 Hello, and welcome to lecture number 11 of this lecture series on turbo machinery aerodynamics. And today we are on the 11th lecture of lecture series. And we are going to discuss a very important aspect of axial flow compressors today. And in today's class, we are going to talk about performance characteristics of single and multi stage axial compressors. And the reason why I said it is very important is because performance characteristics are extremely important information that are required by designers. And obviously, for the engine designers as a whole, because it is the compressor performance which determines the limits of operation of the engine itself, that is something which we are going to discuss in today's class. Which means that the limit of operation of an entire aero engine will in some sense be dictated by the compressor performance itself, that is beyond certain levels the compressor cannot operate. And that puts a limit on the engine operation itself. I am hoping that in the previous few lectures, 7 or 8 odd lectures which you have undergone so far, you must have had a fairly good amount of information about 2 dimensional as well as 3 dimensional aspects of flow through axial compressors. So today's lecture is kind of an overview of the whole thing. We thought of course looking at what is happening on the blade at individual stage level, but trying to find out how a certain compressor which has been designed is going to perform when the operating conditions change. So as the operating conditions of the compressor changes, how does the compressor react to such change in operating condition. So that is the aspect of discussion which we are going to take up in today's class. So we are going to talk about 2 distinct aspects which are related to performance characteristics of axial compressors. We will begin with the discussion on a single stage axial compressor performance and we will extend that to a multi stage axial compressor. The basic trends are identical for both of them, but the distinct differences between what happens in a single stage axial compressor as compared to a multi stage axial compressor. So these are 2 aspects of compressors performance that we are going to discuss in today's class. We will start with a single stage axial compressor. Now you might be aware that a stage of an axial compressor is constituted of a rotor and a stator. So rotor precedes the stator. So a rotor and stator combination together put together is what is known as a stage of an axial compressor and it is this that we are going to analyze in today's lecture on how we can estimate the performance of a single stage and how the single stage performance varies as the operating conditions change. So in today's class we will basically be talking about single and multi stage axial compressor characteristics. Let us start with the single stage axial compressor. So I mentioned that a combination of a rotor and a stator put together is a stage. So this is these 2 put together constitute a stage of an axial compressor and this is what we shall analyze and see how the performance changes as the operating conditions of this particular stage changes. So to understand that better let us construct the velocity triangles. Now this is the inlet velocity triangle for the rotor. The air is entering at an angle of C 1 absolute velocity. Relative velocity is V 1 which enters this blade at a tangential direction. Blade rotates at a peripheral speed of U. This is the velocity triangle at the inlet of the rotor and this is the velocity triangle at the exit of the rotor. Here velocity leaves the rotor at a velocity of V 2 which is again tangential to the rotor. C 2 is the absolute velocity leaving the rotor. U is the blade speed and C 2 is going to be the velocity with which the flow enters the stator. So the velocity leaving the stator is C 3. There is no relative component for the stator because the stator is stationary. Relative component is true only for rotating components like a rotor and that is why we have relative velocities at the inlet and exit of the rotor. So this is a typical axial compressor stage comprising of rotor and stator and we will see how we can estimate the performance or what are the parameters on which the performance of this single stage axial compressor would depend upon and for which we will need to analyze the velocity triangles a little more carefully. So if you look at the velocity triangles little closely and also mark the various velocity components on the velocity triangle, we get velocity triangle combination like this. So this is the velocity triangle at the inlet and exit put together. Since the blade speed is common for both the rotor entry and rotor exit that is the common point here. This is the inlet velocity triangle comprising of C 1 V 1 and angles alpha 1 and beta 1. At the exit we have C 2 V 2 and angles alpha 2 and beta 2. The corresponding velocity components are C w 1 which is the tangential component of C 1. C w 2 is the tangential component of C 2. V w 1 is tangential component of V 1. V w 2 is the tangential component of V 2. C a is the axial component of absolute velocity and delta C w is the difference between C w 2 and C w 1. Delta C w is important as we have seen earlier because the power required to drive the compressor primarily depends upon delta C w. So since the power required depends primarily on delta C w that is why in the analysis that will the change in tangential component will play a significant role. Now from these velocity triangles what we can see is the at the exit C w 2 should be equal to u minus C a tan beta 2 and C w 1 is C a tan alpha 1. Let us take a look at the velocity triangle. C w 2 is this component. C w 2 is u minus C a times tan beta 2 and similarly C w 1 which is this component is basically equal to C a tan alpha 1. And since the net change in enthalpy that is delta H naught is equal to u times delta C w. We have delta H naught as u times delta C w which is u minus C a times tan alpha 1 plus tan beta 2 or delta C w divided by u is equal to delta H naught by u square is 1 minus C a by u into tan alpha 1 plus tan beta 2. So what we see here is that we get a parameter which is in terms of the delta C w to the velocity ratio which is a function of certain angles which come from the velocity triangle tan alpha 1 and tan beta 1 the axial velocity and the peripheral speed. So these are certain parameters which will kind of influence this parameter of delta C w by u. The significance of which is what we will discuss very soon that how this ratio delta C w by u will play any role in the performance characteristics. So what we see from this equation that we have now derived is that the performance of a single stage axial compressor will depend upon certain set of parameters. One of them of course becomes is the axial velocity the blade speed and the angles. So this means that any change in the design mass flow rate obviously will affect axial velocity C a and change in rotor speed obviously affects u. So change in either C a or u will change the inlet angle beta 1. Let us go back to the velocity triangle here. So if either C a or u changes it will cause a change in beta 1 and that means that the blade performance is strongly a function of this ratio C a by u that is this ratio of axial velocity to the blade speed will significantly affect the blade compressor performance. And it can be deduced that the stage performance is a function of 3 parameters the loading coefficient psi the flow coefficient phi and the efficiency. That is there are 3 parameters which of course will also include the flow coefficient which is C a by u. Besides C a by u the performance will depend upon the loading coefficient and also it will depend upon the efficiency of the compressor. So there are 3 parameters significant parameters on which a single stage axial compressor performance will depend loading coefficient which is denoted by psi flow coefficient which is C a by u and the efficiency eta. So these are 3 parameters on which a single stage performance will depend. Now when you go to a multi stage as we will see later on there are a host of other parameters which will also be playing a significant role in the performance which of course we will discuss in later slides. Now if I were to plot the loading coefficient which was delta C w by u with reference to the flow coefficient and also the efficiency with reference to flow coefficient. Let us now look at how that affects the performance. Before I go to that let me also emphasize this particular point that the flow coefficient C a by u is in some sense a measure of the mass flow rate. Because mass flow rate is directly a function of the axial velocity C a and rotational speed of course being fixed the mass flow rate is directly proportional or flow coefficient is directly proportional to the mass flow rate. So as the compressor is throttled or mass flow rate is changed it changes the flow coefficient. Similarly for the same mass flow rate as its speed is changed that also changes the flow coefficient or C a by u. Similarly the loading coefficient will depend upon how this mass flow rate is changing. So there is a dependence of the loading coefficient delta C w by u on the flow coefficient. There is also a dependence of the compressor efficiency on the flow coefficient. So let us now take a look at how these two parameters change as flow coefficient changes. So if I were to plot let us take a look at loading coefficient first. So I have loading stage loading on the y axis which is delta H naught by u square which is also equal to delta C w by u. On the x axis we have C a by u which is the flow coefficient and we have already seen this expression we have derived this delta C w by u is equal to delta H naught by u square is 1 minus C a by u into tan alpha 1 plus tan beta 1. So there is a direct correlation between the loading coefficient and the flow coefficient. If you were to plot this variation you should expect it to be a linear line that is what is given by this dotted line here. One would have expected a linear variation of C w by delta C w by u with the flow coefficient. Of course in an actual practice what is shown here is given by the solid line that is the measured performance. You can see the measured performance is not necessarily a linear variation. It is equal to the design point at which the compressor has been designed which is C a by u design at which both the measured and the actual is the ideal are the same. At any other point you can see that the performance is different from what it has it is supposed to be. And if you were to draw at tangent or basically the actual performance the slope of this line is basically given by this angle tan alpha 1 plus tan beta 2 at the design point. So this is basically giving us the slope which comes from this equation here. So what is to be noted here is that as the flow coefficient changes from the design point it also affects the loading coefficient correspondingly. So there is a change in loading coefficient there is a deviation of the loading coefficient from the design as the mass flow rate or the flow coefficient changes. Now if we similarly look at the stage efficiency the stage efficiency also has a similar variation with reference to the ideal performance. Stage efficiency would have approached an efficiency of 1 for a C a by u design and since when it is actually operating the stage efficiency deviates from the design point and there is a variation in stage efficiency as compared to the design point variation. So there is also a difference of the stage efficiency or dependence of stage efficiency on the flow coefficient. Similarly there is a strong dependence of the loading coefficient on the flow coefficient. So here there are 3 parameters as we have seen which kind of dictates the single stage performance. The first parameter is the stage loading coefficient which also has a dependence on the flow coefficient phi. Then we also have the efficiency which again depends upon phi in some sense that as phi varies that affects the efficiency as well. So these are parameters which drastically affect a single stage performance. Let us now look at what happens as we change the mass flow rate or as the ratio C a by u deviates from the design point at the blade level how does it affect the performance. What happens to the rotor performance as C a by u changes from its design point. So for that we have the velocity triangles for 3 different cases which have been shown. One is a design condition and 2 off design conditions. In the design condition which is normal operation we have C a by u is equal to C a by u design. This is the rotor blade and we have flow entering the rotor at an angle of v 1 which is relative velocity at an angle of beta 1. So this is the velocity triangle at the inlet v 1 at an angle beta 1 to axial direction C 1 at an angle of alpha 1 and a blade speed of u. So this is when the flow coefficient is equal to flow coefficient at design condition. Now if let us say for a constant speed the mass flow rate is reduced then this means that this ratio C a by u also reduces and this is necessarily an off design condition. When C a by u is less than C a by u design that is axial velocity is now decreases for same u. So keeping u fixed if we reduce axial velocity because mass flow rate is reduced it leads to an increase in beta 1 and as beta 1 increases beyond a certain angle it leads to what is known as positive incidence flow separation. So there could be flow separation taking place on the suction surface of the rotor blade. So this is the suction surface of the rotor blade there will be flow separation from the suction surface when beta 1 is greater than beta 1 design which occurs when flow coefficient is less than flow coefficient design. The other a counterpart of this off design condition is the negative incident separation which will occur when C a by u exceeds the C a by u design that is when C a is greater than C a design for constant u beta 1 becomes very low and as beta 1 decreases there could be chances of flow separation from the pressure surface of the blade of the rotor blade you can see that flow has separated from the pressure surface of the rotor blade and this is basically a negative incidence flow separation. So both these cases of flow separation can occur either positive incidence or negative incidence separation when you know the flow coefficient is different from its design value when flow coefficient is lower than design value it leads to what is known as positive incidence separation when flow coefficient is greater than the design value it may lead to negative incidence flow separation. So this is how the performance of a single stage compressor can vary you have dependence on three parameters the loading coefficient the flow coefficient and the efficiency and as flow coefficient changes we have seen how it affects the performance of a rotor and there are two extreme cases possible you may have a positive incidence separation when the flow coefficient is much lower than the design value leading to flow separation from the suction surface of the rotor and negative incidence separation which occurs when the flow coefficient is greater than the design flow coefficient leading to flow separation from the pressure surface of the rotor blades and these are two different possibilities wherein the performance of the compressor can be drastically affected. You also seen how phi versus psi and loading coefficient versus flow coefficient changes and how efficiency changes with the flow coefficient. Now having understood the single stage performance characteristics we will now proceed towards a multi stage axial compressor and see how a multi stage axial compressor performance changes or varies as the operating conditions change or what are the parameters on which a multi stage axial compressor performance will depend upon. So multi stage compressor as we know consists of a series of stages of axial compressor which means you will have several combinations of rotor stator and if you put all of them together that constitutes a multi stage axial compressor and what we are going to do is that we will denote the inlet station of a multi stage compressor by station 1 and exit of the compressor by stator as station 2. Therefore, the overall pressure ratio of the compressor we will denote as p 0 2 by p 0 1 where p 0 2 is the compressor outlet pressure and p 0 1 is the compressor inlet pressure. So, the compressor outlet pressure p 0 2 and the efficiency will depend upon several physical parameters of variables. So, we are going to look at what are these different parameters of variables on which the performance will depend. So, we will list all these different parameters on which the performance of a compressor is going to depend. So, let us list all these parameters and what we see is that p 0 2 which is exit stagnation pressure and efficiency is a function of these many parameters and what are these parameters? We have mass flow rate, inlet stagnation pressure, inlet stagnation temperature, rotational speed omega, the ratio of specific heats gamma, the gas constant r, the viscosity of the air then the design itself and the diameter d. So, these are the different parameters on which the pressure ratio will depend and so now if we non-dimensionalize these parameters we can do that using Buckingham pi theorem or you probably have learned about pi theorem earlier on. So, if you non-dimensionalize this and express these parameters in terms of non-dimensional clusters then we have p 0 2 by p 0 1 and efficiency both of which are functions of these many non-dimensional parameters which is one of them is mass flow rate times square root of gamma r t 0 1 divided by p 0 1 d square then we have omega d divided by square root of gamma r t 0 1 then omega d square by nu then gamma and design. So, these are anyway non-dimensionalized. So, the other non-dimensional parameters are these three. Now, for a particular design we can safely assume that gamma and nu do not affect the performance significantly. Similarly, d and gas constant are fixed. So, since d is fixed gas constant r is fixed design is fixed nu is fixed and gamma are fixed these non-dimensional parameters can be simplified and expressed as p 0 2 by p 0 1 and efficiency are functions of m dot root t 0 1 by p 0 1 and speed as n by root t 0 1. So, here we have the pressure ratio and efficiency expressed in terms of two distinct parameters I will still not call them non-dimensional because if you look at the dimensional if you look at the dimensions of these two parameters they are strictly not non-dimensional that is because we have taken off other parameters which would have indeed made it non-dimensional p 0 2 by p 0 1 pressure ratio and efficiency are functions of two parameters one is mass flow rate times square root of t 0 1 by p 0 1 the other is speed divided by square root of t 0 1. And so that means that there is one parameter which is a function of mass flow rate other parameter which is a function of speed. So, pressure ratio and efficiency are functions of mass flow rate and the rotational speed we will further simplify this and see how the performance changes we will take a look at how the pressure ratio changes as mass flow rate changes and speed changes how efficiency changes as mass flow rate and speed changes. But the bottom line is that the performance of axial compressors multi stage axial compressor in terms of pressure ratio and efficiency can be expressed as functions of two distinct groups one is to do with mass flow rate that is m dot root p 0 1 t 0 1 by p 0 1. And the second parameter is n by root t 0 1 which does not make it non-dimensional because there were other parameters which we have neglected for a given design like the diameter d gas constant and so on. Now let us simplify this expression further what is normally done is that the temperature and pressures are expressed in terms of standard day pressure and temperature we will non-dimensionalize temperatures and pressure with reference to standard day conditions. Therefore, p 0 2 by p 0 1 and efficiency can be expressed as functions of m dot root theta by delta and n by root delta theta. Here theta should be equal to t 0 1 by t 0 1 standard day and delta is stagnation pressure divided by stagnation pressure of a standard day. So, here theta and delta refer to temperature and pressure ratios for a standard day. I will explain the significance of why this non-dimensionalization is also required for temperature and pressure basically because when we are designing a compressor for a particular condition what normally designs it for a certain ambient condition. But this compressor may be operating in an engine which is used in an ambient condition which is entirely different from what it has been designed for. So, what is the guarantee that this compressor is going to perform the same way as it has been designed for a different condition. So, the way to account for this is to express the pressure and temperature in a non-dimensionalized form and so usually this is also referred to as corrected pressure and corrected corrected mass flow rate and corrected speeds because mass flow rate has been corrected for the standard day pressure and temperature. So, even if let us say the engine is operating at a temperature and pressure which is drastically different from what it has been designed for because of this correction it can partly take care of this variation in pressure and temperature. So, with this background that we have had so far on how we can express the performance of multi stage compressors. We will now proceed towards looking at how the variation of a compressor performance itself is expressed. So, basically a compressor performance variation is also referred to as a performance map. So, compressor performance is expressed in the form of a map and performance map forms a significant part of very very significant role in the design of an aero engine or for that matter any gas turbine engine because as I mentioned a compressor performance puts limit on limits on the whole engine performance as a whole because there are certain limits of operation for a compressor beyond which it cannot operate or there are instabilities which are introduced. Therefore, an engine will not be able to operate satisfactorily beyond these operating ranges and that is why a compressor map forms a very significant input for an engine designer who will need to know what are these limits between which this compressor is going to operate. That is if I am designing an engine and I will need to know that this compressor has certain limits in terms of mass flow rate and pressure ratio. So, this is the amount of limit that I have during which between which the engine has to operate because beyond that the compressor cannot operate and therefore, the engine as a whole will not operate at all. And therefore, compressor performance map will form a significant input for an engine designer who will need to know what are these limits. So, compressor performance is expressed in terms of two parameters as a function of the mass flow rate. So, one of the parameters which we are interested is what is the pressure ratio developed by the compressor. So, the pressure ratio p 0 2 by p 0 1 expressed as a function of mass flow rate m dot root theta by delta. And similarly, what is this variation with different speeds as speed changes that is n by root delta or n by root theta how the performance changes. This is one of the parameters which we are interested in. The other parameter is the variation of efficiency with the non-dimensional mass flow rate and the non-dimensional speed how the efficiency changes. So, I will now show you one typical compressor map which basically tells us how this variation can be tracked. Now, if you look at what is shown here I have a typical compressor map which is shown here on the y axis we have p 0 2 by p 0 1 and efficiency x axis we have mass flow rate which is expressed in terms of m dot root theta by delta where this is the temperature standard day ratio and this is the pressure ratio standard day with reference to standard day. And there are series of curves which you will see and each of these lines correspond to constant speed line and the speeds are expressed in terms of n by root theta. And that is as the speed changes as you change the speed of the compressor from very low speed where this ratio will be close to 0 and as you keep increasing that and it approaches the actual speed the design speed which is 1. You can see that initially the speed lines are flatter. So, as you look at compressor performance at lower speeds very low speeds one would see a flat variation rather flat variation of the pressure ratio with mass flow rate that is as mass flow rate changes pressure ratio also changes, but that is over a wide range. But as the speed increases as we go towards higher and higher speeds the speed curves become sharper and sharper. For example, if you look at the speed curve at n by root theta is equal to 1 you can see that the speed curve is extremely sharp. And therefore, the pressure ratio varies drastically with mass flow rate, but that is over a very narrow range of mass flow rate beyond which there is certain line which is shown here a daughter line. Of course, there is a curve beyond this also, but that is a curve which a designer would never want his engine to operate on. I will explain what that curve is a little later. Now, what you notice is that as the speed increases from very low speeds and as we proceed towards higher and higher speeds the performance curve which was initially flat starts becoming sharper and sharper to the extent that at very high speeds that is the design speed of 1 the curve becomes very sharp which means that there is a very narrow range of operation of the compressor here. And the sharp curve here basically means that mass flow rate does not really influence or for the pressure ratio versus mass flow rate is kind of a constant here. And this basically refers to what is known as the choking point where you are trying to pass the maximum mass flow rate which this compressor can generate. So, beyond which mass flow rate does not change much and there is a significant drop in efficiency which I will come to a little later. So, what this line here means is that under this operating condition even if we change the mass flow rate substantially there is the variation in pressure ratio is very drastic for a very narrow change in mass flow rate beyond which the pressure ratio drops as you try to operate the compressor for mass flow rates beyond that because mass flow rate is fixed. You might recall the concept of choking which you would have learnt in your gas dynamics or in fluid mechanics that under certain operating conditions mass flow rate attains a peak level and mass flow rate becomes maximum. This is exactly the case that is happening here that maximum mass flow rate has taken place and no further mass flow rate can be passed through by this compressor. And if you try to pass more and more mass flow what will happen is that it will affect two parameters one is the pressure ratio which drops and also the efficiency which drops drastically. So, the curves which are shown here are the efficiency curves for the corresponding speeds. So, for lower speeds as expected one would see a flatter efficiency curve and as the speeds approach one the efficiency curves also becomes sharp just like the pressure ratio curves and you can see that the change in efficiency is very drastic and very narrow as the speeds approach the design speed. Now, on this each design line or in on each speed line you can see multiple points which have been shown here. These are the different operating points of the compressor. So, compressor may be operating on any of these points or in between these points and so when a designer embarks upon designing a compressor what he does is that he tries to design it for a particular operating condition and then one would also need to evaluate what happens when the compressor is going to operate in conditions which are different from what it has been designed for or what are known as off design conditions. For an aero engine for example, the design the operating conditions can vary so drastically that the designer has to ensure that even if the off design condition is it is extreme the compressor still operates safely because as you have seen here in the curve on the left hand side there is a dotted line which has been shown on the pressure ratio versus mass flow rate. There is a dotted line which is shown and it is indicated as surge line. Now, what is surge is something we will discuss in detail in a later lecture, but let me tell you that compressor operation is affected drastically affected by two instabilities which are likely to occur. One is known as a rotating stall and the other is known as surge. Both of these are instabilities which can drastically affect the performance of the compressor to the extent that if surge occurs the compressor may fail and lead to flame blow out in the combustion chamber and the engine may shut down if the compressor is surging. So, the left the dotted line shown on the left hand side of the pressure ratio versus mass flow rate curve is sort of a limit for operation of the compressor that is though if you extend that line you will still see a line on the left hand side, but that is a line on which you just cannot operate the compressor because that is a region of instability for the compressor. The compressor cannot operate in a stable manner if it is on the left hand side of what is known as the surge line. So, if compressor tries to operate on the left hand side it would undergo what is known as surge during which the entire operation of the compressor breaks down and engine as a whole gets affected drastically and it might lead to failure of the compressor and engine shut down. So, surge is a phenomenon which can affect the performance of a compressor drastically. So, let me take a closer take you to a closer look of what is the surge line and how it affects the performance. Now, the same performance curve that you saw here pressure ratio versus mass flow rate is being shown in a better way here. Now, here we have the speed lines which are these different lines which are shown the surge line is shown here and what is also shown is the engine operating line. Now, engine operating line is the line which is something that the designer would like to use and how does one at arrive at the operating line operating line is the line through which the engine is accelerated from 0 speed all the way to design speed. Operating line would ideally have to be a line where the efficiency is maximum because you would always like to operate the compressor in a condition where it has maximum efficiency. So, if you join all those points ideally if you join all the points of maximum efficiency you can get the ideal operating line, but it may so happen that many a times the maximum efficiency is very close to the surge line which is a risky affair because if you are operating very close to the surge line any off design operation might push your engine into surge that is a risk which the designers are not willing to take will not obviously be willing to take because that is too high a risk to be taken to operate the engine very close to the surge line. So, the engine designer always wants to keep a certain margin between the operating line and the surge line this is known as the surge margin. Surge margin is a certain margin or buffer which the designer wants to put for the engine to ensure that even if the engine undergoes an off design operation a transient operation the engine still does not touch the surge line because if the engine were to indeed go towards the surge and touch the surge line that can lead to catastrophic effects which are something which the designer will always want to avoid. So, surge margin is something which is kind of a protection for the engine provided by the designer to ensure that the engine does not reach the surge condition even if there is an off design operation of the engine and most of the modern day engines have inbuilt surge warning systems and mechanisms which will prevent a pilot from accidentally operating the engine in such a way that it can surge. So, the modern day computer which operates an engine which is also known as the FADEC that is the full authority digital engine control basically has inbuilt functions which will prevent a pilot from making such mistakes that will lead to surge of an engine and even if there is a possibility the surge warning sensors which will give a warning to the pilot saying that there is a possibility that the engine can surge if you operate it so and so. So, in this operating multi stage compressor performance map there are two distinct lines that we should be familiar with one is of course, the surge line the other is the operating line operating line is the line on which the engine is designed to be operating for. There is also another line which may not be that significant that is on the right hand side right most side you may also join all those points on the right hand side to achieve what is known as the choking line. Choking line is not really significant for a compressor, but it may be significant for a turbine which we will see later on because turbines usually operate under choked condition. So, that we will discuss a little later when we take up the turbines for a compressor the choking line is not really a matter of that concern, but there could still be a line which represents choking in an axial compressor. So, surge line and operating line two distinct parameters or lines that we need to be aware of. Now, based on these parameters that we have discussed we will now look at how the performance changes as let us say the mass flow rate changes which we have also seen for a single stage compressor. We have seen that as the flow coefficient changes from the design condition it drastically affects the performance of a stage. Let us also look at how flow coefficient changes can affect the performance of a multi stage axial compressor. Now, in multi stage axial compressor the problem is that a small departure from the design point in the first stage can cause progressively increasing departures from the design from the first stage onwards. That is a small reduction in the ratio C A by U at the design point at the first stage could lead to positive incident separation at the last stage. Similarly, a small increase in C A by U design could lead to negative incident separation in the last stage and the most extreme mismatching of these front and rear stages occurs during starting. Now, during starting what happens is that the compressor has just about begun rotating. So, as that rotation begins the net change in density across the compressor is not very significant. So, change in density from the first stage to the last stage is insignificant. So, there is hardly any change in the densities. So, what happens as a result of that is that as the density changes are not very high whatever changes occurs at the inlet can have a very significant effect on what is happening at the exit. So, there is a very significant effect of the flow from the inlet all the way to the exit and this is especially true during starting when the density development has not really taken place. This is the pressure ratio and the density changes have not really been initiated because the compressor has just about started. So, we will take a look at what happens during starting what happens to the flow conditions or velocity triangles for the inlet or the first stages and what happens to the flow velocity triangles at the exit or the last stages. So, I have two sets of velocity triangles here one corresponding to the first stages and one set corresponding to the last stages. What is shown here by the dotted line correspond to the design velocity triangle. So, these dotted lines are supposed to be the so called design velocity triangles and what happens is that in the first stages axial velocity to then the ratio of C a by u is lower than what it should be for design condition. And what happens in the last stages is that C a by u ratio is higher than what it is for the design condition because mass flow rate is fixed. Now, mass flow rate being fixed the area is fixed and axial velocity is the only parameter which can change because density is also fixed during starting. So, from the inlet to the exit if you have seen an axial compressor geometry the area progressively reduces right. From the inlet you have the inlet you have a large area and that area progressively reduces and you have a smaller and smaller area at the exit. Since the area is reducing density is fixed for a constant mass flow axial velocity has to increase and that is what is happening during starting. Now, once it starts and the compressor operates the density increases and this problem does not take place once the compressor as is fully operational. So, this is basically a problem which occurs during transient operation during starting of the compressor. So, what happens in this case is that there are few stages in the beginning where C a by u ratio is lower than the design ratio which means it first few stages may encounter positive incident separation and towards the last stages C a by u is greater than C a by u design and it may lead to negative incident separation in the last stages. So, there is a huge mismatch between the initial stages and the later stages and that happens just during starting. So, how can we overcome this problem? So, there are different ways of overcoming this problem and what basically happens is that a decreased C a with alpha 1 and beta 2 constant results in increased alpha 2 and beta 1 and similarly it leads to an increased loading in both rotor and stator. Similarly case of increased C a it leads to an opposite defect. So, there are different ways in which designers have configured for self starting of compressors. One of the ways is to use bleed valves which will allow some of the incoming air to escape. So, this is the most common way one of the common ways of elevating this problem that is use a bleed valve somewhere midway between the multi stage compressor that will allow some of the mass flow rate to escape. So, as the mass flow rate escapes later stages will have lower mass flow and therefore, it will have C a by u ratios which are closer to the design value and therefore, the problem of extreme mismatch between the initial stage and the later stage does not really happen. Other way of course, is to use variable guide vanes which can change the inlet angles flow angles to ensure that the flow is matched to the design C a by u values and the third option is of course, to use multi spooling which is probably the most common thing which is used now that is you split the compressor into different stages. So, that the later stages operates at different speeds as compared to the initial stages which is why most of the modern day engines have multiple spools it could be twin spool or a three spool engines and therefore, you have initial fan followed by a low pressure compressor and then a high pressure compressor which are driven correspondingly by different stages of turbines. So, these are different ways of trying to ensure that the compressor operation especially during starting is also taken care of and there are no extreme mismatches taking place between the first few stages and the last stages of a multi stage axial compressor. Now, besides this I mentioned that there are two distinct problems or instabilities which can affect a compressor performance one is known as rotating stall as I mentioned and the other is known as surge. Now, we will discuss in detail about rotating stall and surge in a later lecture. Let me just give you a quick introduction to what is meant by rotating stall and rotating and surge. Rotating stall is basically a non axisymmetric phenomenon and it is a periodic. So, it is not a periodic phenomenon surge on the other hand is axisymmetric which means it affects the entire annulus of the compressor and it is periodic. Rotating stall is basically a progression around the blade annulus of a stall pattern in which one or more adjacent blade passages are instantaneously stalled and then cleared for unstalled flow as the stall cell progresses. Rotating stall obviously causes alternate loading and unloading of blades may be leading to a fatigue failure, but it is not fatal. Surge on the other hand is a low frequency oscillation of the entire annular flow and it is periodic, but it leads to it is a kind of a fatal phenomenon for the engine because onset of surge can almost always lead to engine failure and therefore, surge is much more severe phenomenon as compared to rotating stall. Now, let me just go to the pressure ratio versus mass flow characteristic to explain this little bit more. Now, in this pressure ratio versus mass flow characteristic on the left hand side that you see here is what is meant by surge. Even though theoretically there is a curve which extends if you draw this curve on the left hand side will still extend that way, but practical limitations in terms of initiation of surge will prevent compressor from operating on this side of the curve and that is why this is known as the surge line and it is important that we be clearly demarcate between the surge line and the engine operating line and so that is why there is a clear difference between engine operating line and the surge. Rotating stall is often considered to be a precursor to surge that is as the engine approaches the surge line there is a possibility that rotating stall initiates and rotating stall when allowed to proceed further and propagate further can evolve into surge and the engine can eventually lead to surge, but rotating stall is something which can be prevented and controlled, but surge once initiated is very difficult to control and that is some that is why designers would also always want to avoid that the engine when approaches surge. So, surge is something that will need to be avoided and under all circumstances and that is why it is important that is designer understands the surge margin variable. We will discuss details of surge and stall that is instabilities in detail in one of the later lectures. So, let me conclude today's lecture where we have discussed about very important aspect of performance characteristics of axial compressors. We began our lecture today with discussion on the performance of single stage characteristics and extended that to multi stage characteristics. We have seen that pressure ratio for single stage it is the loading coefficient versus the flow coefficient and efficiency versus flow coefficient that one would be interested in. In a multi stage characteristic we have pressure ratio versus mass flow rate non dimensional mass flow rate as a function of different speeds which is also non dimensionalized with the temperature and efficiency versus mass flow rate at different speeds. So, these are different parameters based on which one can construct the performance characteristics or performance map of multi stage axial flow compressors. So, these were the topics that we have discussed in today's lecture. So, I hope you have been able to grasp some of the effects of aspects of performance characteristics and what is the significance of performance map of an axial flow compressor. So, these were topics which we had discussed in today's lecture and we will continue discussion on instabilities and inflow conditions and their effect on performance of axial compressors in future lectures.