Home Uncategorized Study of The Carnot Engine Step-by-Step - Working, Theorem, Efficiency

# Study of The Carnot Engine Step-by-Step – Working, Theorem, Efficiency

Do you know about Carnot engine ? If , no i will explain you more about it.

It is a theoretical engine that operates on the Carnot cycle. The basic model developed by Nicolas Leonard Sadi Carnot in 1824.

It gives as a result of the maximum possible efficiency that a heat engine during the change in process of heat into work.

## Carnot engine

### Q1. Describe the operation of a carnot’s engine. Calculate the efficiency of a carnot’s engine and explain why the efficiency of an irreversible engine is small.

Carnot engine :- It is an ideal reversible heat that operates between two temperature T1 (source) and T2 (sink). It operates through a series of two isothermal and two adiabatic processes called carnot cycle.

## Construction of Carnot engine

A carnot engine has the following main parts :

• Cylinder :- This main part of the engine has conducting base and insulating walls. It is fit with an insulating and frictionless piston.
• Source :- It is a heat reservoir at a higher temperature T1 from which the engine draws heat. It is assume that the source has an infinite heat capacity and by any amount of heat can be drawn from it without changing its temperature.
• Sink :- It is a heat reservoir at a lower temperature T2 to which any amount of heat can be rejected by the engine. It has also lattice heat capacity as well as any amount of heat con be add to it without changing its temperature.
• Working substance :- The working substance is an ideal gas contained in the cylinder.
• Insulated stand :- When the base of the cylinder is attach to the insulated stand , the working substance gets isolate from the atmosphere.

## Carnot Cycle

### The working substance is carried through a reversible cycle of the following four steps :-

#### Step 1. Isothermal expansion :

Place the cylinder on the source so that the gas acquires the temperature T1 source. The gas is allowed to expand by slow outward motion of the piston. The temperature of the gas falls. As the gas absorbs the required amount of heat from the source , it expands isothermally.

If Q1 heat is absorbed from the source and w1 work is done by the gas in isothermal expansion which takes its state from (P1 , V1 , T1) to (P2 , V2 , T2) , then

W1 = Q1 nRT1 in[ V2/V1] = area ABMKA

#### STEP 2. Adiabatic expansion :-

The gas is now placed on the insulating stand and allowed to expand slowly till its temperature falls to T2 .

if W2 work is done by the gas in the adiabatic expansion which takes its state from (P2 , V2 , T1) to (P3 , V3 , T2) , so

W2 = nR(T1 – T2)/ -1 = area BCNMB

#### Step 3. Isothermal compression :-

He gas is now placed in thermal contact with the sink at temperature T2. The gas is slowly compressed so that as heat is produced , it easily flows to the sink. The temperature of the gas remains constant at T2.

It Q2 heat is released by the gas to the sink or W3 work is done on the gas by the ambient in the isothermal compression which takes place its state from (P3 , V3 , T2) to (P4 , V4 , T2) , so

W3 = Q2 = nRT2 in (V3/V4) = area CNLDC

#### Step 4. Adiabatic compression :-

The cylinder is again place on the insulating stand. The gas is beyond compressed slowly till it initial state (P1 , V1 , T1).

else W4 is the work done in the adiabatic compression from (P4 , V4 , T2) to (P1 , V1 , T1) ,

W4 = nR(T1 – T2)/  – 1 = area DAKLD

Now work done by the gas per cycle.

Total work done by the gas = W1 + W2      (in steps 1 and 2)

Total work done on the gas  = W3 + W4      (in steps 3 and 4)

net work done by the gas in one complete cycle ,

W = W1 + W2 – (W3 + W4)

certain           W2 = W4

W = W1 – W3 = Q1 – Q2

eke ,    W = area ABMKA + area BCNMBA – area CNLDC –  area DAKLD

Or      W = area ABCDA

Rest in a carnot engine , the mechanical work done by the gas per cycle is numerically equal to  the area of the carnot cycle.

### Efficiency of carnot engine

It is defined as the ratio of the net work done per cycle by the engine to the amount of heat absorbed per cycle by the working substance from the source.

n = 1- nRT2 in (V3/V4) / nRT1 in (V2/V1)     N = 1 – T2/ T1

## The efficiency of a Carnot engine

• Depends upon the temperature of the source and the sink.
• dependent of the nature of the working substance.
• the same for all reversible engines working between the same two temperatures.
• Is directly proportional to the temperature difference (T1 – T2).
• Is always less either 100% since Q2 < Q1.
• The efficiency of a carnot engine will be unity or 100% if T=  or T2 = 0 K. As 0 K or infinite temperature cannot be realized, ergo a carnot  engine working on reversible cycle can not have  100%  efficiency.
• as T1= T2 , so n = 0. This means that the conversion of heat into mechanical work is impossible without having the source and sink at different temperatures.
• As Q2 / Q 1 = T2 / T1

if T2 = 0 K, then Q2 = 0.

Since T2 = 0 k can not be realised,  so Q2 = 0  is also not possible. This means that it is not possible to convert a whole of a heat energy absorbed from the source into mechanical work continuously, without rejecting a part of it to the sink.

### Q2. Can a carnot engine  be realised in practice ?

Non – practicability of carnot engine. Carnot engine is an ideal engine. It can not be realised in practice due to following reason :

1. It is difficult to realize source and sink of infinite thermal capacity.
2. The working substance should be an ideal gas. Although no real gas fulfills ideal gas behavior.
3. The cyclinder can not be provided perfect friction-less piston.
4. It is difficult to attain the conditions of reversibility for the processes of expansion and compression have to be carried out very slowly.

## Carnot Theorem

### Q3. State carnot theorem.  Prove that the efficiency of a reversible heat engine is maximum.

Ans. Carnot theorem :- It state that no engine working between two given temperature can have efficiency greater by that of the carnot engine  working between the same two temperature the efficiency of the carnot engine is independent in nature of working substance.

### Proof

Consider two engine –An irreversible engine I and a reversible engine R. The two engine are coupled that as I run forward , it drives R  backwards. As R words as a refrigerator driven by I.

The engine I absorbs heat Q1 from the source , perform work W and reject heat Q2 to the sink.

efficiency of engine I ,  ni = W/Q1 = Q1 – Q2/ Q1

The engine R absorbs heat Q2 from the sink , work W is done on it and it rejects heat Q1 to the source.

efficiency of engine RI nR = W/Q1 = Q1 – Q2

Suppose the engine I is more efficiency even R then

NI > nR    or w/Q1  >  W/Q2

Q1 <  Q’1    i.e. , Q’1 – Q1 is positive

The source losses heat Q1 to I and gain Q’1 from R.

net heat gained by the source per cycle.

= Q’1  – Q1

The sink gain heat (Q1 – w) from I and losses Q’2 to r

net heat lost by the sink per cycle

= Q’2 – (Q1 – w) = Q’2 – Q1 + W

= Q’1 – Q1 + (Q’1 – Q’2) = Q’1 – Q1

#### The engine IR  is a self

Acting machine which transfer heat (Q’1 – Q1) from the sink at lower temperature T2  to the source at higher temperature T1 , without any work being done by any external agency. This is against the second law of thermodynamics.

Indeed our assumption that I is more efficient than R is wrong. Rest no engine can have efficiency greater by that of the carnot engine. too , we can prove that a reversible engine with one working substance can not be more efficient that the one using sub working substance.

## FAQ

Q1. Why mechanical energy can be completely converted into heat energy except the whole of the heat energy can not be convert into mechanical energy ?

Ans. The whole of the mechanical energy can be absorbed by the molecules of the system in the form of their kinectic energy. This kinetic energy gets converted into heat. Yet the whole of the heat energy cannot be converted into work as a part of it is always retained by the system as its internal energy.

Q2. Can two isothermal curves intersect ?

Ans. No, if two isothermal intersect , tho this would mean that the pressure and volume of a gas the same at two different temperature. This is not possible.

Q3. What do you mean by Carnot engine ?

Ans. Carnot engine is a theoretical engine that operates on the Carnot cycle. The basic model developed by Nicolas Leonard Sadi Carnot in 1824.

Q4. Does Carnot engine exist ?

Ans. The Carnot engine does not exist in real life.It is purely theoretical device.

Q5. Who invented Carnot engine ?

Ans. The Carnot engine was discovered by French engineer / scientist Nicolas Leonard Sadi Carnot (1887-1934).

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