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.

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The following equations apply to a perfect gas.

Stream Thrust

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

Mass Flow

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

Stagnation Enthalpy

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

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

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.

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

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

$\overline{h} = {\overline{p} C_p \over \overline{\rho} R}$

2nd Law of Thermodynamics

$\nabla s = C_p \ln({\overline{T}\over T_1}) +R ln({\overline{p} \over p_1})$

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.

$\nabla s = C_p \ln(\overline{T}) +R ln(\overline{p})$

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.