Slow changes in flow

Let me start with the most simple system of just a horizontal pipe connected to a tank of water with a valve at the end of the pipe. It is show diagrammatically in figure 17-1.

Clearly, if the valve is partly open, water will flow through the pipe and the flow will become steady when the friction loss in the pipe plus the loss in the valve is equal to the head above the pipe at inlet. I looked at this system in Chapter 8 by ignoring the loss in the valve and ascribing an area ratio to the valve. That is complicated enough but here we need to consider adding acceleration to the water in the pipe. It makes sense to consider finding a relationship between acceleration and the head that must be impressed on the water to produce the acceleration. In order to do so we make the usual simplifying decision to treat the flow as one-dimensional. However we must also decide what to do about quantifying the acceleration of the water because we do not know whether it is uniform along the length of the pipe. We shall find that the pressures produced in a pipe by accelerations caused by changes in the flow of water in quite ordinary pipe systems is enough to make the compressibility of water become a significant factor. In the first instance we can consider changes in velocity that take place slowly enough for compressibility to be insignificant. Then we can take the acceleration to be uniform along the pipe.

The mass of water in the pipe is  and, if the acceleration of the water is taken to be uniform and equal to  where  is the velocity of flow, we can write that the accelerating force .

 

The accelerating force must be a pressure difference  applied over the length of the pipe. Then   or :- .

It is likely that we shall use this as a head  and then .