Comments on: Goal Seek – Optimization Approach to a Simple Physics Problem

By: scott

scott — Sat, 07 Feb 2009 06:30:17 +0000

Dude, the orifice equation! Find the flow rate based on the head, subtract from bucket volume at discrete time intervals and graph. You shoulda been a civil engineer.

Q=Cd*a*sqrt(2gh)

where Q = flow (cubic metres per second)
Cd = coefficient of discharge
A = area of orifice (square metres)
g = acceleration from gravity (9.81 m/s/s)
h = head acting on the centreline (m)

By: Jon Peltier

Jon Peltier — Thu, 11 Sep 2008 18:32:33 +0000

I told my daughter that a little calculus could solve the problem; she’ll be glad to hear that someone supplied the analytical solution for me.

We assumed that times were in seconds and dimensions (the data was printed out and handed to the class). If times were measured using a stopwatch, then fumbly stopwatch fingers would result in greater percentage errors for the shorter tests.

By: Sjoerd Hoogwater

Sjoerd Hoogwater — Thu, 11 Sep 2008 18:12:40 +0000

It seems like your on to a fun project. I wonder why you use markers for the regressed data in your last graph instead of using them to plot the measured data.
Clearly, the measurement error is largest for the fastest drain times (large height, big hole).
As an engineer, I couldn’t resist checking the empirical formula that you have found. Force balance equations suggest that the flow rate is proportional to the square of the diameter, and the square root of the liquid height. Also, the volume in the bucket is the surface area times the liquid height (I assume that the sides are straight to make it easy). The drain time is the integral of the volume divided by the flow rate, where the liquid height changes from h0 (initial height) to zero: t = integral of V/Q = integral of [ A / ( c *d²*sqrt(h)) dh ] which results in t = 2*A/c / d² * sqrt(h), exactly the exponents that you found. The textbook value for c in this equation is 0.2087 l/min if d is in mm and h in metres. If I plug it into Excel, the diameter of your bucket is 20 cm, assuming that you gave times in minutes, diameter of the hole in cm, and water height in centimeters.
I am glad that physics where you live seems to be the same as the physics elsewhere on the planet!

By: Doug Glancy

Doug Glancy — Thu, 11 Sep 2008 16:08:07 +0000

Jon, Thanks, this topic is very timely to do some work I’m looking at on stormwater retention.

“She thinks I’m a geek.” So do I – it’s what I like most about you :).

By: Jon Peltier

Jon Peltier — Thu, 11 Sep 2008 13:29:42 +0000

Hadley -

This is a good point, and one I thought about after posting this example using Goal Seek. In the next example, which uses Solver to do the same thing, I’ve switched to minimizing the sum of the squared differences. The data are pretty well behaved here, but perhaps I should go back and change this example as well.

By: Hadley

Hadley — Thu, 11 Sep 2008 13:09:03 +0000

Jon, shouldn’t you be minimising the sum of the _squared_ differences (or at least the absolute differences)? If you minimise the sum of the raw differences you can end up with a situation where you have equal numbers of large positive and large negative residuals – the sum is zero, but the fit is very bad.