Simple Mathematical Models

The box model is closely related to the more complex airshed models described below in that it is based on the conservation of mass equation and includes chemical submodels that represent the chemistry more accurately than many plume models, for example. However, it is less complex and hence requires less computation time. It has the additional advantage that it does not require the detailed emissions, meteorological, and air quality data needed for input and validation of the airshed models. However, the resulting predictions are 

FIGURE 16.19 Schematic diagram showing basic elements of a simple box model (adapted from Schere and Demerjian, 1978). 

The kinetic modeling nomenclature arises from the incorporation of chemical kinetic submodels in EKMA. The empirical term comes from the use of observed 03 peaks in combination with the model-predicted ozone isopleths to develop control strategy options. Thus, the approach historically was to use the model to develop a series of ozone isopleths using conditions specific for that area. The second highest hourly observed 03 concentration and the measured 

VOC/NO ratio were located on these iso-pleths. The isopleths were then used to determine how much reduction in VOC or NOx would be required to reduce the peak 03 concentration to the air quality standard. 

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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... 

The first step is to be certain of the basis of the published data and consider in what ways this will be affected by different conditions. Revised figures can then usually be determined. For extensive interpretation work, simple mathematical models of performance can be constructed [69]. 

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. 

May, R. M. (1976). Simple mathematical models with very complicated dynamics review article. Nature, 261, 459-67. 

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 ... 

A relatively simple mathematical model composed of 21 or 23 transcendental and rational equations numbered (7.25) to (7.47) was presented to describe the steady-state behavior of type IV FCC units. The model lumps the reactants and products into only three groups. It accounts for the two-phase nature of the reactor and of the regenerator using hydrodynamics principles. It also takes into account the complex interaction between the... 

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. 

Staggs, J. E. J. Simple mathematical models of char-forming polymers. Polymer International 2000 49 1147. 

If the linear gas velocity approaches the speed of sound, the simple mathematical model used in equation (4) breaks down. The acceleration term must be taken into account, and the steady-state equation of motion for a straight pipeline with constant diameter may be written (8) ... 

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 ... 

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