 Good morning and welcome to all of you to this session of this course. So far we have discussed the basic fluid mechanical principles of some fluid machines like Pelton turbine, Francis turbine, Kaplan turbine and centrifugal pump and reciprocating pump. The characteristic feature of all these machines was that the working fluid was water and we discussed the basic principle of their operations along with the description of different parts of the machines and its performance criteria or performance characteristics. Now, there are several other types of fluid machines are available in practice or found in practice which use air, steam or gases. Gases means the mixture of air and the products of combustion which are generated by the burning of fuel as required and those machines the basic difference is that since use a fluid which are not liquid is compressible in nature in a sense that their bulk modulus of elasticity is relatively much lower compared to that of liquid. Therefore, what happens is that the density changes with pressure as well as with temperature as the fluid flows through the machines and apart from that there are other features of compressible flow found in those machines depending upon the regime of the flow. These machines are usually known as compressible flow machines and in a more acceptable and popular terminologies the turbo machines. Now, a detailed discussions on turbo machines is beyond the scope of this course. We will discuss only few of such machines like centrifugal compressors axial flow compressors fans and blowers. So, first we start with centrifugal compressors. Now, centrifugal compressor is just you think is similar to that of a centrifugal pump which we already discussed which use working fluid water. Now, as I already told in the beginning of this course that a pump or compressor is a machine where energy is being supplied from outside and that energy is again imparted to the fluid by the machines by virtue of which fluid gains its internal energy. That internal energy is gained by the fluid in terms of a rise in static pressure which we can loosely call as pressure energy and in terms of the kinetic energy that means high flow velocity. In pumps and compressors the fluid internal energy is usually obtained in terms of higher static pressure with low flow velocity. While the machine using the liquid or water is termed as pump the machines using air or vapour are termed as compressor. Now, today we will discuss the centrifugal compressor I will come to that discussion why it is called centrifugal compressor. Now, the centrifugal compressor is usually just before starting its description I tell you it has got application today in small turbo jet turbo fan turbo prop engines and small gas turbine plants and in all those engines along with the axial compressors which have been developed later on the centrifugal compressors are also used these are the applications. Now, I come to the basic of centrifugal compressors. Now, you see a centrifugal compressor as I have told that it uses air as the working fluid and energy is being given to the machine to raise the static pressure. Now, centrifugal compressors consist mainly of three parts one is the casing stationary casing all machine basically have stationary casing number two is the important part is the impeller which is known as impeller which is the rotating part of the machine which is known as rotor that is rotating part of the machine rotating part of the machine part of the machine where the energy is being imparted to the fluid and number three is the diffuser diffuser diffuser is the static one sometimes it can be told as stator where what happens when the energy is being imparted in the rotating part or rotor known as impeller the energy is gained both in terms of pressure rise in static pressure and velocity mostly in the terms of velocity along with the rise in static pressure diffuser is that part where the fluid is being decelerated to gain in static pressure by decreasing the velocity because main objective of the centrifugal compressor is to have air at high static pressure. So, therefore, the velocity which is gained by the fluid in terms of the kinetic energy in the impeller is being converted to static pressure by the deceleration which sometimes is called as the diffusion process in fluid mechanics to obtain a rise in static pressure. So, these three are the important parts and comprises a centrifugal compression. Let us now see one by one that how a the parts look like now here we see that this is the impeller you see this is the impeller looks like a looks like this it is just like a rotating disc you see this is the impeller this is the impeller now this is the inlet of the impeller this is known as impeller eye the air is being shot when the impeller rotates the air is being shot like this through the impeller eye this is the impeller eye this is the impeller eye now there are vanes like this, these are the vanes which are curved initially and then at the outlet it is more or less radial flat and radial and the fluid which is shocked change this direction and flows like this try to understand flows like this, this is a radially outward flow machines, this is a radially outward flow machines now what happens I tell you the air is shock actually by the rotating action of the impeller at the impeller eye here it is impeller eye see impeller eye and then the air is ultimately directed radially outwards through the rotating impeller now you see here that since when the air goes through this passage radially outward and at the same time impeller is being rotated for example this is this is the diffuser vane I am sorry this is the impeller this part is rotated this part is rotated like this then what happens as the fluid goes out in the radial location it pressure increases along the radial direction so this can be explained this way that because of this rotation a tangential velocity is imparted on the fluid and therefore fluid obtains a centripetal acceleration which is manifested in terms of a rising pressure radially outward or you can see other way that if you consider a fluid element in this radial gap if you see here if you consider a fluid element like this this is the radial direction or if the fluid particle has a tangential velocity then there has to be a inward radial force due to the pressure on both the side let this side is 2 p2 and this side is 1 p1 so net force should be acting on the fluid in this radially inward direction to balance the centrifugal force in the outward direction so that fluid element can rotate in the tangential path this is the basic requirement for any tangential flow in a fluid that means if the fluid has a tangential component of velocity then the pressure rises radially outward this is because this pressure gradient gives rise to a inward radial force to balance the centrifugal force so therefore the pressure rise obtained in the outward radial direction because of the centrifugal action due to the rotation of the fluid element and that is the reason for which this type of machine is called as centrifugal machines and here it is centrifugal compressor clear so therefore since it is radially outward so pressure is automatically gained by the action of the tangential velocity that is the centrifugal or centripetal acceleration so pressure rise takes place and at the same time by the action of the blade the blade imparts the tangential this velocity to the fluid because since the fluid is flowing through the passage fluids acquires the velocity because of the rotating blade which is basically the impulse action that I already explained while discussing the hydraulic machines so this is basically the impeller part the fluid is the air here is sucked axially by the impeller eye and then goes radially outward through these impeller blade passages then what happens when it comes out of the impeller tip the air has got a high velocity at the same time it has got a high pressure but we want more pressure rather less velocity so what happens therefore it goes through some stationary passages that is being made by these veins these are stationary veins and this is known as the diffuser this passage is known as the diffuser so you see it is written this is the diffuser where what happens when the flow takes place there is no energy exchange only fluid flows in a directions where the area increases so what happens simply the pressure is increased because area increasing means by continuity the fluid velocity decreases and in consequence to that the pressure increases this is the process by which the fluid is decelerated and its pressure is increased so that at the final outlet that the final outward periphery of the diffuser here we have air at a very high static pressure but at a relatively much lower velocity so this is impeller blades as you see and now here another thing I will explain afterwards before entering to the fixed then passages known as diffusers there are some vaneless space which is known as vaneless diffuser vaneless space I will come to that afterwards and why so many number of passages are made that also I will tell afterwards in pump you have seen that the similar thing similar thing happened but similar thing was made by a volute casing where the fluid is decelerated to get high pressure of the water at the expense of its high velocity at the outlet of the runner there the impeller was called they are also impeller sorry not runner impeller at the outlet of the impeller but there we had a single volute casing without number of passages created by this type of vanes but here the vanes are there to create a number of passages to divide the flow into small passages I will come to that afterwards this you can have a look of this diffuser and the impeller if you take a view from this side you will get a view like this so this is the impeller blade this is the diffuser this is the depth of the diffuser now sometimes there are the impellers are both sided that means the air is shock from this side of the impeller from this side of double action so this is also shown like that so this is a double sided impeller that means the air flows from both the sides double sided impeller these are the schematic views of the impeller now after this we come to this figure now we have to find out the equations for energy transfer in this machine that means in air okay so now let us see that this figure is shown in a rather this figure if we have a view from this direction and if you see that front view it looks like this this is the blade which is radial and relatively flat add the outer periphery and curved here so here this is the impeller tip so this the air comes like this axially and it goes then is bent like this and it goes radially outward through the blade passages like this so therefore if one sees so sees from the top the blade looks like this this is the blade this is curved this is the curvature of the blade at the inlet and then goes radially straight at the outlet now the inlet velocity is made this way that under the design condition the fluid as I told earlier in all hydraulic machines we always make the flow in such a way that the fluid whether it is liquid or it is gas or it is air always should flow in a way that it should glide the blade surface that means while it enters the blade it should glide the blade surface which means that the velocity vector relative to the blade because blade is a moving element so if fluid has to glide over that moving blade means that the fluid velocity relative to the blade should be such that the angle should match that means the angle of the velocity vector the relative velocity vector must match the inlet angle of the blade or vane and the blade or vane is designed accordingly the inlet angle is designed such that the relative velocity must that angle of the relative velocity must match the inlet angle of the blade so here also it is true that is this that was it has discussed earlier. So, for smooth entrance of the air here, so the relative velocity angle should be same as that. Here we specify the angle with respect to the tangential direction. We are looking from the top, this is rotating in this direction, the impeller. So, therefore this is the tangential direction, this is this direction is the tangential direction. Now, what happens the impeller is designed in such a way that it draws air I told earlier axially. So, it draws axially. So, at the inlet the velocity the absolute velocity of the air is in axial direction. So, therefore this is the velocity triangle which is in this plane. I tell you if you see here which take place that means this is in axial direction. That means we see from the top we will see this direction which is the axial direction. That means here if you see this direction is the absolute velocity direction which is the axial direction sometime this is referred as the flow velocity. So, flow velocity this is the axial direction this is the absolute velocity. Now, if we have to find out the relative velocity. So, we have to vectorially subtract the velocity of the impeller at the inlet that means the relative velocity v r 1 vector is v 1 minus u 1. So, therefore this is the u 1. So, this if you draw this diagram. So, this will be this is v r 1 this is u 1 this is u 1 and therefore this is v 1. So, this is the relative velocity v 1 minus u 1 and this angle is the angle made by the relative velocity with the tangential direction. While the absolute velocity makes a 90 degree angle because this is axial and axial direction is perpendicular to the tangential direction. So, therefore this is the axial and this is the flow velocity at the inlet that means if this velocity is v 1 this is multiplied by this area frontal area of the impeller i will give you the mass flow rate coming to the volume flow rate times the density is the mass flow rate going to the compressor. So, therefore you have to understand very clearly the inlet velocity triangle already you know the velocity triangle concept earlier also we did it in hydraulic turbines and compressor hydraulic turbines and pumps. Another assumption for this as you know already that we always consider a uniform because the variation in this direction is neglected the circumferential direction. So, therefore we always consider a uniform velocity distribution along the circumference that means it is a azimuthal symmetric flow. So, that any representative point is velocity at any azimuthal location is the representative the velocity of the entire periphery. So, with that we can show the inlet velocity triangle at the for this impeller. Now, what happened what at the outlet now at the outlet what happened the since the blade is made radial what we want why the blade is radial that means we want that the relative velocity that means the velocity radial means if it has to go smooth go out smoothly over this blade the radial velocity sorry the relative velocity of the air with respect to the blade should be in the radial direction. So, this is the therefore here if you write v r 2 is v 2 minus u 2 what is u 2 u 2 is the velocity of the impeller at the outlet. Now, if we want that the relative velocity v r 2 should be such that it must match the angle of the blade at the outlet and since the blade is made radial this is deliberately made radial. So, that for that smooth outlet the radial velocity should also be radial sorry the relative velocity should also be radial. So, this is the relative velocity v r 2 you see this diagram this diagram is better I think you see this this is the relative velocity v r 2 this is the v 2 and this is the u 2 because we can write this one here that v r 2 is v 2 minus u 2. So, you see so with this we can draw this triangle diagram. So, this is the so therefore what happens this is the relative velocity. So, absolute velocity is the since the relative velocity is in the purely radial direction. So, absolute velocity does not have sorry the absolute velocity does have a tangential component the absolute velocity does have a tangential component which is equal to u 2 here you see the absolute velocity does not have a tangential component because absolute velocity is axial perpendicular to the tangential direction here the absolute velocity has a tangential component v w 2 this nomenclature you know earlier that v w 2 is the tangential component or whirling component that is why the w is given at the outlet similarly v w 1 here you can write v w 1 is 0 that is the tangential component of the velocity of the fluid that is air at inlet is 0 whirling component of velocity is 0, but here it is u 2. So, with this blade diagram now what we can write we can write the energy transferred to the fluid now energy which is transferred to the fluid energy transferred to the fluid now energy transferred to the fluid takes place by the action of this rotating wind that already we derived that energy transferred to the fluid per unit mass is given by v this expression with the nomenclature I am telling where v w 2 is the tangential component of the fluid velocity at the outlet of the impeller u 2 is the impeller linear velocity tangential velocity at the outlet v w 1 is the tangential component of velocity of the fluid at the inlet which which call is that whirling component of velocity at inlet and u 1 is the velocity of the impeller at the inlet because of its rotation. So, this expression was derived earlier from the use by making the use of the theorem of angular momentum or conservation of angular momentum theorem of angular momentum that is for a control volume we find out the net a flux of the angular momentum which equals to the torque imparted on the control volume. So, based on this angular momentum or the momentum momentum theorem we derived that the energy which is transferred in this case this expression equals to the energy given by the machine to the fluid v w 2 u 2 minus v w 1 it is a just recollection of the earlier things which we already discussed. Now, in this case particularly in this case v w 1 you see that in this case v w 1 is 0 and v w 2 at the outlet since the relative velocity is radial. So, v w 2 that is the tangential component of the absolute velocity equals to u 2. So, in this case v w 2 equals to u 2 and v w 1 equals to 0. So, therefore, energy given per unit mass of the air can be written as u 2 square simply u 2 square. So, therefore, u 2 square is the nothing but the energy given per unit given by the machine to air per unit mass. Now, let us see there are some other issues I already discussed earlier that there is an phenomena known as slip which is very very important slip. Now, I tell you what is slip when a fluid flows through a curved vane and the vane rotates what happened because of the combined effect of the tangential flow and the radial flow pass the curved vane there is a difference of pressure in two sides of the vane. In the leading edge the pressure becomes high the fluid is dislarated and in the trailing edge the pressure becomes less and the fluid is accelerated. And there becomes a small recirculatory flow probably you can remember which we discussed in case of centrifugal pump here let us discuss here. Now, if we consider this moving in this direction in this side for example, there is a positive pressure positive means higher pressure and this side the pressure is that means this is because of the movement this rotation of the blade and the flow in pass the blade in the radial direction because of the blade curvature it moves like that and at the same time the fluid flows in the radial direction as a combination of that the fluid element here is dislarated in the leading edge this is the leading edge of the blade and fluid here is accelerated. So, what happens there is a higher pressure here there is a lower pressure here. So, what happens here I show you the gives rise to a recirculatory flow in this direction this is the higher pressure region this is the lower pressure region and what happens this recirculatory flow if you see this passage this happens same way in all passages makes a non uniform distribution of velocity and that non uniform is a skewed one. That means this makes a distribution like this this makes this was discussed earlier also this makes a distribution like this the velocity distribution become much skewed here the velocity becomes and this side between this passage is high here this is low. So, this way the velocity distribution changes. So, as a whole what happens because of this small recirculatory flow due to the difference in pressure from the leading and the trailing edge there is a there is a change in the direction of the velocity of the fluid relative to the blade. So, this results in this way that means we wanted that the fluid will go in the radial direction rather it will move in this way that means the direction of the fluid relative to the blade is change like this. So, this is velocity relative to impeller this is the this velocity triangle here you see this velocity triangle this this this triangle this is the triangle. So, this is now the v r 2. So, this is v r 2 and therefore what happens if this becomes v r 2 then this is the change velocity triangle it is just similar if you recall what we discussed in case of centrifugal pump. So, what happens is that in that case this is the tangential component of the absolute velocity now if you compare this two this u 2 this is u 2 obviously u 2 remains same u 2 depends upon the rotational speed and the outer radius of the impeller. So, therefore for the same u 2 at the impeller outlet v 2 is reduced and this component is reduced from that of the u 2. So, therefore as a result what I what we get v w 2 is less than u 2 in this case it was v w 2 u 2 because the absolute the relative velocity is radially outward here the relative velocity is like this is the similar thing what happened in case of centrifugal pump. So, this is the slip but you have to know the fluid mechanical principle of slip it is because of the blade curvature and the rotation of the blade the motion is being imparted to the blade at the same time the fluid flow pass the blade on the two sides of the blade leading edge and the trailing edge which creates it is a sort of a circulation, circulation combined with a linear flow which is in the radial direction which creates this type of difference in pressure and a local recirculatory flow as the combination of this with the radial flow outward makes a skewed non-uniform distribution of the velocity which finally results in this type of velocity triangle where we get the relative velocity at the outlet is not radial as a result of which finally we get v w 2 is less than u 2 which means that if we see the energy transfer. So, therefore this will be now v w 2 is u 2. So, this will be now actually e by m if v w 1 is 0 is v w 2 u 2 since v w 2 equals to u 2 we wrote is u 2 square but now if v w 2 is not equal to u 2. So, it will not be u 2 square and if v w 2 is less than u 2 then it will be less than u 2 square here a terminology is there defines slip factor a factor is defined known as slip factor which is symbolized as sigma I did it for centrifugal pump also which is here defined as v w 2 by the ratio of the outlet welding component or the tangential component of the fluid at outlet divided to the tangential speed of the rotor at the outlet. So, if slip factor is defined like this then with terms in terms of the slip factor we can define that the energy per unit mass is then sigma u 2 square and sigma is less than 1 sigma is less than 1 and therefore we see because of the slip a less amount of energy is being imparted to the fluid as compared to that if there could have been no slip. So, how slip will increase now one thing I tell you again and again which I told you while discussing the centrifugal pump that slip is not a consequence of fluid viscosity slip is a consequence of the motion of the fluid the rotation of the fluid by the action of the blades even the fluid is in visit the slip will occur. So, therefore be careful the slip phenomena is not because of the fluid friction though the slip ultimately reduces the energy from what we could have got if there was no slip there is a formula to find out slip. So, this formula is known as Stanitz formula Stanitz from a potential flow analysis expressed this slip like this you can get some idea from here 0.63 pi by n where n is equal to number of blades where n is equal to number of blades or vanes now you see here very much that if the number of blades. So, therefore you see if the number of blades increases what happened sigma increases the maximum value is 1 when number of blades tends to infinity sigma tends to 1 there is no slip that means if you make small small streams on more number of blades infinite number of blade passages then slip will be totally reduced almost reduced on the other hand when number of blades is decreased slip is decreased which means slip will be much lower than 1 and there will be a much reduction in the energy imparted to the fluid. So, therefore due to the slip finally the energy imparted per unit mass is given by sigma u 2 square where u 2 is the outlet tangential velocity of the impeller or impeller blade and sigma is the slip fact now if we consider of other losses mainly the mechanical frictions and the windage losses this type of losses if you take care of then you have to give more amount of work as given by that expression and therefore the energy per unit mass is written by making another term as a product that is psi which is known as power input factor psi is known as power input factor. So, this is known as power that means this takes care of frictional losses the disc frictions the frictional losses the disc frictions windage losses all these things is taken care of by the power input factor which by physical sense will be greater than 1 definitely that means this will be greater than 1 that means this will be the energy requirement of the machine so that out of this this will be imparted to the fluid which will exhibit the slip phenomena and this value of psi usually lies between lies between between 1.03 to 1.05 like that. So, therefore finally we arrive at this expression that energy per unit mass given to the fluid is sigma psi sigma u 2 square this is the amount of energy given to the fluid now let us find out in terms of the pressure rise let us do some thermodynamics let us consider now here again I come here let us consider this is this will be better here we are not going unnecessarily to this complicated figure let us see that this is the inlet to the impeller section one this is the outlet of the impeller section two this is the outlet of the diffuser section three now let us consider this way that section 1 to 2 when the fluid flows it gains the energy. So, the energy is being imparted here whose value is just now we have written psi sigma u 2 square this is the energy required but actually fluid gets this energy sigma u 2 square psi sigma u 2 square is the total energy required part of which is lost in friction. So, therefore this energy is being imparted in the fluid only here. So, 2 to 3 there is no energy given so total energy between 2 to 3 remains same there is no energy given to the fluid there is no energy interaction. So, therefore what happens the energy level at 2 and 3 remains same, but there is a difference in pressure between 2 and 3 because the pressure rise takes place in the diffuser. So, now if you give this 1 2 3 now if we with this can you see this 1 2 3 yes then we can draw a in thermodynamics the T s diagram let us consider now the pressure here is total pressure is p 1 t and at the end the total pressure is p 3 t and we show this things like this p 1 t and we show this thing p 2 these are diverging lines p 2 t what is total pressure total pressure is the static pressure plus the stagnation plus the velocity pressure that means the pressure velocity head equivalent pressure equivalent of the velocity head or the kinetic energy now I will not show you here it will be difficult now 1 is the inlet of the impeller 2 is the outlet of the impeller now you see one thing that what is this p 1 p 1 now stagnation pressure total pressure I am just recalling again total pressure is the static pressure p s plus rho v square by 2 for any fluid stream if you make the velocity 0 by some way for example you make the increase area infinitely large. So, therefore if you write the Bernoulli's equations at 2 points along a stream line then you can get that p t when the velocity becomes 0 then p t equal to total pressure that means the pressure there equal to p s plus rho v square by 2 and you can write Bernoulli's equation when there is no energy no energy added from outside and there is no dissipation taking place no fluid friction no dissipation. So, therefore under ideal condition for an inviscid fluid this is synonymous to an isentropic flow that the fluid is brought to rest then we can write for example if this is the point 1 if this is the point t for example where the total pressure is obtained. So, therefore p 1 that p 1 is the p s here let this is s p s is this pressure here plus p s by rho plus v square by 2 is p t by rho plus 0. So, which gives rise to this that means the total pressure is obtained by writing the Bernoulli's equation that means that the fluid is decelerated isentropically isentropically means without any internal irreversibility without any heat transfer therefore external irreversibility is also not there purely reversible way and without any other energy added or taken out. So, therefore this is the concept of the total pressure. So, this is the total pressure now let us consider the stagnation temperature what is stagnation temperature similar to that the concept of stagnation temperature is the temperature which is being gained if a fluid stream is brought to rest, but there is no concept of reversible way that is you can bring to rest even in consideration of the friction because the total energy is conserved for example I just give you the recalling this stagnation temperature concept that 1 and t the similar way if you do that a fluid stream is being retarded from a point 1 to a point here t and if we write this energy equation then steady flow energy equation h 1 plus kinetic energy v 1 square by 2 and if we consider that it is done adiabatically that means no energy interaction either in the form of heat or in the form of work takes place then we can write h 2 plus 0. So, this equation is the steady flow energy equation where we neglect the change in potential energy. So, there is no other form of work transfer either from outside or from the system outside to the system or system to the outside then therefore for this control volume system a steady flow energy equation gives rise to this where friction is not coming into picture. So, therefore here this h 2 the h 1 is what c p into t 1 this is the section 1 for example plus v 1 square by 2 and h 2 is c p into t 2. So, this t 2 is correspond to this stagnation temperature of t 1 you understand. So, therefore a stagnation temperature if you write the t t is equal to what it equal to t s this is known as the static temperature plus v square by 2 c p in general that means this is the velocity or kinetic energy equivalent velocity head or kinetic energy equivalent of the temperature this is kinetic energy equivalent of the pressure this kinetic energy equivalent of the pressure is realized if this is being done in a reversible way without any dissipation absence of friction, but this is purely energy conversion. So, therefore this is the temperature this is the velocity equivalent of the kinetic energy equivalent of temperature added with static temperature gives the total temperature where the fluid velocity is reduced to 0 without any restriction whether friction is there or friction is not there. This is because the friction takes care of the pressure, but here the friction does not come into picture I just explain it again for your concept to be clear that h is probably you know u plus p v. So, whenever we make this difference h this balance that h 1 plus v 1 square by 2 is h 2. So, within the h this u and p v adjustment will be made by the friction that means whether friction is there or friction is not there that will depend upon this distribution within u and p v they will adjust the friction is there u will be more the internal energy will be more because of the friction whereas where p v is less the reverse is there if there is no friction this u will not be changed because internal molecular energy will remain same because of the temperature and p v will get as proportion to the h change in h. So, therefore it is the conversion from u and p v that will depend upon the friction, but h 1 minus h 2 is v square by 2 or h 1 plus v 1 square by 2 is h 2 if the velocity is 0 or any other velocity may be there this is a steady flow energy equation which is independent of whether there is friction or not. This is a recapitulation of the total pressure and the total temperature. Now, come to the point if we consider the point 1 that is the inlet to the impeller at the at point 1 where we have got the total temperature is a total temperature t 1 t and if we get a total temperature t 2 t now let us before that let us consider this way sorry let us consider this way that let us consider this way that if t 2 t 3 t is the total temperature total temperature at the outlet of the compressor at the outlet at the outlet of the compressor at the outlet of the compressor that means at the outlet of the at the outlet of the diffuser. Then t 2 t if we write t 2 t is the total temperature or the stagnation temperature total temperature at the outlet of the diffuser since there is no at the outlet of the impeller sorry at the outlet of the impeller that is outlet of the compressor that means at the outlet of the diffuser at the outlet of the diffuser then we can write t 2 t is equal to t 3 t because the total temperature remains same because there is no energy added in the diffuser. So, therefore, we can write that energy added per unit mass that work done per unit mass or energy added per unit mass is equal to c p specific heat into t 3 t minus t 1 t or is equal to c p into t 2 t minus t 1 t clear that we can write very well this one c p t 3 t minus t 1 t is equal to c p t 2 t minus t 1 t now this becomes equal to what again e by m we have got is equal to psi sigma u 2 square. So, therefore, what we can write now we can write we will come to this afterwards I will do something else here we will come it afterwards before that I will do something here that we ultimately get e by m is what now here e by m is what again e by e by m is this yes then we can write from this from this we can write c p let us write here t 3 t minus t 1 t is equal to psi sigma e 2 square. So, therefore, we can write simply t 3 t by one step is equal to this becomes 1 plus psi sigma u 2 square by c p t 1 t. So, a ratio of temperature t 3 t to t 1 t we get now this is the t 3 t outlet temperature this is the t 1 t now you come to this if this is the one now if we have an isentropic expansion we get a point here this is the p 2 t. So, therefore, we can get this point here sorry this is not p 2 t I write it is this is the final pressure of the compressor that is total pressure of the compressor p 3 t this is sorry p 3 t now here is the point this is the isentropic of the actual t 3 t is not at this point because the actual process is not isentropic this point we denote as t 3 t dash that means this temperature. So, therefore, if this temperature I know then I can write the pressure rise the expression for pressure rise which I am interested to know t 3 t dash divided by this is t 1 t that the inlet temperature to the compressor t 1 t to the power gamma by gamma minus one this is the isentropic process relation between pressure and temperature, but actual t 3 that is the temperature at the outlet of the diffuser that is outlet of the compressor is different from that of the that of the isentropic process actual process is not isentropic. So, what will be the actual process actual process will not be vertical. So, there may be two options the process may be like this or process may heal towards left as we know that the adiabatic process that means a process without heat transfer, but with frictions or internal irreversibility entropy always increases what is an isentropic process isentropic process is a process which does not have any heat transfer with the surrounding the system does not have any heat transfer with the surrounding at the same time there is no irreversibility inside there is no friction even no heat transfer within it temperature is almost uniform at any instant. So, that internal dissipation or internal irreversibility is 0 at the same time there is no heat transfer external irreversibility is 0. So, process is totally reversible and at the same time adiabatic entropy remains constant that you know from thermodynamics, but if the process is adiabatic, but you cannot avoid the friction. So, this is known as adiabatic process with internal irreversibility for which we know from the principle of increase of entropy that entropy always increases because of the internal irreversibility. So, therefore the actual process is shown it is recapitulation of your thermodynamics. So, actual process is shown like this that is t 3 t. So, therefore this temperature lies here. So, therefore this is the actual temperature here we define a terminology known as isentropic isentropic efficiency of compressor isentropic efficiency of compressor which is defined as the ratio of these two temperatures in a way that t 3 t dash minus t 1 t divided by t 3 t minus t 1 t. Actually there is a C p this cancels out this ratio is what physically that it is the actual work that is required divided by the ideal work actual work is always more t 3 t is more than that this is because of the friction and temperature increases because of the increase in the entropy due to internal irreversibility. So, therefore actual temperature rise can be written in terms of this in terms of the isentropic efficiency eta c isentropic efficiency of the compressor eta c. So, therefore what we can write t 3 t minus t 1 t is therefore we can write is eta c eta c into C p into what is this t 3 t dash this is eta c is equal to this. So, t 3 t dash t 3 t dash we can write this is t 3 t dash. So, 1 eta c that means we can write now C p t 3 t minus t 1 t is this here we have as t 3 t 3 t dash by t 1 t is this. Therefore, we can relate this t 3 t dash by t 1 t in term t 3 t minus t 1 t in terms of this through this eta c and we can find out the pressure rise. However, today's time is over I think I will discuss it tomorrow next class. Thank you.