Topics in Current Chemistry (2019) 377:22
1 3
Briefly, the discrete ordinate method consists in the discretization of the RTE and
the transformation of the integro-differential equation in a system of algebraic equations that can be solved numerically. It has been applied in systems with different
geometries, such as rectangular (with one and two dimensions) or cylindrical (with
two dimensions), and in radiation fields with various degrees of anisotropy [66].
For example, to calculate the e
a
(x, z) in a flat-plate, bench-scale solar simulator
for 4-chlorophenol photocatalytic degradation, Satuf et al. [75] applied the discrete
ordinate method to solve the RTE in a two-dimensional, two-directional rectangular
coordinate system. On the other hand, to evaluate the e
a
(r,z) in annular slurry photocatalytic reactors, Romero et al. [76–78] and Marugán et al. [79] (Fig. 12) utilized
the discrete ordinate method to solve the RTE in a two-dimensional, two-directional
cylindrical coordinate system.
When applying the finite volume method [67, 68], the computational domain is
divided into a number of control volumes, and the solid angle is discretized into a
number of finite solid angles. The RTE is then integrated over all control volumes
and angles, and finally, the boundary conditions are used to provide the linear algebraic equations necessary to solve the system with an iterative method. Interesting
applications of this technique to calculate the radiation field in photocatalytic slurry
reactors have been published by Camera Roda and Santarelli [80], Pareek al. [81],
Duran et al. [82], and Huang et al. [83].
The Monte Carlo method, in fact, is not based on the discretization of the RTE, but
consists in performing a simulation of the radiation transfer process inside the computer. Basically, this technique uses randomly generated numbers (R i ) to determine the
trajectories and fates of the photons entering through the reactor window or, in some
problems, those emitted by the radiation source as well. Then, by computing the locations where the photons are absorbed, it is possible to calculate the spatial distribution
of the photon absorption rate [84–88].
To calculate the LVRPA in a slurry reactor irradiated through one side, Manassero
et al. [8] performed a Monte Carlo simulation considering a one-dimensional, onedirectional radiation system. Briefly, the following events were considered in the photon tracking in the reactor: (1) calculation of the photon direction at the inner side of
the reactor window from the radiation emitted from the lamp: sin = 2R 1 − 1 ; (2)
determination of the length l of the photon flight in the reacting medium without interactions and, consequently, the new location of the photon after traveling that distance:
l = −
1
ln(1 − R 2 ) ; (3) decisions about the fate of the photon: (3a) if the new position
of the photon lies outside the reactor, the photon is lost and the process is re-initiated;
(3b) if the photon remains inside the reactor after traveling a distance l, a photon–catalyst particle interaction takes place, and two possibilities may occur: (4a) the photon is
absorbed according to the value of the albedo ( =
): 1 − ≥ R 3 ; (4b) the photon
is scattered and a new direction is determined by adopting, for example, the
Henyey–Greenstein
phase
function
(Eq.
44),
where
cos =
1
2g
1 + g
2
−
1−g
2
1+g ( 2R 4 −1)
2
[87, 88]; (5) if the photon is absorbed, it is stored
in the corresponding spatial cell and the trajectory ends. The LVRPA in each cell is
284
Reprinted from the journal
1 3
Briefly, the discrete ordinate method consists in the discretization of the RTE and
the transformation of the integro-differential equation in a system of algebraic equations that can be solved numerically. It has been applied in systems with different
geometries, such as rectangular (with one and two dimensions) or cylindrical (with
two dimensions), and in radiation fields with various degrees of anisotropy [66].
For example, to calculate the e
a
(x, z) in a flat-plate, bench-scale solar simulator
for 4-chlorophenol photocatalytic degradation, Satuf et al. [75] applied the discrete
ordinate method to solve the RTE in a two-dimensional, two-directional rectangular
coordinate system. On the other hand, to evaluate the e
a
(r,z) in annular slurry photocatalytic reactors, Romero et al. [76–78] and Marugán et al. [79] (Fig. 12) utilized
the discrete ordinate method to solve the RTE in a two-dimensional, two-directional
cylindrical coordinate system.
When applying the finite volume method [67, 68], the computational domain is
divided into a number of control volumes, and the solid angle is discretized into a
number of finite solid angles. The RTE is then integrated over all control volumes
and angles, and finally, the boundary conditions are used to provide the linear algebraic equations necessary to solve the system with an iterative method. Interesting
applications of this technique to calculate the radiation field in photocatalytic slurry
reactors have been published by Camera Roda and Santarelli [80], Pareek al. [81],
Duran et al. [82], and Huang et al. [83].
The Monte Carlo method, in fact, is not based on the discretization of the RTE, but
consists in performing a simulation of the radiation transfer process inside the computer. Basically, this technique uses randomly generated numbers (R i ) to determine the
trajectories and fates of the photons entering through the reactor window or, in some
problems, those emitted by the radiation source as well. Then, by computing the locations where the photons are absorbed, it is possible to calculate the spatial distribution
of the photon absorption rate [84–88].
To calculate the LVRPA in a slurry reactor irradiated through one side, Manassero
et al. [8] performed a Monte Carlo simulation considering a one-dimensional, onedirectional radiation system. Briefly, the following events were considered in the photon tracking in the reactor: (1) calculation of the photon direction at the inner side of
the reactor window from the radiation emitted from the lamp: sin = 2R 1 − 1 ; (2)
determination of the length l of the photon flight in the reacting medium without interactions and, consequently, the new location of the photon after traveling that distance:
l = −
1
ln(1 − R 2 ) ; (3) decisions about the fate of the photon: (3a) if the new position
of the photon lies outside the reactor, the photon is lost and the process is re-initiated;
(3b) if the photon remains inside the reactor after traveling a distance l, a photon–catalyst particle interaction takes place, and two possibilities may occur: (4a) the photon is
absorbed according to the value of the albedo ( =
): 1 − ≥ R 3 ; (4b) the photon
is scattered and a new direction is determined by adopting, for example, the
Henyey–Greenstein
phase
function
(Eq.
44),
where
cos =
1
2g
1 + g
2
−
1−g
2
1+g ( 2R 4 −1)
2
[87, 88]; (5) if the photon is absorbed, it is stored
in the corresponding spatial cell and the trajectory ends. The LVRPA in each cell is
284
Reprinted from the journal