# Carnot Cycle: 21 Important Facts You Should Know

## CARNOT CYCLE

Nicolas Léonard Sadi-Carnot, a French mechanical engineer, Scientist, and physicist, introduced a heat engine known as the Carnot Engine in the book “Reflections on the Motive Power of Fire. It leads to being the foundation of the Second law of thermodynamics and entropy. Carnot’s contribution holds a remark which gave him the title of “Father of Thermodynamics.

## Carnot cycle in thermodynamics | working principle of Carnot cycle | ideal Carnot cycle | Carnot cycle thermodynamics | Carnot cycle definition | Carnot cycle working principle | air standard Carnot cycle| Carnot cycle reversible.

Carnot cycle is the theoretical cycle that works under two thermal reservoirs (Th & Tc) undergoing compression and expansion simultaneously.

It consists of four reversible processes, of which two are isothermal, i.e., constant temperature followed alternately by two reversible adiabatic processes.

The working medium used in the Sadi-Carnot cycle is atmospheric air.

Heat addition and Heat rejection are carried out at a constant temperature, but no phase change is considered.

## Importance of Carnot Cycle

The invention of the Carnot cycle was a very big step in the history of thermodynamics. First, it gave theoretical working of heat engine used for the design of an actual heat engine. Then, reversing the cycle, we get refrigeration effect (mentioned below).

Carnot cycle work between two thermal reservoirs (Th & Tc), and its efficiency depends only on this temperature and doesn’t depend on the fluid type. That is Carnot’s cycle efficiency is fluid independent.

## Carnot cycle pv diagram | Carnot cycle ts diagram | pv and ts diagram of Carnot cycle | Carnot cycle pv ts | Carnot cycle graph | Carnot cycle pv diagram explained | Carnot cycle ts diagram explained

Process 1-2: Isothermal expansion

In this process, the air is expanded with constant temperature while gaining heat.

That is, constant temperature heat addition takes place.

Expansion => pressure ↑ => results Temperature ↓

Hence Temperature remain constant

In this process, the air is expanded, keeping entropy constant and with no heat interaction.

That is no change in entropy, and the system is insulated

We get work output in this process

Process 3-4: isothermal compression

In this process, the air is compressed with a constant temperature while losing heat.

That is, constant temperature heat rejection takes place.

Compression => pressure ↓ => results: Temperature ↑

Hence Temperature remain constant

In this process, the air is compressed, keeping entropy constant and no heat interaction.

That is no change in entropy, and the system is insulated

We supply work in this process

## Carnot cycle consists of | Carnot cycle diagram | Carnot cycle steps | 4 stages of Carnot cycle | Carnot cycle work| isothermal expansion in Carnot cycle| Carnot cycle experiment

Process 1-2:

The expansion process is carried out where temperature Th is kept constant, and heat (Qh) is added to the system. The temperature is kept constant as follows: The rise in temperature due to heat addition is compensated by the decrease in temperature due to expansion.

Hence the process carried out results as constant temperature as the start and end temperature of the process is same.

Process 2-3:

As we can see, the process is reversible (change in internal energy = 0) Adiabatic (only work transfer, no heat involvement), the expansion carried out just results in a change in temperature (from Th to Tc), keeping the entropy constant.

System act as being insulated for this part of the expansion.

Sensible cooling is taking place.

Process3-4:

The compression process is carried out where temperature Tc is kept constant, and heat is removed from the system. The temperature is kept constant as follows: The decrease in temperature due to heat rejection is compensated by the increase in temperature due to compression.

Hence the process carried out results as constant temperature as the start and end temperature of the process is same.

Similar to processes 1-2 but in the exact opposite manner.

Process 4-1:

As we can see, the process is reversible (change in internal energy = 0) Adiabatic (only work transfer, no heat involvement), the compression carried out just results in a change in temperature (from Tc to Th), keeping the entropy constant.

System act as being insulated for this part of the compression.

Sensible heating is taking place.

## Carnot cycle equations| Carnot cycle derivation

Process 1-2: Isothermal expansion

as Th is kept constant. [Internal energy (du) = 0] ( PV = K)

Qh = W ,

therefore, $W = \int_{V_{1}}^{V_{2}}PdV$

$P = \frac{K}{V}$

$W = K\int_{V_{1}}^{V_{2}}\frac{dV}{V}$

$W = P_{1}V_{1}\int_{V_{1}}^{V_{2}}\frac{dV}{V}$

$W = P_{1}V_{1}\left ( ln\frac{V_{2}}{V_{1}} \right )$

$W = mRT_{h}\left ( ln\frac{V_{2}}{V_{1}} \right )$

$PV^{\gamma } = K$

$W = \int_{V_{2}}^{V_{3}}PdV$

$PV^{\gamma } = K$

therefore $W = K\int_{V_{2}}^{V_{3}}\frac{dV}{V^{\gamma }}$

$W = P_{2}V^{\gamma }_{2}\int_{V_{2}}^{V_{3}}\frac{dV}{V^{\gamma }}$

$W = P_{2}V^{\gamma }_{2}\int_{V_{2}}^{V_{3}}{V^{-\gamma }{dV}}$

$W = K\int_{V_{2}}^{V_{3}}{V^{-\gamma }{dV}}$

$W = K \left [ \frac{V^{1-\gamma }}{1-\gamma } \right ]_{2}^{3}$

$PV^{\gamma } = K = P_{2}V_{2}^{\gamma } = P_{_{3}}V_{3}^{\gamma }$

$W=\left [ \frac{P_{3}V^{\gamma }_{3}V_{3}^{1-\gamma }-P_{2}V^{\gamma }_{2}V_{2}^{1-\gamma }}{1-\gamma } \right ]$

$W=\left [ \frac{P_{3}V_{3}-P_{2}V_{2}}{1-\gamma } \right ]$

Also

$P_{2}V_{2}^{\gamma } = P_{_{3}}V_{3}^{\gamma } = K$

$\left [ \frac{T_{2}}{T_{3}} \right ] =\left [ \frac{V_{3}}{V_{2}} \right ]^{\gamma -1}$

As process is Adiabatic , Q = 0
therefore W = -du

Process 3-4: isothermal compression

similar to process 1-2, we can get

as Tc is kept constant. [Internal energy (du) = 0] ( PV = K)

Qc = W ,

$W = P_{3}V_{3}\left ( ln\frac{V_{3}}{V_{4}} \right )$

$W = mRT_{c}\left ( ln\frac{V_{3}}{V_{4}} \right )$

similar to process 2-3, we can get

$W=\left [ \frac{P_{1}V_{1}-P_{4}V_{4}}{1-\gamma } \right ]$

$P_{4}V_{4}^{\gamma } = P_{{1}}V{1}^{\gamma } = K$

$\left [ \frac{T_{1}}{T_{4}} \right ] =\left [ \frac{V_{4}}{V_{1}} \right ]^{\gamma -1}$

## Carnot cycle work done derivation

According to first law of thermodynamics

Wnet = Qtotal

Wnet = Qh-Qc

Wnet = $mRT_{h}\left ( ln\frac{V_{2}}{V_{1}} \right ) – mRT_{c}\left ( ln\frac{V_{3}}{V_{4}} \right )$

## Derivation of entropy from carnot cycle | entropy change in carnot cycle | change in entropy carnot cycle | derivation of entropy from carnot cycle | entropy change in carnot cycle

To make cycle reversible, Change in entropy is zero (du = 0).

$ds = \frac{\delta Q}{T} \ + \ S_{gen}$

$S_{gen} \ =\ 0\ , \ for \ reversible \ process$

that means,

$\frac{\delta Q}{T}=\ 0\ , \ for \ reversible \ process$

$ds = \frac{\delta Q}{T} \ = \frac{\delta Q_h}{T_h}+ \frac{\delta Q_c}{T_c} = 0$

For process :1-2

$ds_{1-2} = \frac{mR\ T_{h}\ ln\left ( \frac{P_{1}}{P_{2}} \right )}{T_h}$

$ds_{1-2} = m R \ ln\left ( \frac{P_{1}}{P_{2}} \right )$

For process :1-2

$ds_{3-4} =- \frac{mR\ T_{c}\ ln\left ( \frac{P_{3}}{P_{4}} \right )}{T_c}$

$ds_{3-4} = \frac{mR\ T_{c}\ ln\left ( \frac{P_{4}}{P_{3}} \right )}{T_c}$

$ds_{3-4} = – m R \ ln\left ( \frac{P_{3}}{P_{4}} \right )$

$ds_{3-4} = m R \ ln\left ( \frac{P_{4}}{P_{3}} \right )$

$d_s = ds_{1-2}\ +\ ds_{3-4} = 0$

## carnot cycle efficiency| carnot cycle efficiency calculation| carnot cycle efficiency equation| carnot cycle efficiency formula | carnot cycle efficiency proof | carnot cycle maximum efficiency | carnot cycle efficiency is maximum when | maximum efficiency of carnot cycle

Carnot cycle efficiency has maximum efficiency considering the Th as the hot reservoir and Tc as a cold reservoir to eliminate any losses.

It is a ratio of Amount of work done by the Heat engine to the Amount of  heat required by the heat engine.

$\mathbf{\eta = \frac{Net\ work\ done\ by\ Heat\ engine }{heat\ absorbed\ by\ heat\ engine}}$

$\eta = \frac{Q_{h}- Q_{c}}{Q_{h}}$

$\eta =1- \frac{ Q_{c}}{Q_{h}}$

$\eta =1- \frac{mRT_{c}\left ( ln\frac{V_{3}}{V_{4}} \right )}{ mRT_{h}\left ( ln\frac{V_{2}}{V_{1}} \right )}$

As from above equation we know,

$\left [ \frac{T_{1}}{T_{4}} \right ] =\left [ \frac{V_{4}}{V_{1}} \right ]^{\gamma -1}$

&

$\left [ \frac{T_{2}}{T_{3}} \right ] =\left [ \frac{V_{3}}{V_{2}} \right ]^{\gamma -1}$

but
$\left T_1 = T_2 = T_h$
$\left T_3 = T_4 = T_c$

$\frac{V_{2}}{V_{1}} = \frac{V_{3}}{V_{4}}$

$\eta =1- \frac{T_{c}}{T_{h}}$

We can get an efficiency of 100% if we get to reject heat at 0 k (Tc = 0)

Carnot holds a maximum efficiency of all the engines performing under the same thermal reservoir as Carnot cycle work reversible, making assumptions of eliminating all the losses and making cycle a frictionless cycle, which is never possible in practice.

Hence all practical cycles will have efficiency less than Carnot efficiency.

## Reverse carnot cycle | the reversed carnot cycle | reversed carnot refrigeration cycle

Reverse Carnot cycle:

As all the processes carried out in the Carnot cycle are reversible, We can make it work in a reverse manner, i.e., to take heat from the lower temperature body and dumped to a higher temperature body, making it a refrigeration cycle.

.

In this process, the air is expanded, temperature is reduced to Tc, keeping entropy constant and with no heat interaction.

That is no change in entropy, and the system is insulated

Process 2-3: Isothermal expansion

In this process, the air is expanded with constant temperature while gaining heat. The heat is gain from the Heat sink at low temperature. Heat addition takes place while keep temperature(Tc) is kept constant.

In this process, the air is compressed, rising the temperature to Th, keeping entropy constant and no heat interaction.

That is no change in entropy, and the system is insulated

Process 4-1: isothermal compression

In this process, the air is compressed with a constant temperature while losing heat. Heat is rejected to the hot reservoir. Heat rejection takes place while keep temperature(Th) is kept constant.

## Reverse carnot cycle efficiency

The efficiency of reversed Carnot cycle is termed as Coefficient of performance.

COP is defined as the ratio of the desired output to the energy supplied.

$COP = \frac{Desired\ Output}{Energy\ Supplied}$

## Carnot refrigeration cycle| carnot refrigeration cycle efficiency | coefficient of performance carnot refrigeration cycle | carnot cycle refrigerator efficiency

The refrigeration cycle works on reversed Carnot cycle. The main objective of this cycle is to reduce the temperature of the heat source/ hot reservoir.

$COP = \frac{Desired\ Output}{Energy\ Supplied}=\frac{Q_{c}}{W^{_{net}}}$

$COP =\frac{Q_c}{Q_h-Q_c}=\frac{Q_c}{Q_h}-1$

Application: Air- conditioning, refrigeration system

## Carnot cycle heat pump

The heat pump works on reversed Carnot cycle. The main objective of the Heat pump is to transmit heat from one body to another, most from lower temperature body to higher temperature body with the help of supplied work.

$COP = \frac{Desired\ Output}{Energy\ Supplied}=\frac{Q_{c}}{W^{_{net}}}$

$COP = \frac{Desired\ Output}{Energy\ Supplied}=\frac{Q_{h}}{W^{_{net}}}$

$COP =\frac{Q_h}{Q_h-Q_c}=1-\frac{Q_h}{Q_c}$

$COP_{HP}=COP_{REF}+1$

Comparison:

## Carnot cycle irreversible

When the Carnot cycle runs on Adiabatic and not on reversible adiabatic, it comes under the category of irreversible Carnot cycle.

Entropy is not maintained constant in Process 2-3 and 4-1, (ds is not equal to zero)

as shown below:

Work produce under irreversible cycle is comparatively less than reversible Carnot cycle

Hence, the Efficiency of the irreversible Carnot cycle is less than the reversible Carnot cycle.

## Why Carnot cycle is reversible

According to Carnot, the Carnot cycle is a theoretical cycle that provides maximum efficiency. To get this maximum efficiency, we must eliminate all the losses and consider the system reversible.

If we consider any losses, the cycle will fall under the irreversible category and would not provide maximum efficiency.

## Carnot cycle volume ratio

$\left [ \frac{T_{1}}{T_{4}} \right ] =\left [ \frac{V_{4}}{V_{1}} \right ]^{\gamma -1}$
&

$\left [ \frac{T_{2}}{T_{3}} \right ] =\left [ \frac{V_{3}}{V_{2}} \right ]^{\gamma -1}$

but
$\left T_1 = T_2 = T_h$

$\left T_3 = T_4 = T_c$

$\frac{V_{2}}{V_{1}} = \frac{V_{3}}{V_{4}}$

Hence the volume ratio is maintain constant.

• Carnot cycle is an ideal cycle that gives maximum efficiency among all the cycle available.
• Carnot cycle helps in designing the actual Engine to get maximum output.
• It helps to decide the possibility of any cycle to build. As long as the Engine maintains efficiency less than Carnot, the Engine is possible; otherwise, it is not.

• It is impossible to supply heat and reject the heat at a constant temperature without phase change in the working material.
• It is impossible to construct a reciprocating heat engine to travel a piston at a very slow speed from the beginning of the expansion to the middle to satisfy isothermal expansion and then very rapid to help the reversible adiabatic process.

## Why Carnot cycle is not used in power plant

Carnot cycle has isothermal to adiabatic transmission. Now to carry out isothermal, we have to either make the process very slow or deal with phase change. Next is reversible adiabatic, which must be carried out quickly to avoid heat interaction.

Hence making the system difficult to construct as the half-cycle run very slow and the other half run very fast.

## carnot cycle application | carnot cycle example | application of carnot cycle in daily life

Thermal devices like

• heat pump: to supply heat
• Refrigerator: to produce cooling effect by removal of heat
• Steam turbine: to produce power i.e. thermal energy to mechanical energy.
• Combustion engines: to produce power i.e. thermal energy to mechanical energy.

## Carnot vapor cycle | carnot vapour cycle

In Carnot vapor cycle steam is working fluid

Its impracticalities:

1) It is not difficult to add or reject at constant temperature from two phase system, since maintaining it at constant temperature will fix up the temperature at saturation value. But limiting the heat rejection or absorption process to the mixed phase fluid will affect the thermal efficiency of the cycle.

2) The reversible adiabatic expansion process can be achieved by a well-designed turbine. But, the quality of the steam will reduce during this process. This is not be favorable as turbines cannot handle steam having more than 10% of liquid.

3) The reversible adiabatic compression process involves the compression of a liquid – vapour mixture to a saturated liquid. It is difficult to control the condensation process so precisely to achieve state 4. It is not possible to design a compressor that will handle mixed phase.

## carnot cycle questions | carnot cycle problems | carnot cycle example problems

Q1.) Cyclic heat engine operators between source at 900 K and sink at 380 K. a) what will be the efficiency? b) what will be heat rejection per KW net ouput of the engine?

Ans = given: $T_h = 900\ k$ and $T_c = 380\ k$

$efficiency =1- \frac{T_{c}}{T_{h}}$

$\eta =1- \frac{380}{900}$

$\eta =0.5777=55.77$ %

b) Heat reject (Qc) per KW net output

$\eta =\frac{W_{net}}{Q_h}$

$Q_h=\frac{W_{net}}{\eta }=\frac{1}{0.5777}=1.731\ KW$

$Q_c=Q_h-W_{net}=1.731-1=0.731\ KW$

Heat reject per KW net output = 0.731 KW

Q2.) Carnot engine working at 40% efficiency with heat sink at 360 K. what will be temperature of heat source? If efficiency of the engine is increased to 55%, what will be the effect on temperature of heat source?

Ans = given : $\eta = 0.4,\ T_c=360\ K$

$\eta =1- \frac{T_{c}}{T_{h}}$

$0.4 =1- \frac{360}{T_{h}}$

$T_h=600\ K$

If $\eta = 0.55$

$0.55 =1- \frac{360}{T_{h}}$

$T_h=800\ K$

Q3.) A Carnot engine working with 1.5 kJ of heat at 360 K, and rejecting 420 J of heat. What is the temperature at the sink?

Ans = given: Qh=1500 J, Th= 360 K , Qc= 420 J

$\eta =1- \frac{T_{c}}{T_{h}}=1- \frac{Q_{c}}{Q_{h}}$

$\frac{T_{c}}{T_{h}}=\frac{Q_{c}}{Q_{h}}$

$\frac{T_{c}}{360}=\frac{420}{1500}$

$T_{c}=\frac{420}{1500}*360$

$T_{c}=100.8\ K$

## What is a practical application of a Carnot cycle

• heat pump: to supply heat
• Refrigerator: to produce cooling effect by removal of heat
• Steam turbine: to produce power i.e. thermal energy to mechanical energy.
• Combustion engines: to produce power i.e. thermal energy to mechanical energy.

## carnot cycle vs stirling cycle

Stirling, the Carnot cycle’s isentropic compression and isentropic expansion process are substituted by a constant volume regeneration process. The other two methods are the same as the Carnot cycle it isothermal heat addition and rejection.

## What is the difference between a Carnot cycle and a reversed Carnot cycle

Simple carnot cycle works as power developing while reversed carnot work as power consuming.

Carnot cycle is used to design heat engine, while reversed cycle is used to design Heat pump and refrigeration system.

## Why carnot cycle is more efficient than any other ideal cycles like otto diesel brayton ideal VCR

Carnot cycle work between two thermal reservoirs (Th & Tc), and its efficiency depends only on this temperature and doesn’t depend on the fluid type. That is Carnot’s cycle efficiency is fluid independent.

Carnot holds a maximum efficiency of all the engines performing under the same thermal reservoir as Carnot cycle work reversible, making assumptions of eliminating all the losses and making cycle a frictionless cycle, which is never possible in practice.

## What is the net change in entropy during a Carnot cycle

Net change in entropy during a Carnot cycle is zero.

## why carnot cycle is not possible

Carnot cycle has isothermal to adiabatic transmission. Now to carry out isothermal, we have to either make the process very slow or deal with phase change.

Next is reversible adiabatic, which must be carried out quickly to avoid heat interaction.

Hence making the system difficult to construct as the half-cycle run very slow and the other half run very fast.

## why is the carnot cycle the most efficient

Carnot cycle work between two thermal reservoirs (Th & Tc), and its efficiency depends only on this temperature and doesn’t depend on the fluid type. That is Carnot’s cycle efficiency is fluid independent.

Carnot holds a maximum efficiency of all the engines performing under the same thermal reservoir as Carnot cycle work reversible, making assumptions of eliminating all the losses and making cycle a frictionless cycle, which is never possible in practice.

## Why does the Carnot cycle involve only the isothermal and adiabatic process and not other processes like isochoric or isobaric

The main aim of Carnot Cycle is to achieve maximum efficiency, which leads to make system reversible, so to make system reversible no heat interaction process should me maintain, i.e adiabatic process.

And to get maximum work output we use Isothermal process.

## How is the Carnot cycle related to a Stirling cycle?

Stirling, the Carnot cycle’s isentropic compression and isentropic expansion process are substituted by a constant volume regeneration process. The other two methods are the same as the Carnot cycle it isothermal heat addition and rejection.

## What will happen with efficiency of two Carnot engine works with same source and sink?

Efficiency will be the same, as Carnot cycle efficiency is only dependent on the temperature of the source and sink.

## Combination of Carnot cycle and Carnot refrigerator

The work output of Carnot heat engine supplied as work input for Carnot refrigeration system.

## Is it necessary that refrigerators should only work on Carnot cycle?

To get the maximum Coefficient of performance (COP), theoretically we net refrigeration cycle to work on Carnot.

## The temperature of two reservoirs of a Carnot engine are increased by same amount How will be the efficiency be affected?

The increase in temperature of both reservoirs in same will tend to decrease in efficiency

## Uses of stand in Carnot cycle?

The stand is used to carry out an adiabatic process. It is made up of non-conduction material.

## Important results for Carnot engine cycle?

Any number of engines working under the Carnot principle and having the same source and sink will have the same efficiency.

## Terminal of Carnot engine?

Carnot engine will consist of: Hot reservoirCold sink Insulating stand.

## Definition of insulating stand which is one of the part of Carnot’s engine?

The stand is used to carry out an adiabatic process, and it is made up of non-conduction material.