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Journal of Membrane Science 208 (2002) 171–179

Mass transfer in osmotic evaporation: effect of process parameters V.D. Alves, I.M. Coelhoso∗ Dep. de Qu´ımica-CQFB, Faculdade de Ciˆencias e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal Received 21 March 2002; received in revised form 6 May 2002; accepted 7 May 2002

Abstract The influence of relevant parameters on the osmotic evaporation process, namely temperature, stirring rate and osmotic agent’s nature and concentration, is evaluated. The water flux is expressed as a function of the Reynolds number (Re), characterising the hydrodynamic conditions, and as a function of the water activity difference, which is the driving force of the process. It is our intention to emphasise the need of a systematic presentation of the results, in order to compare different conditions. Mass transfer is described by a series resistance model: the membrane resistance and the boundary layer resistance in both sides of the membrane. The contribution of the boundary layer resistance is determined for two osmotic agents (calcium chloride and glycerol), and a correlation for the mass transfer coefficient, Sh = 1.63Re0.56 Sc0.33 , was obtained. Using as osmotic agents calcium chloride, sodium chloride and glycerol, it was noticed that, the flux increases with the osmotic solution concentration, and is not affected by the osmotic agent’s nature. The water flux increases with temperature according to an Arrhenius type equation and this increase is mainly due to an increase of the water vapour pressure. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Osmotic evaporation; Mass transfer coefficients; Boundary layer; Hydrophobic membrane; Osmotic agent

1. Introduction Osmotic evaporation is an attractive alternative process for the concentration of solutions containing thermally sensitive components, like fruit juices and pharmaceuticals, because it can be operated at low temperature and pressure, with minimal thermal and mechanical damage. In this process, a porous hydrophobic membrane separates a diluted aqueous solution from a concentrated osmotic solution. As long as the penetration pressure is not reached, the membrane is not wetted by the aqueous solutions and the pores are initially ∗ Corresponding author. Tel.: +351-212-948-303; fax: +351-212-948-385. E-mail address: [email protected] (I.M. Coelhoso).

filled with air. Due to the difference of the water activity between the two solutions, water is transported through the membrane involving its evaporation in one side and further condensation in the other side. As the mass transfer proceeds, the air in the pores is replaced by a mixture of air and vapour in equilibrium with the liquid phases. Since it is not a pressure driven process, it is not limited by the osmotic pressure of the solutions, as does reverse osmosis, and higher concentrations (60–70◦ Brix) can be achieved [1,2]. In order to optimise the process, detailed studies about the role of the relevant parameters are necessary. Temperature, stirring rate and osmotic agent’s nature and concentration are important parameters for the osmotic evaporation process in a stirred cell. In this work, the influence of these parameters on the water flux is evaluated, and it is our intention to emphasise

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Nomenclature a A Am b Dw 0 DAB Hv Jw Jw0 k kmp K N P P∗ R Re Sc Sh t T V x

water activity constant of integration (Pa) membrane area (m2 ) stirred cell diameter (m) water diffusion coefficient (m2 s−1 ) diffusion coefficient of solute A in the solvent B at infinite dilution (m2 s−1 ) water latent heat of vaporisation (J mol−1 ) water flux (m3 m−2 s−1 ) parameter of Eq. (12) (m s−1 ) individual mass transfer coefficient (m s−1 ) membrane mass transfer coefficient (m s−1 Pa−1 ) overall mass transfer coefficient (m s−1 ) stirring rate (s−1 ) water vapour pressure (Pa) pure water vapour pressure (Pa) gas constant (J K−1 mol−1 ) Reynolds number Schmidt number Sherwood number time (s) temperature (K) water volume (m3 ) mole fraction

the need of a systematic presentation of the results, in order that, different conditions can be compared. Since water is the compound that is transported, the water flux results are expressed as a function of the driving force of the process, the bulk water activity difference of the two solutions, and the water vapour pressure in the pores is related to the water activity at the membrane interface. In some works presented in the literature [3,4], the authors related the water vapour pressure to the salt concentration in the osmotic solution, however, they only have used one osmotic agent. The approach of using water activities, as done in this work, is preferable because it is possible to compare the effect of different osmotic agents. A series resistance model is developed considering the boundary layer resistance of the solutions and the membrane resistance to water transport. The boundary layer mass transfer coefficient is expressed as a function of the Reynolds number (Re), instead of stirring rate, because in this way, it is possible to normalise the flux obtained for the different osmotic agents used (calcium chloride and glycerol). A correlation for the boundary layer mass transfer coefficient expressed as Sh = αReβ Scγ is obtained, and the boundary layer contribution to the overall resistance is determined. The use of these dimensionless numbers permits the characterisation of the hydrodynamics of the osmotic solution. It is then possible to preview the water flux values for different osmotic agents or different concentrations of the same osmotic solution.

2. Theory Greek α β γ µ ρ ω

letters constant of the Sherwood correlation constant of the Sherwood correlation constant of the Sherwood correlation absolute viscosity (N s m−2 ) density (kg m−3 ) stirring rate (rpm)

Subscripts 0 initial conditions 1 diluted solution 2 osmotic solution m membrane

The driving force of the osmotic evaporation process is the difference in the water activity between two aqueous solutions, due to the different nature and concentration of the solutes. The aqueous solutions, separated by a porous hydrophobic membrane; typically made of polypropylene, polytetrafluoroethylene or polyvinylidenedifluoride; cannot enter the pores and a liquid–vapor interface is formed in each pore end. The water activity difference induces a water vapour pressure difference and, consequently, a water flux (Jw ) is expected to occur across the stagnant film of air within the pores. Water evaporates in the solution of higher water activity (diluted solution) and the vapour is transported across the membrane pores before being

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reduction of the driving force to the water flux. There are two diffusional boundary layers adjoining the membrane and the flux can be related to the respective individual mass transfer coefficients by Jw = k1 (a1 − am1 ) = k2 (am2 − a2 )

(6)

where k1 and k2 are the individual mass transfer coefficients in the boundary layers. Combining Eqs. (4)–(6), we obtain: 1 1 1 1 = + + K k1 km k2 Fig. 1. Water activity profile in osmotic evaporation.

condensed in the solution of lower water activity (osmotic solution) (Fig. 1). The water flux is proportional to the vapour pressure difference across the membrane: Jw = kmp (P1 − P2 )

(1)

where kmp is the membrane mass transfer coefficient. Although the partial pressure of air inside the membrane pores varies with the composition of the aqueous solutions, this variation is not significant and kmp can be assumed as constant. The water vapour pressures P1 and P2 can be related to the water activity at the membrane interface: Pi =

ami Pi∗

(2)

where ami represents the water activity at the membrane interface and Pi∗ is the pure water vapour pressure. In this way, Eq. (1) can be written as Jw = kmp (P1∗ am1 − P2∗ am2 )

(3)

If the temperature is exactly the same at both membrane interfaces, P1∗ = P2∗ = P ∗ , and Eq. (3) can be simplified: Jw = kmp P ∗ (am1 − am2 )

(4)

Since the water activities at the membrane interface are not known, the flux may also be expressed using the bulk water activity difference between the two phases (a1 − a2 ), and the overall mass transfer coefficient K: Jw = K(a1 − a2 )

(5)

Due to the concentration polarisation phenomenon, the water activity at the membrane interface is different from the bulk water activity resulting in the

(7)

where km = P ∗ kmp , is the membrane mass transfer coefficient expressed as a function of a water activity difference. According to Eq. (7), the overall resistance to mass transfer (1/K) is the sum of the three resistances to the water flux: the resistance of the boundary layer in the diluted solution side, the membrane resistance and the boundary layer resistance in the osmotic solution side. If the boundary layers cannot be neglected, the dependence of the individual mass transfer coefficients can be correlated according to Eq. (8), has proposed by several authors for a stirred cell [5–7]. The Sherwood number (Sh) depends on the Re and Schmidt number (Sc): Sh = αReβ Scγ

(8)

with Sh =

kb , Dw

Re =

ρNd2 , µ

Sc =

µ ρDw

(9)

where b is the stirred cell diameter, N the stirring rate, d the stirrer diameter, Dw the water diffusion coefficient, ρ the solution density, µ the solution dynamic viscosity, and α, β and γ are constants. In this work, pure water instead of a diluted solution was used, thus, a1 = am1 . Working in conditions where the resistance to mass transfer in the osmotic solution side is negligible (am2 ≈ a2 ), there is only the contribution of the membrane resistance, and the membrane mass transfer coefficient can be evaluated by Jw = kmp P ∗ (a1 − a2 )

(10)

Temperature is another parameter that affects the water flux. According to Eq. (10), if there are no boundary layer effects, the flux is a function of the membrane mass transfer coefficient, the water activity

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and the pure water vapour pressure. All these parameters are temperature-dependent, however, for the osmotic agents used, the water activity is practically constant over the temperature range of the experiments [8,9]. In addition, temperature can be assumed to have little effect on the mass transfer coefficient itself. On the other hand, pure water vapour pressure increases with temperature, and this dependence can be described using the Clausius–Clapeyron equation [10]: dP ∗ P ∗ Hv = dT RT2

(11)

where T is the temperature and Hv is the water latent heat of vaporisation. Integrating Eq. (11), considering that Hv and R are constant and independent of temperature, and substituting P∗ in Eq. (10), an Arrhenius-type dependence with temperature for the water flux is obtained, as proposed by Mengual et al. [3]: Hv 0 Jw = Jw exp − (12) RT where Jw0 = Akmp (a1 − a2 ), and A is a constant of integration.

3. Experimental 3.1. Materials The compounds employed to prepare the osmotic solutions were glycerol (86–88 wt.%), sodium chloride and calcium chloride dihydrate pro-analysis grade (Riedel-de-Häen, Germany). The osmotic solutions were prepared with de-ionised water. A flat polypropylene membrane (Metricel® , Pall Gelman Laboratory, USA) was used, with nominal pore diameter of 0.1 mm, 90 mm thickness and 55% porosity, as specified by the manufacturer. 3.2. Experimental set-up A stirred cell consisting of two identical cylindrical chambers was used (Fig. 2). The polypropylene membrane with an effective area of 11.3 cm2 was placed between the chambers, one with de-ionised water (chamber 1) and the other with an osmotic solution (chamber 2). Chamber 1 was connected to a water

Fig. 2. Stirred cell: (1) magnetic stirrer, (2) water outlet from thermostat, (3) inlet, (4) thermometer, (5) outlet, (6) water inlet from thermostat, (7) membrane, (8) magnet, (9) stirrer, (10) thermal jacket.

reservoir to avoid hydrostatic pressure differences between both sides of the membrane. The water flux was obtained measuring the variation of the water volume with time in the pipette connected to chamber 2. In order to check reproducibility, three measurements were performed for each case, and the average values are presented. The mean value of the deviations obtained was 2%. The temperature was maintained constant by re-circulating water through the thermal jackets from a water bath and was measured with type J thermopars. Magnetic plates coupled with magnetic bars stirred both chambers. The stirring rate was measured using a digital photo tachometer. 3.3. Calculation methods The fitting calculations were carried out using the software package ScientistTM , from MicroMath® . 4. Results and discussion Three sets of experiments were performed in order to determine the effect of process parameters on mass transfer. In the first set, the contribution of the boundary layer resistance was evaluated. The water flux was measured varying the stirring rate, under constant temperature and solute concentration, for two different osmotic agents. In the second set of experiments the effect of osmotic agent’s nature and concentration was studied. In this case, three osmotic agents were

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used, and the flux was measured for different solute concentrations at the same temperature and stirring rate. Finally, in the last set, both solute concentration and stirring rate were kept constant, and the temperature was varied in order to determine its effect on the water flux. 4.1. Evaluation of the boundary layer resistance The water flux was measured using two osmotic solutions, calcium chloride 3.2 M and glycerol 8.3 M, with the same bulk water activity (a2 = 0.688). The temperature was maintained constant (T = 25 ◦ C) and the stirring rate was varied between 100 and 600 rpm. Fig. 3 shows that the flux increases with the stirring rate for both osmotic agents, until a plateau is reached. Since all variables are maintained constant except the stirring rate, we can assume that this increase is due to a decrease of the boundary layer resistance, which is negligible at high stirring rates leading to a constant water flux. According to Eq. (10), when the boundary layer resistance can be neglected in both sides of the membrane, there is only the membrane resistance, which is not dependent on the osmotic agent used. Since the driving force is the same, it was expected to have a similar water flux value in the plateau for both osmotic agents, which is not the case. As described by Eq. (8), the individual mass transfer coefficient in the boundary layer is dependent, not only on the stirring rate, but also on the properties of the osmotic solution. For the solutions used, the ratio m/r is five times

Fig. 4. Water flux as a function of Reynolds number.

higher for the glycerol solution, so, it is necessary to express the results in terms of Re, which takes into account the difference of the solution properties. As can be seen in Fig. 4, the water flux is similar for both solutions in the range of Re studied. It is clear that, for the glycerol solution, the plateau has not been reached yet, as it seemed when representing the water flux as a function of the stirring rate. It would have been necessary to work at higher stirring rates to achieve the same Re obtained with the calcium chloride solution. In order to obtain a correlation for the boundary layer mass transfer coefficient, Eq. (8) can be rewritten as αDw β γ k2 = Re Sc (13) b Rearranging Eqs. (5), (7) and (13), the following relation for the water flux as a function of Re, is obtained: Jw =

Fig. 3. Water flux as a function of the stirring rate.

kmp Dw αReβ Scγ P ∗ (a1 − a2 ) Dw αReβ Scγ + bP∗ kmp

(14)

By fitting the experimental data (J versus Re obtained for both osmotic agents) using Eq. (14), the values of the constants α and β, and the value of the membrane mass transfer coefficient kmp , were determined. The exponent of the Schmidt number was assumed to be 1/3, as commonly accepted in the literature [5,7]. The water diffusion coefficient in glycerol was estimated using a modification of the Vignes equation [10]: d ln aw 0 xG 0 xw Dw µ = [(DwG µG ) (DGw µw ) ] (15) d ln xw

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Table 1 Parameter values and errors of Eqs. (14) and (17) Parameter

Value and error

Error (%)

r2

Two-step method

α β kmp (×10−10 m s−1 Pa−1 )

1.74 ± 2.83 0.55 ± 0.34 3.33 ± 0.12

163 61.8 3.60

0.999 0.999 0.999

One-step method

α β kmp (×10−10 m s−1 Pa−1 )

1.63 ± 0.99 0.56 ± 0.13 3.32 ± 0.04

60.7 23.2 1.20

0.999 0.999 0.999

where subscripts w refers to water and G to glycerol; D0 is the diffusion coefficient at infinite dilution, x the molar fraction and a is the activity. The value obtained, Dw = 1.90 × 10−10 m2 s−1 , is similar to that measured by Nishijima and Oster [11]. The water diffusion coefficient for the calcium chloride solution, Dw = 1.09 × 10−9 m2 s−1 , was obtained from the literature [12], as well as the values of the water activity, density and viscosity of the solutions [13–15]. The parameter values obtained are presented in Table 1. It can be seen that, even with a good correlation coefficient, the errors associated to the parameters α and β are quite high: 163 and 61.8%, respectively. The method used to determine these constants consists in two-steps: in the first step, the water flux was evaluated by plotting the water volume in the pipette connected to chamber 2 with time, for each experiment. In the second step, the flux values obtained were then used to fit Eq. (14). The disadvantage of using this methodology lies in the fact that there is a loss of information and possible accumulation of errors during the two-step calculation. Viegas et al. [16] analysed the Wilson-plot methodology, which also involves a two-step calculation, to determine mass transfer correlations in membrane extraction. The errors associated with the estimation of their parameter values were also very high. To overcome this problem, they have used a single-step methodology, which involves the fit of the experimental results just to one model equation. A similar approach was adopted in this work. The water flux may be expressed as follows: Jw =

1 dV Am dt

(16)

where Am is the membrane area and V is the water volume transported through the membrane. Using Eqs. (14) and (16), the evolution of water volume with

time can be obtained: kmp Dw αReβ Scγ Dw αReβ Scγ + bP∗ kmp × Am P ∗ (a1 − a2 )(t − t0 )

V = V0 +

(17)

where V0 and t0 are the initial conditions. The values and errors of the constants α and β, and of the membrane mass transfer coefficient kmp , determined fitting the experimental data (V versus t) using Eq. (17), are shown in Table 1. The parameter values are similar to that obtained with the two-step methodology, but the errors associated to α and β are reduced by three times. There was a great improvement just by using this methodology. The values obtained by Sudoh et al. [5] for a stirred cell with a similar geometry (α = 2.00, β = 0.48), are comparable to that determined in this work (α = 1.63 ± 0.99, β = 0.56 ± 0.13), although they have not presented the errors of the parameters. The values of α, β and kmp , obtained with the single-step methodology, are substituted in Eq. (14), which can then be used to preview the evolution of the water flux with the Re for different conditions. The results obtained for other driving forces using glycerol solutions are shown in Fig. 5. It can be seen that, as the driving force increases, more difficult is to cancel the resistance of the boundary layer adjoining the membrane, due to a higher dynamic viscosity and density of the solutions. Using the data presented in Fig. 4, the overall mass transfer coefficient is evaluated by Eq. (5), for each Re. As the value of the membrane mass transfer coefficient is known, the relative contribution of the membrane resistance, (1/km )/(1/K), can be determined. The major resistance to the water flux is due to the membrane itself, even at low Re (Fig. 6).

V.D. Alves, I.M. Coelhoso / Journal of Membrane Science 208 (2002) 171–179

Fig. 5. Water flux prediction for different glycerol solutions: (䉱) experimental values, (—) prediction with Eq. (14).

4.2. Effect of the osmotic agent’s nature and concentration

177

Fig. 7. Flux dependence with osmotic solution concentration.

These experiments were carried out at constant stirring rate (ω = 400 rpm) and temperature (T = 25 ◦ C). Three different osmotic agents were used: calcium chloride, sodium chloride and glycerol. The water flux was measured for various solute concentrations, which correspond to different water activities. Fig. 7 shows that, for all solutions, the water flux increases with the osmotic agent concentration. In these experiments, there is a variation of the Re due to the increase of viscosity and density with

the solution concentration. The Re achieved varied between 1890 and 3950, so we can assume that, for these solutions, the boundary layer effects can be neglected, thus, am2 ≈ a2 . The water activity in the osmotic solution is then related to water vapour pressure in the pores. Representing the flux as a function of the water vapour pressure difference the flux values fall into a straight line (Fig. 8). All osmotic agents cause the same effect being the water flux only dependent on the vapour pressure difference. It is possible to evaluate the membrane mass transfer coefficient value, according to Eq. (1), and the value obtained, i.e. (3.24±0.08)×10−10 m s−1 Pa−1 , is similar to that

Fig. 6. Relative contribution of the membrane and boundary layer resistance.

Fig. 8. Flux variation with the water vapour pressure difference.

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water flux increase with temperature is mainly due to an increase of the driving force (P1 − P2 ). 5. Conclusions

Fig. 9. Flux dependence with temperature: (䊐) experimental values, (—) fit curve to Eq. (12).

determined in Section 4.1 and presented in Table 1, i.e. (3.32 ± 0.04) × 10−10 m s−1 Pa−1 . 4.3. Effect of the temperature The water flux was measured using a 2 M calcium chloride solution (a2 = 0.849). The stirring rate was kept constant at 400 rpm and the temperature was varied between 20 and 45 ◦ C. Fig. 9 shows that, as expected, the flux increases with the temperature according to Eq. (12). From the fit of the experimental data to this equation, the heat of vaporisation of water was obtained: Hv = (3.96 ± 0.38) × 104 J mol−1 . The value is similar to that referred in the literature, Hv = 4.47 × 104 J mol−1 at 20 ◦ C and Hv = 4.35 × 104 J mol−1 at 45 ◦ C [10]. Mengual et al. [3] evaluated the effect of temperature using five different membranes and sodium chloride as osmotic agent. The values of Hv obtained in his study ranged from (2.91 ± 0.17) × 104 to (3.48 ± 0.10) × 104 J mol−1 , which are a little lower than that determined in this work. The individual mass transfer coefficient in the membrane kmp was determined for each temperature using Eq. (1). The value of kmp obtained is nearly constant in the temperature range of the experiments; it varies from (3.15 ± 0.29) × 10−10 m s−1 Pa−1 at 20 ◦ C to (3.71 ± 0.69) × 10−10 m s−1 Pa−1 at 45 ◦ C. Since the temperature effect on the mass transfer coefficient is not significant, it can be concluded that the

The influence of the relevant parameters on the osmotic evaporation process, namely stirring rate, temperature and osmotic agent’s nature and concentration, were studied. The water fluxes were expressed as a function of the Re, characterising hydrodynamic conditions, and as a function of the water activity difference between the solutions, which is the driving force of the process. In this way, it was possible to compare the effect of the different osmotic agents. The contribution of the boundary layer has been evaluated for two osmotic agents (glycerol and calcium chloride). It has been observed that, even for low Re, the membrane is responsible for the major contribution to the overall resistance. From the representation of the flux as a function of Re, it can be concluded that, as long as the osmotic solutions have the same bulk water activity, and the process is carried out under the same hydrodynamic conditions, similar fluxes are obtained regardless of the osmotic agent used. In addition to the water activity, other factors, namely the solution viscosity and density, must be taken into account in the selection of the osmotic agent. A correlation for the individual mass transfer coefficient in the boundary layer has been obtained, and the water flux evolution with the Re could be estimated for different driving forces using that correlation. The flux increases with the osmotic agent concentration and it is not affected by the nature of the osmotic agent for the same water vapour pressure difference, which is the driving force of the process. Temperature has little effect on the membrane mass transfer coefficient. The water flux increases with temperature according to an Arrhenius-type equation, and this increase is mainly due to an increase of the water vapour pressure.

Acknowledgements V. D. Alves acknowledges the research grant PRAXIS XXI/BD/18300/98.

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References [1] A.F.G. Bailey, A.M. Barbe, P.A. Hogan, R.A. Johnson, J. Sheng, The effect of ultrafiltration on the subsequent concentration of grape juice by osmotic distillation, J. Membr. Sci. 164 (2000) 195. [2] F. Vaillant, E. Jeanton, M. Dornier, G.M. O’Brien, M. Reynes, M. Decloux, Concentration of passion fruit juice on an industrial pilot scale using osmotic evaporation, J. Food Eng. 47 (2001) 195. [3] J.I. Mengual, J.M. Ortiz de Zárate, L. Pena, A. Velázquez, Osmotic distillation through porous hydrophobic membranes, J. Membr. Sci. 82 (1993) 129. [4] M. Courel, M. Dornier, G.M. Rios, M. Reynes, Modelling of water transport in osmotic distillation using asymmetric membrane, J. Membr. Sci. 173 (2000) 107. [5] M. Sudoh, K. Takuwa, H. Iizuka, K. Nagamatsuya, Effects of thermal and concentration boundary layers on vapour permeation in membrane distillation of aqueous lithium bromide solution, J. Membr. Sci. 131 (1997)1. [6] D.M. Malone, J.L. Anderson, Diffusional boundary-layer resistance for membranes with low porosity, AIChE J. 23 (2) (1977) 177. [7] L.J. Zeman, A.L. Zidney, Microfiltration and Ultrafiltration: Principles and Applications, Marcel Dekker, New York, 1996.

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[8] J. Ananthaswamy, G. Atkinson, Thermodynamics of concentrated electrolyte mixtures. Part 5. A review of the thermodynamic properties of aqueous calcium chloride in the temperature range 273.15–373.15 K, J. Chem. Eng. Data. 30 (1985) 120. [9] D. Kitic, D.C.P. Jardim, G.J. Favetto, S.L. Resnik, J. Chirife, Theoretical prediction of the water activity of standard saturated salt solutions at various temperatures, J. Food Sci. 51 (4) (1986) 1037. [10] R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, McGraw-Hill, Singapore, 1988. [11] Y. Nishijima, G. Oster, Diffusion in glycerol–water mixture, Bull. Chem. Soc. Jpn. 33 (1960) 1649. [12] J. Tamas, K. Ujszaszy, Diffusion of H2 18 O molecules in certain aqueous salt solutions, Magy. Kem. Folyoirat. 72 (10) (1966) 447. [13] R.A. Robinson, R.H. Stokes, Electrolyte Solutions, Butherworths, London, 1965. [14] E.W. Washburn, International Critical Tables of Numerical Data, Physics, Chemistry and Technology, McGraw-Hill, New York. [15] J.A. Dean, Lange’s Handbook of Chemistry, McGraw-Hill, New York, 1978. [16] R.M.C. Viegas, M. Rodr´ıguez, S. Luque, J.R. Alvarez, I.M. Coelhoso, J.P.S.G. Crespo, Mass transfer correlations in membrane extraction: analysis of Wilson-plot methodology, J. Membr. Sci. 145 (1998) 129.

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Mass transfer in osmotic evaporation: effect of process parameters V.D. Alves, I.M. Coelhoso∗ Dep. de Qu´ımica-CQFB, Faculdade de Ciˆencias e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal Received 21 March 2002; received in revised form 6 May 2002; accepted 7 May 2002

Abstract The influence of relevant parameters on the osmotic evaporation process, namely temperature, stirring rate and osmotic agent’s nature and concentration, is evaluated. The water flux is expressed as a function of the Reynolds number (Re), characterising the hydrodynamic conditions, and as a function of the water activity difference, which is the driving force of the process. It is our intention to emphasise the need of a systematic presentation of the results, in order to compare different conditions. Mass transfer is described by a series resistance model: the membrane resistance and the boundary layer resistance in both sides of the membrane. The contribution of the boundary layer resistance is determined for two osmotic agents (calcium chloride and glycerol), and a correlation for the mass transfer coefficient, Sh = 1.63Re0.56 Sc0.33 , was obtained. Using as osmotic agents calcium chloride, sodium chloride and glycerol, it was noticed that, the flux increases with the osmotic solution concentration, and is not affected by the osmotic agent’s nature. The water flux increases with temperature according to an Arrhenius type equation and this increase is mainly due to an increase of the water vapour pressure. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Osmotic evaporation; Mass transfer coefficients; Boundary layer; Hydrophobic membrane; Osmotic agent

1. Introduction Osmotic evaporation is an attractive alternative process for the concentration of solutions containing thermally sensitive components, like fruit juices and pharmaceuticals, because it can be operated at low temperature and pressure, with minimal thermal and mechanical damage. In this process, a porous hydrophobic membrane separates a diluted aqueous solution from a concentrated osmotic solution. As long as the penetration pressure is not reached, the membrane is not wetted by the aqueous solutions and the pores are initially ∗ Corresponding author. Tel.: +351-212-948-303; fax: +351-212-948-385. E-mail address: [email protected] (I.M. Coelhoso).

filled with air. Due to the difference of the water activity between the two solutions, water is transported through the membrane involving its evaporation in one side and further condensation in the other side. As the mass transfer proceeds, the air in the pores is replaced by a mixture of air and vapour in equilibrium with the liquid phases. Since it is not a pressure driven process, it is not limited by the osmotic pressure of the solutions, as does reverse osmosis, and higher concentrations (60–70◦ Brix) can be achieved [1,2]. In order to optimise the process, detailed studies about the role of the relevant parameters are necessary. Temperature, stirring rate and osmotic agent’s nature and concentration are important parameters for the osmotic evaporation process in a stirred cell. In this work, the influence of these parameters on the water flux is evaluated, and it is our intention to emphasise

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Nomenclature a A Am b Dw 0 DAB Hv Jw Jw0 k kmp K N P P∗ R Re Sc Sh t T V x

water activity constant of integration (Pa) membrane area (m2 ) stirred cell diameter (m) water diffusion coefficient (m2 s−1 ) diffusion coefficient of solute A in the solvent B at infinite dilution (m2 s−1 ) water latent heat of vaporisation (J mol−1 ) water flux (m3 m−2 s−1 ) parameter of Eq. (12) (m s−1 ) individual mass transfer coefficient (m s−1 ) membrane mass transfer coefficient (m s−1 Pa−1 ) overall mass transfer coefficient (m s−1 ) stirring rate (s−1 ) water vapour pressure (Pa) pure water vapour pressure (Pa) gas constant (J K−1 mol−1 ) Reynolds number Schmidt number Sherwood number time (s) temperature (K) water volume (m3 ) mole fraction

the need of a systematic presentation of the results, in order that, different conditions can be compared. Since water is the compound that is transported, the water flux results are expressed as a function of the driving force of the process, the bulk water activity difference of the two solutions, and the water vapour pressure in the pores is related to the water activity at the membrane interface. In some works presented in the literature [3,4], the authors related the water vapour pressure to the salt concentration in the osmotic solution, however, they only have used one osmotic agent. The approach of using water activities, as done in this work, is preferable because it is possible to compare the effect of different osmotic agents. A series resistance model is developed considering the boundary layer resistance of the solutions and the membrane resistance to water transport. The boundary layer mass transfer coefficient is expressed as a function of the Reynolds number (Re), instead of stirring rate, because in this way, it is possible to normalise the flux obtained for the different osmotic agents used (calcium chloride and glycerol). A correlation for the boundary layer mass transfer coefficient expressed as Sh = αReβ Scγ is obtained, and the boundary layer contribution to the overall resistance is determined. The use of these dimensionless numbers permits the characterisation of the hydrodynamics of the osmotic solution. It is then possible to preview the water flux values for different osmotic agents or different concentrations of the same osmotic solution.

2. Theory Greek α β γ µ ρ ω

letters constant of the Sherwood correlation constant of the Sherwood correlation constant of the Sherwood correlation absolute viscosity (N s m−2 ) density (kg m−3 ) stirring rate (rpm)

Subscripts 0 initial conditions 1 diluted solution 2 osmotic solution m membrane

The driving force of the osmotic evaporation process is the difference in the water activity between two aqueous solutions, due to the different nature and concentration of the solutes. The aqueous solutions, separated by a porous hydrophobic membrane; typically made of polypropylene, polytetrafluoroethylene or polyvinylidenedifluoride; cannot enter the pores and a liquid–vapor interface is formed in each pore end. The water activity difference induces a water vapour pressure difference and, consequently, a water flux (Jw ) is expected to occur across the stagnant film of air within the pores. Water evaporates in the solution of higher water activity (diluted solution) and the vapour is transported across the membrane pores before being

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reduction of the driving force to the water flux. There are two diffusional boundary layers adjoining the membrane and the flux can be related to the respective individual mass transfer coefficients by Jw = k1 (a1 − am1 ) = k2 (am2 − a2 )

(6)

where k1 and k2 are the individual mass transfer coefficients in the boundary layers. Combining Eqs. (4)–(6), we obtain: 1 1 1 1 = + + K k1 km k2 Fig. 1. Water activity profile in osmotic evaporation.

condensed in the solution of lower water activity (osmotic solution) (Fig. 1). The water flux is proportional to the vapour pressure difference across the membrane: Jw = kmp (P1 − P2 )

(1)

where kmp is the membrane mass transfer coefficient. Although the partial pressure of air inside the membrane pores varies with the composition of the aqueous solutions, this variation is not significant and kmp can be assumed as constant. The water vapour pressures P1 and P2 can be related to the water activity at the membrane interface: Pi =

ami Pi∗

(2)

where ami represents the water activity at the membrane interface and Pi∗ is the pure water vapour pressure. In this way, Eq. (1) can be written as Jw = kmp (P1∗ am1 − P2∗ am2 )

(3)

If the temperature is exactly the same at both membrane interfaces, P1∗ = P2∗ = P ∗ , and Eq. (3) can be simplified: Jw = kmp P ∗ (am1 − am2 )

(4)

Since the water activities at the membrane interface are not known, the flux may also be expressed using the bulk water activity difference between the two phases (a1 − a2 ), and the overall mass transfer coefficient K: Jw = K(a1 − a2 )

(5)

Due to the concentration polarisation phenomenon, the water activity at the membrane interface is different from the bulk water activity resulting in the

(7)

where km = P ∗ kmp , is the membrane mass transfer coefficient expressed as a function of a water activity difference. According to Eq. (7), the overall resistance to mass transfer (1/K) is the sum of the three resistances to the water flux: the resistance of the boundary layer in the diluted solution side, the membrane resistance and the boundary layer resistance in the osmotic solution side. If the boundary layers cannot be neglected, the dependence of the individual mass transfer coefficients can be correlated according to Eq. (8), has proposed by several authors for a stirred cell [5–7]. The Sherwood number (Sh) depends on the Re and Schmidt number (Sc): Sh = αReβ Scγ

(8)

with Sh =

kb , Dw

Re =

ρNd2 , µ

Sc =

µ ρDw

(9)

where b is the stirred cell diameter, N the stirring rate, d the stirrer diameter, Dw the water diffusion coefficient, ρ the solution density, µ the solution dynamic viscosity, and α, β and γ are constants. In this work, pure water instead of a diluted solution was used, thus, a1 = am1 . Working in conditions where the resistance to mass transfer in the osmotic solution side is negligible (am2 ≈ a2 ), there is only the contribution of the membrane resistance, and the membrane mass transfer coefficient can be evaluated by Jw = kmp P ∗ (a1 − a2 )

(10)

Temperature is another parameter that affects the water flux. According to Eq. (10), if there are no boundary layer effects, the flux is a function of the membrane mass transfer coefficient, the water activity

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and the pure water vapour pressure. All these parameters are temperature-dependent, however, for the osmotic agents used, the water activity is practically constant over the temperature range of the experiments [8,9]. In addition, temperature can be assumed to have little effect on the mass transfer coefficient itself. On the other hand, pure water vapour pressure increases with temperature, and this dependence can be described using the Clausius–Clapeyron equation [10]: dP ∗ P ∗ Hv = dT RT2

(11)

where T is the temperature and Hv is the water latent heat of vaporisation. Integrating Eq. (11), considering that Hv and R are constant and independent of temperature, and substituting P∗ in Eq. (10), an Arrhenius-type dependence with temperature for the water flux is obtained, as proposed by Mengual et al. [3]: Hv 0 Jw = Jw exp − (12) RT where Jw0 = Akmp (a1 − a2 ), and A is a constant of integration.

3. Experimental 3.1. Materials The compounds employed to prepare the osmotic solutions were glycerol (86–88 wt.%), sodium chloride and calcium chloride dihydrate pro-analysis grade (Riedel-de-Häen, Germany). The osmotic solutions were prepared with de-ionised water. A flat polypropylene membrane (Metricel® , Pall Gelman Laboratory, USA) was used, with nominal pore diameter of 0.1 mm, 90 mm thickness and 55% porosity, as specified by the manufacturer. 3.2. Experimental set-up A stirred cell consisting of two identical cylindrical chambers was used (Fig. 2). The polypropylene membrane with an effective area of 11.3 cm2 was placed between the chambers, one with de-ionised water (chamber 1) and the other with an osmotic solution (chamber 2). Chamber 1 was connected to a water

Fig. 2. Stirred cell: (1) magnetic stirrer, (2) water outlet from thermostat, (3) inlet, (4) thermometer, (5) outlet, (6) water inlet from thermostat, (7) membrane, (8) magnet, (9) stirrer, (10) thermal jacket.

reservoir to avoid hydrostatic pressure differences between both sides of the membrane. The water flux was obtained measuring the variation of the water volume with time in the pipette connected to chamber 2. In order to check reproducibility, three measurements were performed for each case, and the average values are presented. The mean value of the deviations obtained was 2%. The temperature was maintained constant by re-circulating water through the thermal jackets from a water bath and was measured with type J thermopars. Magnetic plates coupled with magnetic bars stirred both chambers. The stirring rate was measured using a digital photo tachometer. 3.3. Calculation methods The fitting calculations were carried out using the software package ScientistTM , from MicroMath® . 4. Results and discussion Three sets of experiments were performed in order to determine the effect of process parameters on mass transfer. In the first set, the contribution of the boundary layer resistance was evaluated. The water flux was measured varying the stirring rate, under constant temperature and solute concentration, for two different osmotic agents. In the second set of experiments the effect of osmotic agent’s nature and concentration was studied. In this case, three osmotic agents were

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used, and the flux was measured for different solute concentrations at the same temperature and stirring rate. Finally, in the last set, both solute concentration and stirring rate were kept constant, and the temperature was varied in order to determine its effect on the water flux. 4.1. Evaluation of the boundary layer resistance The water flux was measured using two osmotic solutions, calcium chloride 3.2 M and glycerol 8.3 M, with the same bulk water activity (a2 = 0.688). The temperature was maintained constant (T = 25 ◦ C) and the stirring rate was varied between 100 and 600 rpm. Fig. 3 shows that the flux increases with the stirring rate for both osmotic agents, until a plateau is reached. Since all variables are maintained constant except the stirring rate, we can assume that this increase is due to a decrease of the boundary layer resistance, which is negligible at high stirring rates leading to a constant water flux. According to Eq. (10), when the boundary layer resistance can be neglected in both sides of the membrane, there is only the membrane resistance, which is not dependent on the osmotic agent used. Since the driving force is the same, it was expected to have a similar water flux value in the plateau for both osmotic agents, which is not the case. As described by Eq. (8), the individual mass transfer coefficient in the boundary layer is dependent, not only on the stirring rate, but also on the properties of the osmotic solution. For the solutions used, the ratio m/r is five times

Fig. 4. Water flux as a function of Reynolds number.

higher for the glycerol solution, so, it is necessary to express the results in terms of Re, which takes into account the difference of the solution properties. As can be seen in Fig. 4, the water flux is similar for both solutions in the range of Re studied. It is clear that, for the glycerol solution, the plateau has not been reached yet, as it seemed when representing the water flux as a function of the stirring rate. It would have been necessary to work at higher stirring rates to achieve the same Re obtained with the calcium chloride solution. In order to obtain a correlation for the boundary layer mass transfer coefficient, Eq. (8) can be rewritten as αDw β γ k2 = Re Sc (13) b Rearranging Eqs. (5), (7) and (13), the following relation for the water flux as a function of Re, is obtained: Jw =

Fig. 3. Water flux as a function of the stirring rate.

kmp Dw αReβ Scγ P ∗ (a1 − a2 ) Dw αReβ Scγ + bP∗ kmp

(14)

By fitting the experimental data (J versus Re obtained for both osmotic agents) using Eq. (14), the values of the constants α and β, and the value of the membrane mass transfer coefficient kmp , were determined. The exponent of the Schmidt number was assumed to be 1/3, as commonly accepted in the literature [5,7]. The water diffusion coefficient in glycerol was estimated using a modification of the Vignes equation [10]: d ln aw 0 xG 0 xw Dw µ = [(DwG µG ) (DGw µw ) ] (15) d ln xw

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Table 1 Parameter values and errors of Eqs. (14) and (17) Parameter

Value and error

Error (%)

r2

Two-step method

α β kmp (×10−10 m s−1 Pa−1 )

1.74 ± 2.83 0.55 ± 0.34 3.33 ± 0.12

163 61.8 3.60

0.999 0.999 0.999

One-step method

α β kmp (×10−10 m s−1 Pa−1 )

1.63 ± 0.99 0.56 ± 0.13 3.32 ± 0.04

60.7 23.2 1.20

0.999 0.999 0.999

where subscripts w refers to water and G to glycerol; D0 is the diffusion coefficient at infinite dilution, x the molar fraction and a is the activity. The value obtained, Dw = 1.90 × 10−10 m2 s−1 , is similar to that measured by Nishijima and Oster [11]. The water diffusion coefficient for the calcium chloride solution, Dw = 1.09 × 10−9 m2 s−1 , was obtained from the literature [12], as well as the values of the water activity, density and viscosity of the solutions [13–15]. The parameter values obtained are presented in Table 1. It can be seen that, even with a good correlation coefficient, the errors associated to the parameters α and β are quite high: 163 and 61.8%, respectively. The method used to determine these constants consists in two-steps: in the first step, the water flux was evaluated by plotting the water volume in the pipette connected to chamber 2 with time, for each experiment. In the second step, the flux values obtained were then used to fit Eq. (14). The disadvantage of using this methodology lies in the fact that there is a loss of information and possible accumulation of errors during the two-step calculation. Viegas et al. [16] analysed the Wilson-plot methodology, which also involves a two-step calculation, to determine mass transfer correlations in membrane extraction. The errors associated with the estimation of their parameter values were also very high. To overcome this problem, they have used a single-step methodology, which involves the fit of the experimental results just to one model equation. A similar approach was adopted in this work. The water flux may be expressed as follows: Jw =

1 dV Am dt

(16)

where Am is the membrane area and V is the water volume transported through the membrane. Using Eqs. (14) and (16), the evolution of water volume with

time can be obtained: kmp Dw αReβ Scγ Dw αReβ Scγ + bP∗ kmp × Am P ∗ (a1 − a2 )(t − t0 )

V = V0 +

(17)

where V0 and t0 are the initial conditions. The values and errors of the constants α and β, and of the membrane mass transfer coefficient kmp , determined fitting the experimental data (V versus t) using Eq. (17), are shown in Table 1. The parameter values are similar to that obtained with the two-step methodology, but the errors associated to α and β are reduced by three times. There was a great improvement just by using this methodology. The values obtained by Sudoh et al. [5] for a stirred cell with a similar geometry (α = 2.00, β = 0.48), are comparable to that determined in this work (α = 1.63 ± 0.99, β = 0.56 ± 0.13), although they have not presented the errors of the parameters. The values of α, β and kmp , obtained with the single-step methodology, are substituted in Eq. (14), which can then be used to preview the evolution of the water flux with the Re for different conditions. The results obtained for other driving forces using glycerol solutions are shown in Fig. 5. It can be seen that, as the driving force increases, more difficult is to cancel the resistance of the boundary layer adjoining the membrane, due to a higher dynamic viscosity and density of the solutions. Using the data presented in Fig. 4, the overall mass transfer coefficient is evaluated by Eq. (5), for each Re. As the value of the membrane mass transfer coefficient is known, the relative contribution of the membrane resistance, (1/km )/(1/K), can be determined. The major resistance to the water flux is due to the membrane itself, even at low Re (Fig. 6).

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Fig. 5. Water flux prediction for different glycerol solutions: (䉱) experimental values, (—) prediction with Eq. (14).

4.2. Effect of the osmotic agent’s nature and concentration

177

Fig. 7. Flux dependence with osmotic solution concentration.

These experiments were carried out at constant stirring rate (ω = 400 rpm) and temperature (T = 25 ◦ C). Three different osmotic agents were used: calcium chloride, sodium chloride and glycerol. The water flux was measured for various solute concentrations, which correspond to different water activities. Fig. 7 shows that, for all solutions, the water flux increases with the osmotic agent concentration. In these experiments, there is a variation of the Re due to the increase of viscosity and density with

the solution concentration. The Re achieved varied between 1890 and 3950, so we can assume that, for these solutions, the boundary layer effects can be neglected, thus, am2 ≈ a2 . The water activity in the osmotic solution is then related to water vapour pressure in the pores. Representing the flux as a function of the water vapour pressure difference the flux values fall into a straight line (Fig. 8). All osmotic agents cause the same effect being the water flux only dependent on the vapour pressure difference. It is possible to evaluate the membrane mass transfer coefficient value, according to Eq. (1), and the value obtained, i.e. (3.24±0.08)×10−10 m s−1 Pa−1 , is similar to that

Fig. 6. Relative contribution of the membrane and boundary layer resistance.

Fig. 8. Flux variation with the water vapour pressure difference.

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water flux increase with temperature is mainly due to an increase of the driving force (P1 − P2 ). 5. Conclusions

Fig. 9. Flux dependence with temperature: (䊐) experimental values, (—) fit curve to Eq. (12).

determined in Section 4.1 and presented in Table 1, i.e. (3.32 ± 0.04) × 10−10 m s−1 Pa−1 . 4.3. Effect of the temperature The water flux was measured using a 2 M calcium chloride solution (a2 = 0.849). The stirring rate was kept constant at 400 rpm and the temperature was varied between 20 and 45 ◦ C. Fig. 9 shows that, as expected, the flux increases with the temperature according to Eq. (12). From the fit of the experimental data to this equation, the heat of vaporisation of water was obtained: Hv = (3.96 ± 0.38) × 104 J mol−1 . The value is similar to that referred in the literature, Hv = 4.47 × 104 J mol−1 at 20 ◦ C and Hv = 4.35 × 104 J mol−1 at 45 ◦ C [10]. Mengual et al. [3] evaluated the effect of temperature using five different membranes and sodium chloride as osmotic agent. The values of Hv obtained in his study ranged from (2.91 ± 0.17) × 104 to (3.48 ± 0.10) × 104 J mol−1 , which are a little lower than that determined in this work. The individual mass transfer coefficient in the membrane kmp was determined for each temperature using Eq. (1). The value of kmp obtained is nearly constant in the temperature range of the experiments; it varies from (3.15 ± 0.29) × 10−10 m s−1 Pa−1 at 20 ◦ C to (3.71 ± 0.69) × 10−10 m s−1 Pa−1 at 45 ◦ C. Since the temperature effect on the mass transfer coefficient is not significant, it can be concluded that the

The influence of the relevant parameters on the osmotic evaporation process, namely stirring rate, temperature and osmotic agent’s nature and concentration, were studied. The water fluxes were expressed as a function of the Re, characterising hydrodynamic conditions, and as a function of the water activity difference between the solutions, which is the driving force of the process. In this way, it was possible to compare the effect of the different osmotic agents. The contribution of the boundary layer has been evaluated for two osmotic agents (glycerol and calcium chloride). It has been observed that, even for low Re, the membrane is responsible for the major contribution to the overall resistance. From the representation of the flux as a function of Re, it can be concluded that, as long as the osmotic solutions have the same bulk water activity, and the process is carried out under the same hydrodynamic conditions, similar fluxes are obtained regardless of the osmotic agent used. In addition to the water activity, other factors, namely the solution viscosity and density, must be taken into account in the selection of the osmotic agent. A correlation for the individual mass transfer coefficient in the boundary layer has been obtained, and the water flux evolution with the Re could be estimated for different driving forces using that correlation. The flux increases with the osmotic agent concentration and it is not affected by the nature of the osmotic agent for the same water vapour pressure difference, which is the driving force of the process. Temperature has little effect on the membrane mass transfer coefficient. The water flux increases with temperature according to an Arrhenius-type equation, and this increase is mainly due to an increase of the water vapour pressure.

Acknowledgements V. D. Alves acknowledges the research grant PRAXIS XXI/BD/18300/98.

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