Process Development Without Mathematical Models

The individual steps of a process development procedure not involving a mathematical model are shown in Fig. 2.2. The important characteristic 

The first phase, exploratory research, takes place mainly in the microbiological, biochemical, and chemical laboratories and yields mainly qualitative results concerning the reaction components—the cells and the nutrient medium.Typical results from experiments in Erlenmeyer flasks include 

Optimization of the media can be done in two ways (a) balancing the nutrient solution, the cells, and the remaining fluid (Dostalek et al., 1972 Haggstrom, 1977 Pirt, 1974), or (b) experiments in a chemostat to determine the influence on growth of different concentrations of the components of the growth medium (Kuhn, Friedrich, and Fiechter, 1979 Mateles and Batat, 1974 Tsuchiya, Nishio, and Nagai, 1980). 

After a preliminary estimate of costs to see whether production is economi- 

L [m s ] in case of continuous plug flow reactor (CPFR) 

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Without doubt, the most important result of developing a mathematical model of a chemical engineering system is the understanding that is gained of what really makes the process tick. This insight enables you to strip away from the problem the many extraneous confusion factors and to get to the core of the system. You can see more clearly the cause-and-effect relationships between the variables. 

Those based on strictly empirical descriptions Mathematical models based on physical and chemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinefics) are frequently employed in optimization apphcations. These models are conceptually attractive because a gener model for any system size can be developed before the system is constructed. On the other hand, an empirical model can be devised that simply correlates input-output data without any physiochemical analysis of the process. For... 

The reader will find here a complete mathematical development of the models of chromatography and other physical laws which direct the chemical engineer in the design and scale-up of chromatographic processes. For preparative chromatographic separations, our ultimate purpose is the optimization of the experimental conditions for maximum production rate, minimum solvent consumption, or minimum production cost, with or without constraints on the recovery yield. The considerable amormt of work done on this critical topic is presented in the... 

The equations presented above can be used (with or without modifications) to describe mass transfer processes in cocurrent flow. See, for example, the work of Modine (1963), whose wetted wall column experiments formed the basis for Example 11.5.3 and are the subject of further discussion in Section 15.4. The coolant energy balance is not needed to model an adiabatic wetted wall column and must be replaced by an energy balance for the liquid phase. Readers are asked to develop a complete mathematical model of a wetted wall column in Exercise 15.2.1. 

For mathematical convenience and economy of effort, rate equations in network elucidation and modeling are best written in terms of the minimum necessary number of constant "phenomenological" coefficients, which may be composites of rate coefficients of elementary steps. This not only simplifies algebra and increases clarity, but also lightens the experimental burden With fewer coefficients, fewer experiments are required to determine their numerical values and their temperature dependences, without which a model is worthless for process development and design. 

The result of the above sequence of events will be the occurrence of limit cycles, and mathematical models of continuous crystallizers with perfect and imperfect mixing, with and without product classification have been developed (Nyvlt and Mullin, 1970) to show how periodic changes of supersaturation, solids content, crystal size and production rate can readily occur. Periodic behaviour is most pronounced at the beginning of the crystallization process, and in most cases the fluctuations are subsequently damped. This means that under favourable conditions a steady state is usually reached Figure 9.10a). However, under some operating conditions the damping is not effective and the cyclic behaviour may continue for a considerable time Figure 9.10b). Indeed, in some cases the steady state may not be achieved at all. 

With this type of process development, it is more interesting to establish the special conditions of an essentially given mode of operation in an also given reactor than to attempt optimization. The construction of a mathematical model, and any later systematic optimization of processes in the individual reaction steps, are in practice not done. The time and expense associated with model building are often thought to be economically unjustified Quantification of the experimentally measured conversion rate, or productivity (yield), is sufficient, without separation of phenomena into biological and physical components. In practice, the space-time-yield mode of thought will continue to be justified in most cases. 

In contrast to Fig. 2.2, which demonstrates a process development protocol without modeling. Fig. 2.16 demonstrates the steps involved in an empirical, systematic process development that does operate with mathematical models. 

The term microkinetics is understood to mean the kinetics of a reaction that are not masked by transport phenomena and to refer to a series of reaction steps. For the investigation of intermediary metabolism, idealized conditions are chosen that often do not correspond to the real conditions of engineering processes. This fact makes it difficult to transfer microkinetic data to technical processes. For the purposes of technologically oriented research and the development of a process to technical ripeness, it is often sufficient to know quantitatively how a process runs without necessarily knowing why. (Macrokinetics, however, must be avoided, as they are scale dependent). Mathematical formulations are needed that reproduce the kinetics adequately for the purpose but are as simple and have as few parameters as possible. Today, even when electronic computers greatly reduce the labor of computation, the criterion of simplicity remains important due to the problem of experimental verification. The iterative nature of the process of building an adequate model is an important point that will be considered in greater detail in Sect. 2.4. 

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