Natural exponential family

In fluid dynamics, stream thrust averaging is a process used to convert three dimensional flow through a duct into one 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 second Law of Thermodynamics.

Equations for 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:

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

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

Solutions

Solving for ${\overline {U}}$ 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.

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

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

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

Second 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.

References

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