ChE 4520/5520: Mass Transport. Objective/Introduction. Outline. Gerardine G. Botte

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1 ChE 450/550: Mass Transport Gerardne G. Botte Objectve/Introducton In prevous chapters we neglected transport lmtatons In ths chapter we wll learn how to evaluate the effect of transport lmtatons We wll learn how to account for mass transport n electrochemcal systems usng dlute soluton theory. Dlute soluton theory neglects solute-solute nteractons The transport mechansms nvolved n electrochemcal systems nclude: Dffuson Mgraton Convecton Outlne 3 1

2 Relatonshps A general descrpton of an electrochemcal system takes nto account: Speces fluxes Materal conservaton (materal balance) Current flow Electroneutralty Electro-knetcs Global reactons Hydrodynamcs 4 Speces Flux Usng dlute concentraton theory (consders only nteractons between solute-solvent), the flux of speces s gven by: N = zufc φ D c + cv Eq. 1 mgraton dffuson convecton Comparng to general chemcal engneerng systems the dfference n the flux s gven by the mgraton term (potental generated due to the on nteractons) 5 Speces Flux Where: N : flux of speces, mol/cm s z : charge number of speces, eq/mol u : moblty of speces, cm-mol/js c : concentraton of speces, mol/cm 3 f: electrostatc potental, V D : dffuson coeffcent of speces, cm /s v: flud velocty, cm/s The flux s perpendcular to the surface area (as usual) 6

3 Current Flow Current wll arse from the moton of the charges and s gven by: = F z N Eq. 7 Materal Balance The materal balance s gven by: c =. N + R t Eq. 3 Where R represents a chemcal reacton occurrng n soluton (mol/cm 3 ). Eq. 3 assumes constant volume 8 Electroneutralty Equaton Because the electrcal forces between charged speces are so large, sgnfcant charge separaton cannot occur. Therefore, n the bulk the electroneutralty assumpton s vald: zc = 0 Eq

4 Smulatons To carry out a smulaton of the performance on an electrochemcal system, Eqs. 1-4 need to be solved smultaneously We need a descrpton of the flow pattern to account for v n Eq. 1 Eqs. 1 to 4 apply at the bulk, the electrode knetcs s used as boundary condtons for the soluton of the dfferental equatons 10 Common smplfcatons If bulk concentratons can be gnored we can demonstrate that the gradent of the current s gven by:. = 0 Ths Eq. s known as conservaton of charge When concentraton varatons can be neglected we obtaned ohm s law nstead of Eq. : = k φ Where k (conductvty) s gven by: k = F zuc ChE 490/ Common smplfcatons Another mportant defnton s the transport number: zuc j j j t j = zuc When concentraton varatons are mportant, ohm s law becomes modfed ohm s law: F φ = zd c k k 1 4

5 Common Smplfcatons When mgraton s neglgble (n excess of a supportng electrolyte), the coeffcent of the potental gradent n Eq. 1 s large, then the potental gradent must be small The materal balance can be obtaned from substtutng smplfed Eq. 1 nto Eq Common smplfcatons (supportng electrolyte) Ths equaton s known as the convectve dffuson equaton: c + v. c = D c t The potental dstrbuton can stll be obtaned by solvng the modfed ohm s law equaton 14 boundary layer For systems where the concentraton gradent s sgnfcant, one common smplfcaton s to treat the regon as a regon wth a lnear concentraton gradent Ths s known as Nernst dffuson layer 15 5

6 boundary layer Nernst approxmaton for the concentraton gradent s expressed as: c c c = 0 Eq. 5 x δ Where: c : concentraton at the bulk c 0 : concentraton at the surface d: thckness of the. The thckness of ths layer s between cm 16 Example 1 Use the transport equatons to derve an expresson for the current for a copper deposton reacton as a functon of the surface and bulk concentraton. The electrolyte s composed of copper, water and H SO 4 (added to ncrease conductvty) Wrte down expressons for your transport propertes 17 Consequences of addng supportng electrolytes Reduces ohmc losses Reduces lmtng current densty Increases the vscosty of the soluton and therefore decreases the maxmum velocty Reduces the magntude of the electrc feld 18 6

7 overpotental s assocated wth the mass transport lmtatons It results from: The concentraton dfference between bulk and electrode surface From the potental gradent (see modfed Ohm s law, second term) 19 overpotental Assumng that the varatons n the onc conductvty are small (small current denstes or a supportng electrolyte s used). The concentraton overpotental can be obtaned by: η δ,0 c = ln nf c, k 0 ( c, c,0) RT c F zd dx δ 0 overpotental When the conductvty s large the equaton can be approxmated wth: RT c,0 η ln c = Eq. 6 nf c, Ths s the concentraton overpotental for a cathodc reacton. The concentraton overpotental for a cathodc reacton s negatve (as well as ts surface overpotental) 1 7

8 overpotental For an anodc reacton, the concentraton overpotental s postve and can be estmated (for large conductvtes) by: RT δ ηc = ln 1+ nf nfdc Eq. 7 Lmtng current When the current at the surface s equal to zero, the current measured s known as the lmtng current For the assumpton, the current s defned as (we demonstrated ths n Example 1): nfd( c c0 ) = ChE 490/690 δ 3 Lmtng current Then the lmtng current for the Nernst dffuson layer s gven as: nfdc l = δ The lmtng current densty s a functon of the flow pattern. It s up to 100 order of magntude larger n strred solutons 4 8

9 Lmtng current The overpotental can be expressed as a functon of the lmtng current, for example for hgh conductvtes the cathodc concentraton overpotental s gven by Eq. 6, whch can be expressed as: RT ηc = ln 1 nf l 5 Lmtng current At currents hgher than the lmtng current addtonal reactons takes place After the lmtng current the two reactons take place n parallel wth the secondary reacton takng over the prmary reacton 6 Lmtng current Supportng electrolytes reduce ohmc losses but tend to reduce the lmtng current Supportng electrolytes ncrease the vscosty of the soluton and decreases the moblty of the ons 7 9

10 Dffuson coeffcent The transport equatons requre the use of the dffuson coeffcent. The dffuson coeffcent for onc speces can be calculate by usng the Nernst-Ensten equaton: D = RTu For a bnary electrolyte the dffuson coeffcent becomes: DD + ( z+ z ) D = zd zd ChE 490/ Dffuson coeffcent Because the onc dffuson coeffcent s related to the moblty, t can be calculated usng the equvalent conductances: RTλ D = z F 9 Values of dffuson coeffcents of selected ons at nfnte dluton n water at 5 o C ChE 490/690 Newman J. S., Electrochemcal Systems, Second edton,

11 Estmaton of lmtng current The lmtng current can be estmated from the mass transfer correlatons Usually mass transfer lmtatons are expressed as: N = k ( c c ) m 0 Where k m s the mass transfer coeffcent (cm/s) 31 Estmaton of lmtng current At the lmtng current the surface concentraton s zero, therefore, the lmtng current s related to the mass transfer coeffcent: = l nfkmc The mass transfer coeffcent s related to the Sherwood number (Sh) whch s a functon of the Reynolds (Re) and the Schmdt (Sc) numbers 3 Estmaton of lmtng current The followng correlatons are used kml Sh = D Lv Re = v v Sc = D Where L s the characterstc length, v s the velocty of the flud, and v s the knematc vscosty (cm /s) (v = m/r) 33 11

12 Estmaton of lmtng current Some correlatons requre the use of the Grashof number (Gr). Ths number s used when the mass transport s affected by densty dfferences g( ρ ) 3 ρ0 L Gr = ρ v Δ ρ = ρ ρ 0 Where: g: acceleraton due to gravty r : bulk densty r 0 : surface densty, equal to the solvent densty 34 Estmaton of lmtng current Correlatons for the Sherwood number has been determne for specfc geometres where the flow pattern s well known The correlatons are summarzed n appendx E of the book 35 Example For the cathodc deposton of copper from 0.5 M CuSO 4 and 0.5 M H SO 4 electrolyte, the knetc parameters are a c =0.5 and 0 =1 ma/cm. If the appled potental s 100 mv respect to a SCE. Calculate: A. The current densty for copper deposton expected f only knetcs lmtatons are nvolved B. The lmtng current densty f two plane parallel copper electrodes, cm long are used n a beaker of unstrred electrolyte C. Estmate the thckness of the Nernst dffuson layer 36 1

13 Summary At the end of ths chapter you should be able to: Calculate dffuson coeffcents for onc speces Determne lmtng current for dfferent geometres Calculate currents when knetcs and mass transport lmtatons are nvolved Wrte down the fundamental equatons to model an electrochemcal system assumng dlute soluton theory 37 13

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