Mass transfer in multi-component mixtures

Transcription

1 Chapters -0 ex. 7, of 5 of boo See also Krshna & Wesselngh Chem. Eng. Sc. 5(6) Mass transfer n mult-component mxtures Ron Zevenhoven Åbo Aadem Unversty Thermal and Flow Engneerng Laboratory tel. 33 ; ron.zevenhoven@abo.f februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ /76 Old-school mass transfer februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ /76

2 Mass Transfer as you have learned t J : flux of speces wth respect to the mxture J D dc c dz Fc s law c J D c z dffusvty c mass transfer coeffcent D z z phases α, β c c c februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 3/76 Dffuson wth Drft N : flux wth respect to an nterface dfferental equaton N D dc Nx dz flux of mxture N N dfference equaton N c Nx sc dffuson flux drft flux Stefan or drft correcton februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 4/76

3 Classc - n gases gas: c c c constant fluxes wth respect to mxture only one bnary D, whch s ndependent of composton J D dc dz J D dc dz + J J D dc c 0 dz D 0 x februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 5/76 Three gases () deal gases, 00 Pa, 98 K A N, H N, CO B begnnng: x N 046. x H 054. x N 05. x CO 048. Queston: Does N transfer (a) from A to B? (b) from B to A? (c) not at all? (d) or does t do (a), (b) and (c)? februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 6/76

4 mole fracton x N H A B Three gases () reverse dffuson CO tme h februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 7/76 Two catons caton permeable membrane hgh concentraton H + Na + low concentraton Cl - Cl - H + moves rapdly H + so Na + can move aganst ts concentraton gradent! Na + 3 excess +charge and electrcal feld februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 8/76

5 He 98 K 00 Pa Two gases n a porous plug () Ar 98 K 00 Pa frcton (He / plug) < frcton (Ar / plug) N He the plug, matrx or membrane s a (pseudo)component 3N M M Ar He Ar! februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 9/76 Two gases n a porous plug () He 98 K 00 Pa Ar 98 K N He N 0 Pa(for example) Ar p man reason: vscous flow retards He, accelerates Ar februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 0/76

6 Drvng forces februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ /76 Gravty - a smple potental m g g F Δz the potental dfference s the wor requred to change the condton of the weght here: mg z 98. J ( 98. Nm) or, per mole Mgz the drvng force s the negatve potental gradent: d F Mg dz the force s downwards februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ /76

7 x mxture wor requred: change n the chemcal potental pure (one mole) Chemcal potental chemcal potental const actvty p, T RTlna const p, T a x actvty coeffcent RTlna februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 3/76 n an deal soluton chemcal potental n an deal soluton const ( p, T ) RT ln x n an deal gas p const( p, T ) RT ln p partal pressure februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 4/76

8 Momentum balance forces F momentum n Σ(ṁv) n momentum out Σ(ṁv)out change of momentum d( mv ) dt ((ṁv) mv ) n ((ṁv) mv out n ) out F februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 5/76 Movng through each other H z z dz CO () () u u speces veloctes februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 6/76

9 Forces on hydrogen () drvng dp force dz A z p area A Drvng force F RTp z, z dz ) ζ, A z dz volume Adz februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 7/76, p, frcton force ζ, dp Force balance: pp(u u ) force per volume dz p RT dp wth c gves RTp (u u ) force per mole RT p dz wth frcton coeffcent (u u ζ RT D x (u u ) RT D p p ( u ) u Note: often x and u = 0 Gases: u ~ 0 - m/s Lquds: u ~0-4 m/s Drvng force (per mole of ) F d dz d lna RT dz RT da a dz RT dx x dz for a gven T and p n deal solutons februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 8/76

10 Maxwell-Stefan equaton drvng force on frcton coeffcent between and F, x u u mole fracton of (dffusve) speces veloctes februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 9/76 Multcomponent Dffuson - n Words the drvng force on a speces n a mxture the sum of the frcton forces between and the other speces the frcton exerted by on s proportonal - to the fracton of n the mxture and - to the dfference n velocty between and. februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 0/76

11 Flm theory, thcness of flms two thn, one dmensonal flms next to the phase boundary gases z 0 4 m lquds z 0 5 m eddes & large scale convecton flm : no eddes phase boundary z 0 0 membrane n a sold partcle 7 4 m z z d d 0 februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ /76 Dfference form of force F z lna RT z RT a a z RT x x z for a gven T and p n deal solutons februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ /76

12 + 0 - Approxmaton RT 0 a a exact a a a ln a approxmate a a 0.5 a a - approxmate wors out better n dfference equatons februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 3/76 Forces n a glass of beer growng bubble of CO Note: u = 0 x F RT dx x dz x z 0 5 m mole fracton of CO RT x x z februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 4/76

13 Example (3. from boo) februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 5/76 Example (3. from boo): answer februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 6/76

14 3 Frcton februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 7/76 Frcton coeffcents of spheres coeffcent of a sngle sphere, = A3 d 30 Nmol ms spheres lqud A 60 3 molecules mol - F usng: Stoes Law februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 8/76

15 Dffuson and frcton coeffcents Ð, RT Ð, A 3 d RT, s Maxwell-Stefan dffusvty of large molecules n dlute lquds (not gases), RT Ð each others nverse we use both, 0 9 m februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 9/76 One equaton mssng components: relatve velocty ndependent equaton 3 components: relatve veloctes ndependent equatons n components: n - relatve veloctes n - ndependent equatons februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 30/76

16 Bootstrap () only relatve veloctes bootstrap F x ( u u ), floatng transport relatons: have to be ted to surroundngs februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 3/76 Bootstraps () N H N CO He Ar no net volume flow plug does not move H + Cl - Na+ Cl - membrane does not move (almost) no charge transfer februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 3/76

17 Fluxes n practcal problems we use fluxes: N uc ucx flux form of MS-equaton: F cx cx x ( u u ), f ( x N x N ), F / V ; x = c / c force on per unt volume of mxture februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 33/76 From dfferental to dfference x a x postve drecton RT da RT bnary: x u u u a dz Ð x a nfntesmal layer fnte layer (approxmate) a a da a, u x Ð, x u u, u dz, Ð, z februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 34/76

18 0 c flm u Average velocty average concentraton c speces velocty (depends on poston n flm) 0 u speces velocty at the average composton postve velocty februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 35/76 Dfferences wth fluxes a x( u u a, ) cx c a x ( xn xn a c, ) februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 36/76

19 Multcomponent equatons usng veloctes a a x ( u u for deal solutons, ) usng fluxes x a a x ( x N x N ), c februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 37/76 Transfer coeffcents, Ð, z, m s n pores gases 0 m s lquds 0 4 m s 0-6 n pores februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 38/76

20 Temperature effects MS-equaton F ζ, x (u drvng force d F dz dfference form: RT x F x z T u ) (thermal dffuson terms) small RT dx x dz at constant temperature average flm temperature changes are not very mportant februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 39/76 4 Bnary examples februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 40/76

21 drops on a tray gas: trace of NH 3 () bul of N () Strppng - dlute transport relaton x xn xn c, x N 0 bootstrap x x flux, cx 0..as you already new.. februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 4/76 N Strppng - concentrated x x 05. x N c, x,c x x 0 februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 4/76

22 Vaporsng droplet heat benzene (), volatle toluene () y K x x x x x y K x vapour removed by convecton bootstrap: N y N y februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 43/76 Fluxes from vaporsng droplet x x N x N c, x x N x N c, N N N x c, x x N, c x x x example x x 05. N 4, cx N, cx Δx < 0 Δx > 0 Stefan (drft) correctons februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 44/76

23 Carbon gasfcaton O C CO both components are movng and have a hgh concentraton O () CO( ) C , 0 bootstrap: N ms N c 0 mol m 3 calculate N and N februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 45/76 N Fluxes n gasfcaton N x N xn xn c, c x x x x N c,, x exact exact : mol m N : mol m s N 0.09 s almost the same februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 46/76

24 Bnary dstllaton x x N N heptane () x x hexane () 0 N transport relaton x N x N x c, bootstrap N N (equmolar exchange) x x x N c,, cx N, cx februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 47/ Some bootstraps membrane stagnant bul stagnant (absorpton) trace stagnant (polarsaton) equmolar exchange (dstllaton) nterface determned (vaporsaton) reacton stochometry u M 0 N 0 u 0 N N 0 N y N y N N februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 48/76

25 5 Ternary examples februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 49/76 Ternary - per mole of d, x u u, x u u dz d, x u u, x u u dz forces per mole of forces per mole of februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 50/76

26 Ternary - per mole of mxture x d, xx u u 3, xx 3 u u3 dz x d, xx u u, xx u u dz forces per mole of mxture these should cancel: Ð Ð,,,,,, februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 5/76 More components bnary x x xn xn c, xn xn c, x3n xn c,3 x3n xn c,3 3 3 ternary quaternary februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 5/76

27 Condensor coolng water vapour Mx: NH 3 +H O + H NH 3 () and H O () condense on a tube H (3) does not condense 3, , 3, ms 3 fnd the veloctes n the gas flm ms lqud H O NH 3 februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 53/76 H Condenser () transport (MS) relatons: 0.4N 0.3N 0.3N 0.4N NH 3 : ( 0 )30 (3 0 )30 0.4N 0.3N 0.3N 0.3N H O : ( 0 )30 (3 0 )30 bootstrap N 3 0 three lnear equatons, three unnowns 0.05 exact solutons: N 0.03 N N mol m s N mol m s 3 3 februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 54/76

28 Condenser (3) mxture velocty H O moves down ts gradent HO NH 3 dragged aganst ts gradent H does not move at all NH 3 H februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 55/76 Ternary dstllaton () ethanol 3 water a trace of butanol vapour lqud large frcton between and, 80 ms 0 0 3, 3, m s - bootstrap: equmolar exchange uy uy uy n whch drecton does move? februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 56/76

29 Butanol - whch drecton? y y u u no moton u februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 57/76 Ammona reacton N 3H NH 3 transport relatons: H () N () NH 3 (3) catalytc surface x x x 3 x N x N c x N x N c 3 3, 3, x N x N c x N x N c x N x N c 3 3, 3, 3 3 x N x N c 3 3 3, 3, bootstrap: N N N 3 3 februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 58/76

30 When s: 3 =? a ternary can be approxmated as a bnary when x x x u u x u u 3 3, 3, x u u eff, eff eff februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 59/76 (example: moble speces n many membranes) equal velocty of two speces: 3 one frcton term domnates: x x 3 x u u 3 3, 3, 3, x x 3 x x x eff 3 u u3 u u, 3, eff,, 3, (example: Na + and Cl - n water) equal dffusvtes ( n m- and p-xylene) x x x x x u x u x u, 3, 3 3 3, x x eff 3, eff 3, u eff eff x u x 3 x u x februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 60/76

31 Effectve bnary n a reactve system smplfyng transport equaton of N n ammona formaton: elmnate N and N 3 wth N 3N N3 N x 3x x3 x N, effcx wth, eff, 3, smlarly for H and NH 3 If all fluxes N are related va the same reacton stochometry pseudo - bnary februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 6/76 Example: membrane Water from a saltwater reservor penetrates through a membrane, see the fgure. One result s that the salt concentraton rses near the membrane. a. Gve the Maxwell-Stefan (MS) the equaton for the salt, n dfferental form, as a functon of the fluxes N. b. For a mass transfer coeffcent, = 0.0 m/s, the mean substance concentraton ĉ = mol / m 3 and flux N = 300 mol / (m s) for the water, calculate the gradent Δx f x α = Gve the answer wth decmals accuracy. water () + salt () x α z=α x β x z=β membrane water z u u = 0, x << x water n boundary layer Δx = x β -x α x = x α + ½ Δx februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 6/76

32 Example: membrane membrane u = 0, x << x water u water () + salt () water x β x α x n boundary layer Δx = x β -x α x = x α + ½ Δx z=α z=β z februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 63/76 8 Non-dealty chemcal potental constant RT ln a deal soluton: a x F d dz RT d ln x dz RT z x x non-deal soluton: a x F d dz RT d lna dz RT z a a dfference equaton februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 64/76

33 9 Dffusvtes n Gases molecules are lttle (hard) spheres movng around wth ther thermal velocty and occasonally bumpng nto each other Ð Ð, 3, A 3 / RT pd M M , emprcal modfcaton: / d, 75. T 3 / / p M M 3 d / d d d dffuson volume lqud volume, m 3 mol - M = molar mass g/mol februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 65/76 Gases, emprcal equaton Ð, T 3 / / p M M 3 / dffuson volume lqud volume, m 3 mol m mol H N CO NH 3 H O februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 66/76

34 Gases, dffusvty example N () CO () T 300 K p 0 5 Pa 8 Ð, / 6 3 / / m s februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 67/76 Dlute Speces n a Lqud u d A 3 / ths constant vares wth the sze rato of the speces: Ð RT 9, 0 m s A d d 3 RT Ð, 3 d A d d d d d d 0 4 februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 68/76

35 Non-deal Bnary ethanol () - water () MS and Fc dffusvtes dffer Ð only at x Ð D,, and x s D,, nterpolaton roughly logarthmc for Ð Ð 0 5, D m s,, o C D, Ð, 0 x februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 69/76 0 Dmensonless Groups Sherwood number mass transfer Sh, d, Ð, dameter flm thcness Reynolds number flud flow Re vd Re lamnar Re turbulent Schmdt number mxture property Sc, Ð, Sc n gases Sc 0 3 n lquds februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 70/76

36 Mass transfer coeffcents outsde rgd nterfaces for short tmes for small partcles for large partcles boundary layer theory Ð, 3. t Ð, d g 03. Ð,,, and Ð, are values n the flud outsde,, 3 / februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 7/76 Moble or Rgd? surface actve agents swept to the bac cause surface tenson gradents d these mmoblse the drop or bubble when d f g / f s the foulng (fudge) factor f 0. n clean lquds f n drty lquds the transton s at d mm februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 7/76

37 Mass transfer coeffcents Bubbles wth moble nterfaces bubbles: outsde g 04. 3, Ð, 6 / nsde (?) d g 04. d 3, Ð, 6 / Ð, d Ð, and are for the lqud (outsde) februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 73/76 Mass transfer coeffcents moble drops n a lqud moble drops n a lqud calculate the rgd and moble (bubble) coeffcents separately (these dffer for the outer and nner coeffcents) nterpolate usng d Ð, d Ð, d moble rgd,,, februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 74/76

38 Mass transfer coeffcents drops n a gas d g / d 4 g / rgd sphere moble oscllatng drop drop shatters februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo RZ 75/76 februar 08 Åbo Aadem - Bsopsgatan 8, 0500 Åbo TRP RZ 76/76