The modeling assumptions

In the case of a single solid as product (which excludes the reactions of double decompositions that we will approach in section 14.4.2.4), we can write the reaction between two solids in the general form  

To define the fractional extent of this reaction, we consider as a reference the initial amount o of the least abundant solid reactant (from the stoichiometric point of view). Assume A is the reactant, which means that  

The fractional extent with respect to A will be then  

We assume that the system evolves according to a pseudo-steady state mode and thus the various fractional extents are equal. The common value is the fractional extent a of the reaction (see section 7.4.3). 

We also generally assume that, between two massive solids (or two grains in the case of a powder), the rate of growth is separable (whieh we can verily thereafter) and thus we will have  

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In particular we would like to treat some essential effects of fluctuations where we assume that, for example, thermal fluctuations exist and are localized in space and time. The effects on large lengths and long times are then of interest where the results are independent of local details of the model assumptions and therefore will have some universal validity. In particular, the development of a rough surface during growth from an initially smooth surface, the so-called effect of kinetic roughening, can be understood on these scales [42,44]. 

The following criteria are usually directly applied to the calibration set to enable a fast comparison of many models as it is necessary in variable selection. The criteria characterize the fit and therefore the (usually only few) resulting models have to be tested carefully for their prediction performance for new cases. The measures are reliable only if the model assumptions are fulfilled (independent normally distributed errors). They can be used to select an appropriate model by comparing the measures for models with various values of in. 

Equation 7.1 utilizes exchange coefficients to predict steady-state BCFs and ATswS, and the model assumptions include a uniform lipid phase enclosed in a non-interactive membrane. The model shows that the magnitude of a BMO s BCF or an SPMD s Ksvj is affected by variations in ku and/or ke, unless both constants rise or fall proportionally. In the case of SPMDs, Huckins et al. (1993,2002a) have shown that the uptake and release process is essentially isotropic for HOCs. When residue exchange is isotropic, AfswS will remain relatively constant even when exposure conditions affect SPMD ku and ke values. This is not always the case for BMOs, yet isotropic exchange is a fundamental assumption of EP theory. 

The aim in converting the microhotplate into a geometry model for the FEM simulation was to find a model that is as simple as possible but includes all relevant processes. The model assumptions to be explained in detail in the following section and the steps to arrive at the model are represented in fig. 3.2. The feature on the bottom left-hand side represents the microhotplate schematic. The microhotplate exhibits a symmetric design so that a simulation of one quarter is adequate. Geomet-... 

These equations describe an unheated transistor and were verified for a device with no backside etching (no membrane). The modelling parameters were provided by the manufacturer, whereas the value of the threshold voltage was taken from wafer map data. The channel length modulation parameter. A, had to be extracted from measurement data. The discrepancy between simulated and measured source-drain saturation current, fsd,sat> for a transistor embedded in the bulk silicon was less than 1%, which confirmed the vaHdity of the model assumptions. 

In this simulation the model assumption is that the melt film is transported into the pores of the bed. Thus the melt film is relatively thin even for this mechanism due to the reducing depth of the channel. The model as developed does not account for any conductive heat transfer into the solid particles from the liquid infiltrate. For most extrusions, the center of the solid bed may be porous while the edges exposed to the melt film are not. The sealed edges prevent the melt from infiltrating the solid bed and the melting process occurs via conventional melting. Conventional melting was observed for the case with the measured bulk density data in Fig. 4.1. 

In this section the model for a continuous evaporative crystallizer is discussed. The crystallizer is of the draft tube baffled (DTB) type and is equiped with a fines removal system consisting of a large annular zone on the outside of the crystallizer (see Figure 1). In order to vary the dissolved fines flow without changing the cut-size of the fines removal system, the flow in the annular zone is kept constant and the flow in the dissolving system is varied by changing the recycle flow rate. The model assumptions are ... 

In contrast w DCLS, the ptire spectra in the indirect approach are not measured direcfly, but are estimated from mixture spectra. One reason for using ICLS is that a is not possible to physically separate die components (e.g., when one cd the components of interest is a gas and future prediction samples are mixtures of the gas dissolved in a liquid). Indirect CLS is also used when the model assumptions do not hold if the pure component is run neat. By preparing mixtures, it is possible to dilute a strongly absorbing component so that the modd assumptions hold. 

The statistical prediction error is in concentration units and represents the uncertainty in the predicted concentrations due to deviations from the model assumptions, measurement noise, and degree of overlap of the pure spectra. As the system deviates from the underlying assumptions of CLS, the residual... 

Although the void behavior is calculated through Stage 5, it is likely that the viscosity rises sufficiently high in Stage 4 so that the model assumptions are no longer valid. The void dissolution calculated in Stages 4 and 5 will therefore probably not occur in reality. 

It becomes obvious that a logarithmic plot of the measured Rp as a function of time provides a linear curve if the model assumptions are reasonable, with Rp 0 as the intercept of the ordinate, and kj the negative slope of this curve. 

In order to estimate the systematic errors introduced by the model assumptions, we perform some test calculations. Instead of the velocity-law exponent of 8=1, another fit is obtained with 8=0.5 (Fig. 1). This fit yields an effective temperature of about 3000K higher than with 8=1. In order to simulate the effect of the suspected hydrogen content, a further fit (Fig. 2) is made when one free electron per helium atom is added artificially. This has only marginal influence on the derived temperature (+100K). Thus, we conclude that our model assumptions may introduce a systematic error of the order of 5000K. 

Langmuir-Hinshelwood (LH) kinetics are widely used to quantitatively delineate substrate preadsorption in both solid-gas and solid-liquid reactions. The model assumptions are stated in Table 9.2. Under these... 

In summary, assumptions (1) and (2) are unnecessary and have been avoided in more advanced models. Assumptions (3) and (4) are unavoidable and illustrate the fundamental weakness of most enzymatic sensors, particularly those depending on detection of pH changes. Assumptions (5) and (6) can be avoided to some extent by experimental design, but should be always accounted for in the model. Assumption (7) is easily avoidable. There is another assumption that has not been mentioned, the equality of concentration and activity. As discussed in Chapter 1, that cannot always be a justifiable assumption. 

For the test case, being considered, the optimum coal feed size given by Eq. 22 is 1.6mm. Clearly, the optimum depends upon operating conditions, and also on the model assumptions. 

After the series of metabolic pathways had been elucidated for the three model compounds 1-3, these data were implemented into the mathematical model PharmBiosim. The nonlinear system s response to varying ketone exposure was studied. The predicted vanishing of oscillatory behavior for increasing ketone concentration can be used to experimentally test the model assumptions in the reduction of the xenobiotic ketone. To generate such predictions, we employed as a convenient tool the continuation of the nonlinear system s behavior in the control parameters. This strategy is applicable to large systems of coupled, nonlinear, ordinary differential equations and shall together with direct numerical simulations be used to further extend PharmBiosim than was sketched here. This model already allows more detailed predictions of stereoisomer distribution in the products. 

Comparing the PSDs obtained from the different methods used, we could conclude that DFT seems, at a first glance, to report the most coherent PSD. However, this favorable PSD for DFT merits recalling that the DFT method has several problems derived from the model assumption which are reflected on the PSD shape. For example, PSDs obtained from DFT and also from GCMC, systematically show minima at pore sizes of about 0.6 and l.Onm [76] (see Figure 4.6), which correspond to two and three molecular diameters of the adsorbate. These minima are due to packing effect since two or three molecules of adsorbate could be perfectly packed in pores with these sizes. These minima affect the PSD and give erroneous results. 

Accuracy of information is a result of scientific expertise, the delivery of adequate, complete and unbiased information about results and residual uncertainties. The speed of release is influenced by the organizational culture, to what extent the process to find answers and to acknowledge uncertainties is developed. Empathy is related to the willingness to recognize the situation (the scenario) in which the persons/clients are found. The degree of openness corresponds to the information given about uncertainties and limitations in the exposure assessment, the restrictions with respect to selected scenarios, the model assumptions and... 

Although the level of uncertainty of the model assumptions, structure and details was characterized as Low when considered in isolation, the uncertainty of other components (i.e. model extrapolations, chemical-specific exposure data, non-chemical-specific exposure data and exposure assessment result) was considered to be a function primarily of limitations (i.e. simplicity) of the conceptual model. 

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