Calculating Biomass Yield ($Y_{X/S}$) in Chemostat
The yield coefficient $Y_{X/S}$ represents the amount of biomass produced per unit of substrate consumed. In a chemostat at steady state, this is calculated as:
$Y_{X/S} = \frac{\text{Biomass Concentration}}{\text{Substrate Consumed}}$
We are given:
- Steady-state biomass concentration ($X_{ss}$) = $4.5$ Kg m$^{-3}$
- Incoming sucrose concentration ($S_{in}$) = $10.0$ Kg m$^{-3}$
- Residual sucrose concentration ($S_{res}$) = $2.0$ Kg m$^{-3}$
Determining Substrate Consumed
The amount of substrate consumed is the difference between the concentration in the feed and the residual concentration in the effluent:
$\text{Substrate Consumed} = S_{in} - S_{res}$
Substituting the given values:
$\text{Substrate Consumed} = 10.0 \text{ Kg m}^{-3} - 2.0 \text{ Kg m}^{-3} = 8.0 \text{ Kg m}^{-3}$
Calculating Yield ($Y_{X/S}$)
Now, we can calculate the yield coefficient using the biomass concentration and the calculated substrate consumed:
$Y_{X/S} = \frac{X_{ss}}{\text{Substrate Consumed}}$
$Y_{X/S} = \frac{4.5 \text{ Kg biomass m}^{-3}}{8.0 \text{ Kg substrate consumed m}^{-3}}$
$Y_{X/S} = 0.5625 \text{ Kg biomass/Kg substrate}$
Rounding to three decimal places, the yield is $0.562$ Kg biomass/Kg substrate.
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).
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 ________
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.
