The characteristics of the flow in pipe systems.

The main characteristics of the flow in pipe systems can be illustrated if we choose to ignore the losses in the pipe fittings. When this decision is taken a different form of the Darcy expression may be more useful.

 

This is the form of the Darcy expression which involves  instead of . We have chosen to put  where . From this it follows that . If g is put equal to 9.81 to suit the S.I system this reduces to :-

                                             Joules/Newton or just metres.

 

In practice this is used in the form below because it is easier to remember and because the available data is not of the same order of accuracy as the number 3.025.

                                             Joules/Newton or metres

 

[It is pertinent to note that the original form of the Darcy expression is unusual in that the 4 is not cancelled with the 2. The obvious reason for the decision not to cancel the 4 and the 2 is in order to retain the kinetic energy/unit weight as an identifiable element of the expression. Unfortunately there is another form of this expression that is widely used in the United States of America. There, energy loss/unit weight,  is put equal to . So we have to watch for values of friction factor, both denoted , with one being 4 times the other.]

 

If this pipe system is to be treated as a one-dimensional flow, liquid flowing in a pipe has kinetic energy that can be evaluated as . I think that it is worth plotting this kinetic energy term against velocity.

 

It is given in graph 8-1. It is an obvious sort of graph, just the graph of  but it does give us the magnitudes of the velocity and the kinetic energy head for any liquid.

 

 In fact it is not kinetic energy that concerns us but, if the losses to be inserted into the energy equation are to be evaluated as a fraction of the kinetic energy per unit weight of the liquid flowing in the pipe, it is useful to have some idea of the magnitude of the kinetic energy per unit weight. After all we already know that this kinetic energy is often lost at exit from a pipe simply because no provision is made to recover it and the obvious next step is to draw a graph of energy flow through a pipe of a given diameter in order to put a magnitude to this potential loss. The kinetic energy has units of Nm/N and if we multiply by weight flow we get power in watts, or if it were a loss, the rate of loss of energy in watts.

Potential loss =  where  is the diameter of the pipe.

In graph 8-2 I have plotted this loss for a pipe of 25, 50 and 100 mm diameter.

 

However, we have chosen to express all our losses in pipes as fractions of the kinetic energy of the flow through the pipe. It follows that an interesting graph is that of flow of kinetic energy per unit area of the pipe which is just the graph of . I think that this is one of the many graphs that crop up in this subject that effectively set ranges to go and no go areas on graphs. I know that it is obvious but I found that students often do not see graphs as anything other than y versus x as if they were still plotting their first graph. This is a graph of two real quantities and engineers want x to be as large as possible and y to be as small as possible so that the losses are small. This graph says that this is what you can have and you must choose. A balance has to be found between velocity and possible power loss and the graph says that the loss at 1 m/s is about 1/8 of the loss at 2 m/s and 1/27 of that at 3 m/s and 1/64 of that at 4 m/s. The balance is struck at about 3 m/s as a working maximum.

 

In my view it is important to get a “feel” for energy exchanges and the energy exchanges that take place on the free surfaces of water are the only ones that are visible. They need some idea of speed and as a guide small boats (25 feet long) travel at about 3 m/sec = 10 km/hr or 6 mph or the speed of water in pipes. Ducks make up to 2 mph or 1 m/s. Keep looking at them to find the true character of the flow.