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Theory and Physics
What is Thermal Contact Resistance?
Professor, is it true that even when two metals are joined, there's a temperature jump at the interface?
It's true. Actual solid surfaces are full of microscopic asperities, and the true contact area is only about 1-5% of the apparent area. The remaining gaps are filled with air or grease. This imperfect contact causes a temperature discontinuity. This is thermal contact resistance.
Only a few percent are actually touching?
Yes. Therefore, the thermal contact resistance $R_c$ is defined as the ratio of the interface temperature jump $\Delta T$ to the heat flux $q''$.
Its unit is [m$^2$ K/W]. Its reciprocal is the contact conductance $h_c = 1/R_c$ [W/(m$^2$ K)].
Governing Factors
What determines the magnitude of thermal contact resistance?
There are four main factors.
| Factor | Effect | Typical Value |
|---|---|---|
| Surface Roughness $\sigma_s$ | Rougher increases $R_c$ | Ra 0.1〜10 μm |
| Contact Pressure $P$ | Higher decreases $R_c$ | 0.1〜100 MPa |
| Interstitial Material | Grease can reduce $R_c$ to 1/10 | — |
| Material Hardness $H$ | Softer increases true contact area | — |
The Cooper-Mikic-Yovanovich model is widely used.
Here, $k_s$ is the harmonic mean thermal conductivity $k_s = 2k_1 k_2/(k_1+k_2)$, $m$ is the surface slope, $\sigma$ is the composite roughness $\sigma = \sqrt{\sigma_1^2 + \sigma_2^2}$, and $H$ is the microhardness.
So it's almost proportional to the contact pressure to the power of 0.95. That means bolt clamping force is effective.
Exactly. In heat sink mounting for electronics, bolt torque management directly affects temperature. In some cases, doubling the tightening torque can roughly double the contact conductance.
Three Generations of Micro-Contact Models
The theoretical models for thermal contact resistance have evolved through three generations: (1) GreenWood-Williamson (1966: elastic contact) → (2) Majumdar-Bhushan (1991: fractal surfaces) → (3) Persson (2001: full-scale elasticity). The Persson model originated in the context of car tire friction but gained attention for its application to thermal contact, leading to widespread adoption in precision equipment design in the 2010s.
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 and cools down slowly, while an aluminum pot heats up 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 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 low. Insulation materials (like glass wool) have extremely low $k$, so heat transfer is 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 associated with fluid motion. 【Everyday Example】 Feeling cool under a fan is because the wind (fluid flow) carries away 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 in a PC's CPU cooler also dissipates heat via forced convection. Convection is an order of magnitude more efficient at heat transport 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 where heat is supplied from the outside to the "surface," 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 and 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 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 is thermal contact resistance handled in FEM?
Either by creating a thin virtual layer at the interface or by directly specifying the conductance value to contact elements. The basic discretization is expressed by a thermal conductance matrix between interface nodes.
Assembling this at the element level yields the interface heat transfer matrix.
It's mathematically the same form as the Robin condition for convective boundary conditions.
Good observation. Implementation-wise, it can be handled the same as the Robin condition. However, pairing of contact surfaces (matching master/slave surfaces) is additionally required.
Gap Conductance Setting
In practice, variable conductance depending on the contact state is often used.
| Contact State | Conductance $h_c$ [W/(m$^2$ K)] | Application Scenario |
|---|---|---|
| Perfect Contact | $10^5$〜$10^6$ | Welded joints, shrink fits |
| Grease Filled | $10^3$〜$10^4$ | Heat sink mounting |
| Direct Metal-to-Metal Contact | $10^2$〜$10^4$ | Bolted joint surfaces |
| With Air Gap | $10^0$〜$10^2$ | Loose fits |
There's a difference of over four orders of magnitude. Getting this estimation wrong would completely change the results.
Exactly. The standard practice is to perform a sensitivity analysis, checking the temperature difference with $h_c$ halved and doubled. If the results strongly depend on $h_c$, obtaining measured values should be considered.
Nonlinear Thermal Contact Resistance
Pressure-dependent or temperature-dependent thermal contact resistance becomes a nonlinear problem. It involves coupling with structural analysis to determine contact pressure, calculating $h_c$ from that pressure using the CMY model, and feeding it back to the thermal analysis. This iteration is repeated until convergence.
So coupled structural-thermal analysis is needed.
Ansys Mechanical and Abaqus automate this coupling. In Ansys, you just need to set TCC (Thermal Contact Conductance) for contact elements CONTA174/TARGE170.
Actual Measurement with Laser Flash Method
The laser flash method per ASME D5470 standard is used for precise measurement of thermal contact resistance. It involves stacking samples and calculating the TCR of the contact interface while varying press pressure. The Netzsch LFA 467 (released 2020) can measure in the range of 0.5〜50 mm²·K/W with ±2% accuracy. It is widely adopted for power semiconductor package evaluation.
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 Judgment
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
The explicit method is like "weather forecasting that predicts the next step using only current information"—fast to compute but unstable with large time steps (misses storms). The implicit method is like "prediction that also considers 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
Analysis Flow
Please explain the procedure for analysis including thermal contact resistance.
The standard flow is as follows.
1. Identify Contact Surfaces: Clearly define surface pairs that contact each other in CAD.
2. Determine Conductance Value: From measured values, literature values, or calculated using the CMY model.
3. Set Contact Pairs: Define master/slave surfaces and assign $h_c$.
4. Mesh Conformity: Adjust the mesh so that nodes are aligned on opposing contact surfaces.
5. Solve & Verification: Verify that the temperature jump at the interface is physically reasonable.
Is mesh conformity mandatory?
It can be solved with non-conforming meshes (Mortar method or GGI connection), but conforming meshes tend to yield higher accuracy for contact surfaces. Especially when the temperature jump is small, artifacts from mesh non-conformity can become visible.
Typical Thermal Contact Resistance Values
| Interface | $R_c$ [m$^2$ K/W] | Condition |
|---|---|---|
| Al-Al (Polished, No Grease) | $2 \times 10^{-4}$ | P=1 MPa |
| Al-Al (With Thermal Grease) | $5 \times 10^{-6}$ | k=5 W/(m K) Grease |
| Cu-Cu (Polished) | $1 \times 10^{-4}$ | P=1 MPa |
| Si-Heat Spreader (With TIM) | $1 \times 10^{-5}$ | TIM Thickness 50μm |
| Bolted Flange | $10^{-4}$〜$10^{-3}$ | Varies near bolts vs. far away |
Thermal grease improves it by two orders of magnitude!
That's because grease fills the gaps instead of air (k=0.026 W/(m K)). However, grease degradation over time (pump-out, drying) should also be considered; values after degradation should be used for long-term reliability evaluation.
Result Verification
Here are the verification points for thermal contact resistance analysis.
- Confirm Temperature Jump: Plot node temperatures on both sides of the contact surface and check if $\Delta T = q'' \cdot R_c$ holds.
- Energy Conservation: Check if the heat flow passing through the contact surface matches upstream and downstream.
- Sensitivity Analysis: Quantify the impact on results by varying $h_c$ by $\pm$50%.
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