Natural vs. Forced Cooling Design

Theory: how much heat air can carry

Overview

Natural convection coefficient

For a vertical plate of height $L=0.1$ m at 55 °C in 25 °C air (30 K difference), air properties at the 40 °C film temperature give $Ra_L \approx 2.2\times10^6$, $\overline{Nu} \approx 22.8$ and a heat-transfer coefficient of about 6.2 W/(m²·K).

The radiation share

For the same plate with emissivity $\varepsilon = 0.9$ (black anodising or paint), $h_{rad} \approx 6.3$ W/(m²·K). From one 10 cm square face, about 1.9 W leaves by convection and about 1.9 W by radiation. Bare aluminium (emissivity roughly 0.05–0.1) loses almost all of the radiative part.

Forced convection and the mixed-convection criterion

Same plate as above (0.1 m long, 30 K difference). At around 0.2 m/s the coefficient is barely different from natural convection and buoyancy still matters. Treating a weak fan or leakage flow through an enclosure as pure forced convection leads to large errors.

Design calculations: fins, fans and temperatures

Fin spacing for natural cooling

For 10 cm tall fins at 30 K ($Ra_L \approx 2.2\times10^6$ as above) the optimum spacing is about 7 mm, which is why natural-convection heat sinks usually end up with 6–10 mm gaps. Forced cooling allows tighter spacing, but at the price of pressure drop, so it has to be chosen together with the fan operating point below.

Fan operating point

Take a fan with 40 m³/h free delivery and 50 Pa shut-off pressure (curve approximated as a straight line) on a system with resistance coefficient $K = 0.05$ Pa/(m³/h)², then add a filter that doubles $K$. Power is 50 W and air $\rho c_p \approx 1170$ J/(m³·K).

Thermal resistance to the junction

With 10 W, $R_{jc} = 1.0$ K/W, thermal grease $R_{cs} = 0.3$ K/W and 40 °C ambient, a natural-convection sink ($R_{sa} = 4.0$ K/W) gives $T_j \approx 93$ °C and a forced-air sink ($R_{sa} = 1.2$ K/W) about 65 °C. Both are within the 125–150 °C limit of many semiconductors, but the natural-cooling margin is small and easily lost with higher ambient or altitude.

Derating at altitude

At 3000 m the air density is about 0.74 of sea level. A fan moves the same volume flow, so mass flow drops and the exhaust temperature rise grows by about 1.35×. The natural-convection coefficient scales roughly with the square root of density and falls to about 0.86×. Whichever method you use, recalculate at the specified altitude's density.

Modelling in CFD

Natural-cooling analysis

• Always enable buoyancy (Boussinesq or ideal gas). A wrong gravity direction is a surprisingly common slip.
• Include space outside the enclosure — above it, roughly two to three times the enclosure height — so the rising plume is not pushed back by the boundary.
• Include a radiation model that accounts for view factors between surfaces. Without it, heat dissipation can be underestimated by almost half.
• Flows are slow and driven by small temperature differences, so convergence is slow. If a steady run won't converge, the flow may genuinely fluctuate; run transient and average.

Forced-cooling analysis

• Fans are usually represented by a boundary condition or model driven by the fan curve (flow vs static pressure) rather than detailed geometry. Use a moving reference frame (MRF) when swirl matters.
• Model filters, perforated plates and grilles as porous media with pressure-loss coefficients from the manufacturer or measurements.
• Put several cells across each fin gap so the velocity and temperature profiles can form. One cell per gap gets the heat-transfer coefficient badly wrong.
• Represent components with compact thermal models such as two-resistor models instead of full detail to keep the model size manageable (see thermal resistance (Rth) models).

Common mistakes

Mistakes and fixes