Determination of Antoine Equation Parameters. December 4, 2012 PreFEED Corporation Yoshio Kumagae. Introduction
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1 refeed Soluos for R&D o Desg Deermao of oe Equao arameers Soluos for R&D o Desg December 4, 0 refeed orporao Yosho Kumagae refeed Iroduco hyscal propery daa s exremely mpora for performg process desg ad pla daa aalyss. hyscal properes clude equlbrum physcal properes pure subsace vapor pressure, vapor-lqud equlbrum, specfc hea, hea of evaporao, ec. ad raspor properes vscosy, hermal coducvy, dffuso coeffces. mog hese, he pure subsace vapor pressure s sgfcaly mpora as s he bass for esmag oher physcal propery values. Here we wll roduce a parameer deermao mehod for he oe equao whch s ofe used for pure subsace vapor pressure calculaos. Relaed maerals: O he Value of he oe Equao Tps #08 Soluos for R&D o Desg
2 refeed Smple Deermao Mehods of oe osas Soluos for R&D o Desg 3 refeed Vapor ressure orrelao Equao - oe Equao 3 rd h h ed.: ed.: ed.: l 0 0 [ka [a [mmhg [ 0.870,.49, [, T[K, 73.,, ad he oe equao are called oe cosas parameers, ad her values dffer depedg o he u sysem ha s used. I he case of he Kagaku Kougaku ra Japaese - hemcal Egeerg Hadbook, he forma ad us of he equao dffer accordg o he publcao year as show o he lef. For referece, he coverso of cosas s also show deoed,, for he hrd edo,,, for he ffh edo, ad,, for he sxh edo. I he explaaos ha follow, he oao of he hrd edo wll be used. oe Equao osas for Waer Soluos for R&D o Desg Kagaku Kougaku ra 3 rd ed h ed h ed
3 refeed 0 Lear pproxmao : lausus-lapeyro Equao y approxmag = 73., + ca be replaced wh he absolue emperaure T. The eases way o deerme he oe cosas,, from expermeal daa s o learly approxmae he equao so ha a lear plo ca be made a graph. 0 y plog 0 agas /T, ad ca be deermed from he ercep ad slope of he sragh le. Lear pproxmao : ox Equao pproxmag =30 gves a good approxmao for may subsaces. T 0 30 y plog 0 agas /+30, ad ca be deermed from he ercep ad slope of he sragh le. lhough he above equaos are smple, hey are o suffcely accurae for dsllao calculaos ad he lke. I s ecessary o deerme he oe cosas cludg he value. Soluos for R&D o Desg refeed Dffculy of Deermg oe osas Soluos for R&D o Desg
4 refeed arameer Deermao Mehod by Excel Solver Raw Daa, p, orr oe Equao 0 [mmhg, [ Error Defo Q [ Measured Value [ cal alculaed Value The values of, ad ha mmze he error are deermed wh he Excel solver. Soluos for R&D o Desg 7 refeed Excel Solver Resuls Ial Value fer Opmzao, Ial Value fer Opmzao, Measured Value p, orr Measured Value p, orr alculaed Value pcal, orr = = = alculaed Value pcal, orr = = = Ial Value fer Opmzao, Measured Value p, orr alculaed Value pcal, orr = = = overgece resuls are dffere for each se of al values of,,. The sadard bolg po s he mos mpora daa, bu s grealy affeced by he al value. The reaso for hs s ha he value s wh he deomaor of he fraco ad s a olear parameer. Soluos for R&D o Desg 8
5 refeed Improveme of Deermao of oe osas ar Nolear Leas Squares Mehod Soluos for R&D o Desg 9 refeed Nolear Leas Squares Mehod usg Normal Equao Q [ cal [ The followg codo holds for he values of,, whch mmze Q. Ths s called a ormal equao. Q Q Q 0, 0, 0 s ca be see from he form of he expresso of Q, a lear equao s obaed whe s parally dffereaed by ad. Soluos for R&D o Desg 0
6 Soluos for R&D o Desg refeed Q Q Q cal, [ ad From Eq [ [ [ [ [ [ [ ad ca be calculaed from he measured values, by guessg a value for. I oher words, here s o eed o depedely search for,,, ad s possble o search for oly. Dervao of Normal Equao Soluos for R&D o Desg refeed [ Se a guess value for alculae, from,, alculae cal from,, alculae Q= - cal djus o mmze Q Wh hs mehod, s o ecessary o chage hree parameers. The problem becomes a sgle varable search. lgorhm
7 refeed Opmzao Resuls = 8.8 = 9.03 = Equao for calculag, from Varable o chage p /+ /+ /+^ 0pcal =-/+ square of error base E E E E E E E E E E E-0 Q= 9.3E-07 pcal I s exremely dffcul o oba opmal values smply by applyg a solver o,, ad, bu s possble o oba relavely easly he rue opmal values by usg he ormal equao ad seg oly he search varable. Soluos for R&D o Desg 3 refeed Improveme of Deermao of oe osas ar Mulvarae Lear Equao oeffce Regresso Mehod Soluos for R&D o Desg 4
8 refeed There s a mulvarae lear equao regresso fuco amog he Excel dd- : Daa alyss - Regresso fucos. y a0 ax ax.. a x a 0, a, a, a : osas x, x, x,y: Daa I s possble o deerme lear cosas by usg hs fuco. Soluos for R&D o Desg refeed Trasformao o Mulvarae Lear Equao 0 0 Forma * Forma I s possble o deerme he coeffces sce boh equaos are lear expressos wh wo varables of he followg ype: y a a x a Soluos for R&D o Desg 0 x * s dcaed, for sace, roblem Solvg hemcal ad ochemcal Egeerg wh olymah, Excel ad MTL ulp ad Shacham, rece Hall p37
9 refeed omparso of Mulvarae Lear Regresso Resuls wh Normal Equao Mehod Measured Value Two Varable Learzao Two Varable Learzao, p, orr 0p / 0p/ 0p, *0p oeffce "" Iercep "-" / 37. "-" 0p/ = = = oeffce -/ Iercep / "-/" *0p = = = Soluos for R&D o Desg Measured Value Nolear Learzao Learzao Opmzao y recalculag, ca be see ha all mehods coverge wh suffce accuracy. 7 refeed ocluso Whe aempg o perform a parameer regresso whou rearragg he oe equao, s dffcul o oba he rue covergece values by a smple hree varable regresso by usg he Excel solver because he parameer s olear. I hs case, s cosderably easer o oba he rue covergece values by usg he ormal equao ad opmzg oly he parameer. y rasformg he oe equao o a mulvarae lear equao, s possble o oba he rue covergece values by performg a mulvarae lear regresso by usg he daa aalyss fuco of Excel. 8 Soluos for R&D o Desg
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