Prediction of the Minimum Spouting Velocity by Genetic Programming

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Prediction of the minimum spouting velocity by genetic programming approach Seyyed Hossein Hosseini, Mojtaba Karami, Martin Olazar, Reza Safabakhsh, and Mohammad Rahmati Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie5013757 • Publication Date (Web): 08 Jul 2014 Downloaded from http://pubs.acs.org on July 9, 2014

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Industrial & Engineering Chemistry Research

Prediction of the minimum spouting velocity by genetic programming approach Seyyed Hossein Hosseinia*, Mojtaba Karamib,d, Martin Olazarc, Reza Safabakhshd, Mohammad Rahmatid a

b

Department of Computer and Information Technology, Ilam University, Ilam 69315–516, Iran c

d

Department of Chemical Engineering, Ilam University, Ilam 69315–516, Iran

Department of Chemical Engineering, University of the Basque Country, Bilbao, Spain.

Department of Computer Engineering, Amirkabir University of Technology, Tehran 15914, Iran

Abstract A genetic programming (GP) algorithm is developed to estimate the minimum spouting velocity (Ums) in the spouted beds with cone base. In order to have a general model, five dimensionless variables including seven critical geometric and operating parameters of spouted beds, namely, column diameter, spout nozzle diameter, base angle, static bed height, particle diameter, particle density and gas density, have been taken as model inputs. A general correlation including nearly all fundamental and operating variables has been obtained based on the GP approach. The Ums values predicted by the GP are in fair agreement with those obtained by experiments, with a root mean square error of 0.1329 m/s. The model results show that GP can be used as an effective tool to provide relatively accurate information of minimum spouting velocity in conical spouted beds. Keywords: Minimum spouting velocity, Gas–solid, Spouted bed, Genetic programming.

*

. Corresponding author: Tel: +98-913-7944470; E-mail address: [email protected] (S. H. Hosseini)

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1. Introduction Nowadays the spouted bed reactor has evolved as an efficient fluid-solid contactor with many applications in the chemical, petrochemical and food process industries. Generally, spouted beds have some advantages compared to fixed and fluidized beds, such as smaller pressure drop, better gas-solid contact for large particles (Geldart D), higher capacity for treating solid particles that are thermally sensitive, have different sizes and/or tend to agglomerate, and simpler construction, because it does not require a plate or any other gas distributor device. The minimum spouting velocity is defined as the minimum gas velocity required for maintaining the spouting regime, which determines other important operating parameters, such as residence time and solid movement, among others. The minimum spouting velocity is a crucial variable determining the operating process in spouted beds and is one of the main parameters required in their design and scaling-up. Although numerous studies have approached the minimum spouting velocity, 1–4 there is not yet a widely accepted correlation for accurately predicting this velocity. Most of the correlations proposed based on least square fitting are restricted to limited ranges of the gas and particle properties and bed dimensions.5–9 It should be noted that the least square fitting is basically linear, whereas spouted beds operate according to a nonlinear behavior. Therefore, the available intelligent methods may be useful for these systems. Recently, Zhong et al.10 used a back-propagation neural network to predict the minimum spouting velocity (Ums) in spouted beds with a root mean square error (RMSE) of 0.203 m/s. They used 164 experimental data points for both training and validation of the BP neural network model. Wang et al.11 used the least square support vector machine (LS-SVM) to predict the minimum spouting velocity, and adaptive genetic algorithm to determine the LS-SVM 2


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parameters. They showed that the predicting capacity of the LS-SVM is higher than that of empirical correlations and the neural network. Similarly to the neural network technique the GP approach is a suitable tool that may provide satisfactory results in the treatment of the nonlinear behaviour of spouted beds. In the artificial intelligence, GP is an evolutionary algorithm-based methodology inspired by biological evolution to find computer programs that perform a user-defined task. It is a specialization of genetic algorithms (GA), in which each individual is a computer program. It is a machine learning technique used to optimize a population of computer programs according to a fitness landscape determined by a program’s ability to perform a given computational task. The most widely used genetic programming technique has been reported by Koza.12 This technique has been used in several applications, such as logical design,13 data classification,14 digital circuit area optimization,15 neural network evolution16,17 and image processing.18,19,20,21 The aim of this study is to develop a general correlation by using GP approach based on a wide range of experimental data reported in the literature to predict the minimum spouting velocity in spouted beds with conical base by considering most of the geometric and operating variables. Furthermore, the results predicted for the minimum spouting velocity in spouted beds using different empirical correlations are compared with those obtained by GP model. Finally, the equation proposed is verified by comparing the results predicted with available experimental data. 2. Establishment of the genetic programming technique 2.1. A summary of genetic programming approach

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Genetic programming (GP) is a heuristic evolutionary computing technique and one of the machine learning or artificial intelligence techniques. GP was introduced by Koza12, based on a tree representation of gens, and the original concept was derived from the GA. More recently, several researches have used tree-based GP modelling in order to ascertain process parameters.22,23,24,25,26 Cartesian genetic programming (CGP) was proposed as a general form of genetic programming27 based on graph representation of gens as a two-dimensional node grid. In this paper, a graphbased GP is used to explore a new correlation for predicting the minimum spouting velocity in spouted beds. It should be noted that the GP developed here differs from CGP27 in gen representation and mutation operator.28 In the current GP, expressions are exhibited in the form of directed acyclic graphs as shown in Fig 1a. These graphs are represented as a two-dimensional grid of computational nodes. The proposed graph-based GP has the advantage of size limitation for individuals in the population and resolves the bloat problems in tree-based GPs. The genes that make up the genotype in GP are integers that represent where a node gets its data, what operations the node performs on the data and where the output data required by the user are to be obtained (Fig 1.b). The genotype in GP has a fixed length. However, the size of the phenotype (in terms of the number of computational nodes) can be anything from zero nodes to the number of nodes defined in the genotype. A phenotype would have the same number of nodes as defined in the genotype where every node in the graph is required. The genotype–phenotype mapping used in GP is one of its defining characteristics as showed in Fig 1. The inner nodes are known as function

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(nonterminal); these function nodes consume one or more input values and produce a single output value (e.g., +, −, ×,√,log, exp, etc.). Similarly to the CGP technique18, the crossover operator in the proposed GP approach decreases the diversity of the population and increases the probability of trapping local minima, and therefore is not used in this study. The mutation operator used is a rapid point mutation operator. This mutation can change the input function of a randomly chosen node to another valid random value, delete a node and add a new node to the graph. If a function gene is chosen for mutation, then a valid value is the address of any function in the function set, whereas if an input gene is chosen for mutation, then a valid value is the address of the output of any previous node in the genotype or of any correlation input. Furthermore, a valid value for a program output gene is the address of the output of any node in the genotype or the address of a correlation input. The number of genes in the genotype that can be mutated in a single application of the mutation operator is defined by the user, and is normally a percentage of the total number of genes in the genotype. A total of 356 runs were carried out by GP algorithm, with each run including 2000 generation and 300 populations in each generation. The diversity of the population is an important factor in evolutionary algorithms as it guarantees the escape away from any local minimum. Fig. 2 shows the diversity of the best run among all 356 runs. The mean and variance of the population diversity in 356 runs are 94.6657 and 14.2945, respectively. Consequently, the GP algorithm has a reasonable diversity. Furthermore, the mean and variance for the 356 runs has been calculated and are 0.1763 and 0.0012, respectively. Accordingly, the convergence of the developed GP is achieved. 5



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2.2. Description of the problem for intelligent prediction of Ums Almost all the previous studies report that the minimum spouting velocity is affected by several parameters, such as column diameter (Dc), spout nozzle diameter (Di), static bed height (H0), particle diameter (dp), particle density (ρp), gas density (ρg) and column base angle (γ). The first step in the process for predicting Um lies in finding the dependence of the minimum spouting velocity on theses parameters combined as dimensionless groups. The relevant relationship is as follows:  d H D ρ − ρg U ms γ  = f p, 0, i, p , tan( )  ρg 4  2H 0 g  Dc Dc Dc

(1)

The input parameters used in this paper for proposing the GP are those shown in eq. (1). 3. Results and discussion 3.1. Training of GP The back propagation neural network proposed by Zhong et al.10 is based on two series of experimental data reported in the literature for training and validating the neural network. In the present study, the two series of experimental data used by Zhong et al.10, 164 data points (shown in Table A1 in Appendix 1), are used for training the model based on GP approach. Therefore, our model covers a wide range of experimental data in terms of geometric and operating parameters. Experimental data of the minimum spouting velocity obtained by Epstein et al.29, Fane and Mitchell,30 Day et al.31, Lim and Grace,32 Choi and Meisen,7 Anabtawi8 and Venkatachalam et al.33 are used to train the model.

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Intelligent methods like the artificial neural network do not provide explicit forms of the relationship between Ums and the geometric and operating parameters. Nevertheless, the GP approach provides an applicable correlation, which is the main advantage of this method. Accordingly, the following relationship between the minimum spouting velocity and the significant parameters affecting this velocity is proposed by means of GP: 0.8  D 0.8  d  Di    U ms p i  A ( A − B ) =   +  +  Dc  Dc    2 H 0 g  Dc     

(2)

where d  d  H  A =  p  Log 2  p  /  0  +  Dc   Dc   Dc  d B= p +  Dc 

d p  ρ p − ρg  Dc  ρ g

  

0.3

2

0.8  Di   H 0 +   Dc  Dc   Dc 

dp

(3)

As observed in equation (3), the column base angle does not appear in the correlation. This means that the effect of column base angle is negligible compared to other terms, which is consistent with earlier studies regarding minimum spouting velocity. In order to evaluate the values of Ums predicted by GP or by empirical correlations, they should be compared with the corresponding experimental data, and root mean square error (RMSE) used as a fitting index. RMSE is a measure of the differences between the values predicted by a model and the values actually observed in the experiments. This term is expressed as:

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n

RMSE =

∑ (U i =1

Exp. msi

Calcu. − U msi )

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2

(4)

n

Exp. Calcu. is the ith experimental value and U msi is the corresponding value predicted by the where U msi

GP or by the empirical equation, and n is the number of data. Fig. 3 compares the Ums values predicted by GP with the experimental data points for cylindrical spouted beds. A good agreement is observed between predicted results and experimental data. The best RMSE obtained by the GP algorithm is 0.1329 m/s, which is considerably lower than that of by Zhong et al.[10]. Note that the current model covers twice the number of experimental data points used in the neural network proposed by Zhong et al.10. Therefore, GP is a powerful and promising tool to predict Ums. Predictions by five commonly used expressions to predict the minimum spouting velocity summarized in Table 1 are also compared with experimental data, Fig. 4. As observed, these empirical models follow the same trend as the corresponding experimental values. The RMSE of all equations are also listed in Table 1. The RMSE of the predictions made by the GP, Fig. 3, is 0.1329 m/s, whereas the RMSE values for those calculated by the five empirical correlations, Fig. 4, exceed the value for GP. The Mathur and Gishler5 equation provides the closest RMSE to 0.1329 m/s (0.2191 m/s), whereas those for the others are higher than 0.25 m/s. The Murthy and Singh6 correlation predicts the results with the highest deviation from the experimental measurements. It should be noted that none of the five empirical equations contains the base angle of the bed. Furthermore, GP shows that Ums is independent of the base angle, i.e., its effect is not significant. 8


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3.2. Verification of GP

To verify the GP, the Ums predictions obtained using the current model trained with 164 data points (Equations (2) and (3)) have been compared with the experimental data obtained by San José et al.34 (79 experimental data points) for hydrodynamics of shallow spouted beds. The experimental data by San José et al.34 are listed in Table A2 in Appendix 1. Fig. 5 shows a comparison of Ums values predicted by GP with those predicted by Mathur and Gishler’s equation,5 as well as experimental data points29. The equation by Mathur and Gishler5 has been chosen because it is the one that provides the lowest RMSE (0.2191 m/s) compared to the other empirical equations. As observed, GP provides better predictions than the equation by Mathur and Gishler5. The RMSE of the predictions made by the GP is 0.39 m/s, whereas the RMSE of those calculated using the equation by Mathur and Gishler5 is 0.516 m/s. Therefore, the equation produced by GP is suitable for predicting Ums in shallow spouted beds. Fig. 6 shows the effect of the static bed heigh on Ums in a shallow spouted bed with column diameter Dc = 0.15 m and spout nozzle diameter Di = 0.03 m. The solid particles used belong to Geldart group D and are glass spheres ( ρ p = 2420 kg/m3) with a particle diameter of 3 mm. This figure compares the experimental results34 with those calculated by GP. Both results show a similar trend, i.e., an increase in static bed height involves an increase in minimum spouting velocity. Fig. 7 shows the effect of particle diameter on Ums in a shallow spouted bed with column diameter Dc = 0.15 m, spout nozzle diameter Di = 0.03 m and static bed height 0.1 m. The solid particles used are also glass spheres ( ρ p = 2420 kg/m3) with diameters of 1, 3 and 6 mm. This 9



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figure compares the experimental data34 with the results calculated by GP. The calculated results and experimental ones follow a similar trend, wherein minimum spouting velocity increases by increasing particle diameter. Figs. 6 and 7 show that the equation produced by GP predicts well Ums in shallow spouted beds, and therefore it is suitable for scaling-up and design purposes.

4. Conclusions An intelligent method, namely genetic programming (GP) algorithm was developed for estimating the minimum spouting velocity (Ums) in the spouted beds with conical base. Accordingly, five dimensionless moduli including column diameter, spout nozzle diameter, base angle, static bed height, particle diameter, particle density and gas density have been taken as model inputs. A general correlation including nearly all fundamental and operating variables is suggested based on a GP approach. The GP predictions have been compared with those obtained using the more faithful correlations reported in the literature. It was found that Ums values predicted by GP are in good agreement with those obtained by experiments, with a root mean square error of 0.1329 m/s. A comparison of the Ums predictions obtained using the current model based on GP with the more used equation by Mathur and Gishler5 and with experimental data allows confirming that GP is an effective tool to provide accurate information of minimum spouting velocity in spouted beds with conical base.

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It should be noted that GP is free from the limitations of the other intelligent methods, such as neural network, which do not provide explicit forms of the relationship between Ums and the effective parameters.

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Captions Table 1. Commonly used correlations to predict Ums and their RMSE predictions. Table A1. Training data sets from the literature. Table A2. Experimental data obtained by San José et al.34 Fig. 1. (a) Graph-based (phonotype) representation of the expression

, (b)

Genotype representation of the graph in (a). Fig. 2. Diversity based on the best run. Fig. 3. Comparison of Ums predicted by GP approach with experimental data.7,8,29,30,31,32,33 Fig. 4. Comparison of Ums calculated by empirical equations with experimental data.7,8, 29,30,31,32,33

Fig. 5. Comparison of Ums values predicted by GP with those predicted by Mathur and Gishler’s equation5 and experimental data34. Fig. 6. Effect of static bed heigh on Ums prediction by GP and measured data34. Fig. 7. Effect of particle diameter on Ums prediction by GP and measured data34.

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Table 1. Commonly used correlations to predict Ums and their RMSE predictions. Author

Correlation

Mathur and Gishler 5

1/ 3

U ms

Murthy and Singh 6

RMSE

 d p   Di  =    Dc   Dc 

 d p   Di  U ms = 1.4     Dc   Dc 

Choi and Meisen 7

 ρ p − ρg 2 gH 0   ρ g  0.125

 ρ p − ρg 2 gH 0   ρ g  1.17

 Di     Dc 

0.312

 H0     Dc 

U ms

 dp  = 13.5(2 gH 0 )    Dc 

U ms

 dp  = 0.25    Dc 

U ms

 d p   Di  =    Dc   Dc 

Anabtawi 8

Olazar et al. 9

0.5

0.65

 Di     Dc 

0.1

0.2191

  

0.372

 H0     Dc 

0.254

 ρ p − ρg 2 gH 0   ρ g 

0.8389

   −0.148

 ρ p − ρg   ρg

 ρ p − ρg 2 gH 0   ρ g 

  

17


  

  

0.289

0.3126

0.2561

0.4709


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Fig 1. (a) Graph-based (phonotype) representation of the expression (y-x)*x-(y-x) 61x82mm (300 x 300 DPI)


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Fig. 1(b) Genotype representation of the graph in (a). 33x40mm (300 x 300 DPI)



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Fig. 2. Diversity based on the best run. 148x111mm (96 x 96 DPI)


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Fig. 3. Comparison of Ums predicted by GP approach with experimental data.7,8,29,30,31,32,33 192x128mm (300 x 300 DPI)



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Fig. 4. Comparison of Ums calculated by empirical equations with experimental data.7,8, 29,30,31,32,33 189x147mm (300 x 300 DPI)


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Fig. 5. Comparison of Ums values predicted by GP with those predicted by Mathur and Gishler’s equation5 and experimental data34. 134x101mm (300 x 300 DPI)



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Fig. 6. Effect of static bed heigh on Ums prediction by GP and measured data34. 176x128mm (300 x 300 DPI)


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Fig. 7. Effect of particle diameter on Ums prediction by GP and measured data34. 128x90mm (300 x 300 DPI)


Prediction of the Minimum Spouting Velocity by Genetic Programming

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