Isothermal mathematical models

CFD models of DEFC have been also proposed [188]. Suresh and Jayanti developed an one-dimensional, single phase, isothermal mathematical model for a liquid-feed DEFC, taking into account mass transport and electrochemical phenomena on both the anode side and the cathode side [189]. Tafel kinetics expressions have been used to describe the electrochemical oxidation of ethanol at the anode and the simultaneous ethanol oxidation and ORR at the cathode. The model in particular accounts for the mixed potential effect caused by ethanol cross-over at the cathode and is validated using the data from the literature. Model results show that ethanol crossover can cause a significant loss of cell performance. 

For practical reasons, the blast furnace hearth is divided into two principal zones the bottom and the sidewalls. Each of these zones exhibits unique problems and wear mechanisms. The largest refractory mass is contained within the hearth bottom. The outside diameters of these bottoms can exceed 16 or 17 m and their depth is dependent on whether underhearth cooling is utilized. When cooling is not employed, this refractory depth usually is determined by mathematical models these predict a stabilization isotherm location which defines the limit of dissolution of the carbon by iron. Often, this depth exceeds 3 m of carbon. However, because the stabilization isotherm location is also a function of furnace diameter, often times thermal equiHbrium caimot be achieved without some form of underhearth cooling. 

An isothermal curtain is easily distinguished from the non isothermal but in practice the mathematical models of a free jet are seldom fully representative of the real jet. 

A mathematical model for this polymerization reaction based on homogeneous, isothermal reaction is inadequate to predict all of these effects, particularly the breadth of the MWD. For this reason a model taking explicit account of the phase separation has been formulated and is currently under investigation. 

In this paper we present a meaningful analysis of the operation of a batch polymerization reactor in its final stages (i.e. high conversion levels) where MWD broadening is relatively unimportant. The ultimate objective is to minimize the residual monomer concentration as fast as possible, using the time-optimal problem formulation. Isothermal as well as nonisothermal policies are derived based on a mathematical model that also takes depropagation into account. The effect of initiator concentration, initiator half-life and activation energy on optimum temperature and time is studied. 

Alternative mechanisms have been recently proposed [78,79] based on a kinetic investigation of NO reduction by n-octane under isothermal (200°C) and steady-state conditions in the presence of H2. The authors built up a mathematical model based on supposed reaction pathways, which account for molecular adsorption of NO and CO and dissociative ones for H2 and 02. The elementary steps, which have been considered for modelling their results are reported in Table 10.3. Interesting kinetic information can be provided by the examination of this mechanism scheme in particular the fast bimolecular... 

Using a "home made" aneroid calorimeter, we have measured rates of production of heat and thence rates of oxidation of Athabasca bitumen under nearly isothermal conditions in the temperature range 155-320°C. Results of these kinetic measurements, supported by chemical analyses, mass balances, and fuel-energy relationships, indicate that there are two principal classes of oxidation reactions in the specified temperature region. At temperatures much lc er than 285°C, the principal reactions of oxygen with Athabasca bitumen lead to deposition of "fuel" or coke. At temperatures much higher than 285°C, the principal oxidation reactions lead to formation of carbon oxides and water. We have fitted an overall mathematical model (related to the factorial design of the experiments) to the kinetic results, and have also developed a "two reaction chemical model". 

Since these two types of processes have drastically different effects on the conversion levels achieved in chemical reactions, they provide the basis for the development of mathematical models that can be used to provide approximate limits within which one can expect actual isothermal reactors to perform. In the development of these models we will define a segregated system as one in which the first effect is entirely responsible for the spread in residence times. When the distribution of residence times is established by the second effect, we will refer to the system as mixed. In practice one encounters various combinations of these two limiting effects. 

In the previous section we indicated how various mathematical models may be used to simulate the performance of a reactor in which the flow patterns do not fit the ideal CSTR or PFR conditions. The models treated represent only a small fraction of the large number that have been proposed by various authors. However, they are among the simplest and most widely used models, and they permit one to bracket the expected performance of an isothermal reactor. However, small variations in temperature can lead to much more significant changes in the reactor performance than do reasonably large deviations inflow patterns from idealized conditions. Because the rate constant depends exponentially on temperature, uncertainties in this parameter can lead to design uncertainties that will make any quantitative analysis of performance in terms of the residence time distribution function little more than an academic exercise. Nonetheless, there are many situations where such analyses are useful. 

This section is concerned with analyses of simultaneous reaction and mass transfer within porous catalysts under isothermal conditions. Several factors that influence the final equation for the catalyst effectiveness factor are discussed in the various subsections. The factors considered include different mathematical models of the catalyst pore structure, the gross catalyst geometry (i.e., its apparent shape), and the rate expression for the surface reaction. 

The design q>roblem can be approached at various levels of sophistication using different mathematical models of the packed bed. In cases of industrial interest, it is not possible to obtain closed form analytical solutions for any but the simplest of models under isothermal operating conditions. However, numerical procedures can be employed to predict effluent compositions on the basis of the various models. In the subsections that follow, we shall consider first the fundamental equations that must be obeyed by all packed bed reactors under various energy transfer constraints, and then discuss some of the simplest models of reactor behavior. These discussions are limited to pseudo steady-state operating conditions (i.e., the catalyst activity is presumed to be essentially constant for times that are long compared to the fluid residence time in the reactor). 

Trapaga and Szekely 515 conducted a mathematical modeling study of the isothermal impingement of liquid droplets in spray processes using a commercial CFD code called FLOW-3D. Their model is similar to that of Harlow and Shannon 397 except that viscosity and surface tension were included and wetting was simulated with a contact angle of 10°. In a subsequent study, 371 heat transfer and solidification phenomena were also addressed. These studies provided detailed... 

Verhoyen, 0., Dupret, F. and Legras, R., Isothermal and non-isothermal crystallization kinetics of polyethylene terephthalate mathematical modeling and experimental measurement, Polym. Eng. Sci., 38, 1592-1610 (1998). 

As in the previous chapter, the semi-irrfinite diffusion at a planar electrode is considered, where the adsorption is described by a linear adsorption isotherm. The modeling of reaction (2.173) does not require a particular mathematical procedure. The model comprises equation (1.2) and the boundary conditions (2.148) to (2.152) that describe the mass transport and adsorption of the R form. In addition, the diffusion of the O form, affected by an irreversible follow-up chemical reaction, is described by the following equation ... 

For all reactions, the mass transport regime is controlled by the diffusion of the reacting ligand only, as the mercury electrode serves as an inexhaustible source for mercury ions. Hence, with respect to the mathematical modeling, reactions (2.205) and (2.206) are identical. This also holds true for reactions (2.210) and (2.211). Furthermore, it is assumed that the electrode surface is covered by a sub-monomolecular film without interactions between the deposited particles. For reactions (2.207) and (2.209) the ligand adsorption obeys a linear adsorption isotherm. Assuming semi-infinite diffusion at a planar electrode, the general mathematical model is defined as follows ... 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

The literature on this model reaction is already vast and a complete bibliography would be of great use to the mathematical modeler. Of particular interest are A. d Anna, P. G. Lignola, and S. K. Scott. The application of singularity theory to isothermal autocatalytic open systems The elementary scheme A + mB = (m + 1) B. Proc. Roy. Soc. Lond. A 403, 341-363 (1986) and S. R. Kay, S. K. Scott, and P. G. Lignola. The application of singularity theory to isothermal autocatalytic open systems The influence of uncatalyzed reactions. Proc. Roy. Soc. Lond A 409, 433-448 (1987). 

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... 

Remark 1 The main motivation behind the development of the simplified superstructure was to end up with a mathematical model that features only linear constraints while the nonlinearities appear only in the objective function. Yee et al. (1990a) identified the assumption of isothermal mixing which eliminates the need for the energy balances, which are the nonconvex, nonlinear equality constraints, and which at the same time reduces the size of the mathematical model. These benefits of the isothermal mixing assumption are, however, accompanied by the drawback of eliminating from consideration a number of HEN structures. Nevertheless, as has been illustrated by Yee and Grossmann (1990), despite this simplification, good HEN structures can be obtained. 

Section 10.2 describes the MINLP approach of Kokossis and Floudas (1990) for the synthesis of isothermal reactor networks that may exhibit complex reaction mechanisms. Section 10.3 discusses the synthesis of reactor-separator-recycle systems through a mixed-integer nonlinear optimization approach proposed by Kokossis and Floudas (1991). The problem representations are presented and shown to include a very rich set of alternatives, and the mathematical models are presented for two illustrative examples. Further reading material in these topics can be found in the suggested references, while the work of Kokossis and Floudas (1994) presents a mixed-integer optimization approach for nonisothermal reactor networks. 

This is how simply a mass-balance equation can be turned into a design equation and become a mathematical model for a lumped isothermal system. 

The importance of adsorbent non-isothermality during the measurement of sorption kinetics has been recognized in recent years. Several mathematical models to describe the non-isothermal sorption kinetics have been formulated [1-9]. Of particular interest are the models describing the uptake during a differential sorption test because they provide relatively simple analytical solutions for data analysis [6-9]. These models assume that mass transfer can be described by the Fickian diffusion model and heat transfer from the solid is controlled by a film resistance outside the adsorbent particle. Diffusion of adsorbed molecules inside the adsorbent and gas diffusion in the interparticle voids have been considered as the controlling mechanism for mass transfer. 

Most of the critical effects in oxidation reactions over Pt metals were observed under isothermal conditions. Hence the complex dynamic behaviour can be directly due to the structure of the detailed catalytic reaction mechanism, specifically to the laws of physico-chemical processes in the "reaction medium-catalyst systems. The types and properties of mathematical models to describe critical effects are naturally dependent on those physico-chemical prerequisites on which these models are often based [4, 9], Let us describe the most important factors used in the literature to interpret critical effects. 

Sorption and Desorption Isotherms. To model radionuclide transport in groundwater through geologic media, it is necessary to mathematically describe sorption and desorption in terms of isotherms. The Freundlich isotherm was found to accurately describe sorption and desorption of all radionuclides studied in the interbed-groundwater systems, except when precipitation of the radionuclide occurred. 

In order to assess the feasibility of any nuclear waste disposal concept, mathematical models of radionuclide sorption processes are required. In a later section kinetic descriptions of the three common sorption isotherms (3) are compared with experimental data from the mixing-cell tests. For a radionuclide of concentration C in the groundwater and concentration S on the surface of the granite, the net rate of sorption, by a first-order reversible reaction, is given by... 

In this paper, we will first illustrate the mathematical models used to describe the coke-conversion selectivity for FFB, MAT and riser reactors. The models also include matrix and zeolite contributions. Intrinsic activity parameters estimated from a small isothermal riser will then be used to predict the FFB and MAT data. The inverse problem of predicting riser performance from FFB and MAT data is straightforward based on the proposed theory. A parametric study is performed to show the sensitivity to changes in coke selectivity and heat of reaction which are affected by catalyst type. We will highlight the quantitative differences in observed conversion and coke-conversion selectivity of various reactors. 

A more quantitative analysis of the batch reactor is obtained by means of mathematical modeling. The mathematical model of the ideal batch reactor consists of mass and energy balances, which provide a set of ordinary differential equations that, in most cases, have to be solved numerically. Analytical integration is, however, still possible in isothermal systems and with reference to simple reaction schemes and rate expressions, so that some general assessments of the reactor behavior can be formulated when basic kinetic schemes are considered. This is the case of the discussion in the coming Sect. 2.3.1, whereas nonisothermal operations and energy balances are addressed in Sect. 2.3.2. 

Theories are not used directly, as in the discussion presented in Sect. 3.1, but allow building a mathematical model that describes an experiment in the unambiguous language of mathematics, in terms of variables, constants, and parameters. As an example, when considering the identification of kinetic parameters of chemical reactions from isothermal experiments performed in batch reactors, the relevant equations of mass conservation (presented in Sect. 2.3.1) give a set of ordinary differential equations in the general form... 

Assaf, E.M., Jesus, C.D.F. and Assaf, J.M. (1998) Mathematical modelling of methane steam reforming in a membrane reactor An isothermic model. Brazilian Journal of Chemical Engineering, 15 (2), 160-166. 

R. N. Haward and G. Thackray, The Use of a Mathematical Model to Describe Isothermal Stress-Strain Curves in Glassy Thermoplastics, Proc. R. Soc. London, Series A, 302 (1471), 453 (1967). 

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