(1) The saturation vapor pressure as a function of temperature, often given by the Antoine equation:

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1 CE304, Sprng 2004 Lecture 22 Lecture 22: Topcs n Phase Equlbra, part : For the remander of the course, we wll return to the subject of vapor/lqud equlbrum and ntroduce other phase equlbrum calculatons (lqud/lqud equlbrum, sold/vapor equlbrum, etc.) Frst we wll consder more general formulatons of VLE problems than the Raoult s Law and Modfed Raoult s Law approach used n chapter 0 of SVA. Then we wll go on to consder some of the other phase equlbrum stuatons. The Gamma/Ph Formulaton of VLE: The modfed Raoult s law formulaton that we dscussed n chapters 0 and 2 used actvty coeffcents to provde a realstc treatment of the lqud phase, but were restrcted to an deal gas mxture descrpton of the vapor phase. In some cases, we may need to consder the non-dealty of the vapor phase. In general, we can wrte the vapor phase fugacty of each speces as ˆ v f ˆ φyp and the lqud-phase fugacty of each speces as fˆ l l xf The crteron for phase equlbrum s that these be equal: ˆ l φyp x f As shown n chapter of SVA, we can wrte the pure component lqud fugacty f l as l l V ( P P ) V l ( P P ) f f exp φ P exp RT RT So, we have l V ˆ ( P P ) φyp xφ P exp RT To keep ths compact, we wll defne l ˆ φ V ( ) ˆ P P φ exp φ RT φ As shown above, the exponental term (the Poyntng factor) s often neglected, snce for low to moderate pressures t usually dffers from by less than a percent. Wth ths defnton, the phase equlbrum equaton becomes y P xp To apply ths equaton, we need to have: () The uraton vapor pressure as a functon of temperature, often gven by the Antone equaton: B P exp A T + C p. of 9

2 CE304, Sprng 2004 Lecture 22 (2) A means to compute the fugacty coeffcent ratos,. If the pressure s not too hgh, then the formulaton usng the 2-term vral equaton, as dscussed n chapter of SVA: ˆ P lnφ B + y j y k ( 2δ j δ jk) RT 2 j k where δ j δj 2Bj Bjj B δ jk δkj 2Bjk Bjj Bkk δ δ δ 0 jj kk The fugacty of the urated vapor s, n ths formulaton ln ˆ BP φ RT Combnng these and neglectng the Poyntng factor, we get P ( ) ( 2 ˆ B P P + yjyk δ j δ jk ) 2 φ j k exp φ RT For a bnary system, ths smplfes to 2 B ( P P ) + Py2δ 2 exp RT 2 B22 ( P P2 ) + Py δ 2 2 exp RT (3) A model for the actvty coeffcents (whch can be obtaned from a model for the excess Gbbs energy). That s, we need at the temperature of nterest a model lke the Margules equatons, van Laar equatons, Wlson equaton, etc. of G E /(RT) as a functon of composton at fxed temperature. It s clear that the complex, coupled concentraton dependence of all of the peces of the equlbrum equatons wll requre an teratve soluton method to do thngs lke bubble pont, dew pont, and flash calculatons. Bubble Pont Pressure calculatons: For a bubble pont pressure calculaton, we specfy the temperature and lqud phase mole fractons. Knowng these, we can solve the equlbrum relatonshp for each speces for the (unknown) vapor phase mole fractons as xp y P We can sum these over all the speces to get p. 2 of 9

3 CE304, Sprng 2004 Lecture 22 or x P y P xp P The only thngs on the rght-hand-sde of the equaton that depend on the unknown vapor-phase mole fractons are the { }. An teratve procedure for solvng a bubble pont pressure problem lke ths s llustrated n fgure 4. n SVA, and outlned here: () Collect the nput data: T, lqud phase mole fractons {x }, and all of the parameters for the equatons to be used (Antone coeffcents, crtcal propertes from whch vral coeffcents are computed, parameters for the actvty coeffcent model). (2) Evaluate the {P } and the { }, whch are ndependent of the vapor-phase mole fractons. (3) Compute P from P x P wth all of the { } set to (that s, compute P xp, lke you would n the modfed Raoult s law bubble pont calculaton) (4) Compute the vapor phase mole fractons from teraton, but wll change n later teratons). y xp (the { } wll be n the frst P (5) Compute the { } usng the newly-found mole fractons n the equatons: P ( ) ( 2 ˆ B P P + yjyk δ j δ jk ) 2 φ j k exp φ RT (6) Compute P agan, usng the new values of the { } n xp P. (7) Compare the change n P from the prevous teraton (δp) to some convergence crteron (ε), and f the change s small enough, prnt out the P and {y } and stop. Otherwse, return to step 4 and repeat. Dew Pont Pressure calculatons: For a dew pont pressure calculaton, we specfy the temperature and vapor phase mole fractons. Knowng these, we can solve the equlbrum relatonshp for each speces for the (unknown) lqud phase mole fractons as p. 3 of 9

4 CE304, Sprng 2004 Lecture 22 yp x P We can sum these over all the speces to get yp x P or P y P The only thngs on the rght-hand-sde of the equaton that depend on the unknown lqud-phase mole fractons are the. An teratve procedure for solvng a dew pont pressure problem lke ths s llustrated n fgure 4.2 n SVA, and outlned here: () Collect the nput data: T, vapor-phase mole fractons {y }, and all of the parameters for the equatons to be used (Antone coeffcents, crtcal propertes from whch vral coeffcents are computed, parameters for the actvty coeffcent model). (2) Evaluate the {P }, set the { } to (they depend on the total pressure, whch we don t yet know), and set the { } to (they depend on the unknown lqud-phase mole fractons). (3) Compute P from P y P yp (4) Compute the lqud phase mole fractons from x P (5) Compute the { } usng the known vapor-phase mole fractons and the newly-computed pressure n the equatons: P ( ) ( 2 ˆ B P P + yjyk δ j δ jk ) 2 φ j k exp φ RT yp (6) Compute new lqud phase mole fractons from x, then normalze them (dvde P each x by the sum of all the mole fractons, whch should be, but won t be exactly one, snce we don t yet have the correct pressure). (7) Compute the actvty coeffcents. p. 4 of 9

5 CE304, Sprng 2004 Lecture 22 (8) Compare the change n actvty coeffcents from the prevous values (δ ) to a convergence crteron for actvty coeffcents (ξ). If the changes are all less than the convergence crteron, go on to step (9). Otherwse, go back to step (6). (9) Compute P from P y P usng the new actvty coeffcents. (0) Compare the change n P from the prevous teraton (δp) to some convergence crteron (ε), and f the change s small enough, prnt out the P and {x } and stop. Otherwse, return to step (5) and repeat. Bubble Pont T and Dew Pont T Calculatons: Smlarly, teratve methods are needed to compute bubble pont and dew pont temperatures, and the stuaton s even a bt more dffcult, snce the uraton pressures as well as other coeffcents depend on temperature. Iteratve procedures for carryng out these calculatons are llustrated n fgures 4.3 and 4.4 on pages 530 and 53 of SVA. Flash Calculatons: As you mght expect, flash calculatons are a lttle bt trcker yet, but are smlar. One frst has to do a bubble pont P calculaton and a dew pont P calculaton to confrm that the specfed T and P wll gve a 2-phase mxture. That s, we frst want to make sure that the specfed pressure s between the dew pont pressure and bubble pont pressure for the specfed temperature. If a 2-phase mxture wll be present, then we can estmate the actvty coeffcents and fugacty coeffcent ratos from the values obtaned n the bubble pont and dew pont calculatons. As shown on pp of SVA, we can assemble an equaton for flash calculatons just as we dd when we were usng Raoult s Law where we have a sngle equaton for the fracton of the system n the vapor phase. The most convenent form of ths equaton s: z( K ) 0 + V ( K ) where V s the fracton of the system n the vapor phase and K s the K-value for speces, evaluated as y P K x P The K are complcated functons of both vapor and lqud composton and contan all of the thermodynamc nformaton. An teratve procedure (based on ewton s method) for solvng the above equaton s outlned n fgure 4.5 of SVA. What f I don t want to mplement the algorthms gven n fgures 4. through 4.5 of SVA? p. 5 of 9

6 CE304, Sprng 2004 Lecture 22 Drectly mplementng the soluton algorthms gven n fgures 4. through 4.5 of SVA (n a spreadsheet, short computer program, or whatever) should be wthn your capabltes (gven the outlnes of procedures n SVA and suffcent tme), but s not a trval endeavor. Alternatves that may be more attractve are also avalable. Two alternatves to drect mplementaton of the schemes presented by SVA are: () Use a process smulator (lke HYSYS, Aspen, etc.), whch combnes both the equaton solvng functonalty and databases of thermodynamc propertes and coeffcents for the equatons. (2) Use a software package only for the equaton solvng. Set up the equatons n a package lke MathCad, Maple, Mathematca, or Matlab wth your own values for the parameters n the equatons. Utlze the more general methods for teratve soluton of sets of nonlnear equatons that are bult nto these mathematcal software packages. VLE Predctons from Cubc Equatons of State: Our crteron for vapor/lqud equlbrum s that the fugactes be equal n both phases. Cubc equatons of state can be used to compute fugacty coeffcents n both the lqud and vapor phase. So, we should be able to use cubc equatons of state to do VLE calculatons. The equlbrum crteron s: ˆl ˆv f f for all speces,, 2,, If we wrte both the vapor phase fugacty and the lqud phase fugacty n terms of fugacty coeffcents, we have fˆ l ˆ l ˆ v ˆ v φxp f φ yp or smply ˆl ˆv φ x φ y The smplest applcaton of ths s for pure component VLE. In that case, x y, and we have l v φ φ or alternatvely, l v lnφ lnφ 0 Remember that the Redlch/Kwong, Soave/Redlch/Kwong, and Peng/Robnson equatons of state can be wrtten n general as P RT a ( T) V b ( V + ε b )( V b + σ ) or dvdng by P and substtutng V (Z P)/(RT) qβ Z β Z + εβ Z + σβ ( )( ) p. 6 of 9

7 CE304, Sprng 2004 Lecture 22 ( ) bp a T where β and q RT brt To solve teratvely for a vapor-lke root of ths equaton, we wrote t as qβ( Z β) Z + β ( Z + εβ)( Z + σβ) and to solve teratvely for a lqud-lke root of ths equaton, we wrote t as ( )( ) + β Z Z β + Z + εβ Z + σβ qβ In chapter of SVA, we derved an equaton to compute the fugacty coeffcent of a pure speces usng a cubc equaton of state as q Z + σβ lnφ Z ln ( Z β) ln σ ε Z + εβ We get the lqud phase fugacty from ths equaton when we substtute nto t the lqud compressblty computed from ( )( ) + β Z Z β + Z + εβ Z + σβ qβ and we get the vapor phase fugacty from the equaton when we substtute nto t the vapor compressblty computed from qβ( Z β) Z + β ( Z + εβ)( Z + σβ) At a gven temperature we can fnd the vapor pressure teratvely by tryng dfferent pressures, computng the lqud and vapor compressbltes, computng the fugacty coeffcents, and varyng the pressure untl the lqud and vapor fugacty coeffcents are equal. The pressure at whch they are equal s the uraton pressure. If our guess for the pressure s too hgh, then the lqud fugacty coeffcent wll be less than the vapor fugacty coeffcent (at that pressure, the substance prefers to be a lqud). Conversely, f our guess for the pressure s too low, then the vapor fugacty coeffcent wll be less than the lqud fugacty coeffcent (at that pressure the substance prefers to be a lqud). We can do the same sort of thng to fnd the uraton temperature at gven pressure. Mxture VLE from cubc equatons of state: We can wrte the same equatons of state for a mxture as we dd for a pure component, and all of the composton dependence s taken up n the coeffcents that appear n the equaton (a and b or β and q). The equaton of state for the mxture s exactly as t was for the pure components, and we wll teratvely solve for the vapor and lqud compressbltes from qβ( Z β) Z + β Z + εβ Z + σβ ( )( ) and ( )( ) + β Z β + Z + εβ Z + σβ Z qβ where we have removed the speces subscrpts, snce now Z, β, and q refer to the mxture as a whole. β and q are stll defned by p. 7 of 9

8 CE304, Sprng 2004 Lecture 22 at ( ) bp β and q RT brt but now a and b are composton-dependent propertes of the mxture. Unfortunately, there s no establshed, unversal, theory for ther composton dependence. Usually, emprcal mxng rules are used to combne the pure speces equaton of state parameters to get mxture parameters. The smplest realstc expressons are a lnear mxng rule for b and a quadratc rule for a wth parameters determned as follows: b a a xb j aa j j x x a j j These equatons provde a recpe for evaluatng the mxture parameters from the ndvdual speces parameters wthout any addtonal nformaton. Snce the mxture parameters are functons of composton, we can defne partal molar mxture parameters, whch are of use later n dong the phase equlbrum calculatons: ( nb) b n TPn,, j ( na) a n TPn,, j ( nq) q n TPn,, j ow, we wll skp the long dervaton on pages of SVA. Trustng that ther dervaton s correct, we end up wth an expresson for the fugacty coeffcent of speces n the mxture: ˆ b q Z + σβ lnφ ( Z ) ln ( Z β) ln b σ ε Z + εβ If the lnear mxng rule, b ( nb) n TPn,, j b xb, s used, then the partal molar parameter,, s just b, and we have ˆ b q Z + σβ lnφ ( Z ) ln ( Z β) ln b σ ε Z + εβ The partal molar q s are defned by p. 8 of 9

9 CE304, Sprng 2004 Lecture 22 na ( nq) brt q n n TPn,, j TPn,, j a b q q+ a b where agan we have omtted most of the dervaton. See examples 4. and 4.2 n SVA. p. 9 of 9