3.1 Case of study: state-space nonlinear model

The following assumptions are considered in the EDF-1000 distillation pilot plant to simplify the designing state without significantly affecting the dynamics and precision of the model:


Fuzzy Logic Modeling and Observers Applied to Estimate Compositions in Batch Distillation… DOI: http://dx.doi.org/10.5772/intechopen.83479

Eq. (19) shows the mathematical model of a batch-type 11-plate EDF-1000 distillation pilot plant for an ethanol-water mixture, considering the compositions in each plate ð Þ x1; x2; ⋯ x10; x<sup>11</sup> , where the condenser composition is x<sup>1</sup> and the boiler composition is x11; in addition, the control inputs are the heating power Qf and the reflux Rf :

$$\begin{aligned} \dot{\mathbf{x}} &= A \begin{pmatrix} \mathbf{x}\_1 \\ \mathbf{x}\_2 \\ \vdots \\ \mathbf{x}\_{10} \\ \mathbf{x}\_{11} \\ \mathbf{x}\_{11} \end{pmatrix} + B \begin{pmatrix} R^f \\ Qb \end{pmatrix} \\\\ \mathbf{x}\_{11} \\\\ \mathbf{y} &= C \begin{pmatrix} \mathbf{x}\_1 \\ \mathbf{x}\_2 \\ \vdots \\ \mathbf{x}\_{10} \\ \mathbf{x}\_{11} \end{pmatrix} \end{aligned} \tag{19}$$

where A and B matrixes are defined by

$$A = \begin{pmatrix} & -\frac{(V+D)}{M\_1} & \frac{V \cdot G(x\_{2}a\_{2})}{M\_1} & \cdots & 0 & 0\\ & \frac{L}{M\_2} & -\frac{V \cdot G(x\_{3}a\_{3}) - L}{M\_2} & \cdots & 0 & 0\\ & \vdots & \vdots & & \vdots\\ \vdots & \vdots & \cdots & & \frac{V \cdot G(x\_{10}a\_{10}) - L}{M\_{10}} & \frac{V \cdot G(x\_{11}a\_{11})}{M\_{10}}\\ 0 & 0 & \cdots & & \frac{L}{M\_{11}} & -\frac{L}{M\_{11}}\\ & 0 & 0 & \cdots & & \frac{L}{M\_{11}} & -\frac{L}{M\_{11}} \end{pmatrix}$$

$$B = \begin{pmatrix} \frac{Lx\_{1}}{M\_1} & & 0\\ 0 & & 0\\ \vdots & & \vdots\\ 0 & & 0\\ 0 & & \mathbf{0} \end{pmatrix}$$

$$\mathbf{0} = \begin{pmatrix} \mathbf{x}\_{11}(1 - G(x\_{11}a\_{11}))\\ \mathbf{0} & \mathbf{f}\_{H\mathbf{B}}^{up}(x\_{11} + H\_{H\mathbf{B}}^{up}(1 - x\_{11})M\_{11}) \end{pmatrix}$$

where the output matrix is defined by an 11 � 11 identity matrix as shown in Eq. (20).

$$\mathbf{C} = \begin{pmatrix} \mathbf{1} & \mathbf{0} & \cdots & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} & \cdots & \vdots & \mathbf{0} \\ & \vdots & \vdots & \ddots & \vdots & \vdots \\ \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0} & \mathbf{1} \end{pmatrix} \tag{20}$$

located in the condenser (plate 1); in plates 2, 4, 6, 8, and 10; and in the boiler

The boiler is composed of a 2-L tank to contain the mixture and a side tank to

The following assumptions are considered in the EDF-1000 distillation pilot plant to simplify the designing state without significantly affecting the dynamics

(plate 11).

Figure 2.

contain a 300-watt heating resistance.

• Constant pressure in the column.

• Liquid output flows in the column.

• Adiabatic distillation column.

• Vapor and liquid balance in all the column plates.

and precision of the model:

EDF-1000 distillation pilot plant.

• No vapor retention.

• Batch operation type.

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3.1 Case of study: state-space nonlinear model

Distillation - Modelling, Simulation and Optimization
