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Stream thrust averaging
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<h2 id="equations-for-a-perfect-gas">Equations for a perfect gas</h2> <p>Stream <a href="/facts/Thrust/EM04CqV7">thrust</a>: </p> F = ∫ ( ρ V ⋅ d A ) V ⋅ f + ∫ p d A ⋅ f . {\displaystyle F=\int \left(\rho \mathbf {V} \cdot d\mathbf {A} \right)\mathbf {V} \cdot \mathbf {f} +\int pd\mathbf {A} \cdot \mathbf {f} .} <p><a href="/facts/Mass_flow_(physics)/v2DrbGvq">Mass flow</a>: </p> m ˙ = ∫ ρ V ⋅ d A . {\displaystyle {\dot {m}}=\int \rho \mathbf {V} \cdot d\mathbf {A} .} <p>Stagnation <a href="/facts/Enthalpy/bBou8lCE">enthalpy</a>: </p> H = 1 m ˙ ∫ ( ρ V ⋅ d A ) ( h + | V | 2 2 ) , {\displaystyle H={1 \over {\dot {m}}}\int \left({\rho \mathbf {V} \cdot d\mathbf {A} }\right)\left(h+{|\mathbf {V} |^{2} \over 2}\right),} U ¯ 2 ( 1 − R 2 C p ) − U ¯ F m ˙ + H R C p = 0. {\displaystyle {\overline {U}}^{2}\left({1-{R \over 2C_{p}}}\right)-{\overline {U}}{F \over {\dot {m}}}+{HR \over C_{p}}=0.} <h3>Solutions</h3> <p>Solving for U ¯ {\displaystyle {\overline {U}}} yields two solutions. They must both be analyzed to determine which is the physical solution. One will usually be a subsonic <a href="/facts/Root_of_a_function/mlK3dhoT">root</a> and the other a supersonic root. If it is not clear which value of <a href="/facts/Velocity/Q3o1LVDe">velocity</a> is correct, the <a href="/facts/Second_law_of_thermodynamics/m7SJ2eeH">second law of thermodynamics</a> may be applied. </p> ρ ¯ = m ˙ U ¯ A , {\displaystyle {\overline {\rho }}={{\dot {m}} \over {\overline {U}}A},} p ¯ = F A − ρ ¯ U ¯ 2 , {\displaystyle {\overline {p}}={F \over A}-{{\overline {\rho }}{\overline {U}}^{2}},} h ¯ = p ¯ C p ρ ¯ R . {\displaystyle {\overline {h}}={{\overline {p}}C_{p} \over {\overline {\rho }}R}.} <p>Second law of thermodynamics: </p> ∇ s = C p ln ( T ¯ T 1 ) + R ln ( p ¯ p 1 ) . {\displaystyle \nabla s=C_{p}\ln({{\overline {T}} \over T_{1}})+R\ln({{\overline {p}} \over p_{1}}).} <p>The values T 1 {\displaystyle T_{1}} and p 1 {\displaystyle p_{1}} are unknown and may be dropped from the formulation. The value of entropy is not necessary, only that the value is positive. </p> ∇ s = C p ln ( T ¯ ) + R ln ( p ¯ ) . {\displaystyle \nabla s=C_{p}\ln({\overline {T}})+R\ln({\overline {p}}).} <p>One possible unreal solution for the stream thrust averaged velocity yields a negative entropy. Another method of determining the proper solution is to take a simple average of the velocity and determining which value is closer to the stream thrust averaged velocity. </p> <ul><li>DeBonis, J.R.; Trefny, C.J.; Steffen, Jr., C.J. (1999). <a href="https://ntrs.nasa.gov/api/citations/19990062664/downloads/19990062664.pdf">"Inlet Development for a Rocket Based Combined Cycle, Single Stage to Orbit Vehicle Using Computational Fluid Dynamics"</a> (PDF). <i>NASA/TM—1999-209279</i>. NASA. Retrieved 18 February 2013.</li></ul>