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

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

 v w may be established by physically blowing fluid through a porous solid sphere of same dia => B m  “blowing” parameter  v w is negative in condensation problems, so is B m  Suction enhances Nu m 3 QUASI-STEADY-STATE (QS) DIFFUSION OUTSIDE ISOLATED SPHERE

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

 Stefan-flow effect on Nu m 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 Nu h and Nu m in identical fashion, hence not an analogy-breaker  Corresponding blowing parameters: 5 QUASI-STEADY-STATE (QS) DIFFUSION OUTSIDE ISOLATED SPHERE

and  Nu h same function of B h (or Pe w,h ) & Pr, as Nu m is of B m (or Pe w,m ) & Sc 6 QUASI-STEADY-STATE (QS) DIFFUSION OUTSIDE ISOLATED SPHERE

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 d p  Known:  Gas temperature, T ∞  Vapor mass fraction  A,∞  Unknowns:  droplet evaporation rate  Vapor/ liquid interface conditions (  A,w, T w ) 7

 Assumptions:  Vapor-liquid equilibrium V/L interface  Liquid is pure (  A (l) = 1), surrounding gas insoluble in it  d p >> gas mean free path  Forced & natural convection negligible in gas  Variable thermophysical property effects in gas negligible 8 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, v w, at V/L phase boundary 9 QS EVAPORATION RATE OF ISOLATED DROPLET

 Dimensionless “blowing” (driving force)parameters : where  A,w =  A,eq (T w ; p) 10 QS EVAPORATION RATE OF ISOLATED DROPLET

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

 Since, and We can relate B m directly to mass fractions of A as: 12 QS EVAPORATION RATE OF ISOLATED DROPLET

 Similarly, energy conservation condition at V/L interface leads to relation between B h and T ∞ -T w (neglecting work done by viscous stresses) where L A  latent heat of vaporization 13 QS EVAPORATION RATE OF ISOLATED DROPLET

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

 Driving forces are related by: where Le = D A / [ k/  c p ] = Lewis number  Yields equation for T w  Solution yields B h, B m 15 QS EVAPORATION RATE OF ISOLATED DROPLET

 Single droplet evaporation rate  Equating this to  We find that d p 2 decreases linearly with time: 16 QS EVAPORATION RATE OF ISOLATED DROPLET

 Setting d p = 0 yields characteristic droplet lifetime: 17 QS EVAPORATION RATE OF ISOLATED DROPLET

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 18

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

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

 Then, n A (r) satisfies: Relevant boundary conditions: and 21 STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET

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

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

 By similitude analysis: and, therefore: where the Thiele modulus, , is defined by:  relevant Damkohler number; ratio of characteristic diffusion time (R p 2 /D A,eff ) to characteristic reaction time, (k”’ eff ) STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET

 c (  )  normalized reactant-concentration variable, satisfies 2 nd -order linear ODE : subject to split bc’s: 25 STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET

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

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))

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

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))

 Following additional parameters influence  cat 30 STEADY MASS DIFFUSION WITH SIMULTANEOUS CHEMICAL REACTION: CATALYST PELLET

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

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

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: 33

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

and 35 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: 36 CONVECTIVE MASS TRANSFER IN LAMINAR- AND TURBULENT-FLOW SYSTEMS

 Analogies to Energy Transfer:  Packed duct (by analogy): We find: where if 3 ≤ Re bed ≤ 10 4, 0.6 ≤ Sc, 0.48 ≤  ≤ 0.74 Quantity in square bracket = Bed Stanton number for mass transfer, St m,bed 37 CONVECTIVE MASS TRANSFER IN LAMINAR- AND TURBULENT-FLOW SYSTEMS

 Analogies to Energy Transfer:  Packed duct (by analogy): In terms of St m,bed where, as defined earlier, (= 6(1-  )/d p )  interfacial area per unit volume of bed 38 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 39 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”’St m,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  (Re bed ) correlations 40 CONVECTIVE MASS TRANSFER IN LAMINAR- AND TURBULENT-FLOW SYSTEMS