Diffusivity evaluation parameters

Simulation and predictive modeling of contaminant transport in the environment are only as good as the data input used in these models. Field methods differ from laboratory methods in that an increase in the scale of measurement relative to most laboratory methods is involved. Determination of transport parameters (i. e., transmission coefficients) must also use actual contaminant chemical species and field solid phase samples if realistic values are to be specified for the transport models. The choice of type of test, e.g., leaching cells and diffusion tests, depends on personal preference and availability of material. No test is significantly better than another. Most of the tests for diffusion evaluation are flawed to a certain extent. 

The parameters defined in this chapter are divided into model parameters and evaluation parameters. Model parameters are porosity, voidage and axial dispersion coefficient, type and parameters of the isotherm as well as mass transfer and diffusion coefficient. All of them are decisive for the mass transfer and fluid flow within the column. They are needed for process simulation and optimisation. Therefore their values have to be valid over the whole operation range of the chromatographic process. Experimental as well as theoretical methods for determining these parameters are explained and discussed in Chapter 6. 

Here Djj o is the binary diffusivity evaluated at some reference total pressure pjo-The parameter Ko is the Knudsen flow parameter and like Bq for viscous flow it is also a function of solid structure only. 

Here Tq is some reference temperature and Dpo is the pore diffusivity evaluated at that temperature. The parameter a is equal to 0.5 when Knudsen controls the pore diffusion and to 1.75 where molecular-molecular collision mechanism controls the transport. The volumetric average concentration of the adsorbed species in eq. (10.4-3 8a) is... 

As before, E is defined relative to physical absorption considered to take place with negligible heat effects. The diffusion/reaction parameter M is defined using the reaction rate constant evaluated at the interfacial temperature T. ... 

Where M m) refers to diffusion reaction parameters evaluated at the bulk concentrations of the liquid phase reagents ... 

The relationship between adsorption capacity and surface area under conditions of optimum pore sizes is concentration dependent. It is very important that any evaluation of adsorption capacity be performed under actual concentration conditions. The dimensions and shape of particles affect both the pressure drop through the adsorbent bed and the rate of diffusion into the particles. Pressure drop is lowest when the adsorbent particles are spherical and uniform in size. External mass transfer increases inversely with d (where, d is particle diameter), and the internal adsorption rate varies inversely with d Pressure drop varies with the Reynolds number, and is roughly proportional to the gas velocity through the bed, and inversely proportional to the particle diameter. Assuming all other parameters being constant, adsorbent beds comprised of small particles tend to provide higher adsorption efficiencies, but at the sacrifice of higher pressure drop. This means that sharper and smaller mass-transfer zones will be achieved. 

The overall set of partial differential equations that can be considered as a mathematical characterization of the processing system of gas-liquid dispersions should include such environmental parameters as composition, temperature, and velocity, in addition to the equations of bubble-size and residence-time distributions that describe the dependence of bubble nucleation and growth on the bubble environmental factors. A simultaneous solution of this set of differential equations with the appropriate initial and boundary conditions is needed to evaluate the behavior of the system. Subject to the Curie principle, this set of equations should include the possibilities of coupling effects among the various fluxes involved. In dispersions, the possibilities of couplings between fluxes that differ from each other by an odd tensorial rank exist. (An example is the coupling effect between diffusion of surfactants and the hydrodynamics of bubble velocity as treated in Section III.) As yet no analytical solution of the complete set of equations has been found because of the mathematical difficulties involved. To simplify matters, the pertinent transfer equation is usually solved independently, with some simplifying assumptions. 

Comparison of Eq. (184) with Eq. (183) shows the effect of size distribution for the case of fast chemical reaction with simultaneous diffusion. This serves to emphasize the error that may arise when one applies uniform-drop-size assumptions to drop populations. Quantitatively the error is small, because 1 — is small in comparison with the second term in the brackets [i.e., kL (kD)112). Consequently, Eq. (184) and Eq. (183) actually give about the same result. In general, the total average mass-transfer rate in the disperser has been evaluated in this model as a function of the following parameters ... 

One of the calculation results for the bulk copolyroerization of methyl methacrylate and ethylene glycol dimethacrylate at 70 C is shown in Figure 4. Parameters used for these calculations are shown in Table 1. An empirical correlation of kinetic parameters which accounts for diffusion controlled reactions was estimated from the time-conversion curve which is shown in Figure 5. This kind of correlation is necessary even when one uses statistical methods after Flory and others in order to evaluate the primary chain length drift. 

The electrochemical behavior of niclosamide was described on the basis of d.c. polarography, cyclic voltammetry, a.c. polarography, and differential pulse polar-ography, in the supported electrolytes of pH ranging from 2.0 to 12.0 [32], A tentative mechanism for the reduction of niclosamide is proposed that involves the transfer of 4 e . Parameters such as diffusion coefficients and heterogeneous forward rate constant values were evaluated. 

A small Peclet number assures that diffusive transport is more controlling than convection within the trench. To evaluate the importance of diffusion limitations at small Peclet numbers, Takahashi and Gross define a parameter D. 

Liquid-liquid multiphasic catalysis with the catalyst present in the ionic liquid phase relies on the transfer of organic substrates into the ionic liquid or reactions must occur at the phase boundary. One important parameter for the development of kinetic models (which are crucial for up-scaling and proper economic evaluation) is the location of the reaction. Does the reaction take place in the bulk of the liquid, in the diffusion layer or immediately at the surface of the ionic liquid droplets ... 

Chen K-C, Wu J-Y, Yang W-B et al (2003) Evaluation of effective diffusion coefficient and intrinsic kinetic parameters on azo dye biodegradation using PVA-immobilized cell beads. Biotechnol Bioeng 83 821-832... 

The mechanism of solid catalysis involves processes of diffusion, formation of loose combinations with the solid and reactions of those combinations. Reactions with enzymes also involve intermediate, temporary combinations with the enzymes. The rate equations that may proposed in particular cases depends on what are believed to be controlling mechanisms. Many such eqautions are considered in Chapter 6. Here only one of the simpler forms will be examined for evaluation of the parameters, namely,... 

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