Objective function chemical reactors

The formulation of objective functions is one of the crucial steps in the application of optimization to a practical problem. As discussed in Chapter 1, you must be able to translate a verbal statement or concept of the desired objective into mathematical terms. In the chemical industries, the objective function often is expressed in units of currency (e.g., U.S. dollars) because the goal of the enterprise is to minimize costs or maximize profits subject to a variety of constraints. In other cases the problem to be solved is the maximization of the yield of a component in a reactor, or minimization of the use of utilities in a heat exchanger network, or minimization of the volume of a packed column, or minimizing the differences between a model and some data, and so on. Keep in mind that when formulating the mathematical statement of the objective, functions that are more complex or more nonlinear are more difficult to solve in optimization. Fortunately, modem optimization software has improved to the point that problems involving many highly nonlinear functions can be solved. 

Optimization in the design and operation of a reactor focuses on formulating a suitable objective function plus a mathematical description of the reactor the latter forms a set of constraints. Reactors in chemical engineering are usually, but not always, represented by one or a combination of... 

Optimization problems are by their nature mathematical in nature. The first and perhaps the most difficult step is to determine how to mathematically model the system to be optimized (for example, paint mixing, chemical reactor, national economy, environment). This model consists of an objective function, constraints, and decision variables. The objective function is often called the merit or cost function this is the expression to be optimized that is the performance measure. For example, in Fig. 3 the objective function would be the total cost. The constraints are equations that describe the model of the process (for example, mass balances) or inequality relationships (insulation thickness >0 in the above example) among the variables. The decision variables constitute the independent variables that can be changed to optimize the system. 

In Chapter 3 the basic equations for reactions and reactors are set up the objective function needed to define a realistic optimal problem is discussed in Chapter 4. Subsequently, it is natural to consider separately the main types of chemical reactors and their associated problems. In Chapter 8 three problems with a stochastic element in them are described. Chapter 9 concerns itself with the optimal operation of existing reactors which may be regarded as partial designs in which only some of the variables can be optimally chosen. Some recent advances in optimal control, which, however, lie outside our present considerations, are to be found in the paper by Kalman, Lapidus, and Shapiro (1959) and in that of Rudd, Aris, and Amundson (1961). 

For a chemical process the first of these might involve the concentrations of the different chemical species, and the temperature or pressure of the stream. For the second we might have to choose the volume of reactor or amount of cooling to be supplied the way in which the transformation of state depends on the operating variables for the main types of reactors is discussed in the next chapter. The objective function is some measure of the increase in value of the stream by processing it is the subject of Chapter 4. 

The synthesis problem is the following given a feed state and a number of fundamental processes, as mixing, chemical reactions, heating or cooling, find the best combination of these processes that will built-up the optimal chemical reactor, as well as the optimal operating conditions. Note that the objective function is usually of economic nature, but it may include safety or environmental elements. 

Every engineering text that is intended for use by undergraduates must address two needs. First, it must prepare students to function effectively in industry with only the B.S. degree. Second, it must prepare those students that go to graduate school for advanced coursework in reaction kinetics and reactor analysis. Most of the available textbooks fall short of meeting one or both of these requirements. Chemical Reactions and Chemical Reactors addresses both objectives. In particular ... 

The optimization formulation (presented in Eqs. I.II-I.I6) consists of the objective function (e.g., minimize TAG Eq. l.II) subjected to process constraints, the process models and constraints (Eqs. I.I-I.IO) of the generic model block mentioned earlier (x is a process variable, the mass flow rate), structural constraints (Eqs. 1.12 and 1.13) representing the superstructure which allows selection of only one process alternative in each step, and cost functions (Eqs. 1.14-1.16) to calculate the operating and capital costs using cost parameters (f l", waste treatment cost utility or chemicals cost , reactor... 

Chemical vapor deposition (CVD) of carbon from propane is the main reaction in the fabrication of the C/C composites [1,2] and the C-SiC functionally graded material [3,4,5]. The carbon deposition rate from propane is high compared with those from other aliphatic hydrocarbons [4]. Propane is rapidly decomposed in the gas phase and various hydrocarbons are formed independently of the film growth in the CVD reactor. The propane concentration distribution is determined by the gas-phase kinetics. The gas-phase reaction model, in addition to the film growth reaction model, is required for the numerical simulation of the CVD reactor for designing and controlling purposes. Therefore, a compact gas-phase reaction model is preferred. The authors proposed the procedure to reduce an elementary reaction model consisting of hundreds of reactions to a compact model objectively [6]. In this study, the procedure is applied to propane pyrolysis for carbon CVD and a compact gas-phase reaction model is built by the proposed procedure and the kinetic parameters are determined from the experimental results. 

Wastewater treatment systems can be classified, in addition to pretreatment, as preliminary, primary, secondary, and tertiary (advanced) treatments. Pretreatment of industrial wastewater is required to prevent adverse effects on the municipal wastewater treatment plants. Preliminary treatment is considered as any physical or chemical process that precedes primary treatment. The preliminary treatment processes may consist of influent screening and grit removal. Its function is mainly to protect subsequent treatment units and to minimize operational problems. Primary treatment is defined as the physical or chemical treatment for the removal of settleable and floatable materials. The screened, degritted raw wastewater from preliminary treatment flows to the primary clarification tanks, which are part of the primary treatment facilities. Secondary wastewater treatment is the process that uses biological and chemical treatment to accomplish substantial removal of dissolved organics and colloidal materials. The secondary treatment facilities may be comprised of biological reactor and secondary clarification basins. Tertiary (advanced) wastewater treatment is used to achieve pollutant reductions by methods other than those used in primary and secondary treatments. The objective of tertiary wastewater treatment is to improve the overall removal of suspended solids, organic matter, dissolved solids, toxic substances, and nutrients. 

The objective here is to simulate duct reactor performance with nonuniform catalyst activity and identify optimal deposition strategies when reactant diffn-sion toward the active surface is hindered, particularly in the corners of the flow channel. Both types of power-function profiles, listed in Table 23-3, are evaluated for n = 1,2,4, 8. The delta-function distribution has been implemented by Varma (see Morbidelli et al., 1985) to predict optimum catalyst performance in porous pellets with exothermic chemical reaction. Nonuniform activity profiles for catalytic pellets in fixed-bed reactors, in which a single reaction occnrs, have been addressed by Sznkiewicz et al. (1995), and effectiveness factors for... 

The objective of this study was to determine the total gas yield and gas composition from the various feedstocks as a function of reactor temperature, air-to-feed ratio, and steam-to-feed ratio. The gas components of greatest interest are hydrogen, carbon monoxide, methane, and ethylene. These components contribute not only to the gas heating value, but also to the value of the gas as chemical synthesis feedstock. 

For each of these intensification challenges, the objective to be reached (volume reduction, reduced size/capadty ratio, etc.), and also the constraints (fixed productivity, fixed performance, quality specifications, etc.) can be identified and quantified with respect to technical and economic data. Unfortunately, the means to tackle these issues are much more complex to define since they can be of very different natures operating conditions (temperature, pressure, concentrations, etc.), physical or chemical parameters (solvents, catalysts, etc.), equipment (heat exchangers, mixers, columns, etc.), process parameters (reflux ratio, feed strategy of semi-batch reactors, separate unit operations or multi-functional reactors, separator types, etc.). In... 

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