Two-Equation Model for Catalytic Reactor

Liu [2] used the CMT (P- — two-equation model (abbreviated as two-equation model) for simulating a catalytic reactor with cooling jacket for producing vinyl acetate from acetic acid and acetylene as described below. 

Both the reactant and product are in homogeneous fluid phase, and the operation is steady. 

The fluid-phase flow is axially symmetrical in the catalytic reactor (packed column) and in turbulent state. 

The temperature of outer catalyst surface is equal to the fluid temperature. 

The temperature at the outer wall of the coohng jacket is constant. 

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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... 

They treated the system much like a CSTR, with the balance for the gas-phase concentration substituted by the coverage equation for the catalyst. Ray and Hastings then applied the analytical treatment that they had developed for the CSTR in this same publication. Stability analysis revealed that the critical Lewis numbers for oscillations were in a range that did not allow for oscillations on normal nonporous catalytic surfaces. However, as Jensen and Ray 243) showed, a certain model for catalytic surfaces, the fuzzy wire model, with the assumption of a very rough surface with protrusions is able to produce Lewis numbers in the proper range for the occurrence of oscillations. This model, however, included both mass and heat balances as well as coverage equations, thus combining the two classes of reactor-reaction models discussed above. 

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... 

A major limitation of the present work is that it deals only with well-defined (and mostly unidirectional) flow fields and simple homogeneous and catalytic reactor models. In addition, it ignores the coupling between the flow field and the species and energy balances which may be due to physical property variations or dependence of transport coefficients on state variables. Thus, a major and useful extension of the present work is to consider two- or three-dimensional flow fields (through simplified Navier-Stokes or Reynolds averaged equations), include physical property variations and derive lowdimensional models for various types of multi-phase reactors such as gas-liquid, fluid-solid (with diffusion and reaction in the solid phase) and gas-liquid-solid reactors. 

The development of mathematical models for the simulation of non-adiabatic fixed-bed catalytic reactors has received considerable attention. In previous work, we have analyzed the two-dimensional and one-dimensional versions of the models (1, 2) which, in turn, were classified as (I) pseudohomogeneous, (II) heterogeneous, but conceptually wrong, and (III) heterogeneous, written in the correct way (Table I). Model equations are in the Appendix. 

The abstract models can be divided into two categories, each of which can be further subdivided into three classes (Fig. 5). Some of the models consist of coverage equations only, and these models will be called surface reaction models. The remaining models use additional mass and/or heat balance equations that include assumptions about the nature of the reactor in which the catalytic reaction takes place (the reactor could be simply a catalyst pellet). These models will be called reactor-reaction models. Some of the models mentioned under the heading surface reaction models also incorporate balance equations for the reactor. However, these models need only the coverage equations to predict oscillatory behavior reactor heat and mass balances are just added to make the models more realistic [e.g., the extension of the Sales-Turner-Maple model (272) given in Aluko and Chang (273)]. Such models are therefore included under surface reaction models, which will be discussed first. 

Investigation 10 was a study of fixed-bed reactor models and their application to the data of Hettinger et al. (1955) on catalytic reforming of C7 hydrocarbons. The heuristic posterior density function p 6 Y) proposed by Stewart (1987) was used to estimate the rate and equilibrium parameters of various reaction schemes, two of which were reported in the article. The data were analyzed with and without models for the intraparticle and boundary-layer transport. The detailed transport model led to a two-dimensional differential-algebraic equation system, which was solved via finite-element discretization in the reactor radial coordinate and... 

The model for the H-Oil reactor introduces two complications beyond the axial dispersion model. First, the boundary conditions are modified to account for the recycle and second, the catalyst in the reactor means that both thermal and catalytic reactions are occurring simultaneously. The set of equations given in Eq. (18) are solved numerically with a differential equation solver. This allows the reactor size to be... 

Non-linear two point boundary value differential equations arise in fixed bed catalytic reactors mainly in connection with the diffusion and reaction in porous catalyst pellets. It may also arise in the modelling of axial and radial dispersion in the catalyst bed. In addition they also arise in cases of counter-current cooling or heating of the reactor. For this last case, the use of a shooting technique with an iterative procedure similar to the Newton method (Fox s method) seems to be the easiest and most straightforward technique (Kubicek and Hlavacek, 1983). 

The consideration of thermal effects and non-isothermal conditions is particularly important for reactions for which mass transport through the membrane is activated and, therefore, depends strongly on temperature. This is, typically, the case for dense membranes like, for example, solid oxide membranes, where the molecular transport is due to ionic diffusion. A theoretical study of the partial oxidation of CH4 to synthesis gas in a membrane reactor utilizing a dense solid oxide membrane has been reported by Tsai et al. [5.22, 5.36]. These authors considered the catalytic membrane to consist of three layers a macroporous support layer and a dense perovskite film (Lai.xSrxCoi.yFeyOs.s) permeable only to oxygen on the top of which a porous catalytic layer is placed. To model such a reactor Tsai et al. [5.22, 5.36] developed a two-dimensional model considering the appropriate mass balance equations for the three membrane layers and the two reactor compartments. For the tubeside and shellside the equations were similar to equations (5.1) and... 

Remember that the rate equation must be consistent with the measure of the extent of reaction employed if conversion is used, (—r) must be expressed in terms of x if concentration is used, (—r) is in terms of C. If Fis mass flow rate, (—r) and C must be in mass units, and so on. Also, equation (4-45) is often used as a pseudo-homogeneous model for two-phase PER catalytic reactors. In such cases the rate constant contained in (—r) is usually expressed in terms per volume or weight of catalyst. In the former case, V would refer to catalyst volume and a value for bed porosity would be required to obtain reactor volume. In the latter case, catalyst weight would be... 

In this chapter, the Navier-Stokes equations have been solved in the actual 3D geometry of the reactor, thereby exploiting the full potential of the new approach, and detailed surface kinetics (Visconti et al., 2013) was implemented in the model with two main implications. On a more fundamental level, it demonstrates the power of the CAT-PP approach proposed here, which allows us to perform simulations of complex catalytic reactors characterized by nonideal flow fields, in which multistep reactions take place. On a more applied level, it allows us to assess the extent of the nonidealities of the simulated operando FTIR reaction cell, which is commercially available and is used by many research groups worldwide. This is extremely relevant especially for researchers who ivant to use the cell to collect quantitative information, since it will allow the verification of whether the cell is an ideal reactor or not. This latter hypothesis has been exploited, for example, by Visconti et al. (2013) to develop the first comprehensive and physically consistent spectrokinetic model for NOx storage... 

Borstar is an industrial olefin polymerization plant/technology, which combines different polymerization processes and reactor units, utilizing an advanced catalytic system. In the present work, a detailed model for the dynamic and steady-state simulation of this industrial plant has been developed. A comprehensive kinetic model for the ethylene-1-butene copolymerization over a two-site catalyst was employed to predict the MWD and CCD in the Borstar process. The Sanchez-Lacombe equation of state (S-L EoS) was employed for the thermodynamic properties of the polymerization system and the phase equilibrium calculations in the process units. 

There is a second important reason for introducing the concept of an effectiveness factor. In the ordinary course of events, concentrations within a catalytic reactor packed with catalyst particles will vary both axially in the direction of flow as well as radially within the catalyst pellets. The model mass balance for such a system would consequently lead to a partial differential equation (PDE). By using an effectiveness factor, we reduce the PDE to an equivalent set of two ODEs — one the pellet mass balance in the radial direction and the other the reactor mass balance in the direction of flow. The reaction rate, which previously varied in two directions rj r, z), is now a function of the axial distance only We replace r (z, r) by E r, i(z), where is the so-called intrinsic reaction rate measiued experimentally on a fine powder and excludes diffusional effects. The latter are lumped into the effectiveness factor that now acts as a fractional efficiency on the intrinsic rate r, . This product of Er fz) is used in the reactor mass balance. 

Duducovifi MP, Larachi F, Mills PL (2002) Multiphase catalytic reactors a perspective on current knowledge and future trends. Catal Rev-Sci Eng 44(1) 123-246 Elghobashi S, Abou-Arab TW (1983) A two-equation turbulence model for two-phase flows. Phys Fluids 26(4) 931-938... 

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