The Thermal Core’s time constant falls neatly out of the high power measurements when they’re plotted against time:

Rule of thumb:

- Three time constants get you to 95% of the final value.

The two plateaus for the 6 W and 14 W power inputs give enough information to pull out the time constant. They’re pretty much flat after 30 minutes (which is why I turned up the power then!), so the time constant is on the order of 10 minutes.

Remember that the time constant doesn’t depend on the heater power. Higher power means the Core would stabilize at a higher temperature, but the overall curve would have the same time constant. What you’d see, though, is a faster rise to a given temperature, at which point the controller turns off the power to maintain the Core at that temperature.

In round numbers, a first-order exponential rises to 10% (really 9%) of its final value in the first 0.1 time constant. In this case, guesstimate a 10-minute time constant, apply power, measure the temperature rise after a minute, and the final temperature should be ten times that value above ambient. Various nonlinearities get in the way, but that’ll get you close to the right answer.

Just for the amusement value, I applied 22.5 W to the extruder and recorded the temperature every 10 seconds for the first 1200 seconds. I’m not going to graph it, but the salient points are:

- 21 °C — Ambient temperature
- 199 °C — Final temperature (four consecutive dupes)

So the final temperature rise is 178 °C above the 21 °C ambient. Multiply by 0.09 = 16 °C, so when the temperature passes 37 °C = (21 + 16) it’s 0.1 time constants from the start.

Squint into that table and you’ll find the temperature is 36 °C at 60 seconds and 40 °C at 70 seconds, so the time constant is pretty nearly 625 seconds… call it 10.5 minutes, which I’d say is pretty close to my original eyeballometric guesstimate from that crappy graph up at the top.

Dang, I love it when the numbers work out!

You could do it in your head: 10% is 17.8, truncate to 17, add 21, get 38, interpolate 65 seconds. That’s just shy of 11 minutes, still close enough for what we’re doing.

Now, the tradeoff is that you have a ten-minute delay before your MK5 Extruder can begin squeezing out parts. That may not be a big deal if you’re looking at an hour of extrusion for each part and, after the head is up to operating temperature, it just keeps on running.

Adding more power increases the final equilibrium temperature and increases the initial temperature rise, so the head gets up to operating temperature faster. More power adds more stress to the resistors and shortens their life, which is nasty, brutish, and short even at the power levels I’m using.

You can run the numbers the other way. Measure the temperature rise at 0.09 time constant, multiply by 10, and you’ve got the final temperature rise. If you’re running 60 W into those heaters, a firmware lockup or thermistor failure that leaves the heaters jammed on will stabilize at maybe 350 °C over ambient. Nope, the resistors aren’t going to survive that experience…