Question

A heat pump with a coefficient of performance of 3.5 provides energy at an average rate of 70000 kJ/h to maintain a building at 20 °C on a day when the outside temperature is -5 °C. If the electricity costs 8.5 cents per kW·h. Assume heating is 24 hours a day. (1) Determine the actual operating cost in $/day. (2) Determine the minimum theoretical operating cost in $/day.

Answer 1

The actual operating cost per day for the heat pump is $39.67/day, while the minimum theoretical operating cost per day is also $39.67/day. The average rate of energy consumption for the heat pump is 70000 kJ/h, and the outside temperature is -5 °C. The heat pump has a coefficient of performance (COP) of 3.5.

Step-by-step explanation:

(1) To determine the actual operating cost in $/day, we need to find the total energy consumed by the heat pump in a day. The average rate of energy consumption is given as 70000 kJ/h. Since the heating is 24 hours a day, the total energy consumed in a day is:

Total Energy Consumed = Average Rate × Hours = 70000 kJ/h × 24 h = 1680000 kJ

Since 1 kWh = 3600 kJ, the total energy consumed in a day is 1680000 kJ / 3600 kJ/kWh = 466.67 kWh.

Therefore, the actual operating cost in $/day is:

Operating Cost = Total Energy Consumed × Cost per kWh = 466.67 kWh × $0.085/kWh = $39.67/day

(2) The minimum theoretical operating cost in $/day can be determined by assuming an ideal heat pump that has a coefficient of performance (COP) equal to the maximum possible value (theoretical maximum). The COP is defined as the ratio of heat transfer out of the heat pump to the work input.

COP = Heat transfer / Work input

For an ideal heat pump, the COP is given by:

COP = 1 / (1 - (Tc / Th))

where Tc is the cold reservoir temperature and Th is the hot reservoir temperature.

Given that the COP is 3.5, we can rearrange the equation to find the ratio of Tc to Th:

Tc / Th = 1 - (1 / COP) = 1 - (1 / 3.5) = 0.714

Assuming a hot reservoir temperature of 20°C, we can find the cold reservoir temperature:

Tc = 0.714 × 20 °C = 14.28 °C

Using this cold reservoir temperature and the average rate of energy consumption, we can calculate the total energy consumed in a day:

Total Energy Consumed = Average Rate × Hours = 70000 kJ/h × 24 h = 1680000 kJ

Converting the total energy consumed to kWh:

Total Energy Consumed = 1680000 kJ / 3600 kJ/kWh = 466.67 kWh

Therefore, the minimum theoretical operating cost in $/day is:

Operating Cost = Total Energy Consumed × Cost per kWh = 466.67 kWh × $0.085/kWh = $39.67/day

Answer 2

The actual operating cost of a heat pump with COP 3.5 that provides 70000 kJ/h is calculated as $11.33/day. Without additional data about the maximum performance of the pump, the minimum theoretical operating cost cannot be calculated.

Step-by-step explanation:

To calculate the actual operating cost in $/day for a heat pump with a coefficient of performance (COP) of 3.5 providing an average rate of heat of 70000 kJ/h, first we convert the heat provided per hour to heat provided per day by multiplying 70000 kJ/h by 24h, which gives 1680000 kJ/day. Then, we find the work required per day using the COP formula: Work = Heat Supplied / COP. This gives Work = 1680000kJ / 3.5 = 480000 kJ/day. Next, we convert kJ to kWh (1 kWh = 3600 kJ), Work in kWh/day = 480000 kJ / 3600 kJ/kWh = 133.33 kWh/day. Finally, we multiply this by the cost of electricity 133.33 kWh/day × $0.085/kWh to find the actual operating cost, which is $11.33 per day.

The minimum theoretical operating cost in $/day can be found by considering the COP would be at its maximum under ideal conditions. The best COP is calculated using the temperatures of the hot and cold reservoirs, but since the student's question does not provide information about the maximum performance of the pump, we are unable to calculate the minimum theoretical cost without additional data. In practice, the actual COP already provided would be used to determine costs.