; [Mcrit_Hearsey,Mcrit_Sch] = MCRIT1(rel_ang_in,rel_ang_out,Cm3(N_stg),SONC_OGV,TONC_str(N_stg),kappa3(N_stg)); Mcrit_Hearsey_OGV = Mcrit_Hearsey; Mcrit_Sch_OGV = Mcrit_Sch; %################## OGV entropy rise ############### 86

91 Axial Flow Compressor Mean Line Design Appix [OMEGA_p,OMEGA_ew, K_Re] = WRTMLR(DF_lbl_OGV,Deq_OGV,dH_OGV,rel_ang_out,MC3(N_stg),AR_str(N_stg),EPSONC_str(i),Re_O GV); K_Re_OGV = K_Re; % Correction factor for Reynolds number OMEGA_p_OGV = OMEGA_p; OMEGA_ew_OGV = OMEGA_ew; OMEGA_OGV = OMEGA_p+OMEGA_ew; OMEGA_OGV = OMEGA_OGV*K_Re; dp_ogv = OMEGA_OGV*(P03(N_stg)-P3(N_stg)); % P0_3 - P0_3 ds_ogv_new = -R*log(1-(dP_OGV)/P03(N_stg)); %S_OGV - S_3 ds_ogv_rel_error = ds_ogv_new/ds_ogv - 1; ds_ogv = ds_ogv_new; n_ds = n_ds+1; if n_ds > 20 %Emergency break ds_ogv_rel_error = 0; warning('entropy rise Convergence Error on stator'); % End of entropy loop for OGV if set_react == 1 %#### Calculates the degree of reaction and the slope ##### for k=1:n_stg REACT(k) = (H2(k)-H1(k))/(H03(k)-H01(k)); REACT_slope(k) = Cm1(i)/U1(i)*AVR_rtr(i)*pi/(180*cosd(Alpha3(k))^2); %############### Calculates new Alpha3 values ############# for k=2:n_stg REACT_error(k) = REACT(k)-REACT_stg(k); REACT_rel_error(k) = REACT_stg(k)/REACT(k)-1; Alpha3(k-1) = Alpha3(k-1)+REACT_error(k)/REACT_slope(k)*RLX_REACT; %############### Evaluate RMS error for reaction ########## REACT_rel_RMS_error = 0; for k=1:n_stg REACT_rel_RMS_error = REACT_rel_RMS_error+REACT_rel_error(k)^2; if abs(react_rel_rms_error) < 10^(-4) conv_react = 1; % the itertation has converged n_react = n_react+1; if n_react > 50 %Emergency break REACT_rel_RMS_error = 0; warning('react Convergence Error'); else conv_react = 1; for k=1:n_stg REACT_stg(k) = (H2(k)-H1(k))/(H03(k)-H01(k)); %####################### End of REACT loop ######################## %################################################################## PR(j) = P0_OGV/P01(1); % calculates the pressure ratio %################ breaks the pressure ratio loop ################## if conv_pr == 1 break 87

92 Appix Axial Flow Compressor Mean Line Design % of the "two" calculations for obtaining derivative %############## calculates the PR slope if not converged ############## if conv_pr == 0 PR_slope = (PR(1)-PR(2))/(mean(PSI(1,:))-mean(PSI(2,:))); % the numerical derivative dpr/dpsi PR_rel_error_level = PR_comp/PR(1)-1; PR_error_level = PR_comp-PR(1); %##################### Calculates new PSI values ###################### numbers for i=1:n_stg PSI(1,i) = PSI(1,i)+PR_error_level/PR_slope*RLX_PR; % calculate new PSI PSI(2,i) = PSI(1,i)*factor; n_pr = n_pr+1; if n_pr>20 % Emergency break PR_error_level = 0; warning('pressure ratio Convergence Error'); %####################### End of PR loop ################################ %########################################################################## pressure_ratio = P_OGV/P01(1); %########################## Blade angles ################################## for i = 1:N_stg spacing_rtr(i) = chord_rtr(i)*sonc_rtr(i); diameter_rtr(i) = 2*pi*r_rms_rtr(i); numb_blades_rtr(i) = diameter_rtr(i)/spacing_rtr(i); numb_blades_rtr(i) = ceil(numb_blades_rtr(i)); spacing_str(i) = chord_str(i)*sonc_str(i); diameter_str(i) = 2*pi*r_rms_str(i); numb_blades_str(i) = diameter_str(i)/spacing_str(i); numb_blades_str(i) = ceil(numb_blades_str(i)); %############################## Rotor angles ########################## rel_ang_in = Beta1(i); rel_ang_out = Beta2(i); SONC = SONC_rtr(i); TONC = TONC_rtr(i); M = MW1(i); [incidence_angle,deviation_angle,camber_angle,attack_angle,stagger_angle,blade_angle_in, blade_angle_out] = Bladeangles(rel_ang_in,rel_ang_out,SONC,TONC,M); incidence_angle_rtr(i) = incidence_angle; deviation_angle_rtr(i) = deviation_angle; camber_angle_rtr(i) = camber_angle; attack_angle_rtr(i) = attack_angle; stagger_angle_rtr(i) = stagger_angle; blade_angle_in_rtr(i) = blade_angle_in; blade_angle_out_rtr(i) = blade_angle_out; %########################### Stator angles ############################ rel_ang_in = Alpha2(i); rel_ang_out = Alpha3(i); SONC = SONC_str(i); TONC = TONC_str(i); M = MC2(i); [incidence_angle,deviation_angle,camber_angle,attack_angle,stagger_angle,blade_angle_in, blade_angle_out] = Bladeangles(rel_ang_in,rel_ang_out,SONC,TONC,M); incidence_angle_str(i) = incidence_angle; deviation_angle_str(i) = deviation_angle; camber_angle_str(i) = camber_angle; 88

93 Axial Flow Compressor Mean Line Design Appix attack_angle_str(i) = attack_angle; stagger_angle_str(i) = stagger_angle; blade_angle_in_str(i) = blade_angle_in; blade_angle_out_str(i) = blade_angle_out; %########################### Tip-housing angle ######################## tip_angle_rtr(i) = atand((r_tip_1(i)- r_tip_2(i))/(chord_rtr(i)*cosd(stagger_angle_rtr(i)))); tip_angle_str(i) = atand((r_tip_2(i)- r_tip_3(i))/(chord_str(i)*cosd(stagger_angle_str(i)))); %##################### Blade angles for the OGV ########################### spacing_ogv = chord_ogv*sonc_ogv; diameter_ogv = 2*pi*r_rms_OGV; numb_blades_ogv = diameter_ogv/spacing_ogv; numb_blades_ogv = ceil(numb_blades_ogv); rel_ang_in = Alpha3(N_stg); rel_ang_out = 0; SONC = SONC_OGV; TONC = TONC_str(N_stg); M = MC3(N_stg); [incidence_angle,deviation_angle,camber_angle,attack_angle,stagger_angle,blade_angle_in, blade_angle_out] = Bladeangles(rel_ang_in,rel_ang_out,SONC,TONC,M); incidence_angle_ogv = incidence_angle; deviation_angle_ogv = deviation_angle; camber_angle_ogv = camber_angle; attack_angle_ogv = attack_angle; stagger_angle_ogv = stagger_angle; blade_angle_in_ogv = blade_angle_in; blade_angle_out_ogv = blade_angle_out; %########################## Misc. properties ############################## Power=0; %Resetting stage power summation for i=1:n_stg %####################### static pressure rise ######################### Cp_rtr(i) = (P2(i)-P1(i))/(P01_rel(i)-P1(i)); Cp_str(i) = (P3(i)-P2(i))/(P02(i)-P2(i)); %######################### Stage pressure ratio ####################### PR_stg(i) = P03(i)/P01(i); %###################### Accumulated pressure ratio #################### PR_acc(i) = P03(i)/P01(1); %######################### Stage temp increase ######################## dt0_stg(i) = T03(i)-T01(i); %#################### Polytropic stage efficiency ##################### [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('pt',1,t03(i),0,1); aa = S; % a dummie variable [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('pt',1,t01(i),0,1); bb = S; % a dummie variable poly_eff_stg(i) = R*log(P03(i)/P01(i))/(aa-bb); %################# Accumulated polytropic efficiency ################## [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('pt',1,t03(i),0,1); aa = S; % a dummie variable [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('pt',1,t01(1),0,1); bb = S; % a dummie variable 89

94 Appix Axial Flow Compressor Mean Line Design poly_eff_acc(i) = R*log(P03(i)/P01(1))/(aa-bb); %###################### Isentropic stage efficiency ################### P = P03(i); S = S01(i); [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('ps',p,s,0,1); H03s = H; % the isentropic enthalpy isen_eff_stg(i) = (H03s-H01(i))/(H03(i)-H01(i)); %##################### Stage flow coefficient ######################### PHI_stg(i) = (Cm1(i)+Cm2(i))/(U1(i)+U2(i)); %####################### Compressor Power ############################ Power=Power+flow(i)*(H03(i)-H01(i))/1000; %############################ Stall and Surge ############################# %########################################################################## for i =1:N_stg %##################### constants/variables ###################### % staggered spacing g_rtr = pi*r_rms_rtr(i)*(cosd(blade_angle_in_rtr(i))+cosd(blade_angle_out_rtr(i)))/numb_blades_r tr(i); g_str = pi*r_rms_str(i)*(cosd(blade_angle_in_str(i))+cosd(blade_angle_out_str(i)))/numb_blades_s tr(i); g = (W1(i)^2*g_rtr + C2(i)^2*g_str)/(W1(i)^2+C2(i)^2); % average value of g in the stage % L/g2, L is meanline length of circular-arc profile LONG2_rtr = (1/SONC_rtr(i))/(cosd(blade_angle_out_rtr(i))*cosd(camber_angle_rtr(i)/2)); LONG2_str = (1/SONC_str(i))/(cosd(blade_angle_out_str(i))*cosd(camber_angle_str(i)/2)); LONG2 = (W1(i)^2*LONG2_rtr + C2(i)^2*LONG2_str)/(W1(i)^2+C2(i)^2); % average value of L/g2 in the stage % wall space epsilon_rtr = EPSONC_rtr(i)*chord_rtr(i); epsilon_str = EPSONC_str(i)*chord_str(i); epsilon = (W1(i)^2*epsilon_rtr + C2(i)^2*epsilon_str)/(W1(i)^2+C2(i)^2); % epsilon/g EPSONG = epsilon/g; %Reynolds number Re = (W1(i)^2*Re_rtr(i) + C2(i)^2*2*Re_str(i))/(W1(i)^2+C2(i)^2); % axial spacing axial_spacing_rtr = 0.2*chord_rtr(i); axial_spacing_str = 0.2*chord_str(i); axial_spacing = (W1(i)^2*axial_spacing_rtr + C2(i)^2*axial_spacing_str)/(W1(i)^2+C2(i)^2); dz = axial_spacing; %staggered spacing s_rtr = spacing_rtr(i); s_str = spacing_str(i); s = (W1(i)^2*spacing_rtr(i) + C2(i)^2*spacing_str(i))/(W1(i)^2+C2(i)^2); dzons = dz/s; % dz/s %################## F_ef ################# 90

95 Axial Flow Compressor Mean Line Design Appix % rotor U_in = U1(i); C_in = C1(i); W_rtr_in = W1(i); if (Alpha1(i)+Beta1(i)) >= 90 V_min = C1(i)*sind(Alpha1(i)+Beta1(i)); else V_min = C1(i); F_ef_rtr = (1+2.5*V_min^2+0.5*U_in^2)/(4*C_in^2); % stator U_in = U2(i); C_in = C2(i); V_str_in = C2(i); if (Alpha2(i)+Beta2(i)) >= 90 V_min = C2(i)*sind(Alpha2(i)+Beta2(i)); else V_min = C2(i); F_ef_str = (1+2.5*V_min^2+0.5*U_in^2)/(4*C_in^2); % average F_ef(i) = (W1(i)^2*F_ef_rtr + C2(i)^2*F_ef_str)/(W1(i)^2+C2(i)^2); %######################## ChD ############################### x = LONG2; %polynomial coefficients coeff_ch_d = [ ]; Ch_D(i) = 0; for j=1:6 Ch_D(i) = Ch_D(i)+coeff_Ch_D(j)*(x)^(6-j); % evaluate the polynomial %######################## (Ch/ChD)_eps ############################### x = EPSONG; % epsilon/g %polynomial coefficients coeff_ch_eps = [ ]; Ch_eps(i) = 0; for j=1:6 Ch_eps(i) = Ch_eps(i)+coeff_Ch_eps(j)*(x)^(6-j); % evaluate the polynomial %######################## (Ch/ChD)_dZ ############################### x = dzons; % dz/s %polynomial coefficients coeff_ch_dz = [ ]; Ch_dZ(i) = 0; for j=1:6 Ch_dZ(i) = Ch_dZ(i)+coeff_Ch_dZ(j)*(x)^(6-j); % evaluate the polynomial 91

96 Appix Axial Flow Compressor Mean Line Design %######################## (Cp/CpD)_Re ############################### x = Re; % coefficients a = ; b = ; c = 1.041; Ch_Re(i) = a*x^b+c; %################################################## Ch_max(i) = F_ef(i)*Ch_D(i)*Ch_Re(i)*Ch_dZ(i)*Ch_eps(i); %#################### Ch ##################### kappa_mean = (kappa1(i)+kappa3(i))/2; Cp_mean = (Cp1(i)+Cp3(i))/2; aa = Cp_mean*(T1(i) )*((P3(i)/P1(i))^((kappa_mean-1)/kappa_mean)-1); bb = (U2(i)^2-U1(i)^2)/2; cc = (W1(i)^2+C2(i)^2)/2; Ch(i) = (aa-bb)/cc; omega(i) = Ch(i)/Ch_max(i); %########################## Polytropic efficency ########################## P_ref = 1; % a reference pressure for the calculation [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('pt',p_ref,t03(n_stg),0,1); aa = S; % a dummie variable [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('pt',p_ref,t01(1),0,1); bb = S; % a dummie variable poly_eff = R*log(P03(N_stg)/P01(1))/(aa-bb); %########################## Isentropic efficency ########################## P = P3(N_stg); S = S3(1); [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('ps',p,s,0,1); H2s = H; % the isentropic enthalpy isen_eff = (H2s-H1(1))/(H3(N_stg)-H1(1)); %############################# Compressor cost ############################ mean_diameter = r_rms_1(1)+r_rms_3(n_stg); comp_cost = 1.13*2625*(N_stg^1.155 * PR_comp^0.775 * mean_diameter^ ); %########################### Compressor Length ############################ Comp_length = 0; for i=1:n_stg Comp_length = Comp_length + chord_rtr(i)*cosd(stagger_angle_rtr(i)); Comp_length = Comp_length + chord_rtr(i)*0.2; Comp_length = Comp_length + chord_str(i)*cosd(stagger_angle_str(i)); Comp_length = Comp_length + chord_str(i)*0.2; Comp_length = Comp_length - chord_str(n_stg)*0.2; Comp_angle = atand((r_tip_1(1)-r_tip_3(n_stg))/comp_length); %############ Creates a structure of a properties and values ############# 92

97 Axial Flow Compressor Mean Line Design Appix CompInfo.comp_name = CompInputdata{1,1}; CompInfo.type = comp_type; CompInfo.P0_in = P0_in; CompInfo.T0_in = T0_in; CompInfo.PR_comp = PR_comp; CompInfo.N_stg = N_stg; CompInfo.RPM = RPM; CompInfo.Alpha_in = Alpha_in; CompInfo.PHI_in = PHI_in; CompInfo.HONT_in = HONT_in; CompInfo.poly_eff = poly_eff; CompInfo.isen_eff = isen_eff; CompInfo.comp_cost = comp_cost; CompInfo.area_in = area_in; CompInfo.M_tip_in = M_tip_in; CompInfo.Power = Power; CompInfo.Comp_length = Comp_length; CompInfo.Comp_angle = Comp_angle; OGV.Cm = Cm_OGV; OGV.C = C_OGV; OGV.H0 = H0_OGV; OGV.H = H_OGV; OGV.S = S_OGV; OGV.P = P_OGV; OGV.T = T_OGV; OGV.Cp = Cp_OGV; OGV.rho = rho_ogv; OGV.Visc = Visc_OGV; OGV.kappa = kappa_ogv; OGV.a = a_ogv; OGV.P0 = P0_OGV; OGV.T0 = T0_OGV; OGV.MCm = MCm_OGV; OGV.MC = MC3(N_stg); OGV.dH = dh_ogv; OGV.r_rms = r_rms_ogv; OGV.r_tip = r_tip_ogv; OGV.r_hub = r_hub_ogv; OGV.HONT = HONT_OGV; OGV.height = height_ogv; OGV.chord = chord_ogv; OGV.Re = Re_OGV; OGV.SONC = SONC_OGV; OGV.DF_lbl = DF_lbl_OGV; OGV.Deq_star_lbl = Deq_star_lbl_OGV; OGV.Deq = Deq_OGV; OGV.Deq_star_ks = Deq_star_ks_OGV; OGV.Mcrit_Hearsey = Mcrit_Hearsey_OGV; OGV.Mcrit_Sch = Mcrit_Sch_OGV; OGV.OMEGA_p = OMEGA_p_OGV; OGV.OMEGA_ew = OMEGA_ew_OGV; OGV.numb_blades = numb_blades_ogv; OGV.incidence_angle = incidence_angle_ogv; OGV.deviation_angle = deviation_angle_ogv; OGV.camber_angle = camber_angle_ogv; OGV.attack_angle = attack_angle_ogv; OGV.stagger_angle = stagger_angle_ogv; OGV.blade_angle_in = blade_angle_in_ogv; OGV.blade_angle_out = blade_angle_out_ogv; for i = 1:N_stg STG(i).Alpha1 = Alpha1(i); STG(i).Alpha2 = Alpha2(i); STG(i).Alpha3 = Alpha3(i); STG(i).Beta1 = Beta1(i); STG(i).Beta2 = Beta2(i); STG(i).C1 = C1(i); STG(i).C2 = C2(i); STG(i).C3 = C3(i); STG(i).W1 = W1(i); STG(i).W2 = W2(i); STG(i).Cm1 = Cm1(i); STG(i).Cm2 = Cm2(i); 93

98 Appix Axial Flow Compressor Mean Line Design STG(i).Cm3 = Cm3(i); STG(i).U1 = U1(i); STG(i).U2 = U2(i); STG(i).C_theta1 = C_theta1(i); STG(i).C_theta2 = C_theta2(i); STG(i).C_theta3 = C_theta3(i); STG(i).W_theta1 = W_theta1(i); STG(i).W_theta2 = W_theta2(i); STG(i).MW1 = MW1(i); STG(i).MC2 = MC2(i); STG(i).Mcrit_Hearsey_rtr = Mcrit_Hearsey_rtr(i); STG(i).Mcrit_Sch_rtr = Mcrit_Sch_rtr(i); STG(i).Mcrit_Hearsey_str = Mcrit_Hearsey_str(i); STG(i).Mcrit_Sch_str = Mcrit_Sch_str(i); STG(i).a1 = a1(i); STG(i).a2 = a2(i); STG(i).a3 = a3(i); STG(i).kappa1 = kappa1(i); STG(i).kappa2 = kappa2(i); STG(i).kappa3 = kappa3(i); STG(i).Cp1 = Cp1(i); STG(i).Cp2 = Cp2(i); STG(i).Cp3 = Cp3(i); STG(i).rho1 = rho1(i); STG(i).rho2 = rho2(i); STG(i).rho3 = rho3(i); STG(i).visc1 = Visc1(i); STG(i).visc2 = Visc2(i); STG(i).visc3 = Visc3(i); STG(i).P1 = P1(i); STG(i).P2 = P2(i); STG(i).P3 = P3(i); STG(i).P01 = P01(i); STG(i).P02 = P02(i); STG(i).P03 = P03(i); STG(i).T1 = T1(i); STG(i).T2 = T2(i); STG(i).T3 = T3(i); STG(i).T01 = T01(i); STG(i).T02 = T02(i); STG(i).T03 = T03(i); STG(i).H1 = H1(i); STG(i).H2 = H2(i); STG(i).H3 = H3(i); STG(i).H01 = H01(i); STG(i).H02 = H02(i); STG(i).H03 = H03(i); STG(i).S1 = S1(i); STG(i).S2 = S2(i); STG(i).S3 = S3(i); STG(i).S01 = S1(i); STG(i).S02 = S2(i); STG(i).S03 = S3(i); STG(i).EPSONC_rtr = EPSONC_rtr(i); STG(i).EPSONC_str = EPSONC_str(i); STG(i).SONC_rtr = SONC_rtr(i); STG(i).SONC_str = SONC_str(i); STG(i).HONT_rtr = HONT_rtr(i); STG(i).HONT_str = HONT_str(i); STG(i).TONC_rtr = TONC_rtr(i); STG(i).TONC_str = TONC_str(i); STG(i).AR_rtr = AR_rtr(i); STG(i).AR_str = AR_str(i); STG(i).chord_rtr = chord_rtr(i); STG(i).chord_str = chord_str(i); STG(i).height_rtr = height_rtr(i); STG(i).height_str = height_str(i); STG(i).DF_rtr = DF_lbl_rtr(i); STG(i).DF_str = DF_lbl_str(i); STG(i).Deq_rtr = Deq_rtr(i); STG(i).Deq_str = Deq_str(i); STG(i).Deq_star_lbl_rtr = Deq_star_lbl_rtr(i); STG(i).Deq_star_lbl_str = Deq_star_lbl_str(i); STG(i).Deq_star_ks_rtr = Deq_star_ks_rtr(i); STG(i).Deq_star_ks_str = Deq_star_ks_str(i); STG(i).dH_rtr = dh_rtr(i); STG(i).dH_str = dh_str(i); 94

99 Axial Flow Compressor Mean Line Design Appix STG(i).dS_rtr = ds21(i); STG(i).dS_str = ds32(i); STG(i).area1 = area1(i); STG(i).area2 = area2(i); STG(i).area3 = area3(i); STG(i).Re_rtr = Re_rtr(i); STG(i).Re_str = Re_str(i); STG(i).r_tip_1 = r_tip_1(i); STG(i).r_tip_2 = r_tip_2(i); STG(i).r_tip_3 = r_tip_3(i); STG(i).r_hub_1 = r_hub_1(i); STG(i).r_hub_2 = r_hub_2(i); STG(i).r_hub_3 = r_hub_3(i); STG(i).r_rms_1 = r_rms_1(i); STG(i).r_rms_2 = r_rms_2(i); STG(i).r_rms_3 = r_rms_3(i); STG(i).incidence_angle_rtr = incidence_angle_rtr(i); STG(i).deviation_angle_rtr = deviation_angle_rtr(i); STG(i).camber_angle_rtr = camber_angle_rtr(i); STG(i).attack_angle_rtr = attack_angle_rtr(i); STG(i).stagger_angle_rtr = stagger_angle_rtr(i); STG(i).turning_rtr = Beta1(i)-Beta2(i); STG(i).blade_angle_in_rtr = blade_angle_in_rtr(i); STG(i).blade_angle_out_rtr = blade_angle_out_rtr(i); STG(i).incidence_angle_str = incidence_angle_str(i); STG(i).deviation_angle_str = deviation_angle_str(i); STG(i).camber_angle_str = camber_angle_str(i); STG(i).attack_angle_str = attack_angle_str(i); STG(i).stagger_angle_str = stagger_angle_str(i); STG(i).turning_str = Alpha2(i)-Alpha3(i); STG(i).blade_angle_in_str = blade_angle_in_str(i); STG(i).blade_angle_out_str = blade_angle_out_str(i); STG(i).tip_angle_rtr = tip_angle_rtr(i); STG(i).tip_angle_str = tip_angle_str(i); STG(i).spacing_rtr = spacing_rtr(i); STG(i).numb_blades_rtr = numb_blades_rtr(i); STG(i).spacing_str = spacing_str(i); STG(i).numb_blades_str = numb_blades_str(i); STG(i).PR = PR_stg(i); STG(i).PR_acc = PR_acc(i); STG(i).React = REACT_stg(i); STG(i).PSI = PSI(1,i); STG(i).PHI = PHI_stg(i); STG(i).dT0 = dt0_stg(i); STG(i).dS = ds21(i)+ds32(i); STG(i).OMEGA_p_rtr = OMEGA_p_rtr(i); STG(i).OMEGA_ew_rtr = OMEGA_ew_rtr(i); STG(i).OMEGA_p_str = OMEGA_p_str(i); STG(i).OMEGA_ew_str = OMEGA_ew_str(i); STG(i).poly_eff = poly_eff_stg(i); STG(i).poly_eff_acc = poly_eff_acc(i); STG(i).isen_eff = isen_eff_stg(i); STG(i).flow = flow(i); STG(i).Ch = Ch(i); STG(i).Ch_max = Ch_max(i); STG(i).omega = omega(i); 95

100 Appix B.2 Inlet geometry calculation Axial Flow Compressor Mean Line Design % Inlet geometry calculation %##################################################################### %## ## %## Inlet geometry ## %## ## %## (c) Niclas Falck and Magnus Genrup 2008 ## %## ## %## Lund University/Dept of Energy Sciences ## %## ## %##################################################################### function [r_rms_rtr,r_m_rtr,r_hub_rtr,r_tip_rtr]=inletgeom(p0,t0,flow,rpm,hont,phi,alpha1,blk) %########################################################################## % Inlet state [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV,y_SO2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('pt',p0,t0,0,1); H0 = H; S0 = S; %########################################################################## % Main calculation Cm = 0.6*a; %Initial velocity guess RLX = 0.15; %damping rel_error_cm = 1; n = 0; %Noff iteration steps while abs(rel_error_cm) > 10^(-3) %Solves for correct velocity Cm_old = Cm; C = Cm_old/cosd(alpha1); H = H0-C^2/2; [P, T, H, S, Cp, rho, Visc, lambda, kappa, R, a, crit, FARsto, LHV, y_so2, y_h2o, y_co2, y_n2, y_o2, y_ar, y_he]=state('hs',h,s0,0,1); area = flow/(cm_old*rho*blk); r_tip_rtr = (area/(pi*(1-hont^2)))^0.5; %tip radius r_hub_rtr = HONT*r_tip_rtr; %hub radius r_m_rtr = (r_tip_rtr+r_hub_rtr)/2; %mean radius r_rms_rtr=((r_hub_rtr^2+r_tip_rtr^2)/2)^0.5; %RMS radius U_m = 2*r_m_rtr*pi*RPM/60; %blade speed based om mean radius U_rms = 2*r_rms_rtr*pi*RPM/60; %blade speed based om RMS radius Cm = PHI*U_rms; %Updated meridional velocity rel_error_cm = Cm/Cm_old-1; Cm = Cm+(Cm_old-Cm)*RLX; %damping n = n+1; if n>20 %Emergency break rel_error_cm = 0; warning('velocity Cm Convergence Error'); 96

101 Axial Flow Compressor Mean Line Design Appix B.3 Pitch chord ratio % Pitch-to-chord ratio %##################################################################### %## ## %## Pitch-to-chord ratio ## %## ## %## (c) Niclas Falck and Magnus Genrup 2008 ## %## ## %## Lund University/Dept of Energy Sciences ## %## ## %##################################################################### function[sonc,sonc_hearsey,sonc_mckenzie] = SONC1(rel_ang_in,rel_ang_out,Cm_in,Cm_out,r_in,r_out,DF); rel_vel_in = Cm_in/cosd(rel_ang_in); rel_vel_out = Cm_out/cosd(rel_ang_out); dh = rel_vel_out/rel_vel_in; Cpi = 1-dH^2; C_theta_in = Cm_in*tand(rel_ang_in); C_theta_out = Cm_out*tand(rel_ang_out); SONC = (DF-1+dH)*rel_vel_in*(r_in+r_out)/abs(r_in*C_theta_in-r_out*C_theta_out); %##################### Hearsey solidity method ########################### % The Hearsey method is taken from Concepts ETI's "Practical Compressor % Aerodynamic Design", page 44, eq 44. DF_opt = 0.45; %Init guess j = 0; error_str = 1; while error_str > DF_opt_old = DF_opt; SOL_opt = ( *abs(r_in*C_theta_inr_out*C_theta_out)/(rel_vel_in*(r_in+r_out)))*DF_opt_old^ ; DF_opt = 1-dH+abs(r_in*C_theta_inr_out*C_theta_out)/((r_in+r_out)*SOL_opt*rel_vel_in); error = abs(df_opt/df_opt_old-1); j = j+1; if j>20 error_str = 0; SONC_Hearsey = 1/SOL_opt; %##################### McKenzie solidity method ########################### % The McKenzie method is taken from "Axial Flow Fans and Compressors - % Aerodynamic Design and Performance", figure 4.3 on page 32. SONC_McKenzie = 9*(0.567-Cpi); 97

102 Appix B.4 Diffusion Factor and Diffusion Ratio % Diffusion ratios Axial Flow Compressor Mean Line Design %##################################################################### %## ## %## Diffusion ratios ## %## ## %## (c) Niclas Falck and Magnus Genrup 2008 ## %## ## %## Lund University/Dept of Energy Sciences ## %## ## %##################################################################### function [DF_lbl,Deq_star_lbl,Deq,Deq_star_ks]=Deq_star1(rel_ang_in,rel_ang_out,Cm_in,Cm_out,r_in,r_out,SONC,TONC,AVDR,Max) % rel_ang_in=55.77; % rel_ang_out=39.66; % Cm=167.64; % SONC=1/1.11; % TONC=0.06; % Max=167.64/ rel_vel_in = Cm_in/cosd(rel_ang_in); rel_vel_out = Cm_out/cosd(rel_ang_out); dh = rel_vel_out/rel_vel_in; %DeHaller C_theta_in = Cm_in*tand(rel_ang_in); C_theta_out = Cm_out*tand(rel_ang_out); gamma = (r_in*c_theta_in-r_out*c_theta_out)*2*sonc/(rel_vel_in*(r_in+r_out)); % Lieblein diffusion factor DF_lbl = 1-dH+abs(gamma)/2; % Lieblein equivalent diffusion ratio Deq_star_lbl = 1/dH*( *SONC*(cosd(rel_ang_in))^2*(tand(rel_ang_in)- tand(rel_ang_out))); % Equivalent diffusion ratio Deq = (1-dH+(0.1+TONC*( *TONC))*gamma)/dH+1; % Koch & Smith diffusion ratio Ap_star = ( *TONC/(SONC*cosd((rel_ang_in+rel_ang_out)/2)))*(2+1/AVDR)/3; rho_ratio = 1-(Max^2/(1-Max^2))*(1-Ap_star /SONC*gamma*sind(rel_ang_in)); Deq_star_ks = 1/dH*( *TONC *abs(gamma))*((sind(rel_ang_in) /SONC*gamma)^2+(cosd(rel_ang_in)/(Ap_star*rho_ratio)))^0.5; 98

103 Axial Flow Compressor Mean Line Design Appix B.5 Compressor losses %Wright & Miller Compressor Loss Model %##################################################################### %## ## %## Wright & Miller Compressor Loss Model ## %## ## %## (c) Niclas Falck Magnus Genrup 2008 ## %## ## %## Lund University/Dept of Energy Sciences ## %## ## %##################################################################### function[omega_p,omega_ew, K_Re]=WRTMLR(DF,DR,dH,rel_ang_out,rel_Mach_in,HONC,EPSONC,Re) %############################ Profile losses ############################## M_array = [ ]; %the different Machnumbers for each polynomial %%%%%%%%%%%%%%%%%%%%%%% polynomial coefficients %%%%%%%%%%%%%%%%%%%%%%%%%%% Coeff_Y2 = [ e e e e e e e e e e e e e e e-02]; %%%%%%%%%%%%%%%%%%%%%%%%%%% main calculation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Yp2_array = [0 0 0]; for i=1:3 for j=1:5 Yp2_array(i) = Yp2_array(i)+Coeff_Y2(i,j)*(DR)^(j-1); %calculates each segment/degree of the polynomial LP_p = interp1(m_array,yp2_array,rel_mach_in,'linear','extrap'); %interpolates for the value of M OMEGA_p = LP_p*2*dH^2*(1/cosd(rel_ang_out)); %############################ Endwall losses ############################## EPSONC_array = [ ]; %the different epsilon/c for each polynomial % %%%%%%%%%%%%%%%%%%%%%%%%polynomial coefficients %%%%%%%%%%%%%%%%%%%%%%%%% Coeff_Y1 = [ e e e e e e e e e e e e e e e e e e e e e e e e e01]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%% main calculation %%%%%%%%%%%%%%%%%%%%%%%%%%%% Yp_array = [ ]; for i=1:5 for j=1:5 Yp_array(i) = Yp_array(i)+Coeff_Y1(i,j)*(DF)^(j-1); %calculates each segment/degree of the polynomial if Yp_array(i) > 0.14 %checks if the value is to high Yp_array(i) = 0.14; 99

104 Appix Axial Flow Compressor Mean Line Design LP_ew = interp1(epsonc_array,yp_array,epsonc,'linear','extrap'); %interpolates for the value of EPSONC OMEGA_ew = LP_ew*dH^2*(1/HONC); %%%%%%%%%%%%%%%%%%%% Correction for the Reynolds number %%%%%%%%%%%%%%%%%%% if Re < 10^5 K_Re = 489.8*(Re^-0.5); elseif Re > 10^6 K_Re = 1; else K_Re = 13.8*(Re^-0.19); B.6 Blade angles % Blade angles %##################################################################### %## ## %## Blade angles ## %## ## %## (c) Niclas Falck and Magnus Genrup 2007 ## %## ## %## Lund University/Dept of Energy Sciences ## %## ## %##################################################################### function[incidence_angle,deviation_angle,camber_angle,attack_angle,stagger_angle,blade_a ngle_in,blade_angle_out] = Bladeangles(rel_ang_in,rel_ang_out,SONC,TONC,M) blade = 'DCA'; turning = rel_ang_in-rel_ang_out; %###################### Minimum loss incidence ######################### %######################################################################### %########################### i_0 ############################### i_0_10 = ( /SONC)+( /SONC)*rel_ang_in+(1.64e e- 4/SONC)*rel_ang_in^2; K_i_t = *TONC-122.3*TONC^ *TONC^3; if blade == 'DCA' K_sh = 0.7; elseif blade == 'NACA65' K_sh = 1.0; elseif blade == 'NTGEC' K_sh = 1.1; %####################### Mach # correction ####################### del_i_m = 10*(M-0.7); %########################## Camber influence ########################## n = ( /SONC)+( /SONC)*rel_ang_in-(3.79e e- 5/SONC)*rel_ang_in^2; %###################### Deviation@i = i_ref ######################### %######################################################################### del_0_10 = ( /SONC)+( /SONC)*rel_ang_in+( /SONC)*rel_ang_in^2; K_del_t = *TONC+36.61*TONC^2; 100

105 Axial Flow Compressor Mean Line Design Appix %####################### Camber influence ########################## if blade == 'DCA' m_prime = e-4*rel_ang_in-1.32e-5*rel_ang_in^2+3.16e-7*rel_ang_in^3; elseif blade == 'NACA65' m_prime = e-4*( *rel_ang_in)*rel_ang_in; elseif blade == 'NTGEC' m_prime = e-4*rel_ang_in-1.32e-5*rel_ang_in^2+3.16e-7*rel_ang_in^3; b = e-3*rel_ang_in+4.221e-5*rel_ang_in^2-1.3e-6*rel_ang_in^3; m = m_prime*sonc^b; %####################### Camber iteration ########################## camber = 1.2*(turning); %Start value z = 0; rel_error_level = 1; while rel_error_level > ; camber_old=camber; i_ref = K_i_t*K_sh*i_0_10-1+del_i_M+n*camber_old; del_ref = K_del_t*K_sh*del_0_10+m*camber_old; rel_blade_ang_in = rel_ang_in-i_ref; rel_blade_ang_out = rel_ang_out-del_ref; camber = rel_blade_ang_in-rel_blade_ang_out; rel_error_level = abs(camber/camber_old-1); z=z+1; if z > 10 rel_error_level = 0; error(z) = rel_error_level; camber_angle = camber; incidence_angle = i_ref; deviation_angle = del_ref; blade_angle_in = rel_blade_ang_in; blade_angle_out = rel_blade_ang_out; stagger_angle = rel_ang_in - camber_angle/2; attack_angle = rel_ang_in - stagger_angle; 101

106 Appix Appix C Axial Flow Compressor Mean Line Design A result file from the program LUAX-C based on the input parameters set in chapter 6 Result are shown on the following pages. 102

107 LUAX-C Version 1.0 Niclas Falck & Magnus Genrup Name: Comp 12-Feb :37:58 Type: Contstant Mean Radius Pressure ratio: 20 Number of stages: 15 Inlet massflow: 122 kg/s Rotational speed: 6600 rpm Ambient pressure: 1 bar Ambient temperature: 15 C ===== Compressor Performance ===== ================================================================================================================================================= Polytropic efficiency: % Isentropic efficiency: % Temperature rise: K Inlet tip: : kg/s specific massflow: kg/(s m^2) Compressor cost: Compressor power: MW ===== Koch Surgelimit ===== Stage Ch Ch_max Ch/Ch_max

108 ===== STAGE 1 ===== ================================================================================================================================================= PR REACT Flow PSI PHI dt0 ds Efficiencies Surge (Koch) Poly Isen acc.poly Ch Ch_max Ch/Ch_max point radius Velocities Angles---- tip rms hub U Cm C W C_theta Alpha Beta N/A N/A N/A point ---Pressure--- --Temperatures Enthalpy---- Entropy Cp Kappa rho Visc a Total Static Total Static Total Static e e e dehaller DF Deq Deq* Loss Coefficients Mach H/C S/C T/C H/T Lieblein Koch/Smith Endwall Profile relative critical rotor stator Nr.blades Blade angles inlet outlet incidence deviation camber stagger turning rotor stator ===== STAGE 2 ===== ================================================================================================================================================= PR REACT Flow PSI PHI dt0 ds Efficiencies Surge (Koch) Poly Isen acc.poly Ch Ch_max Ch/Ch_max

109 point radius Velocities Angles---- tip rms hub U Cm C W C_theta Alpha Beta N/A N/A N/A point ---Pressure--- --Temperatures Enthalpy---- Entropy Cp Kappa rho Visc a Total Static Total Static Total Static e e e dehaller DF Deq Deq* Loss Coefficients Mach H/C S/C T/C H/T Lieblein Koch/Smith Endwall Profile relative critical rotor stator Nr.blades Blade angles inlet outlet incidence deviation camber stagger turning rotor stator ===== STAGE 3 ===== ================================================================================================================================================= PR REACT Flow PSI PHI dt0 ds Efficiencies Surge (Koch) Poly Isen acc.poly Ch Ch_max Ch/Ch_max point radius Velocities Angles---- tip rms hub U Cm C W C_theta Alpha Beta N/A N/A N/A point ---Pressure--- --Temperatures Enthalpy---- Entropy Cp Kappa rho Visc a Total Static Total Static Total Static e e e

110 dehaller DF Deq Deq* Loss Coefficients Mach H/C S/C T/C H/T Lieblein Koch/Smith Endwall Profile relative critical rotor stator Nr.blades Blade angles inlet outlet incidence deviation camber stagger turning rotor stator ===== STAGE 4 ===== ================================================================================================================================================= PR REACT Flow PSI PHI dt0 ds Efficiencies Surge (Koch) Poly Isen acc.poly Ch Ch_max Ch/Ch_max point radius Velocities Angles---- tip rms hub U Cm C W C_theta Alpha Beta N/A N/A N/A point ---Pressure--- --Temperatures Enthalpy---- Entropy Cp Kappa rho Visc a Total Static Total Static Total Static e e e dehaller DF Deq Deq* Loss Coefficients Mach H/C S/C T/C H/T Lieblein Koch/Smith Endwall Profile relative critical rotor stator Nr.blades Blade angles inlet outlet incidence deviation camber stagger turning rotor stator

111 ===== STAGE 5 ===== ================================================================================================================================================= PR REACT Flow PSI PHI dt0 ds Efficiencies Surge (Koch) Poly Isen acc.poly Ch Ch_max Ch/Ch_max point radius Velocities Angles---- tip rms hub U Cm C W C_theta Alpha Beta N/A N/A N/A point ---Pressure--- --Temperatures Enthalpy---- Entropy Cp Kappa rho Visc a Total Static Total Static Total Static e e e dehaller DF Deq Deq* Loss Coefficients Mach H/C S/C T/C H/T Lieblein Koch/Smith Endwall Profile relative critical rotor stator Nr.blades Blade angles inlet outlet incidence deviation camber stagger turning rotor stator ===== STAGE 6 ===== ================================================================================================================================================= PR REACT Flow PSI PHI dt0 ds Efficiencies Surge (Koch) Poly Isen acc.poly Ch Ch_max Ch/Ch_max

112 point radius Velocities Angles---- tip rms hub U Cm C W C_theta Alpha Beta N/A N/A N/A point ---Pressure--- --Temperatures Enthalpy---- Entropy Cp Kappa rho Visc a Total Static Total Static Total Static e e e dehaller DF Deq Deq* Loss Coefficients Mach H/C S/C T/C H/T Lieblein Koch/Smith Endwall Profile relative critical rotor stator Nr.blades Blade angles inlet outlet incidence deviation camber stagger turning rotor stator ===== STAGE 7 ===== ================================================================================================================================================= PR REACT Flow PSI PHI dt0 ds Efficiencies Surge (Koch) Poly Isen acc.poly Ch Ch_max Ch/Ch_max point radius Velocities Angles---- tip rms hub U Cm C W C_theta Alpha Beta N/A N/A N/A point ---Pressure--- --Temperatures Enthalpy---- Entropy Cp Kappa rho Visc a Total Static Total Static Total Static e e e

113 dehaller DF Deq Deq* Loss Coefficients Mach H/C S/C T/C H/T Lieblein Koch/Smith Endwall Profile relative critical rotor stator Nr.blades Blade angles inlet outlet incidence deviation camber stagger turning rotor stator ===== STAGE 8 ===== ================================================================================================================================================= PR REACT Flow PSI PHI dt0 ds Efficiencies Surge (Koch) Poly Isen acc.poly Ch Ch_max Ch/Ch_max point radius Velocities Angles---- tip rms hub U Cm C W C_theta Alpha Beta N/A N/A N/A point ---Pressure--- --Temperatures Enthalpy---- Entropy Cp Kappa rho Visc a Total Static Total Static Total Static e e e dehaller DF Deq Deq* Loss Coefficients Mach H/C S/C T/C H/T Lieblein Koch/Smith Endwall Profile relative critical rotor stator Nr.blades Blade angles inlet outlet incidence deviation camber stagger turning rotor stator

114 ===== STAGE 9 ===== ================================================================================================================================================= PR REACT Flow PSI PHI dt0 ds Efficiencies Surge (Koch) Poly Isen acc.poly Ch Ch_max Ch/Ch_max point radius Velocities Angles---- tip rms hub U Cm C W C_theta Alpha Beta N/A N/A N/A point ---Pressure--- --Temperatures Enthalpy---- Entropy Cp Kappa rho Visc a Total Static Total Static Total Static e e e dehaller DF Deq Deq* Loss Coefficients Mach H/C S/C T/C H/T Lieblein Koch/Smith Endwall Profile relative critical rotor stator Nr.blades Blade angles inlet outlet incidence deviation camber stagger turning rotor stator ===== STAGE 10 ===== ================================================================================================================================================= PR REACT Flow PSI PHI dt0 ds Efficiencies Surge (Koch) Poly Isen acc.poly Ch Ch_max Ch/Ch_max

115 point radius Velocities Angles---- tip rms hub U Cm C W C_theta Alpha Beta N/A N/A N/A point ---Pressure--- --Temperatures Enthalpy---- Entropy Cp Kappa rho Visc a Total Static Total Static Total Static e e e dehaller DF Deq Deq* Loss Coefficients Mach H/C S/C T/C H/T Lieblein Koch/Smith Endwall Profile relative critical rotor stator Nr.blades Blade angles inlet outlet incidence deviation camber stagger turning rotor stator ===== STAGE 11 ===== ================================================================================================================================================= PR REACT Flow PSI PHI dt0 ds Efficiencies Surge (Koch) Poly Isen acc.poly Ch Ch_max Ch/Ch_max point radius Velocities Angles---- tip rms hub U Cm C W C_theta Alpha Beta N/A N/A N/A point ---Pressure--- --Temperatures Enthalpy---- Entropy Cp Kappa rho Visc a Total Static Total Static Total Static e e e

116 dehaller DF Deq Deq* Loss Coefficients Mach H/C S/C T/C H/T Lieblein Koch/Smith Endwall Profile relative critical rotor stator Nr.blades Blade angles inlet outlet incidence deviation camber stagger turning rotor stator ===== STAGE 12 ===== ================================================================================================================================================= PR REACT Flow PSI PHI dt0 ds Efficiencies Surge (Koch) Poly Isen acc.poly Ch Ch_max Ch/Ch_max point radius Velocities Angles---- tip rms hub U Cm C W C_theta Alpha Beta N/A N/A N/A point ---Pressure--- --Temperatures Enthalpy---- Entropy Cp Kappa rho Visc a Total Static Total Static Total Static e e e dehaller DF Deq Deq* Loss Coefficients Mach H/C S/C T/C H/T Lieblein Koch/Smith Endwall Profile relative critical rotor stator Nr.blades Blade angles inlet outlet incidence deviation camber stagger turning rotor stator

117 ===== STAGE 13 ===== ================================================================================================================================================= PR REACT Flow PSI PHI dt0 ds Efficiencies Surge (Koch) Poly Isen acc.poly Ch Ch_max Ch/Ch_max point radius Velocities Angles---- tip rms hub U Cm C W C_theta Alpha Beta N/A N/A N/A point ---Pressure--- --Temperatures Enthalpy---- Entropy Cp Kappa rho Visc a Total Static Total Static Total Static e e e dehaller DF Deq Deq* Loss Coefficients Mach H/C S/C T/C H/T Lieblein Koch/Smith Endwall Profile relative critical rotor stator Nr.blades Blade angles inlet outlet incidence deviation camber stagger turning rotor stator ===== STAGE 14 ===== ================================================================================================================================================= PR REACT Flow PSI PHI dt0 ds Efficiencies Surge (Koch) Poly Isen acc.poly Ch Ch_max Ch/Ch_max

118 point radius Velocities Angles---- tip rms hub U Cm C W C_theta Alpha Beta N/A N/A N/A point ---Pressure--- --Temperatures Enthalpy---- Entropy Cp Kappa rho Visc a Total Static Total Static Total Static e e e dehaller DF Deq Deq* Loss Coefficients Mach H/C S/C T/C H/T Lieblein Koch/Smith Endwall Profile relative critical rotor stator Nr.blades Blade angles inlet outlet incidence deviation camber stagger turning rotor stator ===== STAGE 15 ===== ================================================================================================================================================= PR REACT Flow PSI PHI dt0 ds Efficiencies Surge (Koch) Poly Isen acc.poly Ch Ch_max Ch/Ch_max point radius Velocities Angles---- tip rms hub U Cm C W C_theta Alpha Beta N/A N/A N/A point ---Pressure--- --Temperatures Enthalpy---- Entropy Cp Kappa rho Visc a Total Static Total Static Total Static e e e

119 dehaller DF Deq Deq* Loss Coefficients Mach H/C S/C T/C H/T Lieblein Koch/Smith Endwall Profile relative critical rotor stator Nr.blades Blade angles inlet outlet incidence deviation camber stagger turning rotor stator ===== OUTLET GUIDE VANE===== ================================================================================================================================================= radius Velocities tip rms hub Cm Pressure--- --Temperatures Enthalpy---- Entropy Cp Kappa rho Visc a Total Static Total Static Total Static e dehaller DF Deq Deq* Loss Coefficients Mach H/C S/C T/C H/T Lieblein Koch/Smith Endwall Profile relative critical Nr.blades Blade angles inlet outlet incidence deviation camber stagger