Introductory Chemical Engineering Thermodynamics Chapter 9 - - PowerPoint PPT Presentation

Introductory Chemical Engineering Thermodynamics Chapter 9 - Introduction to Multicomponent Systems By J.R. Elliott and C.T. Lira

INTRODUCTION TO MULTICOMPONENT SYSTEMS The primary difference between pure and multicomponent systems is that we must now consider the impacts of changing the composition on the Gibbs energy. Beyond that, the Gibbs energy must still be minimized, the calculus of classical thermodynamics must be applied, the fugacities of the components in the phases must be equal, and, in general, the problem is pedagogically the same as the phase equilibrium problem for pure components. The computational methodology gets more complicated because there is more to keep track of. And the best choice of "equation of state" can change quite a bit depending on the nature of the solution. In this context, "equation of state" refers to any model equation for the thermodynamic properties of mixtures. But computations and equation of state development should not seem unfamiliar at this stage in the course. eg. dU(T,P,n) = ( ∂ U/ ∂ P)T,ndP + ( ∂ U/ ∂ T)P,n dT + Σ ( ∂ U/ ∂ n i )P,T,n j ≠ i dn i eg. dG(T,P,n) = ( ∂ G/ ∂ P)T,ndP + ( ∂ G/ ∂ T)P,n dT + Σ ( ∂ G/ ∂ n i )P,T,n j ≠ i dn i At constant moles of material, the mixture must follow the same constraints as any other fluid. That is, we only have two state variables left if we keep the composition constant. ⇒ ( ∂ G/ ∂ P)T,n = V and ( ∂ G/ ∂ T)p,n = -S ; Fundamental properties therefore dG = VdP - SdT + Σ ( ∂ G/ ∂ n i )T,P,n j ≠ i d n i µ i ≡ ( ∂ G/ ∂ ni )T,P,n j ≠ i ≡ "chemical potential" Elliott and Lira: Chapter 9 - Introduction to Multicomponent Systems Slide 1

Equilibrium Constraint: equality of chemical potentials dG = 0 = Σ µ i dni equilibrium, constant T and P Define system of two components (eg. EtOH+H2O) between two phases (eg. vapor and liquid). µ + µ + µ + µ = L L L L V V V V 0 dn dn dn dn 1 1 2 2 1 1 2 2 but if component 1 leaves the liquid phase then it must enter the vapor phase (and similarly for component 2). = − = − ⇒ dn L V L V and dn dn dn ⇒ ( ) ( ) 1 1 2 2 µ − µ + µ − µ = V V V V L V dn dn 0 1 1 1 2 2 2 Therefore, µ = µ µ = µ V L V L and . 1 1 2 2 Chemical potential of a pure fluid ( ) ( ) ( ) ( ) ( ) µ ≡ ∂ ∂ = ∂ ∂ = ∂ ∂ + ∂ ∂ G / n nG / n G n / n n G / n j n i i i i → → T P n , , → n n n ≠ → n n j i i n n i i i but ( ) ( ) ( ) ( ) ( ) ∂ ∂ → = ∂ ∂ = ≠ ⇒ ∂ ∂ = n G / n n G / n and G G T P , f n n G / n 0 . i n n i Therefore, µ j n → = G n i Note: This means G L = G V for pure fluid, as before. Thus, µ i is a "generalized G" for components in mixtures. Elliott and Lira: Chapter 9 - Introduction to Multicomponent Systems Slide 2

Equilibrium Constraint: equality of fugacities Fugacity - "escaping tendency" For a pure fluid: ln( f/P ) ≡ ( G-G ig )/RT For mixtures, generalize: RT ln( $ ln( f/P ) ≡ ( G-G ig )/RT f i / P ) ≡ ( µ i - µ i ig ) to where $ f i fugacity of component i in a mixture. Equality of fugacities as a criterion of phase equilibrium ( ) ( ) µ = µ V eq eq L eq eq T , P T , P i i [ ] ( ) ( ) ( ) ( ) ln $ / $ µ − µ = V eq eq L eq eq V eq eq L eq eq T , P T , P RT f T , P f T , P By definition, i i i i ( ) ( ) ( ) ln $ / $ µ − µ = ⇒ = V eq eq L eq eq V L T , P T , P 0 f f 0 but, i i i i $ $ = V L f f at equilibrium. i i Elliott and Lira: Chapter 9 - Introduction to Multicomponent Systems Slide 3

Activity, Activity Coefficient, and Fugacity Coefficient f i / fi ° "Activity" ai = $ Ionic solutions, reactions f i /( xi fi ° ) "Activity coefficient" = γ i = $ Low pressure phase equilibria "Fugacity coefficient" = φ i = $ f i /( xi P ) High pressure phase equilibria where fi ° is the value of the fugacity at standard state. The most common standard state is the pure component at the same temperature and pressure as the system of interest. In that case f i ° = f i . Note that the defining equation for activity coefficient can be rearranged to become an equation for calculating the fugacity of a component in a mixture. This is the general procedure used for liquids. $ = γ = γ ≡ γ φ L o o f x f x f x P i i i i i i i i i i Elliott and Lira: Chapter 9 - Introduction to Multicomponent Systems Slide 4

IDEAL SOLUTIONS The simplest solution model is to assume that interaction energiess in the mixture are like interaction energies in the pure fluid. e.g. 35 Cl 2 mixed with 37 Cl 2 is = 0 and H then, ∆ H mix = Σ n is H Tot i i As for the entropy change of mixing, the loss of order due to mixing is unavoidable is / ∆ = − x ln x Σ S R ( ) mix i i ∆ ∆ ∆ is is is G H S = + ∑ = − mix mix mix Combining ⇒ x ln( x ) i i RT RT R The definition of fugacity in terms of a change in free energies gives: ( )   µ − ∆ G $ G ∑ n ∑ f = = i i   mix i i x ln i RT n RT  f  i ( )   µ − $ ∆ G G f ∑ ∑ = = i i   mix i x x ln Thus, for an ideal solution, i i RT RT  f  i $ / $ ⇒ = ⇒ = is is f f x f x f i i i i i i Elliott and Lira: Chapter 9 - Introduction to Multicomponent Systems Slide 5

Raoult's Law = ⇒ Substituting the ideal solution result ⇒ y f V L x f $ V $ L = f f i i i i ( ) i i ( ) = ϕ ω V f / P Tr ,Pr , for the vapor i i i i We correct the fugacity of the saturated liquid for the effect of pressure on the liquid. ( ) = ϕ ω sat sat sat , , f P Tr Pr i i i i i ( ) ( ) = ⇒ = − L sat L sat dG RTd ln f RT ln f / f G G i i i i ( ) ( ) ( ) ( ) ( ) ∂ ∂ = ∂ ∂ − ∂ ∂ = − ∂ ∂ + ∂ ∂ = G / P H / P T S / P V T V / T T V / T V T T T p p ( ) ( ) ( ) ⇒ − = ∫ ≈ − = L sat L L sat L sat G G V dP V P P RT ln f / f i i i i i [ ] ( ) ⇒ = − L sat L sat f / f exp V P P / RT “Poynting Correction” [ ] i i i ( ) ( ) ( ) ϕ − sat sat L sat y P Tr , P r x P Tr ,P r exp V P P / RT Substituting, i i i i i i i i [ ] ( ) ( ) − sat , P V P / = i i i K x ( ) i ϕ   P Tr , r   i Note: at reasonably low pressures, sat [ ] 1 ( ) P i = i ⇒ K ϕ ≈ − ≈ L sat 1 and exp V P P / RT “Raoult’s Law” P i Elliott and Lira: Chapter 9 - Introduction to Multicomponent Systems Slide 6

Classes of VLE calculations Type Information_required Information_computed Convergence BP T,x P,y Easiest DP T,y P,x Not bad BT P,x T,y difficult DT P,y T,x difficult FL P,T x,y,L/F Most difficult Elliott and Lira: Chapter 9 - Introduction to Multicomponent Systems Slide 7

Short-cut Estimation of VLE K-ratios and Initial Guesses for VLE [ ] ( )( ) + ω − 7 / 3 1 1 1 / Tr 10 i i = K i Pr i   ( )( )  7 + −  1 1 1 /Tr i i   = = = sat 3 P P P Pc 10 Bubble Point Pressure: i i i i i 1 ∑ y y y P ⇒ = Σ x i = = Σ = Σ i i i 1   Dew point pressure,:   ( )( ) ( )( ) 7 7 K P + − + ω −  3 1 w 1 1 / Tr   3 1 1 1 / Tr    i   i i i i Pc 10 Pc 10 i i For bubble or dew point temperatures, it is not so straightforward. One guess would be: = Σ sat T x T i i = Σ Σ sat / T y Tc T y Tc i i i i i But these are somewhat cruder guesses than for the pressures. Elliott and Lira: Chapter 9 - Introduction to Multicomponent Systems Slide 8

Example 9.1 Bubble temperature, dew temperature, and flash A distillation column is to produce overhead and bottoms products having the following compositions: z(Overhead) Propane 0.23 Isobutane 0.67 n-Butane 0.10 Total 1.00 a) Calculate the temperature at which the condenser must operate in order to condense the overhead product completely at 8 bars. b) Assuming the overhead product vapors are in equilibrium with the liquid on the top plate of the column, calculate the temperature of the overhead vapors and the composition of the liquid on the top plate when operating at the pressure of part (a). c) Suppose only a partial condensation at 320 K. What fraction of liquid would be condensed? Elliott and Lira: Chapter 9 - Introduction to Multicomponent Systems Slide 9

Example 9.1 Bubble temperature… Solution: Use the short-cut estimates of the K-ratios a) The maximum temperature is the bubble point temperature. Guess T=310K Guess T=320K Ki yi Ki yi C3 1.61 0.370 2.03 0.466 iC4 0.616 0.413 0.80 0.536 nC4 0.438 0.044 0.58 0.058 0.827 1.061   + 1.000 - 0.827 ( )   ⇒ = − = T 310 320 310 317   − 1061 . 0827 . Elliott and Lira: Chapter 9 - Introduction to Multicomponent Systems Slide 10