The T-dS Equations & Diagrams

1. The T-dS Equations & Diagrams Meeting 9Section 4-3 & 4-4

2. Second Law of Thermodynamics This can be viewed as a mathematical statement of the second law (for a closed system). Entropy is a non-conserved property!

3. Entropy Units are s = S/m : intensive property

4. The Entropy Change Between Two Specific States The entropy change between two specific states is the same whether the process is reversible or irreversible

5. We can write entropy change as an equality by adding a new term: entropy change entropy transfer due to heat transfer entropy generation or production Ps 0 ; not a property

6. Entropy generation • Ps  0 is an actual irreversible process. • Ps = 0 is a reversible process. • Ps  0 is an impossible process.

7. Entropy Production • Ps quantifies irreversibilities. The larger the irreversibilities, the greater the value of the entropy production, Ps. • A reversible process will have no entropy production, Ps = 0. • Psdepends upon the process, and thus it is not a property.

8. Entropy transfer and production Entropy production: > 0 when internal irrerversibilities are present; = 0 when no int irr are present; < 0 never. Entropy change of system as it goes from 1 to 2; can be + – or zero depending on the other two terms. Entropy transfer into/out of the system via heat transfer; can be + – or zero, depending on Q and its direction

9. Entropy transfer and production

10. Entropy Increase • Entropy change in a general system (ΔSsys) may be negative(due to heat transfer out of the system) • Entropy production cannot be negative

11. Entropy Transfer • Entropy change is caused by heat transfer, mass flow, and irreversibilities. • Heat transfer to a system increases the entropy, and heat transfer from a system decreases it. • The effect of irreversibilities is always to increase the entropy.

12. Some Remarks about Entropy • Process can occur in a certain direction only, not in any direction such that Ps0. • Entropy is a non-conserved property, and there is no such thing as conservation of entropy principle. The entropy of the universe is continuously increasing. • The performance of engineering systems is degraded by the presence of irreversibilities, and entropy generation is a measure of the magnitudes of the irreversibilities present during that process.

13. The Tds Equations

14. In previous slides, we developed a new property, entropy

15. The Entropy Change of a Pure Substance The entropy of a pure substance is determined from the tables, just as for any other property

16. It’s just like the other properties we’ve encountered in the tables It is tabulated just like u, v, and h. Also, And, for compressed or subcooled liquids,

17. T-s Diagram for Water

19. TEAMPLAY • Use the tables in your book • Find the entropy of water at 50 kPa and 500°C. Specify the units. • Find the entropy of water at 100°C and a quality of 50%. Specify the units. • Find the entropy of water at 1 MPa and 120°C. Specify the units.

20. T-s diagram Recall that the P-v diagram was very important in first law analysis, and that Work was the area under the curve.

21. For a T-s diagram Rearrange: Integrate:

22. d Heat Transfer for Internally Reversible Processes On a T-S diagram, the area under the process curve represents the heat transfer for internally reversible processes

23. Derivation of Tds equations: For a simple closed system: The work is given by: Substituting gives:

24. More derivation…. For a reversible process: Make the substitution for Q in the energy equation: Or on a per unit mass basis:

25. Tds Equations Entropy is a property. The Tds expression that we just derived expresses entropy in terms of other properties. The properties are independent of path….We can use the Tds equation we just derived to calculate the entropy change between any two states:

26. Tds Equations Starting with enthalpy h = u + Pv, it is possible to develop a second Tds equation:

27. Schematic of an h-s Diagram for Water

28. Tds equations • These two Tds relations have many uses in thermodynamics and serve as the starting point in developing entropy-change relations for processes.

29. Entropy change of an Pure substance • The entropy-change and isentropic relations for a process can be summarized as follows: 1. Pure substances: Any process:Δs = s2 - s1 [kJ/(kg-K)] Isentropic process:s2 = s1

30. Let’s look at the entropy change for an incompressible substance: We start with the first Tds equation: For incompressible substances, v  const, so dv = 0. We also know that Cv(T) = C(T), so we can write:

31. Entropy change of an incompressible substance Integrating If the specific heat does not vary with temperature:

32. Entropy Change for Incompressible Substance • The entropy-change and isentropic relations for a process can be summarized as follows: 2.Incompressible substances: Any process: Isentropic process:T2 = T1

33. Sample Problem Aluminum at 100oC is placed in a large, insulated tank having 10 kg of water at a temperature of 30oC. If the mass of the aluminum is 0.5 kg, find the final temperature of the aluminum and water, the entropy of the aluminum and the water, and the total entropy of the universe because of this process.

34. Draw Diagram Insulated wall water AL Closed system including aluminum and water Constant volume, adiabatic, no work done

35. Conservation of Energy Apply the first law But (T2)W = (T2)AL = T2at equilibrium

36. Solve for Temperature mW = 10 kg, CW = 4.177 kJ/kg.K mAL = 0.5 kg, CAL = 0.941 kJ/kg.K T2 = 303.8 K

37. Entropy Transfer Entropy change for water and aluminum

38. Entropy Generation Entropy production of the universe Sgen > 0 : irreversible process

39. The Entropy Change of an Ideal Gas

40. Let’s calculate the change in entropy for an ideal gas Start with 2ndTds equation Remember dh and v for an ideal gas? Substituting:

41. Change in entropy for an ideal gas Dividing through by T, Don’t forget,Cp = Cp(T)….. a function of temperature!

42. Entropy change of an ideal gas Integrating yields To evaluate entropy change, we’ll have to evaluate the integral:

43. Entropy change of an ideal gas • Similarly it can be shown from Tds = du + Pdv that

44. Entropy change of an ideal gas for constant specific heats: Approximation • Now, if the temperature range is so limited that Cp  constant (and Cv  constant),

45. Entropy change of an ideal gas for constant specific heats: Approximation • Note Tds = du + Pdv • Therefore

46. 0 0 Constant lines of v and P for Ideal gases diagrams T v = const. P = const. dT/ds s

47. Summary: Entropy change of an Pure substance 1. Pure substances: Any process: Δs = s2s1 [kJ/(kg-K)] (Table) Isentropic process: s2 = s1

48. Summary: Entropy Change for Incompressible Substance 2. Incompressible substances: Any process: Isentropic process:T2 = T1

49. Summary: Entropy Change for Ideal gases 3. Ideal gases: Constant specific heats (approximate treatment):

50. TEAMPLAY : Air is compressed in a piston-cylinder device from 90 kPa and 20oC to 400 kPa in a reversible isothermal process. Determine: (a) the entropy change of air, (b) the work done and (c) the removed heat. T v2 v1 2 1 293K Q<0 90kPa 293K 400kPa 293 K s Air is ideal gas, R = 287 Jkg-1K-1