Photobioreactor within

The authors would tike to acknowledge the support of the Hydrogen, Fuel Cell, and Infrastructure Technology Energy Biosciences and GTL Programs within the US Department of Energy. We would also tike to thank everyone in our laboratory for many helpful discussions, and in particular, Dr. Dan Blake, who provided us with critical information on photobioreactor materials. 

Analysis of Multiple-Scattering Radiative Transfer Within Photobioreactors ... 

Approximate Solutions for the Radiation Field Within One-Dimensional Cartesian Photobioreactors 22... 

The purpose of a photobioreactor is to absorb incident light in order to convert it into biomass via coupling with photosynthesis. On the one hand, efficient light absorption usually corresponds to heterogeneous radiation fields (x) within the reaction volume (see Section 3). On the other hand, the coupling law (Eq. (4)) is usually a non-linear function of (x) (the law obtained in Section 5 is non-linear, but this is also the case for most of other models reported in the literature). Therefore, the coupling between radiative transfer and photosynthesis must be formulated locally, which implies that determination of the volumetric rate < > requires... 

Thus, the construction of predictive models of photobioreactors requires careful formulation of radiative transfer within the reaction volume, in order to obtain the radiation field (cf step 1 in the earlier procedure). Such analysis is developed this chapter, starting in Section 2 with determination of the light scattering and absorption properties of photosynthetic-microorganism suspensions. Next, these properties are used in Section 3 for analysis of radiative transfer and in Section 4 for rigorous solution of the radiative transfer equation by the Monte Carlo method. Finally, the thermokinetic coupling between radiative transfer and photosynthesis is addressed in Section 5. It should be noted that Sections 2 and 4 mainly summarize works that have been already published elsewhere, whereas Sections 3 and 5 include extensive original work and results. 

The main steps in our model and their organization within this chapter are summarized in Fig. 1. Our practice of the Monte Carlo method extends beyond the solution of the radiative transfer equation in Section 4, we also argue that the Monte Carlo method is well suited for numerical implementation of the entire model, especially in research on photobioreactors with complex geometric structure. 

Our model implies independent scattering (the assumption that we share with the great majority of photobioreactor researchers). Indeed, typical biomass concentrations within the process are low enough to reasonably assume that each microbial ceU interacts with radiation independently. We can therefore define particle cross sections a that characterize the radiative properties of microbial cells independently of their concentration Q and keM.p = where... 

Despite these areas for improvement, the methodological chain that is presented in the present section already yields radiative properties with a fair level of accuracy for standard culture conditions, when the shape of the microorganism is accurately described (see vaHdation in Charon et al., 2015 Dauchet et al., 2015), including all the spectral and angular data that are needed for formulation of radiative transfer within a photobioreactor. 

ANALYSIS OF MULTIPLE-SCATTERING RADIATIVE TRANSFER WITHIN PHOTOBIOREACTORS APPROXIMATE SOLUTIONS FOR THE RADIATION FIELD WITHIN ONE-DIMENSIONAL CARTESIAN PHOTOBIOREACTORS... 

Figure 9 The distribution function atZo = 3 cm within the one-dimensional photobioreactor shown in Fig. 6 where =p = Q, for collimated normal incidence dj — O). (A) Without scattering. (B) With scattering. The results were obtained with the Monte Carlo method (MCM, see Section 4).

Figure 14 The irradiance field 6 within the photobioreactor shown in Fig. 6 pi " — 0 and collimated normal incidence //, = 1. The results were obtained for the equivalent transport problem where a = 0.25, /c, = 110 m" and = 1 /4 r the expression for...

Figure 16 The irradiance field G within the photobioreactor shown in Fig. 6 = 0,...

Figure 17 Angular distribution of the intensity L Zo,d) at the abscissa Zq within the photobioreactor shown in Fig. 6 — 0,p — 0.54, and Lambertian incidence. Comparison...

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).

Substituting Eq. (104) into the radiative transfer equation Eq. (24) (in which we omit the frequency variable) and integrating over propagation directions (O (ie, over the propagation angles ), we obtain the following equation for the irradiance within the photobioreactor shown in Fig. 6 with... 

During the previous radiative-transfer analysis, the following approximate solutions were obtained for the irradiance field within the typical photobioreactor configuration in Fig. 6 ... 

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