Extended Surface Heat Transfer (Fins) — Comprehensive Analysis

It is evaluated by the Overall Surface Efficiency $\eta_o$. It represents the effective heat dissipation capability combining the fin surfaces and the exposed base surface.

Here, $N$ is the number of fins, $A_f$ is the area of one fin, $A_t = N A_f + A_b$ is the total surface area, and $A_b$ is the exposed base area.

Correct. The worse the fin efficiency, the lower $\eta_o$ becomes. For practical heat sinks, $\eta_o$ is typically around 0.7 to 0.9.

The total thermal resistance of a heat sink is:

$R_{\text{spread}}$ is the spreading resistance (occurs when the heat source is smaller than the base), and $R_{\text{base}}$ is the conduction resistance through the base thickness.

When a CPU die (e.g., 30mm square) is mounted at the center of a heat sink base (e.g., 60mm square), this is the resistance encountered as heat spreads laterally within the base. It can be estimated using the Song-Lee-Au formula.

$a$ is the equivalent radius of the heat source, and $\Psi$ is a function of base thickness and the Biot number.

Separate from fin efficiency, fin effectiveness $\varepsilon_f$ is also an important metric.

It represents the heat dissipation multiplier compared to the case without fins (heat dissipation only from the fin root area $A_c$). It is generally considered that there is no benefit to adding fins unless $\varepsilon_f > 2$.

Efficiency is "what percentage of the fin area is effective," while effectiveness is "how many times the heat dissipation increased by adding the fin." One can be high while the other is low.

It changes significantly. Let's compare the efficiency of typical fin shapes.

Triangular fins, which are thick at the root and thin at the tip, have good efficiency due to material distribution matching the temperature profile. The tip area of a rectangular fin is at a lower temperature and contributes little to heat dissipation, making it wasteful.

Annular fins (annular fins) attached to pipes or cylinder outer surfaces are solved using Bessel functions.

$r_1$ is the inner radius (root), $r_2$ is the outer radius (tip). $I_0, I_1, K_0, K_1$ are modified Bessel functions.

Using efficiency charts (Fig. 3.20 in the Incropera textbook) is practical. You read from the graph using two parameters: $r_2/r_1$ and $mL_c$. In Python, you can calculate directly using scipy.special.iv/kv.

Analytical solutions are based on the assumption that $h$ is uniform. In actual fin arrays, $h$ varies significantly due to flow development and vortex generation. Conjugate Heat Transfer (CHT) analysis with CFD, which automatically calculates local $h$, often results in 10–20% lower heat dissipation than analytical solutions.

That's why a two-step approach is practical: initial design using analytical solutions, and final verification using CHT analysis.

The standard flow is as follows.

1. Determine Heat Generation: Confirm TDP (Thermal Design Power)

2. Confirm Allowable Temperature: Junction temperature upper limit (e.g., $T_j \leq 105$°C)

3. Calculate Allowable Thermal Resistance: $R_{\text{hs}} \leq (T_j - T_a)/Q - R_{jc} - R_{\text{TIM}}$

4. Select Heat Sink Candidates: Choose from catalog those with $R_{\text{hs}}$ below the calculated value

5. CFD Verification: Verify reflecting the implementation environment (airflow speed, adjacent components)

Correct. $R_{jc}$ (junction-to-case) is listed in the IC manufacturer's datasheet. $R_{\text{TIM}}$ (Thermal Interface Material) is typically around 0.1 to 0.5 K/W.

Wind tunnel testing is the most reliable method for verifying heat sink performance.