A Detailed Study in the Application of the Benedict Equation to One-Component Systems Citation Opfell, John Burton (1954) A Detailed Study in the Application of the Benedict Equation to One-Component Systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/wpe6-g155. https://resolver.caltech.edu/CaltechTHESIS:07232025-225306219 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Chemical Engineering and Mathematics) Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemical Engineering Minor Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Sage, Bruce H. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1954 Record Number: CaltechTHESIS:07232025-225306219 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:07232025-225306219 DOI: 10.7907/wpe6-g155 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17554 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 25 Jul 2025 18:50 Last Modified: 25 Jul 2025 18:51 Thesis Files PDF - Final Version See Usage Policy. 48MB Repository Staff Only: item control page

A DBTf:_ILED STUDY IN THC APPLICATION OF l'I:F: B8Jrf:0IC T Thesis by John Burton Opfell In Partial Fulf'i U m.ant or the Requirements to~ the Degree ot Doctor 0£ Philosophy California Institute of Teohnology Pasadena, California 1954 Acknowledgment Proi'sssor. B~ Ha Se-:::;e, Dr" Wo G. Scnling0i• 11 ·':o R .. Com1ell 0 Cc Je Pings I Jro 8 and Fe TG Selleck 6 aire genero usly of. t ime imd of ef'fort in the e;ceoution of the research program disc,.:('JSad ill gation into "the extent of the usaf'uln0s s of' t;he Benedict equs.tiou in applied th0r:r1.odyu£U~doso In certain po:r'tim1.s of this invest:1. 6 a= develop techniques b.d5.spsnsable to the pros0cu-cim oi' -that portion of the project d0fior.ib0d here., Professor. -U~ orr; a.n ~-itu·d TeYiewed certain portions r:i:i: the ,f!2.thems.tlos as soolute-:l ·lfdth tho least squares m-:ial~rs1so E·v-el;yn .Anderson typed the 1.m1nuscript:o Vir,~:l.11:la Berry and June G:ray prepared m.a.ny of the figureso The American Petroleum Institute supplied finanoial support for par-b TABLE OF C:J J TFJ{ TS TITLE PART PAGE I In.troduction 1 II Disoussiou of tl1e Mi:,1.the:nat:i. cal Hat'..1X'e of' the Banedioc Equation 6 III The Benediot Equation a.rid Its Sol ution 6 Expressions for Thermodynamic Properties 18 Fitting the Benedict Equa.'Hon to Experimental Voh1metric and Phase Behavior De.ta 22: Volu.,"ll.atrio Fit 22 Pit Along the Two-Phase Boundary 27 Fi ttin.g the Bene<iiot Equation at the Critical State IV Rasul-ts of the Investir;1:1tion of the Utility of' the Benedict; Equation f'or One-Component Systems V 33 36 Experimen te.1 :Program 36 E1tperimental Inf'ox-nation Used in Bvalu= ating t;he Coeff'ioianta 37 Disouss5.on of the ?it of the Benedict Equation to ·volumetric Da•;a 40 Prediotion of Phas e Behavioir 45 Predicted Critioal Properties 56 Conclusions 59 li:omanole.turs 61 66 Tables 69 F:i.gures 91 Appendix 122 ABSTRACT Lk"1 irnrestigation into 'Ghe abil:1ty of: the D8t1.edict ec:r;.w.:tion to describe volum.0tric and phas e behavior in one-component s:rstems of the light ps.:raf:fin hydrocm~bons !·:ias led to the follow ins concl<J,s ions o 1n desc::rfb:1.ng only t:1e volumetric be haviol"' of these com;x:>u.nds f or both -the gas and liqu:td phases, the Benedict equation pe:tforms rerri2,.,x•kahl~,e well "though the eri·o:r :tn the predictions of this equat.ion is c,_.,:,nsidez-0.bly in the :region adjacent to the critical. state. To 1!1£.~e t his equation predict accurate phase oohavior simultaneously with accurate volw.atric behavior for tr.:.e liquid and gas hus been found generally :1.rnpractiro1 :ln the absence of a modification of the equa"c,ion it.self' . tion is proposed. Such a modifica- Accurate predict.ions of t.he cz,itica.1 st.a te and of volumetric behavior fol"' liquids and compressed gases hav0 been found to be incompatible requirements for this eq_uationo Coefficients for the Benedict equation fo:i." t he calculation of volumetric behavio1~ to pressv:res of 10,000 p.s.i.a . for metha11ell ethane, applicatfon of the Benedict equation necessit ates t he use of z,nto1ru:ltic digital computing equipment. This limitation implies -tho:t a..11 equation with a sui'ficit3nt m.1mber of J)3l'amete:rs to permit t he prediction of volUIOOtric b:ahavior within 9:XperjJnental uncertainty would bo a mo:ce useful . starting po:'l.n"t fo:&" developing techniques for per·fo:rr:dne :rapid thermo- dynamic calculations using high...speed automatic dig:1:tal computing equir:iment. A J }::':LULED S'I'u DY IH TaE APPLICATION OF THE BEWIDlC'I' I. .ti.l.'1 equation for INTRODUCTION inter·po1.nt:l.ng volumetric properties of go.sea tion and (::liffe:reritiation of' t hese propcrti0s tl'i':en such opsrutions 2.:w:·o z-equired in ther,nodynamic calculations as well as St1'.:l.marizing conciseJ.y large u..rnounts of experimerrtul infor:r,ationo In t,ho am.i.lys:is of thorrno- dynamic probler:1s involving uriuticomr.icment s;7ste:ns, it s erves as 2 :JOint of int:roduction for the effect.a of composit,ion. Since high-sr-;ced. d.i r.~ ital compv.ters require ti1at the computation :routine be stat ed in j_:1:~ec:tss and non...int,u:!:tive terms, such an equation becomes almost indispensable for the applj_cation of these computers i n t he solution of thermodynamic problems. In recognition of these advantages, :Bene(lict, i.Jebb, and Rubin(l) hive proposed an empirical equation, 3, "Which mee~\ja raa.v of' the most p-ragroo:tic requi.;>ements i'or an analytical desm."':i.ption of volumetric L"'&= ha.vior. .wong these requirements are simplicity su.?fic:tent to make its appllc€rtion f easible_ :ln many- types of calculations without sacrif'icing accuracy in pr,s d:lcting the thermodynrunic properties at states or greatest LTKlt1st!":!.2l :interest, csnvergence to the perfect gae law, Equation 1 9 at infJnite atte nu at,ion, and continuity' in _ p • R'l'v""'l • (1) representing both the liqu.id rul'J gas states nt, tem;_YJ::·a.turas belou ·tl;.O critical temperut urs. Because of the encolli'aG:1.ne rasv.1 ts obtained in several preliminar;1 investigatiom(2,3,4,5,6, 7) ~a-to the efficdenczr of the Benedict-Webb-Rubin equation in describing both °l>"'olu.m-etric D.i,cJ phase behavior in light hydrocarbon. syste:;1s P 3. detail ed i.nvGst.i3ation lnto the utility and limitutiot,s of this equa'&im.1 .:.:.s an e~·1.c,;:L:::0e::Jnc ·tool hw:i lKien undertaken in the Chemical Engineering laboratory a:t the C2.lifm:::ia. Institute of Technology. The firs-t part, of this inv01stig-:1t.ion was con- cerned it1it!1 developing mac:l"h1e m.ethodS: 6, 7) i'o::.~ ho.ndling rout:tno calcu~ lations of t he coefficients* in -this e rnpirics.l equation of sta.to and o:': the thermodyn~ic p!'O}Jerties of materials for whicb such cooff'icfants have been calculated. The second po.rt, th0 subj e ct of tl1is thesisll h~s been concerned larg,aly wHh the study of the ef:f:i.c2.c;1 of this oque.tion in describing the volumetric a."1.d phase bohavior in one-component systena. A t hir:1 part of the i..'l'!Vestigatiou is concerned w:i:ih the applica:tion of the Benedict** equation in the. attomIJt to make c;;;;.lc1.uationci cf th0:c::1od;:mamic behavio:r :L, rnul ticomponent systems more 'tra ct.s.blo . A..'1 equation of state expresses a pprozimo.tely the relationshJ.p which has been found to e..~ist crnong the 4ntensive state proper ties of tempere:ture, pressm·e, and specific volume *** f' or systems corn.n::iser3. of specified moleculru., species. rh.ts relationship is called the vobmet1.,ic The term coefficient denotes those part,s of an e:;;:p:ression which are independent of pressure, ternpero:t ure » and epeci.fic ,101:ume o A coefficient may depend upon composition, molecu1a:r vmie:ht p or aymmet.17 of the molecule. • ** Throughout this thesis ' 1Benedict" will refer -1.io Bene<.!.:lct.Webb , and Rubin. *** Specific volume is the volume of a unit weight of material. of the equation of stnte fro ::rr r.1:i.c:rosco p:J..c cons:LTarations of statist5.cal mecha.'lics and the ld:'.letio theory of gases and of' 1:lqu:tds. From the engineering point of view, these ef'i'o:-cts h.avo beer:. rewa:s""dea with but moderate success to date and are pr.i,n8.r:1.ly of qm1l:tt2t:1.ve :l.ntsrest~ e:npirical equation of state, such us the B0:r.ed:lct 8C{z.,.a.t:i.on. 9 re2, resents an eff.o::..·t to describe the volumetric ooi-:i.~1vio:r a::.d thore:f.'ore tbo t ho:i."tnodynamic behavior for each substance in terms of universal elemcnt,x:.7 flmctions o.f the state properties and of co0ffic:l0:1ts wr.J.ch ch:,,rc.ctc1"ize eac11. molect:tlnr species. Values for t!.1.e8e coefficients mu.st be c.~r;r:l1jec1 from experimental measuro:nents of t he volu.mst:d.e beJ:w.vior c;~- f ~:·om e:-:, ccm--~ bination of heat capacH;r and Joule-Thomson coeffid.cnt i:nfornt,tion. 'I'he empiricnl. equatio:1, -then, roprose1ri:,s an inter;olation device c""'-ic. sheds little lig:1t or. tl:e bchavi◊r cf thermd.y:11:i.'r:ic sys-tens t ro::i. t ll0 n::tcro- scopic pcint of' -view. This llmito:tion is of _pract,icaJ.l3r no s2.snii'ica:nce in erigin.eering thermodynamic calculations hoWGVc~? . Ther:nodynamic calculutions reprr-isent a significant portion of the effort which engineers apply to pr.oble:is i."1 p9troleum r.ef.l,Jr,.roir mech;!nics, distillat,ion, chemical processincj) internal-co:nbustion and jet-propulsion engine desigrl, a11d in the study arid. c1pJ)licat:ton of t }ie tra.11apor·l; phenomena. Equat.ions developed prior(6) to that of' B,,0.M,i0 and Bridgeman(8) are restricted to rather lo'J pressures for J_Fecise application in thee~ calculations and even the Beattie-Bridgeman equation, 2, is of Tather modest accuracy for states near the two-phase*regiono • The two-phase region or the region of heterogeneous equilibri'Ull'l refers to those states at which a liquid and a vapor phase coexist. -4- p Host thermodynamic calculations in the past and. ye'i:, today arr.i parf'ormed by graphical techniques if' high accuracy is desired. Su.ch techniques are, however, very tedious &L-W . are quite involved for mu1:t:l.oom1x,.!lent systems because of the inpossibility of rep-resenting functions or more than two variables on a single plane. An 0quation of state with hif;h precision would be a most useful engineering tool. The Benedict equation is certainly not th3 fina.1. equation to be proposed though the degree to which it deser:ibes volumetric behavior for both gases* a."1.d liquids is rema:t"kn.bl00 Since the B::medict. equation is typical of the more precise, non-1:tnear, expressions vh:i.eh might be employed, a detailed study of its attributes has been nc~eded . ' Probably a r,olynomial or a rational function of polyl)or:d.ale will eventually replace the Benedict equation for high-s:peed d i gital con:p,::ter apl:;lications., but the present need is an extensive test:i.ng of the mnrJ? l-1,y- description of . volumetric behavioro --~nlile the followh"'lf discussion does not exhaust the topic, it does describe several tests which have be~m * Liquid and vapor refer to the dense and to the attenuated phase respecti,rely 1vhe:n they coexist o The distinction is un.11ecessary for temperatures greater than the critical temperature or for pressures greater than the critical pressure for one-component systems h3cn.1.'.S:J the state of a nuid can be ch~nged continuously without passing through two-phase etates. -5- The discussion s-tar-ts 1:ith a consideration of the Wlthe:.nat:i.ca.l nature of the Benedict equation and the consequences of using this empirical ec;uation in devclopi:.1,c thermodyrw.::nic ~:--eLrt:i.ons!dps. :'.'le:d, the evaluation of the emp:t:r-ioal coefficients is eonsiderec1 f:r'om 2. theoretical poiP.t of' Y:lew. critc::ia of tb.e C/..10.l:lty t:~ese evaluations ::re intu1ttvc so th::i.t their applicat:ton 1n pr::,. ctice is largoly throngh tr:te.1 and e:::-ro:r. ' to mako t:.i.e Benod:ict; ~quation L1eet seve'.\."t.1.l of thsne criteria r;re Dr0- sented 2..11.d c.n inte:r•pretation is ntte:':pted. -6- The development of' empir.·ical oqua-cions of' state bas h:1c~ o. long and active history. explieitly as a function of tempm. . nture a.nd molal volu!7!e o the solution of' these oqu.:rtions for molal volt.11110 has been r-:ossihle oni.1~ ·1n power series representations. A-t this point a discussion of tbis unfortunate asp,act of ihe Be:xidiet oquatioa wilJ. bo \.moi'\:l, ~0.n :lllus'Lrating the mathematlct'.l uo.ture c,:' this eq1-1F:J.tiou of sto..to. the mathematical similarit:tes o.nd differences bot-ween tho Boned.lot equation on the one hand and s ove1~a1 simple:,.~ empir:lcal cxprcGsior:.s on. the other will be useful as a basis for iooastc'inc the e~6;ont of :1rozress uhich the Benedict equation represell'ts. Tho IJZ-e,itest potrmtic..l Cl)l'.il:ioa- tion of the Benedict equat:ton is in the comr:tt1t2.tion o:f -ther,10:;.~71am.:tc behavior reprf'i)sented by the Benedict eque:Ucn anr: t.h0 oth0r t l":8r::1cd~71a1r.ic behavior is complicated a,s will be shown ir1 t,he f.ollcr,-Iing .CE~nlys:ts cf the mathematicel nature of the Benedict. equation., T!;\e Benedict Eguaf.;ion anc1.3ts Solutia!_}. form as given in Equation .3 but the term might equ:211ly- weE ,':'.p)lJ t o en;? •5 -) •(" •. . . . 0 c) ri ,- -: '~ + 6ua.90 • (? / 2 ) c T - ;;.'v'd ( y<;r8 ) .:.. + 3cT- •'Jq6 ( Y9'0) 0 +... T'I() ,.,..., = ( 'P p r:, .._Jo ' · ·L lL·--o - • 2 ,... . rri- ) vo- . <' • + 3 (bo'':1 , ,k • I') lt>aa.01.·4 0 a) <!o -~ • • ~ • I"' • <) . '.I: • (°) . + c t •·~go(;3(1 + Y'fo) • 9(yg(5)':; + 2 (yg'6)'-')e:q::i( .. ygo) 1 = (B 0 i:tT .. A0 .. c0 rzi-~?) + ~S (bff}~ • • ,1 .r.\ • • + cr2• '~ ) 9'"0 a n + l Gaa.'?""o - ( 21/2) 0T1• r., <lo ( Y9"0} ° ...J • •2 3 <TO ( YVO) + l ;?cT <J ·3 ,-, ·n ·,, 0 p 0 = (bE'l1 - a) + ~~Oaag-o + cT*"J ( 1 + 'rg-'6 - 1? h'g'6}.:, ·0 . ~ 0 A ? + C:•~J ~,) ( yzo)v .. (4/3){y90) Cl:) exp(- y9<5) ;:: (bET • u 0 - + c'l1-•:.) + 20aa.g8 - 0 -~ {C ) 2 (35/2) c •J:1 - '~ {y91S)., + ~·::Bc•I'- 2 { YC:"£5) 3 • (155/8) c'l1.., 2 ( Yc(Q) 4 + .. .- • • PO ).l (10) (11) {12) . ,(R9" 4 B0 n92 · + bn<?3) 7 c9 . ( ...c 0 2 .,,. + (-P 0 ... A.0 ~ 2 • ai3 + aag-6 ):i-l 3 (1 + yg2)exp(•y,r) )T• 3 = o (J..3) of Equ.aticns* li, , n" /I i :. aro ::;.'.';:tisf:lcd b;;r t ho S 3.:.10 se t s of v e.h:2s for :xressure 11 cc1l r:iolal volum.e, '7 . An explicit form for tl':.e tempera.tu.1>e rr.:tgr1t be obta1ned by solving t he cubic equation/I 1.3, analyt ically(9)o _:rowevcr, the resulting formulae uculd be exceedingl y complicated a.nd t heir forms wo:1lii der--end u1=-on t!le value of t he discriminant of th0 equation in each ca se t hereby restrict .• ine the utility of such forms. The preferred apy:roo.c:':1 would i ~wolve t he numerical solution of Equation 13 by· ite:catfo~ o:t by i"ormu.J.a(9)e Klein(lO)cleseribes a eraphico,l me·i;hod f or solving cubic equations of the type of' Equation 13 whi~h has proven to be perhaps t!~.a most u.s0f'ul met.hod of a.11. This procedurE: involv0s t he use o.f a plot of a eu.hic pe.r:::-,bola which a.pplies univ-ersally to o.11 cubic equatfons wh0:1 t l:ey a.re ree.uced to t }1e f'o:rm in which t::e term of second deeree is absent. This method is illustrated later when it is applied in the calcrtlation or predicted critical state properties. The situation with respect to Equation 9 ;,.ftich 0:z:p:::"0ssoe the relationship of the reciprocal rnolo..1 volu.n1c to the tem~')eratur0 2.nc!. pressure is somewhat less tractable. of The in.:'ini tc series r.cp:::'escnt at i on ([ is not summable into n closed analytic e:i::pression and f or purposes of computing values for q- when P and T are given, an itera.t:1.ve aolut:i.cm. -----------------------------~-------- ~ In these equations a."ld in m.ost of the remainder of this discussion the reciprocal mola.l volume, (!' , replaces t!1e s pec:tf.ic i.rolume as t he volunJetric state property. This substitution l P...rgely a.voids negative exponents in the equations and !)laces n i nf:1.nHe atterrua.tion:: at a convenient point on the graphs describing vol:umetric relationship~. of Equation 3 is much the simpJ.est approach. infinite series re:presenttng . r!". • is 1imited . The convergence of the to an i nterval alone an _isotherm of no greater length than twice the interval between Po a.."ld the branch-point where (oP /a9')T = o. FigUre 1 shows the manner in which the partial sums of Equation 9 approach the isotherm (sol:i.d curve) ·. described by Equation :3 for propane at 190 degrees Fnhr0r2heit. • S~ce ~.11 isotherms for temperatures below the critical temperature have two branch-points, one infinite aeries in P cannot represent both the liquid and. gas phases. The interval of convergence, about Equation 9 is indicated in Figure 1 • .At the :right end of this intei-val the sum of the series oscillates infinitely and at the left end it becomes infinite monotonously • . At least two such series would be re' described analytiquired and in general the volumetric behavior can he cally only b-~ the application of the principle of analvtic continuation(ll,12) if the equation of state is to ho explicit in reciprocal molal volume or in specific volume. The crux of the difficulty in -writing equations explicit 1n the volnroo functions lies no-t w5.th the Benedict equation but with the multivaluedness of these functions with r espGct to t wo-pho.se pressures. Polynomials and integral functions are single-valued every- -where for finite values of' .t he areu:nent and thereby cannot represent the dependence of tho voiume f'tmctions upon pressure though t hey serve admirably in d~scribing the dependence of pressure upon specific volume. Equation 4 reduces to Equation .3 when q0 is chosen to be zero. The point g--= g-o is the center of the interval of convergence of the power series. Since Equation 3 is .the closed form corresponding to the power series represe:itation of' the pressure as e function or reciprocal infinite attenuation, it desc:doos a regula:r f'Tu"1ctio n for n0t:,o.ti1re valuel!Z of (!. Such a .funct:ton is of' no interest thermodynamically since negative values for ~ defines a function · P0 nre physically :tmr:ossihle. Thua, Eque.tion 3 '!Jhich corresponds to the t hermodynamic quai."1tit3r preesu.re for a restricted interval of values of er only. The nlrlher observa-~ion might ba made that Equation 1 implies infinite compreasibiHj;y. In actual [:;ases and liquids a. change in phase to a relatively incompressible type or state probably occurs foT some large value of reciprocal molal volume. Bridgrn.e.n(l3), for example, has f'ou.'1.d wate--.c to exist as e. solid at . temperatures well above the triple-point temperu.ture when the pressm·e becomes sufficiently higb. If the equatfon of state ia efficient for pressures in the interval O ~ P be 10,000 pos.La.., it may be considered generally satis.factocy for most present-day applications in byd.l"Cc- carbon processing calculations. To obtain the temperature depepdent funct:lo:ns s-'1 foT ·i;he reciJ.:iro.; cal-molal-volume-explicit form of t he Benedict ecruation, the reciprocals of the poi-mr series representations of the -tempere.ture dependent functions pi are obtai:;100 by the methods outlined in App:3ndi.x A. Also the ap:pro- priate r-owers of these functicne are obtained and the series of Equation 4 is reverted in the ma.n."1er discussed. :tn t,his S&'ile appendix. Equation 3 is a sufficient description of the volumetric behavior since it can be . sol\red iteratively for temperature or reciprocal molal volume when the need arises. -11The Power Series Eguatio9. The Benedict equation is i> in essence, 1cm extension of the Beatt.ie-Bridgeman equation to permit more accuracy for pred:lctim-is of volumetric properties at t h9 higher values of reciprocal molal volume. The relationship of the Benedict equation to sev~rnl of the simpler equations is illustrated by ~omrr..:i.ring the rower series 0"'~pansiona of these expressions about the point at ·Jnf'in:!.te ~:•. ttemu:>.:tion. Perfect gas law: p RTo- {14) • van der Haals equation: p (15) = RTa •. (16) + Beattie-Bridgeman equation: p (l?) + (-B0 bRT ❖ ~a ... cB RT- 2 )c? 0 " Benedict equation: p RTcr 0 + (bRT - a + ••• The general powor series rep:ressntation of the equation of state ie called the nviria.l aqu0tion of state 11 ' • aJrmbolically as in Eqt1ation 19. This equation 1t1.ay be written (19) -The quantities B are called the virial coei".ficicmts (15} and depend 1 ur,on tempera.turs and composition but not ui::on pressure or t he vo1u~e functions. The virial 9oefficients have proved to 'be e:i; tremely useful in making theoretical st,udies of the behavior of fluills since numerical values for the first · three virial coef.ficien~.-;a can usually be determined from ver; careful experimontul mee.surcmeuts. If Equation 19 is differ- entiated e.nd evaluated at in.finite attentua-tion t hen (80) But since all real gases approach perfect gas behavior as the !Jl.~essure is reduced without limit, it follows that RT (21) Further, at infinite attentuation Oo11Bequently, = (25) lmd ( '.2 4) D....,.) ";'), The tact that per cubic foot, 11 Bj/2 is of the order 10 when ·,:r is in •.oou.n.d-moles ' . P is in pounds per square inch absolute, and T is in degrees Rankine.makes the accurate calculation of the third virial coefficient a particularly delicate oper.e.tion at best and prs.ct.icall,:,'T preclud~s the experimental deierirJ.na:tion of the virial coefficients of higher order from extensions of this method. The second virial co- efficient may also be obtained from measurements of the Joule-Thomson coefficients when they can be extrapolated to in.finite D-ttenno:tion along isothermal paths. From the familiar relationship V " it follows that --Substitution of Equation 19 L~to Equation 27 leads to (28) when evaluated at infinite attentuation. Equation 28 will give more accm-ate information with respect to the dependence of :s,2 u1xm tempera- ture than will Equation 23 primarily because the experimental measurement of the Joule-Thomson coefficient at low pressures is generally more precise than ia t he measurement of specific volume. The arguments of' statistical mechanics O5) have led. . to vi.rial coefficients of the form where the quantities µi ru·e the 11 irreducible cluster integrals a . ( 30) where u (31) Equation 31 represents the Lennard..Jones(16) hypothesis concerning the nature of intermolecular force fields for non-polar. molecules. Stockmayer (16) hae proposed a modification of the wnna..T'Ci-Jones function for describ- ing the i.~termolecular force field for polar molecules. Rice and -15- Eirschfelder(17) have recently calculated values' for the reduced second vi.rial coefficient. At low densities Equation 19 implies that only the first ar1.d second virial coefficients are of irn.po:!..~ance. Since all reason:=.ble equations of state will satisfy Equation 21, a comparison of several proposed forms to describe the dep,zmd0nce of t he second YirioJ. coeff l- cient on tempera tu.re will be of L11terest. Inasmuch as t he .J>0wer seri8S representation of' an analytic function 1.s unique (18), the coefficicn-t.,s of ..., cf~ in F.quations 14, 16, 17, and 18 are upproxim.'),tiom.J of the samf) . quant:lty if P is to be identified with the thermodynamic pressureo Proposed. Rep:resentations of the Second Virial Coef'fiqjent Eguation Perfect gas B2 zero· van der 'Waals Beattie-Bridgeman Benedict Martin-Hou *(19) Hirsohf'elder** * Primes have been added to the notation of Martin and Hou in order to distinguish these coefficients from the virial coefficients where the symbol Bis used Yith a different meaning. **1urscl'>.£elder, Bird, and Spotz(20) have evaluated F(E /kT) numerically for the Lennard-Jones potential function. -16- T:he virial equation assumes that P is a regular f unction of q when q :: o. The validity of' this a osumpt:lon is not subject 't,o either experimental or the:..~::.od:rnamic proof. The virid . 0qua:tion cannot apply -~o the liquid phase as it is written in Bquation 19, si nce the ser.-5.es must diverge for Cf> q-d for el\)r teD1per:J.ture. Figure 2 shows the second viria.l coefficient for pro:pane a.a calculated from Equation 23. The values predicted by the proposed approximation functions are also included. several of The significant fact presented by this .figure is the approximate l:i.nearity of the second virial coefficient with temperature for relatively low va.lu(~S of reduced temperature. This is also •indicated by the cc.lculations of Hirschfelder (6). Hirschfelder, Bird, and Spotz(20) show that the deviation from lineaxity is significant only at temperatures a.bow, the Boyle po:1.1.1t temperature. Ji..'ven for methane at l,,60 degrees Fa.lrcenheit the deviation is not very great. The predictions of the approximation functions are not as good as might be desired. :t>hye.:.~ and 11uyer(21) shoY t hut t he virial equation io the power aeries exr~nsion about the origin of p = where Q is the partition function. Cm1.s equantly, the appropriate power series expansion for representing the pressure of condensed states would • be the expansion of Equation :32 about some '!" > <fb • The virial equation can be considered to be the Laurent series expansion or • P about the singularity at y = O ,, The difficulty of assigning a precise experimentally-confirmed value to the limiting specific volume of a substance as the pressure upon it becomes large uithout. limit introduces some uncertainty into the valtdity of the virial e..~pruision. ~he effect or the assignmsnt of the location of the mL,inro.m molal volume upon the predictions in the gas p.'lass is small, but at temperatures where the virial expansion converges for compressed g~ses it may be of greater importance. The eJrtension of the methods of statistical mechanics to the deduction or the equation or state for comp-~essed r,ases and liquids has not been developed to as satisfactor;,r a point as for dilute gases. Hirschfeldar (16,22) has calculated values for the liquid compressibility factor from the reduced equation of state as deduced from the nrreevolume theory" and the theory of Born, Green, and Yvon. , One objective of the statistical mechanical develop:nent of the equation of state is an understandL""lg of condensation and c:ritica.l state phenooona from the microscopic point or view. The need f o::.· such &"l understanding 13 not apparent for most thermodynamic applications of the equation of state. Thus considerable freedom is gained in the choice of approximating function if the convention(23) is adopted that the volumetric behavior in the.two-phase region need not be predicted accurately or even approximately by the empirical equation. The retention of two restraints- continuity and 1 1/p.d tW -, : P( Yd Yo -is desirable. Under this ,convention, pressure ie a continuous, differ- entiable, single-valued function of either specific volume or specific ~reight. - 18- Empirical equations can, then, approximate t he volumetric bohavior precisely if only enough coefficients are allowed. The statistical mechanical application of the viria.l equation to gaseous mixtures involves certain assumptions(?-4) concerning the nature of the interooleoula.r force field(20). The empirical approach should similarzy allow for the deduction of the volumetric behavior of mixtures from the behavior of the components. Tha statistical argument (24) confirms the possibility of making such a deduction for dilute gases but for compressed gases arl.d liquids the statistical argvroont is not appropriate bocause of the convention discussed above. Benedict, Webb, and Rubin(2,3,4) and Connolly(6) have shown that in. many cases quite good predictions of phase behavior in mixtures are possible with the Benedict equation. This application of' the empirical equations needs a muc..½ more extensive analysis than has yet been supplied. Expressions for Thermodynamic Properties. Many of the moat useful thermodynamic properties or a multi- component system can be expressed algebraically in terms or six . canonical .functions which depend upon t ha equation of' state a.lone. Appropriate expressions for molal chemical potential, partial molal volume, partial molal enthal~J, partial molal entropy, partial molal internal energy, and the Joule-Thomson coefficient may be -written in terms of these six f'unctions. terms or the Ben~iot equation. These functions may be expressed in Appropriate expressions for mixtures are obtained from the partial molal properties by substitution into the tam111er equation -19- co.non:1.c nl :functions are t.h o follo·wlng : P V • ( 35 ) (? I' /o<[)T ,m •:lH(- c::"lP/0 rp) • .,. ~ . • D r;1 • L -~ - v,m l"1"l• ( 3?) f·'tC ( 38 ) • o,t} ~k . ... n V'J:r" ~• k , (40) 1"1 ·"' ] ,. .!.k t:: [ rnn1 ,.t :::i yUY:. z(o ln f /av)T,mJ/[ {cP/ocr)T,m] . ' 1 ( 41) • • H ln I)•::- • .. OL (c§P7cvJ ,-....r 1·11i , • ~ T•l ~[HT l n fk] -+ [ n•:;, 2 (2 ln fk/?T)q, 1nJ + [ :1Tcr ( cJ ln r_1r./?~) r;, \'I,) [ r::·✓ (2 I' /c_ T )" ..,., ]_ • ... • .;,. , -'-'" • .., , il, ! 1-- " 7c:: C O-) 17: ...,.,J' L \ Cl' • - - j - ,~u (44:) ln :e k/0 g) rr ,.lfll (45) (46) (47) (40 ) { SO) (51) fl b e ( 5,1, ) ct ( 55) = y ( 5G ) uhe oe The 1 0 0 d-ok ,. l = ~ AoAok a1~ ·3 f L ol~ = . l'.> ( bfc - ~ 2bk .~0 + ;_;ok n )/0 ,., i:.:: a""ak. ,vok ... ' = "/·:;oc ok Ck = (li = ia. 0.k 1 'fl'C = 1y 2 , 2 I c .... ck Yk :::; bT{r2~ ;~ . .,\ u... o) + ··<· ,... , '") ;.'., • I'") + V Dkl".-1' .. air: '! . + ' 0{ ..!{1 c:i) + ~r--r:l -& 1: ~'J'7"' Cg; + Yi,:;., (62) - ~ ,.;;?) 2 ,.,..1 3 (A t}t/ly) e:ico _ {• "I'~) I. + }tl1 - j. 3 ·~ + 0 a,t..cr.-:, (. :::; ( A6k + ·~ 6 ( ac,'.. + aa1t ) O""' • ro .!:>.• .. 3r;1 (3c 9 3C bkT ... ;~ ) ~ k'~ 2cy ' /y) ... .iC e f~ + (c(l + ycf-?,)y}.J-y( 3 ct ... 2c ·rf.c/ y ) .... { '·( r; ) - 1 .., ( 1/~;) ) ~ exp ( - y f: ) } For one-component eystec11s all of the above equations apply if ·the primes are omitted. Under these ciroumstanoes Equaclon 41 reduces to an i'denti ty and the partial molal properties become the molal properties of the system. ation of the coefficients Ao and B0 fo-r van der Haals 1 equation., 15i simulta..'1eous solution of Equo.tions 79 and CO. By no means does such an evalue:tion z·epresent the on.1J possible one or• even the only useful better be evo.lua:lied from kno,rledge of' -c:,,:; soco;;;.d viz-:lal coefficimt. A similnT sts.t0 of affairs preva.-1.1~ witl:. r.es1-x➔ ct to tb.e Benedict ocp.1ation with tho result -that, the a.pproprinto methods to use in the evG.J:u.e:tions of ts1e coefficients depend to a large me-tent u:pon the intended appli- cation of the equation r;,f state. For p:cesent pttry-,0ses 9 tho assumption is m.J.do that the Benedict equation is to be used to describe tho volumetric behavior for the dense as well a.s t.h0 c.ttentuatod states or hydroca:rbons, cli.'ld similar fluids, for pressures up to 10,000 i::ounds p0r square inch and temperatures in the interval ·jrom 70 to 500 degrees Fa..iu-enheit. That the Benedict equatfo:a is t.o be applied in phase behav- ior calculations involvL~g multicomponent systems is also assumed. Volumetric Fit. The Benedict equation does not describe the volumetric behavior or fluids within the precision of the experimental knoi.rledge. For this reason, ~tatistical argunents . in discussions of the fit of this equation are not appropriate inasmuch as the <lewiatione of the expe:rimontal values from the v:alues calculated from the Benedic·t 0quation are pre- dictable. If the Benedict· equation, with eight coefficients, is asev.m.oo to describe the volumetric bermvior of a fluid, then 1.UOre than eight observations of the volumetric properties over-specify the values for these coefficients. The equations which represent these observations in terms of the Benedict equation are inconsistent w:ith resr,ect to the assumption that each coefficient has one value uniformly fey, this eet of equations. In .Appendix C several criteria for the estimation or· satisfactor-.r values of the coefficients are discussed. The selection of the least squar~s criterion for the present analysis ima on the basis of the usual pragmatic consideration of m;.merical convenience. The method or least squar~s involves the minimization of the sum of the sq.ua.rea or the deviations of the experim-antal values from the predicted values for some function, or eombiriatian of functions, whoso analytic expression contains ull of the coefficients. The selection of a particular !'unction ie usually on the basis of obtaining normal* equations in which the parameters enter. in as riearl;y· a linear maYJ.nor o.s is feasible. Such considerations indice.te very strongly tlmt the e..pµro- prb.te variable to fit the Benedict' equation is the compressibilit;:r factor. The coefficients in the equation for fugacity of a one-compon- ent system, Eque,tion 61, are as linear as they a.re for -the comp:l'.'CS3i=- bility factor but values for f'ugacity e..re not so conveniently obtained * Cf. Ap~ndix C. from experimental . measu.rer:1emts. The comp:l:essib:I.lit,y factor :ls fr1c:}e of dimensions which pr-ovides considerable numerical convenience. From values for the reciprocal molal volume, obtained by iterative solution of Equation 3, and temperature, t he compressibility factor may be ob,.,. tained from the following form of the Benedict equation. 1 + Equation 3, houever, presumes that t he values of the coefficients are known. The use of pressure, then, as a.n independent variable in the least squares calculations is too tedious to be feasible. The desir- ability of' using expreasione more complicated tha...11 tl'-..a.t expressing the dependence or the compressibility factor upon the reciprocal molal volume and the temperature is debatable. If the results of the appli- cation of this expression in the least squares evaluation of tbe coefficients of the Benedict equation are not as good as might be desired in a certain region, then a convenient method of improving the situation would be through the imp,sition of a rigid limitation on the deviation permitted in this region by application of the method of the Lagrange undetermined multi pliers. The appl:i.cation of this method :i.a d:isouseed in Appendix C. The qualit7 of fit of Equation 64 with a particular set of values for its coefficients is reflected in the magnitude or the standard error of estimate(25) of the compressibility factor on temperature and reciproC$l molal volume. . The square of this term is called the variance• or • ~ 2 L (~N- Z) j of experimental _observations. * Variance of . estimate = ·1-::.1 where N is the number estimate of tho compressibility factor on te:n1:J0rature and reciprocal molal volume. These terms will be shortened to simply "standard error of estimate." and 11 vc.riance of estimate" in the ensuing discussion when temperature and reciprocal roolal volv~rne a:to considered t.o be the independent variables which define the state of the thermodynamic system. , These measures or f:i.t are the analogs of' the statistical terms 11 standard devia.tionn and 11 vu:riance 11 • The detailed character or the fit crui..not be described by a single number since it must necessarily represent the result of an averaging process. If the fit is generally very good; :tt may still be quite p.1or in a small region without mak:lng the standard erro:r o:f estimate very large. The standard orroT of estimate gives no indica- tion as to which values of the independent variables lead to the poorest or to the finest predictions. If the exp-srimenta.1 data which a.re used in the least. squares evaluation of t:1e coefficients are concentrated in ' a small region, the standard error of estimate may bG very Si!!all 8.!."1d yet the fit of the equation f.ik1.Y be inadequate for gen01·2.l 1;.pplication. The method of least squares is effect:1.vely an e.nalytice.l device for minimizing the variance of estimato o-ver all poss:i.ble sets of values for the coefficients and it is from this cha.racter:tstic tlw.t it derives its name. The details of' the lineal" pa.rt of the les;st squares a"lalysia or the Benedict equation are presented in Appendix D. The complete linea.rization of the Benedict equation has not been feasible for present purposes though the substitution of a rational-function(26) upproximatfon • for the exponential function in this equo.tion would have much merit from the computational standpoint. The fact that the coefficient gamma enters the normal equations in a non-linear manner ret: '1 1res that the normal equations be solved i.tera"ti.vely. The iteration is st.raightf'o:rward and ,;'( involves t::e computation of -the standard arror of e~rt:f.: ,t0 for several (. ~ values of gamma and the selection of that value uh:lc: . corresp:.mds to the rainimum. A plo-t of' the standard error of estimate for propane as a .function of gamma is given in Figure 2 of Appendi.'( F. Brough(27) has shown that the standard error of estimate must approach a limit as eamrna becomes large without limit and th.at the standm"d error of estioo.te has a relative stationary roint at gamma equal to zero. For an equation to describe the volv.metric behavior of low pressure gases, the coefficients should be evaluated in pa.rt at least from experimental measurements of the Joule-Thomson coefficients. From Equation 27 The quantity 9p (aT,/aP h;,m can be deterwined from integration of ::: and knowledge of the heat capacity, ~p• as a function of tempornture at some pressure. Substitution of the expressions for tho canonical functions from Equations 58 through 63 will yield an expression which is linear in all the coefficients except gamma and which may be used to obtain the coefficients by the method of least squai:-es. To ma.kc a least squares f'it of the Benedict equation which reflects the Joule-Thomson coefficient m.easuremetits as well as t.he volumetric measurements requires a high degree of intuition. In general the elements or the normal 'matrices for the volumetric and for the Joule-Thomson data. are . of different . ordors of mag:1i tude which means that linear co:nbin&tions of these matrices, such as in the method of averages, would tend to emphasize the effect of the matrix with the large elements. The t'l;lo matrices jointly correspond to a system of fourteen simultaneous, linear, inconsistent, equations. The method of least squares can be applied to this system of equations to reduce the number.to seven and thereby permit values for the coefficients to be obtained. Such values have, however, no particular merits over values obtained .from a system of seven equations selected at random from the fourteen except that the least squares evaluation uses an unambiguous procedure which leads to a unique solution to the problem. Such calculations have not been applied to the Benedict equation and will not be discussed further. The methodof-averages approach to an analogous situation which sought to make ·the Benedict equation predict phase behavior in the least squares sense, was singularly unproductive. Fit Along tl1e Two-P'.aase Boundary. ·ii; The equation of state must predict the phenomena associated vtth heterogeneous equilibrium with a .reasonable degree of acc1.1racy. In general, this requireme1,1t presents considerable difficulty from. the - mathematical point of vieT..r. A one-component system a.t heterogeneous equilibrium has one degree of freedom .in the sense or the phase rule. The fa,iiliar a::i.a.lysis 1,"ad:lng to an (~qu.a:tion fo-;: the vc.,.or prcssv.:rc curve from t:r..e Clausius-Clnr1B3r:r·on equation, Eq1,ation 66, d ln P 11 (GG) en, description of phase behavior :1.n one-component systems. If tho Benedict equation with the critsiria of heterogeneous equilib:r:l.um between phases g ar..d £,. , \.!hich t:u:-e n, l:g = Pg C r·r, L l (67) P.t (68) fk,g = fk,i (GG) is to describe r-1hase behavior, the volunetric fit cc, 1 he no better than the best six parameter equat:i.on. The accurate descr:i..ption oi' the volumetric beh1vior or itself is not sufficient for an equation ouch as the Bon.edict eq;_;at:i.011 to permt the aceurute prediction of' phase bohav:':.or or of the t hermoclyna.rnic properties of the liquid phase. Assume that the Benedict equation can accurately describe pressure l'.S the same function of temper::ture and molal volume as that observed experimentally in the regions of homogeneous equilibrium for one-component systems. Then, the temper~ ature and molal volume derivatives and all eombinatfona of pressure, temperature, and molal volume with these derivatives may be predicted accurn:tely by Hie BonecU.ct equation in t his region. Thus the relation- eh:1.p expressed. i n F.,quution ?C determines the slope of t!1e fueacitymola.l volume isotherms accuratol;r in this region of single-phase states. p {70) Equation 70 may oo integrated to obtain an accurate expression for the rntio of tho f'uga.cities at two states lying at the extremes of an inte:rval along a pressure-molal volume isotherm uhich lies entirely within the single-phase region. Thus, in the liquid region, th0 tuga,oity at any stata, for which the temperature ie below the critical temperature, is known accure.tely as a multiple of tho bubble-point fugacity at the tempeTature of the stute. Equation 70 may be rewritten in the following mannero + (71) This equation me.y be rodueed to - 1 (72) ff~ Since the quantity (RT/P .:. V}, called the residual molal volume,. is • defined and is know accurately all the way from dew-point to infinite ·atte~tion for temperatures less than the critiea.l _temperature or for •~ll riiolal volumas for temperatures greater than the critical temperature, The presumption is r.>1e.de the,t the Benc1ict eq_uat:i.on does not. correctly portray the p:ceaaure :is u fu.11etion of t0mpe1"ature a"'.ld cf molal volume in the region of heterogeneous oqui.1:foriu.,_"J. region E,;.ue.tioris 70, 71, ancl '7~ are not a pplicable :o..f In such P :i.s ·t.:.10 sion for (V/RT)dP f-rom Bque:t:ton 3 is irrteg:i:~ated fl"'om c. gaseous state along a.11 isothe:rra -through the region or heteroceneous equilibrium to a liquid state, that part of the integral contributed by the two-phase states must be subtracted from ths total int,egral. In actua.l g-ases the contribution of these states is precisely zero s1nce t he dew-po:i.nt f'ugu.oity equals the bu.bble-!X}int fugr.dty. Thfo contribution to tho intecration of the expression. derived from the Benedict equntfon :ts a function of temperature lLl.one. Cz.lJ.. this ft'"''lction F(T). F(T) w:7.1.1 be precisely zero in the c~seous = P(T) {73) 1n f, end consequcntlJ,, f, ui11 be predicted aoeurately in the region of liquid s·La.tes only 1..f tho f'tmct:to:-1 F{T) is The preceding J..,,'1a.lysis can be re"tTer30d o.r:d tho Benedict t)q_1:,ation may be presu.m.ed to predict the fugaci ties of the states at honogoneous eqnilibriut1 e.ccurately. before. Then Equations 70 and 72 m.e;y be integrated. as In integrating across thia region of het.erofc.meous equilfbrlum • the Benedict equc,t:i..on will predict un :lncreo.se in t.he :~·,ressure by an (f b p, (T) (74) RTF-" (f /f") <l<; % Unless too Benedict equation :ls of such a. 11ature that i·t predicts pressure to be independent of specific volume along isothe?mal paths in the region of heteroceneoua equilibrium, it will not in general pi--e- dict corract liquid phase f'uga.eities if fitted to P-V-T data or, conversely, pr 13dict correct liquid phase preestires if fitted to f-V-T data without special adjustments. :rt is, of course, conceivable -that the Be:nedict equation would predict F(T) and F 1 (T) to be r.:recisely zero for all te:D.pe:ra.tures but this ca,"'.l oo coincidence only. The direct e:rperimen·tal measurement of f.'ugaci ty as a i'unctim-:i' of !"lolal volume and te:J.perutur,a has not boen possible to_ datc.1. For this reason the Boncdict equation will usually be fitted to P-V-T mea.aur9ne11.'iis a.,d the cor·.r.ection fucacities. F(T) will be applied to the predietod After such opo:rationa, on tha Benedict cqua,t::.0:1 it will neoossru."ily pa:~mH t;hc accurate pz,ediction or vulu8s for the ca."'1.onioal fu.nctfons and therefore also the thermodynamic properties of ono-oonponent systems. The Clapeyron equation atatos that ('"/5) Thus the latent heat of vaporization is predicted accurately if tho phase behavio1~ :i.s predict,3d accurately. The Cla.P9yron equation ei'nphasizes the necess.:i:t:r for accurate volumetric predictions a.s Yell as accurate fugacity predictions if' the enthalpy predictions a.re to be accurate in the liquid phase. Equations 67, 6B, and 69 represent three restraints which mig..1-it be imposed upon the solution of the normal equations in order to make th~ prediction of the phase behavior agree with the experimental observations exactly. However 9 the possibility of imposing such restraints to obtail1 perfect proo.ictions or phase behavior is only illusory. The dif'fioultios involved arise first from the fact that ·Equation 9 which expresses the dependence of' reciprocal molal volume upon temperature and pressure cannot be written in closed form in terms of alementar'IJ functionsof pressure a..'1d temperature. Substitution of' EquatioP..s 67 a:-id 68 into the power series representation of reciprocal molal volume, one series each for the dew-point a.i."'1.d bubble-point, yields a pair of functions ea.eh of which may be substituted into Equation 61 to obtain an expression for the logarithm of the fugacity as a function of tempert:iture and pressure, Such a representation for the logarithm of the f'ugaci ty cannot. 1)6 summed into a closed form comprising ala- menta:ry functions of pressure and temperature. When such representations ror the logarithm of the fugacity are combined with Equation 69 there results an infinite aeries .i n pressure~ P, and temperature, T. But, vapor ~ressure CUl""'J'O 1-s i1sc~l to :replace P in this infinite s,3rfos, the final I·esul t will be -:::.n :tnf:trdtc series in T alone. expresses t!1e fact t!:-rs:!; this se~dcs :ts invaria.nt w:tth respect to temperature . The fact that the coef:':'icients or ......!. in this sor:i.es mu::rt he idcnti.c2.lly ze:;:o 1.n ort0r that the series ho tnva.ri.::J.nt forns G. set cf restraint;.; upon t..hc Bonc::dict cceffic:tonts of wldch they arc~ co:npr:.'Lt1(cd. Such eoefi'icionts of T a:re :1.?1~opondent or t he t horr:10dyna.mic variables; however, the~• are at the same tine t'.10 sums of :!.nf'":i.nite nur:ibers o:? terms uhich cm:mot he exp::-essed in simple clcsed fo:rm. Hh:i.le such c, m8thod of nd:ins the Benedict oc;:uat:i.on pl"ocict phase behavior may 2.py::•c cr optim:um i t is s.t tho same time 1.mr.osc'.i.ble . Fitting the Ben~~~Lw.ation at t he Critical St~. Exr,eri:i?9nte,1 m3ast,:'e:,t:mts ho.ve abundantly oonfi.l"111ed that the fo11owins prop-3:r."t.1.es chr:.c'.:i.ctcr1.ze t.he c:d.tica.l state for a. one-component = 0 'l".I. r• ) \, u •? ( o~2 f.;~ o g--J)T,m = 0 (77) ~ P / 0°'.:'lrp ) • ( c~ •• = dP 11 / dT {78) (cf /cq-) 1' ,m o-,m • In Appendix G Equations 76 a,~d 77 are shotm to be equivalent to Equations 79 and 80 for a one-cornr,0nent o~rstem. 0 ('79) 0 (80) Since a one-component system has only t\,ro degrees of fl.•~edom, this set of equations is necessarily inconsistent or redundant. U is redundant for locating the critical state of' a real gas and inconsistent for locating the critical state predicted by the Benedict equation. For convenience, E.quations 79 and 80 will be assumed to define the critical state as predicted by the Benedict equation for a one-component system. This choice i$ also indicated b-/ the isothermal integrations involved in the computation of tuga.city using an empirical equation of state., For the ~nedict equation then, 1 a:v ~r ·Lt.' (~)m m Cfj J.t, = 1 + o.r.•: ~ lu"--1 '; + 3K4q2 + + :::: ov ~n•l ,.,;n.2'¥' .L + 3K592T•l + + cy;~ ~T-3 •Jl'-3 • m~6't°5'l '-l (Bl) Kr,?T- 3 (3(1 + yg'2) - 2(yg2) 2 ) exp(-y~2 ) 0 and + + = 0 I{2p•l + -•·.,. .• 4m•l 1 01':.59" 1.· + 9 (y<12l2 • 3 K3T• + ,. ... 3 I"'?<fr 3IC49" + •0 (3 ( 1 + Yg--') (82) -35- The method of the I.agrange U.."'ldetermined multipliers may be applied to make the Be.."'ledict equation predict the effects described in Equations 81 and 82 at the critical temperature and at the value of the· reciprocal molal volume corresponding to the observed critical state. These restraints could be made to apply at the observed values of' the critical temperature and pressure by an iterative process which would require successive estimates or the reciprocal molal volume predicted by the equation for the observed values of the critical temperature a_~d pressure. Restraints in addition to those described in Equations 81 and 82 might be applied in the same manner. Such a procedure would give greater accuracy to the region adjacent to the critical state at a relatively great cost to the fit in regions removed from it. Martin and Hou(19), hol-rever, imposed seven res,t re.ints at the critical state in the evaluation of the coefficients or their equation and obtained an excellent fit for the low pressure gas phase. Clearly the use of so many cr:J.tice.1 state restraints on the Benedict equation coefficients would preclude the possibility of describing the volumetric behavior of liquids ttlith this equation. Zimrn(28,29) baa supplied an argument in aupp:,rt of a postulate that all isothermal derivatives with resp.~ct to preesu..~ are zero at the critical state for a .,one-comp;ment system. Thus there exist presumably an infinite number of :r;oesible restraints whieh one might impose at the critieal state. Consequently the particular choice of restraints on the evaluation of the coefficients muat be intuitive. -36- RESULTS OF THE INVESTIGATIOU OF THE UTILITY OF THE BENEDICT EQUATION FOR O!tE-COMPONENT SYSTEMS Experimental Program. The present investigation of the applieation of t he Benedict - equation in precise thermodynamic calculations has considered three separate attributes which the equation should possess to a sufficient degree. . Firs·t, the equation should describe the volumetric behavior of tne fluid states in both the gas and liquid phases as Yell as at states . . wll within the region of homogeneous equilibr,ium. This phase of the investigation involved the evaluation of coefficients for the first five light paraffin hydrocarbons from volu.rnetriq information ~lone. The effect of adjusting the data background or the calculations was investigated in detail for propane and the results of this investigation ere discussed in the reprin.t included as . .:\ppendL"-c F. The optimum values for these coafficients are presented in Table lo ?able II presents t he values of the coefficient which were proposed b<J Benedict nnd Table III supplements Tables II, IV, and V of Appendix F to present, all of t he sets of coefficients which have been evaluated in this i nvestigation for purposes of predicting volumetric behavior. Those sets L.'1 1'able III which have been lJlcluded in Table I a.re i ndicated bi; an asterisk. Second, the Benedict equation should be mude to predict the phenomena associated with heterogeneous equilibrium with small error. Some or the difficulties associated with this phase of the-investigation have been discussed in the preceding section. An unsuccessful attempt wat made to formulate an analytic description or the pr.ase behavior in terms of a set o.f coeff:tcierrts differing from those describing the volumetric behavior. This course or ao·tion was sugg0sted by the difficulty of mald.11g the Benedict equation predict correct liqu:i.d phase fugacity when high pressure voltm1et:1:--ic data were :"mcluded .in the evaluation of t he coefficients. The addition of a,1other term to the Benedict equation to improve the vaix>r pressure predictions has been investigated for propane. Third, the Benedict equation should be made to exhibit the characteristics of the critical state, Equations '79 and 80, for a temperature and a pressure reasonably close to -tho observed values. The effect on the volumetric predictions or making the eqmrtion satis~J Equations 78, 79, and 80 at the observed Cl"itical molal volume and temperature has been studied. Experimental Information Used in Eyaj.uating the Coefficients. The volumetric and phase ber.1avior or the light paraffin byd:rocarbons has been carefully measured. These mausurements were made at states which represent most of the regions of temperature at"1d pressure which are of industrial interest, i.e., at states w:1cse pressures are below 10,000 pounda per square inch absolute a11d whose ternpe1..atu.:res are in the intar,ml between 70 and 500 degrees Fahrenheit. With the a.id of.' the concept called 11 the law of corresponding states" these measurements have permitted the general nature of the volumetric and phase behavior of these substances and their mixtures to be inferred for oost of their fluid states. This experimental information is not of uniform density -38over the region of fluid states and in many regions it is e.baent altogether 1.;hile in others the several investigators have duplicated each other. In these regions where tha several L"'lvestigations overlap, the agreement, of the e:>-:perimenta l neasurements is excellent. In Figures 3, 4, 5, and 6 and in Figures 1, 4, and 5 of Appendi.;c r the states w~:ich were actually used in the evaluation of the coefficients of the Benedict equation are shown on presaure-temperatu.."""e projections. tion is noted. In each case the source of the experimental informa- The region or the data used b-y- Benedict (3) is enclosed wi thi....-1 a hac.."1urec1 line in Figures 3, /.,, 5, ~7d 6. In Appendix F, the data points used by Benedict are shown. These figru.~es show two interesting features. First, in ceneral, the region of the data used by Benedict differed significantly from that used in the present investigation. Second, the data used by Benedict represent more nearly a uniform set of reduced states for the .five light hydrocarbons considered here t ht1.."1 do the do.ta used in the evaluutions reported in Tables I and III. The effects of these differences upon the values for the coef'f.:i.cionts is shown in Tables I nnd Il and Figures 7 and 8. The coefficients proposed by Benedict produce the smoother curves when plotted agains·t molecular weight on a linear scale. The effect of intuitive, non-least-squares adjustments performed by Benedict(l) on his coefficients must also be considered when comparing Figures 7 e.nd 8. The fact that the data used by Benedict do not include a significant number of state~ representing the liquid phase would be expected to produce a pronounced effect. The fact that the magnitudes or the corresponding entires in Tables I and II -.39- do not .differ greatly 5.mplies that t he accommodation of the coefficients to the liquid phase data occurs throuch a r0sdJ:.1.stment of the:l.:r· ~ nt0rrelationships. Calculations for states represe:1tine tI:e lic-;dd pr..ase involve the subtraction of relatively large munb-8rs to obta in ,small differences with the result that ver:r sm..~11 percentage chc1nges i?:1 the values o:£' the coefficients result in large changes in the predictions of the volumetric properties for state~ at which the molal volume is small. Table IV presents several sample calculations of the pressure for three valuea of reciprocal molal volume to demonstrate t lie rol.c: t:1:vc contril>u- tion of the several terns in the Benedict equation and to point out, the great dif"ficult:r of .o btqining precisfan in predictfons for compressed states. The data of the Chemical° Engineering Laborato:r7 at the California Institute of Technoloey were used u.:.'lamoothoo while other c1ata i,ere talcen from the literature and eener::,.lly reflected s::noothing operations. A logical d:tf.'ficulty arlses with respect to the relative weighting ·io be assigned to the smooth and to the unsmoothed.dr.rie. :ln vie.w or the £'act that f:rcquently the smoothed infor:iation was bas.eel i,pon a co m:iderably greater or Jssser number of observations than were re;i01°t8d t}·:ercb;;- lncreasing, perhaps, its cred:i.bil:ity per point. In t he 2.bsence of a de- fensible reason for o. different course or action, the deci sion •.ms rrt<'.:,de to weight the information from each source ei:aunll,y. This decision H:l.S perhaps unfortunate for n-pentane because the avullabl e d3.·~o. rro:'l Boattio, Levine, and Douslin(JO) represented a very small region adjacent to the computing the deviation f'or ench state, f.'lquaring nnd St11iJc·rd. nr these deviations, div:tding th.it. sum by the number of statea and extract:tng the square root. Such e. proeedv.re w:tll a.lwa.:,,-s lt~aa to real pcsUive values though the short method may not H' the uncertainties :tn the coefficients are sufficiently great. Figures 15 a.,d 16 show a comparison of' the f'it of the cooff:1oients of Table I and of Table II for n-pentune for two tempera.tm'es. The inforoat:1on in these f:5.gures indicates qual:i.tat:i.veJ.y the manner in which the predictions of cornpressibility factor deviate fro□ the experi- mental values. Benediot(l) shows reeul ts fo·r propane similar to those given for n-pentane in Figure 15 except that he shows density against pressure ra.thm. . than compressibility factor ago.inst pressure. For both of these compounds prod:letions of pressure ror compressed ga.ees become very poor rapidly as the range of extrapolation is inc-.ceased outside of the region in which P,enedict fitted his equati<:m of state. A compar- ison of average standard errors of _estimate for the coefficients proposed by Benedict with t~1ose f5.tted to a broader :i:-ange of press'U!'es for the same value or ga..."l-':l1A is i."lstructive. Coefficient sets 1-B f'or both etha'1e and n-butane represent least squares fits to that portion of the volumetric inform.:.tion for which the oomprsssibility factor is greater than tho.tat the critical stat~ and for which the value of gamma is nem:-ly the same as t.hat proposed b-.r Benedict. applies to ooeff1_cient set l•A for propa.na. The same situation Using this same portion of the volumetric i..'1..fo::-mation the least sq_ua't'eS coef".f.'icients gave 0.0102, o00108, a.'lld o.0068 for the average standard errors or estimate f'or -44ethane, propane, a.nd n-butane respectively while t he corres!;0nding values for Benedict's coeff:i.cients were 0.0299, 0.2158, and 0.3203. This comparison represents c. somew::K'. t extr0:1e: case b"\:t it does illus- trate the nature of extrapolations of the Benedict equation. The coefficients :recorded in Table III show quite clearly thet the seven linear coefficients are related to the coefficient gamma 1n a very complicated way. The existence of a. S!llOoth functional dependence for these coefficients upon gamma ma.y be inferred from the values for the larger gammas. The relationship appears to be oscillatory in the vicinity of gamma. equal to ~ero. This oscillatory behavior could be or creat importance in selecting a set of coefficients for use 1n making calculations on multicomponent systems. It seems·to im- ply very strongly that the individual coefficients themselves are rather elusive when the optimum value for gant.rna is small. freq,iction of Phase Behayio;:. The vai;:or pressure data for pure substances can be interpolated quite adequately from such a simple expression as the A.~toin~)equation (83) so the application of the Benedict equation in this field will be slight indeed. However, in Ol"der for the l30nedict equation to be usef'ul in predicting the phase behavior of multicomponent systems it must predict :reason&bly accurate phase behavior for one-com:r,:onent systems. From the description or heterogeneous equilibrium supplied by Equations 67, 68, and 69 the values for the vapor pressure and tor the fugaci ty a.'1.d molal volUit1!3S of the coexisting phases may be computed. Figure 17 depicts the form of the relationahip.betwoon flugacity and pressure. Thie curve is parametric in reciprocal molal volume. The value of this parameter in- creaees continuously along the curve from the point where both pressure and f'ugacity are zero. For a different temperature or a differsnt compound, of course, the curve would change. The point of intersectio11 of the curve with itself defines the predicted vapor pressure and. the tugacity of the states in heterogeneous equilibrium because at this point tvo differ- ent values of reciprocal molal volume correspond to the same values of pressure, temperature, and f'ugaoity. Clearly, the location of' this point of intersection depends upon temperature. There exists a temperature at which the loop has become so small as to disappear aJ.together. This temperature corresponds to ~he predicted critical temperature. The calculation of t ho w,:1por pressure from t h0 Benedict equation is a straightforward iteration in which the values of the reciprocal molal volumes for the tw9 phases are sst:i.m..':lted ai.~d the subsequent pre- dictions of the pres,suree and f'ugacitiea f"roa Equations J and 46 are compared for equality. Figure 18 shows the deviation of the predictions of vapor pressure from the experimental values f0-r propane and n-pentane using the coefficients of Table I and Figure 19 shows the aa.rne information for the coefficients proposed by Benedict and presen·ted in Table II. Figure 20 shows the manner in which the vapor pressure predictions for n-pentane vary with the value of gamma. An attempt was mde to obta1.n coefficients which would describe the vapor pressure curve in the lea.st squares sense and at the same time describe the volumetric behavior of the liquid and gas states. The method of the attempt involved -the construction of the normal matrices for the mi:alnrl.zation of the sum of the squares of the differences betwoo:::i. the predicted dew-point and bubble-point pressures divided by t.hG observ-ed vapor pI~easure on the basis or the observed values for the orthob.:'"lrio molal oolwoos; for the mini."l'lization of the logarithm of the r atio of the predictod (loo-r-oint a."ld bu.bble-point fugacities again when th0s0 fu.go.ei- ties were calculated from the observed orthobaric molal volumes; and fo~ the minimization of the deviations of the predicted compress1.bility factor from the observed value when the molal volume is conside~od to be the independent state property, respactively. The differences in magni~ tude of t..ri.e corresponding elements in these normal matrices -were t,1-emeri..dous with the result that a linear combination or these matrices on t he ba.eis of equal weighting produced a eet of coefficients which could predict the tvo-phaae fuga.oities reasonably t1ell, uhile t ho predictions of the volumetric behavior were wholly withot1t merit. While these ma.trices might have been combined in accordance \dth the method of least squo.res as outlined in Appendix C such a.n e.pproa.ch will still not permit adequate volumetric predictions if the element~ of the volumetric matrix are appreciably smaller than those of the fugacity matri."C. A second approach to the improvement of the phase behavior predictions problem considered the f'ormulation or the problem in terms of two sets of coefficients--one to describe the volumatric behavior and the other to describe the fugacity as a function of state. Using the set of coefficients from Table III designated 5-A for propane and the experimental values of vapor pressure, values of fugacity a.'l'ld reciprocal molal volume for the dew point a.nd the bubble-point states 'Wero computed from Equations 46 and 3 and may be found in Table VII. These data are presented tor ten degree temperature intervals and represent the predictod properties along tha boundary of the heterogeneous region. Normal matrices were prepared, one for the minimization of the sum or tho squares of the deviations in the compressibility factor at the values or reciprocal molal volume given in Table VII and one for the minimization of the sum or the squares of the deviations in the logarithm or the two-phae~ f'ugacity at the same values of reciprocal molal volume when the TaQle VII value 0£ dew-point fugaeity is assigned to the bubblepoint. The normal matrix for the compressibility factor was reduced to three rows by adding the elements. of adjacent rows and the normal matrix for tugacity was reduced to four rova by a similar process in the man..~er •ot the method of averages. The two reeulting matrices jointly •compris.a a seven row matrix. In addition the restraints expressed L"l Equations 78, 79, and 80 'Were imposed by the method of lagr-a.nge undetermined - multipliers and the solution ua.s then obtained. This set of ooefi'ic:tents is presented in Table VIII and is designated as set 5-C for purposes of future ref'erenoe. A comparison or sets 5-A and 5-C indicates a.l'l appre- ciable difference in the values of the ooefficientsdescr:lbing the volumetric an~ the phase behavior of propane. Figure 21 shows the molal volume or propane at 160 degrees Fahrenheit as a function of pressure. There exists a small difference in the calculated volumetric behavior which is show more clearly 1n Figure 22. Thie figure sho"1s the compressibility factor a.s a function of pressure. The coefficients based upon the volumetric measurements in the region of homogeneous equilibrium yield calculated values in good agreement with experiment except in the low pressure gas region. Tho disagreement in this case results from the absence of direct experimental volumetric measurements in the gas phase at temperatures below 190 degrees Fahrenheit. The experimental points which are sho1,m werG bused upon Joule-Thomson measurements'33) which were not used _L'1 establishing the coefficients of set S-A. In the liquid region the good agreer10nt show,'l • in F~"UI'e 21 for set 5-A at pressures up to 2 ,ooo r;0unda per square i:nch continues up to a pressure or .10,000 potm~s per square 1.neh. If' the Joule-Thomson coefficient data. had been included in the evaluation of the coefficients of eet 5-A significantly better agreement in tho low pressure gas region should have resulted. In the case of set 5-C which was baaed upon only the molal volume a.t dew-point and bubble-point and the dew-point fugacity predictions together w::tth the critical stv.te restraints of E1uations 78 1 79g a11d 80, good agreement was obtained for the volu.netr:!.c: behavior in the gas region. Such a result is similar to that of 14'.lrtin und Ho,;. 11ho fitted their equation to the cr:i.tical state restraints and avoided the volumetric j.nformr.1tion in this ree;ion fo1· the most !)9-.'!'.'t. The '7e.lues for the compressibility factor of the liquid phase are in distinct disag-.ceo- ment with experiment c1.t pressures above 1.,000 pounds pe--.e square 5.neh. Good ~<Teement with e:.rpe:ri...'llent ·w:is obtained iu predicting the molal volume along the dev-r,oint and bubble-point curves up to a temperature of 180 degrees Fahrenheit. Above t!1is temperature, in the vicinity of the critical state., the equation does not describe t he bubble-po:urt volumes satisfactorily. Figure 23 shows t he f'ugaoity of propru-ie as established experimentally and a.e caletllated from the coefficients of' sets 5-A ar:.d 5-C from Table VIII and Equation l,,6. The fugacHy of propane a.t dew-point as calculated from the coefficients of set 5-A ag:."ees rensonably well with the experimental values. However, the f'll 0aci ty at 01--:bble~pcL"'lt ca.lcnla.ted from the coefficients of set 5-C differe wide]~, fr.om exp,3rimen.t . Near the critical ste.te the f'u-zacities established fro!n thc-3 constants of set 5-A of Table VII clo not correspond. with the 0xperim.ental values whereas the fugacities based upon the coefffofo:nts of set 5-C are in good agreement with experiment. Figur0 24 shows the behavior of the critical isotherm. for pro~ as calculated from the ooef.f.ic:le.,"lts of set a 5-A, 5-E, and 5-c • The predictions with set 5-A are not satisfactory but the -predictions from set 5-C are in good agreement up to the critical p.,~essure. -50- describe V1c behavior ·1n the homocon0ous rogic:ns ,iith sc.tisi'aco:-y o.ccurD.ey . Qn the ot'1 . .or ha.."1d, thr.r coe.f'f:teients of set 5-C dcsCTibe the recip- rocal molal volume of tho coo:dst.:!.ng phases and along the critical isotherm withren.sonable accuracy but do not provide 2. useful dssm:-iption of the properties of. l)!'o,ano in the homogeneous z·ec:.tons c.t pressures a'bove 1 1 00'.) r ounds p01"' sq_uai":'0 inch . ?-b r oover, set 5-C does not predict the vapor pressure ~ith acm.racy comparable to that of set ;'-A obtained from iiohmietric da:t.a e.lone . Thus no :real i mprovement results from att0mpting to im1::.ose the restraints of the critical state uz:on the evaluation of tho coefficiente for the Bened:tct equation. Even ~hen only the fugacity at dew-point. and at bub?')lo-potnt were employed aa a bn.sis for determining the coefficfonts as '.rlas done for set 5-C, the descriptio11 of the vapor ' pressure and f"uca.city of the liqtdd phase was unsatisfactor;r a.s :ts shm-m in Figure 22. The acCUT:::i.cy with which en equation based upon volu.rnotric data describGs the ve.por pressure leaves much to oo desired but uppears to be n 0B.2." the opti!ntun that ean be obtained wit:·1 thG Ecenedict O<;_'tiation if :i.t is to be used at pressures in excess of t hose considered by Bene~ ict ( 3). The third approach to ma1d.U£: the Benedict equatlon predict reasonable phase beha.vfor considorscl t !'!.e additlon of' n correction term which would apply priroo.rizy withii"l the region of heterocQ'meoua equili- brium. Before discuss1.ng the correction term itself sevez,al requirements which such a. term should meet will be considered. If the Benedict equa- tion predicts reasonable volumetric behavior then the correction term should not affect these predictions except that it might i~prove them. The correction term. must n::sr::c>o:z:im2.to the f"tt,"lction -F'(T) ::10f:i.ned :tn Equation 73. The £'unction .;..F(T) :ts not defined f'or te1:1per·o.t.m-es greater tha.11 the eri tico.l temperat1.1r0. zero at the cr:i.tica.l stet,e :tf the de-w-po:tnt e.nd bubble-point molal volumes are considered to he the :.rocrt.s of the cr:i.tico.1 :'l.sotherl'.;J. c.t the critical pressure. Conaequently, unless the Benedict equation predicts the critical stat.e exactly, the f.\mct:lon -F(T) rm.mt be d2.ffe::·ent f'z>om zero at states outa:tde the :region of heterogeneous equ.il:ibrium for the opposite hypothesis would impJ.:;r the 0xistenco of a disconM_nuH,;y- in fugaoi ty, entrow, and enth.a:!.pzr for paths crossing the critical isotherm into the liquid region. To the extent that the correct:ton term apr,,r ox- imates -F(T) the predictions in tho region adj.'.lcent to the cr5-t1.ca1 state ulll be ,.mproved. The effect of t!"le correction :i.n the resion o:t' homo- eeneous equilibrium can be eont"ined to the c~itical region l:JJ• suitable choice of. approximating functions. As shown in Figure 18 the ,mr..or pressu't'e rcod:i.ctions for .n- pentane are adequate in the absence of a correction even though tho prediction of the critieal state leaves somcth:tng to bo doaiTed. is not the case u.nfortu..'W.tely for propane. Such Table IX presents the rr.z.gni- tude of the error in the B.quid-phase f.ugo.city predicti.ons. The r nct th.at the error becomes large at the lou temperature is ver~r importai-,t for the computation of liquid-phase f'ugacities, entropies, enthalpies, and ohe.m ioal .potentials, but or seconda't'y imrJOrta.nce with respeot to vapor pressure predictions since the slope of the fugacity-pressm·e curve beoornes so steep for the gas phase o.t lotJ temperatures, that re.ther large errors in the bubble-point fugacity result. ,.n rather small errors Inas:nuch c.s good V3.por ~essure plmse fugacitJ l)red.i ctior!s , tb.c 1):-ec:lsioi;, of th.(=J cc1-,rsct,io11 ter~n i11· 7 i )./ Jappz·oxim':l.tinc -F(T) should be ::..~c.th0::.~ gre2.t. noying Table IX s:1ows a conmon but an- charc1.ctez-istic or the enor in the hubbJ.e-point .fug:i.clty pre- creasL"lG as the c:ri t1.ce,1 sta.te is .'.c.pp-r·oa.choo. A precise app:roxim.ation to -P(T) ,.rill have to be .::-. r ather complicated fu..11ctio!.1 of' t0:-:1:;:o r2.tuxe, the form of which is not cJ.toc,ethcr a:pp:i't"ent .f.rom the res~;lt of the present study. The utilit,y of int:;."oducing 2. very complicated c01"'Z"0ction term or th.i.s charactm:· into the Benedict eq_u:;.tion is open to c021siderable question &"ld the conclusion t l::it the De~1odict equation is or limited use i..."l describ-ing the f ugacit:r of the liquid phase has much ri:orit. The fact t hat the vapor pressure predictions for n-::..x,mtane are satisf'a.cto:r:7 is explained in pnrt in terms of the nature of the e:;::porimental bnckground used in evaluating the coefficients which ua.s discr,ssod exclier and thereby do0s not cont'ro.d:1.ct this concJ.usfon. The cost of cood vapor pressure fit in terms of the volumetric fit fo r eflected in th:c: rolo.tivc o:i.ze of shown i.."'l Table I. Tho no,tt1re of' the erz-or in tho f't.i[c;,'J,cit.y predictions in the close neighborhood of t ho observ0d values f'or cr:i.t:!.coJ. tcrr.i-:,crature e.nd pressure is not known because t•.ro of the pressure roots of t:ie isotherms cone into confluence a..~d loa~,e t he real ro:ia before t:1e cr:.tlcal temperature is reached . Tc.hJ..e IX shows tl:at such confluence occlrr'S below 200 deg..c-eos Fahrenheit for propa..'1e when gamma equals L:.• 24021. ,.. V noncd1.ct molal volume b~comos lu.r c;e. Fro"!l Eq1.w.tfo~ 70 (84) lntegrdion gives - ln f C (85) 0 "' ln HT+ ( Z - 1) + ( 111 cr* • p 8 ) (86) + and so ♦ The function J- (d ) should mecit the following reqnirenients • {89) cont inuity ~ {T) ~ RT ln (rd., ✓rb.e) f or (/,,(T) - 0 T(T 0 (90) T)T •• C (91) From t he s te.ndpoint of crn;1 . u:b1:-::tion a convenient choice of' i ntec;rand is ,,J (~)/( ~ '9' ~0)2 = . ~ . e:iq,(- c~( (j =- 1 0 )..1?(- - _(93) vo)2) c~ s 2 ) fir 'l 'he cooff:lcient Ci is so c:1osen t h.at (94) (95) .,. l + 2 Thus ...n h, f --· ('I • • + 0 -;e He • e ❖ G i--- f (7:) cf exp(-cf( ~ .. <I,)s) ,• \ ) ( 97) 0:: ) ) (98) ~ + T,:J./,/oT )( 1 + erf ( C3. ( O" - 1 + orf( c,._( (j • ~~i) (Y erf'( C ( 1(ut1ar)(1 -o1 -2 - 1< 2 (9G) • - 0: oC ) - ·' • 0- ) .c ) (99) -55- If' t? {T) is assi..uned ·to oo O.0150RT and c1 is assumed to be 6.18 (cu.rt.) per (lb.mole) then the co1Tection terra for pressure has the form shown in Figure 25. Figure 26 shows the effect of 'Ghe corre~ tion term on the vapor preSS"..l!'e _predictions for propane. Fib.rure 27 shows the effect of the oorreotion term on the predictions of volumetric behavior in the vicinity of the critical state. other expressions for t.p (T) which damp out with increasing temperature might be preferable to the one used.for the calculations leading to Figures 25, 26, and 27; however, such expressions would be appreciably more complicated and very likely would prohibit accurate predictions of the fuga.oity for the liquid phase .altogether unless special effort was made to make them fit the error in bubble-point i'ugacity profile illustrated i."l . Tabla IX. present approach is thus more_expedient than anything else. The -56- Prediqted Critigal .froperties. Equations 81 and S2 are simultan.sous cubic equations in recipro- cal temperature and My be conveniently rewritten as 0 (100) 0 (101) vm.ere t U3. ""' V2, 'i:s Uti3 . Va (103) g + K7 ~(3(i + y ) - 2)s')exp(-;;ji!} + 2K1Q" + 3K4sf .. 1 . (104) (105) ;1J ~ainst; (102) . ~ • . . may be roadilv 1n•onnred. V shov,n in Fic;cJ.re 2D. ~ .. . - . (106) An exa:11· :le o:f.' such a plot is 0 T'1is fi[~ure is based upon t he coei'i'icie:::ts i'or !)ropane Y1hich are ;,resen:!-;ed in Table I. X Substi·;:;u'cio:n of {107) "" t y m.sizes ut V 0 {109) uy V 0 t no) 19,quivalent to X + -57- -v. Furt:1.er, t.he set or ya.ra:notric EquG.t:tons 107 and 103 def.:i.nes a cubic po.reibolc,. The lntersection of -the straight J.ines whic}1 results fz-cm Equations 100 ~:rl 101 with the ·oubic parabola w::i.11 locate those values of the parameter t wnich satisfy both Eq:iations 100 and 101 and therefore also 81 and 82. These values of t a.re the roots of 81 and 82. The fact that the cubio para.bola involves temperature a.lone, and therefore need be plotted only once, makes this geometric solution particularly useful. Figu,:,e 29 shows an example of th:ts e;eometr:tc solution for the predicted critical temperature artd is based upon the informn:Uon presented in Ficure 28 . for the va1U•3S of u and of v. The values of the parameter T· are shoi-m along the cubic parabola and are expressed in deerees Rankine. The predicted critical temper- ature corresponds to the point on the cubic parabola where this curve and the two lines are concurrent for the same value of reciprocal molal volume. The point of ~tersection of the two straight lines is a function of reciprocal molal volume alone. Benedict(~ describes the techniques which he applied to the adjustment or the coefficients which he obtained to fit the volumetTic data in order to permit the equation to predict the phase behavior ·1-,ith somewhat greater accUTacJr . These techniques were intuitive and were beyond those associated with the method or least squares as applied to the volumetric in.formation alone. The adjustment of' the coefficients as performed by Benedict was at the sa.crifioe of fit in the liquid region. Table X shows that the predictions of the eritioal properties are not accurate unless special adjustments of the ooef.fioients are made. Table XI -58- illustrateD thci affect of t l:e coefficient g8llll:l1.:1 upon the predicted criticcl. pi·operties. The effect or imposing carte.in rigid restrc.i.nts u,r:on the evaluation of tl-:e coefficients to i:!1p1:·ove the predictions in the vicinity of the critical state has been partially investigated. The effect of. mn!d.ng the cr:iticul isotherm sutiefy both Equations 78 and 79 at. t he critical molal volume is illustrated in Figure 23. That the :restra.:1.nts a1·e snt:i.s-, fied is uppareEt but the ·accuracy of the description of the pressure in the critical region has not been improved. To make the Benedict equation predict the exJY~rimental value of the critical p-.cessu:ce at the critical temperature and molal volume the restraints of the Equations 78, 79, and 80 and P6 , 0 ·= Pc mi.lllt be imposed at the critical temperature a."ld critical molal volume. Si:11ce the Benedict equation can."lot be reverted into a convenient expression for molal volume, these restraints are necessarily indirect. A calculation has show that the restraiat repre- sented by Equation 78 is implied for the Benedict equation b-'/ Equations 79 and 80. A practical limit to the number of restraints which one might desire to impose upon the evaluation of the coeff ieients is set bj· the number of coefficients in the equation. The more rest:rai."!lts are imr..osed at the critical state, the poorer the equation becomes as an extrapolation device. The Benedict equatio..'"l has ·too r ew coefficients to permit a very precise description of the critical state and at the same time give a reasonable representation of the volumetric behavior of the liquid phase. -59- The extension of the Benedict equation to the prediction of the volumetric properties of the liquid phase and the region of high pressure will be of considerable engineering utility. The feet that the predictions of this equation in the region or the critical. state leave much to be desired does not preclude its wide application for thermodynamic calculations for regions removed from ".t he critical state. If latent heats of,vaporization are available then the difficulties associated with the correct prediction of fugacity, enthalpy, and entropy in the liquid phase may be of' minor importance for calculations involving thermodynamic paths lying entirely within the liquid region since the correction needed for the.predicted change or these properties with temperature can be obtained ·from the latent heat of varorization in.formation and the properties of' the gae phase e.t dw-point. I£ such la.tent heat information is not available, the empirical correction term discussed in the pi-eceding section may be usetul. The Benedict equation does not replace graphical procedures for very accurate thermodynamic calculation because the errors in its predictions are much greater than experimental uncertainty. Extensive application of the Benedict equatiqn will necessitate the use of automatic digital computing equipment arid .for this reason the development of a more general .expresaion which can. be used more ettioiently with the high-speed computers for interpolation within the accuracy of the experimental information appears to be worthwhile. ' The methods discussed in the preceding sections for the evaluation of coefficients for the Benedict equation have produeed coefficients whic.~ have a somewhat vague correlation with parameters vhich c.1w.racter1ze -60- gous series to a::-iother. A revision of tho mot!:od of evalu..:1tion to . im- prove -this correla-tio~1 by cva.lun·~ill;; coeff icie:1:cs f or the Benedict equc.tio21 tc ap-_fU-oxime.to t.he r·0tluc0c: equation of sts.te 2.ppeclXs to be desiz·abl0. The 1resent hypothese~) concerning the rela:~ionship of the coefficients :?or m..i..:.durc -to thoso of ·the pv.re comr.,:,onents ir:i.ply regul.:1; .rity in the r·elo.tionship of the coefficients to other pr..ysical :;,:rrope1.,.bies of the syst.era. NOMENCLATURE A A" eoefficie:nt of t,ho Antoine 0qi1.ation coefficient of the equa.tion of state, (p.s.i.a.) (cu.rt.) 2 / (lb.mole) 2 ., a coefficient of the et1uat:ton or state, (p.s.i.a.)(cu.rt.).;i/ (lb.mole)3 ~ element of the matrix A a matrix or an arbitrary function ith virial coefficient coefficient of the Antoine equation .b coefficient of the equation or state, (cu.ft.)/(lb.mole) 2 .... coefficient of tht:J equation or stat,~, (cu.ft.) /(lb.mole)~ bij element of the matrix B Cl coefficient of the heterogeneous correction term, C -.p molal isobaric heat capacity, (B.t.u.)/(lb.mole)(deg.R.) Bo (ou.rt~)/(lb.mole) molal isochoric heat capacity, (B.t.u.)/(lb.mole) (dag.R.) - ~ ? ooef'ficient of the equ.aM.on of state, {p.s.i.a..) (deg.R.)"'(cu.ft.)'.- per (lb.mole) 2 coefficient of' the Antoine equation coefficient of the equation of state, (p.a.i.a.)(deg.R.) 2 (cu.ft.) 3 per (lb.mole)-' · d symbol of differentiation E spooi.f'io internal energy, (B.t.u.)/(lb.) E molal internal energy, (B.t.u.)/(lb.mole) exp exponential function erf error .t'unction • -62- F a function of temperature f f'ugacity, (p.s.i.a.) Q general intensive thermod;ynamie state J'fi."OI.Jerty E specific enthalpy, (B.t.u.)/(lb.) H molal enthalpy, (B.t.u.)/(lb.mole) K a matrix • -1 b -a/R a a./R c/R k Boltama.n constant, 7.263 (B.t.u.)/(deg.R.)(molecule) M molecular weight, (lb.)/(lb.mole) N total number of items in a. se't N Avogadro's number n position number in an ordered set n weight fraction n mole fraotion P absolute pressure, (p.s.i.a.) p coefficient in power series expansion for pressure Q partition function R u..~iversal gas constant, 10.73147 (p.a.i.a.)(cµ.ft.)/ (lb.mole}{deg.R.) .r radial distance, (rt.) rm distance parameter of intermoleculn.r potential i"uncti.on S speoifio entropy, (B.t.u.)/(lb.)(deg.R.) S " molal entropy, (B.t.u.)/(lb.mole)(deg.R.) • -63- s coefficient in I--OYer series expa..~sion for reciprocal m.olal • volume T thermoa"'ynamic temperature, (deg.R.) t a para.meter U intermolecular potential function V specific voluma, (C"J..ft.)/(lb.) V • ·molal volume, (cu.rt.)/(lb.mole) V a .parameter X a. parameter y a parameter z compreseibilitJ factor a. coefficient or th.a equation or statB, (cu.ft.).,/(lb.molef ... ".I a function of temperature alone coefficient or the equation of stato, (crt.1..f-t.)2/{lb.mole) 2 symbol of a pirtia.1 derivative energy parameter or intai"molecula!' }:otentiaJ. function r a parameter a function or reciprocal molal volumo alone chemical potential, (B.t.u.)/(J.b.) molal chemical potentiul, (B.t.u.)/(lb.mole) a parameter lf the transcendental number symbol of summation specific weight, (lb.)/(cu.ft.) -64- '! recipro-0al molal volume, (lb.mole)/(cu.ft.) Z. l i4 7 ~ rpT ,6 1lt3 rl1 a function or temperature alono Superscripts partial or partial molul property 0 in the pure state * in a state at infinite atten'I.W.tion e property has been computed using cort>ected form for Benedict equation two-phase prope1-ty Subsorip'iis - property is extensive b property pertains to the bubble-point. state c property pertains to the critical state d property pertaim; to the dew-point state e property has been computed using the Benedict equation g property pertair1s to the vapor phase i ith element of a set or component i in a mixture H under isenthalpic cor.il.i-t:i.ons j jth element of a set or component j in a mixture k propert7 Ie:rtaina to component k -65,f property pertains to t he condensed phase m under conditions of constant oom:rx,sition P under isobaric conditions r reduced property T under isothermal conditions V,Cf under isoehoric conditions • property is defined in terms of pour.d-moleC"..:tlor \.;-eights -66REFF.RfilTCES 1. M. Benedict, G. B. Webb, and L. c. Rubin. J. Cher:. Phys., ~, 334, {1940). 2. Ibid. !Q, 747, (1942). M. Benedict, G. E. T'lebb 1 L. C. Rubin, and L. Friend. Chem. Eng. Prog., !il.,, 419-422, 449-45h, 571-578, 6J9-620 1 (1951) H. H. Stotler and M. Benedict. Chern. Eng. Prog. Symposium Series, 49, 6, 25, (1953). E. w. Hough and B. H. Sage. Chem. Reviews,§, 193., (1949). 6. T. J. Connolly. ?h. D. Thesis, California Institute ·of Technology, (1950) H. w. Brough. Chemical Engineer Thesis, California Institute of Technology, (19~0) 8. J. A. Beattie and o. C. Bridgeman. Proc. km. Acad. Arts Sci., .fil,· 229, ( 1928). c. D. Hodganan. Mather.iatical Tables from t..r1e Handbook of Chemistry and Phyaics Ninth Edition, Chemical Rubber JSiiblishing Co., Cleve1and, (1949), P• 270. 10. F. Klein. Elementaf, !,{athematics frCl!l an Advanced Standpoint, Vol. 1, Dover, New ork, (1952), P• 88. n. E.T. Copson. --:-,r----...~i---r-:-.--'l""!""--~-.....,,--.._-...,.,~""'" An Int a Coznplex Variab e, p. 85. 12. C. N. Yang and T. D. Lee. Phys. Rev. 1 !!7., 40h, (1952). 13. P. W. Bridgmano J. Chem. Phys., J, 597. (1935). 14. J. E. Mayer and M'.. o. Mayer. Statistical Hechani.cs, John Wiley and Sons, New York, (194()} 1 p. 269. 15. Ibid. • 16. R. B. Bird, J. o. Hirschfelder, and c. F. Curtiss. Survey of the Equation of State and Transport o:f Gases and L1.quids, Univ. of Wisconsin C~J-7~8 m5rc1-9938, (1952). 17. • w. E. Rice and J. o. Hirschfelder. Second Vi rial Coefficients of Oases Obeying a Modified Buckingham Potential, Univ. of Wisconsin ONR-2, (1953). 18. K. Knopp . Theory of F\mc tions, Part 1 1 Dover., New York, (1945), P• 79. 19. J. J. "Martin and Y. c. H~u. "Develop:!lent of a rtew Equation of State for Oases.," Paper presented at Regional !Aeeting of the Amer:l.can Institute of Chemical Gngineers at San Francisco, Calif., September 1953. 20. J. o. Birschfelder., R. B. Bird and E. L. Spotz. 21. J. E. J ayer and ;J . G. :.iayer. op. cit., P• 291. 22. R. J. Buehler, R.H. Wentorf, J. o. Hirschfelder, and c. F. Curtiss. Lennord..Jones and Devenshire Equati on of State for Compressed Oases and Liquids, Un!v. of Wisconsin C?!-565 NOrd- n, 921, (1949). Trans. ASME, 9938, {1950). 23. J. E. \layer and \f. G. Mayer. op. cit., p. 274. 24. R. Fowler and E. A. Guggenheim. Statistical Thermodynami cs, Cambridge, (1949), P• 264 ff. s. s. Wilkes. Elementary Statistical A~~sis, Princeton University Prees, Princeton, N. J., (1949, P• 250. 26. c. Hastings, Jr •• The RAND Corporation Aporoximations in Numerical Analysis• (1952). H. w. Brough, ;;·. G. Schlinger, and B. H. Sage. lu:ier. Doc. Inst., Washington, D. c., Doc. No. 3235, (1951). 28. B. H. Zimm. J. Chem. Phys., 19, 1019, (1951). J.E. Mayer. J. Chem. Phys., 19 1 1024, (1951). 30. J. A. Beattie, s. w. Le·rlne, and D. R. Douslin. J. Amer. Cherr,. Soc. U, 4431, (1951). -31. D.R. Hartree. Numerical Analysis, Oxford University Press, .)2 • . F. D. Rossini and others. Selected Values of Proterties of, Hydrocarbons, N. B. s. Circ. C461. (1947), P• 12 . .33. B. H.' Sage and w. N. Lacey. Thermodynamic Properties of the Lighter Paraffin Hydrocarbons and Nitrogen, K~erican Petroleum London, (1952), P• 15~. Institute, (1950). p.34. -68- A• .:iichels and G. W. Nederbragt. Physica., 2, 1000, (1935), • ibid., ,b 569, (19.36). . R.H. Olds, H. H. Reamer, B. H. Sage, and W. N. Lacey. Ind. Ezis. Chem., 35• 922, (1943). Raw Data: CT.TCHE Data BooklI6, Data Sheets 491-97. H. M. Kvalnes and V. L. Gaddy. J. Amer. Chem. Soc., 53., 394, 36. (1931). 37. J. A. Beattie, G. L. Sj_mard, and G.- J. Su. 9 J. Am. Chem. Soc., .§!, 924-6, (1939). H. H. Reamer, R.H. Olds, B. H. Sage, and W. N. Lacey. Irxi. 38. Eng. Chem., 36, 956, (1944). Raw Data: CITCHE Data Book2°8Ti, •Data Sheets 1204-8. J. A. Beattie, C~ Hadlock,. and N. Poffenberger. J. Arn. Chem. Soc., 3 1 93, {193.5). J. A. Beattie, o. L. Simard, and G• ...J. Su. J. Am. Chan. Soc., r 61, 26, {1939) • R.H. Olds, H. H. Reamer, B. H. Sage, and w. N. Lacey. Ind. ~ • Chem., 361 282 1 (1944). Raw Data, CITCHE Data Book-YY2, a a Sheets 1023-30. B. H. Sage and w. N. Lacey. Thermodynamic Properties of the Li~hter Paraffin rocarbone and Nitro. en American Petroleum nstitute, ew Yor , 9 0 • P• 71. B• .H. Sage and W. N. Lacey. Ind. E,. Chm., 34, 730, (1942). Raw Data: CITCHE Data Books 84 apd 7. H. B. Dwight. Tables of Integrals and Other :fathematical Da~, Macmillan, New fork, (1947), P• ll. . 45. E.T. Copson. op. cit., P• 123. 46. E.T. Copson. op. cit., P• 72. 47. J. A. Beattie. Chem. Reviews, 44, 141, (1949) • • 48. I. E. Sokolnikoff and E. s. Sokolnikoff. Hi~her :fathanatics for Engineers and Physicists, McGraw-Hill, 1941), p. llB. Ibid., P• 5.34. Ibid., P• 114 D. n.· Hartree. ~P• cit., p. 70. LIST OF TABLES I. .Optimum Coefficients for Volumetric ?:,: edi ct ions for Liquids and Gases. II. Coefficients Proposed by Benedict for Describing Ph.use Behavior and Volumetric Behavior for Gaseso III, Coefficients Computed :1.n the Present; Stu.dy for the Description of Volumetric Behavior in the Light Para.f.fin Hydrocarbol'ls. IV O Sample Calculation of P-eessure Using the Benedict Equation and Ooeffioientsl-C for Propane at 190° F. V. Coefficients for n-Buta."1e L'valuated without ~erimental Information for the Gas Phase. • VI. I , Standurd Error of Estimate for n-Butane Coefficients with Respect to Each of the Sets of E,cperimental Data. \'II. Predicted Fugacities and Reciprocal Molal Volumes for Propil'le at Dew-Point and Bubble-Point. VIII. Coefficients for Propane Obtained Using Critical State Restraints and Phase Behavior Data. IX. Ratio of Predicted Dew-Point and Bubble-PoL"lt Fugacities for Propane Using Coefficients of Table I. I. Predicted Critical Properties. XI. Variation of Critical Properties Predictions Yith the ~:tagnitude of Gamma. -70-- TABLE I Optimum Coefficients for Volumetrio Predictions ror Liquids and Gases• Coeffi oient•• Methane E thane Propane n--Butane n•Pentane R 10.73147 10.73147 10.73147 10.73147 10.73147 Ao 4910. 53 7001.40 15913.l 22784.0 60215. 5 Bo 0.455158 0.237607 0.550703 1.00517 3&69003 448. 753 3334.26 6315.35 15491.4 19628.9 a 4551.18 26547.9 67141.l 263507.0 203941.0 b 1.03508 S.43107 7.41650 19.7116 16.0875 619.147 6476.86 19262.l 93708.5 116061.0 a. 0.332260 o.742830 1.60300 2.39047 6 0 6'7703 y· 1.20000 2.60000 4.24021 6.40000 10 .. 5000 M 16.042 30.068 44009 4 58.120 72.146 0.0156 0.0138 0.0094 0.0166 000258 Co X 10• C X lQ•S 6 Aver-J\ge Standard Error of' . Estimate These ooef'f'ioients are reoommended only for volumetric behanor in the homogeneous regions. prediction of The values reoorded are dimensionally oonsistent when used in the Benedict equation with pressure expressed in pounds per square inoh, temperature in degrees Rankine, and molal vo1Ull18 in oubio teet per pound-mole. -71- TABLE II Coefficients Proposed by Benediot for Describing Phase Behavior and Volumetrio Behavior for Gases• Coef'f'ioient .. Methane Ethane Propane n•Butane a•Pentane R 10.73147 10.73147 10.73147 10.73147 10.73147 Ao 7000.06 16681. 7 26934.2 38067.1 45961.9 Bo o.682998 1.00643 l.66027 1.99393 2.51324 . 275.952 2195.80 6214.44 12139.3 26936.9 2988.21 20879.3 57331.2 113870.0 246503.0 b 0.868679 2.858~6 5.78303 10.2804 17.1721 498. 790 6422.ll 25284.5 62016.2 161539.0 Cl, o. 512225 1.00264 2.50121 4.53676 7.45603 y 1. 54172 3.03213 5.66344 8.73709 12.2062 )( 16.042 30.068 44.094 58.120 72.146 C0 X 10· 0 X 10-S 6 • These ooeffioienta were suggested by Benediot(l,3) and have been modified to be oonsistent with the values of the universal gas constant, R, and the moleou~ r weights, Y, indicated in the table• •• The valuea recorded are dimensionally consistent when used in the Benedict equation with pressure expressed in pounds per square 1noh, temperature in degrees Rankine, and molal volume in oubio feet per pound-mole. TABLE III Coefficients Computed in the Present Study for the Description of Volumetric Behavior :1n the Light Pa..-affin 1{1itlroea.rbons •• t"... * (le 11.k"'"ll~ 2-B*~·<* 1-A** · 1..n** 2-A ~ . 7234.58 638.3.54 8930.20 4910.53 B0 o.695761 o.618857 o.,:r,2705 C.455158 127.-301 25L~.L:4? -:>t::? ~---. 14~~ . _.., 7r.;~ ,., 44°c .... a. -195. 741 1069.1.3 -1551.,.(18 4551.18 • b O.57';;09!., o.691890 0.4GOJC'I) ,.• 03508 C -144.017 51.).136 -207.0;0· 619. 1/4.7 u -5.&3086 1.12510 -0.55.t..049 0.332260 o.c.oocc l.54169 o.cncoo 1 .. 20000 0.0147 0.0147 0.0166 0.0156 Coefficient ~ VQ X 1A-6 -U lvarage s't?,.'1dar<l Error of :ss~ti~te • a {KJ. R is 10.7.3147 (p.s.1.a..)(eu.tt.)/(lb.mole)(deg.R.) molecular we:lsht is 1.6 .0.{2. *'-7 s-t2.t,:is at temperatv.:s'.'E;S oolou ,W deg. Fahrenheit were not used in evaluating these coefficients. Cf. Table !. -73- TABLE III (cont.) M3thane Coettioient 2-C 2-D 2-E 2-F Ao 7089.34 7316.35 7080.oo 6472.56 B o.653966 o.657546 o.629442 0.569888 - 218.045 155.073 139.897 124 •.385 a 878.007 • 216.149 518.750 1358.77 b o.715350 0.687ltfl o.P-5277 o.804517 lo-6 240.565 115.791 57.2446 -202.671 1.17173 3.81332 1.57440 0.650130 2.00000 4.00000 6.00000 10.00000 0.0162 0.0169 0.0172 0.0170 0 Cox ~ C X a lverags Standard Error ot Estimate TABLE III (cont.) lth§Wi* Coefficient 1-AH 1-B** 2-A 2-B ~ 22792.l 12768.7 282,33.6 ..;96.,4e5 Bo 1.41875 o.6.34985 1.871% -0.3W457 C0 X 10-6 -.201.267 2006.29 -1165.62 /..;512.84 a -8762.23 1410.3.2 17.334.9 IJ:J964.o b 0.9.36384 2.6.3063 1.208122 4.52568 C X 10-6 -2285.57 3804.50 -.3732.38 8541.92 -1.11402 1.08104 -0.562675 0.539735 0.00000 :3.0.3239 0.00000 2.00000 0.0154 0.0102 0.0209 0.0153 y Average Standard Error or Estimate * R is 10.73]47 (p.s.i.a.)(cu.rt.)/(lb.mole)(deg.R.) and molecular weight is 30.c68. ** States for which the compressibility factor is leas than that at the critical state were not used in evaluating these coefficients. TABLE III (cont.) i;thane Coefficient 2-c* 2-D 2-E 2-F Ao 7001.IJ) 15068.8 17195.8 17175.7 Bo 0.237507 o.787454 o.894960 o.854064 co 3.334.26 1832.58 1386.25 1184.59 a 26547.9 6906.28. 36S0 •.66 206.3.53 b .3.43107 2.30275 2.05581 2,.03067 C 6476.86 3.344,.57 25.32.84 2113.91 a. 0.742830 1.54841. 2.94518 4.32996 2.50000 4.00000 6.00000 10.onooo 0.0138 0.01515 0.0171 0.0206 berage Standard Error or Estimate Cf. Table I. TABLE III (cont.) ptooone* Coefficient 5-A 5-D 5-E 5-F Ao 12559.9 25451.1 26981.9 27052.4 Bo 0.375279 1.04838 1.028?9 0.994030 C0 X lo-6 7294.90 3791.64 ,3181.37 3172.48 a. 77386.o 33056.L~ 20966.2 16122.9 b 7.95732 5.56/.24 5.06201 4.82884 10-6 22120.9 11034.8 84.30.50 8003.38 1.48371 2.47482 3.2826.3 3.94183 4.00000 6.00000 8.30000 10.00000 0.0094 0.0119 0.0128 0.0161 C X a. Ivera.ge Standard Error of Estimate * R is 10.73147 (p.s.i.a..) (cu.rt.)/(lb.mole) (deg.R.) and molecular weight is 44.094. These coefficients are supplementary to those in Appendix F. and represent evaluations based upon the experimental data pi·esented in Figures 4 and 5 of Appendix F •• - 77 - TABLE III (cont.) n:,Butane f: Coefficient 1-A idf 1-B** 2-A 2-B A -8630.07 39213.l -40296.8 -109;327.0 Bo -0.760315 1.31.328 -4.24075 -FJ.59007 r'.'') · 73 62 L•.-:..., 22523.0 L;.1082.l 2.3536.5 116194.0 16'-)537 .. 0 3340ll.O 2.2542~ 13.400/i, 17.0293 27.3220 C x lo-'° 4266 • .38 3943.93 26741.6 74064.2 a. 1.07422 3.26759 1.48772 1.08984 y 0.00000 8.73741 O.OQ,'"'00 2.00000 0.0273 0.0068 0.0294 o.ol!~9 0 C0 X 10-6 3324.% a b , Average Sta.'ri'Jdard Error of Estimate • R ie 10.73147 (p.s.i.a.)(~u.ft.)/{lb.mole)(deg.R.) and molecular weight is 58 .120. • ** States for which the compressibility factor is less than that at the critical state were not used in evaluating these coefficients nor ware the data from reference 33. -78- TABLE III (cont.) n-Butg Coefficient 2-D 2-E* 2-F Ao -.3506.64 1543 •.38 22784.0 . 43021.6 Bo -o.9(:JJ795 0.12005.3 1.00517 1.46812 C0 X lo-6 10077.5 25517.7 15491.4 57.340.1 a 17226.3.0 424~6.o 25.3508.o 87982.2 b 15.6351 28.843.:3 19.7116 11.7515 w-6 37570.0 160034.0 93708.5 35954.5 ct 2.22473 2.0.3867 2.39047 :;.88952 y 4.00000 6.00000 6.40000 10.00JOO 0.0344 0.0377 0.0166 0.0144 0 X Average Standard Error of Estimate * Cf, Tabla I. -79- TABLE III (cont.) n-Pentane* Coefficient 1-A 1-n 1-C 1-D Ao -47185.S -101024.0 55126.4 62714.,2 Bo -.3.32146 -6.26527 6.46500 4.90966 C0 X lcr-6 40987.7 595L'Z.l :31260.1 23914 • .3 a 1879.30.0 :32.3053.0 115555.0 220331.0 b 16.5332 23.4244 4.1284? 14.4783 10-6 48024.6 95.31~.1 76254.2 127847.0 3.99798 2.7.3668 9.9431,,7 6.59891 0.00000 2.00000 6.00000 9 ..40000 0.0396 0.0402 000392 0.0279 C X a. Average Standard Error of Estimate * R is 10. 73147 (p.s.1.a.) (cu.rt.)/(lb.mole) (deg.R.) and molecular weight is 72.1/4.6. -eoTABLE III (cont.) 11-1:fillta.~ Ocef.fiei.ent 1-E l•F 1-G 1-H Ao 61331.8 60215.5 5'7121.7 57441.7 Bo 4.i9623 ,3.69003 2.67363 2.11853 co¼ 10 · 21448.5 19628.9 15750.9 13217.5 a 214,'US.0 203941.0 16268).0 l07!v23.0 b 15.6174 16.0875 15.8:368 l/... 2458 122649.0 116061.0 91659.1 711.32.0 6.58641 6.67703 7.3.36.t,5 9.36790 10.00000 10.50000 12.0000 15.0COO 0.0266 0.0258 • 0.0261 .,..._,,,.13 0 .v,. -6 ex 10·6 a. \veraga • Standard Error of Estimate -81- 0. 5 0 '7J r-r- ~· c,• • .:J .._,, • -- - 1150 . 21 -1577.54 ·,3 57 .on . bffi' CJ J ao- - 1011.3 .05 " J..820 . 67 ~ Oo90 -82- TABLE V Coefficients for n-Butane Evaluated without Experimental 4$ Information for the Gas Phase Coettioient 3-A .3-B Ao -24638.5 -5:3649.l 19592.7 Bo -.'.3.4286,3 -5.12364 -1.23806 15234.5 2.3696.9 9517.60 a 156.047 234205.0 151.301.0 b 17 •.)159 21.6334 15.9336 16912.3 39590.3 18:315.5 a. 1.66727 1.:39535 2.21450 y 0.00000 2 .ooor-.JO 4.00000 C0 0 xio-6 X 10-6 The data of reference 33 wei~ not used in the evaluation of these coefficients. * TABLE V (cont.) .3-F Coefficient 3-D Ao 63U..58 22635.2 37076.9 Bo 0.990863 0.990967 o.887654 C0 X 10-6 29566.4 15466.l 5343.54 a 447496.o 253382.o 102509.0 b 28.7340 19.7130 13.14.31 18.3695.0 934'n.3 33097.2 a. 2.10527 2.38920 3.34430 "{ 6.00000 6.40000 10.0000 0 X 10-6 -84- TABLE VI Standard Error or Estimate for n-Butane Ooeffieients* with Respeot to Each or the Sets of Experimental Data Source of Data Gamma 66 States (3.3) (40) 103 States (41) 113 States 0.00000 0.0173 0.0237 0.0383 2.00000 0.0154 0.0359 0.0462 4.00000 0.0199 0.0326 0.0455 6.00000 0.0141 0.0259 0.0583 6.40000 o.oo:n 0.0116 0.0261 10.00000 0.0086 0.0157 0.0177 * Cf. Table IIIo 200 lW 170 180 . 190 150 140 .311.3 JJcl.9 .314.6 3!.;2.8 474.9 580.l .371.5 401.7 397.5 '367.6 28t,.9 287.9 1;?.7.3 524.• 8 235.7 259.7 2.38.5 262.5 194.5 215.9 343.8 683.8 151.9 27.3.5 .307.l .242.6 )20 13) 155.8 17J.• O 191 • .3 212.9 188.7 2ll~.5 110 Fugacity Dew-Point Bubble-Point (p.a.i.a.) (p.s.i.a.) 171~.6 (p.s.:1..a.) (deg • .F.) 100 Experimental Varx,r Pressure Temperature Coefficie:n.t s of Set 5=-A Dew-1oi11t 0.118.30 0 .138.l):, 0.16951 0.10175 0.07725 0.088.44 0.06766 0.05936 0.05204 0.04562 o.0.39r38 (l o/4.635 O.l,920 0.5121 0.5(9-9 0 .5411. . o.6137 o. 601.'.3 n. 5828 o.6657 o. 6500 o.6351 Recipr."Deal }hlal Voluroo J3ubble-Point (lb.mole)/ (cu.ft.) (lb.mole)/(cu.rt.) at DeY-Point and Bubble-Point Predicted Fugacitiea and Rec1procal Holal Volumes for ~pane TABLE VII ' a> I 01 242.6 ~73.5 307.1 130 l40 /}00.2 52L.,a8 580.1 200 - 37().3 370.l 471..,.9 0.16312 0.13689 O.ll77Ei 0 .5048 0.10166 313.5 341.5 /.,2?.3 o.osg~n 0.404,0 0.4654 . ;·\., 0 •~:;~t.3 ,0.07738 299.4 321., .. 1 350.0 o.6061 0.5856 0.5626 0.624s 275.7 0.06782 0.05951 0.6/i,20 22;6.9 253.1 231.4 173.0 191.1.. 210.9 155.4 1?4.1 193.9 215.2 23?.7 261. 6 - 0.03999 0.04574 0.05218 (p.s.i.a.) o.r>582 Bubble-Point (lb.mole)/(cu.rt.) Dev-Point (lb.mole)/(cu.• rt.) Bubble-Point (p.s.i.a.) Reciprocal i-blal Volume Daw-R:>int Fugacity 150 160 170 180 190 343.8 383.8 188.7 2JJ.-5 100 (p.s.i.a.) (deg. Ji'.) 110 120 EK'perimental Vapor Pressure Temperature Coeffj,eients of Set 5-B TABLE VII (cont.) I I 0) 0) -87- TABLl~ VIII Coefficients .for Propane Obtained Using Critical State Restraints arid Phase Behavior Date Coefficient 5-A* (;. ** :;;-B Ao 12559.9 10553.9 -lli45'--")(; .o Bo 0.'375279 o.U.8506 -14.3549 7?..94.90 7634.24 .30854.6 a 77336.0 108172.0 514882 .. 0 b 7.95782 10.2395 45.3994 C 22J20.9 30260.8 99747.1 a. 1.48371 1 .. 256.33 0.917335 y 4.oocoo 4.00000 4.00000 0 0 X 10-6 * Cf. Table III. ** Set 5-B differs from 5-A only in the respect that the method of Lagrange undetermined multiplie:rs \ms applied to make the coefficients with the Be."ledict oquation satisf"<J F4.uations 78, 79, and 80. -88- 1'A.BLE D: Ratio of Predict,xl Dew-r--o:tnt and }sri.bbJ.e-Point Fu.gacitiea for Pro1~~ne Using Coefficients of Table I Temperature fd/fb ln(fc/fb) 100 1.01095 0.0108905 110 1.00793 0.0078987 120 1.00540 0.0053855 130 1.00420 0.0041912 140 1.0035.3 0.00352:,8 150 1.00371 0.00370.31 160 1.00451 0.0044998 170 1.00571 0.00569,38 180 1.0074.3 0.0074025 190 1.00755 0.0075216 (deg. F) -89- Pr.cd:tctcc1 Cr1.tic.s1 P..t'OJ'JCl-"tios Press-.n-e Temperatu:re Molal Volumo {r, .~.i.a.) (doc. F.) buSt..)/(lb.mole) 716.0 90.1 Table I 1166 100 P.cnedict 718 91 Experimental 614.7 206.J , . 20 Table I '703 224 3.4.8 Benedict 621 207 3.20 ~=' (. ~'!'_;e"r'~tne21· -.,t: .... .. • '~ua, ·- 550.7 305.6 4 .lJ Tablo l 595 315 4 . 56 Benedict 551 306 4 . 27 4,94.2 ,387.4 5.05 Table I 535 ~7 5. 57 ·Benedict 484 384 5.67 Com,_00U1"1.d Ethane fropane n-Butane n-Pentane Experimental -90- TABLE XI Variation of Critioal Properties Predictions with the Magnitude of Gamma Compound Propane n•Pentane Pressure (p.s.i.s..) Temperature (deg.F.) (au.rt.)/(lb.mole) 4.00000 705 225 S.50 4.24021 70Z 224 3o48 10.00000 728 222 3.04 9.40000 572 402 6.35 10.50000 535 397 6.57 12.00000 525 395 5.56 Gamma Molal Volume LIST OF FIGURES 1. Partial Sums of' the Power Series Representation of Reciprooal Molal Volume as a Funotion of Temperature and Prossureo 2. Seoond Virial Coefficient for Propane. 3. Experimental Data 0-sod in Evaluating Coefficients for MethaneD 4. Experimental Data Used in Evaluating Coef'fioieuts for Ethlineo 6. Experimental Data Used in Evaluating Coefficients for n•Butane. 6. Experimental Data Used in Evaluating Coef'tioients for n•Pentane. 1.. ' Effeot of Moleoule.r Weight upon Magnitude of Coefficients of Table I• a. Ef.f'oot of :,:oleoular Weight upon Val_ues of Coefficients Proposed by Benedicto 9. Dependence of Standard Error of Estinate upon Ge.rruna. 10. DeviaUons of Compressibility Factor for Me thane. 11. Deviations of.Compressibility Factor for Ethane. 12. Deviations of Compressibility Faotor for Propane. 13. Deviations or Compressibility Factor for n-Butane. 14. Deviations of Compressibility Factor for n-Pentaneo 15. Predicted and Experimental Co,;,pressibility Fa ctor for n-Pente.ne with Pressure as the Dependent Variable. 16. Predicted and Exr-erime ntal Compressibility Factor for n•Pentana -with Molal Volume as the Dependent Variable • .l7o Calculated and Experimental Values of Fugaoity of Propane at 160° F. 18. Deviations of Vapor Pressure !'or Propane and n-Pentane with Table I Coeffioianta. -92- 19. Deviations of Vapor Pressur e for Propane and n-Pen t ane with Benedict's Coefficients. 20. Deviations in the Vapor Pressure ? re dictions for n-Pentane as Functions ot Gamma. 21. Comparison of Predicted and Experimental .M olal Volumes a.t 160° F. 22. Comparison or Predioted and Experimental Compressibility Faotors e.t 160° F. 23. Comparison of' Observed and Ce.louleted Fugaoities e.t 160° F. 24. Predicted and Experimen tal Isotherms at Critical Temperatureo 25. Pressure Correction Term for Several Temperatures. 26. Deviation of Vapor Pressure for Propane with Heterogeneous Correction Term. 27. Effect of Heterogeneous Correction Term on Prediotion of' Critical Isotherm for Propane. 28. Coefficients of the Critical State Restraint Equations as Ftmotions of Reoiprooal Molal Volume. 29. Geometric Solution of the Critical State Restraint Equations. PROPANE. ~ l90°F. ::) u 0.8 - - - - - - - + - - - - - - f - - - - - - - - + -- ........ - - --+---- - - - - + - - - - - - ' - - - 1 (» 0 E ~ 400 Po 600 000 PRESSURE, p.s.i.a. 1000 -94- Benedid, Table Coefficients Benedict, Benedict Coeffic'1en1s Beatt\e -Bridgeman \IQn der Wools ... --e Hirschf elde 1"( 16) If 0 - 2 8 - - - -- -"-. ~ - --- ~ 1M z:i: - -------cJ co -- 2 4 - - -- .,. .ci.. ""' g -- ~ 2 Q t - - - -- - - + - - -ft - -- + - - - - - - - - -~ IQ I aoo ~-- ....... - ---.--- - ~ 400 Second Viri a l Coe fficient for Fropan~. -95- 2~000,------,----,------.---,----,------, 0 MICHELS AND NEDERBRAG T(34) 0-OLDS, REAMER, SAGE, AND LACEY (35) 9 KVALNES '.A.ND GADDY (36) :-t"'<"<"==- : ~~h 0- <jb- 0- 9 0-~ ---000- 0- 0- 0- 5000~-----.l.'-'---___J:.):=___:O-~----L.r... z d 9 90-- 0 0-- 9 9 (/) a: w a. ro· ~ ca cf'" 0 9 0- 0- 9. 9 9 0- 0- 0 0- 200QIY-....J,.L___J._.1+--Q-M--.~i--cu::L----LO- w' a: ::> (/) V) w a: Q. I000 ,.,.._--Y---+---v--=-f-------+.-~'-0----!..J-----+------l 9 9 0 0 0 9 0 0 0 9 zoo L . . . - - - - ' - - - - ' - - - - - ' - - - - ' - - - - - - - - - - - 0 200 300 400 TEMPERATURE °F • Fig,3. Experimental Data Used in Evaluating Coefficients for Methane 1~000.---~---.----~~-T7""~"""""---.-,---.-----.-----, 00- 00- 00- 00- 00- 0- 0- 0- 0- 0- 0- 0- 0- 0- 0- 0 O 0 O o O-o o~o 0 g_,0 0 =o 0 0 o o o -8 9-0 -0 0--0 z 2000 1-------9==-,,A{)cJ.-~e----'~~i.._;l,,k-~:f--=>-l..--f<?g-.,+-----f 0 c/'l a:: w a.. ai _J I000 1 . 4 - - - - r - 0 > - B - - - - h - J U - - - - + - - - - - = - - - ~ - - - - + - - - - - - - 1 -0 w a: -0 9· :::) (/) ~ a:: a.. -09 -0 500 l---~.--1+--=u.._-+----+--f{.>----+----+-----1 -0 -0 -0 -0 CRITICAL STATE VAPOR PRESSURE -0 -0 -0 -0 -0 0 BEATTIE, SIMARD., AND SU~~ 0- REAMERJ OLD SJ SAGE.J AND LACEY ~8 ) 9 REAMER, OLDS, SAGE, AND LACEY (38) --0 BEATTIE, HADLOCK, AND POFFENBERGER.~) 100 .___ ___.__ _ _--4-_ _ _.1...-_ ____,1._ _ _...J...._ _____,J 100 200 300 400 500 TEMPERATURE Fig. 4. °F Exoerimental Data Used in Evaluatin~ Coefficients for Ethane -97- z 1000 d (/) a: 500 w. a.. dl ..J 200 w"' a: ::::> (/) (/) 100 w 0: Q. 50 10 t - - - - - + - - - - - + - - - - - - + - - - - - - + - - - - c - - - - - - - 1 0 BEATTIE.1 SIMARD.1 AND SU (40} 0- SAGE AND LACEY(33) -0_ _ OLDS.1 REAMER..1 SAGE~ AND (41) 5.___ _ __.__ __.__ _ ___._~_ _..,.::.,._ _ _LACEY _..___ _ ____, 100 200 300 TEMPERATURE Fig, 5. 400 500 °F Experimental Data Used in Evaluating Coefficients for n-Butane -98- 10)000 -<;> -0 rr 5000 -0 -0 --0 --0 -0 -0 -0 -0 -0 =6 -0 r -0 -0 -0 2000 -0 1000 z (j (/) a: 500 w CRITICAL STATE Q. -0 ID ..J 200 ·-O "\ w a: ::) V) 1/) 100 w a: Q. 50 Cr 0- 0- 0- 0- 10 l---- ----+-- - - - + - - -- t - - - - - - - 1 - - - - - - + - - - - - - t BEATTIE) LEVINE) AND DOUSLIN~ SAGE AND LACEY (42) SAGE AND LACEY ( 43) I __ 1 _ _ ___, 5 100 200 300 TEMPERATURE Fig. 6 . 400 0 500 r Experimental Data Used i n Evaluating Coefficients for n-Pentane .. 99 .. 10,000 p-- ___ -Q - / '\ / / / ,o/ +/ V // / / f- z w ,o / t-· / ~/ / 1000 / 7 / / 0 0 I I I 0 I 0 I f- I 100 -~/ / di" d. ...... <( 2 / / / ,o ,,.." / '() 't;.,,,,, / I .,," .,,.. / ...... ...... ...... / / D" ,., ,., / .,, d / .-::£ ,..-:;_\o rY'/ //" ... ,,.g~,,.,. // / Cf / / ... ,,....()--- --__ ,er -- .r, --- - ,.. _o- -;_ -- / 20 60 40 MOLECULAR Fig, 7, // ,.. --✓' / Cf' ,,,,,, ,., ,,,,, c1. ~-'~.:::::..~/ - - - ..,. 3 ,.._,,..fy ;.-~ ,o- / / ,,,;:8 // // / / / ,../ ,"' ,....., / / 0- - ,,()-- ---:;,L ___ -0 .,, ,,, ,.,,.,.D' €lo .,, / ;,-.._-(Y 10 0 / I/,/ (.') ; '1 ,,ff Go /. / z I I _,,, ~- Cf / / I ------ / O" I I w / / p--; / I lL 7 - ---- --g ,,,0-- / / ,o ,.,-'-,.. / w :::> / i. / / / , 0 lL lL /0-- ,,cf WEIGHT Effect of Molecular Weight upon Magnitude of Coefficients of Table I, -100- ,0 / ,,"' 10,000 / / / J::f ,,1 -,..\0,, C, / / / / / --~ / ff / t- zw ,, l) lL 1000 ,, LL w u I I Jf I w I / I 100 ~r- I z 0-~ I l'.) d /, ? <t: ,E ,; ~ - .... Cl' ..... ., ,. / g>., -- ,.. _,, / / --✓ O" (Y ..- ,/ --20 ,. ., --0 - -- - --0 ., _-0 -- .0- -==-----0 'I- ,9__..- -::....o::: .... - ,o-:, cl..,,,.., ..., .,o- ., ., ,.. ..... ., - _-0 -r_ - - - -0_.o--..., ..., 40 MOLECULAR Fig , 8 . .... ,.. .,()" ,.. ., 0-,,,,.,- / / ..... ,.. ,.. ,,. ., ,. .... ,/ \ ...... 'O /,.. ..... ,..0-- --- .... -,,cy f>..o'I- / / / -,.. 0 1--- / ✓, '✓ 10 -2 'C:.--0_o--- -- Bo X - g-/ , / ,; ~ f / {y ,.,,,,, 1/ I 0 ,; -1 I,,-- ✓ 1,, ..... ,g~i;,o I I ........ ---~' / -1-'0 _..... C,o ., -=-z., I i:.... 0 j, / p' 0 ::::, t- ,, ,, 60 WEIGHT Effect of Molecular Weight upon Values of Coefficients Proposed by Benedict -101- w .... < ~ .,_ (/) w LL 0 l'l-PENTANE 0::: 0 a: er: lU E 0 c:: < 0 z < ~ 0.01 w C) < a:: w > < 2.5 5.0 / Fig, 9 . 7.5 10.0 ( cu. FT. PER LB. MOLE) 2 Dependence of Standard Error of Estimate. upon 12.5 7 -102- • 0 I I Ze-Z = P(¥e-¥)/RT Ze - Z = ( Pe - P)':( /RT - 0.015 DEPENDENT VARIABLE p V 0- .. 0 • 0.010 TEMPERA TU RE OF 100° 400° 0- ' QI N -- e= N o.oos 8-- C 4 z 0 I<( i 0 > w .... 0 -0.005 -0.010 • •0 0 8 2500 5000 PRESSURE., Fig. 10. c- 7500 101000 LB. PER SQ.IN. Deviations of Compressibility Factor for Methane -103- 0.01 0 -+0.01 >- • ....• .. .. ~ @ 4L. 0 4~ • 0 .---u- N 0-0'a,-0.02 tN • Z - Z ==(P. - P)¥)V/RT z 0 ~-0 .03 t - - - - - - + - - - - - 1 - < > ~ ze .... Z P( Ye - e e o TEMPERATURE DEPENDENT VARIABLE 0-- p 00 ¥ ► • /RT OF I 00° 0 400 -0.04 r------t-----+----+------+----_j -0.05 r----+----+-----+------+----_j 0-2500 5000 PRESSUREJ Fig. lle 7500 10.,000 LS .. PER SQ. IN. Deviations of Compressibility Factor for Ethane -104- 0,010 0 ( , o.oos .... ~ 0~ re- N t_o~ ·-- I (1.J N 0 ---0 z • ..... --' • > w 0 -0- 0 0 4 -1 -< DEPENDENT VARIABLE p V ~ -0.005 -0.010 ... ~ ... ► • -0- 0 .... :r ...• TEMPERATUP-.E OF - 100° 400° I -0.015 j 2000 4000 PRESSUREJ 6000 8000 LB. PER SQ. IN. Fig.12. Deviations of Compressibility Factor for Propane with Author's Coefficients -104- 0,010 0 ( . o.oos -v- ..... 0~ r+ N t_o~ ·-- I (1J N 0 --0 z ..; • ..... • VARIABLE p V > w 0 -0- C 0 ◄ ~ -< DEPENDENT ~ -0.005 -0.010 ........ ~ ► • -0- 0 .... ► ...• TEMPERATUP-.E OF - 100° 400° I -0.015 j 2000 4000 PRESSUREJ 6000 8000 LB. PER SQ. IN. Fig.12. Deviations of Compressibility Factor for Propane with Author\! Coefficients -105- o.oso 0.02 I- D- .......... ,..--..,. 0 ~- - ;5a~ II N I ~ QI N -0.01 z >o ,---...,. -· 0- I QI 0. ........... ~ >o >o ...__.... 0 .025 ~ 0.01 a: -- -- ·~ • ... ~ • -~ 0.. 0 ....__,, " N f 0 Q) N -0.025 z 0 0- 0 I o..CJJ t- t- 4: > > w 0.050 o 4: w -0.02 0 0 .03 DEPENDENT VARIABLE TEMPERATURE OF &-0 '- I 00° 400° • 0 5000 7500 PRES5URE, LB. PER 2500 Fig. l3e -0.075 10,000 SQ. IN . Deviations of Compressibility Fact or for n-Butane -106- • I • I Ze- Z= P(¥e-':{)/RT Ze- z =(Pe -P) ¥/RT 0 0 .02 ~ a: 0 .01 --'-.. >o I >Cf. ...__.. a.. 0 JI - • • •• B •tboo er- .... N DEPENDENT VARIABLE p "i 0-0 • -- TEMPERATURE OF i 00° 400° ---- ~ 0.. I () - 0 ... a..<JJ ...__,, II N I QI N N -0.01 -- 0- z 0 ~ <{ 0 a: >o QI > w f- '-.._ _ I 0.1 0 t- <{ -- (')... -0 .02 l-0.1 . z > w -0.2 0 ◄..- 0- -0.03 02500 5000 P RE S5 URE J Flg. 14. 7500 t 10.,000 L B. PER !:°>Q. IN , De.vlat:t. 0 11s of Comp 1: 0 r1sJ h :i. I J. 1.,v Fq,,t.,, .1, fc,r p ... 1'011!.A,,., -107- 3.~ 3 .0 a: PP.ESSURE DEPENDENT VARIABLE 0 t--- u <i u.. 2.5 >- !:: _J 2 .0 [I] vi w (/) a: a. ::? u 1.5 0 1.0 2.5 a: 0 f-- 2.0 u <i 0.5 u.. 1.5 0 >!:: _J C!l vi "'a:w 1.0 a. ::;; E XPERIMENTAL 0 u 0.5 2000 4000 PRESSURE, Fig.15, 6000 LB. PER 8000 SQ. IN. Predicted and Experimental Compressibility Factor for n-Pentane with Pressure as the Dependent Variable -108- 2 .5 1 - - - - - f -- - - - 1 - -- - + - - - - +- - - - + - - - - - + - - - - - + - - VOLUME DEPENDENT VARIABLE EXPERIMENTAL AUTHOR'S I >- 1- ..J ID 1.5 1 - - - - l - - - - - - - - - - ! - - - - f - -- - - +- --,~'-------+------+-----L (/) (/) w a: (l. 2.5 :E 0 \.) a: 0 I- 2.0 u ~ >- I- 1.5 :J ID AUTHOR'S (/) Ill w J.O a: D. ~ 0 \.) 0 .5 2000 4000 PRESSUREJ Fig. 16. 6000 LB. PER 8000 SQ. IN. Predicted and Experimental Compressibility Factor for n-Fentane with Molal Volume as the Dependent Variable -109- 0 z EXPERIMENTAL PREDICTED SET A 600 d V) a: w Q. fl:i __J 400 >- I- u <( ~ <.:> 7'\ ::> LL 200 V 1/ 200 v- EXPERIMENTAL \__ PREDICTED 400 PRESSURE Fig. 17e LIQUID l VAPOR VAPOR 600 PRESSURE PRESSURE - 800 LB. PER SQ. IN. Calculated and Experimental Values of Fugacity of Propane at 160° F. -110- . z . CJ 11-PEN1ANE 0 '\ /.,... Cf) a:: w \PROPANE a. . m I. - - -10 1 J VCRITICAL_n ..---._"\ :::: a. :::: I -20 a_GI \...../ 100 Fig.18. 200 300 TEMPERATURE Deviations of Vapor Pressure with Autho~& Coefficients 400 °F for Propane and n-Pentane -m- . z 5 a f/) I a: ...... la.I a. ~ 0 d) ..J "' ~ ~ Cl. I -5 ~Cl) PROPANE ~ ' .......... ~ 6 100 200 n-PENTANE "'~ 400 300 TEMPERATURE °F Fig,19. Deviations of Vapor Pressure.for Propane and n-Pentane with Benedict's Coefficients -112- . . CD 100°f. a: • .. 0.0 t - - - - - \ t - - - - - - + - - - - - - - - - - , ~ = - - - - - - - - - , - - + - - - - - - + - - - - - - 1 "a. I , -10.0t---- 9.& Fig. 20. 10.0 8AMMA •Y• 10.s 11.0 (CU.FT.) 2 PER (LB. M0LE) 2 11.s Deviations in the Vapor Pressure Predictions for n-Pentane as Functions of Gamma. -113- 0 w EXPERIMENTAL --PREDICTED SET A - - PREDICTED SET C ..J 16 0 ~ r:!.l ..J a:: 12 POINT w 0.. i-..: I fJ... ::5 u EXPERIMENTAL VAPOR PRESSURE 8 w ~ :::) ..J 0 4 POINT > 400 800 PRESSURE Fig. 21.. 1200 1600 LB. PER SQ. IN. Comparison of Predicted and Experimental Molal Volumes at 160° F. -114- O - a:: 0 0.8 - -- 4 - - EXPERIMENTAL PREDICTED SET A PREDICTED SET C I- u<{ DEW 0.6 >- f- / _J - (lJ en (/) POINT - -++ ~---• lL -EXPERIMENTAL VAPOR, PRESSURE 0.4 w a: !l. 2 0 0.2 u BUBBLE 400 800 PRESSURE POI NT 1200 1600 LB. PER SQ. IN. Fig. 22. Comparison of Predicted and Experirr,_ental Corr pressit..ilii:y Factors at 160° F. -115- PREDICTED VAPOR PRESSURE SET A z 300 cj (/J PREDICTED PRESSURE a: w VAPOR SET C a. a:i _J I 200 EXPERIMENTAL VAPOR PRESSURE >- f- l) ~ (.!) ::::> lJ... 100 O 200 400 PRESSURE Fig. 23. EXPERIMENTAL - - PREDICTED SET A - - PREDICTED SET 0 600 LB. PER SQ. IN. 800 Comparison of Observed and Calculated Fugacities at 160° F. -116- I 000 r - - - - - t - - - - + - - - - + - - - - ~ - - - l - - 1 - 1 - - - - l i cj U) 800 a: UJ 0.. o::i __J // // -- --0- 600 ;-u w a: :::) ~ w 400 i---------cf'-+----+----+-----+-------+-------1 a: a. O Ex'PERIME NTAL --PREDICTED -,---SET --SET Fig. 24. SET A B O 0.1 0.2 0.3 SPECIFIC WEIGHT LB. MOLE PER CU. FT. 0.4 0,5 Predicted and Experimental Isotherms at Critical Temperature . z 60 0 1/) er: a. w 50 d:i J ... 40 z 0 t- u ILi 30 a: ex: 0 u IJJ 20 a: :::> ~ ILi a: a.. 10 0.1 RECIPROCAL 0.2 0,3 0,4 MOLAL VOLUME, 0.5 0.6 LB. MOLE PER CU. FT. Fig. 25.. Pressure Correction Term for ~everal Temperatures z a 0 0-- - u> ---<> - a:: w a.. crl ..J -10 a. I ::: 4, 6 -20 --WITH HETEROGENEOUS CORP.ECT ION TERM - -WITHOUT HETEROGENEOU5 CORRECTION TERM - - - BENEDICT 150 120 180 TEMPERATURE °F Fig.26e Deviations of Vapor Pressure for Propane with Heterogeneous Correction Term -119- ci. 810 ·- . . Ill ~ 6 0 0 1 - - ---------+---1-t---ccc-+-- :, II) • - en fw 590 wi111,ul t10rr,r;i,,n ,,.r,,, 590 • ~,,,,,..;mNII ,2os°F.:-+-----t o.s 0.2. 0.5 0.4 RECIPROCAL MOLAL VOLUME, lll.11ol, p,r cu.f+. o., Fig. 27. Effeot of Heterogeneous Correction Term on Prediction of Critical Isotherm for Propane.· -120- o.29 o.30 0.31 RECIPROCAL Moi.At.. Vot..UMli Ccu. ft.)/ ( lb.,nole) Fig. 28~ Coeffi.cients of the Cr iti (;al :3tute i-:estrF.d.,:t. E:quatior.s as Functions of Re ciprocal iJ.olal Volu:-:;e. -121- 4.♦ 1 - - - - - - - - - - - t - - - - ' - -- - - t - - - - t - +- 4,0 - - ·-·· --- - ,,0 >.. 3.8 - -- - ---- - ♦ ~-"" ~ . 0 ,c le >< 3.6 ·-··-·-- ······ - -- - · - .. 3.2, - -- '4.5 Bquaticns. l60 16.S -122:- APPE;TDIX . A Aleebra or Pcr'der S~ries B Der:b,a.tion of Thermodynamic Expressions o C Soltition of Systems of Inconsistent, Linear tu.gebraic F.qua.tions • • • • 123 • 126 • •. 131 Least Squ..-u-es .'1.,"'lalysis of the 1inear Part or the Benedict Equation • • • • • • 135 E Description of the Digital Computing Equipment . 140 F Application of Benedict Equation or State to Propsno • • • • • • • • 143 Characteristics of the Critical State 148 D G • • " • 0 • -123- Reciprocal of' Conve1•r;ent Pm"-er Seriea. - • ~- 1 • Let then (112) But goto + io(1x + ~o(zX8 + ••• titer ♦ ~1t1x2 + ~1(~3 + ••• ~~o'X2 + t:a~1x3 + ~t2x6 + (113) ••• ••• The following f'orm.ulae are obtained by equating ooef'fioienta of' like powers of x 1 i. e., sv.rnr:1ing along the diagonal ,e..c"l.d then ,'V?"i ting ~o{o fot1 ::, l + ~1ro ~ote + t1J 1 ••• OJ. + 0 ~,aSo ,,. 0 (0 <# ~-1 0 (1, 0 • S1.f,O rs a -2 ~2~•S l. 0 (114) - f 2 ~-2 O ••• Positive Into 0 ral. ':'0 0,vars of Converge:1.t Power Series,• Sinoe (115) -124- then lim (11G) m-00 e go + E1 x + ~:ax2 + f:y::a 2 ,.. cf~xi) Isl) ,2 + i 0 i 1x ♦ ~~ + i 0 E:,yc3 + ••• ••• • <t~xi) 0 ♦ f:a,foX + + ! 2 to)t2 + f 2 f 1 xa • ♦ + ••• 'sfx2 + ' 1 . ~ + ~ifc;X4 + ~ix4 + s2faXe + ••• (118) (119} ••• (120) ••• n In eve r"IJ c o.se (121) Let y ... £(x) be convergent for (122) Yo Ix - x 0 \ £. r and if' x = f Slj ox X~o x0 -tl1en (123) 0 ns a, i\1.no,don of' y Clearly, the power series defines x well as y Consider a ama.11 neighborhood of -the point as a function of x • y O throughout whi ch x valued ac,d dif'ferentiable. as is defined arid single- In such a neighborhood x nay be -125- (124) Taylor's t},oorem assures that the radius of conver{;cnce or thia 11ower series is greater tha.i.""1 zero. Y • Substi tu.ti on of l:½tua.t ion 124 (125) Yo The radius of oonv~:rgenoe is equul t~o the distance f'rora y 0 the neqres"'c; singularity in the oa:n 1}1ex plane or y • At a Singularity• one of' tho following situations oooursi 1) x beco:,;:as multivalued, 2) x of x with respect to y bec01:1ea in!'inite• 3) -co the derivative does not exist. Since power series may be multiplied within their circles of convereence to give convergent power series, Equo:tion 125 may be oxpan.ded and te:rr.18 oollected to Give = ~1 r1 l ~ 2 (~ + ~1 t3~f + 2~2 r1t2 s's <1 Q t-1 l. .3 "' 0 + ~t3 • 0 t2 • .c2 t,. t3 = (2;: - ~~3)'6;5 (126) -126APPENDIX B DERIVATION OF TEEThKODYNA1.1I C E.Y..FRESSI ONS Beattie(47) has desoribed t.he "general lieut rnethod 11 for deriving expressions for 'thermodynamic properties in terms of heat capacities and volumetric properties. ·rhis method may be a pplied to the Benedict equation direotly as Beattie has done with the Beattie-Bridgeman equation, but a more convenient approach uses the six canonioal fu.~otions defined in Section II. The distinctive feature of the "general limit method" is the applioa.t1on of a limiting process whi oh permits all changes of temperature on tbe system of interest to be considered to aot on a perfeot gas at in.finite attenuation. The fa.ct that real e;ases at infinite attenuation a.re not perfeot gases with respect to residual volume and Joule-Thomson ooeffioient must be observed in applioationa or this method. This difference is of signifioanoe in relatively few appliaatio?l8 or this method. Infinite attenuat~on• literally speaking. does not exist, but certain thermodyna.."1io properties do approach finite limits as the pressure on the system is decreased without limit or the volume oontaining tmit weight of matter is inoreaaed without lL-ut. A system is said to be at "inf'inite attenuation" when the speoifio volume 1s so large that the properties of' the system differ imper• ceptibly from these limiting values. Equations 58 through 63 m.a.y be derived in terms of the Benedict equa-tion simply by partial differentiation. The derivation of Equations 40 thrm1gh 43 ia not so obvious however. In a multi- oomponent system, the fugaoity, fk:, may be defined in terms of' the -127- chemical potantial, equc.t:i.on = RT and the hou.7lda.,·•·•y conditions: isothermal path and fk - (12.7) P ae V ---- 00 a.long an = ~k P·* • This di.ffe:rent:1.a.l eqna:tio:n rit:tj~ be f k* rewritten as an integral equation where the path of :tntegration is an isothermal one. 1pii- p ihlk) dP P Tm , (~) aP dP T,m or (129) RT 1n fk sinca (130) This definition of the relationship between i\~ and H- k avoided the introduction of a temperature dependent constant or integration since all integrations have been isothermal. Equation 129 may be wri-ttan in the form RT 1n p* - RT ln !!.~ (131) -128- uo* r.k = (132) there.fore {133) RT 1n p* and {1.34) or l T sf~dT (1.35) T-0 An expression for the partial molal volume ca."l oo deduoed from the equation de.fining the chemical potential. (136) Differentia.tion -wH.h respect to p3. essure o.t ccm.sta.,.rt tem1:J9ra:l::.ure 0 and composition gives (137) -129Substitution of Equation 129 into E.-:;_uation 137 produces (1.38 ) Fl..om Equation 12:9 c-~'fk) W'" P,m R ,_ ~ LU ... k - - R 1n J.k .,..~ + nm n~ ~ ln "" ,(139) .;.k) { v a T P,m But (140) Hence, Since g_* •K - ,.,o* - R 1n ~ (142) - 1n n, (143) ~c and ln fk* ln p* + •lt ~ - - ~• = R ln p* - ! { RT 1n rk + re? ( a ln r.k) T oT } P,m (141✓) The partial molal enthalp--J is readily obtained from E-quo.tions 129 and -130- At ccnsta.'1t composition s.nd in t he absence of exter nal force fields, a single phuee w..1lticor.n.r,onent system has ·two degrees o.r fz-eedom; hence, dcr • but at constant pressure - (~ oT'P,m dT so that (o P/a'.i.')<r,m ("oP/dg-)T,m Equat::.ona 58 through 6,3 follow from these :relationships. -131- SOLUTIO!J OF SYSTE~,S OF D JCONSISTENT, LINEAR A!1iP.DRAIC EQUATIONS A syat.em of m line~ oque.tions in n vor iablcs ia said t,o be consistent (L~3) if these !'.!. equat:i.ons ha~1e at. l 0ri.st one oon..imon solution. A set fox- wldch no oommon solution ei:ists is called incon- sistent. Let K represent the the m equations, ;p.; mx n mo.t,rix. of t he co(-:lffici0n'ts of represent the veo·tor (o:rdered set or m.2n1b0rs) of the variables , a11d k represent the vector w11ich t!.1e pi~oduct K..1~ is presumed to a.pprox·imnte, for example, e...'\.'1:ierimenta.1 ob.servations. C,-4 9) This system of equ,!f.tions will have at least one non-zero solution for x if ·the rank or berth K an.a. 1.ta augmented ma:l:.r:b:: a.re the sa'1l.e and less than or equal to n. In the problem of f'itttng emp~~ical cxpTossiornJ to e}tP',)rit1crrta.l information, systems of linear equations uhen obtained are generallysuch that t:he rank of the augmented matri:r is g:reatGr than n. Since no non-zero ~a.ct solution :ls then jx)ssiblo, a criterion must be chosen to measure the degree to which vru.~ious sets of values for the variables approach e. solution :in the l"eapect of satisfying the equa- tions preciaelJ·. The least squares critor ion is most genorally ,used tor this p-J..."'1)0se. It selects 'that se·t of values f.or the v111·:i.r:i bles (represented by the vector x) Ill sum which yields the m:1.nimu..'ll value ro1-- the ') ~ (1½_x - k 1 )"'. •=1 A little algebra will ohow z-ead:IJ.y that app11~ea- tion of the least squ~~cs c:riterion is equiv-al0nt to substitut. ·i 11g t he -132- KT K x = KT k (150) Eqtw.tion 1~ has a unique solution, by Cramer' e r·de II if the rv.nk of KT K and its au.gmented matrix are precisely n, whk..h is generally the ease. ~m1tion 150 l"'€lpresents, in matrix notation, the w:mal equations of the lee.st aqua.res method. other criteria can be used equally well f:or csta.blishins the best, fit for an empirical equation to experimental obae:.::'7ations. might oo the minimization or One m fu' \K1 x - 7½_1 • This critm... ion has the advru.1te.ge of app~~ing less weight to the effect or the least typical equation ·l:,han does the method of least squares. I-Iolie·1re:r, this criterion does not imply as direct a method for obtaining values for the eomi-x:,nents of x as does the met.hod of least squm.•es. The general method, usi11g this criterion, would select from the total number. of ways in wr-..ich m equations can rom n 'combinations that ~8:'.f which supplies, b-'J such combinations• minimum valuo of n equations yielding a solution giving the &i IKj_ x - li:j_l • This is substantially tho method of averagee(49,50). Either of these methods yields a U."lique eolut:i.on for the i,"loon- sistent equations problem. 'l'he method of lea.st squares ho..s the oovai1tage of directness in obtaining that solution most appropriate to its criterion. '.t'he method of avora.ges has the adva."ltnge of ease of caloul~J.tion :if' one is satisfied with a fit which is only near the fit most appro:pria.te to its criterion, The latter method becomes exceedingly' -133- statistical met:1ods µrovide for solvinc syster--.1S of iEconsistent l:'l.ne&" equations, once a particular system hae baen choson. · Esoont:lo.lly the same considerations apply for systems of non-1.:1...--iea:.r equations except that. values of those variables which occur aa the non-linear ele111ents s.re established b;r iterative methods. In general, the norcID.l equations will be satisfied b:r more than one set of values for the variables if any of the variables oocur in a non-linear t".lB..,'UlCr s:i.nco the varim1ee of est:1.mate may have several relative stationary p:,ints. The se:tect,ion of the system of equations to be solved in emp:lrical cu:n~e fit-ting must be L--:it,uiti7e for 5.t depends upon non-mathen!ltical cons:~de:r·at:1-ons -o f the reliability of the k terms and the elements l½_j of the matrix K. tagranse UB4etermined Hultiplie;:s. The method of tar.range undetermined multi,lie!"S is o. device for d~term.i.riing constrained rr.iniraa. described by an example. The prL"lciple can be conveniently Let the sum of the aqua.res of the devia'Gione ' of the experimental comr.ressibility factors from those predicted by~~ empirical equation or state be denoted li'J the symbol A. The convontiom.1.l least squru."eS method seeks to determine that ;Jot of values for the coefficients in the erlpirical equation which minlr.J.zG condition that or A necess-2:ry A take its min:trmmi value is that all pa.l"tiul der:tvc:t:tv0s A' 'i:lith respect to t :1-esc coefficientr:: 'c-e neroo -134- of Equation 80 mi::;ht be suet,. a fu.n.c'tion B. If 21 ts a constar:rt,, independent of the values of the coefficients, thN1 (A equal -t,o A for any particular set of values for the pr1.ramct0rs. The equs.tions wh:l.cL equate the partial derivatives to zero have solu-t:1.ons vhich depend upon tl1e undetermined rrrul-tipller dete:rminoo fron J&he r act that m. But r:i oon be B is identlca.lly zero and the para.'1l.et.ers which correspond ·~o 'the cc:.:istrained, minimum can be determined. The method can be extended to several reatrai..'l'lts provided only that the number of restraints does not exceed ti1e ::xu:nher of parameters. 1cs discu~s t hi.s method in de"ba:11. -135- LEAST SQUARES A!:ALYSift') OF THE LINEAR P.:.•RT OF ".;:'EE m.:r:rEDICT EQiJA.TION let .63 be a row vector with comr-.onente .&j1Cr,T), (j=l,2, ... ,m. 'l'he difference between the axperimentally observed value of thG quantit y Z and the value, z8 , predicted b-,1 the equation which hae been µ.~o]'.X)sed t o describe Z in terms of tho exper1.menta.l quantities (J" • and T :ts gh--011 by the following expression. (151) I£ ~ represents the me.tr~ whose rows a.re the vectors .&j then Equation 151 may •be written, (152) and the sum or the squares of the (Ze - Z) .,1 •e iss N ~(zZ8 - where: A .. Z)j = (153) ~ T-!> (154) f!t Thus K1 A K is a positive-definite quadratic form and A is a symmetric positive-definite square matrix of order eight. i::i KTA K/N. The variance or estimate -136- To satisfy the least squares criterion KTA K must be at its minimum value, therefore: <a/oK1 ) KT A K • (aKla~) A K + KT <a~ta~) K + Kr A CoK/aK1 ) • 2 131 K • J 0 (155) (i -=1,2, ••• ,7) Tho matr:lx E is the mat,rix A less the row corresponding +..o i (155) 1:1 o. A·vector which satisfies the normal.equation BK • (156) 0 will also satis.f-3 T (157) KA K Equation 155 is not a sufficient condition flr defining a rela't-,ive minimum. To solve Equation 156 for the vector K, the elem.ants of B .,, D(O) are ~"'Te.nr,ed in the scheme: •• •• l. 0 •• b1,-, •• be-, bu.= l 0 •• •• b.,: ba! •• =0 •• •• •• • •• -137- m1! • • 0 0 l m.12 a = •bil/b2½i;, • •• • 0 0 0 2. !Ila2 = l • •• •• ri" •• •• •• •• 1/o~:s. $ 0 0 ?Dta~ - -~ l bu.* •• m.ri • ~. 0 • Q. 0 -,:;---al. ....... ______ mo:a: • -:ri . o • o 1 . Ms B 0 /b2~ ;r:, m.ra . -b.,~ • •• • 1 22" * ••• 0 ?-iota that boj ,.,, €loj • but is not an element of the matrix B. pivot element of reduction r is indicated by the foregoing process is the matrix ••• - - --,r----,,-----i_--0 rllo/2 Ill -b0Vb22 B7 = M7 l 0 •• 0 K2, 0 l •• 0 Ke •• •• B'l •• •• * . The The result of rfo which may be wr:i.tten ·o It., •• l ----------0 0 •• 0 boi 0 The general formulae for the reduction may be written -138- (158) {i/k) (159) Furthermore (160) A The largest minor 1n each B(r) in whioh no rows or columns conta5.n reduced piv~t elements, is symmetric. Thus, the elements of the ma.trices B(r) will be oo:notly the same if the reduction is dona by the following operations on the IBM~ computer which is described in Appendix E. l} calculate ~)* 2) oalcule.te ~ ); ~)* and punch it into every 3 ca.rd of oolunm j. .3) multiply (j ,. 0,1, ••• ,7) bt~j) / ~f2* by b~) (excepting i =k) 3 blj l) subtract the product from. bf > to obtain Disoa.'l"Cl all . ca.""'tis of column k. (r) bik The multiplier • uni is oa.rried in the computer storee;e this OY'Aration. :-- For i o k, bij (r) / "' b ij . t dur:ing h. (~.) * • -Klt Each of these ,three operations requires a different set of wired ps.nels for the computer. -139- During the reduction arbitrary Tows of B{r) may be multiplied by ten if such_ro-ws do not contain reduced pivot elem.en-ts and if the determinant is divided by ten for each such oparat:1.on after t!1e ce.lct1la- tion is complete. The average variance of estimate times the number of data sources may be computed b--J a short method. Using the pa..l'le:ls or the computer carry Kf as it carried and let the ca.rd cart--y _a 10 as it carriad · left b,--, ~ 1 • Th .•e step · 3) above. b.! ~ ) • . 3 bl~) '!'he cm·d field which bore comput er pu11c1hes blj) 3) let is -K -aiv-i whare it punc ~••ed bi(rj+l) in These quantities are SU!Il!'lled on. the tabulating machine. -140APPEHDIX E DESCRIPJ.'Iot! OF THE DIGITAL COMPUTL1G EQUIP''! E'.NT • ' The semi-automatic digital corn.puting eqt1.ipm.ent usGd in pe::f orm- ing calculations described in t his thesis consists of. t he follouing pieces of machinery; an International .Business Machines Corpoz-u.tion (IBM} Type ffi4 Electronic Calculating Punch (computer), an IBM Type 402 Accounting Machine (tabulator), and an IBM Type 521 Reproduciug Punch · (reproduoor). The computer consists of two electrically oon.nected com- ponents the first of which is called -the puncl1ing unit and the second the calculating section. The punching' unit reads information from • Hollerith-type punched cards at the rate of a.bout one hundred cards per minute and transmits this information by wey of the electrical cor~~ec. . . tion to the calculating unit. The pu.'lching 'l4"'1it c.lso p'l4"lches informa- tion which it receives !'rom the calculating unit i_nto t he 3.25 11 b'<J 7 .375" manila cards. • other functions of this unit include checking for bla~Jc * ~ields which should contain punching, indicating when the calculation is too ·long for _the calculating section to complete during the ullo-tted "calculate time" (unfinished program}, and indicating when insufficient $pace has been allotted on the card to receive the information from t he • caiculating section (product overflow). The calculating section p$rforms . . - the arithmetic operations of addition and subtraction. Multiplication is performed by the high-speed 'addition of the multiplicar1d or s~rae power ••. • ·The teraa field is used here to denote a set of card columns whieh, • when punched, represents a particular number. -141of ten times the multiplicand t,o itself a total nrn<iber of' tir:10s oquaJ. to the sum of the digits in the muUipl1.er. D:1:visfon is J-"Br-fo~~med by subtractions from the dividend in the manner of long di'i.7::ls:!.on . '?he • computer can obtain four separate five digit q_uo·tients during calculate time. The operations which the computer performs during one car d cycle are controlled by two panE)ls, one for each section, which are specially wired to complete the necessary circuits :i.n the machine . For many standard types of' problems these pe.n~la are wired per!lk"mently a.'1d are stored between uses, but for small problems it is frequently desirable. to have the panels wired only when needed in order to reduce storage space requirements. The reproducer will reproduce numbers from one card into a.~other card in the same or in a different field and will at the srune time originate punches which are to be uniform throughout a given deck of cards ·(gang punching). When coupled electrioa.lly with the tabulator this ; machine will punch information rec~ived from the tabulator. Usually such information represents sums which the tabulator has computed and the operation of ptµ1ching such sums with the reproducer is called sum.mary punching. Multiplication by ten is performed on this ma.chine by the obvious method of shifting the number one column to the le.ft in its field. The tabulator prints in.formation which it receives from cards, adde cumulatively either positively or negatively, and prints totals with or without transmitting this information to the reproducer for sum.'l'l.ary pu."l.chi:ng . The arithmetical and printing operations are controlled by wiring a panel similar to those used in the computer. The panel for -142- the tabula.tor may be wired to fit sevora1 d5_i"i'or·on-G pre1::J.0t1s :.:,-:;:J t bo switch". Auxiliary equipment includes a sorte1•, a l:0~7.1..11ch, ,:md 2. ccllutor. The sorter is able to separate the cards in a deck into 'th5.rtoon dif'f.'er- ent classes which oorres:pond to holes 4 n the twelve card rm-m a.'1-1 to no hole at all in a given ~ column. The keypunch is the pr:1.v:iru."y· device by which experimental data is tranelated into punched holea in cards. The collator merges two decks of cards in the sequences required for those calculations which require information from two different cards or for situatione in which the computer result ie to be punched in a ,'card different from the one supplying input information. A tY]:iical ex- ample is the.computation of the exponential function of the variable y. Cards bearing the values of y are merged with part of the exp::ment:lal function master deck in such a manner that the argument x i..11 th0 e1;:- ponential function master card is less than 0.001 greater th&'l "/• The exponential .ftulotion cards carry, in addition to tl->..e argument, exp (x), exp (-x), and the f'irst four central differences of these functions for intervals ot 0.0011n x. From such a. merged deck the exr,t>nentia.1 functions or x oan be computed by Bessel's formula(41) translated into viring 1n the comi;mter control panels. ppl t Equation pane LECK, J-. B. OPFELL, AND B. H. SAGE of Technology, Poaadena, Calif. _, equation of state d bin (6,7) is a satisf ric and phase behavio • urea may be describe "th the "general limit m affords a ready means of eval ties-Of BUch compounds and t jcient- for most industrial needs. tioa of state were originally esta.b tional numerical methods w • Such calculations, although tedio minimum of effort by the use of av put~ equipment (10). Statistim~l efficients for an equation of state for ni ~edict (9). Recently Brough an 'ipplication of these methods to th techniques (17), of the cQefficientsstate for methane and propane. I cients wete-obtained for several valu the equation of state. The purpose o to establish the effects of different sets o the agreement of the Benedict equation of and to determine the feasibility of applying as well as to gaseous states. purposes the Benedict equation of state may be ex- (no - :~ - Rc;a) d + ( b - ;T) d + 2 a- entuitive. d with a. ·tal comating coidered by nded the matrix d6 cd2 aa RT + RT3 (I + -yd2)e--Yt1• (1) conYenient to establish by least squares techniques (9) the f coefficients which corresponds to a minimum root-mean square deviation of predicted from observed pressures for a series of assumed values of the exponent 'Y of Equat· possible effects of 'Y upon the average standard of Equation I from a chosen group of expe,,-i"itfiflfl'te.t been· ·cated (9). Such effects were not inveeti ti nt there are available several sets of equation of state for many of the • • &I valul!II of Benedict ,7) aP, Several ate -144June 1953 0 0 0 6000 • J: u 4000 2 0 ILi a: • 0 0 0 < ::, eo 0 eo Ill a: ILi 0 0 2000 • 0 0 Q. 0 Ill 0 ::> Q. 0 0 •o •o ,i' • • •o 2 0 0 l000 • ul a: ::> "'<IIul 1351 INDUSTRIAL AND ENGINEERING CHEMISTRY 600 • CRITICAL • a: Q. • 400 o - REAMER, SAGE,& LACEY e-BEATTIE, KAY, & KAMINSKY 200 ...__..,...._...._ _ _..__ __.._ _ _....__ __. 100 200 300 400 TEMPERATURE Figure I. Experimental Data for Propane Used to Establish Coefficients the deecription of the volumetric and phase behavior of these compounds and their mixtures at pressures up to about 3000 pounds per square inch. At higher pressures large variations from experiment are realized and these coefficients do not describe the behavior in the liquid region satisfactorily. Marchman and coworkers (1 2, 14 ) computed coefficients for propene and mixtures of propene and ethane. More recently Benedict, Webb, and Rubin (4) reported values of the coefficients for 12 hydrocarbons including propane. In addition, a detailed description of the techniques to be followed in applying this equation of state to the. prediction of the thermodynamic properties is available (5, 8 ). Although these new coefficients extend the applicability of the ~ 0.016 Benedict equation to a variety of the lighter w hydrocarbons and t heir mixtures, the coeffiIL 0 cients are limited to application at pressures bea: low 3000 pounds per square inch. In addi~ 0.012 tion, the available coefficients do not necessarily a: UI yield the minimum standard error of estimate 0 from existing experimental data that 1D11J be ~ 0.008 obtained with the exceedingly flexible relation0 z ship presented in Equation 1. ~ METHOD O F APPROACH "'w 0.004 .., several matrix equations have been solved simul ously to decrease the computing time required. Some tails of such calculations are available (13). Modifications of the methods which have been described result in a decrease in the extent a.nd co ·ty of the numerical calculations when several values of 'Y a be utilized. When utilizing the r vise ethod (13 ) with a commercial electronic calculator ut nded sequential control, the calculation of the elem -original matrix required about 30 hours. The use of 316 card d a period of approximately 2 hours was theu, required the evaluation of a set of coefficients for an assumed value of The calculations reported here are based on appro~ experimental • ts. After the first set of coeffici tained other based upon the sa ta but for 'Y may be calcula n 45 min time. The method of a set volumetric measurements covering range of na which it is desired to apply the eq . Fi pressure-temperature diagram the experim studying the application of the Be propane. 'l'wo sets of data were e measurements of Beattie, Kay, and the unpublished experimental back olumetric behavior of propane at pressures up er square inch (15). A rather large range of pr ratures is covered by both sets of data and thus cq_ppare directly the agreement of the m ~ two sources in the region of overlap. the compressibility factor lees than that at eluded. A total of 83 smoothed points of Beattie et al. (.e) and 131 experimental P,O tit from the more recent study at highw pressures were inclitaed; It should be emphasized that the methods outlined here primarily to the estimation of the optimum coefficients for ~dftltion of thermodynamic data in the homogeneous regioq sarily yield the optimum predictions of va for pure substances or of the heterogeneous qq • mixtures. Benedict utili~4!!:Hhe data of Kemp and Eg-. (11) to adjust the coefficients to'·l'iJiprove the prediction of h ogeneous equilibrium. IDENTIFICATION TABLE I 0 6 (2) ? Qs) 0-0 3 ? 40 -0- < The least squares procedure to establish a set of coefficients for the Benedict equation of state which yields a minimum deviation from a particular set of experimental volumetric data has been described (9) . The present approach Q)Jows the same technique except that a: w ~ -4 -2 0 2 1' (cu.FT/LB.-MOLE) Figure 2. 2 AY~ Standard Error of Estimate of Factor for Propane aa a Function of Exponent -145- DUSTRIAL AND ENGINEERING CHEMISTRY 1352 TABLE I. Vol. 45, No. 6 each of five different sets of coefficients correspondin to widely different values of 'Y· The • ed results of these calculations ar ailable (13, 16). The average rela deviations as functions of temperat or coefficient sets 1-A, Relative 1-C, 1-G, and 1-J are shown in Table Weight III. In addition, the average deviations obtained when using Benedict's 0.5 coefficients (4, 6) were included. Rea0.5 sonable agreement between the deviations recorded in Table II and Table III was obtained. Precise agreement is not to be expected since the deviations shown in Table II were based upon 214 experimental points whereas only 42 points were used to develop the data presented in Table III. These comparisons assume that all the error is in the pressure and that the volume and temperature were established. second part of Table III a similar comparison is made for t's coefficients and set 1-C using volume as the dependent le. A graphical comparison of the deviations from experir coefficient set 1-C is presented in Figure 3. The effect e of dependent variable and the trends with temperature ure are indicated. When volume is the dependent variaf> e deviation is greatest at intermediate pressures for the gas phase, whereas when the pressure is the dependent variable the a.test error is found at low temperatures and pressures for the •d phase. EFFICIENTS t()'R EXPERIMENTAL BACKGROUND FOR BENEDICT EQt;rATION From the experimental data indicated in Figure 1 the coefficients of the Benedict equation (6) were predicted by use of conventional lea.st squares procedures (9, 13, 17). An appropriate solution of, the normal equations was obtained for several values of 'Y• A Jlsting of the several sets of experimental data. and the corresponding values of 'Y that have ~ used is shown in Table I. dentification numbers and lettel'I! have been: used to designate independently the set of data e • a particular calculation and the several values of 'Y c yestigation. It should be realized that the letter de ons are not interchangeable between different sets of data. e fort!gohlg comparison no states were employed for which valuee of the oompressibility factor were smaller than the expe~ta.lly determined compressibility factor at the critical stat,j;, Figure 4 shows the range of conditions covered by Beattie and coworkers (S). These data were employed by Benedict (4, 6} in predicting coefficients for his equation of state. Three e~ental points at low specific volume shown in this figure wi a separate designation were not used by Benedict in his ·one. Table IV records several sets of coefficients for the •ct equation of state which were computed from the data of Table II presents 10 sets of coefficients for propane evaluated by the described method, with several measures of the agreement with the experimental data shown in Figure 1. For comparison the coefficients of Benedict are included. In Figure 2 the average Bt41,ndard error of estimate of the compressibility factor is shown as a function of -y. In this instance a.II the error was assumed to be in pressure, and thus rather large uncertainties would be ext>edted at low temperatures and pressures in the liquid phase. lt is apparent that the behavior of the standard errors for positive and ne~ative values of 'Y is similar to that which was predicted qualitatively (9). Assuming that c is always finiw, for an infinite value of 'Y, EquaTABLE II. COEFFICIENTS FOR BENEDICT EQUATION OF 8•.rATE FOR PROPANE sl reduces to Coefficient 1-A 1-B 4.52289 0. 765331 19.3891 5.56129 6.89394 58.0494 1. 75505 17.3119 0. 01084 0. 0093 b 0. 0121 C 0.0094 'Y Bo Ao X 10-1 d' -a) d 2 + aa( b - RT RT (2) and for a value of 'Y of .o, Equation 1 assumes the following •ial form: z. = 1 + Ao Co) ( Bo - RT - RTa d + (b - J:,T) d 2 + aa ~ + ;~, (3) Co X 10-• b a X 10-1 a c X 10-• Average standard error of estimate Average deviationd Average relative deviationd Average deviation An interpolated minimum value of the standard error of estimate is obtained for a value of 'Y of approximately 4.135. The standard error of estimate for this case is !!J)roximately 80% of that obtained D using Benedict's coefficients (6) for a.ta limited to pressures below 4500 pounds per square inch. It should be emphasized that the standard errors of estimate shown in a part of Table II and in Figure 2 are based upon the premise that all the error is in the pressure .and none in either the temperature or Bil$lific volume. Iii'.:irder to check the standard errors of estimate shown in Table II and Figure 2, a number of values of the compressibility factor were computed from Bo Ao X 10- 1 Co X 10-• b a X 10-1 a c X 10-• Average standard error of estimate Average deviationd Average relative deviationd 0.0092 4 0.0066b 0. 0114• 0.0122 1-E 2. 26144 -0 . 616434 2. 56235 7.04527 7.63062 61.2673 1. 30370 9.18664 1-F 0. 565361 -0 . 511265 5 . 96872 5. 72859 6.65541 42.1686 1. 51553 3.35711 0.0203 0.0206 0.0057b 0.0108b -0, 0025b Coefficient 'Y Identification 1-C 1-D 3. 95753 3 . 39217 0.400735 -0 . 414751 12.7802 0. 520830 7. 39862 10 . 02100 9 . 91381 8.18833 108 . 2314 80.9044 1. 18657 1.44658 23. 3692 28.7030 1.85142 4. 13518 1-H -0. 565361 1. 57611 32.3300 0. 513578 3.57008 2 . 79954 22.0732 -4.50779 0.0205 0.0206 0.0000 -0.824791 2.25654 6 41943 1~85 4ft'QO'l Identification 1-I 1-J -5 . 65361 + a, o. 411348 0.156299 17.5190 15. 2859 3.81056 3.45547 5.45457 5.31753 24 . 7386 25.3555 2.39531 2.50348 -0.048834 0 . 0195 0.0207 Benediotc• +5. 65170 1.55997 25 . 9451 6.21704 5 . 78016 57.3552 2.50044 25.2951 0.2158a 0 0085b f~o, '1'11.:0062/ ....-1[00043/ Average deviation a Average value based on equal/weighting of data sets shown in Figure 1. b Based on data of Beattie et al. ehown in Figure 1 (91. • Based on data of Reamer d al. shown in Figure 1 (15). 4 Without regard to sign. • Coefficients proposed by Benedict et al. on the basis of the following atomic Weights: C = 12.010, H = 1.0080. I Based on data of Beattie et al. shown in Figure 4 (9) as presented by Brouch (9). CHEMISTRY with a-ltbe data from Beattie shown in Figure 4 and the rec data for ~ pressures depicted in Figure 5 whioh.J!1oluded sub !'Dlount of information for the liquid (16) well data (2, 15) for the gas phase which used in coefficients of Table II. The coefficients based u uid and gas phases are recorded in Table V and a stan error of estimate which is but little differen from the minimum value found in Table II. Comparison predicted and observed behavior with this set of c eflioients identified as 4-A is shown in a part of Table III for d 190° F. MPABISON OF Co PRESSIBI 100° F. 190° F. 220° F. Pressure as Dependent Variable 0.5469• 0.1029 0.0262 0.0078 0.0216 O.Oi37 0.0135 0 .0047 0 .0044 0 .0220 0.0094 0.00101 0,0076 0 .0240 0.0458 0.0126 0 .0084 0.0246 0.0454 0.0122 0.0053 0.0460 O.Oi41 0.0289 0 .0037 Benedioi 1-C Benedictb Volume as Dependeni Varia 0.0 2-Cb • FOR BENEDICT EQUATION OF STATE CABLE TO LIQUID AND GA& :PHASES 4. 24021 0. 550703 15 . 9131 6. 31535 7. 41650 67.1411 l. 60300 19.2621 Benedictb 2-Cb • Figures in body of table are averag ei1n. b These oomparisons apply only to Benedict U, 6). 0.0094 Coeffiolente are Beattie, Kay, and Kaminsky (2) s qnly to prel!Sures of 4500 pounds per oiiQipletene\a the coefficients recom ·ct (,IJ 6) and Brough (9) and 'I1 I have been included in Table evaluated from recent e • t 7 v of the volumetri •cted from coe Benedict. set is sh able III for te 482.0°F. Si as the depende other part of much smaller d alues predicted fr In the us region the agr experiment is satisfactory, Co observed and predicted data ( specific weights confirm the n of ~mental data throughou cooaltions for which the Bened" mployed. The mark ts shown in Tables II n of the potentialities o in interpolating the exp • ha~ of hydrocarbon fluids wi accuracy for widely different cients. order to ascertain the potentialit' equation, one set of coefficients was e TEMP. OF 100° 220° 340° 460° I POUNDS . TABLE IV. 1. At c. b •X cir e X 10 -• Averap etandard error of .-tirnate A"9~ deviation• .&,trace relative devia- fifon• Avtra1e deviatidti 0.0062 -0 .0043 9 SQUARE •...• ' INCH Relative Deviation in Compressibility Factor for One Set of Coefficients DICT EQUATION OF STATE FOB PlloPANE Identification 2-E 2-D 4 .24021 4.80557 0.770058 0 . 072770 6 .77038 16.3759 9 . 66786 7 .67489 9 .40853 7.28261 74 . 7481 104. 7969 1.43752 1.81981 26 . 7975 32 . 8976 Benedict 5 . 65170 1 .55997 25 . 9451 6 .21704 5 . 78016 57 .3552 2 .50044 25 . 2951 0 .0085 0 .003~,I)., PER -0 8000 6000 4000 0 0 0 .00340 0.00273 0.00483 -0 .00018 0.0025 0.00191 0 .00401 -0.00011 0.0025 0 .0033 5 .65361 0 . 540336 20 . 7841 3 . 58093 6.66571 44.0121 1.9012 10,379 3-B 4.80557 0 .484874 18.2494 4 .48459 7 .05149 53.9029 1.72068 13.0718 4.24021 0.288876 14.ll05 5.66119 63840 7719 53829 1 8667 0,0113 0.0100 0.0094 3-A No. 6 6000 :r u 0 0 z 4000 0 0 ... a: 0 0 <( ::, .,,0 0: 0 0 0 0 Q. 0 0 0 'o ... .,, 0 0 0 0 0 z::, 0 0 0 0 0 0 0 0 0 0 a. 0 l&I a: 0 ~ 0 0 0 .,,Ill::, 0 0 0 0 0 0 0 0 0 .., <( 0 0 0 a: 0 0 0 :r V ~ 4000 ::, a 0 0 0 0 0 0 Cl) a: 2000 .., Q. 0 0 0 0 Cl) 0 § 0 4 ::, 0 a. 1000 § .., 0 0 0 0 0 0 0 a: ::, Ill ..,Ill 600 0 0 a: "' Q. 0 a: IL 400 REAMER, SAGE, ANO LACEY 200 ....__ _.....__ _ _,___ _......._ _ _,___ __,.__ ___, 200 100 300 200 300 400 500 Figure 5. Experimental Data for Propane at High Pressures and Small Specific Volumes T Experimental Data of Beattie forf ent of the co~ibility factors ca.I ts with those oMained by experiment sures and at ow temperatures~ the liquid phase· than was obtained with the coefficients suggested Benedict (4, 6) and recorded in Table It should be realilp!i that although the new coefficients rec in Tablesil and V g{ve greater accuracy of prediction of the lumetric avior of propane, they do ~t necessarily pre~ the hete neous equilibrium with an ~acy as great as t. obtain from coefficients already av-Able (4, 6). Ho , if ace representation of the volumltric behavior at es up 0,000 po~ per square inch at temperatures n 100 ° 460 ° F. is • ed, the coefficients of set 1-C of able I for a ue of 'Y of 5753 or those of eet 4--A from Table V appear to M111atisfactory. LITERATURE CITED (6) Beattie, J. A., Chem. Revs., 44, 141-92 (1919). Beattie, J. A., Kay, W. C., and Kaminsky, J., J. Am. Chem. Soc., 59, 1589-90 (1937). Benedict, M.,Ibid., 59, 2224--33 (1937). Benedict, M., Webb, G. B., and Rubin, L. C., Chem. Eng. Progr., 41, 419-22 (1951) . Ibid., pp. 449-64. Benedict, M., Webb, G. B., and Rubin, L. C., J. Chem. Phys., (7) 8, 334--45 (1940). Ibid., 10, 747-58 (1942). (8) Be (1) (2) (3) (4) (5) (9) (10) MENT ACKNO Petroleum InThis paper is a contribution from Am stitute Research Project 37 located at the C of Technol6gy. W. G. Schlinger made a num tions for the accomplishment of the numerical work an Connell assisted with the IBM computin Lacey reviewed the manuscript. , and Friend, L., ict, M., Webb, G. B., Rubin, . Eng. Progr., 47, 609-20 (1951 Brough, H. W., Schlinger, W. G., and Sage, B.. H., lNo. ENG. CHEM., 43, 2443-6 (1951) . Connolly, T. J ., Frankel, S. P., 11,n Sage, B. H., Elec. Eng., 70, 47 (1950); Am. Doc. Inst., ·ngton, D.C., Doc. No. 3036 (1951). (11) (12) (13) (14) Ke~{J. D., and Egan, C. J., J . Am. Chem. Soc., 60, 1521-5 (1938). Marchman, H., Prengle, H. W., Jr., and Motard, R. L., INo. ENG. CHEM. , 41, 2658-60 (1949). Opfell, J. B., and Sage, B. H., Washington, D. C., Am. Doc. Inst.. Doc. No. 3849 (1952). Prengle, H. W., Jr., and Marchman, H., hw. ENG. CHEM., 42, 2371--4 (1950). (15) Ao, a, Bo, b, i.. C c, a, 'Y = coefficients of Bene<Uct equation d M P state molal weight, lb. mole/cu. ft. molecular weight, lb./lb. mole = 44.094 absolute pressure, lb. /sq. inch absolute universal gas constant, (lb./sq. inch) mole)( R.) = 10.73147 absolute temperature, R. compressibility factor PV /RT average atandard error of estimate 0 T Z u Subscripts c e - 0 critical state value of property calculated by equation of state (16) (17) Reamer, H. H., Sage, B. H., and La , W. N., IND. ENG. CHEM., 41, 482- 4 (1949). Selleck, F. T ., and Sage, B. H., hington, D. C., Am. Doc. Inst., Doc. No. 3914 (1953). Whittaker, E. T., and Robinson, G., "The Calculus or Observations," London, Blackie and Sons Ltd., 1940. RECEIVED for review DeceJRber 16, 1952. BD March 28, 1953. Material supplementary t<>'lthis article has bee • d as Document 3849 (111) and Document 3914 (16) with the A ry Publications Project, Photoduplication Service, Library of C o ~· Washington 25, D. C. Copies may be secured by citing the document ·nimbers and by remitting 12.50 each for photoprints or $1.76 each for 36-~ Jllicrolih:n. Advaa,,e payment is required. Make chec~ or money 0"""9 payable t9 C9ief, Phot1>duplication Bervioe, Library of Collgle88. -148- APPEHTIX G For a ona•oom.ponant s ystem, two of' -the ,~,rincipal cha:raoterist : cs or the critical s t ate a re the following 0 ('7 G) 0 ! 1')')7 ) 0 (161) 0 (162) \ I • Consequen~.:. ly But (163) ·and so . .~. = (oP ,Ro-)T. ..,., (o2P/o~)T,m · at the oritioal state. = 0 ('79 ) 0 ( ao ) -149Propoaittoml ~tted by John l3urton Optell Ph.ti, ~ ~ rdnation, D e ~ 4, 1953 ut 9s00 A.M.,. Crellin Cont~ Room. ~ttee, Pl"oi"esacn Sage (Chairman), Coreorw,• Mi ~ , $Cho~ and VQ"d. -~Mlt• 1. Nevton•a ttiethod t.o'.r the exti'-.etion ct square n.,ot is but a apecf.41 cGS.e ot t he naore 1 • ~ lt•ative 1'!Dthod which .ia delScr ibed b:/' the fonrula •tn (b1 = Um x. i-+Ot -• • , = 1( (n-l)Xs_-t fl, b/J;:~ 'thU i toration is nooncl ~ @41 co~~q co~• rapidq eftn tw mi.de •&tarting -1.u•• k l ~ e..ppUoat.1on, ot aut~tic dlgital computers, the uae ot IUOh et.ttoiont lttwative ~Hsiu ae tblt arti ~ ~ n t than ti. • • ot t.blo fer t1"al1scendetel functions. 2, The hydNd~• and 1-at 'ban.et@ ~tlollll 4'~t«l v1th ~ :rota.ting WWte tla\ plate have been tolv.S ~eti,2. «?hit dOlutioa lhould btJ ~ended, t~ugh ~i.Ql •tho-di" it M.OeNU'Y, to tbt oaee .b Vbioh a rigid wall 1t loeated at a tWw tietance ~m t.:'16 a.d.• ot rotation, Such a aolut:i<m woul.4 be belptul in the detd.gn ot ndxitig •...ie and 1n t he d.a1SA ot cs,olOM ~ tor heat aensitivo mawrW• ROh as milk., Hartftfi3 p;ro:po,_ that ill-oonditioning or llimult«me0'Wl l ~ _.tion, «m bl given ~1tat!'9'e sip:lt!cance 1n terms or the ratio or ~ l•ses, to· the ..U.n latent ~"'· .Q t the matrb: fJf the ~ t i ed.et.a. Siaoe 114a utru ..,- be •~ by multiplication ot -, iw '• b7 • eot.\8~; 'fdth.out obarlging tm eolution of the 91Btem ot ~tiorle, t:ht value ot tl'd.t ratio-, change also. ~lu . 4fhe tbeot7' ot ~ w t h e ~ • iltpliee a cou:pl.ina .rt•ol4 .,_t_.Jt t.be .r•te ot ~ ' - tr,_.port and tht progrua <>t oheud.-1 ~ ta t1utd -,.tou. Tb.$ e.ftettlt Ma not yet been o ~ n,er~l1'• S-lAH iUOb M .rteot would bfJ ue.tul ~ ecntfflling ;reactions n.tet1 • lavett!g&tlon b.to the magmtucte of Wa .tteot would be ~st vm'bvhll•• s. Kwm ad Silber~ haft reported that in a t1tld ot ~ , they o ~ a ahitt 1n the .ll'L\t.ual eol~'b1llt7 ot two Uq,u1d phrdu. thh p l $ ~ sho\l14 be t a ~ Oil the b&rU.s ot et~tate -150- thermodynamic& rat-her than on the basis of equilibrium thermodynamics. Th(;) .f'aet that steady and non zero temperature gradients exist in a field ot shear in viscous .fluids makes the direct observation of temperature very difi'ieult. It seems likely that in part the reported phenomenon can be explained by the existence of a steady' non-uniform temperature distribution in the e7stem investigated. Chemical Engineering. (,. Coefficients tor · the. Benedict •quation for the reduced volumetric properties would be mo•t useful for extrapolating ths volumetric properties of compounds and mixtures tor which little volumetric information ia available• 'l'he success or the ))#e'Udo-erit1eal concept suggests the possibility of' describing the volut1$trio properties ot a broad claas o.f' compounds in terms of the reduced coettioients for the Benedict equation and ct three numbers vhich correspond to th8 effective critical propertas. 7. Gemant6 baa proposed a correlatlon between the viscosity coefticient and the oompretsib111ty •tor- liquid.a and· plastics. Since, the viscosity and ditf'ua,ivit;y are correlated .in tl:idd sy1;1telnl$ 1 a similar correlation between dittua1•1tr and the eompre~aibility should be possible tor fluid system.a. Such a •:rrelation vould be most useful in practice finee the ♦lastio J)l"opertie• oan ·be obtained easily from velocity or eound measurementa. • • • 8. Pohlbausen7 bae integrated the hfdi'od1namic equations deserib- 1»&. the te~ature distribution in thft.· bt,und~ layer arising from the tlowing ot • viscous in eonq,reaaible fluid along a ·.f'lat plate. A study ,t the results snowed that ~he recover,. rattor was very nearly equal to the square root or the P.randtl number Pr tor Prandtl numbers near unity. The same equatiou ma, be integrated tor the .case of a fluid no~ between parallel plate• to give a recovery raotor 0£ 54/)5 Pr. The indiacrimant use of th& nlation; recovery factor = (Prandtl number)½ 18 to be discouraged. 9. Bernoulll•s theorem 1a a sp&cial case or the more general theorem I '.:>' dh+ ~+V~dx+$-t( 2g h · I =O vhich applies at ever, petnt in a inoying tluid • The function represents the Winiteeimal .&nGWlt of work whioh an Winitea:lmal •lement -0f the tluid, considued as a cloaed system, does upon its surroundings as a re$ult of !te velocity relative to 1te S'Uff0unding8 and the forces aeting at the 'boundaey. • j'/ d e is the Reynold.a . dissipation function and x 1s d!Stance along the streamline. Thia -151- more g$neral theorem is a consequence only of the conservation or raonaen... tum and the definition of work. 10. h elevation (feet) u speed, rt/sec g gi!'avitat1onal constant When stripping a volatile component A , from a relatively nonvolatile absorbent lev pressure operation is usually desirable. Since the equilibi"i'Ufil distribution or the component being stripped is described. by the equation it is apparent that the. stripping gas carrie• a-way the highest eoncen- tration of A when l ~ (K...l)I is zero. For a given teed stock, i.e •• tixed value of X; K should be as nearly unity aa possible. In hydrocar- bon aystema8 , at a given temperature K is unity at the critical state as well as at a pressure which il!J approxiQ'ia.tely the va.p0r pressure or pure A at that temperature.. Consequently, there 1,1hould exist situations 111 which stripping operations might be performed a.t preesures nearer to the critical preasure or the system than at nearly vapor pressure. The advantages te arie,e frotU use of such preasures would :result from the small epecii'ic volume or the gee phaee. ll. When at re1tt many types of powders and finely divided solids are unable to support a ahear stress 1n the bulk.. Thie property is a chai-aete-ri$tic teat'Ul"e ct fluids. '!'he analysis cf the flow of ~ s th1 ough orifices and short tubes might protitably- be tu.de on the baa:is 9 of the equations of motion for fluid and the use of an effective vi► cosity coef'f:leient for the powder. -152- Nomenclature K equillbriu.rn vaporization ratio Pr Pra.ndtl number X lb. moles or A per lb .. mole ot solute-tree absorbent I lb. moles of A per lb. mole or solute-free stripping gas Reference• 1. D. R. P;iartree . N!:Yfflttical £\nalt@~s~ Oxford, London, (1952), P• 192. K. Millsaps, and K. Pohlha.t1s<::m . rtHeat Tr~.nafer by Laminar Flow r:rom a R"tating ¥1a.te., " Wright Ur Development Center, Nav;v· Research Section, U20674, (1951). J. O. Hirschfelder, C. F. Curtisa, R. B. Bird, and E. L. Spotz. 11 The Properties of Gases," Univ. or Wise. 31 July 1952, NOrd 9938, P• 25. W, Kuhn and A. Silberberg. A. Gem.ant., CF'S 1501.A ~md 1510A. Nstm:•h 11Q, 450, (1952). 1'"r!f,ion!:Jr Pb§ngmg. Chemical Publishing Co•, Brook~, (1950, Chapt. XIV. E. Pohlhausen. !tit, W¾ffi!t )fatha Meeu Mt l, 115, (1921) •