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