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

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