Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

1. Advanced Transport Phenomena Module 2 Lecture 6 Conservation Principles: Entropy Conservation Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

2. CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II • Problem Statement: • Continue with atmospheric-pressure combustor problem. • Calculate rate of energy extraction, , necessary to bring product gas to 1000K.

3. CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… • Solution Procedure: values will be needed to calculate which appears in the energy balance below:

4. CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… Apply M-Scopic Energy Balance to Calculate – When the total stress is split into – pl and T, energy conservation equation can be written in the useful form

5. CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… • The convective term now contains the enthalpy e + (p/r). If we now neglect: • KE/mass (v2/2) terms, • accum. term (ss) • term ( no volumetric heat addition ) • body force work

6. CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… the macroscopic energy balance equation then can be written : where and

7. CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… Neglecting

8. CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… To complete calculation of ,we therefore need where ideal gas mixture for each stream, and : Tabulated Tabulated

9. CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… Now is very close to therefore

10. CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… Finally, i.e., Or

11. CONSERVATION OF ENERGY: ILLUSTRATIVE EXERCISE II CONTD… Therefore

12. CONSERVATION OF ENTROPY (2ND LAW OF THERMODYNAMICS) • Inequality: irreversible phenomena (e.g., diffusive momentum transfer, energy transfer, mass transfer, chemical reactions) lead to entropy production (Ds > 0)

13. CONSERVATION OF ENTROPY (2ND LAW OF THERMODYNAMICS) CONTD… • Can be restated as entropy conservation equation:

14. CONSERVATION OF ENTROPY (2ND LAW OF THERMODYNAMICS) CONTD…. • js” = diffusion flux vector for entropy • = local volumetric rate of entropy production due to all irreversible processes occurring within fluid mixture

15. CONSERVATION OF ENTROPY • Integral conservation equation for Fixed CV: • Local PDE for differential CV:

16. CONSERVATION OF ENTROPY CONTD… • “Jump” condition for a “surface of discontinuity”:

17. CONSERVATION OF ENTROPY CONTD… • Inequalities: Or, locally:

18. CONSERVATION OF ENTROPY CONTD… This implies that for any fixed macroscopic region of space & at any instant:

19. CONSERVATION OF ENTROPY CONTD… • Local volumetric entropy production

20. CONSERVATION OF ENTROPY CONTD… • Steady-state => dV is minimum, i.e.: is a minimum compared to all other “eligible” steady-states subject to imposed boundary conditions.

21. CONSERVATION OF ENTROPY CONTD… • Uses: • Set important constraints on otherwise possible physicochemical processes (e.g., maximum work to separate a mixture, maximum possible efficiency of heat engines, etc.)

22. CONSERVATION OF ENTROPY CONTD… • Uses: • Provide basis for numerical solutions of non-equilibrium problems within the domain of “linear irreversible thermodynamics” (principle of minimum entropy production)

23. CONSERVATION OF ENTROPY CONTD… • Uses: • Guide selection of general constitutive laws governing diffusion (of species mass, momentum, energy) in non-equilibrium chemically-reacting mixtures

24. CONSERVATION OF ENTROPY CONTD… • Uses: • Pinpoint sources of entropy production and, hence, inefficiency in proposed or actual engineering devices • Provide insights useful in optimization of such devices

25. CONSERVATION OF ENTROPY CONTD… • Illustrative Exercise: In atmospheric combustor problem, calculate net convective outflow rate of entropy from the combustor– i.e., surface integral rs v . n dA.

26. CONSERVATION OF ENTROPY CONTD… • Solution Procedure: Calculation of (net outflow rate of entropy from M-scopic CV) Known ? ?

27. Note that, whereas stream 1 is pure CH4 for which obtainable from , say, JANAF Thermochemical Tables, Streams 2 and 3 are mixtures; hence, CONSERVATION OF ENTROPY CONTD… “Mixing entropy contribution”

28. CONSERVATION OF ENTROPY CONTD… where Equivalently , (Similarly , stream 2 is a mixture , and this affects calculation of S2.)

29. MATERIAL DERIVATIVE FORM OF CONSERVATION PDES Material derivative (D/Dt) of any function f (x,t) is defined as: Setting f = 1/r = (specific volume), we obtain MDF of total mass conservation equation:

30. INCOMPRESSIBLE FLUID Dr/Dt = 0 (rate of change of density of each moving fluid parcel vanishes) Then, div (v) = 0 (local condition on v(x,t)) = volumetric rate of fluid deformation

31. MDF OF SPECIES & ELEMENT CONSERVATION EQUATIONS Only an inflow by diffusion and/ or local chemical reaction can cause local species mass fraction wi (hence wi /mi) to change for each moving fluid parcel.

32. MDF OF SPECIES & ELEMENT CONSERVATION EQUATIONS CONTD… Chemical reactions are incapable of causing element mass fractions to change within each moving fluid parcel, hence along any streamline in steady flow

33. MDF OF SPECIES & ELEMENT CONSERVATION EQUATIONS CONTD… • Linear momentum conservation: • Set f = v (differences in contact stresses and/ or net body forces cause velocity changes (magnitude and/ or direction) of a moving fluid parcel)

34. MDF OF MOMENTUM, ENTERGY & ENTROPY CONSERVATION EQUATIONS • Energy conservation: • Set f = e + v2/2 • Entropy conservation: (f = s) 34