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Derivation of mass balance equations
The liquid phase mass balance for the system is given as
dM
  F F  Fevap  M O2  M CO2 [kg/h]
dt
where M [kg] is the total weight of the fermentation broth, F is the feed density [kg/L], F is
the substrate feed rate [L/h], Fevap is the water evaporation rate [kg/h], M O2 is the oxygen
transferred to the liquid from the gas phase [kg/h] and M CO2 is the carbon dioxide transferred
from the liquid to the gas phase [kg/h]. The transfer of oxygen is assumed equal to the
consumption of oxygen, and can be described in the following way assuming that all ingoing
carbon in the feed is converted in the fermentor:
M O2  YSO  F  cF [kg/h]
where cF is the substrate concentration in the feed [g substrate/L]. Similarly the produced
carbon dioxide is transferred from the liquid to the gas phase:
M CO2  YSC  F  cF [kg/h]
The mass balance of the broth volume now gives:
dM
dVL
 dt [L/h]
dt

The rate of change of the amount of biomass in the fermentor operated in fed-batch mode is
equal to the specific growth rate, μ [h-1], multiplied with the biomass concentration in the
fermentor, X [g DW/L] and the broth volume, (Nielsen et al. 2003) and with no in- or outflow
of biomass the following equation is derived:
[g/h]
By applying the chain rule of differentiation, and by combining with the mass balance for the
liquid phase, the resulting mass balance for biomass is
X  (  F F  Fevap  M O2  M CO2 )
dX
 X 
[g DW/L/h]
dt
VL  
The last term of this equation is a dilution term that will appear in a similar way in the
following balances.
The substrate concentration can be described by:
S  (  F F  Fevap  M O2  M CO2 )
dS cF  F

     xstrue  ms   X 
[g substrate/L/h]
dt
VL
VL  
where S is the substrate concentration in the broth [g substrate/L],  xstrue is the “true” biomass
yield on substrate [g substrate/g DW] and ms is the maintenance coefficient [g substrate/g
DW/h]. Substrate is added in the feed with a concentration cF and consumed by uptake rates
  xstrue [g DW/g substrate/h] and ms. Uptake of substrate for product formation is included in
the biomass growth as product formation is proportional to biomass growth.
The product concentration can be described by:
P  (  F F  Fevap  M O2  M CO2 )
dP
   YSP  YXS  X 
[g product/L/h]
dt
VL  
The product mass balance is similar to the balance presented for biomass concentration. The
growth related product formation is described by the observed yield coefficients YSP and YXS.
The dissolved oxygen concentration in the liquid is expressed as:
DO  (  F F  Fevap  M O2  M CO2 )
dDO
true
 mo  X 
 k L a  DO*  DO      xo
[moles O2/L/h]
dt
VL  


where DO is the concentration of dissolved oxygen [moles O2/L]. The mass transfer of
oxygen from gas to liquid is the only positive term in the oxygen balance. Two negative terms
add up to equal the oxygen uptake rate of the fungi; the oxygen uptake for growth and product
formation and the oxygen uptake for maintenance.