 Hello and welcome to lecture number 32 of this lecture series on turbo machinery aerodynamics. We have been talking about the different types of turbo machines and with a stronger bias or emphasis towards the axial flow turbo machines, basically the axial compressors and axial turbines. We have already elaborated the reasons why we are sort of giving little more weightage to the axial flow turbo machines, primarily due to the fact that modern day jet engines operate primarily with these type of turbo machines, the axial compressors and the axial turbines due to some of their inherent advantages. Of course, it is not to mention say that the radial flow counterparts are not being used at all. These are also having applications in certain specific areas primarily to do with the smaller sized engines and various other applications. And therefore, with this in mind we also initiated some lectures on the radial flow machines and we started off with the centrifugal compressors. In fact, it must be kept in mind that the earliest jet engines actually operated with centrifugal compressors and for a long time centrifugal compressors were used in most of the jet engines. And of course, once the axial compressors were developed and designed they sort of slowly replaced centrifugal compressors especially in the larger sized engines. But when you look at this smaller class engines, the thrust class smaller thrust class engines centrifugal compressors continue to be used in such applications. So, in with this in mind we had started the previous lecture with discussion on centrifugal compressors. And lecture 31 was devoted towards an introduction towards centrifugal compressors. We discussed the thermodynamics of centrifugal compression process and also the work done and we had looked at power calculations and so on and the governing equations which are involved in centrifugal compressor design and calculations and analysis. We also had quick look at the different components which constitute a centrifugal compressor like the inlet part of the or the intake of the centrifugal compressor, the inducer, the impeller, then the diffuser vanes and so on. So, we also discussed about how one can make calculations and analysis of these different components of a centrifugal compressor. So, that was what we had discussed in the previous class. We what we will do today is to continue some of our discussion which we had had in the last class and sort of wind up our discussions on centrifugal compressor with this lecture. And the next lecture we would obviously be having a tutorial because having undergone two lectures on an overview of centrifugal compressors, it is essential that we understand how one can make analysis and calculations on centrifugal compressors. So, we will devote the next lecture towards a tutorial on centrifugal compressor. In today's class, we will basically be taking up on few important concepts. We will start our discussion with what is known as the Coriolis acceleration. Then we will discuss about the slip factor, the performance characteristics and also stall surge and choking associated with centrifugal compressors. Now, we will start our discussion with Coriolis acceleration. I am sure you must have learnt at least heard about this term called Coriolis acceleration or called Coriolis forces. In your high school classes in your physics, I guess you must have learnt about Coriolis forces and Coriolis acceleration in sort of a very general perspective. We are going to use some of those principles in relation to our current topic of discussion that is the centrifugal compressors. Now, if you remember in one of the slides which I had flashed in my previous class, I mentioned that one of the aspects that distinguishes a centrifugal compressor from an axial compressor is the fact that in a centrifugal compressor, the pressure rise mechanism is slightly different from that of an axial compressor. In the sense that in a centrifugal compressor, pressure rise can also occur because or it primarily occurs because of displacement of the centrifugal or centrifugal force field. And because of that, there is a pressure rise taking place as a result of diffusion in the centrifugal compressor. So, there are these two components which contribute towards the overall pressure rise in a centrifugal compressor. I think I mentioned that the problems associated with the boundary layer flows are not that severe in a centrifugal compressor. This is indeed true that boundary layer separation is not it is still a matter of concern, but it is not the primary matter of concern like we have in axial compressors which also partly explains the fact that axial compressors can develop much lower pressure ratio per stage as compared to centrifugal compressor primarily because of the fact that axial flow performance is impeded by boundary layer characteristics. So, we will also try to explain in the context of centrifugal compressors. Let us also try to look at the pressure rise mechanism and deceleration or diffusion in the passages through Coriolis acceleration as a possible means. So, let us try to analyze what Coriolis acceleration does to this overall performance of a centrifugal compressor. Now, what we will see very soon is the fact that Coriolis acceleration is going to lead towards a certain discrepancy in the velocity triangle from the ideal characteristics that is the velocity triangle at the exit would be slightly different from what it should have been and that is attributed to centrifugal Coriolis acceleration and because of that it leads to certain amount of pressure loss as we will see a little later. So, what we are going to do is that we will consider a certain fluid element which is travelling radially outward in the passage of a rotor. So, before this let me just quickly go through these two bullets I have written here. The one is to do with centrifugal flow field which is primarily not a result of the boundary layer separation and basically that the fact that the pressure change due to centrifugal flow field is not really a cause of boundary layer separation. We will try to explain that with Coriolis forces. So, if you consider a certain fluid element which is let us say travelling radially outward in the passage and we will look at the velocity triangles of this particular fluid element during a certain time period d t. So, this is the fluid element that I am referring to let us consider a fluid element which is passing through these straight radial vanes. Of course, you can see that this is the impeller and these are straight radial vanes a fluid element is passing through the radial vanes which are straight. So, the fluid element obviously has a relative velocity of v and blade speed or rotational velocity omega r where r is the radius at which this fluid element is currently located and omega is the rotational speed. So, if you look at the velocity triangles for this fluid element in addition to v and omega the absolute velocity c is given by the resultant of v and omega r. So, he is the basic velocity triangle is shown by the solid lines and you can see v and omega r and its resultant is the absolute velocity c. Now, after a certain time period well the impeller is rotating that also displaces this fluid element by a certain distance. If we consider the fact that this fluid element is being rotated by this centre that is shown here then after a certain time period d t the fluid element deflects and therefore, the new velocity triangle is shown here by the dotted lines. So, new velocity triangle corresponds to a radial location which is equal to r plus d r where d r is the differential change in the radius with time d t. So, the new speed rotational speed becomes omega omega is unchanged omega multiplied by r times d r and let us assume the fact that there is negligible change in the relative velocity during this time, but because omega has changed c prime also changes that is the absolute velocity changes and it takes a new value which is c prime. Therefore, the net change in absolute velocity is given by d c which is as you can see a tangential component of which is shown by d c subscript w d c w which has two contributions. One contribution is because of the change in the radius omega times d r and the other contribution is because of v times d theta where d theta is this angular deflection. So, here what you need to understand is that there are two distinct aspects which has led to this change in the absolute velocity or let us say the tangential component of the absolute velocity. So, there are two contributions here one of them is from the fact that the radius has now changed to a differential there is a differential change in the radius or radial location of the fluid element which is given by d r and that leads to a change in the peripheral velocity which is omega times d r. There is also a change our contribution from the fact that relative velocity being remaining unchanged for the fact that we will assume that it is unchanged because it is change is much smaller compared to the other components. So, v times d theta where d theta is the angular deflection angle through which the fluid element was deflected. So, these two components put together result in a change in the tangential component of the absolute velocity and at the moment we are going to be looking at this change in the tangential component and it is change with reference to time. So, d c w by d t is basically rate of change of the tangential component of velocity with time that is an acceleration and that is basically the Coriolis acceleration and that is primarily because of the fact that there is a rotation given to the fluid element. If the fluid element was displaced simply radially upward without any angular deflection Coriolis forces would be negligible there would be hardly any Coriolis force acting on the fluid element. But because of the fact that in addition to the fact that the fluid element is getting displaced radially outward there is also an angular displacement of the radial element both of these contributions put together results in this acceleration which we will denote as the Coriolis acceleration. Let me recap the velocity triangles once again. So, here in this velocity triangles the default velocity triangle is this. This is shown by the solid lines v and omega r v is the relative velocity of the fluid element omega r is the tangential component or the plate speed peripheral velocity resultant of that is c. When it is displaced we will assume negligible change in v because this magnitude is much smaller than any other magnitudes here. And so we have a new absolute velocity which we have denoted by c prime which is a resultant of omega times r plus dr and v and this change in the absolute component has tangential component which is dc w which has two contributions omega dr and v d theta. So, the sum of these two gives us the dc theta. So, let us add up these two now the magnitude of this change in the tangential component dc w is basically sum of omega dr plus v d theta or dc w is omega into dr we have expressed as v times d t plus d theta can be expressed as omega into d t. So, v into omega into d t. So, this basically would be equal to 2 into omega v into d t and therefore, dc w by d t is acceleration a theta that is acceleration in the tangential direction or a w or a theta as it is called it is the Coriolis acceleration is equal to twice omega v that is the Coriolis acceleration is directly a function of the relative velocity and the rotational speed. So, higher the rotational speed higher the Coriolis acceleration and higher the relative velocity obviously, that also changes the Coriolis acceleration. Now, this basically requires a certain amount of pressure gradient why is there a Coriolis acceleration in the first phase firstly there is a rotation given to the fluid element there is also a certain amount of pressure gradient which is leading to this Coriolis acceleration. So, if we were to look at the pressure gradient we can express the pressure gradient in the tangential direction as 1 by r del p by del theta that is the rate of change of pressure in the tangential direction which is basically equal to twice rho into omega into v. So, the radial pressure or tangential pressure gradient can be expressed in terms of twice into 2 into minus is basically referring to the fact that the pressure gradient is direction dependent and depends on which direction the rotor is rotating. So, minus 2 rho into omega into v. So, this is the amount of pressure gradient which is acting in the tangential direction and that leads to this much amount of Coriolis acceleration. So, the Coriolis acceleration basically requires certain amount of pressure gradient. So, what we will do is to look at the rate of change of relative velocity in the tangential direction and see how it is related to the or whether it is indeed related to the Coriolis acceleration or not. We have seen that the tangential velocity well Coriolis acceleration is in fact directly proportional to the relative velocity. Let us see the rate of change or is there a rate of change of the relative velocity in the tangential direction. So, if we look at the fact that there is a tangential pressure gradient, then tangential pressure gradient needs to result in a positive gradient of v in the tangential direction because if there is a pressure gradient in a certain direction that needs to also mean the fact that there has to be a velocity gradient in the same direction. And therefore, the pressure gradient in the tangential direction we have expressed in the previous slide. We can equate that with the Coriolis acceleration and then we will see that 1 by rho d p by r d theta is basically equal to minus d into v square by 2 by r d theta which is minus d by r d v by d theta. So, this if you look at if you compare this with the Coriolis acceleration we can basically infer that 1 by r d v by d theta is equal to 2 into omega. And what does this basically tell us? This tells us the fact that there would be a change in the relative velocity in the tangential direction as well. Now, that was in an idealized scenario what would not have expected any change in relative velocity in the tangential direction. But we can now see from our analysis that in addition to the fact that the relative velocity will keep changing in the radial direction because that is what the centrifugal compressor does. There will also be a change in the relative velocity in the tangential direction and that is a very crucial piece of information for us because we will very soon see that this leads to a change in the velocity triangle at the exit of the impeller from what it should have been. So, from our understanding of the Coriolis acceleration we have seen that as a result of the Coriolis acceleration and eventual outcome of the fact that there will be a Coriolis acceleration due to the tangential pressure gradient. Tangential pressure gradient in turn leads to a tangential has to lead to a tangential velocity gradient and this velocity would be affected the component of velocity which is affected is the relative velocity. So, the relative velocity would have a tangential gradient that is it will keep changing in the tangential direction. So, if you look at this aspect in a schematic sense I mentioned that the let us consider the same impeller this is the same impeller we were talking about these are the two straight radial blades and as an outcome of the analysis we just have seen there will be a tangential variation in the relative velocity. So, these vectors which are shown here are basically the velocities relative velocities there is a gradient of this relative velocity in the tangential direction here plus and minus indicate the pressure or higher pressure and this would be the pressure surface and suction surface let us say in the case of an axial compressor blade. So, there is an increasing pressure gradient and correspondingly change in the relative velocity in the tangential direction. So, what happens because of this is that as the fluid element begins to leave the impeller that is if you trace the fluid element which was let us say here and as it leaves the impeller by the time it reaches the tip of the impeller because of this tangential velocity gradient in the relative velocity you can see that the relative velocity is lagging behind the radial direction. So, the relative velocity is actually now pointing in this direction because it is having a gradient in the tangential direction. So, because of the gradient the vector of the relative velocity leaving the impeller is inclined at an angle it is lagging behind the radial direction and therefore, what is the outcome of this? The outcome of this is the fact that ideally one would have assumed a radial velocity relative velocity is being equal to radial velocity that is no longer true therefore, one has tangential component in the absolute velocities scale that is C w 2 which is not equal to u 2. So, if you recall velocity triangles that I had shown in the previous class where I had shown three different types of impellers forward leaning straight radial and the backward leaning blades. For the straight radial blades please go back and take a look at the velocity triangle you will see that I had drawn the velocity triangle with the relative velocity v or v 2 in the radial direction which means that C w 2 would have been equal to u 2 that is an ideal scenario. Now, here we have seen that due to the Coriolis acceleration and its effect and the tangential velocity gradient there will not be a radial of the relative velocity will not really will not necessarily leave the impeller radially. So, there is a change or difference between the tangential component of the absolute velocity that is C w 2 and the blade speed at the impeller exit u 2. So, this change in our difference in C w 2 and u 2 is captured by a parameter which is referred to as the slip factor well slip factor basically tells us the fact that at the exit of the impeller the relative velocity is lagging behind the radial direction resulting in a change or difference in the tangential component of absolute velocity C w 2 as it should have been equal to u 2. So, the ratio of C w 2 to u 2 is called the slip factor and this difference in the relative velocities is basically expressed as a fraction and that is C w 2 by u 2 which is usually denoted by sigma and the sigma is a factor for actual blades it will be less than 1 and one would like to have a slip factor close to 1 because that means C w 2 will be equal to u 2 and it will lead to much higher let us say efficiency and pressure ratios for the same rotational speeds and impeller diameter. Now, slip factor is also a function a strong function of the number of blades you can see I have been I have shown two straight radial blades and you have seen that from one blade to another there is a tangential velocity gradient that means the larger the distance between the blades the velocity gradient will keep increasing and therefore, the slip factor would be or the difference between C w 2 and u 2 would be higher as we increase the blade spacing lower the number of blades the greater would be the difference and as you keep increasing the number of blades the tangential velocity gradient is lower and therefore, the slip factor is also likely to be lower. So, slip factor being strongly related to the number of blades people have come up with empirical correlations for calculating slip factor based on the number of blades one of the most commonly used empirical correlation is given by standards and that is known as the standards slip factor it is equal to 1 minus 2 by n where n is the number of blades which means if there are let us say 10 blades for straight radial blades the slip factor would be 1 by 1 minus 2 by 10 that is 1 minus 0.2 and that is 0.8 and if you increase the number of blades to let us say 20 then you can see that slip factor becomes 0.9 and so on and that is as we increase the number of blades the radial velocity of the change in the tangential direction of the radial relative velocity becomes lesser and therefore, there would be lesser difference between C w 2 and u 2 whereas, for larger spacing or lesser number of blades the variation in the relative velocity in the tangential direction would be larger and larger leading to much larger difference between C w 2 and u 2 leading to poor values of slip factor. And so slip factor being a strong correlation strongly correlated to the number of blades there are different parameters or different researchers have come up with different empirical correlation the most common one is what I have shown which is strictly applicable for straight radial blade for backward leaning blades the slip factor is calculated in a different way and there are different other correlations which are used for such blades and you will find in literature there are many more empirical correlations which are in some way or the other related to the number of blades. So, the basic effect of slip factor is the fact that it reduces the magnitude of the swirl velocity or tangential component of velocity leaving the impeller and since pressure rise is a function of this tangential velocity for lower values of tangential velocity the pressure rise also decreases that for slip factor directly affects the pressure ratio of a centrifugal compressor which obviously is not a good thing that a designer would want to keep for a given rotational speed and impeller diameter try to maximize the pressure ratio and presence of slip factor can bring down the pressure ratio because of the fact that it affects the tangential velocity. So, the out the way out of this is to use more number of blades well more number of blades means you will obviously have higher frictional losses and therefore, it is probably not a good idea to keep increasing the number of blades or other option is to increase the impeller diameter which is for a normal application a land based application may not be a big deal you can keep increasing diameter to some levels, but not for an aero engine application where you know it will increase the drag as well, but larger the diameter of the impeller the stress on the impeller blades also goes up which means then they will have to invest in better materials which can withstand higher stress or the other option is to increase the rotational speed and again if you increase the rotational speed for the same diameter that also affects this stresses on the on the blades and therefore, that again is a constraint. So, you can see that there are all kinds of constraints here in terms of stressing stresses on the blades or frictional losses. So, there is lot of scope for an optimization to be carried out here and to determine what is the best possible configuration for number of blades versus impeller diameter versus the rotational speed and that can give us the best possible efficiency as well as the pressure ratio. So, having understood slip factor and its effect on centrifugal compressor performance let us now talk about the performance of a centrifugal compressor in general. We have already discussed about the performance characteristics of axial compressors as well as axial turbines. Centrifugal compressor performance characteristics would look in some way similar to what we have already discussed for an axial compressor, but problems related to choking is a little more severe in a in a centrifugal compressor as compared to an axial compressor also centrifugal compressors undergo similar problems that we have seen for axial compressors like stall and surge. So, let us look at the performance characteristics of a centrifugal compressor and try to understand how we can estimate the performance of a centrifugal compressor. So, we will evaluate the performance of a centrifugal compressor in a same way as we have done for an axial compressor. We will look at the pressure ratio the dependence of pressure ratio and efficiency on a mass flow rate all of which of course, the pressure ratio and mass flow rate mass flow rate being non dimensionalized for different non dimensionalized operating speeds and we will very soon realize that compressors centrifugal compressors also suffer from instability problems like surge and rotating stall. So, I will make this a little quick because we have already done this for an axial compressor is exactly the same procedure the non dimensional groups are derived from dimensional analysis the exit pressure ratio or exit total pressure p 0 2 and efficiency are functions of variety of parameters like mass flow rate inlet stagnation pressure inlet stagnation temperature rotational speed ratio of specific gamma the gas constant are the viscosity the design as well as diameter. So, if you non dimensionalized this we get these many non dimensional groups p 0 2 by p 0 1 that is the pressure ratio efficiency being functions of m dot root gamma r t 0 1 by p 0 1 d square omega d by square root of gamma r t 0 1 omega d square by nu gamma and design. Now, for a given design and given diameter we can drop a lot of these terms and basically the pressure ratio and efficiency becomes function of mass flow rate m dot root t 0 1 by p 0 1 and n by root t 0 1 this is the non dimensional speed and this is the non dimensional mass flow rate this we will further process for a standard day pressure and temperature. So, we get p 0 2 by p 0 1 efficiency being functions of m dot root theta by delta and n by root theta where theta is t 0 1 by t 0 1 of standard day and delta is p 0 1 by p 0 1 standard day typical values taken for this are t 0 1 standard day is 288.15 Kelvin which is basically 25 degrees Celsius and pressure of standard day is 101.325 kilo Pascal. So, with this set of non dimensional parameters pressure ratio and efficiency being functions of mass flow rate non dimensionalized as well as the non dimensional speed we will now look at how the performance characteristics can be plotted before that let us first look at a very general characteristics which is applicable for any centrifugal compressor and then we will look at a typical performance characteristics in general. So, if you look at the variation of pressure ratio was mass flow rate one can trace characteristic like this ideally one could raise a characteristic like this there are several salient points which I have marked here point a b c d and e. Now, let us take a look at what happens as let us say the compressor was operating at some point e and then as the mass flow is reduced as we throttle the mass of the compressor and decrease the mass flow the pressure ratio across the compressor increases. And as it increases it reaches its peak and eventually you will see that after it reaches its peak the pressure ratio drops and this could this drop of course, could be very drastic as well in some compressors. In the case of axial compressor we have already seen what really happens the fact that when the throttle characteristics become well intersect the pressure ratio characteristics beyond a certain level the compressor undergoes what are known as instabilities. So, the exact same thing happens even in the case of centrifugal compressor beyond this point e where the slope of the pressure ratio mass flow characteristics is positive the centrifugal compressor undergoes instabilities and that is why b is referred to as the surging limit. And any point after the on the left hand side of b between a and b let us say point d would be considered an unstable point and the compressor cannot really operate in a stable condition there. We have discussed the stability in reference to axial compressors in quite detail and all those arguments are very much valid even for a centrifugal compressor. And on the other hand here we have the what is known as the choking limit beyond point e and little later the slope becomes extremely sharp and there is a very sharp drop in the pressure ratio characteristic and mass flow remains more or less constant and that is referred to as the choking limit. So, we will discuss choking in little more detail in relation to centrifugal compressor because these compressors tend to be affected more by choking and as compared to axial compressors and so we will discuss choking in little more detail and limit our discussion on surge and stall because that we have already discussed in relation to axial compressors. So, let us now look at an actual centrifugal compressor map performance map in terms of pressure ratio non-dimensional mass flow rate and for different non-dimensional speeds. So, if you look at the pressure ratio characteristics versus mass flow rate characteristics. So, the previous slide I had shown an idealized curve here we have seen that the curve on the left hand side is not possible and beyond this it chokes. So, the actual performance characteristic is limited between these two points b and e and that is what is shown here by these different lines. So, as you keep changing the speeds the performance characteristics also change and you can see that as a speed reaches its max of the design speeds the performance characteristics become sharper and sharper. For lower speeds one can see one can notice a shallower pressure ratio versus mass flow characteristics that becomes steeper as one proceeds for higher and higher speeds and all these lines are terminated on the left hand side by the surge line and on the right hand side by the choking line. And if you join all the points of maximum efficiency we get the dotted line that is shown here and one would ideally want to operate the compressor very close to this maximum efficiency line provided that of course this line is not very close to the surge line which of course can put the compressor into the risk of surging. Now, if you look at the efficiency characteristic again very similar to the axial compressor that we discussed like efficiency versus mass flow rate for non-dimensional speeds and with increase in speed you can see that the range of high efficiency becomes narrower. For higher speeds we have a narrower band of operation where the efficiency is high and it becomes very sensitive to the mass flow rate. For lower speeds of course efficiency is not is relatively lesser sensitive to the mass flow rate and there is a wider range of operation possible with higher slightly higher efficiencies. In comparison with an axial compressor we can see that even centrifugal compressors have a two aspects of or two lines which basically define the performance on left hand side we have the surge line on the right hand side we have the choking line. Now, between those two points on the map that I shown that is point A and B the compressor may undergo instabilities it could be a rotating stall which eventually leads to surge. Now, we have already discussed these instability mechanisms in fairly good detail I will just quickly mention what happens while a compressor undergoes some of these instabilities. So, basically the operation of a compressor in the positive slope of pressure ratio versus mass flow rate is unstable operation and the compressor cannot operate in that region in a stable manner. One of the instabilities that affect the performance is rotating stall and the more severe one is the surge wherein there is a sudden drop in the delivery pressure and of course, violent aerodynamic pulsations. And in the case of centrifugal compressor it has been noticed that in general the surging begins in the diffuser passages because I think I mentioned in the last class that diffuser passages are significantly affected by boundary layer performance. So, that is one of the weak links in a centrifugal compressor where the performance is very sensitive to boundary layer flow and therefore, surging has been in general observed to initiate get initiated in the diffuser passages. Now, in a centrifugal compressor the pressure ratio or the performance is also a type function of the type of blade that is used in a centrifugal compressor. In the last class we discussed 3 different possible blades blade configurations a forward leaning type a straight radial and backward leaning. Now, if you look at the performance of these 3 different types of blades and analyze the velocity triangles at the exit urge you to go back to the previous lecture and take up that slide where I had shown the velocity triangle for these 3 cases. You will notice that theoretically forward leaning blades produce a higher pressure ratio because if you look at the velocity triangle at the exit you would appreciate this aspect and one would expect forward leaning blades to be much better in terms of performance. But what is interesting to notice is the fact that forward leaning blades have an inherent instability because if you look at pressure ratio versus mass flow characteristic for forward leaning blade the characteristic always has a positive slope and we have just now discussed that operation on the positive slope of pressure ratio mass flow characteristic is inherently unstable and that is the reason I think I mentioned in the class class that forward leaning blades are not really used in centrifugal compressors because of the fact that they are inherently unstable. That is why straight radial and backward leaning blades are commonly used in modern day centrifugal compressor. So, if you look at the performance characteristics either in terms of pressure ratio or the temperature rise versus either mass flow rate or the flow coefficient a forward leaning blade would have a characteristic which is positive slope throughout. So, this is like the left hand side curve of a centrifugal compressor characteristic where of course this is idealized, but one would still get a positive slope throughout for a forward leaning blade which means that this blade is going to have instabilities irrespective of the mass flow rate and therefore, this is not a favorable type of blade that can be used even though the pressure ratio performance is much better than straight radial or backward leaning. So, these two configurations are the ones which are commonly used the straight radial and the backward leaning blades. So, what I will do next is to discuss about problem of choking which is what is probably little more severe in the case of centrifugal compressors as compared to axial compressors whereas, rotating stall and surge are still the limiting performance parameters on one side. On the other side we also have the choking problem associated with trying to increase mass flow rate and beyond a certain level of mass flow rate compressibility effects will prevent us from operating the compressor beyond a certain level of mass flow rate. So, let us take closer look at choking in more detail because we already discussed about the other instabilities in relation to axial compressors. So, as the mass flow increases in the case of centrifugal compressors the pressure ratio will decrease as we have seen in the performance characteristic and therefore, that also reduces the density. After a certain point one would not be able to increase the mass flow beyond a certain value the compressor is then said to have choked that is on this that is probably we have reached the right hand side of the performance characteristic wherein the mass flow rate has reached its maxima and we are not able to increase mass flow rate beyond that level and that is when the compressor is said to have choked. So, in a centrifugal compressor we have seen the different components which constitute a centrifugal compressor we have the inlet the impeller and the diffuser veins. So, calculation of the choking mass flow is different depending upon whether it is the stationary component or the rotating components. So, we will take a look at the choking mass flow as applicable for inlet the impeller and diffuser veins and see their dependence on the upstream parameters. So, choking behavior because it is different for rotating passages from that of stationary passages. So, if we consider the inlet we know that of course, choking irrespective of whether it is inlet or impeller takes place when the Mach number reaches 1. So, when Mach number is unity the ratio of static temperature to total temperature T by T 0 is equal to 2 by gamma plus 1 because Mach number if in the isentropic relation if we equate m is equal to 1 we get this relation between static and stagnation temperature. So, for the moment if we assume an isentropic flow then the choking mass flow rate is basically m dot a per unit area is equal to rho naught a naught into 2 by gamma plus 1 raise to gamma plus 1 by 2 to gamma minus 1. So, this basically comes from mass flow rate being equal to rho a v and we have expressed rho in terms of stagnation density and velocity in terms of the speed of sound. And here we can see that this mass flow rate is expressed purely in terms of parameters which are the upstream parameters or the inlet stagnation conditions which remain constant. So, as we keep changing the operating condition the inlet conditions are still fixed which means that the mass flow rate also has to remain constant and that is why when Mach number becomes 1 what you can see is that the right hand side of the mass flow rate equation has all the parameters which are basically constants. And therefore, when Mach number becomes 1 one can actually calculate the choking mass flow based on the inlet stagnation conditions. The next component that we will take a look at is the impeller in a rotating passage as we have seen the flow conditions are usually referred through the rothalpy which I had discussed in the last class. And during choking in the case of an impeller it is the relative velocity which basically becomes equal to this period of sound when Mach number becomes unity. And so if you look at the expression for rothalpy we have i is equal to h plus half into v square minus u square. And so and we know that stagnation temperature can be expressed in terms of the corresponding static temperature and the speed of sound. So, here we if we express enthalpy in terms of stagnation temperature and v square because when Mach number is equal to 1 v becomes equal to speed of sound. And therefore, that becomes gamma r t divided by 2 c p because enthalpy is t naught times c p minus u square by 2 c p. So, we have divided this by the speed of sound and therefore, we get t naught 1 is equal to t plus gamma r t by 2 c p minus u square by 2 c p. So, if you simplify that we get t by t naught 1 is equal to 2 by gamma plus 1 into 1 plus u square by 2 c p t 0 1. And therefore, mass flow rate which is rho a v again gets expressed as rho 0 1 a 0 1 into t by t 0 1 raise to gamma plus 1 by 2 into gamma minus 1 which can be simplified further. And what you see is that mass flow rate is equal to rho 0 1 a 0 1 into 2 plus gamma minus 1 u square by a 0 1 square by gamma plus 1 raise to gamma plus 1 by 2 into gamma minus 1. So, this rather complex expression for the mass flow rate once again tells us that in addition to the inlet conditions in the case of impeller we also see that it is a function of the rotational speed. And in the case of inlet we have seen that mass flow rate is purely a function of the inlet parameters and that therefore, mass flow rate gets fixed once the inlet conditions are fixed. In the case of impeller besides the inlet conditions you also have the rotational speed. That means that in principle it should be possible for us to operate the compressor at a different mass for at a higher mass flow than the choking mass flow for a higher rotational speed. Of course, this will also require that if once you start operating the compressor at a higher rotational speed the other components do not choke because if the other components choke the compressor will still remain under choke condition. So, other components permitting like the inlet or the diffuser the impeller performance or impeller choking because it is a function also of the rotational speed operating the compressor at a higher rotational speed may permit us to operate or deliver a higher mass flow provided all other components also are able to operate at this new operating point. Choking mass flow in an impeller is a basically a function of the rotational speed and the compressor in principle should be able to handle a higher mass flow with an increase in speed provided that other components like inlet or diffuser does not really undergo choking for this rotational speed. Now, the last component is the diffuser the choking mass flow in a diffuser has an expression the same as that of an inlet. So, it is again a function of the inlet conditions for the diffuser. So, mass flow rate is function of rho naught a naught into 2 by gamma plus 1 raise to gamma plus 1 by 2 into gamma minus 1. Now, here you can see that the stagnation conditions at the inlet of the diffuser depend upon the inlet of the diffuser itself which is the impeller exit and therefore, mass flow rate can basically be related to the rotational speed of the impeller and therefore, you can see that the operation of the diffuser and impeller are sort of coupled because the diffuser choking mass flow is a function of the impeller exit conditions and impeller exit conditions can basically be changed by changing the rotational speed and. So, there is a certain amount of coupling between the impeller operation and the diffuser operation and so with this analysis it should be possible for us to calculate under what conditions centrifugal compressor is likely to choke and how is it that we can operate still operate the compressor at a probably slightly higher mass flow by looking at the other components involved like the inlet and the diffuser and ensuring that all the three components can operate in under the new rotational speed without undergoing a choke. So, let me now quickly recap our discussion in today's class. We had discussed a few aspects which are an extension of what we had discussed in the last class. We started our lecture today with detail discussion on the Coriolis acceleration and we have derived an expression for Coriolis acceleration. We have seen that Coriolis acceleration leads to a tangential velocity gradient or velocity gradient in the tangential direction and that is what leads to a difference in the velocity that is leaving the impeller that is it leads to a certain amount of difference between C w 2 which is the tangential component of absolute velocity and u 2 and that difference is what is referred to as the slip factor. We have seen that slip factor is a strong function of the number of blades lesser the number of blades the greater the spacing between the blades the greater is the tangential velocity gradient and therefore, the slip factor becomes lower and lower. So, this can be that is with higher number of blades one can reduce the tangential variation in relative velocity and therefore, one can achieve higher values of slip factor. So, slip factor and its dependence on the number of blades we have seen the Stanitz formula where one can approximate slip factor as 1 minus 2 by n where n is the number of blades and of course, that is true for a straight radial blade. We have then discussed about the performance characteristics and of a centrifugal compressor of course, we did not discuss too much of the details of the performance characteristics because it is very similar to that of an axial compressor. We spend much less time discussing about surge and stall because we already discussed that and we have discussed slightly more details about the choking condition which is affecting which might affect a centrifugal compressor performance and how one can estimate the choking mass flow for the inlet the impeller and the diffuser and the interrelation between choking conditions for these three different components. So, this were the topics that we had discussed in today's class and so, this will sort of wind up our discussion on centrifugal compressors and therefore, the next lecture as I mentioned in the beginning we will basically take up few problems for solving. So, we will have a tutorial session in the next class. We will discuss about how we can solve problems pertaining to centrifugal compressors and at the end of the tutorial, we will also have a few exercise problems for you to solve based on the discussion during the last few lectures. So, we will take up a tutorial in the next lecture which will be lecture number 33.