 Hello and welcome to lecture number 22 of this lecture series on turbo machinery aerodynamics. The last few lectures we have been talking about axial flow turbines. We started off with an interaction to axial flow turbines and then subsequently we spent about lecture on discussion basically towards the two dimensional flow in axial flow turbines and to do with cascade analysis of axial turbines. We also had some discussion on the performance of axial flow turbines and in today's lecture we going to basically talk about the performance characteristics of axial flow turbines. In the last class we had discussed about the performance parameters which are used in analysis and understanding of axial flow turbines and we will use some of those parameters in today's lecture to look at how the performance of an axial flow turbine can be characterized. And we will also spend some time to look at the different types of blade shapes and the flow characteristics through a turbine blade passages and we will also towards the end of the lecture have some preliminary discussion on matching of the turbine exit flow with downstream component let us say nozzle. And so these are some of the topics that we shall be discussing in today's class. So, we will begin with discussion on axial flow turbine performance characteristics. If you recall some few lectures earlier on when we are talking about axial flow compressors we had spent quite some time in understanding the performance characteristics of axial flow compressors and we will also in today's lecture follow a very similar approach towards understanding of the performance characteristics of axial turbines. But as what we going to see very soon the performance characteristics of compressors axial compressors and turbines are entirely different even though the parameters based on which they are assessed are very much the same basically pressure ratio the mass flow rate non dimensionalized and efficiency. And so though the parameters are the same the way these two components behave are quite different and that is very much to do with the fundamental nature of flow in axial flow compressors and turbines. In the sense that in compressor the flow has to undergo an adverse pressure gradient scenario where as in a turbine the pressure gradient is always favorable and that primarily leads to lot of difference in the performance characteristics. When we are discussing about axial flow compressors and the performance characteristics of compressors we had carried out a kind of an analysis to look at and understand what are the non dimensional parameters which could be used for gauging the performance of axial flow compressors. So, from the dimensional analysis we had identified a few distinct non dimensional groups which can which basically have a significant effect on the performance of axial flow compressors. So, the very same performance parameters are going to be used for judging the performance of turbines as well. And so in the case of axial flow compressor what we had seen was the exit pressure of the compressor and the efficiency are functions of variety of parameters like mass flow rate and the inlet conditions like the inlet stagnation pressure, inlet stagnation temperature, the rotational speed, the ratio of the specific heats, the universal gas constant, the viscosity in design and the diameter and so on. So, this we reduced by dimensional analysis and we arrived at this expression which relates the pressure ratio in the compressor and efficiency to these different non dimensional parameters. The first one that you see here is a function of mass flow rate and a set of other parameters. The second one is a function of the rotational speed and there are of course, a few other parameters. So, for compressors we have seen that for a given design we can assume that this gamma and nu does not change or does not affect the performance significantly. Similarly, the diameter and the gas constant are fixed. If that is the case the set of parameters would now simplify and reduce to the pressure ratio and efficiency being functions of m dot mass flow rate multiplied by the inlet stagnation temperature divided by inlet stagnation pressure and the rotational speed divided by square root of T 0 1. So, this is what we had derived for an axial compressor and what we will see is that the exact same logic can also be applied for a turbine and so we will use similar parameters for plotting the performance characteristics of axial turbines. So, in the case of turbines what we will see is that the exit state the pressure ratio rather of the turbine and the efficiency are again functions of similar parameters mass flow rate times stagnation temperature divided by the stagnation pressure and the speeds n by root of T 0 1. So, we will use these parameters for characterizing the performance of axial flow turbines as well. So, in the case of turbines just like we had done for the compressor we will first take a look at the pressure ratio versus mass flow rate and subsequently as a function of varying non dimensional speed and subsequently we will also look at the variation of efficiency. Now, the interesting thing to observe in the case of turbine that we will very soon see is that unlike in a compressor where we had the performance characteristic which was rather elaborate set of characteristic lines and you also had a surge line and choking line and operating line and so on. In the case of turbines the performance characteristics are much more simpler in the sense that there is only one primarily one limiting line as you could call. In the case of turbines we have the performance which is limited by choking at one end and there is nothing like a turbine surge or instabilities and so on and that is one distinct difference between a turbine performance as compared to a compressor performance. So, if you look at a performance characteristics of a typical axial flow turbine. So, here we have the pressure ratio and mass flow rate as a function of the varying speed. So, as speeds change the performance characteristics also tend to change, but what you see here is that as you keep changing the speed non dimensional speed all these characteristics would kind of bunch and merge towards the single line and that is the choking mass flow. So, there is a certain limiting mass flow that kind of affects the performance of an axial turbine and that is limited by choking. So, this is very much unlike the case of axial compressor where we had seen a distinct variation in performance with non dimensional speed and there was also a limiting line on the left hand side of the compressor characteristic which was to do with the onset of instabilities and that was referred to as a surge line. So, do not really see this kind of a limiting parameter in the case of axial turbine. So, from the performance map of a turbine which is basically the pressure ratio versus mass flow rate for different non dimensional speeds as speeds change you can see that towards a certain mass flow all of them kind of merge and you have a limiting mass flow that is the choking mass flow. So, compressibility in the case of turbine limits the performance and that is one of the limiting performance parameters in the case of turbine. So, we will also look at some of the other limiting parameters like the stresses and temperature which also limit the performance of axial turbines. So, the other parameter that we would be interested in the case of turbine is the efficiency of a turbine and how the efficiency of a turbine is affected by variation of let us say the pressure ratio and what is the effect of pressure ratio on the efficiency of an axial turbine. Now, in the case of efficiency we have plotted efficiency again as a function of the speed as speed changes what really happens to efficiency. So, if you look at the efficiency characteristics here. So, here we have the turbine efficiency as a function of the speeds and on the x axis we have the pressure ratio. You can immediately see that the pressure ratio in the case of turbines can be substantially higher than that of a compressor per stage which we have also discussed in the last lecture that a single stage of a turbine can actually drive multiple stages of compressor because the pressure drop that a single stage of a turbine can handle is much larger than the pressure rise that can be achieved through a single stage of an axial compressor. Now, if you look at the efficiency characteristics you can also see that the efficiency is quite flat and in the sense that the variation of efficiency is very is rather insignificant over a wide range of pressure ratios as well as mass flow rate. So, efficiency as speeds change efficiency characteristics are relatively flatter as compared to a compressor which we had seen is likely to have very sharp change in efficiencies unlike the case of turbines. And it is also known that typically the turbines are operate at much higher efficiency than compressors and turbine efficiencies are usually slightly on the higher side as compared to compressor which is operating under the same pressure ratio range. Now, so there are few aspects that one would one can derive from these performance characteristics of a turbine and one of the important aspects that I just mentioned is the fact that efficiency variation of efficiency is rather the amount of change of efficiency that you see in the case of turbine is very insignificant for a wider range of mass flow rates and pressure ratios. So, efficiency plot basically shows us that the efficiency tends to remain constant over a wide range of rotational speeds and pressure ratios. The main reason for this being the fact that in the case of turbines the flow basically has to operate in a favorable pressure gradient unlike the case of compressors where the lot of limitations which the flow would encounter because it is operating in an adverse pressure gradient and that is one of the reasons why turbines generally tend to have better efficiency and at the same time the pressure ratio that one can achieve is much higher in a turbine as compared to that in a compressor. Now, the other aspect that we also noticed from the mass flow characteristics is that the choking point or choking line in the case of turbine limits the performance of a turbine and that sort of becomes the limiting line in the case of axial turbines and whereas in a compressor you also had a surge line on one side which was limiting the performance of axial flow compressors. And what we also see is that the mass flow characteristics would kind of merge into a single line or a single curve as we keep increasing the rotational speed and we have seen that at higher mass flows all these are individual characteristic lines merge and form a single line which will eventually become the choking limit or choking line for the axial turbine. Now, there are few more aspects that we need to understand here that is where an axial turbine is operating very close to the design point that is at low incidence levels the performance characteristics as one would notice would kind of merge and reduce to a single curve. What is also seen is that as the number of stages increase there is a noticeable change or tendency for these characteristics to become ellipsoidal and this was kind of formulated in way back in 1940s by Strodola who formulated what is known as the ellipse law and which was for a long time being used by a lot of designers. Basically what ellipsoidal law tells us is that as you keep increasing the number of stages the mass flow characteristic kind of sort of resembles an ellipsoid and so that can be you know theoretically or empirically correlated and so that was sort of used as a thumb rule by designers for a long time just as a starting point for the design. So, what is basically being implied here is that as we let us take a look at the performance characteristics once again. Now, this is pressure ratio versus mass flow characteristic and we have seen that we have already seen the performance characteristic which was limited by choking on one end. So, as you keep increasing the number of stages let us say we increase the number of stages from 1 to 3 and tending towards a multi stage turbine the performance characteristic as you can see changes and eventually becomes or resembles an ellipsoid and that is something that was empirically correlated in Strodola ellipse law and which basically relates the mass flow versus with the pressure ratio through an empirical correlation. And so for a long time that was sort of used as a thumb rule in the design preliminary design of axial turbines. And so these are some of the aspects that one can observe from the performance characteristics mass flow versus the pressure ratio and efficiency versus pressure ratio or mass flow. So, here I mentioned that there are few aspects that limit the performance of turbine and so we can identify three distinct aspects which limit the turbine performance. One of them is compressibility and that we have already seen in the performance characteristic that the mass flow is limited by choking mass flow and so that is an effect of compressibility. Besides compressibility there are two other aspects or parameters that limit the performance. The other parameter is the stress that these blades would need to undergo while operating under extremely high temperature as well as rotational speeds. So, stress is the other limiting parameter and the third parameter of course is the inlet temperature. So, we if you have if you recall during your course on let us say propulsion basic propulsion course that I assume you would have carried out a cycle analysis of jet engines. You must have noticed that an engine performance is very sensitive to the turbine inlet temperature which is the maximum temperature in a Brayton cycle. And so higher the temperature better is the performance of the Brayton cycle and so one would always want to use a higher and higher temperature at the turbine entry. So, that the performance of the engine can be improved and therefore, turbine is it is always desires that we use a higher temperature of the turbine inlet for better performance of the whole engine. But as we know that material limitations would prevent us from using extremely high temperatures in the turbine and so inlet temperature also limits the performance of a turbine in the sense that though aerodynamically it would be possible for us to use higher temperature. It is a mechanical limitation because the materials that are currently available have certain amount of temperature limits that would prevent us from using very high temperatures in the turbine. So, if you look at the performance of a turbine these are limited by these three parameters compressibility stress and the inlet temperature. And compressibility as I mentioned limits the mass flow that can pass through a turbine stress on the other hand limits the rotational speed. And since we know that turbine inlet temperature is also a very significant aspect of performance turbine entry temperature also limits the performance. But if you look at the second and the third parameter which is stress and temperature they are quite related in the sense that temperature also affects the stress because it induces a thermal stress on the blade. And so temperature and stress are in some way limited and so there is in a typical design exercise there is always a sort of compromise that is used to arrive at an optimum balance between the stress that a blade can sort of withstand which also limits the rotational speed. And also the maximum temperature that one could use in a turbine and that is why there are lot of technologies which have come in the reset years to use higher turbine entry temperature. Some of these are to do with using artificial cooling techniques to keep the blade cooler and permit higher temperature that could be used. There are also lot of research going on to identify and develop newer materials which can withstand higher temperature as well as stress. And so a turbine performance which is basically limited by compressibility stress and temperature. So, for a turbine designer it is like we have seen also for a compressor there is certain amount of optimization that would be required to be carried out by the designer to arrive at a set of parameters which would give us the best performance. And at the same time it would also enable the materials that are currently available to be used in the current design scenarios. Now, so it is a known fact that the turbine inlet temperature has a significant effect on the engine output. So, it is known that if you look at just a thumb rule 1 percent increase in the turbine inlet temperature can produce up to 2 to 3 percent increase in the engine output if you are looking at the overall engine. And therefore, it is necessary that we identify methods by which one can increase the turbine entry temperature to the extent possible. And which is why there are elaborate mechanisms which are used for keeping the blade cooler than in fact the turbine entry temperature in most of the modern day engines have temperatures which are actually higher than what the materials can withstand. But it is possible for us to use these higher temperatures because of the fact that we employ cooling technologies to keep the blade cooler or the blade surface protected from very high temperatures. We will of course, be discussing about turbine blade cooling techniques in detail in some of the later lectures. And this is just to highlight the significance of the fact that the temperature turbine entry temperature has significant effect on not just the turbine performance, but also on the engine as a whole. So, now that we have been talking about the turbine performance and its effect or the parameters which affect the turbine performance that is also discussed about the turbine blades in little more detail and what are the different types of blades that are used. We have already discussed that there are two fundamental types of turbines axial turbines. And one of them is called the impulse turbine and the other is called the reaction turbine. So, let us now try to understand the aerodynamic shape of these different types of blades. All these blades as we know are fundamentally airfoil sections. But let us also try to look at the different types of blades which are being used in various types of turbines. And so depending upon the application the very shape of the blades can change significantly and that is what I would like to explain in the next few slides. Now, the blades that are used in turbines can be quite different from those that are used in compressors. Something that you have you must have realized by now the last few lectures that we have been discussing. And there are variety of parameters which affect the performance or the shape of these blades. Some of these parameters are to do with the mach number at which these blades have to operate the stress levels and the host of other parameters. Now, depending upon the application the thickness distribution or the curvature of the blade the trailing edge shape all these are varied and these are strong functions of the application for which the turbine is to be used. And a turbine can also be operated or designed for a variety of mach numbers ranging from subsonic to transonic or even super sonic. So, turbine blade shapes can vary substantially as one changes the mach number. This is something you have also seen for compressors. For low speed compressor the blades shapes that are used are quite different from the blades that are used in let us say a transonic axial compressor. So, in the case of turbine as well these blades can be quite different. And there are various ways in which these blade shapes or profiles are derived or developed. Some of these are basically derived from agencies like NACA series. You must have seen NACA series of blades which were discussed during compressor discussions. So, there are series of blades which NACA had developed way back in the 1940s. And so these blades are these are of course some of the sources of blade profiles. And there could be various other profiles that are used in the case of turbines. So, some of these I have listed here. And the most fundamental form of these blades are derived from the NACA series. You could also have turbine blades from the Agard series or one could use profiles with circular arc or parabolic arc camber like what is done in the case of compressors. And profiles could also be derived either graphically or empirically from a specified pressure distribution or a mach number distribution. And most of the engine industries have their own specific proprietary turbine blade shapes which they have developed over the past many years. And so they have their own specific forms or blade shapes which are proprietary to their product. And they have developed these over the past many many years. And there has also been a recent trend towards developing custom designed or custom tailored airfoils. And which are not really related to any of these profiles that I have mentioned like NACA series or any other series. But to develop and design airfoils which are specific to a particular turbine geometry which meets certain set of design parameters. So, turbine profiles blade profiles can be developed or and designed through a variety of these methods which is also very much applicable for compressors. Compressor blades also can be derived either from some of these conventional blades like the NACA series or it could be circular arc or parabolic arc or custom built airfoils which many of these industries have developed over the many years. So, let us take a look at some of these very typical blades which have been developed and used in the last 40 odd years or so. So, some of the very basic turbine profiles which have been derived from the NACA series are shown here. These are two distinct types of turbine blades which are derivatives of the NACA series where they have a certain thickness distribution as you can see here and a certain camber distribution as well. And so you can see here between these two profiles the thickness distribution as well as the camber distribution is quite different and this is from this is along the chord of the airfoil and this is of course, the thickness distribution of the airfoil. One could have profiles which have been designed for a subsonic inlet and a supersonic outlet and this is as I had discussed in the last lecture very similar to a convergent divergent nozzle which has a subsonic inlet a throat and then a supersonic or a divergent section which leads to a supersonic flow at the exit. So, one might have an inlet flow which is subsonic and that accelerates and reaches a throat and then again there is a divergent section which leads to a supersonic flow at the exit. One may have blades blade shapes which are very different from what I just showed for example, a typical let us say a steam turbine blade which is used the one which is shown here is for the tip section of a typical steam turbine blade. And you may also have a blades which have a supersonic inlet and a supersonic outlet like what is shown here. And so as you can see between these 4 or 5 different examples that I just showed the turbine blades can be entirely different from depending upon the kind of application for which it has been developed and designed for. So, turbine blade shapes can be extend quite extremely different blade for example, if you look at the blade which was used for a subsonic inlet to supersonic outlet kind of geometry compare that with a very traditional NACA type of airfoils which generally tends towards the subsonic kind of application. One would see that the blade shapes can be quite different and so how does one arrive at this kind of a blade shape. So, that is something that will basically depend upon the application as I mentioned and that boils down to the kind of work output that the turbine is supposed to generate. And if you look at that work output from an airfoil sense or from the airfoil point of view it basically boils down to the pressure distribution on the airfoil surface or the loading of the blade itself. And so the blade shape basically dictates the loading of the blade or the amount of pressure distribution on the suction surface and the pressure surface. So, the variation of the pressure static pressure on the suction and pressure surface basically gives us the loading that this particular blade profile or the airfoil section has been designed for. So, let us take a look at a typical turbine blade and try to look at understand the pressure distribution around such a typical turbine blade. So, here we have a very standard so called standard turbine profile and we have an inlet velocity absolute velocity of C 1 which is entering at an angle of alpha 1 and it exits at a velocity of C 2 angle alpha 2. So, here as you have seen in the first lecture on the axial turbines there is a pressure surface and the suction surface like any typical airfoil. And so if you look at how the flow behaves as it moves on both these surfaces the suction surface and the pressure surface. So, I have plotted the pressure distribution on what is seen here the static pressure distribution between the inlet and outlet of the turbine blade. So, if you look at the suction surface pressure distribution that is shown here what is seen is that there is a drastic drop in this static pressure up to a certain level and then the static pressure again begins to increase. So, if you look at the flow that as it passes over the suction surface of a turbine blade this basically means that the flow accelerates and the acceleration is quite steep as defined by the slope of this pressure curve. So, this steep curve here or steep drop in the static pressure indicates that there is a rapid acceleration of flow along the suction surface. And that progresses and continues quite for a certain distance along the length of the blade and after a certain distance the static pressure again usually begins to grow of course, this depends very much upon the blade geometry, but in a typical turbine blade one may also have a recovery of static pressure after it reaches its minimum, which means that there could be a certain gain in static pressure towards the exit or towards the trailing edge of the blade. And this is a typical pressure distribution on the pressure surface you can see there is a mild well initially it can kind of remains constant well the slope is much shallower as compared to that of the suction surface and but there is still a steady drop in the suction well static pressure on the pressure surface as well. If you compare the pressure static pressure as at the inlet that is what you see here by let us say denoted by p 1 and compare that with the average static pressure at the exit. Basically, this tells us the amount of this acceleration that has taken place in the blade passage and so the area between these two curves that is the suction surface and the pressure surface gives us the loading that this particular blade has been designed for. And what is shown here as p naught is basically referring to the stagnation pressure, which occurs at the stagnation point of the blade which could be a point somewhere here. And so the maximum pressure that you see here in this particular blade geometry is basically the stagnation pressure which is true for any air foil and exit pressure that is denoted by p 2 is substantially different from what you see at the inlet. And that is the difference between these two is basically given by the dynamic head across this particular blade. So, this is a pressure distribution which is of a typical turbine blade and if you look at the pressure distribution of all those blade geometries I had shown in the few slides earlier. The pressure distributions can be quite different, but the general trend is quite is going to be quite similar to what you see here I have just seen now. And in general you can see that there is an acceleration of flow right from the inlet to the exit of the turbine except for the suction surface where in some cases you might have a local flow deceleration which in some cases might lead to a pockets of flow separation under certain off design conditions something I had mentioned in the last class as well. And unlike in the case of compressors the flow in the case of compressor is going to undergo a steady increase in static pressure and which is basically the adverse pressure gradient which is quite different exactly opposite to that you have seen in the case of a typical axial flow turbine. Now, this also means that there is a significant defect of the way the blades are placed as well. So, that if you have to get a decelerating flow or an accelerating flow that will very much depend upon the stagger of the blades and also the number of blades. So, if you have more number of blades or if you try to put more number of blades in on the annulus this means that there are more number of blades which can take up the loads and so the loading per blade would reduce. And therefore, the stress associated with the blade loading also comes down at the same time more number of blades mean means that you would need the overall weight of the disc and the blades would increase. And therefore, the associated cost and complexity increases at the same time if you look at a configuration which is lesser number of blades this means that each blade has to now take up a load which is more than what it was for the previous case, but at the same time you save on the weight of these blades and complexity and so on. So, is there a certain kind of an optimum number of blades that can be used for a given configuration. So, spacing between these blades because it is a very crucial parameter there is a certain again an empirical correlation which was developed way back in 40s. And of course, this has been modified and there are modern versions of this criteria which is currently available and used by the industry. And so, how does one arrive at an optimum number of blades for a given turbine configuration. So, there is a criteria which was developed in 1945 by Zwefel which relates the blade loading or the blade force to some of the aerodynamic parameters. So, this is again an empirical correlation which is also related to the blade geometric parameters. So, Zwefel's criteria originally was defined as the blade force divided by half rho v e 2 square into the blade chord. So, this can be related to the geometric parameters of the blade. And so, Zwefel's criteria z is also equal to 2 into cos square alpha 2 s by c tan alpha 1 minus tan alpha 2. So, here you can see that this s by c is basically the spacing to chord ratio. And generally accepted optimum Zwefel parameter is taken as around 0.8. So, if you fix Zwefel parameter as 0.8 then one can actually calculate the optimum spacing ratio from assuming Zwefel criteria of around 0.8 and also the blade parameters like the angles alpha 1 alpha 2 and so on. So, this kind of gives us some empirical some idea about how much the blade spacing should be given these parameters like the blade angles alpha 1 alpha 2 and so on. This was of course, developed about 60 years ago or even more than that and there have been several modifications to this and there are modified versions of the Zwefel criteria which are available and used by industry for developing or arriving at certain amount of optimum spacing to chord ratios. And so, what is also noticeable here is the fact that though this parameter was of course, developed way back in the 40s only slightly modified versions of these are available and even though it was developed way back in the 40s it is still being considered as a sort of an empirical thumb rule parameter which can be used at a design point at a starting point of a turbine design level. Now, what I will do now is to summarize the differences between turbine blades and compressor blades and of course, subsequently discuss about turbine and exit nozzle matching. So, before I move on to matching between turbine exit and the nozzle let me just quickly summarize some differences between a turbine blade passage as compared to that of a compressor. One of the obvious differences is the fact that the pressure drop in a turbine is much larger than the pressure rise in a single stage of a compressor. The flow turning in a compressor is usually restricted to around 20 to 35 degrees whereas, in a turbine it can be as high as 160 degrees as you have already seen the flow turning is substantially different from that of a compressor. In a turbine a designer tends to sort of delay the formation of shocks so that the losses across the shocks are minimized whereas, in a typical transonic compressor shocks sort of form one of the modes of deceleration themselves. So, there is no way designer can actually delay the occurrence of shock there because that is one of the modes of deceleration in a transonic compressor and that is one of the reasons why transonic compressors generally have much lower efficiencies than corresponding turbines because of the fact that in a turbine it is still possible for the designer to delay the occurrence of shock till the later part of the blades and towards the trailing edge and so the losses the shocks actually occur in the lower Mach number region and so the losses associated with the flow across the shock is lower than that in the case of a typical transonic compressor wherein deceleration itself is carried out through the occurrence of these shocks. So, these are some fundamental differences between a turbine blade and a compressor blade and I think that that is something that you need to understand very carefully which is of course from the fundamental aerodynamic aerothermodynamics of the flow through a compressor and a turbine that dictates these fundamental differences in the blade shapes themselves. So, let us now take up a slightly different aspect of the turbine flow and we will now try to understand the effect of the turbine exit flow on some of the downstream components let us say the nozzle. So, how is the exit flow of a turbine matched with that of the exit component like a nozzle. So, the operation of a turbine as you already understood that turbine is linked to a variety of other components like the combustion chamber or the compressor and it is actually mechanically directly coupled to the compressor it is also connected to some of the thermo or aerodynamically connected to some downstream components like the nozzle. So, all these upstream and downstream components have a significant effect on the turbine performance and so in a typical design exercise the compressor and turbine performance characteristics would form a very important aspect of engine integration and matching of the performance of the whole engine as well. So, we have seen that in the case of turbine performance very much unlike that of compressors. Turbines do not really exhibit very significant variation in the non-dimensional mass flow with speed which is quite unlike the case of compressors which exhibit a drastic change in the performance characteristics with speed and so the turbine operation. So, if you look at the turbine performance characteristics one might wonder that turbine because the performance does not really change much with the speed it would be rather easy to control the performance of the turbine and match it with the compressor. But what is significant here is the fact that the turbine performance is actually limited by the nozzle which is downstream of the turbine because there is a certain amount of mass flow which the nozzle can handle and even if the turbine is able to operate at different mass flow rate the nozzle downstream would sort of dictate the amount of maximum mass flow that the turbine can actually handle. What we will very soon see is that most of the modern engines for a majority of their operation would be operating with both the turbine as well as the nozzle under choked condition. So, the nozzle exit area has a very significant effect on the turbine performance and its operation and so whether the nozzle is operating under choked or un choked condition would have a significant influence on this matching exercise. And so though if you look at turbine and nozzle from a fundamental thermodynamics point of view and not without actually worrying about what constitutes these components both these components are thermodynamically identical in the sense that both of these components are expanders that is flow expands through a turbine the flow expands through a nozzle as well of course, their functions are entirely different. But if you plot a turbine process and nozzle process on a temperature versus entropy diagram both these are identical they are both expanding processes flow expands through turbines it also expands through a nozzle. And so the matching of turbine with that of a nozzle is very much identical to that of matching between a gas generator turbine with that of a power turbine. If you recall in some of the forms of engines like turboprop engine a separate turbine is used for driving let us say the propeller. And there is a turbine which actually drives a separate turbine which drives the compressor. So, there is also a matching exercise between this kind of a turbine with the power turbine. And the matching between or the matching process between a turbine which is connected to the compressor with that of a power turbine or a nozzle they are quite identical because thermodynamically all these components are very much the same. So, if you look at the nozzle once the nozzle is choked that means the nozzle non dimensional flow has reached its maximum then it is basically independent of the pressure ratio. And therefore, in some sense the flight speed because the nozzle pressure ratio is a function of the pressure build up which has taken place across a variety of components like the intake the compressor the loss of pressure in the combustion chamber and the pressure drop in the turbine. So, all these if you multiply all these pressure ratios that basically dictates the nozzle pressure ratio which means that the once the nozzle is choked it basically becomes independent of this pressure ratio and therefore, the flight speed as well. This means that once the nozzle is choked it sort of fixes the turbine operating point because choked nozzle would mean that the nozzle is operating under a condition where the it has reached its maximum mass flow rate and it is not possible to enhance the mass flow rate any further the nozzle cannot handle any higher mass flow rate. And therefore, the turbine also has to operate under a condition which matches with this nozzle the choked nozzle operating condition. So, a choked nozzle would fix the operating point of a turbine and therefore, when the nozzle is choked or operating under a choked condition with the maximum mass flow rate the equilibrium running line or the operating line would basically be determined or fixed by the turbine operating point and it kind of becomes independent of the flight speed itself. So, I will try to demonstrate this aspect through the performance characteristics of a turbine as well as that of a nozzle. So, here we have the matching characteristics of a turbine as well as that of a nozzle. So, I have denoted station 1 for the turbine inlet station 2 for turbine exit and station 3 for the nozzle entry. So, we have mass flow versus pressure ratio for the turbine and on the right hand side here we have mass flow versus the pressure ratio for the nozzle. Let us take up the nozzle first because this is of course, something you would have learnt in your gas dynamics course as well that the nozzle performance as you change the pressure ratio the mass flow rate increases and then it reaches a peak which is the chocking mass flow after which the mass flow remains the same irrespective of the pressure ratio. So, once the nozzle is choked the mass flow it becomes independent of the pressure ratio and does not change in spite of changing the pressure ratio to any value. So, here if we have a nozzle which is operating under choked condition and if you look at the corresponding turbine characteristics. So, here we have the turbine characteristic here the mass flow versus the pressure ratio. So, this you see here is the mass flow rate of the turbine as a function of its pressure ratio. So, if the nozzle is operating under choked condition which is denoted by let us say this letter a that fixes the mass flow rate of the nozzle and since that fixes the mass flow rate of the nozzle that would also fix the amount of mass flow rate that the turbine can handle. But, you can see here that the mass flow this mass flow rate is higher than the chocking mass flow of the turbine chocking mass flow of the turbine as denoted here as m dot root t naught by p naught is lower than the chocking mass flow of the nozzle. So, the turbine cannot really operate under this condition without operating at a lower pressure ratio which means that the turbine is now operating or is forced to operate at a condition which has a pressure ratio lower than the chocking pressure ratio for which the turbine can is capable of operating. Similarly, if you look at other conditions here in which under which the nozzle is actually un choked. So, here the turbine is actually able to operate at a much higher mass flow rate as well. So, this sort of demonstrate the fact that the nozzle chocking condition would force the turbine to operate under condition which matches with that of the nozzle itself. So, even though the turbine in this example is actually capable of operating at a higher mass flow rate it is forced to operate at a lower pressure ratio because of the fact that the downstream component is choked under that operating condition. And most of the modern day engines as I mentioned would be operating under a choked condition for majority of its operation when it is operating under a high thrust condition that is high mass flow condition like during takeoff or cruise except under very low thrust operating conditions like during landing or taxiing when the engine is not operating under its full mass flow condition these components like the nozzle or the turbine tends to remain un choked under these conditions. But unless it is operating under these conditions most of the time engines would be operating under choked condition. So, only when the aircraft is let us say preparing to land or it is taxiing the nozzle may be un choked and it is also necessary that one would have to ensure the matching even at low speeds. And it is under these operating conditions that when you try to match an un choked nozzle with that of a turbine and eventually with that of a compressor that there could be problems with reference to the compressor operation because under these conditions the operating line might move very close to the surge line which is something that is the designer or the engine integrating team would have to take care. Because if the operation is very close to that of surge there is a risk that the compressor might enter into surge and of obviously that could affect the entire engine operation itself. So, at very low speeds when the operating line tends to become very close to surge and that is where the nozzle and therefore, the turbine tends to remain un choked that lot of care needs to be put in between matching of not just the turbine and the nozzle but also with components upstream like that of the compressor. So, this is just a quick overview of the matching operation there is also an extensive matching that is done between the compressor and the turbine and that is incomplete without taking this nozzle into account because even though a compressor and a turbine might well be matching in their operating range. It is the nozzle which would sort of dictate the or fix the turbine operating point and then the whole matching exercise between the compressor and turbine might require another round of iteration to ensure that all these 3 components operate under a matched condition even at low speeds where the nozzle might be un choked. So, let me now very quickly recap this today's lecture. We had discussion on 3 distinct aspects of turbine operation one was to do with the performance characteristics. We discussed about the different performance characteristics in terms of mass flow and pressure ratio efficiency and pressure ratio and we have seen that the turbine operate performance characteristics is quite different from that of compressor and the turbine performance is usually insensitive rather relatively insensitive to varying speeds as compared to that of compressors which can be quite sensitive to operating speeds. We also discussed about the different blade shapes which are used in turbines and the nature of flow as it passes through some of these different blade shapes and lastly we discussed about matching between the turbine and one of the downstream components like the nozzle. You might have additional components like a power turbine or after burner we have discussed about the nozzle because that is sort of limiting the performance of the turbine itself. So, these were some of the topics that we had discussed in today's lecture and so it is about time that we have a session on solving some numerical problems from axial flow turbine. So, we will devote our next lecture towards a tutorial on axial flow turbines where we will be solving some problems from axial flow turbines and I will also have some exercise problems for you which you can solve based on our discussion in the last few lectures as well as the tutorial session itself. So, we will have a tutorial during the next lecture.