Nozzle exit velocity calculator
Estimate how fast liquid leaves a tank opening: with 5 m of head, Torricelli's rule gives about 9.90 m/s. Add a velocity coefficient such as 0.97 to see the slightly slower real jet beside the ideal one. Free in your browser.
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Worked example
- Head
- 5
- Gravity
- 9.80665
- Velocity coefficient
- 1
- Ideal exit velocity
- 9.9028531242
- Exit velocity
- 9.9028531242
Tank drains, fire-hose nozzles, and laboratory orifices all need a first estimate of how fast liquid leaves an opening under a known free-surface head. Torricelli’s theorem gives that ideal speed without a full CFD model: exit velocity equals the square root of twice gravity times head, the same speed a free-falling drop would reach after falling distance h. This nozzle exit velocity calculator accepts upstream head, optional gravitational acceleration, and an optional velocity coefficient Cv, then returns both the ideal Torricelli speed and the Cv-adjusted exit velocity in consistent length-per-time units. Students check homework against textbook idealizations; plant engineers size relief drains and calibration jets before applying discharge coefficients for volume flow.
Torricelli’s theorem and what ideal nozzle exit velocity means
Torricelli’s theorem is the energy balance for an incompressible inviscid liquid that leaves a small opening under a free surface open to the same ambient pressure. Along a streamline from the free surface (nearly stagnant) down to the vena contracta of the jet, pressure is atmospheric at both ends, elevation drops by head h, and kinetic energy appears as ½ρv². Equating mechanical energy densities yields v equals the square root of two g h—the ideal efflux or nozzle exit velocity.
The same result follows from Bernoulli’s equation when surface velocity is neglected, densities cancel, and the jet discharges to atmosphere. Head h is measured vertically from the free surface down to the centreline of the nozzle or orifice; it is a length, not a gauge pressure, though you can convert a known pressure head as p over ρg if your tank is pressurized and you still want an equivalent Torricelli form.
Gravity g must match the length unit of h: use about 9.80665 m/s² when h is in metres, or 32.174 ft/s² when h is in feet. Ideal velocity ignores wall friction, approach velocity in a finite tank, vena-contracta area reduction, and non-uniform jet profiles; those effects are usually folded into a velocity coefficient Cv near 0.97 to 0.99 for sharp orifices or into a discharge coefficient when you later compute volume flow Q equals Cd A √(2gh). Zero head correctly yields zero exit speed.
Negative head is rejected because it would not describe a free surface above the opening.
Where an exit speed estimate helps
Process and water engineers estimate how fast liquid leaves a tank bottom nozzle, side orifice, or spray lance before specifying pipe size, jet throw, or fill time. Fire-protection and irrigation designers compare ideal Torricelli speeds against nozzle charts when converting residual pressure head at a hydrant or lateral into an expected jet speed. University fluid-mechanics courses and exam generators need a deterministic oracle for √(2gh) that also accepts a Cv factor without students hand-coding floating-point edge cases for zero head.
Content sites that rank for nozzle exit velocity calculator, Torricelli theorem calculator, orifice velocity from head, or tank efflux speed benefit from a live free tool whose arithmetic matches the paid automation path byte-for-byte. Invalid input is not billed.
What you can do with it
Tank drain and orifice jet estimates
Compute ideal exit velocity from measured free-surface head above a bottom nozzle or side orifice before applying area and Cd for discharge.
Nozzle and spray-line design checks
Convert residual pressure head at a hydrant, irrigation lateral, or process lance into an expected jet speed for throw and sizing spreadsheets.
FAQ
How do I calculate nozzle exit velocity from head?
Send non-negative head h and optional gravity g (default 9.80665). Ideal exit velocity is √(2gh).
What is Torricelli’s theorem?
It states that ideal liquid speed leaving an orifice under free-surface head h equals √(2gh)—the same as free-fall impact speed from height h—when surface velocity and losses are neglected.
What units should I use for head and gravity?
Any consistent length and length/time² pair (metres with m/s², feet with ft/s²). Exit velocity is then length/time in that system. Convert pressure to head as p/(ρg) yourself if needed.
When should I change the velocity coefficient Cv?
Leave Cv at 1 for ideal Torricelli. Use a measured or catalogue Cv (often about 0.97–0.99) for real sharp orifices when you want a reduced exit speed without computing full discharge.
Does it also give the flow rate?
No, it gives the exit speed only. For volume flow, multiply that speed by your opening area and a discharge coefficient yourself. It does not follow a tank as it drains, and it does not cover gas nozzles.
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/eng/nozzle-velocity \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"head":5}'const res = await fetch("https://api.kit.forhosting.com/eng/nozzle-velocity", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"head": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/eng/nozzle-velocity",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"head": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/eng/nozzle-velocity", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"head":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"head":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/eng/nozzle-velocity", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"head": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "eng.nozzle_velocity",
"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.
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
Invalid input—missing head, negative head, non-positive gravity, or an out-of-range Cv—fails with invalid_input and is not billed.
Torricelli’s theorem and what ideal nozzle exit velocity means
The calculator returns ideal_exit_velocity as pure √(2gh) and exit_velocity as Cv times that value so you can compare textbook ideal jets with slightly reduced real nozzles without running two separate tools.
Input contract, response fields, units, and error behaviour
Numeric results are rounded to twelve decimal places for stable goldens and automation. Example: head five with default g yields exit speed near 9.90 in metres per second if head was in metres—because √(2 × 9.80665 × 5) is √98.0665. Pair exit velocity with your own orifice area and discharge coefficient if you need volumetric flow; this capability solves velocity only and does not integrate tank drawdown over time or model compressible gas nozzles. POST a JSON body with required head—a non-negative finite number for upstream liquid height above the nozzle or orifice centreline. Aliases include h, liquid_head, water_head, height, and orifice_head. Optional gravity (aliases g, g_accel, gravitational_acceleration) defaults to 9.80665 and must be strictly positive when supplied. Optional velocity_coefficient (aliases cv, Cv, coefficient_of_velocity, C_v) defaults to one for ideal Torricelli; it must be positive and not greater than two, covering ordinary engineering Cv values without inviting absurd amplification. Fields may arrive as JSON numbers or numeric strings. Missing head, non-finite values, negative head, non-positive gravity, or an out-of-range velocity coefficient fail with invalid_input naming the problem; those failures are not billed. On success the body echoes head, gravity, and velocity_coefficient (rounded), and returns ideal_exit_velocity equals square root of two g h, exit_velocity equals Cv times that ideal, a formula string, method set to torricelli, and a short human-readable message.
Where a nozzle exit velocity API earns its keep
Keep requests small, pin the rounded exit_velocity field in tests, and you get reproducible Torricelli speeds for LMS plugins, edge workers, and plant spreadsheets without standing up a full hydraulic solver.
Integration use cases
Homework and fluid-mechanics labs
Check Torricelli √(2gh) results with optional Cv against textbook problems using the same pure module as the free browser widget.
Technical questions
What does the nozzle exit velocity calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.