# Increasing hotend temperature to compensate for increased filament throughput

It seems that when filament throughput is increased (by increasing movement speed or extrusion width/height), printing temperature also has to be increased to compensate, because the filament will have less time to spend in the melting zone. That much seems clear from practical experience. But I have two questions (or to be more precise, one question on two levels):

1. Is there a good rule of thumb for this, to help people calibrate their settings?

2. How much do we know about the formula governing this behavior? Can we calculate the required hotend temperature precisely based on the increased throughput?

For anyone who has studied physics / thermodynamics, this is probably simple stuff. But has the work been done for 3D printing specifically, in a way that is practically applicable?

I share the following train of thought to start off with. Let me know if I make any errors in reasoning.

• Presumably, every material has an optimal printing temperature just above its melting point.

• But the thermistor doesn't read filament temperature. It reads the heat block temperature.

• Below a certain throughput, the temperature of the filament will have time to equalize with the temperature of the heat block before it leaves the nozzle.

• For those slow speeds, heat block temperature should be set exactly to the material's optimal printing temperature.
• For greater speeds, however, heat block temperature will always have to be higher than the mark, because the filament doesn't have time to equalize.

• At that point, it becomes a balancing act. Find the best heat block temperature (°C) given a rate of throughput (mm³/s), the optimal printing temperature for a given material (°C), the volume of the melting zone (mm³) and < some other property of the material >, which determines how fast it heats up. I don't know what that last property is, nor can I come up with the proper unit. The material probably approaches the temperature of the environment asymptotically. This is where thermodynamics comes in, I guess.
• Theoretically, running filament also cools down the heat block, but we can ignore this. If this effect is significant at all (is it?), this is already compensated for by the PID controller.

I'm almost certainly missing some key insights. I'm curious to know what work has been done.