Entropy

1. Entropy Second Law Characteristic

2. Entropy • Clausius Inequality: • For internally reversible cycles:

3. Entropy • Entropy is defined as: dS = (δQ/T)int rev (kJ/K) • S is entropy per unit mass (kJ/kgK) • Entropy is a property of a state not a process • Change of entropy:

5. Entropy • Increase of Entropy Principle: • The entropy of an isolated system during a process always increases or, in the limiting case of a reversible process, remains constant. • dS ≥ (δQ/T) where T is the thermodynamic temperature at the boundary where δQ is transferred to the surroundings

6. Entropy • Process can occur in one direction only, not in any direction. Must proceed in direction of least increase in entropy. Sgen≥ 0 • Entropy is a non-conserving property. • Performance is degraded by irreversibilities, entropy generation is a measure of the magnitude of the irreversibilities during the process

7. Entropy • Sgen > 0 irreversible processes = 0 reversible processes < 0 impossible processes

8. Entropy • Entropy of a fixed mass can be changed by: • Heat transfer • Irreversibilities • If no change, Isentropic • Reversible, adiabatic process

9. Entropy Change of Pure Substance from tables

12. Isentropic Processes • Isentropic process; internally reversible, adiabatic, entropy remains constant Δs = 0 or s2 = s1

13. Entropy • Essentially Isentropic Processes • Pumps • Turbines • Nozzles • Diffusers

16. Temperature-entropy (T-s) diagrams Area under process curve on a T-s diagram equals heat transfer during an internally reversible process Property Diagrams

17. Isentropic processes are a vertical line on T-s diagrams Property Diagrams

18. Enthalpy-entropy diagram, h-s diagram, Mollier diagram Change in h is a measure of work Change in s is a measure of irreversibilities Property Diagrams

20. Entropy is: measure of molecular disorder, molecular randomness Entropy

21. Entropy • Boltzmann equation: S = k ln(p) where k =1.3806*10-23 J/K p = thermodynamic probability, number of possible microscopic states of system

22. Third Law of Thermodynamics • The entropy of a pure crystalline substance at absolute zero temperature is zero, since there is no uncertainty about the state of the molecules at that instant • Provides an absolute reference point for determining entropy

23. There is no entropy transfer associated with energy transferred as work Irreversibilities (friction) will case entropy increase Entropy

24. Work changed to heat increases entropy Entropy

25. During heat transfer net entropy increases Entropy

26. Entropy • To find the change in entropy, need to do the cycle integral of δQ/T. • If isothermal, only need the function for Q • If not isothermal, need functions for Q and T

27. Entropy

28. Entropy

29. Entropy • Can find entropy by integration of either equation • Need to know the relationship between du or dh and temperature • For ideal gases • du = cv dT • Or dh = cp dT • And Pv=RT

30. Entropy Changes of Liquids and Solids • Liquids and solids are incompressible • dv = 0 • Also cv= cp=c and du= c dT

31. Entropy Change of Ideal Gases • In the basic equation, substituting du = cv dT and P = RT/v • Substituting dh = c dT and v = RT/P

32. Entropy Change of Ideal Gases • Need the relationship between the specific heats and temperature • Assume constant specific heats, simpler integration, approximate analysis • Work with variable specific heats, use tables, exact analysis

33. Constant Specific Heats (Approximate Analysis)

34. Variable Specific Heats (Exact Analysis) • If temperature change is large • Specific heats are non-linear with temperature • Need accurate relationships • Calculate integrals with respect to reference entropy (at absolute zero)

35. Variable Specific Heats (Exact Analysis) • So we can find • And substituting into • Get

36. Variable Specific Heats (Exact Analysis)

40. Isentropic Processes of Ideal Gases • Process that has: • No change in entropy, Δs = 0 • Is internally reversible • Is adiabatic

41. Isentropic Processes of Ideal Gases (Approximate Analysis) • Assume constant specific heats, so: • Where k = cp /cv

42. Isentropic Processes of Ideal Gases (Approximate Analysis) • These equations can be stated as • Since the specific heat ratio k, varies with temperature, the average k should be used

43. Isentropic Processes of Ideal Gases (Exact Analysis) • Set up to use tables • Working with • Establish new dimensionless quantities from the equations

44. Isentropic Processes of Ideal Gases (Exact Analysis) • The quantity exp(s°/R) is defined as relative pressure Pr • P is a function of T only, so can be tabulated against T.

45. Isentropic Processes of Ideal Gases (Exact Analysis) • So using table A-17 for air

46. Isentropic Processes of Ideal Gases (Exact Analysis) • When specific volume ratios are given instead of pressure • The quantity T/Pr is a function of T only, defined as relative specific volume vr

50. Reversible Steady-Flow Work • Work done during process depends on: • Properties at the end states • Path between the end states • Quasi-equilibrium work interactions • Max work output • Min work input