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Modeling of Sucrose Permeation through aPectin Gel During Ultrafiltration of DepectinizedMosambi [Citrus sinensis (L.) Osbeck] JuicePPPPPRAMODRAMODRAMODRAMODRAMOD R R R R RAIAIAIAIAI, G.C. M, G.C. M, G.C. M, G.C. M, G.C. MAJUMDARAJUMDARAJUMDARAJUMDARAJUMDAR, S, S, S, S, SURNANDOURNANDOURNANDOURNANDOURNANDO D D D D DASASASASASGGGGGUPTUPTUPTUPTUPTAAAAA, , , , , ANDANDANDANDAND S S S S SIRSHENDUIRSHENDUIRSHENDUIRSHENDUIRSHENDU D D D D DEEEEE

ABSTRAABSTRAABSTRAABSTRAABSTRACTCTCTCTCT: U: U: U: U: Ultrltrltrltrltrafiltrafiltrafiltrafiltrafiltration of synthetic juice (mixturation of synthetic juice (mixturation of synthetic juice (mixturation of synthetic juice (mixturation of synthetic juice (mixture of sucre of sucre of sucre of sucre of sucrose and pectin with vose and pectin with vose and pectin with vose and pectin with vose and pectin with varararararying composition) as wying composition) as wying composition) as wying composition) as wying composition) as wellellellellellas enzymatically treated mosambi [as enzymatically treated mosambi [as enzymatically treated mosambi [as enzymatically treated mosambi [as enzymatically treated mosambi [Citrus sinensisCitrus sinensisCitrus sinensisCitrus sinensisCitrus sinensis (L.) Osbeck] juice is performed in a batch, unstirred mem- (L.) Osbeck] juice is performed in a batch, unstirred mem- (L.) Osbeck] juice is performed in a batch, unstirred mem- (L.) Osbeck] juice is performed in a batch, unstirred mem- (L.) Osbeck] juice is performed in a batch, unstirred mem-brane cell. Thin-film composite polyamide membrane of molecular weight cutoff 50000 was used. The fluxbrane cell. Thin-film composite polyamide membrane of molecular weight cutoff 50000 was used. The fluxbrane cell. Thin-film composite polyamide membrane of molecular weight cutoff 50000 was used. The fluxbrane cell. Thin-film composite polyamide membrane of molecular weight cutoff 50000 was used. The fluxbrane cell. Thin-film composite polyamide membrane of molecular weight cutoff 50000 was used. The fluxdecline mechanism is found to be controlled by the growth of a gel layer of pectin over the membrane surface.decline mechanism is found to be controlled by the growth of a gel layer of pectin over the membrane surface.decline mechanism is found to be controlled by the growth of a gel layer of pectin over the membrane surface.decline mechanism is found to be controlled by the growth of a gel layer of pectin over the membrane surface.decline mechanism is found to be controlled by the growth of a gel layer of pectin over the membrane surface.UUUUUsing standarsing standarsing standarsing standarsing standard cake filtrd cake filtrd cake filtrd cake filtrd cake filtration theoration theoration theoration theoration theoryyyyy, the flux decline and the gr, the flux decline and the gr, the flux decline and the gr, the flux decline and the gr, the flux decline and the grooooowth of gel laywth of gel laywth of gel laywth of gel laywth of gel layer is quantified. er is quantified. er is quantified. er is quantified. er is quantified. The gel layThe gel layThe gel layThe gel layThe gel layer iser iser iser iser ischarcharcharcharcharacteracteracteracteracterizizizizized in tered in tered in tered in tered in terms of specific gel layms of specific gel layms of specific gel layms of specific gel layms of specific gel layer rer rer rer rer resistanceesistanceesistanceesistanceesistance, gel por, gel por, gel por, gel por, gel porosityosityosityosityosity, and gel density, and gel density, and gel density, and gel density, and gel density. A tr. A tr. A tr. A tr. A transient model is pransient model is pransient model is pransient model is pransient model is proposedoposedoposedoposedoposedand solved numerically to quantify the transport of sucrose through the pectin gel for the synthetic juice andand solved numerically to quantify the transport of sucrose through the pectin gel for the synthetic juice andand solved numerically to quantify the transport of sucrose through the pectin gel for the synthetic juice andand solved numerically to quantify the transport of sucrose through the pectin gel for the synthetic juice andand solved numerically to quantify the transport of sucrose through the pectin gel for the synthetic juice andenzymatically treated juice. The hindered diffusion coefficient of sucrose through the gel is calculated by com-enzymatically treated juice. The hindered diffusion coefficient of sucrose through the gel is calculated by com-enzymatically treated juice. The hindered diffusion coefficient of sucrose through the gel is calculated by com-enzymatically treated juice. The hindered diffusion coefficient of sucrose through the gel is calculated by com-enzymatically treated juice. The hindered diffusion coefficient of sucrose through the gel is calculated by com-paring the results with the experimentally obtained permeate concentration profile of sucrose. The calculatedparing the results with the experimentally obtained permeate concentration profile of sucrose. The calculatedparing the results with the experimentally obtained permeate concentration profile of sucrose. The calculatedparing the results with the experimentally obtained permeate concentration profile of sucrose. The calculatedparing the results with the experimentally obtained permeate concentration profile of sucrose. The calculatedpermeate concentration profile agrees well with the experimental data.permeate concentration profile agrees well with the experimental data.permeate concentration profile agrees well with the experimental data.permeate concentration profile agrees well with the experimental data.permeate concentration profile agrees well with the experimental data.

Keywords: pectin gel, sucrose, ultrafiltration, permeate flux, diffusion coefficientKeywords: pectin gel, sucrose, ultrafiltration, permeate flux, diffusion coefficientKeywords: pectin gel, sucrose, ultrafiltration, permeate flux, diffusion coefficientKeywords: pectin gel, sucrose, ultrafiltration, permeate flux, diffusion coefficientKeywords: pectin gel, sucrose, ultrafiltration, permeate flux, diffusion coefficient

Introduction

Mosambi, sweet orange [Citrus sinensis (L.) Osbeck] is a citrusfruit, containing several phytochemicals and/or nutraceuti-

cals including vitamin C. Even though India is the 2nd largest pro-ducer of fruits, the level of processing is only 2% compared withother countries, for example, Brazil: 70%, United States: 60% to70%, Malaysia: 83%, and Israel: 50%. Among almost 180 families offruits grown all over the world, citrus fruits constitute around 20%of the world’s total fruit production.

Membrane separations are an upcoming unit operation for clar-ification and concentration of fruit juices. During filtration of fruitjuice, fouling occurs due to accumulation of pectic substances, tan-nins, proteins, and fibers present in the juice on membrane sur-face or blocking of membrane pores (Todisco and others 1996; Vail-lant and others 1999). It is well known that pectin is a gelling agent(Thakur and others 1997). Therefore, during membrane separationprocess, pectin and its derivatives form a gel-like structure over themembrane surface, thereby reducing the permeate flux, which isthe throughput of the system. To degrade pectin, an enzymatictreatment of the raw juice is usually carried out with pectinase.Pectinase hydrolyses pectin and causes pectin protein complexesto flocculate. The resulting juice after separation has much loweramount of pectins and also of reduced viscosity, which is advanta-geous for the subsequent filtration processes. In a typical raw mo-sambi juice, pectin content (as alcohol insoluble solids) is about0.5% to 0.7%, which is reduced to about 0.25% to 0.3% after pecti-nase treatment. Even after enzyme treatment, remaining amountof pectin in fruit juice plays major role in fouling during ultrafiltra-tion (UF) and microfiltration (MF) (Girard and Fukumoto 2000).

Fruit juices contain mixtures of low-molecular-weight (LMW )and high-molecular-weight (HMW) solutes. The LMW compounds(MW < 1 kDa) consist of sugars, organic acids, amino acids, pig-ments, vitamins, and so forth, and HMW components consists ofproteins, enzymes, pectic substances, and so forth. During filtrationof fruit juice by UF and MF, the nonpermeating class of solutes (con-sisting of HMW) tends to form a gel on the membrane surface andLMW solutes pass though the gel and, thereafter, through mem-brane. Formation of gel layer on the membrane surface during fil-tration of a macromolecular solution is a major problem during UFand MF. Deposition of gel layer and its subsequent growth withtime lead to very low solvent flux.

UF of a mixture of 2 proteins, ovalbumin and �-globulin (bothare gel forming), was carried out by Matsuyama and others (1994).It may be noted that there has been some difference in opinionregarding gel characteristics. Gel may not mean deposition of a dis-tinct solid-like phase, but a highly viscous solution of a “pseudo” gellayer owing to the increasing concentration of macromolecular sol-utes adjacent to the membrane surface (Nakao and others 1986;Bhattacharjee and others 1996). During membrane filtration, theformation of gel layer is unavoidable and as a result, transport ofthe desired permeating solutes gets affected.

De and Bhattacharya (1997) attempted to quantify the retentionof sucrose through a polyvinyl alcohol (PVA) gel formed during UF, bydeveloping a suitable model. They observed that the polymeric gelof PVA acted as a dynamic membrane that rejected sucrose andthereby, increasing the sucrose retention. However, in case of enzy-matically treated fruit juice, the gel formed during UF is due to high-molecular-weight pectin (left over after enzyme treatment), whichrenders the gel more porous. Sucrose (in terms of total soluble solids)permeates through the pectin gel. Because, sucrose is freely perme-able through the high-molecular-weight cutoff UF membrane, thepermeate sucrose concentration continuously increases and after acertain time of filtration, it approaches the feed concentration. Sev-eral researchers reported this observation (Jiraratananon and others

MS 20050365 Submitted 6/20/05, Revised 7/28/05, Accepted 10/20/05. Au-thors Rai and Majumdar are with Dept. of Agriculture and Food Engineer-ing, Indian Inst. of Technology, Kharagpur, India. Authors DasGupta andDe are with Dept. of Chemical Engineering, Indian Inst. of Technology,Kharagpur, Kharagpur- 721 302, India. Direct inquiries to author De (E-mail: sde@che.iitkgp.ernet.in).

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E: Food Engineering & Physical Properties

Modeling sucrose transport . . .

1997; Rai and others 2005). This is a crucial issue for system design.Therefore, the mechanism of transport of sucrose through the pectingel is entirely different from that of sucrose through PVA gel as stud-ied by De and Bhattacharya (1997).

An understanding of the transport of sucrose through the grow-ing pectin gel and the growth dynamics of the gel layer is crucial forcomplete understanding of the process. It is envisaged that thediffusion of sucrose will be hindered due to the presence of pectinlayer and the values of hindered diffusion coefficient will be animportant engineering parameter. These data for sucrose diffusiv-ity through the pectin gel is of utmost importance for the design ofmembrane modules and evaluation of the filtration performance.To the best of the authors’ knowledge, hindered diffusivity data ofsucrose through the pectin gel are not reported in the literature.Therefore, the objectives of the present work are to develop a modelfor the gel layer growth and hindered diffusion process, numericalsolution of the proposed model, and to simulate the process with thehelp of already available experimental data (Rai and others 2005).

TheoryTheoryTheoryTheoryTheoryConsiderable amount of research has been done on the forma-

tion of gel-type layer during ultrafiltration of solutions containingmacromolecules and solutes (pectin and sucrose in this case, re-spectively). It has been generally postulated that at an early peri-od, when there is a low degree of adsorption of macromolecules onthe membrane surface, there is a possibility of an interaction be-tween macromolecules and solute. Such interactions quickly weak-en as more macromolecules are adsorbed on the membrane sur-face. It has also been proposed that the formation of the gel-typelayer starts almost immediately with the beginning of filtration(Jirataranon and others 1997). It has been reported that co-soluteslike sucrose augments the gelation of pectin by replacing waterand thereby increasing interaction among the polymer chains lead-ing to gel-type layer formation (Tsoga and others 2004). Experimen-

tal determination of the actual gel point is not easy and even moredifficult near a membrane surface as determination of viscosity ina very thin region near the membrane surface is practically impos-sible. It has also been reported that the time required for actual gelformation is in the order of seconds (Da Silva and Rao 1999). Inmembrane systems, a gel-type layer will form even earlier. Thesepostulates are supported by the experimental data of Rai and oth-ers (2005), where it was observed that the permeate concentrationimmediately became feed concentration in the case of only sucrosesolution whereas in a sucrose-pectin mixture, the concentration ofsucrose progressively increased with time in the permeate.

A schematic of the process based on the above concepts is pre-sented in Figure 1. Initially (at t = 0), the feed is homogeneous andthus with the application of transmembrane pressure, the perme-ate sucrose concentration will be close to that of the feed (as themolecular weight cutoff of the membrane used herein is 50000)(Figure 1a). This permeate concentration will be extremely difficultto measure in a membrane system as it would take at least a fewminutes to collect sufficient permeate for analysis. However, thebuild up of the gel-type layer will start providing additional resis-tance to the transport of the sucrose molecules (hindered diffu-sion). Therefore, the concentration of the sucrose will be less thanthat of the feed concentration. Experimentally this is the 1st mea-surable concentration of the permeate. This situation is depicted inFigure 1b. After sufficient operating time, with more and more dif-fusion of sucrose through the growing gel-type layer, the permeatesucrose concentration comes close to the feed concentration asshown in Figure 1c. The time required for this is about 15 min forsucrose-pectin mixture and about 25 min for the mosambi juice forthe experiments reported herein.

Determination of permeate flux and gel layer thickness. Determination of permeate flux and gel layer thickness. Determination of permeate flux and gel layer thickness. Determination of permeate flux and gel layer thickness. Determination of permeate flux and gel layer thickness. Usingphenomenological equation, the permeate flux at any instant isgiven as,

(1)

where, the gel layer resistance is given as,

Rg = �(1 – �g)�gL (2)

In Eq. 2, � is the specific gel layer resistance, �g is the porosity ofgel layer, �g is the gel layer density and L is the thickness of the gellayer.

The mass balance of HMW solute (pectin in this case) in the gellayer results in,

LA (1 – �g)�g = (c0 – cp)V (3)

In Eq. 3, A is the membrane area, V is the cumulative volume ofthe filtrate and cp is the permeate concentration of HMW solutes. Inthe synthetic juice, pectin used has a range of molecular weightfrom 30000 to 100000. Because the molecular weight cutoff of mem-brane selected is 50000, some amount of pectin permeate throughthe membrane and this is typically 4% to 12% depending on theoperating conditions. On the other hand, for mosambi juice, cp wasfound out to be zero. Using Eq. 2 and 3, the gel layer resistance canbe expressed as,

(4)

Substituting the expression of Rg from Eq. 4 in Eq. 1 and afterintegration from t = 0 (where, V = 0) to any t (where, V = V), the ex-

Figure 1—Mechanism for solute (sucrose) transportthrough the macromolecular (pectin) gel

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Modeling sucrose transport . . .

pression of the cumulative filtrate volume and the permeate flux asfunction of time of filtration are obtained as,

(5)

and,

(6)

In Eq. 5 and 6, is the pure water flux, that is,

(7a)

and,

(7b)

Using Eq. 3 and 5, the evolution of gel layer thickness with filtra-tion time is obtained as,

(8)

TTTTTrrrrransporansporansporansporansport of the LMW solutet of the LMW solutet of the LMW solutet of the LMW solutet of the LMW solute(sucrose) through the pectin gel(sucrose) through the pectin gel(sucrose) through the pectin gel(sucrose) through the pectin gel(sucrose) through the pectin gel

The transient mass balance of sucrose in the gel layer is writtenas,

(9)

where, cl is the concentration of sucrose in gel layer, and D is thehindered diffusion coefficient of sucrose in that layer. Equation 9is expressed in terms of nondimensional variables. The nondimen-

sional variables are defined as and

, where, L0 is a reference length. Using an order of magnitude

analysis, it is found that the values of gel layer thickness are in theorder of 10–5 and 10–4 m. Therefore, a typical value of 10–4 m is usedfor L0 to make L nondimensional for numerical stability. In terms ofnondimensional variables, Eq. 9 is expressed as,

(10)

where, . The initial and boundary conditions of Eq. 9 are

as follows,at t → 0, cl = � (11)

where, � is small. In the subsequent section, an asymptotic solution isderived and the final results are independent of the value of �.

At y = L, that is, at the edge of gel layer and bulk solution inter-face, the convective flux of sucrose arriving at the interface must beequal to the net flux (comprising of convective as well as diffusive)away from the interface. This condition is mathematically ex-pressed as,

at y = L, (12)

At the gel layer and membrane interface, the sucrose passesthrough the membrane unhindered. Therefore, the permeate con-centration will be same as the membrane surface concentration.Thus the following boundary condition is applicable.

at y = 0, (13)

Eq. 11 to 13 can be expressed in terms of nondimensional vari-ables as,

at � → 0, c* = �/ (14)

at y* = L*, (15)

at y* = 0, (16)

Equation 9 is solved by an integral method, assuming the follow-ing concentration profile of sucrose in the gel layer,

(17)

Identifying the fact that at y* = 0, c* = , Eq. 17 can be written as,

(18)

Using the boundary conditions given in Eq. 15 and 16, the coef-ficients a1 and a2 are evaluated as,

a1 = 0 (19)

a2 = (20)

Once the coefficients of Eq. 18 are determined, the derivatives

and are evaluated from Eq. 18 and substituted in

Eq. 10. Both sides of the resultant equation is multiplied by dy* andintegrated across the gel layer thickness L*. After integration andalgebraic simplification, the following equation results,

(21)

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E: Food Engineering & Physical Properties

Modeling sucrose transport . . .

Equation 21 is coupled with Eq. 6 and 8. Nondimensional formof Eq. 6 and 8 become,

(22)

where, and

(23)

where, . Using Eq. 22 and 23, Eq. 21 becomes

an ODE. It may be noted that, if Eq. 21 is divided by the coefficient

of the 1st term, that is, , then, L* appears in the de-

nominator of the 1st term of right hand side of Eq. 21. Therefore,

the solution of faces a “blow off” at the onset of integration as

L* = 0 at � = 0. Hence an asymptotic solution is sought at � → 0.

Asymptotic solution at Asymptotic solution at Asymptotic solution at Asymptotic solution at Asymptotic solution at ����� →→→→→ 0. 0. 0. 0. 0. As � → 0, → 1 and from Eq. 22

and 23, and . With � → 0, the 2nd term of Eq. 21

on right hand side vanishes as tends to be a constant, the 3rd

term becomes very small compared with the other terms (as it is a

product L* and ). The simplified version of Eq. 21 now becomes,

(24)

Upon integration from �1 and to � and , the expression of

becomes,

(25)

Small values of �1 (corresponding to t = 20 s) and (corre-

sponding to kg/m3) are selected to obtain at � (correspond-

ing to t = 60 s). Now, this value of at 60 s is considered as the ini-

tial condition to integrate Eq. 21. It is tested that the selections of

any lower value of �1 and does not alter the numerical solution

significantly.Numerical solution. Numerical solution. Numerical solution. Numerical solution. Numerical solution. Eq. 21 coupled with Eq. 22 and 23 are

solved by 4th order Runge-Kutta method using the initial condi-tion mentioned earlier, for every set of operating conditions andthe hindered diffusion coefficient of sucrose through pectin gel isestimated by a nonlinear optimization technique (Lvenberg-Marquardt) so that the experimental permeate concentrationprofile of sucrose is matched. For this purpose, optimization sub-

routine BCPOL using “direct search algorithm” from IMSL mathlibrary has been used.

Materials and Methods

MaterialsMaterialsMaterialsMaterialsMaterialsSucrose was procured from BDH Laboratories (Dorset, U.K.). A

high methoxyl pectin (Loba chemie, India, Mumbai) with molecularweight ranging between 30000 and 100000 were used during theexperiments. Pectinase from aspergillus niger with activity 3.5 to 7units/mg (proteins Lowry) was used (SRL Research Chemicals, In-dia) for enzymatic treatment of mosambi juice.

The enzymatic treatment of Mosambi juice was done using op-timum condition of time (100 min), temperature (42 °C) and en-zyme concentration (0.0004%w/v) (Rai and others 2004). UF ex-periments were conducted using a mixture of sucrose and pectin(10% + 0.5%, 12% + 0.3%, and 14% + 0.1%) and enzyme treatedmosambi juice.

Experimental procedureExperimental procedureExperimental procedureExperimental procedureExperimental procedureA batch unstirred membrane cell made of stainless steel and fil-

tration area 15.2 cm2 was used for ultrafiltration. The membraneused in the experiments was a thin film composite polyamide ofmolecular weight cutoff 50000 (Permionics, Baroda, India). Theschematic diagram of the experimental set up is given elsewhere(Rai and others 2005).

The membrane permeability was measured by using distilledwater flux and found to be 1 × 10–10 m/Pa s. After the water run, thecell was charged with 100 mL of the feed solution and was pressur-ized at the operating pressure using a pressure regulator and a ni-trogen cylinder. Permeate from the bottom of the cell was collectedin a small beaker (capacity 100 mL), and its cumulative weight wasmeasured with the help of an electronic balance. The operatingpressures during the experiments were 276 kPa, 414 kPa, and 552kPa. Duration of the experiments was 40 min in case of syntheticjuice and 120 min for the mosambi juice. Freshly prepared feedmaterial was used for each experiment. All the experiments wereconducted at a temperature of 30 ± 2 °C.

After each experimental run, the cell and the membrane werewashed thoroughly using distilled water. The individual parts includ-ing the permeate channels were carefully dried. The membrane wasfurther washed by putting it in distilled water for 12 h. After suchthorough washing, a water run was again taken to measure thechange in the membrane permeability. The permeability of themembrane was found to be within ±5% for the successive runs.

Measurement techniquesMeasurement techniquesMeasurement techniquesMeasurement techniquesMeasurement techniquesThe concentration of pectin in the simulated liquid was deter-

mined using a SmartspecTM spectrophotometer (Biorad, Hercules,California, U.S.A.) at a wavelength of 280 nm. Total soluble solidswere measured using an ABBE-3L Benchtop Refractometer (Ther-mospectronic, Rochester, New York, U.S.A.) and expressed as°Brix.Pectin in the juice was determined in terms of alcohol insoluble sol-ids (AIS). AIS values were determined by boiling 20 g juice with 300mL of 80% alcohol solution and simmering for 30 min. The filteredresidue was then again washed with 80% alcohol solution. The res-idue was dried at 100 °C for 2 h and was expressed in percentage byweight (Hart and Fisher 1971).

Results and Discussion

Flux decline mechanism during UFFlux decline mechanism during UFFlux decline mechanism during UFFlux decline mechanism during UFFlux decline mechanism during UFFrom Eq. 6, it is observed that for a gel layer controlling UF, a plot

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Modeling sucrose transport . . .

versus t results in a straight line. For a typical solution of the syn-

thetic juice consisting of 12% sucrose and 0.3% pectin, the variation

of with time is shown in Figure 2a for various operating pres-

sures. The same plot is presented for mosambi juice in Figure 2b. It

is observed from both the figures that the variation of with time

is fairly linear for all the operating conditions. For other syntheticjuice compositions, the similar trend is observed. Hence, the fluxdecline during unstirred batch UF of juice can be considered to begel layer controlling.

Estimation of various properties of the gel layerEstimation of various properties of the gel layerEstimation of various properties of the gel layerEstimation of various properties of the gel layerEstimation of various properties of the gel layerSpecific gel layer resistance. Specific gel layer resistance. Specific gel layer resistance. Specific gel layer resistance. Specific gel layer resistance. As observed from Eq. 6, the specif-

ic gel layer resistance can be calculated from the slope of versus

time plot. In case of depectinized mosambi juice, AIS (alcohol insol-uble solids) value measured in the permeate is found to be zeroand hence, cp is zero in the equation of � in Eq. 7b. The feed AIS isfound to be 2.5 kg/m3 and is used as c0. For various feed composi-tion of synthetic juice, cp values in the permeate are measured andthey are found to be within 4% to 12% of the feed pectin concentra-tion at various pressures. Corresponding experimental cp valuesare used in Eq. 7b for the synthetic juice to evaluate the specific gellayer resistance. The gel layer resistance is finally related to theoperating pressure drop as follows,

� = �0(�P)n (26)

where, n is the gel compressibility index. For both the synthetic anddepectinized mosambi juice, the values of �0 and n are presentedin Table 1. It may be noted from Table 1 that the value of �0 increas-es with the pectin concentration, as expected.

GGGGGel layel layel layel layel layer porer porer porer porer porosityosityosityosityosity. . . . . The average diameter of the solid constitu-ents in the gel layer is estimated assuming spherical molecules andusing the following empirical relationship (Stryer 1998),

(27)

where, Mw is the average molecular weight of gel forming material.Considering the organic nature of solids in gel layer, the value of zis prescribed as 6 × 1029 g/gmol.m3 (Stryer 1998). Pectins have a widemolecular weight range from 10 to 500 kDa with a global molecularweight average of about 100 kDa (Hoagland and others 1997). Us-ing this value, the equivalent diameter of pectin is calculated fromEq. 27 as 5.5 × 10–9 m, for mosambi juice. For synthetic juice, thepectin used has a molecular weight range from 30000 to 100000and hence 65000 is taken as the average value of molecular weight.The corresponding value of equivalent spherical diameter is foundto be 4.77 × 10–9 m. The gel porosity (assuming gel characteristics tobe the same as a cake of spherical particles) is obtained fromKozney-Carman equation (Oers and others 1992) at a particularpressure, using an iterative method.

(28)

where, �g, �g, and dp are gel porosity, gel layer density and diameterof the gel forming particles. The calculated values of �g at differentpressures are presented in Table 2, both for the synthetic and de-pectinized mosambi juice. It can be seen from the table that theporosity decreases with increase in pressure, as expected.

GGGGGel layel layel layel layel layer densityer densityer densityer densityer density. . . . . The gel layer density is estimated by conduct-ing unstirred UF experiments at the specified composition of syn-thetic juice as well as depectinized mosambi juice. The UF experi-ments are continued until the permeate flux stops. From thedifference in weight of the membrane before and after the filtra-tion, the mass of the gel layer is calculated. The volume of gel lay-er is calculated by the difference of the initial sample volume andthe final volumes of permeate. The average value of the gel layerdensity is found to be 1362 kg/m3 for the synthetic juice and that isfor mosambi juice is about 1300 kg/m3. The effect of the presenceof sucrose solution in the membrane as well as in the gel layer isevaluated, and it is observed that this results a maximum error of±3% in the gel density and volume.

Calculation of flux declineCalculation of flux declineCalculation of flux declineCalculation of flux declineCalculation of flux declineOnce the parameters of the gel layer are estimated indepen-

dently, the variation of permeate flux with time is computed usingEq. 6. The comparison of the calculated flux values with the exper-

Figure 2—(a) Variation of 1/v2w as a function of time for Synthetic juice with operating pressures. (b) Variation of 1/v2

w asa function of time for mosambi juice with operating pressures.

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Modeling sucrose transport . . .

imental data is presented in Figure 3a for a typical synthetic juicecomposition and in Figure 3b for the mosambi juice at various op-erating pressures. The figures show excellent agreement betweenthe calculated and the experimental data.

Calculation of gel layer thicknessCalculation of gel layer thicknessCalculation of gel layer thicknessCalculation of gel layer thicknessCalculation of gel layer thicknessThe gel layer thickness is calculated using Eq. 8, by estimating

various properties of the gel. The growth of the gel layer thicknesswith time for various operating pressures is presented in Figure 4a

for a typical composition of synthetic juice and in Figure 4b for de-pectinized mosambi juice. The figure shows an increase in gel layerthickness with time and pressure. Because the gel layer resistanceincreases with time of filtration and operating pressure, its thick-ness also increases.

Evaluation of profile of permeateEvaluation of profile of permeateEvaluation of profile of permeateEvaluation of profile of permeateEvaluation of profile of permeateconcentration of sucroseconcentration of sucroseconcentration of sucroseconcentration of sucroseconcentration of sucrose

As discussed earlier, Eq. 21 is solved using 4th-order Runge-Kutta

Table 1—Parameters for specific gel layer resistance ofthe synthetic as well as mosambi juice

Feed material �����0 (m/kg) n

Synthetic juice 3.13 × 1013 0.4510% sucrose + 0.5% pectinSynthetic juice 3.19 × 1012 0.6312% sucrose + 0.3% pectinSynthetic juice14% sucrose + 0.1% pectin 1.99 × 1011 0.85Enzymatically treated osambi juice 7.05 × 1013 0.37

Table 2—Gel layer porosity of synthetic and mosambi juiceat different operating pressures

�����gFeed material andoperating pressure (kPa) 276 414 551

Synthetic juice 0.62 0.58 0.5810% sucrose + 0.5% pectinSynthetic juice 0.63 0.59 0.5712% sucrose + 0.3% pectinSynthetic juice 0.63 0.58 0.5514% sucrose + 0.1% pectinEnzymatically treated juice 0.62 0.59 0.59

Figure 4—(a) Gel layer thickness as a function of time for Synthetic juice at various pressures. (b) Gel layer thickness asa function of time for mosambi juice at various pressures.

Figure 3—(a) Permeate flux during ultrafiltration (UF) of Synthetic juice as a function of time. (b) Permeate flux duringUF of mosambi juice as a function of time.

Vol. 71, Nr. 2, 2006—JOURNAL OF FOOD SCIENCE E93URLs and E-mail addresses are active links at www.ift.org

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Modeling sucrose transport . . .

method and the hindered diffusion coefficient is optimized to matchthe experimental permeate concentration profile. The values of dif-fusion coefficient are found to vary in narrow range for the variousoperating pressures at a fixed feed composition. The average valuesof diffusivities of sucrose through the pectin gel for various compo-sitions of synthetic juice as well as the mosambi juice are presentedin Table 3. It is observed from Table 3 that the diffusivity decreasesas the feed sucrose concentration increases. It is well known that thebulk diffusivity of sucrose decreases with concentration (Cussler1998). It is given as D = (0.523 – 0.265c) × 10–9 m2/s where, c is themolar concentration. For the concentrations of sucrose considered inthe synthetic juice, the calculated diffusivity values are 4.47 × 10–10,4.3 × 10–10, and 4.16 × 10–10 m2/s for the concentrations 10%, 12%, and14%, respectively. The same trend of hindered diffusivity of sucroseis observed through the pectin gel with concentration. It may be not-ed that diffusivity of sucrose through the pectin gel is about 2 to 6times less than the bulk diffusivity.

The comparison of the calculated permeate concentration ofsucrose with the experimental data is presented in Figure 5 to 7 forthe synthetic juice at 3 operating pressures. The same comparisonfor the depectinized juice is presented in Figure 8. It is observed

from Figure 5 to 8 that model calculation results in overpredictionof permeate concentration for the first 5 min of operation. This maybe caused by the inaccuracy involved in the measurement of initialpermeate concentration. The permeate sucrose concentration be-comes almost equal to feed concentration within about 15 min ofoperation. For the case of mosambi juice, the model predictions areremarkable as observed from Figure 8. The feed concentration isattained in the permeate within 25 min. It is clear from the Figure8 that the time requirement to attain the feed concentration (in thepermeate) decreases with pressure. At higher pressure, convective

Table 3—Diffusivity of sucrose through gel layer for syn-thetic and mosambi juice at different operating pressures

Feed material Diffusivity × 1010 (m2/s)

Synthetic juice 1.8910% sucrose + 0.5% pectinSynthetic juice 1.3112% sucrose + 0.3% pectinSynthetic juice 0.6314% sucrose + 0.1% pectinEnzymatically treated juice 1.05

Figure 7—Permeate total soluble solid during ultrafiltration(UF) of sucrose and pectin solution at 552 kPa

Figure 8—Permeate concentration of total soluble solid dur-ing ultrafiltration (UF) of mosambi juice at various pressures

Figure 6—Permeate concentration of total soluble solid duringultrafiltration (UF) of sucrose and pectin solution at 414 kPa

Figure 5—Permeate concentration total soluble solid duringultrafiltration (UF) of sucrose and pectin solution at 276 kPa

E94 JOURNAL OF FOOD SCIENCE—Vol. 71, Nr. 2, 2006 URLs and E-mail addresses are active links at www.ift.org

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Modeling sucrose transport . . .

flux of sucrose increases and therefore, the permeate concentrationattains the feed value earlier.

Conclusions

The flux decline in unstirred batch UF of the synthetic juice andenzymatically treated mosambi juice is quantified by the gel fil-

tration theory. The pectin gel is characterized by the specific gel lay-er resistance, gel porosity, and density. The pectin gel is found outto be compressible. For the synthetic juice, the compressibility in-dex of the gel varies in the range of 0.45 to 0.85 for various feed com-position studied here and that for mosambi juice is 0.37. The gellayer thickness as for both the juices increase with the operatingpressure. The transport of the sucrose through the pectin gel ismathematically modeled (for both synthetic and mosambi juice)and is solved numerically. The hindered diffusion coefficient ofsucrose is determined for various operating conditions by optimiz-ing the experimental permeate concentration profile. It is foundthat the diffusion coefficient is reduced by a factor of 2 to 6 for var-ious synthetic juice compositions. The calculated permeate fluxprofiles match well with the experimental data. For the syntheticjuice, the permeate sucrose concentration attains the feed concen-tration after about 15 min and for mosambi juice in about 25 min.

Symbolsa0,1,2 = coefficients in Eq. 17A = membrane area (m2)c0 = feed concentration of pectin (kg/m3)cp = permeate concentration of pectin (kg/m3)cl = concentration of sucrose (kg/m3)c* = nondimensional concentration

= nondimensional concentration of sucrose

dp = equivalent spherical diameter (m)D = diffusivity of sucrose in pectin gel (m2/s)L = gel layer thickness (m)Mw = molecular weight (kg/kmol)n = gel layer compressibility indexRm = membrane hydraulic resistance (/m)Rg = gel layer resistance (/m)t = time (s)V = cumulative volume of filtrate (m3)vw = permeate flux (m3/m2.s)

= water flux through membrane (m3/m2.s)

= nondimensional permeate flux

y = coordinate normal to membrane (m)y* = nondimensional normal coordinatez = parameter in Eq. 27 (kg/kmol/m3)

Greek symbolsGreek symbolsGreek symbolsGreek symbolsGreek symbols� = specific gel layer resistance (m/kg)�0 = coefficient in Eq. 26 (m/kg)� = parameter defined by Eq. 7b (m3)�P = transmembrane pressure drop (Pa)�g = gel layer porosity� = visocity (Pa.s)�g = gel layer density (kg/m3)� = nondimensional time

ReferencesBhattacharjee S, Sharma A, Bhattacharya PK. 1996. A unified model for flux pre-

diction during batch cell ultrafiltration. J Membr Sci 111:243–58.Cussler EL. 1998. Diffusion: mass transfer in fluid systems. Cambridge, U.K.:

Univ. Press. p 580.Da Silva JAL, Rao MA. 1999. Rheological behavior of food gel systems. In: Rao MA,

editors. Rheology of fluid and semisolid foods: principles and applications. 1sted. Maryland: Aspen. p 319–68.

De S, Bhattacharya PK. 1997. Modeling of ultrafiltration process for a two-com-ponent aqueous solution of low and high (gel-forming) molecular weight sol-utes. J Membr Sci 136:57–69.

Girard B, Fukumoto LR. 2000. Membrane processing of fruit juices and beverag-es: a review. Crit Rev Food Sci Nutr 40:91–157.

Hart FL, Fisher HJ. 1971. Modern food analysis. Berlin: Springer.Hoagland PD, Konja G, Clauss E, Fisherman ML. 1997. HPSEC with component

analysis of citrus and apple pectins after hollow fiber ultrafiltration. J Food Sci62:69–74.

Jiraratananon R, Uttapap D, Tangamornsuksun C. 1997. Self-forming dynamicmembrane for ultrafiltration of pineapple juice. J Membr Sci 129:135–43.

Matsuyama H, Shimomura T, Teramoto M. 1994. Formation and characteristicsof dynamic membrane for ultrafiltration of protein in binary protein system.J Membr Sci 92:107–15.

Nakao S, Wijmans JG, Smolders CA. 1986. Resistance to the permeate flux inunstirred ultrafiltration of dissolved macro-molecular solutions. J Membr Sci26:165–78.

Oers OWV, Vorstman MAG, Muijseleer WGHM, Kerkhof PJAM. 1992. Unsteadystate flux decline behavior in relation to the presence of a gel layer. J MembrSci 73:231–46.

Rai P, Majumdar GC, DasGupta S, De S. 2004. Optimizing pectinase usage inpretreatment of mosambi juice for clarification by response surface method-ology. J Food Eng 64:397–403.

Rai P, Majumdar GC, DasGupta S, De S. 2005. Understanding ultrafiltration perfor-mance with Mosambi juice in an unstirred batch cell. J Food Proc Eng 28:166–80.

Stryer L. 1998. Biochemistry. New York, U.S.: Freeman.Thakur BR, Singh RK, Handa AK. 1997. Chemistry and uses of pectin. Crit Rev

Food Sci Nutr 37:47–73Todisco S, Pena L, Drioli E, Tallarico P. 1996. Analysis of the fouling mechanism

in microfiltration of orange juice. J Food Process Pres 20:453–66.Tsoga A, Richardson RK, Morris ER. 2004. Role of cosolutes in gelation of high-

methoxy pectin. Part 1. Comparison of sugars and polyols. Food Hydrocolloid18:907–19.

Vaillant F, Millan P, O’Brien G, Dornier M, Decloux M, Reynes M. 1999. Cross flowmicrofiltration of passion fruit juice after partial enzymatic liquefaction. JFood Eng 42:215–24.