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Next: Unchoked situations in Fanno Up: Isothermal Flow Previous: Figures and Tables   Index

Isothermal Flow Examples

There can be several kinds of questions aside from the proof questions8.6Generally, the engineering'' or practical questions can be divided into driving force (pressure difference), resistance (diameter, friction factor, friction coefficient, etc.), and mass flow rate questions. In this model no questions about shock (should) exist8.7.

The driving force questions deal with what should be the pressure difference to obtain certain flow rate. Here is an example.

Solution

If the flow was incompressible then for known density, , the velocity can be calculated by utilizing . In incompressible flow, the density is a function of the entrance Mach number. The exit Mach number is not necessarily i.e. the flow is not choked. First, check whether flow is choked (or even possible).

Calculating the resistance,

Utilizing the table 8.1 or the program provides

 0.04331 400.00 20.1743 12.5921 0.0 0.89446
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The maximum flow rate (the limiting case) can be calculated by utilizing the above table. The velocity of the gas at the entrance . The density reads

The maximum flow rate then reads

The maximum flow rate is larger then the requested mass rate hence the flow is not choked. It is note worthy to mention that since the isothermal model breaks around the choking point, the flow rate is really some what different. It is more appropriate to assume isothermal model hence our model is appropriate.

To solve this problem the flow rate has to be calculated as

Now combining with equation (8.40) yields

From the table (8.1) or utilizing the program provides

 0.10300 66.6779 8.4826 5.3249 0.0 0.89567
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The entrance Mach number obtained by

 0.04014 466.68 21.7678 13.5844 0.0 0.89442
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The pressure should be

Note that table here above for this example are for

Solution

At first, the minimum diameter will be obtained when the flow is choked. Thus, the maximum that can be obtained when the is at its maximum and back pressure is at the atmospheric pressure.

Now, with the value of either utilizing Table (8.1) or using the provided program yields
 0.08450 94.4310 10.0018 6.2991 0.0 0.87625
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With the value of minimum diameter.

However, the pipes are provided only in 0.5 increments and the next size is or . With this pipe size the calculations are to be repeated in reversed to produces: (Clearly the maximum mass is determined with)

The usage of the above equation clearly applied to the whole pipe. The only point that must be emphasized is that all properties (like Mach number, pressure and etc) have to be taken at the same point. The new is

 0.08527 92.6400 9.9110 6.2424 0.0 0.87627
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To check whether the flow rate is satisfied the requirement

Since the mass flow rate requirements is satisfied.

It should be noted that should be replaced by in the calculations. The speed of sound at the entrance is

and the density is

The velocity at the entrance should be

The diameter should be

Nevertheless, for sake of the exercise the other parameters will be calculated. This situation is reversed question. The flow rate is given with the diameter of the pipe. It should be noted that the flow isn't choked.

Solution

First, the information whether the flow is choked needed to be found. Therefore, at first it will be assumed that the whole length is the maximum length.

with the following can be written

 0.0419 400.72021 0.87531 20.19235 20.19235 12.66915
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From the table ,and

The pressure at point (b) utilizing the isentropic relationship ( ) pressure ratio is 0.52828.

As the pressure at point (b) is smaller the actual pressure than the actual pressure one must conclude that the flow is not choked. The solution is iterative process.

1.
guess reasonably the value of and calculate
2.
Calculate the value of by subtracting
3.
Obtain from the Table ? or using the Potto-GDC.
4.
Calculate the pressure, in mind that this isn't the real pressure but based on the assumption
5.
Compare the results of guessed pressure with the actual pressure. and chose new accordingly.
Now the process has been done for you and is provided in the figure () or in table resulted from the provided program.
 0.0419 0.59338 400.32131 400.00000 0.10000
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The flow rate is

In this chapter, there are no examples on isothermal with supersonic flow.

Next: Unchoked situations in Fanno Up: Isothermal Flow Previous: Figures and Tables   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21