VLE from an Equation of State By J.R. Elliott and C.T. Lira - PowerPoint PPT Presentation

VLE from an Equation of State By J.R. Elliott and C.T. Lira

FUGACITY IN A MIXTURE BY AN EQUATION OF STATE ∂  G  ∑   dn dG = VdP - SdT + ∂ n i   i i T P n , , j i ≠ and noting,  ∂  A ∑   dn dA = -PdV - SdT + ∂ n   i i i T V n , , j i ≠ we may substitute  ∂  G ∑ dn   dA = dG - PdV - VdP = VdP - SdT + - PdV - VdP ∂ n   i i i T P n , , j i ≠  ∂  A ∑  ∂  G ∑ dn   dn   ⇒ - PdV - SdT + = -PdV - SdT + ∂ n ∂ n i   i   i i i T V n , , i T P n , , ≠ j i j i ≠ Equating coefficients of dni ∂ ∂  A   G      = = µ i( T,P,V ) ∂ ∂ n i n i     T V n j i , , T P n j i , , ≠ ≠ Referencing to the ideal gas state:   ig ∂ − ( A A ) / RT   f i/yiP ) = ( µ i ( T,P ) - µ iig ( T,P )) /RT = ln( � - lnZ ∂ n   i T V n j i , , ≠ VLE from an Equation of State Slide 1

K-Values from an Equation of State To apply this, consider the PR EOS as an example.     ig 1 2 1 + 1 + 2 ρ − + + ( ) b A A A Z ( ) B a ( ) ( ) 1 1 = − − −  = − − ρ − ln B Z / ln ln b ln    nRT 8 1 2 8 1 1 2 B Z + ( − ) B + − ρ bRT ( ) b     For “random mixing”, the probability of any “ i-j interaction” is the same and goes as the product of the “i-j A A concentrations”. This suggests that we could define A v = ΣΣ yiyjAij and B v = Σ yiBi letting A ij= jj by ii comparison to the form of the energy equation for mixtures (discussed below). Then differentiation (as detailed below) yields   2 Σ    y A  � v v 1 2 v f B v v + + v B A Z ( ) B ( ) ( ) 1 j ij  = v − − v − v − − ln i i Z ln Z B ln i        v v v y P B 8 + 1 − 2 A B v v v B Z ( ) B       i Note: AL = ΣΣ xixjAij and BL = Σ xiBi but the derivation of the fugacity coefficient would be the same and:   2 Σ   x A 1 2   � L L L L + + L L f B A Z ( ) B B ( ) ( ) j ij 1 i  = i L − − L − L − − i ln Z ln Z B ln        L 8 1 2 L L x P B L L + − L A B B Z ( ) B       i As we saw in the case of pure fluids, there is no fundamental reason to distinguish between the vapor and liquid phases except by the initial guess for Z . The equation of state approach encompasses this lack of distinction in a very direct way. To obtain an expression for Ki , it is convenient to define the fugacity coefficients of a mixture as � v � L f f ϕ L = � ≡ ϕ ≡ ϕ i v i L and � � at equilibrium, we find that K i i ⇒ Recalling that � v = � L f f i i ϕ v y P x P � i i i i i VLE from an Equation of State Slide 2

UNIT III. FLUID PHASE EQUILIBRIA FUGACITY IN A MIXTURE BY AN EQUATION OF STATE: Density Dependent Formulas Example. Fugacity coefficient for the virial equation For pressures to 10 bars, a common method is to use the virial equation given by: B B Z = 1 + B ρ ; where B = ΣΣ yiyjBij and B ij= jj . Develop an expression for the ii fugacity coefficient.   B ρ ig − ig ρ − ig 2 1 ∂ ( A − A ) / RT A A B dB A A Bn ( ) ∑ ∑ ∫ ; ϕ = − = ρ = ρ ⇒ = = ln ln Z B nn B   k i j ij ∂ n ρ nRT B RT V V   0 k T V n , , k ≠ i ) ( ) ( ) ( 2 ∑ ∑ ∑ ∑ ∑ B B = = Note: For B ij= jj , n n B n B n B n B ii i j ij i ii j jj j jj ( ) ( ) ( ) 2 ∑ ∑ ∑ 2 ∂ ∂ n B n B n B   ig ∂ − 1 ( A A ) / RT j jj j jj j jj = =   ∂ ∂ ∂ n V n V n   k k k T V n , , ≠ ( ) k i ∑ ∂ n B ( ) 2 ( ) ⇒ ( ) j jj ∑ ∑ = B ln ϕ = − ln = 2 ρ − ln B n B Z y B Z ∂ kk n k kk j jj j jk V k VLE from an Equation of State Slide 3

UNIT III. FLUID PHASE EQUILIBRIA Example. Fugacity coefficient for the van der Waals EOS The VdW EOS provides a simple but fairly accurate representation of key EOS concepts for mixtures. The main tricks developed for this EOS are the same for other EOS’s but the algebra is a little simpler. 1 ρ a = − Z 1 − ρ b RT a = ΣΣ yiyj a ij ; a ij= a a where ii jj b = Σ yi b i Develop an expression for the fugacity coefficient. Solution   ig ∂ − ( A A ) / RT ( ) ϕ = − ln ln Z   k ∂ n   k T V n , , ≠ k i b ρ b ρ ig   − ρ ρ ρ A A ) ( d b ) b a d b ( ) a ( ) ∫ ∫ 1 1 = − = − ρ = − − ρ − ρ ( Z bRT b ln b bRT b   ρ 1 − ρ ρ nRT b b b   0 0 ∑ ∑ n n a ig 2 − A A an ( ) ( ) 1 1 i j ij = − − ρ − = − − ρ − n ln b n ln b RT V RT V RT VLE from an Equation of State Slide 4

  ∂ ΣΣ   ( n n a ) ig 1 ∂ − ∂ ρ ( A A ) / RT n b ( ) i j ij 1 = − − ρ +  − ln b    1 ∂ − ρ ∂ ∂ n b n VRT n     k k k T V n , , k ≠ i ( ) ( ) ∑ ∑ n y b n b ∂ ρ b b j j j j ρ = = ⇒ = k b ∂ V V n V k ) ( ) ( ) ( 2 ∑ ∑ ∑ ∑ ∑ = = n n a n a n a n a Note: For a ij= a a jj , i j ij i ii j jj j jj ii ( ) 2 ∑ ∂ n a ( ) ∂ ΣΣ n n a ( ) ∑ j jj i j ij 2 = = a n a ∂ ∂ n n kk j jj k k 2 2 Σ ρ Σ n a x a ρ  b  b n ( ) ( ) ( ) j kj j kj k k ln ϕ = − ln 1 − ρ +   − − ln = − ln 1 − ρ + − − ln b Z b Z 1 1 k − ρ   − ρ b V VRT b RT a A b B B a A jk jk ; ; ; k k ρ ≡ ≡ ≡ ≡ b Z bRT B a A b B 2 Σ x A B ( ) ( ) j kj ln ln k ϕ k = − − + − Z B − Z B Z VLE from an Equation of State Slide 5

UNIT III. FLUID PHASE EQUILIBRIA Example. Fugacity coefficient for the PREOS 1 ρ 1 a = − Z ( ) 1 − ρ b RT 1 2 2 2 + ρ − ρ b b a = ΣΣ yiyj a ij ; a ij= a a where ii jj b = Σ yi b i Develop an expression for the fugacity coefficient. Solution   ig ∂ − ( A A ) / RT ( ) ϕ = − ln ln Z   k ∂ n   k T V n , , k ≠ i From our integration for the pure fluid,     ig 1 1 2 − + + ρ A A a ( ) b ( )  = − 1 − ρ − ln b ln    nRT 8 1 + 1 − 2 ρ bRT ( ) b     { }   ig 2 − [ ] [ ] A A an ( ) 1 1 1 2 1 1 2  = − − ρ − + + ρ − + − ρ n ln b ln ( ) b ln ( ) b  RT 8 nbRT   VLE from an Equation of State Slide 6

      ∂ ρ ∂ ρ b b 1 + 2 1 − 2 ( ) ( )         ∂ ∂ ig   2 n n ∂ − ∂ ρ ( A A ) / RT n b an       ( ) k k = − 1 − ρ + −  − − ln b      1 ∂ ρ ∂ n b n 8 1 1 2 1 1 2 + + ρ + − ρ nbRT ( ) b ( ) b       k k T V n , , k ≠ i           ∂ 2 ∂ an nb         ∂ ∂ n n 1 ( 1 2 )   2   + + ρ b an   k k ln − −    ( )  2 1 ( 1 2 ) 8 8  + − ρ  b nbRT RT nb     ( ) ( ) ∑ ∑ n x b n b ∂ ρ b b j j j j ρ = = ⇒ = k b ∂ V V n V k ) ( ) ( ) ( 2 ∑ ∑ ∑ ∑ ∑ = = n n a n a n a n a Note: For a ij= a a jj , i j ij i ii j jj j jj ii ( ) 2 ∑ ∂ n a ( ) ∂ ΣΣ n n a ( ) ( 2 ) ∂ an ∑ j jj i j ij 2 = = = a n a ∂ ∂ ∂ kk j jj n n n k k k VLE from an Equation of State Slide 7