CSTR Cell Recycle Mean Retention Time
This solution details the calculation of the mean cell retention time ($\theta_c$) for a Continuous Stirred Tank Reactor (CSTR) operating under cell recycle conditions, focusing on the steps needed to find the value.
Given Parameters
- Reactor Volume ($V$): $100 \text{ m}^3$
- Volumetric Flow Rate ($Q$): $10 \text{ m}^3 \text{ day}^{-1}$
- Effluent Biomass Concentration ($C_{e}$): $20 \text{ mg L}^{-1}$
- Steady State Biomass Concentration ($C_{ss}$): $200 \text{ mg L}^{-1}$
Mean Cell Retention Time Formula ($\theta_c$)
For a CSTR with cell recycle, the mean cell retention time ($\theta_c$) is the ratio of the total biomass mass in the reactor to the rate at which biomass mass exits the system.
The defining equation is:
$ \theta_c = \frac{V \times C_{ss}}{Q \times C_{e}} $
This formula accounts for the reactor volume, the biomass concentration inside the reactor, the flow rate, and the concentration of biomass in the effluent.
Mean Cell Retention Time Calculation Steps
- Substitute Given Values: Insert the provided reactor volume, flow rate, and biomass concentrations into the formula. Note that the units for volume ($\text{m}^3$) and concentration ($\text{mg L}^{-1}$) will cancel out, leaving the result in days.
$ \theta_c = \frac{100 \text{ m}^3 \times 200 \text{ mg L}^{-1}}{10 \text{ m}^3 \text{ day}^{-1} \times 20 \text{ mg L}^{-1}} $
- Calculate Intermediate Values: Perform the multiplications in the numerator and the denominator.
Numerator: $100 \times 200 = 20000 \text{ m}^3 \cdot \text{mg L}^{-1}$
Denominator: $10 \times 20 = 200 \text{ m}^3 \cdot \text{mg L}^{-1} \text{ day}^{-1}$
- Compute Final Result: Divide the numerator by the denominator to find $\theta_c$.
$ \theta_c = \frac{20000}{200} \text{ days} $
$ \theta_c = 100 \text{ days} $
The calculation yields an integer value as required.
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)
- A microorganism is grown in a batch culture using glucose as a carbon source. The apparent growth yield is $0.5 \frac{\text{g biomass}}{\text{g substrate}}$. The initial concentrations of biomass and substrate are $2 \text{ g L}^{-1}$ and $200 \text{ g L}^{-1}$, respectively. Assuming that there is no endogenous metabolism, the maximum biomass concentration that can be achieved is ________ $\text{g L}^{-1}$.
