Solve 2D steady-state heat conduction with finite differences. Set boundary temperatures and a center heat source — watch the temperature contour and isotherms update instantly.
Boundary Conditions
Top temperature T_top
°C
Bottom temperature T_bottom
°C
Left temperature T_left
°C
Right temperature T_right
°C
Center heat source Q (W/m²)
W/m²
Conductivity k (W/mK)
W/mK
Results
T_max
—
T_min
—
T_center
—
Iterations
—
Temperature Contour Map + Isotherms (10°C intervals)
What exactly is "steady-state" heat conduction? Is it when the temperature stops changing?
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Basically, yes! It's the final, stable temperature distribution after all the transient heating or cooling has settled down. In practice, think of an engine block that's been running for a while—the hot and cold spots aren't shifting anymore. In this simulator, you're directly solving for that final state. Try setting all the boundary temperatures and hitting "Solve" to see the steady-state pattern instantly.
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Wait, really? So the "heat source" Q in the controls is like a constant heater inside the material? How does that show up?
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Exactly! A common case is an electronic chip generating heat on a circuit board. The Q parameter (in W/m²) represents that constant power. If you set it to zero, heat only flows from the hot boundaries to the cold ones. Now, slide the "Center heat source Q" up from zero and solve again. You'll see a hot spot bloom right in the middle of the contour plot.
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What does the conductivity 'k' actually do? If I make it higher, does the plate get hotter or colder?
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Great question. Conductivity is the material's ability to move heat. A high 'k' (like copper) spreads heat out easily, smoothing hot spots. A low 'k' (like plastic) traps heat. For instance, with a strong heat source (high Q), a low conductivity material will have a much more intense, localized hot spot. Adjust the 'k' slider and watch how the temperature contours either spread out or bunch up around the source.
Physical Model & Key Equations
The core physics is described by the Poisson equation for steady-state heat conduction. It states that at any point, the net curvature of the temperature profile (how it bends) is balanced by the internal heat generation.
Here, T is temperature (K), Q is the internal heat generation per unit volume (W/m³), and k is the thermal conductivity (W/m·K). If Q=0, this becomes Laplace's equation, meaning temperature diffuses smoothly from boundaries with no internal sources.
The simulator solves this equation numerically using the Finite Difference Method (FDM). It overlays a grid on the plate and approximates the derivatives using the temperatures at neighboring points. The Gauss-Seidel update formula for each interior grid point is:
Here, Ti,j is the updated temperature at grid point (i,j). The term QΔx²/k is the source contribution. The solver repeatedly applies this formula, using the temperatures you set on the Top, Bottom, Left, and Right boundaries, until the solution converges to the steady state.
Frequently Asked Questions
This simulator uses iterative calculations (Gauss-Seidel method) to converge the solution. After changing the values, the calculation continues until the convergence condition is met, so there is a slight time lag in real-time updates. Please wait a moment, or try adjusting the convergence tolerance.
The larger k is, the more easily heat is transferred, and the temperature distribution becomes more uniform. For example, the difference between high and low temperature regions decreases, and the spacing of isotherms tends to widen. Conversely, when k is small, local temperature changes become more pronounced.
Yes, it is possible. A negative Q value means 'heat absorption' or 'cooling.' For example, it can model the effect of a heat sink that locally removes heat. However, if the negative value is excessively large, the calculation may become difficult to converge.
Currently, only rectangular regions using a Cartesian grid are supported. Complex shapes such as circles or inclined boundaries cannot be analyzed. However, by setting boundary conditions in detail, approximate modeling may be possible in some cases.
Real-World Applications
Electronics Cooling: This is the classic use case. Engineers use this exact 2D analysis to predict "hot spots" on circuit boards and chip packages. By modeling a heat source (Q) at the CPU location and setting boundary temperatures for the heat sink and ambient air, they can design effective cooling solutions.
Building Insulation Design: Analyzing heat flow through walls and windows. The boundaries represent indoor and outdoor temperatures, while different materials (like insulation vs. studs) are modeled with varying conductivity (k). This helps in minimizing energy loss.
Geothermal Heat Flow: Modeling the steady-state temperature distribution in the ground. The top boundary is the surface temperature, the bottom is the Earth's internal heat, and heat sources (Q) could represent buried power cables or waste repositories.
Manufacturing & Heat Treatment: Predicting temperature distribution in parts during processes like annealing or curing. Boundaries represent furnace walls or cooling jets, and the goal is to achieve a uniform temperature (or a specific gradient) throughout the part to ensure material properties.
Common Misunderstandings and Points to Note
First, note that the "Internal Heat Source Q" you set in this tool is simplified as heat generation per unit area. In practice, the standard is per unit volume [W/m³], so when analyzing plates where thickness is important, you need to convert this value, for example, by dividing it by the thickness. Next, boundary condition setting errors are common. If you intend to set all boundaries as "Adiabatic" but accidentally leave one at a fixed temperature, it can lead to an unrealistic distribution where heat seems to be sucked into that point. For instance, if you set all four sides to the "Adiabatic" slider, the entire plate will heat up indefinitely if there's an internal heat source Q (which is impossible in reality). This happens because the heat balance condition (heat generated = heat dissipated) is not satisfied; in actual equipment design, you must always consider heat dissipation paths.
Finally, understand the dependency on grid size. This tool uses a fixed 20x20 grid, but this number actually significantly affects the accuracy of the results. For example, in areas with dense isotherms (where the temperature gradient is steep), a coarse grid will produce a smoother distribution than in reality. Conversely, making the grid too fine will cause computation time to explode. In practice, a "mesh sensitivity analysis" to determine the necessary and sufficient resolution is an essential step.