Question

In a refrigerator the low-temperature coils are at a temperature of $-{13}^{\circ }\mathrm{C}$ and the compressed gas in the condenser has a temperature of ${25}^{\circ }\mathrm{C}$. Find the coefficient of performance of a Carnot refrigerator operating between these temperatures.

Short answer

The coefficient of performance of a Carnot refrigerator operating between the temperature of -13°C in the low-temperature coils and 25°C in the compressed gas in the condenser is the result of the continued fraction 260.15 K divided by the difference of 298.15 K and 260.15 K.

Step-by-step solution

Convert temperatures to Kelvin

Temperatures are given in Celsius. However, in thermodynamic temperature calculations, it's necessary to use the Kelvin scale. Convert the temperatures as: $T_c=-{13}^{\circ }\mathrm{C}+273.15=260.15K$ and $T_h={25}^{\circ }\mathrm{C}+273.15=298.15K$

Calculate Coefficient of Performance

The formula for coefficient of performance (COP) of a Carnot refrigerator is given by: $COP=T_c/(Th-Tc)$. Substituting the values calculated from Step 1, we get: $COP=260.15K/(298.15K-260.15K)$

Simplify and Find Coefficient of Performance

Perform the arithmetic operation in the formula to get the Coefficient of Performance. This value indicates the efficiency of the Carnot refrigerator operating between the given temperatures.

Key concepts

Coefficient of Performance

The coefficient of performance (COP) is a measure of the efficiency of a refrigerator or heat pump. In simple terms, it tells us how well the system uses energy to move heat from one place to another. For a Carnot refrigerator, the COP is determined by the temperatures on the Kelvin scale. The formula is: $$COP=\frac{T_c}{T_h-T_c}$$where:

• $T_c$ is the temperature of the cold reservoir.
• $T_h$ is the temperature of the hot reservoir.

A higher COP means the refrigerator is more efficient, requiring less work to transfer heat. This efficiency depends entirely on the temperature difference between the hot and cold reservoirs. A smaller temperature difference results in a higher COP. Understanding COP is crucial for evaluating the performance of refrigeration systems.

Thermodynamic Cycle

A thermodynamic cycle refers to a series of processes that involve heat transfer, work done by or on a system, and changes in the system's internal energy. Refrigerators typically operate using such cycles to move heat from inside the unit to the outside environment. In the case of a Carnot refrigerator, it follows the Carnot cycle, which is an ideal reversible cycle. It allows us to determine the maximum possible efficiency of a refrigerator. The Carnot cycle consists of two isothermal processes and two adiabatic processes. The efficiency defined by such cycles helps us understand the theoretical limits of refrigerators and how real-world systems can strive towards these limits. Mastering this concept allows students to evaluate how refrigerant transfers energy in cooling systems.

Temperature Conversion

In thermodynamics, working with absolute temperatures is necessary for accurate calculations. This involves converting temperatures from Celsius to Kelvin, the standard unit of temperature in the scientific field.For conversions, you can use the formula:$$T(K)=T{(}^{\circ }C)+273.15$$This step is essential because thermodynamic formulas require absolute temperatures. For instance, in the Carnot efficiency equations, using Kelvin instead of Celsius avoids negative values and ensures accurate energy computations. Understanding temperature conversion is foundational for anyone studying thermodynamic cycles, including refrigeration systems.

Kelvin Scale

The Kelvin scale is the International System of Units (SI) base unit for temperature and is crucial in thermodynamics. Unlike Celsius or Fahrenheit, the Kelvin scale starts at absolute zero, the point at which molecular motion ceases. One Kelvin is equivalent in size to one Celsius degree. Therefore, converting between these two scales involves simply adjusting for the zero-point difference. Using the Kelvin scale in thermodynamics:

• Avoids negative temperatures, simplifying calculations.
• Provides an absolute reference point for temperature-related energy computations.

By adopting the Kelvin scale, scientists and engineers ensure precision and clarity in describing heat exchanges in processes such as those in refrigeration cycles.