Compute the hydraulic diameter of rectangular ducts, annular passages and square ducts. Change the cross-section and dimensions to see the flow area, wetted perimeter, hydraulic diameter D_h = 4A/P and Reynolds number update in real time, and build an intuition for the equivalent diameter that lets non-circular ducts use round-pipe friction and heat-transfer correlations.
The chosen cross-section is filled with fluid (blue). The thick coloured outline is the wetted perimeter P, and the dashed circle is the equivalent circle of diameter D_h. Flow particles travel through the section.
Hydraulic diameter vs aspect ratio / diameter ratio
Reynolds number vs mean velocity
Theory & Key Formulas
$$D_h=\frac{4A}{P}$$
Definition of the hydraulic diameter. A: flow cross-sectional area, P: wetted perimeter (wall length the fluid touches). For a round pipe D_h reduces to D.
D_h for typical sections. Rectangle (width a, height b), annulus (outer diameter D_o, inner diameter D_i). For a square (side a), D_h = a.
$$Re=\frac{U\,D_h}{\nu}$$
Reynolds number. U: mean velocity, ν: kinematic viscosity. For non-circular ducts, Re and the friction correlations use D_h. Re < 2300 is the laminar guideline.
What is the Hydraulic Diameter Simulator?
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Pipe calculations always seem to use round pipes. But real ducts are often square, or shaped like a tube inside a tube. How do you calculate those?
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Good thing to notice. HVAC ducts are square, and the outer passage of a heat exchanger is the gap between two pipes — an annulus. There is a trick that lets you keep using the round-pipe formulas, and it is the hydraulic diameter. We define it as D_h = 4A/P, where A is the flow cross-sectional area and P is the wetted perimeter — the length of wall the fluid is pressed against.
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Why 4A/P, though? That factor of 4 appearing out of nowhere puzzles me.
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Check it against a round pipe and it clicks. For a pipe of diameter D, the area is A = πD²/4 and the wetted perimeter is P = πD. Put those into 4A/P and you get 4·(πD²/4)/(πD) = D. It returns exactly the diameter. So the factor of 4 is the calibration constant chosen so that "for a round pipe, D_h equals D". For a rectangle or an annulus, the value 4A/P then becomes an equivalent length scale — "as if it were a round pipe of about this diameter".
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I see! So if I pick the rectangular duct on the left and change width and height, D_h changes. And the square gives D_h equal to the side length.
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Right — for a square of side a, A = a² and P = 4a, so D_h = 4a²/(4a) = a. It comes out to the side length cleanly. The rectangle gives D_h = 2ab/(a+b), which looks like a harmonic-mean of width and height. Make it flat — say 100 mm wide by 10 mm tall — and D_h shrinks a lot. Move the slider on the "Hydraulic diameter vs aspect ratio" chart below and you will see D_h thin out as the section gets long and narrow.
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Choosing the annular passage gives D_h = outer diameter − inner diameter. Neat shape — any cautions there?
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The derivation for the annulus is satisfying. The area is A = π(D_o²−D_i²)/4, and the wetted perimeter is both the outer and inner walls, so P = π(D_o+D_i). Working out 4A/P gives (D_o²−D_i²)/(D_o+D_i) = D_o−D_i — exactly the gap width. The caution is that the inner diameter must be smaller than the outer one, since otherwise the geometry is impossible, so this tool always checks inner < outer. Also, when the annular gap is extremely narrow the transition Reynolds number drifts away from the usual 2300, so check experimental data for precision design.
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One last thing. It is called a "diameter" — can I use the hydraulic diameter exactly like a real diameter?
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This is the most important pitfall. The hydraulic diameter is an equivalent length scale, not a true diameter. Substituting it for D in friction, Reynolds-number and heat-transfer correlations is fine. But computing the cross-sectional area as π·D_h²/4 to get a flow rate is wrong — it disagrees with the real area A. This tool shows the equivalent-circular-pipe area ratio A/(π·D_h²/4); how far it sits from 1 tells you at a glance that "the D_h circle has a different area". Always use the real area A for flow rate. Remember just that one rule.
Frequently Asked Questions
The hydraulic diameter is defined as D_h = 4A / P, where A is the flow cross-sectional area and P is the wetted perimeter (the wall length the fluid touches). For a rectangular duct (width a, height b) it reduces to D_h = 2ab/(a+b); for an annular passage (outer diameter D_o, inner diameter D_i) it is D_h = D_o - D_i; and for a square duct (side a) it is simply D_h = a. This tool computes A, P and D_h automatically once you choose the cross-section.
Round-pipe correlations for friction loss and heat transfer (the Moody chart, the Dittus-Boelter equation and so on) are organized around the diameter D. They cannot be applied directly to rectangular or annular ducts, but by defining an equivalent length scale D_h = 4A/P you can substitute D_h for D in the round-pipe formulas and get a working estimate. That is the main reason the hydraulic diameter exists.
The Reynolds number is Re = U·D_h / ν, where U is the mean velocity and ν the kinematic viscosity. As with round pipes, Re < 2300 is taken as laminar and higher values as turbulent. The hydraulic-diameter classification is only approximate, though: for extremely flat cross-sections or very narrow annular gaps the transition Reynolds number can shift away from the standard value, so precise design work should also use shape-specific experimental data or numerical analysis.
No. The hydraulic diameter is an equivalent length scale, not a geometric diameter. A round pipe with diameter D_h would have a cross-sectional area of π·D_h²/4, which in general does not equal the area A of the original non-circular duct. This tool shows the equivalent-circular-pipe area ratio A/(π·D_h²/4) to make it visible that the hydraulic diameter does not preserve area. Always use the real flow area A, not D_h, when computing volume flow rate.
Real-World Applications
HVAC and ventilation duct design: Air-conditioning ducts in buildings and factories are mostly rectangular. Selecting a fan requires pressure-loss calculations, but the Darcy-Weisbach equation and the Moody chart are based on round pipes. So engineers find the rectangular duct's hydraulic diameter D_h = 2ab/(a+b), feed the Reynolds number Re = U·D_h/ν into the round-pipe friction chart and estimate the pressure drop. For the same flow area, a flat duct has a smaller D_h and a longer wetted perimeter, so it tends to incur higher friction loss.
Double-pipe and shell-and-tube heat exchangers: The annular passage outside an inner tube is an extremely common heat-exchanger geometry. The annulus hydraulic diameter D_h = D_o - D_i is used with Nusselt-number correlations (such as the Dittus-Boelter equation) to estimate the convective heat-transfer coefficient. Note that in an annulus the actual heat-transfer behaviour depends on whether the heated surface is the inner or outer wall, so the hydraulic-diameter estimate is sometimes multiplied by a correction factor.
Microchannels and electronics cooling: CPU cold plates and microchannel heat exchangers pack many very fine rectangular passages side by side. The smaller the hydraulic diameter, the lower the Reynolds number at the same velocity, so the flow tends to stay laminar; in the laminar regime the balance between heat transfer and pressure drop is the key design trade-off. The hydraulic diameter serves as a single yardstick for comparing the behaviour of such mini- and micro-scale channels.
Open-channel and river engineering: The hydraulic-diameter idea extends beyond fully-flooded ducts to open channels with a free surface. In an open channel the water surface is not a wall, so it is excluded from the wetted perimeter, and the "hydraulic radius R_h = A/P" is used instead (with the relation D_h = 4·R_h). It is a fundamental quantity widely used in river and canal design, for example in Manning's formula for flow rate.
Common Misconceptions and Pitfalls
The biggest misconception is to think that "replacing the duct with a round pipe of the same hydraulic diameter lets you compute the flow rate and pressure loss correctly all at once". The hydraulic diameter is equivalent only as the length scale fed into friction and heat-transfer correlations. When computing the flow rate Q = U·A, you must use the real cross-sectional area A, not π·D_h²/4. The further the equivalent-circular-pipe area ratio shown by this tool sits from 1, the more serious the error from this confusion becomes — the ratio departs from 1 as a rectangle gets flatter and as an annular gap gets narrower.
Next, including a free surface or non-contact face in the wetted perimeter. The wetted perimeter P is strictly the wall length over which the fluid touches a solid and experiences friction. The water surface of an open channel, the gas-liquid interface of a two-phase flow and the walls of dead spaces that do not carry flow are not part of the wetted perimeter. For a fully-flooded rectangular duct all four sides are wetted, but for an open channel only the three sides excluding the water surface count. Getting this distinction wrong shifts D_h badly and throws the pressure-loss and Reynolds-number estimates off at their foundation.
Finally, the assumption that "as long as the hydraulic diameter matches, the laminar-to-turbulent transition is exactly the same as a round pipe". The Re = 2300 transition guideline is an empirical value for round pipes and does not strictly hold for a non-circular duct converted via the hydraulic diameter. For extremely flat rectangles or narrow-gap annuli, the transition Reynolds number can be higher or lower than the standard value. The laminar-regime friction factor f·Re (the Poiseuille number) also takes a different constant for each cross-section and does not equal the round-pipe value of 64. The hydraulic diameter is a convenient estimation tool only; for precise design where shape-dependence matters, use shape-specific experimental data and numerical analysis as well.
How to Use
Select duct geometry type (rectangular, annular, or square) from the dropdown menu
Enter dimension values: for rectangular ducts input width (a) and height (b); for annular passages input outer diameter and inner diameter; for square ducts input side length
Set fluid velocity (m/s) and select working fluid (water, air, oil) to auto-populate density and viscosity
Click Calculate to compute hydraulic diameter D_h, wetted perimeter P, flow area A, Reynolds number Re, and regime classification (laminar/turbulent at Re=2300)
Worked Example
Rectangular air duct: width a=150 mm, height b=100 mm. Flow area A = 15,000 mm². Wetted perimeter P = 500 mm. Hydraulic diameter D_h = 4A/P = 120 mm. At air velocity u=5 m/s with ν=1.81×10⁻⁵ m²/s, Reynolds number Re = (5 m/s × 0.12 m)/(1.81×10⁻⁵) = 33,150, indicating turbulent flow. Equivalent circular duct would need diameter 120 mm to match D_h.
Practical Notes
Annular ducts (concentric pipes) with small gaps experience higher pressure drop per unit length than rectangular ducts of equivalent D_h due to entrance effects and asymmetric velocity profiles
For HVAC design, rectangular ducts with aspect ratio (a/b) near 1 minimize pressure loss; ratios exceeding 4:1 require duct friction factor corrections
D_h applies only to fully developed flow; entrance length L_e ≈ 0.05Re×D_h (laminar) or 4.4D_h⁰·⁶ (turbulent) requires additional head loss calculation
Wetted perimeter excludes any free surfaces; for open channel flow use only the submerged perimeter