Even with all the hype about forced convection cooling, many applications in the electronics cooling industry still depend on natural convection cooling. The natural buoyancy of heated air allows designers to avoid active cooling (fans), increasing reliability and decreasing costs. Unfortunately, due to the viscosity of air, the spacing between fins in natural convection must be a given distance to avoid “choking” of the resulting boundary layers. This fin spacing results in heatsink volumes three to four times larger than equivalent forced convection heatsinks. Let's look at some guidelines for optimizing this fin spacing based on air travel length and temperature rise.
The basic formula governing the heat transfer due to convection is:
Q=hAΔT
Where:
Q=Heat transfer
h=Heat transfer coefficient
A=Area
ΔT=Temperature differential, heatsink over ambient air
The amount of heat removed from a heatsink is directly proportional to the fin surface area and the effect of heat removed per square unit area. More fins mean more heat removal at a given temperature rise, fewer fins mean more heat remains in the sink, increasing temperature. So why not just put as many fins as possible on a heatsink? The answer lies in the other term of the equation “h,” the average amount of heat transferred from each square centimeter of surface area. This coefficient varies based on a wide variety of factors, including temperature rise and fin length. Closely spaced fins require a higher pressure to move heated air away from the fins than natural convection can provide. If fins are closely placed, especially in natural convection, the heatsink will have as much usable surface area as a brick.
How close is too close? What is the optimal spacing? The answer lies in determining the type of airflow available, natural or forced convection, and how much pressure head will be available to move the air.
The optimum spacing that achieves the best thermal performance depends on the thickness of the boundary layer of stagnant air molecules. These molecules retain heat in the heatsink and increase temperature rise. Interference of these layers from both sides of a channel, also known as fully developed flow, slows heat removal and causes a choking effect. The thickness of these layers determines the heat transfer coefficient “h.” This layer of dead air acts like a blanket to insulate the fin surfaces, keeping heat inside the metal and preventing effective dissipation. The thickness of this “blanket” depends on the physical parameters (viscosity, conductivity, and density) of the air at stream temperature as well as the air speed. In natural convection, this layer can be many millimeters thick. With one boundary layer from each side of a pair of fins plus a path of escape for the heated air, the fin spacing can be significant, depending on length and surface temperature. In many cases, open gap between fins can be 12 mm or more. Of course, this is based on the length of fins in the air travel direction in respect to gravity. The figure, on page 52, indicates the effects of various fin spacings.
In natural convection, determining a series of dimensionless numbers helps to give optimum fin spacing in a given surface area. These dimensionless entities are:
 Grashof number — the ratio of heated air buoyancy to viscous forces resisting air movement.
 Prantl number — the ratio of air momentum to thermal diffusivity. This tells the engineer the amount of internal stresses inside an airflow stream. Prandtl is the reciprocal of the Reynolds number used in forcedconvection analysis.
 Rayleigh number — the product of the Grashof and Prantl number. This dimensionless number determines the type of airflow (laminar, transition, or turbulent along a heated fin surface).
To help turn these dimensionless numbers into a useful engineering tool, the following equation is offered ^{[1]}.
(Optimum fin spacing/Length) Grashof No.×Prandtl No.=50
This formula tells us that the maximum heat transfer at a given fin length (or height with respect to gravity) and a set temperature rise equals the Grashof number of 50.
To give more detail, although a more complex formula, the following composite is:
Optimum Spacing=0.29 (L^{0.25}×Dyn. Visc.^{0.5}×T_{air}^{0.25})/(g^{0.25}×density^{0.5}×temp. rise^{0.25})
Where:
L=Length of the fin in air travel direction
Dyn. Visc.=Dynamic Viscosity
T_{air}=Bulk air temperature in absolute degrees  K_{air}
G=Gravitational constant
Density=Air density
Temp. rise=Anticipated temperature rise of the heat dissipation surface
Following is an example using this formula to show optimum fin spacing:
Length of heatsink (direction of gravity)=~5 in. (127 mm)
Dynamic Viscosity=0.144×10^{4} lb/ft sec
Ambient air Temperature: 560 K (38°C)
Estimated fin temperature rise: 100°F (55.5°C)
Gravity=32.2 ft/sec ^{2}
Density=0.06 lb/ft^{3}
In this case, these conditions result in an optimized air gap between fins of 0.0245 ft, or approximately 7.5 mm.
The table shows air gap in inches. By varying the heatsink length and anticipated temperature rise, you can obtain the fin spacing curve for any specific design. This will allow the designer to maximize heat removal in a given heatsink volume.
References

Spalding, D.B. Convective Mass Transfer, McGrawHill New York 1963.

Larson, R., How to Build a Natural Convection Cooled Heat Sink, 1980.

Incropera, F.P, P. Dewitt, Fundamentals of Heat and Mass Transfer 3^{rd} edition, John Wiley and Sons 1990.

Kraus, A. D. and A. BarCohen, Thermal Analysis and Control of Electronic Equipment, MacGrawHill, 1983.
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