Click and drag to place heat sources and experience Fourier's heat equation in real time. Freely change material, boundary conditions, and simulation speed to observe temperature field time evolution.
100×100 Grid Finite Difference Method Jet Colormap Real-time computation Temperature History Plot
Heat
100°C
Legend
0°C
Paint Mode
CanvasClick/Drag：Set temperature by Y position (top=100°C, bottom=0°C)
Thermal penetration depth after 1 s：δ ≈ 2√(αt) = 2√(1.33×10⁻⁵×1) ≈ 7.3 mm
After 5 s: delta ≈ 16.3 mm, exceeding the plate thickness and reaching the back surface
HAZ peak temperature (fusion boundary):> 1400°C(above austenite transformation)
2DSimulator in Heat sourceplace,SteelMode「Velocity10x」do Diffusion behaviorConfirm.
Design Criteria: JIS Z 3001 (welding terminology), AWS D1.1 (structural welding code). Preheat temperature control prevents HAZ hardening and cracking.
What is 2D Heat Diffusion?
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What exactly is "heat diffusion"? I see the grid in the simulator getting colored, but what's physically happening?
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Basically, it's how heat spreads out from a hot spot into cooler areas over time. In this 2D simulator, you're seeing a top-down view of a flat plate. When you click to place a heat source (the red spots), you're adding energy. The simulator then calculates how that energy spreads to neighboring cells, turning them orange, then yellow, and so on. Try clicking and watching the colors flow-that's diffusion in action.
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Wait, really? So the "Velocity" (Speed) slider controls how fast this happens. But what property of the material actually determines that speed in real life?
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Great question. In practice, the speed is governed by a material property called thermal diffusivity, often represented by the Greek letter alpha ($\alpha$). A high $\alpha$ means heat spreads quickly (like in metals), and a low $\alpha$ means it spreads slowly (like in insulation). When you move the "Velocity" slider in the simulator, you're effectively changing this $\alpha$ value. Try setting it very low and placing a source-you'll see the heat barely moves!
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That makes sense! And the "Brush Size"? Is that just for making bigger dots, or does it affect the physics of the simulation too?
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It's both! A larger brush creates a bigger, more powerful initial heat source. In physical terms, you're increasing the initial amount of thermal energy in that region. This affects the physics because a larger, hotter source will diffuse its influence further and for a longer time than a small one. Play with it: make a tiny hot spot and a huge one. You'll see the larger one creates a much wider and more persistent temperature field.
Physical Model & Key Equations
The core physics is described by the Fourier heat equation. For two-dimensional, unsteady (time-dependent) conduction, it governs how temperature T changes at every point (x,y) over time t.
Where:
$T(x,y,t)$: Temperature at position (x,y) and time t [°C or K].
$t$: Time [s].
$\alpha$: Thermal diffusivity [m²/s]. This is the key parameter controlled by the Velocity slider.
$\frac{\partial^2 T}{\partial x^2}+ \frac{\partial^2 T}{\partial y^2}$: The Laplacian, a mathematical operator that sums the curvature (second spatial derivative) of temperature in all directions. It tells us how "bumpy" the temperature field is at a point.
The simulator solves this equation numerically by dividing the domain into a grid of cells. The rate of heat flow between cells is proportional to their temperature difference, a principle known as Fourier's Law of heat conduction.
$$
q = -k \, \nabla T
$$
Where:
$q$: Heat flux vector [W/m²], showing the direction and magnitude of heat flow.
$k$: Thermal conductivity [W/(m·K)].
$\nabla T$: Temperature gradient [K/m]. The negative sign indicates heat flows from hot to cold.
The interplay between this law (driving force) and the conservation of energy leads directly to the heat diffusion equation above.
Frequently Asked Questions
Yes, you can place multiple heat sources simultaneously by clicking or dragging. To delete them, right-click on a heat source, or use the 'Reset' button on the screen to clear all heat sources.
Yes, it is possible. By adjusting the 'Speed' slider on the screen, you can change the time evolution rate in real time. To maintain numerical stability, the step size may be automatically adjusted if the speed is set too high.
This is because each material has a different thermal diffusivity α. Materials with high α, such as copper, conduct heat quickly, while those with low α, such as wood, conduct heat slowly. By switching materials in the simulator, you can directly observe this difference.
Boundary conditions determine the behavior of heat at the edges of the computational domain. 'Fixed temperature' keeps the edges at a constant temperature, while 'Insulated' prevents heat from escaping. This allows you to simulate real experimental environments, such as cooling plates or insulating materials.
Real-World Applications
Electronics Cooling: A primary use of 2D heat diffusion analysis is in designing circuit boards (PCBs). Engineers simulate heat spreading from hot components like CPUs to ensure temperatures stay within safe limits and to optimize heatsink and fan placement. The simulator's click-to-add-source feature mimics placing a hot chip on a board.
Building Science & Insulation: Analyzing heat flow through walls, windows, and roofs is essentially a 2D (or 3D) diffusion problem. Simulators help determine where thermal bridges (areas of rapid heat loss) occur, guiding the placement of insulation to improve energy efficiency in homes and commercial buildings.
Geothermal Studies: Understanding how temperature varies in the Earth's subsurface is crucial for geothermal energy harvesting. The diffusion equation models how heat from deep, hot rocks conducts towards the surface, informing where to drill wells for maximum energy extraction.
Materials Processing: During processes like welding, quenching, or additive manufacturing (3D printing), intense heat is applied to a specific area. Simulating the 2D diffusion of this heat is vital to predict and control the resulting material properties, such as hardness or residual stress, which determine the final product's strength.
Common Misconceptions and Points to Note
When you start using this simulator, there are several points that beginners to CAE often stumble over. A major initial misconception is the image that "heat moves from a high-temperature area to a low-temperature area at a constant speed". Heat does not "flow"; it "diffuses" according to the temperature gradient. For example, if you create a small hot spot in steel, the temperature drops sharply right next to it, whereas in copper, the temperature distribution over the same distance will be much smoother. This is because the difference in thermal diffusivity α determines the "smoothness" of the temperature, a phenomenon that cannot be measured by speed alone.
Next is the tendency to underestimate the importance of boundary condition settings. While the simulator lets you choose "adiabatic" or "constant temperature", in practical work, this setting significantly influences the results. For instance, assuming the edge of a substrate is "adiabatic" will cause heat to build up, resulting in higher temperatures, but in reality, there might be heat dissipation from that edge to the case. Get into the habit of constantly asking, "What is the actual physical boundary condition?"
Finally, be aware of the gap between simulation "resolution" and "reality". This tool calculates using a 100x100 grid, but in practical CAE, the fineness of the mesh (grid) directly impacts result accuracy. For example, in areas with sharp geometry like the tips of a heat sink's fins, failing to use a finer mesh can lead to underestimating the actual temperature. Please remember that this tool is for learning principles, and its results cannot be used directly as design values.