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.
Enter values to calculate.
Live graph
Interactive plot
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Net flux q = ε σ (Th^4 − Tc^4), optional Q = A q, + O3 mpmath tabulated flux.
- Input interpretation
- Enter values to calculate.
- Result
- —
- Verified scope
- Net flux q = ε σ (Th^4 − Tc^4), optional Q = A q, + O3 mpmath tabulated flux.
- Assurance
- Engineering
- Declared partition coverage
- PASS · 6/6 declared partitions (alias, area, awkward, zero-eps, default, invalid-domain) · Matrix
- Deferred
- Not conduction, not convection, and not a heatsink.
- Numerical scope
- O2: net Stefan–Boltzmann flux vs a separate-module identity (≤2 ULP). Not conduction and not a heatsink. ≤2 ULP vs O3 applies only to the published tabulated Stefan–Boltzmann vectors. Not conduction and not a heatsink.
- Known limitations
- Core CVP does not include live graph, viewport, or pointer interaction.
- Model
- Net Stefan–Boltzmann flux, and optional heat rate when area is given.
- Scope
- Gray body
- Verification
- Engine tested · Source checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Engineering assurance · Net flux q = ε σ (Th^4 − Tc^4), optional Q = A q, + O3 mpmath tabulated flux.· View Manifest · CVP overview · Specification
- Versions
- Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-25.o2-o3
- Verification revision
- 2026-09-25.o2-o3 · 1/1 property · digest f910815c6069
- Legacy regression
- 12/12 tests · Production surface contract 6/6
- Trust layers
- Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
- Reference
- O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
- Interfaces
- PASS · UI (SSR) / REST / MCP
- Supplemental domain review
- Not performed
- Named expert review
- Not performed
- CVP suite
- 4/4 golden · 6/6 CVP boundary · 5/5 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 6/6 CVP contract · Manifest
- Sources
- CODATA 2018 — Stefan–Boltzmann constant
- ISO 80000-5 — Thermodynamics
- Evidence
- 1 legacy golden · 6 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 5/5 invalid · Artifact integrity PASS
- Semantic contract
- PASS
Full verification
Formulas
Core equations used by this calculator.
How to use
Enter Th and Tc in kelvin
Both temperatures must be greater than zero. Net flux is signed: Th < Tc gives q < 0.
Enter emissivity
ε in [0, 1]. Leave 1 for a black body.
Optionally enter area
If A is given, the result also includes Q = A·q in watts.
Example calculations
Common configurations with formula and result.
Black body 400 K to 300 K
ε=1, Th=400 K, Tc=300 K
With area
same temperatures, A=2 m²
Stefan–Boltzmann calculator specification
Version 1.0.0 · Engine tested
- Engine tested 12/12 tests · Production surface contract 6/6
- Named expert review Not performed
- Calculation version 1.0.0
- 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 6/6 · 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 — σSupports: σ = 5.670374419×10⁻⁸ W m⁻² K⁻⁴
- ISO 80000-5 — Thermodynamics — Heat flux / thermal radiationSupports: Net radiative flux proportional to T⁴ differences
- CODATA 2018 — Stefan–Boltzmann constant — σ
- 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.
Related tools
Other calculators in this family: Boyle's Law Calculator, Charles's Law Calculator, Composite Wall Calculator, Heat Transfer Calculator, Ideal Gas Law Calculator, Specific Heat Capacity Calculator, Thermal Resistance Calculator . Explore all Thermodynamics.
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.