Estimation of characteristic damage time of food materials in pulsed-electric fields

Estimation of characteristic damage time of food materialsin pulsed-electric fields

N.I. Lebovka a,b, M.I. Bazhal a,c, E. Vorobiev a,*

a D�eepartement de G�eenie Chimique, Universit�ee de Technologie de Compi�eegne, Centre de Recherche de Royallieu,

B.P. 20529-60205 Compi�eegne Cedex, Franceb Institute of Biocolloidal Chemistry named after F.D. Ovcharenko, NAS of Ukraine, 42, blvr.Vernadskogo, Kyiv 252142, Ukraine

c Ukrainian State University of Food Technologies, 68, Volodymyrska str., Kyiv 252033, Ukraine

Received 22 June 2001; accepted 19 November 2001

Abstract

A method based on the electrical conductivity measurements is proposed to estimate the characteristic damage time (s) of a foodmaterial in pulsed-electric fields (PEF). Empirical dependencies of s versus electric field intensity E are obtained for apple, carrot andpotato tissues. The generalized electroporation theory accounting for the sphericity of biological cells and distribution of their

geometric sizes gives a credible description of the experimentally observed dependencies sðEÞ. An optimal electric field intensity Eoptthat corresponds to minimum of power consumption at maximum of material disintegration is estimated theoretically and ex-

perimentally. For investigated tissue materials, the experimentally estimated value of Eopt is approximately 400 V cm�1. The energy

consumption during the PEF treatment at Eopt � 400 V cm�1 decreases in the next order: apple! carrot!potato. � 2002 Elsevier

Science Ltd. All rights reserved.

1. Introduction

Electrical treatment of food materials is used forpreservation and microbial inactivation of liquids(Barbosa-C�aanovas, Pothakamury, Palou, & Swanson,1998; Barsotti & Cheftel, 1998), ohmic heating (Halden,de Alwis, & Fryer, 1990; Lima & Sastry, 1999; Wang,1995), electropermeabilization and electroplasmolysis ofsolid foods (Grishko, Kozin, & Chebanu, 1991; Gulyiet al., 1994; Jemai, 1997; Knorr, Geulen, Grahl, & Sitz-mann, 1994; McLellan, Kime, & Lind, 1991; Miyahara,1985; Papchenko, Bologa, & Berzoi, 1988; Scheglovet al., 1988). Recently, the interest in electropermeabi-lization of solid foods is growing considerably due to de-velopment of the pulsed-electric field (PEF) techniquesand the theory of electrical stability of cellular systems(Angersbach, Heinz, & Knorr, 2000; Barbosa-C�aanovas,Gongora-Nieto, Pothakamury, & Swanson, 1999; Baz-hal, Lebovka, & Vorobiev, 2001; Bazhal & Vorobiev,2000; Knorr & Angersbach, 1998; Lebovka, Bazhal, &Vorobiev, 2000, 2001). The main problem is the choiceof an optimal PEF treatment mode, i.e. electric field

intensity E, pulse shape, number of pulses n, pulse du-ration ti, pulse repetition time Dt.The basic complexity of the optimization procedure is

to obtain correlations between the processing protocoland plasmolysis or damage degree P of biological tis-sue and to estimate the relation between characteristicdamage time s and electric field intensity E.The main method of P value estimation for plant

and animal tissues after electrical treatment is based onconductivity measurements (Knorr & Angersbach, 1998;Rogov & Gorbatov, 1974). The conductivity measure-ments allow to determine the conductivity disintegrationindex Z (Rogov & Gorbatov, 1974):

Z ¼ r � rird � ri

; ð1Þ

where r is the measured electrical conductivity value atlow frequency (1–5 kHz) and the subscripts ‘i’ and ‘d’refer to the conductivities of intact and totally destroyedmaterial, respectively. Eq. (1) gives Z ¼ 0 for intact tis-sue and Z ¼ 1 for totally disintegrated material.A more complex definition of the disintegration index

Z includes the conductivities of the treated (r1) andintact (ri;1) materials measured in the range of 3–50MHz (Angersbach, Heinz, & Knorr, 1999):

Journal of Food Engineering 54 (2002) 337–346

www.elsevier.com/locate/jfoodeng

*Corresponding author.

E-mail address: [email protected] (E. Vorobiev).

0260-8774/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0260-8774 (01 )00220-5

Z ¼ rðri;1=r1Þ � riri;1 � ri

: ð2Þ

Note that at ri;1 ffi r1 and ri;1 ffi rd, the definitions (1)and (2) are equivalent.Unfortunately, a relation between the measured elec-

trical conductivity or disintegration index Z and thevalue of P is unknown and so the conductivity-basedmethod does not permit direct calculation of the bio-logical tissue damage degree P. The relation between Zand P can be approximated with the help of the sim-plified empirical Archie’s equation (Archie, 1942) in thefollowing form:

Z � Pm; ð3Þ

where m is called Archie’s cementation exponent anddepends on the connectivity of phases inside the hetero-geneous material (Glover, Hole, & Pous, 2000).

But it is unclear whether Eq. (3) is applicable for foodtissues (Peters, Hendriks, & Stinstra, 2001). Theoreticalestimation of Z versus P relation is also complicatedbecause food tissues include many air cavities (Aguilera& Stanley, 1999), and the electrical conductivity is verysensitive to spatial distribution of such cavities andheterogeneous structure of biological objects (Angers-bach et al., 1999). Moreover, the PEF treatment inducesdifferent long-lasting transition processes inside the tis-sue, such as mass transfer, moisture and air redistribu-tion among the microscopic pores of the matrixsurrounding the cells (Labuza & Hyman, 1998), reseal-ing of cells (Knorr, 1999). During this period, the hiddenphase of the electric breakdown can become importantwhen the damage degree P is high but the measuredconductivity is low (Bazhal et al., 2001).The damage degree P can be also determined using

other experimental techniques; for example, the value ofP can be estimated from diffusion coefficient measure-

Nomenclature

Cm specific capacity of membrane, F m�2

d fitting parameterDF ¼ px2=c, critical free energy of pore forma-

tion, JE electric field intensity, V cm�1

E ¼ E=E0, normalized electric field intensityE0 ¼ u0=1:5r, electric field intensity parameter,

V cm�1

f ðrÞ normalized size distribution functionk form factorL ¼ 2rN , dimension of a sample, mm Archie’s cementation exponentn number of pulsesN lattice sizeP degree of biological tissue damage (ratio of

the damaged cells number to the total num-ber of cells)

r cell radius, mR universal gas constant, 8:314 J K�1mol�1

S normalized optimization productt time, sti pulse duration, lsDt pulse repetition time, sT temperature, Ku transmembrane voltage, Vu0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2c=ðCmðew=em � 1ÞÞ

p,voltageparameter,V

W ¼ DF =RT , parameterZ conductivity disintegration indexZs saturation conductivity disintegration index

Greek lettersew ¼ 80, dielectric constant of water

em ¼ 2, dielectric constant of membrane� bulk tissue porosity (ratio of the inter-

cellular air space to the total volume of tis-sue)

c surface tension of membrane, N m�1

x linear tension of membrane, Nq correlation coefficientr conductivity, S m�1

rs saturation level of conductivity, S m�1

r1 conductivity measured in the high frequencyrange (3–50 MHz), S m�1

s characteristic damage time of a cellular ma-terial, s

sc characteristic damage time of a cell, ssm characteristic damage time of a membrane,

ssm;1 characteristic damage time of a membrane at

T ¼ 1 or u ¼ 1, ss1 characteristic damage time of a cellular ma-

terial at T ¼ 1 or u ¼ 1, sh angle between the normal to the membrane

and the electric field direction

Subscriptsc cellulard destroyedi intactm membraneopt optimum

AbbreviationPEF pulsed-electric fields

338 N.I. Lebovka et al. / Journal of Food Engineering 54 (2002) 337–346

ments (Jemai, 1997). But, unfortunately, this techniqueis implicit and invasive for biological objects, and itimpacts the structure of the sample. Recently, magneticresonance imaging became a useful technique for foodmoisture map visualization (McCarthy, 1994; Schmidt,Sun, & Litchfield, 1996; Watanabe, Fukuoka, & Wa-tanabe, 1995), but it is not applied yet for quantitativeestimation of the damage degree P.In this work, a method is proposed for estimation of

the characteristic damage time s for PEF-treated foodmaterials. This method is based on the study of the ki-netics of food material electrical conductivity in thecourse of PEF treatment. Computational and experi-mental estimations of Archie’s exponents for food tis-sues are done. Empirical characteristic damage time sversus electric field intensity E relations are obtained forapple, carrot and potato tissues, and condition oftreatment optimization resulting in minimum powerconsumption is formulated.

2. Materials and experimental methods

2.1. Materials

Freshly harvested apples (Golden Delicious), carrots(Nantaise), and potatoes (Agata) purchased from a localmarket were selected for preparation of plant tissue. Amoisture content W was within 80–85% for apples, 87–92% for carrots and 75–80% for potatoes.

2.2. Experimental setup and methods

The experimental set-up for PEF treatment was thesame as described previously (Lebovka et al., 2000,2001). A high voltage pulse generator, 1500V-15A(Service Electronique UTC, France) provided themonopolar pulses of rectangular shape. Pulse protocolsand output current, voltage, impedance and temperaturewere controlled using a data logger and a softwareHPVEE v.4.01 (Hewlett–Packard) adapted by ServiceElectronique UTC, France. The temperature was re-corded by a thermocouple THERMOCOAX type 2 (AB25 NN, 0.1 �C). The conductivity was measured withan LCR Meter HP 4284A (Hewlett–Packard) at a fre-quency of 100 Hz selected as optimal for removing theinfluence of the polarizing effects on electrodes and in-side the chamber with tissue (Bazhal, 2001). The poly-propylene treatment chamber had a size 10� 10�10 mm. A single tissue cube was put inside the chamber.Two stainless steel electrodes were located at both sidesof the tissue cube and connected to the pulse generatorand LCR meter.Experiments were carried out using an electric field

intensity E not exceeding 1500 V cm�1 (with precision

2 V cm�1), pulse duration ti ¼ 10�5–10�3 s (with pre-cision 10�6 s), pulse repetition time Dt ¼ 3 s (withprecision 10�5 s), number of pulses n ¼ 1–1000. Eachexperiment was repeated, at least, five times.

2.3. Computational model

The conductive structure of cellular materials wassimulated using a lattice network computational modeldescribed previously (Lebovka et al., 2001). Electricalproperties of each cell were simulated with account forconductivity of membranes, and both intra- and extra-cellular media. In this work, we used three-dimensionalmodel instead of previously described two-dimensionalone. Biological cells were located at the nodes of asimple cubic lattice and the lattice size was N 3 withN ¼ 50 that corresponds to the total tissue dimensionsL3, where L ¼ 2rN , r is the radius of the biological cell.More information about model and details of calcula-tion can be found elsewhere (Lebovka et al., 2001).

3. Results and discussion

3.1. Theoretical estimation of characteristic damage time

Characteristic damage time of a biological tissuecell depends on a number of factors. Most importantinclude the induced-transmembrane potential and geo-metrical characteristics of cells, e.g. their shapes, distri-bution of sizes.

3.1.1. Characteristic damage time of a single membraneFor a single membrane, the characteristic damage

time of a membrane sm may be estimated on the basis ofthe transient aqueous pore model (Weaver & Chiz-madzhev, 1996):

sm ¼ sm;1 expW

1þ u=u0ð Þ2; ð4Þ

where sm;1 is a membrane characteristic damage time inthe limit of infinite temperature, T ¼ 1, or infinite trans-membrane voltage, u ¼ 1, W ¼ DF =RT , DF ¼ px2=c isthe critical free energy of pore formation, u is thetransmembrane voltage, and u0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2c=ðCmðew=em � 1ÞÞ

pis the voltage parameter related to material properties ofa membrane.Lebedeva (1987) presented the following estimations

for the lipid membranes: s1u3:7� 10�7 s , xu1:69� 10�11 N, cu2� 10�3 N m�2, Cmu3:5� 10�3 F m�2

at 25 �C. From these estimations, the following values ofparameters in Eq. (4) at T ¼ 298 K can be derived:W u109:04 and u0u0:17 V.

N.I. Lebovka et al. / Journal of Food Engineering 54 (2002) 337–346 339

3.1.2. Characteristic damage time of a single cellThe transmembrane voltage induced on a spherical

cell in a homogeneous electrical field is equal to:

uðhÞ ¼ 1:5krE cos h; ð5Þwhere k 1 is a form factor (in a general case k dependson geometry and electrical properties of a cell and amedium (Kotnik, Miklavcic D, & Slivnik, 1998)), E isthe field intensity, r is the radius of the cell, and h is theangle between the normal to the membrane and the fielddirection.The transmembrane potential and, thus, the charac-

teristic damage time of a membrane are h-dependentvalues. The mean characteristic time sc of single cell maybe estimated as (Tomov, 1995):

sc ¼1

2

Z p

0

sin hdhsmðuðhÞÞ

� ��1

; ð6Þ

where smðuðhÞÞ is calculated using Eqs. (4) and (5).

3.1.3. Characteristic damage time of a cellular materialwith cell sizes distributionThe transmembrane potential linearly depends on the

cell diameter (Eq. (5)). A distribution of cell radii existsin cellular materials and influences the characteristicdamage time. In this case, the mean characteristic dam-age time of cellular material can be estimated throughthe averaging of the damage probability that is pro-portional to 1=sðrÞ:

s ¼Z 1

0

f ðrÞdr=sðrÞ� ��1

; ð7Þ

where f ðrÞ is a normalized size distribution function.

3.1.4. Numerical estimations of the characteristic damagetimeFig. 1 presents numerical calculations for the relative

damage time ðs=sm;1Þ versus E. Here, in the case of asingle membrane E ¼ u=u0 and in the case of a cellE ¼ E=E0, E0 ¼ u0=1:5r. When u0u0:17 V (Lebedeva,1987) and r u 10�4 m, E0 ¼ u0=1:5r 10 V cm�1.Squares in Fig. 1 correspond to the single membrane

characteristic damage time calculated from Eq. (4) forW u109:04. Triangles correspond to the single celldamage time (i.e. the case of standard deviation of a cellradius Dr ¼ 0), and are calculated from Eqs. (4)–(6) bynumerical integration. Triangles for a cell denoted byDr ¼ 0:2 and Dr ¼ 0:4 are obtained for a tissue withdistribution of cell sizes. These data are calculated fromEqs. (4)–(7) for the normal distribution function of cellsize:

f ðrÞ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffi2pDr2

p exp

� ðr � hriÞ2

2Dr2

!; ð8Þ

where hri is the mean radius of a cell.

The data represented in Fig. 1 were fitted to equation

s ¼ s1 expðW =ð1þ EdÞÞ ð9Þ

with two fitting parameters W and d. The results offitting are presented in Fig. 1 by solid lines and thecorrelation coefficients were better than 0.997.Note, that for a single membrane s1 ¼ sm;1 and for a

cell s1u2:01sm;1.So, in general case, which includes a single mem-

brane, a single cell and a cellular material with cell sizedistribution, the relation between s and E can be ap-proximated by the same Eq. (9) but with different fittingparameters W and d.Fig. 2 presents dependencies of W and d parameters

versus the standard deviation Dr. For a single membraneW u109:04 (Lebedeva, 1987) and d ¼ 2:0 (Eq. (4)), for asingle cell W u58:01 and d ¼ 1:33, and for a cellularmaterial with cell size distribution, the values ofW and dparameters decrease with Dr increase.

3.2. Computational estimation of Archie’s exponent forcellular materials

Archie’s exponent reflects the porous space configu-ration and the connectivity of phases inside cellularmaterial (Glover et al., 2000; Kuntz et al., 2000). It maybe roughly estimated as m ¼ 3=2 for spherical cells,m ¼ 1 for needle-shape spheroids oriented along theelectric field and m ¼ 2 for needle-shape spheroids ori-ented perpendicularly to the electric field (Peters et al.,

Fig. 1. Relative damage time ðs=sm;1Þ versus E for a single membrane

and for a cell.

340 N.I. Lebovka et al. / Journal of Food Engineering 54 (2002) 337–346

2001). For damaged cellular materials, the value of mmay also depend on the spatial distribution of damagedcells.Archie’s exponent was estimated on the basis of a

resistor network model for a cellular material (Lebovkaet al., 2001). Fig. 3 presents the conductivity disinte-gration index Z versus damage degree P calculated forrandomly distributed damaged cells among the sites of acubic lattice (1, data averaged over 1000 different ran-

dom initial configurations) and for the simulated dam-age of cellular material under the PEF treatment at twodifferent relative electric field intensities E ¼ u=u0 ¼ 3:5(2) and E ¼ 5:5 (3). Probability of a single cell damagewas estimated taking into account the sm versus E re-lation (4) as described earlier (Lebovka et al., 2001).Solid lines are results of the mean square averaging ac-cording to Archie’s Eq. (3), the correlation coefficientswere better than 0.996. The calculations of the damagekinetics under PEF treatment were made for a pulsewidth ti ¼ 10�5 s and the sets of parameters sm;1, W,and u0 presented in Section 3.1.1.Archie’s exponent was numerically estimated for a

random damage model as m ¼ 1:80 0:03. For PEF-damaged material, the m value increases with increase ofthe electric field intensity E. It reflects the changes in theconnectivity of damage cells inside cellular material andthe tendency to formation of more disordered damagedcell clusters with E increase.Fig. 4 presents the simulated conductivity disinte-

gration index Z versus the total time of treatment t ¼ ntiin the double logarithmic presentation for differentvalues of E. In this presentation, the initial sloped lg Z=d lg t is approximately equal to the m value esti-mated from Z versus P dependency similar to thoseshown in Fig. 3. The slopes of by dash lines in Fig. 4correspond to m values estimated from Z versus P de-pendencies at given E. Such behaviour can be explained

Fig. 3. Calculated conductivity disintegration index Z versus damage

degree P.

Fig. 4. Simulated conductivity disintegration index Z versus total time

of treatment t for different values of E. Inset shows the conductivity

disintegration index Z versus reduced time t=s estimated on the basis ofArchie’s law and a simple first-order damage equation (10).

Fig. 2. Dependencies of the two fitting parameters W and d in Eq. (9)

versus S.D. in the normal distribution function of cell size Dr.

N.I. Lebovka et al. / Journal of Food Engineering 54 (2002) 337–346 341

on the basis of material damage kinetics described bythe first-order rate equation:

P ¼ 1� expð�t=sÞ; ð10Þ

where s is the characteristic damage time of a cellularmaterial.At small time values t � s, the approximation

P ’ ðt=sÞ is fulfilled and the following relation can beobtained from Archie’s equation Eq. (3):

Z ’ ðt=sÞm or lnðZÞ ’ m lnðtÞ � m lnðsÞ: ð11Þ

So, in approximation of first-order damage kineticsArchie’s exponent can be estimated from the initial slopeof conductivity disintegration index Z versus time t inthe double logarithmic presentation as it is demon-strated at the insert on Fig. 4.Unfortunately, the applicability of first-order kinetics

equation to describe damage is not yet confirmed, andthis approach may be used only for a rough estimationof Archie’s exponent.

3.3. Energy optimization criteria for PEF treatmentprocedure

For electric field treatment during the characteristicdamage time, t ¼ s, the volume density of the energyinput is equal to

Q ¼Z s

0

rðtÞE2 dt: ð12Þ

For the first-order damage kinetics described by Eq.(10), at t ¼ s the cellular material damage degree is equalto P ¼ 1� e�1u0:632. Taking into account Archie’sequation (3) in approximation of ri � rd, Eq. (12) atconstant E can be rewritten as

Q ¼ sðEÞE2rdgðmÞ; ð13Þ

where gðmÞ ¼R 10ð1� e�xÞmdx is a function of m only;

for example, gð1Þ ¼ 0:368; . . ., gð2Þ ¼ 0:168; . . ., gð3Þ ¼0:084; . . ..So, the energy input during the time t ¼ sðEÞ can be

characterized by a product of sðEÞE2. The sðEÞ valuedecreases with increase of the electric field intensity E,and a product of sðEÞE2 goes through a minimum.Criteria of energy optimization require a minimum ofthis product. Taking into account Eq. (9), the followingnormalized optimization product can be obtained

S ¼ E2ðsðEÞ=s0Þ ¼ E2 expðW =ð1þ EdÞÞ: ð14ÞThe optimum value of E

opt corresponding to the mini-mal value of S can be determined from equationdS=dE ¼ 0. In approximation of W � 1 the followingtheoretical value of E

opt can be obtained:

Eopt � Wd=2ð Þ1=d : ð15Þ

3.4. Experimental investigation of cellular material dam-age under PEF treatment

As demonstrated earlier (Knorr, 1999), the tissueconductivity r may show a rather complex long-lastingbehaviour after PEF application. Such behaviour wasexplained by processes of cell breakage, resealing ofpores and by increase in stress-induced metabolite pro-duction. Tissue conductivity is also sensitive to the po-rous structure of a plant tissue and air content inside thetissue (Angersbach et al., 1999). The bulk porosity offresh tissues � can vary substantially. For example, appletissue bulk porosity is high and falls within the intervalof � � 14–25%. For outer apple parenchyma porosity ishigher than for inner parenchyma (Mavroudis, Gekas,& Sjoholm, 1998). But porosity is small for many otherfresh plant materials, for example, for carrot � � 4% orpotato � � 2% (Karathanos, Kanellopoulos, & Belessi-otis, 1996). After PEF application to the tissue, addi-tional moisture release from the cells develops thelong-lasting transition processes of moisture and air re-distribution, which are very important for tissues(Labuza & Hyman, 1998).In this work, a mode of treatment with an interval

between the pulses Dt ¼ 1–3 s was used. Application oflong pause after each pulse allows the plant tissues toreach equilibrium, when transition processes of resealingand moisture diffusion can be neglected (Bazhal, 2001;Lebovka et al., 2001). Transition time of the order 1 swas also reported in literature (Knorr, 1999). Note, thatthis mode of treatment with long interval between thepulses avoids noticeable increases in the tissue temper-ature.

3.4.1. Saturation levels of the conductivity disintegrationindex ZFig. 5 presents examples of the time dependencies of

the conductivity disintegration index Z during the PEFtreatment of potato tissue at shown electric field inten-sities E. The total time of treatment was determined ast ¼ nti. The disintegration index Z was estimated fromEq. (1).A distinctive feature of the observed ZðtÞ time de-

pendence is the existence of one or more levels of Zsaturation. At relatively small electric field intensities E(E < 300–500 V cm�1) the electrical conductivity r andconsequently disintegration index Z do not increasecontinuously with time t. After an initial increase, theconductivity can stabilize at low saturation value ofrs < rd. The further increase of r is observed after arelatively long period of time as shown in Fig. 5.If the PEF is turned-off at the time when conductivity

gets stabilized at a saturation level r ¼ rs (see arrow atFig. 5) the value of r remains constant during the timeof the order of 103 s. The ratios of rs=rd versus electricfield intensity E for a first saturation level are presented

342 N.I. Lebovka et al. / Journal of Food Engineering 54 (2002) 337–346

in Fig. 6 for apple, carrot and potato tissues. These dataindicate that the described saturation behaviour is morepronounced only for electric field intensities E <300–500 V cm�1.The ‘jamming’ mechanism was proposed earlier for

explanation of the saturation phenomena (Lebovkaet al., 2000). But saturation behaviour may be explainedalso by unfinished resealing processes.

3.4.2. Experimental estimation of the characteristic dam-age timeFigs. 7(a)–(c) show the saturation disintegration index

Zs versus time t for apple (a), carrot (b) and potato (c)tissues. The Zs value is defined as

Zs ¼r � rirs � ri

: ð16Þ

From this definition follows that Zs ¼ 0 for an intacttissue and Zs ¼ 1 for the first saturation level.Archie’s cementation index m determined from the

initial slopes of ln Zs versus ln t curves was in the rangeof 1.8–2.5 for electric fields E > 100 V cm�1 and in therange of 1.0–1.5 for smaller electric fields. The increaseof m value with electric field intensity E increase corre-lates with our simulation predictions as described inSection 3.2.The s value may be estimated from experimental time

dependence of the damage degree P ðtÞ � ZðtÞ1=m on thebasis of known kinetic equation. For the first-order ki-netic equation (10), PðtÞ � ZðtÞ1=m � ðt=sÞ and the svalue can be obtained as a fitting parameter of the ex-perimental P ðtÞ dependence. The P ðtÞ dependence ob-served for food tissues is usually described by a relationmore complex than Eq. (10) and in a general case s valuecannot be obtained using simple fitting procedure.In this work, the order of s magnitude was estimated

as a time when the saturation disintegration index Zsattains one-half of a maximal value, i.e. Zs ¼ 0:5. Suchapproach allows only approximate estimation of s val-ues. The more precise determination of the characteristicdamage time s requires additional information aboutmaterial damage rate equation, but this is beyond thescope of the present study.Dependencies of the characteristic damage time s

versus electric field intensity E for apple, carrot andpotato tissues are presented in Fig. 8. Here, symbolscorrespond to experimental data and solid lines showresults of the least square fitting using Eq. (8). The fittingparameters (s1, WEd

0 , d) and correlation coefficients (q)are presented in Table 1.A characteristic damage time of tissue in the limit of

very high fields s1 reflects material resistance to the PEFtreatment. The higher is the value of s1 more time oftreatment is needed to destroy material (Fig. 8). Theinvestigated materials can be arranged in accordancewith resistance to the PEF treatment in the next order:apple! carrot! potato.Fig. 9 presents the optimization product sðEÞE2 ver-

sus electric field intensity E curves for apple, carrot andpotato tissues (see Section 3.3). Curves sðEÞE2 versus Epass through minimum at E ¼ 200–400 V cm�1. Thisminimum corresponds to the minimum power con-sumption for material treatment during characteristictime t ¼ sðEÞ. The qualitatively similar behaviour ofthe optimization product sðEÞE2 was also described in

Fig. 6. Mean conductivity ratios rs=rd versus electric field intensity E

for apple (a), carrot (b) and potato (c).

Fig. 5. Disintegration index Z versus total time of PEF treatment t for

potato.

N.I. Lebovka et al. / Journal of Food Engineering 54 (2002) 337–346 343

theoretical part 3.3. Putting the parameters W u73,du1:15 for apple from Table 1, and E0 10 V cm�1

(see Section 3.1.4) into Eq. (15), we obtain Eopt 735 V cm�1. This theoretical estimation differs sub-stantially from experimental value Eopt 250 V cm�1

for apple (Fig. 9). It can be explained by very roughestimation of the parameters in Eq. (15).The effective treatment of material that results in high

conductivity disintegration index Z � 1 requires over-coming of the saturation limit, and the minimum valueof E should be of order E ¼ 400 V cm�1. The furtherincrease of E results in progressive increase of optimi-zation product sðEÞE2 and energy input, but give noadditional increase in conductivity disintegration index

Z. So, for the studied materials, the optimal value ofelectric field intensity Eopt that results in maximal ma-terial disintegration at the minimal energy input may beestimated as Eopt � 400 V cm�1. From Fig. 8, thecharacteristic time s at E ¼ Eopt ¼ 400 V cm�1 can beestimated as 2� 10�3 s (apple), 7� 10�4 s (carrot) and2� 10�4 s (potato) and energy consumption decreasesin the next order: apple! carrot! potato.

4. Conclusions

(1) The generalized electroporation theory accountingfor the sphericity of biological cells and distribution of

Fig. 7. Saturation disintegration index Zs versus total time of PEF treatment t for apple (a), carrot (b) and potato (c).

344 N.I. Lebovka et al. / Journal of Food Engineering 54 (2002) 337–346

their geometric sizes provides a credible description ofthe experimentally observed dependencies of character-istic damage time s versus electric field intensity E forapple, carrot and potato tissues.(2) Experimentally estimated cementation index m in

Archie’s relation (3) between tissue damage degree Pand conductivity disintegration index Z falls within therange of 1.8–2.5 for studied biological tissues.(3) There exists some optimal value of electric field

intensity Eopt that results in maximal material disinte-gration at the minimal energy input. The experimentallyestimated value of Eopt is approximately 400 V cm

�1 forinvestigated tissue materials.The energy consumption during the PEF treatment at

the said Eopt decreases in the next order: apple! car-rot! potato.

Acknowledgements

The authors would like to thank the ‘‘Pole RegionalGenie des Procedes’’ (Picardie, France) for providingthe financial support. Authors also thank Dr. N.S.Pivovarova and Dr. A.B. Jemai for helping with thepreparation of the manuscript.

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Fig. 9. Optimization product sðEÞE2 versus electric field intensity E forapple, carrot and potato tissues. Different symbols correspond to ex-

perimental data and solid lines show results of the sðEÞ data fittingusing Eq. (9).

Fig. 8. Characteristic damage time s versus electric field intensity E forapple, carrot and potato tissues.

Table 1

Parameters of the least square fitting of characteristic damage time sversus electric field intensity E dependencies using Eq. (9) for apple,

carrot and potato tissues

s1, s WEd0 ;(V/cm)2

d q

Apple 5:2� 10�5 4821 0.50 0.971

Carrot 4:5� 10�5 2483 0.55 0.993

Potato 5:7� 10�6 3762 0.56 0.976

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