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# Stream thrust averaging

Stream thrust averaging is a process used to convert 3 dimensional flow through a duct into 1 dimensional uniform flow. It makes the assumptions that the flow is mixed adiabatically and without friction. However, due to the mixing process, there is a net increase in the entropy of the system. Although there is an increase in entropy, the stream thrust averaged values are more representative of the flow than a simple average as a simple average would violate the 2nd Law of Thermodynamics.

The following equations apply to a perfect gas.

Stream Thrust:$F = int left\left( ho mathbf\left\{V\right\} cdot d mathbf\left\{A\right\} ight\right) mathbf\left\{V\right\} cdot mathbf\left\{f\right\} +int pd mathbf\left\{A\right\} cdot mathbf\left\{f\right\}$

Mass Flow:$dot m = int ho mathbf\left\{V\right\} cdot d mathbf\left\{A\right\}$

Stagnation Enthalpy:$H = \left\{1 over dot m\right\} int left\left(\left\{ ho mathbf\left\{V\right\} cdot d mathbf\left\{A ight\right) left\left( h+ \left\{|mathbf\left\{V\right\}|^2 over 2\right\} ight\right)$

:$overline\left\{U\right\}^2 left\left(\left\{1- \left\{R over 2C_p ight\right) -overline\left\{U\right\}\left\{Fover dot m\right\} +\left\{HR over C_p\right\}=0$

Solving for $overline\left\{U\right\}$ yields 2 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 2nd Law of Thermodynamics may be applied.

:$overline\left\{ ho\right\} = \left\{dot m over overline\left\{U\right\}A\right\}$

:$overline\left\{p\right\} = \left\{F over A\right\} -\left\{overline\left\{ ho\right\} overline\left\{U\right\}^2\right\}$

:$overline\left\{h\right\} = \left\{overline\left\{p\right\} C_p over overline\left\{ ho\right\} R\right\}$

2nd Law of Thermodynamics:$abla s = C_p ln\left(\left\{overline\left\{T\right\}over T_1\right\}\right) +R ln\left(\left\{overline\left\{p\right\} over p_1\right\}\right)$

The values $T_1$ and $p_1$ are unknown and may be dropped from the formulation. The value of entropy is not necessary, only that the value is positive.:$abla s = C_p ln\left(overline\left\{T\right\}\right) +R ln\left(overline\left\{p\right\}\right)$

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

* [http://gltrs.grc.nasa.gov/reports/1999/TM-1999-209279.pdf Inlet Development for a Rocket Based Combined Cycle, Single Stage to Orbit Vehicle Using Computational Fluid Dynamics]

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