Pipe flow continuity calculator
A pipe narrowing from 0.02 to 0.01 m² with water entering at 3 m/s pushes it out at 6 m/s, carrying 0.06 m³/s throughout. Narrower sections speed the flow up, wider ones slow it down. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Inlet cross-sectional area A1
- 0.02
- Inlet flow speed v1
- 3
- Outlet cross-sectional area A2
- 0.01
- V2
- 6
- Flow rate
- 0.06
The continuity equation is the steady mass-balance statement that, for an incompressible fluid, volumetric flow rate is conserved along a streamline or through a duct: inlet area times inlet speed equals outlet area times outlet speed. In symbols, A1 · v1 = A2 · v2, so the unknown outlet speed is simply v2 = (A1 · v1) / A2. Narrower sections speed the fluid up; wider sections slow it down. This calculator takes three numbers—inlet cross-sectional area, inlet mean speed, and outlet cross-sectional area—and returns the outlet speed, the conserved volumetric flow rate Q = A1·v1, and a clear formula tag. Zero area is rejected because a vanished cross-section makes the product degenerate or divides by zero.
How the continuity equation relates inlet and outlet speed
For a steady stream of incompressible fluid through a pipe, hose, or duct, the volume of fluid that enters a control volume per unit time must leave at the same rate. When density is constant, that statement collapses to equality of volumetric flow rates at two stations: Q1 equals Q2, written A1 · v1 = A2 · v2. Here A is the flow cross-sectional area perpendicular to the mean velocity, and v is the area-averaged speed.
Rearrangement gives the outlet speed v2 = (A1 · v1) / A2. Halving the outlet area doubles the outlet speed for a fixed inlet condition; doubling the outlet area halves it. Areas may be square metres, square centimetres, or square inches as long as A1 and A2 share the same unit; speeds may be metres per second, centimetres per second, or feet per second as long as v1 and the reported v2 are interpreted in that same speed unit.
The calculator does not convert between unit systems, does not introduce density or mass flux for compressible gas, does not apply Bernoulli’s pressure-head relation, and does not account for friction, fittings, or multiphase slip. It answers only the algebraic continuity definition that secondary-school physics, first-year fluid mechanics, and HVAC sizing courses use for incompressible pipe flow. Zero area is the critical validation edge: an inlet or outlet area of zero is rejected with invalid input instead of producing a meaningless infinite speed or a degenerate product.
Where pipe-flow continuity calculations show up in real work
Secondary-school physics and introductory fluid-mechanics courses constantly assign problems that give two areas and one speed and ask for the missing station speed, often for garden hoses, syringe plungers, wind-tunnel contractions, or blood-vessel sketches under the incompressible idealization. University thermo-fluids labs use the same three-number reduction before they introduce Bernoulli, head loss, or compressible mass continuity with density ratios. Process technicians and HVAC trainees sanity-check duct transitions when only a continuity bound is needed before fan curves or friction charts enter the analysis.
If you later need Bernoulli pressure from height and speed, compressible mass flow with density, hydraulic diameter conversion, or Reynolds-number estimation, treat those as separate search intents and landings rather than overloading this pipe-flow continuity calculator page.
What you can do with it
Homework and lab answer keys
Solve A1·v1=A2·v2 for outlet speed so teachers and auto-graders share one ground truth, including the zero-area rejection.
Pipe and duct transition checks
Estimate how much a contraction speeds the fluid up—or a diffuser slows it down—from measured or design areas and inlet speed.
FAQ
What formula does this pipe flow continuity calculator use?
A1 · v1 = A2 · v2 for steady incompressible flow.
What happens if area is zero?
Zero area (a1 or a2) is not allowed under the continuity equation in this calculator.
Can inlet speed be zero?
Yes. A zero inlet speed yields zero outlet speed and zero flow rate. Only areas must be non-zero.
What units should I use?
Any consistent area unit for a1 and a2, and any consistent speed unit for v1 (v2 is returned in that same speed unit). The tool does not convert units; convert before calling if your sheet mixes systems.
What does it cost?
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/continuity-equation \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a1":0.02,"v1":3,"a2":0.01}'const res = await fetch("https://api.kit.forhosting.com/thermo/continuity-equation", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a1": 0.02,
"v1": 3,
"a2": 0.01
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/thermo/continuity-equation",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a1": 0.02,
"v1": 3,
"a2": 0.01
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/thermo/continuity-equation", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a1":0.02,"v1":3,"a2":0.01}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a1":0.02,"v1":3,"a2":0.01}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/thermo/continuity-equation", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a1": 0.02,
"v1": 3,
"a2": 0.01
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "thermo.continuity_equation",
"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
What you send and what you get back
Send three finite numbers under the fields a1, v1, and a2. a1 is the inlet cross-sectional area in any consistent area unit and must be non-zero. v1 is the inlet mean speed in any consistent speed unit; zero speed is allowed and yields zero outlet speed under continuity. a2 is the outlet cross-sectional area in the same area unit as a1 and must be non-zero, because the engine divides by A2. Classic textbook checks include a pipe that narrows from two square centimetres to one square centimetre at three metres per second inlet, recovering six metres per second at the throat, or a diffuser that widens and recovers a proportionally lower exit speed. Missing fields, non-numeric strings, NaN, and Infinity fail with invalid_input and are not charged. On success the response echoes cleaned a1, v1, and a2, reports v2 as (A1 · v1) / A2, reports flow_rate as the conserved product A1 · v1 (volumetric flow rate Q), and includes the formula string A1·v1 = A2·v2 plus method continuity_equation. Pricing for successful API calls is the published base of $0.002 per item; validation failures do not consume prepaid balance.
Where pipe-flow continuity calculations show up in real work
Because this capability is a pure parse engine, API pricing is the published base of $0.002 per successful item and failed validation does not consume balance.
Integration use cases
Curriculum widgets and API fixtures
Drive browser demos and CI golden tests from the same isomorphic solver so classroom tools never disagree with backend grading.