**3. Distillation column dynamics**

6 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

systems, fuzzy logic and neural networks.

function have unstructured uncertainty.

work.

Integral Squared Error (ISE)

Integral Absolute Error (IAE)

Integral Time-weighted Absolute Error (ITAE)

Integral Time-weighted Squared Error (ITSE)

 Heuristic methods—These methods are evolved from practical experience in manual tuning and are coded trough the use of artificial intelligence techniques, like expert

 Frequency response methods—the frequency response characteristics of the controlled process is used to tune the PID controller. Frequently these are offline and academic methods, where the main concern of design is stability robustness since plant transfer

 Optimization methods—these methods utilize an offline numerical optimization method for a single composite objective or use computerised heuristics or, yet, an evolutionary algorithm for multiple design objectives. According to the characteristics of the problem, an exhaustive search for the best solution may be applied. Some kind of enhanced searching method may be used also. These are often time-domain methods and mostly applied offline. This is the tuning method used at the development of this

 Adaptive tuning methods—these methods are based in automated online tuning, where the parameters are adjusted in real-time through one or a combination of the previous methods. System identification may be used to obtain the process dynamics over the

A set of performance indicators may be used as a design tool aimed to evaluate tuning

<sup>2</sup>

( ) *T*

*ISE J e t dt* (3)

*IAE J e t dt* (4)

*ITAE J t e t dt* (5)

*ITSE J t e t dt* (6)

0

0 ( ) *T*

0

0

*T*

*T*

( )

<sup>2</sup>

( )

methods results. These performance indicators are listed from (3) to (6) equations.

use of the input-output data analysis and real time modelling.

**2.2. Measures of controlled system performance** 

In Brazil approximately 50% of vehicle fleet is composed of flex vehicles, resulting in 30 million of vehicles. This kind of vehicle uses fossil fuel and/or ethanol. The ignition system is adjusted automatically depending of the proportion of each fuel kind. To attend the national ethanol demand there are several ethanol distillation facilities across the country. In each of these facilities the fermented sugarcane is distilled, obtaining two products: the anhydrous ethanol and the hydrated ethanol.

The hydrated ethanol is obtained from link between the second and the third column. The anhydrous ethanol is obtained at the base of the third column, see Figure 3. The production process is composed of a series of columns where two variables are controlled to generate the hydrated ethanol and the anhydrous ethanol at the standardized specification: the pressure at the column A and the temperature at the distillation tray A20 (Santos et al., 2010). The hydrated ethanol has to have a concentration of 92,6 oINPM (oINPM is a measurement of the weight of pure ethanol fuel in 100g of ethanol fuel – water mixture). So as near the concentration is about this value, the best will be the quality of the hydrated ethanol and the anhydrous ethanol.

**Figure 3.** Distillation process to produce anhydrous ethanol and the hydrated ethanol.

These variables depend respectively on the steam flow at the basis of the column A and on the flow of fermented mash applied at the column A. The minimization of the variability of the alcoholic content according the brazilian standard NBR 5992-80 is the main design objective of the control system.

The distillation process is characterized by a high coupling through the system variables and by a non-linear relationship between them. According (Santos, 2010) the models that represent the relationship between the main process variables is FOTD (First Order with Time Delay).

$$G(\mathbf{s}) = \frac{\mathbf{K}e^{-\theta\mathbf{s}}}{\tau\mathbf{s}+1} \tag{7}$$

In this work the modeling procedure was developed and the following equation was obtained from the relationship of the pressure variation at the A column and the steam flow valve actuation:

$$\begin{aligned} K &= \frac{\Delta PV}{\Delta MV} = \frac{(0, 36 - 0, 44).110\%}{55 - 85, 91} \\ K &= 0, 26\% bar \text{ /}\% opening \\ \theta &= t\_1 - t\_0 = 3s \\ \tau &= t\_2 - t\_1 = 23s \end{aligned} \tag{8}$$

Where MP is the Manipulated Variable e PV is the Process Variable.

So, the FODT representation is:

$$\mathbf{G}(\mathbf{s}) = \frac{\mathbf{0}\angle 26e^{-3s}}{\mathbf{23s} + 1} \tag{9}$$

The same modeling procedure was developed to obtain the relationship among the variation of the temperature at the distillation tray A20 and the variation of the flow of fermented mash applied at the column A:

$$\begin{aligned} K' &= \frac{\Delta PV}{\Delta MV} = \frac{(97, 8-95, 8)}{150}.110\%\\ K' &= -0, 133\%^\circ C \text{ / /} \text{\(\%\)}\\ \theta' &= t\_1 - t\_0 = 85\text{s} \\ \tau' &= t\_2 - t\_1 = 174\text{s} \end{aligned} \tag{10}$$

So, the FODT representation is:

$$G(\mathbf{s}) = \frac{-0.133e^{-85s}}{174s+1} \tag{11}$$
