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°C. 

• The tub holds 1230 liters of water. 

• Ralph normally keeps the tub temperature at 29°C, raises it to 40°C when he plans to use it, keeps it at 40°C for about one hour, and drops it back to 29°C when he is finished. 

• During heating, it takes about three hours for the water temperature to rise from 29°C to 40°C. When the heat is shut off, it takes eight hours for the water temperature to drop back to 29°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 (kW) from the tub to the outside air? (Consider the period when the tub temperature is dropping from 40°C to 29°C.) 

(b) At what average rate (kW) does the tub heater deliver energy to the water when raising the water temperature? What is the total quantity of electricity (kW•h) 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($) of heating the tub from 29°C to  40°C and the cost($)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).