SENSITIVITY ANALYSIS OF ADSORPTION ISOTHERMS SUBJECT TO MEASUREMENT NOISE IN DATA

Transcription

1 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt SENSITIVITY ANALYSIS OF ADSORPTION ISOTHERMS SUBJECT TO MEASUREMENT NOISE IN DATA Karim Farhat, Mohamd N. Nounou 2 and Ahmd Abdl-Wahab 3 Studnt, Chmical Enginring, Txas A&M Univrsity at Qatar karim.farhat@qatar.tamu.du 2 Assistant Profssor of Chmical Enginring, Txas A&M Univrsity at Qatar Mohamd.nounou@qatar.tamu.du 3 Assistant Profssor of Chmical Enginring, Txas A&M Univrsity at Qatar Ahmd-abdlwahab@tamu.du ABSTRACT Rflcting th importanc of adsorption as a major watr purification mthod, th main objctiv of this rsarch was to prform a snsitivity analysis on som of th common adsorption isothrms subjct to masurmnt nois in data. Evn though most of adsorption isothrms hav bn drivd basd on thortical assumptions about th adsorption mchanism, thy involv modl paramtrs that nd to b stimatd from xprimntal masurmnts of th procss variabls. Spcifically, for th Langmuir isothrm, which can b linarizd in thr forms, it was sought to dtrmin which of ths thr forms would giv th highst accuracy of th adsorption modl paramtrs maximum amount of adsorbat pr unit wight of th adsorbnt and th constant rlatd to th affinity btwn th adsorbnt and adsorbat. Anothr objctiv was to stimat th adsorption paramtrs using th nonlinar Langmuir modl, and to compar thir accuracy to th ons stimatd using th most accurat linar form. Furthrmor, it was dsird to xamin th ffct of nois magnitud on th stimation accuracy for th various Langmuir forms (linar or nonlinar) by varying th nois varianc and th magnitud of th adsorption paramtrs thmslvs. To achiv this aim, MATLAB programming softwar was usd for simulations. Th rsults of this work could b summarizd as follows: On of th linarizd forms of Langmuir modl showd normal distribution and providd most accurat stimation of both modl paramtrs. In addition, it was shown that whn th nois contnt (standard dviation) incrasd on th data, lss accurat stimats wr obtaind for both adsorption paramtrs. Finally, th stimation accuracy was mor snsitiv to th magnitud of th affinity constant than to th maximum amount of adsorbat in adsorbnt; largr valus of affinity constant rsult in highr stimation accuracy of both modl paramtrs. Kywords: Langmuir, linarization, paramtrs, nois

2 2 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt INTRODUCTION During th last fw dcads and as industris wr growing, significant pollution problms ros up, and handling thm bcam a major concrn spcially for scintists and nginrs. In spcific, a grat dal of attntion has bn givn to watr pollution problms as thy thrat on of th most vital natural rsourcs for human lif. For xampl, th rmoval of toxic havy mtals (such as Zinc, Nickl, and Lad) from groundwatr and wastwatr has bn approachd by various works and studis, taking into considration th vry srious implications that ths pollutants can hav on humans and othr living bings halth. Consquntly, many watr purification mthods has bn dvlopd and usd to rmdiat consumabl and wast watr from thos pollutants. Ths mthods includ chmical prcipitation (Hnc []), rvrs osmosis (Ning [2]), lctro dialysis, ion xchang, and finally adsorption. In gnral, adsorption is a mass transfr procss, which involvs th contact of solid calld adsorbnt with a fluid containing crtain pollutants calld adsorbat (Alkan and Dogan [3]). Ths pollutants can b organic compounds, pathogns, and havy mtals. And thir contact with th surfac of th adsorbnt rsults in prmannt bonds, nsuring thir rmoval from th fluid. Th adsorption capacity dpnds on svral factors, such as th adsorbnt typ, its surfac ara, and its intrnal porous structur. Additionally, sinc th attachmnt of th pollutant can b physical or chmical, th physical and chmical structurs as wll as th lctrical charg of th adsorbnt can significantly influnc its intractions with th adsorbats, and thus th ffctivnss of pollutant rmoval. Adsorption procsss ar charactrizd by thir kintic and quilibrium isothrms. Th adsorption isothrms spcify th quilibrium surfac concntration of th adsorbat as a function of its bulk concntration. Svral mathmatical modls hav bn proposd to dscrib th quilibrium isothrms of adsorption. Som of th most popular modls includ Langmuir, Frundlich, Rdlich-Ptrson, and Sips. A summary of ths isothrms is providd by Dabaybh [4]. Evn though most of ths adsorption isothrms wr drivd basd on som thortical assumptions about th adsorption mchanism, thy involv modl paramtrs that nd to b stimatd from xprimntal masurmnts of th procss variabls. Talking about Langmuir modl in spcific, th isothrm has th following form: q Q bc bc c () whr, C is th quilibrium liquid phas concntration (mg/l), q is th quilibrium solid phas concntration (mg/g), Q c is th maximum amount of adsorbat pr unit wight of th adsorbnt to form a complt monolayr, and b is a constant rlatd to th affinity btwn th adsorbnt and adsorbat. In th abov Langmuir modl, Q c and b ar modl paramtrs to b stimatd from masurmnts of q and C.

3 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt 3 Unfortunatly, masurmnts of th adsorption procss variabls, q and C, ar usually contaminatd with nois or masurmnt rrors du to random rrors, human rrors, or malfunctioning snsors. Th prsnc of such masurmnt nois, spcially in larg amounts, can largly dgrad th accuracy of th stimatd isothrm paramtrs, which in turn limits th ability of th isothrm to accuratly prdict th adsorption capacity of a crtain procss. This is bcaus most modling tchniqus stimat th modl paramtrs by minimizing som objctiv function rlatd to th prdiction rrors of th modl output ( q ). Unfortunatly, sinc th isothrm input and output variabls ( q and C ) ar both masurd, only minimizing th output prdiction rrors may not lad to accptabl stimation. Thrfor, th objctivs of this projct wr as follows:. Prform a snsitivity analysis to invstigat th ffct of th prsnc of masurmnt nois on th stimation accuracy of th Langmuir modl isothrms (linar and non-linar). Th ffct of nois on th stimatd paramtrs from ach of ths linarizd modls was assssd, and rcommndations on th bst linarizd modl wr providd. Of cours, th snsitivity analysis rsults will also dpnd on th stimation mthod utilizd. 2. Assss th ffct of masurmnt nois on th paramtrs stimatd from th nonlinar Langmuir modl and compar that to th ffct on th paramtrs of th most accurat linarizd modl. 3. Assss and compar th ffcts of diffrnt nois intnsitis and paramtrs (Qc and b) magnitud on th lattr s accuracy stimation. LITERATURE REVIEW Bing a major mass transfr procss of multipl uss, rsarchs continu about adsorption procss, its mchanisms, and its application in various filds. As it has bn notd, th basic concrn of this study is to prform a gnral snsitivity analysis for adsorption isothrms, using artificially addd nois, to dtct th bst form of Langmuir modl for adsorption paramtrs stimation. In this rgard, this study intrscts with som of th rsarchs don bfor in th sam fild, whil bing uniqu. For xampl, th concpt of comparing adsorption isothrms and paramtrs and valuating thir accuracy by using diffrnt modls has bn discussd for: th rmoval of slctd mtal ions by powdrd gg shll (Otun t al. []), sorption of organic compounds to activatd carbons (Pikaar t al. [6]), adsorptiv rmoval of chlorophnols from aquous solution by low cost adsorbnt (Radhika and Palanivlu [7]), rmoval of boron from aquous solution by clays and modifid clays (Karahan t al. [8]), and many othrs. Howvr, it is clar that ths studis dtrmin th bst fitting modl only for th matrial, substanc or procss undr study. Similarly, som

4 4 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt studis dtrmind th most suitabl modl to xplain th adsorption procss dpnding on spcific typ of instrumnt usd to gt th xprimntal data; Modling in Adsorption-Dsorption Nois in Gas Snsors Using Langmuir and Wolknstin for Adsorption (Gormi t al. [9]) forms a good xampl. In this rgard, th currnt projct provids mor gnral information about th most accurat Langmuir modl, rgardlss of th substanc tstd or th instrumnt usd. On th othr hand, som rsarchs focusd on studying th ffct of som variabls on th accuracy of th adsorption paramtrs stimation by various modls. For xampl, th choic of column hold-up volum, rang and dnsity of th data point was found to hav an impact on systmatic rrors in th masurmnt of adsorption isothrms by frontal analysis (Gritti and Guiochon []). This study shows that th concntration rang within which th adsorption data ar masurd and th way th data points ar distributd ar important factors in rror stimation. Anothr study for Gritti and Guiochon [] shows that th fluctuations of th column tmpratur and th composition and th flow rat of th mobil phas affct th accuracy and prcision of th adsorption isothrm paramtrs masurd by dynamic HPLC mthods. Yt, this study bas its findings on xprimntal data (acquird by frontal analysis FA), and is applid on spcific systm (phnol in quilibrium btwn C8- bondd Symmtry and a mthanol-watr mixtur). In addition, it was noticd that som studis prformd statistical analysis on adsorption isothrms to dtrmin th most accurat modl in stimating th adsorption paramtrs. For xampl, Joshi t al. [2] prformd modl basd statistical analysis of adsorption quilibrium data. Aftr comparing th paramtr stimation by diffrnt linarizd and non-linar adsorption modls, it was shown that Langmuir modl dos not giv a satisfying dscription of th considrd xprimntal data, and that Frundlich isothrm provids th most accurat stimation for th liquid phas concntration rang usd in th xprimnt. Howvr, again, th ffct of nois (found in th xprimntal data du to human or instrumntal rror) on th accuracy of adsorption paramtr stimation has bn ignord. This ffct is highlightd in th currnt papr by adding artificial nois to nois fr data and thn dtcting th chang in th paramtrs valu by diffrnt Langmuir modls. In addition, whil th stimation accuracy in th lattr papr is compard btwn diffrnt modls, our currnt papr aims to compar btwn th accuracy of non-linar Langmuir modl and that of its possibl linarizd forms. EXPERIMENTAL PROCEDURE To start with, th simulation of this study was prformd using MATLAB programming softwar. Part On: Linarizd Langmuir Modls Th Langmuir could b linarizd in thr forms:

5 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt q C q q C Qcb C Qc Langmuir (2) C Q c Qc b Langmuir 2 (3) b q Q b Langmuir 3 (4) c Th abov linarizd modls would provid diffrnt paramtr stimation rsults as thy minimiz diffrnt objctiv functions. Aftr simulating ach of ths modls on MATLAB, nois-fr inputs and outputs ( q and C ) wr usd basd on alrady givn Qc and b valus. Thn, nois was addd to th nois-fr data ( q and C ), and Qc and b of ach linarizd modl wr stimatd to quantify th impact of masurmnt nois on th accuracy of thir stimation. Having randomly distributd nois, th simulation was rpatd tims to gt diffrnt stimatd valus for Qc and b in vry run. Thn, th diffrnt stimats for vry paramtr wr statistically analyzd to show thir distribution pattrn and dviation from th tru valu. Part Two: Non-linar Langmuir Modl To simulat th nonlinar modl on MATLAB, gntic optimization algorithm which is widly usd in nonlinar optimization was usd. In dtails, th objctiv function of th prdiction rror was valuatd ovr a msh of th modl paramtrs, and thn a minimum was slctd ovr th ntir msh. Similar to th linarizd modls, th simulation was rpatd tims to gt diffrnt stimatd valus for Qc and b in vry run. Thn, th diffrnt stimats for vry paramtr wr statistically analyzd to show thir distribution pattrn and dviation from th tru valu, and thy wr plottd togthr with th stimatd paramtrs of th linarizd modls for accuracy comparison. Part Thr: Nois Intnsity and Paramtrs Magnitud Effcts Each of th procdurs xplaind abov was rpatd with thr diffrnt variancs (nois magnitud). In addition, th simulations wr carrid using various combinations of thr diffrnt (progrssivly incrasing) valus of Qc and b. RESULTS AND DISCUSSION Bst Linarizd Langmuir Modl Qc and b stimats distribution pattrn for th thr linarizd modls is rprsntd in Figurs and 2, rspctivly. Figur shows that th Qc distribution pattrn for

6 -2 6 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt Langmuir 2 is closr to th tru valu than that of Langmuir 3 whos distribution pattrn is in turn closr than that of Langmuir. This was provn by th standard dviation valus for th thr modls: Th standard dviation of Qc from its tru valu for Langmuir is.6, smallr than that of Langmuir 3 (2.87) which is in turn smallr than that of Langmuir (2.698). Similar rsults ar dmonstratd in figur 2 whr th standard dviation of b from its tru valu for Langmuir 2 is.64, smallr than that of Langmuir 3 (.68) which is in turn smallr than that of Langmuir (.23). Ths rsults wr confirmd by tsting diffrnt tru valus of Qc and b and undr variabl nois intnsity (as will b shown). As a rsult, Langmuir 2 linarizd form provs to provid th most accurat stimating for th adsorption paramtrs upon applying xprimntal nois Langmuir Langmuir 2 Langmuir 3 STD Langmuir : STD Langmuir 2:.6 STD Langmuir 3:2.87 Qc Tru Valu: Points Figur : Qc Estimation Using th Thr Linar Langmuir Modl Langmuir Langmuir 2 Langmuir 3 STD Langmuir :.23 STD Langmuir 2:.64 STD Langmuir 3:.68 b Tru Valu: Figur 2: b Estimation Using th Thr Linar Langmuir Modls

7 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt 7 Nonlinar Langmuir Modl Qc and b for both th nonlinar and Langmuir 2 modls wr plot as shown in Figurs 3 and 4, rspctivly. Figur 3 shows that whil th nonlinar modl gav bttr stimation for Qc than Langmuir 2 at vry small b, Langmuir 2 provd suprior as b valu incrasd. Thos rsults wr confirmd by th standard dviation for Qc valus of both modls as b incrass: whn b is rlativly vry small, th standard dviation of Qc from its tru valu by Langmuir n was 3.2, smallr than that givn by Langmuir 2 (4.6). Howvr, as b incrass to. and byond, th standard dviation of Qc by Langmuir 2 bcom smallr than that by Langmuir n. Th sam rsults wr shown for b stimation by th two modls in Figur Langmuir 2 Langmuir n STD Langmuir 2: 4.6 STD Langmuir n: 3.2 Tru b =. Tru Qc = Lanmguir 2 Langmuir n STD Langmuir 2:.84 STD Langmuir n:.668 Tru b =. Tru Qc = Langmuir 2 Langmuir n STD Langmuir 2:.76 STD Langmuir n: Langmuir 2 Langmuir n STD Langmuir 2:. STD Langmuir n:.33.2 Tru b =. Tru Qc = Tru b =. Tru Qc = Lanmguir 2 Langmuir n 7 6 Langmuir 2 Langmuir n.6.4 STD Langmuir 2:.726 STD Langmuir n:.992 Tru b =.6 Tru Qc = STD Langmuir 2:.22 STD Langmuir n:.448 Tru b =.6 Tru Qc = Figur 3: Qc stimation by Langmuir n and Langmuir as b dcrass Figur 4: b stimation by Langmuir n and Langmuir as b dcrass

8 8 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt Comparison of Langmuir Linarizd Form Variation of Paramtrs: Effct on Qc Estimation Th rsults of variation of paramtrs magnituds on Qc stimation by thr Langmuir modls ar clarly dmonstratd in Appndix A. - Upon th variation of Qc tru valu (b rmains constant), no significant ffct is noticd on th distribution of Qc xprimntal data obtaind from th thr Langmuir modls. - Whn b tru valu incrass (Qc rmains constant), it is noticd that th standard dviation of data distribution for th thr modls dcrass and th thr modls givs bttr and closr stimation. Also, th data fit according to Langmuir 3 and Langmuir bcoms incrasingly mor similar. On th othr hand, as b tru valu dcrass, it is noticd that th standard dviation of data distribution for th thr modls incrass, th thr modls givs wors and broadr stimation, and distribution pattrns providd by Langmuir and Langmuir 3 bcom mor and mor diffrnt. Finally, th ffct of b variation is most significant on Langmuir stimation whos standard dviation changs significantly with th chang in b valu. - Whn both Qc and b incrass, th ffct of th variation of b rmains significant and dominat th way th modls stimation for paramtrs changs; i.. th rsults ar vry similar to thos obtaind incas of b variation. Variation of Paramtrs: Effct on b Estimation Th rsults of th variation of paramtrs magnituds on b stimation by thr Langmuir modls ar clarly dmonstratd in Appndix B. - Th ffcts of paramtrs magnitud on b stimation by th thr Langmuir modls ar th sam as thos on Qc stimation. Yt, it is worth mntioning that thos ffcts ar mor significant for Qc than for b stimation, undr th sam conditions. Comparison of Nonlinar and Most Accurat Linar Forms Variation of Paramtrs: Effct on Qc Estimation Th rsults of variation of paramtrs magnituds on Qc stimation by Langmuir 2 Langmuir n ar clarly dmonstratd in Appndix C. - Upon th variation of Qc tru valu (b rmains constant), no significant ffct is noticd on th distribution of Qc xprimntal data obtaind from th two modls. - Undr vry small b valus, th nonlinar modl Langmuir n givs bttr stimation for Qc paramtr than th linar modl Langmuir 2. As b incrass, howvr,

9 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt 9 Langmuir 2 stimation improvs and bcoms vn bttr than that of Langmuir n. Also, as b tru valu incrass, it is noticd that th standard dviation of data distribution for th two modls dcrass and thus both modls giv bttr and closr stimation. - Whn both Qc and b incrass, th ffct of th variation of b rmains significant and dominat th way th modls stimation for paramtrs changs; i.. th rsults ar vry similar to thos obtaind incas of b variation. Variation of Paramtrs: Effct on b Estimation Th rsults of variation of paramtrs magnituds on b stimation by Langmuir 2 Langmuir n ar clarly dmonstratd in Appndix D. - Unlik all othr cass, as Qc tru valu incrass (b rmains constant), th stimation of b by both linar and nonlinar modls slightly improvs. - Undr vry small b valus, th nonlinar modl Langmuir n givs bttr stimation for b paramtr than th linar modl Langmuir 2. As b tru valu incrass, its stimation by Langmuir 2 improvs and bcoms vn bttr than that of Langmuir n. Also, as b tru valu incrass, it is noticd that th standard dviation of data distribution for th two modls dcrass and thus both modls giv bttr and closr stimation. Yt, it is worth mntioning that th lattr ffct is mor significant for Qc than for b stimation, undr th sam conditions. - Whn both Qc and b incrass, th ffct of th variation of b rmains significant and dominat th way th modls stimation for paramtrs changs; i.. th rsults ar vry similar to thos obtaind incas of b variation. Nois Variation Effct Th ffct of nois variation on both Qc and b stimation is dmonstratd clarly in Figurs and 6, rspctivly. - As it was xpctd, whn th lvl of nois addd to th tru data incrass, th standard dviation of both Qc and b data distribution by th linar and nonlinar modls incrass. This signifis that ths modls giv broadr and lss accurat stimation. - At vry low lvls of nois th nonlinar Langmuir modl bcoms incrasingly similar to that of Langmuir 2. Yt, as th varianc of nois incrass, th nonlinar modl stimation for Qc and b dtriorats significantly and Langmuir 2 rtains providing th most accurat stimation for th adsorption isothrms paramtrs.

10 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt Figur : Effct of nois variation on Qc stimation Figur 6: ffct of nois stimation on b stimation CONCLUSIONS AND RECOMMENDATIONS From th prvious analysis for th Langmuir modls it could b concludd that: - Langmuir 2 is th most accurat linarizd form of Langmuir modl to stimat th adsorption paramtrs Qc and b. - Langmuir n and Langmuir 2 giv vry clos stimats for th adsorption paramtrs. - Qc valu has no significant ffct on th adsorption paramtrs stimation.

11 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt - As th affinity constant btwn th adsorbnt and adsorbat magnitud incrass: th stimation of Qc and b by all modls improvs and Langmuir 2 provs suprior. Ths rsults allow bttr undrstanding of th adsorption modling by Langmuir. Thus, dpnding on th xprimntal conditions spcifid, it would b asir to dtrmin which Langmuir form would giv thm most accurat adsorption paramtrs and thus th most accurat modling. In addition, th rsults prov to b of significant practical valu. From on sid, now that it is known that th scond linarizd form givs at last th accuracy than th original modl, this linarizd form can b usd with high lvl of confidnc to gt accurat stimations for th final quilibrium solid phas concntration q. On th othr hand, th suggstd Langmuir 2 provids a mor accurat altrnativ for th linarizing th Langmuir modl than Langmuir usd by most industris. Finally, aftr dtrmining th bst linar Langmuir modl, comparing this modl with th original nonlinar form, and dtrmining th ffct of nois and paramtrs magnitud on th lattr s stimation accuracy, it is highly rcommndd to carry similar analysis on othr adsorption modls (Frundlich, Rdlich-Ptrson, Sips tc) and compar thir rspctiv stimation accuracy with that of Langmuir n and Langmuir 2. This would allow having a thorough knowldg about th paramtrs stimation accuracy of various adsorption modls. Evntually, it would prmit bttr adsorption modling and thus mor accurat rsults in th various practical filds of adsorption, spcially watr pollution. ACKNOWLDGEMENTS This publication was mad possibl by a grant from th Qatar National Rsarch Fund. Its contnts ar solly th rsponsibility of th authors and do not ncssarily rprsnt th official viws of th Qatar National Rsarch Fund. REFERENCES [] Hnc, K.R., Watr Environ. Rs., 7, 78 (998). [2] Ning, R.Y., Dsalination, 43, 237 (22). [3] Alkan, M. and Dogan, M., J. Colloid Intrfac Sci, 243, 28 (2). [4] Dabaybh, M. Evaluation of Animal Solid Wastr as a Nw Adsorbnt, M.S. Thsis, Jordan Univrsity of Scinc and Tchnology (2).

12 2 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt [] Otun, J. A., Ok, I. A., Olarinoy, N. O., Adi, D. B., and Okuofu, C. A., Journal of Applid Scincs. ANSInt, Asian Ntwork for Scintific Information, Faisalabad, Pakistan: 26. 6:, rf. [6] Pikaar, Ilj, Albrt A. Kolmans and Paul C.M. van Noort, Sorption of organic compounds to activatd carbons. Evaluation of isothrm modls, Chmosphr, Volum 6, Issu, Dcmbr 26, Pags [7] Radhika, M., and Palanivlu, K., Adsorptiv rmoval of chlorophnols from aquous solution by low cost adsorbnt Kintics and isothrm analysis, Journal of Hazardous Matrials, Volum 38, Issu, 2 Novmbr 26, Pags [8] Karahan, S., Yurdakoc, M., Ski, Y., and Yurdakoc, K., Rmoval of boron from aquous solution by clays and modifid clays, Journal of Colloid and Intrfac Scinc, Volum 293, Issu, January 26, Pags [9] Gormi, S., Sguin, J. L., Gurin, J., and Aguir, K., Erratum to Adsorption dsorption nois in gas snsors: Modlling using Langmuir and Wolknstin modls for adsorption, Snsors and Actuators B: Chmical, Volum 9, Issu, 24 Novmbr 26, Pag 3. [] Gritti, F., and Guiochon, G., Systmatic rrors in th masurmnt of adsorption isothrms by frontal analysis: Impact of th choic of column holdup volum, rang and dnsity of th data points, Journal of Chromatography A, Volum 97, Issus -2, 2 Dcmbr 2, Pags 98-. [] Gritti, F., and Guiochon, G., Accuracy and prcision of adsorption isothrm paramtrs masurd by dynamic HPLC mthods, Journal of Chromatography A, Volum 43, Issu 2, 23 July 24, Pags 9-7. [2] Joshi, M., A. Krmling, and Sidl-Morgnstrn, A., Modl basd statistical analysis of adsorption quilibrium data, Chmical Enginring Scinc, Volum 6, Issu 23, Dcmbr 26, Pags

13 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt 3 Appndix A: Th variation of Qc stimation by th linarizd Langmuir modls as function of Qc and b tru valus Qc= b= Qc= b=. Qc= b= Qc= b= Qc= b= Qc= b= Qc= b= Qc= b= Qc= b=

14 4 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt Appndix B: Th variation of b stimation by th linarizd Langmuir modls as function of Qc and b tru valus Qc= b= Qc= b=. Qc= b= Qc= b=. 2 2 Qc= b= Qc= b= Qc= b= Qc= b= Qc= b=

15 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt Appndix C: Th variation of Qc stimation by Langmuir 2 and Langmuir n as function of Qc and b tru valus Qc= b = Qc= b= Qc= b= Qc = b = Qc= b= Qc= b = Qc = b = Qc= b= Qc= b=

16 6 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt Appndix D: Th variation of b stimation by Langmuir 2 and Langmuir n as function of Qc and b tru valus Qc= b = Qc= b =. Qc= b = Qc= b = Qc= b = Qc= b = Qc= b = Qc= b = Qc= b =

17 Twlfth Intrnational Watr Tchnology Confrnc, IWTC2 28 Alxandria, Egypt 7