Compressible Flow

1) The temperature and pressure at the stagnation point of a high speed missile are ${934^ \circ }$ R and 7.8 atm, respectively.Calculate the density at this point.

Solution:  $$\begin{array}{l}T = {934^ \circ }R\\p = 7.8\,atm\\Density = \rho  = ?\\p = \rho RT\\\rho  = \frac{P}{{RT}} = \frac{{\left( {7.8 \times 2116} \right)}}{{1716 \times 934}} = 0.0103\,slug/f{t^3}\end{array}$$ 2)Calculate ${c_p},{c_v},e\,{\rm{and}}\,h$ for 

a) The stagnation point conditions given in problem (1).

b) Air at standard sea level conditions.

Solution: ${c_p},{c_v},e\,{\rm{and}}\,h$ for stagnation point conditions will be 

a)   $${c_p} = \frac{{\gamma R}}{{\gamma  - 1}} = \frac{{1.4 \times 1716}}{{0.4}} = 6006\frac{{ft}}{{slug}}\frac{{lb}}{{{}^ \circ R}}$$ $${c_v} = \frac{R}{{\gamma  - 1}} = \frac{{1716}}{{0.4}} = 4290\frac{{ft}}{{slug}}\frac{{lb}}{{{}^ \circ R}}$$ $$e = {c_v}T = 4290(934) = 4.007 \times {10^6}\frac{{ft\,lb}}{{slug}}$$ $$h = {c_p}T = 6006(934) = 5.610 \times {10^6}\frac{{ft\,lb}}{{slug}}$$ b) ${c_p}$ and ${c_v}$ For a  calorically perfect gas, ${c_p}$ and ${c_v}$ are constants,  $${c_p} = 6006\frac{{ft}}{{slug}}\frac{{lb}}{{{}^ \circ R}};\,{c_v} = 4290\frac{{ft}}{{slug}}\frac{{lb}}{{{}^ \circ R}}$$ Also at standard sea level $T = {519^ \circ }R$ $$E = 4290 \times 519 = 2.227 \times {10^6}\frac{{ft\,lb}}{{slug}}$$ $$h = 6006\left( {519} \right) = 3.117 \times {10^6}\frac{{ft\,lb}}{{slug}}$$


${\left( {\frac{{{c_l}}}{{{c_d}}}} \right)_{\max }} = \left( {\frac{{{c_l}}}{{{c_f} + {c_d}}}} \right)$