Modeling of Sulfur Dioxide Uptake in Pre-Peeled Potatoes of Different Geometrical Shapes

N. RODRiGUEZ and N.E. ZARITZKY

ABSTRACT Sulfur dioxide uptake by pre-peeled potatoes from dipping solutions was mathematically modeled. Diffusive mass transfer equations in porous medium were experimentally verified; residual levels of sulfur dioxide were measured in the range of industrial operating conditions. Effects of sodium bisulfite solution concentration, immersion time, size, shape, dry matter, density and velocity of the product were analyzed. Three geometric approximations to pre-peeled potato cuts were examined (spheres, cubes and parallelepipeds). The fitting of equations to experimental data determined the effective diffusion coef- ficient of sulfur dioxide in potato tissue, which was compared to theoretical predictions in terms of molecular diffusivity, total solids content and tortuosity factor.

INTRODUCTION

DISCOLORATION of peeled or cut raw potatoes is a result of enzymatic oxidation of injured cells exposed to oxygen. This type of discoloration is of commercial importance in the potato pre-peeling industry. Blackening of raw tubers is due to enzymatic oxidation of tyrosine and other o-dihydric phenols to melanins. The major enzyme system responsible for the discoloration of potatoes after injury is generally accepted to be polyphenol oxidase. Chlorogenic acid also is involved, at least in part, in the enzymatic browning of injured potato tis- sue.

Considerable work has been published on practical methods for preventing the discoloration of peeled potatoes. Most of these investigations have been reviewed by Feinberg et al. (1975) and Smith (1977). The use of salts of sulfur dioxide to control discoloration in pre-peeled potatoes is well established in commercial practice (Nieuwenhuis and Van Nielen, 1974; Keijbets, 1981); one of the first patents was issued to Draper (1934).

Numerous attempts have been made to find a substitute for treating with sulfite (Olson and Treadway, 1949; Anderson and Zapsalis, 1957; Amla and Francis, 1959). The different agents described in trade literature have not been used commercially because of high cost and some undesirable side effects such as leakage and off-flavors. Sulfur dioxide has the advantage of being somewhat antiseptic in addition to its function as an inhibitor of discoloration. Sulfur dioxide and sulfite, bisulfite or metabisulfite salts set up a pH dependent equilibrium mix- ture when dissolved in water. As the pH falls, the proportion of SO* increases; this is important in connection with the an- timicrobial activity of sulfur dioxide that is largely related to the unbound nonionized molecular form (Clark and Taka’cs, 1980).

Excessive treatment with sulfur dioxide causes abnormal ap- pearance of the product, undesirable flavor softening and weeping from potato tissue (Amla and Francis, 1961; Francis and Amla, 1961). There is a marked tendency to decrease the amount of sulfite retained by the product. Vacuum packing has

Authors Rodriguez and Zaritzky are with Centro de lnvestigacidn y Desarrollo en Criotecnologia de Alimentos (CIDCA) - (CON- ICET-CIC-UNLP), Cal/e 47 y 7 16 La Plata (1900), Provincia de Buenos Aires, Argentina.

been reported to reduce sulfite requirements with respect to conventional packaging in gas permeable films (Anderson and Zapsalis, 1957). Several factors such as: color preservation, microbial growth, weep production and softening of the tissue must be taken into account to establish the adequate antioxidant treatment in terms of the optimum level of residual sulfur diox- ide in pre-peeled potatoes.

Limited information has been available in the literature about the effects of dipping solution concentration, immersion time, size, shape and velocity of the product on the amount of sulfur dioxide retained by potato cuts (Ross and Treadway, 1961; Francis and Amla, 1961; Furlong, 1961).

The purpose of this study was to propose a mathematical model describing the uptake of sulfur dioxide in pre-peeled potatoes of different geometry and size as a function of: (1) the product physical properties, and (2) the industrial operating parameters (concentration of dipping solution and sample rec- tilinear velocity). The equations were derived from the volu- metric integration of the concentration profiles and were tested for different experimental conditions.

THEORY

POTATO TISSUE was assumed to consist of an insoluble matrix (starch, cellulose, pectic substances) and an aqueous phase through which sulfur dioxide diffuses. Unsteady state diffusion in symmetric porous materials can be analyzed using a general form of the microscopic mass balance:

where c is the solute concentration in the solid as a whole (weight of sulfur dioxide per unit volume of potato); $ = 1 for an infinite slab, 2 for an infinite cylinder, 3 for a sphere; r is the distance measured from the center of the solid, and D, is the effective diffusion coefficient of sulfur dioxide in the potato tissue which can be expressed in terms of molecular diffusivity (DAB), porosity (E) and tortuosity factor (a) (Sher- wood et al., 1975):

DAB E D, = - R (2)

The porosity is the volume fraction of solid occupied by the occluded liquid; the tortuosity is the ratio of the diffusion path to the nominal distance traversed by the solute (Schwartzberg and Chao, 1982).

Spheres, cubes and parallelepipeds (French fry strips) are common geometrical shapes in the pre-peeled industry. A de- tailed description of equations is only shown for spherical ge- ometry; in the case of spheres, Eq. (1) can be written in the form:

(3)

with the following initial and boundary conditions:

tso c=o O<r<R (4)

618-JOURNAL OF FOOD SCIENCE-Volume 51, No. 3, 1986

. .

t>o &

D, ar = E kL (c’r - c’)

= kL (cr - c) in r = R (5) dC

t>o -=0 in r=O ar

In Eq. (5), kL is the mass transfer coefficient at the potato- fluid interface; c’r, the sulfur dioxide concentrat ion in dipping solution (weight of sulfur dioxide per unit volume of solution); and E, the porosity of the potato t issue (water content on wet basis). The equilibrium distribution ratio between the solute concentrat ion in the liquid bath (c’) and the solute concentra- tion in the solid (c) was given by c = c’ E assuming that the partition coefficient equals one.

The solution of Eq. (3) to (6) in the form of a dimensionless concentrat ion profile is given by (Crank, 1957):

I c*=1-2Bi e-v’. l* sin hn r*) (7)

r* “-1 rz + Bi(Bi - 1) sin 7”

where c* = c/cr; r* = r/R; t* = (D, t)/R*; Biot number, Bi = (kL R)/D,; and yn, n = 1, 2 . . . are the roots of:

Yn cot Y” = 1 - Bi (8)

Eq. (7) was integrated over the total sample volume (V) and the fractional uptake (F) was calculated as follows:

I F+- C

(j Bi* e-?n I* “=i -yz {yz + Bi(Bi - 1)) (9)

m

When product velocities through the surrounding fluid cor- responded to high Biot number (Bi > 200), constant interfacial concentrat ion was assumed; the integration of concentrat ion profile in this case was given by:

Mt 6 = 1 - = ’ - F “:, ;;z exp M,

(10)

In the model ing of sulfur dioxide diffusion in cubes and French fry potato strips (parallelepipeds) concentrat ion profiles for an infinite plane sheet were applied. Three-dimensional contributions have been considered using Newman’s Rule (Carslaw and Jaeger, 1959). Dimensionless concentrat ion pro- files were integrated over the corresponding sample volume.

Fractional uptake for cubes and potato strips were calculated as follows:

exp

(Bi*, + Bi, + Bz) (Bi; + Bi, + Bk) (Biz + Bi, + B;) (‘I)

where a, b, d are the half thickness values of the solid in the x, y, z directions respectively,

Bi, = !!?a; DC

Bi, = !$!? ; Bi, zz kL e D,

and

p, tan p, = Bi,; Bm tan Bm = Bi,; Be tan Ee = Bi, (12)

An average mass transfer coefficient kL was adopted for x, y, z directions.

In cases of constant interfacial concentrat ion approach the following expression was derived for cubes and potato strips:

F2$ 7. 6421 cc cc 3c

=,- - 0 ( exp (- q,,,t)

ll' ii ,T, ,!, “!(l (2 e + 1)2 (2 m + I)2 (2 n + I)2 1

712 D where c+, n = 4

In Eq. (9, 10, 11, 13) M, = c’r E V, where V is the potato sample volume according to its geometrical shape. Residual sulfur dioxide per unit weight of a potato cut (W) for a given time, was calculated as follows:

F c’r E W (w-4 = - x 106

P

Mass transfer coefficients in dipping solution were evaluated using the following equat ions (Sherwood et al., 1975), in terms of Reynolds and Schmidt numbers.

kL 2R - = 2 + 0.6 Re,“2 SC”~ (spheres)

DAB (15)

kL 2L - = 0.664 Re,“* SC”~

DAB

(laminar boundary layer (16) on a flat plate)

Model equat ions for each geometry were fitted to experi- mental data by varying the effective diffusivity D, to minimize least squared differences and residues sum.

MATERIALS & METHODS

SAMPLES of potatoes (Solanum Tuberosum, Kennebec variety from Balcarce, Pcia. de Buenos Aires) with different geometrical shapes and sizes were dipped in solutions of sodium bisulfite, subjected to rectilinear movement at constant velocities through the stationary fluid and analyzed for sulfur dioxide as a function of time.

Percent total solids and specific gravity of the samples were deter- mined according to the methods reported by Porter et al. (1964) and Fitzpatrick et al. (1969). Solids were in the range 20-22% and average density value was 1087 kg/m3. The potatoes were hand peeled, cut into spheres (radius R = 1.1-2.87 cm), cubes (1 X 1 X 1 cm) and French fry strips (parallelepipeds of 1 X 1 X 5 cm and 1 X 1 X 7 cm). Dip solutions were made by dissolving the appropriate amount of sodium bisulfite in distilled water. Experiments were performed at 20°C with solutions of 0.25, 0.5, 0.8, and 1.7% of sodium bisulfite equivalent to 1538, 3076, 4922, and 10466 ppm of sulfur dioxide in dipping solution, respectively. Rectilinear velocities of potato cuts in the antioxidant bath ranged from 0.010 to 0.030 msec; immersion times varied from 30 set to 2 min.

Residual levels of sulfur dioxide in pre-peeled potatoes were de- termined simultaneously by two methods: (a) iodometrically (Hart and Fisher, 1971), and (b) calorimetric method (AOAC, 1980).

The -y,,, Pn,m,f, roots (Eq. 8 and 12) were determined from a computational algorithm using a modified False Position method. More than thirty terms of the series were calculated in each geometry to get convergence, because of the short immersion times involved in the diffusional process.

Theoretical uptake for different geometries obtained from mode1 equations were compared to experimental data of residual sulfur diox- ide to verify the applicability of the diffusive mass transfer model. Percent deviation (%E) was calculated according to:

% E = I x 100 (17) v N-l

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SULFUR DIOXIDE UPTAKE IN PEELED POTATOES

Spheres R = 1.1 cm

V I I I I 30 60 90 120

w

immersion t ime (set)

Fig. l-Comparison of predicted and experimental sulfur diox- ide uptake in spherical pre-peeled potatoes. R = 1.1 cm. Dry matter 22%. Concentration of dipping solution 0.5% sodium bi- sulfite. Experimental A v = 0.028 mlsec; l v = 0.020 mlsec. Predicted - kL = 9.75 x 10-6mlsec, Bi = 99.4; --- kL = 8.24 x 10-6mlsec,Bi = 84.0.

where W, and W, represented the theoretical and experimental residual sulfur dioxide content respectively and N the number of experimental data considered.

RESULTS & DISCUSSION

SULFUR DIOXIDE uptake was strongly dependent on dry matter content of the sample. Density and total solids varied along the tuber; different values corresponded to cortical, peri- medullary and pith tissues. For samples with an average dry solids matter of 22%, the effective diffusion coefficient which minimized differences between theory and experiment was D, = 1.078 x 10e9 m*/sec. This value can also be predicted using Eq. (2) and considering that Dso, - ~~0 = 1.7 X 10m9m2/ set at 25°C (Perry and Chilton, 1973), E = 0.78 (total solids content = 22%) and adopting a tortuosity factor Sz = 1.23 similar to that reported by Califano and Calvelo (1983) and obtained by Stahl and Loncin (I 979).

Spherical samples of R = 1. I cm were immersed in solu- tions of different sulfur dioxide concentration with rectilinear velocities of 0.020 and 0.028 m/set (similar to industrial con- ditions) and analyzed for SO2 residual values. Figure 1 cor- responds to a solution concentration of 0.5% sodium bisulfite (equivalent to 3076 SO2 ppm in the fluid). Theoretical uptake curves were obtained fitting the following values in Eq. (9): D, = 1.078 x 10-9m2/sec, kL = 9.75 x 10-6m/sec, (Bi = 99.4) in the case of v = 0.028 mlsec, and kL = 8.24 X 10-6m/sec, (Bi = 84.0) for v = 0.020nVsec. Mass transfer coefficients were calculated using Eq. (15); physical properties of liquid solution have been included in Re and SC numbers calculations.

A linear relationship between uptake values and dip solution concentration was predicted using Eq. (14) and experimentally verified in spherical geometry for different velocities and im- mersion times (Fig. 2). This linear relationship was also valid iu cases of cubes and parallelepipeds. Similar effects of dip- ping solution concentration on residual SO2 in pre-peeled po- tatoes have been reported (Francis and Amla, 1961; Furlong, 1961; Ross and Treadway, 1961). Percent deviation of the model (Eq. 17) was 1.83% for spherical geometry. Model

1

500 - G :

$A 450- x 0 m

: = 400- x u

5 0 f 350- II 3

300 -

250 -

200 -

150 -

100 -

I : 60 set

conccntrrtion of dippinq solution (%sodium bisulfite)

Fig. 2-Influence of dipping solution concentration on sulfur dioxide uptake. Sphere, R = 1.1 cm. Experimental A v = 0.028 mlsec; l v = 0.020 mlsec. Predicted - kL = 9.75 x 10-G mlsec; --- kL = 8.24 x 1O-6 m/set.

equations were used to analyze the influence of potato cuts velocities and sizes on residual sulfur dioxide values.

Theoretical curves (Fig. 3) show the effects of Biot number in spheres (R = 1.1 cm) with external concentration of 0.5% sodium bisulfite. When these samples were immersed into a stagnant solution and were motionless, Biot number was 1.5; for sample velocities of 0.028 m/set, Bi z 100. Mass transfer has a mixed control in the velocity range of industrial operating conditions. Constant interfacial approach was included in the upper curve denoted by CC.

Size of spherical samples has a marked influence on uptake values in relation to whole pre-peeled potatoes. Figure 4 shows this effect for samples with R = 1.1,2.5 and 3. I cm submitted to a rectilinear velocity of 0.028 m/set in a solution of 0.5% sodium bisulfite. Mass transfer coefficients in each case were indicated in the same figure.

The uptake of sulfur dioxide by cubes and parallelepipeds (French fry potato strips) was calculated using Eq. (11). Com- parison of theoretical and experimental results for cubes (I X 1 X 1 cm), with rectilinear velocities of 0.020 and 0.028 m/set

is observed in Fig. 5. Good agreement with experimental data was obtained calculating the mass transfer coefficients with Eq. (16).

In the case of French fry potato strips, uptake values were measured in parallelepipeds (1 x 1 x 7 cm) with rectilinear motion of 0.010 and 0.020 m/set (Fig. 6). Corresponding mass transfer coefficients were indicated in the same figure. Percent

62CJ-JOURNAL OF FOOD SCIENCE-Volume 51, No. 3, 1986

I 30 60 90 120 150 180

Fig. 3-Effect of Biot number on sulfur dioxide uptake in spher- icalpre-peeled potatoes of R = 1.1 cm. Concentration of dipping solution 0.5% sodium bisulfite. Curve CC corresponds to con- stant interfacial concentration. 0, = 1.078 x lO-9 mz/sec.

250 -

r : 200-

: ._ x 0 - 150-

: =

z loo-

30 60 90 120 150 160

Fig. 4-Size effect on uptake values. Spherical geometry: v = 0.028 mlsec; 0, = 1.078 x 10m9 m2/sec; concentration of dip- ping solution = 0.5% sodium bisulfite - R = 1.1 cm, kL = 9.75 x 1O-6 m/set; -.-. R = 2.5 cm, k, = 6.4 x 1O-6 m/set; --- R =

3.1 cm, kL = 5.77 x 10m6 m/set.

deviation of the model for plane sheet solution was calculated including data for cubes and parallelepipeds (E = 1.14%).

Parallelepipeds of different length are used as potato strips in the pre-peeled potato industry. The effect of length on the amount of sulfur dioxide retained by the product was analyzed comparing results for potato samples of 1 X 1 X 5 cm and 1 x 1 x 7 cm with v = 0.020nUsec. Differences were of 0.02%

at immersion times of 120 set showing the small effect of this variable on uptake values.

Results show that the diffusive mass transfer model for po- rous media described adequately sulfur dioxide uptake in pre- peeled potatoes immersed in bisulfite solutions. The equations obtained can be applied to the engineering optimization of the antioxidant treatment, relating necessary levels of sulfur diox- ide in the product with operating industrial conditions.

NOMENCLATURE

a, b, c

Bi

half thickness values for potato strips in x, y, z directions respectively (m) Biot number

I 30

I 60

1 1 90 120

immersion time (set)

Fig. 5--Comparison of predicted and experimental data for cubes (1 x 1 x 1 cmJ. Concentration of dipping solution = 0.5% so- dium bisulfite. Experimental A v = 0.028 mlsec; l v = 0.02 mlsec. Predicted (0, = 1.078 x lO-9 m2lsecJ - k, = 1.57 x lO-5 mlsec; --- kL = 1.32 x 1O-5 mlsec.

C

C’

Cf

C’f

DAB

D,

%E F kL

L

M,

solute concentration in the solid (weight of sul- fur dioxide per unit volume of potato) solute concentration in the solid (weight of sul- fur dioxide per unit volume of solution; c’ = C/E) solute concentration in the dipping solution (weight of sulfur dioxide per unit volume of potato) solute concentration in the dipping solution (weight of sulfur dioxide per unit volume of solution; cr = c&) molecular diffusivity (m*/sec) effective diffusion coefficient of sulfur dioxide in the potato tissue (m*/sec) percent deviation fractional uptake (F = Mt/M,) mass transfer coefficient at the potato fluid in- terface (m/set) characteristic length in plane sheet geometry (ml amount of diffusing substance in the sample at time t

M,

N r

R R%p

amount of diffusing substance in the sample at infinite time number of experimental data distance measured from the center of the solid (m) sphere radius (m) Reynolds number v up

Re, = p v 2Rlp; Re, = p

Volume 51, No. 3, 19864OURNAL OF FOOD SCIENCE-621

SULFUR DIOXIDE UPTAKE IN PEELED POTATOES

French fry strips (Paralltltpiptd 1x1.7 cm)

250

I 1 I I 30 60 90 120

immersion t ime (set)

Fig. GComparison of predicted and experimental data for French fry potato strips (1 x 1 x 7 cmJ. Concentration of dipping so- lution = 0.5% sodium bisulfite. Experimental A v = 0.02 mlsec; l v = 0.03 m/see. Predicted - kL = 0.50 x 10-5mlsec; --- kL = 0.35 x lo-= mlsec.

.-.-.-. 1 I 1 s5 cm - lal~7cm

60 90 120 immersion t ime (set)

Fig. 7-Effect of French fry strips length on uptake values. v = 0.02 mlsec; -.-. 1 x 1 x 5 cm; - 1 x 1 x 7 cm. Dipping solution concentration = 0.5% sodium bisulfite.

SC Schmidt number SC = p. / p DAB t time (set) V rectilinear velocity of potato cuts (m/set)

Superscripts *

Subscripts e t

potato sample volume (m3) residual sulfur dioxide (ppm) coefficient defined in Eq. (13) positive roots of Eq. (12) positive roots of Eq. (8) porosity, water content on wet basis solution viscosity (kg/m set) density (kg/m3) shape factor in Eq. (2) tortuosity factor

dimensionless variables

experimental values. theoretical values

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the quality of pre-peeled potatoes. Am. Potato J. 38: 121. Anderson, E.E. and Zapsalis, C. 1957. Technique ups quality, shelf life of

repeeled potatoes. Food Eng. 29: 114. A 8 AC. 1980. “Official Methods of Analysis,” 13th ed. Methods 20.109,

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Fitzpatrick, T.J., Porter, W.L., and Houghland, G.V.C. 1969. Continued studies of the relationship of specific gravity to total solids of potatoes. Am. Potato J. 46: 120.

Francis, F.J. and Amla, B.L. 1961. Effect of residual sulphur dioxide on the quality of pre-peeled potatoes. Am. Potato J. 38: 89.

Furlong, CR. 1961. Preservation of eeled potato II. Uptake of sulphite by peeled and chipped potato treate B with sodium metabisulphite. J. Sci. Food Agric. 12: 49.

Hart, F.L. and Fisher, H.J. 1971. “Modern Food Analysis,” Method 17-13. Springer Verlag, New York.

Keijbets, M.J.H. 1981. Vacuum and gas packing, an alternative for sul- phite m pre peeled potatoes. 8th Triennial Conference of the European Association for Potato Research Abstracts, p. 263.

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M S received 5117185; revised 10130185; accepted 10/30/85.

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