 We entered the previous lecture by introducing the concept of relative velocity of fluid with respect to the rotor, so we will explore this concept further in this lecture. So we start by writing down the formal definition of the relative velocity of fluid. So the relative velocity vector c is the absolute velocity vector v minus the blade velocity vector u. We may also write this as absolute velocity vector v is equal to relative velocity vector c plus blade velocity vector u. Since this is a vector addition, we need to show this in this graph or in this form. So this is called a velocity triangle and both these are called velocity triangles. Notice that the triangle is constructed to indicate the factor the absolute velocity vector v is the sum of the relative velocity vector c plus blade velocity vector u. The same relationship is applicable here also as can be ascertained from the direction of this vector. The difference between the two velocity triangles lies in the nature of these angles alpha and beta. The angle alpha is called the flow angle and that is the angle that the absolute velocity vector makes with the reference direction which is indicated here like this. And the angle beta is the blade angle or the angle that the relative velocity vector makes with the reference direction. So that definition holds for both these cases. Notice that the reference direction is the axial direction in case of an axial machine and radial direction in case of a radial machine and it is perpendicular to the blade velocity vector in both cases. Now since this absolute velocity vector is in the clockwise direction from the reference direction by an angle alpha, this angle would be given negative value because it is in the clockwise direction. Alpha would be a number like minus 15 degrees or minus 30 degrees and so on. Now this angle beta because it is in a counterclockwise direction from the reference direction this would be a positive number. So this would be something like plus 30 degrees or 60 degrees and so on. So in the case of this velocity triangle alpha is less than 0 and beta is greater than 0 which is why the triangle looks the way it does. Now in this case it can be seen that both alpha and beta are in the counterclockwise direction I'm sorry in the clockwise direction from the reference direction. So both alpha and beta are negative in this case and because under design condition the velocity is blade velocity in the flow velocity such that the relative velocity is tangential to the blade surface. We discussed this in the previous lecture so it just glides on and off the blade surface. So the angle that the relative velocity vector makes with the reference direction is the blade angle beta and this sign convention is very important. We will use this convention when constructing velocity triangles. A velocity triangle will be constructed taking into account whether alpha and beta are positive or negative but once a triangle is constructed other relationships may be determined or ascertained using a simple algebra. So irrespective of whether the velocity triangle looks like this or looks like this the following relationships holds v theta is always v times sine alpha so that can be seen here. So v theta is always v times sine alpha and vx or vr depending on the case is always v cosine alpha. Similarly c theta is always c sine beta and cx or cr is always c cosine beta and notice that cx is equal to vx or cr is equal to vr in both these cases. So here also you can see that v theta is equal to v sine alpha and vx is equal to v cosine alpha and c theta is c sine beta and cx or cr is c cosine beta which is what we have written here. So these are applicable for all velocity triangles. So as we said before when constructing the velocity triangle we have to make use of the sign attached to the angles alpha and beta but once the construction is complete the relationships are simple algebraic relationships which we can exploit for further calculations. Okay so from the geometry construction velocity triangle it's easy to see that for this triangle or for this triangle Pythagoras theorem applies and for this triangle or for this triangle again Pythagoras theorem applies. So we may write v square equal to v or x square plus v theta square and similarly for relative velocity c. What we said here we have already used the factor c r comma x is equal to v r comma x and we also notice that v theta in this case is equal to u minus c theta this is c theta so u v theta is equal to u minus c theta and in this case v theta is equal to u plus c theta. So we may write v theta equal to u plus minus c theta and if we multiply both sides of this expression by u we get v theta times u equal to u square plus minus c theta times u. Furthermore if we eliminate this v r comma x square between these two relationships we actually end up with a relationship that looks like this which leads to this relationship v theta times u. So if I substitute this into the the expression for u times c theta into this I end up with an expression that looks like this. Now if I replace the v theta times u term in the Euler-Turbine equation with this expression then we end up with the following expression for again Euler-Turbine equation. This is another form of the Euler-Turbine equation which involves only absolute velocity blade velocity and the relative velocity not components of this but absolute velocity blade velocity and relative velocity. Once again the interesting aspect about this equation is that this involves only fluid dynamic quantities namely absolute velocity blade velocity and relative velocity. So what we would like to be able to do is eventually to be able to relate this to changes in thermodynamic properties of the fluid. It's also not very difficult to write the following relationship. So if I take an axial machine so if I take an axial machine like this at any cross-section at any cross-section you can see that the mass flow rate that passes through the entire cross-section is nothing but the density at the cross-section or density at our reference value times axial velocity times the cross-sectional area. So that is what we have written here and a similar relationship may be written for a radial machine as well. So we have written that m dot at the inlet section is density at the inner blade element times axial velocity times the cross-sectional area and same thing for exit and this is the relationship for the radial machine. So again here B is the width of the rotor blade this should not be easy to visualize so you may go back and look at the centrifugal machine and convince yourself that this is indeed the case. So this is how mass flow rate is related to the axial component of the loss. So you can see that the m dot is actually related to vx1 or vr1 and the remaining quantity is the oil turbine equation of the velocity counter. So we can now see why velocity triangles play such an important role in turbo machinery application. What is that? The height of the triangle which is either vr or vx is connected to the mass flow rate that passes through the rotor and the base of the triangle which is either u minus c theta or u plus c theta is related to the work that is produced by the machine. So that is the term that appears in the right hand side of the oil or turbine equation. So it is clear why the velocity triangle plays such an important role in turbo machinery literature. So you can see that the height of the velocity triangle is related to the mass flow rate and the base is related to v theta which is u plus minus c theta and this is what appears in the right hand side of the oil or turbine equation. So two key performance parameters of a turbo machine are controlled by two components of the relative velocity. So this is why both relative velocity as well as the velocity triangle play a very crucial role in turbo machinery theory. Okay so we have been talking about the fact that we need to connect up the oil or turbine equation which as it stands contains only fluid mechanical quantities to thermodynamic quantities which is what we take up next. So if I apply a steady flow energy equation to the rotor assuming no heat loss this is what we get. Notice that this is the static enthalpy. This is the absolute velocity of the fluid or change in total kinetic energy of the fluid. So we neglect elevation changes in the rotor which is quite alright and not a bad assumption to make. Elevation changes will not be very substantial in this case. Now the connection between the thermodynamic equation that we have written here or change in the thermodynamic state and the oil turbine equation is provided by the WX dot term. So the WX dot term here can be replaced using the oil or turbo machine equation and if we do that so we end up with a relation that looks like this. So on the left hand side here we have change in the thermodynamic property of the fluid which is change in enthalpy or on the right hand side we have change in velocity, relative velocity and change in blade velocity. So this provides the link between the fluid mechanics and thermodynamics of the turbo machinery process. In fact we can pull this in even more powerful form by writing this in differential form along a streamline. So let us say that we take a streamline which passes through the rotor from inlet to outlet. So we may write the enthalpy change at any point in the streamline as dH equal to d of u square over 2 minus d of c square over 2. Must bear in mind that dH or incremental change in a property is final minus initial which is why we have introduced a negative sign here. Since u is r omega we may write d of u square over 2 as d of r square omega square over 2. If we use the TDS relationship TDS equal to dH minus vdp and assume the thermodynamic process to be isentropic and then dS becomes equal to 0 and we may write dH equal to vdp or dp over rho and so finally we may write dp over rho equal to d of r square omega square over 2 minus d of c square over 2. This is probably one of the most important and insightful equations that we will encounter in turbo machinery theory. This finally tells us that the pressure rise or change in pressure along a streamline is due to two quantities. One is a centrifugal action. The other one is due to acceleration or deceleration of the fluid in the blade passage. Notice that the relative velocity appears here not the absolute velocity. So we will explore this equation further and then try to get good idea on why this is so important because this actually determines fundamentally how devices operate. And as we just saw if you apply SFE to the rotor we add this equation. In fact, we can recognize the fact that h1 plus v1 square over 2 is nothing but the stagnation enthalpy of the fluid. So we may write this as m dot times h01 minus h02. Now if I go back to this equation and write it for an axial machine for which u2 is equal to u1 then we actually end up with a relationship for an axial machine which looks like this h1 minus h2 like this. And if we rearrange this then I recognize the fact that h1 plus v1 square over 2 is equal to h2 plus c2 square over 2 is equal to h0 related. So this tells us that the stagnation enthalpy in the rotating frame of reference does not change. In other words, there is no work transferring the frame of reference that rotates in the blade and the stagnation enthalpy which is reckoned or calculated using the relative velocity in this frame of reference remains a constant. So the relative stagnation enthalpy remains a constant and remember this is calculated in the frame of reference that rotates in the blade. The total stagnation enthalpy changes across the rotor depending on whether what is supplied to the rotor or power is generated by the rotor. In case power is generated by the rotor then there is a drop in stagnation enthalpy and in case what is supplied or power is supplied to the rotor then there is an increase in the stagnation enthalpy. This is in the frame of reference where the rotor rotates. This is the energy equation for the entire rotor. This is the energy equation for the rotor in the frame of reference where the rotor is stationary and for an axial machine. Now this is a very important fact for an axial machine and let us try to illustrate this on a PS diagram. I am sorry on a HS diagram. So here enthalpy is plotted on the y-axis and specific entropy is plotted on the x-axis and this is state 1 and state 2. This is the static enthalpy at state 1, static enthalpy at state 2. So this is V1 square over 2. So notice that this is H01. H1 plus V1 square over 2 is H01. Similarly, H2 plus V2 square over 2 is H02. So H01 minus H02 is the specific work output from the machine. So that is this H01 minus H02. In addition for an axial machine we may write H1 plus C1 square over 2 is equal to H2 plus C2 square over 2 indicating that the H0 relative for an axial machine is a constant.