pipe-flow

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)

1. Guess flows satisfying continuity at each node
2. For each loop: Δh = Σ(f·L/D·Q²/2gA²) [with sign per direction]
3. Correction: ΔQ = -Δh / (2·Σ|h_f/Q|)
4. 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.