I covered the material in Chapter IX of the notes. This included deriving the relationship between shaft power added to a flow and change in flux of angular momentum. There are a few key points: 1) Without heat addition the only way to change the total enthalpy of a fluid is through an unsteady process (like a moving blade row), 2) In an analogous fashion to the integral form of the linear momentum equation for a steady flow through a control volume of fixed mass (sum of forces is equal to net flux of linear momentum), for a steady flow through a control volume of fixed mass, the sum of the torques is equal to the net flux of angular momentum, 3) Shaft power is equal to angular velocity times torque and is also equal to the change in total enthalpy (from the steady flow energy equation for adiabatic flow). Setting these two relationship equal to one another results in the Euler Turbine Equation. We did two PRS questions (PRS #1, PRS#2).
Responses to 'Muddiest Part of the Lecture Cards'
(21 respondents, 52 students in class)
1) The first PRS question was confusing.(6 students) Yes, and I was not very effective in helping to clarify the confusion. First, this is a right-handed cylindrical coordinate system, so if you align your hand with the direction of r and roll it towards the direction of theta (out of the page) your thumb points in the direction of x. Now consider the cross product of two vectors one only has a component in the radial direction, r, the second vector u, can have components in the radial, tangential and axial directions. If you take the cross product between the two you get a new vector with components in the theta direction (-r times usubx) and the axial direction (r times usubtheta). You do this the same way you do any cross product. The way to show this with your hand is to assume that you only have a velocity component in the theta direction -- then r crossed with usubtheta points in the positive axial direction. You can do the same thing for the theta component of the vector.
2) Can you develop an optimum blade configuration (i.e. angles) for any given number of stages using the theory we have introduced? (1 student) Not yet. You need to be able to connect blade geometry to performance. We do this next lecture. Further, although conceptually we can think about optimum solutions, they are very difficult to establish or verify in practice because of the many varied constraints the system must operate under. So it would take a good deal more knowledge of the details of engine behavior before you could say you found the optimum blade shapes.
3) You said that the rotors add velocity and the stators convert that velocity to pressure. I thought that if you speed up the flow, the pressure decreases. Why doesn't this happen? (2 students) Ask me this again if I don't clarify it for you during the last lecture.
4) If the tangential velocity is increased by the rotors, does the flow actually rotate in the compressor? (1 student) The flow follows a spiral path through the blade rows (axial and tangential velocity components).
5) How do you find the change in stagnation temperature and pressure from the change in stagnation enthalpy? (1 student) For an adiabatic flow with constant specific heats, the stagnation temperature change is related to the stagnation enthalpy change by cpdeltaTt = delta ht. Then for an isentropic flow, the pressure change can be determined using the isentropic relations derived in thermodynamics and/or fluids. This is also discussed in the notes.
6) Can you go over again the various parts of the compressor? (1 student) I will do this at the beginning of the next lecture.
7) Would having compressor blades with variable angles of attack be advantageous to compressor design? (1 student) Indeed it would. Most engines have stators with variable angles of attack. I will show you these in class tomorrow and we will see why they are important.
8) No mud (8 students).