APPENDIX D COMPRESSIBILITY FACTOR EQUATIONS D.1 THE REDLICH KWONG EQUATION

Transcription

1 AENDIX D COMRESSIBILIY FACOR EQUAIONS D.1 HE REDLICH KWONG EQUAION he Redlih-Kwong equation is atually an equation of state. It was fomulated by Otto Redlih and Joseph N. S. Kwong in 1949 [Chemial Review ]. hei equation is oot of a ubi. Robet C. Reid John M. ausnitz and homas K. Shewood [he opeties of Gases and Liquids 3 d Ed. MGaw-Hill Book Company 1977] give the ubi as 3 z z + A* B* B* z A* B* = 0 = nr a V b V V + b a nr a = Ω. bnr b = Ω whee Ωa A* = = / / Ωb B* = = whee and ae the itial tempeatue and pessue of the gas being onsideed. he onstants in the equation ae deived as 1/ 3 1 Ω a = [ 9 1] = / 3 Ω b = [ 1]/ 3 = his equation is quite auate at the itial tempeatue fo / = 4 to 40 eo <.%. At highe tempeatues and at / above about the equation beomes ineasingly inauate. he ompessibility fato may be found expliitly fom the Redlih-Kwong equation. It is the pinipal In the equations below is the edued pessue that is the atio of atual pessue to itial pessue and is the edued tempeatue that is the atio of atual tempeatue to itial tempeatue. Let A B C D and E be onstants defined by A = B = / C = A + A AB ipe Flow: A atial and Compehensive Guide Fist Edition. Donald C. Rennels and Hobat M. Hudson. 01 John Wiley & Sons In. ublished 01 by John Wiley & Sons In. 63

2 64 Compessibility Fato Equations ABLE D.1. Lee Kesle Constants Constant Simple Refeene Constant Simple Refeene b b b d b d β γ If D 3 + E 0 then D = C E C A = + B z = E + D + E + E D + E + 3. If D 3 + E < 0 then 1 1 E z = D 3 D + 1 os os. 3 3 Citial onstants fo seleted gases ae given in able D.3 in Setion D.3. Chemial Enginees Handbook th Ed. MGaw Hill 1973 indiates that the Redlih- Kwong equation of state fits the data fo helium and hydogen only fo edued tempeatues of. and highe when thei itial tempeatues ae ineased by 8 C and thei itial pessues ae ineased by 8 atmosphees. D. HE LEE KESLER EQUAION Anothe equation of state that is muh moe auate than the Redlih Kwong equation is the Lee Kesle equation. his is a genealized Benedit Webb Rubin equation developed by B.I. Lee and M.G. Kesle in 197 fom whih the ompessibility fato may be found. he solution is fomidable but with a ompute it an be obtained without muh diffiulty using the Newton Raphson tial-and-eo solution tehnique. hei equation is: V B C D = V V V 4 γ γ β + exp 3 V V 0 V D.1 whee and V b b3 b B = b1 C = 1 + d D = d + 1 V = R γ / exp. = 0 γ V e 0 V he onstants fo these equations fo a simple fluid ae given in able D.1. he β and γ shown ae fo the Lee Kesle equation and should not be onfused with those in the Nomenlatue. he equation is solved fo V 0 the ideal edued volume fo a simple fluid and then the simple fluid ompessibility fato is alulated: z 0 V =. D. Next using the same edued pessue and tempeatue the equation is solved again fo V 0 but using the efeene fluid onstants fom the table; theefoe all this value V R. hen: z R V = R he ompessibility fato z fo the fluid of inteest is then alulated fom the following fomula: z = z + z z ω R ω R. 0 0 whee ω is itze s aenti fato and fo the efeene gas ω R = he definition of the aenti fato is: ω = log 10 vap =

3 he Lee Kesle Equation 6 whee the pessue tem is the edued vapo pessue at = 0.7. Values of ω ae given fo seleted gases in able D. in Setion D.3. In ode to solve the Lee Kesle equation by the Newton Raphson method we must devise a funtion fom it whose value is zeo. his may be done by moving the V 0 / tem to the ight side of Equation D.1: V B C D 0 = V V V 4β 3 V exp γ + 4γ γ exp. 3 4 V V V Call this funtion f V 0 by substituting f V 0 fo the zeo: V B C f V = V V D 4β + 3 V V exp γ V + γ γ exp V V. D.3 f V 0 is supposed to equal zeo. Of ouse it is not likely to equal zeo if we don t know the oet value fo V 0 but have to guess it instead. Any nonzeo value fo f V 0 is the eo inued by using an inoet value fo V 0 in it. It may be onsideed to be the equied oetion fo the funtion. Using the Newton Raphson method we an efine ou guesses vey easily. In ode to do this we need the deivative of the funtion f V 0. he deivatives of the seven tems of the funtion ae given below: f1 = 0 f = D.4 = 4γ γ γ f7 V exp 3 0 V V. D.9 0 he deivative of the funtion f V is then the sum of the deivatives of its tems o f V = f + f + f + f + f + f D.10 0 he solution tehnique fo the f V equation is to guess an initial V 0 all it V i = 1. An initial guess fo 0 V may be obtained by finding the Redlih Kwong ompessibility fato z RK and assuming that it is appoximately equal to the simple fluid ompessibility fato. hen the z 0 equation Eq. D. may be solved fo V i = 1: V z 0 i= 1 RK. D.11 his guess fo V i = 1 is then inseted into the funtion Eq. D.3 and into the equations fo the tems of its deivative Eqs. D.4 D.11. hese thee values [ V ] i = 1 f[ V 0 ] i = 1 and f [ V 0 ] i = 1 ae then used to find V i =. By dividing the value of the funtion whih is the equied oetion to the funtion that is to the dependent vaiable we tansfom it into an estimate of the equied oetion in the independent vaiable V 0. Equation D.1 applies the oetion. he esult is a muh lose value of the independent vaiable as shown in Figue D.1: f V i V i+ 1 = V i. D.1 f V he poedue is then epeated with this bette estimate of V 0. Afte eah epetition the oetion tem f V 0 / f V 0 will beome smalle and smalle n n i B f 3 = D. V C f 4 = D.6 3 V 0 fv 0 Funtion: fv vesus V 0 Result of Guess 0 V i =1 angent to Cuve D f = D.7 6 V = 4β γ γ f6 1 V exp V V D.8 0 Desied 0 fv Impoved Guess V 0 i = V 0 FIGURE D.1. Solution tehnique fo 0 V. Guessed 0 V i =1

4 66 Compessibility Fato Equations until it beomes small enough as small a value as desied o allowed by the omputational peision of the ompute that the solution may be onsideed to have been found. he solution tehnique desibed above has one aveat it woks well exept in the egion of the itial point. hee the deivative appoahes zeo and the poedue usually gets aught in a loop. o iumvent this a diffeent tehnique must be substituted on the fist ouene of a hange in sign of the oetion tem f V / f V. One method is to intepolate between the V that aused the sign of f V to hange and the one just pevious to it to estimate the 0 V whee the uve osses the f V = 0 line. he eade will note that itze s aenti fato ω is used in the solution fo the ompessibility fato z. Reid et al. he opeties of Gases and Liquids 3d ed. MGaw-Hill Book Company New Yok 1977 state that appliation of oelations employing the aenti fato should be limited to nomal fluids; in no ase should suh oelations be used fo H He Ne o fo stongly pola and/o hydogen-bonded fluids. heefoe fo these nonnomal fluids it is suggested that the ompessibility fato yielded by the Lee Kesle equation be ompaed with atual fluid data ompessibility fatos on a plot suh as Figue 1.3 in Chapte 1. Fom this plot it may be seen what shift in itial onstants will bing the Lee Kesle ompessibility fato into onguene with the eal ompessibility fato fo the lagest egion on the hat. Seveal modifiations of the aenti fato may be neessay to ahieve the best ageement. D.3 IMORAN CONSANS FOR SELECED GASES Following ae two tables of impotant onstants neessay to implement the Redlih Kwong and Lee Kesle ompessibility fato equations. able D. gives the aenti fato ω fo seleted gases. able D.3 gives itial onstants fo the same gases. Data ae inluded fo ammonia hydogen and helium fo use in these equations; the data fo these gases should be amended as desibed in Setion D.1 fo best esults in the Redlih Kwong equation and as desibed above in Setion D. fo best esults in the Lee Kesle equation. able D.3 inludes itial onstants fom vaious authoities. It is suggested that onsensus values o aveages of all the values be used fo eah onstant. ABLE D.. Aenti Fato ω fo Seleted Gases Aetylene CO 0. Methane Ai Helium Nitogen Ammonia 0.0 n-hydogen 0. Oxygen 0.01 Agon p-hydogen 0.19 opane 0.1 See able D.3 footnote d fo bibliogaphy. aahydogen ω is alulated fom footnote g data.

5 Impotant Constants fo Seleted Gases 67 ABLE D.3. Citial Constants fo Seleted Gases Aoding to Vaious Authoities Gas aamete Maks Handbook a ey and oling Chilton b Ražnjević et al. d Handbook of Chemisty and hysis e ASME Fluid Metes f National Bueau of Standads g Aetylene R psia Ai R psia Ammonia R psia Agon R psia CO R psia Helium R psia n-hydogen R psia p-hydogen R 9.9 psia 186. Methane R psia Nitogen R psia Oxygen R psia opane R psia a Baumeiste. E. A. Avallone and. Baumeiste III eds. Maks Standad Handbook fo Mehanial Enginees 8th ed. MGaw-Hill b ey R. H. and C. H. Chilton Chemial Enginees Handbook th ed. MGaw Hill Ražnjević K. Handbook of hemodynami ables and Chats Hemisphee ublishing Copoation d oling B. E. J. M. ausnitz and J.. O Connel he opeties of Gases and Liquids th ed. MGaw-Hill 001. e Lide D. R. ed. CRC Handbook of Chemisty and hysis 8th ed. CRC ess In f Inteim Supplement No. 19. on Instuments and Appaatus Appliation at II of Fluid Metes 6th ed. Ameian Soiety of Mehanial Enginees g MCaty R. D. NBS Standad Database 1 MIROS National Bueau of Standads 1986.