JournalofFoodEngineering 14 (1991) 317-326

Determination of Diffusion Coefficients in Volume-Changing Systems - Application in the Case of

Potato Drying

V. Gekas* & I. Lamberg

Division of Food Engineering, Lund University, PO Box 124, S-221 00 Lund, Sweden

(Received 11 April 1990; revised version received 10 October 1990; accepted 23 November 1990)

ABSTRACT

This work was based on a modified Crank approach in order to deter- mine the di@sion coefficient in systems where volume changes occur during drying. The modified model was applied to the case of potato dry- ing, for both raw and blanched potatoes, at two temperatures: 60 and 80°C.

The degree of volume change (shrinkage) was diflerent for raw and blanched potatoes, but only small differences have been observed for the dimsion coecient of reference, which was evaluated at 2.25 (+ O-13) x lo-I0 m2/s for raw potatoes and 248 (+ 0.28) x 1O-‘o m’ls for blanched potatoes.

NOTATION

4,B Mass of component A and B, respectively (kg) A Mass flow rate of component A (kg/s) c Concentration ( kg/m3) d Exponent in eqn ( 13) D Diffusion coefficient (m*/s) F Flux (kg/m* s)

s” Number of experiments Surface area (m*)

t

V Time (s) Volume ( m3)

*To whom correspondence should be addressed.

317

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

318 I/ Gekas, I. Lamberg

Basic volume (specific) of B ( m3/kg) Length-scale in the constant-volume frame (m) Water content (kg/kg)

1 Thickness (m) 6 Length-scale in the alternative frames (m)

Subscripts A Component A B Component B

f’ Equilibrium Final

i Initial ref Reference

Superscripts A Component A B Component B M Mass V Volume

INTRODUCTION

Potatoes, like other foodstuffs, undergo shrinkage when subjected to drying (Gekas et al., 1987; Motarjemi, 1988). This implies a difficulty in the measurement of the diffusion coefficient using Fick’s second law:

x-x In e

8 Dtn’

Xi-X, =~_--....-

n ~2

because i, the thickness of the studied specimens, changes during the process. Crank (1975) and Fish (1958) have shown a way to overcome this problem.

(4

09

Introduce a moving frame of reference in which 1 remains constant. Then eqn (1) determines a kind of reference diffusion coefficient. Crank uses the terms constant-mass (DAM) and constant-basic-volume-of-B (DAB) coefficients, whereas Fish uses the term pseudo-coefficient ( DLseudo). Derive a relationship between the constant-volume (or the proper) coefficient and the reference diffusion coefficient.

Diffusion coefficients in volume-changing systems 319

(c) Combine steps (a) and (b) to determine the constant-volume (or the proper) diffusion coefficient.

Crank presupposed a one-dimension volume change, while Fish assumed an isotropic, three-dimensional change of volume. Both consider one-dimensional diffusion. The assumptions concerning volume changes represent idealities not usually met in practice. There- fore, a modified approach is suggested in order to cover cases of fractal volume change (Gekas et al., 1987); this approach is discussed in some detail in the next section. Preliminary experimental results on meat and potato drying provide support evidence that volume changes are neither one-dimensional nor isotropically three-dimensional, but follow a fractal-dimension relationship to the change of thickness. The experi- mental part of the present paper focuses mainly on potato drying with both raw and blanched potatoes at two different drying temperatures.

THEORY

The problem of the definition and measurement of diffusion coefficients in a two-component volume-changing (swelling or shrinking) system has been studied by Crank (1975). In that pioneering work, two alternative frames of reference were used, so that a suitably defined basic volume of one component, or two components together, remained constant during diffusion, while the total actual volume did not. Then, the diffusion coef- ficient was defined in each frame of reference and, in a very complicated manner, its relationship to the usual constant-volume diffusion coef- ficient was derived.

Uniaxial diffusion, perpendicular to the surface of a thin specimen, was considered, as in the case of Crank. The following definitions are valid:

dCV, FV= _Dv_ dx

A (1) where CA = v

(2) where CAB = A

*

dC; FM= -DM_ GM

(3) where CAM = A

V;(A + B)

(14

(24

(34

320 I/ Gekas, I. Lamberg

The key to the approach is that the mass flow rate (A) is invariant (or, in other words, the same) independent of the frame of reference used. A is equal to the flux times the surface area in any frame of reference:

where

A=FvSv=FBSB=FMSM (4)

,S”= V;(A + B)

AM

It is also obvious that

(5)

(6)

Substituting S”, S”, SM from eqn (5), and F v, FB, FM from eqns ( 1 ), (2) and (3), and using eqn (6), the following expression is obtained for eqn (4):

DvdC;V=DB dC:V;B

(A”) z A (n”y

=DM dC,MV;(A +B)

(A”)l

Using eqn ( la), (2a) and (3a), this simplifies to

Dv 0; DA” (n”)‘=py=fl

Volume change following the direction of diffusion

In this case, the volume is proportional to the thickness, so that

V V;B V;(A +B)

37=-T= /TM

Equation (9) combined with eqn (8) gives

Dv 0; DM

v’-( V;B)’ (V;(A +B)}2

Written in the form

DM=DV total basic volume ’

actual volume I

(7)

(8)

(9)

(10)

it is at the expression obtained by Crank.

321 Diffusion coeficients in volume-changing systems

Isotropic three-dimensional volume change

In this case, the volume is related to the thickness in the following way:

V VO,B VO,(A + B) --_=-= (I”) 3 (A”)” (lM)3

so that

D” 0; DM 2/3 = ( j,+)2/3 = { I/;(A + B f/3 V

(11)

(12)

Again, rewriting in the form

basic volume of component B 2’3

actual volume

one finds a similar relationship to that given by Fish (1958), except D proper in place of D” and Diseudo in the place of DAB.

Arbitrary volume change

In practice, volume changes are related to changes in thickness via a quasi-fractal dimension exponent d, as follows:

(13)

Evidence of this is provided by experiments carried out in this labora- tory (Gekas et al., 1987). In the general case, eqn ( 12) thus takes the form

V

$!,d = Df: DM

( V;B)2’d= (( V;(A + B)J21d (14)

with d = 1 in the case of a one-dimensional volume change, d = 3 in the case of an isotropic three-dimensional volume change, and d equal to a fractal in the most general case.

Use of DAB is recommended in swelling cases (e.g. when a dry gel takes up moisture), since the initial dry volume of the gel (or the dry thickness) is known.

In cases of shrinkage, use of DM is more convenient, since the total basic volume (or the total basic thickness A”) can be found easily from the initial conditions. The above are summarized in Table 1.

TA

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Diffusion coefficients in volume-changing systems 323

MATERIAL AND METHODS

Bintje potatoes were obtained from a local potato storage (7°C 95% RH) plant in Kavlinge (Sweden) during autumn 1987. The size was approximately 50 x 80 mm and the density ranged between 1080 and 1090 kg/m3. The moisture content varied between 4.50 and 6.29 g water/g dry solid. The potatoes were then kept at 8°C for a week and tempered at 20°C overnight.

For the heat treatment, the potatoes were cut into 6-mm-thick slices with an electrical slicing machine, and from these 40 x 40-mm squares were cut. The slices were kept in a wire net cage and were equilibrated in a water bath at 20°C for 15 min. Blanching took place in a thermo- statically controlled water bath at 75°C for 15 min (see Lamberg & Hallstrom, 1986). Raw and blanched potato slices were dried in a con- vection oven (Skjoldebrand, 1979) at 60 and 80°C. The relative humid- ity was in both cases 30%, and the air velocity 2 m/s.

The dry air and wet-bulb temperatures were measured with Cr/Al thermocouples with a thickness of O-3 mm. The wet-bulb temperature was measured by thermocouples covered with a wick immersed in a beaker containing distilled water. The air velocity was measured with an anemometer. The moisture content was determined by an oven-drying method: 70°C for 24 h, and 105°C for 1 h.

The height and length of the potato slices were measured, and the weight and time registered by a computer every 15 min for 3 h. After this period, the potato slices were dried for 20 h and the height, length and weight were finally noted.

The fractal exponent d was calculated using eqn (13), from the initial and final values of volume and thickness. Dref was calculated using eqn (1 ), where 1 was the initial thickness of the potato square. Dproper was then calculated from eqn ( 18) in Table 1.

RESULTS AND DISCUSSION

The results for the degree of shrinkage (volume change) of raw and blanched potatoes during drying at 60 and 80°C are presented in Tables 2 and 3, respectively. As shown, volume change is related to thickness change via a fractal exponent (d). Raw potatoes show a higher d-value than blanched potatoes. The same trend is observed at both 60 and 80°C.

The results for the diffusion coefficients (D,, and Dproper) are shown in Tables 4 and 5, for drying at 60 and 8O”C, respectively. Whereas the

324 V. Gekas, I. Lamberg

TABLE 2 Volume Change during Potato Drying at 60°C 30% RH, 2 m/s

Kind of potatoes

Raw(m=4)

Average Standard deviation

~I/~, ‘il ‘f d”

2.88 7.10 1.85 3.02 697 1.76 3.33 7.16 1.64 3.18 6.72 1.65

3.10 6.99 1.72 0.20 0.20 0.10

Blanched (M = 4)

Average Standard deviation

“From eqn ( 13).

4.13 7.74 1.44 3.88 6.41 1.37 4.4 1 8G 1.40 3.79 6.65 1.42

4.05 7.20 1.41 0.28 0.79 0.03

TABLE 3 Volume Change during Potato Drying at 8o”C, 30% RH, 2 m/s

Kind of potatoes

Raw(m=3)

Average Standard deviation

&I‘6

3.33 3.15 3.45

3.31 0.15

Blanched (m = 3 ) 4.39 7.81 3.55 6.1 1 397 6.96

Average 3.97 Standard deviation 0.42

6.96 0.85

vil vf d”

6.95 1.61 6.7 I I.66 7.6 I 1.64

7.09 1.64 0.47 0.02

1.39 1.43 1.4 1

1.41 0.02

“From eqn ( 13).

usual diffusion coefficient ( Dproper ) differs for raw and blanched potatoes, D,, is rather constant in the studied range of temperatures, being slightly higher for blanched potatoes.

Based on the average values and their standard deviations reported in Tables 4 and 5, the resulting mean values of D,, were 2.25

Diffusion coefficients in volume-changing systems 325

TABLE 4 Diffusion Coefficients of Water in Potato Dried at 6o”C, 30% RH, 2 m/s

Kind of potatoes D,, x 10’” D

(m’ls)

Raw(m=4) 2.13 0.26 2-18 0.24 2.43 o-22 2.16 0.2 1

Average 2.22 0.23 Standard deviation 0.14 0.02

Blanched (m = 4) 2.47 014 2.65 0.18 2.18 0.11 2.24 0.16

Average 2.38 0.15 Standard deviation O-22 o-03

TABLE 5 Diffusion Coefficients of Water in Potatoes Dried at 8o”C, 30% RH, 2 m/s

Kind of potatoes Dref x 1Oiu D proper x 10’”

Raw(m=4) 2,32 0.21 2.12 0.2 I 2.37 o-20

Average 2.27 0.20 Standard deviation 0.13 0.006

Blanched (m = 4) 3.01 0.16 2.36 0.19 2.46 0.16

Average 2.61 Standard deviation 0.35

0.17 0.02

( f 0.13) x lo- lo m2/s for raw potatoes, and 2.48 ( + O-28) X lo- I” m’/s for blanched potatoes.

As far as Dproper is concerned, the values reflect the expected dif- ferences in volume change discussed above. This may, to some extent, explain the large variations concerning diffusion coefficient values reported in the literature. (For a summary, the reader is referred to Motarjemi, 1988.).

326 K Gekas, 1. Lamberg

In addition, values of D,, for meat (Gekas et al., 1987; see also Motarjemi, 1988), showing similar trends, also were reported. It is there- fore suggested that the reference diffusion coefficient be used, since this is independent of volume change, and the model as discussed in this paper be used in order to calculate the usual (proper) diffusion coef- ficient.

CONCLUSIONS

It was possible to derive relationships between the two kinds of diffusion coefficients in volume-change cases in a manner similar to, yet simpler than, Crank’s, based on the fact that mass flow rate is invariant when using alternative frames of reference. The relationship between Dref and D proper has been extended in cases of arbitrary volume change:

& i

total initial volume ‘ld -= D proper actual volume !

(18)

The fractal exponent (d) for blanched and raw potatoes during drying has been measured at two temperatures, 60 and 80°C. Volume change (shrinkage) was different for the two types of pretreatment of potatoes, and this was reflected in the values of Dproper. Dref, on the other hand, has not shown particular changing trends, being only slightly higher for blanched than for raw potatoes.

REFERENCES

Crank, J. (1975). The Mathematics of Difision, 2nd edn. Oxford University Press, Oxford, pp. 205-9.

Fish, B. P. (1958). Diffusion and thermodynamics of water in potato starch gel. In Fundamental Aspects of the Dehydration of Foodstuffs. Society of Chemical Industry, London, pp. 143-57.

Gekas, V., Motarjemi, Y., Lamberg, I. & Hallstrom, B. (1987). Evaluation of diffusion coefficients in a shrinking system. Paper presented at the Thijssen Memorial Symposium, Eindhoven University of Technology, Eindhoven, The Netherlands, 5-6 November.

Lamberg, I. & Hallstrom, B. (1986). Thermal properties of potatoes and a computer simulation model of a blanching process. J. Food Tech., 21, 577-85.

Motarjemi, Y. (1988). A study of some physical properties of water in food- stuffs. PhD thesis, Division of Food Engineering, Lund University, Lund, Sweden.

Skjoldebrand, C. (1979). Frying and reheating in a forced convection oven. PhD thesis, Division of Food Engineering, Lund University, Lund, Sweden.