Kinetics and Energetics of Photosynthetic Micro-Organisms in Photobioreactors
181
geometries in different systems of coordinates, that has the advantage of offering
analytical solutions. In this paper, the simple two-flux method will be recalled
and other more sophisticated methods will be briefly presented and discussed in
cases where information on incident radiation field and optical characteristics of
the medium is abundantly available.
The radiant energy flux in a non-emitting medium containing cells is given
by the integral-differential equation of radiative transfer in its integrated form
over the entire spectrum and all the directions from the specific intensity, as
follows [36, 37]:
1
-
~ (u.Vl)do) = - V.F = (Ea + Es)Cx ~ Idc0 - EsCx4~
4~
4n
x f ~ p(O, O', qb, qb')Idco' deo
(57)
4n4-n
where Ea and Es are the mean absorption and scattering mass coefficients on the
visible spectrum respectively, co is a solid angle and p(0, 0', qb, qb') the phase
function characteristic of the medium. This phase function is also called scattering diagram, because it describes how the radiation is scattered from the
incident direction.
By calculating the integral terms in Eq. (57) the light energy balance on
a differential volume is established [37, 48] giving the local volumetric rate of
radiant energy absorbed by the cells at each point in the medium ~r
-- V.F = EaCx4nJ = d
(58)
Equation (58) is the general form of Eq. (49) used for the establishment of
Lambert's law in unidirectional applications. It confirms the above considerations concerning the radiative transfer definitions in demonstrating that the
available radiant energy 4nJ multiplied by a constant factor EaCx is equal to the
volumetric rate of radiant energy absorbed d. Introducing the mean volumetric
integral, the volumetric rate of radiant energy absorbed in the total volume is
defined as
1 !41zJ dV
(d) = EaCx V
(59)
The problem lies in the necessity to calculate local 4n J-profiles from integration of the local specific intensity I over the solid angle co (Eq. 57), because
Eq. (58) cannot be directly solved in most cases.
One-dimensional geometries simplify the problem by reducing the knowledge required to the projection of the radiant energy flux vector along one axis
only. If the operating conditions validate this assumption, Eq. (57) may be
simplified and easily solved in different systems of coordinates. This requires
accurate knowledge of the light transfer parameters (absorption and scattering
coefficients, phase function) and the boundary conditions (incident light flux) for
the considered medium.
181
geometries in different systems of coordinates, that has the advantage of offering
analytical solutions. In this paper, the simple two-flux method will be recalled
and other more sophisticated methods will be briefly presented and discussed in
cases where information on incident radiation field and optical characteristics of
the medium is abundantly available.
The radiant energy flux in a non-emitting medium containing cells is given
by the integral-differential equation of radiative transfer in its integrated form
over the entire spectrum and all the directions from the specific intensity, as
follows [36, 37]:
1
-
~ (u.Vl)do) = - V.F = (Ea + Es)Cx ~ Idc0 - EsCx4~
4~
4n
x f ~ p(O, O', qb, qb')Idco' deo
(57)
4n4-n
where Ea and Es are the mean absorption and scattering mass coefficients on the
visible spectrum respectively, co is a solid angle and p(0, 0', qb, qb') the phase
function characteristic of the medium. This phase function is also called scattering diagram, because it describes how the radiation is scattered from the
incident direction.
By calculating the integral terms in Eq. (57) the light energy balance on
a differential volume is established [37, 48] giving the local volumetric rate of
radiant energy absorbed by the cells at each point in the medium ~r
-- V.F = EaCx4nJ = d
(58)
Equation (58) is the general form of Eq. (49) used for the establishment of
Lambert's law in unidirectional applications. It confirms the above considerations concerning the radiative transfer definitions in demonstrating that the
available radiant energy 4nJ multiplied by a constant factor EaCx is equal to the
volumetric rate of radiant energy absorbed d. Introducing the mean volumetric
integral, the volumetric rate of radiant energy absorbed in the total volume is
defined as
1 !41zJ dV
(d) = EaCx V
(59)
The problem lies in the necessity to calculate local 4n J-profiles from integration of the local specific intensity I over the solid angle co (Eq. 57), because
Eq. (58) cannot be directly solved in most cases.
One-dimensional geometries simplify the problem by reducing the knowledge required to the projection of the radiant energy flux vector along one axis
only. If the operating conditions validate this assumption, Eq. (57) may be
simplified and easily solved in different systems of coordinates. This requires
accurate knowledge of the light transfer parameters (absorption and scattering
coefficients, phase function) and the boundary conditions (incident light flux) for
the considered medium.