Final answer:
The Carnot heat pump will need to run continuously to maintain the house temperature at 22 °C. For resistance heating, the daily cost is $32.40, assuming a consumption of 9 kW and electricity cost of $0.15/kWh. Heat pumps are more energy-efficient than resistance heating but may not always result in cost savings depending on the COP and other factors.
Step-by-step explanation:
To determine how long the Carnot heat pump needs to run to maintain the inside temperature at 22 °C when the outdoor temperature remains at 3 °C, we need to calculate the total heat loss for the day and divide it by the heat provided by the heat pump per hour using its power consumption.
The house loses heat at a rate of 76,000 kJ/h. Over 24 hours, the total heat loss is 76,000 kJ/h × 24 h = 1,824,000 kJ.
The heat pump consumes 9 kW of power. Since 1 kW = 1 kJ/s, then 9 kW = 9 kJ/s. Over an hour, which has 3,600 seconds, the pump consumes 9 kJ/s × 3,600 s = 32,400 kJ/h.
To find the number of hours the heat pump needs to run, we divide the total heat loss by the heat provided per hour: 1,824,000 kJ ÷ 32,400 kJ/h = 56.3 hours. Since this exceeds the number of hours in a day, the heat pump must run continuously to maintain the temperature.
For resistance heating, which converts electrical energy directly into heat, the cost is calculated by multiplying the total energy consumed by the cost of electricity. If the resistance heater has the same power consumption as the heat pump, then over 24 hours, it would consume 9 kW × 24 h = 216 kWh. At a cost of $0.15/kWh, the daily heating cost is 216 kWh × $0.15/kWh = $32.40.
Energy efficiency is higher for heat pumps compared to resistance heating because heat pumps move heat rather than generating it. A Carnot heat pump can provide several units of heat for each unit of work, whereas resistance heating can provide only one unit of heat for each unit of electrical energy.
In this scenario, using a heat pump is more energy-efficient, but since it has to run continuously, the actual cost saving would depend on the coefficient of performance (COP) of the pump, which hasn't been given.