Unsteady Mass- Transfer Models

Transcription

1 See T&K Chaper 9 Unseady Mass- Transfer Models ChEn 6603 Wednesday, April 4,

2 Ouline Conex for he discussion Soluion for ransien binary diffusion wih consan c, N. Soluion for mulicomponen diffusion wih c, N. Film heory revisied (surface renewal models) Transien diffusion in droples, bubbles Wednesday, April 4,

3 Conex Film heory - so far we assumed seady sae, no reacion in bulk (only poenially a inerface) Mass Transfer Coefficiens used o simplify problem - don fully resolve diffusive fluxes. Boosrap problem solved (via definiion of [β]) o obain oal species fluxes. Unseady cases? Wha if we wan o know he ransien concenraion profiles? Wha if we wan o consider he effecs of ransien (perhaps urbulen) mixing near he inerface? Here we consider unseady film-heory approaches... Hung Le and Parviz Moin. hp:// Wednesday, April 4, 3

4 Soluion Opions c i = N i + r i Describes evoluion of ci a all poins in space/ime, bu requires Ni, which may involve soluion of he momenum equaions... Solve he problem numerically Allows us o incorporae full descripion of he physics may be quie complex (paricularly if we mus solve for a non-rivial velociy profile...) Could also simplify porions (consan [D], c, ec.) Can solve his for a variey of BCs, ICs Make enough assumpions/simplificaions o solve his analyically Differen BC/IC may require differen form for analyic soluion We already did his for effecive binary and linearized heory for a few simple problems (-bulb problem, Loschmid ube) T&K chapers 5 & 6. Here we show a few more echniques, based on unseady film heory Don resolve mass ransfer compleely - ge a coarser descripion of he diffusive fluxes... (J) = c [k ]( x) (N) = [ ](J) Approximaion for diffusive flux (oal flux). Wednesday, April 4, 4

5 See T&K 9. Binary Formulaion (/3) c c c i = c x i = x i + N N i + r i Wha are he assumpions? N i x i + N x i = x i z = Wha happened here? J i One-dimensional... z J i Problem saemen: semi-infinie diffusion BCs & ICs z 0, =0, xi=xi. (Iniial condiion) z = 0, >0, xi=xi0. (Boundary condiion) z, >0, xi=xi. (Boundary condiion) x + N x c z = D x z (binary) valid for shor conac imes (more laer) Wednesday, April 4, 5

6 Binary Formulaion (/3) c x + N x c z = D x z dx dx d + N z d + N z dx c d D d x d + ( N c p )dx d d x = c D i z d = D d x d = 0 Observaion: since x is dimensionless, z,, D mus appear in a dimensionless combinaion in he soluion. = z p 4 Solve using order reducion chosen for convenience, ζ /D is dimensionless x x,0 x, x,0 = erf + erf = d d z = d d z = d d p D pd = z = d z d = z d d z d d Noe ha his is a funcion of boh z and. x = x,0 =0 x = x, = Wednesday, April 4, 6

7 Binary Formulaion (3/3) x x,0 x, x,0 = erf + erf p D pd Calculae J a z=0, J,0 = c r D exp + erf D pd (x,0 x, ) Wha happens o J as z? Mass Transfer Coefficiens (binary sysem): Low-flux limi (as N 0) r r D D J,0 = c (x,0 x, ) k = = exp + erf D pd J,0 = c k (x,0 x, )=c k (x,0 x, ) N,0 = c 0 k (x,0 x, ) Wednesday, April 4, 7

8 See T&K Mulicomponen Sysem (x) + N c (x) z =[D] (x) z = z p 4 N c p This has he analyic soluion (see T&K ): h h ii h (x x )= [I] erf ( )[D] [I]+erf (J 0 )= c h ih h p [D] exp [D] [I]+erf [D] h [D] ii (x0 x ) ii (x0 x ) remember ha hese are marix funcions! Low-flux limi (as N 0) J 0 = c p [D] (x 0 x ) h [k] =( ) [D] [ ]=exp [D] ih [I]+erf[ [D] i (J 0 ) = c [k][ ](x 0 x ) = c [k ](x 0 x ) (N 0 ) = c [ 0][k ](x 0 x ) Possible approaches: Solve he ransien problem (beware of he shor ime assumpion) Use his informaion o formulae oher seady-sae models (e.g. urbulen mixing from bulk o surface) Wednesday, April 4, 8

9 See T&K 9. Surface Renewal Models (J 0 ) = c [k][ ](x 0 x ) = c [k ](x 0 x ) (N 0 ) = c [ 0][k ](x 0 x ) [k] =( ) [D] (J), (N) are funcions of ime since [k] is a funcion of ime. How would we handle his problem? Idea: develop a model for k ha approximaes he effecs of ransiens near he surface. Concep: Fresh fluid from bulk is ranspored o he inerface, where diffusion occurs for some ime,. Then his is ranspored back o he bulk and replaced by more fresh fluid. Iniial & Boundary condiions: z = 0, x i = x i0 > 0 z 0, x i = x i = 0 z, x i = x i > 0 Assumes ha he bulk is unaffeced by mass ransfer ( shor conac imes) Age disribuion funcion, ψ(), deermines how long a fluid parcel is a he inerface. (will affec expression for [k]) Wednesday, April 4, 9

10 See T&K 9. Surface-Renewal Models Age disribuion funcion, ψ(), deermines how long a fluid parcel is a he inerface. (will affec expression for [k]) k ij () = D ij k ij = 0 k ij () ()d Aemps o model a saisically saionary process (fas mixing, no sauraion o bulk) by a seady sae [k]. Higbie model (935) Assumes ha all fluid parcels say a he inerface for a fixed amoun of ime, e. () = / e e 0 > e [k] = e [D] / Noe ypo in T&K ( vs. e ) Danckwers model (95) Fluid parcels have a greaer chance of being replaced he longer hey are a he inerface. () = s exp( s) [k] = s[d]/ s - rae of surface renewal (/sec) (fracion of surface area replaced by fresh fluid in uni ime) Wednesday, April 4, 0

11 See T&K 9.4 Bubbles, Drops, Jes δ δ δ Spheres Channels Cylinders [Fo] = [D] For small Fo (Fo«), we are safe o use surface renewal conceps. Wha happens a large Fo (Fo )? x i = x ii x i xi - mixing cup average Wednesday, April 4,

12 See T&K 9.4 Fracional Approach o EQ (J 0 ) = c [k][ ](x 0 x ) = c [k ](x 0 x ) (N 0 ) = c [ 0][k ](x 0 x ) x is changing! (his soluion is no valid) F (x 0 x ) (x 0 x I ) (x 0 x ) = [F ](x 0 x I ) Binary Mulicomponen x0 - iniial/boundary composiion (=0) xi - inerface composiion (consan in ime) x - average composiion (changing in ime), use in place of x. For a spherical drople/paricle: 6 [F ] = [I] m exp m Fo ref [D ] m= [D ] = D ref [D] Fo ref D ref r 0 noe: 6 m= m = [Sh] = 3 m= Sh [k] r 0 [D] exp m Fo ref [D ] m= Sherwood number relaed o F/ Fo. m exp m Fo ref [D ] remember ha hese are marix funcions! Limiing cases: Fo ref = Sh = 3 [I] [k] = Fo ref = [k] = [D] / 3r 0 [D] See fig. 9.7 (L Hopial s rule) Wednesday, April 4,