Kinetic models mass transfer resistance

Ahn et al. have developed fibre-based composite electrode structures suitable for oxygen reduction in fuel cell cathodes (containing high electrochemically active surface areas and high void volumes) [22], The impedance data obtained at -450 mV (vs. SCE), in the linear region of the polarization curves, are shown in Figure 6.22. Ohmic, kinetic, and mass transfer resistances were determined by fitting the impedance spectra with an appropriate equivalent circuit model. 

An electrode model is especially advantageous if it can be used to relate the kinetic and mass transfer resistance to electrode geometry and microstructure for instance, to thickness, porosity, pore or particle size, contact areas of phases, and/or grain size of electrode and electrolyte materials. A well-tested and validated electrode model, therefore, may serve to assist in the design of optimised electrode structures or electrode/electrolyte interfaces to minimise polarisation loss. 

OS 63] ]R 27] ]P 46] Experimental results were compared with a kinetic model taking into account liquid/liquid mass transfer resistance [117]. Calculated and experimental conversions were plotted versus residence time the corresponding dependence of the mass-transfer coefficient k,a is also given as well (Figure 4.78). 

A kinetics or reaction model must take into account the various individual processes involved in the overall process. We picture the reaction itself taking place on solid B surface somewhere within the particle, but to arrive at the surface, reactant A must make its way from the bulk-gas phase to the interior of the particle. This suggests the possibility of gas-phase resistances similar to those in a catalyst particle (Figure 8.9) external mass-transfer resistance in the vicinity of the exterior surface of the particle, and interior diffusion resistance through pores of both product formed and unreacted reactant. The situation is illustrated in Figure 9.1 for an isothermal spherical particle of radius A at a particular instant of time, in terms of the general case and two extreme cases. These extreme cases form the bases for relatively simple models, with corresponding concentration profiles for A and B. 

As shown in Example 22-3, for solid particles of the same size in BMF, the form of the reactor model resulting from equation 22.2-13 depends on the kinetics model used for a single particle. For the SCM, this, in turn, depends on particle shape and the relative magnitudes of gas-film mass transfer resistance, ash-layer diffusion resistance and surface reaction rate. In some cases, as illustrated for cylindrical particles in Example 22-3(a) and (b), the reactor model can be expressed in explicit analytical form additional results are given for spherical particles by Levenspiel(1972, pp. 384-5). In other f l cases, it is convenient or even necessary, as in Example 22-3(c), to use a numerical pro-... 

Recently, Pacheco et al developed and validated a pseudo-homogeneous mathematical model for ATR of i-Cg and the subsequent WGS reaction, based on the reaction kinetics and intraparticle mass transfer resistance. They regressed kinetic expressions from the literature for POX and SR to determine the kinetic parameters from their i-Cg ATR experimental data using Pt on ceria. 

A kinetic model was developed based on data obtained over a range of temperatures and hydrogen pressures. The kinetic parameters were expressed as a function of temperature. The kinetic model was applied to the analysis of the trickle-bed data. Predictions of a mathematical model of the trickle-bed reactor were compared with data obtained at two temperatures and a range of pressures. The intraparticle mass transfer resistance was very important. 

The basic approach is to consider the problem in two parts. Firstly, the reaction of a single particle with a plentiful excess of the gaseous reactant is studied. A common technique is to suspend the particle from the arm of a thermobalance in a stream of gas at a carefully controlled temperature the course of the reaction is followed through the change in weight with time. From the results a suitable kinetic model may be developed for the progress of the reaction within a single particle. Included in this model will be a description of any mass transfer resistances associated with the reaction and of how the reaction is affected by concentration of the reactant present in the gas phase. 

Because of their multicomponent nature, RSPs are affected by a complex thermodynamic and difihisional coupling, which, in turn, is accompanied by simultaneous chemical reactions (57-59). To describe such phenomena adequately, specially developed mathematical models capable of taking into consideration column hydrodynamics, mass transfer resistances, and reaction kinetics are required. 

The rate-based models usually use the two-film theory and comprise the material and energy balances of a differential element of the two-phase volume in the packing (148). The classical two-film model shown in Figure 13 is extended here to consider the catalyst phase (Figure 33). A pseudo-homogeneous approach is chosen for the catalyzed reaction (see also Section 2.1), and the corresponding overall reaction kinetics is determined by fixed-bed experiments (34). This macroscopic kinetics includes the influence of the liquid distribution and mass transfer resistances at the liquid-solid interface as well as dififusional transport phenomena inside the porous catalyst. 

Another enzyme that was studied extensively in microreactors to determine kinetic parameters is the model enzyme alkaline phosphatase. Many reports have appeared that differ mainly on the types of enzyme immobilization, such as on glass [413], PDMS [393], beads [414] and in hydrogels [415]. Kerby et al. [414], for example, evaluated the difference between mass-transfer effects and reduced effidendes of the immobilized enzyme in a packed bead glass microreactor. In the absence of mass-transfer resistance, the Michaelis-Menten kinetic parameters were shown to be flow-independent and could be appropriately predicted using low substrate conversion data. 

Reactions of solids are typically feasible only at elevated temperatures. High temperatures are achieved by direct contact with combustion gases. Often, the product of reaction is a gas. The gas has to diffuse away from the reactant, sometimes through a solid product. Thermal and mass-transfer resistances are major factors in the performance of solids reactors. There are a number of commercial processes that utilize solid reactors. Reactor analysis and design appear to rely on empirical models that are used to fit the kinetics of solids decomposition. Most of the information on commercial reactors is proprietary. 

A few reactor models have recently been proposed (30-31) for prediction of integral trickle-bed reactor performance when the gaseous reactant is limiting. Common features or assumptions include i) gas-to-liquid and liquid-to-solid external mass transfer resistances are present, ii) internal particle diffusion resistance is present, iii) catalyst particles are completely externally and internally wetted, iv) gas solubility can be described by Henry s law, v) isothermal operation, vi) the axial-dispersion model can be used to describe deviations from plug-flow, and vii) the intrinsic reaction kinetics exhibit first-order behavior. A few others have used similar assumptions except were developed for nonlinear kinetics (27—28). Only in a couple of instances (7,13, 29) was incomplete external catalyst wetting accounted for. 

The evaluation of catalyst effectiveness requires a knowledge of the intrinsic chemical reaction rates at various reaction conditions and compositions. These data have to be used for catalyst improvement and for the design and operation of many reactors. The determination of the real reaction rates presents many problems because of the speed, complexity and high exo- or endothermicity of the reactions involved. The measured conversion rate may not represent the true reaction kinetics due to interface and intraparticle heat and mass transfer resistances and nonuniformities in the temperature and concentration profiles in the fluid and catalyst phases in the experimental reactor. Therefore, for the interpretation of experimental data the experiments should preferably be done under reaction conditions, where transport effects can be either eliminated or easily taken into account. In particular, the concentration and temperature distributions in the experimental reactor should preferably be described by plug flow or ideal mixing models. 

We now look at the mathematical equations for a general isothermal steady-state model for the trickle-bed reactor, which takes into account external mass-transfer resistances, i.e., gas-liquid and liquid-solid, axial dispersion, and the intraparticle mass-transfer resistances, along with the intrinsic kinetics occurring at the catalyst surface. Since many practical reactions can be characterized as... 

Yan et al. [7] proposed a model based on the model of Ho et al. [3], where the possibUity of kinetic control of the stripping reaction is added as well as the mass transfer resistance inside and outside of the emulsion globules. Correia and de Carvalho [29] have presented a comparison between several models for the recovery of 2-chlorophenol from aqueous solutions by ELM, where they proposed the effective upgrade of the Bunge and Noble [16] model. The same authors present an application of the Yan et al. [7] model for the recovery of 4-hydroxybenzoic acid from aqueous solutions by ELM [29,30]. 

The overall effectiveness factor of a catalyst pellet can be characterized by the ratio of the observed reaction rate to the rate in the absence of poisoning or external mass transfer resistance. It is expressed in the form of a power-law kinetic model for benzene hydrogenation as... 

Several models use the mass balance in Eq. 2.2 (ideal and equUibrimn-disper-sive models. Sections 2.2.1 and 2.2.2) as derived here without combining it with kinetic equations. In the latter case, Di in Eq. 2.2, which accounts only for axial diffusion, bed tortuosity, and eddy diffusion, is replaced with Da, which accoimts also for the effect of the mass transfer resistances. This is legitimate imder certain conditions, as explained later in Section 2.2.6. Other simple models account for a more complex mass transfer kinetics by coupling Eq. 2.2 with a kinetic equation (lumped kinetic models. Section 2.2.3) in which case Di is used. More complex models write separate mass balance equations for the stream of mobile phase percolating through the bed and for the mobile phase stagnant inside the pores of the particles (the general rate model and the lumped pore diffusion or FOR model, see later Sections 2.1.7 and 2.2.4). 

The preparative separations of certain polar (e.g., strongly basic) compounds and of many large molecular compotmds e.g., peptides and proteins) usually involve a complex mass transfer mechanism that is often slower than the mass transfer kinetics of small molecules. This slow kinetics influences strongly the band profiles and its mechanism must be accovmted for quantitatively. The accurate prediction of band profiles for optimization purposes requires a correct mathematical model of the various mass transfer processes involved. The piupose of the general rate model (GRM) is to accormt for the contributions of all the sources of mass transfer resistances to the band profiles [52,62,94,95]. The mass transfer of molecules from the bulk of the mobile phase percolating through the bed to the surface of an adsorbent or the mass of a permeable resin particle involves several steps that must be identified. 

A simpler version of the GRM model assumes that the mass transfer kinetics is controlled by mere pore diffusion and by the external film mass transfer resistance. The mass balances in the two fractions of the mobile phase are then written [52,62,101] ... 

In the equilibrium-dispersive model, we assume that the mobile and the stationary phases are constantly in equilibrium. We recognize, however, that band dispersion takes place in the column through axial dispersion and nonequilibrium effects e.g., mass transfer resistances, finite kinetics of adsorption-desorption). We assume that their contributions can be lumped together in an apparent dispersion coefficient. This coefficient is related to the experimental parameters by... 

In this model we assume that the contributions of the mass transfer resistances to band broadening are negligible, but that the kinetics of adsorption-desorption is slow. So the behavior of the chromatographic system is described by the mass balance Eq. 6.40. If we assume now that the kinetics of adsorption-desorption is of first order, we have the kinetic equation ... 

The two Eqs. 6.57a and 6.57b are classical relationships of the most critical importance in linear chromatography. Combined, they constitute the famous Van Deemter equation, which shows that the effects of the axial dispersion and of the mass transfer resistances are additive. This is the basic tenet of the equilibrium-dispersive model of linear chromatography. We will assume that this rule of additivity and Eqs. 6.57a remain valid when we apply the equilibrium-dispersive model to nonlinear chromatography. In this case, however, it is only an approximation because the retention factor, k = dq/dC, is concentration dependent. These equations have been derived from the lumped kinetic model. Thus, they show that the kinetic model and the equilibrium-dispersive model are equivalent as long as the rate of the equilibrium kinetics in the chromatographic system is not very slow. 

A closed-form analytical solution of this system of partial differential equations and relations (Eqs. 6.58 to 6.64a) is impossible to derive in the time domain. This is due to the extreme complexity of the general rate model, which accormts for the axial dispersion, the film mass transfer resistance, the pore diffusion and a first-order, slow kinetics of adsorption-desorption. 

Lee also extended the non-equilibrium theory developed originally by Gid-dings [10] to obtain H in/ the plate height contribution due to the mass transfer resistances and to axial dispersion, the non-equilibrium contribution. He started from the kinetic equation of the lumped rate constant kinetic model ... 

---