Mass transfer modeling in osmotic dehydration: Equilibrium characteristics and process dynamics under variable solution concentration and convective boundary

food and bioproducts processing 9 7 ( 2 0 1 6 ) 88–99

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Mass transfer modeling in osmotic dehydration: Equilibrium characteristics and process dynamics under variable solution concentration and convective boundary H. Pacheco-Angulo a , E. Herman-Lara b , M.A. García-Alvarado a , I.I. Ruiz-López c,∗ a

Unidad de Investigación y Desarrollo en Alimentos, Departamento de Ingeniería Química y Bioquímica, Instituto Tecnológico de Veracruz, Av. M.A. de Quevedo S/N. Col. Formando Hogar, C.P. 91860 Veracruz, Veracruz, México b Coordinación de Posgrado e Investigación, Departamento de Ingeniería Química y Bioquímica, Instituto Tecnológico de Tuxtepec, Av. Dr. Víctor Bravo Ahuja S/N. Col. 5 de Mayo, C.P. 68350, Tuxtepec, Oax. , México c Facultad de Ingeniería Química, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y 18 Sur. Ciudad Universitaria, Puebla, Puebla, México

a r t i c l e

i n f o

a b s t r a c t

Article history:

The aim of this study was to model both the dynamic and equilibrium mass transfer periods

Received 22 April 2015

for water, osmotic solute and food solids interchange between product and solution during

Received in revised form 10

an osmotic dehydration (OD) process. The OD model is able to represent situations where

November 2015

concentration of osmotic media changes during the process or where interfacial resistance

Accepted 25 November 2015

to mass transfer cannot be neglected. Water and solute are considered to move within the

Available online 8 December 2015

product by a diffusion mechanism based on Fick’s second law, while external convective mass transfer is considered in the fluid. The state-space form of the model is analyti-

Keywords:

cally solved for one-dimensional mass transfer in products with flat slab, infinite cylinders

Interfacial resistance

and sphere geometries. The developed theory was applied to the analysis of equilibrium

Mass Biot number

and OD dehydration curves of carrot slices obtained at 40 ◦ C in sodium chloride solutions

Mass transfer

with and without stirring and different ratios between solution volume and product mass.

Osmotic dehydration

Water and NaCl diffusivities were identified in the narrow ranges of 6.0–7.6 × 10−10 m2 /s

Variable solution concentration

and 3.5–4.1 × 10−10 m2 /s, respectively, demonstrating the applicability of the proposed model

Equilibrium distribution

under a wide range of operating conditions. © 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

1.

Introduction

Osmotic dehydration is a solid–liquid contact operation involving the immersion of food products, especially fruits and vegetables, in hypertonic solutions such as brines or syrups. When the food is immersed in the solution, an osmotic pressure gradient is developed between the involved phases originating a dynamic mass transfer period in which water

∗

is removed from food toward the liquid media with a simultaneous solute gain by the product (Khan et al., 2008; Goula and Lazarides, 2012; Herman-Lara et al., 2013; Souraki et al., 2013, 2014). If processing is performed for a long enough time, then both water loss and solute gain reach a stationary state, where the driving potential for mass transfer between food and solution becomes zero (Sablani and Rahman, 2003; Herman-Lara et al., 2013). This operation involves several mass

Corresponding author. Tel.: +52 222 2295500x7250. E-mail addresses: [email protected], [email protected] (I.I. Ruiz-López). http://dx.doi.org/10.1016/j.fbp.2015.11.002 0960-3085/© 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

food and bioproducts processing 9 7 ( 2 0 1 6 ) 88–99

Nomenclature A A aw Bi C d D F Fo g Gr kc Ke L, Lc m m M n N n+ , n− R Re Sc Sh t T u U v V X X* X Y Y* x1 , x2 W

surface area available for mass transfer (m2 ) state-space matrix water activity mass Biot number (dimensionless) concentration (g/L) maximum inner diameter of flask (m) diffusion coefficient (m2 /s) Faraday constant (96485.34 C/mol) Fourier number for mass transfer (dimensionless) gravitational constant (9.81 m2 /s) mass Grashof number (dimensionless) convective mass transfer coefficient (m/s) equilibrium partition coefficient (kg product/kg osmotic media) characteristic lengths for diffusion and convection, respectively (m) solute molality (mol solute/kg solvent) mass in product (kg) mass in osmotic media (kg) number of nodes number of components valences of cation and anion, respectively (dimensionless) gas constant (8.314 J/mol K) Reynolds number (dimensionless) Schmidt number (dimensionless) Sherwood number (dimensionless) time (s) temperature (K) axial or radial coordinate (m) food solids-to-(osmotic solute + water) mass ratio in product (kg/kg) orbital velocity (1/s) volume (m3 ) mass fraction of a given component in product (kg/kg product) mass fraction in product free of food solids (kg/kg osmotic solute + water) state-space vector mass fraction of a given component in solution (kg/kg solution) mass fraction in solution free of food solids (kg osmotic solute + water) coded temperature and time, respectively food solids-to-(osmotic solute + water) mass ratio in osmotic media (kg/kg)

Greek letters ˛ parameter defining product geometry mass ratio between osmotic media and product ı volume fraction occupied by osmotic media (m3 ε solution/m3 solution + product) thermodynamic factor (dimensionless) ϕ  dimensionless concentration in solution mean ionic activity coefficient of solute (dimen± sionless) ◦+ , ◦− limiting (zero-concentration) ionic conductances of cation and anion, respectively (S m2 /mol)

 

 ,

89

dynamic viscosity (Pa s) dimensionless group related with product-toosmotic media mass ratio density (kg/m3 ) chemical potential (J/mol) dimensionless coordinate dimensionless concentration in product: local and average, respectively

Subscripts 0 at beginning of the process at infinite dilution ◦ denotes food solids f at equilibrium e i at the product–solution interface j denotes either water or osmotic solute denotes node numbering k denotes osmotic media o p denotes product denotes osmotic solute s t at time t denotes water w

transfer mechanisms; however, diffusion of water and solute within product are usually considered the controlling factors, usually described by Fick’s second law (Kaymak-Ertekin and ˘ 2000; Rastogi and Raghavarao, 2004; Goula and Sultanoglu, Lazarides, 2012; Souraki et al., 2012; da Silva et al., 2013, 2014; Rodríguez et al., 2013). Depending on modeling assumptions, non-steady-state diffusion equation may be solved under analytical or ˘ numerical techniques (Kaymak-Ertekin and Sultanoglu, 2000; Porciuncula et al., 2013; Rodríguez et al., 2013); however, the use of analytical models remains widespread due to several practical advantages, including easier implementation and lower computational effort. Analytical solutions for OD process in well-stirred systems (i.e., those with negligible external resistance to mass transfer) can be classified in two main groups based on the available amount of working solution (Crank, 1975). Most studies consider an infinite volume of osmotic media (i.e., of constant concentration; Rastogi and Raghavarao, 2004; Rodríguez et al., 2013; Souraki et al., 2012, 2013, 2014), experimentally achieved by using mass ratios between solution and product of 10 and above (Herman-Lara et al., 2013). The use of lower ratios between osmotic media and product may result in an appreciable decrease of solute concentration during product impregnation, affecting water loss and solute gain rates as well as final dehydration and impregnation levels at equilibrium. Therefore, special analytical solutions considering a finite volume of osmotic media (i.e., of variable concentration) should be applied in these cases (Singh et al., 2007; Bellary et al., 2011), or alternatively, resort to a numerical solution allowing the use of additional assumptions such as variable mass ˘ 2000). As of involved phases (Kaymak-Ertekin and Sultanoglu, a diffusion-controlled process is presumed in the aforementioned models (Crank, 1975), these may not be applicable to describe experimental conditions where convective resistance to mass transfer may occur (for example, processes conducted without or with mild stirring), which may be required in products with fragile structures.

90

food and bioproducts processing 9 7 ( 2 0 1 6 ) 88–99

Analytical solutions considering both diffusion and convection at product surface have also been developed and are expressed in terms of a mass Biot number (Crank, 1975), whose value may be used to identify the controlling mechanism (da Silva et al., 2013, 2014). A well-known fact in mass transfer operations is that any solute will not distribute equally between two or more phases due to different chemical and ˘ physical affinities (Corzo and Bracho, 2004; Togrul and I˙spir, 2008). Consequently, an equilibrium relationship is required to completely characterize a multiphase system. In drying, for example, the equilibrium water contents in product and air are related through the sorption isotherm, and a mean partition constant should be used to obtain the correct mass Biot definition to apply the existing analytical solutions. In order to consider a convective boundary condition in the OD model, some studies have considered that product and solution equilibrate at the same concentration (da Silva et al., 2013, 2014). However, the use of models with convective boundary in OD processes is close to non-existent because: (i) equilibrium data are scarce and should be experimentally obtained, (ii) existing analytical solutions are only applicable for solutions of infinite volume (Crank, 1975) and (iii) physical properties necessary to evaluate convective mass transfer coefficient (such as viscosity and density of solution, or diffusivity of solute in osmotic media) are not readily available in comparison with other wellcharacterized systems such water vapor–air mixtures. The objective of this study was to develop a rigorous model to describe the simultaneous water and solute interchange between product and solution during an OD process, applicable under any ratio between product and osmotic media and any convective mass transfer condition. To fully achieve this purpose the following topics were covered: (i) mathematical description of mass equilibrium during OD, (ii) development of a general model for unsteady state mass transfer of diffusing substances, (iii) evaluation of diffusion coefficients of water and solute and (iv) assessment of dominant mass transfer phenomena under different experimental conditions. The model was validated using experimental OD data of carrot slices in NaCl solutions.

2.

Theory

2.1.

Basic definitions

In order to develop an equilibrium relationship for OD process, it will be considered that product (p) and osmotic (o) media are made up from water (w), osmotic solute (s) and a mixture of solids (f) which are naturally present in the product and may leach into the osmotic solution, such as carbohydrates, fat and proteins. According to this assumption, Xf + Xs + Xw = 1

(1)

Yf + Ys + Yw = 1

(2)

where, X and Y represent the mass fractions of a given component in product and osmotic media, respectively. Let us define the fractions of osmotic solute or water free of leaching food solids in involved phases as (for j = s, w) Xj∗ =

Xj Xs + Xw

and

Yj∗ =

Yj Ys + Yw

(3)

Similarly, the mass ratio between food solids and the sum of osmotic solute and water in product and osmotic media are defined as U=

Xf Xs + Xw

and

W=

Yf Ys + Yw

(4)

Above expressions are related with regular mass fractions either of water or osmotic solute by simply algebra (for j = s, w) Xj =

Yj =

Xj∗

(5)

1+U Yj∗

(6)

1+W

2.2.

Description of mass transfer equilibrium

According to equilibrium thermodynamic theory (Gibbs, 1876), a necessary and sufficient condition for mass transfer equilibrium in OD process is given by,

 

j

 

product

= j

osmotic media

(7)

or, equivalently, in terms of water activity (Barbosa-Cánovas and Vega-Mercado, 2010) (aw )product = (aw )osmotic media

(8)

According to Ross (1975), water activity in a multicomponent system can be developed as the product sequence of water activities of each involved solute in a series of binary mixtures. In this case, N

aw = ˘ awi = aws awf

(9)

i=1

Components such as proteins and carbohydrates, etc., are all lumped in Eq. (9) in the mixture of food solids. Since the effect of a given solute on water activity depression mainly depends upon its molar fraction (Barbosa-Cánovas and Vega-Mercado, 2010), it is clear that high-molecular weight substances originally present in product such as proteins, oligo- and polysaccharides will have a negligible effect on water activity depression and thus in mass transfer equilibrium. On the other hand, the impact of mono and disaccharides such as fructose, glucose and sucrose on OD equilibrium will depend on the original amount of these substances in product and their lixiviated fraction, and the type and concentration of osmotic solute. However, their effect on water activity depression may be also considered negligible under most OD processes performed with brines or syrups, as will be later demonstrated. Therefore, if food solids do not contribute to the water activity depression (awf ≈ 1) and the osmotic solution is not in saturation, then mass transfer equilibrium will be reached when the compositions of the binary mixtures between water and osmotic solute (i.e., excluding food solids) in involved phases satisfy (for j = s, w) ∗ ∗ Xje = Yje = (1 + Ue ) Xje = (1 + We ) Yje

(10)

From Eq. (10) it is clear that mass compositions Xje and Yje are related by (for j = s, w) Yje = Ke Xje

(11)

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food and bioproducts processing 9 7 ( 2 0 1 6 ) 88–99

Ke =

1 + Ue 1 + We

(12)

Two important results are derived from Eqs. (11) and (12) under the given assumptions: (i) Partition coefficient (Ke ) is the same for both water and osmotic solute. (ii) Ke is always greater than unity (since Ue > We ). Additionally, if Ke does not change with the concentration of osmotic solution, then the following statements apply: (i) Equilibrium compositions for water and osmotic solute distribute along a straight-line equation with zero intercept. (ii) Mass fraction of water in osmotic media at equilibrium only depends on its corresponding fraction in product. The same statement holds for solute. (iii) Unsteady-state equations for water and osmotic solute concentrations can be solved independently from each other. The previous statements still apply if it is considered that food solids are not lost during OD process (We = 0 and Ke = 1 + Ue ). By assuming a negligible net mass change between involved phases, the following steady-state mass balance can be written (for j = s, w) mp Xj0 + Mo Yj0 = mp Xje + Mo Yje

(13)

equilibrium relationship relating mass fraction of specie j in osmotic solution with those of the other components in product and may have multiple mathematical representations.

2.4.

In this study, OD model is solved in dimensionless form under the following assumptions: (1) Product has a constant density. (2) Apparent diffusivities of both solute and water in product are constants. (3) Mass transfer occurs in one direction in products with flat slab, infinite circular cylindrical or spherical geometries. (4) Equilibrium relationship Eq. (19) is given by Eq. (11). Thus, Eqs. (16)–(19) can be rewritten as (for j = s, w), ∂

Xj0 − Xje Xje =

mp 1 =− Mo ı

=−

(14)

Xj0 + ıYj0

(15)

1 + ıKe

2.3.

General model for osmotic dehydration process

The following equation system describes the mass transfer of specie j (j = 1, . . ., n) between disperse and continuous phases in an OD process (without chemical reactions):



∂ p Xj



∂t







= ∇ · Djp ∇ p Xj

d εVop o Yj



dt







= kc A o Yji − Yj





−Djp ∇ p Xji = kc o Yji − Yj Yji = fj (Xki )

(16)

for





k = 1, . . ., n

(17) (18) (19)

where, Vop is the combined volume of product and osmotic media, kc is the convective mass transfer coefficient, A is the surface area available for mass transfer, ε is the volume fraction occupied by osmotic media, and the subscript i indicates at the product–solution interface. Eqs. (16) and (17) represent both the concentration change of specie j within a homogeneous and isotropic solid by diffusion and in osmotic solution by convection, respectively, while Eq. (18) express the mass flux between product surface and bulk solution. Eq. (19) is an

j

=

∂Foj dj

∂

ji

∂

∂2

j

∂ 2

+

˛∂ j  ∂

= − (˛ + 1) Bij

dFoj



= −Bij

ji

(20)

 ji

+ j

+ j



(21)



(22)

where, ˛ takes value 0, 1, or 2, for infinite slabs, infinite cylinders or spheres, respectively, while the following dimensionless variables and groups were introduced

Finally, the combination of Eqs. (11) and (13) produce the following key relationships Yj0 − Yje

Dimensionless form of osmotic dehydration model

j

=

Xj − Xje Xj0 − Xje

Bij = Ke 

;

j =

kc o



Djp /L p

;

Yj − Yje Yj0 − Yje =

;

1 mp ; Ke Mo

Foj =

Dj t L2

;

=

u L

AL = (˛ + 1) (1 − ε) Vop

(23)

(24)

Here u and L represent the mass transfer direction (axial or radial) and the characteristic length for diffusion, respectively. It can be observed that solution of dimensionless Eqs. (20)–(22) only depends on parameters ˛, Bij and . Eq. (22) satisfies the following limits: ∂

ji

∂

= −Bij

ji

for  → 0

for  → 0 , Bij → ∞

ji

=0

ji

= −j

for

Bij → ∞

(25) (26) (27)

Eqs. (25)–(27) represent the boundary conditions for (i) convective boundary with infinite surrounding media, (ii) diffusion-controlled process with infinite surrounding media and (iii) diffusion-controlled process with finite surrounding media, respectively. Thus, Eqs. (20)–(22) share the analytical solutions provided by Crank (1975) in these circumstances. For simplicity but without loss of generality, the subscript j is hereafter dropped from equations.

2.5. State-space representation of OD model and analytical solution Eq. (20) can be expressed as an equivalent system of ordinary differential equations by using a discrete representation of spatial derivatives. If central finite differences are used, then

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food and bioproducts processing 9 7 ( 2 0 1 6 ) 88–99

the following model is obtained for 1 ≤ k ≤ N, where k represents a point in space d k = dFo

k+1

−2

k+ 2

k−1

()

− 2

˛ (k − 1) 

+

k+1

k−1

(28)

Points outside of the boundary where k < 1 or k > N, i.e., k+1 for k = N (interface) and k−1 for k = 1 (center), can be estimated by applying convective boundary Eq. (22) and a symmetry condition (only half domain is solved). Thus, k+1

=

k−1

k−1

=

k+1

− 2Bi (

+  )

k=N

for

(29)

same advantages and limitations of the analytical ones provided by Crank (1975) (such as constant partition coefficient, constant product density, etc.), with the added benefit that it can move between limiting cases. However, product shrinkage or variable diffusivity could be also considered in the current model by updating matrix A during time.

3.

Methods

3.1.

Osmotic dehydration experiments

Eq. (31) is an homogeneous linear time-invariant system of ordinary differential equations, which accepts the well-known solution (Ogata, 2010)

The developed theory was applied to the analysis of OD data sets using carrot as the food model system. All experiments were conducted in NaCl solutions at a temperature of 40 ◦ C. Fresh carrots, approximately 10 cm-long and 2.7 cm-diameter, were locally purchased (Puebla, Pue., México) and processed the same day. Carrots were washed, dried with a cloth and then mechanically cut into circular 0.3 cm-thick slices without removing the peel. On average, 4 slices were obtained from the central part of each root while remaining portions were reserved for the initial water/dry solids analysis. Carrot slices were OD-processed in separated dilution bottles placed in a controlled-temperature water bath (±0.1 ◦ C), which were allowed to equilibrate before introducing samples. Samples were osmodehydrated in separated bottles to eliminate variations in the solution-to-product ratio and avoid the mixing of solution (in free convection experiments) that are introduced by sampling removal when all samples are processed in a single recipient. The following variables were recorded from these experiments at the beginning and during OD process: water fraction in product (Xw ), volumetric NaCl concentration (g/L) in osmotic media (Cs ), product mass (mp ) and solution mass (Mo ). Water fraction in product was determined by oven-drying the samples until constant weight was attained (when mass change was less than 0.001 g over and 8-h period). On the other hand, NaCl concentration in osmotic media was determined with a sodium ion meter (HI 931101, Hanna Instruments, Woonsocket, RI, USA). Corresponding NaCl fractions in osmotic media (Ys ) were calculated from the volumetric solution concentration (Cs ) and its density ( o ). Several experiments were completed in order to fulfill different objectives as follows. A brief description of all experiments is given in Table 1.

X (t) = eAFo X (0)

3.2.

k

for k = 1

(30)

Eqs. (21) and (28)–(30) can be combined to form the statespace representation dX = AX dFo

(31)

where the state-vector X ∈ RN+1 is defined as XT =

 1

...

2



N



(32)

and the non-zero elements of state-space matrix A ∈ R(N+1)×(N+1) are a (1, 1) = − a (i, i) = −

2 ; ()2

a (1, 2) =

(33)

⎫ ⎬

2 ()2

a (i, i ± 1) =

2 ()2

for 1 < i < N

⎭

2(i−1)±˛ 2

(34)

2(i−1)()

a (N, N − 1) =

2 ; ()2

a (N, N + 1) =

−[2(N−1)+˛]Bi

a (N, N) = −

[2(N−1)+˛]Bi+2(N−1)

(35)

(N−1)()2

(36)

(N−1)()2

a (N + 1, N) = − (˛ + 1) Bi;

a (N + 1, N + 1) = − (˛ + 1) Bi

(37)

(38)

Mean concentration of water and osmotic solute in product can be estimated by applying a spatial-averaging of local values (Whitaker, 1977)





=



˛

dVp =

dVp / Vp

Vp



1

0

1

 ˛ d

 d/

(39)

0

An explicit solution to Eq. (39) was developed in this study from the trapezoidal rule, giving the expression (˛ + 1)  = 2

˛

0

N−1 

1

+2

[(k − 1) ]

˛

k

+ [(N − 1) ]

˛

N

k=2

(40) The state-space solution in Eq. (38) corresponds to a constant state-matrix A. In this case, the given solution shares the

Estimation of equilibrium relationship

The first set of OD experiments was conducted to characterize the effect of brine concentration on the mass equilibrium relationship between product and osmotic media by identifying parameter Ke . Carrot slices were processed without stirring under several brine concentrations (20, 40, 60, 80, 100, 120, 140, 160, 180 and 200 g/L NaCl solution) with a fixed ratio between solution volume and product mass (Vo /mp ) of 4:1. In these experiments, carrot slices were immersed in brines for a total of 24 h. This time was considered long enough for reaching mass equilibrium in the product within the proposed solutions according to previous similar studies with other plant materials (Herman-Lara et al., 2013). Mass balances for water, osmotic solute and food solids allowed the straightforward estimation of Xse , Ywe , Ue , We and Ke by assuming Xs0 = 0 (product initially has a negligible amount of osmotic solute) and Yf0 = 0 (osmotic solution does not have food solids at the beginning of the OD process).

93

food and bioproducts processing 9 7 ( 2 0 1 6 ) 88–99

Table 1 – Summary of all experiment sets. Experiment set

Experimental conditionsa

Purpose

Stirring (rpm) 1 2 3 4 5 a

Evaluate equilibrium coefficients Estimate diffusion coefficients Appraise external resistance to mass transfer Appraise external resistance to mass transfer Appraise external resistance to mass transfer

0 190 0 0 0

Cs (g/L)

Vo /mp (L/kg)

20–200 100 100 100 150

4:1 20:1 4:1 20:1 20:1

All experiments were conducted at 40 ◦ C.

3.3. Estimation of effective water and solute diffusivities in carrot slices Carrot slices were processed with a single brine concentration (100 g/L NaCl solution) in separated dilution bottles (50 mm of inner diameter) with a fixed Vo /mp ratio of 20:1. Bottles were then placed in a thermal bath with orbital stirring (190 rpm). The Vo /mp ratio was selected to avoid a significant dilution of osmotic solution, while the high stirring speed was used to allow for a negligible external resistance to mass transfer (Herman-Lara et al., 2013). Samples were withdrawn from the solution at predefined immersion times (5, 10, 15, 20, 25, 30, 40, 50, 60, 80, 100, 120, 150, 180, 210 and 250 min), gently blotted dry with a paper towel to remove adhering osmotic solution. Product and osmotic solution were analyzed for their water/total solids content and NaCl as previously described. Experimental OD curves (Xs vs t, Xw vs t) were fitted to OD model formed by Eqs. (38) and (40) in order to estimate Ds and Dw , where carrot samples were considered to be slabs (˛ = 0) due to their dimensions. Parameter  was calculated from experimental variables Ke , Vo , mp0 and Cs0 as well as corresponding physical property o , estimated by solving Eq. (A6). Equilibrium compositions were estimated with Eq. (15). Osmotic solute fraction in product (Xs0 ) and food solids fraction in solution (Yf0 ) were assumed to be negligible in the same way as during the estimation of equilibrium relationship. In this case, two scenarios were considered; OD model was solved with (i) Bis = Biw = 1000 in order to consider a diffusion-controlled process and (ii) Bis and Biw were simultaneously estimated with Ds and Dw .

3.4. Appraisal of external resistance to mass transfer under no stirring conditions Carrot slices were processed without stirring under the following conditions: (i) 100 g/L NaCl solution with a Vo /mp ratio of 20:1, (ii) 100 g/L NaCl solution with a Vo /mp ratio of 4:1 and (iii) 150 g/L NaCl solution with a Vo /mp of 20:1 (experiment sets 3–5 in Table 1). Remaining experimental details are the same as above. Experimental OD curves were fitted to the proposed model in order to estimate Bis and Biw using the values for Ds and Dw determined as detailed in Section 3.3. Moreover, these data were also used to evaluate Ds and Dw , where Bis and Biw were determined in advance with the aid of empirical mass transfer correlations.

3.5. Estimation of mass Biot number from empirical relationships The value of external mass transfer coefficient kc can be calculated from empirical correlations for the Sherwood number (Sh). For natural convection, these expressions are given in

terms of mass Grashof (Gr) and Schmidt (Sc) numbers. However, under forced convection conditions, the mass Grashof (Gr) is replaced by Reynolds (Re) number. These dimensionless groups are given below (Geankoplis, 1993; Büchs et al., 2000):

Gr =

L3c sw g sw 2sw

,

Re =

vd2 sw , sw

Sc =

sw , sw Dsw

Sh =

Lc kc Dsw (41)

The Reynolds number (Re) definition in Eq. (41) is a special form given by Büchs et al. (2000) for rotary shaking systems. The following correlations have been recommended for mass transfer around flat surfaces under natural (Geankoplis, 1993) and forced convection (Basmadjian, 2004), respectively Sh = 0.54(GrSc)1/4

(42)

Sh = 0.66Re1/2 Sc1/3

(43)

Convective mass transfer coefficients were calculated under studied experimental conditions and further used to estimate a theoretical Bi number according to the definition given in Eq. (24) and properties summarized in Appendix A.

3.6.

Data analysis

Nonlinear regression (based on ordinary least squares) was used to estimate Ds , Dw , Bis and Biw according to the described experiments (Table 1). Diffusion coefficients Ds and Dw were individually fitted to experimental solute or water fraction evolution curves from experiment set 2 by assuming a negligible external resistance to mass transfer (Bis and Biw were set to 1000) or a convective boundary (Bis and Biw were also fitted during regression procedure). External resistance to mass transfer in experiments conducted without stirring was appraised by simultaneously fitting Bis to experimental solute and water fraction evolution curves from experiment sets 3–5, where corresponding value for water transport was calculated as Biw = Ds Bis /Dw with Ds and Dw values obtained from experiment set 2. In addition, Ds and Dw were also estimated by fitting experiment sets 3–5 with mass Biot numbers calculated with empirical mass transfer relationships. A summary of all properties used in unsteady-state model solution is presented in Table 2. The fitness quality of the identified model was quantified by the generalized determination coefficient (R2 ) and statistical significance of parameter estimates was evaluated through their 95% confidence intervals (95% CI). Nonlinear regression procedures as well as statistical analysis were performed using Matlab Statistics Toolbox 7.3 (MathWorks Inc., Natick, MA, USA).

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food and bioproducts processing 9 7 ( 2 0 1 6 ) 88–99

Table 2 – Summary of variables used during solution of unsteady-state OD modela . Experiment set

Xw0 (g/g)b

2

0.903

100.0

1058.1

0.094

0.0436

3 4 5

0.904 0.904 0.909

100.3 100.3 150.2

1058.2 1058.2 1089.5

0.094 0.094 0.137

0.2181 0.0436 0.0424

a b c d e

Cs0 (g/L)b

o (g/L)c

Ys0 (g/g)d

e

Bij (j = s, w) Estimated from experimental data or set to 1000 Estimated from experimental data or calculated with Eqs. (41)–(43) and (A1)–(A6)

In all experiment sets: Xs0 = 0, Yf0 = 0, Ke = 1.08, ˛ = 0, N = 50, Xse and Xwe were calculated with Eq. (15). Value determined during experiments. Calculated by solving Eq. (A6). Calculated from Cs0 . Calculated with Eq. (24).

4.

Results and discussion

4.1.

Equilibrium and distribution data

Eq. (15) provides a simple way to estimate final dehydration or impregnation levels in product based on experimental conditions and equilibrium characteristics of product and osmotic solute. This equation reduces to that previously reported by Beristain et al. (1990) if Ke = 1, i.e., by considering that osmotic media and product equilibrate at the same composition. However, in some cases the model proposed by Beristain et al. (1990) has failed to provide a satisfactory prediction of experimental mass equilibrium values (Parjoko et al., 1996). A quick inspection of Eq. (15) reveals that overestimation of Xje will occur for increasing values of the difference Ke − 1, a fact that was previously observed by Parjoko et al. (1996) and is now explained by the lack of a proper equilibrium term in the model reported by Beristain et al. (1990). Equilibrium data for water, NaCl and leached food solids are summarized in Table 3. According to these data, Ke values did not exhibit a significant change with osmotic media concentration (p < 0.05) (Fig. 1). Thus, Ke can be simply estimated as the slope of the Yje vs Xje plot simultaneously considering data for both NaCl and water, as shown in Fig. 2. An excellent agreement was found between experimental and predicted results (R2 = 0.998). Appreciable deviations of this linear behavior have been reported by other authors for NaCl

Fig. 1 – Effect of NaCl fraction in osmotic media on equilibrium partition coefficient for osmodehydrated carrot at 40 ◦ C. Experimental data were calculated with Eq. (12). Ke in continuous line is the slope of the plot Yje vs Xje (for j = s, w).

concentrations of about 0.15 g NaCl/g solution and above (del Valle and Nickerson, 1967; Corzo and Bracho, 2004). It is readily observable that these water and solute data distribute along a straight-line with approximately the same slope in both cases, confirming the fact that a single Ke value is able to describe the mass equilibrium of these species according to the theoretical analysis. In this case, Ke was identified as 1.083 with 95% CI ranging from 1.075 to 1.092, i.e., the distribution constant is significantly higher than unity, which was an anticipated result from the developed theory (Section 2.2). It was considered that OD process reaches equilibrium when Xe∗ = Ye∗ , as stated in Eq. (10). A necessary condition to validate this assumption is that food solids do not contribute to water activity depression. On average, carrots slices lost about the 33.6 ± 13.3% of their initial solids content. Sucrose, glucose and fructose account for about the 53% of the dry matter content of carrot (Soujala, 2000; Nyman et al., 2005), the rest being both soluble and insoluble fiber, proteins and fat. According to Nyman et al. (2005), sucrose, glucose and fructose represent the 3.8%, 1.25% and 1.14% of the fresh weight of carrot, respectively. By assuming that all lixiviated solids are sucrose, glucose and fructose in their initial proportions, estimated differences in water activity calculated with Ross equation and Raoult’s law are less than 0.1% than that obtained by neglecting these components. A comparable result was obtained in product where differences in water activity with and without considering sucrose, glucose and fructose in the calculations were lower than 0.5%. Overall, predicted differences in water activity between both phases at

Fig. 2 – Distribution data for NaCl and water during osmotic dehydration of carrot at 40 ◦ C.

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food and bioproducts processing 9 7 ( 2 0 1 6 ) 88–99

Table 3 – Equilibrium dehydration and impregnation levels of osmotic dehydration of carrot slicesa . Cs0 (g/L) 20 40 60 80 100 120 140 160 180 200 psdb

Ue × 102 (g/g) 9.7 ± 0.8 7.9 ± 0.6 7.5 ± 0.0 6.5 ± 0.9 7.6 ± 0.7 7.8 ± 0.8 7.7 ± 3.1 4.8 ± 1.6 6.8 ± 2.4 9.6 ± 0.8 1.5

We × 103 (g/g) 3.9 ± 1.0 7.6 ± 0.3 8.5 ± 0.7 10.2 ± 1.6 9.2 ± 2.4 8.9 ± 3.0 7.0 ± 3.0 11.6 ± 0.8 7.7 ± 1.6 3.2 ± 0.4 1.8

Xse × 101 (g/g) 0.05 ± 0.01 0.32 ± 0.04 0.52 ± 0.04 0.76 ± 0.05 0.88 ± 0.12 0.98 ± 0.13 1.05 ± 0.09 1.40 ± 0.07 1.41 ± 0.08 1.34 ± 0.11 0.08

Yse × 101 (g/g) 0.17 ± 0.00 0.29 ± 0.01 0.41 ± 0.01 0.52 ± 0.01 0.68 ± 0.03 0.84 ± 0.03 1.00 ± 0.02 1.11 ± 0.01 1.25 ± 0.02 1.43 ± 0.11 0.04

Xwe × 101 (g/g)

Ywe × 101 (g/g)

Ke (g/g)

9.07 ± 0.07 8.95 ± 0.04 8.79 ± 0.04 8.62 ± 0.03 8.41 ± 0.09 8.30 ± 0.11 8.24 ± 0.19 8.14 ± 0.18 7.95 ± 0.14 7.78 ± 0.13 0.12

9.79 ± 0.01 9.64 ± 0.01 9.51 ± 0.01 9.38 ± 0.01 9.22 ± 0.01 9.07 ± 0.03 8.93 ± 0.01 8.78 ± 0.01 8.67 ± 0.00 8.54 ± 0.03 0.01

1.09 ± 0.01 1.07 ± 0.01 1.07 ± 0.00 1.06 ± 0.01 1.07 ± 0.01 1.07 ± 0.01 1.07 ± 0.03 1.04 ± 0.02 1.06 ± 0.03 1.09 ± 0.01 0.02

Temperature = 40 ◦ C, Vo /mp = 4 L/kg, slice diameter = 2.7 cm, slice thickness = 0.3 cm. a b

Data expressed as means ± standard deviation of three independent experiments. Pooled standard deviation.

equilibrium were in the order of the 1.34%. These results give support to the theoretical analysis presented in Section 2.2. Several authors have used partition constants, such as that defined by Eq. (11), albeit in their reciprocal form ( K1e =

Xje Yje ),

to describe OD equilibrium (del Valle and Nickerson, 1967; Parjoko et al., 1996). These values were transformed when necessary to allow a direct comparison with our results. For example, del Valle and Nickerson (1967) obtained distribution constants for NaCl in swordfish muscle ranging from 1.09 to 1.43 (5–37 ◦ C, 0.01–0.31 g/g NaCl solution). Sarang and Sastry (2007) reported distribution coefficients for NaCl from 1.04 to 1.09 during OD of Chinese water chesnut (25–80 ◦ C, 50–100 g/L NaCl solution). A similar study was conducted by Corzo and Bracho (2004), reporting separated distribution coefficients for water (from 1.60 to 2.00) and NaCl (from 1.28 to 1.89) during osmotic dehydration of sardine sheets (32–38 ◦ C, 0.15–0.27 g/g NaCl solution). Corzo and Bracho (2004) considered that mpe = mp0 − (mw0 − mwe ) + (mse − ms0 )

(44)

ignoring the lixiviation of food solids and, according to Eq. (5), leading to the underestimation of Xse and Xwe and to the overestimation of Kse and Kwe . Our research group proposed a method to estimate Kse from total solids gain data by estimating Xse as (Herman-Lara et al., 2013) Xse = 1 − Xf 0 − Xwe

these cases, deviations from the theoretical behavior may be due to an increased mass transfer resistance caused by higher syrup viscosities and solute deposition on product surface, which may lead to a pseudo-equilibrium state.

4.2.

Dehydration and impregnation curves

Fig. 3 shows the evolution of solute and water fractions during OD of carrot slices under forced convection and the ratio between solution volume and product mass (experiment set 2, Table 1). These data were used to evaluate diffusion coefficients assuming both a negligible external resistance to mass transfer (Bis and Biw were set to 1000) or a convective boundary (Bis and Biw were also fitted during regression procedure). By assuming a diffusion-controlled process, water and NaCl diffusivities were estimated as 6.0 × 10−10 m2 /s and 4.1 × 10−10 m2 /s, respectively (Table 4), allowing a good reproduction of experimental dehydration and impregnation curves (R2 > 0.96, Fig. 3). These diffusion coefficients remained mostly unaffected (values changed between 4% and 6% of their original value) when Bis and Biw were also estimated with data from the same experiment achieving a similar fitness quality (R2 = 0.97) with overlapping model curves. However, while estimated Biot numbers were consistent with a diffusion-controlled process (Bis = 74.4 and Bis = 50.7), their 95% CI indicated a non-significant parameter estimation.

(45)

This procedure, while neglecting leaching of food solids as well, leads to an overestimation of Xse and the underestimation of Kse . This fact may explain some distribution for NaCl and water during OD of radish (40 ◦ C, 0.05–0.25 g/g NaCl solution) with values as low as 0.66 and 0.88, respectively (Herman-Lara et al., 2013). Several studies have also investigated the distribution coefficients of diffusing substances during OD process in sugar solutions (Parjoko et al., 1996; Rahman et al., 2001; Sablani ˘ et al., 2002; Togrul and I˙spir, 2008; Ruiz-López et al., 2011b). However, in all cases osmotic solute has been grouped with solids originally present in foods. Thus, a direct comparison with the results obtained in this work is not attainable. On the other hand, distribution coefficients for water have shown a different trend to that expected in this study, with Kwe values very often lower than unity, especially at high syrup concentrations, indicating that water cannot be eliminated even if osmotic media provides the driving force for mass transfer. In

Fig. 3 – Experimental and fitted evolution of water and sodium chloride mass fractions in product during osmotic dehydration of carrot slices using a 10% NaCl solution at 40 ◦ C with stirring (190 rpm) and a high Vo /mp ratio (20:1).

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food and bioproducts processing 9 7 ( 2 0 1 6 ) 88–99

Table 4 – Estimated diffusion coefficients and mass Biot numbers during OD of carrot slices. Valuea

Cs0 (g/L)

Stirring (rpm)

Vo /mp (L/kg)

Parameters

100

190

20:1

Dw (m2 /s)d Ds (m2 /s)d Dw (m2 /s) Ds (m2 /s) Bis (Biw )

6.0 4.1 6.3 4.3 74.4 50.7

100

0

4:1

Bis (Biw ) Dw (m2 /s)d Ds d

26.4 18.3 7.6 3.5

Bis (Biw ) Dw (m2 /s)d Ds (m2 /s)d

89.0 61.7 7.0 3.7

Bis (Biw ) Dw (m2 /s)d Ds (m2 /s)d

4148 2877 6.1 3.6

100

150

a b c d e f

0

0

20:1

20:1

95% CIb

R2 0.966 0.976 0.972e

Biw = 87.9 Bis = 126.7

0.9/53.7

0.862e,f

Bis = 21.3 Biw = 14.8

5.3/10.0 3.3/3.9

0.787 0.991

113.4/291.4

0.932e,f

5.5/8.4 3.6/3.9

0.891 0.991

5.4/6.6 3.8/4.5 4.5/8.2 3.4/5.3 − 266.2/414.9 −

−

Predicted Bic

−

4.1E6/4.2E6 0.925e,f 4.9/7.2 3.3/3.9

Bis = 27.0 Biw = 18.8

Bis = 28.3 Biw = 19.7

0.878 0.975

Diffusivity values × 1010 . Bold numbers indicate non-significant parameter estimates at the given confidence level. Estimated from empirical correlations. Diffusivity values estimated by considering a negligible external resistance to mass transfer (Bis = Biw = 1000). Parameters estimated by simultaneously fitting Xs and Xw curves during nonlinear regression with Biw = Ds Bis /Dw . Dw and Ds values were obtained from experiment set 2.

Empirical correlations allowed the estimation of Biot number for water and NaCl transport under these experimental conditions as 87.9 and 126.7, respectively. In both cases, these values indicate a strictly diffusion-controlled process, which is reached for practical purposes when mass Biot number is higher than 30 or 40 in analytical solutions for invariable properties of continuous phase (Córdova et al., 1996; Yanniotis, 2008; Ruiz-López et al., 2011a). Under such conditions, theoretical model curves overlap no matter the mass Biot value used to generate them. In this case, as long as Bi > 30 (process is diffusion-controlled) water and solute diffusivities will be estimated free of convective mass transfer effects, but any Bi value will produce a good fit of experimental data, explaining the lack of significance of this parameter. OD experiments conducted without stirring were presumed to show an important resistance to mass transfer

outside product (Figs. 4 and 5). In this case, minimum Bi numbers for water and solute transfer were identified as 18.30 and 26.38 (Table 4), respectively. These values, obtained with the lowest Vo /mp ratio (4:1) (experiment set 3, Table 1), are near the limit of a diffusion-controlled process. On the other hand, the analysis of remaining experimental data produced mass Biot numbers estimations well in the zone of a diffusioncontrolled process (Biw > 61, Bis > 88). However, under natural convection conditions, all regression procedures produced non-significant estimations of Bi numbers (p = 0.05). The Bi values estimated with empirical correlations allow gaining a better insight into the mass transfer phenomena and the lack of significance of regression parameters. Under natural convection conditions Bi values are in the ranges of 14.8–19.7 and 21.3–28.3 for water and NaCl transfer, respectively. Again, these values are near the limit of a diffusion-controlled

Fig. 4 – Effect of solution concentration on dehydration (j = w) and impregnation (j = s) curves of carrot slices (40 ◦ C, no stirring and Vo /mp = 20 : 1 L/kg).

Fig. 5 – Effect of solution mass-to-product volume ratio on dehydration (j = w) and impregnation (j = s) curves of carrot slices (40 ◦ C, no stirring and 10% NaCl).

food and bioproducts processing 9 7 ( 2 0 1 6 ) 88–99

Fig. 6 – Effect of mass Biot number on dehydration (j = w) and impregnation (j = s) curves of carrot slices (40 ◦ C, 10% NaCl and a Vo /mp ratio of 20:1). Arrows indicate the direction in which Bi is increased. Numbers refer to the Bi values used in model solution.

process where a change in Bi value will not alter the model behavior in an important way. This fact jointly with experimental data dispersion did not allow a reliable identification of Bi. In fact, the rate at which mass transfer occurs under natural convection conditions is practically the same to that observed under forced convection conditions, causing experimental data to overlap, as shown in Fig. 6. Moreover, this figure also exhibits the minimal separation between simulated OD curves obtained under different Bi values ranging from 10 to 1000, which is well between the observed experimental dispersion. In this way, diffusion coefficients could be also estimated from experiments conducted without stirring as shown in Table 4. This time, water diffusivities were estimated in the range of 6.1–7.6 × 10−10 m2 /s, while corresponding values for NaCl are between 3.5 × 10−10 and 3.7 × 10−10 m2 /s. In this case, the 95% CI for these values overlap with those previously calculated under forced convection conditions. Figs. 4 and 5 allow observing the effect of solution concentration and solution-to-product ratio on impregnation and dehydration curves. From Eq. (15) it is clear that the use of a higher solute concentration (Ys0 ) will led to an increase of water loss and solute gain at equilibrium. On the other hand, decreasing Vo /mp (and thus Mo /mp ) will led to a reduction of water loss and solute gain at equilibrium. Moreover, as Mo /mp → ∞, Xje → Yj0 /Ke and when Mo /mp → 0, Xje → Xj0 . All these cases can be predicted with current equilibrium and unsteady-state models. A notable feature of proposed model is that, in addition to describing water and solute fractions in the product, it also allows the prediction of osmotic media concentration evolution, which may change in a significant way especially in experiments conducted with a low Vo /mp ratio. For example, by lowering the Vo /mp ratio from 20:1 to 4:1, the final dilution of osmotic media increases from 4% to 18% (Fig. 7). Estimated diffusivities for water were smaller than the corresponding NaCl values, which may be very likely due to its lower molecular weight as reported elsewhere (Panagiotou et al., 1999). Diffusivity values are comparable to those obtained during OD of other foodstuffs. For example, water and NaCl diffusivities during OD of potato (25–55 ◦ C, 0.10–0.18 g solute/g solution) have been estimated in the ranges of 8.7–12.3 × 10−10 and 8.2–12.2 × 10−10 m2 /s, respectively (Khin

97

Fig. 7 – Effect of Vo /mp ratio on dilution of osmotic media. et al., 2006). Comparable results were obtained during OD of cherry tomato (25 ◦ C, 0.10 and 0.25 g solute/g solution) with water and NaCl diffusivities values in the ranges of 1.2–1.8 × 10−9 and 0.2–0.5 × 10−9 m2 /s (Azoubel and Murr, 2004). Sarang and Sastry (2007) reported NaCl diffusivities in the range of 7.5–18.5 × 10−10 during OD of Chinese water chesnut (25–80 ◦ C, 50–100 g/L NaCl solution). There are few reports regarding the estimation of Bi number during OD of foods. Selected studies include that of da Silva et al. (2013), who performed the OD of coco parallelepipeds (6 mm × 10 mm × 31 mm) in sucrose solutions under forced convection conditions (magnetic stirring at 60 Hz, 35 ◦ Bx, 40 ◦ C) and reported fitted values as Biw = 7.75 and Bis = 5.00. In a later work, da Silva et al. (2014) studied the OD of pineapple cubes (side length of 15 mm) in sucrose solutions without stirring (40 and 70 ◦ Bx, 30 ◦ C) estimating Bi values in the ranges of 1.5–2.3 and 0.62–0.95 for water and solute transport, respectively. Unlike current study, Bi values for solute transfer were smaller than those estimated for water; however, it should be highlighted that da Silva et al. (2013, 2014) analyzed solute transfer as total solids and did not report a statistical analysis of estimated parameters to allow a further comparison of observed differences with values obtained in this study.

5.

Conclusions

Proposed models allowed the satisfactory description of both dynamic and equilibrium mass transfer periods for water, osmotic solute and food solids interchange between product and solution during an osmotic dehydration (OD) process. The mathematical description of OD equilibrium helped to explain differences between previous studies, describing the necessary adjustments to eliminate them and providing wellfounded bases for further investigations. A better insight of mass transfer phenomena occurring during OD of carrot slices in NaCl solutions was gained, where convection was almost negligible even in non-stirred osmotic media. Further studies are required to evaluate governing mechanism for mass transfer in other systems.

Acknowledgments The authors wish to thank the Consejo Nacional de Ciencia y Tecnología (CONACYT) for providing financial support through project 130011. Hermelinda Pacheco Angulo acknowledges her doctoral scholarship from CONACYT. Miguel Ángel

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food and bioproducts processing 9 7 ( 2 0 1 6 ) 88–99

García Alvarado acknowledges his sabbatical financial support from CONACYT.

Appendix A. Properties used in empirical prediction of mass Biot numbers Diffusion coefficient for NaCl–water mixture. Poling et al. (2001) proposed the following equation to describe the effect of solute concentration on mutual diffusion coefficients in electrolyte solutions of a single salt: Dsw = D◦sw

w ϕ sw

(A1)

where, the diffusion coefficient at infinite solution (D◦sw ) and thermodynamic factor (ϕ) are given by (Poling et al., 2001) D◦sw =

RT 1/n+ + 1/n− F2 1/◦+ + 1/◦−

ϕ =1+m

∂ ln ± m ∂± =1+ ∂m ± ∂m

(A2)

(A3)

with + = 5.01 × 10−3 and − = 7.63 × 10−3 S/m2 mol for Na+ cation (n+ = 1) and Cl− anion (n− = 1), respectively (Pitzer et al., 1984). Mean ionic activity coefficient ( ± ) of solute. Tabulated data (100 kPa) in the ranges of 0.1–6 molal NaCl and 0–100 ◦ C provided by Pitzer et al. (1984) were fitted in this study with predictions within the ±4% of the original data (R2 = 0.9895):

± = 0.7358 + 0.2170x2 − 0.0591x12 − 0.0472x12 x2 − 0.0303x1 x22 + 0.0098x13 − 0.0972x23 + 0.1393x24

(A4)

where, x1 = [T (◦ C) − 50] /50 and x2 = (m − 3.05) /2.95. Dynamic viscosity of solvent (w ) and solution (sw ). Tabulated data (100 kPa) in the ranges of 0–6 molal NaCl and 20–100 ◦ C provided by Kestin et al. (1981) were fitted in this study with predictions within the ±3% of the original data (R2 = 1.0000): sw × 103 = 0.6487 − 0.3626x1 + 0.2179x2 − 0.1289x1 x2 + 0.1745x12 + 0.0466x22 + 0.0782x12 x2 − 0.0499x1 x22 − 0.1094x13 + 0.0092x23 − 0.0367x13 x2 + 0.0300x12 x22 + 0.0491x14

(A5)

where, x1 = [T (◦ C) − 60] /40 and x2 = (m − 3) /3. Eq. (A5) becomes w when x2 = −1. Density of solvent ( w ) and solution ( sw ). Tabulated data (100 kPa) in the ranges of 0–6 molal NaCl and 0–100 ◦ C provided by Pitzer et al. (1984) and Geankoplis (1993) were fitted in this study with predictions within the ±3% of the original data (R2 = 1.0000): sw = 1093.4898 − 26.8911x1 + 94.6352x2 − 2.5050x1 x2 − 3.8475x12 − 8.9248x22 + 3.2483x12 x2 + 1.3048x1 x22 + 0.7381x13 + 1.4752x23 − 0.9954x13 x2 − 0.5638x1 x23 − 1.3408x12 x22 − 0.3065x14 − 0.3796x24

(A6)

where, x1 = [T (◦ C) − 50] /50 and x2 = (m − 3) /3. Eq. (A6) becomes w when x2 = −1.

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