pipe-network-analysis

name	pipe-network-analysis
description	Pipe network analysis — Hardy-Cross, Newton-Raphson, series/parallel pipes, head loss (Darcy-Weisbach, Hazen-Williams), pump curves, water hammer, flow measurement, ASME B31.
metadata	{"priority":7,"promptSignals":{"phrases":["pipe network","pipe flow","Hardy Cross","Darcy-Weisbach","Hazen-Williams","pipe pressure drop","pipe sizing"],"minScore":3}}

Pipe Network Analysis — Complete Skill

Darcy-Weisbach (Universal)

Head loss in pipe: h_f = f × (L/D) × V²/(2g) [m]

f = Darcy friction factor; L = pipe length; D = diameter; V = velocity Q = V × A = V × π D²/4

In terms of Q: h_f = 8 f L Q² / (g π² D⁵) = R_pipe × Q²

R_pipe = 8 f L / (g π² D⁵) — pipe resistance coefficient

Friction factor f: Laminar (Re < 2300): f = 64/Re Turbulent smooth (Blasius, 4000 < Re < 100,000): f = 0.316/Re^0.25 Colebrook-White (general turbulent): 1/√f = -2 log₁₀(ε/(3.7D) + 2.51/(Re√f)) [implicit; iterate] ε = roughness [mm]: commercial steel 0.046, galvanized 0.15, cast iron 0.26, concrete 0.3–3 mm

Swamee-Jain (explicit approximation): f = 0.25 / [log₁₀(ε/(3.7D) + 5.74/Re^0.9)]²

Hazen-Williams (Water Distribution, Empirical)

V = 0.8492 C R_h^0.63 S^0.54 [m/s; SI] or: Q = 0.2785 C D^2.63 S^0.54 [m³/s]

S = head loss gradient = h_f/L; R_h = hydraulic radius = D/4 (full pipe) C = Hazen-Williams roughness coefficient: New PVC: C = 150; New ductile iron: C = 140; 10-year-old iron: C = 120; Old iron: C = 80

Advantage: no friction factor iteration; preferred for water distribution Limitation: valid only for water near 20°C; turbulent flow

Minor Losses

h_m = K × V²/(2g)

Fitting	K
Sharp-edge entrance	0.5
Well-rounded entrance	0.04
Gate valve (fully open)	0.2
Globe valve (fully open)	10
Ball valve (fully open)	0.05
90° elbow (standard)	0.9
90° long radius elbow	0.6
Tee (branch flow)	2.0
Check valve	2.5
Sudden expansion (A₁→A₂)	(1-A₁/A₂)²

Series Pipes

Same Q through all pipes: h_total = Σ h_{f,i} = Σ R_i Q² R_series = ΣR_i

Parallel Pipes

Same head loss across all branches: h_f,1 = h_f,2 = ... = h_f,n = R_i Q_i² Q_total = Σ Q_i

Solving: Q_i = √(h_f / R_i); Q_total = Σ Q_i; iterate for h_f until continuity satisfied

Hardy-Cross Method (Loop Networks)

Iterative method for complex networks with loops

Governing equations:

1. Continuity at each node: Σ Q_in = Σ Q_out
2. Energy around each loop: Σ h_f (loop) = 0

Correction: ΔQ = -Σ(R_ij |Q_ij| Q_ij) / (2 Σ(R_ij Q_ij²)) [per loop] Update Q_ij = Q_ij + ΔQ (signs per direction convention) Repeat until |ΔQ/Q| < 0.001 for all loops

Newton-Raphson (Matrix Method)

Set up system: [A]{h} = {Q_ext} (sparse linear system for node heads) Linearize nonlinear resistance; iterate: [J] {Δh} = -f(h) (Jacobian × correction = residual) More robust than Hardy-Cross for large networks; basis for EPANET

Pump in Network

Operating point: intersection of pump H-Q curve and system curve H_pump(Q) = H_static + R_system × Q²

Pump H-Q curve (approximate polynomial): H = H_shutoff - A Q² - B Q From manufacturer curve: fit polynomial to catalog data

Cavitation (NPSH) check: NPSH_A = (P_s - P_v)/ρg + V_s²/2g > NPSH_R (from pump curve)

Pipe Sizing (Engineering Process)

1. Define design flow Q and allowable ΔP (or h_f)
2. Select preliminary D from V_design = 1–3 m/s (liquids) or Q-velocity chart
3. Calculate Re and f (Colebrook); compute h_f
4. Check h_f vs. available pressure; if excessive → increase D
5. Add minor losses; verify total head
6. Select standard pipe size from ANSI B36.10/B36.19 nominal sizes

Standard Pipe Schedules (ASME B36.10)

Nominal pipe size (NPS) vs. schedule number Wall thickness t = Sch × D_nom / 1000 (approximately) Sch 40 standard; Sch 80 extra strong; Sch 160 heavy; XXS double extra strong ID depends on OD and wall thickness from tables

Common NPS (OD fixed per NPS): NPS 1": OD = 33.4 mm; NPS 2": 60.3 mm; NPS 4": 114.3 mm; NPS 6": 168.3 mm; NPS 8": 219.1 mm; NPS 12": 323.9 mm

Output

Provide: pipe diameter D [mm], flow velocity V [m/s], Re, f (Darcy), h_f [m per 100m], total system head H [m], pump operating point (Q, H), NPSH check, pipe schedule/class selection, minor loss summary.