Your next-door neighbor, Josephine Rackstraw, surprised her husband last
January by having a hot tub installed in their back yard while he was away on
an ice-fishing trip. It surprised him, all right, but instead of being pleased
he was horrified.
"Have you lost your mind, Josephine?" he sputtered. "It will cost a fortune to
keep this thing hot, and you know what the President said about conserving
energy."
"Don't be silly, Ralph," she replied. "It can't cost more than a few pennies a
day, even in the dead
of winter."
"No way -just because you have a PhD, you think you're an expert on
everything!" They argued for a while, bringing up several issues that each had
been storing for just such an occasion. After calming down and using the tub
for a week, they remembered their neighbor (i.e., you) had a chemical
engineering education and came to ask if you could settle their argument. You
asked a few questions, made several observations, converted everything to
metric units, and arrived at the following data, all corresponding to an
average outside temperature of \(5^{\circ} \mathrm{C}\).
\- The tub holds 1230 liters of water.
\- Ralph normally keeps the tub temperature at \(29^{\circ} \mathrm{C}\), raises
it to \(40^{\circ} \mathrm{C}\) when he plans to use it, keeps it at \(40^{\circ}
\mathrm{C}\) for about one hour, and drops it back to \(29^{\circ} \mathrm{C}\)
when he is finished.
\- During heating, it takes about three hours for the water temperature to
rise from \(29^{\circ} \mathrm{C}\) to \(40^{\circ} \mathrm{C}\). When the heat is
shut off, it takes eight hours for the water temperature to drop back to
\(29^{\circ} \mathrm{C}\).
\- Electricity costs 10 cents per kilowatt-hour.
Taking the heat capacity of the tub contents to be that of pure liquid water
and neglecting evaporation, answer the following questions.
(a) What is the average rate of heat loss ( \(k W\) ) from the tub to the
outside air? (Hint: Consider the period when the tub temperature is dropping
from \(40^{\circ} \mathrm{C}\) to \(29^{\circ} \mathrm{C}\).)
(b) At what average rate ( \(\mathrm{kW}\) ) does the tub heater deliver energy
to the water when raising the water temperature? What is the total quantity of
electricity (kW\cdoth) that the heater must deliver during this period?
[Consider the result of Part (a) when performing the calculation.]
(c) (These answers should settle the argument.) Consider a day in which the
tub is used once. Use the results of Parts (a) and (b) to estimate the cost
(S) of heating the tub from \(29^{\circ} \mathrm{C}\) to \(40^{\circ} \mathrm{C}\)
and the cost \((\mathrm{S})\) of keeping the tub at a constant temperature.
(There is no cost for the period in which \(T\) is dropping.) What is the total
daily cost of running the tub? Assume the rate of heat loss is independent of
the tub temperature.
(d) The tub lid, which is an insulator, is removed when the tub is in use.
Explain how this fact would probably affect your cost estimates in Part (c).
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