# Symbolic Expression for Speed of Draining a Tank

1. Nov 22, 2015

### cruckshank

Hi, I've completed an experiment in which I measured the height of the water in the tank with time. I plotted my results on graph paper as a graph of height against time, resulting in a curve of decreasing gradient, slightly resembling that of the e^(-kx) graph.

1) I am asked to symbolically suggest an expression for the speed at which the water level falls.

2) Additionally I am asked to use a simple technique to determine the average discharge velocity, u(t), and head of water h(t).

Thanks.

2. Nov 22, 2015

### MexChemE

Hi, welcome to PF!

1) Try solving the following differential equation, which is an unsteady state mass balance on the tank (works for rectangular or cylindrical tanks)
$$A_b \frac{dh}{dt} = - A_o c \sqrt{2gh}$$
Where h is the height of water in the tank, Ab is the area of the tank and Ao is the area of the orifice from which the water drains, c is called the discharge coefficient (usually 0.62 for this kind of systems), and g is the acceleration of gravity. The discharge velocity in this case, according to Torricelli's law is $u=c\sqrt{2gh}$.

2) For this case I would just use the numeric data to calculate the arithmetic mean of u using Torricelli's law. It is not clear to me what does the problem ask with average head of water.