SEC.Models.GrmLinear

SEC.Models.GrmLinear#

SEC.Models.GrmLinear.py

PDF of the General Rate Model (GRM) with a linear isotherm, computed via characteristic function (CF) inversion using FFT (PDE-based).

The GRM is the most physically detailed of the three linear-isotherm SEC transport models available in this package:

EDM (EdmLinear) — axial dispersion only (k_MT → ∞ limit) LKM (LkmLinear) — axial dispersion + lumped mass transfer GRM (this file) — axial dispersion + film mass transfer + pore diffusion

Assumptions#

  • Linear isotherm: q = a_star * c inside the pore (no overloading)

  • Axial dispersion parametrised by the Peclet number Pe = u*L/D_ax

  • External (film) mass transfer with rate constant k_ext [length/time]

  • Intraparticle pore diffusion with effective diffusivity D_eff [length²/time]

  • Spherical particles of radius R_p

  • Intraparticle porosity eps_p absorbed into a_star:

    a_star = eps_p + (1 - eps_p) * a_henry

    where a_henry is the Henry coefficient for adsorption on the pore wall. For non-porous particles set eps_p = 0; then a_star = a_henry.

Model parameters#

Pe : Peclet number Pe = u*L/D_ax t0 : dead time t0 = L/u (mobile-phase transit time) k_ext : external film mass-transfer coefficient [length/time] R_p : particle radius [length] D_eff : effective intraparticle pore diffusivity [length²/time]

Large D_eff (film-only limit) → k_MT_eff = 3*F*k_ext / (R_p * R)

a_star : effective intraparticle retention parameter (see Assumptions above) F_ratio : phase ratio F = (1 - ε) / ε (ε = interstitial porosity)

Retention factor#

R_eff = 1 + F_ratio * a_star

Transfer function (Laplace domain, Qamar 2014 eqs. 22–24, 37)#

φ(s) = s + (3*F*k_ext/R_p) * (1 − f(s))

f(s) = B_p / (B_p − 1 + ξ·coth(ξ))
where ξ = sqrt(a_star * s / D_eff) * R_p

B_p = k_ext * R_p / D_eff

H(s) = exp( Pe/2 * (1 − sqrt(1 + 4*t0*φ(s)/Pe)) )

Moment formulas (Qamar 2014 Table 1, Dirichlet BCs, linear isotherm)#

μ₁ = t0 * R_eff σ²_GRM = 2*t0²*R_eff²/Pe + 2*t0*(R_eff−1)²/(R_eff * k_MT_eff)

where k_MT_eff = 3*F*k_ext / (R_p * R_eff) [film-only limit, large D_eff]

GRM ↔ LKM moment matching (Qamar 2014, Appendix C)#

Given GRM k_MT_eff and R, the equivalent LKM k_MT is:

k_MT_LKM = R / (R−1) * k_MT_eff

LKM initialisation from GRM result:
k_ext = k_MT_LKM * (R−1)/R * R_p / (3*F/R)

= k_MT_LKM * (R−1) * R_p / (3*F)

References

  • Qamar et al. (2014), Chem. Eng. Sci. 107:192–205 (GRM CF formula)

  • Validated in molass-researcher/experiments/27_qamar_2014_paper/27a, 27b - μ₁ error: 0.00%, σ² error: 0.00% vs Qamar 2014 Table 1 - Timing: ~1.1× LKM overhead

grm_linear_cf(w, Pe, t0_s, k_ext_s, R_p, D_eff_s, a_star, F_ratio)#

Characteristic function of the GRM elution profile (scaled coordinates).

Parameters are in scaled time units (see grm_pdf for the scaling convention). Do not call this directly; use grm_pdf instead.

grm_pdf(x, Pe, t0, k_ext, R_p, D_eff, a_star, F_ratio, timescale=None)#

PDF of the GRM (General Rate Model) elution profile with linear isotherm.

Uses FFT-based characteristic function inversion via FftInvPdf.

Parameters:

  • x (array_like) – Time array (physical time units, e.g. seconds or minutes).

  • Pe (float) – Peclet number Pe = u*L/D_ax. Controls axial dispersion width. Typical SEC range: 100–1000.

  • t0 (float) – Dead time — mobile-phase transit time (same units as x).

  • k_ext (float) – External film mass-transfer coefficient [length/time]. Must use the same length unit as R_p.

  • R_p (float) – Particle radius [length].

  • D_eff (float) –

    Effective intraparticle pore diffusivity [length²/time]. Use a large value (e.g. 1e3 cm²/min) to approximate the film-only (no pore diffusion resistance) limit where:

    k_MT_eff = 3 * F_ratio * k_ext / (R_p * R_eff)

  • a_star (float) – Effective intraparticle retention parameter. a_star = eps_p + (1 - eps_p) * a_henry where eps_p is the intraparticle porosity and a_henry is the Henry coefficient. For non-porous particles (eps_p=0): a_star = a_henry.

  • F_ratio (float) – Phase ratio F = (1 - ε) / ε (ε = interstitial/column porosity).

  • timescale (float or None, optional) – Time rescaling factor for the internal FFT grid. If None (default), chosen automatically as 80 / (t0 * max(1, R_eff)) where R_eff = 1 + F_ratio * a_star.

Returns:

Normalised PDF evaluated at each point in x (integral ≈ 1). Multiply by the peak area (c_inj × t_inj) to obtain absolute concentration units.

Return type:

ndarray

Notes

Model hierarchy (linear isotherm)

EDM (EdmLinear.edm_pdf) — narrowest, axial dispersion only GRM (this function) — intermediate width LKM (LkmLinear.lkm_pdf) — broadest, lumped mass transfer

LKM initialisation
Given a GRM result (k_ext, R_p, R), the equivalent LKM k_MT is:

k_MT_LKM = R / (R−1) × k_MT_eff

where k_MT_eff = 3 * F * k_ext / (R_p * R). Inverse: k_ext = k_MT_LKM * (R−1) * R_p / (3 * F)

Units

All length parameters (k_ext, R_p, D_eff) must use consistent units. Typical choices: [cm] and [min] (k_ext in cm/min, D_eff in cm²/min).

Examples

>>> import numpy as np
>>> from molass.SEC.Models.GrmLinear import grm_pdf
>>> t   = np.linspace(1.0, 50.0, 3000)
>>> eps = 0.4; F = (1 - eps) / eps  # = 1.5
>>> R   = 3.0; a = (R - 1) / F      # Henry coeff
>>> y = grm_pdf(t, Pe=400, t0=5.0, k_ext=0.00533, R_p=0.004,
...             D_eff=1e3, a_star=a, F_ratio=F)
>>> np.trapz(y, t)   # ≈ 1.0