1997_ruiz-cabrera_the effect of path diffusion on the effective moisture diffuslvlty in carrot slabs

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The Effect of Path Diffusion on The Effective Moisture

Diffuslvlty in Carrot SlabsM.A. Ruiz-Cabrera

a, M.A. Salgado-Cervantes

a, K.N. Walislewski-Kubiak

a& M.A. y Garc

Alvaradoa

aDepartamento de Ingeniea Qumica y Bioqumica, Instituto Tecnologico de Veracruz P.O

Box 1420 , Veracruz, Ver, 91870, Mexico

Published online: 07 May 2007.

To cite this article:M.A. Ruiz-Cabrera , M.A. Salgado-Cervantes , K.N. Walislewski-Kubiak & M.A. y Garca-Alvarado (1997)

The Effect of Path Diffusion on The Effective Moisture Diffuslvlty in Carrot Slabs, Drying Technology: An International Journ

15:1, 169-181, DOI: 10.1080/07373939708917224

To link to this article: http://dx.doi.org/10.1080/07373939708917224

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DRYINGTECHNOLOGY S(I). 169-181 (1997)

T H E E F F E C T O F P A T H D I FF US IO N O N T H E E F F E C T I V E M O I S T U R ED I F F U S l V l T Y I N C A R R O T S L A B SRu izC abr era , M.A.. Salgado-Cervantes. M.A.,Walislewsh -Kubiak, K.N. y Garcia.

Alvarado. M.A.'Depanam ento d e lngenierfa Quimica y Bioquimica, Institute Tecnologico de VeracluzP.O. Box 1420. Veracruz. Ver. 91870 Mexico

Key words: effective water diffusivity, food drying.A B S T R A C T

In order to evaluate the effect of path diffusion on the average moisture diffusivity incarrot. drvinp. curve s for different shaves (slices and cvlinders) and tem veratures of 50 60and 7 0 0 ~ ere ohtained lakine into consideration the us o f an ver he leneth of carrot~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~s a mp le ( s l ~ c e hickne ss or the inder radio). Tht esulLs showe d significant differencesbetuecn radial and axial average dilfusr\rues. SignKmnl diiicrencds were also obscnedbetween core and annular diffu s~vity . he expenmenla1 drying curve s did not show enoughevidence on the effect of drying temperature on the average moismre diffusivity

I N T R O D U C T I O NTo de velop an understanding of the mechanism of moisture movem ent in food

cons iderab le research information has te en reponcd. But limited data on moistu re diffusivityis available, w i h a w ide variation of the reporled values. due to the structure comp lexity olfoods and different m ethods of water diffusivity determination. M oisture diffusivity is animportant transport property needed for c o m t modeling and calculations in the f w d dryingprocess and is generally supposed s an indepedent of mass transfer path (Pakowski and

Author to whom all correspondence should be addressed169

Copyright 997 by Marrcl Dekkcr Ins


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170 RUIZEABRERA El AL.Muju mda r. 1987: Karathanos e l al.. 1990). Nevenheless, the directio n path o f watermovement may affect h e water diffusivity due t ,the anisotropic nature o f some fwd s. Th ispath effect may be mathem atically described using h e tensorial fo rm o f mass transferequation, as shown in equation I):

Phenomenon relationships of w aler movement i n food c ellular stluctures are groupedin h e moisture diffusiv ity lensor De,q), includin g molecular water diffu sion i n the differen tccllular structures, water permeability across cellular membranes, vapor diffusion ininle rce llula r spaces and mechanical waler movem ent between capillar ies Crap iste et al..198 8). Some authors Hsu . 1983. Balaban and Pigo t. 1988. Kara than os et al., 1990. Mu lct .I99 4 reported that the effective diffus ivity is a function of food m oisture content howeverthe water diffusivity depends also on structural changes in rood tissue during dryingCrapiste ct al.. 1988). Ncvenhclcss, other authors Ch irife and Cachero. 1970. Suarer e l

al.. 1980. Igbeka. 1982. Pillaga et al.. 1984. Kirano udis et al.. 1992 )s ho we d that dryin gbehavior may be described using an avenge effective moisture diffusivity, that is constantduring the dryin g period. In our work an avarage effeclive m oisture diffu sivity was lakeninto consideration and while employing normal tensor nolation. The average effectivemoisture diff usiv ity tensor could be expressed as follows B ird et al..1960):

Considering cylindrical coordinates r. 0 z) of food and if cross moisture diffusi vityis absent. the elem cnls of tensor cou ld be expressed as fol lows :

Considering only axial z) and radial r) directions equation 1 can be rearranged sfol lows :For axial dimt ion :

For radial direction:


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PATH DIFFUSION IN CARROT SLABS 171ax D J a x=

For spaual direction. equations 4) and 5) could be employed considering theappropriate bounda ry conditions. Some foods, inclu din g carrols. have anisotropy in theirstructure and i t is possible that the moisture dif fusivity in the axial direction D may bedifferent compared to the radial direction Dm).Nevertheless obtained experimental cu rvesduring dry ing have non-controlled variability residuals). and the difference between axialand radial dryin g direction curvcs may not be statistically significant. The objecd ve o f thisstudy was to analyre i f he axial and radial moisture diffusivities on carrot drying curves areswlislica lly different. considering illso h e dillu sivit y differences between the carrot core andann ular space.

METHODOLOGYExperimental drying curves

Fresh carrots were purchased fm m a local market and processed the same day. Thcrools were cut into four shapes for sample types: shape I lab of on ly the core shape ofcarrot w ith 1 cm o f thickness: shape 2: slab o f annular space shape of c a m 1 withou t corewi th I m o f thickness: shape 3: cylindrica l shape core of carrot with 7 cm in leng tht: shape: cylindrica l shape of c a ~ ro t ithout core o f7 cm in length. The samples were sealed with l

high vacuum grade silicon to assure dehydration from di fk re nt lhces o f the slab: shapes Iand 2 were sealed on the lateral face and dried on ihe top and bouo m faces: shapes 3 andwere sealed on the top and bouom faces and dried on their lateral faces. This way h e dry ingprocess took place on ly in the desired direction Igteka . 1982). The samples were placed inine n suppons held with pins in four replicates and were dried in a cabinet drier using an airveloc ity o f 2.5 m/s at 50.6 0 and 7043 A ir temperature and air velocity were measured by ;melectric thermometer and an memomeler respectively which were placed in the m e ocationin the drier as the samples. A sample w a withdrawn from the drying chamber at a regularinterva l and rapidly weighed. The air velocity was szlecled to leach a B i number gr at er than100 so that h e d rying prccess was convolled by water diffusion Cordova et al.. 1W 6 ).Dr yin g curves were prepared by mo nitoring the weight loss of the carrot slabs every 15minutes during the first hour, and every 30 minules for the remainder of the dr yin g period.The eq uilibrium food m oisture content of different slabs was calculated fm m carrot sorption


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172 RUIZCABRERA T AL.isotherms reported previously by Co rdova et al, (1996). The dry solids o f each slab weredetermined after each experiment by a vaccum oven dryin g method at W for 24 hours.Est i mat i on of average water diffusivities

The av enge water diffusivity was estimated by fitting with non-linar regression theoblained results to the ana lytical solutions of the mass vansfer equations (4 o r 5) in theexperimental dry ing curves. The boundary conditions (focd surface) were considered to be inequilibrium with the drier air, because the Bin um ber was always greater lhan 100 and thedry ing process was assumed to be contro lled by moisture diffusion. The analytical solutionsused re expressed by the follo win g equations:

for shapes and 2 fla t slab dried on two sides)

for shape 3 Imge cylinder dried on the lateral side )

n are all the positive ro o6 of the followin g equation:J,(A ) =

for shape 4 Imge cylind er dried on the inner and external lateral sides )


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PATH DIFFUSION IN CARROT LABSk n are all the positive mots of the following qu ati on :

Equation (8) tends to equation 7) when ends to ler o.The temperature eff ects over diffusivit ies were modeled using an Arhenius type

function:

Th e equation (10) were sustituted in equations 6). 7)and 8) respxtivelly and fittedwith non-linear regression to the whole of experimental drying curves in each shape. Theparameters Dm. Dm,E nd E were evaluated with the joint con fidenc es interval for non-linear pa nm ete rs reported by Alkinson and Hunter (1 8).

RESULTS A N D DISCUSSIONAll carrot slabs showed shrinkage during drying and this appears to be an imponnnt

aspect to consider in modelling. If this is neglected, as is often the case in the literature. thesim ple diffusional model can offer a good description of the experimental d ata. Although it isuseful for describing the drying rates. the effective diffusivity reflects the shrinkage process.s o effective diffusivity is not exaclly showing moisture uans pon properties. Fo r this reasonadditional drying curves were obtained to observe the shrinkage behavior. The shrinkageshow ed an exponential b ehavior with respect to time. Therefore the next model was fittedusing non-linear regression to the experimental data of shrinkage employing the simplifiedequation:

Where L is a characteristic length: the slab thickness L) or axial diffusivity in core andannular: the cylind er n d io (R) for radial diffusivity in the carrot core ; and the ratio or radialdiffusivily in thc annular carrot. The average charactetist ic length was calculated byintegn lion of equation ( I I) with respect l lime:


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174 RUIZC BRER ET AL.

Table 1.Low er and upper limits for 99 oint interval confidence of shrinkage paramelers (h). andaverage length

Thc esulcr of confidence intervals for parameters h a re show n in Table I The resullsof fittcd and experimental shrinkage behavior arc shown in Figures I and 2 and parameter bshowcd a s ip il k a n t comelalion to lemperature during drying. like is showed in Table I. Theavcragc length ohwincd at different temperature ar e show loo in Table I.

The drying resulcr showed that the parameters E and Err of qu al io n (10) had notstatistical significant (p=O.l). which m a n s that there is not enough evidencc o f lhe dryintemperature clfcct on moisture dil lusivity in carrot slabs. This could he conlirmcd hycalculating the eflect of the drying temperature dcpendance of the average characterislicshape s of carrot slahs which would predict the temperature effect on thc drying urves for


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P TH DIFFUSION IN C RROT SL BS


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RUIZ CABRERA T AL


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P TH Dl FN Sl ON IN C RROT SL BS


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RUlZ CABRERA T AL


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PATH DIFFUSION IN CARROT SLABS 179

Table 2.L ow er w d upper l imits of9996 confidence interval o f avenge moisture diffusivities i n cmI I Shape 2 Shape3 Shape4

each shape. Therefore. the resulv* of average moisture din'usivity were cstimatcd by fittingresults of the experimcnul drying curve to non-linear regression at the whole temperaturesfor every carm t shape em ploying e qu ali on 6.7 and 8 respeciively and the average length ofTable I

Experimental and fit lcd drying curves are shown i n Figures 3 and 4 A deviat ion inlinear behavior was prcscnted in the semi-log experimental graphics which was prohablycaused by: carrot shrinkage. slrong variation of effective m oisture diffu sivit y from an averageresult due the hcat iransfcr cffcct. The experimental r e s u l ~ ~ehavior appam nly cu ntirmcd thelast hypothesis because showed the greatest variation of l inear behavior al the dryingtemeprature of 70C his deviat ion in l inear khavior may be predicted with 3 t imedependenl moisture diffus ivity and we hope to obtain a betler model with analylical solutionin w hich thc cffcct ol't ime dependent moisture diffusivity can c considc~ed nd thc dryinglcmp enlurc can be considcm l on the effcct o f moisture diffusivity in carrot slahs.

The r cs ul~ s f the estimaled moisture diffusivity in differem i o m s of carrut slahs areshown in Tahlc 2. The radial and annular space calrot slabs diffusivities were lower(p 4. 01 compared to the axial and core shapes This meens rhal waler transpon in the ax ialdirection and carrot core presents a lower resistance than in the r ~ d i a l irection and carrotann ular space. Proba bly the carrot core has more exu-acc lluar space than the annu lar andwater d iffuses eas ily across this space as a vapm (Crapiste et al.. 1988). The diffcrencesbetween radial and longitudinal diffusivity can probably be explained by tissue fiberorientation. I t s possible that in the radial direction water movement requieres more cellularmem brane permeate than in the axial dirccuon .

ON LUSIONS

t was showed ihat the average moisture diffu sivit y in carrot dry ing arc slatislicallydifferent in a xial and radial dire ction and in core and carrot annular. The m odels used fitted


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180 RUIZ-CABRERA ET AL .the expe rim enu l data wit h moderate accurale. bu l showed enought evidence for the aa tisuca ldifference between the elcmenles of the tensor of average effective m oisture diffus ivity incarrols slabs.

L I S T OF SYMBOLSD fl= kc on d order r e n w o f avengc mo lclure dd fusnw~ycm-s)L = cnrnn rlah thsknes\ (cm)R = ca rr ot c y l ~n d c r d ~ ocm)Rc = carrot core n d io (cm)I time.

.X e =equ ilibriu m moisture of carrot (idem)t =dimensionless avenge moisture of the carrot (Xavg-Xe)/(Xo-Xe)

=rat io between c o n and slab radio (RIRc).

R E F E R E N C E SAlkinso n A.C. and W.G. Hunter. 1968. The design of experiment for p arm ete r estimrlion .nchnomrrrics 10 (2): 271-289.Bird. R B Slewart W.E., and Ligh tfoo t E.N. 1960. Trans pon Phenomena . Wi ley .Balaban M.. and Pigot. G.M. 1988. Mathematical model of simulwneous heat and masstransfer in food wi?h dimensional changes and variable transpon pardmelers.53 53 : 935-939.Ch irife. J. and Cachero, R.A. 1970. Th mu ght circul;hion dry ing of tapioca root. ,hua&fF od 35: 364-368.C6rdova-Quiroz. A.V.. Ruiz-Cabrera. M.A. y Garcia-Alvarado. M.A. 1996. Analyticalsolution o f mass transfer equation with in lerfacial resistance i n food d rying. Dry ingTechnology 14 (7-8). I n Press.Crap isle G.H.. W hitaker. S. and Rou tein. E 1988. Dr yin g of cellular ma terial I mass.vansfer theory. - (1 1): 2919-2928.Hsu. H. 1983. A diffusion m odel with a concentration deoenden t difusion coefficient fordescribing waler movem ent in legumes during soaking. 48:61 8-622Igbeka. J.C. 1982. Simu lation of moisture movem ent dur ing dryin g a swrc hy food produ ct-cassava.loum l 17: 27-36,Karathanos. V.T.. Villalobos. G nd Saravacos. G.D. 1990. Com paratio n of tw o methodsof estimation o f the effective moisture diffusivity from dryin g daw.55 (I) : 218-223.Kiranoudis. C.T.. M aroulis. Z.B. and Marinos-Kouris. D . 1992. Dr yin g kinetics of onionand green pepper. Ninp 10 (4): 995- 101 1.Mulet . A. 1994. Dry ing modell ing and waler d if fusiv ity in c m l s and powtoes.

: 329P&owski. 2 and Mujumda r. AS . 1987. Basic Process Calculations i n Drying . I nHandbook of Industrial Drying A S . Mujumdar.cd. Marcel Dekker. New York.


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PATH DIFFUSION IN CARROT SLABS IS1Piiiaga. F.. Carb onell . J.V. Peiia. J.L. and M iquel. J.J. 1984. Experimenlal simula~ion fsolar dlyin g of garlic using and adsorken1 energy storage bed.1 P Ina. ,o,-& ,.San vac os. G.D.1986. Mass lransfer propenies of foods. In Engineering Prop enies o lFoods.M.A.Rao and S.S.Rizvi ed ed. Marcel Dekker New YorkSuArez. C.. Viollaz. P.. and Chirile J. 1980. Diffusional analysis of air drying of grainsorghum. loum l 5: 553 5531.

1997_ruiz-cabrera_the effect of path diffusion on the effective moisture diffuslvlty in carrot slabs

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