cap 22 absorption and ionic exchange

7/25/2019 Cap 22 Absorption and Ionic Exchange

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2003 by CRC Press LLC

823

22Adsorption and Ionic Exchange

22.1 Introduction

Adsorption and ionic exchange are mass transfer unit operations in whicha solute contained in a fluid phase is transferred to the solid phase due toretention on the solids surface or to a reaction with the solid.

22.1.1 Adsorption

The solute retained in adsorption processes is called adsorbate, whereas thesolid on which it is retained is called adsorbent. Solids with a large contactsurface are used as adsorbents and are generally porous. Activated carbonis widely used as adsorbent, although there are synthetic polymers calledmolecular sieves also used for adsorption processes.

The force with which the solute is retained can be of three types: electric,Van der Waals forces, and chemical. Electric forces are due to attractions

between a solute with a certain charge and points of the adsorbent with anopposite charge. The adsorption is called physical when the forces are ofVan der Waals type, and adsorption is usually reversible. However, adsorp-

tion can be due to a chemical reaction between the solute and the adsorbent a chemisorption. While in the physical adsorption the solute can be retainedon any point of the surface of the adsorbent, in chemisorption the adsorbentpresents active points on which the adsorbate is retained.

Adsorption is used in many cases of purification of fluids containingcontaminants that give them unpleasant flavors or aromas. Limonene is acompound that confers bitter flavor to orange juice, and it can be eliminated

by adsorption on polymers. In the same way, melanins and melanoidins,formed by enzymatic and nonenzymatic browning, can be eliminated byadsorption on activated carbon.

22.1.2 Ionic Exchange

Ionic exchange consists of replacing ions of a solution with others containedin a solid, which is called exchange resin. The ionic exchange can be consid-


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824 Unit Operations in Food Engineering

ered a chemical adsorption, where ion exchange occurs at defined points ofthe resin. The mathematical treatment for calculation of exchange columnsis similar to that for adsorption. Depending on the type of ions that they can

exchange, resins can be anionic or cationic. One of the most importantapplications of ion exchange is desalination and conditioning of water.

22.2 Equilibrium Process

22.2.1 Adsorption Equilibrium

When a solid adsorbent and a fluid containing a solute come into contact,the system evolves in such a way that the solute is transferred to the surfaceof the solid and retained there. This process continues until reaching adynamic equilibrium between both phases. At that moment, the fluid phasehas a concentration of solute C, while in the solid phase the amount of soluteper unit mass is m. The values of Cand mat equilibrium depend on tem-perature, and the function that gives the variation of the amount of soluteretained by the adsorbent (m) with the concentration of solute in the fluidphase (C) is called adsorption isotherm. This type of isotherm can havedifferent forms. Figure 22.1 shows the typical isotherms that can be presentedin different types of soluteadsorbent systems.

FIGURE 22.1Types of adsorption isotherms.

Irreversible

Linear

g

adsorbed

/g

solid

g dissolved / 1C solutionOO

C

m


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Adsorption and Ionic Exchange 825

The theoretical obtainment of the adsorption isotherms can be based onkinetic or thermodynamic considerations, the former more intuitive. Also, itdepends on considering whether the solute is retained by the adsorbent in

one or in various molecular layers.One intuitive and simple case is adsorption of one adsorbate on one molec-

ular layer. Thus, if it is supposed that soluteAin the fluid phase is adsorbedby solid S, according to the kinetic mechanism:

The adsorption rate ofA is expressed by the equation:

(22.1)

where:rA = adsorption rate ofACA = concentration ofAin the fluid phase

m0 = maximum concentration ofAretained by the adsorbentmA= concentration ofAretained by the adsorbent

At the adsorption equilibrium, rA= 0, hence:

The adsorption equilibrium constant can be defined as:

(22.2)

Obtaining:

(22.3)

This equation is called isotherm of Langmuir, and experimental data ofmany different systems fit well. However, another equation frequently usedis the isotherm of Freundlich, which is an empirical equation of the form:

(22.4)

A S S AF

K

K( ) +

1

2

r k C m m k mA A A A= ( ) 1 0 2

k C m m k mA A A1 0 2( )=

Kk

k

m

C m mA

A A

= =( )

1

2 0

m

m

K C

K C

A A

A0 1

=

+

m K CA An= ( )


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where Kand nare parameters, the values of which are a function of the typeof adsorbate-adsorbent system and of temperature. The determination ofthese parameters should be made experimentally.

Another isotherm is that of Brunauer, Emmett, and Teller (BET), used foradsorption of one solute on multilayers (Brunauer et al., 1938; Emmett andde Witt, 1941):

(22.5)

where mAis the amount of adsorbed solute per unit mass of adsorbent forthe concentration CA, C

SA is the saturation concentration of the solute, m

1A is

the amount of solute adsorbed per unit of adsorbent that forms a monolayeron the surface of the solid, and Bis a constant representing the interactionenergy with the surface.

For solutes contained in a gas phase, the BET isotherm for nlayers can beexpressed according to the equation:

(22.6)

where:

When the number of layers n= 1, it is obtained that:

(22.7)

which coincides with the isotherm of Langmuir.

If the number of layers is high, the BET isotherm is transformed into:

For low values of x, the last equation is transformed as:

mBC m

C C BC

C

AA A

AS

AA

AS

=

( ) ( )

1

1 1

m

m

Bx

x

n x nx

B x b X A

n n

n0

1

11

1 1

1 1=

+ ++

+

+

( )

( )

xP

P

A

AA

A

= =0

partial pressure of

vapor pressure of pure

m

m

B x

xA

0 1=

mm

B xx B x

A

0 1 1 1=

( ) + ( )( )

m

m

Bx

B xA

0 1 1=

+ ( )


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22.2.2 Ionic Exchange Equilibrium

If a fluid that contains an anion An+ with charge n+ is available and comes

into contact with a resin that can exchange a cationBn+

with the same chargeas A, it can be considered that the following cationic exchange reaction iscomplied with:

The disappearing rate of the cation An+is:

When equilibrium is reached, the disappearing rate of A is null, so thefollowing is complied with:

Substitution in the last equation allows expression of the disappearing rateofAas a function of the equilibrium constant K.

The concentration in the liquid phase of the ions involved in the ionicexchange is usually expressed in equivalents/liter of solution; while in thesolid phase, the concentration is expressed in equivalents of ions per unitmass of resin. In this way, if C0A is the initial concentration ofAin the solutionand C

A is the concentration of A at a determined instant, and if C

B is the

concentration of ions Bthat have been exchanged forA, then it is compliedthat CB= C

0A CA. For the resin, EAare the equivalents ofAper unit mass of

dry resin and EMis the maximum capacityof the resin, expressed as equiv-alents ofAper unit mass of dry resin that it can exchange with ions A. Theconcentration of Bin the resin is:

Therefore, at equilibrium, it is complied that:

(22.8)

If the equivalent fractions ofAin the liquid and resin phases are defined as:

A R B B R AnK

K

n+ ++ + 1

2

( )= r k C C k C CA A R B B R A1 2

Kk

k

C C

C C

B R A

A R B

= =

1

2

E E EB M A=

KC C E

C E E

A A A

A M A

= ( )

( )

0


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liquid phase:

resin phase:

the equilibrium constant is expressed as:

(22.9)

This equilibrium constant is called the separation factor.

22.3 Process Kinetics

22.3.1 Adsorption Kinetics

In every adsorption process, three solute transfer stages can be considered:

1. External transfer: The solute at the fluid phase with a concentrationCis transferred to the fluidsolid interphase in which the concen-tration is Ci. The mass flux is given by the equation:

(22.10)

where kFis the mass transfer coefficient in the external phase.

2. Diffusion inside the solid: The mass flux on the wall of the solidfor a spherical solid particle with radius riis expressed as:

(22.11)

where CS is the concentration of solute in the solid and De is theeffective diffusivity of the solute.

3. Adsorption stage: For the simpler case, expressed before, it can bestated that:

YC

CA

A

=0

X

E

EA

M

=

KY X

Y X

= ( )

( )

1

1

N k C CF i= ( )

N D

C

reS=

r k C m m k mS S S= ( ) 1 0 2


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and at equilibrium:

Generally, the first of these stages controls the process, so it can logicallybe assumed that equilibrium is reached during the adsorption stage. There-fore, if the mass transfer stage controls the process, it is complied that Ci=CS, and its value is constant along the whole solid.

22.3.2 Ionic Exchange KineticsAs they occur in physical adsorption processes, different stages of masstransfer can be considered during ionic exchange:

1. External mass transfer of ion A from the solution to the resinssurface

2. Diffusion of ionAthrough the pores of the resin until reaching theexchange points

3. Ionic exchange reaction in which ion A is exchanged by ion B insuch a way thatAis bound to the resin while Bpasses to the fluidphase

4. Diffusion of ion Bthrough the pores of the resin until reaching theresins surface

5. External mass transfer of B from the surface of the resin to thesolution

The slowest stage is the one that controls the ionic exchange process.Generally, the diffusion stages or the external mass transfer stages controlthe global process:

where the superscript iindicates interface concentrations, and it is compliedthat C i

A

is in equilibrium with C iB

.

22.4 Operation by Stages

As in other mass transfer unit operations, adsorption and ionic exchangeprocesses can operate in stages, in batch as well as in continuous operations.

m

m

K C

K C

S S

S0 1=

+

( r) k C C k C CA A A

iB B

iB = ( )= ( )


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The most common ways of operation are single stage or contact and multi-stage, which can be repeated single contact or countercurrent multiple stage.

22.4.1 Single Simple Contact

This is the simple method of operation. As shown in Figure 22.2a, a fluidstream containing the solute and the solid stream are contacted in one stage.The solute passes to the solid stream, thereby decreasing its concentrationin the fluid phase. It is assumed that the fluid and solid that leave the stageare in equilibrium; this means that an ideal stage is supposed.

The mass flows of the fluid and solid streams are F and S, respectively.Also, Cand mare the concentrations in the fluid and solid streams, respec-

tively. Generally, the mass flows of the fluid and solid streams present a

FIGURE 22.2Single contact: (a) operation sketch; (b) operation in the equilibrium diagram.

S

mE

FCS

S

mS

F

CE

CS C

(CE, mE )

a)

b)

m

mS

Isotherm


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slight variation between the inlet and outlet, so they are considered constant.When performing a solute balance in the system it is complied that:

where the subscript Emeans inlet and Smeans outlet. Rearranging:

(22.12)

which, in the mCdiagram, is the equation of a straight line with slope F/S.Therefore, if the conditions of the inlet streams are known, the concentrationsof the streams that leave the stage can be determined in the mCdiagram(Figure 22.2b). For this reason, a straight line with slope F/Sis traced fromthe point that represents the inlet streams (CE,mE), and the concentrations ofthe outlet streams (CS,mS) are obtained where the straight line crosses theequilibrium curve (adsorption isotherm).

For a batch operation, it is convenient to use total amounts and concen-trations:

(22.13)

where V is the total volume of the fluid and S is the total amount of solid,while the subscripts Iand Fdenote initial and final concentration, respectively.

22.4.2 Repeated Simple ContactThis is a multistage operation in which the fluid phase that leaves a stage isfed to the following stage (Figure 22.3a). It is considered that all the stagespresent an ideal behavior, so the concentrations of the fluid and solid streamsthat leave any stage are in equilibrium. Generally, the solid fed to each stagecontains no solute; therefore, mE= 0. In addition, the amount of solid usedin each stage is the same (S1= S2= = SN= S).

When performing a solute balance around any stage i:

Rearranging the equation:

(22.14)

F C C S m mE S S E( )= ( )

mF

SC m

F

SCS S E E= + +

V C C S m mI F F I ( )= ( )

F C C S m mi i i E ( )= ( )1

mF

SC m

F

SCi i E i=

+ +

1


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If it is desired to determine the concentrations of the streams that leaveeach stage, one should operate as follows. The point that represents the inletstreams (mE,C0) is represented in the mCdiagram. A straight line with slopeF/Sis traced from this point. The point where this straight line crosses theequilibrium curve determines the concentrations m1and C1. A straight line

with the same slope is traced from the point (C1,mE); the point where it crossesthe equilibrium curve determines the composition C2and m2of the streamsthat leave the second stage. The process continues until N, which allows oneto obtain the concentration of solute CNin the fluid phase (Figure 22.3b).

When the final concentration of the fluid phase is known and it is desiredto determine the number of stages required, the way to operate is similar.The graphical process begins at point (C0,mE), and straight lines with slopeF/Swill be traced as described previously until exceeding the final con-

centration CN. The number of straight lines of slope F/Straced is exactlythe number of stages Nneeded to decrease the solute content in the fluidstream from C0to CN.

22.4.3 Countercurrent Multiple Contact

Nstages are used in this type of operation, and the fluid and solid streamscirculate in opposite directions. The outlet stream of each stage is fed to the

FIGURE 22.3Repeated simple contact: (a) sketch of the operation; (b) operation in the equilibrium diagram.

S1

1F

b)

a)

m

C 0 C

Isotherm

Slope = - F/S

CN

CE

m E m E mE

S1

S 2

S2

SN

SNm 1 m 2 m N

F1 F2 FN

C1C 2

CN

2 N

C1

....


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following one. The fluid stream enters the system by the first stage, whilethe solid stream is introduced in the last stage (Figure 22.4). As in the othercases, it is supposed that the stages are ideal and that equilibrium is reached

in each one.The following equation is obtained from the global balance:

In the mCdiagram, this is the equation of a straight line with slope F/Sthat passes through the points (C0,m1) and (CN,mN+1), called the operatingstraight line. The number of stages required to decrease the concentration

in the fluid phase from C0to CN+1is obtained by plotting steps between theoperating line and the equilibrium curve (Figure 22.4).

The maximum fluid flow to treat per unit of solid is obtained by drawingthe straight line that passes through the point (CN,mN+1) and the point on theequilibrium curve with abscissa C0. This line has a slope equal to (F/S)MAX(Figure 22.5a). It can occur that, when tracing the straight line with maximumslope, it crosses the equilibrium curve. In this case, the tangent to the curveshould be traced and its slope is given by the relationship (F/S)MAX(Fig. 25.5b).

FIGURE 22.4Countercurrent multiple contact.

a)

F

C0 1

b)m

C0 CCN

S

mN+1

m1m 2 mN

F

CNC1C2

2 N

m1

....

mN+1

Isotherm

Operatingstraight line

F C C S m m N N0 1 1( )= ( )+


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22.5 Movable-Bed Columns

In certain cases, the adsorption or ionic exchange stage consists of a cylin-drical column in which the fluid and solid phases are fed under countercur-rent (Figure 22.6). It is assumed that the solid moves along the column under

plug flow.A solute balance around a differential section of height dzyields:

where vis the fluids circulation linear velocity,Ais the transversal sectionof the column, is the porosity of the solid in the column, and aS is the

FIGURE 22.5Countercurrent multiple stage. Conditions of maximum treatment flow: (a) convex isotherm;(b) concave isotherm.

m

C0

Isotherm

(F/S) max

CN

m N+1

m

C 0

Isotherm

(F/S) max

C N

m N+1

a)

b)

C

C

v A C v A C dC N A dz aS = +( )+ ( )1


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specific surface of the solid bed, while Nis the flux of the transferred solute,its value being:

In this equation, kFis the mass transfer constant and Ciis the concentrationof the solute at the fluidsolid interphase. The calculation of Cican be diffi-cult, so it is convenient to express this transfer as a function of the concen-tration of the fluid Ce, which is in equilibrium with the concentration of thesolid:

Therefore, when substituting this expression in the solutes balance, theheight of the column can be obtained if the resulting equation is integrated:

(22.15)

FIGURE 22.6Moving-bed column.

N

Cm

C- dC m-dm

1

0

S

dz

C

C

F

N k C CF i= ( )

N k C CF e= ( )

z dzv

k a

dC

C C

z

F S eC

C

= =( )

0 1

0

1


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Generally, the integral does not have an analytical solution and should besolved by numerical or graphical methods.

22.6 Fixed-Bed Columns

Fixed-bed columns are the most used equipment for adsorption and ionicexchange processes. The adsorbent solid or exchange resin is contained inthe column, and the fluid that contains the solute to retain or exchange iscirculated through the column (Figure 22.7).

If a solute balance is performed at a column differential of height dz, it isobserved that the inlet term is equal to the outlet term plus the part accu-mulated in the liquid retained in the porous fraction of dzand that accumu-lated in the solid:

In this equation, vis the fluids circulation linear velocity, Ais the trans-versal section of the column, is the porosity of the solid bed, C is theconcentration of solute in the fluid phase, m is the concentration of solutein the solid phase, and Pis the density of the adsorbent particles or exchangeresins.

When developing the accumulation term, and rearranging all the terms,the last equation can be expressed as:

(22.16)

This is a basic equation that allows calculation of the height of the column,although the solution method varies according to the operation conditions.Three methods to calculate the height of solid particle beds are presentednext.

FIGURE 22.7Fixed-bed column.

A

dz

C0v

C C-dC

v A C v A C dC

d

dtA dz C A dz m

P = ( ) + + ( )( )1

vdC

dz

dC

dt

dm

dt =

1

P


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22.6.1 Fixed-Bed Columns with Phase Equilibrium

To solve Equation 22.16, it will be assumed that the equation does not control

the mass transfer stage and also that the fluid and solid phases are inequilibrium. The rate at which a point with constant concentration movesalong the column can be obtained from this equation:

(22.17)

where the relationship dm/dCis the slope of the equilibrium curve.If it is considered that a volume Vof a fluid that passes at a rate vthrough

a column with transversal sectionAhas been treated in a time t, it is compliedthat: V=Avt.

For constant concentration, integration of Equation 22.17 allows calcula-tion of the height of the solid bed contained in the column:

(22.18)

Sometimes it is desired to know the amount of solid that should be chargedto the column to carry out the operation or the amount of fluid that can betreated per kg of solid contained in the column. If Vis the amount of fluidto treat and Sis the amount of solid in the column, it is complied that:

(22.19)

22.6.2 Rosens Deductive Method

This is another method that allows solution of problems related to soluteadsorption by fixed beds of adsorbent. It is assumed that resistance to masstransfer occurs inside the adsorbent.

When performing a solute balance for the whole column, it is supposedthat all the solute that enters is accumulated on the surface of the solid. Thesolutes inlet and accumulation terms are:

Inlet:

dz

dt

v

dm

dCP

=

+ ( )

1

1

z

v t

dm

dC

V

Adm

dCP P=

+ ( )

=+

( )

11

11

V

S

A v t

z A

dm

dtP P

=( )

=( )

+

1

1

1

1

N Az a k C C A z aS F e S= ( )


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Accumulation:

The following expression is obtained when equaling and rearranging theinlet and accumulation terms:

(22.20)

where Ce is the concentration of the fluid phase in equilibrium with the

concentration in the solid phase m and is obtained from the equilibriumcurve.

In the case that the isotherm is linear, m = KaC, the equation has ananalytical solution (Rosen, 1952; 1954). Thus, the concentration of the fluidstream that leaves the column is calculated by the expression:

(22.21)

where erfis the error function of Gauss, while the parameters , , and are:

Time parameter: (22.22)

Length parameter: (22.23)

Resistance parameter: (22.24)

where De is the effective diffusivity, Kais the slope of the equilibrium line,kFis the mass transfer, dPis the diameter of the particles in the bed, vis thelinear circulation velocity of the fluid, andzis the height of the column.

22.6.3 The Exchange Zone Method

The exchange zone is defined as the part of the bed of solids where the massexchange is produced. When a fluid stream contains a solute with a concen-

d

dtm A z Az

dm

d tp

p1 1( )

= ( )

dm

dt

k aC CF S

pe= ( )

( )1

C C erf S= +

+

0

1

2 1

3

21

21

5

=

81

D

d

z

ve

P2

=

12 12

D K z

v de

P p

a

( )

=2D K

d ke a

p F


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tration C0and is introduced in a column with an adsorbent bed or exchangeresin, the solute passes to the solid phase and the fluid stream leaves thecolumn free from this solute. The upper layers of solid will fill first, and aconcentration profile of the fluid phase is created in the column as shownin Figure 22.8. It can be observed that when the first layer of adsorbent isfull, there is a layer at certain heightzCthat still has not retained or exchangedsolute, and the fluid is free from solute. The heightzCis called height of theexchange zone. The time required to form this exchange zone is the formationtime tF.

Once the zone is formed with the corresponding concentration profile, the

concentration front crosses the whole column until it reaches the outlet,which occurs when this front has crossed the total column height zT. The

breakpoint is defined as the instant in which the fluid stream leaving thecolumn starts leaving solute, although in practice it is considered that thispoint is reached when the concentration of the fluid stream is 5% of the inletstream concentration. At the instant when the breakpoint is reached, thevolume of fluid treated is VR, while the elapsed time is the breakpoint timetR. When the fluid stream that leaves the column has the same concentrationas the inlet stream, the solid is completely full and the saturation point isreached, although in practice it is considered that this point has been reachedwhen the concentration of the fluid at the columns outlet is 95% of the inletconcentration. At the saturation point, a volume of fluid VThas been treatedin a total time tT.

Figure 22.9is obtained by plotting the concentration of solute in the fluidstream against the volume of fluid treated. It is easy to deduce that thevolume treated to form the exchange zone is the difference between the

FIGURE 22.8Exchange zone and concentration profile of the fluid stream in this zone.

Z

C

Z

a) b)

C

Z C

C0

C0

F

F


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volume treated to reach the saturation point and that required to reach thebreakpoint: VC= VTVR.

The amounts of solute retained or exchanged by the solid at differentoperation points can also be obtained. The amount of solute retained orexchanged by the solid bed to form the exchange zone is defined as MC,whileMRis the solute retained or exchanged by the solid until the breakpoint:MR= VRC0. In this equation, C0 is the concentration of solute in the fluid

stream at the columns inlet. In adsorption processes these amounts areexpressed in grams or moles of solute, while in ionic exchange processesthey are given in equivalents of solute.

Time tF has been previously defined as the time needed to develop theexchange zone; however, definitions of new parameters are required. Thus,the time needed by the exchange zone to cross its own height zCis definedas the relationship between the volume of fluid to form the exchange zoneand the circulation volumetric flow rate of the fluid stream:

(22.25)

where vis the linear velocity andAis the transversal section of the column.In the same way, the total time required to reach the saturation point is

given by the relationship between the total volume treated and the volumet-ric flow rate:

(22.26)

The rate with which the exchange zone moves is:

(22.27)

FIGURE 22.9

Volume of fluid treated under breakpoint and saturation conditions.

0

VR VT V

SATURATION

BREAKPOINT

C

C

MR MC

t

V

q

V

v ACC C= =

tV

q

V

ATT T= =

v

vz

t tT

T F

=


19/31



It is easy to deduce from all these definitions the relationship between thetotal height of the bed and the height of the exchange zone:

(22.28)

On the other hand, the relationship between the total time and the timerequired by the exchange zone to cross its own height is obtained by com-

bining the last equations:

(22.29)

The amount of solutes retained or exchanged from the breakpoint to thesaturation point is obtained by integrating the variation of the concentrationin this range:

(22.30)

It is easy to observe that the maximum amount of solute that can beretained or exchanged in this zone is:

A new parameter i is defined as a fraction of the exchange zone withcapacity for retaining or exchanging:

(22.31)

The formation time of the exchange zone and the time required by theexchange zone to cross its own height can be related according to the equation:

(22.32)

The concentration profile of the exchange zone is different according tothe value of the parameter i (Figure 22.10). In the case when i= 0.5, it is saidthat the breakthrough curve is symmetrical, complying that tF= 0.5 tC.

z t v t

z

t tC C CT

T F

= =

t t

V

VT CT

C=

M C C dVCV

V

R

T

= ( )

0

M C V V C VC MAX T R C( ) = ( )=0 0

iM

M

C C dV

V C

C dV

V CC

C MAX

V

V

C

V

V

C

R

T

R

T

=( )

=

( )

= 0

0 0

1

t i tF C= ( )1


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22.6.3.1 Calculation of Height of Exchange Zonein an Adsorption Column

The calculation of the height of the exchange zone is made assuming thatthis zone is similar to a countercurrent unit of infinite height that operatesunder stationary conditions. Therefore, it is considered that, in this zone, theconditions represented in Figure 22.11 are given, where a fluid stream Fwitha solute concentration C0is fed by the top of the column, while a stream Sof the adsorbent solid with a concentration m1 is fed by the bottom of thecolumn. The fluid that leaves the column has a solute concentration equalto C1,while the solute content of solid stream is m1.

FIGURE 22.10Concentration profile (breakthrough curve) of the exchange zone as a function of the value of i.

FIGURE 22.11Conditions of the exchange zone.

a) b) c)

C0 C0

i 1

C0

i = 0.5

ZCZCZC

i 0

1

2

C

C + dC m + dm

mN

Sm1

FC2

FC 1

Sm2

BREAKPOINT

SATURATION


21/31



Since the columns height is infinite, the saturation conditions will be givenin zone 2 while the breakpoint conditions are in section 1. Therefore it iscomplied that C1= 0 and m2= mMAX, where mMAXis the maximum concen-

tration of solute that the solid can adsorb.When performing a solute balance between the bottom of the column and

any column section, it is obtained that:

(22.33)

But C1= 0 and if, in addition, the adsorbent solid is free from solute whenit enters into the column, m1= 0. Hence:

If the solute balance is made for a dzof column, then:

This equation allows calculation of the height of the exchange zone. If it

is integrated with the boundary conditions marked inFigure 22.11:

(22.34)

This equation is generally solved by graphical or numerical integration.

When the equilibrium isotherm is linear (m= KaC), the equilibrium con-

centration Ceis:

Substitution into Equation 22.34 yields an expression easy to integrate, inwhich the height of the exchange zone is:

(22.35)

where C1and C2are the concentrations of the fluid phase at the breakpointand saturation point, respectively, while Kis a parameter defined as:

F C C S m m( )= ( )1 1

F C Sm=

F dC k C C a AdzF e S= ( )

zF A

k a

dC

C CC F S eC

C

=

1

2

C

m

K

F C

S Ke a a= =

z F AK k a

CCC F S

=

ln 2

1

= K

F

SKa1


22/31



If it is considered that the following is complied at the breakpoint andsaturation point:

then Equation 22.35 is transformed into:

(22.36)

22.6.3.2 Calculation of Height of Exchange Zone in an IonicExchange Column

The way to operate is similar to the case of the adsorption column. Fromthe solute balance between the bottom of the column and any column section,it is obtained that:

(22.33)

The equivalent fractions in the fluid phase (Y) and the resin (X) can bedefined as:

In this equation the concentration C of the fluid phase is expressed asequivalents of solute per liter and the concentration in the resin m isexpressed as equivalents of solute per kg of resin. Also, mMAXis the maximumretention capacity of the resin. The equation of the solute balance is:

(22.37)

which is the so-called operating line in the equilibrium diagram.If the balance is performed between the bottom and the top of the column,

this equation is transformed into:

(22.38)

Since sections 1 and 2 of the column correspond to the breakpoint andsaturation point, respectively, it should be complied that:

C C C C1 0 2 00 05 0 95= = . .and

zF A

K k aC F S=

( )ln 19

F C C S m m( )= ( )1 1

YC

C

Xm

mMAX

=

=

0

FC Y Y Sm X X MAX0 1 1( )= ( )

FC Y Y Sm X X MAX0 2 1 2 1( )= ( )


23/31



Breakpoint: X1= 0 Y1= 0

Saturation point: X2= 1 Y2= 1

Combination of Equations 22.37 and 22.38 yields that the operating line(Equation 22.33) is a straight line with slope 1 and ordinate to the origin 0;this means Y= X.

When performing a solute balance around a dzof the column, it is obtainedthat:

This equation allows calculation of the height of the exchange zone:

(22.39)

In this case, this equation has an analytical solution, since from the oper-

ating line (Y= X) and the definition of the separation factor (Equation 22.9),a relationship between Yeand Yis obtained:

(22.40)

Substitution in Equation 22.39 allows one to obtain the height of theexchange zone by integration as:

(22.41)

Since in the exchange zone it is complied that the equivalent fractions ofthe fluid phase Y1and Y2correspond to the breakpoint and saturation point(Y1 = 0.05 and Y2 = 0.95), then the height of the exchange zone can be

calculated using the following equation:

(22.42)

Besides this method, the height of the exchange zone can also be calculatedusing the equation of Wilke:

FC Y k C Y Y a A dzF e S0 0= ( )

zF A

k a

dY

Y YC F S eY

Y

=

1

2

YY

K Y Ye=

( ) +1

zF A

k a KK

Y

Y

Y

YCF S

=( )

11

12

1

2

1

ln ln

zF A K

k a KC F S=( ) +( )

( ) ( )

1

119ln


24/31



(22.43)

where vis the linear velocity expressed in cm/s, and bis a parameter thatdepends on the type of transfer. In case of an exchange of ions Ca2+and Mg2+

by Na+ions, the value of this constant is 37.4.The height of the exchange zone can also be calculated by experimentation.

For this reason, different experiments are performed with different bedheights zT, and the volume of fluid VT required to fill the column is deter-mined; then, the height of the columnzTis plotted against VT. Data are fittedto a straight line, and the value of the ordinate to the origin is the height zCof the exchange zone.

Problems

22.1

One of the causes of deterioration of clarified juices of fruits is nonenzymaticbrowning due to the formation of melanoidins that can be eliminated fromthe juice by adsorption on activated carbon. The degree of nonenzymatic

browning of a juice can be evaluated by measuring its absorbance at awavelength of 420 nm (A420). In an experimental series at the laboratory,different amounts of activated carbon (particles of 1 mm of mean diameter)are mixed with loads of 10Brix juice, whose A420 is 0.646, until reachingequilibrium. Data obtained are given in the following table:

in whichA420is expressed as absorbance/kg of solution, while bis the kg ofcarbon/kg of solution. Determine: (a) data of the equilibrium isotherm as atable; (b) the number of stages required, operating under repeated singlecontact, if it is desired to decrease the A420of the juice down to a value of0.200, using in each stage 0.025 kg of carbon per each kg of 10Brix juice;and (c) the flow of carbon that should be fed to a countercurrent moving-

bed column with a juice flow of 1000 kg/h, if it operates with a carbon flowthat is double the minimum, and it is desired to obtain juice with an A420

value not higher than 0.2.

(a) The equilibrium data are obtained from the problem statement, since theconcentration of melanoidins in the liquid phase is given by the absorbanceat A420. A measure of the melanoidins adsorbed by the carbon is given bythe difference between the initial absorbance of the juice and the absorbancethat it has at a determined instant. For this reason, the melanoidin concen-tration in the liquid and solid phases is:

A420 0.646 0.532 0.491 0.385 0.288 0.180b 0 0.01 0.02 0.06 0.12 0.26

z bvC=0.51


25/31



Liquid phase:

Solid phase:

Hence, the data of the equilibrium isotherm are:

(b) When performing a solute balance for the first stage, in which it issupposed that the carbon that enters has no solute, it is obtained that:

This equation indicates that, when a straight line with slope (L/S) istraced in the equilibrium diagram from the point with abscissa C0, the valuesof C1and m1 are obtained from the equilibrium curve.

The slope of this straight line can be easily obtained from the data in theproblem statement:

Once C1 has been obtained, a straight line with the same slope as beforeis drawn from this abscissa, which allows one to obtain the values of C2andm2 on the equilibrium curve (Fig. 22.P1a). The process is repeated untilexceeding the value of Cn= 0.2. Table 22.P1 presents the values obtained forthe outlet streams of each stage.

0.532 0.491 0.385 0.288 0.180

11.4 7.75 4.35 2.98 1.79

TABLE 22.P1

Concentrations in the Stages

Stage

1 0.470 7.12 0.365 4.03 0.285 2.94 0.225 2.25 0.180 1.8

C = A

absorbance

kg juice

m

A A

b=

0 melanoidines adsorbed

kg carbon

Cabsorbance

kg juice

m

melan. adsorb.

kg carbon

mL

SC

L

SC

1 1 0= +

( )= = L S 1 0 025 40. kg juice kg carbon

C

absorbance

kg juice

mmelan. adsorb.

kg carbon


26/31



FIGURE 22.P1(a) Graphical calculation of the stages and maximum slope; (b) movable column.

a)

b)

5 4 3

2

1

LS

LS

( )max

- = - 40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7c1

C

m

10

5

0

c m

c1 m1

c2 m2

m1

LS

LS


27/31



Since Cn= 0.2, five stages will be required.(c) For the column that works with a moving bed, solute balance between

the top of the column and any section yields:

It is assumed that the carbon entering the column is free from the solute(m2= 0) and that the concentration of the liquid stream that leaves the columnis C2= 0.2. Therefore, substitution of data yields the following operation line:

The maximum slope for this straight line is obtained for C = C1 in theequilibrium curve. However, this is not possible since the straight line that

joins point 2 with the point at the equilibrium isotherm for C= C1will crossthe curve. The value of (L/S)MAXis obtained by tracing a straight line tangentto the equilibrium curve from point 2, in such way that its value is:

Since the juice stream that circulates by the column is L = 1000 kg, theminimum amount of carbon is SMIN= 39.82 kg carbon. Since the amount ofcarbon required is double the minimum:

22.2

An industry that processes navel oranges obtains juice that contains 110 ppmof limonene that gives it a bitter taste. With the objective of eliminating the

bitter taste from the juice, a 100 kg/h juice stream is fed to a cylindricalcolumn (0.15 m2of cross section) that contains a synthetic molecular sievethat adsorbs limonene. The adsorbent solid has a density of 950 kg/m3ofpacking, with a volumetric transfer coefficient equal to 1.5 102 h1, and

maximum retention capacity of 10 mg of limonene per kg of adsorbent. Thedensity of the juice can be considered 1000 kg/m3. The adsorption isothermin the concentration range at which the column operates is linear andexpressed by the equation: m= 0.12 C, where Cis the limonene content inthe juice in mg/kg, while m is the concentration in the solid expressed inmg/kg of adsorbent. It can be assumed that the breakthrough curve issymmetrical. Calculate: (a) the height of the exchange zone; and (b) if 25 minare required for limonene in the juice stream to begin leaving the column,calculate the height that the adsorbent bed should have.

m m

L

SC C= + ( )2 2

m LS C= ( )0 2.

LS

MAX

= 25 11. kg juice kg carbon

S SMIN= =2 79 64. kg carbon


28/31



(a) The exchange zone behaves as a moving bed column with infinite height,so the saturation conditions are given on the columns top, while breakpointconditions are on the bottom: m1= 0 and C1= 0. From the global balance ofsolute, it is obtained that S= LC2/m2. If the balance is performed betweensection 1 and any section: S= L C/m. Since the adsorption isotherm is linear(m = K Ce), it is possible to obtain a relationship between the equilibriumconcentration and the composition of juice by combining the last equations:

The height of the exchange zone is obtained from the expression:

FIGURE 22.P2Conditions of an exchange zone.

SL

SL

m2

C2

m1C1

Saturation

Breakpoint

dz

m + dm

m

C + dc

C

2

1

C mK C

Ce=2

2

zL A

K a

dC

C C

L A

K a

K C

KC m

C

CC L S eC

C

L S

=

=( )

1

2

2

2 2

2

1

ln


29/31



From data in the statement:

C0= 110 mg limonene/kg juice

C1= 0.05 C0= 5.5 mg limonene/kg juice

C2= 0.95 C0= 104.5 mg limonene/kg juice

m2= m0= 10 mg limonene/kg adsorbent

Therefore:

(b) The volume of the exchange zone is Vc=zCA= 0.00969 m3. The amount

of limonene that enters the column with the juice stream is:

The breakpoint is reached when limonene begins to be present in thestream leaving the column, which occurs at 25 min (tR= 25 min). The amountof limonene that entered during 25 min has been retained in the adsorbentin the column. If V is the volume occupied by the adsorbent in the wholecolumn, the volume of saturated adsorbent is (VVC). Therefore, the fol-lowing is complied at the breakpoint:

Substituting data:

Thus, the volume that the adsorbent occupies in the column is obtained:

The height of the adsorbent isz= V/A= 3.12 m.

22.3

In a stage of a certain food process, 8000 kg/h of water with a magnesiumsalts content of 40 meq/kg are treated in an ionic exchange cylindricalcolumn. The exchange capacity of the resin is 15 eq/kg, with a separation

z

Z

C

C

=100 kg h

1000 0.15 150

0.12 104.5

0.12 104.5 10

ln0.95

0.05

m( )( )( )

( ) ( )

( )( ) ( )

= 0 0646.

wC0 100 1100=( )( )=110 mg limonene h

wC t V V m V m iR C a C0 0 0= ( ) + a

100 110 25 60 0 00969 950 10 0 00969 950 10 0 5( )( )( )= ( )( )( ) + ( )( )( )( )V . . .

V= 0 468. m3


30/31



factor of 50 and an apparent density of 550 kg/m3. The column has a 30 cmdiameter and contains 110 kg of resin. From previous experiments, it has

been obtained that the mass transfer coefficient is KL aS= 2.5 106l/(h.m3).

If it is a fixed-bed column and the breakthrough curve is symmetric, deter-mine: (a) height of the exchange zone; (b) the breakpoint and saturationtimes; and (c) the magnesium salts content after 75 min of operation.

If it is supposed that the density of the liquid stream is 1000 kg/m3, thenthe volumetric flow rate is q= 8 m3/h.

The mass transfer coefficient is expressed in proper units to facilitate latercalculations:

(a) The height of the exchange zone is calculated by Equation 22.42:

The volume of the exchange zone can be obtained from the previousequation:

which corresponds to an exchange zone height equal to zC= 0.139 m.The volume occupied by the resin is the total of the column:

and a total column height equal toz= 0.643 m.(b) The breakpoint time is produced when salts begin to exit with the liquid

stream leaving the column. At the saturation volume, no more ions can beexchanged. Also, the exchange zone will have half of its exchange capacity.

Saturation volume:

The amount of ions that entered during the breakpoint time have beenretained in the volume of saturated resin and in the exchange zone, comply-ing with the following:

K aL S= ( ) 2 5 10 25006 1. l h m = h3

zq A

K a

K +

KC L S=

1

119ln

V z AC C= = ( )( )

+

=8

2500

50 1

50 119 0 00981 3ln . m

V RESIN= =

25

550 0 045463kg

kg m m3 .

V V VS C= = 0 035653. m

wC t V E V E i tR S a M C a M R0 320

0 03565 550 15 0 00981 550 15 0 5

= + ( )

=( )( )( ) + ( )( )( )( )

eq Mg h

eq Mg eq Mg

2+

2+ 2+. . .


31/31


Thus, the breakpoint time is:

At saturation time, the resin of the column would be saturated; also, it haspassed the exchange zone:

Hence, the saturation time is:

(c) Concentration after 75 min in the water stream leaving the column isobtained by linear interpolation between the breakpoint and saturationtimes:

tR

= =1 046 63. h minutes

wC t V E V E i tR RESIN a M C a M S0 320

0 04546 550 15 0 00981 550 15 0 5

= + ( )

=( )( )( ) + ( )( )( )( )

eq Mg h

eqMg eqMg

2+

2+ 2+. . .

tS= =1 298 78. h minutes

C Ct t

t tS

S R

= = meq kg0 1 32

cap 22 absorption and ionic exchange

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