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Hydraulic Diameter: D_h = 4A/P for Rectangular Ducts, Annuli, and Non-Circular Flow

Published July 3, 2026 · by Industrial Unit Converter Editorial Team

Hydraulic diameter: D_h = 4A/P for rectangular ducts, annuli, and non-circular flow

In 2018, a high-rise retrofit in Chicago replaced a 14-inch round supply duct with a 24×12 inch rectangular duct to clear a structural beam. The hydraulic diameter of the rectangular duct is 16.0 inches, larger than the round duct, so the engineer expected less pressure drop. The installed system ran 11% over the design fan static pressure. The airflow had to be derated from 4,000 CFM to 3,650 CFM to stay within the air-handler's external static budget. The cause: a 2:1 aspect ratio does not behave like a 16-inch round pipe, even after the hydraulic diameter substitution, because the long-wall and short-wall boundary layers grow at different rates.

The hydraulic diameter D_h = 4A/P is the standard way to apply Reynolds number, Moody chart, and Darcy-Weisbach to non-circular ducts. The substitution is exact for circular pipes and within a few percent for square, near-square, and 2:1 rectangular ducts, plus concentric annuli. For high-aspect-ratio ducts, finned tubes, and some other geometries, the answer is 10-50% off.

The definition

For any duct with cross-sectional flow area A and wetted perimeter P:

D_h = 4A / P

The area is the area of the flow. The wetted perimeter is the length of wall in contact with the fluid. For a closed duct running full, these are the full geometric values. For a partially filled pipe or open channel, only the wetted portions count.

The sanity check: for a circular pipe, A = πD²/4 and P = πD, so D_h = 4(πD²/4) / (πD) = D. The formula reduces to the actual diameter in the only case where the circular-pipe correlation is exact. The substitution is not a general-purpose characteristic length. It is a placeholder that works in the Darcy-Weisbach framework under specific conditions.

The Reynolds number calculator accepts D_h as an input. The friction factor calculator then takes the modified Re and ε/D_h into Colebrook-White or Swamee-Jain. For circular pipes, use the pipe flow variant instead.

Common geometries

Rectangular duct (width W, height H)

A = W × H, P = 2(W + H), so D_h = 2WH / (W + H).

For a square duct (W = H): D_h = W. For 2:1 (W = 2H): D_h = 1.33H. For 4:1: D_h = 1.6H. For 6:1: D_h = 1.71H. For 8:1: D_h = 1.78H. As the aspect ratio grows, D_h approaches 2H, the limit set by the short wall.

Concentric annulus (outer ID D_o, inner OD D_i)

A = π(D_o² - D_i²) / 4, P = π(D_o + D_i), so D_h = 4A / P = D_o - D_i.

The annulus D_h simplifies to the gap between the walls. A 6-inch carrier (ID 6.065 in) with a 1-inch internal tracer (OD 1.049 in) has D_h = 5.016 in. This is the most useful single result of the formula for heat-exchanger and pipe-in-pipe work.

For an eccentric annulus, the wetted perimeter is unchanged but the area decreases as the inner pipe moves off-center. ASHRAE Handbook Fundamentals (2021, Chapter 21) covers the corrections.

When D_h works and when it fails

D_h gives friction factors within 5% of measured values for square and near-square ducts (W/H up to 1.5:1), rectangular ducts with W/H up to 2:1, and concentric and near-concentric annuli. The 2,300 and 4,000 Reynolds number boundaries still apply when D_h is the characteristic length, as covered in the flow regimes guide.

For rectangular ducts with W/H above 2:1, the long walls develop a thinner boundary layer than the short walls, and D_h underestimates the friction factor. The ASHRAE Handbook Fundamentals (2021, Chapter 21: Duct Design) tabulates a correction:

W/H f correction Notes
1:1 to 2:1 1.00 Use D_h directly
3:1 1.05 Interpolated
4:1 1.10
6:1 1.20
8:1 1.30
10:1 1.50

The corrected friction factor is f_corr = k × f, where f comes from the Moody chart using D_h and Re based on D_h. For W/H above 10:1, the ASHRAE correction is unreliable and specialized correlations (Shah and London 1978, Idelchik 1994) should be used instead.

Finned tubes are outside the D_h framework. The fin pitch, fin height, and fin thickness dominate the flow, and D_h gives meaningless answers. Use the manufacturer's correlations or the Shah and London (1978) tabulated data. Non-Newtonian fluids require a generalized Reynolds number (Metzner-Reed) with apparent viscosity, not the Newtonian Re with D_h. Transitional flow (Re 2,300-4,000) is sensitive to specific geometry, and the circular-pipe boundary values may not apply.

Worked example 1: 24×12 inch rectangular HVAC duct

A commercial air handler supplies 4,000 CFM of air at 20°C, 1 atm, through a 24-inch × 12-inch galvanized steel rectangular duct. Aspect ratio 2:1, so D_h is used directly.

A = 24 × 12 = 288 in² = 2.0 ft² = 0.1858 m² P = 2(24 + 12) = 72 in = 6.0 ft = 1.829 m D_h = 4A / P = 4 × 0.1858 / 1.829 = 0.406 m = 16.0 in Q = 4,000 CFM = 1.888 m³/s V = Q / A = 1.888 / 0.1858 = 10.16 m/s = 33.3 ft/s

Air at 20°C: ρ = 1.205 kg/m³, μ = 1.81 × 10⁻⁵ Pa·s.

Re = ρVD_h / μ = (1.205 × 10.16 × 0.406) / (1.81 × 10⁻⁵) = 275,000

Turbulent. For galvanized steel (ε = 0.00015 m), ε/D_h = 0.00037. Per Swamee-Jain: f ≈ 0.018. Pressure drop per 30 m: h_f = 0.018 × (30/0.406) × (10.16²/(2 × 9.81)) = 7.0 m of air, or about 85 Pa. ASHRAE recommends total duct static pressure below 1.0 in.wc (250 Pa) for low-pressure systems, so 4,000 CFM through a single 24×12 run leaves very little budget for fittings and dampers.

A 16-inch round duct has the same D_h but smaller area (201 in² vs 288 in²) and higher velocity (56.5 ft/s). The 24×12 duct has 43% more area for the same D_h, which is why rectangular ducts are preferred when D_h is the design target and material cost matters.

Worked example 2: 12×6 inch rectangular branch

A 12-inch × 6-inch galvanized branch duct off the main supply carries 1,000 CFM at 20°C. Aspect ratio 2:1, D_h used directly.

A = 72 in² = 0.0465 m², P = 36 in = 0.914 m, D_h = 0.203 m = 8.0 in Q = 1,000 CFM = 0.472 m³/s, V = 0.472 / 0.0465 = 10.16 m/s = 33.3 ft/s Re = (1.205 × 10.16 × 0.203) / (1.81 × 10⁻⁵) = 137,000

Turbulent. For galvanized steel, ε/D_h = 0.00074. Per Swamee-Jain: f ≈ 0.021. Pressure drop per 30 m: h_f = 16.3 m of air, about 197 Pa = 0.79 in.wc. At this size and flow rate, a single 30 m run already pushes the pressure drop close to the ASHRAE budget for the entire branch.

Worked example 3: concentric annulus for steam tracing

A 6-inch Schedule 40 steel carrier pipe (ID = 6.065 in) carries saturated steam condensate at 150°C. A 1-inch Schedule 40 steam tracer line (OD = 1.049 in) is centered inside. The condensate flows in the annulus between carrier and tracer at 20,000 lb/hr. A refinery steam-tracing application.

A = π(6.065² - 1.049²) / 4 = 28.03 in² = 0.01808 m² P = π(6.065 + 1.049) = 22.35 in = 0.568 m D_h = 4A / P = D_o - D_i = 5.016 in = 0.1274 m

The cleanest result of the formula: for a concentric annulus, D_h equals the gap.

Q = 20,000 lb/hr × 0.4536 / 3600 = 2.52 kg/s V = 2.52 / (970 × 0.01808) = 0.144 m/s = 0.47 ft/s

Condensate at 150°C: ρ = 970 kg/m³, μ = 1.8 × 10⁻⁴ Pa·s.

Re = (970 × 0.144 × 0.1274) / (1.8 × 10⁻⁴) = 98,600

Turbulent. For new commercial steel (ε = 0.045 mm), ε/D_h = 0.000353. Per Swamee-Jain: f ≈ 0.020. Pressure drop per 100 m: h_f = 0.020 × (100/0.1274) × (0.144²/(2 × 9.81)) = 0.017 m of water column, low for a 100 m run, which is why internal steam tracing works for long process lines without booster pumps.

Six common mistakes

Mistake 1: Using D_h for very high aspect ratios (W/H > 8:1) without the ASHRAE correction. At 8:1 the correction is 1.30, meaning 30% higher friction than the circular-pipe Moody chart predicts. A designer who uses the Moody chart directly underpredicts the duct static pressure by 30%.

Mistake 2: Using D_h for finned tubes. The fin pitch, fin height, and fin thickness dominate the flow, none of which the D_h formula captures. A finned tube with D_h = 5 mm can have a real friction factor 5-10× higher than the Moody chart value. Use the manufacturer's pressure-drop curves or Shah and London (1978) data.

Mistake 3: Forgetting the wetted perimeter for partially filled ducts. A circular pipe at 25% full has wetted perimeter about 1.21 × D, not π × D. Using the full perimeter gives a D_h 2.6× too large and a Re 2.6× too high, putting the flow into turbulent when the actual flow is laminar.

Mistake 4: Mixing D_h with the wrong Re definition. Heat-transfer correlations that use D_h in Nu = hD_h/k require Re = VD_h/ν with the same D_h. Mixing D in one place and D_h in another gives nonsense.

Mistake 5: Using D_h for non-Newtonian fluids. Polymer solutions, slurries, food products, and drilling muds have viscosity that depends on shear rate. The Newtonian Re = ρVD_h/μ does not apply. The Metzner-Reed generalized Re uses apparent viscosity from a consistency index and power-law index. A 1% polymer solution in water can shift the effective Re by 10×.

Mistake 6: Confusing the annulus D_h with the carrier ID or the tracer OD. A 6-inch pipe with a 1-inch internal tracer has D_h = 5.0 in (the gap), not 6.065 in (carrier ID) and not 1.049 in (tracer OD). A designer who uses 6.065 in will calculate Re 21% too high and a pressure drop 20% below the actual value.

Standards and reference data

  • White, F.M. (2016). Fluid Mechanics, 8th ed., McGraw-Hill.
  • ASHRAE (2021). ASHRAE Handbook Fundamentals, Chapter 21: Duct Design. Source of the aspect-ratio correction table.
  • Shah, R.K. and London, A.L. (1978). Laminar Flow Forced Convection in Ducts. Academic Press.
  • Idelchik, I.E. (1994). Handbook of Hydraulic Resistance. Begell House.
  • Munson, B.R., Young, D.F., Okiishi, T.H. (2018). Fundamentals of Fluid Mechanics, 8th ed., Wiley.
  • Cengel, Y.A. and Cimbala, J.M. (2018). Fluid Mechanics: Fundamentals and Applications, 3rd ed., McGraw-Hill.
  • Engineering Toolbox. Hydraulic Diameter. https://www.engineeringtoolbox.com/hydraulic-diameter-d_444.html

D_h for rectangular ducts as a function of aspect ratio

W/H D_h / H Use ASHRAE correction?
1:1 1.00 No
1.5:1 1.20 No
2:1 1.33 No
3:1 1.50 Yes (1.05)
4:1 1.60 Yes (1.10)
6:1 1.71 Yes (1.20)
8:1 1.78 Yes (1.30)
10:1 1.82 Yes (1.50)

Frequently asked questions

Q: What is the hydraulic diameter of a square duct with side a?

A: D_h = a. A 12-inch square duct has D_h = 12 in, the same as a 12-inch round pipe. The square duct has 27% more area (144 in² vs 113 in²) for the same D_h, which is why square and rectangular ducts are preferred where D_h and material cost both matter.

Q: What is the hydraulic diameter of an annulus between two concentric pipes?

A: D_h = D_o - D_i, the gap between the outer and inner walls. A 6-inch Schedule 40 carrier (ID 6.065 in) with a 1-inch internal tracer (OD 1.049 in) has D_h = 5.016 in. The derivation follows from the area and perimeter formulas, and the result simplifies to the gap.

Q: When does the hydraulic diameter fail?

A: Three cases. First, rectangular ducts with W/H above 2:1, where the ASHRAE correction (1.05 to 1.50 depending on aspect ratio) must be applied to the friction factor. Second, finned tubes, where the fin geometry dominates and D_h is meaningless. Third, non-Newtonian fluids, where a generalized Reynolds number is required.

Q: Does D_h apply to open channel flow?

A: No, not directly. Open channel flow uses the hydraulic radius R_h = A/P, which is D_h/4. The Manning and Chezy equations use R_h, not D_h. For a pipe flowing partially full, the wetted perimeter is the length of wall in contact with the fluid, not the full circumference.

Q: How do I convert from a rectangular duct to an equivalent round duct for the same airflow?

A: The equivalent round duct is the one with the same D_h. For a 24×12 rectangular duct, D_h = 16.0 in, so the equivalent round duct is 16 inches. The round duct has less area (201 in² vs 288 in²) and higher velocity, so the pressure drop is different. ASHRAE's "equal friction" method sizes the duct for the same friction loss per unit length, not the same D_h.

Q: Why does the annulus D_h simplify to D_o - D_i?

A: Substituting A = π(D_o² - D_i²)/4 and P = π(D_o + D_i) into D_h = 4A/P gives D_h = (D_o² - D_i²)/(D_o + D_i) = (D_o - D_i)(D_o + D_i)/(D_o + D_i) = D_o - D_i. The D_o + D_i factors cancel. One of the cleanest algebraic results in fluid mechanics, and the reason the D_h formula is so widely used in heat-exchanger design.

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