GATE 2026 BT Q.46: solution and answer

Official answer: B, CSource: Official answer key, GATE 2026 BT (IIT Guwahati)

1. Oxygen transfer rate: $\text{OTR}=k_{L}a(C^{\ast }-C_L)$. Here $k_L$ is the liquid-film mass transfer coefficient, $a$ is the gas-liquid interfacial area per unit volume of the gas-liquid dispersion, $C^{\ast }$ is the saturation (equilibrium) dissolved-oxygen concentration and $C_L$ is the actual dissolved-oxygen concentration. The question asks only about the product $k_{L}a$, not about the OTR as a whole.

2. Interfacial area of a bubble swarm: $a=\frac{6{\epsilon }_G}{d_b}$, where ${\epsilon }_G$ is the gas holdup (volume fraction of gas in the dispersion) and $d_b$ is the mean (Sauter) bubble diameter. So $a$, and with it $k_{L}a$, goes up when ${\epsilon }_G$ goes up or $d_b$ goes down.

3. Empirical correlation (van't Riet type) for a sparged, stirred vessel: $k_{L}a=K{\left(\frac{P_g}{V}\right)}^{\alpha }{u}_G^{\beta }$, with $P_g/V$ the gassed power input per unit volume, $u_G$ the superficial gas velocity and $K$ a constant for the system. Both exponents $\alpha$ and $\beta$ are positive, so $k_{L}a$ rises with more power input and with more gas flow.

4. (A) Pure oxygen raises the partial pressure of ${\text{O}}_2$ in the gas about 4.8-fold ($1/0.21\approx 4.8$), so by Henry's law $C^{\ast }$, and with it the driving force $(C^{\ast }-C_L)$, rises about 4.8-fold. $k_L$ and $a$ depend on hydrodynamics and fluid properties (bubble size, holdup, turbulence, diffusivity, viscosity), not on the composition of the gas, so $k_{L}a$ does not change. The OTR rises, but $k_{L}a$ does not. Not correct.

5. (B) A higher volumetric air flow rate means a higher superficial gas velocity $u_G$, which raises the gas holdup ${\epsilon }_G$ and therefore $a$; in the correlation $k_{L}a\propto u_G^{\beta }$ with $\beta >0$. $k_{L}a$ increases. Correct.

6. (C) A smaller $d_b$ gives more interfacial area per unit volume ($a\propto 1/d_b$). Small bubbles also rise more slowly and stay longer in the liquid, which raises ${\epsilon }_G$ and increases $a$ further. Very small bubbles can have a somewhat lower $k_L$ (they behave like rigid spheres), but the rise in $a$ outweighs it, so $k_{L}a$ increases. Correct.

7. (D) Less agitation means a lower $P_g/V$, so bubbles are broken up less (larger $d_b$), holdup falls and there is less turbulence at the interface. $k_{L}a$ decreases. Not correct.

8. Options B and C raise $k_{L}a$.