Rev Chem Eng 2016; 32(2): 149–180
Elena Niculina Dragoi and Silvia Curteanu* The use of differential evolution algorithm for solving chemical engineering problems
DOI 10.1515/revce-2015-0042 become more available, and the industry is on the lookout Received July 10, 2015; accepted December 11, 2015; previously for increased productivity and minimization of by-product published online February 8, 2016 formation (Gujarathi and Babu 2009). In addition, there are many problems that involve a simultaneous optimiza- Abstract: Differential evolution (DE), belonging to the tion of many competing specifications and constraints, evolutionary algorithm class, is a simple and power- which are difficult to solve without powerful optimization ful optimizer with great potential for solving different techniques (Tan et al. 2008). In this context, researchers types of synthetic and real-life problems. Optimization have focused on finding improved approaches for opti- is an important aspect in the chemical engineering area, mizing different aspects of real-world problems, includ- especially when striving to obtain the best results with ing the ones encountered in the chemical engineering a minimum of consumed resources and a minimum of area. Optimizing real-world problems supposes two steps: additional by-products. From the optimization point of (i) constructing the model and (ii) solving the optimiza- view, DE seems to be an attractive approach for many tion model (Levy et al. 2009). researchers who are trying to improve existing systems or The model (which is usually represented by a math- to design new ones. In this context, here, a review of the ematical description of the system under consideration) most important approaches applying different versions has three main components: (i) variables, (ii) constraints, of DE (simple, modified, or hybridized) for solving spe- and (iii) objective function (Levy et al. 2009). The scope cific chemical engineering problems is realized. Based on of optimization is to determine the values of the system the idea that optimization can be performed at different parameters (variables) for which the overall performance levels, two distinct cases were considered – process and is the best (measured using an objective function or a model optimization. In both cases, there are a multitude fitness function) under some given conditions (Das and of problems solved, from different points of view and with Suganthan 2011). The variables of the model correspond various parameters, this large area of successful applica- to the specific characteristics of the studied system, rep- tions indicating the flexibility and performance of DE. resenting components that can be changed to create Keywords: chemical engineering processes; differential different possibilities. The objective function and the evolution; modeling; neural networks; optimization. constraints values can be obtained from (i) analytically known formulae (phenomenological models), (ii) the outcome of a computational process (such as modeling 1 Introduction approaches) and (iii) measurements of the actual process (Snyman 2005). Optimization is an important aspect, the efficient explo- Concerning the problem of solving the optimization ration of systems workflow and reduction of consumed model, different approaches can be encountered. The resources being always desirable. As technology evolves, traditional optimization algorithms are based on gradi- highly sophisticated and well-controlled chemical plants ent methods. These methods have a high probability of getting trapped in local optima (especially in high nonlin- ear cases) and do not guarantee identification of optimum *Corresponding author: Silvia Curteanu, Faculty of Chemical Engineering and Environmental Protection, Department of Chemical (Angira and Babu 2006). As nonlinearity is an important Engineering/Applied Informatics, “Gheorghe Asachi” Technical aspect (as it is introduced by equipment design relations, University Iasi, Bd. Prof.dr.doc. Dimitrie Mangeron, No. 71A, equilibrium relations, or by combined heat and mass bal- 700050, Iasi, Romania, e-mail: [email protected] ances, among other things) (Angira and Babu 2006), new Elena Niculina Dragoi: Faculty of Chemical Engineering and techniques that can deal well with these problems have Environmental Protection, Department of Chemical Engineering/ Applied Informatics, “Gheorghe Asachi” Technical University Iasi, been proposed and tested (Bastos-Filho et al. 2011, Precup Bd. Prof.dr.doc. Dimitrie Mangeron, No. 71A, 700050, Iasi, Romania. et al. 2012, Mavrovouniotis and Yang 2013, Yacoub et al. http://orcid.org/0000-0001-5006-000X 2014). In recent years, population-based optimization 150 E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering procedures (with biological inspiration) have been dis- 5 refers to some practical aspects, being a short over- tinguished as powerful approaches that can be efficiently view that connects the DE algorithm (taking into account applied for solving different optimization problems from characteristics and possibilities of improvement) with various areas. This group includes algorithms like genetic its application for model and process optimization. The algorithm (GA), differential evolution (DE), ant colony opti- last section concludes the paper. The chemical engineer- mization, or particle swarm optimization (PSO). Among ing applications are selected to emphasize all the above these algorithms, DE is relatively new, being proposed in aspects. 1995 (Storn and Price 1995). It is a powerful approach, in many theoretical and practical situations outperforming its counterparts. It is a simple, new-generation evolution- ary algorithm (EA), developed for solving optimization 2 Differential evolution problems over continuous domains (Abbass 2002, Hu and Yan 2009a). Due to its characteristics – simplicity, few The DE inspiration source is represented by the Darwinian control parameters, and ability to handle nondifferenti- principle of evolution, in which only the fittest individuals able, nonlinear, and multimodal objective functions – it survive to the next generations. It was proposed in 1995, gradually became more popular, in the field of chemical and over the years, as its strong and weak points were engineering being used for a large area of applications identified, it underwent a series of improvements and (Pant et al. 2009b). hybridizations. However, the base elements remained the In this article, a review of the most relevant works same. applying classic or modified DE versions, in stand-alone As with every EA, DE starts with an initial pool of applications or in combination with other specific soft- potential solutions that are evolved (using principles like ware or algorithms, is performed. The applications dis- mutation, crossover, and selection) until a stop criterion is cussed in the next sections are dated after 2009, but DE reached. The mutation and selection steps (which are sto- has been used to solve chemical engineering problems chastic processes) are the main mechanism used for new from the early days. A few examples are given in Table 1. individual generation, while selection is (as its name sug- The paper is organized as follows: Section 2 presents gests) the step in which a competition for survival takes the main characteristics of the DE algorithm, its steps, the place (Greenwood and Tyrrell 2006). stop criteria usually employed, and the principal mecha- These DE operations have similar terminology with nisms for improving its performance. In Section 3, specific the ones from other EAs, but their application, role, and aspects related to the application of DE for the problem motivation are different. Another difference between DE of process optimization are discussed. A clear distinction and other EAs (such as GA) consists in the parameter between the use of simple variants (also known as classic encoding. DE encodes all the parameters using a floating approaches) and the hybrid versions is made. Section 4 point, while classic GA uses a binary encoding scheme. tackles the problem of model optimization, and Section This real value encoding and the fact that DE parameters
Table 1: A few examples of DE applications in chemical engineering prior to 2009.
Domain References Application
Reactors and fermentation Chiou and Wang (2001) Estimation of Monod model parameters of a bioreactor Wang et al. (2001) Estimation of kinetic parameters of the ethanol fermentation process Chiou and Wang (1999) Feed batch fermentation Wang and Cheng (1999) Feed batch fermentation Kapadi and Gudi (2004) Feed batch fermentation Thermal engineering Babu et al. (2004) Optimal design of an ammonia synthesis reactor Babu et al. (2005) Optimization of adiabatic styrene reactor Babu and Sastry (1999) Estimation of heat transfer parameters in a trickle bed reactor Babu and Munawar (2001) Optimal design of shell and tube heat exchanger Babu and Angira (2006) Heat exchanger network design Fuel engineering Chen et al. (2002) True boiling point curve of crude oil; effect of pressure on entropy Babu and Chaurasia (2003) Estimation of optimal time of pyrolysis and heating rate Angira and Babu (2006) Alkylation process optimization Babu and Angira (2006) Alkylation process optimization E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering 151
are manipulated with arithmetic operators have several where bk,L and bk,U are the lower and upper limits of each advantages: efficient memory utilization, ease of use, k characteristic and rand(0, 1) is a random generated lower computational effort, and complexity (Price et al. number in the interval (0, 1). 2005). Along with efficiency, other advantages are flex- In a review about the initialization procedures for ibility (the algorithm adapts to modifications) and funda- large-scale optimization problems (Kazimipour et al. mentality (the principle of differentiation synthesizing in 2013), the initialization techniques were classified into itself the fundamentals concepts of the solution search) (i) stochastic, (ii) deterministic, (iii) two steps, (iv) other (Feoktistov 2006). 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 2.1 Initialization opposition-based learning (OBL) and quasi-OBL.
In this step, the initial population is created, usually by employing a random-number-generating procedure, the 2.2 Mutation algorithm selecting which locations should be explored and which should be discarded (de Melo and Botazzo In the mutation step, in order to create new individu- Delbem 2012). Although it is an important step (as the per- als, the selected individuals are modified by introducing formance is affected by initialization and improper initial- more genetic material into the population (Salman et al. ization can lead to high errors), initialization is somewhat 2007). The role of mutation is to add diversity and to avoid discarded, the majority of researchers mentioning in their getting trapped into local optima. In the DE case, a new works a series of general elements without any additional individual is created by adding a scaled differential term details (Ali et al. 2013). to a base vector (individual) (Price et al. 2005, Das and In order to perform a proper initialization procedure, Suganthan 2011). This mechanism [described in its simple different aspects must be taken into consideration. The form by Eq. (2)], also called “differentiation”, is the main most important ones are represented by (i) the position characteristic separating DE from other EAs: of the global minimum (considering that the problem is a minimization one) and (ii) boundary determination. ωi =+αβF⋅ , (2) Before solving an optimization problem, the position of where ω is the ith element from the mutated population, α is the global minimum is usually not known, and in this i the base vector, β is the differential term, and F is the scaling context, it is indicated to start the algorithm by using factor. The differential term is defined as the difference of equally dispersed points from the search space (Lei et al. two distinct, randomly chosen vectors: β = x -x . The base 2010, Peng and Wang 2010). Some algorithms start from k p vector is also randomly chosen, and in order to achieve good a single point, but DE needs a population composed of convergence speed and probability, Price et al. (2005) indi- diverse individuals (Price et al. 2005). cate that all vectors used in the mutation step must be dis- Although there are problems that have unconstrained tinct. This mechanism – differentiation – has the advantage parameters, for the most real-world problems, the exist- of not producing a disruption in the hyper-plan in which the ence of natural physical limits or logical constraints vectors exist (Feoktistov 2006). imposes different values for each parameter, and their ini- Initially, Storn (1996) presented 10 variants of the tialization is a straightforward process. For example, the algorithm, with a coding “Mode/DiffTerm/Cross” being time for studying a phenomenon cannot be negative or used to indicate each variant. “Mode” represents the the pressure of a chemical reactor takes values from a pre- mode in which the base vector of the mutation step is specified interval, which is dependent on the characteris- chosen, “DiffTerm” represents the number of differential tics of the reactor. In this context, initialization cannot be terms used for mutation, and “Cross” is the type of crosso- performed until all the boundaries are known. ver. The first two terms (“Mode” and “DiffTerm”) refer to Eq. (1) describes the formula used for initializing the characteristics of the mutation step, and the last one, each k characteristic of each i individual from the popula- “Cross”, is related to the crossover step. The base vector tion. This equation corresponds to the classic variant of for the mutation step can be chosen randomly (Rand), as DE, which, in the literature, has different variations and the best individual in the population (Best), or as a vector improvements. that lies on a line between the target and the best individ-
xbik,,=+kL rand(0, 1)⋅(-bbkU,,kL), (1) ual (Rand-to-Best). Regarding the number of differential 152 E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering terms used in the mutation phase (information coded the differential term is scaled up. A larger F can increase with “DiffTerm”), one [Eq. (2)] or two terms are usually (i) the probability of escaping the local optimum and employed, which are denoted with 1 and 2. (ii) the number of function evaluations (Deng et al. 2013). The first variant of the DE algorithm, which uses the random method of vector selection, along with one differ- 2.3 Crossover ential term and binomial crossover, is called Rand/1/Bin. The other versions included in the initial 10 are Rand/2/ In this step, an operation for population diversity Bin, Rand/1/Exp, Rand/2/Bin, Rand/2/Exp, Best/1/Bin, improvement is applied. Using two populations (current Best/2/Bin, Best/1/Exp, Best/2/Exp, Rand-to-Best/1/Bin, and mutated), a new trial population is created. Generally, and Rand-to-Best/1/Exp. two variants of crossover are used in the DE algorithm: Eq. (2) describes the variant in which only one dif- binomial [Eq. (5)] and exponential. ferential term is used. However, in literature, a series of variants that use more differential terms can be encoun- ωij, if ((rand 0, 1)
The classic crossover versions can generate only function evaluations reaching a maximum value (FEmax). one solution, limiting the search ability of DE (Wang Figure 1 presents the general workflow of the DE algo- et al. 2012). Consequently, new crossover variants were rithm, in a simple/classic variant where the stop criterion proposed or borrowed from other algorithms. One such is represented by the number of generations reaching the example is represented by the orthogonal design method, prespecified upper limit. with different studies presenting the advantages of apply- ing an orthogonal crossover for generating offspring (Gong et al. 2006, 2008). 2.5 Algorithm improvement
In order to improve the algorithm performance and to 2.4 Selection and stop criteria make it more flexible for the theoretical and real-life applications, different modifications at various levels In the last step of the algorithm, a mechanism for selecting have been performed. All these modifications followed the individuals forming the next generation is employed three main directions (Brest et al. 2011): (i) replacing the (Rahnamayan and Tizhoosh 2008). The classic version of hand tuning of control parameters with adaptive or self- DE uses a “one-to-one” competition, the trial and current adaptive mechanisms, (ii) hybridizing DE by combining it individuals being compared based on their objective with other optimization techniques, and (iii) introducing values. The ones with the lowest value (when consider- more mutation strategies during the optimization process. ing a minimization problem) are selected to form the next Each control parameter influences specific aspects generation. This comparison determines in DE a tighter of the algorithm, its effectiveness, efficiency, and robust- integration of recombination and selection than in other ness being dependent on their correct values, which are EAs (Price et al. 2005). This survival criterion is also called problem specific and can vary substantially (Brest 2009, “objective function based criterion” or “greedy criterion”. Hu and Yan 2009a). Due to their influences, finding the In literature two other mechanisms can be encountered: optimal values of the control parameters is not a straight- age criterion and age-objective function criterion (Price forward process, and the methods used for their correct et al. 2005, Li and Li 2010). settings can be classified into parameter tuning and As the majority of problems have specific constraints, parameter control (Eiben and Schut 2008). Parameter special attention must be given to the parameters that tuning consists of finding good values before running the exceed the boundaries. When applying the selection cri- algorithm. On the other hand, in the case of parameter terion, because DE does not have an innate mechanism control, the values are changed dynamically during the for dealing with constraints, a constraint handling mech- run based on a set of defined rules. Taking into account anism must be used in order to evaluate, and if neces- the “how” criterion of Eiben and Schut (2008) indicating sary, to move the exceeding parameters back into their how the change is performed, four subclasses are encoun- feasible interval. Depending on the characteristics of the tered: (i) deterministic control, where the parameters are problem being solved, different approaches can be used. determined using a deterministic law, without feedback For example, in the case of single-objective problems, from the system; (ii) adaptive control, where the param- the selection process is somewhat simple (as it is easy to eters are determined using feedback from the system. Two decide which between the two individuals is better), but categories are encountered here: parameter adaptation, in the case of multioptimization problems, the decision is which obeys the state of the population and refresh of not so straightforward (Qin et al. 2010). population; (iii) self-adaptive control, where the param- The steps of the algorithm are repeated until a stop eters are dependent of the algorithm being encoded into criterion is reached. An optimal selection of this criterion it; and (iv) hybrid control. ensures a good tradeoff between solution performance The second direction of improvement – hybridization and consumed resources. In an ideal case, the optimiza- – is the process of combining the best features of two or tion is stopped when the convergence to the optimum is more algorithms in order to create a new algorithm that is achieved (Zielinski and Laur 2008). As the determination of expected to outperform the parents (Das and Suganthan 154 E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering
Figure 1: Schema of the simple DE variant.
2011). Depending on the type of algorithm DE can be The principles from all these directions can be com- hybridized with, three situations encountered: (i) DE and bined, with DE being modified at different levels, using dif- other global optimization algorithms, (ii) DE with local ferent principles. For example, opposition-based chaotic search methods, and (iii) DE with global optimization and DE algorithm (Thangaraj et al. 2012) has an OBL-based ini- local search methods. Depending on the level of interac- tialization and the crossover factor is adapted using chaos tion, the hybridization can be performed at the (i) individ- theory; DE-CCS (Wang and Gao 2014) (DE with coopera- ual level, describing the behavior of an individual from the tive coevolution) has a jumping DE (jDE)-based approach population; (ii) population level; (iii) external level; and for the control parameters, a crossover scheme that (iv) meta-level, where a superior metaheuristics includes employs both binomial and exponential crossover and a the algorithm as one of its strategies (Feoktistov 2006). local search using cooperative coevolution; in hybrid self- In case of the third direction of improvement, differ- adaptive DE with neural networks (Curteanu et al. 2014), ent approaches related to the mutation were employed. neural networks are optimized using a DE-based version The classic variant regarding mutation in DE is to create, that includes OBL initialization, a modified mutation and for each individual in the population, a single mutated a local search procedure based on BackPropagation and individual based on one or more vectors. The alterations Local Search algorithms. proposed by researchers refer to the (i) introduction of more differential terms, (ii) selection of the individuals participating in the mutation process using specific prin- 3 Process optimization ciples, (iii) introduction of other mechanisms (evolution- ary game theory, Pant et al. 2009a, and replacement of Process optimization refers to the action of adjusting dif- control parameter with Laplace distribution, Thangaraj ferent aspects of the process in order to make it more effi- et al. 2010), and (iv) employment of multiple mutation cient and to minimize costs. There are three areas in which strategies (Qin et al. 2009, Zamuda and Brest 2012). the process optimization can be performed: equipment, E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering 155 operating procedures, and control optimization. In order in the predictions, the online approach exhibits properties to perform such a task, a series of mathematical systems of graceful degradation. However, when noisier settings describing the process is necessary. As, often, control opti- were employed, the DE algorithm performance decreased. mization is closely related to the equipment and operation Using the same case studies and objectives (from Rocha procedures, and the boundaries between all these areas et al. 2007), Mendes et al. (2008) performed online and are somewhat blurry, all the works related to the process offline optimizations of input variables by applying three optimization will be presented in the same section. 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/ 3.1 Simple DE versions used in process bin has the best results, followed closely by DE/best/2/ optimization bin and FIPS. Similar findings were obtained when online optimization was performed. In this section, different optimization procedures based Khademi et al. (2009) studied the optimization of on the classic DE versions (without any improvements methanol synthesis and cyclohexane dehydrogenation such as adaptation or hybridization) applied to specific in a thermally coupled reactor by means of a Best/1/bin chemical engineering processes are presented. strategy with fixed control parameters (Cr = 1, F = 0.8, and In Moonchai et al. (2005), the effect of pH and temper- Np = 100, with Np being the number of individuals in the ature on the bacteriocin production in Lactococcus lactic population). Their main objective was to maximize the C7 was studied and a series of models for five variables methanol and benzene mole fractions based on three were developed. In order to determine the optimal temper- parameters: inlet temperature of the exothermic and ature profile (which, in this case, was considered a control endothermic sides and initial molar flow rate of endo- variable), the temperature parameter was introduced in thermic side. It was shown that suitable values of the the models created for cell growth, lactic acid production, considered parameters can provide the necessary heat glucose consumption, and bacteriocin production. After to drive the endothermic process and, in the same time, that, using a weighted sum method, the two objective to enhance the methanol mole fraction with 3.67% and problems (determination of temperature control leading methanol yield with 3.1%. In another study (Khademi to the maximization of productivity and final bacteriocin et al. 2010), the same research group proposed a new titer) were converted into a single objective problem. It membrane thermally coupled reactor for methanol syn- was then solved using the DE algorithm, the simulation thesis, cyclohexane dehydrogenation, and hydrogen pro- results indicating that the bacteriocin production rate and duction. The new system was optimized with the same DE titer are at the highest level when starting with a 30°C tem- version, having the same control parameter values, the perature and then lowering it quickly at 22°C. scope being the maximization of outlet mole fractions of As the dynamic optimization of the fermentation methanol, benzene, and hydrogen. Six decision variables processes is a problem requiring not only powerful algo- were chosen: inlet temperature and molar flow rate of the rithms but also carefully chosen methodologies, Rocha endothermic and exothermic sides, inlet pressure of the et al. (2007) proposed an online optimization procedure exothermic side, and inlet temperature of the permeation based on DE. Three different case studies were consid- side. Compared with the conventional methanol reactor ered: (1) fed-batch recombinant Escherichia coli fermen- (CMR), the optimized system had a reduced size of reactor, tation, where microorganisms follow three different enhanced equilibrium conversion, and an increased metabolic paths (oxidative growth on glucose, fermenta- methanol production with 16.3%. tive growth on glucose, and oxidative growth on acetic A novel methanol synthesis loop with hydrogen acid); (2) a hybridoma reactor; and (3) fed-batch bioreac- permselective membrane reactor was modeled and opti- tor for the production of ethanol by Saccharomyces cer- mized using a Rand/2/bin version (Parvasi et al. 2009). evisiae. In case 1, the scope was to determine the optimal The behavior of the system was described by a set of ordi- feeding rate profile maximizing productivity (which was nary differential and algebraic equations, which was then represented by the recombinant protein formed per unit of used as a model for the optimization procedure with the time). For case 2, the objective was to increase the amount final scope of improving methanol production based on of monoclonal antibodies produced by the system, and in the inlet temperatures of the membrane tube and steam case 3, the aim was to determine the substrate feed profile drum. The new design had a 40% increase in production, leading to the maximization of ethanol amount. If in case which indicated that the applied approach was suitable of the offline optimization small noise leads to disruption for the considered process. 156 E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering
Another type of reactor optimized with DE was rep- Np = 100), the production process of para-diethylbenzene resented by a radial-flow, spherical bed methanol syn- through disproportionation of ethylbenzene was modeled thesis reactor in the presence of catalyst deactivation and optimized (Rahman et al. 2010). The selectivity and (Rahimpour et al. 2009). The Rand/2/bin version was used conversion of para-diethylbenzene were modeled as a in four cases for maximizing the methanol production function of temperature, liquid hourly space velocity, and based on (i) the inlet temperature of reactors, (ii) tempera- pressure. ture profile, (iii) radius ratios of reactors, and (iv) catalyst Dimethyl ether (DME) is another fuel that has cap- deactivation parameters. In each case, the optimization tured the attention of researchers, as it is a clean multi- was successful, with the considered parameters being purpose and multisource fuel (Vakili et al. 2011). In the determined with great accuracy. work of Vakili et al. (2011), the operation conditions of Catalytic naphtha reforming is a technology of utmost a thermally coupled direct DME heat exchanger (TCDR) importance for the petrochemical industry, with effort were optimized using a Best/1/bin version of DE with and energy being spent on improving or replacing the Cr = 0.8, F = 0.8, and Np = 40. The scope was to maximize classic approaches and systems such as the conventional the DME and benzene production based on a set of deci- tubular reactor (CTR) with newer and improved reactors. sion variables represented by the inlet temperature and The dynamic of a multistage spherical, radial flow reactor initial molar flow rates of the exothermic and endother- for the naphtha reforming process in the presence of cat- mic sides. Compared with the conventional DME synthe- alyst deactivation was performed using a DE approach sis reactor (CDR), the optimized system had a reduced (Best/1/bin) (Rahimpour et al. 2010). Based on seven deci- inlet feed of exothermic side with about 4300 kmol/ sion variables (inlet temperature and catalyst distribution day and a 1.6 ton/day increased DME production. Con- of each reactor, outlet gas pressure), the maximization cerning the endothermic side, the inlet feed is reduced of production of aromatics and hydrogen was followed. with 25,700 kmol/day and the hydrogen production is The results indicated that by using higher temperature increased with 0.8 ton/day. furnace (allowing gas to enter at 840 K), the efficacy of the In another work, the DME synthesis and cyclohex- system rises. ane dehydrogenation reaction taking place in a thermally In another study, the CTR was optimized via DE coupled reactor were optimized using classic Best/1/bin method in three cases and the reactor was discretized version (Khademi et al. 2011). The optimization goal was in 20 segments (Iranshahi et al. 2011). The optimiza- to determine the operating conditions (temperature pro- tion objective was the maximization of hydrogen and files, endothermic and exothermic sides) that maximize aromatics productions based on (i) temperature profile, the DME and benzene mole fractions. The decision vari- (ii) amount of hydrogen removed, and (iii) a combination ables considered were inlet temperatures of the exother- of temperature profile and hydrogen removal. The best mic and endothermic sides and initial flow rates of the results were obtained in the third case, an improvement endothermic and exothermic sides. The final result had an of 24% for aromatics and 10% for hydrogen production improvement of 2.43% for the DME fraction and of 2.78% being predicted. for methanol conversion. The electrical parameters (six temperature-depend- Rahimpour et al. (2011) used the DE algorithm for opti- ent and current-dependent nonlinear resistors) of a biohy- mizing the operating conditions of a thermally coupled drogen real-time power-generating system were optimized membrane reactor for decalin dehydrogenation and FT using a DE version with fixed control parameters (F = 0.8, synthesis in gas-to-liquid (GTL) technology. The gaso- Np = 300) (Huang et al. 2010). The proposed proton line yield (exothermic side) and hydrogen model fraction exchange membrane fuel cell system (with a rated power (endothermic side) were maximized and hydrogen molar of 3 W) had bipolar plated designed for low-pressured bio- flow rate (recycled stream) was minimized based on five hydrogen. Using the optimization results, the V-I charac- decision variables (inlet temperature and flow rate of the teristics were determined based on the Kirchhoff’s law. In endothermic and exothermic sides and exothermic pres- addition, a comparison between the DE predictions and sure). The Best/1/bin variant with fixed control parame- those generated by GA was performed, with the optimal ters (F = 1, Cr = 0.8, and Np = 100) had good results, with an temperature estimated by DE being 336.89 K and by GA increase of 14.28% of gasoline yield. being 303 K. A fixed-bed tri-reformer reactor (combining endother- Using a multi-input, multioutput orthogonal least- mic reforming reaction and exothermic oxidation reac- square-based radial basis function model and a classic tion) for methanol production was optimized in order to DE approach with fixed parameters (F = 0.5, Cr = 0.9, and determine the optimal operating condition leading to a E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering 157 maximization of methane conversion, hydrogen produc- a series of applications of the DE algorithm. What can be tion, and desired H2/CO ratio (Arab Aboosadi et al. 2011). observed is that GAs and ANNs are widely used and the The classic DE-based optimization procedure with penalty number of works is considerable, with these algorithms function used eight decision variables: inlet composition establishing themselves as “to go tools” when specific of CH4, CO2, CO, H2O, H2, O2, and N2 and inlet tempera- aspects cannot be solved with classic approaches. On the ture of gas feed. The optimized system had an increase other hand, the application of DE for fermentation pro- in methane conversion from 94.3% to 97.9%, a H2 yield of cesses is still in its infancy.
1.84%, and a H2/CO ratio of 1.7%. In the study of Vakili et al. (2012), a new DE optimized The estimation of the biomass concentrations at a thermally coupled membrane configuration (TCMDR) product scale feed batch process was studied by five dif- simultaneously producing hydrogen and DME was pro- ferent estimation methods: (i) kinetic model of overflow posed. This system was based on the TCDR developed in metabolism, (ii) metabolic black-box model, (iii) observer, Vakili et al. (2011) to which a membrane concept is added. (iv) artificial neural networks (ANNs), and (v) an adap- The main role of the optimization procedure (represented tive DE (RandToBest/1/bin version) (Hocalar et al. 2011). by the Best/1/bin version with fixed control parameters Two different technical scale bubble column fermenters Np = 50, F = 0.8, and Cr = 0.8) was to identify the optimum represent the system modeled, where S. cerevisiae yeast is design of the configuration and maximization of hydro- used as host organisms for recombinant proteins and for gen recovery and of production yield (DME, benzene). producing baker yeast and ethanol. In order to evaluate Compared with the CDR and TCDR, the DME production of the performance of the selected approaches, four types of the optimized TCMDR was improved by 10.3% and 11.4%, fermentation were considered. The algorithms were com- respectively. Also, the cyclohexane conversion was raised pared in terms of difficulty of implementation, number of by 8.3% (compared with TCDR) and the endothermic flow parameters measured, and process knowledge. The mean- rate was reduced, with the operation cost of the system squared errors (MSE) obtained with all “fermentation being lower. type-modeling” combinations are presented in Table 2. Another type of reactor proposed for direct DME pro- The results pointed out that each procedure has advan- duction is represented by the industrial scale dual-type tages and disadvantages, in the DE case, a few measure- reactor, combining two fixed-bed reactors (one water ments being required and the convergence being good for cooled and one gas cooled) (Vakili and Eslamloueyan different fermentations. On the other hand, DE has a long 2012). In order to achieve the best performance, the reactor response time that depends on the control parameters. design was combined with an optimization procedure, In this case, one possible improvement, in both response with a set of seven decision variables (number of tubes time and efficiency, is represented by the use of a DE self- in the water and gas cooled reactors, temperature of feed adaptive procedure in which the control parameters (espe- gas, reactor length, coolant temperature water in the first cially the crossover probability and the scaling factor) are reactor, second reactor diameter) being considered. The internally adapted to the specific characteristics of the optimization procedure employed a Best/1/bin version search space problem. with Np = 70, F = 0.8, and Cr = 0.8. Compared with CDR, In a review related to the use of EAs for optimization the optimized proposed system had an increased DME of a fermentation process, Adeyemo and Enitan (2011) list production, with 60 ton/day. In addition, by eliminating
Table 2: MSEs obtained with different estimation methods in the case of four fermentation types of Saccharomyces cerevisiae.
Fermentation Estimation method
Kinetic model of Metabolic black- ANNs Observer DE overflow metabolism box model
Fermentation producing minimum ethanol, in the 100 m3 0.1848 0.1296 0.2638 0.3777 0.1213 reactor Fermentation producing and then consuming a small amount of 0.4108 0.0697 0.2608 0.3580 0.2107 ethanol in the first hour of fermentation, in the 100 m3 reactor Fermentation producing around 1% ethanol during the 0.3315 0.0875 0.2964 0.3328 0.1725 exponential phase, in the 100 m3 reactor Fermentation producing and consuming around 0.2% ethanol 0.2597 0.2181 0.4311 0.6145 0.2010 in the first hours of fermentation, in the 25 m3 reactor 158 E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering the need for a separate unit for methanol production and The operational conditions in both sides of a cata- purification, the production costs were reduced. lytic thermally coupled fluidized bed naphtha reactor In Arabpour et al. (2012), in order to enhance the gas- (TCFBNR) were optimized by using a DE technique oline production, the optimization of the Fischer-Tropsch (Pourazadi et al. 2013). Based on a penalty function (with (FT) synthesis reactions in GTL technology was performed a parameter equal to 108), a set of constraints guarantee- using Best/1/bin version in case of five discretized reactor ing an increase in aromatic and hydrogen production models. For each model, different aspects were followed: were imposed. The results obtained indicated that the temperature optimization (first model, D1), optimization of number of tubes can be decreased by 29% compared to injected hydrogen (second model, D2), effect of optimum the non-optimized TCFBNR. Also, through optimization, removed water (third model, D3), effect of temperature an increase in aromatic production (464.4 kg/h), hydrogen and hydrogen removal (fourth model, D4), and simultane- production (3.03 kg/h), and aniline flow rate (14%) was ous optimization of temperature, injected hydrogen, and realized. removed water (fifth model, D5). The reactor length was Iranshahi et al. (2013) proposed an alternative to the divided into 20 elements, with the optimized values for CTR by developing a combination of tubular membrane each model and for each element being determined by DE, and radial-flow spherical reactors (TM-RSR). In addition, starting from the first segment from the top of the reactor. the new system was optimized by DE, the optimal design The results indicated that the highest yield of gasoline is parameter, and the operating conditions being used. Two obtained in the D5 model. Compared to the conventional configurations, combining a radial flow spherical reactor reactor, in D1 to D5 cases, the CO2 production is reduced by with one tubular membrane reactor (case I) or with two 6.2%, 21.6%, 27.6%, 24.1%, and 31.4%, respectively. tubular membrane reactors (case II), were considered. In Rahimpour et al. (2012), a process intensification The number of parameters and their meaning vary in both procedure of the catalytic naphtha reforming unit was cases, with a list of all the parameters used for optimiza- performed by optimizing a system based on a thermally tion in this work (Iranshahi et al. 2013) and the ones pre- coupled naphtha reforming reactor (TCR). The scope was viously discussed (Iranshahi et al. 2012, Rahimpour et al. to maximize the aromatic hydrogen and aniline molar 2012, Pourazadi et al. 2013) being presented in Table 3. flow rates and the aromatic in reformate mole fractions by The configurations used for case I were MSS, SMS, SSM, using several decision variables, from which 10 belong to and SSS, and the configurations for case II were MMS, the endothermic side (naphtha reforming) and 16 belong to MSM, SMM, and MMM, where S denotes the spherical the exothermic side (nitrobenzene hydrogenation). Com- reactor and M denotes the tubular membrane. The best pared with the nonoptimized TCR, the optimized system variants were SMS and SMM, with their optimized variant has a 26.3% increase in aromatic yield (endothermic side) (compared with CTR) having a considerable increase in and a 39.21% increase in aniline production (exothermic the hydrogen (23 kmol/h) and aromatic (6.5 kmol/h) pro- side). The higher thermal efficiency and lower operational duction rates. cost of the optimized system indicated that the proposed Samimi et al. (2013a) proposed a DE optimized ther- approach is a good way to boost production. mally coupled reactor with a dual membrane (TCDMR) for In another study, the operating conditions of a radial- simultaneous production of DME, hydrogen, and naph- flow tubular membrane reactor (RF-TMR) are optimized thalene. After determining the system models using the (Iranshahi et al. 2012). As in the previously presented study kinetic equations, energy, and mass balances in addition (Rahimpour et al. 2012), a relatively high number of param- with several auxiliary correlations, the operation con- eters were used for optimization. Some of them are specific ditions of the thermally coupled reactor (case 1) and of to the first, second, and third reactors (sweep gas pressure, TCDMR (case 2) were optimized using a Best/1/bin version membrane thickness, inner radius flow of the collector, with fixed control parameters (Np = 100, F = 0.8, and Cr = 1). catalyst mass distribution, fraction of the total sweep gas, In the first case, the scope was to maximize the DME mole number of sweep sections, ratio of permeation angle, and fractions on the outlet exothermic side based on three ratio of length per diameter), whereas some are specific to decision variables (inlet temperature, initial molar flow the sweep gas (total molar flow and hydrogen molar frac- rate, and inlet pressure of the exothermic side). In the tion), recycle gas (hydrogen mole fraction), and to the first second case, the maximization of the outlet exothermic reactor (total fresh naphtha feed). Compared with the con- DME mole fraction and endothermic hydrogen mole frac- ventional, dimensionless tubular reactor, the optimized tion was considered, eight decision variables being used radial flow tubular membrane reactor had an increase of (inlet temperature of water permeation, hydrogen perme- 11.6% for hydrogen and 8.9% for aromatic yields. ation, endothermic and exothermic sides, inlet pressure, E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering 159
Table 3: Parameters optimized for catalytic naphtha reactors using the DE algorithm.
Parameter/Study Reactors TCR RF-TMR TCFBNR TM-RSR (Rahimpour et al. 2012) (Iranshahi (Pourazadi (Iranshahi et al. 2013) et al. 2012) et al. 2013) Endoth. side Exoth. side Case I Case II
Length to diameter 1, 2, 3 * * * *3 * Sweeping gas pressure 1, 2, 3 * *3 * Hydraulic diameter 1, 2, 3 *3 * Membrane thickness 1, 2, 3 * *4 * Catalyst mass distribution 1, 2, 3 * * * * * Inner radius 1, 2, 3 * * *5 The fraction of the total sweep gas 1, 2, 3 * Number of sweep sections 1, 2, 3 * The ratio of permeation angle to the 1, 2, 3 * reaction one’s Exothermic side molar flow rate 1, 2, 3 * Inlet pressure of the exothermic side 1, 2, 3 * *2 Inlet molar flow rate of exothermic side 1, 2 * Number of tubes 1, 2, 3 * *2 Naphtha feed pressure 1 *1 * * * Sweeping gas molar flow rate * * * Hydrogen mole fraction in recycled reactor * * * * Fresh naphtha feed molar flow rate * * Hydrogen mole fraction in sweeping gas * * * Naphtha inlet temperature * * * Sweep gas inlet temperature * * Compressor discharge pressure 1 * * Total molar flow rate for the exothermic side * Inlet temperature of endothermic side *
*1 Total naphtha feed. *2 For reactors 1 and 2. *3 Parameters of tubular reactor. *4 Membrane thickness (not applied to the three reactors). *5 Inner radius of the spherical reactor.
and feed rates of the endothermic and exothermic sides). multitubular reactor where two exothermic reactions rep- Compared with CDR, for DME mole fraction, a 3.42% resented by methanol production and FT synthesis provide (case 1) and a 34.2% (case 2) improvement was observed. the heat required for the endothermic decalin dehydroge- Concerning the hydrogen mole fraction, a 6.4% improve- nation) is optimized using Best/1/bin variant with fixed ment of case 2 over case 1 was realized. control parameters (F = 0.8, Cr = 1 and Np = 100). The objec- The operating conditions and the length per radius of tive was to maximize the summation of the methanol mole a novel axial flow spherical packed bed membrane reactor fraction (endothermic) and outlet gasoline yield (exother- for DME production through methanol dehydration were mic) based on five decision variables: inlet temperature optimized using the Best/1/bin version (Samimi et al. and inlet pressure at the FT and methanol synthesis sides 2013b). The scope was to maximize the DME mole frac- and inlet flow rate at the FT side. Compared with CMR, the tions in the reactor outlet, using a set of 11 variables: inlet methanol production rate was increased by 10.52%. temperatures, molar flow rates, composition of compo- In Rahimpour and Mirvakili (2013), a decalin and nents in the reaction and permeation sides, reaction side hydrogen looping approach named thermally coupled pressure, and length per radius. The kinetic model had a membrane ternary reactor (TCMTR) for application in GTL
2.15% maximum absolute error and the optimization leads technology is proposed. In order to minimize the H2 mole to a 13.5% increase compared with CDR. fraction in the outlet of the third reactor and to maximize In Samimi et al. (2013c), a novel thermally double the gasoline yield and hydrogen mole fraction in the endo- coupled reactor is proposed for hydrogen, methanol, thermic side, the operating conditions of the TCMTR are and high-octane gasoline production. The system (a optimized using a DE approach. A set of seven decision 160 E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering variables were considered, including inlet temperatures deactivation, was proposed in the work of Bayat et al. of the endothermic and exothermic sides, inlet molar flow (2014). In addition, an evaluation of the optimal operating and pressure of the exothermic side, initial flow rate of conditions was performed using DE, the results indicating the endothermic side, temperature of the shell side, and that under the optimal values of gas phase inlet tempera- length of the third reactor. The results obtained indicated tures, flowing solids phase, and shell side, a high meth- that in the new proposed configuration, decalin can be anol production can be achieved. A very similar study used as a hydrogen carrier, with only a small amount of related to the same type of reactor and reaction was also this substance being required for injection in the first performed in Dehghani et al. (2014). reactor. Also, compared with a thermally coupled mem- As can be observed, different aspects of problems brane dual-type reactor, the DE-optimized TCMTR has a belonging to the chemical engineering area can be solved 92% decrease in decalin consumption. effectively with simple DE variants. The problems range In Bayat et al. (2013), the industrial fixed-bed ethyl- from fermentation to catalytic naphta reforming to fuel ene oxide reactor was optimized (at both steady-state and and biofuel production. For the same process, various dynamic conditions and taking into account the cata- aspects and conditions are studied from multiple points of lyst deactivation) using DE algorithm. Distinctively from view. In this manner, the findings of different studies can the Lahiri et al. (2008) work, a mathematical model was be combined, and useful conclusions for the experimental employed, and two different cases were investigated with practice derived. the scope of maximizing the ethylene oxide production Concerning the DE algorithm, the researchers did not rate during 1100 days of catalyst life. In the first case, four focus only on a single variant, the literature providing parameter were optimized: inlet pressure of the reaction examples where not only the type of mutation or crossover side, inlet feed gas temperature, inlet cooling saturated were changed, but also the selected values for the control water, and inlet molar flow rate. For the second case parameters. As each process has its own characteristics, study, six parameters were optimized, from which three which change even when different parameters are taken were related to the inlet gas temperature and three to the into consideration, and as DE successfully solved all cooling water temperature. Based on the results obtained aspects, it can be concluded that even in the simplest in the first case, in the second situation, the optimal forms, the algorithm is flexible enough to provide satis- behavior of cooling water and feed gas temperature were factory solutions. determined. In the end, an improvement of 1.726% (first case) and 4.22% (second case) was obtained. A design method for simultaneous optimal determi- 3.2 Modified and hybrid DE variants in nation of process configuration, equipment design, and process optimization operating conditions of a membrane-assisted separation process was developed by Koch et al. (2013). The consid- Kapadi and Gudi (2004) extended the optimal feed policy ered case study was represented by separation through of a fed-batch fermentation process proposed in Chiou and distillation and membrane separation of the ternary Wang (1999) by including (i) systems with multiple feeds, system acetone/isopropyl alcohol/water. The steps of the (ii) nonuniform parameterization (the time duration of new approach consisted of (i) generation of process alter- each interval is not the same), and (iii) path and end- natives through thermodynamic analysis, (ii) experimen- point constraints. In order to demonstrate the effective- tal investigations in a laboratory-scale membrane plant, ness of the methodology, the same case study as the one (iii) generic modeling based on mass-transfer models, from Chiou and Wang (1999) was employed. In addition, and (iv) process superstructure optimization with DE. The the dynamic optimization of simultaneous saccharifica- variables optimized include distribution feed, distribution tion and fermentation of starch to lactic acid was studied. side, distribution recycle, number of membrane modules The goal was to maximize the final concentration of lactic in series, number of membrane sheets in a module, total acid, productivity, and yield. In the case of a single feed weight of active packing for sections 1 to 4, reflux ratio, approach, DE in combination with the Lagrange-like flow rate of side stream, and molar recycle ratio. The scope method, including dynamic penalty functions, provided was to reduce costs, with the objective function contain- better results than Pontryagin’s maximum principle. ing investments and operation costs. When using the nonuniform control vector parameteriza- For methanol synthesis, a dynamic mathematical tion, the optimization was performed twice, first setting model of a membrane-gas-flowing solid-fixed-bed reactor the fermentation time as a decision variable, and second, with in situ water absorption, in the presence of catalyst considering it fixed at 80 h. The results of the optimization E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering 161 problem (with 79 decision variables, from which 40 were The catalytic industrial ethylene oxide reactor was constant feed rates and 39 were time interval parameters) modeled and optimized using a DE-ANN approach (Lahiri indicated that nonuniform control vector parameteriza- et al. 2008). As a rigorous mathematical model of the tion and endpoint constraints used in DE are more effi- considered process (quantifying all the effects of param- cient for end-product concentration and production rate. eters on selectivity) does not exist, the role of the neural Good results were also obtained in case of multiple feed network was to act as a model, correlating all the process optimization. data and performance variables. DE determined the In the work of Babu et al. (2005), a multioptimization process input variables (which were also the model input DE version (MODE) was employed to optimize the styrene variables) leading to a maximization of yield and selec- production by dehydrogenation of ethyl benzene in an adi- tivity. The considered inputs of the model were cycle gas abatic reactor. Five case studies corresponding to different flow and pressure; reactor temperature; reactor inlet con- combinations of objectives were studied based on ethyl centration of ethylene, CO2, oxygen, and ethane; oxygen benzene feed temperature, pressure, steam over reactant flow in the reactor; inhibitor flow; and catalyst running ratio, and initial flow rate of ethyl benzene. Maximization hours, while the outputs were ethylene oxide production, of styrene selectivity (SST) and styrene productivity (FST) selectivity, and CO2 production. These parameters were represents the first case, maximization of SST and styrene chosen based on a set of criteria, which included reduced yield (YST) represents the second case study, and maximiza- input number leading to the best prediction, high corre- tion of FST and YST is the third case study. In the fourth case, lation of input-output, and minimum cross-correlation the maximization of FST and minimization of H2O produc- between inputs and minimum of network complexity. tivity are considered, while the last case is represented by After determining the model, a simple DE version (F = 0.5, maximization of YST, FST, and SST. The same MODE approach Cr = 0.2, and Np = 100) was successfully used to optimize proposed in Babu et al. (2005) was applied in another the model inputs corresponding to the set of process study for optimization of two styrene reactor configura- operation conditions. The optimized solutions resulted tions (single bed adiabatic operation and stream injected in a significant improvement in the ethylene oxide pro- pseudo-isothermal operation) (Gujarathi and Babu 2010a). duction rate and catalyst selectivity. The DE optimization Five decision variables were considered for the adiabatic program can be integrated with an online, real-time APC operation (feed temperature, pressure, steam over reac- application, or the optimum value may be calculated on tant ratio, initial ethyl benzene flow rate, and steam tem- an off-line computer and further used for optimizing the perature), while for the steam injected reactor, two more reactor. are added (steam fraction at the reactor inlet and fraction Three DE-based versions were employed to optimize between the reactor length and location of the injection the liquid phase oxidation of p-xylene to purified there- port). Four optimization cases were considered, including phthalic acid in four different cases (Gujarathi and Babu
(i) maximization of FST and SST, (ii) maximization of SST and 2009). The scope of this work was to maximize the total
YST, (iii) maximization of FST and YST, and (iv) maximization feed rate (first objective) and to minimize the concentra- of FST, SST, and YST. For both configurations, in all four cases, tion of the 4-carboxybenzoic acid as intermediary product Pareto optimal sets were obtained. (second objective). The first objective plays as a deci- A simple self-adaptive approach (utilizing a co-evo- sion variable along with (i) catalyst concentration (CO), lutionary proves running on a fractional time scale with (ii) CO and water content in the solvent (W II2O), (iii) CO, respect to the main process and working similarly to a high- (WII2O), unconverted reactant (PX), and vent oxygen from level co-evolutionary strategy) was applied for designing the reactor and crystallizer merged into one variable VO2, a block decentralized PI controller of an ALSTOM gasi- and (iv) CO, WH2O, PX, VO2, and temperature of the reactor. fier (Nobakhti and Wang 2008). The control system was The DE versions used are represented by MODE (a variant required to minimize the integral absolute error of the fuel proposed in Babu et al. 2005); a modified MODE approach gas pressure and caloric value even when disturbances called MODE-II, in which, at each generation, the popu- appear (a 0.2 bar step reduction in the sink pressure and lation is randomly generated (Babu 2007); and an elitist a sinusoidal disturbance of 0.2 bar amplitude at 0.04 Hz). MODE version (E-MODE), in which the concept of elitism These requirements were formulated into an optimization with crowded distance sorting is included. The results problem with 20 design parameters and 26 constraints, showed that E-MODE provided widely spread, equally with the DE optimization providing near-optimum results, spaced Pareto fronts, while MODE covered a wide area a noticeable improvement over previously reported solu- of Pareto fronts in detriment of nonuniformly distributed tions being observed. solutions. 162 E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering
The same process was also studied by Xu et al. (2013), In Yuzgec (2010), the feeding profile of an industrial a self-adaptive DE strategy for solving constrained mul- scale baker’s yeast fermentation is optimized (maximiza- tiobjective problems proposed in Qian et al. (2012) being tion of the amount of biomass at the end of the process and used to improve the product yield. A set of conflicting minimization of ethanol formation during the process). objectives (minimization of combustion loss and maximi- Different DE variants were used: opposition-based DE zation of therephthalic acid yield) were considered, based (ODE), adaptive DE (ADE), and adaptive opposition DE on nine process variables (concentrations of cobalt and (AODE), which is a modified ODE version with an adaptive manganese catalysts, concentration of bromide promoter, jumping rate. As the conditions in the reactor (substrate and temperature and residence time of the three reactors). feeding into the fluidized bed) and the initial conditions The system model was developed in Aspenplus 2006.5, (IC) are the main factors influencing productivity and and the DE-based algorithm (with two versions, Rand/1 product yield, in order to achieve a compromise between and Best/1) was implemented in Matlab, with an interface objectives and to optimize the process, four IC were con- between the two environments being created to perform sidered: (i) high initial biomass and glucose concentration, the optimization. In order to determine the performance (ii) high initial biomass concentration and low glucose, (iii) of this approach, a comparison with NSGA-II and fuzzy low initial biomass and high glucose, and (iv) low biomass immune algorithm was realized, with the results indicat- and glucose concentration. The results obtained for cases ing that DE is effective in solving the considered problem. (i) and (ii) are presented in Table 4. It was observed that A hybrid strategy combining support vector regression the initial seed values of the DE algorithm do not have an (SVR) and DE was used to model and optimize ultravio- influence on the overall performance and that the Rand- let (UV) transmittance of the monoethylene glycol (MEG) to-best/1/Bin strategy is the best. Among all variants, the product of a commercial petrochemical plant (Lahiri and AODE approach had the best performance, with its predic- Khalfe 2010). The procedure was designed to maximize tions being the closest to the experimental data. the UV based on a set of process parameters (reflux ratio In a study related to the stabilization of feedstock and flow, MEG column top and condenser pressure, MEG properties in refineries (by combining different types of column control temperature and feed flow, drying column crude oil), a two-level optimization procedure based on control and bottom temperature, and off-spec glycol a hybrid algorithm combining Tabu search (TS) and DE reprocessing flow). After the SVR model was determined, was proposed (Bai et al. 2010). As the true boiling point DE was employed to optimize the N-dimensional model influences the yield and the operational stability of the input in three temperature case limits, which were then crude distillation unit, this parameter was used as a main verified in an actual plant. quality index for evaluating the crude oil properties. The
Table 4: Results obtained when optimizing the industrial scale baker’s yeast fermentation (Yuzgec 2010).
Case Strategy Algorithm Final biomass Total ethanol concentration formation (g l-1) (g l-1)
IC 1 (high initial biomass and Rand/1/Bin DE 67.0329 0.8203 glucose concentration) ODE 67.4065 0.9422 ADE 68.2067 0.6010 AODE 68.3400 0.6040 Rand-to-best/1/Bin DE 67.4237 0.9414 ODE 67.9857 0.6033 ADE 68.2493 0.5928 AODE 64.4213 0.6052 IC II (high initial biomass and low Rand/1/Bin DE 66.8828 0.6873 glucose concentration) ODE 67.4065 0.5612 ADE 68.1247 0.2642 AODE 68.0394 0.2736 Rand-to-best/1/Bin DE 67.0906 0.6066 ODE 67.8838 0.2775 ADE 68.0413 0.2709 AODE 68.1466 0.2742 E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering 163 main constraints imposed to the system were related to column coupled with pressure-swing distillation, mem- the yield of lights components, trace element concentra- brane-assisted reactive distillation process, reactive divid- tion, flow rate, and due date. The model used, which is ing wall column coupled with pressure-swing distillation, a mixed-integer nonlinear programming model, had two and membrane-assisted reactive dividing wall column groups of decision variables: the blending sequence of (Holtbruegge et al. 2015). imported crude ranks and the corresponding flow rates. Nian et al. (2013) used a combination of DE with group When only blending sequence is considered, the optimi- search optimization for optimizing an ethylene-cracking zation becomes continuous, and when the flow rate is furnace. The system was described by a model determined taken into account, then the optimization is combinato- using a cracking reaction software named COILSIMID. The rial. Therefore, a two-level optimization was performed considered inputs were the mole percentages of n-alkanes, and the proposed algorithm (called HIHI) effectively and iso-alkane, cycloalkane, and aromatics in the feedstock; efficiently determined the scheduling strategy. feed flow rate; coil output temperature (COT); and steam In case of batch processes, a very important aspect hydrocarbon ratio (SHR), while the output variable was is related to the zero-wait scheduling problems. In order represented by the yield of ethylene and propylene. The to solve this problem, Dong and Wang (2012) applied a optimization procedure was applied for three kinds of permutation-based DE (PDE) hybridized with two insert- feedstock, with the scope of maximizing the model output based methods for local search: remove-insert method by tuning COT and SHR, when the properties of feedstock and fast complex heuristic (HCF). In order to test the per- changed. After applying the algorithm (called DEGSO), the formance of this new approach, three case studies (from yield of ethylene and propylene increased remarkably, a the literature) were considered. Compared with GA, TS, fact that indicates that the method was able to accurately PDE, and HCF, the new hybrid versions not only had the find the optimum. best performance but also were also simple to use, as In order to optimize the retrofitting of heat exchanger there were no parameters to adapt, except the probability network in industrial plants, a one-step approach based of local search. on integrated DE (IDE) was used (Zhang and Rangaiah In a study related to the optimization of distillation 2013). This algorithm was first proposed in Zhang et al. with reactor-side for hydrodesulphurization process of (2011a), and Zhang and Rangaiah modified it and applied diesel, minimization of the total annual cost, CO2 emis- it to three case studies represented by (i) part of a petro- sions, and the amount of sulfur compounds was per- chemical process including a distillation column and an formed using a multiobjective approach based on DE and exothermic reactor, (ii) crude oil preheat train of a petro- Taboo list (Miranda-Galindo et al. 2012). The algorithm, leum refinery with six hot and one cold streams, and (iii) called MODE-TL, was proposed in Sharma and Rangaiah preheating crude in an atmospheric crude unit with nine (2010) and was used in combination with Aspen ONE hot and three cold streams. In a single step, both con- Aspen Plus software. tinuous and discrete variables were optimized (structure, Other systems where DE is applied in combina- heat exchanger areas, and split ratios). The scope of this tion with the Aspen software are as follows: (i) produc- optimization was represented by the determination of a tion of biodiesel, pretreatment (of cooking oil using minimum operation cost and annualized investment cost.
H2SO4), and transesterification (based on alkali catalyst) In the work of Gujarathi and Babu (2013), the indus- (Sharma and Rangaiah 2013a). As in the previous study trial low-density polyethylene tubular reactor was opti- (Miranda-Galindo et al. 2012), the DE-based variant was mized using three DE modified versions represented by represented by the MODE-TL; (ii) simultaneous reaction MODE, MODE III (Gujarathi and Babu 2012), and hybrid and separation in a reactive distillation column (Safe MODE (Gujarathi and Babu 2010b). Two distinct optimi- et al. 2013). The role of DE was to fit the collected data zation cases were considered: (i) maximization of conver- of the optimal surface of the response surface method- sion and of the weighted average value of the undesirable ology (RSM); (iii) design and optimization of an ethanol side products and (ii) independent maximization on con- dehydration process (Vázquez-Ojeda et al. 2013). Two DE version and minimization of side products (methyl, vinyl, optimized sequences were used: conventional arrange- and vinylidene side chain content). The results showed ment (CSS) and alternative arrangement employing liq- that both hybrid MODE and MODE III variants converged uid-liquid extraction (OSS-I); (iv) sweetening the natural to the same Pareto front. gas process (Ortegaa et al. 2013); (v) p-xylene oxidation In a study of the effect of carburizing and oxidizing process (Fan and Yan 2014); (vi) optimal design of biore- flames on surface roughness in turning of aluminum fineries (Geraili et al. 2014); and (vii) reactive distillation metal matrix composite, the optimization was performed 164 E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering using a multiobjective DE approach (Sudheer and Pavan most common models: statistical, alternatives (ANNs or 2014). The procedure was applied for determining the support vector machines [SVMs]), and deterministic. values of speed, feed, and depth of cut that lead to a mini- mization of surface roughness and maximization of mate- rial removal rate. 4.1 Statistical/stochastic models In combination with the sequential design method, a multiobjective DE variant was applied to solve the problem Statistical models are very important in chemical engi- of distillation column design for a five-compound mixture neering, as they can be applied for systems in which (i) (Errico et al. 2014). In addition, the Aspen Economic Ana- the known information is not complete and the design lyzer was used to assess the capital cost. It was observed of a deterministic model is very difficult or impossible or that when the decision variables in the reference simple (ii) the existence of complex relations between process column is high, initializing DE with the sequential design parameters translates into very complex models. A series method solutions can produce a final result in a reason- of examples are listed and discussed below. able amount of time. 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, 4 Model optimization and Np = 10). These values were then used as inputs to the coupled ordinary differential equations to find the optimal The optimization of mathematical models consists of parameters based on the Runge-Kutta fourth-order determining the optimal parameters (of a model which can method. The results indicated that as the heating rate is be statistical, mechanistic, or a black/gray box, usually increased, the pyrolysis time first decreases, reaches an represented by an ANN) that lead to an optimal state. This optimum, and then increases. can be represented by (i) estimation of the model input In Garlapati and Banerjee (2010), the input space of parameters (model optimization from the input point of a RSM model of an extracellular lipolytic enzyme produc- view), (ii) estimation of the model internal parameters tion based on Rhizopusoryzae NRRL 3562 was determined (tuning with experimental data), and (iii) minimization of using a classic variant of DE. The most important para the model errors compared with the experimental data. In meters of the study were represented by temperature, liq- all cases, the general workflow is similar and is based on uid-to-solid ratio, pH, and incubation time, with the scope the idea that the characteristics of the process model are of the optimization being represented by the maximiza- optimized in order to generate information (predictions) tion of lipase activity. The results obtained with DE were for further use or to optimize the process itself. Figure 2 experimentally validated, with a maximum of 96.52 U/gds points out the three directions related to the model of lipase activity being observed at 35.59°C, 1.5 liquid-to- optimization. solid ratio, a pH of 5.28, and 4.83 days of incubation. As the literature provides different types of models The same RSM approach was used to model the reac- for chemical engineering processes and since the combi- tive extraction of glycolic acid from aqueous solutions nation of optimizer-model is often considered a hybridi- using tri-n-octylamine (Datta and Kumar 2011). After zation, this section is organized around the types of determining the influence of different process param- models. The cases considered in this work include the eters (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 Figure 2: General workflow of the model optimization approaches. engineering. E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering 165
4.2 DE optimization applied to ANNs and optimized using either a classic DE approach or a hybrid SVMs combination of DE with chaotic system theory in which the chaotic sequences generated based on an Ikeda map When other approaches fail or are difficult to use, the arti- tune the F and Cr parameters. The experimental data were ficial intelligence tools with modeling behavior come as gathered using two thermocouples (one at the base, close an appropriate alternative. In this context, the most used to the motor, and one in the upper part) and a humidity modeling approaches are represented by SVMs and ANNs. sensor. In Luo et al. (2014), the optimization of the hyper- When combining an ANN with an optimization proce- parameters of the LS-SVM model was performed using dure, two aspects can be considered: what is optimized a novel hybrid algorithm combining the invasive weight and at what level? The combination of EA with ANNs (also optimization and DE. The system was then applied to known as neuroevolution) can be applied at roughly four predict the carbon content in spent catalyst in a continu- different levels: connection weights, architecture, learn- ous catalytic reforming plant. ing rules, and node behavior (Montana et al. 2009). In the work of Xu et al. (2012), a hybridization of PSO and DE (HPSODE) is applied to identify the hyperplanes (regulation parameter and width of the kernel function) 4.2.1 Support vector machines of the least square SVR (LSSVR). The combination, called HPSODE_LSSVR was applied to model the ammonia con- Lahiri and Ghanta (2008) studied the critical velocity of version rate in ammonia synthesis. In order to model the a slurry flow in a pipeline by a SVM, whose parameters process, five operational parameters were considered: were optimized by the DE algorithm. The determined SVM hydrogen concentration in the recycle gas and four-spot model had six inputs (particle diameter, solid and liquid temperatures in the four-stage catalyst bed. A compari- density, liquid viscosity, velocity, and solid concentra- son with other modeling approaches (back propagation tion). The absolute relative and standard deviation errors neural networks, LSSVR, PSO-LSSVR, and DE-LSSVR) were minimized, while the cross-correlation coefficient showed that the performance of the HPSODE algorithm for was kept around unity. The parameters of the model con- determining optimal model parameters (and, therefore, sidered for optimization were represented by C (cost func- better models) is higher than that the other approaches. tion measuring the empirical risk), ε (precision parameter representing the tube location around the regression func- tion), loss function type (e-insensitive or Hubber loss func- 4.2.2 Artificial neural networks tions), kernel types (linear, polynomial, Gaussian radial basis, exponential radial basis, splines, b splines), and the In the work of Lahiri and Ghanta (2009b), an ANN model degree of polynomials of the kernel function. In addition to for the hold-up in solid-liquid slurry flow in a pipeline the model optimization, an exhaustive search of the SVM is optimized using a classic variant of DE. Nine inputs parameters was performed; the results indicated that this (pipe diameter, particle diameter, solid concentration and approach is difficult and does not provide the same level of density, liquid density and viscosity, maximum packing performance as the SVM-DE combination does. concentration) and one output (hold-up ratio) were con- A similar approach was used for studying the pressure sidered, and the objective function of the optimization drop in slurry flow pipelines (Lahiri and Ghanta 2009a). was to obtain minimum average absolute relative error on Based on a set of experimental points, a SVM model for the test set. From the multitude of neural network para regression with seven inputs, six from the previous study meters, only five were chosen for optimization: number of (Lahiri and Ghanta 2008) plus pipe diameter, was deter- nodes in the hidden layer, input and output layer activa- mined. The optimization had the same objectives and the tion function, learning rate, and training algorithm. The same model parameters as in Lahiri and Ghanta (2008). A number of activation functions the layer can have was five comparison with literature data indicated that over a range (log-sigmoid, tan-sigmoid, radial basis, linear transfer, of operation conditions (physical properties and pipe and triangular basis), and the training algorithm can be diameters), the prediction performances were improved. one of the following: batch gradient decent, resilient back In order to estimate and control the temperature propagation, conjugate gradient (Fletcher-Reeves update, of a thermal process, a hybrid approach based on least Polka-Ribiere update, Powell-Beale restarts), quasi-New- squares SVM (LS-VM) and pulse width modulation (acting ton algorithm (BFGS, Levenberg-Marquardt [LM]), and in an electrical resistance) was employed (dos Santos one-step secant algorithm. The comparison between the et al. 2012). The parameters of the LS-SVM model were simple ANN model determined after 10,000 runs and the 166 E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering hybrid ANN-DE model (after 2000 runs) indicated that using a feed-forward multilayer perceptron ANN with the proposed methodology, ANN-DE, is more suited to the parameters optimized using two DE variants; (i) classic problem at hand. and (ii) self-adaptive approach (Dragoi et al. 2011). The The simultaneous topological and structural optimi- process parameters based on which the ANN was created zation of a neural model for the styrene polymerization and optimized were represented by viscosity, superficial was performed using three methods: a six-step optimiza- speed of air, specific power, and oxygen-vector volumetric tion methodology based on a systematized trial and error fraction. The model parameters considered for optimiza- applied for ANN parameters (OMP method), a classic tion were the number of hidden layers and neurons in DE variant, and a GA approach (Curteanu et al. 2010). each hidden layer, weights, biases, and activation func- A model able to predict monomer conversion, numeri- tions for each neuron. For each specific goal (prediction or cal average molecular weight, and gravimetrical average classification), different neural models were determined. molecular weight was created using initiator concentra- In the prediction case, a stack approach (stacked neural tion, temperature, and reaction time as input variables. networks) was also applied because the individual neural The considered parameters for GA and DE optimization network (generated with both classic and self-adaptive were represented by the number of hidden layers, number versions) did not provide acceptable results. In the clas- of neurons in each hidden layer, the weights between net- sification case (where the classes were generated using a work’s neurons, and biases of each neuron. In addition, discretization method applied to the mass transfer coef- DE also optimized the activation function of each neuron. ficient), different DE variants were tested, with the best On the other hand, the OMP approach determined the results being obtained with a simple DE version (mean number of hidden layers and neurons in each layer, acti- squared error [MSE] = 0.096 in the testing phase). vation functions for the neurons in the output layer, learn- The prediction of the liquid crystalline property of ing rate, and momentum term. The results showed that some organic compounds (bis aromatic and azo aromatic the three methods have a similar performance, as can be types) was performed using ANNs optimized with two observed from Figure 3. different DE self-adaptive versions (Dragoi et al. 2012a). The oxygen mass transfer in the presence of oxygen One version is proposed in Brest et al. (2006) and is called vectors of an aerobic biosynthesis system was modeled jDE. The other one (SADE-NN) is a simple self-adaptive
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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). E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering 167 approach. The process parameters considered were struc- a local search procedure (Dragoi et al. 2012b). Using the tural and molecular descriptors: the length of the rigid model of the freeze-drying process and given the values core, the length of the flexible core, molecular weight, and of the operating conditions (temperature of the heating an asymmetry factor (represented by the ratio of diameter shelf and pressure in the drying chamber), both the tem- to the total length of molecule). Different percentages for perature and the residual ice content in the product vs. splitting the available data into training and testing were time can be determined offline. This makes possible to tested, with the results obtained indicating that special understand if the maximum temperature allowed by the attention must be given to this aspect. In addition, a product is trespassed and when the sublimation is com- comparison with some previous studies was performed, plete, thus providing a valuable tool for recipe design and and the conclusion drawn was that when using the right optimization. The algorithm called SADE-NN-2 (Dragoi technique, ANNs can reach the same level of performance et al. 2012b), which is an improvement of SADE-NN-1 as specific classifiers algorithms. The difference in per- (Dragoi et al. 2013a), was specially designed to work with formance obtained between SADE-NN and jDE pointed the considered process specifics (freeze-drying process). out that, for the considered case study, the simplest self- The workflow of the application, compared with the adaptive approach was better. The maximum average general approach, has some additional steps (1, 2, and 3 percentage of correct answers was 87.09% for the training from Figure 4), in accordance with the complexity of the data and 85.15% for the testing data. This means that the process and the final goals considered. error of prediction is around 15%, which is considered an In the monitoring case, based on the system’s state, acceptable value. predictions for specific parameters at a future time were ANNs and a modified DE version were applied to performed. The measurement of product temperature is model the oxygen transfer when n-dodecane is added in used as input variable of the ANN in order to provide an aerobic fermentation systems of bacteria (Propionibacte- inline estimation of the state of the product (temperature rium shermanii) and yeasts (Saccharomyces cerevisiae) and residual amount of ice). In the modeling case, taking (Dragoi et al. 2013a). The process parameters considered into account that the pressure and temperature remain for model input were biomass concentration, superficial unchanged, the algorithms predict the duration of the air velocity, specific power, and oxygen-vector volumet- primary phase and maximum temperature product. In this ric fraction, and the output was the mass transfer coef- manner, the chemical engineer can avoid the apparition ficient. The algorithm, called SADE-NN-1, represents an of specific conditions leading to changes in the product improved version of SADE-NN proposed in Dragoi et al. properties and ensure final product quality. In order to (2012a), the alteration consisting in the introduction of generate the dynamic of the system, a recurrent ANN was the following elements: (i) OBL to improve initializa- developed, with different combinations of delayed inputs tion, (ii) a new mutation strategy in order to improve the and specific parameters of the process being tested. The offspring generation, and (iii) increasing the number of best results were obtained with a set of nine inputs rep- activation functions that a neuron could have during the resented by time; pressure chamber; fluid temperature; evolution (linear, hard limit, bipolar sigmoid, logistic product temperature at the interface of sublimation sigmoid, tangent sigmoid, sinus, radial basis, and trian- delayed one, two, and three times; and dried layer thick- gular basis functions). Simple ANNs, with one hidden ness delayed one, two, or three times. As this process is layer and small number of intermediate neurons, accu- slow, special attention was given to the time interval con- rately model the process considered as case study. After sidered for delay, a value of 600 s (as a fraction of the time the best model was determined (using different combina- constant of the process) being used. tions of mutation strategies and crossover), a sensitivity A comparison between the predictions (in both moni- analysis was performed. As a result, the process param- toring and modeling cases) with a complex phenomenolog- eters having the most influence on the system (model) ical model indicated that there is little difference between were determined. A good correlation between model and the two. On the other hand, the phenomenological model is process is obtained, which translates in good prediction very complex and requires many computational resources, performance as the comparison between the ANN and a which makes it undesirable in control systems where speed phenomenological model (solved with a multiregression and low-resource consumption are critical. As performance approach) showed. is crucial in the modeling and monitoring phases of free- The freeze-drying of pharmaceutical products was drying process, the ANN determined was further tested, studied from multiple points of view using a hybrid and its robustness and flexibility were verified using combination of DE with ANNs and backpropagation as various process settings (Dragoi et al. 2013b). 168 E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering
Figure 4: Workflow of the SADE-NN-2 approach designed for the freeze-drying process.
The SADE-NN-2 (Dragoi et al. 2012b) algorithm was caffeine concentration, NaF concentration, type of alloy also employed for the prediction of partition coefficients of (Ti content), and solution pH. Based on the considered guanidine hydrochloride in a poly(ethylene glycol) (PEG) process characteristics, the optimal model parameters 4000/phosphate/guanidine hydrochloride/water system that can efficiently perform predictions with specific (Pirdashti et al. 2015). The process parameters considered values were determined. The initial test showed that the were the PEG/phase forming salt ration, guanidine hydro- use of single ANNs does not provide acceptable solutions, chloride concentrations, and pH. The aqueous two-phase with the most important aspect influencing these results system was studied using this approach because the lack being the inability of individual ANNs to efficiently cover of knowledge related to the partitioning equilibrium of the entire search space and the significant differences in macromolecules translated into a lack of comprehensive the behavior of the system at low and high pH. In order to theory able to predict the data. The selected model had solve this problem, a series of ANNs, organized in stacks one hidden layer with 19 neurons and predicted the spe- (Figure 5), were used in two cases: low pH with values of cific values with an average relative error < 1.4%. 3, 4, and high pH with values of 5, 6, 7, and 8. For the low Another process studied with SADE-NN-2 was rep- pH, the absolute relative error for the testing values was resented by the corrosion resistance of TiMo alloys in 14.3%, while for the high pH, the absolute relative error acidic artificial saliva with NaF and/or caffeine, at 37°C was 10.47%. (Mareci et al. 2015). The polarization resistance of the The modeling of the depollution of some gaseous TiMo alloys was modeled based on the immersion time, streams containing n-hexane was performed by our group E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering 169
Figure 5: The general structure of the ANN stack employed for modeling the corrosion resistance of TiMo alloys.
using a neuroevolutive technique, based on a DE modified streams). On the other hand, in combination with SVM, DE self-adaptive version (Curteanu et al. 2014). Taking into was scarcely applied to solve chemical engineering prob- account the type of absorbent, contact type, temperature, lems. One of the aspects that can explain this difference bed length, and initial concentration of the volatile com- consists of the fact that ANNs (in simple or in stacked ver- pounds, predictions of the report between the outlet con- sions, stand-alone, or hybridized with other approaches) centration at an arbitrary time and the inlet concentration are more used than SVM. were generated. The role of DE was to simultaneously opti- mize the topology and internal parameters of the neural network. DE was hybridized with a local search procedure 4.3 DE optimization applied to deterministic based on two algorithms (BackPropagation and Random models Search), with their role being to improve the best solu- tion found in each generation. The results obtained were In combination with the orthogonal collocation method, better than those provided by nonhybridized versions, a DE was applied for the estimation of the heat transfer fact that indicated that the improvements in the algorithm parameters (effective radial thermal conductivity of the were correlated to the performance enhancement. In bed and the effective wall to bed heat transfer coefficient) addition, a comparison with a classic model (Yoon Nelson of a gas-liquid co-current down flow through trickle bed model) was performed, with the ANN based model having reactors (Babu and Sastry 1999). A comparison with the superior performances. The DE approach had a MSE for radial temperature profile method (RTP) based on Pow- the training data of 0.002565, while the classic model had ell’s method for optimization indicated that DE is a pow- a MSE of 0.007680. erful and less expensive computational alternative to the The examples previously described show that DE in classic approaches. In order to converge (irrespective of combination with ANNs was widely applied in the area of the initial population), it required only 10 generations, chemical engineering for modeling, predicting and some- while RTP (which needs an initial guess relative near to times optimizing multiple processes and aspects of pro- the global optimum) took 32 iterations. cesses (hold-up and pressure drop in slurry pipes, styrene Chiou and Wang (2001) applied a hybrid DE algorithm polymerization, oxygen mass transfer, freeze drying, cor- for parameter estimation of an unstructured kinetic model rosion resistance of alloys, and depollution of gaseous based on the Monod model. The experimental data were 170 E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering
gathered from a Biostat ED bioreactor with E. coli HB101 SO2 oxidation with Cs-Rb-V sulfuric acid catalyst (case study containing plasmid pBR329. The optimal parameters of 1) and diesel oil catalytic cracking reaction (case study the kinetic model (cell growth, substrate consumption, 2) (Wu et al. 2008). In the first case, the objective was to and product formation) were successfully determined in increase the SO2 rate conversion. In order to determine the 5000 generations with a set of fixed control parameters performance of the new method, called immune theory DE, (Cr = 0.5 and Np = 5), a population diversity of 0.05, and a a comparison with the classic DE version was performed. tolerance for gene diversity of 0.05. A model variation of For both case studies, 60% of the solutions obtained with ±50% for the initial concentration of glucose fitted the the DE based variant were better, with the near-optimal experimental data, and therefore, the model parameters solution being determined much faster. were in the optimal area. A hybrid variant of DE in which the individuals Moonchai et al. (2005) studied the effect of pH and were arranged hierarchically in a nearly regular tree was temperature on the bacteriocin production in L. lactic C7. applied to optimize the kinetic models of two processes: (i) For a set of five variables (cell growth, lactic acid produc- pyrolysis and dehydrogenation of benzene and (ii) super- tion, substrate consumption, and bacteriocin production), critical water oxidation (Shi and Zhong 2008). In the first simple mathematical models were developed in order to case, the reaction rate constants were estimated, while in describe and predict their kinetic behavior. DE was then the second case, a set of parameters (order of reaction for used to estimate the model parameters, with the minimi- dichlorophenol, preexponential factor, and apparent acti- zation of a weighted absolute error being considered as vation energy) were considered for optimization. A com- an objective function. The comparison between simula- parison with the classic DE version and PSO showed that tion and experimental data showed a good agreement, a the new hybrid approach was best suited to the consid- fact that indicates that DE was able to determine optimal ered processes as it provided the best results. values for the considered case study. In the work of Hu and Yan (2009b), an adaptive DE Efficiently removing acid gas impurities from gas version was tested on a series of noisy and no-noisy mixture is an important aspect in natural gas process- benchmark functions and on a process of supercritical ing and hydrogen purification, and therefore, DE was water oxidation. For the considered process, the goal of employed for parameter estimation of modified Clegg- the optimization was represented by the minimization of Pitzer equation used for designing a sour-gas treating the error between the 2-clorphenol conversion predicted plant with alkanolamine solvents (Anil and Kundu 2006). with the model and experimental data. The parameters In order to determine their efficiency at parameter estima- taken into account were represented by optimal Arrhenius tion of the vapor-liquid equilibrium of (CO2+AMP+H2O) parameters and the reaction order for 2-clorphenol, O2, and system, a comparison between DE, simulated annealing H2O. The results obtained with the adaptive DE version (in (SA), and LM was performed, with the strategy DE/rand/1/ which the control parameters were computed based on bin providing the best results. a specific relation dependent on the current generation) In Mariani et al. (2008), the apparent thermal diffusiv- indicated an improvement over other reported solutions. ity of banana (nanicao variety) was determined by apply- In the study of Liu and Wang (2009), the HDE version ing a simple DE version to an inverse problem based on a proposed by Chiou and Wang (1999) was used for param- mathematical model, considering the effects (at the surface eter estimation of two bioreactors. The algorithm, called of the fruit) of shrinkage and convective heat transfer. The hybrid DE with geometric mutation (HDE-GM), contains a parameters of the two functions (one dependent on mois- series of modifications including three additional opera- ture content and one dependent on moisture content and tions (restriction, acceleration, and migration) and a temperature) were determined by modeling the apparent geometric mean mutation strategy. The first case study thermal diffusivity. To perform the required predictions, is represented by a three-step biochemical pathway the transient temperature taken at specific moments of described by a set of eight ordinary differential equa- time was considered as decision variable. The comparison tions having 36 parameters. The search was divided into between predicted data and experimental profiles of tem- two classes (Hill coefficients and other), and distinctively perature at the thermal center showed no significant dif- from the base HDE variant, HDE-GM was able to generate ferences, and therefore, the determined parameters of the optimal solutions, the highest error obtained being 7%. two functions can be considered optimal. The second case study was represented by the production A new combination of DE with the vaccination principle of ethanol based on the fermentation of Saccharomyces used in immune theory was applied for the kinetic model distaticus LORRE 316. Using some sets of experimental parameter estimation of two processes: low-temperature data, 19 parameters (including yield coefficients) were E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering 171 estimated efficiently, resulting in a pruned kinetic model calculated using DE and modified Esmailzadeh-Roshan- with a more compact formulation. In another study (Liu feker, Peng-Robinson, and Soave-Redlich-Kwong equa- and Wang 2010), the same HDE-GM was applied for the tions of state (Nourbakhsh et al. 2011). In this case, the same problems as in Liu and Wang (2009), along with a most important parameters considered were interaction, series of unconstrained and constrained benchmark opti- solvent rations, molecular weights of asphaltene and mization problems. solvent, and solubility. A comparison between the opti- The stability analysis of separation processes and mized model and experimental data shows the concord- the phase equilibrium calculation are challenging prob- ance between the two sets of data, with the model having a lems due to the highly nonlinear characteristics of the good potential for predicting the asphaltene precipitation. thermodynamic models used. Bonilla-Petriciolet et al. An IDE version was applied for parameter estima- (2010a) performed a series of parameter tuning simula- tion of the VLE model (Zhang et al. 2011a). The algorithm tion of DE and TS on a set of reactive separation processes (which introduces an adaptation of mutation parameter, (the reaction of butyl acetate production at 298.15 K and crossover parameter, Tabu list and Tabu check, a novel 1 atm, methyl tert-butyl ether reaction system with inert stopping criterion, and a local optimizer) was also tested at 10 atm and 373.15 K, and the reactive system for tert- on a set of benchmark functions having two to 20 vari- amyl methyl ether synthesis at 335 K and 1.5 atm). Then, ables and multiple local optima. In addition, a compari- the algorithms were applied on two examples represented son with DETL (Mekapati and Gade 2010) was performed. by (i) a hypothetical multicomponent reactive mixture After the superiority of IDE was established, 20 VLE prob- undergoing three reactions and (ii) esterification reaction lems (from which 10 are based on least squared approach of ethanol and acetic acid to form ethyl acetate and water. and 10 were based on error-in-variable) were solved. In The results showed that although DE is better than TS, it addition, a comparison with SA, PSO, a classic DE variant, requires more CPU time and function evaluations. a hybrid version (DETL), and Branch and Reduce Optimi- In another study performed by Bonilla-Petriciolet et al. zation Navigator (BARON) was performed, with IDE being (2010b), the parameter estimation of vapor-liquid equilib- the best. Only BARON provided comparable results. rium models was performed using (i) DE with Tabu List Zhang et al. (2011b) applied three bioinspired algo- (DETL), a version proposed by Mekapati and Gade (2010); (ii) rithms, unified bare-bones PSO, IDE, and IDE without SA; (iii) GA; (iv) classic DE; and (v) PSO. After generalizing Tabu list and radius (IDE_N), respectively, to perform the relative performance of the algorithms by applying them and optimize phase equilibrium calculations, phase sta- on a set of benchmark functions, three vapor liquid equi- bility, and chemical equilibrium problems of 16 different librium (VLE) systems in the classic least squares variable systems. The results indicated that the IDE version per- formulation (tert butanol+1 butanol; water+1,2 ethanediol; formed best. In addition, for the chemical equilibrium benzene+hexafluorobenzene) and one in the error-in-vari- problems, a comparison of IDE with SA, GA, and DETL able formulation (benzene+hexafluorobenzene) were con- points out that IDE has the best performance. sidered. On both formulations, DE and DETL proved to be In a review about the use of different global optimi- the best, with these algorithms being viable alternatives to zation methods for phase equilibrium and calculations, the parameter optimization of VLE systems. different approaches, including DE, are presented (Zhang The same group of authors solved the problem of et al. 2011c). In addition, a set of specific problems solved constrained and unconstrained Gibbs free energy mini- with DE and published prior to year 2011 are detailed. mization in a reactive system by applying GA and DETL The kinetic parameters of model of three independent (Bonilla-Petriciolet et al. 2011). The optimized reac- parallel reactions of sugarcane pyrolysis (based on ther- tive system was represented by a set of eight systems mogravimetric analysis) are optimized with DE and then modeled with one of the following thermodynamic a series of relations (activation energy, preexponential models: nonrandom two liquid model (NRTL) and ideal factor of Arrhenius equation, and mass fraction of each gas, Wilson model and ideal gas, universal quasi-chem- subcomponent of biomass) are computed and compared ical model, Margules solution model, or NRTL. The with data from literature (Santos et al. 2012). A sensitiv- scope was to determine the parameters of these models ity analysis of the models was performed with respect to that lead to a global minimum. A comparison between the perturbations of 1% of the model’s inputs. The results the three algorithms pointed out that DETL and SA are indicated that the most influential parameter on bagasse better than GA. conversion is represented by the activation energies. The amount of adjustable parameters of the Flory- Kumar et al. (2011) studied the determination of Huggins models of the asphaltene precipitations was optimal stoichiometric and equilibrium constants of the 172 E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering reactive extraction of propionic acid using tri-n-butyl minimize the isobutane recycling. The decision variables phosphate dissolved in n-decane and 1-decanol. The main considered were isobutane makeup, olefins feed, and variables influencing the equilibrium extraction charac- fresh feed. The new stop criteria forced the algorithm to teristics are represented by (i) the nature of the extracted stop earlier than the maximum allowed generation, the acid, (ii) concentration, and (iii) type of diluent. Therefore, performance indexes indicating that optimal solutions for the main parameters determined were distribution coeffi- engineering application were obtained. In the case of the cient, degree of extraction, loading ratio, and equilibrium Williams-Otto process (with both reaction and separation complexation constant. The results obtained pointed out sections), the objective was to maximize the net present that the extraction efficiency is proportionally dependent worth and payback before tax and minimize the payback on the concentration of the active diluent. In addition, period. The decision variables used were the feed with the increase in temperature affects the extraction equilib- reactants. Although the solutions obtained converged rium, with the enthalpy and entropy being -12.47 kJ mol-1 near the known pareto-optimal fronts, the distribution of and -32.42 J mol-1 K-1, respectively. solutions was not good; for the Williams-Otto process, a
The solubility of CO2 in ionic liquid 1-alkyl-3 methyl higher number of generations is required. In the fermen- imidazolium bis (trifluoromethyl-sulfonyl) imide was tation case, a three-stage fermentation process integrated modeled using a set of phenomenological rules (Peng- with cell recycling was considered. The objectives were Robison equation of state, Wong-Sandler mixing rule, and to maximize ethanol productivity and overall glucose van Laar model for excess Gibbs free energy). After that, conversion based on the dilution rate, glucose concentra- the model parameters (binary interaction and activity tion in feed, and bleed ration for each stage. The results coefficients) were optimized with DE algorithm in order obtained showed that unnecessary computations were to obtain better predictions (Yazdizadeh et al. 2011). The eliminated, with the quality of solutions remaining at a optimized parameters were temperature dependent, and a high level. comparison between the considered model and one based The parameter determination of the kinetic model on Peng-Robison equation of state and Wong-Sandler of the cassava alcoholic fermentation hydrolyzed in a mixing rule showed that the former is better. batch reactor was performed by a series of four stochas- Chakraborty et al. (2012) used two variants of DE- tic optimization algorithms: artificial bee colony (ABC), based algorithm proposed in Das et al. (2009) for param- DE, PSO, and SA (Da Ros et al. 2013). From the kinetic eter optimization of a mechanicist model of the proton model (composed of three ordinary differential equa- membrane fuel cell (PEMFC). The idea was to determine tions describing the variation of cell, substrate, product the model’s optimal seven parameters so that it best fits concentration, and reaction time), 10 parameters were the PEMFC stack. The algorithm, called DEGL (DE with chosen for optimization. In addition, in the optimization global and local neighborhoods), had a self-adaptive procedure, a set of 12 ICs were included, with the final weight factor. The results showed that DEGL with expo- number of parameters optimized being 22. All the algo- nential crossover is better than the one with binomial rithm’s parameters were set in such a manner so that the crossover, standard versions of DE and GA, from three objective function evaluation was equal to 500,000. The points of view: (i) quality of solution when an imposed ABC prediction had a poor performance, despite its effec- computational cost is given, (ii) computational cost when tiveness on a set of benchmark functions. The SA perfor- a given performance is required, and (iii) frequency with mance was dispersive, as the initial random selection of which an acceptable solution is found. solution has a very high impact on the algorithm’s perfor- In Sharma and Rangaiah (2013b), a complex MODE mance. PSO had a satisfactory performance, but the best with a new termination criterion (using nondominated results were obtained by the DE variant with the mutation solutions and several performance metrics) is used to based on the best-so-far solution. solve three process applications represented by alkyla- In combination with the Gauss-Newton method with tion, Williams-Otto, and a fermentation process. The algo- orthogonal collocation (for computing the sensitive matrix rithm (called I-MODE), which is an improvement of DETL and state vector), DE was applied on a set of six test prob- proposed by Mekapati and Gade (2010), is used to maxi- lems for parameter estimation (first-order reversible chain mize the productivity by determining the optimal model reaction, first-order irreversible chain reaction, catalytic parameters. In the case of the alkylation process (where cracking of gas oil, Bellman’s problem, methanol-hydro- products are combined with refined petroleum products carbon process, and Lotka-Volterra problem) and then used in order to enhance the octane number), the objective for estimating the kinetic parameters of a batch fermenter was to maximize the profit and the octane number and to for penicillin production (Zhao et al. 2013). Based on the E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering 173 parameters determined with the proposed algorithm (called i) There are cases when the application of a general HDE), the biomass concentration, product, substrate, and optimizer to solve a specific problem is not efficient. reactor volume were computed with high efficiency. As discussed in the previous sections, process optimi- In order to model the chlorine decay in bulk water, zation requires the existence of a model. This model a phenomenological model with DE optimized param- can be a mathematical model expressed by systems of eters (reactive constituents, rate constants for reactions equations with complex derivatives, a mathematical of the two forms of the free chlorine with the reactive model containing one equation with complex deriva- constituents, corresponding rates constants at 20°C, and tives and one (or more) ordinary system(s) of differ- Arrhenius rations of each reaction’s activation energy to ential equations, mathematical models promoted by the ideal gas constant) was employed (Liu et al. 2014). a group of ordinary systems of differential equations, This model incorporated the effect of pH and temperature a mathematical model with one set of ordinary differ- so that it can be applicable to everyday situations like the ential equations complete with algebraic parameters chlorine decay when heating tap water. For the process and relationships between variables, or a mathemati- model, two cases were considered: when two reactive con- cal model given by algebraic equations relating the stituents are assumed and when three reactive constitu- variables of the process (Dobre and Marcano 2007). In ents are assumed. The simulation results indicated that the cases when the model is based on complex deriva- the two-site reactive model having 10 parameters is the tives, optimization is difficult to implement because most suited for the process approached. numerically solving the derivatives is computation- When DE is applied to determine the optimal param- ally difficult (even when employing specialized soft- eters of the deterministic models, the multitude of good ware) and the EAs in general (and DE in particular) results indicates the flexibility and performance of this require a high number of these calculations (as the optimizer. The aspects considered for optimization are gen- fitness function is based on the model predictions). erally represented by heat transfer parameters, parameters ii) DE works with individuals represented by vectors of kinetic models and of vapor-liquid equilibrium models, of real numbers. The evolution process is based on phase equilibrium, phase stability, and chemical equilib- the idea that each parameter of the vector can have rium. On the other hand, the systems taken into account any continuous value. As the majority of real-world vary considerably, ranging from alcoholic fermentation and processes have characteristics limited to specific bioreactor systems to sour gas treatment plants, pyrolysis intervals, a constraint handling technique must be dehydrogenation, and supercritical water oxidation. implemented. Another aspect that must be taken into consideration is the existence of discrete values. In this case, a proper methodology for transforming the continuous values into discrete ones must be 5 Practical aspects introduced. iii) As discussed in Section 2, the performance of DE As it can be observed from the previous sections, DE has depends not only on the characteristics of the prob- been applied to a multitude of problems (model improve- lem being solved but also on the settings and variants ment, process optimization, process control, system used. The main aspects that can be varied to change design, and parameter determination) from the chemical the behavior of DE when tackling different types of engineering area. However, there are situations where the search spaces are type of mutation (how the base results can be significantly improved. Although DE is easy individual is chosen, how many individuals partici- to implement and use, this simplicity is deceiving because pate in the mutation phase), type of crossover (bino- there are a lot of aspects that influence the performance mial or exponential), how the control parameters and, therefore, the final results. In this context, in order to are selected (trial and error, deterministic, adaptive, obtain good solutions and to use the algorithm at its full self-adaptive), initialization (type of random-gener- potential, the chemical engineer should take the follow- ator algorithm, improvement with other approaches ing into consideration: (i) Is a global optimizer required such as OBL), hybridization (with other global algo- for the specific problem? (ii) What are the main character- rithms, with local algorithms, or with local-global istics of the system that can pose difficulties for a global combinations). In the majority of cases, the empirical optimizer and what are the modifications that need to be and statistical studies showed that the self-adaptive performed so that the algorithm can be applied? (iii) Is the improved versions (hybridized) are better than the version selected the best one? classic variants. However, some of the hybridization 174 E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering
performed by the researchers are problem specific, to be implemented in real situations (lab scale setups or and therefore, special attention must be given when plants) are obtained. Although, as an optimizer, DE is not trying to apply that version to a chemical engineering a panacea, having its share of problems and drawbacks system. For example, DE hybridized with BK is appli- (which researchers try to minimize by employing differ- cable only in cases where the process model is rep- ent types of approaches as self-adaptation, introduction resented by an ANN and the scope is to improve the of new mutation strategies, or hybridization), its use in model performance. Also, the hybridizations are more the chemical engineering field is beneficial, with multiple complex and can have an exponential requirement for systems being improved with its help. computational resources, which makes them unsuit- In the last years, the DE applications in the chemi- able when dealing with systems with low resources. cal engineering field had a rising trend, with this review presenting the main research performed in this area. If for the hybrid versions, extended studies to determine Different types of processes and different types of prob- the exact characteristics of the problems where each lems were solved, the main classes being represented by variant performs better have not yet been performed, in process optimization and model optimization. In case of the case of simple variants, the most effective strategies process optimization, the scope was to identify the values are Rand/1 and Best/2 (Hu and Yan 2011). The experimen- of process parameters (operating conditions) that lead tal results showed that the strategies based on the best to maximization/minimization of specific objectives. For solution found so far have a fast convergence speed and model optimization, the intention was to determine the perform well on unimodal problems but are more likely to optimal values of model inner parameters that determine get stuck at local optimum when solving multimodal prob- a good correlation between model predictions and experi- lems (Qin et al. 2009). Concerning the choice of crossover mental data. type, studies show that, in general, the binomial version A clear distinction between the variants of DE used tends to be a little more efficient than exponential (Lin was made: (i) simple, unmodified versions (also called et al. 2010). Also, the initialization type used has an influ- classic approaches) and (ii) modified versions (which ence on the general performance: the closer the initial include, among others, aspects such as self-adaptation solutions to the global minima, the higher the probability and hybridization with other optimization techniques or of determining good solutions. The most used approaches with special chemical engineering modeling approaches of improving initialization are the ones based on OBL or software). principles (Thangaraj et al. 2012, Dragoi et al. 2013a). Concerning model optimization, the number of Except the aspects directly related to applicabil- works combining DE with statistical models is relatively ity, similar to the main directions of research concern- small when compared with the ones tackling the problem ing the DE algorithm, its use for chemical engineering of other types of model optimization (artificial-intelli- problems must focus on how parameters are set and gence-based approaches, ANNs or support vectors, and how the performance can be improved though different deterministic). mutation types and hybridization. Therefore, even if the Related to the aspect of which chemical engineering classic approaches provide acceptable solutions, the use areas are the most active (in the last 5 years) in applying of improved versions that show higher performance for DE as a state-of-the-art optimization techniques, the fuel benchmark functions can lead to better results if proper and energy fields are in top, especially in case of process care is taken and all aspects linking the algorithm to the optimization with simple versions. process parameters are considered (type of parameter, An important conclusion is the fact that modeling limits, interdependency). Another important point that and optimization procedures based on DE algorithm are must be mentioned is the fact that in order to have a repro- promising tools, efficient, and reliable, due to the accu- ducible study, all parameters and characteristics of the rate results generally provided. process and of the algorithm must be clearly mentioned.
Nomenclature
6 Conclusions ANN artificial neural network CDR conventional DME synthesis reactor By applying DE to solve specific aspects of chemical engi- CMR conventional methanol reactor neering problems, optimal solutions with a great potential Cr crossover probability E.N. Dragoi and S. Curteanu: Differential evolution in chemical engineering 175
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In: Swarm and evolutionary computation. Berlin Heidelberg: Silvia Curteanu Springer, 2012: 154–161. Faculty of Chemical Engineering and Zhang J, Sanderson AC. Adaptive differential evolution: a robust Environmental Protection, Department of approach to multimodal problem optimization. Berlin: Chemical Engineering/Applied Informatics, Springer, 2009. “Gheorghe Asachi” Technical University Iasi, Zhang H, Bonilla-Petriciolet A, Rangaiah GP. A review on global Bd. Prof.dr.doc. Dimitrie Mangeron, No. 71A, optimization methods for phase equilibrium modeling and 700050, Iasi, Romania, calculations. Open Thermodynam J 2011a; 5: 71–92. [email protected] Zhang H, Fernandez-Vargas JA, Rangaiah GP, Bonilla-Petriciolet A, Segovia-Hernandez JG. Evaluation of integrated differential Silvia Curteanu has been a Professor and PhD supervisor in evolution and unified bare-bones particle swarm optimization Chemical Engineering since 2005 at “Gheorghe Asachi” Technical for phase equilibrium and stability problems. Fluid Phase University of Iasi, Romania, Faculty of Chemical Engineering and Equilib 2011b; 310: 129–141. Environmental Protection. She is coordinator of the Applied Infor- Zhang H, Rangaiah GP, Bonilla-Petriciolet A. Integrated differential matics Laboratory and Research Center “Chemical and Biochemical evolution for global optimization and its performance for Process Engineering and Advanced Materials”. Her professional modeling vapor-liquid equilibrium data. Ind Eng Chem Res experience and research interests are artificial intelligence tools 2011c; 50: 10047–10061. applied in chemical engineering, neural networks methodologies Zhang H, Rangaiah GP. One-step approach for heat exchanger used for modeling purposes, and evolutionary algorithms (genetic, network retrofitting using integrated differential evolution. differential evolution, artificial immune algorithms) applied for Comput Chem Eng 2013; 50: 92–104. process optimization. Prof. Curteanu has more than 150 publica- Zhao C, Xu Q, Lin S, Li X. Hybrid differential evolution for estimation tions (scientific papers and books). Web page address: http://www. of kinetic parameters for biochemical systems. Chin J Chem ch.tuiasi.ro/cv/ic/curteanusilvia/index.html. Eng 2013; 21: 155–162. Zielinski K, Laur R. Stopping criteria for differential evolution in constrained single-objective optimization. In: Chakraborty U, editor. Advances in differential evolution. Berlin/Heidelberg: Springer, 2008: 111–138.
Bionotes
Elena Niculina Dragoi Faculty of Chemical Engineering and Environmental Protection, Department of Chemical Engineering/Applied Informatics, “Gheorghe Asachi” Technical University Iasi, Bd. Prof.dr.doc. Dimitrie Mangeron, No. 71A, 700050, Iasi, Romania
Elena Niculina Dragoi received a BA degree in Computer Science and a MA degree in Distributed Systems from the “Gheorghe Asachi” Univer- sity of Iasi, Faculty of Automatic Control and Computer Engineering, in 2008 and 2009, respectively, and a PhD degree in Chemical Engineer- ing from the “Gheorghe Asachi” University of Iasi, Faculty of Chemical Engineering and Environmental Protection, in 2012. Her research interests include evolutionary computation, bio-inspired algorithms, artificial neural networks, and their application for solving modeling and optimization problems related to chemical engineering.