Heat conduction calculator
A 12 m² wall 0.2 m thick, with a conductivity of 0.8 W/(m·K) and 25 K between its faces, lets 1200 W of heat through. Enter the four values to get the steady heat flow and each substitution step. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- K
- 0.8
- A
- 12
- Wall thickness L
- 0.2
- Dt
- 25
- Thermal conductivity
- 0.8
- Conductivity
- 0.8
- Area
- 12
- A
- 12
A heat conduction calculator turns thermal conductivity, wall area, thickness, and temperature difference into the steady heat transfer rate predicted by Fourier’s one-dimensional plane-wall law. Enter k for the material, A for the face area normal to the flux, L for wall thickness, and ΔT for the temperature drop across the wall; the tool returns Q equals k times A times ΔT over L, a readable equation string, substitution steps, and mirrored field names for scripts and worksheets. Homework checks, building-envelope estimates, insulation sizing demos, and coding-kata expected values all need that same quotient without opening a full finite-element thermal stack.
What Fourier’s plane-wall law says about heat transfer rate
Thermal conductivity k is a material property—about zero point six to zero point seven watts per metre-kelvin for liquid water near room temperature, far higher for metals, far lower for good insulators—so the same geometry and ΔT produce very different heat leaks. Area A is the wall face normal to the heat flux; thickness L is the path length of conduction through the solid.
Temperature difference ΔT is usually written as the hot-side temperature minus the cold-side temperature; a positive ΔT with positive k and A yields positive Q in the chosen sign convention, while a negative ΔT flips the reported flow direction without changing the physics magnitude. Classroom and automated-grader paths care about domain checks as much as the arithmetic: thickness must be strictly positive because the formula divides by L, and a zero or negative thickness is almost always a data-entry or modelling mistake.
Zero area or zero conductivity is allowed and yields zero heat rate, which matches “no path” or “perfect insulator” idealizations. The check path recomputes k times A times ΔT over L with the same fixed rounding used for published fields, so a successful response always ships with steps you can assert in tests or lab notebooks.
Practical tips, limits, and related searches
Keep k, A, L, and ΔT inside the published absolute range so the product and quotient cannot overflow into a non-finite heat rate 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 consistent units: watts per metre-kelvin, square metres, metres, and kelvin (or Celsius difference) yield watts; imperial inputs need a coherent unit set if you want BTU per hour.
This tool does not convert units, stack multi-layer R-values, add convection or radiation resistances, or look up material properties from names—those need a composite-wall model or a material table. Prefer this capability when you already have k, A, L, and ΔT and only need Q; if your source is a word problem, extract those four quantities first.
What you can do with it
Pre-flight wall heat-loss estimates in engineering scripts
Pipelines compute steady conduction through a known slab before comparing against HVAC capacity or insulation upgrade branches.
FAQ
What formula does this heat conduction calculator use?
Steady one-dimensional Fourier conduction through a plane wall: Q = k · A · ΔT / L, where k is thermal conductivity, A is area, ΔT is temperature difference, and L is thickness.
Can ΔT be negative?
Yes. A negative temperature difference yields a negative heat transfer rate, which means heat flows opposite the direction you chose as positive.
Does this handle multi-layer walls or convection?
No. It is a single homogeneous slab with constant k. Composite walls, contact resistance, convection, and radiation need other models.
Is the heat conduction calculator free?
The browser tool is free and runs locally.
Can I choose the wall material by name?
No. You enter the thermal conductivity yourself, taken from a table or a datasheet. Metals conduct far more than good insulators, so the same wall and temperature difference can give very different heat flows.
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/thermal-conduction \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"k":0.8,"a":12,"l":0.2,"dt":25}'const res = await fetch("https://api.kit.forhosting.com/thermo/thermal-conduction", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"k": 0.8,
"a": 12,
"l": 0.2,
"dt": 25
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/thermo/thermal-conduction",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"k": 0.8,
"a": 12,
"l": 0.2,
"dt": 25
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/thermo/thermal-conduction", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"k":0.8,"a":12,"l":0.2,"dt":25}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"k":0.8,"a":12,"l":0.2,"dt":25}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/thermo/thermal-conduction", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"k": 0.8,
"a": 12,
"l": 0.2,
"dt": 25
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "thermo.thermal_conduction",
"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
Zero thickness is rejected with a clear invalid_input error so a missing depth never silently becomes infinity or a divide-by-zero crash.
What Fourier’s plane-wall law says about heat transfer rate
Fourier’s law for steady conduction through a flat slab of constant thermal conductivity states that the heat transfer rate Q is proportional to conductivity, area, and temperature difference, and inversely proportional to thickness: Q equals k times A times ΔT divided by L.
How to call the API and what comes back
Send a JSON body with four required finite numbers: k (aliases thermal_conductivity, conductivity, lambda), a (aliases area, A, surface_area), l (aliases thickness, L, length, delta_x, dx), and dt (aliases delta_t, dT, temperature_difference, deltaT, temp_diff). Absolute values must stay within the published limit of one trillion so intermediate products and the quotient remain finite in IEEE double precision on the Worker and in the browser. Thermal conductivity and area must be non-negative; thickness must be strictly positive—zero or non-positive L triggers invalid_input mentioning thickness. On success the response echoes rounded k, a, l, and dt under primary and alias keys, sets q, heat_transfer_rate, and Q to the same rounded heat rate, formula to the string Q equals k times A times ΔT over L, equation to a human-readable substituted quotient, law to Fourier, message to a short English summary, and steps to an ordered array of given-values and substitution lines. Failed validation—missing fields, non-finite numbers, values outside absolute limits, negative k or area, zero thickness, or a non-finite heat rate—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: k equals zero point eight, A equals twelve square metres, L equals zero point two metres, and ΔT equals twenty-five kelvin yields one thousand two hundred watts; a deliberate L equals zero demonstrates the hard invalid_input path. This landing targets people searching for a heat conduction calculator, thermal conductivity wall heat transfer rate, or Fourier law Q equals k A ΔT over L tool online; it is not a CFD package, not a U-value building code checker, and not a unit converter. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when learners deliberately send zero thickness. Round-trip checks are easy: recompute k times A times ΔT divided by L by hand and confirm it equals the published heat_transfer_rate within the same twelve-decimal rounding.
Integration use cases
Homework checks for Fourier conduction problems
Students enter textbook k, A, L, and ΔT, then compare hand-written Q and substitution steps against a deterministic API or browser result before submitting.
Unit tests for student thermal solvers
Generate known-good Q = k A ΔT / L cases plus deliberate zero-thickness rejections for coding kata expected values and continuous-integration fixtures.
Technical questions
What happens when thickness is zero?
The formula would divide by zero. This tool returns invalid_input instead of infinity or a silent NaN.