Any number of prongs

A breeches has one crotch. A fitting with three prongs has three, and with four, four or more, depending on how the prongs are arranged. Each crotch is where two neighbouring prongs meet, and each is found exactly as a breeches’ crotch is. What is new is where the crotches meet each other. In a manifold whose prongs are arranged round the main’s axis, every crotch runs inward and upward from the main’s edge, and they all meet at one point on the axis. That point, and the way the crotches approach it, is the heart of the fitting.

This chapter develops G6, the FX-4 three-way manifold, using nothing that the earlier chapters have not already set out. It arranges the prongs round the axis, finds the three crotches and the point where they meet, develops one prong that serves for all three, compares a manifold with its prongs in a row, and ends with a row manifold set out as if it were a ring.

G6 in coordinates

This is an illustration. At the far end of FX-4, a Ø400 duct from the filter unit’s inlet plenum must collect from three branches that arrive from different directions. G6 divides the Ø400 duct into three Ø230 prongs, sized by the area rule of Chapter 1.

The main end is at the origin, square to z. The three prong ends are parallel to it, 450 mm along z, with their centres on a circle of 160 mm radius round the axis, 120° apart: at 0°, 120° and 240°. The first is at (160, 0, 450). The centres are 160 × √3 = 277.1 mm apart, so the Ø230 prong ends stand 47.1 mm clear of each other.

Every prong is a frustum of an oblique cone through the main circle and its end. For the prong at 0°, k = 115 ÷ 200 = 0.575, 1 − k = 0.425, and the apex is at (160 ÷ 0.425, 0, 450 ÷ 0.425) = (376.5, 0, 1058.8). The other prongs’ cones are the same cone turned 120° and 240° about the axis.

Three flat crotches

Each pair of neighbouring prongs is a pair of cones on the main circle with parallel ends, so Chapter 6’s proof applies: their crotch lies in a plane. By symmetry, the plane is the one halfway between them, through the main’s axis. The crotch between the prongs at 0° and 120° lies in the vertical half-plane at 60°; the others at 180° and 300°.

Each crotch starts at a foot on the main circle, where its half-plane meets the circle: for the crotch at 60°, the foot is (100, 173.2, 0). It rises inward and reaches its peak at (35.8, 62.0, 402.6), 71.6 mm from the axis. Then it falls a little as it reaches the axis, at (0, 0, 367.3).

The peak’s height agrees with Chapter 6’s formula for a pair of prongs: with the main’s radius 200 mm, the length 450 mm, the prongs’ centres 277.1 mm apart and their radii summing to 230 mm, the peak is 2 × 200 × 450 ÷ (277.1 + 400 − 230) = 402.6 mm.

The triple point

All three crotches reach the axis at the same point, (0, 0, 367.3), because the fitting is the same seen from each prong. That point is the triple point. Below it, inside the fitting, is the main’s undivided stream; above it, between the three prongs, is open space outside the fitting. The three crotch seams meet there in a three-way joint, the most demanding point of the fitting to seal.

Each crotch, from its foot over its peak to the triple point, is 531.4 mm long. The crotch’s last part, from the peak down to the triple point, is where three prongs come together. Seen from above, the region between the prongs’ inner sides is a three-cornered hollow, dipping from the three peaks at 402.6 mm to the triple point at 367.3 mm.

Figure 1Points on G6's crotch at 60°

Element of the 0° prong from main-circle point at θ	Crotch point x, y, z	Distance from the axis
60° (foot)	100.0, 173.2, 0	200.0
75°	96.3, 166.7, 145.1	192.6
90°	88.4, 153.1, 248.5	176.8
105°	77.8, 134.7, 320.3	155.6
120°	65.3, 113.1, 367.3	130.6
135°	51.3, 88.8, 394.0	102.6
150° (peak)	35.8, 62.0, 402.6	71.6
165°	18.8, 32.5, 394.0	37.5
180° (triple point)	0, 0, 367.3	0

Why the crotch dips at the axis

A breeches’ crotch rises from foot to peak and falls again to the other foot, symmetric about its peak. G6’s crotch has only one foot. It rises from the foot at the main’s edge to its peak and then falls, not to another foot, but to the axis, where it meets the other two crotches.

The reason is in the elements. Each crotch point is where an element of the prong’s cone crosses its half-plane. Every element rises at the same rate along its length, so the higher a crossing, the further along its element it lies. The element from the main circle’s point at 150° reaches the half-plane 0.380 of the way to the apex, at 402.6 mm. The element from 180°, the main’s point directly opposite the prong, runs straight through the main’s axis, where all three half-planes meet, and reaches it sooner, 200 ÷ 576.5 = 0.347 of the way, at 367.3 mm.

The dip is small, 35.3 mm over the last 71.6 mm of the crotch, but it decides the shape of the prongs’ inner corners. A worker who expected each crotch to rise all the way to the axis, as a breeches’ crotch rises to its peak, would cut the prongs’ notches too shallow near the triple point and leave a three-cornered hole there.

One prong serves for all

Because the three prongs are identical, one prong’s pattern serves for all three. The prong at 0° owns the main circle from −60° to 60°, 120° of it. Its elements from those points run their full frustum length. Its elements from 60° to 180°, and from −60° to −180°, start at the crotch in the half-plane on their side and run to the prong end.

Figure 2G6's prong, elements from its apex

Main-circle point at θ	Pattern angle from the outer element	Length used
0° (outer)	0°	456.2
30°	5.54°	460.2
60° (foot)	10.86°	470.9
90°	15.92°	217.2
120°	20.84°	91.6
150° (under the peak)	25.77°	53.6
180° (triple point)	30.74°	94.1

The half pattern spans 30.74° at the apex, and the whole prong 61.5°. The element lengths behave differently from a breeches’: they fall from the foot to a minimum of 53.6 mm under the peak, then rise again to 94.1 mm at the triple point, because the crotch falls from its peak towards the axis. That dip shows on the pattern as a notch whose deepest point is not at its centre line.

Each prong’s seam goes on its element through the triple point, 94.1 mm long, so all three prongs’ seams and all three crotch seams meet at the triple point. Each prong’s area is about 0.233 m², and the three together about 0.698 m², about 4.4 kg at 0.8 mm.

Prongs in a row

The prongs could instead be arranged in a row: three Ø230 prong ends with their centres at x = −260, 0 and 260 mm, all at 450 mm along. That suits branches arriving side by side along a wall. The geometry changes in an instructive way.

Each crotch is still flat, because each neighbouring pair is still two cones on the main circle. Intuition says that the crotch between the middle and right prongs should start midway between their centres, at x = 130 mm. It does not. Chapter 6’s foot formula, R × (r₁ − r₂) ÷ (c₂ − c₁), gives zero for equal prongs, and that zero is measured from the main’s centre, not from the midpoint of the prong centres. Both crotches, left and right, therefore start at the same feet, (0, ±200, 0), and lean outward, each at 16.1° from vertical.

The consequence is striking. The middle prong owns none of the main end’s circle: its two crotches start from the same two points and spread apart as they rise, so the middle prong begins as a line across the main end and widens from there. The two outer prongs share the main circle between them, half each. Each crotch’s peak, by the peak formula with centres 260 mm apart, is at 2 × 200 × 450 ÷ (260 + 400 − 230) = 418.6 mm.

The error: a row manifold set out as a ring

This is an illustration. Late in the job, the branches at G6 were rerouted to arrive side by side along a wall, and the manifold was changed to the row arrangement. A worker reused the ring layout’s assumptions: each prong owning 120° of the main circle, and each crotch on a vertical plane.

Neither holds for a row. The middle prong’s patterns, cut to own 120° of the main circle, could not be fitted: the outer prongs’ patterns, also cut for 120° each, left a gap of 120° at the main end that the middle prong’s edges did not match, and the crotches, cut vertical, did not meet the cones’ true intersections, which lean 16.1°. The prongs were offered together on the bench and the mismatch was obvious.

The fix was to recalculate from the cones: the feet from the foot formula, the peaks from the peak formula, and each prong’s crotch edges from its elements. The shop’s note on multi-way manifolds now reads: “Find every crotch from its pair of cones. Never assume a share of the main circle.”

Figure 3Share of the main end's circle owned by each prong

Ring, each prong120 degrees

Row, each outer prong180 degrees

Row, middle prong0 degrees

Assembly and the triple point

A manifold is assembled from the inside out. The three prongs are rolled, each with its seam on the element through the triple point, and the first two are brought together along their shared crotch, riveted from the foot up to about the peak. The third prong is then offered in, and its two crotch seams are riveted from their feet upward. The last few rivets on each crotch, near the triple point, are set last, after all three prongs have been pulled together and the triple point checked on the axis, 367.3 mm above the main end, with a rule held up through the main.

The triple point is sealed from inside where possible, by reaching up through the main end before the fitting is installed, with a generous bead of sealant over the three-way joint. On a fitting this size an arm reaches; on a smaller one, the triple point is sealed as each prong is added, before the joint is closed.

Checking a manifold

The checks follow the same pattern as a breeches’, applied to each prong and each crotch.

• The main end: round to its Ø400 template, with the three feet marked 120° apart.
• Each prong end: Ø230, parallel to the main end, its centre 160 mm from the axis at its angle and 450 mm along. A plywood gauge with the three centres marked checks all three at once.
• Each crotch: against a single template cut to the crotch’s true shape, which serves all three because they are identical, held in each 60°, 180° and 300° half-plane in turn.
• The triple point: on the axis, 367.3 mm above the main end.

Four or more prongs

The method extends to any number of prongs. In a ring of n equal prongs, each owns 360° ÷ n of the main circle, each crotch lies in the half-plane midway between its neighbours, and all n crotches meet at one point on the axis. The peak of each crotch follows from the peak formula with the distance between neighbouring centres. As n grows, the prong ends crowd together round their circle, the distance between neighbours shrinks, and the crotches’ peaks rise towards the prong ends; at some point, the prong ends collide, which the formula shows as a peak at or above the prong ends’ level.

Exercise: a four-way ring

Suppose G6 were a four-way manifold: a Ø400 main to four Ø200 prongs, their ends parallel to the main, 450 mm along, with centres on a 150 mm radius round the axis, 90° apart. Find the distance between neighbouring centres, the clearance between neighbouring prong ends, and each crotch’s peak.

Worked answer.

• Neighbouring centres are 150 × √2 = 212.1 mm apart.
• The prong ends’ radii sum to 200 mm, so the ends stand 12.1 mm clear of each other: tight, but they do not collide.
• Each crotch’s peak is 2 × 200 × 450 ÷ (212.1 + 400 − 200) = 436.8 mm, only 13.2 mm below the prong ends.

The peaks are so close to the prong ends that the branch collars would have almost no room between them. Moving the centres out to 170 mm would give 240.4 mm between neighbours, a peak at 2 × 200 × 450 ÷ 440.4 = 408.7 mm, and room to work.