uses Pr function Calculate the air temperature and pressure at the stagnation point right in front of a meteorite entering the atmosphere (-50 °C, 50 kPa) with a velocity of 2000 m/s. Do this assuming air is incompressible at the given state and repeat for air being a compressible substance going

Question

uses [latex]P _{ r }[/latex] function
Calculate the air temperature and pressure at the stagnation point right in front of a meteorite entering the atmosphere (-50 °C, 50 kPa) with a velocity of 2000 m/s. Do this assuming air is incompressible at the given state and repeat for air being a compressible substance going through an adiabatic compression.

Accepted Answer

Kinetic energy: [latex]\frac{1}{2} V ^{2}=\frac{1}{2}(2000)^{2} / 1000=2000 kJ / kg[/latex]

Ideal gas: [latex]v _{ atm }= RT / P =0.287 imes 223 / 50=1.28 m ^{3} / kg[/latex]

a) incompressible

Energy Eq.4.13: [latex]\Delta h =\frac{1}{2} V ^{2}=2000 kJ / kg[/latex]

If A.5 [latex]\Delta T =\Delta h / C _{ p }=1992 K[/latex] unreasonable, too high for that [latex]C _{ p }[/latex]

Use A.7:

[latex]\begin{array}{l}h _{ st }= h _{o}+\frac{1}{2} V ^{2}=223.22+2000=2223.3 kJ / kg \T _{ st }=1977 K\end{array}[/latex]

Bernoulli (incompressible) Eq.7.17:

[latex]\begin{array}{l}\Delta P = P _{ st }- P _{ o }=\frac{1}{2} V ^{2} / v =2000 / 1.28=1562.5 kPa \P _{ st }=1562.5+50=1612.5 kPa\end{array}[/latex]

b) compressible

[latex]T _{ st }=1977 K[/latex] the same energy equation.

From A.7.2: Stagnation point [latex]P _{ r st }=1580.3[/latex] ; Free [latex]P _{ r o }=0.39809[/latex]

[latex]\begin{aligned}P _{ st } &= P _{ o } imes \frac{ P _{ r st }}{ P _{ r o }}=50 imes \frac{1580.3}{0.39809} \&=198485 kPa\end{aligned}[/latex]

Notice that this is highly compressible, v is not constant.

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Eq.4.13 : [latex]q+h_{i}+\frac{ V _{i}^{2}}{2}+g Z_{i}=h_{e}+\frac{ V _{e}^{2}}{2}+g Z_{e}+w[/latex]

Eq.7.17 : [latex]v P_{i}+\frac{1}{2} V _{i}^{2}+g Z_{i}=v P_{e}+\frac{1}{2} V _{e}^{2}+g Z_{e}[/latex]

Question 7.80: uses Pr function Calculate the air temperature and pressure ...

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