Lecture T8: The Steady Flow Energy Equation--Shaft work and Flow work

Lecture T9: Steady Flow Energy Equation--Stagnation Temperature


General comments

This was a double lecture so I asked for only one set of mud cards. So the responses are combined here. The lectures focused on the steady flow energy equation (SFEE). The steady flow energy equation is an expression of conservation of mass and energy for an open thermodynamic system. It is common for people to have difficulty both with units and with signs when applying the equation to solve a problem. Therefore, I used the first PRS question to emphasize the importance of getting this right. We then moved on to one of the more confusing concepts -- the distinction between flow work and other forms of external work (shaft work being the most common example). When we express the SFEE in terms of shaft work and flow work, the flow work can be combined with the internal energy terms to arrive at enthalpy. Remember the enthalpy is the energy transfer to an open system when flow crosses the system boundary. Part of this energy transfer is the internal energy carried by the flow (u), the other part is the work done by the system when the flow crosses the boundary (pv). The latter we call the flow work. For a steady flow system, with mass flowing both in and out, the flow work is the sum of the work done by the system when the gas flows in (-pinvin) and the work done by the system when the gas flows out (+poutvout). This is also equal to R(Tout-Tin). We did the second and third PRS questions to highlight the distinction between shaft work and flow work. The second PRS question was intended to emphasize that anytime there is a temperature change there is flow work, even if we are commonly accustomed to thinking of the device producing shaft work (e.g. a compressor or a turbine). The third PRS question focus on applying some of the equations (the SFEE and the expression for flow work) to determine the relative magnitudes of shaft work, flow work and work for a turbine.

And between the 2nd and 3rd PRS questions, WE BROKE FOR DONUTS -- Remember, you each probably ate about a MJ of energy.

We then moved on to discuss what has historically been the most confusing part of the thermo lectures in Unified: stagnation quantities and in particular issues regarding frame dependence of stagnation quantitites. The stagnation (or total) temperature is the temperature the flow would reach if it were decelerated to zero speed in a particular reference frame via an adiabatic process with no external work. The acceleration and deceleration of gases can very frequently be well-modeled with such a process. We began with a fourth PRS question to highlight why it is important to understant the stagnation temperature (it sets the skin temperature on a high speed vehicle). Note that even moving flows have a stagnation temperature (i.e., they don't have to be stopped, the stagnation temperature is the temperature they would reach IF you stopped them via this particular process.) The total or stagnation enthalpy (cpTtotal) is a quantity that is conserved for these processes. We also related the stagnation temperature to the Mach number (instead of the flow speed in m/s). The stagnation temperature is dependent on the speed of the reference frame. It is not a thermodynamic property. The thermodynamic property is T (which we sometimes make confusing by calling it the "static" temperature. Here "static" has nothing to do with stopping the flow. It is just the word people have historically used to distinguish the thermodynamic properties. So yes, our "p" for pressure, is often called the "static" pressure.)

Remember the following points. 1) You can only apply the steady flow energy equation in a reference frame where the flow appears as steady. 2) Flows stagnate on surfaces. 3)If two points are not moving relative to one-another (i.e. they are in the same reference frame), and the flow moves from one point to another without heat addition or external work, then the total enthalpy is constant and the stagnation temperatures at the two points are the same. 4) If a flow moves without heat addition or external work, the enthalpy is reduced ( and the temperature along with it) as the kinetic energy is increased (because total enthalpy is constant). 5) If a reference frame is moving relative to a stationary frame, flows that stagnate in the moving frame have a higher stagnation temperature (the stagnation or total enthalpy in the moving frame is higher). The stagnation temperature is dependent on the speed of the reference frame.

We concluded with a fifth PRS question to test your knowledge of the stagnation temperature and frame dependence. Most of you got this wrong. This is to be expected. Once we try a few more questions and you have some more time to think about it, you will do better.


Responses to 'Muddiest Part of the Lecture Cards'

(60 respondents out of 74 students)

1) It would seem to me that the flow work would affect the shaft work. (1 student) The detailed fluid mechanical design of the turbine and compressor sets both the shaft work and the flow work, and the two forms of work are interdependent. That is, changing the shape or speed of the blades will effect both flow work and shaft work. But if you go to the inlet and outlet of the device you can still measure two different quantities (flow work and shaft work).

2) Can we set a precendent for naming the static temperature something else, something better? (1 student) Probably not, this naming convention is set pretty deep.

3) What is shaft work? (4 students) A flow of energy into or out of a system via a shaft (such as a shaft attached to a compressor or turbine). Note I use this in the equations in the notes because it is the most common situation, but in general ws should be replaced with all external forms of work other than flow work. You could measure the shaft power by knowing the torque and the rate of rotation of the shaft. Doesn't the shaft work in a compressor have a negative sign? (1 student) Yes. What is flow work? (3 students) For a steady flow system, with mass flowing both in and out, the flow work is the sum of the work done by the system when the gas flows in (-pinvin) and the work done by the system when the gas flows out (+poutvout). This is also equal to R(Tout-Tin). Think of this as how much energy would be required to push a kilogram in one end of a control volume while at the same time pushing a kilogram out the other end of the control volume, but where the volumes occupied by each kilogram and the pressure at each end were different. If there is only flow work from the combustor, then what drives the turbine? (1 student) Energy is transferred to the turbine inlet in two forms, the internal energy carried by the flow, and the work required to push the flow into the turbine (the flow work). This energy is then typically converted to kinetic energy in a nozzle which then directs a high speed flow against the turbine blades, causing them to spin (creating mechnical energy in the spinning rotor). You will learn all the details of this in the spring propulsion lectures.

4) Can you explain the significance of dividing by mdot to get ? (2 students) It is often useful to solve problems on a per kilogram basis instead of a per second basis. When you divide J/s by kg/s you get J/kg. I am not sure how you can substitute cp(T2-T1) for h2-h1. (1 student) For an ideal gas cpdT=dh. I have further assume that cp is a constant (i.e. not a function of temperature).

5) How do you not confuse h "height" and h "enthalpy" in the SFEE? (1 student) I neglect changes in potential energy so I don't have to worry about the changes in height. Other people use "z" for height. This is what I use in the notes. I just wasn't paying enough attention on the board. I need a bigger alphabet.

) In the Anderson book it mentions that the air surrounding the vehicle heats up. Is that air directly around the vehicle (stuck to it) or flowing--and related questions (3 students) Some of the air molecules are touching the surface of the vehicle. These move at the speed of the vehicle (and are the hottest). Others a small distance away are pulled along with the vehicle but lag behind. These are a little cooler. A more detailed discussion if provided below.Why can/do molecules stop? Why don't they drip across the surface like water on a car window when it rains? (1 student) The drops form (on a window) because of surface tension in the liquid. This only occurs if there is a easily deformable surface between the air and the liquid. For water that flows around a fish or air that flows around an airplane, there is no surface that can be deformed as a result of the surface tension. It is a single phase flow against a fairly rigid body.

13) Isn't the exhaust gas temperature of a turbine engine greater than the inlet temperature? (2 students). Yes. There is some confusion about the words I use. A turbine is a device for extracting work from a fluid. A gas turbine engine contains a turbine (following the combustor) as well as a compressor. The PRS question asked about a turbine, not a gas turbine engine. What is a compressor? (1 student) A device for raising the pressure of a fluid.

14) How can the stagnation temperature be calculated? (1 student) Using the SFEE with q=0 and ws=0.

14) Do the blades inside a gas turbine really go at M=2? (1 student) It depends on the design. Most run at tip Mach numbers a little less than this (say Mach 1.2 to Mach 1.5), but some run faster (Mach 2.5).

15) What formulae do we need to care about/know/use? (1 student) Through doing the homework problems, you will get a better feel for the commonly used equations. In know that it is difficult when you are seeing them for the first time to determine which equations are most useful and which are just steps in a derivation. Arrgh! Too many equations, too many variables (2 students). Practice will help. Also, feel free to see me or the TA's for help.

) The last PRS question was confusing (14 students) This is discussed in detail in the notes. And we will have more examples tomorrow.

) General confusion about static and stagnation quantities--and related questions (16 students) More discussion and examples are on the way. In the mean time, here are two additional discussions from the old mud:

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.

13) No mud (16 students). Good.