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Physics calculator

Stefan–Boltzmann Calculator

Compute net gray-body flux q = εσ(Th⁴−Tc⁴) and optional Q = A·q. σ from CODATA 2018. Runs locally. Not Fourier/Newton heat-transfer and not a view-factor solver.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Source checked · 12/12 tests · Production surface contract 4/4 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
Net Stefan–Boltzmann flux, and optional heat rate when area is given.
Scope
Gray body
Verification
Engine tested · 12/12 tests · Production surface contract 4/4 · Source checked · v1.0.0
Named expert review
Optional · Not performed
Sources
Sources
Evidence
1 golden · 6 boundary · 5 property · Production surface contract 4/4 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status PASS (0 stale; 164 CURRENT) @ 2026-09-16T06:44:24.909Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

Net fluxq = εσ(Th⁴ − Tc⁴)
Heat rateQ = A · q
iTemperatures in kelvin. Gray body with view factor F=1. This page is not conduction or convection.

How to use

1

Enter Th and Tc in kelvin

Both temperatures must be greater than zero. Net flux is signed: Th < Tc gives q < 0.

2

Enter emissivity

ε in [0, 1]. Leave 1 for a black body.

3

Optionally enter area

If A is given, the result also includes Q = A·q in watts.

Example calculations

Common configurations with formula and result.

ϟ

Black body 400 K to 300 K

ε=1, Th=400 K, Tc=300 K

q = σ(400⁴ − 300⁴)
q≈992.316 W/m²
ϟ

With area

same temperatures, A=2 m²

Q = 2 · q
Q≈1984.631 W

Stefan–Boltzmann calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Net Stefan–Boltzmann flux: q = εσ(Th⁴−Tc⁴), with σ = 5.670374419×10⁻⁸ W/(m²·K⁴). Optional heat rate Q = A·q.
What it calculates
Net Stefan–Boltzmann flux, and optional heat rate when area is given.
Inputs
  • Th
  • Tc
  • epsilon?
  • A?
Outputs
  • q_W_m2
  • Q_W?
  • Th
  • Tc
  • epsilon
  • sigma
Formula
q = εσ(Th⁴−Tc⁴)
Assumptions
  • Gray body
  • View factor F=1
  • T in kelvin
  • σ = 5.670374419e-8 (CODATA 2018)
  • Not Fourier conduction or Newton cooling
  • Not heatsink Rth or steam
Units
  • T in kelvin
  • q in W/m²
  • Q in W
  • A in m²
Boundary conditions
  • missing Th or Tc → MISSING_REQUIRED_INPUT
  • Th ≤ 0 or Tc ≤ 0 → VALUE_MUST_BE_POSITIVE
  • epsilon outside [0, 1] → VALUE_OUT_OF_RANGE
  • A given and A ≤ 0 → VALUE_MUST_BE_POSITIVE
Example
Th=400 Tc=300 ε=1 → q=σ(400⁴−300⁴)
Validation cases

3 published on this page · 12/12 tests · Production surface contract 4/4 · View evidence

  • Th=400 Tc=300 → q≈992.315523325 W/m²
  • Th=400 Tc=300 A=2 → Q≈1984.63104665 W
  • Th=0 Tc=300 → VALUE_MUST_BE_POSITIVE
Sources
  • CODATA 2018 — Stefan–Boltzmann constant — σ
    Supports: σ = 5.670374419×10⁻⁸ W m⁻² K⁻⁴
  • ISO 80000-5 — Thermodynamics — Heat flux / thermal radiation
    Supports: Net radiative flux proportional to T⁴ differences
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Compute net thermal radiation from Stefan–Boltzmann.

Supported and not supported

Supported — net flux q = εσ(Th⁴−Tc⁴); optional Q = A·q · API physics.thermo.stefan_boltzmann

Not supported — view factors, spectrally selective surfaces, Fourier conduction, Newton cooling, heatsink Rth, steam

Agent / API notes

Capability id: physics.thermo.stefan_boltzmann · tool id: stefan-boltzmann · pin 1.0.0.

{ "Th": 400, "Tc": 300, "epsilon": 1 }

With area: { "Th": 400, "Tc": 300, "A": 2 }. Errors: MISSING_REQUIRED_INPUT, INVALID_NUMBER, VALUE_MUST_BE_POSITIVE, VALUE_OUT_OF_RANGE.

Calculator URL stays /calc/physics/stefan-boltzmann. There is no /calc/thermal.

Other calculators in this family: Heat transfer, Specific heat .

Q Heat transfer › c Specific heat ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this the heat-transfer calculator?

No. /calc/physics/heat-transfer is Fourier conduction and Newton cooling. This page is net thermal radiation only.

Does this include a view factor?

No. F = 1. Two-surface enclosure geometry is out of scope.

Must Th be greater than Tc?

No. Both must be positive kelvin. If Th < Tc the net flux is negative.

Where does this run?

Locally in the browser by default. REST and MCP call the same thermal-engine Stefan–Boltzmann wrapper.