Several studies have focused on modeling of the RO process. The two most impor-tant models for the prediction of permeate flux and retention of aroma compounds during concentration of fruit juice by RO are, namely, (1) solution–diffusion model combined with film model (Lonslade, 1972) (2) preferential sorption–capillary flow model (Kimura and Sourirajan, 1967).
4.3.1.3.1 Solution–Diffusion Model
Solution–diffusion model assumes that sorption of both the solvent and the solute occurs at the upstream surface of the membrane in accordance with phase equi-librium consideration, followed by diffusion through a nonporous and homoge-neous diffusive barrier under chemical potential gradient in an uncoupled manner.
According to this model, the permeate flux is given by
Vw=L [ Pp∆ −∆π], (4.21)
where
Vw is the permeate flux ΔP is the TMP
Lp is the permeability coefficient of water
Δπ is the osmotic pressure difference between the retentate and the permeate sides of the membrane, defined as
∆π π= (cm)−π(cp), (4.22)
where cm and cp are the solute concentration on the membrane at the feed side and permeate side, respectively. At steady state, the concentration polarization phenom-enon can be written by the film theory as
V k lnc c
c c
w m p
o p
= −
− , (4.23)
where k is the mass transfer coefficient defined as D/d, D being the solute diffusiv-ity and d the boundary layer thickness. For a fully retentive RO membrane, cp, in Equations 4.22 and 4.23, can be neglected to obtain:
Vw=Lp
[
∆P−π(c )m]
(4.24)V k lnc
w cm
o
= . (4.25)
Fruit juice contains sugar, salt, acid, and a large number of aroma and flavor com-pounds. For the fruit juice containing n solutes, the process can be explained with the following n + 1 equation system:
Vw=Lp
[
∆P−π(c )mi]
(4.26)V k lnc
w i cmi
oi
= , (4.27)
where the subscript “i” denotes the solute present in the juice. The mass transfer coefficient, ki, can be estimated using Sherwood number relationship obtained from heat and mass transfer analogies for flow through a nonporous geometry. The per-meability coefficient of water (Lp) is obtained from the slope of the experimental curve of permeate water flux vs. TMP (using distilled water as feed). In fruit juices, the major contributors to osmotic pressure are sugars (hexoses, disaccharides) and organic acids (Merson and Morgan, 1968). Various relationships are available in the literature for the calculation of osmotic pressure. The simplest method is the van’t Hoff equation:
π = RTc
M
mi n
. (4.28)
Thussen (1970) presented an empirical equation for the osmotic pressure of fruit juices:
Osmotic pressure (MPa) 13.375C (1 C) ,
= − (4.29)
where C is the weight fraction of dissolved solids. According to Matsuura et al.
(1974), for the fruit juice solution, the plot of π (osmotic pressure) vs. π/Xc is a straight line. The following correlation is proposed:
π π
X a b,
c
= + (4.30)
where a and b are the constant characteristics of the fruit juice solution and the rela-tionship is valid up to a carbon weight fraction (Xc) of about 12 × 10−2. Values of a, b, and Xc are given in Table 4.7 for the calculation of osmotic pressure of various juices.
4.3.1.3.2 Physicochemical Properties of Fruit Juice
Constenla et al. (1989) have proposed empirical relationships based on their experi-mental data for the estimation of viscosity and density of apple juice as a function of concentration and temperature as
ln AX
(100 BX)
w
µ µ
=
− , (4.31)
where A = −0.258 + 817.11/T and B = 1.891 − 3.021 × 10−3 T
ρ =0.8278+0.347 8exp0 ( .0 01X)−5 479 1. × 0−4 T, (4.32) where
μ and μw are the apple juice and water viscosities (kg/m s), respectively ρ is the apple juice density (kg/m3)
X is the apple juice concentration in °Brix A and B are constants
T is the absolute temperature
During the study of apple juice concentration by RO, Alvarez et al. (1997) have used the following expression reported by Gladdon and Dole (1953) to estimate the effect of concentration on the diffusion coefficient of the solute:
Dn Don
w 0.45
=
µ
µ , (4.33)
where Dn and Don are the diffusion coefficients (m2/s) for the solute, in the mix-ture and in the dilute water, respectively. From the knowledge of Lp, transmembrane
TABLE 4.7
Data of a, b, and Xc for Calculation of Osmotic Pressure π(psi)/Xc = aπ + b
Name of Juice a b Xc × 102
Apple, pineapple, orange, grape fruit, and grape
3.94 3560 —
Lime juice 3.31 3997 3.51
Lemon juice 2.59 4442 3.21
Prune juice 3.31 4217 7.56
Carrot juice 4.93 3088 3.87
Tomato juice 8.95 4187 2.59
Source: Reprinted from Matsuura, T. et al., J. Food Sci., 39, 704, 1974. With permission.
pressure, cross-flow velocity, physicochemical properties of fluid (density, viscosity, diffusivity), dimension of flow geometry (equivalent diameter, length), and osmotic pressure, the permeate flux can be estimated.
4.3.1.3.3 Preferential Sorption-Capillary Flow Model
The preferential sorption-capillary flow model assumes that the transport of sol-ute and solvent occurs through the pores in the membrane permselective layer.
According to this mechanism, RO is governed both by the porous structure of the membrane surface and preferential sorption at the membrane–solution interface under the given experimental conditions. Preferential sorption at the membrane–solution interface is a function of the solute–solvent–membrane interactions. These interactions arise in general from polar, steric, nonpolar, and ionic character of each one of the three components in the RO system. The polar parameter gives measures of the solute’s acidity or basicity. The nonpolar parameter quantifies the extent of hydrophobic interactions between the non-polar part of the membrane and solute. These two factors would influence the
where DAM/K lm is the solute transport parameter, which is shown to be constant for a particular membrane–solute system; DAM the diffusivity of the solute in the mem-brane; K the distribution ratio of the solute between the aqueous solution and the membrane; lm the membrane thickness. The solute concentration on the membrane surface at the feed side can be written by the film theory as
where
c1 is the molar concentration (mol/m3) of the bulk solution
xo, xm, xp are the mole fraction of the solute in the bulk solution, on the membrane at the feed side and permeate side, respectively
Assuming constant molar concentration, c1 permeate flux (v) can be written as
v V V
c
s w
1
= + . (4.40)
Therefore, from Equations 4.39 and 4.40, solvent flux (Vw) can be written as
V c k(1 x )lnx x
The solute transport parameter (DAM/Klm) for different membrane materials can be written in terms of the polar and steric Taft numbers and the Small’s number of the
The coefficient δ* also depends on the porous structure of the membrane surface and the chemical nature of both solute and membrane material. The Small’s number gives a measure of the nonpolar character of the compound. It is only considered when the molecular structure of the solute contains a straight chain involving more than three carbon atoms not associated with a polar functional group. The Taft number gives measure of the electron withdrawing capacity of the substituent group in a polar molecule. Basicity of the molecule increases with increasing the negativity of the Taft number, resulting in an increase in solute rejection by the membrane. The quantities ρ*σ*, w*s*, and δ*Es* in Equation 4.42 represent the contributions of polar effect, non-polar effect, and steric effect to the solute transport parameter. The solute transport parameter can also be expressed in terms of appropriate free energy and steric parameters (Matsuura and Sourirajan, 1973):
ln D
KlAM ln c* *Es* w*s*
m
= + −
+ +
∆∆G
RT δ , (4.43)
where the term (−ΔΔG/RT) is the polar free energy parameter for the solute. The quantity (ΔΔG) can be obtained from the following relation:
∆∆G=∆G1−∆G2, (4.44)
where
ΔG denotes the free energy of the solute–solvent interaction
The subscripts 1 and 2 refer to the membrane solution interface and the bulk phase, respectively
RO separations of completely ionized inorganic solutes are governed by electrostatic interactions. Hence, the steric and nonpolar terms are neglected in Equation 4.43 for the calculation of the term (DAM/Klm). However, for nonionized polar organic solutes, the term (DAM/Klm) is usually estimated by estimating the corresponding value for a completely ionized inorganic solute taken as a reference such as NaCl (Matsuura et al., 1974).
For the known values of membrane permeability, operating conditions (feed concentration, TMP, and cross-flow velocity), composition and physicochemical properties of feed, experimental values of permeation rate, and solute rejection, it is possible to calculate the flux of solute, solvent, and membrane surface concentra-tion using Equations 4.38 through 4.44. Assuming solute transport parameter to be constant, the solute rejection can be predicted.
Alvarez et al. (1997) used the solution–diffusion model for the prediction of per-meate flux during concentration of apple juice. Variations of perAlvarez et al. (1997) used the solution–diffusion model for the prediction of per-meate flux with cross-flow velocity for different feed concentrations are represented in Figure 4.11.
The solid lines represent the values of permeate flux predicted by the model and the points the experimental values.
4.3.2 direCt osMosis ConCentration