🎓 Lesson 22
D2
Continuity, Momentum, and Energy Balances
These three balances are like accounting rules for fluids: continuity says mass can’t vanish or appear; momentum says forces change how fast and in what direction fluid moves; energy says heat, work, and flow energy must add up correctly.
🎯 Learning Objectives
- ✓ Calculate mass flow rate and velocity profiles using the continuity equation for steady and unsteady pipe and open-channel flows
- ✓ Analyze pressure drop and force exerted on structures (e.g., blast header nozzles, slurry pipelines) using the momentum equation
- ✓ Apply the steady-flow energy equation (SFEE) to determine pump power requirements or thermal losses in dewatering and slurry transport systems
- ✓ Explain physical meaning and assumptions behind each balance (e.g., incompressibility, steady state, negligible heat transfer) in mining fluid applications
📖 Why This Matters
In mining, fluid mechanics governs everything from dewatering flooded pits and pumping abrasive slurries to designing high-velocity airblast systems for dust suppression and pneumatic conveying of fines. Misapplying continuity, momentum, or energy balances leads to undersized pumps (causing system failure), overpressurized pipelines (risking rupture), or inaccurate energy estimates (wasting 15–30% of operational power). These balances aren’t abstract—they’re the foundation of safe, efficient, and compliant fluid-handling design.
📘 Core Principles
Continuity arises from mass conservation: for incompressible flow, velocity × area is constant—so narrowing a pipe increases speed. Momentum balance tracks how pressure, friction, gravity, and external forces accelerate fluid or act on boundaries—critical for sizing anchor bolts on slurry elbows or predicting jet impact force on rock faces. Energy balance extends this by including work (pumps/compressors), heat (frictional heating in long pipelines), and enthalpy—essential when modeling temperature rise in high-pressure water jets used for rock cutting or evaluating efficiency of mine dewatering stations. Each balance assumes specific conditions (e.g., steady vs. unsteady, inviscid vs. turbulent), and recognizing their limits prevents dangerous oversimplification.
📐 Steady-Flow Energy Equation (SFEE)
The SFEE quantifies total energy per unit mass across a control volume under steady-state conditions, incorporating shaft work, heat transfer, and mechanical/thermal energy changes. It’s indispensable for sizing pumps, compressors, and heat exchangers in fluid transport systems.
Steady-Flow Energy Equation (SFEE)
q − wₛ = Δh + ΔKE + ΔPEPer-unit-mass energy balance for steady flow, where q is heat added, wₛ is shaft work done *on* fluid, and Δh, ΔKE, ΔPE are enthalpy, kinetic energy, and potential energy changes.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| q | Heat transfer per unit mass | J/kg | Positive if added to fluid |
| wₛ | Shaft work per unit mass | J/kg | Positive if done *on* fluid (e.g., pump); negative if done *by* fluid (e.g., turbine) |
| Δh | Enthalpy change | J/kg | h₂ − h₁; includes internal + flow work |
| ΔKE | Kinetic energy change | J/kg | (V₂² − V₁²)/2 |
| ΔPE | Potential energy change | J/kg | g(z₂ − z₁) |
Typical Ranges:
Mine dewatering pump (100 m head): wₛ ≈ 980–1,150 kJ/kg depending on efficiency
💡 Worked Example
Problem: A mine dewatering pump lifts water vertically 85 m at 0.45 m³/s. Pipe friction loss is 12 m head, and the pump efficiency is 72%. Calculate required shaft power (kW). Assume ρ = 1000 kg/m³, g = 9.81 m/s².
1.
Step 1: Compute total head H_total = elevation head (85 m) + friction head (12 m) = 97 m
2.
Step 2: Apply SFEE for ideal (lossless) pump work: W_s,ideal = ρgQH_total = (1000)(9.81)(0.45)(97) = 429,721.5 W ≈ 429.7 kW
3.
Step 3: Account for efficiency: W_shaft = W_s,ideal / η = 429.7 / 0.72 = 596.8 kW
Answer:
The required shaft power is 597 kW, which exceeds typical medium-duty mine pump ratings (400–600 kW), confirming need for a high-efficiency, duty-rated unit.
🏗️ Real-World Application
At the Cadia East underground copper mine (NSW, Australia), engineers applied the momentum equation to redesign a 300-mm-diameter slurry pipeline bend carrying abrasive copper-gold tailings (ρ = 1,320 kg/m³, v = 3.1 m/s). Initial supports failed after 8 months due to underestimated reaction force. Using the momentum equation with vector resolution, they calculated a resultant force of 4.8 kN normal to the bend centerline—37% higher than prior scalar estimates—and specified reinforced anchoring with 22-mm A325 bolts spaced at 0.6-m intervals, extending service life to >5 years.
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