Variational Methods Applied to Problems of Diffusion and Reaction PDF
Preview Variational Methods Applied to Problems of Diffusion and Reaction
Springer Tracts in Natural Philosophy Volume 24 Edited by B. D. Coleman Co-Editors: S. S. Antman . R. Aris . L. Collatz . J. L. Ericksen P. Germain· W. Noll· C. Truesdell w. Strieder . R. Aris Variational Methods Applied to Problems of Diffusion and Reaction With 12 Figures Springer-Verlag New York Heidelberg Berlin 1973 William Strieder University of Notre Dame, Department of Chemical Engineering Notre Dame, Indiana 46556/ U.S.A. Rutherford Aris University of Minnesota, Department of Chemical Engineering and Materials Science Minneapolis, Minnesota 55455/U.S.A. AMS Subject Classifications (1970): Primary 49H05, 76R99, 76P05, 60170 Secondary 82A40, 60J65, 60J60 ISBN-13: 978-3-642-65626-2 e-ISBN -13: 978-3-642-65624-8 DOl: 10,1007/978-3-642-65624-8 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, repro duction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer Verlag Berlin Heidelberg 1973. Softcover reprint of the hardcover 1st edition 1973 Library of Congress Catalog Card Number 73-80873. Monophoto typesetting: Keter Press Ltd., Jerusalem. w. Strieder . R. Aris Variational Methods Applied to Problems of Diffusion and Reaction With 12 Figures Springer-Verlag Berlin Heidelberg New York 1973 William Strieder University of Notre Dame, Department of Chemical Engineering Notre Dame, Indiana 46556/ U.S.A. Rutherford Aris University of Minnesota, Department of Chemical Engineering and Materials Science Minneapolis, Minnesota 55455/US.A. ~~~-.--.. ---~~- AMS Subject Classifications (1970): Primary 49H05, 76R99, 76P05, 60170 Secondary 82A40, 60165, 60160 ISBN-13: 978-3-642-65626-2 e-ISBN -13: 978-3-642-65624-8 DOl: 10.1007/978-3-642-65624-8 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, repro duction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer Verlag Berlin Heidelberg 1973. Softcover reprint of the hardcover 1st edition 1973 Library of Congress Catalog Card Number 73-80873. Monophoto typesetting: Keter Press Ltd., Jerusalem. To Stephen Prager amicitiae et admirationis ergo Preface This monograph is an account of some problems involving diffusion or diffusion with simultaneous reaction that can be illuminated by the use of variational principles. It was written during a period that included sabbatical leaves of one of us (W.S.) at the University of Minnesota and the other (R.A.) at the University of Cambridge and we are grateful to the Petroleum Research Fund for helping to support the former and the Guggenheim Foundation for making possible the latter. We would also like to thank Stephen Prager for getting us together in the first place and for showing how interesting and useful these methods can be. We have also benefitted from correspondence with Dr. A. M. Arthurs of the University of York and from the counsel of Dr. B. D. Coleman the general editor of this series. Table of Contents Chapter 1. Introduction and Preliminaries . 1.1. General Survey 1 1.2. Phenomenological Descriptions of Diffusion and Reaction 2 1.3. Correlation Functions for Random Suspensions 4 1.4. Mean Free Path Statistics . 8 1.5. Void Point-Surface Statistics . 11 1.6. Variational Principles Applied to the Diffusion Equation. 12 1.7. Notation. 16 Chapter 2. Diffusion Through a Porous Medium . 18 2.1. Introduction 18 2.2. Diffusion Through an Isotropic Porous Medium 18 2.3. Variational Formulation for De . 20 2.4. Bounds on De for an Isotropic Suspension 22 2.5. Best Possible Bounds on De 26 2.6. Bounds on De for a Random Bed of Solid Spheres 30 2.7. Knudsen Diffusion Through a Porous Medium . 35 2.8. Variational Formulation for Knudsen Diffusion 37 2.9. Upper Bounds on the Knudsen Permeability. 39 Chapter 3. Diffusion Limited Reaction 42 3.1. Introduction 42 3.2. Diffusion Limited Precipitation . 43 3.3. The Spherical Cell Approximation 45 3.4. Variational Principle and Upper Bound on the Precipitation Rate . . 46 3.5. Diffusion Controlled Quenching 49 3.6. Upper Bound on ke . 50 3.7. Lower Bound on ke . 52 3.8. Random Quencher Particles 56 Chapter 4. Heterogeneous Catalysis 59 4.1. Introduction 59 Table of Contents IX 4.2. Variational Principles for Heterogeneous Catalysis: the Homogeneous Model 60 4.3. Application to First Order Kinetics in a Slab. 65 4.4. Application to Langmuir-Hinshelwood Kinetics in a Slab 68 4.5. Effects of Pellet Shape on the Effectiveness Factor . 78 4.6. Variational Principles for Heterogeneous Catalysis: the Discrete Model 83 4.7. Application to Linear Kinetics in a Slab: Discrete Model 89 4.8. Analysis of Experimental Data 91 4.9. Non-monotonic Kinetics 95 Bibliography . 97 Notation 100 Author Index 107 Subject Index 108 Chapter 1 Introduction and Preliminaries 1.1. General Survey The calculus of variations has been patient of many and varied interpretations and applications to physical problems. Indeed the notion of a principle of least activity has at times exercised an almost mystical fascination, as if it were the counterpart in natural philosophy of the dictum of Ockham in metaphysical. In mathematics the calculus of variations has a place as a discipline in itself with applications in the proving of existence theorems and the provision of estimates. In applied mathematics it serves to unify the basis of certain sets of equations and leads to numerical approximations to their solutions or bounds on certain important functionals. It is with the last aspect that this mono graph is concerned. Attention is restricted to a class of problems in volving diffusion and reaction, for, desirable as it might be to review the whole scope of variational principles in natural philosophy, the severely modest ambition of this monograph better meets the limitations of our ability and the compass of a tract. The broader aspect of the subject and applications in other areas may be explored elsewhere in the literature, which is indeed vast. The calculus of variations is a standard element of all books of "Mathematical Methods in ..." of which pride of place must be given to Courant and Hilbert's classic (1937). There is a wealth of introductory texts among which it is invidious to make a selection, while for applications in various areas one may turn to Funk (1962), Lanczos (1966), Lauwerier (1966), Gould (1966), Young (1969), Biot (1970), to mention but a handful. The volume of papers given at a conference in Chicago in 1965 (Donnelly, Herman and Prigogine 1966) is interesting in bringing together several aspects of applied mathematics. The contribution of Finlayson and Scriven in this volume is a valuable reminder that the approximation scheme that arises from a variational formulation may really be no different than that which comes from a more direct approach to the basic equations.
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