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Journal of Membrane Science 237 (2004) 8795

Mathematical modeling of the membrane separationof nutmeg essential oil and dense CO2

Cinthia Bittencourt Spricigo a,, Ariovaldo Bolzan b,Ricardo Antonio Francisco Machado b, Jos Carlos Cunha Petrus b

a Centro de Ciencias Exatas e de Tecnologia, Pontifcia Universidade C atlica do Paran, Rua Imaculada Con ceio,

1155 Prado Velho, Curitiba, Paran CEP 80215-901, Brazilb Departamento de Engenharia Qumica e Engenharia de Alimentos, Centro Tecnolgico,

Universidade Federal de Santa Catarina CP 476, Florianpolis, Santa Catarina CEP 88010-970, Brazil

Received 1 July 2003; received in revised form 30 January 2004; accepted 9 February 2004

Abstract

This work presents the application of a mathematical model to describe the membrane separation of nutmeg essential oil from supercriticalCO2 mixtures. The phenomenological analysis led to a mathematical description of the process based on the irreversible thermodynamicsapproach. The carbon dioxide permeate flux was modeled by associating the dependence of flux on the transmembrane pressure gradient to amathematicalequation whichrepresents the concentration polarizationphenomenon at the membrane surface. Thethickness of the polarizationlayer was employed as fitting parameter, and the value which best-fitted the experimental data was 400 m. The experimental observationof convective solvent transport mechanisms in the case of a such a dense membrane was related to the plasticization effect associated toan increase in the polymer chain mobility of the membrane due to the presence of dense CO 2. The essential oil permeation was modeledby relating the proportionality between the essential oil permeate flux and the solvent permeate flux to the logarithmic mean concentrationdifference between the essential oil concentration in the feed and permeate sides.

2004 Elsevier B.V. All rights reserved.

Keywords: Concentration polarization; Dense carbon dioxide; Liquid permeability and separations; Membrane transport

1. Introduction

Different approaches are employed to the mathematicalmodeling of membrane separation processes. The choicedepends on the characteristics of the process, of the mem-brane and of the substances which are involved. Mathemat-ical models based on irreversible thermodynamics require

little information on membrane structure and on transportmechanisms. The membrane is seen as a black box separat-ing two phases far away from equilibrium. The parameterswhich are employed are coefficients of global transport suchas the membrane hydraulic permeability [1,2].

The irreversible thermodynamics approach assumes thatslow permeation processes may be treated as processes closeto the equilibrium. It also accepts that different simultaneous

Corresponding author. Tel.: +55-41-271-1567;fax: +55-41-271-1567.

E-mail address: [email protected] (C.B. Spricigo).

processes can be separated in non-interacting single mech-anisms. Eq. (1) is a linear law which assumes that any flux

Ji is proportional to its conjugated driving force Fi in a sys-tem with simultaneous fluxes. Lii is the proportionality co-efficient and Lik are the cross coefficients that relate eachflux to its non-conjugated forces. Onsager has established arelation for quasi-equilibrium processes known as Onsager

reciprocity relation (Eq. (2)). This relation can reduce signif-icantly the number of parameters of the mathematical pro-cess representation:

Ji =

nk=1

LikFk (i = 1, 2, 3, . . . , n) (1)

Lik = Lki (2)

The first model proposed for a membrane permeation sys-tem based on irreversible thermodynamics is the Kedem andKatchalsky model, elaborated for aqueous electrolytic solu-

0376-7388/$ see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.memsci.2004.02.024


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90 C.B. Spricigo et al. / Journal of Membrane Science 237 (2004) 8795

are presented by Eqs. (8) and (9):

y = 0, DABdw

dy

y=0

= pvpwf (8)

y = , w = w0 (9)

f

=

wm wp

wm (10)

where f (Eq. (10)) is the intrinsic retention index of themembrane, vp the permeation velocity (m s1), w the so-lute concentration (wt.%), w0 the feed solute concentration(wt.%), wm the solute concentration at the membrane sur-face (wt.%), wp the permeate solute concentration (wt.%),

y the spatial coordinate, the thickness of the polarizationlayer (m), the density of the feed solution (kg m3) andp the density of the permeate solution (kg m3). It was as-sumed that the density of the solutions were equal to thedensity of the pure CO2.

The boundary condition presented by Eq. (8) establishesthe flux continuity of solvent and solute at the interfacemembrane/feed solution through a mass balance that equalsthe total feed flux to the permeate flux. The analytical solu-tion ofEq. (7) is presented in Eq. (11). A constant boundarylayer thickness is assumed in this work. The CO2 flux inthe test cell was opposed to gravity so that natural convec-tion phenomena could compensate for solute accumulationat the membrane surface:

J=DAB

ln

wm wp

w0 wp

(11)

Eq. (7), that represents the film theory, explains the influence

of concentration polarization on the flux, but it does notrelate that to the transmembrane pressure gradient.

The relation with P is given by Eq. (6), which mustbe somehow connected to Eq. (11). In their work, Geraldeset al. [5] observed that the permeate fluxes calculated byEq. (6) were systematically larger than the correspondingexperimental values. They also have observed that the dif-ferences between the values increased with the increase infeed solute concentration. The differences were explained asa deviation of the phenomenological equation (6), that wasthen corrected with a factor dependent on solute concentra-tion in the solution adjacent to the membrane, as described

by the following equation:J= (wm)Lp(P ) (12)

where (wm) is the correction factor of Eq. (6), which isdependent on solute concentration at the membrane surface.

DAB was estimated through the WilkeChangs equationfor supercritical fluids (Eq. (13)) [8]:

DAB =7.4 108(MB)1/2T

BV0.6A

(13)

where MB is the CO2 molecular mass (g moll), T the tem-perature (K), VA the molar volume of the solute (cm3 moll),

Table 2Calculated DAB and B values (working pressure: 12 MPa)

Temperature (C) DAB (m2 s1) B (cP)

23 7.66 109 0.082240 11.24 109 0.059250 15.70 109 0.0436

the association factor of the solvent (1.0 to CO2) and Bthe solvent viscosity at the system temperature and pressureconditions (cP). Table 2 presents the values of DAB calcu-lated by Eq. (13) and the values of B calculated by theAltunin and Sakhabetdinovs correlation [9].

Calculations were performed following a simple compu-tational sequence. Based on the experimental values of Jobtained for each set of experimental conditions (tempera-ture, pressure gradient and feed oil concentration), the val-ues ofwm were calculated for each one of those conditionsby assuming different values of (Eq. (11)). Afterwards, by

Eq. (12), the correction factors for each wm were calcu-lated to each . The solution ofEq. (12) was also based onexperimental values of J and on the mean permeability ofthe membrane to CO2 (Lp = 31.1kgh1 m2 MPa1). Themean values of(wm) were plotted against the mean valuesofwm calculated to each feed oil concentration. A potentialtype relation was found between these values, and the valueof was adjusted until the best relation between (wm) andwm was found. The mean retention index (92.5 wt.%) wasapplied for all the experimental conditions. The differencein osmotic pressure was considered negligible, as explainedearlier. The experimental flux data that were obtained to the

same conditions of P and w0, but at different tempera-tures, were used together to represent the variability of theresults, as the temperature did not exert significant effectson the permeation process [6].

2.2. Essential oil permeate flux

Regarding the mathematical modeling of the essential oilpermeation through the membrane, Figs. 3 and 4 present, re-spectively, the dependence of the average experimental sta-tionary essential oil fluxes across the membrane CF withthe pressure gradient and with the essential oil concentra-tion gradient. The flux values presented refer to the meanvalues obtained to the same P and to the same essentialoil feed concentration, but at different temperatures, as sta-tistical analyses of the experimental data demonstrated thatthe temperature had not a significant influence on essentialoil permeate flux [6]. The results indicate that besides thesolvent flux contribution, which is directly proportional tothe pressure gradient, there is a diffusive contribution to theoil transport across the membrane.

Based on experimental information and on the theory ofirreversible thermodynamics, the average essential oil fluxcan be expressed by Eq. (4), where is assumed to benegligible and JV is considered equal to the CO2 flux (J)


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C.B. Spricigo et al. / Journal of Membrane Science 237 (2004) 8795 91

Fig. 3. Dependence between the mean stationary oil flux and the trans-membrane pressure to different feed oil concentrations. Feed constantpressure: 12 MPa; mean flux values at 23, 40 and 50 C.

due to the high retention indexes observed experimentally. Inthis way, the permeate flux of essential oil (Js) is describedby the following equation:

Js = Cs,av(1 )J (14)

In the above equation, the values of Cs,av were calculatedbased on essential concentration at the membrane surface(wm) and not on the feed oil concentration. Each feed oilconcentration had a corresponding value ofwm presented inFig. 5. The values ofJwhich were employed were calculatedby the mathematical model described to the pure CO2 flux.The calculations of the mean logarithmic concentration usedthe average retention index of 92.5% to all the experimentalconditions of pressure gradient, temperature and feed oilconcentration. The value was fitted to the experimental

Fig. 4. Dependence between the nutmeg essential oil permeate flux and the oil concentration gradient to different transmembrane pressure gradients.Constant feed pressure: 12 MPa; mean values of flux at 23, 40 and 50 C.

data for minimizing the square error between the calculatedand the experimental values of permeate oil flux.

3. Results and discussion

3.1. CO2 permeate flux

By minimization of the square error, the best relation be-tween (wm) and wm was found for = 400m, and it isrepresented in the following equation and in Fig. 5:

(wm) = 0.1164 w0.6686m (15)

It was verified that the correction factor becomes smalleras the feed oil concentration increases at the membrane sur-face. A similar result was found by Geraldes et al. [5] inthe permeation modeling of PEG1000 through a nanofiltra-tion membrane. It can be observed at Fig. 5 that, due tothe concentration polarization, the oil concentration at the

membrane surface reaches an estimated value around 4 wt.%when the feed oil concentration is 1 wt.%, and approximately17 wt.% when 10 wt.% of oil are introduced in the feed.

The relation represented in Eq. (15) was used in Eq. (12)for the calculation of the values ofJ. Following the method-ology employed in the elaboration of the graphics ofFig. 5,the average correction factor (wm), obtained to a samefeed oil concentration and different temperatures and trans-membrane pressure gradients, was employed in the calcula-tions. Table 3 and Fig. 6 present the calculated values andthe comparison with the experimental ones.

Considering the experimental variability observed amongthe solvent mass fluxes obtained with different samples ofthe membrane, the relation found among the calculated andthe experimental values ofJwas very good. The good resultsobtained with the application of the mathematical model in-


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Fig. 5. Relation between mean values of (wm) and wm (averages taken among the values at the feed oil concentrations and different temperatures and

transmembrane pressures).Table 3Comparison between the calculated and experimental values of J

Feed oil concentration P (MPa) Calculated J (kgs1 m2) Experimental J (kgs1 m2) Mean ratio between calculatedand experimental J

1 2 0.0173 0.0174 0.993 0.0260 0.0258 1.014 0.0346 0.0345 1.00

5 2 0.0094 0.0076 1.243 0.0141 0.0148 0.954 0.0187 0.0215 0.87

10 2 0.0068 0.0059 1.153 0.0102 0.0110 0.93

4 0.0136 0.0142 0.96

dicate that the hypothesis formulated regarding the natureof the mass transfer phenomena presented in this work areconsistent. Consequently, the process that has been studiedis predominantly convective, and the occurrence of concen-tration polarization can explain the reduction of the CO2

Fig. 6. Comparison between experimental and calculated J values.

flux observed experimentally as the feed oil concentrationincreases. Eq. (6) can be corrected by a dependent factoron oil concentration on the membrane surface so that thepermeate flux caused by the pressure gradient is associatedto the flux reduction caused by concentration polarization(Eq. (11)). The occurrence of the plasticization phenomenain the cellulose acetate membrane by the presence of denseCO2 allows for the swelling of the polymeric chains. Thisfacilitates the transport of substances across the membrane[10]. This phenomenon can be responsible for the possibil-

ity of mathematical representation of the CO2 permeate fluxby purely convective phenomena.

3.2. Essential oil permeate flux

Fig. 7 presents the dependence between the square errorand the value of . The minimization of the square erroroccurs in the region of equal to 0.89, that is, a value closeto the experimental apparent rejection coefficient (0.925).Table 4 and Fig. 8 present the experimental and calculated(Eq. (14)) values of Js with = 0.89. The model did notfit well the experimental data, overestimating Js for the feed


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Fig. 7. Relation between the square error and the value of .

Table 4Comparison between the experimental Js and calculated Js solute fluxes (= 0.89)

Feed oil concentration P (MPa) Calculated Js(105 kg s1 m2)

Experimental Js(105 kg s1 m2)

Mean ratio between calculatedand experimental Js

1 2 3.12 1.04 3.003 4.69 1.73 2.714 6.25 1.38 4.53

5 2 3.86 3.02 1.273 5.79 5.99 0.974 7.68 9.47 0.81

10 2 4.54 5.28 0.86

3 6.81 9.70 0.704 9.08 11.24 0.81

Fig. 8. Relation between experimental and calculated solute fluxes to equal to 0.89.


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oil concentration of 1 wt.%. These results indicate that theestimate of the essential oil concentration at the membranesurface performed by the mathematical modeling of CO2permeate flux presents deviations from the real values, whichcould not be measured.

4. Conclusions

By applying a mathematical equation relating the linearproportionality of the CO2 mass flux to the transmembranepressure associated to the equation that represents the con-centration polarization phenomenon, it was possible to de-scribe the permeation of the solvent through the membrane,including the effects of permeate flux reduction due to theincrease in feed oil concentration. Both equations were con-nected by means of a correction factor that was dependenton the oil concentration at the membrane surface, and ap-plied as fitting parameter the polarization layer thickness. Itwas established that an exponential type mathematical rela-tion between the correction factor and the essential oil con-centration at the membrane surface. The polarization layerthickness that best-fitted the experimental data was 400 m.

The mathematical modeling of the essential oil perme-ation through the membrane was done based on irreversiblethermodynamics, by direct proportionally relating the oilpermeate flux to the total permeate flux and to the loga-rithmic mean concentration difference of the solute at themembrane surface and in the permeate sides. The parame-ter was employed as fitting parameter, for minimizing thesquare error between the experimental and calculated values

of the permeate flux. The minimum error was obtained fora value equal to 0.89, similar to the apparent rejection co-efficient (0.925). However, the model did not fit so well theexperimental data, exposing a limitation of the mathemati-cal approach regarding the estimate of the oil concentrationat the membrane surface, specially to the feed oil concen-tration of 1 wt.%.

Acknowledgements

This work was financially supported by CAPES (Brasilia,

Brazil).

Nomenclature

Cs solute concentration (kg m3)Cs,av solute logarithmic mean concentration across

the membrane (kgm3)DAB binary diffusion coefficient of nutmeg

essential oil and CO2 in the feed side(m2 sl)

f intrinsic retention index of the membrane

Fi generalized driving forceJ CO2 permeate flux (kgs1 m2)Ji generalized fluxJs solute permeate flux

(kgm2 s1, mol m2 s1)

JV total volumetric flux (m

3

m

2

s

1

)Lii proportionality coefficientLik cross coefficientLp membrane permeability to the solvent

(kgs1 m2 MPa1)LV permeability coefficient (m3 m2 s1 Pa1)MB CO2 molar mass (g mol1)P transmembrane pressure gradient (MPa)T temperature (K)v CO2 feed velocity (m s1)vp CO2 permeation velocity (m s1)VA molar volume of the solute (cm3 mol1)w solute concentration (wt.%)wm solute concentration at the membrane

surface (wt.%)wp permeate solute concentration (wt.%)w0 feed solute concentration (wt.%)

y spatial coordinate (m)

Greek symbols

polarization layer thickness (m)(wm) correction factorB CO2 viscosity (cP) transmembrane osmotic pressure gradient

(MPa)

density of the feed solution (kgm

3)p density of the permeate solution (kg m3) Staverman reflection coefficient association factor of the solvent membrane permeability to the solute at

zero total volumetric flux (kg m2 s1 Pa1)

References

[1] P.M. Bungay, Transport principlesporous membranes, in: P.M.Bungay, H.K. Londsale, M.N. de Pinho (Eds.), Synthetic Mem-branes: Science, Engineering and Applications, Reidel, Dordrecht,1986, pp. 57107.

[2] J.G.A. Bitter, Transport Mechanisms in Membrane Separation Pro-cesses, Plenum Press, New York, 1991.

[3] S. Sarrade, tude du couplage de lextraction par CO2 supercritiqueavec la separation par membrane de nanofiltration, Ph.D. Thesis,Universit de Montpellier, Montpellier, 1994.

[4] A. Kargol, A mechanistic model of transport processes in porousmembranes generated by osmotic and hydrostatic pressure, J. Membr.Sci. 191 (2001) 61.

[5] V. Geraldes, V. Semio, M.N. Pinho, Flow and mass transfer mod-eling of nanofiltration, J. Membr. Sci. 191 (2001) 109.


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[6] C.B. Spricigo, A. Bolzan, R.A.F. Machado, L.H.C. Carlson, J.C.C.Petrus, Separation of nutmeg essential oil and dense CO2 with acellulose acetate reverse osmosis membrane, J. Membr. Sci. 188(2001) 173.

[7] S. Angus, B. Armstrong, KM. de Reuck, International Thermody-namic Tables of the Fluid State Carbon Dioxide, vol. 3, PergamonPress, New York, 1973.

[8] M.R. Riazi, C.H. Whitson, Estimating diffusion coefficients of densefluids, Ind. Eng. Chem. Res. 32 (1993) 3081.

[9] H. Sovov, J. Prochzka, Calculation of compressed carbon dioxideviscosities, Ind. Eng. Chem. Res. 32 (1993) 3162.

[10] J.S. Chiou, J.W. Barlow, D.R. Paul, Plasticization of glassy polymersby CO2, J. Appl. Polym. Sci. 30 (1985) 2633.

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