Oxygen Transfer 1

Oxygen Transfer Through Gas-Liquid Interface 1Oxygen and Carbon Dioxide Concentration in Medium 2Oxygen Concentration in Culture and Oxygen Transfer 3Oxygen Diffusion and Order of Magnitude Analysis 4Oxygen Transfer through Interface of Two Phases Not covered in class 7Oxygen Transfer in bioreactors 8Oxygen Consumption and CO2 Production 9Experimental Measurement of KLa and OUR 10

Mass Transfer Resistance in Cell Immobilization Reactor 11Intraparticle Diffusion Limitation 11

Oxygen Transfer

Oxygen Transfer Through Gas-Liquid Interface Oxygen transfer rate (OTR) = KLa (c-c) =KLa ΔC[mmolelhr] [=] [cmhr] [1cm] [mmolel]c solubility of oxygen ΔC oxygen concentration gradient (ie the driving force)

Three factors affecting OTRbull Mass transfer coefficient (KL)bull Specific transfer area (ie the interfacial area)bull Driving force (ie the gradient across interface)

To improve oxygen transfer bull Increase KL (make the interface more ldquoturbulentrdquo)bull Increase a (use smaller bubbles for sparging or use

silicone tubing)bull Increase ΔC (use oxygen enriched air or donrsquot maintain

dissolved oxygen (C) at an unnecessarily high level

In cell cultivation oxygen is transferred from the gas phase (it could be gas bubbles or the gas space in the flask) through gas liquid interface into the medium In order for oxygen molecules to diffuse to the liquid phase there must be a concentration difference the gas phase and liquid phase must not be at equilibrium If the two phases are in equilibrium (ie the medium is already saturated with oxygen) there will not be more net transfer of oxygen into the medium The concentrations of oxygen in the gas phase and liquid phase in a culture are depicted in the figure In the bulk liquid (ie some distance away from the interface) the concentration of oxygen is CL Customarily oxygen levels in the gas phase are described as percentages mole percent or mole fraction (YO2) If the liquid phase is at equilibrium with the gas phase its oxygen concentration is denoted as C Closer to the interface there is a boundary layer or an imaginary film around each side of the interface wherein the concentration is different than the bulk At the edge of the gas film (or gas side boundary layers) the concentration of oxygen is thus YO2 or C On the other side of the gas bubble the concentration of oxygen is CL If the two phases are at equilibrium (ie C = CL) then there is no oxygen transfer Conversely the speed at

2 Oxygen Transfer

which oxygen gets transferred is dependent on three factors First the magnitude of the difference between C and CL has a major effect on the oxygen transfer rate The difference in concentration is a measure of how far the system is from equilibrium It is the driving force for the mass transfer The second factor is the area of gas-liquid interface that is available for oxygen transfer The larger the interfacial area is the larger the transfer rate Normally the surface area is expressed as interfacial area per volume of liquid It thus has a unit of inverse of length The third factor is the mass transfer coefficient This is a measurement of the degree of turbulence at the interface The magnitude of resistance at the boundary layer is also affected by the molecules that are being transferred The mass transfer coefficient (KL) has the same units as velocity

To be able to estimate how much oxygen can be transferred it is important to calculate the saturation level or equilibrium concentration of the gas species being transferred For a sparingly soluble solute the equilibrium relationship between gas phase and liquid phase is described by Henryrsquos Law It says that concentrations in the liquid phase and gas phase are linearly related One can use a variety of ways to describe the concentration in each phase Many handbooks use partial pressure (which is mole fraction of the gas solute multiplied by total pressure) for gas phase and mole fraction for liquid phase For general use it is more convenient to express liquid phase concentration in mmolel In the medical profession the gas phase composition is often expressed in mmHg (760 mmHg= 1atm)

Henryrsquos Law applies well to an ideal solution Cell culture medium contains salts and other nutrients In spite of all the components in it cell culture medium is still close to an ideal solution However the cell culture medium is frequently used under 1-20 CO2 environment Furthermore in a bioreactor the air pressure is usually greater than 1 atm Thus corrections on oxygen solubility under culture conditions need to be made

Calculation from Henryrsquos Law

Oxygen and Carbon Dioxide Concentration in Medium

Henryrsquos Law

xA mole solute Amole solutionPA partial pressure of AH Henryrsquos law constant

AA

PxH

=

= 022 mmole O2ℓ ℓH20

Henryrsquos Law Constants for O2 (atmmole 02mole H20)

T degC 5 20 25 30 35 37

10-4 x H 291 401 438 475 507 518 Example at 37deg C in 1 atm air (P02 = 021 atm) O2 concentration in H20 is

24

22 2

555 mole H 0021atmH 0518 x 10 atmmoleO moleH O

c = bull

In ambient air (oxygen partial pressure = 021 atm) c ~ 022 mmolel in H20 at 37 degC = 160 mmHg = 58 mgl (58 ppm)

Oxygen Transfer 3

bull Oxygen solubility is not affected by other dissolved species in medium Its solubility is virtually identical to PBS and water

bull Solubility in medium should be adjusted by CO2 used in gas phase (ie the fraction of O2 in gas phase is usually not 021)

Blood Gas Analyzer Measurement

Temp (degC) Water PBSCulture Medium (15 mm NaHCO3

and 10 mm HEPES)PO2 mmHg pH PO2 mmHg pH PO2 mmHg PCO2 mmHg pH

215 1944a 646 1948a 73737 1580a 651 1571a 737 1319b 464 727

a In open air without CO2b In 5 CO2 humidified incubator

2 2

2 2 2 3

2 3 3

( ) ( )( ) ( )

CO g CO aqCO aq H O H CO aqH CO HCO Hminus +

+

+

Example Using10 CO2 in air CO2

concentration at 37degC is 2COPC

H=

23

2 2 2

555 mole H O010atm22 10 atmmole CO mole H 0 0H

= bulltimes bull

=22mmolel

Henryrsquos Law Constants for CO2 (atmmole CO2mole H2O)

T degC 0 5 10 15 20 25 30 35 40

10-3 x H 0728 0876 104 122 142 164 180 209 233

Oxygen Concentration in Culture and Oxygen Transfer

The solubility of CO2 can be calculated in the same way using Henryrsquos law However CO2 associates with water-

molecules and then dissociates into 3HCOminus and HT The value calculated using Henryrsquos law is the sum of CO2 (aq)

and 3HCOminus in pure water One can see that the solubility of CO2 will be affected by the pH of the solution and the counter ion It is not like oxygen for which the solubility is very similar in pure water and in medium

Oxygen has a low solubility in aqueous medium as seen in the previous section Its solubility under ambient air is only about 7 mgl as compared to over 900 gl for glucose at room temperature If cells are grown in a medium satu-rated with glucose they would have had to undergo osmotic shock Therefore typical glucose concentration for cell cul-tivation is a few grams per liter The specific oxygen con-sumption rates in terms of mole of oxygen and glucose are typically a few fold higher for oxygen In a solution with saturated oxygen levels and with a typical glucose concen-tration oxygen will get depleted by cell consumption much

4 Oxygen Transfer

faster than glucose Thus oxygen will need to be continu-ously supplied for an aerobe while glucose and other nu-trients can only be supplied at the beginning of the culture

The supply of oxygen to cultured organisms serves not only to prevent it from depleting but also to maintain it in an optimal range Most aerobic microorganisms grow well in a very wide range of dissolved oxygen concentration up to saturation with ambient air However some cells from higher organisms grow better under unsaturated concen-tration For example stem cells from mammalian species have been reported to grow better when the gas used for cultivation contains 5 O2 instead of 21 Some tissues such as bone marrow have a lower concentration and cells from such tissues may prefer somewhat lower oxygen con-centration However except obligative or facultative anaer-obes all other organisms require oxygen levels be sustained about certain level otherwise adverse effects on growth and survival is seen This critical level of oxygen is typically about 20 of saturation with air for aerobic microorgan-isms (or about 001 mM) For mammalian cells such lung fibroblasts 3T3 cells Chinese Hamster Ovary Cells the critical level is around 10 of saturation with air at 1atm

The supply of oxygen to cells in culture is typically achieved through a gas phase to the liquid phase (ie the medium in which cells are growing) There are a number of steps that oxygen molecules need to traverse through in order to reach the cells and eventually the intracellular sites where the reaction cosuming oxygen occurs This is described in the panel A question to ask is tha among all the steps in-volved which one is rate limiting or is the controlling step

A very effective way of identifying the rate controlling step without resorting to solving a set of differential equations is by order of magnitude analysis of time constant In this type of analysis instead of considering the detailed concentration changes over time in a system of fixed geometry one tries to examine the amount of time (called characteristic time or time constant) it takes to change a set of concentrations of a chemical species (characteristic concentration) over a distance (characteristic length) In such order of magnitude analysis one compares the ldquocharacteristic timerdquo needed for mass (or heat) transfer in different systems that have different characteristic lengths or a different transfer mechanism The step that has the longest time constant is the slowest one

Consider a solute that is soluble in two different ldquophasesrdquo (liquid-gas gas-solid liquid-solid etc) Given the two phases co-exist for a long enough time the concentrations in the two phases will be in equilibrium If suddenly the concentration in one phase is increased to a new level the solute will be transferred through the interface into the other phase and the

Oxygen Diffusion and Order of Magnitude Analysis

bullemsp ResistancesemspforemspO2emsptransferbullemsp Diffusionemsptoemsptheemspgas-liquidemspinterfacebullemsp Solutionemspofemsptheemspgasemspintoemsptheemspliquidemspatemsptheemspemsp emspemsp interfacebullemsp Diffusionemspfromemsptheemspliquidemspsideemspofemsptheemspinterfaceemspemspemsp intoemsptheemspliquidemspbulkbullemsp Diffusionemspfromemsptheemspbulkemspliquidemspnearemsptheemspsurfaceemspemspemsp ofemspgasemspbubbleemsptoemspbulkemspliquidemspnearemsptheemspemsp emspemsp surfaceemspofemspcellsbullemsp Diffusionemspfromemsptheemspliquidemspbulkemspthroughemspliquidemspemsp emspemsp filmemspsurroundingemsptheemspcellbullemsp Diffusionemspthroughemspcellemspclumpemsppelletemsporemspmyceliabullemsp Consumptionemspreactionemspinternalemspdiffusionbullemsp Isemspintracellularemspoxygenemspdiffusionemspimportant

Oxygen Transfer 5

concentration in the other phase will continue to increase For simplicity let us assume that the concentration at the interface will remain constant In other words the transfer of solute into the other phase does not cause the concentration in the first phase to decrease This would be the case that the first phase has a very large capacity (eg large volume) relative to the second phase so that even some solute is being transferred into the second phase the amount transferred is relatively small to the total in the first phaseIf we look at a slice of the second phase perpendicular to the direction of transfer and do a material balance on this slice the balance will be

Slice is referred to as ldquocontrol volumerdquo

Its equation form is shown in the panel In general whenever a solute is carried by a fluid (this type of transport is convection) its rate of transport in the direction of fluid flow is far greater (by orders of magnitude) than that from diffusion unless the flow rate is extremely slow We will consider only the case that convection is not involved in transfer and especially in the case of transfer across the interface The convection caused by solute transfer is generally negligible So the case we consider is the transport into a stagnant liquid of a low solubility chemical species (so that the transport of solute does not cause convection) or in a solid

By neglecting convection the equation is simplified We first examine a condition that in the medium (ie the stagnant liquid or the solid) no reaction occurs ie as solute is traveling down the medium it does not get reacted (ie consumed) nor does the medium produce any solute by itself

The diffusion term is generally described by Fickrsquos Law The diffusion is driven by concentration difference between two points in space If the concentration in position 2 is greater than position 2 the solute will diffuse in the direction toward 2 or in the ldquonegativerdquo direction In equation form it says

C x∆ ∆ is the concentration difference or the concentraton gradient J is the flux of the solute The negative sign accounts for the direction of transfer being opposite the sign of the slope The proportionality between flux and the concentration gradient is the diffusion coefficient as shown belowJ is a measure of transfer rate (per surface area)

rate of solute rate of change of solute concentration rate of solute transported rate of solute transported

concentration change in the control volume by fluid flow by diffusion

due

= + +

to reaction

2 1

2 1

(flux of solute) - c c CJx x x

α minus ∆= minus

minus ∆

3 2( ) ( ) ( )m L L T M L Tbull = bull

6 Oxygen Transfer

normal to the direction of transfer It has units of

The diffusion coefficient has units of 2 L t Its magnitude is dependent on the molecular species and the medium it diffuses in For example oxygen and glucose and small proteins have very different diffusion coefficients in water (approximately 10-4 10-6 10-8 cm2sec for oxygen glucose and protein respectively) which the diffusion coefficient of oxygen in air lipid and water decreases progressively Considering only diffusion in the medium the change of the oxygen concentration in the control volume is the difference between the diffusion at positions x and x x+ ∆ (assume diffusion is one dimensional thus ignoring bottom and top

surface) At surface x the diffusion rate is -D D xx

∆∆

While at x+ox it is CD x oxx

∆+

∆

Taking differential between the two planes and assuming D

is constant we obtain CD

x x∆ ∆

∆ ∆

The equation for the control volume thus becomes

2

2

2 2 02 2c CDt x

= = or 22 2

2 2c CDt x

=

Instead of solving the partial differential equation we can do an order of magnitude estimation on each term The left

hand side 22ct

can be estimated as

The order of magnitude for x∆ is C1-C0 Again that can be estimated as S the penetration depth or the characteristic length So overall combining right and left hand sides

2d

C CDt g

∆ ∆

So 2

dgtD

( )2t

x∆

∆

3 2( ) ( ) ( )m L L T M L Tbull = bull

Oxygen Transfer 7

Oxygen Transfer through Interface of Two Phases

The characteristic length is usually chosen to represent the ldquopenetration depthrdquo For a long slab with diffusion from the two surfaces a long cylinder (both neglecting the diffusion from the two ends) and a sphere the characteristic length is shown in the table One can calculate the concentration of the solute as of the final equivalent to a characteristic time As can be seen after one characteristic time the concentration at a characteristic length is very close to equilibriumOne can estimate the time constant for convection and different types of reaction kinetics similarly as shown in the Table

Going back to oxygen transfer problems in bioreactors the step that incurs most resistance is the transfer between gas phase and liquid phase Microscopically the transfer can be depicted as oxygen molecules passing through two stagnant films one on the gas side the other on the liquid side as shown in Figure 1 We have introduced the equation for the rate of transfer This overall transfer is the net result of transfer through the two films For each film we can write an equation to describe the rate using kL and kg for liquid-side and gas-side transfer coefficient respectively A can be considered the same for both gas film and liquid film At the interface between the two films the two phases can be considered to be in equilibrium ie Cli=HPgi (recall that at equilibrium the concentration at the two phases is related by Henryrsquos Law constant)

One can write the equation for oxygen transfer rate (OTR) in three different forms overall liquid-side and gas-side Since there is no accumulation in the films the rate expressed by the three equations should be equal As shown in the panel by combining liquid side and gas concentration gradient to give the overall concentration gradient one can show that the inverse of Kl is the sum of the inverse of kl and hg If one considers K kl kg as the conductivity the 1KL 1kl 1kg gives the resistance In other words it shows that the overall mass transfer resistance 1kl is the sum of the two film resistances 1kl and 1kg

In using kl or kg for calculating OTR one has to use the interface concentration (Cli Pgi) for calculating concentration gradient The concentration at interface cannot be determined readily By using overall transfer the driving force used becomes the Por C or the concentration of the two phases were at equilibrium For this case it is a measure of how far away the system is from the equilibrium P and C can be readily calculated

Note that one can also use Kg for overall mass transfer coefficient that gives OTR=Kga(Pg-P)But traditionally Kl is used We just discussed that the overall resistance 1kL is the sum of 1kL and 1kg But what is the

8 Oxygen Transfer

Oxygen Transfer in bioreactors

relative magnitude of 1kL and 1kg The magnitude of mass transfer coefficient is dictated by the diffusion coefficient and the film thickness (Dδ) The film thickness for gas and liquid side is similar While the diffusion coefficient on the gas side and liquid side are orders of magnitude dififerent It can be shown that the liquid side resistance dominates the overall resistance It is thus unusual that kLis used in place of kL even when overall driving force is used

KL has units of Lt (like velocity) and the interfacial area has units of 1L After multiplying by a concentration the mass transfer equation gives a concentration perimeter that is the rate of transfer The interfacial area term is interfacial area for oxygen transfer per liquid volume aV A is the total amount of transfer area in the reactor V is the total liquid volume In a large bioreactor aerated by sparging air bubbles the bubbles are not necessarily evenly distributed and local kL is not necessarily the same The oxygen transfer rate discussed in literature is mostly an average over the reactor

OUR is the product of specific oxygen concentration and cell concentration (q0χ) Obviously if OTR is lower than OUR then dCdt will be negative and dissolved oxygen level will decrease In a culture the dissolved oxygen level decreases as cell concentration increases With a dissolved oxygen the controller manipulates the agitation rate air flow rate or the oxygen content (ie by influence kL or triangle C0 to maintain dissolved oxygen levels at the set point But even under uncontrolled conditions that change of dissolved oxygen level is not fast all the time Consider the three terms in liquid phase oxygen balance equations If the accumulation term (dcdt) is small compared to OTR and OUR one can consider the system is at a quasi steady state ie dCdt ~) thus OTR=OUR

In a typical bioreactor the oxygen level in the gas phase (Pg or C) decreases from the level at the inlet to the level at the outlet while it traverses through the reactor as oxygen is being transferred into liquid phase The question thus arises as to which C should be used the one corresponding to gas at the inlet or something else If the reactor is a small stirred tank that one can consider to be well mixed then CL will be the same in the entire liquid phase and so is the gas phase concentration In a well mixed reactor one assumes that the sample taken at any point is the same as that in the well mixed phase Therefore C should be that of the exit gas stream not the inletIn a large reactor the gas phase is likely to behave closer to plug flow One then uses logrithmic mean to calculate the driving force This is analogous to using logrithmic mean driving force in heat transfer in a double-pipe heat exchanger Basically it is an average (but a simple geometrical

Oxygen Transfer 9

OTRV=(oxygen concentration at inlet)(inlet airflow rate)-(oxygen concentration in outlet)(outlet airflow rate)

average) of the driving force at the inlet and the outlet

In addition to balance on the liquid the gas supplying the oxygen also conforms to material balance The balance on the amount of oxygen supplied to the reactor at the intlet and the amout at the outlet is the amount that has been transferred to the liquid phase In principle in most reactors above the liquid there is also a headspace with gas and should be considered for balance but in general the gas composition in the head space is considered to be the same as that in the exit Thus oxygen transferred into the liquid phase by gas phase is one line

OTR (dot V-(oxygen concentration at inlet) dot (inlet air flow rate) ndash(oxygen concentration at outlet)dot (outlet airflow rate)=NO2 in dot Qin-NO2out-Qout (asme as in slide)

With a quasi steady state assumption the oxygen transferred into liquid phase by gas phase is the same as that received by liquid phase both being OTR Furthermore OTR can be assumed equal to OUR

If the exit gas (or falled off gas) composition is measured on line using mass spectrometer or other oxygen gas sensor one can easily obtain the volumetric oxygen uptake rate In some cases kLa is relative constant such in the case of some cells can be calculated from C if C is kept constant If off-gas is not measured one can resort to a dynamic method The method involves switching gas to a composition that is an equivalent to the current dissolved oxygen leve At that moment no net transfer of oxygen may occur (because ∆ C=0) As a result the change of dissolved oxygen is only caused by cell consumption From the slope one can obtain OUR

The objective in oxygen transfer is to supply oxygen at a rate that is enough to maintain the dissolved oxygen at a level suitable (or optimal) for cell growth and production For most mammalian cells a 30 saturation with ambient air is sufficient for optimal growth The amount of oxygen cells consumed per culture volume is referred to as Oxygen Uptake Rate (OUR) As you may recall from the kinetics chapter OUR is dependent on the specific oxygen consumption rate (qO2) and the cell concentration In culture the change of oxygen concentration with time is thus the balance between oxygen transfer rate (OTR) and oxygen uptake rate

Most oxygen taken up by cells is used to oxidize carbon sources The most important carbon sources are glucose and glutamine For both the oxidation reaction yield a

Optimal Oxygen Concentrationbull Most cells grow well at a dissolved oxygen level of 30

of saturation with air bull When oxygen-enriched gas is used sometimes the

dissolved oxygen level goes above saturation A short duration of above saturation level of oxygen is usually not detrimental to cells

bull The OTR efficiency is higher with a larger oxygen concentration gradient between gas and medium (driving force) Maintaining DO at an unnecessarily high level decreases the driving force

Oxygen Consumption and CO2 Production

10 Oxygen Transfer

Integrate

ln( ) LC C K a tminus = sdot

2 2( )L O

dCV OTR V OUR Vdt

K a V C C q x V

= sdot minus sdot

= sdot sdot minus minus sdot sdot

Oxygen Demand by Cellsbull On the average q02 asymp 10 x 10-10 mmolecell-hr

CO2 Production by Cellsbull Respiratory quotient of cells is about 10 ie 1 mole CO2

is produced for every mole of O2 consumed or qCO2 asymp 10 x 10-10 mmolecell-hr

bull A high CO2 concentration is inhibitorybull Inhibitory level usually starts at 15 to 20 CO2

Experimental Measurement of KLa and OUR

stoichiometric ratio of one for CO2 to O2 (called respiratory quotient [RQ]) The CO2 produced by cells will need to be removed from the culture to avoid excessive accumulationExcess amount of either dissolved O2 or CO2 can be growth inhibiting or toxic to the cells However the growth inhibitory effect of O2 and CO2 take an exposure time to be seen A short exposure of high concentration (gt100 saturation of O2 or nearly 150 mmHg of CO2) up to a few hours may not be detrimental

The mass transfer coefficient can be estimated by using some empirical correlations developed in the form of dimension-less numbers There has been a report that KL is relatively insensitive to bubble size except in small scale reactors and under the conditions that the liquid flow pattern is relatively simple like most bubbles rise in a simple pattern the esti-mated KL is probably not reliable Furthermore except un-der those conditions a is also difficult to estimate as relied on direct experimental measurement of KLa is often lumped together and called volumetric mass transfer coefficient in-stead of having KL and a estimated or measured separately Many bioprocess plants measure gas composition from the biotreactor inlet and outlet From the balance equa-tion one can see that the OUR and OTR calculated from the gas balance can also be used to calculate the average KLa in the reactor since Cand C are both measured One can also use a dynamic method to measure KLa One way to do this is by stripping the dissolved oxygen off using N2 to replace air (or the original gas phase) In this case

Oxygen Transfer 11

Mass Transfer Resistance in Cell Immobilization Reactor

Intraparticle Diffusion Limitation

Criteria for assessing the magnitude of mass-transfer effects on overall kinetics

Criterion η value

Limiting rate process

Extent of mass-transfer

limitation

Φ lt 03 ~1 Chemical reaction Negligible

Φ gt 3 αФ-1 Diffusion Large

Or Φ critical the critical value at which η = 1

12 Oxygen Transfer

bull The shape of the curve varies with reaction kinetics and the shape of the particles

bull Zero order kinetics is often a good assumptionbull Oxygen is usually the first compound to become

limiting because of its low solubility

• Oxygen Transfer Through Gas-Liquid Interface
• Oxygen and Carbon Dioxide Concentration in Medium
• Oxygen Concentration in Culture and Oxygen Transfer
• Oxygen Diffusion and Order of Magnitude Analysis
• Oxygen Transfer through Interface of Two Phases
• Oxygen Transfer in bioreactors
• Oxygen Consumption and CO2 Production
• Experimental Measurement of KLa and OUR
• Mass Transfer Resistance in Cell Immobilization Reactor
• Intraparticle Diffusion Limitation

Oxygen Transfer 3

bull Oxygen solubility is not affected by other dissolved species in medium Its solubility is virtually identical to PBS and water

bull Solubility in medium should be adjusted by CO2 used in gas phase (ie the fraction of O2 in gas phase is usually not 021)

Blood Gas Analyzer Measurement

Temp (degC) Water PBSCulture Medium (15 mm NaHCO3

and 10 mm HEPES)PO2 mmHg pH PO2 mmHg pH PO2 mmHg PCO2 mmHg pH

215 1944a 646 1948a 73737 1580a 651 1571a 737 1319b 464 727

a In open air without CO2b In 5 CO2 humidified incubator

2 2

2 2 2 3

2 3 3

( ) ( )( ) ( )

CO g CO aqCO aq H O H CO aqH CO HCO Hminus +

+

+

Example Using10 CO2 in air CO2

concentration at 37degC is 2COPC

H=

23

2 2 2

555 mole H O010atm22 10 atmmole CO mole H 0 0H

= bulltimes bull

=22mmolel

Henryrsquos Law Constants for CO2 (atmmole CO2mole H2O)

T degC 0 5 10 15 20 25 30 35 40

10-3 x H 0728 0876 104 122 142 164 180 209 233

Oxygen Concentration in Culture and Oxygen Transfer

The solubility of CO2 can be calculated in the same way using Henryrsquos law However CO2 associates with water-

molecules and then dissociates into 3HCOminus and HT The value calculated using Henryrsquos law is the sum of CO2 (aq)

and 3HCOminus in pure water One can see that the solubility of CO2 will be affected by the pH of the solution and the counter ion It is not like oxygen for which the solubility is very similar in pure water and in medium

Oxygen has a low solubility in aqueous medium as seen in the previous section Its solubility under ambient air is only about 7 mgl as compared to over 900 gl for glucose at room temperature If cells are grown in a medium satu-rated with glucose they would have had to undergo osmotic shock Therefore typical glucose concentration for cell cul-tivation is a few grams per liter The specific oxygen con-sumption rates in terms of mole of oxygen and glucose are typically a few fold higher for oxygen In a solution with saturated oxygen levels and with a typical glucose concen-tration oxygen will get depleted by cell consumption much

4 Oxygen Transfer

faster than glucose Thus oxygen will need to be continu-ously supplied for an aerobe while glucose and other nu-trients can only be supplied at the beginning of the culture

The supply of oxygen to cultured organisms serves not only to prevent it from depleting but also to maintain it in an optimal range Most aerobic microorganisms grow well in a very wide range of dissolved oxygen concentration up to saturation with ambient air However some cells from higher organisms grow better under unsaturated concen-tration For example stem cells from mammalian species have been reported to grow better when the gas used for cultivation contains 5 O2 instead of 21 Some tissues such as bone marrow have a lower concentration and cells from such tissues may prefer somewhat lower oxygen con-centration However except obligative or facultative anaer-obes all other organisms require oxygen levels be sustained about certain level otherwise adverse effects on growth and survival is seen This critical level of oxygen is typically about 20 of saturation with air for aerobic microorgan-isms (or about 001 mM) For mammalian cells such lung fibroblasts 3T3 cells Chinese Hamster Ovary Cells the critical level is around 10 of saturation with air at 1atm

The supply of oxygen to cells in culture is typically achieved through a gas phase to the liquid phase (ie the medium in which cells are growing) There are a number of steps that oxygen molecules need to traverse through in order to reach the cells and eventually the intracellular sites where the reaction cosuming oxygen occurs This is described in the panel A question to ask is tha among all the steps in-volved which one is rate limiting or is the controlling step

A very effective way of identifying the rate controlling step without resorting to solving a set of differential equations is by order of magnitude analysis of time constant In this type of analysis instead of considering the detailed concentration changes over time in a system of fixed geometry one tries to examine the amount of time (called characteristic time or time constant) it takes to change a set of concentrations of a chemical species (characteristic concentration) over a distance (characteristic length) In such order of magnitude analysis one compares the ldquocharacteristic timerdquo needed for mass (or heat) transfer in different systems that have different characteristic lengths or a different transfer mechanism The step that has the longest time constant is the slowest one

Consider a solute that is soluble in two different ldquophasesrdquo (liquid-gas gas-solid liquid-solid etc) Given the two phases co-exist for a long enough time the concentrations in the two phases will be in equilibrium If suddenly the concentration in one phase is increased to a new level the solute will be transferred through the interface into the other phase and the

Oxygen Diffusion and Order of Magnitude Analysis

bullemsp ResistancesemspforemspO2emsptransferbullemsp Diffusionemsptoemsptheemspgas-liquidemspinterfacebullemsp Solutionemspofemsptheemspgasemspintoemsptheemspliquidemspatemsptheemspemsp emspemsp interfacebullemsp Diffusionemspfromemsptheemspliquidemspsideemspofemsptheemspinterfaceemspemspemsp intoemsptheemspliquidemspbulkbullemsp Diffusionemspfromemsptheemspbulkemspliquidemspnearemsptheemspsurfaceemspemspemsp ofemspgasemspbubbleemsptoemspbulkemspliquidemspnearemsptheemspemsp emspemsp surfaceemspofemspcellsbullemsp Diffusionemspfromemsptheemspliquidemspbulkemspthroughemspliquidemspemsp emspemsp filmemspsurroundingemsptheemspcellbullemsp Diffusionemspthroughemspcellemspclumpemsppelletemsporemspmyceliabullemsp Consumptionemspreactionemspinternalemspdiffusionbullemsp Isemspintracellularemspoxygenemspdiffusionemspimportant

Oxygen Transfer 5

concentration in the other phase will continue to increase For simplicity let us assume that the concentration at the interface will remain constant In other words the transfer of solute into the other phase does not cause the concentration in the first phase to decrease This would be the case that the first phase has a very large capacity (eg large volume) relative to the second phase so that even some solute is being transferred into the second phase the amount transferred is relatively small to the total in the first phaseIf we look at a slice of the second phase perpendicular to the direction of transfer and do a material balance on this slice the balance will be

Slice is referred to as ldquocontrol volumerdquo

Its equation form is shown in the panel In general whenever a solute is carried by a fluid (this type of transport is convection) its rate of transport in the direction of fluid flow is far greater (by orders of magnitude) than that from diffusion unless the flow rate is extremely slow We will consider only the case that convection is not involved in transfer and especially in the case of transfer across the interface The convection caused by solute transfer is generally negligible So the case we consider is the transport into a stagnant liquid of a low solubility chemical species (so that the transport of solute does not cause convection) or in a solid

By neglecting convection the equation is simplified We first examine a condition that in the medium (ie the stagnant liquid or the solid) no reaction occurs ie as solute is traveling down the medium it does not get reacted (ie consumed) nor does the medium produce any solute by itself

The diffusion term is generally described by Fickrsquos Law The diffusion is driven by concentration difference between two points in space If the concentration in position 2 is greater than position 2 the solute will diffuse in the direction toward 2 or in the ldquonegativerdquo direction In equation form it says

C x∆ ∆ is the concentration difference or the concentraton gradient J is the flux of the solute The negative sign accounts for the direction of transfer being opposite the sign of the slope The proportionality between flux and the concentration gradient is the diffusion coefficient as shown belowJ is a measure of transfer rate (per surface area)

rate of solute rate of change of solute concentration rate of solute transported rate of solute transported

concentration change in the control volume by fluid flow by diffusion

due

= + +

to reaction

2 1

2 1

(flux of solute) - c c CJx x x

α minus ∆= minus

minus ∆

3 2( ) ( ) ( )m L L T M L Tbull = bull

6 Oxygen Transfer

normal to the direction of transfer It has units of

The diffusion coefficient has units of 2 L t Its magnitude is dependent on the molecular species and the medium it diffuses in For example oxygen and glucose and small proteins have very different diffusion coefficients in water (approximately 10-4 10-6 10-8 cm2sec for oxygen glucose and protein respectively) which the diffusion coefficient of oxygen in air lipid and water decreases progressively Considering only diffusion in the medium the change of the oxygen concentration in the control volume is the difference between the diffusion at positions x and x x+ ∆ (assume diffusion is one dimensional thus ignoring bottom and top

surface) At surface x the diffusion rate is -D D xx

∆∆

While at x+ox it is CD x oxx

∆+

∆

Taking differential between the two planes and assuming D

is constant we obtain CD

x x∆ ∆

∆ ∆

The equation for the control volume thus becomes

2

2

2 2 02 2c CDt x

= = or 22 2

2 2c CDt x

=

Instead of solving the partial differential equation we can do an order of magnitude estimation on each term The left

hand side 22ct

can be estimated as

The order of magnitude for x∆ is C1-C0 Again that can be estimated as S the penetration depth or the characteristic length So overall combining right and left hand sides

2d

C CDt g

∆ ∆

So 2

dgtD

( )2t

x∆

∆

3 2( ) ( ) ( )m L L T M L Tbull = bull

Oxygen Transfer 7

Oxygen Transfer through Interface of Two Phases

The characteristic length is usually chosen to represent the ldquopenetration depthrdquo For a long slab with diffusion from the two surfaces a long cylinder (both neglecting the diffusion from the two ends) and a sphere the characteristic length is shown in the table One can calculate the concentration of the solute as of the final equivalent to a characteristic time As can be seen after one characteristic time the concentration at a characteristic length is very close to equilibriumOne can estimate the time constant for convection and different types of reaction kinetics similarly as shown in the Table

Going back to oxygen transfer problems in bioreactors the step that incurs most resistance is the transfer between gas phase and liquid phase Microscopically the transfer can be depicted as oxygen molecules passing through two stagnant films one on the gas side the other on the liquid side as shown in Figure 1 We have introduced the equation for the rate of transfer This overall transfer is the net result of transfer through the two films For each film we can write an equation to describe the rate using kL and kg for liquid-side and gas-side transfer coefficient respectively A can be considered the same for both gas film and liquid film At the interface between the two films the two phases can be considered to be in equilibrium ie Cli=HPgi (recall that at equilibrium the concentration at the two phases is related by Henryrsquos Law constant)

One can write the equation for oxygen transfer rate (OTR) in three different forms overall liquid-side and gas-side Since there is no accumulation in the films the rate expressed by the three equations should be equal As shown in the panel by combining liquid side and gas concentration gradient to give the overall concentration gradient one can show that the inverse of Kl is the sum of the inverse of kl and hg If one considers K kl kg as the conductivity the 1KL 1kl 1kg gives the resistance In other words it shows that the overall mass transfer resistance 1kl is the sum of the two film resistances 1kl and 1kg

In using kl or kg for calculating OTR one has to use the interface concentration (Cli Pgi) for calculating concentration gradient The concentration at interface cannot be determined readily By using overall transfer the driving force used becomes the Por C or the concentration of the two phases were at equilibrium For this case it is a measure of how far away the system is from the equilibrium P and C can be readily calculated

Note that one can also use Kg for overall mass transfer coefficient that gives OTR=Kga(Pg-P)But traditionally Kl is used We just discussed that the overall resistance 1kL is the sum of 1kL and 1kg But what is the

8 Oxygen Transfer

Oxygen Transfer in bioreactors

relative magnitude of 1kL and 1kg The magnitude of mass transfer coefficient is dictated by the diffusion coefficient and the film thickness (Dδ) The film thickness for gas and liquid side is similar While the diffusion coefficient on the gas side and liquid side are orders of magnitude dififerent It can be shown that the liquid side resistance dominates the overall resistance It is thus unusual that kLis used in place of kL even when overall driving force is used

KL has units of Lt (like velocity) and the interfacial area has units of 1L After multiplying by a concentration the mass transfer equation gives a concentration perimeter that is the rate of transfer The interfacial area term is interfacial area for oxygen transfer per liquid volume aV A is the total amount of transfer area in the reactor V is the total liquid volume In a large bioreactor aerated by sparging air bubbles the bubbles are not necessarily evenly distributed and local kL is not necessarily the same The oxygen transfer rate discussed in literature is mostly an average over the reactor

OUR is the product of specific oxygen concentration and cell concentration (q0χ) Obviously if OTR is lower than OUR then dCdt will be negative and dissolved oxygen level will decrease In a culture the dissolved oxygen level decreases as cell concentration increases With a dissolved oxygen the controller manipulates the agitation rate air flow rate or the oxygen content (ie by influence kL or triangle C0 to maintain dissolved oxygen levels at the set point But even under uncontrolled conditions that change of dissolved oxygen level is not fast all the time Consider the three terms in liquid phase oxygen balance equations If the accumulation term (dcdt) is small compared to OTR and OUR one can consider the system is at a quasi steady state ie dCdt ~) thus OTR=OUR

In a typical bioreactor the oxygen level in the gas phase (Pg or C) decreases from the level at the inlet to the level at the outlet while it traverses through the reactor as oxygen is being transferred into liquid phase The question thus arises as to which C should be used the one corresponding to gas at the inlet or something else If the reactor is a small stirred tank that one can consider to be well mixed then CL will be the same in the entire liquid phase and so is the gas phase concentration In a well mixed reactor one assumes that the sample taken at any point is the same as that in the well mixed phase Therefore C should be that of the exit gas stream not the inletIn a large reactor the gas phase is likely to behave closer to plug flow One then uses logrithmic mean to calculate the driving force This is analogous to using logrithmic mean driving force in heat transfer in a double-pipe heat exchanger Basically it is an average (but a simple geometrical

Oxygen Transfer 9

OTRV=(oxygen concentration at inlet)(inlet airflow rate)-(oxygen concentration in outlet)(outlet airflow rate)

average) of the driving force at the inlet and the outlet

In addition to balance on the liquid the gas supplying the oxygen also conforms to material balance The balance on the amount of oxygen supplied to the reactor at the intlet and the amout at the outlet is the amount that has been transferred to the liquid phase In principle in most reactors above the liquid there is also a headspace with gas and should be considered for balance but in general the gas composition in the head space is considered to be the same as that in the exit Thus oxygen transferred into the liquid phase by gas phase is one line

OTR (dot V-(oxygen concentration at inlet) dot (inlet air flow rate) ndash(oxygen concentration at outlet)dot (outlet airflow rate)=NO2 in dot Qin-NO2out-Qout (asme as in slide)

With a quasi steady state assumption the oxygen transferred into liquid phase by gas phase is the same as that received by liquid phase both being OTR Furthermore OTR can be assumed equal to OUR

If the exit gas (or falled off gas) composition is measured on line using mass spectrometer or other oxygen gas sensor one can easily obtain the volumetric oxygen uptake rate In some cases kLa is relative constant such in the case of some cells can be calculated from C if C is kept constant If off-gas is not measured one can resort to a dynamic method The method involves switching gas to a composition that is an equivalent to the current dissolved oxygen leve At that moment no net transfer of oxygen may occur (because ∆ C=0) As a result the change of dissolved oxygen is only caused by cell consumption From the slope one can obtain OUR

The objective in oxygen transfer is to supply oxygen at a rate that is enough to maintain the dissolved oxygen at a level suitable (or optimal) for cell growth and production For most mammalian cells a 30 saturation with ambient air is sufficient for optimal growth The amount of oxygen cells consumed per culture volume is referred to as Oxygen Uptake Rate (OUR) As you may recall from the kinetics chapter OUR is dependent on the specific oxygen consumption rate (qO2) and the cell concentration In culture the change of oxygen concentration with time is thus the balance between oxygen transfer rate (OTR) and oxygen uptake rate

Most oxygen taken up by cells is used to oxidize carbon sources The most important carbon sources are glucose and glutamine For both the oxidation reaction yield a

Optimal Oxygen Concentrationbull Most cells grow well at a dissolved oxygen level of 30

of saturation with air bull When oxygen-enriched gas is used sometimes the

dissolved oxygen level goes above saturation A short duration of above saturation level of oxygen is usually not detrimental to cells

bull The OTR efficiency is higher with a larger oxygen concentration gradient between gas and medium (driving force) Maintaining DO at an unnecessarily high level decreases the driving force

Oxygen Consumption and CO2 Production

10 Oxygen Transfer

Integrate

ln( ) LC C K a tminus = sdot

2 2( )L O

dCV OTR V OUR Vdt

K a V C C q x V

= sdot minus sdot

= sdot sdot minus minus sdot sdot

Oxygen Demand by Cellsbull On the average q02 asymp 10 x 10-10 mmolecell-hr

CO2 Production by Cellsbull Respiratory quotient of cells is about 10 ie 1 mole CO2

is produced for every mole of O2 consumed or qCO2 asymp 10 x 10-10 mmolecell-hr

bull A high CO2 concentration is inhibitorybull Inhibitory level usually starts at 15 to 20 CO2

Experimental Measurement of KLa and OUR

stoichiometric ratio of one for CO2 to O2 (called respiratory quotient [RQ]) The CO2 produced by cells will need to be removed from the culture to avoid excessive accumulationExcess amount of either dissolved O2 or CO2 can be growth inhibiting or toxic to the cells However the growth inhibitory effect of O2 and CO2 take an exposure time to be seen A short exposure of high concentration (gt100 saturation of O2 or nearly 150 mmHg of CO2) up to a few hours may not be detrimental

The mass transfer coefficient can be estimated by using some empirical correlations developed in the form of dimension-less numbers There has been a report that KL is relatively insensitive to bubble size except in small scale reactors and under the conditions that the liquid flow pattern is relatively simple like most bubbles rise in a simple pattern the esti-mated KL is probably not reliable Furthermore except un-der those conditions a is also difficult to estimate as relied on direct experimental measurement of KLa is often lumped together and called volumetric mass transfer coefficient in-stead of having KL and a estimated or measured separately Many bioprocess plants measure gas composition from the biotreactor inlet and outlet From the balance equa-tion one can see that the OUR and OTR calculated from the gas balance can also be used to calculate the average KLa in the reactor since Cand C are both measured One can also use a dynamic method to measure KLa One way to do this is by stripping the dissolved oxygen off using N2 to replace air (or the original gas phase) In this case

Oxygen Transfer 11

Mass Transfer Resistance in Cell Immobilization Reactor

Intraparticle Diffusion Limitation

Criteria for assessing the magnitude of mass-transfer effects on overall kinetics

Criterion η value

Limiting rate process

Extent of mass-transfer

limitation

Φ lt 03 ~1 Chemical reaction Negligible

Φ gt 3 αФ-1 Diffusion Large

Or Φ critical the critical value at which η = 1

12 Oxygen Transfer

bull The shape of the curve varies with reaction kinetics and the shape of the particles

bull Zero order kinetics is often a good assumptionbull Oxygen is usually the first compound to become

limiting because of its low solubility

• Oxygen Transfer Through Gas-Liquid Interface
• Oxygen and Carbon Dioxide Concentration in Medium
• Oxygen Concentration in Culture and Oxygen Transfer
• Oxygen Diffusion and Order of Magnitude Analysis
• Oxygen Transfer through Interface of Two Phases
• Oxygen Transfer in bioreactors
• Oxygen Consumption and CO2 Production
• Experimental Measurement of KLa and OUR
• Mass Transfer Resistance in Cell Immobilization Reactor
• Intraparticle Diffusion Limitation

4 Oxygen Transfer

faster than glucose Thus oxygen will need to be continu-ously supplied for an aerobe while glucose and other nu-trients can only be supplied at the beginning of the culture

The supply of oxygen to cultured organisms serves not only to prevent it from depleting but also to maintain it in an optimal range Most aerobic microorganisms grow well in a very wide range of dissolved oxygen concentration up to saturation with ambient air However some cells from higher organisms grow better under unsaturated concen-tration For example stem cells from mammalian species have been reported to grow better when the gas used for cultivation contains 5 O2 instead of 21 Some tissues such as bone marrow have a lower concentration and cells from such tissues may prefer somewhat lower oxygen con-centration However except obligative or facultative anaer-obes all other organisms require oxygen levels be sustained about certain level otherwise adverse effects on growth and survival is seen This critical level of oxygen is typically about 20 of saturation with air for aerobic microorgan-isms (or about 001 mM) For mammalian cells such lung fibroblasts 3T3 cells Chinese Hamster Ovary Cells the critical level is around 10 of saturation with air at 1atm

The supply of oxygen to cells in culture is typically achieved through a gas phase to the liquid phase (ie the medium in which cells are growing) There are a number of steps that oxygen molecules need to traverse through in order to reach the cells and eventually the intracellular sites where the reaction cosuming oxygen occurs This is described in the panel A question to ask is tha among all the steps in-volved which one is rate limiting or is the controlling step

A very effective way of identifying the rate controlling step without resorting to solving a set of differential equations is by order of magnitude analysis of time constant In this type of analysis instead of considering the detailed concentration changes over time in a system of fixed geometry one tries to examine the amount of time (called characteristic time or time constant) it takes to change a set of concentrations of a chemical species (characteristic concentration) over a distance (characteristic length) In such order of magnitude analysis one compares the ldquocharacteristic timerdquo needed for mass (or heat) transfer in different systems that have different characteristic lengths or a different transfer mechanism The step that has the longest time constant is the slowest one

Consider a solute that is soluble in two different ldquophasesrdquo (liquid-gas gas-solid liquid-solid etc) Given the two phases co-exist for a long enough time the concentrations in the two phases will be in equilibrium If suddenly the concentration in one phase is increased to a new level the solute will be transferred through the interface into the other phase and the

Oxygen Diffusion and Order of Magnitude Analysis

bullemsp ResistancesemspforemspO2emsptransferbullemsp Diffusionemsptoemsptheemspgas-liquidemspinterfacebullemsp Solutionemspofemsptheemspgasemspintoemsptheemspliquidemspatemsptheemspemsp emspemsp interfacebullemsp Diffusionemspfromemsptheemspliquidemspsideemspofemsptheemspinterfaceemspemspemsp intoemsptheemspliquidemspbulkbullemsp Diffusionemspfromemsptheemspbulkemspliquidemspnearemsptheemspsurfaceemspemspemsp ofemspgasemspbubbleemsptoemspbulkemspliquidemspnearemsptheemspemsp emspemsp surfaceemspofemspcellsbullemsp Diffusionemspfromemsptheemspliquidemspbulkemspthroughemspliquidemspemsp emspemsp filmemspsurroundingemsptheemspcellbullemsp Diffusionemspthroughemspcellemspclumpemsppelletemsporemspmyceliabullemsp Consumptionemspreactionemspinternalemspdiffusionbullemsp Isemspintracellularemspoxygenemspdiffusionemspimportant

Oxygen Transfer 5

concentration in the other phase will continue to increase For simplicity let us assume that the concentration at the interface will remain constant In other words the transfer of solute into the other phase does not cause the concentration in the first phase to decrease This would be the case that the first phase has a very large capacity (eg large volume) relative to the second phase so that even some solute is being transferred into the second phase the amount transferred is relatively small to the total in the first phaseIf we look at a slice of the second phase perpendicular to the direction of transfer and do a material balance on this slice the balance will be

Slice is referred to as ldquocontrol volumerdquo

Its equation form is shown in the panel In general whenever a solute is carried by a fluid (this type of transport is convection) its rate of transport in the direction of fluid flow is far greater (by orders of magnitude) than that from diffusion unless the flow rate is extremely slow We will consider only the case that convection is not involved in transfer and especially in the case of transfer across the interface The convection caused by solute transfer is generally negligible So the case we consider is the transport into a stagnant liquid of a low solubility chemical species (so that the transport of solute does not cause convection) or in a solid

By neglecting convection the equation is simplified We first examine a condition that in the medium (ie the stagnant liquid or the solid) no reaction occurs ie as solute is traveling down the medium it does not get reacted (ie consumed) nor does the medium produce any solute by itself

The diffusion term is generally described by Fickrsquos Law The diffusion is driven by concentration difference between two points in space If the concentration in position 2 is greater than position 2 the solute will diffuse in the direction toward 2 or in the ldquonegativerdquo direction In equation form it says

C x∆ ∆ is the concentration difference or the concentraton gradient J is the flux of the solute The negative sign accounts for the direction of transfer being opposite the sign of the slope The proportionality between flux and the concentration gradient is the diffusion coefficient as shown belowJ is a measure of transfer rate (per surface area)

rate of solute rate of change of solute concentration rate of solute transported rate of solute transported

concentration change in the control volume by fluid flow by diffusion

due

= + +

to reaction

2 1

2 1

(flux of solute) - c c CJx x x

α minus ∆= minus

minus ∆

3 2( ) ( ) ( )m L L T M L Tbull = bull

6 Oxygen Transfer

normal to the direction of transfer It has units of

The diffusion coefficient has units of 2 L t Its magnitude is dependent on the molecular species and the medium it diffuses in For example oxygen and glucose and small proteins have very different diffusion coefficients in water (approximately 10-4 10-6 10-8 cm2sec for oxygen glucose and protein respectively) which the diffusion coefficient of oxygen in air lipid and water decreases progressively Considering only diffusion in the medium the change of the oxygen concentration in the control volume is the difference between the diffusion at positions x and x x+ ∆ (assume diffusion is one dimensional thus ignoring bottom and top

surface) At surface x the diffusion rate is -D D xx

∆∆

While at x+ox it is CD x oxx

∆+

∆

Taking differential between the two planes and assuming D

is constant we obtain CD

x x∆ ∆

∆ ∆

The equation for the control volume thus becomes

2

2

2 2 02 2c CDt x

= = or 22 2

2 2c CDt x

=

Instead of solving the partial differential equation we can do an order of magnitude estimation on each term The left

hand side 22ct

can be estimated as

The order of magnitude for x∆ is C1-C0 Again that can be estimated as S the penetration depth or the characteristic length So overall combining right and left hand sides

2d

C CDt g

∆ ∆

So 2

dgtD

( )2t

x∆

∆

3 2( ) ( ) ( )m L L T M L Tbull = bull

Oxygen Transfer 7

Oxygen Transfer through Interface of Two Phases

The characteristic length is usually chosen to represent the ldquopenetration depthrdquo For a long slab with diffusion from the two surfaces a long cylinder (both neglecting the diffusion from the two ends) and a sphere the characteristic length is shown in the table One can calculate the concentration of the solute as of the final equivalent to a characteristic time As can be seen after one characteristic time the concentration at a characteristic length is very close to equilibriumOne can estimate the time constant for convection and different types of reaction kinetics similarly as shown in the Table

Going back to oxygen transfer problems in bioreactors the step that incurs most resistance is the transfer between gas phase and liquid phase Microscopically the transfer can be depicted as oxygen molecules passing through two stagnant films one on the gas side the other on the liquid side as shown in Figure 1 We have introduced the equation for the rate of transfer This overall transfer is the net result of transfer through the two films For each film we can write an equation to describe the rate using kL and kg for liquid-side and gas-side transfer coefficient respectively A can be considered the same for both gas film and liquid film At the interface between the two films the two phases can be considered to be in equilibrium ie Cli=HPgi (recall that at equilibrium the concentration at the two phases is related by Henryrsquos Law constant)

One can write the equation for oxygen transfer rate (OTR) in three different forms overall liquid-side and gas-side Since there is no accumulation in the films the rate expressed by the three equations should be equal As shown in the panel by combining liquid side and gas concentration gradient to give the overall concentration gradient one can show that the inverse of Kl is the sum of the inverse of kl and hg If one considers K kl kg as the conductivity the 1KL 1kl 1kg gives the resistance In other words it shows that the overall mass transfer resistance 1kl is the sum of the two film resistances 1kl and 1kg

In using kl or kg for calculating OTR one has to use the interface concentration (Cli Pgi) for calculating concentration gradient The concentration at interface cannot be determined readily By using overall transfer the driving force used becomes the Por C or the concentration of the two phases were at equilibrium For this case it is a measure of how far away the system is from the equilibrium P and C can be readily calculated

Note that one can also use Kg for overall mass transfer coefficient that gives OTR=Kga(Pg-P)But traditionally Kl is used We just discussed that the overall resistance 1kL is the sum of 1kL and 1kg But what is the

8 Oxygen Transfer

Oxygen Transfer in bioreactors

relative magnitude of 1kL and 1kg The magnitude of mass transfer coefficient is dictated by the diffusion coefficient and the film thickness (Dδ) The film thickness for gas and liquid side is similar While the diffusion coefficient on the gas side and liquid side are orders of magnitude dififerent It can be shown that the liquid side resistance dominates the overall resistance It is thus unusual that kLis used in place of kL even when overall driving force is used

KL has units of Lt (like velocity) and the interfacial area has units of 1L After multiplying by a concentration the mass transfer equation gives a concentration perimeter that is the rate of transfer The interfacial area term is interfacial area for oxygen transfer per liquid volume aV A is the total amount of transfer area in the reactor V is the total liquid volume In a large bioreactor aerated by sparging air bubbles the bubbles are not necessarily evenly distributed and local kL is not necessarily the same The oxygen transfer rate discussed in literature is mostly an average over the reactor

OUR is the product of specific oxygen concentration and cell concentration (q0χ) Obviously if OTR is lower than OUR then dCdt will be negative and dissolved oxygen level will decrease In a culture the dissolved oxygen level decreases as cell concentration increases With a dissolved oxygen the controller manipulates the agitation rate air flow rate or the oxygen content (ie by influence kL or triangle C0 to maintain dissolved oxygen levels at the set point But even under uncontrolled conditions that change of dissolved oxygen level is not fast all the time Consider the three terms in liquid phase oxygen balance equations If the accumulation term (dcdt) is small compared to OTR and OUR one can consider the system is at a quasi steady state ie dCdt ~) thus OTR=OUR

In a typical bioreactor the oxygen level in the gas phase (Pg or C) decreases from the level at the inlet to the level at the outlet while it traverses through the reactor as oxygen is being transferred into liquid phase The question thus arises as to which C should be used the one corresponding to gas at the inlet or something else If the reactor is a small stirred tank that one can consider to be well mixed then CL will be the same in the entire liquid phase and so is the gas phase concentration In a well mixed reactor one assumes that the sample taken at any point is the same as that in the well mixed phase Therefore C should be that of the exit gas stream not the inletIn a large reactor the gas phase is likely to behave closer to plug flow One then uses logrithmic mean to calculate the driving force This is analogous to using logrithmic mean driving force in heat transfer in a double-pipe heat exchanger Basically it is an average (but a simple geometrical

Oxygen Transfer 9

OTRV=(oxygen concentration at inlet)(inlet airflow rate)-(oxygen concentration in outlet)(outlet airflow rate)

average) of the driving force at the inlet and the outlet

In addition to balance on the liquid the gas supplying the oxygen also conforms to material balance The balance on the amount of oxygen supplied to the reactor at the intlet and the amout at the outlet is the amount that has been transferred to the liquid phase In principle in most reactors above the liquid there is also a headspace with gas and should be considered for balance but in general the gas composition in the head space is considered to be the same as that in the exit Thus oxygen transferred into the liquid phase by gas phase is one line

OTR (dot V-(oxygen concentration at inlet) dot (inlet air flow rate) ndash(oxygen concentration at outlet)dot (outlet airflow rate)=NO2 in dot Qin-NO2out-Qout (asme as in slide)

With a quasi steady state assumption the oxygen transferred into liquid phase by gas phase is the same as that received by liquid phase both being OTR Furthermore OTR can be assumed equal to OUR

If the exit gas (or falled off gas) composition is measured on line using mass spectrometer or other oxygen gas sensor one can easily obtain the volumetric oxygen uptake rate In some cases kLa is relative constant such in the case of some cells can be calculated from C if C is kept constant If off-gas is not measured one can resort to a dynamic method The method involves switching gas to a composition that is an equivalent to the current dissolved oxygen leve At that moment no net transfer of oxygen may occur (because ∆ C=0) As a result the change of dissolved oxygen is only caused by cell consumption From the slope one can obtain OUR

The objective in oxygen transfer is to supply oxygen at a rate that is enough to maintain the dissolved oxygen at a level suitable (or optimal) for cell growth and production For most mammalian cells a 30 saturation with ambient air is sufficient for optimal growth The amount of oxygen cells consumed per culture volume is referred to as Oxygen Uptake Rate (OUR) As you may recall from the kinetics chapter OUR is dependent on the specific oxygen consumption rate (qO2) and the cell concentration In culture the change of oxygen concentration with time is thus the balance between oxygen transfer rate (OTR) and oxygen uptake rate

Most oxygen taken up by cells is used to oxidize carbon sources The most important carbon sources are glucose and glutamine For both the oxidation reaction yield a

Optimal Oxygen Concentrationbull Most cells grow well at a dissolved oxygen level of 30

of saturation with air bull When oxygen-enriched gas is used sometimes the

dissolved oxygen level goes above saturation A short duration of above saturation level of oxygen is usually not detrimental to cells

bull The OTR efficiency is higher with a larger oxygen concentration gradient between gas and medium (driving force) Maintaining DO at an unnecessarily high level decreases the driving force

Oxygen Consumption and CO2 Production

10 Oxygen Transfer

Integrate

ln( ) LC C K a tminus = sdot

2 2( )L O

dCV OTR V OUR Vdt

K a V C C q x V

= sdot minus sdot

= sdot sdot minus minus sdot sdot

Oxygen Demand by Cellsbull On the average q02 asymp 10 x 10-10 mmolecell-hr

CO2 Production by Cellsbull Respiratory quotient of cells is about 10 ie 1 mole CO2

is produced for every mole of O2 consumed or qCO2 asymp 10 x 10-10 mmolecell-hr

bull A high CO2 concentration is inhibitorybull Inhibitory level usually starts at 15 to 20 CO2

Experimental Measurement of KLa and OUR

stoichiometric ratio of one for CO2 to O2 (called respiratory quotient [RQ]) The CO2 produced by cells will need to be removed from the culture to avoid excessive accumulationExcess amount of either dissolved O2 or CO2 can be growth inhibiting or toxic to the cells However the growth inhibitory effect of O2 and CO2 take an exposure time to be seen A short exposure of high concentration (gt100 saturation of O2 or nearly 150 mmHg of CO2) up to a few hours may not be detrimental

The mass transfer coefficient can be estimated by using some empirical correlations developed in the form of dimension-less numbers There has been a report that KL is relatively insensitive to bubble size except in small scale reactors and under the conditions that the liquid flow pattern is relatively simple like most bubbles rise in a simple pattern the esti-mated KL is probably not reliable Furthermore except un-der those conditions a is also difficult to estimate as relied on direct experimental measurement of KLa is often lumped together and called volumetric mass transfer coefficient in-stead of having KL and a estimated or measured separately Many bioprocess plants measure gas composition from the biotreactor inlet and outlet From the balance equa-tion one can see that the OUR and OTR calculated from the gas balance can also be used to calculate the average KLa in the reactor since Cand C are both measured One can also use a dynamic method to measure KLa One way to do this is by stripping the dissolved oxygen off using N2 to replace air (or the original gas phase) In this case

Oxygen Transfer 11

Mass Transfer Resistance in Cell Immobilization Reactor

Intraparticle Diffusion Limitation

Criteria for assessing the magnitude of mass-transfer effects on overall kinetics

Criterion η value

Limiting rate process

Extent of mass-transfer

limitation

Φ lt 03 ~1 Chemical reaction Negligible

Φ gt 3 αФ-1 Diffusion Large

Or Φ critical the critical value at which η = 1

12 Oxygen Transfer

bull The shape of the curve varies with reaction kinetics and the shape of the particles

bull Zero order kinetics is often a good assumptionbull Oxygen is usually the first compound to become

limiting because of its low solubility

• Oxygen Transfer Through Gas-Liquid Interface
• Oxygen and Carbon Dioxide Concentration in Medium
• Oxygen Concentration in Culture and Oxygen Transfer
• Oxygen Diffusion and Order of Magnitude Analysis
• Oxygen Transfer through Interface of Two Phases
• Oxygen Transfer in bioreactors
• Oxygen Consumption and CO2 Production
• Experimental Measurement of KLa and OUR
• Mass Transfer Resistance in Cell Immobilization Reactor
• Intraparticle Diffusion Limitation

Oxygen Transfer 5

concentration in the other phase will continue to increase For simplicity let us assume that the concentration at the interface will remain constant In other words the transfer of solute into the other phase does not cause the concentration in the first phase to decrease This would be the case that the first phase has a very large capacity (eg large volume) relative to the second phase so that even some solute is being transferred into the second phase the amount transferred is relatively small to the total in the first phaseIf we look at a slice of the second phase perpendicular to the direction of transfer and do a material balance on this slice the balance will be

Slice is referred to as ldquocontrol volumerdquo

Its equation form is shown in the panel In general whenever a solute is carried by a fluid (this type of transport is convection) its rate of transport in the direction of fluid flow is far greater (by orders of magnitude) than that from diffusion unless the flow rate is extremely slow We will consider only the case that convection is not involved in transfer and especially in the case of transfer across the interface The convection caused by solute transfer is generally negligible So the case we consider is the transport into a stagnant liquid of a low solubility chemical species (so that the transport of solute does not cause convection) or in a solid

By neglecting convection the equation is simplified We first examine a condition that in the medium (ie the stagnant liquid or the solid) no reaction occurs ie as solute is traveling down the medium it does not get reacted (ie consumed) nor does the medium produce any solute by itself

The diffusion term is generally described by Fickrsquos Law The diffusion is driven by concentration difference between two points in space If the concentration in position 2 is greater than position 2 the solute will diffuse in the direction toward 2 or in the ldquonegativerdquo direction In equation form it says

C x∆ ∆ is the concentration difference or the concentraton gradient J is the flux of the solute The negative sign accounts for the direction of transfer being opposite the sign of the slope The proportionality between flux and the concentration gradient is the diffusion coefficient as shown belowJ is a measure of transfer rate (per surface area)

rate of solute rate of change of solute concentration rate of solute transported rate of solute transported

concentration change in the control volume by fluid flow by diffusion

due

= + +

to reaction

2 1

2 1

(flux of solute) - c c CJx x x

α minus ∆= minus

minus ∆

3 2( ) ( ) ( )m L L T M L Tbull = bull

6 Oxygen Transfer

normal to the direction of transfer It has units of

The diffusion coefficient has units of 2 L t Its magnitude is dependent on the molecular species and the medium it diffuses in For example oxygen and glucose and small proteins have very different diffusion coefficients in water (approximately 10-4 10-6 10-8 cm2sec for oxygen glucose and protein respectively) which the diffusion coefficient of oxygen in air lipid and water decreases progressively Considering only diffusion in the medium the change of the oxygen concentration in the control volume is the difference between the diffusion at positions x and x x+ ∆ (assume diffusion is one dimensional thus ignoring bottom and top

surface) At surface x the diffusion rate is -D D xx

∆∆

While at x+ox it is CD x oxx

∆+

∆

Taking differential between the two planes and assuming D

is constant we obtain CD

x x∆ ∆

∆ ∆

The equation for the control volume thus becomes

2

2

2 2 02 2c CDt x

= = or 22 2

2 2c CDt x

=

Instead of solving the partial differential equation we can do an order of magnitude estimation on each term The left

hand side 22ct

can be estimated as

The order of magnitude for x∆ is C1-C0 Again that can be estimated as S the penetration depth or the characteristic length So overall combining right and left hand sides

2d

C CDt g

∆ ∆

So 2

dgtD

( )2t

x∆

∆

3 2( ) ( ) ( )m L L T M L Tbull = bull

Oxygen Transfer 7

Oxygen Transfer through Interface of Two Phases

The characteristic length is usually chosen to represent the ldquopenetration depthrdquo For a long slab with diffusion from the two surfaces a long cylinder (both neglecting the diffusion from the two ends) and a sphere the characteristic length is shown in the table One can calculate the concentration of the solute as of the final equivalent to a characteristic time As can be seen after one characteristic time the concentration at a characteristic length is very close to equilibriumOne can estimate the time constant for convection and different types of reaction kinetics similarly as shown in the Table

Going back to oxygen transfer problems in bioreactors the step that incurs most resistance is the transfer between gas phase and liquid phase Microscopically the transfer can be depicted as oxygen molecules passing through two stagnant films one on the gas side the other on the liquid side as shown in Figure 1 We have introduced the equation for the rate of transfer This overall transfer is the net result of transfer through the two films For each film we can write an equation to describe the rate using kL and kg for liquid-side and gas-side transfer coefficient respectively A can be considered the same for both gas film and liquid film At the interface between the two films the two phases can be considered to be in equilibrium ie Cli=HPgi (recall that at equilibrium the concentration at the two phases is related by Henryrsquos Law constant)

One can write the equation for oxygen transfer rate (OTR) in three different forms overall liquid-side and gas-side Since there is no accumulation in the films the rate expressed by the three equations should be equal As shown in the panel by combining liquid side and gas concentration gradient to give the overall concentration gradient one can show that the inverse of Kl is the sum of the inverse of kl and hg If one considers K kl kg as the conductivity the 1KL 1kl 1kg gives the resistance In other words it shows that the overall mass transfer resistance 1kl is the sum of the two film resistances 1kl and 1kg

In using kl or kg for calculating OTR one has to use the interface concentration (Cli Pgi) for calculating concentration gradient The concentration at interface cannot be determined readily By using overall transfer the driving force used becomes the Por C or the concentration of the two phases were at equilibrium For this case it is a measure of how far away the system is from the equilibrium P and C can be readily calculated

Note that one can also use Kg for overall mass transfer coefficient that gives OTR=Kga(Pg-P)But traditionally Kl is used We just discussed that the overall resistance 1kL is the sum of 1kL and 1kg But what is the

8 Oxygen Transfer

Oxygen Transfer in bioreactors

relative magnitude of 1kL and 1kg The magnitude of mass transfer coefficient is dictated by the diffusion coefficient and the film thickness (Dδ) The film thickness for gas and liquid side is similar While the diffusion coefficient on the gas side and liquid side are orders of magnitude dififerent It can be shown that the liquid side resistance dominates the overall resistance It is thus unusual that kLis used in place of kL even when overall driving force is used

KL has units of Lt (like velocity) and the interfacial area has units of 1L After multiplying by a concentration the mass transfer equation gives a concentration perimeter that is the rate of transfer The interfacial area term is interfacial area for oxygen transfer per liquid volume aV A is the total amount of transfer area in the reactor V is the total liquid volume In a large bioreactor aerated by sparging air bubbles the bubbles are not necessarily evenly distributed and local kL is not necessarily the same The oxygen transfer rate discussed in literature is mostly an average over the reactor

OUR is the product of specific oxygen concentration and cell concentration (q0χ) Obviously if OTR is lower than OUR then dCdt will be negative and dissolved oxygen level will decrease In a culture the dissolved oxygen level decreases as cell concentration increases With a dissolved oxygen the controller manipulates the agitation rate air flow rate or the oxygen content (ie by influence kL or triangle C0 to maintain dissolved oxygen levels at the set point But even under uncontrolled conditions that change of dissolved oxygen level is not fast all the time Consider the three terms in liquid phase oxygen balance equations If the accumulation term (dcdt) is small compared to OTR and OUR one can consider the system is at a quasi steady state ie dCdt ~) thus OTR=OUR

In a typical bioreactor the oxygen level in the gas phase (Pg or C) decreases from the level at the inlet to the level at the outlet while it traverses through the reactor as oxygen is being transferred into liquid phase The question thus arises as to which C should be used the one corresponding to gas at the inlet or something else If the reactor is a small stirred tank that one can consider to be well mixed then CL will be the same in the entire liquid phase and so is the gas phase concentration In a well mixed reactor one assumes that the sample taken at any point is the same as that in the well mixed phase Therefore C should be that of the exit gas stream not the inletIn a large reactor the gas phase is likely to behave closer to plug flow One then uses logrithmic mean to calculate the driving force This is analogous to using logrithmic mean driving force in heat transfer in a double-pipe heat exchanger Basically it is an average (but a simple geometrical

Oxygen Transfer 9

OTRV=(oxygen concentration at inlet)(inlet airflow rate)-(oxygen concentration in outlet)(outlet airflow rate)

average) of the driving force at the inlet and the outlet

In addition to balance on the liquid the gas supplying the oxygen also conforms to material balance The balance on the amount of oxygen supplied to the reactor at the intlet and the amout at the outlet is the amount that has been transferred to the liquid phase In principle in most reactors above the liquid there is also a headspace with gas and should be considered for balance but in general the gas composition in the head space is considered to be the same as that in the exit Thus oxygen transferred into the liquid phase by gas phase is one line

OTR (dot V-(oxygen concentration at inlet) dot (inlet air flow rate) ndash(oxygen concentration at outlet)dot (outlet airflow rate)=NO2 in dot Qin-NO2out-Qout (asme as in slide)

With a quasi steady state assumption the oxygen transferred into liquid phase by gas phase is the same as that received by liquid phase both being OTR Furthermore OTR can be assumed equal to OUR

If the exit gas (or falled off gas) composition is measured on line using mass spectrometer or other oxygen gas sensor one can easily obtain the volumetric oxygen uptake rate In some cases kLa is relative constant such in the case of some cells can be calculated from C if C is kept constant If off-gas is not measured one can resort to a dynamic method The method involves switching gas to a composition that is an equivalent to the current dissolved oxygen leve At that moment no net transfer of oxygen may occur (because ∆ C=0) As a result the change of dissolved oxygen is only caused by cell consumption From the slope one can obtain OUR

The objective in oxygen transfer is to supply oxygen at a rate that is enough to maintain the dissolved oxygen at a level suitable (or optimal) for cell growth and production For most mammalian cells a 30 saturation with ambient air is sufficient for optimal growth The amount of oxygen cells consumed per culture volume is referred to as Oxygen Uptake Rate (OUR) As you may recall from the kinetics chapter OUR is dependent on the specific oxygen consumption rate (qO2) and the cell concentration In culture the change of oxygen concentration with time is thus the balance between oxygen transfer rate (OTR) and oxygen uptake rate

Most oxygen taken up by cells is used to oxidize carbon sources The most important carbon sources are glucose and glutamine For both the oxidation reaction yield a

Optimal Oxygen Concentrationbull Most cells grow well at a dissolved oxygen level of 30

of saturation with air bull When oxygen-enriched gas is used sometimes the

dissolved oxygen level goes above saturation A short duration of above saturation level of oxygen is usually not detrimental to cells

bull The OTR efficiency is higher with a larger oxygen concentration gradient between gas and medium (driving force) Maintaining DO at an unnecessarily high level decreases the driving force

Oxygen Consumption and CO2 Production

10 Oxygen Transfer

Integrate

ln( ) LC C K a tminus = sdot

2 2( )L O

dCV OTR V OUR Vdt

K a V C C q x V

= sdot minus sdot

= sdot sdot minus minus sdot sdot

Oxygen Demand by Cellsbull On the average q02 asymp 10 x 10-10 mmolecell-hr

CO2 Production by Cellsbull Respiratory quotient of cells is about 10 ie 1 mole CO2

is produced for every mole of O2 consumed or qCO2 asymp 10 x 10-10 mmolecell-hr

bull A high CO2 concentration is inhibitorybull Inhibitory level usually starts at 15 to 20 CO2

Experimental Measurement of KLa and OUR

stoichiometric ratio of one for CO2 to O2 (called respiratory quotient [RQ]) The CO2 produced by cells will need to be removed from the culture to avoid excessive accumulationExcess amount of either dissolved O2 or CO2 can be growth inhibiting or toxic to the cells However the growth inhibitory effect of O2 and CO2 take an exposure time to be seen A short exposure of high concentration (gt100 saturation of O2 or nearly 150 mmHg of CO2) up to a few hours may not be detrimental

The mass transfer coefficient can be estimated by using some empirical correlations developed in the form of dimension-less numbers There has been a report that KL is relatively insensitive to bubble size except in small scale reactors and under the conditions that the liquid flow pattern is relatively simple like most bubbles rise in a simple pattern the esti-mated KL is probably not reliable Furthermore except un-der those conditions a is also difficult to estimate as relied on direct experimental measurement of KLa is often lumped together and called volumetric mass transfer coefficient in-stead of having KL and a estimated or measured separately Many bioprocess plants measure gas composition from the biotreactor inlet and outlet From the balance equa-tion one can see that the OUR and OTR calculated from the gas balance can also be used to calculate the average KLa in the reactor since Cand C are both measured One can also use a dynamic method to measure KLa One way to do this is by stripping the dissolved oxygen off using N2 to replace air (or the original gas phase) In this case

Oxygen Transfer 11

Mass Transfer Resistance in Cell Immobilization Reactor

Intraparticle Diffusion Limitation

Criteria for assessing the magnitude of mass-transfer effects on overall kinetics

Criterion η value

Limiting rate process

Extent of mass-transfer

limitation

Φ lt 03 ~1 Chemical reaction Negligible

Φ gt 3 αФ-1 Diffusion Large

Or Φ critical the critical value at which η = 1

12 Oxygen Transfer

bull The shape of the curve varies with reaction kinetics and the shape of the particles

bull Zero order kinetics is often a good assumptionbull Oxygen is usually the first compound to become

limiting because of its low solubility

• Oxygen Transfer Through Gas-Liquid Interface
• Oxygen and Carbon Dioxide Concentration in Medium
• Oxygen Concentration in Culture and Oxygen Transfer
• Oxygen Diffusion and Order of Magnitude Analysis
• Oxygen Transfer through Interface of Two Phases
• Oxygen Transfer in bioreactors
• Oxygen Consumption and CO2 Production
• Experimental Measurement of KLa and OUR
• Mass Transfer Resistance in Cell Immobilization Reactor
• Intraparticle Diffusion Limitation

6 Oxygen Transfer

normal to the direction of transfer It has units of

The diffusion coefficient has units of 2 L t Its magnitude is dependent on the molecular species and the medium it diffuses in For example oxygen and glucose and small proteins have very different diffusion coefficients in water (approximately 10-4 10-6 10-8 cm2sec for oxygen glucose and protein respectively) which the diffusion coefficient of oxygen in air lipid and water decreases progressively Considering only diffusion in the medium the change of the oxygen concentration in the control volume is the difference between the diffusion at positions x and x x+ ∆ (assume diffusion is one dimensional thus ignoring bottom and top

surface) At surface x the diffusion rate is -D D xx

∆∆

While at x+ox it is CD x oxx

∆+

∆

Taking differential between the two planes and assuming D

is constant we obtain CD

x x∆ ∆

∆ ∆

The equation for the control volume thus becomes

2

2

2 2 02 2c CDt x

= = or 22 2

2 2c CDt x

=

Instead of solving the partial differential equation we can do an order of magnitude estimation on each term The left

hand side 22ct

can be estimated as

The order of magnitude for x∆ is C1-C0 Again that can be estimated as S the penetration depth or the characteristic length So overall combining right and left hand sides

2d

C CDt g

∆ ∆

So 2

dgtD

( )2t

x∆

∆

3 2( ) ( ) ( )m L L T M L Tbull = bull

Oxygen Transfer 7

Oxygen Transfer through Interface of Two Phases

The characteristic length is usually chosen to represent the ldquopenetration depthrdquo For a long slab with diffusion from the two surfaces a long cylinder (both neglecting the diffusion from the two ends) and a sphere the characteristic length is shown in the table One can calculate the concentration of the solute as of the final equivalent to a characteristic time As can be seen after one characteristic time the concentration at a characteristic length is very close to equilibriumOne can estimate the time constant for convection and different types of reaction kinetics similarly as shown in the Table

Going back to oxygen transfer problems in bioreactors the step that incurs most resistance is the transfer between gas phase and liquid phase Microscopically the transfer can be depicted as oxygen molecules passing through two stagnant films one on the gas side the other on the liquid side as shown in Figure 1 We have introduced the equation for the rate of transfer This overall transfer is the net result of transfer through the two films For each film we can write an equation to describe the rate using kL and kg for liquid-side and gas-side transfer coefficient respectively A can be considered the same for both gas film and liquid film At the interface between the two films the two phases can be considered to be in equilibrium ie Cli=HPgi (recall that at equilibrium the concentration at the two phases is related by Henryrsquos Law constant)

One can write the equation for oxygen transfer rate (OTR) in three different forms overall liquid-side and gas-side Since there is no accumulation in the films the rate expressed by the three equations should be equal As shown in the panel by combining liquid side and gas concentration gradient to give the overall concentration gradient one can show that the inverse of Kl is the sum of the inverse of kl and hg If one considers K kl kg as the conductivity the 1KL 1kl 1kg gives the resistance In other words it shows that the overall mass transfer resistance 1kl is the sum of the two film resistances 1kl and 1kg

In using kl or kg for calculating OTR one has to use the interface concentration (Cli Pgi) for calculating concentration gradient The concentration at interface cannot be determined readily By using overall transfer the driving force used becomes the Por C or the concentration of the two phases were at equilibrium For this case it is a measure of how far away the system is from the equilibrium P and C can be readily calculated

Note that one can also use Kg for overall mass transfer coefficient that gives OTR=Kga(Pg-P)But traditionally Kl is used We just discussed that the overall resistance 1kL is the sum of 1kL and 1kg But what is the

8 Oxygen Transfer

Oxygen Transfer in bioreactors

relative magnitude of 1kL and 1kg The magnitude of mass transfer coefficient is dictated by the diffusion coefficient and the film thickness (Dδ) The film thickness for gas and liquid side is similar While the diffusion coefficient on the gas side and liquid side are orders of magnitude dififerent It can be shown that the liquid side resistance dominates the overall resistance It is thus unusual that kLis used in place of kL even when overall driving force is used

KL has units of Lt (like velocity) and the interfacial area has units of 1L After multiplying by a concentration the mass transfer equation gives a concentration perimeter that is the rate of transfer The interfacial area term is interfacial area for oxygen transfer per liquid volume aV A is the total amount of transfer area in the reactor V is the total liquid volume In a large bioreactor aerated by sparging air bubbles the bubbles are not necessarily evenly distributed and local kL is not necessarily the same The oxygen transfer rate discussed in literature is mostly an average over the reactor

OUR is the product of specific oxygen concentration and cell concentration (q0χ) Obviously if OTR is lower than OUR then dCdt will be negative and dissolved oxygen level will decrease In a culture the dissolved oxygen level decreases as cell concentration increases With a dissolved oxygen the controller manipulates the agitation rate air flow rate or the oxygen content (ie by influence kL or triangle C0 to maintain dissolved oxygen levels at the set point But even under uncontrolled conditions that change of dissolved oxygen level is not fast all the time Consider the three terms in liquid phase oxygen balance equations If the accumulation term (dcdt) is small compared to OTR and OUR one can consider the system is at a quasi steady state ie dCdt ~) thus OTR=OUR

In a typical bioreactor the oxygen level in the gas phase (Pg or C) decreases from the level at the inlet to the level at the outlet while it traverses through the reactor as oxygen is being transferred into liquid phase The question thus arises as to which C should be used the one corresponding to gas at the inlet or something else If the reactor is a small stirred tank that one can consider to be well mixed then CL will be the same in the entire liquid phase and so is the gas phase concentration In a well mixed reactor one assumes that the sample taken at any point is the same as that in the well mixed phase Therefore C should be that of the exit gas stream not the inletIn a large reactor the gas phase is likely to behave closer to plug flow One then uses logrithmic mean to calculate the driving force This is analogous to using logrithmic mean driving force in heat transfer in a double-pipe heat exchanger Basically it is an average (but a simple geometrical

Oxygen Transfer 9

OTRV=(oxygen concentration at inlet)(inlet airflow rate)-(oxygen concentration in outlet)(outlet airflow rate)

average) of the driving force at the inlet and the outlet

In addition to balance on the liquid the gas supplying the oxygen also conforms to material balance The balance on the amount of oxygen supplied to the reactor at the intlet and the amout at the outlet is the amount that has been transferred to the liquid phase In principle in most reactors above the liquid there is also a headspace with gas and should be considered for balance but in general the gas composition in the head space is considered to be the same as that in the exit Thus oxygen transferred into the liquid phase by gas phase is one line

OTR (dot V-(oxygen concentration at inlet) dot (inlet air flow rate) ndash(oxygen concentration at outlet)dot (outlet airflow rate)=NO2 in dot Qin-NO2out-Qout (asme as in slide)

With a quasi steady state assumption the oxygen transferred into liquid phase by gas phase is the same as that received by liquid phase both being OTR Furthermore OTR can be assumed equal to OUR

If the exit gas (or falled off gas) composition is measured on line using mass spectrometer or other oxygen gas sensor one can easily obtain the volumetric oxygen uptake rate In some cases kLa is relative constant such in the case of some cells can be calculated from C if C is kept constant If off-gas is not measured one can resort to a dynamic method The method involves switching gas to a composition that is an equivalent to the current dissolved oxygen leve At that moment no net transfer of oxygen may occur (because ∆ C=0) As a result the change of dissolved oxygen is only caused by cell consumption From the slope one can obtain OUR

The objective in oxygen transfer is to supply oxygen at a rate that is enough to maintain the dissolved oxygen at a level suitable (or optimal) for cell growth and production For most mammalian cells a 30 saturation with ambient air is sufficient for optimal growth The amount of oxygen cells consumed per culture volume is referred to as Oxygen Uptake Rate (OUR) As you may recall from the kinetics chapter OUR is dependent on the specific oxygen consumption rate (qO2) and the cell concentration In culture the change of oxygen concentration with time is thus the balance between oxygen transfer rate (OTR) and oxygen uptake rate

Most oxygen taken up by cells is used to oxidize carbon sources The most important carbon sources are glucose and glutamine For both the oxidation reaction yield a

Optimal Oxygen Concentrationbull Most cells grow well at a dissolved oxygen level of 30

of saturation with air bull When oxygen-enriched gas is used sometimes the

dissolved oxygen level goes above saturation A short duration of above saturation level of oxygen is usually not detrimental to cells

bull The OTR efficiency is higher with a larger oxygen concentration gradient between gas and medium (driving force) Maintaining DO at an unnecessarily high level decreases the driving force

Oxygen Consumption and CO2 Production

10 Oxygen Transfer

Integrate

ln( ) LC C K a tminus = sdot

2 2( )L O

dCV OTR V OUR Vdt

K a V C C q x V

= sdot minus sdot

= sdot sdot minus minus sdot sdot

Oxygen Demand by Cellsbull On the average q02 asymp 10 x 10-10 mmolecell-hr

CO2 Production by Cellsbull Respiratory quotient of cells is about 10 ie 1 mole CO2

is produced for every mole of O2 consumed or qCO2 asymp 10 x 10-10 mmolecell-hr

bull A high CO2 concentration is inhibitorybull Inhibitory level usually starts at 15 to 20 CO2

Experimental Measurement of KLa and OUR

stoichiometric ratio of one for CO2 to O2 (called respiratory quotient [RQ]) The CO2 produced by cells will need to be removed from the culture to avoid excessive accumulationExcess amount of either dissolved O2 or CO2 can be growth inhibiting or toxic to the cells However the growth inhibitory effect of O2 and CO2 take an exposure time to be seen A short exposure of high concentration (gt100 saturation of O2 or nearly 150 mmHg of CO2) up to a few hours may not be detrimental

The mass transfer coefficient can be estimated by using some empirical correlations developed in the form of dimension-less numbers There has been a report that KL is relatively insensitive to bubble size except in small scale reactors and under the conditions that the liquid flow pattern is relatively simple like most bubbles rise in a simple pattern the esti-mated KL is probably not reliable Furthermore except un-der those conditions a is also difficult to estimate as relied on direct experimental measurement of KLa is often lumped together and called volumetric mass transfer coefficient in-stead of having KL and a estimated or measured separately Many bioprocess plants measure gas composition from the biotreactor inlet and outlet From the balance equa-tion one can see that the OUR and OTR calculated from the gas balance can also be used to calculate the average KLa in the reactor since Cand C are both measured One can also use a dynamic method to measure KLa One way to do this is by stripping the dissolved oxygen off using N2 to replace air (or the original gas phase) In this case

Oxygen Transfer 11

Mass Transfer Resistance in Cell Immobilization Reactor

Intraparticle Diffusion Limitation

Criteria for assessing the magnitude of mass-transfer effects on overall kinetics

Criterion η value

Limiting rate process

Extent of mass-transfer

limitation

Φ lt 03 ~1 Chemical reaction Negligible

Φ gt 3 αФ-1 Diffusion Large

Or Φ critical the critical value at which η = 1

12 Oxygen Transfer

bull The shape of the curve varies with reaction kinetics and the shape of the particles

bull Zero order kinetics is often a good assumptionbull Oxygen is usually the first compound to become

limiting because of its low solubility

• Oxygen Transfer Through Gas-Liquid Interface
• Oxygen and Carbon Dioxide Concentration in Medium
• Oxygen Concentration in Culture and Oxygen Transfer
• Oxygen Diffusion and Order of Magnitude Analysis
• Oxygen Transfer through Interface of Two Phases
• Oxygen Transfer in bioreactors
• Oxygen Consumption and CO2 Production
• Experimental Measurement of KLa and OUR
• Mass Transfer Resistance in Cell Immobilization Reactor
• Intraparticle Diffusion Limitation

Oxygen Transfer 7

Oxygen Transfer through Interface of Two Phases

The characteristic length is usually chosen to represent the ldquopenetration depthrdquo For a long slab with diffusion from the two surfaces a long cylinder (both neglecting the diffusion from the two ends) and a sphere the characteristic length is shown in the table One can calculate the concentration of the solute as of the final equivalent to a characteristic time As can be seen after one characteristic time the concentration at a characteristic length is very close to equilibriumOne can estimate the time constant for convection and different types of reaction kinetics similarly as shown in the Table

Going back to oxygen transfer problems in bioreactors the step that incurs most resistance is the transfer between gas phase and liquid phase Microscopically the transfer can be depicted as oxygen molecules passing through two stagnant films one on the gas side the other on the liquid side as shown in Figure 1 We have introduced the equation for the rate of transfer This overall transfer is the net result of transfer through the two films For each film we can write an equation to describe the rate using kL and kg for liquid-side and gas-side transfer coefficient respectively A can be considered the same for both gas film and liquid film At the interface between the two films the two phases can be considered to be in equilibrium ie Cli=HPgi (recall that at equilibrium the concentration at the two phases is related by Henryrsquos Law constant)

One can write the equation for oxygen transfer rate (OTR) in three different forms overall liquid-side and gas-side Since there is no accumulation in the films the rate expressed by the three equations should be equal As shown in the panel by combining liquid side and gas concentration gradient to give the overall concentration gradient one can show that the inverse of Kl is the sum of the inverse of kl and hg If one considers K kl kg as the conductivity the 1KL 1kl 1kg gives the resistance In other words it shows that the overall mass transfer resistance 1kl is the sum of the two film resistances 1kl and 1kg

In using kl or kg for calculating OTR one has to use the interface concentration (Cli Pgi) for calculating concentration gradient The concentration at interface cannot be determined readily By using overall transfer the driving force used becomes the Por C or the concentration of the two phases were at equilibrium For this case it is a measure of how far away the system is from the equilibrium P and C can be readily calculated

Note that one can also use Kg for overall mass transfer coefficient that gives OTR=Kga(Pg-P)But traditionally Kl is used We just discussed that the overall resistance 1kL is the sum of 1kL and 1kg But what is the

8 Oxygen Transfer

Oxygen Transfer in bioreactors

relative magnitude of 1kL and 1kg The magnitude of mass transfer coefficient is dictated by the diffusion coefficient and the film thickness (Dδ) The film thickness for gas and liquid side is similar While the diffusion coefficient on the gas side and liquid side are orders of magnitude dififerent It can be shown that the liquid side resistance dominates the overall resistance It is thus unusual that kLis used in place of kL even when overall driving force is used

KL has units of Lt (like velocity) and the interfacial area has units of 1L After multiplying by a concentration the mass transfer equation gives a concentration perimeter that is the rate of transfer The interfacial area term is interfacial area for oxygen transfer per liquid volume aV A is the total amount of transfer area in the reactor V is the total liquid volume In a large bioreactor aerated by sparging air bubbles the bubbles are not necessarily evenly distributed and local kL is not necessarily the same The oxygen transfer rate discussed in literature is mostly an average over the reactor

OUR is the product of specific oxygen concentration and cell concentration (q0χ) Obviously if OTR is lower than OUR then dCdt will be negative and dissolved oxygen level will decrease In a culture the dissolved oxygen level decreases as cell concentration increases With a dissolved oxygen the controller manipulates the agitation rate air flow rate or the oxygen content (ie by influence kL or triangle C0 to maintain dissolved oxygen levels at the set point But even under uncontrolled conditions that change of dissolved oxygen level is not fast all the time Consider the three terms in liquid phase oxygen balance equations If the accumulation term (dcdt) is small compared to OTR and OUR one can consider the system is at a quasi steady state ie dCdt ~) thus OTR=OUR

In a typical bioreactor the oxygen level in the gas phase (Pg or C) decreases from the level at the inlet to the level at the outlet while it traverses through the reactor as oxygen is being transferred into liquid phase The question thus arises as to which C should be used the one corresponding to gas at the inlet or something else If the reactor is a small stirred tank that one can consider to be well mixed then CL will be the same in the entire liquid phase and so is the gas phase concentration In a well mixed reactor one assumes that the sample taken at any point is the same as that in the well mixed phase Therefore C should be that of the exit gas stream not the inletIn a large reactor the gas phase is likely to behave closer to plug flow One then uses logrithmic mean to calculate the driving force This is analogous to using logrithmic mean driving force in heat transfer in a double-pipe heat exchanger Basically it is an average (but a simple geometrical

Oxygen Transfer 9

OTRV=(oxygen concentration at inlet)(inlet airflow rate)-(oxygen concentration in outlet)(outlet airflow rate)

average) of the driving force at the inlet and the outlet

In addition to balance on the liquid the gas supplying the oxygen also conforms to material balance The balance on the amount of oxygen supplied to the reactor at the intlet and the amout at the outlet is the amount that has been transferred to the liquid phase In principle in most reactors above the liquid there is also a headspace with gas and should be considered for balance but in general the gas composition in the head space is considered to be the same as that in the exit Thus oxygen transferred into the liquid phase by gas phase is one line

OTR (dot V-(oxygen concentration at inlet) dot (inlet air flow rate) ndash(oxygen concentration at outlet)dot (outlet airflow rate)=NO2 in dot Qin-NO2out-Qout (asme as in slide)

With a quasi steady state assumption the oxygen transferred into liquid phase by gas phase is the same as that received by liquid phase both being OTR Furthermore OTR can be assumed equal to OUR

If the exit gas (or falled off gas) composition is measured on line using mass spectrometer or other oxygen gas sensor one can easily obtain the volumetric oxygen uptake rate In some cases kLa is relative constant such in the case of some cells can be calculated from C if C is kept constant If off-gas is not measured one can resort to a dynamic method The method involves switching gas to a composition that is an equivalent to the current dissolved oxygen leve At that moment no net transfer of oxygen may occur (because ∆ C=0) As a result the change of dissolved oxygen is only caused by cell consumption From the slope one can obtain OUR

The objective in oxygen transfer is to supply oxygen at a rate that is enough to maintain the dissolved oxygen at a level suitable (or optimal) for cell growth and production For most mammalian cells a 30 saturation with ambient air is sufficient for optimal growth The amount of oxygen cells consumed per culture volume is referred to as Oxygen Uptake Rate (OUR) As you may recall from the kinetics chapter OUR is dependent on the specific oxygen consumption rate (qO2) and the cell concentration In culture the change of oxygen concentration with time is thus the balance between oxygen transfer rate (OTR) and oxygen uptake rate

Most oxygen taken up by cells is used to oxidize carbon sources The most important carbon sources are glucose and glutamine For both the oxidation reaction yield a

Optimal Oxygen Concentrationbull Most cells grow well at a dissolved oxygen level of 30

of saturation with air bull When oxygen-enriched gas is used sometimes the

dissolved oxygen level goes above saturation A short duration of above saturation level of oxygen is usually not detrimental to cells

bull The OTR efficiency is higher with a larger oxygen concentration gradient between gas and medium (driving force) Maintaining DO at an unnecessarily high level decreases the driving force

Oxygen Consumption and CO2 Production

10 Oxygen Transfer

Integrate

ln( ) LC C K a tminus = sdot

2 2( )L O

dCV OTR V OUR Vdt

K a V C C q x V

= sdot minus sdot

= sdot sdot minus minus sdot sdot

Oxygen Demand by Cellsbull On the average q02 asymp 10 x 10-10 mmolecell-hr

CO2 Production by Cellsbull Respiratory quotient of cells is about 10 ie 1 mole CO2

is produced for every mole of O2 consumed or qCO2 asymp 10 x 10-10 mmolecell-hr

bull A high CO2 concentration is inhibitorybull Inhibitory level usually starts at 15 to 20 CO2

Experimental Measurement of KLa and OUR

stoichiometric ratio of one for CO2 to O2 (called respiratory quotient [RQ]) The CO2 produced by cells will need to be removed from the culture to avoid excessive accumulationExcess amount of either dissolved O2 or CO2 can be growth inhibiting or toxic to the cells However the growth inhibitory effect of O2 and CO2 take an exposure time to be seen A short exposure of high concentration (gt100 saturation of O2 or nearly 150 mmHg of CO2) up to a few hours may not be detrimental

The mass transfer coefficient can be estimated by using some empirical correlations developed in the form of dimension-less numbers There has been a report that KL is relatively insensitive to bubble size except in small scale reactors and under the conditions that the liquid flow pattern is relatively simple like most bubbles rise in a simple pattern the esti-mated KL is probably not reliable Furthermore except un-der those conditions a is also difficult to estimate as relied on direct experimental measurement of KLa is often lumped together and called volumetric mass transfer coefficient in-stead of having KL and a estimated or measured separately Many bioprocess plants measure gas composition from the biotreactor inlet and outlet From the balance equa-tion one can see that the OUR and OTR calculated from the gas balance can also be used to calculate the average KLa in the reactor since Cand C are both measured One can also use a dynamic method to measure KLa One way to do this is by stripping the dissolved oxygen off using N2 to replace air (or the original gas phase) In this case

Oxygen Transfer 11

Mass Transfer Resistance in Cell Immobilization Reactor

Intraparticle Diffusion Limitation

Criteria for assessing the magnitude of mass-transfer effects on overall kinetics

Criterion η value

Limiting rate process

Extent of mass-transfer

limitation

Φ lt 03 ~1 Chemical reaction Negligible

Φ gt 3 αФ-1 Diffusion Large

Or Φ critical the critical value at which η = 1

12 Oxygen Transfer

bull The shape of the curve varies with reaction kinetics and the shape of the particles

bull Zero order kinetics is often a good assumptionbull Oxygen is usually the first compound to become

limiting because of its low solubility

• Oxygen Transfer Through Gas-Liquid Interface
• Oxygen and Carbon Dioxide Concentration in Medium
• Oxygen Concentration in Culture and Oxygen Transfer
• Oxygen Diffusion and Order of Magnitude Analysis
• Oxygen Transfer through Interface of Two Phases
• Oxygen Transfer in bioreactors
• Oxygen Consumption and CO2 Production
• Experimental Measurement of KLa and OUR
• Mass Transfer Resistance in Cell Immobilization Reactor
• Intraparticle Diffusion Limitation

8 Oxygen Transfer

Oxygen Transfer in bioreactors

relative magnitude of 1kL and 1kg The magnitude of mass transfer coefficient is dictated by the diffusion coefficient and the film thickness (Dδ) The film thickness for gas and liquid side is similar While the diffusion coefficient on the gas side and liquid side are orders of magnitude dififerent It can be shown that the liquid side resistance dominates the overall resistance It is thus unusual that kLis used in place of kL even when overall driving force is used

KL has units of Lt (like velocity) and the interfacial area has units of 1L After multiplying by a concentration the mass transfer equation gives a concentration perimeter that is the rate of transfer The interfacial area term is interfacial area for oxygen transfer per liquid volume aV A is the total amount of transfer area in the reactor V is the total liquid volume In a large bioreactor aerated by sparging air bubbles the bubbles are not necessarily evenly distributed and local kL is not necessarily the same The oxygen transfer rate discussed in literature is mostly an average over the reactor

OUR is the product of specific oxygen concentration and cell concentration (q0χ) Obviously if OTR is lower than OUR then dCdt will be negative and dissolved oxygen level will decrease In a culture the dissolved oxygen level decreases as cell concentration increases With a dissolved oxygen the controller manipulates the agitation rate air flow rate or the oxygen content (ie by influence kL or triangle C0 to maintain dissolved oxygen levels at the set point But even under uncontrolled conditions that change of dissolved oxygen level is not fast all the time Consider the three terms in liquid phase oxygen balance equations If the accumulation term (dcdt) is small compared to OTR and OUR one can consider the system is at a quasi steady state ie dCdt ~) thus OTR=OUR

In a typical bioreactor the oxygen level in the gas phase (Pg or C) decreases from the level at the inlet to the level at the outlet while it traverses through the reactor as oxygen is being transferred into liquid phase The question thus arises as to which C should be used the one corresponding to gas at the inlet or something else If the reactor is a small stirred tank that one can consider to be well mixed then CL will be the same in the entire liquid phase and so is the gas phase concentration In a well mixed reactor one assumes that the sample taken at any point is the same as that in the well mixed phase Therefore C should be that of the exit gas stream not the inletIn a large reactor the gas phase is likely to behave closer to plug flow One then uses logrithmic mean to calculate the driving force This is analogous to using logrithmic mean driving force in heat transfer in a double-pipe heat exchanger Basically it is an average (but a simple geometrical

Oxygen Transfer 9

OTRV=(oxygen concentration at inlet)(inlet airflow rate)-(oxygen concentration in outlet)(outlet airflow rate)

average) of the driving force at the inlet and the outlet

In addition to balance on the liquid the gas supplying the oxygen also conforms to material balance The balance on the amount of oxygen supplied to the reactor at the intlet and the amout at the outlet is the amount that has been transferred to the liquid phase In principle in most reactors above the liquid there is also a headspace with gas and should be considered for balance but in general the gas composition in the head space is considered to be the same as that in the exit Thus oxygen transferred into the liquid phase by gas phase is one line

OTR (dot V-(oxygen concentration at inlet) dot (inlet air flow rate) ndash(oxygen concentration at outlet)dot (outlet airflow rate)=NO2 in dot Qin-NO2out-Qout (asme as in slide)

With a quasi steady state assumption the oxygen transferred into liquid phase by gas phase is the same as that received by liquid phase both being OTR Furthermore OTR can be assumed equal to OUR

If the exit gas (or falled off gas) composition is measured on line using mass spectrometer or other oxygen gas sensor one can easily obtain the volumetric oxygen uptake rate In some cases kLa is relative constant such in the case of some cells can be calculated from C if C is kept constant If off-gas is not measured one can resort to a dynamic method The method involves switching gas to a composition that is an equivalent to the current dissolved oxygen leve At that moment no net transfer of oxygen may occur (because ∆ C=0) As a result the change of dissolved oxygen is only caused by cell consumption From the slope one can obtain OUR

The objective in oxygen transfer is to supply oxygen at a rate that is enough to maintain the dissolved oxygen at a level suitable (or optimal) for cell growth and production For most mammalian cells a 30 saturation with ambient air is sufficient for optimal growth The amount of oxygen cells consumed per culture volume is referred to as Oxygen Uptake Rate (OUR) As you may recall from the kinetics chapter OUR is dependent on the specific oxygen consumption rate (qO2) and the cell concentration In culture the change of oxygen concentration with time is thus the balance between oxygen transfer rate (OTR) and oxygen uptake rate

Most oxygen taken up by cells is used to oxidize carbon sources The most important carbon sources are glucose and glutamine For both the oxidation reaction yield a

Optimal Oxygen Concentrationbull Most cells grow well at a dissolved oxygen level of 30

of saturation with air bull When oxygen-enriched gas is used sometimes the

dissolved oxygen level goes above saturation A short duration of above saturation level of oxygen is usually not detrimental to cells

bull The OTR efficiency is higher with a larger oxygen concentration gradient between gas and medium (driving force) Maintaining DO at an unnecessarily high level decreases the driving force

Oxygen Consumption and CO2 Production

10 Oxygen Transfer

Integrate

ln( ) LC C K a tminus = sdot

2 2( )L O

dCV OTR V OUR Vdt

K a V C C q x V

= sdot minus sdot

= sdot sdot minus minus sdot sdot

Oxygen Demand by Cellsbull On the average q02 asymp 10 x 10-10 mmolecell-hr

CO2 Production by Cellsbull Respiratory quotient of cells is about 10 ie 1 mole CO2

is produced for every mole of O2 consumed or qCO2 asymp 10 x 10-10 mmolecell-hr

bull A high CO2 concentration is inhibitorybull Inhibitory level usually starts at 15 to 20 CO2

Experimental Measurement of KLa and OUR

stoichiometric ratio of one for CO2 to O2 (called respiratory quotient [RQ]) The CO2 produced by cells will need to be removed from the culture to avoid excessive accumulationExcess amount of either dissolved O2 or CO2 can be growth inhibiting or toxic to the cells However the growth inhibitory effect of O2 and CO2 take an exposure time to be seen A short exposure of high concentration (gt100 saturation of O2 or nearly 150 mmHg of CO2) up to a few hours may not be detrimental

The mass transfer coefficient can be estimated by using some empirical correlations developed in the form of dimension-less numbers There has been a report that KL is relatively insensitive to bubble size except in small scale reactors and under the conditions that the liquid flow pattern is relatively simple like most bubbles rise in a simple pattern the esti-mated KL is probably not reliable Furthermore except un-der those conditions a is also difficult to estimate as relied on direct experimental measurement of KLa is often lumped together and called volumetric mass transfer coefficient in-stead of having KL and a estimated or measured separately Many bioprocess plants measure gas composition from the biotreactor inlet and outlet From the balance equa-tion one can see that the OUR and OTR calculated from the gas balance can also be used to calculate the average KLa in the reactor since Cand C are both measured One can also use a dynamic method to measure KLa One way to do this is by stripping the dissolved oxygen off using N2 to replace air (or the original gas phase) In this case

Oxygen Transfer 11

Mass Transfer Resistance in Cell Immobilization Reactor

Intraparticle Diffusion Limitation

Criteria for assessing the magnitude of mass-transfer effects on overall kinetics

Criterion η value

Limiting rate process

Extent of mass-transfer

limitation

Φ lt 03 ~1 Chemical reaction Negligible

Φ gt 3 αФ-1 Diffusion Large

Or Φ critical the critical value at which η = 1

12 Oxygen Transfer

bull The shape of the curve varies with reaction kinetics and the shape of the particles

bull Zero order kinetics is often a good assumptionbull Oxygen is usually the first compound to become

limiting because of its low solubility

• Oxygen Transfer Through Gas-Liquid Interface
• Oxygen and Carbon Dioxide Concentration in Medium
• Oxygen Concentration in Culture and Oxygen Transfer
• Oxygen Diffusion and Order of Magnitude Analysis
• Oxygen Transfer through Interface of Two Phases
• Oxygen Transfer in bioreactors
• Oxygen Consumption and CO2 Production
• Experimental Measurement of KLa and OUR
• Mass Transfer Resistance in Cell Immobilization Reactor
• Intraparticle Diffusion Limitation

Oxygen Transfer 9

OTRV=(oxygen concentration at inlet)(inlet airflow rate)-(oxygen concentration in outlet)(outlet airflow rate)

average) of the driving force at the inlet and the outlet

In addition to balance on the liquid the gas supplying the oxygen also conforms to material balance The balance on the amount of oxygen supplied to the reactor at the intlet and the amout at the outlet is the amount that has been transferred to the liquid phase In principle in most reactors above the liquid there is also a headspace with gas and should be considered for balance but in general the gas composition in the head space is considered to be the same as that in the exit Thus oxygen transferred into the liquid phase by gas phase is one line

OTR (dot V-(oxygen concentration at inlet) dot (inlet air flow rate) ndash(oxygen concentration at outlet)dot (outlet airflow rate)=NO2 in dot Qin-NO2out-Qout (asme as in slide)

With a quasi steady state assumption the oxygen transferred into liquid phase by gas phase is the same as that received by liquid phase both being OTR Furthermore OTR can be assumed equal to OUR

If the exit gas (or falled off gas) composition is measured on line using mass spectrometer or other oxygen gas sensor one can easily obtain the volumetric oxygen uptake rate In some cases kLa is relative constant such in the case of some cells can be calculated from C if C is kept constant If off-gas is not measured one can resort to a dynamic method The method involves switching gas to a composition that is an equivalent to the current dissolved oxygen leve At that moment no net transfer of oxygen may occur (because ∆ C=0) As a result the change of dissolved oxygen is only caused by cell consumption From the slope one can obtain OUR

The objective in oxygen transfer is to supply oxygen at a rate that is enough to maintain the dissolved oxygen at a level suitable (or optimal) for cell growth and production For most mammalian cells a 30 saturation with ambient air is sufficient for optimal growth The amount of oxygen cells consumed per culture volume is referred to as Oxygen Uptake Rate (OUR) As you may recall from the kinetics chapter OUR is dependent on the specific oxygen consumption rate (qO2) and the cell concentration In culture the change of oxygen concentration with time is thus the balance between oxygen transfer rate (OTR) and oxygen uptake rate

Most oxygen taken up by cells is used to oxidize carbon sources The most important carbon sources are glucose and glutamine For both the oxidation reaction yield a

Optimal Oxygen Concentrationbull Most cells grow well at a dissolved oxygen level of 30

of saturation with air bull When oxygen-enriched gas is used sometimes the

dissolved oxygen level goes above saturation A short duration of above saturation level of oxygen is usually not detrimental to cells

bull The OTR efficiency is higher with a larger oxygen concentration gradient between gas and medium (driving force) Maintaining DO at an unnecessarily high level decreases the driving force

Oxygen Consumption and CO2 Production

10 Oxygen Transfer

Integrate

ln( ) LC C K a tminus = sdot

2 2( )L O

dCV OTR V OUR Vdt

K a V C C q x V

= sdot minus sdot

= sdot sdot minus minus sdot sdot

Oxygen Demand by Cellsbull On the average q02 asymp 10 x 10-10 mmolecell-hr

CO2 Production by Cellsbull Respiratory quotient of cells is about 10 ie 1 mole CO2

is produced for every mole of O2 consumed or qCO2 asymp 10 x 10-10 mmolecell-hr

bull A high CO2 concentration is inhibitorybull Inhibitory level usually starts at 15 to 20 CO2

Experimental Measurement of KLa and OUR

stoichiometric ratio of one for CO2 to O2 (called respiratory quotient [RQ]) The CO2 produced by cells will need to be removed from the culture to avoid excessive accumulationExcess amount of either dissolved O2 or CO2 can be growth inhibiting or toxic to the cells However the growth inhibitory effect of O2 and CO2 take an exposure time to be seen A short exposure of high concentration (gt100 saturation of O2 or nearly 150 mmHg of CO2) up to a few hours may not be detrimental

The mass transfer coefficient can be estimated by using some empirical correlations developed in the form of dimension-less numbers There has been a report that KL is relatively insensitive to bubble size except in small scale reactors and under the conditions that the liquid flow pattern is relatively simple like most bubbles rise in a simple pattern the esti-mated KL is probably not reliable Furthermore except un-der those conditions a is also difficult to estimate as relied on direct experimental measurement of KLa is often lumped together and called volumetric mass transfer coefficient in-stead of having KL and a estimated or measured separately Many bioprocess plants measure gas composition from the biotreactor inlet and outlet From the balance equa-tion one can see that the OUR and OTR calculated from the gas balance can also be used to calculate the average KLa in the reactor since Cand C are both measured One can also use a dynamic method to measure KLa One way to do this is by stripping the dissolved oxygen off using N2 to replace air (or the original gas phase) In this case

Oxygen Transfer 11

Mass Transfer Resistance in Cell Immobilization Reactor

Intraparticle Diffusion Limitation

Criteria for assessing the magnitude of mass-transfer effects on overall kinetics

Criterion η value

Limiting rate process

Extent of mass-transfer

limitation

Φ lt 03 ~1 Chemical reaction Negligible

Φ gt 3 αФ-1 Diffusion Large

Or Φ critical the critical value at which η = 1

12 Oxygen Transfer

bull The shape of the curve varies with reaction kinetics and the shape of the particles

bull Zero order kinetics is often a good assumptionbull Oxygen is usually the first compound to become

limiting because of its low solubility

• Oxygen Transfer Through Gas-Liquid Interface
• Oxygen and Carbon Dioxide Concentration in Medium
• Oxygen Concentration in Culture and Oxygen Transfer
• Oxygen Diffusion and Order of Magnitude Analysis
• Oxygen Transfer through Interface of Two Phases
• Oxygen Transfer in bioreactors
• Oxygen Consumption and CO2 Production
• Experimental Measurement of KLa and OUR
• Mass Transfer Resistance in Cell Immobilization Reactor
• Intraparticle Diffusion Limitation

10 Oxygen Transfer

Integrate

ln( ) LC C K a tminus = sdot

2 2( )L O

dCV OTR V OUR Vdt

K a V C C q x V

= sdot minus sdot

= sdot sdot minus minus sdot sdot

Oxygen Demand by Cellsbull On the average q02 asymp 10 x 10-10 mmolecell-hr

CO2 Production by Cellsbull Respiratory quotient of cells is about 10 ie 1 mole CO2

is produced for every mole of O2 consumed or qCO2 asymp 10 x 10-10 mmolecell-hr

bull A high CO2 concentration is inhibitorybull Inhibitory level usually starts at 15 to 20 CO2

Experimental Measurement of KLa and OUR

stoichiometric ratio of one for CO2 to O2 (called respiratory quotient [RQ]) The CO2 produced by cells will need to be removed from the culture to avoid excessive accumulationExcess amount of either dissolved O2 or CO2 can be growth inhibiting or toxic to the cells However the growth inhibitory effect of O2 and CO2 take an exposure time to be seen A short exposure of high concentration (gt100 saturation of O2 or nearly 150 mmHg of CO2) up to a few hours may not be detrimental

The mass transfer coefficient can be estimated by using some empirical correlations developed in the form of dimension-less numbers There has been a report that KL is relatively insensitive to bubble size except in small scale reactors and under the conditions that the liquid flow pattern is relatively simple like most bubbles rise in a simple pattern the esti-mated KL is probably not reliable Furthermore except un-der those conditions a is also difficult to estimate as relied on direct experimental measurement of KLa is often lumped together and called volumetric mass transfer coefficient in-stead of having KL and a estimated or measured separately Many bioprocess plants measure gas composition from the biotreactor inlet and outlet From the balance equa-tion one can see that the OUR and OTR calculated from the gas balance can also be used to calculate the average KLa in the reactor since Cand C are both measured One can also use a dynamic method to measure KLa One way to do this is by stripping the dissolved oxygen off using N2 to replace air (or the original gas phase) In this case

Oxygen Transfer 11

Mass Transfer Resistance in Cell Immobilization Reactor

Intraparticle Diffusion Limitation

Criteria for assessing the magnitude of mass-transfer effects on overall kinetics

Criterion η value

Limiting rate process

Extent of mass-transfer

limitation

Φ lt 03 ~1 Chemical reaction Negligible

Φ gt 3 αФ-1 Diffusion Large

Or Φ critical the critical value at which η = 1

12 Oxygen Transfer

bull The shape of the curve varies with reaction kinetics and the shape of the particles

bull Zero order kinetics is often a good assumptionbull Oxygen is usually the first compound to become

limiting because of its low solubility

• Oxygen Transfer Through Gas-Liquid Interface
• Oxygen and Carbon Dioxide Concentration in Medium
• Oxygen Concentration in Culture and Oxygen Transfer
• Oxygen Diffusion and Order of Magnitude Analysis
• Oxygen Transfer through Interface of Two Phases
• Oxygen Transfer in bioreactors
• Oxygen Consumption and CO2 Production
• Experimental Measurement of KLa and OUR
• Mass Transfer Resistance in Cell Immobilization Reactor
• Intraparticle Diffusion Limitation

Oxygen Transfer 11

Mass Transfer Resistance in Cell Immobilization Reactor

Intraparticle Diffusion Limitation

Criteria for assessing the magnitude of mass-transfer effects on overall kinetics

Criterion η value

Limiting rate process

Extent of mass-transfer

limitation

Φ lt 03 ~1 Chemical reaction Negligible

Φ gt 3 αФ-1 Diffusion Large

Or Φ critical the critical value at which η = 1

12 Oxygen Transfer

bull The shape of the curve varies with reaction kinetics and the shape of the particles

bull Zero order kinetics is often a good assumptionbull Oxygen is usually the first compound to become

limiting because of its low solubility

• Oxygen Transfer Through Gas-Liquid Interface
• Oxygen and Carbon Dioxide Concentration in Medium
• Oxygen Concentration in Culture and Oxygen Transfer
• Oxygen Diffusion and Order of Magnitude Analysis
• Oxygen Transfer through Interface of Two Phases
• Oxygen Transfer in bioreactors
• Oxygen Consumption and CO2 Production
• Experimental Measurement of KLa and OUR
• Mass Transfer Resistance in Cell Immobilization Reactor
• Intraparticle Diffusion Limitation

12 Oxygen Transfer

bull The shape of the curve varies with reaction kinetics and the shape of the particles

bull Zero order kinetics is often a good assumptionbull Oxygen is usually the first compound to become

limiting because of its low solubility

• Oxygen Transfer Through Gas-Liquid Interface
• Oxygen and Carbon Dioxide Concentration in Medium
• Oxygen Concentration in Culture and Oxygen Transfer
• Oxygen Diffusion and Order of Magnitude Analysis
• Oxygen Transfer through Interface of Two Phases
• Oxygen Transfer in bioreactors
• Oxygen Consumption and CO2 Production
• Experimental Measurement of KLa and OUR
• Mass Transfer Resistance in Cell Immobilization Reactor
• Intraparticle Diffusion Limitation