download the script: The Reversed Carnot Cycle

Unlike the Carnot heat engine, the Carnot refrigeration cycle undergoes a process with opposite direction. We see from the model, heat Q_{L} is absorbed from the low-temperature reservoir (T_{L}=constant) and heat Q_{H}* *is rejected to a high-temperature reservoir (T_{H}=constant). In this case a work input in the amount of W_{rev} is required to achieve this process. And we know from the 1^{st} law of thermodynamics, the required work can be determined in W_{rev}=- Q_{H }– Q_{L}. Here Q_{H}<0 and Q_{L}>0.

The reversed Carnot cycle also consists of two isentropic and two isothermal processes. The process undergoes in direction 3-2-1-4-3

Process 3-2: Reversible Adiabatic Compression

This process is isentropic. The engine is perfect insulated so that no heat is lost and absorbed. Gas is compressed slowly until the temperature rises from T_{L} to T_{H}.

Process 2-1: Reversible Isothermal Compression (T_{H}=constant)

During this process, heat is rejected. Gas is compressed reversibly at the constant temperature T_{H}.

Process 1-4: Reversible Adiabatic Expansion

This process is isentropic. The engine is perfect insulated so that no heat is lost and absorbed. Gas expands slowly until the temperature drops from T_{H} to T_{L}.

Process 4-3: Reversible Isothermal Expansion (T_{L}=constant)

During this process, heat is absorbed. Gas is compressed reversibly at the constant temperature T_{H}.

The coefficient of performance (COP) of any refrigerator or heat pump, reversible or irreversible, can be calculated with the general expression:

- For a Carnot refrigerator:

- For a Carnot heat pump:

From the T-s-diagram, we obtain:

Q_{ab}=Q_{H }= T_{H}∙ ΔS_{2→1}

Q_{zu}=Q_{L }= T_{L}∙ ΔS_{4→3}

ΔS_{2→1}= – ΔS_{4→3}

Therefore

- For a Carnot refrigerator:

- For a Carnot heat pump:

According to coefficient of performance (COP), we can also draw the following conclusions:

- The coefficient of performance (COP) of a reversed Carnot cycle only depends on the highest and lowest temperature.
- Normally T
_{H}>T_{L}, so that means COP_{R}>1 and COP_{HP}>1 - Both coefficients of performance (COP) have a relationship: COP
_{HP}= COP_{R}+1 - If T
_{L}decreases, both COP_{R}and COP_{HP}decrease.

As mentioned above, the reversed Carnot cycle is a reversible process. Hence, if a real refrigerator has the coefficient of performance of COP, then:

- COP
_{R}< COP_{R,rev}: irreversible refrigerator - COP
_{R}= COP_{R,rev}: reversible refrigerator - COP
_{R}> COP_{R,rev}: unrealistic refrigerator