Conjugate Heat Transfer Analysis
Theory: the heat transfer coefficient isn't known in advance
Overview
In thermal analysis, a heat transfer coefficient on the surface represents cooling. Why solve the fluid as well?
Because the coefficient isn't a material constant — it depends on the flow and the wall temperature distribution. In the same flow, the leading edge with its thin boundary layer cools well and downstream cools less; and if wall temperature varies, air warmed upstream weakens cooling downstream. If the solid spreads heat well, the wall is nearly uniform and an average coefficient suffices, but for poorly conducting solids the local peak changes a lot. Conjugate analysis solves solid and fluid together to handle this. Let's compare plate materials.
Interface conditions
Both temperature and heat flux are continuous at the solid–fluid boundary; the heat transfer coefficient comes out of the solution afterwards.
Fluid side: response to wall temperature
Superposing the laminar flat-plate solution for a temperature step starting at $\xi$ (the unheated-starting-length solution) gives the heat flux for any wall temperature distribution. Here this relation was solved simultaneously with 2D conduction in the solid.
The name 'conjugate'
Solving heat in solids and fluids together is usually traced to Perelman and colleagues in the Soviet Union, who treated it systematically in the 1960s as the 'conjugate problem'. Analytical solutions were then limited, but computers made solving solid and fluid on one mesh routine. Conjugate analysis is now standard in electronics thermal design and turbine blade cooling.
Worked example: a heated plate cooled by air
A 100 mm plate is heated uniformly from below at 2000 W/m² and cooled on top by 20 °C air at 2 m/s (laminar, Reynolds number 11,100). No other heat enters or leaves.
Simple estimates
| Method | Coefficient | Surface temperature |
|---|---|---|
| Uniform average coefficient | 17.4 W/m²K | 135 °C (uniform) |
| Local coefficient (trailing edge) | 11.9 W/m²K | 188 °C |
Conjugate results (2 mm thick)
| Material | Conductivity | Top min / max | Spread | Bottom max |
|---|---|---|---|---|
| Aluminium | 200 W/mK | 134 / 141 °C | 7 K | 141 °C |
| Steel | 16 W/mK | 104 / 159 °C | 55 K | 159 °C |
| FR-4 | 0.3 W/mK | 47 / 186 °C | 139 K | 199 °C |
On FR-4 the top reaches 75 °C at x = 10 mm, 140 °C at 50 mm and 180 °C at 90 mm — hotter downstream. Aluminium stays nearly uniform at 134–141 °C. In every case heat in at the bottom equalled heat out at the top.
With aluminium the average coefficient is fine, but for FR-4 it underestimates the peak by 50 K.
Aluminium spreads heat sideways toward the well-cooled leading edge, so the whole plate sits at nearly one temperature — close to the uniform wall the average coefficient assumes. FR-4 barely moves heat sideways, so heat escapes upward where it's generated; the poorly cooled trailing edge gets hot, approaching the 188 °C set by the local coefficient alone. That's why downstream components on low-conductivity boards run hotter, and an analysis with an average coefficient misses it. Steel sits in between, with results sensitive to thickness and material.
How to run the analysis
- Match meshes at the interface or connect them by interpolation.
- Resolve the thermal boundary layer near walls (check wall-function conditions for turbulence).
- For transients with very different time scales, solving fluid steady and solid transient is an option.
- Compare with a constant-coefficient analysis to see where they differ.
- Include surface radiation where temperatures make it significant.
“Only the downstream parts ran hot”
In a power-board thermal analysis with an average coefficient on the fan-cooled face, every component had margin. On the prototype, only the downwind components exceeded limits: air warmed by upstream parts flowed downstream, and the board couldn't spread heat sideways, so downstream cooling fell well below assumptions. A conjugate analysis of board and air reproduced the trend, and rearranging components fixed it.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Uniform average coefficient | Local peak missed | Check with conjugate analysis |
| Coarse interface mesh | Heat-flux error | Resolve boundary layer |
| Ignoring contact resistance | Temperature underestimated | Add interface resistance |
| Not checking energy balance | Unconverged results unnoticed | Compare heat in and out |
| Ignoring upstream heating | Downstream underestimated | Solve the flow too |
I'd like to learn related topics.
Try the forced convection tool and heat sink simulator. Related pages include natural convection–conduction coupling, mixed convection, heat pipes and boiling heat transfer.
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