Mathematical Processing Operations

Conceptually, the problem of going from the time domain spectra in Figures 3.7(a)-3.9(a) to the frequency domain spectra in Figures 3.7(b)-3.9(b) is straightforward, at least in these cases because we knew the result before we started. Nevertheless, we can still visualize the breaking down of any time domain spectrum, however complex and irregular in appearance, into its component waves, each with its characteristic frequency and amplitude. Although we can visualize it, the process of Fourier transformation which actually carries it out is a mathematically complex operation. The mathematical principles will be discussed only briefly here. 

The next two steps after the development of a mathematical process model and before its implementation to "real life" applications, are to handle the numerical solution of the model s ode s and to estimate some unknown parameters. The computer program which handles the numerical solution of the present model has been written in a very general way. After inputing concentrations, flowrate data and reaction operating conditions, the user has the options to select from a variety of different modes of reactor operation (batch, semi-batch, single continuous, continuous train, CSTR-tube) or reactor startup conditions (seeded, unseeded, full or half-full of water or emulsion recipe and empty). Then, IMSL subroutine DCEAR handles the numerical integration of the ode s. Parameter estimation of the only two unknown parameters e and Dw has been described and is further discussed in (32). 

Bunch, P. R. D. L. Watson and J. F. Pekny. Improving Batch Manufacturing Process Operations Using Mathematical Programming Based Models. FOCAPO Conference Proceedings, AIChE Symp Ser 320, 94 204-209 (1998). 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

Bunch, P.R., Rowe, R.L., and Zentner, M.G. (1998) Large scale multi-facility planning using mathematical programming methods. AIChE Symposium Series, Proceedings of the Third International Conference of the Foundations of Computer-Aided Process Operations. Snowbird, Utah, USA, July 5-10, American Institute of Chemical Engineering, 94, p. 249. 

The process design space should be expressed in the form of a mathematical model which quantitatively links process capability, quality of input materials, and process operating parameters. 

One of the major uses for mathematical models is in the simulation of process operations. This application is readily appreciated by chemical engineers because of the similarity of the model to a pilot plant. In spite of its importance, this subject has not received much attention in the literature. There are probably two main reasons for this. First, process models tend to be quite specific for a particular application and there is little to be said about the subject from a general viewpoint. Second, detailed accounts of specific examples would include information which most firms regard as being of competitive value. 

This chapter introduces the reader to elementary concepts of modeling, generic formulations for nonlinear and mixed integer optimization models, and provides some illustrative applications. Section 1.1 presents the definition and key elements of mathematical models and discusses the characteristics of optimization models. Section 1.2 outlines the mathematical structure of nonlinear and mixed integer optimization problems which represent the primary focus in this book. Section 1.3 illustrates applications of nonlinear and mixed integer optimization that arise in chemical process design of separation systems, batch process operations, and facility location/allocation problems of operations research. Finally, section 1.4 provides an outline of the three main parts of this book. 

A universal method of handling the problem is mathematical modelling, i.e., a quantitative description by means of a set of equations of the whole complex of interrelated chemical, physical, fluiddynamic, and thermal processes taking place concurrently or consecutively in a reactor. Constants of these equations are determined in laboratory experiments. If the range of determining factors (reactive mass compositions, temperature, reaction rates, and so on) in an actual process lie within or only slightly outside the limits studied in laboratory experiments, the solution of the determining set of equations provides a reliable idea of the process operation. 

For the description of the particle property distribution, a population balance approach is recommended which is mathematically challenging but which provides valuable insight into the steady-state and dynamic process operating behavior. 

These reasons that make the overall problem difficult suggest that a concerted attack is needed, from both the engineering and business ends of the problem. It should be stressed that, while there may be some common mathematical tools used in both engineering and business, the bottleneck in computer integration of process operations is not the lack of solution to a given mathematical problem, but rather the need for the formulation of a mathematical problem that both corresponds to physical reality and is amenable to solution. 

Another situation of interest in applications is one where the product purities are fixed, and the objectives for optimal process operation are to reduce operating costs and increase production. Hence, in this case, for a fixed target product purity of both extract and raffinate streams, we seek the optimal process parameters for a SMB and a 4-subinterval Varicol unit, which maximize production using minimum amount of eluent, for another chiral separation system taken from literature [6,15]. The optimization problem is represented mathematically as follows ... 

As the plant to be optimized considers a process operating at steady state, then the variation of the phase concentrations with time is zero. For this reason, the mathematical model that describes the plant is a set of ordinary differential equations, as the phase concentrations depend only on the module axial position. In the tanks, the concentrations are constant. The differential-algebraic nonlinear optimization (DNLP) problem PI to be solved includes the ordinary differential equations that represent the mass balances for the phases in the membrane module. The objective function to be maximized is the amount of metal processed FeC , where Fe is the effluent flow rate whose Cr(VI) concentration after dilution from wastewaters is C . The problem has the following form ... 

For continuous processes operating at steady state, optimization typically consists in determining the operating point that minimize or maximize some performance of the process (such as minimization of operating cost or maximization of production rate), while satisfying a number of constraints (such as bounds on process variables or product specifications). In mathematical terms, this optimization problem can be stated as follows ... 

Numerical Solvers The mathematical process models of a unit operation or a complete plant are large and highly non-linear. As analytical solutions are impossible, iterative, numerical approaches are used to solve the equations. 

J. Szekely, J. W. Evans, and J. K. Brimacombe, The Mathematical and Physical Modeling of Primary Metals Processing Operations, John Wiley, New York, 1988. 

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