Boundary Layer Theory:

Transcription

1 Mass Transfer Bondar Laer Theor: Mass and Heat/Momentm Transfer Letre,..7, Dr. K. Wegner

2 9. Basi Theories for Mass Transfer Coeffiients 9. Flid-Flid Interfaes (letre of 5..7) Flid-flid interfaes are tpiall not fied and are strongl affeted b the flow leading to heterogeneos sstems that make it diffilt to deelopment a general theor behind the MT orrelations (Cssler Table 8.3-). 9. Flid-Solid Interfaes Solids tpiall hae fied and well-defined srfaes, allowing to deelop theoretial fondations for the empirial MT orrelations (Cssler Table 8.3-3). In ontrast to flid-flid interfaes the models are far more rigoros bt, nfortnatel, omptationall INTENSIVE. Mass Transfer Bondar Laer Theor 9-

3 In addition to this, flid-solid interfaes hae been inestigated intensel with respet to heat transfer. We an make se of this de to the analog between heat momentm and mass transfer. 9.. MT from a plate (bondar laer theor) Eample: A sharp-edged, flat plate that is sparingl dissolable is immersed in a rapidl flowing solent. MT orrelation (Table 8.3-3): Sh kl D L.646 D 3 L: plate length; : blk flid eloit Mass Transfer Bondar Laer Theor 9-3

4 Prandtl first introded the onept of bondar laers. The transition from zero eloit at the plate to the eloit of the srronding free stream takes plae in the bondar laer. Goal: Callate the MTC for this flid solid interfae Literatre: R.B. Bird, W.E. Steward, E.N. Lightfoot, Transport Phenomena, nd ed. J. Wile&Sons. H. Shlihting, K. Gersten, "Bondar Laer Theor", 8th ed., Springer 999. Mass Transfer Bondar Laer Theor 9-4

5 Proedre:. Callate the eloit profile in the B.L.. Callate the onentration profile in the B.L. 3. Callate the fl at the interfae j D and set it eqal to k to obtain k. Mass Transfer Bondar Laer Theor 9-5

6 The relatie magnitde of the flid flow (momentm) B.L. and the onentration B.L. is gien b S D kinemati isosit diffsiit S > : Momentm B.L. > Conentration B.L. (tpial) S : Momentm B.L. Conentration B.L. S < : Momentm B.L. < Conentration B.L. (rare ase) Mass Transfer Bondar Laer Theor 9-6

7 Mass Transfer Bondar Laer Theor 9-7 Laminar B.L. Continit eqation: () Momentm eqation in (for stead flows): z w ( dim.) ρ p, onst. shear stress on plate () for fll deeloped, inisid flow with onst.

8 Momentm eqation in (for stead flows): p ρ As tpiall is small in the BL (for lam. ase here ), this redes to: p (Negligible pressre hanges in (thin laer) (3) Bondar onditions: : and : (flid adheres to plate) Mass Transfer Bondar Laer Theor 9-8

9 Eqations () and () are soled b introding the ariable Digression: How did Prandtl ome p with that? The B.L. thikness, δ, is related to Re b δ ~ Re δ ~ Re Mass Transfer Bondar Laer Theor 9-9

10 Eq. () an be simplified b formlating it in terms of a stream fntion Ψ. In order to satisf the ontinit eqation (), the eloit omponents mst be: Ψ Ψ Ψ Ψ The orresponding stream fntion is: Ψ f( ) where f() is the dimensionless stream fntion. Ψ Ψ 3 Ψ 3 Ths, Ψ Ψ f f Mass Transfer Bondar Laer Theor 9-

11 3 f f f and f Ψ f f 3 f f ( f f ) Mass Transfer Bondar Laer Theor 9-

12 Now the eqation of motion () beomes: f f f at : f and f : f (see Table) Blasis obtained the soltion in the form of a series epansion at and ( Blasis series ) and the two forms are mathed at a sitable. See attahed Table from H. Shlihting ( Bondar Laer Theor ), whih gies the omplete ales of and for eer and. The eloit profiles hae been allated. Mass Transfer Bondar Laer Theor 9-

13 Sore: H. Shlihting, Bondar Laer Theor Mass Transfer Bondar Laer Theor 9-3

14 Blasis similarit soltion of the eloit distribtion in a laminar bondar laer on a flat plate. P.K. Knd, I.M. Cohen, Flid Mehanis nd ed., Aademi Press Mass Transfer Bondar Laer Theor 9-4

15 Mass Transfer Bondar Laer Theor 9-5 Mass Transfer in Bondar Laers The diffsion eqation in the B.L. is: D B.C. s: : : Introde dimensionless ariables:,, negligible sall D B.C.s: : : (4) (5)

16 Mass Transfer Bondar Laer Theor 9-6 Compare to the momentm eqation (): negligible sall B.C. s: : and : The two eqations hae the same form! S : In this speial ase where D the soltion to the onentration profile is gien b or D The bondar laers for flow and onentration are the same.

17 j D For this ase we write Fik s first law j j D ( ) with : ( ) ( ) j j f.33 f from Table ( ) ( ) Mass Transfer Bondar Laer Theor 9-7

18 As j k ( - ), k.33 Sh k D.33 D.33.33S Re D Here, S : Sh.33Re This is the Loal Mass Transfer Coeffiient to be distingished from the aerage MTC oer a plate of length L (Cssler Table 8.3-3): Sh.646 Re S 3 Mass Transfer Bondar Laer Theor 9-8

19 A more realisti ase is with S >> whih is tpial in liqids and partile / gas sstems. S >> : (S.K. Friedlander, Smoke, Dst and Haze, st ed., 977) Eamples: Dissoltion of solid omponds in flowing liqid streams. Flow (deposition) of polltants oer lakes or leaes Mass Transfer Bondar Laer Theor 9-9

20 More speifiall, if the B.L. (displaement) thikness is: δ.7 / (Blasis, Shlihting) For a wind blowing at a speed of km/h, δ.5 mm. E.g., at m from the leading edge of the flat srfae: δ 6 5 Pa s.m 3 kg m.7 m 36s m.56mm Goal: To obtain the mass transfer oeffiient Mass Transfer Bondar Laer Theor 9-

21 Mass Transfer Bondar Laer Theor 9- Again at stead-state D Epand and near the wall ( ) into Talor series: ( ) f ( ) ) f( ) ( f a () f () f () f disregard higher order terms 3 4 a () f 3 () f f() (5)

22 Mass Transfer Bondar Laer Theor 9- Sbstittion in the aboe differential eqation (5) gies: Assme that is a ft onl of : ) g( Then: g D 4 a a g g g / with

23 Mass Transfer Bondar Laer Theor 9-3 Introding the aboe ariables in the diffsion eqation: or g g Soltion: Set g P where g D g 4 a g a g D 4 a g with P S 4 a P g P Integrate: S a P P ln 3

24 g P P 3 ep a S From the B.C s.: at, g : g P ep a S r 3 dr From the B.C s.: at, g, hene: P a 3 ep S r dr m Mass Transfer Bondar Laer Theor 9-4

25 and 3 a S g 3 ep -m dm a S Γ( 3) Remember (Gamma fntion): Γ(n) (n ) Γ(n ) ep [ ] n d Mass Transfer Bondar Laer Theor 9-5

26 j D ( ) D - ( ) g D - 3 D a S -.89 ( ) S Re -.89 k ( ) Mass Transfer Bondar Laer Theor 9-6

27 So k.34 S / 3 Re / This is the loal mass transfer oeffiient while the oerall is fond b: L j A - D d L -/3 /.68 S Re ( - ) Mass Transfer Bondar Laer Theor 9-7

28 Let s allate Sh to ompare it with or Table of orrelations kl L D D /3 / LD.68 /3 / / D L /3 /.68 S Re /3 / L.68 D.68 S Re /3 / So we obtained the mass transfer oeffiient from theor! Mass Transfer Bondar Laer Theor 9-8

29 In general j k onst S / 3 Re / Appliable for flow arond spheres, linders et. Mass transfer orrelation oer a wide range of Re: Sh.6Re / S / 3 Mass Transfer Bondar Laer Theor 9-9

30 9.3 Theories for onentrated soltions The MTC s are based tpiall on dilte soltions bt the are VERY sessfl een in onentrated ones as, tpiall, the olme aerage eloit normal to the flid-flid interfae is rather small. When the MTC s fail, it is obsered tpiall that the k depends strongl on onentration espeiall when mass transfer is fast. The fl is written as ( ) N k i i ( ) ( V N V ) k i i N Mass Transfer Bondar Laer Theor 9-3

31 This framework an be sed to bild new film, penetration and srfae-renewal theories. Howeer, this is diffilt and we also hae the preios nknowns again (film thikness, ontat or residene times). Instead we are looking for orretions to the dilte mass transfer oeffiient b the film theor: k k MTC in onentrated soltion MTC in dilte soltion Mass Transfer Bondar Laer Theor 9-3

32 The total fl for a onentrated soltion aross a film: d n D with B.C. s: z : i dz z L For onentrated soltions, n and are onstant, so the aboe eqation an be integrated from z to L to gie Rearranging: n / ep n / i Compare: N k( ) i i [ L / D] n ep N z i i [ ] L /D ( ) Mass Transfer Bondar Laer Theor 9-3

33 Then the mass transfer oeffiient is k ep [ L /D] For dilte soltions: k D L Taking the ratio k/k, we eliminate the dependen in L: k k ep / k [ / k ] Mass Transfer Bondar Laer Theor 9-33

34 k k an be smaller or larger than k depending on the diretion of. k k e k k Fl awa from interphase Fl toward interphase k k Sore: Bird, Stewart, Lightfoot, "Transport Phenomena" k Mass Transfer Bondar Laer Theor 9-34