What must be the pressure difference between the two ends of a 1.6-km section of pipe, 29 cm in diameter, if it is to transport oil (ρ = 950 kg/m³, η=0.20 Pa⋅s) at a rate of 650 cm³/s?

Verified step by step guidance

1. Convert all given quantities to SI units for consistency. The length of the pipe is 1.6 km = 1600 m, the diameter is 29 cm = 0.29 m, and the flow rate is 650 cm³/s = 6.5 × 10⁻⁴ m³/s.
2. Determine the radius of the pipe from the diameter: radius r = diameter / 2 = 0.29 m / 2 = 0.145 m.
3. Use the Hagen-Poiseuille equation for laminar flow to relate the pressure difference (ΔP) to the flow rate (Q): $Q=\frac{\mathrm{\pi }}{8}\frac{r^4}{\mathrm{\Delta P}}/\eta L$, where η is the dynamic viscosity, L is the length of the pipe, and r is the radius.
4. Rearrange the equation to solve for the pressure difference ΔP: $\mathrm{\Delta P}=\frac{8}{\mathrm{\pi }}\frac{\eta }{L}\frac{Q}{r^4}$.
5. Substitute the known values into the equation: η = 0.20 Pa⋅s, L = 1600 m, Q = 6.5 × 10⁻⁴ m³/s, and r = 0.145 m. Perform the calculations to find the pressure difference ΔP.

Key Concepts

Bernoulli's Principle

Bernoulli's Principle states that in a flowing fluid, an increase in the fluid's speed occurs simultaneously with a decrease in pressure or potential energy. This principle is essential for understanding how pressure differences drive fluid flow in pipes, particularly in applications involving varying diameters and flow rates.

Continuity Equation

The Continuity Equation is a fundamental principle in fluid dynamics that asserts that the mass flow rate must remain constant from one cross-section of a pipe to another. For incompressible fluids, this means that the product of the cross-sectional area and the fluid velocity is constant, which helps in determining flow rates and velocities in different sections of a pipe.

Viscosity and Flow Rate

Viscosity is a measure of a fluid's resistance to deformation or flow. In the context of the question, the viscosity of oil affects how easily it can flow through the pipe, influencing the pressure difference required to maintain a specific flow rate. The relationship between viscosity, flow rate, and pressure drop is often described by the Hagen-Poiseuille equation for laminar flow.