Chem. Prod. Process Model. 2015; 10(3): 203–210

Ricardo Durán*, Aída L. Villa, Rogers Ribeiro and José A. Rabi

Pectin Extraction from Mango Peels in Batch Reactor: Dynamic One-Dimensional Modeling and Lattice Boltzmann Simulation DOI 10.1515/cppm-2015-0014

Abstract: A dynamic one-dimensional model accounting for pectin generation from protopectin in the solid matrix of mango peels and its degradation in both interstitial and extra-particle (i.e. reactor-filling) acid solution is proposed. The model assumes that pectin diffusive transport occurs in the interstitial fluid while eventual diffusive, thermal and pH influences in the solid phase were lumped into the kinetic coefficient of protopectin-pectin conversion. Firstorder kinetic was assumed to pectin degradation. Differential equations were numerically solved by adapting an in-house simulator of bioprocesses via the lattice Boltzmann method (LBM). As part of the LBM method, particle distribution functions were assigned to the pectin concentration in interstitial and reactor-filling fluid as well as assigned to the protopectin concentration in the solid phase. Equilibrium distribution functions were adopted by considering stationary solid phase, diffusive transport in interstitial fluid, and no spatial dependence in the reactorfilling fluid. Model parameters were assessed by comparing numerically simulated extraction yield curves with existing experimental data of pectin extraction using a batch reactor under either conventional or microwave heating. While the expected behavior of extraction yield curves was fairly reproduced in LBM simulations, discrepancies with respect to the experimental data can be assigned to assumptions in this preliminary model (e.g. first-order degradation kinetic and/or lumping effects into the protopectin-to-pectin kinetic). Prospective influence of slab thickness on extraction yields was also examined in LBM simulations.

*Corresponding author: Ricardo Durán, Agroindustrial Optimization Research Group, Universidad Popular del Cesar, bloque F, Lab 201, sede Sabanas, Valledupar, Colombia, E-mail: [email protected] Aída L. Villa, Chemical Engineering Department, Environmental Catalysis Research Group, Universidad de Antioquia, Cra. 53 No. 61–30, Medellín, Colombia, E-mail: [email protected] Rogers Ribeiro, José A. Rabi, Faculty of Animal Science and Food Engineering, University of São Paulo, Av. Duque de Caxias Norte 225, Pirassununga, SP, 13635–900, Brazil, E-mail: [email protected], [email protected]

Keywords: reactor design, physics-based modeling, bioprocess simulation, lattice Boltzmann method

1 Introduction Pectin is a long-chain molecule that can be de-esterified and degraded (i.e., reduced in chain length) by heat, acidity, alkalinity, or enzyme action [1]. Chain length is a vital characteristic for pectin quality (namely, its gelling ability) as longer chains render higher quality. As extracted from natural products (e.g. orange or mango peels), pectin quality and concentration depend on not only fresh-peel pectin quality and farming agro-ecological conditions [2] but also prevailing conditions within the reactor [3]. Most pectin in fresh mango peels is fairly insoluble in cold water so that hot dilute acid is used to make it extractable. The amount of recovered pectin increases as both temperature and acidity increase but these agents also lead to higher rates of pectin degradation, thus affecting extract quality. Therefore, optimal values of process parameters such as extraction time, temperature, acid type and concentration must be pursued for favorable compromise between pectin yield and degradation [4]. Pectin can be degraded (and form reducing groups) by acid hydrolysis as well as by β-elimination. With respect to pectin extraction from mango peels using hydrochloric acid, hydrolysis of methyl ester groups is lower for conventional heating while β-elimination is lower for microwave heating [5]. Microwave assisted extraction (MAE) is an interesting alternative to conventional extraction methods and it has advantages as, for instance, shorter extraction time, less solvent demand, higher extraction rate, and better products with lower costs [6]. Based on comprehensive models, numerical simulation may help one achieve optimal operation [7]. By relying on known extraction and degradation kinetics, dynamic models of pectin extraction have invoked process parameters such as pectin intraparticle diffusivity and external mass transfer coefficient, besides the necessary kinetic coefficients [1, 8, 9]. The present works proposes a phenomenological model of pectin extraction from mango peels where

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protopectin-to-pectin conversion takes place in the solid phase whereas pectin degradation occurs in both interstitial and reactor-filling acid solution. Parameters were grouped into temperature-dependent and temperature-independent ones, with the former entailing kinetic coefficients of protopectin-pectin conversion and pectin degradation, pectin diffusivity in interstitial fluid and pectin transfer coefficient from peels to reactor-filling fluid. In order to numerically solve the model equations, a computational simulator based on the lattice Boltzmann method (LBM) was adapted from [10]. As part of ongoing research on computational modeling of food and bioprocesses via LBM, the original in-house LBM simulator was adapted to batch-reactor extraction rather than in continuous-flow equipment as in Ref. [11] for extraction of natural compounds or Ref. [12] for biospecific affinity chromatography. Launched in Ref. [13] and inspired by the kinetic gas theory, LBM has become an encouraging numerical technique to simulate food and bioprocesses [14] while rendering relatively simpler computer codes [15]. There is wide research niche to be explored as far as LBM simulation of food and bioprocesses is concerned. Bearing in mind the 17th World Congress of Food Science & Technology (IUFoST 2014), only three communications (from approximately one thousand) dealt with LBM [16–18]. LBM simulations were performed in view of ongoing experimental research on pectin extraction from chopped mango (Mangifera indica L.) peels in batch reactor with hot acid solution. Mango peels have been deemed a promising, but not yet exploited, alternative pectin source [19], achieving pectin yields between 6.4% and 21% in experimental extractions [3, 20]. Extraction yields simulated via LBM were compared with experimental data for distinct conditions bearing in mind the influence of extraction temperature and slab thickness.

2 Theory 2.1 Phenomenological model for pectin extraction from mango peels Adapted from Ref. [10] and at its current development stage, the LBM simulator can deal with dynamic (i.e. time-dependent) one-dimensional bioprocesses invoking as many chemical species as necessary. Species concentrations depend on time t and a coordinate z suitably identified in view of the

process to be simulated. In the present model of pectin extraction from mango peels in batch reactor, chemical species refer to pectin as well as protopectin. Chopped mango peels were modeled as small thin slabs whose dimensions (in mm) are 10.0
10.0
2.0. It is assumed that a large number Nslabs of such slabs uniformly fill up the reactor volume while randomly oriented therein. For spatial dependence purposes, coordinate axis z was aligned to the smallest dimension of the square slabs so that slab surfaces for 1-D mass transfer were located at z ¼ 0 and z ¼ L ¼ 2.0 mm. In order to adapt the LBM simulator towards pectin extraction from mango peels, the following assumptions were introduced: – The solid phase comprises a single layer (i.e. mango peel) with uniform porosity ε; – Pectin transport is diffusion-dominant with mass diffusivity Dpf; due to the small thickness of the slabs (L ¼ 2.0 mm), mass diffusion takes place in the interstitial fluid (acid solution) exclusively along coordinate z; – Eventual diffusive, thermal and pH effects in the solid phase are lumped into the kinetic coefficient of protopectin-pectin conversion process (which is introduced ahead); and – Due to continuous stirring, pectin concentration in the reactor-filling fluid (acid solution) is allegedly uniform while still depending on time t (i.e. zero-order spatial dependence). Protopectin is only found in the solid phase, where it remains immobile. Let cpp ¼ cpp(z,t) be protopectin concentration at time t and position z and let r_pp be the instantaneous rate at which it is converted into pectin. If kp is the kinetic coefficient of such conversion, protopectin concentration is ruled by the following governing differential equation: @cpp ¼ r_pp ; @t

r_pp ¼ kp cpp

ð1Þ

Coefficient kp is assumed to lump not only temperature and pH effects but also those related to pectin diffusion in the solid phase. Similar to further rates defined in this model, r_pp > 0 by definition so that a minus sign is introduced in eq. (1) in line with the fact that protopectin concentration decreases along the process. Pectin concentration at time t and position z in interstitial fluid (which percolates mango peels) is indicated as cpf ¼ cpf(z,t). By accounting for pectin generation from protopectin at rate r_pp (i.e. source term) while assuming that pectin degrades in the fluid

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R. Durán et al.: Pectin Extraction from Mango Peels in Batch Reactor

phase at rate r_pd (i.e. sink term), the following governing equation is put forward: "

@cpf @ 2 cpf ¼ "Dpf þ ð1  "Þr_pp  " r_pd @t @z 2 @cpf @ 2 cpf 1  " ¼ Dpf , þ r_pp  r_pd " @t @z 2

and

cpf ðz; 0Þ ¼ cpf;0 ;

ð3Þ

for 0  z  L ð4Þ

Due to constant stirring, mango slabs are assumed to be evenly distributed throughout the reactor while zeroorder spatial dependence is presumed to the pectin concentration in the reactor-filling fluid, cpr ¼ cpr(t). Hence, eq. (1) lacks partial derivatives with respect to the spatial coordinate z. Boundary conditions are solely required to eq. (2) and they are imposed at slab surfaces (at z ¼ 0 and z ¼ L) as: 
 @cpf  km cpf ð0; tÞ  cpr ðtÞ ¼ j0 ðtÞ ¼ þDpf and @z z¼0 
 @cpf  km cpf ðL; tÞ  cpr ðtÞ ¼ jL ðtÞ ¼ Dpf @z z¼L ð5Þ with the same mass transfer coefficient km being assigned to bottom and top surfaces. In eq. (5), mass fluxes j0 and jL are both positive, i.e. towards the reactor-filling fluid. Pectin concentration in reactor-filling fluid cpr ¼ cpr(t) is assessed by adding up pectin fluxes from all Nslab slabs while considering its degradation, assumed to follow a linear kinetic as in eq. (3). If Vrf is the reactor volume occupied by the fluid phase, the following governing equation holds: Vrf

dcpr ¼ Nslab Aslab ½j0 ðtÞ þ jL ðtÞ dt dcpr ¼ r_pr ;  Vrf kd cpr , dt Nslab Aslab ½j0 ðtÞ þ jL ðtÞ  kd cpr r_pr ¼ Vrf

ð7Þ

ð2Þ

At start-up, protopectin concentration cpp,0 and pectin concentration cpf,0 are respectively assigned to solid and fluid phases in the slab. Those two initial conditions are expressed as: cpp ðz; 0Þ ¼ cpp;0

slab. As initial condition to eq. (6), null pectin concentration is imposed: cpr ð0Þ ¼ 0

where Dpf is pectin diffusivity in interstitial fluid and ε is mango peel porosity. Similar to eq. (1), pectin degradation rate r_pd is assumed to comply with the following kinetics: r_pd ¼ kd cpf

205

ð6Þ

where Aslab is surface area for pectin transfer, which is the same at bottom (z ¼ 0) and top (z ¼ L) sides of the

2.2 Rationale of lattice Boltzmann method (LBM) Envisaged as a spin-off of lattice gas cellular automata [13], LBM is more recent than long-established methods such as finite differences (FDM), finite elements (FEM) or finite volumes (FVM) [21]. From the chronological viewpoint, one may then consider LBM as a cutting-edge technique, which has proved to suitably simulate food and bioprocesses [14]. Aiming at developing in-house simulation software as engineering tools and for computational modeling education, research on LBM simulation of compounds extraction and bioseparation processes has been pursued [10]. While there is no doubt about the ability of FDM, FEM or FVM to perform simulations as those in this work, by means of LBM one may simulate multiphase phenomena, moving boundaries, and fluid flow without directly solving Navier-Stokes equations [22]. For those who have programmed their own CFD (computational fluid dynamics) simulators, aforesaid features sound quite attractive as they lead to relatively simpler codes [15] by dismissing, for instance, computer-consuming Galerkin formulations [23] or pressure-velocity coupling [24]. Furthermore, as claimed in [15], LBM may prevent false (i.e. numerical) diffusion resulting from traditional discretization schemes of convective terms (e.g. upwind scheme) [24]. LBM is bottom-up approach whose fundamentals can be found in Ref. [22, 25]. LBM treats any medium as comprised by fictitious particles in a fictitious lattice structure. According to their velocities and during discrete time steps, those particles travel between adjacent sites via lattice links (streaming step). As they arrive at sites, particles mutually collide so that their velocities are rearranged for subsequent streaming (collision step). By imposing conservation principles to such repetitive particle dynamics, one may then simulate macroscopic medium behavior through a sequence of streaming-collision steps. Pictorially speaking, one may see aforesaid particle dynamics as similar to bumper cars (dodgems) typically found in amusement parks. People in those cars can be interpreted as the observable property (e.g. species concentration, temperature or momentum) transported within the medium. However, instead of being able to freely displace in any direction, LBM restricts bumper cars to stream

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R. Durán et al.: Pectin Extraction from Mango Peels in Batch Reactor

in a discrete set of directions defined by the fictitious lattice structure. In other words, lattice links behave as rails onto which cars are forced to move. In LBM the basic mathematical entity is the particle distribution function f ¼ f ð~ r; ~ c; tÞ. Ruled by Boltzmann transport, function f provides at time t the probability of finding particles about position ~ r with velocities between ~ c and ~ c þ d~ c. By taking appropriate moments of function f one may assess the observable properties of interest [14, 15, 22]. In the absence of external forces and under the so-called BGK approach (after Bhatnagar, Gross and Krook), Boltzmann transport equation reads: @f 1 þ~ c~ Ñf ¼ ðf eq  f Þ @t τ

ð8Þ

ð9Þ

being ck ¼ Δzk/Δt with Δz1 ¼ þ Δz (forward streaming) and Δz2 ¼ −Δz (backward streaming) whereas Δt is the advancing time step. Space-time discretization of eq. (9) leads to an algebraic equation whose numerical evolution is performed in two steps [14, 15, 22]. During collision (time evolution), particle distribution functions fk are updated from instant t to t þ Δt for each lattice link k and at each lattice site according to: fk ðz; t þ ΔtÞ ¼ ½1  ω fk ðz; tÞ þ ω fkeq ðz; tÞ

ð10Þ

where ω ¼ Δt/τ is known as relaxation parameter. During streaming (spatial evolution), collision results are propagated to from site to site simply as: fk ðz þ Δzk ; t þ ΔtÞ ¼ fk ðz; t þ ΔtÞ

In order to simulate pectin extraction from mango peels as modeled by eqs (1)–(7), a LBM simulator was adapted from [10]. Particle distribution functions fpp,k(z,t), fpf,k(z,t), and fpr,k(t) were assigned to protopectin concentration in solid phase cpp(z,t), pectin concentration in interstitial fluid phase cpf(z,t), and pectin concentration in reactor-filling fluid phase cpr(t), respectively. As D1Q2 lattice (i.e. k ¼ 1, 2) was used, protopectin and pectin concentrations were retrieved at any time t and position z (if pertinent) according to: cpp;k ðz; tÞ ¼

X

fpp;k ðz; tÞ ¼ fpp;1 ðz; tÞ þ fpp;2 ðz; tÞ

k

where τ and feq are respectively known as relaxation time and equilibrium distribution function. In LBM, eq. (8) is expressed in view of a fictitious lattice structure, when it becomes referred to as lattice Boltzmann equation (LBE). Sketched elsewhere [14, 15, 22], lattice structures are identified as DnQm with n and m respectively referring to the problem dimensionality (e.g. n ¼ 1 ¼ 1-D) and the speed model ( ¼ number of particle distribution functions to be numerically solved for each observable property). In the present work, D1Q2 lattice is used because the model framework is dynamic 1-D. By writing eq. (8) for a given lattice link k, with k ¼ 1 and k ¼ 2, respectively, referring to forward and backward streaming (i.e. particle displacement) one arrives at 1-D LBE, namely: eq @fk ðz; tÞ @fk ðz; tÞ fk ðz; tÞ  fk ðz; tÞ þ ck ¼ @t @z τ

3 Calculation

ð11Þ

LBM simulations are linked to macroscopic properties via the equilibrium distribution function feq and the relaxation parameter ω. The former sets the transport phenomenon while the later prescribes the corresponding transport coefficient.

cpf;k ðz; tÞ ¼

X

fpf;k ðz; tÞ ¼ fpf;1 ðz; tÞ þ fpf;2 ðz; tÞ

k

cpr;k ðtÞ ¼

X

ð12Þ

fpr;k ðtÞ ¼ fpr;1 ðtÞ þ fpr;2 ðtÞ

k

In line with Ref. [10], the streaming step concerning functions fpp,k(z,t) and fpr,k(z,t) was suppressed from the LBM code as eqs (1) and (6) explicitly lack partial derivatives with respect to coordinate z. In contrast, streaming step for functions fpf,k(z,t) concerning pectin concentration in interstitial fluid was implemented according to eq. (11). Bearing in mind source and sink terms in eqs (1), (2), (3) and (6), the collision steps were implemented as follows: eq fpp;k ðz; tþΔtÞ ¼ ½1  ωpp fpp;k ðz; tÞþωpp fpp;k ðz; tÞ wk Δt r_ pp eq fpf;k ðz; t þ ΔtÞ ¼ ½1  ωpf fpf;k ðz; tÞ þ ωpf fpf;k ðz; tÞ   1" þ wk Δt r_ pp  r_ pd " eq ðtÞ þ wk Δt r_ pr fpr;k ðt þ ΔtÞ ¼ ½1  ωpr fpr;k ðtÞ þ ωpr fpr;k

ð13Þ where w1 ¼ w2 ¼ ½ are the weighting factors wk for D1Q2 lattice [14, 15]. By recalling that convective mass transfer is absent in the model framework herein proposed, equilibrium distribution functions were set as [15]: eq fpp;k ðz; tÞ ¼ wk cpp ðz; tÞ; eq fpf;k ðz; tÞ ¼ wk cpf ðz; tÞ; eq fpr;k ðtÞ

ð14Þ

¼ wk cpr ðtÞ

As eq. (2) entails pectin diffusive transport in interstitial fluid, relaxation parameter ωpf and diffusivity Dpf were connected to each other as [10]:

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R. Durán et al.: Pectin Extraction from Mango Peels in Batch Reactor

  ðΔzÞ2 1 1 Dpf ¼  ωpf 2 Δt

,

Dpf Δt 1 1 ¼ þ ωpf ðΔzÞ2 2

ð15Þ

Conversely, because eqs (1) and (6) explicitly lack diffusive terms, the remaining relaxation parameters simplified to ωpp ¼ ωpr ¼ 2 (i.e. as if for null diffusivity). Initial conditions were imposed to particle distribution functions as [15]: fpp;k ðz; 0Þ ¼ wk cpp ðz; 0Þ; fpf;k ðz; 0Þ ¼ wk cpf ðz; 0Þ;

207

Table 1: Parameters concerning pectin extraction from mango peels [5]. Process parameter

Value

Slab thickness of mango peel (axial length) Slab cross-sectional area of mango peel Mango peel porosity (allegedly uniform) Total mass of mango peels (dry basis) Total number of mango peel slabs Volume of reactor-filling acid solution

L ¼ . m A ¼ .
. m ε ¼ . mtotal ¼ . kg Nslabs ¼  Vrf ¼ − m

ð16Þ

fpr;k ð0Þ ¼ wk cpr ð0Þ where initial protopectin and pectin concentrations are those in eqs (4) and (7). In terms of boundary values, fpf,2 (0,t) was obtained at z ¼ 0 via streaming from the adjacent lattice site whereas a similar rationale applied to fpf,1(L,t) at z ¼ L [10, 15]. As defined in eq. (5), boundary conditions were imposed by relying on flux conservation and finite-differences discretization [15] so that distribution functions fpf,1(0,t) and fpf,2(L,t) resulted as: fpf;1 ð0; tÞ ¼ fpf;1 ðΔz; tÞ þ fpf;2 ðΔz; tÞ  fpf;2 ð0; tÞ 

j0 ðtÞ Δz Dpf

fpf;2 ðL; tÞ ¼ fpf;1 ðL  Δz; tÞ þ fpf;2 ðL  Δz; tÞ  fpf;2 ðL; tÞ jL ðtÞ Δz  Dpf ð17Þ with instantaneous mass fluxes j0(t) and jL(t) being assessed in line with eq. (5).

4 Results and discussion LBM simulations were performed in view of an existing experimental work on pectin extraction from mango

peels [5], whose process parameters are shown in Table 1. In those experiments, mango peels treated with acidulated water (10:1 volume ratio) were first heated at 80°C either for 10 min by conventional heating or for 5 min by microwave heating (500 W) so as to inactivate enzymes. Subsequent heating followed at controlled temperature for 55 min by conventional heating or 20 min by microwave heating (500 W) in order to extract pectin. Finally, pectin was precipitated with 96% alcohol in a 1:1 ratio, filtered, and then washed with ethanol [5]. While experimental data solely refer to slabs 2.0 mm thick, LBM simulations were carried out for three distinct slab thickness (1.0 mm, 1.5 mm, and 2.0 mm) as an attempt to examine the prospective influence of peel thickness on the extraction performance. Figure 1 presents the time evolution of extraction yields as simulated for pectin extraction under conventional heating and pH ¼ 2.0 respectively at (a) 90°C and (b) 85°C. Figure 2 shows analogous LBM simulations for pectin extraction under microwave heating at pH ¼ 1.5 and 85°C. Experimental data are also shown in Figures 1(a)-(b) and 2 for comparison purposes. By recalling that experiments exclusively refer to slabs 2.0 mm thick, correlation

Figure 1: Pectin extraction yield from mango peels under conventional heating and pH ¼ 2.0: experimental data (slab thickness ¼ 2 mm) and LBM simulations (slab thickness ¼ 2, 1.5 or 1 mm) for extractions at (a) 90°C and (b) 85°C.

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R. Durán et al.: Pectin Extraction from Mango Peels in Batch Reactor

experimentally studied [5]. Parameters suggested in Ref. [8] for similar studies on pectin extraction from apple are also shown in Table 3 for comparison purposes. Similarly reported in Ref. [1], slightly higher kinetic coefficients kp and kd were obtained at 90°C as far as conventional heating is concerned. Moreover, the coefficients kp obtained for conventional heating are consistent with values in Ref. [8] and the coefficients kd have the same magnitude as in Ref. [9] for pectin extraction from apples at 80°C. The mass transfer coefficient km is lower in the present study possibly to compensate the absence of pectin degradation in Ref. [9]. Differences concerning the order of magnitude obtained for pectin diffusivity Dpf in the interstitial fluid might be due to the assumption that pectin diffusion in the solid phase was lumped into the protopectin-to-pectin kinetic coefficient kp. One may claim that such lumping assumption might have caused discrepancies between experimental data and numerical simulations as evidenced in Figure 1(b). Moreover, those discrepancies can be equally assigned to (relatively simpler) first-order degradation kinetic as presently assumed. Not only differences between experimental and simulated yields but also best-fitting model parameters in Table 3 might be different if additional phenomena become accounted for in upcoming extended (and improved) versions of the model framework and of the corresponding LBM simulator. The influence of the heating mechanism (at the same temperature) can be inferred by comparing curves in Figure 1(b) with their counterparts in Figure 2. Microwave extraction led to higher pectin yields with lower extraction times, i.e. microwave extraction is a faster process [6]. Degradation coefficient was basically the same for both heating mechanisms (kd ¼ 5.1
10−4 s−1), suggesting it is more linked to temperature (kept at 85°C in both scenarios) than pH (which varied from 2.0 to 1.5). Finally, one may analyze the prospective influence of slab thickness with the help of Figures 1(a)-(b) and 2. As previously mentioned, it was assumed that parameters kp, kd, Dpf and km do not depend on slab thickness. LBM

Figure 2: Pectin extraction yield from mango peels under microwave heating at 85°C and pH ¼ 2.0: experimental data (slab thickness ¼ 2 mm) and LBM simulations (slab thickness ¼ 2, 1.5 or 1 mm).

coefficients were assessed by solely considering LBM simulations carried out for slabs with the aforesaid thickness. Accordingly, a statistical analysis was performed and correlation coefficients are presented in Table 2. Table 2: Correlation coefficients between experimental data (i.e. extraction yields) [5] and corresponding LBM simulations performed in this work concerning slabs 2.0 mm thick. Extraction scenario (slab thickness ¼  mm) Correlation coefficient

Conventional heating at °C, pH ¼ .

Conventional heating at °C, pH ¼ .

Microwave heating at °C, pH ¼ .

.

.

.

Together with pectin diffusivity Dpf in interstitial fluid and pectin transfer coefficient km, kinetic coefficients kp and kd were assessed by fine-tuning LBM simulations against each set of experimental data. Once those parameters were fine-tuned, they were kept the same in trial LBM simulations for slabs with lower thickness (namely, 1.5 mm and 1.0 mm). Table 3 shows the fine-tuned values of parameters kp, kd, Dpf and km for each pectin extraction scenario

Table 3: Fine-tuned pectin extraction parameters obtained by comparing LBM-simulated extraction yield profiles with experimental data [5]. Pectin extraction parameter

Conventional heating at: °C, pH ¼ .

kp (s−) km (m · s−) kd (s−) Dpf (m · s−)

. . . .







− − − −

°C, pH ¼ . . . . .







− − − −

Microwave heating at:

Minkov et al.,  °C,

°C, pH ¼ .

pH ¼ .

− − − −

.
− .
− Not considered .
−

. . . .







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R. Durán et al.: Pectin Extraction from Mango Peels in Batch Reactor

simulations suggest that thinner slabs lead to shorter extraction times but lower extraction yields, as one would expect.

209

weighting factors for lattice Boltzmann method (dimensionless) axial position within the peel slab (m)

w z

Greek symbols

5 Conclusion Even when mathematically described by means of onedimensional dynamic models, pectin extraction invokes coupled differential equations so that lattice Boltzmann method (LBM) emerges as an interesting numerical solution technique. In this work, an existing in-house LBM code for bioprocess [10] was adapted so as to simulate pectin extraction from mango peels in acid solution under either conventional or microwave heating. While the expected behavior of extract yield curves was reproduced in LBM simulations, discrepancies between experimental data and numerical simulations can be assigned to assumptions in the preliminary version of the model such as first-order degradation kinetic and lumping thermal and pH influences effects into the protopectin-topectin kinetic coefficient. LBM simulations suggested that pectin degradation rates mostly depend on temperature (for the same pH) whereas the heating mechanism influences the extraction kinetics.

c cpf cpp cpr Dpf f j0 jL kd km kp L Nslabs r_pd r_pp r_pr t Vrf

mango peel porosity (dimensionless) relaxation time (s) relaxation parameter (dimensionless)

Subscripts and superscripts 0 1 2 eq k pf pp pr

referring referring referring referring referring referring referring referring

to to to to to to to to

initial condition forward streaming backward streaming particle distribution function at equilibrium streaming directions pectin in interstitial fluid phase protopectin in solid phase pectin in reactor-filling fluid phase

Funding: Authors acknowledge the financial support from Colciencias, SENA and Universidad de Antioquia (UdeA) through project 1115-479-22043 and from UdeA through “Estrategia de Sostenibilidad 2013-2014”. R.D. acknowledges to Universidad Popular del Cesar, Colciencias and Gobernación del Cesar, his doctoral fellowship.

References

Nomenclature Aslab

ε τ ω

surface area of peel slab for pectin transfer to reactor-filling fluid (m2) particle velocities for lattice Boltzmann method (m s−1) pectin concentration in interstitial fluid (kg m−3) protopectin concentration in solid phase (kg m−3) pectin concentration in reactor-filling fluid (kg m−3) pectin diffusivity in interstitial fluid (m2 · s−1) particle distribution function (units depend on the model framework) mass (pectin) flux from peel bottom surface to reactorfilling fluid (kg m−2 s−1) mass (pectin) flux from peel top surface to reactor-filling fluid (kg m−2 s−1) kinetic coefficient of pectin degradation (s−1) mass (pectin) transfer coefficient from interstitial to reactor-filling fluid (m · s−1) kinetic coefficient of protopectin-to-pectin conversion (s−1) mango peel thickness (m) number of peel slabs uniformly filling up the reactor volume (dimensionless) instantaneous rate of pectin degradation (kg m–3 s–1) instantaneous rate of protopectin-to-pectin conversion (kg m–3 s–1) instantaneous rate of pectin (net) accumulation in reactorfilling fluid (kg m–3 s–1) time (s) reactor volume occupied by the fluid phase (m3)

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