Regulation of the Reacted Layer Thickness in a Gas-Solid Reacting System
- 1 Departamento de Materiales, Universidad Autonoma Metropolitana Azcapotzalco, Mexico, D.F., Mexico
- 2 Departamento de Materiales, Universidad Autonoma Metropolitana Azcapotzalco, Mexico, D.F., Mexico
- 3 Centro de Investigacion e Innovacion Tecnologica-IPN, Mexico, D.F., Mexico
Abstract
A mathematical model for the analysis of a gas-solid reacting system is presented. This model is an alternative to the classical shrinking-core model. The model has a structure that can be easily transformed into a canonical control form, which is proper for controller synthesis. Analytical solution of the model to describe the open-loop behavior is expressed in terms of the Lambert function. The Lambert function is evaluated from aTaylorexpansion series. Besides, a controller is proposed to regulate the reacted layer thickness using initially the diffusion coefficient as control input. The control law is synthesized employing the feedback linearization technique. Main contributions of this work are the synthesis of the layer thickness controller, and the employment of the process temperature as substitute of the diffusion coefficient as the control input.
- Levenspiel, O. (1999) Chemical Reaction Engineering. 3rd Edition, Wiley, Hoboken.
- Son, H.Y. and Wadsworth, M.E. (2012) Rate Processes of Extractive Metallurgy. Springer, London.
- Braun, A., Bartsch, M., Schnyder, B. and Kotz, R. (2000) A Model for the Film Growth in Samples with Two Moving Reaction Frontiers an Application and Extension of the Unreacted-Core Model. Chemical Engineering Science, 55, 5273-5282. http://dx.doi.org/10.1016/S0009-2509(00)00143-3
- Braun, A., Wokaun, A. and Hermanns, H.-G. (2003) Analytical Solution to a Growth Problem with Two Moving Boundaries. Applied Mathematical Modeling, 27, 47-52. http://dx.doi.org/10.1016/S0307-904X(02)00085-9
- Jeffrey, D.J., Hare, D.E. and Corless, R.M. (1996) Unwinding the Branches of the Lambert W Function. The Mathematical Scientist, 21, 1-7.
- Chapeau-Blondeau, F. (2002) Numerical Evaluation of the Lambert W Function and Application to Generation of Generalized Gaussian Noise with Exponent 1/2. IEEE Transactions on Signal Processing, 50, 2160-2165. http://dx.doi.org/10.1109/TSP.2002.801912
- Corless, R.M., Jeffrey, D.J. and Knuth, D.E. (1997) A Sequence of Series for the Lambert W Function. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 197-204. http://dx.doi.org/10.1145/258726.258783
- Chapra, S.C. and Canale, R.P. (2010) Numerical Methods for Engineers. 6th Edition, McGraw-Hill, New York.
- Slotine, J.J. and Li, W. (1991) Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs.
- Khalil, H.K. (2002) Nonlinear Systems. 3rd Edition, Prentice-Hall, Upper Saddle River.
- Murakami, T., Fukuyama, H. and Nagata, K. (2001) Mechanisms of Carburization and Melting of Iron by CO Gas. ISIJ International, 41, 416-421. http://dx.doi.org/10.2355/isijinternational.41.416