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Stream thrust averaging

Equations for a perfect gas[edit]

Stream thrust:

${\displaystyle F=\int \left(\rho \mathbf {V} \cdot d\mathbf {A} \right)\mathbf {V} \cdot \mathbf {f} +\int pd\mathbf {A} \cdot \mathbf {f} .}$

Mass flow:

${\displaystyle {\dot {m}}=\int \rho \mathbf {V} \cdot d\mathbf {A} .}$

Stagnation enthalpy:

${\displaystyle H={1 \over {\dot {m}}}\int \left({\rho \mathbf {V} \cdot d\mathbf {A} }\right)\left(h+{|\mathbf {V} |^{2} \over 2}\right),}$

${\displaystyle {\overline {U}}^{2}\left({1-{R \over 2C_{p}}}\right)-{\overline {U}}{F \over {\dot {m}}}+{HR \over C_{p}}=0.}$

Solutions[edit]

Solving for ${\displaystyle {\overline {U}}}$ yields two solutions. They must both be analyzed to determine which is the physical solution. One will usually be a subsonic root and the other a supersonic root. If it is not clear which value of velocity is correct, the second law of thermodynamics may be applied.

${\displaystyle {\overline {\rho }}={{\dot {m}} \over {\overline {U}}A},}$

${\displaystyle {\overline {p}}={F \over A}-{{\overline {\rho }}{\overline {U}}^{2}},}$

${\displaystyle {\overline {h}}={{\overline {p}}C_{p} \over {\overline {\rho }}R}.}$

Second law of thermodynamics:

${\displaystyle \nabla s=C_{p}\ln({{\overline {T}} \over T_{1}})+R\ln({{\overline {p}} \over p_{1}}).}$

The values ${\displaystyle T_{1}}$ and ${\displaystyle p_{1}}$ are unknown and may be dropped from the formulation. The value of entropy is not necessary, only that the value is positive.

${\displaystyle \nabla s=C_{p}\ln({\overline {T}})+R\ln({\overline {p}}).}$

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.

References[edit]

• DeBonis, J.R.; Trefny, C.J.; Steffen, Jr., C.J. (1999). "Inlet Development for a Rocket Based Combined Cycle, Single Stage to Orbit Vehicle Using Computational Fluid Dynamics" (PDF). NASA/TM—1999-209279. NASA. Retrieved 18 February 2013.