Two vertical cylindrical tanks, of diameters \(2.5 \mathrm{~m}\) and \(1.5 \mathrm{~m}\) respectively, are connected by a \(50 \mathrm{~mm}\) diameter pipe, \(75 \mathrm{~m}\) long for which \(f\) may be assumed constant at \(0.01\). Both tanks contain water and are open to atmosphere. Initially the level of water in the larger tank is \(1 \mathrm{~m}\) above that in the smaller tank. Assuming that entry and exit losses for the pipe together amount to \(1.5\) times the velocity head, calculate the fall in level in the larger tank during 20 minutes. (The pipe is so placed that it is always full of water.)

Bernoulli's equation is fundamental in fluid dynamics, relating the pressure, velocity, and elevation head in a flowing fluid. For an incompressible fluid flowing along a streamline, it states:\[P + \frac{1}{2}\rho v^2 + \rho g z = constant\]Here, \(P\) is the pressure, \(\rho\) is the density, \(v\) is the flow velocity, \(g\) is the gravitational acceleration, and \(z\) is the elevation height. This equation helps us understand how energy is conserved in a fluid system. When applied between two points in the connected tanks, it relates the head differences – including height difference due to gravity – and the velocity of water.In the given problem, the initial height difference provides the driving force for water flow.

The Hagen-Poiseuille equation describes the flow of a viscous liquid through a circular pipe. This relationship helps us understand how factors like pipe length and diameter, as well as fluid viscosity, affect flow rate. The equation is:\[ Q = \frac{\pi r^4}{8\mu L} \Delta P \]where:

• \(Q\): volumetric flow rate
• \(r\): radius of the pipe
• \(\mu\): dynamic viscosity
• \(L\): length of the pipe
• \(\Delta P\): pressure difference across the pipe

Although the exercise doesn't directly use the Hagen-Poiseuille equation, it helps us intuitively understand how increasing pipe length (\(L\)) or reducing its diameter (\(r\)) increases resistance to flow, just as increasing pressure difference (\(\Delta P\)) boosts flow rate.

The volumetric flow rate, \(Q\), quantifies how much fluid is passing through a section of a pipe over time. It's essential for determining how quickly the water levels in the tanks change. The flow rate is given by:\[Q = A_p v\]where:

• \(A_p\): cross-sectional area of the pipe
• \(v\): velocity of the fluid

To find the velocity, we use both Bernoulli's equation and head loss considerations, and then we multiply it by the pipe's cross-sectional area. For our exercise, once we know the volumetric flow rate, calculating the fall in water level in a certain time, like 20 minutes, becomes straightforward.

Head loss represents the loss of energy due to friction and other resistances as fluid flows through a pipe. It is a crucial factor in determining the actual velocity of flow and involves factors like pipe roughness and flow velocity. The head loss due to friction (\(h_f\)) in a pipe can be calculated using Darcy-Weisbach equation:\[h_f = f \frac{L}{D} \frac{v^2}{2g}\]where:

• \(f\): friction factor
• \(L\): length of the pipe
• \(D\): diameter of the pipe
• \(v\): velocity
• \(g\): gravitational acceleration

In the problem, the head loss coefficient (\(K\)) also includes entry and exit losses, sum up as:\[h_f = \left(1 + K + 4f \frac{L}{D}\right) \frac{v^2}{2g}\]These loss terms help us refine the velocity calculation and, thus, the flow rate through the pipe.