What does the heat diffusion equation simulator calculate?

This tool numerically solves the 1D transient heat conduction equation ∂T/∂t = α ∂²T/∂x² using the FTCS (Forward-Time Central-Space) explicit finite difference method. It computes how the temperature distribution evolves over time in a 1D domain such as a rod or wall, given thermal diffusivity α, length, boundary conditions, and an initial temperature profile.

How do I use the heat diffusion simulator?

Select a material preset (steel, aluminum, copper) or manually enter the thermal diffusivity α in the left panel. Set the rod length, number of grid nodes, boundary conditions (fixed temperature or insulated), and initial temperature distribution. Press Play to start the simulation. Use the time scrubber to inspect the temperature profile at any time step, and view the heatmap for a complete overview of spatial and temporal temperature change.

What is the FTCS stability condition (Courant number)?

The FTCS explicit method is conditionally stable. The stability criterion is the diffusion Courant number r = α·Δt/Δx² ≤ 0.5. Exceeding this limit causes the numerical solution to diverge. This tool displays r in real time and warns when the condition is violated. To restore stability, decrease the time step Δt or increase the grid spacing Δx.

What are practical engineering applications of the heat diffusion equation?

The heat diffusion equation is used in heat treatment process design (quenching, welding, casting), transient thermal analysis of electronic circuit boards, non-steady thermal response of building envelopes, and fire safety engineering. The FTCS method is ideal for concept verification and education; practical designs typically use implicit methods (Crank-Nicolson) or finite element software.

Heat Diffusion Equation Simulator — Free Online Calculator

Material & Geometry

Thermal Diffusivity α 1.00×10&supmin;&sup5;

m²/s — Iron≈8e-6, Aluminum≈8e-5, Stainless Steel≈4e-6

Rod Length L 0.10 m

Boundary Conditions

Left-End Temperature T_L 0 °C

Right-End Boundary Fixed Temp.

Right-End Temperature T_R 100 °C

Initial Condition

Initial Temperature Profile

Solver Settings

Stable (r = 0.45)

t = 0.0 s

Speed:

🌡 Temperature Heatmap (Rod Cross-Section) t = 0.000 s

        x = 0 m (left end)
        x = 0.05 m
        x = 0.10 m (right end)
      

📈 Temperature Profile T(x, t) t = 0.000 s

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Max Temp T_max

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Min Temp T_min

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Avg Temp T_avg

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Left-End Heat Flux q″

⏱ Center-Point Temperature History T(L/2, t) Time Series

Numerical Method Notes

$$\frac{T_i^{n+1} - T_i^n}{\Delta t} = \alpha \frac{T_{i+1}^n - 2T_i^n + T_{i-1}^n}{\Delta x^2}$$

Uses the explicit (FTCS) scheme. Stability condition: diffusion number r = α·Δt/Δx² ≤ 0.5. Spatial nodes N = 100; the time step is automatically set from the stability condition (r = 0.45). Boundary conditions: Dirichlet (fixed temperature) or Neumann (insulated: first-order ghost-node method).