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I came upon this when reading "Introduction to Rocket Science and Engineering, he said "by differentiating" , my question is: differentiate with respect to what? and how do you get such result in the end?

This is the original equation and This is the final Result

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  • $\begingroup$ Is that the Travis Taylor book? $\endgroup$ Oct 14 '20 at 19:05
  • $\begingroup$ By the same passage, this is apparently "non-trivial" to show. $\endgroup$ Oct 14 '20 at 19:13
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    $\begingroup$ @OrganicMarble Yes, that's the one $\endgroup$ Oct 14 '20 at 19:16
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    $\begingroup$ What page is this on? I'll need some more context to give a good response. $\endgroup$ Oct 15 '20 at 1:23
  • $\begingroup$ I'm guessing it's with respect to gamma (ratio of specific heats, itself a function of P, T, and composition). After differentiating, you'd need to solve for a zero of the differentiated expression, as that would be an extrema (likely, the minimum represented by the throat). You'd then substitute it back in. $\endgroup$ Oct 15 '20 at 1:26
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The equation shows that the cross-sectional area of the nozzle depends on the local pressure $P$ and values that are constant. So, he expects you to differentiate with respect to $P$, set the result to zero and solve for the value of $P$. Sounds like a fun project. I would let $x=P/P_c$ and take derivative wrt x.

Some books go through equations with Mach number and finding the throat condition is rather easier. See Mattingly Elements of Gas Turbine Propulsion (1996) in Chapter 3, or Sutton Rocket Propulsion Elements, also Chapter 3.

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