Subjects Science Technology Chemistry Nonfiction. CP Process Engineering Research Project 3 credits Pre-requisites: A project proposal with the work plan and prior approval of the Head of the Department for the selected project proposal and the work plan. Assesment Percentage Marks Stage 1: Assessment on the project proposal, work-plan, and the literature survey submitted before the 4th week via presentation by the student or student group during the 4th week.
Premixed and diffusion flames; properties, theory, laminar flame structure, stability limits, flame propagation, shock waves, detonation. Spray combustion; properties, atomization, combustion of droplets. Coal; characteristics, physical and chemical properties. Combustion of coal; particle ignition, flames, gasification of coal. Boilers, furnaces, burners, efficiency of combustion. Biomass; biomass conversion, synthetic fuels. Energy conservation, cogeneration. Pollutants formation and control. Crude oil classification; properties of petroleum products and their respective uses. Transportation of crude oil and gas.
Storage of petroleum products. Gas liquefaction, storage and transportation. Purification of petroleum products; lubrication oil refining. Petrochemical industry. Total 30 15 eq. Fluid flow in food processing; liquid handling, pumping equipment, pump selection and performance evaluation, flow and pressure measurements.
Energy for food processing; fuels, fuel utilization, boilers, generation of steam, electric power utilization. Heat transfer in food processing; heat transfer modes, steady state heat transfer, thermal properties of food, systems for heating and cooling of food products. Principles of food process design. Refrigeration; components of a refrigeration system, vapour compression refrigeration, refrigerants selection, cooling and cold storage of foods.
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Freezing; freezing systems, freezing of foods, freezing time, frozen food storage, thawing. Food dehydration, evaporation. Size reduction; introduction, mechanisms, size reduction equipment. Food packaging, controlled or modified atmosphere storage. Water supply and waste disposal; water quality, water purification, waste treatment and disposal. ISO Food safety management system.
Total 30 7. Industrial pollution with examples. Impact of pollution on ecosystems. The need for pollution prevention. Environmental standards for emission of pollutants. Industrial effluent pollution in major process and chemical industries with special emphasis on Sri Lankan industries.
CP and GP methodologies. CP and GP tools such as product modification, raw material substitution, good housekeeping, process control, brain-storming, eco mapping, fishbone diagram and Pareto diagram. Eco design concept. Tools for eco design. Economic development indices and their critique. Human development index and its critique. Discussion on sustainable development indices. Successful application of what are learnt in the said independent course modules to deal with practical process engineering systems require the following skills in an undergraduate Modelling of practical process engineering systems in a unified framework where momentum, energy and mass transport phenomena often occur simultaneously.
Concept of unit operations. Chemical thermodynamics. Mathematical modelling of steady-state and transient-state processes using mass and energy balances. Flow-charting of industrial processes using computer software. Equilibrium between phases.
Mass transfer: Diffusion. CP Reaction Engineering 3 credits Pre-requisites: none. Intended Learning Outcomes Upon successful completion of this course, students should be able to, Determine the parameters in kinetic expressions for different types of reactions eg. Kinetics of chemical and biochemical reactions.
Design of batch, semi-batch, continuous stirred tank and plug flow reactors with isothermal and non-isothermal operations. Design of dryers, absorption and stripping columns, binary and multi-component distillation columns, adsorption columns, extractors, crystallisers, cooling towers, mixers, settlers, plate, packed and spray columns, and their operations. Computational approaches in design. Boiling and condensation with industrial applications. Reviewing the use of psychometric charts. Problem identification and project formulation; search for, and retrieval of, information required such as literature survey; identification and optimum utilisation of available resources; project execution; cost analysis, socio-economic and ethical evaluations of the project, analysis of political and environmental consequences, and safety evaluations when applicable ; elements of technical report writing; communicating the results of the project study with the outside world via a report, a web-page, etc.
Intended learning outcomes Upon successful completion of this course, students should be able to, Explain the need and the importance of paying due attention to safety in industry Describe Sri Lankan legislations pertaining to industrial safety Discuss the causes of industrial accidents Size relief and vent valves Discuss the different types of strategies and procedures adopted in industry to reduce and mitigation of risks Perform risk assessment on a given process using different risk identification and analysis tools Describe the components of safety management systems.
Chemical hazards. Study in depth of a topic not available through other course work. Stage 1: Assessment of the work carried out on the study via presentation of the study results by the student during the 3rd week. Stage 3: Assessment of the work carried out via presentation of the study results by the student during the 11th week. Stage 4: Assessment of the final study report submitted during the 14th week or a written examination, as may be necessary. Process engineering design of a chemical, food or other process industry; Mechanical engineering outline design; Optimisation of process design; Outline of control system design; Operability study including start-up and shut-down; Material selection; Design codes; Determination of capital and operating costs; Study of environmental and other hazards; Process equipment selection, specification and design; Safety and loss prevention; Mechanical design of process equipment; Costing and project evaluation; Utilities; Environmental considerations waste management; noise; visual impact; legislation; environmental auditing.
Rather than presenting many facts about reactors, we focus on the framework for how to think about reactors — a framework for thinking that enables one, with some experience, to establish any of these facts for oneself, and discover new facts given new situations.
All engineering and science textbooks do this to some extent; in this text, we will do it to a rather large extent. Computations matter in this subject. Reactor fundamentals, like the fundamentals in any subject, are few in number. Rut the diversity of the consequences of these fundamentals is enormous. We attempt to exploit the significant advances in computing algorithms, software, and hardware in order to revise and streamline the presentation of reactor fundamentals. For example, all calculations required for the figures in this text were performed with Octave.
The goal is to develop sufficient expertise so that students can set up appropriate models from the basic principles for each new problem they encounter. That is the time-tested way to Instill confidence that one can analyze a new situation, which we fully expect to be the experience of practicing engineers. Newly practicing engineers certainly will need to learn the economics of manufacturing their main products, main reactor configurations, de- taiicd energy-recovery schemes, contacting patterns, new catalysts, and reactor monitoring and control systems. Rather than make inadequate efforts to include all of this information as a survey during the reactor design course, we feel this material deserves separate coverage, or can be learned during the early years of engineering practice.
Grounding in the fundamentals enables you to explore with confidence and be creative. More recently, Warren also provided-many helpful comments on computational issues involving reaction-diifusion, and parameter estimation.
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Bob Bird provided many helpful comments on presenting the material related to transport phenomena. He also provided detailed feedback on a review copy of the text. His feedback and encouragement are very much appreciated. John Falconer also provided many helpful comments. Their feedback was very helpful. Several graduate students contributed to the software, which is an essential part of this text. Ken Muske developed some examples for the chapter on mixing.
Chris Rao provided helpful feedback on mixing Issues, and Matt Temiy and Eric Haseltine provided helpful input on the cover design. Eric also helped prepare the solution manual. We have benefited from the feedback of many classes of undergraduates both at the University of Wisconsin and The University of Texas. We appreciate the enthusiasm with which they received some very early and incomplete drafts of the notes.
We could not have written this text in the way we planned without Octave. S Boundary-Value Problems and Collocation colloc I Choosing a reaction with appropriate probability. Stochastic simulation of the first-order reactions A -B -C starting with A molecules. SS fact0r versus Thiele modulus for a first- 7 7 charart eaCtl0n. C measurement versus time for reduced model; early time measurements have been added. Kinetic and mass-transfer parameters for the catalytic converter example. Feed flowrate and heat-transfer bed catalytic converter..
Mass-transfer and kinetic parameters for micromixing prob- Reactor and kinetic parameters for feed-mixing example Parameters for the dispersed PF. R example. Temperature control in a CSTR. PFR and interstage cooling. Oxidation of o-xylene to phthalic anhydride Ammonia synthesis. First-order, isothermal fixed-bed reactor Mass-transfer limitations in a fixed-bed reactor Second-order, isothermal fixed-bed reactor Hougen-Watson kinetics in a fixed-bed reactor Multiple-reaction, nonisothermal fixed-bed reactor 8. A-4 A simple fixed-bed reactor problem 1 Setting the Stage 1.
The chemical reactor Is that essential component in which the feed is converted into the desired products. The chemical reactor is the place in the process where the most value Is added: lower-value feeds are converted into higher-value products. To accomplish these goals, we construct and illustrate a set of reactor analysis and design principles. These principles can be applied at many size scales to many different types of chemically reacting systems.
We will illustrate throughout the text, using :2 3 Setting the Stage many different examples, the diversity of systems and issues addressed by chemical reactor analysis and design. Batch, continuous-stirred-tank, and plug-flow reactors. We model many chemical reactors using three main reactor archetypes: batch, continuous-stirred-tank, and plug-flow reactors. Concentration, temperature and pressure are therefore the usual dependent variables that are solved as functions of time or distance along the reactor as the independent variable.
More often the design involves many reactions and nonisothermal operation, and coupled sets of ODEs or algebraic equations are needed to describe the temperature, pressure and species concentrations. Regardless of the complexity, the design problem is approached in the same manner using the same set of principles.
Fortunately, high-level programming languages are readily available and easily can be used to solve the complex models. The batch, continuous-stirred-tank, and plug-flow reactors are defined by certain idealized assumptions on the fluid flow. The batch and continuous-stirred-tank reactors are assumed to be ideally well mixed, which means that the temperature, pressure and species concentrations are independent of spatial position within the reactor.
The plug-flow reactor describes a special type of flow in a tube in which the fluid is well mixed in the radial direction and varies 1. Plug flow often describes well the limit of fully developed turbulent flow, be. If the flow in a tubular reactor is not turbulent Reynolds Number less than 10 3 4 , then the flow may not be well modeled as ideal plug flow, and other models are needed to describe the reactor. Homogeneous and heterogeneous reactions. The reactants and products of homogeneous chemical reactions are in a single phase. Multiphase reactions need not involve heterogeneous catalysts.
Another two-phase, noncatalytic reaction is the low-pressure epitaxial growth of Si films from gas-phase disilane Si 2 H 6. Batch, semi-batch and continuous operation. The operation of the reactor can be classified as batch, semi-batch and continuous. In batch operation, the reactor is charged with reactants, the reaction takes place, and after some processing time the contents of the reactor are removed as product, Batch reactors, depicted in Figure 1. Hatch reactors also are used in 4 c j T Vr Figure 1. Figure 1. This unit is placed in a cylindrical vessel that comprises the exterior of the batch reactor.
Products may also be removed during the semi-batch process. Reactants and products flow into and out of the reactor continuously, and the contents of the reactor are assumed to be well mixed. The well- mixed assumption can be realized more easily for liquids than gases, so CSTRs are often used for liquid-phase reactions.
Polymerization reactions are sometimes conducted in CSTRs. The cascade permits one to realize high conversion of reactant, while minimizing total reactor volume. The plug-flow reactor PFR is a constant cross-section, tubular re- actor as depicted in Figure 1. The velocity, composition and temperature 6 Setting the Stage Figure 1. In this text, we are usually interested in only the steady-state profile in the tube and neglect the dynamics.
If the PFR is filled with a porous catalyst and the fluid flowing in the void space Is turbulent, the reactor is referred to as a fixed-bed reactor. We will see that the isothermal PFR usually leads to higher con- 1. Copyright , Symyx Technologies, Inc. For this reason, PFRs are employed in situations that require high capacity and high conversion. Patent Net. Additional ll. Setting the Stage 1. By changing the gas flowrates, we!
Tht Uniform flow esign challenge ree flow d is tribe the :iis- Figure 1. Batch times and residence times can be as short as milliseconds for high-temperature, gas-phase reactions, Such as ammonia oxidation and HCN synthesis in flow reactors, or as long as days for liquid-phase fermentation reactions in batch reactors. The reactor volume is determined by the reaction rate and amount of product to be manufactured.
Chances are one or more of the articles of clothing you are wearing are made from synthetic fibers. We are surrounded by and regularly use items that are made from plastics and polymers. The automobiles we drive and the fuels we use to power them depend on products made in chemical reactors. Electronic devices and computer chips require a myriad of chemical reaction steps in their manufacture. One solution that permits the traditional silicon-based circuit design to be retained is to use silicon-germanium alloys in place of silicon in the active region of the device. Silicon-germanium alloys have higher mobilities than silicon and a lower bandgap enabling faster speeds and lower power.
When the pressure is low enough, the diffusion lengths of the gas-phase components are long enough that one can assume the gas phase to be well mixed, uniform and independent of position. This means the gas phase can be modeled as a CSTR in which gas enters and leaves the reactor in the flow streams and undergoes reaction at the surface of the wafer. Much, is known about the chemistry that takes place during film growth [5, 19, 15, 2, 12, 13 , This information can be. Courtesy of Applied Materials. One such kinetic model consists of eight reactions among nine species.
The total sulfur content of crude oil can easily be several percent and it must be reduced to several parts per million ppm. The feed is heated and mixed with hydrogen in the inlet to a fixed-bed catalytic reactor. HDS is an exothermic reaction and if the reactor temperature becomes too high, undesired side reactions occur, such as the hydrogenation of the unsaturated bonds and hydrogenolysis of the C—-C bonds. Adapted from McCulloch [8j, bed cooling, is one way to control the temperature and to ensure the complete removal of the sulfur.
Separate reactors in series permit the reactor sequence to be changed as catalyst activity degrades with time. Figures 1. Wilhin each section containing catalyst, the reactor is modeled as a hxed bed. After cooling the outflow of one of the beds, the reaction is allowed to proceed in the this way, the reactor temperature can be controlled by the heat removal. These concepts are discussed in Chapter 6. Commercial HDS reactors such as shown in Figure 1. The catalytic reaction chemistry is quite complex.
As can be seen from Figure 1. The organosulfur molecule must adsorb and orient itself so that the sulfur atom is over an active site, such as an exposed molybdenum atom on a molybdenum disulfide Catalyst. LDPE is a homopolymer of ethylene with, side-chain branching at a frequency 16 Setting the Stage of per carbons in the chain.
The temperature Figure 1. High-density polyethylene is produced at lower pressures, from atm. HDPE requires a catalyst [22, 17, 6 ]. HDPE processes can employ a slurry of catalyst, polymer and diluent. The cylindrical reactor can be J 8 Setting the Stage 2. The reactor typically operates at psig and S F. Consider Figure 1. The understanding and modeling of these biochemical events is an area of current research activity [14, 20,21, 16J.
Reaction LI accounts for Steps 5, Reaction 1. Reaction 1,3 accounts for Steps Reactions 1. Courtesy of ASM Press . The chemical reaction modeling principles remain valid. In Chapter 4 we use this simple model to make quantitative predictions about the evolution of the viral species concentrations. We also show how to model systems that have small concentrations of species, down to less than a few hundred molecules.
We should not underestimate the complexity of some of the systems of interest to chemical engineers. This caution is perhaps especially true for biological systems. Simple models often can explain complex system behavior, especially when feedback mechanisms or autocatalytic steps are Involved. The remaining text is divided into eight chapters. Since most processes involve multiple chemical reactions, we make free use of matrices and linear algebra to summarize compactly the reaction stoichiometry.
Chapter 3. The conditions for equilibrium are developed using the Gibbs energy and the chemical potential or species activity. We also briefly review phase equilibrium so that we are prepared for multi-phase reactions. Chapter 4. In Chapter 4 we develop the material balances for the three reactor types: batch and semi-batch , conlinuous-stirrcd-tank, equation of state to complement the species material balances m situ ations in which the fluid density is not constant.
We feel it works best to introduce the chemical reactor as soon as possible in Chapter 4, so we delay a comprehensive study of reaction rates and reaction rate expressions until Chapter 5. We illustrate the use of these assumptions in developing kinetic expressions, such as free-radical polymerization kinetics.
The chapter concludes with a discussion of mechanisms for reactions occurring at the surfaces of solid catalysts. We show how to couple the fluid balances to the catalyst particle balances in order to predict the overall fixed-bed reactor behavior. Chapter 8. The tradition in introductory reactor design courses is to neglect a careful treatment of the fluid flow and use the simplified, ideal reactors for modeling. In Chapter 8 we explore what to do when the reactor flow pattern is not well represented by these ideal mixing assumptions.
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We introduce the reactor residence time distribution and describe the general issues of mixing in chemical reactors. We describe the limitations of these approximate mixing models. Although that topic is beyond the scope of this text, computational fluid dynamics software is evolving to the point that this approach is becoming tractable for many problems of interest. Chapter 9. We understand which phenomena cause which observed reactor behaviors, and which design variables should be changed if we wish to alter the reactor performance.
But when we want to make quantitative predictions of reactor performance, we require values for the model parameters. It is a simple fact that most of the parameters needed for the chemistries and reactor configurations of interest are not available in the literature. Chapter 9 covers this important topic of parameter estimation, which is not usually addressed in a systematic manner in introductory treatments of reactor analysis and design.
Appendix A: Computational Methods. Finally, in Appendix A we summarize the numerical methods that have been necessary to solve Bibliography i[l] Facts and figures for the chemical industry. Bramblett, Q. Lu, T. Karasawa, M. Hasan, S. Jo, and J. Appl Phys. Farrauto and C. Fundamentals of Industrial Catalytic Processes. Flint, L W. Enquist, R. Krug, V. Racaniello, and A. Gates, C. Greenlief, D.
Beach, and P. Kissin and L. Fundamental of Microelectronics Processing. Catalytic hydrotreating in petroleum refining. Leach, editor, Applied Industrial Catalysis, volume one, pages L Academic Press, New York, Occelli and R. MarcelL Dekker, Inc. Plummer, M, Deal, and P. Russell and W. A surface kinetics model for the growth of Sii,.. Russell and. Kinetics of hydrogen desorption from germanium-covered SiU Surface Science, , Seeger and W.
Hepatitis B virus biology. Bibliography monohyriride phase on Sii loO. Sinniah, M. Sherman, I Lewis. Weinberg, [17 M. Polymer Chemistry, on Introduction. Oxford University Press, New Ynrk, Streetrnan and S.
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So ltd Stole Lie. Greene, Adsorption and thermal dissociation of disdane on SMI00 2x Summers, P. Smith, and A. L Horwich. Hepadnavirus envelope proteins regulate covalently closed circular DMA amplification. Smith, M. Huang, and M. Younkin, E. Henderson, S. Friedrich, R. Science, ,0 , The Stoichiometry of Reactions 2. Section 2. References for further study are provided at the end of the chapter. Exercises are provided in Section 2. The reactor analysis book by Aris [ 1 influenced several sections of this chapter. It consists of a single reaction among three species.
The third example illustrates the complexity of common industrial reactions of interest, consisting of 20 reactions among 14 species. For the first example, consider two molecules of nitric oxide and one molecule of oxygen reacting to form two molecules of nitrogen dioxide. In this example, there is a single chemical reaction and three different chemical species taking part in the reaction, NO, 0 2 , and N0 2. The second example is known as the water gas shift reaction. There are three chemical reactions and six different chemical species taking part in the three reactions, H, H 2 , OH, H 2 0, CO, and C species taking part in a reaction.
Using the rules of matrix multiplication, one can express Equations 2. The index i runs from 1 to n r , the total number of reactions in the network, and the index j runs from 1 to n s , the total number of species in the network. Reactions 2. Therefore one can make the connection between the columns of v and the species taking part in the reactions. More precisely, the jth column of the v matrix supplies the stoichiometric numbers of the jth species in all of the reactions.
What is the impact of this change on the v matrix? We can therefore make the connection between the rows of v and the reactions. Since there is no reason to prefer one ordering of species and reactions over another, one may permute the columns and rows into any order and maintain a valid stoichiometric matrix.
We now introduce the third example, which is a more complicated reaction network. The only difference is that Reactions 2. Polymerizations and long-chain-producing reactions consist of thousands of species and associated reactions. Developing and understanding that procedure is the topic of the next several sections. Example 2. We then look through all of the reactions and identify the different species taking part.
With an A chosen, it is a simple matter to look through Reactions 2. Do not forget the convention that species appearing as products in a given reaction have positive coefficients and those appearing as reactants have negative coefficients. Practice filling out a few rows of the v matrix and check it with the values given in Equation 2. Notice that for this example v is a 20 x 14 matrix, and it contains many zero entries. A matrix with many zero entries is called sparse. The large number of zeros simply reflects the physical fact thai very few molecules can take part In a particular reaction.
All of the reactions in the CVD chemistry, for example, are unimolecular or bimolecular. More will be said about this issue in the discussion of mechanisms in Chapter 5. Solution In a chemical reaction, the number of molecules is not conserved in general. The mass, however, is conserved in chemical i. It is clear in the above example that the atoms N and O are conserved, so the mass is conserved.
Another way to state conservation of mass involves molecular weights of the species. The transpose of the matrix means to exchange the rows for columns and vice versa. Mco, which is the first row of v in Equation 2. The second and third reactions simply fill out the second and third row's of v so that again, for multiple reactions vM - 0 o 2. It is not necessary to eliminate extra reactions and work with the smallest set, but it is sometimes preferable. Before making any of these statements precise, we explore the question of whether or not the three reactions listed in Reactions 2.
Can we express the first reaction as a linear combination of the second and third reactions? By linear combination we mean multiplying a reaction by a number and adding it to the other reactions. It is clear from inspection that the first reaction is the sum of the second and third reactions, so the set of three reactions is not independent.
Likewise there is no way to produce H 2 or H 2 0 from only Reaction 2. This discussion is not meant to imply that there is something wrong with the first reaction in Reactions 2. Indeed if we focus attention on the second reaction, we can again ask the question whether or not it can be written as a linear combination of the first and third reactions.
The answer is yes because the second reaction is the first reaction minus the third reaction, il. Finally, the third reaction in Reaction 2. For this example then, any two of the reactions comprise an independent set. We now- consider the stoichiometric matrix for the water gas shift reaction presented in Equation 2. The question of whether or not the ith reaction can be written as a linear combination of the other reactions is the same as the question of whether or not the tth row of the v matrix can be written as a linear combination of the other rows. The linear independence of the reactions in a reaction network is equivalent to the linear independence of the rows in the corresponding stoichiometric matrix.
There are efficient numerical algorithms available for finding the rank of a matrix and a set of iinearly independent rows. The focus of our attention is not on the algorithm, but on how we can exploit the results of the algorithm to analyze sets of chemical reactions. Notice that one must consider linearly independent reactions for the statement in the example to be true. That is possible as long as one of the Mj multipliers in Equation 2. In our case, all of the multipliers are nonzero.
Up to this point, we have started with a set of reactions and investigated constructing subsets of these reactions that are linearly independent. Now consider the reverse problem. In other words, a given set of reactions may be linearly independent, but we want to be sure we have not left out some valid reactions. The following describes a systematic approach to this problem.
First, list formation reactions for every species in the list from its elements. We may use molecules of pure elements O 2 rather than atoms O to save work as long as the atoms themselves do not appear in the species list. Then eliminate through linear combinations of reactions any new elements that were introduced in the formation reactions and that do not appear in the original species list. The remaining set is a maximal linearly independent set of reactions for the original species list.
This procedure is perhaps best illustrated by example. It is necessary to write these reactions in the order in which the species 40 The Stoichiometry of Reactions 2,4 Reaction Rates and Production Rates appear in the species list.
See Exercise 2. We now wish to eliminate carbon from the species list. The approach is to replace formation reactions involving C with independent linear combinations of the four reactions that eliminate C from the set. For example we could replace Reaction 2. Equivalently we replace row 1 in Equation 2. We always add linear combinations of rows below the row on which we are making the zero in order not to disturb the pattern of zeros below the diagonal in the first columns of the matrix.
Because these portions of the first four rows are. Independent, so are the entire rows, and, therefore, the reactions. Proceeding down the rows we replace row 3 by twice row 3 minus row 4. When we reach the last row, either a zero already exists in the last column or we remove this last row because we have no rows below the last with which to zero that element.
If we now multiply out these equations, the last column of zeros removes the C from the species list and we have vA - 0 0 2 - 10 " 0 - -2 -4 1 0 in which v -. See also Exercise 2. As an example, we consider the third reaction to the CVD chemistry, Reactions 2. We define the reaction extent, t, to keep track, of the number of times this reaction event occurs. Imagine that we could somehow count up the net number of times an S 1 H 4 molecule hit an SiHj molecule and turned into an SUHe molecule during a short period of time. The change in the reaction extent, Ac, is the net number of reaction events that occur in the time interval At.
Under this continuum assumption, we can speak of the reaction rate as defined at a point in space within some larger reacting system or physical reactor equipment. We are postulating that these collision and transformation events are taking place at the molecular level. These literal reactions are known as elementary reactions. We delay a more complete discussion of elementary' reactions and reaction mechanisms until Chapter 5. We will also see that for complex reacting systems, it may be difficult to know whether or not a reaction is an elementary reaction.
But that is a separate issue, which we take up later, and that issue does not prevent us from defining the reaction rate. It is difficult to measure reaction rates directly, because we do not directly sense molecular transformation events. Each time the reverse reaction occurs, an SUHc, molecule is consumed. Recall the water gas shift reaction. We notice that :H does not take part in the first reaction, is consumed in the second reaction, and is produced in the third reaction.
We therefore write 1? It is produced in the first and second reactions and does not take part in the third reaction. Compare the matrices in Equations 2. Notice that the first row of the matrix in Equation 2,7 is the same as the first column of the matrix in Equation 2. Moreover, each row of the matrix in Equation 2.
In other words, the two matrices are transposes of each other. We can therefore summarize Equation 2. Equa- f fion 2. That computation is a simple matter of matrix multiplication. The reverse problem, deducing the reaction rates from j the production rates, is not so simple as it involves solving a set of I equations. We will see in the next section under what conditions that solution can be found.
You may wish to choose another component such as SiH 4 and check another column of the v matrix. In Appendix A, we briefly summarize Octave and Matlab as high-level programming languages for numerical solution of reactor analysis and design problems. Octave is freely available for a variety of hardware platforms and can be down-loaded from wvm. Matiab is commercially available from The MathWorks, Inc. Please perform this calculation for yourself. Computing the rank of v confirms that only two reactions are independent.
We explore subsequently what happens when this restriction is violated by the production-rate measurements. Because r 3 is arbitrary, we cannot deduce the reaction rates from production rates, which is characteristic of using sets of reactions that are not linearly independent. So we conclude that measuring production rates is not enough to tell us reaction rates. We require more information. We take up that important problem in Chapter 9. A system with more equations than unknowns is called over-determined.
Normally we would not expect to find a solution for arbitrary R, only those R that are generated by multiplying an r by v r. Computing least-squares solutions to over-determined equations is a useful computation in linear algebra. If one is given J? If the reactions are linearly independent, then the matrix product v v T has an inverse and the least-squares solution is unique. If the reactions are not linearly independent this inverse does not exist and the least-squares solution is not unique as before.
If we add small amounts of random noise to the data given in Equation 2. Next consider what happens when we have many measurements available. Figure 2. Notice that the estimated, rates, are again. We can summarize the reaction stoichiometry with one vector equation A set of reactions is linearly independent if no reaction in the set can be written as a linear combination of the other reactions in the set. The reaction rate is a fundamental concept that allows quantitative prediction of rates of conversions of reactants to products. In this situation, one is normally interested in finding the reaction rates that most closely satisfy Equation 2.