Okay, enough speculation. Time for some actual science.
Test Setup
Six LM317s configured as 0.125 A current sources propped up by silicon diodes. The diodes are there to make the numbers come out nice: with 10 V across the circuit, 1.25 V is dropped across the current-setting resistor, ~0.7 V is dropped across the diode, leaving a nice, round 8 V across the regulator. Since the current through the regulator is nominally 0.125 A, that makes for a very easy to handle 1 W. Ya might think I planned this or something.
From left to right, front to back, the configurations are:
1a. Naked regulator; b. paper clip heat sink. (It was easy enough to change from one to the other that I didn't bother making separate board instances for these two.)
2. 4x penny heat sink.
3. Aavid-Thermalloy model 577202B00000G heat sink.
4. Penny bolted to regulator.
5. Penny epoxied to regulator.
6. Penny paper-clipped to regulator.
Results
1a: 0.127 A through regulator, 75.0° C
1b: 0.127 A, 71.3° C
2: 0.125 A, 55.5° C
3: 0.126 A, 54.5° C
4: 0.126 A, 60.0° C
5: 0.126 A, 61.0° C
6: 0.126 A, 58.0° C
Discussion
1: The naked regulator is the baseline, and as you can see, it gets pretty hot. This is about as hot as I'd let a chip get before I'd say a heat sink is no longer optional. Adding a paperclip knocked a few degrees off. I'm surprised it did anything at all, actually. It's not terribly effective, but it's literally better than nothing.
2: This is four pennies bolted together, with nuts in between each penny to give some room for air to get in there. The penny closest to the heat sink is oriented to give maximum surface contact, and the others are oriented higher to get them more into the open air. You might instead orient the pennies in a rosette sort of pattern, but I didn't have enough room here to get away with that. As-is, it's nearly as good as the purpose-made heat sink. It takes some work to put it together, but if it's late at night, you don't have a heat sink, and you can't get down to your local Radio Shack, dig up some pre-1982 pennies, fire up the drill press and get crafty.
3: Bog standard heat sink. Available lots of places. Works well. Will probably cost you less than what it's protecting. Standard lore says parts will last twice as long for every 10° C drop, so this gives a nice round 4x lifetime increase. It'll literally pay for itself unless you're using some extremely cheap parts.
4: #2's little brother, and it works nearly as well. You might be able to equal its effectiveness by lapping the mounting surface flat to maximize the surface contact area.
5: Again, I'm a bit surprised. I would have expected to give up more than 1 degree by using glue instead of heat sink compound. Now, if you're dealing with more than 1 W of dissipation, the penalty here will get larger.
6: After seeing #1b, this makes sense. It's just adding the surface area of the paper clip to the penny, so you get a few degrees additional drop. You might get another degree or two of drop if you use some heat sink compound instead of allowing there to be an air gap.
Test Details
The regulators are all Fairchild LM317Ts. They're probably even from the same tube.
Each heat generator was allowed to cook for at least 15 minutes to get it up to thermal equilibrium.
Temperatures are approximations. I would often see the regulator swing back and forth over about a 1 degree range. The number I give above is my off-hand estimate of the center of the range.
Where possible, the temperature was measured in the same location. Regardless of actual location, I spent a few minutes on each part looking for the hot spot. The location I used most is down and to the left of the mounting hole, with the temperature probe jammed in the corner between the IC encapsulation and the tab.
All of the pennies are 1981 or earlier vintage, that being the last year US pennies were made mostly of copper. After that, they're mostly zinc, with some copper plating. (
Source) The pennies in cases 4, 5, and 6 are all 1981 pennies made in the same mint. I didn't feel a great need to find identical pennies for case 2.
In cases 2, 3, and 4, I used a silver-bearing heat sink compound. Not Arctic Silver ®©℠, but something of that sort. In cases 1 and 6, there's nothing added between the heat sink and the regulator. And in case 5, it's a fast-dryng epoxy.
In all penny heat sink cases, the surface towards the regulator is the reverse side, as it looks like the obverse has more relief. Also, Miss Manners says it's not nice to apply sticky substances to ex-Presidents' faces. (Miss Manners has nothing to say on the topic of power tools, so I'm also okay in cases 2 and 4.)
All pennies were cleaned in an acid bath to remove dirt and corrosion. I used
CLR initially, but switched to a vinegar and salt solution that was recommended on a coin collector's site. I flushed the acid off thoroughly with tap water.
Holes drilled through the pennies are the same size as the holes through the Aavid heat sink and the regulators' tabs. I used the thinnest screws I had on hand and clipped their length as well as I could using my wire stripper's bolt cutting holes to minimize their contrubution to the thermal bulk. (#2-56 size.) Normally you'd use something more like #6-40, or maybe even #8-32.