Age-based models

The IEM model is a simple example of an age-based model. Other more complicated models that use the residence time distribution have also been developed by chemical-reaction engineers. For example, two models based on the mixing of fluid particles with different ages are shown in Fig. 5.15. Nevertheless, because it is impossible to map the age of a fluid particle onto a physical location in a general flow, age-based models cannot be used to predict the spatial distribution of the concentration fields inside a chemical reactor. Model validation is thus performed by comparing the predicted outlet concentrations with experimental data. 

The relationship between age-based models and models based on the composition PDF can be understood in terms of the joint PDF of composition ( p) and age (A) 103 

The marginal PDF of the age, /a( ), will depend on the fluid dynamics and flow geometry.104 On the other hand, the conditional PDF of p given A is provided by the age-based 

The age of the fluid particle located at a specific point in the reactor is now treated as a random variable. In the CRE literature, [a ( ) is known as the internal-age distribution function. 

Note that even if fA(a) were known for a particular reactor, the functional form of UmifW) will be highly dependent on the fluid dynamics and reactor geometry. Indeed, even if one were only interested in ( / at the reactor outlet where fA(a) = E(a), the conditional PDF will be difficult to model since it will be highly dependent on the entire flow structure inside the reactor. 

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The multimedia model present in the 2 FUN tool was developed based on an extensive comparison and evaluation of some of the previously discussed multimedia models, such as CalTOX, Simplebox, XtraFOOD, etc. The multimedia model comprises several environmental modules, i.e. air, fresh water, soil/ground water, several crops and animal (cow and milk). It is used to simulate chemical distribution in the environmental modules, taking into account the manifold links between them. The PBPK models were developed to simulate the body burden of toxic chemicals throughout the entire human lifespan, integrating the evolution of the physiology and anatomy from childhood to advanced age. That model is based on a detailed description of the body anatomy and includes a substantial number of tissue compartments to enable detailed analysis of toxicokinetics for diverse chemicals that induce multiple effects in different target tissues. The key input parameters used in both models were given in the form of probability density function (PDF) to allow for the exhaustive probabilistic analysis and sensitivity analysis in terms of simulation outcomes [71]. 

For higher-order reactions, the fluid-element concentrations no longer obey (1.9). Additional terms must be added to (1.9) in order to account for micromixing (i.e., local fluid-element interactions due to molecular diffusion). For the poorly micromixed PFR and the poorly micromixed CSTR, extensions of (1.9) can be employed with (1.14) to predict the outlet concentrations in the framework of RTD theory. For non-ideal reactors, extensions of RTD theory to model micromixing have been proposed in the CRE literature. (We will review some of these micromixing models below.) However, due to the non-uniqueness between a fluid element s concentrations and its age, micromixing models based on RTD theory are generally ad hoc and difficult to validate experimentally. 

Figure 5.15. Two examples of age-based micromixing models. In the top example, it is assumed that fluid particles remain segregated until the latest possible age. In the bottom example, the fluid particles mix at the earliest possible age. Numerous intermediate mixing schemes are possible, which would result in different predictions for micromixing-sensitive reactions. 

The U-Pb system has been a chronometer of choice for the Earth s age since the pioneering study of Clair Patterson in 1956, as discussed in Chapter 8. Virtually all U-Pb model ages of the Earth (reviewed by Allegre et al., 1995) are younger than the ages of chondritic and achondritic meteorites. The oldest Pb-Pb age, based on ancient terrestrial... 

The likelihood can be used to estimate not only the secular terns (birth cohort and period effects), but also the parameters of the biologically based model. In addition, if the data can be broken further by carcinogen(s) exposure (dose and duration) by age and period, one can also estimate in principle dose-response parameters. 

Matrix models are sets of mostly linear difference equations. Each equation describes the dynamics of 1 class of individuals. Matrix models are based on the fundamental observation that demographic rates, that is, fecundity and mortality, are not constant throughout an organism s life cycle but depend on age, developmental stage, or size. Ecological interactions, natural disturbances, or pesticide applications usually will affect different classes of individuals in a different way, which can have important implications for population dynamics and risk. In the following, I will only consider age-structured models, but the rationale of the other types of matrix models is the same. For an example of this approach applied to pesticide risk assessment, see Stark (Chapter 5). 

Individual-based models describe the life cycle of individual (discrete) organisms. The organisms can differ and display autonomous behavior (DeAngelis and Mooij 2005 Grimm and Railsback 2005). The entities of an IBM — individuals, habitat units, and the abiotic environment — are characterized by sets of state variables, for example, sex, age, body mass, location (individuals) vegetation cover, soil moisture, food level (habitat units) or temperature, rainfall, and disturbance rate (environment). 

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