Question

An inventor claims to have developed a heat pump that produces a 200 -kW heating effect for a \(293 \mathrm{K}\) heated zone while only using \(75 \mathrm{kW}\) of power and a heat source at \(273 \mathrm{K} .\) Justify the validity of this claim.

Short answer

Answer: Yes, the claim is valid as the calculated COP (2.67) is less than the maximum possible COP (14.65).

Step-by-step solution

Calculate the Coefficient of Performance (COP) of the heat pump

To calculate the COP of the heat pump, use the formula: COP = Heating Effect / Power Input. Plugging in the given values, we can calculate the COP: COP = (200 kW) / (75 kW) = 2.67

Calculate the Maximum Possible Coefficient of Performance (COPmax)

To calculate COPmax, we'll use the formula: COPmax = T_high / (T_high - T_low). Plugging in the given values, we can calculate the COPmax: COPmax = (293 K) / (293 K - 273 K) = 293 K / 20 K = 14.65

Compare the COP and COPmax values

Now we'll compare the COP and COPmax values calculated in step 1 and step 2: COP = 2.67 COPmax = 14.65 Since the calculated COP (2.67) is less than the maximum possible COP (14.65), the heat pump claim by the inventor is valid.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics concerning heat, work, temperature, and energy. In the context of heat pumps, the second law of thermodynamics is particularly significant. This law states that heat cannot spontaneously flow from a colder location to a hotter location without external work being performed. In a heat pump, this external work is provided by the power input, such as electricity, which drives the heat pump's compressor.

The performance of a heat pump is often analyzed using the concept of the Coefficient of Performance (COP), which is a ratio of the heat output to the work input. By understanding the principles of thermodynamics, we can assess the efficiency of a heat pump and determine the maximum amount of heating effect that can be achieved for a given amount of input energy.

Heat Pump Efficiency

The efficiency of a heat pump is commonly measured by its Coefficient of Performance (COP), which indicates how effectively the pump transfers heat compared to the energy it consumes. A higher COP value represents a more efficient heat pump. The actual COP calculated from the heat pump's operation gives us real-world insight into the unit’s efficiency.

To improve the exercise explanation, it's important to note that the COP can vary based on different factors such as the temperature differential between the heat source and the heated space, and the specifics of the heat pump's design. In practice, no heat pump operates at 100% efficiency due to real-world constraints such as frictional losses and imperfections in the system's components.

Maximum Possible Coefficient of Performance (COPmax)

The maximum possible Coefficient of Performance (COPmax) for a heat pump operating between two temperatures can be determined theoretically using the Carnot efficiency. Based on thermodynamic principles, the Carnot COP is calculated as the ratio of the temperature of the heated zone (in kelvins) to the difference in temperature between the heated zone and the heat source (also in kelvins).

Considering the given example, the COPmax represents the most efficient scenario, where all processes are reversible, and there are no energy losses. While this is an ideal and not practical scenario, COPmax serves as a benchmark to measure the actual performance of a heat pump. In the exercise improvement advice, understanding that the actual COP will always be less than the COPmax due to practical inefficiencies is crucial for evaluating the claims of a heat pump's performance accurately.