Non-isothermal model

Modelling non-isothermal crystallization is the next important step in a quantitative description of reactive processing. This is particularly important, because crystallization determines the properties of the end product. Therefore, the development the spatial distribution of crystallinity, a, and temperature, T, with time throughout the volume of the reactive medium must be calculated. It is also noteworthy that crystallization and polymerization processes may occur simultaneously. This happens when polymerization proceeds at temperatures below the melting point of the newly formed polymer. A typical example of this phenomenon is anionic-activated polymerization of e-caprolactam, which takes place below the melting temperature of polycaproamide. 

From the redundant species analysis it is clear that all reactions which consume H2O2 and O3 are redundant and can be removed automatically from the mechanism. In order to identify other redundant reactions the techniques of rate sensitivity analysis coupled with a principal component analysis of the resulting matrix can be used. The principal component analysis of the rate sensitivity matrix containing only the remaining important and necessary species will reveal the important reactions leading to reduced mechanisms applicable at various ambient temperatures. In principle it may be possible to produce a reduced scheme which models non-isothermal behaviour from analysis carried out on an isothermal model. An isothermal system is easier to model since thermodynamic and heat-transfer properties can be excluded from the calculations. However,... 

This paper presents a numerical simulation of the swellin shrinking processes using the numerical model RF-TH/M - a fully coupled thermo-hydraulic model with mechanical coupling - which has been developed for the purpose of modelling non-isothermal multiphase flow in swelling porous media. In this paper details of the mathematical and numerical multiphase-multicomponental formulation for bentonite, as well as code implementation are described. As an example a test case is investigated. 

Therefore, the equations shown in Table 2.10, either in transient or steady-state, are usually applied to industrial reactors as well. To these equations may be added an energy balance to model non-isothermal reaction operation d T... 

Bruschke, M. V. and Advani, S. G., A numerical approach to model non-isothermal viscous flow through fibrous media with free surfaces . International Journal for Numerical Methods in Fluids, 19, 575-603, 1994. 

Bruschke, M.V. Advani, S.G. A numerical approach to model non-isothermal. 

In this chapter the general equations of laminar, non-Newtonian, non-isothermal, incompressible flow, commonly used to model polymer processing operations, are presented. Throughout this chapter, for the simplicity of presentation, vector notations are used and all of the equations are given in a fixed (stationary or Eulerian) coordinate system. 

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. 

S. Olive, U. Grafe, I. Steinbach. The modelling of Ostwald-ripening during non-isothermal heat treatments resulting in temperature dependent matrix solubihty of the precipitate forming elements a further development of the LSW-theory. Comput Mater Sci 7. 94, 1996. 

Empirical grey models based on non-isothermal experiments and tendency modelling will be discussed in more detail below. Identification of gross kinetics from non-isothermal data started in the 1940-ties and was mainly applied to fast gas-phase catalytic reactions with large heat effects. Reactor models for such reactions are mathematically isomorphical with those for batch reactors commonly used in fine chemicals manufacture. Hopefully, this technique can be successfully applied for fine chemistry processes. Tendency modelling is a modern technique developed at the end of 1980-ties. It has been designed for processing the data from (semi)batch reactors, also those run under non-isothermal conditions. 

Figure 1.19. Information flow diagram for modelling a non-isothermal, chemical reactor, with simultaneous mass and energy balances.

The coupling of the component and energy balance equations in the modelling of non-isothermal tubular reactors can often lead to numerical difficulties, especially in solutions of steady-state behaviour. In these cases, a dynamic digital simulation approach can often be advantageous as a method of determining the steady-state variations in concentration and temperature, with respect to reactor length. The full form of the dynamic model equations are used in this approach, and these are solved up to the final steady-state condition, at which condition... 

The dispersion model of example DISRE is extended for non-isothermal reactions to include the dispersion of heat from a first-order reaction. 

Xu T. and Pruess K. Coupled modeling of non-isothermal multiphase flow, solute transport and reactive chemistry in porous and fractured media 1. Model develop-... 

The kinetics of the CTMAB thermal decomposition has been studied by the non-parametric kinetics (NPK) method [6-8], The kinetic analysis has been performed separately for process I and process II in the appropriate a regions. The NPK method for the analysis of non-isothermal TG data is based on the usual assumption that the reaction rate can be expressed as a product of two independent functions,/ and h(T), where f(a) accounts for the kinetic model while the temperature-dependent function, h(T), is usually the Arrhenius equation h(T) = k = A exp(-Ea / RT). The reaction rates, da/dt, measured from several experiments at different heating rates, can be expressed as a three-dimensional surface determined by the temperature and the conversion degree. This is a model-free method since it yields the temperature dependence of the reaction rate without having to make any prior assumptions about the kinetic model. 

Sklavounos, S. and Rigas, F., Simulation of Coyote series trials—Part I CFD estimation of non-isothermal LNG releases and comparison with box-model predictions, Chem. Eng. Sci., 61,1434, 2006. 

The basic physical model is that of fast reactions occurring in thin quasi-one-dimensional reaction sheets with thickness less than the size of the smallest eddies in the flow. As discussed in Section 5.5, this picture will be valid for non-isothermal one-step reactions when the reaction rate constant k (T) is near zero at the ambient temperature, but very... 

Despite these difficulties, the multi-environment conditional PDF model is still useful for describing simple non-isothermal reacting systems (such as the one-step reaction discussed in Section 5.5) that cannot be easily treated with the unconditional model. For the non-isothermal, one-step reaction, the reaction-progress variable Y in the (unreacted) feed stream is null, and the system is essentially non-reactive unless an ignition source is provided. Letting Foo(f) (see (5.179), p. 183) denote the fully reacted conditional progress variable, we can define a two-environment model based on the E-model 159... 

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). 

Tadmor, Z. and Broyer, E., Solids Conveying in Screw Extruders Part 11 Non Isothermal Model, Polym. Eng. Set, 12, 378 (1972)... 

The non-isothermal viscoelastic cell model was used to study foam growth in the continuous extrusion of low density foam sheet. Surface escape of blowing agent was successfully incorporated to describe the foaming efficiency. Reasonable agreement was obtained with experimental data for HCFC-22 blown LDPE foam in the sub-centimetre thickness domain. 11 refs. 

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 equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

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