Lecture T9: The Steady Flow Energy Equation--Shaft work and Flow work

People seemed a little tired today and there were many late arrivals. Based on the muddy cards, it also seems that some people may not be doing the reading before class. Previously it had seemed that most of the people were doing the reading so I haven't set any for-credit reading assignments or quizzes. I will probably set a short for-credit reading assignment for Monday's lecture.

I thought things went well with the discussion of flow work and shaft work based on the lack of questions in class, but there were still many muddy points (see below). The introduction to the stagnation temperature and enthalpy was rough. It is a difficult topic and hard to introduce in 10 minutes at the end of class. Please carefully read through the material prior to T10 when we will spend the whole day on the topic. And remember, these are difficult concepts, so if they aren't clear to you instantly, that is okay. Keep working at it.

An addition to the mud art gallery:

Responses to 'Muddiest Part of the Lecture Cards'

(61 respondents)

1) Could you please re-explain the answer to the 1st PRS question? (1 student) For an adiabatic steady flow compressor (neglecting changes in kinetic and potential energy) the shaft work is equal to the change in enthalpy as seen from crossing off the appropriate terms in this equation:

2) Mud. 2nd PRS question--please re-explain. (2 students) For a quasi-static, steady flow process (neglecting changes in kinetic and potential energy) the work is equal to q + u1-u2, which you arrive at by crossing off the appropriate terms in the equation shown below. This was not one of the choices, so the answer is "none of the above".

3) I am still a bit muddy on the concept of shaft work. Is there some sort of equation that defines it? (4 students) A better word for it is external work, but it is commonly referred to as shaft work (I think I will change it in the notes next year). External work is everything that is not flow work. I need practice with shaft work and flow work. (1 student) You will get plenty. Can you give me an example of shaft work without flow work? (1 student) and What was the question asked by the student about the flow work for an isothermal process? (1 student) I could put shaft work into a control volume and then extract heat so that the temperature was the same at the inlet and outlet (because if RT is the same then pv is the same too). So as was pointed out in class, the flow work is zero for an isothermal process. So an isothermal compression or expansion process would be an example of a situation without flow work but with shaft work. Or a control volume that had a compressor in it followed by a cooler that got the flow back to the inlet temperature. How do you determine the shaft work? Is it like the total work we were using in non-flow problems? (2 students) You can determine shaft work using the equation below. The distinction between flow work and shaft (external) work is not necessary for non flow problems, because there is no flow work. So the work is the work for a closed system!

4) How does a difference in p1v1 & p2v2 do flow work? Why is it merely the difference and not some sort of integral like piston work? (1 student) It is not an integral because we assume that at each end of the control volume the work that the flow does is done at constant pressure. This is a very good assumption since the flow into and out of the control volume is steady. So the pressure at the inlet will not vary over time.

5)For flow work, is the "chunk" of air coming out of the system larger because it is moving at a faster rate? (1 student) and Won't changes in volume force changes in speed (kinetic energy) between the input and output? (1 student) Good question. The answer is: not necessarily because there is one more variable we can play with--the duct area. What matters is the quantity Dpv (or DRT) between the inlet and the outlet of the control volume. Mass flow is equal to density x velocity x duct area (rcA). So it is possible for the flow that exits the c.v. to be hotter (less dense, larger specific volume) than at the inlet, but to have the same velocity as the inlet because it is flowing through a duct of larger area.

6) You mentioned that the flow work in = flow work out in an answer to one student's question (1 student).I am not sure what you are referring to. This is certainly not generally true. See response (3) above for situations where the flow work is zero.

8) In does the pressure at (1) and (2) matter? (1 student) Not for an ideal gas. Enthalpy is only a function of temperature for an ideal gas.

9) How do the units on cpT + c^2/2 work out to enthalpy? (1 student) J/kg. Work it out and see.

10) In recitation, you once asked us to find the enthalpy in a room. I thought we had determined we couldn't b/c we didn't have a changine temp. Yet in this lecture, you write h in absolute terms as h=u+pv or h=cpT. (1 student). Great question. Sloppiness on my part. dh=cpdT (for an ideal gas), h does not equal cpdT. I have corrected it in my notes. Since we are looking for the change in enthalpy (h2-h1) we can substitute in cp(T2-T1). And in absolute terms h=u+pv. So that part is right. One note about finding the enthalpy in the room. We certainly can do it. Go to the back of your thermo text and look up what the enthalpy for air is at standard atmospheric conditions (usually T=296K), and then add to that Dh=cp(Troom-296K).

11) For stagnation temperature, why did we assume it is quasi-static adiabatic? (2 students) Great question. My mistake--I get so used to saying "quasi-static adiabatic" that it rolls off my tongue (and keyboard) even when it is not supposed to. The new medication I am on should help with this. In the notes it was correct in one place and wrong in the other. I have corrected it in the notes. The only conditions necessary for the stagnation temperature to be a constant are that the process be adiabatic, steady, and involve no external work. When we get to stagnation pressure and its relationship to stagnation temperature it will be necessary to assume a quasi-static process, but it is not necessary to do this for stagnation temperature.

12) What does stagnation mean? What does it mean when flow is stagnated? (2 students) It means it is not moving (relative to some coordinate frame), or its velocity (relative to some coordinate frame) is zero.

13) What is the significance of stagnation temperature, what is it used for? (3 students) Stagnation quantities are very useful estimates of the conditions that the flow would arrive at if it was brought to zero speed (relative to some reference frame) by a steady, adiabatic process with no external work (add quasi-static as a requirement for stagnation pressure). It doesn't mean that the flow will pass through such a process, but often it does. Flow always stagnates on bodies. So this tells us (for many situations) the temperature on the surface of the body--a very useful thing to know if you have to design that body to withstand those thermal loads. The stagnation temperature and stagnation pressure can also be related to the maximum amount of work that the flow is capable of producing although you will not learn about that in Unified (wait for 16.050). But think of a bottle of compressed gas (stagnated). The higher the pressure, the more potential there is to get work out of it. Stagnation pressure is the analogous quantity for moving gases (which are what we often deal with in aerospace engineering).

14) What is T? (4 students) Temperature. This is distinct from stagnation temperature which is the temperature the flow would reach if it were brought to zero speed (relative to some reference frame) by a steady, adiabatic process with no external work.

15) Overall confused about stagnation (7 students). You have a right to be. Let me hold off explaining more of it for now since we have a whole lecture devoted to it on Friday. Please read the notes!!

16) The last PRS question was confusing. (1 student) Read the notes it should help. If the airplane was going slower, would the wings be cooled down relative to the atmosphere? (1 student) No. Anytime it had some finite speed relative to the atmosphere, the skin temperature (i.e. the stagnation temperature of the flow on the skin) would be hotter than the atmospheric temperature. The faster the airplane flies the hotter the temperature gets.

17) What happens when you blow on a piece of paper? (1 student) The pressure drops so it lifts up the paper. This is a demonstration of the Bernoulli equation which you will hear much more about in Unified Fluids. We will also discuss it from the perspective of thermodynamics in the next lecture.

18) No mud (13 students). Good.