Carnot Cycle – Definitions and Principles

Irreversibilities are very important in thermodynamic cycles. Because they are very important in the calculation of efficiencies. Also, we use fully reversible calculations in the cycles. One of them is the Carnot cycle which is very important to us to understand the general working principles of thermodynamic systems. Here, we explain the Carnot cycle in detail.

What is Carnot Cycle?

You remember that in heat engines, there is a working fluid in a thermodynamical cycle. This fluid takes heat energy which we call input energy or work. And in one cycle of this heat engine, fluid gives an amount of work energy. So, the ratio between these two works gives us the efficiency of the system.

In irreversible systems, there are always irreversible losses because of friction, heat leakages, and unintended chemical reactions. So, the total efficiency of the system will be lower. But for a system that does not have any irreversibilities, has bigger efficiencies.

But in reality, there is no reversible processes. We can not escape from the irreversibilities. So, there is a reversible heat engine application that we call it Carnot cycle. It is a very important thermodynamical cycle to understand the general working principles of irreversibilities.

In the Carnot cycle, there are two isothermal and two adiabatic steps. The very basic system includes a piston-cylinder system. Let’s check the steps below.

Steps of Carnot Cycle

To understand how the Carnot cycle works, you need to understand all the steps below. After understanding all these steps, you will understand why we can not produce any devices or systems that have a Carnot efficiency.

1 – Reversible Isothermal Expansion

Firstly, we need to think that the piston has two middle of the cylinder and bottom dead center or position 1 for the bottom and position 2 for the middle. We call the first process 1-2. In this reversible isothermal process, there is no insulation in the cylinder heat. Because we need a heat transfer between the gas inside the cylinder and the surroundings.

In this step, we call the temperature of the gas T1. And the piston goes from the bottom dead center to the middle of the cylinder very slowly. It is very slow that when the gas expands, the temperature of the gas decreases an infinitesimal amount. And the heat transfer from the surroundings increases the temperature of the gas to T1 again. The first reversible thermal expansion process takes place up until the piston reaches the middle of the cylinder.

At the upper dead center, the gas is expanded but at the same T1 temperature. And the gas took heat from the surroundings which are Q1.

Position 1; Temperature = T2, Volume = V1, Internal Energy = 0.

Position 2; Temperature = T2, Volume = V2, Internal Energy = Q1.

2 – Reversible Adiabatic Expansion

In the second step which we also call as 2-3 step, the expansion continues. But, there is no heat transfer between the surroundings and the cylinder. Because all the sides are insulated and the process is adiabatic. The cylinder goes to the upper dead center which we call this position 3. So, the temperature of the gas drops to the T1 again because of this expansion.

There are no frictions or other kinds of irreversibilities in the systems. All the work and energy interactions are between the surroundings and the gas in the Carnot cycle.

Position 3; Temperature = T1, Volume = V3, Internal Energy = Q1

3 – Reversible Isothermal Compression

In the third stage of the Carnot cycle, the compression of the gas starts. The piston starts to move very slowly. And the insulation between the heat sink and the cylinder is removed. So, there is a heat exchange between the gas and the heat sink. The piston moves downward and the temperature of the gas begins to rise. But, a portion of internal energy Q is released to the heat sink and the temperature of the gas remains T1.

And you understand that the temperature of the gas remains constant but, there is a heat rejection to the outside. We call this heat rejection as Q2. Also, we call this process 3-4.

Position 4; Temperature = T1, Volume = V2, Internal Energy = 0.

4 – Reversible Adiabatic Compression

In this step of the Carnot cycle, the compression of the gas continues up to the bottom dead center of the piston-cylinder mechanism. There is insulation to obtain the reversible adiabatic compression. And the temperature of the gas returns to the first stage. So, we can call this process a 4-1 process.

Position 1; Temperature = T2, Volume = V1, Internal Energy = 0.

So, this is the complete cycle of the Carnot cycle and we can make our efficiency calculations upon it.

The efficiency of the Carnot Cycle

According to the calculations above, we can calculate the efficiency of the Carnot cycle. The total efficiency is very straightforward. We just need to divide the outlet Q which is the total work to the inlet Q which is the work inlet.

The efficiency of the Carnot cycle is the maximum attainable efficiency that we can have from a thermodynamical cycle between two temperature limits. Because we eliminate all the irreversibilities in the Carnot cycle.

As you see above, we can directly calculate the Carnot efficiency by using the total rejected energies. And also, we can calculate it with the temperature states of the gas.

In real life, we can not design a system that works in the efficiency levels of the Carnot cycle. Because it is impossible to obtain an irreversible system.

So, we call the maximum attainable efficiency limit as Carnot efficiency of machines.

This means that we can use the maximum temperature limits of the thermodynamics cycles to calculate the Carnot efficiency.

Carnot Refrigeration Cycle

This is also a very important cycle that we can obtain maximum efficiency from refrigeration. The only difference in this cycle is that we take heat from the cold-temperature reservoir and reject this heat to the high-temperature reservoir.

Again in real, no cycle works at this level of efficiency. All the refrigeration cycles work with irreversibilities. If we consider these irreversibilities, we obtain a system that has a lower COP.

So, by looking at this cycle, we can conclude very important statements related to the second law of thermodynamics.

Carnot Refrigerator and Heat Pump Performance Calculations

Also, we can calculate the maximum coefficient of performance o the Carnot refrigerators. As we stated above the only difference is that we are rejecting heat from the higher energy source and taking heat from the lower energy source. So, the Carnot COP calculation is like this;

This is the Carnot COP calculations for refrigerators. Q2 is the total heat that we take from the cold environment. And Q1 is the total heat that we rejected in the hot environment. With the increasing difference between these heat values, their Carnot COP of them increases.

Also for heat pumps;

The working principle of the heat pump is the same of the refrigerator. But the formula changes according to our purpose. In the refrigerator, our purpose is to take heat from the cold environment. But in heat pumps, our purpose is to reject heat from the hot environment. So, the COP formulation is re-arranged according to that.

We can also write the temperature values at the places of heat. They give us the maximum Carnot COPs which are reversible processes.

Carnot Principles

First of all, we cannot obtain a device that has a thermodynamic cycle whişch works with only heat and energy sources. As you see above, Carnot cycles are working with one source of heat and energy sink.

And we can conclude that the efficiency of irreversible engines which are working between two heat reservoirs that in different temperatures, is lesser than the reversible engines which are working with only one energy reservoir.

Also, the efficiencies of all types of reversible engines between two energy sinks in two temperatures are the same.

Carnot Efficiencies of Certain Systems

We the examples below, you will understand calculations of the maximum efficiencies of different thermodynamical systems.

Maximum Efficiencies of Steam Power Plants

We can apply this principle to the steam power plant systems. So, we can see the possibility of maximum efficiency if we minimize the irreversibilities of the steam power plants. Also, as you see above, with the increasing difference in temperature, the total efficiency that we can take from these systems increases. And if the temperature difference is infinite, the maximum Carnot efficiency increases to 100.

For example, if we know the maximum temperature difference of the steam in the whole thermodynamical cycle, we can calculate the maximum efficiency a steam power plant can have.

At the maximum pressure and energy situation of steam in the cycle, the temperature is equal to = 250°C or 523.25 K.

And the minimum temperature that the water has = 30 °C or 303.25 K. If we put these Kelvin temperature values at the Carnot efficiency formula above;

1 – (303.25/523.25) = 0.43

Carnot efficiency = 43%.

So, 43% is the maximum efficiency of a steam power plant that can achieve in the working temperatures stated above if there are no irreversibilities.

Maximum COP of Refrigerators

For example, you can think of a refrigerator where the internal temperature is 1 °C and the outside temperature is 30°C. So, we can easily calculate the maximum COP of this refrigerator by putting these temperature values on the formulation above.

COP = 1 / ((303.25/274.25)-1) = 9.52

As you see above, we can attain a maximum of 9.52 COP from a system that works between 1°C and 30°C. But because of the irreversibilities, this value will much be lower than usual.

Maximum COP of Air Conditioners

Also, for example, we can think the heat pumps are air conditioner systems. Because in winter, we use air conditioners to take the heat from the outside to give this heat to the inside environment which is warmer than the outside. So, we can directly calculate the maximum efficiency of the air conditioner that can have by putting the temperature values in the formula.

By using the same value with refrigerators, we can make our calculations for an air conditioner that takes heat from 1°C outside temperature and rejects heat to the 30°C temperature. Let’s calculate according to the Carnot COP formula for heat pumps.

COP = 1 / (1-(274.25/303.25)) = 10.41.

Also as you see above, this value is a very high COP value. But the real value will be very low from this value because of the irreversibilities.

Conclusion

These are the general statements about the Carnot cycle and principles that emerge from the Carnot machines.

These statements are very important to see the maximum attainable efficiency values that we can obtain from the different thermodynamical devices. It is a very important principle that states the second law of thermodynamics. We can easily calculate the maximum efficiencies and coefficient of performance of different devices such as refrigerators, heat pumps, and engines.

It gives engineers very big insight that if they minimize the irreversibilities, which efficiency or coefficient of performance values they can attain.

You can understand the general working principle of the Carnot cycles by looking at the examples above. So, we can not obtain 100% efficiency at any time with finite temperature differences.

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