1. Introduction

Nowadays, process simulation [1] plays an important role in the chemical engineering field as an indispensable tool to gain precise knowledge about the process units. The use of powerful mathematical-computational tools allows to accomplish an optimal final design of chemical processes. An important matter is to ensure that the mathematical model correctly represents the quantities that reflect the state of the studied system. Then, what is a model? A model is a mathematical relationship that links the state variables of a system, such as temperature, pressure, or

compositions, influencing the process performance. Therefore, the accuracy of the selected model is essential and greatly affects the final results of the simulation and design processes. Two milestones should be considered.

testing \$ modeling steps, and having obtained the corresponding result fronts, each

A Practical Fitting Method Involving a Trade-Off Decision in the Parametrization Procedure…

selected is carried out using a parametric equation already used by us [8], which is

2 i¼0 gi ð Þ <sup>p</sup>; <sup>T</sup> <sup>z</sup><sup>i</sup>

tested, by adapting it according to the availability of experimental data. Thus, different "sub-models" are defined by neglecting terms in Eq. (2) which give rise to

M1 ! <sup>g</sup>i2; <sup>g</sup>i3; <sup>g</sup>i4; <sup>g</sup>i5 <sup>¼</sup> <sup>0</sup>; M2 ! <sup>g</sup>i2; <sup>g</sup>i3; <sup>g</sup>i5 <sup>¼</sup> <sup>0</sup>;

From Eq. (1), the expression that characterizes the activity coefficients is

k<sup>2</sup>‐<sup>1</sup>

Eqs. (1) and (5) assume that the different sub-models established by Eq. (3) impose certain behavior hypothesis in the mixture. Thus, sub-model M1 implies

To obtain the optimal parameters of the different sub-models that represent the

) is the root of the mean square error (or RMSE) calculated by the

E :

<sup>z</sup><sup>1</sup> <sup>þ</sup> <sup>3</sup> <sup>g</sup><sup>2</sup> � <sup>g</sup><sup>1</sup>

Likewise, the excess enthalpies h<sup>E</sup> are calculated considering.

1 <sup>þ</sup> ð Þ <sup>1</sup> � <sup>i</sup> � <sup>x</sup><sup>1</sup>

z<sup>2</sup>

} data for each one of the two systems

(1)

<sup>1</sup>z<sup>1</sup> <sup>¼</sup> <sup>x</sup><sup>1</sup> <sup>x</sup><sup>1</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup>‐<sup>1</sup> <sup>g</sup> x<sup>2</sup>

ð Þ¼ <sup>p</sup>; <sup>T</sup> <sup>g</sup>i1 <sup>þ</sup> <sup>g</sup>i2 <sup>p</sup><sup>2</sup> <sup>þ</sup> <sup>g</sup>i3 pT <sup>þ</sup> <sup>g</sup>i4=<sup>T</sup> <sup>þ</sup> <sup>g</sup>i5T<sup>2</sup> (2)

<sup>2</sup>–<sup>1</sup> is a fitting parameter. In this study, several forms of Eq. (2) are

M3 ! <sup>g</sup>i2; <sup>g</sup>i5 <sup>¼</sup> <sup>0</sup>; M4 ! <sup>g</sup>i2 <sup>¼</sup> <sup>0</sup> (3)

<sup>1</sup> � <sup>4</sup>g2z<sup>3</sup> 1

<sup>x</sup>,p; h<sup>E</sup> <sup>¼</sup> <sup>z</sup>1ð Þ <sup>1</sup> � <sup>z</sup><sup>1</sup> <sup>h</sup><sup>0</sup> <sup>þ</sup> <sup>h</sup>1z<sup>1</sup> <sup>þ</sup> <sup>h</sup>2z<sup>2</sup>

, the multi-objective optimization (MOO) procedure is

OF <sup>¼</sup> s g<sup>E</sup>=RT s h<sup>E</sup> (6)

<sup>E</sup> 6¼ <sup>h</sup><sup>E</sup>

<sup>g</sup> ð Þ z1=x<sup>1</sup>

1 (5)

, allowing different functions to

<sup>2</sup> (4)

candidate model is analyzed on its suitability for process simulation.

2. Modeling procedures

2.1 Thermodynamic model

and as described in [8]:

where k<sup>g</sup>

that g

<sup>E</sup> = h<sup>E</sup>

where s(y

49

s.

data set iso-p VLE and h<sup>E</sup>

E

2.2 Calculation of the efficient fronts

used, which is characterized by the vector

model when representing the generic property y

express the h<sup>E</sup>

the following four cases:

gi

The correlation of the iso-p {VLE + h<sup>E</sup>

DOI: http://dx.doi.org/10.5772/intechopen.85743

applied to the excess Gibbs function of a binary system:

<sup>g</sup>Eð Þ¼ <sup>p</sup>; <sup>T</sup>; <sup>x</sup> <sup>z</sup>1ð Þ <sup>1</sup> � <sup>z</sup><sup>1</sup> <sup>∑</sup>

RTln <sup>γ</sup><sup>i</sup> <sup>¼</sup> <sup>z</sup>1ð Þ <sup>1</sup> � <sup>z</sup><sup>1</sup> <sup>g</sup><sup>0</sup> <sup>þ</sup> <sup>g</sup>1z<sup>1</sup> <sup>þ</sup> <sup>g</sup>2z<sup>2</sup>

g<sup>0</sup> þ 2 g<sup>1</sup> � g<sup>0</sup>

<sup>h</sup><sup>E</sup> <sup>¼</sup> <sup>g</sup><sup>E</sup> � <sup>T</sup> <sup>∂</sup>g<sup>E</sup>=∂<sup>T</sup>

. For the remaining assumptions, g

The first one is the selection of the model, built-in with the mathematical relationships that best represent the variables of the system. There is no a priori a procedure to make this choice, so heuristic or experience-based criteria are generally used [2]. The second question refers to get the best parameters that complete the definition of the model for a given data series. For this latter case, there are several numerical procedures [3, 4] that allow to address the problem to optimize the parameter set considering the starting hypothesis.

The thermodynamic properties having the greatest influence on the simulation of separation processes are those related to phase equilibria (vapor-liquid equilibrium, VLE, in the scope of this work), as well as other thermodynamic quantities that arise in the mixing process. These properties are associated with the excess Gibbs energy g E , which is written as g <sup>E</sup> = g E (xi,p,T). As such, the goal of the modeling is to achieve a functional type of f = f(θ,xi,p,T) that minimizes the norm | g <sup>E</sup> <sup>f</sup> |. The vector <sup>θ</sup> represents a set of parameters in the model, which must be optimized. However, due to the existing relation between phase and mixing properties, former approach may not be enough. In the first place, g <sup>E</sup> values can only be obtained from VLE, which satisfies a dependency of the type, F(xi,p,T) = 0, preventing us from obtaining individual relations of g <sup>E</sup> with each one of the variables. On the other hand, defects in the experimental data could give place to incoherencies between experimental activity coefficients γ<sup>i</sup> = γi(xi,yi,p,T) and the excess Gibbs function g E . Thus, the VLE fitting process becomes a bi-objective optimization problem; hence, two error functions are included in the correlation problem. The complexity grows as other properties, such as h<sup>E</sup> = h<sup>E</sup> (x,p,T), are included in the modeling, since new error functions should also be minimized. However, one of the benefits of this approach is to use a unique thermodynamic model that avoids the issues caused by possible discontinuities or inconsistencies between partial models of some properties. Even more relevant is that this allows the model to describe better the physics of the system under study. This supposes a mean to verify the coherence of the mathematical formalism imposed by thermodynamics, validating the different properties. A drawback of this practice is the increase of the complexity of the procedure, but this is just a numerical issue of relative complexity. Therefore, addressing the global modeling of the thermodynamic behavior of solutions as a problem with multiple objectives is a notable contribution in chemical engineering.

It is known that the resolution of multi-objective problems does not produce a single result; on the contrary, a set of non-dominated results that constitute the socalled Pareto front [5] is obtained. There is no precise mathematical criterion that allows the selection of a unique result. However, the process simulation requires a single result to define the intended design, having to resort to external criteria different from those used to obtain the front.

This study evaluates the effect of choosing the different results from the Pareto front in the simulation task. To achieve this, some partial goals are proposed such as (a) to establish a rigorous methodology to carry out the optimization procedure with the suggested modeling and (b) to check the real impact of the chosen model on the simulation, with the purpose of proposing a selection criterion. Thus, the designed methodology should include different stages, like the data selection used in the procedure, obtaining the result front and the election of the final result. Two systems, considered as standard in many studies on thermodynamic behavior of solutions, are selected since the necessary experimental information (VLE and h<sup>E</sup> ) is available in literature. After checking the data sets [6, 7], making up the

A Practical Fitting Method Involving a Trade-Off Decision in the Parametrization Procedure… DOI: http://dx.doi.org/10.5772/intechopen.85743

testing \$ modeling steps, and having obtained the corresponding result fronts, each candidate model is analyzed on its suitability for process simulation.
