|Thermodynamics and Propulsion
The expressions we develop will allow us to define a particular mission and then determine the optimum component characteristics (e.g. compressor, combustor, turbine) for an engine for a given mission. Note that ideal cycle analysis addresses only the thermodynamics of the airflow within the engine. It does not describe the details of the components (the blading, the rotational speed, etc.), but only the results the various components produce (e.g. pressure ratios, temperature ratios). In Chapter 12 we will look in greater detail at how some of the components (the turbine and the compressor) produce these effects.
where the capital subscript will refer to the turbine. Stagnation properties, and , are more easily measured quantities than static properties ( and ). Thus, it is standard convention to express the performance of various components in terms of stagnation pressure and temperature ratios:
To get started with a simple example (no turbomachinery), we will reexamine the ideal ramjet, picking up where we left off in Section 3.7.3. (Note that we will continue to use station 5 as the exit station, consistent with Figure 3.25.)
The coordinate system and control volume are chosen to be fixed to the ramjet. The thrust, , is given by:
where and are the exit and inlet flow velocities, respectively. The thrust can be put in terms of nondimensional parameters as follows:
Using , in the expression for stagnation pressure,
The ratios of stagnation pressure to static pressure at inlet and exit of the ramjet are
The ratios of stagnation to static pressure at exit and at inlet are the same, with the consequence that the inlet and exit Mach numbers are also the same.
To find the thrust we need to find the ratio of the temperature at exit and the temperature at inlet. This is given by
where is the stagnation temperature ratio across the combustor (burner). The thrust is thus
11.13, we get:
From the steady flow energy equation:
The exit mass flow is not greatly different from the inlet mass flow, , because the fuel-air ratio is much less than unity (generally several percent). We thus neglect the difference between the mass flows and obtain
Thus, the fuel-air ratio is
The fuel-air ratio, , depends on the fuel properties ( ), the desired flight parameters ( ), the ramjet performance ( ), and the temperature of the atmosphere ( ).
The specific impulse can be written in terms of fuel properties and flight and vehicle characteristics as
We wish to explore the parameter dependency of the above expression, which is a complicated formula. How can we do this? What are the important effects of the different parameters? How do we best capture the ramjet performance behavior?
To make effective comparisons, we need to add some additional information concerning the operational behavior. An important case to examine is when we have stoichiometric conditions and all the fuel burns (denoted by ). What happens in this situation as the flight Mach number, , increases? is fixed so increases, but the maximum temperature does not increase much because of dissociation: the reaction does not go to completion at high temperature. A useful approximation is therefore to take constant for stoichiometric operation. In the stratosphere, from 10 to 30 km, . The maximum temperature ratio is
For the stoichiometric ramjet,
Using the expression for , the specific impulse is
A plot of the performance of the stoichiometric ramjet is shown in Figure 11.14. The figure shows that for the parameters used, the best operating range of a hydrocarbon-fueled ramjet is . The parameters used are , in the stratosphere, for hydrocarbons, such that .
Between Section 3.7.3 and this section we have:
How is found for rockets in space where ? (MP 11.2)
Why does industry use rather than ? Is there an advantage to this? (MP 11.3)
Why isn't mechanical efficiency an issue with ramjets? (MP 11.4)
How is thrust created in a ramjet? (MP 11.5)
What is the relation between and the existence of the maximum value of ? (MP 11.6)
Why didn't we have a 2s point for the Brayton cycle with non-ideal components? (MP 11.7)
What is the variable ? (MP 11.8)
We now examine an engine with turbomachinery, as shown in Figure 11.15. Methodology:
where is the fuel/air mass flow ratio,
Now we have to do a little algebra to manipulate these expressions into more useful forms. First we write an expression for the exit velocity:
we can write
which expresses the exit temperature as a function of the inlet temperature, the Mach number, and the temperature changes across each component. We now write the pressure at the exit in a similar manner:
and then equate this to our expression for the temperature in (11.1):
We now continue on the path to our expression for .
Now we have two steps left. First we write in terms of , by noting that they are related by the condition that the power used by the compressor is equal to the power extracted by the turbine. Second, we put the burner temperature ratio in terms of the exit temperature of the burner, ( or more specifically ) since this is the hottest point in the engine and is a frequent benchmark used for judging various designs.
The steady flow energy equation states
Assuming that the compressor and turbine are adiabatic, then
Since the turbine shaft is connected to the compressor shaft
Assuming the mass flow and specific heats are the same between compressor and turbine, this can be rewritten as
That was the first step relating the temperature rise across the turbine to that across the compressor. The remaining step is to write the temperature rise across the combustor in terms of .
and for an engine with an afterburner
Now substituting our expressions for , and into our expression for , and finally into the first expression we wrote for thrust, we get:
This is what we were seeking, an expression for thrust in terms of important design parameters and flight parameters:
By adding and substracting the quantity
we may write this write in another form which is often used,
A recap of the important variables:
Our final step involves writing the specific impulse and other measures of efficiency in terms of these same parameters. We begin by writing the First Law across the combustor to relate the fuel flow rate and heating value of the fuel to the total enthalpy rise.
where again, is the fuel/air mass flow ratio. The specific impulse becomes
where is expressed in terms of typical design parameters, flight conditions, and physical constants
Similarly, we can write our overall efficiency as
Using the definition from before, the ideal thermal efficiency is
and the propulsive efficiency can be found as
We can now use these equations to better understand the performance of a simple turbojet engine. We will use the following parameters (with ):
Note it is more typical to work with the compressor pressure ratio ( ) rather than the temperature ratio ( ) so we will substitute the isentropic relationship:
into the equations before plotting the results in Figures 11.16, 11.17, and 11.18.
To conclude this chapter, we will now improve our estimates of cycle performance by including the effects of irreversibility. We will use the Brayton cycle as an example. What are the sources of non-ideal performance and departures from reversibility?
We take into account here only irreversibility in the compressor and in the turbine. Because of these irreversibilities, we need more work, (the changes in kinetic energy from inlet to exit of the compressor are neglected), to drive the compressor than in the ideal situation. We also get less work, , back from the turbine. The consequence, as can be inferred from Figure 11.19, is that the net work from the engine is less than in the cycle with ideal components.
To develop a quantitative description of the effect of these departures from reversible behavior, consider a perfect gas with constant specific heats and neglect kinetic energy at the inlet and exit of the turbine and compressor. We define the turbine adiabatic efficiency as
where is specified to be at the same pressure ratio as . There is a similar metric for the compressor, the compressor adiabatic efficiency:
again for the same pressure ratio. Note that for the turbine the ratio is the actual work delivered divided by the ideal work, whereas for the compressor the ratio is the ideal work needed divided by the actual work required. These are not thermal efficiencies, but rather measures of the degree to which the compression and expansion approach the ideal processes.
We now wish to find the net work done in the cycle and the efficiency. The net work is given either by the difference between the heat received and rejected or the work of the compressor and turbine, where the convention is that heat received is positive and heat rejected is negative and work done is positive and work absorbed is negative.
The thermal efficiency is
We need to calculate , .
From the definition of ,
Similarly, by the definition
we can find :
The thermal efficiency can now be found:
There are several non-dimensional parameters that appear in this expression for thermal efficiency. We list these in the two sections below and show their effects in accompanying figures.
In addition to efficiency, net rate of work is a quantity we need to examine,
Putting this in a non-dimensional form,
Trends in net power and efficiency are shown in Figure 11.20 for parameters typical of advanced civil engines. Some points to note in the figure:
A scale diagram of a Brayton cycle with non-ideal compressor and turbine behaviors, in terms of temperature-entropy ( - ) and pressure-volume ( - ) coordinates is given below as Figure 11.21.
Isn't it possible for the mixing of two gases to go from the final state to the initial state? If you have two gases in a box, they should eventually separate by density, right? (MP 11.9)
How can be the maximum turbine inlet temperature? (MP 11.10)
When there are losses in the turbine that shift the expansion in - diagram to the right, does this mean there is more work than ideal since the area is greater? (MP 11.11)
For an afterburning engine, why must the nozzle throat area increase if the temperature of the fluid is increased? (MP 11.12)
Why doesn't the pressure in the afterburner go up if heat is added? (MP 11.13)
Why is the flow in the nozzle choked? (MP 11.14)
What's the point of having a throat if it creates a retarding force? (MP 11.15)
Why isn't the stagnation temperature conserved in this steady flow? (MP 11.16)