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Explore a homogeneous 1D rod with constant thermal diffusivity using explicit finite differences. Compare fixed-temperature, insulated and right-end convective boundaries while following the physical time. Composite materials, internal heat generation, side cooling, radiation and phase change are not modeled.
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Why do different diffusivities give similar temperatures after the same number of steps?
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The automatic time step changes with diffusivity. With the same grid and boundaries, doubling α halves the physical time represented by each step. At the same physical time, the higher diffusivity spreads heat further. Compare seconds, not just step counts.
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Does a 500mm rod with requested spacing 2mm reach the actual end?
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Yes: 251 nodes span 0–500mm. When the requested spacing does not divide the length, the interval count is rounded up and the actual spacing becomes L divided by that count. For L=200mm and requested spacing 7mm, 29 intervals give about 6.897mm. The input remains the request; the grid note shows the actual spacing.
Model and boundary conditions
Interior nodes use forward Euler with central spatial differences. Insulated and convective endpoints use half-cell heat balances, with no external heat added at an insulated end. Right-end convection has fixed full-length Bi=10; the grid coefficient is b=Bi·Δx/L.
The interval count is ceil(L/requested spacing), and actual spacing is L divided by that count. The full length is preserved. Supported inputs use 2–251 nodes. Finer grids resolve gradients better but require more time steps.
There is no manual time-step slider. Fixed and insulated boundaries use r=0.45; right convection uses r=0.45/(1+b), where b=10·Δx/L. The solver sets Δt=r·Δx²/α. The step stability number r is different from elapsed-time Fo=αt/L²; Fo does not need to stay below 0.5.
T_init specifies a uniform initial temperature. Choose boundaries with the buttons. Fixed endpoints take their specified temperature from the start. An insulated end ignores its endpoint-temperature setting. Right convection uses T_R as ambient T_∞. There is no per-position initial-temperature drawing tool.
No separate h or conductivity k input is provided. Full-length Bi=hL/k is fixed at 10. This demonstrates convection at the rod end; it does not directly predict real cooling capacity or a fin with heat loss along its entire side.
The required duration depends on L, α, boundaries and initial temperature; neither 100 steps nor Fo=0.2 is a universal completion rule. Compare refined grids at the same physical time. A stable coarse grid can still be inaccurate. This constant-property educational model is not a safety-design certificate.
It shares inputs and both boundary choices. Opening the URL restarts at physical time zero; elapsed time and comparison curves are not transferred. During animation, up to eight tab-local profiles are recorded every 20 steps. Input changes and reset clear them.
They load the stated inputs and boundaries and advance a defined number of steps. The fixed-end example shows 675 seconds; the insulated/convective example shows 60 seconds. Displayed finite-difference approximations differ slightly from the continuum reference values in the explanation.
Two independently checkable examples
Fixed ends: α=15mm²/s, L=100mm, requested spacing 5mm, left 500°C, right 300°C and initial 300°C. With Δt=0.75s, 900 steps represent 675s. The continuum midpoint is about 399.994°C; compare with steady T=500−200x/L.
Insulated left, convective right: α=15mm²/s, L=100mm, requested spacing 5mm, initial 150°C, ambient 25°C and full-length Bi=10. With Δt=0.5s, 120 steps represent 60s. Continuum temperatures are about 147.233°C at the left, 129.051°C at the midpoint and 47.375°C at the right.
Before practical use
Enter diffusivity in mm²/s, lengths in mm and temperature in °C. Material properties depend on temperature and composition. Presets are interaction examples, not certified material data. The high-temperature weld preset is a hypothetical case without melting or latent heat. Building and heat-treatment applications illustrate modeling concepts; composite walls, radiation and safety thresholds require additional models.
Use a preset or the numeric controls to set α(mm²/s), length L(mm), requested spacing(mm) and initial temperature.
Select left and right boundary buttons and set fixed or right-end ambient temperatures. Check actual spacing and time step in the grid note.
Start/pause the animation or load a checkable example. Match physical time and Fo when comparing temperatures.
Two independently checkable examples
Fixed ends: α=15mm²/s, L=100mm, requested spacing 5mm, left 500°C, right 300°C and initial 300°C. With Δt=0.75s, 900 steps represent 675s. The continuum midpoint is about 399.994°C; compare with steady T=500−200x/L.
Insulated left, convective right: α=15mm²/s, L=100mm, requested spacing 5mm, initial 150°C, ambient 25°C and full-length Bi=10. With Δt=0.5s, 120 steps represent 60s. Continuum temperatures are about 147.233°C at the left, 129.051°C at the midpoint and 47.375°C at the right.
Interpreting results
Separate step stability r from elapsed-time Fo. The convective endpoint needs a stricter time-step condition than interior nodes.
Reset and input changes restart time and history. Shared URLs also start from the initial state.
Agreement with a steady solution does not prove transient accuracy; check grid convergence and the appropriateness of initial and boundary conditions.