5.2 Volumetric Oxygen Mass Transfer Coefficient
In a typical aeration system, oxygen from the air bubble is transferred through the gas-liquid interface followed by liquid phase diffusion/bulk transport to the cells. Although this is a multi-step serial transport, in a well dispersed systems, the major resistance to oxygen transfer is in the liquid film surrounding the gas bubble. Consider the oxygen concentration profiles in the region near the interface illustrated in Figure 5-1.
Figure 5-1 Oxygen Concentration Profile at Air Bubble-Medium Interface
The transport of oxygen through the gas and liquid films are equal at steady state. They can be expressed by
where subscript G and L refer to gas and liquid phases respectively. The terms, NO2G and NO2L are oxygen transfer expressed in g O2 h-1, A is interfacial area and CDO is oxygen concentration expressed in g O2 per unit volume. At the interface, equilibrium between the liquid and gas phase oxygen is reached. That is
Because of low oxygen solubility and the fact that kG is much higher than kL,
Hence, Eq (5-1a) can be written as
The subscript L in NO2 has been dropped to note that the above represents overall transfer of oxygen. The driving force in the above consists of the difference between bulk oxygen concentrations in the two phases; the first term represents the concentration of oxygen in the liquid which is in equilibrium with the bulk gas phase oxygen. If air is the gas medium, this term will equal to 7 mg/L at 35 C.
When the above oxygen transfer is applied to an entire volume of a bioreactor, A will represent the total interfacial area and kL will represent an average mass transfer coefficient. The concentrations will be bulk gas and liquid phase oxygen concentrations. If we divide the above equation by volume of liquid phase, V, the resulting term will represent the amount of oxygen transfered per unit volume per unit time --- which is in the same units as the rate expressions we saw in last chapter. Since the rate is due to a physical phenomena, let us distinguish it by the symbol, RO2. That is,
The term, kL A represents the product of mass transfer coefficient and interfacial area available for mass treansfer. In a bioreactor, air is sparged and the liquid is agitated to break up the bubbles so that interfacial area can be kept high to enhance rate of oxygen transfer. In such systems, the area, A, is not easily measured or estimated. But, the term consisting of the product - mass transfer coefficient and interfacial area - is more readily measured. Further more, it is convenient to use interfacial area per unit volume, a, rather than total area, A because rate of oxygen transfer is expressed per unit volume of bioreactor, similar to rate of cell growth, which is reported on a volumetric basis. Hence, area per unit volume, a, is combined with the mass transfer coefficient, kL and is given by the term, kLa. In Eq(5-5) the term, can be replaced by oxygen solubility at bioreactor conditions, .
The above will be our working equation for describing transfer of oxygen from gas phase to growth medium. In order for us to calculate oxygen transfer rate (OTR), we need the mass transfer coefficient, kLa , solubility of oxygen in the medium, and the dissolved oxygen concentration in the medium, CDOL. In the last chapter we had used the notation, CDO to describe dissolved oxygen concentration. In the discussion above, there was a need to make a distinction between gas and liquid phase concentration. In Eq (5-5), one notes that both concentrations are expressed on the basis of liquid phase. Hence, from here on we will drop the subscript L. In situations where we need to make a distinction between the two phases, we will re-introduce the subscript L and G.