Calculations

Calculations

• Mixture parameters

  a, da/dT and b of a mixture of given composition.

  a = Σᵢ Σⱼ yᵢ yⱼ √(aᵢ aⱼ) (1 − kᵢⱼ); b = Σ yᵢ bᵢ

  Cubic equations of state process/eos.js

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• Compressibility roots

  The physical compressibility factors of a mixture at a temperature and pressure.

  β = bP ÷ RT; A = aP ÷ (RT)²; Z³ + c₂Z² + c₁Z + c₀ = 0 with c from β, A, ε, σ; keep Z > β (1 + 1e-12)

  Cubic equations of state process/eos.js

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• Fugacity coefficients

  ln φ of every component for one compressibility root.

  ln φᵢ = bᵢ⁄b (Z − 1) − ln(Z − β) − a ÷ (bRT) × (2 Σⱼ yⱼ aᵢⱼ ÷ a − bᵢ⁄b) × I, I = ln((Z + σβ) ÷ (Z + εβ)) ÷ (σ − ε)

  Cubic equations of state process/eos.js

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• Enthalpy and entropy departures

  How far a real phase's enthalpy and entropy are from the ideal gas at the same temperature and pressure.

  H − H_ig = RT (Z − 1) + (T da/dT − a) ÷ b × I; S − S_ig = R ln(Z − β) + (da/dT) ÷ b × I

  Cubic equations of state process/eos.js

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• Vapor pressure of a pure component

  The pressure at which the liquid and vapor roots of one component have equal fugacity.

  P ← P × exp(ln φ_L − ln φ_V) until |ln φ_L − ln φ_V| < 1e-12

  Cubic equations of state process/eos.js

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• Ideal-gas enthalpy

  The ideal-gas enthalpy of a component, with its formation enthalpy, at a temperature.

  H = h_f + R (A (T − T₀) + B⁄2 (T² − T₀²) + C⁄3 (T³ − T₀³) − D (1⁄T − 1⁄T₀))

  Property package and flashes process/thermo.js

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• Ideal-gas entropy

  The ideal-gas entropy of a component at a temperature, relative to 298.15 K.

  S = R (A ln(T ÷ T₀) + B (T − T₀) + C⁄2 (T² − T₀²) − D⁄2 (1⁄T² − 1⁄T₀²))

  Property package and flashes process/thermo.js

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• Vapor pressure for the ideal and activity models

  The vapor pressure of a component from its Antoine constants, or an estimate when it has none.

  P = exp(A − B ÷ (T + C)); without Antoine: P = P_c exp(5.373 (1 + ω)(1 − T_c ÷ T))

  Property package and flashes process/thermo.js

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• Heat of vaporization

  The heat of vaporization of a component at a temperature.

  ΔH(T) = ΔH(T_b) × ((1 − T⁄T_c) ÷ (1 − T_b⁄T_c))^0.38

  Property package and flashes process/thermo.js

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• Liquid molar volume

  The saturated liquid volume of a component for the ideal and activity models.

  V = R T_c ÷ P_c × Z_ra^(1 + (1 − T_r)^(2⁄7)), Z_ra = 0.29056 − 0.08775 ω

  Property package and flashes process/thermo.js

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• Enthalpy, entropy and volume of one phase

  The molar enthalpy, entropy and volume of a liquid or vapor of given composition.

  EOS: H = Σ xᵢ H_ig,ᵢ + departure; V = ZRT ÷ P. Activity liquid: H = Σ xᵢ (H_ig,ᵢ − ΔH_vap,ᵢ) + H^E, H^E = −RT² Σ xᵢ ∂ln γᵢ⁄∂T

  Property package and flashes process/thermo.js

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• K-values

  ln K of every component between a liquid and a vapor composition.

  EOS: ln Kᵢ = ln φᵢ(liquid) − ln φᵢ(vapor); activity: ln Kᵢ = ln γᵢ + ln(P_sat,ᵢ ÷ P), or ln(Hᵢ ÷ P) for a Henry solute

  Property package and flashes process/thermo.js

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• Vapor fraction (Rachford–Rice)

  The vapor fraction that balances a feed for given K-values.

  Σ zᵢ (Kᵢ − 1) ÷ (1 + V (Kᵢ − 1)) = 0, 0 ≤ V ≤ 1

  Property package and flashes process/thermo.js

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• Phase stability test

  Whether a mixture at a temperature and pressure stays one phase or splits.

  Yᵢ ← exp(ln zᵢ + ln φᵢ(z) − ln φᵢ(y)); unstable when ΣY > 1 + 1e-8

  Property package and flashes process/thermo.js

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• Two-phase flash iteration

  The vapor fraction and the two compositions at a temperature and pressure.

  V from Rachford–Rice; xᵢ = zᵢ ÷ (1 + V (Kᵢ − 1)); yᵢ = Kᵢ xᵢ; K ← exp(ln K(x, y)) until max |Δ ln K| < 1e-13

  Property package and flashes process/thermo.js

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• Liquid–liquid split

  Two liquid phases and their fractions for an activity model.

  trial phases from each component at 0.9; split when the tangent-plane distance < −1e-9; K ← exp(ln γ(x₂) − ln γ(x₁)) until max |Δ ln K| < 1e-12

  Property package and flashes process/thermo.js

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• Saturation residual

  How far a mixture is from its bubble or dew point at a trial temperature or pressure.

  bubble: Σ Kᵢ zᵢ − 1; dew: 1 − Σ zᵢ ÷ Kᵢ

  Property package and flashes process/thermo.js

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• Bubble and dew points

  The bubble or dew temperature at a pressure, or pressure at a temperature.

  scan a geometric grid for a sign change of the residual, then bisect on √(a b) until b ÷ a − 1 < 1e-13

  Property package and flashes process/thermo.js

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• Temperature for a given enthalpy or entropy

  The temperature at which a PT flash has the wanted enthalpy or entropy.

  false position inside a bracket; stop when b − a < 1e-10 K or |f| < 1e-9 (|target| + 1)

  Property package and flashes process/thermo.js

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• Flash at a vapor fraction

  The state at a pressure and a wanted vapor fraction.

  bisect T between the bubble and dew temperatures until the bracket is below 1e-9 K

  Property package and flashes process/thermo.js

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• Saturation temperature of a pure component

  The temperature at which a pure component's vapor pressure equals the pressure.

  bisect T on [20 K, T_c − 1e-6] until the bracket is below 1e-11 K

  Property package and flashes process/thermo.js

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• Gas constant

  R, used by the property package, the equations of state, the reactors and the dynamic simulation.

  R = 8.314462618 J/(mol K)

  Property package and flashes process/data.js

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• Component data to SI

  A component's critical pressure, molar mass and Antoine constants in the units the engine works in.

  P_c (Pa) = P_c (bar) × 1e5; M (kg/mol) = M ÷ 1000; A′ = A + ln 1000; C′ = C − 273.15

  Property package and flashes process/data.js

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• 2×2 solve

  The Newton step for the two phase fractions.

  det = a d − b c; x = (r₁ d − b r₂) ÷ det; y = (a r₂ − c r₁) ÷ det

  Three-phase flash process/vlle.js

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• Phase fractions

  The vapor and second-liquid fractions for the current K-values.

  gⱼ = Σ zᵢ (Kⱼᵢ − 1) ÷ Dᵢ = 0, Dᵢ = 1 + β_v (K_v,ᵢ − 1) + β_l (K_l,ᵢ − 1)

  Three-phase flash process/vlle.js

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• Composition update

  The liquid compositions and the new K-values after each pass.

  x₁,ᵢ = zᵢ ÷ Dᵢ; x₂,ᵢ = x₁,ᵢ K_l,ᵢ; error = max |ln(K_new ÷ K)|

  Three-phase flash process/vlle.js

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• Residuals of a result

  The largest iso-fugacity difference between the phases and the largest component-balance error, as the calculator reports them.

  iso = max |ln(xᵢγᵢ)₁ − ln(xᵢγᵢ)₂|, |ln(yᵢP) − ln(xᵢγᵢP_sat,ᵢ)|; balance = max |Σ_phases frac × xᵢ − zᵢ|

  Three-phase flash process/vlle.js

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• Reference-state offsets

  The shifts that put CoolProp's enthalpy and entropy on the engine's reference state.

  ΔH = h_f − H(298.15 K, 1e-4 mol/m³); ΔS = −(S(298.15 K, 1e-4 mol/m³) − R ln(1e5 ÷ (1e-4 R T₀)))

  Pure-fluid properties from CoolProp process/coolprop.js

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• Properties at a temperature and pressure

  Enthalpy, entropy, heat capacity, molar density, molar mass and phase of a pure fluid.

  H = H_CoolProp + ΔH; S = S_CoolProp + ΔS; M (g/mol) = M (kg/mol) × 1000

  Pure-fluid properties from CoolProp process/coolprop.js

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• Saturation values

  Vapor pressure, saturated liquid and vapor densities and heat of vaporization at a temperature.

  ΔH_vap = H(T, vapor) − H(T, liquid)

  Pure-fluid properties from CoolProp process/coolprop.js

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• Ideal-gas heat-capacity fit

  A quadratic C_p ÷ R = A + B T + C T² fitted to the dilute-gas enthalpy slope over a temperature range.

  C_p(T) ≈ H(T + 0.5) − H(T − 0.5) at 13 temperatures; normal equations M s = v solved by 3×3 determinants

  Pure-fluid properties from CoolProp process/coolprop.js

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• Component from a fluid

  A component for the property package from CoolProp's critical constants, acentric factor, boiling point and the heat-capacity fit.

  P_c (bar) = P_c (Pa) ÷ 1e5; M (g/mol) = M × 1000; ΔH_vap(T_b) = H(1 atm, vapor) − H(1 atm, liquid)

  Pure-fluid properties from CoolProp process/coolprop.js

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• Minimum stages (Fenske)

  The least number of stages at total reflux for a split of two key components.

  N_min = ln((d_LK ÷ d_HK)(b_HK ÷ b_LK)) ÷ ln α

  Distillation and absorption columns process/column.js

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• Gilliland correlation (Eduljee)

  The stage ratio Y for a reflux ratio X above the minimum.

  Y = 0.75 (1 − X^0.5668)

  Distillation and absorption columns process/column.js

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• Gilliland correlation (Molokanov)

  The same stage ratio by the fit of Molokanov and others.

  Y = 1 − exp((1 + 54.4X) ÷ (11 + 117.2X) × (X − 1) ÷ √X)

  Distillation and absorption columns process/column.js

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• Number of stages

  The stages needed at a reflux ratio, from the minimum stages and minimum reflux.

  X = (R − R_min) ÷ (R + 1); N = (N_min + Y) ÷ (1 − Y)

  Distillation and absorption columns process/column.js

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• Minimum reflux (Underwood)

  The Underwood root between the key volatilities and the minimum reflux ratio from it.

  Σ αᵢ z_F,ᵢ ÷ (αᵢ − θ) = 1 − q; R_min = Σ αᵢ x_D,ᵢ ÷ (αᵢ − θ) − 1

  Distillation and absorption columns process/column.js

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• Kremser equation

  The fraction of a component absorbed or stripped in N stages.

  (A^(N+1) − A) ÷ (A^(N+1) − 1); N ÷ (N + 1) when |A − 1| < 1e-12

  Distillation and absorption columns process/column.js

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• Feed thermal condition

  q of a feed: the fraction that joins the liquid below the feed stage.

  q = (H_dew − H_feed) ÷ (H_dew − H_bubble)

  Distillation and absorption columns process/column.js

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• Shortcut split of the non-key components

  How every component divides between distillate and bottoms, and the volatilities used for it.

  dᵢ ÷ bᵢ = (d_HK ÷ b_HK) αᵢ^N_min; αᵢ = √((Kᵢ ÷ K_HK)_top (Kᵢ ÷ K_HK)_bottom)

  Distillation and absorption columns process/column.js

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• Feed stage and duties of the shortcut column

  The split of stages above and below the feed (Kirkbride) and the condenser and reboiler duties.

  N_R ÷ N_S = ((z_HK ÷ z_LK)(B ÷ D)(x_B,LK ÷ x_D,HK)²)^0.206; Q_c = (R + 1) D (H_liquid − H_vapor); Q_r from the overall enthalpy balance

  Distillation and absorption columns process/column.js

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• Component balances on every stage

  The liquid flow of each component on each stage of the rigorous column.

  l_{j−1} − (1 + S_j) l_j + S_{j+1} l_{j+1} = −f_j, S_j = K_j V_j ÷ L_j; solved by the Thomas algorithm

  Distillation and absorption columns process/column.js

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• Vapor and liquid flows from the energy balances

  The vapor flow leaving each stage and the liquid flow below it, and the test that ends the iteration.

  V_{j+1} = (V_j (H_j − h_j) − L_{j−1} (h_{j−1} − h_j) − (H_F,j − F_j h_j)) ÷ (H_{j+1} − h_j); stop when max |ΔT| < tol and max |ΔV| ÷ |V| < 1e-9

  Distillation and absorption columns process/column.js

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• Duties and closure of the rigorous column

  The condenser and reboiler duties and how well the component and energy balances close.

  Q_c = V₁ (H₁ − h₀); component closure = max |Σf − d − b| ÷ F; energy closure = |in − out| ÷ (|in| + 1)

  Distillation and absorption columns process/column.js

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• Tridiagonal solve

  The solution of a tridiagonal system, used for the component balances.

  forward: m = a_j ÷ b_{j−1}; b_j −= m c_{j−1}; r_j −= m r_{j−1}; back: x_j = (r_j − c_j x_{j+1}) ÷ b_j

  Distillation and absorption columns process/column2.js

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• Simple models fitted in the outer loop

  The local K-value and enthalpy models the inner loop uses in place of the property package.

  ln K_b = A + B ÷ T; B = (ln K_b(T + 0.5) − ln K_b(T)) ÷ (1 ÷ (T + 0.5) − 1 ÷ T); h = h_a + h_b T with h_b = (h(T + 0.5) − h(T)) ÷ 0.5

  Distillation and absorption columns process/column2.js

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• Stage temperatures in the inner loop

  The temperature and vapor composition of each stage from the simple K model.

  K_b = 1 ÷ Σ αᵢ xᵢ; T = B ÷ (ln K_b − A); yᵢ = αᵢ xᵢ, normalized

  Distillation and absorption columns process/column2.js

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• Inside-out stopping tests

  When the inner and outer loops of the inside-out column end.

  inner: max |ΔT| < 1e-10 K and max |ΔV| ÷ |V| < 1e-11; outer: max |ΔT| < tol and max |ΔV| ÷ |V| < 1e-8

  Distillation and absorption columns process/column2.js

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• Absorber flows by sum-rates

  The vapor and liquid flow leaving each stage of an absorber or stripper.

  V_j = Σᵢ l_ij K_ij V_j ÷ L_j; L_j = F(1..j) − V₁ + V_{j+1}

  Distillation and absorption columns process/column2.js

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• Absorber stage temperatures

  The temperature of every stage from the stage energy balances.

  E_j = L_{j−1} h_{j−1} + V_{j+1} H_{j+1} + H_F,j + Q_j − L_j h_j − V_j H_j = 0; J ΔT = −E with heat capacities by a 0.5 K difference

  Distillation and absorption columns process/column2.js

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