Calculations

Calculations

• Minimum design metal temperature

  The stress ratio and the temperature reduction read from points a person types.

  ratio = tr·E ÷ (tn − c); MDMT = base − ΔT(ratio)

  Pressure vessel thickness and support formulas process/equip/vessel.js

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• Data sheet values

  The rows and text of the vessel data sheet.

  Pressure vessel thickness and support formulas process/equip/vessel.js

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• Equipment parameters for the plant model

  The diameter, length and support height handed to the plant equipment.

  Pressure vessel thickness and support formulas process/equip/vessel.js

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• External pressure from chart factors

  The allowable external pressure of a cylinder from the chart factor A and the material curve a person types.

  chart: P_a = 4B ÷ (3·D_o ÷ t); elastic: P_a = 2A·E ÷ (3·D_o ÷ t)

  Pressure vessel thickness and support formulas process/equip/vessel2.js

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• Stiffening ring inertia

  The moment of inertia a stiffening ring must have.

  I_s = D_o² × L_s × (t + A_s ÷ L_s) × A ÷ 14

  Pressure vessel thickness and support formulas process/equip/vessel2.js

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• Head outline

  The outline of a hemispherical, ellipsoidal or torispherical head from the axis to the tangent line.

  ellipsoidal: (R·cos a, (R ÷ ratio)·sin a), a from π ÷ 2 to 0 in n steps

  Vessel outline and inside volume process/equip/vessel2.js

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• Torispherical head outline

  The crown and knuckle arcs of a torispherical head and where they join.

  c = R − r_k; dz = √((R_c − r_k)² − c²); junction angle = atan2(dz, c)

  Vessel outline and inside volume process/equip/vessel2.js

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• Vessel outline, height and volume

  The full inside outline with both heads, the overall height and the inside volume.

  V = Σ π·(r₁² + r₁·r₂ + r₂²) ÷ 3 × (z₂ − z₁); m³ = mm³ × 10⁻⁹

  Vessel outline and inside volume process/equip/vessel2.js

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• Nozzle position on the body

  The position and axis of a nozzle on the shell or on a head.

  position = (R·cos φ, R·sin φ, z_tangent + at)

  Vessel outline and inside volume process/equip/vessel2.js

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• Liquid release through a hole

  The mass flow and exit velocity of a liquid leaving through a hole.

  v = √(2ΔP ÷ ρ + 2g·h); ṁ = Cd × A × ρ × v

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

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• Gas release through a hole

  The mass flow of a gas leaving through a hole, choked or not.

  choked: ṁ = Cd·A·P₀·√(k·M ÷ (Z·R·T) × (2 ÷ (k + 1))^((k+1)/(k−1)))

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

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• Flash fraction

  The fraction of a pressurized liquid that turns to vapor on release.

  F = 1 − exp(−cp·(T − T_b) ÷ h_fg)

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

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• Pool evaporation rate

  The evaporation rate from a liquid pool in wind.

  k_g = 0.004786 × U^0.78 × d^(−0.11) × Sc^(−0.67); ṁ = k_g × A × M × P_v ÷ (R·T)

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

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• Radiation distance from a point source

  The distance at which the heat flux from a fire falls to a given level.

  r = √(τ·χ·Q ÷ (4π·q))

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

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• 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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