A mathematical model to describe potato chemical (NaOH) peeling. Energy and mass transfer model resolution

PII:

SO260-8774(97)00091-X

Journal of Food Engineering 32 (1997) 20’9-233 0 1997 Else&r Science Limited. All rights rcservcd Printed in Great Britain 0260-8774197 $17.00 + 0.00

ELSEVIER

A Mathematical Model to Describe Potato Chemical (NaOH) Peeling. Energy and Mass lkansfer Model Resolution M6nica S. Chavez,” * Julio A. Lunaa & Ralil L. Garrote” “lnstituto de Desarrollo Tecnolbgico para la lndustria Quimica, (INTEC), Consejo National de lnvestigaciones Cientificas y Tknicas, Universidad National de1 Litoral, Giiemes 3450 (3000), Santa Fe, Argentina “lnstituto de Tecnologia de Alimentos (FIQ-UNL), Ciudad Universitaria, C.C. No 438 (3000), Santa Fe, Argentina (Received

lh September

1995; revised 13 September

1996; accepted 28 October

1996)

ABSTRACT One of the oldest and most commonly used industrial processes for peelin,g potatoes is chemical peeling. Solute (NaOH) diffusion, energy transfer and chemical kinetics are involved in the process. The Shrinking Core Model and the second Fick’s law were selected to formulate the mathematical model, according to mechanisms involved during the peeling process. Heats of reactions were neglected as a proposition of the model, so energy and mass balances were unfastened. Consequently, a temperature profile inside the potato was obtained. Hypotheses were assumed and three simplified models for mass transfer were deduced; they were then solved using different Biot mass numbers (30, SO, 100, 200 and 300) which were used as the fitting variable of models. The temperature profile, previously obtained, was included in the solute mass balance using the NaOH effective diffusion coeficient correlation in potato ,fTesh. The mathematical system was analytically and numerically solved and results were compared to experimental data and showed good agreement. 0 1997 Elsevier Science Limited

NOTATION a Bi c, CH

Number of moles of solute per mole of suberin. Dimentionless Biot number. NaOH concentration (mol/cm3). Solid reactant concentration (molig).

To whom correspondence

should be addressed. 209

h4. S. Chavez et al.

210

CP Def e

h” k kg Ik RZ rc T T t

Heat capacity (Cal/g “C). Solute effective diffusion coefficient (cmzOJcm&id set). Potato skin thickness (cm). Dimentionless number. Heat transfer coefficient (cal/sec cm2 “C). Thermal conductivity (cal/sec cm “C). External mass transfer coefficient (cmi,iJcm&iid set). Radial position. Potato radius (cm). Surface reaction rate (mole/set cm,‘). Moving front position (cm). Average temperature (“C). Temperature (“C). time (s).

Greek Symbols Density (g/cm”) P Porosity (CI&&3Il~,j~) E Superscripts Potato flesh f Initial condition. i Bath condition. 0 Potato skin S

INTRODUCTION The potato chemical peeling process involves energy and mass transfer with simultaneous chemical reaction. Sodium hydroxide (NaOH) diffuses from the surface of the potato to its centre, reacting with potato skin and flesh constituents and promoting cell breakage (Floros et al., 1987). Moreover, it frequently moves deeper causing undesirable flesh tissue damage. Additionally, the NaOH solution temperature increases the process of reactions and solute diffusion transport. Consequently, a mathematical model to describe the chemical peeling process is needed to obtain a solution that allows the minimisation of processing losses of the edible part of the vegetable, together with minimum energy and NaOH solution consumption. Potato chemical peeling can be identified as a heterogeneous, non-catalytic (liquid-solid) reaction in a porous medium. McFarland and Thomson (1972) concluded that the process was limited by mass transfer until the potato skin had been completely penetrated and, subsequently, the process became controlled by reaction rate. On the other hand, Chavez et al. (1996) proposed that the main chemical reactions in the chemical peeling process were that between NaOH with the suberin of the skin and the potato starch hydrolysis inside the flesh. Previous information gives a basis upon which to propose two different mathematical models according to the medium in which the solute moves forward: the shrinking core model, which was selected by McFarland and Thomson (1972) was applied to describe solute transport in the skin; whereas the second Fick’s law was applied to define the NaOH concentration profile in potato flesh.

Potato chemical (NaOH) peeling

71 I

Previous work, dealing with heat transfer through potato during blanching (Califano and Calvelo, 1983; Lamberg and Hallstriin, 1986), cooking (Verlinden et al., 1995) and drying processes (Wang and Brennan, 1995), assumed that it occurred by conduction. Garrote et al. (1993) reported that experimental heat ring thickness was greater than that of NaOH penetration. These results suggested that the temperature profile developed earlier and that the heat transport problem might be solved initially and separately from mass transference. The objectives of the present work were to propose and solve a mathematical model to describe simultaneous solute (NaOH) and heat transport in potato skin and flesh during vegetable chemical peeling. In order to carry on the objectives, the heat of reaction was neglected; therefore, the mass and energy equations could be unfastened and solved separately. Then, pseudo steady state approximation was applied in the mass balance. Later, three assumptions leading to three simplified models were proposed. The temperature profile, previously obtained, was included in the model resolutions using the solute effective diffusion coefficient correlation. Finally, solute concentration and temperature profiles were compared to solute volume average concentration provided by the surface response methodology (Garrote et al., 1993) and experimental temperature profile, respectively.

MATHEMATICAL

MODEL

Potato sodium content was very low and can be neglected (Chavez et al., 1996), so any change of its content inside the vegetable during the peeling process could be directly associated with sodium hydroxide solution penetration; therefore, the sodium ion was selected to calculate mass balance. The present study did not go further into involved physicochemical phenomena descriptions, since such details were not needed. Moreover, all ion effects during the peeling process were included in the model through effective transport coefficients (Chavez et al., 1996). Mass balance

The shrinking core model (SCM) is applicable to nonporous solid reactants and assumes that the reaction takes place only at the outer edge of an unreacted core. Chemical reaction defines a moving boundary or moving front, in which the rate #
was (Froment

212

hf. S. Chavez

et al.

Pseudo steady-state approximation was applied in eqn (l), which assumed that there was no solute accumulation in the reacted solid left behind the moving front. So, eqn (I) became:

6 (Def”(T)-ac*)=o

(2)

ax

subjected to the following boundary conditions:

r = R,

r =

ac.4

Dey(T)

x(t),

-

r=R =

ar

-

Def"(T)

k$“?. - CA>

(3)

=R:,

(4)

ah

ar r=F-t

where: solute (NaOH) is represented by A, I-: is the porosity, Def is the solute effective diffusion coefficient, kg is the external mass transfer coefficient and Cl is the NaOH bath concentration. External mass transfer was included in eqn (3) and RA of eqn (4) represented the kinetics of the reaction between NaOH and the suberin (Chavez et al., 1996). The moving front rate, at any distance rc inside the skin, was obtained from the mass balance of solid reactant (named B) expressed as follows: Def'@)

drc r = rc(t),

-

-

=

dt

XA

ar

aCg

T=i-C

(5)

where a is the number of moles of the solute reacting with one mole of the suberin. Equation (5) showed that the front propogation rate depended on the solute diffusive flux. Initial conditions were: t=O, rc=R,

GA=0

(6)

The previous mathematical system, eqns (2)-(6) was analytically solved assuming that solute diffusive flux was lower than chemical reaction rate. Thus, the solute concentration profile inside the potato skin and the moving front position were obtained:

c&

(’- rc(t))

Def”

(R - rc(t))+ -

x = rc(t),

5:/m))

(7)

>

(8)

21.3

Potato chemical (NaOH) peeling Second stage: the moving front enters the potato flesh

The skin left behind represented an exhausted medium to NaOH chemical reaction once the moving front reached the flesh. Chavez et al. (1996) proposed starch hydrolysis as the most important reaction in potato edible tissue. NaOH catalysed starch hydrolysis, so there was no actual solute consumption. Consequently, there was no moving front promoted by NaOH consumption inside the flesh, but solute diffusion through skin and then into the flesh. Equation (1) was, therefore, simultaneously applied to the solute inside the skin and the flesh. Boundary conditions at the edge of the potato were eqns (3) and (4) the initial condition for solute concentration was eqn (6) and the following equations complete the mathematical model: r=O,

r= R-e,

ac*

De.fYT) pv

ac.4

-

&

=o

ac,

= Def’( 7) ,-=K-_c, ar

r=K-*,

(10)

where e is the skin thickness. The latter equation defines the solute diffusive flux continuity at the skin-flesh interface. The time required for the solute to reach the flesh was obtained during first stage of the calculation, and then included in the second. Three simplified models were obtained when the hypotheses were applied: Simplified first model It was proposed that there was no NaOH accumulation in the skin during second stage; a linear profile was obtained, which was coupled to mass balance in the flesh. Fick’s first law was applied to solute mass balance in the flesh subjected to the following boundary conditions:

r=O,

ac.4

-

ar

ZO

(12)

In this model, the skin was included as a resistance to solute diffusion together with external mass resistance in eqn (11). The solute initial condition was defined by eqn (5). Simplified second model

Due to the small thickness of the skin, it was assumed that the skin-flesh interface solute concentration and skin-bulk solution were equal. The skin was included as a

M. S. Chavez et al.

214

resistance to solute diffusive flux when solute diffused into the flesh in the second stage of the calculation. Simplified third model

It was assumed that the potato skin acted as a resistance to solute diffusive flux into the flesh from the very beginning of the potato chemical peeling process. Consequently, this model did not consider chemical reaction in the skin and, therefore, the definition of stages was not required. This model was defined by Ficks’s first law and eqns (5), (11) and (12). Energy balance Proposed mass transport equations and the analogous energy mathematical model were coupled by means of transfer coefficients and the kinetic of the reaction introduced in the energy balance. In order to solve the energy transfer problem separately from the mass transfer without losing information, the following hypothesis was proposed: heat of reaction, which included heat of starch gelatinization, were supposed to have less influence on whole energy transfer than the heat transferred by conduction. This assumption removed the moving heat front term from the energy balance; however, there still existed the "heat ring" in the potato caused by heat conduction only. Although heat transfer occurred in two media, information about thermal properties presented in Table 1 gave a basis on which to consider both media as one, a consideration only valid in the heat transport study. In the present model, potato thermal conductivity was considered constant with temperature as shown by Wang and Brennan (1992). Potato heat capacity was assumed to be constant, as Lamberg and Hallstron (1986) verified that it may be considered as temperature independent when potato moisture content is greater than 85%.

TABLE 1

Thermal Properties of Huincul Potato Potato skin Water content (%I) k (calisec “C cm)

0.85 1.355 x 1om3

x ( cm2/sec)

1.50x lo- 3

P (g/cm’) Cp (cal/g”C)

Potato flesh 0.80 1.312 x IO-?

1.010

(1) Heldman and Singh (1981) Hough and Alzamora (1994) (2)

0.894

(3)

0.894

1.46 x lo-”

1.068

(1): Experimentally determined for Huincul potato. (2): Estimated using displacement method. (3): Estimated using previous thermal properties and r definition.

(1) Califano and Calvelo (1983) Luna and Garrote (1987) Garrote et al. (1993) (3)

Potato chemical (NaOH) peeling

215

Assuming that heat was transferred only by a conduction mechanism and its flux defined by Fourier’s law, the heat balance equation inside the potato for a spherical coordinate was:

3T PCP iit

Boundary conditions

1 = 7

a 8T gr2k z r l

were: r=R,

h(TO-T)=kE

r=O,

ar =0 i3r

(15)

The initial conditions were: t=O, T=T’

(16)

where h is the extermal heat transfer coefficient, T” is the NaOH bath temperature and T’ is the initial temperature inside the potato. MATERIALS

AND METHODS

Potatoes, Huincul variety, weight 238.004 +537 g (n = 130, P~0.05) and density 1.068 kO.006 Kg/cm” (n = 10, P
216

Zkt.S. Chavez et al.

RESULTS

AND DISCUSSION

The external heat transfer coefficient value obtained was h = (3.439&0*791) x lop3 cal/sec cm2 “C a mean value for different temperatures and concentrations of NaOH. The obtained coefficient value (h) was as expected; Lamberg and Hallstron (1986) tested their model for a potato blanching process using an external heat transfer coefficient between 17.91 x 10V3 and 3 x 10W3 cal/sec cm2 “C. The mathematical system defined by eqns (13),(16) was solved analytically, using the separation of variables method proposed by Ozisick (1980) and numerically by means of the finite-difference method. For that purpose, a uniform grid (150 nodes), the implicit scheme and Thomas algorithm were selected to solve the problem. Theoretical temperature profiles obtained analytically and numerically were very similar (Table 2). The numerical method parameter (F) value, 0.25, was proposed as the most convenient after considering the compromise between good results and computer times. Figure 1 compared the temperature distribution predicted by the numerical method, using the thermal properties of potato flesh, with experimental temperature profile as a function of time and a bath temperature of 75°C. Inside the potato flesh and near the surface, percentages of deviation were acceptable (Table 3) indicating that heats of reactions and heat of gelatinization were correctly ignored. Percentages of deviation were determined using the equation proposed by Heldman (1974): 2

TV-Ev ’

Ev

i=l

i

%D=

x

J’

’

N-l

100

(17)

TABLE 2

Temperature Profile obtained from Analytical and Numerical Mathematical Methods for Bath Temperature 75°C NaOH Concentration 12% (wt/wt) and 4 Minutes of Potato Peeling Process Heat penetration thickness (mm)

Numerical method according to value (F = (Ata)/(

Analytical method

F=0.05

F=O.l

F=0.25

F parameter

F=0.5

F=l

0

60.14

60.376

60.376

60.375

60.374

60.371

:‘: 016 0.8 1 1.2 1.4 1.6

59.37 58.61 57.85 57.09 56.33 55.57 54.82 54.07

58.843 59.061 58.079 57.317 56.558 55.802 55.049 54.301

59.609 58.843 58.079 57.317 56.557 55.801 55.048 54.300

59.608 58.842 58.077 57.316 56.556 55.800 55.047 54.298

58.840 59.606 58.076 57.314 56.555 55.798 55.046 54.297

58.837 59.603 58.073 57.311 56.551 55.795 55.042 54.293

“b: Time increment.

b: Spatial increment.

217

Potato chemical (NaOH) peeling

~

Theoretical

Expenmental (cm) . 3 56 ’

41

0

3.16

.

2.66

100

200

300

400

500

Time (sec.) Fig. 1. Numerical

Percentages

and experimental

of Deviations

between

temperature profile at different of the potato.

TABLE 3 Experimental Temperature perature Profile

positions from the centre

Profile and Numerical

Tem-

Radial coordinate” (cm) 3.56

3.16 2.66 “From the centre of the potato.

2.7?i 5.38 6.44 -.__

218

M. S. Chavez et al.

One of the most important factors that affected the efficiency of the chemical peeling was the depth of the heat ring generated during the process (Huxsoll and Smith, 1975). The heat ring is a consequence of potato starch gelatinization caused by heat and humidity transport, which was not desirable inside the flesh. The potato starch gelatinization process starts at approximately 55-60°C (Kubota et al., 1979; Verlinden et al., 1995), so, according to temperature profiles calculated from the model proposed (Fig. 2), potato starch gelatinization penetration depth may be predicted at different times, keeping the temperature of the bath constant. It was assumed that the main solute consumption in skin during chemical peeling was attributed to the saponification reaction of the skin suberin. This meant that 7 moles of solute per mole of suberin were needed to generate the saponification reaction (Chavez et al., 1996). The initial concentration of solid reactant (suberin) was then estimated, extrapolating solute concentration decrease to 2 h from Fig. 3, when all the solid reactant should have been consumed. According to this, the minimum dimentionless solute concentration value was O-8775. This value, the stoichiometry relation previously proposed and equivalent neutralisation relation between solute and solid reactant were used to calculate the dimentionless initial concentration of solid reactant: (C#(C”,) = 0.0175. External mass transfer coefficient, kg = 2.84 x lo-” cm/s, was evaluated using the Chilton and Colburn analogy: 2/.?

kg=

h

C&I,

CPYDN~OH

PfihCPfih

~ H,O

k

(18)

film

SO-

80

T,=9S°C

TD = 75°C

Th =SS"C

70

Posmon

Fig. 2. Temperature

profile numerically

(cm)

obtained

for different bath temperatures.

219

Potato chemical (NaOH) peeling

,z l.oo8i

l

1

OSO.%-

BO.Y .s 0.92-

B

0.901

j

0.88-

cs

Fig. 3. Dimentionless

. . l

.

.

0

.

.

IO

20

30

Time(rnin) solute concentration (C.&i)

.

40

50

60

as a function of peeling time.

when fluid film temperature was 66_5”C, NaOH solute concentration was 12%, bath temperature was 75°C and external heat transfer coefficient h = 3.439 x lob-” cal/sec cm* “C. Low values of solute effective diffusion coefficients obtained experimentally in the potato skin (Chavez et al., 1996) gave basis to suppose that chemical reaction effect was included in them. The same behaviour and hypothesis were reported by Floros and Chinnan (1990) when NaOH effective diffusion coefficients in tomato and pimento pepper skin were measured. Therefore, as it was not possible to discriminate diffusion mass transport phenomena from the reaction effect, the solute diffusive coefficient experimental correlation in potato skin could not be used in the present model. The Biot mass number, whose definition included the solute effective diffusion coefficient in skin, was therefore taken as the fitting variable of the proposed models. The range of the Biot mass number was from 23 to 300. The first value was selected since the solute effective diffusion coefficient in the skin could not be greater than the solute diffusion coefficient in water (De = 4801 x lo-’ cm2/sec at 70°C estimated according to Perry et al., 1973). Schwartzberg and Chao (1982) reported that Bi > 200 produced an error less than 1% in Def when neglecting external resistance. On the other hand, Luna and Garrote (1987) used Bi = 300 as the highest value of the Biot number to ensure that external mass transport resistance was considered when testing their mathematical model for the prediction of the vitamin C retention of potato strips during blanching. In the first stage calculus of the simplified models, solute concentration profile inside the skin as well as the time required by the solute to reach the flesh were solved analytically according to eqns (7) and (8) for Biot mass numbers 30, 50, 100, 200 and 300. Low Biot mass number caused higher values of the solute diffusion coefficient, producing the increase of the solute diffusive flux through the skin. In this case, a greater amount of solute was available to the chemical reaction at the moving front, increasing its rate. The solute therefore required less time to reach the flesh (Table 4, the delaying time). The second stages of the first, second and third simplified models were solved using the finite difference method, in the mathematical domain of radius coordinates (1.87, 3.74), divided into 301 nodes. Parameter F = (Def(T)tcAt)/(R2~‘A?) of the numerical method was selected as 0.25 due to results obtained during the

220

M. S. Chavez et al.

TABLE 4 Delaying Time caused by Solute Diffusive Flux Resistance during the Simplified Stage of Calculus in Skin

Model-First

Biot Mass Number

Delaying Time (sec.)

300 200 100 50 30

15.22 12.95 10.71 9.57 9.13

temperature profiles calculation and because of the mathematical analogy between the equations system. Mathematical systems were solved using the Thomas algorithm and the implicit scheme. First and second simplified models took into account the delaying time (Table 4), whereas it was not needed in the third model calculation. The numerical temperature profile, previously obtained, was introduced into the concentration profile calculation using the solute effective diffusion coefficient correlation equation in flesh, proposed by Chavez et al. (1996): Def; = 3.65 x lop3

exp

- 1872.06 +(4’13 Tj

X

10~4 Tj-0’101)

C”, >

(19)

where Tj represents the average temperature at the node j obtained from the temperature numerical profile. Solute volumetric average concentration was estimated for every obtained NaOH concentration profile and were compared to results provided by surface response methodology (Fig. 4). According to percentages of deviation (Table 5), determined using Heldman’s equation (Heldman, 1974), the simplified second model fitted best when the Biot mass number was 30. This could be explained considering that Biot mass number 30 produced the highest solute diffusive flux needed to satisfy the hypothesis proposed in this model, which was that there were no important differences between solute concentration in the potato surface with respect to the skin-flesh interface. The main improvement this model had with respect to the third was to consider the moving front in the skin, which produced the delaying time. On the other hand, the second simplified model neglected significant solute concentration changes between the surface of the potato and the skin-flesh interface, which were included in the first model. We can, therefore, conclude that during the first minute of the potato chemical peeling process, the skin offered a chemical reaction resistance; then, when the solute overtook it, the skin caused diffusive resistance. On the other hand, results provided by the second simplified model showed that differences between theory and experimental data increase with process time. This may be explained as a result of secondary chemical reactions between solute and skin constituents and the possibility that solute was occluded in the starch matrix, both reasons would reduce measured solute concentrations in the potato. Finally, if the Biot mass numbers 30 and 50 are considered as the most suitable numbers for the potato chemical peeling process at 75°C it can be concluded that the solute effective diffusion coefficient in the skin varied from 3.643 x lo-” to 2.136 x lop5 cm’/sec. These values are similar to the one experimentally obtained in potato flesh (2.418 x lo-” cm2/sec, 70°C and 12% w/w, Chavez et al., 1996). However, lower values of solute effective diffusion

221

Potuto chemical (NaOH) peeling

Matbematml Model for Potato Chemical Peeling

cl

b)

a)

ll-

II 10

1

l.O09OS-

07-

0.77

06-

0.6-

05.

05-

0.4-

04-

03-

03-

02-

02Ol-

Ol-

4.

I.,

O

50

I,

loo

IS0

I.

200

I-

250

A,’ ,’ ’ ,’

4

0

.,

50

I.

I.

I.

I

G

100

150

200

250

T. 0

50

I, loo

I. 150

8. 200

8. 250

Time of PeelingProcess(set) Fig. 4. Volumetric average concentration of solute vs peeling process time as a function of Biot mass number. (a) First simplified model, (b) second simplified model and (c) third simplified model. S.R., surface response methodology values.

Deviation

TABLE 5 Percentages of Volume Average Solute Concentration obtained from several Simplified Models compared to the Response Surface Methodology Deviation Percentages of Volume Average Solute Concentration (%)

Biot Mass Number

300 200 100 50 30

--

Simplified First Model

Simplified Second Model

Simplified Third Model

20.17 24.34 37.61 48.69 54.07

48.93 37.24 21.21 11.33 8.83

43.98 34.06 21.45 15.78 14.88

222

M. S. Chavez et al.

coefficient are expected to occur in the skin; latest results can be explained suberin-NaOH reaction is considered, since it opens the skin pore structure.

if the

CONCLUSION A mathematical model that considered simultaneous mass and energy transfer with chemical reaction was proposed to describe potato chemical peeling. Mass and energy balances were unfastened and the temperature profile inside the potato was first obtained. The key to our approach lay in two hypotheses. The first was to assume that heats of reaction and heat of gelatinization inside the potato could be neglected. Additionally, it was realised that skin thermal properties were quite similar to those of flesh and, consequently, it was assumed that skin and flesh behaved as one medium in the energy study. The proposed heat transfer mathematical model could be easily solved analytically and numerically; theoretical temperature profiles thus obtained showed good agreement with experimental ones. The results allowed us to conclude that heat conduction may be taken as the most important mechanism involved in energy transport during potato chemical peeling. Three simplified models were deduced from the mass balance model and they were solved with inclusion of the temperature profile through effective diffusion coefficient correlation. Obtained results gave a basis on which it could be concluded that: the skin causes a delaying time at the very beginning of the process and, later, when the solute overtakes the skin, it behaves as a resistance to NaOH diffusive flux into the potato. The saponification of the suberin and potato starch hydrolysis were correct approximations of the solute chemical action during the peeling process. Pseudo steady state simplification was suitably applied. Methodology developed to introduce the temperature profile into the concentration distribution, using the solute effective diffusion coefficient correlation, was appropriate. It was also verified that the finite difference method not only provided accuracy and reproducibility in results, but it also made it easier to introduce solute diffusion coefficient temperature dependence into the mathematical system. Finally, one of the three simplified models reproduced the peeling process better than the others, providing an easy tool for calculation.

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Potato chemical (NaOH) peeling

223

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