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

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

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

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

Solving for $overline\{U\}$ 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.

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

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

:$overline\{h\}\; =\; \{overline\{p\}\; C\_p\; over\; overline\{\; ho\}\; R\}$

2nd Law of Thermodynamics:$abla\; s\; =\; C\_p\; ln(\{overline\{T\}over\; T\_1\})\; +R\; ln(\{overline\{p\}\; over\; p\_1\})$

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.:$abla\; s\; =\; C\_p\; ln(overline\{T\})\; +R\; ln(overline\{p\})$

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

* [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]