A plumbing experiment

We have been conducting experiments over the past few days on the time it takes to fill a standard bathroom sink with hot water.

We suppose that we require A gallons of hot water at B degrees Fahrenheit (from the German glassblower, born a long time ago in what is now Gdańsk, but who mostly lived in what is now the Netherlands), in the sink.

We further suppose that at the start of the experiment the hot tap is set, instantaneously, to a position on a scale of [0,1] where 0 is completely off and 1 is completely open.

We then conduct a series of experiments to determine the value of P on that scale which minimises the time T to fill the sink in the way specified above.

Exogenous variables include the type of the hot water supply, the length & bore of the pipe run between that supply & the tap, the thermal properties of the sink and the ambient temperature, this last being assumed for today's purposes to be a lot less than B.

Considerations include the following. First, that if the flow is too high, we may exceed the thermal capacity of the supply and the hot water starts out from supply below the target temperature B. Second, given that during the fill we are losing heat from both the pipe and the sink, if the flow is too low, we lose too much heat and the temperature in the sink is lower to that extent. In any event, given that there will always be some loss, the hot water must leave the supply at a temperature greater than B.

Question 1: does the temperature of the hot water at exit from the supply vary with P? And if it does vary, how does it vary?

Question 2: is the flow of hot water simply proportional to P or is it some more complicated function of P?

Question 3: does the flow of hot water at any particular setting of P vary with time? If, for example, someone does something which affects the pressure in the cold water system from which the hot water system is derived?

Question 4: how does the temperature of the hot water vary after exit from the supply? Does it vary along or across the pipe? Does it vary in the sink? Is there a temperature gradient from top to bottom of the sink? Do we care?

Main question: what sort of a function of P (horizontal axis) is the time T (vertical axis)? Is it a nice U-shaped curve with a single, well defined minimum? Can we be sure that there will be just one minimum? Can we arrange things so that T is arbitrarily large, or is there always a maximum?

Follow up question: what changes if we allow the setting of the tap to vary during the filling? Can we get a better result this way?

All of which seems far too complicated for the time of day. To be resumed in due course.