Pi approximation

As Ax approaches 0, PI (Ax, xk) approaches P(xk), so PI approximates P arbitrarily well if Ax is small enough. We ensure that Ax is small enough by imposing the step bounds... 

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... 

Of a different nature is the PI approximation (Modest, 2003), also known as the diffusion approximation (Ishimaru, 1997). Here the number of propagation directions is not restricted, but instead it is assumed that energy distributes quite uniformly over all these directions, as will be described in the next section. This approximation is the lowest order of the spherical harmonics method (also known as the Pn approximation). It is more versatile than two-and four-flux models, because it lends itself more easily to different geometries. 

Taking the divergence of Equation (35) and substituting it into Equation (37), a second-order partial differential equation of the Helmholtz type is finally obtained for the PI approximation... 

Note that, within the assumptions of the PI approximation, solving Equation (38) is entirely equivalent to solving the RTE, because once Gx is known, I can be evaluated by using Equations (34) and (35). 

To solve Equation (38) boimdary conditions which describe the reflection and transmission of radiation at the boimdaries are required. In principle, boimdary conditions can only be established in a rigorous manner for the radiative intensity, not for G, because the optical properties of the interfaces depend on the direction of incidence of radiation. Because the PI approximation solves for an integrated quantity like G instead, approximate boundary conditions must be established (Modest, 2003). One possibility is the Marshak boundary condition (Marshak, 1947), which comes from considering the continuity of the radiative flux through the interface. If this continuity is considered together with the assumption (34) of the PI approximation and Equation (37), the following equation is obtained (Spott and Svaasand, 2000)... 

General solution of the PI approximation for solar tubular reactors... 

As discussed previously, several solar photoreactor geometries can be reduced to cylindrical glass tubes externally illuminated by different types of reflectors, like parabolic troughs, CPC, V-grooves, or without reflector, directly illuminated by the sun. In this section the general solution of the PI approximation for this t)q5e of photo reactors is reported. This general solution is applicable to any particular reactor if the flux distribution impinging on the wall of the tubular reaction space is known. 

Even though this chapter is devoted mostly to solar photocatalytic reactors, we would like to discuss the modeling of an annular lamp reactor, as a different example of the application of the PI approximation. This problem was studied (Cuevas et al., 2007) with reference to a particular reactor known as photo CREC-water II (Salaices et al., 2001, 2002). Equation (38) is again written in cylindrical coordinates. Nevertheless in this case the... 

The main limitation of the PI approximation seems to be its applicability mainly to participating media with high optical depths (Cuevas et al., 2007). Optical depth is defined as... 

It would appear that the above requirement imposes strong limitations to the use of the PI approximation, but this is really not so in practice, the prevailing situation is operation of the reactors at high optical depths, based on the characteristic dimensions of reacting spaces and the typical catalyst concentrations used (Cuevas et al., 2007). This seems contradictory at first, because from the point of view of radiative transfer the optimal catalyst... 

Chapter 6 describes solar-powered photocatalytic reactors for the conversion of organic water pollutants. Nonconcentrating reactors are identified as some of the most energetically efficient units. It is reported that the absorption of radiation is a critical parameter in the efficiency reactor evaluation. The radiative transfer equation (RTE) solution under the simplified conditions given by the PI approximation is proposed for these assessments. 

The / -approximation is more difficult to solve in multidimensional geometries. It yields improved predictions over those obtained from the Pi-approximation, and its accuracy is comparable to the similar-order discrete ordinates approximation (S4), which is computationally more efficient. For one-dimensional systems, it is possible to develop higher-order PN approximations [51,52] however, for multidimensional geometries, even the P5-approximation is extremely complicated. 

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.). 

In addition to Eq. (34) (that is exact), the construction of macroscopic models consisting of a closed set of equations for the moments of the distribution function (or the intensity), usually requires to formulate approximations. In fluid mechanics, this approximation leads, for example, to the Navier-Stokes equation. The most common approximate macroscopic radiative models describe radiative transfer with heat-like equations (eg, see the Rosseland approximation and the PI approximation). Among them, the PI approximation leads to Pick s equation of the flux density vector Jr (see Eq. (73) substituting Pick s equation into Eq. (34) yields the following heat-like equation for the irradiance G (in the absence of a source term) ... 

The radiative transfer equation is not invariant with this transformation, but we find this invariance in various situations for example, the diffusion equation obtained with the PI approximation is invariant with this transformation (see Section 3.4). In addition, we fbrmd that solution of this equivalent problem usually provides results that are very close to those obtained by solution of the original problem in the case of a photobioreactor. The approximate solutions that are derived and validated in Sections 3.3 and 3.4 are obtained by addressing this equivalent problem. Note that this transformation is also useful for comparison of very different situations, regardless of the form of the phase function in the field of transport theory research, when mentioning optical thickness, we are generally referring to c rather than Cj. 

The PI approximation consists of truncating the spherical-harmonic expansion of the intensity at order 1. In the one-dimensional configuration shown in Fig. 6, for Lambertian or collimated normal incidence, this method is equivalent to fixing the following functional form for the angular... 

From the mesoscopic point of view, the PI approximation (according to... 

Figs. 16 and 17 represent respectively the irradiance field and the angular distribution of the intensity obtained with the PI approximation. Although the situation under study is far from equilibrium, the irradiance field produced by the approximation is in good agreement with the reference solution. This correspondence is surprising because PI should be unsuitable for such a situation with intermediate optical thickness. The next paragraph is focused on the vahdity conditions of the PI approximation. 

In the angnbr distributions presented in Fig. 17, we use the same scale for the PI approximation and the Monte Carlo method. Note that the area under the curve does not represent the irradiance because the element of solid angle sin( ) i is not taken into consideration here G = 2 r fj dQ sin( )L ). 

Figure 18 The irradiance field G within the photobioreactor shown in Fig. 6, with the same parameters as in Fig. 16, but the Lambertian emission is replaced by collimated emission at 6, — 0. Comparison between the PI approximation (Eq. (88)) and the reference solution (Monte Carlo method, MCM). For collimated incidence, only the boundary condition atz = 0 is modified, in comparison with the solution used in Fig. 16. We still have g(°)(z = 0) =qn/ but the ballistic irradiance becomes G ° z 0) —qn/ni- Therefore, the same solution as in Fig. 16 can be used, but with replacement of 4gn with 2+ /fl )qn in Eq. (88).

In Section 3.3, Fig. 13, we saw that the equivalent transport problem allows us to separate our radiative study into two simple systems the ballistic photons, for which the exact solution is analytical, and the scattered photons, which correspond to intensity close to isotropy. This relative isotropy of the scattered intensity in the equivalent transport problem suggests that the PI approximation is relevant. Therefore, to formulate the coUimated incidence phenomena, we will address the equivalent transport problem and separate the analysis of ballistic photons from that of scattered photons only scattered photons will be subjected to the PI approximation. In the rest of the chapter, the baUistic population is denoted as (0), whereas the scattered photons wiU be called the diffuse population and denoted as (d). 

Figure 21 The irradiance field G within the photobioreactor shown in Fig. 6 p and collimated normal incidence //, = 1. The results were obtained by means of the PI approximation of the equivalent transport problem where a =0.25, = 110 m ...

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