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Fluid Mechanics & Hydraulics

Hazen–Williams Pipe Flow

Hazen–Williams water-pipe friction, V = 0.849·C·R^0.63·S^0.54 (SI; 1.318 in US units), closed-form for head loss, flowrate, or required diameter of a full circular pressure pipe. Use for: waterworks/distribution pipe sizing with a given C, 'find the head loss by Hazen–Williams', friction-slope problems. Not for: fluids other than water near ordinary temperatures, partially full pipes, or problems that specify Darcy–Weisbach/Colebrook.

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Worked examples

These are the solver's own reference problems — the answers come from published sources or independent hand computation, never from the solver itself. The solution below is the live solver output for each.

Example 1

SI head loss: C = 130, D = 0.3 m, Q = 0.1 m³/s, L = 500 m

Given

solve for
headLoss
hazen williams c
130
length
500 m
diameter
0.3 m
flowrate
0.1 m^3/s

Assumptions

  • Hazen–Williams is an empirical fit for WATER at ordinary temperatures flowing full in a pressure pipe — it does not apply to other fluids or partially full pipes.
  • Turbulent flow in the range the correlation was calibrated for.

Solution steps

  1. Velocity from continuity

    Mean velocity from the flowrate and the full-pipe area.

    V = Q/A = 4Q/(πD²)

    V = 0.1 m^3/s / 0.0707 m^2 = 1.41 m/s

    = 1.41 m/s

  2. Friction slope from Hazen–Williams

    Invert the velocity form for the friction slope; a full circular pipe has hydraulic radius R = D/4.

    V = 0.849·C·R^(0.63)·S^(0.54) with R = D/4 ⇒ S = (V/(0.849·C·R^(0.63)))^(1/0.54)

    R = 0.075 m; C = 130; S = 0.006433

    = 0.006433

  3. Head loss over the run

    The loss is the slope times the length.

    hL = S·L

    h_L = 0.006433 × 500 m = 3.216 m

    = 3.216 m

Results

Friction head loss hL

3.216m

Mean pipe velocity V

1.41m/s

Friction slope S = hL/L

0.006433

Where this answer was checked
source
Hazen–Williams SI velocity form V = 0.849·C·R^0.63·S^0.54 (NCEES FE Reference Handbook) — solve S from V, then hL = S·L
verified by
hand-recomputed
derivation
A = π·0.3²/4 = 0.0706858 m²; V = 0.1/0.0706858 = 1.414711 m/s; R = D/4 = 0.075 m. S = (V/(0.849·130·0.075^0.63))^(1/0.54) = 0.0064326. hL = 0.0064326·500 = 3.2163 m. Cross-check with the monomial hL = 10.67·L·Q^1.852/(C^1.852·D^4.87) = 3.2102 m (−0.19%, the rounded 10.67 constant).

Example 2

US flowrate: C = 100 cast iron, D = 12 in, hL = 10 ft over L = 1000 ft

Given

solve for
flowrate
hazen williams c
100
length
1000 ft
diameter
12 in
head loss
10 ft

Assumptions

  • Hazen–Williams is an empirical fit for WATER at ordinary temperatures flowing full in a pressure pipe — it does not apply to other fluids or partially full pipes.
  • Turbulent flow in the range the correlation was calibrated for.

Solution steps

  1. Friction slope

    Available head loss per unit length.

    S = hL/L

    S = 10 ft / 1000 ft = 0.01

    = 0.01

  2. Hazen–Williams velocity

    Velocity from the SI form with R = D/4 for a full circular pipe.

    V = 0.849·C·R^(0.63)·S^(0.54) with R = D/4

    R = 0.25 ft; C = 100; V = 4.576 ft/s

    = 4.576 ft/s

  3. Flowrate from continuity

    Multiply by the full-pipe area.

    Q = V·A = V·πD²/4

    Q = 4.576 ft/s × 0.785 ft^2 = 3.594 cfs

    = 3.594 cfs

  4. US-customary form (footnote)

    The identical equation in US units — the 1.318 constant is just 0.849 carried through the ft↔m conversion of R and V.

    V(ft/s) = 1.318·C·R(ft)^(0.63)·S^(0.54)

Results

Flowrate Q

3.594cfs

Mean pipe velocity V

4.58ft/s

Friction slope S = hL/L

0.01

Where this answer was checked
source
Hazen–Williams US velocity form V = 1.318·C·R^0.63·S^0.54 — S = 0.01; V then Q = V·A
verified by
hand-recomputed
derivation
R = 1/4 ft; R^0.63 = 0.417544; S^0.54 = 0.01^0.54 = 0.0831764. V = 1.318·100·0.417544·0.0831764 = 4.57739 ft/s. A = π/4 = 0.785398 ft²; Q = 4.57739·0.785398 = 3.59507 cfs. (SI cross-check: 0.849·0.3048^−0.37 = 1.31772, within 0.02% of the 1.318 constant.)

Example 3

SI diameter: C = 150 PVC, Q = 0.05 m³/s, hL = 5 m over L = 1000 m

Given

solve for
diameter
hazen williams c
150
length
1000 m
flowrate
0.05 m^3/s
head loss
5 m

Assumptions

  • Hazen–Williams is an empirical fit for WATER at ordinary temperatures flowing full in a pressure pipe — it does not apply to other fluids or partially full pipes.
  • Turbulent flow in the range the correlation was calibrated for.

Solution steps

  1. Friction slope

    Available head loss per unit length.

    S = hL/L

    S = 5 m / 1000 m = 0.005

    = 0.005

  2. Required diameter from the discharge form

    Substituting A = πD²/4 and R = D/4 into the velocity form collapses Hazen–Williams to a single monomial in D, solved directly.

    Q = 0.278·C·D^(2.63)·S^(0.54) ⇒ D = (Q/(0.278·C·S^(0.54)))^(1/2.63)

    discharge constant = 0.2784; C = 150; D = 0.2299 m

    = 0.2299 m

Results

Required inside diameter D

0.2299m

Mean pipe velocity V

1.2m/s

Friction slope S = hL/L

0.005

Notes

  • A computed diameter is a hydraulic minimum — real designs round UP to the next commercial pipe size.
Where this answer was checked
source
Hazen–Williams discharge form Q = 0.2784·C·D^2.63·S^0.54 — invert the monomial for D
verified by
hand-recomputed
derivation
S = 5/1000 = 0.005; S^0.54 = 0.0572063. Discharge constant 0.849·(π/4)·4^−0.63 = 0.2784196. D = (0.05/(0.2784196·150·0.0572063))^(1/2.63) = (0.0209653)^(0.380228) = 0.229877 m ≈ 230 mm.