The clearance that is the seal

Rotors that mesh and cannot drive each other computed a pair of identical lobed rotors that are each other’s conjugates. Give half a lobe — the tip, the part outside the pitch circle — and the meshing equation returns the root the mate’s tip needs; put that root on the rotor between its own tips, and the rotor is its own mate by construction. The pair then holds a one-to-one ratio exactly, at every instant, and cannot drive itself: whenever a tip crosses the line of centres the contact normal lies along that line and its moment about the driven shaft is nought.

That essay ended with a question about what conjugacy is worth to a machine that is not driving anything. A Roots blower’s rotors do not touch. They are timed by gears on their shafts and cut slightly undersize, and the thin gap left between them is the seal that makes the machine a pump. The question was whether conjugacy makes that gap a number or a range.

It makes it a number, and the comparison is stark enough to be a design rule.

They touch at every angle #

The first measurement is the one everything else rests on, and it is worth making explicitly rather than inferring from the construction.

Two conjugate rotors at their own centre distance are in contact at every angle of the mesh. Measured as the closest the two outlines come, over a whole lobe pitch — which is the whole of the period, since the rotor repeats every $2\pi/n$ and the pair is symmetric about the line of centres — the worst gap is $5.2 \times 10^{-5}$ on rotors of pitch radius 50, at four hundred points along each tip. That is the outline’s own sampling and not a gap; it falls as the sampling is refined.

Continuous contact is what a conjugate pair is for. The second shape is not a choice: given one profile and a ratio, the other is the envelope, and an envelope touches its family member at every instant by definition. What is unusual here is that the envelope is the same shape again, so the contact is continuous without either rotor being the designed one.

It is also the thing that separates this pair from the other continuous-contact families this field has built. Two shapes stop touching when a profile runs out of arc, and a gear pair’s contact is continuous only over the stretch where both flanks still exist — which is why a contact ratio is a number a designer has to keep above one. A Roots rotor has no such stretch: every point of its boundary is either a tip in mesh or a root that some tip is in mesh with, and the two halves are computed from each other, so the mesh never hands over and never runs out.

Cut both back and the gap is twice it #

Now shrink. Erode each rotor by a disc of radius δ — move every point of its boundary δ inward along the boundary’s own normal, and drop the points a fold has eaten. Two bodies each eroded by δ are $2\delta$ further apart than they were, wherever their nearest points face each other. And these two were nowhere apart at all.

Cut back by 0.5 and the gap runs from 1.0000 to 1.0000. Cut back by 1 and it runs from 1.9998 to 2.0000. Cut back by 2 and it runs from 3.9990 to 4.0000. The departures are the erosion’s own sampling — the boundary is a polygon and its offset is a polygon — and they scale with the sampling rather than with anything about the rotors.

So a conjugate pair can be given any seal a designer likes, uniform to a part in a thousand, by one number on a drawing. That is the sentence the machine’s efficiency rests on, and it is worth saying why. A leak goes through the tightest place: the flow past a gap depends on the gap, and one slot half as wide as the rest leaks a fraction as much. What matters is not the average clearance but the worst one, and a profile that is flat has no worst one.

There is a second reading of the same fact, and it is about where the tolerance goes. A rotor is cut by a tool, and the tool is the definition: a rotor cut undersize is the same tool held a stated distance further out, so the undersize is a machine setting rather than a new shape to compute. The seal is therefore specified where a tolerance ought to be specified — on one dimension of one operation — and the conjugate geometry is what guarantees that the one dimension controls the whole gap.

That is not the usual arrangement. In most mechanisms a clearance is distributed and only its total matters: a four-bar’s play enters the loop as a sum of four displacement vectors, so the linkage cannot tell which bearing is loose, and a designer can put the tolerance wherever it is cheapest. Here the opposite holds. The clearance is not a sum over joints but a distance between two surfaces at every angle, and where it is taken from decides its shape rather than only its size. The next two sections are what happens when it is taken from the wrong place.

The same clearance, bought the other way #

A designer has a second way to open a running gap, and it costs nothing to machine: leave both rotors full size and set the shafts a little further apart. It is the obvious move for anybody who has met the involute, whose whole advertised virtue is that moving the shafts changes nothing about the ratio.

Nothing is flat, and the shape of the failure is the same at every opening. The widest place is the whole of the centre-distance error — at the position where a tip faces a root across the line of centres, the two bodies separate by exactly what the shafts were opened by. The tightest place is a small fraction of it.

At an opening of 0.1 the gap runs from 0.0042 to 0.1000 — a ratio of 23.7 to one, and the tightest place is 4.2% of what the drawing says. At an opening of 2 the ratio has fallen to 5.7 to one and the tightest is 17.7%, which is the general behaviour: the larger the opening the more nearly uniform it becomes, because a large opening is dominated by the translation and a small one by the shapes’ curvature mismatch. That is exactly backwards from what a designer wants. The smaller the clearance asked for, the worse this way of buying it is.

Reading the table as a purchase: a pair run at a centre distance 0.1 too large has a seal of 0.0042 and has given up 0.1 of tip clearance to get it, so twenty-four twenty-fifths of what was spent is doing nothing. Cutting the rotors back by 0.0021 each buys the same 0.0042 seal and costs nothing else.

The involute’s indifference to centre distance does not transfer, and it is worth being precise about which half fails. A Roots pair’s ratio is indifferent: it is one to one because the rotors are identical and timed by gears, and no centre distance changes that. What is not indifferent is the shape of the gap, because conjugacy is a statement about two profiles at a stated centre distance and moving the shafts breaks it. An involute pair moved apart stays conjugate and loses backlash; a Roots pair moved apart keeps its ratio and loses its seal. Two different properties, and each mechanism keeps the one its own geometry protects.

There is a geometric reason for the ratio, and it is readable off the rotors. The tightest place appears when a tip is crossing the line of centres — the same position at which the pair’s own push goes through nought, because the contact normal lies along the line of centres there. At that position the two bodies meet nearly tangentially, curved surface against curved surface with their centres of curvature on the same side, so a translation of the shafts along the line of centres opens them by only the component of that translation along their common normal — which is small, because the normal is nearly square to the line the shafts moved along. At the other extreme, a tip facing a root across the line of centres has its normal along the translation, and the whole of it is spent.

So the ratio between the widest and tightest places is a ratio of direction cosines, and it is worst where the contact normal is most nearly perpendicular to the line of centres, which is exactly where the pair’s own driving moment is largest. The place a Roots pair pushes hardest is the place its seal is thinnest when the shafts are opened.

A pair that is not its own mate #

The last case is the one that says what conjugacy is actually doing, and it is more emphatic than a varying gap.

Take the rotor with its cycloidal tip and its computed root, and run it against a rotor whose tip is a circular arc of the same height — a perfectly good rotor, with its own computed root, which has a mate of its own and is simply not this one’s.

Over one lobe pitch the clearance swings from 2.4044 inside each other to 2.3305 apart, a spread of 4.73 on rotors of pitch radius 50. The pair does not have a varying seal; it has an interference at some angles and a hole at others.

Cutting both back to clear the interference costs 1.203 each and leaves a seal of nought where they were deepest and 4.73 where they were widest. There is no undersize that makes this pair seal uniformly, and no centre distance either, because the two are not a translation apart — they are different shapes.

That is the reading worth carrying. Conjugacy is not a refinement of a rotor’s shape. It is the condition under which the clearance between two rotors is a number instead of a range, and a pump’s seal is the one quantity that needs to be a number.

It also draws a line under a question the meshing field keeps meeting from different sides. Any shape has a partner: given a profile and a ratio, the envelope exists and is computable, so the space of workable pairs is enormous. What is scarce is not a partner but a partner with a property, and here the property is that the pair be identical, which is what makes one undersize serve both and one erosion the whole seal. Eleven lobes from twelve pins is the same demand with a different answer: a cycloidal disc and its pins are not identical, and their clearance is specified pin by pin.

What the three measurements are, together #

Three ways of putting a gap between two rotors, and they are not three settings of one knob.

Erode both by δ. Every point of both boundaries moves inward along its own normal. The nearest-point pairs are unchanged — erosion does not move them, because both surfaces retreat along the same common normal — so the gap is the old gap plus $2\delta$, and the old gap was nought. One number, flat.

Translate one body by ε. The nearest-point pairs move, and the gap they leave is the component of the translation along each pair’s own normal. That component varies with the angle because the normal turns as the rotors turn, and a profile whose contact normal sweeps through ninety degrees over a lobe pitch gets almost all of the translation at one end and almost none at the other.

Change one of the shapes. The nearest-point pairs are no longer even pairs: the two boundaries cross, so there are angles at which the bodies are inside each other and angles at which they are apart, and no rigid motion or uniform erosion can turn that into a constant.

The distinction between the first two is the general one: an erosion is normal to the surface and a translation is not. It holds for any pair of bodies in continuous contact, and it is why a running clearance is specified as a size allowance on the part rather than as a position allowance on the assembly wherever a surface has to seal. What is particular to this pair is that the first option is available at all, which needs the two bodies to touch everywhere — and that is conjugacy.

What a seal is, and what this is not #

Nothing here is fluid. A leak is a flow and flow past a gap depends on the gap, the pressure, the viscosity and the length of the path. The claim made here is geometric — that a flat clearance profile has no worst angle — and the step from that to a leakage rate is a step nothing here takes.

The rotors are two-dimensional. A real Roots rotor is helical, so its lobes twist along the shaft and the clearance at a section is not the clearance of the machine. The twist was named as an open question and it is still one; a twisted pair’s seal is a sum along its length and the section-by-section profile measured here is its integrand.

The erosion is of a sampled boundary. The outlines are polygons of several hundred points a tip, and their offsets are polygons too. The departures from $2\delta$ reported above are that sampling, which is why they grow with the offset and not with anything about the rotors. A rotor cut back by more than the tightest bend in its root loses the bottom of the root altogether, and nothing here says at what undersize that starts to matter for the gap rather than for the shape.

The centre-distance comparison holds the rotors at the ratio. Both are still turning one to one, because the timing gears say so. A pair whose centres are opened and whose timing is not corrected is a different and worse problem.

The undersize is uniform. Both rotors are eroded by the same amount, which is what a single fit on a drawing produces and what the argument above needs. A pair with one rotor cut back and the other left alone has the same total gap and a different distribution of material, and whether that matters is a question about wear and about which rotor is the cheaper to replace rather than about geometry.

Only one lobe count and two tip shapes. Two lobes with a cycloidal tip is the classical Roots profile and it is the pair measured. Three lobes were checked for the contact claim and behave the same; the ledger and the mismatch are drawn on the two-lobe pair only.

Still open: the seal a helical pair actually has #

The measurement here is a section’s, and a real blower’s rotors are twisted so that no section is at the same phase as its neighbour. That is the whole reason a helical pair runs quietly, and it is also what turns the flat profile above into a statement about a family of sections rather than about a machine.

Its distinct argument would be the leak path rather than the gap: on a twisted pair the tightest places at different sections are at different angles, so the shortest route from the high-pressure side to the low one is no longer a straight line across one section but a path that wanders along the rotor. The question is whether that path’s narrowest point is the section profile’s narrowest point, which the flat profile above would make trivially true for a conjugate pair and interestingly false for one that is not — and whether a twist can make a non-conjugate pair seal as well as a conjugate one by never letting its worst angle occur at more than one section at a time.