Chapter From Fundamentals to Propertes 6 Table. Total propertes from an excess approach V U H A G S Pure component Real mxture Ideal mxture Mxng contrbuton Excess property = * v + 0 + * v v (.0) = * u + 0 + * u u (.0) = h* + 0 + (.06) h h* = a* + RT + (.07) ln( x ) a a* ( ) = g* + RT + (.08) ln( x ) g g* ( ) = s* R x + (.09) ln( ) s s* ( ).. Phase equlbrum... Some basc prncples for phase calculatons A. The phase equlbrum condton Accordng to the thermodynamc prncples, equlbrum s reached, consderng the constrants on the system and at gven pressure and temperature, when the Gbbs energy s lowest (see equaton (.9)). It can be shown that ths mnmum leads to the statement that the chemcal potental s dentcal n all phases, for any component. For two phases α and β, ths s wrtten as: μα β = μ (.0) For a two-phase equlbrum, equaton (.0) provdes as many relatonshps as components n the mxture ( ). For an equlbrum wth φ phases, ( φ ) relatonshps can be wrtten. Usng the defnton of the fugacty (.), the same rule can be wrtten as: fα β = f (.)
6 Chapter From Fundamentals to Propertes a. The vapour phase fugacty The vapour phase fugacty s always expressed usng the resdual approach. f V = Pyϕ V where (.) P total pressure; y vapour mole fracton of component ; ϕv vapour phase fugacty coeffcent of component. Ths property s computed wth an equaton of state usng (.7). In the lmt of low pressure (below 00 kpa), the deal gas approxmaton can be used, whch states: ϕv =. b. The lqud phase fugacty The fugactes n the lqud phase can be computed usng ether the resdual or the excess approach (see also the FFF Famous Fugacty Formulae of O Connell [, 7]). Usng the resdual approach,.e. wth an equaton of state, the same expressons can be wrtten as for a vapour phase: f L = Pxϕ L (.) wth x lqud mole fracton of component ; ϕl lqud phase fugacty coeffcent of component. Ths property s computed wth an equaton of state usng (.7). Usng the excess approach, the fugacty s calculated usng the defnton of the actvty coeffcent (.97): f L f L * T, P γx (.) where = The most general expresson for the lqud phase fugacty s gven by: f L = f L* γx Pσ ϕσ γx (.) P = s the Poyntng correcton that uses L P L σ v P P exp v dp exp σ RT RT v L, the lqud molar volume of component at T and assumed to be ndependent of P; ϕ σ s the fugacty coeffcent of component at saturaton calculated usng the vapour phase (equaton (.7); γ s the actvty coeffcent of component n the lqud. When the vapour pressure of component, Pσ s lower than 00 kpa (.e. most often, except for very lght components), the pure component lqud fugacty can be approxmated as: f L* = P σ (.6)
Chapter From Fundamentals to Propertes 6 c. Excess approach wth the asymmetrc conventon: The use of expresson (.) assumes that component exsts as a pure lqud n the pressure and temperature condtons of the mxture (or at least that ts propertes can be calculated n these condtons). Ths may be a strong restrcton (e.g. supercrtcal gases or onc speces). Ths s why the use of the excess approach s extended by defnng a generalsed reference state. f L xf L, = ref γ ( ref ) (.7) Ths reference state can be the pure component, n whch case equaton (.) s recovered, but t also can be user-defned (often at nfnte dluton). The exponent (ref) s added to the actvty coeffcent n order to ndcate that ts value depends on the reference state chosen. Fgure.a shows, for the example of a bnary mxture, how the fugacty vares wth composton. Each component has ts own fugacty (n the same way as each component has ts own chemcal potental). In the lmt of zero concentraton of component, ts fugacty * tends to zero, whle the fugacty of the other component s f (and vce-versa). If the plot had been lnear between these two end-ponts, the mxture would have been deal accordng to Raoult s law ( γ = n equaton (.) yeldng equaton (.8)). f ref = f * Fugacty of solvent f ref = f * Fugacty of solute f ref = f * Fugacty of solvent Henry's law of solute n solvent f ref = H f * Raoult's law for solute Fugacty of solute Raoult's law for solvent 0 x Solvent Solute Raoult's law for solvent 0 x Solvent Solute a) Symmetrc conventon b) Asymmetrc conventon Fgure. Fugactes for symmetrc and asymmetrc conventons, at fxed temperature and pressure, and varable composton for a bnary mxture. In case one of the components s supercrtcal (say component ), the rght sde of the dagram becomes unphyscal (fgure.b). Only the left sde of the dagram has a physcal sgnfcance: component s a lqud, and s called the solvent. Component, that does not exst n the lqud state at the gven temperature and pressure condtons, s called the solute. As a result, t s mpossble to determne f L* n equaton (.). Equaton (.7) needs to be used, and a specfc reference state needs to be defned for ths solute. An alternatve
66 Chapter From Fundamentals to Propertes method, based on the slope at nfnte dluton, may be used to calculate ts fugacty: the Henry constant, defned as: (.8) In other words, the reference state used for defnng the component s propertes s taken at nfnte dluton (as shown n table.6). ote that the Henry constant s necessarly defned at the solvent s vapour pressure (ths s where the solute concentraton s zero) and s therefore not a functon of pressure. In order to account for pressure, a Poyntng correcton must be consdered. Ths s why the lqud fugacty for the solute s now calculated as: (.9) where, n addton to the Henry constant, the followng factors are used: the dlute Poyntng correcton: v ( ) P Pσ s ( T, P) = exp (.0) RT It requres the nfnte dluton partal molar volume of component n the soluton ( v ). Ths factor s close to one f the pressure s less than MPa above the solvent vapour pressure ( Pσ s ), the asymmetrc actvty coeffcent, γ H ( T,x), that s computed usng the same type of models as the symmetrc actvty coeffcent. It s related to the symmetrc actvty coeffcent n such a way that at nfnte dluton n the solvent ( x = 0 ), the asymmetrc actvty coeffcent becomes one: (.) where γ T, x = 0 s the actvty coeffcent of the solute at nfnte dluton of component n the solute gven by the symmetrc actvty conventon. Expresson (.9) s only used for the solutes. For the solvents, equaton (.) remans vald. Ths s why, n a solvent + solute mxture, ths approach s called asymmetrc conventon (the defnton of the reference state, as used n (.7), s dfferent dependng on the type of component). In fact, the use of the asymmetrc conventon can be extended to all cases where the solutes are n low concentraton n a phase: gases dssolved n a lqud (they do not exst as pure lquds under the pressure and temperature condtons of the system) [8]; all solutes (ncludng lquds) n an aqueous phase (ther propertes are very dfferent from those of the pure component) [9, 0]; ons n an aqueous phase (they do not exst as onc speces n a pure state). s f L T, P Hσ Δ s, ( T) = lm x 0 x P Pσ f L = Hσ s, γ H x = γ H T, x γ T, x γ T, x = 0
Chapter From Fundamentals to Propertes 67 Table.6 Possble reference states of components Conventon Component Temperature Pressure Composton Phase Solute T System P System Pure Lqud Symmetrc Solvent T System P System Pure Lqud Solvent T System P System Pure Lqud Asymmetrc Solute T System Pσ Solvent Infnte dluton n Lqud the solvent It s mportant to note that as a result of ts defnton (.8), the Henry constant s not a pure component property, but rather a bnary, as t depends both on the solute () and on the solvent (n whch s dluted). Ths s why n mxed solvents, an exact defnton as gven by (.8) becomes qute dffcult, and a mxng rule s requred. Carroll [] suggests the use of solvents solvents solvents ln Hm, = xjln H, j+ ajk xx j k (.) j j k j where H j, s the Henry constant of solute n solvent j. The a jk parameters must be ftted on expermental values. It becomes very smple when a jk = 0. A useful summary of the use and ms-use of Henry s law s gven by Carroll []. Some addtonal comments on ths ssue are provded n secton.. (p. 89). d. The sold phase fugacty Calculatng ths fugacty requres a specfc approach whch depends on the phase beng looked for. The most general s the excess approach: f S = x S f S* γ S (.) In cases where the sold phase s a pure component, no actvty coeffcent γ S s requred. However, the sold phase s sometmes a mxture, n whch case ths coeffcent must be taken nto account. The models for the sold phase actvty coeffcent are generally dentcal to that for the lqud phase (Coutnho []). The pure component sold fugactes are calculated usng the fact that at ther crystallsaton temperature, the phase equlbrum condton holds (.): = f L * TF,, PF f S *, TF,, PF, (.) where the subscrpt F stands for the crystallsaton or fuson condtons. In order to calculate f L* ( T, P) at the system pressure and temperature, the fundamental equatons (.0) and (.6) are used. The dervaton can be found n numerous thermodynamcs handbooks (e.g. Vdal []). The resultng expresson s: * f S T, P hf, ln (.) f * T,, P RT L ( F F, ) = Δ T F, c PF,, T F, T F, ln F, T + Δ R T T + ΔvF, P PF, RTF, where ΔhF, = h L h S s the enthalpy of fuson of component (taken at T F, and P F, );
68 Chapter From Fundamentals to Propertes Δc cl cs PF,, = P, P, and the sold phase. s the dfference n molar sobarc heat capacty between the lqud ΔvF, = v L v S s the molar volume dfference upon fuson, taken at T F, and P F,. Once agan, ths property s consdered constant wth respect to pressure and temperature. Ths volume s generally small and has lttle nfluence on the fnal result when P-P F, s not too large. Often, t s consdered ndependent of temperature. B. The dstrbuton or partton coeffcent In practce, vapour-lqud phase equlbra are often calculated usng the so-called dstrbuton coeffcent, equlbrum coeffcent or equlbrum rato, whch descrbes, for each component, the rato of molar fracton n the vapour phase and n the lqud phase: y K = Δ (.6) x Accordng to the phase equlbrum relatonshp (.), the dstrbuton coeffcent can be computed usng ether a smlar (resdual) approach n both phases, or a dfferent (resdual and excess) approach n each phase. The former case s called a homogeneous method (or ph-ph), the latter a heterogeneous method (or gamma-ph). Ths s summarsed n table.7. Table.7 omenclature of the thermodynamc methods Approach Vapour fugacty calculaton Lqud fugacty calculaton homogeneous ( ϕ ϕ ) Resdual approach heterogeneous, symmetrc ( γ ϕ ) Excess approach, Resdual approach symmetrc conventon heterogeneous, asymmetrc ( γ ϕ ) Excess approach, asymmetrc conventon Usng equatons (.) and (.), the homogeneous method results n: ϕ L ( T, P, x) K = ϕv T, P, y (.7) In the heterogeneous approach, equaton (.) s combned wth (.) (symmetrc conventon), to yeld: Pσ σ T T P T Pσ T (, ) ϕ (, ) γ(, x) K = (.8) P ϕv T, P, In cases where the pure component has no vapour pressure, or when ts pure component propertes are dffcult to fnd, the asymmetrc conventon s used for the solutes (equaton.9), to yeld: ( y)
Chapter From Fundamentals to Propertes 69 Hσ T s, T, P γ H T, x K = P ϕv T, P, y (.9) It s worth notng that, dependng on the mxture and pressure condtons, the heterogeneous approach allows a number of smplfcatons that are of great use for understandng phase behavour and dentfyng trends: When the process pressure s low (below 0. MPa), all fugacty coeffcents may be consdered equal to one. In addton, the Poyntng correctons may then also be neglected. Ths consderably smplfes equatons (.8) and (.9). They become: Pσ ( T) K = γ ( T,x) (.0) P and Hσ s, ( T) K = γ H ( T, x) (.) P Equaton (.0) s very often used for low pressure non-deal mxture, and ts effect on the phase dagram wll be dscussed n secton.. (p. 60). Equaton (.) s generally further smplfed assumng that the solute s very dluted, and as such that γ H T,x s very close to unty: Hσ s, T K = (.) P Ths last equaton s sometmes known as the smplfed Henry s law []. When the lqud phase forms an deal mxture, then the actvty coeffcent n equaton (.0) becomes unty, and Raoult s law s found: P T K = P Usng (.6), ths law s also wrtten as: σ = (.) xp T yp (.) where the rght hand sde of the equaton s sometmes called the partal pressure of component. Snce the sum of the partal pressures equals the total pressure, (.) leads to: P= xp σ ( T) (.) Equaton (.) clearly shows the consequence of Raoult s law on the vapour-lqud equlbrum on an sothermal Pxy plot (as shown n fgure. n secton..., p. 0): the bubble pressure s a straght lne. Ths s n fact what s observed for mxtures of lke components (e.g. alkanes of smlar sze). It s mportant to note that Raoult s law mples that the composton be gven n mole fracton, whch s the second reason why molar fractons, rather than weght fractons are used n thermodynamc calculatons. σ
70 Chapter From Fundamentals to Propertes For lqud-lqud equlbra the dstrbuton coeffcent requred s defned by: α αβ x K = Δ β x If the excess approach s used, we obtan from equaton (.) the expresson: f L β β * β αβ γ γ ( x ) K = f L = * γα α α γ x (.6) (.7) The pure component lqud fugacty (the vapour pressure) has no effect on lqud-lqud phase splt. The actvty coeffcent model s often dentcal n both phases. The only dfference between nomnator and denomnator n equaton (.7) s due to the composton dfference between the two phases. It may be of nterest to note that the heterogeneous approach (dfferent model n dfferent phases) can also be used for lqud-lqud phase splt (n partcular when the phases are very dfferent, as for example an aqueous and an organc phase). The asymmetrc conventon can then be employed, usng a Henry constant for calculatng the fugacty of dlute components (e.g. hydrocarbons n the aqueous phase). In ths case, equaton (.7) becomes, n the same way as (.) (where a specfc actvty coeffcent model s used n each phase): Hσ T T s H, β β αβ γ (, x, ) K = (.8) Pσ T γ α T α, x The rato of actvty coeffcents s n ths case often neglected, as the concentraton n the dlute phase (β) s very small, and the other phase (α) s consdered almost deal. C. Flash calculaton (the set of equatons and unknowns) The dscusson proposed below focuses on the case of two-phase vapour-lqud equlbra. The same prncples are also vald, however, for lqud-lqud, lqud-sold or vapour-sold equlbrum calculatons. The equlbrum coeffcent must then be defned as the rato of molar compostons of the two phases present. When more than two phases are present, the number of equatons and number of unknowns ncreases but the basc prncples reman the same. More detals concernng the algorthmc mplementaton of the calculatons are avalable elsewhere (Rachford and Rce [], Mchelsen [6, 7]). In any of the equatons (.7), (.8) or (.9), clearly the lqud and vapour compostons (x and y) must be known before the dstrbuton coeffcent can be calculated. But snce the objectve of the phase equlbrum calculaton s precsely to calculate these compostons, t s clear that an teratve algorthm should be used. Duhem s phase rule (secton.., p. ) ndcates that two state varables are suffcent to calculate the composton and propertes of all phases present, provded that the feed composton s known. In the remander of ths document, the feed compostonal vector s wrtten as z n order to dfferentate t from the lqud (x) and the vapour (y) compostons. Consequently, we wll call the phase equlbrum calculaton (flash) dependng on the type of the two state varables gven (see also Table.7).
Chapter From Fundamentals to Propertes 7 a. PT, Tθ or Pθ flash Any phase property can be calculated knowng ts composton, pressure and temperature. The vapour fracton (rato of mole number n the vapour phase wth respect to the total mole number, ( θ=v ) must also be known n order to evaluate the materal balance. Hence, the basc equatons are dentcal for PT, Tθ or Pθ flash calculatons: The so-fugacty condton (.): f L = f V (.9) whch, usng one of equatons (.7), (.8) or (.9) for calculatng K, can be wrtten as: xk = y (.0) ote that there are two phases n equlbrum, resultng n as many equatons as components n the mxture ( ). If there had been three phases, the so-fugacty condton would have gven rse to equatons, and so on ( more equatons for each addtonal phase). The mass balance equatons: Fz & & & = Lx + Vy z = ( θ) x + θy (.) where &F, L & and V& are respectvely the molar feed flow, the lqud flow and the vapour flow. If more phases had been present, there would have been more terms on the rght hand sde of the equaton, but the number of equatons would be the same. As unknowns, we have the composton vector of the phases (lqud x, and vapour y), n addton to the vapour fracton θ (n the case of a PT flash). Ths makes + unknowns n the case of a two-phase flash. An addtonal equaton s requred to solve the problem. It s found by the smple consderaton that the sum of all molar fractons must be one. Ths sum can be appled to the lqud ( x = ), or to the vapour ( y = ). In order to keep the equaton general, t s replaced by: x y = 0 (.) Rachford and Rce (9) [] propose to substtute (.0) nto (.) resultng n: z x = + θ( K ) (.) Kz y = + θ( K ) These values of x and y can now be substtuted n (.), whch yelds: ( K ) z (.) + ( K ) = 0 = θ whch s known as the Rachford-Rce equaton very often used nsde computer algorthms.
7 Chapter From Fundamentals to Propertes Summng up, the unknowns n a PT flash are K and θ (.e. to be solved are: + unknowns), and the equatons ϕ K L = V ϕ ( K ) z = + θ K = (.) One possble procedure to solve the equatons (known as successve substtuton) s as follows:. Estmate the mssng pece of nformaton, θ. Estmate the dstrbuton coeffcents K. Use (.) to calculate x and y. Improve the evaluaton of K usng equatons (.7), (.8) or (.9). Evaluate a better θ from (.) a 6. If θ s dfferent from ts prevous value, return to, otherwse the answer s reached. a A ewton-raphson method s sutable for ths equaton [8], [9]. 0 Smlarly, a Pθ flash (bubble or dew temperature calculaton for example), the unknowns are K and T (.e. + unknowns), and the equatons to be solved are agan: ϕ K L = V ϕ ( K ) z = + θ K = 0 (.6) ote that, for bubble pont calculatons, θ = 0, and z = x ; so (.) becomes: Kx = (.7) Smlarly, for dew pont calculatons, θ =, and z = y ; so (.) becomes: ( y K) = (.8) One possble procedure to solve the equatons s as follows:. Estmate the mssng pece of nformaton, T. Estmate the dstrbuton coeffcents K. Use (.) to calculate the unknown phase composton. Solve for the temperature usng smultaneously equatons for K calculatons (for example (.7), (.8) or (.9)) and (.7) or (.8).
Chapter From Fundamentals to Propertes 7 A Tθ flash s conceptually dentcal to the Pθ flash, except that the unknown s the pressure P nstead of the temperature T. Example.7 Dstllaton column A mxture of lght hydrocarbons s processed n a dstllaton column (see fgure.). Compostons of dstllate obtaned from the total condenser and of the resdue at the bottom of the column are as follows (see table.8): V n Stage n L n + D F B Fgure. Sketch of a dstllaton column. Table.8 Dstllate and resdue composton for example.7 Component Dstllate Resdue Propane 0. so-butane 0.67 0.0 n-butane 0.0 0.6 so-pentane 0. n-pentane 0.7 a. What s the column pressure to obtan the specfed dstllate f temperature n the condenser drum s 0 F? (The pressure drop n column, exchanger and drum wll be neglected). b. What s the temperature n the upper stage and the composton of the lqud pourng off ths plate (stage n on the fgure)? c. What s the temperature n the reboler? Use the Schebel and Jenny dagram (fgure.6) to evaluate the dstrbuton coeffcents.
7 Chapter From Fundamentals to Propertes Analyss: a. In the condenser, the lqud composton s known. Snce a lqud n equlbrum wth a vapour s at ts bubble pont, the pressure s the bubble pressure. The pressure s a constant n the column. b. The overall mass balance around the condenser ndcates that all flows have the same composton. Hence, the vapour leavng the top plate of the column has the dstllate composton. In addton, ths vapour s saturated and s therefore at ts dew pont. As pressure s known (as calculated n a.), temperature has to be computed (dew temperature) and lqud composton wll be part of the answer. c. The resdue from the reboler s also a lqud phase. Pressure s unchanged, so bubble temperature must be calculated. Components are all lght hydrocarbons and we are told to use the Schebel and Jenny procedure. Ths procedure assumes an deal mxture (equlbrum coeffcents are ndependent of composton) and s more sutable than Raoult s law at expected workng pressure (0 bar). All calculatons are relatve to vapour-lqud equlbrum and none of the hydrocarbons n the mxture s at supercrtcal condtons. Soluton: a. A bubble pressure s calculated usng (.7), Kx =. In order to be able to read the dstrbuton coeffcents K n the Schebel & Jenny dagram, both pressure and temperature are needed. A frst pressure must be guessed, so dstrbuton coeffcents can be read on the nomograph. Raoult s law can be used as a frst approxmaton: P = P σ x. The estmated pressure s 8. atm. = ow a lne must be drawn between the ponts T = 0 F and P = 8. atm. The three K values are read on the fgure. On the frst attempt, the sum s not equal to (see table.9). A dfferent guess must be made. A basc trck conssts n multplyng the old ( n) ( n ) pressure by the sum of the vapour phase compostons ( P = P K x ). Due to = the poor accuracy of the graphcal method, the soluton can be consdered to be reached n the second teraton. The bubble pont of the mxture s 8. atm. Ths procedure s summarsed n table.9. Table.9 Bubble pressure calculaton procedure at 0 F Component Propane x 0. Pσ 6. K ( 8. atm).68 y = Kx 7 K ( 8. atm).7 y = Kx 0.9 so-butane 0.67 6. 0. 0. n-butane 0.0.6 0.6 0.06 0.6 0.06 Sum/Result 8. 0.99.00
Chapter From Fundamentals to Propertes 7 00 PSIA Atm. abs. 0 00 00 0 0 00 80 60 0 0 9 0 8 7 00 90 6 80 70 60 0 0 0 0 0.9 0 9 0.6 8 0. 7 6 0. 0. Methane 6 7 8 9 0 Ethylene.. 0 Ethane 0 0 0 60 70 80 90 00. 0 0 00 00 Methane 00 00 600 700 0 6 7 8 0 0 0 0 0.6 0.7 0. 0. 0.. 60 70 80 0 00 0 00 00 Ethylene Propylene 6 6 0 7 Ethane Propane. 8 9 0 0 0.6 0.7 7 0 8 0. 0. 0. 0. 0 0 60 70 80 0 00 0 0 0 60 70 80 00 00 0. 0.6 0.7. 0. 0. 6 0 0. 0. 7 8 0 0 Propane Propylene 0 0. 0 0 Isobutane 60 0 70 80 00 0. 0.6 0.7. 0. 0. 6 7 8 n-butane 0. 0 0 0 0 0 Isobutane n-butane Isopentane n-pentane 0. 0. 0. 0. 0. 0. 0. 0.6 0.7. 6 7 0..06.07.08.0 0.0 0. 0. 0. 0. 0. 0.6 0.7.0.0.06.07.08 0. Hexane Isopentane.0. 0.6 6 7 8 0.0 8 0 0. 0. 0. 0.7. 6 7 0. 0..0.06.07.08 0. 8 6 7 Heptane.0.0.0 n-pentane.0.0 0.0 0. Octane onane Hexane.0.0.06.08 0..0.0 Decane Undecane.00.00.006.007.008 0.0.0 0.0 0.0 0.0 0.06 0.08 0. 0. 0. 0. 0.6 0.0 6.007.008 0. 0. 0. 0. 0.6 0.0..006.008.00.00.00.6.8 Dodecane.00 0. 0.6.6.8.00 0....6 Trdecane 0...7.... Heptane Tetradecane Octane 0. 0....6.....08 0..08 0..0.06 0..0.06 0.0.08 0.0 0. 0.06 0.08.... 0. 0.6.08.0.06 0..0.0.0 0..0.07.0.0.0.0.0 onane 0.0.0.0 0.0.0.0 0.06 0.07.008 0.0.008.0.0 0.0 0.0.006.006 Decane.006.008.00.00.00.00.00 Undecane.00.00 Dodecane.00 Trdecane Tetradecane C - -0-0 0 0 0 0 0 60 70 80 90 00 0 0 0 0 0 60 70 80 90 00 0 F 0 0 0 0 0 0 60 70 80 90 00 0 0 0 0 0 60 70 80 90 00 0 0 60 80 00 0 0 60 80 00 0. Equlbrum constant for hydrocarbons (Schebel and Jenny) 0 Fgure.6 Schebel and Jenny [0] nomograph for lght hydrocarbons. b. Snce the pressure drop s neglected, the entre column s assumed to be at 8. atm. We wll start the calculaton wth the dstrbuton coeffcents estmates from a. A dew pont s calculated and the crtera n y K = (equaton.8) must therefore be satsfed. =
76 Chapter From Fundamentals to Propertes At the frst teraton (table.0), as the composton of the lqud phase s not correct (sum equals.), the composton s normalsed, allowng us to calculate a hypothetcal so-butane dstrbuton coeffcent (K so-butane = 0.67/0.7 = 0.9). A lne s drawn at the same pressure through ths pont and the other values of K are read from the dagram and ntroduced n the table. Convergence s reached n almost two teratons, wth a fnal temperature of F, as summarsed n table.0. Table.0 Dew temperature calculaton procedure at 8. atm Component y K ( 0 F) x = y K x = x xj K ( F) x = y K j= Propane 0..7 0. 0..8 0. so-butane 0.67 7 0.7 0.9* 0.76 n-butane 0.0 0.6 0.6 0. 0.7 0.9 Sum/Result. 0.999 ' * calculated usng K = y x = 0.67/0.7 c. The reboler s the equpment where lqud at the bottom of the column exts the system whle vapour n equlbrum s renjected. Ths s a new bubble pont problem where the pressure s known, and the temperature s to be calculated. In ths case agan, convergence s reached when Kx =. After the frst teraton (table.), the vapour composton s normalsed and a K (n ths case n-butane s used) s estmated. The lne s drawn through ths value and the pressure of the column. In ths case, two teratons are necessary. The dstrbuton coeffcents are gven n table. for the two ntermedate calculatons and the fnal result. Table. Bubble temperature calculaton procedure at 8. atm, for the resdue composton Component x K ( F) y = Ky y = y yjk ( 07 F) K ( 97 F) y = Ky j= so-butane 0.0 0.9 0.08 0.07.78.70 0.0 n-butane 0.6 0.7 0. 0.67.6 *.7 0.69 so-pentane 0. 0. 0.08 0.097 0.77 0.7 0. n-pentane 0.7 0.6 0.096 0.9 0.66 0.6 0. Sum/Result 0.9.000.007 * calculated usng K = y ' x = 0.67/0.6 The temperature of the reboler s found to be 97 F. Ths example s dscussed on the webste: http://books.fpenergesnouvelles.fr/ebooks/thermodynamcs
Chapter From Fundamentals to Propertes 77 b. Flash where ether P, T or θ s provded plus another property In all cases, f the basc equatons (.) along wth the K calculaton methods (.7), (.8) or (.9) are to be used, pressure, temperature and vapour fracton need to be known. In ths type of problem, however, only one of the three varables s gven. An addtonal unknown must therefore be taken nto account. In order to solve the problem, an addtonal equaton must also be avalable. Ths extra equaton can be wrtten as a balance equaton usng the addtonal property that s gven: If volume s gven (sochorc flash): ( θ) vl + θvv = v (.9) If enthalpy s gven (senthalpc flash): ( θ) hl + θhv = h If entropy s gven (sentropc flash): ( θ) sl + θsv = s If nternal energy s gven (closed system flash): ( θ) ul + θuv = u (.0) (.) (.) ote that ths means that a method must be provded for the calculaton of volume, enthalpy or entropy for each phase as a functon of pressure, temperature and phase composton (see secton., p. ). As an example, f we can llustrate a PH flash (senthalpc flash as n a dstllaton column), the unknowns are K, T and F (.e. + unknowns), and the equatons to be solved are: ϕ K L = V ϕ ( K ) z (.) + ( K ) = 0 = θ θ hl + θhv ( ) = h One possble procedure to solve the equatons s as follows:. Estmate the mssng pece of nformaton, T and θ. Estmate the dstrbuton coeffcents K. Use (.) to calculate x and y. Solve for the temperature usng the full system (.): + equatons wth + unknowns. The K values are calculated as shown wth equaton (.7) or wth (.8) or (.9) dependng on the chosen method.
78 Chapter From Fundamentals to Propertes Example.8 Enthalpy balance n a column An adabatc dstllaton column s used to separate a mxture of n-butane () and n-heptane (). The lqud feed s ntroduced drectly on the thrd stage as shown on fgure.7. The column operates at an sobarc pressure of.6 atm. Some addtonal peces of nformaton concernng the characterstcs of the feed and the surroundng stages are also gven n table.: Table. Data of example.8 Poston Feed Stage Stage Temperature ( C) 7. Vapour molar flow (mol s ) 0. Lqud molar flow (mol s ) 00 0.6 8. Vapour molar composton y 0. 0.6 0.990 Lqud molar composton x 0.969 For the smplcty of the model, some basc expressons have been selected for enthalpy calculaton and for dstrbuton coeffcent predctons: the expressons are h = AT + B (wth T n C, and h n cal mol ) for both phases and Ln( K ) = a / T + b (wth T n K). These equatons apply to each component and the values of the coeffcents are found n table.: Table. Parameters for example.8 Vapour enthalpy Lqud enthalpy Dstrbuton coeffcent Component n-butane AV. BV 70 AL BL 0 a 0. b 8.6 n-heptane 9.7 98 0.6 0. a. What s the temperature of stage number? b. What s the temperature of stage number? c. What are the temperature and compostons of stage number? Analyss: The propertes to be evaluated are the temperature of the varous equlbra on the three stages around the feed. Pressure s fxed and part of the compostons s gven. a. For the nd stage, the lqud and vapour composton are known, so the dstrbuton coeffcents are drectly found. The temperature can be calculated usng the relatonshp between K and T. b. For the th stage, composton of the lqud s known, meanng that a bubble pont has to be determned. c. On the rd stage, all ncomng flows are known, so the overall composton and the total enthalpy may be calculated. A PH flash must be solved usng the set of equatons (.). The fluds are regular lght hydrocarbons. All propertes are well known. Partcular expressons are recommended and no bnary nteractons are consdered (deal mxture, * *.e. h= hx + hx ).
Chapter From Fundamentals to Propertes 79 F L L V V Fgure.7 Secton of the column wth stream and plate nomenclature Soluton: a. Stage number s defned n both phases. So the dstrbuton coeffcents of each component can be calculated (t s a bnary mxture!). y K = 0 969 and x =. 0 6 =.. 7 y K = 0 0 x =. 0 =0.. 07 ormally, knowledge of T on the nd stage wll gve both K and K, so there are two equatons for one unknown. They lead to: a a T, = = 6.98 K and T, = =.98 K. The average s 6.8 K. ln( K) b Ln( K) b b. For the th stage, a bubble pont calculaton has to be undertaken. The teratve procedure must start wth an ntal value (perhaps the temperature of stage number, or any other chosen by the user). If the temperature of stage s used, equaton (.7) yelds: 0.. 6 Kx = exp + 8. 6 0. 9 + exp +... 6. 08 0. 0 07 = 96 6 08 The temperature must be lowered to reduce the sum. After a few teratons, a value of 98.66 K s found for stage number. c. Materal and energy balances around the rd stage must be wrtten. Fx & F L& xl V& yv L& xl V& yv + + = + Fx & F + L & xl V & yv L & xl V & yv + = + Fh & LF, + L& hll, VV LL VV V h, L h,, + & = & + V& h Ths mples that enthalpes of each flow must be calculated. For example, the enthalpy of the vapour leavng stage s expressed as: VV, V VV, V VV, h = y h + y h = 0. 969 670 + 0. 0 08 = 660 cal mol
80 Chapter From Fundamentals to Propertes Usng ths technque, the materal and enthalpy content of each of the streams enterng the rd plate ( F &, L & and V& ) can be calculated and summed, as shown n table.. Table. Materal and enthalpy content of the stream enterng plate Component n-butane (mol s ) &F 0 V&. L & 09.9 F& + L& + V& 0.08 n-heptane (mol s ) 0. 8.7 9.6 Total nc + nc 7 (mol s ) 00. 8. 6.7 h (cal mol ) 090 660 90 0 H (cal s ) 09000 876 0670 60077 Introducng the equlbrum condton n the above materal balances ( frst equatons), they can be treated as equaton (.) leadng to the Rachford-Rce equaton. For a bnary mxture, the vapour fracton can be calculated drectly: ( K ) z ( K ) z ( K 0 + θ K + θ( K ) = θ = ) z+ ( K ) z K K where z and z are the global mole fractons on plate. They can be found from the global flow rates shown n table.: z = 0.08/6.7 = 0.77 and z = 9.6/6.7 =. We are now left wth two equatons (the enthalpy balance and the above Rachford-Rce equaton) wth two unknowns (θ and T). The complete algorthm becomes very smple: assume a temperature, calculate K and K, solve the Rachford-Rce equaton to fnd θ, calculate the enthalpes of streams V and L, compare the sum of these two outlet enthalpes wth the nlet enthalpy calculated n table.. A complete teraton s shown n table.: Table. Results of one teraton, usng an assumed temperature of 0 C, yeldng a vapour fracton θ = 0.66, and a total enthalphy of 989. kcal s Component + ( ) F& + L& + V& z K x y L & V& L & V & + n-butane (mol s ) 0.08 0.77.87 0.6 0.976 90.6.6 0.08 n-heptane (mol s ) 9.6 0.06 0.0 6.8.79 9.6 Total nc + nc 7 (mol s ) 6.7 7.. 6.7 h (cal mol ) 0 668 607 78 H (cal s ) 60077 076 6078 989 As observed n the results above, calculated at a temperature of 0 C, the vapour fracton of the flash s 0.66 and the total enthalpy of L& + V& s 989 cal s, greater than the value of 6078 cal s for F& + L& + V&. The temperature must therefore be less than 0 C. A few addtonal teratons lead to a fnal value of.9 C wth the correspondng compostons and vapour fracton as seen n table.6:
Chapter From Fundamentals to Propertes 8 Table.6 Fnal teraton, at.9 C and vapour fracton θ = 0.90 Component F& + L& + V& z K n-butane (mol s ) 0.08 0.77.67 0.7 0.986.9 9. 0.08 n-heptane (mol s ) 9.6 0.00 0.78 0.0 8.9 0.69 9.6 Total nc + nc 7 (mol s ) 6.7.87 9.8 6.7 h (cal mol ) 0 7 67 0 H (cal s ) 60077 076 6078 60077 Ths example s dscussed on the webste: http://books.fpenergesnouvelles.fr/ebooks/thermodynamcs x y L & V& L & V & + c. Flash where P, T and θ are unknown When none of the basc varables needed n the basc equatons (.) along wth the K calculaton methods (.7), (.8) or (.9) s avalable, they all must be calculated, resultng n a total of + unknowns (K, P, T et θ). Hence, + equatons are requred, or one more compared wth (.). Two addtonal balance equatons (from (.9) through (.)) are therefore requred: one for each of the known state varables. As an example, for solvng a flash at fxed volume and nternal energy (UV flash ths case s found when solvng ppe transport problems), the unknowns are K, T, P and θ (.e. + unknowns), and the equatons to be solved are (provded that (.7) s used for calculatng K ): ϕ K L = V ϕ n ( K ) z + ( K ) = 0 = θ θ ul + θuv ( ) = u L V ( θ) v + θ v = v (.) One possble procedure to solve the equatons s as follows:. Estmate the mssng peces of nformaton, T, P and θ. Estmate the dstrbuton coeffcents K. Use (.) to calculate x and y. Solve for the temperature usng the full system (.): + equatons wth + unknowns... Practcal applcatons of phase equlbrum In process engneerng, phase equlbrum must be calculated n dfferent types of problems. In ths secton, we show how the practcal applcatons wll lead to the use the concepts developed above.
8 Chapter From Fundamentals to Propertes A. Dstllaton or strppng: separaton accordng to volatlty (VLE) When the problem requres separaton accordng to the relatve volatlty of the components, the most mportant property to be calculated s the equlbrum coeffcent (see (.7), (.8) or (.9)) of each component. The relatve volatlty s then defned as: K T, P, x α j = Kj T, P, x (.) The concept of relatve volatlty s essental n dstllaton processes. When pressure s low, but non-deal mxtures are consdered, equaton (.0) yelds: Pσ γ T, x αj = Pσ j γ j T, x (.6) The actvty coeffcents can become sgnfcant and modfy the relatve volatlty between two components consderably. An example s dscussed n example. (p. 6). The relatve volatlty between n-butane and - butadene changes sgnfcantly upon addng acetontrle, whch s therefore used as an extracton solvent. Ths solvent has a low volatlty, and therefore remans n the lqud phase throughout the dstllaton column. Whle butane and butadene are almost nseparable by classcal dstllaton (, butadene s very slghtly more volatle than n-butane), they can be separated due to ther dfference n volatlty usng a technque known as extractve dstllaton. Butane s then sgnfcantly more volatle than,-butadene. For low pressures and deal mxtures, the relatve volatlty may be approxmated as (usng (.)): Pσ αj = (.7) Pσ j B. Extracton: separaton accordng to chemcal affnty (LLE) Lqud-lqud extracton methods are also based on phase separaton and as such equatons (.0) and (.) reman vald. evertheless, two lqud phases are present smultaneously, and the dstrbuton coeffcent now s defned by (.6). The fugactes are calculated ether usng the resdual approach (.), or usng the excess approach (.6). In prncple, both methods can be used for calculatng a lqud-lqud phase splt. Consderng the complexty of lqud-lqud phase behavour (results are very senstve to small changes n parameters), the excess property models are often more accurate for ths type of calculaton. The reason why a lqud phase splt appears s not related to the vapour pressure of the pure components, but rather to the chemcal affnty among the components. Ths has been dscussed above: the lqud-lqud dstrbuton coeffcent defned n equaton (.7). On the condton of havng good mxng rules, equatons of state can also be used. ote that, as a consequence of ths, the correspondng states prncple secton...c (p. 7) (whch provdes a very convenent method to calculate hydrocarbon propertes and uses the crtcal pont to descrbe ts volatlty) s not adapted to predct lqud-lqud phase
Chapter From Fundamentals to Propertes 8 splts. Consequently, the tradtonal pseudo-component descrpton s not adapted to the calculaton of these types of equlbrum. The lqud phase separaton s n fact senstve to chemcal affnty, whch s more accurately modelled usng the actvty coeffcent models, further dscussed n secton.. (p. 7). C. Separaton usng crystallsaton (LSE or VSE) Crystallsaton or sold phase formaton s based on another physcal phenomenon. When the temperature s suffcently low, molecular vbratons can no longer keep the molecules apart n the lqud phase. Instead, they start plng up n a confguraton that depends on the molecular structure. Ths s why crystallsaton temperatures are totally unrelated to the volatltes of the components (except when comparng smlar famles where the structure remans dentcal). The pure component crystallsaton propertes are further dscussed n secton.. (p. 0). Crystallsaton s a very convenent separaton technque as ts product s generally very pure: n the sold phase, components do not mx as n a lqud. In secton... (p. 70), the types of solds that may be encountered are dscussed. Phase equlbrum n the presence of a sold phase uses the same equatons as for all phase equlbra,.e. the so-chemcal potental (.0) or so-fugacty (.) condton along wth mass balances. The flud phase fugacty s calculated usng ether the resdual (equaton of state) or the excess (actvty coeffcent) approach. However, for the sold phase fugacty, the most general approach s the excess approach (.) wth (.), as dscussed above. The resdual approach can not be used here. D. Rsk of appearance of a new phase On some occasons, t s essental to know the rsk of appearance of an addtonal, unwanted phase. As an example, the appearance of a lqud phase n a vapour flow can strongly perturb the operaton of compressors, turbnes, heat exchangers, or, n reservor condtons, decrease the permeablty n porous materal. In addton, f ths lqud s an aqueous phase, t tends to concentrate corrosve components (acds, etc.) and as a result creates mportant rsks of corroson. If the ncpent phase s sold, the rsk of pluggng equpment s evdent. From a thermodynamc pont of vew, ths queston s nvestgated lookng at the phase dagram boundares. Phase boundares or solubltes are essentally two dentcal ways to envson the very same phenomenon. Ths can be understood when lookng at a Txy dagram (see fgure. for example). The bubble curve s the boundary between the pure lqud phase and the two phase vapour-lqud regon. The queston can be nvestgated n four dfferent ways:. It s useful to draw the full phase dagram to vsualse the pressure-temperature (and perhaps composton) path that the system encounters n the process. Specfc algorthms for phase dagram calculaton exst. An example of such curves s gven n fgure.6 and fgure.. If presented n the pressure-temperature plane, they often nclude a crtcal pont, and as such requre the use of a homogeneous method (equaton of state). If presented as Txy or Pxy plots, n the absence of a crtcal pont, they can be obtaned qute easly for vapour-lqud equlbra. For lqud-lqud or lqud-sold equlbrum, the calculatons are more complex and automatc tools are not readly avalable.
8 Chapter From Fundamentals to Propertes. The smplest approach conssts n searchng the boundary of the phase envelope by performng ether Tθ (search for the bubble or dew pressure) or Pθ (search for the bubble or dew temperature) calculatons. The actual system pressure and temperatures are then compared wth those resultng from the calculaton. If the ncpent phase s a second lqud or a sold, bubble or dew calculatons cannot be used. If t s a sold phase, the temperature when the frst crystal appears s calculated. In the case of paraffnc crudes, the termnology used s the Wax Appearance Temperature (WAT).. When the ncpent phase s assumed to be pure, t s also possble to compare the fugacty of the component n the ncpent phase wth ts fugacty n the bulk system. Whenever the former fugacty drops below the latter, phase separaton should occur (n partcular for sold phases, but also water condensaton).. The most elegant approach, when the system composton and two propertes correspondng to those lsted n table.7 are provded (for example pressure and temperature), s to perform a so-called stablty calculaton. Its purpose s to evaluate whether the sngle phase s thermodynamcally stable. Ths algorthm (Mchelsen []) evaluates whether, consderng the constrants of the system, a second ncpent phase can be formed, whch would result n a lower Gbbs energy of the system. It s generally mathematcally rather ntrcate, snce beng close to a phase boundary, the set of equatons to be solved presents several mnma that may be close to each other. The true soluton s the global mnmum, but the solver may stop wth a local soluton. Obvously, n all cases, a sutable model s requred both for the bulk phase and the ncpent phase. As wll be dscussed n secton.. (p. 6), ths type of calculaton s rather senstve to the correct representaton of the components at both ends of the phase behavour consdered: For vapour-lqud calculatons, these are the lowest volatlty and hghest volatlty components. For lqud-lqud calculatons, these are the components that most readly separate (wth the lowest affnty for each other). The presence of a thrd component that acts as a co-solvent may nfluence the phase behavour dramatcally however. For flud-sold calculatons, these are the components wth the hghest crystallsaton temperature on the one hand and the majorty component on the other hand. Example.9 Rsk of water condensaton n a gas stream A lght hydrocarbon mxture (CH : 80%; C H 6 : %; C H 8 : % n molar percent) s contamnated wth water. The mxture s avalable at 00 kpa but due to severe weather condtons there s a rsk of low temperature. Is there a real rsk of condensaton of water n the lne? Analyss: Ambent temperature s n all possble cases greater than the crtcal temperature of methane. Ethane and propane could condense but the vapour pressure, even at 0 C s greater than 00 kpa, so no hydrocarbons are expected to condense. The pressure s gven and a condensaton temperature must be found. It s a dew pont calculaton.
Chapter From Fundamentals to Propertes 8 Components are lght hydrocarbons and water. Water s known not to mx wth hydrocarbons n the lqud phase. Phases are vapour and lqud. The lqud phase wll contan only water. Soluton: If water condenses the aqueous phase can be consdered as pure, so wth a composton equal to unty. Pressure s low so the Raoult approxmaton s vald. Hence, the fugacty of the lqud, ncpent phase may be approxmated wth the vapour pressure of pure water. The fugacty of water n the bulk (vapour) phase s equal to ts partal pressure. Hence, f the partal pressure of water reaches the vapour pressure at a gven temperature, water wll condense: partal P Py P T HO = HO = σ HO The form of the DIPPR equaton for vapour pressure cannot be solved analytcally n T. A tral and error procedure has to be mplemented.. ln P sat = 7. 69 78 7. 07ln( T ) +. 6. 0 6T HO T σ P HO The maxmum water content before lqud drops out s then found as yho=. P Table.7 shows some results. If the temperature s below 0 C then the sublmaton pressure of water should be used and ce would appear. Table.7 Maxmum water content of a gas at the total pressure of 00 kpa Temperature ( C) Temperature (K) Vapour pressure of water (Pa) Maxmum water content (%) 0 7. 60 0.% 78. 87 0.% 0 8. 7 0.6% 88. 70 % 0 9. 9.7% 98. 70.9% 0 0. 8.% 08. 60.8% 0. 786.69% 8. 996.80% 0. 6.8% 8. 760 7.88% 60. 990 9.97% 6 8. 00.% 70. 8.9% 7 8. 86 9.8% Ths example s dscussed on the webste: http://books.fpenergesnouvelles.fr/ebooks/thermodynamcs
86 Chapter From Fundamentals to Propertes E. Relatve amounts of the phases For a number of ndustral applcatons, the composton of the phases s of lesser mportance, but the true ssue s the relatve amount of each phase. An example where ths s the true property to be looked for s found n multple phase flow calculatons, whether n ppelnes or fludsed bed. The procedure used to calculate the relatve amounts of the phases s dentcal to that explaned above (.) for vapour-lqud equlbrum. For lqud-lqud equlbrum the same knd of equaton would be used αβ n K z θ α αβ + ( K ) = 0 = where θ α = α α + β s the rato of matter n phase α. (.8) The choce of the correct model wll essentally depend on the components and ther nteractons, as dscussed n chapter. In the case of vapour-lqud equlbrum, the vapour fracton wll be more senstve to the correct solublty calculaton of the lght components n the lqud than to the descrpton of the volatlsaton of the heavy end. Indeed, the vapour phase generally contans only a very small quantty of heavy components, whle gases can make up a large fracton of the lqud phase. As a result, the engneer s attenton should focus on the nteractons between the gases and the bulk of the lqud. In the case of lqud-lqud equlbrum, the same ssue should be consdered as for the phase boundary calculaton (hereabove secton D, p. 8): the relatve concentratons of the components that show the least affnty should be examned frst. The presence of a co-solvent wll be the next mportant ssue. The calculaton of the relatve amounts of the sold and flud phases requres, n addton to the crystallsaton propertes of the pure components, a good descrpton of the sold phase. Consderng that many types of sold phase may exst (see secton..., p. 8), ths ssue may become complex, n partcular when several sold phases coexst. For ths ssue, we refer to more detaled textbooks [])... Chemcal equlbrum In some processes, thermodynamc calculatons are needed n order to determne how the equlbrum composton changes wth pressure and temperature as a result of one or several chemcal reacton(s). Ths assumes that equlbrum s quckly reached, or that no knetc lmtatons exst. Yet even when the reacton rate s not large, and must therefore be evaluated, t may be useful to descrbe t as the sum of postve and negatve contrbutons that cancel out at the equlbrum condtons. These condtons must therefore be known.