One-dimensional flow with friction: Difference between revisions

\noindent Figure~\ref{fig:friction:Ts} shows the Fanno flow process in a $Ts$-diagram. The dashed line represents the sonic temperature, which means that the flow states along the process line above the dashed line are subsonic flow states and the part of the line below the dashed line represents supersonic flow states. In both subsonic and supersonic flow addition of friction leads to a change in temperature in the direction towards the sonic temperature, i.e. the flow approaches sonic conditions ($M=1$). When the length of the pipe through which the fluid flows is equal to the length at which the flow is sonic, the flow is choked (friction choking) and further pipe length cannot be added without a change in the flow conditions. For an initially subsonic flow, a pipe longer than $L^\ast$, the change in flow conditions is analogous to the what happens for addition of heat to a subsonic flow that has reached sonic state discussed in the previous section. The inlet conditions will change such that the massflow is reduced without changing the inlet total conditions such as the pipe length is equal to $L^\ast$ for the new inlet conditions.

M_{1'} & = f(L^\ast)\\

 

 

\noindent For a choked supersonic flow, addition of more friction (increasing the length of the pipe such that $L>L^\ast$) may lead to the generation of a shock inside the pipe. In contrast to the one-dimensional flow with heat addition where a shock does not change $q^\ast$, $L^\ast$ is increased over a shock. The internal shock will be generated in an axial location such that $L^\ast$ downstream of the shock equals the remaining pipe length at the shock location (see Figure~\ref{fig:friction:choking:sup}). As more length is added to the pipe, the shock will move further and further upstream in the pipe until it stands at the pipe entrance. If the pipe is longer than $L^\ast$ after o shock standing at the inlet, the shock will move to the upstream system and the pipe flow will be subsonic and the massflow will be adjusted such that $L=L^\ast$ according to the process described for subsonic choking above.  

\noindent From prvevious derivations, we know that $L^\ast$ is a function of mach number according to\\

\[\dfrac{4\bar{f}L^\ast}{D}=\dfrac{1-M^2}{\gamma M^2}+\left(\dfrac{\gamma+1}{2\gamma}\right)\ln\left(\dfrac{(\gamma+1)M^2}{2+(\gamma-1)M^2}\right)\]\\

\noindent by dividing both the numerator and denominator in the fractions by $M^2$ it is easy to see that the choking length (Figure~\ref{fig:friction:factor}) approaches a finite length for great Mach numbers and thus the upper limit for the choking length $L^\ast_1$ is given by\\

\[\left.\dfrac{4\bar{f}L_1^\ast}{D}(M_1)\right|_{M_1\rightarrow \infty}=-\dfrac{1}{\gamma}+\left(\dfrac{\gamma+1}{2\gamma}\right)\ln\left(\dfrac{\gamma+1}{\gamma-1}\right)\]\\

\noindent From the normal shock relations we know that the downstream Mach number approaches the finite value $\sqrt{(\gamma-1)/2\gamma}$ large Mach numbers and thus the choking length downstream the shock is limited to \\

\[\left.\dfrac{4\bar{f}L_2^\ast}{D}(M_2)\right|_{M_1\rightarrow \infty}=\left(\dfrac{\gamma+1}{\gamma(\gamma-1)}\right)+\left(\dfrac{\gamma+1}{2\gamma}\right)\ln\left(\dfrac{(\gamma+1)(\gamma-1)}{4\gamma+(\gamma-1)^2}\right)\]\\

\noindent From the relations above we get \\

\[\left.\left(\dfrac{4\bar{f}L_2^\ast}{D}(M_2)-\dfrac{4\bar{f}L_1^\ast}{D}(M_1)\right)\right|_{M_1\rightarrow \infty}=\left(\dfrac{2}{\gamma-1}\right)+\left(\dfrac{\gamma+1}{2\gamma}\right)\ln\left[\left(\dfrac{(\gamma+1)(\gamma-1)}{4\gamma+(\gamma-1)^2}\right)\left(\dfrac{\gamma-1}{\gamma+1}\right)\right]\]\\

\noindent Figure~\ref{fig:friction:factor:shock} shows the development of choking length $L_1^\ast$ in a supersonic flow as a function of Mach number in relation to the corresponding choking length $L_2^\ast$ downstream of a normal shock generated at the same Mach number. As can be seen from the figure, a normal shock will always increase the choking length.