in the last few columns, i described methods to calculate pressure drop and heat transfer in rectangular channels, as well as spreading resistance. these are various aspects of basic heat sink performance prediction.
in this column, i thought i would pull together an overview of the whole procedure. this applies to a straightforward heat sink with a base and uninterrupted vertical fins (see diagram/photo).
- find air flow rate through the heat sink. use the fan curve and the pressure drop calculations in earlier columns (january & february). take into account any flow paths other than through the heat sink ("bypass"). calculate air-side thermal performance by using procedure in march column, with a fin modification if necessary. usually this takes the form of a "fin efficiency" (see below).
- use the procedure outlined in april's column to get a spreading thermal resistance, if that scenario applies to you.
- add the air-side and spreading thermal resistances to get the total heat sink thermal resistance.
- for design work, try some parametric variations to see general trends. that is actually the most powerful reason to go the effort of calculation: to get design guidance.
- take the whole thing with a grain of salt. there are many simplifying assumptions we've made along the way. also, if you're going to "validate" the calculation with experimental data, keep in mind the many, many possible sources of error in generating that data. the biggest one is usually the fact that heat from the chip (or heater) is going to take any possible escape route. the heat sink might be the most efficient one, but it isn't the only one, so not all the heat will go that way. this has the effect of making the experimental data look "better" than your calculation. in my experience, even if you go to quite a bit of effort to make all the heat go through the heat sink, about 10% of it still escapes by other routes. if you don't expend that effort (and you don't always need to), even more will leak away. actually, this "leakage" of heat is usually a fine thing, because it effectively reduces your chip temperature without any effort on your part!
this efficiency is a calculational trick, and not a design goal in itself. it ranges from 0 to 1. essentially it's a correction factor for the fact that the tip of the fin is cooler than the base, being thermally farther away from the source of the heat than the root of the fin.
calculating the fin efficiency is not always necessary. extruded heat sinks have thick fins for extrudability; it's rarely worth the extra effort to calculate fin efficiency. thin fins in natural convection have such a low heat transfer coefficient that the fin efficiency is very high, and again it's typically not worth the calculation effort.
"thin" is a relative term, of course; for heat sinks, if the fin is made of a sheet of material, it's probably thin. skiving also produces thin fins. whether you want to expend the calculation effort to include fin efficiency depends on what kind of accuracy or design guidance you're looking for. generally speaking, you'll want to include it for thin fins in forced convection.
calculation formulae for fin efficiency are in most introductory heat transfer texts. look for "extended surfaces" in the index if "fins" aren't listed. once you get to the right section, you probably want to look at the simplest case, a fin of constant cross-sectional area (as opposed to tapered, pin, or wrapped around a cylindrical core).
skip the whole bit about the temperature profile within the fin, and go right to fin efficiency. the simplest version is "insulated tip", meaning that the calculation ignores the heat transferring from the itty-bitty area at the tip of your thin fin. (you do have thin fins, don't you? otherwise, why are you bothering?)
so, you're wishing now i would just save you from wading through texts and give the formula? all right, but beware of the greek letters…
fin efficiency where l is the fin height andnote that k is the conductivity of the fin material. a is the cross-sectional area that's transferring heat up the fin. usually this is the fin thickness times the flow length of the heat sink. moving air produces heat transfer coefficient h (remember the nusselt number?) and p is the "perimeter" of the fin.
the perimeter is basically two times the flow length of the heat sink plus two times the thickness of the fin (negligible for thin fins); think of it as the length of a string wrapped around the fin at a constant height.
so now that you have the fin efficiency, what do you do with it? the way to apply it is to use instead of just (ha) in the log-mean temperature difference (see march column). check that for a given heat flux, the surface temperature increases when you use the fin efficiency; that way, you'll know you wrote the formula correctly. refrain, however, from placing too much emphasis on what the actual fin efficiency value is.
it's not a design goal, but a calculation tool, and possibly a diagnostic to help you figure out why the heat sink performance isn't quite as high as you'd like.
fin efficiency calculator (if your calculator-maker supports hyperbolic tangent…)
fin height l
heat sink flow length
heat transfer coefficient h
material conductivity k (default = 205, extruded aluminum)
area a = (fin thickness)*(heat sink flow length)
perimeter p = 2*(heat sink flow length + fin thickness)
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 cross-functional 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.