 Hello and welcome to lecture number 31 of this lecture series on turbo machinery aerodynamics. We have discussed quite a lot on different aspects of turbo machines over the last several lectures. We actually started off with very introductory topics and very fundamentals of the turbo machines starting with thermodynamics and its analysis as applicable to turbo machines. We then moved on to discuss about axial compressors and we had quite a lot of discussion, detailed discussions on axial compressors and the various design aspects of axial compressors and how an analysis, detailed analysis of axial compressors can be carried out. Subsequently, we spent some time on discussion on turbines and we first started off with the axial turbines, the design issues related to axial turbines, performance analysis of axial turbines and so on. What we are going to discuss today is a slightly different topic in the sense that the component that we are talking now about is not really an axial turbo machine. We are going to discuss about centrifugal compressors starting from today's lecture and we will continue with this on two more lectures. We will discuss some details about centrifugal compressors but in contrast to axial compressors, our discussion would be rather superficial in the sense that we will not be really taking up very detailed discussion on the design aspects of centrifugal compressors. The basic reason being that centrifugal compressors are not really widely used in modern day aircraft engines as compared to the axial compressors. They are still used but their usage has been limited to rather smaller sized aero engines and some other applications and that is the reason why we shall not really be discussing too many details about especially to do with design aspects of centrifugal compressors. But of course, we will spend some time discussing about the fundamental issues related to centrifugal compressors, the different aspects of analysis of centrifugal compressors as well as the performance characteristics in quite some detail over the next two or three lectures or so. Of course, we will also be having a tutorial session where we will get a chance to solve some problems associated with centrifugal compressors. So, in today's lecture we will start with the fundamentals, we will talk about the thermodynamics of centrifugal compressors and subsequently we will be discussing about the different elements of centrifugal compressors. What are the different components which constitute a centrifugal compressor and what is the flow characteristic associated with the flow through these different components. So, let us begin our discussion with the thermodynamics of centrifugal compressors. But of course, we will also have a quick introduction to centrifugal compressors as a whole and see what centrifugal compressors are and why is it that these compressors are not really used in that much popularly as compared to the axial compressors. So, there are two distinct aspects of centrifugal compressors as you can see thermodynamics as well as the components of centrifugal compressors. So, it will be very interesting to know that centrifugal compressor in fact, has a longer history than axial compressors. In fact, the earliest jet engines that flew one by Frank Bittel in England and the other by a German engineer named Franz Ohain, both of them developed the jet engine independently and both these initial developments of jet engines used centrifugal compressors. In fact, that continued for a very long time and even the very common aircraft which were used the fighter aircraft which were used during the second world war all of them had in fact, most of them had centrifugal compressors. There are certain inherent advantages with centrifugal compressors, but of course, the disadvantages are also equally substantial and that is the reason why as we look at larger and larger sized aero engines use of centrifugal compressors become rather disadvantages in the sense that centrifugal compressors require a larger frontal area that is probably the most important disadvantage of the centrifugal compressor that the pressure ratio that is developed per stage from a centrifugal compressor very much depends upon the overall diameter of the jet of the centrifugal compressor which means that if you look at a larger sized aero engine which is used in let us say modern day passenger or fighter aircraft they all require the thrust requirement from such an engine is tremendous and to be able to develop such a high level of thrust it is necessary that the compressor develops the corresponding pressure ratio required for developing or generating this kind of a thrust. From a centrifugal compressor if you were to develop such a high level of thrust the frontal area required by such an aircraft of such a compressor would be substantial and obviously, such an engine would also have a huge amount of drag. So, an aero engineer an aircraft airframe engineer would not really want an aero engine with a large frontal area because that is going to increase the overall drag of the aircraft and obviously, that is not a good idea. And so if you compare this with an axial compressor to develop the same pressure ratio an axial compressor requires a smaller frontal area, but of course, axial compressor requires multiple stages to develop the same pressure ratio. Centrifugal compressors can generate far higher pressure ratio per stage as compared to axial compressor and that is of course, one advantage huge advantage of centrifugal compressor that they can generate a large much larger pressure ratio per stage than an axial compressor. We will of course, be exploring the reasons for this and little later when I will explain thermodynamics of centrifugal compressor. We will see that centrifugal compressors have a slightly different mechanism of pressure rise than axial compressor and that is the reason why they can generate much higher pressure ratios per stage. Yet there is yet another difference or disadvantage so to say between centrifugal and axial compressor. For larger sized engines axial compressors have slightly higher or better efficiencies than centrifugal compressor and that is of course, if you look at an aero engine perspective the efficiency is of at most importance and even a slight improvement in efficiency means a big deal in terms of the overall engine performance and therefore, that is where axial compressor scores yet again over centrifugal compressor and these are primarily the two reasons why centrifugal compressors are not really used in larger sized aero engines. They are used very much in smaller sized engines because for smaller engines axial compressors in fact, have a certain disadvantage because as you reduce the engine overall diameter to smaller and smaller levels the losses associated with tip clearance and so on increase substantially and therefore, in fact for smaller sized engines centrifugal compressors may have performance or efficiencies as high or in fact, even better than axial compressors and so for smaller engines it is a common practice to use centrifugal compressors and therefore, smaller sized engines which are used in let us say business jet or smaller airplane they still have engines which have centrifugal compressors. So, to sum up centrifugal compressors obviously can generate much higher pressure ratios per stage they obviously, have larger frontal area and therefore, they are not as commonly used as axial compressors and of course, they are little less efficient and centrifugal compressors are used in auxiliary power units the APUs of many aircraft. Some aircraft also have centrifugal compressors as part of the air conditioning system which is used in the aircraft and in some engines centrifugal compressors are used in combination with an axial compressor that is centrifugal compressor would form the last stage of a set of axial compressor stages and such engines are also quite popularly used which are of course, the medium or smaller sized engines. Some examples being the T 700 from G P T 6 from Pratt & Whitney or the Honeywell T 53 engines. So, these are some engines which have centrifugal compressors which are used in combination with an axial compressor and the inherent advantage here is that one axial compressor can replace multiple centrifugal compressor stages because it is possible to generate a higher pressure ratio per stage by from an axial from a centrifugal compressor as compared to axial compressors and that is why having a centrifugal compressor as the last stage would actually replace several axial compressor stages and that is a big advantage because it leads to lot of saving possible savings in weight and part count and so on. So, there are some advantages of trying to do that kind of a configuration. So, let us now take a look at some typical centrifugal compressor rotors. So, I have here two different types of rotors. The one you see on the left hand side is a so called classical centrifugal compressor rotor which has a straight radial blade you can see this is the impeller of the centrifugal compressor and you can see that these blades are radial and straight and of course, there is a bend here at what is known as an inducer we will discuss little more details about this later on. So, an inducer actually turns an axial flow and guides it into the impeller and makes the flow radial. On the right hand side you can see the rotor of a centrifugal compressor which is much more complicated than what you see on the left hand side and this is a rather recent development and you can see that these blades are much more complicated than what you see in a conventional centrifugal compressor rotor and you can also see that at the exit of the impeller these are the diffuser veins of course, that is not really shown here these also would have diffuser veins. So, these are the veins of the diffuser and of course, we will discuss more details on why a diffuser is used in a centrifugal compressor. So, having understood or at least taken look at two different classes of centrifugal compressor rotor let us take a look at the schematic and understand what constitutes a centrifugal compressor. So, there are primarily three distinct components in centrifugal in a typical centrifugal compressor it has an inlet and followed by a rotor which is also referred to as an impeller in a centrifugal compressor and the impeller exhausts or discharges the flow into what is known as a diffuser. And from the diffuser there is a collector or a volute which guides the flow towards the outlet. So, the flow enters the impeller in an axial direction and then the impeller deflects the flow and turns it into a radial flow and this radial flow exits or exhaust the rotor exhaust the compressor through the collector or volute. And most of the impellers also have what are known as inducers. Inducers basically guide an axial flow and allow the flow to enter the impeller smoothly. The absence of inducer as we will see later the flow can actually say there is a tendency for the flow to separate from the impeller wings if one does not have an inducer. So, inducer is sort of like an inlet guide wing that we have seen in the case of an axial compressor. So, there are there is a there is a set of stationary components as well as one rotating component in a centrifugal compressor. The impeller forms the rotating component of the rotor of a centrifugal compressor. The inlet and the diffusers are the stationary components of the stator of a centrifugal compressor. So, once we have and so as now that we have as understood the working or basically the components or constituents of a centrifugal compressor. Let us now try to understand the operation of a centrifugal compressor from a little more fundamental sense from a thermodynamics perspective. We will try to understand what really happens as the flow passes through these different components of a centrifugal compressor. Now, I had shown the there are three distinct components which I had marked as 1, 2 and 3. Let us take a look at them once again. One is here being referred to as the inlet of the centrifugal compressor. Two is the diffuser and three is the exit of the diffuser or the volute or collector of the centrifugal compressor. Now, on a T s diagram this is temperature entropy diagram. You can see that of course, it looks quite complicated here. Let us try to understand what are what is basically being implied by these set of constant pressure lines that are shown here. So, station one as we have seen is this inlet of the compressor to corresponds to the diffuser and three is for the exit of the diffuser or the volute. So, there are static pressure lines as well as the corresponding total pressure lines which have been shown. Energy is added as you know in the impeller and so, the stagnation pressure rise takes place between 0 1 and 0 2 between stations P 0 1 and P 0 2 and in the volute there is a stagnation pressure loss. That is why you can see that there is a certain amount of loss which is being associated here because of the loss of stagnation pressure in the volute. Now, from station one static pressure rises between station one P 1 to P 0 1 and this is the corresponding stagnation parameter C 1 square by 2 C P. So, T 1 plus C 1 T 0 1 is T 1 plus C 1 square by 2 C P. There is no change in stagnation temperature between station two and three because there is no energy added after the impeller. So, energy is added between station one and two and that is why we see a change in stagnation temperature from T 0 1 to T 0 2 because that is where the energy is added. And actual process they are shown in two different lines here. There is a thicker one which is showing the process between 0 1 and 0 2 and there is another line which is showing the process between 0 2 and 0 1 to 0 3. So, it is possible of course, that in some centrifugal compressor there may not be a vaneless diffusion could be one may not really have this vaneless space between the inducer impeller exit and the diffuser and that is why these two distinct lines have been shown. But of course, they both lead to the same stagnation temperature there is no change in stagnation temperature here. If you want to look at the losses which are incurred in a centrifugal compressor then the total losses as shown here is between the stagnation temperature which it would normally achieve minus the stagnation temperature that it should have achieved if the process was isentropic. So, this is corresponding to the process if the whole process were to be isentropic and in which case there are of course, no losses taking place. And so the corresponding dynamic parameters are also shown here between station for let us say station 3 between P 3 and P 0 3. We have C 3 square by 2 C p and C 2 square by 2 C p between which basically takes the temperature from D 2 to T 0 2. So, if you look at if you compare this with the temperature entropy diagram that we had discussed for an axial compressor you will see you can quickly figure out the similarities between both these compressor operation at least in a thermodynamic sense. Even in an axial compressor we have seen that energy is added in the rotor and you may actually have a total pressure loss taking place in the stator. Even though static pressure continues to rise in the stator one may have total pressure loss taking place because of frictional effects. So, they are quite similar in the sense that in the case of centrifugal compressors also we have an impeller or the rotor where energy is added and that is followed by a diffuser where there is obviously no more energy addition taking place. One may have one continues to have a static pressure rise which is why P 3 is actually greater than P 2, but there could be some amount of total pressure loss taking place in the diffuser due to frictional losses and that is why we have P 0 3 which is less than P 0 2. So what we will do next is to look at the working of the centrifugal compressor as well as trying to estimate the work done or work required for driving a centrifugal compressor rotor and we will relate that to the velocity components very similar to the analysis we have done for an axial compressor as well. We will try to relate the work done or work required for driving the compressor to the velocity components because it would be easy for us to construct the velocity triangles and develop and calculate the work done or work required for a centrifugal compressor. So, let us look at the governing equations for centrifugal compressor stage. Now, in a centrifugal compressor rotor the torque required for or torque applied on the fluid by the rotor is a function of the mass flow rate of course and components of the tangential velocity. So, here we have torque required is equal to m dot into r times C w at station 2 minus r times C w at station 1 and so here the stations 1 and 2 are denoting compressor inlet and outlet respectively. Unlike an axial compressor where the those hardly any change in these tangential velocities between for a given station here what we will see little later as well that even the blade speed is not really a constant because the flow is radial the blade speed also changes with every radial location and so we have mass flow rate m dot times the r the velocity component tangential velocity C w at station 2 minus r times C w at station 1. So, the total work done per unit mass is w is basically a function of the rotational speed omega times the torque divided by the mass flow rate. This is in turn equal to omega times r times C w at station 2 minus r times C w at station 1. So, if you multiply omega times r we get the blade speed u therefore, work is equal to u times C w at station 2 minus u at u times C w at station 1. And so in axial compressors we had actually written this as u times delta C w because u at inlet and exit of the compressor was assumed to be the same here it does not remain the same and it changes with because the flow is indeed radial. Now, if you now look at the energy equation the steady flow energy equation and compare that with what we have written here for a centrifugal compressor rotor we have work is equal to the change in enthalpy stagnation enthalpy h 0 2 minus h 0 1. This is equal to the static conditions and the dynamic conditions. So, h 2 minus h 1 plus C 2 square by 2 minus C 1 square by 2. We have already calculated the change in enthalpy stagnation enthalpy in the previous equation here. So, if you substitute that here we have h 2 minus h 1 is equal to w minus this velocity changes where w has already been calculated as shown here. So, h 2 minus h 1 change in static enthalpy is equal to u times C w at station 2 minus u times C w at station 1 minus C 2 square by 2 plus C 1 square by 2. Where C 1 and C 2 are the absolute velocities entering the rotor entering the compressor and leaving the compressor respectively. Now, if you look at the impeller for example, let us take a look at the schematic of an impeller and we have an impeller inlet tip radius as shown here as r 1 and impeller outlet radius as r 2 and the corresponding blade speeds would be u 1 at this location and u 2 at the exit of the impeller. The velocity components for an impeller then the above equation which we have written earlier that is h 2 minus h 1 is the difference of u times C w at station 2 minus u C w at station 1 and the velocity components that gets transformed into h 2 minus h 1 is u 2 square by 2 minus u 1 square by 2 minus the relative velocities v 2 square by 2 minus v 1 square by 2. So, this in differential form we can write as d h is equal to d into omega square r square by 2 minus d into v square by 2. Now, if you recollect the basic thermodynamics the T d s equations T d s is also equal to d h minus d p by rho and therefore, we have on the left hand side d h we shall replace by d p by rho for an isentropic process T d s is 0. So, d p by rho is equal to d into omega square r square by 2 minus d v square by 2 minus T d s which for an isentropic flow becomes 0. Therefore, d p by rho is d into omega square r square by 2 minus d into v square by 2. So, what we have just now written down is an expression which relates the pressure change in pressure across a compressor to do two distinct parameters or terms. One is proportional to the rotational speed and the radius or radial location and the other is proportional to the change in relative velocity. Now, this is a generalized form of an equation that I have written down which also could be extended to an axial compressor in which case we have assumed that there is no change in the axial well the radial location for a given analysis. So, if you take up one radial plane then there is no change the d omega square r square term becomes basically 0 for an axial compressor which means that the pressure rise would now be equal to minus d v square by 2 for an axial compressor that is the pressure rise in an axial compressor is proportional to the amount of deceleration taking place in the compressor in the relative velocity in terms of the relative velocity. But in a in a centrifugal compressor as we have just seen the pressure rise is a function of one additional parameter which is d of omega square r square by 2 that is even if there is no deceleration taking place in a centrifugal compressor rotor which means that if the second term is equal to 0 one can still attain a certain amount of pressure rise simply because of the first term that is d omega square r square by 2 that is it is possible to achieve pressure rise in a centrifugal compressor even if there is no deceleration taking place which means that centrifugal compressors ideally should not be affected by boundary layer flows because we are saying that even if there is no deceleration taking place one can still achieve pressure rise because of the centrifugal effect and that is why it is called a centrifugal compressor. But most of the modern centrifugal compressors also have a certain amount of deceleration taking place that is pressure rise also because of deceleration or diffusion as well as because of displacement of the centrifugal flow field. So, there is a component of both in a centrifugal compressor and which is why even centrifugal compressors are indeed affected by boundary layer flows and that is not possible to eliminate the boundary layer effects all together even in a centrifugal compressor. But of course, it is possible to achieve much higher pressure ratios per stage in a centrifugal compressor as compared to an axial compressor. So, in an axial compressor where we can assume we have assumed the d r that is change in the radius is equal to 0 the equation that we have written here d p by rho is d omega square r square by 2 minus d v square by 2 will simply reduce to d p by rho as minus d v square by 2 that is in an axial compressor rotor the pressure rise can be obtained only by decelerating the flow. In a centrifugal compressor the first term is basically greater than 0 because omega square r square by 2 change of that is always greater than 0. Therefore, pressure rise can be obtained even without any change in relative velocity and which means that it is possible to have a rotor which does not have any deceleration and still develops a certain amount of pressure. But most of the modern compressors as I mentioned do have deceleration and also this means that centrifugal compressors are indeed affected by flow separation, but not to the extent that axial compressors are and therefore, it is possible to achieve much higher pressure ratio per stage from a centrifugal compressor as compared to axial compressors. Now, that is one set of governing equation that we have just discussed there is one more aspect of centrifugal compressor which is also true with some of these radial flow machines we shall discuss about what that equation of conservation is and that is also valid for a centrifugal compressor. So, this is to do with what is known as Rothalpy. Now, if you let us say assume a steady viscous flow without any heat transfer then in a radial flow scenario we have this particular conservation equation which is also valid for these radial flow machines that is h 1 plus c 1 square by 2 minus u 1 c w 1 is equal to h 2 plus c 2 square by 2 minus u 2 c w that is h plus c square by 2 minus delta u c w it is basically conserved parameter in a centrifugal compressor or even in any other radial flow machines. So, this is usually denoted by symbol I and this is known as the rotational lengthal p or Rothalpy for short it is called rotational lengthal p because it combines enthalpy in the conventional sense that is h plus c square by 2 and the component which is associated with u times c w which is to do with the tangential velocity and therefore, it has been observed that this parameter generally is conserved as it as the flow takes place through an impeller and any change in Rothalpy in some cases one does see that there is a change in in Rothalpy is primarily because of fluid friction which is acting on the stationary shroud. So, that is if you consider the impeller plus this shroud there could be certain amount of losses taking place as a result of this shroud and that probably would explain the amount of loss in Rothalpy which might be observed in some analysis. But in general in radial flow machines Rothalpy is a conserved quantity. So, in centrifugal compressors also in most of the analysis conservation of Rothalpy is assumed to be satisfied as the flow takes place through the impeller. So, having understood some of the fundamental governing equations of centrifugal compressor let us now look at the different components of centrifugal compressor and also look at how the flow develops as it passes through these different components. So, we will be discussing about three distinct components one is the impeller second is the inducer which is also a part of the impeller. In fact, and the third is the diffuser which is the other stationary component of the stator in a centrifugal compressor. So, let us begin with the impeller which is the rotor of a centrifugal compressor and the most crucial component in a centrifugal compressor. So, impeller is the component which draws the working fluid and it is like the rotor as we have discussed for an axial compressor. And impeller has diverging passages which diffuses the flow to a lower static pressure and higher static pressure and lower relative velocities and there are different ways or different configurations of an impeller it could be either single sided or double sided shrouded or unshrouded and so on. Now, in impeller the working fluid besides deceleration it will also experience centripetal forces because of the rotation itself and displacement of the rotational of the fluid elements. And therefore, besides the fact that the flow decelerates and diffuses there is also a certain amount of centripetal force acting on the fluid elements as it passes through an impeller. Now, there are three different types of impellers or impeller blades that are possible. Now, the simplest type is the straight radial type or one might have forward leaning blades or backward leaning blades. Now, forward leaning blades are considered to be inherently in unstable and we will see the reason for its instability probably in the next class where we will be talking about performance analysis or performance characteristics of centrifugal compressors. So, we will see the reason for why forward leaning blades are not really used that is probably something which I will be explaining in next class. The two other configurations which are commonly used are the straight radial blades and the backward leaning blades. So, both these configurations are commonly used in most modern day compressors centrifugal compressors. So, straight radial backward leaning and forward leaning blades. Let us take a look at what these three different blading configurations are. We will start with the straight radial first. So, this the one that is shown in the center is a typical straight radial blade. This is the direction of rotation and this is the velocity triangle of the flow as it leaves the impeller. So, V 2 is the relative velocity leaving the impeller, C 2 is the absolute velocity and U 2 corresponds to the blade speed at the tip of the impeller. So, as you can see the flow leaves the blade radially that is V 2 is indeed radial that is why it is called a straight radial blade. If you look at the forward leaning blade, V 2 leaves the blade tangentially that is why V 2 is at a certain angle to the impeller vanes at the exit and it leaves at an angle of beta 2 which is negative. So, forward leaning blades have a negative blade angle or beta and the velocity triangle is what you see here C 2 and U 2. Backward leaning blades on the other hand have a positive beta 2 and so you have V 2 in this direction and which has a positive blade angle beta 2 and C 2 is the absolute angle and U 2 is the blade speed at the tip. So, these are the three different configurations of impellers which are possible and the two of them as I mentioned straight radial and backward leaning are the ones which are commonly used. Of course, the straight radial blades are have conventionally been commonly used because it is simpler to construct as well as the fact that the blades do not have to undergo a lot of stress because as you bend the blades the stress on the blades becomes substantial, but modern day design hand materials permit us to use these complicated shapes as well. In fact, some of the blades have a combination of straight radial and backward leaning geometry. So, there are blades which have a combination of backward and straight radial as well. So, that is about the impeller where which is probably the more crucial component of a centrifugal compressor and we have seen the velocity triangle corresponding to these three different configurations of a centrifugal compressor impeller. Now, there is another component which is usually part of the impeller itself. We have already seen that in the pictures I had shown in the earlier probably the third slide in today's class where the initial part of the impeller as you have probably have noticed is bent. So, there is a curvature given to the initial part of the impeller and that is known as an inducer and an inducer is basically a component which is just ahead of the impeller or in fact, it is almost always initial part of the impeller itself. The basic function of an inducer is to guide the flow smoothly into the impeller. In the absence of any inducer when the impeller is rotating there is a relative component at the impeller inlet and in the absence of inducer the relative velocity enters the impeller at a certain angle and there could be flow separation taking place from the impeller veins. To avoid that one uses set of blades which are like guide veins and these are known as inducers. So, inducer is the inlet impeller entrance section where the tangential motion of the fluid is basically changed to the radial direction and one may have either acceleration or some amount of acceleration within the inducer as we will see later. Inducer the basic function of an inducer is to ensure that the flow enters the impeller smoothly. Without inducers the flow operation is likely to suffer from flow separation and high levels of noise because of flow separating from the impeller. So, this is a schematic of an inducer and there are two views of the inducer shown. So, the initial part of the impeller as you can see this part of the impeller is the inducer and if you take a cross section of the inducer one would see veins like what is shown here. So, the inlet velocity triangle of an impeller would look something like this that the flow assuming that the flow is coming at an axial in an axial direction C 1 and because of the inducer the relative velocity is v 1 and this is at an angle of beta 1. So, this is the inlet blade speed which is u 1 and the flow leaving the inducer could be at a velocity of v 1 prime let us say at the tip of the inducer. So, that is why it is called v 1 denoted by v 1 prime with a subscript t which corresponds to the tip of the inducer. Now, if you look at the velocity triangles you can see that since the blade speed changes all the way from hub to the tip the velocity triangle and the angles at the hub beta h beta mean and beta tip they are quite different that is because of the fact that the velocity triangle the blade speed continuously changes from the hub to tip. And therefore, if you take the velocity triangles at the hub the mean and the tip velocity triangles are different because of the fact that the blade angles are changing which means that there is a certain twist to the inducer itself as it as we look at the inducer from the hub and trace it all the way to the tip. So, there could be a certain amount of twist now from the velocity triangles we have seen that the velocity leaving the inducer, relative velocity leaving the inducer could be is denoted by v prime t. And therefore, v prime t is a component it is related to the inlet relative velocity through the blade angle. So, v prime t is basically equal to v 1 t times cos beta 1 t and so you can see that since v 1 v prime t is equal to v 1 t times cos beta 1 t there is a certain amount of diffusion taking place even in the inducer that is because v 1 v prime will always be less than v 1 because v prime t is equal to v 1 times cos beta 1. And that means that there is a certain amount of diffusion also taking place in the inducer. And similarly, the Mach number is related to the inlet Mach number through the blade angle beta 1. Now, the third component that we will be discussing is the diffuser. So, we have seen the impeller the inducer and now we will talk about the diffuser. Now, diffuser is has a function very similar to that of a stator of an axial compressor that is the deceleration which was taking place in the impeller continues. And that continues in the diffuser where the flow is further decelerated. Of course, there is neither energy addition or of course, there is certain amount of pressure loss that is as I mentioned total pressure loss taking place in the diffuser vanes, but there is static pressure rise continuously taking place in the diffuser as well. So, the flow leaving the impeller is at a very high speed and so one can convert part of that kinetic energy into pressure rise of static pressure and that is basically through the diffuser. And there are different types of diffusers which have been used in different types of applications and there are vaned types diffusers and vaneless types pipe diffusers and channel diffusers and so on. So, we will discuss some of these aspects and different types of diffusers in the next few slides. So, the high velocity that is of the flow leaving the impeller is basically decelerated using a diffuser. And the diffuser basically decelerates the flow and thereby reduces the absolute velocity of the working fluid and how much deceleration takes place depends upon firstly the application for which the compressor is being used as well as the efficiency of the diffusion process itself. And since the diffusion is as we have discussed many times before is a process wherein the flow tries to encounter an adverse pressure gradient, the chances of the risk of flow separation is always there when the flow encounters an adverse pressure gradient. And therefore, the amount of diffusion that one can achieve in a stationary component like a diffuser will depend upon how much the flow can withstand the pressure gradients. Therefore, the diffuser flow is kind of always affected by or limited by the fact that there could be chances of flow separation taking place. And so, the flow basically leaves from the impeller in a radial direction and then there is a certain space or gap between the impeller exit and the diffuser beginning and that is called the vaneless space. And actually the diffuser as we will see little later that diffusion continues even in the vaneless space and after that the flow enters the diffuser or the vaned space and diffusion continues even further in the vaned space. So, the fluid which leaves the impeller, it leaves the impeller radially outward and then it passes through a vaneless region and subsequently through a vaned diffuser. Now, there are different types of diffusers as I mentioned, vaned type diffusers, vaneless type diffusers. These are diffusers which are conventionally being used and if you look at compressors, centrifugal compressors which have applications in aero engines, these conventional types of diffusers may not really serve the purpose, one needs to employ better diffusion mechanisms of the flow exiting the impeller. So, in aero engine centrifugal compressors one might encounter diffusers which are known as pipe diffusers or channel diffusers which are much more efficient and more amenable to integration with the combustion chambers. For example, if you have can type combustion chambers then it is easy for distributing the flow from these pipe diffusers and directing them towards these can type of combustors. So, these type of diffusers which are used in aero engines are normally not the vaned type diffusers which are used in other applications, diffusers used in aero engines are usually the pipe type of diffusers. We will do now a very simple analysis of the flow through a diffuser, we will restrict our discussion to diffuser the vane type of diffuser because it is much more simpler to in understanding and analysis and we will restrict our discussion to these conventional type of vane diffusers. So, this is the impeller vane and the flow exiting the impeller first enters into a certain region known as the vaneless space and then it is guided out of the compressor into the volute through the diffuser vanes. So, as the flow leaves the impeller it basically leaves the impeller with a radial velocity and this is the radial direction as you can see and so the absolute velocity that actually leaves the impeller is at an angle of c absolute velocity c and so as the flow leaves the impeller and it moves towards the vane diffuser because of the velocity triangle as you see here the flow actually follows what is known as a logarithmic spiral because of the fact that the flow leaves the impeller in a radial direction and then it suddenly encounters a vaneless space. So, if you consider an incompressible let us for the moment consider an incompressible flow in the vane less region which has a constant axial width the mass flow through the incompress through the vane less space let us represent that by m dot. So, from the continuity equation we have m dot is equal to rho times the annular area that is 2 pi r h where h is the width of the vane less space times the radial velocity c r. So, this is equal to a constant. Now, from the conservation of angular momentum we also know that r times c w is also a constant. Therefore, this means that this ratio c w by c r is also a constant that is velocity ratio the tangential velocity to the radial velocity as we have seen in the velocity triangle this is also a equal to a constant and the angle is given by tan alpha. So, here the angle alpha is basically the angle between the velocity and the radial direction the absolute velocity leaving the impeller to the radial direction. So, this means that this velocity is basically inversely proportional to radius and. So, as we increase the radius the velocity would kind of reduce. So, in the vane less space we have velocity which is inversely proportional to radius and therefore, with increase in the radius the velocity absolute velocity actually reduces which means that there is diffusion taking place even in the vane less space and once the flow leaves the vane less space and enters the diffuser vanes since the flow is guided through a diffusing passage which has an increasing area in the radial direction the flow continues to decelerate and there is further static pressure rise taking place even in the vane section of the diffuser. So, what we can see here is the fact that the vane less space also contributes to the diffusion overall diffusion process which begins right from the inducer as we have seen that diffusion also takes place in the inducer vanes it continues in the impeller. Of course, in the impeller there is another mechanism which contributes towards the pressure rise which is the centripetal forces which are coming there. Once the flow leaves the impeller and enters the vane less space the flow continues deceleration and that is because the velocity is inversely proportional to r and it follows a logarithmic spiral. From there from the vane less space the flow enters the vane diffuser space where the flow is further decelerated and the flow exiting the diffuser vanes then goes into the volute or the collector and then it is exhausted from the centrifugal compressor. So, this is the overall working of a centrifugal compressor. So, what I will do is quickly take a recap of what we had discussed in today's class we had discussed about two distinct aspects of centrifugal compressors. We started off our discussion today with the thermodynamics of a centrifugal compressor where we had looked at how diffusion is indeed achieved in a centrifugal compressor from a thermodynamics perspective in terms of temperature and entropy. We have seen that in a centrifugal compressor energy is actually added in the impeller where we have an increase in the stagnation temperature which also contributes to which also gets converted into the stagnation pressure rise taking place in the centrifugal compressor. After the impeller we have the vane less space and the diffuser where there is no energy addition taking place, but static pressure rise continues to take place through these components as well. So, after the temperature entropy diagram we also wrote down the governing equations for centrifugal compressor and how one can analyze the flow through a centrifugal compressor. After this we took up the individual components of a centrifugal compressor and how one can analyze the flow through these different components. We have seen that in impeller there are different configurations of an impeller, the straight radial, the backward face backward leaning blades and the forward leaning blades. I made a passing remark that forward leaning blades are unstable. We will discuss details of that in the next class where we take up the instabilities in centrifugal compressor and then we discussed about the flow through an impeller and the velocity triangles at the inlet of an impeller as well as the flow exits the impeller and how the velocity triangle gets modified as we change the configuration of the impeller from straight radial to backward leaning blades. We then discussed about inducer and the flow analysis of the as the flow passes through the inducer we have seen that it undergoes a deceleration and as the flow leaves the impeller it enters into a vaneless space. Diffusion continues in the vaneless space as well as in the diffuser or the vaned space which follows the vaneless space. So, this was in a nutshell about what we had discussed in the various slides we had for discussion today and we will continue with discussion of different aspects of centrifugal compressors in the next class as well. We will take up an important aspect of centrifugal compressors that is to do with Coriolis acceleration first and then we will define what is known as the slip factor associated with the centrifugal compressor. After that we will take up the performance characteristics of centrifugal compressors and we will also talk about details regarding stall, surge and choking associated with centrifugal compressors. So, we will take up these some of these topics for detailed discussion in the next class on centrifugal compressors. Subsequent to that of course, we will have one session which would be a tutorial on centrifugal compressors we will solve some problems which are related to centrifugal compressors and we will also have some exercise problems for you which you can solve based on our discussions. So, in the next class we will basically have these topics for discussion we will talk about Coriolis acceleration, the slip factor, the performance characteristics and stall surge and choking associated with centrifugal compressors. So, we will take up these topics for discussion in the next class which would be lecture number 32.