Mass Transfer 2. Diffusion in Dilute Solutions

Transcription

1 Mass Transfer. iffusion in ilute Solutions. iffusion aross thin films and membranes. iffusion into a semi-infinite slab (strength of weld, tooth deay).3 Eamples.4 ilute diffusion and onvetion Graham (85) monitored the diffusion of salt (NaCl) solutions in a larger ar ontaining water. Every so often he removed the bottle and analyed it. Mass Transfer iffusion in ilute Solutions_ Fik s Laws -

2 Initial salt Relative Flu onentration, Weight-% of NaCl He postulated that a) The quantities diffused appear to be proportional to the salt onentration. b) iffusion must follow diminishing progression. Fik (855) analyed these data and wrote The diffusion of the dissolved material... is left ompletely to the influene of the moleular fores basi to the same law... for the spreading of warmth in a ondutor and whih has already been applied with suh great suess to the spreading of eletriity. Mass Transfer iffusion in ilute Solutions_ Fik s Laws -

3 Fik s first law: d d This is analogous to Newton s law This is analogous to Fourier s law q T or y q dv dy dt d These equations imply no onvetion (dilute solutions!). Mass Transfer iffusion in ilute Solutions_ Fik s Laws -3

4 . iffusion aross thin films and membranes Eample..: iffusion aross a thin film C Goal: onentration profile in the film, and the flu aross it at steady state. Mass balane aross arbitrary thin layer : solute rate of diffusion aumulation into the layer at Steady state A( ) rate of diffusion out of layer at C Mass Transfer iffusion in ilute Solutions_ Fik s Laws -4

5 A( ) ivide this equation by the film volume A ( ) As d Fik s d first law d d () Mass Transfer iffusion in ilute Solutions_ Fik s Laws -5

6 If we solve this equation we have the onentration profile of in and then we an alulate the flu from Fik s first law () d d by estimating the d at d or The boundary onditions are Then the solution to eq. is a b Mass Transfer iffusion in ilute Solutions_ Fik s Laws -6

7 and using the boundary ondition gives: ( ) d d d d or Mass Transfer iffusion in ilute Solutions_ Fik s Laws -7

8 Eample..: Membrane diffusion erive the onentration profile and the flu for a single solute diffusing aross a thin membrane. The analysis is the same as before leading to d A but the boundary onditions differ: d, H, H where H is a partition oeffiient (the onentration in the membrane divided by that in the adaent solution e.g. Henry s or Raoult s law). Mass Transfer iffusion in ilute Solutions_ Fik s Laws -8

9 Then the onentration profile beomes: H H The solute is more soluble in the membrane than in the adaent solution The solute is less soluble in the membrane than in the adaent solution Mass Transfer iffusion in ilute Solutions_ Fik s Laws -9

10 Eample..3: Conentration dependent diffusion oeffiient The diffusion oeffiient an vary with onentration. (water aross films and in detergent solutions) Assumption: slow diffusion (small ), S fast diffusion (large ), s Consider two-films in series. C At steady state =onst in both films. Z l-z Mass Transfer iffusion in ilute Solutions_ Fik s Laws -

11 In film : Large s d d d d In film : Small d d d d () = () small s d large d s d s ( ( d ) ) ( s ) s ( ) () ( ) () s ( ( ) ) The flu beomes then: If then s( ( ) ) s Mass Transfer iffusion in ilute Solutions_ Fik s Laws -

12 In the following film two ompounds A and B diffuse from to through the film. Whih one diffuses faster or whih one has the largest iffusivity? A B A A B B Mass Transfer iffusion in ilute Solutions_ Fik s Laws -

13 A ompound diffuses through two films in series. When it diffuses faster in film A than in film B, whih onentration profile best desribes this proess,, or 3 and why? A B 3 Mass Transfer iffusion in ilute Solutions_ Fik s Laws -3

14 . iffusion in a Semi-infinite Slab Fik s Seond Law iffusion is the net migration (mass transfer-transport) of moleules from regions of HIGH to LOW onentration. A d B d dy d d C X : flu of partiles in the -diretion Mass Transfer iffusion in ilute Solutions_ Fik s Laws -4

15 Mass Transfer iffusion in ilute Solutions_ Fik s Laws -5 Rate at whih partiles enter the elemental volume ddyd aross the left side of that volume y X X d d d IN IN - OUT = y d d d gradient of at the enter of ddyd Net rate of transport into that element y d d d A B C d dy d d d Similarly for the dd fae: y y y d d d y d d d and for ddy fae: OUT

16 Mass Transfer iffusion in ilute Solutions_ Fik s Laws -6 The rate of hange of the number of partiles per unit volume (& sie), n, in the elemental volume ddyd is: d d d d d d y t y y y y t y From eperimental observations: (Fik s first law without onvetion, dilute solutions). Substituting it in the above gives Fik s seond law: y t

17 Eample..: Unsteady diffusion in a semi-infinite slab Consider that suddenly the onentration at the interfae hanges. Goal: To find how the onentration and flu varies with time. Very important in diffusion in solids (tooth deay, orrosion of metals). This is the opposite to diffusion through films. Everything else in the ourse is in between. Mass Transfer iffusion in ilute Solutions_ Fik s Laws -7

18 At t : but at t > : Mass balane: ivide by A: rate of diffusion rate of diffusion solute aumulation into the layer out of the layer involume A at at (A t t ) A( Combine this with Fik s first law ) d d gives: t t () Mass Transfer iffusion in ilute Solutions_ Fik s Laws -8

19 t () How to solve Fik s nd law? efine a new variable (Boltmann): Boundary Conditions: t = all : t > () 4t =: =: (It requires the wild imagination of mathematiians) So eqn. () beomes: d d t d d or using eqn. : d d d d (3) Mass Transfer iffusion in ilute Solutions_ Fik s Laws -9

20 The B.C. beome: Set d dy y so eqn.(3) beomes: y d d or: dy y integrate d lny lna y a ep( ) d d Resubstitution: a ep( ) a ep( s )ds k (4) Mass Transfer iffusion in ilute Solutions_ Fik s Laws -

21 at a ep( s )ds k k at so a / a ep( s ) ds ( ) s ) ds / in (4): ep( ep( s )ds erf Mass Transfer iffusion in ilute Solutions_ Fik s Laws -

22 So the flu an be obtained as : - 4t - / t e ( - ) and the flu aross the interfae beomes (=) : ( ) t This is the flu at time t. Total flu at time t t dt Mass Transfer iffusion in ilute Solutions_ Fik s Laws -

23 .3 Eamples Eample.3.: Steady dissolution of a sphere Consider a sphere that dissolves slowly in a large tank. The sphere volume does not hange. Find the dissolution rate and the onentration (r) profile away from the sphere at steady-state. Mass Transfer iffusion in ilute Solutions_ Fik s Laws -3

24 Mass balane on a spherial shell: solute aumulation within this shell diffusion into the shell diffusion out of the shell ( 4 r r ) ( 4 r ) t r ( 4 r ) r r ivide both sides by the spherial shell s volume, note that LHS= at steady-state and take the limit as r d r r dr () () Combine this with Fik s law at spherial oordinates and = onst: d r dr r d dr (3) Mass Transfer iffusion in ilute Solutions_ Fik s Laws -4

25 Boundary Conditions: r R (sat) (4) r (5) d a (6) Integrating eqn. 3 gives: dr r where a is an integration onstant. a Integrating eqn. 6 again gives: b r where b is another integration onstant. (7) Using the B.C. gives b= from eqn. 5 and a = (sat)r from eqn. 4 so eqn. 7 beomes R (sat) (8) r Mass Transfer iffusion in ilute Solutions_ Fik s Laws -5

26 The dissolution flu an be found from Fik s law: d dr d dr R (sat) r (sat)r r whih at the sphere s surfae is (sat) R If you double the sphere (partile) sie, the dissolution rate per unit area is only half as large even though the total dissolution rate over the entire surfae is doubled. Very important in pharmaeutis!! Also in the growth of fog droplets and spraying, as well as in growth of partiles by ondensation or by surfae reation limited by transport. Mass Transfer iffusion in ilute Solutions_ Fik s Laws -6

27 Challenging Mathematis: Tet:.4. eay of a Pulse.4.3 Unsteady iffusion into Cylinders eay of a Pulse Unsteady iffusion into Cylinders Mass Transfer iffusion in ilute Solutions_ Fik s Laws -7

28 .4 ilute iffusion and Convetion Till now we did not onsider any flow. Here we address a speial ase where diffusion and onvetion our normal to eah other: Convetion iffusion Mass Transfer iffusion in ilute Solutions_ Fik s Laws -8

29 .4. Steady iffusion aross a falling film Assumptions:. The solution is dilute (no diffusion-driven flow). The liquid is the only resistane to mass transfer. 3. Mass transfer by diffusion in -diretion and flow in -diretion Mass Transfer iffusion in ilute Solutions_ Fik s Laws -9

30 Mass balane on volume ( w ) (w = width of film wall) solute aumulation solute diffusinginat - in w solute diffusing out at ( t w ) solute flowinginat - solute flowing out at w w v w v w as and v are onstant in varies in but not in! (The film is long) v varies in but not in! (Couette flow, no pressure drop in ) Mass Transfer iffusion in ilute Solutions_ Fik s Laws -3

31 Now we an write d d Combining it with Fik s law gives: Boundary onditions: d d The solution is: ( ( ) ) Unbelievable! The flow has no effet. That s right! When solutions are dilute this is orret. Mass Transfer iffusion in ilute Solutions_ Fik s Laws -3

32 .4. iffusion into a falling film A thin liquid film flows slowly without ripples (waves) down a flat surfae. One side of the film wets the surfae while the other is in ontat with the gas whih is slightly (sparingly) soluble in the liquid. Goal: Find how muh gas dissolves in the liquid. (important to Penetration Theory ) Mass Transfer iffusion in ilute Solutions_ Fik s Laws -3

33 Assumptions:. The solution is dilute. Mass transfer in -diretion and flow (onvetion) in -diretion 3. The gas over the film is pure (no resistane to diffusion) 4. Short ontat between liquid and gas (for onveniene) Mass balane: mass aumulation in w mass diffusinginat - mass diffusing out at mass flowinginat - mass flowing out at Mass Transfer iffusion in ilute Solutions_ Fik s Laws -33

34 t ( w ) w w w v w v At steady state and after dividing by the volume (w ) and taking the limit as this volume goes to ero: v We ombine this with Fik s law and set v = v ma (fluid veloity at the interfae) as the gas-liquid ontat time is short (based on our bold (too strong) assumption #4) () Mass Transfer iffusion in ilute Solutions_ Fik s Laws -34

35 The impliation here is that the solute barely has a hane to ross the interfae so slightly diffuses into the fluid. So equation () beomes: ( / v ma ) () Boundary onditions: (sat) (3) (4) (5) Mass Transfer iffusion in ilute Solutions_ Fik s Laws -35

36 Now revoking (realling) again assumption #4 the last B.C. is replaed by (6) meaning that the solute diffuses only shortly into the liquid. As a result, the solute does not see the wall. In this ase this problem redues to that of diffusion in a semiinfinite slab with t / v ma and the solution is the same: (slide -3) (sat) erf 4 / v ma and the flu at the interfae is: v / (sat) ma Mass Transfer iffusion in ilute Solutions_ Fik s Laws -36

37 What did we learn so far?. iffusion of dilute solutions. Aross thin film and steady-state. Aross a thik slab and no steady-state How to hoose between these two? length diffusion oeffiient time Fo This is the variable in the error funtion of the semiinfinite slab problem. Fo: Fourier number if / Fo >> => semi-infinite slab / Fo << => steady-state / Fo ~ => detailed analysis Mass Transfer iffusion in ilute Solutions_ Fik s Laws -37

38 Eample: Membrane for industrial separation: Thikness =. m = - m /s If the duration of the eperiment is a) t= s Fo 4 m 7 m s s This is a semi-infinite slab problem! b) t=3 hrs 4 s Fo 7 m 4 m s 4 s. This is a thin film, steady-state problem. The value of Fo = indiates that mass transfer is signifiantly advaned in a given proess. As a result it an be used to estimate the EXTENT (or EGREE) of advanement (or progress) for unsteady-state proesses. Mass Transfer iffusion in ilute Solutions_ Fik s Laws -38

39 For eample: a) Guess how far gasoline has evaporated into the stagnant air in a regular glass-fiber filter. Say that evaporation is going on for min and = -5 m /s. Fo -5 length m s 6s length 8m b) Consider H diffusion in nikel making it rather brittle. If = - m /s estimate how long it will take for H to diffuse mm through the Ni speimen. -6 m - Fo m s t 6 t s days Mass Transfer iffusion in ilute Solutions_ Fik s Laws -39

40 Another important differene of the two limiting ases stems from the interfaial flues. (thin film) t (thik slab) t Note that both and have veloity units (dimensions), some people even all them the veloity of diffusion. In fat these are equivalent to the mass transfer oeffiients we talked earlier on!! Mass Transfer iffusion in ilute Solutions_ Fik s Laws -4

41 Eample: iaphragm-ell diffusion Goal: To measure the diffusion oeffiient Cell: Two well-stirred volumes and a thin barrier (or diaphragm, e.g. sintered glass frit or even a piee of paper). Combination of a steady-state (inside diaphragm) and a transient problem (in liquid reservoirs). Mass Transfer iffusion in ilute Solutions_ Fik s Laws -4

42 Proedure: Upper ompartment = solvent, Lower ompartment = solution,, upper, lower After time t, measure new at the upper and lower ompartment Assumptions: Rapid attainment of steady state flu aross the diaphragm. Note that this says the flu is steady even through the onentrations are hanging! Can we get away with that? Mass Transfer iffusion in ilute Solutions_ Fik s Laws -4

43 At this pseudo steady-state the flu aross the diaphragm (membrane) is: H (,lower,upper ) (H an also be regarded as the fration of the diaphragm area available for diffusion) () Mass balane on eah ompartment d,lower Lower: Vlower A d dt,upper Upper: Vupper A dt () (3) where A is the area of the diaphragm. Mass Transfer iffusion in ilute Solutions_ Fik s Laws -43

44 ividing eqs. () and (3) by V lower and V upper, respetively, followed by subtrating eqn. (3) from () and substituting eqn. (), gives: d dt ( ) (,lower,upper,upper,lower where the geometri onstant is ) ) AH ( V lower V upper (4) Boundary ondition:,lower,upper,lower,upper t=: (5) Integrating eqn. (4) subet to (5) gives,lower,lower,upper,upper e t (6) ln t,lower,upper Mass Transfer iffusion in ilute Solutions_ Fik s Laws -44,lower,upper or (7) is obtained by alibration with solute of known. Now as we an measure t and the solute onentration at the two ompartments, an be obtained.

45 Let s go bak to our assumptions: a) is affeted by the diaphragm and its tortuosity (internal hannel-like struture) This an be aounted for by rewriting eqn. (7) as: t ln,lower,lower,upper,upper Where is a new alibration onstant that inludes tortuosity. Surprisingly this works well as agrees with that measured by other tehniques. Mass Transfer iffusion in ilute Solutions_ Fik s Laws -45

46 b) Pseudo steady-state (steady-state flu aross a diaphragm with unsteady-state onentrations in the ompartments) Compare the volume of material (solvent and solute) in the diaphragm voids (empty spae) with that of eah ompartment. The solute onentrations in the ompartments hanges slooooowly beause they are very large ompared to the diaphragm. The solute onentration in the diaphragm hanges muh faster as it has little volume. Thus the onentration profile in the diaphragm will reah a (pseudo) steady-state before the orresponding onentrations hange muh. Thus the flu will reah steady-state! Mass Transfer iffusion in ilute Solutions_ Fik s Laws -46

47 Now more quantitatively and professionally: We an ompare the harateristi (or relaation) times of the two units: iaphragm: Compartment: t t C (/e) efinition: The relaation time is the time at whih the distane to equilibrium has been redued to the fration /e of its initial value. Fo (8) (9) So set:,lower,upper (, lower e, upper ) t And ompare with eq. (6): e,lower,upper,lower,upper Mass Transfer iffusion in ilute Solutions_ Fik s Laws -47

48 So eqn. (6) an be written as: so t t e e R t C e e t Now the above analysis is aurate when t C t or A volume diaphragm H V lower V upper or V diaphragm voids V lower V upper This is engineering MAGIC!!! Mass Transfer iffusion in ilute Solutions_ Fik s Laws -48