Thermal Conduction in Composite Materials
Theory and Physics
Thermal Conductivity Characteristics of Composite Materials
How does the thermal conductivity of composite materials like CFRP and GFRP differ from that of homogeneous materials?
Fiber-reinforced composites exhibit strong anisotropy. The thermal conductivity in the fiber direction is largely contributed by the fibers, while the transverse direction is dominated by the matrix (resin). As a result, the thermal conductivity (k) can differ by more than 10 times depending on the direction.
Theory of Effective Thermal Conductivity
The effective thermal conductivity for the fiber direction (parallel model) and transverse direction (series model) are as follows.
Here, $V_f$ is the fiber volume fraction, $k_f$ is the thermal conductivity of the fiber, and $k_m$ is the thermal conductivity of the matrix.
What is the approximate difference in concrete numerical terms?
For PAN-based carbon fiber ($k_f=10$ W/(mK)), epoxy resin ($k_m=0.2$ W/(mK)), and $V_f=0.6$:
- $k_{\parallel} = 0.6 \times 10 + 0.4 \times 0.2 = 6.08$ W/(mK)
- $k_{\perp} \approx 0.45$ W/(mK)
There is a difference of more than 13 times. Ignoring this and analyzing with isotropic properties will completely change the temperature distribution.
Hashin-Shtrikman Bounds
For more precise evaluation, the Hashin-Shtrikman upper and lower bounds are used.
Measured values fall within these upper and lower bounds. More precise predictions are possible using the Halpin-Tsai model or finite element homogenization (RVE analysis).
What happens with short fiber materials where the fiber orientation is random?
With random orientation, the material becomes nearly isotropic. However, in injection-molded parts, fibers align in the flow direction, resulting in partial anisotropy. Practical methods exist that export the fiber orientation tensor from injection molding simulations like Moldflow and map it to the thermal conductivity tensor.
Composite Material Laws, Since the 1850s
Maxwell (1873) first theorized the equivalent thermal conductivity of composites with dispersed spherical particles. His formula is still used today in designing TIMs (Thermal Interface Materials) made by mixing copper particles into polymer substrates, and remains in textbooks in the form λeff ≈ λm(λp+2λm+2φ(λp−λm))/(λp+2λm−φ(λp−λm)).
Physical Meaning of Each Term
- Heat Storage Term $\rho c_p \partial T/\partial t$: Rate of thermal energy storage per unit volume. 【Everyday Example】 An iron frying pan heats up slowly and cools down slowly, while an aluminum pot heats up quickly and cools down 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 experience slower temperature changes. Water has a very high specific heat (4,186 J/(kg·K)), which is why temperatures near the sea are more stable than inland. In transient analysis, this term determines the rate of temperature change over time.
- Heat 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 causes the handle to become hot—because metal has a high thermal conductivity $k$, heat transfers quickly from the high-temperature side to the low-temperature side. A wooden spoon doesn't get hot because its $k$ is low. Insulation materials (e.g., glass wool) have extremely low $k$, making heat transfer difficult even with a temperature gradient. This term mathematically expresses the natural tendency of "heat flowing 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 when facing a fan is because the wind (fluid flow) carries away warm air near the body surface and supplies fresh, cold 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 in a PC's CPU cooler also dissipates heat via forced convection. Convection is an order of magnitude more efficient heat transport method than conduction.
- Heat Source Term $Q$: Internal Heat Generation (Joule heat, chemical reaction heat, radiation absorption, etc.). Unit: W/m³. 【Everyday Example】 A microwave oven 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 supply heat from the "surface" externally, the heat 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, single crystals, etc.)
- 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: View factor method for surface-to-surface radiation; DO method or P1 approximation for participating media
- Non-Applicable Cases: Consideration of latent heat is necessary for phase change (melting/solidification). Thermal-stress coupling is essential for extreme temperature gradients
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Temperature $T$ | K (Kelvin) or Celsius | Be careful not to confuse absolute temperature and Celsius. 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 specific heat at constant pressure and constant volume (important for gases) |
| Heat Flux $q$ | W/m² | Neumann condition as a boundary condition |
Numerical Methods and Implementation
Homogenization via RVE Analysis
Do you model down to the fiber level for analysis?
Multi-scale analysis using a Representative Volume Element (RVE) is the standard method. An RVE is created with carbon fibers (7μm diameter) arranged in a hexagonal array, and an effective thermal conductivity tensor is obtained by applying temperature differences in each direction.
How is the RVE size determined?
A guideline is 10 to 20 times the fiber diameter. If the results don't change when the RVE size is increased, it's sufficient. COMSOL and Digimat allow for automatic RVE generation and parametric homogenization.
Laminate Modeling
For a laminate (e.g., [0/90/45/-45]s), the thermal conductivity tensor of each ply is rotated according to the stacking direction and superimposed. In Abaqus, the ply orientation angle is defined using ORIENTATION andSHELL SECTION.
| Laminate Configuration | In-plane k [W/(mK)] | Through-thickness k [W/(mK)] |
|---|---|---|
| UD [0]8 | 6.0 / 0.45 | 0.45 |
| Cross-ply [0/90]2s | 3.2 / 3.2 | 0.45 |
| Quasi-isotropic [0/45/90/-45]s | 3.2 / 3.2 | 0.45 |
So cross-ply and quasi-isotropic become uniform in-plane, right?
Correct. However, the through-thickness direction remains low and matrix-dominated for all configurations. Ensuring thermal pathways in the through-thickness direction is the biggest challenge in thermal design of CFRP structures. Research is progressing on improvements through Z-pinning or through-thickness introduction of carbon nanotubes.
Difference in Accuracy Between Parallel and Series Rules
The in-plane thermal conductivity of fiber-reinforced plastics (CFRP) falls within ±5% using the parallel rule (rule of mixtures), but the through-thickness direction often has errors exceeding 20% even with the series rule. A report from NASA Langley in the 1990s (NASA TM-4756), based on measurements of carbon fiber/epoxy, recommended using the Hashin-Shtrikman bounds model for the through-thickness direction.
Linear Elements vs. Quadratic Elements
In thermal 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
Upwinding stabilization (e.g., SUPG) is needed when the Peclet number is high (convection-dominated). 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 nonlinearity from radiation.
Steady-State Analysis Determination
Convergence is determined 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
The explicit method is like "predicting the next step using only current information, like a weather forecast"—fast to compute but unstable with large time steps (misses storms). The implicit method is like "a prediction that also considers future states"—stable even with large time steps but requires solving equations at each step, which is more computationally intensive. For problems without rapid temperature changes, using the implicit method with larger time steps is more efficient.
Practical Guide
Points to Note in Practice
What should I be especially careful about in thermal analysis of composite materials?
The most important thing is the correct definition of the material coordinate system. Since the fiber direction differs for each ply, the coordinate system for each element must be set accurately.
Integration with Ansys ACP
Using Ansys Composite PrepPost (ACP) automates the process from defining composite laminates to transferring data to the thermal analysis model.
1. Define the laminate configuration (ply sequence, orientation angle, thickness) in ACP
2. The material coordinate system is automatically generated
3. Data is transferred to Steady-State Thermal
4. The anisotropic k for each ply is automatically applied
Not having to set the coordinate system manually is a big advantage.
In Abaqus, equivalent functionality is available via the Composite Layup feature in Abaqus/CAE. In COMSOL, the Composite Materials Module supports this.
Comparison with Measured Values
The thermal conductivity of composite materials varies depending on the measurement method.
| Measurement Method | Application | Accuracy |
|---|---|---|
| Laser Flash Analysis (LFA) | Through-thickness direction | ±5% |
| Steady-State Method (Guarded Hot Plate) | Through-thickness direction | ±3% |
| Angstrom Method | In-plane direction | ±10% |
So the measurement methods differ for in-plane and through-thickness directions.
The Laser Flash method measures the thermal diffusivity in the through-thickness direction and converts it to thermal conductivity using $k = \alpha \rho c_p$. Measuring the in-plane direction is difficult due to sample preparation and has lower accuracy. This uncertainty must be considered when comparing analysis results with measurements.
Evolution of Smartphone Heat Dissipation Materials
Early 2010s smartphones simply used copper foil graphite sheets (λ≈400 W/m·K), but high-end models since 2019 (e.g., Samsung Galaxy S10 and later) adopted vapor-grown carbon fiber (VGCFs) composites, achieving in-plane thermal conductivity exceeding 1500 W/m·K.
Analogy for the Analysis Flow
Think of the thermal analysis flow as "designing a bath reheating system." Determine the bathtub shape (analysis target), set the initial water temperature (initial condition) and outside air temperature (boundary condition), and adjust the reheater output (heat source). Predicting "whether the water will become lukewarm after 2 hours" through calculation—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. Radiative heat transfer 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.
Boundary Conditions
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