Part B | 9.4
284 Part B Tools and Methods in Marine Biotechnology
actor configuration is assessed by performing a material
balance on CO 2 dissolved in the liquid culture. The CO 2
material balance is
Â
rate of CO 2
added to culture
Ã
Â
rate of CO 2
removed from culture
Ã
C
Â
rate of generation of
CO 2 by culture
Ã
D
Â
rate of accumulation of
CO 2 within culture
Ã
:
(9.43)
Liquid-phase CO 2 material balances are developed for
batch and continuous photobioreactors. The batch and
continuous photobioreactors are well mixed and continuously aerated. Although the cell density (C x ) increases
with time in batch culture, CO 2 consumption and mass
transfer rates are at a nominal steady state for a given
value of C x . At these conditions, the CO 2 material balance for the culture in the batch photobioreactor is
k L A i .C A
C A /
C x V
Y X=CO2
D 0 ;
(9.44)
where A i is the gas/liquid interfacial surface area of
all the bubbles, and k L is the liquid phase mass transfer coefficient for dissolved CO 2 flux across the liquid
film surrounding the bubble (m s
1 ). The first term in
(9.44) represents the interphase mass transfer flux of
CO 2 from the gas to the liquid phase. Note that C A
is dependent on the gas phase CO 2 partial pressure P A
by (9.41). The second term in (9.44) represents the consumption rate of CO 2 by the culture. Note the sign is
negative to denote consumption rather than generation.
Similarly, the CO 2 material balance on the continuous
photobioreactor is
v o C A;o C k L A i .C A
C A / v o C A
C x V
Y X=CO2
D 0 :
(9.45)
Usually, the v o C A;o term is small in magnitude with respect to the other terms and is neglected.
At the point of CO 2 limitation, the CO 2 that is delivered to the liquid culture is immediately consumed
by the cells, and so C A will be near zero, but not at zero.
In this case, equations (9.44) and (9.45) both reduce to
CO 2 TR D k L a C A
C x
Y X=CO2
;
(9.46)
where a is the gas–liquid interfacial area per unit volume of culture A i =V (m
2 m
3 ). Therefore, to avoid
CO 2 -limited growth, the CO 2 -TR must always be
higher than the CO 2 demand. If the growth is CO 2 -
limited, then the biomass productivity C x is set by the
CO 2 -TR.
The CO 2 demand given by (9.42) is a function of
both specific growth rate and cell density C x . Recall from (9.36) that light attenuation lowers as C x
increases and so
0 is also implicitly a function of C x .
At a fixed aeration rate and CO 2 partial pressure in the
aeration gas, as the cell density increases, the process
moves closer to CO 2 limitation. Therefore, the critical cell density (C x;c ) at which CO 2 limitation occurs
in a well-mixed, continuously aerated photobioreactor
is found by combination and rearrangement of equations (9.46) and (9.36) to yield
C x;c D
k L aC A
Y X=CO2
max I m .C x;c /=.I k C I m .C x;c //
;
(9.47)
where C N K N . Since I m is a function of C x , (9.47)
must be solved implicitly for C x;c using the appropriate
relationship for I m , e.g., (9.33).
Effects of Aeration on CO 2 Transfer Rate
Equation (9.46) contains a new parameter, a, which is
defined as
a D
A i
V
D
gas–liquid interfacial area .m
2
/
volume of culture.m 3 /
: (9.48)
The parameter a is difficult to determine independently.
Therefore, a and k L are usually lumped together into
a single parameter called the volumetric mass transfer
coefficient k L a, which has units of reciprocal time (e.g.,
h
1
/. Generally, k L a increases with increasing aeration
rate and decreasing bubble size. Many correlations exist for estimation of k L a in aerated systems, including
enclosed algal photobioreactors [9.41–44]. The k L a for
CO 2 transfer can be readily scaled from k L a for O 2
transfer. Specifically, k L a for CO 2 is related to the k L a
for O 2 using penetration theory for mass transfer, for
example,
.k L a/ CO2 D .k L a/ O2
 D CO 2
D O2
à 1
2 ;
(9.49)
where D CO2 and D O2 are the liquid-phase diffusion
coefficients for dissolved CO 2 and O 2 dissolved in seawater, respectively. The diffusivity ratio D CO2 =D O2 is
284 Part B Tools and Methods in Marine Biotechnology
actor configuration is assessed by performing a material
balance on CO 2 dissolved in the liquid culture. The CO 2
material balance is
Â
rate of CO 2
added to culture
Ã
Â
rate of CO 2
removed from culture
Ã
C
Â
rate of generation of
CO 2 by culture
Ã
D
Â
rate of accumulation of
CO 2 within culture
Ã
:
(9.43)
Liquid-phase CO 2 material balances are developed for
batch and continuous photobioreactors. The batch and
continuous photobioreactors are well mixed and continuously aerated. Although the cell density (C x ) increases
with time in batch culture, CO 2 consumption and mass
transfer rates are at a nominal steady state for a given
value of C x . At these conditions, the CO 2 material balance for the culture in the batch photobioreactor is
k L A i .C A
C A /
C x V
Y X=CO2
D 0 ;
(9.44)
where A i is the gas/liquid interfacial surface area of
all the bubbles, and k L is the liquid phase mass transfer coefficient for dissolved CO 2 flux across the liquid
film surrounding the bubble (m s
1 ). The first term in
(9.44) represents the interphase mass transfer flux of
CO 2 from the gas to the liquid phase. Note that C A
is dependent on the gas phase CO 2 partial pressure P A
by (9.41). The second term in (9.44) represents the consumption rate of CO 2 by the culture. Note the sign is
negative to denote consumption rather than generation.
Similarly, the CO 2 material balance on the continuous
photobioreactor is
v o C A;o C k L A i .C A
C A / v o C A
C x V
Y X=CO2
D 0 :
(9.45)
Usually, the v o C A;o term is small in magnitude with respect to the other terms and is neglected.
At the point of CO 2 limitation, the CO 2 that is delivered to the liquid culture is immediately consumed
by the cells, and so C A will be near zero, but not at zero.
In this case, equations (9.44) and (9.45) both reduce to
CO 2 TR D k L a C A
C x
Y X=CO2
;
(9.46)
where a is the gas–liquid interfacial area per unit volume of culture A i =V (m
2 m
3 ). Therefore, to avoid
CO 2 -limited growth, the CO 2 -TR must always be
higher than the CO 2 demand. If the growth is CO 2 -
limited, then the biomass productivity C x is set by the
CO 2 -TR.
The CO 2 demand given by (9.42) is a function of
both specific growth rate and cell density C x . Recall from (9.36) that light attenuation lowers as C x
increases and so
0 is also implicitly a function of C x .
At a fixed aeration rate and CO 2 partial pressure in the
aeration gas, as the cell density increases, the process
moves closer to CO 2 limitation. Therefore, the critical cell density (C x;c ) at which CO 2 limitation occurs
in a well-mixed, continuously aerated photobioreactor
is found by combination and rearrangement of equations (9.46) and (9.36) to yield
C x;c D
k L aC A
Y X=CO2
max I m .C x;c /=.I k C I m .C x;c //
;
(9.47)
where C N K N . Since I m is a function of C x , (9.47)
must be solved implicitly for C x;c using the appropriate
relationship for I m , e.g., (9.33).
Effects of Aeration on CO 2 Transfer Rate
Equation (9.46) contains a new parameter, a, which is
defined as
a D
A i
V
D
gas–liquid interfacial area .m
2
/
volume of culture.m 3 /
: (9.48)
The parameter a is difficult to determine independently.
Therefore, a and k L are usually lumped together into
a single parameter called the volumetric mass transfer
coefficient k L a, which has units of reciprocal time (e.g.,
h
1
/. Generally, k L a increases with increasing aeration
rate and decreasing bubble size. Many correlations exist for estimation of k L a in aerated systems, including
enclosed algal photobioreactors [9.41–44]. The k L a for
CO 2 transfer can be readily scaled from k L a for O 2
transfer. Specifically, k L a for CO 2 is related to the k L a
for O 2 using penetration theory for mass transfer, for
example,
.k L a/ CO2 D .k L a/ O2
 D CO 2
D O2
à 1
2 ;
(9.49)
where D CO2 and D O2 are the liquid-phase diffusion
coefficients for dissolved CO 2 and O 2 dissolved in seawater, respectively. The diffusivity ratio D CO2 =D O2 is