Solve 2D steady-state heat conduction through wall cross-sections using the finite difference method. Visualize temperature contours, isotherms, and compute ψ-value and condensation risk in real time.
Parameters
Wall Preset
Indoor Temp T_H
°C
Outdoor Temp T_C
°C
Concrete k₁
W/mK
Insulation k₂
W/mK
Steel Bridge k₃
W/mK
Results
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Heat Flux Q (W/m)
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ψ-value (W/mK)
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U-value (W/m²K)
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Cond. Risk fRsi
Thermal Bridge Temperature
ColdHot
Temperature contour map with isotherms (blue=cold, red=hot). Click "Solve" to run 100 Gauss-Seidel iterations.
What exactly is a "thermal bridge"? I've heard it makes buildings less efficient, but I don't get the physics.
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Basically, it's a localized spot in a building envelope where heat escapes much faster than the surrounding area. In practice, it happens where materials with high thermal conductivity, like steel or concrete, create a shortcut for heat to flow from the warm inside to the cold outside. Try selecting the "Steel Bridge" preset in the simulator above—you'll instantly see a bright red "hot spot" of heat flux piercing through the wall.
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Wait, really? So the 2D temperature map isn't uniform? Why can't we just use a simple 1D calculation?
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Great question! A 1D calculation assumes heat only flows straight through layers. But at corners, window frames, or structural connections, heat flows in two or three dimensions, creating complex patterns. For instance, in the simulator, slide the "Concrete k₁" and "Insulation k₂" conductivities far apart. You'll see the isotherms (lines of equal temperature) bend sharply at the material interface—that's the 2D effect a 1D model completely misses, leading to underestimated heat loss.
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So how do engineers quantify this "extra" heat loss? Is there a standard number to look for?
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Yes! They use a metric called the linear thermal transmittance, or psi (ψ). It measures the additional heat loss per meter of bridge length. The simulator calculates it using the formula ψ = Q₂D/ΔT − U₁D×L. A common target, like the Passive House standard, requires ψ < 0.01 W/(m·K). Try changing the "Steel Bridge k₃" to a very high value and watch the calculated ψ value skyrocket, showing how bad the bridge is.
Physical Model & Key Equations
The core physics is governed by the steady-state heat conduction equation. For 2D, with no internal heat generation, this simplifies to Laplace's equation for temperature.
$$\nabla^2 T = \frac{\partial^2 T}{\partial x^2}+ \frac{\partial^2 T}{\partial y^2}= 0$$
This equation states that at any point (except boundaries), the net flow of heat into a tiny volume is zero. The simulator solves this numerically using the Finite Difference Method across the grid of materials you define.
Once the temperature field T(x,y) is known, we calculate the heat flux (heat flow per unit area) using Fourier's Law, and from that, derive key performance metrics.
q is the heat flux vector (W/m²), k is thermal conductivity (W/m·K). ψ quantifies the bridge's severity. f_Rsi predicts surface condensation risk; a value below 0.7 is often a problem.
Real-World Applications
Building Envelope Design: Architects and engineers use 2D thermal analysis to design wall-to-roof, wall-to-floor, and window-to-wall junctions. By simulating different insulation wraps and structural details, they can eliminate costly thermal bridges before construction begins, ensuring compliance with energy codes like Passive House.
Condensation & Mold Prevention: A critical application is predicting interior surface temperatures. A cold spot (low f_Rsi) leads to condensation and mold growth. This simulator helps identify these risky areas—for instance, where a concrete balcony slab penetrates the insulation layer—so designers can add thermal breaks.
HVAC Load Calculation: Accurate heating and cooling load estimates depend on knowing the building's real heat loss. Using 1D calculations for walls and adding ψ-values for all linear bridges (like perimeter edges) gives a far more accurate total load, leading to correctly sized and more efficient HVAC systems.
Retrofit and Renovation Planning: When upgrading old buildings, thermal bridges are often the weakest link. Simulators like this help prioritize interventions. For example, analyzing an uninsected concrete column in a brick wall shows whether adding interior insulation alone is sufficient or if exterior insulation is needed to cover the bridge.
Common Misunderstandings and Points to Note
When you start using this simulator, there are a few points you should be careful about. First, "the thermal conductivity value is not determined solely by the material name." For example, even something broadly called "concrete" can have a thermal conductivity that varies greatly depending on its density and moisture content. The tool's default values are merely representative. For real projects, the golden rule is to check and input the catalog values or standard values (like JIS A) for the specific material you're using. Using rough values can sometimes lead to psi-values differing from reality by as much as 30%.
Next, understand the limitations of 2D analysis. Since this tool looks at a cross-section, it cannot fully capture the complete picture of "three-dimensional thermal bridges" like corners or column heads/bases. For instance, a concrete corner becomes a "3D thermal bridge" where heat escapes concentrated from three directions: the two inner surfaces and the ceiling surface. Even if the 2D analysis shows a high surface temperature, there might still be a condensation risk at the actual corner. For particularly critical areas, it's best to verify with a 3D simulation.
Finally, boundary condition setting errors. The internal and external surface heat transfer resistances (Rsi, Rse) are often set to ISO standard values by default, but this assumes "normal natural convection." For example, if you place large furniture in front of an external wall obstructing airflow, or if there's forced ventilation indoors, the actual heat transfer changes. Don't over-rely on simulation results; get into the habit of always clearly stating the premise: "given these conditions, the result is this."
Enter boundary temperatures: hot side (vTH, °C) and cold side (vTC, °C). Typical range: 20–50°C differential for building envelopes.
Define material thermal conductivity (vK1, W/mK). Common values: concrete 1.4, insulation 0.04, brick 0.6.
Set mesh density (sTHNum, sTCNum) to 20–50 nodes per dimension for convergence without excessive computation.
Run solver to generate isotherms and extract heat flux Q, linear thermal transmittance ψ (W/mK), and condensation risk factor fRsi.
Worked Example
Simulate a timber-framed wall: hot interior 21°C, cold exterior −5°C (26 K difference). Layer 1 timber frame (vK1=0.12 W/mK, 50 mm). Mesh 40×40 nodes. Solver computes heat flux Q≈1.2 W/m through the thermal bridge, ψ-value≈0.046 W/mK, U-value≈0.58 W/m²K at bridged section. Surface temperature fRsi=0.82 (low mould risk: fRsi>0.75 acceptable per EN ISO 13788).
Practical Notes
Refine mesh near material interfaces and corners where thermal gradients peak; coarse grids underestimate heat flux by 15–25%.
Account for air film resistance (add 0.13 m²K/W per side) when comparing to design U-values in building codes.
Wood-stud thermal bridges in timber frames reduce effective insulation by 8–12%; use multi-layer stacking (vK1 switching) to model realistic cavity fill.
Validate fRsi against EN ISO 13788 limits: residential target fRsi>0.75 to prevent surface condensation at design conditions.