Two-flux approximation

Initially, we will focus on the mesoscopic description associated with the radiative transfer equation. Then, we will introduce the single-scattering approximation and two macroscopic approximations the PI approximation and two-flux approximation. AH of these discussions are based on the configuration shown in Fig. 6. Collimated emission and Lambertian emission wiU also be considered in the discussion later they correspond to the direct component and the diffuse component of solar radiation, respectively. Throughout our study, the biomass concentration Cx is homogeneous in the reaction volume V (assumption of perfect mixing), and the emission phenomena in V are negligible. The concentration Cx is selected close to the optimum for the operation of the photobioreactor the local photon absorption rate. 4 at the rear of the photobioreactor is close to the compensation point A.C (see Section 5 and chapter Industrial Photobioreactors and Scale-up Concepts by Pruvost et al.). 

Figure 23 The two-flux approximation different angular distributions of the intensity that can be postulated. (A) Isotropic distribution for each hemisphere. (B) Distribution (Eq. (104)) for 0 e [0, r/2] at different values of n. (C) Collimated distribution for one hemisphere and isotropic distribution for the other.

In Fig. 24, the irradiance field obtained with the two-flux approximation for fi —> 00 is compared with the Monte Carlo reference solution in the case of collimated solar-light incidence. The two-flux approximation wiU be used in Section 5.6 to analyze the coupling between radiative transfer and photosynthesis thermokinetics in photobioreactors with simple geometric structure. 

Figure 24 The irradiance field G in the photobioreactor shown in Fig. 6 and collimated normal incidence pi = 1. Comparison between the two-flux approximation (Eq. (105)) at b = 0.008 and the reference solution obtained by the Monte Carlo method (MCM see Section 4).

It should be noted that the approximations explored in this section can also be used to obtain analytical solutions for one-dimensional cyHndrical configurations (eg, see Pruvost and Comet, 2012 regarding the case of the two-flux approximation). Moreover, in the case of solutions 1 and 3 in the earher list, we chose to focus on non-reflecting surfaces in order to simpHfy the mathematical expressions. These approximations, however, are not restricted to non-reflecting surfaces. For example, when extended to reflecting surfaces on both sides, solution 3 is stiU analytical. 

Umitation by Hght (luminostat y = 1, or photo-limitation y < 1, see Comet, 2010) are presented in Table 2. They are compared with the predictive model calculations presented in this chapter, where the radiative transfer equation was solved using the one-dimensional two-flux approximation for all the simple geometric stmctures of photobioreactors except for reactor PBR 2 (as indicated in Table 2), for which we used the three-dimensional finite element method developed by Comet et al. (1994). As shown in the table, the mean deviation between the experimental results and the model calculation is less than 5% (ie, within the range of the experimental standard deviation), thus confirming the ability of the proposed predictive approach to quantify photobioreactor performance under many conditions of operation. 

Conveniently, simple analytical solutions of the RTE have been derived for G i(r), based on the two-flux approximation, for one-dimensional flat-plate... 

In present study two-flux approximation is used for radiation modeling. This approximation imlike the others is not limited to specific ranges of optical thickness. This feature will become more important when we know optical thickness of each layer of multilayer insulation may vary from optical thickness of the other layers. So if we use optically thick or thin approximations, optical thickness of two different layers may fall into both thick and thin ranges. 

Another benefit of using of two-flux approximation is that the radiativepro-perties of the boundaries are spotted in formulation. The other approximations neglect the radiative properties of the boundaries in their formulation. 

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... 

Note that the zero-flux surface condition for this species, after discretisation with the two-point approximation, establishes that... 

This flux approximately equals the volume flux in a fictitious suspension of the same particles at the same mean concentration but without fluctuations. However, these two fluxes by no means identically coincide. Pseudo-turbulent fluctuations cause the appearance of an additional component that is added to the total flux and that usually differs from zero. 

Under these circumstances the flux near the boundary is highly directional, and the two-term approximation of diffusion theory is inadequate [see (5.30)]. 

The general principles for the calculation of the quantities defined in Eqs. (5.275) through (5.281) in the two extreme approximations may now be presented for the case of an absorber of general geometric configuration placed in an isotropic flux. 

The first Chapman-Cowling approximation to the thermal conductivity of a dilute polyatomic gas within the two-flux approach leads to a total value that is the sum of two contributions related to translational and internal degrees of freedom respectively ... 

The result in the second-order approximation is exactly equivalent to equation (4.28), derived from the two-flux approach, as can be seen by substituting relations (4.19)-(4.21) into equation (4.36). Practical experience (Thijsse et al. 1979 Millat et al 1988a van den Oord Korving 1988) and model calculations for nitrogen (Heck Dickinson 1994) demonstrate that... 

On the other hand, the two-flux approach to the thermal conductivity of a polyatomic gas offers a rather greater opportunity for its prediction from viscosity and other experimental data because it is possible to make use of exact and approximate relationships between various effective cross sections and, in some cases, theoretically known behavior in the high temperature limit. The starting point for such an approach is... 

Equation (14.33) can be used to analyze the different levels of approximation within the two-flux approach. The Monchick-Pereira-Mason (MPM) approximation (Monchick et al. 1965) considers only the isotropic, first-order result the Kagan-Afanas ev (Kagan Afanas ev 1961) or Viehland-Mason-Sandler (>fiehland et al. 1978)... 

Because only fluxes are physically significant in our problem, the two-stream approximation is deemed adequate. Hence the radiation fields are restricted to the directions fio = l i = —At-i = 1 J i- It can be shown (Samuelson, 1983) that a solution of Eq. (9.1.4) leads to a flux divergence... 

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