Example of linear regression: Hot water heater for a mass casualty decontamination facility

Here's some data from the AOSMITH 199,000 BTU/H hot water heater in the experimental mass casualty decontamination facility, which I generated after the heater had been off long enough for the tank to cool:

Fri Jan 24, 2003 I switched on the hot water heater at 8:03:48pm.

It kicked in at 8:04:30, showing 48degF, beyond which I made the following table by noting the time and temperature: 

time temp
806  50
807  52
808  55
809  58
810  60
811  61
812  64
813  68
814  70
815  73
816  76
817  79
818  81
819  84
820  87
821  89
822  92
823  95
824  97
825  100
826  102
827  105
828  107
829  109
830  112
831  115
832  118
833  120
834  123
835  125
836  128
837  130
8:37:30 it just shut off. At the instant it shut off the temperature indicator was reading 132F.

8:38 it's still off, but rose to 133F 

octave:7> A=[sum(x.*x) sum(x); sum(x) length(x)]
A =
21598320     26288
26288        32
octave:8> b=[sum(x.*y); sum(y)]
b =
2377163
2885
octave:9> A\b
ans =
2.6157
-2058.6023
Slope of graph is 2.6157 deg F/minute.

1 BTU is 1 pound 1 degree.

There are 10 pounds in a gallon, and the tank is 100 gallons, so the tank holds 1000 pounds of water that is being raised 2.6157 deg F/minute, 2.6157*60*1000BTH/hour = 156,942 BTU/hour.

The unit is rated at 200,000 (199,000) BTU/hour, so it's running around 79% of its rating.

To put this into perspective, if we have 52 deg F water (the temperature it's coming out of the ground), and want a 105F shower, we mix hot and cold, but that's equiv. to running hot water heater up to 105F only, so it's 105-52 = 53 deg F difference,
53/2.6157 = 20.262 minutes to heat the whole tank, e.g. therefore produces 4.9353 gallons per minute, therefore we get approximately enough to run two showers for an infinitely long duration without ever running out of hot water.

The tank of course serves as a "buffer", so, for example, it can run the firehose through the Navigator (25GPM) for only 4 minutes on a full tank if at 105F but if at higher temperature, it stores more energy, and can then run the firehose for longer and still recover.

Another interesting question: what's the cost (dollars per minute) to keep the firehose running with 105 deg F water?

I did google search: 61.18 cents per 100,000 BTU natural gas;

$1.07 per 100,000 BTU propane

$2.43 per 100,000BTU electric.

That means our 200,000 btu /h unit costs 122.36 cents (i presume per hour) to run continuously.

A full tank to goto showering temp (or equiv. that portion of a tank that would be used for showering temp if mixed from a higher temp) is therefore 40.786cents

That means for less than 41 cents worth of natural gas, a person can be decontaminated with 100 gallons of comfortably warm water dumped onto their body through a firehose (which takes 4 minutes at 25 GPM).

Cold water costs approx. 1.2317$/cubic meter approx. .4662491960 cents per gallon, e.g. approx. 47 cents for a full (100 gallon) tank.

Thus the true cost to decon someone is approximately 87 cents.