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over most of the operating dilution rates. This high conversion of substrate is a key attribute of the ideal CSTR system, improving the economics of processes that depend on efficient substrate utilization and minimizing the effects of substrate inhibition. Also shown in Figure 1 is the high concentration of cells in the vessel until washout at the critical dilution rate. The maximum biomass productivity is located close to the washout point, making operation at the maximum productivity point very sensitive to deviations in dilution rate.

The dilution rate associated with maximum biomass productivity for a fixed limiting nutrient concentration in the feed (DM) can be calculated by setting the derivative of (DX) with respect to D to zero and solving for the dilution rate. DM is then given as

By substituting DM into equation 8, we can solve for XM, the cell concentration corresponding to DM.

Considering the limiting nutrient concentration (S0) as an independent design variable, equation 11 suggests that the substrate concentration in the feed can be increased arbitrarily to achieve extraordinary cell densities and productivities. In reality, the productivity of an aerobic reactor system is ultimately limited by the rates of heat and/ or mass transfer when the reaction kinetics are fast (i.e., high Xand D) (see "Mass Transfer"). Oxygen transfer, and not the carbon substrate, is often growth limiting because oxygen is an essential nutrient for aerobic metabolism. It is poorly soluble in the medium (typically around 7 mg/L for air at 1 atm), and its transfer rate is restricted by the physical capabilities of the oxygenation system. Oxygen-limited growth may be expressed as a steady-state balance between the oxygen uptake rate (OUR) and the oxygen transfer rate (OTR). Oxygen transfer is usually limited by transfer from the gas to liquid phases, leading to the steady-state balance:

in which Fx/Oz is the cell yield on oxygen (g DCW/g O2), mO2 is the maintenance coefficient for oxygen (g O2/g DCW X h), kL is the liquid phase mass transfer coefficient (cm/ h) a is the specific interfacial area for mass transfer (cm2/ cm3), kLa is the mass transfer coefficient (h-1), and (C* — CL) is the driving force for mass transfer where C* is the equilibrium oxygen concentration (mmol/L) and CL is the dissolved oxygen concentration (mmol/L). Typical values for Yx/O2 are located in Table 2. The ability of the heat transfer system to remove heat generated during microbial growth can also limit Rcstr, as discussed in "Energy Balance". These limitations will be important in evaluating and comparing CSTR performance.

Table 2. Yield Coefficients for Bacteria on Different Carbon Substrates

Substrate

YX/S

^X/O;,

Yhi

(i)

[g DCW/g substrate]

[g DCW/g O;]

[g DCW/kcal]

Acetate

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