Calculations

Calculations

• Stress at each node

  The bending, torsion and axial stress at each node for the sustained, expansion and occasional cases.

  expansion: S_E = √(S_b² + 4·S_t²), S_b = √((i_i·M_i)² + (i_o·M_o)²) ÷ Z, S_t = T ÷ 2Z; M_i = √(M_b² − M_o²)

  Pipe stress checks process/equip/pipestress.js

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• Stress ratios against the allowables

  Sustained, expansion and occasional stress as a fraction of what is allowed.

  S_L = bending + P·D ÷ 4t; S_A = f·(1.25·S_c + 0.25·S_h) + max(0, S_h − S_L); occasional limit 1.33·S_h

  Pipe stress checks process/equip/pipestress.js

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• Free thermal growth

  How much the developed length of the run would grow if it were free.

  growth = ΔT × α × (s_last − s_first)

  Pipe stress checks process/equip/pipestress.js

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• Support lift-off and friction

  Releases rests that would have to pull the pipe down and applies sliding friction at the rests that stay in contact.

  release when R_z(weight) + R_z(thermal) < −10⁻⁶ N; F = −µ·N·(u ÷ |u|), relaxed by 0.6 per pass

  Pipe stress checks process/equip/pipestress.js

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• Nozzle loads

  The force and moment a support passes to the equipment, with a check against a limit.

  F = √(Fx² + Fy² + Fz²); M = √(Mx² + My² + Mz²)

  Pipe stress checks process/equip/pipestress.js

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• Fire coefficient in SI units

  Converts the pool-fire coefficient from Btu/h per ft^1.64 to W per m^1.64.

  C_SI = C_US × 0.29307107 × (1 ÷ 0.3048²)^0.82

  Relief valve sizing, depressuring and flare loads process/equip/relief.js

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• Gas and vapor orifice area

  The orifice area a relief valve needs for a gas or vapor flow.

  A = W ÷ (Kd·Kb·Kc × G); critical: G = P₁·√(k·M ÷ Z·R·T) × (2 ÷ (k + 1))^((k+1)/(2(k−1)))

  Relief valve sizing, depressuring and flare loads process/equip/relief.js

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• Liquid orifice area

  The orifice area for a liquid flow, with the viscosity correction.

  A = Q ÷ (Kd·Kw·Kc·Kv) × √(ρ ÷ 2ΔP); Kv = 1 ÷ (0.9935 + 2.878 ÷ √Re + 342.75 ÷ Re^1.5)

  Relief valve sizing, depressuring and flare loads process/equip/relief.js

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• Critical pressure ratio, omega method

  The pressure ratio at which a flashing two-phase flow chokes.

  η² + (ω² − 2ω)(1 − η)² + 2ω²·ln η + 2ω²(1 − η) = 0

  Relief valve sizing, depressuring and flare loads process/equip/relief.js

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• Two-phase mass flux

  The mass flux of a flashing mixture through a nozzle.

  G = √(P₀·ρ₀) × √(2[ω·ln(1 ÷ η) − (ω − 1)(1 − η)]) ÷ (ω(1 ÷ η − 1) + 1)

  Relief valve sizing, depressuring and flare loads process/equip/relief.js

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• Fire heat input

  The heat a wetted vessel absorbs in a pool fire.

  Q = C × F × A^0.82

  Relief valve sizing, depressuring and flare loads process/equip/relief.js

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• Wetted surface of a vessel

  The vessel surface in contact with liquid up to the fire height.

  vertical: π·D·level + k·D²; horizontal: r·θ·L + 2k·D²·(h ÷ D), θ = 2·acos(1 − h ÷ r)

  Relief valve sizing, depressuring and flare loads process/equip/relief.js

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• Tube rupture flow

  The flow through broken tube ends from the high-pressure side to the low-pressure side.

  Q = Cd × A × √(2(P_h − P_l) ÷ ρ), A = n × π·d² ÷ 4

  Relief valve sizing, depressuring and flare loads process/equip/relief.js

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• Standard orifice letter

  The smallest standard orifice whose area covers the required area.

  in² = m² ÷ 0.0254²

  Relief valve sizing, depressuring and flare loads process/equip/relief.js

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• Flare radiation distance

  The distance from a flare at which the heat flux falls to an allowable level.

  D = √(τ·F·Q ÷ (4π·K))

  Relief valve sizing, depressuring and flare loads process/equip/relief.js

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• Flare header back pressure

  The back pressure at each relief valve as a fraction of its set pressure.

  fraction = (P_back − 101325) ÷ P_set

  Relief valve sizing, depressuring and flare loads process/equip/relief.js

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• Steam orifice area

  The orifice area for steam relief.

  A[in²] = W[lb/h] ÷ (51.5 × P₁[psia] × Kd·Kb·Kc·Kn·Ksh)

  Relief valve sizing, depressuring and flare loads process/equip/relief.js

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• Discharge reaction force

  The thrust on a valve outlet discharging a sonic gas jet to the atmosphere.

  F = W·v + (P_e − P_a)·A_e; v = √(2k·Z·R·T ÷ ((k + 1)·M))

  Relief valve sizing, depressuring and flare loads process/equip/relief.js

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• Outlet line back pressure

  The pressure at the valve outlet for a gas flow through the discharge pipe, and the exit Mach number.

  P₁² − P₂² = (G²·Z·R·T ÷ M) × (f·L ÷ D + K + 2·ln(P₁ ÷ P₂))

  Relief valve sizing, depressuring and flare loads process/equip/relief.js

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• Vessel depressuring curve

  Pressure, temperature and vent flow of a gas-filled vessel over time as it is vented, and the time to reach a target pressure.

  dm/dt = −ṁ; m·cv·dT/dt = Q − ṁ·Rs·T; P = m·Rs·T ÷ V

  Relief valve sizing, depressuring and flare loads process/equip/relief2.js

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• Control valve failure load

  The relief load when a control valve fails open, less the normal flow.

  liquid: Q[m³/h] = Kv × √(ΔP[bar] ÷ SG), ṁ = Q ÷ 3600 × SG × ρ_w; load = max(flow − normal, 0)

  Relief valve sizing, depressuring and flare loads process/equip/relief2.js

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• Header load per case

  The total load on each flare header for each case, and the governing case.

  load = Σ device loads of the scenarios in the case; governing = largest load

  Relief valve sizing, depressuring and flare loads process/equip/relief2.js

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• Cylindrical shell thickness

  The required thickness of a cylindrical shell under internal pressure.

  t_circ = P·R ÷ (S·E − 0.6P); t_long = P·R ÷ (2S·E + 0.4P); t = larger of the two

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

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• Formed head thickness

  The required thickness of an ellipsoidal, torispherical or hemispherical head.

  ellipsoidal: t = P·D·K ÷ (2S·E − 0.2P), K = (2 + (D ÷ 2h)²) ÷ 6; torispherical: t = P·L·M ÷ (2S·E − 0.2P), M = (3 + √(L ÷ r)) ÷ 4

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

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• Conical section thickness

  The required thickness of a cone.

  t = P·D ÷ (2·cos α × (S·E − 0.6P))

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

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• Maximum allowable working pressure

  The pressure a shell of given thickness may carry.

  P_c = S·E·t ÷ (R + 0.6t); P_l = 2S·E·t ÷ (R − 0.4t); P = smaller of the two

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

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• Hydrostatic test pressure

  The test pressure from the allowable working pressure.

  P_test = 1.3 × MAWP × stress ratio

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

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• External pressure, elastic formulas

  The critical and allowable external pressure of a plain cylinder.

  long: P = 2E·(t ÷ D)³ ÷ (1 − ν²); short: P = 2.42E·(t ÷ D)^2.5 ÷ ((1 − ν²)^0.75 × (L ÷ D − 0.45·√(t ÷ D))); P_a = P_cr ÷ 4

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

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• Limits of reinforcement

  How far along and normal to the wall material counts as reinforcement of an opening.

  along wall = max(d, Rn + tn + t); normal = min(2.5t, 2.5tn + te)

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

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• Opening reinforcement areas

  The area an opening removes and the areas available to replace it.

  A = d·tr + 2tn·tr·(1 − fr₁); available = A₁ + A₂ + A₄₁ + A₅; margin = available ÷ A − 1

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

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• Saddle bending stresses

  Longitudinal bending moments and stresses of a horizontal vessel on two saddles.

  M₁ = Q·a·(1 − (1 − a ÷ L + (R² − H²) ÷ 2aL) ÷ (1 + 4H ÷ 3L)); σ = M ÷ (π·R²·t)

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

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• Wind velocity pressure

  The wind pressure on a vessel from the wind speed.

  q = 0.613 × Kz × Kzt × Kd × V² × I

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

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• Wind force and overturning moment

  The wind force on a vertical vessel and its moment at the base.

  F = q × 10⁻⁶ × Cf × D × H; M = F × H ÷ 2

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

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• Seismic base shear and moment

  A first estimate of the seismic shear and overturning moment.

  V = Cs × W; M = Cs × W × 2H ÷ 3

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

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• Skirt stresses

  Compressive and tensile stress at the base of a skirt from weight and overturning.

  A = π·D·t; Z = π·D²·t ÷ 4; σ = M ÷ Z ± W ÷ A

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

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• Flat head thickness

  The required thickness of a circular flat head or cover.

  t = d × √(C·P ÷ (S·E))

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

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• 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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