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Mass Transport in Composite Planar Slabs

Learn about concentration fields, surface mass transfer rates, and coefficients in composite planar slabs. Understand conservation equations and boundary conditions for solving simultaneous PDEs. Discover analogies and analogy-breakers in mass transfer phenomena.

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Mass Transport in Composite Planar Slabs

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  1. Advanced Transport Phenomena Module 6 Lecture 25 Mass Transport: Composite Planar Slab Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

  2. CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS • Conservation Equations: • Concentration fields are coupled by the facts that: • Homogeneous reaction rates involve many local species • All local mass fractions must sum to unity (only N-1 equations are truly independent) • Species i mass conservation condition may be written as: •  local mass rate of production of species i

  3. CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS • Conservation Equations: • Sinceri = rwi, by virtue of total mass conservation: and the species balance becomes: LHS  proportional to (Lagrangian) rate of change of wifollowing a fluid parcel RHS  is “forced” diffusion flux

  4. CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS • Conservation Equations: • PDEs for coupled to each other, and to PDEs governing linear momentum density rv(x,t) & temperature field, T(x,t) • All must be solved simultaneously, to ensure self-consistency • Simplest PDE governing is Laplace equation:

  5. CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS • Conservation Equations: • Laplace eq. holds when there are no: • Transients • Flow effects • Variations in fluid properties • Homogeneous chemical reactions involving species A • Forced diffusion (phoresis) effects • In Cartesian coordinates:

  6. CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS • Conservation Equations: • Boundary conditions on are of two types: • wi along each boundary surface, or • Some interrelation between flux (e.g., via heterogeneous chemical kinetics) • Solution methods: • Numerical or analytical • Exact or approximate • Solutions could be carried over from corresponding momentum or energy transfer problems

  7. ANALOGIES & ANALOGY-BREAKERS • Heat-Transfer Analogy Condition (HAC) applies when: • Fluid composition is spatially uniform • Boundary conditions are simple • Fluid properties are nearly constant • All volumetric heat sources, including viscous dissipation and chemical reaction, are negligible • known as functions of Re, Pr, Rah, etc.

  8. ANALOGIES & ANALOGY-BREAKERS • Corresponding temperature field: • Under HAC, the rescaled chemical species concentration • And corresponding coefficients, , will be identical functions of resp. arguments. A

  9. ANALOGIES & ANALOGY-BREAKERS • MACs: • Species A concentration is dilute ( ) • specified constant along surface • Negligible forced diffusion (phoresis) • No homogeneous chemical reaction • Analogy holds even when: • Fluid flow is caused by transfer process itself (e.g., natural convection in a body force field) • Analogy to linear momentum transfer breaks down due to streamwise pressure gradients

  10. ANALOGIES & ANALOGY-BREAKERS • MAC: • Schmidt number plays role that Pr does for heat transfer • Mass-transfer analog of Rah is: where bwdefines dependence of local fluid density on wA: w

  11. ANALOGIES & ANALOGY-BREAKERS • Correction Factors for Analogy-Breaking Phenomena: • Two analogy breaking mechanisms: • Phoresis • Homogeneous chemical reaction • Have no counterpart in energy equation T(x,t) • BL situation: Substance A being transported from mainstream to wall • wA,w << wA,∞

  12. ANALOGIES & ANALOGY-BREAKERS • Correction Factors for Analogy-Breaking Phenomena: • Phoresis toward the wall: • Distorts concentration profile • Affects wall diffusional flux, where Num,0 mass transfer coefficient without phoretic enhancement; analogous to Nuh F(suction)  augmentation factor

  13. ANALOGIES & ANALOGY-BREAKERS • Correction Factors for Analogy-Breaking Phenomena: • F(suction) is a simple function of a Peclet number based on drift speed, -c, boundary-layer thickness, dm,o, and diffusion coefficient, DA: or In most cases

  14. ANALOGIES & ANALOGY-BREAKERS • Correction Factors for Analogy-Breaking Phenomena: • Homogeneous reaction within BL: • wA profile distorted, diffusional flux affected or

  15. ANALOGIES & ANALOGY-BREAKERS • Correction Factors for Analogy-Breaking Phenomena: • F(reaction) depends on Damkohler (Hatta) number: where k”’  relevant first-order rate constant; time-1 can be rewritten using:

  16. ANALOGIES & ANALOGY-BREAKERS • Correction Factors for Analogy-Breaking Phenomena: • For an irreversible reaction with wA,w << wA,∞ (F(reaction)  1 when Dam  0) Reaction augmentation factors  Hatta factors • F(Reaction) = dm,o/dm • If only heterogeneous reactions occur, analogy is intact: Num = Num,0

  17. QUIESCENT MEDIA • Above conditions not sufficient in nondilute systems • Mass transfer itself gives rise to convection normal to surface, Stefan flow • vwfluid velocity @ interface • Additional condition for neglect of convective transport in mass transfer systems: • Inevitably met in dilute systems

  18. QUIESCENT MEDIA • Stefan flow becomes very important when wA,w ≠ wA,∞, andwA,w 1 • e.g., at surface temperatures near boiling point of liquid fuel

  19. COMPOSITE PLANAR SLAB • T continuous going from layer to layer, but not wA • Only chemical potential is continuous • Two unknown SS concentrations at each interface • Linear diffusion laws to be reformulated using a continuous concentration variable • Applies to nonplanar composite geometries as well

  20. COMPOSITE PLANAR SLAB x Mass transfer of substance A across a composite barrier: effect of piecewise discontinuous concentration (e.g., mass fraction wA(x))

  21. COMPOSITE PLANAR SLAB • Continuous composition variable = a-phase mass fraction of solute A • Corresponding mass flux through a composite solid (or liquid membrane) ka,lconcentration-independent dimensionless equilibrium solute A partition coefficients, (wA(a)/wA(l))LTCE, between phaseaand phase l (=a, b, g, d, …) dm,eff stagnant film (external) thickness (resistance)

  22. COMPOSITE PLANAR SLAB • Dilute solute SS diffusional transfer between two contacting but immiscible fluid phases a, b  in separation/ extraction devices • Modeled as through two equivalent stagnant films of thicknesses dm,eff(a) and dm,eff(b) • In series, negligible interfacial resistance between them • “Two-film” theory (Lewis and Whitman, 1924)

  23. COMPOSITE PLANAR SLAB • KA(a)  overall interface mass transfer coefficient (conductance) • Satisfies “additive-resistance” equation (symmetrical replacement ofaandbyields KA(b))

  24. COMPOSITE PLANAR SLAB • Gas absorption/ stripping: • One phase (say b) vapor phase • Kab relevant partition coefficient; inversely proportional to Henry constant, H: where M  solvent molecular weight pA  partial pressure of species A in vapor phase

  25. COMPOSITE PLANAR SLAB • H  dimensional inverse partition (distribution) coefficient (if b-phase (vapor mixture) obeys perfect gas law)

  26. COMPOSITE PLANAR SLAB • Addition of reagents to solvent phase a: • Reduces dm,eff(a) • Simultaneous homogeneous chemical reaction increases liquid-phase mtc’s, accelerates rate of uptake of sparingly-soluble (large H) gases • Additive (B) in sufficient excess => pseudo-first-order reaction ( linearly proportional to rwA, with rate constant k”’)

  27. COMPOSITE PLANAR SLAB where and

  28. COMPOSITE PLANAR SLAB • When reaction is so rapid that the two reagents meet in stoichiometric ratio at a thin reaction zone (sheet): • Distance between reaction zone & phase boundary plays role of wB,b concentration of additive B in bulk of solvent wA,i concentration of transferred solute A at solvent interface • b gms of B are consumed per gram of A

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