Functional mathematical model, development

The mathematical model developed to describe the nitric acid absorption process isdiscussed in detail in Section G.2. It uses a tray-by-tray approach that incorporates reaction-mass balance corrections and heat balance calculations. Tray efficiency calculations are also included in the model, the efficiency being a function of the tray geometry and gas velocity. Rate equations and other data specific to the nitric acid/nitrous gas system are applied. 

Step 5 in Table 8-6 involves the computation of the optimum point. Quite a few techniques exist to obtain the optimal solution for a problem. We describe several classes of methods below. In general, the solution of most optimization problems involves the use of a digital computer to obtain numerical answers. Over the past 15 years, substantial progress has been made in developing efficient and robust computational methods for optimization. Much is known about which methods are most successful. Virtually all numerical optimization methods involve iteration, and the effectiveness of a given technique can depend on a good first guess for the values of the variables at the optimal solution. After the optimum is computed, a sensitivity analysis for the objective function value should be performed to determine the effects of errors or uncertainty in the objective function, mathematical model, or other constraints. 

One of the characteristics of basic devices is that they cannot be split up into parts. A basic device can also be a signal transformer (a function tvhich transforms the input into output, such as the thermocouple that transforms the input temperature into an electrical tension). The process phases are connected and characterized quantitatively, from the vievrpoint of characteristic relations (equations), as in Fig. 1.4 [1.11-1.13]. This structured mathematical modelling development corre-... 

Prediction of the log reduction of an inoculated organism as a function of acid concentration, time, and temperature can also be done by a mathematical model developed for this purpose, using the second-order polynomial equation to fit the data. The following tests justified the reliability of the model the analysis of variance for the response variable indicated that the model was significant (P < 0.05 and R2 = 0.9493) and had no significant lack of fit (P > 0.05). Assumptions underlying the ANOVA test were also investigated and it was demonstrated that with the normal probability plot of residuals, plot of residuals versus estimated values for the responses, and plot of residuals versus random order of runs, that the residuals satisfied the assumptions of normality, independence, and randomness (Jimenez et al., 2005). 

The goal of the electrochemical modelhng in this chapter is to solve the mathematical model developed in the previous chapter in order to obtain the form of the algebraic (containing no derivatives) function C X,T), i.e., to determine how the concentration of the chemical species varies in space and in time. From this, other information, such as the current passed at the electrode, can be inferred. A munber of analytical techniques exist that may be used for solving partial differential equations (PDEs) of the type encountered in electrochemical problems, including integral transform methods such as the Laplace transform, and the method of separation of variables. Unfortunately these techniques are not applicable in all cases and so it is often necessary to resort to the use of numerical methods to find a solution. 

The kinetics of quality deterioration in dried onion flakes (NEB and thiosulfinate loss) and dried green beans (chlorophyll-u loss) were studied as a function of water activity and temperature and empirical equations and mathematical models developed that successfully predicted the shelf life of the dried products as a function of temperature and (Table 25.11 and Table 25.12) [88]. Above the mono-layer (flw, 0.32-0.43) for onion, increasing moisture contents resulted in greater reaction rates for browning and thiosulfinate loss. Very little browning was observed over a storage period of 631 d at 20 C and... 

A mathematical model of the operating characteristics of a modem HLW storage tank has been developed (60). This model correlates experimental data for the rate of radiolytic destmction of nitric acid, the rate of hydrogen generation owing to radiolysis of water, and cooling coil heat transfer. These are all functions of nitric acid concentration and air-lift circulator operation. 

No single method or algorithm of optimization exists that can be apphed efficiently to all problems. The method chosen for any particular case will depend primarily on (I) the character of the objective function, (2) the nature of the constraints, and (3) the number of independent and dependent variables. Table 8-6 summarizes the six general steps for the analysis and solution of optimization problems (Edgar and Himmelblau, Optimization of Chemical Processes, McGraw-HiU, New York, 1988). You do not have to follow the cited order exac tly, but vou should cover all of the steps eventually. Shortcuts in the procedure are allowable, and the easy steps can be performed first. Steps I, 2, and 3 deal with the mathematical definition of the problem ideutificatiou of variables and specification of the objective function and statement of the constraints. If the process to be optimized is very complex, it may be necessaiy to reformulate the problem so that it can be solved with reasonable effort. Later in this section, we discuss the development of mathematical models for the process and the objec tive function (the economic model). 

Many references can be found reporting on the mathematical/empirical models used to relate individual tolerances in an assembly stack to the functional assembly tolerance. See the following references for a discussion of some of the various models developed (Chase and Parkinson, 1991 Gilson, 1951 Harry and Stewart, 1988 Henzold, 1995 Vasseur et al., 1992 Wu et al., 1988 Zhang, 1997). The two most well-known models are highlighted below. In all cases, the linear one-dimensional situation is examined for simplicity. 

Non-ideal reactors are described by RTD functions between these two extremes and can be approximated by a network of ideal plug flow and continuously stirred reactors. In order to determine the RTD of a non-ideal reactor experimentally, a tracer is introduced into the feed stream. The tracer signal at the output then gives information about the RTD of the reactor. It is thus possible to develop a mathematical model of the system that gives information about flow patterns and mixing. 

A first principle mathematical model of the extruder barrel and temperature control system was developed using time dependent partial differential equations in cylindrical coordinates in two spatial dimensions (r and z). There was no angular dependence in the temperature function (3T/30=O). The equation for this model is (from standard texts, i.e. 1-2) ... 

Different reactor networks can give rise to the same residence time distribution function. For example, a CSTR characterized by a space time Tj followed by a PFR characterized by a space time t2 has an F(t) curve that is identical to that of these two reactors operated in the reverse order. Consequently, the F(t) curve alone is not sufficient, in general, to permit one to determine the conversion in a nonideal reactor. As a result, several mathematical models of reactor performance have been developed to provide estimates of the conversion levels in nonideal reactors. These models vary in their degree of complexity and range of applicability. In this textbook we will confine the discussion to models in which a single parameter is used to characterize the nonideal flow pattern. Multiparameter models have been developed for handling more complex situations (e.g., that which prevails in a fluidized bed reactor), but these are beyond the scope of this textbook. [See Levenspiel (2) and Himmelblau and Bischoff (4).]... 

These two types of deviations occur simultaneously in actual reactors, but the mathematical models we will develop assume that the residence time distribution function may be attributed to one or the other of these flow situations. The first class of nonideal flow conditions leads to the segregated flow model of reactor performance. This model may be used... 

The development of new polyanhydrides has sparked researchers to developed new device fabrication and characterization techniques, instrumentation, and experimental and mathematical models that can be extended to the study of other systems. The growing interest in developing new chemistries and drug release systems based on polyanhydrides promises a rich harvest of new applications and drug release technologies, as well as new characterization techniques that can be extended to other materials. Future endeavors will likely focus on multicomponent polyanhydride systems, combining new chemical functionalities to tailor polyanhydrides for specific applications. 

However, if you extend this notion to an extreme and make 100,000 production runs of one unit each (actually one unit every 315 seconds), the decision obviously is impractical, since the cost of producing 100,000 units, one unit at a time, will be exorbitant. It therefore appears that the desired operating procedure lies somewhere in between the two extremes. To arrive at some quantitative answer to this problem, first define the three operating variables that appear to be important number of units of each run (D), the number of runs per year (n), and the total number of units produced per year (Q). Then you must obtain details about the costs of operations. In so doing, a cost (objective) function and a mathematical model will be developed, as discussed later on. After obtaining a cost model, any constraints on the variables are identified, which allows selection of independent and dependent variables. 

Constraints in optimization arise because a process must describe the physical bounds on the variables, empirical relations, and physical laws that apply to a specific problem, as mentioned in Section 1.4. How to develop models that take into account these constraints is the main focus of this chapter. Mathematical models are employed in all areas of science, engineering, and business to solve problems, design equipment, interpret data, and communicate information. Eykhoff (1974) defined a mathematical model as a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in a usable form. For the purpose of optimization, we shall be concerned with developing quantitative expressions that will enable us to use mathematics and computer calculations to extract useful information. To optimize a process models may need to be developed for the objective function/, equality constraints g, and inequality constraints h. 

Part I comprises three chapters that motivate the study of optimization by giving examples of different types of problems that may be encountered in chemical engineering. After discussing the three components in the previous list, we describe six steps that must be used in solving an optimization problem. A potential user of optimization must be able to translate a verbal description of the problem into the appropriate mathematical description. He or she should also understand how the problem formulation influences its solvability. We show how problem simplification, sensitivity analysis, and estimating the unknown parameters in models are important steps in model building. Chapter 3 discusses how the objective function should be developed. We focus on economic factors in this chapter and present several alternative methods of evaluating profitability. 

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