Numerical solutions including transfer calculation

Analytic solutions for flow around and transfer from rigid and fluid spheres are effectively limited to Re < 1 as discussed in Chapter 3. Phenomena occurring at Reynolds numbers beyond this range are discussed in the present chapter. In the absence of analytic results, sources of information include experimental observations, numerical solutions, and boundary-layer approximations. At intermediate Reynolds numbers when flow is steady and axisym-metric, numerical solutions give more information than can be obtained experimentally. Once flow becomes unsteady, complete calculation of the flow field and of the resistance to heat and mass transfer is no longer feasible. Description is then based primarily on experimental results, with additional information from boundary layer theory. 

Morbidelli et al. [41] discussed a numerical procedure for the calculation of numerical solutions of the GRM model in the case of an isothermal, fixed-bed chromatographic column with a multicomponent isotherm. These authors considered two different models for the inter- and intra-particle mass transfers. These models can either take into account the internal porosity of the particles or neglect it. They include the effects of axial dispersion, the inter- and intra-particle mass transfer resistances, and a variable linear mobile phase velocity. A generalized multicomponent isotherm, initially proposed by Fritz and Schliider [34] was also used ... 

When the sample is introduced into the column, usually in the form of a zone of vapor, it takes the form of a narrow band. During transit through the column, various factors influence the width of this band, which is continuously increased due to various dispersion processes. These include diffusion of the solute, resistance to mass transfer between and within phases, and the influence of flow irregularities and pertur-bations.f A simple concept, the theoretical plate, carried over from distillation processes, has been used to compare columns and account for the degree of dispersion that influences bandwidth. A chromatographic column may be considered to consist of numerous theoretical plates where the distribution of sample components between the stationary and mobile phase occurs. Hence, a measure of the efficiency of a GC column may be obtained by calculating the number of theoretical plates, N, in the column from ... 

As we saw in Eq. (18), description of each frequency of the spectrum is completely independent of other frequencies. Thus, all the derivation of equations in the rest of this section, including the approximate solutions developed in Sections 3.3—3.5, are valid regardless of the wavelength of incident radiation. For this reason, we decided to omit the spectral dependencies in our notations. For numerical calculations, however, one should use the value of the radiative properties and the incident flux n,t/ corresponding to the wavelength in question. In this approach, the radiative transfer equation is solved for each frequency, and the spectral solution thus obtained is integrated over PAR in order to calculate the local absorption rate A according to Eq. (31). This approach can be implemented with approximate solutions from Sections 3.3-3.5. 

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