Solution approach with green's functions for predicting the concentration of the slurry within a stirred tank reactor with nonlinear kinetics

Método de solución con funciones de Green para predecir la concentración de lodos dentro de un reactor tanque agitado con cinética no lineal

J. A. Ochoa-Tapia* and F.J. Valdés -Parada

Departamento de Ingeniería de Procesos e Hidráulica. División de Ciencias Básicas e Ingeniería. Universidad Autónoma Metropolitana-Iztapalapa. Av. San Rafael Atlixco 186 col. Vicentina, CP. 09340, México D.F., México. * Corresponding author. E-mail: jaot@xanum.uam.mx Tel. 58-04-46-00.

Abstract

The purpose of this work is to set the basis for a numerical scheme to solve the model that describes the diffusion and reaction, with nonlinear kinetics, in the dispersed catalytic pellets part of the slurry contained in a reactor tank. The method presented is based on the use of Green's functions for the solution of the linear problem. However, different analytical solution approaches can lead to identical expressions for the solution; some of these results are compared and discussed. The numerical solution for the nonlinear case relies on the use of an iterative procedure. At this point, it is evident that the main drawback of the method proposed for the solution of the nonlinear transient problem is the infinite Fourier series that represent the Green's function. For such reason, the presented method is also used to obtain fluid and pellet concentration profiles for the quasi-steady state and steady-state cases. The resulting expressions for such two simpler cases are used to predict the concentration profiles that are also compared with those resulting from the numerical solution of the problem using finite differences. The good agreement of the predictions indicates that more compact expressions for the Green's function will improve the efficiency of the new numerical scheme.

Keywords: Green's function, analytical solution, iterative scheme, nonlinear kinetics, stirred tank reactor.

Resumen

El propósito de este trabajo es sentar las bases de un esquema numérico para resolver el modelo que describe la difusión y reacción, con cinética no lineal, en la parte del lodo que contiene pellets catalíticos dispersos en un reactor tanque agitado. El método presentado se basa en el uso de funciones de Green para la solución del problema lineal. Sin embargo, diferentes métodos de solución analítica pueden llevar a expresiones idénticas de la solución; algunos de estos métodos son comparados y discutidos. La solución numérica del caso no lineal se basa en el uso de un procedimiento iterativo. En este punto, es evidente que la principal desventaja del método propuesto para la solución del problema no lineal transitorio son las series de Fourier infinitas que representan a la función de Green. Por estas razones, el método presentado es usado también para obtener los perfiles de concentración en el fluido y las partículas para los casos de estados cuasi-estacionario y estacionario. Las expresiones resultantes para estos casos más simples se usan para predecir los perfiles de concentración que son comparados con los resultantes de la solución numérica usando diferencias finitas. La buena concordancia de las predicciones indica que expresiones más compactas para las funciones de Green mejorarán la eficiencia del esquema numérico.

Palabras clave: funciones de Green, solución analítica, método iterativo, cinética no lineal, reactor tanque agitado.

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Acknowledgments

FVP expresses his gratitude to Fondo Sectorial de Investigación para la educación from CONACyT (Project number: 12511908; Arrangement number: 112087) for the financial aid provided.

References

Álvarez-Ramírez J., Valdés-Parada F.J., Álvarez J., Ochoa-Tapia J.A. (2007). A Green's function formulation for finite-differences schemes. Chemical Engineering Science 62, 3083-3091.         [ Links ]

Boe K., Angelidaki I. (2009). Serial CSTR digester configuration for improving biogas production from manure. Water Research 43, 166-172.         [ Links ]

Castrillón L., Fernández-Nava Y., Ormaechea P., Marañon E. (2013). Methane production from cattle manure supplemented with crude glycerin from the biodiesel industry in CSTR and IBR. Bioresource Technology 127, 312-317.         [ Links ]

Fan K.S., Kan N.R., Lay L.J. (2006). Effect of hydraulic retention time on anaerobic hydrogenesis in CSTR. Bioresource Technology 97, 84-89.         [ Links ]

Gargouri B., Karray F., Mhiri N., Aloui F., Sayadi S. (2011). Application of a continuously stirred tank bioreactor (CSTR) for bioremediation of hydrocarbon-rich industrial wastewater effluents. Journal of Hazardous Materials 189, 427-434.         [ Links ]

Hernández-Martínez E., Álvarez-Ramírez J., Valdés-Parada F.J., Puebla H. (2011a). An Integral formulation approach for numerical solution of tubular reactors models. International Journal of Chemical Reactor Engineering 9, Note S12.         [ Links ]

Hernández-Martínez E., Valdés-Parada F.J., Álvarez-Ramírez J. (2011b). A Green's function formulation of nonlocal finite-difference schemes for reaction-diffusion equations. Journal of Computational and Applied Mathematics 235, 3096-3103.         [ Links ]

Hernández-Martínez E., Puebla H., Valdés-Parada F.J., Álvarez-Ramírez J. (2013). Nonstandard finite difference schemes based on Green's function formulations for reaction-diffusion-convection systems. Chemical Engineering Science 94, 245-255.         [ Links ]

Kim D., Park S.W., Jun S. (2008). Axial Green's function method for multi-dimensional elliptic boundary value problems. International Journal for Numerical Methods in Engineering 76, 697-726.         [ Links ]

Kim J.K., Josic K., Bennet M.R. (2014). The validity of the quasi-steady-state approximations in discrete stichastic simulations. Biophysical Journal 107, 783-793.         [ Links ]

Mandaliya D.D., Moharir A.S., Gudi R.D. (2013). An improved Green's function method for isothermal effectiveness factor determination in one- and two-dimensional catalyst geometries. Chemical Engineering Science 91, 197-211.         [ Links ]

Mansur W.J., Vasconcellos C.A.B., Zambrozuski N.J.M., Rotunno-Filho O.C. (2009). Numerical solution for the linear transient heat conduction equation using an explicit Green's approach. International Journal of Heat and Mass Transfer 52, 694-701.         [ Links ]

Marroquín de la Rosa, J.O., Morones Escobar, R., Viveros García, T. and Ochoa-Tapia, J.A. (2002). An analytic solution to the transient diffusion-reaction problem in particles dispersed in a slurry reactor. Chemical Engineering Science 57, 1409-1417, 2002.         [ Links ]

Pedersen M.G., Bersani A.M., Bersani E., Cortese G. (2008). The total quasi-steady-state approximation for complex enzyme reactions. Mathematics and Computers in Simulation 79, 1010-1019.         [ Links ]

Reungsanga A., Sreela-or C., Plangklang P. (2013). Non-sterile bio-hydrogen fermentation from food waste in a continuous stirred tank reactor (CSTR): Performance and population analysis. International Journal of Hydrogen Energy 38, 15630-15637.         [ Links ]

Sales-Cruz, C.G., Valdés-Parada, F.J., Goyeau, B. and Ochoa-Tapia, J.A. (2012). Effect of reaction and adsorption at the surface layer of porous pellets on the concentration of slurries. Industrial and Engineering Chemistry Research 51, 12739-12750.         [ Links ]

Smith J.M. (1981). Chemical Engineering Kinetics, McGraw-Hill Chemical Engineering Series.         [ Links ]

Truskey G.A., Yuan F., Katz D.F. (2009). Transport Phenomena in Biological Systems, 2nd. Edition, Pearson, Prentice Hall.         [ Links ]

Tzafriri A.R. and Edelman E.R. (2007). Quasi-steady-state kinetics at enzyme and substrate concentrations in excess of the Michaelis? Menten constant. Journal of Theoretical Biology 245, 737-748.         [ Links ]

Valdés-Parada F.J., Álvarez-Ramírez J., Ochoa-Tapia J.A. (2005). An approximate solution for a transient two-phase stirred tank bioreactor with nonlinear kinetics. Biotechnology Progress 21, 1420-1428.         [ Links ]

Valdés-Parada F.J., Álvarez-Ramírez J., Ochoa-Tapia J.A. (2007). Análisis de problemas de transporte de masa y reaccioín mediante funciones de Green. Revista Mexicana de Ingeniería Química 6, 283-294.         [ Links ]

Valdés-Parada F.J., Sales-Cruz M., Ochoa-Tapia J.A., Álvarez-Ramírez J. (2008a). On Green's function methods to solve nonlinear reaction-diffusion systems. Computers and Chemical Engineering 32, 503-511.         [ Links ]

Valdés-Parada F.J., Sales-Cruz M., Ochoa-Tapia J.A., Álvarez-Ramírez J. (2008b). An integral equation formulation for solving reaction-diffusion-convection boundary-value problems.         [ Links ]