176
J.-F. Cornet et al.
In Eq. (39), 0 is the angle from the normal n of a surface reference dS to the
direction of Ix. Since O is in the integral, the flux Fx depends on the direction of n;
hence Fx is a vector.
For the entire spectrum, the following expressions for specific intensity I,
mean specific intensity J, and radiant energy flux F are
I, J, F = ~ Ix, Jx, Fxdk
(41)
0
The integrated variables Jx(J) or Fx(F) may be used to describe the availability of light in the medium; but given the vectorial nature of the radiative flux F,
which does not represent the microorganism's light energy environment, it is
preferable to work with the mean specific intensity Jx. More exactly, the total
available radiant energy at each point of the medium is described by the scalar
quantity 4=Jx (or 4rcJ over the entire spectrum). This is justified a posteriori by
the calculation of the local volumetric rate of radiant energy absorbed by the
cells ~4x or d at each point in the medium in which this quantity appears (this
calculation will be presented in Sects. 3.3 and 3.4).
Moreover, if a bounded volume V is defined, such as a photoreactor or an
open pond, the mean volumetric rate of radiant energy absorbed in this volume
for the wavelength k or the entire spectrum must be defined as follows:
1
(42)
This mean volumetric rate in radiant energy absorbed is crucial for the characterization of absorbing and scattering media, as it represents the radiative
energy exchanged from the photonic phase to the material phase in the medium
[38]. Also, this rate appears as a macroscopic quantity that must be experimentally determined on a given volume to validate theoretical calculations for
local available radiant energy profiles [39].
However, <~4) is a macroscopic term and therefore unsuitable for modeling
coupling between kinetics and radiative transfer, which is typically a
local problem because the quantum yield of the reaction on the local available
light energy and parameters of the model have to be identified from local
kinetic laws and volumetric integration over the working illuminated volume.
This is an important special feature of photobioreactors compared to
chemical photoreactors. Hence the use of must be restricted to energy
aspects [38].
3.2.2 Kinetic Laws for Photosynthesis and Photoinhibition:
The kinetic laws for photosynthesis saturation curves vs available radiant light
energy are well-known and have been thoroughly discussed [40, 41]. The most
often encountered law in the literature is the Monod-type law [42-46] in which
J.-F. Cornet et al.
In Eq. (39), 0 is the angle from the normal n of a surface reference dS to the
direction of Ix. Since O is in the integral, the flux Fx depends on the direction of n;
hence Fx is a vector.
For the entire spectrum, the following expressions for specific intensity I,
mean specific intensity J, and radiant energy flux F are
I, J, F = ~ Ix, Jx, Fxdk
(41)
0
The integrated variables Jx(J) or Fx(F) may be used to describe the availability of light in the medium; but given the vectorial nature of the radiative flux F,
which does not represent the microorganism's light energy environment, it is
preferable to work with the mean specific intensity Jx. More exactly, the total
available radiant energy at each point of the medium is described by the scalar
quantity 4=Jx (or 4rcJ over the entire spectrum). This is justified a posteriori by
the calculation of the local volumetric rate of radiant energy absorbed by the
cells ~4x or d at each point in the medium in which this quantity appears (this
calculation will be presented in Sects. 3.3 and 3.4).
Moreover, if a bounded volume V is defined, such as a photoreactor or an
open pond, the mean volumetric rate of radiant energy absorbed in this volume
for the wavelength k or the entire spectrum must be defined as follows:
1
(42)
This mean volumetric rate in radiant energy absorbed is crucial for the characterization of absorbing and scattering media, as it represents the radiative
energy exchanged from the photonic phase to the material phase in the medium
[38]. Also, this rate appears as a macroscopic quantity that must be experimentally determined on a given volume to validate theoretical calculations for
local available radiant energy profiles [39].
However, <~4) is a macroscopic term and therefore unsuitable for modeling
coupling between kinetics and radiative transfer, which is typically a
local problem because the quantum yield of the reaction on the local available
light energy and parameters of the model have to be identified from local
kinetic laws and volumetric integration over the working illuminated volume.
This is an important special feature of photobioreactors compared to
chemical photoreactors. Hence the use of
aspects [38].
3.2.2 Kinetic Laws for Photosynthesis and Photoinhibition:
The kinetic laws for photosynthesis saturation curves vs available radiant light
energy are well-known and have been thoroughly discussed [40, 41]. The most
often encountered law in the literature is the Monod-type law [42-46] in which