A Simple Mathematical Model

Assume that a mathematical model for a motor vehicle is required, relating the accelerator pedal angle 6 to the forward speed u, a simple mathematical model might be... 

It has been possible to obtain a good measure of agreement between the experimental results, and those predicted by even a simple mathematical model of the system, assuming ideal stirred tank behaviour. One typical result is presented here. 

Previous reports on FMSZ catalysts have indicated that, in the absence of added H2, the isomerization activity exhibited a typical pattern when measured as a function of time on stream [8, 9], In all cases, the initial activity was very low, but as the reaction proceeded, the conversion slowly increased, reached a maximum, and then started to decrease. In a recent paper [7], we described the time evolution in terms of a simple mathematical model that includes induction and deactivation periods This model predicts the existence of two types of sites with different reactivity and stability. One type of site was responsible for most of the activity observed during the first few minutes on stream, but it rapidly deactivated. For the second type of site, both, the induction and deactivation processes, were significantly slower We proposed that the observed induction periods were due to the formation and accumulation of reaction intermediates that participate in the inter-molecular step described above. Here, we present new evidence to support this hypothesis for the particular case of Ni-promoted catalysts. 

Oosawa (1971) developed a simple mathematical model, using an approximate treatment, to describe the distribution of counterions. We shall use it here as it offers a clear qualitative description of the phenomenon, uncluttered by heavy mathematics associated with the Poisson-Boltzmann equation. Oosawa assumed that there were two phases, one occupied by the polyions, and the other external to them. He also assumed that each contained a uniform distribution of counterions. This is an approximation to the situation where distribution is governed by the Poisson distribution (Atkins, 1978). If the proportion of site-bound ions is negligible, the distribution of counterions between these phases is then given by the Boltzmann distribution, which relates the population ratio of two groups of atoms or ions to the energy difference between them. Thus, for monovalent counterions... 

Kozlowski J, Wojcik A. 1987. Accumulation and elimination of orally administered lead in laboratory mice Experimental studies and a simple mathematical model. Ekologia Polska 35 355-371. 

Even a simple mathematical model for transport on colloids in an aquifer must include dynamic equations for the dissolved phase and for the colloids. The latter equation describes the migration, immobilization, and detachment of the colloids. More sophisticated models include dynamic equations for sorption and desorption of the chemical onto colloids and the stationary solid phase. 

It is useful to examine the consequences of a closed ion source on kinetics measurements. We approach this with a simple mathematical model from which it is possible to make quantitative estimates of the distortion of concentration-time curves due to the ion source residence time. The ion source pressure is normally low enough that flow through it is in the Knudsen regime where all collisions are with the walls, backmixing is complete, and the source can be treated as a continuous stirred tank reactor (CSTR). The isothermal mole balance with a first-order reaction occurring in the source can be written as... 

We can describe this process by a simple mathematical model developed for steady-state conditions. The total heat release at the interface, q, can be expressed in terms of the latent heat of vaporization and heat of solution, qs ... 

Although different models of diffusion rate constants have been proposed in the literature, one must bear in mind that it is very difficult to capture the evolution of the system in the glassy state with a simple mathematical model. The main problem is the physical aging that occurs in parallel with the advance in conversion. Physical aging produces a densifica-tion of the glass, which brings two consequences ... 

Heim and Olejnik [1-3,9] proposed a simple mathematical model based on the theory of statistical moments, whose applicability was confirmed by the results of laboratory-scale studies. 

The main effect of the presence of gases and vapors in the reactor head is the relevance of an additional variable, the pressure. In a simple mathematical model, each phase may be described as a well-mixed volume (even if no mechanical stirrer is present in the gaseous phase), and it is also possible to consider thermal and mechanical equilibrium between the phases, i.e., to set equal values of temperature and pressure in the two phases. 

A simple mathematical model is used for quantitative description of the process and consists of a set of equations relating inputs, outputs, and key parameters of the system. The model for an alcoholic fermentation fed-batch process developed by Mayer (10) and adapted with the Ghose and Tyagi (11) linear inhibition term by the product was used as the starting point for the development of a model-based substrate sensor with product (ethanol) and biomass on-line measurements. 

In this section we derive a simple mathematical model for the single screw pump. In such a model, we seek relationships between performance and operating variables with the geometrical variables as parameters. 

We now derive a simple mathematical model for calculating the rate of melting in a melting chamber. The outer and inner radii of the disk are Ra and If, respectively, and the gap between them is H the solid bed occupies 2k r. of the circumference the disk speed N is constant, the disk temperature is Td the molten polymer is Newtonian with constant viscosity /< the solids and melt have densities and specific heat ps,cs and pm,cm, respectively the melting point is Tm and the heat of fusion is X. We now turn to Eq. 5.7-36, with W replaced by 2nn , and the velocity Vo by 2nrN, to get the rate of melting per unit distance in the r direction (kg/(sm))... 

Very recently, new descriptors have successfully been derived from 3-D molecular fields. These descriptors were correlated with the experimental permeation data by discriminant partial least-squares methods. The training set consisted of 44 compounds. The authors were able to deduce a simple mathematical model that allows external prediction. More than 90% of blood-brain permeation data were correctly predicted [77]. 

The outline of this chapter is as follows First, some basic wave phenomena for separation, as well as integrated reaction separation processes, are illustrated. Afterwards, a simple mathematical model is introduced, which applies to a large class of separation as well as integrated reaction separation processes. In the limit of reaction equilibrium the model represents a system of quasilinear first-order partial differential equations. For the prediction of wave solutions of such systems an almost complete theory exists [33, 34, 38], which is summarized in a second step. Subsequently, application of this theory to different integrated reaction separation processes is illustrated. The emphasis is placed on reactive distillation and reactive chromatography, but applications to other reaction separation processes are also... 

In the remainder of the chapter, wave dynamics in integrated reaction separation processes will be studied in more detail. The analysis is based on a simple mathematical model, which will be discussed in the following section. 

Dry gaseous deposition is a partitioning process between the plant and the gas phase. A simple mathematical model with a plant compartment and an air compartment can be written to describe this process.37 While there is evidence that more than one plant compartment is necessary to describe the uptake behaviour in some species,38 the one compartment assumption would appear to be reasonable for ryegrass.39 The structure of the model is illustrated in Figure 3. The defining equations are as follows ... 

These results suggest a competitive interaction between the active and nodal substituents. The geometries of these transition states support this competition their values are quite similar to the distance found in the parent 1,5-hexadiene. Computational examinations of the substituent effects on the Cope rearrangement conclude that the centauric model does not apply. The chameleonic model makes a better accounting of the cooperative and competitive ways the substituents affect the Cope rearrangement. Borden has proposed a simple mathematical model that allows for the prediction of the stabilization of the transition state by substituents solely on the change in... 

Oscillations in the number of polymer particles, the monomer conversion, and the molecular weight of the polymers produced, which are mainly observed in a CSTR, have attracted considerable interest. Therefore, many experimental and theoretical studies dealing with these oscillations have been published [328]. Recently,Nomura et al. [340] conducted an extensive experimental study on the oscillatory behavior of the continuous emulsion polymerization of VAc in a single CSTR. Several researchers have proposed mathematical models that quantitatively describe complete kinetics, including oscillatory behavior [341-343]. Tauer and Muller [344] proposed a simple mathematical model for the continuous emulsion polymerization of VCl to explain the sustained oscillations observed. Their numerical analysis showed that the oscillations depend on the rates of particle growth and coalescence. However, it still seems to be difficult to quantitatively describe the kinetic behavior (including oscillations) of the continuous emulsion polymerization of monomers, especially those with relatively high solubility in water. This is mainly because the kinetics and mech-... 

The objective of this paper is to present a simple mathematical model for clustering in dry materials. We refer to a more detailed version of the calculation (5). The strongly hydrated state, when the water molecules and the mobile cations form kinds of vesicles (or inverted micelles) will be examined later in this paper. As in any model, our mathematical model is rather ideal in the sense that it... 

Mathematical Modeling. The results obtained by various dilatomet-ric techniques were compared to theoretical predictions of thermal expansion using a simple mathematical model. Thermal properties of Kevlar fabric/epoxy lamina were simulated by considering the fabric to be made of two consecutive 0° and 90° lamina of... 

In the event of adequate vascularization of the collagen-GAG layer, following migration of endothelial cells in it, nutrient transport can proceed via blood capillaries. A simple mathematical model based on the analysis of Thiele (4), and modified by Wagner (5) and Weisz (6), relating the reactivity and diffusive flow in porous catalyst particles can be used (I) to gain insight into this process. 

A Simple Model of Coal Pyrolysis. In this section, a simple mathematical model of coal pyrolysis is formulated to qualitatively describe the pyrolysis of coal in a fluidized bed. This model is based on the assump-... 

Finegood D T, Scaglia L, Bonner-Weir S (1995). Dynamics of beta-cell mass in the growing rat pancreas. Estimation with a simple mathematical model. Diabetes. 44 249-256. 

In many cases, it is useful to have a simple mathematical model of the shear stress-shear rate relationship. Equation (4.3) defines Newtonian fluids. For non-Newtonian fluids, with data over a wide range of shear, a good fit gives the Ellis model (Middleman, 1975) ... 

Under the assumptions about the kinetics presented above the construction of a simple mathematical model for this process is quite strai tforward. A balance equation for glucose over an infinitesimal slice of the fixed bed (see figure 1) can be formulated as follows ... 

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