An agitated batch bioreactor is aerated with O2-containing gas (C*= 10 mM).  At a certain point during exponential phase, the dissolved oxygen concentration is CL = 0.2 mM; when the sparging is turned off briefly, CL drops to 0.1 mM in 30 seconds. Estimate the value of kLa (s-1). dissolved oxygen balance:
 dCL/dt = OTR - OUR. BUT sparge is shut off so OTR+ 0 over the 30s time frame. dCL/dt= -OUR

We are tasked with estimating the value of \(k_La\) (the volumetric mass transfer coefficient) in an agitated batch bioreactor. The dissolved oxygen concentration drops from \(C_L = 0.2 \, \text{mM}\) to \(C_L = 0.1 \, \text{mM}\) over 30 seconds when the sparging is turned off. During this time, oxygen transfer from the gas phase (OTR) is zero, leaving only oxygen consumption by the organisms (OUR) as a factor.

### Key Equation:
The general dissolved oxygen balance is:
\[
\frac{dC_L}{dt} = OTR - OUR
\]
Since sparging is turned off, \(OTR = 0\), leaving:
\[
\frac{dC_L}{dt} = -OUR
\]
Given that \(OUR\) can be described by first-order kinetics, we express oxygen uptake as:
\[
\frac{dC_L}{dt} = -k_La \cdot (C_L - C^*)
\]
However, in this case, when the sparge is shut off, \(C^*\) is irrelevant since there is no gas-liquid oxygen transfer. We are just considering the decay of \(C_L\) with respect to time due to cellular consumption. Thus, the equation simplifies to:
\[
\frac{dC_L}{dt} = -k_La \cdot C_L
\]

### Solution Approach:
The differential equation can be solved using the separation of variables approach:

\[
\frac{dC_L}{C_L} = -k_La \, dt
\]

Integrating both sides from \(C_L = 0.2 \, \text{mM}\) to \(C_L = 0.1 \, \text{mM}\) over \(t = 0 \, \text{s}\) to \(t = 30 \, \text{s}\):

\[
\int_{0.2}^{0.1} \frac{dC_L}{C_L} = -k_La \int_0^{30} dt
\]

\[
\ln \left( \frac{0.1}{0.2} \right) = -k_La \cdot 30
\]

\[
\ln (0.5) = -k_La \cdot 30
\]

\[
-0.693 = -k_La \cdot 30
\]

Solving for \(k_La\):

\[
k_La = \frac{0.693}{30} = 0.0231 \, \text{s}^{-1}
\]

### Conclusion:
The estimated value of \(k_La\) is \(0.0231 \, \text{s}^{-1}\).

Would you like a further breakdown or explanation of any step?

### Follow-up Questions:
1. What is the significance of \(k_La\) in a bioreactor and how can it be improved?
2. How would the process change if the sparging wasn't completely shut off?
3. How does the value of \(C^*\) affect the oxygen transfer rate during normal operation?
4. How would you measure the value of \(k_La\) experimentally in different types of reactors?
5. How do OUR and OTR interact when the system reaches steady state?

### Tip:
To improve oxygen transfer in a bioreactor, increasing the agitation speed or using smaller bubbles in the sparging process can significantly enhance the value of \(k_La\).

An agitated batch bioreactor is aerated with O2-containing gas (C* = 10 mM). At a certain point during exponential phase, the dissolved oxygen concentration is CL = 0.2 mM; when the sparging is turned off briefly, CL drops to 0.1 mM in 30 seconds. Estimate the value of kLa (s-1). Dissolved oxygen balance: dCL/dt = OTR - OUR. But sparge is shut off so OTR = 0 over the 30s time frame. dCL/dt = -OUR.

Estimate the volumetric mass transfer coefficient (kLa) in a batch bioreactor by solving a differential equation for oxygen decay. In this example, the oxygen concentration drops from 0.2 mM to 0.1 mM in 30 seconds. Suitable for undergraduate chemical engineering or biotechnology students.

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