Ind. Eng. Chem. Res. 2004, 43, 8369-8377
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Correlating Multicomponent Mixture Properties with Homogeneous Rational Functions Walter W. Focke* and Barend Du Plessis Institute of Applied Materials, Department of Chemical Engineering, University of Pretoria, P.O. Box 35285, Menlo Park, Pretoria 0102, South Africa
Multivariate homogeneous rational functions, analogous to Pade´ approximants, are proposed for the modeling of mixture properties. They are defined as the ratio of two homogeneous Scheffe´ K polynomials. This choice offers a flexible functional form, attractive symmetries, parameter parsimony, and consistent expressions for multicomponent mixtures. The proposed rational functions satisfactorily correlate the nonlinear behavior of diverse multicomponent physical property data. They also extend the application of classic binary activity coefficient models such Porter, Margules, Scatchard, etc., to multicomponent mixtures. Introduction Accurate knowledge of physical properties is vital for the reliable design and optimization of chemical plants. Relevant properties include density, thermal conductivity, viscosity, heat of mixing, etc. Typically, they depend on the system pressure, temperature, and composition. Owing to the dominant position of distillation as a separation technique, much research has focused on theoretical and empirical models for the (excess) Gibbs free energy. The historical development started in the 19th century (van Laar and Margules) and culminated in predictive versions of semitheoretical local composition models.1 Although there are some outstanding issues, the success of the latter in industrial applications is incontrovertible. Progress in representing the heat of mixing has been less impressive.2 Formulation chemistry is a rudimentary form of chemical product design. In essence, it entails the mixing of compounds in order to get a product with the required attributes. Desired characteristics are legion but include effects such as adhesion, weather resistance, texture, shelf life, biodegradability, controlled biological activity, etc. Key to chemical product formulation is the selection of appropriate ingredients and the optimization of their relative proportions. In the past, chemical product formulation was regarded as a “black art” because it relied on a trial-anderror approach.3 Once a basic formulation was established, it was refined by a process of successive approximation. Nowadays, a systematic experimental design process is followed.4-6 Sound experimental mixture designs are employed to minimize the number of formulations that need to be tested.7,8 The factor space defines the range of possible compositions. Mixture experiments are conducted whereby predetermined compositions are prepared and their properties measured. The resultant set of discrete data points is used to generate a continuous and analytical representation of the variation of a given property F in factor space. Unfortunately, the theoretical knowledge available about the system is, more often than not, insufficient * To whom correspondence should be addressed. Tel.: +27 12 420 2588. Fax: +27 12 420 2516. E-mail:
[email protected].
to generate a sound mechanistic model for the response surface. This necessitates the use of empirical models that, hopefully, will at least be locally satisfactory. Scheffe´ polynomials are the most common models applied in the context of experimental mixture designs.7-9 When such second or higher order polynomials are fitted to data, they generate convoluted response surfaces that feature stationary points corresponding to “tops of hills”, “troughs of valleys”, and saddle points. If the model fits the data well, these should correspond to best or worst possible formulations. Clearly, algebraic correlating equations are more convenient to use than tabulations or graphs. The question is, will advances in molecular simulation and modeling ultimately supplant experimental data generation and eliminate the need for correlating equations? In Churchill and Zajic’s10 judgment, the dependability of current numerical solutions is still suspect and conditional on the inherent need for idealization and simplification. Therefore, results obtained from molecular and computer simulations may not yet be as accurate as good experimental data. Usually the computed values are discrete and differ from experimental data only in their regularity and in the precision of their determination. Thus, for the moment, correlation equations will remain convenient because they provide a reliable and efficient data interpolation method for the optimization of chemical processes and product formulations. This paper addresses the common need of formulation chemists and chemical engineers to correlate physical properties with mixture composition. It combines concepts from experimental mixture designs with those developed for solution thermodynamics. Rational functions based on revised canonical Scheffe´ polynomials are presented that should facilitate empirical data correlation and the prediction of multicomponent mixture data. Ideal Solutions and the “Blending” Rule The concentrations of the components are conveniently expressed in terms of fractions (e.g., by mass, volume, or mole). No matter what is chosen as the basis, all proportions have to be nonnegative and their sum
10.1021/ie049415+ CCC: $27.50 © 2004 American Chemical Society Published on Web 11/25/2004
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Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004
must equal unity. Mathematically, these requirements are expressed as follows:
xi g 0
for i ) 1, 2, ..., q
(Taylor) series. The coefficients Ck are related to the first derivatives of the function f evaluated at z ) 0:
(1) Ck )
and
∑i xi ) x1 + x2 + ... + xq ) 1
(2)
In formulations, the “blending rule” states the general expectation that mixture properties vary linearly with composition:
Fideal ≡
∑i Fixi ) F1x1 + F2x2 + ... + Fqxq
(3)
In eqs 1-3, q is the number of components in the mixture, Fi denotes the pure-component property value, and xi is the mole fraction of component i in the mixture. The mixing function (property) is defined as the difference in the actual mixture property value and the expected value according to the blending rule:
∆F ≡ F - Fideal
(4)
In thermodynamics, the ideal solution concept serves as a standard to which the properties of real mixtures can be compared. For ideal solutions, some properties, e.g., the entropy of mixing, include an additional term that is proportional to ∑xi ln xi. Excess properties are defined as the difference between the actual mixture property and the corresponding value for an ideal solution:
FE ≡ F - FIS
(5)
With the exception of entropy- and entropy-dependent properties, e.g., the Gibbs and Helmholtz free energies, eqs 3 and 4 also express the properties of the ideal solution. Thus, the heat of mixing and volume of mixing are equivalent to the corresponding excess properties. Although the “blending rule” and the ideal solution model provide useful first approximations, deviations are often significant. Both the excess and mixing properties equal zero in the two concentration limits for binary mixtures. This implies a leading behavior of the form FE ∼ fx1x2 and justifies the following definition of a peeled property for binary mixtures:
f ≡ FE/x1x2
(6)
Plotting f ) f(x1) reveals the complexity of the system behavior, making it a useful vehicle for model discrimination.11
f = C0 + C1z + C2z2 + ... + Cnzn
Scheffe´ Polynomials The ordinary second-order polynomial with three independent composition variables is
S3(2) ) R0 + R1x1 + R2x2 + R3x3 + R12x1x2 + R13x1x3 + R23x2x3 + R11x12 + R22x22 + R33x32 (9) Scheffe´ polynomials emerge from ordinary polynomials when the mixture constraint, eq 2, is taken into account. Thus, when x1 + x2 + x3 ) 1 is utilized, eq 9 simplifies to7,9
S3(2) ) β1x1 + β2x2 + β3x3 + β12x1x2 + β13x1x3 + β23x2x3 (10) This is the canonical second-order Scheffe´ form for a ternary mixture. The coefficients βi and βij are combinations of the corresponding Ri and Rij in eq 9. The coefficient βi is the property value assumed by pure component i. The values that βij assume quantify the nonlinear mixing effects between components i and j. Comparing eqs 9 and 10 reveals that Scheffe´ polynomials are trimmed-down versions of Taylor polynomials. They have fewer adjustable parameters and feature unambiguous multivariate forms.7-9 The first-order Scheffe´ model is equivalent to the “blending rule” described by eq 3. The canonical form of the secondorder Scheffe´ polynomial for multicomponent mixtures is7,8 q
Sq(2) )
(7)
Equation 7 is equivalent to a truncated Maclaurin
(8)
z)0
The use of polynomials to approximate functions in a small interval is justified by Taylor’s theorem.13 Taylor polynomials are partial sums or truncated versions of a Taylor series expansion. They tend to provide good approximations for the value of the function near the point of expansion. However, the approximation error of a given Taylor expansion may increase rapidly at points further away: The errors tend to “bunch up” near the ends of the interval.14 Approximations based on orthogonal polynomials, e.g., Chebyshev polynomials, are generally more efficient: They can “distribute” the approximation error more uniformly over the interval of interest.
Polynomial Models for Excess Properties Taking eq 2 into consideration, a single composition variable is sufficient for a binary blend. Redlich and Kister12 proposed z ) x1 - x2 as the composition variable. Note that this maps the composition range into the finite interval z ∈ [-1, 1]. They next proposed polynomial expansions for the peeled property of the form
( )
1 dkf k! dzk
q
βixi + ∑∑βijxixj ∑ i)1 i
