Calculating Maximum Biomass Concentration
The maximum biomass concentration ($X_{max}$) in a batch culture, assuming no endogenous metabolism, can be determined using the apparent growth yield ($Y_{XS}$) and the initial concentrations of biomass ($X_0$) and substrate ($S_0$). The apparent growth yield relates the increase in biomass produced to the amount of substrate consumed.
Growth Yield Concept
The apparent growth yield ($Y_{XS}$) is defined as the change in biomass ($\Delta X$) divided by the change in substrate ($\Delta S$):
$Y_{XS} = \frac{\Delta X}{\Delta S}$
In this scenario, $\Delta X = X_{max} - X_0$ and $\Delta S = S_0 - S_{final}$. Since maximum biomass is achieved when the substrate is nearly depleted (assuming it's the limiting nutrient), $S_{final} \approx 0$. Therefore, $\Delta S \approx S_0$.
The relationship simplifies to:
$Y_{XS} = \frac{X_{max} - X_0}{S_0}$
Step-by-Step Calculation
-
Identify Given Values:
- Apparent growth yield, $Y_{XS} = 0.5 \frac{\text{g biomass}}{\text{g substrate}}$
- Initial biomass concentration, $X_0 = 2 \frac{\text{g}}{\text{L}}$
- Initial substrate concentration, $S_0 = 200 \frac{\text{g}}{\text{L}}
-
Rearrange the formula to solve for $X_{max}$:
$X_{max} - X_0 = Y_{XS} \times S_0$
$X_{max} = X_0 + (Y_{XS} \times S_0)$
-
Substitute the given values into the formula:
$X_{max} = 2 \frac{\text{g}}{\text{L}} + \left( 0.5 \frac{\text{g biomass}}{\text{g substrate}} \times 200 \frac{\text{g substrate}}{\text{L}} \right)$
-
Calculate the substrate consumed contribution to biomass:
$0.5 \times 200 = 100 \frac{\text{g biomass}}{\text{L}}$
-
Calculate the final maximum biomass concentration:
$X_{max} = 2 \frac{\text{g}}{\text{L}} + 100 \frac{\text{g}}{\text{L}} = 102 \frac{\text{g}}{\text{L}}$
The maximum biomass concentration that can be achieved is $102 \frac{\text{g}}{\text{L}}$.
Important Questions from Kinetics of Cell Growth Substrate Utilization and Product Formation
If the rate at which $E. coli$ divides is $0.5 \text{ h}^{-1}$, then its doubling time is _______________ h.
- Which of the following factors can affect the growth of a microbial culture in a batch cultivation process?
Let $y(t)$ be a bacterial population whose growth is given by
$ \frac{dy}{dt} = \lambda(y + 2) $
where $ \lambda $ is the growth rate constant. If $y(0) = 1$ and $y(1) = 4$, then the value of $ \lambda $ is
If the doubling time of a bacterial population is 3 hours, then its average specific growth rate during this period is _________ $h^{-1}$.
(Round off to two decimal places)
- Which one of the following represents non-growth associated product formation kinetics in a bioprocess system? X and P denote viable cell and product concentrations, respectively.
