Rheology

Rheology is the study of the flow of matter and that is what I have mainly been doing for the past few weeks. When I made my experimental set-up to measure flow rate versus extrusion force I expected to be able to produce some nice graphs for different plastics and different temperatures. I found this excellent page which derives the formula for flow rate I in a pipe in terms of pressure P, radius a, viscosity η and length L.

I = πΔPa⁴ / 8ηL

A cylindrical section of flow is considered. Since it flows at a constant speed the force pushing it forwards, which is the pressure plus the viscous drag from the faster inner cylinder, must equal the force retarding it, which is the viscous drag from the slower outer cylinder. Integrating twice yields the formula.

Until recently I had assumed that the large amount of force required to extrude was due to pushing viscous plastic through a tiny hole. The equation shows that for a given flow rate and viscosity, the force is proportional to the length and inversely to the fourth power of the bore.

The RepRap V1.1 extruder has a heater barrel that is 45mm long with a 3mm bore and a nozzle with a 0.5mm hole that is about 3mm long, so that would mean that it is about (3/0.5)⁴×3/45 = 86.4 times harder to push the plastic through the nozzle than the heater. However, that assumes the viscosity is constant. At the point where the plastic melts the viscosity tends towards infinity, so the actual force required to push the filament through the heater is much higher. I have had some extruder configurations where it was hard to push the filament even without the nozzle attached. This simple experiment showed that cutting off 5mm of the heater barrel from the cold end made a significant difference.

Despite these observations I expected the flow rate to be directly proportional to pressure and, with a constant pressure provided by gravity, I expected the flow rate to be constant. In fact the flow rate varies wildly from one run to the next and often increases towards the bottom of the fall. Flow is not directly proportional to pressure, it increases faster than pressure does, and lower pressures seem to give more erratic results.

I tried improving my test equipment to see if I could get more consistency. I reduced the size of the opto tab to record just the last 20mm of the fall, so things had plenty of time to reach equilibrium. I also made a piece to guide the tab into the slot as the weights have a tendency to rotate and make it catch.



I also tried force cooling the heatsink with a small fan. I made a cowling to stop the fan cooling the heater.



This is probably the most complicated shape I have modelled so far. The only mistake I made was not leaving enough room for two of the nuts to hold the fan. I used self tapping screws instead. If I were designing it again I would put tubular bosses behind the screw holes and use four self tappers. It takes some time to get used to designing in plastic. I tend to use a lot of nuts and bolts, and so do RepRap designs, but they are rarely used in commercial plastic products.



The fan didn't seem to make much difference when extruding ABS, either in the variability or the flow rate. If it did affect the flow rate its effect was lost in the variability.

So after some thought about where the variability was coming from I came to realise that it is an inherently unstable experiment. A lot of the force required is pushing the solid plastic plug through the entrance to the extruder.

For ABS and PLA, which both have glass transitions, the situation in the thermal transition zone looks like this.



When the filament meets the point in the thermal barrier where the temperature is above Tg (the glass transition temperature) the filament transitions from its glassy brittle state to a soft rubbery state. In this state it will change shape as force is applied, but it will not flow. Further down it gets to the point where it melts and becomes a very viscous fluid until it warms up to extrusion temperature, where the viscosity is much less. The soft plug gets compressed length-wise by the extrusion pressure, which makes it expand outwards and grip the wall of the insulator. This greatly increases the force required to push the filament, which in turn causes even more outwards force. If the plug is long enough, relative to the coefficient of friction with the wall, it can become impossible to slide it along. Applying more force simply exerts more force against the tube wall, increasing the friction to match the extra push. This is the condition that causes the extruder to jam.

A plug is formed even in plastics without a glass transition, like HDPE and PCL. Molten plastic simply flows backwards until it freezes.

The plug acts like a piston pushing the molten plastic out of the nozzle. Its front face is continually consumed by melting, but the back is replaced by new plastic that is softening.

To prevent the jam, either the coefficient of friction has to be low, or the thermal transition, and hence the plug, has to be short. An outward taper seems to help a lot.

I was asked for a drawing of my tapered stainless steel transition zone, so I drew one from measurements and extrapolation of the taper. The result was scary: -

I hadn't realised I got so close to rupturing the pipe, although it may not actually be as close as the drawing implies. It does work well though.

The reason the plug leads to an unstable result is that the slower the filament travels, the longer the plug is and so the resistance increases and the flow slows further. I.e. a positive feedback effect. It is also why increasing the force gives a disproportionate increase in flow rate. The faster flow reduces the plug length (because the plastic has less time to absorb heat) reducing the resistance, so more pressure gets to the nozzle, increasing the flow rate.

One implication of this effect is that an open loop DC motor is never going to work well. Another is that measuring the force applied to the filament is not a good guide to the nozzle pressure.

I think a more consistent experiment would be to extrude at the desired rate and measure the force applied. The plug would then have a fairly constant length and hence the force should be fairly constant.

Although I cannot get any accurate measurements from the experiment, I did get a rough idea of the force required to extrude various plastics at the extrusion speed I use. I.e. I added weights to get the flow rate around π mm³.

Material	Diameter	Temperature	Nozzle	Weight	Flow rate
HDPE	3.1 mm	240 C	0.5mm	4.60 Kg	3.81 mm3
HDPE	3.1 mm	200 C	0.5mm	4.60 Kg	2.39 mm3
PCL	2.8 mm	150 C	0.5mm	4.60 Kg	3.44 mm3
ABS	2.7 mm	240 C	0.5mm	2.32 Kg	3.67 mm3
PLA	2.9 mm	200 C	0.5mm	3.32 Kg	6.95 mm3

The viscosity of PCL and PLA drops rapidly with temperature, for example PLA would not extrude at all at 180°C but was very fast at 200°C.

The next thing to try is putting a taper in my PEEK extruder and evaluating the copper welding nozzles.

Three times better

I made a plastic mounting plate to allow me to test the new stainless steel extruder in my test rig.



Here it is under test: -



I felt that this design was working well. Now I have the figures to back it up. It is three times better! I.e. with the same weight the extrusion rate is three times faster through the same nozzle and at the same temperature. With 8.27Kg I am now extruding HDPE at 9.43 mm³/s.

This is a dramatic improvement, especially considering it did not work at all until I added the taper to the end of the transition zone. It shows that the design of the entrance to the extruder is critical and at least as important as the exit.

It is really good news because using stainless steel as the insulator really simplifies the extruder and at the same time extends its temperature range and makes it strong and reliable.

By replacing the heater block with one made with a vitreous enamel resistor and a screw-in welding nozzle, I should have a design that can be made with a drill press, a couple of taps and a die.



I haven't tried the welding tip yet, but now I have a means of comparing it against the standard nozzle.

If at first you don't succeed ...

Remember this?



It was my last attempt to get a high temperature extruder idea working. ABS jammed in it, so I put it to one side. This morning I made a slight modification and got it to work extremely well.

The filament was getting stuck in the end of the stainless steel tube where it enters the heater block. I removed the PTFE tape from the threaded joint as I thought that may have been insulating it. That made a small improvement. I could push ABS through it by hand, but only just.

I then flared the hole with this tapered reamer so that it has a 5mm inside diameter at the end, tapering back to 3.6mm, which is the internal diameter of the stainless steel tube.



That made all the difference, I can now extrude my oversized ABS very easily and even HDPE only requires moderate force.

I am not sure why it made so much difference. It makes the wall thinner, so the heat from the heater can get to the plastic easier. It also reduces the friction of the plastic against the inside pipe wall because any downwards motion causes the plastic to come away from the wall.

The next step is to connect it to my test rig to get some comparative pressure figures. My feeling is this extrudes more easily than my PEEK version. That may well benefit from a taper as well.