Final answer:
To determine a heat pump's balance point, one must calculate its coefficient of performance (COPhp), which relates the heat transfer to the work input. The COPhp can be determined using the formula W = Qh/Kp or, for a Carnot pump, by the temperatures of the reservoirs. The economic comparison for the use of heat pumps is also made with traditional heating methods based on cost-effectiveness.
Step-by-step explanation:
Determining a Heat Pump's Balance Point
To determine a heat pump's balance point, one must first understand the concept of a heat pump's coefficient of performance (COPhp). The COPhp is a measure of the efficiency of the heat pump, which compares the amount of heat transfer into the warm space (Qh) with the work input (W) required. Mathematically, the COP is expressed as COPhp = Qh/W. This is rooted in the thermodynamics of the heat pump, which functions by transferring heat from a cold reservoir to a hot reservoir. Specifically, when calculating the balance point, one would typically have to calculate the amount of work needed (W) using the heat delivered (Qh) and the performance coefficient (Kp) of the pump through the formula W = Qh/Kp.
In the case of an ideal Carnot pump, the performance coefficient can be given by Kp = Th/(Th - Tc), where Th is the temperature of the hot reservoir and Tc is the temperature of the cold reservoir. By knowing the hot reservoir's temperature and desiring a certain COP, we can rearrange this equation to solve for the unknow cold reservoir temperature (Tc).
Additionally, when considering the economic aspect of implementing a heat pump system, one compares the cost of using the heat pump to traditional heating methods, like burning natural gas, and evaluates the cost-effectiveness based on the energy content and price per unit.