# Hotend theory

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).

To sum up this work in one equation:

<math>V_{max} = k(T_{HotEnd}-T_{softening})</math>

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

Where

- V
_{max}is the maximum velocity achievable by a given extruder. (aka, nozzle pressure for the max torque the extruder motor can handle) - T
_{HotEnd}is the temperature of the hot end. Note that the filament temperature is somewhat lower than this, especially in the center. - T
_{softening}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 Vicant 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 T
_{HotEnd}.

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).