| name | pipe-flow |
| description | Pipe flow analysis — Darcy-Weisbach, Moody chart, minor losses, pipe networks, series/parallel systems, pump curves, water hammer, compressible pipe flow, non-Newtonian fluids. |
| metadata | {"priority":8,"promptSignals":{"phrases":["pipe flow","friction loss","head loss","Darcy","Moody","Reynolds","pipe network","pressure drop","flow rate"],"minScore":4}} |
Pipe Flow Analysis — Complete Skill
Flow Classification
Reynolds number: Re = ρVD/μ = VD/ν
Re < 2300: laminar
2300 < Re < 4000: transitional
Re > 4000: turbulent (assume turbulent in practice if Re > 4000)
Hydraulic diameter (non-circular): D_h = 4A/P_wetted
Bernoulli + Energy Equation
p₁/γ + V₁²/2g + z₁ = p₂/γ + V₂²/2g + z₂ + h_L - h_pump + h_turbine
Where γ = ρg, h_L = total head loss [m or ft]
Darcy-Weisbach (Major Losses)
h_f = f·(L/D)·V²/2g = f·(L/D)·Q²/(2gA²)
ΔP = f·(L/D)·ρV²/2
Friction factor f:
Laminar: f = 64/Re
Turbulent: Colebrook equation (implicit):
1/√f = -2.0·log(ε/D/3.7 + 2.51/(Re·√f))
Swamee-Jain (explicit approximation, ±2%):
f = 0.25/[log(ε/(3.7D) + 5.74/Re^0.9)]²
Moody chart regions:
Smooth pipes (ε/D → 0): f = 0.316/Re^0.25 (Blasius, Re < 10⁵)
Fully rough: f = 1/(-2.0·log(ε/D/3.7))² [f independent of Re]
Roughness ε:
| Material | ε (mm) |
|---|
| Smooth drawn copper/plastic | 0.0015 |
| Commercial steel | 0.046 |
| Galvanized steel | 0.15 |
| Cast iron | 0.26 |
| Concrete | 0.3-3.0 |
Minor Losses
h_m = K·V²/2g (K = loss coefficient)
| Fitting | K |
|---|
| Sharp entrance | 0.5 |
| Well-rounded entrance | 0.04 |
| Sharp exit | 1.0 |
| Gate valve (open) | 0.2 |
| Globe valve (open) | 10 |
| Ball valve (open) | 0.1 |
| Check valve | 2.0 |
| 90° elbow (standard) | 0.9 |
| 90° elbow (long radius) | 0.4 |
| 45° elbow | 0.4 |
| Tee (flow-through) | 0.4 |
| Tee (branch) | 1.8 |
| Sudden expansion (A₁→A₂) | (1-A₁/A₂)² |
| Sudden contraction | 0.5(1-A₂/A₁) |
Series Pipe System
Same Q through all pipes: Q₁ = Q₂ = Q_n
Total head loss: h_L,total = h_L1 + h_L2 + ... + h_Ln
Parallel Pipe System
Same head loss across each branch: h_L1 = h_L2 = ... = h_Ln
Total flow: Q_total = Q₁ + Q₂ + ... + Q_n
Solve iteratively or use equivalent pipe method
Pipe Network (Hardy Cross Method)
- Guess flows satisfying continuity at each node
- For each loop: Δh = Σ(f·L/D·Q²/2gA²) [with sign per direction]
- Correction: ΔQ = -Δh / (2·Σ|h_f/Q|)
- Update flows, repeat until ΔQ < tolerance
Pump Selection
Pump curve: H = a - bQ² (parabolic approximation)
System curve: H_sys = static head + K·Q² (K from pipe losses)
Operating point: intersection of pump and system curves
Specific speed: N_s = N·√Q/(H^0.75) [rpm, gpm, ft]
N_s < 1000: centrifugal (radial)
N_s = 1000-4000: mixed flow
N_s > 4000: axial flow
Similarity laws (affinity laws):
Q ∝ N, H ∝ N², P ∝ N³ (same pump, different speeds)
NPSH (cavitation prevention):
NPSH_available = (p_atm - p_vapor)/(ρg) + z_s - h_f,suction ≥ NPSH_required
Water Hammer
Pressure rise: ΔP = ρ·c·ΔV (Joukowski equation)
Wave speed: c = √(K/ρ) / √(1 + KD/(Et)) [for elastic pipe]
c ≈ 1400 m/s (rigid pipe, water), ≈ 1000-1300 m/s (steel pipe)
Closure time t_c < 2L/c: rapid closure (worst case)
Mitigation: surge tank, slow-closing valves, pressure relief valves
Compressible Pipe Flow (Adiabatic — Fanno Flow)
Mach number change along duct with friction:
(4f·L_max/D) = (1-M²)/γM² + (γ+1)/2γ · ln((γ+1)M²/(2+(γ-1)M²))
Choking occurs at M=1 at exit
Non-Newtonian Fluids
Power-law fluid: τ = K·(du/dy)^n
Effective viscosity: μ_eff = K·(8V/D)^(n-1)·(3n+1/4n)^n
Laminar: f = 64/Re_gen where Re_gen = ρV²⁻ⁿDⁿ/(K·8^(n-1))
Output
Provide: Re, flow regime, f, h_L (friction + minor), ΔP, Q or V (whichever unknown), pump operating point if applicable.
