Improved Inheritance Algorithm for the Assembly of Coal Fragments

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Improved Inheritance Algorithm for the Assembly of Coal Fragments Zhao Lei and Bolun Yang* Department of Chemical Engineering, State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China

Jianwei Li School of Chemistry and Chemical Engineering, Xi’an University of Science and Technology, Xi’an 710054, China ABSTRACT: There are many methods to simulate coal structure that might be beneficial to coal technologies. However, they often suffer from producing an impractical structure and a large calculation time and could not be applied to most rank coal. Hence, the hybrid inheritance algorithms combined with Elitist strategy (HIA-ES) and Monte Carlo and Elitist Strategy (HIA-MC-ES) are proposed to simulate the coal structure. The reported fragment structures of the coal macromolecule are summarized, and the fragment structure is accurately quantified with an adjacency matrix, an atom identification vector, bonding parameters, a bond characteristic index, and a functional group characteristic parameter, respectively. The humic acid (HA) is used to test the effectiveness of two algorithms since HA has the similar fragment structures with coal. With the use of the method of HIA-ES, the numbers of fragment structures is gained. Then, the parameters of the simulated structures are obtained with the HIA-MC-ES method. Simulated HA and Shenfu coal structures are highly similar with the reported results, indicating that the two novel hybrid inheritance algorithms are effective to simulate a molecular structure for the most rank coal.

1. INTRODUCTION Coal, a plentiful fossil fuel, will be the dominating material in the future and plays an extremely important role in many developing countries all over the world.1 Coal is considered as the heterogeneous substance formed after lignocelluloses or plant remains including lipids, lignin, protein, resins, pigments, polysaccharides, and less amounts of other materials were buried and have undergone a wide range of chemical transformations over geological time-scale periods. It is desirable to be able to explore and utilize the coal structure to aid in improving the efficiency of coal technology.2 Insights at a molecular level, processes with simulating the macromolecular structure of coal might be helpful in the development of the process.3 Coal consists of complex organic molecules,4,5 and different coals show different molecular structures but similar fragment structures for certain rank coal, which are bonded through various cross-links such as alkyl and ether. Generally, models for the description of coal structure can be classified into two categories. One is the coal macromolecular structure model68 and the other is the associated structure model of coal.9,10 For the coal macromolecular structure model, some structures of coal are artificially worked on the single molecular simulation software platform according to chemical knowledge.1115 Thereby, molecular construction was relevant to computer-aided molecular design16 since they could produce the reasonable macromolecular coal structure. Faulon17 developed an efficient program called SIGNATURE to enumerate the structure of large molecules, which can retain nonredundancy according to graph theory. However, this method could generate impractical structures because of having no mechanism to choose preferred structures. For this reason, Faulon18 further proposed a stochastic algorithm based on the r 2011 American Chemical Society

simulated annealing method that searched constitutional isomers with desired properties. This method might take much calculation time to construct large structures of coal since it required estimating all structures in each step of the annealing process. The associated structure model of the coal method can avoid producing an impractical structure. For this model, an initial structure of coal is obtained by experiment first, and then the coal structure is obtained based on a try-and-error approach. Takanohashi et al.19 obtained an initial structure using a solidstate 13C NMR spectrum and then iterated the coal model structure until spectra calculated by the model matched well with the experimental spectrum data to estimate the physical density of Upper Freeport coal. This method is in good agreement with analytical data from 13C NMR; however, the result might depend on the initial structure. Therefore, it might be necessary to explore a new approach method that is independent of the initial structure and could be applied to most rank coal. In this work, two new methods are proposed to simulate the coal macromolecular structure. A fragment structure taken from the reported literature13,20,21 is selected first based on the fact that different coals have a similar fragment structure for certain rank coal. Then, these fragment structure are quantified with an adjacency matrix, an atom identification vector, bonding parameters, a bond characteristic index, and functional group characteristic parameters. The hybrid inheritance algorithm21,22 combined with Elitist strategy (HIA-ES) is developed to get Received: August 3, 2011 Accepted: October 11, 2011 Revised: October 7, 2011 Published: October 11, 2011 12392

dx.doi.org/10.1021/ie201715q | Ind. Eng. Chem. Res. 2011, 50, 12392–12399

Industrial & Engineering Chemistry Research the suitable numbers of fragment structures, and then the coal structures parameters are obtained by the hybrid inheritance algorithms combined Monte Carlo with Elitist strategy (HIAMC-ES) method.2325 Finally, the simulated coal structure is drawn with these parameters. These two methods could simplify input parameters, which are the molecular weight, the aromatic carbon number, and the total carbon. Moreover, the proposed methods are not dependent on the initial coal structure obtained from experimental results. Also, Elitist strategy can optimize fragment structures which are stochastically chosen based on the Monte Carlo method and can save computational time to produce practical structures. More importantly, the fragment structure could be supplemented for simulating most rank coal. Thus, these methods could be applied to simulating the coal macromolecular structure in most rank coal. The humic acid (HA) is extracted from peat, leonardite, and lignite and is regarded as a part of coal.26 Moreover, the HA has similar fragment structures with low rank coal. For this reason, the steelink model of the HA monomer27 is taken to test the simulation process in our work. Then, input datum from the literature20 is applied to simulating Shenfu coal structure.

2. METHODS 2.1. Quantify the Fragment Structure in Coal Molecule. Fragment structures, summarized in Table 1, are selected from the literature13,20,21 and used to form coal molecular structure in this work. As shown in Table 1, structures 17 are aromatic ring compounds, structures 815 represent heterocyclic compounds, structures 1621 are aliphatic ring compounds, structures 2227 can only bond with the aromatic carbon atom linkage, structures 2835 can only bond with the aliphatic carbon atom linkage, structures 3642 can both bond with the aromatic carbon atom and aliphatic carbon atom. The fragment structure must be quantified in order to carry on numerical simulation. Thus, the fragment structure is described separately through the adjacent matrix, the atom identification vector, the bonding parameter, the bond characteristic index, and the functional group characteristic parameter. Adjacency Matrix. For representation of the fragment structure, the adjacency relation among atoms in a structural fragment is represented as an n n matrix, where n is the number of the atoms other than hydrogen. If two atoms are adjacent, the matrix element is 1, otherwise it is 0. As shown in Figure 1, the structure may be represented as an m matrix. Atom Identification Vector (AI). All sorts of atoms in a fragment structure are distinguished by an atom characteristic vector, which is an m dimension vector. Aliphatic carbon atom, aromatic carbon atom, sulfur atom, oxygen atom, nitrogen atom, and total carbon atom are labeled as 0, 1, 2, 3, 4, and 5, respectively. 8 > 0 aliphatic carbon atom > > > > 1 aromatic carbon atom > > > < 2 sulfur atom AIi ¼ 3 oxygen atom > > > > > 4 nitrogen atom > > > : 5 total carbon atom

Bonding Parameters (BP). The bonding parameter of a fragment structure is defined as an n dimension vector. If the

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atom can be bonded with other fragment structures, the vector element is 1, otherwise it is 0. Bond Characteristic Index (BCI). The kinds of structure in which one or more atoms can be connected with the aromatic carbon atom or the aliphatic carbon atom may be described as following. The atom denoted / can only bond with the aromatic carbon atom linkage. The atom denoted # can only bond with the aliphatic carbon atom linkage. Thus, the BCI gives information on a fragment structure connecting with one or more carbon atoms in other fragment structures. For aromatic carbon atoms, the array is expressed as a vector named Nar, in the number of fragment structures that could be bonded with the aromatic carbon. Also, for aliphatic carbon atoms, the array is expressed as a vector named Nal, where the number of fragment structure could be bonded with the aliphatic carbon. These two vectors have supplied the principle information. According to this principle and chemical knowledge, the fragment is stochastically selected and judged as bonding or not. The vector value scope between 0 and 3 indicated the atomic number links to the i position, as shown in Figure 2. Functional Group Characteristic Parameters (FGCP). A row vector with eight elements figures the fragment structure characteristic parameters. The first element denotes the molecular weight of the fragment structure; elements 28 represent, respectively, the number of C, H, O, N, S, aromatic C, and aliphatic C atoms, as shown in Figure 3. 2.2. Coal Simulation Process. Two methods (HIA-MC-ES and HIA- ES) have been developed to obtain a coal structure in the coal molecular structure simulation. The simulation flowchart is shown in Figure 4. First, we input the original data including the molecular weight, the aromatic carbon number, and the carbon number, respectively. Then, all of the satisfied fragment structures on the under- mentioned conditions are selected by the method of a genetic algorithm combined with HIA-ES. Then, each selected fragment structure is connected. In this way, one of the isomers is obtained with the genetic algorithm combined with HIA-MC-ES. The HA molecular weight, the aromatic carbon number, and the total carbon number is 769, 18, and 38, respectively.27 The HA structure is shown in Figure 5. 2.3. Establishment of Model for Selecting the Number of Fragment Structures. The integer programming is performed because the number of the fragment structure is an integer. Mathematically, the problem for choosing the numbers of fragment structures is formulated as follows: 42 Ai xi m1 min ð1Þ i¼1

∑

8 > > > > > > > > > > < subject to

> > > > > > > > > > :

42 Ai xi m1 e 0:5m1 i¼1 42 Bi xi m2 e 6 i¼1 42 Ci xi m3 e 1xi ∈ Z i¼1

∑

∑

ð2Þ

∑

These two formulas express the problem of objective optimization with constraints, and they can be rewritten as an objective optimization without constraint which is used in the HIA-ES and 12393

dx.doi.org/10.1021/ie201715q |Ind. Eng. Chem. Res. 2011, 50, 12392–12399

Industrial & Engineering Chemistry Research

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Table 1. Fragment Structure

expressed as eq 3: 0 min@

42

∑ Ai xi m1 i¼1

42 þ

42

!2 ð0:5m1 Þ2 þ

∑ Ci x i m 3

i¼1

!2

42

∑ Bi xi m2 i¼1

!2

!1=2 1

ð3Þ

where 0 e xi e [m1/Ai],i = 1, 2, ..., 42, xi is the integer value of [m1/Ai], Ai is the ith fragment structure molecular weight, Bi is the aromatic carbon number of the ith fragment structure, Ci is the aliphatic carbon number of the ith fragment structure, xi is the number of ith obtained fragment structure, m1 is the input molecular weight, m2 is the input aromatic carbon number, m 3 is the inputted aliphatic carbon number, and Z is an integer. 12394



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Figure 3. Functional group characteristic parameters.

The mutation is performed by the following equations. yj ¼ ½αymax þ ð1 αÞymin

Figure 1. Molecular and adjacency matrix demonstration.

Figure 2. Bond characteristic index.

2.4. Solving the Model by GA Combined with HIA-ES. The traditional methods cannot solve the integer optimization problem in selecting the number of fragment structures, which is because, first, the variety of molecular isomers can be constructed by the fragment structures, and second, it takes a huge computational workload to construct coal macromolecules.28 For example, the branch-and-bound solution strategy could not be adopted for the presence of an isomer.29 The cutting plane method does not work due to the aforementioned two reasons.30 The enumeration method does not work on account of the second reason.31 Thus, the hybrid inheritance algorithm combined with HIA-ES32 is adopted in this work. The proposed HIA-ES procedure woks though the following steps: (step 1) initialization: The procedure of creating the size of the initial population33 corresponds to the decision variable within d = [m1/g], the fragment structure maximal number, where g is the minimum molecular weight of fragment structures. The evaluation of each individual in terms of the objective function, eq 3, without constraints is required during the simulation. Step 2, codification: In the procedure, the population is coded in a decimal vector named D with 2 d, where D1, i is the number of the ith fragment structure and D2,i is the serial number of the ith fragment structure. Step 3, fitness function: Each fragment structure in the population is evaluated by a fitness function.34 The fitness value describes a measure of each fragment structure performance, which will be used on the later procedure. The fitness function, fi, is determined by the following equations. 0 ! !

fi ¼ @

42

∑

i¼1

2

A i xi m1

42 þ

42

ð0:5m1 Þ2 þ

∑ Ci x i m 3 Þ i¼1

42

∑

i¼1

2

Bi xi m2

where yj is the jth fitness functional value, ymax is the maximal fitness functional value of fragment structures, ymin is the jth minimal fitness functional value of fragment structures, and α is a random value in [0, 1] . Step 5, crossover: Fitness functional values of two fragment structures having the same fitness value are submitted to produce two new fitness functional values.35 The generated values are expressed as 8 < x0 ¼ ½αx1j þ ð1 βÞx2j 1j ð6Þ : x02i ¼ ½βx2j þ ð1 αÞx1j where x1j is the fitness functional value of the jth fragment structure, x2i is the fitness functional value of the ith fragment structure, x1j0 is the new fitness functional value of the jth fragment structure, x2i0 is the new fitness functional value of the ith fragment structure, and β is a random value in [0, 1]. Step 6, register: Mathematically, the proposed ES method is conducted in this step. The minimum value of fragment structures are recorded using the ES method named x with 1 42, in which the number of the ith fragment structure is at the ith position. The x could store the best fragment structure that has the perfect fitness functional value. If the fragment structure is selected, the fitness functional value is changed according to mutation. If there are two same fitness functional values, those values are changed according to crossover. Step 7, repeating these steps until the fitness function value is lower than 0.0001. 2.5. Bonding Fragment Structures Using GA Combined with HIA-MC-ES. The optimal solutions are obtained with the method of GA combined with HIA-ES. In other words, a sequence number of fragment structures can be gained. Furthermore, these fragment structures must be linked to simulate the structure. Hence, the method of GA combined with HIA-MC-ES is applied to combine selected fragment structures into one structure. The procedure is described as follows. Step 1, initialization: Generate a population for a 42-dimensional problem. Step 2, codification:36 Each fragment structure is regarded as a category, is corded as structural variable, and is described by an adjacency matrix, an atom identification vector, and functional group characteristic parameters (FGCP). Step 3, evalulation and ranking: The fitness of each fragment structure is evaluated, expressed as

!1=2 2

1

f ðiÞ ¼

ð4Þ

Step 4, mutation: The genetic mutation is introduced by changing the fitness value of the selected fragment structure.

ð5Þ

xi

∑ xi i¼1

ð7Þ

where ∑i=1xi is the total number of the ith fragment structure. The fragment structure is ranked according to the fitness. 12395



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Figure 4. Flowchart for constructing macromolecular coal.

Step 4, crossover: The first fragment structure is chosen as the maximal fitness value of fragment structures. The second is

randomly selected in the population with the MC method. Also then, the second is judged based on BCI and BP whether it could 12396



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Figure 5. HA structure.

Figure 6. Simulated HA structure.

be bonded with the first one or not. If they could be bonded, two fragment structures were carried on the linkage to produce the third. Otherwise, the second is chosen in other fragment structures except the first and the second. The rest may be deduced by analogy, until the one which can be bonded is found. The fragment structure after bonding is recorded using the ES method and also described by the adjacency matrix, atom identification vector, and FGCP. The best fragment structure is also registered during each generation by introducing the ES method. The new fragment structure parameters are referred to update the fragment structure parameters. Then, the population size is plus 1. Step 5, mutation: The fitness value of the selected fragment structure in step 4 is changed, and carried out by the equation z0j ¼ αzj

ð8Þ

where zj0 is the new fitness functional value of the jth fragment structure and zj is the fitness functional value of the jth fragment structure. Then, all of the fragment structures are renewedly ranked according to the fitness. Step 6, repeating these steps until x is empty. 2.6. Update Fragment Structure Parameters. When two fragment structures are connected to form a new fragment structure, the aforementioned parameters will be updated as follows (1) The connection of two fragment structures with p p and n n adjacency matrices will result in a new (n + p) (n + p) matrix, in which the main diagonal blocks are n n and p p matrices (the sum of the two matrices) and the element at the connecting atom is 1 and the others are 0. (2) The new atom identification vector is n + p dimensional and can be obtained by putting one atom identification vector behind another. (3) The BP of the composite structure is similarly formed by locating one array behind another, and the parameter of the original bonding atom becomes zero. (4) Linking the BCI vectors of two fragment structures into a new BCI array is accomplished by putting one BCI vector behind another one. But the Nar value (or Nal value) at the bonding position should decrease by 1. (5) A new row vector of fragment structure parameters for the new structure is the sum of two row vectors for the fragment structure parameters, namely, the new vector elements are obtained by adding corresponding elements of the two vectors.

3. RESULTS AND DISCUSSION In this program, the solution of integer programming selecting the number of fragment structures is unique because of the method combined with HIA-ES, while the simulating structures are diverse because the method combined with HIA-MC-ES is introduced. Also, when the bonding is performed between two functional groups, many isomers could be produced due to the

Figure 7. Another simulated HA structure.

different linkage positions, resulting in a heavy computational load. Additionally, the number of hydrogen atoms is presumably ignored.37 Thus, the optimal solution is single and can be expressed as x=[3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2 0 1 0 8 5 4 0 0 1 0 3 0 0 2 0 1 0 0], where the number in the ith position is the number of the ith fragment structure. In order to test HIA-MC-ES, two simulated structures are produced. The simulated molecular formulas of the structure are C38H43NO16 and are shown in Figures 6 and 7, respectively. It takes approximately 4.7 h to complete 157 generations with a population of size of 60 on the E255 processor. In Figures 6 and 7, the molecular weight, the aromatic carbon number, and the total carbon number are the same as the HA. The molecular weight, the aromatic carbon number, and the total carbon number are 769, 18, and 38, respectively, and shown in Figures 57. It can be obtained by comparing the HA structure with two simulating structures that the molecular weight and the carbon atom are in agreement with the HA. The simulated structures in Figures 6 and 7 are both isomers of the HA. With the application of HIA-MC-ES, a fragment structure is randomly selected with MC and judged whether it is bonded or not. One of isomers could be produced when running the program. Thus, two simulated structures are somewhat different with the HA. In Figure 6, two hydroxyl groups are both in the α position. Although this is different with the HA, this structure is reasonable in chemistry. In Figure 7, the no. 33 fragment structure is in the β position in cyclohexa-1,3-diene. However, the no. 33 fragment structure is in the α position in cyclohexa-1,3-diene in Figure 5. However, the structure in Figure 7 is a reasonable structure. Moreover, the composition of C (59.29%), H (5.63%), N (1.82%), and O (33.26%) by mass for simulated structures in Figures 6 and 7 are well in agreement with the HA structure parameters of C (59.29%), H (5.63%), N (1.82%), and O (33.26%). Thus, it can be claimed that the two proposed methods are both reliable to construct the coal macromolecular structure. The molecular weight, the aromatic carbon number, and the total carbon number of a fragment of Shenfu coal based on 12397



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Figure 8. Segment of Shenfu coal macromolecular structure.

elemental analysis and 13 C NMR is 3000, 120, and 189, respectively.20 The optimal solution is expressed as x = [4 2 1 0 0 1 0 0 2 0 2 0 0 1 0 0 0 0 0 1 0 6 7 0 6 13 10 20 0 6 0 0 0 3 0 0 1 1 15 2 0 0]. The simulated molecular formula of the structure is C197H188N3O31S. The result is expressed in Figure 8. It takes approximately 7.3 h to complete 1128 generations with a population of size 234 on the E255 processor. The simulated formula weight is 3003 with a relative error of 1%, which is compared with the experimental formula weight 3000. The simulated composition of C (74.73%), H (6.30%), N (1.40%), O (16.50%), and S (1.07%) by mass is very close to the ultimate analysis result of C (75.70%), H (5.208%), N (1.348%), O (16.60%), and S (0.305%).20

4. CONCLUSIONS New methods (HIA-ES and HIA-MC-ES) are proposed and applied to simulate the macromolecular structure. Parameters of simulated molecule structures are obtained with the proposed methods. The relative errors of the simulated HA structures are below 1.4%. Results show that the two simulated structures are similar with the HA structure, which indicate that the proposed methods are reliable and effective to apply in simulating coal macromolecular structure. It can be used as a reference for the macromolecular structure of coal processes. A segment of the Shenfu coal macromolecular structure is obtained by those methods, and the simulated results agree well with the analytical one, with a relative error of 1.4%. Two proposed methods could be applied for most coal rank to simulate macromolecular structure with updating the fragment structures. ’ AUTHOR INFORMATION Corresponding Author

*Phone: +86-29-82663189. Fax: +86-29-82668789. E-mail: [email protected].

’ NOTATION α = a random value in [0, 1] β = a random value in [0, 1] in crossover Ai = the ith fragment structure molecular weight AI = atom identification vector Bi = the ith fragment structure aromatic carbon number BCI = bond characteristic index BP = bonding parameters Ci = the ith fragment structure aliphatic carbon number d = the size of the initial population D = the population vector D1,i = the number of the ith fragment structure D2,i = the serial number of the ith fragment structure FGCP = functional group characteristic parameters m = a dimension of the atom identification vector m1 = the input molecular weight m2 = the input aromatic carbon number m3 = the inputted aliphatic carbon number n = a dimension of adjacency matrix g = the minimum molecular weight of fragment structures yj = the jth fitness functional value in mutation ymax = the maximal fitness functional value of fragment structures ymin = the jth minimal fitness functional value of fragment structures fi = the fitness function in HIA-ES f(i) = the fitness function in HIA-MC-ES p = a dimension of other adjacency matrix x = the number of obtained fragment structure x(i) = the obtained number of the ith fragment structure x1j = the fitness functional value of the jth fragment structure in crossover x2j = the fitness functional value of the ith fragment structure in crossover x1j0 = the new fitness functional value of the jth fragment structure x2i0 = the new fitness functional value of the ith fragment structure 12398


Industrial & Engineering Chemistry Research zj = the fitness functional value of the jth fragment structure zj0 = the new fitness functional value of the jth fragment structure Z = an integer

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Improved Inheritance Algorithm for the Assembly of Coal Fragments

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