Bernoulli equation calculator
Use it to estimate pressure at a bearing or spray nozzle that sits above or below the pump, or after a pipe size change.
P + ½ρv² + ρgh energy balance between two points. Solve for any one unknown, or provide every value for an energy-balance check.
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ — Solve for any one unknown.
How this calculator works
In plain words: between two points in the same line, the oil’s energy stays the same if nothing is lost on the way. That energy has three parts: its pressure, its speed, and its height. Lifting oil costs pressure, and so does speeding it up into a smaller pipe. Dropping down or slowing down gives pressure back.
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂, with ρ = specific gravity × 1,000 kg/m³ and g = 9.81 m/s². When you enter a flow rate and a pipe ID, the velocity at that point is flow ÷ pipe area. Pipe friction and losses in filters, fittings and valves are left out on purpose, so the answer is a best case. Method: NASA Glenn: Bernoulli’s equation.
Worked example
An ISO VG 68 hydraulic oil, Shell Tellus S2 MX 68, whose data sheet lists a density of 0.860 kg/l at 15 °C (Shell technical data sheet, PDF), so specific gravity 0.86. The pump puts out 40 psi at point 1. The oil is pumped 15 ft up to a gearbox, and the line steps from 1 in to ½ in ID. Flow is 5 GPM.
Inputs: P₁ 40 psi, h₁ 0 ft, pipe ID₁ 1 in · P₂ blank, h₂ 15 ft, pipe ID₂ 0.5 in · specific gravity 0.86 · flow 5 GPM.
Calculator output: P₂ = 34.04 psi (2.347 bar / 234.7 kPa) · ΔP (P₁ − P₂) 5.96 psi · v₁ 2.04 ft/s → v₂ 8.17 ft/s · Δh 15.00 ft · energy: P₁ 40.0 psi + KE₁ 0.02 psi + PE₁ 0.00 psi = P₂ 34.0 psi + KE₂ 0.39 psi + PE₂ 5.59 psi.
So the 15 ft lift takes 5.59 psi and the faster flow in the ½ in line takes about another 0.37 psi. A gauge at the gearbox will read less than 34 psi because of friction.
Frequently asked questions
Where do I get the specific gravity?
From the oil’s product data sheet. Look for density at 15 °C in kg/l (or g/ml): that number is, for practical purposes, the specific gravity. If the sheet gives kg/m³, divide by 1,000. For example, Shell Tellus S2 MX 68 lists 0.860 kg/l. The 0.87 already in the box is only a starting value; enter your oil’s figure.
Why does my pressure gauge read lower than this?
The calculator leaves out friction on purpose. Pipe walls, filters, fittings, elbows and valves all take pressure, so a real gauge reads lower than this answer. Treat the result as the best case.
When does ignoring friction matter most?
On long runs, in small lines, and with cold, thick oil, because that is where friction losses are largest. Check the flow with the Reynolds number tool before relying on this estimate. This page does not calculate friction loss.
Can it solve for something other than downstream pressure?
Yes. Leave exactly one of P₁, P₂, v₁ or v₂ blank and fill in the other three; a velocity counts as filled when you enter a flow rate and that point’s pipe ID. It solves the blank one. If you fill in all four, it checks the energy balance instead. Heights, pipe IDs and specific gravity are inputs only; it does not solve for them.