Mathematical transport

The papers on which the articles in this volume are based, were prepared at the invitation of the organizing committee, for presentation at the Conference on Stochastic Processes in Chemical Physics which was held at the University of California at San Diego, La Jolla, March 18-22, 1968. The purpose of this meeting was to bring together selected experts in the fields of probability theory, applied mathematics, transport processes, statistical mechanics, chemical kinetics, polymer chemistry, and molecular biochemistry for an exchange of ideas and to stimulate interest and activity in the application of the theory of stochastic processes to problems in chemical physics. 

Soil column experiments with conservative and reactive tracers are used for the development of reactive transport models in soil and groundwater and for the determination of model parameters. The influence of the real structure of the solid layers on the transport and reaction processes is very important and has to be taken into consideration for the development of mathematical transport models (Chin and Wang, 1992). Several methods exist for the investigation of layer structures (ultrasound and electrical tomography, computer tomography with X-rays) (Just et al., 1994 Meyer et al. 1994), but generally these methods give no information about the dynamic processes. 

Department of Applied Mathematics, Transport and Telecmnmnnication histitule, Lomonosov Street 1, LV-1019 Riga, Latvia e-mail k(mstan tsi.lv... 

Pintauro PN, Bennion DN (1984) Mass-transport of electrolytes in membranes. 1. Development of mathematical transport model. Ind Eng Chem Fundam 23 230-234... 

Mathematically, transport through a channel can be viewed as an FP process whose starting point is me entrance of me particle into the channel and its final point is me particle exit from me channel. Figure 2 illustrates me different representations of me channel transport problem, discussed later. 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

Computer simulation of the reactor kinetic hydrodynamic and transport characteristics reduces dependence on phenomenological representations and idealized models and provides visual representations of reactor performance. Modem quantitative representations of laminar and turbulent flows are combined with finite difference algorithms and other advanced mathematical methods to solve coupled nonlinear differential equations. The speed and reduced cost of computation, and the increased cost of laboratory experimentation, make the former increasingly usehil. 

Transport Models. Many mechanistic and mathematical models have been proposed to describe reverse osmosis membranes. Some of these descriptions rely on relatively simple concepts others are far more complex and require sophisticated solution techniques. Models that adequately describe the performance of RO membranes are important to the design of RO processes. Models that predict separation characteristics also minimize the number of experiments that must be performed to describe a particular system. Excellent reviews of membrane transport models and mechanisms are available (9,14,25-29). 

Figure 16-9 depicts porous adsorbent particles in an adsorption bed with sufficient generality to illustrate the nature and location of individual transport and dispersion mechanisms. Each mechanism involves a different driving force and, in general, gives rise to a different form of mathematical result. 

A mathematical model describing how material is transported and dispersed from a release... 

In its simplest form, a model requires two types of data inputs information on the source or sources including pollutant emission rate, and meteorological data such as wind velocity and turbulence. The model then simulates mathematically the pollutant s transport and dispersion, and perhaps its chemical and physical transformations and removal processes. The model output is air pollutant concentration for a particular time period, usually at specific receptor locations. 

Burns, R.S., Richter, R. and Polkinghorne, M.N. (1995) A Multivariable Neural Network Ship Mathematical Model. In Marine Technology and Transportation, Graczyk, T., Jastrzebski,... 

A model can be defined as a set of relationships between the variables of interest in the system being investigated. A set of relationships may be in the form of equations the variables depend on the use to which the model is applied. Therefore, mathematical equations based on mass and energy balances, transport phenomena, essential metabolic pathway, and physiology of the culture are employed to describe the reaction processes taking place in a bioreactor. These equations form a model that enables reactor outputs to be related to geometrical aspects and operating conditions of the system. 

In modeling an RO unit, two aspects should be considered membrane transport equations and hydrodynamic modeling of the RO module. The membrane transport equations represent the phenomena (water permeation, solute flux, etc.) taking place at the membrane surface. On the other hand, the hydrodynamic model deals with the macroscopic transport of the various species along with the momentum and energy associated with them. In recent years, a number of mathematical... 

The analysis of the consequences of nuclear accidents began with physical concepts of core melt, discussed the mathematical and code models of radionuclide release and transport within the plant to its release into the environment, models for atmospheric transport and the calculation of health effects in humans. After the probabilities and consequences of the accidents have been determined, they must be assembled and the results studied and presented to convey the meanings. 

Daviskas, E., Gonda, I., and Anderson, S. D. (1990). Mathematical modeling of heat and water transport in human respiratory tract. /. Appl. Physiol. 69, 362-372. 

Scherer, P. W., and Hanna, L. M. (1985). Heat and water transport in the human respiratory tract. In Mathematical Modeling m Medicine and Biology (A. Shitzer and R. C. Eberh.trt, Eds. , pp. 287-306. Plenum Press, New York. 

The mathematical model developed by Stegmaier for horizontal transport... 

General solution of the population balance is complex and normally requires numerical methods. Using the moment transformation of the population balance, however, it is possible to reduce the dimensionality of the population balance to that of the transport equations. It should also be noted, however, that although the mathematical effort to solve the population balance may therefore decrease considerably by use of a moment transformation, it always leads to a loss of information about the distribution of the variables with the particle size or any other internal co-ordinate. Full crystal size distribution (CSD) information can be recovered by numerical inversion of the leading moments (Pope, 1979 Randolph and Larson, 1988), but often just mean values suffice. 

To understand the mathematics, consider a large empty space into which a number of production units are to be placed, and assume that the major variable to be optimized is the cost of transporting materials between them. If the manufacturing process is essentially a flow-line operation, then the order in which units should be placed is clear (from the point of view of transport costs), and the problem is simply to fit them into the space available. In a job-shop, where materials are flowing between many or all the production units, the decision is more difficult. All the potential combinations of units and locations... 

Using the mathematical technique of dimensionless group analysis, the rate of mass transport (/ m) in terms of moles per unit area per unit time can be shown to be a function of these variables, which when grouped together can be related to the rate by a power term. For many systems under laminar flow conditions it has been shown that the following relationship holds ... 

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