Problem Description:

A 12-hour water clock is to be designed with the dimensions shown in the sketch below. The shape of the clock is obtained by revolving the curve y = f(x) around the y axis. Define the shape function, f(x), and the radius of the circular hole at the bottom that gives a constant water level decrease of 4 in/hr.

Problem Solution:

where A_e is the exit area, c is the discharge coefficient (assumed to be unity), and g is the gravitational acceleration.

To solve this ODE, we first must relate the container volume, V, to the height of fluid within the water clock. The volume of water in the container is given by

where A(y) is the top surface area for any height, y, and the variable is just a dummy variable of integration. We can form the time derivative of the volume as

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Leibnitz Rule

The above implied relationship, , can be developed formally using Leibnitz Rule, which is a theorem for treating derivatives of integrals with variable limits of integration. Leibnitz’s rule states that, given

then

if a(t), b(t) and are continuous functions.

For the case of interest here, we have

and

or simply

as implied in the above relationship.

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Now, Torricelli’s law as expressed in eqn. (1) becomes

From the original problem statement, however, the clock shape is obtained by revolving the curve y = f(x) around the y-axis. Thus, for any y value, , where x is the radius at height y. Substitution into eqn. (4) gives

But, based on the design statement, we require dy/dt to be constant. We can achieve this goal if

With this functional relationship, the balance equation gives

Thus, the specification that the rate of decrease in water height be constant leads to the shape function given by eqn. (6) - this defines the shape of the container for the water clock.

To explicitly determine the constant, b, we note that at time zero the clock is full of water, and the above diagram gives

Putting these values into eqn. (6) gives

Now if we desire a water level decrease of 4 in/hr, we have

or

Now, let's work with the units a little to compute a numerical value for the exit radius. The constant can be written as

and with , we have

or

Therefore, the radius of the exit hole is very small indeed - only 0.0288 in!

Thus, we have finally completed the design of the water clock per the above specifications. In particular, our final design has a shape defined by and an initial fluid height and radius of 4 ft and 1 ft, respectively. If this clock has a small circular hole at the bottom with a radius of about 0.0288 in for draining the fluid, the water level will decrease by 4 in per hour - which gives the desired 12-hr clock!