AT U = superficial gas velocity X ( t ) , Y(t) = random variables = time interval during which X ( t ) locates itself betweenx and x + Ax, Eq. 1 = superficial gas velocity at minimum fluidization Greek Letters P = mean Pr U = standard deviation U2 = variance +xx = auto-correlation function of X ( t ) +, , , = cross-correlation function between X ( t ) and Y(t) E = voidage of the bed E , , = voidage at minimum fluidization 0 = angular frequency = r-th central moment about the mean 7 = time shift variable, Eqs. 9 and 10

A new mathematical model is presented model is presented for use in the design and optimization of fluidized-bed fermenters. Unlike previous models, the biomass particle size is not a required input parameter, but is predicted as a consequence of the process by which the fermenter reaches a steady state. Both tower fermenters and supported-film bioreactors are included in the analysis. The differences between them are explained as a consequence of the different effects of added biomass on the particle settling velocity and the tendency of a fluidized bed to stratify. A detailed qualitative treatment of solids mixing allows the model to predict the varying biomass concentration through a tower fermenter and the more constant concentration in the supported film reactor. Other features of this analysis are the inclusion of an axial dispersion term to allow for different liquid mixing conditions, and the introduction of a variable transformation that eliminates the need for a computer solution. A sample design problem is included.

Although the fast filter worked well for cases where more concentration measurements were available (for example, Soliman and Ray, 1978), for the results shown here, the fast filter did not converge. This is not surprising in case 2, where there are no concentration measurements, since one would not expect the fast concentration dynamics to be observable without direct concentration measurements, For case 1, it is likely that a single exit concentration sensor is not sufficient for system observability.The slow filter performance, shown in Figures 1 and 2, was quite acceptable in both cases. Note that both the temperature and the quasi-steady-state concentration estimates converge quickly to the true profiles in case 1, while for case 2 the estimates are less accurate but reasonable.This example suggests that slow state estimates will perfoim acceptably, even in the case of no direct concentration measurements, but will perform significantly better with one or more composition sensors. The fast state estimator, which would be needed to follow fast concentration dynamics, requires more than one concentration sensor for good performance. However, in many applications the fast estimator would not be required, particularly if feed concentration were not a control variable.Presently work is underway in our laboratory to test these algorithms on-line in real time with pilot scale reactors.

The balance equations for carbon, reduction potential, and energy during cell growth and product formation are rederived in a general form. Cells are treated simply as a very complex product, and the Y(ATP) concept is extended to products. Limitations on the theoretical yield are discussed for different product types. Simple aerobic products cannot be energy limited unless the maintenance requirement is large, while complex products cannot be reduction limited. A maximum yield is defined for products much more oxidized than their substrate (carbon limited) because the theoretical yield conditions may violate the energy balance. For reduced complex products the yield on available electrons is related to Y(ATP), the P/O ratio, and the product composition. Narrow bounds are established on the actual yields in simple anaerobic fermentations, and the significance of the yields in the linear growth equation is discussed.

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