Stream thrust averaging


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]


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