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

Abstract

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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References

  1. 1.

    del Valle JM (2015) Extraction of natural compounds using supercritical CO2: going from the laboratory to the industrial application. J Supercrit Fluids 96:180–199

    Google Scholar 

  2. 2.

    del Valle JM, de la Fuente JC (2006) Supercritical CO2 extraction of oilseeds: Review of kinetic and equilibrium models. Crit Rev Food Sci Nutr 46:131–160

    PubMed  Google Scholar 

  3. 3.

    del Valle JM, Germain JC, Uquiche E, Zetzl C, Brunner G (2006) Microstructural effects on internal mass transfer of lipids in prepressed and flaked vegetable substrates. J Supercrit Fluids 37:178–190

    Google Scholar 

  4. 4.

    Toledo FR, del Valle JM, Opazo ÁP, Núñez GA (2020) Supercritical CO2 extraction of pelletized oilseeds: representation using linear driving force model with nonlinear sorption isotherm. J Food Eng

  5. 5.

    Núñez GA, Gelmi CA, del Valle JM (2011) Simulation of a supercritical carbon dioxide extraction plant with three extraction vessels. Comput Chem Eng 35:2687–2695

    Google Scholar 

  6. 6.

    del Valle JM, Núñez GA, Aravena RI (2014) Supercritical CO2 oilseed extraction in multi-vessel plants. 1. Minimization of operational cost. J Supercrit Fluids 92:197–207

    Google Scholar 

  7. 7.

    Núñez GA, del Valle JM (2014) Supercritical CO2 oilseed extraction in multivessel plants. 2. Effect of number and geometry of extractors on production cost. J Supercrit Fluids 92:324–334

    Google Scholar 

  8. 8.

    Núñez GA, del Valle JM, Navia D (2017) Supercritical CO2 oilseed extraction in multivessel plants. 3. Effect of extraction pressure and plant size on production cost. J Supercrit Fluids 122:109–118

    Google Scholar 

  9. 9.

    Fiori L, Basso D, Costa P (2008) Seed oil supercritical extraction: particle size distribution of the milled seeds and modeling. J Supercrit Fluids 47:174–181

    CAS  Google Scholar 

  10. 10.

    del Valle JM, Carrasco CV, Toledo FR, Núñez GA (2019) Particle size distribution and stratification of pelletized oilseeds affects cumulative supercritical CO2 extraction plots. J Supercrit Fluids 146:189–198

    Google Scholar 

  11. 11.

    del Valle JM, Calderón D, Núñez GA (2019) Pressure drop may negatively impact supercritical CO2 extraction of citrus peel essential oils in an industrial-size extraction vessel. J Supercrit Fluids 144:108–121

    Google Scholar 

  12. 12.

    Zabot GL, Moraes MN, Petenate AJ, Meireles MAA (2014) Influence of the bed geometry on the kinetics of the extraction of clove bud oil with supercritical CO2. J Supercrit Fluids 93:56–66

    CAS  Google Scholar 

  13. 13.

    J.M. del Valle, C. Lorca, L. Fiori, G.A. Núñez, 2018 Temperature gradients within the packed bed affect cumulative supercritical CO2 extraction plots for oilseeds, in: XII Int. Symp. Supercrit. Fluids, Jean Les Pins, France,

  14. 14.

    Brunner G (1994) Gas extraction: an introduction to fundamentals of supercritical fluids and the application to separation processes. Springer, New York, NY

    Google Scholar 

  15. 15.

    Sovová H, Kučera J, Jež J (1994) Rate of the vegetable oil extraction with supercritical CO2—II. Extraction of grape oil. Chem Eng Sci 49:415–420

    Google Scholar 

  16. 16.

    del Valle JM, Rivera O, Mattea M, Ruetsch L, Daghero J, Flores A (2004) Supercritical CO2 processing of pretreated rosehip seeds: effect of process scale on oil extraction kinetics. J Supercrit Fluids 31:159–174

    Google Scholar 

  17. 17.

    Fiori L, Costa P (2010) Effects of differential permeability on packed bed supercritical extractors: a theoretical insight. J Supercrit Fluids 55:176–183

    CAS  Google Scholar 

  18. 18.

    Schwartz CE, Smith JM (1953) Flow distribution in packed beds. Ind Eng Chem 45:1209–1218

    CAS  Google Scholar 

  19. 19.

    Yuan QS, Rosenfeld A, Root TW, Klingenberg DJ, Lightfoot EN (1999) Flow distribution in chromatographic columns. J Chromatogr A 831:149–165

    CAS  Google Scholar 

  20. 20.

    Shalliker RA, Broyles BS, Guiochon G (2000) Physical evidence of two wall effects in liquid chromatography. J Chromatogr A 888:1–12

    CAS  PubMed  Google Scholar 

  21. 21.

    Benenati RF, Brosilow CB (1962) Void fraction distribution in beds of spheres. AICHE J 8:359–361

    CAS  Google Scholar 

  22. 22.

    Thadani MC, Peebles FN (1966) Variation of local void fraction in randomly packed beds of equal spheres. Ind Eng Chem Process Des Dev 5:265–268

    CAS  Google Scholar 

  23. 23.

    Vortmeyer D, Schuster J (1983) Evaluation of steady flow profiles in rectangular and circular packed beds by a variational method. Chem Eng Sci 38:1691–1699

    CAS  Google Scholar 

  24. 24.

    Negrini AL, Fuelber A, Freire JT, Thoméo JC (1999) Fluid dynamics of air in a packed bed: velocity profiles and the continuum model assumption. Braz J Chem Eng 16:421–432

    CAS  Google Scholar 

  25. 25.

    Alazmi B, Vafai K (2004) Analysis of variable porosity, thermal dispersion, and local thermal nonequilibrium on free surface flows through porous media. J Heat Transf 126:389–399

    Google Scholar 

  26. 26.

    Khan LM, Hanna MA (1983) Expression of oil from oilseeds—a review. J Agric Eng Res 28:495–503

    Google Scholar 

  27. 27.

    Savoire R, Lanoisellé J-L, Vorobiev E (2013) Mechanical continuous oil expression from oilseeds: a review. Food Bioprocess Technol 6:1–16

    CAS  Google Scholar 

  28. 28.

    Urrego FA, Núñez GA, Donaire YD, del Valle JM (2015) Equilibrium partition of rapeseed oil between supercritical CO2 and prepressed rapeseed. J Supercrit Fluids 102:80–91

    CAS  Google Scholar 

  29. 29.

    E.W. Lemmon, M.L. Huber, M.O. McLinden, (2013) N.I.S.T. Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties (R.E.F.P.R.O.P.), version 9.1

  30. 30.

    T. Funazukuri, M. Toriumi, K. Yui, C.Y. Kong, S. Kagei, 2009 Correlation for binary diffusion coefficients of lipids in supercritical carbon dioxide, in: 9th Int. Symp. Supercrit. Fluids, Arcachon, France

  31. 31.

    del Valle JM, de la Fuente JC, Uquiche E (2012) A refined equation for predicting the solubility of vegetable oils in high-pressure CO2. J Supercrit Fluids 67:60–70

    Google Scholar 

  32. 32.

    Benyahia F, O'Neill KE (2005) Enhanced voidage correlations for packed beds of various particle shapes and sizes. Part Sci Technol 23:169–177

    CAS  Google Scholar 

  33. 33.

    M.B. King, O. Catchpole, (1993)Physico-chemical data required for the near-critical fluid extraction process, in: M.J. King, T.R. Bott (Eds.), Extraction of Natural Products Using Near-Critical Solvents, Blackie Academic & Professional: pp. 184–231

  34. 34.

    Eggers R (1996) Supercritical fluid extraction (SFE) of oilseeds/lipids in natural products. In: King JW, List GR (eds) Supercritical fluid Technology in oil and Lipid Chemistry. AOCS Press, Champaign, IL, pp 35–64

    Google Scholar 

  35. 35.

    Aguilera JM, Stanley DW (1999) Microstructural principles of food processing and engineering, 2nd. edn. Aspen Publishers, Gaithersburg, MD

  36. 36.

    Sovová H (1994) Rate of the vegetable oil extraction with supercritical CO2—I. Modelling of extraction curves. Chem Eng Sci 49:409–414

    Google Scholar 

  37. 37.

    Puiggené J, Larrayoz MA, Recasens F (1997) Free liquid-to-supercritical fluid mass transfer in packed beds. Chem Eng Sci 52:195–212

    Google Scholar 

  38. 38.

    Germain JC, del Valle JM, de la Fuente JC (2005) Natural convection retards supercritical CO2 extraction of essential oils and lipids from vegetable substrates. Ind Eng Chem Res 44:2879–2886

    CAS  Google Scholar 

  39. 39.

    Goto M, Roy BC, Kodama A, Hirose T (1998) Modeling supercritical fluid extraction process involving solute-solid interaction. J Chem Eng Japan 31:171–177

    CAS  Google Scholar 

  40. 40.

    Fiori L, Calcagno D, Costa P (2007) Sensitivity analysis and operative conditions of a supercritical fluid extractor. J Supercrit Fluids 41:31–42

    CAS  Google Scholar 

  41. 41.

    Fiori L (2007) Grape seed oil supercritical extraction kinetic and solubility data: critical approach and modeling. J Supercrit Fluids 43:43–54

    CAS  Google Scholar 

  42. 42.

    Gunn DJ (1987) Axial and radial dispersion in fixed beds. Chem Eng Sci 42:363–373

    CAS  Google Scholar 

  43. 43.

    Delgado JMPQ (2007) Longitudinal and transverse dispersion in porous media. Chem Eng Res Des 85:1245–1252

    CAS  Google Scholar 

  44. 44.

    del Valle JM, de la Fuente JC, Uquiche E, Zetzl C, Brunner G (2011) Mass transfer and equilibrium parameters on high-pressure CO2 extraction of plant essential oils. In: Aguilera JM, Barbosa-Cánovas GV, Simpson R, Welti-Chanes J, Bermúdez-Aguirre D (eds) Food Eng. Interfaces. Springer, New York, NY, pp 393–470

    Google Scholar 

  45. 45.

    Roblee LHS, Baird RM, Tierney JW (1958) Radial porosity variations in packed beds. AICHE J 4:460–464

    CAS  Google Scholar 

  46. 46.

    Mueller GE (1992) Radial void fraction distributions in randomly packed fixed beds of uniformly sized spheres in cylindrical containers. Powder Technol 72:269–275

    CAS  Google Scholar 

  47. 47.

    Sharma S, Mantle M, Gladden L, Winterbottom J (2001) Determination of bed voidage using water substitution and 3D magnetic resonance imaging, bed density and pressure drop in packed-bed reactors. Chem Eng Sci 56:587–595

    CAS  Google Scholar 

  48. 48.

    de Klerk A (2003) Voidage variation in packed beds at small column to particle diameter ratio. AICHE J 49:2022–2029

    Google Scholar 

  49. 49.

    Giese M, Rottschäfer K, Vortmeyer D (1998) Measured and modeled superficial flow profiles in packed beds with liquid flow. AICHE J 44:484–490

    CAS  Google Scholar 

  50. 50.

    Morales M, Spinn CW, Smith JM (1951) Velocities and effective thermal conductivities in packed beds. Ind Eng Chem 43:225–232

    CAS  Google Scholar 

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Funding

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

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Correspondence to José M del Valle.

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Appendix

Appendix

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:

$$ \varepsilon (R)={\varepsilon}_{\mathrm{o}}\left[1+C\ \exp \left(1-2\frac{R-R}{d_{\mathrm{p}}}\right)\right]=1 $$
(A1)

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:

$$ \varepsilon (r)={\varepsilon}_{\mathrm{o}}\left[1+\exp \left(-1\right)\left(\frac{1}{\varepsilon_{\mathrm{o}}}-1\right)\ \exp \left(1-2\frac{R-r}{d_{\mathrm{p}}}\right)\right], $$
(or)
$$ \varepsilon (r)={\varepsilon}_{\mathrm{o}}+\left(1-{\varepsilon}_{\mathrm{o}}\right)\ \exp \left(-\frac{D}{d_{\mathrm{p}}}\right)\ \exp \left(\frac{2r}{d_{\mathrm{p}}}\right), $$
(11a)

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 (\( \overline{\varepsilon} \)):

$$ \overline{\varepsilon}=\frac{1}{A}\underset{0}{\overset{R}{\int }}\varepsilon (r) dA. $$
(A2)

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:

$$ \overline{\varepsilon}=\frac{2}{R^2}\left[{\varepsilon}_{\mathrm{o}}\underset{0}{\overset{R}{\int }} rdr+\left(1-{\varepsilon}_{\mathrm{o}}\right)\exp \left(-\frac{2R}{d_{\mathrm{p}}}\right)\underset{0}{\overset{R}{\int }}\exp \left(\frac{2r}{d_{\mathrm{p}}}\right) rdr\right] $$
(A3)

where, in turn:

$$ \underset{0}{\overset{R}{\int }} rdr={\left(\frac{r^2}{2}\right|}_0^R=\frac{R^2}{2} $$
(A4)

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

or

$$ \underset{0}{\overset{R}{\int }}\exp \left(\frac{2r}{d_{\mathrm{p}}}\right) rdr=\frac{d_{\mathrm{p}}^2}{4}\left[\exp \left(\frac{D}{d_{\mathrm{p}}}\right)\left(\frac{D}{d_{\mathrm{p}}}-1\right)+1\right] $$
(A5)

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

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

or

$$ \overline{\varepsilon}={\varepsilon}_{\mathrm{o}}+2\left(1-{\varepsilon}_{\mathrm{o}}\right)\left[\frac{d_{\mathrm{p}}}{D}-{\left(\frac{d_{\mathrm{p}}}{D}\right)}^2+{\left(\frac{d_{\mathrm{p}}}{D}\right)}^2\exp \left(-\frac{D}{d_{\mathrm{p}}}\right)\right] $$
(A6)

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:

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

or

$$ \overline{U}=\frac{2}{R^2}\underset{0}{\overset{R}{\int }}U(r) rdr $$
(A7)

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

$$ \underset{0}{\overset{R}{\int }}U(r) rdr={U}_{\mathrm{o}}\underset{0}{\overset{R}{\int }}\left[1-\left(1-m\frac{R-r}{d_{\mathrm{p}}}\right)\ \exp \left(a\frac{R-r}{d_{\mathrm{p}}}\right)\right] rdr $$
(A8)

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

$$ {U}_{\mathrm{o}}=\frac{R^2\overline{U}}{2\underset{0}{\overset{R}{\int }}\left[1-\left(1-m\frac{R-r}{d_{\mathrm{p}}}\right)\ \exp \left(a\frac{R-r}{d_{\mathrm{p}}}\right)\right] rdr} $$
(A9)

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:

$$ \underset{0}{\overset{R}{\int }}\left[1-\left(1-m\frac{R-r}{d_{\mathrm{p}}}\right)\ \exp \left(a\frac{R-r}{d_{\mathrm{p}}}\right)\right] rdr=A+B+X $$
(A10)

where:

$$ A=\underset{0}{\overset{R}{\int }} rdr=\frac{R^2}{2}\ \left[\mathrm{from}\ \mathrm{Eq}.\left(\mathrm{A}4\right)\right] $$
(A10a)
$$ B=\left(1-\frac{mR}{d_{\mathrm{p}}}\right)\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)\underset{0}{\overset{R}{\int }}r\ \exp \left(-\frac{ar}{d_{\mathrm{p}}}\right) dr,\mathrm{and} $$
(A10b)
$$ X=\frac{m}{d_{\mathrm{p}}}\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)\underset{0}{\overset{R}{\int }}{r}^2\exp \left(-\frac{ar}{d_{\mathrm{p}}}\right) dr. $$
(A10c)

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:

$$ \underset{0}{\overset{R}{\int }}r\ \exp \left(-\frac{ar}{d_{\mathrm{p}}}\right) dr={\left(\frac{d_{\mathrm{p}}}{a}\right)}^2\left[1-\exp \left(-\frac{aR}{d_{\mathrm{p}}}\right)\right]-\frac{d_{\mathrm{p}}R}{a}\exp \left(-\frac{aR}{d_{\mathrm{p}}}\right) $$
(A10d)

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

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

or

$$ B=\left[{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2-\frac{mR{d}_{\mathrm{p}}}{a^2}\right]\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)-{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2-\frac{d_{\mathrm{p}}R}{a}+\frac{mR{d}_{\mathrm{p}}}{a^2}+\frac{m{R}^2}{a} $$
(A10e)

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

(A10f)

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

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

or

$$ X=\frac{2m{d}_{\mathrm{p}}^2}{a^3}\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)-\frac{2m{d}_{\mathrm{p}}^2}{a^3}-\frac{m{R}^2}{a}-\frac{2 mR{d}_{\mathrm{p}}}{a^2} $$
(A10g)

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

$$ \dots -\frac{mR{d}_{\mathrm{p}}}{a^2}-\frac{2m{d}_{\mathrm{p}}^2}{a^3}+\left[{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2-\frac{mR{d}_{\mathrm{p}}}{a^2}+\frac{2m{d}_{\mathrm{p}}^2}{a^3}\right]\exp \left(\frac{aR}{d_{\mathrm{p}}}\right) $$
(A10h)

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

$$ {U}_{\mathrm{o}}=\frac{\frac{R^2}{2}\overline{U}}{\frac{R^2}{2}-{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2-\frac{R{d}_{\mathrm{p}}}{a}-\frac{mR{d}_{\mathrm{p}}}{a^2}-\frac{2m{d}_{\mathrm{p}}^2}{a^3}+\left[{\left(\frac{d_{\mathrm{p}}}{a}\right)}^2-\frac{mR{d}_{\mathrm{p}}}{a^2}+\frac{2m{d}_{\mathrm{p}}^2}{a^3}\right]\exp \left(\frac{aR}{d_{\mathrm{p}}}\right)}. $$
(A10i)

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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Keywords

  • 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