PRINCIPLES OF MASS TRANSFER

Transcription

1 PRIIPLES OF MSS TRSFER ITROUTIO Mass Tansfe When a component n a mxtue mgates n the same phase o fom phase to phase because of a dffeence n concentaton Examples of mass tansfe Evapoaton of wate n the open pal to the atmosphee offee dssolves n wate Oxgen dssolves n the soluton to the mcoogansm n the fementaton pocess Reacton occus when eactants dffuse fom the suoundng medum to the catalst suface Possble dvng foce fo mass tansfe oncentaton dffeence Pessue dffeence Electcal gadent, etc TYPES OF MSS TRSFER Tpes of Mass Tansfe. Molecula dffuson Tansfe of ndvdual molecules though a flud b andom movement Fom hgh concentaton to low concentaton E.g. a dop of blue lqud de s added to a cup of wate the de molecules wll dffuse slowl b molecula dffuson to all pats of the wate. To ncease ths ate of mxng of the de, the lqud can be mechancall agtated b a spoon and convectve mass tansfe wll occu.. onvectve mass tansfe Usng mechancal foce o acton to ncease the ate of molecula dffuson E,g sted the wate to dssolve coffee dung coffee makng FE 3- PRT: MOLEULR IFFUSIO-

2 PRT I- MOLEULR MSS TRSPORT. Intoducton to mass tansfe. Popetes of mxtues.. oncentaton of speces.. Mass veaged veloct.3 ffuson flux.3. Fck s Law.3. Relaton among mola fluxes.4 ffusvt.4. ffusvt n gases.4. ffusvt n lquds.4.3 ffusvt n solds.5 Stead state dffuson.5. ffuson though a stagnant gas flm.5. Pseudo stead state dffuson though a stagnant gas flm..5.3 Equmola counte dffuson..5.4 ffuson nto an nfnte stagnant medum..5.5 ffuson n lquds.5.6 Mass dffuson wth homogeneous chemcal eacton..5.7 ffuson n solds.6 Tansent ffuson.. Intoducton to Mass Tansfe When a sstem contans two o moe components whose concentatons va fom pont to pont, thee s a natual tendenc fo mass to be tansfeed, mnmzng the concentaton dffeences wthn a sstem. The tanspot of one consttuent fom a egon of hghe concentaton to that of a lowe concentaton s called mass tansfe. The tansfe of mass wthn a flud mxtue o acoss a phase bounda s a pocess that plas a majo ole n man ndustal pocesses. Examples of such pocesses ae: () () () (v) (v) speson of gases fom stacks Removal of pollutants fom plant dschage steams b absopton Stppng of gases fom waste wate euton dffuson wthn nuclea eactos condtonng FE 3- PRT: MOLEULR IFFUSIO-

3 Man of da-b-da expeences also nvolve mass tansfe, fo example: () () () lump of suga added to a cup of coffee eventuall dssolves and then eventuall dffuses to make the concentaton unfom. Wate evapoates fom ponds to ncease the humdt of passng-asteam Pefumes pesent a pleasant fagance whch s mpated thoughout the suoundng atmosphee. The mechansm of mass tansfe nvolves both molecula dffuson and convecton.. Popetes of Mxtues Mass tansfe alwas nvolves mxtues. onsequentl, we must account fo the vaaton of phscal popetes whch nomall exst n a gven sstem. The conventonal engneeng appoach to poblems of multcomponent sstem s to attempt to educe them to epesentatve bna (.e., two components) sstems. In ode to undestand the futue dscussons, let us fst consde defntons and elatons whch ae often used to explan the ole of components wthn a mxtue... oncentaton of Speces: oncentaton of speces n multcomponent mxtue can be expessed n man was. Fo speces, mass concentaton denoted b ρ s defned as the mass of, m pe unt volume of the mxtue. m ρ () V The total mass concentaton denst ρ s the sum of the total mass of the mxtue n unt volume: ρ ρ whee ρ s the concentaton of speces n the mxtue. Mola concentaton of,, s defned as the numbe of moles of pesent pe unt volume of the mxtue. defnton, FE 3- PRT: MOLEULR IFFUSIO-3

4 umbe of moles mass of molecula weght of m M n () Theefoe fom () & () n V ρ M Fo deal gas mxtues, P V n [ fom Ideal gas law PV nrt] RT n V P RT whee P s the patal pessue of speces n the mxtue. V s the volume of gas, T s the absolute tempeatue, and R s the unvesal gas constant. The total mola concentaton o mola denst of the mxtue s gven b.. Veloctes In a multcomponent sstem the vaous speces wll nomall move at dffeent veloctes; and evaluaton of veloct of mxtue eques the aveagng of the veloctes of each speces pesent. If ν s the veloct of speces wth espect to statona fxed coodnates, then mass-aveage veloct fo a multcomponent mxtue defned n tems of mass concentaton s, FE 3- PRT: MOLEULR IFFUSIO-4

5 ν ρ ν ρ ρ ν ρ smla wa, mola-aveage veloct of the mxtue ν * s ν * V Fo most engneeng poblems, thee wll be ttle dffeence n ν * and ν and so the mass aveage veloct, ν, wll be used n all futhe dscussons. The veloct of a patcula speces elatve to the mass-aveage o mola aveage veloct s temed as dffuson veloct (.e.) ffuson veloct ν - ν The mole facton fo lqud and sold mxtue, x,and fo gaseous mxtues,, ae the mola concentaton of speces dvded b the mola denst of the mxtues. x (lquds and solds) (gases). The sum of the mole factons, b defnton must equal ; (.e.) x b smla wa, mass facton of n mxtue s; w ρ ρ FE 3- PRT: MOLEULR IFFUSIO-5

6 Example. The mola composton of a gas mxtue at 73 K and.5 x 5 Pa s: etemne O 7% O % O 5% 68% a) the composton n weght pecent b) aveage molecula weght of the gas mxtue c) denst of gas mxtue d) patal pessue of O. alculatons: Let the gas mxtue consttutes mole. Then O O O.7 mol. mol.5 mol.68 mol Molecula weght of the consttuents s: O O O x 6 3 g/mol g/mol + x 6 44 g/mol x 4 8 g/mol Weght of the consttuents s: ( mol of gas mxtue) O O O.7 x 3.4 g. x 8.8 g.5 x g.68 x g Total weght of gas mxtue g omposton n weght pecent:.4 O x 7.3% 3.68 FE 3- PRT: MOLEULR IFFUSIO-6

7 .8 O x 9.3% O x.5% x 6.6% 3.68 Weght of gas mxtue veage molecula weght of the gas mxtue M umbe of moles 3.68 M g mol ssumng that the gas obes deal gas law, PV nrt n V P RT n mola denst ρ m o V Theefoe, denst (o mass denst) ρ mm Whee M s the molecula weght of the gas. 5 PM (.5x ) x3.68 enst ρ m M kg RT 834x73 m 3.3 kg/m 3 Patal pessue of O [mole facton of O ] x Total pessue 7 x 5 (.5x ).7 (.5 x 5 ).5 x 5 Pa FE 3- PRT: MOLEULR IFFUSIO-7

8 .3 ffuson flux Just as momentum and eneg (heat) tansfes have two mechansms fo tanspotmolecula and convectve, so does mass tansfe. Howeve, thee ae convectve fluxes n mass tansfe, even on a molecula level. The eason fo ths s that n mass tansfe, wheneve thee s a dvng foce, thee s alwas a net movement of the mass of a patcula speces whch esults n a bulk moton of molecules. Of couse, thee can also be convectve mass tanspot due to macoscopc flud moton. In ths chapte the focus s on molecula mass tansfe. The mass (o mola) flux of a gven speces s a vecto quantt denotng the amount of the patcula speces, n ethe mass o mola unts, that passes pe gven ncement of tme though a unt aea nomal to the vecto. The flux of speces defned wth efeence to fxed spatal coodnates, s ν () Ths could be wtten n tems of dffuson veloct of, (.e., ν - ν) and aveage veloct of mxtue, ν, as ( ν ν ) + ν () defnton ν ν * ν Theefoe, equaton () becomes ( ν ν ) + ν ( ν ν ) + ν Fo sstems contanng two components and, ν ν ) + ( ν + ( ν ( ν ν ) + ( + ) ( ν ) + ν (3) ) The fst tem on the ght hand sde of ths equaton s dffusonal mola flux of, and the second tem s flux due to bulk moton. FE 3- PRT: MOLEULR IFFUSIO-8

9 .3. Fck s law: n empcal elaton fo the dffusonal mola flux, fst postulated b Fck and, accodngl, often efeed to as Fck s fst law, defnes the dffuson of component n an sothemal, sobac sstem. Fo dffuson n onl the Z decton, the Fck s ate equaton s J d d Z whee s dffusvt o dffuson coeffcent fo component dffusng though component, and d / dz s the concentaton gadent n the Z-decton. moe geneal flux elaton whch s not estcted to sothemal, sobasc sstem could be wtten as J d (4) d Z usng ths expesson, Equaton (3) could be wtten as d (5) d Z +.3. Relaton among mola fluxes: Fo a bna sstem contanng and, fom Equaton (5), o Smlal, J + J (6) J (7) ddton of Equaton (6) & (7) gves, J + J + ( (8) ) defnton + and +. Theefoe equaton (8) becomes, FE 3- PRT: MOLEULR IFFUSIO-9

10 J + J J -J d d (9) d Z Fom + d - d Theefoe Equaton (9) becomes, () Ths leads to the concluson that dffusvt of n s equal to dffusvt of n..4 ffusvt Fck s law popotonalt,, s known as mass dffusvt (smpl as dffusvt) o as the dffuson coeffcent. has the dmenson of L / t, dentcal to the fundamental dmensons of the othe tanspot popetes: Knematc vscost, νη (µ / ρ) n momentum tansfe, and themal dffusvt, α ( k / ρ ρ ) n heat tansfe. ffusvt s nomall epoted n cm / s; the SI unt beng m / s. ffusvt depends on pessue, tempeatue, and composton of the sstem. In table, some values of ae gven fo a few gas, lqud, and sold sstems. ffusvtes of gases at low denst ae almost composton ndependent, ncease wth the tempeatue and va nvesel wth pessue. Lqud and sold dffusvtes ae stongl concentaton dependent and ncease wth tempeatue. Table. : Geneal ange of values of dffusvt: Gases : 5 x x -5 m / s Lquds : m / s Solds : 5 x x - m / s In the absence of expemental data, sem theoetcal expessons have been developed whch gve appoxmaton, sometmes as vald as expemental values, due to the dffcultes encounteed n expemental measuements. FE 3- PRT: MOLEULR IFFUSIO-

11 .4. ffusvt n Gases: Pessue dependence of dffusvt s gven b (fo modeate anges of pessues, up to 5 atm) P and tempeatue dependenc s accodng to T 3 ffusvt of a component n a mxtue of components can be calculated usng the dffusvtes fo the vaous bna pas nvolved n the mxtue. The elaton gven b Wlke s mxtue 3 n n whee -mxtue s the dffusvt fo component n the gas mxtue -n s the dffusvt fo the bna pa, component dffusng though component n n s the mole facton of component n n the gas mxtue evaluated on a component fee bass, that s n Example.. etemne the dffusvt of O (), O () and (3) n a gas mxtue havng the composton: O : 8.5 % O : 5% : 56.5% The gas mxtue s at 73 K and. x 5 Pa. The bna dffusvt values ae gven as: (at 73 K) P.874 m Pa/s 3 P.945 m Pa/s 3 P.834 m Pa/s FE 3- PRT: MOLEULR IFFUSIO-

12 alculatons: ffusvt of O n mxtue m whee Theefoe m P Snce P. x 5 Pa,.93 m.pa/s.93 5 x m.6 m 5.x s ffusvt of O n the mxtue, m Whee. 335 (mole facton on- fee bans) and and P P.874 m.pa/s.665 FE 3- PRT: MOLEULR IFFUSIO-

13 Theefoe m P.847 m.pa/s m.539x m sec 5.x smla calculatons dffusvt of n the mxtue can be calculated, and s found to be, 3m.588 x 5 m /s..4. ffusvt n lquds: ffusvt n lqud ae exemplfed b the values gven n Table.. Most of these values ae neae to -5 cm / s, and about ten thousand tmes lowe than those n dlute gases. Ths chaactestc of lqud dffuson often lmts the oveall ate of pocesses accung n lquds (such as eacton between two components n lquds). In chemst, dffusvt lmts the ate of acd-base eactons; n the chemcal ndust, dffuson s esponsble fo the ates of lqud-lqud extacton. ffuson n lquds s mpotant because t s slow. etan molecules dffuse as molecules, whle othes whch ae desgnated as electoltes onze n solutons and dffuse as ons. Fo example, sodum chlode (al), dffuses n wate as ons a + and l -. Though each on has a dffeent moblt, the electcal neutalt of the soluton ndcates the ons must dffuse at the same ate; accodngl t s possble to speak of a dffuson coeffcent fo molecula electoltes such as al. Howeve, f seveal ons ae pesent, the dffuson ates of the ndvdual catons and anons must be consdeed, and molecula dffuson coeffcents have no meanng. ffusvt vaes nvesel wth vscost when the ato of solute to solvent ato exceeds fve. In extemel hgh vscost mateals, dffuson becomes ndependent of vscost..4.3 ffusvt n solds: Tpcal values fo dffusvt n solds ae shown n table. One outstandng chaactestc of these values s the small sze, usuall thousands of tme less than those n a lqud, whch ae n tun, tmes less than those n a gas. ffuson plas a majo ole n catalss and s mpotant to the chemcal/ food engnee. Fo metallugsts, dffuson of atoms wthn the solds s of moe mpotance. FE 3- PRT: MOLEULR IFFUSIO-3

14 .5 Stead State ffuson In ths secton, stead-state molecula mass tansfe though smple sstems n whch the concentaton and mola flux ae functons of a sngle space coodnate wll be consdeed. In a bna sstem, contanng and, ths mola flux n the decton of z, as gven b Eqn (5) s d ( ) ().5. ffuson though a stagnant gas flm The dffusvt o dffuson coeffcent fo a gas can be measued, expementall usng nold dffuson cell as shown n Fg.. Fg. nold dffuson cell The naow tube of unfom coss secton whch s patall flled wth pue lqud, s mantaned at a constant tempeatue and pessue. Gas whch flows acoss the open end of the tub, has a neglgble solublt n lqud, and s also chemcall net to. (.e. no eacton between & ). omponent vapozes and dffuses nto the gas phase; the ate of vapozaton ma be phscall measued and ma also be mathematcall expessed n tems of the mola flux. FE 3- PRT: MOLEULR IFFUSIO-4

15 onsde the contol volume S z, whee S s the coss sectonal aea of the tube. Mass balance on ove ths contol volume fo a stead-state opeaton elds [Moles of leavng at z + z] [Moles of enteng at z]. (.e.) S S () z + z z vdng though b the volume, S Z, and evaluatng n the lmt as Z appoaches zeo, we obtan the dffeental equaton d () Ths elaton stpulates a constant mola flux of thoughout the gas phase fom Z to Z. smla dffeental equaton could also be wtten fo component as, d d Z, and accodngl, the mola flux of s also constant ove the ente dffuson path fom z an. onsdeng onl at plane z, and snce the gas s nsoluble s lqud, we ealze that, the net flux of, s zeo thoughout the dffuson path; accodngl s a stagnant gas. Fom equaton () (of secton.5) FE 3- PRT: MOLEULR IFFUSIO-5

16 FE 3- PRT: MOLEULR IFFUSIO-6 ( ) z d d + + Snce, z d d + Reaangng, z d d (3) Ths equaton ma be ntegated between the two bounda condtons: at z z Y Y nd at z z Y ssumng the dffusvt s to be ndependent of concentaton, and ealzng that s constant along the dffuson path, b ntegatng equaton (3) we obtan Z Z d z d ln Z Z (4) The log mean aveage concentaton of component s defned as, ln lm Snce,, ln ln ) ( ) ( lm (5) Substtutng fom Eqn (5) n Eqn (4),

17 Z z ( (6), lm ) Fo an deal gas n p, and V RT fo mxtue of deal gases P P Theefoe, fo an deal gas mxtue equaton. (6) becomes RT ( z z ) ( p p p, lm ) Ths s the equaton of mola flux fo stead state dffuson of one gas though a second stagnant gas. Man mass-tansfe opeatons nvolve the dffuson of one gas component though anothe non-dffusng component; absopton and humdfcaton ae tpcal opeatons defned b these equaton. Example.3 Oxgen s dffusng n a mxtue of oxgen-ntogen at atm, 5. oncentaton of oxgen at planes mm apat ae and volume % espectvel. togen s non-dffusng. (a) eve the appopate expesson to calculate the flux oxgen. efne unts of each tem cleal. (b) alculate the flux of oxgen. ffusvt of oxgen n ntogen.89 x 5 m /s. Soluton: Let us denote oxgen as and ntogen as. Flux of (.e.) s made up of two components, namel that esultng fom the bulk moton of (.e.), x and that esultng fom molecula dffuson J : x + J () FE 3- PRT: MOLEULR IFFUSIO-7

18 Fom Fck s law of dffuson, J d () Substtutng ths equaton () d x (3) Snce + and x / equaton (3) becomes ( + ) d Reaangng the tems and ntegatng between the planes between and, c ( + ) d (4) Snce s non dffusng. lso, the total concentaton emans constant. Theefoe, equaton (4) becomes z d ln Theefoe, ln (5) z Replacng concentaton n tems of pessues usng Ideal gas law, equaton (5) becomes P P t t ln (6) RTz P P P t whee FE 3- PRT: MOLEULR IFFUSIO-8

19 Gven: molecula dffusvt of n P T total pessue of sstem R unvesal gas constant T tempeatue of sstem n absolute scale z dstance between two planes acoss the decton of dffuson P patal pessue of at plane, and P patal pessue of at plane.89 x 5 m /s P t atm.35 x 5 /m T K z mm. m P. x. atm (Fom Ideal gas law and addtve pessue ule) P. x. atm Substtutng these n equaton (6) 5 5 (.89x )(.35x ) ( 834)( 98)(.) 4.55 x 5 kmol/m.s. ln..5. Pseudo stead state dffuson though a stagnant flm: In man mass tansfe opeatons, one of the boundaes ma move wth tme. If the length of the dffuson path changes a small amount ove a long peod of tme, a pseudo stead state dffuson model ma be used. When ths condton exsts, the equaton of stead state dffuson though stagnant gas can be used to fnd the flux. If the dffeence n the level of lqud ove the tme nteval consdeed s onl a small facton of the total dffuson path, and t t s elatvel long peod of tme, at an gven nstant n that peod, the mola flux n the gas phase ma be evaluated b ( ) () z, lm whee z equals z z, the length of the dffuson path at tme t. The mola flux s elated to the amount of leavng the lqud b FE 3- PRT: MOLEULR IFFUSIO-9

20 ρ, L () M d t whee ρ M, L s the mola denst of n the lqud phase unde Psuedo stead state condtons, equatons () & () can be equated to gve ρ, L ( ) M d t z, lm (3) Equaton. (3) ma be ntegated fom t to t and fom z z t to z z t as: t t dt ρ, L, lm ( ) M Zt Zt z dz eldng t ρ ( ) z z, L, lm t t M (4) Ths shall be eaanged to evaluate dffusvt as, M ρ ( ) t z z, L, lm t t Example.4 vetcal glass tube 3 mm n damete s flled wth lqud toluene to a depth of mm fom the top opened. fte 75 hs at 39.4 and a total pessue of 76 mm Hg the level has dopped to 8 mm fom the top. alculate the value of dffusvt. ata: vapo pessue of toluene at k / m, denst of lqud toluene 85 kg/m 3 Molecula weght of toluene 9 ( 6 H 6 H 3 ) M ρ Z Z, L lm t t ( ) t FE 3- PRT: MOLEULR IFFUSIO-

21 whee, l m ln p (76 mm Hg.3 k/m ) P Theefoe, lm. 968 ln P.35x.39 k mol /m 3 RT 834x ( ) Theefoe 85x.968 9x.39x.8. (.754 ) x75x36.56 x 3 (.8. ) 9.57 x -6 m /s.5.3 Equmola counte dffuson: phscal stuaton whch s encounteed n the dstllaton of two consttuents whose mola latent heats of vapozaton ae essentall equal, stpulates that the flux of one gaseous component s equal to but actng n the opposte decton fom the othe gaseous component; that s, -. The mola flux, fo a bna sstem at constant tempeatue and pessue s descbed b d ( ) + + o d + ( + ) () wth the substtuton of -, Equaton () becomes, FE 3- PRT: MOLEULR IFFUSIO-

22 d () Fo stead state dffuson Equaton. () ma be ntegated, usng the bounda condtons: at z z an z Gvng, fom whch Z Z d ( ) (3) z z n p Fo deal gases,. V RT Theefoe Equaton. (3) becomes ( P P ) (4) RT ( z z ) Ths s the equaton of mola flux fo stead-state equmola counte dffuson. oncentaton pofle n these equmola counte dffuson ma be obtaned fom, d ( ) (Snce s constant ove the dffuson path). nd fom equaton. () d. Theefoe FE 3- PRT: MOLEULR IFFUSIO-

23 d d. d o. Ths equaton ma be solved usng the bounda condtons to gve z z z z (5) Equaton, (5) ndcates a lnea concentaton pofle fo equmola counte dffuson. Example.5. Methane dffuses at stead state though a tube contanng helum. t pont the patal pessue of methane s p 55 kpa and at pont,.3 m apat P 5 kpa. The total pessue s.3 kpa, and the tempeatue s 98 K. t ths pessue and tempeatue, the value of dffusvt s 6.75 x 5 m /s. alculaton: ) calculate the flux of H 4 at stead state fo equmola counte dffuson. ) alculate the patal pessue at a pont. m apat fom pont. Fo stead state equmola counte dffuson, mola flux s gven b ( p p ) () RT z Theefoe; x 8.34x98x.3 ( 55 5) kmol m.sec 5 kmol 3.633x m sec nd fom (), patal pessue at. m fom pont s: 3.633x x 8.34x98x. ( 55 ) p p 8.33 kpa FE 3- PRT: MOLEULR IFFUSIO-3

24 Example.6. In a gas mxtue of hdogen and oxgen, stead state equmola counte dffuson s occung at a total pessue of kpa and tempeatue of. If the patal pessues of oxgen at two planes. m apat, and pependcula to the decton of dffuson ae 5 kpa and 5 kpa, espectvel and the mass dffuson flux of oxgen n the mxtue s.6 x 5 kmol/m.s, calculate the molecula dffusvt fo the sstem. Soluton: Fo equmola counte cuent dffuson: ( p p ) () RTz whee mola flux of (.6 x 5 kmol/m.s): molecula dffusvt of n R Unvesal gas constant (8.34 kj/kmol.k) T Tempeatue n absolute scale ( K) z dstance between two measuement planes and (. m) P patal pessue of at plane (5 kpa); and P patal pessue of at plane (5 kpa) Substtutng these n equaton ().6x 5 ( )( )( ) ( 5 5) Theefoe, * 5 m /s Example 3.7. tube cm n nsde damete that s cm long s flled wth O and H at a total pessue of atm at. The dffuson coeffcent of the O H sstem unde these condtons s.75 cm /s. If the patal pessue of O s.5 atm at one end of the tube and.5 atm at the othe end, fnd the ate of dffuson fo: ) stead state equmola counte dffuson ( - ) ) stead state counte dffuson whee -.75, and ) stead state dffuson of O though stagnant H ( ) FE 3- PRT: MOLEULR IFFUSIO-4

25 ) + ( + ) Gven - d Theefoe d d p (Fo deal gas mxtue RT whee p s the patal pessue of ; such that p + p P) ( p RT ) d Theefoe Fo sothemal sstem, T s constant Theefoe RT d p (.e.) Z Z RT ( p p ) P P d p () RT z whee Z Z Z Gven:.75 cm /s.75 x 4 m /s ; T 73 K 4.75x 834 x 73 x. 6 k mol.38x m sec 6 Rate of dffuson S Whee S s suface aea 5 5 (.5x.35x.5x.35x ) Theefoe ate of dffuson 6.38 x -6 ( π ) 6.38 x 6 π (.5 x ) 4.8 x k mol/s FE 3- PRT: MOLEULR IFFUSIO-5

26 ) + ( + ) d gven: x 3 mol/h. Theefoe + (. 75 ) d d +. 5 d. 5 d.5 fo constant and Z Z d.5 ( ) [ ln (.5 )] d x ln a + b x b z ln () z.5 ( a + b x) Gven: 5 p x.35x.893 k mol RT 834x73 p.5.75 P p.5.5 P Substtutng these n equaton (), m 3 4x.893(.75x. 4 ).5x.5 ln.5x.75 FE 3- PRT: MOLEULR IFFUSIO-6

27 6 kmol 7.8x m sec Rate of dffuson S 7.8 x 6 π (.5 x ) 5.5 x kmol/s.987 x 3 mol/h. ) + ( + ) Gven: d Theefoe Z Z d + d ln Z 4.893(.75x ).5 ln kmol.349 x m.sec Rate of dffuson π (.5 x ).59 Kmol / s 3.84 mol/h.5.4 ffuson nto an nfnte standad medum : Hee we wll dscuss poblems nvolvng dffuson fom a sphecal patcle nto an nfnte bod of stagnant gas. The pupose n dong ths s to demonstate how to set up dffeental equatons that descbe the dffuson n these pocesses. The solutons, obtaned ae onl of academc nteest because a lage bod of gas n whch thee ae no convecton cuents s unlkel to be found n pactce. Howeve, the solutons developed hee fo these poblems actuall epesent a specal case of the moe common stuaton nvolvng both molecula dffuson and convectve mass tansfe. FE 3- PRT: MOLEULR IFFUSIO-7

28 a) Evapoaton of a sphecal oplet: s an example of such poblems, we shall consde the evapoaton of sphecal doplet such as a andop o sublmaton of naphthalene ball. The vapo fomed at the suface of the doplet s assumed to dffuse b molecula motons nto the lage bod of stagnant gas that suounds the doplet. onsde a andop, as shown n fgue. t an moment, when the adus of the dop s, the flux of wate vapo at an dstance fom the cente s gven b d + ( + ) d Hee (snce a s assumed to be stagnant) Theefoe, d + d Reaangng, d () d The flux s not constant, because of the sphecal geomet; deceases as the dstance fom the cente of sphee nceases. ut the mola flow ate at and + δ ae the same. Ths could be wtten as, () + δ Whee suface aea of sphee at o + δ. Substtutng fo 4 π n equaton (), o 4π 4π lm δ d d Integatng, + δ + δ as ( ) δ (3) constant (4) Fom equaton (4), Substtutng fo fom equaton (), FE 3- PRT: MOLEULR IFFUSIO-8

29 d d d d (5) ounda condton : t S nd t Theefoe equaton (5) becomes, [ ln ( )] S Smplfng, ln (6) S Tme equed fo complete evapoaton of the doplet ma be evaluated fom makng mass balance. Moles of wate dffusng molesof wate leavng the doplet unt tme unt tme d ρ 4 3 L 4 π π dt 3 M ρ d L 4π (7) M d t Substtutng fo fom equaton (6) n equaton (7), ρ d L ln (8) S M d t Intal condton : When t Integatng equaton (8) wth these ntal condton, t ρ d t L d M ln S ρ L t (9) M ln S Equaton (9) gves the total tme t equed fo complete evapoaton of sphecal doplet of ntal adus. FE 3- PRT: MOLEULR IFFUSIO-9

30 b) ombuston of a coal patcle: The poblem of combuston of sphecal coal patcle s smla to evapoaton of a dop wth the excepton that chemcal eacton (combustons) occus at the suface of the patcle. ung combuston of coal, the eacton + O O cccus. ccodng to ths eacton fo eve mole of oxgen that dffuses to the suface of coal (maxmum of cabon), eact wth mole of cabon, eleases mole of cabon doxde, whch must dffuse awa fom ths suface. Ths s a case of equmola counte dffuson of O and O. omall a (a mxtue of and O ) s used fo combuston, and n ths case does not takes pat n the eacton, and ts flux s zeo. (.e. ). The mola flux of O could be wtten as d O O ( ) O gas + O O + O + () d whee s the dffusvt of O n the gas mxtue. O gas Snce, and fom stochomet O O, equaton () becomes d O O O gas d () Fo stead state condtons, d ( O ) d (3) Integatng, constant (4) O O s Whee s the adus of coal patcle at an nstant, and suface of the patcle. Substtutng fo O s O fom equaton () n equaton (4), d O O gas O s (5) d ounda condton : t O O s nd t O O Wth these bounda condton, equaton (5) becomes O d O gas d O O s s the flux of O at the FE 3- PRT: MOLEULR IFFUSIO-3

31 whch elds O gas O ( ) s O s O (6) Fo fast eacton of O wth coal, the mole facton of O at the suface of patcle z zeo. (.e.,). O s nd also at some dstance awa fom the suface of the patcle. (because a s a mxtue of mole % O and 79 mole % ) O O Wth these condtons, equaton (6) becomes,. O gas O s (7) Example 3.6. sphee of naphthalene havng a adus of mm s suspended n a lage volume of shell a at 38 K and atm. The suface pessue of the naphthalene can be assumed to be at 38 K s.555 mm Hg. The of naphthalene n a at 38 K s 6.9 x 6 m /s. alculate the ate of evapoaton of naphthalene fom the suface. alculaton Stead state mass balance ove a element of adus and + δ leads to S S + δ whee S s the suface ae ( 4 π ) () dvdng () b Sδ, and takng the lmt as δ appoaches zeo, gves: ( ) d d Integatng constant (o) 4 π constant We can assume that thee s a flm of naphthalene vapo / a flm aound naphthalene though whch molecula dffuson occus. ffuson of naphthalene vapo acoss ths flm could be wtten as, d ( ) d (snce a s assumed to be stagnant n the flm) + + FE 3- PRT: MOLEULR IFFUSIO-3

32 d + d d d d ln d [ ( )] W Rate of evapoaton 4 π R constant. W 4π d d ( ln ( ) d 4π d ln ( ) W ounda condton: t R 7.33x 76 ln ( ) x 4 t ln (- ) d Theefoe W 4π [ ln ( )] d R 4 7.3* [ ln ( )] 7.3 x 4 [ x ] W 4 4 π R W + 4π R W 4 π R ( 7.3 x 4 ) 5 P.35x RxT 834x kmol/m 3 Intal ate of evapoaton: Theefoe W 4 x 3.4 ( x 3 )( 6.9 x 6 ).383 ( 7.3 x 4 ) x kmol/s.75 x 5 mol/h. FE 3- PRT: MOLEULR IFFUSIO-3

33 3.5.5 ffuson n Lquds: Equaton deved fo dffuson n gases equall apples to dffuson n lquds wth some modfcatons. Mole facton n lqud phases s nomall wtten as x (n gases as ). The concentaton tem s eplaced b aveage mola denst, ρ M av a) Fo stead state dffuson of though non dffusvt : constant, z x M ρ M av ( x x ) whee Z Z Z, the length of dffuson path; and X X X M X ln X b) Fo stead state equmola counte dffuson : - const Z ρ ( ) ( x x ) Z M av Example 3.7. alculate the ate of dffuson of butanol at unde undectonal stead state condtons though a. cm thck flm of wate when the concentatons of butanol at the opposte sdes of the flm ae, espectvel % and 4% butanol b weght. The dffusvt of butanol n wate soluton s 5.9 x 6 cm /s. The denstes of % and 4% butanol solutons at ma be taken as.97 and.99 g/ml espectvel. Molecula weght of utanol ( 4 H 9 OH) s 74, and that of wate 8. alculatons Fo stead state undectonal dffuson, ( x x ) z x, lm whee s the aveage mola denst. M ρ avg FE 3- PRT: MOLEULR IFFUSIO-33

34 onveson fom weght facton the Mole facton: x x (. 74) ( ) (.4 74) ( ).6 veage molecula weght at & : ( ). M kg M kg ( ) ( ρ M ) ρ + ρ M M avg gmol / cm kmol/m 3 ( x ) ( x ) x x x, lm ln ( x x ) x ln x (. ) (.6 ) (.e.) x,lm. ln ρ x x Theefoe M x avg ( ), lm kmol kmol 6 4 (5.9x )( )(5.7).x 7 kmol 4.97* m sec gmol.789 m. h. x (.6.).98 FE 3- PRT: MOLEULR IFFUSIO-34

35 g.789 x 74 m. h. g 3.4 m. h. Mass dffuson wth homogeneous chemcal eacton: bsopton opeatons nvolve contact of a gas mxtue wth a lqud and pefeental dssoluton of a component n the contactng lqud. ependng on the chemcal natue of the nvolved molecules, the absopton ma o ma not nvolve chemcal eacton. The followng analss llustates the dffuson of a component fom the gas phase nto the lqud phase accompaned b a chemcal eacton n the lqud phase. onsde a lae of absobng medum (lqud) t the suface of the lqud, the composton of s. The thckness of the flm, δ s so defned, that beond ths flm the concentaton of s alwas zeo ; that s δ. If thee s ve lttle flud moton wthn the flm, d + ( ) + () If concentaton of n the flm, s assumed small, equaton () becomes d () The mola flux changes along the dffuson path. Ths change s due to the eacton that takes place n the lqud flm. Ths changes could be wtten as d ( ) (3) whee s the ate of dsappeaance of. Fo a fst ode eacton, k k (4) wth the substtuton fom equaton (4) and () n equaton (3), d d + k Fo constant ffusvt, FE 3- PRT: MOLEULR IFFUSIO-35

36 d + k (5) whch s a second ode odna dffeental equaton. The geneal soluton to ths equaton s k + k cos h z h z sn (6) The constants of ths equaton can be evaluated fom the bounda condtons: at Z nd at Z δ. The constant s equal to, and s equal to substtuton equaton (6) becomes, k δ wth ths tan h k sn h z k cos h z (7) tan h k z Ths equaton gves the vaaton of concentaton of wth z (.e concentaton pofle of n the lqud). The mola flux at the lqud suface can be detemned b dffeentatng equaton (7), and evaluatng the devatve d at z ffeentatng wth espect to z, k k cos h z d k k sn h z (8) tan h k δ Substtutng z n equaton (8) and fom equaton (), k δ (9) Z δ tan h k δ FE 3- PRT: MOLEULR IFFUSIO-36

37 Fo absopton wth no chemcal eacton, the flux of s obtaned fom equaton () as () δ whch s constant thoughout the flm of lqud. On compason of equaton (9) and (), t s appaent that the tem k δ tan h δ k shows the nfluence of the chemcal eactons. Ths tems a dmensonless quantt, s often called as Hatta umbe ffuson n solds In cetan unt opeaton of chemcal engneeng such as n dng o n absopton, mass tansfe takes place between a sold and a flud phase. If the tansfeed speces s dstbuted unfoml n the sold phase and foms a homogeneous medum, the dffuson of the speces n the sold phase s sad to be stuctue ndependent. In ths cases dffusvt o dffuson coeffcent s decton ndependent. t stead state, and fo mass dffuson whch s ndependent of the sold matx stuctue, the mola flux n the z decton s : d constant, as gven b Fck s law. Integatng the above equaton, ( ) z whch s smla to the expesson obtaned fo dffuson n a stagnant flud wth no bulk moton (.e. ). ffuson n pocess solds: In some chemcal opeatons, such as heteogeneous catalss, an mpotant facto, affectng the ate of eacton s the dffusons of the gaseous component though a poous sold. The effectve dffusvt n the sold s educed below what t could be n a fee flud, fo two easons. Fst, the totuous natue of the path nceases the dstance, whch a molecule must tavel to advance a gven dstance n the sold. Sond, the fee coss stonal aea s estcted. Fo man catalst pellets, the effectve FE 3- PRT: MOLEULR IFFUSIO-37

38 dffusvt of a gaseous component s of the ode of one tenth of ts value n a fee gas. If the pessue s low enough and the poes ae small enough, the gas molecules wll collde wth the walls moe fequentl than wth each othe. Ths s known as Knudsen flow o Knudsen dffuson. Upon httng the wall, the molecules ae momental absobed and then gven off n andom dectons. The gas flux s educed b the wall collsons. use of the knetc flux s the concentaton gadent s ndependent of pessue ; wheeas the popotonalt constant fo molecula dffuson n gases (.e. ffusvt) s nvesel popotonal to pessue. Knudsen dffuson occus when the sze of the poe s of the ode of the mean fee path of the dffusng molecule. 3.6 Tansent ffuson Tansent pocesses, n whch the concentaton at a gven pont vaes wth tme, ae efeed to as unstead state pocesses o tme dependent pocesses. Ths vaaton n concentaton s assocated wth a vaaton n the mass flux. These geneall fall nto two categoes: ) the pocess whch s n an unstead state onl dung ts ntal statup, and ) the pocess whch s n a batch opeaton thoughout ts opeaton. In unstead state pocesses thee ae thee vaables-concentaton, tme, and poston. Theefoe the dffuson pocess must be descbed b patal athe than odna dffeental equatons. lthough the dffeental equatons fo unstead state dffuson ae eas to establsh, most solutons to these equatons have been lmted to stuatons nvolvng smple geometes and bounda condtons, and a constant dffuson coeffcent. Man solutons ae fo one-dectonal mass tansfe as defned b Fck s sond law of dffuson : t z () Ths patal dffeental equaton descbes a phscal stuaton n whch thee s no bulk moton contbuton, and thee s no chemcal eacton. Ths stuaton s FE 3- PRT: MOLEULR IFFUSIO-38

39 encounteed when the dffuson takes place n solds, n statona lquds, o n sstem havng equmola counte dffuson. ue to the extemel slow ate of dffuson wthn lquds, the bulk moton contbuton of flux equaton (.e., ) appoaches the value of zeo fo dlute solutons ; accodngl ths sstem also satsfes Fck s second law of dffuson. The soluton to Fck s second law usuall has one of the two standad foms. It ma appea n the fom of a tgonometc sees whch conveges fo lage values of tme, o t ma nvolve sees of eo functons o elated ntegals whch ae most sutable fo numecal evaluaton at small values of tme. These solutons ae commonl obtaned b usng the mathematcal technques of sepaaton of vaables o Laplace tansfoms. FE 3- PRT: MOLEULR IFFUSIO-39