Chapter 13  Part  2  Flow of gases in pipes

 

Introduction

It is common engineering practice to convey gases, often at high pressure, through pipes of circular cross section. I think that the pneumatic systems that most of us see are those in garages specialising in tyre and exhaust replacement. There a compressor runs somewhere in the background in order to reduce the noise level at the workplace where power tools are used for nut running, tyre removal, vehicle lifts and so on. Similar systems are used in factories for nailing, stapling, spraying paint, abrasion and many other purposes. In every case a pipe or a pipe system must be installed to convey compressed air to suitable connectors to the flexible pipes to the air tools. The system would have to be very extensive to justify much design input to the pipe system. However there are other much longer pipes that convey gas, e.g. methane, from country to country and then there must be a serious design input. A working knowledge of the physics of the pipe carrying a gas under pressure has value to the engineer as a necessary part of having a grasp of the wider physics of engineering.

 

In the garage the conveying pipe is a small part of the total cost but for conveying gas over a long distance the goal must be to design pipes that give the best compromise between first cost and running cost. The running cost in any system is really that of pumping to overcome the friction loss that occurs as the gas flows through the pipe. We might expect the friction loss to increase in a non-linear way with velocity and low running costs are associated with low velocities. Inevitably some compromise has to be made between the cost of large and therefore expensive pipes for conveying at low velocities and running cost. Too often the cost of the installation dominates the decision and running costs are too high.

 

So let us have a look at the physics of such pipes.

 

All the gases are compressible fluids. To date we have considered the flow of incompressible fluids typified by water. We found that when any real fluid flows in, say, a pipe of uniform bore, the pressure drops and we attributed this drop to fluid friction. We must expect a similar pressure drop in pipes carrying gas. If the fluid is incompressible no change in volume per unit mass occurs even though the pressure falls but, if the fluid is compressible, the gas expands as the pressure falls. This means that the steady mass flow of a gas in a pipe of uniform bore will be associated with a steadily increasing velocity of flow along the pipe. This is a major difference to the uniform velocity of steady incompressible flow and it has ramifications where the velocity becomes high. This means that we have to have a new way of designing pipe systems when they are to carry gases. I shall follow the traditional approach and that starts by exploring the implications of the continuity equation.

 

The continuity equation applied to steady flow in pipes.

We need a new form of the continuity equation to take account of the change in volume per unit mass. If we form this equation for one-dimensional flow it becomes:-

                                       where   is the mass flow,  is the density =  where  is the volume per unit mass,  is the cross-sectional area of the pipe and  is the uniform velocity.

Clearly, for a given pipe with a steady mass flow :-   and as  decreases with falling pressure  must increase.

 

We can go no further until we relate the pressure and volume of the gas as it expands during its passage through the pipe. The most commonly used expression relating absolute pressure and volume per unit mass for a gas is the polytropic expression . It is pertinent to have an idea where this comes from. Once more I disclaim any historical knowledge of its origin but I did meet an experiment to relate the volume and pressure of air trapped in glass-ware in physics classes at school. We drew a graph that looked a bit like a rectangular hyperbola. It was all very slow and took place at room temperature. Subsequently, when I met the engine indicator attached to a gas engine cylinder I was faced with another graph, only 50 mm high, of the gas expanding in the engine cylinder. The graph was processed manually to find out whether it could be represented by some simple relationship. The processing involved the addition of the pressure and volume axes to the diagram taken from the engine and this needed considerable draughtsmanship to be accurate. The hope was that the relationship would be that the , that is, a simple power law that would be found by plotting logs.

 

At the time that the whole science of the behaviour of gases was evolving the engine indicator was the normal means of recording the behaviour of gases when changing volume quickly, or at least what was then thought to be quickly[1]. It could never have been sufficiently accurate in itself, let alone the error caused by the way in which it was connected to the engine or other apparatus, to provide anything better than a power law. Those who sought to pursue their science had to find ways to manage with this simple relationship.

 

In order to use it we need some values for the index . We have one value for it because we already have  for a process at constant temperature. I have drawn attention to the idea of fully-resisted, partially-resisted and un-resisted expansion. It turns out[2] that for a fully-resisted expansion with no heat input during the expansion, that is a fully-resisted adiabatic process, we can use .

 

This leaves us with a need to put a value to  for the other partially resisted processes. Many practical processes involve some heat exchange and we have  =1 for a constant temperature process, that is, for one where the equivalent to the work done during compression is lost during the compression and  for a fully-resisted adiabatic process. It is reasonable to expect  to lie between 1 and  for partially-resisted processes that involve heat exchange. These days we have to go to  =1.5 for compressions in rotary machines.

 

The use of these figures have been remarkably successful and engineers would have to be faced with some very important application before they sought to apply some more accurate relationship found by modern equipment and abandon the methods used in the past.

I think that it is important to hve an idea of where a plot of  lies on the p-V plane relative to the plots of . I have re-worked graph 13-2 to limit its range to 30 bar 1 cubic metre and shown all the isothermal lines in red. I have added two arbitrary  lines in black. The most obvious thing is that the lines intersect at very small angles.

 

(As an aside to the main thrust of this text I have cross-hatched a Carnot cycle. This is what a Carnot cycle really looks like not the cupid’s bow that most textbooks offer. It tells us immediately that the Carnot cycle is not practical because of the miserably small area enclosed in the cycle compared with the range of pressure and volume involved. This is not a criticism of Carnot because he did not know the properties of air and could not have drawn this diagram.)

 

So let us suppose[3] that it would be reasonable to use the polytropic relationship  to relate pressure  and volume/unit mass  of the gas flowing along a pipe. This decision allows for some heat transfer between the flowing gas and its surroundings. If we let the inlet pressure be , the volume per unit mass at inlet be , and the pressure at a typical point in the pipe be  we can say that -

                                                   

 or that :-

 

As  we get the following expression for velocity :-

                                 

As           or  this expression can be re-written :-

                                          .

We could explore this if we put some sensible starting values to  and to  and some typical values to .

A typical working pressure for a pneumatic system is 7 bar gauge or 100 psi gauge and we must choose an inlet velocity. I chose 50 m/s, knowing that in Mathcad it can be changed at will, and plotted a family of graphs of velocity in the pipe versus the pressure in the pipe for values of n = 1.1, 1.2, 1.3, 1.4 and 1.5.

 

The equation to the graph is:-

            

I have given the graph in graph 13-3. When you look at this graph keep in mind that this is derived from nothing more than the steady flow energy equation and the polytropic relationship between pressure and volume.

 

In a real installation it would be pointless to have a drop in pressure from 8 bar absolute  to 2 bar absolute just in friction so we have to think what the likely delivery pressure would be or, if it were to be a really long pipe, how low the pressure might be allowed to drop before a booster pump would return it to 8 bar.

 

I suppose that a drop to 7 bar absolute and a rise in velocity of flow to about 60 m/s would be acceptable but even if it went to 6 bar absolute and about 70 m/s our graph tells us that the choice of the value of the lowest pressure seems not to be greatly dependent on the value of the polytropic index n.

 

However the graph also tells us that, if the pipe were to be open at its end, the physics of this system predicts very high speeds and we may be certain that serious changes would take place in the character of the flow as speeds became sonic. That is another field of activity and will form another section of this chapter. Here we are concerned with the practicalities of conveying gas through a pipe.

 

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[4] 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.

 

See below for worked examples.

 

Solutions

1 We have :-

   

 

from this we can get:-

This expression must be transferred to Mathcad and the mass flow plotted against n

The outcome is that the flow as calculated is not dependent on the value of n. This is extraordinarily convenient

 

2 From the Moody diagram  for copper =0.0015 .

Now  will be large  and then .  and we can put  (simply chosen) and then :-

.  

.    .

At this moss flow

 and a better value of  =.0045

This means that the friction term becomes  and  and

From this  which is near enough.

( Note that where the pipe is long compared with its diameter the pressure term is small compared with the friction term. In most cases of practical pneumatic transport the pressure term can be ignored and the expression much simplified.)

 

3 7.5 bar gauge = 8.5 bar absolute;  7 bar gauge = 8 bar absolute.

 and then:-

.   and :-

                                 .

 

4 8.5 bar gauge = 9.5 bar absolute;  7 bar gauge = 8 bar absolute

 and then:-

.  and