An Improved State Equation In The Vicinity Of The Critical .
JO URNAL O F RESEARCH of the Nationa l Bureau of Standards - A. Physics and Chemi stryVo l. 76A, No. 3, May- June 1972An Improved State Equation in the Vicinityof the Critical PointOlav B. Verbeke *Institute for Basic Standards, National Bureau of Standards, Boulder, Colo. 80302(December 21, 1971)An im prove d s ta te equa ti o n for th e vi c init y of th e c riti c a l po int is propose d. An a na lys is o f th e e xperime nt a l da ta o n he lium and xe no n has bee n ca rri ed ou t in o rde r to inves ti ga te th e influ e nce of th enumbe r of con s tant s in th e eq uati on a nd th e PpT ra nge on th e c riti c a l con s ta nt s T,. a nd p,. a nd on th ec riti c a l e xpo ne nt s ex , {3 , y. a nd 8. No s uc h inAu e nce has bee n d e tec ted. Th e mod e l fo r th e c riti ca l point.rece ntl y p roposed by Widolll . has bee n c h ec ke d regardi n g it s co nse qu e nces for th e rec tilin ea r d iam e te r.No d e fi nit e c o nnrm a ti o n but indi c ati ons for it s correc tn ess have bee n fu u nd.Ke y wo rd s: C ompress ibi lit y; c riti ca l po int ; equation of s tat e; Auid ; low te mpera ture ; me th a ne ; sca lin gla w; s pecifi c hea t ; he lium ; xenon.1. Introduction'flle d'JVe rge nce 0 (. dplII. t he rect '1dT ' wIl e re Pili ISi 'lI1 ea rRecently an equ atio n of s tate for th e vicinity of th ec riti cal point has bee n proposed by Ve rbe ke e t aI. [1).1Thi s eq uati on was appli ed to the data for xenon ofHabgood a nd Schn eid er [3]. In a s ubseq ue nt paper th eeq uati on was appli ed to d ata for me th a ne of Jan soo neet al. [2]. It is th e purpose of thi s paper to report improvements in th e eq uati on which have b een madesin ce th e n.di a me ter , is a co nseq ue nce of a mode l for the c riti ca ls tate, whi c h has rece ntl y bee n proposed by Widoma nd Row lin so n [51. Thi s dive rge nce is giv e n by a noth e rpowe r law ,;;: (Tc- T) - " ,with 0' 1 0' (as deri ve d by W ido m). Th e sa me mode l. Id s a SImi" 1ar d'Iv ergence 0 {' dp,dT ' w Ilere p, .IS t he I'In e2. Formulation of the Modified State Equation Yleof sy mm e try in th e C v(P, T) s urface wh e re C v is th eSeveral questions co ncernin g the ori ginal equations pecifi c hea t a t co ns ta nt vo lum e. A div e rge nce ofremained to be a nswered.2 swas in corporated in our origin al eq (1) in ord er to'fh e tem perature denvatl. .ve, ddT2'P werehP's IS t heprovide a si mil ar dive rge nce in all term s of th at equavapor pressure a nd T is th e te mpe rature , is beli e vedtion ; sin ce A in that equation was not necessa rilyto dive rge ase qual to L the co rres pondin g 0'1 was r estri c ted tod 2 PS (T. · - T) - "2(1)e0' 1 1- {3.C;;dPclose to the criti cal poi nt. The value of 0'2 mu st , accord in g to th e weak in equa lity of Griffiths whi c h is based onth ermodynami cs, be less than or equal to 0' {3 , where{3 is th e ex pon e nt of th e power law for the diffe re nceof th e coexis tin g de nsiti es a nd 0' is th e expo ne nt of th epower law for Ce. It has bee n proved [7] for a n equati onof state of th e type proposed by M. S. Green et al. [4],that 0' mu s t be equal to 0'2. Th e present equa ti on canalso, but mu s t n ot necessaril y, imply the latter equ ality.*On deta il to th e Nati o nal Bureau of S tand a rd s. Con trac t CST 8109. from the Katholiek eUni versiteit of Leuvcn. H ev e rl ee . Be lgi um .1It was found, howe ve r, that a va lu e of A diffe re nt from1 does introdu ce a di sco ntinuit y ac ross the criti ca lisoc hore.In ord e r to tes t th e co nj ectures on 0' 1 a nd O' made byM. S . Green et aL and Widom res pec tive ly, ouroriginal eq uation [11 was modified as follo ws:with(3)Figures in brac ke ls indic at e t he literature references a t th e e nd of th is paper.207
(4)Pili pe Pili I (Te - T.,)I - a"(5)(6)andPI pc plo( T - T8 ) I- a,.(7)It can be s hown from eq (2) that th e ex pon e nt , 0', ofthe power law for C L' ca n be expressed in te rm s of/-I- and {3 as follow s:0' press ure within a selected perce nta"e around th ecirti cal point parameters. The m e th d of d efinin othe various ranges is illustrated in figure 1. Th e actualran ges indi ca ted by th e c ro ss hatc hin g in fi gure 1are given in tabl e 1. A sin gle s ignifi ca nt parameter, N,the numb e r of data points in th e ran ge, res ults fromthi s limitin g procedure; N will be tak e n as a quanti·tative meas ure of th e ran ge . In tabl e 1 th e boundari esare defin ed for each range and for eac h substance,and the corres pondin g numbe r of data points in theran ge is give n. Ran ge No.1 for he lium ex te nd s farth e rthan any oth e r range, and for thi s ran ge th e numbe r ofcoe ffi cie nts, B, in eq (6) was varied in order to inv es ti·gate th eir relevance.1 - (2{3 /-1-).f)Several questions can now be formulated.1. I s 0' 0'1 O' ?2. Is Pili I equ al to - PIO?3. Is there any PpT range·dependence for theexponents {3, 'Y, 0 for the followin g powerlaws pe - pg - (Ye - T)13 (the indices landg re fer to liquid and gas),/ max1p//--- e;II4. How man y terms are relevant in eq uation (6)?Answers to th ese questions hav e bee n obtain ed byapplying eq (2) to two s ubs tan ces for which exce ll e ntdata is available, i.e., the data on helium by Roach [6]and the data on xenon by Habgood and Schneider[3]. As far as th e latt er data are concern ed , a rathe rlarge difference was detected previously be tween th emeas ure me nts on isochores and on isoth e rms [11. Th eerror (for version 2) with the isotherm data includedamounted to 0.00311 atmospheres versus 0.00227without. In the s ub seq ue nt analysis only meas ure mentson isochores hav e bee n con si dered.3. Analysis and ResultsBefore estimating the PpT range dependence ofth e c riti cal point parameters, th e ranges hav e to bede fin ed. Th e c hoi ce of th e ran ges will always be moreor less arbitrary. In this case the ranges are defin edb y confinin g th e variables de nsity, te mpe rature andNo. ofrangeTminTc.TmaxTFIGURE1.The variation of the error and th e apparent valu esof th e scaling exponents 0', {3, 'Y, and 0 as a fun ctionof th e number of coe ffi cie nts in eq (6) is s hown infi gure 2. Bo and B I were fo und to provide a satisfac·tory desc ripti on for data of the give n accuracy withinthe give n ran ge. Beyo nd th ese two cons tants , node penden ce on the number of B's was found for 0'and 'Y and only a slight d ependen ce for {3 and o.Consequently in further calculations all other B·shav e been assumed to be zero. Equation (2) can beapp li ed on th e data in two alternative versions; version1 whe re 0'1, 0'2, PIIII and PIO are indepe ndent adjustableBoundaries of the different PpT ranges for xeno n and helium andthe n mber of data points leji in each range( IPma pel ) (I T ma;c- T'I)0.50.35.30.10.Schematic P- T diagram showing boundaries of a" range" around the critical point.(lfm PI' l)ax -PcNu mber of datapoints, NXenon1. . .2 . .3 . . . .4 .c/IP-PeI8 1.p/P fip T(at saturation or at th e c riti cal isoc hore),TABLEI/1 (fiP) ITe- TI-YIP-P eIT Tc-/0.05.035.03.012080.08.05.05.02173142114He lium414295
param e te rs and 'ver sion 2 , where a , at 1 - (2 f3 /-t )and Pili I - Ptu. Th e tw o ve rsio ns are co mpared forhe lium a nd for xe non in ta bl e 2. Th e diffe re nce inth e qua lit y of fit betwee n th e two versions is not s ubs tanti a l, a nd th e refore furth e r s pecu la tion s on th erange depe nd e nce of th e criti cal parameters havebe e n carri ed o ut with vers ion 2. Th e de pendence ofth e criti ca l expo ne nt s a , 8, y and f3 on th e ran ge isillu s tra te d in fi gure 3 The co ns ta nt s for ra nge 1 a ndfor ve rsion 2 of equatio n (2) a re give n for he lium inta bl e 3, a nd for xenon in tab le 4.Th e estim a tion of th e parameters has bee n carri edout throu gh a n ite rativ e procedure. Initial co ns tantshave bee n c ho se n and th e eq uation ha s been lin earized with respect to th e dev iat ions of th e con s tants.These de viations wer e th e n ca lc ulated by th e we llknown leas t- squares m eth ods for th e lin ear ,--r--r---,-,--,--,.15 ---- - ---- -------- - .' 0.37.36.3 5.34'2yC --- j ----- --9I. ,4544HE LI U M(Range I , Vers ian 2)434 .284'Number of constants 840Th e e rror analysis is carried out on the basis ofth e correlation matrix for the lin ear case. There maybe so me do ubt wheth er this is e ntirely justified inth e case of a nonlin ear equation.3.9 ,L -L-l.---'----lL-!:---'---.l.--L-'----:----'.---" L--'----:---'--'--'--L F,G URE2.C .f3.'Y and Il and their dependence on the namber aJB's in eq (2).TABLE 2.Comparison oj the relevant parameters Ja r the two versions oj eq (2) and Jo r helillm and xenonHeliumRange No.Version 11Il'Yf3C c ,O:lp",Pili I---* "i.6. PiN - NC atm2( He) and 3(Xe)XenonParamelerIl'Yf3C c ,O :lPIVPili!4.281.128.343.184.270.400- .00100 .00208 0.09.004.003.008.045.026.00015.00022Version 24. 16 1.119 .353 .173 .173 .173 - .00132 - PIO0.000724.24 1.137 .351 .161 .277 .338 - .000314 .00168 006.006.006.000150.000744.13 1.138 .363 .135 .135 .135 . 000l3 0.08.009·.006.015.015.015.0004- proVersion 1Version 24.45 0.144.39 0.1011.2321.191 .01 2 .006.356.351 .005 .005.054.108 .014 .020.287.108 .014 .116.470.108 .045 .014- .0000417 .000013] - .0000145 .000004 .0000453 .0000186- PIQ0.002240.002273.734.72 0. 39 O. 761.1781. 208 .016 .012.430.324 .052 .012- .039.142 .033 .106-. 149.142 .389 .033.060.142 1.591 .033- .0000105 .0000070 - .000025 1 .0000059 .000228 .000057- pro- - -* "i. 6.PiN-NC40.000590.00059Il0.001980.00203PtvPili I5.73 8.191.284 .074.271 .129.1 72 .283.172 .283.172 .283 .0000171 .0000425- pro- --* "i. !::.Pi0.00209yf3C c ,c ,N - NC*N is th e number of data points; NC is th e number of co nstant s.209
TABLE 3.Constantsfor eq (2) applied to helium data [6)"Constants of th e equationConstantAValue2 -a2Cpc0.32043251E 04. 17091894E 01.33758179E 03.72719432E 00- .66253524E 01- .13174329E - 02 I 2,8 !L0.74227144E 02. 17232286E - 01pm!- PtOTe,8!LI - a,.51909878E 01. 35366522E 00. 1l915742E 00 2,8 !LPSI8,Ps2PsoPtOErrorThe standard deviati on is 0.000740.10131254E 03. 19034584E - 02.12381564 02.34948345E - 01.98947437E 02. 15183463E - 03Maximum error in press ure is - 0.0025 atm.At T 5.26370 andP 2.40487 atm.45068578E 00.52370344E-05.27286947E - 03.23593233E - 02.33542782E - 02Derived ConstantsConstantaYIlValu eError0.17351214E 00.1l191574E 01.41644543E 010.60508733E - 02.33542782E - 02.62999387E - 01a Th e dens ity range is 50, the tem perature range is 5 and th e pressure r ange is 8 perce nt around th e critical point. Th e nu mber of datapoints is 414.TABLE 4.Constants for eq (2) applied to xenon data [3] aConstants of th e equ ationConstan tValue2 - a2Cpc0.94959356E 04.1l860651E OJ.13935997E 04.15611421E - OJ- .28601355E 03-.14547437E - 04 1 2,8 !L0.47872048E 03.84971537E-02pml- PIOAPSI8,P stPsoPt OTe,8!LI - a, ErrQrThe standard dev iation is 0.00227.0.21903837E 03.83952672E - 03.53686691E 02.23977382E - 02.24339522E 00.3999001OE - 05Maximum error in pressure is 0.0067 atm .At T 289. 13990 K andP 56.3430 atm .12869899E 01.39913666E - 05.31386983E - 02.46079306E - 02.63731809E-02.28976481E 03.350500350 00.19110181E 002,8 !LDerived ConstantsConstantCIYIlValue0.10789750E 00.1l911018E 01.43982899E 01Error0.13966633E - Ol.6373 1809E-02.10541327E 00a The d ensity range is 50, the temperature range is 5 an d the press ure range is 8 percent around the c ritica l point, the number of datapoints is 165.210
0.4.(JaI)0--- - - - - - 0(Jf3I)f---- - -- (JyI)f-----0.2It can be seen from th e error- b-a r-S- ]-'n- figU, eXENON (Version 2 )HELI UM (Version 2)heli um is the more likely candidate for interpretationof the resulting exponents:The agree me nt betwee n th e ex pone nts for the differ·en t ranges and for the diffe re nt version s of the stateequati on for helium is almost within the error bars.Noteworthy is the fac t th at 0' 1 te nd s to be higher·(- 0. 3 0.1 ) than 0' and that a2 see ms to be evenhi gher (- 0.45) for version 1 and for both helium andxe non.W e do not exclude the possibility that th ese valuesare s uggested by version 1 but it is note worthy that an0'2 equal to 0.45 is still lower th an 0' {3 (0.15 0.355).Thi s value (at 0.45) however, would imply a thermo·dynamic pote ntial diverge nt at th e criti cal point , apossibility which is doubted by many.A weak point in eq (2) is the nonanalyti city intro'duce d by the secon d term on the right sid e of eq (6)0030 .11. 31.1'"0- - - - 0o------- ---- 9.0B.O7.0(ap)2. m. a d'Iv erge nt ap2res uItll1g6.0"-,5.04.0 ---- ------ (J8'" ,I)300400'", ----- 1001501"f or P equ a1 to Pili . 'fh isnonanalyticity ca n be avoid ed but we doubt that itsinAu ence on th e criti cal param ete rs is important.More difficult to avoid is the nonanalyticity intro·duced by th e third term of eq (2). Thi s n nan l yti c itydisappears when y/ {3 is a n integer, say y/{3 3, forinstan ce. This effect may have had th e odd result ofproducing rather high vaiues for a.3.02.03;' -200NUMBER OF DATA POINTS3. c/. {3. Y and 8for helium. and xenon and th.eir dependenceon the number of data poin.ts (error bars are indicated by the cross·h.atched strips).FIGUHE4. ConclusionsOnly qualitative co nclu sion s can be drawn fromtable 2. The values obtained for a l and at, ind eed, areconsistent with the diverge nces as claimed by Widom'and by Green e t al. Th e valu es obtai ned , however , arelarger th an a . Furthermore th e equality PIIII - PIOseems to chec k out qualitatively. It should be e m·phasize d howe ver that th e accuracy of the data ,although being wi thin th e limits of th e present stateof the art , do es not allow definite conclusions. Thesmall differences obtain ed for a , {3 , y , and 8 in versions1 and 2 and by other authors [7] could be explained b ythe fact that equation s of this type can be very sug·gestive with respect to these values.It should also be noted that th e value for a obtainedfor helium is mu ch higher th an th e values obtained inthe past by other authors [7].In a more recent analysis, however , hi gher valueshave bee n obta ined [8]; th er rece nt experiments onhelium also s u gges t high er valu es for a (e.g. , 0' 0.14 [9]).Weare ind ebted to J. M. H. Levelt Sengers forstimulating di sc ussions on thi s subject , a nd to V.J ansoone who suggested the error analysis. Most ofof the calc ulations were performed on th e CDC3800computer , the use of which was granted by the Cryo·ge nics Di vis ion of th e National Bureau of Standards ,Bould er, ColoJ:ado 80302. W eare also inde bted to the" Rekencentrum " of th e University of Leuven, Belgiumwhere part of the computation s were performed.5. References[IJ Verbeke. O. B., Jan soo ne . V., Gielen. R., J. DeBoe lpaep. J ourna lof Physi ca l Che mi stry 73, 4076 (1969).[21 J ansoo ne, V., Giele n , H. , 1. DeBoelpaep, O . B. Ve rbeke, Physica46,213 (1970).[3] Habgood, H. W., Sc hn ei der. W. G., Ca n. J . C hem. 32,98 (1954).[4] Green. M. S., Vi ce ntini·Missoni. M., Lev ell Se nge rs. 1. M. H. ,Phys. Rev. Letters 18, 111 3 (1967).[5] Wido·m . B. , Row lin son. J. S. , J . Chem. Phys. 52 , 1670 (1 970).[6] Roach. P. R., Physical Re v. 170,213 (1968).[71 Vincentini·Missoni , M. , Levell Se ngers, J. M. H., Gree n, M. S.,l Res. Nat. Bur. Stand. (U.S.), 73A (phys. a nd Che m.), No.6,563-583 (Nuv. - Dec. 1969).[8] Green . M. S. , Coo per. M. J . Leve lt Sengers. J . M. H., Phys.Rev. Letters 26, 492 (1971).[9J Mo ld over. M. R. , Phys. Rev. 182, 342 (1969).211(paper 76A3-717)I
number of constants in the equation and the PpT range on the critical constants T,. and p,. and on the c ritical exponents ex , {3 , y. and 8. No such inAuence has been detected. The model for the critical point. recently proposed by Widolll. has been checked
phase concentrations and volumes by Equations 8 to 10. Substituting Equations 8 to 10 into Equation 7 gives Equation 11. The compound concentrations in each phase may be related to the partition coefficient by Equation 12, which is a re-arrangment of Equation 1. Substituting Equation 12 into Equation 11 gives Equation 13 C S M S V S .
Chapter 5 Flow of an Incompressible Ideal Fluid Contents 5.1 Euler’s Equation. 5.2 Bernoulli’s Equation. 5.3 Bernoulli Equation for the One- Dimensional flow. 5.4 Application of Bernoulli’s Equation. 5.5 The Work-Energy Equation. 5.6 Euler’s Equation for Two- Dimensional Flow. 5.7 Bernoulli’s Equation for Two- Dimensional Flow Stream .
Page 6 of 18 A radical equation is an equation that has a variable in a radicand or has a variable with a rational exponent. ( 2) 25 3 10 3 2 x x radical equations 3 x 10 NOT a radical equation Give your own: Radical equation Non radical equation To solve a radical equation: isolate the radical on one side of the equation and then raise both sides of the
The Manning Equation is a widely used empirical equation that relates several uniform open channel flow parameters. This equation was developed in 1889 by the Irish engineer, Robert Manning. In addition to being empirical, the Manning Equation is a dimensional equation, so the units must be specified for a given constant in the equation.
The Manning Equation. is a widely used empirical equation that relates several uniform open channel flow parameters. This equation was developed in 1889 by the Irish engineer, Robert Manning. In addition to being empirical, the Manning Equation is a dimensional equation, so the units must be specified for a given constant in the equation.
Euler equation and Navier-Stokes equation WeiHan Hsiaoa aDepartment of Physics, The University of Chicago E-mail: weihanhsiao@uchicago.edu ABSTRACT: This is the note prepared for the Kadanoff center journal club.We review the basics of fluid mechanics, Euler equation, and the Navier-Stokes equation.
Chapter 7 The Energy Equation 7.1 Energy, Work, and Power When matter has energy, the matter can be used to do work. A fluid can have several forms of . 7.5 Contrasting the Bernoulli Equation and the Energy Equation The Bernoulli equation and the energy equation are derived in different ways.
Chapter Outline 1. Fluid Flow Rate and the Continuity Equation 2. Commercially Available Pipe and Tubing 3. Recommended Velocity of Flow in Pipe and Tubing 4. Conservation of Energy –Bernoulli’s Equation 5. Interpretation of Bernoulli’s Equation 6. Restrictions on Bernoulli’s Equation 7. Applications of Bernoulli’s Equation 8 .
