Low Flow Shower Delay

When I start up the shower it's the wrong temperature and adjusting it to the right temperature takes longer in this apartment than it has in any home in which I've previously lived. I wanted to blame the problem on the low flow shower head, but I'm having a hard time doing it. My thinking was that the time delay from when I adjust the shower to when I actually feel the change is unusually high due to the shower head's reduced flow rate.

In setting out to measure that delay I first found the flow rate in liters per second for the shower. I did this three times using a bucket and a stopwatch finding the shower filled 1.5 qt in 12 s, or 1.875 gpm, or 0.118 liters per second. A low flow shower head, according to the US EPA's WaterSense program, is under 2.0 gpm, so that's right on target.

That let me know how quickly the "wrong temperature" water leaves the pipe, so next to see how much of it there is. From the hot-cold mixer to the shower head there's 65 inches of nominal 1/2" pipe, which has an inner diameter of 0.545 inches. The volume of a cylinder (I'm ignoring the curve of the shower arm) is just pi times radius squared times length:

$V_{cylinder} = \pi r^{2} L$

Converting to normal units gives 1.3843 centimeters diameter and 165.1 centimeters length, which yeilds 248 cm², or 0.248 liters, of wrong temperature water to wait through until I get to sample the new mix.

With a flow rate of 0.118 liters per second and 0.248 liters of unpleasant water I should be feeling the new mix in 2.1 seconds. I recognize that time drags when you're being scalded, but it still feels like longer than that.

I've done some casual reading about linear shift time delays in feedback-based control systems and the oscillating converge they show certainly aligns with the too-hot then too-cold feel of getting this shower right. This graph is swiped from MathWorks and shows a closed loop step response with a 2.5 second delay:

Closed loop step response with 2.5 second delay.

That shows about 40 seconds to finally home-in on the right temperature and doesn't include a failure-prone, over-correcting human. I'm still not convinced the delay is the entirety of the problem, but it does seem to be a contributing factor. Some other factors that may affect perception of how show this shower is to get right are:

non-linearity in the mixer
water in the shower head itself
water already "in flight" having left the head but not yet having hit me

Mercifully, the water temperature is consistent -- once you get it dialed in correctly it stays correct, even though a long shower.

I guess the next step is to get out a thermometer and both try to characterize the linearity of the mixing control and to try to measure rate of change in the temperature as related to the magnitude of the adjustment.

Update: I got a ChemE friend to weigh in with some help.

Published: May 23, 2012, 3:35:51 PM - Modified: May 30, 2012, 6:32:56 PM by Ry4an Brase
Tags: funny, ideas-built
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