**2.1. Nonlinear mathematical model**

process of high importance. The industrial scale production of 2 ethyl‐hexanol is made either through liquid phase 2 ethyl‐hexanal hydrogenation or through gaseous phase 2 ethyl‐hexanal hydrogenation. The liquid phase hydrogenation is preferred on industrial scale due to its

328 New Advances in Hydrogenation Processes - Fundamentals and Applications

The development of complex, accurate mathematical models is an essential step in the dynamic behaviour analysis without expensive experiments and last but not least in the development and testing of control strategies. To this end, for the 2 ethyl‐hexanal hydrogenation process was developed at first a distributed parameter mathematical model validated using experi‐ mental data. It consists of a system of partial differential equations based on mass (total and component) and energy conservation laws. In order to analyse the dynamic behaviour and to emphasize the interactions between the parts of a hydrogenation process a dynamic study was performed. Several scenarios have been carried out in order to evaluate the dynamic behaviour. The one presented in this chapter is the study of the catalyst deactivation effect. The effect of a variation of the input flow temperature of the streams as well as the effect of a change in the reactor load: the volumetric ratio between the 2 ethyl‐hexanal flow rate and the recirculated 2 ethyl‐hexanol flow rate are worth to study. The dynamic behaviour study shows the complex‐ ity of the hydrogenation process due to heat, mass and kinetic interactions, which are de‐ pendent on the operating conditions, reactor loading as well as on the trajectory from one state

Unfortunately, despite their accuracy, detailed, nonlinear mathematical models are too complex for efficient use in controller design so the considered approach is the use of a simple model of the process, which describes its most important properties in combination with an advanced control algorithm which takes into account the model uncertainties, the disturbances

To this end, another mathematical model–operational model–is determined based on the main connections between input and output parameters of the process and was obtained based on the result analysis from both simulations step responses related to a distributed parameter

For the hydrogenation process presented in this case study, various methods of control where designed: conventional PID control, internal model control (IMC) and robust control in order

The two main control objectives of all the applied control strategies are: (a) to maintain the inlet reactor temperature below an imposed critical value; (b) to ensure a high 2 ethyl‐hexanol (product) concentration. From a technological point of view the reactor inlet temperature can be controlled by modifying the 2 ethyl‐hexanol recirculated flow rate. The product concentra‐ tion is influenced by the reactant flow rates and also by the catalyst degree of activity which acts as a variable disturbance. The catalyst degree of activity will continuously decrease as the hydrogenation reaction takes place up to the point it needs to be replaced. However, during this period, this effect can be compensated by continuously increasing the input temperature

advantages [1].

to another.

and command signal limitations.

model and the experimental data.

to find the optimal solution.

of the reactants.

Based on laws of mass conservation and energy conservation, the mathematical model determined in this section for the hydrogenation process 2 ethyl‐hexanal is nonlinear. A dynamic mathematical model can be used to simulate complex mass transfer phenomena and to understand processes occurring inside the reactor. So far there are various models proposed in the literature [2, 3] for the hydrogenation reaction kinetics, but there is no mathematical model to describe the processes taking place inside the hydrogenation reactor. There is a need to develop equations, to determine the parameters and boundary conditions that form the mathematical model consisting of differential equations with partial derivatives. The spatially distributed nature of the process is generally unnoticed or ignored and the control techniques are applied using conventional approximate models with concentrated parameters, identified by experiments input/output. Because these simple models ignore the spatial nature of the process, they often suffer from the close interaction and apparent delays due to diffusion and convection phenomena inherent in such processes. Hence, the need for a modelling procedure that generates a general model, a model that takes into account the spatial structure of the process variable and is deducted from the input and output measured data.

The production of 2 ethyl‐hexanol through the liquid phase 2 ethyl‐hexenal hydrogenation is depicted in **Figure 1**.

**Figure 1.** Hydrogenation reaction: A—2 ethyl‐hexanal, B—2 ethyl‐hexanal, C—2 ethyl‐hexanol.

As it can be observed, the production of 2‐ethyl hexanol is in fact two successive hydrogenation reactions. Thus, the reaction product of the first hydrogenation reaction—2‐ethyl‐hexanal—is a reactant in the next one. In the hydrogenation process n‐butanol and iso‐butanol can be obtained as side products of some side reactions. Also, symmetric or asymmetric C8 ethers can be obtained through the etherification reaction of butanols. These side reactions are favoured by operating parameters like: input flows through the reactor, inlet temperature. However, for mathematical modelling purposes only the main reaction will be considered and the following assumptions are made: the model parameters are considered to be constant on the radial section of the reactor; both liquid and gas velocity are considered constant; adiabatic reactor; perfect mixing is considered and the chemical reactions occur only at the catalyst surface. Also, in the reaction zone the following phenomena occur: mass transfer through the volume element dz (theoretical plate); 2 ethyl‐hexanal hydrogenation on the catalyst surface and heat transfer through volume element dz.

The component mass transfer is essential in a heterogeneous reactor with several phases (gas, liquid) because the reactants have to pass from one phase to another making the modelling of gas‐liquid‐solid mass transfer process a crucial step. Substances from a gas phase (hydrogen) and a liquid (2 ethyl‐hexanal) are transformed on the surface of a solid catalyst (nickel on silicon). One of the most important factors in the chemical reaction is the reaction rate (reaction kinetics). In the present case there are two rates of reaction: r1, which relates to the hydroge‐ nation of 2 ethyl‐hexanal and r2, which relates to the hydrogenation of 2 ethyl‐hexanal (intermediate product). For this particular case the chosen catalyst is nickel (Ni) based catalyst on a silica (Si) support. Considering the literature studies [2] for the Ni catalyst supported on Si, the best model to express the reaction rate of the proposed model is:

$$r\_1 = \frac{k\_1 \cdot \mathcal{K}\_{\text{enal}} \cdot \mathcal{K}\_H \cdot \mathcal{C}\_{\text{enal}} \cdot \mathcal{C}\_{H\_2}}{\left\{1 + \mathcal{K}\_{\text{enal}} \cdot \mathcal{C}\_{\text{enal}} + \mathcal{K}\_{\text{anal}} \cdot \mathcal{C}\_{\text{anal}} + \sqrt{\mathcal{K}\_H \cdot \mathcal{C}\_{\text{H}\_2}}\right\}^3} \tag{1}$$

$$r\_2 = \frac{\kappa\_1 \cdot \kappa\_{anal} \cdot \kappa\_{M'} \cdot \mathcal{C}\_{anal'} \mathcal{C}\_{H\_2}}{\left(1 + \mathcal{K}\_{\text{crank}} \cdot \mathcal{C}\_{\text{crank}} + \mathcal{K}\_{\text{anal}} \cdot \mathcal{C}\_{\text{crank}} + \sqrt{\mathcal{K}\_{H'} \mathcal{C}\_{H\_2}}\right)^3} \tag{2}$$

where enal is 2 ethyl‐hexanal, anal is 2 ethyl‐hexanal, oct is 2 ethyl‐hexanol, H is hydrogen, *Ki* (m3 /mol) is the adsorption equilibrium constant for the component *i* (*i:* enal, anal, H) , *ki* (m3 /s kg) is rate constant of the surface reaction *i*,(*i*:1,2), *Ci* (kmol/m3 ) concentration of component *i*.

In industrial scale heterogeneous reactors, the catalyst ages gradually and gets deactivated to the point it becomes inefficient and needs to be replaced. The dependence between the catalyst activity degree and the reaction rate can be expressed as follows:

$$r\_l = r\_l \cdot e^{-k\_D t} \tag{3}$$

where *kD* is the catalyst degree of deactivation.

**Figure 1.** Hydrogenation reaction: A—2 ethyl‐hexanal, B—2 ethyl‐hexanal, C—2 ethyl‐hexanol.

330 New Advances in Hydrogenation Processes - Fundamentals and Applications

transfer through volume element dz.

As it can be observed, the production of 2‐ethyl hexanol is in fact two successive hydrogenation reactions. Thus, the reaction product of the first hydrogenation reaction—2‐ethyl‐hexanal—is a reactant in the next one. In the hydrogenation process n‐butanol and iso‐butanol can be obtained as side products of some side reactions. Also, symmetric or asymmetric C8 ethers can be obtained through the etherification reaction of butanols. These side reactions are favoured by operating parameters like: input flows through the reactor, inlet temperature. However, for mathematical modelling purposes only the main reaction will be considered and the following assumptions are made: the model parameters are considered to be constant on the radial section of the reactor; both liquid and gas velocity are considered constant; adiabatic reactor; perfect mixing is considered and the chemical reactions occur only at the catalyst surface. Also, in the reaction zone the following phenomena occur: mass transfer through the volume element dz (theoretical plate); 2 ethyl‐hexanal hydrogenation on the catalyst surface and heat

The component mass transfer is essential in a heterogeneous reactor with several phases (gas, liquid) because the reactants have to pass from one phase to another making the modelling of gas‐liquid‐solid mass transfer process a crucial step. Substances from a gas phase (hydrogen) and a liquid (2 ethyl‐hexanal) are transformed on the surface of a solid catalyst (nickel on silicon). One of the most important factors in the chemical reaction is the reaction rate (reaction kinetics). In the present case there are two rates of reaction: r1, which relates to the hydroge‐ nation of 2 ethyl‐hexanal and r2, which relates to the hydrogenation of 2 ethyl‐hexanal (intermediate product). For this particular case the chosen catalyst is nickel (Ni) based catalyst on a silica (Si) support. Considering the literature studies [2] for the Ni catalyst supported on

(1)

(2)

Si, the best model to express the reaction rate of the proposed model is:

The reaction rates expression should also include the temperature dependence of the reaction rate constants and absorption constants, which must comply with the Arrhenius law [4] given by:

$$k\_i = k\_0^i \cdot e^{\frac{E}{\pi T}} \tag{4}$$

where *E* is the activation energy and *T* is the temperature.

The developed total and component mass balance equations are as follows:

$$\begin{cases} \frac{\partial Q\_L}{\partial t} = -\upsilon\_L \frac{\partial Q\_L}{\partial x} + \upsilon\_G \cdot \mathbb{S} \cdot M\_{H\_2} \cdot a\_v \cdot N\_{H\_G} \\ \frac{\partial Q\_G}{\partial t} = -\upsilon\_G \frac{\partial Q\_G}{\partial x} - \upsilon\_G \cdot \mathbb{S} \cdot M\_{H\_2} \cdot a\_v \cdot N\_{H\_G} \end{cases} \tag{5}$$

$$\begin{cases} \begin{aligned} \frac{\partial c\_{H}^{G}}{\partial t} &= -\nu\_{L} \frac{\partial c\_{H}^{G}}{\partial x} + a\_{\upsilon} \cdot N\_{H\_{G}} - vph\\ \frac{\partial c\_{H}^{G}}{\partial t} &= -\nu\_{L} \frac{\partial c\_{H}^{G}}{\partial x} - a\_{\upsilon} \cdot N\_{H\_{G}}\\ \frac{\partial c\_{metal}}{\partial t} &= -\nu\_{L} \frac{\partial c\_{metal}}{\partial x} - r\_{1} \cdot \rho\_{sol}\\ \frac{\partial c\_{metal}}{\partial t} &= -\nu\_{L} \frac{\partial c\_{metal}}{\partial x} + (r\_{1} - r\_{2}) \cdot \rho\_{sol} \end{aligned} \tag{6}$$
 
$$\begin{aligned} \frac{\partial c\_{oct}}{\partial t} &= -\nu\_{L} \frac{\partial c\_{oct}}{\partial x} + r\_{2} \cdot \rho\_{sol} \end{aligned} \tag{7}$$

and the heat balance equations for the liquid and gas phases are

$$\begin{cases} \frac{\partial T\_L}{\partial t} = -\nu\_L \frac{\partial T\_L}{\partial x} - \sum \frac{\Delta\_R H\_l \cdot r\_l}{c\_{pL}}\\ \frac{\partial T\_G}{\partial t} = -\nu\_G \frac{\partial T\_G}{\partial x} - \sum \frac{\Delta\_R H\_l \cdot r\_l}{c\_{pG}} \end{cases} \tag{7}$$

All the parameters are detailed in the nomenclature section.

An important feature of the three‐phase reactors is the hydrodynamic characteristic. One of the hydrodynamics parameters with high influence on the performance of the three‐phase 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 be calculated using equation ERGUN [5]:

$$\delta = \frac{1 - \varepsilon}{\varepsilon^3} (1.75 + 150 \frac{1 - \varepsilon}{\text{Re}\_p}) \frac{\rho \cdot v^2}{d\_p} \tag{8}$$

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

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 implemented as a function‐s: S‐function.

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 candidate for the numerical solution of differential equations.
