The use of differential evolution algorithm for solving chemical engineering problems

2.1 Initialization

In this step, the initial population is created, usually by employing a random-number-generating procedure, the algorithm selecting which locations should be explored and which should be discarded (de Melo and Botazzo Delbem 2012). Although it is an important step (as the performance is affected by initialization and improper initialization can lead to high errors), initialization is somewhat discarded, the majority of researchers mentioning in their works a series of general elements without any additional details (Ali et al. 2013).

In order to perform a proper initialization procedure, different aspects must be taken into consideration. The most important ones are represented by (i) the position of the global minimum (considering that the problem is a minimization one) and (ii) boundary determination. Before solving an optimization problem, the position of the global minimum is usually not known, and in this context, it is indicated to start the algorithm by using equally dispersed points from the search space (Lei et al. 2010, Peng and Wang 2010). Some algorithms start from a single point, but DE needs a population composed of diverse individuals (Price et al. 2005).

Although there are problems that have unconstrained parameters, for the most real-world problems, the existence of natural physical limits or logical constraints imposes different values for each parameter, and their initialization is a straightforward process. For example, the time for studying a phenomenon cannot be negative or the pressure of a chemical reactor takes values from a prespecified interval, which is dependent on the characteristics of the reactor. In this context, initialization cannot be performed until all the boundaries are known.

Eq. (1) describes the formula used for initializing each k characteristic of each i individual from the population. This equation corresponds to the classic variant of DE, which, in the literature, has different variations and improvements.

where b_k,L and b_k,U are the lower and upper limits of each k characteristic and rand(0, 1) is a random generated number in the interval (0, 1).

In a review about the initialization procedures for large-scale optimization problems (Kazimipour et al. 2013), the initialization techniques were classified into (i) stochastic, (ii) deterministic, (iii) two steps, (iv) other methods, and (v) application specific. These methods can be also used in the DE case; the most employed variant (except the deterministic approaches, which are the classic choice) is the two-step variant, which includes opposition-based learning (OBL) and quasi-OBL.

2.2 Mutation

In the mutation step, in order to create new individuals, the selected individuals are modified by introducing more genetic material into the population (Salman et al. 2007). The role of mutation is to add diversity and to avoid getting trapped into local optima. In the DE case, a new individual is created by adding a scaled differential term to a base vector (individual) (Price et al. 2005, Das and Suganthan 2011). This mechanism [described in its simple form by Eq. (2)], also called “differentiation”, is the main characteristic separating DE from other EAs:

where ω_i is the ith element from the mutated population, α is the base vector, β is the differential term, and F is the scaling factor. The differential term is defined as the difference of two distinct, randomly chosen vectors: β=x_k-x_p. The base vector is also randomly chosen, and in order to achieve good convergence speed and probability, Price et al. (2005) indicate that all vectors used in the mutation step must be distinct. This mechanism – differentiation – has the advantage of not producing a disruption in the hyper-plan in which the vectors exist (Feoktistov 2006).

Initially, Storn (1996) presented 10 variants of the algorithm, with a coding “Mode/DiffTerm/Cross” being used to indicate each variant. “Mode” represents the mode in which the base vector of the mutation step is chosen, “DiffTerm” represents the number of differential terms used for mutation, and “Cross” is the type of crossover. The first two terms (“Mode” and “DiffTerm”) refer to the characteristics of the mutation step, and the last one, “Cross”, is related to the crossover step. The base vector for the mutation step can be chosen randomly (Rand), as the best individual in the population (Best), or as a vector that lies on a line between the target and the best individual (Rand-to-Best). Regarding the number of differential terms used in the mutation phase (information coded with “DiffTerm”), one [Eq. (2)] or two terms are usually employed, which are denoted with 1 and 2.

The first variant of the DE algorithm, which uses the random method of vector selection, along with one differential term and binomial crossover, is called Rand/1/Bin. The other versions included in the initial 10 are Rand/2/Bin, Rand/1/Exp, Rand/2/Bin, Rand/2/Exp, Best/1/Bin, Best/2/Bin, Best/1/Exp, Best/2/Exp, Rand-to-Best/1/Bin, and Rand-to-Best/1/Exp.

Eq. (2) describes the variant in which only one differential term is used. However, in literature, a series of variants that use more differential terms can be encountered [Eqs. (3) and (4)]. Therefore, the search space is extended, existing ∑i=0k∗2(Np-i) possibilities of combination of 2 * k randomly chosen individuals (Feoktistov 2006). A high number of individuals leads to high diversity, and Feoktistov (2006) recommends the use of three to five individuals, which is a good compromise between computational time and quality of exploration.

where xbest,gp is a randomly chosen solution from the top 100 * p% individuals from the g population, r1 and r2 are two uniformly chosen indexes, and F_i is the scale factor associated with the ith individual.

where r1, r2, and r3 are three randomly and mutual different individuals from the g population and x_pbest is a randomly chosen individual from a S^p set generated at the beginning of each generation according to a specific procedure.

The variant described by Eq. (3) was proposed in Zhang and Sanderson (2009), being called DE-current-to-pbest/1. The one represented by Eq. (4) is an improvement of the variant of Zhang and Sanderson (2009) and was proposed by He and Yang (2012).

Except the number of differential terms, another aspect that influences mutation in particular, and the algorithm performance in general, is the scaling factor F applied to the differential terms. Its role is to control the rate at which the population evolves by determining the perturbation size of the differential term. Usually, the interval (0, 1) is considered for this parameter, but a fixed upper limit was not determined. However, a limit of F=1.2 was empirically determined, optimization problems requiring F to trespass this threshold not being encountered (Mallipeddi et al. 2011). In addition, there are only a few optimization problems that need F>1, a case in which the differential term is scaled up. A larger F can increase (i) the probability of escaping the local optimum and (ii) the number of function evaluations (Deng et al. 2013).

2.3 Crossover

In this step, an operation for population diversity improvement is applied. Using two populations (current and mutated), a new trial population is created. Generally, two variants of crossover are used in the DE algorithm: binomial [Eq. (5)] and exponential.

where u_i is the trial vector and x_i is the ith individual from the current population.

In the binomial version, the number of mutant components used has a binomial distribution, the principle behind it being simple: a random number in the (0, 1) interval is generated and, if it is smaller than the crossover control parameter (also known as Cr), then the element is taken from the mutated individual. If the generated value is higher than Cr, then the element from the current individual is copied into the trial one. On the other hand, the exponential crossover is somewhat different. An initial start point (sp) is generated, and all the characteristics with an index lower than sp are copied from the mutation vector. When the index is higher than sp, a random value between 0 and 1 is generated, and until it is bigger than Cr, the characteristics from the current individual are copied to the trial vector. After that, the remaining characteristics are taken from the mutation vector.

The difference between the two mechanisms consists in the position of characteristics taken from the parents. In the binomial version, the components from the mutant individual are randomly selected, while in the exponential one they are organized into groups of compact sequences (Zaharie 2002). Analysis of the differences between these two variants showed that the use of exponential crossover to the detriment of the binomial one increases efficiency only for a small number of problems, and therefore, researchers did not focus on the crossover as a mean to improve performance (Tvrdik 2008).

A control parameter (Cr) is employed to control which and how many components of each individual are copied (Mallipeddi et al. 2011). It can take values in the [0, 1] interval, with its optimal value influenced by the type of problem and type of crossover. For example, when the objective function is nonseparable, a [0.9, 1] interval is considered suitable and a [0, 0.2] interval when it is separable (Mohamed et al. 2013). Concerning the optimal value of Cr based on crossover type, knowledge is limited to the distribution of the number of mutated components (Alguliev et al. 2012).

The classic crossover versions can generate only one solution, limiting the search ability of DE (Wang et al. 2012). Consequently, new crossover variants were proposed or borrowed from other algorithms. One such example is represented by the orthogonal design method, with different studies presenting the advantages of applying an orthogonal crossover for generating offspring (Gong et al. 2006, 2008).

2.4 Selection and stop criteria

In the last step of the algorithm, a mechanism for selecting the individuals forming the next generation is employed (Rahnamayan and Tizhoosh 2008). The classic version of DE uses a “one-to-one” competition, the trial and current individuals being compared based on their objective values. The ones with the lowest value (when considering a minimization problem) are selected to form the next generation. This comparison determines in DE a tighter integration of recombination and selection than in other EAs (Price et al. 2005). This survival criterion is also called “objective function based criterion” or “greedy criterion”. In literature two other mechanisms can be encountered: age criterion and age-objective function criterion (Price et al. 2005, Li and Li 2010).

As the majority of problems have specific constraints, special attention must be given to the parameters that exceed the boundaries. When applying the selection criterion, because DE does not have an innate mechanism for dealing with constraints, a constraint handling mechanism must be used in order to evaluate, and if necessary, to move the exceeding parameters back into their feasible interval. Depending on the characteristics of the problem being solved, different approaches can be used. For example, in the case of single-objective problems, the selection process is somewhat simple (as it is easy to decide which between the two individuals is better), but in the case of multioptimization problems, the decision is not so straightforward (Qin et al. 2010).

The steps of the algorithm are repeated until a stop criterion is reached. An optimal selection of this criterion ensures a good tradeoff between solution performance and consumed resources. In an ideal case, the optimization is stopped when the convergence to the optimum is achieved (Zielinski and Laur 2008). As the determination of this condition is tricky, this aspect usually cannot be used. Therefore, researchers focused on finding a good solutions for this problem, with the best variants proposed including (i) the number of current generations reaching a predefined maximum number (G) (as in Figure 1) or (ii) the number of function evaluations reaching a maximum value (FE_max).

Figure 1:

Schema of the simple DE variant.

Figure 1 presents the general workflow of the DE algorithm, in a simple/classic variant where the stop criterion is represented by the number of generations reaching the prespecified upper limit.

2.5 Algorithm improvement

In order to improve the algorithm performance and to make it more flexible for the theoretical and real-life applications, different modifications at various levels have been performed. All these modifications followed three main directions (Brest et al. 2011): (i) replacing the hand tuning of control parameters with adaptive or self-adaptive mechanisms, (ii) hybridizing DE by combining it with other optimization techniques, and (iii) introducing more mutation strategies during the optimization process.

Each control parameter influences specific aspects of the algorithm, its effectiveness, efficiency, and robustness being dependent on their correct values, which are problem specific and can vary substantially (Brest 2009, Hu and Yan 2009a). Due to their influences, finding the optimal values of the control parameters is not a straightforward process, and the methods used for their correct settings can be classified into parameter tuning and parameter control (Eiben and Schut 2008). Parameter tuning consists of finding good values before running the algorithm. On the other hand, in the case of parameter control, the values are changed dynamically during the run based on a set of defined rules. Taking into account the “how” criterion of Eiben and Schut (2008) indicating how the change is performed, four subclasses are encountered: (i) deterministic control, where the parameters are determined using a deterministic law, without feedback from the system; (ii) adaptive control, where the parameters are determined using feedback from the system. Two categories are encountered here: parameter adaptation, which obeys the state of the population and refresh of population; (iii) self-adaptive control, where the parameters are dependent of the algorithm being encoded into it; and (iv) hybrid control.

The second direction of improvement – hybridization – is the process of combining the best features of two or more algorithms in order to create a new algorithm that is expected to outperform the parents (Das and Suganthan 2011). Depending on the type of algorithm DE can be hybridized with, three situations encountered: (i) DE and other global optimization algorithms, (ii) DE with local search methods, and (iii) DE with global optimization and local search methods. Depending on the level of interaction, the hybridization can be performed at the (i) individual level, describing the behavior of an individual from the population; (ii) population level; (iii) external level; and (iv) meta-level, where a superior metaheuristics includes the algorithm as one of its strategies (Feoktistov 2006).

In case of the third direction of improvement, different approaches related to the mutation were employed. The classic variant regarding mutation in DE is to create, for each individual in the population, a single mutated individual based on one or more vectors. The alterations proposed by researchers refer to the (i) introduction of more differential terms, (ii) selection of the individuals participating in the mutation process using specific principles, (iii) introduction of other mechanisms (evolutionary game theory, Pant et al. 2009a, and replacement of control parameter with Laplace distribution, Thangaraj et al. 2010), and (iv) employment of multiple mutation strategies (Qin et al. 2009, Zamuda and Brest 2012).

The principles from all these directions can be combined, with DE being modified at different levels, using different principles. For example, opposition-based chaotic DE algorithm (Thangaraj et al. 2012) has an OBL-based initialization and the crossover factor is adapted using chaos theory; DE-CCS (Wang and Gao 2014) (DE with cooperative coevolution) has a jumping DE (jDE)-based approach for the control parameters, a crossover scheme that employs both binomial and exponential crossover and a local search using cooperative coevolution; in hybrid self-adaptive DE with neural networks (Curteanu et al. 2014), neural networks are optimized using a DE-based version that includes OBL initialization, a modified mutation and a local search procedure based on BackPropagation and Local Search algorithms.

3.1 Simple DE versions used in process optimization

In this section, different optimization procedures based on the classic DE versions (without any improvements such as adaptation or hybridization) applied to specific chemical engineering processes are presented.

In Moonchai et al. (2005), the effect of pH and temperature on the bacteriocin production in Lactococcus lactic C7 was studied and a series of models for five variables were developed. In order to determine the optimal temperature profile (which, in this case, was considered a control variable), the temperature parameter was introduced in the models created for cell growth, lactic acid production, glucose consumption, and bacteriocin production. After that, using a weighted sum method, the two objective problems (determination of temperature control leading to the maximization of productivity and final bacteriocin titer) were converted into a single objective problem. It was then solved using the DE algorithm, the simulation results indicating that the bacteriocin production rate and titer are at the highest level when starting with a 30°C temperature and then lowering it quickly at 22°C.

As the dynamic optimization of the fermentation processes is a problem requiring not only powerful algorithms but also carefully chosen methodologies, Rocha et al. (2007) proposed an online optimization procedure based on DE. Three different case studies were considered: (1) fed-batch recombinant Escherichia coli fermentation, where microorganisms follow three different metabolic paths (oxidative growth on glucose, fermentative growth on glucose, and oxidative growth on acetic acid); (2) a hybridoma reactor; and (3) fed-batch bioreactor for the production of ethanol by Saccharomyces cerevisiae. In case 1, the scope was to determine the optimal feeding rate profile maximizing productivity (which was represented by the recombinant protein formed per unit of time). For case 2, the objective was to increase the amount of monoclonal antibodies produced by the system, and in case 3, the aim was to determine the substrate feed profile leading to the maximization of ethanol amount. If in case of the offline optimization small noise leads to disruption in the predictions, the online approach exhibits properties of graceful degradation. However, when noisier settings were employed, the DE algorithm performance decreased. Using the same case studies and objectives (from Rocha et al. 2007), Mendes et al. (2008) performed online and offline optimizations of input variables by applying three distinct algorithms, namely DE, a real value EA, and a fully informed particle swarm (FIPS). The results obtained for the offline optimization indicated that DE/Rand/1/bin has the best results, followed closely by DE/best/2/bin and FIPS. Similar findings were obtained when online optimization was performed.

Khademi et al. (2009) studied the optimization of methanol synthesis and cyclohexane dehydrogenation in a thermally coupled reactor by means of a Best/1/bin strategy with fixed control parameters (Cr=1, F=0.8, and Np=100, with Np being the number of individuals in the population). Their main objective was to maximize the methanol and benzene mole fractions based on three parameters: inlet temperature of the exothermic and endothermic sides and initial molar flow rate of endothermic side. It was shown that suitable values of the considered parameters can provide the necessary heat to drive the endothermic process and, in the same time, to enhance the methanol mole fraction with 3.67% and methanol yield with 3.1%. In another study (Khademi et al. 2010), the same research group proposed a new membrane thermally coupled reactor for methanol synthesis, cyclohexane dehydrogenation, and hydrogen production. The new system was optimized with the same DE version, having the same control parameter values, the scope being the maximization of outlet mole fractions of methanol, benzene, and hydrogen. Six decision variables were chosen: inlet temperature and molar flow rate of the endothermic and exothermic sides, inlet pressure of the exothermic side, and inlet temperature of the permeation side. Compared with the conventional methanol reactor (CMR), the optimized system had a reduced size of reactor, enhanced equilibrium conversion, and an increased methanol production with 16.3%.

A novel methanol synthesis loop with hydrogen permselective membrane reactor was modeled and optimized using a Rand/2/bin version (Parvasi et al. 2009). The behavior of the system was described by a set of ordinary differential and algebraic equations, which was then used as a model for the optimization procedure with the final scope of improving methanol production based on the inlet temperatures of the membrane tube and steam drum. The new design had a 40% increase in production, which indicated that the applied approach was suitable for the considered process.

Another type of reactor optimized with DE was represented by a radial-flow, spherical bed methanol synthesis reactor in the presence of catalyst deactivation (Rahimpour et al. 2009). The Rand/2/bin version was used in four cases for maximizing the methanol production based on (i) the inlet temperature of reactors, (ii) temperature profile, (iii) radius ratios of reactors, and (iv) catalyst deactivation parameters. In each case, the optimization was successful, with the considered parameters being determined with great accuracy.

Catalytic naphtha reforming is a technology of utmost importance for the petrochemical industry, with effort and energy being spent on improving or replacing the classic approaches and systems such as the conventional tubular reactor (CTR) with newer and improved reactors. The dynamic of a multistage spherical, radial flow reactor for the naphtha reforming process in the presence of catalyst deactivation was performed using a DE approach (Best/1/bin) (Rahimpour et al. 2010). Based on seven decision variables (inlet temperature and catalyst distribution of each reactor, outlet gas pressure), the maximization of production of aromatics and hydrogen was followed. The results indicated that by using higher temperature furnace (allowing gas to enter at 840 K), the efficacy of the system rises.

In another study, the CTR was optimized via DE method in three cases and the reactor was discretized in 20 segments (Iranshahi et al. 2011). The optimization objective was the maximization of hydrogen and aromatics productions based on (i) temperature profile, (ii) amount of hydrogen removed, and (iii) a combination of temperature profile and hydrogen removal. The best results were obtained in the third case, an improvement of 24% for aromatics and 10% for hydrogen production being predicted.

The electrical parameters (six temperature-dependent and current-dependent nonlinear resistors) of a biohydrogen real-time power-generating system were optimized using a DE version with fixed control parameters (F=0.8, Np=300) (Huang et al. 2010). The proposed proton exchange membrane fuel cell system (with a rated power of 3 W) had bipolar plated designed for low-pressured biohydrogen. Using the optimization results, the V-I characteristics were determined based on the Kirchhoff’s law. In addition, a comparison between the DE predictions and those generated by GA was performed, with the optimal temperature estimated by DE being 336.89 K and by GA being 303 K.

Using a multi-input, multioutput orthogonal least-square-based radial basis function model and a classic DE approach with fixed parameters (F=0.5, Cr=0.9, and Np=100), the production process of para-diethylbenzene through disproportionation of ethylbenzene was modeled and optimized (Rahman et al. 2010). The selectivity and conversion of para-diethylbenzene were modeled as a function of temperature, liquid hourly space velocity, and pressure.

Dimethyl ether (DME) is another fuel that has captured the attention of researchers, as it is a clean multipurpose and multisource fuel (Vakili et al. 2011). In the work of Vakili et al. (2011), the operation conditions of a thermally coupled direct DME heat exchanger (TCDR) were optimized using a Best/1/bin version of DE with Cr=0.8, F=0.8, and Np=40. The scope was to maximize the DME and benzene production based on a set of decision variables represented by the inlet temperature and initial molar flow rates of the exothermic and endothermic sides. Compared with the conventional DME synthesis reactor (CDR), the optimized system had a reduced inlet feed of exothermic side with about 4300 kmol/day and a 1.6 ton/day increased DME production. Concerning the endothermic side, the inlet feed is reduced with 25,700 kmol/day and the hydrogen production is increased with 0.8 ton/day.

In another work, the DME synthesis and cyclohexane dehydrogenation reaction taking place in a thermally coupled reactor were optimized using classic Best/1/bin version (Khademi et al. 2011). The optimization goal was to determine the operating conditions (temperature profiles, endothermic and exothermic sides) that maximize the DME and benzene mole fractions. The decision variables considered were inlet temperatures of the exothermic and endothermic sides and initial flow rates of the endothermic and exothermic sides. The final result had an improvement of 2.43% for the DME fraction and of 2.78% for methanol conversion.

Rahimpour et al. (2011) used the DE algorithm for optimizing the operating conditions of a thermally coupled membrane reactor for decalin dehydrogenation and FT synthesis in gas-to-liquid (GTL) technology. The gasoline yield (exothermic side) and hydrogen model fraction (endothermic side) were maximized and hydrogen molar flow rate (recycled stream) was minimized based on five decision variables (inlet temperature and flow rate of the endothermic and exothermic sides and exothermic pressure). The Best/1/bin variant with fixed control parameters (F=1, Cr=0.8, and Np=100) had good results, with an increase of 14.28% of gasoline yield.

A fixed-bed tri-reformer reactor (combining endothermic reforming reaction and exothermic oxidation reaction) for methanol production was optimized in order to determine the optimal operating condition leading to a maximization of methane conversion, hydrogen production, and desired H₂/CO ratio (Arab Aboosadi et al. 2011). The classic DE-based optimization procedure with penalty function used eight decision variables: inlet composition of CH₄, CO₂, CO, H₂O, H₂, O₂, and N₂ and inlet temperature of gas feed. The optimized system had an increase in methane conversion from 94.3% to 97.9%, a H₂ yield of 1.84%, and a H₂/CO ratio of 1.7%.

The estimation of the biomass concentrations at a product scale feed batch process was studied by five different estimation methods: (i) kinetic model of overflow metabolism, (ii) metabolic black-box model, (iii) observer, (iv) artificial neural networks (ANNs), and (v) an adaptive DE (RandToBest/1/bin version) (Hocalar et al. 2011). Two different technical scale bubble column fermenters represent the system modeled, where S. cerevisiae yeast is used as host organisms for recombinant proteins and for producing baker yeast and ethanol. In order to evaluate the performance of the selected approaches, four types of fermentation were considered. The algorithms were compared in terms of difficulty of implementation, number of parameters measured, and process knowledge. The mean-squared errors (MSE) obtained with all “fermentation type-modeling” combinations are presented in Table 2. The results pointed out that each procedure has advantages and disadvantages, in the DE case, a few measurements being required and the convergence being good for different fermentations. On the other hand, DE has a long response time that depends on the control parameters. In this case, one possible improvement, in both response time and efficiency, is represented by the use of a DE self-adaptive procedure in which the control parameters (especially the crossover probability and the scaling factor) are internally adapted to the specific characteristics of the search space problem.

In a review related to the use of EAs for optimization of a fermentation process, Adeyemo and Enitan (2011) list a series of applications of the DE algorithm. What can be observed is that GAs and ANNs are widely used and the number of works is considerable, with these algorithms establishing themselves as “to go tools” when specific aspects cannot be solved with classic approaches. On the other hand, the application of DE for fermentation processes is still in its infancy.

In the study of Vakili et al. (2012), a new DE optimized thermally coupled membrane configuration (TCMDR) simultaneously producing hydrogen and DME was proposed. This system was based on the TCDR developed in Vakili et al. (2011) to which a membrane concept is added. The main role of the optimization procedure (represented by the Best/1/bin version with fixed control parameters Np=50, F=0.8, and Cr=0.8) was to identify the optimum design of the configuration and maximization of hydrogen recovery and of production yield (DME, benzene). Compared with the CDR and TCDR, the DME production of the optimized TCMDR was improved by 10.3% and 11.4%, respectively. Also, the cyclohexane conversion was raised by 8.3% (compared with TCDR) and the endothermic flow rate was reduced, with the operation cost of the system being lower.

Another type of reactor proposed for direct DME production is represented by the industrial scale dual-type reactor, combining two fixed-bed reactors (one water cooled and one gas cooled) (Vakili and Eslamloueyan 2012). In order to achieve the best performance, the reactor design was combined with an optimization procedure, with a set of seven decision variables (number of tubes in the water and gas cooled reactors, temperature of feed gas, reactor length, coolant temperature water in the first reactor, second reactor diameter) being considered. The optimization procedure employed a Best/1/bin version with Np=70, F=0.8, and Cr=0.8. Compared with CDR, the optimized proposed system had an increased DME production, with 60 ton/day. In addition, by eliminating the need for a separate unit for methanol production and purification, the production costs were reduced.

In Arabpour et al. (2012), in order to enhance the gasoline production, the optimization of the Fischer-Tropsch (FT) synthesis reactions in GTL technology was performed using Best/1/bin version in case of five discretized reactor models. For each model, different aspects were followed: temperature optimization (first model, D1), optimization of injected hydrogen (second model, D2), effect of optimum removed water (third model, D3), effect of temperature and hydrogen removal (fourth model, D4), and simultaneous optimization of temperature, injected hydrogen, and removed water (fifth model, D5). The reactor length was divided into 20 elements, with the optimized values for each model and for each element being determined by DE, starting from the first segment from the top of the reactor. The results indicated that the highest yield of gasoline is obtained in the D5 model. Compared to the conventional reactor, in D1 to D5 cases, the CO₂ production is reduced by 6.2%, 21.6%, 27.6%, 24.1%, and 31.4%, respectively.

In Rahimpour et al. (2012), a process intensification procedure of the catalytic naphtha reforming unit was performed by optimizing a system based on a thermally coupled naphtha reforming reactor (TCR). The scope was to maximize the aromatic hydrogen and aniline molar flow rates and the aromatic in reformate mole fractions by using several decision variables, from which 10 belong to the endothermic side (naphtha reforming) and 16 belong to the exothermic side (nitrobenzene hydrogenation). Compared with the nonoptimized TCR, the optimized system has a 26.3% increase in aromatic yield (endothermic side) and a 39.21% increase in aniline production (exothermic side). The higher thermal efficiency and lower operational cost of the optimized system indicated that the proposed approach is a good way to boost production.

In another study, the operating conditions of a radial-flow tubular membrane reactor (RF-TMR) are optimized (Iranshahi et al. 2012). As in the previously presented study (Rahimpour et al. 2012), a relatively high number of parameters were used for optimization. Some of them are specific to the first, second, and third reactors (sweep gas pressure, membrane thickness, inner radius flow of the collector, catalyst mass distribution, fraction of the total sweep gas, number of sweep sections, ratio of permeation angle, and ratio of length per diameter), whereas some are specific to the sweep gas (total molar flow and hydrogen molar fraction), recycle gas (hydrogen mole fraction), and to the first reactor (total fresh naphtha feed). Compared with the conventional, dimensionless tubular reactor, the optimized radial flow tubular membrane reactor had an increase of 11.6% for hydrogen and 8.9% for aromatic yields.

The operational conditions in both sides of a catalytic thermally coupled fluidized bed naphtha reactor (TCFBNR) were optimized by using a DE technique (Pourazadi et al. 2013). Based on a penalty function (with a parameter equal to 10⁸), a set of constraints guaranteeing an increase in aromatic and hydrogen production were imposed. The results obtained indicated that the number of tubes can be decreased by 29% compared to the non-optimized TCFBNR. Also, through optimization, an increase in aromatic production (464.4 kg/h), hydrogen production (3.03 kg/h), and aniline flow rate (14%) was realized.

Iranshahi et al. (2013) proposed an alternative to the CTR by developing a combination of tubular membrane and radial-flow spherical reactors (TM-RSR). In addition, the new system was optimized by DE, the optimal design parameter, and the operating conditions being used. Two configurations, combining a radial flow spherical reactor with one tubular membrane reactor (case I) or with two tubular membrane reactors (case II), were considered. The number of parameters and their meaning vary in both cases, with a list of all the parameters used for optimization in this work (Iranshahi et al. 2013) and the ones previously discussed (Iranshahi et al. 2012, Rahimpour et al. 2012, Pourazadi et al. 2013) being presented in Table 3. The configurations used for case I were MSS, SMS, SSM, and SSS, and the configurations for case II were MMS, MSM, SMM, and MMM, where S denotes the spherical reactor and M denotes the tubular membrane. The best variants were SMS and SMM, with their optimized variant (compared with CTR) having a considerable increase in the hydrogen (23 kmol/h) and aromatic (6.5 kmol/h) production rates.

Samimi et al. (2013a) proposed a DE optimized thermally coupled reactor with a dual membrane (TCDMR) for simultaneous production of DME, hydrogen, and naphthalene. After determining the system models using the kinetic equations, energy, and mass balances in addition with several auxiliary correlations, the operation conditions of the thermally coupled reactor (case 1) and of TCDMR (case 2) were optimized using a Best/1/bin version with fixed control parameters (Np=100, F=0.8, and Cr=1). In the first case, the scope was to maximize the DME mole fractions on the outlet exothermic side based on three decision variables (inlet temperature, initial molar flow rate, and inlet pressure of the exothermic side). In the second case, the maximization of the outlet exothermic DME mole fraction and endothermic hydrogen mole fraction was considered, eight decision variables being used (inlet temperature of water permeation, hydrogen permeation, endothermic and exothermic sides, inlet pressure, and feed rates of the endothermic and exothermic sides). Compared with CDR, for DME mole fraction, a 3.42% (case 1) and a 34.2% (case 2) improvement was observed. Concerning the hydrogen mole fraction, a 6.4% improvement of case 2 over case 1 was realized.

The operating conditions and the length per radius of a novel axial flow spherical packed bed membrane reactor for DME production through methanol dehydration were optimized using the Best/1/bin version (Samimi et al. 2013b). The scope was to maximize the DME mole fractions in the reactor outlet, using a set of 11 variables: inlet temperatures, molar flow rates, composition of components in the reaction and permeation sides, reaction side pressure, and length per radius. The kinetic model had a 2.15% maximum absolute error and the optimization leads to a 13.5% increase compared with CDR.

In Samimi et al. (2013c), a novel thermally double coupled reactor is proposed for hydrogen, methanol, and high-octane gasoline production. The system (a multitubular reactor where two exothermic reactions represented by methanol production and FT synthesis provide the heat required for the endothermic decalin dehydrogenation) is optimized using Best/1/bin variant with fixed control parameters (F=0.8, Cr=1 and Np=100). The objective was to maximize the summation of the methanol mole fraction (endothermic) and outlet gasoline yield (exothermic) based on five decision variables: inlet temperature and inlet pressure at the FT and methanol synthesis sides and inlet flow rate at the FT side. Compared with CMR, the methanol production rate was increased by 10.52%.

In Rahimpour and Mirvakili (2013), a decalin and hydrogen looping approach named thermally coupled membrane ternary reactor (TCMTR) for application in GTL technology is proposed. In order to minimize the H₂ mole fraction in the outlet of the third reactor and to maximize the gasoline yield and hydrogen mole fraction in the endothermic side, the operating conditions of the TCMTR are optimized using a DE approach. A set of seven decision variables were considered, including inlet temperatures of the endothermic and exothermic sides, inlet molar flow and pressure of the exothermic side, initial flow rate of the endothermic side, temperature of the shell side, and length of the third reactor. The results obtained indicated that in the new proposed configuration, decalin can be used as a hydrogen carrier, with only a small amount of this substance being required for injection in the first reactor. Also, compared with a thermally coupled membrane dual-type reactor, the DE-optimized TCMTR has a 92% decrease in decalin consumption.

In Bayat et al. (2013), the industrial fixed-bed ethylene oxide reactor was optimized (at both steady-state and dynamic conditions and taking into account the catalyst deactivation) using DE algorithm. Distinctively from the Lahiri et al. (2008) work, a mathematical model was employed, and two different cases were investigated with the scope of maximizing the ethylene oxide production rate during 1100 days of catalyst life. In the first case, four parameter were optimized: inlet pressure of the reaction side, inlet feed gas temperature, inlet cooling saturated water, and inlet molar flow rate. For the second case study, six parameters were optimized, from which three were related to the inlet gas temperature and three to the cooling water temperature. Based on the results obtained in the first case, in the second situation, the optimal behavior of cooling water and feed gas temperature were determined. In the end, an improvement of 1.726% (first case) and 4.22% (second case) was obtained.

A design method for simultaneous optimal determination of process configuration, equipment design, and operating conditions of a membrane-assisted separation process was developed by Koch et al. (2013). The considered case study was represented by separation through distillation and membrane separation of the ternary system acetone/isopropyl alcohol/water. The steps of the new approach consisted of (i) generation of process alternatives through thermodynamic analysis, (ii) experimental investigations in a laboratory-scale membrane plant, (iii) generic modeling based on mass-transfer models, and (iv) process superstructure optimization with DE. The variables optimized include distribution feed, distribution side, distribution recycle, number of membrane modules in series, number of membrane sheets in a module, total weight of active packing for sections 1 to 4, reflux ratio, flow rate of side stream, and molar recycle ratio. The scope was to reduce costs, with the objective function containing investments and operation costs.

For methanol synthesis, a dynamic mathematical model of a membrane-gas-flowing solid-fixed-bed reactor with in situ water absorption, in the presence of catalyst deactivation, was proposed in the work of Bayat et al. (2014). In addition, an evaluation of the optimal operating conditions was performed using DE, the results indicating that under the optimal values of gas phase inlet temperatures, flowing solids phase, and shell side, a high methanol production can be achieved. A very similar study related to the same type of reactor and reaction was also performed in Dehghani et al. (2014).

As can be observed, different aspects of problems belonging to the chemical engineering area can be solved effectively with simple DE variants. The problems range from fermentation to catalytic naphta reforming to fuel and biofuel production. For the same process, various aspects and conditions are studied from multiple points of view. In this manner, the findings of different studies can be combined, and useful conclusions for the experimental practice derived.

Concerning the DE algorithm, the researchers did not focus only on a single variant, the literature providing examples where not only the type of mutation or crossover were changed, but also the selected values for the control parameters. As each process has its own characteristics, which change even when different parameters are taken into consideration, and as DE successfully solved all aspects, it can be concluded that even in the simplest forms, the algorithm is flexible enough to provide satisfactory solutions.

3.2 Modified and hybrid DE variants in process optimization

Kapadi and Gudi (2004) extended the optimal feed policy of a fed-batch fermentation process proposed in Chiou and Wang (1999) by including (i) systems with multiple feeds, (ii) nonuniform parameterization (the time duration of each interval is not the same), and (iii) path and end-point constraints. In order to demonstrate the effectiveness of the methodology, the same case study as the one from Chiou and Wang (1999) was employed. In addition, the dynamic optimization of simultaneous saccharification and fermentation of starch to lactic acid was studied. The goal was to maximize the final concentration of lactic acid, productivity, and yield. In the case of a single feed approach, DE in combination with the Lagrange-like method, including dynamic penalty functions, provided better results than Pontryagin’s maximum principle. When using the nonuniform control vector parameterization, the optimization was performed twice, first setting the fermentation time as a decision variable, and second, considering it fixed at 80 h. The results of the optimization problem (with 79 decision variables, from which 40 were constant feed rates and 39 were time interval parameters) indicated that nonuniform control vector parameterization and endpoint constraints used in DE are more efficient for end-product concentration and production rate. Good results were also obtained in case of multiple feed optimization.

In the work of Babu et al. (2005), a multioptimization DE version (MODE) was employed to optimize the styrene production by dehydrogenation of ethyl benzene in an adiabatic reactor. Five case studies corresponding to different combinations of objectives were studied based on ethyl benzene feed temperature, pressure, steam over reactant ratio, and initial flow rate of ethyl benzene. Maximization of styrene selectivity (S_ST) and styrene productivity (F_ST) represents the first case, maximization of S_ST and styrene yield (Y_ST) represents the second case study, and maximization of F_ST and Y_ST is the third case study. In the fourth case, the maximization of F_ST and minimization of H₂O productivity are considered, while the last case is represented by maximization of Y_ST, F_ST, and S_ST. The same MODE approach proposed in Babu et al. (2005) was applied in another study for optimization of two styrene reactor configurations (single bed adiabatic operation and stream injected pseudo-isothermal operation) (Gujarathi and Babu 2010a). Five decision variables were considered for the adiabatic operation (feed temperature, pressure, steam over reactant ratio, initial ethyl benzene flow rate, and steam temperature), while for the steam injected reactor, two more are added (steam fraction at the reactor inlet and fraction between the reactor length and location of the injection port). Four optimization cases were considered, including (i) maximization of F_ST and S_ST, (ii) maximization of S_ST and Y_ST, (iii) maximization of F_ST and Y_ST, and (iv) maximization of F_ST, S_ST, and Y_ST. For both configurations, in all four cases, Pareto optimal sets were obtained.

A simple self-adaptive approach (utilizing a co-evolutionary proves running on a fractional time scale with respect to the main process and working similarly to a high-level co-evolutionary strategy) was applied for designing a block decentralized PI controller of an ALSTOM gasifier (Nobakhti and Wang 2008). The control system was required to minimize the integral absolute error of the fuel gas pressure and caloric value even when disturbances appear (a 0.2 bar step reduction in the sink pressure and a sinusoidal disturbance of 0.2 bar amplitude at 0.04 Hz). These requirements were formulated into an optimization problem with 20 design parameters and 26 constraints, with the DE optimization providing near-optimum results, a noticeable improvement over previously reported solutions being observed.

The catalytic industrial ethylene oxide reactor was modeled and optimized using a DE-ANN approach (Lahiri et al. 2008). As a rigorous mathematical model of the considered process (quantifying all the effects of parameters on selectivity) does not exist, the role of the neural network was to act as a model, correlating all the process data and performance variables. DE determined the process input variables (which were also the model input variables) leading to a maximization of yield and selectivity. The considered inputs of the model were cycle gas flow and pressure; reactor temperature; reactor inlet concentration of ethylene, CO₂, oxygen, and ethane; oxygen flow in the reactor; inhibitor flow; and catalyst running hours, while the outputs were ethylene oxide production, selectivity, and CO₂ production. These parameters were chosen based on a set of criteria, which included reduced input number leading to the best prediction, high correlation of input-output, and minimum cross-correlation between inputs and minimum of network complexity. After determining the model, a simple DE version (F=0.5, Cr=0.2, and Np=100) was successfully used to optimize the model inputs corresponding to the set of process operation conditions. The optimized solutions resulted in a significant improvement in the ethylene oxide production rate and catalyst selectivity. The DE optimization program can be integrated with an online, real-time APC application, or the optimum value may be calculated on an off-line computer and further used for optimizing the reactor.

Three DE-based versions were employed to optimize the liquid phase oxidation of p-xylene to purified therephthalic acid in four different cases (Gujarathi and Babu 2009). The scope of this work was to maximize the total feed rate (first objective) and to minimize the concentration of the 4-carboxybenzoic acid as intermediary product (second objective). The first objective plays as a decision variable along with (i) catalyst concentration (CO), (ii) CO and water content in the solvent (W_II2O), (iii) CO, (W_II2O), unconverted reactant (PX), and vent oxygen from the reactor and crystallizer merged into one variable V_O2, and (iv) CO, W_H2O, PX, V_O2, and temperature of the reactor. The DE versions used are represented by MODE (a variant proposed in Babu et al. 2005); a modified MODE approach called MODE-II, in which, at each generation, the population is randomly generated (Babu 2007); and an elitist MODE version (E-MODE), in which the concept of elitism with crowded distance sorting is included. The results showed that E-MODE provided widely spread, equally spaced Pareto fronts, while MODE covered a wide area of Pareto fronts in detriment of nonuniformly distributed solutions.

The same process was also studied by Xu et al. (2013), a self-adaptive DE strategy for solving constrained multiobjective problems proposed in Qian et al. (2012) being used to improve the product yield. A set of conflicting objectives (minimization of combustion loss and maximization of therephthalic acid yield) were considered, based on nine process variables (concentrations of cobalt and manganese catalysts, concentration of bromide promoter, and temperature and residence time of the three reactors). The system model was developed in Aspenplus 2006.5, and the DE-based algorithm (with two versions, Rand/1 and Best/1) was implemented in Matlab, with an interface between the two environments being created to perform the optimization. In order to determine the performance of this approach, a comparison with NSGA-II and fuzzy immune algorithm was realized, with the results indicating that DE is effective in solving the considered problem.

A hybrid strategy combining support vector regression (SVR) and DE was used to model and optimize ultraviolet (UV) transmittance of the monoethylene glycol (MEG) product of a commercial petrochemical plant (Lahiri and Khalfe 2010). The procedure was designed to maximize the UV based on a set of process parameters (reflux ratio and flow, MEG column top and condenser pressure, MEG column control temperature and feed flow, drying column control and bottom temperature, and off-spec glycol reprocessing flow). After the SVR model was determined, DE was employed to optimize the N-dimensional model input in three temperature case limits, which were then verified in an actual plant.

In Yuzgec (2010), the feeding profile of an industrial scale baker’s yeast fermentation is optimized (maximization of the amount of biomass at the end of the process and minimization of ethanol formation during the process). Different DE variants were used: opposition-based DE (ODE), adaptive DE (ADE), and adaptive opposition DE (AODE), which is a modified ODE version with an adaptive jumping rate. As the conditions in the reactor (substrate feeding into the fluidized bed) and the initial conditions (IC) are the main factors influencing productivity and product yield, in order to achieve a compromise between objectives and to optimize the process, four IC were considered: (i) high initial biomass and glucose concentration, (ii) high initial biomass concentration and low glucose, (iii) low initial biomass and high glucose, and (iv) low biomass and glucose concentration. The results obtained for cases (i) and (ii) are presented in Table 4. It was observed that the initial seed values of the DE algorithm do not have an influence on the overall performance and that the Rand-to-best/1/Bin strategy is the best. Among all variants, the AODE approach had the best performance, with its predictions being the closest to the experimental data.

In a study related to the stabilization of feedstock properties in refineries (by combining different types of crude oil), a two-level optimization procedure based on a hybrid algorithm combining Tabu search (TS) and DE was proposed (Bai et al. 2010). As the true boiling point influences the yield and the operational stability of the crude distillation unit, this parameter was used as a main quality index for evaluating the crude oil properties. The main constraints imposed to the system were related to the yield of lights components, trace element concentration, flow rate, and due date. The model used, which is a mixed-integer nonlinear programming model, had two groups of decision variables: the blending sequence of imported crude ranks and the corresponding flow rates. When only blending sequence is considered, the optimization becomes continuous, and when the flow rate is taken into account, then the optimization is combinatorial. Therefore, a two-level optimization was performed and the proposed algorithm (called HIHI) effectively and efficiently determined the scheduling strategy.

In case of batch processes, a very important aspect is related to the zero-wait scheduling problems. In order to solve this problem, Dong and Wang (2012) applied a permutation-based DE (PDE) hybridized with two insert-based methods for local search: remove-insert method and fast complex heuristic (HCF). In order to test the performance of this new approach, three case studies (from the literature) were considered. Compared with GA, TS, PDE, and HCF, the new hybrid versions not only had the best performance but also were also simple to use, as there were no parameters to adapt, except the probability of local search.

In a study related to the optimization of distillation with reactor-side for hydrodesulphurization process of diesel, minimization of the total annual cost, CO₂ emissions, and the amount of sulfur compounds was performed using a multiobjective approach based on DE and Taboo list (Miranda-Galindo et al. 2012). The algorithm, called MODE-TL, was proposed in Sharma and Rangaiah (2010) and was used in combination with Aspen ONE Aspen Plus software.

Other systems where DE is applied in combination with the Aspen software are as follows: (i) production of biodiesel, pretreatment (of cooking oil using H₂SO₄), and transesterification (based on alkali catalyst) (Sharma and Rangaiah 2013a). As in the previous study (Miranda-Galindo et al. 2012), the DE-based variant was represented by the MODE-TL; (ii) simultaneous reaction and separation in a reactive distillation column (Safe et al. 2013). The role of DE was to fit the collected data of the optimal surface of the response surface methodology (RSM); (iii) design and optimization of an ethanol dehydration process (Vázquez-Ojeda et al. 2013). Two DE optimized sequences were used: conventional arrangement (CSS) and alternative arrangement employing liquid-liquid extraction (OSS-I); (iv) sweetening the natural gas process (Ortegaa et al. 2013); (v) p-xylene oxidation process (Fan and Yan 2014); (vi) optimal design of biorefineries (Geraili et al. 2014); and (vii) reactive distillation column coupled with pressure-swing distillation, membrane-assisted reactive distillation process, reactive dividing wall column coupled with pressure-swing distillation, and membrane-assisted reactive dividing wall column (Holtbruegge et al. 2015).

Nian et al. (2013) used a combination of DE with group search optimization for optimizing an ethylene-cracking furnace. The system was described by a model determined using a cracking reaction software named COILSIMID. The considered inputs were the mole percentages of n-alkanes, iso-alkane, cycloalkane, and aromatics in the feedstock; feed flow rate; coil output temperature (COT); and steam hydrocarbon ratio (SHR), while the output variable was represented by the yield of ethylene and propylene. The optimization procedure was applied for three kinds of feedstock, with the scope of maximizing the model output by tuning COT and SHR, when the properties of feedstock changed. After applying the algorithm (called DEGSO), the yield of ethylene and propylene increased remarkably, a fact that indicates that the method was able to accurately find the optimum.

In order to optimize the retrofitting of heat exchanger network in industrial plants, a one-step approach based on integrated DE (IDE) was used (Zhang and Rangaiah 2013). This algorithm was first proposed in Zhang et al. (2011a), and Zhang and Rangaiah modified it and applied it to three case studies represented by (i) part of a petrochemical process including a distillation column and an exothermic reactor, (ii) crude oil preheat train of a petroleum refinery with six hot and one cold streams, and (iii) preheating crude in an atmospheric crude unit with nine hot and three cold streams. In a single step, both continuous and discrete variables were optimized (structure, heat exchanger areas, and split ratios). The scope of this optimization was represented by the determination of a minimum operation cost and annualized investment cost.

In the work of Gujarathi and Babu (2013), the industrial low-density polyethylene tubular reactor was optimized using three DE modified versions represented by MODE, MODE III (Gujarathi and Babu 2012), and hybrid MODE (Gujarathi and Babu 2010b). Two distinct optimization cases were considered: (i) maximization of conversion and of the weighted average value of the undesirable side products and (ii) independent maximization on conversion and minimization of side products (methyl, vinyl, and vinylidene side chain content). The results showed that both hybrid MODE and MODE III variants converged to the same Pareto front.

In a study of the effect of carburizing and oxidizing flames on surface roughness in turning of aluminum metal matrix composite, the optimization was performed using a multiobjective DE approach (Sudheer and Pavan 2014). The procedure was applied for determining the values of speed, feed, and depth of cut that lead to a minimization of surface roughness and maximization of material removal rate.

In combination with the sequential design method, a multiobjective DE variant was applied to solve the problem of distillation column design for a five-compound mixture (Errico et al. 2014). In addition, the Aspen Economic Analyzer was used to assess the capital cost. It was observed that when the decision variables in the reference simple column is high, initializing DE with the sequential design method solutions can produce a final result in a reasonable amount of time.

4.1 Statistical/stochastic models

Statistical models are very important in chemical engineering, as they can be applied for systems in which (i) the known information is not complete and the design of a deterministic model is very difficult or impossible or (ii) the existence of complex relations between process parameters translates into very complex models. A series of examples are listed and discussed below.

The optimum values of pyrolysis time and heating rate of a pyrolysis process of biomass were determined by Babu and Chaurasia (2003) using a classic version of DE with predefined control parameter values (Cr=0.7, F=0.5, and Np=10). These values were then used as inputs to the coupled ordinary differential equations to find the optimal parameters based on the Runge-Kutta fourth-order method. The results indicated that as the heating rate is increased, the pyrolysis time first decreases, reaches an optimum, and then increases.

In Garlapati and Banerjee (2010), the input space of a RSM model of an extracellular lipolytic enzyme production based on Rhizopusoryzae NRRL 3562 was determined using a classic variant of DE. The most important parameters of the study were represented by temperature, liquid-to-solid ratio, pH, and incubation time, with the scope of the optimization being represented by the maximization of lipase activity. The results obtained with DE were experimentally validated, with a maximum of 96.52 U/gds of lipase activity being observed at 35.59°C, 1.5 liquid-to-solid ratio, a pH of 5.28, and 4.83 days of incubation.

The same RSM approach was used to model the reactive extraction of glycolic acid from aqueous solutions using tri-n-octylamine (Datta and Kumar 2011). After determining the influence of different process parameters (initial concentration of glycolic acid in the aqueous phase, initial amine composition in the organic phase, and modifier composition) on the degree of extraction, the process was optimized using a classic DE version with constant control parameters (F=0.8, Cr=0.7, and Np=30). Through simulation, a 73.13% degree of extraction was determined, with the difference between prediction and experimental verification being 5.7%.

As can be observed, in case of statistical approaches, DE was scarcely applied. To the authors’ knowledge, in the last 5 years, no other research combining DE and statistical models was published in the area of chemical engineering.

4.2 DE optimization applied to ANNs and SVMs

When other approaches fail or are difficult to use, the artificial intelligence tools with modeling behavior come as an appropriate alternative. In this context, the most used modeling approaches are represented by SVMs and ANNs. When combining an ANN with an optimization procedure, two aspects can be considered: what is optimized and at what level? The combination of EA with ANNs (also known as neuroevolution) can be applied at roughly four different levels: connection weights, architecture, learning rules, and node behavior (Montana et al. 2009).

4.2.2 Artificial neural networks

In the work of Lahiri and Ghanta (2009b), an ANN model for the hold-up in solid-liquid slurry flow in a pipeline is optimized using a classic variant of DE. Nine inputs (pipe diameter, particle diameter, solid concentration and density, liquid density and viscosity, maximum packing concentration) and one output (hold-up ratio) were considered, and the objective function of the optimization was to obtain minimum average absolute relative error on the test set. From the multitude of neural network parameters, only five were chosen for optimization: number of nodes in the hidden layer, input and output layer activation function, learning rate, and training algorithm. The number of activation functions the layer can have was five (log-sigmoid, tan-sigmoid, radial basis, linear transfer, and triangular basis), and the training algorithm can be one of the following: batch gradient decent, resilient back propagation, conjugate gradient (Fletcher-Reeves update, Polka-Ribiere update, Powell-Beale restarts), quasi-Newton algorithm (BFGS, Levenberg-Marquardt [LM]), and one-step secant algorithm. The comparison between the simple ANN model determined after 10,000 runs and the hybrid ANN-DE model (after 2000 runs) indicated that the proposed methodology, ANN-DE, is more suited to the problem at hand.

The simultaneous topological and structural optimization of a neural model for the styrene polymerization was performed using three methods: a six-step optimization methodology based on a systematized trial and error applied for ANN parameters (OMP method), a classic DE variant, and a GA approach (Curteanu et al. 2010). A model able to predict monomer conversion, numerical average molecular weight, and gravimetrical average molecular weight was created using initiator concentration, temperature, and reaction time as input variables. The considered parameters for GA and DE optimization were represented by the number of hidden layers, number of neurons in each hidden layer, the weights between network’s neurons, and biases of each neuron. In addition, DE also optimized the activation function of each neuron. On the other hand, the OMP approach determined the number of hidden layers and neurons in each layer, activation functions for the neurons in the output layer, learning rate, and momentum term. The results showed that the three methods have a similar performance, as can be observed from Figure 3.

Figure 3:

Some values for conversion: desired vs. experimental conversion in the testing phase through the neural networks designed with the three methods (OMP, GA, DE).

The oxygen mass transfer in the presence of oxygen vectors of an aerobic biosynthesis system was modeled using a feed-forward multilayer perceptron ANN with parameters optimized using two DE variants; (i) classic and (ii) self-adaptive approach (Dragoi et al. 2011). The process parameters based on which the ANN was created and optimized were represented by viscosity, superficial speed of air, specific power, and oxygen-vector volumetric fraction. The model parameters considered for optimization were the number of hidden layers and neurons in each hidden layer, weights, biases, and activation functions for each neuron. For each specific goal (prediction or classification), different neural models were determined. In the prediction case, a stack approach (stacked neural networks) was also applied because the individual neural network (generated with both classic and self-adaptive versions) did not provide acceptable results. In the classification case (where the classes were generated using a discretization method applied to the mass transfer coefficient), different DE variants were tested, with the best results being obtained with a simple DE version (mean squared error [MSE]=0.096 in the testing phase).

The prediction of the liquid crystalline property of some organic compounds (bis aromatic and azo aromatic types) was performed using ANNs optimized with two different DE self-adaptive versions (Dragoi et al. 2012a). One version is proposed in Brest et al. (2006) and is called jDE. The other one (SADE-NN) is a simple self-adaptive approach. The process parameters considered were structural and molecular descriptors: the length of the rigid core, the length of the flexible core, molecular weight, and an asymmetry factor (represented by the ratio of diameter to the total length of molecule). Different percentages for splitting the available data into training and testing were tested, with the results obtained indicating that special attention must be given to this aspect. In addition, a comparison with some previous studies was performed, and the conclusion drawn was that when using the right technique, ANNs can reach the same level of performance as specific classifiers algorithms. The difference in performance obtained between SADE-NN and jDE pointed out that, for the considered case study, the simplest self-adaptive approach was better. The maximum average percentage of correct answers was 87.09% for the training data and 85.15% for the testing data. This means that the error of prediction is around 15%, which is considered an acceptable value.

ANNs and a modified DE version were applied to model the oxygen transfer when n-dodecane is added in aerobic fermentation systems of bacteria (Propionibacterium shermanii) and yeasts (Saccharomyces cerevisiae) (Dragoi et al. 2013a). The process parameters considered for model input were biomass concentration, superficial air velocity, specific power, and oxygen-vector volumetric fraction, and the output was the mass transfer coefficient. The algorithm, called SADE-NN-1, represents an improved version of SADE-NN proposed in Dragoi et al. (2012a), the alteration consisting in the introduction of the following elements: (i) OBL to improve initialization, (ii) a new mutation strategy in order to improve the offspring generation, and (iii) increasing the number of activation functions that a neuron could have during the evolution (linear, hard limit, bipolar sigmoid, logistic sigmoid, tangent sigmoid, sinus, radial basis, and triangular basis functions). Simple ANNs, with one hidden layer and small number of intermediate neurons, accurately model the process considered as case study. After the best model was determined (using different combinations of mutation strategies and crossover), a sensitivity analysis was performed. As a result, the process parameters having the most influence on the system (model) were determined. A good correlation between model and process is obtained, which translates in good prediction performance as the comparison between the ANN and a phenomenological model (solved with a multiregression approach) showed.

The freeze-drying of pharmaceutical products was studied from multiple points of view using a hybrid combination of DE with ANNs and backpropagation as a local search procedure (Dragoi et al. 2012b). Using the model of the freeze-drying process and given the values of the operating conditions (temperature of the heating shelf and pressure in the drying chamber), both the temperature and the residual ice content in the product vs. time can be determined offline. This makes possible to understand if the maximum temperature allowed by the product is trespassed and when the sublimation is complete, thus providing a valuable tool for recipe design and optimization. The algorithm called SADE-NN-2 (Dragoi et al. 2012b), which is an improvement of SADE-NN-1 (Dragoi et al. 2013a), was specially designed to work with the considered process specifics (freeze-drying process). The workflow of the application, compared with the general approach, has some additional steps (1, 2, and 3 from Figure 4), in accordance with the complexity of the process and the final goals considered.

Figure 4:

Workflow of the SADE-NN-2 approach designed for the freeze-drying process.

In the monitoring case, based on the system’s state, predictions for specific parameters at a future time were performed. The measurement of product temperature is used as input variable of the ANN in order to provide an inline estimation of the state of the product (temperature and residual amount of ice). In the modeling case, taking into account that the pressure and temperature remain unchanged, the algorithms predict the duration of the primary phase and maximum temperature product. In this manner, the chemical engineer can avoid the apparition of specific conditions leading to changes in the product properties and ensure final product quality. In order to generate the dynamic of the system, a recurrent ANN was developed, with different combinations of delayed inputs and specific parameters of the process being tested. The best results were obtained with a set of nine inputs represented by time; pressure chamber; fluid temperature; product temperature at the interface of sublimation delayed one, two, and three times; and dried layer thickness delayed one, two, or three times. As this process is slow, special attention was given to the time interval considered for delay, a value of 600 s (as a fraction of the time constant of the process) being used.

A comparison between the predictions (in both monitoring and modeling cases) with a complex phenomenological model indicated that there is little difference between the two. On the other hand, the phenomenological model is very complex and requires many computational resources, which makes it undesirable in control systems where speed and low-resource consumption are critical. As performance is crucial in the modeling and monitoring phases of free-drying process, the ANN determined was further tested, and its robustness and flexibility were verified using various process settings (Dragoi et al. 2013b).

The SADE-NN-2 (Dragoi et al. 2012b) algorithm was also employed for the prediction of partition coefficients of guanidine hydrochloride in a poly(ethylene glycol) (PEG) 4000/phosphate/guanidine hydrochloride/water system (Pirdashti et al. 2015). The process parameters considered were the PEG/phase forming salt ration, guanidine hydrochloride concentrations, and pH. The aqueous two-phase system was studied using this approach because the lack of knowledge related to the partitioning equilibrium of macromolecules translated into a lack of comprehensive theory able to predict the data. The selected model had one hidden layer with 19 neurons and predicted the specific values with an average relative error <1.4%.

Another process studied with SADE-NN-2 was represented by the corrosion resistance of TiMo alloys in acidic artificial saliva with NaF and/or caffeine, at 37°C (Mareci et al. 2015). The polarization resistance of the TiMo alloys was modeled based on the immersion time, caffeine concentration, NaF concentration, type of alloy (Ti content), and solution pH. Based on the considered process characteristics, the optimal model parameters that can efficiently perform predictions with specific values were determined. The initial test showed that the use of single ANNs does not provide acceptable solutions, with the most important aspect influencing these results being the inability of individual ANNs to efficiently cover the entire search space and the significant differences in the behavior of the system at low and high pH. In order to solve this problem, a series of ANNs, organized in stacks (Figure 5), were used in two cases: low pH with values of 3, 4, and high pH with values of 5, 6, 7, and 8. For the low pH, the absolute relative error for the testing values was 14.3%, while for the high pH, the absolute relative error was 10.47%.

Figure 5:

The general structure of the ANN stack employed for modeling the corrosion resistance of TiMo alloys.

The modeling of the depollution of some gaseous streams containing n-hexane was performed by our group using a neuroevolutive technique, based on a DE modified self-adaptive version (Curteanu et al. 2014). Taking into account the type of absorbent, contact type, temperature, bed length, and initial concentration of the volatile compounds, predictions of the report between the outlet concentration at an arbitrary time and the inlet concentration were generated. The role of DE was to simultaneously optimize the topology and internal parameters of the neural network. DE was hybridized with a local search procedure based on two algorithms (BackPropagation and Random Search), with their role being to improve the best solution found in each generation. The results obtained were better than those provided by nonhybridized versions, a fact that indicated that the improvements in the algorithm were correlated to the performance enhancement. In addition, a comparison with a classic model (Yoon Nelson model) was performed, with the ANN based model having superior performances. The DE approach had a MSE for the training data of 0.002565, while the classic model had a MSE of 0.007680.

The examples previously described show that DE in combination with ANNs was widely applied in the area of chemical engineering for modeling, predicting and sometimes optimizing multiple processes and aspects of processes (hold-up and pressure drop in slurry pipes, styrene polymerization, oxygen mass transfer, freeze drying, corrosion resistance of alloys, and depollution of gaseous streams). On the other hand, in combination with SVM, DE was scarcely applied to solve chemical engineering problems. One of the aspects that can explain this difference consists of the fact that ANNs (in simple or in stacked versions, stand-alone, or hybridized with other approaches) are more used than SVM.

4.3 DE optimization applied to deterministic models

In combination with the orthogonal collocation method, DE was applied for the estimation of the heat transfer parameters (effective radial thermal conductivity of the bed and the effective wall to bed heat transfer coefficient) of a gas-liquid co-current down flow through trickle bed reactors (Babu and Sastry 1999). A comparison with the radial temperature profile method (RTP) based on Powell’s method for optimization indicated that DE is a powerful and less expensive computational alternative to the classic approaches. In order to converge (irrespective of the initial population), it required only 10 generations, while RTP (which needs an initial guess relative near to the global optimum) took 32 iterations.

Chiou and Wang (2001) applied a hybrid DE algorithm for parameter estimation of an unstructured kinetic model based on the Monod model. The experimental data were gathered from a Biostat ED bioreactor with E. coli HB101 containing plasmid pBR329. The optimal parameters of the kinetic model (cell growth, substrate consumption, and product formation) were successfully determined in 5000 generations with a set of fixed control parameters (Cr=0.5 and Np=5), a population diversity of 0.05, and a tolerance for gene diversity of 0.05. A model variation of ±50% for the initial concentration of glucose fitted the experimental data, and therefore, the model parameters were in the optimal area.

Moonchai et al. (2005) studied the effect of pH and temperature on the bacteriocin production in L. lactic C7. For a set of five variables (cell growth, lactic acid production, substrate consumption, and bacteriocin production), simple mathematical models were developed in order to describe and predict their kinetic behavior. DE was then used to estimate the model parameters, with the minimization of a weighted absolute error being considered as an objective function. The comparison between simulation and experimental data showed a good agreement, a fact that indicates that DE was able to determine optimal values for the considered case study.

Efficiently removing acid gas impurities from gas mixture is an important aspect in natural gas processing and hydrogen purification, and therefore, DE was employed for parameter estimation of modified Clegg-Pitzer equation used for designing a sour-gas treating plant with alkanolamine solvents (Anil and Kundu 2006). In order to determine their efficiency at parameter estimation of the vapor-liquid equilibrium of (CO₂+AMP+H₂O) system, a comparison between DE, simulated annealing (SA), and LM was performed, with the strategy DE/rand/1/bin providing the best results.

In Mariani et al. (2008), the apparent thermal diffusivity of banana (nanicao variety) was determined by applying a simple DE version to an inverse problem based on a mathematical model, considering the effects (at the surface of the fruit) of shrinkage and convective heat transfer. The parameters of the two functions (one dependent on moisture content and one dependent on moisture content and temperature) were determined by modeling the apparent thermal diffusivity. To perform the required predictions, the transient temperature taken at specific moments of time was considered as decision variable. The comparison between predicted data and experimental profiles of temperature at the thermal center showed no significant differences, and therefore, the determined parameters of the two functions can be considered optimal.

A new combination of DE with the vaccination principle used in immune theory was applied for the kinetic model parameter estimation of two processes: low-temperature SO₂ oxidation with Cs-Rb-V sulfuric acid catalyst (case study 1) and diesel oil catalytic cracking reaction (case study 2) (Wu et al. 2008). In the first case, the objective was to increase the SO₂ rate conversion. In order to determine the performance of the new method, called immune theory DE, a comparison with the classic DE version was performed. For both case studies, 60% of the solutions obtained with the DE based variant were better, with the near-optimal solution being determined much faster.

A hybrid variant of DE in which the individuals were arranged hierarchically in a nearly regular tree was applied to optimize the kinetic models of two processes: (i) pyrolysis and dehydrogenation of benzene and (ii) supercritical water oxidation (Shi and Zhong 2008). In the first case, the reaction rate constants were estimated, while in the second case, a set of parameters (order of reaction for dichlorophenol, preexponential factor, and apparent activation energy) were considered for optimization. A comparison with the classic DE version and PSO showed that the new hybrid approach was best suited to the considered processes as it provided the best results.

In the work of Hu and Yan (2009b), an adaptive DE version was tested on a series of noisy and no-noisy benchmark functions and on a process of supercritical water oxidation. For the considered process, the goal of the optimization was represented by the minimization of the error between the 2-clorphenol conversion predicted with the model and experimental data. The parameters taken into account were represented by optimal Arrhenius parameters and the reaction order for 2-clorphenol, O₂, and H₂O. The results obtained with the adaptive DE version (in which the control parameters were computed based on a specific relation dependent on the current generation) indicated an improvement over other reported solutions.

In the study of Liu and Wang (2009), the HDE version proposed by Chiou and Wang (1999) was used for parameter estimation of two bioreactors. The algorithm, called hybrid DE with geometric mutation (HDE-GM), contains a series of modifications including three additional operations (restriction, acceleration, and migration) and a geometric mean mutation strategy. The first case study is represented by a three-step biochemical pathway described by a set of eight ordinary differential equations having 36 parameters. The search was divided into two classes (Hill coefficients and other), and distinctively from the base HDE variant, HDE-GM was able to generate optimal solutions, the highest error obtained being 7%. The second case study was represented by the production of ethanol based on the fermentation of Saccharomyces distaticus LORRE 316. Using some sets of experimental data, 19 parameters (including yield coefficients) were estimated efficiently, resulting in a pruned kinetic model with a more compact formulation. In another study (Liu and Wang 2010), the same HDE-GM was applied for the same problems as in Liu and Wang (2009), along with a series of unconstrained and constrained benchmark optimization problems.

The stability analysis of separation processes and the phase equilibrium calculation are challenging problems due to the highly nonlinear characteristics of the thermodynamic models used. Bonilla-Petriciolet et al. (2010a) performed a series of parameter tuning simulation of DE and TS on a set of reactive separation processes (the reaction of butyl acetate production at 298.15 K and 1 atm, methyl tert-butyl ether reaction system with inert at 10 atm and 373.15 K, and the reactive system for tert-amyl methyl ether synthesis at 335 K and 1.5 atm). Then, the algorithms were applied on two examples represented by (i) a hypothetical multicomponent reactive mixture undergoing three reactions and (ii) esterification reaction of ethanol and acetic acid to form ethyl acetate and water. The results showed that although DE is better than TS, it requires more CPU time and function evaluations.

In another study performed by Bonilla-Petriciolet et al. (2010b), the parameter estimation of vapor-liquid equilibrium models was performed using (i) DE with Tabu List (DETL), a version proposed by Mekapati and Gade (2010); (ii) SA; (iii) GA; (iv) classic DE; and (v) PSO. After generalizing the relative performance of the algorithms by applying them on a set of benchmark functions, three vapor liquid equilibrium (VLE) systems in the classic least squares variable formulation (tert butanol+1 butanol; water+1,2 ethanediol; benzene+hexafluorobenzene) and one in the error-in-variable formulation (benzene+hexafluorobenzene) were considered. On both formulations, DE and DETL proved to be the best, with these algorithms being viable alternatives to the parameter optimization of VLE systems.

The same group of authors solved the problem of constrained and unconstrained Gibbs free energy minimization in a reactive system by applying GA and DETL (Bonilla-Petriciolet et al. 2011). The optimized reactive system was represented by a set of eight systems modeled with one of the following thermodynamic models: nonrandom two liquid model (NRTL) and ideal gas, Wilson model and ideal gas, universal quasi-chemical model, Margules solution model, or NRTL. The scope was to determine the parameters of these models that lead to a global minimum. A comparison between the three algorithms pointed out that DETL and SA are better than GA.

The amount of adjustable parameters of the Flory-Huggins models of the asphaltene precipitations was calculated using DE and modified Esmailzadeh-Roshanfeker, Peng-Robinson, and Soave-Redlich-Kwong equations of state (Nourbakhsh et al. 2011). In this case, the most important parameters considered were interaction, solvent rations, molecular weights of asphaltene and solvent, and solubility. A comparison between the optimized model and experimental data shows the concordance between the two sets of data, with the model having a good potential for predicting the asphaltene precipitation.

An IDE version was applied for parameter estimation of the VLE model (Zhang et al. 2011a). The algorithm (which introduces an adaptation of mutation parameter, crossover parameter, Tabu list and Tabu check, a novel stopping criterion, and a local optimizer) was also tested on a set of benchmark functions having two to 20 variables and multiple local optima. In addition, a comparison with DETL (Mekapati and Gade 2010) was performed. After the superiority of IDE was established, 20 VLE problems (from which 10 are based on least squared approach and 10 were based on error-in-variable) were solved. In addition, a comparison with SA, PSO, a classic DE variant, a hybrid version (DETL), and Branch and Reduce Optimization Navigator (BARON) was performed, with IDE being the best. Only BARON provided comparable results.

Zhang et al. (2011b) applied three bioinspired algorithms, unified bare-bones PSO, IDE, and IDE without Tabu list and radius (IDE_N), respectively, to perform and optimize phase equilibrium calculations, phase stability, and chemical equilibrium problems of 16 different systems. The results indicated that the IDE version performed best. In addition, for the chemical equilibrium problems, a comparison of IDE with SA, GA, and DETL points out that IDE has the best performance.

In a review about the use of different global optimization methods for phase equilibrium and calculations, different approaches, including DE, are presented (Zhang et al. 2011c). In addition, a set of specific problems solved with DE and published prior to year 2011 are detailed.

The kinetic parameters of model of three independent parallel reactions of sugarcane pyrolysis (based on thermogravimetric analysis) are optimized with DE and then a series of relations (activation energy, preexponential factor of Arrhenius equation, and mass fraction of each subcomponent of biomass) are computed and compared with data from literature (Santos et al. 2012). A sensitivity analysis of the models was performed with respect to the perturbations of 1% of the model’s inputs. The results indicated that the most influential parameter on bagasse conversion is represented by the activation energies.

Kumar et al. (2011) studied the determination of optimal stoichiometric and equilibrium constants of the reactive extraction of propionic acid using tri-n-butyl phosphate dissolved in n-decane and 1-decanol. The main variables influencing the equilibrium extraction characteristics are represented by (i) the nature of the extracted acid, (ii) concentration, and (iii) type of diluent. Therefore, the main parameters determined were distribution coefficient, degree of extraction, loading ratio, and equilibrium complexation constant. The results obtained pointed out that the extraction efficiency is proportionally dependent on the concentration of the active diluent. In addition, the increase in temperature affects the extraction equilibrium, with the enthalpy and entropy being -12.47 kJ mol^-1 and -32.42 J mol^-1 K^-1, respectively.

The solubility of CO₂ in ionic liquid 1-alkyl-3 methylimidazolium bis (trifluoromethyl-sulfonyl) imide was modeled using a set of phenomenological rules (Peng-Robison equation of state, Wong-Sandler mixing rule, and van Laar model for excess Gibbs free energy). After that, the model parameters (binary interaction and activity coefficients) were optimized with DE algorithm in order to obtain better predictions (Yazdizadeh et al. 2011). The optimized parameters were temperature dependent, and a comparison between the considered model and one based on Peng-Robison equation of state and Wong-Sandler mixing rule showed that the former is better.

Chakraborty et al. (2012) used two variants of DE-based algorithm proposed in Das et al. (2009) for parameter optimization of a mechanicist model of the proton membrane fuel cell (PEMFC). The idea was to determine the model’s optimal seven parameters so that it best fits the PEMFC stack. The algorithm, called DEGL (DE with global and local neighborhoods), had a self-adaptive weight factor. The results showed that DEGL with exponential crossover is better than the one with binomial crossover, standard versions of DE and GA, from three points of view: (i) quality of solution when an imposed computational cost is given, (ii) computational cost when a given performance is required, and (iii) frequency with which an acceptable solution is found.

In Sharma and Rangaiah (2013b), a complex MODE with a new termination criterion (using nondominated solutions and several performance metrics) is used to solve three process applications represented by alkylation, Williams-Otto, and a fermentation process. The algorithm (called I-MODE), which is an improvement of DETL proposed by Mekapati and Gade (2010), is used to maximize the productivity by determining the optimal model parameters. In the case of the alkylation process (where products are combined with refined petroleum products in order to enhance the octane number), the objective was to maximize the profit and the octane number and to minimize the isobutane recycling. The decision variables considered were isobutane makeup, olefins feed, and fresh feed. The new stop criteria forced the algorithm to stop earlier than the maximum allowed generation, the performance indexes indicating that optimal solutions for engineering application were obtained. In the case of the Williams-Otto process (with both reaction and separation sections), the objective was to maximize the net present worth and payback before tax and minimize the payback period. The decision variables used were the feed with reactants. Although the solutions obtained converged near the known pareto-optimal fronts, the distribution of solutions was not good; for the Williams-Otto process, a higher number of generations is required. In the fermentation case, a three-stage fermentation process integrated with cell recycling was considered. The objectives were to maximize ethanol productivity and overall glucose conversion based on the dilution rate, glucose concentration in feed, and bleed ration for each stage. The results obtained showed that unnecessary computations were eliminated, with the quality of solutions remaining at a high level.

The parameter determination of the kinetic model of the cassava alcoholic fermentation hydrolyzed in a batch reactor was performed by a series of four stochastic optimization algorithms: artificial bee colony (ABC), DE, PSO, and SA (Da Ros et al. 2013). From the kinetic model (composed of three ordinary differential equations describing the variation of cell, substrate, product concentration, and reaction time), 10 parameters were chosen for optimization. In addition, in the optimization procedure, a set of 12 ICs were included, with the final number of parameters optimized being 22. All the algorithm’s parameters were set in such a manner so that the objective function evaluation was equal to 500,000. The ABC prediction had a poor performance, despite its effectiveness on a set of benchmark functions. The SA performance was dispersive, as the initial random selection of solution has a very high impact on the algorithm’s performance. PSO had a satisfactory performance, but the best results were obtained by the DE variant with the mutation based on the best-so-far solution.

In combination with the Gauss-Newton method with orthogonal collocation (for computing the sensitive matrix and state vector), DE was applied on a set of six test problems for parameter estimation (first-order reversible chain reaction, first-order irreversible chain reaction, catalytic cracking of gas oil, Bellman’s problem, methanol-hydrocarbon process, and Lotka-Volterra problem) and then used for estimating the kinetic parameters of a batch fermenter for penicillin production (Zhao et al. 2013). Based on the parameters determined with the proposed algorithm (called HDE), the biomass concentration, product, substrate, and reactor volume were computed with high efficiency.

In order to model the chlorine decay in bulk water, a phenomenological model with DE optimized parameters (reactive constituents, rate constants for reactions of the two forms of the free chlorine with the reactive constituents, corresponding rates constants at 20°C, and Arrhenius rations of each reaction’s activation energy to the ideal gas constant) was employed (Liu et al. 2014). This model incorporated the effect of pH and temperature so that it can be applicable to everyday situations like the chlorine decay when heating tap water. For the process model, two cases were considered: when two reactive constituents are assumed and when three reactive constituents are assumed. The simulation results indicated that the two-site reactive model having 10 parameters is the most suited for the process approached.

When DE is applied to determine the optimal parameters of the deterministic models, the multitude of good results indicates the flexibility and performance of this optimizer. The aspects considered for optimization are generally represented by heat transfer parameters, parameters of kinetic models and of vapor-liquid equilibrium models, phase equilibrium, phase stability, and chemical equilibrium. On the other hand, the systems taken into account vary considerably, ranging from alcoholic fermentation and bioreactor systems to sour gas treatment plants, pyrolysis dehydrogenation, and supercritical water oxidation.