Laboratory Experiment

Volumetric Mass Transfer Coefficient

BCHE 3180
Introduction
Mass transfer in the context of biological reactors typically involves the transport of a
gaseous species such as oxygen from a gas phase to a liquid phase. The rate at which gas dissolves
in a liquid (flux) at any particular time is proportional to the difference between the equilibrium
concentration and the concentration at that time. A departure from equilibrium can be considered a
"driving force" for mass transfer. When the equilibrium concentration is reached, the liquid is
saturated with the gas. Under this condition, no additional gas will dissolve, and the rate of transfer
is therefore zero. If the liquid is devoid of dissolved gas, then the rate of dissolution will be at a
maximum. Well come back to this point. The proportionality between the concentration difference
and the transfer rate is called a mass transfer coefficient.
A simple approach is to consider two important resistances to mass transfer from a gas to a
liquid: a gas film and a fluid film. The most convenient type of mass transfer coefficient is an
overall mass transfer coefficient (K), which encompasses both film resistances and uses both liquid
phase and gas phase concentrations in calculating an overall driving force. For gases like oxygen
which are sparingly soluble in water, the liquid film is the principal resistance to mass transfer, and
the overall mass transfer coefficient is approximately equal to the liquid phase mass transfer
coefficient (kL). The subscript L denotes that liquid phase concentrations will be used to describe
the overall driving force. The concentration difference used for the driving force is the concentration
of oxygen in the liquid at saturation (c*O2) and the actual liquid phase concentration of oxygen (cO2).
Thus, the molar oxygen flux (O2) can be written:
O2 = k L cO2 - cO2
*

(1)

Equation 1 can be though of as providing a definition for the mass transfer coefficient.
Performing a mole balance in the liquid phase of a bioreactor yields:
Accumulation of O2 = Moles O2 In - Moles O2 Out + Moles O2 Generated
In the absence of microbial cells, the two terms Moles O2 Out and Moles O2 Generated are each
equal to zero. The material balance equation thus becomes:

d V cO2
= O2 A
dt

(2)

where A is the surface area available for transfer and V is the system (fermentor) volume. With the
system volume constant, substituting the expression of the molar flux (Equation 1) into the molar
balance (Equation 2) and letting the specific exchange surface A/V equal a results in the following
equation:

dcO2
= k L a c*O2 - cO2
dt

(3)

The rate at which oxygen is supplied is also called the oxygen transfer rate OTR. That is, this same
equation indicates how fast oxygen is being supplied to the bioreactor (e.g., units in mg/Lh):

OTR = k L a c*O2 - cO2

(4)

Equation 3 may be integrated from a concentration of cO2i to cO2 and a time of 0 to t to yield the
following equation:

ln c*O2 - cO2 ln c*O2 - ciO2 = - k L at

(5)

Dissolved oxygen probes which are used in bioreactors typically are calibrated to read 100% when
the liquid phase is saturated with oxygen at a given temperature and 0% when oxygen is absent
from the liquid phase at the same temperature. In other words, the relationship between the
fractional dissolved oxygen (DO) and the actual concentrations is:

DO cO2

c
*
O2

(6)

Combining Equations 5 and 6 leads to:

c* - ci
ln 1 - DO ln O2 * O2 = - k L at
cO2

(7)

As oxygen is dissolving in the solution, the value for DO obviously increases. Equation 7 indicates
that a plot of ln 1 - DO versus time will be linear with a negative slope equal to kLa. The value of
the specific exchange surface (a) is difficult to determine for small bubbles found in a bioreactor.
So, the entire term kLa is often called the volumetric oxygen transfer coefficient (units of h-1).
Strictly speaking this statement is incorrect, since only the "kL" is the mass transfer coefficient.
Also, sometimes kLa is called the overall mass transfer coefficient. However, kL is really just the
liquid phase mass transfer coefficient, not the overall mass transfer coefficient, KL. Calling kL the
overall mass transfer coefficient is not an egregious statement though since the values for the two
mass transfer coefficients approach each other for a sparingly soluble gas, like oxygen in water. As
is the common practice, in this document the term kLa will be simply referred to as the mass transfer
coefficient. We now have a way to calculate the mass transfer coefficient for a certain set of
physical conditionsjust measure how the concentration of dissolve oxygen changes with time.
U

How is kLa correlated to physical parameters? The mass transfer coefficient depends on the
fluid properties, type of impeller, the geometry of the bioreactor, the agitation, and the air flow
conditions. For a given bioreactor with a fixed set of impellers agitating a given fluid like water, kLa
can be correlated simply to agitation and air flow, typically by the following empirical relationship:

k L a ( P / V ) (U )

(8)

In Equation 8, P/V (e.g., units of W/m3) is the specific power or the power-per-volume,

while U is the superficial gas velocity (e.g., units of m/h). The superficial gas velocity is equal to the
volumetric flowrate of the gas divided by the cross-sectional area of the bioreactor. Both P/V and U
are important parameters for scale-up of bioreactors. At small scale, typical operations have higher
P/V and lower U than are usually feasible at large scale. The terms , , are empirical "constants"
which depend on quite a few other system characteristics like viscosity of the fluid. (Scale up of
bioreactors is made particularly challenging by other strongly interrelated factors including impeller
type, impeller location, impeller number, and the effect the fluid conditions have on cells and their
morphology!)
Lets reconsider that the oxygen transfer rate OTR (Equation 4) is proportional to the driving
force for dissolution. As introduced in the first paragraph above, for a constant kLa as the solution
approaches saturation, the rate at which oxygen dissolves slows. If the solution is already saturated
with oxygen, then the OTR is zero. On the other hand, if the bioreactor is oxygen-free, then the rate
of dissolving oxygen will be at its maximum possible value:

OTR MAX = k L a c*O2

(9)

If aerobic cells are present in the bioreactor, then they will be withdrawing oxygen at a rate,
too. This rate of microbial consumption is called the volumetric oxygen uptake rate (OUR or QO).
Imagine that very few cells are present in the bioreactor, or that those which are present are not
consuming oxygen quickly. In this case, the OUR is relatively low, the rate of supplying oxygen
will be able to "keep up" with the rate of oxygen withdraw, and the concentration of oxygen in the
bioreactor will remain close to saturation (DO = 100%). If more cells are in the bioreactor and the
OUR is comparatively high, then the consumption will tend to reduce the dissolved oxygen
concentration. But remember, the lower the DO, the greater the OTR. Thus, the OUR and OTR
tend to balance each other, and more specifically they tend to equalize.
The metabolism of cells is affected by the level of dissolved oxygen. At some point, the
concentration of oxygen can become so low that oxygen limits aerobic metabolism. If the cells are
strictly aerobic, then cell growth would become limited by the availability of oxygen. If one knows
the critical oxygen concentration necessary to prevent cell growth from being limited by oxygen
availability (cO2CRIT), then one can calculate the kLa necessary to sustain that level of oxygen for a
given OUR. Specifically:
CRIT
OUR = k L a c*O2 cO2

(10)

In order to calculate the kLa needed to keep the culture above cO2CRIT, we just need to know how fast
the bioreactor is consuming the oxygen (OUR). Note that cases could easily arise in dense cultures
where it is simply not reasonable to have a very high OTR, and instead one is indeed forced to limit
growth by oxygen or by the availability of some other nutrient. Limiting growth rate is not really a
bad thing, as slowing the process slightly can have many advantages (e.g., more controlled
metabolism, lower heat requirement).
The objective of this experiment is to understand how the volumetric oxygen transfer
coefficient varies with operational conditions (e.g., agitation and gas flowrate).

Procedure
Set-Up
U

1. Assembly of the Bioreactor

a. Use a standard volume of water (1.25 L)
b. Use a standard temperature (37C)
2. Calibration of the Dissolved Oxygen Electrodes
3. For each variable studied, set the agitation and air flowrate (etc.) to the desired values, sparge
nitrogen into the fermentor until the DO reads zero, switch the gas flow from nitrogen to air, record
the DO at time intervals of 5 seconds starting a 15 seconds and ending when the DO reaches about
90% saturation.
Other Data Needed:
Diameter of Vessel:

________________cm

Height of Impeller from Shaft End:

________________cm

Distance between Impellers:

________________cm

Length of Shaft from Headplate:

________________cm

Height of Reactor:

________________cm

(Note: Experiments A, B, C, and D together require about 3 hours.)

Day # 1
U

A. Agitation Rate

Study the effect of agitation rate on the value of the mass transfer coefficient.
With the air flowrate set at 1.00 L/min (0.8 vvm), perform the experiment
with several agitation rates: 100 rpm, 200 rpm, 300 rpm, 400 rpm, 500 rpm,
600 rpm, 700 rpm. We are not set up to measure P/V.

B. Air Flowrate

Study the effect of air flowrate on the value of the mass transfer coefficient.
With the agitation rate set at 400 rpm, perform the experiment with several
air flowrates: 0.20 L/min, 0.35 L/min, 0.50 L/min, 0.65 L/min and 0.80
L/min. Convert these volumetric flowrates into superficial gas velocities.

C. Impeller Position Use an air flowrate of 1.00 L/min. Move the upper impeller higher on the
shaft so that it no longer is submerged in the water. Examine 300 rpm, 400
rpm, and 500 rpm.
D. Glycerol

Instead of using 1.25 L water, use a mixture of 20% glycerol in water (i.e.,
250 mL glycerol plus 1.0 L water). Using 2 impellers and 1.00 L/min
flowrate, perform experiment using 200 rpm, 300 rpm, 400 rpm, 500 rpm,
600 rpm. How does kLa compare? What has changed by using glycerol?
Visual observations can be helpful.

Day #2
E. Oxygen Uptake

We will grow some microbial cells in the bioreactor and measure the oxygen
uptake rate. How do we do that?

F. Predicting DO

With some microbial cells in the bioreactor and knowledge of their oxygen
uptake rate, we should be able to set the kLa (from data obtained in sections
A and B above), and predict where the DO settles. Given the OUR, try a few
values for the agitation rate and air flowrate. Do the values of DO agree with
your prediction? What are the problems one could encounter when doing
this with living cells?

Report
BE ORGANIZED. In addition to the comments above,
1. Calculate kLa for all the experiments.
0B

2. Plot kLa versus agitation rate. Can you find a correlation to predict how kLa varies with
agitation rate (as opposed to P/V)? How do your results compare with this correlation?
3. Plot kLa versus air flowrate and with superficial gas velocity. Can you calculate the
parameter in Equation 8?
4. How important is it to have two impellers in this fermenter when conducting an aerobic
process? Do you think 3 impellers would be beneficial?
5. We just cant increase agitation of flowrate enough to get the kLa we need. The reason we
want to increase kLa, though, is usually to provide a high OTR. Looking at Equation 4, how
else can OTR be increased? Explain.

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