Two-flux model

On diffuse irradiation, Eqs. (8.10) through (8.15) become much simpler since all terms with the factor (3/m - 2) vanish, j (3/m - 2)fiod/xo = 0. Helpwise, collimated irradiation under //o = 2/3 (ao = 48.2°) has the same effect, but only for weak absorption. With increasing absorption the light fluxes inside the sample deviate more and more from the condition of diffuse irradiation. It has been often shown that the two-flux model derived first by Schuster<30) and then by Kubelka and Munk(28) has formally the same analytical solutions as the Pi-approximation under diffuse irradiation. Kubelka... 

As several researchers have shown empirically, the use of —log(reflectance) can provide, analogous to a transmittance measurement, a linear relationship between the transformed reflectance and concentration, if the matrix is not strongly absorbing as can be found for many samples studied by near-infrared spectroscopy. This issue is presented in detail below. A different approach based on a physical model was considered for UV/VIS measurements and later also applied within the mid-infrared. A theory was derived by Kubelka and Munk for a simple, onedimensional, two-flux model, although it must be noted that Arthur Schuster (1905) had already come up with a reflectance function for isotropic scattering. A detailed description of theoretical and practical aspects was given by Korttim. The optical absorption... 

The simplest multiflux model can be developed for a one-dimensional planar medium or a one-dimensional axisymmetric cylindrical geometry by dividing the entire solid angle range (4k sr) into two components one in the forward and the other in the backward hemisphere. It is assumed that the incident radiation and flux are proportional to each other in each of these hemispheres. Then, the question is finding a closure condition, or, in other words, a proportionality factor between flux and integrated intensity in each half-sphere. This choice yields different two-flux methods, such as Schuster-Hamaker, Schuster-Schwarzchild or modified two-flux models [1,19, 51]. 

A solution to this problem based on the two-flux model is given by Tien and Drolen [59] ... 

Brucato, A., Rizzuti, L., 1997b, Simplified modeling of radiant fields in heterogeneous photoieactois. 2. Limiting two-flux model for the case of reflectance greater than zero. Ind. Eng. Chem. Res., 36 4748-4755. 

The Kubelka-Munk two flux model predicted significantly different magnitudes of photon flux within the layers of a coating than a model based on the Lambert-Beer law. Equations and calculation methods are described and results are given that illustrate the effect of substrate reflectance, layer thickness and the absorption and scattering of the layer components on the photon flux and the light absorbed at various levels within the coating. 

This is a two flux model and it has been found of value when calculating the reflectance of a layer containing a mixture of components. Computer programs that use this approach have found practical application in predicting the amounts of pigment required in a layer to match the colour of a standard material. 

A two flux model would be reasonable for such a layer receiving light froa a wide range of directions. The scattering would tend to maintain the diffuse nature of the radiation for both the fluxes. 

The Kubelka-Munkli.) two flux model appears to be superior to a single flux model based on the Lambert-Beer law as it is capable of taking into account the scattering by components in the layer and the reflectance of the substrate. The model can provide estimates of the amount of light absorbed at any point within the coating and predict how this varies with film thickness, concentration of materials and reflectance of the substrate. 

The culture system described earlier is based on cylindrical tubes, which makes it difficult to calculate radiative transfer in the culture volume, which has to be solved numerically (Lee et al, 2014). As already described, the one-dimensional hypothesis where light attenuation occurs along only one main direction serves to obtain analytical relations to represent the light attenuation field (as with the two-flux model, Eq. 12). This enables accurate and easy determination of light attenuation conditions for any operating conditions and thus greater system control. Based on this statement, researchers designed a specific PBR. Like the multimodule external-loop airlift PBR, this system is of industrial size (130 L), but the unit is a flat panel with front illumination so as to respond to the one-dimensional hypothesis. It is also illuminated on both sides to increase specific illuminated area... 

While the primary failings of the theory were fixed with a modification by Simmons [65], it was Simmons himself who pointed out that simple scattering models seem to be as useful as more complex ones. Recent work in discontinuous theories has been dominated by using a two-flux model and the mathematics of plane parallel layers (sheets). 

Experimentation in relation to flux models is therefore of two kinds. ... 

In general, tests have tended to concentrate attention on the ability of a flux model to interpolate through the intermediate pressure range between Knudsen diffusion control and bulk diffusion control. What is also important, but seldom known at present, is whether a model predicts a composition dependence consistent with experiment for the matrix elements in equation (10.2). In multicomponent mixtures an enormous amount of experimental work would be needed to investigate this thoroughly, but it should be possible to supplement a systematic investigation of a flux model applied to binary systems with some limited experiments on particular multicomponent mixtures, as in the work of Hesse and Koder, and Remick and Geankoplia. Interpretation of such tests would be simplest and most direct if they were to be carried out with only small differences in composition between the two sides of the porous medium. Diffusion would then occur in a system of essentially uniform composition, so that flux measurements would provide values for the matrix elements in (10.2) at well-defined compositions. 

Though the solution procedure sounds straightforward, if tedious, practice difficulty is encountered immediately because of the implicit nature of the available flux models. As we saw in Chapter 5 even the si lest of these, the dusty gas model, has solutions which are too cumbersc to be written down for more than three components, while the ternary sol tion itself is already very complicated. It is only for binary mixtures therefore, that the explicit formulation and solution of equations (11. Is practicable. In systems with more than two components, we rely on... 

Ishii (1977) One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase regimes. AML Report ANL-77-47 Ide H, Matsumura H, Tanaka Y, Fukano T (1997) Flow patterns and frictional pressure drop in gas-liquid two-phase flow in vertical capUlary channels with rectangular cross section, Trans JSME Ser B 63 452-160... 

In reality, the slip velocity may not be neglected (except perhaps in a microgravity environment). A drift flux model has therefore been introduced (Zuber and Findlay, 1965) which is an improvement of the homogeneous model. In the drift flux model for one-dimensional two-phase flow, equations of continuity, momentum, and energy are written for the mixture (in three equations). In addition, another continuity equation for one phase is also written, usually for the gas phase. To allow a slip velocity to take place between the two phases, a drift velocity, uGJ, or a diffusion velocity, uGM (gas velocity relative to the velocity of center of mass), is defined as... 

Figure 9.9 Summary of rate or flux expressions for gas-liquid reactions (two-film model)...

This simple two component model for the Fe isotope composition of seawater does not consider the effects of the Fe isotope composition of dissolved Fe from rivers or from rain. Although the dissolved Fe fluxes are small (Fig. 19) the dissolved fluxes may have an important control on the overall Fe isotope composition of the oceans if they represent an Fe source that is preferentially added to the hydrogenous Fe budget that is ultimately sequestered into Fe-Mn nodules. In particular riverine components may be very important in the Pacific Ocean where a significant amount of Fe to the oceans can be delivered from rivers that drain oceanic islands (Sholkovitz et al. 1999). An additional uncertainty lies in how Fe from particulate matter is utilized in seawater. For example, does the solubilization of Fe from aerosol particles result in a significant Fe isotope fractionation, and does Fe speciation lead to Fe isotope fractionation ... 

The Hamiltonian in Eq. (39) has bear used to calculate the adiahatic free energy as a function of the solvent coordinate using the umbrella sampling method, and reactive flux correlation function calculations have been used to determine the adiabatic rate constant. The results were qualitatively similar to the results based on the two-state model. 

While a one-dimensional model for steady-state heat conduction might seem an oversimplifying assumption, values for a typical heat flux range from 0 up to 10 kW/m With a one-dimensional model, this translates to about a maximum 0.5 °C temperature gradient within the barrel metal in the radial direction. If a two-dimensional model is used, the temperature gradient will decrease even further, and thus have virtually no impact on the interfacial temperature calculations. 

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