The following schematic diagram shows a chemostat with cell recycle where $F_0$ and $F_r$ are the volumetric flow rates (in $L.h^{-1}$) of feed and recycle streams, respectively. $X_1$, $X_0$ and $X$ are the cell concentrations (in $g.L^{-1}$) in the reactor, recycle-stream and product-stream, respectively. If $\frac{X_0}{X_1}=1.5$, $\frac{F_r}{F_0}=0.7$ and $X_1$ is $7.3 g.L^{-1}$, the value of $X$ (in $g.L^{-1}$, rounded off to one decimal place) is ________
To find the cell concentration $X$ in the product stream, we use the given parameters and the mass balance around the system.
The mass balance equation for the reactor is:
\[ (F_0 + F_r)X_1 = F_r X_0 + F_0 X \]
We are given:
- \(\frac{X_0}{X_1}=1.5\) meaning \(X_0=1.5X_1\).
- \(\frac{F_r}{F_0}=0.7\) meaning \(F_r=0.7F_0\).
- \(X_1=7.3\ g.L^{-1}\).
Substituting the relationships into the mass balance equation:
\[ (F_0 + 0.7F_0)X_1 = 0.7F_0(1.5X_1) + F_0 X \]
Simplifying:
\[ 1.7F_0X_1 = 1.05F_0X_1 + F_0 X \]
Cancel \(F_0\) from all terms:
\[ 1.7X_1 = 1.05X_1 + X \]
Rearrange to solve for \(X\):
\[ X = 1.7X_1 - 1.05X_1 \]
\[ X = 0.65X_1 \]
Substitute \(X_1 = 7.3\) into the equation:
\[ X = 0.65 \times 7.3 = 4.745\ g.L^{-1} \]
Rounding off to one decimal place gives:
\[ X = 4.7\ g.L^{-1} \]
Finally, verifying the range, \(X = 4.7\ g.L^{-1}\) lies within the range (4.6, 4.9).
Important Questions from Batch Fed Batch and Continuous Processes
- Under complete cell washout condition in a chemostat with sterile feed, which of the following statements is/are correct?
A fed batch process is running at quasi-steady state with respect to substrate and biomass concentration. At $2 \text{ h}$, the culture volume is $500 \text{ L}$ with a constant sterile inlet feed at $50 \text{ L } h^{-1}$ of glucose. The culture kinetic parameters $ \mu_m$ and $K_s$ are $0.2 \text{ } h^{-1}$ and $0.1 \text{ } g \text{ } L^{-1}$, respectively.
The substrate concentration in the reactor will be ________ $g \text{ } L^{-1}$ (rounded off to one decimal place).
A $2 \text{ L}$ bioreactor is being operated as a chemostat, at a flow rate of $0.8 \text{ L/h}$ and sterile feed of $10 \text{ g/L}$ substrate. The bacterial growth follows Monod kinetics at a maximum specific growth rate of $0.6 \text{ h}^{-1}$ with a Monod constant of $0.5 \text{ g/L}$ and a biomass yield coefficient of $0.4 \text{ g/g}$. The exit biomass concentration is __________ $\text{g/L}$.
(Round off to one decimal place)
- The amount of biomass in a reactor at the end of the batch process is 50 g. Fed- batch operation is initiated by feeding the substrate solution at a constant rate of $1 \text{ L h}^{-1}$. The concentration of substrate in the feed is $50 \text{ g L}^{-1}$. The maximum biomass yield ($Y_{XS}^M$) is $0.4 \frac{\text{g biomass}}{\text{g substrate}}$. Assuming the system is at quasi-steady state, the maximum amount of biomass after 5 h of feeding is ________________ g.
- In a chemostat with a dilution rate of $0.8 \text{ h}^{-1}$, the steady state biomass concentration and the specific product formation rate are $8 \text{ mol m}^{-3}$ and $0.2 \text{ (mol product) (mol biomass)}^{-1} \text{ h}^{-1}$, respectively. The steady state product concentration in $mol \text{ m}^{-3}$ is ________
