Adsorption pseudo-steady state

For several cases, e.g. for linear pseudo-steady-state equations (linear mechanisms), the steady state is certain to be unique. But for non-linear mechanisms and kinetic models (which are quite common in catalysis, e.g. in the case of dissociative adsorption), there may be several solutions. Multiplicity of steady-states is associated with types of reaction mechanisms. 

Fairly recently it has been established that a set of pseudo-steady-state equations for complex catalytic reactions can have several solutions only when their detailed mechanisms involve as one step an interaction between various intermediates [22], The simplest catalytic mechanism possessing this property is an adsorption mechanism. For example... 

Later, it became clear that the concentrations of surface substances must be treated not as an equilibrium but as a pseudo-steady state with respect to the substance concentrations in the gas phase. According to Bodenstein, the pseudo-steady state of intermediates is the equality of their formation and consumption rates (a strict analysis of the conception of "pseudo-steady states , in particular for catalytic reactions, will be given later). The assumption of the pseudo-steady state which serves as a basis for the derivation of kinetic equations for most commercial catalysts led to kinetic equations that are practically identical to eqn. (4). The difference is that the denominator is no longer an equilibrium constant for adsorption-desorption steps but, in general, they are the sums of the products of rate constants for elementary reactions in the detailed mechanism. The parameters of these equations for some typical mechanisms will be analysed below. 

The empty-site requirement in Eq. (28) can be physically interpreted in one of two different ways either the adsorbed A and B have to rearrange prior to reaction, or they are bound to more than one adsorption site. For the latter case, the intermediate concentration is low, thus allowing a pseudo-steady-state assumption. Through the application of bifurcation analysis and catastrophe theory this model was found to predict a very rich bifurcation and dynamic behavior. For certain parameter values, sub- and supercritical Hopf bifurcations as well as homoclinic bifurcations were discovered with this simple model. The oscillation cycle predicted by such a model is sketched in Fig. 6c. This model was also used to analyze how white noise would affect the behavior of an oscillatory reaction system... 

In photochemical experiments, this very simple approach may be compromised if desorption of the reactants is fast, in that reactant adsorption-desorption equilibrium is not established during the reaction [then equation (13.5) does not hold]. In addition, active center reactivity is continuous because of continuous illumination thus, no equilibrium is established. This may lead to the derivation of a pseudo-steady-state kinetic model [200,201] with a rate expression slightly different from equation (13.4), the discussion of which is, however, out the scope of this work. 

For instance, if it is supposed that sites can be considered in pseudo-steady state it suffices to set the parameters Cj to zero, so that the differential equations (2) are transformed into algebraic ones. If simple adsorption equilibrium is to be considered for all the species over all the sites, this would be equivalent to write down equations (4) as ... 

The gas phase balance includes a convective flow term for the mass flow of species i into the system. The pressure would rise were it not for the rate of adsorption, that is, the process that removes i from the gas phase and locates it in the second phase, the adsorbent. Now we can make progress in the analysis even before we substitute in the rate expression. The reason is this in the experiment the rate of adsorption must be equal to the rate of delivery. Therefore we have a pseudo-steady state in that the gas phase concentration remains constant all the while the surface concentration is changing ... 

The curve shown in Figure 1 obtained at 25 °C, corresponds to the blank, since at this temperature there is no interaction between the lanthanum oxide and the CO2. The first adsorption at 500°C indicate that there is a strong interaction between the CO2 and the support, and it does not reach the pseudo-steady state with a constant amplitude during the first 20 pulses of the experiment, although it is apparent that it is approaching that state. 

A similar model that specifically considers the poison deposition in a catalyst pellet was presented by Olson [5] and Carberry and Gorring [6], Here the poison is assumed to deposit in the catalyst as a moving boundary of a poisoned shell surrounding an unpoisoned core, as in an adsorption situation. These types of models are also often used for noncatalytic heterogeneous reactions, which was discussed in detail in Chapter 4. The pseudo-steady-state assumption is made that the boundary moves rather slowly compared to the poison diffusion or reaction rates. Then, steady-state diffusion results can be used for the shell, and the total mass transfer resistance consists of the usual external interfacial, pore diffusion, and boundary chemical reaction steps in series. 

With kx = k2Ct- From equation 68, it is clear that the assumption of the occurrence of a RDS in addition to the pseudo-steady-state approximation for the surface intermediates results in a significant simplification of the expressions for the surface intermediate concentrations and, hence, for the rate of the global reaction. Similar equations can be derived for the reactant adsorption or product desorption as RDS. 

Tronconi and co-workers (98,116) have validated against experiment a more complex, heterogeneous, transient model, accovmting also for diffusion and reaction of NO and NH3 inside the porous walls of extruded honeycomb SCR catalysts. The model equations are presented in Table 5 x and z are the intraporous and axial coordinate, respectively is the ammonia adsorption capacity of the catalyst 6 is the NH3 surface coverage is the effective intraporous diffiisiv-ity s is the monolith wall half-thickness i is the gas velocity in the monolith channels are gas-solid mass transfer coefficients and dh is the hydraulic diameter of the monolith channels. Notably, a pseudo-steady-state assumption... 

In corrosion, phenomena other than mass transport in the electrolyte can slow down the establishment of steady state conditions, including adsorption, precipitation or film growth. Especially, solid state transport processes in passive oxide films are generally slow (Chap. 6) and as a consequence the measured current density will depend on the sweep rate, even if from a solution mass transport point of view steady state prevails (t 1). Polarization curves measured under these conditions are sometimes called pseudo-steady-state polarization curves. When reporting such data one should always indicate the sweep rate used. 

R. S. Barlow, Analysis of the Adsorption Process and Desiccant Systems-A Pseudo Steady State Model for Coupled Heat and Mass Transfer, Solar Energy Research Institute, Golden, CO, Report No. SERI/TR-631-1330, 1982. 

We can now apply the method of resolution of the pure kinetics, with one of the preceding reactions as the rate determining step in pseudo-steady state mode. For each type of solid MG, we will obtain three solutions according to whether the determining step is adsorption, the reaction at internal interface i or at the external interface e (we exclude the modes limited by the heart reaction that we have never encountered). Table 15.4 provides the results obtained in conditions far from equilibrium. To obtain the expression of the reactivity in the opposite case, closed to equilibrium conditions, we have to multiply the preceding relations by the term ... 

In order to use this rate expression, we require the concentration of component A adsorbed on the catalyst surface. This is obtained by assuming that the adsorption reaction is in pseudo-steady state. This means that the rate of the forward reaction... 

At the same time, current mechanistic understanding of photocatalysis speaks in favor of applying the pseudo-steady-state model for virtually all photocatalyzed reactions, where adsorption/desorption equilibrium cannot be achieved. 

The apparent constant Xj pp degenerates to the adsorption equihbrium constant for the first step at dark conditions (/ = 0) Xj app,dark = /k-. The pseudo-steady-state results in... 

Reaction rates for the start-of-cycle reforming system are described by pseudo-monomolecular rates of change of the 13 kinetic lumps. That is, the rates of change of the lumps are represented by first-order mass action kinetics with the same adsorption isotherm applicable to each reaction step. Following the same format as Eq. (4), steady-state material balances for the hydrocarbon lumps are derived for a plug-flow, fixed bed catalytic reformer. A nondissociation, Langmuir-Hinshelwood adsorption model is employed. Steady-state material balances written over a differential fractional catalyst volume dv are the following ... 

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