Question

A house is losing heat at a rate of \(1800 \mathrm{kJ} / \mathrm{h}\) per \(^{\circ} \mathrm{C}\) temperature difference between the indoor and the outdoor temperatures. Express the rate of heat loss from this house per \((a) \mathrm{K},(b)^{\circ} \mathrm{F},\) and \((c) \mathrm{R}\) difference between the indoor and the outdoor temperature.

Short answer

Question: Express the rate of heat loss from a house in terms of temperature differences in Kelvin, Fahrenheit, and Rankine if the rate of heat loss is given as 1800 kJ/h per Celsius difference. Answer: (a) 1800 kJ/h per K difference (b) 1000 kJ/h per °F difference (c) 1000 kJ/h per R difference

Step-by-step solution

Converting to Kelvin

To express the heat loss per K difference between the indoor and outdoor temperatures, note that the difference between temperatures in Celsius and Kelvin is the same, as the scales are identical except for the offset. Thus, the heat loss per K difference is the same as for Celsius: 1800 kJ/h per K difference.

Converting to Fahrenheit

To convert the heat loss per Celsius difference to Fahrenheit difference, we can use the formula: \( \Delta T_{F} = \frac{9}{5} \Delta T_{C} \) where \(\Delta T_{F}\) is the temperature difference in Fahrenheit and \(\Delta T_{C}\) is the temperature difference in Celsius. Taking the reciprocal, we have: \( \Delta T_{C} = \frac{5}{9} \Delta T_{F} \) So, the heat loss per °F difference is: \( \frac{1800 \mathrm{kJ/h}}{^\circ \mathrm{C}} \times \frac{5}{9} = 1000 \mathrm{kJ/h} \) per \(^\circ \mathrm{F}\) temperature difference.

Converting to Rankine

To convert the heat loss per Celsius difference to Rankine difference, we can use the formula: \( \Delta T_{R} = \frac{9}{5} \Delta T_{C} \) where \(\Delta T_{R}\) is the temperature difference in Rankine and \(\Delta T_{C}\) is the temperature difference in Celsius. Since Rankine is the Fahrenheit equivalent of Kelvin, the relationship between the temperature differences is the same as for Fahrenheit. Taking the reciprocal, we have: \( \Delta T_{C} = \frac{5}{9} \Delta T_{R} \) So, the heat loss per R difference is: \( \frac{1800 \mathrm{kJ/h}}{^\circ \mathrm{C}} \times \frac{5}{9} = 1000 \mathrm{kJ/h} \) per R temperature difference. In conclusion, the rate of heat loss from the house is: (a) \(1800 \mathrm{kJ/h}\) per K difference (b) \(1000 \mathrm{kJ/h}\) per \(^\circ \mathrm{F}\) difference (c) \(1000 \mathrm{kJ/h}\) per R difference