Wednesday, July 7, 2010

Heated Bed Theory

Heated Bed Theory

As this is the first post in a while, I should mention that the SpoolHead project is on hiatus for now. Our team has opted to stick together for another open-ended project course next year, APSC 479, in which we are quite likely to pursue RepRap-related work. But no decisions have been made as of yet, nor have we decided if we'll continue developing the SpoolHead or whether we'll pursue another development.

In the meantime, unrelated to the subject of wire-printing, I've been observing how much interest there's been in heated beds. I thought I'd post an observation I've had regarding the selection of materials for these, because there's a fundamental tradeoff to wrestle with. Namely, if you're building a heated bed out of a slab of Aluminum, how thick should it be, from a thermal point of view?

Materials all have three important thermal quantities: Conductivity (if two ends are held at different temperatures, how much heat will flow across the material), heat capacity (if heat is dumped into a cold object, how long will it take to warm up), and operating temperature range (will it burn?). It's really important to consider both of these properties when choosing a material. The third one is generally pretty obvious; for example, it rules out using PLA as a primary material for a heated bed, because it would melt.

A steel rail feels colder than a wooden one in the morning because it conducts heat much better than wood, so when it touches your warm hand, the heat moves through the rail quickly. Although lots of heat is flowing into the steel from your hand, it quickly gets pulled away from the surface and distributed through the rest of the rail because of the high conductivity. But conductivity is not the only important effect here. The steel rail is also massive enough, with a high enough heat capacity, that the heat flowing in from your hand does not bring it rapidly up to your body temperature. If it did, it wouldn't feel cold for very long. On the other hand, if a substance had an extraordinarily high heat capacity and a poor conductivity, it might also feel quite cold because the heat drawn from your hand wouldn't warm the surface much, even though that heat stays put. So it's important to see how both of these properties can work together.

Anisotropic materials have different conductivities in different directions. Aluminum, an isotropic material, has the same conductivity in every direction, but materials with a grain (like wood) or a sheetlike crystal structure (like graphite) do not.

Aluminum is a very good conductor of heat - one of the best that's cheaply available. Copper is a bit better, but you wouldn't want to use it for a heated bed because it has a higher heat capacity (per unit volume, since it's so dense), so it would take a long time to heat up the bed.

For a heated bed, the lowest possible thermal mass is desirable, because then it will take less time to achieve the target temperature. There's two ways to reduce the thermal mass: Make the bed thinner, and choose a material with a lower heat capacity.

Making the bed thinner, besides introducing structural concerns, has another problem against it. It reduces the ability of the heated bed to distribute the heat, which can result in hot spots. A sheet of Aluminum foil would do little to distribute the heat.

The conductance along the plane of a sheet of material is:


where t is the thickness and hp is the material's thermal conductivity along the plane.

The conductance per unit area through the thickness is


Where ht is the material's thermal conductivity through the thickness. (ht = hp for an isotropic material like Aluminum).

To achieve heat spreading and avoid hot spots, we want heat to go across the bed but not through it. So a good measure of the heat-spread-ability of our bed is to take the ratio of these two:

t*hp/(ht/t) = t^2*hp/ht

For an isotropic material, this is just t^2. So the heat spreading power increases with the square of the thickness: if you double the thickness of the bed, the temperature difference between hot and cold spots will be 1/4 as much as before. And this measure of heat-spread-ability actually *doesn't depend* on the conductivity of an isotropic material at all! This is a little counter-intuitive, but it's a relatively simple analysis. A high heat conductivity has other advantages: It reduces the temperature difference between the hot (bottom) and cold (top) sides of the heated bed, meaning when the desired temperature on the build surface is achieved, it won't be scorching hot below.

Still, this is a very important conclusion to consider. We can't go boosting the material thickness forever. The thermal mass of the bed is proportional to the thickness, and it's desirable to minimize this. So we want to get away with the thinnest bed we can. In order to do that and still spread heat effectively, we want a very high ratio of hp/ht - an anisotropic material. Ideally one that can still conduct heat reasonably well through the thickness, but not *too* well.

Consider a laminated structure of thin Aluminum sheets, like the ones available from McMaster-Carr. I haven't tested anything like this, but if the laminations have less than stellar thermal contact, I think it would be much better for our purpose. The in-plane conductivity would still be very high but the through-thickness conductivity would be a fair bit lower, because the heat would have to flow through successive contacts. I don't know how rigid this kind of material is, though.

As a very enticing alternative, consider an anisotropic stone like slate. I don't know the ratio of hp/ht for slate, although I do know that it's greater than one. Slate is cheap and machinable; it can be milled, drilled, tapped, cut with a bandsaw, and it's naturally very flat (because it has shear planes). Slate has a low coefficient of thermal expansion: about 9x10-6 per C, which is less than half of that of Aluminum. It has a lower specific heat capacity too, but its significantly higher density offsets this advantage somewhat. We might try using this material for our print bed upgrade perhaps...

Of course, one other way to reduce hot spots is to use a distributed heater, like nichrome wire. This method highly favours heated beds that are not electrically conductive, because then there's no risk of electrical shorts.

I think it's worth looking into other materials for heated beds than just Aluminum, because there certainly are other options worthy of consideration. And thermal anisotropy is a good quality to look for.