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.