Equations of Extrusion

When I first tested my extruder I found that the filament diameter varied with the flow rate and temperature. This was contrary to what others have experienced so I decided to investigate further. It turns out that this is known as die swell and is caused by non Newtonian fluids expanding after they have been squeezed through a hole. Apparently it is a very complicated subject.

To get an idea of what was going on I designed my extruder controller to be able to make measurements. Rather than drive the motor with open loop PWM I used a shaft encoder with proportional feedback. Instead of specifying what PWM setting to use, the host specifies how many shaft encoder steps to move and at what rate. The extruder controller then adjusts the PWM to maintain the correct shaft position at any given instant. Assuming the filament does not slip against the drive screw, that means I can extrude a known volume of plastic in a known time to the tolerance of the the original feed material. The host can then ask the controller what the total on time and off times have been so that it can calculate the average power that has been used.

My temperature control works in a similar way. The host calculates the resistance the thermistor should have at the desired temperature, and from that, what voltage reading the ADC should produce. It sends that setting to the controller which turns the heater on and off. Again it keeps track of the total on and off times so that the host can calculate the average power.

My heater has a resistance of 8.5Ω and has 11.8V across it after the drop in the MOSFET switch and the wiring. That gives a power of 16.4W. This is a graph of the temperature reading from the thermistor plotted against the heater duty cycle :-

As you can see it is not quite a straight line. This is because the resistance of the nichrome heating element increases slightly as it gets hotter, so power does not quite rise in line with the duty cycle. I measured the resistance at 200°C to be 9.7Ω. Using the formula:

R = R₀[1 + α(T − T₀)]

that gives a temperature coefficient α of 7.8 × 10^-4 which is about twice the figure I found on the web for nichrome. I expect that it varies widely according to the exact alloy being used.

Here is a graph of temperature against power, calculated using the above formula for resistance :-

It is a lot closer to the straight line I was expecting.

I decided to investigate how much extra power is needed to heat the incoming plastic when extruding. I found that while feeding the filament in at 1mm/s, which is about the maximum my motor can do, the PWM to maintain 200°C increased from 44.6% to 61.2%. An increase of 16.6% corresponding to an extra 2.4W. Feeding a 3mm filament at 1mm/s gives a flow rate of 7.1 × 10^-3 cc/s. HDPE has a density of around 1 so that is 7 × 10^-3 g/s. The specific heat capacity is 2.2 J/g-°C which gives 2.8W to heat 7 mg from 20°C to 200°C per second. I think that is reasonably close to the value I measured, given that HDPE has quite a wide range of densities.

Next I decided to look at the effect of temperature on the motor power required to extrude at a given rate :-

I concluded that temperature has little effect on the motor power required, except when it gets close to the melting point, where it rises rapidly as expected. That was how I broke my extruder!

Next I looked at filament diameter against temperature :-

No real correlation, so it seems temperature is not very important as long as it is above the melting point. This was a surprise to me as I imagined molten plastic would get less viscose as temperature increased. It may become more critical when I start laying down filament as it will effect how it fuses together and shrinks. I did all the subsequent measurements at 200°C.

Feed rate (in mm/s) against PWM was another surprise. I expected power to rise rapidly with feed rate but, in fact, it is quite proportional :-

Presumable 30% is the power required to overcome static friction in the system.

Here you can see the output rate versus the feed rate :-

It does not increase in proportion, so if conservation of matter is true then it must be getting bigger in diameter. Indeed it does, here is output diameter against output rate :-


Either it is a very complex relationship with multiple inflexions or it is just linear with lots of measurement error. I made three measurements per test with digital calipers and took the average but the deviation between samples was quite high.

I prefer to think it is a simple linear relationship which means I can make a simple mathematical model of my extruder. As you can see it will hit the Y axis at about 0.93 mm. I think that must be the size of the hole in my nozzle. I drilled it 0.5mm but perhaps I drilled the hole from the back too far and opened it out a bit. It seems to have got bigger with use because I could get 0.8mm filament when I first tested it but I don't seem to be able to now, even at very low extrusion rates.

So if the filament diameter equals hole size plus a constant times extrusion rate then from conservation of volume I can relate the output rate to the feed rate.

d_o = d_h + kv_o

v_od_o² = v_id_i²

So: v_o(d_h + kv_o)² = v_id_i² a cubic equation!

Where d_o is the output filament diameter, d_i is the input filament diameter, d_h is the nozzle hole diameter and v_o is the output filament speed, v_i is the input filament speed.

With these equations I can calculate the output rate to get a particular filament diameter. That also tells me how fast to move the head. From the output rate I can also calculate the feed rate required.

Conclusion? Well I definitely have die swell which increases with extrusion rate but other people have reported constant die swell. The only explanation I can think of is that I drilled my nozzle too deep from the back so the aperture has almost zero thickness instead of the 0.5 to 1mm expected.

I have a simple mathematical model which allows me to exploit the variable filament width if I need to. This may all become irrelevant when I start laying down filament to build things because the filament can be stretched or compressed if the head movement does not match the output rate.

Tomorrow I will try laying down the filament.

Self destruction

Not managed any self replication yet but my machine has done a bit of self destruction!

While doing some experiments running my extruder at different speeds and temperatures, I managed to run it at too low a temperature such that it forced the PTFE barrel out of its clamp. That broke off one of the heater wires under the JB Weld. Fortunately I was able to dig out the end of the nichrome and reconnect it. I soldered the joint but that is not the best idea as solder melts at 183°C and I am running my barrel at about 200°C. The heater gets a bit hotter than that. Presumably molten solder is still a good conductor. The ideal way to make the connection would be with a miniature barrel crimp but I don't know if they exist. Here it is repaired :-



Clamping a very slippery plastic rod with a clamp made out of a slightly less slippery plastic is probably not the best design.

I seem to spend as much time stripping down and rebuilding my extruder as I do running it. Looking on the bright side at least the thermistor didn't fall off again despite some rough treatment. Here it is all back together again and working :-

Temperature drop

My extruder controller is working much better after I cured the motor noise problem. The 10 bit SAR ADC also seems to work better than the 16 bit sigma-delta version did. With the 16 bit one there was a lot of noise on the readings, even when the motor wasn't running. I had to average over many samples to get a consistent reading which delayed the response. With the 10 bit ADC I just read it and compare it with the temperature set point value to decide if the heater should be on or off. That gives a temperature swing of about ± 3°C with the heater going on and off every four or five seconds.

The temperature is calculated from the ADC reading and vice versa by the PC with the following Python class :-

from math import *

class Thermistor:
 "Class to do the thermistor maths"
 def __init__(self, r0, t0, beta, r1, r2):
     self.r0 = r0                        # stated resistance
     self.t0 = t0 + 273.15               # temperature at stated resistance, e.g. 25C
     self.beta = beta                    # stated beta
     self.vref = 1.5 * 1.357 / 1.345     # ADC reference, corrected
     self.vcc = 3.3                      # supply voltage to potential divider
     self.vs = r1 * self.vcc / (r1 + r2) # effective bias voltage
     self.rs = r1 * r2 / (r1 + r2)       # effective bias impedance
     self.k = r0 * exp(-beta / self.t0)  # constant part of calculation

 def temp(self,adc):
     "Convert ADC reading into a temperature in Celsius"
     v = adc * self.vref / 1024          # convert the ADC value to a voltage
     r = self.rs * v / (self.vs - v)     # resistance of thermistor
     return (self.beta / log(r / self.k)) - 273.15        # temperature

 def setting(self, t):
     "Convert a temperature into a ADC value"
     r = self.r0 * exp(self.beta * (1 / (t + 273.15) - 1 / self.t0)) # resistance of the thermistor
     v = self.vs * r / (self.rs + r)       # the voltage at the potential divider
     return round(v / self.vref * 1024)    # the ADC reading

It is instantiated as follows :-

thermistor = Thermistor(10380, 21, 3450, 1790, 2187)

10380 is the resistance of the thermistor measured by my multimeter at a room temperature of 21°C. 3450 is the beta of the thermistor taken from the data sheet. The last two values are the two resistors forming a potential divider with the thermistor wired across the second one, again the values are measured with a multimeter. The fudge factor of 1.357 / 1.345 corrects the MSP430 internal reference voltage so that it agrees with the multimeter.

The result seems to track the temperature measured by a thermocouple to within about 5°C, good enough for me. Just as I had finished checking it, I knocked the thermistor leads with my thermocouple and it fell off. It was stuck to the brass nozzle with JB Weld but I forgot to roughen the surface first. I am now waiting 16 hours for it to set again.

The extruder controller firmware is only about 400 lines of C. As well as temperature control it also controls the DC motor precisely using the shaft encoder and handles the I²C protocol.

I have now completed all the mechanical parts, the electronics and the firmware. I just need to get the RepRap host code to talk to my non standard hardware to complete the machine.