Multi-Layer Wall Heat Conduction
Theory and Physics
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.
Physical Meaning of Each Term
- Storage Term $\rho c_p \partial T/\partial t$: Rate of thermal energy storage per unit volume. 【Everyday Example】An iron frying pan is slow to heat up and cool down, while an aluminum pot heats and cools quickly—this is due to the difference in the product of density $\rho$ and specific heat $c_p$ (Heat Capacity). Objects with large heat capacity have slower temperature changes. Water has a very high specific heat (4,186 J/(kg·K)), so temperatures near the sea are more stable than inland. In transient analysis, this term determines the rate of temperature change over time.
- Conduction Term $\nabla \cdot (k \nabla T)$: Heat conduction based on Fourier's law. Heat flux proportional to the temperature gradient. 【Everyday Example】Putting a metal spoon in a hot pot makes the handle hot—metal has a high thermal conductivity $k$, so heat transfers quickly from the hot side to the cold side. A wooden spoon doesn't get hot because its $k$ is small. Insulation materials (like glass wool) have extremely small $k$, making heat transfer difficult even with a temperature gradient. This is the mathematical formulation of the natural tendency: "Heat flows where there is a temperature difference."
- Convection Term $\rho c_p \mathbf{u} \cdot \nabla T$: Heat transport accompanying fluid motion. 【Everyday Example】Feeling cool under a fan is because the wind (fluid flow) carries away the warm air near your skin and supplies fresh, cool air—this is forced convection. The ceiling area of a room becoming warm with heating is due to natural convection where heated air rises due to buoyancy. The fan on a PC's CPU cooler also dissipates heat via forced convection. Convection is an order of magnitude more efficient at heat transport than conduction.
- Source Term $Q$: Internal Heat Generation (Joule heating, chemical reaction heat, radiation absorption, etc.). Unit: W/m³. 【Everyday Example】A microwave heats food via microwave absorption inside the food (volumetric heating). The heater wire in an electric blanket warms up via Joule heating ($Q = I^2 R / V$). Heat generation during lithium-ion battery charging/discharging and friction heat from brake pads are also considered as heat sources in analysis. Unlike boundary conditions that apply heat from the "surface" externally, the source term represents energy generation "inside" the material.
Assumptions and Applicability Limits
- Fourier's Law: Linear relationship where heat flux is proportional to temperature gradient (non-Fourier heat conduction is needed for extremely low temperatures or ultra-short pulse heating).
- Isotropic Thermal Conductivity: Thermal conductivity is independent of direction (anisotropy must be considered for composite materials or single crystals).
- Temperature-Independent Material Properties (Linear Analysis): Assumption that material properties do not depend on temperature (temperature dependence is needed for large temperature differences).
- Treatment of Thermal Radiation: Surface-to-surface radiation uses the view factor method; for participating media, the DO method or P1 approximation is applied.
- Non-Applicable Cases: Phase Change (melting/solidification) requires consideration of latent heat. Extreme temperature gradients necessitate thermal-stress coupling.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Temperature $T$ | K (Kelvin) or Celsius | Be careful not to confuse absolute and Celsius temperatures. Always use absolute temperature for radiation calculations. |
| Thermal Conductivity $k$ | W/(m·K) | Steel: ~50, Aluminum: ~237, Air: ~0.026 |
| Heat Transfer Coefficient $h$ | W/(m²·K) | Natural Convection: 5〜25, Forced Convection: 25〜250, Boiling: 2,500〜25,000 |
| Specific Heat $c_p$ | J/(kg·K) | Distinguish between constant pressure and constant volume specific heat (important for gases). |
| Heat Flux $q$ | W/m² | Neumann condition as a boundary condition. |
Numerical Methods and Implementation
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.
Linear Elements vs. Quadratic Elements
In heat conduction analysis, linear elements often provide sufficient accuracy. Quadratic elements are recommended for regions with steep temperature gradients (e.g., thermal shock).
Heat Flux Evaluation
Calculated from the temperature gradient within an element. Smoothing may be required, similar to nodal stresses.
Convection-Diffusion Problem
When the Peclet number is high (convection-dominated), upwinding stabilization (e.g., SUPG) is needed. Not required for pure heat conduction problems.
Time Step for Transient Analysis
Set a time step sufficiently smaller than the characteristic thermal diffusion time $\tau = L^2 / \alpha$ ($\alpha$: Thermal Diffusivity). Automatic time step control is effective for rapid temperature changes.
Nonlinear Convergence
Nonlinearity due to temperature-dependent material properties is often mild, and Picard iteration (direct substitution method) is often sufficient. Newton's method is recommended for strong nonlinearities like radiation.
Steady-State Analysis Convergence Criterion
Convergence is judged when the temperature change at all nodes falls below a threshold (e.g., $|\Delta T| / T_{max} < 10^{-5}$).
Analogy for Explicit and Implicit Methods
Explicit methods are like "weather forecasting that predicts the next step using only current information"—fast to compute but unstable with large time steps (misses storms). Implicit methods are like "predictions that also consider future states"—stable even with large time steps but require solving equations at each step, which is more work. For problems without rapid temperature changes, using an implicit method with larger time steps is more efficient.
Practical Guide
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.
Analogy for Analysis Flow
Think of the thermal analysis flow as "designing a bath reheating system." Decide the bathtub shape (analysis target), set the initial water temperature (initial conditions) and outside air temperature (boundary conditions), and adjust the reheater output (heat source). Predict via calculation "whether it will become lukewarm after 2 hours?"—this is the essence of transient thermal analysis.
Common Pitfalls for Beginners
"Can I ignore radiation?"—Usually OK around room temperature. But it's a different story above several hundred degrees. Heat transfer by radiation is proportional to the fourth power of temperature, so it overwhelms convection at high temperatures. Have you ever experienced how different the perceived temperature is in the sun versus in the shade on a sunny day? That's the power of radiation. Ignoring radiation in the analysis of industrial furnaces or engine components is like insisting "sunlight doesn't matter" on a scorching hot day.
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