186
J.-F. Cornet et al.
This system of ordinary differential equations was integrated in different
systems of coordinates with boundary conditions expressing that a rectangular
reactor was illuminated on one or two sides with a homogeneous incident
radiant energy flux Fo or that a cylindrical reactor was radially illuminated with
homogeneous flux FR [51].
The obtained solutions giving the available radiant energy profiles are
summarized in Table 3 by Eqs. (66-69, 74 77, 82-85) and plotted in Fig. 4. The
mean volumetric rate of radiant energy absorbed in the medium can be expressed from Eqs. (70, 78, 86) in Table 3. This ratio reaches an asymptotic value
as the biomass concentration increases into the medium given by Eqs. (71, 79,
87) confirming that light energy transfer may be a limiting step when biomass
concentrates in the medium [46].
The illuminated fraction of the reactor y can be deduced from Eqs. (72, 80,
88) for different geometries by taking the roots of Eqs. (73, 87, 89) respectively
[51].
Equations (47, 48) coupled to Eqs. (66-73) or (74-84) for rectangular geometries and to Eqs. (82-89) for cylindrical geometries may be used to calculate
volumetric biomass growth rates in different operating conditions, when the
one-dimensional approximation is made. It must be noticed that situations in
Table 3 are not exhaustive and that the system of Eq. (65) can be integrated with
a wide range of boundary conditions.
3.4.3 The P-N and Discrete Ordinate Methods for Solving the One-dimensional
Radiative Transfer Equation
If the incident radiant light energy distribution and the scattering characteristics
of the medium are well known, the one-dimensional radiative transfer equation
can be solved with high accuracy without restrictive hypotheses as above.
A thorough treatment of numerical methods is available in the literature
[37, 53-56, 65, 71]. In this case, the one-dimensional equation of radiative
transfer is solved in terms of specific intensity I and written in the form [36]
1
dlz
1
+ 1
-- Ix (Ea + ES)Cx dz = Iz - ~ ~5o -1S P~ 0, 0')IzdIx'
(90)
with Ix = cos0, 0 being the angle with the normal vector, 6)0 the albedo of single
scattering equal to Es/(Ea + Es) and the phase function
1 2~ 0
p~ 0') = ~ I P( , qb, 0', qb') dqb'.
(91)
b
Fig. 4a-c. Profiles in normalized radiant energy available 4n J/F0. R, VS the dimensionless thickness
of the medium; ct = 0.655 (corresponding to Spirulina), 8 as parameter: a rectangular reactor
illuminated on one side; b rectangular reactor illuminated on both sides; e cylindrical reactor radially
illuminated. (Permission from Elsevier)
J.-F. Cornet et al.
This system of ordinary differential equations was integrated in different
systems of coordinates with boundary conditions expressing that a rectangular
reactor was illuminated on one or two sides with a homogeneous incident
radiant energy flux Fo or that a cylindrical reactor was radially illuminated with
homogeneous flux FR [51].
The obtained solutions giving the available radiant energy profiles are
summarized in Table 3 by Eqs. (66-69, 74 77, 82-85) and plotted in Fig. 4. The
mean volumetric rate of radiant energy absorbed in the medium can be expressed from Eqs. (70, 78, 86) in Table 3. This ratio reaches an asymptotic value
as the biomass concentration increases into the medium given by Eqs. (71, 79,
87) confirming that light energy transfer may be a limiting step when biomass
concentrates in the medium [46].
The illuminated fraction of the reactor y can be deduced from Eqs. (72, 80,
88) for different geometries by taking the roots of Eqs. (73, 87, 89) respectively
[51].
Equations (47, 48) coupled to Eqs. (66-73) or (74-84) for rectangular geometries and to Eqs. (82-89) for cylindrical geometries may be used to calculate
volumetric biomass growth rates in different operating conditions, when the
one-dimensional approximation is made. It must be noticed that situations in
Table 3 are not exhaustive and that the system of Eq. (65) can be integrated with
a wide range of boundary conditions.
3.4.3 The P-N and Discrete Ordinate Methods for Solving the One-dimensional
Radiative Transfer Equation
If the incident radiant light energy distribution and the scattering characteristics
of the medium are well known, the one-dimensional radiative transfer equation
can be solved with high accuracy without restrictive hypotheses as above.
A thorough treatment of numerical methods is available in the literature
[37, 53-56, 65, 71]. In this case, the one-dimensional equation of radiative
transfer is solved in terms of specific intensity I and written in the form [36]
1
dlz
1
+ 1
-- Ix (Ea + ES)Cx dz = Iz - ~ ~5o -1S P~ 0, 0')IzdIx'
(90)
with Ix = cos0, 0 being the angle with the normal vector, 6)0 the albedo of single
scattering equal to Es/(Ea + Es) and the phase function
1 2~ 0
p~ 0') = ~ I P( , qb, 0', qb') dqb'.
(91)
b
Fig. 4a-c. Profiles in normalized radiant energy available 4n J/F0. R, VS the dimensionless thickness
of the medium; ct = 0.655 (corresponding to Spirulina), 8 as parameter: a rectangular reactor
illuminated on one side; b rectangular reactor illuminated on both sides; e cylindrical reactor radially
illuminated. (Permission from Elsevier)