Material Balances for Chemical Engineers (03-24-2014)

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Unformatted text preview: See discussions, stats, and author profiles for this publication at: Material Balances for Chemical Engineers Book · April 2014 CITATION READS 1 8,509 3 authors: Ramon L Cerro Brian G Higgins University of Alabama in Huntsville University of California, Davis 108 PUBLICATIONS 1,608 CITATIONS 67 PUBLICATIONS 1,787 CITATIONS SEE PROFILE SEE PROFILE Stephen Whitaker University of California, Davis 220 PUBLICATIONS 12,625 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Diffusion of Charged Species in Liquids View project All content following this page was uploaded by Stephen Whitaker on 04 April 2014. The user has requested enhancement of the downloaded file. Material Balances for Chemical Engineers d dt òc V A dV + òc v A A × n dA òR = A é k I cBr2 ù ê ú é RBr2 ù é -1 0 -1 0 1 2 ù ê k II cBr cH 2 ú ê ú ê 0 -1 0 1 ú êk c c ú = 0 R ê H2 ú ê ú ê III H Br2 ú êR ú êë 0 1 1 -1 0 úû ê k IV cH cHBr ú ë HBr û 14444 4244444 3ê ú 2 stoichiometric matrix ëê k V cBr ûú A dV V 5 6 4 mixer reactor 3 1 2 a b s o r b e r 8 7 R.L. Cerro, B.G. Higgins & S. Whitaker Ramon L. Cerro Department of Chemical Engineering University of Alabama at Huntsville Brian G. Higgins and Stephen Whitaker Department of Chemical Engineering and Material Science University of California at Davis Dedication To our wives Frances, Sandra and Suzanne for their patience and support ii Historical Perspective More than half a century ago, the typical chemical engineering program began with a course devoted to material and energy balances. The dominant text was Chemical Process Principles (1943) by Hougen and Watson. Stoichiometry was covered in four pages and restricted to single independent reactions. More than sixty years later the typical chemical engineering program still begins with a course on material and energy balances. But much has changed in those intervening years. For example, students now have access to powerful computational software (Matlab, Mathematica) and simulators (Aspen). Yet the discussion of stoichiometry in standard undergraduate textbooks is still restricted to mostly single independent reactions, with little if any methodology for handling real world stoichiometry involving multiple independent reactions. Paradigm Shift Motivated by the “sketches” of Aris in Introduction to the Analysis of Chemical Reactors (1965), the “foundation” provided by Amundson in Mathematical Methods in Chemical Engineering (1966), and the “perspective” of Reklaitis in Introduction to Material and Energy Balances (1983), we have focused on providing a rigorous treatment of material balances for reacting systems. It is a treatment that will not require revision and upgrading in a subsequent course. Our presentation is based on the two axioms for the mass of multicomponent systems, and it is not limited to single independent reactions. The latter are dominant in the academic world and almost non‐existent in the real world where our students must practice their profession. The philosophy behind this text is that the axioms and the associated proved theorems are the tools that we use to solve material balance problems, and they are also the basis for the use of computational software to solve more demanding problems. We have purposely avoided any extensive use of software in this text because we believe that having a firm grasp of the theory is what is needed most. There is ample opportunity for students in subsequent courses to focus on numerical solutions of specific problems. In this text we make a clear distinction between global stoichiometry, local stoichiometry, and the stoichiometry related to elementary reaction steps. It is of no value to the students to shield them from the underlying theory, thus we propose a paradigm shift from the approach initiated by Hougen and Watson to a new and rigorous analysis of chemical engineering fundamentals. iii Preface This text has been written for use in the first course in a typical chemical engineering program. That first course is generally taken after students have completed their studies of calculus and vector analysis, and these subjects are employed throughout this text. Since courses on ordinary differential equations and linear algebra are often taken simultaneously with the first chemical engineering course, these subjects are introduced as needed. Chapter 1 introduces students to the types of macroscopic balance problems they will encounter as chemical engineers, and Chapter 2 presents a review of the types of units (dimensions) they will need to master. While the fundamental concepts associated with units are inherently simple, the practical applications can be complex and chemical engineering students must be experts in this area. Chapter 3 treats macroscopic balance analysis for single component systems and this provides the obvious background for Chapter 4 that deals with the analysis of multicomponent systems in the absence of chemical reactions. Chapter 5 presents the analysis of two‐phase systems and equilibrium stages. This requires a brief introduction to concepts associated with phase equilibrium. Chapter 6 deals with stoichiometry and provides the framework for the study of systems with reaction and separation presented in Chapter 7. Chapter 8 treats steady and transient batch systems with and without chemical reactions. Chapter 9 provides a connection between stoichiometry as presented in Chapter 6 and reactor design as presented in subsequent courses. Throughout the text one will find a variety of problems beginning with those that can be solved by hand and ending with those that benefit from the use of computer software. The problems have been chosen to illustrate concepts and to help develop skills, and many solutions have been prepared as an aid to instructors. Students are encouraged to use the problems to teach themselves the fundamental concepts associated with macroscopic balance analysis of multicomponent, reacting systems for this type of analysis will be a recurring theme throughout their professional lives. Many students and faculty have contributed to the completion of this text, and there are too many for us to identify individually. However, we would be remise if we did not point out that Professor Ruben Carbonell first introduced this approach to teaching material balances at UC Davis in the late 1970’s. iv CONTENTS 1. Introduction 1 1.1 Analysis versus Design . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . 5 1.2 Representation of Chemical Processes . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2. Units 12 2.1 International System of Units . . . . . . . . . . . . . . . . .. . . . . . . . . . 13 2.1.1 Molecular mass . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 15 2.1.2 System of units . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 16 2.2 Derived Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Dimensionally Correct and Dimensionally Incorrect Equations . . . . . . . 19 2.4 Convenience Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Array Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5.1 Units . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 27 2.6 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3. Conservation of Mass for Single Component Systems 39 3.1 Closed and Open Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.1.1 General flux relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.1.2 Construction of control volumes . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Mass Flow Rates at Entrances and Exits . . . . . . . . . . . . . . . . . . . . 56 3.2.1 Convenient forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3 Moving Control Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4. Multicomponent Systems 96 4.1 Axioms for the Mass of Multicomponent Systems . . . . . . . . . . . . . . . 96 4.1.1 Molar concentration and molecular mass . . . . . . . . . . . . . . . . . . . . 99 4.1.2 Moving control volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.2 Species Mass Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.2.1 Mass fraction and mole fraction . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.2.2 Total mass balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 v 4.3 Species Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.4 Measures of Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.5 Molar Flow Rates at Entrances and Exits . . . . . . . . . . . . . . . . . . . 113 4.5.1 Average concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.6 Alternate Flow Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.7 Species Mole/Mass Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.7.1 Degrees‐of‐freedom analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.7.2 Solution of macroscopic balance equations . . . . . . . . . . . . . . . . . . . 124 4.7.3 Solution of sets of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.8 Multiple Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.9 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.9.1 Inverse of a square matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.9.2 Determination of the inverse of a square matrix . . . . . . . . . . . . . . . . 142 4.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5. Two‐Phase Systems & Equilibrium Stages 156 5.1 Ideal Gas Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.2 Liquid Properties and Liquid Mixtures . . . . . . . . . . . . . . . . . . . . 163 5.3 Vapor Pressure of Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.3.1 Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.4 Saturation, Dew Point and Bubble Point of Liquid Mixtures . . . . . . . . . 170 5.4.1 Humidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.4.2 Modified mole fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.5 Equilibrium Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5.6 Continuous Equilibrium Stage Processes . . . . . . . . . . . . . . . . . . . 193 5.6.1 Sequential analysis‐algebraic . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5.6.2 Sequential analysis‐graphical . . . . . . . . . . . . . . . . . . . . . . . . . 203 5.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 6. Stoichiometry 227 6.1 Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 6.1.1 Principle of stoichiometric skepticism . . . . . . . . . . . . . . . . . . . . . 230 6.2 Conservation of Atomic Species . . . . . . . . . . . . . . . . . . . . . . . . 231 6.2.1 Axioms and theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 6.2.2 Local and global forms of Axiom II . . . . . . . . . . . . . . . . . . . . . . 236 vi 6.2.3 Solutions of Axiom II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 6.2.4 Stoichiometric equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 6.2.5 Elementary row operations and column/row interchange operations . . . . . 241 6.2.6 Matrix partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 6.3 Pivots and Non‐Pivots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 6.3.1 Rank of the atomic matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 6.4 Axioms and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 6.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 7. Material Balances for Complex Systems 269 7.1 Multiple Reactions: Conversion, Selectivity and Yield . . . . . . . . . . . . 269 7.2 Combustion Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 7.3 Recycle Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 7.3.1 Mixers and splitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 7.3.2 Recycle and purge streams . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 7.4 Sequential Analysis for Recycle Systems . . . . . . . . . . . . . . . . . . . . 318 7.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 8. Transient Material Balances 354 8.1 Perfectly Mixed Stirred Tank . . . . . . . . . . . . . . . . . . . . . . . . . . 355 8.2 Batch Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 8.3 Definition of Reaction Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 8.4 Biomass Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 8.5 Batch Distillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 8.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 9. Reactions Kinetics 395 9.1 Chemical Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1. Local and elementary stoichiometry . . . . . . . . . . . . . . . . . . . . . 9.1.2 Mass action kinetics and elementary stoichiometry . . . . . . . . . . . . . . 9.1.3 Decomposition of azomethane and reactive intermediates . . . . . . . . . . 395 403 404 407 9.2 Michaelis‐Menten Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 9.3 Mechanistic Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 9.4 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 9.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 vii Appendix A 439 A1. Atomic Mass of Common Elements Referred to Carbon‐12 . . . . . . . 439 A2. Physical Properties of Various Chemical Compounds . . . . . . . . . . 442 A3. Constants for Antoine’s equation . . . . . . . . . . . . . . . . . . . . . . 445 Appendix B: Iteration Methods 448 B1.Bisection Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 B2. False Position Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 B3. Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 B4. Picard’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 B5. Wegstein’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 B6. Stability of Iteration Methods . . . . . . . . . . . . . . . . . . . . . . . . 456 Appendix C: Matrices 462 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 Appendix D: Atomic Species Balances 470 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 Appendix E: Conservation of Charge 482 Appendix F: Heterogeneous Reactions 487 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 viii Chapter 1 Introduction This text has been prepared for use in what is normally the first chemical engineering course in a typical chemical engineering program. There are essentially two major objectives associated with this text. The first objective is to carefully describe and apply the axioms for conservation of mass in multicomponent, reacting systems. Sometimes these ideas are stated as mass is conserved or mass is neither created nor destroyed and in this text we will replace these vague comments with definitive mathematical statements of the axioms for conservation of mass. Throughout the text we will use these axioms to analyze the macroscopic transport of molecular species and their production or consumption owing to chemical reaction. The macroscopic mass and mole balances presented in this text are often referred to as material balances. A course on material balances is generally taken after students have completed courses in calculus, vector analysis, and ordinary differential equations, and these subjects will be employed throughout the text. Since a course on linear algebra is often taken simultaneously with the first chemical engineering course, the elements of linear algebra required for problem solving will be introduced as needed. The second objective of this text is to introduce students to the types of problems that are encountered by chemical engineers and to use modern computing tools for the solution of these problems. To a large extent, chemical engineers are concerned with the transport and transformation (by chemical reaction) of various molecular species. Although it represents an oversimplification, one could describe chemical engineering as the business of keeping track of molecular species. As an example of the problem of “keeping track of molecular species” we consider the coal combustion process illustrated in Figure 1‐1. Coal fed to the combustion chamber may contain sulfur, and this sulfur may appear in the stack gas as SO 2 or in the ash as Ca SO 3 . In general, the calcium sulfite in the ash does not present a problem; however, the sulfur dioxide in the stack gas represents an important contribution to acid rain. 1 2 Introduction Figure 1‐1. Coal combustion The sulfur dioxide in the stack gas can be removed by contacting the gas with a limestone slurry (calcium hydroxide) in order to affect a conversion to calcium sulfite. This process takes place in a gas‐liquid contacting device as illustrated in Figure 1‐2. There we have shown the stack gas bubbling up through a limestone slurry in which SO 2 is first absorbed as suggested by SO 2 gas SO 2 liquid (1‐1) The absorbed sulfur dioxide then reacts with water to form sulfurous acid H 2O SO 2 H 2SO 3 (1‐2) which subsequently reacts with calcium hydroxide according to Ca(OH)2 H 2SO 3 CaSO 3 2H 2O (1‐3) Here we have used Eq. 1‐1 to represent the process of gas absorption, while Eqs. 1‐2 and 1‐3 are stoichiometric representations of the two reactions involving sulfurous acid. The situation is not as simple as we have indicated in Eq. 1‐3 for the sulfurous acid may react either homogeneously with calcium hydroxide or heterogeneously with the limestone particles. This situation is also illustrated in Figure 1‐2 where we have indicated that homogeneous reaction takes place in the fluid surrounding the limestone particles and that heterogeneous reaction occurs at the fluid‐ solid interface between the particles and the fluid. It should be clear that “keeping Introduction 3 track of the sulfur” is a challenging problem which is essential to the environmentally sound design and operation of coal‐fired power plants. Figure 1‐2. Limestone scrubber for stack gases There are other mass balance problems that are less complicated than those illustrated in Figures 1‐1 and 1‐2, and these are problems associated with the study of a single chemical component in the absence of chemical reaction. Consider, for example, a water balance on Mono Lake which is illustrated in Figure 1‐3. It is not difficult to see that the sources of water for the l...
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