Heat Simulation of Heat Pipes
Theory and Physics
Operating Principle — Four Heat Transport Processes
Heat pipes are in laptops too, right? It's amazing that such a thin tube can carry so much heat. How does it work?
A heat pipe is a heat transport device that repeats four processes—evaporation → vapor transport → condensation → capillary return—inside a sealed tube. It's driven solely by the phase change of the working fluid and capillary forces, without using any pumps or fans. Its greatest strengths are zero moving parts and being maintenance-free.
Could you explain the four processes in more detail?
Let's look at them in order.
- Evaporation (Evaporator): The working fluid (usually water) evaporates at the end in contact with the heat source (like a CPU), absorbing latent heat $h_{fg}$.
- Vapor Transport (Adiabatic Section): The vapor moves rapidly to the condenser section due to the pressure difference. Compared to copper's thermal conductivity of 380 W/(m·K), the effective thermal conductivity of a heat pipe can reach thousands to tens of thousands of W/(m·K).
- Condensation (Condenser): The vapor condenses at the heat sink or fan side, releasing latent heat.
- Capillary Return (Capillary Return): The condensed liquid returns to the evaporator via capillary forces in the wick (sintered metal powder, mesh, grooves, etc.).
For example, in a laptop, it transports 15–30W of heat from the CPU (evaporator, ~95°C) to the heat sink near the hinge (condenser, ~55°C) with just a 40°C temperature difference.
The liquid returns without a pump? Can it work against gravity?
Good question. As long as the wick's capillary pressure $\Delta P_c = 2\sigma/r_\text{eff}$ exceeds the gravitational head $\rho_l g L \sin\phi$, the liquid can return even upwards. However, this becomes the major design constraint known as the "capillary limit." Especially in thin heat pipes, a smaller $r_\text{eff}$ is advantageous, but it also reduces permeability $K$, creating a trade-off.
Equivalent Thermal Resistance Network
So, how do you simulate this? Do you directly solve the two-phase flow with CFD?
In practice, the first step is the equivalent thermal resistance network. This approach treats the entire heat pipe as a circuit of series resistances, predicting the temperature distribution at less than 1/100th the computational cost of CFD.
The overall thermal resistance $R_\text{hp}$ is decomposed as follows:
The meaning of each term is as follows:
| Thermal Resistance | Physical Meaning | Formula | Typical Value (K/W) |
|---|---|---|---|
| $R_\text{e,wall}$ | Tube wall conduction at evaporator | $\ln(r_o/r_i)/(2\pi k_w L_e)$ | 0.01–0.05 |
| $R_\text{e,wick}$ | Wick conduction at evaporator | $\ln(r_i/r_v)/(2\pi k_\text{eff} L_e)$ | 0.05–0.5 |
| $R_\text{e,evap}$ | Evaporation interface | $1/(h_e A_e)$ | 0.001–0.01 |
| $R_\text{vapor}$ | Pressure loss in vapor flow path | $\Delta T_\text{sat}/Q$ | 0.001–0.01 |
| $R_\text{c,cond}$ | Condensation interface | $1/(h_c A_c)$ | 0.001–0.01 |
| $R_\text{c,wick}$ | Wick conduction at condenser | $\ln(r_i/r_v)/(2\pi k_\text{eff} L_c)$ | 0.05–0.5 |
| $R_\text{c,wall}$ | Tube wall conduction at condenser | $\ln(r_o/r_i)/(2\pi k_w L_c)$ | 0.01–0.05 |
How do you determine the wick's effective thermal conductivity $k_\text{eff}$? The wick itself is porous, right?
A representative method is Maxwell's effective medium approximation:
Here, $k_l$ is the liquid's thermal conductivity, $k_s$ is the wick solid's thermal conductivity, and $\varepsilon$ is the porosity. For a sintered copper powder wick ($\varepsilon \approx 0.5$, $k_s = 380$ W/(m·K), water $k_l = 0.65$ W/(m·K)), $k_\text{eff} \approx 30\text{--}50$ W/(m·K).
Capillary Limit
You mentioned the "capillary limit" earlier. How is it calculated quantitatively?
The condition for a heat pipe to operate is that the wick's capillary pressure must exceed all pressure losses:
Expanding each term:
- Capillary Pressure (Driving Force): $\displaystyle \Delta P_\text{cap} = \frac{2\sigma}{r_\text{eff}}$
- Liquid Phase Pressure Loss: $\displaystyle \Delta P_l = \frac{\mu_l L_\text{eff}}{K A_w \rho_l h_{fg}} Q$
- Vapor Phase Pressure Loss: $\displaystyle \Delta P_v = \frac{128 \mu_v L_\text{eff}}{\pi d_v^4 \rho_v h_{fg}} Q$
- Gravitational Head: $\displaystyle \Delta P_g = \rho_l g L \sin\phi$
Here, $L_\text{eff} = L_a + (L_e + L_c)/2$ is the effective length. The maximum heat transport $Q_\text{max}$ is found from the condition where equality holds:
Laptops use thin heat pipes, right? For thicknesses like 0.4mm, which term becomes the bottleneck?
Sharp observation. In thin (including Vapor Chamber) designs, the vapor flow path cross-sectional area is extremely small, so the $d_v^4$ term in $\Delta P_v$ becomes dominant. With a 0.4mm thickness, the equivalent diameter of the vapor flow path is only about 0.2mm, and vapor pressure loss can account for over 80% of the total. This means the "vapor flow path limit" often becomes the practical bottleneck, not the capillary limit. This is why recent gaming laptops use heat pipes thicker than 0.6mm.
Merit Number (Performance Index M) and Working Fluid Selection
Are there working fluids other than water? How do you choose?
The Merit Number (Performance Index M) is a single metric to compare the heat transport capability of working fluids:
A larger $M$ means more heat can be transported with the same wick structure and dimensions. Comparison of Merit numbers for typical working fluids:
| Working Fluid | Operating Temp. Range | Merit Number M (W/m²) | Typical Applications |
|---|---|---|---|
| Water | 30–200°C | $5 \times 10^{11}$ (60°C) | Electronics cooling, CPU, LED |
| Methanol | 10–130°C | $9 \times 10^{10}$ (60°C) | Low-temperature environments, plastic enclosures |
| Ammonia | -60–100°C | $5 \times 10^{11}$ (25°C) | Satellite thermal control, cryogenic |
| Acetone | 0–120°C | $6 \times 10^{10}$ (60°C) | LED lighting |
| Sodium | 600–1200°C | $2 \times 10^{12}$ (800°C) | Nuclear reactors, solar thermal power |
So water is the strongest in the normal temperature range. Is it the only choice for electronics?
Pretty much. However, water can react with tube materials and generate non-condensable gas (NCG), so copper tubes are standard. Putting water in an aluminum tube generates hydrogen, causing the heat pipe to fail within months. Checking material compatibility is a prerequisite before any simulation.
Other Operating Limits
Are there limits other than the capillary limit?
Heat pipes have five operating limits. All must be checked during the design phase:
- Capillary Limit: Limit of the wick's liquid return capability. Most critical in normal temperature ranges.
- Boiling Limit: Bubble formation and growth within the wick at the evaporator, hindering liquid supply. Occurs at high heat flux.
- Sonic Limit: Vapor flow velocity chokes (reaches sonic speed). A concern during startup or at low temperatures.
- Entrainment Limit: High-speed vapor flow shearing off the opposing liquid film. Judged by the Weber number.
- Viscous Limit: When vapor pressure is extremely low at very low temperatures. Relevant for liquid metal heat pipes during startup.
For electronics (water-copper, 30–100°C), the capillary and boiling limits are typically the design frontiers.
Numerical Methods and Implementation
Implementation of 1D Thermal Resistance Model
How do you actually incorporate the equivalent thermal resistance network into CAE?
The simplest method is to embed a 1D thermal resistance network into a system-level thermal circuit simulator (Ansys Icepak, Flotherm, 6SigmaET, etc.). The heat pipe is treated as a single "two-terminal element":
Here, $R_\text{hp}$ is obtained from a datasheet or calculated using the thermal resistance decomposition formula mentioned earlier. Practical points:
- $R_\text{hp}$ is nonlinear with respect to input heat $Q$ (high at low Q, rises sharply near the capillary limit).
- For transient response, add thermal capacitances $C$ for the wall, wick, and vapor in parallel to create an RC circuit.
- Iterative calculation is needed if temperature-dependent properties are used.
Equivalent Thermal Conductivity Model (Anisotropic $k_\text{eff}$)
If you want to see the temperature distribution across the entire board, a 1D model isn't enough, right?
Exactly. When there are multiple chips on a board or you need to evaluate in-plane heat spreading, model the heat pipe as a solid with anisotropic equivalent thermal conductivity. This method is often used for Vapor Chambers:
$k_\text{axial}$ is typically determined by fitting to measured values. It's adjusted so the difference from analysis results is within 3–5°C. Note that $k_\text{axial}$ depends on the input heat. It drops sharply near dryout, so parameters need to be switched depending on the operating point.
CFD Two-Phase Flow Model (VOF / Mixture)
In what situations do you actually solve the two-phase flow rigorously with CFD?
During R&D phases or when you want to study local evaporation/condensation phenomena in detail. There are mainly two methods:
- VOF Method (Volume of Fluid): Directly tracks the gas-liquid interface. Shows interface shape but strongly depends on mesh resolution. Used for analyzing liquid film behavior within the wick at the evaporator.
- Mixture / Euler Two-Fluid Model: Treats gas and liquid by volume fraction. Coarser mesh than VOF is acceptable, but interface details are lost.
In both, the Lee model is widely used as the evaporation/condensation source term:
Here, $r_i$ is the mass transfer coefficient [1/s]. The value of $r_i$ varies widely in literature from 0.1 to $10^6$, so calibration against experiments is essential. If $r_i$ is too large, it diverges; if too small, the interface temperature deviates significantly from $T_\text{sat}$.
Wick Porous Media Model (Darcy Flow)
How do you model the liquid flow inside the wick?
Since the wick is a porous medium, approximating with Darcy's law instead of the Navier-Stokes equations is standard:
$K$ is the permeability [m²], which depends on the wick structure:
| Wick Structure | Permeability $K$ [m²] | Effective Capillary Radius $r_\text{eff}$ [m] | Porosity $\varepsilon$ |
|---|---|---|---|
| Sintered Powder | $10^{-13}$ – $10^{-11}$ | $0.21 d_p$ (0.21 times particle diameter) | 0.4–0.6 |
| Mesh Screen | $10^{-11}$ – $10^{-9}$ | $1/(2N)$ (depends on mesh number $N$) | 0.6–0.7 |
| Axial Grooves | $10^{-9}$ – $10^{-7}$ | $w/2$ (half the groove width) | 0.1–0.3 |
Sintered powder has a small $r_\text{eff}$, so capillary driving force is large, but $K$ is also small, leading to high liquid pressure loss. Grooves are the opposite: $K$ is large but capillary force is weak. Sintered powder is mainstream in thin Vapor Chambers for smartphones, while axial grooves are common in space heat pipes.
Practical Guide
Analysis of Thin Heat Pipes (Vapor Chamber)
Recent smartphones have Vapor Chambers, right? I hear they're like 0.4mm thick. How do you analyze those?
A practical approach for Vapor Chamber (flat-plate heat pipe) analysis is a three-phase process:
- Phase 1 — 1D/2D Screening (1 hour): Thermal resistance network + capillary limit check. Narrow down candidate materials and dimensions.
- Phase 2 — Anisotropic $k_\text{eff}$ Model (1 day): System-level 3D thermal analysis. Evaluate temperature distribution on the board and thermal impact on other components.
- Phase 3 — Porous Two-Phase Flow CFD (1–2 weeks): Predict liquid distribution within the wick, dryout location. Only for developing new wick structures.
80% of practical work is completed in Phases 1–2. Phase 3 is for the R&D department.
Mesh Strategy
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