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:

1. The coefficient of performance (COP) of a reversed Carnot cycle only depends on the highest and lowest temperature.
2. Normally T_H>T_L, so that means COP_R>1 and COP_HP>1
3. Both coefficients of performance (COP) have a relationship: COP_HP = COP_R +1
4. 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