The conditions for steady flow.

Consider the sequence of events that follows the opening of a tap in a domestic plumbing system that is supplied from a storage tank. Before the tap is opened the water in the pipes will be at rest and have the usual hydrostatic pressure distribution up to the free surface in the tank. From the instant of opening the tap, the water leaving the tap and the water in the free surface of the tank will be at atmospheric pressure, and the gravitational force acting on the water will cause the water to flow through the pipe work. In a time that is not much longer than it takes to open the tap, the rate of flow of water becomes constant and the transition from being at rest to flowing steadily will be complete. (I shall examine this transition process is much more detail when we deal with water hammer.)

 

We could consider the condition of steady flow by considering the much-simplified system shown in figure 7-1 where the pipe is straight and horizontal with an open end and the tank is a simple open-topped tank. In Chapter 4 we established the conditions required for a flow pattern to be stable. Now that we have the total head equation we can use it to decide on the conditions for the flow in this pipe in figure 7-1 to be steady.

 

We can apply the total head equation to this system by taking point 1 as being in the free surface and point 2 in the flow just outside the end of the pipe. Then we can say that :

                                  .

As the pressures at 1 and 2 are both atmospheric pressure and  is zero this expression reduces to:

                                                      .

This tells us that the potential head between 1 and 2 is used to produce the kinetic head at 2 and to overcome the friction loss. If now we consider the water before it reaches the condition of steady flow the only difference is that the velocity of flow is increasing. We can amend the total head equation to take this into account:-

   + the loss between 1 and 2 + head required to accelerate the water between 1 and 2.

 

Clearly the condition of steady flow is reached when the acceleration has ceased.

 

Not all systems depend on gravity to cause the flow, a pump or a compressor may be used. We can regard figure 7-2 as being representative of such systems. Here a pump maintains a flow from the lower tank to the upper one. Again we can apply the energy equation to 1 and 2, which are points in the free surfaces. Now we have to take account of the input of energy to the water by the pump.

 

 

 

 

 

 

 + the energy per unit weight given to the water by the pump.

                                                  =  +loss of head between 1 and 2.

Here  and  are atmospheric pressure and  and  are both zero and the expression becomes:

        + the energy given to the water per unit weight by the pump =  + the loss between 1 and 2

 

Once more we could deal with the transient phase when the flow is increasing or decreasing to the steady condition by including a term to allow for the acceleration of the water. It becomes clear that, if we want to make predictions about either steady flow or changing flow, we must have some way of predicting the loss of energy to friction. This cannot be done by analysis so we must experiment and make suitable measurements and find some way of storing the resulting data and retrieving it for use.

 

We know that pipe work is made up of straight lengths of pipe joined by fittings that have various functions. It follows that the loss due to friction will be caused by both the pipe and by the fittings and we must expect that the loss in the pipes is not wholly independent of the loss in the fittings. Fortunately, in practical pipe systems in which liquid or gases flow, (as distinct from, say, a pipe carrying powder suspended in a liquid or gas) the loss in the fittings is small compared with that in the pipe. It then becomes useful to find ways of measuring the loss of head to friction in the pipe independently from the loss in any fittings that may be associated with it.

 

This problem taxed engineers and physicists for more than 100 years from the early nineteenth century until 1944. We have ended up with a chart, called the Moody diagram, which is used for estimating the friction loss in a pipe and the form of that chart is totally dependent of the history of its origins. We must therefore look carefully at that history which really means that we have to study the contributions made by those people whose work happened to suit the end result. Others, whose contemporary work may have been carried out just as competently, turned out to have no part to play in the evolution of the Moody diagram. We can usefully take each contribution in turn.