**General comments**

Wow! The 16.070 problem set must have taken its toll. When I started the lecture there were 23 people in the room. When I finished there were only 42. This will cause some people to struggle with the application of the integral momentum theorem. This is one of the most important concepts in the propulsion material. If you are one of these people: 1) read the notes, 2) start your homework early so you can ask me questions, and 3) come to the recitation on Tuesday--I will do some examples.

I think the discussion of the integral momentum equation went pretty well. I was pleased that the class identified problems in each of the 3 PRS questions (see my notes on these PRS #1, PRS#2, PRS#3). It is gratifying when you folks know enough to not only answer the problem, but also to find what is wrong with the problem statement.

**Responses to 'Muddiest Part of the Lecture Cards'**

(21 respondents)

1)* How did you introduce
(pe-po)Ae into the thrust equation?* (3 students) Read
over the section of
the notes and the explanation of the
PRS question. These should help. Pressure forces are one of the external
forces (along with drag, thrust, gravity, etc.) that we typically work with.
They are captured in the SFx
term in the integral momentum equation. Since they are applied on a surface
(e.g. the surface of the control volume), you need to do the surface integral
around the control volume and add up the net force. Remember
that pressure always acts normal to a surface (so if you know the orientation
of the surface, you know the orientation of the force.

2)** In the second
concept question, if inlet and exhaust velocities are the same regardless
of back pressure, how can you manipulate back pressure? **(1 student).
They are not the same regardless of back pressure, but I wanted the class to
focus on arriving at the pressure term in the thrust equation through simplified
pressure balance arguments. As I noted, I was so intent on this objective, that
I didn't even include in the problem statement the proviso that everything else
was constant.

3)* For that PRS
question with the plane taking off, the answer makes sense, but I thought
when we were talking about the streamtubes that you said inlet velocity was
constant. That would lead to constant thrust.* (1 student) Good
question. The velocity in the

4) * I am confused about where
the terms come from (mduo/dt). (Notations written on card--yes?).*
(1 student) The question refers to the example
of the falling eraser. You have it exactly correct as written on your card.

5) * In the equation SFx
- Fox = ...., does the Fx mean just the forces in the x-direction?*
(1 student). Yes. This equation
is one of three (x, y, z components) from the integral momentum equation. The
momentum equation is a vector relation and expresses force and momentum balances
in three coordinate directions. To get the version for the y-component, just
replace all the little "x's" with "y's".

6) * Can we stop using screwed up coordinate systems
(drawing of axes with x-direction pointed up) and just stick with the usual
ones?* (1 student). We don't design bridges in this department.
We design things that fly and spin. Coordinate frames can appear in all sorts
of orientations.

7) * [On the 3rd
PRS question] Since the boundary line is parallel to the y-axis why is the
flux in the y-direction not zero? * (1 student) Good question.
The problem asks for the flux of y-momentum across the interface. The fluid
crossing the interface (in the x-direction) carries with it some y-momentum
(i.e. it is moving up and over at the same time).

8) * Can we see some more examples
of applying the integral momentum equation?* (1 student) Yes!

9)* Why is an engine most efficient
when the back pressure is equal to the atmospheric pressure?* (1 student). This is a super question. The thrust force produced by the change
in momentum flux of the gas flowing through the control volume is transmitted
to the engine surfaces through pressure and viscous forces. It is easier for
us to calculate thrust by looking at the change in momentum flux + pressure
on all of the surfaces of the gas that goes through the engine, but we could
just as well integrate up all the surface forces on every internal part of the
engine and sum all these up and get the same answer. However, given the complexity
of the geometry and flowfield within an engine, this is not practical. But nonetheless,
when the nozzle exit velocity and/or pressure changes, there is a corresponding
change in pressure on the surfaces within the engine and this in turn leads
to a change in thrust. If the flow is subsonic at the exit of the nozzle, then
the pressures are matched (the ambient pressure and the exhaust pressure equilibrate--just
like opening a high pressure air bottle in a room). It is only for the case
of supersonic nozzle flow that this term can be important. For this situation,
the typical case is one where the flow has not fully expanded to atmospheric
pressure (pe>po). The flow does expand after leaving the engine, but there
is no structure for it to push against, so the expansion produces no thrust.
There is a pretty good discussion of this on page 79 of Kerrebrock's book "Aircraft
Engines and Gas Turbines" (available in the library--it is the text for
16.50).

10) * No mud* (9 students).
Good!