School of Mathematics and Actuarial Science, Jaramogi Oginga Odinga University of Science and Technology, Bondo, Kenya
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1 N XXXX XXXX 6 E esearc Arcle olume 6 ssue No. Two mesoal Maemacal Models for ovecve-spersve Flow of Pescdes Porous Meda e H. W. Adams N. Omolo Oga M. E. Oduor Okoa 3 T.. O. Amer cool of Maemacal ceces Appled ascs ad Acuaral cece Maseo Uvers Prvae Bag Maseo 3 cool of Maemacs ad Acuaral cece aramog Ogga Odga Uvers of cece ad Tecolog Bodo Kea Absrac: Te raspor of solues roug porous meda were cemcals udergo adsorpo or cage process o e surface of e poro us maerals as bee a subec of researc over e ears. Use of pescdes as resuled produco of dverse qua ad qual for e marke. sposal of ecess maeral as also become a acue problem. Te cocep of adsorpo s esseal de ermg e moveme paer of pescdes sol order o assess e effec of mgrag cemcals from er dsposal ses o e qual of groud waer. s paper we derve a wo dmesoal equao accoug for bo laeral ad aal pes cde flow a porous meda b covecve- dspersve raspor w sead sae waer flow. Te model s derved from e frs prcple ad solved usg Alerao-rec-mplc A meod. Ke Words: covecve-dspersve adsorpo pescdes porous meda solues. NTOUTON ovecve-spersve equaos ave bee solved usg mplc meods. Ts s due o er ucodoal sabl bu e calleges assocaed w e marces ave become a cocer ad a lmao obag soluos [ 9]. mplc fe dfferece meods oba e soluo for e e sep from e sae of bo e curre ad e e seps wle eplc meods oba e soluo from e curre sep ol. mplc meods requre compuao per me sep ad ca mpleme log me sep ervals wou sufferg umercal sables. O e corar eplc umercal meods suffer fro m sabl es. mplc umercal meods are sable oedmeso problems bu e do o guaraee sabl muldmesoal problems. verso of marces produced b eplc umercal are easer o solve compared o ose of mplc umercal meods bu requre smaller me erval us creasg compuao me. s paper we adop A meod. umercal aalss e Alerag reco mplc A meod s a fe fferece meod for solvg parabolc ad ellpc paral dffereal equaos. Te advaage of e A meod s a e equaos a ave o be solved eac sep ave a smpler srucure ad ca be solved effcel w e rdagoal mar algorm. also called Tomas Alogarm ws s user fredl [6] Peacema ad acford [] meod s e mos deal for. Tese meods bled mplc umercal meods w eplc umercal meods. A meod solves e frs dmeso mplcl ad e secod dmeso eplcl ad e e sep e frs dmeso eplcl ad e secod dmeso mplcl ad so o. Ts meod s ucodoall sable ad sce apples mplc sceme o oe dmeso a a me e o-zero erms are prese ol ree dagoal le mar wc s smple ad fredler o solve compared o e mar creaed b e full mplc meod [7]. Advaages of A meod s a preves umercal problems ecouered b e full mplc scemes ad sores compug me b a facor of compared o e mplc meod ad does o ecouer umercal problems suc as egave dsrbuo fucos or crases durg mar verso [6] a are see mplc meods.. EATON OF ONETE-PE E OLUTE TANPOT EQUATON WTH TEAY TAT WATE FLOW ONTON Le e average pore waer veloc be LT - = q [5] H.e. q K s e flu des w z s w s volume of waer e porous meda ad s wc s volume of solds used sead flow medum before e flow akes place. s paper we appl e cocep of dsperso roug a cldrcall packed sol vessel o derve e covecve dspersve equao for pescde adsorpo a porous meda. ee Fgure below Fgure.. ldrcall packed sol vessel eraoal oural of Egeerg cece ad ompug Ocober 6 78 p://esc.org/
2 A ver low flow rae e dsperso s dffere logudal ad radal drecos. Te sperso coeffces are deoed b L for logudal ad for radal dff. ds were dff L T - s molecular dffuso coeffce ds L T - s e drodamc dsperso ad s e mg or spreadg of e solue durg raspor due o dffereces veloces w a pore ad bewee pores. Te volumerc waer coe deoed b wc we ca assume o be e vod age for sauraed sols. Te eleme eg s deoed b. er radus s r ad ouer radus s r r s e cocerao of e maeral o be dspersed ad s a fuco of aal poso l radal poso r me ad dsperso coeffces ad L radal ad aal respecvel. Te rae of er of referece maeral due o flow aal. Te correspodg efflu rae s dreco s q rr q rr l l.. Te e accumulao rae eleme due o flow aal dreco s : q rr l l.. ae of dffuso aal dreco across le boudar s: - rr l Te correspodg rae a oule boudar s: L.3 Te e accumulao rae due o dffuso from boudares s:. rr l r r r l r r r..8 f we gore e las erm because we are cosderg fe small cages ad a makes e secod dervave eglgble becomes: lr r r r..9 For a represeave elemear volume of sol e oal amou of a gve cemcal speces X ML -3 s represeed b e sum of e amou reaed b e sol mar ad e amou prese e sol as X b. were b s e bulk des ad s e amou of solue adsorbed ffereag. w respec o elds X b Now e oal accumulao rae s: rrl = rrl.. X b.. rr L l l l. Te e accumulao rae due o dffuso from boudares aal dreco s: rr L l l ffuso radal dreco a r s: - rr Te correspodg rae a radus r r r l r r r s r r l From equaos. o. we ave: b rrl qrr l l rr L l l r r l r r.3 vdg roug b rr l we ge L r l r r r. Takg l= ad r= equao. becomes q q l.5. eraoal oural of Egeerg cece ad ompug Ocober 6 79 p://esc.org/
3 eraoal oural of Egeerg cece ad ompug Ocober 6 7 p://esc.org/ Bu q pore waer veloc erefore equao.5 comes o *..6 From e Freudlc equao [] N K N KN KN N..7 Pug equao were N KN. Equao.8 s our model equao descrbg wodmesoal moveme of solue e sol or porous meda.. OLUTON OF THE EQUATON UNG NUMEAL METHO Te epaded equao.8 s Te fe dfferece meod s deal for solvg olear equaos. We replace e dffereal w s fe dfferece equvale. We sall esabls grds based o dmesos we are o cosder. We use e oao a s used o desgae e pvo po for wo-dmesoal space dreco ad beg e couers e drecos. Te paral dervave of w respec o mples a s kep cosa ad vce versa. Te al codo s a e cocerao of pescde a all posos e sol a me zero s cosa ad equal o. Ta s = for > = Boudar codos: wo codos are ecessar:. e frs case e cocerao of e pescdes a e poso = = s specfed for a perod of me e cocerao a e surface s zero. Ta s = for < o = for > o. e secod case e cocerao of e pescdes e soluo eerg e sol ssem a poso or = s specfed for a perod me. Followg a me e cocerao a e surface s zero. Tus for d d for >. Assumpos. Te pore waer veloc s cosa me ad space. Ts codo ca be me for a uform sol f e flu des of waer veloc ad volumerc waer coe are cosa for all posos all e mes.. Te spread of solue s domaed b draulc dsperso raer a dffuso.. Te drodamc dsperso ca be appromaed as e produc of e dspersv ad pour waer veloc. v. Te adsorpo process s saaeous ad reversble ad e adsorpo soerm ca be descrbed b e model.e e cocerao of pescde absorbed o e sol solds s proporoal o e cocerao e soluo [8] Te secod order accurac me ca be obaed b usg e rak-ncolso Meod Fgure..Grd sowg secod order accurac me s Oba usg rak Ncolso Meod.. f.
4 eraoal oural of Egeerg cece ad ompug Ocober 6 7 p://esc.org/ Usg Mar s equao s epesve o solve. Te mos praccal soluo o s came w e developme of Alerao-rec-mplc A Meod b Peace ma ad cford[]. Ts cosss of frs reag oe row mplc w backward Euler ad reversg e roles ad reag e oer oe b backward Euler.. OMPUTATON MOLEULE FO THE A METHO + +/ Fgure.3.ompuao Molecule for A Meod Tese meods volve solvg oe se of lear equaos for wo dmesoal ssems solve equao for grd le. also provdes for solvg b alerag dreco o preve bas.. Te mar form for eac row s ep Te mar form for eac row s
5 eraoal oural of Egeerg cece ad ompug Ocober 6 7 p://esc.org/ Te equaos ca be solved o avod forward elmao ad backward subsuo. abl aalss Usg o Neuma Aalss ep os os os os Le os os os os ep os os os os os os os os.5 We we combe equaos. ad.5 we ge os os os os
6 os os os os Fromm s sceme keeps rack of weer e wave speed s posve or egave ad alers e dreco of formao rasfer accordgl. Ts sceme s sable for [7].. HANG ad G. OOPE: A praccal dfferece sceme for Fokker-Plack equaos.. ompu. Ps. ol. 6 ssue Augus 97 [8] Faus. ad Osma Al.: 983. emsr of waer reame. Publsed b Amerca Book oce [9] T. Welsk ad M. Bowe A cemes for ger order olear dffuso equaos App. Numer. Ma. 5: Freudlck H. 96. ollod ad apllar emsr Lodo Meue ad ompa A A cos os os os A A A A Ucodoall For =.EFEENE A A [] Peacema. W.; acford r. H. H. 955 "Te umercal soluo of parabolc ad ellpc dffereal equaos" oural of e oce for dusral ad Appled Maemacs 3 : 8 [] ouglas r. m 96 "Alerag dreco meods for ree space varables" Numersce Maemak : 63. [3]. ouglas r ad. E. Gu A geeral formulao of alerag dreco Meods-Par. Parabolc ad perbolc problems Numersce Maemak vol. 6 pp [] a Geuce M. T avdso.m & Werega P. : 97. A Evaluao of Kec ad equlbrum equao for predcg pescde moveme roug porous meda. ol c. oc. Am. [5] Wel Morrs.. 99 aulso ad cardso Edo emcal Egeerg. Ame. em soce 7: 66. [6] Tomas L.H. 99 Ellpc Problems Lear ffereal Equaos over a Nework Waso c. ompu. Lab epor olumba Uvers New York. eraoal oural of Egeerg cece ad ompug Ocober 6 73 p://esc.org/
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