Single process model

Thus a single process model for manufacturing cost, M can be formulated as ... 

These models must include the presence of the two processes of nucleation and growth. At the begiiming, researchers tried to simplify the problem by analyzing borderhne cases in which one of the two processes was taking place veiy quickly with the respect of the other. These are called borderline cases or single process models since either the single growth or nucleation process evolution over time is taken into account, while the other is considered to have an infinite speed. 

Laws issued from the single-process model... 

The construction of models has shown that the following data are required to estabhsh the rate law in a single-process model ... 

The establishment of two-process models requires the same data as for single-process models (grain shape and orientation of the layer development) by adding the chosen mode of growth (isotropic or anisotropic). Note that in models involving an instantaneous nucleation process, the growth is anisotropic by default. 

It is experimentally useful to know whether the search for a morphological model should be directed towards single- or double-process models. We have seen that in the case of a single-process model the rate at a given time depends on the values of variables at that moment, while in a nucleation growth with two processes the previous evolutions of these variables affect the rate, i.e. the rate depends on the history of the system. It is this property that we will use to choose the family of models suitable for the reaction being studied. 

In the higher pressure sub-region, which may be extended to relative pressure up to 01 to 0-2, the enhancement of the interaction energy and of the enthalpy of adsorption is relatively small, and the increased adsorption is now the result of a cooperative effect. The nature of this secondary process may be appreciated from the simplified model of a slit in Fig. 4.33. Once a monolayer has been formed on the walls, then if molecules (1) and (2) happen to condense opposite one another, the probability that (3) will condense is increased. The increased residence time of (1), (2) and (3) will promote the condensation of (4) and of still further molecules. Because of the cooperative nature of the mechanism, the separate stages occur in such rapid succession that in effect they constitute a single process. The model is necessarily very crude and the details for any particular pore will depend on the pore geometry. 

The problems involved in finding random process models for particular sources and channels are, of course, very difficult. Such models can hardly ever be more than crude approximations to physical reality. Even the simplest random process model, however, makes it possible to consider a class of inputs rather than a single input and to consider the frequency with which the inputs are used. 

Figure 10-4. The double- and single-site titration models for His and Asp groups [42]. (A) In the double site model, only one X is used for describing the equilibrium between the protonated and deprotonated forms, while the tautomer interversion process is represented by the variable x. (B) In the single-site model, protonation at different sites is represented by different X variables. HSP refers to the doubly protonated form of histidine. HSD and HSE refer to the singly protonated histidine with a proton on the h and e nitrogens, respectively. ASP1 and ASP2 refer to the protonated carboxylic acid with a proton on either of the carboxlate oxygens...

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). 

Simpler optimization problems exist in which the process models represent flow through a single pipe, flow in parallel pipes, compressors, heat exchangers, and so on. Other flow optimization problems occur in chemical reactors, for which various types of process models have been proposed for the flow behavior, including well-mixed tanks, tanks with dead space and bypassing, plug flow vessels, dispersion models, and so on. This subject is treated in Chapter 14. 

In principle, any type of process model can be used to predict future values of the controlled outputs. For example, one can use a physical model based on first principles (e.g., mass and energy balances), a linear model (e.g., transfer function, step response model, or state space-model), or a nonlinear model (e.g., neural nets). Because most industrial applications of MPC have relied on linear dynamic models, later on we derive the MPC equations for a single-input/single-output (SISO) model. The SISO model, however, can be easily generalized to the MIMO models that are used in industrial applications (Lee et al., 1994). One model that can be used in MPC is called the step response model, which relates a single controlled variable y with a single manipulated variable u (based on previous changes in u) as follows ... 

The program THERMFF solves the same dynamic process model equations as THERM, where it was shown that all the parameters, including the inlet temperature and concentration will influence the steady state. In the case of multiple steady states the values of the steady state parameters cannot be set, because they are not unique. This example should, therefore, be mn under parameter conditions that will guarantee a single steady state for all expected values of the CA0 and T0. These can be selected with the aid of the programs THERMPLOT and THERM. 

In some mixture design problems (such as formulations), it may not be necessary to consider processing issues and hence we would not have the process model constraints. In this case the problem becomes a simple mixing problem, which would already have been addressed by the miscibility criteria in sub-problem 4M. Hence, for these problems, we will not need sub-problem 5M. Also in some cases we might have to identify a mixture whose constituents perform different functions such as solvents and anti solvents for crystallization. In such cases we would have to formulate and solve more than one single compound design problems to identify the constituents and then solve the final two sub-problems to identify the optimal mixture. In certain cases we may not have process model constraints, however, we may still have to solve an optimization problem with other constraints, in sub-problem 4 and sub-problem 5m respectively. 

For a viscoelastic solid the situation is more complex because the solid component will never flow. As the strain is applied with time the stress will increase continually with time. The sample will show no plateau viscosity, although there may be a low shear viscous contribution. This applies to both a single Maxwell model and one with a spectrum of processes ... 

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