Capstone: laying out a transition and a nozzle set for a vessel

The last job of the winter is the one Joanne has been saving. Northgate is adding a stripper column, C-305, and Ironbridge has the order for its middle section: a lower shell course of 96 in inside diameter, a concentric cone up to an upper shell course of 60 in inside diameter, and two nozzles on the lower course. The shell plate is 3/4 in, and so is the cone. The section is built to ASME Section VIII, Division 1, with Northgate's specification on top. Joanne gives Kiera the drawing and the job file and tells her she is the lead hand on the layout and fit-up, with Joanne and Harjit checking. "Everything you've done this winter is on this one drawing," Joanne says. "A cone, two nozzles on a curved shell, a CNC file, a head-type fit at both ends of the cone. Take it from the drawing to a fitted section, and show me every number." This capstone follows Kiera through the review, the development, the file, the cutting, the forming, the fit-up and the record.

Reading the drawing

Kiera starts by reading the drawing and listing what it fixes. The cone's half apex angle is 30 degrees, with no knuckle at either end. Section VIII allows a cone-to-cylinder junction without a knuckle up to a half apex angle of 30 degrees, provided the junction is checked by the designer for the reinforcement it needs; the design engineer has done that, and the drawing shows the junction welds as full penetration with no extra reinforcing ring. The lower course carries two nozzles: N1, an NPS 8 radial nozzle at 90 degrees, and N2, an NPS 4 hillside nozzle at 180 degrees, offset 24 in from its radial line to line up with a reflux line. The cone's two seams are to be at 45 and 225 degrees, staggered from the shells' long seams at 0 and 180 degrees. Everything is dimensioned to the inside.

Developing the cone

The cone's pattern comes from lessons 3 and 4. Kiera works on the mean line: the mean radius of the lower end is the inside radius plus half the thickness, and the same for the upper end. With a 30 degree half apex angle, the pattern radii are the mean radii divided by the sine of 30 degrees, which is a half, so each pattern radius is twice its mean radius. The sector angle is 360 degrees times the sine of the half apex angle, exactly 180 degrees. The cone's pattern is half a ring. The cone's height between the two ends, from the difference in mean radii divided by the tangent of 30 degrees, is the dimension she will check when it is fitted.

Half a ring with an outer radius over 96 in is far too big for one plate, so she splits it along elements into two pieces of 90 degrees, one for each seam position. The code works out the pieces, checks them against the shop's plate size, and gives the nozzle marks for the lower course.

# C-305 capstone: cone pattern, plate check and nozzle marks (US units)
import math
from math import gcd
MODE = "in"

def frac(x, den=16):
    sign = "-" if x < 0 else ""
    n = round(abs(x) * den); whole, rem = divmod(n, den)
    if rem == 0:
        return f"{sign}{whole} in"
    g = gcd(rem, den)
    return f"{sign}{whole} {rem // g}/{den // g} in" if whole else f"{sign}{rem // g}/{den // g} in"

def show(x):
    x = 0.0 if abs(x) < 1e-6 else x
    return f"{x:,.0f} mm" if MODE == "mm" else frac(x)
Db, Ds, t = 96, 60, 0.75        # inside diameters of the lower and upper shells, cone plate thickness
half = math.radians(30)         # half apex angle of the cone
Rb, Rs = (Db + t) / 2, (Ds + t) / 2   # mean radii (thickness measured square to the surface, simplified)
h = (Rb - Rs) / math.tan(half)
Lb, Ls = Rb / math.sin(half), Rs / math.sin(half)
ang = 360 * math.sin(half)
print("cone height", show(h), " pattern radii", show(Lb), "/", show(Ls), f" sector {ang:.1f} deg")
pieces = 2
a = math.radians(ang / pieces)                 # angle of each piece
# lay each piece symmetric about a vertical line, chord along the plate's length
pts = [(R * math.sin(f), R * math.cos(f)) for R in (Lb, Ls) for f in [a * (k / 40 - 0.5) for k in range(41)]]
xs, ys = [p[0] for p in pts], [p[1] for p in pts]
long_, wide = max(xs) - min(xs), max(ys) - min(ys)
print(f"each of {pieces} pieces is {ang / pieces:.0f} deg and needs {show(long_)} by {show(wide)} of plate")
plate = (240, 96)
print("plate available", show(plate[0]), "by", show(plate[1]), "->", "one piece per plate fits" if long_ <= plate[0] and wide <= plate[1] else "nest differently")
# nozzle N1 (radial) on the lower shell, outside radius
Ro = Db / 2 + 0.75
for name, d, e in [("N1 NPS 8 radial, lower shell", 8.625, 0), ("N2 NPS 4 hillside, lower shell", 4.5, 24)]:
    c = Ro * math.asin(e / Ro)
    print(f"{name}: ", end="")
    marks = []
    for k in range(4):
        th = math.pi / 2 * k
        lat = e + d / 2 * math.sin(th)
        marks.append(Ro * math.asin(lat / Ro) - c)
    print("around-shell marks at 0/90/180/270:", ", ".join(show(m) for m in marks))

cone height 31 3/16 in  pattern radii 96 3/4 in / 60 3/4 in  sector 180.0 deg
each of 2 pieces is 90 deg and needs 136 13/16 in by 53 13/16 in of plate
plate available 240 in by 96 in -> one piece per plate fits
N1 NPS 8 radial, lower shell: around-shell marks at 0/90/180/270: 0 in, 4 5/16 in, 0 in, -4 5/16 in
N2 NPS 4 hillside, lower shell: around-shell marks at 0/90/180/270: 0 in, 2 5/8 in, 0 in, -2 9/16 in

Each cone piece fits on one plate, with its chord along the plate's length. The two pieces would not both fit on one plate even turned end for end, so she orders two plates, and the offcuts go to the nozzle reinforcing pads and clips. The nozzle marks show N1's opening very slightly wider around than its diameter, as lesson 8 described, and N2's lopsided opening, wider on its high side.

The file and the first check

The CNC programmer, Dustin, draws the cone pieces from Kiera's sketch and sends her the nest for checking. Following Joanne's four steps from lesson 11, she measures the pieces in the drafting program. The outer radius is 96 in and the inner 60 in: the inside radii doubled, not the mean radii. The file would cut a cone whose large end was about 2 3/8 in short around its circumference. Dustin had entered the drawing's inside diameters. He corrects the radii to 96 3/4 in and 60 3/4 in, Kiera measures again, checks the kerf setting and runs the simulation, and the first piece is cut. It measures within 1/16 in on both radii and the straight edges. Joanne adds the mistake to her list of why every file gets a hand check.

The nozzle openings

N1 and N2 are laid out on course 1's flat plate before rolling, by the method of lesson 8. Kiera transfers the orientation lines onto the plate: 90 degrees for N1 and 180 degrees for N2, as arc lengths on the outside radius from the plate's 0 degree edge, and the elevation lines from the course's lower edge. At N2, she marks the center of its opening not on the 180 degree line but at the arc for its offset from that line, then lays out both openings point by point and joins the points with a batten. Neither opening is larger than a quarter of the shell's diameter, so both are cut flat. She writes the nozzle number, size and the words "hillside, high side up" inside N2's outline, so no one mistakes which way the lopsided hole should face.

Their necks are developed by the parallel-line method, with each line's length from the shell surface to the flange face. N2's lines vary by nearly a quarter of its projection from one side to the other. Following Harjit's practice, she cuts both necks 1/8 in long.

Forming

Course 1 is rolled with its two nozzle openings already cut. On the first pass, the roll operator, Gus, watches the plate near N1, where the opening makes the plate weaker, and adds a light extra pass there to stop a flat spot forming. The cone pieces are rolled on the cone-rolling attachment with their element lines kept parallel to the rolls, as lesson 3 described. Each is checked with a template cut to the cone's large-end arc and a straightedge along the elements. The two pieces are fitted to each other on the layout table, their seams tacked, and the cone's large and small ends measured by tape: both within 1/16 in of the mean circumferences.

Fitting the cone

Course 1 stands on the fitting table with its top end up. Kiera checks its roundness at the top: within 3/16 in, well inside the out-of-roundness limit. She measures its outside circumference and the cone's large-end outside circumference and works out the even offset per side, 1/32 in, within the quarter-thickness limit. The cone is lifted on, turned until its seams are at 45 and 225 degrees, and its large end fitted to course 1's top with the offset spread evenly, as in lesson 10. Then she checks the small end: the height from course 1's top to the cone's small end at eight stations. The readings vary by 1/8 in, within the drawing's limit for the upper shell to stand plumb.

Course 2, the upper shell, is fitted to the cone's small end the same way. The final check is plumb: with the section standing, Kiera hangs a plumb line from the top of course 2 at four stations and measures to the bottom of course 1. The upper shell is concentric with the lower within 1/8 in, and the overall height from course 1's bottom to course 2's top matches the drawing.

Fitting the nozzles

N1's neck goes into its opening with its flange held level and square to the vessel's axis by a magnetic level and a square from the shell's reference line. Kiera measures its projection: 1/16 in long, so she trims the neck and tacks it. N2 takes longer. Its lopsided neck must sit with its long side down, and its flange face must be parallel to N1's direction but 24 in off the radial line. She sets the flange with a straightedge laid across from the 180 degree reference marks, checks the offset at the flange face with a tape and a square, and finds the neck needs 3/16 in more trimming on its high side to bring the flange level. After trimming, both nozzles' bolt holes straddle their center lines, the projections match the drawing, and the reinforcing pads are fitted around both necks with their telltale holes at the bottom.

The record and the release

Harjit checks every point on Kiera's fit-up sheets, measures the nozzle projections and offsets himself, and signs the hold points for the two cone seams, the two cone-to-shell seams and the two nozzles. The record includes the hand calculations for the cone and the nozzle openings, the corrected CNC file with Kiera's check sheet, the heat number transfer for every part, and the readings at each step. Welding starts the next morning.

At the end of the week, Joanne sits down with Kiera to go over the winter. Kiera has developed parallel-line patterns, cones, eccentric cones and transitions; found the lines where cylinders meet cylinders and cones; laid out nozzle openings on flat plate; checked, marked and fitted heads; and turned patterns into CNC files and caught the mistakes in them. The common thread, Joanne says, is the one she started the winter with: every pattern is a set of true lengths on the right line of the plate, and every pattern is checked before it is cut. In Pressure Equipment Codes, the next course, Kiera goes into the rules behind the drawings she has been building to: how the codes set thicknesses, joint categories, inspection and the stamps on the nameplate.