Two-process models

Reliable scale-up of the desired operating conditions from the model to the full-scale plant. According to the theory of models, two processes may be considered similar to one another if they take place under geometrically similar conditions and all dimensionless numbers, which describe the process, have the same numerical value. 

Dimensional analysis is a method for producing dimensionless numbers that completely characterize the process. The analysis can be applied even when the equations governing the process are not known. According to the theory of models, two processes may be considered completely similar if they take place in similar geometrical space and if all the dimensionless numbers necessary to describe the process have the same numerical value [2], The scale-up procedure, then, is simple express the process using a complete set of dimensionless numbers, and try to match them at different scales. This dimensionless space in which the measurements are presented or measured will make the process scale invariant. 

Abstract In this work the ion exchange kinetics of at Mg -montmorillonite have been investigated. These kinetics are fast, i.e., the exchange in diluted suspension is complete after 30 s, therefore, the stopped flow method is applied. The data were analyzed by the kinetic spectrum method, because the reaction caimot be described by classical kinetic models. Two processes were observed i) A fast one with a mean pseudo-first-order rate constant of 20 s . This process is assigned to the exchange at easily accessible sites at the outer surface, ii) A group of slow processes with... 

Various mathematical concepts and techniques have been used to derive the functions that describe the different types of dispersion and to simplify further development of the rate theory two of these procedures will be discussed in some detail. The two processes are, firstly, the Random Walk Concept [1] which was introduced to the rate theory by Giddings [2] and, secondly, the mathematics of diffusion which is both critical in the study of dispersion due to longitudinal diffusion and that due to solute mass transfer between the two phases. The random walk model allows the relatively simple derivation of the variance contributions from two of the dispersion processes that occur in the column and, so, this model will be the first to be discussed. 

We will not go any further into the interesting speculations Minsky offers in his 1982 paper [minsky82]. The point of these speculations was not to propose a serious alternative model of fundamental physics, per se, but to stimulate thinking along the. lines of What if physics were, fundamentally, discrete How would we describe the processes we now think we iinderstaiul with our continuous models Two questions that we will repeatedly come back to in this concluding chapter. 

Borbely AA (1982) A two process model of sleep regulation. HumNeurobiol 1 195-204... 

There are two ways to evade this problem especially for modelling polymerization processes ... 

There is some debate about what controls the magnesium concentration in seawater. The main input is rivers. The main removal is by hydrothermal processes (the concentration of Mg in hot vent solutions is essentially zero). First, calculate the residence time of water in the ocean due to (1) river input and (2) hydro-thermal circulation. Second, calculate the residence time of magnesium in seawater with respect to these two processes. Third, draw a sketch to show this box model calculation schematically. You can assume that uncertainties in river input and hydrothermal circulation are 5% and 10%, respectively. What does this tell you about controls on the magnesium concentration Do these calculations support the input/removal balance proposed above Do any questions come to mind Volume of ocean = 1.4 x 10 L River input = 3.2 x lO L/yr Hydrothermal circulation = 1.0 x 10 L/yr Mg concentration in river water = 1.7 X 10 M Mg concentration in seawater = 0.053 M. 

The following model of the corrosion process can be proposed based on the wealth of data provided by the combined application of SPFM, contact AFM, and IRAS At low RFl, the principal corrosion prodnct, hydrated alnminnm snlfate, is solid. It acts as a diffn-sion barrier between the acid and the alnminnm snbstrate and prevents fnrther corrosion. The phase separation observed between the acid and the salt at low RH strongly snggests that the salt inhibits fnrther corrosion once it precipitates. At high RH, on the other hand, alnminnm snlfate forms a liqnid solntion. Snlfnric acid mixes with this solntion and reaches the nnderlying snbstrate, where fnrther reaction can occnr. The flnid snlfate solntion also wets the snrface better and thns spreads the snlfnric acid. The two processes assist each other, and the corrosion proceeds rapidly once the critical RH of 80-90% is reached. 

In order to develop intelligent, computer-aided systems with systematic and sound methodologies for the automatic creation of mental models of process operations, we need to resolve the following two and interrelated issues ... 

Mathematical modeling of these two processes led to the development of the model (4) that forms the basis of most mechanistic models of nutrient uptake from soil. 

Preliminary work showed that first order reaction models are adequate for the description of these phenomena even though the actual reaction mechanisms are extremely complex and hence difficult to determine. This simplification is a desired feature of the models since such simple models are to be used in numerical simulators of in situ combustion processes. The bitumen is divided into five major pseudo-components coke (COK), asphaltene (ASP), heavy oil (HO), light oil (LO) and gas (GAS). These pseudo-components were lumped together as needed to produce two, three and four component models. Two, three and four-component models were considered to describe these complicated reactions (Hanson and Ka-logerakis, 1984). 

For turbulent flow in single-phase systems, the predicted temperature profile is not changed significantly if the Peclet number is assumed to be infinite. Therefore, in turbulent two-phase systems the second-order terms in Eqs. (9) probably do not have a significant effect on the resulting temperature profiles. In view of the uncertainties in the present state of the art for determining the holdups and the heat-transfer coefficients, the inclusion of these second-order terms is probably not justified, and the resulting first-order equations should adequately model the process. 

---