Calculations

Calculations

• Pool fire

  Flame height, heat release and ground distances to each flux level for a burning pool.

  H = 42·D × (m″ ÷ (ρ_a·√(g·D)))^0.61; ground = √(r² − (H ÷ 2)²)

  Release, fire, explosion and toxic screening process/safety/consequence.js

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• TNT-equivalent mass

  The mass of TNT with the same explosion energy as the fuel.

  W = η × m × ΔH_c ÷ E_TNT

  Release, fire, explosion and toxic screening process/safety/consequence.js

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• Blast overpressure

  The side-on overpressure at a distance from a TNT charge.

  Z = R ÷ ∛W; P ÷ P_a = 808·(1 + (Z ÷ 4.5)²) ÷ √((1 + (Z ÷ 0.048)²)(1 + (Z ÷ 0.32)²)(1 + (Z ÷ 1.35)²))

  Release, fire, explosion and toxic screening process/safety/consequence.js

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• Distance to an overpressure

  The distance at which the blast overpressure falls to a given level.

  Release, fire, explosion and toxic screening process/safety/consequence.js

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• Probit of a dose

  The probit value for a concentration and exposure time.

  Pr = a + b × ln(Cⁿ × t)

  Release, fire, explosion and toxic screening process/safety/consequence.js

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• Error function

  The error function used to turn a probit into a probability.

  erf(x) ≈ 1 − (a₁t + a₂t² + a₃t³ + a₄t⁴ + a₅t⁵)·e^(−x²), t = 1 ÷ (1 + 0.3275911·|x|)

  Release, fire, explosion and toxic screening process/safety/consequence.js

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• Probability from a probit

  The probability that corresponds to a probit value.

  P = ½ × (1 + erf((Pr − 5) ÷ √2))

  Release, fire, explosion and toxic screening process/safety/consequence.js

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• Blast curve interpolation

  Reads a scaled overpressure or duration from the curve points a person typed.

  ln y = ln y₀ + (ln y₁ − ln y₀) × (ln R − ln R₀) ÷ (ln R₁ − ln R₀)

  Release, fire, explosion and toxic screening process/safety/blast.js

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• Sachs scaling length

  The length that scales distances for an explosion of a given energy.

  L = ∛(E ÷ P₀)

  Release, fire, explosion and toxic screening process/safety/blast.js

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• Multi-energy overpressure

  The overpressure and positive duration at a distance from a blast source of given energy and strength.

  R̄ = R ÷ L; ΔP = P̄(R̄) × P₀; duration = t̄(R̄) × L ÷ 340

  Release, fire, explosion and toxic screening process/safety/blast.js

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• Distance to a multi-energy overpressure

  The distance at which the overpressure of a blast source falls to a given level.

  Release, fire, explosion and toxic screening process/safety/blast.js

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• Overpressure from several sources

  The overpressure at a point from each blast source and which one governs.

  E = volume × energy density; R = max(distance, 0.01 m)

  Release, fire, explosion and toxic screening process/safety/blast.js

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• Blast contour circles

  Circles on the site plan at given overpressure levels.

  Release, fire, explosion and toxic screening process/safety/blast.js

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• Heavy-gas cloud spreading

  Radius, height, concentration and drift of a dense gas cloud over time, and when it dilutes to a given concentration.

  dR/dt = Cg·√(g′·H); dV/dt = π·R²·a_top·u* + 2π·R·H·a_edge·dR/dt; g′ = g·(ρ − ρ_a) ÷ ρ_a

  Release, fire, explosion and toxic screening process/safety/heavygas.js

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• Dilution flow, volume and radius

  The ventilation flow needed to dilute a release, the hypothetical flammable volume and the radius of a sphere of that volume.

  LFL_m = LFL × M × P ÷ (R·T); Q_min = rate ÷ (k × LFL_m) × (T ÷ 293); V_z = f × Q_min ÷ C; r = ∛(3V_z ÷ 4π)

  Hazardous area extents and zone geometry process/safety/area.js

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• Zone extent and shape

  The extent of each zone and its shape: a sphere, or a ground-to-top cylinder for a heavy vapor.

  sphere: center (x, y, z), radius r; cylinder: z from 0 to z + r

  Hazardous area extents and zone geometry process/safety/area.js

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• Zone sections

  The outline of each zone where a vertical plane cuts it.

  half-width = √(r² − d²)

  Hazardous area extents and zone geometry process/safety/area.js

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• Point inside a zone

  Whether an item's position lies inside a zone volume.

  sphere: |p − c| ≤ r; cylinder: plan distance ≤ r and z₀ ≤ z ≤ z₁

  Hazardous area extents and zone geometry process/safety/area.js

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• Circle as a polygon

  The ring of points that stands for a circular zone in exported files.

  (cx + r·cos(2π·k ÷ 48), cy + r·sin(2π·k ÷ 48)), rounded to 6 decimals

  Hazardous area extents and zone geometry process/safety/zones.js

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• Zone footprints as GeoJSON

  The plan footprints of the zones as a GeoJSON file in local meters.

  Hazardous area extents and zone geometry process/safety/zones.js

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• Zone footprints as DXF

  The same footprints as DXF polylines, one layer per level.

  Hazardous area extents and zone geometry process/safety/zones.js

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• Zone volume mesh

  A closed triangle mesh of a sphere, hemisphere, cylinder or extruded polygon zone.

  Hazardous area extents and zone geometry process/safety/zones.js

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• Mesh volume

  The volume enclosed by a closed triangle mesh.

  V = Σ p · (q × r) ÷ 6

  Hazardous area extents and zone geometry process/safety/zones.js

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• Release sources from the diagram

  Release sources with position, pressure and temperature taken from the equipment on a process diagram.

  P₀ = 101325 + dp × 10⁵; T = dt + 273.15; position = drawing coordinate × factor, rounded to 6 decimals

  Hazardous area extents and zone geometry process/safety/zones.js

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• Mitigated event frequency

  The frequency of a scenario after its protection layers, against the target.

  f = Σ f_IE × Π p_modifier × Π PFD_IPL; required PFD = target ÷ f

  Layer of protection and SIL arithmetic process/safety/lopa.js

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• Required PFD of a new layer

  The probability of failure a safety function must reach to close the gap.

  required PFD = target ÷ (frequency without that layer)

  Layer of protection and SIL arithmetic process/safety/lopa.js

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• SIL band of a PFD

  The safety integrity level a probability of failure on demand falls in.

  SIL n when 10^−(n+1) ≤ PFD < 10^−n

  Layer of protection and SIL arithmetic process/safety/lopa.js

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• Unavailability of one channel

  The probability that one channel has failed dangerously at a time since new.

  u(t) = 1 − exp(−λ_t·(t mod TI) − λ_n·t)

  Layer of protection and SIL arithmetic process/safety/lopa.js

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• PFDavg by the simplified equations

  The average probability of failure on demand for each voting architecture.

  1oo1: λ·TI ÷ 2; 1oo2: ((1 − β)λ·TI)² ÷ 3 + β·λ·TI ÷ 2; 2oo3: ((1 − β)λ·TI)² + β·λ·TI ÷ 2

  Layer of protection and SIL arithmetic process/safety/lopa.js

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• PFDavg by averaging over the life

  The average probability of failure on demand with imperfect proof tests, by numerical averaging.

  PFDavg = (1 ÷ life) × ∫₀^life [P(at least N − K + 1 channels failed) + common cause] dt

  Layer of protection and SIL arithmetic process/safety/lopa.js

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• Safety function total

  The PFDavg of each subsystem, their total and the SIL of the total.

  PFDavg = Σ subsystem PFDavg

  Layer of protection and SIL arithmetic process/safety/lopa.js

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• Line fit

  The best straight line through a run of center-line pixels, and how far the worst pixel lies from it.

  θ = ½ × atan2(2·Sxy, Sxx − Syy); ends = feet of the first and last points on the line through the centroid

  Tracing a scan into lines and arcs raster/trace.js

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• Circle fit

  The best circle through a run of center-line pixels, its sweep and the angles of an arc.

  Kåsa: solve the 2 × 2 system for the center about the centroid, r = √(uc² + vc² + (Suu + Svv) ÷ n); then Gauss-Newton on d − r

  Tracing a scan into lines and arcs raster/trace.js

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• Corner detection

  Where a chain of pixels turns sharply, so it can be cut there before fitting.

  turn = |atan2(r − q) − atan2(q − p)| over ± w steps; corner when turn ≥ 0.6 rad and largest nearby

  Tracing a scan into lines and arcs raster/trace.js

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• Splitting a chain into lines and arcs

  Fits a line, else an arc, else splits the chain at the point farthest from its chord, and joins neighbors that fit as one.

  distance to chord = |(P − A) × (B − A)| ÷ |B − A|

  Tracing a scan into lines and arcs raster/trace.js

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• Joining collinear lines

  Merges two lines that lie on the same straight line and nearly touch.

  Tracing a scan into lines and arcs raster/trace.js

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• Squaring level and plumb lines

  Makes nearly horizontal and vertical lines exactly so, and gives lines of one wall the same coordinate.

  y = (y₁ + y₂) ÷ 2; shared coordinate = Σ(coordinate × length) ÷ Σ length

  Tracing a scan into lines and arcs raster/trace.js

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• Closing corners

  Moves a line end to the point where its line meets a nearby line.

  s = ((B₁ − E) × v) ÷ (u × v); new end = E + s·u

  Tracing a scan into lines and arcs raster/trace.js

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• Lines and arcs in image coordinates

  The traced lines and arcs with the window offset added and the arc end points worked out.

  start = (cx + r·cos a₀, cy + r·sin a₀); end = (cx + r·cos a₁, cy + r·sin a₁)

  Tracing a scan into lines and arcs raster/trace.js

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• Crossing of two traced lines

  The point where two traced line segments cross.

  d = (a₂ − a₁) × (b₂ − b₁); t = ((b₁ − a₁) × (b₂ − b₁)) ÷ d; point = a₁ + t·(a₂ − a₁)

  Tracing a scan into lines and arcs raster/snap.js

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• Snap points of a trace

  The ends, middles, arc centers, corners and crossings offered for snapping.

  middle = (p₁ + p₂) ÷ 2; arc middle at angle (a₀ + a₁) ÷ 2; corner = mean of the ends within 3 px

  Tracing a scan into lines and arcs raster/snap.js

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• Black-and-white level (Otsu)

  The grey level that best separates ink from paper for the whole sheet.

  maximize w_B × w_F × (m_B − m_F)² over the 256 levels

  Scan clean-up and JPEG decoding raster/image.js

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• Local threshold (Sauvola)

  Decides ink or paper for each pixel from the mean and spread of the pixels around it.

  ink when v ≤ mean × (1 + k × (sd ÷ 128 − 1)); sd = √(Σv² ÷ n − mean²)

  Scan clean-up and JPEG decoding raster/image.js

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• Sheet skew angle

  The angle by which the long lines of a scanned sheet are rotated.

  score(angle) = Σ rows² + Σ columns² of the ink projected at that angle; best angle on a 0.25° then 0.02° grid

  Scan clean-up and JPEG decoding raster/image.js

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• Rotating the image

  Turns a grey image by an angle about its middle.

  source = c + R(angle) × (pixel − c); value = bilinear mix of the 4 nearest source pixels

  Scan clean-up and JPEG decoding raster/image.js

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• JPEG inverse cosine transform

  Turns the stored frequency coefficients of an 8 × 8 block back into pixel values.

  pixel = round(Σ_u Σ_v C(x,u)·C(y,v)·coef(u,v) + 128), clamped to 0–255

  Scan clean-up and JPEG decoding raster/jpeg.js

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• JPEG color conversion

  Converts the decoded luminance and color-difference values to red, green and blue.

  R = Y + 1.402·Cr; G = Y − 0.344136·Cb − 0.714136·Cr; B = Y + 1.772·Cb

  Scan clean-up and JPEG decoding raster/jpeg.js

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• Table values in millimeters

  Converts each published row to millimeters, mm² and kg/m.

  mm = in × 25.4; mm² = in² × 25.4 × 25.4; kg/m = lb/ft × 1.48816394; r = (k_des − t_f) × 25.4

  Steel section dimensions, areas and outlines shapes/sections.js

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• Exact section area

  The cross-section area worked out from the dimensions, with fillets and rounded corners included.

  I: 2·b_f·t_f + (d − 2t_f)·t_w + (4 − π)·r²; RHS: 2t(h + b − 2t) − (4 − π)(r_o² − r_i²); CHS: π(D − t)·t

  Steel section dimensions, areas and outlines shapes/sections.js

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• Section outline

  The outline of a section as loops of points, centered on its bounding box.

  arc point = (cx + r·cos a, cy + r·sin a), a = a₀ + (a₁ − a₀)·i ÷ n

  Steel section dimensions, areas and outlines shapes/sections.js

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