Question 4 of 5: k L and k L a — Definitions and the Air-On/Air-Off Method

Question 4: k_L and k_La — Definitions and the Air-On/Air-Off Method (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Definitions. By two-film theory, the resistance to oxygen transfer from a rising gas bubble into the bulk liquid is dominated by a thin stagnant liquid film at the bubble surface (the gas-side resistance is negligible for a sparingly soluble gas like O₂). k_L (units: m/s or cm/h) is the liquid-film mass-transfer coefficient — the local rate constant describing how fast oxygen diffuses across that film per unit interfacial area per unit concentration driving force. Because the total interfacial area a (m² of gas–liquid interface per m³ of broth) in a real sparged/agitated bioreactor is enormous, irregular, and essentially impossible to measure bubble-by-bubble, k_L and a are always determined and reported together as the lumped volumetric mass-transfer coefficient k_La (units: h⁻¹ or s⁻¹), which appears directly in the oxygen-transfer rate equation $$OTR=k_La\,(C^{*}-C_L)$$ where C* is the dissolved-oxygen concentration in equilibrium with the sparge-gas oxygen partial pressure (from Henry's law) and C_L is the actual bulk dissolved-oxygen concentration. k_La is the single most important design/scale-up parameter for an aerobic bioreactor because it sets the maximum sustainable oxygen-uptake rate (and hence maximum cell density/productivity) the vessel can support.

The Air-On/Air-Off (dynamic gassing-out) method

The dynamic gassing-out method exploits an unsteady dissolved-oxygen balance on the broth, $$\frac{dC_L}{dt}=k_La\,(C^{*}-C_L)-q_{O_2}X$$ where q_O2 is the specific oxygen-uptake rate (mass O₂ per unit biomass per unit time) and X is the biomass concentration. The method is run in two steps on an actively respiring culture, with a dissolved- oxygen probe recording C_L(t) throughout:

1. Air OFF — isolate the consumption term. Aeration is briefly stopped (or switched to pure N₂) while agitation continues. With no gas supply, k_La(C*−C_L)→0 and the balance collapses to a pure linear decay: $$\frac{dC_L}{dt}=-q_{O_2}X \quad\Rightarrow\quad C_L(t)=C_{L,0}-q_{O_2}X\,t$$ The measured slope of the (straight) DO-vs-time trace during this window is read directly as −q_O2X. Since X is measured independently (dry weight, OD, or cell count), q_O2 is obtained immediately — provided the DO stays above a critical concentration C_crit throughout the window, below which oxygen itself becomes limiting and the decay would curve rather than stay linear, corrupting the reading.
2. Air ON — recover k_La with q_O2 now known. Aeration is restored before DO falls below C_crit. The full balance dC_L/dt=k_La(C*−C_L) −q_O2X now applies, with q_O2 already fixed from Step 1 as the only remaining unknown besides k_La. Rearranging, $$k_La=\frac{dC_L/dt+q_{O_2}X}{C^{*}-C_L}$$ k_La is obtained either from the initial recovery slope (dC_L/dt at a single instant) or, more robustly, by linear regression of dC_L/dt+q_O2X against C_L over the whole recovery transient — the regression slope is −k_La and the intercept is k_LaC*, so a single recovery curve over-determines and cross-checks both C* and k_La.

Because both q_O2 and k_La come from the same short experiment on the actual culture at its actual operating conditions (not a sterile-water proxy), the dynamic method is prized for giving in-situ, biologically realistic values without needing separate chemical (sulfite oxidation) or gas-balance methods, and it can be repeated at intervals through a fermentation to track how k_La and q_O2 evolve as the broth's rheology and cell density change.