Calculations

Calculations

• Compressor head, power and outlet temperature

  Isentropic and polytropic head, both efficiencies, shaft work, power and discharge temperature.

  Hs = (Z·R·T₁ ÷ M) × (k ÷ (k − 1)) × (r^((k−1)/k) − 1); Hp = (Z·R·T₁ ÷ M) ÷ m × (r^m − 1); T₂ = T₁·r^m

  Piping network, pumps and compressors process/equip/compressor.js

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• Operating point against the surge line

  The surge margin of a compressor run, with the head converted to the unit of the map.

  H(m) = Hp ÷ g

  Piping network, pumps and compressors process/equip/compressor.js

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• Pressure wave speed

  The speed of a pressure wave in a liquid inside an elastic pipe.

  a = √((K ÷ ρ) ÷ (1 + (K·D ÷ E·e)·c₁))

  Water hammer and pump trip process/equip/network.js

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• Joukowsky pressure rise

  The pressure rise for an instant change of velocity.

  ΔP = ρ × a × ΔV

  Water hammer and pump trip process/equip/network.js

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• Valve closure transient

  Head and flow at the valve over time, and the highest and lowest head, when a valve at the end of a pipe opens or closes.

  H_P = (C⁺ + C⁻) ÷ 2; Q_P = (C⁺ − C⁻) ÷ 2B; valve: q = (−B·c + √(B²c² + 4c·C⁺)) ÷ 2

  Water hammer and pump trip process/equip/network.js

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• Fitting loss, 2-K form

  The loss coefficient of a fitting from its two constants, the Reynolds number and the pipe diameter in inches.

  K = K₁ ÷ Re + K∞ × (1 + 1 ÷ D)

  Water hammer and pump trip process/equip/transient.js

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• Fitting loss, 3-K form

  The loss coefficient of a fitting from its three constants.

  K = K₁ ÷ Re + Kᵢ × (1 + K_d ÷ D^0.3)

  Water hammer and pump trip process/equip/transient.js

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• Pump trip transient

  Head, flow and pump speed over time after a pump loses power, and when the check valve closes.

  H = α²·h(Q ÷ α), α = ω ÷ ω₀; I·dω/dt = −ρ·g·Q·H ÷ (η·ω)

  Water hammer and pump trip process/equip/transient.js

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• Pipe section properties

  Metal area, second moment of area, torsion constant and section modulus of a pipe.

  A = π(D² − d²) ÷ 4; I = π(D⁴ − d⁴) ÷ 64; J = 2I; Z = I ÷ (D ÷ 2)

  Pipe stress checks process/equip/pipestress.js

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• Tee stress intensification

  The stress intensification factors of a welding, reinforced or unreinforced tee.

  i_o = max(0.9 ÷ h^(2/3), 1); i_i = max(0.75·i_o + 0.25, 1); welding tee h = 4.4·T ÷ r

  Pipe stress checks process/equip/pipestress.js

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• Bend stress intensification and flexibility

  The flexibility characteristic, stress intensification factors and flexibility factor of a bend.

  h = t·R₁ ÷ r²; i_i = 0.9 ÷ h^(2/3); i_o = 0.75 ÷ h^(2/3); k = 1.65 ÷ h

  Pipe stress checks process/equip/pipestress.js

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• Bend tangent points

  Replaces each corner of the run by the two tangent points of a bend of the given radius.

  θ = acos(u₁ · u₂); tangent length a = R₁ × tan(θ ÷ 2)

  Pipe stress checks process/equip/pipestress.js

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• Node stations along the run

  Places a node at every corner, support and tee by its distance along the run.

  s = Σ segment lengths; key = s rounded to 0.001 mm; point = a + f × (b − a)

  Pipe stress checks process/equip/pipestress.js

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• Weight and thermal loads

  The contents and insulation weight per length and the forces that represent free thermal expansion.

  w = ρ_c × 10⁻⁹ × (π ÷ 4)·d² × g + (kg/m ÷ 1000) × g; F = E × A × α × ΔT

  Pipe stress checks process/equip/pipestress.js

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• 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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