Hydraulic Diameter Calculator

D_h = 4A/P for a round pipe, a rectangular or square duct, an annulus, parallel plates, a part-full pipe, a trapezoidal channel or any section you can measure — with the hydraulic radius beside it, a plain statement of which of the two your next formula wants, and the exact laminar friction constant for the section, because it is not 64/Re unless the section is round. For engineers setting up an internal-flow simulation or checking one.

One definition, and the closed form it collapses to for each section

Every section, its 4A/P, and its exact laminar friction constant

What the factor of four does to a threshold

How wrong the plausible substitutes are, for one 200 × 100 mm duct

Channel size classification, for readers working below a millimetre

Which of D_h and R your next formula wants, and why 64/Re is the wrong laminar answer

The hydraulic diameter is 4A/P and nothing else, and almost every mistake made with it is one of three. Using the wrong perimeter, confusing it with the hydraulic radius, or forgetting that it is a similarity shortcut rather than an equivalence. The 4 is not decoration: it exists so that 4A/P collapses to D for a round pipe, and therefore so that the transition Reynolds number of 2,300 and every friction and heat-transfer correlation ever measured in a circular tube can be carried across to a duct. Take the 4 away and you have the hydraulic radius, which is a perfectly good quantity used by a different profession for the same job.

Which of the two your next formula wants, stated once and properly. Pipe and duct work — Reynolds numbers, Darcy–Weisbach f·L/D, Colebrook, Blasius, Dittus–Boelter, Gnielinski, y+ estimates, entrance lengths — is written in the hydraulic DIAMETER. Open-channel hydraulics — Manning, Chézy, the Darcy–Weisbach form used for channels, and the laminar and turbulent thresholds of 500 and 2,000 — is written in the hydraulic RADIUS. D_h = 4R exactly, for every section, with no exceptions and no approximations. So a Reynolds number quoted on R is a quarter of the same flow’s Reynolds number quoted on D_h, and a threshold moved from one convention to the other without the factor of 4 misplaces the regime by a whole band. It is the factor-of-four error that makes a turbulent channel look laminar, and someone then turns the turbulence model off. Both numbers are printed above, every time, for exactly this reason.

The wetted perimeter is where the arithmetic goes wrong. Three rules cover it. A free surface is not wetted: air above water exerts shear four orders of magnitude smaller than the bed does, so in a part-full pipe or a channel the surface width is simply not in P. A symmetry plane is not wetted: if you have cut your model in half you must not count the cut. And every solid surface IS wetted, including the inner tube of an annulus, both faces of a splitter, and the full developed length of a fin — which is why a finned passage can have a hydraulic diameter several times smaller than its apparent size. Get P wrong and D_h, the Reynolds number and the friction term are all wrong together, in the same direction, and none of them looks odd.

Now the honest part: D_h is a similarity shortcut, and in laminar flow it is a poor one. The claim being made when you substitute D_h into a round-pipe correlation is that the section’s shape does not matter once its area-to-perimeter ratio is fixed. In laminar flow that is measurably false, and the size of the error is not a matter of opinion — fully developed laminar flow satisfies ∇²u = (1/μ)dp/dx over the section, which has an exact solution for every shape on this page, and the friction factor that comes out of it is f = C/Re with C a pure number fixed by the shape alone. C is 64 for a circle. It is 56.908 for a square duct, so 64/Re reads 12.5 per cent HIGH there. It is exactly 96 between parallel plates, so 64/Re reads 33.3 per cent LOW. It is 72.93 for a 4:1 rectangle, 82.34 for 8:1, 95.25 for an annulus with a half-diameter inner tube, and 160/3 = 53.33 for an equilateral triangle, where 64/Re is 20 per cent high. Those are analytic results, so the error survives any mesh you like. This page prints C for the section you chose and the percentage error beside it.

Turbulent flow is far kinder, and it is worth knowing why. In a turbulent duct the velocity profile is set by the local wall shear over most of the section rather than by the global shape, so substituting D_h works much better — and the residual sensitivity is small for a reason you can quantify. The friction factor depends on the Reynolds number roughly as Re^−0.2 near Re = 10⁴ and more weakly still above it: the measured logarithmic slope of the Haaland correlation is −0.275 at Re = 10⁴ and −0.169 at Re = 10⁶ for a smooth wall. So even a 20 per cent error in the effective Reynolds number moves f by under 5 per cent, and the shape effects that cause a 33 per cent laminar error cause a few per cent turbulent one. The corners are the exception: secondary flows drive fluid into the corners of a rectangular duct and the local shear there is well below the section mean, which matters if you are predicting local heat transfer rather than a pressure drop.

The part-full pipe is the section with the counter-intuitive answer, and it is worth a moment. Sweep the depth of a circular pipe from empty to full and the hydraulic diameter does not rise monotonically. It equals the bore exactly at half depth, because both A and P are then exactly half their full-pipe values; it keeps rising to a maximum of 1.2172 D at 81.3 per cent depth; and then it falls back to exactly D when the pipe runs full and the whole circumference is wetted. Everything about circular-sewer hydraulics follows from that curve: the maximum velocity at 81.3 per cent depth is where R^(2/3) peaks, and the maximum discharge at 93.8 per cent depth is where A·R^(2/3) peaks. Both numbers drop out of the geometry in the table above, and both are worth checking against your own solver if you are modelling a partly filled pipe with a volume-of-fluid method.

Where this page fits. The hydraulic diameter it gives you is the length for the Reynolds number and flow regime calculator, the D_h in the pressure-drop calculation on the pipe friction factor and pressure drop page, and the length scale behind the turbulence inlet conditions on the turbulence inlet conditions calculator, which takes the mixing length as a fraction of D_h. It also sets the wall-normal spacing you will need, through the friction velocity, on the y+ first cell height calculator. Four pages, one number, and if it is wrong they are all wrong the same way.

Frequently asked questions

Related calculators

References

1. R. K. Shah and A. L. London, Laminar Flow Forced Convention in Ducts, for the tabulated laminar friction constants f·Re on the hydraulic diameter. Not reproduced here. Every value on this page was recomputed from the governing equation instead: the rectangular-duct constants from the double sine-series solution of ∇²φ = −1 with f·Re = 2D_h²/̅φ, summed to 800 odd terms in each direction and checked for convergence to six figures; the annulus from the exact logarithmic solution; the equilateral triangle from the product of the three side distances, which gives 160/3 analytically. The recomputed values agree with the published tabulation to better than 0.07 per cent (square 56.9083, parallel plates 96 exactly, 2:1 rectangle 62.1922, 4:1 72.9311, 8:1 82.3386, annulus at D_i/D_o = 0.5 95.2502).
2. The closed-form rectangular-duct fit used by this page, f·Re = 96(1 − 1.3657266α + 2.0664904α² − 2.2212156α³ + 1.9770629α⁴ − 1.1741859α⁵ + 0.3103697α⁶) with α = short side / long side, was fitted here to the exact series values under the constraints that it be exact at α = 1 (56.908308) and at α → 0 (96). Worst deviation from the exact series over the whole range 0 < α ≤ 1 is 0.0102 per cent. For comparison, Shah and London’s own published polynomial deviates from the same exact series by up to 0.063 per cent, so the two agree with each other to within 0.07 per cent everywhere — which is the cross-check that matters.
3. V. T. Chow, Open-Channel Hydraulics, and standard open-channel teaching material, for the hydraulic radius convention R = A/P and the transition thresholds quoted on it (laminar below Re_R = 500, transitional to 2,000). Converted here to the hydraulic-diameter basis by the exact factor of four and both bases printed side by side, because mixing the two is the commonest error in this subject. The part-full circular-pipe geometry — A = D²(θ − sin θ)/8 and P = Dθ/2 with θ = 2 arccos(1 − 2y/D) — is elementary and was verified here against three identities that must hold: D_h = D exactly at y/D = 0.5 and again at y/D = 1, maximum D_h/D = 1.21723 at y/D = 0.8128 coinciding with the maximum of R^(2/3) and therefore with the maximum Manning velocity, and maximum A·R^(2/3) at y/D = 0.9382, the textbook maximum-discharge depth.
4. S. G. Kandlikar and W. J. Grande, Evolution of Microchannel Flow Passages — Thermohydraulic Performance and Fabrication Technology (2003), for the channel size classification used in the last table: conventional above 3 mm, minichannel 200 µm to 3 mm, microchannel 10 to 200 µm, transitional microchannel 1 to 10 µm. These are conventions for describing size, not physical transitions, and the page says so — 4A/P is geometry and holds wherever the continuum does.
5. B. S. Petukhov’s explicit smooth-tube friction factor and S. E. Haaland’s explicit Colebrook approximation, used on the companion pressure-drop page and quoted here only for the statement about how weakly turbulent friction depends on Reynolds number. The logarithmic slope d ln f / d ln Re of the Haaland form was differentiated numerically here at a smooth wall and comes out −0.275 at Re = 10⁴, −0.209 at 10⁵, −0.168 at 10⁶ and −0.141 at 10⁷, which is where the “a 20 per cent Reynolds-number error moves f by under 5 per cent” figure comes from.
6. Y. A. Çengel and J. M. Cimbala, Fluid Mechanics: Fundamentals and Applications, and F. M. White, Fluid Mechanics, for the hydraulic diameter convention itself and the conventional internal-flow transition bands that make the factor of 4 matter. Cited, not reproduced.

Setup guidance, not validation. Correlations have ranges of validity and cell-count estimates are order-of-magnitude. A converged simulation is not a correct one. Full disclaimer at calcengines.com/disclaimer/