Fluid Velocity Calculator from Pressure

This calculator determines the theoretical velocity of a fluid flowing through an orifice or nozzle based on the pressure difference driving the flow. You enter the pressure drop across the opening and the fluid density; it returns the velocity in meters per second (or feet per second) using Bernoulli’s equation. The result is the ideal, frictionless velocity, which is a useful upper bound for real systems.

Formula

The calculator uses Torricelli’s law, a special case of Bernoulli’s equation for a fluid exiting a small opening:

v = √(2·ΔP / �)

• v = fluid velocity (m/s)
• ΔP = pressure difference across the opening (Pa, or N/m²)
• Ï� = fluid density (kg/m³)

Note that the calculator uses kPa for pressure input, so the formula includes a conversion factor: 1 kPa = 1000 Pa. The equation comes from Bernoulli’s principle, which states that for an incompressible, inviscid fluid, an increase in velocity occurs simultaneously with a decrease in pressure. When a fluid accelerates from a stagnant reservoir (velocity ≈ 0) to an orifice, the pressure energy converts to kinetic energy.

Worked Example

Using the default values: pressure drop = 100 kPa, fluid density = 1000 kg/m³ (about the density of water).

1. Convert pressure to pascals: 100 kPa × 1000 = 100,000 Pa
2. Multiply by 2: 2 × 100,000 = 200,000
3. Divide by density: 200,000 / 1000 = 200
4. Take the square root: √200 ≈ 14.14 m/s

The calculator shows 14.14 m/s (46.4 ft/s). This is the speed of water exiting a hole if there were no friction or turbulence.

How Pressure Relates to Velocity

Pressure and velocity in a moving fluid are linked by Bernoulli’s equation, which is essentially an energy balance. For a horizontal pipe with no height change, the sum of static pressure and dynamic pressure (½Ï�v²) remains constant. When fluid passes through a constriction, it speeds up and the static pressure drops. This is the principle behind venturi meters and carburetors.

In the case of a tank with a hole, the pressure at the hole is atmospheric, while the pressure at the top of the tank is higher (due to hydrostatic head or an applied gas pressure). The pressure difference accelerates the fluid. The velocity depends only on the pressure difference and density, not on the size or shape of the hole (for an ideal orifice).

How to Choose the Right Values

To get a realistic result, you need accurate inputs:

• Pressure drop: This is the difference between the upstream pressure (inside the tank or pipe) and the downstream pressure (usually atmospheric). For a tank open to the atmosphere, the pressure at the water surface is atmospheric, so the pressure at the hole is just the hydrostatic pressure: ΔP = Ï�·g·h, where h is the height of water above the hole.
• Fluid density: Use the actual density of your fluid at its operating temperature. Water is about 1000 kg/m³, but gasoline is around 750 kg/m³, and air is about 1.2 kg/m³ at room temperature. For air, the formula still applies but compressibility effects become significant at high pressure ratios.

If you are calculating flow from a pipe, measure the pressure drop across the orifice or nozzle using a pressure gauge. For a tank, measure the height of liquid above the outlet and compute the pressure drop as �·g·h.

Common Mistakes and Misconceptions

• Unit mix-ups: The most common error is using pressure in kPa but density in g/cm³. Always convert to consistent SI units (Pa and kg/m³) before calculating.
• Confusing pressure drop with absolute pressure: Use the difference between upstream and downstream, not the absolute pressure. If a tank is pressurized to 200 kPa gauge and discharges to atmosphere, ΔP = 200 kPa, not 200 + 101.3.
• Ignoring friction: The formula gives ideal velocity. Real pipes and orifices have friction, so actual velocity is 60-90% of the ideal value. This is captured by a discharge coefficient (Cd). For a sharp-edged orifice, Cd ≈ 0.62; for a well-rounded nozzle, Cd ≈ 0.97.
• Thinking velocity is independent of hole size: The velocity is the same for any size hole (for the same pressure), but the flow rate (volume per time) is velocity × area. A bigger hole gives more flow at the same velocity.

Limits of the Formula

Torricelli’s law assumes the fluid is incompressible, inviscid, and the flow is steady. It works well for liquids like water, but for gases the density changes with pressure, so the formula overestimates velocity at high pressure ratios. For gases, use the compressible flow equations (e.g., isentropic flow) instead. Also, the formula assumes the upstream velocity is negligible, which is true for a large tank but not for a pipe where the fluid is already moving.

Another limitation: the formula gives the velocity at the vena contracta, the narrowest point of the jet, which may be slightly downstream of the hole. For a sharp-edged orifice, the vena contracta is about 0.6 times the hole area, so the actual jet velocity is higher than the average velocity through the hole.

Practical Applications

This calculation is used in many engineering and everyday contexts:

• Designing sprinkler systems and drip irrigation
• Sizing relief valves and safety vents
• Estimating leak rates from pressurized tanks
• Calculating flow from a fire hose nozzle
• Determining the velocity of water exiting a dam’s spillway
• Understanding the operation of a venturi carburetor

In each case, the ideal velocity is a starting point. Engineers apply a discharge coefficient to account for real-world losses, and for long pipes they use the Darcy-Weisbach equation or Hazen-Williams formula, which consider friction over the pipe length.

Frequently Asked Questions

Why does the calculator show a velocity even if the pressure is low?

Because the formula gives the theoretical velocity for any pressure difference. Even a small pressure difference produces some velocity. For example, 1 kPa pressure drop on water gives √(2×1000/1000) = 1.41 m/s. In practice, very low pressures may not produce a steady jet due to surface tension and viscous effects.

Can I use this for air or other gases?

For low pressure drops (less than about 10% of absolute pressure), the incompressible formula is a good approximation. For higher pressure ratios, you need to account for compressibility. The calculator will give a value, but it will be an overestimate for gases at high pressure.

How do I convert the velocity to flow rate?

Multiply the velocity by the cross-sectional area of the opening. For a circular hole of diameter d, area = π(d/2)². Flow rate Q = v × A. Remember to use consistent units (m/s × m² = m³/s). For a 10 mm hole (0.01 m), area = π(0.005)² = 7.85×10�� m². With v = 14.14 m/s, Q = 0.00111 m³/s = 1.11 L/s.

What is a discharge coefficient and how do I use it?

The discharge coefficient (Cd) is the ratio of actual flow to ideal flow. It accounts for friction and turbulence. For a sharp-edged orifice, Cd ≈ 0.62; for a rounded nozzle, Cd ≈ 0.97. To get actual velocity, multiply the ideal velocity by Cd. So for a sharp-edged orifice with ideal velocity 14.14 m/s, actual velocity ≈ 8.77 m/s.

Does the shape of the opening affect the velocity?

In the ideal formula, no. But in reality, a rounded nozzle produces a higher velocity than a sharp-edged hole because it reduces turbulence and vena contracta effects. The discharge coefficient captures this difference. For the same pressure drop, a well-rounded nozzle will have a higher actual velocity than a sharp hole.

What if the fluid is not water?

You need to know the density of your fluid. The formula uses density directly, so a lighter fluid (like oil) will have a higher velocity for the same pressure drop. For example, with a pressure drop of 100 kPa and a density of 800 kg/m³, the velocity would be √(2×100000/800) = 15.81 m/s, compared to 14.14 m/s for water.

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in p | Pressure drop | kPa | 100 | 0
in rho | Fluid density | kg/m³ | 1000 | 0
out v | Velocity | m/s | sqrt(2 * p * 1000 / rho) | 2
out v_fps | Velocity | ft/s | sqrt(2 * p * 1000 / rho) * 3.28084 | 2