The thermal machines

The following figure shows the stroke internal combustion engine.

The experimental diagrams (left) and theoretical (right) are available in the following figure.

A video that shows the internal combustion engine :

Simplifying assumptions :

During the cycle, the fluid properties change. We do not take it into account and considering the gas as a perfect gas.

This is always the same gas that undergoes the cycle.

The transformations are reversible :

• : adiabatic compression

• : isochoric (spark from the spark plug)

• : adiabatic expansion

• : isochoric cooling

Cycle efficiency :

The cycle is motor ( ) : the fluid during transformation receives heat and gives heat to the external medium during transformation.

means the sum of the works received by the gas during the cycle (that is to say during the transformations).

Cycle efficiency (of the motor) is defined by :

Where :

• is the work done by the system (energy exiting the system as work).

• is the heat put into the system (heat energy entering the system).

Either here :

According to the first law :

Is :

The efficiency expression becomes :

Yet (isochoric transformations) :

Whence :

For both isentropic (reversible adiabatic), we can write :

And noting that and , we obtain :

We deduce the expression of the efficiency :

Or, finally :

We note the compression ratio :

For between and and with , .

General principle of a heat engine :

The figure below shows the principle of a Carnot engine (dithermal machine).

A fluid undergoes transformations of cycles in which it exchanges work and heat with the outside.

If the fluid "effectively" provides work to the outside, the machine is a motor.

If the fluid receives work and takes heat from the cold source, the thermal machine is a refrigerator (or air conditioner).

If the fluid receives work and provides heat to the hot source, the thermal machine is a heat pump.

If the fluid exchanges heat with two heat sources, the machine is dithermal.

The cycle of reversible Carnot engine (dithermal motor cycle) :

The block diagram is shown in the following figure.

The fluid receives heat from the heat source, provides work to the external environment and rejects a portion of the heat energy received from the cold source (impossibility of monothermal engine).

The cycle consists of two reversible adiabatic (no heat exchange) and two isotherms (in contact with two heat sources).

It is drawn into the Clapeyron plan in the following figure.

Yield (motor efficiency) calculation :

The yield is defined as :

So :

Where represents the total work received by the fluid during the cycle.

According to the first law :

Whence :

Entropy balance for the fluid in one cycle is :

With :

Whence :

We deduce that :

And :

This yield is always less than ; for example, with :

Note that this performance does not depend on the nature of the fluid which undergoes the cycle (perfect gas, real gas, water, ...), but only the temperatures of hot and cold sources.

Engine of irreversible Carnot :

The Carnot cycle is now irreversible (for example, heat transfer is no longer reversibly in contact with two heat sources).

We will show that the efficiency of this irreversible cycle is lower than the reverse cycle, operating between the same two sources.

Entropy balance for the fluid in one cycle is :

With :

Whence :

And :

Is the inequality of Clausius.

The efficiency is always defined by :

But Clausius gives :

Where : (Carnot's theorem)

Carnot's refrigerating machines :

The Carnot cycle is now gone in the other direction (counterclockwise wise).

If we look at the hot source, this refrigerating machine is a heat pump.

If we look at the cold source, the refrigerating machine is a refrigerator (or air conditioner).

Efficiency of a heat engine (reversible case) :

Numerical applications :

For a refrigerator :

For heat pump :

This result shows that spent to operate the heat pump supplies as much heat as Joule dissipation of of electrical work in an electric heater !

In the case of an irreversible operation :