 In the previous lecture, we mentioned that H plus C square over 2 is a constant for an axial machine and that this may be thought of as the energy equation for an observer moving with the rotor. It is important to note that there is no working traction in this case in this frame of reference and the other important point has to do with the manner in which H1 the specific enthalpy has to be evaluated. It must be borne in mind that H1 is the static enthalpy and hence it is frame independent. So it may be evaluated in the moving frame of reference in the same manner as it is evaluated in a stationary frame of reference. That is a very important point to keep in mind. The two or the most important fact that comes out of this is for this for an axial machine that H plus C square over 2 is a constant and this has very important consequences in the actual design and working of an axial machine as we will see in a short while. Now the SFE applied to the rotor led to an expression that look like this. So if you look at the terms inside the square bracket, it can be seen that there are basically two effects that cause work transfer to the rotor, whether it is power produced by the rotor or power being supplied to the rotor. There is a change in enthalpy and there is a change in specific kinetic energy. So change in specific enthalpy and change in specific kinetic energy of the fluid. So these two effects cause the work transfer to the rotor. Now this naturally leads to the definition of a very important performance parameter of a turbo machine, namely the degree of reaction. Simply put, the degree of reaction is a quantity that signifies how much the enthalpy change contributes to the term in the square bracket. Notice that the term in the square bracket is nothing but H01 minus H02. So the degree of reaction basically quantifies how much the change in specific enthalpy contributes to the change in stagnation enthalpy. So you can see the definition of the degree of reaction R here, H1 minus H2 divided by H01 minus H02. So it is clear from SFE applied to the rotor that given amount of work transfer can be accomplished either through a change in specific enthalpy or a change in specific kinetic energy or both. Now for an axial machine since H plus C square over 2 is the constant in addition, any change in specific enthalpy necessarily leads to a change in relative velocity also. So if we combine this with the earlier statement and we can say that given amount of work transfer can be accomplished through a change in relative velocity or absolute velocity or both. That is the insight that we gain from the application of SFE to the rotor and the definition of the degree of reaction. Notice that degree of reaction can vary between 0 and 1. For instance, if the degree of reaction is 0 then there is no change in specific enthalpy across the rotor and such machines are usually called as 0 reaction or impulse machine. Now if the working fluid is compressible then R equal to 0 simply implies that the static enthalpy or specific enthalpy remains constant across the rotor. Now in the case of an incompressible working fluid such as water, it follows from the definition of specific enthalpy that a change in specific enthalpy is equivalent to a change in pressure. So if the change in specific enthalpy is 0 across the rotor then it implies for an incompressible working fluid that the pressure remains constant across the rotor. The Pelton deal that we saw earlier is an example of such an impulse rotor where the pressure remains constant. Let us take a look at the figure for an impulse machine. So here the water that comes from the reservoir at the higher elevation is actually accelerated to a high velocity in the nozzle and when it comes out of the nozzle the water comes out at atmospheric pressure and the air inside the casing is also at atmospheric pressure so the entire turbine operates at constant pressure and hence it is an impulse machine. It must be understood clearly that a 0 reaction machine and an impulse machine are not the same. A 0 reaction machine implies that enthalpy is constant whereas an impulse machine generally implies that pressure remains constant across the rotor. But it is customary to actually use these two terms interchangeably although the context must pick up very clear when these terms are used. Let us now revisit the equation relating pressure change across the rotor and change in the flow properties along a streamline. So this is captured in this expression which states that any change in the pressure along a streamline is due to change in u square over 2 and change in c square over 2. So this term captures centrifugal effect and this term captures deceleration or acceleration of the fluid in the blade passage. Now let us look at this expression for the two machines that we have seen so far namely the radial machine. It may be recalled that we looked at both radial and axial machines in the introductory part of the series. If we take this equation and apply it to a radial machine by design the change in relative velocity in the radial machine is usually very small. So we may actually set this term to 0. So it then becomes clear that the pressure change in the impeller of radial machine is due to centrifugal action only. So if the pressure increases along the streamline that is dp is positive then it is clear from here that dr must also be positive which means that the flow is radially outward in case the pressure increases along the streamline. So this explains why the flow is radially outward in a centrifugal pump and in a centrifugal compressor that we saw earlier and it will be recalled that this was one of the questions that we had posed earlier. Namely why must the flow be radially outward in a centrifugal compressor of pump and this answers that question. By the same token if dp is negative along a streamline that is pressure decreases along the streamline then dr must also be negative. So in the case of a centrifugal turbine then the flow is radially inward as we saw in the case of the centrifugal turbine in an automotive turbocharger. So this equation explains quite clearly why the flow direction is the way it is in radial machines where the pressure change is due entirely to centrifugal action. The work that is supplied to a pump pushes the fluid element radially outward thereby increasing its pressure. So that is a very simple interpretation that is possible from this equation. Now if we apply the same equation to an axial machine to the rotor of an axial machine then we can actually set this term to zero in this case because the radius of the streamline changes very little across the rotor of an axial machine. It may change from one rotor to the next one set of rotor blades to the next set of rotor blades but within a rotor blade itself the change in radius is usually very small so this term may be set to zero. So then it becomes clear that the pressure change in the rotor of an axial machine is due entirely to a change in the relative velocity. So if a DP is positive meaning pressure increases along the streamline for example in an axial compressor then the relative velocity must decrease that is seen DC is negative and so seen decreases. So in this case the blade passage actually acts like a diffuser and decelerates the fluid and the deceleration of the fluid results in an increase in pressure of the fluid in the case of an axial compressor in the case of an axial turbine where the pressure decreases along the streamline. From this expression we can see that the velocity relative velocity must increase DC is positive so the relative velocity must increase in this case the blade passage acts like a nozzle. So we can actually then gain further insights from the blade profiles that we had looked at earlier. So for instance this is the rotor of an axial turbine and it is clear that the passage area between the blades decreases and hence this acts like a nozzle and the relative velocity increases from inlet to exit in the case of a axial turbine. In the case of an axial compressor whose rotor is shown here the relative velocity decreases from inlet to exit and the passage area increases from inlet to exit. So the flow is actually diffused in this case and the diffusion results in an increase in pressure in the in the case of an axial compressor. Now the blade shapes of for an axial turbine and the axial compressor are also quite strikingly different. Now in the case of a turbine because the pressure gradient is favorable along the along the directional flow it is possible to have large pressure drops across the blade passage and the large amount of work transfer is also possible which is why the blades are so highly curved and they are also thicker to withstand the large amount of work transfer. In the case of a compressor because the pressure rise is accomplished by a diffusion of the flow there is always a danger of the boundary layer on the wall of the blade or on the blade surface separating. So if the boundary layer separates then the compressor undergoes what is known as a stall and in a compressor design such as this where the pressure rise is accomplished through diffusion of the flow there is always a danger of flow separation so the pressure rise across such a compressor stage cannot be very high. It is typically 1.15 to 1.2 no more than that because the pressure rise is more than that there is a possibility of boundary layer separation. So since the pressure rise cannot be very high in across the blade passage in the rotor the amount of work transfer that can be accomplished is also less in this case so consequently the blades of an axial compressor or slender and the curvature is also not very high. In a sense we can summarize and say that the blade passage of an axial turbine resembles a nozzle so the rotor of an axial turbine is actually a set of rotating nozzles and similarly the blade passage in an axial compressor acts like a diffuser and so the rotor of an axial compressor is actually a set of rotating diffusers. So this sort of interpretation is possible from the equations that we just described and the comments that we made about the work transfer being very high in the case of a turbine and work transfer being relatively small in the case of a compressor is also evident from the picture of the axial machine that we saw earlier. So if you look at this gas turbine engine the turbofan engine it can be seen that a single ring of turbine blades here the high pressure turbine drives about five or six of the high pressure compressor blades which clearly demonstrates that the work produced by a single row of turbine blades here is sufficient to run the work I mean is sufficient to run five or six rows of compressor blades or it is equal to the work requirement of five or six rows of compressor blades. Similarly the two rows or even one row of the intermediate pressure turbine here is sufficient to run about seven or eight stages of the intermediate pressure compressor. So the work requirement of seven or eight stages of the intermediate pressure compressor can be met by just one or two rows of intermediate pressure turbine blades. So it is clear this is possible because the work transfer per turbine blade passage can be very high because of the favorable pressure gradient across these blade passages when compared to the adverse pressure gradient across the blade passages of the axial flow compressor. Although the process of diffusion that is utilized in in an axial flow compressor is through diffusion and it is not really good from a fluid mechanic perspective going to the dangers of flow separation. It is still a much preferred design and widely used when compared to centrifugal machines. So the pressure rise in the axial machine is due to diffusion of the flow which as we said is always accompanied by a danger of flow separation whereas the pressure rise in the case of a centrifugal compressor is accomplished through centrifugal action and because the relative velocity remains constant across the impeller there is no danger of flow separation here. Nevertheless the axial flow compressor is used quite extensively in many installations because there are other advantages to this design which offset the disadvantage that we have just mentioned.