Mixture mixing model

Figure 5.24. A four-environment mixing model can be developed for reactors with three feed streams. In environment 1, the two components of the mixture-fraction vector are null = 0. In environment 2, fi = 1 and 2 = 0, while, in environment 3, = 0 and 2 = 1- Chemical...

As will be shown for the CD model, early mixing models used stochastic jump processes to describe turbulent scalar mixing. However, since the mixing model is supposed to mimic molecular diffusion, which is continuous in space and time, jumping in composition space is inherently unphysical. The flame-sheet example (Norris and Pope 1991 Norris and Pope 1995) provides the best illustration of what can go wrong with non-local mixing models. For this example, a one-step reaction is described in terms of a reaction-progress variable Y and the mixture fraction p, and the reaction rate is localized near the stoichiometric point. In Fig. 6.3, the reaction zone is the box below the flame-sheet lines in the upper left-hand corner. In physical space, the points with p = 0 are initially assumed to be separated from the points with p = 1 by a thin flame sheet centered at... 

It is important to emphasize here that, theoretically, if a solid mixture is ideal, intracrystalline distribution is completely random (cf section 3.8.1) and, in these conditions, the intracrystalline distribution constant is always 1 and coincides with the equilibrium constant. If the mixture is nonideal, we may observe some ordering on sites, but intracrystalline distribution may still be described without site interaction parameters. We have seen in section 5.5.4, for instance, that the distribution of Fe and Mg on Ml and M3 sites of riebeckite-glaucophane amphiboles may be approached by an ideal site mixing model—i.e.. 

Immiscibility phenomena in silicate melts imply positive deviations from ideality in the mixing process. Ghiorso et al. (1983) developed a mixing model applicable to natural magmas adopting the components listed in table 6.12. Because all components have the same standard state (i.e., pure melt component at the T and P of interest) and the interaction parameters used do not vary with T, we are dealing with a regular mixture of the Zeroth principle (cf sections 2.1 and 3.8.4) ... 

In another effort, by Nickolay Smirnov s group at the M. V. Lomonosov Moscow State University, Moscow, a model for theoretical investigation of turbulent mixing and combustion of polydispersed mixtures in confined volumes was developed (Chapter 14). The numerical model and the software created make it possible to determine the combustion and ignition characteristics of polydispersed mixtures. The model has been validated with experiments. 

In the general case, individual particles have differing compositions and refractive indices and to take this into account in detail is not possible from a practical point of view. To allow for a variation of refractive index, a convenient model is that of a mixture of aerosols from the several sources, each with its own extinction cross-section. The particles are assumed not to coagulate so that the aerosol is not mixed on the individual particle basis. Such an aerosol is known as an external mixture. This model would also be applicable, approximately, to an aerosol mixture whose particles are growing in size by gas-to-particle conversion. 

Second, there are biometrical requirements. Various exposure response models may be used and compared. The models need to be clearly defined, and goodness of fit should be reported, both for the separate exposures as well as for the mixtures. Concentration addition, response addition, and mixed-model results may be compared as possible alternatives, especially when underpinning of mechanistic assumptions is weak. Results at one exposure level (e.g., EC50) do not necessarily predict results at other exposure levels due to different slopes and positions of the curves for separate compounds and the mixtures. Statistical tests should be executed properly to compare predicted and observed responses. If any statements about the significance of results are made, the methods of dose-response analysis need to be reported. 

In our opinion, the data are sufficiently clear to suggest that when it is not feasible to test the mixture in question, mixture extrapolation is the preferred option compared to no extrapolation. Indeed, all literature observations suggest that applying mixture extrapolation is to be preferred over not applying mixture extrapolation. Technical options for extrapolation are concentration addition, response addition, and the mixed-model approach, of which concentration addition is most often applied. Exceptions may apply in cases that are more specific. For example, when it is clear that 2 compounds precipitate (a situation of no exposure due to chemical interactions in the environment), one should acknowledge this prior to assessing mixture risks by mixture extrapolation approaches. When the data of a study allow, refined conclusions are possible. For example, when the study design is appropriate and the mathematical models are appropriate, researchers are able to discriminate between concentration addition and response addition, and (with sufficient experiment efforts) between these models and the mixed-model approach. 

In the fourth step of extrapolation, specific mixture extrapolation protocols are needed. Below, some details on the theories and the associated protocols are given for concentration addition, response addition, and mixed-model approaches, and for the species and assemblage levels separately (this section and, next section, respectively). 

BOX 5.1 Example of a spreadsheet calculation of the expected combined defined effect for a multiple mixture using different amounts of information. Note Tier-1 prediction relies on exposure and EC50 information (toxic unit summation), Tier-2 needs additional concentration response information for calculation of expected combined effects according to the reference models of response addition or concentration addition, and Tier-3 calculation (mixed models) requires information on the relevant mode of action. The sample is based on real analytical and effect data. Source Redrawn from data from Altenburger et al. (2004). 

FIGURE 5.1 Mixed-model mixture risk assessment approach. Note This illustrates the calculation of steps for combined effects of mixtures with similarly (e.g., Substances 1 to 3) and dissimilarly (e.g., Substances 1 to 3 versus Substances 6 to 8) acting components. 

A proposed stepwise protocol for calculating an expectation for the combined effect of a mixture with components that act similar and groups or components that act dissimilar is presented in Figure 5.1. In the first step, evaluation of the concentration addition responses to individual modes of action is required. This calculation needs to be performed for a dilution series, which can subsequently be fitted to an expected concentration response function for the groups of similarly acting compounds. In the third step, the protocol requires evaluation of the response-additive effect of different modes of action. In the mixed-model case, the protocols for concentration addition are applied within groups of compounds that share the same mode of action, and response addition is applied across these groups. 

Mixed model with concentration addition for all compounds that share 1 mode of action In current practice (e.g., Traas et al. 2002 Mulder et al. 2004 De Zwart and Posthuma 2005), concentration addition is also applied to mixtures of nonnarcotic compounds with the same mode of action, such as photosynthesis inhibition or acetyl-cholinesterase inhibition. The protocol is shown in Figure 5.2. 

The proposed mixed-model approach for assemblages, preceded by an exposure analysis, is in line with Ashford s ideas the ecological interactions need further attention. Whether Ashford s ideas can be fully worked out conceptually, tested experimentally, and applied in a validated predictive framework remains to be solved by mixture ecotoxicologists. 

A tiered system for mixture extrapolation is proposed. The lowest tier is based on extrapolation using toxicological point-estimate information such as EC50 values. This translates into the use of toxic units, toxic equivalencies, and similar techniques. The use of the entire concentration-response relationships of the separate compounds is recommended for Tier-2, in conjunction with the use of either concentration or response addition as a modeling approach. In Tier-3, a mixed-model approach can be considered, to more specifically address considerations on toxic modes of action. In the latter case, the approach may be extended to allow incorporation of the responses of different ecological receptors (Tier-4). Research needs have been clearly identified in community-level mixture assessments. 

Extrapolation by Prediction of mixture effects SSD mixed-model split... 

Tier 3 involves the use of both CA and R A models together (mixed-model approaches). This approach differs from the previous tiers by using detailed information on the modes of action for the different mixture components as well as full-curve-based modeling approaches. Mixed models are used in human as well as ecological assessment. An example of mixed-model approaches in ecological risk assessments is the approach proposed for assemblages (De Zwart and Posthuma 2005) a similar approach has been proposed by Ra et al. (2006) see Chapter 4 and Figure 4.2. 

---