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

1. Advanced Transport Phenomena Module 6 Lecture 26 Mass Transport: Diffusion with Chemical Reaction Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

2. QUASI-STEADY-STATE (QS) DIFFUSION OUTSIDE ISOLATED SPHERE • In completely quiescent case, diffusional mass transfer from/ to sphere occurs at a rate corresponding to Num = 2 • If Bm ≡ vwdm/D is not negligible, then: and • Results from radial outflow due to net mass-transfer flux across phase boundary

3. QUASI-STEADY-STATE (QS) DIFFUSION OUTSIDE ISOLATED SPHERE • vw may be established by physically blowing fluid through a porous solid sphere of same dia => Bm “blowing” parameter • vw is negative in condensation problems, so is Bm • Suction enhances Num

4. QUASI-STEADY-STATE (QS) DIFFUSION OUTSIDE ISOLATED SPHERE • Pew,m alternative blowing parameter, defined by: and • Equivalent to correction factor for “phoretic suction”

5. QUASI-STEADY-STATE (QS) DIFFUSION OUTSIDE ISOLATED SPHERE • Stefan-flow effect on Num very similar to phoresis effect, but with one significant difference: • Phoresis affects mass transfer of dilute species, but not heat transfer • Stefan flow affects both Nuh and Num in identical fashion, hence not an analogy-breaker • Corresponding blowing parameters:

6. QUASI-STEADY-STATE (QS) DIFFUSION OUTSIDE ISOLATED SPHERE and • Nuh same function of Bh (or Pew,h) & Pr, as Num is of Bm (or Pew,m) & Sc

7. QS EVAPORATION RATE OF ISOLATED DROPLET • Droplet of chemical substance A in hot gas • Energy diffusion from hotter gas supplies latent heat required for vaporization of droplet of size dp • Known: • Gas temperature, T∞ • Vapor mass fraction wA,∞ • Unknowns: • droplet evaporation rate • Vapor/ liquid interface conditions (wA,w, Tw)

8. QS EVAPORATION RATE OF ISOLATED DROPLET • Assumptions: • Vapor-liquid equilibrium (VLE) @ V/L interface • Liquid is pure (wA(l) = 1), surrounding gas insoluble in it • dp >> gas mean free path • Forced & natural convection negligible in gas • Variable thermophysical property effects in gas negligible

9. QS EVAPORATION RATE OF ISOLATED DROPLET • Assumptions: • Species A diffusion in vapor phase per Fick’s (pseudo-binary) law • No chemical reaction of species A in vapor phase • Recession velocity of droplet surface negligible compared to radial vapor velocity, vw, at V/L phase boundary

10. QS EVAPORATION RATE OF ISOLATED DROPLET • Dimensionless “blowing” (driving force)parameters: where wA,w = wA,eq(Tw; p)

11. QS EVAPORATION RATE OF ISOLATED DROPLET • Species A mass balance: • continuous, and, in each adjacent phase, given by: • SincewA(l)= 1, in the absence of phoresis: (applying total mass balance condition = 0)

12. QS EVAPORATION RATE OF ISOLATED DROPLET • Since , and We can relate Bm directly to mass fractions of A as:

13. QS EVAPORATION RATE OF ISOLATED DROPLET • Similarly, energy conservation condition at V/L interface leads to relation between Bh and T∞-Tw (neglecting work done by viscous stresses) where LA latent heat of vaporization

14. QS EVAPORATION RATE OF ISOLATED DROPLET • Heat Flux: • Mass Flux: • Relating the two:

15. QS EVAPORATION RATE OF ISOLATED DROPLET • Driving forces are related by: where Le = DA/ [k/rcp] = Lewis number • Yields equation for Tw • Solution yields Bh, Bm

16. QS EVAPORATION RATE OF ISOLATED DROPLET • Single droplet evaporation rate • Equating this to • We find that dp2 decreases linearly with time:

17. QS EVAPORATION RATE OF ISOLATED DROPLET • Setting dp = 0 yields characteristic droplet lifetime:

18. STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET • Catalyst impregnated throughout with porous pellets • To avoid having to separate catalyst from reaction product • Pellets are packed into “fixed bed” through which reactant is passed • Volume requirement of bed set by ability of reactants to diffuse in & products to escape • Core accessibility determined by pellet diameter, porosity & catalytic activity

19. STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET • Assumptions in continuum model of catalytic pellet: • SS diffusion & chemical reaction • Spherical symmetry • Perimeter-mean reactant A number density nA,w at R = Rp • Radially-uniform properties (DA,eff, k”’eff, r, …) • First-order irreversible pseudo-homogeneous reaction within pellet

20. STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET • SS nA(r) profile within pellet satisfies local species A mass-balance: since and

21. STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET • Then, nA(r) satisfies: Relevant boundary conditions: and

22. STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET • Applying species A mass balance to a “microsphere” of radius e, and taking the limit as e  0: which, for finite leads to:

23. STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET • Once nA(r) is found, catalyst utilization (or effectiveness) factor can be calculated as: or

24. STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET • By similitude analysis: and, therefore: where the Thiele modulus, f, is defined by: relevant Damkohler number; ratio of characteristic diffusion time (Rp2/DA,eff) to characteristic reaction time, (k”’eff)-1

25. STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET • c (z; f) normalized reactant-concentration variable, satisfies 2nd-order linear ODE: subject to split bc’s:

26. STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET Solution to this two-point BVP: or, explicitly: Catalyst-effectiveness factor is explicitly given by:

27. STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET Catalyst effectiveness factor for first-order chemical reaction in a porous solid sphere (adapted from Weisz and Hicks (1962))

28. STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET • Reaction only in a thin shell near outer perimeter of pellet • Alternative presentation of : based on dependence on • Independent of (unknown)

29. STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET Catalyst effectiveness factor vs experimentally observable (modified) Thiele modulus (adapted from Weisz and Hicks (1962))

30. STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET • Following additional parameters influencehcat

31. STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET Representative Parameter Values for Some Heterogeneous Catalytic Reactions (after Hlavacek et al (1969))

32. TRANSIENT MASS DIFFUSION: MASS TRANSFER (CONCENTRATION) BOUNDARY LAYER • Discussion for thermal BL applies here as well • Thermal BL “outruns” the MTBL: • D << a (Le << 1) for most solutes in condensed phases (especially metals) • Ratio holds for time-averaged penetration depth in periodic BC case as well

33. CONVECTIVE MASS TRANSFER IN LAMINAR- AND TURBULENT-FLOW SYSTEMS • Analogies to Energy Transfer: • When “analogy conditions” apply, heat-transfer equations can be applied to mass-transfer by substituting:

34. CONVECTIVE MASS TRANSFER IN LAMINAR- AND TURBULENT-FLOW SYSTEMS • Analogies to Energy Transfer: • Mass transfer of dilute species A in straight empty tube flow (by analogy): where

35. CONVECTIVE MASS TRANSFER IN LAMINAR- AND TURBULENT-FLOW SYSTEMS and

36. CONVECTIVE MASS TRANSFER IN LAMINAR- AND TURBULENT-FLOW SYSTEMS • Analogies to Energy Transfer: • Packed duct (by analogy): Since, in the absence of significant axial dispersion:

37. CONVECTIVE MASS TRANSFER IN LAMINAR- AND TURBULENT-FLOW SYSTEMS • Analogies to Energy Transfer: • Packed duct (by analogy): We find: where if 3 ≤ Rebed ≤ 104, 0.6 ≤ Sc, 0.48 ≤ e≤ 0.74 Quantity in square bracket = Bed Stanton number for mass transfer, Stm,bed

38. CONVECTIVE MASS TRANSFER IN LAMINAR- AND TURBULENT-FLOW SYSTEMS • Analogies to Energy Transfer: • Packed duct (by analogy): In terms of Stm,bed where, as defined earlier, (= 6(1-e)/dp) interfacial area per unit volume of bed

39. CONVECTIVE MASS TRANSFER IN LAMINAR- AND TURBULENT-FLOW SYSTEMS • Analogies to Energy Transfer: • Packed duct (by analogy): • Height of a transfer unit (HTU) is defined by: • HTU  bed depth characterizing exponential approach to mass-transfer equilibrium

40. CONVECTIVE MASS TRANSFER IN LAMINAR- AND TURBULENT-FLOW SYSTEMS • Analogies to Energy Transfer: • Packed duct (by analogy): In the case of single-phase fluid flow through a packed bed, HTU = (a”’Stm,bed)-1 • Widely used in design of heterogeneous catalytic-flow reactors and physical separators • No chemical reaction within fluid • Also to predict performance of fluidized-bed contactors, using e(Rebed) correlations