Mass balance equation, catalytic

We summarize the rates we need in a catalytic reactor in Table 74. We always need r to insert in the relevant mass-balance equation. We must be given i- or P as functions of Cj and T from kinetic data. 

To summarize the goal of this section, we must start with the microscopic description of the catalytic reaction, then consider diffusion in pores, and then examine the reactant composition around and within the pellet, in order finally to describe the reactor maSS-balance equations in terms of z alone. The student should understand the logic of this procedure as we go from micrscopic to macroscopic, or the following sections will be unintelligible (or even more unintelligible than usual). 

Thus we see that environmental modeling involves solving transient mass-balance equations with appropriate flow patterns and kinetics to predict the concentrations of various species versus time for specific emission patterns. The reaction chemistry and flow patterns of these systems are sufficiently complex that we must use approximate methods and use several models to try to bound the possible range of observed responses. For example, the chemical reactions consist of many homogeneous and catalytic reactions, photoassisted reactions, and adsorption and desorption on surfaces of hquids and sohds. Is global warming real [Minnesotans hope so.] How much of smog and ozone depletion are manmade [There is considerable debate on this issue.]... 

As with catalytic reactions, our task is to develop pseudohomogeneous rate expressions to insert into the relevant mass-balance equations. For ary multiphase reactor where reaction occurs at the interface between phases, the reactions are pritnarily surface reactions (rate r ), and we have to find these expressions as functions of concentrations and rate and transport coefficients and then convert them into pseudohomogeneous expressions,... 

To change the above mass-balance equations into design equations we just replace r - by V r j and fj by V r), where r) and r are the rates of reaction per unit volume of reaction mixture or per unit mass of catalyst for catalytic reactions, etc. 

The mass balance equation, expressed in moles, for the catalytic reactor is given by (see Figure 9.16) Input - Output + Production = Accumulation where, in a steady state, the accumulation is zero. 

The molecular weight distribution can be calculated by solving the mass balance equations for monomer(s), initiator (catalytic sites), and polymeric species with different chain lengths. When quasisteady state assumption is applied to live polymers or propagating active centers, the molecular weight distribution of live polymers is often represented by the Schultz-Flory most probable distribution. However, the calculation of the chain length distribution of dead polymers is in general quite complicated. For some special cases such as... 

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... 

The concentration of ArOQ depends on the amounts of ArOK, KX, and KI, and the initial usage of the catalyst QX. Combining the mass balance equation for initial RX in the organic phase with Eq. (134), a deactivation function cj) can be introduced in the situation under decline of catalytic efficiency, leading to the following equations ... 

The interplay between the diffusion of the starting material in the pores and the reaction of the starting material on the catalytically active walls of the pores can be derived from the general mass balance (Equation 2.1-49). 

These rates have dimensions of [g T" h" ] or [kJ T h ] and are intensive quantities that may be substituted directly into the mass balance equation. On the other hand, these absolute rates may take on any value and are therefore not characteristic of the system. Comparable variables, which are biologically representative, are the so-called specific rates of production or utilization, which refer to the catalytically active mass (to a first approximation, the concentration of biomass, x, is taken as the dry cell weight). With this, one has the definitions of the specific rates for bioprocesses of Equs. 2.5a-f where, in each case, the specific rate has the dimension [h ]. For growth, the specific rate is p... 

To change the above mass balance equations into design equations, we just replace rj by Frj and Tj by Vr j, where rj and 7j are the rates of reaction per unit volume of reaction mixture (or per unit mass of catalyst for catalytic reactions, etc.). More details about the design equations is given in the following sections. Note that these terms can also be Vj r j (where Vj is the volume of phase I and r j is the rate of reaction per unit volume of phase I) and (where Vjj is the volume of phase II and 7y is the rate of reaction per unit volume of phase II). 

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... 

Figure 9.38, for example, shows the application of the chemical mass balance approach to the fine particle fraction of particles collected at a location in Philadelphia (Dzubay et at., 1988 Olmez et al., 1988 Gordon, 1988). If the set of equations (II) fitted the data perfectly, the sum of the contributions of the various sources would be 100% for each element. Clearly, from the top frame, this is not the case for a number of elements, and both positive and negative deviations from 100% can be seen. However, the contributions of several sources are clear Si and Fe from soil, Ni, V, and Ca from oil-fired power plants, Ti from a paint pigment plant, La, Ce, and Sm from a catalytic cracker, K, Zn, and Sn from an incinerator, Sb from an antimony roaster, and Pb and Br from motor vehicles. 

The computer-reconstructed catalyst is represented by a discrete volume phase function in the form of 3D matrix containing information about the phase in each volume element. Another 3D matrix defines the distribution of active catalytic sites. Macroporosity, sizes of supporting articles and the correlation function describing the macropore size distribution are evaluated from the SEM images of porous catalyst (Koci et al., 2006 Kosek et al., 2005). Spatially 3D reaction-diffusion system with low concentrations of reactants and products can be described by mass balances in the form of the following partial differential equations (Koci et al., 2006, 2007a). For gaseous components ... 

A classic chemical engineering problem of the form under consideration here is that of a non-isothermal reaction occurring in a catalytic particle or packed bed into which a single gaseous participant diffuses from a surrounding reservoir (Hatfield and Aris 1969 Luss and Lee 1970 Aris 1975 Burnell et al. 1983). This scenario is also appropriate to the technologically important problem of spontaneous combustion of stockpiled, often cellulosic, material in air (Bowes 1984). If we represent the concentration of the gaseous species as c, the mass- and heat-balance equations for reaction in an infinite slab are... 

If the reaction rate is a function of pressure, then the momentum balance is considered along with the mass and energy balance equations. Both Equations 6-105 and 6-106 are coupled and highly nonlinear because of the effect of temperature on the reaction rate. Numerical methods of solution involving the use of finite difference are generally adopted. A review of the partial differential equation employing the finite difference method is illustrated in Appendix D. Figures 6-16 and 6-17, respectively, show typical profiles of an exothermic catalytic reaction. 

Given any complex system of heterogeneous catalytic first order reactions the mass balance on a differential volume element of the reactor at the height h yields the following system of differential equations for the j-th reaction component i) for the bubble phase... 

The main reaction i.e, benzene hydrogenation occuring inside porous Ni-catalyst pellets is accompanied by poisoning reaction in which the thiophene presented in the feed stream reacts irtevcrsibly with the catalytic active sites. An analysis was made assuming isothermal behaviour [7], the same effective diffusivity for reactant and poison and that the steady-state continuity equation represents a good approximation at all times [8,9]. Under these conditions the mass balances for benzene, thiophene and catalyst activity are... 

The mass transfer equation is written in terms of the usual assumptions. However, it must be considered that because the concentration of the more abundant species in the flowing gas mixture (air), as well as its temperature, are constant, all the physical properties may be considered constant. The only species that changes its concentration along the reactor in measurable values is PCE. Therefore, the radial diffusion can be calculated as that of PCE in a more concentrated component, the air. This will be the governing mass transfer mechanism of PCE from the bulk of the gas stream to the catalytic boundaries and of the reaction products in the opposite direction. Since the concentrations of nitrogen and oxygen are in large excess they will not be subjected to mass transfer limitations. The reaction is assumed to occur at the catalytic wall with no contributions from the bulk of the system. Then the mass balance at any point of the reactor is... 

The above discussions pertain to models assuming three regions the dense phase, bubble phase and separation side of the membrane. The membrane is assumed to be inert to the reactions. There are, however, cases where the membrane is also catalytic. In these situations, a fourth region, the membrane matrix, needs to be considered. The mass and heat balance equations for the catalytic membrane region will both contain reaction-related terms. 

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. 

Cybulski and Moulijn [27] proposed an experimental method for simultaneous determination of kinetic parameters and mass transfer coefficients in washcoated square channels. The model parameters are estimated by nonlinear regression, where the objective function is calculated by numerical solution of balance equations. However, the method is applicable only if the structure of the mathematical model has been identified (e.g., based on literature data) and the model parameters to be estimated are not too numerous. Otherwise the estimates might have a limited physical meaning. The method was tested for the catalytic oxidation of CO. The estimate of effective diffusivity falls into the range that is typical for the washcoat material (y-alumina) and reacting species. The Sherwood number estimated was in between those theoretically predicted for square and circular ducts, and this clearly indicates the influence of rounding the comers on the external mass transfer. 

To derive the overall kinetics of a gas/liquid-phase reaction it is required to consider a volume element at the gas/liquid interface and to set up mass balances including the mass transport processes and the catalytic reaction. These balances are either differential in time (batch reactor) or in location (continuous operation). By making suitable assumptions on the hydrodynamics and, hence, the interfacial mass transfer rates, in both phases the concentration of the reactants and products can be calculated by integration of the respective differential equations either as a function of reaction time (batch reactor) or of location (continuously operated reactor). In continuous operation, certain simplifications in setting up the balances are possible if one or all of the phases are well mixed, as in continuously stirred tank reactor, hereby the mathematical treatment is significantly simplified. 

Consequently, the reactor model is constituted by a system of N+1 equations, where N is the number of chemical species present in the system (NO, NO2, N2 and O2, neglecting the presence of N2O N = 4) and another unknown variable is pressure. The equations are one momentum balance (in the form of simplified Ergun Law), and four mass balance relationships. The presence of NO2 among the reaction products has been related to the catalytic activity of Cu-ZSM5 towards the oxidation of NO to NO2, as revealed by our previous investigation in similar experimental conditions [7], as well as by the present results (Fig. 1). It has been hypothesised that reaction (2) proceeds in parallel to NO decomposition, having not assumed that NO2 formation is responsible for copper reduction from Cu (inactive in decomposing NO) to Cu (the active site), as also proposed by some author [20-21,23]. 

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