The physics of conveying gas through a pipe.

The exploration of the continuity equation may give some idea of the relationship between pressure and velocity but it cannot give any idea of how the pressure falls along a pipe. In order to do that, we have to take friction into account.

 

Before we can do that we must consider which of the relevant physical quantities we can actually measure. The density of the gas and the viscosity are not at all easy to measure but they can be inferred from measurements of pressure and temperature. In many cases temperature is measurable but with some uncertainty. Velocity is never easy to measure so we must try to avoid it in any expressions that we may form.

 

The drop in pressure has been attributed to friction, that is, to the interaction between the flowing fluid and the wall of the pipe and the subsequent “loss” into the molecular structure of the gas. The only effective method that we have to predict the effect of friction is the Moody chart and fortunately this is applicable to both liquids and gases. This necessarily means using the Darcy expression and we must remember that there is some inherent inaccuracy. In the end it is the use to which the outcome is put that makes the result acceptable or unacceptable.

 

Of course, as soon as you contemplate using Moody, the problem of finding Reynolds number crops up. In incompressible flow of, say, water, Reynolds number is pretty much the same throughout the pipe. For compressible flow there is a progressive change in velocity, and, as the temperature changes, in density and viscosity. Reynolds number also changes along the pipe. We must have this in mind when trying to find a value for the friction coefficient  f.

 

If everything that can change varies along the pipe we shall have to consider what goes on over a short length of pipe, find some relationships, and integrate. So let figure 13-3 represent a short length  of the pipe of cross-sectional area A and of overall length l. In the light of the fact that both Darcy and Moody are based on one-dimensional flow we must treat the flow as one-dimensional and use uniform properties over any given section of the pipe. Then we can let the pressure at inlet to the short section in figure 13-3 be  and the outlet pressure be  (The sign will permit us to interpret signs during integration.) the corresponding velocities will be  and .

 

We have from Darcy that the head lost to friction in a pipe is given by:-

                                                           

 and this can be converted to suit this application to:-

                                               

where  and  are average values over the short length of pipe and  is the pressure drop over the short length  to overcome the friction.

 

Then we can say that the net force in the direction of flow equals the increase in momentum per second, that is:-

                         .

 

This reduces to:-

                                                   

or to:-

                                             

The basic equation is:-

                                                                    

                                            

 

The best result in the end[1] comes from dividing though by  noting that  always equals . And we get:-

                                              .

The best we can do with this is to put  which, when substituted, gives:-

                                            .

This could be integrated if we could eliminate the inconvenient variable  and take some decision about f. We can make progress by using pressures instead of densities because :-

                                           from which:-

                                                          

This can now be substituted to give:-

             

At this point it seems that we might move to Mathcad to explore this equation but the inconvenient velocity term is there to get in the way. The traditional approach is to go ahead and integrate. After some decision has been taken about f .

 

We have to address the obvious problem that, at this stage, we expect f to vary as Reynolds number changes as the physical properties of the gas change along the pipe. We do not know that this is the case and, as engineers, it pays us to find out likely values of Reynolds number for practical pneumatic systems, in other words, proceed by trial. I suppose that if we are dealing with air it will be supplied to the pipe at or above 20°C and a pressure of 7 bar absolute. Then  will be about . As  we have to offer a typical diameter and an inlet velocity. Suppose the diameter to be 0 75 mm and the velocity 50 m/s. The Reynolds number at inlet is of the order of 560,000 which the Moody diagram shows is just at the start of the region where, for this and higher vales of , the value of f is dependent on the equivalent sand grain roughness and not on the value of . Further inspection of the Moody chart shows that this will be true for almost any likely combination of speed and pipe diameter and, indeed, as the value of  will rise as the air flows along the pipe so the value of f  will become dependent on equivalent sand grain roughness and nothing else.

 

This puts us in the position where we know that the flow is not greatly dependent on the value of the polytropic index and that we can use a single value of f. So let us integrate. We get:-

                                                                   

                     

Where the suffices 1 and 2 indicate supply and delivery values.

 

The velocity term must go in favour of pressures and we already have   which can be re-arranged to give  and substituted to give the most convenient form of this equation when it is to be used in engineering.

                   

From this the mass flow can be calculated from a knowledge of the dimensions of the pipe and its equivalent sand grain roughness, the supply and delivery pressures, the temperature at supply and n.  If  the only uncertain item in this is n and we do not expect that to matter much. Now it can be explored with Mathcad. If you enter this expression to Mathcad and substitute all sorts of combinations of diameter length etc. the graphs of mass flow against n in the range 1.01  to 1.5 will all be horizontal straight lines or nearly so as shown in graph 13-4. The expression is obviously sensitive to values of pressure ratio between supply and delivery, to the equivalent sand grain roughness and, of course, to diameter but not to the polytropic index.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Then a useful form of the expression will come from putting n equal to 1.2 and .

         

This can be rearranged to give mass flow versus diameter or vice versa.

 

It is as good as engineering expressions get but there is still a problem with it. Engineers work in free air per minute and not in mass flow. The rating of a compressor is quoted in delivery gauge pressure and the flow given in litres per minute or cubic feet per minute at standard conditions of 1 bar absolute and 20°C. This raises the problem of changing this volumetric rate of flow to mass flow to go into our expression. We can use  to get  to make the conversion.