6.2 Separation processes - Treccani · VOLUME V / INSTRUMENTS 319 6.2.1 Characteristics of separation processes The separation of solutions and mixtures ... refining tower, where

319VOLUME V / INSTRUMENTS

6.2.1 Characteristics of separation processes

The separation of solutions and mixtures into their singlecomponents is an operation of great importance for thechemical, petrochemical, and oil industries. Almost allchemical processes need preliminary raw materialpurification or the separation of primary from secondaryproducts. These operations go against the tendency ofsubstances to intimately and spontaneously mix, which is, asis well-known, a consequence of the second law ofthermodynamics. For instance, salt mixed in water dissolvesto give a homogeneous solution and the separation of thecomponents in the mixture requires the use of energy. In thiscase separation can be performed in one of the followingways: • By heating the solution to make the water evaporate, and

subsequently condensing it at a lower temperature.• By cooling the solution in order to separate the water in

the form of ice. • By exploiting the selective properties of a membrane;

water passes through this membrane more easily thansalt. Separation processes are of paramount importance

in oil plants. Crude oil, in fact, contains a very highnumber of hydrocarbons, which go from light gases toheavy fractions which are difficult to distill even in avacuum. The most important classes are alkanes andcycloalkanes (naphtenes), in particular, and aromaticcompounds in various proportions. In refineries, thevarious fractions are separated by distillation and thentreated further in order to supply different products ofspecific interest. Usually, first crude oil undergoeswashing with water to remove salts and possiblesuspended particles, and then is evaporated in an oven,which takes it to a temperature of about 400°C. Crudeoil vapours are then sent to a distillation column, orrefining tower, where the separation of the differenthydrocarbon fractions is obtained: in the lower point ofthe column combustion oils are condensed, togetherwith lubricating oils, paraffins, waxes, and bitumens;gas oil condenses between 350°C and 250°C and is usedas a fuel in heating plants and in Diesel motors;

kerosene, which is an oily combustible material used forheating plants and as a propellant for jet planes,condenses between 250°C and 160°C; naphtha, which isused as a fuel and as a raw material for pesticides,fertilizers, and plastic materials, condenses between160°C and 70°C. Gasoline, mainly used as a fuel forplanes and cars, condenses between 70°C and 20°C. At20°C only gaseous products like methane, ethane,propane, and butane remain. Butane and propane, inparticular, form the fuel called LPG (LiquefiedPetroleum Gas).

The above example shows how a separation processmakes it possible to transform a mixture of substancesinto two or more products with different compositions. Aseparation process is fed by one or more streams,whereas streams of products of different compositionsleave the separation equipment. Separation is caused bya separation agent that can be another mass stream, or anenergy flux, or both. Often, separation processesgenerate the formation of a further phase, different fromthat of the feed. For instance, by feeding a liquid stream,products can be made up of two streams, one liquid andone vapour.

Based on the above, it is possible to formulate a generalclassification of the most commonly used separationprocesses in industry. This is shown in Table 1, whichsummarizes their essential characteristics.

It is convenient to characterize separation processes bymeans of a separation factor, defined as follows:

[1]

where xi indicates the molar fraction of component i and xjindicates the molar fraction of component j, whereasindexes 1 and 2 indicate the two streams of separationproducts. Therefore, as

ij represents the ratio between themolar fractions of the two components i and j in the twostreams 1 and 2. Therefore, if as

ij�1, the process does notallow any separation of components i and j. If as

ij�1,component i tends to concentrate in stream 1, whereas ifas

ij�1, this behaviour is manifested in component j.Conventionally, the two components are chosen so that as

ijis always greater than one.

αijs i j

i j

x xx x= 1 1

2 2

6.2

Separation processes

6.2.2 Mass and energy balances in separation equipment

A continuous separation plant can be considered as athermodynamic system open to mass and energy exchange. Itis possible to associate to a separation equipment a series ofmass fluxes corresponding to feed streams and separationproducts as well as a series of energy fluxes necessary for theseparation to take place. In normal conditions, continuousequipment works in stationary regime, so that the values of theintensive parameters of the system (the pressure, temperatureand concentrations of the different components) are not time-dependent but vary according to position. Their gradients, infact, determine the rates at which mass and energy transferstake place in the different regions of the system.

Mass balances The mass conservation principle should be applied to

separation equipment, which in general terms(non-stationary conditions) can be expressed as follows:

The balance can be applied to all the equipment or toany portion of it arbitrarily chosen. In the following, it willbe assumed that no chemical reaction takes place inside theseparation equipment and therefore mass generation andconsumption terms are equal to zero. The balance equation,therefore, assumes the following simplified form:[accumulation]�[mass in]�[mass out].

In order to formulate the above in quantitative terms,the mass of component i contained in the system will beindicated with mi and the mass flow rates of the samecomponent in and out of the equipment with Fi

(e) andFi

(u), respectively. Material balance can be therefore bewritten as:

[2]

Summations at the second member are to be performed on all in and out streams. The preceding equation expresses the mass balance on component i. By summing balance equations relative to the differentcomponents, the following total mass balance equation is obtained:

[3]

In stationary regime, mi and m do not vary with time andtherefore [2] and [3] become:

dmdt F Fe

e

u

u

= −( ) ( )∑ ∑

dmdt F Fi

ie

iu

ue

= −( ) ( )∑∑

consumption inside t− hhe system[ ]generation+ inside the system[ ] −

transport outwards throughthe

− system surface boundary

+

trans= pport inwards throughthe system surface bounndary

−

mass accumulationin the system

=

PROCESS ENGINEERING ASPECTS

320 ENCYCLOPAEDIA OF HYDROCARBONS

Table 1. Classification of separation processes

Type of process Feed Separating agent Products Separation principle

Evapouration Liquid Heat Liquid and vapour Volatility difference

Distillation Liquid Heat Liquid and vapour Volatility difference

Absorption Gas Non volatile liquid Liquid and gas Preferential solubility

Extraction Liquid Immiscible liquid Two liquids Different solubilities

Crystallization Liquid Heat (heating or cooling) Liquid and solid Difference in crystallizationtemperature

Adsorption Gas or liquid Adsorbing solid Fluid and solid Difference in adsorptioncharacteristics

Ionic exchange Liquid Solid resin Liquid and solid Adsorption equilibrium

Solid-liquid extraction Solid Liquid Liquid and solid Diffusion and osmosis

Drying Solid Heat Solid and vapour Volatility difference

Sedimentationand centrifugation Slurry, dispersion Gravitational force Solid and liquid Density difference

Filtration Suspension Filter Solid and liquid Dimensional difference

Membrane processes Gas or liquid Membrane Gas or liquidDifference in dimensions

or difference in membranesolubility

Flotation Suspension Collector agents Solid and liquid Surface tension

[4]

Similarly, it is possible to write molar balance equationsreferring to species i and to total moles. If the number ofmoles of component i contained in the system is indicatedwith ni and the molar flow rate of the component with Fi oneobtains:

[5]

which in stationary regime becomes:

[6]

Energy balances A characteristic of chemical equipment is the presence

of movements of fluid streams in which transformations cantake place. In order to write energy balance equations for theseparation equipment it is therefore necessary to combinefluid mechanics and thermodynamics. Consider, forinstance, a system where a fluid flows continuously in a tubebetween two sections 1 and 2, at different heights withrespect to a reference plane. A pump supplies work, W, and aheat exchanger delivers (or subtracts) heat quantity Q. Byreferring to the unit mass, the energy entering or exiting thetwo considered sections is given by the sum of the followingterms: internal energy referred to the unit mass U; potentialenergy relative to the reference plane (if referring to the unitmass it is expressed by F�gz, where g is the acceleration ofgravity and z is the height of the section evaluated along avertical axis); kinetic energy, which referring to the unitmass is K�(1/2)u2, where u is the average velocity at theconsidered section.

The energy variation for mass dm when it goes fromsection 1 to section 2 is given by:

[7]

where operator D indicates the difference between the valuescorresponding to the two sections.

Following the energy conservation principle, such adifference in stationary conditions must be equal to the sumof energy that the system receives from the environment andtherefore:

[8]

In the third member of [8], dW is split into the sum oftwo terms. P1 and P2 represent the pressures at the sectionstaken into consideration, whereas V1 and V2 are the volumesper unit mass. (P1V1�P2V2)dm therefore represents the workassociated with the pressure variation the fluid undergoes

when it moves from section 1 to section 2; dWs is, on theother hand, the work performed on the system by mechanicalequipment, or subtracted by a turbine. Equation [8] cantherefore be written in the following form:

[9]

By recalling the definition of the enthalpy functionH�U�PV, and dividing all terms of the equation by dt oneobtains:

[10]

where m� is the mass flow rate and W�s and Q� respectively

represent the quantities of mechanical and thermal energydelivered to the system in the unit time. By dividingequation [10] by m� one obtains:

[11]

where Ws and Q are the work and heat exchanged per unitmass of flowing fluid respectively.

Equation [11] can be extended to systems with severalinlet outlet streams and in this case it is necessary tocalculate the difference between the sum of the values of thevariables for all the outlet streams and the sum of the valuesof the variables for all the inlet streams. Often, whenanalysing chemical equipment, the potential and kineticenergy terms are neglected and [11] simply becomes:

[12]

Expressed in this way, the energy balance is calledenthalpy or thermal balance.

6.2.3 Distillation

Distillation is the most important and most widely usedmethod for separating the components in a liquid mixture. Itis based on their distribution between the liquid phase andthe vapour phase when the mixture is brought to boilingconditions. The feasibility and economic interest of adistillation process depend on many factors, among which itis important to mention the favourable characteristics of theliquid-vapour equilibrium, the feed composition, the numberof components to be separated, the required purity, theabsolute pressure needed to perform the operation, heatstability and the corrosive power of the mixtures.

The first of these aspects is expressed by the values ofthe relative volatility of components and this is thepredominating factor since it significantly affects the energyand dimensions of the equipment required to obtain therequired degree of purity. In actual fact, the relative volatilityof two components can be modified by adding a thirdcomponent (in this case, the operation is called extractive orazeotropic distillation) or by decreasing the absolutepressure.

Continuous distillation in one stage (flash)The process normally called ‘flash’ is given here as a

first example of a separation process, where a liquid mixtureis partially evaporated in a single stage. A diagram of atypical process like this is shown in Fig. 1. Liquid feed isheated, for instance, by passing through a tubular exchangerso that when the pressure is reduced, vapour adiabaticallyforms at the expense of the thermal content of the liquid.

∆ � � �H W Qs= +

∆ ∆ ∆� � � � �H K W Qs+ + = +Φ

∆ ∆ ∆� � � � � �H K m W Qs+ +( ) = +Φ

∆ ∆ ∆ ∆� � � �U K PV dm W Qs+ + + ( ) = +Φ δ δ

� �Q PV PV dm= + −( ) +δ δ1 1 2 2 WWs

∆ ∆ ∆� � �U K dm Q W+ +( ) = + =Φ δ δ

�U= ∆ ++ +( )∆ ∆� �Φ K dm

� �U U g z z u u dm2 1 2 1 2

2

1

21

2−( ) + −( ) + −( )

=

� �

� �

F F

F F

ie

iu

ue

e

e

u

u

( ) ( )

( ) ( )

=

=

∑∑∑ ∑

dndt F F

dndt F F

iie

iu

ue

e

e

u

u

= −

= −

( ) ( )

( ) ( )

∑∑

∑ ∑

� �

� �

F F

F F

ie

iu

ue

e

e

u

u

( ) ( )

( ) ( )

=

=

∑∑∑ ∑

SEPARATION PROCESSES


The mixture is then sent to a vessel where the separationbetween the two phases, liquid and vapour, takes place.

In an ideal model of this process the two phases presentin the separator are assumed to be in thermal equilibrium. Inthis case, the flash process is an example of separation bydistillation that takes place in a single ideal stage; it ispossible to define a partition parameter Ki, which can beevaluated by means of thermodynamics:

[13]

where xi,1 and xi,2 are the molar fractions of the componentin phases 1 and 2 respectively. If, as in the present case, thetwo phases are a vapour and a liquid phase, parameter Ki iscalled the ‘evaporation factor’:

[14]

where yi and xi are the molar fractions of component i in thevapour and liquid. respectively. In the case of mixtures ofhydrocarbons, the values of the Kis are often reported asnomograms.

The separation factor defined in [1] becomes:

[15]

also called relative volatility of component i with respect tocomponent j, which depends only on the thermodynamiccharacteristics of the mixture to be separated.

Returning to the problem of flash, the total materialbalance equation for the equipment in Fig. 1 can be written as:

[16]

where F is the molar flow rate (moles/time) of the feed, V isthe molar flow rate of the vapour, and L that of the liquid.On the other hand, the material balance for the genericcomponent i is:

[17]

where zi represents the molar fraction of i in the feed, yi themolar fraction of i in the vapour phase and xi the molarfraction of i in the liquid phase. By combining [16] and [17]one obtains:

[18]

from which derives:

[19]

By replacing [14] in [17] one obtains:

[20]

which solved with respect to xi gives:

[21]

The values of yi can be obtained by combining this lastequation with [14]:

[22]

When Ki depends only on T and P, by operating at anassigned pressure it is possible to calculate the value of thetemperature present in the equipment by recalling that thesum of the molar fractions of the different componentspresent in both phases must be equal to 1:

[23]

By replacing [21] in the first of [23] one obtains:

[24]

The temperature must then have a value that satisfies theprevious equation.

The heat quantity necessary for the examined processcan finally be calculated through an energy balance, givenby [12], where Ws is equal to zero, as there are nomechanical devices present in the system. Thus, one obtains:

[25]

where H is the vapour molar enthalpy, h the liquid molarenthalpy, hF the feed molar enthalpy, and Q the heat to besupplied in the unit time.

Multistage continuous distillationIf the distillation of a mixture of two components is

performed in a single stage, or flash, two phases are obtainedas products: a liquid rich in the less volatile component, anda vapour rich in the more volatile one, but generally with afairly low degree of separation. The purity of the morevolatile product can be increased if part of the vapourproduced is condensed, to be subsequently evaporated, inthis way realizing a two-stage process. If the procedure isrepeated several times, it is possible to obtain a head product(the most volatile) with a high degree of purity. The sameoperation can be performed on the liquid of the firstevaporation process, by performing several evaporations insubsequent stages. This process, however, supplies smallproduct quantities, since subsequent evaporation stagescontinuously impoverish the liquid stream leaving the firstevaporator. In the same way, subsequent evaporation stagescontinuously impoverish the vapour stream leaving the firstevaporator. This problem can be avoided by operatingaccording to the process shown in Fig. 2 in which, in theportion above the feed, the liquid produced in generic stagem feeds the preceding stage (m�1), and in the same way, inthe portion below the feed, the vapour produced in genericstage n feeds stage (n�1). This scheme illustrates the seriesof operations taking place in a continuous distillationprocess, which in fact is implemented by means of a seriesof subsequent stages, and each one of them can beconsidered similar to a flash.

� � � �Q VH Lh FhF= + −

FV

zK T L V

i

ii ( ) + =∑ 1

x

y

ii

ii

=

=

∑∑

1

1

y FV

K zK L Vi

i i

i

=+

x FV

zK L V

LV

zK L Vi

i

i

i

i

=+

= +

+

1

Fz x VK Li i i= +( )

LV

z yx zi i

i i= −

−

Vz Lz Vy Lxi i i i+ = +

Fz Vy Lxi i i= +

F L V= +

αiji j

i j

y xx y=

Kyxii

i

=

Kxxii

i

= ,

,

1

2



vapour

liquid

feed

heater

Fig. 1. Scheme of a single-stage continuous distillation process (flash).

In industrial practice, the overall operation is performedin a distillation column made up of a vertical cylindrical unitin which there are several stages and which is structuredaccording to the diagram shown in Fig. 3. At each stage,

close contact takes place between the vapour rising from theplate below and the liquid situated in it. In this case as well,the stage is called ideal if it is described by a model in whichthermodynamic equilibrium conditions between the twophases involved are assumed.

Usually, stages are identified by the plates in a column,which have a geometrical configuration that makes itpossible for the liquid to cross the plate and then descendthrough a duct towards the lower plate after having surpassedan overflow weir. Vapour rising against the current bubblesthrough the liquid by going through, for example, a series ofholes on the surface of the plate itself. The contact betweenliquid and vapour obtained in this way allows the exchangebetween the most volatile compounds which accumulate inthe vapour phase, and those less volatile which accumulatein the liquid phase. The process can also be exemplified byassuming that on each plate, adiabatic condensation of thevapour coming from the plate below takes place togetherwith partial evaporation of the liquid present in it andarriving from the upper plate

The feed is introduced at about the midpoint of thecolumn, and the part of the column situated above it isusually called the rectifier section. The vapour leaving fromthe top of the column is condensed by a shell-and-tube heatexchanger which has water running through it. A part of thecondensed fluid represents the head product, whereasanother part, called reflux, is sent back to the column inorder to guarantee the presence of a fluid flowcountercurrent to the vapour rising through the column,necessary for the previously described mass exchange totake place. The part of the column situated below the feed iscalled the stripper section, and the liquid descending fromthe bottom of the column partially evaporates in ashell-and-tube boiler, while a portion of it is withdrawn as abottom product.

Material balances Material balances in a distillation column are

appropriately developed by considering the rectifier and thestripper section separately. For the rectifier section, theprocedure follows the diagram in Fig. 3, where D indicates



V0

V1

L1V'1

L'0

L'1

V'2

L'2

V'n�1

L'n�1

V'n

L'n

V2

L2

Vm�2

Lm�1

Vm�1

Vm

Lm

Fig. 2. Multistage distillation process with recycling.

condenser

refluxaccumulator

feed

liquidW

V'

L'heater

DLD

vapour

Fig. 3. Scheme of a continuous distillation column.

the molar flow rate of the head product and LD is the molarflow rate of the condensed component sent back into thecolumn.

Furthermore, by indicating a generic plate in therectifying section by m, the molar flow rate of the liquidfalling from stage m by Lm while the molar feed rate of thevapour rising from plate m is Vm, the global material balancefor the portion of the column above is given by:

[26]

The same balance for component i takes the followingform:

[27]

where xD,i is the molar fraction of component i in the headstream. By combining the last two equations one obtains:

[28]

from which:

[29]

The stripping section proceeds similarly, by isolating aportion of the column as shown in Fig. 3, where W indicatesthe molar flow rate of the bottom product.

By indicating a generic plate of the stripping section byn, the global material balance and that for component i takethe following form:

[30]

[31]

By eliminating W one obtains:

[32]

Equations [29] and [32] alone, however, are notsufficient to characterize the global behaviour of the columnand they must be associated with two further series ofequations. The first set establishes a relationship between thevapour and the liquid compositions at a certain stage, thesecond set relates the flow rate of the liquid falling from acertain stage and the flow rate of the vapour rising from thestage below.

Assuming that each stage behaves ideally and by usingthe evaporation ratio one obtains:

[33]

The second series of equations, instead, is formulated byperforming an enthalpy balance for each plate of the column.By considering a generic plate m and applying equation [12],accounting for the fact that Q and W are both zero, oneobtains:

[34]

The previous equation, therefore, provides a relationshipbetween the molar flow rates of the vapour and liquidstreams on the m plate.

The set of aforementioned equations can be simplified.Indeed, since H��h, it follows that in [34] it is acceptable toignore the heat flow associated with the liquid stream withrespect to that of the vapour stream, so that:

[35]

In this approximation, the heat flux associated with thevapour rising through the column is essentially constant. Afurther approximation consists of assuming:

[36]

This is acceptable if the molar heat of evaporation of thedifferent substances are mutually comparable. In this case,indeed, the enthalpy of a vapour mixture is virtuallyindependent of its composition. The last two equations arecompatible only if:

[37]

From a global balance on the different plates, if V isconstant, it follows that:

[38]

Therefore, it is assumed that both the liquid and thevapour flow rate are constant in the two sections of thecolumn, a hypothesis which is also called ‘constant molarflow’.

Previous approximations make it possible to simplify theanalysis of the behaviour in a distillation columnsignificantly; this can be performed without accounting forthe enthalpy balance on each plate. Material balanceequations are also simplified significantly. Equation [29], infact, can be written as:

[39]

By solving the previous equation with respect to ym�1,recalling that V�L�D, one obtains:

[40]

By indicating the reflux ratio (L /D) by R, [40] becomes:

[41]

known as the ‘working equation’ of the rectifying equation.It is possible to develop the analysis for the stripping

section in a similar way; by indicating the correspondingmolar flow rates of the liquid and of the vapour with L� andV�, different from those in the rectifying section, [39]becomes:

[42]

which, solved with respect to yi,n, and recalling thatW�V��L, gives:

[43]

also called the working equation of the stripping section. Thevalues of the fluxes involved in the last equation can becalculated from the global balance of the column:

[44]

Moreover, if the feed is represented by a liquid stream atthe same temperature as that of the stage on which it is fed, itis possible to write:

[45] L L F RD F�= + = +

F D W= +

y LL W x W

L W xn i n i w i, , ,= − − −+�

� �1

LV

x yx x

w i n i

w i n i

�

�=

−− +

, ,

, ,1

y RR x

xRm i m i

D i+ = + + +1 1 1, ,

,

y L V x D V xm i m i D i+ =( ) +( )1, , ,

LV

x yx xD i m i

D i m i=

−−

+, ,

, ,

1

L L L Lm m m+ − =1 1≈ ≈

V V V Vm m m+ −1 1≈ ≈ =

� � �H H Hm m m+ −1 1≈ ≈ ≈...

� � �H V H V H Vm m m m m m+ + − −1 1 1 1≈ ≈ ≈...

� � � �h L H V V H h Lm m m m m m m m− − + ++ = +1 1 1 1

yx

K T Pi

ii= = ( )Φ ,

LV

x yx x

n

n

w i n i

w i n i

+

+=

−−

1

1

, ,

, ,

V y Wx L xn n i w i n n i, , ,+ = + +1 1

V W Ln n+ = +1

LV

x yx x

m

m

D i m i

D i m i+

+=−−1

1, ,

, ,

V y L x V x L xm i m i m m i m D i m D i+ + += + −1 1 1, , , , ,

V y L x Dxm m i m m i D i+ + = +1 1, , ,

V L Dm m+ = +1



Within the approximations previously discussed it ispossible, by using equations [41] and [43] together withequilibrium relationships, to calculate the number of stagesnecessary for a distillation column to be able to provide acertain performance. One calculation procedure is thefollowing: • The composition in the boiler xw,i is set and the

temperature in the boiler is determined by imposing thecondition:

[46]

• yw,i is calculated through equation [33].• The compositions of the liquid falling from the plate

directly above the heater, xi, are determined throughequation [43].

• The temperature of the mixture on plate 1 is determinedby solving equation

[47]

• The composition of the vapour rising from this plate andin equilibrium with the liquid on the same plate, yi, isthen determined, by using equation [33] again.The steps described above are then applied to the

subsequent stage, proceeding iteratively until a liquidcomposition similar to that of the feed mixture is met.

A similar calculation method can be applied to therectifying section, although by using equation [40] tocalculate the liquid composition falling from a stage throughthat of the vapour rising from the plate below. Thecalculation proceeds until the vapour composition risingfrom a certain stage is comparable to that of the distillationhead product. In fact, if condensation is total, vapourcomposition rising from the highest stage at the top of thecolumn is equal to that of the distilled product, which is:

[48]

A global enthalpy balance finally makes it possible tocalculate the quantity of heat necessary to perform thedistillation. The following relation can be derived.

[49]

where QC and QD indicate the quantity of heat supplied tothe boiler and subtracted at the condenser per unit time,respectively.

Two component mixtures The calculations for the distillation of a two-component

mixture can be further simplified compared to the generalprocedure described above. It is even possible to use graphicmethods. In describing these methods it will be assumed, forthe sake of simplicity, that the feed is liquid with atemperature equal to that of the feed plate.

If liquid-vapour equilibrium conditions are shown in anx-y diagram, in which the working equations of the twosections, [41] and [43], are shown by a straight line (Fig. 4),the line relative to the rectifying section can be drawn,considering that for xm,i�xD,i one obtains ym�1,i�xD,i and forxm,i�0 one obtains

[50]

When drawing the straight line for the stripping sectionit is necessary to account for the fact that for xn�1,i�xw,i oneyn,i�xw,i obtains and for xn�1,i�0 one obtains

[51]

The number of stages in the column can be evaluated bymeans of a McCabe and Thiele graph (McCabe and Thiele,1925), as shown in Fig. 5 for a seven-stage column and a

yWxL Wn i

w i,

,= −−�

yxRm i

D i+ =

+1 1,

,

� � �H h R h DD D D= −( ) +( )+1 ++ −� �h W h FW F

� � � � �Q Q h D h W h FC D D W F= + + − =

y xi D i1, ,=

K T xi ii

( ) =∑ 1

K T xi w ii

( ) =∑ , 1



rectificationsectionoperating line

strippingsectionoperating line

isobaricequilibrium

line

x D/(

R�

1)

0

1

xW xF xD0 1

y

Fig. 4. Equilibrium and operating lines for a distillation column in a binary system.

y

0

1

xW xF xD0 1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8 D

W

Fig. 5. McCabe and Thiele calculation graph of the number of stages in a distillation column for a two component system. Note that the first platecorresponds to the boiler.

total head condenser. Starting from the top, the equationxD,i�x8,i stands, whereas the composition of the falling liquidcan be identified on the equilibrium curve. Through avertical segment it is possible to identify, on the operatingline, the value of y7, which is the composition of the vapourrising from the seventh stage and so on in the same way forall subsequent stages. After the fourth stage, which is thefeed stage, the graphic construction is performed by usingthe operating line of the stripping section. In this way, it ispossible to obtain a composition comparable to that of thebottom product.

If all the vapour rising from the last stage after completecondensation is sent back to the column, the distillation issaid to take place in total, or infinite, reflux conditions, wherethe entire condensed vapour is sent back to the column. Thereflux ratio R�L �D is infinite, since D�0, and bothoperating lines identify with the diagonal in the diagram. Inthis case, as the number of plates in the column is assigned,the maximum concentration difference of the compoundunder examination between top and bottom is obtained.

If the intersection point between operating lines islocated on the equilibrium curve, the number of platesnecessary for distillation tends to infinity, since at theintersection point the composition on each plate remains, forpractical purposes, constant: the corresponding reflux valueis called the ‘minimum reflux’ ratio and is indicated by Rm.

Analytical methods to calculate the number of stagesDetermining the number of stages in a two-component

distillation column can sometimes be performed analytically.These methods, in general, produce results that are lessaccurate than those obtained graphically as discussed above,since their application depends on the possibility ofdescribing liquid-vapour equilibrium via a relatively simpleanalytical function. For a binary system this goal can bereached by using relative volatility:

[52]

which is a particular case of the separation ratio, defined in[1], corresponding to a liquid-vapour system at equilibrium.Often, in particular temperature ranges, it is possible toattribute an average constant value, a, to relative volatilitya12. For ideal mixtures, whose equilibrium constant isexpressed as:

[53]

where pi0 represents the liquid vapour pressure, [52]

becomes:

[54]

and is therefore expressed by the ratio of the vapourpressures of the two components. Recalling that x1�x2�1and y1�y2�1, it is possible to derive the following equation,which gives the relationship between y and x, or, in otherwords, the y�f(x) curve at equilibrium:

[55]

By applying this equation it is possible to derive thefollowing relationships providing the minimum number ofstages of the column, not considering the boiler, Nm (which

means at infinite reflux) and Rm (which corresponds to aninfinite number of stages). The first, called a Fenskeequation (Fenske, 1932), takes this form:

[56]

whereas the second, called an Underwood equation(Underwood, 1948), can be written as:

[57]

The molar fractions reported refer to the most volatilecomponent.

Once the values of Nm and Rm are known, it is possible toroughly calculate the number of stages in the columncorresponding to a generic value of the total reflux ratiothrough the correlation, called a Gilliland correlation, builton empirical bases through the analysis of a certain numberof operating distillation columns. The results obtained bythis method, called ‘short cuts’, are reasonably accurate andmake a good starting point for more detailed calculations.

Precise calculation of a multicomponent distillation column

The calculation methods applied to solve problemsrelated to the simulation of the operation of a continuousdistillation column with N components and M stages isdiscussed below. This situation frequently occurs in themodelling of petrochemical and oil industry plants in which itis necessary to deal with mixtures with a high number ofcomponents by using units having tens if not hundreds ofstages. In this context, superfractionation columns, whichseparate compounds with boiling temperatures very close toeach other as, for instance, in the case of xylenes, are typical.From a mathematical point of view, the equations necessaryto deal with these problems are those expressing the materialbalances at each ideal stage together with the thermodynamicrelationships expressing liquid-vapour equilibriumconditions. The problem is usually solved by simulating thebehaviour of a column once the number of stages andoperating conditions have been assigned. The results of thecalculation must supply the temperatures and compositions ateach stage as well as the flow rates and compositions of thestreams leaving the column itself. Therefore, from amathematical point of view, the problem is well defined butrather complex, particularly when the values of N and M arehigh. In actual fact, the availability of powerful computershas made it possible to perform precise calculations, thelimitations of which, today, are essentially attributed to theaccuracy of the available information on the physico-chemical properties of the substances and mixtures involved.

In order to give a wider perspective to the problem, it isimportant to consider that mixtures are often introduced orwithdrawn on certain plates of the column, since thisprocedure helps the separation of specific cuts of themixture having the desired composition. Therefore, thepossibility should be considered that a feed or a withdrawalor a heat exchange might be present at each stage. Inaddition, to the symbols already introduced, the feed at stagen will be indicated with Fn, the composition of the ith

component in F with zn,I and the vapour withdrawn fromstage n with wn. Un is the liquid withdrawn from stage n and

Rxx

xxm

D

F

D

F

=−

−−−

1

1

1

1αα

N

x xx x

m

D m

w D+ =

−( )−( )1

11

ln

lnα

y xx1

12 1

1 12 1 1=

−( )+αα

α12 10

20= p p

K pPii=0

α121 2

2 1= y xy x



Q is the heat exchanged at stage n. Obviously, the values ofthese variables will differ from zero at only a few stagesbecause only at a few of them is it possible to externallyexchange mass and heat.

The configuration of the column taken intoconsideration is indicated in Fig. 6 and it is composed of Nstages including a condenser, partial or total (stage 1), and aboiler (stage N ). All the values of the variables previouslymentioned must be known, together with the liquid-vapourequilibrium conditions.

With this premise, it is possible to demonstrate that if thematerial balance of each component is performed at eachstage, it is possible to derive a series of equations that can besummarized as follows: • Stage 1:

[58]

• Stages 2�n�N�1:

[59]

• Stage N:

[60]

where the expressions for each one of the various parameterspresent in the equations are reported in Table 2. Furthermore,by inserting Wn�Ln+Un, it is possible to demonstrate thatthe energy balances conducted at each stage take thefollowing form:

[61]

The previous relations form a system of (N�1)Malgebraic equations, expressing material balances at eachstage, with M expressing the energetic balances. To these,the following 2N equations should be added:

[62]

[63]

where [62] makes it possible to evaluate the liquid boilingtemperature and [63] the vapour condensation temperature.

Therefore, globally there are NM�2N equations with thecorresponding number of unknown quantities, for the sake ofprecision NM expresses the compositions of the liquidpresent at each stage and 2N expresses the liquid and vapourtemperature at each stage.

The dependence of K (T, p, xi, yi) on the differentvariables involved in the process gives a non-linear characterto the previous equations. Therefore, the solution of thesystem can be obtained only numerically by using iterativecalculation methods. There are several ways to deal with thisproblem, which approximate numerical methods for thesolution of systems of non-linear algebraic equations withthe tridiagonal characteristics like those under examination.For instance, it is possible to adopt iteration methods bywhich, in subsequent calculation steps, trial values oftemperatures, compositions and liquid and vapour flows aremodified. As an alternative, it is possible to use theNewton-Raphson technique to linearize the equations at eachcalculation stage. It is possible to adopt recursive formulaewhich can be used to solve linear equations systems in bothcases.

Approximated global methods for multicomponent systems

For multicomponent systems, too, there areapproximated methods making it possible to determinethe number of stages based on the assumption that therelative volatilities of the various components can beconsidered constant. Usually, these methods are based onthe identification of two key components in the feed. The

yK xi

n iii

i,∑ ∑= =1

K x yn i n i iii

, , = =∑∑ 1

� �h hn n−+ − 1(( ) + −( )+−L F h H Qn n n F n n1� �

,

� � � �H h V H h V Wn n n n n n n+ +−( ) = −( ) +( )+1 1

A x B xN N i N N i− + =1 0, ,

A x B x C x F zn n i n n i n n i n n i− ++ + + =1 1 0, , , ,

B x C xi i1 1 1 1 0, ,+ =



V2

V1

U1

f2

fn

fW

fD

fn�1

V2

V3

Vn

VH

U2

Un�1

Ln�1

Un

L1

Ln

W2

W3

Wn

W�LN�UN

D�

V1�

U1

Wn�1

Vn�1

F2

F3

Fn

Fn�1

Fig. 6. Scheme of a multicomponent rectifying column.

Table 2. Expressions of parameters in equations [58-60]

B V K U Li1 1 1 1 1= − + +( ),

A L V F W U Dn n n k k kk

n

= = + − −( ) −−=

−

∑12

1

B V W K V F W U D Un n n n i n i k k k nk

n

= − +( ) + + − −( ) − + =

∑, ,2

D V U= +1 1

A V WN N= +

B V K WN N N i= − +( ),

W L UN N= +

WN = 0

C V Kn n n i= + +1 1,

(Kn�1,i is the equilibrium constant on plate i)

light key component is the most volatile among bottomcomponents, whereas the heavy key component is theleast volatile among the head components. The Fenskeequation [56] can therefore be applied by referring to thekey components, thus deriving the minimum number ofstages Nm:

[64]

where xL and xP respectively indicate the molar fractions ofthe light and heavy key components and indices D and Windicate the head and bottom products.

An approximated calculation of the reflux ratio can beperformed by applying the following Underwood equations:

[65]

where ai represents the relative volatility of component icompared to a generic one taken as a reference. Bydeveloping the first relation, an equation of degree n(equal to the number of components) is obtained as afunction of variable ÿ. Among the solutions of theequation, it is necessary to choose from among the relativevolatilities of key components. By replacing the valueobtained in this way in the second of [65], the value of Rmcan be derived. From the values of Nm and Rm obtained inthis way, it is then possible to derive the number of platescorresponding to a certain reflux by means of a Gillilandcorrelation.

Mass transfer in the stages of vapour (gas)-liquidequipment

The description of distillation processes, up to this point,has been based on the use of suitable stages in which the twophases, liquid and vapour, are in close contact, thus helpingmutual processes of mass transfer. To be specific, the stagescan be obtained by using the aforementioned sieve trays,even if the use of particular packing, called ‘structured’packing (see below), has proven to be particularly efficient.Setting aside this aspect for the moment, it is useful todevelop the analysis of distillation columns by making ageneral assumption that each stage behaves ideally so thatthe vapour and liquid streams separating from it are inthermodynamic equilibrium conditions. Obviously, inpractice this does not occur because the vapour compositionleaving the stage is less rich in volatile components thanpredicted by equilibrium conditions, especially because thecontact time between the two phases, in general, is notsufficient for equilibrium to be reached. Basically, in orderto evaluate the correct composition of the two phasesinvolved it is necessary to extend the analysis by explicitlyaccounting for the influence of the velocity at which themass transfer processes between the liquid and the vapourphase present in the stage itself take place. Obviously, thiswill depend on both the geometrical characteristics of theplate or the packing and on the fluid dynamic conditionsestablished on it.

In order to express the velocity at which mass transferprocesses between two phases in contact take place(vapour-liquid in the case of distillation and gas-liquid in the

case of absorption; see section 6.2.4), it is possible to use thedouble film theory, assuming that within each phase, thetransfer velocity of each component is proportional to thedifference between its partial pressure or concentration in thebulk of the phase and its value at the interphase surface(identified by the index i), which means:

[66]

[67]

where N1 represents the moles of species 1 transferred perunit time and per unit of contact surface, kg and kc aresuitable coefficients, called mass transfer coefficients, thevalues of which mainly depend on the aforementioned fluiddynamic characteristics on a plate, whereas theconcentrations C1 and C1i are generally expressed in molesof species 1 per unit volume.

If it is supposed that equilibrium conditions stand at theinterphase surface, it is possible to assume:

[68]

where Hi is a suitable parameter by means of which it ispossible to express the equilibrium between the two phaseswith a linear relationship.

In stationary conditions, the mass transfer rates in bothphases must be the same:

[69]

from which:

[70]

Thus, once kc and kg are known it is possible to derive p1iand C1i by solving the two equations [68] and [70].

However, in general it is better to express the rate ofthe mass transfer process by using variables that areoperatively measurable as concentrations or partialpressures present in the bulk of the two phases. It istherefore appropriate to introduce two new global masstransfer coefficients defined as:

[71]

where p1* is the partial pressure that would be in the gas if it

were in equilibrium with C1, whereas C1* is the concentration

that would be in the liquid if it were in equilibrium with p1.Obviously, the following relationships stand:

[72]

Then, it is possible to derive that:

[73]

and

[74]

Applying these equations implies that the geometric andfluid dynamic conditions of the plates used in the process areknown, since they are obviously necessary to evaluate the

1 1 1

K m k kc i g c

= +

1 1K k

mkg g

i

c

= +

p f C

p f C1 1

1 1

= ( )= ( )

∗

∗

N K p p

N K C Cg

c

1 1 1

1 1 1

= −( )= −( )

∗

∗

p pC C

kk

i

i

c

g

1 1

1 1

−− =−

k p p k C C k C Cg i c i c i1 1 1 1 1 1−( )= −( )=− −( )

p f C H Ci i i i1 1 1= ( )=

N k C Cc i1 1 1= −( ) (for the liquid phase)

N k p pg i1 1 1= −( ) (for the gas or vapour phasse)

αα ϑαα ϑ

i F i

ii

i D i

iim

x

xR

,

,

− =

− = +

∑

∑

0

1

N

xx

xx

m

L

P D

P

D W

LF+ =

1ln

lnα



mass transfer coefficient in the two single phases. Theproblem is, however, highly complex and can be dealt withby using suitable flow models. For instance, in a simplifiedapproach it is assumed that both phases, liquid and vapour,are perfectly mixed so that the concentrations of the differentcomponents are uniform on the whole plate.

Leaving aside more detailed analyses, which give areasonably accurate description of the deviations fromideality established on the plates of a distillation column, itis important to recall that departures from equilibriumconditions are usually expressed by means of a globalparameter called plate efficiency, or Murphree efficiency(Murphree, 1925), defined by the following relation:

[75]

where yn* indicates the composition in equilibrium with

composition xn of the liquid leaving the plate and yn is thecomposition of the vapour leaving the plate.

6.2.4 Absorption of gases in liquids

Absorption is an operation during which a gaseous mixture isbrought in contact with a liquid with the aim of separating oneof its components by dissolving it in the liquid itself. There areseveral industrial cases in which this operation is carried out;of particular interest is the removal of the carbon dioxide fromsynthesis gas and from combustion products by washing withwater under pressure, with ethanolamine solutions or otherspecific agents. Another example is washing the gas from acoke oven, first treated with water to remove ammonia andthen with a mineral oil to remove benzene and toluenevapours. The objective of absorption therefore can be, forinstance, gas purification but it can also be product recoveryor the production of suitable gas mixtures.

The choice of the most appropriate liquid for an absorptionoperation is generally guided by the criteria listed below.

Gas solubility. This should be high in order to increasethe process rate and decrease the required solvent quantities.It is possible to obtain good solubility by using solvents witha chemical nature similar to that of the solute to be removed.If a chemical reaction takes place between solvent andsolute, very high solubility is obtained but for the solvent tobe recovered, the reaction must be reversible.

Volatility. The solvent should have a low vapourpressure, since the gas leaving absorption is usuallysaturated with the solvent, therefore there is the risk oflosing significant quantities of it. Sometimes, a second, lessvolatile, liquid is also used to extract the evaporated portionfrom the first.

Low corrosiveness. In order to lower the total cost, thematerials required to build the plant do not necessarily haveto be highly resistant to corrosion.

Solvent cost. This should be low, so that possible lossesdo not become too economically relevant.

Viscosity of the solvent liquid. This must be low in orderto increase absorption rates, minimize pressure losses andimprove thermal exchange properties.

Other characteristics. The solvent should be non-toxic,chemically stable, and should have a low freezing point.

Usually, an absorption operation is performed in verticalcolumns in which two countercurrent fluxes are present, one

of gas from high to low, and the other of liquid in theopposite direction. Contact between the two phases is carriedout through stages or more commonly in a continuous wayalong the whole column by using packing, as shown in Fig. 7.Operating conditions in an absorption column are derivedfrom its material balance. It is possible to assume that thefeed mixture is composed of a soluble and an insolublecomponent in a liquid solvent, the evaporation of which canbe disregarded. In this case, molar fluxes of the insoluble gasand of the solvent are considered constant along the wholecolumn.

By indicating the concentration in the gas of thesoluble component, expressed as moles of gas/moles ofinsoluble gas with Y, and the concentration in the liquid ofthe soluble component expressed as moles/mole of solventwith X, obviously X�x�(1�x) and Y�y�(1�y), where xand y are the molar fractions of the soluble component inthe gas and in the liquid respectively. By indicating themolar flow rate of the insoluble gas per column unitsection with Gs and the molar flow rate of the pure solventper column unit section with Ls, the material balance ofthe whole column referring to the soluble material has thefollowing form:

[76]

whereas between one extreme of the column, for instance itsbase, and a generic section the material balance is given by:

G Y Y L X Xs s1 2 1 2−( )= −( )

Ey yy yMVn n

n n

=−−

+∗

+

1

1



z

dzZ

G2,y2

G1,y1

L2,x2

L1,x1

liquid

gas

Fig. 7. Scheme of gas-liquid absorption column. In the detail, a part of the packing formed by Raschig rings is shown, wet with the liquid and run through countercurrently by gas.

[77]

or by:

[78]

This last equation expresses the link between the molarfraction of the soluble compound in the gaseous mixture andthe corresponding molar fraction in the liquid phase. Byplotting y as a function of x in a graph, a curve is obtained,called the operating line, which must be located above theequilibrium line, because only under this condition is theconcentration of the soluble compound in the gaseous phasehigher than the concentration corresponding to equilibriumconditions, thereby allowing a transfer of this componentfrom the gaseous to the liquid phase. It is important to bearin mind that the equilibrium line provides, at the pressureand temperature at which the column operates, therelationship between y and x in thermodynamic equilibriumconditions; when dealing with diluted gases the equilibriumrelationship can be expressed by using Henry’s law statingthat, at a constant temperature, the partial pressure of acomponent in the gaseous phase is assumed proportional toits molar concentration through the equilibrium constant H.

If the absorption operation is performed in a multi-stagecolumn, the number of stages is evaluated by means of agraph like that of McCabe and Thiele’s, described for thecalculation of stages in a distillation column. In a packingcolumn, the rate at which the absorption process takes placeis calculated, for instance, by using the first of equations[71], according to which the driving force of the process isgiven by the difference between the partial pressure of thesoluble component actually present in the gaseous phase andthe partial pressure that this component would have if it werein equilibrium with the liquid phase.

At this point, take an element of a column with height dzand area A into consideration (see Fig. 7 again). Byindicating the liquid/gas contact surface per column unitaryvolume with a, the moles of gas transferred in the unit timefrom the gaseous to the liquid phase in the element ofvolume considered, recalling [71], are given by the followingrelation:

[79]

By equating the second member of this expression withthat of the material balance performed on the volumeelement under investigation, it is possible to derive:

[80]

By insulating dz and integrating the expression obtainedin this way between the two extremities of the column, it ispossible to derive the total height Z of the column necessaryto obtain the separation:

[81]

If the concentration of the soluble component in thegaseous phase is low, it is acceptable to assume that(1�y)2�1 and Gs�G, so that the previous equationbecomes:

[82]

Usually, the integral at the second member of theprevious equation is called the number of transfer units andis indicated by NOG. It can be seen that if y�y* is low onaverage, NOG is high whereas, on the other hand, if thisdifference is high, NOG is a low number. In other words, NOGexpresses how difficult the absorption operation is. Equation[81] can therefore be written as follows:

[83]

where the expression HOG�G�KgaP, having the dimensionsof a length, is called HTU (Height of a Transfer Unit). If[73] is rewritten by inserting Hi equal to Henry’s constantand multiplying the various terms by G�aP, one obtains:

[84]

This expression accounts for the fact that Henry’sconstant can be written as H�mP/C, where m represents theequilibrium constant when molar fractions are used toexpress the concentrations of the soluble components; C isexpressed in total moles per unit volume. Therefore, it ispossible to decompose the transfer unit into twocontributions, each relative to the phases present in thesystem, by the relation:

[85]

seeing that and HG�G�kgaP e HL�L�kcaC.

Equipment for gas-liquid contact In both distillation and absorption operations specific

types of equipment are used, such as plates and packing, to

H H mGL

HOG G L= +

GK aP

Gk aP

H Gk aP

Gk aP

mGL

Lk aCg g

i

c g c= + = +

Z GK aP N H N

gOG OG OG= =

Z GPK a

dyy yg y

y=

− ∗∫2

1

Z GPK a

dyy y y

s

g y

y=

−( ) −( )∗∫ 1 21

2

AG yy

AG dy

yPK y y aAdzs s gd

1 12−

=

−( )= −( )∗

NaAdz PK y y aAdzg= −( )∗

G yy

yy L x

xxxs s

1

1

1

11 1 1 1− − −

= − − −

G Y Y L X Xs s1 1−( )= −( )



liquid descendingfrom the tray aboveweir

foam

vapour risingfrom the tray below

Fig. 8. Scheme of sieve tray.

obtain the contact necessary for mass, heat and momentumtransfer between the phases involved. As has already beenmentioned, a typical example of a plate is the sieve tray,where the liquid moves transversally and then falls into theplate below through a downcomer, after having passed anoverflow weir (Fig. 8), while the vapour risingcountercurrently bubbles through the liquid. The contactobtained on each plate between liquid and vapour allows theexchange between the most volatile components whichaccumulate in the vapour phase, and the less volatilecomponents that gather in the liquid phase. The vapour fluxthrough a series of holes on the plate surface guaranteeseffective bubbling. In order to stop the liquid from fallingthrough the holes it is necessary for the vapour flow rate tobe higher than a certain limit value.

The dispersion of vapour in the liquid on a plate can alsobe obtained by means of suitable equipment such as a bubblecap or valve. Obviously, for the plate to be able to workproperly it is necessary that the vapour, or the gas flow rate,be suitably calibrated so that the entrainment of liquiddroplets is limited and, at the same time, its stability isguaranteed.

In packing columns, on the other hand, gas-liquidcontact is obtained by filling the column with suitablematerials, like those shown in Fig. 9. Their averagedimensions are 12-50 mm. In the upper part of the columnthe liquid is fed by using suitable distributors whichguarantee good dispersion. The gas flowing countercurrentlyto the liquid is fed at the base of the column. From a fluiddynamic point of view, packing columns must satisfyspecific constraints in order prevent flooding the columnwith liquid. Meanwhile, it should be remembered that boththe gas and the liquid flow rates greatly influence the valuesof the mass transfer coefficients between the phases. Thisaspect has been the object of extended investigation, bothexperimental and theoretical, thanks to which suitablecorrelations have been formulated making it possible toestimate the values of these mass transfer coefficients.

In recent years, structured packing systems have becomewidespread, often also used for revamping already existingcolumns. Usually, they are made of corrugated metal, plastic

or ceramic sheets. Adjacent sheets are perpendicular to eachother so as to form a very open honeycomb structure, whichcreates inclined flow channels with high surface areas byvirtue of which it is possible to obtain close contact betweenthe two phases involved by means of different possibleconfigurations. This type of solution offers severaladvantages, in particular it causes small drops in pressureand effective contact between the phases due to the fact thatthe vertical orientation of the corrugated sheets makes itpossible to eliminate any horizontal surface which createsresistance to fluid flow. Moreover, structured packing helpsto prevent the creation of channels through which one of thetwo phases preferentially flows, a phenomenon that is quitecommon in conventional packing systems, thereforeproviding a significant improvement in mass transferefficiency. However, it is important to consider the fact thatstructured packing can give problems with fluids thatgenerate scaling because it is extremely difficult to cleanthem. Furthermore, the behaviour of structured packing inhigh pressure operations has proved to be decidedly lesssatisfactory since, as the vapour density increases, the liquidcan be pushed upwards thereby causing backmixingphenomena and consequent capacity and efficiencyreductions.

Using packing systems in a distillation column createsthe problem of their simulation which must be developed onthe basis of balance equations for every component at eachelementary stage of the column, integrated on the entirecolumn height.

Alternatively, it is possible to proceed by means ofevaluating theoretical plates, using the concept of HeightEquivalent to a Theoretical Plate (HETP), representing theheight of packing which, in identical operating conditions,produces the same effect as an ideal stage. It is a parameterthat obviously depends on the nature and characteristics ofthe packing, on the properties of the fluids involved and onthe fluid dynamic conditions in the column. All these factorsare summarized in semiempirical correlations which, in thecase of structured packing, are provided by the packingmanufacturers. In actual fact, since the parameters used toexpress HETP are the same as those for mass transfer



A B C

D E F

Fig. 9. Packing materials. A, Raschig rings; B, Lessingrings; C, Berl saddles; D, Intalox saddles; E, tellerettes; F, Pall rings.

coefficients, the two described methods can be considered tobe equivalent.

6.2.5 Liquid-liquid extraction

Solvent extraction, or liquid-liquid extraction, is anoperation that makes it possible to separate a component in aliquid mixture by using a solvent in which the differentcomponents of the original mixture have different degrees ofsolubility. A simple example helps to identify its objective,and some of its characteristics. If an aqueous solution ofacetic acid is brought into contact with a suitable liquid, forinstance an ester such as ethyl acetate, and stirred, a certainquantity of acid is transferred into the ester. Since thedensities of the aqueous layer and of the ester are different,once stirring is stopped the two phases separate. Theaqueous phase is thus depleted by the soluble componentwhich separates from it. Residual water can be repeatedlytreated with the ester, so that the quantity of acid is reducedthrough subsequent stages. This process can be carried outthrough a cascade of countercurrent stages, or in equipmentworking in direct contact, always countercurrently. In thiscase, too, as in distillation, a reflux can help to improve thefinal degree of separation. At the end of the operation thesolvent can be recovered by distillation.

To apply this separation method, it is necessary tooperate in a field of compositions which make it possible fortwo liquid phases to be present in the system, onesolvent-rich called extract (E) and the other called raffinate(R). Sometimes, it is necessary to work on multicomponentmixtures but for the sake of simplicity it will be assumedthat only the following phases are present: the extract phase,containing the component of interest, or solute, in thesolvent; the mixture to be refined; the selective solvent forthe component to be extracted.

Often solvent extraction is used to replace distillation,particularly when the substances to be separated are ratherdifferent chemically. In this case, it is important to estimatethe costs of operation, accounting for the fact that extractionproduces a new solution which, in turn, must be purified bydistillation. For instance, it is difficult to separate acetic acidin a diluted aqueous solution, whereas it is easier to useextraction followed by extract distillation, the more dilutedthe solution the more convenient it is where great quantitiesof water must be separated. Another case where extraction isan important alternative to distillation is when it is necessaryto separate substances that are thermally unstable and whichtherefore cannot undergo the relatively high temperaturesrequired for distillation; this is typical in the case of theextraction of penicillin and many other substances in thepharmaceutical industry.

Sometimes, on the other hand, extraction is the onlyviable solution, since separation cannot be obtained in anyother way. For instance, the separation of aliphatic andaromatic hydrocarbons with very similar molecular weightsis very difficult to perform by distillation, since their vapourpressures are very close but it can be done relatively easilyby using a series of solvents such as sulphur dioxide ordiethylene glycol.

As has been mentioned, liquid-liquid extraction can beobtained by a stage procedure or by continuous contact. Inthe first case, the mixture to be refined, F, and the solvent, S,

are kept in close contact by stirring so that the systemapproaches thermodynamic equilibrium conditions. In Fig. 10a diagram of an extraction stage is illustrated. In mixer M,one of the two liquids is dispersed in small droplets(dispersed phase) immersed in the other liquid (continuousphase), to accelerate the mass transfer process between thetwo phases. The mixture then goes into the decanter D whereit resides for the time necessary for the two phases to formlayers E and R which are separated subsequently.

A continuous contact process in its simplest form can beobtained by sending the two liquids flowing in verticaltowers, usually countercurrently. The flow of the two liquidsis generated by gravity, exploiting the difference betweentheir specific weights.

The solvents used in liquid-liquid operations usuallysatisfy the following characteristics: a) the ability to dissolvelarge quantities of the species to be separated; b) selectivitywith respect to the component to be separated; c) sufficientvolatility so as to be easily distilled in the subsequentrecovery phase; d ) low cost, non-toxic and non-corrosive; e) low viscosity so that they can easily flow in the tubes ofthe extraction plant.

Obviously, no solvent simultaneously satisfies all theaforementioned characteristics and therefore the choice of asuitable solvent implies a compromise between the differentfactors.

The factor of separation defined by [1] for liquid-liquidtakes the following form:

[86]

where the indices a and b indicate the two phases, whereas gare the activity coefficients of the two components in the twophases, and Sa

ij�gia�gj

a is a parameter called phaseselectivity.

In many cases, selectivity in extraction processes is dueto the ability of the solvent to give hydrogen bonds with theextracted molecules or to produce, through chemicalinteractions, adducts which make a certain substancepreferentially soluble in a particular solvent. For instance,dimethylene glycol has selective properties in the extractionof aromatic hydrocarbons through the formation of hydrogenbonds. Hydrocarbon solubility with respect to differentsolvents has been the subject of many detailed studies.

Stage extraction The liquid-liquid stage extraction process is carried out

by using a series of stages with a countercurrent set-up. Eachextractor is fed with two streams, one of the raffinate and theother of the extract. One important problem is the choice ofthe number of stages necessary to make the composition of

αγ γγ γ

α β

β α

α β

β α

α

βiji j

i j

i j

i j

ij

ij

x xx x

SS

= = =



F

SM

D

E

R

Fig. 10. Scheme of single-stage extraction equipment.

the substance to be extracted from current F lower than acertain value.

As an example, an approximated case where the solventis virtually immiscible in the raffinate will be considered. Inthat case, it is convenient to indicate the mass of the dilutantpresent in the feed stream (F) with A and the mass of the puresolvent present in stream S with B. It is also supposed that thesubstance to be extracted separates between the two phasesthrough an equilibrium relation like K�X/Y, in which it ispossible to attribute an average constant value to the partitionconstant, where X�(substance mass)/( solvent mass) andY�(substance mass)/(dilutant mass).

In this case, material balances between the differentstages can be written as follows: • Stage 1:

[87]

or: Y2�Y1�(BK�A)Y1• Stage 2:

[88]

or: Y3�Y2�(BK�A)(Y2�Y1)�(BK�A)2Y1• Stage N:

[89]

or: Yn�1�Yn�(BK�A)(Yn�Yn�1)�(BK�A)nY1By summing all the previous expressions, member to

member, one obtains:

[90]

The term between parentheses is a geometricprogression and therefore it is possible to derive:

[91]

which is a relation that makes it possible to calculate thenumber of stages necessary to reduce the concentration ofthe substance of interest in the raffinate from Y1 to Yn�1�YF.

When the partition law [91] is not valid because it is notpossible to assign a constant value to K, the calculation of

the number of stages can be performed by means of a graphlike the one proposed by McCabe and Thiele for distillationcolumns. The operating line is given by the followingequation:

[92]

shown in the diagram in Fig. 11 together with the equilibriumline describing the partition of the substance underexamination between the two solvents as a function of X.The composition of the extract and of the raffinate leaving astage are located on the equilibrium curve, whereas thecompositions of the two streams moving countercurrentlybetween the two successive stages are located on theoperating line. Therefore, the aforementioned step graph,shown in Fig. 11, can be easily applied.

The composition of the two fluids leaving a non-idealextraction stage are different from those corresponding tothe thermodynamic equilibrium conditions since theresidence time of the two liquid phases in the mixingequipment is not sufficient to reach equilibrium. Thedeviations from equilibrium, analogous to the plates in adistillation column, can be expressed by an efficiency which,by indicating the raffinate with R and the extract with E canbe defined as follows:

[93]

[94]

where the inlet and outlet from a stage are indicated with aand b respectively.

Continuous contact extraction The extraction operation can be performed using spray

or packing columns where the extract and raffinate phaseflow in countercurrent. R and E represent the total molarflow rates per unit section of the raffinate and extractphase; xR and xE are the molar fractions in the two phasesof the extracted component, and finally a and b indicate theextremes of the column. Obviously, for an extractionprocess to take place the inequalities Ra�Rb and Ea�Ebshould be satisfied. In a column element with unit area andheight dz, indicating the contact surface per unit volumebetween the two phases with s, the contact surface rises tosdz. It is then possible to equate the material balances forboth phases to the mass transfer rate, expressed by theglobal coefficients K referring to the different phases. It ispossible to obtain:

[95]

where xR* is the composition of the raffinate in equilibrium

with composition xE in the extract and xE* is the

concentration in the extract in equilibrium with thecomposition xR in the raffinate.

To develop subsequent calculations it is necessary toconsider both terms of the previous equation, each relative toa specific phase. For instance, the refined phase can beconsidered here; if the mutual solubility of the raffinate andof the solvent does not vary along the column, the quantityR(1�xR) expressing the diluent flow rate can also beconsidered constant along the column. In this case it is:

d Rx d Ex K x x sdz K x x sdzR E E E E R R R( )= ( )= −( ) = −( )∗ ∗

EX XX XME

E b E a

E b E a

=−−∗

, ,

, ,

EX XX XMR

R b R a

R b R a

=−−∗

, ,

, ,

Y Y BA

X XS− = −( )1

YY

BK ABK A

nn

++

=( ) −( )−

1

1

1 11

Y BK A BK A BK A Yn

n

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

1

2

11 ...

A Y Y B X X BK Y Yn n n n n n+ − −−( ) = −( ) = −( )1 1 1

A Y Y B X X BK Y Y3 2 2 1 2 1−( )= −( )= −( )

A Y Y B X X BKYS2 1 1 1−( )= −( )=



Xs Xn�2 Xn�1

Yn�1

Yn

Yn�2

Y1

YF�Yn�1

Xn

operating line

equilibriumline

Fig. 11. Equilibrium and operating curves in an extraction operation.

[96]

which combined with [95] gives the following relationship:

[97]

By insulating dz and integrating, it is possible to derivethe following expression of the column height:

[98]

In many practical cases, xR is not high and therefore it isacceptable to assume that (1�xR)�1. Moreover, assumingthat both R and KR are constant along the column, theprevious equation becomes:

[99]

which can be written as:

[100]

being defined in a similar way to [83]:

[101]

called ‘number of transfer units’, whereas HOR�(R�KRs) iscalled ‘height of a transfer unit’.

The calculation of the integral appearing in [101]can be performed by evaluating the difference (xR�xR

*)in a diagram similar to that shown in Fig. 11, whereboth the operating curve, relating xR to xE at each pointof the column and obtained from the material balance,and the equilibrium curve relating the compositions ofxR and xE in thermodynamic equilibrium conditions areshown.

It is also important to point out that when the solvent andthe raffinate are mutually insoluble, the only process takingplace in the column is the transfer of soluble componentfrom phase R to phase E, so that the operating line is givenby the relation:

[102]

similar to [78] for the absorption column. Rs and Es expressthe flow rates of the pure raffinate and solvent, free fromother components.

6.2.6 Solid-liquid extractionand extraction with supercriticalsolvents

The extraction of a soluble component from a solid materialby using a solvent was first applied long before the birth ofthe chemical industry, as this technique has been used sinceantiquity to obtain alkaline salts from wood ash.

A simple widely-used method consists of covering thesolid with the solvent inside a large steel container andleaving the system in this state for a long period. Theextraction usually occurs at environmental temperature, andtherefore without the risks of decomposition forthermolabile compounds; however, the long periods

necessary are due to the slowness of diffusive processes,especially in the solid. Temperature increase, with obviousproblems for thermolabile substances, and ultrasound can beused to accelerate the extractive process. To reduce the timeneeded for extraction, if the quantity of solid to be treated isvery large, it is advisable to load the solid into a steelcolumn in which the extracting fluid flows. Since yields perpassage are not very high, this liquid is usually recirculatedseveral times. In this way, the extraction time is significantlyreduced.

A relatively recent technique to extract components in acondensed phase uses gases at a temperature above thecritical temperature, which means that they are in that regionof the state diagram where a pure compound cannot beliquefied no matter how much pressure is applied. Thecritical point, in fact, is located at the limit of the curvedefining the two-phase zone in which liquid and vapourcoexist (see Chapter 2.3).

A supercritical fluid has intermediate properties betweena liquid and a gas, as it has the solvent characteristics of aliquid and the transport characteristics of a gas. Therefore, itcan be described as a very mobile liquid. This makes itpossible to obtain extraction rates and efficiencies that aremuch higher than those obtained in conventional processes.Furthermore, extraction conditions can be varied so that it ispossible to obtain well controlled separations. Extractionwith supercritical fluids depends on the fluid density whichcan be modified by varying the system pressure andtemperature. The solvent power of a supercritical fluid growsat a constant temperature as density increases, and at aconstant density as temperature increases. A supercriticalfluid can be used to extract a solute from a solid with theadvantage that, contrary to what happens in a conventionalextraction, when environmental conditions are establishedagain, the very volatile solvent almost completely leaves theextracted material.

The fundamental principle of supercritical fluidextraction lies in the fact that the solubility of a solventvaries according to pressure and temperature. Inenvironmental conditions, the solubility of a solute in a gasis correlated to the solute vapour pressure and is generallynegligible, but in a supercritical solvent, solubility is 10times higher than that predicted on the basis of the ideal gaslaw.

Supercritical extraction is not yet widely used in industrybut it is becoming more and more attractive both for the highlevels of purity that it is possible to obtain and forenvironmental impact reasons, since no toxic solvents areused. Moreover, supercritical extraction makes it possible toobtain separations which would be rather difficult by meansof conventional processes. Furthermore, it generally operatesat low temperatures without, therefore, creatingdecomposition problems when dealing with thermolabileproducts. The most significant disadvantages of supercriticalextraction are: the need to operate at a high pressure; thepresence of complex recycling operations in order to reduceenergy costs due to solvent compression; high investment forthe necessary equipment.

Carbon dioxide is the most common solvent, especiallybecause its critical parameters are rather low (31°C; 74 bar);moreover, it is economical and non-toxic. Organic solvents,widely used in petrochemistry, are, on the other hand,explosive and as a result need expensive equipment.

Rxx

xx E

xx

xs

R a

R a

R

Rs

E a

E a

E,

,

,

,1 1 1 1− − −

= − − −xxE

N dxx xOR

R

R Rx

x

R a

R b=−( )∗∫

,

,

Z H NOR OR=

Z RK s

dxx xR

R

R Rx

x

R a

R b=−( )∗∫

,

,

Z RK s

dxx x xRx

xR

R R RR a

R b=−( ) −( )∫ ∗

,

,

1

Rdxx K x x sdzR

RR R R1− = −( )∗

d RxRdx

xRR

R

( ) = −1



Chlorofluorocarbons would have excellent properties assupercritical solvents but their use is limited due to theirharmful effects on the ozonosphere.

Supercritical extraction is used in the food industry, forinstance in decaffeinization or in the extraction of aromasand essential oils from spices, and in the pharmaceuticalindustry where it is used to obtain the active ingredientsfrom plants without the potentially destructive action of heatand preventing the presence of residual solvent in theproduct. In the petrochemical industry the most common useof supercritical extraction is in the treatment of the residuefrom crude oil extraction.

6.2.7 Drying

In drying, the separation of a liquid from a solid is obtainedby evaporation, involving heat and mass transfer processes.Drying equipment can be classified according to the methodby which heat is transferred to the mass to be dried:• In direct dryers, the most common in the chemical

industry, heat is transferred by convection through directcontact between the solid to be dried and a gas.

• In indirect dryers heat is transferred by conduction to thewet solid through a wall.

• In radiation dryers, heat is transferred to the solid byelectromagnetic radiation, in particular infraredradiation, which can come from a wall or a hot gas. Dryers can use more than one of the transfer

mechanisms described above, even if only one is usuallypredominant. Mass transfer in a drying process involvesvapour removal from the surface of the solid and thetransport of internal humidity towards the surface. Dryerscan work continuously or discontinuously.

In every drying process, the rate at which liquidevaporation takes place obviously depends on theconcentration of the vapour in the gas used as a dryingagent. Since, in many processes, air is used as a dryingagent, whereas the liquid to be removed is water, in thefollowing it is convenient to refer to air-water mixtures.

The quantity of water contained in the air, i.e. itshumidity, can be defined in the following different ways.

Absolute humidity H. This is defined as the ratio betweenthe mass of water per unit of dry air and therefore can beexpressed as:

[103]

where p is the partial pressure of water, ptot is the total oratmospheric pressure; 18 and 28.9 are the molecular weightsof the water and air, respectively. In saturation conditions pcoincides with the vapour pressure of water at the mixturetemperature, so that [103] becomes:

[104]

If air, having a certain absolute humidity, is gradually cooled,saturation temperature is reached, called the ‘dew point’.

Relative humidity. This is given by:

[105]

Percentage humidity. This is given by:

[106]

For many applications it is convenient to make useof a psychrometric diagram, as shown in Fig. 12, whichcharacterizes the state of humidity of the air. The y axisreports absolute humidity H and the x axis,thermodynamic temperature T. The diagram shows thesaturation line, or Hs, as a function of temperature (thecurve where H�100%); the lines of constant relativehumidity (the set of curves with H between 10% and90%); and ‘adiabatic saturation curves’ (the set ofcurves with a negative slope).

The latter are represented by the following expression:

[107] T TH Hcss

s= +

−( )λ

H HHA

s=100

H ppRs

=

H pp ps

s

tot s= −

1828 9.

H pp ptot

=−

18

28 9.



volu

me

(ft3

/lb

dry

air)

hum

idit

y (l

b w

ater

/lb

dry

air)

13

12

14

15

16

17

18

19

20

21

22

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.15

temperature (°F)

humid heat (Btu/lb dry air)

hum

id h

eat

satured volume

dry volume

25 40

0.24 0.26 0.28 0.30

10%20

%30%

40%50

%

60%

70%

80%

90%

100%

60 80 100 120 140 160

135°140°

130°

125°

120°

115°110°

105°100°

95°90°85°80°75°70°

65°60°

55°50°

45°

180 200 220 240

Fig. 12. Psychrometricdiagram.

where l is the heat of evaporation of water and cs is thespecific heat of humid air. Equation [107] relates absolutehumidity to temperature in an adiabatic and, as such,isoenthalpic process.

The ‘wet bulb lines’ are also particularly important. If aliquid mass is evaporated by passing a continuous gas flowover the surface, the temperature of the surface decreasesuntil a stationary condition is established at which, in theunit time, the heat transferred from the gaseous mass to thesurface is equal to the energy involved in liquid evaporation.Temperature Ts of the liquid surface in this case is called wetbulb temperature, whereas the partial pressure of theevaporating fluid at the surface itself is equal to the vapourpressure of the liquid at temperature Ts. The processdescribed satisfies the equation:

[108]

where the first member represents the heat transmitted bythe gas by convection and the second is the energy involvedin evaporation. To be specific, l is the molar heat ofevaporation, h is the heat transfer coefficient from the gas tothe liquid, kg is the mass transfer coefficient, T is the gastemperature, Ts is the liquid surface temperature, ps is thewater vapour pressure at temperature Ts, (ps�p) is thedriving force of the evaporation process expressed as thedifference between the partial pressure of water at theevaporating surface and its partial pressure in air.

By recalling [107], it is possible to write [108] as:

[109]

where kg��kg(28.9�18)p. The lines describing the latterequation are called wet bulb lines. Actually, for the air-watersystem it can be verified that the ratio h�kg� happens to bevery close to cs. For this reason, the adiabatic saturation linesand the wet bulb lines in Fig. 12 overlap.

Wet bulb lines are very useful to determine the humidityof a gaseous mixture. In order to do so, it is necessary toevaluate the mixture temperature (called dry bulbtemperature) and the temperature obtained with athermometer in contact with a gauze soaked in a liquid, onwhich the gas is blown. The absolute humidity of the mixturecan then be found, via the diagram in Fig. 12, by means of asimple construction in which the intersection point betweenthe straight line describing [109] and the saturation curve isidentified.

In order to measure and establish the operative capacityof a dryer it is necessary to know the rate at which the dryingprocess takes place. This, in fact, makes it possible to derivethe time necessary to bring the solid to the desired degree ofhumidity and then establish the residence time of the soliditself in the equipment. If drying experiments on a solid areperformed by measuring its liquid content W (in kg/kg of drysolid) as a function of time, it is possible to calculate thevalues of the evaporation rate (dW�dt)�(DW/Dt) as afunction of time. It is possible to observe that the rate ofevaporation keeps constant up to a critical time after which itdecreases regularly. This result can be interpreted, assumingthat during the initial period the evaporation process involvesonly the external part of the solid, and therefore can bedescribed as a simple evaporation process of a liquidcompletely covering the external surface of the solid. Duringthe subsequent period, however, the process involves the

liquid held inside the solid particles and thereforeevaporation involves more complex processes. In the initialperiod, evaporation kinetics identifies the mass transfer ratefrom the solid surface to the bulk of the fluid mass acting asthe drying agent. To describe the kinetics of the process it ispossible to use an equation where, referring to unit time andto solid unit volume, the energy involved with waterevaporation is assumed equal to the heat transmitted by thefluid acting as drying agent:

[110]

where Ts is the temperature of the evaporation surface, rs isthe density of the dry solid, a is the surface per solid unitvolume; l refers to mass rather than moles. From theprevious relation it is possible to derive:

[111]

After the critical point, the drying rate depends on theliquid motion within the solid and consequently theinfluence of external variables becomes less important. Theevaporation process is controlled by liquid diffusion withinthe solid and therefore the rate of evaporation can be derivedby integrating the diffusion equation. For a layer of depth lthe following relation stands:

[112]

where Dl is the liquid diffusion coefficient and We indicatesthe value of humidity in the solid, called equilibriumhumidity, which can be brought into contact with a gas of agiven humidity and a certain temperature. By integrating[112], the following expression of the drying time isobtained:

[113]

where Wc indicates the liquid content in the solid at thecritical point.

In a bed of highly porous particles, a liquid can movefrom one region to another under the effect of capillaryaction. In fact, when the liquid is progressively removedfrom the bed, the curvature of the liquid surface towards theexternal parts of the porous interstices increases, generatinga pressure that tends to suck the liquid from the insidetowards the external surface. As the drying process goes on,the curvature of the meniscus increases, until the ‘sucking’force reaches such a high value that it causes a pore break atthe surface; therefore, air can come in, and the internalhumidity of the solid redistributes at a lower level.

The drying rate in processes not conditioned by diffusioncan be expressed by the relation:

[114]

Constant K1 can be evaluated by applying the followingrelation to the critical drying point:

[115]

By substituting the value of (dW�dt)c into [115], derivedfrom [111], it is possible to obtain:

KdW dtW Wl

c

c e

= −( )

−

dWdt

K W Wl e= − −( )

t lD

W WW W

l

c e

e

=−−

4 2

2πln

dWdt

Dl

W Wle= −( )π2

24

dWdt

ha T Ts

s= −( )r λ

rs sdWdt ah T Tλ = −( )

H H hk

T Tsg

s− = −( )�λ

h T T k p ps g s−( )= −( )λ



[116]

By integrating the relation thus obtained it is possible toobtain the following expression for drying time:

[117]

Equation [117] makes it possible to deduce the timenecessary to bring the content of a solid to the desireddegree, which is the residence time of the material in thedryer necessary to carry out the process under examination.Obviously, to use [117] it is necessary to know the criticalhumidity content Wc. For common materials such a value isknown, whereas for others it must be derived through dryingexperiments.

6.2.8 Crystallization

Crystallization is one of the oldest unit operations used inchemical production. Each year, large quantities of chemicalsubstances are produced in crystalline form (sodiumchloride, sodium and ammonium sulphates, etc.). Moreover,many products of pharmaceutical and fine organic chemistryare solids and so can be separated by crystallization.Purification of organic liquids by crystallization is quitecommon as an alternative to distillation when azeotropicmixtures or mixtures with boiling points very close to eachother must be separated. Enthalpy variations associated withcrystallization processes are, in general, much lower thanthose of evaporation and the process can be performed attemperatures close to environmental temperature, which ismuch lower than distillation temperature. Crystallization is aprocess involving simultaneous mass and heat transfers inmultiphase and multicomponent systems. Moreover, crystalgrowth significantly depends on the mechanical propertiesof the crystallizer itself.

If crystallization is performed by progressively cooling asaturated solution, mass transfer from the solution bulktowards the crystal surface takes place. In many cases, thesolute and the solvent cannot be mixed in the solid state and

produce a state diagram like that shown in Fig. 13, where Mindicates the melting point of component B, and P themelting point of component A. Curves MU and PDUindicate the solubility limit of such components in the liquidsolution. If mixture C is cooled, the precipitation of pure Astarts at point D. A further temperature decrease causes asubsequent crystallization of A, whereas the composition ofthe remaining liquid varies along the DU curve until itreaches the eutectic composition point, at which furthercooling causes the solidification of a mixture, calledeutectic, of crystals A and B which cannot be mechanicallyseparated.

If the solution had initially been at point E, coolingwould have produced a parallel phenomenon, except for thefact that in the first phase, in other words before reaching theeutectic temperature, the crystallization of pure B would takeplace. In practice, the crystals of a component whichseparate in the first phase might mechanically obstruct partof the solution and thereby remain impure. In this case it isnecessary to wash them with the pure solvent.

If a concentrated solution of a solid substance is slowlycooled, it is possible to exceed the solubility limits withoutseparation of the solid taking place. This condition, knownas ‘supersaturation’, plays a fundamental role incrystallization processes. Ostwald (1896) and Miers (1906)suggested the existence of two types of supersaturation, onecorresponding to a metastable state and the othercorresponding to a labile state. They can be represented on atemperature-concentration diagram as shown in Fig. 14. Thecontinuous line represents the normal solubility curve,whereas the dashed line is the supersolubility curve, showingthe relationship between temperatures and concentrations atwhich spontaneous nucleation phenomena are likely tooccur. In actual fact, this line is not as well-defined as thesolubility curve since its position in the diagram depends onthe stirring intensity and on the possible presence ofimpurities.

Therefore, the diagram can be subdivided into threezones: the lower solubility zone, where the formation ofcrystallization nuclei does not take place, nor the growth ofany possible nuclei introduced; the intermediate metastablezone where nuclei formation does not take place but theirgrowth can occur; the upper supersaturation zone, unstable

tW W

ha T TW WW W

s c e

s

e

e

=−( )−( )

−−

rλln

0

Kha T T

W Wls

s c e

= −−( )−( )r λ



temperature

eutecticpoint

solid A�liquid

composition

solid A�solid B

U

C

EM

P

A B

D

solid B�liquid

TU

Fig. 13. Example of a diagram of crystallization of a binary mixture.

conc

entr

atio

n

temperature

stable

metastable

unstableE

D

ABC

Fig. 14. Graph of the solubility zones of a solid.

or labile, where spontaneous nucleation is likely, but notinevitable.

If a solution represented by point A (see Fig. 14 again) iscooled by keeping the quantity of solvent constant,spontaneous nucleation occurs only at the moment when theconditions represented by point C are reached. The tendencytowards nucleation increases as the supersaturation region isentered, unless the solution becomes so viscous as to preventit. On the other hand, it is also possible to reachsupersaturation conditions by evaporating solvent from thesolution. Line ADE represents an operation of this type,performed at constant temperature. By operating in this way,however, it is difficult to enter the supersaturation regionsince the part of the solution located close to the evaporationsurface is more supersaturated than that in the bulk ofsolution, so that the crystals generated at the surfacepenetrate the solution, inducing other nucleation phenomena,even if the bulk of the solution has not yet reached theconditions represented by point E.

However, nucleation is a change from athermodynamically unstable situation to a metastablesituation. The process can be schematized in this way:homogeneous phase A�� crystalline nuclei dispersed in A.

This process takes place spontaneously and is associatedwith a Gibbs free energy decrease which, in turn, can bedivided into the summation of two terms, the first of which(positive), represents the work of formation of the surfacebetween the two phases, whereas the second (negative),represents the decrease of free energy associated with theformation of the new phase. It is possible to derive thefollowing expression of the global variation of free energy:

[118]

where s is the interfacial tension, r is the nucleus radius(assumed to be spherical) and rc is its critical value, whichmust be reached for nucleation to proceed spontaneously. Itis easy to demonstrate that a maximum of DG correspondsto rc, which is an unstable state at which a particle has anequal chance of growing or shrinking, since in both cases afree energy decrease takes place.

From the kinetic point of view, the phenomenon ofcrystallization shows two typical situations: the first is therate of nuclei formation in the supersaturation zone, whereasthe second is their growth rate in both regions,supersaturated and metastable.

Nuclei formation is a typical activated process, the rateof which, Rn (number of formed nuclei/time), is given by:

[119]

where W is the isothermal work of formation of a nucleus,which is infinite at the supersaturation curve, and decreasesas supersaturation increases. It follows that as the degree ofsupersaturation rises, the formation of a high number ofsmall crystals increases exponentially. Therefore, if theformation of large crystals is desired, it is necessary tooperate at low supersaturation so that the process proceedsessentially by growth.

The growth rate in stationary conditions can be written as:

[120]

where m is the deposited mass of solid, D is the diffusioncoefficient of the component crystallizing in solution, A isthe crystal surface, d is the depth of the surface liquid layerin which diffusion resistance is located. k indicates the rateconstant of the surface process by which an ion is insertedon the surface of the reticule. Finally C, Ci� and Cs representthe concentrations of the solute in the bulk of solution, at theinterface and at saturation, respectively. Therefore, [120]expresses the equality between the transport rate of thematerial towards the surface of solid particles and that of theformation of the crystalline layer. By eliminating Ci� [120]can be rewritten as:

[121]

where (D�d)�kc expresses a mass transfer coefficient. Itsvalue depends on the depth of the surface boundary layer,which in its turn depends on the fluid dynamic conditionsunder which the crystallization process takes place. In theliterature, several relations have been proposed, expressedthrough the use of adimensional parameters which make itpossible to evaluate this coefficient for specific equipment insuitable physical conditions. In practice, the value of kc isoften high compared to that of k, so that the global kineticenergy of the crystallization process is controlled by crystalgrowth rate.

However, the simulation of a crystallization processdepends on the formulation of a physical model whichmakes it possible to describe and combine nucleation andgrowth processes, so that not only the amount of crystallizedmass can be determined, but also the distribution of crystalsof different dimensions. A rather useful general model is thatof Randolph and Larson (1962). By indicating the totalnumber of crystals contained in a determined volume with acharacteristic length L, by N, it is possible to define avariable n, called ‘population density’:

[122]

The n∆L product expresses the number of crystalsper unit volume with a length included in a shortinterval ∆L�L1�L2. Its values can be experimentallydetermined by a mechanical classification process or byoptical and microscopic analysis, in the case of smallcrystals.

By referring, then, to a continuous crystallizer with avolume, V, and performing a balance referring to theparticles with a particular value of L, in stationary conditionsit is possible to obtain the equation:

[123]

where v is the volumetric flow rate of the exit stream. Thefirst term of the previous equation represents the increase inthe number of particles with length L in the system, thesecond their transport towards the outside. To be specific, ris the rate of production of particles with L dimensions,expressing the number of particles of dimension L, producedper unit time and unit volume which can be rewritten in thefollowing form:

[124] r dndL

dLdt

dndLG=− =−

rV nv− =0

n NL

dNdLL

= =→

lim∆

∆∆0

dmdt

A C C

k D

s=−( )+1 δ

dmdt

D A C C kA C Ci i s= −( ) = −( )δ� �

R enW k TB∝ −

∆G r rrc

= −

4

2

3

23

πσ



where G�dL�dt is the growth rate of crystals since itexpresses the increase of their linear dimension L as timevaries. The minus sign in the previous relation reflects thefact that population density variation must be expressed asthe difference between the number of particles enteringdimension L and those leaving it, both as an effect of growth.It is now possible to assume that the value of G for a groupof crystals geometrically similar, of the same material,suspended in the same solution, is the same for all crystals,irrespective of their initial dimensions. This proposal isknown as McCabe’s ‘delta L law’, and has proved to be validfor many materials with crystals having dimensions smallerthan 50 mesh.

By applying this law to [123] it is possible to derive

[125]

where ÿ�V�v indicates the average residence time of thesuspension in the crystallizer. By indicating the populationdensity of particles with dimensions equal to thecrystallization nuclei by n0 and by integrating the previousequation it is possible to derive:

[126]

from which:

[127]

establishing a linear relationship between lnn and L.Different systems satisfy equation [127]; among whichpotassium sulphate, sugar, sodium chloride, CaSO4�0,5H2Oand urea. In other cases, the relationship between and L isnot linear and it is necessary to apply more sophisticatedmodels to describe the phenomenon under examination.

The usefulness of [127] is due to the fact that, by meansof crystallization experiments performed on a small scale, itis possible to evaluate the model parameters, n0 and G, sothat it is possible to use the results obtained to calculate forlarger scale equipment.

As mentioned above, crystallization can be obtained bycooling or by evaporation, and crystallizers can be dividedinto two categories depending on the type of processadopted. Fig. 15 shows the scheme of a crystallization bycooling unit, in which the feed to the crystallizer flowsthrough a refrigerant and then goes back to the main tankwhere growth crystallization takes place. Liquid circulation,obtained by using a pump, helps to keep the mixture in themain tank uniformly stirred.

Crystallization by evaporation, on the other hand, isperformed in units like that shown in Fig. 16, in which thefeed can be preheated before being pumped into theevaporator to partially remove the solvent. Crystals arewithdrawn at the base of the tank, whereas at its top theconcentrated solution is cleaned out.

In operations performed on a large scale, heat is suppliedby vapour condensation. Since most of the energy requiredby an evaporator is that corresponding to the energy used toremove the solvent, it is convenient to recover the thermalcontent of the vapour produced. This can be done by meansof multiple effect plants where the vapour produced in thefirst evaporator is used as heating agent in the secondevaporator and so on. In order for this to be possible, it isnecessary for the evaporation pressure to decrease whenpassing from one evaporator to the next, and therefore thelast evaporator in the series is kept under vacuum by abarometric device.

n n e L G= −0 ϑ

dnn

dLGn

n L

0 0∫ ∫=− ϑ

dndL

nVG

nG=− =−ν

ϑ



refrigerator

motherliquid

feed

circulatingpump

crystalexit

Fig. 15. Operating scheme of a crystallizer with refrigeration.

coolingwater

vapour

exch

ange

r

condensedproduct exit

non-condensablegases

product

baro

met

ric

cond

ense

r

feed

Fig. 16. Crystallization by forced circulation evaporation.

6.2.9 Adsorption

Adsorption is the phenomenon that allows the selectiveaccumulation on a solid surface of particular types ofmolecules contained in a fluid in contact with the surfaceitself. In other words, adsorption concerns the distribution ofdifferent components at the solid-fluid interphases which isdifferent from the absorption phenomenon in which thedistribution of the components between the bulks of twophases, one liquid and gaseous, takes place. Usuallyadsorbent solids are made up of granular porous particleswith extensive surfaces, usually between 102 and 103 m2/g.Their adsorbing capacity can be greater than 20% of themass of the solid itself.

In industry, adsorption is a separation technique adoptedin many different types of process among which are worthmentioning: a) decoloration of liquids and of oil fractions; b)water deodorization; c) removal of harmful substances fromgases; d ) extraction of vitamins or other components fromfermentation mixtures; e) extraction of solvents fromgaseous mixtures; f ) separation of components in mixtureswhen it is difficult, or even impossible, to separate bydistillation (for instance, the separation of the isomericxylenes and the extraction of light olefins from oil-crackinggas).

There are several industrial applications where adsorbentparticle beds are used and in order to classify them it isadvisable to consider both the phase in which the operationis performed (gas or liquid) and its final objective(purification or separation). In actual fact, adsorption is alsoused to remove undesired products from a process stream orfrom polluted waste. In this case, the fluid stream flows overa bed of particles, careful attention being given to thebreakthrough point of the impurity which is the instant atwhich its concentration at the exit of the adsorption columngoes above a tolerable value.

Separation by adsorption, on the other hand, exploits theselectivity that certain solids display with respect to thecomponents present in fluid mixtures. In fact, if thesemixtures are made to flow in a column containing a fixedbed of adsorbing particles, it is possible to observe that thedifferent components leave the column at different times,depending on their degree of adsorbability. An example ofselective adsorption is that used in the laboratory whenperforming a gas chromatographic analysis.

Table 3 lists a few typical adsorbing materials and alsoreports some of their particular characteristics such as thoseof the surface and porosity. Among them, zeolites are

particularly important. Their structure, in fact, ischaracterized by cavities with well defined geometricalcharacteristics which can host molecules selectively which,in turn, due to their geometrical configuration, have amorphological correspondence with the cavity.

For a long time, the use of separation processes byadsorption on a wide scale was limited by design difficultiesand by the problems encountered in the management of thiskind of unit, as it is necessary to have a profound knowledgeof both the physico-chemical aspects of the phenomenainvolved and the modelling problems for their simulation.The most common adsorption separation procedures involvethe use of fixed beds of particles operating in transitoryconditions, alternating an adsorption stage with aregeneration stage. The situation can be illustrated in anarrangement with three columns. When columns A and Boperate in adsorption mode, column C works in regenerationmode. Subsequently, column A is no longer fed andundergoes regeneration by being treated with a desorbent orby being heated, whereas feed flows through columns B andC. In the following phase, column B works in regenerationmode, while the feed flows through columns C and A.Finally, the initial configuration is restored. Obviously, aprocess of this type can also be performed by using a greaternumber of columns than taken in to consideration, but tomanage the process, it is necessary to be able to simulate thetransient behaviour of each column adequately. Therefore,this confirms the importance of having models available toevaluate the variation of the composition over the time of thestream leaving the column for a particular feed and theconcentration of the different components adsorbed on thesolid at different points of the column itself.

All these difficulties can be overcome if separation isperformed continuously, with fluid and solid movingcountercurrently. An example can be found in the processknown as Hypersorption which makes it possible to separateethylene from methane. Fig. 17 is a diagram of a plantbasically composed of a fractionation column, situated on avapour desorption, or stripping, column. Gaseous feed(composed of ethylene and methane) is inserted near thecentre of the fractionation column and rises upwards withthe adsorption of the ethylene and part of the methane. Anethylene reflux is fed at the base in order to help the methanedesorb from the adsorbent, especially at the lower part of thecolumn. Water vapour helps the ethylene to desorb from theadsorbent, while humidity is removed from the solid bybeing washed with methane. The descending movement ofthe solid is the weak point in this process, since it requires



Table 3. Materials used in separation processes by adsorption

Material Chemical composition Surface area(m2/g) Porosity Apparent

density

Zeolites Y Na12[(AlO2)12(SiO2)12]�27H2O (faujasites) 824 0.45 1.1

Zeolites X Na86[(AlO2)86(SiO2)106]�H2O (faujasites) 410 0.57 1.18

Silica-gel SiO2 515 0.46 1.09

Activated alumina Al2O3 354 0.61 1.25

Activated carbon C 700-1,200 0.56 0.76

expensive mechanical operations; moreover, the solidparticles are subject to crumbling and contamination. Forthis reason, mobile beds have been not adopted very oftenand the use of fixed beds is preferred.

An important alternative to the previous process, whichis a compromise between the use of a fixed bed and therealization of a continuous process, is the operation knownas simulated mobile bed (Broughton et al., 1970), which wassuccessfully applied in the separation of isomerichydrocarbons (xylenes in particular) and which consists ofkeeping the adsorbing bed fixed, and periodically varyingthe positions of liquid feed and withdrawal.

This approach is carried out industrially as shown in Fig. 18, in which a pump circulates the liquid from low tohigh externally with respect to the column. A device,indicated with RV (Rotating Valve), operates in such a waythat each of the different streams is sent to the various lineslinked to the adsorption column. At each moment, only four

lines are active; in the example in the figure, these are lines2, 5, 9 and 12. This configuration makes it possible toperform regeneration operations in the upper part of thecolumn and adsorption operations in the lower partsimultaneously. When the rotating element of the valvemoves towards the subsequent position, each stream istransferred to the adjacent line. In this way, the desorbententers line 3 rather than 2, the extract is withdrawn from line6 rather than 5, feed enters from 10 rather than 9 and theraffinate is extracted from 1 rather than 12. Adsorption andregeneration operations, therefore, follow the evolution overtime of the concentrations of the different components on thesolid bed. This is an ingenious device, but its realization iscostly, so much so that its use is limited to high productionplants, like those for xylene production, as mentioned above.

Usually, small and medium scale adsorbing beds areoperated in a semicontinuous manner. Much simpler plantshave also been built, exploiting an impulse feed technique,obtained by a finite quantity of absorbable mixture in adesorbent stream (Seko et al., 1979). In this case, theconcentration over time of two components A and B showsthe typical impulse movement of chromatography. They canbe separated by means of appropriate withdrawals, and byexploiting multiple columns it is possible to make thisoperation continuous. In this way, the process looks verymuch like laboratory chromatography. Previous descriptionsshow that the industrial application of separation byadsorption methods requires expertise in using themathematical models which make it possible to simulate thetransient behaviour of these units. When formulating thesemodels, it is important to identify the following fundamentalpoints: • The study of the fluid-solid thermodynamic equilibrium.• The formulation of kinetic equations expressing the

adsorption rates on the single particles.• The formulation of material balances for adsorption

columns and the identification of methods for solvingthe equations so obtained.

• The analysis of the behaviour of operation units as afunction of operating variables, such as feed compositionand flow rate, operating temperature and so on.

Adsorption equilibria Adsorption equilibrium can be described by means of

suitable isotherms expressing, at a particular temperature,the relationship existing between the partial pressures andthe concentration of the different components in the fluidphase and those existing on the solid surface. In separationprocesses, only physical adsorption (generated by van derWaals forces) plays a role and no chemical bonds areformed. In actual fact, studying these phenomena is quitecomplicated, particularly when the adsorbent material has acomplex structure as in the case of zeolites. The mostcommonly used equation to describe adsorption isotherms isthe Langmuir equation:

[128]

where the indexes i and j refer to the different components inthe mixture; qi is an adimensional variable expressing thecoverage degree of the surface by the i component, whereasCi is the concentration of i in the fluid phase; bi are

θii i

j jj

nbC

b C=

+=∑11



CH4

C2H4

R

C

C

CH4�C2H4

C2H4

H2O

H2O

CH4

solidtransportation

vapour

Fig. 17. Scheme of the Hypersorption plant to separate ethylenefrom methane. C, condenser; R, vaporizer.

1

234567

R

F

F

E

D

89101112

R R

D

E

RVE

Fig. 18. Simulated mobile bed process. E, extract; R, raffinate; D, desorbent; F, feed.

appropriate parameters, called adsorption constants, whichare usually evaluated experimentally. The coverage degree isexpressed by the ratio qi�Gi �Gi

, where Gi is theconcentration of i in the solid phase, and Gi

represents theload capacity of the adsorbent. The widespread use of theLangmuir equation is due to the fact that it is able todescribe how solid saturation conditions are reached and toaccount for the fluid mixture composition. This effect is, infact, important to condition the selectivity of an adsorbentwith respect to the various components involved. Fowler’sisotherm is an improvement compared to the Langmuirisotherm since it accounts for variation in the heat ofadsorption with the degree of coverage, due to surfaceheterogeneity and to the interactions between the adsorbedmolecules (ci):

[129]

Therefore, this equation makes it possible to account forthe influence of the degree of coverage on selectivity.

Adsorption process kinetics As far as kinetic aspects are concerned, it is necessary to

consider that adsorption is a typical mass transfer processwhich takes place through the following sequence of stages:• Stage 1: external mass transfer, from the fluid bulk to the

particle surface.• Stage 2: diffusion inside the particle pores.• Stage 3: surface adsorption.• Stage 4: diffusion inside the solid.

This situation could refer to a material like zeolite whoseparticles are made up of microcrystals (1-3 mm) joinedtogether. In this way, it is possible to identify a bimodalporous structure where both crystal microporosity, site ofdiffusive stage 4, and the intercrystalline macroporositystructure, site of diffusive stage 2, are present. Moreover,concentration gradients appear within the particles, whichobviously influence the process global rate. In calculatingthe adsorption rates, it is possible to assume a fewsimplifying hypotheses: • The influence of stage 4, where internal diffusion takes

place, can be ignored if the ratio between the particlediameter and that of the microcrystals is high (as in thecase of many zeolites).

• Stage 3 can be considered at equilibrium.• In the calculation, it is possible to use the average values

of the concentrations inside the particles2

Ci therebyavoiding the cumbersome calculation of concentrationgradients. Under these hypotheses, the kinetic model of the single

particle is reduced to three equations only. The first,

[130]

expresses the impoverishment of the external fluid phase asan effect of mass transfer from the fluid bulk to the particles,where the rate of this process is proportional to the differencebetween the external concentration of the component and theaverage concentration in the particle through a globalcoefficient Ki, reflecting the presence of two resistances inseries 1�Ki�1�ke�1�ki, where ke is the external transfercoefficient and ki the internal transfer coefficient. The

external coefficient can be evaluated from the fluid dynamiccharacteristics of the system, whereas the internal coefficientdepends on the diffusion coefficient Di of the component, onthe tortuosity of the internal channels of the particles and onthe radius of the particles themselves Rp; ee is the externalporosity, C

_i is the average concentration in the particle, Ci

e

the concentration in the core of the particle. The second equation,

[131]

on the other hand, expresses the variation of the internalconcentration given as the difference between the masstransfer rate examined above and the rate of adsorption ofthe component on the solid; rs represents the density of thesolid and ei the internal porosity.

Finally, the third equation,

[132]

expresses the relationship between the concentration in themacropores and on the solid as given by the adsorptionisotherm; feqi

is, in fact, the variable that describes theadsorption equilibrium.

By integrating equations [130-132] numerically it ispossible to describe the evolution over time of the speciesadsorbed within the particles.

Balances in adsorption columnsThere are numerous studies on criteria and

approximations in the literature which can be used to writethe equations expressing the dynamic balance for adsorptioncolumns. In a general description it is necessary to considerthat systems containing several components described by anon-linear adsorption isotherm are usually involved. Theequation

[133]

expresses the transitory balance of component i on a volumeelement of the column, taking into account axial andconvective dispersion effects. Variable u represents gasvelocity, evaluated taking into account the fluid velocityvariation due to the flow rate variation taking place along thecolumn as an effect of adsorption processes. This effect canbecome important when describing separation processes ofconcentrated mixtures in which, as an effect of adsorption, asignificant impoverishment of one or more components ofthe mixture can take place. Equation [133] should thereforebe associated with single-particle balance equations [131]and [132], and to the stoichiometric congruency equation:

[134]

The simplified equations are the following:

[135]

[136]

where y is the molar fraction in the gaseous phase, e is thevoid fraction in the column, X

_the average concentration on

∂∂ = ( )− Xt R

X y Xeff152

δ

∂∂ + −( ) ∂

∂ =− ∂∂

yz

Xt

yts

s1 ε εr

M x Vi i

ii

n

r=∑ =1

�

ε εeie

ie

Lie

ip

e iCt

uCz D C

zK R C∂

∂ +∂( )∂ − ∂

∂+ −( )

2

23 1 ee

iC−( )=0

ddt

f

CCt

i eq

j

j

j

ni

Γ=

∂

∂∂∂=

∑1

ε εei

ip

ie

i i sidC

dtK

RC C

ddt

= −( ) − −( )31 r

Γ

ε εei

ip

e ie

idCdt K K C C=− −( ) −( )3 1

b Ci i i jj

ni

jj

n0

1

1

1exp −

=

−=

=

∑∑

χ θ θ

θ



the solid, X(y) the average concentration on the solid inequilibrium with y, deff the effective coefficient of diffusionin the solid, z the axial coordinate, R the particle radius.

Their application is helped by the analytical solutionproposed by Rosen, which simplifyies the exact solutionsproposed by Anzelius and Nusselt in a similar problem ofheat transfer. This analytical solution is still widely used inpurification problems (Kast, 1981).

6.2.10 Ionic exchange

Ionic exchange is an operation in which a reversible ionexchange is made between a solid and a liquid, withoutcausing permanent variation to the solid structure. This typeof operation is often used in water softening anddeionization processes, but it is also a separation methodwhich is extremely useful in many chemical processes. Itsutility is based particularly on the possibility of reutilizingthe materials used to perform it.

For instance, in water softening operations the followingreaction takes place:

Material R_

, usually a resin containing sodium ions, iscapable of exchanging the calcium ion with the aqueousphase. Subsequently, the resin full of calcium can be treatedwith a sodium chloride solution so as to regenerate it into itsoriginal form and make it available for a new cycle ofoperations by means of a reversible regeneration reaction. Ingeneral, it is possible to soften millions of litres of waterwith 1 m3 of resin, operating over a period of several years.There are several substances capable of exchanging ions, forinstance silicates, phosphates, cellulose. Ionic exchange,industrially applied since about 1910, used natural zeolites atfirst; around 1935 synthetic organic resins were introduced,made of polymers containing sulphonic, carboxylic orphenolic groups, for instance, which can be considered asbeing made of an extremely large anion and an exchangeablecation which facilitate exchanges like:

Na��HR��NaR �H�

Similarly, polymeric resins containing aminic groups andanions can be used to exchange anions in solution:

RNH3OH �Cl��RNH3Cl �OH�

H��OH��H2O

where RNH3 represents the immobile cationic portion in theresin. These resins can be regenerated by contact withcarbonate solutions or with sodium hydroxide. In general,there are different types of synthetic ion exchange resinsavailable, with different exchange capacities, usually in theform of granular solids.

Operational techniques commonly used for adsorptionare also used for ionic exchange. The operation is usuallyperformed by using fixed beds of resin in which the usefulionic exchange phase is alternated with regeneration.Fluidized beds are used less frequently. Chromatographicmethods have been used to fractionate multicomponent ionicmixtures. Besides the water softening methods mentionedabove, complete water deionization can be obtained byperforming percolation first through a cationic resin andthen through an anionic resin. By using a bed formed of a

mixture of equivalent quantities of a strong cationic resinand strong anionic resin in close contact with each other, it ispossible to simultaneously remove all the ions untilneutrality is reached. The two resins must be then separated(through hydraulic classification, by exploiting the differentparticle dimensions and their densities) in order to beregenerated.

The equilibrium distribution of an ion between a resinand a solution can be described in a graph by isothermalcurves very similar to those used for ordinary adsorption. Todescribe these isotherms, several empirical equations can beused, among which the Freundlich equation.

Furthermore, ionic exchange reactions are reversible. Bywashing a resin with an excess of electrolyte it is possible toconvert it completely to the desired ionic form:

However, if the quantity of B� in solution is limited,equilibrium depending on the proportions of A� and B� andon the selectivity of the resin is established. The selectivityKB

A coefficient for this reaction is given by:

[137]

where m and m_

refer to the ionic concentrations in solutionand in solid phase respectively.

The rate at which ionic exchange takes place depends, asfor conventional adsorption, on the rates of the stagesthrough which the process develops: a) ion diffusion fromthe liquid phase bulk to the external surface of a particle ofthe exchange resin; b) ion diffusion within the solid towardsthe site at which the exchange takes place; c) ion exchange;d ) diffusion of ions released towards the external surface ofthe solid; e) diffusion of ions released from the solid surfacetowards the bulk of the liquid phase.

In some cases, the kinetics of stage d ) is decisive, but itis often much faster than the rates of the diffusive stages.

6.2.11 Sedimentationand centrifugation

Sedimentation is an operation that makes it possible toseparate, by means of the gravitational field, a heterogeneousmixture made up of solid particles suspended in a liquid, intotwo distinct phases, namely a solid phase made up of asuspension in which the concentration of the suspended solidparticles is higher than that of the initial suspension, and aliquid phase where the concentration of the suspended solidparticles is much lower. The objective of this operation canbe to thicken the solid phase or to clean the liquid phase.Drag forces act on a particle moving in relation to the liquidin which it is suspended:

[138]

where v is the particle-liquid relative velocity, rL is the liquiddensity, A is the particle surface projected into the directionof motion and CD, called the drag coefficient, is a functionof the Reynolds number

[139] RePLx

=νµr

F C Av

D DL=

r 2

2

K mm

mmA

B B

A

A

B= ⋅

RA B RB A+ + + ++ +��➤�

2RNa Ca R Ca 2Na22

2+ + + ++ +��➤�



where x represents the particle dimension, m is the liquidviscosity, and the CD coefficient depends on the flow regime.Generally, sedimentation takes place in a laminar regime, ata low Reynolds number (Rep�0.2). In these conditions CDcan be calculated by using the following relation:

[140]

A particle of a suspension, in the gravitational field, issubject to the drag force, the weight force and Archimedeanbuoyancy, so that the following relation stands:

[141]

where m is the particle mass, g the acceleration of gravity, rsthe density of the particle, and t is the time. When steadystate conditions are reached (dv/dt�0), from [141] oneobtains:

[142]

In the case of spherical particles with diameter d it is:

[143]

Therefore, by inserting the relations [138], [140] and[143] into [142], it is possible to derive Stokes’ law:

[144]

where vg is also called terminal velocity. In general, terminalvelocity is reached quickly and the transient phase can beignored. Equation [144] is derived by assuming isolatedparticles and can be considered valid for sufficiently dilutedsuspensions. As concentration grows, particles approacheach other and start interfering reciprocally, causing aslowing down of sedimentation which can be expressed asfollows:

[145]

where vp is the rate of sedimentation of the concentratedsuspension, e is the volumetric fraction of the fluid and f(e)its function. There are several ways of expressing f(e),among which the most well-known is the Barman-Kozenyequation:

[146]

A fairly common way to increase sedimentation velocityvg consists of increasing the particle dimensions. This isgenerally done by inducing aggregation processes, alsocalled coagulation, between the particles, usually triggeredby adding additives, for example, polyelectrolites. Industrialsedimentators can operate continuously or discontinuously.Sedimentators of the first kind are very simple, but relativelyseldom used except when it is necessary to treat smallquantities of suspensions.

The components of a mixture are separated bycentrifugation, imposing a centrifugal field, the intensity ofwhich is given by w2r, where w is the angular velocity and r

the distance from the rotation axis. The field acts in a similarway to the gravitational field (the intensity of which is g),but its intensity can be varied by changing the rotation speedor the equipment dimensions, whereas the gravitational fieldis constant. Industrial equipment is capable of producingaccelerations as much as 20,000 times greater than theacceleration of gravity, while laboratory equipment canreach 300,000 g.

Centrifugation is used especially to separate immiscibleor insoluble components in a liquid medium. In general,centrifugation makes it possible to perform more efficientlythe same operations as can be performed under the action ofa gravitational field. Different types of equipment withvarious configurations and geometries can be used toseparate a mixture by centrifugation, for instance bottle,tubular and disk centrifugation.

In centrifugation equipment, rather complicated flowsare produced and this has so far hindered the development ofa mathematical model allowing rational dimensioning. Asimplified approach consists of using Stokes’ law also forthe centrifugal field, therefore arriving at the followingequation:

[147]

where vs is the fall velocity of a particle in a centrifugalfield, w is the angular velocity of the centrifuge, r is thedistance from the rotation axis at which this velocity isdetermined, vg is the terminal velocity of the particle in thegravitational field, rs is the particle density, r the density ofthe medium, d the particle diameter, m the viscosity of themedium. It is supposed that the particle is found at an initialposition corresponding to distance r from the rotation axis.By applying equation [147] to this particle and assumingvs�dr�dt, one obtains:

[148]

where rc is the radius of the sedimented cake and t is thetime interval during which the particle is submitted to thecentrifugal field. By integrating, it is possible to obtain:

[149]

The previous equation is valid only if the followingseries of simplifying assumptions are made:• The particles composing the suspension are all spherical

and all equal, and do not change shape or dimension as aresult of coalescence or flocculation, duringcentrifugation.

• Particles are uniformly distributed in the suspension, andtheir concentration is sufficiently low to make themdeposit as if they were isolated particles.

• The fall velocity is such that the Reynolds number isbelow 100, so that the application of Stokes’ law doesnot cause an error greater than 10%.

6.2.12 Filtration

Filtration is the operation by which the particles of asuspension (liquid or solid) are separated by passing themthrough a permeable sect, also called a filter. The filter holds

lnrr

vgtc

g= ω 2

drr

vgdtg

t

r

rc = ∫∫ω 2

0

vd r

v rgs

sg=

−( )=

r r 2 22

18

ω

µω

f ε εε( )= −( )10 1

vv fp

g= ( )ε ε2

vg d

gs=−( )r r

18

2

µ

m d

A d

s=

=

π

π6

4

3

2

r%

gFms

D1−

=r

r

m dvdt

mg mg Fs

D≈ − −r

r

CDP

= 24

Re



the solid particles and allows the fluid to go through. Whenproceeding with the operation, a cake of solid particles,forming a porous mass, which can be compressible or not,accumulates on the filter. In the cake, particles transmitmechanical stress by reciprocal contact, also called effectivepressure. The cake, when incompressible, has a rigidstructure, and therefore cannot be deformed, even if theeffective pressure increases. If the cake is made ofnon-flocculated solid particles, like sand or salt crystals, itcan be considered incompressible. If, on the other hand, thecake is made of flocculated particles, as in waste slurries,metallic hydroxides or in polyelectrolytic floccules, itsstructure is deformable, and therefore compressible. Solidconcentration in a compressible cake is variable, anddepends on the local values of effective pressure. Usually, ina filtration process, a constant hydraulic pressure differenceis kept between the chambers on the two sides of the filter.The suspension flows through the filter, thereby depositingthe solid particles on its surface. The applied pressure alsoforces the suspension through the cake which, as the processproceeds, leaves deposits on the filter.

Usually, the particles forming the porous bed are verysmall, and therefore also the diameters of the channels arevery small. Consequently, the fluid flow rates through thefilter are generally low, and the fluid motion is of the laminartype. A drop in pressure Dpa in a porous medium iscalculated by applying the Blake-Kozeny equation:

[150]

where e is the void fraction in the porous medium, m is thefluid viscosity, Deq is the equivalent diameter of particles, Lis the width of the porous medium and us is the rate at whichthe fluid passes through the filter, defined by the ratiobetween the volumetric flow rate of the fluid and the totalarea of the filter. Since, in a solid bed obtained from crystalsuspension it is practically impossible to determine Deq ande, it is therefore appropriate to write [150] in the followingform, called a Darcy equation:

[151]

Coefficient a, called permeability, depends on thecharacteristics of the porous bed, and can be empiricallydetermined by directly applying the Darcy equation. Bycalculating us from it, one obtains:

[152]

from which

[153]

where A is the total section of the filter and dV is the liquidvolume filtered in time dt.

After time t, if V is the volume of the filtered liquid, Lthe width of the solid cake, rs the density of the solidcomposing it, and r the liquid density, a simple materialbalance makes it possible to obtain the following relation:

[154]

where e is, as usual, the porous sect porosity and x is the

solid fraction in the suspension. By solving [154] withrespect to V and L it is possible to obtain:

[155]

[156]

By replacing [155] in [153], one obtains:

[157]

where Kv indicates a constant summarizing all characteristicproperties of the porous mass and of the feed, given by:

[158]

If porosity and permeability are kept constant duringfiltration, as in the case of incompressible cakes, Kv is alsoconstant. Furthermore, if Dpa is also constant [158] can beeasily integrated:

[159]

by defining

[160]

It is possible to derive a relation between the thickness ofthe solid cake at time t by writing [156] in differential formand replacing it into [153]:

[161]

By introducing constant KL defined by the relation:

[162]

Equation [161] becomes:

[163]

If the pressure difference is constant, [163] can be easilyintegrated giving:

[164]

Constants Kv and KL can be calculated from feed andsolid cake properties. In practice, however, this evaluation isparticularly difficult, especially where the estimation of cakeporosity and the shape and dimensions of the particlesforming it are concerned; therefore, it is preferable to use adirect experimental evaluation using equations [157] and[164] by measuring the volume of filtered liquid, and thethickness of the cake as a function of time.

When deriving the above equations it is assumed that theresistance met by the fluid is only due to the solid cake. Inreal cases, it is necessary to take into consideration the factthat there are other sources of resistance: first of all, theporous sect and then all the ducts, connections and valvesmaking up the filter. This set of resistances can be expressedthrough an ‘equivalent thickness of the cake’, Leq and an‘equivalent volume of filtration’ Veq.

t K LpL

a=

2

∆

dLdt

pK L

a

L= ∆2

Kx x

xLs=−( ) −( )− µ ε εα

r rr

%1 12

dLdt

x pL x x

a

s

=−( ) −( )−

αµ ε ε

r

r r

∆

%1 1

t K VA p

v

a

=2

2∆

22

00

KA p

VdV dtv

a

tV

∆= ∫∫

K xx xv

s

=−( ) −( )−

µα ε ε

r

r r2 1 1%

dVdt

A x x pV x

A ps a a=−( ) −( )− =

α ε εµ

2 21 12

r rr

% ∆ ∆KK Vv

L V xA x xs

=−( ) −( )−

r

r r% 1 1 ε ε

Vx x

x LAs=−( ) −( )−r r

r% 1 1 ε ε

1 1−( ) =+( )−ε ε

LAV LA x

xrr

dVdt

p ALa=αµ

∆

u AdVdt

pLs

a= =1 αµ∆

∆pL ua

s= 1α µ

∆pL

u

Da s

eq

=−( )

( )1 1502

3 2εε

µ



Equations [160] and [163] then assume the followingform:

[165]

[166]

Integrating at constant Dpa it possible to obtain:

[167]

[168]

Equation [165] can be written in the following form:

[169]

By experimentally determining the series of values ofDV, fractions of filtered liquid in different time intervals Dt,if reported on a graph Dt/DV as a function of V, a straightline representing equation [169] is obtained. Its inclinationgives the value of , whereas its intercept on the x-axis givesthe value of 2KvVeq �A2Dpa. As A and Dpa are known, it istherefore possible to determine Kv and Veq.

If the cake is compressible, its porosity, e, depends onpressure and therefore Kv varies with Dpa. Moreover, solidparticles can infiltrate and clog the interstices of the sect, byvarying Veq in a way that also depends on Dpa.

6.2.13 Membrane separation processes

It is possible to obtain the separation of fluid mixtures byusing membranes that allow the passage of one or morecomponents and slow down the passage of others. Thecomposition at the two sides of the membrane istherefore different. There is a great variety of membraneprocesses and in a certain sense filtration, which consistsof the separation of large and visible particles from afluid or a gas, can also be considered as a particular typeof membrane separation. At the opposite end of thespectrum there are all those other processes which makeit possible to separate ions or molecules on the basis oftheir molecular weight. Membrane processes can beclassified on the basis of particle dimensions or on themolecules to be separated (Table 4). The various

mechanisms which can be involved in separations arelisted below.

Dimensional exclusion mechanism. Membranes havepores allowing the passage of some species and not others.

Knudsen diffusion mechanisms. Membranes have poresclose to molecular dimensions, therefore causing someselective delays in the passage of some species. This processtakes place above all in gaseous system separations, wheremembranes with very small pores can induce separation dueto this type of mechanism, in which diffusion rates of thevarious molecules vary as the inverse of the square root oftheir molecular weight.

Diffusion mechanism by solution. It is possible for somespecies to dissolve in the membrane, subsequently migratingthrough it by molecular diffusion, to emerge again on theother side.

The industrial importance and economical relevance ofmembrane processes is growing, with applications rangingfrom water treatment to the food and pharmaceuticalindustries.

Membrane processes have a long series of arguments infavour and against, when compared to more conventionalprocesses. Among the most significant advantages thefollowing should be listed: • No phase variation is required and therefore energy costs

are low. • Process schemes are very simple, with little auxiliary

equipment.• Sometimes they are effective in cases where

conventional systems fail, as in the separation of gaseousmixtures that form azeotropes.

• They do not cause significant damage to theenvironment.

• There is a wide choice of membranes and therefore it ispossible to exert excellent control over processselectivity.However, there are also conditions in which these

processes are difficult to apply, for instance when there ischemical incompatibility between the membrane and thesystem to be treated, or when scales form on the membranesurface significantly decreasing the process potential.Furthermore, it is not possible to use high temperatureswhich might damage the membrane.

Usually, two different flow configurations are used,perpendicular to the membrane or parallel to the membrane.In the first case, mass accumulation on the membranesurface is created and therefore the operation must beperiodically stopped in order for it to be washed. In the

dtdV

KA p

V KA p

Vv

a

v

aeq= +2 2

2 2∆ ∆

tC L LL

pL eq

a=

+( )2 2∆

tK V VV

A pv eq

a

=+( )2

2

2∆

dLdt

pK L L

a

L eq

=+( )

∆2

dVdt

A pK V V

a

v eq

=+( )

2

2∆



Table 4. Membrane process classification

Process Separation mechanism Pores dimensions (Å) Transport regime

Filtration Dimensional exclusion �50,000 Macropores

Microfiltration Dimensional exclusion 500-50,000 Macropores

Ultrafiltration Dimensional exclusion 20-50 Mesopores

Nanofiltration, inverse osmosis Dimensional exclusion;solution/diffusion �20 Micropores

Gas separation Solution/diffusion �5 Molecular

second case, the possibility of accumulation decreases, sincedeposited material tends to be dragged away by the flow.

Driving forces acting in membrane separations can be ofvarious types: pressure difference, concentration gradient, orelectrical potential difference.

The flow NA, of component A through a membrane isgiven by the equation:

[170]

where PA is the permeability of A, L the thickness of themembrane and D/A is the driving force inducing the passagethrough the membrane. For gas phase separations, D/A isgiven by DpA, which is the difference between the partialpressures of A on the two sides of the membrane. For liquidphase separations the situation is more complex. For a pureliquid D/A would simply be given by the pressure differencep1�p2 on the two sides of the membrane; however, if a smallquantity of solute which cannot pass through the membraneis added, a flux reduction can occur due to osmotic pressurep effects. Indeed, the flux of A in this case is given by:

[171]

Osmotic effects can be ignored in the separation ofspecies with molecular weights higher than a few thousanddaltons, but if the solute has a low molecular weight they canbecome surprisingly relevant, so much so that they can limitseparation capabilities. For sufficiently low soluteconcentrations the osmotic pressure can be expressed by thevan’t Hoff equation:

[172]

where CA is the solute concentration, R is the gas universalconstant, and T the thermodynamic temperature. The effectof osmotic pressure is to oppose the increase of soluteconcentration, and therefore to oppose the creation of astream of pure liquid. Note that, for low molecular weightmaterials, the osmotic pressure is very high and it resists theconcentration of the species which do not pass through themembrane. For instance, for molecular weights in the orderof 100 daltons, if the pressure difference is equal to 35 bar,the maximum solute concentration obtainable is equal toabout 22%. At this concentration, osmotic pressure is equalto the pressure difference on the two sides of the membraneand the flux goes to zero.

In order to evaluate the selectivity of a membrane it ispossible to use equation [170]. If it is written for a species Aand a species B, and the two equations written in this wayare divided member by member, one obtains:

[173]

If DpA�DpB, the membrane selectivity is given by theratio of permeabilities.

By applying an electric potential difference to the twosides of a membrane it is possible to cause ion transport.This type of process is called electrodialysis which makes itpossible to separate or concentrate salts, or acids and basesfrom solutions. Membranes used in electrodialysis can be ofthe cationic or anionic type. Both are generally made fromcrosslinked polystyrene, but those of the first type aresubsequently sulphonated, whereas those of the second type

only contain quaternary ammonic groups. Membranes of thefirst type only allow the passage of anions, whereas those ofthe second type only permit the passage of cations.

6.2.14 Flotation

In separation by flotation, a suspension of the material to betreated is prepared and to which are added chemicalcompounds, called collectors, which have the power of makingthe surface of the material hydrophobic. By bubbling airthrough the system, the gas bubbles remain attached to theparticles of mineral which, therefore, rise towards the surfaceforming a foam. Flotation is widely applied in mineralenriching. The mineral’s surface usually has polar, andtherefore hydrophilic, characteristics; it is therefore necessaryto add collectors which greatly reduce the affinity of themineral particles with water. The most common collectors areorganic salts with long hydrocarbon chains. The polar sideorientates towards the mineral surface, which is thereforeexternally protected by a hydrophobic layer and dragged by theair bubbles. This kind of process is controlled by many factors,among which the pH, the dimensions and the type of thesurface to be treated. Collectors can be of the anionic orcationic type, depending on whether the charge is positive ornegative. Anionic collectors are either oxydrilic orsulphydrylic. Cationic collectors are generally quaternaryammonic salts used for minerals with a strong surface chargedensity. In flotation operations depressing agents are also used,and these are substances that modify the surface of the materialby making it more hydrophilic; they are used especially in theselective flotation of two minerals which have an affinity forthe same collector. These agents are used to diversify thesurface of minerals which can be separated into two flotationstages. Another class of additives used in flotation are foamers,which help the formation of stable foams and favour theadhesion of air bubbles to the mineral-collector complex.

In practice, flotation is performed on material milled anddispersed in water; this is sent to a vertical cell in which astream of very small air bubbles is generated. The number offlotation cells varies according to number and the species ofminerals present.

Bibliography

King C.J. (1983) Separation processes, New York, McGraw-Hill.

McCabe W.L. et al. (2005) Unit operations of chemical engineering,Boston (MA), McGraw-Hill.

Seader J.D., Henley E.J. (2005) Separation process principles,Etobicoke (Canada), John Wiley.

References

Broughton D.B. et al. (1970) Parex process for recovering paraxylene,«Chemical Engineering Progress», 66, 70-75.

Fenske M.R. (1932) Fractionation of straight-run Pennsylvaniagasoline, «Industrial and Engineering Chemistry», 24, 482-485.

Gilliland E.R. (1940) Multicomponent rectification. Minimum refluxratio, «Industrial and Engineering Chemistry», 32, 1101-1106.

Kast W. (1981) Adsorption aus der gasphase - Grundlagen unVerfahren, «Chemie Ingenieur Technik», 53, 160-172.

NN

PLPL

pp

PP

pp

A

B

A B A

B

A

B

A

B

= =∆∆

∆∆

πA AC RT=

NPL

p pAA

A= − −( )1 2π

NPLAA

A= ∆φ



McCabe W.L., Thiele E.W. (1925) Graphical design of fractionatingcolumns, «Industrial and Engineering Chemistry», 17, 605-611.

Miers H.A. (1906), «Journal of the Chemical Society», 89, 413.

Murphree E.V. (1925) Rectifying column calculations with particularreference to n-component mixtures, «Industrial and EngineeringChemistry», 17, 747-750.

Ostwald W. (1896) Lehrbuch der Allgemeinen Chemie, Leipzig,Engelmann, 2v.; v.II, Part 1.

Randolph A.D., Larson M.A. (1962) Transient and steady-state sizedistributions in continuous mixed suspension crystallizers,«American Institute of Chemical Engineers Journal», 8, 639-645.

Seko M. et al. (1979) Economical p-xylene and ethylbenzene separatedfrom mixed xylene, «Engineering Chemistry Process Design andDevelopment», 18, 263-268.

Underwood A.J.V. (1948) Fractional distillation of multicomponentmixtures, «Chemical Engineering Progress», 44, 603-614.

List of symbols

a liquid/gas contact surface per volume of gas in anabsorption column

a surface per unit volume of solid in a dryingoperation

A cross section of an absorption columnA mass of diluent in the feed stream of a liquid-liquid

extractorA surface of the particle projected in the direction of

motionA surface of a crystal B mass of pure solvent in the solvent stream in a

liquid-liquid extraction bi Langmuir equation parameters CD drag coefficientC1 concentration of species 1 in the bulk of the liquid

phase C1i concentration of species 1 at the interphase surfaceC1

* concentration in the liquid were it in equilibriumwith p1

Ci concentration of i in the fluid phaseCi� concentration of the solute at the interface with the

solid in a crystallization Ci

e average concentration in the bulk of the fluid phaseof an adsorption column

C_

i average concentration in the particles of anadsorption column

Cs concentration of the solute in the bulk of solution,in a crystallizer

cs specific heat of humid air d particle diameter D molar flow rate of the head product of a distillation

column D diffusion coefficient of the component crystallizing

in solution Deq equivalent particle diameter Di diffusion coefficient of component iDl diffusion coefficient in a liquid E molar flow rate of the extracted phase per unit cross

section in a continuous extractor EME efficiency of a liquid-liquid extraction operation

referring to the extract EMR efficiency of a liquid-liquid extraction operation

referring to the raffinate

EMV Murphree efficiency Es flow rate of a pure solvent F molar flow rate of the feedFD drag force F(e) inlet mass flow rate F(u) outlet mass flow rateFi

(e) inlet of mass flow of component i Fi

(u) outlet mass flow rate of component i Fn feed rate onto stage nFe

(i) inlet mass flow rate of component i Fu

(i) outlet mass flow rate of component iF(u) outlet mass flow rate g acceleration of gravity G crystal growth rate Gs molar flow rate of the insoluble gas per unit column

section Hi Henry constant H absolute humidity HA percentage humidity HR relative humidityHs absolute humidity in saturation conditions HOG transfer unit height, in gas absorption operations HOR transfer unit height, in liquid-liquid extraction

operations H vapour molar enthalpy h heat transfer coefficient from gas to liquid h liquid molar enthalpy hF feed molar enthalpy Ki vaporization ratio of component iK kinetic energy referring to the unit mass kc mass transfer coefficient in crystallization kg mass transfer coefficient in the gas phase Kg global mass transfer coefficient in the gas phase kl mass transfer coefficient in the liquid phase Kl global mass transfer coefficient in the liquid phaseL liquid flow rate L crystal characteristic length L membrane thickness Leq equivalent cake thicknessLD molar flow rate of the condensed component sent

back to the column Ls molar flow rate of the pure solvent per unit section

of the column Ln molar flow rate of the liquid downflowing from

plate ml thickness of a drying solid m mass of solid deposited in crystallization mi mass of component i in equipment N total number of crystals in a specified volume N moles of soluble component transferred from gas to

liquid per unit timeNA flow of component A through a membrane Nm minimum number of stages in a column NOR number of transfer units N1 moles of species 1 transferred per unit time and per

unit contact surface n number of crystals per unit volume with a specified

length ni number of moles of component i contained in the

system P total pressure



PA permeability of Ap partial water pressure ps water vapour pressurepi

0 liquid vapour pressurep1i partial pressure of component 1 at the interphase p1

* partial pressure in the gas were it in equilibriumwith C1

Q heat exchanged by a thermodynamic system Qn heat exchanged on stage nr global velocity of crystallization process r distance from the rotation axis at which the velocity

in a centrifugal field is determined r radius in a crystallization nucleus R reflux ratio R total molar rate of the refined phase per unit sectionrc critical radius of a crystallization nucleus Rm minimum reflux ratio Rp particle radius in an adsorption column Rep Reynolds number of a particle in a sedimentation Sij phase selectivity Ts temperature of the liquid surface U internal energy of a system referring to the unit

mass Un liquid withdrawn from stage nus flow rate through a filter V vapour flow rate V volume per unit massVeq equivalent filtration volumeVm vapour molar flow rate uprising from plate mv volumetric flow rate of the stream leaving a

crystallizer v particle-fluid relative velocity vg terminal velocity vp sedimentation velocity of the concentrated

suspension vs fall rate of a particle in a centrifugal field W thermodynamic workW isothermal work of formation of a crystallization

nucleus W bottom product flow rate W humidity of a solidWc humidity of a solid at the critical point We equilibrium humidity in a solid wn vapour withdrawn from stage nX ratio between the mass of substance and mass of

solvent in a liquid-liquid extraction x dimension of a particle in a sedimentation xi molar fraction of component i in the liquid phasexn composition of liquid leaving plate nxE molar fraction of the extracted component in the

raffinate phase xR molar fraction of the extracted component in the

raffinate phase

yi molar fraction of component i in the vapour phase Y gas concentration of the soluble component Y ratio between the mass of substance and mass of

solvent in a liquid-liquid extraction yn composition of the vapour leaving a plate yn

* composition in equilibrium with composition xn ofthe liquid leaving the plate

z height of the section of a duct zi molar fraction of i in the feed Z total height of an absorption column necessary to

obtain a certain separation

Greek lettersa permeability coefficient of a filter as

ij separation factor of components i and jci interaction parameter between adsorbed species gi activity coefficient of component iGi concentration of i in the solid phase of an

adsorption column Gi

load capacity of the adsorbent d depth of the liquid layer where the resistance to

diffusion in the growing phase of a crystallization islocated

D difference between the values of an inlet stream andan outlet stream

Dp pressure drop in an adsorption column e fluid volumetric fraction e vacuum fraction in the porous mediumee external porosity in adsorption columnsei internal porosity of the particles in adsorption

columnsÿ average residence time in the crystallizerÿ Underwood equation parameter qi degree of surface coverage by component il water evaporation heat m liquid phase viscosity rl liquid density rs solid density rv vapour density s surface tensionw centrifuge angular velocity F potential energy

Sergio Carrà

Dipartimento di Chimica, Materiali eIngegneria chimica ‘Giulio Natta’

Politecnico di MilanoMilano, Italy

Stefano Carrà

MAPEI

Milano, Italy



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