Reaction-dispersive model

Akashi and coworkers prepared small platinum nanoparticles by ethanol reduction of PtCl in the presence of various vinyl polymers with amide side chains [49]. These authors studied the effects of molecular weight and molar ratio [monomeric unit]/[Pt] on the particle sizes and size distributions by electron microscopy, and in some cases by the dispersion stability of the Pt colloids. The hydrogenation in aqueous phase of allyl alcohol was used as a model reaction to examine the change in catalytic activity of polymer-stabilized Pt colloids upon addition of Na2S04 to the reaction solution. The catalytic tests were performed in water or in Na2S04 aqueous solution at 25 °C under atmospheric pressure of... 

The hydrothermal carbons obtained in the end from soluble, non-structural carbohydrates are micrometer sized, spherically shaped particle dispersions, containing a sp2 hybridized backbone (also responsible for the brown to black color) decorated with a dense layer of polar oxygenated functionalities still remaining from the original carbohydrate. The presence of these surface groups offers the possibility of further functionalization and makes the materials more hydrophilic and well-dispersible in water. The size of the final particles depends mainly on the carbonization time and precursor concentration inside the autoclave, as well as additives and stabilizers potentially added to the primary reaction recipe. An SEM image of a model reaction illustrating this dispersion state is shown in Fig. 7.1. 

An L-L-S system based on PEG, is, for example, the one described by Leitner and co-workers, where the ability of PEGs to stabilize dispersed Pd-nanoparticles was coupled with the use of SCCO2. The model reaction that was investigated in this system was the oxidation of alcohols with oxygen in the presence of Pd-clusters of structure [Pd56i-phen6o(OAc)igo] (phen = 1,10-phenanthroline) (Figure 6.14)." °... 

Fig. 28. Comparison of performance of reactors for the plug flow and dispersed plug flow models. Reaction is of first order, aA- products, and constant density, occurring in a closed vessel (L14, L15).

In this section, we will obtain the non-dimensional effective or upscaled equations using a two-scale expansion with respect to the transversal Peclet number Note that the transversal P let number is equal to the ratio between the characteristic transversal timescale and longitudinal timescale. Then we use Fredholm s alternative to obtain the effective equations. However, they do not follow immediately. Direct application of Fredholm s alternative gives hyperbolic equations which are not satisfactory for our model. To obtain a better approximation, we use the strategy from Rubinstein and Mauri (1986) and embed the hyperbolic equation to the next order equations. This approach leads to the effective equations containing Taylor s dispersion type terms. Since we are in the presence of chemical reactions, dispersion is not caused only by the important Peclet number, but also by the effects of the chemical reactions, entering through Damkohler number. 

For adsorption rate, LeVan considered four models axial dispersion (this is not really a rate model but rather a flow model), external mass transfer, linear driving force approximation (LDF) and reaction kinetics. The purpose of this development was to restore these very compact equations with the variables of Wheeler equation for comparison. 

BIOPLUME III is a public domain transport code that is based on the MOC (and, therefore, is 2-D). The code was developed to simulate the natural attenuation of a hydrocarbon contaminant under both aerobic and anaerobic conditions. Hydrocarbon degradation is assumed due to biologically mediated redox reactions, with the hydrocarbon as the electron donor, and oxygen, nitrate, ferric iron, sulfate, and carbon dioxide, sequentially, as the electron acceptors. Biodegradation kinetics can be modeled as either a first-order, instantaneous, or Monod process. Like the MOC upon which it is based, BIOPLUME III also models advection, dispersion, and linear equilibrium sorption [67]. 

The surfactant-aided Lewis acid catalysis was first demonstrated in the model reaction shown in Table 13.1 [22]. While the reaction proceeded sluggishly in the presence of 10 mol% scandimn triflate (ScfOTOs) in water, a remarkable enhancement of the reactivity was observed when the reaction was carried out in the presence of 10 mol% Sc(OTf)3 in an aqueous solution of sodium dodecyl sulfate (SDS, 20 mol%, 35 mM), and the corresponding aldol adduct was obtained in high yield. It was found that the type of surfactant influenced the yield, and that Triton X-100, a non-ionic surfactant, was also effective in the aldol reaction (but required longer reaction time), while only a trace amount of the adduct was detected when using a representative cationic surfactant, cetyltrimethylammonium bromide (CTAB). The effectiveness of the anionic surfactant is attributed to high local concentration of scandium cation on the surfaces of dispersed organic phases, which are surroimded by the surfactant molecules. 

Tanks-in-Series Model Versus Dispersion Model. We have seen that we can apply both of these one-parameter models to tubular reactors using the variance of the RTD. For first-order reactions the two models can be applied with equal ease. However, the tanks-in-series model is mathematically easier to use to obtain the effluent concentration and conversion for reaction orders other than one and for multiple reactions. However, we need to ask what would be the accuracy of using the tanks-in-series model over the dispersion model. These two models are equivalent when the Peclet-Bodenstein number is related to the number of tanks in series, n, by the equation ... 

There is similar electrochemical promotion behavior of Rh films on YSZ and similar metal-support interaction-induced behavior of dispersed Rh on different supports for the model reaction of C2H4 oxidation on Pt (Figure 40). In particular, there are very similar p values (p ep]y[sj e 120) upon increasing the potential and work function of the Rh film or upon increasing the work function (or absolute potential) of the support of the dispersed Rh catalyst (Figure 40). 

The same reaction was also investigated over Rh nanoparticles dispersed over pure and doped Ti02 and other porous supports of known work functions [136]. It was established that the influence of electrochemical promotion on kinetic parameters of the model reaction is identical to the influence of metal-support interactions, under conditions at which the change of the work function of the catalyst is the same, regardless of the means by which the alteration in the work function is achieved. 

The other subgroup of the lumped rate approach consists of the reaction dispersive model where the adsorption kinetic is the rate-limiting step. It is an extension of the reaction model (Section 6.2.4.3). Like the mass transfer coefficient in the transport dispersive model, the adsorption and desorption rate constants are considered as effective lumped parameters, kads,eff and kdes.eff- Since no film transfer resistance exists (Cpi = q), the model can be described by Eq. 6.79 ... 

Depending on the main cause of sluggishness in reaching equilibrium in the column, we can distinguish several kinetic models. If the kinetics of the retention mechanism (e.g., the kinetics of adsorption-desorption) is slower than the other steps of the chromatographic process, we use the reaction-dispersive model. If the slowest step in the chromatographic process is the mass transfer kinetics, we have the transport-dispersive model. 

Numerical Solutions of the Reaction-Dispersive and the Transport-Dispersive Models for a Pulse Injection... 

The reaction-dispersive model is an extension of the Thomas model with a Langmuir kinetics... 

The effect of slow mass transfer on the band profile is illustrated in Figure 14.11 which shows a series of profiles calculated with the reaction-dispersive model, for... 

Figure 14.11 Numerical solutions of the reaction-dispersive model obtained for various values of the rate constant of the adsorption-desorption kinetics. Calculation conditions = 1%. Npisp = 1000. Chro-...

The band profiles for overloaded elution calculated with the reaction-dispersive, transport, and transport-dispersive models can be fitted to Thomas model [1]. 

If the effect of dispersion is not taken into accoimt in the apparent number of reaction units (N ), there will be a large difference between the solutions of the Thomas model and the transport-dispersive or the reaction-dispersive models. This is illustrated in Figure 14.14, which compares the analytical solution of the Thomas model and the numerical solution of the reaction-dispersive model. The front of the latter solution is less steep than that of the former because the Thomas model does not take into accoimt axial dispersion, but only the kinetics of adsorption-desorption. 

Figure 14.14 Comparison of the numerical solution of the reaction-dispersive model (solid line) and the analytical solution of the Thomas model (dotted line). Both models kg = 5 Nrea = fcgfcdL/M = 2000. Thomas model Noisp = co. Reaction-dispersive model 2NDisp[fc()/(l + = 2000. (Left Lf =1%. (Right) Lf = 1, 1% 2, 5% 3, 10% 4, 20%.

The values of the parameters of the Thomas model (see Table 14.1) were obtained by fitting the profiles obtained as numerical solutions of the reaction-dispersive model (solid lines) to Eq. 14.65, for the different values of the loading factor. Reproduced with permission from S. Golshan-Shirazi and G. Guiockon,. Chromatogr., 603 (1992) 1 (Figs. 1 and 2). 

The best values of the parameters obtained when fitting the numerical solution of the reaction-dispersive model to Eq. 14.65 for the Thomas model are given in Table 14.1. The results in Table 14.1 show a very small error, of the order of 0.1% for the retention factor. For the loading factor, the error is approximately 1%. Both errors are independent of the loading factor. On the other hand, the value obtained for is not constant. It is significantly lower than 1000 and it decreases... 

Table 14.1 Best Values of the Parameters Obtained by Fitting the Band Profiles Calculated with the Reaction-Dispersive Model to the Thomas Model "...

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