It is important to recognize that in this analysis one must take into account the very marked decrease (32%) in the volume of the growth medium during the course of the fermentation. We can utilize the data provided in the problem statement, together with an Excel spreadsheet, to calculate the total amounts of substrate and biomass present at any of the indicated times, recognizing that at any time these totals are equal to the product of the volume of the cellfree growth medium and the corresponding concentration of biomass or glucose. The results of these calculations are summarized in Table I13.1-2, which also contains estimates of the total rates of consumption of substrate (−R_{S}) and production of biomass (R_{X}) that were obtained by simple numerical differentiation of the appropriate entries in Table I13.1-2: for example,
This simple approach was adopted in order to circumvent the complications that are introduced by the fact that the volume of the liquid phase in the reactor varies with time. When the volume of the aqueous growth medium varies during the course of the reaction, an approach based on integration of a proposed rate law is problematic, although numerical integration would be possible. An additional reason for employing the differential approach below is that for rate laws that are other than those of the simple nth-order form (such as a Monod rate expression) a differential method of data analysis is often adequate for preliminary considerations involved in the design of a bioreactor that is intended to operate in a batch mode.
If one examines the entries in Table I13.1-2 for the total quantities of substrate and biomass present as functions of time, it is evident that the total amount of substrate declines continuously while the total biomass increases as the reaction proceeds. Note that on a mass basis the total rate of consumption of substrate always significantly exceeds the total rate of formation of biomass. Moreover, the magnitudes of both of these rates increase as time elapses, an indication that this reaction is indeed autocatalytic.
Although there are entries in Table I13.1-2 for the rate at which the total amount of biomass is increasing, it is preferable to use the entries in Table I13.1-1 to calculate the biomass specific growth rate (μ \text{ or } q_{X/S}) using the following relation derived from equation (13.1.28):
\mu=\frac{1}{ X } \frac{d X }{d t}=\frac{d \ln X }{d t} \approx \frac{\ln X _i-\ln X _{i-1}}{t_i-t_{i-1}}=\frac{\ln \left( X _i / X _{i-1}\right)}{t_i-t_{i-1}} (B)
From the entries for the rate of consumption of substrate in Table I13.1-2, one can estimate the biomass specific rate of consumption of substrate (q_{S}) by dividing the entries for the former rate by the amount of biomass present at the corresponding time. Estimates for both q_{X} \text{ and } q_{S} are presented in Table I13.1-3 together with the corresponding values of the yield of biomass in grams of dry weight per gram of substrate consumed (Y_{X/S}).
Inspection of the values of q_{X} \text{ and } q_{S} in Table I13.1-3 indicates that even in terms of our unsophisticated approach to the determination of these parameters, the percentage variations are surprisingly small. The fact that the variations are small can be attributed to a situation in which the Monod parameter K_{S} in equation (13.1.13)
-r_{ S , m}=\frac{-1}{X} \frac{d s}{d t}=\mu=\frac{\mu_{\max } s}{K_S+s} (13.1.13)
is much less than the substrate concentration (s). Consequently, the Monod rate law reduces to a pseudo-zero-order form
| Table I13.1-3 Determination of Biomass Specific q-Rates (μ and q_{S}) and Yield Coefficients for a Trial Involving Growth of a Generic Microorganism on Glucose |
|
Time
(min)
|
q-rate glucose,
q_{ S } \times 10^3
[(g/g) min^{−1}] |
q-rate biomass,
q_X(\text { a.k.a. } \mu) \times 10^4
[(g/g) min^{−1}] |
Yield coefficient
Y_{X/S}
(g/g)
|
| 0 |
|
|
|
| 40 |
− 1.142 |
5.937 |
0.52 |
| 80 |
− 1.191 |
6.194 |
0.52 |
| 160 |
− 1.168 |
6.199 |
0.53 |
| 320 |
− 1.172 |
6.245 |
0.53 |
| 640 |
− 1.171 |
6.241 |
0.53 |
| 1280 |
− 1.156 |
6.262 |
0.54 |
| Average |
− 1.17 |
6.18 |
0.53 |
\mu \approx \mu_{\max } (C)
Such results are observed sufficiently often that q-rate parameters may be employed in preliminary considerations of the design of a bioreactor.
The biomass specific growth μ can be written as
\mu=q_{ X / S }=Y_{ X / S }\left(-q_{ S }\right) (D)
where Y_{X/S} can be regarded as the actual yield of biomass per unit of substrate consumed. Note from the entries in Table I13.1-2 that the yield coefficients are positive (the biomass increases as substrate is consumed). Values of the biomass yield coefficients, Y_{X/S}, for many fermentations
are typically about 0.5.