The 2D heat equation $\partial T/\partial t=\alpha\nabla^2T$ is solved by relaxation (Gauss–Seidel). At steady state $\nabla^2T=0$ and every interior node satisfies $T_{ij}=\tfrac14(T_{i+1,j}+T_{i-1,j}+T_{i,j+1}+T_{i,j-1})$. By the maximum principle there are no interior extrema.
Results
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Center temperature
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Max temperature span
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Heat-flux index
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Boundary bias
Temperature field
Centerline temperature
Heat-flux direction
Model and equations
$$\nabla\cdot(k\nabla T)=0$$
This page approximates steady conduction in a rectangular plate with a Laplace-equation field. Internal heat generation and contact resistance require additional modeling.
How to read it
The heatmap shows smooth temperature decay from hot to cold boundaries.
The profile plot reads the left-to-right centerline gradient.
The flux view highlights where local gradients are stronger.
Learn 2d Conduction Temperature by dialogue
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When reading 2d Conduction Temperature, where should I look first? Moving Left temperature changes both the plots and the result cards.
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Start with Center temperature, but do not treat the number as the whole answer. Use Temperature field to confirm the assumed state, then read Centerline temperature for the distribution or trend. The heatmap shows smooth temperature decay from hot to cold boundaries.
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I can see why Left temperature changes Center temperature. How should I judge the influence of Right temperature?
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Move Right temperature in small steps and watch Max temperature span. That reveals which term is controlling the result. This page approximates steady conduction in a rectangular plate with a Laplace-equation field. Internal heat generation and contact resistance require additional modeling. A single operating point is not enough; sweep the realistic scatter range.
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What is Heat-flux direction for? It feels like the ordinary curve already tells the story.
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Heat-flux direction is for finding boundaries where the condition becomes risky or margin collapses quickly. The profile plot reads the left-to-right centerline gradient. In Early comparison of insulation or heat spreaders, the important question is often what happens after a small change, not only the nominal value.
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So if Center temperature is within the target, can I accept the condition?
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Treat this as a first-pass review. It helps with Checking enclosure boundary temperature differences and Understanding boundary sensitivity before detailed FEM, but final decisions still need standards, measured data, detailed analysis, and vendor limits. The flux view highlights where local gradients are stronger.
Practical use
Early comparison of insulation or heat spreaders.
Checking enclosure boundary temperature differences.
Understanding boundary sensitivity before detailed FEM.
FAQ
Start with Center temperature and Max temperature span. Then use Temperature field to confirm the assumed state and Centerline temperature to read distribution or bias. The heatmap shows smooth temperature decay from hot to cold boundaries
Move Left temperature alone, then move Right temperature by a comparable amount and compare the change in Center temperature. Heat-flux direction shows combinations where margin or performance changes quickly.
Use it for Early comparison of insulation or heat spreaders. Instead of trusting a single point, widen the input range and check whether Center temperature keeps enough margin before moving to detailed analysis.
This page approximates steady conduction in a rectangular plate with a Laplace-equation field. Internal heat generation and contact resistance require additional modeling. Final decisions still require standards, measured data, detailed analysis, and vendor limits.
How to Use
Enter boundary temperatures (°C) for left, right, top, and bottom edges using leftVal, rightVal, topVal, and bottomVal fields
Set material thermal conductivity (W/m·K) for the domain—typical values: aluminum 237, steel 50, concrete 1.4
Click simulate to solve the 2D Laplace equation; observe centerline temperature and heat-flux vectors pointing from hot to cold regions
Review output statistics: center temperature, maximum temperature span across domain, heat-flux index magnitude, and boundary bias ratio
Worked Example
Steel plate (k=50 W/m·K), 0.5m × 0.5m domain: left=100°C, right=20°C, top=80°C, bottom=40°C. Solver yields center temperature ≈62°C, maximum span=80°C, heat-flux index ≈450 W/m² pointing southeast (dominant rightward and downward flow), boundary bias=1.8 indicating asymmetric thermal gradient. Heat flows preferentially toward the cooler right edge.
Practical Notes
High boundary bias (>2.0) reveals strong directional heat flow; use to optimize cooling-channel placement opposite dominant flux
Center temperature should approximately equal the arithmetic mean of boundaries only if all four are equal; asymmetry confirms non-uniform conduction
For composite materials, run separate simulations with different k values to compare thermal response; aluminum designs see 4× faster heat diffusion than steel
Heat-flux index scales with conductivity and temperature gradient; use to size heat sinks in electronics cooling applications