Implementation of the Pertubed-Chain Statistical Association Fluid Theory Model

Transcription

1 Implementation of the Pertubed-Chain Statistical Association Fluid Theory Model Bjørn Tore Løvfall Olaf Trygve Berglihn February 20, 2004 Abstract This report represents a workbook from the subject KP8108, Phase equilibrium for fluids. First a review of the theory is presented, and afterward the theory is applied on the Perturbed-Chain Statistical Association Fluid Theory model (PC-SAFT). PC-SAFT is implemented giving Helmholtz free energy derivatives as a result. Using the Helmholtz free energy derivatives, phase equilibrium are calculated for examples of simple non-associating fluids. 1 Introduction Thermodynamics is the science concerned with the conversion of different forms of energy. A simple thermodynamic system is limited to describing a continuum with well defined system boundaries. A thermodynamic equlibrium state is characterized by being in a mode where an infitesimal imposed change can reversibly alter the state of the system. Heat and work terms are energy transfer modes. When you are doing thermodynamic calculations, it is convenient to divide the universe into two different parts. The specific part you are interested in, is called a system, while the rest is called surroundings. The system can be bounded with both physical and imaginary boundaries. This is unimportant from thermodynamic point of view, since one here only is interested in the amount of energy transported through the boundary. A system can be open, closed or isolated, depending on how it interacts with its surroundings. An open system can freely exchange both matter and energy with its surroundings. A closed system can only exchange energy, while an isolated system have no interaction with the surroundings. The state of the system is defined when all the thermodynamic properties (here state variables) of the system is defined. The state variables only relate to instantaneous properties, and thereby the state does not depend on previous 1

2 history. When a system change from one state to another, the system may pass through a series of intermediate states. The succession of intermediate states a system passes through when changing from one state to another, is called a path. A fundamental concept of thermodynamics is equilibrium. A system in a state of equilibrium cannot change without interactions with the surroundings. A system can still be in equilibrium with the surroundings even if the surroundings influence the system. For this to be possible, a quasistatic change in the surroundings must result in an equal change in the system. A system in a state of equilibrium cannot change its state without external interactions, but a system which is not in a state of equilibrium can change to such a state without external interactions. In order to reduce the complexity of a system, it is sometimes helpful to look at a simplified version of the system. A system where effects from gravitation, electric and magnetic fields, surface tension and so on is disregarded, is called a simple system.you can now look at internal energy as a function of entropy, volume and mole numbers U(S,V,n). The energy of a system is defined as the systems ability to perform work. Work is a macroscopic movement of matter by an external force. If one system performs work on an other system, energy is transfered from one system to another. Energy can also be transfered from one system to another without doing any work. This energy is transfered as a consequence of temperature differences in the two systems, and is called heat. This means it is two ways to transfer energy from one system to another; work and heat. The change in internal energy of a system, can be expressed by: du = ( ) U dt + T V 2 Theoretical background 2.1 Euler theorem ( ) U dv V T The euler theorem is used in thermodynamics to determine the homogenity of the functions used to describe the system. If you define a thermodynamic function as f, where f def = f(x 1,..., x n, ξ n+1,..., ξ m ) (1) Where the variables ξ n+1,..., ξ m are extensive variables, and x 1,..., x n are intensive. If this is true, one can write F = λ k f, if all the intensive variables are multiplied with λ. The k will reflect the order of homogenity of the intensive variables. If one takes a Helmholz function as an example, where A(T, V, n) 2

3 here V and n are intesnive, and T extensive. If you calculate the state with F 1 = A(T, V, n) and F 2 = A(T, λv, λn), F 2 = λ k F 1 These calculations have been done, for a system of NO x The order of homogenity of the elements of the gradient vector and the hessian matix can be compared. From the following MATLAB transcript, one can see how the homogenity of the gradient and the hessian are distributed. >> a=nox([300;1;1;1;1]); >> b=nox([300;2;2;2;2]); >> log(b.g./a.g)./log(2) ans = >> log(b.h./a.h)./log(2) ans = This also show that when a thermodynamic function is derived by an intensive variable, the homogenity of the functions remains the same, while when it is derived with respect to an extensive variable, the homogenity is reduced by one for each derivation. This can also be shown mathematically. If you have a function F = f(x, ξ) the differensial df = ( F X ) ξdx + ( F ξ ) Xdξ (2) Here X is defined as X = λx and the differensial will thus be dx = λdx + xdλ. The differensial in equation 2 will now be: df = ( F X ) ξxdλ + ( F X ) ξλdx + ( F ξ ) Xdξ (3) As an other example of homogenity one can make a contour diagram. Given Gibbs energy equation for a two component system (with constant T and p) where G = an 1 x 2 + N 1 ln(x 1 ) + N 2 ln(x2) (4) x 1 = x 2 = N 1 N1 + N2 N 2 N1 + N2 3 (5) (6)

4 N N 1 Figure 1: Gibbs Energy Isoplet Diagram Contour diagram for a = 2.4 is given in figure 1. From this figure you see that the distance between the contour lines is constant for constantly increasing system size. 2.2 Legendre transformations Legendre transformations is a mathematical tool that is useful in thermodynamics. The energy of a system, can be defined from the canonic U(S, V, n). Entropy, volume and mole numbers is not always available when calculations shall be done. Legendre transformations can then be used to transform the energy function into a function of something else. A Legendre transform f i of the function f is defined as: Where, by definition f i (ξ i, x j, x k,..., x n ) = f (x i, x j, x k,..., x n ) ξ i x i (7) ξ i def = ( f x i ) xi,x k,...,x n (8) In words, this can be summed up to; If you have a function with undesirable variables, you can transform this function into another function of useful variables. This is done by subtracting the partial derivative of the function with respect to the unwanted variable, multiplied by the unwanted variable (with all other variables constant) from the original function. If you for instance have 4

5 an internal energy function (U), but information for the entropy is missing, and you have the temperature, the internal energy function can be transformed to a helmhotz energy function by Legendre transforms. Table 1 show a complete set of transformations. What makes Legendre transformations especially useful in thermodynamics, is that no information is lost in the transformation. It is possible to transform back to the original function afterward. Table 1: Table of transformations Func var deriv Euler Legendre U S,V,N T,-P,µ T S P V + µn U A T,V,N -S,-P,µ P V + µn U - TS H S,-P,N T,-V,µ TS + µ N U + PV X S,V,µ T,-P,-N TS -PV U - µ N G T,-P,N -S,-V,µ µn U -TS + PV Ω T,V,µ -S,-P,-N -PV U - TS + µ N Y S,-P,µ T,-V,-N TS U + PV + µn 0 T,-P,µ -S,-V,-N U-TS +PV -µn 2.3 Multicomponent phase equlibrium For any given thermodynamic system in a state of equlibrium there is a minimization of energy. For a closed system at constant temperature and pressure, the Gibbs free energy can be minimized. For a closed system at constant volume, the Helmholz free energy can be minimized. Other sets of conditions can give rise to a minimum of their corresponding energy functions. In general: the energy functions are minimized in their canonical variables. At equlibrium for the phases, the system internal energy is at a minimum. At this minimum, du must be zero for all possible variations in S j, V j and n j, where superscript j denotes the phases j {α, β, }: (du) S,V,n = = ω du j ( ω T j ds j p j dv j + ) n µ j i dn j i i=1 = 0 (9) The above equation require constant entropy, volume and total composition (closed system), so the following constraints must be active: ω ds j = 0 (10) 5

6 ω dv j = 0 (11) ω dn j i = 0 i [1, n] (12) Using equations (10-12), ds ω dv ω and dni ω can be eliminated from (9): ( ) ψ n (du) S,V,n = T j ds j p j dv j + µ j i dn j i = ψ ds j T ω 1 + i=1 ψ p ω dv j ψ i=1 [ ψ (T j T ω )ds j (p j p ω )dv j + n µ ω i dn j i ] n (µ j i µω i )dn j i = 0 (13) Given that all the phases α,, ω exist in the system, ds j, dv j and dn j i are all indepedant variables near equlibrium. In order to satisfy (du) S,V,n) = 0, the following must hold: i=1 T j = T k j, k {α, β,, ω} (14) p j = p k j, k {α, β,, ω} (15) µ j i = µ k i j, k {α, β,, ω} (16) (17) Thus, equlibrium can be split into thermal equilibrium, mechanical (pressure) equilibrium, and chemical equilibrium. 2.4 Solving for phase equilibria The traditional way of solving vapor-liquid equilibrium (VLE) is the K-value method where the equlibrium equations are given as x j i = K ix k i i [1, n] (18) where x k i and k k i are the mole fraction in the phases j (liquid) and k (vapor). Combined with total mass balance and after some manipulations, the resulting equations x j i = x k i = N i N j + K i (N N j = K i N i N j + K i (N N j = 6 z i z j + K i z k (19) K i z i z j + K i z k (20)

7 Table 2: Stability tests δu 0 for any y stable δu < 0 for any y metastable δ 2 U < 0 for some y unstable δ 2 U = 0 for some y unstable δ 3 U = 0 for some y critical where z j = N j /(N j + N k ) and z k = N k /(N j + N k are the phase fractions and z i = N i /N the total fraction of component i. Using that f = n i=1 (xj i xk i ) = 0 leads to the following equation that can be solved by e.g. a Newton search: f = n (1 K k )z n i z j + K i z k = f i z i = 0 i=1 A more general aproach is to iterate on the relevant parts of the Hessian for du. i=1 2.5 Material stability Given a system of constant entropy, volume and composition and internal energy U(x) where x = (S, V, n) T. The phase assembly can be investigated for stability by introducing or removing trial phases and examining the effect on the system total internal energy. The energy change associated with the change in the phase assembly can be expressed as an Euler integration U/ y U y : Ũ(x, y) (U y U x ) y + 1 k! U x k ( y) k k=2 δu + ( 1) k δ k U (21) k! k=2 Depending on the values of δ k U U k x y k, the phase ensemble is said to be stable, metastable, unstable or critical. A complete stability search (by adding or subtracting trial phases) would require a massive search in all the dimentions of the system, and a smaller number of weaker tests given in table 2 can be used within a calculatory feasible set of search dimentions. 3 The PC-SAFT Model The original Statistical Associating Fluid Theory (SAFT) equation of state (EOS) was first introduced by Chapman et al. in 1990[1] and has inspired much derived work on the modeling of polymer-solvent systems. In 2001 Gross and Sadowski published a modified SAFT-EOS applying the perturbation theory of 7

8 Baker and Henderson to a hard-chain reference fluid known as PC-SAFT[2]. This model is applicable to mixtures of small spherical molecules such as gases, non-spherical solvents and chainlike polymers. The initial PC-SAFT model of Gross and Sadowski assume limited association and model the interaction between components with a dispersion contribution only. In 2003 both Solms, Michelsen and Kontogeorgis and Gross and Sadowski have published versions of the PC-SAFT including a association contribution term to account for highly asymmetric and associating mixtures and copolymer systems[5, 3]. In this work, the initial PC-SAFT equations and their derivatives have been implemented in Matlab and applied to phase equilibrium calculations of a simple system of methane in isobutane solvent. 3.1 The Equations The PC-SAFT equations are based on the reduced Helmholtz free energy contributions relative to ideal gas. Expressions for the residual Helmholtz free energy from chain of hard-spheres and dispersion are added to the ideal gas energy. We use here lower case with tilde, ã to denote dimensionless Helmholtz free energy pr. molecule. A = A id + A res (22) ã res = Ares (23) NRT ã res = ã hc + ã disp (24) The residual Helmholtz free energy is composed of two parts. A hard-chain (ã hc ) part based on temperature dependant hard-spheres that form chains, and a perturbation contribution which account for the attractive interactions (ã disp ). The original SAFT also has a association term (ã assoc ) for strongly asymmetric and associating fluids. We will look briefly on the different terms in the following sections. For a more in-depth look at the equations, we refer to the original PC- SAFT article by Gross and Sadowski[2]. Note that the equations have been restated in the free variables T, V, n in this work. In the original PC-SAFT paper, the authors make use of T, ρ, x, where ρ and x is the particle density and mole fractions respectively Some common terms and mixing rules m = 1 n i m i (25) N i 8

9 σ ij = 1 2 (σ i + σ j ) (26) ɛ ij = ɛ i ɛ j (1 k ij ) (27) Here m is the average number of segments per chain in the mixture of i different components. N is the total number of moles. Simple mixing rules are used for the pure component parameters for segment diameter σ and pair potential ɛ. The k ij is a fitted binary interaction parameter. This implies that the model only requires three pure component parameters m, σ and ɛ, plus the interaction parameter k ij. The segment number m is available in more general form for polymers by specifying the parameter ( m M w ), where M w is the mole weight of a polymer composed of a monomers of the same type. The specific segment number is then available by multiplying ( m M w ) by the mole weight Hard-Sphere and Hard-Chain Term The hard chain contribution to the Helmholtz free energy use a hard-spere reference as suggested by Mansoori et al[4]. The hard-chain is then built using thermodynamic perturbation theory of first order. ã hc = mã hs 1 n i (m i 1) ln gii hs (σ ii ) (28) N i [ ã hs = 1 3 ( 3 ) ] 3ζ 1 ζ 2 ζ 2 + ζ 0 1 ζ 3 ζ 3 (1 ζ 3 ) 2 + ζ2 2 ζ ζ 0 ln(1 ζ 3 ) (29) 3 ( ) ( ) 2 2 gij hs 1 di d j 3ζ 2 = + 1 ζ 3 d i + d j (1 ζ 3 ) 2 + di d j 2ζ 2 d i + d j (1 ζ 3 ) 3 (30) ζ p = π 1 6 V N Av n i m i d p i, p {0, 1, 2, 3} (31) i ( d i = σ i [ exp 3 ɛ )] i (32) kt Dispersion Term The dispersion term uses perturbation theory of Baker and Henderson to calculate the attractive part of the chain interactions. The total interaction of two chains is calculated by considering the sum of all square well potentials of the individual chain segments. ã disp = 2πN Av 1 V N I 1( mη) i j ( ɛij ) n i n j m i m j σij 3 kt 9

10 C 1 = I 1 (η, m) = I 2 (η, m) = N Av 1 V N π mc 1I 2 ( m, η) i ( 8η 2η2 1 + m j ( ɛij ) 2 n i n j m i m j σ 3 kt ij (33) ) 1 (34) (1 η) 4 + (1 m)20η 27η2 + 12η 3 2η 4 [(1 η)(2 η)] 2 6 a i ( m)η i (35) i=0 6 b i ( m)η i (36) i=0 η = ζ 3 (37) a i ( m) = a 0i + m 1 m a 1i + m 1 m 2 m m a 2i (38) b i ( m) = b 0i + m 1 m b 1i + m 1 m 2 m m b 2i (39) 4 Derivation of the model equations The model equations given describe a Helmholtz free energy surface in the free variables temperature, volume and mole numbers, A = A(T, V, n). Boldface n denotes the array of mole numbers [n 1,, n i ] T. From this function and its first and second order derivatives, all thermodynamic properties needed for phase equilibrium calculations can be found by noting that ( ) A = S (40) T V,n ( ) A = p (41) V T,n ( ) A = µ (42) n where p denotes pressure, and µ is the array of chemical potentials. T,V Second order derivatives (Hessian) can be used to check that the set of equations is homogeneous and for material stability analysis. Access to a symbolic Hessian is also very convenient when doing numerical iterations involving Newton search. One might argue that transformation to a enthalpy plane H = H(T, P, n) is more convenient for phase-equilibrium calculations. However, flash calculations based on an enthalpy surface would not yield information about mechanical stability and will pose problems when doing simulations of time variant systems (dynamic simulation). First and second order derivatives were calculated for the equations (22)-(39) with help from the Maple program. The result from derivation of the equations 10

11 with respect to all free variables gave an enormous equation set. All equations and the derivatives were coded partly by hand and partly using the Maple code generation routines and stated in matrix algebra form in a Matlab function m-file. 4.1 Checking Consistency The semi-manual derivation of the equations of such a large equation set can be very error-prone. All first and second order derivatives were therefore checked by first order perturbation. The reduced properties ã were checked for homogeneity by calculating ( g([t, 2V, 2n] T ) ) ln g([t, V, n] T (ln 2) 1 = k e (43) ) ln ( H([T, 2V, 2n] T ) H([T, V, n] T ) ) (ln 2) 1 = k e T e (44) where g is the gradient and H is the Hessian of A, e is a dim([t, V, n] T ) array of ones (i.e. [1, 1, 1, ] T ), and k e and k e T e are vectors and matrices giving homogeneity for partial first and second derivatives. The elements of the vector and matrix should correspond to the (reduced) homogeneity if the model equations are derived correctly, as described in section 2. Another property which is useful for checking the model is that the Hessian is singular in the direction of which it is calculated, i.e. ( 2 ) F x = 0 (45) x x ξ All the diagnostic checks above were applied when developing and debugging the model equations. The checks indicate that the final system of equations and derivatives are coherent and correctly implemented. 5 Results Using equation 41 a pressure-volume diagram was calculated for the system of 1 mol methane in 10 mol isobutane for temperatures in the range K. The results are shown in figure 2(a). A log-log plot of the same data is given in figure 2(b). The solid lines in the figures are calculated from the PC-SAFT model in this work. The dotted lines are calculated from a similar model using the Peng-Robinson EOS. The PC-SAFT EOS show good agreement with the Peng-Robinson EOS for lower pressure vapor phase. Deviation in the two phase region is significant. Deviation is also significant for liquid at high pressures. 11

12 7 x Pressure [Pa] Volume [m 3 ] x 10 3 (a) normal scale Pressure [Pa] Volume [m 3 ] (b) log-log scale Figure 2: Pressure-volume diagrams for 1 mol methane in 10 mol isobutane for temperatures in the range K with 10K increments. Solid lines are based on the PC-SAFT EOS, dotted lines on the Peng-Robinson EOS 12

13 For the log-log plot in figure 2(b), the results get non-physical meaning in the extreme pressures. The Peng-Robinson model goes to infinity for pressure, the PC-SAFT model becomes log-log linear. In this region the packing fraction η in PC-SAFT exceed the maximum possible value of Flash calculations for pure isobuane and the above system of methane in isobutane have been made for temperature range K in 30K increments. The calculations are based on iteration in volume, testing at each iteration the stability of the system to find possibly more stable set of phases. The results are given in figures 3(a) and 3(b). Note that the reference enthalpy was chosen to 0J at 298K and 1atm for both models. The result of flash calculations show the same tendency as in the p-v diagrams with deviation between the models in the two phase region and in liquid. Deviation is especially noticeable when closing in on the critical temperature. The simple flash code used in this work had problems in converging the Newton search for both Peng-Robinson and PC-SAFT in the two phase region when close to critical temperature. This can be seen by the broken 370K and 400K lines. 6 Conclusion and suggestions for further work The equations for the Perturbed-Chain Statistical Association Fluid Theory have been implemented together with their first and second order partial derivatives in a Matlab function. The function calculates and returns the first and second order derivatives of the Helmholtz free energy. By means of numerical perturbation and diagnostic checks utilizing the nature of thermodynamic functions, the system of equations have been tested and found to be coherent. Pressure-volume and enthalpy-pressure diagrams have been calculated to compare the PC-SAFT equation of state implemented with the Peng-Robinson equation of state for pure isobutane and a methane-isobutane system. The PC- SAFT and Peng-Robinson EOS differ most noticeably for two-phase systems close to critical temperature and for liquids at high pressure. The PC-SAFT model as published by Gross and Sadowski[2] does not include the association term as presented in the original SAFT-paper by Chapman et al. Gross and Sadowski have later presented a version of PC-SAFT for strongly associating co-polymer systems[3]. Further work could be done to extend the model implemented in Matlab to include the associating term and handling of co-polymers. As a first approach, the simplifications suggested by Solms, Michelsen and Kontogeorgis[5] should be considered. With available parameters, the PC-SAFT EOS could be used together with discritization of mole weight distribution of polymer to calculate more complex phase equilibrium. Further work on the flash routine used in this work would be required to handle multiple liquid phases and cases of failures in Newton search for multiphase region. 13

14 Pressure [Pa] Enthalpy [J] x 10 5 (a) Pure isobutane Pressure [Pa] Enthalpy [J] x 10 5 (b) 1 mol methane in 10 mol isobutane Figure 3: Enthalpy-pressure diagrams for pure isobutane and 1 mol methane in 10 mol isobutane for temperatures in the range K with 30K increments. Solid lines are based on the PC-SAFT EOS, dotted lines on the Peng-Robinson EOS. Reference enthalpy is chosen to 0J at 1 atm and 298K 14

15 Nomenclature m Average number of segments ɛ Depth of pair potential [J] g Gradient array (Jacobian) H Second derivative matrix (Hessian) µ Chemical potential [J mole -1 ] σ Segment diameter [m] ã Reduced (dimensionless) Helmholtz free energy ζ n disp hc hs id res A d g Abbreviation. Note however that η = ζ 3 equals the packing fraction Dispersion Hard chain Hard sphere Ideal gas reference Residual Helmholtz free energy [J] Temperature dependant segment diameter [m] Hard sphere radial distribution function k Boltzmann s constant, 1, JK -1 k ij m i N n i Binary interaction parameter Number of hard sphere segments in chain for component i Total number of moles Number of moles for component i N Av Avogadro s number, p Pressure [Pa] R Gas constant, JK -1 mol -1 S Entropy [J] T Temperature [K] 15

16 References [1] Walter G. Chapman, Keith E. Gubbins, George Jackson, and Maciej Radoz. New reference equation of state for associating liquids. Ind. Eng. Chem Res., 29: , [2] Joachim Gross and Gabrielle Sadowski. Perturbed-chain saft: An equation of state based on a pertubation theory for chain molecules. Ind. Eng. Chem. Res, 49: , [3] Joachim Gross, Oliver Spuhl, Feelly Tumakaka, and Gabrielle Sadowski. Modelling copolymer systems using the perturbed-chain saft equation of state. Ind. Eng. Chem. Res., 42: , [4] G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland. Equilibrium thermodynamic properties of the mixtures of hard speres. J. Chem. Phys, 54: , [5] Nicolas von Solms, Michael L. Michelsen, and Georgios M. Kontogeorgis. Computational and physical performance of a modified pc-saft equation of state for highly asymmetric and associating mixtures. Ind. Eng. Chem. Res, 42: ,