Four-stage model

Figure 21. The four-stage model of the circulating fluidized-bed reactor (adapted from Ref. 116).

A four-stage model has been used to describe the development of a bacterial biofilm. The stages comprise, first, primary attachment of cells to the underlying biomaterial, second, accumulation in multiple bacterial layers, third, maturation of the biofilm, and fourth, detachment of planktonic cells from the biofilm, which may then initiate a new cycle of biofilm formation elsewhere [11], In the staphylococci clear evidence exists for the attachment and accumulation/maturation stages of the biofilm model [12] (Fig. 1), while active detachment mechanisms are not well... 

Figure 3 A Four-Stage Model of Carcinogenesis With Clonal Expansion and Repair...

Fig. 3. Simple four-stage model of human information processing [6]...

For convenience, only four stages were used in this model. An iterative solution is required for the bubble point calculations and this is based on the half-interval method. A FORTRAN subroutine EQUIL, incorporated in the ISIM program, estimates the equilibrium conditions for each plate. The iteration routine was taken from Luyben and Wenzel (1988). The program runs very slowly. 

In this fashion, we extend our deterministic model with a prediction horizon of H = 2 to a multi-stage model. The multi-stage tree of the possible outcomes of the demand within this horizon (starting from period i = 1) with four scenarios is shown in Figure 9.5. Each scenario represents the combination fc out of the set of all combinations of the demand outcomes within the horizon. The production decision x has to be taken under uncertainty in all future demands. The decision xj can react to each of the two outcomes of d i, but has to be taken under uncertainty in the demand di. The corrective decisions are explicitly modeled by replacing xj by two variables 2,1 and 2.2 ... 

Originally ASTRA was developed on the base of existing models that have been converted into a dynamic formulation feasible for implementation in system dynamics and allowing for closure of the feedbacks between the models. Among these models have been the macroeconomic model, ESCOT (Schade et al., 2002) and the classical four-stage transport model, SCENES (ME P, 2000). The ASTRA model then runs scenarios for the period 1990 until 2030 using the first 12 years for calibration of the model. Data for calibration stem from various sources, with the bulk of data coming from the EUROSTAT (2005) and the OECD online databases (OECD, 2005). A detailed description of ASTRA is provided by Schade (2005). 

It is difficult to build these four stages into a mathematical model because the kinetics of each processes is not generally known, and therefore more empirical methods have to be adopted. 

These four stages are identical to the four steps of the risk assessment process set out in a key 1983 US publication on risk assessment (NRC, 1983), demonstrating the influence of the US approach on Europe. It is recognized, however, that this model is not always appropriate, with ozone depletion being quoted as an effect for which stages 2 and 3 do not apply. In these cases regulators have to assess risks on a case-by-case basis and give a full description and justification of their assessments in their report to the Commission. In Chapter 7 I discuss how risk from chemicals are assessed in practice. 

With 11 stages and 5 components the equilibrium-stage model has 143 equations to be solved for 143 variables (the unknown flow rates, temperatures, and mole fractions). Convergence of the computer algorithm was obtained in just four iterations. Computed product flows are shown in Fig. 13-37. 

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