🎓 Lesson 5 D3

Calculation Methods and Formulas

Pressure loss is how much pressure drops as water or slurry flows through pipes, hoses, and fittings due to friction and changes in direction.

🎯 Learning Objectives

✓ Calculate total pressure loss in a multi-segment slurry pipeline using the Darcy–Weisbach equation
✓ Design pipe diameter and pump head requirements to meet target flow rate and allowable pressure loss
✓ Analyze the impact of flow velocity, pipe roughness, and slurry concentration on pressure loss
✓ Explain the difference between laminar and turbulent flow regimes and their influence on friction factor selection
✓ Apply correction factors for non-Newtonian behavior in high-concentration mine tailings slurries

📖 Why This Matters

In mining operations, moving water, process fluids, or tailings slurries over long distances consumes significant energy — often 30–50% of total site power. Underestimating pressure loss leads to undersized pumps, system failure, unplanned shutdowns, and costly retrofits. Overestimating wastes capital on oversized infrastructure. This lesson equips you to predict, control, and optimize hydraulic performance across dewatering, paste fill, and tailings transport systems — directly impacting safety, cost, and sustainability.

📘 Core Principles

Hydraulic pressure loss arises from two primary components: major (frictional) loss along straight pipe lengths and minor (local) loss at valves, bends, and transitions. Flow regime (laminar vs. turbulent) dictates which friction factor correlation applies — the Hagen–Poiseuille law for laminar flow (Re < 2,000), and Colebrook–White or Swamee–Jain approximations for turbulent flow (Re > 4,000). For slurries, particle concentration, size distribution, and rheology shift effective viscosity and density, requiring empirical corrections (e.g., Thomas, Wilson, or Sellgren models). Pipe roughness (ε), Reynolds number (Re), and relative roughness (ε/D) govern the Moody diagram behavior — a foundational tool for industrial hydraulics design.

📐 Darcy–Weisbach Major Loss Formula

The Darcy–Weisbach equation is the most physically rigorous method for calculating frictional pressure loss in circular pipes. It accounts for flow regime, pipe geometry, and fluid properties via the dimensionless friction factor f. It is preferred over Hazen–Williams for non-water fluids and slurries when accurate rheological data are available.

Darcy–Weisbach Equation

ΔP = f × (L/D) × ½ρV²

Calculates major (frictional) pressure loss in a straight pipe segment.

Variables:

Symbol	Name	Unit	Description
ΔP	Pressure loss	Pa	Energy loss per unit volume due to friction
f	Darcy friction factor	dimensionless	Function of Reynolds number and relative roughness
L	Pipe length	m	Length of straight pipe segment
D	Internal pipe diameter	m	Hydraulic diameter for circular pipe
ρ	Fluid density	kg/m³	Effective density of slurry or fluid
V	Average flow velocity	m/s	Volumetric flow rate divided by cross-sectional area

Typical Ranges:

Turbulent water flow in steel pipe: 0.012 – 0.025

8–15% w/w mineral slurry in HDPE: 0.014 – 0.032

💡 Worked Example

Problem: A 300-mm-diameter HDPE pipeline (ε = 0.0015 mm) carries 8% by weight iron ore slurry (ρ = 1,060 kg/m³, μ = 1.8 × 10⁻³ Pa·s) at 2.1 m/s over 1,200 m. Calculate major pressure loss.

1. Step 1: Compute Reynolds number: Re = ρVD/μ = (1060)(2.1)(0.3)/(0.0018) ≈ 371,000 → turbulent flow.

2. Step 2: Determine relative roughness: ε/D = 0.0015 mm / 300 mm = 5.0 × 10⁻⁶.

3. Step 3: Use Swamee–Jain to estimate f: f = 0.25 / [log₁₀((ε/D)/3.7 + 5.74/Re⁰·⁹)]² ≈ 0.0142.

4. Step 4: Apply Darcy–Weisbach: ΔP = f (L/D) (½ρV²) = 0.0142 × (1200/0.3) × 0.5 × 1060 × (2.1)² ≈ 112,400 Pa (≈ 1.14 bar).

Answer: The major pressure loss is 112.4 kPa (1.14 bar), well within typical allowable gradients of 0.1–0.2 bar/100 m for slurry pipelines.

🏗️ Real-World Application

At the BHP Olympic Dam dewatering system (South Australia), engineers redesigned the 4.2 km tailings transfer line after repeated pump tripping. Field measurements revealed 28% higher pressure loss than predicted using clean-water Hazen–Williams. Switching to Darcy–Weisbach with a Wilson-type slurry correction factor (Cₛ = 1.32 for 15% w/w hematite slurry) brought model predictions within ±3% of measured values — enabling precise pump selection and eliminating cavitation risk at the booster station.