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Theory and Physics
What is Thermal Contact Resistance?
Why does the temperature jump at the interface even when two solids are pressed together?
At the microscopic level, the contact surface only makes point contacts (true contact points) due to surface roughness. The actual contact area is about 1% of the apparent area, with the rest being an air gap. This contact imperfection causes the temperature jump.
Fundamental Equation
The defining equation for thermal contact resistance is as follows.
The contact conductance is $h_c = 1/R_c$ [W/(m2K)].
What are typical values?
They vary by orders of magnitude depending on conditions.
| Contact Condition | $h_c$ [W/(m2K)] |
|---|---|
| Aluminum to Aluminum・Polished Surface・High Pressure | 10000〜25000 |
| Steel to Steel・Machined Surface・Medium Pressure | 2000〜5000 |
| With Thermal Grease | 5000〜50000 |
| Air Gap (0.1mm) | 250 |
| In Vacuum・Low Pressure | 100〜500 |
Cooper-Mikic-Yovanovich Model
A representative theoretical model is the Cooper-Mikic-Yovanovich (CMY) correlation.
Here, $k_s = 2k_1k_2/(k_1+k_2)$ is the harmonic mean thermal conductivity, $m$ is the surface slope, $\sigma$ is the composite roughness, $P$ is the contact pressure, and $H_c$ is the microhardness.
So increasing the pressure reduces the thermal contact resistance.
Exactly. In bolted joint design, the tightening torque determines the performance of the thermal path. Insufficient torque becomes a critical risk in thermal design.
The Discovery of Thermal Contact Resistance Came from NASA
Thermal Contact Resistance (TCR) gained engineering attention in the 1950s during the space race. In a vacuum, where there is no convection, TCR at bolted joints becomes the dominant thermal resistance. NASA's Glenn Research Center built the first systematic experimental database in 1959. Even today, that data is used as a reference value in early-stage design.
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 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)), which is why temperatures near the ocean 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】When you put a metal spoon in a hot pot, the handle gets hot—because metal has a high thermal conductivity $k$, 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 term mathematically expresses 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 in front of 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 getting warm with heating is due to natural convection where heated air rises due to buoyancy. The fan in a PC's CPU cooler also uses forced convection for heat dissipation. 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 through microwave absorption inside the food (volumetric heating). The heater wire in an electric blanket warms up through 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 heat source term represents energy generation "inside" the domain.
Assumptions and Applicability Limits
- Fourier's Law: Linear relationship where heat flux is proportional to the 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: 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 System
| 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
Modeling Thermal Contact Resistance in FEM
How do you represent thermal contact resistance in FEM?
There are three main methods.
| Method | Overview | Accuracy |
|---|---|---|
| Gap Conductance | Set $h_c$ between surfaces | Practical |
| Thin Layer Element | Model TIM with equivalent k, t | Requires thickness |
| Pressure-Dependent Conductance | Calculate $h_c(P)$ via structural coupling | High Accuracy |
In Ansys Mechanical, it's set via TCC (Thermal Contact Conductance) for contact elements. In Abaqus, you can define a pressure-dependent table using the *GAP CONDUCTANCE keyword.
If making it pressure-dependent, coupling with structural analysis is necessary, right?
Yes. In Ansys Workbench, you can link "Static Structural → Steady-State Thermal" to transfer contact pressure and reference the $h_c(P)$ table from the CMY model. This is an essential workflow for thermal path design in bolted joints.
TIM Modeling
Thermal Interface Material (TIM) is represented by the combination of bulk k and contact resistance.
It's the sum of the bulk resistance and the contact resistance on both sides. For thermal grease, even with bulk k=3 W/(mK) and thickness 50um, the bulk resistance is often small, and the contact resistance on both sides tends to dominate.
That's why the application method and pressure are so important.
Exactly. Even with the same grease, the effective $R_{TIM}$ can vary by 2 to 5 times depending on the implementation conditions.
Experimental Correlation Between Applied Pressure and TCR
The Cooper-Mikic-Yovanovich (CMY) model (1969) predicts the contact heat transfer coefficient hc from contact pressure p and surface roughness σ. For metal-to-metal contact, doubling the pressure often increases hc by about 1.5 times. This model is referenced in ISO/TS 22007 and serves as the theoretical basis for mount pressure design in CPU heat sinks.
Linear Elements vs. Quadratic Elements
In heat conduction analysis, linear elements often provide sufficient accuracy. For areas with steep temperature gradients (e.g., thermal shock), quadratic elements are recommended.
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 (SUPG, etc.) 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 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
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). Implicit method is like "predicting while considering future states"—stable even with large time steps, but requires solving equations at each step. For problems without rapid temperature changes, using the implicit method with larger time steps is more efficient.
Practical Guide
Measurement Methods
How do you obtain values for thermal contact resistance?
The most reliable method is actual measurement. Measure using the steady-state method conforming to ASTM D5470.
1. Sandwich the TIM sample between two copper blocks
2. Heat one block and cool the other
3. Extrapolate the interface temperature difference from the temperature gradient within each block
4. $R_{TIM} = \Delta T_{interface} / q''$
Can I use the values from commercial TIM datasheets as-is?
Caution is needed. Datasheets often measure under ideal conditions (high pressure, perfect wetting). Values under actual implementation conditions (pressure, surface roughness, application amount) can be 1.5 to 3 times worse. For initial design, it's safe to estimate using 50% of the datasheet value.
Thermal Design of Bolted Joints
As a practical example, consider the bolted joint of a heat sink plate. For 4 M4 bolts, tightening torque 1.5 Nm:
- Bolt axial force: ~2600N per bolt
- Contact surface pressure: ~52 MPa at the seat surface (φ8mm)
- From the CMY model: $h_c \approx 8000$ W/(m2K) (Aluminum・Polished surface)
So you can estimate $h_c$ from the contact pressure.
Yes. However, pressure decreases between bolts, so a precise approach is to obtain the pressure distribution via structural analysis first, then map $h_c$ accordingly.
TIM Thermal Conductivity and Effective TCR
Intel's Core i9-13900K (2022) uses InFusion liquid metal TIM between the CPU die and IHS, reducing contact thermal resistance by about 50% compared to solid grease. Liquid metal (GaInSn-based) has a thermal conductivity of about 40 W/m·K, significantly exceeding that of high-end silicone grease (~12 W/m·K). However, it corrodes aluminum IHS, so it was changed to copper.
Analogy for Analysis Workflow
Think of the thermal analysis workflow as "designing a bathtub reheating system." Decide 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 "Will it 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. 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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