Investigate the effect of tube roughness on flow rate by...

Investigate the effect of tube roughness on flow rate by computing the flow generated by a pressure difference $\Delta p=100 \mathrm{kPa}$ applied to a length $L=100 \mathrm{m}$ of tubing, with diameter $D=25 \mathrm{mm}$. Plot the flow rate against tube relative roughness $e / D$ for $e / D$ ranging from 0 to 0.05 (this could be replicated experimentally by progressively roughening the tube surface). Is it possible that this tubing could be roughened so much that the flow could be slowed to a laminar flow rate?

Key Concepts

Flow Regimes (Laminar vs Turbulent Flow)

Flow regimes distinguish between orderly laminar flow—typically characterized by smooth, parallel layers—and chaotic turbulent flow, which involves mixing and eddies. Each regime displays particular frictional behaviors; for example, laminar flow follows the Hagen-Poiseuille law, while turbulent flow requires empirical or semi-empirical relationships for friction factor determination.

Friction Factor

The friction factor is a dimensionless number that quantifies the resistance due to friction in a pipe. It depends on both the flow regime (laminar or turbulent) and the relative roughness of the pipe. In turbulent flows, its accurate determination (often via the Colebrook-White equation) is essential for connecting the measured pressure drop to the flow rate.

Reynolds Number

The Reynolds number is a key non-dimensional parameter that compares inertial forces to viscous forces in a flowing fluid. It is instrumental in predicting the transition between laminar and turbulent regimes, influencing the choice of friction factor correlations and overall flow behavior in pipe systems.

Darcy-Weisbach Equation

The Darcy-Weisbach equation relates the pressure drop across a pipe to the friction factor, fluid density, flow velocity, pipe length, and diameter. It forms the cornerstone for calculating losses in pressure-driven pipe flows regardless of whether the regime is laminar or turbulent.

Relative Roughness

Relative roughness is the ratio of the characteristic height of the surface irregularities (roughness) to the pipe diameter. This non-dimensional quantity is critical in determining how wall roughness affects the friction factor in turbulent flows, thus influencing the overall pressure drop and flow rate in the system.

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