Simulation and Optimization in Process Engineering: The Benefit of Mathematical Methods in Applications of the Chemical Industry

Simulation and Optimization in Process Engineering: The Benefit of Mathematical Methods in Applications of the Process Industry brings together examples where the successful transfer of progress made in mathematical simulation and optimization has led to innovations in an industrial context that created substantial benefit. Containing introductory accounts on scientific progress in the most relevant topics of process engineering (substance properties, simulation, optimization, optimal control and real time optimization), the examples included illustrate how such scientific progress has been transferred to innovations that delivered a measurable impact, covering details of the methods used, and more.

With each chapter bringing together expertise from academia and industry, this book is the first of its kind, providing demonstratable insights.

• Recent mathematical methods are transformed into industrially relevant innovations.
• Covers recent progress in mathematical simulation and optimization in a process engineering context with chapters written by experts from both academia and industry
• Provides insight into challenges in industry aiming for a digitized world.

• Language: English
• Publisher: Elsevier Science
• Release date: Apr 16, 2022
• ISBN: 9780323850445

Book preview

Preface

Knowledge- and data-based modeling, in conjunction with setting up new technical solutions, form the basis of progress driven by engineers. A high degree of domain expertise is necessary to arrive at new insights and ideas that have a substantial practical impact.

As opposed to highly specialized engineers, mathematicians aim to find context-free structural relations, prove the existence of solutions, and design algorithms to identify them. The more general mathematical statements are, the higher their considered value.

The idea for this book is rooted in the conviction that substantial benefits can arise when these two worlds—the domain-driven engineering world and the method-driven mathematical world—come into contact. These benefits can be twofold as explained in the following.

On the one hand, from a scientific point of view, mathematical understanding of models and their structures can generate new solutions for engineering problems. The way how a model is formulated can have a significant impact on whether and how existing solutions can be revealed algorithmically. Furthermore, real-world problems posed by engineers can challenge mathematical methods and go well beyond what are considered as examples in mathematical textbooks.

On the other hand, from a practical point of view, significant progress can result from this pairing. If the application of mathematical methods leads to more reliable, more versatile, and more applicable model predictions, less trial and error is necessary, and the time needed to arrive at improvements—of whatever kind these may be—can be reduced dramatically. Eventually, this is the very promise of model-based simulation and optimization: that predictions can be made for highly complex, nonlinear systems on how to achieve high utility at not more than the necessary costs.

This book highlights applications from chemical engineering that show how mathematical advances in simulation and optimization lead to substantial impact in the real world. The order of the chapters follows the different scales involved in chemical engineering, ranging from substance properties over unit and flowsheet models, both in the steady state and dynamically, up to organizational questions when it comes to scheduling and production planning.

Throughout the book, the reader will discover how mathematical structure hidden in models, in simulation and optimization tasks, is exploited to arrive at new insights. We hope that this book will be an inspiration for both engineers and mathematicians challenged by real-world problems.

Michael Bortz; Norbert Asprion, Kaiserslautern and Ludwigshafen

Chapter 1: Prediction and correlation of physical properties including transport and interfacial properties with the PC-SAFT equation of state

Jonas Mairhofera; Joachim Grossb    a BASF SE, Ludwigshafen, Germany

b Institute of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Stuttgart, Germany

Abstract

This chapter discusses thermodynamic properties and phase equilibria using equations of states. Focus is thereby placed on the perturbed-chain statistical associating fluid theory (PC-SAFT) as a representative for physically-based equations of state. It is a model that is suitable for nonspherical molecules and hydrogen-bonding species or mixtures. Thermodynamic models are regularly used in process simulations to provide properties like enthalpies, densities, pressures, coexisting compositions of mixtures, chemical potentials for chemical equilibria. With modern equations of state, it is also possible to predict interfacial tensions, adsorption isotherms, or contact angles using density functional theory or density gradient theory. Moreover, transport coefficients, such as shear viscosity, thermal conductivity, and diffusion coefficients can be correlated or predicted using excess entropy scaling approaches. This chapter is also concerned with determining meaningful model parameters. The predictive capability of the model, once pure-component and binary interaction parameters were adjusted to a given set of thermodynamic properties, for predicting further properties relevant for a flowsheet simulation is discussed. Group-contribution approaches are presented to address the frequently encountered case of absent experimental data for parameter adjustment.

Keywords

Physical property models; Equation of state; Parameter estimation; Group-contribution methods; Excess entropy scaling; Density gradient theory; Classical density functional theory

The reliability of a process simulation, to a large extent, is determined by the fitness of the applied physical property model. Adequate models are required for many different physical properties: the simulation of a distillation column using the equilibrium stage approach may require a model for activity coefficients, vapor pressure, enthalpy, and possibly vapor phase fugacity coefficients. Rate-based process models, in addition, need transport properties such as viscosity, thermal conductivity, or diffusion coefficients as input. In order to determine volume flows from mass or molar flows, for example for equipment sizing, the density of a stream needs to be known. Furthermore, entropy plays an important role in the simulation of compressors or pumps, etc. High demands are placed on the physical property models: they have to accurately correlate available experimental data, but also show robustness in extrapolating over large ranges of temperature and pressure. They have to be able to model the thermodynamic behavior of species and mixtures exhibiting complex molecular interactions. At the same time, they cannot be arbitrarily complex in order to ensure reliable numerical solutions and to avoid prohibitively long run times of a process simulation.

Several alternatives exist for providing the required physical properties. One of the most widely used approaches in a process simulation is in applying an activity coefficient model, such as the nonrandom two-liquid model (NRTL) [1] for modeling the liquid phase nonideality in combination with dedicated, usually empirical correlations for the pure-component vapor pressures, liquid and vapor phase heat capacities, liquid density and further required properties. The gas phase is most often approximated as an ideal gas. This approach shows several drawbacks: it is not suited for high-pressure applications because the pressure dependence of the activity coefficients is usually neglected, and the ideal gas approximation will not be justified. The nonideality of the gas phase then also has to be addressed in calculating the vapor enthalpy. Furthermore, special treatment is necessary for components with a critical temperature lower than the simulation temperature.

An equation of state model complemented with expressions for ideal gas heat capacities of all pure species can provide all static thermodynamic properties. Furthermore, models for transport coefficients, such as viscosity, thermal conductivity, or diffusion coefficients, exist which build on outputs of an equation of state. However, simple cubic equations of state such as the Peng-Robinson [2] or Soave-Redlich-Kwong [3] equations of state can only be applied to simple molecules and in general, do not produce results that are accurate enough for all required properties in most real-world processes simulation applications. The need for more accurate thermodynamic models applicable also to molecules showing complex molecular interactions such as hydrogen-bonding or polar interactions led to the development of more sophisticated, physically-based equations of state such as the statistical associating fluid theory (SAFT) developed by Chapman et al. [4,5] and Jackson et al. [6] based on the work of Wertheim [7–10]. SAFT leads to an expression for the residual Helmholtz energy Ares(T,ρ,x) = A(T,ρ,x) − Aig(T,ρ,x), where Aig denotes the Helmholtz energy of the ideal gas at temperature T, number density ρ and mole fractions x. The value of Ares is obtained as the sum of different contributions. Each contribution takes into account a specific type of molecular interaction. The different Helmholtz energy contributions are developed using perturbation theory which (under suitable conditions) allows to obtain the thermodynamic properties of a target fluid with specified interaction potential from the properties of a reference fluid with a simpler interaction potential and known properties. The advantage of SAFT is that it allows to include a contribution for highly directional attractive interactions such as hydrogen-bonding which makes it a suitable choice for modeling the properties of molecules (and their mixtures) exhibiting such complex molecular interactions. Different SAFT-type equations of state can be derived depending on the choice of reference fluid and interaction potential of the target fluid.

A detailed description of the fundamentals of SAFT can be found in the book by Solana [11]. A comprehensive list of successful applications of SAFT-type equations of state to complex systems including polar and associating molecules, polymers, ionic liquids, pharmaceuticals as well as bio-molecules can be found in review articles [12–18].

The scope of this chapter is to introduce the perturbed-chain statistical associating fluid theory (PC-SAFT). The equations to implement the model are given in Section 1. The determination of meaningful pure-component and binary interaction parameters is the topic of Section 2. Section 3 presents group-contribution methods for PC-SAFT which allow to predict the model parameters in the frequently encountered situation that not enough experimental data is available for a given molecule, thus preventing parameter regression. Finally, Sections 4 and 5 are concerned with correlating or predicting the transport and interfacial properties, respectively.

1: Model equations of PC-SAFT

In the case of PC-SAFT developed by Gross and Sadowski [19–21], the hard-chain fluid is used as the reference. Each chain is made up of m bonded hard-sphere segments of diameter parameter σ. Attractive interactions such as dispersive interactions or hydrogen bonding between the chains are then added as perturbations to this purely-repulsive reference fluid. The dimensionless residual Helmholtz energy ares ≡ Ares/NkT with molecule number N and Boltzmann’s constant k is obtained as

si1_e

   (1.1)

The Helmholtz energy has several contributions, namely the hard-chain reference fluid, ahc, the change in Helmholtz energy due to dispersive interactions between the chains, adisp, highly directional attractive interactions such as hydrogen-bonding, aassoc, and dipole-dipole, quadrupole-quadrupole as well as dipole-quadrupole interactions. With polar contributions included, the model is usually referred to as perturbed-chain polar statistical associating fluid theory (PCP-SAFT). In the following section, the various terms of Eq. (1.1) will be presented for a mixture at temperature T, number density ρ, and mole fraction of component i, xi. Using basic thermodynamic relationships, all other thermodynamic properties can be obtained as derivatives of ares.

The Helmholtz energy of the hard-chain fluid is given by

si2_e

   (1.2)

where ahs denotes the Helmholtz energy of the hard-sphere fluid. The average segment number of the mixture is calculated as si3_e with mi as the number of segments on a chain of components i. We note that parameter mi is in SAFT models relaxed to be a real-valued (rather than integer-valued) parameter.

The value of ahs is obtained from the accurate equation of state for the hard-sphere fluid presented by Boublík [22] and Mansoori et al. [23]

si4_e

   (1.3)

with density measures ζn defined as

si5_e

   (1.4)

Here, di denotes the temperature-dependent segment diameter of component i

si6_e

   (1.5)

with the dispersive interaction energy ɛi/k between segments within the chain of component i. Furthermore, the radial distribution function of the hard-sphere fluid at contact distance is calculated as

si7_e

   (1.6)

The contribution to the Helmholtz energy due to dispersive interactions between the chain molecules is developed as a perturbation to the hard-chain reference fluid using the second-order perturbation theory of Barker and Henderson [24,25] extended to chain fluids [19] and a perturbation potential of Lennard-Jones type, with

si8_e    (1.7)

The contributions of first and second-order are obtained as

si9_e

   (1.8)

si10_e

   (1.9)

with

si11_e

   (1.10)

and packing fraction η = ζ3. The values for ɛij and σij are determined from the Lorentz-Berthelot combining rules as si12_e and σij = 0.5(σi + σj). As discussed in Section 2.2, it is often necessary to introduce adjustable binary interaction parameters (BIP) for calculating ɛij and σij in order to improve the description of mixture properties. The perturbation approach requires the evaluation of integrals over the pair-correlation function of the reference fluid and the perturbation potential. In PC-SAFT, these integrals are approximated as power-series in density

si13_e    (1.11)

si14_e    (1.12)

with coefficients si15_e and si16_e that depends on the average segment number of the mixture. A simple but accurate [19] dependence on segment-number was taken in analogy to a chain-formation theory by Liu and Hu [26], as

si17_e

   (1.13)

si18_e

   (1.14)

The model constants a0i, a1i, a2i as well as b0i, b1i, and b2i were adjusted to experimental vapor pressure and PvT-data of n-alkanes. Their values can be found in the original publication [20].

In order to include highly directional attractive interactions such as hydrogen bonds, association sites are placed on the chain molecules. These sites can be of different types and only interactions between certain site types are allowed to occur. Site types can for example represent electron donors or acceptors. Interactions are then allowed between a donor and an acceptor site but not between two donor or two acceptor sites. Early classification of association-site schemes for several important chemical families can be found in the work of Huang and Radosz [27]. The final expression for the Helmholtz energy contribution due to association is

si19_e

   (1.15)

where Γi denotes the set of association sites located on a molecule of species i and χiA is the fraction of nonbonded association sites A on molecules of type i. The value of χiA has to be determined by solving the set of nonlinear equations given by

si20_e

   (1.16)

with

si21_e

   (1.17)

Here, ɛAi, BjHB and κAi, BjHB denote the association strength between site A on molecules of component i and site B on molecules of component j and the dimensionless volume in which association between the two sites can occur, respectively. Their values are obtained from the following combining rules suggested by Wolbach and Sandler [28]

si22_e    (1.18)

and

si23_e

   (1.19)

Strictly speaking, combining rules have no sound justification for cross-associations between two (self-)associating molecules i and j (as opposed to dispersive interactions, where approximations can be made to derive the Berthelot-Lorentz combining rules). Cross-associations can be determined through quantum mechanical calculations. In many practical applications, however, the simple combining rules, Eqs. (1.18) and (1.19), can produce suitable approximations of the cross-association parameters. Clearly, binary interaction parameters may be introduced in Eqs. (1.18) and (1.19) to improve results, see Section 2.2. Efficient schemes to solve Eq. (1.16) and to determine derivatives of aassoc have been developed for example by Michelsen [29], Michelsen and Hendriks [30], Tan et al. [31], and Langenbach and Enders [32].

In mixtures of self-associating molecules i with molecules j that do not self-associate which, however, provide proton donor or acceptor sites that can form cross-hydrogen bonds with molecules of type i, it is often advisable to assign a nonzero value for κAi, BjHB to component j [33]. Pure-component results for component j remain unchanged because the value for ɛAi, BjHB is set to zero. However, the model now takes the cross-association between molecules i and j into account because both, cross-association volume, κAi, BjHB, as well as cross-association energy, ɛAi, BjHB, are nonzero. To fine-tune mixture results after a (more or less arbitrary) value for association volume was assigned to component j, BIP may be introduced as presented in Section 2.2.

Helmholtz energy contributions for polar molecules were developed, among others, by Gross [34] for quadrupole-quadrupole interactions, Gross and Vrabec [35] for dipole-dipole interactions, and Vrabec and Gross [36] for dipole-quadrupole interactions. In all three cases, the resulting expressions are obtained from a third-order perturbation approach presented by Stell et al. [37,38] extended to the two-center Lennard-Jones fluid. The three polar Helmholtz energy contributions are similar in structure and only the expression for the dipole-dipole contribution add is presented here. The reader is referred to the original publications for details on the quadrupole-quadrupole term, aqq, and the dipole-quadrupole contribution, adq.

For dipole-dipole interactions, the Helmholtz energy is given as the result of the third-order perturbation in Padé approximation

si24_e    (1.20)

The second and third-order perturbation terms are

si25_e

   (1.21)

and

si26_e

   (1.22)

The dimensionless squared dipole moment is obtained as μi⁎ 2 = μi²/(miɛiσi³), with dipole moment of component i, μi. The perturbation approach requires the evaluation of integrals over the pair-correlation and three-body correlation functions of the reference fluid. These integrals are approximated as simple power functions of dimensionless packing fraction η:

si27_e    (1.23)

si28_e    (1.24)

The coefficients an,ij, bn,ij, and cn,ijk depend on chain length m as

si29_e

   (1.25)

si30_e

   (1.26)

and

si31_e

   (1.27)

The following combining rules are used for mij and mijk

si32_e

   (1.28)

si33_e    (1.29)

The model constants a0n, a1n, a2n, b0n, b1n, b2n, c0n, c1n, c2n were adjusted to the results of molecular simulations of Stoll et al. [39]. Their values can be found in the original publication. In summary, a molecule of component i that is modeled as nonassociating and nonpolar is characterized by three pure-component model parameters: segment number mi, segment diameter parameter σi, and dispersive energy ɛi. To include association, two more parameters are required: association strength ɛAi, BjHB and association volume κAi, BjHB. The polar contributions require the value for the dipole or quadrupole moment. For both, literature values can be used.

2: Parameterization

2.1: Pure-component parameters

The goal in the adjustment of pure-component parameters is, of course, the accurate description of the properties of interest. The simplest way to achieve this is to include experimental data for the target properties in the parameter regression. If only data for other properties are available, one has to make sure, that parameters adjusted to this data allow satisfying predictions of the primary properties of interest.

When choosing the properties to include for parameter optimization, it has to be ensured that these properties indeed define all model parameters well. For example, in the PC-SAFT model for a nonassociating, nonpolar molecule, the values for enthalpy of evaporation and critical temperature do not depend on the value of the parameter σ and the Pitzer acentric factor ω is only a function of segment number m [40].

Also, limitations of equations of state such as PC-SAFT to accurately reproduce the critical region have to be kept in mind, and data in the vicinity of the critical point is usually excluded from the parameter optimization.

Most parameter sets found in the literature for PC-SAFT were adjusted to experimental data for vapor pressure, ps, and liquid density, ρl. Other studies include data for speed of sound, surface tension [41,42], fraction of nonbonded association sites [43], or results of molecular simulations [41]. To address the conflict that exists between the subobjectives of describing different thermodynamic properties accurately, the parameter adjustment has been performed as a multicriteria optimization problem [42,44].

A recent study [45] on the parameter adjustment for nonassociating, nonpolar molecules, concluded that it is indeed sufficient to use combined data for ps and ρl in order to obtain parameter values for m, σ, and ɛ that allow the accurate calculation of the target properties vapor pressure, saturated-liquid density, ρls, enthalpy of evaporation, Δ hlv, as well as saturated-liquid heat capacity, cpl, s, a set that is most relevant for process simulation. Reported average deviations for a database of more than a thousand nonassociating, nonpolar molecules with parameters adjusted to vapor pressure and saturated-liquid density account to only 0.97% for ps, 0.85% for ρls, 3.04% for Δ hlv and 3.35% for cpl, s [45]. The literature survey of Zhu et al. [46] evaluates the performance of several equations of state to reproduce heat capacities and speed of sound for nonassociating as well as associating molecules such as alcohols and water. The study shows that predictions of PC-SAFT with parameters adjusted to vapor pressure and liquid density for these derivative properties can still be in satisfactory agreement with experimental data. However, larger deviations have to be expected than for nonassociating molecules. Furthermore, results are sensitive to the choice of the association scheme [47,48]. The remedy in cases with larger deviations of the predicted derivative properties is to include these properties in the parameter optimization as demonstrated, e.g., in the work of Oliveira et al. [49]. Furthermore, a considerable effort has been undertaken to reduce the number of model parameters that need to be adjusted, especially in the case of molecules modeled as associating fluids, where five model parameters need to be determined and different local and shallow minima are often observed. These approaches include the use of generalized pure-component parameters for members of the same chemical family [50], establishing correlations for the parameters of molecules of homologous series [51–53], performing sensitivity analysis and fixing the least-sensible parameter [54], determining the value of single parameters, often the association energy, by independent methods such as ab initio calculations [55–57], molecular simulation [58] or COSMO-RS [53] or establishing a link between the parameters of PC-SAFT and those of cubic EOS [59]. Special routes to determine PC-SAFT parameters have been developed for polymers, where vapor pressure data are unavailable [60,61] or for petroleum fractions with unspecified components [62]. Maybe the most important approach to alleviating the need for determining pure-component parameters is in group-contribution methods, as described below, in Section 3.

2.2: Binary interaction parameters

In mixtures where dispersive interactions are the dominant part of the attractive intermolecular potentials, often accurate results can be obtained without introducing binary interaction parameters (BIP) in the combining rules for ɛij, σij, ɛAi, BjHB, and κAi, BjHB. However, for many mixtures of industrial interest, BIP is necessary to improve mixture results. In this section, commonly applied BIP is presented and it is discussed how meaningful values can be obtained.

The dispersive energy between molecules of type i and molecules of type j, ɛij, can be corrected with BIP kij, as

si34_e    (1.30)

A further BIP for dispersive energy, lij, which is especially useful for correlating liquid-liquid equilibria is sometimes used as a correction to the value of σij as

si35_e    (1.31)

Alternatively, Tang and Gross [63] proposed an asymmetric mixing rule where a BIP acts on the double sum in the first-order perturbation term of the dispersive contribution, a1disp, Eq. (1.8).

Between two species modeled as associating molecules, additional BIP, kijHB and lijHB, can be introduced in the combining rules for the association energy and volume:

si36_e

   (1.32)

And

si37_e

   (1.33)

If results over wide temperature ranges are relevant, temperature-dependent BIP can be introduced, for example, as

si38_e

   (1.34)

In many cases, however, a temperature-dependent kij(T) can be avoided by using two or more constant BIP of different kinds, for example, a constant kij and a constant lij. Analogous to pure-component parameters, the binary interaction parameters, adjusted for members of the same chemical family, often exhibit well-behaved trends which can be exploited in order to obtain parameters in the absence of experimental data. Fig. 1.1 illustrates this behavior for binary mixtures of methane with n-alkanes: despite some outliers, a clear trend can be observed for the optimized kij value with the chain length of the n-alkane.

Fig. 1.1

Fig. 1.1 Constant k ij adjusted to binary VLE data [64] for methane with several n-alkanes as a function of carbon atoms on the n-alkane. The dotted line represents a cubic polynomial adjusted to the results (excluding values for tetradecane and hexadecane).

Similar trends were also observed for more complex mixtures: Haarmann et al. [65] demonstrated that a correlation for BIP obtained from parameters adjusted to LLE of certain mixtures of n-alkane with water with increasing carbon-number of the alkane, can be extrapolated to further n-alkanes. For mixtures of benzyl-ethanoate with n-alkanes, it was shown that only one value for kij was required to obtain accurate results for excess enthalpy for all considered mixtures with n-alkanes [66].

In many applications, the accurate description of phase equilibria is of high priority. Thus, the BIP is most often adjusted to experimental data for binary vapor-liquid or liquid-liquid equilibria. However, in process simulation, the physical property model also needs to provide accurate results for other mixture properties, such as densities and enthalpies. Several studies [67–72] show that predicted mixture density, that is with all BIP set to zero, is sufficiently accurate in many cases. Furthermore, mixture densities obtained with BIP adjusted to phase equilibrium data were also shown to be in good agreement with experimental results [73–75].

It is furthermore desirable that the BIP of a model can be adjusted to such thermodynamic properties that are obtained by simple measurements and are prevalently available, whereas the model will be subsequently applied to another property of interest. The work of Schacht et al. [76] demonstrates that BIP adjusted to data for activity coefficients at infinite dilution, γ∞, extrapolate well to describe phase equilibria over the entire composition range. Satisfactory results for excess enthalpy were obtained for mixtures of ionic liquids with alcohols [77] and solvents [78], again, using BIP adjusted to γ∞. Soo [79] shows that for some mixtures, it is possible to obtain good predictions of the phase equilibrium with a single BIP adjusted to data for excess enthalpy of the system. However, systems are also presented where this approach fails. In this case, it is of course possible to adjust several BIP to phase equilibrium data as well as results for excess enthalpy. A further frequent requirement in process simulation is the satisfactory description of vapor-liquid as well as liquid-liquid equilibria using the same set of binary interaction parameters. Tumakaka et al. [16] presented convincing results for the simultaneous correlation of VLE and LLE data for the system methanol cyclohexane using a single, constant value for kij. On the other hand, the results of Grenner et al. [50] indicate that for water-alcohol systems, a constant kij is not sufficient to describe VLE and LLE well.

A case study for the binary mixture of water with 1-butanol addresses several of the issues discussed in this section. Results for this mixture are presented in Figs. 1.2 and 1.3. In order to test the capability to simultaneously correlate VLE and LLE data, BIP is simultaneously adjusted to both types of data: four temperature-independent BIP are adjusted to experimental VLE and LLE results at p = 1 bar only, corresponding to the black line in Fig. 1.2. VLE results are correlated very accurately and the composition of both liquid phases of the LLE is also in good agreement with the experimental data. To assess the extrapolation capability for varying temperature and pressure (not considered in the parameter optimization), the model is then applied to VLE at 2, 5, and 8 bar and to LLE at 2500 bar. Considering the large temperature and pressure range of the data, model results are very satisfactory, as seen in Fig. 1.2. Mixture density as well as excess enthalpy obtained with these BIP is presented in Fig. 1.3.

Fig. 1.2

Fig. 1.2 Phase equilibria of the binary mixture of water and 1-butanol. Values for the four temperature-independent BIP, k ij (− 0.0223), l ij [63] (− 0.0102), k ij HB (− 0.1256) as well as l ij HB (4.1437e-3), are adjusted to experimental data for VLE and LLE at p  = 1 bar only (larger black crosses and black circles) . Results for VLE at higher pressures ( green crosses, gray in print version : 1.98 bar, blue crosses, dark gray in print version : 4.92 bar, and red crosses, gray in print version : 7.87 bar) and LLE at 2482 bar (magenta circles, light gray in print version) are calculated using these BIP values. Pure-component model parameters for water and 1-butanol are taken from Stavrou [80] and Lötgering-Lin et al. [81]. Experimental data is taken from Aoki and Moriyoshi [82], Boublik [83] as well as Hessel and Geiseler [84].

Fig. 1.3

Fig. 1.3 (A) Liquid densities of the mixture of water and 1-butanol at 1 bar at T  = 283.15 K and at 303.15 K. Results with temperature-independent BIP ( k ij ,  l ij ,  k ij HB ,  l ij HB ) adjusted to VLE and LLE data at p  = 1 bar (solid lines) and results with all BIP set to zero (dashed lines) . Experimental data (symbols) is taken from Cristino et al. [85]. (B) Excess enthalpy for mixtures of water and 1-butanol at 1 bar at T = 303.15 K and at 328.15 K. Results with temperature-independent BIP (kij, lij, kijHB, lijHB) adjusted to VLE and LLE data at p = 1 bar (solid lines) and results with all BIP set to zero (dashed lines). Experimental data (symbols) is taken from Lim and Smith