Modelling using properties

I will start with the treatment using properties.  The basic argument is that, in a nozzle, the work done by the gas as it expands goes to increase the kinetic energy of the gas in one direction. This means that a thermodynamic process takes place in which some of the random energy that is stored in the molecular structure is converted to mechanical work in the form of kinetic energy. This is done in a flow process and we must write :-

        .

In figure 13-6 I have drawn a pressure-volume graph for a fully resisted adiabatic expansion of air. The terms in the equation above can be picked out as areas on the diagram.  is the area under A®1,  is the area under the curve 1®2 and  is the area under 2®B. Summing these as in the equation shows that the work done on the gas as it flows through the nozzle is the area A®1®2®B®A.

 

Then it is evident that:-

           . This is a regular integral in thermodynamics and the equation becomes :-

            

In fact this is a steady flow of air from some region where the velocity  can be regarded as very small and then the temperature is the stagnation temperature. I have shown this in principle in figure 13-7. Then the measurable stagnation conditions are .

 

Now, if we choose to consider an expansion from stagnation conditions of  to some state where the conditions are  equation  can be rearranged to give velocity :-

                                             .

Now this must be manipulated to eliminate the volume terms.

                                            

But  and then:-

                                            

This expression for  can be made more convenient if  to give

                                          

 

(It is worth noting that there is a maximum velocity for the air obtained by putting . For a stagnation temperature0f 293°K it is 767 m/s or 2,760 km/h or 1716 mph.)

 

A graph of  could be plotted against  very easily and  we can find the diameter of the nozzle at the point where the pressure is  at the same time.

We have from continuity  or . This requires another manipulation

       :             .

Then :-                                  and .

Whilst I am plotting graphs I can add the graphs of  and . This generates the two graphs 13-7 and 13-8.

 

Graph 13-8 shows the profile of a nozzle that could, in principle, have a uniform pressure drop from inlet to outlet. It has a divergence that is hopelessly impractical.

 

I think that this topic of flow in a nozzle is complicated and it leads to a complicated result and it needs to be explored just to get a feel for the numbers. What is really needed is a graph showing to give the property distribution in a nozzle with a practical profile.I made a couple of attempts just to get practical shapes and start conditions. In the end I settled to look at a nozzle having a throat of 12 mm diameter with air supplied at 10 bar gauge and 100°C with a mass flow of 0.26 kg/s. What I thought that I would like to know is the way that the velocity and the pressure would change along the length of a nozzle having the sort of profile shown in figure 13-7. I let my nozzle have an inlet radius of 15 mm and a divergence of 16° included angle for no other reason than it is much the same as that for the divergence in a Venturi meter.

 

The result is graph 13-9.

 

It should be noted that in order to draw graph 13-7 and 13-8 I had to start with values for the stagnation conditions. At one time this would have meant that the calculations for drawing the graphs would have been very time consuming. That problem has disappeared with the emergence of mathematics packages on computers. We can draw as many graphs as we like almost instantly.

 

I think that this graph gives a good idea of what goes on in a convergent-divergent nozzle. It is surprising how short the divergence needs to be for the Mach number to reach 2 and then how much more of the divergence is required to reach Mach 4 but the increase in Mach number from 2 to 4 takes place at absolute pressures less than 1.5 bar. This reflects the shape of the  curve in graph 13-2 where small changes in low pressures produce large changes in volume.