Non-isothermal Reactor Models

In this section, we present the low-dimensional multi-mode models for different types of non-isothermal homogeneous reactors. Flere, we skip the details of the averaging process and summarize the results. 

The dimensionless temperature variables for the fluid and solid phases (0f and 0S) and the dimensionless reaction rate (r(c, Of)) are given by 

Using the above dimensionless parameters and variables, Eqs. (188)—(198) are written in dimensionless form and spatially averaged over transverse dimensions to obtain the low-dimensional model for non-isothermal homogeneous tubular reactors, which is given to order p by Eqs. (130) (134) with r((c)) being replaced by r((c), (0f and 

In Eqs. (202)-(203), tjH is the characteristic dimensionless local thermal mixing time, which is given by 

The coefficients / i, / 2, and j03 depend on the flow profile and the local shear rates of the system, and j04 depends on the reactor geometry and for a tubular geometry, 

---

C. Non-isothermal Reactor Models 1. Wall-catalyzed Reactions... 

The qualitative analysis of intraparticle heat transport suggests severe limitations of packed-bed laboratory reactors compared with the thin-film catalyzed microchannel, as discussed previously. It is imperative that a quantitative study of reactor heat transfer limitations is performed. With PrOx as a model reaction, this study was realized through the non-isothermal reactor modeling of the microreactor and the packed-bed reactors with both 2 and 4 mm radii. In the model, the operating... 

Double-click on the FEMLAB 3.1 ECRE icon on your desktop. In the Model Navigator, select model denoted "4-Non-Isothermal Reactor 11" and press Documentation". This will open your web browser and display the documentation of this sped lie model. You can also review the detailed documentation for the whole scries of models listed on the left-hand side in the web browser window. Use chapter 2 in the online documentation to answ cr the questions below using the model 4-Non-Isothermai Reactor 11". Select the Model Navigator and press "OK" to open the model,... 

Model 3 - Pseudo-homogeneous non-isothermal reactor with reaction rate correction... 

The previous description of the processes considered in a fixed-bed reactor leads to a heterogeneous non-isothermal 2D model (symmetry with respect to the angular coordinate is assumed). The unsteady state mass and energy balances for a single reactant species with dimensional concentration (subscript i is omitted for convenience) and appropriate boundary conditions are given in Table 3.1. The independent variables are normalized as follows (see Figure 3.2) ... 

In this chapter, the dynamics of ideally stirred tank reactors will be analyzed. First, the assumptions, required to limit model complexity, will be discussed. Next, various types of reaction will be considered such as simple first-order reactions, equilibrium reactions, parallel reactions, etc. Subsequently, the analysis will be expanded to include non-isothermal reactors. Numerical examples of chemical reactors are given and the non-linear model descriptions are compared with the linearized model descriptions. 

At the entrance and exit of the coil reactor, the temperatures are 326.5°C and 439 C, and the pressure is 2.62 and 0.90MPa (1.72MPa of delta-P), respectively. A typical feed flowrate of 124,300kg/h (about 20,000 barrel per day of nominal feed flowrate) is assumed for simulation purposes. One weight percentage of water steam was also fed to the coil reactor. Delta-P and steam flowrate are typical values reported in the literature (Joshi et al., 2008). Since the coil reactor is modeled as non-isothermal reactor, the required heat transfer rate is 4.33 x lO kJ/h. For the case of the soaker reactor, it is modeled as adiabatic reactor, so that temperature will reduce due to the endothermic nature of the visbreaking reactions. 

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... 

For isothermal, first-order chemical reactions, the mole balances form a system of linear equations. A non-ideal reactor can then be modeled as a collection of Lagrangian fluid elements moving independe n tly through the system. When parameterized by the amount of time it has spent in the system (i.e., its residence time), each fluid element behaves as abatch reactor. The species concentrations for such a system can be completely characterized by the inlet concentrations, the chemical rate constants, and the residence time distribution (RTD) of the reactor. The latter can be found from simple tracer experiments carried out under identical flow conditions. A brief overview of RTD theory is given below. 

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)]... 

This chapter deals with the design of reactors which do not conform to these ideal models its attention is restricted to constant volume, single phase, isothermal reactors which are operated in the steady state. It is not intended to be a state of the art review of non-ideal reactor design methods, but rather an introduction to basic ideas and techniques frequently, the reader will be referred to more extended or specific coverage of the material being considered. 

The CSTR is, in many ways, the easier to set up and operate, and to analyse theoretically. Figure 6.1 shows a typical CSTR, appropriate for solution-phase reactions. In the next three chapters we will look at the wide range of behaviour which chemical systems can show when operated in this type of reactor. In this chapter we concentrate on stationary-state aspects of isothermal autocatalytic reactions similar to those introduced in chapter 2. In chapter 7, we turn to non-isothermal systems similar to the model of chapter 4. There we also draw on a mathematical technique known as singularity theory to explain the many similarities (and some differences) between chemical autocatalysis and thermal feedback. Non-stationary aspects such as oscillations appear in chapter 8. 

In chapters 2-5 two models of oscillatory reaction in closed vessels were considered one based on chemical feedback (autocatalysis), the other on thermal coupling under non-isothermal reaction conditions. To begin this chapter, we again return to non-isothermal systems, now in a well-stirred flow reactor (CSTR) such as that considered in chapter 6. 

Maximum Release. The analytical model described above assumes that the liquid phase is completely stagnant. While this may be true in an ideal laboratory experiment where a small sample can be kept isothermal at a specified temperature, in large scale systems where non-isothermal conditions exist, both natural convection and molecular diffusion will contribute to mass transfer. This combined effect, which is often very difficult to assess quantitatively, will result in an increase in fission-product release rate. Therefore, in making reactor safety analyses, it is desirable to be able to estimate the maximum release under all possible conditions. 

Non-isothermal and non-adiabatic conditions. A useful approach to the preliminary design of a non-isothermal fixed bed reactor is to assume that all the resistance to heat transfer is in a thin layer 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 approximate design of reactors. Neglecting diffusion and conduction in the direction of flow, the mass and energy balances for a single component of the reacting mixture are ... 

Transfer of the theoretical results obtained for isothermal conditions to the non-isothermal case is rather simple provided the temperature dependencies of the constants are known. A theoretical analysis of non-isothermal processes may be limited by thermal runaway, which is equivalent to the thermal instability already thoroughly studied for chemical reactors.80,81 There are two limiting situations, known as the Semenov and Frank-Kamenetzky models 80,82 these correspond to periodic ideal stirred and periodic ideal plug reactors, respectively. 

We have used CO oxidation on Pt to illustrate the evolution of models applied to interpret critical effects in catalytic oxidation reactions. All the above models use concepts concerning the complex detailed mechanism. But, as has been shown previously, critical. effects in oxidation reactions were studied as early as the 1930s. For their interpretation primary attention is paid to the interaction of kinetic dependences with the heat-and-mass transfer law [146], It is likely that in these cases there is still more variety in dynamic behaviour than when we deal with purely kinetic factors. A theory for the non-isothermal continuous stirred tank reactor for first-order reactions was suggested in refs. 152-155. The dynamics of CO oxidation in non-isothermal, in particular adiabatic, reactors has been studied [77-80, 155]. A sufficiently complex dynamic behaviour is also observed in isothermal reactors for CO oxidation by taking into account the diffusion both in pores [71, 147-149] and on the surfaces of catalyst [201, 202]. The simplest model accounting for the combination of kinetic and transport processes is an isothermal continuously stirred tank reactor (CSTR). It was Matsuura and Kato [157] who first showed that if the kinetic curve has a maximum peak (this curve is also obtained for CO oxidation [158]), then the isothermal CSTR can have several steady states (see also ref. 203). Recently several authors [3, 76, 118, 156, 159, 160] have applied CSTR models corresponding to the detailed mechanism of catalytic reactions. 

In a recent survey [19] it was noted that a realistic model for catalytic oxidation reactions must include equations describing the evolution of at least two concentrations of surface substances and account for the slow variation in the properties of the catalyst surface (e.g. oxidation-reduction). For the synchronization of the dynamic behaviour for various surface domains, it is necessary to take into consideration changes in the concentrations of gas-phase substances and the temperature of the catalyst surface. It is evident that, in the hierarchy of modelling levels, such models must be constructed and tested immediately after kinetic models. On the one hand, the appearance of such models is associated with the experimental data on self-oscillations in reactors with noticeable concentration variations of the initial substances and products (e.g. ref. 74) on the other hand, there was a gap between the comprehensively examined non-isothermal models with simple kinetics and those for the complex heterogeneous catalytic reactions... 

Most commonly, distributed parameter models are applied to describe the performance of diesel particulate traps, which are a part of the diesel engine exhaust system. Those models are one- or two-dimensional, non-isothermal plug-flow reactor models with constant convection terms, but without diffusion/dispersion terms. 

---