GATE 2022 BT – Question 60

Fundamentals of Biological Engineering · Bioreaction Engineering: Kinetics of cell growth, substrate utilization and product formation, structured and unstructured models · 2 marks · Numerical answer

Oxygen bulk and saturation concentrations are $10^{-4}$ and $10^{-3}$ mol/L, uptake is $9\times10^{-4}$ mol/(L s), and interfacial area is $100\ \mathrm{m^2/m^3}$. The steady liquid mass-transfer coefficient $k_L$ (cm/s) is ___________.

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Correct answer: 1

Explanation

**Step 1: oxygen balance.** At steady state the oxygen transferred from the gas equals the oxygen taken up by the cells:
$$k_La\,(C^*-C_L)=OUR .$$

**Step 2: solve for $k_La$.**
$$k_La=\frac{9\times10^{-4}}{10^{-3}-10^{-4}}=\frac{9\times10^{-4}}{9\times10^{-4}}=1\ \text{s}^{-1}.$$

**Step 3: remove the area.** The interfacial area per unit volume is $a=100\ \text{m}^2/\text{m}^3=100\ \text{m}^{-1}$:
$$k_L=\frac{k_La}{a}=\frac{1\ \text{s}^{-1}}{100\ \text{m}^{-1}}=0.01\ \text{m/s}=\mathbf{1\ cm/s}.$$