188
J.-F. Cornet et al.
In this phase function, qb is the azimuthal angle.
If the radiation field is considered as isotropic, the phase function is equal to
1 and Eq. (90) reduces to
1
dlz
1
+ 1
- g(Ea +Es)Cx dz = Iz-~o_~
I~dg'
(92)
3.4.3.1 P-N Methods
In such an approximation, Eq. (92) is reduced with a finite set of moment
equations. The intensity I is expanded in an orthogonal series of spherical
harmonics truncated after a finite number of terms N [37].
3.4.3.2 The Discrete Ordinate Method
This is an extension of the two-flux method of Schuster discussed above. The
general equation at Eq. (92) is represented by a discrete set of equations for the
average intensity over a finite number of ordinate directions. Integrals over solid
angles are replaced by sums over the ordinate directions. For the direction
~ti Eq. (92) becomes [36, 37]
1
dlz(~i)
1
-- gi (Ea + Es)Cx dz = Iz(l-ti) - ~ ~o ,_, ajlz(gj)
(93)
j=l
The aj's coefficients are weighting coefficients, functions of the divisions of the
interval [- 1; + 1]. Gaussian quadrature is usually used for their calculation.
The discrete ordinate method is currently considered as a reference method for
testing accuracy of results obtained with other methods [37, 65].
In these numerical methods, the local available radiant energy 4nJz is obtained by the following double integral:
2x +1
4 J. = I I I.dgd,
(94)
0 -1
The working illuminated length L 2 cannot be obtained from an analytical
expression, but must be numerically calculated from the value of the compensation point (1 W.m-2).
3.4.4 Radiative Transfer Theory for Three-Dimensional Applications:
General Case
The general equation of radiative transfer in terms of specific intensity may be
used on a finite medium such as a photobioreactor. In the general case,
a one-dimensional approximation is inappropriate and the equation must be
J.-F. Cornet et al.
In this phase function, qb is the azimuthal angle.
If the radiation field is considered as isotropic, the phase function is equal to
1 and Eq. (90) reduces to
1
dlz
1
+ 1
- g(Ea +Es)Cx dz = Iz-~o_~
I~dg'
(92)
3.4.3.1 P-N Methods
In such an approximation, Eq. (92) is reduced with a finite set of moment
equations. The intensity I is expanded in an orthogonal series of spherical
harmonics truncated after a finite number of terms N [37].
3.4.3.2 The Discrete Ordinate Method
This is an extension of the two-flux method of Schuster discussed above. The
general equation at Eq. (92) is represented by a discrete set of equations for the
average intensity over a finite number of ordinate directions. Integrals over solid
angles are replaced by sums over the ordinate directions. For the direction
~ti Eq. (92) becomes [36, 37]
1
dlz(~i)
1
-- gi (Ea + Es)Cx dz = Iz(l-ti) - ~ ~o ,_, ajlz(gj)
(93)
j=l
The aj's coefficients are weighting coefficients, functions of the divisions of the
interval [- 1; + 1]. Gaussian quadrature is usually used for their calculation.
The discrete ordinate method is currently considered as a reference method for
testing accuracy of results obtained with other methods [37, 65].
In these numerical methods, the local available radiant energy 4nJz is obtained by the following double integral:
2x +1
4 J. = I I I.dgd,
(94)
0 -1
The working illuminated length L 2 cannot be obtained from an analytical
expression, but must be numerically calculated from the value of the compensation point (1 W.m-2).
3.4.4 Radiative Transfer Theory for Three-Dimensional Applications:
General Case
The general equation of radiative transfer in terms of specific intensity may be
used on a finite medium such as a photobioreactor. In the general case,
a one-dimensional approximation is inappropriate and the equation must be