An ideal refrigerator or ideal heat pump is equivalent to a...

An ideal refrigerator or ideal heat pump is equivalent to a Carnot engine running in reverse. That is, energy $Q_{c}$ is absorbed from a cold reservoir, and energy $Q_{h}$ is rejected to a hot reservoir. (a) Show that the work that must be supplied to run the refrigerator or heat pump is $$ W=\frac{T_{h}-T_{c}}{T_{c}} Q_{c} $$ (b) Show that the coefficient of performance (COP) of the ideal refrigerator is $$ \mathrm{COP}=\frac{T_{c}}{T_{h}-T_{c}} $$

An ideal refrigerator or ideal heat pump is equivalent to a Carnot engine running in reverse. That is, energy $Q_{c}$ is absorbed from a cold reservoir, and energy $Q_{h}$ is rejected to a hot reservoir. (a) Show that the work that must be supplied to run the refrigerator or heat pump is
$$
W=\frac{T_{h}-T_{c}}{T_{c}} Q_{c}
$$
(b) Show that the coefficient of performance (COP) of the ideal refrigerator is
$$
\mathrm{COP}=\frac{T_{c}}{T_{h}-T_{c}}
$$

Key Concepts

Carnot Cycle

The Carnot cycle is a theoretical thermodynamic cycle that establishes the maximum possible efficiency any heat engine can achieve during the conversion of heat into work, or vice versa, when operating between two thermal reservoirs. This cycle is composed of two isothermal processes and two adiabatic processes and serves as the benchmark for comparing real-world engine performance due to its reversible nature.

Reversed Carnot Cycle

When the Carnot cycle is operated in reverse, it functions as a refrigerator or heat pump. In this mode, rather than producing work from heat, work is input to extract heat from a cold reservoir and reject it to a hotter reservoir. This process enables the theoretical derivation of work relationships and performance limits for cooling systems, based on the same thermodynamic principles as the forward Carnot engine.

Coefficient of Performance (COP)

The coefficient of performance (COP) is a measure used to evaluate the efficiency of refrigerators and heat pumps. Unlike efficiency measures that relate work output to energy input in heat engines, the COP for refrigerators is defined as the ratio of the amount of heat removed from the cold reservoir to the work input required to achieve that removal. This makes it a critical parameter in assessing the performance and energy efficiency of cooling devices.

Thermodynamic Work and Energy Transfer

This concept focuses on the balance of energy in thermal systems, where the work supplied to a system is related to the heat transfers between thermal reservoirs. In ideal cyclic processes such as the Carnot refrigerator or heat pump, energy flows and work are governed by the temperature differences between the reservoirs. The work input and the corresponding heat transfers are determined by conservation of energy and the second law of thermodynamics, leading to specific mathematical relationships that define the system's behavior.

Transcript

Please give Ace some feedback