A process variable is any quantity that is under the direct control of the designer or operator of the process. For gas treatment bioreactors, the variables to be fixed by the designer include the bioreactor type, shape, and size, whereas those fixed by the operator include the addition of water, nutrients, and chemicals for pH control. The objective in fixing these variables is to satisfy a set of process requirements that specify the flow rate and composition of the gas stream, the nature and concentration of the contaminant, and the fractional removal required. The difficulty is that many designs, that is, many combinations of the process variables, will meet the process requirements for a particular case. The proper goal of engineering design is not simply to pick one of these designs at random, but to find the single design that meets the requirements at the lowest possible cost. The design of gas treatment bio-reactors, like that of any other process, is an economic optimization problem, but one that is so complex that the exact solution cannot be found within the constraints of time and money normally imposed on the design process. The best solution for a particular case will certainly depend on the scale of the problem, the solubility and possible inhibitory effects of the contaminant, the type of mi-crobial metabolism (aerobic or anaerobic, growth associated or cometabolic, etc.) needed to degrade it, and the temperature and pressure of the gas stream. Even coming close to the best solution requires a careful scale-up program in which the insights available from education and experience are used to integrate experiments at different scales with mathematical models that can extrapolate the results from one scale to the design of the equipment at the next scale. The bioreactor analysis presented in this section is intended as a guide to this program, sug gesting which bioreactor types be chosen for experimental evaluation in each case, specifying the operating conditions around which the experiments should be designed, and showing how performance may be expected to vary with scale.
Because the design goals are essentially economic, the relationship of the variables shown in Figure 1 to the process costs must be kept in mind. The volume of the commercial-scale bioreactor equals the actual gas flow to be treated (a process requirement) divided by g, a large value for g translates directly into a smaller and, all other things being equal, cheaper bioreactor. The liquid flow, f should be kept as small as possible because providing clean water and disposing of any process wastewater both cost money, as do any nutrients and chemicals that must be added. The pressure drop, h, through the bioreactor may be significant, particularly on a large scale, and it determines the capital cost of the gas compressors and the energy costs for running them. Compressing the feed gas is not only expensive in itself, it heats the feed such that heat exchangers may be needed to cool it to the desired temperature.
It will be assumed that the biophase is completely mixed and that the gas approximates plug flow through the bio-reactor. This provides a realistic description of some bio-reactors (bubble columns, some biotrickling filters), and a reasonable starting point for quantitative thinking about the others. More detailed mathematical models are available in the literature for specific types of bioreactors and metabolism (31,32). The equilibrium solubility of the contaminant in the biophase, S*, will be described by Henry's law, S* = p/H, where p= partial pressure of the contaminant in the gas and H= Henry's law constant. Although this is adequate for relatively insoluble and dilute contaminants, it is a considerable oversimplification for others. For example, the dissolution of oxides of nitrogen (NOX: a mixture of NO and NO2) from stack gases into water is an extremely complex process (34,35) that involves the gasphase chemical oxidation of NO to NO2 and a dissociation reaction that generates H + , NO3, and NO2 when the gas dissolves. In such cases, Henry's law can only be interpreted in the purely qualitative sense that low H means a soluble gas.
Any complete bioprocess model combines the mass conservation equations that describe the bioreactor with equations that try to approximate the complexities of microbial metabolism. Because the objective here is to compare different bioreactor configurations, it is essential to have consistent descriptions of the metabolism. In most gas treatment applications the contaminant is a major microbial nutrient, either the electron donor or the electron acceptor. Its consumption provides metabolic energy for growth and cell maintenance, so its specific consumption rate, q is given by the yield equation where 1 is the specific growth rate of cells, Y is the cell-yield coefficient, and k is the cell-maintenance coefficient.
In a well-operated bioreactor, the pH and temperature are carefully controlled and all required nutrients are provided in adequate amounts, making the specific consumption rate a function only of the concentrations of contaminant, S, and any inhibitory product, Sm, dissolved in the biophase. The exact form of this function, written q(S, Sm), will not be specified because it varies among cases, but it must describe the effects on the metabolic rate ofsubstrate limitation, product inhibition, and possibly (when ni/H is large) substrate inhibition. It is more common to describe the specific cell growth rate, 1, by a Monod-type function of the concentration S, but this is clearly incorrect at the low concentrations found in gas treatment bioreactors, because it predicts that growth stops and substrate uptake continues when no substrate is available 1 = 0 and q = kwhen S = 0). In fact, when a major nutrient is exhausted, its consumption must stop, and the microorganisms go into an endogenous state in which the viable biomass declines, a phenomenon correctly approximated by requiring that q(S, Sm) = 0, and thus 1 = kY (equation 1) when S = 0. Cell growth now stops at a nonzero "stationary phase" concentration, Ss, corresponding to the point where substrate uptake is just sufficient to satisfy the maintenance requirement, and defined by q(Ss, 0) = k.
For cometabolic processes, the contaminant is neither the main electron donor nor the acceptor; two rate equations are needed, and equation 1 contains an extra parameter whose value may be found from knowledge of the deg-radative pathway. See, for example, the analysis by Andrews of chloroform degradation by methanotrophic bacteria (36). The design of cometabolic bioreactors follows many of the principles described here but is too complex and case specific to be analyzed in detail.
At steady state the conservation of mass requires that the rate of growth of microorganisms in the bioreactor, iebSx, equals the rate at which they flow out, fzSx (see Fig. 1 for nomenclature). The loss of microorganisms in mist suspended in the effluent gas may be significant in some bioreactors without demisting devices but is not included explicitly in the analysis. The specific growth rate, 1, a measure of the physiological state of the microorganisms, thus equals zf/eb, the reciprocal of the mean cell-residence time (the number of cells in the bioreactor divided by the cell outflow rate), which is a quantity under our direct control. This remarkable result, although well known for chemostats, has not been fully appreciated for gas treatment. It follows from equation 1 that the concentrations in the biophase, S and Sm, are constrained by q(S, Sm) = k(1 + D
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