First, my base assumption is that we want to replicate - as closely as possible with a cylinder/tube and weight/plunger - the manufacturer's intended grinding behavior given a reasonably loaded stock hopper. The measurements and tests in this post used my M7K's stock hopper dimensions and capacity. Other Titan grinders will have different hoppers of course, but I believe the ideas here can be readily adapted.
- • The capacity of this particular hopper is specified at three pounds. Although it holds, more, filling it to a higher level may not be best, however (I'll get to that in a minute). Due to the funnel shape, a pound of coffee (the least dense types of beans taking up the most room per pound) fills the stock hopper to 5 5/8" which is half the height of the total hopper. The second pound brings the level up to 7 5/8". Subsequent pounds raise the level about 1 3/4" per pound up to the limit of the full hopper height of 10 3/4 inches. So the capacity is really about four pounds of light beans, and close to five pounds of "heavy" beans with a denser makeup.
• The behavior of the grinder with a half-full (height-wise) hopper with a pound of beans (or something in that approximate range such as 500g) is a reasonable target for replication with a cylinder and weight. It's common for cafés to have their Roburs and other large grinders filled around that level, and a pound of beans would need to be within any manufacturers intended or even optimum range.
• I weighed several different coffees within a 2" (inside diameter) straight-sided acrylic cylinder which I've used when single-dosing to replace the huge stock hopper. The heaviest coffee weighed 22.5 grams per inch, and the lightest beans (very dark roast, very little residual moisture) weighed only 16 grams per inch, a significant difference.
• My initial direction was simply to calculate the weight of a 2" cylindrical column of beans about 5 1/2" tall (the height of a pound of median-density beans when loaded in the grinder's stock hopper) and create a plunger/weight to replicate that column of beans bearing down upon my single dose. This would theoretically replicate the mass pressing beans into the burrs at the bottom of a hopper filled with a pound of coffee.
• The problem is that it doesn't really work out that way because a mass of coffee beans does not constitute a "Newtonian fluid" - there are several other forces and factors acting inside a hopper that complicate things.
• To explain with a thought experiment (or a real one if you like) imagine a hopper full of styrofoam packing peanuts. You'll see that once the hopper is filled, no peanuts will escape from the bottom: fundamentally the friction/drag between the peanuts overwhelms gravity and whatever lightweight mass of peanuts is pressing down from above.
• An example with the opposite results would be a hopper filled with small, polished steel ball bearings. These will pour freely right out the bottom, since the minimal drag between the ball bearings is easily overcome by their own gravity that of the other bearings pressing from above.
• Coffee beans are somewhere in the middle of these extremes. In a funnel-shaped hopper, the beans do not exit with the same ease as would be indicated by their mass. For example, I measured the weight (at the bottom of the Macap hopper) of the beans pressing down upon a scale from a pound of beans in a hopper suspended above the scale, and over many, many samples the average was only 45 grams - that's only about half the mass of the theoretical cylindrical column of beans directly above the opening at the hopper base. This "packing factor" - the friction/drag of the beans in the hopper - causes the beans to behave to some degree like the styrofoam peanuts, and this factor reduces the pressure of the beans at the base below their normal weight.
• Furthermore, the more beans you add to the hopper, the lower the proportion of the beans' weight will be evident at the bottom of a funnel-shaped hopper. As another thought experiment, consider what would happen if you extended this out using a very, very tall hopper: you'll eventually hit a point where no beans whatsoever come out the hopper bottom and there is zero pressure. The crushing weight of the pounds and pounds of beans above creates pressure vectors and resulting friction pressing inward from the sides of the constricting hopper shape, choking any possible flow. Given enough beans in a tall enough funnel shape, the hopper of beans behaves the same as a hopper full of styrofoam peanuts and drag exceeds gravity.
• An interesting sidebar here - the Macap hopper funnels have convex rounded sides which increase this "clogging" effect since the inward vectors become progressively stronger as inward curve of the hopper wall accelerates as it approach the base. If you had a concave shape for the funnel walls (like a Pilsener or old Coca Cola soda fountain glass) the effect would be decreased. A straight-sided purely conical funnel - such as with most Mazzers - is more neutral and between these extremes.
• Now, all of this gets even more interesting when you reverse your frame of reference to examine the opposite forces and vectors, where the hopper of beans acts as a critical damping force against beans pop-corning up from the burrs. To understand this, one has to start by measuring the force required to move the beans upward in contrast to measuring the downward pressures discussed above.
• Interestingly, the upward differences scale the other way around from the downward forces: it takes a lot MORE force to press upward than would be indicated by the simple the weight of the beans directly above the piston you're pushing upward.
• This sounds counter-intuitive at first, but not so much if you again consider the friction/drag of the beans being pushed upwards as they engage with other beans surrounding them in the hopper. Basically, our theoretical cylinder of beans above the piston is interlocked with the surrounding beans in the hopper. So when a coffee bean pop-corns, it's hitting a lot more effective weight above it than the theoretical column directly above that bean. It's not a precise analogy, but consider raising a tent from the inside - the weight on the pole is not just from the little bit of fabric at the top - it's weighed down by the entire top cover. Similarly, when you press upward from the bottom of a hopper, you'll be raising a mound that's much larger than the cross section of the original cylinder you're pressing upward.
• The results I got, in fact, were 3x. That is, with a pound of beans total in the hopper and the 5 1/2" column of beans above the 2" diameter opening weighing 87g, it actually took 272g of force to press those beans upward (averaged over many tries).
• Two other factors complicate things further. One is vibration from the grinding action and the motor. The vibration compacts the beans and interlocks them tightly, increasing the friction/drag. On the other hand, the vibration ALSO causes the beans to act more like a fluid, since the vibration helps an otherwise "locked" pack of beans to fill in any hole created (behaves more like water or a true Newtonian fluid) or allowed to freely fall out the bottom unimpeded by other beans. The vibration would therefore appear to mitigate at least some and maybe most of the 2x "packing factor" (average measurement I got using a static, not vibrating, hopper).
• I'm not sure how vibration affects the upward pressure factor. On one hand, the vibration might even increase and solidify the interlock making the static upward-pressure factor of 3x even more than that. But it could also help "shed" beans off the theoretical column too, although pop-corning is more of an instant effect and I'm not sure at all that vibration could meaningfully affect momentary impacts.
• The other complicating factor is "damping" - like shock-absorption. Imagining beans pop-corning upward into the mass of beans just entering the hopper throat, that mass absorbs and dampens the energy in a different way than if it were simply a solid unyielding wall. You can see that "boiling" look in the surface of a loaded hopper when the grinder is running.
• To control pop-corning, then, I think you need more mass in a static plunger than what you'd calculate given just the column of beans directly above the burrs: this because any popping fragments are actually lifting far more than the simple theoretical column directly above that fragment moving upward.
So where does this leave us on creating a plunger weight to mimic a pound of beans in the hopper? Again, assuming for this exercise that I'm trying to replace the effect of one pound in the hopper by a 2" column of beans plus a weight, then how heavy should this weight be? (If we're using the lightest beans, recall that the 2"D x 5 ½"H column (in my case) weighs 87g, while the heaviest such column of beans weighed 120g.
After mulling over all that measuring, I concluded that the vibration factor will likely compensate for most of the "packing factor" that limits the escape of beans out the bottom. However, the 3x upward pressure factor will be meaningful (from physics anyway, if possibly not from the standpoint of taste), since after all one primary purpose of the weight is to control pop-corning.
So this was resolved with a weight that is three times heavier than the 2" diameter column of beans at the one-pound hopper level. In my case, that was a 6" cylindrical rod of UHMW polyethylene. This was expertly milled by Danetrainer (thank you again, Pat!) to perfectly match the inside diameter of the tube.
I've been very pleased with the results. This is easy to use, and I myself have been unable to taste any difference when using the weight/cylinder versus using the standard hopper. This has greatly simplified single-dosing and made it successful in my situation.
I have to relate, however, that we also created another weight/plunger about half that length and weight. As many might suspect, it doesn't seem to matter taste-wise which one is used (see Single dose versus hopper grinding: an experiment by Sherman Chong and Jim Schulman). The behavior of the grinder and the taste of the results don't seem to change at all when I use one weight or the other. It may well be - as some have written here - that you just need some[x] weight to control pop-corning, but beyond a point, how much you use doesn't matter much.
It's probably more important that the weight/plunger is able to go all the way down to the burrs so that it controls more of the grind. This usually means drilling a recess in the bottom of your weight to accommodate the center nut. This is what Pat created in the end for both of our grinders, and I want to thank him again for his advice and experience in discussing these ideas - not to mention his outstanding machinist capabilities.