Boundary conditions and phase functions in a Photo-CREC Water-II reactor radiation field
Graphical abstract
Introduction
Heterogeneous photocatalysis is a promising wastewater treatment alternative for the removal of organic contaminants in both water and air (de Lasa et al., 2005). This technology belongs to the so-called advanced oxidation processes, which are characterized by the production of the hydroxyl radical. The hydroxyl radical is a powerful oxidant capable of oxidizing virtually any organic compound present in water (Chong et al., 2010). Photocatalytic processes are the result of the interaction of light and solid semiconductors. UV-light, with the proper energy, is absorbed by the TiO2 generating electron/hole charges (Fujishima et al., 2008). These charges promote the formation of OH radicals, which in turn are responsible for the complete mineralization of organic compounds into CO2, water and some mineral acids.
TiO2 Degussa P25 is the most active catalyst in the near-UV-region (Zhou et al., 2006). Degussa P25 is characterized by its high stability, good performance and relatively low cost (Fujishima and Zhang 2006). Annular photoreactors with suspended photocatalyst are a popular choice in photocatalysis. These type of reactors have shown the largest photocatalytic activity when compared to reactors with TiO2 immobilized on a support (de Lasa et al., 2005)
Photocatalytic reactions are only plausible when UV light is present in the system. Therefore, the accurate estimation of the interactions of light with the TiO2 particles is crucial in designing or scaling-up an annular photoreactor (Pasquali et al., 1996). In addition, the correct estimation of the radiation field inside the photoreactor allows for accurately determining energy efficiencies. Most efficiency definitions, such as the QY and the PTEF, involve a key variable: the rate of photons absorbed by the TiO2 (Moreira, 2010). The estimation of the photons being absorbed by a solid semiconductor requires solving the radiative transfer equation (RTE) inside the photoreactor. From the solution of the RTE, one can obtain values for the local volumetric rate of energy absorption (LVREA) (Yokota et al., 1999).
Nevertheless, the analytical solution of the RTE for heterogeneous media is a rather complex task and is so far only achievable in a restricted number of idealized reactor models (Grčić and Li Puma, 2013). TiO2 particles inside the photoreactor annular section cause near-UV radiation scattering. Thus, establishing scattering mechanisms poses extra challenges given that this phenomenon is a function of many variables such as lamp emission spectra, reactor geometry, type of TiO2 (concentration and agglomeration), nature of reactor walls and flow rate (Salaices et al., 2002, Pareek et al., 2003)
To address these issues, the RTE has been solved numerically using the discrete ordinate method (DO), the finite volume method (FV) and the Monte Carlo method (MC). It has been demonstrated that the MC method is preferable over deterministic methods to calculate the LVRPA with complicated geometries (Changrani and Raupp, 1999). The MC method consists of tracing individual photons from their generation by the UV-light source until they are absorbed or scattered inside the slurred system. In addition, it has also been shown that a statistical method provides a very effective approach in predicting the absorbing and scattering phenomena in slurry systems (Yokota et al., 1999).
When solving the RTE in photocatalytic reactors the following is required:
- (1)
TiO2 optical properties such absorption and scattering coefficients are to be provided. Romero et al. (1997) reported values for the absorption and scattering coefficients of Degussa P25 for a concentration range of 0.05–0.5 g L−1. Absorption and scattering coefficients for Degussa P25 as a function of wavelength were also given by Romero et al. (2003) for a range from 275 to 405 nm.
- (2)
Photoreactor boundary conditions have to be carefully selected. The light emitted by the radiation source has to be accurately defined and the optical properties of the reactor walls must also be properly identified.
- (3)
A scattering phase function has to be established (Piskozub and McKee, 2011). Phase functions calculate the angle at which photons are scattered from one direction to another. As result selecting a phase function in heterogeneous photocatalysis allows calculating multiple scattering events (Binzoni et al., 2006).
Different authors reported the solution of the RTE in annular photoreactors. For example, Pareek et al. (2008) divided the annular section into small cubical cells. When a photon was determined to be absorbed by a TiO2 particle, its position inside the reactor was stored. Successive events of this type allow an LVREA profile to be established. Yokota et al. (1999) also developed a MC simulation model. These authors consider an attenuation coefficient, a probability of photon absorption and isotropic and anisotropic scattering modes. These authors also found that the simulation results agreed with the experimental data. Changrani and Raupp (1999) used alumina foam structures to support the photocatalyst. Cabrera (1995) studied the effect of different models of the scattering phase function on the solution of the radiative transfer equation. Cabrera (1995) also considered three cases for the phase function: (1) specular with partial reflection, (2) isotropic scattering and (3) scattering centers with diffuse type of reflection. This study covered phase functions accounting from backwards to forwards type of scattering.
The specific selection of the adequate phase functions still remains until today an area of uncertainty. Both isocratic and Henyey–Greenstein phase functions were used in finding the solution of the RTE in photocatalytic systems (Moreira, 2010, Moreira, 2011, Satuf et al., 2005, Marugan et al., 2006). However, one should notice that an isocratic phase function was used in most of the simulations. Furthermore, the effect of the scattering mode in the calculated LVRPAs is not reported nor discussed. In this respect, it has been suggested by different authors that a precise evaluation of the scattering mode (or phase function) appears not to be critical for a good TRPA representation of experimental values (Moreira, 2010, Pasquali et al., 1996). Moreira (2010) concluded that for the Henyey–Greenstein phase function scattering modes the g values in the range −0.7<g<0.7 are satisfactory. Pasquali et al. (1996) also studied isocratic and diffuse phase functions concluding that both scattering modes render close results in the modeling of the radiation field in a photoreactor. Besides solving the RTE, semi-empirical approaches like the two-flux and the six-flux absorption-scattering models (Li Puma and Brucato, 2007) have proven to predict radiation as well as degradation rates for specific reactor geometries (Grčić and Li Puma, 2013).
However, in order to refine the selection of radiation models in photocatalytic reactors, it is of critical importance to consider the difference between local volumetric rate of photon absorption (LVRPA) and total rate of photon absorption (TRPA). The LVRPA represents a scalar field accounting for the gradients within the reaction media, whereas the TRPA is a single scalar proportional to the integral value of the LVRPA as described in the upcoming sections. Furthermore, in order to perform a true discrimination of the phase function and the associated “g” parameter, one has also to account for the applicable BC in the photocatalytic reactor unit.
In order to advance the art of photocatalytic reactor simulation this manuscript addresses several closely linked issues vis-à-vis of phase functions and BCs selection. The key of this study is a parametric sensitivity analysis showing the influence of phase functions and BCs selection on LVRPA. Furthermore, this also allows the determination of the statistical limits for both phase functions and LVRPA distributions. With this end in view isocratic, binomial and Henyey–Greenstein phase functions describing forward, isotropic and backward scattering are used to simulate the radiation profile inside a photoreactor with TiO2 Degussa P25. It is also proven that a MC method using a one parameter Henyey–Greenstein phase function (within a narrow “g” range) and properly selected BCs are able to describe TT in a wide range of conditions. We are not aware of a similar contribution for photocatalytic reactor numerical simulation in this critical area of phase function definition.
In a photocatalytic process, the RTE is obtained by performing a radiation balance across a thin slab. The resulting equation has already been presented as: (Pareek et al., 2003, Brandi et al., 2003, Martin et al., 1996):where Iλ is the spectral specific intensity of radiation having a wavelength λ (Einsteins m−2 s−1 sr−1), and are the absorption and scattering coefficients, respectively (m−1), and represents the phase function for the scattering phenomena. In developing this equation, it was assumed that emission due to high temperatures is negligible, given that photocatalysis occurs at low temperatures. The sum of the + is called the extinction coefficient βλ, and therefore, Eq. (1) is rearranged as:
Furthermore, the local incident intensity at any point inside the reactor is given by:
Therefore, a LVREA can be obtained by:
Finally, a TRPA can be obtained via the integration of LVRPA according to the following equation:
The MC method offers the flexibility of using the TiO2 and coefficients either as a function of wavelength of the light source or by using the wavelength-averaged values. Determining the and coefficients, however, is not a trivial exercise. In this study, the adopted influence of wavelength on scattering and absorption coefficients in various MC simulations is the one reported by Romero et al. (1997) and Toepfer (2006). These coefficients are applicable for a photocatalyst concentration range of 0.05–0.5 g L−1.
Multiple scattering events present in photocatalysis are described using a phase function, in Eq. (2). This parameter accounts for the probability of a photon being scattered from an incident direction to a Ω scattered direction. It is important to mention that in MC simulations, the most computer intensive step is frequently the one of establishing the new direction of the scattered photon (Satuf et al., 2005). As a result, the use of phase functions requiring intensive computations should be limited as much as possible.
Section snippets
Reactor setup and experimental measurements
The RTE was solved inside an annular photoreactor previously described by Moreira, 2011, Moreira, 2010. The Photo-CREC Water-II photoreactor measurement section is reported in Fig. 1. The Photo-CREC Water-II photoreactor consists of the following: a Pyrex glass inner tube, a black-light lamp (BL lamp), silica windows, a polyethylene outer tube, a reactor inlet and an outlet and an annular section where the TiO2 slurry flows in a semi-batch recirculation mode. The complete reactor setup allows
MC simulation for an isocratic phase function at a constant photocatalyst concentration
The MC method of this study was considered for a photocatalyst concentration of 0.05 g L−1. In this simulation, the isocratic phase function was used in order to assess the different results that one can obtain with the MC approach. As described in Fig. 6, the MC code was intended to be used to calculate 3D profiles of the LVRPA inside the annular photoreactor. Furthermore, the MC method was set to deliver the total rate of energy absorption (TRPA), i.e. the light absorbed by the TiO2 catalyst in
Conclusions
- (a)
It is shown that a MC is able to simulate the three dimensional field for the LVRPA inside an annular photoreactor. The proposed stochastically based MC model accounts for radiation absorption through the inner Pyrex glass, reflection of photons reaching the inner and outer walls of the concentric unit and photons absorption and scattering inside the photocatalyst slurry.
- (b)
It is proven that MC calculations using isocratic, H–G and binomial scattering models play a significant role in the
Nomenclature
- a
gap between lamp and reactor
- Ebg
energy band gap (eV)
- ex,ey,ez
direction cosines
- G
total incident radiation (Einstein m−2 s−1)
- g
asymmetry factor of the Henyey–Greenstein phase function (dimensionless Henyey–Greenstein scattering parameter).
- Iasym
asymmetry factor of the binomial phase function
- L
length of the photon flight (m)
- P(a)
absorption probability of the photocatalyst particle
- P(θ)
scattering probability density function
- PabsLamp
probability of absorption (no reemission) of the lamp (inner boundary)
- P
Acknowledgments
The scholarship granted to Patricio Valades Pelayo by the Consejo Nacional de Ciencia y Tecnologia (CONACyT), Mexico (Ref. File: 213979) is gratefully acknowledged. The authors also express their appreciation to the Natural Sciences and Engineering Research Council of Canada for the financial support in this investigation. Benito Serrano would also like to specially thank the Mexican Federal Program P/PIFI-2009-32MSU0017H-07 and the grant CONACYT-Mexico Ciencia Basica 2007-83144.
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