This value can only be approached, since if D = 0, then Q = 0, and the system becomes a batch, not a chemostat.

On the other hand, if D is too high, the organisms cannot grow fast enough to maintain themselves in the reactor (Figure 11.28). At this critical dilution rate (Dc), or washout rate, XS = 0 and thus SS = Si. In other words, as D (and hence, growth rate) increases, SS also increases until it reaches the influent concentration. Since it cannot go any higher, neither can growth rate, and the culture washes out. The highest net growth rate that can be approached (at steady state) is thus

Determining Microbial Kinetic Coefficients in Chemostats One use of chemostats is for determining the microbial kinetic coefficients £ and Ks. For this purpose, chemostats are run several times at different dilution rates and the steady-state substrate concentration is determined for each. A plot of S vs. D is then made (Figure 11.28). Note in the figure that at values of D > Dc, X = 0 and S = Si. Also, at low values of D, the importance of decay (b) leads to a decrease in X.

From the development of equation (11.28), we can see that

Such nonlinear equations are becoming readily usable on computers, but a traditional linearization method, the Lineweaver-Burke plot, may still be useful to know. (It can also be applied to data collected from batch cultures.) This involves inverting equation (11.30):

For values of D ^ b, a plot of 1/D vs. 1/S now gives a straight line with an intercept of 1/(1 and a slope of Ks/p. (Figure 11.29). The value of p. is then obtained by taking the inverse of the y-intercept value. Ks can also be calculated from the slope once ( is known; however, for imperfect data with their variability, it is usually better (particularly if b is not very small; see the "apparent slope'' in Figure 11.29) to obtain it from the plot of S vs. D (Figure 11.28). This is done by noting the D value that corresponds to 21, then finding the corresponding S value that gives this growth rate (by definition, Ks). (To improve the quality of the estimates of the coefficients obtained, the slope Ks/m can then be calculated and used to better draw the Lineweaver-Burke plot, with the new intercept giving a better value of (1, which can then be used in turn to better estimate Ks from the plot of ~ vs. D. After two or three iterations, the values will stabilize on the best estimate for the data.)

Figure 11.29 Lineweaver-Burke plot. In part (a) the same coefficients are used as in Figure 11.28 to generate the theoretical curves. In part (b) the value of b — 0.15, showing its effect on the nonlinearity of the 1/D plot.

Figure 11.30 Determining Y and b. The same coefficients were used to generate the plot as in Figure 11.28, except that b = 0.15.

Figure 11.30 Determining Y and b. The same coefficients were used to generate the plot as in Figure 11.28, except that b = 0.15.

One criticism of the use of Lineweaver-Burke plots is that with some data they do not give very good estimates of the coefficients (particularly for Ks values obtained from the slope only rather than the ~ vs. D plot). Nonlinear regression is now a preferred approach if sufficient data points have been collected.

The coefficients Y and b can also be found from chemostat data. Since mn = D in a chemostat, we can rewrite equation (11.19) as

Thus, by plotting D vs. U, Y (the slope) and b (the negative of the y-axis intercept) can be determined (Figure 11.30). For this purpose, U can be determined as

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