What Mathematical Principles Govern the Functioning of Water Clocks?

[hanson]

Hi all!
I'm finding information about the ancient time counting machine --- water clock, as shown in the figure.
I know that the water clock invented by Ktesibios has a self-regulating ability. This is achieved by a float.
Anyone got more information about the mathematics involved?

 

[FredGarvin]

Since it is a cone shaped level float, the level changes the effective flow area of the supply line coming into the tank. This is exactly the same principle as a modern day flow regulator.

The Bernoulli Equation would be an appropriate place to start looking.

 

[hanson]

can anyone help to derive the fluctuation of height h, if the float is absent? This is a typical control problem I suppose.

 

[Danger]

Can't help with the math... sorry. I assume, however, that since the float is there to keep the tank from overflowing, that's exactly what would happen if it wasn't there.

 

[enigma]

Looks like just a differential equation problem to me. There is no control system if the float is removed.

Rate of flow IN is constant (if the inlet is always above the surface of the water). Rate of flow OUT is a function of height. So the height is a function of geometry and the two flows.