Kinetics and Energetics of Photosynthetic Micro-Organisms in Photobioreactors
209
This system of equations is a simplified reaction model compared with the
complete system written by Curless [85] and Erickson et al. [95], but these
approximations are justified by many authors. The equation providing the
accumulation rate for the bicarbonate gives the HCO~ concentration time
course in the medium with growth of S. platensis. If only light limitation is
present, the mass volumetric rate (rxx) is given by Eqs. (47) and (48); however, if
a mineral limitation by HCO~ occurs, the model requires an additional kinetic
equation to obtain (rx@.
If the carbon dioxide/bicarbonate (Eq. 116) and bicarbonate/carbonate
(Eq. 118) reactions are considered to be instantaneous and equilibrated, the
system of equations at Eq. (123) can be simplified to a system of only one
differential equation on the CO2 transfer and two algebraic equations (Eqs. 117
and 119) giving the bicarbonate and carbonate concentrations at thermodynamic equilibrium [7].
To complete this kinetic model, the gas balance equation is required:
P ~Yco~
EkLa 1 - ~ ,,
C'
~(Gyco~)
-
(Cco2
-
co2)
(125)
RT St
e
~V~
where Yco2 is the molar fraction of CO/and G is the total molar flow rate. If the
difference in molar fraction between the input and the output of the reactor is
low, the gas phase can be considered as perfectly mixed so that ~(Gyco)/~VG
YcEo:G E -- Yco~G. Moreover, as previously discussed in Sect. 2, the accumulation term in Eq. (125) may often be neglected.
The system of equations at Eqs. (123) and (125) must be simplified for
different operating conditions:
for a continuous culture, the system of equations at Eq. (123) has to be
solved entirely, except if a physical limitation by CO2 transfer rate occurs or if
a steady state is reached. In these cases, the accumulation terms in d/dt vanish;
for a batch culture, the dilution rate D is equal to zero. In addition, for
limitation by the CO2 transfer rate, the terms in d/dt also vanish.
In other general cases, experiments showed that the time constant of such
processes was very high [93] (three months at pH 10), and depended mainly on
pH and the ratio G/V.
5.4 Effects on Spirulina Growth
As discussed above, the Spirulina growth rate is limited in two ways as regards
the carbon source:
1 - the culture may be limited by the physical CO2 transfer rate between the
gas and liquid phases. In this case, the mass volumetric growth rate is fixed by
the technical characteristics of the apparatus and is equal to
MHCO~
(rxv) = rxx -
kLa Cc*o2
(126)
YC/XT
209
This system of equations is a simplified reaction model compared with the
complete system written by Curless [85] and Erickson et al. [95], but these
approximations are justified by many authors. The equation providing the
accumulation rate for the bicarbonate gives the HCO~ concentration time
course in the medium with growth of S. platensis. If only light limitation is
present, the mass volumetric rate (rxx) is given by Eqs. (47) and (48); however, if
a mineral limitation by HCO~ occurs, the model requires an additional kinetic
equation to obtain (rx@.
If the carbon dioxide/bicarbonate (Eq. 116) and bicarbonate/carbonate
(Eq. 118) reactions are considered to be instantaneous and equilibrated, the
system of equations at Eq. (123) can be simplified to a system of only one
differential equation on the CO2 transfer and two algebraic equations (Eqs. 117
and 119) giving the bicarbonate and carbonate concentrations at thermodynamic equilibrium [7].
To complete this kinetic model, the gas balance equation is required:
P ~Yco~
EkLa 1 - ~ ,,
C'
~(Gyco~)
-
(Cco2
-
co2)
(125)
RT St
e
~V~
where Yco2 is the molar fraction of CO/and G is the total molar flow rate. If the
difference in molar fraction between the input and the output of the reactor is
low, the gas phase can be considered as perfectly mixed so that ~(Gyco)/~VG
YcEo:G E -- Yco~G. Moreover, as previously discussed in Sect. 2, the accumulation term in Eq. (125) may often be neglected.
The system of equations at Eqs. (123) and (125) must be simplified for
different operating conditions:
for a continuous culture, the system of equations at Eq. (123) has to be
solved entirely, except if a physical limitation by CO2 transfer rate occurs or if
a steady state is reached. In these cases, the accumulation terms in d/dt vanish;
for a batch culture, the dilution rate D is equal to zero. In addition, for
limitation by the CO2 transfer rate, the terms in d/dt also vanish.
In other general cases, experiments showed that the time constant of such
processes was very high [93] (three months at pH 10), and depended mainly on
pH and the ratio G/V.
5.4 Effects on Spirulina Growth
As discussed above, the Spirulina growth rate is limited in two ways as regards
the carbon source:
1 - the culture may be limited by the physical CO2 transfer rate between the
gas and liquid phases. In this case, the mass volumetric growth rate is fixed by
the technical characteristics of the apparatus and is equal to
MHCO~
(rxv) = rxx -
kLa Cc*o2
(126)
YC/XT