where G = k1 aRT/gH is the dimensionless mass transfer coefficient; F = YS'mlzXym is the dimensionless product inhibition factor; X = gnJkebRT is the maximum possible viable cell concentration in the biophase; and S'm is the product concentration that completely inhibits cell growth.
Rather than solving equations 2 and 4 exactly, which would require the form of the inherent kinetic function q(S, Sm), consider some limiting cases in order of decreasing liquid outflow rate. These cases are used to summarize the bioreactor performance in Figure 2, a graph that can be universal, at least within the limitations of the assumptions, because the parameters describing the contaminant (H, n), the reactor ( f, kja), and the metabolism (k, S'm) in individual cases have been incorporated into the dimen-sionless numbers x, D, F, and G.
• Washout. D > (q(nJH,0)Ik) - J: Because the dissolved concentration, S, must obviously be less than that in equilibrium with the feed gas, ni H, equation 2 now has no solution, meaning that all of the biomass is washed out of the reactor and there can be no contaminant removal.
• Substrate limitation. D > (J — e-G)IF: Equation 4 shows that under this condition, S r 0, while Sm <
/ Decreasing F
Figure 2. Operating regime for a completely mixed bioreactor.
j Decreasing G /
/ Decreasing F
I Product inhibited \ Sx = 0-8X
1 -HSJnj Fractional contaminant removal, x
Figure 2. Operating regime for a completely mixed bioreactor.
S'm, meaning that microbial kinetics are limited by the availability of substrate rather than product inhibition and the exact value of the inhibition number, F, is irrelevant. If gas-biophase mass transfer is very rapid, the dissolved contaminant is in equilibrium with the outlet gas, giving the best possible bioreac-tor performance. The shape of this G = <» line in Figure 2 is simply a reflection of the relationship between i( = zfl eb) and S( = p/H) from the inherent microbial kinetics. Mass-transfer limitation (G < <») leads to a progressive reduction in the contaminant removal, X.
• Product inhibition. D < (J — e—G)/ F: Now the concentration of metabolic products, Sm reaches inhibitory levels, S'm, while dissolved contaminant is still available, meaning that metabolism is limited by product inhibition rather than substrate limitation, and the exact value of G is irrelevant. Increasingly stronger inhibition (smaller F) produces a series of curves in Figure 2 whose exact shape depends on the details of the inherent inhibition kinetics. All of these curves pass through the origin because, with no liquid outflow (D = 0), any soluble, nonvolatile metabolic products must eventually accumulate to inhibitory levels.
Any pair of values of F and G define a regime in Figure 2 within which the bioreactor can operate. The most important result is that for any such pair of values there exists a value of D that maximizes the contaminant removal, X. If the form of the rate equation q(S, Sm) were known, the pointed maximum seen in Figure 2 would be replaced by a smooth curve that defined this optimum design more accurately. In the absence of such detailed modeling the optimum must be found experimentally, the analysis providing a starting point and overall guidance. Because the gas flow rate, g, appears in the denominator of both F and G, the first step is to establish the maximum g for which the desired contaminant removal, X, falls safely within the operating regime. It is clear from equation 4 that a high re-
moval rate (e.g., x = 0.95) with reasonable values of S(>Hn2) and Sm(<0.5 S'm) requires both G ~ 3 and FD ~ 2, and such values are reasonable, if approximate, general design goals. Much larger values represent inefficient design because they imply excessive consumption of energy and water.
For processes that produce large amounts of very inhibitory, nonvolatile, soluble metabolic products, the problem is to wash the products out of the bioreactor without washing out the biomass. Mathematically, ymIS'm r œ, making it difficult to keep FD > 2 without D exceeding the washout criterion. These are the processes that require cell retention (z r 0), either by cell recycle or by immobilization as a biofilm. For processes with more moderate values of ymI S'm, the analysis shows that there exists an optimum set of values of the process variables g, f and z. Some processes, the aerobic mineralization of hydrocarbons for example, generate only innocuous (H2O) or gaseous (CO2) products, so ym/S'm r 0, and the analysis suggests operating with no liquid outflow. This not only gives the best effluent-gas quality (Fig. 2 with F = œ) but, because the biomass is in a maintenance condition (m = 0 when D = 0), it also avoids the cost of adding nutrients, such as phosphates, that are needed for microbial growth (see "Nutrient Addition and Start-Up"). In practice there are reasons to avoid this condition. Even if ymIS'm is too small to be observed in short-term laboratory experiments, any nonvolatile product must eventually accumulate to inhibitory levels in a continuous bioreactor with no liquid outflow. Salts in the water added to compensate for evaporation or for pH control will also accumulate in the biophase. Also, it must not be forgotten that complete mixing of the biophase is an abstraction never achieved in practice and sometimes, as in the biofilter, not even attempted. The discussion prior to equation 2 shows that any bioreactor with D = 0 must have a cell growth rate m = 0 as an average, but it will contain microniches where cells are dying because conditions are worse than average, and others where conditions are slightly better and cells are growing. Although the growing cells may assimilate some of the material released by the lysis of the dying cells, there must be some long-term accumulation of dead biomass, with consequent reduction in bioreactor performance. Keeping D on the of order of 1, a very low flow rate in most cases can wash out the dead biomass and ensure stable operation for the cost of a few growth nutrients.
The analysis assumed the usual case in which the cell concentration in the bioreactor is kept as high as possible, with sufficient amounts of other nutrients provided to ensure that the contaminant is the limiting nutrient. The quantity X, defined under equation 4, is then the maximum possible viable-cell concentration in the biophase, corresponding to the situation where all of the contaminant flowing into the bioreactor, gnJRT, is being used to satisfy the cell maintenance requirement, kebX. The actual concentration at any given operating condition, Sx, can be calculated from the fourth term of the equation and is shown in Figure 2 as a function of D and x. There is, however, no guarantee that this concentration corresponds to a dilute, easily mixed cell suspension. If the maintenance requirement, k, is small, and the contaminant loading gnt is large, it may be a paste of cells too dense to be, for example, pumped around a biotrickling filter. Although it is possible to reduce Sx by restricting the amounts of growth nutrients provided, such artificial reductions in the microbial "catalyst" concentration are generally poor bioprocess design. The alternative is to choose the right type of bioreactor, recognizing that, although the mass of microorganisms in the bioreactor is fixed by the process requirements and the microbial metabolism, the amount of water is a process variable that can be controlled. The concentration, X, is an order of magnitude higher in a trickle-bed type bioreactor where eb ^ 0.1, than in a bubble column where eb ^ 1.
In some processes, the aerobic mineralization of organics in beds of compost for example, the air provides the electron acceptor and the media provides what little growth nutrients are needed. In other processes these substances, and sometimes the electron donor, must be provided in the liquid inflow. The amounts can be estimated from the "stoi-chiometry" of metabolism, a procedure best illustrated by example. Ongcharit et al. (37) have shown that hydrogen sulfide can be removed from industrial gas streams by the acidophilic, autotrophic, sulfur-oxidizing, denitrifying bacterium Thiobacillus denitriicans. Even such esoteric metabolism can be approximated by a pseudo-chemical reaction in which all of the main nutrients and products are written in the ionic form in which they enter or leave the reactor. After performing all of the element and charge balances the result is
- yCHhNnOoSsPp + (0.8 - 0.1y(y + 5n))N2 + (1 - ys)SO4- + (0.4 + y(0.2y - p - 2s))H+ + wH2O
The first term on the right-hand side represents the dry weight of the biomass produced. y = 4 + h - 2o + 6s + 5p is a measure of its oxidation/reduction state and is remarkably consistent between species (36) (note that the base oxidation state for nitrogen here is N2). The actual cell yield is y, the ratio of cells produced to H2S consumed. For a completely mixed biophase this equals 1 Iq or, from equations 1 and 2, y=
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