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:

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

Mass flow:

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

Stagnation enthalpy:

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

$\stackrel{2}{\stackrel{‾}{U}}\left(1-\frac{R}{2C_p}\right)-\bar{U}\frac{F}{\dot{m}}+\frac{HR}{C_p}=0.$

Solutions

Solving for $\bar{U}$ 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.

$\bar{\rho }=\frac{\dot{m}}{\bar{U}A},$

$\bar{p}=\frac{F}{A}-\bar{\rho }\stackrel{2}{\stackrel{‾}{U}},$

$\bar{h}=\frac{\bar{p}{C}_p}{\bar{\rho }R}.$

Second law of thermodynamics:

$\mathrm{\nabla }s=C_p\ln (\frac{\bar{T}}{T_1})+R\ln (\frac{\bar{p}}{p_1}).$

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.

$\mathrm{\nabla }s=C_p\ln (\bar{T})+R\ln (\bar{p}).$

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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