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\Large \bf  The Hot Water Tank Problem\\ \vspace{.5cm}
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 Newpark Shipyards manufactures and repairs large barges.
Many of the barges brought in for repair are tankers with substantial
amounts of fuel vapor in the tanks. In the past this explosive gas was
vented to the atmosphere but clean air standards now require that it be
burned cleanly. The shipyard had to choose between a \$250,000 flare
system or a \$500,000 hot water system which would use the fuel vapor to
produce hot water which would then be used to clean the barge. The
decision on which alternative to use boiled down to whether the hot water
produced would on one day would be hot enough to use the next. This was
because the vapor had to be removed but the tanks would not be safe for
workers to enter for up to eight hours more.

Here is a modified version of one the problems that was solved to help with the
decision.

\begin{quote}

At six a.m. a tank contains 20,000 gallons of water at 90 degrees
Fahrenheit. One hour earlier the water in the tank had been at At six
a.m. fuel vapor from a barge is burned and the heat transferred to the
water in the tank via a heat exchanger at a rate of 80,000 BTU per
minute for four hours. Assuming that the ambient air temperature remains
at a constant 80 degrees and that only Newton's Law of Cooling applies
(ignore radiant heat loss) what will be the temperature at the end of
the four hours and how long (if at all) will the water in the tank
remain above 170 degrees Fahrenheit. 

\end{quote}

Investigate the  problem above, answering the question posed.  Then
investigate the  sensitivity of the answer to the "constant ambient
temperature" assumption by 
assuming that the temperature is a constant during each clock hour,
starting with 40 degrees from 6:00 to 7:00 and that it increases two
degrees per hour until 6:00 p.m and then decreases two degrees per hour
until it gets back to 40 degrees at 6:00 a.m. at which time the cycle repeats.






