This is a traditional topic to appear in degree courses. It is really quite mundane but the need for examinations has to be taken into account and this does require the use of the energy equation and some proficiency in calculus. In practice, as distinct from examination conditions, software like Mathcad have removed the problem of integration leaving just the energy equation.
The only way to illustrate the method is by a worked example.
There is a model boating lake at Maldon in Essex in England. The lake lies alongside the tidal estuary of the river Chelmer from which it is separated by a sea wall. The lake is surrounded with banks and is effectively in a bowl. For over 100 years the lake has been filled from the river through an open pipe that let water in when the tide was high and let it out again when the tide was low. Somehow someone found a pipe size that kept the level steady within a few centimetres. During a recent redevelopment this arrangement was changed. The pipe was replaced by a much larger pipe set well below the required surface level in the lake and fitted with a gate valve. This permits the controlled filling of the lake and of course the controlled emptying. It also creates a problem when there are extra high tides that top the river wall and then the lake is filled by up to one metre above normal and there is no automatic outlet.
Someone has to open the valve fully and keep watch to ensure that the valve is closed again when the normal level is attained. That someone could go home for several hours if the time taken to reach the normal level is long enough. The effective size of the lake is 50 metres by 20 metres, the pipe is 10 metres long and metres in diameter set metres below the normal surface level. The top of the river bank is about 1 metre above normal level in the lake and metres above the outlet.
It is possible to get an idea of the time taken for the level to fall using the energy equation and integration. If I take the friction coefficient to be and the loss at entry to this pipe to be equal to I can make an estimate of the time taken for the level in the lake to fall from the level of the top of the bank to normal level.
Suppose that at some instant in time the level in the lake is above the outlet level of the pipe. Applying the energy equation to the free surface and the outlet gives :- + the loss at inlet and to friction. This reduces to :-
I can substitute in this to give :- .
Then and the instantaneous flow is :-
In a time of the outflow is m3. This outflow will cause the level in the lake to drop by :-
where is the area of the free surface.
So, . Then the time taken to fall from to above the outlet is . Mathcad gives this result in hours :-
This result is much dependent on the value chosen for the friction coefficient. The range of value for friction coefficient is from about and this gives a range of about 9 hours to 18 hours. Regardless this is a long time and may involve supervision of this process during the night if the lake is to be available for use from say 10 am.
The calculation above is for a real situation and is probably quite typical. It is not very complicated mathematically but some ingenuity might be required in some cases. It can generate all sorts of examination questions.