The results obtained in Illustration 13.1 indicate that for the biochemical reaction of interest the biomass specific growth rate is essentially constant. This fact, in turn, implies that the parameter K_{S} in the Monod equation is sufficiently small that it can neglected over the large majority of the time that the biochemical transformation is taking place. Under these circumstances a mass balance for the microorganism leads to the following developments for circumstances when the volume of the growth medium remains constant:
\frac{1}{ X } \frac{d X }{d t}=\mu=\mu_{\max } (A)
Separation of variables and integration yield
\int_{X_0}^X \frac{d X }{ X }=\int_0^t \mu_{\max } d t (B)
or
\ln \left(\frac{ X }{ X _0}\right)=\mu_{\max } t (C)
where X_{0} is the mass of cells present at time zero, when the cells are presumed to be in the exponential stage of the growth cycle. Because the microorganism has been acclimated to the growth medium prior to initiation of the “experiment,” there are no substantive lag-time effects. Exponentiation of both sides of this equation followed by rearrangement produces the relation
X = X _0 e^{\mu_{\max } t} (D)
We can now proceed to write a material balance for the substrate in the form of equation (13.1.27) as
d S =q_s X d t (E)
where we may again take the appropriate value of q_{S} as that corresponding to the maximum biomass-specific growth rate. Substitution of equation (D) into equation (E) yields
\frac{d S }{d t}=q_{S \max } X =q_{S \max } X _0 e^{\mu_{\max } t} (F)
Integration then gives
\int_{S_0}^S d S =\int_0^t q_{S \max } X _0 e^{\mu_{\max } t} d t (G)
or
S = S _0+\frac{q_{S \max } X _0}{\mu_{\max }}\left\langle e^{\mu_{\max } t}-1\right\rangle (H)
Readers should note from the entries in Table I13.1-3 that q_{smax} is negative (the total amount of substrate in the batch reactor declines as fermentation proceeds). Combination of equations (C) and (H) yields
| 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 |
S = S _0+\frac{q_{S \max }\left( X – X _0\right)}{\mu_{\max }} (I)
Substitution of the average values of the biomass specific q-rates into equations (D) and (I) yields relations that may be employed in a spreadsheet format to generate plots of the total masses of substrate and cells present in the bioreactor as functions of time. These plots are shown in Figure I13.2 together with “points” representing the data employed in Illustration 13.1 to determine the q-rate values used here in Illustration 13.2.
Examination of Figure I13.2 indicates that the model fits the biomass data very well; by contrast, the quality of the fit of the model to the substrate data is not nearly as good, especially at long fermentation times. For fermentation times less than 10 h, the quality of the substrate fit is good, but for longer times the model significantly overpredicts the amount of substrate remaining. Readers are cautioned, however, that the q-rate model requires analysis of experimental data to generate the parameters of the model and that these parameters are specific to a particular microorganism and the environment in which the organism finds itself. The model is a well-tested semiempirical approach rather than a correlation based on theoretical considerations.
The shapes of the plots for substrate consumption and biomass production, as well as both the entries in the underlying spreadsheet and equations (D) and (H) of this illustration, indicate that not only is the rate of growth of the microorganism exponential but so too is the rate at which the mass of substrate present declines.
The approach utilized in this illustration can be employed to ascertain the time at which one may regard a batch culture as being completed (i.e., the time at which further production of biomass is no longer possible). For example, equation (I) indicates that when substrate is no longer present in the growth medium,
X _F= X _0+\frac{\mu_{\max }}{q_{s \max }}\left(- S _0\right) (J)
Substitution of the numerical values from the problem statement into equation (D) gives a numerical value for the final quantity of biomass in the reactor (X_{F}):
X _F=6.00+\frac{6.18 \times 10^{-4}}{-1.17 \times 10^{-3}}(-20.17)=16.65 g (K)
From equation (C),
t_{\text {final }}=\frac{\ln \left( X _{\text {final }} / X _0\right)}{\mu_{\max }}=\frac{\ln (16.65 / 6.00)}{6.18 \times 10^{-4}}=1652 min
Both this time and the final concentration of biomass are consistent with the plots in Figure I13.2.
