Download the script: Carnot Cycle

In practice reversible cycles are unrealistic because each process is associated with irreversibilities, for example friction. However, the upper limit on the performance of real cycles can be obtained by using the corresponding reversible cycles. And the most well-known reversible cycle process is the Carnot cycle.

An heat engine with Carnot cycle, also called Carnot heat engine, can be simplified by the following model:

A reversible heat engine absorbs heat Q_H from the high-temperature reservoir at T_H und releases heat Q_L to the low-temperature reservoir at T_L. The temperatures T_H and T_L remain unchanged. And we know from the 1^st law of thermodynamics, work is done by the heat engine, W=Q_H+Q_L. Here Q_H>0 and Q_L<0.

The Carnot cycle in this heat engine consists of two isentropic and two isothermal processes.

Process 1-2: Reversible Isothermal Expansion (T_H=const)

During this process, heat is absorbed. Gas expands reversibly at the constant temperature T_H.

Process 2-3: Reversible Adiabatic (Isentropic) Expansion

This process is isentropic. The engine is perfect insulated so that no heat is lost and absorbed. Gas continues expanding slowly until the temperature drops from T_H to T_L.

Process 3-4: Reversible Isothermal Compression (T_L=const)

After gas reaches the low temperature T_L, some external force is applied on the engine in order that gas can be compressed. Since the temperature remains constant at T_L, no change of internal energy of gas occurs, if we assume that gas is ideal gas. Knowing from the 1^st law of thermodynamics, we obtain a conclusion that heat must be transferred from engine to low-temperature reservoir.

Process 4-1: Reversible Adiabatic (Isentropic) Compression

This process is isentropic. The engine is perfect insulated so that no heat is lost and absorbed. Gas continues being compressed slowly until the temperature rises from T_L to T_H. The process comes to an end when reaching its initial state (state 1).

The thermal efficiency of the Carnot heat engine can be calculated with the general expression:

From the T-s-diagramm, we obtain:

Q_zu=Q_H=T_H·ΔS_1-2

Q_ab=Q_L= T_L·ΔS_3-4

ΔS_1-2= – ΔS_3-4

 Therefore

This equation is also referred to as Carnot efficiency.

 

According to the Carnot efficiency, we can also draw the following conclusions:

1. The Carnot efficiency only depends on the highest and lowest temperature.
2. If we want to increase the Carnot efficiency, we can increase the highest temperature T_H or reduce the lowest temperature T_L.
3. The efficiency of a Carnot heat engine is always smaller than 1. Only when T_H→∞ or T_L→0 could η_th,C→1. But both methods are impossible in the practice.
4. For T_H=T_L we have η_th,C =0. That means, if we only one reservoir, system is not able to undergo a cyclic process and, of course, no work can be done.

As mentioned above, the Carnot cycle is a reversible process which delivers the most work output for a heat engine operating between the same temperature limits. Hence, if a heat engine has the thermal efficiency of η_th, then:

• η_th < η_th,C : irreversible heat engine
• η_th = η_th,C : reversible heat engine
• η_th > η_th,C : unrealistic heat engine