There are two options for heating a hunting cabin in the north woods of Minnesot

Question

There are two options for heating a hunting cabin in the north woods of Minnesota - a wood burning fireplace and a heat pump. There are no powerlines available, so a generator set (a heat engine) must be used to power the heat pump. The electricity produced by the generator set is used to run the compressor of the heat pump. The wood for the generator set (heat engine) bums at approximately 1000 K. Unlike the heat pump, the wood burning stove does not require any electricity to function, however, 15% of the thermal energy released from the wood goes out the chimney. The amount of heat needed by the cabin is approximately 3 kWh/day for every degree C difference between the outdoor and indoor temperature. So for example, if it is 4 C outside and 7 C inside, then 9 kWh/day are required, but if it is 4 C outside and 25 C inside, then 63 kWh/day are needed. Wood contains approximately 5 kWh of energy per kilogram, a cord of wood costs about $225, and a cord has about 2000 kg of wood. The heat pump is not reversible, but it is found to perform with a COP that is 40% of the reversible value. The heat engine is also not reversible, but is found to perform with a thermal efficiency that is 25% of the reversible value. The generator set (heat engine) is installed outdoors. In the winter, the temperature is -20 C outside and 25 C inside. In the spring, the temperature is 14 C outside and 25 C inside. How much does the stove cost per day to operate in the spring and how much in the winter? How much does the heat pump/heat engine cost to operate in the spring and how much in the winter?

Explanation / Answer

In Spring, Outside Temp = 14 C ; Inside Temp = 25 C ; Temp Difference = 11 C; Heat needed = 3kWh/day*11 = 33kWh/day

Cost of Wood = 225/2000=$0.1125/Kg

Stove: 85% is used , implies 33/0.85 = 38.823kWh/day ; Wood reqired = 38.823/5=7.76kf, implies cost = 7.76*0.1125= 0.8735 $

Heat pump works at 40% efficiency; implies energy conumed by it = 33/0.4 = 82.5 kWh/day

Now this is result of 25% efficiency; implies energy recieved from wood = 82.5/ 0.25 = 330kWh/day

Wood needed = 330/5 = 66kg ; implies 66*0.1125 = 7.4$

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WInter:

In Winter, Outside Temp = -20 C ; Inside Temp = 25 C ; Temp Difference = 45 C; Heat needed = 3kWh/day*45 = 135kWh/day;

Stove: 85% is used , implies 135/0.85 = 158.82Wh/day ; Wood reqired = 158.82/5=31.76kf, implies cost =31.76*0.1125= 3.57 $

Heat pump works at 40% efficiency; implies energy conumed by it = 135/0.4 = 337.5 kWh/day

Now this is result of 25% efficiency; implies energy recieved from wood = 337.5/ 0.25 = 1350kWh/day

Wood needed = 1350/5 = 270kg ; implies Wood cost: 270*0.1125 = $30.375

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