In extremely cold weather conditions where the inside thermostat is set to 27 C

Question

In extremely cold weather conditions where the inside thermostat is set to 27 C and the outside temperature is -20 C, your 1200 square ft house might lose heat through the walls at a rate of 4030 W.

A. How much energy might you lose in one week?

B. Electric heat can be 100% efficient.  At 10 cents per kilowatt-hour how much would it cost to heat your house for one week under such cold conditions?

C. For an air conditioner used as a heat pump, the coefficient of performance can be as high as the inverse of the Carnot efficiency which is run between the temperature inside (27 C) and the temperature outside (-20 C).  How much would it cost to heat your house for one week using such a heat pump?  (note that a real air conditionier usually has a COP which is half that for a Carnot cycle)

D. An even more efficient system is a geothermal air conditioner, where the heat pump runs between the inside temperature (27 C) and pipes which are buried underground below the frost line where the temperature is always 59 F or 15 C.  How much would it cost to heat your house for one weak using such a geothermal system?

Explanation / Answer

Part A)
For the total enegry for a week, we have 7 days times 24 hrs times 3600 sec, so

3980(7)(24)(3600) = 2.41 X 10^9 J

Part B
2.41 X 10^9/1000 = 2.41 X 10^6 kWs/3600s/hr = 668.64 kWh

668.64(.1) = $66.86 (6686 cents)

Part C)
eff = Th/(Th - Tc) in Kelvin

eff = 300/(300 - 253) = 6.38

eff = Qh/W

6.38 = 2.41 X 10^9/W

W = 3.77 X 10^8 J

3.77 X 10^8/(1000)(3600) = 104.8 kWh

104.8(.1) = $10.48 (1048 cents)

Part D
eff = Th/(Th - Tc) in Kelvin

eff = 300/(300 - 288) = 25

eff = Qh/W

25 = 2.41 X 10^9/W

W = 9.63 X 10^7 J

9.63 X 10^7/(1000)(3600) = 26.74 kWh

26.74(.1) = $2.67 (267 cents)

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