# Oxygen Mass Transfer

To meet the oxygen demand of the cells (*OD*), oxygen has to be continuously supplied because of its limited solubility in the media. In the case of animal cells, the speciﬁc oxygen demand (*SOD*), which is independent of scale, is low (typical values of about 10 16 to 10 17 mol s 1 cell 1 (Nienow et al. 1996; Xing et al. 2009) for CHO cells) and though cell density, *X*, has steadily increased over the years to about 107 cells mL 1, relative to many other organisms, the overall *OD* ¼ (*SOD. X*) is also low. Provided the cell density achieved is the same across the scales, then so is the *OD*. This low *OD* means that the oxygen transfer rate required is low and therefore because it is linked to the speciﬁc power input from the agitator when aerated, *Pg/*ρ*V*, is also typically low (< ~ 0.075 W kg 1) as is the sparge rate (< ~ 0.01 vvm) (Nienow 2006). These factors impact on the way the basic mass transfer model is modiﬁed in order to calculate the mass transfer coefﬁcient from the basic mass transfer equation which will now be addressed.

## Basic Oxygen Mass Transfer Concepts and Equations

Figure 5.2 shows the steps by which oxygen passes from a bubble at a partial pressure *pg* (partial pressure of oxygen in air, oxygen-enriched air or pure oxygen also allowing for back pressure) to the animal cell. The steps consist of transport through the gas ﬁlm inside the bubble, across the bubble-liquid interface, through the liquid ﬁlm around the bubble, across the well-mixed bulk liquid (media plus product in solution) through the liquid ﬁlm around the cell where it is utilized within the cell. Each step offers a resistance to oxygen transfer; and each step itself is dependent on the mass transfer coefﬁcient for that step, the area available over which mass transfer can occur and the relevant driving force. For very small entities

**Fig. 5.2 Oxygen mass transfer steps**

like animal cells, the mass transfer coefﬁcient is very high (Nienow 1997a) and the speciﬁc surface area, *aP* (both α 1/particle size) is very large so that step is very fast. At the gas liquid interface, the area is the same for both the gas ﬁlm and the liquid ﬁlm but the mass transfer coefﬁcient increases with an increase in the diffusion coefﬁcient. Since the latter is much higher for the gas phase than for the liquid, the rate limiting step is in the liquid ﬁlm. In addition, as a result, *pg* � *pi*. In addition for all mass transfer processes, it is assumed that equilibrium exists at the interface between the two phases. This assumption implies that, at the interface, the concentration of the gas in the liquid, *Ci,* is equal to its solubility at its partial pressure in the gas phase, *pi*. Since, for sparingly soluble gases such as oxygen,

there is a direct proportionality between the two,

where *H* is the Henry's law constant.

The rate of mass transfer, *J*, at the gas liquid-interface is then assumed to be proportional to the concentration differences existing within each phase, the surface area between the phases, *A*, and a coefﬁcient (the gas or liquid ﬁlm mass transfer coefﬁcient, *kg* or *kL*, respectively) which relates the three. Thus

and

where *Cg** is the solubility of oxygen in the media that is in equilibrium with the gas phase partial pressure of oxygen. Thus the aeration rate per unit volume of bioreactor, *N*, is given by

where *a* is the speciﬁc area of bubbles Thus, for satisfactory operation, the maximum oxygen demand of the cells (*ODmax*) must be met and it is related to the maximum cell concentration (*Xmax) by*

Thus, for stable operation, *OD* (or Oxygen Uptake Rate, *OUR*) needs to be met by the oxygen transfer rate, *OTR*. Thus,

For satisfactory operation, *CL* > *(CL)crit* where *(CL)crit* is the critical oxygen concentration below which the performance of the bioreactor begins to deteriorate whilst above it, the performance is zero order with respect to dissolve oxygen concentration, i. e. independent of it. It may in some cases, need to be below some upper level that inhibits growth. Neither the upper or lower levels have been well established for animal cells though values between 5 % and 95 % of saturation with respect to oxygen in air (between 5 % and 95 % dO2) have been reported not to have any impact on cell growth (Nienow 2006). Nevertheless, it is more common to try to control dO2 in the range 30–50 % of saturation.

*Cg** depends on the partial pressure of oxygen in the gas phase, *pg* and this can be

related to the total pressure, *Pg* from Dalton's law of partial pressures

where *y* is the mole (volume) fraction of oxygen in the gas phase and *Pg* is the total pressure, i. e. back pressure plus static head. *Cg** is also a function of the liquid composition (and is lower in solutions containing electrolytes than in pure water) and reduces with increasing temperature. Thus the driving force for mass transfer can be increased by enhancing Cg*; and lowering *CL* provided it is > *Ccrit*. However, in a large scale bioreactors (which may be up to 25 m3) where circulation times of the liquid are sufﬁcient for signiﬁcant oxygen depletion by the respiring cells (Sweere et al. 1987), the average value may need to be kept well above *Ccrit* so that local values below it can be avoided. Hence, the choice of values in the range 30–50 %.

Nevertheless, in a stirred bioreactor, the liquid is generally considered wellmixed, i. e. *CL* is spatially constant. This is a reasonable assumption for the liquid phase in animal cell culture except at the larger scales. The measurement of the local concentration is done by a polarographic electrode and the reading obtained is dependent on the velocity over the probe. Since the velocity varies so much, establishing the exact concentration ﬁeld is generally not possible. Assumptions also have to be made regarding the mixing of the gas phase. It may be well mixed so that

where *(pg)out* is the partial pressure of oxygen in the exit gas and

On the other hand, for ease of determining *kLa* from experimental data, it is often assumed that the gas is well mixed and *pg* ¼ *(pg)in*, i.e. no oxygen is utilized. This is the so-called no-depletion model. In this case,

For larger-scale bioreactors (Pedersen 1997), the gas phase is generally considered as being in plug ﬂow, so that a log mean value of driving force is obtained, ΔClog:

where

and

The assumption made is not very important in cell culture except at large scale and high cell densities because the amount of oxygen removed compared to that introduced is quite small. On the other hand it is very important in high oxygen demanding bacterial and mycelial fermentations.