## 4 Applications in Optimization
### Differential Evolution (DE)
#### 4.2.2 Chemical Reactor Optimization  
Senkerik et al. used Differential Evolution in the optimization task of the batch reactor geometry [74]. The novelty of this approach is that Lozi map was not only used as a chaotic pseudo-random number generator to drive the mutation process as in [58], it was used for the crossover process in DE. The obtained results were compared with original reactor geometry and process parameters adjustment. The statistical analysis of the results provided by chaos-driven DE was compared with canonical DE strategy. The authors used a simplified mathematical model of the reactor consisting of two physical inputs: one for chemical substances **Chemical FK (Filter Cake)**, with parameters temperature \(T_{FK}\), mass flow rate \(\dot{m}_{FK}\), and specific heat \(c_{FK}\), and one for the cooling medium of the temperature \(T_{VP}\), mass flow rate \(\dot{m}_{V}\), and specific heat \(c_{V}\). The reactor had one output, with the cooling medium flowing through the jacket inner space of the reactor (with the volume related to the mass \(m_{VR}\)) and flowing out through the second output, with the following parameters: mass flow rate \(m_{V}\), temperature \(T_{V}\), and specific heat \(c_{V}\). There was an initial batch inside the reactor, with the mass parameter \(m_{P}\) introduced at the beginning of the process. The chemical FK was then added to this initial batch, resulting in a reaction mixture inside the reactor with the following parameters: total mass \(m\), temperature \(T\), and specific heat \(c_{R}\). In addition, the partially unreacted portions of the chemical FK were described by the concentration parameter \(a_{FK}\). This \"batch\" technique allowed for partial control of the temperature of the reaction mixture through controlled feeding of the input chemical FK. In general, this reaction is highly exothermic; thus, the most important parameter is the temperature of the reaction mixture. This temperature must not exceed 100 \({}^{\circ}\)C because of safety and product quality requirements. The original design of the reactor was based on standard chemical-technological methods as part of a proposal involving the physical dimensions of the reactor and the parameters of the chemical substances. In the paper, these values were called the **original parameters**. The main objective of the optimization was to achieve the processing of large amounts of the chemical FK in a very short time. A system of four balance equations (49) was used to describe the reactor process, while another Equation (50) defined the term \(k\):  
\[\left\{ \begin{array}{l}\dot{m}_{FK}=m^{\prime}[t],\\ \dot{m}_{FK}=m[t]a_{FK}^{\prime}[t]+km[t]a_{FK}[t],\\ \dot{m}_{FK}c_{FK}T_{FK}+\Delta H_{r}km[t]a_{FK}[t]=KS(T[t]-T_{V}[t])+m[t]c_{ R}T^{\prime}[t],\\ \dot{m}_{V}c_{V}T_{VP}+KS(T[t]-T_{V}[t])=\dot{m}_{V}c_{V}T_{V}[t]+m_{VR}c_{V}[t ]T_{V}^{\prime}[t],\end{array} \right. \tag{49}\]  
\[k=Ae^{-\frac{E}{RT[t]}}. \tag{50}\]Similar to dos Santos Coeholo (2018), Senkerik et al. implemented the \"DE/rand/1/bin\" strategy; for the mutation operation, they used  
\[u_{j,i,G_{+1}}=x_{j,r_{1},G}+F\cdot\left(x_{j,r_{2},G}-x_{j,r_{3},G}\right) \tag{51}\]  
to minimize the cost function (CF)  
\[f_{cost}=\sum_{t=0}^{t_{1}}\lvert w-T[t]\rvert+\sum_{t=0}^{t_{1}}a_{FK}[t]+pen. 1+pen.2 \tag{52}\]  
with two penalizations:  
\[pen.1=\left\{ \begin{array}{l}0,Max(T[\tau])\leq 323.15,\\ 50,000\;else,\end{array} \right. \tag{53}\]  
for \(\tau\in(t_{2},t_{3})\) and  
\[pen.2=\left\{ \begin{array}{l}0,Max(T[\tau])\leq 373.15,\\ 50,000\;else,\end{array} \right. \tag{54}\]  
for \(\tau\in(0,t_{3})\). The time intervals were set as follows: \(t_{1}\) = 15,000 s, \(t_{1}\) = 20,000 s, \(t_{1}\) = 25,000 s.  
The first penalization helps to find solutions in which the temperature of reaction mixture quickly cools down to its initial state, which shortens the process duration. The second corresponds to the critical temperature. The best optimization results are shown in Tables 3 and 4.