8.97 Water flows at 25 L / s through a gradual contraction, in which the pipe diameter is reduce

step 1 In this question we have to find the pressure after contraction and compare to sudden contractio

8.97 Water flows at 25 L / s through a gradual contraction,...

8.97 Water flows at $25 \mathrm{L} / \mathrm{s}$ through a gradual contraction, in which the pipe diameter is reduced from $75 \mathrm{mm}$ to $37.5 \mathrm{mm}$ with a $150^{\circ}$ included angle. If the pressure before the contraction is $500 \mathrm{kPa}$, estimate the pressure after the contraction. Recompute the answer if the included angle is changed to $180^{\circ}$ (a sudden contraction).

Key Concepts

Effect of Contraction Geometry

The geometry of the contraction, particularly the included angle, strongly influences the loss coefficient. A gradual contraction generally produces lower losses because the flow has a smoother transition, while a sudden contraction (180° included angle) induces higher losses due to flow separation and turbulence. This concept is crucial for comparing pressure drops in different contraction scenarios.

Minor Losses and Loss Coefficients

Minor losses refer to energy losses in a piping system due to fittings, bends, contractions, or expansions. These losses are commonly quantified using loss coefficients that vary depending on the geometry and abruptness of the change. Understanding and applying these coefficients allows for estimation of additional pressure drops beyond those predicted by idealized Bernoulli analysis.

Dynamic Pressure and Flow Acceleration

When a fluid flows from a larger to a smaller pipe, the acceleration of the fluid increases its velocity, which in turn raises the dynamic pressure (½?v²). This increase in kinetic energy at the expense of static pressure plays a significant role in the pressure recovery or drop observed in such contractions.

Continuity Equation

In fluid flow, the continuity equation states that for an incompressible fluid, the mass flow rate must remain constant throughout the system. When the cross-sectional area of a pipe changes, the flow velocity adjusts accordingly such that the product of area and velocity remains constant. This concept is critical when examining how the increase in velocity in a smaller diameter section leads to changes in the fluid’s dynamic pressure.

Bernoulli’s Principle

Bernoulli’s Principle is based on the conservation of energy in a flowing fluid, incorporating pressure energy, kinetic energy, and potential energy. In the absence of significant frictional losses, an increase in fluid velocity (kinetic energy) results in a decrease in static pressure. This relationship is key to estimating pressure variations before and after pipe contractions.

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