 Hello and welcome to lecture number 21 of this lecture series on turbo machinery aerodynamics. We have been discussing about turbines and in particular axial flow turbines in the last few lectures. And we have had a chance to discuss about quite a few things about axial turbines. We started off with the very basics of turbines in general which is basically to do with the thermodynamics of turbines. And after having a very detailed discussion on the thermodynamics of flow through turbines. We have also discussed about the types of turbines. And we have seen that there are basically three types of turbines possible. One is the axial flow turbine, the radial flow and the mixed flow turbines. And we have been talking about the axial flow turbines in the last couple of lectures. And in the last class we have discussed about the fact that there are basically two configurations of axial turbines which are possible. One is the impulse turbine and the other is the reaction turbine. And when I discussed about these two different types of turbines, I happen to mention that the reason why there are these distinct classes of turbines is the fact that the way stagnation or enthalpy drop takes place in a turbine is different in these two different types of turbines. Now, in a turbine stage, we know that turbine stage basically consists of a stator or a nozzle followed by a rotor. There is certain amount of pressure drop taking place in the nozzle and there could be certain amount of pressure drop taking place in the rotor as well. But there are certain types of turbines where the entire pressure drop or the enthalpy drop takes place only in the nozzle and the flow simply undergoes a turn as it passes through the rotor. So, these types of turbines are called impulse turbines. And there are also types of turbines where the pressure drop or enthalpy drop is split or shared by both the nozzle as well as the rotor. And these are known as reaction turbines. So, these are the two different or distinct types of turbines which we had talked about. In today's class, when we begin with, we will basically be talking about a parameter which can be used as a parameter to distinguish between these two types of turbines. So, that is one of the main things that we are going to discuss about. We will start the lecture with discussion on what is known as degree of reaction. After that, we will be talking about the losses encountered in turbine, what are the different forms of losses. And subsequently, we will also be talking about the efficiency or different forms of efficiency in a turbine. Now, when we talk about losses, there are well we have already had a very detailed discussion on well 2D as well as 3D losses in the context of axial compressors. It is the exact same concept which can be also used in a turbine. Therefore, I will not really go into the details of the different types of losses and how it can be estimated and so on because we have already talked about that in the case of compressors. We can simply extend that to turbines. And so, I would rather not repeat the same thing here. But I will of course, go through the essentials of losses in a turbine and also how it can be estimated in a very generic sense and without going into too much of details because they have already been covered in compressors. And then we will talk about efficiency and what are the different types of efficiencies in fact, in turbines you can have different forms of efficiency. There are at least 4 or more different types of efficiencies that can be defined for a turbine. But we will restrict our discussion to 2 forms of efficiencies which are more commonly used. One is known as the total to static efficiency and the other is known as the total to total efficiency. There are also static to static and there are other types of efficiency which are not really used commonly and so we will not discuss those in a great detail. So, let us start our discussion with degree of reaction and what we mean by degree of reaction and how it can be used in a turbine. Now, degree of reaction as you have seen in the case of compressors is a concept which is used to kind of understand how much amount of work sharing is done by the rotor as a comparison of the entire work done in a stage. And so in the context of a turbine here the flow basically undergoes acceleration as you already know by now that it is an accelerating flow in a turbine and acceleration takes place both in the nozzle as well as in the rotor. And therefore, as a consequence of that there is an enthalpy drop taking place both in rotor well in the nozzle as well as the rotor. Degree of reaction gives us some idea about well it is basically an indicator of the amount of enthalpy drop that is taking place in the rotor or in the rotor as a as compared to the enthalpy drop taking place across the entire stage. So, that is the basic significance of degree of reaction. So, but before we go into details of degree of reaction let us take a look at the typical velocity triangle which I had discussed in detail in the last class. Let me quickly recap what this velocity triangle means. This is a typical stage of an axial turbine which consists of a nozzle and a rotor. So, flow enters the nozzle at an absolute velocity of C 1 which is at an angle of alpha 1 leaves the nozzle at velocity of C 2 which is the absolute velocity and making an angle of alpha 2 with the axial direction V 2 is the relative velocity which makes an angle of beta 2 with the axial direction. And then flow from this enters into the rotor and leaves the rotor with the relative velocity of V 3 making an angle of beta 3 with the axial direction and C 3 which is the absolute velocity makes an angle of alpha 3 with the axial direction. The blade speed in both at the inlet as well as the exit of the rotor is assumed to be the same and equal to u. So, this is a very typical or a generic velocity triangle applicable to any axial flow turbine. And so our definition of degree of reaction is with reference to since it is with reference to a very generic axial turbine it can be used in the case of impulse turbines as well as for reaction turbines. And what we will see very soon is that impulse turbine is a special case of a 0 degree of reaction turbine that is when the degree of reaction is 0 then that turbine refers is basically an impulse turbine. So, as I had mentioned earlier degree of reaction is defined as static enthalpy drop in the rotor divided by stagnation enthalpy drop in the stage. So, if you look at the rotor these are stage it is between station 2 and 3. So, static enthalpy drop is h 2 minus h 3 divided by h 0 1 minus h 0 3 that is for the stage well of course, we can always say that h 0 1 is also equal to h 0 2 because in the stator there is no enthalpy change stagnation enthalpy change. Now, so if you look at a coordinate system which is fixed on the rotor or in the relative frame of reference the apparent stagnation enthalpy is basically a constant. And so we have h 2 minus h 3 is equal to b 3 square by 2 minus v 2 square by 2. So, if the axial velocity is assumed to be the same upstream and downstream of the rotor then this can be reduced to h 2 minus h 3 which is stagnation or static enthalpy drop in the rotor is one half of v w 3 square minus v w 2 square which is half of v w 3 minus v w 2 multiplied by v w 3 plus v w 2. We also know that the stagnation enthalpy change across a stage which is given by h 0 1 minus h 0 3 is basically a function of the blade speed and the change in the tangential component of the absolute velocity that is u times delta c w is basically the stagnation enthalpy drop. Therefore, h 0 1 minus h 0 3 is also equal to u times c w 2 minus c w 3. So, let us simplify the degree of reaction here. So, degree of reaction would become v w 3 minus v w 2 multiplied by v w 3 plus v w 2 divided by 2 u into c w 2 minus c w 3. Now, if you go back to the velocity triangle let us go back to the velocity triangle here if you look at the components or the difference between c w 3 and c w 2 that is basically equal to the difference between v w 2 and v w 3 that is in their tangential direction. Therefore, v w 3 minus v w 2 is basically c w 3 minus c w 2 and so, this degree of reaction will basically reduce to minus v w 3 plus v w 2 divided by 2 u. Now, from the velocity triangle you can also see that v w 3 which is the tangential component of relative velocity at the exit of the rotor is c a times tan beta 3. Similarly, v w 2 is c a tan alpha 2 minus u. So, degree of reaction basically reduces to half of 1 minus c a by u tan alpha 2 plus tan beta 3. So, this is one form of defining the degree of reaction that you can relate degree of reaction. We have seen this definition even for compressors and we have also seen that degree of reaction is a function of a few parameters. One of them of course, it is the ratio of axial velocity to the blade speed c a by u besides that there are the angles alpha 2 and beta 3 in this case. So, it is a function of the angles as well as the axial velocity and the blade speed. So, we can also simplify this in the sense that if you look at 0 degree of reaction and also look at 50 percent degree of reaction turbine, we will see what are these special cases of axial turbines where we can look at an impulse turbine and a 50 percent degree of reaction turbine. So, degree of reaction starting from the fundamentals it is basically ratio of static enthalpy drop in the rotor divided by stagnation enthalpy drop in the stage which we can simplify as we have seen and relate degree of reaction to the flow coefficient which is c a by u and the angles in this case it is the absolute angle at the inlet of the rotor and the relative angle or the blade angle at the exit of the rotor. So, it is tan alpha 2 plus tan beta 3. So, let us look at some special cases of the degree of reaction. Now, if you look at a symmetrical velocity triangle configuration where alpha 2 is equal to minus beta 3, what we will see is that we get the degree of reaction as 0.5. So, this is known as a 50 percent degree of reaction turbine, we have seen this in the previous lecture as well. Another special case is when V w 3 is equal to minus V w 2 then we get degree of reaction as 0 which is basically an impulse turbine. So, if we were to look at an impulse turbine little more carefully and compare that with the 50 percent stage. For a given stator outlet angle that is alpha 2 the impulse turbine stage requires a much higher axial velocity than the 50 percent reaction stage. In the impulse turbine it is generally seen that all the flow velocities are higher and therefore, it is generally also seen that the efficiency of an impulse turbine is usually lower than that of a 50 percent reaction stage for 2 turbines which are generating the same power. That is of course, a general observation that because of velocity components are higher the losses are likely to be higher and therefore, efficiency is usually slightly lower than that of a 50 percent reaction turbine stage. So, let us look at these 2 special cases. This is the impulse turbine stage we can see that the V w 3 or V w 2 will be equal to minus V w 3. If that is so then degree of reaction becomes 0 and such a turbine is an impulse turbine stage. So, we have V w 3 and V w 2 which are opposing each other and equal in magnitude. That is when we have degree of reaction as 0 and what is the physical implication of this in degree of reaction 0 it means that there is nothing much happening in the rotor as far as enthalpy drop is concerned. The rotor simply deflects the flow and there is no change in the enthalpy in the rotor and that is why degree of reaction is 0 because if there is no change in enthalpy in the rotor the numerator is 0 degree of reaction is 0. Now, if you look at a 50 percent reaction turbine stage then we have the angle it is a symmetrical velocity triangle and therefore, alpha 2 will be equal to beta 3. So, if those angles are equal in magnitude then you get velocity triangles which are symmetrical you can see that this is the velocity triangles are basically mirror images what you have at the inlet is mirrored at the exit. So, that is why you have a symmetrical or a mirror image velocity triangle if the reaction turbine is a 50 percent reaction stage and what it means is that the enthalpy drop is shared equally between the rotor and the stator and that is what a 50 percent reaction stage basically means. So, what we have defined in the last few minutes is this very important concept of degree of reaction where which basically tells us the amount of enthalpy drop which is shared between the rotor and the stator and how we can you know use that as a parameter to distinguish between these two types of turbines impulse turbine where you have degree of reaction as 0 which means that there is no enthalpy drop taking place in the rotor and we have seen that such a velocity triangle we have the tangential component of relative velocity V w 3 is equal to minus V w 2 they are equal in magnitude, but they are opposite in direction and that is why in the velocity triangle you can see that they will oppose each other when you take up their components. 50 percent reaction stage velocity triangles are symmetrical and you have alpha 2 is equal to minus beta 3. So, these angles are same making the velocity triangles symmetrical or mirror images across the rotor. So, now that we have discussed about degree of reaction let us move on to a very important aspect of performance of turbines that is the efficiency. I think I mentioned in the beginning of the lecture that efficiency in the case of turbine unlike in compressors we have in turbines defined in different ways basically depending upon the application for which the turbine is being used. Now, there are certain applications let us say in a land based gas turbine power plant where you are generating you are using a gas turbine to generate power. So, here the application is such that you would not want the turbine exhaust to have any very high levels of kinetic energy because that is getting wasted. So, you would like to use up as much kinetic energy as possible from the turbine itself without having to waste kinetic energy. So, here we would like to expand it to the minimum possible enthalpy static enthalpy and therefore, any kinetic energy that is there at the exit is considered a waste. So, in such turbines we usually define efficiency in the form of what is known as total to static enthalpy and the other form of enthalpy that we are going to define is known as total to total enthalpy which is what is of interest to aero engineers because in a gas turbine engine which is used in an aircraft for example, the there is enough kinetic energy available at the turbine exhaust which can be again expanded or further expanded through a nozzle to generate thrust and you would not want the turbine exhaust to get or turbine to exhaust itself to the minimum possible kinetic energy because you would also like to expand the flow further in a nozzle to generate thrust. So, in such applications you one would prefer to define efficiency based on total to total or stagnation enthalpy. So, these are the two commonly used forms of efficiencies as I mentioned there are also other forms of efficiencies which are not very commonly used like static to static and so on. We will restrict our discussion to these two types of efficiencies total to static and total to total efficiencies. Now, so some general comments which I had meant let me just list them down here. So, the aerodynamic losses in a turbine as we have seen differ with stage configuration or the degree of reaction and so improved efficiency is associated with a higher amount or level of reaction which implies less work per stage and therefore, higher number of stages for a given overall pressure ratio. So, the reason why we need to understand efficiency or the sources of losses is that it firstly helps us in making a choice between different configurations either impulse or a reaction, but the other advantage is that it will also tell us how one can control these different forms of losses. So, based on our understanding we can define two types of efficiencies total to static efficiency and the total to total efficiency and which efficiency definition to use will basically be determined by the application for which the turbine is being used. So, in let us say the land based power plant as I mentioned the turbine output is basically in the form of shaft power that is the turbine is connected to a generator which generates work out electricity and therefore, exhaust kinetic energy is basically considered as a loss. Therefore, in such a case the ideal turbine process would be isentropic such that there is no exhaust kinetic energy that is the exhaust itself is static and there is no kinetic energy associated with that exhaust and that is where we would define what is known as total to static efficiency. In aero engines the turbine exhaust is required to have certain amount of energy which will further be expanded in a nozzle to generate thrust. So, there you would not want to expand the turbine to such a level that it is static at the exit and then with very little kinetic energy, but you would like that some more kinetic energy left which can be expanded further in a nozzle. So, there we would normally define the total to total efficiency in such applications. So, let us take a look at a general turbine process or expansion through a turbine and then we will come up with the efficiency definitions. So, this is an expansion process in a turbine stage well where station one is the nozzle entry, two is nozzle exit and three is the rotor exit. So, two is also the rotor inlet. So, the flow initially has pressure at the inlet stagnation pressure at p 0 1 and static pressure at the entry is p 1. So, p 0 1 plus p 1 plus the dynamic head gives us p 0 1. So, we have flotted this on a temperature entropy scale. Now, if this entire process were to be isentropic then the expansion takes place along these dotted lines. So, p 0 1 all the way up to the exit which is 3 s if it were if you are considering a static condition at the exit and. So, the actual turbine process of course, is defined by this solid line the bolt line here between static pressure p 1 static pressure p 2 at the nozzle exit or rotor entry and p 3 at the rotor exit. The corresponding stagnation pressure at the rotor exit is p 0 3 which is basically what you have the temperature at station 3 plus the dynamic head c 3 square by 2 c p will give us the stagnation temperature there. Now, the corresponding conditions in the isentropic case would be t 0 3 subscript s or at the rotor exit which is the stagnation. And so, when we are defining efficiency in two different ways that we are going to discuss about. Let us first take up the total to static efficiency. Now, in this case we are talking about an ideal turbine work with no exhaust kinetic energy which means that we have expanded all the way up to the station which is given by this particular state. So, from 0 1 all the way up to the station and which means there is no more kinetic energy at the exit of the turbine. So, we have the ideal turbine work in this case would be c p times t 0 1 minus t 3 s. So, the total to static efficiency in this case is defined as the it is denoted by symbol eta t s which is total to static t 0 1 minus t 0 3 divided by t 0 1 minus t 3 s that is t 0 1 minus the temperature corresponding to this t 0 3 divided by t 0 1 minus t 3 s. So, that is the total to static efficiency. The denominator we are going to simplify because this is an isentropic temperature here. So, this can be expressed in terms of the corresponding pressure ratios. And so, we have t 0 1 minus t 0 3 at the numerator divided by t 0 1 into 1 minus p 3 by p 0 1 raise to gamma minus 1 by gamma this follows from the isentropic relation. So, this is basically 1 minus t 0 1 by t 0 3 divided by 1 minus p 3 by p 0 1 raise to gamma minus 1 by gamma. So, this is the basic definition of total to static efficiency. Now, if you look at applications typical application being turbojet engines where the exhaust kinetic energy is not really a loss it can be converted to thrust using a nozzle. So, in such cases the ideal turbine work is not equal to the static conditions at the exit, but this stagnation conditions. So, the ideal work in such cases would be C p times t 0 1 minus t 0 3 s. And therefore, we define total to total efficiency which is eta t t that is t 0 1 minus t 0 3 divided by t 0 1 minus t 0 3 s. And again the denominator we will express in terms of pressure ratios because that is isentropic. We have 1 minus t 0 3 by t 0 1 divided by 1 minus p 0 3 by p 0 1 raise to gamma minus 1 by gamma. So, we have defined two forms of efficiencies the total to static efficiency and the total to total efficiency. We can also now relate these two types of efficiencies and see how these efficiencies can be compared for the same type of or for the same configuration. If you were to compare these two different forms of efficiency of course, with certain assumptions we can still compare total to static efficiency and total to total efficiency. And we will also see how using these efficiencies we can calculate work done by a given turbine. So, if you are to make an approximation that t 0 3 s minus t 3 s is approximately equal to t 0 3 s minus t 3 is equal to c 3 square by 2 C p which is let me go back to the diagram here. So, what we are saying is the difference between t 0 3 s minus t 3 s and this and the t 3 s is c 3 square by 2 C p. So, which means that effectively t 0 3 s and t 3 they are not much different as you can see from this t s diagram itself. So, it is a very much a valid assumption that we can make. And so, if this were to be the case if you make this assumption then we can relate total to total efficiency as equal to eta t s divided by 1 minus c 3 square multiplied by 2 C p t 0 1 minus t 3 s. So, what you can see here is that if this assumption were to be true and you calculate the total to total efficiency and total to static efficiency for a turbine. We could see that the total to total efficiency is likely to be greater than the total to static efficiency which is also obvious from the t s diagram that I had shown. If you look at the expansion process for the same turbine if you calculate both these efficiencies the total to total efficiency is likely to be higher than the total to static efficiency. Now, so using these definitions one can also calculate or make use of these definitions to calculate the corresponding work done by the turbine depending upon the application itself. So, if you were to use the total to total efficiency then we have the work done or specific work done as in the case where let us say in an application of a gas turbine engine used in an aero aircraft engine like in a turbo jet. Then the work done by the turbine is related to the efficiency which is total to total efficiency multiplied by C p into t 0 1 into 1 minus p 0 3 by p 0 1 raise to gamma minus 1 by gamma. And similarly the work specific work related to the total to static efficiency as eta t s into C p t 0 1 1 minus p 3 divided by p 0 1 raise to gamma minus 1 by gamma. So, using the efficiency definitions and the specific application for which these efficiencies have been defined for we can use these efficiencies to calculate the corresponding work done by the turbine under these are different applications. So, let me give you one example to just indicate the effect of reaction. I think I mentioned when I was talking about impulse and reaction turbines that both these turbines in the last class as well I mentioned that there is a difference in the specific work done and loading that both of these different types of turbines can handle. And also the fact that there is a certain difference in the efficiencies that one would get by using these two different configurations of turbines. So, let us take a look at the influence of loading on the efficiency we will in this case calculate total to static efficiency. So, you have reaction on the x axis the efficiency total to static efficiency on the y axis and three different values of loading. So, one can see that as you increase the loading and keep changing the reaction what happens to the efficiency. So, let us take a look at one of these cases let us say loading factor is equal to 1. So, as you change the reaction on the extreme right we have an impulse turbine which has a reaction of 0. So, as we start from an extreme which is an impulse turbine and we move towards let us say 50 percent reaction case. You can see that there is a steady increase in the efficiency and after that of course, there is a drop in the efficiency. This is for loading factor is equal to 1. Now, if you look at a loading factor greater than 1 that is say loading factor of 2 or 3 then the trends are slightly different. In fact, you get the highest efficiency when the reaction is equal to 0 that is for an impulse turbine stage that is with higher amounts of loading your impulse turbine stage has a better efficiency than any other case of reaction. Because the moment you have any amount of reaction it is no longer an impulse it it basically becomes a reaction turbine and that is also true for higher values of loading between 2 and 3 and so on. And so, this is just to give some idea about what happens as we change the amount of loading with increased levels of loading how does reaction influence the efficiency. This is also linked to a comment I had made earlier I would want you to think about why is it that as you increase the loading an impulse turbine seems to at least perform better in terms of efficiency. And what is the effect of increasing loading on let us say the efficiency of the turbine as you keep changing the level of reaction from impulse which has 0 reaction. Let us say to 50 percent reaction where the reaction is the enthalpy drop is equally shared by the nozzle and the rotor. So, just give it a thought on why there should be a drop in efficiency as you move from impulse towards higher levels of reaction. So, let us move on to the next topic we have for discussion in today's class that is to do with losses in a turbine. I mentioned in the beginning that I will restrict the discussion to just the basics of losses because I have already had a detailed discussion on losses when we are talking about compressors. And so, most of the concepts we had discussed there is applicable for the turbine as well. Of course, the magnitude of losses will be quite different for compressors and turbines, but the concept is still the same. So, I will not repeat the estimation of losses that we had discussed in detail with reference to a compressor because it is also applicable for a turbine. Now, when I had discussed about compressors and losses in a compressor I had mentioned that there are distinct forms of losses there are basically we could classify losses as 4 sets of losses. One is on account of viscous effects are known as the viscous losses. Then there are 3 dimensional effects like triplicate flows and secondary flows. One may have shock losses and also mixing losses. And so, if you were to isolate these losses if because if you have to estimate losses in a turbine and one would like to target let us say different forms of these losses and see if we can minimize these losses. One would need to know let us say what is the contribution of viscous loss, what is the contribution of 3D losses like secondary flows or triplicate flows and so on. But it is not very easy to segregate these different losses. There are empirical correlations for estimating all these different forms of losses. We had discussed some of them in the context of compressors. One could extend the same for turbines as well. Total losses in a turbine obviously is some total of all these different forms of losses whether it is viscous loss or 3D losses which includes secondary flows and triplicate flows, shock losses and the mixing losses. So, let us look at these losses in little more detail, but not too much as I had mentioned. Just some preliminary discussion on these losses. If you look at viscous losses there are again different components of viscous losses. There is one on account of the profile or the nature of the airfoil cross section and that is known as the profile loss. Annual loss loss would refer to growth of boundary layer along the axis and end wall losses on account of boundary layer effects in the corner or junction between the blade surface and the casing or hub. Now, in 3D effects we have secondary flows which is on account of flow through curved blade passages, triplicate flows which is basically the flow leaking from the pressure surface to the suction surface. And what is generally observed is that if you look at the 3D effects the losses are likely to be higher for a turbine primarily because of the fact that the flow turning is much higher in a turbine as compared to a compressor. Secondary flows for example, is directly related to the amount of flow turning and if you compare a compressor with that of a turbine the flow turning in a typical turbine blade is much higher than that of a compressor. Secondary flows are likely to be much higher in the case of turbine. This is also true for the tip leakage flows basically because tip leakage is on account of the difference between the pressure surface and the suction surface and blade loading is usually much higher in a turbine than in a compressor. And therefore, leakage flows are also likely to be higher in a turbine. And what complicates the matter in a turbine is the fact that you also have a higher temperature and it is no longer just pure air you also have a combustion product coming in from the combustion chamber which might complicate the flow behavior in the case of a turbine. Now, let me just give you one example of profile loss I will as I mentioned I will not go into details of estimating all these losses we have done that for the compressor and you could easily extend that to the turbine as well. Now, if you look at let us say the profile loss and look at what happens as you keep changing the incidence. Now, I have these profile loss distribution for two distinct cases the impulse turbine and the reaction turbine. The solid line refers to the impulse turbine and the dotted line is for the reaction turbine. So, one can see that there is a significant difference between what happens in an impulse turbine and reaction turbine the losses as you can see are much higher for an impulse turbine case and that varies significantly with the incidence. So, the sensitivity of impulse blades to incidence is much higher especially a positive incidence you can see that after around 8 or 9 degrees the losses increase substantially. There is a very sharp increase in losses that positive incidence around 8 or 9 degrees. Of course, this is for a very typical case of a turbine blade. On the other hand if you look at a reaction blade it is probably little better adjusted to higher changes in incidence of course with very high incidence exceeding 20 degrees there is of course, a very steep increase in the losses even in a reaction stage. But, if you look at the performance of a typical reaction blade it is not very sensitive to incidence between let us say plus minus 10 degrees. Whereas, an impulse blade is quite sensitive to incidence and especially at positive incidence angles. We also have alpha 2 plotted for both these cases what we can see is that alpha 2 remains more or less well behaved whether it is impulse or reaction turbine even though the incidence is different. The basic reason for this being true is the fact that in both impulse as well as reaction blades the flow is encountering an accelerating flow I mean a favorable pressure gradient. So, even if there is a higher level of incidence of the flow entering the nozzle because it is an accelerating flow the flow is generally well behaved which is unlike in a compressor where the flow encounters an adverse pressure gradient. So, the outflow would be extremely sensitive to the incidence angle as well that is if the incidence varies between a beyond a certain range the outflow angle also correspondingly changes drastically because of the fact that it the flow is encountering an adverse pressure gradient and. So, the chances of flow separation is substantially higher in the case of a compressor which is not true for a turbine where the flow is almost always encountering a favorable pressure gradient. And therefore, that partly explains why the gas outflow angle alpha 2 really does not change much and the insensitivity of outflow angle is larger much larger for a turbine as compared to that of a compressor. Now, if you now we come back to the types of losses in a turbine if you recall when we discussed about losses in a compressor we had classified them into two distinct sets of losses one is to do with 2 D losses and one is to do the 3 D effects like secondary flows and so on. Now, if you look at just the two dimensional losses for which there are lot of empirical correlations available 2 D losses basically are relevant to axial flow turbo machines and we have seen that in a compressor as well that when we were discussing about axial compressors that if you look at 2 D losses they are mainly associated with the blade boundary layers shock boundary layer interaction separated flows and wakes some of these of course are not really that significant for a turbine for example, the separation or blade boundary layers which are fairly well behaved in the case of turbines and separation on the other hand in certain operating conditions one might have a leading edge separation bubble in the rotor. But that is the chances of such occurrences are very rare unlike in the case of compressors where boundary layer behavior is always a concern because of adverse pressure gradients. So, mixing of these wakes that come from the rotor blade with the nozzle downstream of the second next stage obviously, creates a certain amount of losses and that is of course something that can be estimated by mixing loss models which are available from which one can estimate to a certain amount of accuracy what is the effect of these wakes shed from the rotor on subsequent stages. So, if you look at 2 D losses in particular we can classify 2 dimensional losses into different forms we have profile loss due to the boundary layer and its effect and whether you have laminar or boundary layer separation which is of course rather rare in the case of turbines one may have wake mixing losses because of the wake from the rotor interacting with the subsequent stages. One may also have shock losses which I will also discuss in little more detail in a later slide and of course the trailing edge loss due to the blade because trailing edge is usually rounded as we have seen in compressors one may have to provide a certain rounding at the trailing edge and there is certain amount of loss associated with that rounding as well. So, the total loss in a turbine like in the case of compressor is obviously, a sum total of all these different components. So, without going into details of how to estimate these losses we can just summarize that the overall losses in a turbine is a sum total of all these different forms of losses one may have profile loss on account of the nature of the blade surface itself one may have shock losses and secondary flow losses which can be quite significant in turbines triplicate loss and of course, end wall losses. So, if you make a comparison of this with a compressor let us say a transonic compressor where also you may have shock losses. The major distinguishing factor between turbine and the compressor in terms of losses would be the 3 D effects which are likely to be much more significant in a turbine like secondary flows and triplicate flows as compared to a compressor where of course, these losses are still present, but if you were to make a one to one comparison the losses in the case of turbine when it comes to secondary flows and triplicate flows are likely to be higher. But of course, there are methods of controlling some of these in a turbine because many of the turbine blades also have cooling mechanisms which again we will discuss in detail in later lectures. So, some of these cooling holes are also sometimes used to minimize let us say the triplicate flows or secondary flows in some way or the other we will discuss that in some of our later lectures. Now, there is another aspect I wanted to have some discussion on that is to do with deviation and I will probably spend a couple of slides on this aspect as well. Now, this is also true with in the case of compressors and that it is a known fact that the flow exiting the rotor does not really leave the blade at the angle for which it has been designed for. But it of course, in the case of turbines it is easier to estimate the outflow angle because the flow is encountering an accelerating flow passage and if the flow is not basically choked or if there are no shocks present at the exit of the rotor then it is at the exit of the nozzle it is relatively easier to estimate the amount of the outflow gas outflow angle from the nozzle. But of course, if there are shocks present that something which I will show you little later then the outflow angle can be quite different from what it has been basically designed for. So, what is basically found from experience is that the actual exit angle at the design pressure issue can be fairly well estimated by cos inverse of d by s as long as the nozzle is not choked. So, let me just explain what I mean by cos inverse of d by s. So, if you look at a typical nozzle exit flow when I had shown a picture of the cascade I had mentioned that this is basically the throat of the nozzle and the flow exits the nozzle at an angle of alpha 2. So, if you look at the pitch at the trailing edge which we have denoted by s and this is the throat and which is denoted by d then one can estimate the outflow angle the gas outflow angle alpha 2 as what is shown here that is cos inverse of d by s. That is if you take an inverse of of course, this is still an approximation, but it is found that it is fairly well the approximation fairly well captures the angle at which the flow exits the nozzle. Now, this is true as long as the nozzle is not choked because once it choked once their nozzle is operating under choked condition then there is a possibility that the outflow is supersonic which means that there is a possibility of the presence of shocks at the exit or the trailing edge of the nozzle presence of shocks can deflect the flow and cause a certain amount of deviation and in such cases of course, one cannot really estimate the angle exiting the nozzle as just cos inverse d by s. So, if you look at a case where there is the flow is not choked it is un choked then there is obviously, there is no the flow is not supersonic at the exit of the nozzle and which means that the flow angle can be basically well estimated by just taking the inverse of d by s. Now, if you look at the other case where there is a possibility of shock in which case it is basically choked flow after the throat because it is if you let me take you back here after the throat we might have an expansion here which might cause the flow to become supersonic. If the back pressure is low enough then you may have it basically acts like a converging diverging nozzle and one may have shocks emanating from downstream of the throat. If there is a trailing edge shock exiting at the flow exiting the nozzle then the presence of these shocks obviously, can deflect the flow to a different angle. So, as you can see here the flow is not really exiting at an angle that it was designed for the presence of these shocks there is a trailing edge shock or a reattachment shock and so on. The presence of these series of shocks can cause the flow to get deflected or deviated at an angle which is quite different from what it has been designed for. So, in such cases as what you see here the angle alpha 2 is not well established or estimated by simply taking a d by cos inverse of d by s. Of course, there are also a empirical correlations in this case to estimate the flow angle because if you know the shock structure of the flow leaving the nozzle from the analysis of the flow through these different shocks one can sort of estimate what is the angle at which the flow is leaving the nozzle. But that requires a much more complicated analysis than simply taking considering just the geometric parameters and estimating the exit flow angle to be a function of these geometric parameters. So, in the presence of supersonic flow where there are shocks present the flow structure is obviously, more complicated and the flow undergoes deviation which is more quite different from what one can otherwise easily estimate. So, I just brought up this aspect of deviation because of the fact that in nozzle flow there is a possibility that the flow exiting the nozzle can be supersonic and therefore, the flow might undergo deviation which is quite different from what it has been primarily designed for which means that this flow entering into the rotor basically has a different angle than what it has been designed for and therefore, one needs to consider take this aspect into account when estimating the flow through the entire stage. So, let me now quickly recap our discussion in today's class we had discussion on three distinct topics. We started off with degree of reaction and I spent some time discussing about degree of reaction its significance and how one can estimate degree of reaction and based on this estimation how one can determine the configuration of the turbine whether it is impulse or reaction and so on. We then spend some time discussing about losses the different types of losses the 2 D losses and the 3 D losses and I mentioned that there are certain aspects of losses which are the contribution of these different sources of losses is different in the case of turbine and a compressor because of the very nature of flow passing through a turbine or a compressor. We also discussed about the efficiencies and the different definitions of efficiency the total to static efficiency and total to total efficiency which is what we discussed in detail today. And of course, towards the end I also discussed about the aspect of deviation which is of significance especially when the flow is unchoked and well especially when the flow is choked and the flow exiting the nozzle is supersonic. We will continue our discussion on axial flow turbines in the next lecture. We will basically be talking about the performance characteristics of an axial flow turbine and how one can match the exit flow from a turbine with a downstream component like a nozzle. So, these are 2 aspects that we will be discussing in the next lecture which will be lecture number 22.