Prior to examining the time profile of the dissolved carbon dioxide concentration during an entire fermentation, the effect of the agitation rate on the dissolved CO2 is first considered in isolation. During the fed-batch phase of an E. coli fermentation, step changes to the agitation rate were made (700rpm —> 500rpm 400rpm -4 360rpm), at a constant aeration rate (Q=l/3 min *) and glucose feedrate
on a molar basis. In Figure 49 are shown the calculated K^°^a and Ç resulting from these changes, plotted against time. The K^^^a were calculated assuming a well- mixed gas phase, but this assumption has little effect on the numerical result, as the change in gas phase oxygen mole fraction across the fermentor is small. The significant amount of noise in the ^ data in particular are caused by the noise introduced into the CTR by pH control action (Buckland et al, 1985), and also reflect the resolution limit of the dissolved carbon dioxide probe. In Figure 50 the average value of ^ in Figure 49 at each agitation rate is plotted against the corresponding average K^^^a for this agitation rate. Also plotted are the theoretical results for Ç for the well-mixed gas and plug-flow gas cases (Equations 21 & 22), with H^°^=4000 Pa.m^.mol \ corresponding to the operating temperature of 37°C. There is good agreement between the experimental results and the theoretical results for a well- mixed gas phase, indicating that in the 42L fermentor used, the gas is highly backmixed for all agitation rates of practical interest, as might be expected in a small fermentor.
In Figure 51 are presented the time profiles of the partial pressures of CO2 in the liquid phase and exit gas during the course of an E. coli fermentation. Also presented is the oxygen mass transfer coefficient, K^°^a, which only varied slightly during the course of the fermentation, from 0.04s * to 0.055s *. As the mass transfer coefficient only varied slightly, and the aeration rate was constant throughout the fermentation, the theoretical work already presented in Section 4.3.2 would suggest that the carbon dioxide excess (the ratio of the partial pressures of dissolved CO2 to that in the exit gas) should be constant throughout the fermentation. This ratio is evaluated in Figure 52, and is indeed fairly constant during the course of the fermentation, with a value around 1.25. Early on in the fermentation, however, the ratio is close to unity. This may be because, at the low encountered early in the fermentation, the dissolved CO2 probe has a slow response time, at a time when p ^ ^ n g fast.
The ratio of p^^°^ to is even constant during periods of unsteady-state carbon dioxide transfer. In Figure 53, unsteady-state CO2 transfer was generated between 4h and 9h, by changes in the pH. The partial pressures of carbon dioxide in the liquid
0 . 1 6
0.12
.m 0.08
O - I0.04
800
ê 600
400
O)200
1.6
1.4
1.2
01
2
Tim e (h)
Figure 49: Effect of changes in the agitation rate on the mass transfer coefficient, KL°^a, and the carbon dioxide excess, during the fed-batch phase of an E. coli fermentation (Vavvm, 3TC)
1.6
■
E, (measured)
— E, (well-mixed gas)
E (plug flow gas)
1.5
1.4
1.3
1.2
1.1
10.04
0.08
0.12
0.16
K
la (s ')
Figure 50: Plot of theoretical (well-mixed gas) and experimental (Figure 49) carbon dioxide excess, against the mass transfer coefficient, K^°^a (V^vvm, 37°C)
5000
C02 C02 C02, out(Pa)
C02, out 0 24
6 810
Tim e (h)
Figure 51; The partial pressure of dissolved carbon dioxide, Pl^^“ and of carbon dioxid
3TC)
œ
(0
0.06
0 .04
0.02
01.8
1.6
1.4
1.2
Time (h)
Figure 52: The mass transfer coefficient, K^°^a and the carbon dioxide excess, ^ for the E. coli fermentation data presented in Figure 51
and gas phases can be seen to fluctuate as a result of the changing pH, but their ratio appears qualitatively to be approximately constant. This is confirmed in Figure 54 which presents the carbon dioxide excess, for this fermentation.
The limited accuracy of p^^°^ measurements precludes the explicit validation of the mass transfer coefficient ratio proposed in Equation (18). However, the values of the carbon dioxide excess obtained experimentally, being in fair agreement with the theoretical predictions of Section 4.3.2, are at least consistent with a mass transfer coefficient ratio of around 0.89.
4.3.4 Sum m ary of Section 4.3
Table 15 summarizes the main results and conclusions from Section 4.3. To support the view that carbon dioxide transfer at steady state is a purely liquid-film limited physical process, it was shown that the resistance in the liquid-film is at least seventy times that in the gas film. Further, an expression describing the extent of any mass transfer enhancement by reaction in the liquid film was derived, and shown to indicate negligible enhancement. A new ratio of the mass transfer coefficients for carbon dioxide to that of oxygen of 0.89 was obtained, being equal to the diffusion coefficient ratio raised to the two-thirds power, and suggested by noting that bubbles in agitated fermentors are "small" (i.e. <2.5mm diameter). The above results were implicitly used to investigate the accuracy of the commonly-made assumption that the partial pressure of dissolved carbon dioxide is approximately equal to that in the exit gas. A new theoretical expression was obtained which shows that the maximum relative error in this assumption can be large, but only when the dissolved carbon dioxide partial pressure is small (and therefore of little concern). The exact error is typically in the range 20-30% for fermentors in which the gas phase is well backmixed (i.e. small fermentors) but considerably smaller when the gas phase is in plug flow (corresponding larger fermentors). This error decreases with decrease in aeration rate, or with an increase in the oxygen mass transfer coefficient.
5000
C02 C02 C02, out(Pa)
C02, out 0 6 810
24
0Time (h)
Figure 53: The partial pressure of dissolved carbon dioxide, Pl*^^ and of carbon dioxide in the exit gas, Pg“ ^°“‘ for an E. coli fermentation (%vvm, 37°C). Unsteady- State CO2 transfer generated between time 4h and 9h by changing the pH doesn’t appear to affect the relationship between p^^°^ and p^co2.oui
CO
0.06
0.04
0.02
0 0 24
6 810
1.8
1.6
1.4
1.2
Figure 54: The mass transfer coefficient, K^°^a and the carbon dioxide excess, ^ for E. coli fermentation data presented in Figure 53
1. The ratio of the partial pressure gradient in the liquid film to that in thae gas film is at least 1700 for oxygen transfer and 70 for carbon dioxide transfer, indicating the mass transer of both these components is liquid-film limited
2. A theoretical expression for the extent of enhancement of carbon dioxide mass transfer by reaction in the liquid film was derived and shown to indicate negligible enhancement
3. As bubbles in fermentors are small (<2.5mm diameter as defined by Calderbank, 1959), the ratio of the mass transfer coefficients for carbon dioxide and oxygen depends on the ratio of their liquid diffusivities to the two-thirds power, yielding a coefficient ratio of 0.89 in water
4. The maximum error in the assumption of equilibrium between dissolved CO2, and CO2 in the exit gas is limited by practical constraints
5. The exact value of the error in the equilibrium assumption increases with increase in aeration rate, or decrease in K^°^a, and is typically around 20-30% when the gas phase is well mixed (small fermentors) and less when the gas phase is in plug flow
6. Experimental measurements using a sterilizeable dissolved CO2 probe are consistent with theoretical results
Table 15: Principal results and conclusions arising from a study of steady-state CO2 transfer presented in Section 4.3
4.4 THE DEVELOPMENT AND EXAMINATION OF THE BEHAVIOUR