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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]].
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| ==Equations for a perfect gas==
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| Stream [[thrust]]:
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| :<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>
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| [[Mass flow]]:
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| :<math> \dot m = \int \rho \mathbf{V} \cdot d \mathbf{A}.</math>
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| Stagnation [[enthalpy]]:
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| :<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>
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| :<math> \overline{U}^2 \left({1- {R \over 2C_p}}\right) -\overline{U}{F\over \dot m} +{HR \over C_p}=0.</math>
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| ===Solutions===
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| 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.
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| :<math> \overline{\rho} = {\dot m \over \overline{U}A},</math> | |
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| :<math> \overline{p} = {F \over A} -{\overline{\rho} \overline{U}^2},</math>
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| :<math> \overline{h} = {\overline{p} C_p \over \overline{\rho} R}.</math>
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| Second law of thermodynamics:
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| :<math> \nabla s = C_p \ln({\overline{T}\over T_1}) +R \ln({\overline{p} \over p_1}).</math>
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| 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.
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| :<math> \nabla s = C_p \ln(\overline{T}) +R \ln(\overline{p}).</math>
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| 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.
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| ==References==
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| * {{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}}
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| [[Category:Equations of fluid dynamics]]
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| [[Category:Fluid dynamics]]
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