Posts
Introduction-
Thus far I have not been totally clear on the specific physics, and set of process models to which we should appeal for guidance in designing and determining the appropriate extrusion settings for Rotoforge... The process parameters are sufficently divergent from those typically used in friction stir welding, and additive friction stir deposition, due to the change in scale and the general opacity of the literature, that I have not found existing literature of the most help.
However, after further research in wire drawing, reading a variety of papers and guides written in the 30s, 50s, 70s, and 90s, and running a few experiments most of which failed, but a few of which resulted in stable extrusion of aluminum 6061-T6 wire, from a 1.6mm OD wire feedstock, without a shaft liner, without a guide tube, and without any special modification to the hollow shafted brushless DC drone motors I have been using, nor any special materials, beyond the stainless steel acorn nut nozzle of yore, I believe I have a reasonable set of theoretical assumptions from basic physical first principals which can guide our choice of printing parameters with some reliability.
In the sections below I will first describe which constitutive physical equations I use to determine appropriate printing conditions, then use them to arrive at a set of acceptable printing parameters for aluminum 6061 for Rotoforge, which I have tested and know to be at least valid for stable extrusion.
The First Principles-
Formula for axial force in wire drawing:
F = K * σy * (do/di) * (1/tan(α/2))
Where F is the force required to push the wire through the die, σy is the yield strength of the wire material, do is the outer diameter of the wire, di is the inner diameter of the die, α is the angle of entry, and K is a constant that depends on the frictional coefficient between the wire and the die.
I have opted for this description because it seems to produce more realistic results than cutting force equations in machining or friction stir welding, and is a relevant description of a system nearly identical to Rotoforge, but without the rotation. I have also found it very difficult to obtain stable extrusion with diameter reductions below about ~30% as per typical wire drawing intuition in the literature.
Formula for power due to friction:
Q = F * V * μ
Where Q is the power due to friction, F is the force required to push the wire through the die, V is the tangential velocity of the wire at the die entrance, and μ is the coefficient of friction between the wire and the die material.
This equation probably ignores the resulting heat from sliding friction of the extruding material moving perpendicular to the axis of rotation through the die, but I suspect, based on the velocity of rotation compared to the extrusion speed, that the friction due to linear sliding friction is negligible... I have also opted, due to the surface area to volume ratio of Rotoforge extrudate and feed stocks, to ignore the heat contribution due to plastic deformation. My reasoning for this, is that the surface area of our system per unit length of material extruded, is more than 5X larger, than the volume. This indicates that the vast majority, ~80% or more, of the total heat evolved in the system is due to surface friction, not bulk plastic deformation.
Formula for temperature rise in a solid due to heating:
ΔT = Q / (ρ * c * A * L)
Where ΔT is the temperature rise of the wire due to frictional heating, Q is the power due to friction, ρ is the density of the wire material, c is the specific heat of the wire material, A is the cross-sectional area of the wire, and L is the length of wire that passes through the die in one second.
A Working Example-
To calculate the temperature rise in a 1.6 mm OD, 6061-T6 aluminum metal wire, if it is forced through a 1.4 mm internal diameter X 0.5mm long, 316L stainless steel die, with a 60 degree entry angle, which is spinning at 15,000 RPM, at a wire length feed rate of 2 mm / second, we need to determine the amount of heat generated due to friction between the wire and the die.
Figure 1. Initial conditions of interest, before the wire touches the entrance of the rotating die.
This heat generation can be calculated using the following equation:
Q = (μ * F * v) / (2π * r) * l
where Q is the heat generated per unit length of the wire (in watts/meter), μ is the coefficient of friction between the wire and the die, F is the force required to push the wire through the die (in Newtons), v is the velocity of the wire (in meters/second), and r is the radius of the wire (in meters), l is the length of the wire in contact with the die.
First, let's calculate the force required to push the wire through the die. This can be determined using the following equation:
F = (π/4) * (D^2 - d^2) * L * σ
where D is the outer diameter of the wire, d is the inner diameter of the die, L is the length of wire passing through the die per unit time, and σ is the yield strength of the wire material.
Plugging in the given values, we get:
D = 1.6 mm = 0.0016 m
d = 1.4 mm = 0.0014 m
L = 2 mm/s
σ = 276 MPa (for 6061-T6 aluminum)
F = (π/4) * (0.0016^2 - 0.0014^2) * (0.002 m/s) * 276e6 Pa
= 88.4 N
This is within our extruder's capabilities, so we can be sure that we can start the wire through the die when the system starting cold, at a minimum.
Next, we need to determine the velocity of the wire. Since the die is spinning around the wire at 15,000 RPM, we can use the peripheral velocity at the wire surface, which is given by:
v = ω * r
where ω is the angular velocity of the die (in radians/second) and r is the radius of the die orifice (in meters). Since the wire is entering the die at a semi-cone 60 degree angle, the effective radius of the die can be calculated as:
r_eff = r / sin(60)
where r = 0.0014 m is the radius of the die.
Plugging in the given values, we get:
r_eff = 0.0014 m / sin(60) = 0.0016 m
ω = 2π * 15000 / 60 = 157.08 rad/s
v = 157.08 rad/s * 0.0016 m = 0.51 m/s
Finally, we need to determine the coefficient of friction between the wire and the die. This can vary depending on the materials and lubrication used, but a typical value for dry friction between aluminum and stainless steel is around 0.5. and where l is the length of wire in contact with the die entrance and interior,
Plugging in all the values into the heat generation equation, we get:
Q = (0.5 * 88.4 N * 0.51 m/s) / ((2π * 0.0008 m) * 0.0005 m)
= 8,969,176 W/m
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| Figure 2. Heat is generated according the heat generation equation which includes the friction in the die at the applied constant extruder force, the constant RPM, and the geometric parameters of the die. This "evolved heat" is primarily a result of mass flow at a constant frictional loss, and so scales with feed rate, or with RPM independently, so long as one keep these two values within the tolerable levels for the material being extruded. Similar to the machining envelope of table feed(linear wire feed in our case) and spindle speed(die RPM in our case) in conventional turning of materials on lathes and mills. |
To calculate the temperature rise, we need to know the specific heat capacity of aluminum, which is approximately 900 J/kg°C.
Assuming the wire has a density of 2.7 g/cm^3 (the density of 6061-T6 aluminum), we can calculate the mass of the wire passing through the die per unit time as:
m = π * (0.0008 m)^2 * 2.7 g/cm^3 * 1000 cm^3/m^3 * 2 mm/s
= 1.23 g/s
Multiplying the heat generated per unit length by the length of wire passing through the die per unit time and the specific heat capacity of aluminum, we get:
ΔT = Q * t / (m * c)
where t is the time interval over which the heat is generated. Let's assume a conservative value of t = 1E-5 seconds, (10 microseconds) for this calculation, since the temperature will tend to change less on short timescales, and we are only considering a lossless system, so we do not want to allow too much time for temperature rise in a perfectly insulating environment.
Plugging in the values, with the specific heat and mass flows in units compatible with watts, we get:
ΔT = (8,969,176 W/m) * 1E-5s / (0.000123 kg/s * 900 J/kg°C)
= 433°C
In reality there will be large losses due to thermal conduction through the wire feedstock, the die walls, and convection to the air. I am not really equipped to characterize these losses fully just yet but will return to them in future.
What It All Means-
Thus, the temperature rise in the 1.6 mm OD, 6061-T6 aluminum wire forced through a 1.4 mm internal diameter, 316L stainless steel die, 0.5mm long, with a 60 degree entry angle, spinning at 15,000 RPM, and at a wire length feed rate of 2 mm / second, at a time of 10 microseconds after contact would be approximately 433 °C .... under lossless conditions. Below is a real experiment based on the above calculated values, which resulted in very stable continuous extrusion of aluminum 6061 wire through a stianless steel die.
Figure 3. Microscope images of 316 L 1.4 mm ID X 0.5 mm long stainelss steel die, extruding Aluminum 6061-T6 to 1.4mm OD, from a 1.6mm OD wire feedstock. In this test, we obtained stable extrusion until we ran out of wire to feed, with an extrudate surface that was smooth and uniform. This was obtained with no shaft liner, no guide tube, and not external thrust assembly. we observed the smoking of the 10W-40 full synthetic motor oil used to lubricate the interior of the shaft off of the extrudate surface.
Considering the smoking off of the 10W-40 synthetic motor oil on the wire surface, My best guess for our real wire temperature without a direct measurement(complicated by emissivity, and target size) would be a minimum of ~250 °C based on the smoke point of such oils typically. This temperature is substantially below our estimate, which makes sense under steady state conditions with real world conduction and convection losses. By comparing the real value, as well as we can measure it, with our calculated values, we can start to estimate the magnitude of those losses and, for a given configuration make reasonable accommodation for them in future predictions and thus improve the consistency with which we can hit the target print parameters for stable material extrusion and deposition.
Fortunately, as a result of the physics of wire drawing, the force required to push a wire through a die of smaller diameter does not typically increase with pushing speed, although the heat evolved does increase, and this can change the friction conditions at the die orifice, which can effect the pushing force requirement.
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| Figure 4. If the system can sustain a doubling the feed rate at a constant extruder force and die rpm, the evolved heat should double as well. Similarly, doubling the RPM at constant feed rate should also double the evolved heat. In the feed rate case, the motor torque and power at the target RPM are the limiting factor, while in the RPM doubling case the friction properties of the die and feedstock are the limiting factor. In the balance of these factors to obtain a target extrudate temperature, IE viscosity, is the ideal condition for Rotoforge printing found. |
Basically, extrusion ratio impacts extrusion force, but total material mass flow rate (linear wire feed rate) does not, while motor torque demand depends on mass flow rate, and this directly influences the evolved heat at the wire surface.
So 2X the feed rate to 4 mm/s would make the evolved heat roughly double, to ~16,000,000 W/m... this means we can make up for the thermal loss paths by increasing wire feed rate to increase heat evolved at some constant extrusion ratio, and some constant motor torque capacity(power output) at the target die RPM, which can be selected based on the literature machining parameters of the wire feedstock material. The E3D Hemera, our current extruder is capable of maximum feed rates of ~250 mm/second, principally. This would represent a truly enormous evolved heat...
But What About Actually Printing?-
Since we now have a first principles understanding of and some reasonable models for estimating printing parameters, how do we pick a target that allows us to not just stably extrude, but to deposit material onto a surface on onto itself, IE, PRINTING!
All materials have a temperature dependent yield strength. Al-6061 is no exception to this rule, as seen below in figure
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| Figure 5. the temperature dependent yield strength of Al-6061 and other alloys. |
Since we can now target extrudate temperatures with some precision by controlling wire feed rate, motor RPM, and layer height, we should be able to adjust the extrudate temperature to some temperature(say between 350-500°C) where the material extruding from the die, will deform into the all too familiar flash, on the build surface / previously deposited layer, and form a deposit on the underlying surface. Except, instead of the flash galling against the inside of the motor shaft and jamming things, it will expand, and rub against the outside of the die, which will flatten and stir it onto the underlying surface.
Figure 6. A cross section of what the formation of a flash and a deposit might look like, as well as cartoon representation of secondary heat sources from friction between die and extrudate flash which should form under the constant extrusion force, once an appropriately high extrudate temperature is obtained to facilitate viscoplastic flow of the extrudate under the applied extruder force.
figure7. one can imagine this situation, but occurring just outside the tip of the die and the deposit as shown, being constrained between the outside of the die and the build surface.
Rather fortunately, the die reducing the diameter of the feedstock as it extrudes should help to concentrate the extruder force on a small area of hot extrudate, which will increase the effective compressive stress on the extrudate and hopefully make generating a deposit easier....
All that remains is to start trying to deposit the aluminum onto a build surface, and adjust our feeds and speeds until we can obtain sufficient extrudate temperature to produce reliable deposition.
