Bernoulli equation calculator
Water at 101325 Pa flowing at 2 m/s, 10 m up, reaches 144358.25 Pa at 4 m/s and 5 m: the drop in height adds more pressure than the faster flow removes. Every step is shown for homework and pipe estimates. Free in your browser.
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Worked example
- Fluid density ρ
- 1,000
- V1
- 2
- V2
- 4
- Elevation of point 1
- 10
- Elevation of point 2
- 5
- P1
- 101,325
- Rho
- 1,000
- Pressure1
- 101,325
- P2
- 144,358.25
- Pressure2
- 144,358.25
A Bernoulli equation calculator turns density, two flow speeds, two elevations, and one known static pressure into the missing pressure along a streamline. Gravity defaults to standard Earth 9.80665 metres per second squared so SI pipe problems work without an extra field, while you may pass another g when the height unit or planetary field differs. Homework checks, exam prep, coding-kata expected values, and quick pipe-network estimates all need that same rearrange without opening a full CFD package.
What Bernoulli's equation means between two points
Bernoulli's equation for steady, inviscid, incompressible flow along a streamline states that the sum of static pressure, dynamic pressure one-half ρ v squared, and hydrostatic term ρ g h is the same at every point on that streamline. Written between stations one and two it reads P one plus one-half ρ v one squared plus ρ g h one equals P two plus one-half ρ v two squared plus ρ g h two.
The density multiplies both the speed-squared term and the elevation term, so a consistent unit system is mandatory: with ρ in kilograms per cubic metre, speeds in metres per second, heights in metres, and g in metres per second squared, pressures come out in pascals. Speeds enter only as squares, so the dynamic contribution is always non-negative for real speeds; elevations are measured from a shared datum and may be negative if a station sits below the reference.
When speed rises in a constriction the dynamic term grows and static pressure falls if elevation is fixed—the classic venturi intuition—while a drop in elevation can raise pressure even when speed is unchanged because the hydrostatic term transfers into P. This tool assumes the ideal Bernoulli conditions: no shaft work, no heat addition, negligible viscous losses, and constant density. Real pipe networks add friction head losses and fittings; use this calculator for textbook streamline problems and first-order estimates, not as a substitute for a full energy equation with losses.
The check path recomputes both Bernoulli totals with the same fixed rounding used for published fields, so a successful response always ships with steps you can assert in tests or worksheets.
Practical tips, limits, and related searches
Keep density non-negative and inputs inside the published absolute range so products such as one-half ρ v squared and ρ g h cannot overflow into a non-finite pressure that the solver must reject. Prefer sending numbers rather than pre-formatted strings when you control the client; string numbers are accepted only when they match a strict finite numeric pattern. Use a single pressure unit system end to end: if you mix gauge and absolute values without converting, the algebraic answer is still consistent but may not match field gauges.
Round-trip checks are easy: recompute the rearranged expression by hand and confirm it equals the published unknown within the same twelve-decimal rounding, and confirm that the two Bernoulli totals agree for a fully consistent rounded state.
What you can do with it
Homework checks for two-point Bernoulli
Students enter density, speeds, heights, and one pressure from textbook streamline problems and compare hand-written P₁ or P₂ against a deterministic result before submitting.
Quick pipe and venturi estimates
Estimate the missing static pressure at a constriction or elevation change when ideal Bernoulli conditions are a fair first approximation.
FAQ
What form of Bernoulli's equation does this use?
The two-point steady form P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ for incompressible, inviscid flow along a streamline. Given one pressure, it solves for the other.
What happens if density is negative?
Negative fluid density is not physical for this model and is never accepted.
Do I need both pressures?
No. Provide exactly one of p1 or p2 and omit the unknown. Providing both or neither is a validation error.
What gravity value is used by default?
g defaults to 9.80665 (standard Earth gravity in SI). Pass g explicitly for other fields or unit systems.
Is the Bernoulli equation calculator free?
The browser tool is free and runs locally.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/thermo/bernoulli-pressure \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"density":1000,"v1":2,"v2":4,"h1":10,"h2":5}'const res = await fetch("https://api.kit.forhosting.com/thermo/bernoulli-pressure", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"density": 1000,
"v1": 2,
"v2": 4,
"h1": 10,
"h2": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/thermo/bernoulli-pressure",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"density": 1000,
"v1": 2,
"v2": 4,
"h1": 10,
"h2": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/thermo/bernoulli-pressure", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"density":1000,"v1":2,"v2":4,"h1":10,"h2":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"density":1000,"v1":2,"v2":4,"h1":10,"h2":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/thermo/bernoulli-pressure", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"density": 1000,
"v1": 2,
"v2": 4,
"h1": 10,
"h2": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "thermo.bernoulli_pressure",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Limits
max_abs | 1000000000000 |
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Negative density is rejected with a clear invalid_input error because a negative ρ is not physical for this model.
What Bernoulli's equation means between two points
Teachers and automated graders care about the negative-density rejection as much as the success path: a learner who accidentally flips the sign of ρ should see an explicit invalid_input message instead of a nonsense negative dynamic pressure scale.
How to call the API and what comes back
Density must not be negative; gravity must be non-zero when provided so the elevation term is well defined. Send a JSON body with required finite numbers density (aliases rho, fluid_density), v1 (aliases speed1, velocity1, u1), v2 (aliases speed2, velocity2, u2), h1 (aliases height1, z1, elevation1), and h2 (aliases height2, z2, elevation2), plus exactly one of p1 (aliases pressure1, P1) or p2 (aliases pressure2, P2). Optionally include g (aliases gravity, g_accel, gravitational_acceleration); when omitted the solver uses 9.80665. Absolute values must stay within the published limit of one trillion so intermediate products remain finite in IEEE double precision on the Worker and in the browser. On success the response echoes rounded density, v1, v2, h1, h2, p1, p2, and g, mirrors them as rho, pressure1, pressure2, and gravity, sets solved_for and unknown to p1 or p2, formula and rearrange to the algebraic form used, equation to a filled two-point Bernoulli string, dynamic_pressure_1 and dynamic_pressure_2 to one-half ρ v squared at each station, hydrostatic_1 and hydrostatic_2 to ρ g h at each station, bernoulli_total_1 and bernoulli_total_2 to the full P plus dynamic plus hydrostatic sums, message to a short English summary, and steps to an ordered array of substitution lines. Failed validation—missing fields, both or neither pressure, non-finite numbers, values outside absolute limits, negative density, or zero g—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item and zero marginal unit price.
Practical tips, limits, and related searches
When teaching, pair examples: water at one thousand kilograms per cubic metre with known P one, a speed increase from two to four metres per second, and a height drop from ten to five metres to solve for P two; reverse by omitting P one instead; deliberately send a negative density to demonstrate the hard invalid_input path; and compare bernoulli_total_1 with bernoulli_total_2 after rounding to show conservation of the mechanical energy density. This landing targets people searching for a Bernoulli equation calculator, pressure from fluid velocity and height, or pipe-flow pressure drop online; it is not a compressible-flow isentropic nozzle tool, not a head-loss friction calculator, and not a multi-stream junction solver. Prefer this capability when you already have density, two speeds, two elevations, and one pressure under ideal Bernoulli assumptions; if viscous losses dominate, add a head-loss term outside this API. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when learners deliberately send bad density.
Integration use cases
Unit tests for student fluid solvers
Generate known-good rearrangements for each unknown pressure plus deliberate negative-density rejections for coding kata expected values.