11.7.3: The Entrance Limitation of Supersonic Branch

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This section deals with situations where the conditions at the tube exit have not arrived at the critical condition. It is very useful to obtain the relationships between the entrance and the exit conditions for this case. Denote {m 1} and {m 2} as the conditions at the inlet and exit respectably. From equation (24)

\[ \dfrac{4\,f\,L}{D} = \left. \dfrac{4\,f\,L}{D}\right|_

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=
{ 1 - k\,{M_{1}}^{2} \over k\,{M_{1}}^{2}} -
{ 1 - k\,{M_{2}}^{2} \over k\,{M_{2}}^{2}} +
\ln \left( {M_{1} \over M_{2}} \right)^{2}
\label{isothermal:eq:workingFLD}
\]

For the case that \(M_1 > > M_2\) and \(M_1 \rightarrow 1\) equation is reduced into the following approximation

\[ \dfrac{4\,f\,L}{D} = 2 \ln \left( M_{1}\right) -1 -
\overbrace{\dfrac{ 1 - k\,{M_{2}}^{2} }{ k {M_{2}}^{2}}}^{\sim 0}
\label{isothermal:eq:workingFLDApprox}
\] Solving for \(M_1\) results in

\[ M_1 \sim \text{e}^{\dfrac{1 }{ 2}\,\left(\dfrac{4\,f\,L}{D} +1\right)}
\label{isothermal:eq:workingFLDAppSol}
\] This relationship shows the maximum limit that Mach number can approach when the heat transfer is extraordinarily fast. In reality, even small \(\dfrac{4\,f\,L}{D} > 2\) results in a Mach number which is larger than 4.5. This velocity requires a large entrance length to achieve good heat transfer. With this conflicting mechanism obviously the flow is closer to the Fanno flow model. Yet this model provides the directions of the heat transfer effects on the flow.

Example 11.14

Calculate the exit Mach number for pipe with \(\dfrac{4\,f\,L}{D} = 3\) under the assumption of the isothermal flow and supersonic flow. Estimate the heat transfer needed to achieve this flow.

Contributors and Attributions

Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.