In this summary of the theory of the course Bioreactor Design (BPE-21306), which is given at Wageningen University, all the material discussed during the lectures and tutorials is included. In addition, all the material from the reader that relates to these lectures and tutorials is included in th...
Summary lectures and reader Bioreactor Design (BPE-21306)
Lecture 1 – Stoichiometry
Some frequently used bioreactors:
1. Stirred-tank bioreactors: bioreactors in which the mixing is caused by a stirrer.
2. Bioreactors for algae cultures (e.g. plastic bags).
3. Kimchi pots
4. Bubble columns: bioreactors in which small gas bubbles are introduced in the bioreactor by a gas
distributor at the top or the bottom of the bioreactor (these cause the mixing). -> e.g. used for:
Stirrer of stirred-tank bioreactor has 2 functions:
1. Homogenize the liquid in the bioreactor to give each cell the same physico-chemical environment.
2. Distribute gas bubbles and break gas bubbles into smaller gas bubbles to increase the surface area
of the gas and thereby increase the absorption rate of the gas.
Reasons for the use of very big bioreactors in industry:
1. The bigger the bioreactor is (higher volume), the more product can be produced using the
bioreactor (bigger bioreactors enable a bigger production).
2. The bigger the bioreactor is (higher volume), the lower the costs per unit volume of the bioreactor
become (bigger bioreactors have relatively lower costs).
For analysis of all bioreactors (regardless of the type) mathematical models are used, reasons to use
mathematical models for bioreactors are:
1. Mathematical models are necessary to translate findings from the laboratory to the industry,
because experiments can only be done in the laboratory and not in the big industrial scale reactors,
because this would be way too expensive. -> the physical behaviour of big industrial scale reactors
differs from lab reactors, in big industrial scale reactors cells are exposed to different concentrations
and due to this scaling-up (from lab-scale to industrial scale) requires extra equations.
,2. Mathematical models allow simpler and quicker testing of different operating conditions of the
reactors.
3. Mathematical models give insight in what is going on in the bioreactor and in the effects of this.
Mathematical models for bioreactors are used to find (for different operating conditions!):
1. All component flow rates entering or leaving the bioreactor.
2. The bioreactor size needed for a given product formation rate OR the product formation rate
which can be acquired with a bioreactor with a certain size.
The ingredients of a mathematical model for a bioreactor are:
1. Balances over the cells in the bioreactor:
- Atom balances and/or electron balance
- Energy balance
- Stoichiometric law for substrate consumption rate (e.g. Pirt’s law)
2. Balances over the reactor:
- Total mass balance
- Mass balances of individual components = component balances
- Energy balance
3. Kinetic laws:
- For product production rate (e.g. Luedeking-Piret’s law)
- For cell production rate (e.g. Monod’s law)
- For transport
4. Additional laws which are necessary in some cases:
- Ideal gas law
- Physical equilibria
- Chemical equilibria
3 different types of rates:
1. Production rate (ri):
a. i = subscript to denote to which component the rate belongs.
b. Unit = (C)moli s-1 -> Cmole used when component i contains C-atoms.
c. Value determines whether there is production or consumption:
- Positive value? -> production
- Negative value? -> consumption
2. Volumetric production rate: production rate per unit volume (riν).
a. i = subscript to denote to which component the rate belongs.
b. Unit = (C)moli m-3 s-1 -> Cmole used when component i contains C-atoms.
c. Value determines whether there is production or consumption:
- Positive value? -> production
- Negative value? -> consumption
d. Connected to production rate (ri) via the following formula:
ri = riν * VL
3. Specific production/consumption rate (qi): the production/consumption rate per unit biomass/cell
mass.
a. i = subscript to denote to which component the rate belongs.
b. The specific production/consumption rate of biomass/cells is not called q x but is called μ (specific
growth rate).
c. Unit = (C)moli Cmolex s-1 -> Cmole used when component i contains C-atoms.
d. Always a positive number!
,e. Connected to production rate (ri) via the following formulas:
- When component i is a reactant/is consumed: ri = -qi * Mx = -qi * cx * VL
- When component i is a product/is produced: ri = qi * Mx = qi * cx * VL
f. Connected to the volumetric production rate (riν) via the following formulas:
- When component i is a reactant/is consumed: riν = -qi * cx
- When component i is a product/is produced: riν = qi * cx
Most simple model of a cell in a bioreactor:
Reaction equation:
6 different components and thus 6 different production rates to be calculated, therefore 6 equations
(which all define reaction stoichiometry) are required:
1. Atom balances -> 1 atom balance for every type of atom in reaction equation = 4 equations:
Composition of a component can be read in each
column of the set of atom balances -> good way to
check if all atom balances are correct.
Notes about atom balances:
- The 3 different types of rates can all be used to construct the reaction equation and the associated
atom balances.
E.g. reaction equation using specific production/consumption rate (qi):
Associated atom balances:
- The atom balances can always be combined into the electron balance: linear combination of all
atom balances in which the number of electrons coming from reactants is compared with the
number of electrons ending up in products and biomass/cell mass.
γi = degree of reduction -> can be defined in any way you want, but the most smart definition is:
γi > 0 = the number of electrons which can be donated by component i to intracellular electron
carriers like NAD(P)H.
γi < 0 = the number of electrons which is received by component i from intracellular electron carriers
like NAD(P)H.
By using the smart definition of γi mentioned above the electron balance can be simplified a lot
because using this definition yields γN = γC = γW = 0:
,This short and simplified electron balance is especially convenient for calculating the oxygen
production rate (rO) when all other production rates are known:
Electron balance -> NOT an independent equation, because it is a linear combination of the atom
balances -> electron balance cannot be used together with all atom balances, at least 1 atom balance
must be omitted from the model if the electron balance is used!
2. Stoichiometric law for sugar consumption rate:
E.g. Pirt’s law: law which states that cells use the sugar they take up in 2 ways:
a. As carbon source in anabolism -> sugar used for construction of biomass/cell mass
b. As energy source in catabolism -> sugar used for:
- Synthesis of ATP which is necessary for constructing biomass/cell mass in anabolism.
- Synthesis of ATP which is necessary for maintenance in the cell.
ms = maintenance coefficient
(Cmols Cmolx-1 s-1)
3. Equation which depends on the type of bioreactor used:
- Chemostat? -> cell balance (= component balance of biomass)
- Batch reactor? -> simplified kinetic law
- Fed-batch reactor? -> simplified sugar balance (= component balance of sugar)
Calculating heat production rate/heat of reaction (rQ)? -> energy balance over cells necessary!
2 ways of setting up energy balance over cells:
1. Way used by chemistry = energy balance using standard enthalpy of formation (ΔhiF):
Component i = element? -> ΔhiF = 0 J/mol
2. Way used by biotechnology = energy balance using standard enthalpy of combustion (ΔhiC):
Component i = element OR CO2 (g) OR H2O (l)? -> ΔhiC = 0 J/mol
rQ = heat production rate/heat of reaction (J s -1 = W).
rQν = volumetric heat production rate/heat of reaction (J s -1 m-3 = W m-3).
Reasons to use an energy balance using standard enthalpy of combustion (ΔhiC) instead of an energy
balance using standard enthalpy of formation (ΔhiF):
1. H2O and CO2 are not included in this energy balance, because their ΔhiC = 0.
2. It gives a positive instead of a negative number for the heat production rate/heat of reaction (rQ)
of exothermic reactions (reactions in which energy is released as heat).
,Both ways of setting up the energy balance over the cells yield the same result:
The small difference is observed due
to the fact that the tabulated
standard enthalpies of formation
and standard enthalpies of
combustion have been rounded of.
Combustion reactions with oxygen as electron acceptor generate almost the same amount of heat
per mole of electrons transferred, regardless of the electron donor which is used. Therefore, for
calculating the heat production rate (rQ) of aerobic reactions the following shortcut exists:
General form of a component balance over a bioreactor:
Nearly complete model for a simple bioreactor:
E.g. anaerobic yeast culture which produces ethanol (CH3O0.5) and which is in a well-mixed
continuous reactor (chemostat):
Reaction equation:
Assumptions:
a. FLIN, VL and ciLIN can all be chosen by us.
b. FLOUT = FLIN = FL
c. The gas leaving the bioreactor only contains CO2.
d. Steady state -> dMi/dt = 0
Using these assumptions the model becomes:
,14 unknowns:
a. 1 gas flow rate (FG)
b. 1 gas-phase concentration (cCG)
c. 6 liquid-phase concentrations (ciL)
d. 6 volumetric production rates (riν)
Because there are 14 unknowns, 14 equations are necessary to calculate all unknowns (among these
is the volumetric production rate of product (rpν) in which you are most interested):
a. 6 component balances
b. 4 atom balances
c. Pirt’s law
d. Kinetic law
e. Ideal gas law
f. Equation for the gas-liquid equilibrium of CO2
Electron balance should be counted
as 2 equations here, because there
are 4 atom balances in total from
which 2 are used individually (N-
atom balance and C-atom balance)
and the other 2 are used in the form
of the electron balance.
FLOUT = FLIN = FL is assumed but is in reality only almost true, because some water will also leave the
reactor via the gas phase. -> water balance does NOT close and is therefore NOT used in solving the
equations.
The bioreactor is a chemostat, so the cell balance is the key for solving the equations:
1. Cell balance:
2. Removing cX and subsequent rearranging yields:
DL = dilution rate of the chemostat: the amount of liquid phase flowing into/out of the chemostat per
unit time (FL) relative to the volume (VL) of the chemostat.
3. When μ is known all q’s can be calculated from Pirt’s law and the atom balances and/or electron
balance:
,4. The other component balances can be rearranged and substituted into the kinetic law:
Notes about the found equation for cX:
- This equation can only be solved by trial and error.
- This equation will be even more complex if the cells use oxygen (aerobic process instead of current
anaerobic process), because then more balances are necessary in the bioreactor model.
- This equation will be even more complex if heat removal is a problem, because then more balances
are necessary in the bioreactor model.
- This equation will be even more complex if the bioreactor is a lot bigger, because then more
balances are necessary in the bioreactor model because it is not ideally mixed.
These disadvantages of the found equation for cX outline that simplification is necessary. -> simplify
by simplifying the kinetic law used by assuming that there is a rate-limiting component: the one
component which has the biggest effect on the reaction rate and therefore also has the biggest
effect on the specific growth rate (μ).
Originally used kinetic law:
Example of kinetic law used after simplification:
Sugar is assumed to be the rate-
With fs = Monod’s law OR one of many other laws for fs. limiting component when this kinetic
Monod’s law: law is used.
,After simplifying the kinetic law used by assuming that there is a rate-limiting component solving the
equations for the unknowns is less difficult:
FLOUT = FLIN = FL is assumed but is in reality only almost true, because some water will also leave the
reactor via the gas phase in reality. -> water balance does NOT close and is therefore NOT used in
solving the equations.
Outlined model is nearly complete and not totally complete, because:
- Steady state is assumed.
- Constant liquid flow rate (FLOUT = FLIN = FL) is assumed and therefore the water balance is omitted.
- Perfect temperature control is assumed and therefore no energy balance over the reactor is used
(heat is not taken into account).
- Perfect pH control is assumed and therefore the chemical equilibrium of CO2 for dissolving into
water and the charge balance are not used.
- Ideal mixing is assumed, but this is in big industrial reactors seldom the case.
The more complete the model is made, the more complex it will become and the harder it thus
becomes to solve.
Types of products distinguished in biotechnology:
1. Luxury products: products which the cell does not have to make to grow and maintain itself, but
which are made because we give it the opportunity or because we have modified its genome.
E.g. penicillin, monoclonal antibodies
- Overflow products: luxury products which are formed because cells get an excess of carbon and
energy source when compared to for example oxygen or nitrogen source.
E.g. citric acid produced by fungi
2. Anabolites/biomass/cell mass: compounds that are part of a normal cell.
E.g. cell wall fragments which are used as vaccines.
3. Catabolites/necessary products: waste products from maintenance and/or from the synthesis of
anabolites and/or luxury products.
E.g. CO2, ethanol, lactic acid
2 types of yields (Y):
1. Theoretical yields: the yields which are found in laws (e.g. in Pirt’s law) and which are never
reached in reality because they are only reached if a cell uses energy source for making only one
specific product and uses no energy source for maintenance.
,2. Observed yields (YOBS): the yields which are measured and which are always lower than the
corresponding theoretical yields.
Lecture 2 – Sugar-limited bioreactors (Chemostat and batch reactors)
Strategy for solving bioreactor design problems:
1. Make assumptions to simplify the problem.
2. Make a schematic drawing in which all relevant quantities are indicated and in which the made
assumptions are incorporated.
3. Set up the required component balances:
- Cell balance
- Balance for the rate-limiting component
4. Mark and count the unknown variables in the required component balances.
5. Add extra equations if the number of equations is not equal to the number of unknowns to make
the number of equations equal to the number of unknowns.
6. Solve the equations algebraically.
7. Calculate the asked values.
8. Check the assumptions made in the beginning:
Assumption which is always made is that one component is the rate-limiting component, this is
assumption is checked in the following way:
- Find the production rate of a reactant or product for which no component balance was required by
using an atom balance.
- Subsequently find the concentration of this reactant or product in the broth by using a component
balance for this reactant or product.
- Repeating this for all reactants and products for which no component balances were required to
find out all concentrations of these. If one of these reactants or products turns out to have an
unrealistic concentration in the broth probably this reactant or product is the rate-limiting
component instead of the assumed rate-limiting component. The calculated values are then thus not
a good indication of reality, so the assumptions should be altered by assuming that this other
reactant or product is the rate-limiting component and the problem should be solved again with this
assumption. Then you should check again if this assumption was correct, etcetera.
All types of bioreactors used can be divided in 3 classes of bioreactors which differ in the way they
are operated:
1. Chemostat/CSTR (Continuous Stirred-Tank Reactor): reactor in which medium/feed is continuously
pumped into and out of the reactor.
2. Batch reactor: reactor which has NO inflow and outflow of medium/feed.
3. Fed-batch reactor: reactor which has inflow but NO outflow of medium/feed.
Use of the different classes of bioreactors:
1. Chemostat:
, - Popular lab tool, because you can choose μ yourself because you can choose the dilution rate (DL)
yourself.
- Only used in industry when the reactor is used a long time (e.g. a year) to produce the same
product, because a chemostat has to reach a steady state to start producing and this can take a long
time.
2. Batch reactor:
- Popular lab tool, because it is a lot cheaper than a chemostat.
- Only used in industry for food fermentation and biopharma.
3. Fed-batch reactor:
- Not frequently used as lab tool, but emerging as lab tool at the moment.
- Most used class of reactors in industry, because it is the workhorse for multi-purpose plants
because:
a. Sugar inhibition and/or unwanted by-product formation due to the high concentration of sugar
present at the start of the reaction such as in a batch-reactor is avoided.
b. It has a longer running time than a batch reactor because sugar is not completely depleted
because it is continuously added.
c. In a chemostat you have to wait until steady state is reached (which can take quite a long time),
this is not necessary for a fed-batch reactor.
Growth by cell division? -> described by the following formula:
rX = μcXVL
With:
rX = production rate of biomass/cells (CmolX s-1).
μ = specific growth rate (CmolX CmolX-1 s-1).
cX = cell concentration (CmolX m-3).
VL = liquid volume of bioreactor (m3).
μcX = rXν = volumetric production rate of biomass/cells (Cmol X m-3 s-1).
cXVL = MX = mass of biomass/cells (CmolX).
Different kinetic laws for μ exist, the most used kinetic laws for μ are:
1. Monod’s law:
Variables, because the values of these quantities depend on how the reactor is operated:
μ = specific growth rate (CmolX CmolX-1 s-1).
cS = concentration of sugar inside the reactor (CmolS m-3). -> usually assumed that reactor is ideally
mixed, in that case the concentration of sugar inside the reactor is equal to the concentration of
sugar in the outlet of the reactor and cS is then thus also the concentration of sugar in the outlet of
the reactor.
Parameters/constants, because these are cell properties which must be measured:
μMAX = maximum specific growth rate (CmolX CmolX-1 s-1).
KS = Monod constant (CmolS m-3): cS at which μ is half of μMAX.
2. Blackman’s law
3. Equation for μ from the Han-Levenspiel model
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