entropy

Prepared By :Aakash Singh

Enrollment No. : 150410119112Mechanical -2 – C

Sardar Vallabhbhai Patel Institute of Technology, Vasad

Entropy

Content :Inequality Of Clausius

Entropy Change For Open System

Reversible & Irreversible Processes

Inequality Of Clausius

HistoryThe Clausius Theorem is a mathematical

explanation of the Second Law of Thermodynamics.

Also referred to as the “Inequality of Clausius”, the theorem was developed by Rudolf Clausius who intended to explain the relationship between the heat flow in a system and the entropy of the system and its surroundings.

The Clausius Theorem was first published in 1862 in Clausius’ sixth memoir, “On the Application of the Theorem of the Equivalence of Transformations to Interior Work”. 1

Inequality of Clausius“ When a system undergoes a complete cyclic

process, the integral of around the cycle is less than zero.”

Mathematically : ( ) ≤ 0δQ is energy flow into the system due to heating

and T being absolute temperature of the body when that energy is absorbed.

The following equation must be found true for any cyclical process that is possible, reversible or not.

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ProofConsider a reversible engine R and irreversible engine I

working between two thermal reservoirs at temperatures TH and TL.

Efficiency of reversible engine is : where QH = heat added, QL = heat rejected.

Efficiency of irreversible engine is : We know that efficiency of reversible engine is more than

that of irreversible engine under same temperature limit.

∴ ηR > ηI ∴ ( )R > (

)I

∴ ( )R > ( )I ( for reversible ∵engines )

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Proof ∴ ( ) < ( )I

∴ ( )I < ( )I

∴ ( )I - ( )I < 0We know that, heat added (Q H ) should be positive and

heat rejected (Q L ) should be negative.

∴ ( )I - ( - )I < 04

Proof ∴ ( )I + ( )I < 0Considering complete original irreversible cycle :

∴

∴ ∮ ( ) < 0 for an irreversible cycle.According to Clausius Theorem ∮ = 0 for

reversible cycle.Combining results for reversible and irreversible cycle, we

get : This expression is known as Clausius Inequality.

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Entropy Change For

Open System

Entropy Change For Open SystemA closed system involves

no mass flow across its boundaries, and its entropy change is simply the difference between the initial and final entropies of the system.

The entropy change of a closed system is due to the heat transfer.

In an open system, as compared with closed system, there is additional change of entropy due to the mass crossing the boundaries of the system.

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Entropy Change For Open SystemThe general entropy

balance equation is :

[Rate of change of C.V.] = [Rate of entropy transfer with heat] + [Rate of entropy transport with mass]

To = temperature of surroundingsSi = specific entropy of the inletSo = specific entropy of the outletdmi = mass entering the systemdmo = mass leaving the system

2

Entropy Change For Open SystemThe small change of entropy of the system during a small

interval is given by :For reversible process

In above equation, entropy flow into the system is considered positive and entropy out-flow is considered negative.

This equation is applicable to reversible process in which the heat interactions and mass transport to and from the system is accomplished reversibly.

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Entropy Change For Open System

For Irreversible process -

For Reversible &

Irreversible - Process

Rate of Entropy Change -

The equality sign is applicable to Reversible Process and the inequality sign is applicable to the irreversible process.

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Entropy Change For Open SystemIn case of steady state, steady flow process, the time rate

of entropy change of system is Zero and the time rate of the mass entering is equal to that of leaving system,

So equation become

or

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Entropy Change For Open SystemFor Adiabatic Steady Flow Process,

If the process is Reversible Adiabatic steady flow ,

then

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Reversible &

Irreversible Process

Reversible ProcessA reversible process is

defined as :

“A reversible process is a process whose direction can be "reversed" by inducing infinitesimal changes to some property of the system via its surroundings, while not increasing entropy.”

Or“A process that can be

reversed without leaving any change on the surroundings.”

1

Reversible ProcessA reversible process passes

through a continuous series of equilibrium states.

It can be stopped at any stage and reversed so that the system and surroundings are exactly restored to their initial states.

Consider the system in fig. The process take place from state 1 to state 2 by following path 1-2.

If process is reversed, path 2-1 will be followed and system will reach its initial state. So the process 1-2 is called Reversible Process. 2

Reversible ProcessConsider expansion of a gas as shown in figure.The expansion of the gas takes place by removing infinitesimal

weights slowly from the piston one by one, therefore process passes through equilibrium states and tending to reversible process.

The gas can be brought back by compression after putting weights on the piston.

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Reversible ProcessSome of the processes that can be

idealized as reversible process are : Frictionless relative motion Expansion and compression of

spring Frictionless adiabatic expansion or

compression of fluid Isothermal Expansion or

compression Elastic stretching of a solid Electrolysis process

A reversible process produces the maximum work in engines and requires minimum work in devices such as heat pumps 4

Irreversible Process“A process that is not reversible

is called an Irreversible Process.”In irreversible process, system

passes through a series of non-equilibrium states.

It is difficult to locate properties on property diagram as they don’t have a unique value.

When irreversible process is made to proceed in backward direction, it does not reach its original state.

The system reaches a new state.Irreversible processes are

usually represented by dotted lines. 1

Irreversible ProcessThe factors that cause a

process to be Irreversible are :i. Frictionii. Free Expansioniii. Mixing of two gasesiv. Heat transfer between

finite temperature difference

v. Electric resistancevi. Inelastic deformationvii. Chemical reactions

The presence of any of these effects makes a process irreversible. 2

Irreversible ProcessExamples of irreversible

processes are :1. Relative motion with friction2. Combustion3. Diffusion of gases : mixing of

dissimilar gases4. Chemical reactions5. Free expansion and throttling

process6. Plastic deformation7. Electricity flow through a

resistance.

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entropy

Engineering