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

1. Advanced Transport Phenomena Module 3 Lecture 10 Constitutive Laws: Energy & Mass Transfer Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

2. VISCOUS LIQUID SOLUTIONS, TURBULENT VISCOSITY m intrinsic viscosity of (non-turbulent) fluid mt  turbulent contribution; more dependent on local condition of turbulence than on nature of fluid

3. ENERGY DIFFUSION FLUX VS TEMPERATURE GRADIENT • Fourier’s Heat Flux Law: • For energy diffusion (conduction) in pure isotropic solids • Vector equation, equivalent to 3 scalar components in, say, cylindrical polar coordinates:

4. ENERGY DIFFUSION FLUX VS TEMPERATURE GRADIENT • k  local thermal conductivity • Non-isotropic materials => k is a tensor (vector operator)

5. SPECIES DIFFUSION CONTRIBUTION TO ENERGY FLUX • In multi-component systems (e.g., reacting gas mixtures), each diffusing species also transports energy in accordance with its enthalpy, hi • Hence, Fourier’s Law must be generalized:

6. SPECIES DIFFUSION CONTRIBUTION TO ENERGY FLUX • Consistent with requirement of locally positive entropy production for k > 0, irrespective of sign of grad T • Radiative energy transport (“action at a distance”) cannot be treated as a diffusion process, must be dealt with separately.

7. ENTROPIC ASPECTS • Energy diffusion (conduction) contributes additively to local rate of entropy production: • Quadratic in gradient of relevant local field density • Positive for any flux direction

8. ENTROPIC ASPECTS • In the absence of multi-component species diffusion, entropy diffusion flux vector is given by: • Entropy flows by diffusion as well as convection!

9. THERMAL CONDUCTIVITY COEFFICIENT • Experimentally obtained by matching results of steady-state or transient heat-diffusion experiments with predictions based on energy conservation laws & constitutive relations (in the absence of convection) • Unit of k: W/ (m K) • , thermal diffusivity; m2/s • k has modest temperature dependence

10. k FROM KINETIC THEORY OF GASES • Chapman – Enskog - Herschfelder Expression:  viscosity  molar specific heat R  universal gas constant

11. k FROM KINETIC THEORY OF GASES Mixture: cube-root law

12. CORRESPONDING STATES CORRELATION FOR THERMAL CONDUCTIVITY OF SIMPLE FLUIDS

13. THERMAL CONDUCTIVITY OF LIQUID SOLUTIONS, TURBULENT FLUIDS • No simple relations for thermal conductivity of liquid solutions • Greater dependence on direct experimental data • Gases & liquids in turbulent motion display augmented thermal conductivities

14. THERMAL CONDUCTIVITY OF LIQUID SOLUTIONS, TURBULENT FLUIDS k  intrinsic thermal conductivity of quiescent fluid kt  turbulent contribution; more dependent on local condition of turbulence than on nature of fluid

15. EQUIVALENCE OF THERMAL & MOMENTUM DIFFUSIVITIES Due to additional terms chemically reacting mixtures in LTCE also exhibit augmented thermal conductivities.

16. MASS DIFFUSION FLUX VS COMPOSITION GRADIENT • Fick’s diffusion-flux law for chemical species: • In pure, isothermal, isotropic materials, species mass diffusion is linearly proportional to local concentration gradient • Directed “down” the gradient

17. MASS DIFFUSION FLUX VS COMPOSITION GRADIENT where is local mass fraction of species i Di = Fick diffusion coefficient (scalar diffusivity) for species i transport in prevailing mixture • Valid for trace constituent i, and • When mixture has only two components (N = 2)

18. OTHER CONTRIBUTIONS TO MULTI-COMPONENT DIFFUSION • Other forces, such as pressure & temperature gradients • - grad p, • - grad (lnT), etc. • Interspecies “drag” or “coupling”, i.e., influence on flux of species i due to fluxes (hence, composition gradients) of other species • - grad , where j ≠ i

19. CHEMICAL ELEMENT DIFFUSION FLUXES Example: local diffusional flux of element oxygen in a reacting multi-component gas mixture

20. ENTROPY PRODUCTION & DIFFUSION ASSOCIATED WITH CHEMICAL SPECIES DIFFUSION General form of driving force for chemical species diffusion: where  chemical potential, dependent on mixture composition via “activity” ai:

21. ENTROPY PRODUCTION & DIFFUSION ASSOCIATED WITH CHEMICAL SPECIES DIFFUSION and gradT,p  spatial gradient, holding T & p constant

22. ENTROPY PRODUCTION & DIFFUSION ASSOCIATED WITH CHEMICAL SPECIES DIFFUSION General form of Multi-component Diffusion Flux Law where  scalar coefficients, directly measurable Reciprocity relation (L Onsager):

23. DIFFUSIONAL FLUX OF ENTROPY • For the case of multi-component species diffusion in a thermodynamically ideal solution (ai = xi): Each bracketed quantity = , partial specific entropy of chemical species i, such that

24. DIFFUSIONAL FLUX OF ENTROPY Convective flux of entropy: Mixing entropy contributions (  origin of minimum work required to separate mixtures into their pure constituents

25. SOLUTE DIFFUSIVITIES IN GASES, LIQUIDS, SOLIDS– REAL & EFFECTIVE • Di,eff effective mass diffusivity of species i in prevailing medium • May be a tensor for solute diffusion in: • Anisotropic solids (e.g., single crystals, layered materials) • Anisotropic fluids (e.g., turbulent shear flow)

26. SOLUTE DIFFUSIVITIES IN GASES, LIQUIDS, SOLIDS– REAL & EFFECTIVE • In such cases, diffusion is not “down concentration gradient”, but skewed wrt –grad • Can often be treated as single scalar coefficient, valid in any direction

27. DILUTE SOLUTE DIFFUSION IN LOW-DENSITY GASES and yj mole fraction of species j yi<< 1 Di not very temperature-sensitive, varies as Tn, n ≥ 3/2, ≈ 1.8

28. BINARY INTERACTION PARAMETERS

29. MOMENTUM – MASS – ENERGY ANALOGY • For mixtures of similar gases, Di is always of same order of magnitude as momentum diffusivity, (kinematic viscosity) and energy diffusivity, • Reason: for gases, mechanisms of mass, momentum and energy transfer are identical • viz., random molecular motion between adjacent fluid layers

30. MOMENTUM – MASS – ENERGY ANALOGY Both dimensionless ratios are near unity for such mixtures. Sci can be >> 1 for solutes in liquids, aerosols in a gas

31. DILUTE SOLUTE IN LIQUIDS & DENSE VAPORS • Di estimated using a fluid-dynamics approach • Each solute molecule viewed as drifting in the host viscous fluid in response to • Net force associated with gradient in its partial pressure

32. DILUTE SOLUTE IN LIQUIDS & DENSE VAPORS Stokes-Einstein Equation:  effective molecular diameter of solute molecule i  Newtonian viscosity of host solvent Also applies to Brownian diffusion of particles in a gas, when

33. SOLUTE DIFFUSION IN ORDERED SOLIDS Di calculated from net flux of solute atoms jumping between interstitial sites in the lattice  energy barrier encountered in moving an atom of solute i from one interstitial site to another

34. SOLUTE DIFFUSION THROUGH FLUID IN PORES • Interconnected pores of a solid porous structure where solid itself is impervious to solute • Solute mfp << pore diameter => Di,eff < Di-fluid • Reduction depends on pore volume fraction,

35. SOLUTE DIFFUSION THROUGH FLUID IN PORES Denominator  correction for “tortuosity” (variable direction & variable effective dia of pores) • usually determined experimentally • Can be computed theoretically for model porous materials • e.g., for impermeable spheres, = 1 + 0.5 (1- )

36. SOLUTE DIFFUSION THROUGH FLUID IN PORES • When solute mfp > mean pore diameter: • e.g., gas diffusion through microporous solid media at atmospheric pressure • Solute rattles down each pore by successive collisions with pore walls • For a single straight cylindrical pore (Knudsen, 1909): (pore diameter plays role of solute mfp)

37. SOLUTE DIFFUSION THROUGH FLUID IN PORES • For Knudsen diffusion in a porous solid, • Independent of pressure when fluid is an ideal gas • Interpolation formula, rigorous for a dilute gaseous species at any mfp/ pore size combo:

38. SOLUTE DIFFUSION THROUGH FLUID IN PORES Widely used to describe gas diffusion through porous solids (e.g., catalyst support materials, coal char, natural adsorbents, etc.)

39. SOLUTE DIFFUSION IN TURBULENT FLUID FLOW • Effective diffusivity, Di,t, unrelated to molecular diffusivity, but closely related to prevailing momentum diffusivity, , in local flow = number near unity (turbulent Schmidt number) • e.g., tracer dispersion measurements near centerline of ducts containing a Newtonian fluid in turbulent flow ( > 2,000) reveal that:

40. SOLUTE DIFFUSION IN TURBULENT FLUID FLOW Peeff(Re)  weak function of Re, 250-1000

41. SOLUTE DIFFUSION THROUGH FIXED BED OF GRANULAR MATERIAL • Similar to turbulent flow in a homogeneous medium • Deff nearly proportional to product of average interstitial velocity, ui, and particle size, dp • In packed cylindrical duct with Rebed > 100:

42. SOLUTE DIFFUSION THROUGH FIXED BED OF GRANULAR MATERIAL • Peclet numbers weakly dependent on bed Reynolds number, near 10 & 2, resp. • Time-averaged solute mixing, apparently anisotropic, much more rapid than expected based on molecular motions alone