Your boss, from the "old school," claims that for pipe flow...

Your boss, from the "old school," claims that for pipe flow the flow rate, $Q \propto \sqrt{\Delta p}$, where $\Delta p$ is the pressure dif ference driving the flow. You dispute this, so perform some calculations. You take a 1 -in.- -diameter commercial steel pipe and assume an initial flow rate of 1.25 gal/min of water. You then increase the applicd pressure in equal increments and compute the new flow rates so you can plot $Q$ versus $\Delta p,$ as computed by you and your boss. Plot the two curves on the same graph. Was your boss right?

Key Concepts

Scaling Relationships in Pipe Flow

The relationship between flow rate and pressure difference depends on the flow regime. In laminar flow, Q is proportional to ?p, while in turbulent conditions a different scaling, often approximating Q proportional to the square root of ?p, may arise due to energy losses from turbulence and friction effects. Recognizing these different scaling laws is essential for accurately predicting flow behavior in practical applications.

Flow Regime Identification

Understanding whether the flow is laminar or turbulent is crucial in fluid dynamics. Laminar flow, characterized by smooth, orderly fluid motion, follows the Hagen–Poiseuille law and exhibits a linear dependency of flow rate on pressure difference. In contrast, turbulent flow, which involves chaotic eddies and mixing, leads to a different, often non-linear, dependency. Determining the flow regime typically involves the Reynolds number, a dimensionless parameter that compares inertial and viscous forces in the fluid.

Hagen–Poiseuille Law

This principle governs laminar flow in pipes, stating that the volumetric flow rate (Q) is directly proportional to the pressure difference (?p) across the pipe. The law assumes a steady, viscous, and incompressible fluid under laminar conditions and results in a linear relationship between Q and ?p when all other factors, such as fluid viscosity and pipe dimensions, remain constant.

Transcript

Please give Ace some feedback