A length factor artificial neural network method for the numerical solution of the advection dispersion equation characterizing the mass balance of fluid flow in a chemical reactor

In this article, a length factor artificial neural network (ANN) method is proposed for the numerical solution of the advection dispersion equation (ADE) in steady state that is used extensively in fluid dynamics and in the mass balance of a chemical reactor. An approximate trial solution of the ADE is constructed in terms of ANN using the concept of the length factor in a way that automatically satisfies the desired boundary conditions, regardless of the ANN output. The mathematical model of ADE is presented adopting a first-order reaction, and the steady-state case for the same is examined by estimating the numerical solution using the ANN technique. Numerical simulations are performed by choosing the best ANN ensemble, based on a combination of numerous design parameters, random starting weights, and biases. The solution obtained using the ANN method is compared to the existing finite difference method (FDM) to test the reliability and effectiveness of the proposed approach. Three cases of ADE are considered in this study for different values of advection and dispersion. The numerical results show that the ANN method exhibits a higher accuracy than the FDM, even for the smaller number of training points in the domain, and eliminates the instability issues for the case where advection dominates dispersion.

[1]  Naveen Kumar.  Unsteady flow against dispersion in finite porous media , 1983 .

[2]  Alaeddin Malek,et al.  Numerical solution for high order differential equations using a hybrid neural network - Optimization method , 2006, Appl. Math. Comput..

[3]  Anupam Yadav,et al.  An efficient algorithm based on artificial neural networks and particle swarm optimization for solution of nonlinear Troesch’s problem , 2015, Neural Computing and Applications.

[4]  J. B. Riggs,et al.  Basic Principles and Calculations in Chemical Engineering , 1974 .

[5]  E. O'Loughlin,et al.  DILUTION AND DECAY OF AQUATIC HERBICIDES IN FLOWING CHANNELS , 1975 .

[6]  Anders Krogh,et al.  Neural Network Ensembles, Cross Validation, and Active Learning , 1994, NIPS.

[7]  Y. H. Chen,et al.  Finite element analysis of contaminant transport in groundwater , 2002, Appl. Math. Comput..

[8]  Kevin Stanley McFall,et al.  Artificial Neural Network Method for Solution of Boundary Value Problems With Exact Satisfaction of Arbitrary Boundary Conditions , 2009, IEEE Transactions on Neural Networks.

[9]  Raja Muhammad Asif Zahoor,et al.  Neural network optimized with evolutionary computing technique for solving the 2-dimensional Bratu problem , 2012, Neural Computing and Applications.

[10]  J. H. Bowen,et al.  Basic Principles and Calculations in Chemical Engineering. : 2nd edn, 1967. Prentice-Hall New Jersey. 483 pp. 96s. , 1968 .

[11]  Dimitris G. Papageorgiou,et al.  Neural-network methods for boundary value problems with irregular boundaries , 2000, IEEE Trans. Neural Networks Learn. Syst..

[12]  S. M. Hassanizadeh,et al.  Analytical solutions of the convection–dispersion equation applied to transport of pesticides in soil columns , 1998 .

[13]  Behzad Ataie-Ashtiani,et al.  Numerical errors of explicit finite difference approximation for two-dimensional solute transport equation with linear sorption , 2005, Environ. Model. Softw..

[14]  H. S. Fogler,et al.  Elements of Chemical Reaction Engineering , 1986 .

[15]  Arjun Singh Yadaw,et al.  B-Spline collocation method for a two-parameter singularly perturbed convection-diffusion boundary value problems , 2008, Appl. Math. Comput..

[16]  Robert McKibbin,et al.  A comparison of some numerical methods for the advection-diffusion equation , 2006 .

[17]  C. Zheng,et al.  Applied contaminant transport modeling , 2002 .

[18]  V. Guvanasen,et al.  Numerical solutions for solute transport in unconfined aquifers , 1983 .

[19]  Dimitrios I. Fotiadis,et al.  Artificial neural networks for solving ordinary and partial differential equations , 1997, IEEE Trans. Neural Networks.

[20]  Kevin Stanley McFall,et al.  Automated design parameter selection for neural networks solving coupled partial differential equations with discontinuities , 2013, J. Frankl. Inst..

[21]  Behzad Ataie-Ashtiani,et al.  Error analysis of finite difference methods for two-dimensional advection-dispersion-reaction equation , 2005 .

[22]  Mohsen Hayati,et al.  Multilayer perceptron neural networks with novel unsupervised training method for numerical solution of the partial differential equations , 2009, Appl. Soft Comput..

[23]  James B. Rawlings,et al.  Chemical Reactor Analysis and Design Fundamentals , 2002 .

[24]  Md. Akram Hossain,et al.  Accuracy of the Taylor-Galerkin model for contaminant transport in groundwater , 1999, Appl. Math. Comput..

[25]  Hristo V. Kojouharov,et al.  Nonstandard methods for the convective-dispersive transport equation with nonlinear reactions , 1999 .

[26]  W. J. Alves,et al.  Analytical solutions of the one-dimensional convective-dispersive solute transport equation , 1982 .

[27]  Manoj Kumar,et al.  An Introduction to Neural Network Methods for Differential Equations , 2015 .

[28]  N. Smaoui,et al.  Modelling the dynamics of nonlinear partial differential equations using neural networks , 2004 .