Question

A freezer is maintained at \(20^{\circ} \mathrm{F}\) by removing heat from it at a rate of 75 Btu/min. The power input to the freezer is 0.70 hp, and the surrounding air is at \(75^{\circ}\) F. Determine \((a)\) the reversible power, \((b)\) the irreversibility, and \((c)\) the second-law efficiency of this freezer .

Short answer

Answer: The reversible power is 134.94 W, the irreversibility is 387.26 W, and the second-law efficiency is 25.84%.

Step-by-step solution

Convert given information to consistent units

First, we need to convert the given information about the power input and the heat transfer rate into consistent units. In this case, we will use Watt (W) for power and Joule/second (J/s) for the heat transfer rate. 1 hp = 746 W 75 Btu/min = (75 * 1055.06 J/Btu) / 60 s = 1312.593 J/s

Find the reversible power

To calculate the reversible power, we need to find the rate of heat transfer between the freezer and its surroundings. We can use the Carnot efficiency (η) formula to find the efficiency of a reversible heat engine operating between the freezer and the surroundings: η = 1 - (T_low / T_high) Where: T_low = Temperature of the freezer in Kelvin T_high = Temperature of the surrounding air in Kelvin First, convert the Fahrenheit temperatures to Kelvin: T_freezer = (20°F - 32) × 5/9 + 273.15 = 266.483 K T_surrounding = (75°F - 32) × 5/9 + 273.15 = 297.039 K Next, we can find the Carnot efficiency: η_carnot = 1 - (266.483 / 297.039) = 0.1027 Assuming a perfect heat engine, the reversible power (W_rev) is equal to the product of the Carnot efficiency and the rate of heat transfer: W_rev = η_carnot * (rate of heat transfer) W_rev = 0.1027 * 1312.593 J/s = 134.94 W (a) The reversible power is 134.94 W.

Calculate the irreversibility

To find the irreversibility, we can use the formula for irreversibility (I): I = P_input - W_rev Given the power input (0.7 hp), we first convert it to watts: P_input = 0.7 * 746 W/hp = 522.20 W Then, we can find the irreversibility: I = 522.20 W - 134.94 W = 387.26 W (b) The irreversibility is 387.26 W.

Calculate the second-law efficiency

Finally, we can determine the second-law efficiency (η_II) by dividing the reversible power by the actual power input: η_II = W_rev / P_input η_II = 134.94 W / 522.20 W = 0.2584 (c) The second-law efficiency of the freezer is 25.84%.

Key concepts

Reversible Power

Reversible power is a theoretical concept referring to the maximum amount of power that could be generated if a system operated without any thermodynamic inefficiencies or irreversibilities. In practice, no system is perfectly reversible, but calculating the reversible power enables engineers to gauge the performance of actual systems against the ideal.

For instance, consider a freezer working to maintain its internal temperature. If this process was entirely reversible, the freezer would operate with the maximum possible efficiency, transforming all input energy into useful cooling without losses. To compute this reversible power for a real freezer, it's necessary to apply the theory of Carnot efficiency, which provides a benchmark for the highest efficiency that any heat engine could theoretically achieve given the temperatures at which it operates.

Ultimately, the reversible power value serves as a part of the foundation for calculating other metrics, such as irreversibility and second-law efficiency, both crucial for assessing the true performance of thermodynamic systems.

Carnot Efficiency

Carnot efficiency, named after the French physicist Sadi Carnot, is a measure of the maximum possible efficiency that a heat engine could achieve during the conversion process of heat into work, operating between two temperatures. It sets the upper limit for the efficiency of all real engines. The formula for Carnot efficiency is:

\[\begin{equation}\eta_\text{carnot} = 1 - \frac{T_\text{low}}{T_\text{high}}\end{equation}\]
Where \(T_{low}\) and \(T_{high}\), are the absolute temperatures (in Kelvin) of the cold and hot reservoirs, respectively. Though no real engine can be Carnot efficient due to unavoidable losses and irreversibilities, this efficiency is still a crucial standard for judging the performance of actual heat engines, including those found in refrigerators and freezers like our example.

Carnot efficiency not only plays a role in energy conservation strategies but also in the design of more sustainable systems by highlighting the potential areas of improvement through comparing real systems with the idealized, reversible ones.

Thermodynamic Irreversibility

Thermodynamic irreversibility is inherent in all real processes and systems. It represents the energy losses due to factors such as friction, unrecoverable heat transfer, and other inefficiencies that prevent the system from achieving perfect reversibility. Simply put, it's the difference between the actual work output (or input) of a system and the work that would be required if the process were entirely reversible.

Irreversibility can be quantified as the excess energy usage that results from these inefficiencies. Understanding the concept of irreversibility is essential because it allows engineers to identify where energy losses occur and how they can be minimized to improve the overall efficiency of thermal systems.

The concept ties directly into calculating the second-law efficiency, which gives a realistic measure of how well the system performs compared to the ideal, reversible process. It's a pivotal part of diagnosing and enhancing energy systems to reduce waste, save costs, and move toward more sustainable practices in energy management.