We must start by applying the energy equation to any two points 1 and 2 in the flow through the nozzle. There is no work done to the surroundings so we can sum all the forms of energy at the two sections and equate them. The forms of energy are internal energy, kinetic energy, pressure energy and, to be pedantic, potential energy. In this analysis it is better to work, not in head (energy per unit weight) as we did for liquids, but in energy per unit mass.
Then the equation becomes :- where is the internal energy per unit mass and the other symbols their usual meanings.
Nozzles are short so the potential energy terms are pointless and the equation becomes :-
We have and . From this as is well known in thermodynamics and called the enthalpy[1]. Then the energy equation reduces to :-
I have argued for the flow through a convergent-divergent nozzle to be continuous and this means that this equation can be applied to any two points in the flow. Then, if as before, we refer to the stagnation conditions we can proceed.
For any section in the nozzle we can write :- where is the absolute temperature at the section.
If we work with ratios between of the properties in the nozzle and the stagnation properties we shall end up with a set of ratios that are the same for every convergent-divergent nozzle. In order to use these ratios we shall have to find values of the reference properties for every set of stagnation properties and mass flow.
Now can be changed to Mach number form by dividing through by to give :-
and then by using to give .
Then substituting in the energy equation gives :-
.
The first of these ratios is between absolute temperatures .
This gives us in terms of and we may need in terms of and , and, in terms of and .
We have and . One gives and the other gives :-
. Then
Therefore :-
Finally and then
These expressions allow us to find out how the properties vary with Mach number but we have no way as yet to link the values of the properties with the nozzle. We really want some link with the area of cross-section so that, if we knew the profile of the nozzle, we can plot properties against length. We cannot continue with the stagnation conditions because the idea of having a cross-section at the stagnation conditions is meaningless. But we can use the throat where we not only have the minimum diameter but also conditions that can be related to the stagnation conditions. This leads us to find a ratio between the cross-sectional area at any point and that at the throat.
We have the continuity equation and we can write where the subscript indicates throat conditions. From this and we have
.
This does not match the other equations and we want it in terms of so another manipulation is required. We can use and we can write :-
which reduces to :-
and that deals with the density term. Now we need to eliminate the temperature term. We can use :-
Now we can substitute to find the ratio of areas.
and this reduces to:-
In the top of figure 13-9 I have gathered the four resulting expressions. They are in the most convenient form that is known and each of the ratios is expressed in terms of and . In the lower part I have substituted 1.4 for to give much more manageable expressions for air. They can easily be plotted using, say, Mathcad and the result is shown in graph 13-11 and in graph 13-12. It is also easy to evaluate any of the ratios accurately using the calculating facility.
It should be noted that this is the graph for a reversible, adiabatic, one-dimensional, expansion of air through a convergent-divergent nozzle. There will be similar graphs for other gases and there must be some similar graph, probably based on an approximation, for steam although that is outside of the scope of this text.
[1] At one time this was called total heat but as it was not total and not heat there was a good case for changing it.