Diffusion Mass Transfer

Transcription

1 Dffuson Mass Transfer

2 General onsderatons Mass transfer refers to mass n transt due to a speces concentraton gradent n a mture. Must have a mture of two or more speces for mass transfer to occur. The speces concentraton gradent s the drvng potental for transfer. Mass transfer by dffuson s analogous to heat transfer by conducton. Physcal Orgns of Dffuson: Transfer s due to random molecular moton. onsder two speces and B at the same T and p, but ntally separated by a partton. Dffuson n the drecton of decreasng concentraton dctates net transport of molecules to the rght and B molecules to the left. In tme, unform concentratons of and B are acheved.

3 : Molar concentraton of speces. ρ : Densty (kg/m 3 ) of speces. M : Molecular weght (kg/kmol) of speces. ρ M * : Defntons 3 ( kmol/m 3 ) J Molar flu kmol/s m of speces due to dffuson. Transport of relatve to molar average velocty (v*) of mture. N : bsolute molar flu kmol/s m of speces. Transport of relatve to a fed reference frame. j : Mass flu kg/s m of speces due to dffuson. Transport of relatve to mass-average velocty (v) of mture. n : bsolute mass flu kg/s m of speces. Transport of relatve to a fed reference frame. : Mole fracton of speces ( / ). m : Mass fracton of speces ( m ρ / ρ ).

4 Property Relatons 4 Mture oncentraton: Mture Densty: ρ ρ ρ m m ρ Mture of Ideal Gases: p R R T p ρ RT p p p p R Unversal deal gas constant -gas constant

5 Molar and Mass Flues of Speces due to Dffuson n a Bnary Mture of Speces and B 5 Molar Flu of Speces : By defnton: ( J v v ) v v + v B B From Fck s law (mass transfer analog to Fourer s law): D J B Bnary dffuson coeffcent or mass dffusvty (m /s) Mass Flu of Speces : By defnton: j ρ v v v m v + m v B B From Fck s law: j ρd m B

6 bsolute Molar and Mass Flues of Speces n a Bnary Mture of Speces and B 6 Molar Flu of Speces : N J + v v N J + v + v B B N D + N + N Mass Flu of Speces : B B n ρ v j + ρ v ( v v ) n j + ρ m + m B B n ρd m + m n + n B B Specal ase of Statonary Medum: v 0 N J v 0 n j cheved to a good appromaton for (or m ) << and N (or n ) 0. B B

7 onservaton of Speces pplcaton to a ontrol Volume at an Instant of Tme: dm M & &,n M&,out + M&,g M&,st dt M &, M& M,n n,, M,out out, M & M M & M g,,g,st st, rate of transport across the control surfaces rate of generaton of due to homogeneous chemcal reactons occurrng n the control volume rate of accumulaton of n the control volume 7 pplcaton n artesan oordnates to a Dfferental ontrol Volume for a Statonary Medum of onstant D B and or ρ : Speces Dffuson Equaton on a Molar Bass: + y + z + N& D B D Speces Dffuson Equaton on a Mass Bass: ρ + ρ y + ρ z + n& D B B D B t ρ t

8 Boundary ondtons (Molar Bass): onsder a Gas () / qud (B) or Gas () / Sold (B) Interface. 8 Known surface concentraton: 0, s For weakly soluble condtons of a gas n lqud B, p s, H (Henry s law) H Henry's const ant (Table. 9) For gas n a unform sold B, S ( 0) Sp ( kmol/m 3 bar) solubl ty (Table.0) Heterogeneous (surface) reactons (atalyss) N '', (0) N& '' D B d d 0

9 9 Specal ases for One-Dmensonal, Steady-State Dffuson n a Statonary Medum Dffuson wthout Homogeneous hemcal Reactons For artesan coordnates, the molar form of the speces dffuson equaton s d 0 d () Plane wall wth known surface concentratons: (, s,, s, ) +, s, d D N, J, DB d D N N R mdff, D (,,,, ) B s s ( ) B,, s,, s,, B Results for cylndrcal and sphercal shells Table 4.

10 Planar medum wth a frst-order catalytc surface: 0 ssumng depleton of speces at the catalytc surface ( 0), N '', (0) N & d D k B '' k d 0 '' Reacton rate constant (m/s) ( 0) ( o)

11 ssumng knowledge of the concentraton at a dstance from the surface,, Soluton to the speces dffuson equaton () yelds a lnear dstrbuton for : + k / DB, + / B k D Hence, at the surface, ( 0), + k / D ( 0) B d k, N DB d 0 k D mtng ases: Process s reacton lmted: k D << k / B 0 + ( ) / B k k / DB Process s dffuson lmted: ( 0) ( ) N ( 0 ) N ( 0 ) k, ( 0) 0 >> DB,

12 Equmolar counterdffuson: Occurs n an deal gas mture f p and T, and hence, are unform. N N N, B, D p p T, 0,, 0, B, DB R

13 Dffuson wth Homogeneous hemcal Reactons For artesan coordnates, the molar form of the speces dffuson equaton s DB + N& 0 For a frst-order reacton that results n consumpton of speces, N& k and the general soluton to the dffuson equaton s m + e e m / m k D / B 3

14 onsder dffuson and homogeneous reacton of gas n a lqud (B) contaner wth an mpermeable bottom: 4 Boundary condtons ( 0 ), 0 d d 0 Soluton ( ) m ml m, 0 cosh tanh snh ( 0) 0 N, DB, m tanh ml

15 Evaporaton n a olumn: Nonstatonary Medum 5 Specal Features: Evaporaton of from the lqud nterface (, 0, sat( v) >, ) Insolublty of speces B n the lqud. Hence downward moton by dffuson must be balanced by upward bulk moton (advecton) such that the absolute flu s everywhere zero. N 0 B, Upward transport of by dffuson s therefore augmented by advecton.

16 6 Soluton:,, 0, 0 / D B, N, n, 0

17 7 One-Dmensonal, Transent Dffuson n a Statonary Medum wthout Homogeneous hemcal Reactons Speces Dffuson Equaton n artesan coordnates D B t Intal and Boundary ondtons for a Plane Wall wth Symmetrcal Surface ondtons (, 0) (, ) t,, s 0 0 Non dmensonal form γ, s, s, t m D B t Fo m Mass transfer Fourer number

18 Speces Dffuson Equaton γ γ Fo m 8 Intal and Boundary ondtons γ γ γ (, ) ( Fo ) 0 0 0, m 0 nalogous to transent heat transfer by conducton n a plane wall wth symmetrcal surface condtons for whch B, and hence T. Hence, the correspondng one-term appromate soluton for conducton may be appled to the dffuson problem by makng the substtutons θ γ Fo Fo m Table 4. summarzes analogy between heat and mass transfer varables. s T

19 N J + v v J * * ( v v ) 9 N J + v + v B B v * v + B v B J * D B ( v + v ) ( v + v ) B * ( v + v ) v B B B N B B + N '' B v * v * N D + N + N B B