CE 6333, Levicky 1 Multicompoet Systems MSS TRNSFER. Mass tasfe deals with situatios i which thee is moe tha oe compoet peset i a system; fo istace, situatios ivolvig chemical eactios, dissolutio, o mixig pheomea. simple example of such a multicompoet system is a biay (two compoet) solutio cosistig of a solute i a excess of chemically diffeet solvet. 1. Itoductio ad asic Defiitios. I a multicompoet system, the velocity of diffeet compoets is i geeal diffeet. Fo example, i Fig. 1 pue gas is peset o the left ad pue gas o the ight. Whe the wall sepaatig the two gases is emoved ad the gases begi to mix, will flow fom left to ight ad fom ight to left clealy the velocities of ad will be diffeet. Fig. 1 The velocity of paticles (molecules) of compoet (elative to the laboatoy fame of efeece) will be deoted v. The, i this fame of efeece, the mola flux N of species (uits: moles of /(aea time) ) is N = c v (1) whee c is the mola cocetatio of (moles of /volume). Fo example, (1) could be used to calculate how may moles of flow though a aea C pe uit time (Fig. ). I Fig., the flux is assumed to be omal to the aea C. The the amout of caied acoss the aea C pe uit time is mout of caied though C pe uit time = N C = c v C (moles / time) Sice the volume swept out by the flow of pe uit time equals v C (see Fig. ), the above expessio is see to equal this ate of volumetic "sweepig" times c, the amout of pe volume.
CE 6333, Levicky Fig. Moe geeally, fo abitay diectio of N ad a diffeetial aea elemet d, the ate of taspot though d would be (Fig. 3), flux of though d = c v d (moles / time) () is the outwad uit omal vecto to d. Oe ca udestad equatio () by ealizig that v d is the volumetic flowate of species (volume/time) passig acoss d fom "outside" to "iside", whee "outside" is poited at by the uit omal vecto. Multiplyig the volumetic flowate v d by the umbe of moles of pe volume, c, equals the moles of passig though d pe uit time. "outside" d - v { "iside" v Fig. 3 volume swept out pe uit time = - v d c is elated to the total mola cocetatio c (c is moles of paticles, iespective of paticle type, pe volume) via
CE 6333, Levicky 3 c = x c (3) whee x is the mole factio of. Summig ove the mole factios of all species must poduce uity ( equals the total umbe of diffeet species peset i solutio), x i 1 (4) i1 Similaly, we ca also defie a mass flux of, (uits: mass of /(aea time) ), = v (5) Hee, v is still the velocity of species, exactly the same as i equatio 1. is the mass cocetatio of (mass of pe volume of solutio), = (6) whee is the total desity ( is the summed mass of all paticles, iespective of paticle type, pe volume) of the solutio ad is called the mass factio of (i.e. = /). Summig the mass factios of all species must equal uity ω i 1 (7) i 1 s peviously stated, i geeal each chemical species "i" i a multicompoet mixtue has a diffeet velocity v i. Howeve, it will evetheless pove coveiet to defie a aveage velocity of the bulk fluid, a velocity that epesets a aveage ove all the v i 's. I geeal, thee types of aveage velocities ae employed: mass aveage velocity v (v is what is usually dealt with i Fluid Mechaics), mola aveage velocity V, ad volume aveage velocity v o. We will oly deal with the fist two aveage velocities, defied as follows: v = ivi (8) i1 V = i1 x i v i (9)
CE 6333, Levicky 4 Fom its defiitio, v is a mass factio based aveage of the idividual species' velocities, while V is a mole factio based aveage. It ca be show that if total desity = i is costat iespective of i1 compositio, ad if the total mola cocetatio c = c i is costat iespective of compositio, the i1 v = V. It ca futhe be show that if all paticles have the same mass m, so that m i = m fo all i whee m i is mass of i type paticles, the x i = i ad theefoe v = V. Why bothe with two diffeet aveage velocities? The mass aveage velocity is what is eeded i equatios such as the Navie Stokes equatios, which deal with mometum, a popety that depeds o how much mass is i motio. Thus, the amout of mometum pe uit volume of a flowig multicompoet mixtue is v ( v = mv/volume, whee m is the total mass tavelig with velocity v; m/volume = ); thus mometum must be calculated usig the mass aveage velocity v. Similaly, the Equatio of Cotiuity expesses cosevatio of mass, ad is similaly witte i tems of v. The physical laws expessed by these equatios (cosevatio of mometum, cosevatio of mass) do ot deped o the moles of paticles ivolved, but they do deped o the mass of the paticles. O the othe had, whe dealig with mass tasfe, we will see that it is commo to wite some of the basic equatios i tems of V as well as v. The easo fo usig V, i additio to v, is coveiece. Fo istace, if i a paticula poblem thee is o bulk flow of paticles fom oe locatio to aothe so that, duig the mass tasfe pocess the umbe of paticles at each poit i space stays the same, the V = 0. Settig V to zeo simplifies the mathematics. Figue 1 at the begiig of this hadout povides a example. Imagie that, i thei sepaated state as daw, ad ae both ideal gases at the same pessue p ad tempeatue T. The, fom the ideal gas equatio, the mola cocetatio of ad is the same, c = c = c = p/rt (R = gas costat) The equality of c ad c to the total cocetatio c is appopiate because the gases ae pue; thus i each compatmet the cocetatio of the gas ( o ) must also equal the total cocetatio c. fte the sepaatig wall is emoved, paticles of ad will mix util a uifom compositio is achieved thoughout the vessel. I the fial state, assumig the gases emai ideal whe mixed, the value of p ad T will emai the same as i the umixed state ad theefoe the total cocetatio c also emais the same, c = p/rt (p is ow the total pessue, a sum of the patial pessues of ad ). Thus, i the fial mixed state, the umbe of paticles pe volume c (hee a sum of paticles of ad types) is the same as the umbe of paticles pe volume i the iitial umixed state. Thus mixig poduced o et tasfe of paticles fom oe side of the vessel to the othe, it oly mixed the diffeet paticle types togethe. Ude these coditios, whe thee is ot et tasfe of paticles fom oe pat of a system to aothe, V = 0. I cotast, fo the same mixig pocess, i geeal v will ot be zeo. Fo example, imagie that mass of paticles is twice as lage as that of paticles. The i the iitial umixed state the left had side of the vessel (filled with ) cotais moe mass, ad the desity (mass/volume) of the gas is highe tha that of eve though its cocetatio (paticles/volume) is the same. Oce ad mix, howeve, the desity eveywhee will become uifom. Fo this uifomity to be achieved mass must
CE 6333, Levicky 5 have bee tasfeed fom the side to the side; theefoe, i cotast to the mola aveage velocity V, the mass aveage velocity v was ot zeo duig the mixig pocess.. Itegal ad Diffeetial alaces o Chemical Species. We will efe to the species ude cosideatio as species. Followig a deivatio that paallels that employed fo the othe cosevatio laws, the fist step i the deivatio of a cosevatio law o the amout of species is to pefom a balace fo a closed cotol volume V'. V' is eclosed by a closed suface (Fig. 4). Fig.4 The amout of species iside V' ca chage eithe due to covectio though the bouday, o by geeatio/cosumptio of due to a chemical eactio. I wods, the cosevatio fo species ca stated as: ccumulatio of i V' = covectio of ito V' + geeatio of by chemical eactios (10) itegal mola balace o species, pefomed ove the cotol volume V', is witte d dt c dv ' = c V ' v d + R dv ' (11) V ' is the outwad uit omal vecto to suface, ot to be cofused with the mass flux i = i v i of species i. O the left side, c is cocetatio of i moles pe volume; thus c dv' is the umbe of moles of i a diffeetial volume dv'. Itegatig (i.e. summig) this tem ove the etie cotol volume V' yields the total umbe of moles of i V'; the time deivative of this itegal is the ate of chage of moles of iside V' (uits: moles/time). Thus, the left had tem is just the ate of accumulatio of i V', expessed i mola uits. The accumulatio tem equals the ate at which is covected ito V' (1st tem o ight) plus the ate at which is geeated iside V' by a homogeeous chemical eactio (d tem o ight). The covectio tem ca be udestood by efeig to Fig. 5. v is the compoet of the species
CE 6333, Levicky 6 velocity pepedicula to, so that v d is the volumetic flowate acoss the aea elemet d fo paticles tavelig with a velocity v. Multiplyig this volumetic flowate by the moles of pe volume esults i c v d, the mola flowate of though d. Summig all the mola flowates ove the etie suface the leads to the covectio tem (1st tem o ight) i equatio (11). Fig. 5 The d tem o the ight i equatio (11) epesets poductio of by homogeeous eactios. "homogeeous" eactio is oe that occus thoughout the iteio of V'. I cotast, a heteogeeous chemical eactio would be oe that occus oly at a iteface fo istace, betwee a solid ad a liquid phase ad is ot distibuted thoughout the etie volume. The mola eactio ate R has uits of moles/(volume time) ad epesets the ate at which moles of ae poduced o cosumed by all homogeeous eactios. R dv' is the umbe of moles of poduced iside a volume elemet dv' pe uit time (uits: moles/time). Summig this poductio ove the etie cotol volume leads to the total mola ate of poductio of, iside V', due to homogeeous chemical eactios. Equatio (11), by assumptio, did ot iclude ay geeatio of due to heteogeeous eactios. Clealy, if i V' thee was a lage iteface at which a heteogeeous eactio leads to poductio of, oe would have to add that tem to equatio (11). The tem would typically have the fom of a ate of poductio of pe aea (moles / (aea time)) times the total aea of the eactig suface. Howeve, it may also be that a heteogeeous eactio is actually moe coveietly modeled as homogeeous. Fo example, imagie that small catalyst paticles (e.g. platium powde) ae suspeded i a liquid iside V', ad that a eactio that poduces occus o the suface of these powde paticles. ecause the eactio occus oly at the iteface betwee a paticle ad the liquid, it is heteogeeous. Howeve, sice the paticles ae dispesed thoughout V', oe could thik of the eactio ate o a pe volume basis (i.e. moles poduced pe volume of solutio pe time) as opposed to a pe aea basis (moles poduced pe suface aea of paticles pe time). s doe peviously fo the othe balaces, oe ca (1) use the Divegece Theoem to covet the suface itegal of the covectio tem (1st tem o ight) ito a volume itegal, () move the d/dt deivative iside the accumulatio itegal sice the itegatio limits ae time idepedet (the limits do ot deped o time because a fixed cotol volume is cosideed, whose shape ad locatio do ot chage; this assumptio ca be elaxed at the expese of a somewhat moe complicated mathematical expessio), ad (3) combie all tems ude a commo volume itegal to obtai,
CE 6333, Levicky 7 V ' c t c v R dv ' = 0 (1) The oly way to esue that equatio (1) evaluates to zeo fo a abitay cotol volume V' is to equie that c t c v R = 0 (13) c t = cv R (14) Equatio (14) is the diffeetial mola balace o species. It states that the ate of accumulatio of moles of at a poit i space (left had side) equals the ate at which moles of ae covected ito that poit (1st tem o ight), plus the ate at which moles of ae poduced at that poit by chemical eactios (d tem o ight). These physical itepetatios ca be veified by tacig the oigi of the tems back to the coespodig tems i the itegal balace, equatio (11). Multiplicatio of equatio (11) by the mola mass M (mass / mole of ) of species, ad ecogizig that, the mass of pe volume, is give by = M c (15) leads to the itegal mass balace o species, d dt dv ' = V ' v d + d V ' (16) V ' I equatio (16), the mass eactio ate has uits of mass/(volume time) ad epesets the poductio o cosumptio of mass of species by all chemical eactios. is give by = M R (17) Though maipulatios aalogous to those that led to (14), equatio (16) ca be coveted to a diffeetial mass balace o species, t = v (18) Recallig that N = c v (mola flux) (1) = v (mass flux) (5)
CE 6333, Levicky 8 equatios (14) ad (18) ca be witte as c t = N R (19) t = (0) Note that, i deivig these equatios, o assumptios wee made as to which compoet of a solutio (i.e. a solute, the solvet, etc.) is epeseted as species theefoe, these equatios apply to each solute species as well as the solvet. Thus, if oe chooses to label the solvet as species, the a solute species could be labeled as species. The equatios that would be used fo the solute ae exactly as i (19) ad (0) except that the subscipt would be eplaced by the subscipt. The diffeetial species' balaces wee deived idepedet of ay paticula coodiate system. To apply them to solvig a paticula poblem, oe must fist choose a coodiate system suited to descibig the poblem ad the tascibe the equatios ito that coodiate system. Fo example, equatio (18) becomes t = v v v x 1 1 x x 3 3 (Catesia "CCS" coodiates) (1) t t 1 v v 1 = v θ θ z Z 1 vφ θ = v 1 siθ v 1 siθ θ (cylidical coodiates) () siθ φ (spheical coodiates) (3) Compaed to the CCS expessio, the moe complex fom of the cylidical ad spheical coodiate expessios fo the divegece tem (the covectio tem) esults fom the cuviliea atue of these coodiates; i.e., the agula coodiate vaiables (such as ad ) chage alog cuves, ot lies (see late hadout o cuviliea coodiate systems). The diffeetial equatio of cotiuity (total mass balace) deived i fluid mechaics fo sigle compoet systems also applies to multicompoet systems i which chemical eactios happe. To pove this is staightfowad, ad begis by summig equatio (18) ove all species peset i solutio, i i t 1 = i v i i (4) i 1 i 1
CE 6333, Levicky 9 Itechagig the summatio ad /t opeatios o the left had side ad makig use of the elatio i i 1 (5) the left had side of equatio (4) becomes i i 1 t = t (6) Futhemoe, usig the defiitio of the mass aveage velocity v, v = ivi (7) i1 ad the fact that the mass factio i = i /, v = ivi i1 = i1 i v i (8) Usig equatio (8), the covectio tem i v i i equatio (4) becomes i 1 i v i = - v (9) i 1 Fially, the chemical eactio tem i equatio (4) is equied to evaluate to zeo i = 0 (30) i 1 sice mass is ot poduced o destoyed i chemical eactios (uclea eactios ca itecovet mass ad eegy, but this case is ot beig cosideed). Substitutig equatios (30), (9), ad (6) ito equatio (4) leads to
CE 6333, Levicky 10 = - v (31) t Equatio (31) is the diffeetial equatio of cotiuity familia fom fluid mechaics. This equatio states the law of mass cosevatio; eve i multicompoet systems, eve if chemical eactios ae peset, the total accumulatio of mass at a poit (left had side) ca oly occu by covectio of mass to that poit (ight had side). Fo multicompoet systems whose desity is costat (i.e. does ot vay fom poit to poit iespective of vaiatios that may be peset i tempeatue, pessue, o compositio), equatio (31) agai simplifies to the coditio fo icompessible systems, v = 0 (3) 3. Diffusio Fluxes. The diffusio flux of species is that potio of its total flux that is ot attibuted to bulk flow (as epeseted by the mass o mola aveage velocities). Moe pecisely, j, the mass diffusive flux of, is defied as j = total mass flux of mass flux of due to bulk motio = - v = v - v = (v v) (33) Similaly, J, the mola diffusive flux of, is defied by J = total mola flux of mola flux of due to bulk motio = N - c V = c (v - V) (34) Fo istace, i equatio (34), the mola diffusio flux J is see to be the diffeece betwee the total mola flux of (N ) ad mola flux of (c V) attibutable to a bulk flow of mola aveage velocity V. The total fluxes, ad N, ae the the sums of the fluxes of due to bulk motio ad diffusio, = j + v (35) N = J + c V (36) Note that the diffusive fluxes j ad J have diffeet uits, mass/(aea time) fo j ad moles/(aea time) fo J. lso, j ad J i geeal possess diffeet umeical values. 4. Causes of Diffusio. Why would a species move with a flux that is diffeet fom bulk motio? Oe cause of diffusio is cocetatio diffeeces. Fo example, usig a slight vaiatio o Fig. 1, imagie a costat desity ( = costat; iespective of compositio) mixtue of liquids ad, eclosed i a
CE 6333, Levicky 11 appaatus that cosists of a pai of vessels coected by a aow eck (Fig. 6). Iitially, the stopcock o the eck is closed, ad the amout of is geate i the left vessel tha i the ight. The vessels have equal volume ad, sice the desity of the mixtue does ot vay with compositio by assumptio, ae filled with equal mass of liquid. The, at some poit, the stopcock is opeed ad the mixtue is allowed to achieve a ew state of equilibium. The amout of mass i each vessel is fixed sice is assumed ot to deped o compositio, so opeig the stopcock to mix up ad does ot esult i a et tasfe of mass. If thee is o et, bulk flow of mass i the system, the v = 0. Nevetheless, fom expeimet it is kow that opeig of the stopcock does esult i a et tasfe of ad such that, at equilibium, thee ae o diffeeces i compositio betwee the two vessels. I othe wods, a et taspot of occued fom a egio of geate cocetatio (left vessel) to oe of lesse cocetatio (ight vessel), while flowed i the opposite diectio (ight to left i.e. fom high cocetatio of to low cocetatio of ). This mass taspot elimiated the cocetatio diffeeces that wee peset iitially, ad took place i absece of ay bulk flow (v = 0); it occued etiely due to diffusive fluxes. The diffusio occued spotaeously because it was themodyamically favoed. Fig. 6 T = costat T iitial state fial state Let's look a little close at the themodyamic oigis of diffusio. The discussio will focus o biay mixtues, whee the fluid of iteest cosists of two compoets, so that the umbe of species =. Oe ca imagie that such a mixtue udegoes a pocess i which its extesive iteal eegy U plus extesive exteal potetial eegy Y ae chaged ifiitesimally. Deotig the extesive etopy of the mixtue by S, its volume by V, the chemical potetial pe mass of species i by i, the exteal potetial eegy pe mass of species i by y i, ad the total mass of species i by m i, themodyamics states that d(u + Y) = TdS pdv + dm + y dm + dm + y dm (37) Rewitig equatio (37) fo a uit mass of the solutio yields d(u + y) = Tds pdv + d + y d + d + y d (38) whee s is etopy pe uit mass of the solutio, v is volume occupied by uit mass of solutio, u is iteal eegy pe uit mass of solutio, ad y is exteal potetial eegy of a uit mass of solutio. Now, ad ae subject to the costait + = 1 (39)
CE 6333, Levicky 1 Fom (39), d = - d (40) Usig (40) i (38) ad slightly eaagig, d(u + y) = Tds pdv + {( + y ) - ( + y )}d (41) The tem i paetheses, {( + y ) - ( + y )}, is efeed to as the "exchage potetial" T. Physically, T d epesets a diffeetial chage i iteal + potetial eegy of a uit mass of the solutio whe some species is exchaged fo species, iceasig the mass factio of by d. Thus, (41) ca be witte d(u + y) = Tds pdv + T d (4) T = ( + y ) - ( + y ) (43) Themodyamics also states that, fo a system with compoets, + 1 itesive vaiables ae sufficiet to fully specify the equilibium state (this statemet is subject to some estictios, such as absece of mateials whose iteal state is depedet o thei histoy, fo istace past mechaical defomatio). Sice biay fluid mixtues ae beig cosideed, fo which =, thee itesive vaiables ae eeded. It will be coveiet to choose tempeatue T, pessue p, ad T. I the absece of equilibium, oe o moe of these vaiables will vay with locatio i a way that o-equilibium gadiets i T, p, ad T exist. To move towad equilibium, the system will tasfe heat ad masses of the diffeet species aoud so as to elimiate these gadiets. Fo istace, heat flux will occu fom hot to cold to equalize the tempeatue, ad mass fluxes of idividual chemical species will occu so as to equalize each species' total ( i + y i ) potetials. No-equilibium pessue gadiets ca be omalized by bulk flow of mateial fom high to low pessue egios. Oe possible way the system ca elimiate o-equilibium gadiets i T, p, ad T is by diffusio of the vaious chemical species; it is the logical to assume that the diffusive fluxes will be fuctios of these gadiets, with steepe gadiets poducig geate fluxes. Let's coside the diffusive mass flux j, assumed to be a fuctio of the gadiets such that j = j (T, p, T ). Whe the gadiets T, p, ad T ae ot too lage, oe could pefom a Taylo seies expasio of j (aoud equilibium) i the gadiets ad tucate it afte the fist ode tems. Such a expasio would lead to the followig mathematical elatio fo j : j = - C 1 T - C T - C 3 p (44) The popotioality factos C i ae fuctios of T, p, ad T, but ot of the oequilibium gadiets i these quatities (this is because, ecallig Taylo Seies expasios, these factos ae to be evaluated at equilibium, the "poit" aoud which the expasio is beig fomed. Howeve, at equilibium, the oequilibium gadiets ae zeo). Equatio (44) must be costaied to obey equiemets imposed by themodyamics. I paticula, usig the secod law of themodyamics, it ca be show that C 3 must
CE 6333, Levicky 13 equal 0 (Ladau & Lifshitz, Fluid Mechaics, Pegamo Pess, pgs 187 ad, 1959). Theefoe, (44) simplifies to j = - C 1 T - C T (45) species' chemical potetial, at some poit i the mixtue, ca be viewed as a fuctio of the pessue, tempeatue, ad compositio at that poit. Fo a biay mixtue, this meas that = (T, p, ) ad = (T, p, ). [ implicit assumptio is beig made that themodyamic elatios such as = (T, p, ), which ae stictly applicable to systems at equilibium, apply eve though equilibium does ot exist thoughout the system. Qualitatively, this assumptio ca be expected to hold ove sufficietly shot legth scales ove which oly isigificat vaiatios i tempeatue, pessue, ad compositio occu, so that the values of these quatities ae well defied. Sice themodyamic quatities ae oly eeded at a poit (i.e. ove vey shot legths), fom a pactical pespective this cosideatio is ot limitig. lso, ote that is ot a idepedet themodyamic vaiable sice, fo a biay mixtue, = 1 -.] Takig the chemical potetials to be fuctios of T, p, ad, usig them i the defiitio fo T (equatio (4)), ad applyig the chai ule of diffeetiatio to obtai a expessio fo T yields T = ω T p ω p T p T p T,,, + y y (46) Isetig (46) ito (45) esults i j = - 1, C p T C T - C 1 T p, p - C 1 p T ω, - C 1 y y (47) The uwieldy pefactos i fot of the vaious gadiets ae usually expessed moe succictly by defiig thee quatities D, k T, ad k P as follows, k T D / T = 1, C p T C (48)
CE 6333, Levicky 14 k P D / p = C 1 p T, (49) D = C 1 ω T, p (50) If the diffusio coefficiet D (equatio (50)) ad chemical potetials ae kow (i.e. fom expeimetal measuemet), C 1 ca be evaluated fom (50) ad substituted ito (48) ad (49). With equatios (48) though (50), (47) becomes j = - (D k T / T ) T - (D k P / p)p - D - C 1 y (51) y The poduct D k T is called the themal diffusio coefficiet, k T the themal diffusio atio, ad D k P may be called the baodiffusio coefficiet. Equatio (51) shows that diffusive flux of mass of species, i a biay solutio of ad, ca aise fom fou diffeet cotibutios. (i). Odiay Diffusio. Odiay diffusio aises fom vaiatio i compositio, ad is epeseted by the thid tem o the ight of equatio (51). This tem ivolves, ad would be zeo if the mass factio of was uifom (costat). Odiay diffusio is the most commo cause of diffusio. Whe diffusio aises oly fom vaiatios i compositio, equatio (51) simplifies to j = - D (5) Equatio (5) is kow as Fick's Law, a cetal equatio i mass taspot that states that the diffusive mass flux is equal to a mateial paamete (D ) times a gadiet i compositio. s witte i equatio (5), diffusio occus fom highe to lowe mass factios; i othe wods, dow a compositio gadiet. The "-" sig esues that the diffusive flux j poits i the opposite diectio of the gadiet ; j poits i the diectio of steepest decease of. While ofte D is assumed costat, i geeal it vaies with T, p, ad. costat D is usually a good assumptio if the solutio is sufficietly dilute; that is, oe species (the solute) is peset at much lowe cocetatios tha the othe species (the solvet). Ude dilute coditios, the solute molecules ae fa apat ad "do ot see" each othe ad theefoe thei diffusio is ot iflueced by chages i thei cocetatio as log as dilute coditios pesist. The uits of D ae legth /time (eg. cm /sec). It is useful to biefly commet o the molecula basis of odiay diffusio. Imagie havig 1000 blue molecules ad 1000 ed molecules. You the exploe all the ways of aagig these molecules i a patte o a suface. You would discove that may moe aagemets exist i which blue ad ed molecules ae well mixed o the suface tha oes i which sigificat sepaatio of ed ad blue molecules is peset. I othe wods, the pobability of fidig a well mixed state, i which ed molecules ae itespesed with blue oes, is much highe tha that of a pooly mixed oe i which the two colos ae well sepaated. I themodyamic temiology, the etopy of a well mixed state is geate. Theefoe, a system pepaed i a less pobable state, oe that icludes sigificat sepaatio of
CE 6333, Levicky 15 ed ad blue paticles (i.e. cocetatio gadiets) will spotaeously evolve to a moe pobable, wellmixed state. (ii). Foced Diffusio. Foced diffusio aises due to vaiatio i the exteal potetial of species ad, ad is epeseted by the fouth tem o the ight of equatio (51). Oe commo example of foced diffusio occus with chaged species i a electic field. Fo example, if a solutio with Na + catios ad Cl - aios is placed i a exteal electic field, the catios will migate i the diectio of the electic field while the aios will migate opposite to it. This is oe possible maifestatio of foced diffusio. Fluids cotaiig chaged paticles ae dealt with extesively i electochemisty. iteestig obsevatio is that gavitatioal potetial does ot lead to mass diffusio. Rathe, this tem evaluates to zeo. This ca be udestood as follows. Imagie a paticle of with gavitatioal potetial eegy m gz, whee z is height i the gavitatioal field, m is the mass of the paticle, ad g is the gavitatioal costat. The y, the gavitatioal potetial pe uit mass of, is Theefoe, y = m gz / m = gz y = g Similaly, fo a paticle with a diffeet mass m, ad y = m gz / m = gz y = g. Sice y = y = g, y - y = g g = 0. Fo gavitatioal potetial, theefoe, the foced diffusio tem i equatio (8) evaluates to zeo. The easo fo the ull esult is that gavitatioal potetial of a paticle, whe omalized by the paticle's mass, is always simply gz, idepedet of whethe it is a paticle, paticle, etc. Thus, o a pe uit mass basis, diffeet desity mateials do ot possess diffeet gavitatioal potetials ad o foced diffusio esults. Havig said this, oe may be wodeig why do dese paticles settle i a gavitatioal field does that ot epeset mass tasfe due to a exteal field? Fo diffusive mass fluxes such as j, this effect is accouted fo by the secod tem (pessue diffusio) i equatio (51) (see below). (iii). Themal Diffusio (Soet effect). Themal diffusio aises due to spatial vaiatio i tempeatue. Themal diffusio is epeseted by the fist tem o the ight of equatio (51). I most poblems of pactical iteest, this cause of mass diffusio is quite small. Refeig back to equatio (47), the pefacto i fot of the themal diffusio tem has two cotibutios: (1) vaiatio of chemical potetials with tempeatue ad () diectly fom the pesece of a tempeatue gadiet (the "C "
CE 6333, Levicky 16 cotibutio). y ispectio, the chemical potetial cotibutio, -C1 T, will T p, poduce a flux of species i the diectio of iceasig tempeatue if is egative T p, (ote: C 1 ca be show to be positive). Usually, the deivative of a chemical potetial with espect to tempeatue is egative. The will cocetate i (diffuse to) hotte egios if its chemical potetial deceases with tempeatue faste tha the chemical potetial of. The effect of such themal diffusio is to take advatage of existig vaiatios i tempeatue so as to decease the chemical potetials of the vaious species as much as possible, leadig to the lowest possible total fee eegy of the solutio. The udelyig physical causes of themal diffusio oigiatig diectly fom the pesece of a themal gadiet, -C T, deped o the paticula situatio cosideed ad ae difficult to explai i geeal. Oe cause has to do with the fact that, at a give tempeatue, the mass flux associated with a heavie molecule (the mass flux is popotioal to the mass of the molecule times its velocity) is lage tha fo a light molecule. This depedece of mass flux o molecula mass, whe combied with a gadiet i tempeatue ca lead to diffeet ates of themal mass diffusio due to diffeeces i molecula masses. (iv). Pessue Diffusio. Pessue diffusio aises fom vaiatios i pessue, ad is epeseted by the secod tem o the ight i equatio (51). y ispectio (see equatio (47)), pessue diffusio of to a egio of highe pessue will esult if is egative (ude this coditio j poits i p T, the same diectio as p). Themodyamics tells us that the deivative = V p T, whee is the volume occupied by a uit mass of i solutio, efeed to as the patial mass volume of. Thus = V - V, ad will be egative if a uit mass of occupies a p T, smalle volume i solutio that a uit mass of. I othe wods, this tem will be egative if is dese tha, ude which coditio will pefeetially diffuse to egios of highe pessue. Fom a themodyamic pespective, pessue diffusio occus because dese fluids have lowe pessue-volume eegy pe uit mass (i.e. it costs less wok to iset a uit mass of a dese fluid ito a egio of high pessue because the dese fluid occupies less volume), ad by iceasig the mass cocetatio of dese fluids i high pessue egios the total fee eegy of the system is miimized. Sice gavity iduces pessue vaiatios iside a liquid, it ca lead to pessue diffusio. I devices such as cetifuges i which pessue diffeeces coespodig to thousads of g's ca be attaied, pessue V
CE 6333, Levicky 17 dive mass diffusio is used to sepaate diffeet solutio compoets based o miute diffeeces i thei desities. 5. Fick's Law. Havig biefly outlied the fou causes of mass diffusio spatial vaiatios i compositio, exteal potetial, tempeatue, ad pessue it is useful to highlight the most commo sceaio i which oly odiay diffusio is of impotace. The discussio will be specialized to biay solutios that obey Fick's Law, equatio (5). The estictio to biay solutios is ot as limitig as it may seem. Ideed, eve whe moe tha two compoets ae peset, as log as the solutios ae sufficietly dilute the diffusio of solute species ca be modeled as fo a biay system. This is because whe oe of the species (the solvet) is peset i vast excess, with all the est (the solutes) i tace amouts, the diffusio of each solute species ca be teated as if it was i pue solvet aloe. Ude these dilute coditios a solute paticle will ot "see" ay of the othe solute paticles, ad so its diffusio will ot be affected by thei pesece but oly by the solvet. Such a situatio is effectively a two compoet poblem, the solute of iteest plus the solvet. Fick's Law ca be witte i seveal commo foms: Mass diffusio flux; efeece bulk aveage velocity v (equatio 5 above): j = (v - v) = - D (5) If the total mass desity = costat thoughout the solutio, iespective of compositio, the ca be moved iside the gadiet opeato: j = - D ( = costat) (53) Mola diffusio flux; efeece bulk aveage velocity V: J = c (v - V) = j M/(M M ) = - cd x (54) I equatio (54), M is the aveage mola mass (mass pe mole of paticles, also efeed to as the aveage molecula weight) of the solutio, M = /c = (c M + c M )/c = x M + x M (55) M ad M ae the mola masses of ad paticles, espectively (e.g. the mola mass of wate is 18 gams pe mole of wate molecules). Equatio (54) ca be deived diectly fom equatio (5) though a athe legthy algebaic pocedue ivolvig covesios betwee mass factios ad mole factios, desities ad mola cocetatios. If the total mola cocetatio c is costat thoughout the solutio, iespective of compositio, c ca be bought iside the gadiet opeato ad equatio (54) becomes: J = - D c (c = costat) (56)
CE 6333, Levicky 18 Fially, it should be emphasized that the diffusio coefficiet D i all of the above Fick's Law expessios is the same (has the same umeical value ad uits of legth /time), whethe the expessios ae fo a mass diffusive flux o a mola diffusive flux. I the stictest itepetatio, the above expessios ae specialized to biay solutios i which the oly gadiets peset ae those i compositio. Howeve, as aleady metioed, they i fact wok quite well fo dilute solutios cotaiig moe tha oe solute. 6. Covective Mass Tasfe. pplicatio of Fick's Law to the calculatio of a diffusive mass flux equies kowledge of the compositio gadiet at each poit i space at which the mass flux is to be detemied. Futhe complicatig the pictue, thee ae may situatios i which covective mechaisms ae impotat i additio to diffusive oes. y covective mechaisms oe meas mass tasfe that aises by vitue of bulk motio. Fo example, i the case of the total mass flux of species, equatio (35) states that = j + v (35) I equatio (35), j is the diffusive pat of the total mass flux (aisig fom the pesece of themodyamically ufavoable gadiets i popeties such as cocetatio), while v is mass tasfe of due to bulk motio. It is this v pat that oe is efeig to as "covective" mass tasfe. I the case of the mola flux N, the covective pat is c V. This is evidet fom equatio (36), N = J + c V (36) To obtai the total mass tasfe of a species, oe would eed to calculate ot oly the diffusive potio (j o J ) but also the covective cotibutio. The sum of these two tems the leads to the total mass tasfe (whethe expessed i mass o mola uits). The touble is that, fo a abitay situatio i which mass tasfe may occu by diffusio as well as by covectio, the ecessay calculatio of compositio ad velocity at each poit iside a mixtue is ofte simply too had. The difficulties make calculatio of mass tasfe fom Fick's Law ad equatios such as (35) o (36) vitually impossible. Theefoe, aothe appoach, based o so-called mass tasfe coefficiets, is vey commoly implemeted. I geeal, oe is iteested i calculatig mass tasfe acoss the bouday of some system (e.g. fom a liquid ito a gas, fom oe liquid ito aothe liquid, etc.). The oe way to defie a mass tasfe coefficiet k C (uits: legth/time) is as follows, N = k C (c S - c ) (57) N is the mola mass tasfe of acoss the bouday (moles /(aea time)), c is mola cocetatio of (moles / volume) i the bulk of the system fa away fom the bouday (suface), ad c S is the mola cocetatio of ight at the bouday but still o the same side as the bulk of the system.
CE 6333, Levicky 19 Fig. 7 Fig. 7 depicts this situatio. I the figue, flux of, N, is occuig fom the bulk of a solutio phase (system) to its bouday. The bouday could epeset, fo example, the iteface betwee two immiscible fluids o betwee a fluid ad a solid wall. The simplicity of equatio (57) is achieved by two key assumptios: (i) cocetatio i the bulk of the system is uifom (hece c is well-defied), so that cocetatio of oly vaies acoss a "bode" (o "film") egio ext to the bouday, ad (ii) flux of species fom the bulk to the iteface is popotioal to the magitude of the cocetatio chage acoss the bode film egio (this chage beig equal to c s - c ). Howeve, ote that the mass tasfe coefficiet k C is itself i geeal a fuctio of flow geomety, tempeatue, cocetatio, ad possibly othe paametes. Still, elatively simple expessios fo k C ca be ofte developed diectly fom expeimetal measuemets, o fom theoy whe possible. Use of equatio (57) is suppoted by its ofte excellet success i applicatios. 7. Species' alaces fo Systems That Obey Fick's Law. Equatios (14) ad (18), the diffeetial mola ad mass balaces o a species, ae geeal i the sese that they ae idepedet of the umbe of compoets peset o ay models of diffusio. Fo example, they do ot equie odiay diffusio to obey Fick's Law. These equatios ae oly based o the statemet that the ate at which the amout of ca chage at a poit i space equals the ate at which is covected ito that poit plus the ate at which it is geeated by chemical eactios. Howeve, i thei peset fom, c t = cv R (14) t = v (18) these equatios ca be icoveiet because they cotai the species' velocity v, which must be kow i ode to accomplish the usual goal of solvig fo the mass o mola cocetatios. Oe way to model v is to coside it as cosistig of the two cotibutios peviously ecouteed: (i) oe due to bulk flow of the fluid mixtue ad (ii) oe due to the diffusio of species.
CE 6333, Levicky 0 = v = j + v (35) N = c v = J + c V (36) Sepaatio of the total flux of ito diffusive ad bulk covectio cotibutios is motivated by coveiece. The mass aveage velocity v is easy to measue, ad ca be obtaied by diect calculatio fom the diffeetial equatios of fluid mechaics (e.g. Navie Stokes equatios fo Newtoia fluids with ad costat). Typically, the diffusive fluxes ae modeled well by Fick's Law fo solutios i which oly odiay diffusio is peset ad which ae eithe biay o, as discussed ealie, dilute. Equatios (14) ad (18) ca be specialized to fluids that follow Fick's Law. Oe ca begi by isetig (35) ito (18) to obtai t j v + (58) = j v v + (59) = vecto idetity was used i deivig equatio (59) fom equatio (58) (equatio (30d) fom Hadout 1). Substitutig Fick's Law fo j, j = - D (60) ito (59) ad eaagig t = D - v v + (61) The left had side tem ad the d tem o the ight ca be ewitte usig the mateial deivative otatio, D + v t =, ad (61) becomes Dt D Dt = D v + (6) Equatios (61) ad (6) ae commo foms of the diffeetial mass balace o species with Fick's Law diffusio. These equatios simplify fo mixtues assumed to possess a costat desity, ad fo which the diffusio coefficiet ca be egaded as costat. Fo costat v = 0, so the 3d ight tem i equatio (61) ad the d ight tem i (6) dop out. If i additio D is also assumed costat, ad ecallig that =, equatio (61) eaages to t = D - v + (, D costat) (63)
CE 6333, Levicky 1 I physical tems, equatio (63) states that the ate at which mass desity of chages at a poit (left had side) equals the ate at which mass of diffuses to that poit due to cocetatio gadiets (fist tem o ight), plus the ate at which mass of is covected to that poit by bulk motio v of the fluid (secod tem o ight), plus the ate at which mass of is geeated at that poit by all chemical eactios (thid tem o ight). I the mateial deivative otatio, equatio (63) becomes Dt D = D + (, D costat) (64) Recall that the tem Dt D icludes a local ate of chage, t, plus chage due to motio though a gadiet of, v. Equatio (64) states: the ate at which the mass desity chages iside a fluid elemet movig with the mass aveage velocity v (left had side) equals the ate at which mass of diffuses ito the fluid elemet (fist tem o ight) plus the ate at which mass of is geeated iside the fluid elemet by chemical eactios (secod tem o ight). Compaed to equatio (63), the diffeece lies i itepetatio of the tem v. Fo a statioay poit though which a fluid flows, as i equatio (63), this tem epesets the chage i amout of species due to bulk covectio that bigs to that poit species at a diffeet ate tha the ate at which it emoves. I cotast, whe thikig ot i tems of a statioay poit but a fluid elemet that moves with the flow, v epesets chage i compositio that a obseve i the fluid elemet would obseve due to motio though a gadiet of ; i.e. fom a aea of oe cocetatio of to aothe. Expessed i Catesia, cylidical, ad spheical coodiate systems, equatio (63) becomes x v x v x v x x x D t 3 3 1 1 3 1 (Catesia) (65) Z θ z v θ v v z θ D t 1 1 (cylidical) (66) φ θ φ θ v θ v v φ θ θ θ θ θ D t si si 1 si si 1 1 (spheical) (67) Equatios (65) to (67) ae specialized to the case of costat ad D. The above equatios wee obtaied by takig the geeal diffeetial mass balace o species, sepaatig the total flux v of species ito bulk covectio ad diffusio cotibutios, ad usig Fick's Law to model the diffusio flux. Istead, had oe stated fom the diffeetial mola balace, equatio (14),
CE 6333, Levicky c t = cv R (14) ad sepaated the total mola flux c v ito bulk ad diffusive cotibutios accodig to equatio (36), c v = J + c V, the esult would be c = J c V + R = J c V V c + R (68) t Isetig i Fick's Law fo J, J = cd x, leads to c t = cdx c V V c + R (69) If the total cocetatio c ad diffusio coefficiet D ae costat (i.e. c ad D do ot vay fom poit to poit iespective of vaiatios that may be peset i tempeatue, pessue, o compositio), equatio (69) becomes c t = D c c V V c + R (c, D costat) (70) Eve though the total mola cocetatio c was assumed costat, the divegece of the mola aveage velocity, V, will i geeal ot be zeo ad theefoe the d ight tem i (70) caot be dopped. This is i cotast to the case of costat, fo which v equals zeo. Oe could show that whee V = R/c (c = costat) (71) R = R i i1 (7) R is the total poductio ate of paticles, i uits of moles/volume, due to chemical eactios. Equatio (71) was stated hee without poof, but ca be deived by summig equatio (70) ove all of the compoets of a mixtue. What is the value of R? Fo a eactio such as R = R + R = 0 sice oe molecule is coveted to exactly oe molecule, so that R = - R ad the total umbe of molecules does ot chage. Howeve, fo a eactio such as
CE 6333, Levicky 3 R = R + R = R / sice oly oe molecule of is cosumed fo two molecules of poduced (i.e. R = - R / ). Equatios (61) ad (69) assumed that the solutio is eithe biay o, if moe tha two compoets ae peset, sufficietly dilute. The key poit hee is that the diffusio of a molecule of should occu as if though pue ; this will be tue eve if othe species ae peset (i.e. i additio to ad ) as log as the cocetatios of the additioal species ae vey dilute. If the cocetatios ae ot dilute, the the diffusio tem is usually modified by defiig additioal diffusio coefficiets esposible fo couplig the diffusio flux of a species to compositioal gadiets of all the othe compoets. Of couse, fo a stictly biay solutio of ad, equatios (61) ad (69) hold ove the etie age of compositio, dilute as well as cocetated. 8. The ulk Refeece Velocity ad the Dilute Solutio Limit. The pefeece fo choosig to wok i mass uits (, j, v, ) o mola uits (N, J, V, c ) is based o coveiece. Ofte, coveiece has to do with hadlig of the bulk covectio tems; these ae the tems ivolvig the bulk aveage velocities v o V. Fo example, if oe expects V but ot v to be zeo, oe may choose (69) istead of (61) because the species mass balace would be simple. example of a sceaio i which V is zeo is equimola coutediffusio. This situatio ofte aises i the itediffusio of ideal gases i the absece of foced o atual covectio. I equimola coutediffusio, the two compoets have mola fluxes that ae equal i magitude but opposite i diectio, so that the total mola flux is zeo, N = - N (73) total mola flux = N + N = cv = 0 (74) I equatio (74), idetificatio of cv as the total mola flux, which equals the sum of the mola fluxes N i, follows fom the defiitio V = x i v i : i1 total mola flux = N i = c i v i = c x i v i = cv (75) i1 i1 i1 Similaly, the total mass flux, give by the sum of all the mass fluxes i, equals v, total mass flux = i = i v i = ivi = v (76) i1 i1 i1 Relatios (75) ad (76) ofte come i useful whe maipulatig mass tasfe expessios. While V is a good choice fo some poblems, i othe situatios v may be costat o zeo, e.g. if = -. I geeal, decidig whethe v o V is moe coveiet as the bulk aveage velocity may ot always be taspaet ad may equie pio expeiece with solvig mass tasfe poblems. Fotuately, this
CE 6333, Levicky 4 decisio is etiely elimiated i a impotat class of mass tasfe poblems i which bulk covectio is egligible. Let s coside the equiemets fo eglectig bulk covectio. ulk covectio is epeseted by the tems ivolvig the bulk aveage velocities i the above equatios (fo example, d tem o the ight i equatios (58) ad (68)). These tems will be zeo if v = 0 ad V = 0. I geeal, both the mass ad mola bulk aveage velocities will be zeo whe (i) thee is o foced covectio, (ii) thee is o fee covectio, ad (iii) the solutio is dilute. Foced covectio efes to covectio "foced" by pessue gadiets, shaft wok, ad foces such as may aise fom exteal potetials such as gavity. Most ofte, foced covectio is dive by machiey e.g. pumps, impelles. Fee covectio efes to the specific case whe covectio is caused by vaiatio of fluid desity. Fee covectio occus because, i the pesece of a gavitatioal field, less dese fluid will ise to the top. The isig of hot ai though colde ai is a example of fee covectio. Fee covectio is also kow as atual covectio. Eve if foced ad fee covectio ae ot peset, diffusio itself ca still egede covectio. This ability of diffusio to ceate bulk covectio is why the solutio is stipulated to be dilute (coditio (iii) i pevious paagaph). To illustate how diffusio ca ceate bulk covectio, let us coside agai the two bulb appaatus ecouteed ealie (Fig. 8). Fo the puposes of this illustatio it is assumed that pue species (i the left bulb) is dese tha pue species (i the ight bulb), ad the questio is whethe bulk covectio of mass is (v 0) o is ot (v = 0) peset whe ad mix. Iitially thee must be moe mass o the left tha o the ight, sice is dese tha. Whe the valve is opeed, a fial state is eached i which the compositio iside both bulbs will be the same. I the fial state, ight ad left bulbs both cotai equal amouts of mass theefoe, mass was moved fom left to ight o passage fom the iitial state (i which moe mass was peset i the left bulb) to the fial state (i which both bulbs cotai equal mass). This movemet was by bulk covectio, sice thee was ot just mixig of ad, but also a et tasfe of mass fom left to ight. This covectio was ot dive by a pessue gadiet, shaft wok, o body foces it was egedeed by the itediffusio of ad species. Thus, the above thought expeimet poves that diffusio ca egede bulk covectio of mass. This was a example of mass tasfe i a cocetated solutio sice the amout of ad peset i the two bulb appaatus was compaable. T iitial state Fig. 8. fial state Next, imagie the same appaatus. Howeve, while the ight bulb is agai filled with pue, the left bulb is filled with a dilute solutio of solute i a vast excess of solvet. Fo coceteess, imagie that the ight bulb has 10000 molecules, while the left bulb has 9900 molecules ad 100 molecules. Sice is agai dese tha, moe mass is peset o the left tha o the ight. Howeve,
CE 6333, Levicky 5 because the cocetatio is so dilute, this iitial diffeece i mass betwee the two bulbs is vey slight, almost impeceptible. O passage fom the iitial to the fial state, we expect about 50 molecules of to flow fom left to ight so that at equilibium, o aveage, the compositio iside both bulbs is the same (9950 molecules ad 50 molecules i each bulb ote that has to flow fom ight to left to "make oom" fo the molecules goig fom left to ight). Clealy, the bulk covectio of mass will be much, much less tha fo the cocetated case discussed above, sice the two bulbs wee vey simila i mass to begi with. The impotat questio is whethe bulk covectio ca be made so small, ude sufficietly dilute coditios, that it ca be eglected i compaiso with diffusio. Moe pecisely, ca the d ight tem (bulk covectio) i equatio (58) (ote: thee ae o eactios i the example ude discussio so = 0) t = D v (58) be igoed to yield the simple equatio t = D (59) which etais oly the diffusive tem o the ight. The aswe is YES; i.e. bulk covectio ca be eglected fo sufficietly dilute solutios. Fo example, fo biay mixtues of solute i solvet, the total flux of,, is give by = v = j + v = (v v) + v s stated ealie, j = (v v) is diffusive mass flux ad v is flux of due to bulk covectio. The atio of bulk covectio to diffusio is (bulk covectio) / diffusio = v / j = v / [ (v v)] = v / (v v) Usig the defiitio of v fo a biay solutio, v = v + v, leads to (bulk covectio) / diffusio = v / j = ( v + v )/(v - v - v ) I the dilute limit, solute (species ) vaishes. Theefoe, 0 ad 1, leadig to (bulk covectio) / diffusio = v / j = v / (v - v ) (77) If v << v, the (77) simplifies to v / j v / v << 1 (78)
CE 6333, Levicky 6 Iequality (78) shows that, fo v << v, the bulk covectio tem is much less tha the diffusio tem. This iequality will always hold fo sufficietly dilute solutios. Let's etu to the pevious thought expeimet i which 50 molecules of wee taspoted fom the left to the ight bulb. To make oom fo the icomig molecules, the solvet molecules must be simultaeously displaced i the opposite diectio. Howeve, because thee ae so may moe molecules tha molecules, o aveage a molecule oly eeds to tavel a shot distace; i othe wods, it is the summed displacemet of 10000 molecules that must coutebalace the additio of just 50 molecules. O aveage, a molecule will oly eed to be displaced by a factio, 50/10000 = 1/00, i the diectio opposite to that of the molecules. This meas that the velocity of a typical molecule will oly be about 1/00 that of a typical molecule duig the mixig pocess. Thus, v / v 0.005 << 1, satisfyig the iequality (78) ad demostatig that bulk covectio could ideed be eglected compaed to diffusio. Put diffeetly, if oe adds a gai of suga (species ) to a bathtub of statioay wate (species ), the wate is ot displaced vey much eve as the suga dispeses though the etie bathtub. Thus the aveage velocity of the suga, v, must be much geate tha the aveage velocity of a typical wate molecule, v, duig this mixig pocess. Theefoe, i this dilute, biay suga solutio it would also be appopiate to eglect bulk covectio sice v / v << 1. I summay, whe (i) thee is o foced covectio, (ii) thee is o fee covectio, ad (iii) the solutio is dilute, bulk covectio tems i the species' balace equatios ca be eglected. Thus equatios (68) ad (58) become c t = cdx + R (o bulk covectio) (79) t = D + (o bulk covectio) (80) If c, D, ad ae costat, the c t = D c + R (c, D costat; o bulk covectio) (81) t = D + (, D costat; o bulk covectio) (8) Fo the case of o chemical eactios (R ad = 0), equatios (81) ad (8) ae kow as Fick's Secod Law of Diffusio. Thee ae some additioal, special coditios ude which bulk covectio ca be eglected. Fo istace, equatio (63), which holds fo the case of costat ad D, will educe to equatio (8) if the vectos v ad ae pepedicula so that v = 0. Thus icompessible flows i which bulk covectio (foced o atual) is pepedicula to the diectio of diffusio also obey (8). 9. Commo ouday ad Iitial Coditios i Mass Taspot. 9.i Coditios o Cocetatio. The cocetatio of a chemical species ca be specified at a poit i space o time. Examples ae:
CE 6333, Levicky 7 (x 1 = 0) = (83) (t = 0) = (84) whee the value of is give. Ofte, a coditio o cocetatio may appea i the fom of a patitio coefficiet o a Hey's Law costat. Fo example, imagie two phases i cotact, both of which cotai species (Fig. 9). I phase I, the cocetatio of is kow, ad the goal is to solve fo the cocetatio pofile of i phase II. commo pactice i these types of poblems is to assume that, at the iteface betwee the two phases, local equilibium applies. Local equilibium meas that the cocetatios of, immediately o the two sides of the iteface, ae i equilibium. This coditio ca be specified by a equilibium patitio coefficiet H c, fo example Z phase II phase I iteface ( z = 0) Fig. 9 II (z = 0) = H c I (z = 0) (85) I equatio (85), II (z = 0) is the cocetatio of at the iteface immediately iside phase II ad I (z = 0) is the cocetatio of at the iteface immediately iside phase I. I essece, the patitio coefficiet accouts fo the elative pefeece of paticles to be dissolved iside the two phases. If H c ad I (z = 0) ae kow, the II (z = 0) ca be calculated ad used as the bouday coditio o iside phase II. closely elated cocept is Hey's law, p = Hx (86) whee p is patial pessue of i a gas phase, x is its mole factio i a liquid that is i equilibium with the gas, ad H is Hey's Law costat. Hey's law woks well fo dilute solutios; i.e. small x. Fo example, if a gas mixtue cotaiig oxyge at a patial pessue p O is i cotact with a wate phase the, assumig local equilibium, the mole factio of oxyge i the wate phase immediately at the wate-gas iteface would be x O (z = 0) = p O /H (87) If p O ad H ae kow, the mole factio x O (z = 0) ca be calculated ad used as a bouday coditio i solvig fo the oxyge cocetatio pofile i the wate phase. Note that the value of H will deped
CE 6333, Levicky 8 o the specific system cosideed. Fo ideal solutios obeyig Raoult's Law, H equals the satuatio pessue p Sat of the patitioig species. 9.ii Coditios o Fluxes. ouday ad iitial coditios ca also expess statemets about the flux of a species. The most obvious example is pehaps that of a impemeable bouday, acoss which thee ca be o taspot of a species. Sice diffusio of species acoss the bouday must be zeo, j (z = 0) = 0 (impemeable bouday) (88) The "" subscipt idicates diffusio pepedicula to (i.e. acoss) the bouday. Fo odiay diffusio obeyig Fick's Law, equatio (88) becomes ω j (z = 0) = - D z = 0 so that z 0 ω z = 0 (89) z 0 z is the coodiate pepedicula to the bouday, as i Fig. 9. While equatio (89) is witte i tems of the mass diffusive flux, a aalogous statemet ca be made egadig the mola diffusive flux. Fo the mola flux, the gadiet of mole factio x with espect to z vaishes at z = 0. I cotast, at a pemeable bouday species ca coss fom phase I to phase II. If steady state applies, the caot accumulate at the bouday. Ude these coditios,,i (z = 0) =,II (z = 0) (steady state; pemeable bouday) (90),I ad,ii ae mass fluxes of pepedicula to the pemeable bouday. Equatio (90) states that the mass flux,i of to the bouday fom phase I must equal the flux,ii of fom the bouday ito phase II othewise would accumulate at the bouday ad the coditio of steady state would be violated. Whe bulk covectio pepedicula to the iteface is egligible (this implies dilute solutios see ealie discussio), = j ad, assumig diffusio obeys Fick's Law, bouday coditio (90) ca be ecast as ω D I ω I = D II II z z0 z z 0 (steady state; pemeable bouday; o bulk covectio acoss bouday) (91) D I ad D II ae diffusio coefficiets of i phase I ad II, espectively, ad I ad II ae the coespodig total desities of the two phases. Equatio (91) efoces cotiuity of the diffusio flux of acoss the iteface the diffusive flux of to the bouday fom phase I equals that away fom the bouday ito phase II.
CE 6333, Levicky 9 9.iii Coditios Ivolvig Heteogeeous Reactios. Whe a chemical eactio occus at a bouday o a iteface, it is efeed to as a heteogeeous eactio. Heteogeeous eactios typically appea i mass taspot poblems i the fom of bouday coditios. Fo example: (a). Ifiitely apid, ievesible cosumptio of at a suface located at z = 0: (z = 0) = 0 (91) I (91), the eactio is assumed to occu so fast that cocetatio of at the suface is effectively zeo. I othe wods, as soo as a paticle of aives at the suface, it eacts ad theefoe is elimiated. (b). Steady-state, slow cosumptio of at a suface located at z = 0: (z = 0) = S (9) Sice steady state applies, caot accumulate at the suface. Theefoe, as stated i equatio (9), the ate S (uits: mass / (aea time)) at which mass of is cosumed by the heteogeeous suface eactio must equal the flux at which mass of aives at the suface. The above examples of bouday coditios wee expessed usig the mass uits. y of these coditios could just as well be witte i mola uits. Fo example, i mola uits equatio (9) would become N (z = 0) = R S (93) whee R S is the ate (uits: moles / (aea time)) at which is cosumed by the suface eactio ad N is the mola flux of pepedicula to the suface.