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Chapter 7
where the specific rate vector (r3)consump,ion contains the
N
specific substrate uptake rates, rp the
M
specific product formation rates, and rx the net, specific formation rates of the Q biomass
constituents.
7.2.3 Dynamic Mass Balances
In Section 3.1 we specified mass balances at steady state conditions for substrates, metabolic
products and biomass - equations (3.1)-(3.3). In these mass balances the specific rates derived
above can be inserted and hereby steady state conditions can be calculated. In many cases
fermentation processes are, however, operated at non-steady-state conditions, i.e., at dynamic
growth conditions. A t these conditions the equations are based on the general mass balance,
which for the case of a substrate takes the form:
= r(c)xV + vfttdcf -
v„, c
(7.10)
c is the vector of medium concentrations of
N
substrates,
M
metabolic products and
Q
biomass
components. vfeed is the volumetric flow rate to the bioreactor, veilit is the volumetric flow rate out
of the reactor. Different parts of the production rate vector r are taken from (7.7)-(7.9). In
analogy with the steady state balances presented in Section 3.1 transfer of substrates and
metabolic products from the gas to the liquid phase can if desired be included in the dynamic
mass balance (7.10).
The mass balances for the biomass and the
Q
biomass constituents
X,
that together make up the
biomass require special attention. The specific rate of formation of biomass is the sum of
contributions from each of the /reactions as specified in eq. (7.6), which also can be written as:
^ = ^ = E r rv
(7.11)
1=1
where n is the ith column of r . Consequently, with a sterile feed into a bioreactor the mass balance
for biomass is:
=
r, x v ~
x = qxV - vexi! x
(7.12)
at
The concentrations of biomass constituents in the cell are normally expressed in units of g (g DW)'1
and consequently the sum of all the concentrations equals 1. Furthermore, a mass balance for
component
i
in a cell of mass
m
is:
d(m X t)
= r rvm
dt
(7.13)