# Hot End Design Theory

The hot end is arguably the most complex aspect of 3d printers as it deals with the tricky business of melting and extruding plastic filament. To understand the design features of hot ends, you must have a basic knowledge of the thermal properties of thermoplastics, specifically the way they behave at their glass transition temperature (Tg).

## Contents

### Glass Transition Temperature (Tg)

At temperatures below Tg, thermoplastics retain their hard, solid consistency (as we see in plastic filament). As the temp rises above the Tg of the thermoplastic, its consistency changes from solid to rubbery and it begins to expand.

### Melting Temperature (Tm)

If you continue to increase the temperature, the filament will eventually hit its melting temperature (Tm). At the melting temperature, the plastic becomes a liquid. Once the plastic is in the liquid phase, it can be extruded.

### The Critical Transition Phase

The transition phase between the Tg and Tm temperatures is the most critical point of the extrusion process. Just before hitting the liquid phase, the consistency of the filament is rubbery.

In this rubbery transition state, the plastic will expand and grip the inside of the hot end and will resist extrusion/retraction and thus increase the likelihood of the hot end jamming. As a result, the hot end developer makes an effort to mitigate this problem by reducing the area that the rubbery plastic can grip and cause jams (by shortening the transition zone), and by reducing the friction between the rubbery plastic and the interior walls of the hot end (by polishing the internal pathway within the hot end). This rubbery filament problem is more apparent when extruding PLA which has a very low Tg (about 60C).

## RESEARCH

### Temperature vs extrusion speed

Willy has done a number of interesting measurement series: http://forums.reprap.org/read.php?252,217620 . He adjusted his extruder to loose steps at some specific torque, then he tested how fast he could extrude at different temperatures. The result is, the hotter the heater is, the faster one can extrude (not surprising) and also, that this relation is pretty much linear (a bit unexpected) over the entire 170 °C to 260 °C range tested on a piece of PLA.

To sum up this work in one equation:

$V_{max} = k(T_{HotEnd}-T_{softening})$

Vmax = k (THotEnd - Tsoftening)

Where

• Vmax is the maximum velocity achievable by a given extruder. (aka, nozzle pressure for the max torque the extruder motor can handle)
• THotEnd is the temperature of the hot end. Note that the filament temperature is somewhat lower than this, especially in the center.
• Tsoftening is the softening temperature of the filament. This is the lowest temperature at which it is possible to extrude; around 153°C for PLA. This should be approximately equal to the Vicat softening point.
• k is some empirically determined constant. It is a property of the extruder. In theory, k should scale with both nozzle area (aka, Pi*R^2) and the torque the motor can produce. More efficient hot ends should also contribute to a higher k, since the filament temperature should be closer to THotEnd.

It would be interesting to conduct these sorts of tests for different nozzle diameters and filament sizes. Thinner filament should heat more quickly, allowing it to be extruded more rapidly. Smaller nozzle apertures would create higher back pressure, limiting extrusion speed.

One researcher speculates that: Perhaps the ABS in this experiment isn't really getting heated up all the way to 260 degrees C. Perhaps the thermistor is measuring 260 degrees C at one point, but the rapid injection of cold ABS plastic is keeping the actual temperature of the ABS plastic at the tip at some lower temperature, creating a strong temperature gradient. (Assuming a constant thermal resistance, the amount of heat energy per second flowing down that temperature gradient is proportional to the difference in temperatures).

### ideal hot end

What *should* happen in the extruder, independent of how this is mechanically implemented?

#### shape

Is there an optimum shape inside the nozzle to transition from the input feedstock to the output filament? In other words: Is it better to have a blunt, sharp transition, or is it better to have a very gradual taper from the 3 mm or 1.75 mm feedstock as it comes from Printing Material Suppliers, to the output filament exiting a hole typically 0.5 mm diameter?

#### thermal conductivity

What Thermal Conductivity does the hot end really need to have? Researchers initially thought that the hot end needed to have a high thermal conductivity -- so the first RepRap, Darwin, used lots of 109 W/(m*K) brass in the hot end. More recent researchers seem to think lower thermal conductivity would be better -- 16 W/(m*K) stainless steel in the Strong Nozzle, 1 W/(m*K) Glass Nozzles, etc.