One could argue that when R = KLaf ( Cs - C), the fugacity (the maximum transferable rate) has been reached and so the OTR or Rd is zero at that gas flow rate representing the energy input. However, this is a state that could not exist at that given gas flow rate, because OTR can never be zero in a respiring system. The case in which R = KLaf (Cs - C) is not a stable situation (i.e., not a steady state and C ≠ CR), which implies that the actual transfer rate, as opposed to total potential oxygen transfer rate, to the liquid is less than R and the DO concentration is decreasing because the consumption rate has exceeded the transfer rate so that the system is now outgassing oxygen. However, this does not mean Rd is negative. That is, the actual transfer rate, the net oxygen transfer rate given by Eq. (10a), i.e., OTR = KLaf (Cs – C) – r, is not KLaf (Cs - C). To maintain a given DO set point, the air flow rate (AFR) would have to be increased to a new KLa value such that ultimately R equals the actual transfer rate to the liquid. A change in air flow rate would result in a different transfer efficiency, at least in a fine bubble system where an increase in air flow decreases efficiency, and so the true C* also would be nominally different (higher, actually, due to lower gas side depletion), but eventually a steady state would be reached in which the oxygen consumption (R) would exactly match the oxygen transfer rate. For any further increase in the organic loading rate (OLR) and if C > 0, the system can respond by lowering C so that the driving force increases, giving more impetus to transfer. However, when the consumption rate exceeds the oxygen transfer rate, CR approaches C, which itself is ever-decreasing, such that dC/dt is a decreasing function of the consumption rate, i.e., dC/dt < 0. Therefore, the system is no longer in a steady state in such an event. Eventually a point is reached at which C becomes vanishingly small such that even the maximum fugacity is not enough to satisfy, and so the only remedy is to increase the gas flow rate again to match the demand. The conclusion of this exercise is that, for submerged aeration in which the gas loss rate from the system is significant, the rate of transfer under the action of microbial respiration must be given by Eq. (12b) i.e., dC/dt = Kla (Cs – C) – 2Ru, in which both the associated liquid phase oxygen equilibrium concentration (Cs) and the apparent oxygen saturation CR will decrease accordingly (such a phenomenon can be experimentally verified in a converse manner by a reduction in the microbial GDP (the resistance due to biochemical reactions), the net result of a dilution is that both the associated liquid phase oxygen equilibrium concentration Cs and the apparent oxygen saturation CR will increase accordingly. (It is notable that the latter increases faster than the former, so that at R = 0, the rise of CR catches up with the rise of Cs, and so both become one and the same, C*inff).
Mines' paper began on the right track by citing Bartholomew, Albertson and DiGregorio, and some others like Eckenfelder, that there is definitely a relationship between KLa and OUR and so Mines conducted his experiment. Herein lies the difference: Those previous researchers used plant operation data, where the DO is maintained constant. One can only have either constant DO or constant AFR (aeration gas flow rate), but not both. Mines' attempt to verify the dependency of Kla on OUR is premised on constant AFR which is exactly right but he used the wrong equation, resulting in Table 3 and Table 4 that yield the strange result that at steady-state, the OUR is not the same as the OTR. Had he used the right equation, he would have got a consistent result that would support my theory. The consequence of an increase of Ru can only be a reduction of OTR for a constant AFR. It can never by an enhancement! Mines' equation 6, stating that Rd = KLa (Cs- C) - Ru + Ri is therefore insupportable.
His experiment needs to be repeated, but with the following caveat:
equations must be correct, i.e., equation 7 must be written OTR = alpha KLa(beta Cs - C) – Ru resulting in the accumulation term as:
dC/dt = Kla (Cs – C) – 2Ru
OUR of the mixed liquor suspended solids as determined by Method 213B in Standard Methods must be modified to eliminate the shaking effect;
the OTR should be independently measured by the offgas method to compare with the modified Equation 7, since the offgas method is widely considered the best way to determine OTR.
It is important to recognize that the transfer equation given by Equation 1 in Mines' paper, is only valid when R = 0. When R changes, both Cs and OTR will change, even though C changes, (decrease to increase the driving force, or increase if the AFR increases).
Here are the thoughts of one reviewer in which he advocated an equation (eq. E) that says the OTR is given by OTR* = KLa (Css - C) where Css is the saturation concentration at the steady state. However, this equation cannot be logical since, when C = Css the OTR* would become zero. In a respiring system there is always some oxygen transfer even at the SS. The reviewer then explained that OTR* is not the real OTR, which is not logical since there is only one OTR at any particular condition. My reply is given in the attached file.