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Next: Subsonic Fanno Flow for Up: The Practical Questions and Previous: The Practical Questions and   Index

Subsonic Fanno Flow for Given 4fL/D and Pressure Ratio

Figure: Unchoked flow calculations
Image subBranch
This pair of parameters is the most natural to examine because, in most cases, this information is the only information that is provided. For a given pipe $ \left(\frac{4fL}{D}\right)$ , neither the entrance Mach number nor the exit Mach number are given (sometimes the entrance Mach number is give see the next section). There is no exact analytical solution. There are two possible approaches to solve this problem: one, by building a representative function and find a root (or roots) of this representative function. Two, the problem can be solved by an iterative procedure. The first approach require using root finding method and either method of spline method or the half method found to be good. However, this author experience show that these methods in this case were found to be relatively slow. The Newton-Rapson method is much faster but not were found to be unstable (at lease in the way that was implemented by this author). The iterative method used to solve constructed on the properties of several physical quantities must be in a certain range.. The first fact is that the pressure ratio $ P_2/P_1$ is always between 0 and 1 (see Figure (9.18)). In the figure, a theoretical extra tube is added in such a length that cause the flow to choke (if it really was there). This length is always positive (at minimum is zero).

The procedure for the calculations is as the following:

  1. Calculate the entrance Mach number, $ {M_1}^{'}$ assuming the $ \frac{4fL}{D} = {\left.\frac{4fL}{D}\right\vert _{max}}^{'}$
    (chocked flow);
  2. Calculate the minimum pressure ratio $ \left(P_2/P_1\right)_{min}$ for $ {M_1}^{'}$ (look at table (9.1))
  3. Check if the flow is choked:
    There are two possibilities to check it.
    Check if the given $ \frac{4fL}{D}$ is smaller than $ \frac{4fL}{D}$ obtained from the given $ P_1/P_2$ , or
    check if the $ \left(P_2/P_1\right)_{min}$ is larger than $ \left(P_2 / P_1\right)$ ,
    $\textstyle \parbox{0.9\textwidth}{
continue if the criteria is satisfied.
Or if not satisfied abort this procedure and continue
to calculation for choked flow.}$
  4. Calculate the $ M_2$ based on the $ \left(P^{*} / P_2\right) = \left(P_1 / P_2\right)$ ,
  5. calculate $ \Delta \frac{4fL}{D}$ based on $ M_2$ ,
  6. calculate the new $ \left(P_2 / P_1\right)$ , based on the new $ f\left(\left(\frac{4fL}{D}\right)_1, \left(\frac{4fL}{D}\right)_2\right)$ ,
    (remember that $ \Delta\frac{4fL}{D} = \left(\frac{4fL}{D}\right)_2$ ),
  7. calculate the corresponding $ M_1$ and $ M_2$ ,
  8. calculate the new and ``improve'' the $ \Delta \frac{4fL}{D}$ by

    Note, when the pressure ratios are matching also the $ \Delta \frac{4fL}{D}$ will also match.

  9. Calculate the ``improved/new'' $ M_2$ based on the improve $ \Delta \frac{4fL}{D}$ fanno flow!entrance Mach number calculations
  10. calculate the improved $ \frac{4fL}{D}$ as $ \frac{4fL}{D} = \left(\frac{4fL}{D}\right)_{given} + \Delta

  11. calculate the improved $ M_1$ based on the improved $ \frac{4fL}{D}$ .

  12. Compare the abs ( $ \left(P_2/P_1\right)_{new} - \left(P_2/P_1\right)_{old}$ ) and if not satisfied
    returned to stage (6) until the solution is obtained.

To demonstrate how this procedure is working consider a typical example of $ \frac{4fL}{D}=1.7$ and $ P_2/P_1 = 0.5$ . Using the above algorithm the results are exhibited in the following figure.

Figure 9.19: The results of the algorithm showing the conversion rate for unchoked Fanno flow model with a given $ \frac{4fL}{D}$ and pressure ratio.
Image subFLDP2P1    
Figure (9.19) demonstrates that the conversion occur at about 7-8 iterations. With better first guess this conversion procedure will converts much faster (under construction).

next up previous index
Next: Subsonic Fanno Flow for Up: The Practical Questions and Previous: The Practical Questions and   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21