Radial Variations in Axial Velocity Affect Supercritical CO2 Extraction of Lipids from Pre-pressed Oilseeds


Packed beds of spherical particles in a cylindrical vessel have a high porosity region next to the vessel wall that allows preferential fluid flow. Consequently, there are radial variations in porosity (ε) and superficial fluid velocity (U) that depend on the vessel-to-particle diameter ratio (D/dp) and the flow regime of the fluid. This work ascertained if these radial variations affected SuperCritical (SC) CO2 extraction curves of oil from pre-pressed seeds at 40 °C and 28 MPa, as compared with the commonly adopted plug flow condition. It focused specifically on comparing extraction curves as a function of the controlling mass transfer mechanism (characterized by the dimensionless Biot number, Bi) and D/dp ratio. A predictive model was adopted to describe the SC-CO2 extraction of oil from sheared seeds comparing plug flow with radial variations in superficial CO2 velocity, U(r), from literature correlations. Selected independent variables included the initial oil content of the substrate (132.7 ≤ Co ≤ 397.2 g/kg), dp (1 or 2 mm), U (1–4 mm/s), and vessel volume (0.038–495 L). Co markedly affected the effective diffusivity of the oil (0.780 ≤ De ≤ 6.24 × 10−10 m2/s), whereas dp and U moderately affected the film mass transfer coefficient (2.44 ≤ kf ≤ 7.40 × 10−5 m/s). Radial variations in superficial CO2 velocity decreased extraction rates, with differences between extraction curves when considering plug flow or adopting U(r) diminishing as Bi increased for D/dp = 20, or as D/dp increased for Bi = 18. Bi increased by increasing U and kf, or decreasing Co and De, whereas D/dp increased by increasing vessel volume. The radial variations in porosity in a packed bed and associated changes in superficial CO2 velocity may have a more pronounced negative impact in laboratory or pilot plant extraction vessels (small D) than industrial vessels (large D), mainly when extracting small particles and applying large superficial CO2 velocities. A proxy for the SC-CO2 extraction of oil from pre-pressed seeds in an industrial extraction vessel (495-L capacity, D/dp = 270) would be plug flow using the porosity, and superficial CO2 velocity predicted for the axis of the extraction vessel (εo and Uo, respectively). Literature correlations predict a value of εo slightly less than ε, and value of Uo slightly less than U. The remainder of the CO2 bypassing the vessel along a high porosity region near the vessel wall, containing a small fraction of the loaded substrate.

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The Chilean scientific agency FONDECYT (Project #1150623) funded this work.

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Average interparticle bed porosity

As noted before, parameter C in Eq. (11) is defined so that the porosity next to vessel wall, ε(R), equals unity:

$$ \varepsilon (R)={\varepsilon}_{\mathrm{o}}\left[1+C\ \exp \left(1-2\frac{R-R}{d_{\mathrm{p}}}\right)\right]=1 $$

From Eq. (A1) the value of C provided by Eq. (14b) can be derived, that results in the following expression for the radial variations in porosity:

$$ \varepsilon (r)={\varepsilon}_{\mathrm{o}}\left[1+\exp \left(-1\right)\left(\frac{1}{\varepsilon_{\mathrm{o}}}-1\right)\ \exp \left(1-2\frac{R-r}{d_{\mathrm{p}}}\right)\right], $$
$$ \varepsilon (r)={\varepsilon}_{\mathrm{o}}+\left(1-{\varepsilon}_{\mathrm{o}}\right)\ \exp \left(-\frac{D}{d_{\mathrm{p}}}\right)\ \exp \left(\frac{2r}{d_{\mathrm{p}}}\right), $$

where D = 2 R.

On the other hand, εo is defined in such a way that the average porosity, given by Eq. (A2), equals the required bed porosity (\( \overline{\varepsilon} \)):

$$ \overline{\varepsilon}=\frac{1}{A}\underset{0}{\overset{R}{\int }}\varepsilon (r) dA. $$

where A = π R2 and dA = 2 π r dr. Considering the definition of ε(r) in Eq. (11a), the expression for \( \overline{\varepsilon} \) is as follows:

$$ \overline{\varepsilon}=\frac{2}{R^2}\left[{\varepsilon}_{\mathrm{o}}\underset{0}{\overset{R}{\int }} rdr+\left(1-{\varepsilon}_{\mathrm{o}}\right)\exp \left(-\frac{2R}{d_{\mathrm{p}}}\right)\underset{0}{\overset{R}{\int }}\exp \left(\frac{2r}{d_{\mathrm{p}}}\right) rdr\right] $$

where, in turn:

$$ \underset{0}{\overset{R}{\int }} rdr={\left(\frac{r^2}{2}\right|}_0^R=\frac{R^2}{2} $$

and \( \underset{0}{\overset{R}{\int }}\exp \left(\frac{2r}{d_{\mathrm{p}}}\right) rdr={\left(\frac{d_{\mathrm{p}}}{2}\right)}^2{\left[\exp \left(\frac{2r}{d_{\mathrm{p}}}\right)\left(\frac{2r}{d_{\mathrm{p}}}-1\right)\right|}_0^R \),


$$ \underset{0}{\overset{R}{\int }}\exp \left(\frac{2r}{d_{\mathrm{p}}}\right) rdr=\frac{d_{\mathrm{p}}^2}{4}\left[\exp \left(\frac{D}{d_{\mathrm{p}}}\right)\left(\frac{D}{d_{\mathrm{p}}}-1\right)+1\right] $$

By replacing Eq. (A4) and Eq. (A5) in Eq. (A3), the following definition for \( \overline{\varepsilon} \) results:

$$ \overline{\varepsilon}={\varepsilon}_{\mathrm{o}}+\left(1-{\varepsilon}_{\mathrm{o}}\right)\frac{d_{\mathrm{p}}^2}{2{R}^2}\left[\frac{D}{d_{\mathrm{p}}}-1+\exp \left(-\frac{D}{d_{\mathrm{p}}}\right)\right] $$


$$ \overline{\varepsilon}={\varepsilon}_{\mathrm{o}}+2\left(1-{\varepsilon}_{\mathrm{o}}\right)\left[\frac{d_{\mathrm{p}}}{D}-{\left(\frac{d_{\mathrm{p}}}{D}\right)}^2+{\left(\frac{d_{\mathrm{p}}}{D}\right)}^2\exp \left(-\frac{D}{d_{\mathrm{p}}}\right)\right] $$

A value of εo can be derived as a function of \( \overline{\varepsilon} \) as already presented in Eq. (14a).

Average axial superficial CO 2 velocity

As in the case of porosity, parameter Uo in Eq. (12) can be defined from the required average CO2 velocity:

$$ \overline{U}=\frac{1}{A}\underset{0}{\overset{R}{\int }}U(r) dA, $$


$$ \overline{U}=\frac{2}{R^2}\underset{0}{\overset{R}{\int }}U(r) rdr $$

Considering the definition of U(r) in Eq. (12), the following definition is obtained:

$$ \underset{0}{\overset{R}{\int }}U(r) rdr={U}_{\mathrm{o}}\underset{0}{\overset{R}{\int }}\left[1-\left(1-m\frac{R-r}{d_{\mathrm{p}}}\right)\ \exp \left(a\frac{R-r}{d_{\mathrm{p}}}\right)\right] rdr $$

Combining Eq. (A7) and Eq. (A8), the following definition of Uo emerges:

$$ {U}_{\mathrm{o}}=\frac{R^2\overline{U}}{2\underset{0}{\overset{R}{\int }}\left[1-\left(1-m\frac{R-r}{d_{\mathrm{p}}}\right)\ \exp \left(a\frac{R-r}{d_{\mathrm{p}}}\right)\right] rdr} $$

Considering the following equality:

$$ \left[1-\left(1-m\frac{R-r}{d_{\mathrm{p}}}\right)\ \exp \left(a\frac{R-r}{d_{\mathrm{p}}}\right)\right]r=r-\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)\left[\left(1-\frac{mR}{d_{\mathrm{p}}}\right)r+\frac{m{r}^2}{d_{\mathrm{p}}}\right]\ \exp \left(-\frac{ar}{d_{\mathrm{p}}}\right), $$

the integral term in Eq. (A9) can be divided into three definite integral terms:

$$ \underset{0}{\overset{R}{\int }}\left[1-\left(1-m\frac{R-r}{d_{\mathrm{p}}}\right)\ \exp \left(a\frac{R-r}{d_{\mathrm{p}}}\right)\right] rdr=A+B+X $$


$$ A=\underset{0}{\overset{R}{\int }} rdr=\frac{R^2}{2}\ \left[\mathrm{from}\ \mathrm{Eq}.\left(\mathrm{A}4\right)\right] $$
$$ B=\left(1-\frac{mR}{d_{\mathrm{p}}}\right)\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)\underset{0}{\overset{R}{\int }}r\ \exp \left(-\frac{ar}{d_{\mathrm{p}}}\right) dr,\mathrm{and} $$
$$ X=\frac{m}{d_{\mathrm{p}}}\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)\underset{0}{\overset{R}{\int }}{r}^2\exp \left(-\frac{ar}{d_{\mathrm{p}}}\right) dr. $$

Considering that\( \int x\ \exp (cx) dx=\exp (cx)\left(\frac{x}{c}-\frac{1}{c^2}\right) \), the integral term in Eq. (A10b) is as follows:

$$ \underset{0}{\overset{R}{\int }}r\ \exp \left(-\frac{ar}{d_{\mathrm{p}}}\right) dr={\left\{\exp \left(-\frac{ar}{d_{\mathrm{p}}}\right)\left[-\frac{d_{\mathrm{p}}r}{a}-{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2\right]\right|}_0^R=\dots \dots =\exp (0){\left(\frac{d_{\mathrm{p}}}{a}\right)}^2-\exp \left(-\frac{aR}{d_{\mathrm{p}}}\right)\left[\frac{d_{\mathrm{p}}R}{a}+{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2\right] $$

from which:

$$ \underset{0}{\overset{R}{\int }}r\ \exp \left(-\frac{ar}{d_{\mathrm{p}}}\right) dr={\left(\frac{d_{\mathrm{p}}}{a}\right)}^2\left[1-\exp \left(-\frac{aR}{d_{\mathrm{p}}}\right)\right]-\frac{d_{\mathrm{p}}R}{a}\exp \left(-\frac{aR}{d_{\mathrm{p}}}\right) $$

Replacing Eq. (A10d) in Eq. (A10b) provides the final expression for the term Β:

$$ B=\left(1-\frac{mR}{d_{\mathrm{p}}}\right)\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)\left\{{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2\left[1-\exp \left(-\frac{aR}{d_{\mathrm{p}}}\right)\right]-\frac{d_{\mathrm{p}}R}{a}\exp \left(-\frac{aR}{d_{\mathrm{p}}}\right)\right\}=\dots $$
$$ \dots =\left(1-\frac{mR}{d_{\mathrm{p}}}\right){\left(\frac{d_{\mathrm{p}}}{a}\right)}^2\left[\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)-1\right]-\left(1-\frac{mR}{d_{\mathrm{p}}}\right)\frac{d_{\mathrm{p}}R}{a}=\dots $$
$$ \dots =\left(1-\frac{mR}{d_{\mathrm{p}}}\right){\left(\frac{d_{\mathrm{p}}}{a}\right)}^2\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)-\left(1-\frac{mR}{d_{\mathrm{p}}}\right)\left[{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2+\frac{d_{\mathrm{p}}R}{a}\right], $$


$$ B=\left[{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2-\frac{mR{d}_{\mathrm{p}}}{a^2}\right]\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)-{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2-\frac{d_{\mathrm{p}}R}{a}+\frac{mR{d}_{\mathrm{p}}}{a^2}+\frac{m{R}^2}{a} $$

On the other hand, considering that\( \int {x}^2\exp (cx) dx=\exp (cx)\left(\frac{x^2}{c}-\frac{2x}{c^2}+\frac{2}{c^3}\right) \), the integral term in Eq. (A10c) is as follows:

$$ \underset{0}{\overset{R}{\int }}{r}^2\exp \left(-\frac{ar}{d_{\mathrm{p}}}\right) dr={\left\{\exp \left(-\frac{ar}{d_{\mathrm{p}}}\right)\left[-\frac{d_{\mathrm{p}}{r}^2}{a}-2{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2r-2{\left(\frac{d_{\mathrm{p}}}{a}\right)}^3\right]\right|}_0^R=\dots, $$
$$ \dots =2\ \exp (0){\left(\frac{d_{\mathrm{p}}}{a}\right)}^3-\exp \left(-\frac{aR}{d_{\mathrm{p}}}\right)\left[\frac{d_{\mathrm{p}}{R}^2}{a}+2{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2R+{\left(\frac{d_{\mathrm{p}}}{a}\right)}^3\right], $$

from which: \( \underset{0}{\overset{R}{\int }}{r}^2\exp \left(-\frac{ar}{d_{\mathrm{p}}}\right) dr=2{\left(\frac{d_{\mathrm{p}}}{a}\right)}^3\left[1-\exp \left(-\frac{aR}{d_{\mathrm{p}}}\right)\right]-\left[\frac{d_{\mathrm{p}}{R}^2}{a}+2{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2R\right]\exp \left(-\frac{aR}{d_{\mathrm{p}.}}\right) \)


Replacing Eq. (A10f) in Eq. (A10c) provides the final expression for the term Χ:

$$ X=\frac{m}{d_{\mathrm{p}}}\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)\left\{2{\left(\frac{d_{\mathrm{p}}}{a}\right)}^3\left[1-\exp \left(-\frac{aR}{d_{\mathrm{p}}}\right)\right]-\left[\frac{d_{\mathrm{p}}{R}^2}{a}+2{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2R\right]\exp \left(-\frac{aR}{d_{\mathrm{p}}}\right)\right\}=\dots $$
$$ \dots =2\frac{m}{d_{\mathrm{p}}}{\left(\frac{d_{\mathrm{p}}}{a}\right)}^3\left[\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)-1\right]-\frac{m}{d_{\mathrm{p}}}\left[\frac{d_{\mathrm{p}}{R}^2}{a}+2{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2R\right]=\dots $$
$$ \dots =\frac{2m{d}_{\mathrm{p}}^2}{a^3}\left[\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)-1\right]-\frac{m{R}^2}{a}-\frac{2 mR{d}_{\mathrm{p}}}{a^2} $$


$$ X=\frac{2m{d}_{\mathrm{p}}^2}{a^3}\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)-\frac{2m{d}_{\mathrm{p}}^2}{a^3}-\frac{m{R}^2}{a}-\frac{2 mR{d}_{\mathrm{p}}}{a^2} $$

Replacing Eq. (A10a), Eq. (A10e), and Eq. (A10g) in Eq. (A10), the following definition results:

$$ \underset{0}{\overset{R}{\int }}\left[1-\left(1-m\frac{R-r}{d_{\mathrm{p}}}\right)\ \exp \left(a\frac{R-r}{d_{\mathrm{p}}}\right)\right] rdr=\frac{R^2}{2}+\dots $$
$$ \dots +\left[{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2-\frac{mR{d}_{\mathrm{p}}}{a^2}\right]\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)-{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2-\frac{R{d}_{\mathrm{p}}}{a}+\frac{mR{d}_{\mathrm{p}}}{a^2}+\frac{m{R}^2}{a}+\dots $$
$$ \dots +\frac{2m{d}_{\mathrm{p}}^2}{a^3}\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)-\frac{2m{d}_{\mathrm{p}}^2}{a^3}-\frac{m{R}^2}{a}-\frac{2 mR{d}_{\mathrm{p}}}{a^2} $$

from which: \( \underset{0}{\overset{R}{\int }}\left[1-\left(1-m\frac{R-r}{d_{\mathrm{p}}}\right)\ \exp \left(a\frac{R-r}{d_{\mathrm{p}}}\right)\right] rdr=\frac{R^2}{2}-{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2-\frac{R{d}_{\mathrm{p}}}{a}+\dots \)

$$ \dots -\frac{mR{d}_{\mathrm{p}}}{a^2}-\frac{2m{d}_{\mathrm{p}}^2}{a^3}+\left[{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2-\frac{mR{d}_{\mathrm{p}}}{a^2}+\frac{2m{d}_{\mathrm{p}}^2}{a^3}\right]\exp \left(\frac{aR}{d_{\mathrm{p}}}\right) $$

Replacing Eq. (10 h) in Eq. (A9), Eq. (A10i) results:

$$ {U}_{\mathrm{o}}=\frac{\frac{R^2}{2}\overline{U}}{\frac{R^2}{2}-{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2-\frac{R{d}_{\mathrm{p}}}{a}-\frac{mR{d}_{\mathrm{p}}}{a^2}-\frac{2m{d}_{\mathrm{p}}^2}{a^3}+\left[{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2-\frac{mR{d}_{\mathrm{p}}}{a^2}+\frac{2m{d}_{\mathrm{p}}^2}{a^3}\right]\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)}. $$

Eq. (A10i) coincides Eq. (16), with factoring for 2/dp2 in both the numerator and denominator being the first step to achieve a closer look. It is relevant to point out that Eq. (16) for Uo/U does not coincide with Eq. (23) of Vortmeyer and Schuster [23] for their so-called β parameter, but did represent (unlike Eq. (23) in [23]) the radial variations in superficial velocity presented in the original reference. Thus, the authors assumed that there was a typo in Eq. (23) in the manuscript of Vortmeyer and Schuster [23], as also noted before, with regard to Eq. (13b) (Eq. (18c) in [23]).

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del Valle, J.M., Núñez, G.A., Díaz, J.F. et al. Radial Variations in Axial Velocity Affect Supercritical CO2 Extraction of Lipids from Pre-pressed Oilseeds. Food Eng Rev (2020). https://doi.org/10.1007/s12393-020-09232-1

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  • Linear driving force model
  • Mathematical simulation
  • Nonlinear solute partition
  • Packed bed extraction vessel
  • Pre-pressed oilseed
  • Radial variations in bed porosity
  • Radial changes in CO2 velocity
  • Supercritical CO2 extraction