Lecture T10: Stagnation Quantities

I concluded the discussion of shaft work and flow work. Based on the performance on the first PRS question, I seems that people have a good understanding of these concepts. I then introduced the concept of stagnation quantities (enthalpy, pressure temperature) using a second PRS question. I find it easiest to understand stagnation properties and whether temperatures and pressures go up or down with moving and stationary flow by making arguments about changes in internal energy and kinetic energy. We will spend all of the next lecture on these concepts. I also recommend you read over the notes as well as the old mud responses (T9, T10, T11)

Responses to 'Muddiest Part of the Lecture Cards'

(25 respondents out of 67 students)

1)I don't really understand how the temperature of a peice of fluid that gets stuck to an airplane goes up because it gains k.e. by moving (I guess I don't understand the trade-off). (1 student) When a moving airplane moves a particle in the air somewhere above it, where does the energy come from to move it? (1 student) I still don't understand how you can move or stop air molecules without doing work on them. (1 student) Still confused how one can make a particle move without inputting any energy to it -- does creating pressure waves require energy? (1 student) Why is internal energy traded on k.e. on particles outside the wing? Doesn't the pressure wave impart k.e. on the particles so i.e. doesn't need to be sacrificed? (1 student) These are all good questions. First note that the quantity that is conserved for these types of flow processes (adiabatic and no external work) is total or stagnation enthalpy, not energy. Total enthalpy has in it a term related to flow work, internal energy and kinetic energy. So when I was describing trade-offs between internal energy and kinetic energy, I was being a little loose (with the intention of having the class understand the overall concept). We will talk more about these issues during the next lecture, but several of the questions may be resolved by reading through some of my old mud responses. I have put two relevant ones in below:

OLD MUD #1: How can a particle be accelerated to a given speed without work? How did the chunk of gas in the example get to position (1) if no work was done on it? and related questions (3 students) There is work, but it is flow work, not external work. Remember, our control volume is defined by a set of streamlines. Between the inlet and the outlet of the streamlines the velocity of the flow changes, but it does this without heat transfer and without external work. Remember work is the transfer of energy across a system boundary. It is when it crosses the system boundary that we label and identify it.

OLD MUD #2: I am unclear as to why the leading edge of the wing heats up in the first example, then later you said temperature drops over the wing causing the condensation trails. How can it get hotter and colder at the same time? (1 student) This is a very good question. It does get hotter and colder at the same time. Just not in the same place. The flow very close to the body gets hotter, the flow farther from the body gets cooler. You need to know a little more about fluids before the answer is clear. There are two different processes going on. The first is like the example of the engine sitting motionless on the ground drawing in air--the (static) temperature and (static) pressure drop. The second is like the example of the skin temperature of a supersonic airplane being significantly elevated above the ambient atmospheric temperature. In the first case, no energy is added to the flow; energy is just converted from internal energy to kinetic energy. In the second case, energy is added to the flow. Now let us discuss why this happens.

PROCESS 1:When a body moves through a fluid it creates a disturbance. That is, it changes the velocity, pressure and temperature of the flow around it. It tells the flow "Get out of my way, I am coming through!" This disturbance is felt some distance away from the body (a distance of about two times the characteristic physical dimension of the body). You can think of this like the bow wave in front of a boat, the water starts to move out of the way before the boat gets there. This information is transmitted upstream of the body at the speed of sound. So for a supersonic body (one traveling faster than the speed of sound), the flow doesn't know to get out of the way sometimes until the body has passed (it happens when the shockwave passes through the region of flow). There is no external work done in causing the flow to move (there can be flow work). So the total energy of the flow is the same. When it moves to get out of the way, the kinetic energy must come from somewhere. It comes from the internal energy (or more appropriately, the enthalpy since stagnation enthalpy is the conserved quantity for these processes) and thus the temperature and pressure are reduced.

PROCESS 2: For the flow very close to the body (within an inch or so of the surface for a large airplane), the body adds energy to the flow. That is it pulls it along with it. We discussed this in lecture T1 and in a PRS question. Some number of molecules very close to the surface of the body stick to the body. They in turn pull on the particles next to them. This is exactly the same mechanism by which honey sticks to a spoon. The more viscous the fluid, the stickier it is and the more molecules get pulled along with the body (compare how much honey sticks to a spoon--to how much water sticks to a spoon--to how much air sticks to a spoon). What develops is something called a boundary layer. So the velocity very near the surface of the body looks like the sketch shown below. The thickness of the boundary layer depends on how viscous the fluid is, on how fast the body is moving, and on the distance from the leading edge of the body. The boundary layer is thicker the more viscous the fluid is, thinner the faster the body is moving, and thicker the longer the distance from the leading edge. By dimensional analysis you can see that the thickness of the boundary layer is proportional to sqrt(nx/c) where n is the kinematic viscosity (units=m^2/s), x is the distance from the leading edge of the body and c is the speed of the body. So to cause the flow to stagnate on a body (move at the same speed as the body) kinetic energy must be added, thus raising the total energy of the flow.

2) What are the conditions for the stagnation temperature equation to be true? and related questions (5 students) The stagnation temperature is the temperature you would reach if you stagnated an ideal gas via an adiabatic process with no external work.

3) Are we doing PRS questions on this stuff? (1 student) Yes we will.

4) How do we use these things? (1 student) You will see several examples in the next lecture and on the homework.

5)Why is "c" used for velocity? It is confusing because it is also used for light. (1 student) It is confusing, but we don't have enough letters in the alphabet. The other common letters used for velocity are "v" and "u" and "w" (you will see a lot of these in fluids). But in thermo we use these for volume, internal energy and work.

6)Would be clearer if you emphasized external work=shaft work. (1 student) I tried to say it several times, but I do need to add more words to this effect in the notes.

7) What do flow work and shaft work mean when we are talking about small bits of air on or around an airplane? (1 student) The same thing they mean for any control volume (note shaft work is really a placeholder for all forms of external work).

8) How do you determine the signs of wf and ws? (1 student) The same way we do for all forms of work. If energy flows out of the system (i.e. the system does work on its surroundings) the sign is positive. If energy flows into the system (i.e. the surroundings do work on the system) the sign is negative.

9) Would the temperature drop you talked about lead to accelerated ice formation on an airfoil under the right atmospheric conditions? (1 student) Perhaps. I don't realy know. Professor Hansman is an expert on icing and he may know the ansewr to this.

10) With the turbine engine, is it best to maximize wshaft and minimize wflow? (1 student) In general, one would like to extract as much work from the flow as possible. But note that in the case of the turbine, the flow work is negative, causing ws > w. Note also that in reality there are a variety of fluid mechanical and mechanical design considerations that go into determining the best flow speeds, pressures and temperatures in various parts of the device.

11) No mud (7 students). Good.