Question

Two cylindrical reservoirs 'A' and 'B' are connected by a 30 m long pipe of 250 mm internal diameter as shown. Darcy friction factor for the pipe is 0.025. Initially reservoir 'A' was full and reservoir 'B' was empty. Neglecting entrance and exit losses, find the time required to empty reservoir 'A' in hours (rounded off to 3 decimal places). \begin{center} \includegraphics[width=0.5\textwidth]{03.jpeg} \end{center}

Hint:

Always apply unsteady flow equation for two-reservoir problems with head difference and frictional losses. Integration gives the emptying time.

Answer 1

Step 1: Cross-sectional areas. Reservoir A diameter = 8 m \(\Rightarrow r=4\). \[ A_A = \pi r^2 = 3.14 \times 16 = 50.24 \, m^2 \] Reservoir B diameter = 10 m \(\Rightarrow r=5\). \[ A_B = 3.14 \times 25 = 78.5 \, m^2 \] Pipe cross-sectional area: \[ A_p = \frac{\pi d^2}{4} = \frac{3.14 \times 0.25^2}{4} = 0.0491 \, m^2 \]

Step 2: Discharge coefficient due to friction. Darcy–Weisbach: \[ Q = A_p \sqrt{\frac{2gh}{f\frac{L}{D}}} \] Effective coefficient: \[ C = A_p \sqrt{\frac{2g}{fL/D}} = 0.1255 \]

Step 3: Governing equation. For change in water level between reservoirs: \[ \frac{dy}{dt} = -C \Bigg(\frac{1}{A_A} + \frac{1}{A_B}\Bigg) \sqrt{\Delta h + y} \] Where \(\Delta h = 8 \, m\).

Step 4: Integration. \[ t = \frac{2}{CS} \Big[\sqrt{\Delta h + y_0} - \sqrt{\Delta h + y_{end}}\Big] \] Here, \(y_0 = 10\), \(y_{end} = -6.4\). \[ t = \frac{2}{0.1255 \times 0.03264} \times ( \sqrt{18} - \sqrt{1.6}) \] \[ t = 1453 \, s = \frac{1453}{3600} = 0.404 \, h \] \[ \boxed{0.404 \, h} \]