Diffusion, mass reaction systems

Example 9.11 Modeling of a nonisothermal plug flow reactor Tubular reactors are not homogeneous, and may involve multiphase flows. These systems are called diffusion convection reaction systems. Consider the chemical reaction A -> bB described by a first-order kinetics with respect to the reactant A. For a nonisothermal plug flow reactor, modeling equations are derived from mass and energy balances... 

If the reactant solid is porous, the reactant fluid would diffuse into it while reacting with it on its path diffusion and chemical reaction would occur in parallel over a diffuse zone. The analysis of such a reaction system is normally more complex as compared to reaction systems involving nonporous solids. Here also it is important to assess the relative importance of chemical reaction kinetics and of mass and heat transport. 

In a conventional relaxation kinetics experiment in a closed reaction system, because of mass conservation, the system can be described in a single equation, e.g., SCc(t) = SCc(0)e Rt where R = ((Ca) + ( C b)) + kh- The forward and reverse rate constants are k and k t, respectively. In an open system A, B, and C, can change independently and so three equations, one each for A, B, and C, are required, each equation having contributions from both diffusion and reaction. Consequently, three normal modes rather than one will be required to describe the fluctuation dynamics. Despite this complexity, some general comments about FCS measurements of reaction kinetics are useful. 

The solution procedure to this equation is the same as described for the temporal isothermal species equations described above. In addition, the associated temperature sensitivity equation can be simply obtained by taking the derivative of Eq. (2.87) with respect to each of the input parameters to the model. The governing equations for similar types of homogeneous reaction systems can be developed for constant volume systems, and stirred and plug flow reactors as described in Chapters 3 and 4 and elsewhere [31-37], The solution to homogeneous systems described by Eq. (2.81) and Eq. (2.87) are often used to study reaction mechanisms in the absence of mass diffusion. These equations (or very similar ones) can approximate the chemical kinetics in flow reactor and shock tube experiments, which are frequently used for developing hydrocarbon combustion reaction mechanisms. 

Similar to the pure surface electrode reaction, the response of reaction (2.146) is characterized by splitting of the net peak under appropriate conditions. The splitting occurs for an electrochemically quasireversible reaction and vanishes for the pure reversible reaction. Typical regions where the splitting emerges are 3 < m < 10 and 0.1 < r < 10 for a = 0.5 and i sw = 50 mV. Contrary to the surface electrode reaction where the ratio of the split peak currents is solely sensitive to a, in the present system this ratio depends additionally on r. For instance, if a = 0.5 and r = 1 the ratio is = 1 for r = 10, > 1 and r = 0.1, < 1. Finally it is worth mentioning when experimentally possible, the electrode mechanism represented by (2.145) to (2.147) has to be simplified to a simple surface reaction (Sect. 2.5.1) in order to avoid the complexity arising from the effect of diffusion mass transport. 

Similar considerations concern the irreversible processes of diffusion and reaction in mixtures [5]. A system of M different molecular species is described by the three components of velocity, the mass density, the temperature, and (M — 1) chemical concentrations and is ruled by M + 4 partial differential equations. The M — 1 extra equations govern the mutual diffusions and the possible chemical reactions... 

Treatment of systems in which gas-phase diffusion, mass accommodation, liquid phase diffusion, and reaction both in the bulk and at the interface must be taken into account is discussed in Section E.l. 

Analysis of Systems with Gas- and Liquid Phase Diffusion, Mass Accommodation, and Reactions in the Liquid Phase or at the Interface... 

The model (Fig. 23.6) consists of three compartments, (a) the surface mixed water layer (SMWL) or epilimnion, (b) the remaining open water column (OP), and (c) the surface mixed sediment layer (SMSL). SMWL and OP are assumed to be completely mixed their mass balance equations correspond to the expressions derived in Box 23.1, although the different terms are not necessarily linear. The open water column is modeled as a spatially continuous system described by a diffusion/advection/ reaction... 

In these types of laboratory reactor, the flow of the liquid is very carefully controlled so that, although the mass transfer step is coupled with the chemical reaction, the mass transfer characteristics can be disentangled from the reaction kinetics. For some reaction systems, absorption of the gas concerned may be studied as a purely physical mass transfer process in circumstances such that no reaction occurs. Thus, the rate of absorption of C02 in water, or in non-reactive electrolyte solutions, can be measured in the same laboratory contactor as that used when the absorption is accompanied by the reaction between C02 and OH ions from an NaOH solution. The experiments with purely physical absorption enable the diffusivity of the gas in the liquid phase DL to be calculated from the average rate of absorption per unit area of gas-liquid interface NA and the contact time te. As shown in Volume 1, Chapter 10, for the case where the incoming liquid contains none of the dissolved gas, the relationship is ... 

In the previous section the effects of poisons on reaction rates were related to the active component surface, while the influence of mass transfer was disregarded. It has long been recognized, of course, that the overall rate and selectivity of chemical reactions in porous systems involves the coupling of chemical reactions with convective and diffusive mass transfer processes. Beginning with the pioneering work of Thiele (67), an entire discipline has evolved in which model systems are used to... 

Cooper, J.A. and Compton, R.G. (1998) Channel electrodes a review. Electroanalysis, 10, 141. Cussler, E.L. (1984) Diffusion Mass Transfer in Fluid Systems. Cambridge University Press, New York. Dankwerts, P.V. (1970) Gas-Liquid Reactions. McGraw-Hill, New York. 

When a constant potential, E, is applied to the electrode immersed in the solution containing species A and L such that the electron transfer reactions take place, the mass transport supposed by pure diffusion to and from the electrode surface, in the presence of an excess of supporting electrolyte, is described by the following differential diffusive-kinetic equations system ... 

Another classification involves the number of phases in the reaction system. This classification influences the number and importance of mass and energy transfer processes in the design. Consider a stirred mixture of two liquid reactants A and B, and a catalyst consisting of small particles of a solid added to increase the reaction rate. A mass transfer resistance occurs between the bulk liquid and the surface of the catalyst particles. This is because the small particles tend to move with the liquid. Consequently, there is a layer of stagnant fluid that surrounds each particle. This results in reactants A and B transferring through this layer by diffusion in order to reach the catalyst surface. The diffusion resistance gives a difference in concentration between... 

Transport to the electrode surface as described in Chapter 5 assumes that this occurs solely and always by diffusion. In hydrodynamic systems, forced convection increases the flux of species that reach a point corresponding to the thickness of the diffusion layer from the electrode. The mass transfer coefficient kd describes the rate of diffusion within the diffusion layer and kc and ka are the rate constants of the electrode reaction for reduction and oxidation respectively. Thus for the simple electrode reaction O + ne-— R, without complications from adsorption,... 

Basically, reactant and product selectivities are mass transfer effects, where the diffusivities of the various species in practice frequently do not differ that extremely as indicated above. Instead, in most cases only a preferred diffusion of certain species is observed, a fact which often hinders a clear understanding of product shape selectivity. This is because the various products, during their way through the pore system, may be reacted when contacting the catalytically active surface of the wall. This combined effect of diffusion and reaction will be discussed in detail in the following, as it is of great importance for the product distribution in zeolite-catalyzed reactions. 

Solution of the coupled mass-transport and reaction problem for arbitrary chemical kinetic rate laws is possible only by numerical methods. The problem is greatly simplified by decoupling the time dependence of mass-transport from that of chemical kinetics the mass-transport solutions rapidly relax to a pseudo steady state in view of the small dimensions of the system (19). The gas-phase diffusion problem may be solved parametrically in terms of the net flux into the drop. In the case of first-order or pseudo-first-order chemical kinetics an analytical solution to the problem of coupled aqueous-phase diffusion and reaction is available (19). These solutions, together with the interfacial boundary condition, specify the concentration profile of the reagent gas. In turn the extent of departure of the reaction rate from that corresponding to saturation may be determined. Finally criteria have been developed (17,19) by which it may be ascertained whether or not there is appreciable (e.g., 10%) limitation to the rate of reaction as a consequence of the finite rate of mass transport. These criteria are listed in Table 1. 

The kinetics found for the reactions at the solution/solid interface show some marked similarities with those at gas/solid [9, 49], gas/liquid, and liquid/liquid interfaces [268]. Whenever one of the phases is a liquid rather than a gas, mass transport is apt to become rate-controlling because of the smaller diffusion coefficients of species in liquids. Many of the catalysed redox reactions in Sect. 4 were indeed partly or wholly diffusion-controlled. These systems could be converted to surface-controlled ones simply by reducing the size of the catalysing material by using colloidal catalysts, for... 

The most efficacious catalysts to date have been the noble metals, carbons, and some insoluble oxides and salts. As was emphasized in Sect. 1.8, tests should always be carried out to confirm that heterogeneous catalysis is the true reason for any observed rate increase. One of these tests requires the catalytic rate to rise proportionately with the mass or area of the catalyst. While most reaction systems have satisfied this criterion, the rates of several carbon-catalysed reactions in the literature have been reported as increasing either much more or much less than expected when larger amounts of the solid were added. Pore diffusion could account for only some of these results. Since carbons are cheap catalysts with large surface areas, their aberrant behaviour in this respect would be worth serious investigation. 

In this text, the conversion rate is used in relevant equations to avoid difficulties in applying the correct sign to the reaction rate in material balances. Note that the chemical conversion rate is not identical to the chemical reaction rate. The chemical reaction rate only reflects the chemical kinetics of the system, that is, the conversion rate measured under such conditions that it is not influenced by physical transport (diffusion and convective mass transfer) of reactants toward the reaction site or of product away from it. The reaction rate generally depends only on the composition of the reaction mixture, its temperature and pressure, and the properties of the catalyst. The conversion rate, in addition, can be influenced by the conditions of flow, mixing, and mass and heat transfer in the reaction system. For homogeneous reactions that proceed slowly with respect to potential physical transport, the conversion rate approximates the reaction rate. In contrast, for homogeneous reactions in poorly mixed fluids and for relatively rapid heterogeneous reactions, physical transport phenomena may reduce the conversion rate. In this case, the conversion rate is lower than the reaction rate. 

Internal and external mass transfer resistances are important factors affecting the catalyst performance. These are determined mainly by the properties of the fluids in the reaction system, the gas-liquid contact area, which is very high for monolith reactors, and the diffusion lengths, which are short in monoliths. The monolith reactor is expected to provide apparent reaction rates near those of intrinsic kinetics due to its simplicity and the absence of diffusional limitations. The high mass transfer rates obtained in the monolith reactors result in higher catalyst utilization and possibly improved selectivity. 

In the glucose oxidase system, dissolved oxygen concentration as well as glucose levels will influence dehvery response requiring close control of mass-transfer limitations. For both systems containing a protein component, stabihty factors may limit operational hfetime. This may be particularly severe in the case of glucose oxidase where the reaction product H2O2 will accelerate enzyme denaturation unless it is rapidly removed by diffusion or reaction with a second enzyme (peroxidase or catalase). 

Next, a mathematical model that allows description of the separation and concentration of the components of a metallic mixture will be detailed the principal assumptions of the model are (1) convective mass transfer dominates diffusive mass transfer in the fluid flowing inside the HFs, (2) the resistance in the membrane dominates the overall mass transport resistance, therefore the overall mass transfer coefficient was set equal to the mass transfer coefficient across the membrane, and (3) chemical reactions between ionic species are sufficiently fast to ignore the contribution of the chemical reaction rates. Thus, the reacting species are present in equilibrium concentration at the interface everywhere [31,32,58,59]. For systems working under nonsteady state, it is also necessary to describe the change in the solute concentration with time both in the modules and in the reservoir tanks. The reservoir tanks will be modeled as ideal stirred tanks. 

There is a large body of literature that deals with the proper definition of the diffusivity used in the intraparticle diffusion-reaction model, especially in multicomponent mixtures found in many practical reaction systems. The reader should consult references, e.g.. Bird, Stewart, and Lightfoot, Transport Phenomena, 2d ed., John Wiley Sons, New York, 2002 Taylor and Krishna, Multicomponent Mass Transfer, Wiley, 1993 and Cussler, Diffusion Mass Transfer in Fluid Systems, Cambridge University Press, 1997. 

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