Multi-Layer Wall Heat Conduction
Multi-Layer Wall Heat Conduction: Theoretical Foundations
Fundamentals of Multi-Layer Walls
Teacher, building walls and furnace walls are made of multiple layers, right? How do we do the heat calculation for those?
Treat each layer as a thermal resistance in series and sum them up. The steady-state heat flow rate for an $n$-layer flat multi-layer wall is
The temperature drop in each layer is $\Delta T_i = q \cdot L_i/(k_i A)$, and the layer with a smaller $k$ has a larger temperature drop.
That's the same concept as series resistance in electrical circuits.
Exactly. It's the most basic form of a thermal resistance network. Including convective boundaries gives:
Temperature Distribution
The temperature distribution within each layer is linear (when $k$ is constant).
The temperature is continuous at the layer interface, but the temperature gradient becomes discontinuous, proportional to the inverse of $k$.
Typical Multi-Layer Wall Configurations
| Structure | Layer Composition | Overall U-Value [W/(m$^2$ K)] |
|---|---|---|
| Residential Exterior Wall | Gypsum Board + GW Insulation + Plywood + Siding | 0.3~0.5 |
| Furnace Wall | Firebrick + Insulating Brick + Steel Plate | 0.5~2.0 |
| Refrigerator | Steel Plate + PU Insulation + Steel Plate | 0.2~0.4 |
| LNG Storage Tank | SUS + Perlite Insulation + CS | 0.02~0.05 |
The U-value of an LNG tank is less than 1/10th of a house's.
Because it stores liquefied natural gas at -162°C, requiring extremely high insulation performance. Vacuum perlite insulation achieves an effective $k \approx 0.002$ W/(m K).
Series Rule for Thermal Resistance of Multi-Layer Walls
The total thermal resistance of a multi-layer wall is the series sum of each layer's Rth=L/(kA). This principle corresponds exactly to Ohm's law in electrical circuits and was adopted in the 1940s building insulation standard ISO 6946. In Japan, it is mandated as the basis for UA value calculation in the 2025 Energy Conservation Standards.
Computational Methods for Multi-Layer Wall Heat Conduction
When to Use Hand Calculation vs. FEM
Are hand calculations sufficient for multi-layer wall calculations?
For 1D cases, hand calculation is completely sufficient. FEM becomes necessary in the following situations.
| Condition | Hand Calculation | FEM |
|---|---|---|
| 1D Flat Multi-Layer | Sufficient | Not Needed |
| Temperature-Dependent $k(T)$ | Approximable via iteration | Recommended |
| 2D/3D Effects (Corners, Openings) | Not Possible | Essential |
| Thermal Bridge | Not Possible | Essential |
| Contact Thermal Resistance | Possible by hand | Recommended if pressure-dependent |
Treatment of Thermal Bridges
The most important 2D effect in building multi-layer walls is the thermal bridge. Studs in wooden houses have a higher $k$ than insulation, causing heat leakage through the studs.
Can we treat the insulation part and the stud part as parallel resistances?
Simplistically, yes. It can be roughly estimated using the upper/lower bound method (Series-Parallel method) specified in ISO 6946. However, accurate evaluation requires 2D FEM analysis, assessed using the linear thermal bridge coefficient $\Psi$ [W/(m K)].
A larger $\Psi$ indicates a greater influence of the thermal bridge.
Multi-Layer Wall Modeling in FEM
Points to note when modeling multi-layer walls in FEM.
- Assign different materials to each layer.
- Share nodes (Merged) between layers or use Bonded Contact.
- Thin layers (adhesive layers, vapor barriers, etc.) can be approximated with Shell or Interface elements.
- Air layers can be replaced with an equivalent thermal conductivity (including convection + radiation).
Do we also model air layers as solids?
For sealed air layers, they can be approximated with an equivalent $k$. ISO 6946 has a table of thermal resistance values for air layers by thickness. Ventilated layers (if there is airflow) should be modeled with CFD or treated as a boundary condition.
UA Value Calculation Procedure
The UA value (thermal transmittance W/m²K) of a building exterior wall is the reciprocal of the sum of the thermal resistances of each layer. For a standard composition of 100mm concrete + 100mm glass wool + 12.5mm gypsum board, including surface heat transfer resistance, UA ≈ 0.35 W/m²K.
Multi-Layer Wall Heat Conduction in Practice
Building Exterior Wall Calculation Example
Please show me an actual calculation example for a building exterior wall.
Let's use a wooden house exterior wall (filled insulation) as an example.
| Layer | Material | $L$ [mm] | $k$ [W/(m K)] | $R$ [m$^2$ K/W] |
|---|---|---|---|---|
| Indoor Convection | — | — | — | 0.11 |
| Gypsum Board | PB12.5 | 12.5 | 0.22 | 0.057 |
| Insulation | GW16K | 105 | 0.038 | 2.763 |
| Plywood | Structural 9mm | 9 | 0.16 | 0.056 |
| Ventilated Layer | — | 18 | — | 0.09 |
| Siding | Ceramic-based | 14 | 0.35 | 0.040 |
| Outdoor Convection | — | — | — | 0.04 |
| Total | 3.156 |
$U = 1/R_{\text{total}} = 0.317$ W/(m$^2$ K). This meets the energy conservation standard requirement (regions 4-7) of $U \leq 0.53$.
The insulation accounts for 87% of the total thermal resistance.
Yes. The other layers contribute virtually no thermal resistance. The performance of the insulation almost entirely determines the overall wall performance.
Furnace Wall Design Example
A typical steel heating furnace wall has a 3-layer structure.
| Layer | Material | $L$ [mm] | $k$ [W/(m K)] |
|---|---|---|---|
| Firebrick | SK34 Equivalent | 230 | 1.3 |
| Insulating Brick | B-2 | 115 | 0.3 |
| Steel Plate | SS400 | 6 | 50 |
For a furnace interior of 1200°C and ambient air at 25°C, the steel plate temperature is about 80°C. Design the insulating brick thickness to meet the worker safety standard (steel plate ≤ 80°C).
The thermal resistance of the steel plate is almost zero, so it's essentially just the two layers of firebrick and insulating brick.
Exactly. The steel plate is a structural element, not an insulator.
Insulation Design Practice for ZEH Houses
The revised Building Energy Conservation Act enacted in 2023 requires ZEH-equivalent UA values of 0.6 or less (in warm regions). Using Asahi Kasei Construction Materials' Neomaform α (k=0.020 W/m·K) at 60mm thickness can achieve insulation performance equivalent to conventional glass wool 16K-100mm at half the thickness.
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