Question

An inventor claims to have created a heat pump that draws heat from a lake at \(3.0^{\circ} \mathrm{C}\) and delivers heat at a rate of \(20 \mathrm{~kW}\) to a building at \(35^{\circ} \mathrm{C}\), while using only \(1.9 \mathrm{~kW}\) of electrical power. How would you judge the claim?

Short answer

The claim is likely inaccurate, as the claimed efficiency of the heat pump is greater than the maximum efficiency allowed by the second law of thermodynamics (the Carnot efficiency).

Step-by-step solution

Convert temperatures to Kelvin

We first need to convert all temperatures to Kelvin. The scale of Kelvin starts at absolute zero and each degree Kelvin is exactly the size of a degree Celsius. The difference is that 0 Kelvin is equivalent to -273.15 Celsius. Thus, to convert a temperature in Celsius to Kelvin, add 273.15. So, \(T_{hot} = 35^{\circ} \mathrm{C} + 273.15 = 308.15 \mathrm{K}\) and \(T_{cold} = 3^{\circ} \mathrm{C} + 273.15 = 276.15 \mathrm{K}\)

Calculate the Carnot efficiency

The maximum possible efficiency for a heat pump (the Carnot efficiency) is \((T_{hot}-T_{cold})/T_{hot}\). Substituting the values that we got in the first step we get \((308.15 - 276.15) / 308.15 = 0.104\). So, the Carnot efficiency is \(0.104\), or \(10.4\%\)

Calculate the claimed efficiency of the heat pump

The claimed power output of the heat pump is \(20 \mathrm{~kW}\), and the electrical power input is \(1.9 \mathrm{~kW}\). Thus the claimed efficiency is \(20 / 1.9 = 10.53\)

Compare the Carnot and claimed efficiencies

The claimed efficiency, \(10.53\), is greater than the Carnot efficiency, \(10.4\%\). Since Carnot efficiency is the maximum theoretically possible, having a higher efficiency breaks the second law of thermodynamics. Therefore, the claim may not be accurate.

Key concepts

Carnot Efficiency

Carnot efficiency is a fundamental concept in thermodynamics. It represents the maximum possible efficiency that any heat engine or heat pump can achieve when working between two temperatures. This ideal efficiency is named after Sadi Carnot, a French physicist who first introduced the concept in the 1820s.

The efficiency is calculated based on the temperature difference between the hot reservoir and the cold reservoir of the system. For a heat pump, the formula is:

• Carnot Efficiency = \( \frac{T_{hot} - T_{cold}}{T_{hot}} \)

Here, \(T_{hot}\) and \(T_{cold}\) should be measured in Kelvin. This efficiency shows how well the heat pump can move heat between these two reservoirs. Remember, no real engine or pump can be more efficient than the Carnot efficiency, as it would require no energy loss and perfect, frictionless operation.

In our scenario, the inventor claims a pump efficiency of 10.53%, which surpasses the calculated Carnot efficiency of 10.4%. This would suggest the claim is impossible, as it goes against the fundamental laws of physics regarding heat engine efficiencies.

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. It explains how thermal energy is converted to and from other forms of energy and affects matter.

Thermodynamics is guided by several laws. However, when studying heat pumps, we often focus on these laws:

• The First Law of Thermodynamics: Energy cannot be created or destroyed, only transformed from one form to another. In the case of a heat pump, electrical energy is transformed into heat energy, which is then transferred from a colder area to a warmer area.
• The Second Law of Thermodynamics (which we discuss in the next section): Highlights the direction of heat transfer and limits efficiency.

Thermodynamic principles dictate the design and efficiency of machines like heat pumps, engines, and refrigerators. Understanding these can help in analyzing the feasibility of systems, like judging an inventor's claim for a highly efficient heat pump.

Second Law of Thermodynamics

The Second Law of Thermodynamics introduces the concept of entropy, often generalized as the measure of disorder within a system. This law states that in any energy transfer, some energy becomes less available to perform work, usually lost as waste heat.

In practical terms, this means no machine can be 100% efficient. For heat pumps, this law implies that they cannot transfer all input energy perfectly into heat. Some energy will always be wasted, often making the system work less efficiently than the ideal Carnot cycle.

When the claimed efficiency of a heat pump is higher than its Carnot efficiency, it violates this law. The claim implies higher than possible efficiency, suggesting an error in the claim or understanding. The second law thus serves as a checkpoint for the validity of efficiency claims and the feasibility of thermal machines in real scenarios.