| 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:
- Continuity at each node: Σ Q_in = Σ Q_out
- 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)
- Define design flow Q and allowable ΔP (or h_f)
- Select preliminary D from V_design = 1–3 m/s (liquids) or Q-velocity chart
- Calculate Re and f (Colebrook); compute h_f
- Check h_f vs. available pressure; if excessive → increase D
- Add minor losses; verify total head
- 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.
