Bioprocess Engineering of Phototrophic Marine Organisms 9.3 Basic Elements of Photobioreactor Design and Operation 269
Part B | 9.3
Simplification of (9.16) and combination with (9.12)
yields
dC x
dt
D C x D
0
C N
K N C C N
C x ;
(9.17)
where C x is the cell density in the liquid suspension culture (g cells L
1 culture), V is the total culture volume,
and /describes the dependence of specific growth rate
on light intensity as presented earlier in (9.11). Equation (9.17) is subject to an initial condition C x D C x;i at
t D 0, the beginning of the cultivation cycle. For now,
(9.17) is left in differential form because also depends
on the delivery of light and CO 2 to the photobioreactor. Detailed methods on how to estimate growth rate
in light-limited or CO 2 -limited growth are described in
Sects. 9.4.2 and 9.4.3.
In a batch bioreactor, the cell mass will not grow
forever because at some point the limiting nutrient will
be exhausted (C N D 0). The cell density at nutrient depletion is determined by an integral material balance on
the cell mass, which is described in words as
Â
final cell mass
at end of cultivation
Ã
Â
initial cell mass
at beginning of cultivation
Ã
Â
amount of cell mass produced
by limiting nutrient consumption
Ã
D 0 :
(9.18)
Mathematically, (9.18) is expressed as
C x V C x;i V .C N;i C N /V Y X=N D 0 ;
(9.19)
where C N;i is the initial concentration of the limiting nutrient (mol N L
1 ), and C x;i is the initial
biomass concentration (g cells L
1 ), and Y X=N is the
biomass yield coefficient for the limiting nutrient (g
cell produced mol
1 N consumed). At nutrient depletion (C N D 0), (9.19) reduces to
C x;f D C x;i C C N;i Y X=N ;
(9.20)
where C x;f is the final cell density at C N D 0.
Well-Mixed Continuous Cultures:
Steady-State Material Balances
In a continuous bioreactor, the biomass suspension is
continuously removed from the culture vessel. Fresh
liquid medium enters the culture vessel at a constant
volumetric flow rate equal to the volumetric flow rate of
the suspension culture exiting the vessel so that the total
liquid volume within the vessel is constant. The aeration gas and light are also continuously delivered to the
culture vessel. Continuous bioreactors are designed to
operate at steady state, meaning that the rate of biomass
production within the vessel equals the rate of biomass
removal from the vessel. In steady-state operation, the
cell density in the vessel does not change with time, and
there is no accumulation of cell mass within the culture
vessel. If the liquid suspension culture is well mixed
within the vessel, then the cell density inside the vessel equals the cell density of liquid suspension culture
exiting the vessel.
Material balances on cell biomass and the dissolved
limiting nutrient describe the rates and amounts of
biomass production within the continuous bioreactor.
Again, the cell suspension within the culture vessel is
well mixed and the total culture volume is constant.
Using (9.15) as a template, the mathematically stated
material balance on the cell mass in the continuous
bioreactor is given by
C x;o v o C x v C
0 C x C N V
K N C C N
D 0 :
(9.21)
In a similar manner, the material balance on the limiting
nutrient is
C N;o v o C N v C
0 C x C N V
.K N C C N / Y X=N
D 0 :
(9.22)
In continuous culture, D is the dilution rate (h
1 ), defined as
D D
v o
V
:
(9.23)
Usually, there are no cells in the liquid inlet (C x;o D 0)
and the volumetric flow rates of the inlet and outlet are
equal (v o D v). In this case, (9.21) and (9.22) are rearranged to yield the following:
D D D
K N
K N C C N
0
;
(9.24)
C x D Y X=N
Â
C N;o
DK N
0 D
Ã
;
(9.25)
C N D
DK N
0 D
:
(9.26)
Note that (9.25) and (9.26) depend on the dilution rate
D. However, D cannot exceed
0 . If D exceeds
0 then
Part B | 9.3
Simplification of (9.16) and combination with (9.12)
yields
dC x
dt
D C x D
0
C N
K N C C N
C x ;
(9.17)
where C x is the cell density in the liquid suspension culture (g cells L
1 culture), V is the total culture volume,
and /describes the dependence of specific growth rate
on light intensity as presented earlier in (9.11). Equation (9.17) is subject to an initial condition C x D C x;i at
t D 0, the beginning of the cultivation cycle. For now,
(9.17) is left in differential form because also depends
on the delivery of light and CO 2 to the photobioreactor. Detailed methods on how to estimate growth rate
in light-limited or CO 2 -limited growth are described in
Sects. 9.4.2 and 9.4.3.
In a batch bioreactor, the cell mass will not grow
forever because at some point the limiting nutrient will
be exhausted (C N D 0). The cell density at nutrient depletion is determined by an integral material balance on
the cell mass, which is described in words as
Â
final cell mass
at end of cultivation
Ã
Â
initial cell mass
at beginning of cultivation
Ã
Â
amount of cell mass produced
by limiting nutrient consumption
Ã
D 0 :
(9.18)
Mathematically, (9.18) is expressed as
C x V C x;i V .C N;i C N /V Y X=N D 0 ;
(9.19)
where C N;i is the initial concentration of the limiting nutrient (mol N L
1 ), and C x;i is the initial
biomass concentration (g cells L
1 ), and Y X=N is the
biomass yield coefficient for the limiting nutrient (g
cell produced mol
1 N consumed). At nutrient depletion (C N D 0), (9.19) reduces to
C x;f D C x;i C C N;i Y X=N ;
(9.20)
where C x;f is the final cell density at C N D 0.
Well-Mixed Continuous Cultures:
Steady-State Material Balances
In a continuous bioreactor, the biomass suspension is
continuously removed from the culture vessel. Fresh
liquid medium enters the culture vessel at a constant
volumetric flow rate equal to the volumetric flow rate of
the suspension culture exiting the vessel so that the total
liquid volume within the vessel is constant. The aeration gas and light are also continuously delivered to the
culture vessel. Continuous bioreactors are designed to
operate at steady state, meaning that the rate of biomass
production within the vessel equals the rate of biomass
removal from the vessel. In steady-state operation, the
cell density in the vessel does not change with time, and
there is no accumulation of cell mass within the culture
vessel. If the liquid suspension culture is well mixed
within the vessel, then the cell density inside the vessel equals the cell density of liquid suspension culture
exiting the vessel.
Material balances on cell biomass and the dissolved
limiting nutrient describe the rates and amounts of
biomass production within the continuous bioreactor.
Again, the cell suspension within the culture vessel is
well mixed and the total culture volume is constant.
Using (9.15) as a template, the mathematically stated
material balance on the cell mass in the continuous
bioreactor is given by
C x;o v o C x v C
0 C x C N V
K N C C N
D 0 :
(9.21)
In a similar manner, the material balance on the limiting
nutrient is
C N;o v o C N v C
0 C x C N V
.K N C C N / Y X=N
D 0 :
(9.22)
In continuous culture, D is the dilution rate (h
1 ), defined as
D D
v o
V
:
(9.23)
Usually, there are no cells in the liquid inlet (C x;o D 0)
and the volumetric flow rates of the inlet and outlet are
equal (v o D v). In this case, (9.21) and (9.22) are rearranged to yield the following:
D D D
K N
K N C C N
0
;
(9.24)
C x D Y X=N
Â
C N;o
DK N
0 D
Ã
;
(9.25)
C N D
DK N
0 D
:
(9.26)
Note that (9.25) and (9.26) depend on the dilution rate
D. However, D cannot exceed
0 . If D exceeds
0 then