Lecture P8: Energy Exchange with Moving Blades

If the flow spins too fast does it get dizzy? (1 student)

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).

There was one comment on the cell phone technology. Keep these coming (that is why we are trying these, to get student and instructor feedback):

I like the phones. (1 student)

Responses to 'Muddiest Part of the Lecture Cards'

(34 respondents, 52 students in class)

1) What are the b and c components? (1 student) These are the notation I will use when we get to axial flow compressors. For now just think of them as "inlet" nad "exit".

2) Maybe I'm missing something, but I didn't see the integrals & equation in the notes ....where are they in there? (1 student) You are referring to the integral form of the angluar momentum equation. A link to this appears in two places (near the linear momentum theorem and also where we discuss the Euler Turbine Equation). Maybe it is just some terminology I missed, but how did M^2 suddenly turn into a bunch of u^2's? Isn't M Mach #? (same student) I think you are referring to this equation which just expresses the magnitude of the velocity vector in terms of its components. I don't think I used Mach number in class.

4) On the first PRS I did not follow r x u = ruq. (1 student). The vector r x u has three components. We are interested in the axial component. To form the axial component, the relevant flow component must be perpendicular to both r and x --- hence uq.

5) How does the flow leave a centrifugal pump? (1 student) The flow leaves a centrifugal pump into a volute (a large chamber at the exit of the spinning blade rows). In some designs there are vanes that help convert the swirling kinetic energy to internal energy. Hill and Peterson, Mechanics and Thermodynamics of Propulsion, 2nd edition (available in the library) has a much more extensive discussion (and pictures -- if I get a chance I will scan a few in).

6) Could we find a better way to mark variables for T=temperature, T=thrust and T=torque? (1 student) Well, we could probably start using other alphabets, but this is a common problem--too many variables for the 26 letters we have (even when we use lower and upper case and Greek). As you become more familiar with the various equations, you will be able to recognize which is which (e.g. by comparing units).

7) You can just multiply omega (the angular velocity) by the radius to get the exit velocity of the turbopump? (on the PRS) I was uncertain how to find vc. (1 student). Yes, since the problem says that the tangential velocity at the exit is approximately equal to the rim speed (r times omega).