The Chernrcal Engmeermg Journal, 43 (1990) B53 - B66

ModelIing and Optimization of the Cyclic Regime of an Ion-exchange Process for Sugar Juice Softening

YUDONG SUN, GEORGES GREVILLOT and DANIEL TONDEUR*

Loboratowe des Srlences du Genw Chrmque, CNRS-ENSIC. B P 451, 54001 Nancy Ccdex (France)

B53

ABSTRACT

The cychc opera&on of a sugar decalclfica- tlon system by ion exchange is studied Although the saturation step can be con- sldered as a pure ion-exchange process, the regeneration step rnvolves complexatron and adsorption in add&on to ion exchange A mathematical model taking znto account both equrhbrrum and kmetlc effects is devel- oped and used to srmulate the system rn the saturation-regeneratron cycle m unrdrrectlon- al flow The effect of mcompleteness of saturation and regeneration 1s analysed and some quahtatrve rules are obtained Process optrmzatlon IS made with respect to the durations of the saturation and regeneration steps, the approach 1s not a numerical optl- mlzatlon calculation using the model, but rather a functional approach based on the qualltatrve rules above which leads to some optlmlzatlon cnterra that can be described by the system’s effluent concentration his- tories Different optimal pohcles of regenera- tion and saturation are identified, depending on the speclflcatlon on the effluent composl- tlon, and on operatmg modes

1 INTRODUCTION

Most mdustrlal sorption processes work on a cyclic regime, I e a succession of satura- tion and regeneration steps, because of the hmlted capacity of sorbent materials An analysis of cychc operation 1s therefore of practical mterest Much of the previous work has studied smgle-step operation, 1 e operation m which there 1s only a saturation step, or m which the saturation and regen-

*Author to whom correspondence should be addressed

0300-9167/90/$3 50

eratlon steps are Independent. Theoretically, this 1s the problem of descrlbmg the re- sponse of a uniformly saturated bed to a feed of constant composltlon

Approaches to the sunulatlon of cychc sorption processes have been essentially numemcal, or have rehed on the “equlhbrlum theory” for Langmulr equlllbrla [l - 31 m which mass-transfer resistances and hydro- dynamic dispersion are neglected Attempts at comprehensive approaches to the cychc regune have been based on the semi-emplrlcal “superposltlon method” [ 4 - 71, other seml- emplncal methods [ 81 or lumped approaches [9] All these efforts were restricted to predlctmg column behavlour and none attempted process optunlzatlon

In the present study, we shall be concerned with predlctmg and optlmlzmg the cychc operation, with some degree of mteractlon between the saturation and the regeneration steps. Studied here 1s a sugar Juice softenmg process by ion exchange recently developed and patented by Akzo [lo]

Cychc operation 1s first sunulated by nu- merically solvmg the mathematical model of the system, taking mto account mass-action equlhbrla and first-order transfer kmetlcs These slmulatlons are used to analyse the calculated concentration hEtomes, and some quahtatlve rules concemmg the cychc regune are obtamed The optunlzatlon 1s then made, with the arm of maxlmlzmg the regenerant efficiency of the switch between the saturation and regeneration steps The approach 1s functional rather than a detailed numerical calculation using the model, m that it 1s based on the quahtatlve rules gov- erning the system The result 1s some optunl- zatlon criteria which may be vlsuahzed on the effluent concentration histories For this reason, the emphasis here 1s not put on the detailed analysis of the hmltmg mass-transfer

0 Elsevler Sequola/Prlnted In The Netherlands

B54

mechamsm, nor on the precise vahdatlon of the model We have shown that the results presented are not sensltnre to the mass- transfer model used

We have considered only umdlrectlonal flow, I e the feed 1s m the same dlrectlon m both the saturation and the regeneration steps, although the model has proved capable of sunulatmg the case where the two feeds are m opposite dlrectlons A brief outline of the Akzo process 1s given below

Industrial sugar Juice contams catlomc species, essentially Ca*+, KC and Na+ The softening process cons&s m removmg part of the Ca*+ ions, to avoid scale formatlon m the subsequent evaporation and crystal- hzatlon steps This IS done partly m a carbo- nation step, m which the calcium carbonate 1s ehmmated, entrammg other unpurltles This softenmg 1s contmued m an Ion-exchange step, m which calcium ions are exchanged with sodium ions Ca2+ has a greater affmlty for the ion-exchange resins (a strong cation exchanger of polystyrene-sulfonate type) than Na+ and 1s thus easily removed from the sugar Juice In conventional processes, the regeneration 1s done with an aqueous NaCl solution, and requires large quantltles of salt, which ultunately constitute waste streams

In contrast, the Akzo regeneration process mvestlgated here uses as regenerant sugar Juice to which sodium hydroxide has been added In this basic environment, calcium forms complexes with saccharose, and this weakens the apparent affmlty of Ca*+ for the resm, thus conslderably faclhtatmg regeneration In addltlon, the regeneration effluent 1s recycled to the carbonation step, thus ehmmatmg the problem of liquid waste The prmclple of the Akzo cycle 1s sum- marized m Fig 1

The description of the physlcochemlcal processes needs to account for ternary ion

NnOH sugar tscLi

?+ sugar C.I NJ K’

ettlmlr

2 3 Equdrbrlum relataonshlps Three types of equlhbrlum relatlonshlps

are mtroduced here, descrlbmg complexa- tlon, ion exchange and adsorption

The complexatlon takes place m the hquld phase and may be considered as

Fig 1 Sugar softenmg process nCa*+ + 2nOH- + S --+ (SnCaO) + nH20

exchange, the formation of calcmm-saccha- rose complexes, sorption of these complexes on the resm, and possibly preclpltatlon Expenmental studies of these phenomena were carried out on a laboratory column 45 cm long and 1 5 cm m diameter loaded with Duohte C20, a gel-type, strong catlon exchanger Detailed experunental studies will be presented elsewhere Some experunental effluent concentration histories will be shown below m comparison with model calculations

2 MATHEMATICAL FORMULATION

2 1 Assumptions We make the followmg assumptions m

our mathematical formulation (1) The system works m an isothermal

envvonment with piston flow and neghglble radial and axial dlsperslon

(2) The complexatlon reaction between saccharose and calcmm IS sufficiently rapid to be considered m constant equlilbnum

(3) Ion exchange and adsorption kmetlcs are governed by a lmear transfer law

2 2 Mass balance The above assumptions lead to the mass

conservation equations

ac, ac, 1-e acq,+a,) =. -+-+- a2 a7 f a7 (1)

where C, 1s the total fluid-phase concentra- tion of species 1, q, and a, are its solid-phase concentrations due to Ion exchange and adsorptlon respectively, and E 1s the total void fraction of the packed bed

It should be noted that because of ad- sorption the number of mdependent species IS equal to the number of lon-exchangable species m the system, m contrast with pure ion exchange, where the number of mde- pendent species 1s reduced by one

B55

A

1 OLII c I LollcclIII:IIIoIl N.I concenv;luon

lo-

Fig 2 PredIcted effluent concentration hIstorIes for the regeneration step -, calculated with the slmphfled representation of the complexatlon reac- tion equlhbrlum, , calculated with the mass- action representation

where S mdlcates the sugar molecule and (SnCaO) 1s the complex formed, n bemg the stolchlometrlc number Actually, the com- plexatlon process 1s not as sunple as this; several complexes are formed, with more or less uncertam stolchlometrles Here, we restrict ourselves to the smgle, overall reac- tion above, with n = 3 [ll]

Smce calcium (and sugar) 1s mvolved m complexatlon, its total fluid-phase con- centration 1s composed of two parts the con- centration of lonlc calcium and the concen- tration of complexed calcium. A normal representation of the complexatlon equlhb- rlum would be a mass-action equation m- volvmg the concentrations of all relevant species as they appear m the chemical reac- tlon above. However, for the numerical procedure to be less tune-consummg and stable for cychc operation, we have used a sunpler representation a proportionality between the concentration of lomc calcium and the concentration of complexed calcium

[SnCaO] = K[Ca”]

This 1s Justlfied because the concentrations [OH-], [S] and [H,O] are relatively high and vary little, compared with [Ca2+] and [SnCaO] Figure 2 shows the comparison of two calculated effluent concentration histories of the regeneration step Both are solutions of the mathematical model

presented here, but one (dashed lme) 1s calculated usmg the mass-action representa- tion of the complexatlon whereas the other (solid lme) employs the sunple proportlonal- lty mentloned above, with the proportlonal- lty factor calculated from the concentrations of the first plateaux of the first sunulatlon The sunphflcatlon 1s successful, because of the sharpenmg (compressive) nature of the calcium front

Adoptmg this slmphflcatlon, we need not dlfferentlate between the lomc form and the complexed form of calcium, but will mstead use the total fluid-phase calcium concentration In this case, the coefflclents m the formulation of Ion exchange and adsorption equlhbrla and kmetlcs assume unplicitly new values

We adopt the constant separation factor form for the ion-exchange isotherm and the Langmulr form for the adsorption isotherm, which prove to be suitable for the present problem They are

q,

qJ a[J (2)

aO,KICla* a1 = 1 + K,C,“*

(3)

where C,* and C,a* are the fluid-phase con- centrations of species 1 III ion exchange and adsorption equlhbnum respectively with the solid-phase concentrations ql and a, In the present problem, the only ion species mvolved m adsorption LS Ca2+ (m its complexed form)

2 4 Kmetlcs For ion exchange and adsorption kinetics,

the lmear drlvmg force form 1s used

- = St,(C, - c,*) a7

aa, - = st,yc, - Cl**) a7

(4)

where St, and St,” designate the Stanton numbers for ion exchange and adsorption of species 1

2 5 Total sohd-phase capacity and total fluld- phase concen tratlon

cq, = Qt

There 1s a selectlvlty reversal between satura- tlon (where Q, IS the largest) and regenera- tion (where aCa 1s the smallest) AdsorptIon exists only m the regeneration step and only for calcium, where the parameters of the Langmuv isotherm are

(7) aoc,=08

where 1 stands for all lon-exchangable species It should be noted that C, 1s not meant to be a constant here, m contrast to pure ion exchange In view of eqn (6), eqn (4) should be omitted for one of the ion species

2 6 Sugar Sugar concentration 1s also governed by

eqn (1) where Its solid-phase concentration q due to Ion exchange 1s zero and that due to adsorption 1s proportional to that of calcium (it 1s thev complex that 1s adsorbed)

2 7 Numerrcal solution The finite difference method 1s used to

solve the above equations Equations (2) and (3) with eqns (6) and (7) are used to calculate equlllbrlum fluld-phase concentra- tions needed m eqns (4) and (5) When the whole process proves numerically stable, adaptation for cychc sunulatlon can be read Ily made

3 RESULTS AND DISCUSSION

3 1 Model parameters and operatmg conditions

The equlllbrlum parameters are estunated through an analysis of experimental effluent histories using the local equlllbrlum model m the presence of ion exchange and adsorp- tion [12] The values for the regeneration step are

Q&i = l(calcium reference)

CVK = 3 77

(llNa = 2 58

and for the saturation step

%a = 1

Q K = 0 0231

(XNa = 0 0346

Kc, = 7

For kmetlcs, the followmg values of Stanton numbers are chosen for a good fit to the experiments (see, for example, Fig 3(a))

StCaa = St,, = StK = 400

for both the regeneration and saturation steps These values merely refine the cal- culated results, and large values may com- pensate for too strong a numerical dlsper- sion

The composltlon of the feed (sum&ted mdustrlal sugar Juice) during the saturation step IS Ca2+ 0 00142, Na+ 0 00450; K+ 0 0321, equlv 1-l

The composltlon of the regenerant 1s 0 99 equlv 1-l NaOH, 0 24 mol 1-l saccha- rose

We take mto account four sequential steps m one cycle (1) saturation, (2) wash- mg, with or without fluldlzatlon of the bed, (3) regeneration, and (4) rmsmg

By fluldlzatlon of the bed, we mean the complete muting of the resm (and thus, a umform concentration profile along the column) Washmg and rmsmg are assumed to be done with a solution neutral to the system (a decalclfled sugar solution) and to be long enough to stabilize the effluent concentration In the mdustrlal process, washmg and rmsmg are used to remove the mterstltlal liquid from the previous step.

The system 1s supposed to be m umduec- tlonal flow. We shall study the effect of the duration of the regeneration step (regenera- tion following a complete saturation) on the next saturation step, and the effect of the duration of the saturation step (saturation followmg a complete regeneration) on the next regeneration step As we shall see later, the study of these two sltuatlons will suffice to understand and optlmlze the system m unldlrectlonal flow

B57

0 5 10 (a) Dlmenslonless Time ‘c

(cl

32 Regeneration and saturatron as two isolated steps

For comparison, we show m Fig 3 the calculated effluent concentration hlstorles for the regeneration of a completely saturated bed and the saturation of a completely regenerated bed, I e regeneration and satura- tion as two mdependent steps. An experi- mental result for the regeneration IS also presented, and the agreement 1s good We have no experimental result for the satura- tion step under the same operatmg condl- tlons In Fig 3(c), an experiment on satura- tion at a different temperature (60 “C, rather than 30 “C at which the present work 1s carried out) IS shown, which was mterrupted before the calcium breakthrough

Quahtatnrely, the behavlour of each species m Fig 3 can be mterpreted usmg the equllb- rmm theory. The sn-nulatlon of Fig 3(b) agrees even quantltatlvely well with the result of local equlllbrlum theory that predicts two shock waves The first corre-

00 4 Fii

g K

,_______________-______________ >

-__

2 S - A

: I :

z 002- 1 c , 6

_ i

E .

2 5 N3

CT.? 0 L..........wK. a

0 500 1000 (b) Dmjenslonless Time T

Fig 3 (a) Regeneration followlng complete satura- tlon (experlmental and calculated effluent concen- tration hIstorIes) (b) Calculated saturation followmg complete regeneration (c) Expenmental saturation followmg complete regeneration

sponds to the breakthrough of K+ and the second to that of Ca’+ and exhaustlon of the resin capacity The selectlvlty order 1s Ca2+ > K+ > Na+

For Fig 3(a), an equlllbrlum analysis accounting for ion exchange, complexatlon and adsorption predicts three mass-transfer waves, m addltlon to the hydrodynamic breakthrough. In Fig. 3(a), we thus observe, at 7 = 1, a decrease m sugar concentration that corresponds to the hydrodynamic breakthrough The first mass-transfer wave, observed at 7 shghtly larger than umty, 1s a shock wave correspondmg to the break- through of calcium and an mcrease m sugar concentration The plateaux (constant con- centration zones) predicted by the equlhb- rlum analysis are well observed on the concen- tration profile A second wave 1s predicted and observed, m which Ca’+ 1s exhausted while Na+ and K+ mcrease This second wave 1s a shock or possibly a “combined wave”, I e comprlsmg a dlsperswe tall The third

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Sug.lr _\ r-w-.

OS- L.

K

(a)

b

004. $

2 NCI

2

_________________-------_______.

3

0 .

(b) ’ 500 1000 Dlmenblonless Time r

Fig 4 (a) Regeneration (78 = 1 5) and rinsmg follow- mg a complete saturation (calculated) (b) Saturation foliowmg an mcomplete regeneration with TR = 1 5 (calculated)

wave, accordmg to the equlhbrmm analyus, mvolves only exchange between Na’ and Kt and 1s a dispersive wave

As usually, a non-equlllbrlum model, such as used m this paper, does not change the quahtatlve behavlour of the effluent history (number and types of waves, dlrectlon of concentration vacations of all species) as predicted by the equlllbrmm analysis

3 3 Saturation followmg mcomplete regeneration

In this case, regeneration 1s recomplete and we are to study the effect of its duration on the saturation followmg It Let us first examine the effluent concentration hlstorles of such mcomplete regenerations

Figures 4(a) and 5(a) show the calculated effluent concentration hlstorles for two partial regenerations followed by rmsmg

(a)

Sugar Z\ -_ --

OS- \’

A

004-

b K E ,________________---------____ --_ > P I

Fig 5 (a) Regeneration (TR = 1 0) and rmsmg foilow- mg a complete saturation (calculated) (b) Saturation following an Incomplete regeneration with TR = 1 0 (calculated)

with regeneration duration respectively at 7 a = 1.5 and Ta = 1.0 The correspondmg hlstones of the followmg saturation steps are shown m Figs. 4(b) and 5(b) We are mterested m TR + 1, rather than ra, smce a regeneration tipped at TR actually ter- mmates at TR + 1, If the rmsmg 1s assumed to occur m piston flow In fact, Figs 4(a) and 5(a) can be deduced from Fig 3(a) by makmg cuts on it at T = 78 + 1 = 2 5 and 7 = 2 respectively, defmmg the end of the regen- eration step After this value of 7, rmsmg takes place, resultmg m a purely hydro- dynamic washout curve of the cations

We can now examme the effluent hlstorles of subsequent saturations. For snnphclty, we shall restrict the discussion to the calcium concentration, which 1s the most Important vanable m the present case

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* 500 1000

Dmw~~~onless 1 IIIIZ 5

Fig 6 Saturations followlng different regenerations

(calculated)

In Fig 6, the two calcmm curves of Figs 4(b) and 5(b) are superunposed on the calcium curve of Fig 3(b), 1 e the calcium history of the saturation followmg a com- plete regeneration. For saturation followmg part& regeneration, the saturation step fast shows the effect of a continued regeneration calcium 1s eluted from the column with decreasmg concentration for some tnne before breakthrough. From the point of view of saturation, however, this constitutes a “leakage” of calcium, a very nnportant factor m the mdustrlal process, determmmg the quality of the product It 1s seen that this leakage is stronger for less complete pre- cedmg regeneration We can also see that the breakthrough waves of calcium are nearly superposed m the three cases, this super- posltlon 1s more exact, the more complete 1s the precedmg regeneration

The compresswe nature of this break- through calcium wave leads to the followmg nnportant quahtative result in a cyclic regnne, saturations that are always stopped somewhere on the ascendant part of the calcium wave (1 e durmg breakthrough) leave quite snnllar wave patterns mslde the column for the followmg regeneration, m- dependently of the extent of the previous regeneration

3 4 Regeneration followmg mcomplete saturation

We dlstmgulsh between washmg the resm with fluldlzatlon after saturation, which unphes complete mutmg and uniform con-

Dlmenslonless Time ‘c

Fig ‘7 Regenerations foliowmg different saturations

(calculated) - TS > 1000, , TS = 850, - - -,

7s = 700

centratlon profiles along the bed, and washmg without fluldlzatlon In mdustrlal processes, murmg of the bed can occur to some extent during the up-flow washmg required to remove fme particles and avoid cloggmg and mcreasmg pressure drops

As m the study of recomplete regeneration above, we are also theoretically mterested m 7s + 1 (at which the saturation actually termmates), rather than 7s However, smce the order of magnitude of 7s 1s m the hundreds, we shall not dlstmgulsh between 7s and 7s + 1 but mstead use Just rs

3 4 1 Washmg wlthout fluldlzatlon of the bed after saturation Figure 7 shows the calculated effluent

concentration h&ones of two regenerations correspondmg to precedmg saturation of durations 7s = 850 and 7s = ‘700, compared with complete saturation (TV > 1000)

For large 7s (complete or almost complete saturation), the concentration hM.ory IS that of Fig 3(a) For lower 7s, part or all of the Ca*+ breakthrough wave of the saturation remams inside the bed at the end of the saturation (Fig 3(b)) Therefore, this wave will interact with the waves of the subsequent regeneration. In Fig 7, the quahtatlve behav- lour before 7 = 2 is explamed m this way After this, the curves correspondmg to different presaturatlons supernnpose very well with one another

The mcompleteness of the precedmg satu- ration 1s reflected, among other thmgs, by a

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Dlmenslonless Tune T

Fig 8 Regeneration followmg an Incomplete pre- saturation (experlmental) ~5 = 645

Dmlenslonless Time z

Fig 9 Schematic comparison between the regenera- tlons followmg -, a complete saturation, , an mcomplete saturation wth fluldlzatlon, - - -, an mcomplete saturation wrthout fluldlzatlon

“delay” m the calcium breakthrough, com- pared with complete saturation. Figure 8 shows the experunental effluent concentra- tion history of the regeneration followmg an mcomplete presaturatlon, where we can fmd the general patterns predicted by the model, especially the delay of calcuun break- through (compared with Fig 3(a)) and the mclmed plateau of K+ This regeneration corresponded to the presaturatlon shown m Fig 3(c)

3 4 2 Washmng with fluldrzatron of the bed after saturation Fluldlzatlon of the bed makes the concen-

tration profile uniform along the column after saturation Incomplete saturation followed by fluldlzatlon leads to the same

behavlour as a complete presaturatlon with less calcium m the saturant It can be shown through an equlhbrmm calculation that the followmg regeneration tends to give a lower and narrower calcium peak than for pre- saturation which goes to completion

A comparison between regenerations fol- lowing a complete presaturation, an incom- plete presaturatlon followed by fluldlzatlon of the bed, and an mcomplete presaturatlon not followed by fluldlzatlon of the bed, 1s shown schematically m Fg 9 The effect of fluldlzatlon 1s to displace the calcium peak towards lower values of 7, identical to that of complete saturation

Qualltatwe rules From the above dlscusslons, we can draw

some qualitative rules concerning cychc operation that wfl be needed m the next section

(1) The breakthrough front of calcium m saturation effluent hlstorles IS very sharp, and m a cychc regune its posltlon 1s relatively independent of regeneration durations, when these are not too short

The sharpness of the calcium front leads to another rule

(2) If the saturation step 1s always stopped on the ascendant part of the calcium break- through curve, the wave pattern of the regeneration step wfl depend little on 7s

(3) For no fluldlzatlon of the bed after saturation, the wave patterns of the regenera- tion effluent hlstorles depend on 7s only at the begmmng of regeneration

This also leads to another rule (4) For no fluldlzatlon of the bed after

saturation, the wave pattern of the saturation step is completely determmed by TR and barely depends on 7s, If TR + 1 1s not too small

(5) Let us suppose we are m a steady cycle characterized by TR and 7s Then an mcrease m 7s wfl cause, m the next cycle, a larger amount of calcium to be regenerated m the regeneration step, whether or not fluldlzatlon takes place after saturation Graphlcally, m the case of fluldlzatlon, it can be asserted that If the dotted lme m Fig 9 shows the calcium effluent history of the regeneration before the mcrease m TV, the solid lme will depict quahtatwely the calcium effluent history after the mcrease m 7s

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1 ANALYSIS OF CYCLIC UNSTEADY REGIME

Before studymg the optnnlzatlon problem of the system m a cychc steady regune, we shall look at the system m a cychc un- steady regune, 1 e the response of the system to variations m the durations 7a and 7s of the regeneration and saturation steps

To facilitate explanation, we make the followmg convention the cycles before cycle 1 are m cychc steady regune with speclfled 78 and 78, and a perturbation 1s unposed on cycle 1, the cycles begmnmg from cycle 1 are numbered 2, 3, sequenttiy; a cycle begms with regeneration and ends with saturation

4 1 No fluldlzatlon of the bed after so turatron

4 1 1 Perturbatron on 7s

Let us consider a variation m 7s 7S’ = 7s +

A7s with 78 unchanged. From Rule (4) of the previous section, the wave pattern of the saturation of cycle 2 will not change with respect to that of cycle 1, smce 7R is not changed That 1s to say, the wave pattern 1s the same m the saturation m both cycle 1 and cycle 2 Moreover, their durations are also the same (~~‘3 This means that the wave pattern of the regeneration step of cycle 3 IS the same as that of cycle 2 There- fore, cycle 2 1s the first cycle of the new cychc steady regime This means that the new steady regime 1s unmedlately attamed after the perturbation

Moreover, Rule (5) tells us that m the new cychc steady regnne, the amount of calcium removed from the bed m the regen- eration step, and thus the amount of calcium removed from the saturation feed, IS larger than m the former cychc steady regnne If Ars > 0

4 1 2 Perturbation on TR Let us consider a perturbation on 78 such

that 7R’ = 7R + ATR < 7~ with 7s unchanged Accordmg to the calculated result of Fig. 6, the calcium effluent history of the saturation step of cycle 1 will be graphically above that of the precedmg cycles when they are plotted on the same graph With the satura- tion step of cycle 1 also stopped at 7s, the column will be more saturated m calcium than m the precedmg cycles. However, since

1 I I

Dlmenslonless Time r 5 3

Fig 10 SchematIc saturation In a steady cychc regime with 7~ (-) and saturation In the new steady cychc regime (- - -) obtained when 7~ IS Increased to ~8’

the wave pattern of the regeneration step of cycle 2 E mfluenced only at the begmmng accordmg to Rule (3), the wave pattern of the saturation step of cycle 2 will be the same as m cycle 1 Therefore, we conclude that the new cychc steady regime 1s attamed m the saturation step of cycle 2 The amount of calcium removed from the saturation feed 1s smaller than m the former cychc steady regnne when A7R < 0

4 2 Flurdrzatron of the bed after saturatron 4 2 1 Perturbatron on rs Let 7s’ = rs + A7s > 7s after a step pertur-

bation with 7R unchanged The bed is more saturated than previously after the saturation m cycle 1 As a result, m the regeneration step of cycle 2, the calcium peak will tend to get hgher and wider graphically, accordmg to Rule (5) With 7a unchanged, there tends to be more calcium left m the bed after this regeneration Therefore the calcium curve of the saturation step of cycle 2 will tend to ruse higher graphically (I e more “leakage” of calcium) Smce the saturation step of cycle 2 also stops at 7s’, the bed will tend to become still more saturated The con- clusion 1s that m the regeneration step, the amount of calcium regenerated from the bed tends to mcrease and the calcium curve m saturation tends to rise higher graphically

This shows that when ~~1s mcreased, the amount of calcium removed from the bed m the regeneration step will mcrease m the new cychc steady regune, or equivalently, the amount of calcium removed from the saturation feed 1s larger than m the former cychc regune In Fig 10, this 1s expressed by

B62

Dlmenslonless Time z

Fig 11 SchematIc regeneration In a steady cyclic

reglme with 7~ ( -) and regeneration In the new

cychc regime (- - -) obtamed when 7~ is decreased

to TR’

Dmlenslonless Time 5

Fig 12 SchematIc saturatlon m a steady cychc regime with TR (- ) and saturation m the new re- glme (- - -) obtamed when 7~ IS decreased to 7~’

area AB’C’D’ > area ABCD

le

(8)

area D’ECD < area BB’C’E (9)

This can be used to estnnate the location of the new steady cychc regime when 7s 1s mcreased to 7s ’

4 2 2 Perturbation on 78 Let rR’ = TR + ATE < ~a after a step per-

turbation with 7s unchanged There will then be more residual calcium m the bed after the regeneration By reasonmg m a snmlar manner to the perturbation on rs above, we can conclude that the calcium saturation effluent history tends to become higher graphlcally and that of the regenera- tlon effluent history higher and wider. There- fore m the new cychc steady regune, the amount of calcium removed from the satura- tion feed or that removed from the bed m the regeneration step will be less than m the

former cychc regnne This 1s expressed m Fig 11 by

area AC’B’ < area ACB

ie

(10)

area AC’DA < area B’DCB

for the regeneration step

(11)

Let 6 = area B’DCB - area AC’DA We then have for the saturation step

S<6 (12)

as shown m Fig. 12 The mequality of eqns (11) or (12) gives

an estunatlon of the location of the new cychc steady regnne

5 OPTIhlIZATION OF CYCLIC STEADY REGIME

The optlmlzation obJectwe is to mmnne the cost of a unit quantity of treated saturant, which can be formulated as

c= RTR + STY + A

=R’% +S+ fl (13) TS 7s 7.S

where R 1s the cost of passing a unit volume of regenerant mcludmg the cost of regenerant, S 1s the cost of passing a unit volume of saturant (sugar Juice to be treated), and A 1s the cost of mtermedlate treatments between the saturation and regeneration steps such as rmsmg and washing The factors R, S and A are functions of operatmg condltlons such as temperature and NaOH concentration of regenerant

When the cycle 1s long enough (large TV), it can be expected that the term A/T~ m eqn (13) 1s neghglble compared with the two other terms This means that the operation cost lies essentially m the regeneration and saturation steps Equation (13) can then be snnphfled to

C=& +S

7s (14)

Thus the optlmlzatlon problem consists m frst mmm=mg Ta/Ts at constant operatmg condltlons

O,=mm’A (15) 78

with respect to ~a and TV, the constramt being that the effluent of the saturation step must be of acceptable quality in calcium content

B63

The overall optunlzation of the operating con- ditions will then be 0 = mm C = mm (RO, + S)

In this work, we restrict ourselves to the optunlzatlon problem of eqn (15). and m what follows, we shall treat it separately accordmg to different constramts on the quality of the saturation effluent

5 1 Maxzmrzrng regenerant effzclency without constram t on the saturation effluent

Here the optunlzation ObJectwe IS to remove from the saturation feed the largest amount of calcium possible per unit volume of regenerant wlthout speclfymg the quality of the saturation effluent

Accordmg to our dlscusslon m the previous section, it 1s favourable to mcrease 7s so that the saturation 1s complete, as an mcrease m rs will cause a greater amount of calcium removed by the system m the new cychc steady regune Therefore at a given Ta, the best situation 1s complete saturation How- ever, such a sltuatlon always gives the same regeneration wave pattern, mdependently of TR, so that the optunal Ta can be easily determmed Figure 13 shows the calcium effluent history of the regeneration step correspondmg to complete saturation Let us use the function C,,(T) to represent it, then the value of ra maxunlzmg the amount of calcium eluted per unit volume of regener- ant 1s given by

rR+l

J Cca(T)d7 d o -_ = 0

dra (17)

713

which 1s equlvlaent to Ta + 1

s Cca(~W = 7RCCatTR + 1)

0

(W

where Cca(~R + 1) 1s the value of Cc. at 7 = Ta + 1 The correspondmg pomt on the regen- eration curve IS easily located If one notices the property of equal areas, as G.&rated m Fig 13

5 2 Constramt on the average calcium concen tratlon of satura tlon effluent

This 1s the optunlzatlon defmed by eqn (15)

5 2 1 No fluldlzatlon of the bed after sa tura tron Let us first study the best 7s correspondmg

to a given Ta The best 7s should be the

Dm~enslonless Time T

Fig 13 Schematic calcium curve In regeneration with optlmal TV Indicated

Are.1 S, = .ved S,

?I

Dlmenslonless Time r k2

Fig 14 Schematic calcmm curve In saturation with two values of 7s correspondmg to the maxlmum aver- age calcium concentration

largest possible as long as the constramt 1s satisfied Accordmg to Rule (4), for a given Ta, the saturation wave pattern is completely determmed So the best rs IS such that the average calcium concentration of the satura- tion effluent between zero and 7s 1s equal to the concentration llmlt unposed There may be two such values of TV, as shown m Fig. 14, but it 1s the larger one 7s2 that should be taken, 1 e the one on the ascendant part of the calcium curve

Let C,,, be the maxmum average calcium concentration unposed and Co be the calcium concentration of the saturation feed. Then

Ts(Co - Cnl,,)

gives the amount of calcium removed from the saturation feed, which 1s equal to the amount of calcium eluted m the regeneration step The optumzatlon problem of eqn. (15) remams the same when its right-hand side 1s dlvlded by the constant (Co - C,,,) There- fore the optunlzatlon problem 1s equivalent

B64

Dlmenslonless Time r

Fig 15 SchematIc calclum curve m saturation with one TS correspondmg to the average calclum con- centratlon

to that of maxunlzmg the amount of calcium eluted from the bed per unit volume of regenerant.

Now let us consider 7R Smce the best rs for a given 78 corresponds to the ascendant part of the calcium curve of the saturation effluent history, Rule (2) suggests the approx- unatlon that the regeneration wave pattern 1s the same for different VdUeS Of ra It follows that, by reasonmg m the same way as above, the best ~a should approxunately be given by eqn (18) and G&rated graph- ically m Fig 13 However, it can be that for a given 712, no value of Ts satisfies the con- stramt To flu&rate this, let us consider the sltuatlon of Fig 15 where there 1s only one rs such that the average calcium con- centration 1s C,,,, unhke the sltuatlon m Fig. 14 where there are two such values of rs If we lower the value of C,,,, we shall fmd no value of 7s that wfi satisfy the constramt. Therefore, when Ta 1s determmed m the way described above, the saturation step should be checked, 1 e d such a ~a corresponds to no 7s satlsfymg the constramt, then Ta should be mcreased until the con- stramt 1s met as shown m Fig. 15

5 2 2 Fluldlzatzon of the bed after saturation We take it for granted that the saturation

step should be stopped on the ascendant part of the calcium curve of the saturation ef- fluent history Then Rule (2) leads to the approxlmatlon that the wave pattern of the regeneration step is not mfluenced by the duration of the saturation step Therefore,

7s should be such that the average calcium concentration of the saturation effluent 1s equal to the concentration lunlt unposed

We can then reach the same conclusion as for no fluldlzatlon, I e the Optlmd 78

1s given by eqn (18) or graphically m Fig 13, and If such a 7a corresponds to no 7s satis- fymg the constramt, then 78 should be m- creased until a sltuatlon analogous to Fig 15 1s reached.

It should be noted that although the same conclusion 1s obtamed as for no fluldlzatlon, it 1s somewhat rougher when used here

5 3 Constramt on the local calcium concentratron of saturatron effluent

Now the constramt 1s the maxunum local calcium concentration of the saturation effluent

5 3 1 No flurdrzatlon of the bed after saturation Accordmg t0 Rule (4), for a given 78,

the wave pattern of the followmg saturation 1s completely determmed It follows that the best 7s should be on the ascendant part of the calcium effluent concentration history, at which the calcium concentration IS equal to the maxunum local concentration unposed

Accordmg to Rule (l), such a 7s does not vary much when ra changes. Therefore the optunal 7a should be such that the calcium concentration of the saturation effluent at the begmnmg of the saturation step (the first calcium concentration maxunum) is equal to the concentration lunlt unposed

5 3 2 Fluldlzatwn of the bed after saturation We take it for granted that the saturation

step should be stopped on the ascendant part of the calcium curve of the saturation effluent history Then by reasonmg m the same way as for fluldlzatlon, the same con- cluslon can be obtamed, 1 e the best 7s corresponds to the pomt on the calcium curve (ascendant part) of the saturation effluent history, at which the calcium concentration IS equal to the concentration hmlt imposed, and the optimal 78 is such that the calcium concentration of saturation effluent at the begmnmg of the saturation step (the first calcium concentration maxunum) 1s equal to the concentration lunlt imposed

B65

6 CONCLUSIONS

A model 1s proposed for the cyclic soften- mg of sugar Juice, usmg the Akzo process The speclflclty of this process lies m the regeneration, which uses sugar Juice at a basic pH (obtamed by addmg sodium hydroxide) Under these condltlons, calcium forms com- plexes with saccharose, and the ion exchange of Ca2+ and Na+ IS therefore displaced m favour of Na+, the reverse of the softening step The complexatlon equlhbnum may be described by a sunphfled relatlonshlp The resultmg model 1s shown to represent pro- perly experimental results on smgle-step operation It 1s then used to simulate co- current (unldlrectlonal) cychc operations.

The mam concern of this work is to op- tunlze the cychc operation, especially with respect to the switch between the regenera- tion and saturation steps, 1 e their respective durations 7a and TV, as defined by eqn (15) The approach adopted IS not a numerical optunlzatlon calculation usmg the model, but 1s functional The model 1s used to calculate the effluent concentration hlstorles of the system m a cychc regune; observations and analyses are then made, which lead to some general rules concemmg the system Based on these rules, optunlzatlon IS carried out vm a reasomng process. The result IS some cnterla that can be described by the system’s effluent concentration h&ones (I e the effluent concentration h&ones of the system when optumzed satisfy these cntena) These criteria are valid masmuch as the numerical sunulatlon 1s qualitatively correct

A character&c of the system 1s that both saturation and regeneration steps are favour- able (to takmg up Ca2+ m saturation and to removmg it m regeneration), which leads to compressive (sharp) waves m both steps

The followmg properties are shown (1) In a cychc steady regune, when a

change is effected on TR or TV, the new steady regune to be attamed may be estimated in terms of its effluent concentration hlstones from those of the former steady regime.

(2) When there 1s no strmgent constramt on the calcium content of the effluent, the operational oblectlve may be to maxumze the regenerant efficiency (amout of calcium removed per cycle and per unit amount of

regenerant) with saturation carried to com- pletlon. The optunal Ta 1s shown VI Fig 13

(3) When the constramt 1s on the average

calcium content of the softened Juice, the optunal 7s yields an average calcium con- centration equal to the constramt, as shown m Fig 14 The optunal 78 1s determmed as m the previous case, or, when such a Ta corresponds to no 7s satlsfymg the constramt, Ta should be such that it yields a saturation curve as 1s shown m Fig 15 These con- cluslons are somewhat crude when the bed 1s fluldlzed and mured after saturation

(4) When the constramt 1s on the local calcium concentration (z e the calcium concentration must never be above a certam hmlt), the optunal 7s 1s such that the corre- spondmg calcium concentration (on the ascendant part of the calcium curve) 1s equal to the specified hmit In turn, Ta should be chosen so that the first effluent of the satura- tion step (after rmsmg) 1s at the speclfled lunlt Agam, this 1s rather crude when the bed is remured after saturation

It seems that optnnlzatlon could be con- ducted, usmg these cntena, on a system m cychc operation m an iterative way, provided that on-lme analytical follow-up 1s available In each cycle, we stop the regeneration and saturation steps at 7R and rs values as speclfled above, and then, after several cycles, the system should converge to a steady regune that 1s optnnal

ACKNOWLEDGMENTS

The authors would hke to thank the So&t6 G&&ale Sucn6re for financial sup- port, and especially Mr Plever and Mr Rous- seau for helpful dlscusslons

REFERENCES

1 F Helfferlch and C Klem, ~fultlcomponen~ Chromatography. Theory of Interference, Marcel

Dekker. New York, 1970 2 G Grevlllot, D Tondeur and J A Dodds, J

Chromntogr , 102 (1974) 421 3 H -K Rhee, Equlhbrlum theory of multlcom-

ponent chromatography In A Rodrlgues and

D Tondeur (eds ). Percotatlon Processes Theory and Appbcot~ons, SlJthoff & Noordhoff, Alphen

aan den kjn, Netherlands, 1981, pp 285 - 328

8

9

10

11

12

J A Dodds and D Tondeur, Chum Eng Scr , 27 (1972) 1267 - 1281 J A Dodds and D Tondeur, Chem Eng Scl , 27 (1972) 2291 - 2298 J A Dodds and D Tondeur, Chem Eng Scr , 29 (1974) 611 - 619 C Klem, Design and development of cyclic

operations In A Rodrlgues and D Tondeur (eds ), Percolation Processes Theory and Apph- cations, SlIthoff & Noordhoff, Alphen aan den RlJn, Netherlands, 1981, pp 427 441 C Grewllot, These de 3eme cycle, Umverslte de

Nancy I, Nancy, France, 1973

D Tondeur. Dual-step countercurrent processes In A Rodrlgues and D Tondeur (eds ), Percola- tron Processes Theory and Applrcatlons, Sljthoff

& Noordhoff, Alphen ann den Rijn, Netherlands,

1981, pp 517 - 538 Akzo Company, Patent EP 0 032 263 A 1, Euro-

pean Patent, 1981 H Desmorleux, L Aranyl, S Marques and G Grewliot, Sugar Juice softenrng usmg sochum hydroxide m thm Juice for regeneration, m P

Ha&i and A Marton (eds ), 5th Int Svmp on Ion Exchange. Slofok, Hungary, 1986, Unwerslty of Veszprem, Hungary, 1986

Y Sun, itfemowe de DEA, ENSIC, Instltut Na- tlonal Polytechnlque de Lorrame, Nancy, France,

1936

APPRENDIX A NOMENCLATURE

A cost of mtermedlate treatment (be- tween the regeneration and saturation steps) per un’t volume of saturant

=01 coefflclent of Langumulr isotherm for the adsorpt’on of species 1 (equlv 1-l)

a, sohd-phase concentrat’on of species I due to adsorption (equlv 1-l)

c cost of treatmg unit volume of sat- urant

c, fluid-phase concentration of species I (equlv I-‘)

c,* fluid-phase concentration of species 1 m Ion-exchange equlllbrlum with the solid phase (equlv 1-l)

c,a*

C max

co

Ct

K

K,

9,

9t

R

s

St,

St,”

z

Greek

a,

E 7

TR

7.s

fluid-phase concentiatlon of species I III adsorption equlllbrlum with the solid phase (equlv I-‘) maxunum concentration lunit un- posed on the average calcium con- centration of the saturation effluent calcium concentration of the satura- tion feed (equlv 1-l) total fluid-phase concentration (equlv 1-1) proportlonahty factor m the slmph- fled representation of complexatlon equlhbrlum coefficient of Langmulr isotherm for the adsorptlon of species 1 (1 equlv-‘) solid-phase concentration of species I due to ion exchange (equlv 1-l) total solid-phase capacity for ion exchange (equlv 1-l) cost of passmg unit volume of regen- erant mcludmg the cost of regenerant cost of passing unit volume of sat- urant Stanton number of ion exchange for species 1 (dunenslonless) Stanton number of adsorption for species I (dunenaonless) distance from column mlet over column length (dunenwonless)

symbols separation factor for species 2 us a reference species total void fraction of the packed bed dlmenslonless tune, I e passed solu- tion volume over mterstltlal bed volume dunenslonless tune at the end of the regeneration step dunenslonless tune at the end of the saturation step