Heat Transfer in Pin-Fin Arrays
Theory: pin fins as a bank of thin cylinders
Overview
What's good about pin-fin heat sinks compared with plate fins?
Thin cylinders have thin boundary layers and high heat transfer coefficients, and they work with flow from any direction. But thin pins conduct poorly from root to tip, so the tip runs cooler and contributes less — captured by fin efficiency. Heat transfer comes from the Zukauskas correlation treating pins as a tube bank, and efficiency from fin theory. Let's see how dissipation changes with material and arrangement.
Zukauskas correlation
For Re = 10³ to 2×10⁵: in-line C = 0.27, m = 0.63; staggered C = 0.35(S_T/S_L)^0.2, m = 0.60. C₂ corrects for the number of rows, about 0.97 for ten rows.
Fin efficiency
Tip heat loss is approximated by the corrected length $L_c$, extended by d/4. Larger $mL_c$ means lower efficiency.
Heat transfer in tube banks
Tube-bank heat transfer has been studied through many experiments since the early 20th century for boilers and heat exchangers. In 1972 Lithuania's Zukauskas reorganised those experiments over a wide Reynolds range, giving correlations for both in-line and staggered banks. A pin-fin heat sink is that tube-bank idea scaled down; pin fins are also used for internal cooling of aircraft-engine turbine blades.
Worked example: a 10×10 pin-fin array
Pins 2 mm diameter, 15 mm tall, 4 mm pitch both ways, on a 40×40 mm base; 25 °C air at 5 m/s; base 40 K above air.
Heat transfer coefficient
| Arrangement | Max velocity | Re_max | Nu | h |
|---|---|---|---|---|
| In-line | 10.0 m/s | 1,274 | 20.9 | 275 W/m²K |
| Staggered | 10.0 m/s | 1,274 | 21.9 | 288 W/m²K |
Material, efficiency and dissipation (in-line)
| Material | Conductivity | Fin efficiency | Dissipation |
|---|---|---|---|
| Copper | 390 W/mK | 0.90 | 111 W |
| Aluminium | 200 W/mK | 0.83 | 103 W |
| Steel | 16 W/mK | 0.35 | 51 W |
| Plastic | 1 W/mK | 0.09 | 24 W |
A bare 40 mm square plate gives h = 44 W/m²K and 2.8 W. Pins multiply area by 6.9, and with the high coefficient of thin cylinders, aluminium dissipates 37 times as much.
The same shape in steel halves the dissipation.
Pin h is high, about 280 W/m²K, so heat leaves the surface quickly. If conduction from root to tip can't keep up, the tip cools and contributes little. Aluminium pins work at 83% to the tip, steel at 35% and plastic at only 9% — plastic pins are effectively used only near the root. So the higher the heat transfer coefficient, the thicker, shorter and more conductive pins must be. Conversely, with low coefficients as in natural convection, low-conductivity materials lose much less efficiency.
Design tips
- Compute Reynolds number with the maximum velocity between pins (check diagonal gaps for staggered).
- Check fin efficiency; if below about 0.7, make pins thicker and shorter or change material.
- Account for bypass over and around the array; ducting improves performance.
- Compute pressure drop and match the fan curve to find the operating point.
- Pins too close put downstream pins in the wake of upstream ones.
“The flow went around the pins”
One heat sink failed to cool as well as its catalogue value. Investigation found a large gap above it: much of the fan air bypassed the pins through the low-resistance gap. Pin fins have high pressure drop, so air takes any escape route available. Covering the top with a plate to close the gap increased flow through the pins and lowered temperature considerably.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Re from approach velocity | h underestimated | Use max velocity |
| Efficiency = 1 | Dissipation overestimated | Compute efficiency |
| Ignoring bypass | Poor real performance | Duct the array |
| Ignoring pressure drop | Insufficient airflow | Find fan operating point |
| Forgetting row correction | Overestimate for few rows | Apply C₂ |
I'd like to learn related topics.
Try the finned heat sink tool. Related pages include cylinder in crossflow, jet impingement, microchannel cooling and natural convection in vertical channels.
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