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

2.  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 2 CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS

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

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

5.  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: 5 CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS

6.  Conservation Equations:  Boundary conditions on are of two types:   i 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 6 CONCENTRATION FIELDS, SURFACE MASS- TRANSFER RATES & COEFFICIENTS

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, Ra h, etc. 7

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

9.  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 9

10.  MAC:  Schmidt number plays role that Pr does for heat transfer  Mass-transfer analog of Ra h is: where   defines dependence of local fluid density on  A : 10 ANALOGIES & ANALOGY-BREAKERS 

11.  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   A,w <<  A,∞ 11 ANALOGIES & ANALOGY-BREAKERS

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

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

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

15.  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: 15 ANALOGIES & ANALOGY-BREAKERS

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

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

18.  Stefan flow becomes very important when  A,w ≠  A,∞, and  A,w  1  e.g., at surface temperatures near boiling point of liquid fuel 18 QUIESCENT MEDIA

19. COMPOSITE PLANAR SLAB  T continuous going from layer to layer, but not  A  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 19

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

21.  Continuous composition variable =  -phase mass fraction of solute A  Corresponding mass flux through a composite solid (or liquid membrane)  ,l  concentration-independent dimensionless equilibrium solute A partition coefficients, (  A (  ) /  A (l) ) LTCE, between phase  and phase l (=  …)  m,eff  stagnant film (external) thickness (resistance) 21 COMPOSITE PLANAR SLAB

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

23.  K A (  )  overall interface mass transfer coefficient (conductance)  Satisfies “additive-resistance” equation (symmetrical replacement of  and  yields K A (  ) ) 23 COMPOSITE PLANAR SLAB

24.  Gas absorption/ stripping:  One phase (say  ) vapor phase  K    relevant partition coefficient; inversely proportional to Henry constant, H: where M  solvent molecular weight p A  partial pressure of species A in vapor phase 24 COMPOSITE PLANAR SLAB

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

26.  Addition of reagents to solvent phase  :  Reduces  m,eff (  )  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  A, with rate constant k”’) 26 COMPOSITE PLANAR SLAB

27. where and 27 COMPOSITE PLANAR SLAB

28.  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  B,b  concentration of additive B in bulk of solvent   i  concentration of transferred solute A at solvent interface  b gms of B are consumed per gram of A 28 COMPOSITE PLANAR SLAB