A expeimetal detemiatio of te effective oxyge diffuio coefficiet fo a ig deity polypopylee geomembae Micael G. Tefy ad Badley M. Patteo CSIRO ad ad Wate Tecical Repot 37/01, Septembe 001 CSIRO AN ad WATER
A expeimetal detemiatio of te effective oxyge diffuio coefficiet fo a ig deity polypopylee geomembae Micael G. Tefy ad Badley M. Patteo CSIRO ad ad Wate, Pivate Bag No. 5 PO Wembley WA 6913, Autalia Tecical Repot 37/01 Septembe 001
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet Abtact aboatoy expeimet wee pefomed to detemie te effective oxyge diffuio coefficiet fo a polyme geomembae. Suc geomembae ae commoly employed to limit oxyge upply to eactive tailig depoit, teeby educig acidificatio ad ote udeiable cemical tafomatio. Te expeimet ued a imple cambemembae appaatu, togete wit a o-lie oxyge eo poitioed witi te cambe. Oxyge beaktoug cuve, meaued ove eveal week fo a ig deity polypopylee HPP membae, wee fitted wit a ew aalytical model paameteied by te membae diffuio coefficiet, cambe leakage ad oxyge eo coumptio. Te aalytical model povided cloe fit to te data, ad wa alo able to yield etimate of te oxyge coumptio ate of te o-lie eo. Te idicated value fo te oxyge diffuio coefficiet of te HPP geomembae wa 4.6 ± 1.3 x 10-8 cm -1. Ti value wa coupled wit a imple pyical model to deduce effective diffuio coefficiet fo compoite HPP-betoite geoytetic clay lie. Te eult ow ig eitivity to betoite wate atuatio, i lie wit ealie fidig. 1
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet Itoductio Te poblem of maagig te geocemical fate of eactive oil ad idutial tailig i of global igificace. Fo example, tee ae a etimated 1 millio ectae of acid ulpate oil i te wold [1], ad i Caada aloe tee ae 00 millio toe of ulpidic idutial tailig []. Wee poible, ecet maagemet pla fo uc eactive tailig ivolve emplacig te tailig i dewateed oil pofile ad uig geotecical o ote mea to limit oxyge ige to te tailig, teeby educig acid poductio to level deemed to be acceptable i a eviometal ee. Sigle-mateial geolie, e.g. lie coitig of oil, clay o ytetic mateial, epeet a ueful tecology fo limitig upply of oxyge ad ote cemical compoud, ad tei pyicocemical popetie ae eaoably well tudied [3,4]. Howeve tee lie may be ieffective ad cotly, ad tee ae till quetio uoudig te ageig caacteitic of te popula polymeic geomembae [5], altoug tei pyical popetie may be table ove te ot to medium tem [6]. We emplaced ove a tailig facility, tey act to educe vetical oxyge fluxe to te tailig; te effect may be augmeted by te itiic oxyge miealiatio caacteitic of te uoudig oil pofile. I developig epeetative model of te eactio dyamic of ulpidic tailig facilitie, quatifyig te oxyge patway ad fluxe i citical. Wee geomembae ae employed, umeical imulatio become poblematic due to te dipaity i patial ad tempoal cale betwee te dyamic i te ti geomembae ad te muc lage tailig ytem. Special imulatio tecique may be equied ad, accodigly, it i impotat to upply tee tecique wit accuate etimate of geomembae popetie. Recetly, mot iteet wa focued o te ydaulic popetie of geomembae [7-9], oweve ga pemeability caacteitic ae ow alo aliet [10-1]. Ti pape peet eult of a expeimetal tudy of te diffuio caacteitic of a polyme geomembae ued i a idutial ite ea Pet, Wete Autalia. Etimate of effective diffuio coefficiet fo
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet te geomembae ae obtaied uig a imple diffuio cambe appaatu, togete wit log-tem o-lie moitoig of oxyge cocetatio ad oliea egeio. Figue 1: Scematic of te diffuio cambe appaatu. Te cylidical cambemembae aembly i ow i co ectio. Te aembly i immeed i a cotolled atmopee ad coected to a exteal data logge. Expeimetal Appaatu ad Metod A appaatu wa cotucted to detemie te flux ate of oxyge toug diffeet membae mateial. Te mateial coideed wee taile teel eeafte efeed to a te metal membae ad a ig deity polypopylee HPP membae. A cematic of te appaatu i ow i Fig. 1. Te appaatu wa cotucted fom tick acylic platic polymetylmetacylate wit a Figao oxyge eo KE 5 couteuk o te eo wa flu wit te bae of te ga cambe. Te ga cambe volume wa 18.3 ± 0.3 cm 3 gavimetically detemied, wit a expoed uface aea to te atmopee of 38.5 ± 0.5 cm. Te cambe dept wa 0.48 ± 0.05 cm. A ga-tigt eal betwee te ga cambe ad te mateial teted wa cotucted uig a vito O-ig. Te appaatu ad membae 3
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet mateial wee toed aaeobically fo 1 to week pio to expeimet, wic wee caied out at oom tempeatue. Fo te expeimet eite a 1.0 mm tick taile teel eet o a 0.75 ± 0.15 mm tick HPP eet Nylex Milleium polypopylee wa attaced to te appaatu ude aaeobic coditio, ad te te appaatu wa expoed to a aeobic 1% oxyge atmopee. I bot expeimet metal ad HPP, oxyge eo epoe data wee collected at mi iteval ad toed uig a CSIRO data logge. Figue : Beaktoug data quae ad leat-quae etimate cuve olid lie fo te metal membae expeimet. Fo claity, oly evey 30 t data poit i ow. Te fit 1099 data poit wee excluded fom te egeio. Aalyi of Reult evelopmet of te Beaktoug Model Figue ad 3 ow oxyge beaktoug cuve meaued i te ecloed cambe fo te cae wee te membae i impeviou metal membae, Fig., ad pemeable HPP membae, Fig. 3. All cambe cocetatio, deoted by v, ae omalized by te exteal atmopeic equlibium coetatio, deoted by u atm. All odiate axe ae omalized wit epect to atmopeic oxyge cocetatio. Figue ad 3 ow almot 4
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet immediate beaktoug of oxyge ito te cambe. Te log time cale of equilibatio i impotat, a i te fact tat i bot Figue 3: Beaktoug data quae ad leat-quae etimate cuve olid lie fo two HPP membae expeimet. Fo claity, oly evey 00 t data poit i ow. Pat a ow data ad fitted cuve fo a five-week expeimet, wile pat b ow data fo a epaate tee-week expeimet togete wit te fitted cuve fom pat a. expeimet te equilibium cambe cocetatio i le ta atmopeic cocetatio. Tee obevatio idicate tat te beaktoug cuve embody effect aociated wit i oxyge flux toug te membae take to be zeo fo te metal membae expeimet, ii a lumped leakage flux epeetig te combied effect of all ote patway by wic oxyge may ete te cambe, ad iii a oxyge lo mecaim due to coumptio by te eo. Wile tee fluxe ad ate ae vey low, a i evideced by te log 5
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet equilibatio time of te beaktoug cuve, te ate of diffuive editibutio of oxyge oce it i iide te cambe i elatively apid ad te oxyge cocetatio i te cambe ca be egaded a well-mixed. Appedix A peet a model of beaktoug dyamic baed o ti well-mixed cambe appoximatio. Te eultig beaktoug fuctio eq A.16 i paameteied by a goup of tapot paamete te membae diffuio coefficiet,, te leakage ate paamete,, ad te eo coumptio ate, a well a two geometic paamete te membae ticke,, ad cambe legt,. A featue of te beaktoug fuctio i tat fo low, zeo ad lage te beaktoug i igificatly delayed. A dawback of te model i tat it i pyically ill-poed, i tat te leakage ad membae diffuio tem ae cloely elated. Geatet dicimiatio betwee tee fluxe occu wee membae beaktoug i low i.e. wee membae ticke i lage ad diffuio coefficiet i mall, i wic cae ealy cambe beaktoug may be attibuted to leakage. Ufotuately, te oxyge eo i leat accuate fo mall cocetatio i te viciity of te detectio limit, wic may lead to difficulty i coectly appotioig te obeved cambe iflux betwee te membae diffuio ad cambe leakage pocee. Matematically, ti ill-poed beaviou i maifeted i te peece of a umbe of feaible, if ot alway ealitic, extema i te leat-quae objective fuctio ued i egeio of te beaktoug fuctio agait meaued data. Fotuately, exitig liteatue value may be ued a a guide i cooig te appopiate extemum. Noliea Regeio Metodology I piciple, te beaktoug fuctio pemit imultaeou etimatio of te effective membae diffuio coefficiet ad te leakage ad coumptio ate. I pactice ti wa acieved by te followig umeical mea. Fit, te evaluatio of te beaktoug fuctio vt/u atm wa automated by employig a Newto-Rapo metod [13] ove a iegula gid to locate te fit twety pole poit. I pactice, oly te pole wee 6
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet equied due to te apidly coveget atue of te ummatio fo t > 0 i eq A.16. Te, ti umeical beaktoug fuctio wa coupled to te expeimetal data i a leat-quae objective fuctio, wic wa miimied uig a tadad oliea egeio outie. All calculatio wee pefomed wit double peciio Fota aitmetic i ode to maitai acceptable peciio, epecially i te detemiatio of pole poit te Fota code i available fom te auto o equet. Te o-lie atue of te oxyge eo povided may beaktoug data poit at egligible cot, leadig to excellet tatitic i te egeio. Cofidece iteval fo te p etimated paamete ove data poit wee cotucted uig te geeal liea appoximatio θ ˆ θ P t 1 i i ii p, / wee θˆ i i te leat-quae etimate of paamete θ i, P ii i te quae oot of te i-t diagoal of te aymptotic covaiace matix ad etimate te aymptotic paamete tadad eo σ i, ad t p, / i te uppe / quatile fo te t-ditibutio wit -p degee of feedom. Peciio of Regeio I ti ectio te ize of ucetaity meaue fo te oliea egeio ae exploed fo eac of te tee expeimet fo fixed mea value of te geometic quatitie ad. Fo te metal membae expeimet, wa aumed to be zeo ad te beaktoug cuve of eq A.18 wa fitted to te data. Te pupoe of ti wa to gai a appoximate etimate of te leakage ate fo te Vito ig eal metod, wic i ued i bot metal ad HPP membae expeimet. Figue ow tat at ealy time te beaktoug data quae ae oiy, wic may be due to two effect. Fitly, te oxyge eo may expeiece detectio limit poblem fo uc low cocetatio level. Secodly, te aw eo voltage output i coveted to oxyge cocetatio uig a igle calibatio 7
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet cuve detemied a a bet fit ove te age fom o-detect to atmopeic oxyge cocetatio. Ti bet-fit cuve i mot accuate fo ige cocetatio. Accodigly, te fit 1100 data poit of te beaktoug cuve wee igoed duig te leat-quae poce, leavig 345 poit fo egeio. Table 1 ow tat te 95% cofidece iteval o te eultig ad paamete ae extemely tigt. Te coepodig beaktoug cuve i alo daw o Fig. olid lie, owig good ageemet wit te meaued data. Membae egee of Feedom,p cm -1 10-7 -1 10-7 -1 Metal Eq A.18 345, - 1.01 ± 0.00 50.71 ± 0.1 HPP, Eq A.16,, 7359, 7359,3 4.49 ± 0.01 x 10-8 0.00 a 3.8 ± 0.09 x 10-9 11.5 ± 0.0.18 ± 0.01.14 ± 0.01 Table 1: Etimated tapot paamete value, ad fo te two expeimet. Reduced-paamete egeio fo HPP ae explaied i te text. iea 95% cofidece iteval ae upplied fo eac paamete etimate. a Value eld cotat. Fo te HPP expeimet all tee paamete,, may be o-zeo. Seveal egeio metod wee ued. Pefomig a igle paamete fit fo wit ad fixed at te metal expeimet value eulted i a uacceptably poo eult ot ow. A a alteative, te poblem wa educed to a two-paamete egeio a follow. Sice te vito O-ig i i cotact wit te HPP membae at te eal, it i aumed tat ude mecaical peue agait te igid cambe tuctue te eal will effectively act a a compeed HPP membae. Tu, te leakage ate i aumed to be zeo, wit all flux ito te cambe attibuted to diffuio toug te HPP membae. I ti way, te poblem i educed to a egeio ove te ad paamete. A a fial alteative, a full tee-paamete egeio wa attempted diectly. Table 1 give eult fo tee vaiou 8
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet egeio fo te HPP expeimet. Agai, te cofidece iteval ae tigt, a i evideced by te cloe ageemet betwee te two- ad tee-paamete fit cuve ad te meaued data ee Fig. 3. Figue 3 alo ow te idetical two-paamete bet-fit cuve plotted agait a epeat of te HPP expeimet ove a ote time. Ageemet i eauigly cloe, altoug te paamete etimate fo te diffeet egeio ow iteetig ted. Table 1 ow tat te fitted eo coumptio ate,, ae imila fo te two- ad tee-paamete egeio, at appoximately.15 ± 0.05 x 10-7 -1. Howeve, te coefficiet fo HPP diffe by a ode of magitude, wit icluio of te leakage tem i te egeio leadig to a muc malle diffuio coefficiet, beig imila to a value ecetly epoted fo a acylic polyme [14]. Te fitted value of of 4.5 x 10-8 cm -1 two paamete fit ad 3.8 x 10-9 cm -1 tee paamete fit may be compaed wit liteatue value. Electocemical [15] ad lumiecece-quecig [16,17] tudie idicate te typical age 10-8 to 10-6 cm -1 fo oxyge diffuio coefficiet i vaiou polyme. I a tudy elated to food packagig, ibelo et al. [18] foud a value fo te oxyge diffuio coefficiet of tadad-deity polypopylee film of.9 x 10-8 cm -1, ad Eke et al. [19] epoted value agig fom 0.9 to 7. x 10-7 cm -1 fo γ-iadiated polypopylee film. Howeve, fo HPE, Epace et al. [0] upplie a eie of efeece datig fom 1961 uppotig a value of 1.7 x 10-7 cm -1, wic i igificatly ige ta te value fo tadad deity polypopylee fom ibelo et al. [18]. Va Kevele [1] povide activatio eegie fo polypopylee, wic may be ued i a Aeiu elatio to obtai a age of 0.67 to 8.1 x 10-7 cm -1 fo te oxyge diffuio coefficiet. O ti bai, te tee-paamete etimate fo ad tu fo may be ejected a beig ueaoably low, pobably due to te commo paameteiatio of te diffuio ad leakage tem metioed ealie. Te two-paamete etimate lie jut below te age idicated by va Kevele 1, but ige ta te value of ibelo et al. [18]. Hece te two-paamete etimate 4.5 x 10-8 cm -1 i coguet wit peviou aalye ad i take a epeetative 9
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet fo HPP i ti expeimet. Alo, te table ow tat te eo coumptio ate i te HPP expeimet i twety-five time lowe ta i te metal expeimet. Ti may igify tat te fit-ode coumptio model i iaccuate ove te full cocetatio age ecouteed i te metal ad HPP expeimet. It i eld tat te eo coumptio ate value.15 ± 0.05 x 10-7 -1 i epeetative fo omalized oxyge cocetatio of te ode of uity, i.e. at atmopeic oxyge cocetatio. Expeimet Tapot Paamete Vayig Vayig Vayig, 4.43, 5.11 0.60, 0.90 HPP 10-8 cm -1 4.1, 4.80 3.53, 5.49 3.31, 5.86 10-7 -1.0,.16.1,.3.11,.6 Table : Effect of geometic paamete ucetaity o leat-quae etimate of te tapot paamete, ad fo te HPP expeimet. Tee-paamete egeio wee pefomed uig ad at te extema of tei meauemet ucetaitie to obtai abolute eo meaue fo te tapot paamete. Ifluece of Ucetaity i te Geometic Paamete Te peviou ectio demotated tat te popoed beaktoug model wa able to epoduce te featue of te obeved HPP data eaoably peciely, wit te lage umbe of data poit cotibutig to tigt cofidece iteval i te leat-quae etimate fo te tapot paamete. Ufotuately, te tue eo meaue fo te expeimet ae moe ubtatial. Abolute meauemet eo i te cambe legt paamete,, i etimated at 5%, wilt typical toleace fo polyme aemblie ae betwee 10 ad 0%. I ode to accout fo te effect of tee ucetaitie, HPP egeio wee caied out ove te full age of geometic paamete value, take idividually ad togete. Reult ae peeted i Table fo te HPP expeimet. Note tat fo te lat colum i te table, te quoted paamete age wee detemied fom te fou coe poit of te 10
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet mi, max x mi, max egio ad te idividual paamete age peviou colum. ue to te olieaity i tee paamete of te leat-quae objective fuctio, tee age may ot tuly epeet te full age of tapot paamete value ove te geometic egio. Mappig te tue iteval i computatioally expeive, o Table cotai te eigt-poit appoximatio to te iteval. Fo te HPP expeimet, te egeio ow tat i eitive to te vaiatio i membae ticke, but ot to te vaiatio i cambe legt, wile i ieitive to cage i ad. Allowig ad to vay imultaeouly make little diffeece to te iteval calculated by vayig aloe, wit te ucetaity i etimated at appoximately 30% fo ti expeimet. Te ecommeded oxyge diffuio coefficiet value fo HPP i 4.6 ± 1.3 x 10-8 cm -1. Tee geometic ucetaitie ae ot ueaoable, epecially coideig tat pactical field geomembae mut be fabicated to cove aea of te of ectae o moe. I uc cicumtace it would be ulikely tat fie fabicatio toleace could be met. Compoite Betoite-HPP Geomembae Compoite lie ae pecially fabicated to impede pecific compoud. Fo example, a compoite lie compiig a ti polyme laye ad a ticke clay laye may impede oxyge flux weeve te clay i at leat patially atuated. We emplaced ove a tailig facility, tey act to educe vetical oxyge fluxe to te tailig; te effect may be augmeted by te itiic oxyge miealiatio caacteitic of te uoudig oil pofile. Hee te effective diffuio coefficiet fo a compoite betoite-hpp geolie i coideed. I paticula, data fo a Betofix 000X betoite-impegated membae 7. ± 1.4 mm tick, pooity 0.6 ± 0.05, deity 1.17 ± 0.05 g cm -3 ae ued. Of citical impotace i te atuatio tate of te betoite. Hige wate atuatio lead to malle diffuio coefficiet. idie et al. [11] ow tat modeate vaiatio i betoite wate cotet ca 11
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet lead to vaiatio i pemeability of eveal ode of magitude, wile Aubeti et al. [10] ue a empiical totuoity fomulatio to map te vaiatio of diffuio coefficiet wit wate atuatio. Tei model educe to betoite = mol a θ x a K H mol w θ y w wee te wate atuatio S = θ w / = -θ a / i expeed i tem of te volumetic betoite ai ad wate factio θ a ad θ w, ad te pooity. K H i Hey coefficiet 0.03, ad te molecula oxyge diffuio coefficiet i ai ad wate ae aumed to be [10] mol a = 1.8 x 10-1 cm -1 ad mol w =.5 x 10-5 cm -1, epectively. Te expoet x ad y ae foud to lie i te age 0.6 to 0.75, ad may be aumed to be equal. Figue 4 ow tat te betoite diffuio coefficiet deceae by tee ode of magitude a wate atuatio cage fom 0 to 1, ad i appoximately 10-6 cm -1 we fully atuated. Te quoted ucetaitie i pooity yield vaiatio of imila ize to toe fo te expoet age i Fig. 4. Figue 4: Vaiatio of betoite diffuio coefficiet wit wate atuatio S. Cuve ae ow fo te epeetative age of totuoity expoet x = y. 1
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet Figue 5: Vaiatio of te compoite diffuio coefficiet wit wate atuatio S. Te olid lie ow te mea etimate, wile te daed lie epeet te boud of te expeimetal accuacy, baed o abolute meauemet eo. Te aymptotic lowe boud diffuio coefficiet fo te betoite membae i tu at leat a ode of magitude ige ta te HPP value, but te uually geate ticke of te betoite membae will impove it pefomace a a oxyge baie. Te cotibutio of te betoite ad HPP laye to te effective lie diffuio coefficiet ca be etimated uig te paallel elatioip [, p.47] compoite compoite HPP betoite = 3 HPP betoite wee =. Wit te mea value compoite = 7.95 mm, HPP = 0.75 compoite HPP betoite mm, betoite = 7. mm togete wit te ecommeded value fo HPP ad te totuoity model of Aubeti et al. [10], lead to etimated value fo te diffuio coefficiet of oxyge i te compoite geomembae a ow i Fig. 5 olid lie. A obviou eult i tat ule te betoite atuatio i vey ig 1, te betoite laye doe ot mateially ifluece te diffuio coefficiet of te compoite eet, a a bee etablied elewee [10]. Te daed lie i Fig. 5 ow te abolute vaiatio i te etimated value due to ucetaitie i te geometic paamete, i HPP ad te x ad y expoet. 13
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet Summay I ummay, expeimetal diffuio data wa itepeted i tem of a cambe beaktoug model paameteized by membae diffuio coefficiet, eo coumptio ad cambe leakage. It wa foud tat te membae diffuio ad cambe leakage wee matematically depedet, leadig to iaccuate etimate fo te aociated paamete duig egeio. Elimiatig te cambe leakage paamete fom te egeio yielded a diffuio coefficiet fo HPP i lie wit liteatue value. A a by-poduct of te egeio, te itiic oxyge coumptio ate of a popula electocemical eo wa alo detemied. Majo cotibutio to ucetaity of tee oxyge diffuio coefficiet aie fom ucetaitie i te geometic paamete of te expeimet, e.g. membae ticke ad cambe legt. Takig tee ito accout lead to eaoably accuate effective diffuio coefficiet fo te HPP mateial. Tee value wee applied to a imple compoite diffuio model to gai etimate of te effective diffuio coefficiet fo a compoite HPP-betoite geoytetic clay lie. Ti model igligt te eitivity of te compoite geolie diffuio coefficiet to betoite wate atuatio. Fo mot atuatio le ta uity, betoite i ig, ad te compoite diffuio coefficiet i appoximately equal to te HPP x betoite / HPP. Oly at te iget atuatio doe te declie i betoite lead to eductio i compoite. Ackowledgemet Ti wok wa pat fuded by WMC Reouce td. 14
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet Appedix A etailed Beaktoug Model Te poblem of a oe-dimeioal lab i.e. te baie i te peet cotext i cotact wit a well-mixed cambe i coideed, uig te appoac of Calaw ad Jaege [3, pp. 311-313]. Cocetatio i te lab ad cambe ae deoted ux,t ad vt, epectively. Te lab, of ticke ad iitially at zeo cocetatio, i i cotact at x = 0 wit cotat atmopeic cocetatio u atm ad at x = wit a well-mixed cambe legt ad co ectio A iitially at zeo cocetatio. Te lab baie diffuio coefficiet i. Oxyge diffue toug te lab ito te cambe, ad te cambe cocetatio i affected by ote ouce leakage ad ik eo coumptio tem. eakage i modeled a a diffuive poce, wit leakage flux popotioate to te diffeece i cocetatio betwee te cambe ad te exteal atmopee. Te leakage ate cotat i. Similaly, te coumptio dyamic of te electocemical eo ae epeeted by a fit-ode tem wit ate cotat. Pyically, ti decibe a eo tat utilize oxyge i popotio to te local oxyge cocetatio, i.e. at low cocetatio oly mall amout of oxyge ae coumed pe uit time, ad at ig cocetatio ige amout ae coumed pe uit time. No aalytical olutio fo ti ytem ae extat i te liteatue. Te equatio goveig te dyamic ae u u = t x ; u x,0 = 0 ; u0, t = u atm A.1 A dv u = A dt x x= A v v A v ; v0 = 0 atm A. A oly te effective diffuio coefficiet of te baie i of iteet, patitioig effect ae eglected ee ef. 4 fo a imila applicatio ivolvig patitioig ad te equatio ytem i completed by pecifyig cotiuity elatioip betwee u ad v, i.e. 15
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet u, t = v t ; u atm = v atm A.3 Te olutio of te equatio ytem poceed by ubtitutig eq A.3 ito A., ad takig aplace tafomatio [5] of eq A.1 ad A., yieldig d u pu = ubject to u 0 = uatm / p A.4 dx du p u = u u dx x= atm / p A.5 wee u x i te aplace tafomatio of ux,t. I ti otatio, u x a implicit depedece o te tafomatio vaiable p. Fo 0, te olutio of eq A.4 i u u x = Bi atm qx co qx A.6 p wit q = p /. Te itegatio cotat B i fixed by atifyig eq A.5, yieldig, afte algeba, uatm q co q x p i q x i qx u x = A.7 p p i q q co q I ode to ecove te tempoal olutio ux,t, it emai to pefom te ivee aplace tafomatio o eq A.7. Ti i mot eaily accomplied by ivokig te Iveio Teoem [3, p. 30], wic tate 1 pt u x, t = e u x dp π i Γ A.8 wee Γ i a cotou i te complex plae coe lage eoug to eue tat it ecloe all pole of te itegad. Wit ti coice, ad uig covegece popetie of te itegad fo lage p, eq A.8 educe via te Reidue Teoem to [6, p. 403] 16
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet pt e u x u x, t = Re A.9 wee te ummatio i ove all te pole of te itegad, ad Re i te eidue opeato. Te pole ae mot eaily idetified by makig te tafomatio p =. Eq A.7 te become q = i, ad tu uatm i x i x co x u x = A.10 co i Pole occu at = 0, ad wee co i = 0 A.11 o, equivaletly, ta = A.1 Te oot of eq A.1 ae tacedetal ad fom a deumeably ifiite et of pole poit { }, =1,, Tey ae eaily detemied umeically fo give value of te ytem paamete,,,,. It emai to calculate te eidue of eq A.7 at te pole poit. Ti ofte laboiou tak i facilitated toug te ue compute algebaic tecique. Coide te teady tate pole at = 0. Ti ca be ow to be a imple pole ad, ece, te coepodig eidue i limit p 0 p e pt u x = u atm { [ x ]} A.13 17
CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet 18 Fo te -t pole at p = -, te itegad of eq A.7 may be expeed a gp/p wee - = 0, but eite g o ' ae zeo at te pole. Teefoe [6, p. 13] te eidue at te pole i give by g- /'-. I ti way te eidue of te itegad of eq A.7 at te pole p = - ca be ow to be [ ] [ ] [ ] x x x x e u t atm i co i i i A.14 Hece, ummig te eidue A.13 wit te ifiite et of eidue A.14, ad applyig eq A.11, te cocetatio ditibutio i te baie lab i [ ] { } [ ] [ ] [ ] = = x x x x e u x u t x u t atm atm i co i i i, 1 A.15 Te quatity meaued i te expeimet i vt, te cambe cocetatio, wic, by eq A.3, i fially [ ] [ ] = = = e u t u v t v t atm atm i i, 1 A.16 If te baie i impemeable, = 0, te above expeio doe ot old. I ti cae u i eveywee zeo ad te dyamical equatio fo te cambe cocetatio i 0 0 ; = = v v v v dt dv atm A.17
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CSIRO ad ad Wate Tecical Repot 37/01 Geomembae Oxyge iffuio Coefficiet 8. Cazaux,., ad idie, G., field evaluatio of ydaulic pefomace of geoytetic clay lie by mall ad lage-cale tet, Geotext. Geomembae 18-4, 163-178, 000. 9. Fox, P. J., e Battita,. J., ad Mat,. G., Hydaulic pefomace of geoytetic clay lie ude gavel cove oil, Geotext. Geomembae 18-4, 179-01, 000. 10. Aubeti, M., Aacib, M., ad Autie, K., Evaluatio of diffuive ga flux toug cove wit a GC, Geotext. Geomembae 18-4, 15-33, 000. 11. idie, G., Bouazza, A., ad Cazaux, A., Ga pemeability of geoytetic clay lie, Geotext. Geomembae 18, 35-50, 000. 1. Sa, H.-Y., ad Yao, J.-T., Meauemet of ai pemeability of geoytetic clay lie, Geotext. Geomembae 18, 51-61, 000. 13. Matew, J. H., Numeical Metod fo Matematic, Sciece ad Egieeig, Secod Editio, Petice Hall: New Jeey, 199, p.84. 14. Katicioglu, T. Y., Kapta, H. Y., ad Guve, O., etemiatio of oxyge diffuio coefficiet of polymetacyloitile II ad te calculatio of diffuio activatio eegy, J. Appl. Polyme Sci. 74, 1108-1118, 1999. 15. Haug, A. T., ad Wite, R. E., Oxyge diffuio coefficiet ad olubility i a ew poto excage membae, J. Electocem. Soc 1473, 980-983, 000. 16. Rabi, Y., Yekta, A., ad Wiik, M. A., A metod fo meauig oxyge diffuio ad oxyge pemeatio i polyme film baed o fluoecece quecig, Aal. Cem. 71, 5045-5053, 1999. 17. Jayaaja, C. N., Yekta, A., Mae, I., ad Wiik, M. A., Oxyge diffuio ad pemeability i alkylamiotioylpopazee film iteded fo popoecece baomety applicatio, Macomolecule 33, 5693-5701, 000. 0
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