## Friday, 6 March 2009

### 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.

1. Very impressive.

2. Your observations about the glass plug are very interesting Nophead. I think this may also explain why a rapid thermal transition leads to a better extruder.

3. Fascinating! Thank you.

4. I'm not sure wether I should praise you for your work, or rather hate you for making me shelve one extruder design after another, fearing that, should I finally make one, you'd post some new findings the following day that would reveal its crappiness ;)

Seriously, it really looks like no one before did so much experimentation and serious analysis on the extruder design. This could get rather amusing actually, when it turns out that your designs could match or maybe even outperform some commercial ones (on the second thought, isn't your machine already a bit faster than some of the cheaper Strats?)

On a more technical note, I think I can see a potential problem with those welding nozzles, as they would screw into the heater barrel, not onto it like every other nozzle variant. That would introduce a potential leak (the molten plastic would be pushed down straight into the thread) and a noticeable bore reduction right in the middle of the melt chamber. I'm not entirely sure what consequences such a reduction could have, but I guess they won't be very good. Other than that, making an upwards taper inside the melt chamber without damaging the inner thread seems pretty difficult, even with the 3,6-5mm reamer you have there.

5. Well done!

6. As always, thanks Nop!

7. This comment has been removed by the author.

8. "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."

Using your expression with the extruder barrel and orifice on T1 I had...

((.127/.5)^4)x3/25 = ~5

9. Forrest,
You have lost me with those figures. The first ratio is pipe bore over nozzle hole and the second ratio is nozzle hole depth over the pipe length.

IIRC your pipe was quite long and the hole very shallow so you would get a lower number meaning less of the force is pushing plastic through the nozzle.

But my point was that the formula predicts the force to push plastic down the barrel is a lot less than the hole, but that is not actually the case because the viscosity is very high at the top end of the barrel.

Where I think is is useful is at the exit of the extruder, where for example, it shows that drilling out the welding tip makes a massive difference to its contribution.

Also if the entrance of the heater opens out, as is the case when the insulator screws into the heater, then the place where the viscosity is very high is a larger diameter so resistance drops with the fourth power. That also makes the taper doubly beneficial.

10. Just a curiosity question, but has anyone considered a pressure vessel for the molten plastic? A small vessel that has filament as an input with a secondary pressure source to drive the output (ie. the extrusion bead) would bypass the GTT plug back force on the filament, and allow a more even extrusion rate, as it wouldn't be fighting non-fluid friction to extrude the fluid. Also, it would be easy to modify to accept the plastic in pretty much any format (filament, beads, or recycled scrap), as long as it would fit in the heating neck before the pressure vessel. It would most definitely not be a simpler solution, as it would require some sort of compressor or other pressure source, but it would give some interesting flexibility in terms of being able to switch extruder nozzles on a single print head for different bead widths, and also would allow a wider choice of feedstock.

11. No I don't think anybody has suggested that for extruding to make objects but there have been various suggestions along those lines for making filament from granules or recycled plastic.

Extruding at an even rate is not too big a problem, you simply feed the filament at an even rate. The plug causes the force to vary but as long as your motor's speed is not affected by torque it is OK. The instability in this experiment is because I am using a constant force, not a constant feed rate.

12. Hey hey hey, hold on there! You've got some serious misunderstandings.

First off, the equation that you are using (the Poiseuille equation) assumes that the viscosity is constant. That's great for "everyday" liquids for horrible for molten polymers. The polymers that you are working with exhibit "shear-thinning" meaning that the viscosity decresases as the shear rate increases. Since you have a range of shears in the flow channel, you do not have the parabolic profile that the Poiseuille equation predicts.

Additionally, the equation assumes that the flow profile was completely developed, meaning that it had "forgotten" that it was earlier in a larger cross-section. This typically takes about a length of about 20 times the diameter of the exit section. There are also entrance effects in going from the a large cross-section to a small cross-section that you are completely ignoring and that add to the pressure drop.

Now for the die swell: Most polymers when sheared, exhibit "normal forces" perpendicular to the direction of flow. In this case, in the radial direction. The retention of the pressure after exiting the die is the cause of the swelling. Your explanation doesn't explain why water and other "everyday" fluids show the opposite behavior - flow contraction after leaving a die.

Rheology is a very complicated subject that unfortunately cannot be quickly assimulated, even in small pieces. You really need to look deeper into the subject.

13. Hello John,
Thanks for pointing out the errors in my theory. I have removed the section about die swell, as clearly nonsense.

Given that viscosity reduces with shear rate in polymers, does that mean I should expect flow rate to increase faster than linear with pressure drop, assuming a long uniform pipe?

Does flow rate still vary with the fourth power of pipe bore?

Chris

14. It would, but given we have a variety of internal diameters, expansions, contractions, varying shear rates leading to variation in the viscosity throughtout the nozzel, coupled with varying temperature... Oh no, I've gone cross-eyed!

This is a very difficult problem to derive an analytical solution to. Most equations in rheology relate to simple geometry, with variables such as temperature maintained constant, especially when dealing with non-newtonian fluids.

15. I better stick to the empirical approach then. Now that I have tapered the transition zone, the results seem a lot more consistent so I should be able to get some decent graphs.

16. You can further narrow the transition zone with a Peltier-cooled barrel a few mm above the heater, concentrating temperature differential to the gap between barrels, or rather the pipe in that region.

You might adapt your current barrel design by machining a flat face in its side, sandwiching a small Peltier cooler between the flat face and a heat sink; or you might place the cooler somewhere between your current heat sink and a barrel affixed beneath.

Here in the States I believe the cost would be between $15 and$35 for a 15x15mm or 20x20mm junction.