Water flows through a 25 -mm-diameter tube that suddenly...

Water flows through a 25 -mm-diameter tube that suddenly enlarges to a diameter of $50 \mathrm{mm}$. The flow rate through the enlargement is 1.25 Liter/s. Calculate the pressure rise across the enlargement. Compare with the value for frictionless flow.

Key Concepts

Pressure Recovery in Pipe Expansions

This concept involves the rise in static pressure that occurs when the fluid decelerates in the enlarged section. While frictionless flow models predict a full recovery of the pressure based on energy conservation (as given by Bernoulli’s equation), in real systems the presence of energy losses due to separation and turbulence means that the actual pressure rise will be lower than the ideal prediction.

Continuity Equation

This principle, based on the conservation of mass, states that the mass flow rate must remain constant in a steady flow. In the context of a pipe with a sudden change in diameter, it relates the velocities in the smaller and larger sections through their respective cross?sectional areas. It is essential for determining how the fluid accelerates or decelerates when the pipe diameter changes.

Bernoulli's Equation

This is an energy conservation principle for inviscid, incompressible flow which relates pressure, velocity, and elevation between two points along a streamline. In the analysis of a sudden enlargement under the frictionless assumption, Bernoulli's equation predicts an ideal conversion of kinetic energy into pressure energy without accounting for losses due to turbulence or separation.

Sudden Pipe Expansion and Energy Losses

When a fluid flows from a smaller to a larger diameter pipe in a sudden expansion, the abrupt change in geometry causes flow separation and the formation of recirculation zones. This leads to energy losses as some of the fluid's kinetic energy is dissipated rather than fully converted into static pressure. The associated head loss can be quantified by introducing a loss coefficient, which highlights the difference between the ideal frictionless case and the actual flow with viscous effects.

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