Hugo Cloudt Advanced Bioreactor Design (BPE-36306)
Summary lectures & reader Advanced Bioreactor Design (BPE-36306)
Lecture 1 - Reactors with cell retention
Cell retention: keeping cells in a bioreactor by using a to the bioreactor coupled separator (e.g.
sedimentation tank, filtration unit) which divides the outgoing flow of the bioreactor in two flows:
1. Flow coming out at the bottom of the separator (𝐹𝐵 + 𝐹𝑅 ) from which a part is recycled back into
the bioreactor (𝐹𝑅 ) and from which a part leaves the system (𝐹𝐵 ).
2. Flow coming out at the top of the separator which leaves the system (𝐹𝑇 ).
Schematic drawing of a bioreactor with cell retention:
𝜇
Cell retention in a bioreactor -> increases 𝑐𝑋
Bioreactors with cell retention are usually used when getting a high 𝑐𝑋 is not possible because of a
low concentration of carbon source/energy source/electron donor (e.g. sugar, low 𝑐𝑆 ) in the feed
stream of the bioreactor (e.g. often the case in bioreactors used for wastewater treatment). The low
concentration of carbon source/energy source/electron donor (e.g. sugar, low 𝑐𝑆 ) in the feed stream
of the bioreactor then prevents getting a high 𝑐𝑋 and thus prevents getting much conversion in the
bioreactor, by increasing the 𝑐𝑋 in the bioreactor by using cell retention more conversion can be
achieved.
No details about the separator in a bioreactor with cell retention have to be known, it is just assumed
that it has a known efficiency which is described by the cell retention factor (𝑅𝑋 ). -> definition of the
cell retention factor (𝑅𝑋 ):
𝑐
𝑅𝑋 = 1 − 𝑋𝑇
𝑐𝑋𝐵
𝑅𝑋 = cell retention factor (no unit): for the separator measured factor which describes which fraction
of the cells is retained and therefore end up in the bottom flow leaving the separator.
a. 𝑅𝑋 = 1? -> all cells are retained in the separator, no cells are not retained in the separator, so all
cells leave the separator via the bottom flow.
b. 0 < 𝑅𝑋 < 1? -> NOT all cells are retained in the separator, not all cells leave the separator via the
bottom flow but there are also cells which leave the separator via the top flow.
𝑐𝑋𝑇 = concentration of cells in the top flow leaving the separator (𝐹𝑇 ) (CmolX m-3).
𝑐𝑋𝐵 = concentration of cells in the bottom flow leaving the separator (𝐹𝐵 ) (CmolX m-3).
Mathematical model of a bioreactor with cell retention:
1. Assumptions:
- The separator only retains cells and not solutes.
- There is only cell growth or cell decay in the bioreactor, there is no cell growth or cell decay in the
separator.
- The feed stream is cell-free (𝑐𝑋𝐼𝑁 = 0).
- The bioreactor is a chemostat which is in steady state.
2. Cell balance:
2 sequential transport steps for cells (transport of the cells out of the bioreactor into the separator &
transport of the cells from the separator out of the system). -> 3 possible cell balances from which
,Hugo Cloudt Advanced Bioreactor Design (BPE-36306)
only 2 should be used for solving the model:
a. Cell balance over the reactor:
𝐹𝑅 𝑐𝑋𝐵 − (𝐹𝐿 + 𝐹𝑅 )𝑐𝑋 + 𝜇𝑐𝑋 𝑉𝐿𝑅 = 0
b. Cell balance over the separator:
(𝐹𝐿 + 𝐹𝑅 )𝑐𝑋 − (𝐹𝐵 + 𝐹𝑅 )𝑐𝑋𝐵 − 𝐹𝑇 𝑐𝑋𝑇 = 0
c. Cell balance over the complete system = overall cell balance (sum of the cell balance over the
reactor & the cell balance over the separator):
−𝐹𝑇 𝑐𝑋𝑇 − 𝐹𝐵 𝑐𝑋𝐵 + 𝜇𝑐𝑋 𝑉𝐿𝑅 = 0
3. Balance rate-limiting component:
E.g. overall sugar balance:
𝐹𝐿 𝑐𝑆𝐼𝑁 − 𝐹𝑇 𝑐𝑆 − 𝐹𝐵 𝑐𝑆 + 𝑟𝑆𝜈 𝑉𝐿𝑅 = 0
𝐹𝐿 𝑐𝑆𝐼𝑁 − (𝐹𝑇 + 𝐹𝐵 )𝑐𝑆 − 𝑞𝑆 𝑐𝑋 𝑉𝐿𝑅 = 0
4. Overall/total mass balance:
𝐹𝐿 𝜌𝐿 − 𝐹𝑇 𝜌𝑇 − 𝐹𝐵 𝜌𝐵 = 0
Usually all streams have more or less the same density (𝜌) and thus 𝜌𝐿 ≈ 𝜌𝑇 ≈ 𝜌𝐵 , when that is the
case the overall/total mass balance can thus be simplified:
𝐹𝐿 𝜌𝐿 − 𝐹𝑇 𝜌𝐿 − 𝐹𝐵 𝜌𝐿 = 0
𝐹𝐿 − 𝐹𝑇 − 𝐹𝐵 = 0
Solving the mathematical model of a bioreactor with cell retention for the usual case in which:
- Known variables (because they are chosen) = 𝑉𝐿𝑅 , 𝐹𝐿 , 𝐹𝐵 , 𝐹𝑅 and 𝑅𝑋 .
- Unknown variables (which have to be found) = 𝐹𝑇 , 𝑐𝑋 , 𝑐𝑋𝑇 , 𝑐𝑋𝐵 and 𝜇.
- Rate-limiting component = sugar.
1. Find 𝐹𝑇 from the overall/total mass balance:
𝐹𝐿 − 𝐹𝑇 − 𝐹𝐵 = 0 -> 𝐹𝑇 = 𝐹𝐿 − 𝐹𝐵
2. Find 𝜇 by combining 2 of the cell balances with the definition of R X and subsequently combining
the resulting equations:
- Definition of 𝑅𝑋 :
𝑐𝑋𝑇
𝑅𝑋 = 1 −
𝑐𝑋𝐵 When doing this you use in principle 3
This can be rearranged to the following equation for 𝑐𝑋𝑇 : equations and 4 unknowns (𝜇, 𝑐𝑋 ,
𝑐𝑋𝑇 , 𝑐𝑋𝐵 ), despite this finding the
𝑐𝑋𝑇 = (1 − 𝑅𝑋 )𝑐𝑋𝐵 unknown 𝜇 is still possible because
after combining the equations an
- Cell balances combined with the equation for 𝑐𝑋𝑇 (use 2 of
equation in which the unknown 𝑐𝑋
these):
can be removed is found.
a. Cell balance over the reactor:
𝐹𝑅 𝑐𝑋𝐵 − (𝐹𝐿 + 𝐹𝑅 )𝑐𝑋 + 𝜇𝑐𝑋 𝑉𝐿𝑅 = 0
b. Cell balance over the separator:
(𝐹𝐿 + 𝐹𝑅 )𝑐𝑋 − (𝐹𝐵 + 𝐹𝑅 )𝑐𝑋𝐵 − 𝐹𝑇 (1 − 𝑅𝑋 )𝑐𝑋𝐵 = 0
c. Overall cell balance:
,Hugo Cloudt Advanced Bioreactor Design (BPE-36306)
−𝐹𝑇 (1 − 𝑅𝑋 )𝑐𝑋𝐵 − 𝐹𝐵 𝑐𝑋𝐵 + 𝜇𝑐𝑋 𝑉𝐿𝑅 = 0
So 2 of the cell balances should be combined with the from the definition of 𝑅𝑋 found equation for
𝑐𝑋𝑇 and these cell balances should subsequently be combined to eliminating 𝑐𝑋𝐵 , which results in the
following equation for 𝜇:
𝐹𝐿 + 𝐹𝑅 𝐹𝐿 (1 − 𝑅𝑋 ) + 𝐹𝐵 𝑅𝑋
𝜇=
𝐹𝐿 (1 − 𝑅𝑋 ) + 𝐹𝐵 𝑅𝑋 + 𝐹𝑅 𝑉𝐿𝑅
3. Find 𝑞𝑆 by using Pirt’s law for sugar:
𝜇
𝑞𝑆 = + 𝑚𝑆
𝑌𝑋𝑆
4. Find 𝑐𝑆 by using Monod’s law for sugar:
𝑐𝑆
𝑞𝑆 = 𝑞𝑆𝑀𝐴𝑋
𝐾𝑆 + 𝑐𝑆
𝑞𝑆 (𝐾𝑆 + 𝑐𝑆 ) = 𝑞𝑆𝑀𝐴𝑋 𝑐𝑆
𝑞𝑆 𝐾𝑆 + 𝑞𝑆 𝑐𝑆 = 𝑞𝑆𝑀𝐴𝑋 𝑐𝑆
𝑞𝑆𝑀𝐴𝑋 𝑐𝑆 − 𝑞𝑆 𝑐𝑆 = 𝑞𝑆 𝐾𝑆
𝑐𝑆 (𝑞𝑆𝑀𝐴𝑋 − 𝑞𝑆 ) = 𝑞𝑆 𝐾𝑆
𝑞𝑆 𝐾𝑆 𝑞𝑆
𝑐𝑆 = = 𝐾𝑆
𝑞𝑆𝑀𝐴𝑋 − 𝑞𝑆 𝑞𝑆𝑀𝐴𝑋 − 𝑞𝑆
5. Find 𝑐𝑋 from the overall sugar balance:
𝐹𝐿 𝑐𝑆𝐼𝑁 −(𝐹𝑇 +𝐹𝐵)𝑐𝑆
𝐹𝐿 𝑐𝑆𝐼𝑁 − (𝐹𝑇 + 𝐹𝐵 )𝑐𝑆 − 𝑞𝑆 𝑐𝑋 𝑉𝐿𝑅 = 0 -> 𝑐𝑋 =
𝑞𝑆 𝑉𝐿𝑅
6. Find 𝑐𝑋𝐵 and 𝑐𝑋𝑇 from the cell balances:
- Cell balance over the reactor:
(𝐹𝐿 +𝐹𝑅 )𝑐𝑋 −𝜇𝑐𝑋 𝑉𝐿𝑅 (𝐹𝐿 +𝐹𝑅−𝜇𝑉𝐿𝑅 )𝑐𝑋
𝐹𝑅 𝑐𝑋𝐵 − (𝐹𝐿 + 𝐹𝑅 )𝑐𝑋 + 𝜇𝑐𝑋 𝑉𝐿𝑅 = 0 -> 𝑐𝑋𝐵 = =
𝐹𝑅 𝐹𝑅
- Overall cell balance:
𝜇𝑐𝑋 𝑉𝐿𝑅−𝐹𝐵 𝑐𝑋𝐵
−𝐹𝑇 𝑐𝑋𝑇 − 𝐹𝐵 𝑐𝑋𝐵 + 𝜇𝑐𝑋 𝑉𝐿𝑅 = 0 -> 𝑐𝑋𝑇 =
𝐹𝑇
Cell retention -> increases 𝑐𝑋 , which has an advantage and a disadvantage:
a. Advantage = more cells in the bioreactor, so more conversion in the bioreactor.
b. Disadvantage = more cells in the bioreactor, so a higher viscosity of the broth in the bioreactor,
which leads to a slower breakup of for oxygen supply in the bioreactor introduced gas bubbles, thus
to bigger gas bubbles in the bioreactor, which leads to a lower 𝑘𝑂𝐿 𝑎, so the rate of the oxygen
transfer from the gas into the broth decreases and thus less oxygen is dissolved in the broth and
available for the cells, therefore oxygen can become limiting for the cell growth when 𝑐𝑋 is increased
by cell retention.
It can thus be concluded that in a bioreactor with cell retention 𝑐𝑋 is increased because cells are
recycled back into the bioreactor, but 𝑐𝑋 also decreases somewhat again because the high 𝑐𝑋 leads
to a lower 𝑘𝑂𝐿 𝑎 and thus less oxygen dissolved in the broth and available for the cells.
For taking into account the effect of the cells on 𝑘𝑂𝐿 𝑎 empirical equations are used, to keep the math
as simple as possible we use the following linear empirical equation for 𝑘𝑂𝐿 𝑎 when the effect of the
cells on 𝑘𝑂𝐿 𝑎 is taken into account:
𝑘𝑂𝐿 𝑎 = 𝛼𝑋 ∗ 𝑘𝑂𝐿 𝑎𝑊
𝛼𝑋 = factor which expresses the effect of the cells on 𝑘𝑂𝐿 𝑎 (no unit):
, Hugo Cloudt Advanced Bioreactor Design (BPE-36306)
𝑐3
𝛼𝑋 = 𝑐1 − 𝑐2 𝑐𝑋 +
𝜇
𝑐1 , 𝑐2 , 𝑐3 = constants which are determined by measurements (CmolX m-3 s-1).
𝑘𝑂𝐿 𝑎𝑊 = the 𝑘𝑂𝐿 𝑎 of pure water (s-1) = the 𝑘𝑂𝐿 𝑎 when the effect of the cells on 𝑘𝑂𝐿 𝑎 is not taken
into account in the mathematical model of the bioreactor.
This linear empirical equation for 𝑘𝑂𝐿 𝑎 when the effect of the cells on 𝑘𝑂𝐿 𝑎 has to be taken into
account is only a good approximation of reality and thus only valid at low 𝑢𝐺 (𝑢𝐺 up to roughly 0.04
m s-1).
The flow rates which you can choose when operating a bioreactor with cell retention = 𝐹𝐿 , 𝐹𝑅 and 𝐹𝐵
(because in the schematic drawing of the bioreactor with cell retention the pumps are within the
arrows which represent only one of these flow rates). -> these flow rates can among others be set as
such that there is one of the following extreme situations:
1. 𝐹𝑅 >> 𝐹𝐵 >> 𝐹𝐿 :
- Situation in which the cell recycle of the bioreactor with cell retention is extensively used, because
𝐹𝑅 is made very high (e.g. 𝐹𝑅 ≈ ∞).
- When this is the case and 𝑅𝑋 ≈ 1:
a. From the equation for 𝜇 it follows that:
𝐹𝐵
𝜇≈
𝑉𝐿𝑅
This equation shows that in a bioreactor with cell retention NOT 𝐹𝐿 , like in the same bioreactor
without cell retention, but 𝐹𝐵 determines 𝜇. Therefore in a bioreactor with cell retention a higher 𝐹𝐿
can be used than in the same bioreactor without cell retention, in a bioreactor with cell retention
even an 𝐹𝐿 which would give washout in the same bioreactor without cell retention can be used.
b. The residence time/retention time of the cells (𝜏𝑋 ) is given by:
1
𝜏𝑋 =
𝜇
𝐹𝐵
With 𝜇 ≈ this becomes:
𝑉𝐿𝑅
𝑉𝐿𝑅
𝜏𝑋 ≈
𝐹𝐵
Comparing this expression to the definition of the residence time/retention time of water (hydraulic
𝑉𝐿𝑅
residence time/hydraulic retention time (𝜏𝐿 )), 𝜏𝐿 ≈ , shows that the residence times/retention
𝐹𝐿
times of cells and water are decoupled in the bioreactor with cell retention (in contrast to the same
bioreactor without cell retention), so a cell residence time/retention time which is sufficiently long to
avoid washout and a water residence time/retention time which is a lot shorter can be used in
parallel.
2. 𝐹𝑅 << 𝐹𝐵 << 𝐹𝐿 :
- Situation in which the cell recycle of the bioreactor with cell retention is hardly used at all, because
𝐹𝑅 is made very low (e.g. 𝐹𝑅 ≈ 0).
- When this is the case and 𝑅𝑋 ≈ 1:
a. From the equation for 𝜇 it follows that:
𝐹𝐿
𝜇≈ = 𝐷𝐿
𝑉𝐿𝑅
