Water at \(10^{\circ} \mathrm{C}\left(\rho=999.7 \mathrm{~kg} / \mathrm{m}^{3}\right.\) and \(\mu=1.307 \times\) \(10^{-3} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\) ) is flowing in a \(0.20\)-cm-diameter 15 -m-long pipe steadily at an average velocity of \(1.2 \mathrm{~m} / \mathrm{s}\). Determine \((a)\) the pressure drop and (b) the pumping power requirement to overcome this pressure drop. Assume flow is fully developed. Is this a good assumption? Answers: (a) \(188 \mathrm{kPa}\), (b) \(0.71 \mathrm{~W}\)

Understanding the Reynolds number is crucial for predicting the flow regime in a pipe. It's a dimensionless quantity that helps engineers determine whether the flow will be laminar or turbulent.

The Reynolds number is computed using the formula:
\[ Re = \frac{D\rho V}{\mu} \]
Here, \(D\) represents the pipe diameter, \(\rho\) the density of the fluid, \(V\) the average velocity, and \(\mu\) the dynamic viscosity. For instance, in a pipe with water at \(10^\circ C\), if the Reynolds number is below 2000, the flow is considered laminar, meaning it's smooth and orderly. Between 2000 and 4000, it's transitional, and above 4000, it becomes turbulent, which is chaotic and mixes more intensely.

The physical characteristics of laminar and turbulent flow are distinctive and critical to understanding fluid dynamics. Laminar flow is smooth and orderly, with fluid particles moving in parallel layers. It's typically found at low velocities and with less viscous fluids.

On the other hand, turbulent flow is characterized by erratic changes in pressure and flow velocity. It's common at high velocities, in larger pipes, or with fluids of low viscosity. In a turbulent flow, different layers of fluid mix together due to these variations, leading to more friction and energy loss.

The friction factor represents the resistance to flow within a pipe, which is a consequence of the contact between the fluid and the pipe's inner surface. It's denoted by \(f\) and plays a pivotal role in the calculation of pressure drop.

For laminar flow, the friction factor is precisely calculated by:
\[ f = \frac{16}{Re} \]
For turbulent flow, approximations like the Blasius equation can be used:
\[ f = 0.079 Re^{-0.25} \]
The smaller the friction factor, the less resistance the fluid encounters, affecting both the energy required to maintain the flow and the efficiency of the fluid transport.

The Darcy-Weisbach equation is a well-established formula used to estimate the pressure drop due to friction in a pipe. It integrates the friction factor, pipe length, diameter, fluid density, and velocity into a single equation.

Here's the equation at work:
\[ \Delta P = f \frac{L}{D} \frac{1}{2} \rho V^2 \]
Where \(\Delta P\) is the pressure drop, \(L\) is the length of the pipe, and \(D\), \(\rho\), and \(V\) maintain their previous definitions. By knowing the pressure drop across the pipe, engineers can design appropriate pumping systems to ensure adequate fluid delivery.

The volumetric flow rate, which represents how much fluid passes through a section of a pipe per unit of time, is pivotal in engineering applications. It's denoted as \(Q\) and is calculated by multiplying the pipe's cross-sectional area by the fluid's velocity.

You can find \(Q\) using the formula:
\[ Q = A \cdot V \]
With \(A\) being the pipe's cross-sectional area, which for a circular pipe is \[ A = \frac{1}{4} \cdot \pi \cdot D^2 \] The control of the volumetric flow rate is essential for processes in industries such as water treatment, chemical manufacturing, and irrigation systems.