Diffusion through living systems: Sugar loss from sugar beets

DIFFUSION SUGAR

THROUGH LIVING LOSS FROM SUGAR

SYSTEMS BEETS

ANDREA SODDUt and FRANCE!320 GIOIA Istltuto dl Chmuca Apphcata e Metallur@a, Facolta dl Ingegnena, Umverslta degh Stud1 & Caghar~. Caghan, Italy (Recewed 20 Apd

1978, accepted 29 May 1978)

Abstract-The

drffuslon rate of sugar from sugar beets has been measured and Interpreted in the hght of the dual sorptron theory, namely it has been assumed that the sugar contamed m sugar beet IS present as lmmobdlzed sugar inside the cells and as mobile sugar m the vascular bundles Langmuv eqmhbrmm IS assumed to hold between these two states of sugar The dlscrepancles between theory and experimental results are well wlthm the lunlts of the uncertamty affecting the many physical parameters which play a role m the dlffuslon process and the complexity of the sugar beet’s microstructure The mterpretatlve theory proposed might be applied Just as well to the hvmg tissues of other vegetables th”l’RODUcTION

The object of this work has been to extend the dual sorption theory to the study of dlffuslon through a bulky hvmg system The experunental system adopted for this purpose has been suggested by its consrderable unportance m practice The dual sorption theory has previously been apphed with satisfactory results to the study of dlffuslon through many mlcroheterogeneous morgamc systems for which two concurrent modes of penetrant sorption take place [S, 10,27-291 Recently[41 this theory has also found successful apphcatlon in the study of ddfusion through hvmg systems However, its apphcatlon has been hmlted to the diffusion of penetrants through very thin layers In the present work, for the first time, the theory has been extended to the study of dlffuslon through a bulky hvmg system In particular, the process concernmg sugar loss from sugar beets immersed m water has been studied m detail However, the transport mechamsm proposed here might also be applied to the hvmg tissue of any other vegetable, the general basic cellular structure of many plants berng slmllar Besides the above theoretIcal interest, the study of sugar dlffuston from sugar beets unmersed In water 1s m itself of interest m mndustial practice for at least two reasons In fact, the sugar lost during the washmg and hydraulic transportation of the beet Influences both the efficiency of the sugar production process and the treatmg capacity of the waste water disposal system The greater this loss, the greater the amount of unrecovered sugar and the BOD content of the “flume water” However, the fust reason IS relatively less important Inasmuch as the sugar lost by mteger unpeeled beets during the washmg process ranges, as will be shown in this work and as reported in other works[5,21.26] between 0 2 and 2% of the total sugar content The tPresent address Rutgers Umverslty, Chemical and BiochemEnmeermg Department, Busch Campus, New Brunswick, New Jersey, U S A ical

763

difference depends on the uutlal state of the beet, the sod ongm, the cultivation procedure and the superficial damage produced m stockmg However it will be shown, m agreement with other results[ 1,121 that these losses can become quite significant when the cellular structure 1s badly damaged, as happens for Instance m northern countnes when mcldental hard-frosts occur before the harvest In these cases the sugar loss may be as much as 6%[1,12] On the contrary, the second reason (water disposal system) 1s most Important In fact, the sugar content of the flume waste water IS mamly due[5] to sugar losses durmg the washmg and hydrauhc transport of the sugar beets and produces a pollutmg load of about 300mg BODs per hter which may represent 60-90% of the total BODs load These losses therefore play a leading role m the economy of the water punficatlon system In order to be able to appreciate the Importance of this problem, It 1s worthwhile mentlonmg that the world sugar productron from sugar beets m 1975-76 was estimated to be about 33 x IO” metnc tons This corresponds to about 9 x 10” gallons of flume waste water to be treated The numerous works pubhshed concerning sugar dlffuslon from sugar beets are mainly directed towards the understandmg of the apparatus where the sugar IS extracted for production purposes Therefore they refer, for the most part, to thin shces of sugar beet whose internal cellular structure has been destroyed either by temperature or by chemical attack Only a few works, previously cited, deal with the study of sugar losses from integer beets either fresh or damaged But they are for hmited ranges of residence times and are substantially reports of experimental observations without provldmg any satisfactory mterpretatlve theories capable of leading to useful generallzatrons The reason for this IS that the current approach tends, roughly, not to take mto account the sugar losses durmg the washing process of integer beets This holds good m fact when, havmg only m mind the production process and when dealing with really fresh and undamaged beets,

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one IS mclmed to neglect losses of sugar of the order of 05% However, this rough approach, as shown before, loses any slgmficance If, instead of on sugar production, the attention IS focused on the waste waters resulting from the process, and on the consequent polluting load of the water treatment plant In the present work an attempt 1s made to study the mechamsm of dlffuslon through integer sugar beets, also accountmg for the state of the cellular structure The results, as previously mdlcated, whdst duected essentially to the prevlslon of the sugar lost dunng the washmg process, will also be valuable in giving a better insight mto the performance of the dlffuslon apparatus In fact the dlffuslon mechanism proposed may be easily applied to slices of beet, Inasmuch as the equations presented in this paper are wr&ten locally m the beet and theu extension to sugar beet shces requues only a change of coordinate system and boundary condltlons The present work can be outlmed as follows Fustly the experimental apparatus and procedure adopted to obtain measured dlffuslon rate data vs time are described Secondly the features of a physical-mathematical model describing the diffusion process at hand are discussed Anally a comparison between theoretical and experimental results is given EXPEXMENTAL

SECTION

The determmatlon of the amount of sugar lost by the sugar beets was accomphshed by lmmersmg a few beets into a stirred tank containing pure water and measuring the change of the sugar concentration m the water with time Four sets of experiments were carried out m order to obtam, by theu complementary results, a complete picture of the process mvestlgated Each set of tests concerned the foIlowmg uutlal states of the sugar beet I, fresh unpeeled beets, 2, fresh peeled beets, 3, beets whose internal cellular structure was somewhat damaged, 4, small pieces of beet A bnef description of the apparatus and experlmental technique follows Further details are reported by Soddu [23] Apparatus and expenmental technique A sketch of the apparatus IS shown m Ftg 1 The cyhndrlcal tank had a total capacity of 200 1 and a diameter of 52 cm

Rg 1 Sketch of the experlmental set up

and F

GIOIA

The stlrrmg of the hquld was assured by a pump The sampling section was placed on the pump line as shown m the figure All experiments were carried out with 6 = 11, m fact, as reported by Fordyce and Cooley [7] the ratio between water and sugar beet volumes whrch IS realized m sugar factories’ hydraulic transportatron system IS about 11 The followmg procedure was adopted to obtain the experlmental data 1 Fresh unpeeled beets Nme or ten sugar beets (density 1 1 g/cm3) from the stock were brushed and cleaned with compressed au, in order to remove the mould, and weighed A volume of water (19-20°C) ten times the weight of the beets was poured into the tank and the pump was started The beets were then placed m the tank and the timer was started From this Instant, which was time zero of the run, periodic samples of about 100 cm3 of water were drawn off A concentration of lOC~300 ppm HgC12 was added to these samples lmmedlately after they were taken m order to prevent any sugar loss due to bacterIaI action The sampling was always preceded by a purging of the samplmg duct and the water resulting from this purging was poured back mto the tank The energy dissipated by the pump tended to slowly increase the water temperature When this Increase reached 10°C (after about 30-60 mm) the pump was operated exclusively durrng samplmg This apparently questionable procedure IS lustlfied, as ~111 be shown later, m the light of the fact that the dlffusron process 1s hmlted by internal dlffuslonal resistances, the external mass transfer coefficient plays a limited role on the overall process especially after long dlffuslon times For the experiments regarding the other mltlal states of the beets the foltowmg operations were carried out pnor to theu lmmerslon rnto the tank 2 Fresh peeled beets The sugar beets were cIeaned as before and peeled by removmg a layer about 0 5 cm thick The reason for peehng so much off, was to ehmmate with the peel any damaged spots which were created durmg stocking and where bactetial action might be present This action may be quite important as reported by McGmms[14] 3 Damaged beets The beets were peeled as before and then left for 36 hr at 21°C This caused a weight loss, due to evaporation of theu water content, of 16% on average (22% of theu water content), thus mdlcatmg that damage of the cellular structure took place For this case 6 = 9 4 was adopted 4 Small pieces of beet The beets were peeled and then cut by means of a slicing machme mto pieces the dimensions of which ranged between 1 and 3 mm These experiments were carried out m a mechamcally agitated tank, m order to avoid cuculatlon of small pieces through the pump Chemical analysts of the samples The usual techniques adopted for sugar analysis m sugar factories are intended for the measurement of large sugar concentrations and unfortunately are unrehable when applied to the small sugar concentrations (ranging between 1 and 50 ppm) encountered m this work

Dlffuslon through hvmg systems Therefore, It was first of all necessary to devise a rehable method of analysis for measuring sugar concentrations m water of the order of magnitude mentioned above The method of analysis adopted, which permitted the determination of sugar concentrations as low as I ppm was a colorlmetrlc one based on resorcmol Details of this method are reported by Soddu[23]

assuming that the total sugar content of the beet 1s free to diffuse This implies that the local driving force for diffusion IS given by the total sugar concentration gradlent VcT Eventually, the cellular structure of the beet may be roughly accounted for by employmg an effective diffusion coefficient which, as done m the study of dlffuslon mslde porous sohds[19] may be defined as D. = D.(&) In particular this last approach was applied by Brumche_Olsen[3] to a thin slice of beet m order to make an estimate of E A prehmmary attempt to interpret our experimental data was done just using this approach The data were compared with the solutions of the differentml mass balance equations pertaining to the above model In practice, we resorted to the avahable analytical solutions of analogous heat transfer problems [2] Detals of this procedure are reported by Soddu[23] The comparison was quite unsatisfactory especmlly for the expertmental data regarding fresh sugar beets In partlcular these data were smaller than theoretical predlctlons even for one order of magnitude while the experimental process time was much longer A better agreement was found, on the contrary, for badly damaged beets (Fig 2) By damaged beets we intend here peeled beets whose internal cellular structure was somewhat damaged This damage was obtained m our case by evaporating, at room temperature for 36 hr about 22% of their water content The experimental sugar losses of these sugar beets as shown m Fig 2, were much higher than those of fresh beets but very similar to those reported m literature for other causes of damage, 1 e mechanical damagmg[6,26] and freezing [ 11 The above results were interpreted as glvmg a strong mdlcahon that locally m the fresh beets the sugar dlffuslon LSregulated by a driving force smaller than that based on the total sugar concentration cT This lmphes that not all the sugar contained rn the tissue participates directly m the dlffuslon process Part of It IS somehow bound to the cellular structure and its amount IS strictly dependent on the state of this structure, the fresher the beet, the greater this amount In order to account for this phenomenon It IS necessary to resort to a physical model

Expenmental results The raw experimental results were obtained as dtagrams of c, vs t They were worked out m terms of the fraction (Q/Q”) of sugar lost by the sugar beets at any time t, by eqn (22) In Fig 2, for comparison purposes, examples of experimental results, typical of each set of experiments, are reported Overall results are reported m Figs 3, 5-7 FwYsK!AL

MODEL

A first attempt to model the process of sugar diffusing from a beet immersed m water, may be accomplished by

l

. r------------,

:03 .‘k_: 0

-1

I 04 102 :01

I

+ 3

@

1

.**

1

.

0,

L_____‘_2_3_l*_m!P~ I I

4 r, hr

.

I

I

5

6

765

I

Fig 2 Example of experrmental results worked out as (Q/Q?% and (Q/W)% V, small pieces of sugar beet, 0, damaged peeled beets,

I, fresh peeled beets, A, fresh unpeeled beets

zo-

0 I

01

I

IlIllrl

I

I

I1111111

IO

I t.rmn

I IIllrll

100

I

I

I Illlll

Ftg 3 Plot of sugar lost by fresh peeled sugar beets vs tune The curves are calculated for BI = m, y* = 14 and for the lndlcated values of 0 Contmuous curves are for 8 = 11 Dotted curves are for 6 = 0~Calculated asymptotic values are for /3 = 10, (Q/Q’)_ = 0 63, for p = 100, (Q/Q”).. = 0 28, for fi = 1000, (Q/Q”)- = 0 11

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which represents more closely the mlcroscoplc structure of the beet For the reader who IS not farmhar wtth sugar beets, a synthetic descnption of Its structure IS provided m the foot-note t According to this descnpbon the mlcrostructure of a sugar beet may be ldeahzed as formed by storage cells of dunenslons rc 4 R separated by mterstlces (vascular bundles) formmg a porous structure filled by intercellular hquld which, for the purpose of the followmg analysis, may be assumed to have the same physical properties as water The bundles may be consldered as a connected system of tubes not blocked at any pomt by cell wails The sugar IS contamed both m the cells (at a concentration cr) and m the mtercellular hquld (CM) The eqmhbnum relatlonshlp between the two sugar concentrations may be assumed, as done by Chandrasekaran et al [4], for the study of dtiuslon through an analogous hvmg system, to be represented synthetically by an adsorptlon Isotherm of the Langmulr form

s$=-

KCM l+KcM

and F GIOIA

freely diffusible molecules The second IS formed by non-moble molecules, bound to the cells, which do not parttcipate duectly m the free dlffuslon process Furthermore, havmg ascertamed that the dkffuslon IS a slow process as compared with the sugar exchange between cells and mtercellular hquld, It may be assumed that, locally m the beet, eqmhbnum exists between those moblle and lmmoblhzed sugar molecules Namely, eqn (1) holds dunng the dlffuslon process The problem m many respects IS analogous to that reported m papers pubhshed[8-101 In particular It has been demonstrated m these works that a diffusion process combined with an eqmhbnum adsorption IS mdeed a slow process because It may be regarded, m many Instances, as free diffusion takmg place with a pseudodlffuslvlty smaller than the free dlffuslvlty D. This approach, which has proved useful m descnbmg drffuslonal processes m many systems m which penetrant molecules are mteractmg with sites of different activity, has found, m many respects, m the most recent literature [23,29] a systematic settlement m the “dual sorptton theory”

(1) MATHEMATICAL MODEL

When the eqmhbrlum condltlons are altered somewhere m the beet, sugar transport takes place According to the descnptlon even m the foot-note this transport follows essentmlly the pattern--cells to mtercellular hqmd (through the cell wall) foIlowed by free dlffuslon mto the vascular bundles, wluch, as sard above, are at no point blocked by cell walls Any dvect cell to cell transport may be excluded Furthermore works pubhshed[ll, 14,18,20] concernmg the study of the transport mechanism through the cell wall, allow to postulate that this sugar exchange may be assumed to be fast as compared to the free dlffuslon takmg place m the rnterstlces These conslderatlons allow to formulate the followmg synthetic physical model m order to descnbe the sugar transport mslde a beet the two concentrations cM and CI, m which the total sugar concentration CT has been divtded, may be characterized as follows[4] the first, relative to the mtercelluIar hquld, IS represented by

With the above physIcal model, the prevlslon of the sugar lost by a beet (nutlally having a total sugar concentration CT? Immersed m a stu-red tank contaming a volume Vf of pure water, may be described by the equations reported below

Sugar mass balance rnsrde the beet

which,

being

by eqn (1) 8Cl

Kc”

at =(1 + Kcr+,)’ 2

be of m r,

wluch may be estlmated[24.25] to range between 3 x 10e3 and 5 x 10m3cm and they form 2/3 to 3/4 of the beet tlssue[25], the rest IS chtefly vascular bundles The parenchuna ceils have thm walls and are separated from each other by mterceIlular spaces formmg a porous structure filled by mtercellular hqmd[lS] The vascular system IS formed ma& of xylem vessel cells land tracheld cells) and phloem sieve t&s[l3, IS] These two

systems are both morpholomcally and physlolomcallv different

The xylem vessel cells have a tibular form and whdn they are completeIy developed theu vacuoles disappear along with the cell walls at the ends of the cells The result IS a tubular structure that may extend for meters The lateral walls of vessel cells (thuzkened and h@ly bgmfied) have many pits thrnugh wtuch movement of water mto and out IS posslble[lS] The phloem sieve tubes are tubular living cells but thev ends are not mlssmg. however they are perforated and thus the substance contained m each of them IS contiguous with that of their nelghbours[13,15] A sketch of the structure of a beet may be found m [22]

at

becomes

++f+-$ tin the sugar beet, as m other plants, two mam tissues may dlstmgulshed the parenchuna and the vascular system One the chief functions of parenchuna cells is food storage, which ttus case IS aImost completely sucrose They have a radms

acM

l

+ Kc*(l (1

+

-E)

&

KcM)’ > at

(4)

Equations (2) and (4) have been wrltten for sphencal geometry although the beet does not have this shape This has been done m order to slmphfy the mathematical treatment However, analogously to the analysis of gassolld reactlon processes, the results may be apphed to the shghtly ddferent geometnes shown by the beets, upon definltlon of a smtabIe characterlstlc dimension (R = 3V/S) and, If necessary, by the mtroductlon of a shape factor The mltlal condltlon IS

t = 0,

CM = CM”

where CM”IS related to the known the sugar balance equation

cro concentration,

CT = ECM+ (1 - E)CI which at t = 0 and by eqn (1) gives

(5) by (6)

Diffusion

cM” = KCT” -

l

through livmg systems

- (1 - E)Kc* + d/((KcrO - c - (1 2eK

The hypothesis of an 1Ntd sugar concentration CT” umform m the beet IS confirmed by the results of other works [22,25,30] In dtmenslonless vartables, eqns (S)-(7) become

T =

$J = /3 - E - (1 - &?y*

0,

Y =

Y0

(9)

+ d{(B - E - (1 - EMT*)2 + 4e/3} 2G3 (10)

Sugar mass balance rnsrde the sugar beet peel The peel of the beet may be supposed to have dunenstons of the order of 1 mm Therefore flat-plate geometry may be adopted Furthermore, rt may be assumed that no storage cells are present m tlus hssue Therefore the sugar would be contamed exclusively as mobde sugar m the mterstlces Neglecting the differences between the radms R of the unpeeled beet and that of the peeled beet, the sugar mass balance m the peel, m drmenslonless vmables 1s (11) with the untial condltlon r = 0,

‘y. = Y”

(12)

Sugar mass balance m the tank and boundary condrtrons In order to write the sugar mass balance equation m the tank and the boundary conditions for eqns (8) and (11) we must dlstmnwsh between the two cases for which expenmental results were obtamed 1, peeled sugar beets, 2, unpeeled sugar beets The reason for these two sets of experunents 1s related to the observation that m sugar factory practice the sugar beets, when washed, have some extent of surface damage This unavoidable partml peeling takes place dunng harvesting, transportation and stockmg Therefore It was considered important to determine the role played by the sugar beet peel on the sugar loss rate For a completely mixed fluid in the tank the sugar balance equations may be written as Peeled sugar beets (13) Unpeeied sugar beets

767

l)Kcf)2

+ 4EKcTq

(7)

For the mltial condltlons of eqn (13) we must account for some mdlcatlons which were Ipven by the expenmental results The expenmental q vs t data mdrcated that about time zero, m a few seconds (w&h 1s a tune comparable with the tank residence tune), an abrupt change of cf took place This sudden change cannot be explatned m the hght of the above dtiuslon equahons The following explanation was assumed for thus phenomenon when the sugar beets are peeled, a layer of cells under the skm are unavoidably cut Therefore, the sugar contamed m these damaged cells IS no longer lmmobtilzed by the cell wall and 1s rapidly washed away when the beet IS unmersed into the tank An analogous reasoning, as will be seen later on, holds for unpeeled beets Actually De Vletter[S] attnbutes the en&e sugar loss of both peeled and unpeeled beets to the above phenomenon But hzs experrments were camed out exclusively for residence tunes shorter than those reported in the present work Consequently, the mltud conditions for both eqns (13) and (14) may be written as 7 =

0,

(15)

Yf = YfO

The values assumed by y; for the cases mvestlgated will be discussed further on Boundary condrttons The boundary conditions for eqn (Q-peeled may be wntten as 5-o.

5-1,

beets-

?L,

ae

$=$cut-Yl,=d

(17)

Analogously, for unpeeled beets (eqn 11) they are

(W (1% Values of parameters appeanng m equahons Ranges of parameters have been determined by assuming accepted values of the physical variables which enter their definition CT”,5 c * , y*, y,O The total sugar content of beets has been assumed constant for all experiments and equal to the measured value of 16% by weight This figure for sugar beet density of 1 1 g/cm’ fives Go= 0 176

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and F

for s, according to Tulhn’s results [25] (see foot-note), an average value of l = 0 3 was assumed c* may be estimated by assuming c P = cTo This Imphes that the sugar concentration of npe sugar beets comcides with theu maximum permissible sugar content As a matter of fact, this LSa reasonable assumption in the light of the results reported [31] Here, the sugar content of sugar beet 1s reported as being a function of npemng This content tends to an asymptotzc value, which on average for the years l-74 1s 18% Therefore one may write

Neglecting

c&e

with respect

to c*(l - E) one obtains

c*=_&=025 and y*=14 Takmg into account what has been said previously, yf” may be calculated by assuming that half of the sugar contained m the outermost layer of the beet cells IS responsible for the formatron of yf” in the water, I e 7,“ = 1 4 x lo-’ This value IS well comparable with the value which would be estunated by assuming that yf” 1s given by all the sugar lost up to the bendmg of the experlmental curves From Fig 3 one reads at about 0 2 mm 7,’ = 1 6 x 10d4 In the calculations this last figure was adopted R, 6. 0s. 7, 0, De, k, BI In order to evaluate the equivalent radius defined as R = 3( MS) it was necessary to measure the voIume and the external surface of the sugar beets This was accomphshed by shcmg the beets and measurmg the penmeter, thickness and weight of the shces The measurements were carried out on 25 beets chosen at random from the stock Then, by hnear regresslon it was possible to Cal&late the followmg average values V, = 773 cm3, S,,, = 556 cm’ and R,,, = 42cm It was also found that the average external surface of a sugar beet may be fairly well related to the average weight by the relatlonshlp

For the free dlffuslvlty D, of sugar m water a value of 4 4 X toe6 was assumed This IS the average of the values reported by Brumche-Olsen [3] for sugar concentrations ranging between 0 and 15% Then, by assuming f = 2[19] one calculates

An estimate of the mass transfer coefficient resulted m a value for k of the order of magmtude of low4 From the above figure a Blot number of the order of lo3 results RESULTSOF

CAL4XJLATIONS AND COMPARISON

WITB

EXPERIMENT AL DATA

Peeled sugar beets The set of equations previously discussed was set up to be solved numerlcally by a computer But, preliminary calculation runs showed that the convergency was qmte crltlcal particularly for BI > 100 and consequently the Moreover, these computmg times were qmte large d&c&es increased as Bt+w However, these prehmmary runs, necessardy hmited to short dfiusion tunes, showed that the calculated values of Q/Q” were not too sensitive to the value of the Blot number when this number was larger than 10’ In fact, In the range 50 < BI < 103, Q/Q” changed for about 20%, while for 103< BI < 10” the change was about 7% This Implies that for BI > 103, the external resistances play a hmlted role on the rate of the overall process The above constderatlons, together with the previously estimated values of the Blot number, suggested limiting the solution of the probem to the case BI = 00 In this instance eqns (13) and (17) are modified (as shown below) and the computmg times are reduced conaderably It 1s worthwhlle addmg that BI > 10’ IS the most reahstlc case that may be encountered m practice durmg any sugar beet washmg process For BI = Q) B C (17) must be substituted by

Yle-1= Yf and eqn (13) must be substituted

(20) by

(21) Equations (8) and (21) together with I C (9). (15) and B C ‘s (16) and (20) were solved numerically by a fimte difference procedure The results of these calculations are reported for different values of /3 in Fig 3 Both the theoretical results and the duectly measured expenmental data were worked out m terms of fractron Q/Q” of sugar lost by the beet at any time according to the relatIonshIp

Q = sy, Q" Of course for S = 00 in place relation holds

(22)

of eqn (22) the followmg

(23)

D=4=22x,O-” and D c =%=66x 7

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,O-7

The solution of the above equations also allowed to calculate the concentration profiles mslde the sugar beet It 1s worthwhile pointing out that these profiles (not reported here) show that for p = 100, after 60 mm the

DdTwon

through

S = 11, have a hmlted Influence on the dlffuslon rate m the beet This unphes that an approximate, but still satrsfactory solution of the proposed equations could be obtamed by substltutmg for B C (20) the equation

ddfuslon penetration has mvolved only the layer between 5=095 and e= 1 That IS, the concentration profiles are stdl flat and undisturbed for 95% of the sugar beet radms In Fig 3 the amount of sugar lost by fresh peeled beets vs time IS plotted and the asymptotic values (Q/Q?are also reported They were calculated by the relatlonshlps The balance equation sy+ + cya.cTo + (1 - E)-yk = 1

z‘= 1,

=

(24)

(YIE--0..= Y-

y=o

(28)

Peeled damaged beets As shown m Fig 2, the sugar lost by these sugar beets 1s much larger than that lost by fresh beets Furthermore, the data show an mltml large dtffuslon rate followed by a definitely smaller rate The &a&t forward apphcatlon of the above equations cannot explam both rates together Some further considerations are necessary Certamly,

with eqn (1) and the condltlon Yf-

769

hwng systems

(25)

gives

yf_=B-(l-c)Bv*-6-E+~((B-(1-C)BY*-S--)*+4B(S+c)} 2B@ + Q)

(P-Q1=

sy,,

(27)

Exammatlon of Fig 3 shows that the agreement between theory and experiment IS quite satisfactory for times up to lOOmm The /3 value which fits the experlmental data may be estimated to be about 100 On the contrary, for tunes longer than IOOmm the expertmental data tend to be smaller than prevlslons However times of this length are of no Interest m practical sltuatlons No waslung process can have a residence tune longer than 20 mm At any rate many reasons, not accounted for m the model, may be Invoked to explam dlscrepancles at longer tnnes, among these sugar consumption m the tank due both to chemical reactions and bactenal actlon to the In Fig 4 (@~/6E>le-I, which IS proportlonal dlffuslon rate, IS reported vs time A further analysis of the theoretical results reported in Fas 3 and 4 show that the dependency of Q/Q” and (Sy/S&-, on 8 (for 8 > 11) IS slight Tins means that the low ‘yf values whch bmld up m the tank, already for

I

Fig 4 Plot of the rnterface dunenslonless

the damagmg of cells has loosened part of the imtrally (before the damagmg) nnmobtilzed sugar On the other hand, the damagmg procedure, adopted m the expenmerits. was not so drastic as to damage all the cells Therefore, It can be assumed that only some of the cells were damaged, presumably the outermost ones According to ttus hypothesis it can be considered that the mitral diffuston takes place both by free diffusion (loosened sugar) and by the “unmobthzation-diffusion” mechanism For longer tnnes, the loosened sugar bemg exhausted, the dlffuston process proceeds accordmg to the much slower “lmmobtizatlon-d&ufuslon” process Neglectmg m the untlal tunes (up to 30 mm) the second ddfuslon mechanism and assuming umform dBtnbutlon of the damaged cells, one may apply the free diffusion theory to correlate the nuual data, I e by solvmg the previous equations but with fi = 0, or, by resortmg as we did to the solutions avatlable[2] for analogous heat transfer problems These solutions contam as a parameter the Blot number which, by the way, for the system at hand could not be evaluated ~th the necessary accuracy Therefore

IO

100

t, min

concentration gradlent vs tune for fi = 100 Dotted curve IS for 8 = 03. Contmuous curve is for 6 = 11

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and F GIOIA

r, min Fig 5 Plot

of sugar lost by damaged peeled sugar beets vs tune Contmuous hne ISfor BI’ = 200. Dotted he ISfor B=lOO

it was left as an adJustable parameter for the data fittmg The best fittmg of the data obtamed was for BI’ = 200 curve (a) of Fig 5 wluch represents fatly well the data pomts up to 30 mm It 1s mterestmg to note that the value of the Blot number resulting from the fittmg procedure IS of the ngbt order of magmtude The discrepancy. Indeed not very large, with the previously independently es& mated value, may be attnbuted to the fact that the procedure adopted load onto this group alI uncertamties (hke for example the unknown dlstnbutlon of damaged cells) &ectmg the model From 30mm onwards, accordmg to the above hypothesis, the sugar may be considered to be lost mamly by cells stall alive Therefore this second part of the phenomenon should be described by the same equatsons vahd for fresh peeled beets By applymg these equations one obtams curve (b) which agam describes quite well the final expenmental data Of course, a more sophisticated approach to the above mterpretation should have resorted to a knowledge (presently mlssmg) of the dlstnbutlon of damaged cells as a function of radius However, such a comphcation would not have added a sub&anti unprovement to the descnption of the phenomenon ~th respect to the assumption of a umform &stnbuuon of damaged cells mslde the beet Smallpieces of beet Prehmmary calculations were performed by utdlung the same equations wntten for peeled sugar beets, changmg of course the value of R,,, to R, = 0 1 The theoretical results showed to be consistently lower than the expenmental data Tlus was taken as a strong m&ction that the shcmg procedure mechamcally damaged to some extent also the cells mslde the pieces Therefore, tt was appropnate to resort to the same mterpretahon aven for the data pomts relative to damaged beets The two curves of Fu 6 have indeed the same rneanmg as those reported m Fu 5

Unpeeledfresh beets The experunental data, reported m FQS 2 and 7 agree substantmlly wth those reported by De Vletter[S] and Radbruch[l6] and show that sugar losses are present, although quite contamed Indeed, contranly to what has been assessed by many authors (among themr3.221) but m agreement with Tullm’s sugar beet structure descnptionI251 and the above mentioned experunental results, the sugar can pass through the tissue that makes up the peel From examination of Fig 7, it can be observed that a notable ddference exists between the results of the three expenments Thrs difference, also present m De Vletter’s data[51, may be attnbuted to the dtierent orwnal condotions of the beets utilzed m the expenmental runs In fact, for one experunent (the mtermedlate one) beets picked dvectly from the land were ut&ed The other two expenments refer to beets from stock, unavoidably damaged to a different extent both mechamcally and by bactenal action[14] Other causes that may explam the above dlscrepancles are the different sods and cultlvatmg conditions [ 171

50

“0

x

IO

opo

I/

-v

B-a-u-o

Bi.20

Fv 6 Plot of sugar lost by small pieces of sugar beet vs tune Contmuous Ime ISfor BI’ = 20 Dotted he 1s for #I= 100

771

Diffusion through hvrng systems

I

OJ

lo

t,mln

FIN 7 Plot of sugar lost by fresh unpeeled sugar beets vs tune Upper curve IS for QJQ” = lo-’ QJQ’ = 5 x IO+

For these data, an abrupt change in the water concentration m the first seconds 1s also present This may be attrzbuted to free sugar stickmg to the external surface in correspondence Hrlth micro and macro damaged spots, which bemg washed off rapldly when the beets are placed m the tank, create the uutlal concentration rjo In order to interpret the expenmental data reported m Fig 7 m the hght of the proposed model, the set of dlfferentlal equations (8). (11) and (14) with the approprlate I C ‘s and B C ‘s should be solved However, It LS possible to avold troublesome mathematical procedures and thus save computmg tune by assummg that 1 The concentration ‘yf m the tank IS constantly equal to rfo masmuch as for the complete run, rf/ro never exceeds 6 x 10e4 2 For short diffusion times the concentration mslde the beet and then at the interface sugar beet tlssuelpeel1s undisturbed Then the dlffuslon model reduces to the balance eqn (11) with the conditions

r=o.

ys =

y”

0%

f=O,

rs =

YfO

(30)

5=-.

ys =

y”

(31)

The solution of eqn (11) then sves

Assuming from the experunental data (Fig 7) that Q,lQ” ranges between 5 x lo-’ and 10e4 (consequently 4 5 x 10d < yfo < 9 x 103, eqn (33) applied to the expenmental data allows to estunate that ls ranges between 005 and 0 1 Namely (l/6) < (E./E) < (l/3) B C (31) implies that eqn (33) is valid up to tunes 7 for whch the penetration A 1s less than the peel thickness Namely, estunatmg thrs thickness to be equal to 1 5 mm, eqn (33) IS valid for 7181~10-~ that 1s fS 10min However, as can be seen from Fu 7, the above equation (dotted part) correlates with the data points also for times much longer than 10 mm, i e up to = 100 mm which IS much longer than any residence time reahzed in sugar factones For longer tunes, which are then of minor importance m practice, the diffusion rate becomes much smaller than the correspondrng value for peeled beets At about 100 nun, m fact, the ratio between the two rates IS about 0 1 CONCLUSIONS

application of the dual sorption theory has led to a simple but useful model for interpreting experunental data regardmg sugar dtffuslon from integer sugar beets The apphcation was based on the mlcroscoplc descnphon of the sugar beet’s structure In order to obtam as broad as possible a view of the rehabhty of the proposed model, several cases were expllcltly tested 1, fresh peeled beets, 2, damaged beets, 3, small pieces of fresh beets, 4, fresh unpeeled beets In all cases, with very few assumptions, the theoretlcal prediction correlated satlsfactortiy urlth the experunental data Both theory and expenmental data showed that sugar losses are strongly dependent on the state of the cellular structure of the sugar beet The

7s - Y0 m=erfc&

(32)

from which (33) the penetration IS A=2t/7

Lower curve LSfor

(34)

A

772 A

few

words

must

be

satd

about

the

SODDU and F GIOIA

practtcal

the theories of this work to the problem of sugar extractton In the sugar mdustry the sugar beets are cut into thm shces (only a few mm) and the sugar is extracted from them m a dii%.ion apparatus The operation IS carried out at temperatures as high as 75°C This procedure causes the destruction of the cellular structure Therefore, accordmg to the results previously reported the sugar may be assumed to be completely loosened (1 e #i = 0) and to dtffuse freely (wrth 0,) to reach the extractive medmm surroundmg the slices Furthermore, m most sugar beet plants, the msoluble pulp IS pressed to remove as much water and sugar as possible Fmally, rt must be pomted out that the transport mechanism proposed m this work might be applied lust as well to other vegetables In fact all plants have a simtlar microstructure stgntficance

of applymg

Acknowledgements-Mr Soddu destres to express hts grateful thanks to Ir A Pot, Duector of “Centraal Ontwtkkelmgs Laboratonum” for MS consent to carry out thts work and for his sttmulatmg dtscusston Mr Soddu IS also indebted to all the staff of the Central Laboratory and also to the Dtrector of the Msrobtologtcal Laboratory for theu prectous help The expertmental part of thus work was accomphsbed at the “Centraal Ontwtkkehngs Laboratonum Cooperatieve Vereorgmg SUIKER UNIE U A, Roosendaal Nederland” dunng a student trauuog penod spent there by Mr Soddu

Greek svmbols Kc,-“, dimensionless concentration = cdc,“, dimensronless concentration m the stirred tank, c,lc~‘, drmensionless concentration m sugar beet peel, dimensionless nntial concentration = cMo/cTo, dimensionless maximum rmmobllrzable sugar concentration = c*/cTo, dimensionless stirred flurd volume to beet volume ratio 3 Vf/4N?rR3, dimensionless void fractron (mtercellular volume fraction) position variable m the peel = x/R, dimensionless peel thickness = 1/R, dimensionless penetration, dimensionless radial position = n’R, dimensionless time = tD/R’, dimensionless tortuosity factor, dimensionless

Subscripts f m the stirred fluid I M m s T 03

immoblhzed mobile average peel total asymptotic value

NOTATION

Bt Bl’ c co c* Cf c*

D De D, K

k 1 :

Q Q" Qt R r r, s t V Vf W X

Blot number = kR/D, dimensionless Blot number = kR/D,, dunensionless sugar concentration, g/cm3 imtial sugar concentration, g/cm3 maxtmum immoblltzable sugar concentration, g/cm3 sugar concentration m sttrred tank, g/cm3 maxtmum total sugar concentration, g/cm3 diffustvtty DJ?, cm2/s effective dlffuslvity, (D,e)/T, cm*ls free diffusivity of sugar m water, cm2/s Langmuir’s isotherm constant, cm3/g mass transfer coefficient, cm/s peel thickness, cm (1 - l)/e, dnnenslonless number of sugar beets m stirred tank sugar lost by sugar beets pr’esent m the tank at any time f, g initial total sugar content of sugar beets present in the tank, g sugar untmlly lost by wash out, g sugar beet radius, peel not mcluded Defined as 3V/S, cm radial positton, cm cell’s dimension, cm external surface of sugar beet, cm’ tnne, s volume of sugar beet, cm” volume of water m the stirred tank, cm3 weight of sugar beets, g positton vanable m the peel, origin on the external suriace, cm

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[i8] izje Von W Zert frir dte Zuckenndustne 1965 15 506 [19] Satterfield, C N , Mass Transfer m Heterogeneous Catalysts M I T Press, London 1970 [20] Schbephake Von D and Wolf A, Zucker 1968 21 489 [21] Schneider F, TechnoJogte des Zucker Verlag M & H Schaper, Hannover 1%8

Ihffuslon

through

P M , Technology of Beet Sugar Productron and R&nmg Israel Program for Screntdic Translations, Jerusalem

[22] Sihn

1964 1231 Soddu A , Chem Engng Thesw. Cagllu~ Unlverslty. ar,, Italy 1977 [24] Terry N , J fip Botany 1970 21 477 (251 Tullm Von Vagen, Zucker 1952 5 433 [26] Uhlenbrock W , Zucker 1972 25 771

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Cagb-

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K J , I Collard Scl 1%5 u) 1014 A, Laudolph R A,

, Costantmtdes

Ind Engng Chem Fundls 1977 16 82 [29] Vleth W R .Howell J M and Hsleh J H , J Membrane SCI 1976 1 177 [30] Winner Von C and Feyerabend I311 Institut fttr Zuckerrubenforschung. 27(9) 504

I , Zucker 1971 24 35 Gottmgen. Zucker

1974