Mass Transfer Chapter 3. Diffusion in Concentrated Solutions

Transcription

1 Mass Trasfer Chapter 3 Diffusio i Coetrated Solutios. Otober 07

2 3. DIFFUSION IN CONCENTRATED SOLUTIONS 3. Theor Diffusio auses ovetio i fluids Covetive flow ours beause of pressure gradiets (most ommo) or temperature differees (buoa or free or atural ovetio). However eve i isothermal ad isobari sstems, ovetio a our due to diffusio. This is alled diffusioidued ovetio I geeral, diffusio ad ovetio alwas our together i fluids. Maxwell (860): Mass trasfer is due partl to the motio of traslatio ad partl that of agitatio. Mass Trasfer Diffusio i Coetrated Solutios 3-

3 A first example to illustrate diffusio-idued ovetio: Evaporatio of Beee At 6 C the beee vapor is dilute ad evaporatio is limited b diffusio. At 80. C beee boils (p atm). Evaporatio is otrolled b ovetio. At 60 C a itermediate ase ours i whih both diffusio ad ovetio are importat. Mass Trasfer Diffusio i Coetrated Solutios 3-3

4 Aalsis of the ase at 60 C: Coetratio of beee (speies ) ad air (speies ) at h:,h, ( 0); p,h p, ( 0);,h max p,h p (air blows awa ad dilutes beee vapor). At 0:,0 f (T,p) >,h ad,0 <,h p,0 f (T,p) > p,h ad p,0 < p,h Mass Trasfer Diffusio i Coetrated Solutios 3-4

5 The differee i oetratios (or partial pressures) betwee 0 ad h gives rise to upward diffusio of beee from the liquid surfae ad dowward diffusio of air. Sie the beee surfae is osidered impermeable to air, a ovetive upwardpoitig flux must ompesate the dowward diffusive flux of air! This is the Stefa flow! NOTE: This ovetio also trasports beee moleules! Mass Trasfer Diffusio i Coetrated Solutios 3-5

6 Determie oetratio profile ad flux for the 60 C ase Mass balae for speies i at stead state: J i,ov () + J i,diff () J i,ov ( + ) + J i, diff ( + ) Diffusive flux: J i, diff D A d d Covetive flux with veloit u: J i,ov i u A Mass Trasfer Diffusio i Coetrated Solutios 3-6

7 u + D Divide b A Δ ad Δ 0: i i 0 () For pratiabilit we trasform molar oetratios ito mole fratios: M, ( ) M, + M, + V where M,i is the umber of moles of speies i. If is give as a mass oetratio the trasformatio ito mole fratios is: V M with the average molar mass ρ M k M i M i Mass Trasfer Diffusio i Coetrated Solutios 3-7

8 Mass Trasfer Diffusio i Coetrated Solutios 3-8 So () a be rewritte as: 0 + D u i i () Itegratio of eq. () for speies (beee): C D u (3) The two terms are the ovetive ad diffusive molar flux desities. The sum of the molar fluxes is ostat. Itegratio of eq. () for speies (air) with ( ) C D u + (4)

9 Sie air does ot aumulate ad aot peetrate the beee surfae, the total (overall) molar flux of air is ero. Thus C 0 Eq. (4) is rewritte as: ( ) Itegratio of (5) with B.C.,0 at 0: ( ) ( ) u D D (5) u e (6), 0 Determie veloit u with B.C.,h at h: u D h l, h,0 (7) Mass Trasfer Diffusio i Coetrated Solutios 3-9

10 Isertig eq. (7) ito (6) gives the oetratio profile: ( ) ( ),0, h,0 h (8) With the oetratio profile ad eq. (3) we a obtai the total molar flux: j tot D h l, h,0 (9) This tpe of approah leads to the Stefa-Maxwell equatios for multi-ompoet diffusio i oetrated solutios. Mass Trasfer Diffusio i Coetrated Solutios 3-0

11 Separatig Covetio from Diffusio Cussler s Approah Now, Cussler approahes this topi slightl differet but gets to the same results: Agai, assume that the two trasport effets are additive: total mass trasported mass trasported + b diffusio mass trasported b ovetio If the total mass flux is, the mass trasported per area per time relative to fixed oordiates: v where v is the average solute veloit (veloit due to ovetio ad superimposed diffusio). Mass Trasfer Diffusio i Coetrated Solutios 3-

12 The total average solute veloit a be split ito oe part due to diffusio ad oe due to ovetio, alled referee veloit v a : ( a ) a a a v v + v j + v diffusive flux ovetio The art is to selet v a i suh a wa that the ovetio term is simplified or ideall: v a 0. For example v a is the veloit of the solvet beause the solvet is usuall i exess so its trasfer is miimal (i other words the differee i solvet oetratio is too small aross the solutio). That wa we elimiate ovetio ad deal with a SIMPLER problem. Mass Trasfer Diffusio i Coetrated Solutios 3-

13 Two-bulb apparatus (Diaphragmell) for uderstadig differet defiitios of referee veloities. Volume average veloit 0 Molar average veloit 0 Mass average veloit 0 Fial eter of moles Iitial eter of moles Volume average veloit 0 Molar average veloit 0 Mass average veloit 0 Mass Trasfer Diffusio i Coetrated Solutios 3-3

14 For gases (e.g. H ad N ) at equal T ad p the umber of moles is alwas the same i both sides. Similarl the volume i both sides is the same. As a result, the v 0 0 volume average veloit v * 0 molar average veloit v 0 mass average veloit, beause the masses of N ad H are differet. As a result, as time goes b the eter of mass i the two-bulb apparatus moves awa from the bulb otaiig N iitiall. Thus the mass average veloit v is ot ero. Mass Trasfer Diffusio i Coetrated Solutios 3-4

15 For liquids: The volume is earl alwas ostat. v 0 0 volume average veloit v 0 mass average veloit. This is usuall orret as liquid desities differ little. e.g. ρ HO g/m 3 ρ Glerol. g/m 3 However, the molar oetratio is usuall quite differet followig large differees i moleular weight. e.g. MW HO 8 g/mol ad MW Glerol 9 g/mol So v * 0 molar average veloit for liquids. I olusio: For gases use as referee veloit v a the v 0 or v *, while for liquids use v 0 or v. Mass Trasfer Diffusio i Coetrated Solutios 3-5

16 Mass Trasfer Diffusio i Coetrated Solutios 3-6

17 Table leged: ω i : mass fratio of speies i i : mole fratio of speies i i V i: volume fratio of speies i, where is the partial speifi volume of speies. Vi Preisel: Partial speifi volume: V i V mi p, T, m j Partial molar volume: V i V M,i p,t, M, j The partial speifi or molar volume expresses how muh a volume hages upo additio of a ertai mass or umber of moles of a give ompoet. Mass Trasfer Diffusio i Coetrated Solutios 3-7

18 For ideal gases (pv RT), V a be expressed as: i V V M, ( + ) RT p M, M, R T M, M, p p,t, p,t, M, Mass Trasfer Diffusio i Coetrated Solutios 3-8

19 3. Examples for Parallel Diffusio ad Covetio Example 3..: Fast diffusio through a stagat film Goal: Calulate the flux ad the oetratio profile Remember that at itermediate temperatures both diffusio ad ovetio affet the evaporatio of beee (or a other solute). Now both diffusio ad ovetio are importat! Mass Trasfer Diffusio i Coetrated Solutios 3-9

20 . Step: Mass balae solute aumulated i volume A solute trasported i at solute trasported out at + t ( A ) A A + Divide b A ad as volume 0 t At stead state: 0 Mass Trasfer Diffusio i Coetrated Solutios 3-0

21 . Step: Choose ad simplif mass trasport equatio Now the flux is affeted b both diffusio ad ovetio. For simpliit we hoose v a v 0 (volume average veloit) 0 d j + v D + (Vv + V v ) d Note that v ad v The total average flux of the solvet (air) is ero (it seems to be stagat), sie it aot peetrate ito the liquid phase ad does ot aumulate. Therefore 0 ad v 0. Mass Trasfer Diffusio i Coetrated Solutios 3-

22 d + d D V So or ( V ) D d d If the vapor is a ideal gas, the V V ( + ) V ad ( ) D d d () 3. Step: Boudar oditios 0: 0 () L: L (3) Mass Trasfer Diffusio i Coetrated Solutios 3-

23 Solve eq. () subjet to BC s to determie D l 0 Total Flux of beee (4) Note that doublig the oetratio differee DOES NOT double the flux, as i dilute sstems. Compare to our iitial result for ombied diffusio ad ovetio of beee (page 3-0): j tot C D l h, h,0 The diret approah ad the oe usig the referee veloit give the same results! Mass Trasfer Diffusio i Coetrated Solutios 3-3

24 Itegratig eq. () also for 0 to ad 0 to ad assumig that does ot hage with height (whih is a fair assumptio here as the ross-setioal area does ot hage) gives: 0 0 Coetratio profile (5) With (5) ad Fik's law we a determie the diffusive flux: j d D d D 0 0 l 0 Diffusive Flux of beee (6) Mass Trasfer Diffusio i Coetrated Solutios 3-4

25 Now does this result (eq. 5,6) redue to that for dilute solutios? Expasio ito series for small (dilute sstem small o. ): 0 0 a a( a ) a( a )( a ) 3 ± ± a + ± + ± a (7)! 3! ( ) ( ) ( + ) a( a + )( a + ) a a a + +! 3! ± a 3 Here for l, thus a: + (8) 3 ( ) l 3 (9) Mass Trasfer Diffusio i Coetrated Solutios 3-5

26 Let s appl eq. (8) to eq. (5) [( )( + )] [ + ] ( ) + ( 0) 0 ( ) + + ( ) (0) If we rearrage ad multipl both sides of eq. (0) with 0 + ( 0) () Mass Trasfer Diffusio i Coetrated Solutios 3-6

27 Likewise for the flux from eq. (4) D D [ l( ) l( )] D [ + 0] (0 ) Eq 9 0 () Eq. () ad () are idetial to the dilute limit oes! Mass Trasfer Diffusio i Coetrated Solutios 3-7

28 Example 3..: Calulate the error assoiated with the eglet of diffusio-drive ovetio whe estimatig the evaporatio rate of 6 C 60 C. a) At 6 C the saturatio vapor pressure is p (sat) 37 mmhg p (sat) 37 Mole fratio p 760 Total flux at stead-state for oetrated solutio: D D 0 l l Total flux for dilute solutio: 0.05 D D D D j ( 0 ) ( ) Ol % error! Mass Trasfer Diffusio i Coetrated Solutios

29 b) At 60 C the saturatio vapor pressure is p (sat) 395 mmhg Mole fratio Coetrated solutio: D D 0 l l / D Dilute solutio: D ( ) D j D There is 40% error!! Mass Trasfer Diffusio i Coetrated Solutios 3-9

30 Mass Trasfer Diffusio i Coetrated Solutios l D 0 0 (6): Phsial piture for speies : (4): (5): l D d d D j

31 Example 3..3: Hdroge produtio b atalti rakig of CH 4 Methae gas is raked at the surfae of a solid atalst formig hdroge ad a solid arbo deposit. Goal: Total methae (molar) flux per uit area at stead state? Catalst surfae CH 4 CH 4(g) C (s) + H (g) Carbo deposit 0 H Mass Trasfer Diffusio i Coetrated Solutios 3-3

32 Choose ad simplif mass trasport equatio: Note: For proesses with hemial reatios, it is best to use the molar flux ad the molar average veloit! Thus, from Table 3..: with d d * * j + V D + v () v + v () v * + Now mole of CH 4 gives moles of H, flowig i the opposite diretio. Therefore, i eq. (): So: * v d D d Mass Trasfer Diffusio i Coetrated Solutios 3-3

33 Use molar fratios d d D ( + ) d D (3) d B.C.: 0:,0 0 (due to deompositio) L:,L (some measured o. at L) Itegratio of (3) subjet to B.C.s ields: l ( + ) D L L Or the geeral form if,0 0: D L + l +, L,0 Mass Trasfer Diffusio i Coetrated Solutios 3-33

34 Example 3..4: Fast Diffusio ito Semi-Ifiite Slab A volatile liquid solute evaporates ito a log apillar. There is o solvet (air) flow aross the apillar, blowig the solute awa. As a result, the solute aumulates i the apillar. Iitiall the apillar otais o solute. As the solute evaporates the iterfae betwee the vapor ad the liquid solute drops. Goal: Calulate the solute evaporatio rate aoutig for diffusio-idued ovetio ad the effet of movig iterfae. Mass Trasfer Diffusio i Coetrated Solutios 3-34

35 Mass balae: solute aummulatio i A solute trasport i solute trasport out t ( ) A ( A ) ( A ) + Divide b A ad as 0: t Choose ad simplif mass trasport equatio: 0 j + v 0 with v V v + V v V + V 0 I (): D v () t Now fid a expressio for v 0! () Mass Trasfer Diffusio i Coetrated Solutios 3-35

36 Mass Trasfer Diffusio i Coetrated Solutios 3-36 I the ustead ase, the solvet flux varies with positio ad time but the solvet gas does ot dissolve i the liquid, thus at the iterfae (0): 0. ( ) 0 0 D V (3) ( ) 0 0 V D V V D ad ( ) 0 0 V D ( ) ost V V D V v 0 0 So,

37 Mass Trasfer Diffusio i Coetrated Solutios 3-37 t 0 > 0 0 t > 0 0 (sat) 0 Boudar oditios: Defie ombied variable (as i the dilute ase): t D ζ 4 with B.C. ζ 0 (sat) ζ 0 V D V D t 0 + (4) Isert i ():

38 (4) beomes: d d + ( ζ Φ) 0 (5) dζ dζ V ζ where Φ V (6) ζ 0 I eq. (5) Φ is a dimesioless veloit harateriig the ovetio b diffusio ad the movemet of the iterfae. Note that if Φ 0 the problem redues to that of diffusio i dilute oetratios!! ζ Eq. (5) is itegrated to give: ( ) [ ] os tat exp ( ζ Φ) Mass Trasfer Diffusio i Coetrated Solutios 3-38

39 d itegratio ad isertio of B.C.: ( sat) erf + erf ( ζ Φ) ( Φ) (7) eq. (6) (7): V (sat) + π ( ( )) [ ] + erf Φ Φ exp Φ Calulate ow also the iterfaial flux (see eq. 3) N eq. 4 0 D V D / π t (sat) 0 dilute limit V (sat) exp + erf [ ] Φ ( Φ) Mass Trasfer Diffusio i Coetrated Solutios 3-39

40 Mass Trasfer Diffusio i Coetrated Solutios 3-40