MASS DIFFUSION MASS TRANSFER WHAT IT IS AND WHAT IT ISN'T

Transcription

1 MASS DIFFUSION Mass Transfer... 1 What t s and what t sn't... 1 What t s for. Applcatons... How to study t. Smlartes and dfferences between Mass Transfer and Heat Transfer... Forces and fluxes... 4 Specfyng composton. Nomenclature... 4 Specfyng boundary condtons for composton... 6 Speces balance equaton... 9 Dffuson rate: Fck's law The dffuson equaton for mass transfer Some analytcal solutons to mass dffuson Instantaneous pont-source Sem-nfnte planar dffuson Dffuson through a wall Summary table of analytcal solutons to dffuson problems Evaporaton rate... References... 3 MASS TRANSFER WHAT IT IS AND WHAT IT ISN'T The subject of Mass Transfer studes the relatve moton of some chemcal speces wth respect to others (.e. separaton and mxng processes), drven by concentraton gradents (really, an mbalance n chemcal potental, as explaned n Entropy). Flud flow wthout mass transfer s not part of the Mass Transfer feld but of Flud Mechancs. Heat transfer and mass transfer are knetc processes that may occur and be studed separately or jontly. Studyng them apart s smpler, but t s most convenent (to optmse the effort) to realse that both processes are modelled by smlar mathematcal equatons n the case of dffuson and convecton (there s no mass-transfer smlarty to heat radaton), and t s thus more effcent to consder them jontly. On the other hand, the subject of Mass Transfer s drectly lnked to Flud Mechancs, where the snglecomponent flud-flow s studed, but the approach usually followed s more smlar to that used n Heat Transfer, where flud flow s manly a boundary condton emprcally modelled; thus, the teachng of Mass Transfer tradtonally follows and bulds upon that of Heat Transfer (and not upon Flud Mechancs). In fact, development n mass-transfer theory closely follows that n heat transfer, wth the poneerng works of Lews and Whtman n 194 (already proposng a mass-transfer coeffcent hm smlar to the thermal convecton coeffcent h), and Sherwood's book of 1937 on "Absorpton and extracton". Even more, snce the mlestone book on "Transport phenomena" by Brd et al. (1960), heat transfer, mass transfer, and momentum transfer, are often jontly consdered as a new dscplne. Mass dffuson page 1

2 As usual, the basc study frst focuses on homogeneous non-reactng systems wth well-defned boundares (not only n Mass Transfer, but n Heat Transfer and n Flud Mechancs), touchng upon movng-boundary problems and reactng processes only afterwards. As for the other subjects, t s based on the contnuum meda theory,.e. wthout accountng for the mcroscopc moton of the molecules (so that feld theory and the flud-partcle concept are appled here too). Dffuson theory only apples to molecular mxtures (d<10-8 m); for collods and suspensons ( m), Brownan theory must be appled, and for larger partcles (>10-5 m) Newtonan mechancs apples. Notce that we only consder here mass dffuson due to a concentraton gradent, what mght be called concentraton-phoress n analogy to other mechansms of mass dffuson lke thermo-phoress (Soret effect), pezo-phoress (dffuson due to a pressure gradent), or electrophoress (dffuson due to a gradent of electrcal potental appled to onc meda). Tradtonally, the feld of Mass Transfer has been studed only wthn the Chemcal Engneerng currculum, except for humd-ar applcatons (evaporaton) and thermal desalnaton processes, whch has been always studed n Mechancal Engneerng. But mass transfer problems are prolferatng n so many crcumstances, especally at hgh temperatures (dryng, combuston, materals treatment, pyrolyss, ablaton...), that the subject should be covered on dfferent grounds to encourage effectve nterdscplnary team-work WHAT IT IS FOR. APPLICATIONS Applcatons of Mass Transfer nclude the dsperson of contamnants, dryng and humdfyng, segregaton and dopng n materals, vaporsaton and condensaton n a mxture, evaporaton (bolng of a pure substance s not mass transfer), combuston and most other chemcal processes, coolng towers, sorpton at an nterface (adsorpton) or n a bulk (absorpton), and most lvng-matter processes as respraton (n the lungs and at cell level), nutrton, secreton, sweatng, etc. A common process to separate a gas from a gaseous mxture s to selectvely dssolve t n an approprate lqud (ths way, carbon doxde from exhaust gases can be trapped n aqueous lme solutons, and hydrogen sulfde s absorbed from natural-gas sources; when water vapour s removed, the absorpton process s called dryng. Strppng s the reverse of absorpton,.e. the removal of dssolved components n a lqud mxture. Dstllaton s the most mportant separaton technque. HOW TO STUDY IT. SIMILARITIES AND DIFFERENCES BETWEEN MASS TRANSFER AND HEAT TRANSFER Mass Transfer educaton tradtonally follows and bulds upon that of Heat Transfer because, on the one hand, mass dffuson due to a concentraton gradent s analogous to thermal-energy dffuson due to a temperature gradent, and thus the mathematcal modellng practcally concdes, and there are many cases where mass dffuson s coupled to heat transfer (as n evaporatve coolng and fractonal dstllaton); on another hand, Heat Transfer s mathematcally smpler and of wder engneerng nterest than Mass Transfer, what dctates the precedence. But there are mportant dfferences between both subjects. Mass dffuson page

3 Radaton. Frst of all, from the three heat transfer modes (conducton, convecton, and radaton), only the two frst are consdered n mass transfer (dffuson and convecton), radaton of materal partcles (as neutrons and electrons) beng studed apart (n Nuclear Physcs). Notce, by the way, that the word dffuson can be appled to the spreadng of energy (heat dffuson), or speces (mass dffuson), or even momentum n a flud or electrc charges n conductors, but the word conducton s more commonly used than heat dffuson (whereas mass conducton s rarely used). Solds versus fluds. Heat Transfer starts wth, and focuses on, heat dffuson n solds, whch have hgher thermal conductvtes than fluds, the latter beng consdered globally through emprcal convectve coeffcents, whereas Mass Transfer focuses on gases and lquds, whch have hgher mass dffusvtes than solds. The explanaton for such a dfference s that heat conducton propagates by partcle contact (for the same type of partcles, the shortest separaton the better), whereas mass dffuson propagates by partcles movng through the materal medum (for the same type of partcles, the largest vods the better). Moreover, Heat Transfer problems n solds are smple and relevant to many applcatons, whereas Mass Transfer problems n solds are of much lesser relevance, and Mass Transfer problems n fluds are much more complcated because the smplest mass-dffuson problems are of lttle practcal nterest, convecton wthn fluds beng the rule (fluds tend to flow). When dffuson n solds s wanted, as n dopng slcon substrates n mcroelectroncs, or n surface dffuson of carbon or ntrogen n steel hardenng, hgh temperature operaton s the rule (dffuson coeffcents show an Arrhenus' type dependence wth temperature). Slowness. Thermal dffusvtes decrease from solds to fluds, wth typcal values of a 10-4 m /s for metals and a 10-5 m /s for non-metals, down to a 10-5 m /s for gases and a 10-7 m /s for lquds. On the contrary, mass dffusvtes decrease from fluds to solds, wth typcal values of D 10-5 m /s for gases and D 10-9 m /s for lquds, down to D 10-1 m /s for solds. Bulk flow. There s no bulk flow n heat dffuson (ether wthn solds or fluds), whereas there s always some bulk flow assocated to dffuson of speces (except n the rare event of counterdffuson of smlar speces);.e. mass dffuson generates mass convecton, n general. Number of feld varables. One may consder just one heat-transfer functon, the temperature feld T (the heat flux s bascally the gradent feld), but several mass-transfer functons must be consdered, one mass fracton, y, for each speces 1..C (C beng the number of dstnct chemcal speces), although most problems are modelled as a bnary system of just one speces of nterest dffusng n a background mxture of averaged propertes. Contnuty at nterfaces. Mass-transfer boundary condtons at nterfaces are more complex than thermal boundary condtons, because there are always concentraton dscontnutes, contrary to the contnuous temperature dctated by local equlbrum (chemcal potentals are contnuous at an nterface, not concentratons). Dffuson 'uphll'. Besdes the effect of coupled fluxes, t s mportant to realse that mass dffuson can be from a low concentraton wthn a condensed medum towards a hgh concentraton wthn a more dsperse medum, because, as sad, t s not concentraton-gradent but chemcal-potentalgradent, what drves mass dffuson (e.g. see Dffuson through a wall, below). Mass dffuson page 3

4 Forces and fluxes Mxng,.e. decreasng dfferences n composton (really, n chemcal potental) or temperature, s a natural process (.e. t does not requre an energy expendture), drven by the gradents of temperature, relatve speed and chemcal composton (wth the natural stratfcaton n the presence of gravty or another force feld). It s nterestng to realse that the thermal and mechancal forces towards equlbrum have been harnessed to yeld useful power (heat engnes, wnd and water turbnes), but the chemcal forces that drve mass transfer have not yet been rendered useful as energy source, no doubt because of ts low specfc energy (there has been proposals to bult power plants drven by the dfference n salt concentraton at rver mouths). The gradent of temperature, momentum and concentratons, gve rse to correspondng fluxes n thermal energy, momentum and amount of speces. The relaton between forces and fluxes are the transport consttutve equatons: Fourer law for Heat Transfer, Newton (or Stokes) law for Flud Mechancs, and Fck law for Mass Transfer (to be presented below), and the purpose of the subject s to solve generc feld balance equatons (energy balance, momentum balance, and speces balance), wth the help of consttutve equatons, and the partcular boundary condtons and ntal condtons. But before developng the theory, t must be understood that mxng s a slow physcal process, f not forced by convecton and turbulence, and even so. Many practcal processes are lmted by the dffculty to ncrease the mass transfer rate. An order of magntude analyss shows that the relaxaton tme for dffuson-controlled phenomena (thermal, momentum, speces) across a dstance L s trelaxl /a, where a s the dffusvty that, as explaned below, s of order 10-5 m /s n gases, what teaches that dffuson across a 1 m dstance takes some 10 5 s,.e. one whole day. Of course, everybody knows that heatng one metre of ar doesn't take one day, nether t takes so long for odours to travel one metre, or for puttng n moton or arrestng a gas; the explanaton s that fluds are very dffcult to keep at rest when perturbed, and the convecton that develops greatly ncreases the mxng rate and lowers the requred tme. Thermodynamcs teaches that, wthn an solated system n absence of external forces, temperature, relatve moton and chemcal potental tend to get unform, by establshng a thermal-energy flux, a momentum flux and a mass-dffuson flux, proportonal (to a frst approxmaton) to the gradents of temperature, velocty and concentraton, that tend to equlbrate the system. Notce however that, besdes those drect fluxes, other smaller cross-couplng fluxes may appear, as mass-dffuson due to a temperature gradent n a unform concentraton, or heat transfer due to a concentraton gradent n an sothermal feld, whch, n the lnear approxmaton, are related among them by Onsager's recprocal relatons. Specfyng composton. Nomenclature Mass transfer may take place wthn gases, lquds, solds or through ther nterfaces, always nvolvng a mxture, but mass dffuson n a gas s of man nterest for two reasons: frst, t s the best understood, and Mass dffuson page 4

5 second, t s the best dffusng medum (dffuson n lquds and solds s much slower). For that reason, and for smplcty, we start here wth a gaseous (sngle phase) mult-component mxture. A mxture s any mult-component system,.e. one wth several chemcal speces. The thermodynamcs of mxtures n general (gaseous, lqud or sold) has been consdered under the headng Mxtures, manly devoted to deal mxtures. We assume true solutons,.e. homogeneous solutons, and do not consder collods and suspensons, treated under the headng Mxture settlng. Although, from the theoretcal pont-of-vew, molar fractons and concentratons should be preferred, the most common composton determnant n a sngle-phase mxture s the mass fracton, y, or the mass densty ρ. Only one of those parameters s needed, but all of them are made use of n practce, so a common (an tedous) task n mass-transfer calculatons s to pass from one varable to another, based on ther defntons: mass fractons: y m m x M x M m mass denstes (or mass concentratons): ρ yρ V () n ρ x molar denstes (or molar concentratons): c V M x M ρ (3) n y / M molar fractons: x n y / M (4) IGM Ru partal pressure of a speces n a gas mxture: p xp crt u ρ T. (5) M The molar mass of the mxture s defned as Mm m/nρ/cσxm, although t s only used for gas mxtures. There are stll other specal varables n use to defne a mxture composton, as ar-to-fuel rato and rchness (equvalence rato) n combuston problems. (1) Exercse 1. Dry ar can be approxmated as a mxture of 79% N and 1% O by volume (meanng that, by lettng 79 volumes of pure ntrogen to mx wth 1 volumes of pure oxygen, wthout changes n pressure and temperature,.e. by just removng the partton, we obtan 100 volumes of a mxture closely resemblng dry ar). Determne other possble specfcatons of dry ar composton, from (1-5). Soluton. Assumng deal gas behavour,.e. pvnrt, at constant p and T, volumes V are proportonal to amounts of substance, n, and thus volume percentage concdes wth molar fractons (4);.e., we can consder as data xn0.79 and xo0.1 (mnd that subndces are just labels, not meanng atoms but molecules). From (1), wth MN0.08 kg/mol and MO0.03 kg/mol, one gets ΣxM kg/mol, yn / and yn0.3, ndcatng that the molar mass for the mxture s a weghted average of those of the Mass dffuson page 5

6 components, Mm m/nρ/cσxm0.09 kg/mol, and that the heaver speces shows a larger concentraton-value n terms of masses than n terms of amounts of substance. From () we get mass concentratons (mass denstes) n terms of the mxture densty, whch depends on temperature and pressure; for T88 K and p100 kpa, we get for the densty of ar ρ1.1 kg/m 3, and for the speces ρn kg/m 3, and ρo kg/m 3. Notce that some authors use m or w nstead of y for mass fractons. From (3) we can get molar concentratons, agan dependng on actual p-t values; wth the prevous choce, cnρn/mn0.93/ mol/m 3, and coρo/mo0.8/0.039 mol/m 3 (n total, ccn+cop/(rt)10 5 /(8.3 88)4 mol/m 3 ). Fro (5) we get the partal pressures, pnxnp kpa, and poxop kpa. Fnally notce that we can equally say that ar has 0.79/ tmes more ntrogen than oxygen, by volume (or amount of substance), or tmes more ntrogen than oxygen, by mass. The fndng of qualtatve or quanttatve composton n a mxture s known as chemcal analyss, or smply 'the analyss'. We focus here on quanttatve analyss, assumng the substances are already known. Most methods of concentraton analyss are based on measurng mxture densty (provded the densty dependence on speces concentraton, ρmρm(t,p,x), s known beforehand by calbraton), by one of the dfferent technques: Absorpton radometry. By lght transmttance (n the vsble, nfrared, or monochromatc). Refractometry. By ray tracng. Refractve ndex vares almost lnearly wth densty. Gravmetry. Weghtng a known volume of lqud. Ths s perhaps the easest and quckest method to measure soluton concentraton, but requres samplng. Resonant vbraton. The natural frequency of an encapsulated lqud sample precsely metered depends on ts mass. May be appled to a lqud flowng along a bend connected by soft bellows to the ppes. Sonc velocmetry. Densty s obtaned from ρe/c, where E s the bulk modulus of the soluton and c the sound speed through t. Electrc conductvty. Ths s the best method for very low concentraton of electrolytc solutons. The measurng electrodes may be generc, or selectve for some specfc on (e.g. Ca +, NH4 +, Cl -, NO3 - ). Specfyng boundary condtons for composton Composton at boundares or nternal nterfaces n a mxture usually shows a dscontnuty, contrary to temperature n heat transfer problems, when contnuty s the rule (except for the specal topc of thermal jont conductance). The typcal boundary condtons for a speces concentraton are, as for heat transfer, a known value of the functon (mposed concentraton or temperature, respectvely), or a known value of ts gradent (mposed speces flux or heat flux, respectvely), the specal case n the latter beng the mpermeable nterface or adabatc wall, respectvely; here: Mass dffuson page 6

7 mpermeable nterface: x x n 0, or n 1D 0 x x xnterface (6) n beng the unt normal vector to the nterface. Imposng a non-zero mass flux, or a gven concentraton value, s done as n Heat Transfer,.e. by provdng large sources of the chemcal speces (a sold chunk, a lqud pool, a gas reservor), smlarly to large metal blocks to specfy the temperature at a wall. Local thermodynamc equlbrum then teaches that the temperature of the system near the wall s equal to that of the wall, but the same s not true for concentratons, where local equlbrum mples equalty of ts chemcal potental, not of ts concentraton. The boundary condton n a gas mxture may be another gas phase, as when mxng along a tube connected to a large reservor of a gven gas; f one assumes that the large reservor s well-strred, thence, the boundary condton for the gas mxture n the tube may be approxmated by the known concentraton at the reservor. For a gas mxture n contact wth a condensed phase, the typcal boundary condtons for a speces concentraton, assumng deal mxtures, s Raoult's law (deduced n Mxtures): * * x,gas p ( T) pure condensed phase p ( T) p u B x,vap exp A (7) x,condensed p p p C+ T / Tu where Antone's fttng coeffcents for the vapour pressure curve have been explctly shown (see Phase Change for an explanaton). Notce that sublmaton vapour-pressure data should be used when the source s sold, e.g. when ce s the source of water vapour, nstead of lqud water. For nstance, the boundary value for water-vapour dffuson n ambent ar close to a water pool at 15 ºC s x,vap0.017, correspondng to the two-phase equlbrum pressure of pure water at 15 ºC: 1.7 kpa. When gases are sparngly soluble, Henry's law must be used nstead of Raoult's law (see Solutons). Exercse. Fnd the concentraton of carbon doxde at a water surface at 5 ºC, when exposed to a gas stream wth a partal pressure of CO of 300 kpa. Soluton. Henry's law data can be found n a bewlderng varety of manners, and wth dfferent unts, usually under the common name of 'Henry constant', KH. For solublty of CO n water at 5 ºC, we may fnd, from the solublty data (Table 3) n Solutons, KHc,lq/c,gas0.80, meanng that, for CO to be at equlbrum between the aqueous phase and the gas phase, there must be 0.80 mol/m 3 of CO dssolved n water per each 1 mol/m 3 of CO dssolved n the gas phase (or pure). We mght fnd the same number but referrng to mass concentratons, snce they are just proportonal wth the factor MCO0.044 kg/mol, (3), KHρ,lq/ρ,gas0.80, meanng that, for CO to be at equlbrum between the aqueous phase and the gas phase, there must be 0.80 kg/m 3 of CO dssolved n water per each 1 kg/m 3 of CO dssolved n the gas phase (or pure). Those are the only non-dmensonal 'constants' (constant n Henry's law, and other equlbrum laws n Chemstry, means that t only depends on temperature, not on pressure). Mass dffuson page 7

8 We mght fnd KHc,lq/p,gas3 (mol/m 3 )/bar, meanng that, for CO to be at equlbrum between the aqueous phase and the gas phase, there must be 3 mol/m 3 of CO dssolved n water per each 1 bar (100 kpa) of partal pressure of CO dssolved n the gas phase (or pure); of course, we can check for consstency: c,lq/c,gasrtc,lq/p,gas, but t s prone to trval errors on unt converson (e.g. the 10 5 n c,lq/c,gasrtc,lq/p,gas /10 5. We mght fnd KHx,lq/p,gas580 ppm_mol/bar, meanng that, for CO to be at equlbrum between the aqueous phase and the gas phase, there must be 580 parts-per-mllon n molar base of CO dssolved n water per each 1 bar (100 kpa) of partal pressure of CO dssolved n the gas phase (or pure); we can check for consstency: c,lq/c,gas(ρmrt/mm)x,lq/p,gas, where subndex m referrng to the soluton, whch can be approxmated as pure water, and thence c,lq/c,gas(ρmrt/mm)x,lq/p,gas0.80 ( /0.018) /10 5. We mght fnd KHc,lq/c,gas,STP0.73 m 3 (STP)/bar, meanng that, for CO to be at equlbrum between the aqueous phase and the gas phase at 5 ºC, the amount of CO dssolved n 1 m 3 of soluton, per each 1 bar (100 kpa) of partal pressure of CO dssolved n the gas phase (or pure), would occupy 0.73 m 3 at STP-condtons of 0 ºC and 100 kpa; we can check for consstency: c,lq/c,gas,stp(c,lq/c,gas)/(tstp/t ) /980.73, where subndex m referrng to the soluton, whch can be approxmated as pure water, and thence c,lq/c,gas(ρmrt/mm)x,lq/p,gas0.80( /0.018) /10 5. In summary, f we assume that pure carbon doxde at 300 kpa (or a gas mxture wth that partal pressure of CO) s at equlbrum wth water at 5 ºC, the CO concentraton n the gas phase s c,gasxp/(rt) /(8.3 98)11 mol/m 3, and the CO concentraton n soluton s c,lqkhc,gas mol/m 3,.e. 0.17% of the molecules n the lqud phase are CO, and 99.8% are HO molecules (assumng no other solute s present); t can also be concluded that, f all the CO dssolved n 1 m 3 of water at equlbrum at 5 ºC and 100 kpa, were extracted and put at STP-condtons (0 ºC and 100 kpa), t would occupy a volume of. m 3. For a lqud mxture n contact wth another condensed phase (a sold or an mmscble lqud), the boundary condton for a speces concentraton,, called a solute, cannot be modelled n a smple form as Raoult's law; at most, n the deal case, from the equalty of the solute chemcal potental n both phases one gets: x µ ( T) µ ( T) g ( T) h ( T) s ( T) + (8) x RT RT RT R,lq,sol,lq,sol-lq,sol-lq,sol-lq ln,sol u u u u where the other phase has been labelled 'sol' both for the case of a sold or an mmscble lqud. In the case of a pure sold as a source of solute, the boundary condton (8) yelds x,lqexp((µ,sol µ,lq)/(rut)), and t s known as the solublty of the sold solute n the lqud solvent specfed (.e. the maxmum molar fracton of solute the lqud can hold). Solublty data for sold and lqud solutes n a lqud solvent can be found asde. Mass dffuson page 8

9 Dffuson of speces wthn a sold s much more ntrcate, partcularly when the sold s porous or s n a granular state, where hydrodynamc flow appears (seepage). Dffuson through one-pece solds s nearly neglgble n most cases at room temperature, but can be studed wth Henry's law (some values are gven n Solutons). Gas solublty n solds ncreases wth temperature, contrary to what happens n lquds, and subsequent degassng on coolng may be a nusance (may even run a castng process by creatng porosty and vods). Besdes, chemcal reactons may occur at room temperature (e.g. oxdaton) but partcularly when the temperature s ncreased to enhance mass transfer. SPECIES BALANCE EQUATION For a gven speces n a mxture (sold, lqud or gaseous), ts mass balance for a control volume s (accumulaton flux + producton): dm dt surfaces m + m j n da + w dv (9), gen A V where m s the mass of speces n the volume V, j s the local mass-flux of speces at the surface area A, and w a possble local speces generaton densty due to chemcal reactons. For a control-volume system of dfferental volume dxdydz, wth the contnuum model: ρ x ddd xyz ρvxdd yz ρvx + dx dd yz+ ρvydd zx ρvy + dy dd zx wddd xyz t x y ( ρ v ) ( ρvy ) ρ t ( ρ v ) + w (10) where ρ s ts mass densty and v the local velocty of the -component flud n a fx reference frame. For ρ/ t+ ρv 0, the well-known contnuty a one-component flud, the mass balance (10) reduces to ( ) equaton of Flud Mechancs, that can be recovered by summaton n (10) for all the speces n the mxture;.e.: Σρ wth v Σ( ρv) / ρ ρ + Σ ( ρv) Σw + ( ρv) 0 t t (11) Notce that a smlar argument mght have been followed wth molar denstes nstead of mass denstes, and a molar-averaged velocty defned that would not concde n general wth the mass-averaged velocty v, that s tradtonally used. Besdes the speces balance n a generc dfferental volume (10), the speces balance n a generc nterface must be establshed n many problems: (1) 0 m, out m, n + m, surface gen Mass dffuson page 9

10 where the last term, speces generaton at the nterface, only appears n the case of heterogeneous reactons at the nterface. Dffuson rate: Fck's law Actual mxng of chemcal speces s governed by mass-transfer laws very smlar to heat-transfer laws, establshng a lnear proporton between forces and fluxes: n Heat Transfer, a lnear proporton between the temperature-gradent, and the energy flow as heat; n Mass Transfer, a lnear proporton between the speces densty-gradent, and the relatve velocty of the speces-flud to the mean-flud. The basc knetclaw for mass dffuson was proposed n 1855 by the German physologst A. Fck for a homogeneous meda wthout phase changes or chemcal reactons, namely: m d n ρ ( v v ) ρ v d j d D ρ, wth j yj j + d A (13) that reads: the mass-flow-rate of speces dffusng per unt area n the normal drecton n (massdffuson flux of speces ), jd, whch s ts densty tmes the relatve velocty of the speces-flud to the mean-flud (the latter dfference smply called dffuson speed v d v v ), s proportonal and opposes to the speces densty-gradent, ρ, wth the proportonalty constant D named mass-dffusvty for speces n the gven mxture, and ρyρ the mass-densty for speces n the gven mxture. Notce that Fck's law, jd D ρ, only accounts for mass-flow-rates and fluxes due to dffuson (by a gradent n concentraton); f there s a convectve flux j ρv (not assocated to gradents n concentraton but to bulk transport at speed v ), then the net flux of speces s j yj + j, or d ρv ρv+ ρv, whch was d used n (13). The orgnal Fck's law (13), whch he proposed just emulatng Fourer's law (of 18), perfectly matches experments wth dlute solutons,.e. when the propertes of the medum can be assumed ndependent of the speces concentraton, and (13) can also be wrtten as j ρd y, the most general Fck's law statement, extendng (13) to cover dffuson at hgh concentratons. Even n the orgnal case he tred, salt dffuson along a test tube from a saturated brne below to a fresh-water-swept zero-concentraton at the mouth, wth a densty jump from 100 kg/m 3 at the salt-brne nterface and 1000 kg/m 3 at the top surface, devatons from the lnear densty profle correspondng to the one-dmensonal steady-state problem wth constant D are less than a 1% at most; he found Dsalt,water m /s. Fck's law s smlar to Fourer s law for heat transfer q k T (or q a ( ρct p ) for a constantproperty medum), and apples to gases, lquds and sold mxtures, wth D dependng on the dffusng speces, the medum and ts thermodynamc state. Fck's law s also smlar to Darcy's law of mean flud velocty through porous-meda v κ ( h+ p/( ρg) ), and to Newton's law of momentum transport by vscosty τ ν ( ρv ) for a constant-densty flud of knematc vscosty ν, where τ s the stress tensor. In fact, for gases, a smplfed analyss dctates that Dαν. Notce that only the flux assocated to the man drvng force s consdered n Eq. (13),.e. mass-dffuson due to a speces-concentraton gradent (as for heat-dffuson due to a temperature gradent). There are Mass dffuson page 10

11 also secondary fluxes assocated to other possble gradents (e.g. mass-dffuson due to a temperature gradent, known as Soret effect, and mass-dffuson due to a pressure gradent; alternatvely, there may be heat-dffuson due to a speces-concentraton gradent, known as Dufour effect, and heat-dffuson due to a pressure gradent), but most of the tmes those cross-couplng fluxes are neglgble. Besdes, selectve force felds may yeld dffuson (e.g. ons n an electrc feld). Typcal values for D (and the thermal dffusvty ak/(ρcp)) are gven n Table 1, wth Schmdt numbers, Scν/D, to evaluate non-dealty n gases (knetc theory of deal gases predcts Sc1); for ar solutons, the dynamc vscosty s practcally that of ar, ν m /s at 300 K. for aqueous solutons, the dynamc vscosty s practcally that of water, ν m /s at 300 K. Table 1. Typcal values for mass and thermal dffusvtes, D and a, and Schmth number, Sc, all at 300 K (extracted from Mass dffusvty data). Substance Dffusvty Typcal values Example Scν/D Gases a) a 10 5 m /s aar 10 6 m /s ach m /s D 10 5 m /s Dwaterapour,ar m /s DCO,ar m /s ( m /s at 000 K) DCH4,ar m /s 0.99 Lquds b) a 10 7 m /s awater m /s D 10 9 m /s DN,water m /s 40 DO,water m /s 340 Solds c) a 10 6 m /s asteel m /s ace m /s afresh food m /s D 10 1 m /s DN,rubber m /s DH,polyethylene m /s DH,steel m /s a) For gas dffuson, both for a and D, a general dependence wth temperature and pressure of the form T n /p can be used, wth 1.5<n< (accordng to smple knetc gas theory, n3/. b) Mass dffuson n lquds grows wth temperature, roughly nversely proportonal vscosty-varaton wth temperature. c) Mass dffuson n solds s often not well represented by Fck's law, so that dffuson coeffcents mght not be well-defned, and other (emprcal) correlatons are appled nstead of Fck's law Notce that the defnton of Fck`s law n (13) has been establshed n mass terms, but an analogous development could have been made n molar terms: A nd n c ( v v ) cv d j d, molar D c (14) Fnally, notce that all the above expressons of Fck's law (13-14) assume a constant densty medum, and wll gve good predctons for dffuson n dlute mxtures, e.g. when x<0.1 all around (x,max0.104 n Fck's orgnal experment). But what happens n mxtures wth large densty gradents lke the dffuson Mass dffuson page 11

12 through a tube connectng two large reservors of hydrogen and ntrogen (or ar) where x0 at one end and x1 at the other? As sad above, the real drvng force for mass dffuson s not c as n (14) nether ρ as n (13), but µ. The only explct relaton between concentraton varables and the chemcal potental corresponds to deal mxtures, where µrut lnx. Even f we assume for smplcty that the flux s proportonal to x (and not to lnx),.e. jd1 K1 x1 wth K1 ndependent of x1, the followng dentty apples for a non-dluted bnary mxture: y1 M M ( z) j K x K K y ρ ( zd ) y 1 m d m M M m ( z) (15) showng that the assumpton of K1 ndependent of z s stll equvalent to the assumpton of D1 ndependent of z only for gaseous non-dluted mxtures, where the varaton of mxture densty along the length, ρm(z), compensates wth the varaton of mxture molar-mass, Mm(z). Thence, we may use jd1 ρ ( ) m zd1 y1 for dlute mxtures n any physcal state (sold, lqud or gas), and for non-dlute mxtures n the gaseous state. The bnary dffuson model just descrbed (one speces dffusng n an ndependent medum) requres some averagng when several speces dffuse n a medum, as for exhaust gases n ambent ar; n some cases, consderng an equvalent global dffusng speces of molar fracton x,globalσx and an equvalent average dffusvty D,avrg gven by x,global/d,avrg Σ(x/D), has gven good results. Exercse 3. Fnd the speces dffuson speed n the complete combuston of sold carbon n ar at 300 K and 100 kpa, knowng that the reacton C+OCO takes place at the surface, whch attans 1500 K, consumng. grams of carbon per second, per square meter, and that mass fractons n the gas close to the surface are yn0.75, yo0.15, and yco0.10. Soluton. The am of ths exercse s to make clear some common msconceptons, as thnkng that, because 1 mol of gas s released by each mol of oxdser consumed, there would be no macroscopc veloctes but just dffuson. We only deal here wth the mass transfer process, and only partally, snce we do not compute the composton at the surface (we assume we know them), what really comes from a combned heat and mass transfer nteracton, as well as the 1500 K at the surface. We only work here wth fluxes,.e. flow-rates per unt area, snce we keep close to the surface, although the real problem may correspond to the burnng of a carbon slab or of a small quassphercal carbon partcle. Of course, the data would be constant n the deal planar case, but may change wth tme for other geometres. The applcable equatons are (9-15). Let us start wth the fuel flow-rate. The suppled data s the carbon flux jc m C A 0.00 (kg/s)/m, whch can be nterpreted as a consumpton of carbon n the real unsteady process for a fxed control volume (fxed reference frame), or as a steady snk of carbon at the combuston front n the quas-steady process for a thn control volume centred at the surface and movng wth t at the recedng speed (movng reference Mass dffuson page 1

13 frame); the latter can also be thought of as a source of carbon n an magnary strctly-steady process n whch the front does not move because new fuel s njected nto the system (that can now be ether of nterfacal or volumetrc sze). In the fx-frame case, there s no veloctes wthn the fuel (t s only de nterface that s recedng), whereas n the movngframe case the fuel has a postve speed vcjc/ρc0.00/ m/s, havng placed the combuston front at the orgn, the fuel to the left-hand-sde of the front, and takng the densty of carbon 00 kg/m 3 from Sold data tables. In movng axes, the stochometry C+OCO ndcates that the requred flux of oxygen s the same n molar bass, or, n mass terms jo jcmo/mc / kg/(s m 3 ), where the mnus sgn takes account of the drecton of the oxygen flow (from the ar at rght to the front at the orgn). Smlarly, jcojcmco/mc / kg/(s m 3 ); of course, a global mass balance dctates that jc + jo jco (here ; notce the extreme care needed to deal wth the sgn of fluxes, whch are postve n the geometrcal sense f they pont to the rght, but postve n the thermodynamc sense f they enter the system). Ntrogen has no net bulk moton and thus jn0. Notce that yn+yco+yo1 at every stage n the gas phase, where yc0. But the orgnal queston was on dffuson speeds. Frst of all, we must realse that all the above fluxes are net fluxes n the movng frame, not dffuson or convecton fluxes. The global convecton flux s obtaned by averagng net fluxes for all speces at a pont, j j. Thence, on the left of the front, where there s only pure fuel (sold carbon) the convectve flux concdes wth the net flux, and there s no dffuson, j jjc0.00 kg/(s m 3 ). On the rght sde of the front, the sum of fluxes s (there s no fuel) j jjo+jco+jn kg/(s m 3 ), as can be expected from the global mass balance n movng axes (0.00 kg/(s m 3 ) enter the front and 0.00 kg/(s m 3 ) ext t). We conclude then that the dffuson fluxes are jdj y j; jd,ojo yo j kg/(s m 3 ), jd,cojco yco j kg/(s m 3 ), and jd,njn yn j kg/(s m 3 ). We fnally get from (13) the dffuson speeds sought, jdρvd, wth ρyρ and ρp/(rt)10 5 /( )0.3 kg/m 3 wth the deal gas model and the gas constant for standard ar (we can compute the molar mass of the mxture, snce we know the composton, but the effect s mnmal snce ntrogen s always domnant). Thence, the dffuson speeds are vd,ojd,o/(yoρ) 0.006/( ) 0.18 m/s and vd,cojd,co/(ycoρ) /( )0.34 m/s, and vd,njd,n/(ynρ) /( ) m/s. The latter result s worth analysng: s then ntrogen dffusng, beng an nert component n ths combuston process? Yes, ntrogen dffuses towards the combuston front (were ts concentraton s smaller), to compensate the carry-over by the global convectng flow. Notce that there s an overall convecton speed v j/ρ0.00/ m/s (.e. to the rght), so that the 'absolute' speeds (stll n the movng frame) for each speces are vov+vd,o m/s, vcov+vd,co m/s, and vnv+vd,n Besdes, n the fxed frame, the movng frame has a recedng Mass dffuson page 13

14 speed opposte the fuel-feedng speed above computed, vcjc/ρc0.00/ m/s, whch s nsgnfcant to the others. THE DIFFUSION EQUATION FOR MASS TRANSFER The substtuton of Fck's law (13) n the speces mass balance (10), and the assumpton of constant dffusvty, gves the mass-dffuson equaton: ρ ρ + ( ρv+ ρvd ) w + ( ρv) D ρ w t t (16) entrely smlar to the heat equaton, that s here presented together to better grasp ther smlarty. For a unt-control-volume system, the speces balance (n terms of mass fractons yρ/ρ) and the heat balance (n terms of temperature), adopt the followng form: Balance of Accumulaton Producton Dffusve flux Convectve flux y w mass of speces + D y ( y v) (17) t ρ T φ thermal energy + a T ( Tv ) (18) t ρc p where, agan, w s mass-producton rate per unt volume by chemcal reacton, φ s heat-producton rate per unt volume (e.g. by nternal energy dsspaton or external energy deposton), D s speces dffusvty, and ak/(ρcp) thermal dffusvty. The constancy of overall densty, ρconstant, has been ntroduced to pass from (16) to (17), a good approxmaton for dlute mxtures. Notce that wth ths approxmaton the contnuty equaton reduces to v 0. Another useful form of the mass and energy balances s obtaned usng the convectve dervatve D() / D t () / t+ v () : Dy w D Dc w Dx Mw D y D w D c D x Dt + ρ Dt ρ + Dt + M Dt + M ρ (19) ρ smlarly to the heat equaton: DT Dt φ ρc a T + (0) The dffuson equaton, (0) or (19), s a second order parabolc partal dfferental equaton (PDE), to be solved wth the partcular boundary and ntal condtons of the problem at hand. There are only a few cases where analytcal solutons can be found, mostly for problems wth very smple geometry (e.g. unbounded condtons) n steady state, or when the unsteady state has a self-smlar soluton reducng the dffuson equaton to an ordnary dfferental equaton (ODE), as presented n Heat conducton. Otherwse,.e. n most practcal problems, the dffuson equaton has to be solved numercally (usually by fnte-element or fnte-dfference methods), as presented n Heat conducton too. Mass dffuson page 14

15 Notce also that, for lnear equatons, the superposton prncple apples and a seres of solutons can be assembled to meet partcular boundary condtons. SOME ANALYTICAL SOLUTIONS TO MASS DIFFUSION Although solutons to the mass dffuson equaton are smlar to those of the heat equaton, we gve here some partcular applcatons of mass dffuson, as an example of how easy t s to convert from one formulaton to the other. As any other tme-dependant mult-dmensonal phenomena, mass-dffuson models may be classfed accordng to ther dmensonalty: steady, 1D problems (planar, cylndrcal or sphercal), D problems and 3D problems. We start by consderng one of the smplest cases, the nstantaneous pont-source deposton, a key problem n mass transfer (as the nstantaneous pont-source release n heat conducton). Instantaneous pont-source Consder the self-smlar dffuson, n tme and space, whch can be planar, cylndrcal or sphercal, of a pulse deposton of a fnte amount of mass, m of speces, n an unbound non-movng medum of dfferent composton;.e. f at tme t<0 there were no speces, and at tme t0 a fnte amount m s deposted at r0; how wll t dffuse for t>0? The soluton s the prncpal soluton (.e. a pont-source n an unbound medum) of the dffuson equaton, whch, n terms of the mass-densty of speces, ρ m/m, s: r m exp ρ 4 Dt D r rt t r r r 1 n ρ (,) n ρ 1+ n ( 4π Dt) (1) wth n0 for the planar case (m s then the mass released per unt nterface area), n1 for cylndrcal case (m s then mass released per unt axal length), and n for the sphercal case. Ths pont-source soluton s plotted n Fg. 1 for three tme nstants, and has the followng propertes: Fg. 1. Pont-source dffuson. Speces dstrbuton at three tme nstants. It s only vald for t>0, where t s a Gauss-bell shape (t s a Drac delta functon at t0, and does not exsts for t<0). The mass of the speces dffusng s conserved: Mass dffuson page 15

16 for n0 ρ(,) r t dr m, for n1 ρ( r, t)πrdr m, for n 0 ρ (, )4 0 r t πr dr m () The maxmum densty occurs at the orgn and decays wth tme as: m ρ (0, t) (3) + n ( 4π Dt) 1.e. as 1/t 1/ n the planar case, as 1/t n the cylndrcal case, and as 1/t 3/ n the sphercal case. Of course, the model cannot be vald for very short tmes; the densty of speces cannot be larger than n ts pure state (e.g. ts lqud densty for a drop dffusng n a lqud meda, or ts gas densty for a puff dffusng n a gas meda). Sem-nfnte planar dffuson Another key problem s the nter-dffuson when two quescent sem-nfnte meda (e.g. two dfferent gases) are brought nto contact, ether by removal of a separatng wall, or by parallel njecton at the same speed at the end of a sem-nfnte wall, what s the same f we change the reference frame, as sketched n Fg.. Fg.. One-dmensonal, planar nter-dffuson: a) ntal and generc mass fracton n two quescent meda, b) mass fracton n two meda movng at the same speed, before mxng, and whle beng mxed.. As there s no characterstc length (the two meda beng sem-nfnte), there s a self-smlar soluton n the combned varable x/((dt) 1/ ), whch, n terms of the mass fracton of speces, y, takes the form: x η Dt η y y y y x D + 0 y erf A+ B t x η η Dt (4) to be appled to each of the meda by mposng the partcular ntal and boundary condtons. Wth subscrpt '-' for the left-hand-sde medum and '+' for the rght one, one gets: y ( ) y A B y x y ( x) A1 + B1 erf y (0) y,0 A1 y,0 Dt 1 1 j,0 ρ1db 1 1 Dt 1,1 1 1,1 Mass dffuson page 16

17 y ( ) y A + B y x y+ ( x) A + B erf y+ (0) y,0 A y,0 Dt 1 j,0 ρdb Dt +,, (5) what allows fndng the sx unknowns (A1, B1, A, B y,0, j,0) from the sx equatons, n terms of the data (y,1, y,). Of great mportance s the value of mass-fracton at the contact, y,0, whch happens to be nvarant wth tme, and results n: y,0 ρ D y ρ D + ρ D y 1 1,1, + ρ D k 1 1 (6) Several other dffuson solutons are presented n Table, both n terms of speces varables y and D, and n terms of thermal varables T and a, or just usng the latter for ease of wrtng (to be appled to specesdffuson problem by changng T to y, a to D, and Q/(ρc) to m. Exercse 4. A mld steel wth yc0.% (0.% carbon percent n weght) s to be surface-hardened by exposure to a carbonaceous atmosphere at 1000 K. Assumng an equlbrum concentraton of yc1% at that temperature (from solublty data for that carbonaceous mxture condtons), fnd the requred exposure tme to acheve yc>0.8% at a depth of 1 mm from the surface. Soluton. From Table 1 we get the dffuson coeffcent for carbon n ron, DC,ron m /s at 1000 K. The dffuson equaton (4), wth the boundary condtons yc(0)yc,01%, yc)yc,0.%, becomes yc(x,t)yc,0 (yc,0 yc,)exp(x/(4dt) 1/ ), whch yelds t4300 s (1. h) for yc0.8%, yc,01%, yc,0.%, DC,ron m /s, and x10-3 m. Notce that the assumpton of equlbrum at the surface s acceptable because dffuson n the gas phase s much more effcent than n the sold phase. Dffuson through a wall We present now a fnal example of mass dffuson, namely, the smple problem of steady leakage of a gas through a wall, manly amng at nsstng on the fact that what drves mass dffuson s chemcal potental and not concentraton, as explaned n Entropy. Exercse 5. Consder gas dffuson through the rubber wall of a ntrogen-flled balloon n ar. Assume pure N nsde, a 0.01 mm thck rubber wall of 0.5 m n dameter (.e. a rubber mass of kg), and 300 K and 100 kpa both outsde and nsde (neglgble elastc force). Make a sketch of the ntrogen concentraton everywhere, and estmate the relaxaton tme (e.g. the tme for the gradents to dump half-way to equlbrum). Soluton. Frst of all, notce that we focus just on ntrogen dffuson, but, contrary to heat dffuson where there s only one varable dffusng (thermal energy), here there s ntrogen dffusng outwards to the ambent but at the same tme oxygen dffusng nwards. Mass dffuson page 17

18 A second comparson wth heat transfer s that mass dffuson through solds s much less effcent than heat dffuson: for a typcal elastomer, wth data from Solds data, thermal dffusvty s a k/(ρc)0.1/( ) m /s, whereas dffusvty for ntrogen n rubber s m /s, and for oxygen n rubber m /s. Ths reason alone would explan why for most practcal problems solds can be consdered mpermeable to fluds (.e. "contaners"), but there s more on that. In fact, the most radcal dfference between heat dffuson and speces dffuson s the abrupt jump on speces concentraton through an nterface, contrary to the contnuty of the temperature feld. In effect, full thermodynamc equlbrum mposes unform temperature, unform velocty, and unform chemcal potental of each speces across the nterface, but ths does not mples unform speces concentraton except for unform phases; at a phase-change nterface, equalty of chemcal potental gves way to a jump n concentraton that depends on the materals propertes, wth two mportant deal cases deduced under Mxtures: Raoult's law for the equlbrum of an deal mxture of gases wth an deal condensed phase, and Henry's law for the equlbrum of an deal mxture of gases wth an deal dlute condensed phase. The latter s the case here, where a solute (N and O) dffuses through a dluted condensed phase (bascally rubber macromolecules wth very lttle N and O). From the solublty data (Table 3) n Solutons, we can get the Henry constants: KH,Nc,sol/c,gas0.04 for the solublty of ntrogen n rubber at 98 K, and KH,Oc,sol/c,gas0.08 for the solublty of oxygen. You mght have notced that, unfortunately, there s a huge varety on Henry's law data presentaton; the one used here was advocated by Ostwald, and means for nstance that, for ntrogen to be at equlbrum between both phases, there must be 0.04 mol/m 3 of ntrogen dssolved n rubber per each 1 mol/m 3 of ntrogen dssolved n the gas phase. Notce, by the way, that all nterfaces are selectve to some extent (e.g. rubber lets oxygen to flow more readly than oxygen, what can be advantageously used for separaton of speces from a mxture. In our case, nsde the balloon, ntrogen s pure, wth a concentraton of cp/(rt)10 5 /( )40 mol/m 3 of ntrogen, whereas n the ar outsde there s a concentraton of cnxnp/(rt) /( )3 mol/m 3 of ntrogen. We can assume that these equlbrum concentratons apply also close to the rubber even durng non-equlbrum (.e. whle dffuson s takng place), due to the small fluxes mpled. Wthn the balloon matter tself, the equlbrum concentratons at each end are the followng. At the nternal nterface, cn,solkh,nc,gas mol/m 3 of ntrogen, whereas at the external nterface, cn,solkh,nc,gas mol/m 3 of ntrogen. What results n the concentraton profle shown n Fg. 3a. Mass dffuson page 18

19 Fg. 3. Gas dffuson through a rubber wall: a) Intal concentraton profle for ntrogen dffuson from a ntrogen-flled balloon (left) to ambent ar (rght); notce that ntrogen dffuses aganst the concentraton jump n the outer nterface. b) Tme evoluton of the amounts of ntrogen, oxygen, and the sum, nsde the balloon. Fnally, for the estmaton of the relaxaton tme, we must compute the mass-dffuson flux jn D,N ρn, (13), or n molar terms, (14), jn,molar D,N cn D,N(cN,ext cn,nt)/lth ( )/ (mol/s)/m of ntrogen. Wth a balloon area of AπD 0.79 m (VπD 3 / m 3 ), and an ntal content of nnpv/(rt) /( ).6 mol, we have a rough estmate thalfnn/(jn,molar A).6/( ) s,.e. of the order of 8 days (no wonder why one always starts neglectng dffuson through solds). But ths analyss s too crude: what happens to nternal pressure, or balloon volume, when ntrogen dsappears? What about the oxygen flux? The latter s jo,molar D,O co D,O(cO,ext co,nt)/lth (0 0.67)/ (mol/s)/m of oxygen. You can observe that oxygen flux s three tmes that of ntrogen, what mght have been expected from the hgher solublty of oxygen n rubber, and the hgher dffusvty, the jump n concentraton beng equal (from 40 mol/m 3 to 3 mol/m 3 for ntrogen, and from 0 to 8 mol/m 3 for oxygen; the sum beng 40 mol/m 3 n each sde as expected for deal gases at the same temperature and pressure). So, what happens to nternal pressure and volume? It depends on the elastc law for the rubber. In any case, the equaton pvnrt shows that, at Tconstant (very slow process) and wth n ntally ncreasng because the nflow of O s larger than the outflow of N, the product pv should ncrease ntally but decrease afterwards because fnal equlbrum must be wth the same composton everywhere, and thus recoverng the same ntal pv-values. The maxmum can be found to be 8% larger than ntal condtons (pvmax1.08pvn), and to take place about one day from the begnnng, as shown n Fg. 3b. Summary table of analytcal solutons to dffuson problems Table. Analytcal solutons to some dffuson problems. Problem Sketch Soluton Notes Instantaneous pontsource deposton, one-, two-, trdmensonal r Q exp 4 at Trt (,) 1+ n ρc ( 4πat) T relatve to T(t<0). Planar case: n0 and Q [J/m ]. Cylndr. case: n1 and Q [J/m].. Sphercal case: n and Q [J].. Mass dffuson page 19

20 Instantaneous fnte lne-source, onedmensonal deposton of wdth L T(0, t) ρc Q + n ( 4πat) 1 r m exp 4 Dt ρ (,) rt ρ (0, t) ( 4π Dt) m 1+ n ( 4π Dt) 1 + n t0, Delta(x). t>0, Gauss bell. Q ( T T0 ) dv V ρc V ρ dv m L L t>0. x x Q + T( xt, ) erf erf Tends to a pontsource (Gauss bell) + ρcl 4πat 4πat for t Q L T(0, t) erf Lρc 4 at L L x x y + L y erf erf + 4πDt 4πDt Contnuos planarsource, one- Q x t exp 4 at dmensonal T(,) xt deposton ρc πa Q t T(0, t) ρc πa Contnuos lne-source deposton r Q E 4at Trt (,) 4πρca Sngular at r0. For r << 4at, E(-x) γ ln(x), wth γ Contnuos pontsource, trdmensonal deposton Contnuos sphercalsource kept at fxed-t by controllng Qt () r Q erfc 4at Trt (,) 4πaρcr Q T(0, t) 4ρc πa t ( ) 3 R r R Trt (, ) TR erfc r 4 at 1 1 Qt () 4π RTR + R π at Only vald for r>r. Mass dffuson page 0

21 Movng planar-source one-dmensonal deposton xv T ( T0 T ) exp a xv y ( y0 y ) exp D t<0: T(x)T, y(x)y. t>0: T(0)T0, y(0)y0. Movng pont-source tr-dmensonal deposton Planar contact wth forced jump at the surface U ( r x) q exp a T 4π ar x T ( T0 T ) erfc at x y ( y0 y ) erfc Dt Only vald for steady state (t ). r x +y +z. U s the constant relatve speed. t<0: T(x)T, y(x)y. t>0: T(0)T0, y(0)y0. Planar contact Contnuos onedmensonal planar plate mmerson One-dmensonal planar contact, steady n a movng frame (e.g. mxng layer of two equal-speed streams) T 0 y x T A+ Berf at x y A+ Berf Dt ρ cp kt+ ρ cp kt ρ c k + ρ c k 0 1,1 1 1, 1 p,1 1 p, ρ D y ρ Dk + ρ + ρ D y Dk ( 0 ) ( ) n+ 1 π y y y D exp t 4 L 1 π n 0 n + 1 sn ( n + 1) L π x x y A+ Berf az / v Contact value s T0 for heat transfer, or y0 for mass transfer. t0, Heavsde(x). If equal propertes: T0(T1+T)/ and y0(y1+y)/. t<0: T(x)T, y(x)y. t>0: T(0)T0, y(0)y0, T(L)T0, y(l)y0. It s the same as the latter changng tz/v, where v s the common speed Mass dffuson page 1