Topic B — Supercritical CO₂ Extraction in a Packed Bed

packed-bed flow · axial and radial dispersion · linear-driving-force desorption · tier: standard

Motivation

Supercritical CO₂ is a green solvent for extracting oils, flavours and drugs from packed plant material, and for drying aerogels without capillary collapse. The extraction curve (extracted mass against time or solvent used) decides the vessel size, the flow rate and the cycle time. It is controlled by three time scales: convection through the bed, dispersion, and mass transfer out of the particles.

Related fields: supercritical fluid technology, phase equilibria, supercritical extraction, aerogels. Course weeks: 12 (advection–diffusion–reaction, SUPG), 13 (porous media: Darcy/Brinkman, wall channelling).

Model

Cylindrical extractor, axisymmetric (r,z), length L, radius R, bed porosity \varepsilon, particles of radius a.

Flow: Darcy (or Brinkman with a wall region of higher porosity, Week 13), \mathbf u=-\frac{\kappa}{\mu}\nabla p, \nabla\cdot\mathbf u=0, given flow rate at the inlet.

Fluid phase (solute concentration c) and solid phase (solute loading q, per particle volume): \varepsilon\,\partial_tc+\mathbf u\cdot\nabla c-\nabla\cdot(\varepsilon\mathbf D\nabla c)=-(1-\varepsilon)\,\partial_tq,\qquad \partial_tq=-k_f\,a_p\,\big(c^*(q)-c\big)\ \text{(linear driving force)}, with an equilibrium relation c^*(q) (linear c^*=q/K to start, then a solubility cap c^*\le c_{\rm sat} for the “easily accessible solute” stage), c=0 at the inlet, natural outflow, q(\cdot,0)=q_0. Dispersion \mathbf D with axial and radial coefficients.

Design outputs: the extraction curve m(t)=\int_0^t\int_{\rm outlet}u\,c\,ds\,dt, the time to 90 % recovery, and its dependence on the flow rate.

Suggested parameters

L=0.2 m, R=0.025 m, \varepsilon=0.4, interstitial velocity 1–5 mm/s, axial Péclet number uL/D_{ax} = 100–1000, K=1–10, k_fa_p such that the mass-transfer Damköhler number k_fa_pL/u ranges over 0.1–10. Report all results against the dimensionless groups.

Required tasks

  1. Flow field (uniform bed, then with a wall layer of higher porosity); check the flow rate.
  2. Tracer test without desorption (q\equiv0): breakthrough curve; Péclet number and SUPG.
  3. Coupled (c,q): θ-method or BDF2 in time, extraction curves for three flow rates.
  4. Time-step and mesh independence of the 90 %-recovery time.

Verification specific to this topic

  • Ogata–Banks: in 1D (or a 2D bed with uniform flow), a step inlet with no desorption has \frac{c}{c_0}=\frac12\Big[\operatorname{erfc}\frac{x-vt}{2\sqrt{Dt}}+e^{vx/D}\operatorname{erfc}\frac{x+vt}{2\sqrt{Dt}}\Big] with the interstitial velocity v=u/\varepsilon and the axial dispersion D of the model above (semi-infinite domain — use a long domain).
  • Mass balance: extracted + remaining in the bed = initial, at every time step, to the time-integration error.
  • Local equilibrium limit: k_fa_p\to\infty with linear c^*=q/K gives a retarded tracer with retardation factor 1+\frac{(1-\varepsilon)K}{\varepsilon} — your breakthrough must shift accordingly.
  • MMS for the coupled (c,q) system in time (a smooth space–time solution), with predicted rates in space and time.

Stretch goals

  1. Two-stage extraction curve (solubility-controlled, then diffusion-controlled: Sovová’s broken-and-intact-cells model).
  2. Intraparticle diffusion instead of the linear driving force (a 1D radial particle model at every bed point, or a second coordinate).
  3. Pressure- and temperature-dependent CO₂ density and viscosity (Span–Wagner tables) in the flow.

Milestones

M1 flow · M2 tracer with Ogata–Banks · M3 coupled linear model with mass balance · M4 equilibrium limit and the solubility cap · M5 flow-rate study.

Pitfalls

  • Oscillations at the front for \mathrm{Pe}_h>1: SUPG (Week 12), and check that the front speed is unchanged.
  • Interstitial vs superficial velocity: \mathbf u in Darcy is superficial; the advection term in \varepsilon\partial_tc+\mathbf u\cdot\nabla c is then correct as written — mixing the two conventions shifts the breakthrough by the factor \varepsilon.
  • The solubility cap makes the problem nonlinear (non-smooth): use a smooth approximation or a semismooth Newton and report the iterations.

References

  • A. Ogata, R. B. Banks, A solution of the differential equation of longitudinal dispersion in porous media, USGS Prof. Paper 411-A (1961).
  • H. Sovová, “Rate of the vegetable oil extraction with supercritical CO₂ — I. Modelling of extraction curves”, Chem. Eng. Sci. 49 (1994) 409–414.
  • D. M. Ruthven, Principles of Adsorption and Adsorption Processes, Wiley 1984 — linear driving force, retardation.