Carnot’s Ideal Heat Engine

What is Carnot’s ideal heat engine?

Carnot’s ideal heat engine is an ideal heat engine that operates on the Carnot cycle. The model for this engine was developed by Nicolas Leonard Sadi Carnot in 1824. Its working is similar to the heat engine and it’s based on the second law of thermodynamics.

The second law of thermodynamics

The direction of heat movement is determined by this law. It is impossible to build a cyclic machine that collects heat from a source, turns all of that heat into work, and rejects no heat to sink. The Kelvin-Planck statement refers to the above expression of the second law.

Mechanical labour can be totally converted into heat, but the opposite is not true. Heat and labour are not similar in this regard. We will now look at several applications of the second law of thermodynamics.

Heat engine

The device, used to convert heat energy into mechanical energy, is called a heat engine. For conversion of heat into work with the help of a heat engine, the following conditions have to be met. There should be a body at a higher temperature ‘T1’  from which heat is extracted. It is called the source. The body of the engine should contain a working substance. There should be a body at a lower temperature ‘T₂’ which can reject heat. This is called the sink.

Working of heat engine

The engine derives an amount ‘Q1’ of heat from the source.

A part of this heat is converted into work ‘W’. The remaining heat ‘Q’ is rejected to the sink.

Thus,   

Q1 = W+Q₂

or the work done by the engine is given by

W=Q₁-Q₂

The efficiency of heat engine

The efficiency of a heat engine (𝛈) is defined as the fraction of total heat supplied to the engine which is converted into work. 

Mathematically,

Since 𝛈=  W/Q₁

Or 𝛈= Q₁-Q2/Q₁ = 1-Q2/Q₁

Carnot’s ideal heat engine

The Carnot engine is an ideal heat engine that operates on the Carnot cycle. The model for this engine was developed by Nicolas Leonard Sadi Carnot in 1824. It consists of different parts.

Source: It is a reservoir of heat energy with a conducting top maintained at a constant temperature T1K. The source is so big that extraction of any amount of heat from it does not change its temperature.

Body of heat engine: It is a barrel having perfectly insulating walls and conducting bottom. It is fitted with an air-tight piston capable of sliding within the barrel without friction. The barrel contains some quantity of an ideal gas.

Sink: It is a huge body at a lower temperature T₂ having a perfectly conducting top. The size of the sink is so large that any amount of heat rejected to it does not increase its temperature.

Insulating stand: It is a stand made up of perfectly insulating material such that the barrel when placed over it becomes thoroughly insulated from the surroundings.

Carnot’s ideal heat engine working & its processes 

When the Carnot engine works, the working substance of the engine undergoes a different process known as the Carnot cycle and this cycle consists of four different stages.

1. The First stage:- Known as Isothermal expansion process

In this stroke, the barrel is placed over the source. The piston is gradually pushed back as the gas expands. Fall of temperature, due to expansion, is compensated by the supply of heat from the source and consequently, temperature remains constant. The conditions of the gas change from A(P₁, V1) to B(P₂, V2). If W1 is the work done during this process, then heat Q₁, derived from the source is given by

Q₁ = W1 = -nRT1 loge(V2 / V1)

2. The Second stage:- Known as Adiabatic expansion process

The barrel is removed from the source and is placed over the insulating stand. The piston is pushed back so that the gas expands adiabatically resulting in a fall of temperature from T1 to T₂. The conditions of the gas change from B(P₂, V₂) to C(P3, V3). If W₂ is the work done in this case, then:

W₂ = nCv(T2 – T1)

3. The Third stage:- Known as the isothermal compression process

The barrel is placed over the sink. The piston is pushed down thereby compressing the gas. The heat generated due to compression flows to the sink maintains the temperature of the barrel constant. The state of the gas changes from  C(P3, V3) to D(P4 , V4). If W, is the work done in this process and Q is the heat rejected to the sink, then:

W3 = -nRT₂ loge(V4 / V3)

4. The Fourth stage:- Known as Adiabatic compression process

The barrel is placed over the insulating stand. The piston is moved down thereby compressing the gas adiabatically till the temperature of gas increases from T₂ to T1  The state of gas changes from D(P4 , V4) to   A(P₁, V1). If W4 is the work done in this process, then:

W4 =  nCv(T1 – T2)

Heat converted into work in Carnot’s engine

Wcycle = W1 + W₂ + W3 + W4 

⇒ – nRT1 loge(V2 / V1) +  nCv(T2 – T1) – nRT₂ loge(V4 / V3) +  nCv(T1 – T2)

⇒ -nR[ T1  loge(V2 / V1) + T2 loge(V4 / V3) ] 

For BC, T1V2𝜸 – 1 = T2V3𝜸 – 1 

For DA, T1V1𝜸 – 1 = T2V4𝜸 – 1

(V2 / V1)𝜸 – 1 = (V3 / V4)𝜸 – 1  ⇒  V2 / V1 = V3 / V4

Thus, the net work done by the engine during one cycle is equal to the area enclosed by the indicator diagram of the cycle. Analytically:

Wcycle = -nR(T1 – T2) loge(V2 / V1) 

The efficiency of Carnot’s ideal heat engine

Efficiency (𝛈) of an engine is defined as the ratio of useful heat (heat converted into work) to the total heat supplied to the engine. Thus:

𝛈 = mod of W/Q1 = (Q1 – Q2)/Q1

𝛈 = nR(T1 – T2) loge(V2 / V1) / nRT1 loge(V2 / V1)  = (T1 – T2) / T1

𝛈 = 1 – Q2 / Q1 = 1 – T2 / T1

Some important points regarding Carnot’s ideal heat engine

1. The efficiency of an engine depends upon the temperatures between which it operates.
2. 𝛈 is independent of the nature of the working substance.
3. 𝛈 is one only if T2  = 0. Since absolute zero is not attainable, hence even an ideal engine cannot be 100% efficient.
4. 𝛈 is one only if Q2 = 0 But 𝛈 = 1 is never possible even for an ideal engine. Hence Q2 ≠ 0.
5. Thus, it is impossible to extract heat from a single body and convert the whole of it into work.
6. If T2  = T1 , then 𝛈 = 0
7. In actual heat engines, there are many losses due to friction, etc., and various processes during each cycle are not quasistatic, so the efficiency of actual engines is much less than that of an ideal engine.

Conclusion

In this article, we studied the Carnot engine. These are any device that converts heat into mechanical work. Carnot engines idealise these heat engines. The Carnot theorem states that no engine operating between two known temperatures can be more efficient than a reversible engine operating between the same two temperatures. All reversible engines operating between the same two temperatures, regardless of the working material, have the same efficiency. The Carnot Cycle is a theoretical ideal thermodynamic cycle. In Isothermal Expansion, constant temperature is maintained in the gas. The Carnot Engine is the system that runs on the Carnot Theorem