Math Help - Calculating water heights in a rain gauge

1. Calculating water heights in a rain gauge

I'm making an application where I can automatically record rainfall using a very sensitive water pressure sensor at the bottom of a rain gauge. The pressure is directly proportional to the vertical height of the water, and because rain gauges are conical, this is not directly proportional to actual amount of rainfall.

I'm trying to work out how I could relate the water pressure (which will essentially be a number that increases linearly with the water pressure) to the actual rainfall, in mm.

I have that the volume of water in a cone is V = (1/3)*pi*r^2*h. By equating that to V=pi*r^2*h, where r is the radius of the top of the rain gauge, I can get h which is the actual rainfall. The only variable I know is the h in the first equation, which is the vertical height of the water which is given to me by the pressure reading (a scaled factor of the height).

By calibrating the application initially, which would amount to recording the water pressures at various rainfall amounts, would I be able to come up with a quadratic (or cubic) function that would relate the pressure reading to the actual rainfall amount? I thought this would be the case but I couldn't get it to work out with some test numbers on an Excel speadsheet.

I could of course solve my problem by just using a cylindrical container, but I like the fact that I can get more accurate readings at lower rainfall amounts using a conical container.

Any help is appreciated! Thanks.

2. Re: Calculating water heights in a rain gauge

I think the problem here is that you define' 'r as the radius at the top of the gauge, but the 'r' variable as you used it in the formula for volume of the cone is the raidus where the top of the water is, not the top of the gauge. Let's define variables a little more explicitly:

Let:
h_g = depth of water in the conical gauge
r = radius at water level
V = volume of water in the gauuge
R = radius at opening of the gauge
H_g = height of the gauge
h_cyl = height of water in a cylindrical gauge

The volume of water in the conical gauge is $V_g = (1/3) \pi r^2 h$, but r is function of h, where r = h(R/H). So the volums as a function of 'h' is: $V = \frac 1 3 \pi h^3 \frac {R^2}{H^2}$

In a cylindrical gauge this volume would come to a height of: $h_{cyl} = \frac {V} {\pi R^2}$ From this we can set h_cyl as follows:

$h_{cyl} = \frac {\frac 1 3 \pi h^3 \frac {R^2}{H^2}} {\pi R^2} = \frac 1 3 \frac {h^3}{H^2}$

So given height 'h' (which you obtain from your pressure sensor) you can calculate the rain fall in inches with this equaton.