Heterogeneous concentration profiles

Figure 2 Relative concentration profiles of oxidation products during thermal degradation as a function of depth for a plaque with thickness 3 mm Nearly homogeneous profile when the degradation depth, a, is 3.0 mm, similar to sample thickness, and heterogeneous profile when the degradation depth, a, is 0.1 mm, smaller than the sample thickness. The profiles were calculated from Equation (4).

Recall that for a heterogeneous catalytic reactor we had to consider the microscopic concentration profiles around and within a catalyst particle and then elirninate them in terms of the macroscopic position variable z in the reactor design ecpaations (mass transfer limits and effectiveness factors). 

Solution times using the homogeneous model are 15 to 25% less than that for the full two-phase analysis, although the accuracy of the results may be somewhat in question. Figure 17 shows the axial temperature and concentration profiles under type I operating conditions. Although these simulations appear similar to those obtained with the heterogeneous analysis, no direct comparison is possible, as was explained earlier. 

The differences between BV and MH also have implications in the concentration profiles of the electro-active species. Thus, whereas the BV model predicts a zero surface concentration of the oxidized species at the electrode surface, in the Marcus-Hush model the surface concentration of species O also depends on the electrode kinetics such that for small values of the heterogeneous... 

A comparison of the marked terms het and horn in Eq. (3.13) shows that for a small Thiele modulus the term het is also small whereas the term horn is small only if the product of ()C and the time t is small. This means that for small t the concentration profile is dominated by the heterogeneous character. On the other hand, for times, which are not too short, the influence of the exponential term horn predominates. Then, the dependence of the dimensionless parameters p and q on d is weak and d is assumed to be well enough fitted by the value 0.2. 

Despite the fact that the skin is a heterogeneous membrane, Fick s laws of diffusion have been successfully used to analyze skin permeation data. Solutions to the second law have been used in mechanistic interpretations (see later) and in considering concentration profiles within the skin. Fick s first law has been used to analyze steady-state diffusion rates and in the development of predictive models for skin permeability. 

In such a three-phase system, a different spatial concentration profile occurs, as shown in Figure 4.1.11, as compared to a two-phase system, cf. Figure 4.1.2. The gaseous reactant molecules first have to cross the boundary layer between the gas and liquid phase. In the liquid phase, they have to diffuse through the boundary layer between the liquid and the solid particle. The remaining way is the same as for a heterogeneous catalyst in the gas phase, as described in a previous Section 4.1.1.2. 

A heterogeneous reaction A -> 2B with nth order kinetics. /rA = k( A (n > 0) takes place on a catalyst surface. The component A with initial concentration CA0 diffusses through a stagnant film on the catalyst surface at isothermal and isobaric conditions. Assume one-dimensional diffusion, and determine the concentration profile of component A within the film of thickness 8 if the k is constant. 

Network thermodynamics has also been applied to nonstationary diffusion through heterogeneous membranes concentration profiles in the composite membrane and change of the osmotic pressure have been calculated with the modified boundary and experimental conditions. 

We have developed several new measurement techniques ideally suited to such conditions. The first of these techniques is a High Pressure Sampling Mass Spectrometric method for the spatial and temporal analysis of flames containing inorganic additives (6, 7). The second method, known as Transpiration Mass Spectrometry (TMS) (8), allows for the analysis of bulk heterogeneous systems over a wide range of temperature, pressure and controlled gas composition. In addition, the now classical technique of Knudsen Effusion Mass Spectrometry (KMS) has been modified to allow external control of ambient gases in the reaction cell (9). Supplementary to these methods are the application, in our laboratory, of classical and novel optical spectroscopic methods for in situ measurement of temperature, flow and certain simple species concentration profiles (7). In combination, these measurement tools allow for a detailed fundamental examination of the vaporization and transport mechanisms of coal mineral components in a coal conversion or combustion environment. 

The second question is more difficult to answer. Indeed, since the electrical perturbation extends only over a few angstroms from the electrode surface, the medium in which the electrogenerated intermediates react is identical to the bulk of the solution, which favors a positive answer. On the other hand, they react under essentially nonisotropic conditions because of the existence of concentration profiles, that is, of concentration changes with distance of the electrode (see Fig. 15). Yet this is a situation identical to that encountered when heterogeneous reagents or polyphasic conditions are used in usual homogeneous chemistry. A more serious difference is related to the nature of the media... 

FIG 7-12 Typical concentration profiles for the volume reaction model. [From Wen, Noncatalytic Heterogeneous Solid-Fluid Reaction Models, Ind. Eng. Chem. 60(9) 34-54 (1968), Fig. 3.]... 

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