GATE 2026 BT Q.65: solution and answer

Official answer: 0.31 to 0.32Source: Official answer key, GATE 2026 BT (IIT Guwahati)

1. Symbols: $Q$ = volumetric flow rate, $d$ = inside diameter of the column, $A$ = cross-sectional area of the empty column, $u_0$ = superficial velocity, $u_i$ = interstitial velocity, $\epsilon$ = void fraction of the bed.

2. Put everything in consistent units: $Q=0.025{\text{L s}}^{-1}=0.025\times 1000{\text{cm}}^3{\text{s}}^{-1}=25{\text{cm}}^3{\text{s}}^{-1}$, $d=10\text{cm}$, $u_i=1{\text{cm s}}^{-1}$.

3. Cross-sectional area of the empty column: $A=\frac{\pi d^2}{4}=\frac{\pi (10{)}^2}{4}=78.54{\text{cm}}^2$.

4. Superficial velocity (the flow rate spread over the whole cross-section, as if the column were empty): $u_0=\frac{Q}{A}=\frac{25}{78.54}=0.3183{\text{cm s}}^{-1}$.

5. Dupuit relation: the liquid can pass only through the voids, which occupy a fraction $\epsilon$ of the cross-section, so $u_i=\frac{u_0}{\epsilon }$, which gives $\epsilon =\frac{u_0}{u_i}$.

6. $$\epsilon =\frac{0.3183{\text{cm s}}^{-1}}{1{\text{cm s}}^{-1}}=0.318\approx 0.32$$

7. The bed length (1 m) is not needed for the void fraction. It would matter only for the bed volume or the liquid residence time in the voids, $\tau =L/u_i=100\text{cm}/1{\text{cm s}}^{-1}=100\text{s}$.