over the last several months, we've looked at several different hand (or rather spreadsheet) calculations that can be used to size or analyze heat sinks in forced convection. it's a long and iterative process.
fortunately, for heat sinks in natural convection, the process is much simpler as long as the heat sink base and fins form vertical chimneys (see illustration). in fact, there is a definite optimum for the fin spacing, where the temperature difference (sink to ambient) is minimized. or, to think of it another way, the optimum intends to maximize the heat flux for a given temperature difference.
for most applications where the heat flux is fixed, there's still a bit of iteration involved, but nothing you couldn't do by hand (not to mention you can make your spreadsheet do it). in fact, coolingzone already has a little calculator to do this for you; see the problem statement here.
(the nomenclature shown there is a little different from that used below.)
here's the "recipe" as described in white (ref. 1):
calculate the rayleigh number using the vertical length l. it is extremely important to use the correct length  the one aligned with gravity is the one you want.
where g is the acceleration of gravity, is the thermal expansivity of the fluid (for an ideal gas, you can use 1/t in a pinch, where t is the absolute temperature), is the temperature difference between the fin channel and the incoming ambient air, pr is the prandtl number of the fluid, and is the kinematic viscosity of fluid. many heat transfer texts have the quantity () listed in the physical properties tables.
the optimum spacing . that's all there is to it. notice the scaling with l in this relationship: actually s varies as (in the definition of ra, l is raised to the power of 3), a pretty weak dependence. this weak dependence is rather good news if you don't know exactly how big the heat sink needs to be, or if you are limited to instock heat sink material.
if you have optimum spacing, you can get the heat transfer coefficient from the sbased nusselt number, . (nus = hs/k, where h is the heat transfer coefficient and k is the conductivity of the fluid  not the heat sink!) then, to get the total heat transfer from the array, use .
remember to count both sides of the fins for the area a. i usually also include the base area that is exposed to the induced flow in the channel.
if for some reason you end up far off the optimum, you'll need to use other naturalconvection correlations to get the heat flux. kraus and barcohen (ref. 2) has several variations. one look at the many greek letters there will convince you that maybe the using the optimum is a good idea!
a couple of comments are in order here. most importantly, this correlation and optimum only applies to heat sink fin channels that form vertical chimneys. both the heat sink base and the heat sink fins must align with the gravity vector. i cannot emphasize this enough! heat sinks with horizontal bases and vertical fins have a completely different (and usually unstable) flow pattern. there are correlations for that geometry out there, but my experience with them has been unsatisfactory for sizes not included in the original study.
another comment concerns general design direction. in a previous article, we discussed fin efficiency and when it's necessary to use it. my experience has been that for natural convection, the fin efficiency of even very thin fins is quite high. with that in mind, design the fins to be as thin as the manufacturing technology will allow within your budget. and keep it simple. there is little to be gained from the extra trouble of making pinfins, wavy fins or other "enhanced" geometries for natural convection applications.
references: 1. f. m. white, heat and mass transfer, addisonwesley, 1991. 2. a. d. kraus and a. barcohen, design and analysis of heat sinks, j. wiley, 1995.
about cathy biber
dr. catharina biber is senior thermal engineer at infocus corporation where she works with product design teams to solve optical and electronic cooling issues in advanced digital data/video projection systems. she particularly enjoys collaborating with crossfunctional team members to address all the aesthetic, manufacturability and regulatory aspects of design needed for a successful product.
previously, she was a technical staff member at wakefield engineering, inc., where she was involved in the design, analysis, and optimization of high performance heat sinks. she has taught seminars on electronics cooling and basic thermal analysis throughout the u.s. and in europe.


