Solve ∂T/∂t = α∇²T using finite difference method in real time. Freely set boundary conditions, internal heat sources, and thermal diffusivity, then watch temperature distribution evolve in a Canvas animation.
Material properties
Thermal diffusivity alpha
Boundary conditions
Top edge T_top
°C
Bottom edge T_bottom
°C
Left boundary
Right boundary
Color map
Preset
Results
T_max
-
T_min
-
Fourier number Fo
0.00
State
Transient
Heat
0°C
Colorbar
100°C
Hover over the map to display temperature
Profile
History
Theory & Key Formulas
Governing equation for two-dimensional transient heat conduction with internal heat generation:
∂T/∂t = α(∂²T/∂x² + ∂²T/∂y²) + q̇/(ρcp)
Here α = k/(ρcp) is thermal diffusivity and q̇ is volumetric heat generation. The simulator advances the field with an explicit finite-difference update.
TemperatureDiverged. Temperature, NaN. SimulatorStable $\Delta t \leq (\Delta x)^2/(4\alpha)$ from , α $\Delta t$ 1AnimationFre . . CAE「」 .
Frequently Asked Questions
The explicit method (forward difference) directly computes the next time step from the current values. It is easy to implement but has a stability condition $\Delta t \leq (\Delta x)^2/(4\alpha)$, requiring very small time steps for fine meshes or large α. Implicit methods (backward difference, Crank-Nicolson, etc.) require solving a system of equations but are stable for arbitrary time steps. Commercial CAE thermal solvers typically adopt implicit methods.
This simulator computes on a 60×60 = 3,600 node grid. Since it calculates multiple steps per frame in browser JavaScript, it is sufficient for visual understanding of phenomena. Actual CAE thermal analysis uses tens of thousands to millions of nodes to capture finer temperature gradients and localized heat concentrations. Grid dependency (mesh convergence) verification is a must-do in practical engineering.
Approximate values (×10⁻⁶ m²/s): Copper ≈ 117, Aluminum ≈ 84, Iron (mild steel) ≈ 12, Silicon ≈ 88, Concrete ≈ 0.7, Wood ≈ 0.1–0.2, Water ≈ 0.14, Air ≈ 20. Metals are 2–3 orders of magnitude higher than resins and building materials, allowing heat to spread quickly. Use the slider in this simulator as a relative "speed comparison."
It handles various internal heat generation phenomena such as Joule heating from electric current ($\dot{q} = \rho_e J^2$), chemical reaction heat (catalysis, combustion), and absorption heat from radiation or ultrasound. In this simulator, heat-generating cells are placed as square regions with a constant $\dot{q}$. In actual power semiconductor ICs, localized current concentration can cause extremely high-density heat generation (hot spots) at specific locations, and their transient response is key to design.
This simulator solves only heat conduction in solids. For real electronic device cooling, conjugate heat transfer (CHT) is needed, which simultaneously solves heat conduction in solids (chips, substrates) and convection in fluids (air, cooling water). CHT couples the Navier-Stokes equations for fluids with the heat conduction equation for solids, drastically increasing computational cost. Solvers like OpenFOAM and Ansys Fluent handle this.
The blue line shows T_max (maximum temperature across all nodes), the orange line shows the center point temperature, and the green line shows T_min (minimum temperature across all nodes) over time. In the IC chip cooling preset, you can observe T_max and T_center rising sharply at first, then gradually leveling off as heat flows to the cooling plate, eventually settling to steady-state values. The difference between this "transient peak temperature" and "steady-state temperature" is a key metric in thermal design.
What is Heat Conduction Transient?
2DTransient Heat Conduction Simulator (FDM) is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.
By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition - making it an effective learning tool for students and a rapid-verification tool for practicing engineers.
Physical Model & Key Equations
The simulator is based on the governing equations of 2D. Understanding these equations is key to interpreting the results correctly.
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Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.
Real-World Applications
Engineering Design: The concepts behind 2D are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.
Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.
CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.
Common Misconceptions and Points of Caution
Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.
Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.
Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.
Set thermal diffusivity α (m²/s) using alphaSlider or alphaValNum. For steel, use 1.2×10⁻⁵; for aluminum, 9.7×10⁻⁵.
Define boundary conditions: ttopSlider sets top surface temperature (°C), tbotSlider sets bottom surface temperature (°C).
Click "Run Simulation" to solve the transient heat equation ∂T/∂t = α∇²T using explicit finite difference method over your specified domain and time steps.
Worked Example
Copper rod, L=0.1m, initial uniform T=20°C. Set α=1.17×10⁻⁴ m²/s (copper), ttop=100°C, tbottom=20°C. After 10 seconds, FDM discretization (Δx=0.01m, Δt=0.1s) yields temperature profile: x=0m→100°C, x=0.05m→65°C, x=0.1m→20°C. Thermal penetration depth δ≈√(4αt)=0.0217m confirms front reaches mid-section in under 5 seconds.
Practical Notes
Stability criterion: Ensure Fourier number Fo=αΔt/(Δx)²≤0.25 to prevent oscillations. Simulator auto-adjusts Δt if α is very large.
For transient verification in electronics cooling: ceramic substrate α≈3×10⁻⁶ m²/s shows slower diffusion; increase simulation time proportionally.
Boundary layer approximation breaks down near sharp transitions; use finer mesh (smaller Δx) if gradients exceed 50°C per millimeter.