Calculate the energy loss that occurs at a "sudden expansion" where a pipe widens abruptly. Adjust the small and large pipe diameters and the velocity to see the Borda-Carnot loss coefficient K, head loss and pressure loss, together with the static pressure rise from the momentum balance, all update in real time.
Parameters
Small pipe diameter D1
mm
Inner diameter of the thin pipe upstream of the step
Large pipe diameter D2
mm
Inner diameter of the wide pipe downstream (larger than D1)
Upstream mean velocity U1
m/s
Mean velocity in the thin upstream pipe
Fluid density ρ
kg/m³
Water at 20 °C is about 998 kg/m³
Results
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Loss coefficient K
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Large-pipe velocity U2 (m/s)
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Head loss h_L (m)
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Pressure loss ΔP_loss (kPa)
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Area ratio A2/A1
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Pressure rise ΔP (kPa)
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Sudden-expansion flow — separation-eddy animation
The jet leaving the small pipe cannot follow the abrupt step, so it separates and forms turbulent recirculation eddies in the corners. Those eddies dissipating kinetic energy is the Borda-Carnot loss.
Head loss h_L of a sudden expansion (Borda-Carnot equation) and the loss coefficient K referred to the upstream velocity head, with hL = K·U1²/(2g). g: gravitational acceleration.
$$U_2=U_1\frac{A_1}{A_2}$$
Downstream velocity U2 from continuity. A1 and A2 are the small- and large-pipe cross-sectional areas. The loss is largest when the area ratio is large (a small pipe discharging into a big one) — K approaches 1.
$$\Delta P_{rise}=\rho\,U_2(U_1-U_2)$$
Static pressure rise ΔP across the expansion from the momentum balance. ρ: fluid density. The expansion converts part of the kinetic energy to pressure, but the loss makes the rise smaller than the ideal.
What is a sudden expansion loss?
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I heard that if a pipe suddenly gets wider, you lose pressure just from that. But it only gets wider — why would that cost anything?
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Good question. The key is that "flow cannot turn a sharp corner". The stream leaving the small pipe is a fast jet. So even when the pipe wall suddenly vanishes outward, the flow cannot spread out and hug that step. Its inertia keeps it going straight, and it peels off the wall. We call that separation.
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If it peels off, what happens in that gap? Does it become empty?
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It does not go empty — a swirling eddy forms in the corner between the jet and the large-pipe wall. We call it a recirculation eddy. See the eddies drawn at the step corners in the animation above? They take kinetic energy from the jet, stir it around, and finally turn it into heat through turbulent friction. So it is energy that never comes back. That is the Borda-Carnot loss.
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But the slider on the left shows "Pressure rise ΔP" as positive. Losing energy yet the pressure goes up — isn't that a contradiction?
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That is the interesting part of a sudden expansion. When the pipe gets wider, the velocity drops (continuity gives U2 = U1·A1/A2). As the velocity drops, kinetic energy decreases and part of it converts to static pressure — so the pressure does rise. But if it converted ideally and completely, it would rise more. In reality it falls short by exactly the Borda-Carnot loss. So the right picture is "pressure rises, but rises less because of the loss".
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I see... so what condition makes the loss the largest?
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Since the loss coefficient is K = (1−A1/A2)², the smaller A1/A2 gets — that is, the thinner the small pipe is relative to the big pipe — the closer K gets to 1. The extreme case is a thin pipe discharging straight into a large tank or room. There A1/A2 is about 0, so K is about 1 and the whole upstream velocity head is lost. This is also called the exit loss, and it is often missed in piping calculations. Drag the slider on the "loss coefficient vs diameter ratio" chart below and you'll see K push hard toward 1 as the diameter ratio grows.
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When I want to reduce the loss, what should I do?
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Don't widen it "suddenly". Instead of a step, use a gentle taper — a "diffuser". With a half-angle of about 7°, the flow can decelerate while staying attached to the wall, so there is almost no separation and no eddies. For the same area ratio, a diffuser can cut the sudden-expansion loss to a fraction. That is exactly why diffusers are used at pump discharge ports and duct expansions — to recover that loss.
Frequently Asked Questions
For a sudden expansion where a pipe widens abruptly from area A1 to A2, the head loss is given by the Borda-Carnot equation hL = (U1−U2)²/(2g). U1 is the upstream (small-pipe) velocity and U2 the downstream (large-pipe) velocity, with U2 = U1·A1/A2 from continuity. Expressed as a loss coefficient referred to the upstream velocity head, K = (1−A1/A2)², so that hL = K·U1²/(2g). This tool computes K, hL and the pressure loss from your inputs in real time.
The jet leaving the small pipe cannot follow the abrupt step where the wall suddenly disappears outward, so it separates from the wall. Turbulent recirculation eddies form in the corners between the jet and the large-pipe wall. These eddies stir the jet's kinetic energy and dissipate it as heat through turbulent friction, which is exactly the Borda-Carnot loss (U1−U2)²/(2g). The sharper the step, the larger the separation and the loss.
The loss coefficient K = (1−A1/A2)² approaches 1 as the area ratio A1/A2 gets small, that is, as the small pipe becomes thin relative to the large pipe. In the extreme case of a thin pipe discharging into a very large tank or room, A1/A2 is about 0, so K is about 1 and almost the entire upstream velocity head U1²/(2g) is lost. This is also called the exit loss and is a large loss source that is easy to overlook in piping systems.
A sudden expansion has a step change in cross-section, so separation and eddies always occur and the Borda-Carnot loss is incurred almost in full. A diffuser widens the section with a gentle taper; with a half-angle of around 7° the flow stays attached to the wall as it decelerates and recovers most of the kinetic energy as static pressure. For the same area ratio, a diffuser has a loss many times smaller than a sudden expansion.
Real-World Applications
Pressure-loss calculation in piping systems: In water-supply, HVAC and process piping, a local loss occurs at every fitting where the pipe diameter changes. Where a reducer enlarges the pipe, the geometry is close to a sudden expansion and the Borda-Carnot loss must always be included. When sizing a pump head or fan static pressure, the total loss is built up by summing these local losses through their K values, not just the straight-pipe friction.
Pipe outlets and tank inlets: Where a pipe discharges into a large tank, reservoir or the atmosphere, the area ratio is effectively infinite, so it behaves as a sudden expansion with K ≈ 1, the "exit loss". The entire upstream velocity head U1²/(2g) is lost, which is not negligible for high-velocity pipes. Overlooking it in siphon or gravity-flow design causes the real flow rate to fall well below the calculation.
A baseline for diffuser design: In pump discharge diffusers, blower outlets and wind-tunnel expansions, performance hinges on how much kinetic energy is recovered as static pressure. The Borda-Carnot loss of a sudden expansion is the "worst-case" benchmark, and efficiency is judged by how much a designed diffuser cuts the loss below it. This tool's pressure rise and recovery efficiency are a starting point for that comparison.
Verification of CAE/CFD analysis: The sudden expansion is a fundamental problem involving flow separation and reattachment, often used to benchmark turbulence models and wall functions. Whether the pressure loss from CFD matches the Borda-Carnot value at the right order of magnitude is the first step in checking mesh resolution and boundary conditions. This tool's analytical solution can serve as a reference for that verification.
Common Misconceptions and Pitfalls
The most common misconception is "the pressure drops at a sudden expansion". Used to straight-pipe friction loss, one tends to equate loss with a pressure drop, but at a sudden expansion the velocity falls, so the Bernoulli effect actually makes the static pressure rise. What drops is the total pressure (pressure plus kinetic energy); the static pressure increases. This is why this tool reports the pressure loss ΔP_loss (the decline in total pressure) and the pressure rise ΔP (the increase in static pressure) separately. Confusing the two leads you to read the pipe pressure distribution backwards.
Next is the mix-up of "which velocity head to base the coefficient on". The Borda-Carnot loss hL = (U1−U2)²/(2g) matches the loss coefficient K = (1−A1/A2)² when K multiplies the upstream (small-pipe) velocity head U1²/(2g). Some handbooks list a different K value based on the downstream velocity U2, and getting the reference velocity wrong shifts the value by the square of the area ratio. When using a K, always check "which velocity head it refers to". This tool consistently uses the upstream U1 basis throughout.
Finally, the overconfidence that "the Borda-Carnot equation works for any expansion". This equation is an idealised model that assumes a step change in cross-section and that the flow fully reattaches and becomes uniform downstream. For a tapered diffuser, a gentle expansion where separation is suppressed, or a high-speed flow with strong swirl or compressibility effects, the real loss departs from this equation. For a gentle expansion the real loss is actually smaller, so applying the sudden-expansion value overestimates it. Judge whether the geometry is truly "sudden" and use the equation within its valid range.
How to Use
Enter the inlet pipe diameter (d1) in mm and upstream velocity (u1) in m/s—typical values: d1=50 mm, u1=3 m/s for process piping.
Set the outlet (expanded) pipe diameter (d2) in mm; sudden expansions commonly range d2=75–150 mm for industrial applications.
Input fluid density (rho) in kg/m³—use 1000 for water, 850 for mineral oil, or 1.2 for air at standard conditions.
The simulator calculates loss coefficient K (dimensionless), velocity drop to U2, head loss h_L in meters, and pressure loss ΔP_loss in kPa using Borda-Carnot equations.
Worked Example
Water (ρ=1000 kg/m³) flows through a sudden pipe expansion: inlet d1=40 mm at u1=2.5 m/s expands to d2=80 mm. Area ratio A2/A1=4.0. Loss coefficient K≈0.56 (from (1−A1/A2)² approximation). Outlet velocity U2≈0.625 m/s. Head loss h_L≈0.163 m (163 mm). Pressure loss ΔP_loss≈1.59 kPa. Static pressure rise ΔP≈9.8 kPa due to velocity recovery downstream.
Practical Notes
Borda-Carnot loss dominates when expansion ratio exceeds 2:1; gradual diffusers (15° half-angle) reduce K by 40–60% compared to sudden steps in HVAC and cooling systems.
In low-Reynolds-number regimes (Re <1000), friction losses at the corner become significant; add 10–20% to theoretical K for laminar pipe expansions.
High-speed expansions (u1 >5 m/s) in compressible flow require Mach-corrected formulas; assume incompressible for Ma <0.3.
Valve seat openings and diffuser outlets in pump discharge lines are common sudden-expansion sites; monitor cavitation risk when pressure drop approaches vapor pressure.