Choked Flow: Throat Radius should not change any more below critical pressure ratio?

Question

I am trying to understand some calculations of a student colleague:

The pressure inside the combustion chamber is 200 bar. This means, the critical pressure ratio should be p*/p0 with p0=101325 Pascal and k=1.18 -- >

p*/(200*10^5 Pa) = (2/1.18+1)^(1.18/0.18)

--> p* = 1.13489×10^7 Pa

This means, as long as my atmosphere pressure is lower than p* I will have choked flow condition. So my throat radius should not change anymore?

Because what I see in the calculations of my student colleague is, that the lower I make p* (so like 10000 Pascal) the throat diamter is still reducing?

Where does this come from?

Thanks for your help

Lucas

Answer 1

I will assume this is a rocket problem with the use of combustion chamber pressure, stated to be 200 bar (which is 2 x 10^7 Pa). The ratio p*/p0 would be throat pressure divided by chamber pressure (which we assume is stagnation pressure). Standard sea level pressure in 101325 Pa, but is irrelevant to the equation. Just for completeness, the equation is:

$$\frac{p*}{p_0}=\left(\frac{2}{k+1}\right)^\frac{k}{k-1} $$

With k=1.18 I get 0.56839, or p* = 1.1368 $\cdot10^7$ Pa. (close enough to above)

Rockets use converging-diverging nozzles, the flow will adjust itself to be Mach 1 at the minimum area, that will be the throat. Only if the combustion chamber pressure gets real low (1/.56839 times the ambient pressure) will the minimum area go below mach = 1.

Reference note: equation 2.35a of Elements of Gasdynamics by Liepmann and Roshko, or equation 3-20 Rocket Propulsion Elements, 7th ed., by Sutton.