Water is conveyed through a uniform tube of 8cm in diameter and 3140m in length at the rate of 2×10⁻³ m³ per second. The pressure required to maintain the flow is (viscosity of water = 10⁻³ SI unit)

To determine the pressure required to maintain the flow of water through a uniform tube, we use the Hagen–Poiseuille equation, applicable for laminar flow in circular pipes:

Formula:

ΔP = (8 × μ × L × Q) / (π × r⁴)

Where:

• ΔP: Pressure drop
• μ: Dynamic viscosity of the fluid
• L: Length of the pipe
• Q: Volumetric flow rate
• r: Radius of the pipe

Given Data:

• Diameter of the tube: d = 8 cm = 0.08 m
• Radius: r = d / 2 = 0.04 m
• Length of the tube: L = 3140 m
• Flow rate: Q = 2 × 10⁻³ m³/s
• Viscosity of water: μ = 10⁻³ Pa·s

Step-by-Step Calculation:

1. Calculate the denominator:

r⁴ = (0.04 m)⁴ = 2.56 × 10⁻⁷ m⁴
π × r⁴ ≈ 8.042 × 10⁻⁷ m⁴

2. Calculate the numerator:

8 × μ × L × Q = 8 × 10⁻³ × 3140 × 2 × 10⁻³ = 0.05024 Pa·m³/s

3. Calculate the pressure drop:

ΔP = 0.05024 / 8.042 × 10⁻⁷ ≈ 62500 Pa

Convert to N/m² (since 1 Pa = 1 N/m²):

ΔP ≈ 62500 N/m²

Final Answer

The calculated pressure drop is approximately 62500 N/m².