Mass Transfer in Laminar & Turbulent Flow. Mass Transfer Coefficients

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1 Mass Transfer in Laminar & Turbulent Flow Mass Transfer Coefficients 25 MassTransfer.key - January 3, 204

2 Convective Heat & Mass Transfer T T w T in bulk and T w near wall, with a complicated T profile in between! Note: h=λ only if T is linear. q = rt Fourier s law requires us to resolve T. This, in turn, requires detailed knowledge of u since it will cause non-trival T profiles. Concept: if we only care about an overall q, and not about T(z), then we can approximate: dt q z = dz T T w = h (T w T ) ` q h (T w T ) Heat transfer coefficient, h, is empirical, and contains a lot of physics (must be correlated for different situations). x x w Note: k c =D AB only if c A is linear. J A = c A D AB rx A Fick s law requires us to resolve x A. This, in turn, requires detailed knowledge of u since it will cause non-trival x profiles. Concept: if we only care about an overall J A, and not about x A (z), then we can approximate: J Az cd AB x x w ` J A k c (c Aw c A ) = k c (c Aw c A ) Mass transfer coefficient, k c, is empirical, and contains a lot of physics (must be correlated for different situations). 26 MassTransfer.key - January 3, 204

3 Mass vs. Heat Transfer Fick s Law for a binary mixture of A and B: Only really valid for binary mixtures (2 species). There are several diffusive fluxes, depending on the frame of reference that you choose (molar, mass)! Need an appropriate choice for DAB. Dependent on temperature, pressure and (sometimes) composition. x A is not the only possible driving force for J A. Could include other species (for more than 2 components) or even T and p! See SHR 3.8 for more information. Lots of cool physics here... Fourier s Law : J A = cd AB rx A molar diffusive flux relative to a molar-averaged velocity q = krt need a model for k. Dependent on temperature, pressure & composition. Heat can be transferred by radiation! No analogous mechanism for mass (until you invent teleportation). Heat can diffuse by other mechanisms as well. x can also cause heat diffusion! Convective mass transfer: J A = k c c A Convective heat transfer: q = h T 27 MassTransfer.key - January 3, 204

4 SHR 3.5 Analogies in Diffusive Fluxes ū x x-momentum diffusive flux diffusive heat flux diffusive mass flux q z = zx = µ du x dz f ū2 z=0 2 k dt dz z=0 h(t i T ) J Az = D AB dc A dz z=0 k c (c A c A ) z x Only gradients are in z-direction No velocity (bulk flow) in z-direction Diffusion in z-direction Constant properties (c, ρ, μ, k, DAB, etc.) N StH = N Nu N Re N Pr = Reynolds analogy: f 2 = N St H = N StM h C p ū x = N StM = N Sh = k c = k c N Re N Sc ū x G h C p G (valid only when N Pr = N Sc = ) f = Fanning friction factor G =ū x mass velocity See SHR Table 3.3 for summary of some dimensionless groups See equations (3-66)- (3-7) in SHR Chilton-Colburn analogy: j M f 2 {z } momentum = j H N StH N 2 /3 Pr {z } heat = j D N StM N 2 /3 Sc {z } mass Other analogies also exist. See SHR MassTransfer.key - January 3, 204

5 Mass Transfer Coefficients Fick s law: J A = cd AB dx A dz c A c A 2 dx A dz x A 2 x A ` If kc is chosen just right then: J A cd AB x A2 x A ` ck c (x A x A2 ) J A = k c c A = D AB dc A dz N Sh = `k c D AB l is an appropriate length scale z The mass-transfer coefficient, k c Units of length/time Analogous to the heat transfer coefficient: q = hδt Useful when we don t know (or want to know) xa(z). Need to choose kc just right to get the correct flux. kc=dab/δz is usually not even close (because xa is not linear in general and Δz is typically not known). Usually evaluated at one location (e.g. phase interface) since JA might vary over z. Approaches to get k c : If we can get an analytic solution for xa(z) then we can get an exact form for kc using Fick s law. (frequently not possible) Often, kc is correlated with nondimensional groups like Reynolds number, Schmidt number and Peclet number. Analogies with heat transfer. 29 MassTransfer.key - January 3, 204

6 Accounting for Bulk Flow Examples: Evaporation Condensation Absorption Stripping N A = x A N + J A = x A (N A + N B )+J A J A = k c c A = NB NA N A = J A x A k 0 c = k c x A = k c x B Note: k is just a convenience definition. N A = k 0 c c A Typically, correlations give you k, not k. For C > 2 components, this gets much more complicated. 30 MassTransfer.key - January 3, 204

7 SHR 3.6 Two-Phase Mass Transfer Models for kc... p A c Ai c Ab Phase equilibrium at gas-liquid interface to relate p A to c A (discontinuity in mole fractions) diffusion through a liquid film of thickness δ well-mixed bulk region (fluid motion) we want N A. gas liquid film z=0 z=δ bulk liquid N A = J A = k c (no bulk flow) c A 3 MassTransfer.key - January 3, 204

8 SHR 3.6. Film Theory p A cai Neglecting bulk flow N A = J A = cd AB dx A dz Accounting for bulk flow N A = x A (N A + N B ) +J A {z } N c Ab = cd AB dx A x A dz gas liquid film z=0 z=δ bulk liquid c is constant N A is constant with z. N B =0 Separate this ODE & integrate it to find xa(z). N A = cd apple AB xab N A = DAB (c Ai c Ab ) ln x Ai cd = cdab (x AB Ai x Ab ) = (x Ai x Ab ) ( x A ) LM ( x A ) LM = x Ai x Ab ln [ ( x A b )/( x Ai )] =(x B) LM Film theory is often not very accurate, but is widely used nonetheless. Force fit into form N A = k c (c Ai c Ab ) to find k c. k c = D AB kc 0 D AB = = ( x A ) LM k c ( x A ) LM Since δ is unknown, kc is typically replaced by an empirical correlation. 32 MassTransfer.key - January 3, 204

9 SHR Other models Concept: p A cai c Ab A pocket of fresh fluid arrives from the bulk to the film and stays there for some period of time, whereafter it is replaced by a new packet of fresh fluid. Essentially diffusion into an infinite slab (but only for a short time tc). We solved this problem already! gas liquid film z=0 z=δ bulk liquid N A = k c (c Ai c Ab ) Higbie Model: fluid parcels are replaced at uniform interval tc. k c =2 r DAB t c Choosing t c : Bubbles: ratio of bubble diameter to its relative velocity in the fluid Droplets: residence time (assuming no internal circulation) Packed tower: typically around s. Correlations...? Surface Renewal Model: old fluid parcels are more likely to be replaced than young parcels k c = p D AB s s - rate of surface renewal (/sec) (fraction of surface area replaced by fresh fluid in unit time) It is difficult to determine s MassTransfer.key - January 3, 204

10 SHR 3.7 Two-Film Theory Overall Mass Transfer Coefficients idealized picture more realistic picture 34 MassTransfer.key - January 3, 204

11 Two-Film Theory Assumptions & Formulation No reaction. Phase equilibrium at the interface & Henry s law applies. This has the effect of being a contact resistance to mass transfer (analog to heat transfer). Henry s law implies a linear relationship between the gas & liquid phases compositions. Typically only valid for small ranges of x A or y A. Bulk flow is negligible (N A = J A ). trace species diffusing (N A 0, N B = 0) equimolar counterdiffusion (N A = -N B ). Constant total molar concentration, c. Gas phase: Liquid phase: Phase Equil.: p A = H A x A p A = c A/H A y A = H A x A N A = k p (p Ab p Ai ) N A = k c (c Ai c Ab ) c Ai = H A p Ai Several forms of Henry s law exist: distinguish by units of H A. Combine to eliminate interface compositions. N A = K L (c A c Ab ) K L H A c A = H A p Ab N A = p A b H A c Ab ( H A/k p )+( /k c ) overall mass transfer coefficient based on liquid phase. k p + k c fictitious c A in equilibrium with p A b. 35 MassTransfer.key - January 3, 204

12 Variations on the Theme Liquid phase concentration N A = K L (c A c Ab ) c A = H A p Ab K L H A k p + k c Gas phase concentration N A = K G (p Ab p A) p A = c A b H A = + K G k p H A k C Liquid mole fraction Gas mole fraction N A = K x (x A x Ab ) = K y (y Ab y A) x A = y A b K A y A = x Ab K A = + K x K A k y k x = + K A K y k y k x K A is the K-value from equilibrium thermo: K A = y A i x Ai SI AE k m/s ft/h k kmol/(s-m lbmol/(h-ft k kmol/(s-m lbmol/(h-ft 36 MassTransfer.key - January 3, 204

13 SHR Large Driving Forces N A = k y (y Ab y Ai )=K y (y Ab y A) = k x (x Ai x Ab )=K x (x A x Ab ) For large variation in xa (or ya) across a phase, the change in KA is important. Henry s law is not valid here! = + m x K y k y k x = + K x k x m y k y See derivation in SHR m = dy A dx A mx evaluate on the liquid side. my evaluate on the gas side. 37 MassTransfer.key - January 3, 204

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