Two reservoirs are joined by a sharp-ended flexible pipe 100 mm diameter and 36 m long. The en

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Two reservoirs are joined by a sharp-ended flexible pipe 100...

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 $=101.3 \mathrm{kPa}$.)

Key Concepts

Bernoulli's Principle

Bernoulli's Principle is a statement of energy conservation for flowing fluids. It relates the pressure, kinetic energy, and potential energy per unit weight along a streamline, allowing one to analyze changes in fluid flow when there are variations in elevation, velocity, and pressure. This principle is foundational in determining how energy is distributed in a piping system and in making adjustments to ensure that flow conditions meet specific pressure requirements.

Darcy-Weisbach Equation

The Darcy-Weisbach equation is used to quantify the frictional head loss in a pipe due to viscous effects. By incorporating factors such as the friction factor, pipe length, and diameter, it helps estimate the energy loss resulting from friction along the flow path. This equation is essential in pipeline hydraulics to predict pressure drops and therefore plays a crucial role when determining maximum allowable elevations in a piping network.

Pressure Head

Pressure head represents the height of a fluid column that corresponds to a given pressure. It is a measure of the energy stored in the fluid as pressure energy. Understanding pressure head is critical in designing fluid systems because it helps ensure that the pressure within a pipe remains above a specified minimum, thereby preventing issues such as cavitation or vapor formation.

Gravitational (Elevation) Head

Gravitational or elevation head is the potential energy per unit weight of a fluid due to its elevation above a reference point. In a piping system, differences in elevation lead to differences in gravitational head, which affect the overall energy balance of the flow. This concept is vital for analyzing systems where the pipe follows a variable route with elevation changes, as it directly influences the pressure distribution along the pipe.

Energy Loss Due to Friction

Energy loss due to friction in a pipe is a critical consideration in flow analysis. As fluid flows through a pipe, friction between the fluid and the pipe wall dissipates energy, resulting in a drop in pressure. This energy loss, often calculated using the Darcy-Weisbach equation, must be accounted for when applying Bernoulli's equation to ensure that the system maintains adequate pressure levels and meets the design constraints.

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