What the rod reference explains
A rod can carry heat along its length, lose heat from its surface, or store heat as its temperature rises. These are different physical questions. Lemonergy uses the retained inputs from this reference in four separate examples: steady conduction, surface convection, insulated heating and a lumped thermal transient.
The attributed source is MathWorks, Heat Conduction Through Iron Rod, specifically Examples 1–3 and the “Lumped Mass Rod Model” equation. The tables below reproduce the retained contract, not a newly inspected version of that webpage. Download the reference contract from the comparison campaign; its thermal.source section records the inputs, equations and printed answers.
Retained geometry, properties and boundaries
In the Basis column, S1 identifies that frozen thermal contract and its source attribution. Property values are held constant in these examples. No material grade, uncertainty or temperature-dependent property curve is established by this record.
| Quantity | Symbol | Value | Unit | Conditions and use | Basis |
|---|---|---|---|---|---|
| Rod diameter | D | 0.025 | m | Uniform cylindrical geometry | Retained input · S1 |
| Rod length | L | 0.2 | m | Axial length | Retained input · S1 |
| Density | ρ | 7800 | kg/m³ | Thermal storage cases | Retained input · S1 |
| Specific heat | c | 447 | J/(kg·K) | Thermal storage cases | Retained input · S1 |
| Thermal conductivity | k | 80.2 | W/(m·K) | Axial conduction and lumped model | Retained input · S1 |
| Convection coefficient | h | 32.1 | W/(m²·K) | Surface loss; fixed, not calculated from flow | Retained input · S1 |
| Base temperature | Tbase | 100 | °C | Conduction and lumped-model boundary | Retained input · S1 |
| Air temperature | Tair | 20 | °C | Convection and lumped-model boundary | Retained input · S1 |
| Initial rod temperature | T0 | 20 | °C | Transient initial condition | Retained input · S1 |
| Uniform surface temperature | Tsurface | 55 | °C | Convection-only calculation | Retained input · S1 |
| Conducted heat flow | Qconduction | 18 | W | Imposed steady axial heat flow | Retained input · S1 |
| Insulated heat input | Qheating | 18 | W | Constant input with no environmental loss | Retained input · S1 |
Use only the inputs belonging to the selected problem. Density and specific heat do not affect the steady conduction result. The 20 °C air boundary does not apply to its tip: that case prescribes heat flow and base temperature. Likewise, the convection example fixes the rod at 55 °C rather than predicting its cooling.
Equations and printed comparison values
Cross-sectional area is A = πD²/4; lateral surface area is As = πDL. Thermal resistance is L/(kA), and heat capacity is Cth = ρcAL. These quantities are derived from the retained inputs, not independent measurements.
| Quantity | Symbol | Value | Unit | Conditions and equation | Basis |
|---|---|---|---|---|---|
| Conduction tip temperature | Ttip | 8.5554 | °C | Tbase − Qconduction L/(kA); no side loss | Rounded printed answer · S1 |
| Lateral convection heat | Qsurface | 17.6479 | W | hAs(Tsurface − Tair); excludes end faces | Rounded printed answer · S1 |
| Insulated heating rate | dT/dt | 0.0526 | K/s | Qheating/Cth; uniform temperature | Rounded printed answer · S1 |
The contract allows an absolute difference of 0.00005 in each scalar's listed unit, half the last printed decimal. This numerical rounding allowance is not a material tolerance or measurement uncertainty. The solver stores absolute temperatures in kelvin; temperature differences have the same numerical magnitude in kelvin and degrees Celsius.
Lemonergy authors the linear insulated-temperature curve and exponential lumped response from the disclosed balances. They are not exported traces from another application. The 0–1 s steady observation windows and 0–100 s insulated window are authored test choices; the lumped comparison uses 0–2500 s to cover the retained source plot horizon. The single-mass model does not reproduce a spatial fin field or segmented rod.
Publication date and retained provenance
The publication date is unknown. Campaign notes record a source check on 30 September 2026; the frozen thermal contract also records a recheck on 1 October 2026. Those are historical provenance dates, not a claim that the live page was visited when this wiki article was edited.
No complete source-page copy or measured rod trace is retained, and the external application was not executed for this comparison. Documentation URLs can change. Use the frozen contract to reproduce the numerical problem, then follow the individual worked examples for software pictures, block inventories and fresh run instructions.