I~VESTIGAClÓN REVISTA MEXICANA DE FíSICA ~7 (1) 37-42 FERRERO 2001

Kinetics of absorption of the environmental moistllre in grainy materials

E. Villar-Cociña, E. Valencia-Morales. and R. González-RodríguezDepartamento de Física. UlIirersidlld Central de 11'.vVillas

Santa Clara 54830, Vil/aclara. Cuba

Recibido el 25 de mayo de 2000: aceptado el 6 de octuhrc de 2000

The moisturc absorption in granulalcd ma(crials used in foundry tcchnologics is analyzcd. Thc absorption proccss has a diffusivc behaviormainly. A simple cxperimental technique, in \,vhich Ihc wet weigh! increlllcn! \Vas rerordctl as the cxperimental variable and an analyticmethod with computing proccLlurc to hml the parnmeters charac(crizing the proccss \Vas lIscd. The dctcnnination of thesc parameters hytradilional lllcthods is a vcry dirncuh task. vcry refined amI cxpcnsive Irials arc needed, The fitting 01' the moJel pcrmits (o determine thediffusillll cocfficient and the moislUre concentration in lhe scparntion surfarc hcl\ •...ecn the samplc and the cnvironment. Thc concentrationprollles are estahlished for diffcrent times. f'inally. the possihility 01'occurrcl1cc 01' superimposetl diffusive processes in some materials is;lllalY7ed amlthc dilTusion cocfficicnt antlthe amount 01'moisturc incorporated hy each process are calculated.

K('.\,\\'(mls: ~10isture dilTllsion; humidity; \•...et climate; iron foundry

Se .ln<lliz<lla absorción de humedad en materiales granulados utilizados en lenlOlogías de fundición cuyo proceso liene un comportamientodifusivo principalmente. Se utiliza una técnica experimental simple en la cual el incremenlO en peso húmedo es registrado t'OIllOvariableexperimental. Un método analítico con procesamiento computacional es usado para determinar los parámetros que caracterizan el proceso.cuya determinación por métodos tradicionales es difícil y requiere de ensayos muy retinados y costosos. El ajuste del modelo permitedeterminar el coeficiente de difusión y la concentración de humedad en la supcrticie de separaci6n de la muestra y la atmósfera circundante.Se eSlahlel'l.,'n los perfiles de concentraciones para diferentes instantes de tiempo. Finalmente. la posihilidad de ocurrencia de procesosdifusivos superpuestos en algunos materiales es analizada y son calculados los coeficientes de difusión y la cantidad de humedad incorporJi.bpor cada prot'eso.

/Jescri¡)/ot'C,S: Difusión de humedad; humedad; clima hlímedo: fundición férrica

PACS: 66.~(); 92.60.J

\, Introdllction

II is knowll that SOIl1Cporous malerials are highly hygro-scopic and \",'hen pUl in Iypical \Vel tropical c1imates lile)' in-corporate moislure and Ihis influenccs Ihcir mechanical prop-erties.

An eXilmple 01' this. is the negalive inl1uence 01' lllois-tme in granulated Illaterials hased on silica sand-sodium sili-cate ami silica sand-sugary aggregales (molasses). bOlh lIsedin foundry. This negati\'e inlluence on Ihe properties 01' thematerials (Ioss of mechanical strength. superfkial rcsistance,ele.) [1-:l[ is due lo lheir highly hygroscopic hehavior.

Dcvkes made of these materials acquire, during long ex-POSiliolls in \Vel climates. a dangerous incrernent of moislure.As a result nI' this. during the casting process. a high gas gen-eralioll lakes place, which produces defects on lhe surface 01'the casting pieces [1, :~I.I:or that rcason. Ihe kinelics 01' mois-lure ahsorplion in the ahove granulaled materials has scien-litk <\mllechnological importance.

1I is reporled I1,2] Ihat. for Ihe same raw materials lIsed inmixtures for foundry. the hygroscopk properties are direcllyre¡ated with the hehavior 01' the agglulinating substances.

Hygroscopicity. as physical phenolllenon, is very corn-plex ami il can he fralllcJ within the general principies of atl-sorplioll antl diffusion of a gas in a soliJ ahsorbent. The mois-lure atlsorption process in Ihese solids elapses praclically ina short time [.11. The physical phase formed can he consti-

luted by only one layer 01' moleclllcs nf waler or there can tK'a slltlden condcnsatioll of the gas. forming several layers 01'1ll0lcclIles on Ihe ahsorhent surface. The lime in which thisprocess occurs can he negleclcd compared with the laler dif-fllsive processes [.1l. For that reason. an initialmoisture con-centralioll on Ihe surface (for fixed relative hlllllidities 01' Ihecnvirolllllenl) can he consitlercd conslant during the wholedilTlIsive proccss.

According to (his. which is rigorously veriIJeo in Ihe ex-perimenlal practicc, Ihe hygroscopicity research in Ihe ahovcmaterials is sirnplillcJ lo a dilTlIsive problem consideringphysical diffusion uf waler 11].

For nUll1crnus industrial applications, simplc but accuralclIlodels descrihing Ihe natural phcnol11ena are rcquircd. Suchl1ludels have practical and cconomical imporlanu:: (5]. Esre~c¡aHy nccessary is Ihe devclopmcnt nI' models and methods10 dctcnnine tht~ pruccss parameters: Ihe di ffusioll coefficient.the moisture conccntration on Ihe separation surface (samplc~sUlTounding atlllosphere (interface») and the alllount of rnois-tme incorporaled lo the sample.

In Ihis work the wet wcighl increment is recorded ex.pcrimcntally. which can be easily determined. An analylicalIllethod with computational procedure allowed calculation 01'aH the prnccss paramcters. including moislure concentration011 lhe interface. \.I,.'hich is not easy lo obtain hy traditionallrcatmcnts (6,71 and rcquires very expensivc lrials for theirdetermination (e.g. microanalysis l1lethods).

38 E. VILLAR.COCIÑA. E. VALENClA.MORALES. AND R. GONZÁLEZ.RODRiGUEZ

L (mm)(hcigh')

7.5

5.2

5.2

5R

50

50

D (mm)(diarncter)

5

10

9590

TABLE 1. Chcmical composition of ¡he materials amI dimcnsions of (he capsulcs.

Mixture Silica sand SodiulTl Molasscs(%) silicatc (%) (%)

94 6Silica sand-molasses

Si lica sand-molasscs

Sil¡ca sand-sodium silicatcI

34

No.

FIGURE l. Diagram 01"the linilc rastille permeahle only for a faee.

matical prohlcrn to determine the diffusion eoeflicicnl, whichnot depends on thc geornelry and houndary condilions. Thiseffeclive dilTusion coeffkient (D) is more likely govcrned bythe nalure of the substance. lhe particle sizcs, al1l! by the wayhow Ihese are eornpactcJ.

By choosing this gcometry. lhe Ihree-dirncnsional Jiffu-sive prohlelll is rcduced to a unidircctionallrealrnent (Fig. 1),whcrc D is assumed constanl for eaeh lest at a lixed rclativchumidity. The mathernatieal prohlcm is given hy the cquation

The llrst houndary condilion [Eq. (3)J shows the invari-a!lee 01' rnoisture concenlration (Co) in Ihe surfacc and isacccptcd for t'lOth mixtures (silica sand-1ll01asses and silicasand-sodilllll silicate), although the siliea sand-sodium sil-icate mixture has some amount 01' water (nol environrnen-tal rnoisture) held in the gel structure initially. which is ue-termined hy the Illodulus 01' soJiulll silicate. This IlloisturecontaincJ in ahove mixtures does not affec! lhe moisturc ab-sorption process qualitatively, hut il changes lhe gradient ofwater cOllcenlration on Ihe interface hctween lhe sample andthe surroullding almosphcrc, which could change (he Cn val-ues for Ihe samc outdoors Illoisture conditions compareJ wilhthe sanJ-molasscs mixtures. The second bOllndary condition[Eq. (4) I rellects ,he impermeahi lily of thc wall uf the capsulein.1' = L.

In onler lo solvc Ihc bOllndary prohlcm lhe fouricrrnelhod [S-lO] or lhe Illethod 01' Laplaee transform 161 canoc lIscd. Tlle result is

(1)

(4)

(2)

(3 )

\i l.

\i.rf (O. L)

OC 02C-=D--."01 O.r-

C(O. t) = en, \i t.

OCDI = O 111.1: = L.

C(c,., O) = O.

with ¡nitial and houndary eonditions given by

1xThe diffusive model fitted by compulerized methods per-

miucd 10 determine lhe parametcrs, as Ihe diffusion cocffi-cienl and rhe ¡niria! concentration of moisture on lhe inter-face. Thcrcaftcr. lhe conccntration profilcs al diffcrcnl limeinstanls in lhe matcrials were estahlishcd. Por samplcs whcrclhe supcrimposcd diffusive proccsscs took place, a composcddilTusive modcl \Vas arr1icd amlthc diffusion coefficicnts as\Vell as Ihe waler amount ¡nCafroratco hy c3eh proccss \VeTedctcfmincd.

This proccdure constitutcs a ehcap and salisfaelory dc-scription 01' a phcnomenon willl scicntifie and teehnologicalinterest ami it can also be used for tcaching purposes to studylile kinetics of Illoisture absorption in lhesc materials.

]. FormulatiulI uf the pruhlcm

2. l\laterials and methods

~loislure ahsorption is simpliflcu lo a dilTlIsive problem COI1-

sidering the physical Jiffusion 01' water, thercfore il is neces-sary lO sol\'e Ihe continuily equation or sccond Fick's law, inIlon-slalionary s!alCs with Ihe initial amI houndary condilionsimposed by Ihe givell silllation. Samples, penneable only fora race. were linite round sheels (paslilles) with zcro cnviroll-llIelltal moistlln: as initial eondition. Jt simplifics Ihe Illalhe-

The mixtures analYl.ed werc composcd fundamentally by sil-ica sand and sodiulll silicate or molasscs as agglutinallts.Thcir compositions are shown in Table 1.

BOlh cOlllponenls werc mixed in a rollcr mixerduring 1.510 ~ minutes (silica sand-sodiulTl silicate mixlures) and 4-5 minutes (silica sand-molasses mixtures). Afterward stan-dan! cylindrieal samples werc made and compaeted hy thrc(~hits. Thus, lhe samplcs were put in porcelain capsules amicxposcJ lO heat trcatmcnt (silica sand-lIlolasscs mixturcs) orblown with CO;! during 60 seconds (silica sand-sodium sili-cate mixtures). Thc hcaling 01' the silica sand-molasses mix-tures \Vas perforllled in a drying-chambcr at 2100e :1: loeduring 50 minutes and subsequently the samplcs \\'crc eooledto cnvironmcnl IClllperature in a hermelic dryer. Thcrcaftcrthe initial weights of Ihe samples wcre measured ami lheywere plaeed in a climaric chamher al constant relative hu-midity (RH) amI rcmpcralure [1,31.

The v.'cl wcight was de!ermined by using an analyticalhalance (,",0.0001 g) al different limes.

Re\'. ¡\kx. Fú ...l7 (1) (2()()I) :'1-l2

KINETICS OF AHSORPTION OF THE ENVIRONMENTAL f\1OISTURE IN GRA1NY MATERIALS 39

r = 0.99771

0.10

0.15

__.1.1--.--..-- ••.•.•...., ;:.•• 0. r= 0,99662'oI

!0.05MI (g).- -..-- .

••• e_•."...'¡•

"•;•/••••

I•

0,02

004

0.06

0.06

M '(9)

0.00O 20 40 60 80 100 120 140 160

Time (x 1035)Time (x 10'5)

(a) ¡h)

,=0,987460.10

006

•••• _o_o_~ • ._ •.....•

/0"loo

••0.06 .'

0.04fofo,'o,0.02 h'

0.14Mt(g)

0.12_ •• e_ •••••••••••

/"O.~../..

••• r = 0.98997••.'

./

0.01

0.02

0.05

004

003

0.07MI (g)

006

0.00O 20 40 60 80 lOO

Time (x 10'5)

0.00L-~-'_~-'-~_-"-~_,-_-,---,o 20 40 60 80 100

Time(x 10'5)

(e) (<1)

FI(;URE 2. Gain of wet wcight of silica sand-sodium silicatc salllplc in cl1\'ironmcnl 01'(a) XOo/r KH and (b) 90% RH: ane! silica sand-lOo/rmolasscs samplc in environmcnt of (e) ~O% RIt ane! (el) 90% RII. (e) Expcrimcnt<tl (--) :"lodd

whcre AGuL = J[ex;, is the alllount orahsorhed moisturc in-l'(Jrporated (o Ihe smnple in a suflidently long lime t -+ oo.

{4 ~ 1 [(2" + 1)' " ]e = e" I - ;;: ¿ 211+ 1 (,xi> - 4L' 7[-DI

11=0

2n + 1 }x :-;in ----;¡¡;-7r,r . (S)

This solUlion cxprcsscs lhe Illoisturc concentration pro-files accon.ling 10 the (.r) coortlinate antllhe (t) time in al! thescclion 01'Ihe pastille. In Ihe cxpcrimenlal practice it is easicrlo work ",ilhout measuring the concentrations but the amount01" Illoisture that is incorporaled inlo the sample. This amount01' ahsorbed moislure Ihrollgh Ihe A arca 01' the penneablesurfacc is

MI = I/I C (:1', 1) dn/y dz\'

8001,1ft = ACoL{1 - "" '\' (' r

¡;- L 111 + 1 -n=O

,,[ (21!+1)'7[' ]}x LXI' - clL2 Dt.

(6)

(7)

TraJilionally, the analysis 01'lhe diffusive behavior 01'wa-ler into some Illaterials is Ireated in terms ol' the relative gain01'wel weight, obtaining Ihe experimcntal curves ol' JUt /J.'l!ooI'S. f [6,71. This proccdurc gives the possibility 01'dClermin-ing the D diffusion coeflkient without knowing Co' thereforeit is casier this way. Howe\'er. working dircctly wilh the dalanI' J[ t I'S. t it is possible lo determine (through lhe adjustmelltnI' Ihe I11odellEq, (7)], lhe C" I11(Jislu,e concenl,ation (Jn lheinlerface and lo get lhe real concentration proflles, \\:hich isvery importan!.

4. Results ami disclIssion

Shown in Fig. 2 are lile alllollnts 01' Ihe environlllenlal Illois-lure incorporaled into the sample Al t \'erslts time for the sil-¡ca sand-sodiull1 sil ¡cate ami silica s3nd-1 0% molasscs mix-tures. which whcrc cxposed respectively to cllvironlllents ol'relativc humidilies of HO and 90%. Sulid lines rcpresent thecurves 01' the fllted model. The fitting 01' the model pcrmit-led lo delermine D ami CIl in each case. The values 01'the DdilTusioll coeflkicnls and lhe Cn moisturc concentration onlile interface are givell in Table 11. In Fig. 2 the corrclalioncodfkienls r are shown.

Rel'. Mex. F(,. ~7 (1) (2(X)I) .17-42

E. VILLAR-COCIÑA. E. VALENCIA-MORALES. ANO R_GO~ZÁLEZ.ROI)RíGUEZ

'["OLE 11. Diffusion coefficients and moisturc concenlralion in the interface for different matcrials and relativc humiditics.

J'vlixturc

Silica s<Jnd-sodiulTlsilicate

Silica sand-sodiulTl s¡licate

Silica sand.lOo/t-lTlolasses

Silica sand-IOo/t-molasses

RII (%)

xo90XO90

Diffusioll cocflicicnt D( I1Im1/s)

8.40 x 10-1 Oc0.20 x 10-.'

1.04 x 111-' Oc003 X 111-3

7.7 X lO-1 ::i:0.3 X lO-1

6.9 X lO--1 ::i: 0 ..1 x 10--1

Conccntratiol1 inlhcintcrfacc (glmm1)

HI2 x 10-6 Oc002 X 111-6

9.11 X 10-6 Oc0.06 x lO-o

8..1~1X 10-6 ::i: 0.06 X 10-6

1.0 X 1O~5 ::f::: 0.02 X 10-,';

0.10

MI (g)

2

3. -._-.- -./ ••• -. r = 0.99952

.'/

.l:•••

0.06

0.08

0.04

0_02

2

3...- .....•-.-. -..-.-..-.••• -.- r = O 99932.' ..'í,•,

0.005

0.020

0_025

0.035Mt(g)

0.030

O.()()()O 10 20 30 40 50 60 70 80

Time(x 10's)

0.00O 20 40 60 80 100 120

Time (x 10's)

(a) (h)

FIGURE J. Gain 01' ",ct wcight oí"silica sand-5% molasscs samplc in cnvironlllcnt of (a) ~O% RII ami (h) 90% RH. considcring 2 sllpcrim-poseo dilTllSivc proccsscs. (a) Experimental (--) Modcl.

4.5 6 7.5Penetration deplh (nm)

15h I

10h

O.5h

30h

4h

31.5

g 4~¡¡ 2g"O

O

t5h

tOh

4h

O.5h

3 4.5 6 75Penetration deplh(rrrn)

"~45r--~ 4o, 3.5

" 3; 2.5

g 2~ 1.5e 1~o 0.5

" O o 1.5

(a) (h)

F1<;l1I-U;4. Conccl1tratioll protiles of silica sand-sodium si¡ieate sample in environmcnt of (a) RO%>RH amI (h) 90% RII at dífferenl timeinstants.

In Fig. 2, .\lt s~luratjol1 values (JI:::.•::,) are larger for sall1-pies exposcd undcr <)0% RH lhan for lhe s~mc samplc at 80%RH. sOlIlething cxpcctcd due to the faet that lhe grcaler thewater concentralion uf l!le environment. the greatcr lhe Co\ViII he in Ihe inlerface (Table II), anJ therefore the gain ofwet \\'cight wilI he grcaler. Prom thcse ligures the hygro-scopic po\\'cr that l1lolasscs give lo lhe mixtures is also de-ducl'd, sinee lhe relativc gains of Illoislurc in lhe equilibriulllare grc~ler in Ihe 10% lllo1asses mixtures Ihan in Ihe SCk mo-lasscs mixtures (Fig. 3) al the same environmcnlal eonditions.

In Fig ....L for cxampll', the l1loisture conccntralion pro-files for siliea sand ami sodiulll silicale samples are shown. Aconsiderahle variation of lhe concel1lration wilh Ihe paslillcdepth is apprcciatcd in thc initial instants. gctting lhe rellee-tion on lile impermeable faec al shOrl times, until. close 10saturatioll, the moislure concenlralion on the samplc is prac-Ikally Ihe samc :md equal 10 Ihe Co surfaee concenlration.Thl' llloislurc conccntralioJl in the sample can be known atallY depth from the surface at difrercnt timc inslants throughIhese proliles.

1/"1'. Mex. Fú. 47 (1) (2(XI)) 37-42

KINETICS OF ABSORPTION OF THE ENVIRONMENTAL MOISTURE IN GRAINY MATERIALS

TABLE 111. Diffusion cocfficients and amount 01'environmental moisture incorporate 10 Silica sand-5% molasses.

41

RII (%)

XO90

D1 (rnm2/s)

1.03 x 1Il-' Oc0.05 X lIl-J

6.9 X lIl-' Oc0.2 x 10-'

.H~

0.1l1892 Oc0.00092

0.05888 Oc0.00067

D'l (mm2/s)

(j.55 x 10-3 ::f: 0.8 X 10-3

1.G X 10-2 ::f: 0,01 X 10-20.01328 Oc0.00096

0.02734 Oc0.00072

A peculiar situation is found in lhe silica sand-5% mo-lasscs mixtures. According lo lhe rcsults shown in Fig. 3, il isclear thal spccifically in lhe case al' thesc mixtures, it shouldhe assulllcd that therc are {wo ¡ndependent plaees in lhe slruc~

lure 01' lhe mixture which can be lhe cause 01' lhe supcr-posilion 01'lwo diffcrcnt proccsses w¡lh diffusion cocfficicntsDI < D2.

In Ihe ¡iterature [11-17] sume examplcs are rcportcdwhcrc difflculties during lhe cvaluation 01' lhe diffusivc pro-ccss throllgh a simple model appear. These were overcomehy considering two different places 01'diffusion or diffcrcntrl1ases in lhe malerial and, therefore, 2 diffusive superposedprocesses with different diffusion coefficients.

The amounl of moisture incorporaled lo the sarnple istilen an addilive su\")erposition 01'solulions 01'type (7) whcre(..lCoL)II) = M;;' and (ACoL)(2) = M~) are Ihe amollnls01"moislure incorparated to lhe sample by each proccss aflera sllrt1cienlly long limc. Thc supraindex l and 2 correspond toproccsscs I and 2 respcctively. Observe that the surn 01"Jf~)and AJ~) coincides with the corresponding experimental sat-uration value of (he mixture.

The sllperposition 01' diffusive processes could be ex-plained phenomenologically in this way: lhe mixture is COtl-stituted by silica sand and the agglulinant, the last one joinsthe grains strongly hy making a layer around them. This layerincreases its thickness with the incrcase 01'the molasses con-tent for thc same grain size in the mixturc. Purthermore, itis also known Ihat molasscs are highly hygroscopic [11. \Vecan presuppose then a principal Jiffllsive way (principal pro-cess) through the molasses and another secondary diffusion\Val' (secondary process) lhrough the houndary between sandgrains and lllolasscs, which acts as agglutinant.

The consideration of a simple ditTllsive process in somenI' these mixtures leads to results lhal are not as satisfactoryas in lhe case 01'the eomposed process. The faet lhat in sOlllecases hoth proccsses are more noticeahle and in olher casesonly a simple proccss is naticed, is givcn hy the magnitudc ofone process eompared wilh the other one. \Vhen the magni-tude 01'olle process (secondary) is very small compared withlhe othcr onc (principal) this last one overlaps the 11rstone.In (his case Ihc principal process is almost nol alTeeted bythe supcrposition and il seems that only one process is hap-pcning. An approximation lo a simple process (DI = D) inthis casc lioes not lead to Iarge mistakes in Ihe liescriplionprocess. Several authors have analY7.ed this prohlcm in othersyslems [1I-]7J.

0.10

Mt (g)0.08

0.04

0,02

0.00O 10 20 30 40 50 60 70 80

Time(s) x 10'

FIGURE 5. Gain uf wel weight 01'silica sand-5% molasscs samplein cllvironment of90% RIl. considering a simplc di1'fusive process.

In the analyzed case 01'silica sand-molasses mixtures, thisphenolllcnon is more lllanifcsted in 5% of 1110lasscs(IcastIllolasses), where the consideralion 01'a simple diffllsive pro-cess leads (o rcslIlts tha! are not so salisfaetory as ohserved inFig. 5 comparcd v,'ith Ihe consideration of a compound pro-cess (see Fig. 3h).

The values of Ihe diffllsion coeffieients and lhe amount afenvironmental moisturc incorporatcd to the salllple by bothprocesscs are shown in Table 111.In the case of 10% mo-lasses mixtures, the principal process prevails and it can bedeseribed by a simple process. (Pigs. 2e and 2d). The values01' diffusion coerticients and Illoisturc concentralion un theinterface are shown in Table 11.

The ahove is explained laking into account lhe fael that,reducing Ihe Illolasses %, lhe agglutinant layer thieknessis reduced and lhe elTcclive diffusion arca is also reduecd.This cvidently causes Ihe Illagnitude 01' lhe diffusive processlhrough molasses to he smaller and so it <.loesnot overlap theseeondary process. Figs. 3a and 3b give an idea af the mag-nilude 01'both proeesses ami of lhe addition of processes.

5. Conc!usions

Assullled lhe ditTusive mollel pennits to describe the mois-ture ahsorptioll process lhat is evideneed in the experiment inthese materials. delermining the diffusion coeflkicnt valucsand Ihe moisture conccnlration in lhe interface (surroundingalmosphere-sample) in the fitting model process wilh enoughaccuracy.

Rel'. Mex. Fí.,. ~7 (1) (2001) 37-42

42 E. VILLAR-COCINA. E. VALENCIA-MORALES. ANO R. GONZÁLEZ-RODRíGUEZ

It a1l pcnnits lo determine the conccntratioll profilesCer. f) in a diffcrcnt way than other lreatments in whichC(.I',I)/Co is ohtaincd, and thcrcforc it shows the amount01' Illoisturc in the sample al any depths from the surfacc alt1ilTcrcnt time inslants.

1. E. Valencia. Ph.D. Thcsis, Central Univcrsity of Las Villas.CUC". 1992.

2. A.P. r..lakaricvich. Mezclas Au(ojragualltes para Moldes y Ma-chos. (Mir, Kiev, 1985).

3. E. Valencia and N.J. Galeano, Rel'. Soldadura 23 (1993) 94.Madrid.

4. Th" f)Y"(/lIlical Characler of Adsorplicm Mechanism, cditcd byJ.H. Dc Boer. (Oxford University Press. Oxford. 1928).

,jo 1. Dincer and S. Dosl.. /lIt. 1. Energy Resmrch 20 (1996) 53!.G. J. Crank, TI/(' Mathematics of Dijfusiml, (Clarendon Press, Ox-

ford. 1975).l. J. Crank and G.S. Park. Tram. Faraday Soco 47 (1951) 1072.

S. \V. Jos!. DiffllSioll in Solids, Liquids, amI Gmes. (AcademicPrcss. Ncw York. 1960).

Thc occurrcncc of supcrimposcd diffusivc processes isevidcnccd for silica sand-5% molasscs mixtures. and the val-ucs 01' the Jiffusion cocflkients and thc amount of moistureincorporated to the samplc for both proccsscs are dctcrmincdaccuratcly in the fitting proccss of a composcd moJel.

n. H.S. Carslaw and J.e. Jacgcr. COlldl/uioll of Hmt ill Solid~.(Oxford University Press. Oxford. 1970).

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