A Carnot heat pump is to be used to heat a house and...

A Carnot heat pump is to be used to heat a house and maintain it at $25^{\circ} \mathrm{C}$ in winter. On a day when the average outdoor temperature remains at about $2^{\circ} \mathrm{C},$ the house is estimated to lose heat at a rate of $55,000 \mathrm{~kJ} / \mathrm{h}$. If the heat pump consumes $4.8 \mathrm{~kW}$ of power while operating, determine $(a)$ how long the heat pump ran on that day; $(b)$ the total heating costs, assuming an average price of $\$ 0.11 / \mathrm{kWh}$ for electricity; and $(c)$ the heating cost for the same day if resistance heating is used instead of a heat pump. Answers: (a) $5.90 \mathrm{~h},$ (b) $\$ 3.11,$ (c) $\$ 40.3$

Key Concepts

Resistance Heating Efficiency

Resistance heating converts almost all electrical energy directly into heat, resulting in nearly 100% energy conversion efficiency. However, while it is straightforward and reliable, it does not provide the additional efficiency benefits that come with heat pumps capable of moving more heat per unit of electrical energy consumed. Understanding this efficiency difference is crucial when comparing the operational costs of resistance heating with those of a heat pump.

Electrical Energy Cost Calculation

Calculating the cost of electrical energy is a straightforward application of energy consumption and electricity pricing. Once the energy usage (typically measured in kilowatt-hours) is determined, multiplying this value by the unit price of electricity yields the total cost. This concept is essential for comparing the cost-effectiveness of different heating methods, such as a heat pump versus resistance heating.

Carnot Heat Pump

A Carnot heat pump is an idealized model that operates on the reversible Carnot cycle. It represents the maximum theoretical efficiency that any real heat pump can achieve when transferring heat from a cold source (the outdoor environment) to a hot reservoir (inside the house). This concept is underpinned by the second law of thermodynamics and sets a benchmark for evaluating the performance of practical heat pumps.

Coefficient of Performance (COP)

The Coefficient of Performance (COP) is a key metric for heat pumps, defined as the ratio of heat delivered to the work input. In an ideal Carnot heat pump, the COP can be determined using the temperatures of the hot and cold reservoirs, showing the efficiency advantage of heat pumps over direct heating methods. A higher COP indicates that less electrical energy is needed to transfer a given amount of heat, leading to lower operational costs.

Energy and Unit Conversions

Energy conversion and unit conversion are critical in problems involving heating loads and power consumption. For instance, converting power (in kilowatts) and time into energy (in kilojoules or kilowatt-hours) allows for the calculation of how long the heat pump must run to meet the heating requirements. This concept ensures that consistent units are used throughout the analysis, facilitating accurate energy calculations.

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