# Heat Conduction Simulator - 1D Rod, 2D Plate, U-Value & Newton Cooling

> Simulate heat conduction: animated 1D rod (Crank–Nicolson) with real materials, paintable 2D plate, composite wall U-value calculator and Newton cooling fit.

- URL: https://qilingtools.com/en/math/heat-conduction
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Four classic heat-conduction problems in one place. On the 1D rod, pick copper, aluminium, steel, glass, wood, water or air, set each end to a fixed temperature, insulated or convective, and watch the colour bar and T(x) profile spread out while a draggable probe records T(t). On the 2D plate you paint heat sources, cold sinks and insulating walls with a finger or mouse and see the field flow around obstacles towards steady state. The composite-wall calculator adds up thermal resistances in series to give the U-value, heat loss and every interface temperature, with brick, insulation and glazing presets. Newton's law of cooling takes two readings and returns the cooling constant, so you can work out when your coffee is drinkable or back-calculate a forensic-style time.

## How to use

1. Choose the 1D rod, 2D plate, composite wall or Newton cooling panel
2. Set the material, boundary conditions and initial temperatures — or draw the initial profile, sources and insulators
3. Press play to watch heat diffuse and drag the probe to read T(t)
4. For walls and cooling, type your numbers and read the U-value, heat flux or predicted time

## Features

- 1D rod with explicit, Crank–Nicolson and fully implicit schemes (Thomas algorithm) and an analytic sine-mode check
- Fixed, insulated and convective boundaries with live end heat fluxes and the steady-state profile
- 2D plate solved with ADI; paint sources, sinks and insulators; banded isotherm colouring
- Composite wall: R = R_si + Σd/k + R_se, U-value, heat loss per day and interface temperatures
- Newton cooling fit from two readings, works across midnight, forwards or backwards in time
- Nine common materials with k, ρ, c and diffusivity; touch-friendly on phones

## FAQ

### Why does a copper rod warm through quickly while a wooden one takes ages?

How fast temperature changes depends on the thermal diffusivity α = k/(ρc), not on the conductivity alone. Copper has α ≈ 1.2×10⁻⁴ m²/s, pine about 1.7×10⁻⁷ m²/s — nearly a thousand times smaller. The time for heat to spread over a length L scales as L²/α, so doubling the length quadruples the time: a 20 cm copper rod settles within minutes, the same length of wood takes days.

### Why does the explicit scheme blow up, and what's the catch with Crank–Nicolson?

The explicit FTCS scheme needs the mesh ratio r = αΔt/Δx² ≤ ½ (¼ in 2D); beyond that, errors grow every step and the profile turns into a sawtooth that diverges, so the tool sub-steps automatically. Crank–Nicolson is unconditionally stable and second-order accurate, but with a large r and a sharp initial step it shows slowly decaying wiggles. Fully implicit never oscillates but is only first-order accurate in time.

### How is a wall's U-value calculated, and why does a thin insulation layer matter most?

In steady state the layers act like resistors in series: each has R = thickness/conductivity, plus inside and outside surface resistances (typically 0.13 and 0.04 m²·K/W for walls). U = 1/R_total and the heat flux is q = U·ΔT. The same flux passes through every layer, so each layer's temperature drop is proportional to its resistance: 80 mm of EPS (k ≈ 0.038) gives R ≈ 2.1, while 240 mm of brick gives only about 0.3, so the insulation takes most of the temperature difference.

### Is Newton's law of cooling good enough to estimate time of death?

Newton's law, dT/dt = −k(T − Te), holds for an object with a nearly uniform internal temperature (Biot number below 0.1) in constant surroundings; two readings fix k and let you extrapolate back to 37 °C. Real bodies break these assumptions — there is an early post-mortem temperature plateau, and body mass, clothing, humidity and air movement all matter — so forensic practice uses the Henssge nomogram with error bands. Treat the result here as a teaching example.

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