Functions used in the modelling

Heat and mass balance equations are used in all aspects of process modelling however, what is key to this model is an understanding of the electrolytic process behind the cell. For example, the model must be able to predict current efficiency and k-factor if it is to predict electricity consumption. Most of these electrolytic parameters are calculated using empirical relationships derived from experimental data both from test cells and the full-scale plant. Considering k-factor, this is primarily a function of brine strength and temperature. Figure 20.5 illustrates the experimentally derived function used in the model. 

With those functions, assuming the same deactivation for all reactions and an injection time of 90 s, several simulations using the 12 lump model were made. It allowed one to obtain values for the constants A, B, K and a that produce similar results regardless of the deactivation function used in the model. Figs. 3 and 4 show gasoline and coke yields. 

One question that has not been discussed so far concerns the metal dielectric functions used in the modelling. It appears in fact that the dielectric functions of the bulk metals, either gold or silver dielectric functions as taken from P.B. Johnson and R.W Christie for example, do not yield a good agreement of the models with the experimental data [57]. Hence, the models used in Figures 9 and 10 are calculated with the use of Drude type dielectric functions of the form (< >) = , (w) + such that [58] ... 

Deactivation functions used in the modeling of the deactivation of the US-Y-zeolite catalyst in the catalytic cracking of vacuum gas oil [Moustafa and Froment, 2003]. 

A predictive macromolecular network decomposition model for coal conversion based on results of analytical measurements has been developed called the functional group, depolymerization, vaporization, cross-linking (EG-DVC) model (77). Data are obtained on weight loss on heating (thermogravimetry) and analysis of the evolved species by Eourier transform infrared spectrometry. Separate experimental data on solvent sweUing, solvent extraction, and Gieseler plastometry are also used in the model. 

No assistance is provided to guarantee that the data used in the modeling process is complete and accurate. Therefore, the comprehensiveness of the final OAET will be a function of experience of the analyst. (This criticism applies to all HRA techniques.)... 

The objective function used in the mathematical model is either the maximisation of profit or the minimisation of effluent. This is dependent on the nature of the data given for a problem. If the production, e.g. number of batches of each product or total tonnage, is not given then the objective function is the maximisation of profit. However, if this is given then the objective function is the minimisation of effluent. 

The difference between the expected demands d used in the model and the realization of an actual demand d causes a model mismatch. The scheduler corrects this error after the observation of this demand in a reactive manner by the production decisions taken at the beginning of the next period. For example, the decisions taken in period i = 1 take the expected value of the demand di into account (di = 6) while the decisions taken in period i = 2 take the true value into account. When di = 0, the large storage causes a relatively low production in the next period [xiih) = 5) whereas in case of d = 12, a deficit results and the production of the next period is larger (x2(t2) = 12). The sequence of decisions is a function of the observed demands and thus the sequence varies over the scenarios. 

The turnover function is used in the model when optimizing profit including the respective turnover. 

Stine et al. proposed a simple analytic model for patterned feature removal incorporating an effective density determination step [54]. Figure 10 defines terms used in the model, which reformulates Preston s equation [Eq. (2)] as a function of blanket rate K and effective pattern density... 

In Equation 5.28, s is a function of the concentration residuals observed during calibration, r is tlie measurement vector for the prediction sample, and R contains the calilxation measurements for the variables used in the model. Because the assumptions of linear regression are often not rigorously obeyed, the statistical pret ion error should be used empirically rather than absolutely. It is useful for validating the prediction samples by comparing the values for... 

Root Mean Square Error of Prediction (RMSEP) Plot (Model Diagnostic) The validation set is employed to determine the optimum number of variables to use in the model based on prediction (RMSEP) rather than fit (RMSEO- RM-SEP as a function of the number of variables is plotted in Figure 5.7S for the prediction of the caustic concentration in the validation set, Tlie cuive levels off after three variables and the RMSEP for this model is 0.053 Tliis value is within the requirements of the application (lcr= 0.1) and is not less than the error in the reported concentrations. 

Figure 2. Fractionation factors as a function of temperature as used in the model based on empirical fits determined by (Majoube 1971b Horita and Wesolowski 1994) and (Merlivat and Nief 1967 Majoube 1971a) See Rowley et al. (2001), Rowley and Garzione (2007) for original publications for the equations graphed in Figure 2.

The guidelines of linear-free energy relationships have also been used to capture not only the hydrocarbon stmcture/function but also catalyst structure/function relationships. Thus Liguras et al. (39) have fashioned a model where the rate constant is a function of the reactant, the reaction family, and the catalyst silicon to aluminum ratio. This fledgling approach considerably reduces the number of kinetic parameters and appears to be quite useful in the modelling of complex kinetics of hydrocarbon feedstocks. 

A simple relation between the steric factor of the collision model and the partition functions used in the transition-state theory may be made by employing the relation derived in Eq. (XII.3.14) between frequency... 

With this in mind, it seems to be a reasonable compromise to consider a FEM implementation of the modelling of stress-assisted diffusion over the previously (or simultaneously) performed stress analysis taking the nodal values of stresses, obtained with a post-processing technique, as the entry data for diffusion, i.e., constructing a finite-element approximation of the stress field with the aid of the same finite-element shape functions used in the mechanical analysis to approximate the displacement fields. 

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