Distributed Constants Models

It is well known that the behavior of electrical transmission lines can be represented in terms of distributed passive elements. As we mentioned at the beginning of this chapter, there exists an analogy between the electrical and mechanical behavior of the systems. Returning to the Maxwell model, one has 

On the other hand, according to Ohm s law, the electrical admittance is given by 

The electromechanical analogy indicates that e( ) can be identified with I(t) (electrical intensity of current) and j t) with V(t) (electrical voltage). Therefore, se s) and d(j ) correspond, respepctively, to I s) and V s) so that the mechanical admittance can be written as 

We notice that the elements in series in the mechanical model are transformed in parallel in the electrical analogy. The converse is true for the Kelvin-Voigt model. The electrical analog of a ladder model is thus an electrical filter. 

We can generalize the analogy by considering the viscoelastic materials as a continuum where the theory of transmission lines can be applied. In this way, a continuous distribution of passive elements such as springs and dash-pots can be used to model the viscoelastic behavior of materials. Thus the relevant equations for a mechanical transmission line can be written following the same patterns as those in electrical transmission lines. By representing the impedance and admittance per unit of length by g and j respectively, one has 

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The activity of antioxidants in food [ 1 ] emulsions and in some biological systems [2] is depends on a multitude of factors including the localisation of the antioxidant in the different phases of the system. The aim of this study is determining antioxidant distributions in model food emulsions. For the purpose, we measured electrochemically the rate constant of hexadecylbenzenediazonium tetrafluorborate (16-ArN,BF ) with the antioxidant, and applied the pseudophase kinetic model to interpret the results. 

Figure 4.17 General phenonenaloglcal retention model for a solute that participates in a secondary chemical equilibrium in liquid chromatography. A - solute, X - equilibrant, AX analyte-equilibrant coeplex, Kjq - secondary chemical equilibrium constant, and and are the primary distribution constants for A and AX, respectively, between the mobile and stationary phases.

Nevertheless, these modeling efforts are of little value when the practical implementation does not corroborate the above-calculated results. Ensuring the constancy of any parameter in the catalytic testing workflow, the reactor performance with regard to temperature distribution, gas distribution, constant feed... 

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.. 

It was shown by Roux (1974) that the two ideal sites model applies perfectly to partitioning of Rb distribution between nepheline and hydrothermal solutions. Based on the experimental work of Roux (1974), deviation from ideality in the normalized Rb/Na distribution between nepheline (X.) and hydrothermal solution (Xaq) is detectable for X, values higher than 10 (figure 10.3A) Distribution constant Xin the Nemst s law range is Kq = 0.82 (Roux, 1974), and the modification of K with increasing X, is well described by equation 10.20, within experimental approximation (figure 10.3B). 

Figure 10.4 shows normalized Ba/K and Sr/K distributions between sanidine and a hydrothermal solution (liyama, 1972), Li/K between muscovite and a hydrothermal solution (Voltinger, 1970), and Rb/Na between nepheline and a hydrothermal solution (Roux, 1971b), interpreted through the local lattice distortion model, by an appropriate choice of the Nernst s law mass distribution constant K and the lattice distortion propagation factor r. 

Medium-chain alcohols such as 2-butoxyethanol (BE) exist as microaggregates in water which in many respects resemble micellar systems. Mixed micelles can be formed between such alcohols and surfactants. The thermodynamics of the system BE-sodlum decanoate (Na-Dec)-water was studied through direct measurements of volumes (flow denslmetry), enthalpies and heat capacities (flow microcalorimetry). Data are reported as transfer functions. The observed trends are analyzed with a recently published chemical equilibrium model (J. Solution Chem. 13,1,1984). By adjusting the distribution constant and the thermodynamic property of the solute In the mixed micelle. It Is possible to fit nearly quantitatively the transfer of BE from water to aqueous NaDec. The model Is not as successful for the transfert of NaDec from water to aqueous BE at low BE concentrations Indicating self-association of NaDec Induced by BE. The model can be used to evaluate the thermodynamic properties of both components of the mixed micelle. 

The chemical equilibrium model of Roux et al (6) is a powerful tool for the study of the thermodynamics of mixed micellar solutions. It can estimate the distribution constant of the surfactant 3 between water and micelles of the surfactant 2 and the thermodynamic properties of the surfactant 3 in the mixed micelles. For this it is necessary to obtain reliable data over a large concentration range of solute 2. 

Most commonly, distributed parameter models are applied to describe the performance of diesel particulate traps, which are a part of the diesel engine exhaust system. Those models are one- or two-dimensional, non-isothermal plug-flow reactor models with constant convection terms, but without diffusion/dispersion terms. 

The mathematical model for char combustion described in the previous two sections is applicable to a bed of constant volume, i.e., to a fluidized bed of fixed height, Hq, and having a constant cross-sectional area, Aq. The constant bed height is maintained by an overflow pipe. For this type of combustor operating for a given feed rate of char and limestone particles of known size distributions, the model presented here can predict the following ... 

A combination of the two last properties is, however, expressed by the respective distribution constants (or Henry law constants). Hence a change in the oxidation state implies a change in the distribution of the compound in question between air and water. This in its turn usually results in a transfer of this compound from one box in the model to another. 

The use of distributed pharmacokinetic models to estimate expected concentration profiles associated with different modes of drug delivery requires that various input parameters be available. The most commonly required parameters, as seen in Equation 9.1, are diffusion coefficients, reaction rate constants, and capillary permeabilities. As will be encountered later, hydraulic conductivities are also needed when pressure-driven rather than diffusion-driven flows are involved. Diffusion coefficients (i.e., the De parameter described previously) can be measured experimentally or can be estimated by extrapolation from known values for reference substances. Diffusion constants in tissue are known to be proportional to their aqueous value, which in turn is approximately proportional to a power of the molecular weight. Hence,... 

Reaction rate parameters required for the distributed pharmacokinetic model generally come from independent experimental data. One source is the analysis of rates of metabolism of cells grown in culture. However, the parameters from this source are potentially subject to considerable artifact, since cofactors and cellular interactions may be absent in vitro that are present in vivo. Published enzyme activities are a second source, but these are even more subject to artifact. A third source is previous compartmental analysis of a tissue dosed uniformly by intravenous infusion. If a compartment in such a study can be closely identified with the organ or tissue later considered in distributed pharmacokinetic analysis, then its compartmental clearance constant can often be used to derive the required metabolic rate constant. 

Campanella and Peleg (1987) presented stress growth and decay data on mayonnaise at shear rates of 1.8, 5.4,9.9, and 14.4s with a controlled shear rate viscometer and a concentric cylinder geometry. They modeled the data by a three-constant model that was a modification of Larson s (1985) model that was successfully employed for polyethylene melts with a wide distribution of molecular weights. The model employed by Campanella and Peleg was ... 

Figure 4 compares several of these models with respect to the nature of the constants that each uses. The simplest model (linear sorption or Ai ) is the most empirical model and is widely used in contaminant transport models. values are relatively easy to obtain using the batch methods described above. The Aid model requires a single distribution constant, but the Aid value is conditional with respect to a large number of variables. Thus, even if a batch Aid experiment is carefully carried out to avoid introduction of extraneous effects such as precipitation, the Aid value that is obtained is valid only for the particular conditions of the experiment. As Figure 4 shows, the radionuclide concentration, pH, major and minor element composition, rock mineralogy, particle size and solid-surface-area/solution volume ratio must be specified for each Aid value. 

Figure 4 Comparison of sorption models. Several commonly used sorption models are compared with respect to the independent constants they require. These constants are vahd only under specific conditions, which must be specified in order to properly use them. In other words, the constants are conditional with respect to the experimental variables described in the third column of the figure. is the radionuclide distribution constant K and n are the Freundlich isotherm parameters and are surface complexation constants for protonation and deprotonation of surface sites K-, are surface complexation constants for sorption of cations and anions in the constant...

Opponents of the nuclear winter theory argue that there are many problems with the hypothesized scenarios either because of the model s incorrect assumptions (e.g., the results would be right only if exactly the assumed amount of dust would enter the atmosphere, or because the model assumes uniformly distributed, constantly injected particles). Other critics of the nuclear winter scenario point out that the models used often to not include processes and/or feedback mechanisms that may moderate or mitigate the initial effects of nuclear blasts on the atmosphere (e.g., the moderating effects of the oceans). 

Fig. 5 illustrates a physical model of the chromatography process. Initially, there is a dynamic equilibrium of molecules between the phases. Then, one phase is moved relative to the other with an average velocity, v. In the stationary phase, molecules do not move while in the mobile phase, molecules move with a velocity equal to v. Provided that the interphase mass transfer rate is fast relative to the flow rate of the mobile phase, the time-average distribution of a molecule between the phases is statistically equal to the equilibrium distribution as determined by the distribution constant. 

The most commonly used adsorption model in contaminant transport calculations is the distribution coefficient, model. In large part this reflects the simplicity of including a value in transport calculations (cf. Freeze and Cherry 1979 Domenico and Schwartz 1990 Stumm 1992). Nevertheless, such applications should be limited to conditions where values can be expected to remain near constant during transport (cf. Reardon 1981). Alternatively, if a Kj can be confidently shown to be a maximal or minimal possible value, such calculations can provide bounding or conservative information on contaminant transport. The bounding minimum approach has become standard in the modeling radionuclide transport, for example (cf. Meijer 1992). 

A number of studies have been reported in the literature for the development and testing of mathematical descriptions for solute transport through hquid membranes. The existing models can be broadly classified into (i) the membrane film model in which the entire resistance to mass transfer is assumed to be concentrated in a membrane film of constant thickness and (ii) distributed resistance model which considers the mass transfer resistance to be distributed throughout the emulsion drop. 

On a second level, opportunities exist because the current level of geochemical studies in the environmental field is low. We know that a distribution constant Kj is usually insufficient to describe the chemical processes (see 10.3), but most fate and transport models have employed this concept. Geochemical modeling is generally under-used. 

Figure 4.8 Scatter plot of simulated data from a Michaelis-Menten model with Vmax — 100 and Km — 20 (top) and Lineweaver-Burke transformation of data (bottom). Stochastic variability was added by assuming normally distributed constant variability with a standard deviation of 3.

A conceptual and mechanistic model of particle interactions in silica-iron binary oxide suspensions is described. The model is consistent with a process involving partial Si02 dissolution and sorption of silicate onto Fe(OH)3. The constant capacitance model is used to test the mechanistic model and estimate the effect of particle interactions on adsorbate distribution. The model results, in agreement with experimental results, indicate that the presence of soluble silica interferes with the adsorption of anionic adsorbates but has little effect on cationic adsorbates. 

The neutral species are partitioned between the liquid stationary and mobile phases (KD is the relevant distribution constant). Separation is based upon the relative values of the distribution coefficient of the different neutral species. This model most closely explains the experimental results obtained with non-bonded reversed-phase columns (e.g. n-pentanol coated onto silica gel), in which the stationary phase behaves as a bulk liquid. The ion-pair model is, however, unable to explain ion-pair interactions with chemically bonded reversed-phase columns, and the working of these clumns is more appropriately explained by a dynamic ion-exchange model. 

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