Entropy

This presentation gives you information om Clausius Statement, its proof, Entropy change for Open System and reversible and irreversible processes with simple explanation and day to day examples.

1. Prepared By : Aakash Singh Enrollment No. : 150410119112 Mechanical -2 – C Sardar Vallabhbhai Patel Institute of Technology, Vasad

2. Entropy

3. Content : Inequality Of Clausius Entropy Change For Open System Reversible & Irreversible Processes

4. Inequality Of Clausius

5. History The 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

6. Inequality of Clausius “ When a system undergoes a complete cyclic process, the integral of around the cycle is less than zero.” Mathematically : ( ) ≤ 0 δ Q  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. 2

7. Proof Consider a reversible engine R and irreversible engine I working between two thermal reservoirs at temperatures T H and T L . Efficiency of reversible engine is : where Q H = heat added, Q L = 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. ∴ η R > η I ∴ ( ) R > ( ) I ∴ ( ) R > ( ) I (∵ for reversible engines ) 3

8. Proof ∴ ( ) < ( ) I ∴ ( ) I < ( ) I ∴ ( ) I - ( ) I < We know that, heat added ( Q H ) should be positive and heat rejected ( Q L ) should be negative. ∴ ( ) I - ( - ) I < 4

9. Proof ∴ ( ) I + ( ) I < Considering complete original irreversible cycle : ∴ ∴ ∮ ( ) < for an irreversible cycle . According to Clausius Theorem ∮ = for reversible cycle . Combining results for reversible and irreversible cycle, we get : This expression is known as Clausius Inequality. 5

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12. Entropy Change For Open System

13. Entropy Change For Open System A 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. 1

14. Entropy Change For Open System The general entropy balance equation is : [Rate of change of C.V.] = [Rate of entropy transfer with heat] + [Rate of entropy transport with mass] T o = temperature of surroundings S i = specific entropy of the inlet S o = specific entropy of the outlet dm i = mass entering the system dm o = mass leaving the system 2

15. Entropy Change For Open System The 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. 3

16. 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. 4

17. Entropy Change For Open System In 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 5

18. Entropy Change For Open System For Adiabatic Steady Flow Process, If the process is Reversible Adiabatic steady flow , then 6

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20. Reversible & Irreversible Process

21. Reversible Process A 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

22. Reversible Process A 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

23. Reversible Process Consider 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. 3

24. Reversible Process Some 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

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26. 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

27. Irreversible Process The factors that cause a process to be Irreversible are : Friction Free Expansion Mixing of two gases Heat transfer between finite temperature difference Electric resistance Inelastic deformation Chemical reactions The presence of any of these effects makes a process irreversible. 2

28. Irreversible Process Examples of irreversible processes are : Relative motion with friction Combustion Diffusion of gases : mixing of dissimilar gases Chemical reactions Free expansion and throttling process Plastic deformation Electricity flow through a resistance. 3

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