Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 4817565, 11 pages

https://doi.org/10.1155/2018/4817565

## Solubility Optimal System for Supercritical Fluid Extraction Based on a New Nonlinear Temperature-Pressure Decoupling Model Constructed with Unequal-Interval Grey Optimal Models and Peng-Robinson Models

School of Electrical and Electronic Engineering, Changchun University of Technology, Changchun 130012, China

Correspondence should be addressed to Wen You; nc.ude.tucc.liam@newuoy

Received 25 December 2017; Accepted 6 March 2018; Published 17 April 2018

Academic Editor: Ivan Giorgio

Copyright © 2018 Binglin Li and Wen You. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a new solubility optimal system to improve the efficiency of supercritical fluid extraction (SFE). The major contribution is a nonlinear temperature-pressure decoupling model constructed with unequal-interval grey optimal models (UEIGOMs) and Peng-Robinson models (PRMs). The linear parts of temperature and pressure process can be constructed with UEIGOM, respectively. The nonlinear parts of temperature and pressure process can be described by PRMs, respectively. The whole nonlinear model cannot be input-output decoupled resulting from the singularity of decoupling matrix for PRM. This problem on input-output nondecoupling can be transformed to the problem on disturbance decoupling for a class of MIMO nonlinear systems. Therefore, the whole nonlinear coupling model can be disturbance decoupled. Furthermore, solubility optimal method is presented in the paper; it can calculate the optimal pressure according to the given temperature, namely, optimal working points, to maximize solubility for SFE process. The feasibility, effectiveness, and practicality of the proposed nonlinear temperature-pressure decoupling model constructed with UEIGOMs and PRMs are verified by SFE experiments in biphenyl. Experiments using the designed solubility optimal system are carried out to demonstrate the effectiveness in control scheme, simplicity in structure, and flexibility in implementation for the proposed solubility optimal system based on a new nonlinear temperature-pressure coupling model constructed with UEIGOMs and PRMs.

#### 1. Introduction

Extraction of a material using a supercritical fluid is called supercritical fluid extraction (SFE). SFE, which is a contamination-free extraction technology in food science and chemical industry, is of central importance in biomaterial processing. During the separation process, the solvency of SFE can be modified by adjusting temperature, pressure, moisture contents, and so on [1–4]. Temperature and pressure play a crucial role in SFE process. The model of temperature and pressure process, which is nonlinear, is composed of linear and nonlinear parts. The linear part, which can be obtained easily, is SISO model of temperature or pressure. The nonlinear part is the coupling relationship between temperature and pressure. Numerous methods and models have been proposed to describe SFE process. Cubic equation of state (EoS) with simplified inner structure and generalized form is one of the most widely used models to describe the temperature, pressure, and time behaviors for fluid. vdW EoS, which is used in calculation of vapor-liquid equilibrium, was proposed by van der Waals in 1873 [5]. RK EoS was proposed by Redlich and Kwong in 1949 [6]. RK EoS was improved by Soave, and RKS EoS was proposed in 1979 [7]. RP EoS was proposed by Peng and Robinson in 1976 [8]. PR EoS is widely used and contrasted with the other three models. A hybrid model, which is constructed with a radial basis function (RBF) model and RP EoS, was proposed to keep all the physical information in PR model and optimize the binary interaction parameter in the PR model [9, 10]. Combined with operating cost, safety index, and yield rate of extraction calculated by hybrid model, an optimal control system can be designed. However, temperature has a momentous effect on pressure, in temperature and pressure control process, and vice versa. The coupling relationship between temperature and pressure in SFE process is not considered in the proposed optimal control system. The performance of temperature and pressure control has an effect on yield rate and solubility of extraction. Therefore, combining with PR EoS, study on modeling of temperature and pressure coupling and decoupling model has great significance with new theories and methods for improving SFE work efficiency to obtain maximal yield rate and solubility of extraction.

In this work, the linear parts of temperature and pressure models can be described through UEIGOMs with grey technology, respectively. The nonlinear parts of temperature and pressure models are modeled with PR EoS. Discussing the decoupling conditions of input-output of temperature-pressure nonlinear model, the decoupling system is given through state and output transforms. Furthermore, solubility optimal method is presented in the paper; it can calculate the optimal pressure according to the given temperature, namely, optimal working points, to maximize solubility for SFE process. The rest of the paper is organized as follows: In Section 2, temperature-pressure process and solubility modeling are discussed. In Section 3, temperature-pressure decoupling control system and solubility optimal system are designed. Computer simulation results of UEIGOMs of temperature and pressure process, temperature-pressure decoupling control, and solubility optimal system are presented and discussed, respectively, in Section 4. Finally, a conclusion regarding research and future works is made in Section 5.

#### 2. Temperature-Pressure Process and Solubility Modeling

Due to the reaction character of SFE process, the nonlinear strong coupling relationship between temperature and pressure is the main factor affecting the efficiency of SFE. In order to improve the performance of SFE control system effectively, it is necessary to model temperature-pressure process. In this work, the temperature and the pressure processes can be modeled by grey technology, respectively, and their UEIGOMs can be given; the coupling parts can be obtained by PRMs.

##### 2.1. UEIGOM for Temperature-Pressure Process

Grey system theory is based on fewer samples, which are made of some known and unknown information. It can work on extracting the valuable information from the fewer samples. Grey generating technology can provide intermediate information and weaken the randomness of original data; therefore, the model using grey generating technology can give the correct description of the reaction character of SFE process. In grey calculation, accumulated generating operation (AGO) and inverse accumulated generating operation (IAGO) are utilized [11–13]. They can be defined as follows.

Let be original series and be -AGO series, if there is an accumulated generated matrix that satisfies (1); let be -IAGO series, if there is an inverse accumulated generated matrix that satisfies (2).where

###### 2.1.1. Transforming from Unequal-Interval Series to Equal-Interval Series

AGO and IAGO operations can all aim at equal-interval series; however, actual process signals are always unequal-interval series. Therefore, it is necessary to transform to equal-interval series before AGO or IAGO operation.

Let , , be an unequal-interval series; then its average interval can be calculated as follows:

The coefficient between every interval and average interval is given.

The total difference value of every interval can be calculated.

Above all, the equal-interval series , , can be obtained, where

###### 2.1.2. Grey Optimal Model

Series is obtained by 1-AGO with series ; namely, . Define as the black ground value of series , where , . Generally, ; then differential equation can be described aswhere is developing coefficient and is grey input. Utilizing least square method, and can be obtained:where

Time response function of GOM can be expressed as follows: where is translation value.

###### 2.1.3. Reversion from Equal-Interval Series to Unequal-Interval Series

Considering , (11) can be transformed for unequal-interval series, namely, UEIGOM:where ; is residual error. Therefore, the temperature and the pressure processes can be described as first-order inertia system by UEIGOM, respectively.

##### 2.2. PRM Modeling

Peng-Robinson model is the most commonly exploited model for treating solubility in SFE. The nonlinear strong coupling relationship between temperature and pressure is reflected in the Peng-Robinson equation of state. Therefore, the coupling parts of temperature-pressure process can be embodied by PRM in this work. PRM is given aswhere is gas constant, is absolute temperature, is the molar volume of pure solvent, is the parameter describing attractive interactions between molecules, is critical temperature, is critical pressure, is acentric factor, is normal boiling point, and is the parameter describing volume exclusion and repulsive interactions. The parameters in PR model are listed in Table 1. Subscript 1 represents the solvent and subscript 2 represents the solute. Considering that the molar volume of pure solvent is invariable, (13) can be simplified as follows:where Furthermore, the inverse function of (14) can be obtained as follows: