drying food

Journal ofFood Engineering 13 (1991) 103-l 14

Shrinkage Effect on Drying Behavior of Potato Slabs

Constantino Suarez & Pascual E. Viollaz

Departamento de Industrias, Facultad de Ciencias Exactas y Naturales. Ciudad Universitaria, ( 1428) Buenos Aires, Argentina

(Received 26 September 1989; revised version received 30 May 1990: accepted 6 June 1990)

ABSTRACT

The rate of drying of potato slabs at different initial moisture contents and thicknesses was investigated. The variation of surface area with time was measured in order to consider the degree of shrinking of the samples during drying. Experimental drying curves were interpreted in terms of Ficks law for shrinking bodies and by means of the classical difSctsiona1 model, without shrinking. It was found that for relatively short drying time Ficks model with shrinking correlates adequately with the experimental results, with a di@sion coeficient independent of moisture content. However, in terms of the classical diffiaional model the experimental data predict a strong dependence of the difSusion coeficient on moisture content. Such dependence was attributed to a shortcoming of the classical Fickian model and not to physical reasons.

NOTATION

1 A j: D

D,r FO,

R

R RI, RII

Solution of eqn ( 14) Initial surface area of natural sample ( cm2) Initial surface area of predried sample (cm) Mutual diffusion coefficient (cm/s) Effective diffusion coefficient (cm/s) = DO/R:, Fourier number based on the totally dried half- thickness of the slab Half-thickness of the slab (cm) Initial half-thickness of the slab (natural samples) (cm) Initial half-thickness of the slab (predried samples) (cm) Defined by eqn ( 13)

103

Journal of Food Engineering 0260-8774/91/$03.50 - 0 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain

104

Ri

Rs

U

UO

14:)

ui

Uf

X

Z

E

PA

PA,

PA,,

PB

Pb, Ph

P, Pf PW

C. Suarez, P. E. Viollaz

Half-thickness of the slab at the equilibrium moisture content (cm) Half-thickness of the totally dried solid assuming unidirectional shrinkage (cm) Moisture content (g water/g dry solid) Initial moisture content of natural samples (g water/g dry solid) Initial moisture content of predried samples (g water/g dry solid) Equilibrium moisture content (g water/g dry solid) =(U-Ui)/(U"-Ui)

Coordinate along the diffusion path (cm) = E /R, (dimensionless coordinate along the diffusion path)

Time (s) Coordinate along the diffusion path measured from the centre of slab and equal to the volume of totally dried solid divided by transversal area (0 G 6 G R,) (cm) Mass concentration of water by unit volume (g/cm) Equilibrium interfacial mass concentration of water (g/cm) Initial mass concentration of water (g/cm) Mass density of dried solid (g/cm) Mass density of dried solid at initial moisture content (g/cm) Mass density of dried solid at equilibrium moisture content (g/cm) Mass density of fully dried solid (skeletal density) (g/cm) =(PA,, - PA,)/(P~ - PA,) WmensionW Mass density of water (g/cm)

INTRODUCTION

Considerable research is reported in the literature to develop an understanding of the mechanism of moisture movement in natural products, but findings are not yet conclusive. Ficks second law for diffusion was used by many investigators to describe the drying process, accepting the hypothesis that the resistance to moisture flow is distributed throughout the material (Vaccarezza et al., 1974; Young & Whitaker, 1971; Chen & Johnson, 1969; Alzamora & Chirife, 1980), among others. Other solutions have been postulated considering variable diffusion coefficients but neglecting shrinking of the body during drying (Hall & Rodriguez-Arias, 1958).

Shrinkage effect on drying behavior ofpotato slabs 105

Shrinking of biological products during drying takes place simultane- ously with moisture diffusion and thus may affect the moisture removal rate. Hence, a study of the shrinking phenomena is of importance for better understanding of the drying process. Consideration of shrinking in drying models is generally difficult because of the lack of information about shrinking coefficients and their relationship with moisture dif- fusivity. Viollaz and Suarez (1984) obtained a mathematical expression of Ficks second law for drying of shrinking bodies, assuming unidirectional shrinking and volume additivity for water and dry solids.

The objective of the present study is to analyze the drying curves of potato slabs in the light of the classical diffusional model and by means of Ficks equation for shrinking bodies. During the development of this work some of the difficulties derived from application of the diffusional model to the drying curves of products which change their volume during drying will be presented.

THEORETICAL CONSIDERATIONS

The classical approach to the analysis of the drying process under isothermal nonsteady diffusion is by means of the equation:

au/a6 = D,,@u/i3x2 (1) Equation (1) was proposed by Sherwood (1929) to analyze the drying process in an infinite slab and used by various investigators to interpret experimental drying curves, even for those cases where shrinking is important. A different approach was proposed by Viollaz and Suarez (1984) who postulated the following equation for an infinite slab:

apAlae = Da2p,lax (2J According to the above-mentioned authors eqn (2) can be transformed as follows assuming volume additivity and unidirectional shrinking:

au/aFO,=a/az(l/(i +P,lI~PH)~au~az) (3) Equation (2) was numerically solved by means of Landaus transforma- tion in order to fix the integration domain (Viollaz & Suarez, 1984). Equation (3), which has a fixed domain of integration, was also numerically solved by Viollaz and Suarez ( 1985), its solution being coincident with the solution of eqn (2). Furthermore, an analytical solution of eqn (2) was obtained for a semi-infinite body by Viollaz (1985). Such solutions can be used to describe the drying process in

106 C. Suarez, P. E. Viollaz

finite slabs during relatively large periods of time, owing to the low values of the diffusion coefficients usually found in food dehydration.

MATERIALS AND METHODS

The potatoes used for the dehydration experiments were bought in a supermarket and stored in a refrigerator at 5C for about one week. Samples of different thickness were mechanically sliced and the corresponding thicknesses measured with a dial micrometer. The edges of the slab were sealed with a resin in order to assure dehydration from the two major faces of the slab.

To investigate the effect of the initial moisture content on drying rate, some potato slabs were predried to different moisture content levels. In order to assure uniform moisture throughout the slab, the predried samples were left to equilibrate at room temperature for about three days prior to their use in the drying experiments. The surface area corresponding to the samples having initial moisture content were immediately measured after they were sliced. Similar measurements were made with the predried samples, after they reached uniform moisture distribution, and during the drying process. The surface area measurements were made by drawing the edges of the slab on a piece of paper and weighing the paper on a precision balance ( + O-1 mg).

To estimate the mass density of the fully dried solid, ,os, volume measurements were performed using a pycnometer and chlorobenzene as the fluid. The solid density was calculated as the fully dried weight sample divided by its volume; the resulting value was l-3 g/cm.

The dry solids of each sample were determined after each drying run by the vacuum oven method at 70C for 72 h. The average moisture of the samples was expressed as g water/g dry solid. The equilibrium

OJt, I 100 200 300

Elimid

Fig. 1. Drying curves of potato slabs at 51C; u,, = 396 g/g and R,, = 1.02 cm. 0, Natural sample; A, u;, = 1.87 g/g and RI, = 0.69 cm; 0, ui, = O-88 g/g and RI, = 0.49 cm.

Shrinkage effect on dtying behavior ofpotato slabs 107

moisture content was determined by equilibration against saturated salt solutions which provided known constant relative humidities.

The drying equipment used in this work was described in detail in a previous work (Suarez et al., 1980). It consists, basically, of a centrifugal

0.011 200 3cxl CJImtn)

Fig. 2. Drying curves of potato slabs at 5 1C; u,, = 4.63 g/g and R,, = 0.4 1 cm. A. Natural sample; 0, ui, = 2-45 g/g and RI, = 0.26 cm; 0, ui, = 153 g/g and RI, = 1.2 1 cm.

9lmtn)

Fig. 3. Drying curves of potato slabs at 5 1C; L+~ = 4.63 g/g and R,, = 0.20 cm. A, Natural sample; 0, u;, = 2-60 g/g and RI, = 0-I 5 cm; 0, u;, = l-5 1 g/g and Ri, = O-1 1 cm.


fan which blows the air through a heating section and then upwards through a vertical duct at the end of which was placed the sample to be dried. The sample was withdrawn from the drying chamber at regular intervals and rapidly weighed. All drying experiments were carried out in air at 51C, flowing parallel to the evaporation surfaces at a rate of 11 m/s.

RESULTS AND DISCUSSION

The time-moisture content variation of potato slabs having different initial moisture contents and thicknesses are shown in Figs 1, 2 and 3 (the curves correspond to natural and predried samples). For the sake of clarity it must be mentioned that the curves plotted in Figs 1, 2 and 3 correspond to samples with initial thickness of about 2 cm, 0.8 cm and 0.4 cm, respectively; these values correspond to the natural samples (not predried). It is interesting to observe that the drying behavior between samples with different initial thicknesses differed markedly. The drying curves plotted in Fig. 1 (2R,, 2: 2 cm) for natural and predried samples, show the typical Fickian behavior found by other investigators (Vaccarezza et al., 1974). The plot of log u + versus time gives the straight line characteristic of the first falling rate period and the intercept values extrapolated at large enough time were OS-0.9, approximately. The drying curves of the thinner samples (2R,, = O-8 and 0.4 in Figs 2 and 3, respectively) show a different behavior. As can be seen from these figures most of the drying curves deviate from the straight line, which can be attributed to the appearance of a second falling rate.

Given that the thickness of the predried samples was different from the natural ones owing to the lateral contraction resulting from the moisture losses during the predried treatment, the initial thickness of the predried samples was corrected according to the following equation:

(4)

Equation (4) can be obtained assuming that the specific volume of water and solid are constant and additive. In Fig. 4 several drying runs were plotted in terms of log u + versus 19/Rif (the water density was taken as equal to 1 g/cm3). This kind of plot is in agreement with the analytical solution of eqn (1) for a semi-infinite slab of half thickness RI,, written here for convenience of the reader, and is of the form (Luikov, 1968)


It is clear that if the drying data are interpreted in terms of eqn (5) they must fall in a single curve, a fact that was not observed. As can be seen in Fig. 4, natural and predried samples behave in a quite different way.

Given that the experimental data seems not to obey eqn (1) or the corresponding solution given by eqn (5), three different hypotheses were formulated in order to explain such disagreement.

(a) The diffusion coefficient depends on moisture content. (b) The surface area variation with 6/R{f varies from one sample to

another. (c) The model based on eqn ( 1) is not appropriate.

In order to test these hypotheses the surface area variation of some samples were plotted in Fig. 5 in the form A /A,, versus 0/R; (the curves plotted in this figure correspond to natural samples). It can be seen that the curves are not smooth, increasing or decreasing with time, although with a net tendency to decrease. It is interesting to observe that for relatively short drying times, O/R,', < 300 approximately, the values of A / A, differing slightly from unity, a fact particularly observed for samples of large thickness. However, for increasing values of B/R,',, the different samples show considerable irregularities in the form of the curves, as can be seen in Fig. 5 for some of the used samples. (Other curves not shown here corroborate this behavior.)

OlL- - 0 500 loo0

0/R:

Fig. 4. Drying curves of potato slabs corrected by the initial half-thickness (log 14 + versus e/R{f) (R,,, RI, for natural and predried samples). A, u,,= 3.96 g/g, ai,= O-88 g/g, R,,= 1.02 cm and R:, =0.49 cm; 0, u,, = 4.63 g/g, ui,= 1.53 g/g, R;, =0.39 cm and RI, =0.21 cm; A, u,, =4.63 g/g, R0=0.20 cm; H, u,)= 4.63 g/g, R,,= 0.41 cm 0,

L+, = 3.96 g/g, R,, = 1.02 cm.


Fig. 5. Surface area variation of the slabs. Curve 1: 0, u,, = 4.63 g/g and R,, = 0.20 cm; Curve 2: A, u,, = 3.96 g/g and R,, = 1.02 cm; Curve 3: 0, u,, = 4.63 g/g and R,, = 0.41 cm.

At this point in the analysis the curves shown in Fig. 4 will be analyzed in the light of the results shown in Fig. 5. The sample whose drying behavior is shown in Fig. 5 (curve 1) is expected to show the largest drying rate as it has the largest evaporation area for relatively short drying times (8/R; < 300). However, as the sample shows the smallest drying rate (see Fig. 4) the area variations do not explain the drying behavior of the samples plotted in Fig. 4.

Taking into account that for short drying times (O/R; < 300) the area variation of the samples was relatively small, it was decided to reduce the present analysis to short drying times. Under this circumstance the sample can be considered as a semi-infinite slab, the relationship between u + and t3'12/R,, being a straight line (Luikov, 1968). The resulting curves are shown in Fig. 6, where it can be seen that the curves 1, 2 and 3 practically present the same drying rate (the three curves have similar slopes). The corresponding surface area variations are given in Fig. 5 and correspond to samples that were not predried. It can also be observed that curves 4 and 5 of Fig. 6, which correspond to predried samples, show lower drying rates. It can be concluded that both groups of samples, natural and predried, should have different diffusion coefficients. On the other hand, the different intercept values of the curves at u + = 1 can be attributed to different time lags. The appearance of the time lag in the drying curves is due to the fact that the samples of different thicknesses also have different temperature evolution. It must be noticed that even though the thicker samples have longer time lag, in terms of 01/2/Ro the effect is opposite, i.e., double thickness corresponds to double time lag but 8 /Ri reduces to half.

The analysis of the experimental data, based on the validity of eqn (l), and represented in Fig. 6 shows certain variations of the diffusion coefficient with moisture content. An alternative analysis will be undertaken


Fig. 6. Dimensionless moisture content variation for short drying times. 0. u,, = 3.96 g/g and R,, = 1.02 cm; I[;, = 0.88 g/g and RI, = 0.49 cm; A , ~4,~ = 4.63 g/g and R,, = 0.40 cm; u:,= 1.53 g/g and RI, = 0.12 cm; 0, u,, = 3.63 g/g and R,, = RI,= 0.20 cm; A.

u,, = 4.63 g/g and R,, = RI, = 0.41 cm; n , u,, = 3.96 g/g and R,, = RI, = 1.02 cm.

U*

Fig. 7. Dimensionless moisture content variation for short drying times (characteristic length RI;). 0, I~,~ =4.63 g/g and R,,=0.20 cm; A, u,, =4,63 g/g and R,, = 0.41 cm; 0, u,, = 3,96 g/g and R,, = 1.02 cm; 0, q, = 4.63 g/g and R,, = W40 cm; u:, = 153 g/g and R:, = 0.2 1 cm; n , u,, = 3.96 g/g and R,, = 1.02 cm; u;, = 1.87 g/g and RI, = 0.69 cm; A.

H,, = 3.96 g/g and R,, = 1.02 cm; u:, = 0.88 g/g and RI, = 0.49 cm.


based on eqn (2). For this purpose we will make use of the analytical solution derived by Viollaz (1985) from eqn (2) and that corresponds to eqn (12) in the work previously mentioned. For convenience, that solution is written here in terms of moisture content, taking into account that there exists a linear relationship between u and the half thickness R of the slab, which varies with time as a consequence of the shrinkage effect. The solution is (Viollaz, 1985):

u +=1_2a(D8)*/(R,,-Ri) (6)

Equation (6) can also be written in the following form:

U + = 1 -(L'~)"'I(Ro(R,, -Ri)/(2a&)) (7)

On the other hand, if it is assumed that the drying is unidirectional, this results in

RiPB, =Rc~PB,, (8)

and

(Ro -Ri)IRo = 1 - (PS,,/PB, )

Assuming additivity of volumes allows for the relation:

(&I/P,) + (&JP,) = 1

and eqn (9) results in

(9)

(10)

(Ru -Ri)IR,, = (PA,, - P*,)/(P~ - PA,) = P + (11)

Based on these transformations of variables, eqn (7) can be put in the form

with

u + = 1 - (DB)/R;; (12)

R;I= Rl,p/2a (13)

The variable Ri, resulting from the product of p /2a times the initial half-thickness (R, for natural samples and Ri, for predried samples) can be considered as the effective initial half-thickness. The main advantage of this variable is the possibility of reducing to a single curve the drying behavior of samples having different initial and equilibrium moisture contents. The value of a is obtained from the expression (Viollaz, 1985)

an/* erfc(a)exp(a*)=p+ (14)

Shrinkage effect on drying behavior ofpotato slabs II3

Some drying curves were analyzed by means of eqns ( 12) and ( 13) and the results plotted in Fig. 7 in terms of u+ versus t9/2/R:[. From this figure it is observed that the drying curves are straight parallel lines during certain time intervals. According to eqn (12) the present result means that the diffusion coefficient can be considered independent of moisture content. The fact that a single straight line was not obtained can be due to the different thermal history of the samples with the con- sequent different time lag. The departure of the straight line as drying time increases, particularly notorious in the thinner samples, is not necessarily due to diffusion-coefficient variation with moisture content but to surface area variations that samples may undergo during drying.

CONCLUSIONS

Drying curves of potato slabs having different initial moisture contents and thicknesses were analyzed in the light of the classical diffusion equation (Ficks second law) with constant diffusion coefficient and no shrinking. The resulting curves, plotted in the form of II + versus 8 /Ri, show a strong dependence of the diffusion coefficient on moisture content.

A different result was obtained using Ficks law of diffusion for a shrinking body. Assuming unidirectional shrinkage and constant diffusion coefficient, the drying behavior of potato samples was described satisfactorily using the analytical solution for diffusion with volume change in a semi-infinite slab, for relatively short drying times. It can be supposed that the deviation of the experimental data from the model, observed in some samples for large drying times, is not necessarily due to the variation of the diffusion coefficient with moisture but to other factors such as the lateral contraction of the samples, which may vary considerably from one sample to another in a rather unpredictable way.

It was also observed that the experimental data do not fall in a single drying curve but form a group of parallel straight lines with different time lags. This effect was attributed to the different thermal history of the samples.

REFERENCES

Alzamora, S. M. & Chirife, J. (1980). Some factors controlling the kinetics ot moisture movement during avocado dehydration. .I. Food Sci., 45, 1649.

Chen, S. C. & Johnson, W. H. ( 1969). Kinetics of moisture movement in hygro- scopic materials. Trans. ASAE, 12 (l), 109.

114 C. Suarez, Z? E. Viollaz

Hall, C. N. & Rodriguez-Arias, J. H. ( 1958). Application of Newtons equation to moisture removal from shelled corn at 40-140F. J. Agri. Engng Res., 3, 275.

Luikov, A. V. ( 1968). Analytical Heat Diffusion Theory. Academic Press, New York.

Sherwood, T. K. ( 1929). The drying of solids. Zndust. and Engng Chem., 21, 12. Suarez, C., Viollaz, P. E. & Chirife, J. ( 1980). Diffusional analysis of air drying of

grain sorghum. J. Food Tech., 15,523. Vaccarezza, L. M., Lombardi, J. L. & Chirife, J. (1974). Heat transfer effects on

drying rate of food dehydration. Canadian J. Chem. Engng, 52,576. Viollaz, P. E. (1985). An analytical solution for diffusion in a shrinking body. J.

Polym. Sci.: Polym. Phys. Edn, 23, 143. Viollaz, P. E. & Suarez, C. ( 1984). An equation for diffusion in shrinking or

swelling bodies. J. PoIym. Sci. Polym. Phys. Edn, 22,875. Viollaz, P. E. & Suarez, C. (1985). Drying of shrinking bodies. Am. Inst. Chem.

EngngJ., 31, 1566. Young, J. H. & Whitaker, T. B. ( 197 1). Numerical analysis of vapor diffusion in a

porous composite sphere with concentric shells. Trans. ASAE, 14 (6), 1051.

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