Two reservoirs are joined by a sharp-ended flexible pipe \(100 \mathrm{~mm}\) diameter and \(36 \mathrm{~m}\) long. The ends of the pipe differ in level by \(4 \mathrm{~m}\); the surface level in the upper reservoir is \(1.8 \mathrm{~m}\) above the pipe inlet while that in the lower reservoir is \(1.2 \mathrm{~m}\) above the pipe outlet. At a position \(7.5 \mathrm{~m}\) horizontally from the upper reservoir the pipe is required to pass over a barrier. Assuming that the pipe is straight between its inlet and the barrier and that \(f=0.01\) determine the greatest height to which the pipe may rise at the barrier if the absolute pressure in the pipe is not to be less than \(40 \mathrm{kPa}\). Additional losses at bends may be neglected. (Take atmospheric pressure \(=101.3 \mathrm{kPa}\).)

In fluid dynamics, the term 'pressure head' represents the potential energy of a fluid due to the pressure exerted by the fluid column above a specific point. This can be particularly useful when comparing points within the same fluid system. To calculate the pressure head (represented by 'h'), we use the equation: \[ h = \frac{P}{\rho g} \] In this equation, \ \( P \) is the pressure in Pascals (Pa), \( \rho \) is the density of the fluid (for water it is \( 1000 \) \( \text{kg/m}^3 \)), and \( g \) is the acceleration due to gravity (\( 9.81 \) \( \text{m/s}^2 \)).
For instance, if the pressure is \( 61300 \) Pa, the pressure head will be:
\[ h = \frac{61300 \times 1000}{1000 \times 9.81} \]
This calculation helps us understand how pressure differences translate into fluid column heights within pipe systems.

The Darcy-Weisbach equation is vital for calculating the head loss due to friction in a pipe system. Friction affects how fluid flows and creates energy losses which need to be accounted for in engineering calculations. The equation is:
\[ h_f = f \frac{L}{D} \frac{v^2}{2g} \]
Here’s what each symbol represents:

• \( h_f \): Head loss due to friction.
• \( f \): Friction factor (dimensionless), which depends on the pipe's characteristics and the flow conditions.
• \( L \): Length of the pipe (m).
• \( D \): Diameter of the pipe (m).
• \( v \): Flow velocity (m/s).
• \( g \): Acceleration due to gravity (\( 9.81 \) \( \text{m/s}^2 \)).

For an example, if we have a pipe with a length \( L=36 \) \( \text{m} \), diameter \( D=0.1 \) \( \text{m} \), and friction factor \( f=0.01 \), we can calculate the head loss using the known velocity \( v \). By calculating this friction head loss, we can ensure that the design of the pipe system meets necessary efficiency and safety standards.

Bernoulli’s equation is a principle of fluid dynamics that relates pressure, velocity, and height in a flowing fluid. It helps us understand how energy is conserved in a fluid flow. The equation is expressed as:
\ \frac{P}{\rho g} + \frac{v^2}{2g} + z = constant \
Here:

• \( P \): Pressure energy per unit volume (Pa).
• \( \rho \): Density of the fluid (\( \text{kg/m}^3 \)).
• \( v \): Velocity of the fluid (\( \text{m/s} \)).
• \( z \): Elevation head or height relative to a reference point (m).

Bernoulli’s equation ensures that at any two points in a streamline, the sum of the pressure energy, kinetic energy, and potential energy is the same.
By applying Bernoulli's equation to points in the pipe system, we can solve for unknowns like flow velocity or pressure differences. This helps us to design pipe systems that efficiently transport fluid while maintaining desired pressure and flow rates.

When fluid flows through a pipe, energy losses occur primarily due to frictional forces between the liquid and the pipe walls. These losses must be accounted for to ensure proper fluid transport. To calculate the head loss due to friction, we can use the Darcy-Weisbach equation discussed earlier:
\[ h_f = f \frac{L}{D} \frac{v^2}{2g} \].
In our example, the calculated parameters include:

• Friction factor \(f \ = 0.01 \).
• Length of pipe \(L \ = 36 \; \text{m} \).
• Diameter of pipe \(D \ = 0.1 \; \text{m} \).

Head loss due to friction is vital for designing effective pipe systems, as high friction losses can result in inadequate pressure and flow rates. To minimize this friction, engineers may use smoother pipe materials or optimize the pipe diameter and length.