Natural exponential family: Difference between revisions

Revision as of 11:23, 14 April 2013

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 (ρ V \cdot d A) V \cdot f + \int p d A \cdot f .

Mass flow:

\dot{m} = \int ρ V \cdot d A .

Stagnation enthalpy:

H = \frac{1}{\dot{m}} \int (ρ V \cdot d A) (h + \frac{| V |^{2}}{2}),

\overset{2}{\overline{U}} (1 - \frac{R}{2 C_{p}}) - \overline{U} \frac{F}{\dot{m}} + \frac{H R}{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{ρ} = \frac{\dot{m}}{\overline{U} A},

\overline{p} = \frac{F}{A} - \overline{ρ} \overset{2}{\overline{U}},

\overline{h} = \frac{\overline{p} C_{p}}{\overline{ρ} R} .

Second law of thermodynamics:

\nabla s = C_{p} \ln (\frac{\overline{T}}{T_{1}}) + R \ln (\frac{\overline{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.

\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

Template:Cite web

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+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 [[adiabatic]]ally 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]]:
+:<math> F = \int \left(\rho \mathbf{V} \cdot d \mathbf{A} \right) \mathbf{V} \cdot \mathbf{f} +\int pd \mathbf{A} \cdot \mathbf{f}.</math>
+[[Mass flow]]:
+:<math> \dot m = \int \rho \mathbf{V} \cdot d \mathbf{A}.</math>
+Stagnation [[enthalpy]]:
+:<math> H = {1 \over \dot m} \int \left({\rho \mathbf{V} \cdot d \mathbf{A}} \right) \left( h+ {|\mathbf{V}|^2 \over 2} \right),</math>
+:<math> \overline{U}^2 \left({1- {R \over 2C_p}}\right) -\overline{U}{F\over \dot m} +{HR \over C_p}=0.</math>
+===Solutions===
+Solving for <math> \overline{U}</math> yields two solutions.  They must both be analyzed to determine which is the physical solution.  One will usually be a subsonic [[Root of a function|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.
+:<math> \overline{\rho} = {\dot m \over \overline{U}A},</math>
+:<math> \overline{p} = {F \over A} -{\overline{\rho} \overline{U}^2},</math>
+:<math> \overline{h} = {\overline{p} C_p \over \overline{\rho} R}.</math>
+Second law of thermodynamics:
+:<math> \nabla s = C_p \ln({\overline{T}\over T_1}) +R \ln({\overline{p} \over p_1}).</math>
+The values  <math> T_1</math> and  <math> p_1</math> are unknown and may be dropped from the formulation.  The value of entropy is not necessary, only that the value is positive.
+:<math> \nabla s = C_p \ln(\overline{T}) +R \ln(\overline{p}).</math>
+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==
+* {{cite web |url=http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19990062664_1999094040.pdf |title=Inlet Development for a Rocket Based Combined Cycle, Single Stage to Orbit Vehicle Using Computational Fluid Dynamics |first1=J.R. |last1=DeBonis |first2=C.J. |last2=Trefny |first3=C.J. |last3=Steffen, Jr. |publisher=NASA |year=1999 |work=NASA/TM—1999-209279 |accessdate=18 February 2013}}
+[[Category:Equations of fluid dynamics]]
+[[Category:Fluid dynamics]]

Natural exponential family: Difference between revisions

Revision as of 11:23, 14 April 2013

Equations for a perfect gas

Solutions

References

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