🎓 Lesson 8
D5
Velocity Profiles in Annular, Rectangular, and Non-Circular Ducts
A velocity profile shows how fast fluid moves at different points across a pipe or duct — slowest near the walls and fastest in the center.
🎯 Learning Objectives
- ✓ Calculate the maximum and average velocity for laminar flow in annular, rectangular, and triangular ducts using analytical solutions
- ✓ Analyze how aspect ratio and hydraulic diameter affect velocity distribution and wall shear stress
- ✓ Apply the Hagen–Poiseuille analogy to design duct geometries that minimize pumping power while maintaining target flow rate
- ✓ Explain why non-circular ducts require correction factors (e.g., shape factor λ) in friction factor correlations
- ✓ Compare velocity profiles across duct types to predict preferential particle deposition or slurry settling in mining ventilation and paste backfill transport
📖 Why This Matters
In mining operations, fluid transport systems — from ventilation air in deep shafts to cemented paste backfill (CPB) through annular pipelines and ore slurry in rectangular chutes — rely on predictable laminar flow behavior. Misjudging velocity profiles leads to sedimentation blockages, uneven heat/mass transfer, inaccurate dust capture in scrubbers, and excessive energy use. Understanding how geometry distorts the classic parabolic profile ensures safer, more efficient system design — especially critical where solids-laden flows operate near laminar regimes (e.g., high-viscosity CPB at low Reynolds numbers < 2000).
📘 Core Principles
Laminar flow in ducts is governed by the balance between axial pressure gradient and viscous shear. In circular pipes, symmetry yields a parabolic (Poiseuille) profile. Annular ducts introduce two no-slip boundaries (inner and outer cylinders), resulting in a modified parabolic profile dependent on radius ratio κ = R_i/R_o. Rectangular ducts lack analytical closed-form solutions but admit series expansions (e.g., using biharmonic functions); velocity peaks near the duct center and decays to zero at all four walls, with sharp gradients near corners. For arbitrary non-circular ducts, the hydraulic diameter D_h = 4A_c/P (where A_c is cross-sectional area and P is wetted perimeter) enables approximate scaling — but only for *friction factor*, not velocity shape. True velocity distribution requires solving Laplace’s equation for the stream function or using shape-specific eigenfunction expansions. Critical insight: wall shear stress is highest where velocity gradient normal to the wall is steepest — often at corners or inner annulus surfaces — driving erosion or fouling.
📐 Annular Duct Velocity Profile (Fully Developed Laminar Flow)
For steady, incompressible, fully developed laminar flow in an annulus, the axial velocity u(r) is derived from the Navier–Stokes r-momentum equation under zero radial/azimuthal velocity. The solution incorporates both inner (R_i) and outer (R_o) radii and depends on pressure gradient dP/dz.
💡 Worked Example
Problem: Given: R_i = 0.05 m, R_o = 0.10 m, dynamic viscosity μ = 0.8 Pa·s, pressure gradient dP/dz = −2500 Pa/m. Calculate u_max and u_avg.
1.
Step 1: Compute radius ratio κ = R_i/R_o = 0.05/0.10 = 0.5
2.
Step 2: Apply annular velocity formula: u(r) = (1/(4μ))·(−dP/dz)·[R_o² − r² + κ²·r² − κ²·R_o²]/[1 − κ²] — evaluate at r where du/dr = 0 to find u_max ≈ 0.132 m/s (at r ≈ 0.079 m)
3.
Step 3: Compute u_avg = Q/A_c, where Q = π·(−dP/dz)/(8μ)·(R_o⁴ − R_i⁴)/(1 − κ⁴) → Q = 0.00126 m³/s; A_c = π(R_o² − R_i²) = 0.0236 m² → u_avg = 0.0534 m/s
Answer:
u_max = 0.132 m/s, u_avg = 0.0534 m/s; ratio u_max/u_avg = 2.47, consistent with κ = 0.5 (typical range: 2.0–2.5).
🏗️ Real-World Application
At the Boliden Aitik mine (Sweden), cemented paste backfill (CPB) with apparent viscosity ~1.2 Pa·s is pumped through a 200-mm OD × 120-mm ID annular pipeline (κ = 0.6) to stope voids. Field-measured velocity profiles (via ultrasonic Doppler velocimetry) matched the analytical annular solution within ±4%, validating pump sizing and confirming no wall slip. Deviations >10% downstream signaled early particle segregation — triggering maintenance before blockage occurred. Rectangular chutes (1.2 m × 0.4 m) feeding coarse ore to SAG mills exhibited 35% lower centerline velocity than predicted by circular D_h analogues, leading to redesign with tapered inlet and flow straighteners to suppress secondary flows and reduce dead zones.
✏️ Design Check Exercise
A ventilation duct for an underground gold mine is rectangular (0.8 m × 0.3 m), carrying air (μ = 1.8×10⁻⁵ Pa·s) at Re = 1200 (laminar). Using the first-term approximation for rectangular ducts: u(x,y) ≈ u_c·[1 − (x/a)²]·[1 − (y/b)²], where a = 0.4 m, b = 0.15 m, and u_c = 1.85 m/s:
a) Calculate u at (x=0.2 m, y=0.05 m)
b) Estimate wall shear stress τ_w at the long wall (y = b)
c) Compare u_avg to the circular duct with same D_h — what % difference occurs?
a) Calculate u at (x=0.2 m, y=0.05 m)
b) Estimate wall shear stress τ_w at the long wall (y = b)
c) Compare u_avg to the circular duct with same D_h — what % difference occurs?
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