**2.2. Validation and dynamic behaviour analysis of the nonlinear mathematical model**

The developed nonlinear, distributed parameter mathematical model was calibrated and tested based on experimental data acquired from a functional 2 ethyl‐hexanal hydrogenation reactor at Oltchim S.A. company, Romania.

The calibration process was performed taking into account the constructive characteristics of the hydrogenation reactor. The height of the reactor is approximately 20 m, with a diameter of 1 m, while the height of the catalyst is around 18 m. The reactants (hydrogen – gaseous phase – and 2 ethyl‐hexanal—liquid phase) are fed at the top of the reactor. Perfect mixing can be considered because the reactor is equipped with a fine sifter at the top. The product (2 ethyl‐ hexanol) is extracted at the bottom of the reactor. Part of the product is cooled to 90–100 °C in a heat exchanger and recirculated at the top to maintain the inlet temperature below 160 °C. The operating temperature of the reactants is around 100 °C. The plant is also equipped with a heater, to be able to increase the input temperature of the reactants as the catalyst gets deactivated. The flow ratio between the 2 ethyl‐hexanal flow and the recirculated 2 ethyl‐ hexanol flow must be maintained at a specific value in order to maintain the output tempera‐ ture below the critical value. The main reactor operating point is characterized by the following parameter values: hydrogen flow of 1250 – 1300 (m3 /h), 2 ethyl‐hexanal flow of 4 (m3 /h), recirculated 2 ethyl‐hexanol flow of 24 –32 (m3 /h).

The accuracy of the developed mathematical model was tested in different operating points and the comparison between the simulated results and the experimental data in the main operating point is presented in **Table 1**.


**Table 1.** Mathematical model validation.

reactor is the pressure loss of the two‐phase mixture. The pressure loss in the functional layer is an important parameter, which depends on the amount of energy required to operate and which also correlates the interphase mass transfer coefficients. The friction pressure loss can

where *ε* is layer void fraction, and *v*, *υ*, *ρ* are considered for the phase for which the pressure

The dynamic process simulator was implemented in MATLAB programming environment along with graphical extension to SIMULINK. The process simulator, being represented by a mathematical model comprising a system of nonlinear partial differential equations, was

To solve the partial differential equations the finite difference method [6] was used. According to this method, the derivatives where written as finite differences. The solution domain must be covered with a network of nodes in order to apply the proposed method. Theoretically, the approximation of the exact solution will be better if the number of nodes included in the network is greater. To approximate the concentrations, temperature and pressure over time and space (height of the reactor) 100 discretization points (the number of theoretical plates) were chosen, being considered the best choice between the complexity of the model and the accuracy of the results. This method is reasonably simple, robust and is a good general

**2.2. Validation and dynamic behaviour analysis of the nonlinear mathematical model**

The developed nonlinear, distributed parameter mathematical model was calibrated and tested based on experimental data acquired from a functional 2 ethyl‐hexanal hydrogenation

The calibration process was performed taking into account the constructive characteristics of the hydrogenation reactor. The height of the reactor is approximately 20 m, with a diameter of 1 m, while the height of the catalyst is around 18 m. The reactants (hydrogen – gaseous phase – and 2 ethyl‐hexanal—liquid phase) are fed at the top of the reactor. Perfect mixing can be considered because the reactor is equipped with a fine sifter at the top. The product (2 ethyl‐ hexanol) is extracted at the bottom of the reactor. Part of the product is cooled to 90–100 °C in a heat exchanger and recirculated at the top to maintain the inlet temperature below 160 °C. The operating temperature of the reactants is around 100 °C. The plant is also equipped with a heater, to be able to increase the input temperature of the reactants as the catalyst gets deactivated. The flow ratio between the 2 ethyl‐hexanal flow and the recirculated 2 ethyl‐ hexanol flow must be maintained at a specific value in order to maintain the output tempera‐ ture below the critical value. The main reactor operating point is characterized by the following

/h).

/h), 2 ethyl‐hexanal flow of 4 (m3

/h),

(8)

be calculated using equation ERGUN [5]:

332 New Advances in Hydrogenation Processes - Fundamentals and Applications

implemented as a function‐s: S‐function.

reactor at Oltchim S.A. company, Romania.

parameter values: hydrogen flow of 1250 – 1300 (m3

recirculated 2 ethyl‐hexanol flow of 24 –32 (m3

candidate for the numerical solution of differential equations.

loss is computed.

The standard deviation is between 5 and 10 % in all cases. This indicates the presence of an acceptable systematic error. The evolution of the main parameters can also be observed in **Figures 2**–**4**.

By choosing 100 discretization points the height of the theoretical plates of the reactor is 0.2m. Taking into account the diameter of the reactor (∼1m) it can be considered that the concentra‐ tion of 2 ethyl‐hexanol and the temperature of the product are constant along the theoretical plate. However, to prove this assumption, a study was performed on the influence of the number of discretization points on the accuracy of the results. The simulation results are presented in the **Figures 5** and **6**, considering 80, 100 and 125 discretization points. As can be observed from the figures, there are no significant changes in the accuracy of the model, the differences being only at the second decimal.

**Figure 2.** 2 ethyl‐hexanal and 2 ethyl‐hexanal concentration evolution: simulated vs. plant data.

Another important study that needs to be performed to prove that the developed mathematical model captured the hydrogenation mechanism is a dynamic behaviour study. To this end, it is necessary to evaluate that the model captures the effect of catalyst deactivation [7]. By analysing the experimental data it was concluded that the catalyst degree of activity decreases up to 50% after 4 months of continuous functioning. **Figure 7** presents the effect of catalyst deactivation on the product concentration by maintaining the reactants input and temperature constant.

**Figure 3.** 2 ethyl-hexanal and 2 ethyl-hexanol concentration evolutions: simulated vs. plant data.

**Figure 4.** Product temperature evolution: simulated vs. plant data.

**Figure 5.** 2 ethyl-hexanol concentration evolution: 80, 100 and 125 discretization points.

Dynamic Mathematical Modelling and Advanced Control Strategies for Complex Hydrogenation Process http://dx.doi.org/10.5772/65336 335

**Figure 6.** Outlet temperature evolution: 80, 100 and 125 discretization points.

deactivation on the product concentration by maintaining the reactants input and

**Figure 3.** 2 ethyl-hexanal and 2 ethyl-hexanol concentration evolutions: simulated vs. plant data.

**Figure 4.** Product temperature evolution: simulated vs. plant data.

**Figure 5.** 2 ethyl-hexanol concentration evolution: 80, 100 and 125 discretization points.

temperature constant.

**Figure 7.** Catalyst deactivation effect on the product concentration.

It is obvious that the catalyst deactivation influences considerably the quality of the product increasing the production costs. This effect can be diminished if the input temperature of the reactants is gradually increased as the catalyst gets older.

#### **2.3. Operational mathematical model: development and validation**

Based on the previously presented studies, it can be concluded that the hydrogenation process dynamics ethyl 2‐hexanal is very complex; thus, it is only normal that the resulting model is nonlinear, higher order with distributed parameters. The only problem is that highly complex models are difficult to use in the development of most control strategies, being more appro‐ priate for control strategy testing. For this reason it is necessary to design a simpler, linear operation model to be used in control design. There are two possible approaches. The first one infers model reduction methods and linearization which can be troublesome. In this section is presented a more unconventional approach, developing a simpler operational model based on the main connections between input and output parameters, using experimental identifi‐ cation methods and the developed nonlinear mathematical model. The results will be validated by simulation.

To this end, by analysing the hydrogenation process and based on the process engineers experimental knowledge the main input variables are considered to be: input flow of 2 ethyl‐ hexanal, recirculated input flow of 2 ethyl‐hexanol, input temperature of the reactants and hydrogen pressure. The main output variables are considered to be the 2 ethyl‐hexanol concentration and the output temperature which is critical. Nevertheless, the dependence between the input and the output variables is considered to be of second order as follows:

$$
\begin{vmatrix} T\_{out} \\ C\_{out} \end{vmatrix} = \begin{vmatrix} \frac{K1\,\mathrm{v}^{-\mathrm{ev}}}{T1\,\mathrm{s}^{2} + T2\,\mathrm{s}\,\mathrm{s} + 1} & \frac{K2\,\mathrm{v}^{-\mathrm{ev}}}{T3\,\mathrm{s}^{2} + T4\,\mathrm{s}\,\mathrm{s} + 1} \\\frac{K3\,\mathrm{s}^{-\mathrm{ev}}}{T5\,\mathrm{s}^{2} + T6\,\mathrm{s} + 1} & \frac{K4\,\mathrm{s}\,\mathrm{e}^{-\mathrm{ev}}}{T7\,\mathrm{s}^{2} + T9\,\mathrm{s}\,\mathrm{s} + 1} \end{vmatrix} \begin{vmatrix} Q\_{\mathrm{oct}} \\ T\_{\mathrm{lin}} \end{vmatrix} + \begin{vmatrix} \frac{K5\,\mathrm{v}^{-\mathrm{ev}}}{T1\,\mathrm{s}^{2} + T10\,\mathrm{s}\,\mathrm{s} + 1} & \frac{K6\,\mathrm{v}^{-\mathrm{ev}}}{T1\,\mathrm{s}^{2} + T12\,\mathrm{s}\,\mathrm{s} + 1} \\\frac{K7\,\mathrm{s}^{2} + T10\,\mathrm{s}\,\mathrm{s} + 1}{T1\,\mathrm{s}\,\mathrm{s}^{2} + T14\,\mathrm{s}\,\mathrm{s} + 1} & \frac{K9\,\mathrm{v}^{-\mathrm{ev}}}{T1\,\mathrm{s}\,\mathrm{s}^{2} + T16\,\mathrm{s}\,\mathrm{s} + 1} \end{vmatrix} \begin{vmatrix} Q\_{\mathrm{real}} \\ P \end{vmatrix} \tag{9}
$$

The parameter values are determined using experimental identification methods and consid‐ ering step variations of the recirculated 2 ethyl‐hexanol flow (*Q*oct), input temperature of reactants (*T*in). The same step variations are considered for the 2 ethyl‐hexanal input flow (*Q*enal) and the hydrogen input pressure (*P*) even if these parameters are considered to be constant in normal mode of operation. For validation **Figure 8** presents the simulated values obtained for the output temperature and considering a step variation in the recirculated 2 ethyl‐ hexanol flow.

**Figure 8.** Output temperature evolution: operational model vs. nonlinear model.

#### **3. Design and implementation of conventional control strategy**

Conventional PID controllers are the most common control solution in the industry. Most of the research deals with mono‐variable (single input single output—SISO) processes. However, most industrial processes are by their nature multi‐variable (multi‐input multi‐output—‐ MIMO). Using mono‐variable controllers for each output variable, even if it is a solution easy to apply, it will lead to inferior performances. It is possible that despite the fact that each individual PID loop control works, the overall PID control structure to fail. For this reason there is a demand for the development of multi‐variable PID control strategies to compensate the effect of functional interactions between variables from many companies that consider the interactions between variables in multi‐variable systems as the main common problem in the industry.

The hydrogenation process is characterized by the presence of time delays of approximately 30 minutes. The difference between the dead time of each input‐output pair is about 1–2 minutes. For this reason, the dead time is considered to be identical for all input‐output pair. A typical approach to deal with time delay is the non‐delayed output prediction [8, 9]. The non‐delayed output may be estimated and the controller can be computed as for a process without delay. The most popular output predictor is the Smith predictor.

Currently, the 2 ethyl‐hexanal hydrogenation plant is operated using the feed forward control (indirect, open‐loop control). Thus, using methods sometimes simple, sometimes complicated (even in closed loop), the following parameters are controlled and adjusted to the desired level: the 2 ethyl‐hexanal input flow, the recirculated 2 ethyl‐hexanol flow, hydrogen input pressure and the reactant input temperature. At present time, an operator decides whether or not to manually modify the control loops set points to maintain the same process parameters and product specifications. To achieve a more effective operation of the hydrogenation process, both conventional and advanced control methods require a closed‐loop control structure by including the reaction from the output.

In order to develop a multi‐variable PID control, the operational model described in the previous section determined by the Eq. (9) will be used. The desired multi‐variable controller matrix has the following form:

$$H\_R\text{(s)} = \begin{vmatrix} H\_{R11}\text{(s)} & H\_{R12}\text{(s)}\\ H\_{R21}\text{(s)} & H\_{R22}\text{(s)} \end{vmatrix} \tag{10}$$

where *H*R11, HR22 intended for direct adjustment of output variables and *H*R12, *H*R21 are intended to counter act the interactions between input and output channels. The four controllers are computed by imposing a phase margin *γk* = 60° [9].

The obtained controllers are described by:

To this end, by analysing the hydrogenation process and based on the process engineers experimental knowledge the main input variables are considered to be: input flow of 2 ethyl‐ hexanal, recirculated input flow of 2 ethyl‐hexanol, input temperature of the reactants and hydrogen pressure. The main output variables are considered to be the 2 ethyl‐hexanol concentration and the output temperature which is critical. Nevertheless, the dependence between the input and the output variables is considered to be of second order as follows:

336 New Advances in Hydrogenation Processes - Fundamentals and Applications

The parameter values are determined using experimental identification methods and consid‐ ering step variations of the recirculated 2 ethyl‐hexanol flow (*Q*oct), input temperature of reactants (*T*in). The same step variations are considered for the 2 ethyl‐hexanal input flow (*Q*enal) and the hydrogen input pressure (*P*) even if these parameters are considered to be constant in normal mode of operation. For validation **Figure 8** presents the simulated values obtained for the output temperature and considering a step variation in the recirculated 2 ethyl‐

**Figure 8.** Output temperature evolution: operational model vs. nonlinear model.

**3. Design and implementation of conventional control strategy**

Conventional PID controllers are the most common control solution in the industry. Most of the research deals with mono‐variable (single input single output—SISO) processes. However, most industrial processes are by their nature multi‐variable (multi‐input multi‐output—‐ MIMO). Using mono‐variable controllers for each output variable, even if it is a solution easy to apply, it will lead to inferior performances. It is possible that despite the fact that each individual PID loop control works, the overall PID control structure to fail. For this reason there is a demand for the development of multi‐variable PID control strategies to compensate

hexanol flow.

(9)

$$\begin{aligned} H\_{R11} &= 1.3259 \left( 1 + \frac{1}{102.3016 \text{s} \text{s}} \right); H\_{R12} = 2.733 \left( 1 + \frac{1}{106.667 \text{s}} \right); \\ H\_{R21} &= 595.6621 \left( 1 + \frac{1}{90.4977 \text{s}} \right); H\_{R22} = 223.8721 \left( 1 + \frac{1}{119.0476 \text{s}^2} \right) \end{aligned} \tag{11}$$

**Figure 9** presents the closed loop Smith predictor control structure using a PID control.

The next step is to test and evaluate the performances of the developed control strategy by analysing its ability to reject disturbance effects, respectively, the set point tracking capability. In the first scenario, a reference variation of 10 K for the first output value (outlet temperature of the product, Tout) is considered. The simulation results are presented in **Figure 10**.

The second scenario is designed to test the control system capability to counteract the dis‐ turbance effects. Hence, a 6 % step variation in the 2 ethyl‐hexanal input flow is considered for the results presented in **Figure 11**.

**Figure 9.** Closed loop control structure: conventional PID control in MIMO‐SP structure.

**Figure 10.** Output temperature reference step variation of 10 K: (a) output temperature evolution and (b) 2 ethyl‐hexa‐ nol concentration evolution.

**Figure 11.** (a) Output temperature evolution and (b) 2 ethyl‐hexanol concentration evolution considering a step varia‐ tion in the 2 ethyl‐hexanal input flow.

Based on the above results, the effectiveness of the proposed control is emphasized, presenting acceptable overshoot, response time and deviation values, but with opportunities for im‐ provement. Thus, in the third scenario is considered an evolution for a period of 4 months and the catalyst activity degree is decreased up to 50%. **Figure 12** shows the temperature and 2 ethyl‐hexanol concentration evolutions in this situation. It can be observed a steady error of 0.06% for the temperature and an error of 1% for the 2 ethyl‐hexanol concentration.

The last simulation scenario is conceived in order to test the robustness of the designed control system for process parameter variations: gain variation and time constants variations (**Figure 13**).

**Figure 12.** Catalyst deactivation: MIMO‐SP PID controls.

The second scenario is designed to test the control system capability to counteract the dis‐ turbance effects. Hence, a 6 % step variation in the 2 ethyl‐hexanal input flow is considered for

**Figure 9.** Closed loop control structure: conventional PID control in MIMO‐SP structure.

**Figure 10.** Output temperature reference step variation of 10 K: (a) output temperature evolution and (b) 2 ethyl‐hexa‐

the results presented in **Figure 11**.

338 New Advances in Hydrogenation Processes - Fundamentals and Applications

nol concentration evolution.

**Figure 13.** Output temperature evolution for reference step variation—nominal case vs. uncertain case (multivariable PID control).

A decrease in the control system performance can be observed, inferring reduced robustness, an aspect that may be improved by using advanced control strategies.
