Distillation and Rectification

c 2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim10.1002/14356007.b03 04

Distillation and Rectication 1

Distillation and RecticationJohann Stichlmair, Universitat Gesamthochschule Essen, Essen, Federal Republic of Germany

1. Introduction . . . . . . . . . . . . . . 52. Vapor Liquid Equilibrium . . . . 62.1. Empirical Findings in

Vapor Liquid Equilibrium . . . . 62.1.1. Binary Mixtures . . . . . . . . . . . . 62.1.2. Ternary Mixtures . . . . . . . . . . . . 82.2. Empirical Laws of Vapor Liquid

Equilibrium . . . . . . . . . . . . . . 132.2.1. Equilibrium Constant . . . . . . . . . 142.2.2. Equilibrium Curve . . . . . . . . . . . 142.2.3. Distillation Curve . . . . . . . . . . . 142.3. Calculation of Phase Equilibrium

from Excess Enthalpy . . . . . . . . 152.4. Calculation of Phase Equilibrium

from Equations of State . . . . . . 162.5. Practical Determination of Phase

Equilibria . . . . . . . . . . . . . . . . 183. Distillation . . . . . . . . . . . . . . . 183.1. Continuous Distillation . . . . . . . 193.1.1. Continuous Distillation of Binary

Mixtures . . . . . . . . . . . . . . . . . 193.1.2. Continuous Distillation of Multi-

component Mixtures . . . . . . . . . 203.2. Batch Distillation . . . . . . . . . . . 213.2.1. Batch Distillation of Binary Mix-

tures . . . . . . . . . . . . . . . . . . . 213.2.2. Batch Distillation of Multicompo-

nent Mixtures . . . . . . . . . . . . . . 223.3. Semicontinuous Distillation . . . . 233.3.1. Semicontinuous Distillation of Bi-

nary Mixtures . . . . . . . . . . . . . . 233.3.2. Semicontinuous Distillation of Mul-

ticomponent Mixtures . . . . . . . . 234. Continuous Rectication (Multi-

ple Distillation) . . . . . . . . . . . . 244.1. Principles . . . . . . . . . . . . . . . . 244.1.1. Equilibrium-Stage Concept . . . . . 244.1.2. Transfer-Unit Concept . . . . . . . . 264.1.3. Comparison of Equilibrium-Stage

and Transfer-Unit Concepts . . . . . 284.2. Rectication of Binary Mixtures . 284.2.1. Prediction of Rectication Based on

Material Balance . . . . . . . . . . . . 294.2.2. Prediction of Rectication Based on

Material and Enthalpy Balances . . 314.2.3. Rectication of Binary Mixtures at

Innite Reux Ratio . . . . . . . . . 32

4.2.4. Rectication of Binary Mixtures atMinimum Reux Ratio . . . . . . . . 33

4.2.5. Heat Requirements for Recticationof Binary Mixtures . . . . . . . . . . 34

4.3. Rectication of Ternary Mixtures 364.3.1. Rectication of Ternary Mixtures at

Innite Reux Ratio . . . . . . . . . 364.3.2. Rectication of Ternary Mixtures at

Minimum Internal Loads . . . . . . . 374.3.2.1. Minimum Reux Ratio and Mini-

mumReboil Ratio for Preferred Sep-aration . . . . . . . . . . . . . . . . . . 37

4.3.2.2. Minimum Reux Ratio for Separa-tion of the Low Boiler . . . . . . . . 39

4.3.2.3. Minimum Reboil Ratio for Separa-tion of the High Boiler . . . . . . . . 40

4.3.3. Rectication of Ternary Mixtures atFinite Reux Ratio . . . . . . . . . . 41

4.3.4. Problems Arising in Column Calcu-lations . . . . . . . . . . . . . . . . . . 42

4.4. Rectication of MulticomponentMixtures . . . . . . . . . . . . . . . . 43

5. Batch Rectication . . . . . . . . . . 455.1. BatchRectication of BinaryMix-

tures . . . . . . . . . . . . . . . . . . . 455.1.1. Operation at Constant Reux Ratio 455.1.2. Operation at Constant Distillate

Composition . . . . . . . . . . . . . . 465.1.3. Operation with Minimum Energy

Requirement . . . . . . . . . . . . . . 475.2. Batch Rectication of Multicom-

ponent Mixtures . . . . . . . . . . . 476. Industrial Rectication Processes 496.1. Binary Rectication . . . . . . . . . 496.2. Multicomponent Separation . . . 496.3. Separation of Heterogeneous

Azeotropic Mixtures . . . . . . . . . 516.4. Separation of Binary Azeotropic

Mixtures by Changing SystemPressure . . . . . . . . . . . . . . . . . 52

6.5. Separation of AzeotropicMixtures with an Entrainer(Azeotropic and Extractive Dis-tillation) . . . . . . . . . . . . . . . . . 53

6.5.1. Process Description . . . . . . . . . . 536.5.2. Choice of Entrainer . . . . . . . . . . 54

2 Distillation and Rectication

6.5.3. Entrainment Rectication ofAzeotropic Mixtures in Combina-tion with Other Separation Methods 55

7. Energy Economization in Recti-cation . . . . . . . . . . . . . . . . . . 60

7.1. Single Columns . . . . . . . . . . . . 617.1.1. Reduction of Heat Requirement . . 617.1.2. Reduction of Exergy Losses . . . . . 617.2. Optimal Separation Sequences for

Multicomponent Mixtures . . . . . 627.3. Column Coupling . . . . . . . . . . 667.3.1. Direct Coupling of Columns . . . . 667.3.2. Indirect Coupling of Columns . . . 687.3.2.1. Binary Mixtures . . . . . . . . . . . . 697.3.2.2. Ternary Mixtures . . . . . . . . . . . . 697.4. Design of Heat-Exchanger Net-

works . . . . . . . . . . . . . . . . . . 728. Design andDimensioning ofMass-

Transfer Equipment . . . . . . . . . 72

8.1. Types of Construction . . . . . . . . 728.1.1. Plate Columns . . . . . . . . . . . . . 738.1.2. Packed Columns . . . . . . . . . . . . 748.1.3. Criteria for Use of Plate or Packed

Columns . . . . . . . . . . . . . . . . . 768.2. Dimensioning of Plate Columns . 778.2.1. Operating Region of Plate Columns 778.2.2. Two-Phase Flow in Plate Columns . 808.2.3. Mass Transfer in the Two-Phase

Layer on Column Plates . . . . . . . 848.3. Dimensioning of Packed Columns 868.3.1. Operating Region of Packed

Columns . . . . . . . . . . . . . . . . . 898.3.2. Two-Phase Flow in Packed Columns 908.3.3. Calculation of Mass Transfer in

Packed Columns . . . . . . . . . . . . 959. References . . . . . . . . . . . . . . . 97

Symbolsa low boilera relative area of liquid vapor interface,

m1a ratio of surface area to volume of column

packing, m1a coefcient (equation of state),

Nm4mol2ae interface effective in mass transfer, m1A area, mass-transfer area, m2A Wilson coefcientA 1 absorberb medium boiler/high boilerb coefcient (equation of state), m3/molB amount of bottom product, kmolB second virial coefcient, kmol/m3B bottom stream (ow rate), kmol/sc medium boiler/high boilerc exponent of Reynolds number in Equa-

tion (8.56)C constantC third virial coefcient, (kmol/m3)2CG capacity factor, m/sCh constant in Equation (8.62)CG factor in Equation (8.80)C 1 columnc.p. critical pointd high boilerd diameter, mdN nominal diameter, mD diameter, m

D diffusion coefcient, m2/sD amount of overhead product, kmol (dis-

tillate)D overhead stream (ow rate), kmol/sDE dispersion coefcient, m2/se entrainerE efciencyE exergy, kJ/kmolE 1 heat exchangerf fugacity, PaF factor in Equation (8.3), kg1/2m1/2 s1F amount of feed, kmolF feed stream (ow rate), kmol/sFa factor in Equation (8.79)Fr Froude numberg acceleration due to gravity, m/s2g specic free enthalpy, kJ/kgG gas or vapor stream (ow rate), kmol/sG gas or vapor stream (ow rate) below the

feed point, kmol/sh height, mh holduph specic enthalpy, kJ/kghp height of pressure drop, mh0 holdup below the loading pointHij Henry coefcient for component i in sol-

vent j, PaH plate spacing, height of packing, mHOG height of a transfer unit with reference to

the vapor side, mk binary interaction coefcient


K equilibrium constantK mass-transfer coefcient, kmolm2 s1l length, path length, mL amount of liquid, kmolL liquid stream (ow rate), kmol/sL liquid stream (ow rate) below the feed

point, kmol/sm coefcient (equation of state), slope of

equilibrium curveM mixing stream (ow rate), kmol/sM molar mass, kg/kmoln mole quantity, kmoln number of equilibrium stages; exponent

in Equation (8.46)N number of transfer units, number of nodesN molar ow rate, kmol/sp pressure, Papi partial pressure of component i, Papi0 vapor pressure of pure component i, PaPe Peclet numberQ heat, kJQ heat ow rate, kJ/sQ molar heat, kJ/kmolr molar latent heat of vaporization, kJ/kmolR external reux ratioR gas constant, kJ kg1K1R universal gas constant 8.31451,

Jmol1K1R external reboil ratioRe Reynolds numbers plate thickness, mS number of saddle pointsS molar ow rate after the decanter, kmol/sS 1 decantert spacing, mt temperature, Ct time, sT absolute temperature, Kv specic volume, m3/kgV volume, m3V volumetric (ow rate), m3/sV molar volume, m3/kmolw supercial velocity, m/sW molar work, kJ/kmolWe Weber numberx mole fraction (liquid phase)x mole fraction (liquid phase) below the

feed pointy mole fraction (vapor phase)y mole fraction (vapor phase) below the

feed point

z mole fraction (vapor liquid mixture),number of spheres

Z compressibility factor

Greek Symbols contraction number, relative volatility,

coefcient (equation of state) mass-transfer coefcient, m/s porosity, holdup orice coefcient activity coefcientgE free excess enthalpy, kJ/kmolgid free enthalpy change for mixing of ideal

uids, kJ/kmolp pressure drop, N/m2 density difference, kg/m3 dynamic viscosity, N s/m2 contact angle kinematic viscosity, m2/s pole on the enthalpy concentration dia-

gram, kJ/mol density, kg/m3 surface tension, N/m contact time, sL liquid residence time in the in the two-

phase layer, s fugacity coefcient, caloric factor, rela-

tive free hole area empirical packing constant acentric factor

Subscripts and Superscriptsa low boilerac activeb medium boiler/high boilerB bottom productc medium boiler/high boiler, columnC condenser (cooler)cl clearance under downcomer (skirt clear-

ance)cr criticald downcomer, high boiler, dryD overhead productE entrainmentf froth layerF feedG vapor (gas)h holei component iid ideal


irr irrigatedj component jk highest boiling componentl component lL liquidm intermediatemax maximummin minimumn number of plates or stepsN nominalo orice, single particleOG overall gas-phase stateOL overall liquid-phase stateR reboilers solid matter, particle swarmsp spheret plate (tray)V valvew weir0 initial state innity equilibrium state liquid state, irrigated packing vapor state molar average, below the feed point

1. IntroductionSeparation of individual substances in a homo-geneous liquid mixture or complete fractiona-tion of such mixtures into their components isan important step in many production processes.Different separation procedures can be used forthis purpose, but distillation is the most impor-tant industrial method.

Distillation utilizes a very simple separationprinciple based on the development of intimatecontact between the homogeneousmixture and asecond phase, which thereby allows mass trans-fer to occur between phases. The thermody-namic conditions are chosen so that only thecomponent to be separated enters the secondphase. The phases are subsequently separated;one of them contains the desired substance, andthe other consists of a mixture that is largely freeof this substance.

Three steps are always involved in industrialimplementation of this separation principle:1) Creation of a two-phase system

2) Mass transfer between phases3) Separation of the phases

A large number of separation techniques uti-lize this very effective principle or modica-tions thereof. Absorption, desorption, evapo-ration, condensation, and distillation involve agaseous and a liquid phase; solvent extractionuses two liquid phases. Separating techniquesthat utilize a uid phase and a solid phase areadsorption, crystallization, drying, and leaching.

In most of these separations, the necessarytwo-phase system is created by adding an aux-iliary phase to the mixture; the diluted sub-stances to be separated collect with the auxiliaryagent. However, in distillation, the second phaseis produced by partial vaporization of the mix-ture. Hence, the use of an auxiliary substance,which usually requires laborious recovery, canbe avoided, and the components to be separatedcan be recovered as pure substances. Indeed, dis-tillation requires energy only in the form of heat,which can subsequently be removed from thesystem, this is another important advantage.

Historical. Distillation is a very old sepa-ration method; chemists pioneered its use inAlexandria in the rst century a.d. [7]. Aroundthe eleventh century, distillation was used forthe rst time in northern Italy to produce alco-holic beverages. The development of distillationequipment has been inuenced tremendously bythis eld of application. Distillation equipmenttaken from The Alchemy of Andreas Libav-ius dating from 1597 is illustrated in Figure 1[8]; it was used for the batch distillation of al-cohol. Heat was applied to the liquid contentsof the boiler (a) built into the oven (b), and thevapor formed was allowed to condense in twocoolers (c). Cooling water was changed period-ically. The only visible process was the conden-sate dripping into the receiver (d). Hence, thisseparation technique was named after the Latinword destillare, which means dripping or trick-ling down.

Even in early times, the fact was known thatan even higher alcohol content could be attainedby using a second distillation step. In the appa-ratus shown in Figure 1 two distillations couldbe carried out simultaneously. Condensate fromthe rst distillation was returned to the head-piece (e), the so-called Recticatorium, which


was heatedwith vapor rising from the boiler. Thevapor thus produced was also condensed in thetwo coolers (c). A liquid with a higher alcoholcontent was then collected in a second receiver(f). The term rectication is derived from thisprocess which, especially in Europe, is used todescribemultistage distillation. The Latinwordsrecte faceremean to rectify or improve. Indeed,to this day, rectication refers to the process bywhich a further concentration change is achievedafter the rst evaporation step.

Figure 1. Distillation and rectication equipment takenfrom The Alchemy of Andreas Libavius, 1597 [8]a) Boiler; b) Oven; c) Coolers; d) Receiver; e) Headpiece;f) Receiver

The vapor and liquid are brought into inti-mate contact by countercurrent ow and massexchange occurs because the two phases are notin thermodynamic equilibrium. The phases pro-duced during rectication are formed by evap-oration and condensation of the initial mixture.The separation of a liquid mixture into its purecomponents can be controlled solely via the heatsupply.

The basis for planning rectication is aknowledge of the vapor liquid equilibrium.The separation achieved depends primarily onthe concentration of the individual substancesin the vapor and liquid phases. The principlesof vapor liquid equilibrium are discussed inChapter 2; special attention is paid to the equi-libria of ternary and higher mixtures.

Thermodynamic analysis of distillation andrectication is essential to establish the optimalconditions for mass transfer. The decisive fac-tor is the driving force for mass transfer, i.e., thedifference between the actual concentrations ofthe substances and their equilibrium concentra-tions. Operating conditions should ensure thatthis difference is sufciently large. Appropri-

ate relationships and methods are described inChapters 3, 4, and 5. Examples of industriallyimportant separation processes and energy re-quirements are discussed in Chapters 6 and 7,respectively.

In distillation and rectication, separation isachieved by bringing two phases into intimatecontact. Hence, in practice, the problems causedby multiphase ow and mass transfer betweenphases must be confronted. Although the math-ematical interrelationships of multiphase owhave been studied systematically from the view-point of uid mechanics, often only empiricalresults are available for practical equipment de-sign, as described in Chapter 8.

2. Vapor Liquid EquilibriumIf vapor and liquid are in intimate contact for along period of time, equilibrium is attained bet-ween the two phases. Thismeans that no net owof heat, mass, or momentum occurs across thephase boundary. The complete formulation fora system at equilibrium is as follows:

Thermal equilibrium T =T Mechanical equilibrium p =pComposition equilibrium yi = f (xi. . .)

where T is the absolute temperature, p thepressure, yi the mole fraction of component iin the vapor phase at phase equilibrium, and xithe mole fraction of i in the liquid phase. Singleand double primes denote the liquid and vaporphases, respectively. The mass equilibrium bet-ween vapor and liquid is of primary importancein separation by distillation and rectication.

2.1. Empirical Findings inVapor Liquid Equilibrium

The most important empirical ndings pertain-ing to the vapor liquid equilibrium of binaryand ternary mixtures are discussed below.

2.1.1. Binary Mixtures

Experimental phase equilibriameasured atmod-erate pressure for binary systems composed ofsubstances a and b are shown in Figure 2 [9].


In the upper row, total and partial pressures areplotted vs. themole fraction of substance a in theliquid (xa) at constant temperature. Substance ais more volatile than substance b and thus has ahigher vapor pressure and a lower boiling point.The middle row shows boiling points (boilingpoint or bubble point line) and condensationtemperatures (dew point line) plotted against theliquid and vapor compositions at constant pres-sure. The lower row is the concentration of a inthe vapor phase (ya) plotted vs. its concentrationin the liquid phase (xa) at constant pressure.

Signicant differences in the behavior ofthese binary systems at equilibrium are due tothe varying forces of interaction between thetwo molecular species a and b. Mixtures inwhich molecules of a and b repel each otherare shown on the left side of Figure 2. Bothsubstances undergo an increase in partial pres-sure p of the liquid phase. According to theClausius Clapeyron relationship, this increasein vapor pressure leads to a decrease in boilingpoint. In the 1-butanol water system, repulsiveforces between the molecules are so strong thatseparation into two liquid phases occurs at in-termediate concentration ranges.

Binary mixtures shown on the right side ofFigure 2 exhibit exactly the opposite behavior.Molecules of substances a and b attract eachother and tend to form complexes. This leadsto a decrease in partial pressure p and thus to anincrease in boiling point.

In the binary mixture benzene tolueneshown in the center of Figure 2, forces of in-teraction between the molecular species a andb are negligible. A molecule a does not knowwhether it is collidingwith an identicalmoleculeor with a molecule b of the other substance.Suchmixtures are known as ideal mixtures; theirphase equilibria can be represented by very sim-ple equations (see Section 2.2).

The conditions at which the vapor and liq-uid compositions are equal (i.e., y= x) are veryimportant. Such mixtures, known as azeotropicmixtures, have a minimum or a maximum boil-ing point. In a minimum-boiling azeotrope themolecular species repel each other, whereas ina maximum-boiling azeotrope they attract eachother. If the azeotrope occurs in the composi-tion range in which the two liquids are immisci-ble, phase splitting occurs and a heterogeneous

azeotrope is formed. A heterogeneous azeotropeis always minimum boiling.

Amixture of close-boiling components tendsto form an azeotrope when the chemical struc-tures of the components are dissimilar. Binarymixtures with two azeotropes (one minimumand one maximum) have been observed occa-sionally, e.g., peruorobenzene benzene (cf.Fig. 6H).

The inuence of pressure on the phase equi-librium of a mixture of nitrogen and methane isshown in Figure 3A. At higher pressures, differ-ences between the concentrations of the vapor yand the liquid x decrease markedly. When thecritical pressure of the pure substances is ex-ceeded, they can no longer be liqueed in thesupercritical region. However, mixtures of su-percritical vapors can split into two phases, asshown in Figure 3B. The region in which thiscan occur is restricted by the convergence pres-sure curve, which represents the critical pointsof mixtures of the two substances and may passthrough a maximum.

2.1.2. Ternary Mixtures

The phase equilibrium of ternary mixtures is de-picted conveniently on a triangular diagram, inwhich the corners represent the pure substancesa, b, and c.

In Figure 4, the mole fraction of component iin the liquid, xi, is represented by the basic gridand the vapor mole fraction yi in equilibriumwith the liquid is shown as parameter curves. Forideal mixtures with constant relative volatilitiesik (see Section 2.2.2), these parameter curvesare straight lines. However, for nonideal systemsplots of yi do not yield straight lines, and theentire representation becomes confusing. In ad-dition, the azeotropic points, which are very im-portant in rectication, are difcult to recognize.

The use of distillation curves is more effec-tive. If an initial liquid concentration of com-ponent i, xi0, is assumed, the equilibrium vaporconcentration yi0 is determined. The vapor isthen assumed to be totally condensed. Thus, anew liquid of composition xi1= yi0 is obtainedandmarked on the triangular diagram.The vaporstate yi1 in equilibriumwith this liquid state xi1is, in turn, determined and so on. A series of liq-uid states xi1, xi2, xi3, . . . , is nally obtained; the


Figure2.Ph

aseequilib

riaof

vebinary

mixturesc

omposedofsubstances

aandb[9]

Substanceaismore

volatilethan

substanceb.

Thedashed

lines

intheupper

row

representideal

behavior.Temperature

andvapor

concentrationequilib

ria(m

iddleandbotto

mrows)weremeasuredatconstantp

ressure(0.

1MPa).Sh

aded

area

representstheliq

uidim

misc

ibilitycompositionrange.A

zeotropicpointsareindicatedby

opencircles.

p a=partialpressure

ofa

;pb=partialpressure

ofb

;xa=molefra

ctionofa

inliq

uidphase;y a=molefra

ctionofa

invapor

phase


Figure 3. Inuence of pressure p on the phase equilibrium of nitrogen and methaneA) Boiling point lines () and dew point lines ( ) of nitrogen methane mixtures; xN2 =mole fraction of nitrogen inthe liquid phase; yN2 =mole fraction of nitrogen in the vapor phase; B) Critical points of nitrogen (a), methane (b), and theirmixtures (convergence pressure curve )

composition difference in each case representsan equilibrium step. These liquid states can bejoined together to give a distillation curve. Thisprocess leads to step-by-step enrichment of themore volatile component, which is accompaniedby a decrease in boiling point.

Several distillation curves are shown in Fig-ure 5. The direction of boiling point decrease isindicated by an arrow. The sides of the triangle,which represent binary mixtures, can also be in-terpreted as distillation curves. The dots on oneof the curves represent the sequence of distinctliquid states from one equilibrium stage to thenext. Large spaces between points depict regionswith large differences in concentration betweenthe liquid and the corresponding vapor.

Distillation curves for nine ternary systemsare shown in Figure 6. In each case, the lowestboiling substance (a) is at the top of the triangle,the medium boiler (b) is at the right side, and thehigh boiler (c) is at the left.

Mixture A exhibits almost ideal behavior.The distillation curves proceed from the high-boiling substance c to the low boiler a. At thesame time, themedium-boiling component b ex-hibits a more or less pronounced concentrationmaximum.

A binary minimum azeotrope between thelow- and medium-boiling substances is formedin mixture B. Indeed, the lowest boiling tem-perature of the system is at this azeotrope. Asa result, all distillation curves starting from thehigh boiler c tend toward this binary azeotrope.Analogous behavior is observed with mixtureC in which a binary maximum azeotrope occursbetween themedium-boiling substance b and thehigh boiler c. The highest boiling temperatureof the system is at this azeotrope. All distillationcurves start with this temperature maximum andend at the temperature minimum, i.e., at the lowboiler a.

Mixture D shows a binary a b maximumazeotrope. The distillation curves, starting from


Figure 4. Representation of the phase equilibrium of the ternary mixture nitrogen argon oxygen at 131 kPaThe uniformly divided grid of the triangular diagram represents the liquid composition xAr, xO2 , and xN2 . The bold anddashed lines indicate equilibrium vapormole fractions of oxygen (yO2 ) and argon (yAr), respectively. Temperatures indicateboiling points of the pure components.

the high boiler c, rst proceed toward thisazeotrope and then turn toward the low boilera or the medium boiler b. Thus, this ternarysystem is divided into two areas whose distil-lation curves have different end points. The dis-tillation curve that forms the boundary betweenthese two areas is called the distillation bound-ary. In this case, the boundary runs from the purehigh-boiling substance c to the binarymaximumazeotrope.

A distillation boundary also arises when abinary mixture of a medium-boiling substanceb and a high boiler c produces a minimumazeotrope (mixture E). In this case, a heteroge-neous azeotrope is formed because it lies withinthe liquid immiscibility range. The distillationcurves for mixture E have two starting points,but they all end at the low-boiling substance a.The distillation boundary runs from the hetero-geneous azeotrope to the low boiler a.

Two binary minimum azeotropes occur inmixture F where the distillation boundary runsbetween the two azeotropes. This ternary sys-tem is thus divided into two areas, the distilla-tion curves in each area having different startingpoints.

In mixtures G and H in Figure 6, a saddle-point ternary azeotrope is formed in addition tothe two azeotropes. Distillation curves do notstart or end at the saddle point. Instead, they pro-ceed from different directions toward this pointand then turn away. Distillation boundaries di-vide both ternary systems into four distillationelds. The binary mixture of the low boiler aand the medium-boiling substance b in mixtureH forms a minimum- and a maximum-boilingbinary azeotrope.


Figure 5. Representation of the phase equilibrium of theternarymixture nitrogen argon oxygen at 131 kPa by dis-tillation curvesThe distillation curves start at the component with the high-est boiling point and end at the component with the lowestboiling point. The direction of decreasing temperature is in-dicated by arrows. Distinct equilibrium states are shown onone curve as dots.

An immiscibility range with a heterogeneousazeotrope and two binary minimum azeotropesoccur in system I. In addition, a ternary min-imum azeotrope with the lowest boiling tem-perature of the entire system is formed. Thus,all distillation curves originate at the pure sub-stances and converge at the ternary azeotrope.The azeotropes are connected by distillationboundaries giving rise to three distillation elds.

Distillation curves facilitate clear repre-sentation of the phase equilibria of ternary sys-tems. Reliable vapor liquid equilibrium dataare essential for planning distillation and recti-cation processes. As shown in Figure 5, determi-nation of the distance fromone equilibriumstageto the next is also very useful. However, knowl-edge of the boiling temperatures of the points in-dicated (pure substances and azeotropes) and ofthe directional arrows on the distillation curvesis usually sufcient for process design. The po-sition of the distillation boundaries, which di-vide the ternary system into different elds, is ofconsiderable importance. These boundaries al-ways connect points of minimum or maximumtemperature, which are usually the high boiler,

the low boiler, and binary or ternary azeotropes.Knowledge of these points alone may be suf-cient to estimate the approximate positions ofdistillation boundaries; phase equilibrium dataare not always required.

The boiling points of a ternary mixture canbe represented three-dimensionally where thedistillation boundaries form either a ridge ora valley. Graphical representation of the con-tours of the temperature surface as a func-tion of liquid composition in the systemacetone chloroform methanol (G in Fig. 6) isshown in Figure 7. The distillation boundariesintersect at a saddle-point indicating a ternaryazeotrope.

The existence of ternary azeotropes is of con-siderable importance and can be checked by us-ing the following equation [10]:

2 (N3S3) +N2S2+N1= 2 (2.1)

whereN is the number of nodes atwhich distilla-tion curves either start or end, and S is the num-ber of saddle points at which distillation curveseithermeet or split, usually combinedwith a sud-den change of direction. The subscripts 1, 2, and3 indicate a pure substance, a binary mixture,and a ternarymixture, respectively. For example,mixture A in Figure 6 exhibits only two nodesfor pure substances. Thus N1= 2 and Equation(2.1) balances. The distillation curves in mix-ture I start at all three pure substances; thusN1= 3. All three binary azeotropes are saddlepoints at which two distillation curves meet andthen break away by changing direction. There-fore, S2= 3 and Equation (2.1) becomes

2 (N3S3) +0 3 + 3 = 2 (2.2)

In other words, ternary azeotropes must exist.The simplest solution to this equation is N3= 1and S3= 0; i.e., one ternary azeotrope exists. Sev-eral ternary azeotropes can occur in this systemonly if ternary saddle points exist at the sametime, which however is not the case.

Analogously, for mixture G in Figure 6,N1= 1 and N2= 3; the following equation is ob-tained:

2 (N3S3) +3 0 + 1 = 2 (2.3)

The simplest solution to this equation isS3= 1. In other words, one ternary saddle point


Figure 6. Phase equilibria of nine ternary mixtures at 101.3 kPaThe distillation curves proceed from a temperature maximum to a temperature minimum. Distillation boundaries are repre-sented by bold lines. Shaded areas indicate the presence of two liquid phases. Open circles denote azeotropic points.


exists, which corresponds to the experimentalndings.

The preceding examples show that Equation(2.1) can be employed to predict the existenceof ternary azeotropes by using a knowledge ofbinary marginal mixtures. Although several so-lutions may exist, the simplest is probably cor-rect.

Figure 7. Boiling point surface as a function of liquidcomposition for the ternary acetone chloroform metha-nol systemTemperatures indicate boiling points of pure substances andazeotropic points.

2.2. Empirical Laws of Vapor LiquidEquilibrium

Vapor liquid equilibria are best understoodwith the help of simple, empirical laws whichmay, however, only apply within certain limits.

Daltons law describes vapor systems:

pi /p=ni /n

=V i /V (2.4)

where p is the pressure, n the number ofmolesof substance, and V the volume of the vapor.The subscript i denotes component i. Becauseni/n = yi (where yi is themole fraction of com-ponent i in the vapor), this law can be expressedas:

pi =yi p (2.5)

Daltons law generally holds at low pressure(p pcr, where pcr is the critical pressure). At

higher pressure, however, it may require modi-cation. Instead of using pressure p, the equationis written in terms of fugacity f :f i =yi f (2.6)

The fugacity f can be determined from the fu-gacity coefcient = f/p (Section 2.4).

According to Raoults law, the partial pres-sure pi of component i from a liquid mixtureis

pi=xi p0i (2.7)

Hence, the partial pressure pi depends only onthe vapor pressure pi0 of pure component i andits liquid mole fraction xi; it is not affectedby the nature of other substances in the liquidmixture. Raoults law holds only as long as theforces of interaction between different types ofmolecules are equal (ideal mixture). Nonidealliquid-phase behavior (i.e., unequal interactionforces) is described by using the activity coef-cient i, which is dened as follows:

pi=i xi p0i (2.8)

The activity coefcient i is highly concentra-tion-dependent. Methods used to determine ac-tivity coefcients are described in Section 2.3.

Equations (2.7) and (2.8) apply only attemperatures below the critical temperature(T T cr, Henrys law is applied insteadof Raoults law:

pi=Hij xi (2.9)

whereHij is the Henry coefcient for substancei in solvent j. Again, a correction must be madefor nonideal mixtures by using the activity coef-cient i:

pi=Hij i xi (2.10)

This relationship is analogous to Raoults law,except that the proportionality constant is notthe vapor pressure pi0 , but Henrys coefcientHij .


2.2.1. Equilibrium Constant

The equilibrium constant for component i,Ki, iscommonly used to represent phase equilibriumdata, especially in multicomponent mixtures. Itis dened as

Ki = yi /xi (2.11)

Because at phase equilibrium

pi =pi (2.12)

then from Equations (2.5) and (2.7),

Ki=p0i /p or K=p0i i/p (2.13)

Similarly, at temperatures above the criticalpoint,

Ki=Hij/p or K=Hij i/p (2.14)

Both the vapor pressure pi0 and Henrys coef-cientHij are strongly dependent on temperature.Hence, the equilibrium constant Ki can be em-ployed only at known temperatures.

2.2.2. Equilibrium Curve

For binary systems consisting of components aand b,

ya=pa

pa+pb=

xa p0axa p0a+xb p0b

(2.15)

Since the vapor pressures of the pure com-ponents p0a and p0b are highly temperature-dependent, this equation applies only at constanttemperature.However, the ratiop0a/p0b is often in-dependent of temperature; therefore, vapor liq-uid equilibria are frequently expressed in thisform as the relative volatility :

ab=p0a/p0b (2.16)

From the equation xb= 1 xa,

ya=ab xa

1+ (ab1) xa(2.17)

This relationship is called the equilibrium curveand is valid when the temperature is constant.If ab does not vary with temperature, Equation

(2.17) can also be employed at other temper-atures at constant pressure; this is particularlyimportant in distillation and rectication.

Similarly, for ternary systems consisting ofcomponents a, b, and c:

ya=pa

pa+pb+pc=

xa p0axa p0a+xb p0b+xc p0c

(2.18)

Substituting ac = p0a/p0c , bc = p0b/p0c , andxc = 1 xa xb givesya=

xa ac1+ (ac1) xa+ (bc1) xb

(2.19)

yb=xb bc

1+ (ac1) xa+ (bc1) xbThe general equation for multicomponent mix-tures is

yi=ik xi

1+k1j=1

(jk 1

) xj(2.20)

for component 1 to k 1 where k refers to thehighest boiling substance. In nonideal systems,the activity coefcient i must be incorporatedinto the equations.

2.2.3. Distillation Curve

The distillation curves of ideal mixtures are veryimportant in representing the phase equilibria ofternary mixtures and can be expressed by a sim-ple equation. Rearrangement of Equation (2.17)gives the equilibrium curve of ideal mixtures inthe following form:

y01y0

= x01x0

(2.21)

where the subscript 0 denotes the initial state.Total condensation of the vapor arising from

the liquid is assumed in determining the paths ofdistillation curves (see Section 2.1.2). A new liq-uid with the concentration x1= y0 is obtained.Thus:

x1

1x1= x0

1x0(2.22)

Similarly, x2= y1 is valid for the second step,giving:

x2

1x2= x1

1x1=2 x0

1x0(2.23)


For the nth step,xn

1xn=n

x0

1x0(2.24)

Rearrangement gives

xn=n x0

1+ (n 1) x0(2.25)

This equation is similar to the equilibrium rela-tionship (Eq. 2.17), except that here the term nreplaces . Thus, for ternary mixtures,

xan=nac xa0

1+ (nac1) xa0+(nbc1

) xb0 (2.26)xbn=

nbc xb1+ (nac1) xa0+

(nbc1

) xb0Similarly, formulticomponentmixtures, the dis-tillation curve is represented by

xin=nik xi0

1+k1j=1

(njk 1

)xj0

(2.27)

The distillation curves of ideal mixtures with aconstant relative volatility ik can easily be cal-culated by using these equations.

2.3. Calculation of Phase Equilibriumfrom Excess Enthalpy

Deviations from ideal behavior occur in the liq-uid phase rather than in the gas phase. As a resultof smaller intermolecular distances, the forcesof interaction between molecules in the liquidare considerably stronger. In contrast, the vaporphase can be assumed to be ideal at moderatepressure.

Real liquid-phase behavior is usually de-scribed by means of the activity coefcient (Eq. 2.8). However, the activity coefcients ofindividual components i of a liquid mixture arenot independent of each other, as formulated bythe Gibbs Duhem equation [11]:

xi dlni= 0 (2.28)

For binary mixtures, the Gibbs Duhem equa-tion requires that both activity coefcients areeither< 1 or> 1 and that they exhibit oppositelydirected gradients when plotted as a function ofthe concentration.

Equation (2.28) is a fundamental thermody-namic equation which must always be satisedand has long been used to check the thermody-namic consistency of vapor liquid equilibriumdata [11].

To obtain a relationship for the concentrationdependency of the activity coefcient i, start-ing with one value of state of the entire mix-ture is useful. In this way, a thermodynamicallyconsistent formulation of phase equilibrium isattained automatically. Mixing processes can bedescribedmost simply by using the free enthalpyg [11]. The free enthalpy change for the mixingof ideal liquids, gid, is

g = RT

xi lnxiid (2.29)

where R is the universal gas constant. Analo-gously, for nonideal mixtures,

g = RT

xi lnxi+RT

xi lni (2.30)

The second term accounts for the nonideal be-havior of themixture and is called the free excessenthalpy gE:

g = RT

xi lniE (2.31)

Hence,

g= gid+gE (2.32)

Differentiation of the equation for the free ex-cess enthalpy gE gives relationships for theactivity coefcients i of the individual compo-nents of a mixture. In general,

lni=1

RT

(gE+

gE

xi

nk=1

xk gE

xk

)(2.33)

For a binary mixture of substances a and b,

lna=1

RT

(gE+

gE

xa xa g

E

xa

)(2.34)

Using xb= 1 xa nally gives

lna=RT(gE+xb

gE

xa

)(2.35)

Similarly,

lnb=RT(gExa g

E

xa

)(2.36)


Relationships for the free excess enthalpygE,which satisfy the condition gE = 0 for xi = 1,are required to calculate the activity coefcienti of a mixture from Equation (2.35). Empiricalcorrelations containing two or three correlatingparameters are generally used for binary mix-tures [11]. However, improved theoretical for-mulations, usually based on the concept of localcomposition or on the incrementmethod, are be-ing employed increasingly. An example is theWilson equation for a binary mixture [11]:gE

RT = [xa ln (xa+Aab xb) +xb ln (xb+Aba xa)] (2.37)

where Aab and Aba are the Wilson coefcientsfor the binary pair a b. Hence, the activity co-efcients can be represented by

lna= 1 ln (xa+Aab xb) xaxa+Aab xb

Aba xbxb+Aba xa

lnb= 1 ln (xb+Aba xa) xbxb+Aba xa

Aab xaxa+Aab xb

(2.38)

The forces of interaction between molecules ofa and b can be determined from a few measuredvalues and are described by the coefcients Aaband Aba; these coefcients depend only slightlyon pressure.

The Wilson equation for a ternary mixture is

gE

RT = xa ln (xa+Aab xb+Aac xc)xb ln (Aba xa+xb+Abc xc)xc ln (Aca xa+Acb xb+xc) (2.39)

Therefore, the activity coefcient of substance ais

lna= 1 ln (xa+Aab xb+Aac xc) xaxa+Aab xb+Aac xc

Aba xbAba xa+xb+Abc xc

Aca xcAca xa+Acb xb+xc

(2.40)

The activity coefcients of substances b and care obtained analogously.

The Wilson equation for a multicomponentmixture consisting of n components is

gE

RT = ni

xi ln n

j

Aij xj ;

whereAii= 1 (2.41)

Hence, the activity coefcients are given by

lni= 1 ln n

j

Aij xj

nl

Ali xlnjAlj xj

(2.42)

Even in ternary and higher systems, the Wilsonequation contains only the interaction parame-tersAij for the binarymixtures involved. Hence,prediction of the phase equilibria of multicom-ponent systems frombinary data is possible. TheWilson equation has proved itsworth in practice;the majority of distillation curves presented inFigure 6 were calculated by using this equation.However, unlike the Uniquac (UNiversal QUA-siChemical) equation [12], it is not applicable tosystems that undergo phase splitting. Systems Eand I in Figure 6 were calculated with the Uni-quac equation.

2.4. Calculation of Phase Equilibriumfrom Equations of State

The method for calculating phase equilibriumfrom the excess enthalpy, described in Section2.3, can be applied to systems that are nonidealin the liquid phase only. However, because Dal-tons law does not hold at higher pressures, thevapor phase may also deviate from ideal behav-ior. An effective way of calculating phase equi-libria at high pressure is based on equations ofstate for the uid, which describe both the vaporand the liquid phases with sufcient accuracy.Partial fugacities fi are used instead of partialpressures pi. At equilibrium,

f i =fi (2.43)

where f i and f i are the fugacities of componenti in the vapor and liquid phases, respectively. Byanalogywith Equation (2.6) and using the fugac-ity coefcientsi = fi/pi, the following equationis obtained:


i yi p=i xi p (2.44)

Hence, the equilibrium constant Ki is

Ki=i/i (2.45)

Thus, the equilibrium constant can be calculatedfrom fugacity coefcients determined in the twophases. The following rigorous relationship isobtained by using the laws of thermodynamics[11]:

RT lni=p0

(Vi RT/p

)dp (2.46)

where the partial molar volume

Vi= (V /ni)p,T ,nj =i

and the total volume

V=Vi

ni

The molar volume V and the derivative V /n1are determined by using a suitable equation ofstate.

The real behavior of pure uids can be de-scribed by the compressibility factor Z:

Z=pV /(RT

)(2.47)

For ideal vapors, Z = 1. Deviations from thisvalue are directly proportional to the degree ofnonideal behavior.

Two different types of equations of state de-scribe the real behavior of pure uids. One isbased on the van der Waals equation:

Z=V /(V b

)a/

(RT V

)(2.48)

The coefcients a and b can be interpreted as re-presenting interaction forces and molecular vol-ume, respectively.

The second type of expression is based on thevirial equation

Z= 1+B/V +C/V 2+... (2.49)

Intermolecular forces are described by the virialcoefcients B, C, etc., which for pure uids area function only of temperature.

Both types of equations have been improvedby numerous modications to obtain a pres-sure volume temperature equation that is ap-plicable to both vapor and liquid phases. Equa-tions of state used to determine phase equilibriamust correctly represent mixtures of substancesand must include equations for the pure com-ponents as special cases. Equations of state formixtures are generally derived fromequations ofstate for the pure substances by using concen-tration-dependent coefcients. However, theo-retical methods are not available, and empiri-cal rules of combination are used. In principle,equations for excess enthalpy (see Section 2.3)can be applied successfully to the formulationof mixing rules [13].

The procedure can be illustrated by usingthe Soave modication of the Redlich Kwongequation (Eq. 2.48) [14]:Z=V /

(V b

)a/

(RT

(V +b

))(2.50)

The coefcients a and b can be determined fromthe critical values pcr, T cr, and the acentric fac-tor of the pure substances.

a= 0.42747R2 T 2cr/pcrwhere =

[1+m

(1T/Tcr

)]2m= 0.48 + 1.574 0.1762

b= 0.08664RTcr/pcr (2.51)

The following mixing rules apply to mixturesof components i and j:a=

i

j

xi xj aij

Here,

aij=aiiajj (1kij) , where kii= 0 (2.52)

kij for i = j can be found in tables [15].b=

xi bi

Substituting Equation (2.50) in Equation (2.46)gives

lni=[bi/(bRT

)][pv RT

]

ln[p/(RT

) (v b)]

[a/(bRT

)] [2/a (xj aij)bi/b

] ln [(v + b) /v] (2.53)


Trial values for the concentration andmolar vol-ume of each phase must be estimated to solvethis equation for each phase. Equation (2.50) isused to improve the values during iteration. Theequilibrium constant K for each component canthen be calculated by using Equation (2.45) tocheck the assumed phase concentrations. Thisiteration method often converges to trivial solu-tions. However, other available procedures re-quire good starting values close to those of thesolution [16].

2.5. Practical Determination of PhaseEquilibria

Reliable phase equilibrium data are essentialfor the design of distillation and recticationoperations. Despite the highly advanced stateof mixed-phase thermodynamics, good exper-imental data should be used when possible.Several thousand publications on vapor liquidequilibria are available which include completedescriptions of the p, v, T, x behavior of a sys-tem. However, the majority provide equilibriumconcentrations only over a certain range of tem-perature and pressure. Useful data on azeotropiccompositions are readily available [17], mainlyin the form of data collections [15], [1823].Older data collections consist merely of compi-lations of measured values; however, newer dataare subjected to a consistency test. In addition,coefcients for several equations of state or forequations for excess enthalpy are presented in[15], [23].

Data for many binary mixtures can be foundin the above collections. Available data forternary or higher mixtures over the concentra-tion range of interest are seldom adequate. Atthe very best, data are provided for binary mix-tures present in the system. Usually, equilibriumdata are far from complete. In all these cases,the methods of thermodynamic representationof phase equilibria described in Sections 2.3 and2.4 must be used.

Equations for excess enthalpy are useful atlow pressure and when nonideal behavior oc-curs, mainly in the liquid phase. Equations thatuse binary interaction parameters to calculatethe phase equilibrium of multicomponent mix-tures are especially useful. These are primarily

equations based on the concept of local com-position. Furthermore, the incremental methodpermits approximation of the interaction param-eters from the chemical structure of the sub-stances involved [24].

In high-pressure systems, the liquid phaseand especially the vapor phase exhibit nonidealbehavior. The convergence pressuremethodwasoften used in the past [25] but has now beensuperseded by the more effective equations ofstate. Equations of state based on the virial equa-tion have the advantage of being theoreticallybased for mixtures. However, these equationsare valid only at reduced pressure, i.e., p/pcr< 0.5. Equations of the van der Waals type arepreferred at higher pressure. Apart from empir-ical mixing rules, almost all equations of statecontain a further binary correlating parameter(kij in Eq. 2.52), which greatly inuences phaseequilibrium.

Equations of state are preferred to equationsfor excess enthalpy, because the former per-mit calculation not only of phase equilibria butalso of other values of state, e.g., density andenthalpy. Empirical mixing rules are not com-pletely satisfactory because they are usually ap-plied to both phases in the same way; this doesnot correspond fully to real conditions in multi-componentmixtures. The vapor liquid equilib-rium cannot yet be reliably predicted. Progressin mixed-phase thermodynamics permits only athermodynamically consistent representation ofequilibria based on relatively few experimentaldata.

3. Distillation

As discussed in Chapter 2, the composition ofthe vapor usually differs from that of the liquidfromwhich it originates. This concentration dif-ference between the two phases is the basis ofdistillation, which is a simple method of sepa-rating substances. The method involves partialevaporation of the liquid followed by conden-sation of the vapor. The resulting condensate isenriched in the more volatile components, in ac-cordance with the phase equilibrium for the sys-tem at hand. Thus, the starting mixture or feed isseparated into two fractions with different com-positions.


3.1. Continuous Distillation

Figure 8 represents the continuous distillation ofa liquid mixture. The feed (molar ow rate F) isseparated into an overhead product or distillate(molar ow rate D) and a bottom product (mo-lar ow rate B). The composition difference bet-ween these two products depends on the phaseequilibrium.

Figure 8. Continuous distillation of a binary mixture a bA) Flow diagram showing symbols for total molar streamsor ow rates ( F, L, etc.) and mole fractions of the morevolatile component a (xF, xD, etc.); B) A y x diagramshowing equilibrium and operating linesThe y x diagramcan be used to determine the composition of the overheadand bottom product streams ( D and B, respectively).

3.1.1. Continuous Distillation of BinaryMixtures

The material balance around the boiler (seeFig. 8) yields

F=G+L (3.1)

where F, G, and L are the molar ow ratesof the feed, vapor, and bottom liquid, respec-tively. Similarly, the material balance for themore volatile feed component a of a binary mix-ture a b gives

F xF=Gy+Lx (3.2)

where xF is the mole fraction of a in the feed, yis the mole fraction of a in the vapor, and x is themole fraction of a in the bottom liquid. Thus,

y= L/Gx+(1+L/G

)xF (3.3)

This is the equation for a straight line with slope L/ G.When plotted on the y x diagram, it fur-nishes the operating line. The operating line in-tersects the y= x diagonal at x= xF (Fig. 8 B).

With the closelymet assumption that equilib-rium exists between the vapor G and the liquid Lin the boiler, (i.e., y= y), the following equationholds for ideal mixtures (see Section 2.2.2):

y=x

1+ ( 1) x (3.4)

where is the relative volatility.The parameters x and y can be determined by

equating relationships (3.3) and (3.4) to obtaina quadratic equation which can be solved eas-ily. However, a graphical solution based on they x diagram is usually preferred because it is avisual method that can also be easily applied tononideal mixtures. The desired concentrationsof vapor y and liquid x can be determined fromthe point of intersection of the operating lineswith the equilibrium line. Subsequent total con-densation of the vapors is not accompanied by achange in concentration; i.e., the distillate con-centration xD is the same as the vapor concen-tration y. This is represented graphically in they x diagram by the intercept of the horizontalline between y and xD (i.e., y is constant) withthe diagonal. Figure 8B clearly shows the large


concentration difference between the heads xDand the bottoms xB. The decisive factor here isthe position of the equilibrium curve, i.e., forideal systems, the relative volatility . Gener-ally, only a limited difference between the con-centrations of the two fractions can be obtainedby simple continuous distillation.

3.1.2. Continuous Distillation ofMulticomponent Mixtures

The calculations for multicomponent systemsare identicalwith those used for binarymixtures.The material balance for the apparatus gives theoperating line for each substance i:

yi= L/Gxi+(1+L/G

)xFi (3.5)

where xFi is the mole fraction of i in the feed.The phase equilibrium is effectively dened byusing the equilibrium constant Ki. Thus,

yi=Ki xi (3.6)

For ideal mixtures, the equilibrium constant Kiis a function only of absolute temperature T,which is at rst unknown. Combining Equations(3.5) and (3.6) gives the following relationshipfor each component i:

yi=xFi

(1+L/G

)1+ L/G

Ki(T )

(3.7)

From the mole fraction constraint

yi = 1,

ki=1

xFi (1+L/G

)1+ L/G

Ki(T )

= 1 (3.8)

This set of equations must be solved iteratively.A trial value for the boiling temperature is rstassumed, and the equilibrium constant Ki foreach component is then determined. The estima-tion is checked and corrected by using Equation(3.8). The individual summation terms of Equa-tion (3.8) give the vapor concentration yi of eachcomponent involved.

Iterations involving the boiling temperaturecan be avoided by using the relative volatil-ity ik, instead of the equilibrium constant Ki.Equation (3.7) is then solved for Ki, the equilib-rium constant, as follows:

Ki=L/G

xFi/yi (1+L/G

)1

(3.9)

This relationship applies to each component ofthe mixture; it also applies to the highest boilingsubstance k.

Kk=L/G

xFk/yk (1+L/G

)1

(3.10)

Using the relationship ik =Ki/Kk gives

ik=xFk/yk

(1+L/G

)1

xFi/yi (1+L/G

)1

(3.11)

Solving for yi and employing the mole fractionconstant

yi = 1 yield

ki=1

ik xFi(1+L/G

)xFk/yk

(1+L/G

)+ik 1

= 1 (3.12)

This relationship is equivalent to Equation (3.8)but employs the gas concentration yk of the high-est boiling component as the iteration variableinstead of the boiling temperatureT . Calculationof this method is identical with that of Equation(3.8).

Figure 9.Continuous distillation of a mixture of butane (a),pentane (b), hexane (c), and heptane (d)xDi = liquidmole fraction of i in distillate; xFi = liquidmolefraction of i in feed; D = distillate stream; F = feed stream

Analysis of a quaternary system (a, b, c, d)is illustrated in Figure 9. The ratio of the liquiddistillate concentration to the feed concentration


xDi/xFi (i.e., enrichment) is plotted against therelative amount of distillate D/ F. The best sepa-ration is achieved at low D/ F values. Obviouslyseparation diminishes as the concentration ratioD/ F approaches unity. Both medium boilers band c pass through a concentration maximumduring distillation.

3.2. Batch Distillation

Aschematic of batch distillation is shown in Fig-ure 10. The vapor stream G produced by heatingL moles of liquid is continuously removed andcondensed. The distillate (D) is usually collectedin several receivers.

Figure 10. Schematic of batch distillation

3.2.1. Batch Distillation of Binary Mixtures

Material balance around the apparatus permitscalculation of the batch distillation of a binarymixture of substances a and b. For a differentialtime element dt, the following holds:

Gdt+dL= 0 (3.13)

and

Gy dt+d (Lx) = 0

where the vapor and liquid mole fractions x andy refer to the more volatile component a. By us-ing G = dG/dt, the Rayleigh equation is obtained[26]:

dLL

=dx

y x (3.14)

For ideal mixtures, the equilibrium concentra-tion of a in the vapor phase, y, can be repre-sented by

y=x

1+ ( 1) x (3.15)

An analytical solution to the differential equa-tion is obtained by substitution into Equation(3.14):L

L0=[x

x0

]1/(1)[1x01x

]/(1)(3.16)

where the subscript 0 represents the liquid in itsinitial state.

Typical changes in the concentrations of themore volatile component a in the vapor and inthe liquid are plotted against the relative amountof distillateD/L0= 1L/L0 inFigure 11. Duringthe operation, both phases are increasingly de-pleted of substance a. The intermediate distillateconcentration xDm (= ym) is given byxDm=x+

x0xD/L0

(3.17)

This equation can be interpreted visually withthe help of Figure 11. Even for nonideal mix-tures, the intermediate distillate concentrationxDm can be determined easily.

Figure 11.Batch distillation of a binary mixture a b show-ing concentrations of the more volatile component a in theboiler (x) and in the vapor or distillate (y) as a function ofthe relative amount of distillate D/L0The intermediate distillate concentration xDm can be deter-mined graphically (see Eq. 3.17).


3.2.2. Batch Distillation of MulticomponentMixtures

The Rayleigh equation (Eq. 3.14) can be applieddirectly to multicomponent mixtures. The fol-lowing holds for each component i of a system:

dLL

=dxi

yi xi(3.18)

Numerical methods are generally used to solvethis set of equations because the equilibriumconcentration yi is usually inuenced by all thesubstances present in the mixture.

An analytical solution is obtained under idealconditions by employing the equilibrium con-stant Ki = yi/xi which is then independent ofthe other components of the system. Hence, thefollowing equation applies:

dLL

=dxi

xi (Ki 1)(3.19)

Rearrangement gives

(Ki1) dlnL= dlnxi (3.20)

Integration of each component gives

(Ki1) ln (L/L0) = ln (xi/xi0) (3.21)

where the index 0 denotes the starting condition.Solving for the concentration xi gives

xi=xi0 (L/L0)K1i (3.22)

For the mole fraction constraint xi= 1, the fol-lowing is obtained:

ki=1

xi0 (L/L0)K1i = 1 (3.23)

To solve Equation (3.23) for given L/L0 val-ues, the boiling temperature of the mixture mustbe estimated. The vapor pressure of each sub-stance p0i , and consequently the equilibriumconstantKi = p0i/p, can then be determined. Thetemperature estimation is checked and correctediteratively by employing the above relationship,whereby convergence is usually attained rapidly.

Alternatively, temperature iteration can beavoided by using the relative volatility ik in-stead of the equilibrium constantKi. By analogywith Equations (3.9) (3.12):

ki=1

[xi0 (L/L0)ik1 (xk/xk0)ik

]= 1 (3.24)

In this case, instead of absolute temperature T,the mole fraction of the high boiler xk is the it-eration variable. Equations (3.23) and (3.24) de-scribe the changes in concentration of the liquidin the boiler. Thus, the distillate composition canbe determined with the help of the equilibriumconstant Ki.

Variation of liquid and vapor concentrationwith the relative amount of distillate D/L0 for aquaternary mixture is shown in Figure 12. Theconcentration of themost volatile component (a)decreases rst because it is released preferen-tially into the vapor phase. Concentrations ofthe higher boiling substances in the boiler con-sequently increase. When most of substance ais separated (D/L0 0.7 in Fig. 12), substanceb is the most volatile component, so its con-centration decreases similarly. Both medium-boiling substances b and c pass through a con-centrationmaximum in the course of distillation.The distillate is collected in several receivers,which enable the recovery of fractions of vary-ing composition. However, complete separationof a liquid mixture into pure components is sel-dom achieved. Batch rectication must usuallybe employed for this (see Chap. 5).

Figure 12. Course of concentration changes during thebatch distillation of a mixture of butane (a), pentane (b),hexane (c), and heptane (d)Mole fraction of component in the liquid in the boiler,xi; Mole fraction of component in the distillate, yi


3.3. Semicontinuous Distillation

Semicontinuous distillation can often be usedto advantage with liquids containing smallamounts of impurities with low volatility. Asshown schematically in Figure 13, feed is fedcontinuously into the boiler at a rate F, andhead product is withdrawn continuously at arate D. During the course of distillation, low-volatility impurities collect in the boiler andmust be drained off periodically.

Figure 13. Schematic of semicontinuous distillation

3.3.1. Semicontinuous Distillation of BinaryMixtures

As in Equation (3.14), a material balance for abinary mixture yields

dFL0

=dx

xFy (x)(3.25)

where mole fractions refer to the more volatilecomponent a. For ideal mixtures, integrationgives

F

L0=

1xF ( 1)

(x xF)

[xF (1)]2

ln{

x [xF (1)] +xFxF [xF ( 1)] +xF

}(3.26)

Hence, the concentration changes of impuritiesin the boiler and in the distillate can be deter-mined. Variationwith time t of the concentrationis given in the relationship F = F t. In the spe-cial case of nonvolatile impurites (i.e., ),Equation (3.26) is simplied as follows:F/L0=

x xFxF1

(3.27)

Rearrangement gives

x=xF+F/L0 (xF1) (3.28)Thus, the concentration x of the more volatilesubstance in the boiler decreases linearly withthe amount of feed, while that of the less volatilesubstance (1 x) increases linearly.

3.3.2. Semicontinuous Distillation ofMulticomponent Mixtures

In combination with a suitable equilibrium re-lationship, Equation (3.25) can also be used todescribe multicomponent mixtures. In general,a set of differential equations is obtained, whichare coupled bymeans of the equilibrium concen-trations. However, this does not apply to idealmixtures because the equilibrium constant Kiof each component is independent of other sub-stances in the mixture. Individual equations arerelated to each other only via temperature.

Thus, for ideal solutions,dFL0

=dxi

xFi yi=

dxixFi Ki xi

(3.29)

Rearrangement givesF

F=0

dFL0

= 1Ki

xixi=xFi

dln (Ki xi xFi) (3.30)

After integration, the following equation is ob-tained:F

L0= 1

Kiln Ki xi xFi

xFi (Ki 1)(3.31)

Solving for the concentration of component iand using the mole fraction constraint xi = 1yields


Ki=1

1Ki

[xFi+xFi (Ki 1) exp

( FL0Ki

)]= 1 (3.32)

This equation can be solved iteratively, the tem-perature T being the iteration variable. The indi-vidual summation terms give the concentrationsxi in the boiler.

4. Continuous Rectication(Multiple Distillation)4.1. Principles

As discussed in Chapter 3, only limited separa-tion of components is possible in simple distil-lation. The composition difference between thefractionation products is often very small; there-fore, pure substances are seldom obtained.

Figure 14 represents a multiple distillation,i.e., rectication. The overhead product of therst distillation is subjected to a second distil-lation, whose overhead product, in turn, is sub-jected to a third distillation and so on. Like Fig-ure 8, the vapor and liquid concentrations at eachstep can be determined easily on the y x dia-gram (Fig. 14B).

This process can yield a very pure fraction,but it has two serious disadvantages:

1) Only a small part of the starting mixture isrecovered in a highly concentrated state.

2) Byproducts are obtained at each stage,whichoften cannot be processed further.

Both disadvantages can be avoided if the liquidfrom each distillation step is returned to the pre-vious step, as shown in Figure 15A. The vapordoes not have to be condensed before each stepnor does the liquid have to be subsequently re-vaporized (Fig. 15B).On the contrary, vapor andliquid can be brought into direct contact. In fact,intensive countercurrent contact between the va-por and liquid streams at each stage is importantso that intensive mass transfer can take place.The concentrations attainable at each step canbe determined on the y x diagram by means ofmaterial balances around the top of each column,as shown in Figure 16.

Two methods are usually employed to eval-uate countercurrent separation efciency: theequilibrium-stage and the transfer-unit concepts.

4.1.1. Equilibrium-Stage Concept

The equilibrium-stage concept assumes that thevapor and liquid phases are indistinct mass-transfer elements, where they are brought intosuch intimate contact that they are in equilibriumwith eachother at each stage.Thephases are sub-sequently separated; vapor ows up to the nextstage and liquid runs down to the stage below(countercurrent phase ow), as shown in Fig-ure 16. A material balance for the more volatilecomponent around the top of the column givesthe following relationship for the vapor and liq-uid concentrations (x, y) and streams ( G, L) bet-ween the individual stages:

yn1=Ln

Gn1xn+ G0

Gn1y0 L0

Gn1x0 (4.1)

The subscripts 0, n, and n 1 refer to the stagesfrom which the mass stream originates, 0 beingthe initial stage. When Equation (4.1) is plottedon the y x diagram, it furnishes the operatingline (Fig. 16B). The phase equilibrium can alsobe represented by a line on the y x diagram. Theliquid concentration xn is connected to the vaporconcentration yn 1 by a balance line; the con-centrations xn and yn 1 dene a point on the op-erating line. Equilibrium exists between yn, thevapor above stage n, and xn the liquid concentra-tion in the same stage. Therefore, a vertical lineat the liquid concentration xn must intersect theequilibrium curve at point yn. Likewise, a hor-izontal line at the vapor concentration yn mustpass through the point xn+1 on the operatingline. The point of intersection of xn+1 with theequilibrium curve gives the vapor concentrationyn+1 of the stage above and so on. Alternate useof the equilibrium curve and the operating lineto step from one equilibrium stage to the otheris illustrated on the y x diagram in Figure 16B.The operating line is the locus of states betweenthe individual stages.

The number of equilibrium stages n requiredto achieve a desired concentration change isa good index of the difculty of separation.The number of equilibrium stages, representedby steps, can easily be determined graphically


Figure 14.Multiple distillation of a binary mixture a bA) Flow diagram; B) Determination of vapor and liquid mole fractions of component a at each stage on a y x diagram

Figure 15. Improvement of multiple distillation by returning liquid from each distillation step to the previous step with (A) orwithout (B) condensation


Figure 16. Equilibrium-stage concept for determining countercurrent separation efciencyA) Schematic showing equilibrium stages and associated parameters: dotted dashed line indicates the boundary for the ma-terial balance described in Equation (4.1).B) McCabe Thiele diagram: concentrations at each equilibrium stage and number of stages can be determined by steppingoff the stages between the equilibrium line and the operating line.

on the y x diagram; this is often called theMcCabe Thiele method. Analytical solutionsare available for a few special cases. A particu-larly important case occurs when both the equi-librium curve and the operating line are straightlines; the number of equilibrium stages n is givenby [27]

n= 1ln m

L/G

ln[(

m

L/G 1

)y y0y0y0

+1]

(4.2)

where m is the slope of the equilibrium line.When m/ (L/G) = 1,

n= y y0y0y0

(4.3)

Equation (4.2) can often be used to determinethe number of stages in the separation of minuteimpurities.

4.1.2. Transfer-Unit Concept

The concept of transfer units, conceived byChilton and Colburn, considers a differen-tial transfer area dA of the column [28]. It isillustrated schematically in Figure 17A. Masstransfer can be formulated as follows:dN=KOG (yy) dA (4.4)

where KOG is the overall mass transfer coef-cient. The molar mass stream d N causes achange in vapor composition dy:


Figure 17. Transfer-unit concept for determining countercurrent separation efciencyA) Flow diagram showing important parameters and material balance boundaries (I, II)B) A y x diagram: the number of transfer units NOG is determined by integrating between the equilibrium line and theoperating line (shaded area)

Gdy= dN (4.5)

Thus,

Gdy=KOG (yy) dA (4.6)

Separating variables givesdy

yy=KOG dA

G(4.7)

The termon the right-hand side ofEquation (4.7)contains factors pertaining mainly to the appa-ratus, which are of interest only for column de-sign (see Chap. 8). The difculty of separationis characterized sufciently by the term on theleft-hand side, which contains factors pertainingmainly to the process itself. It is dened as theoverall number of gas-phase transfer unitsNOG:

dNOG=dy

y y (4.8)

This term can be interpreted as the ratio of thedesired concentration change dy to the drivingforce (y y) available for mass transfer. At agiven liquid state x, y lies on the operating lineand y on the equilibrium curve. The shadedarea in Figure 17B represents the driving force(y y).

The difculty of the separation to be accom-plished is characterized in terms of transfer unitsNOG. If NOG is very large a very tall column isrequired.

The number of transfer units NOG is usuallydetermined by integrating between the operatingline and the equilibriumcurve, either graphically


or numerically. Analytical solutions are avail-able in a few special cases. Again, the solutionwhen both the equilibrium curve and the oper-ating line are straight lines on the y x diagramis important [27]:

NOG= 1mL/G

1

ln[(

m

L/G 1

)y y0y0y0

+1]

(4.9)

When m/(L/G) = 1,

NOG= y y0y0y0

(4.10)

Equation (4.9) is comparable to Equation (4.2)for the number of equilibrium steps n.

4.1.3. Comparison of Equilibrium-Stage andTransfer-Unit Concepts

The equilibrium-stage concept for characteri-zation of the difculty of separation providesan adequate measure of separability for dis-tinct mass-transfer elements in countercurrentcolumns, e.g., in a highly efcient plate column.On the other hand, the transfer-unit concept ispreferred when vapor and liquid are in contin-uous contact, e.g., in packed columns and platecolumns which have an efciency well below1. The equilibrium-stage concept is used almostexclusively in practice because graphical deter-mination of the number of equilibrium steps onthe y x diagram is much easier than integrationwith respect to the driving force.

The equilibrium curve and the operating linecan almost always be linearized in narrow con-centration ranges. The following relationship isthen obtained:

NOG=nln m

L/GmL/G

1 (4.11)

In the special casewhenm/( L/ G) 1,NOG n.This condition is often satised to a close ap-proximation in difcult separations that requiremany equilibrium stages or transfer units.

Figure 18. Schematic of a continuously operated rectica-tion column used to separate a binary mixturePart of the overhead product and part of the bottom productare returned to the column to maintain countercurrent ow.Boundaries for material balances discussed in the text areindicated by dotted dashed lines (I IV).

4.2. Rectication of Binary Mixtures

A continuously operated rectication column isshown schematically in Figure 18. The binaryfeed material (molar ow rate F) is introducedinto themiddle of the column.Thedistillate (mo-lar ow rate D), whichmainly contains the lowerboiling component a, is withdrawn from the topof the column. The bottom product (molar owrate B), which contains the higher boiling com-ponent b in greater concentration, is removedfrom the bottom of the column. Some overheadand bottom products are returned to the columntomaintain countercurrent ow of vapor and liq-uid in the column. As discussed in Chapter 3,this is a prerequisite for the separation of feed


into substances a and b with maximum purity.The sections of the column above and below thefeed are called the rectication section and thestripping section, respectively.

4.2.1. Prediction of Rectication Based onMaterial Balance

Rectication can be predicted simply with thehelp of material balances by assuming that thevapor and liquid streams ( G and L, respectively)in the column do not change (see Section 4.2.2).

The overall column balance (boundary I inFig. 18) yields

F=D+B

F zF=D xD+B xB (4.12)

where zF, xD, and xB are the mole fractions ofthe more volatile substance a in the feed, dis-tillate, and bottom product, respectively. Rear-rangement gives

D=FzFxBxDxB

and B=F xDzFxDxB

(4.13)

These values can be used to determine the quan-tities of distillate D and bottom product B forgiven product specications. Amaterial balancefor the top of the column (boundary II in Fig. 18)gives a relationship for the states in the column:

G=L+D

Gy=Lx+D xD (4.14)

where L is the molar stream of the reux liquorreturned to the column. Thus,

y=L

Gx+

(1 L

G

)xD (4.15)

The exact position in the column to which bal-ance II applies is not dened. Hence, Equation(4.15) gives the relationship between the vaporconcentration y and the liquid concentration x atany point in the column above the point of feedentry, i.e., in the rectifying section. Because L/ Gis constant, Equation (4.15) represents a straightlinethe rectifying operating linewith a slopeof L/ G on the y x diagram, which intersects they= x diagonal at xD (Fig. 19).

Figure 19. McCabe Thiele diagram for a binary mixturea bShaded area between the operating line and the equilibriumcurve is decisive for operation. Equilibrium stages are alsoshown.a) Equilibrium line; b) Feed or line; c) Rectifying oper-ating line; d) Stripping operating line

The ratio L/ G is known as the internal reuxratio. The use of the external reux ratio R ismore advantageous:

R=L

D(4.16)

Thus Equation (4.15) becomesy=

R

R+1x+ 1

R+1xD (4.17)

Analogously, material balance III around thebottom of the column yields the following re-lationship:

y=LGx+

(1

LG

)xB (4.18)

where the horizontal bar (y, x, etc.) denotes pa-rameters below the feed point, i.e., in the strip-ping section. Use of the external reboil ratio R,according to the denition R = G/B R= G / B,gives

y=R+1R

x 1RxB (4.19)

Equation (4.19) denes the relationship betweenthe vapor concentration y and the liquid concen-tration x. If the ow rates G and L are assumedto be constant, it represents a straight line, called


Figure 20. Rectication column and McCabe Thiele diagram for multiple feed entries or side-stream withdrawalA) Flow diagram; B) McCabe Thiele diagram for two feeds; C) McCabe Thiele diagram for one feed and one side stream

the stripping operating line, on the y x diagram(Fig. 19). Feed is introduced into the column atthe point of intersection of the rectifying andstripping operating lines dened by Equations(4.17) and (4.19) (see Fig. 19). Material balanceIV around the feed point gives the followingequation for the point of intersection [29]:

y=

1 x 1

1 zF (4.20)

The caloric factor characterizes the thermalcondition of the feed and is dened by

=Enthalpy required tovaporize 1mol of feedMolar latentheat of vaporization of feed

If the feed is introduced as a boiling liquid, thevalue of is 1 and Equation (4.20) gives a ver-tical line on the y x diagram; i.e., the operating

lines intersect at x= zF. However, if the feed isintroduced as a saturated vapor, = 0 and Equa-tion (4.20) yields a horizontal line; the two op-erating lines intersect at y= zF. If the feed isintroduced into the column as a vapor liquidmixture, the point of intersection of the twooper-ating lineswill be between the two extreme casesdescribed above, as shown in Figure 19. The line can also represent a subcooled liquid or asuperheated vapor feed.

According to Equation (4.20), the line orfeed line denes the locus of all points of inter-section of the two operating lines in the rectify-ing and stripping sections of the column. Thus, ifthe feed conditions and the composition of over-head and bottom products are specied, and oneoperating line is established, the location of thesecond operating line is also known. This means


that a relationship exists between the external re-ux ratio R and the external reboil ratio R. If thefeed enters at the boiling point, i.e., = 1,

R=D/F

1D/F (R+1) (4.21)

Analogously, for a saturated-vapor feed,

R=D/F

1D/F (R 1) (4.22)

The term D/ F is the relative quantity of over-head product, which can be determined by usingEquation (4.13) after the product concentrationshave been established.

The calculation of a rectication columnwithmultiple feed entry and side stream withdrawalis conducted in an analogous manner (Fig. 20).The operating line bends at each entry or with-drawal point. Figure 20Bshows the course of theoperating lines when F1 is a vapor feed stream( F1> 0); Figure 20C shows the correspondinglines for a vapor side stream ( F1< 0).

Figure 21.Enthalpy concentration diagramof binarymix-ture a bThe points of intersection of the lines through poles D andB with the dew point line and the boiling point line givethe concentrations x and y on the operating line.

If the positions of the operating lines andthe equilibrium curve on the y x diagram areknown, the difculty of separation can be deter-mined by using the number of equilibrium stagesn (shown as steps in Fig. 19, cf. Section 4.1.1) or

the number of transfer units NOG (shaded areain Fig. 19, cf. Section 4.1.2). The desired sepa-ration is possible only when the operating linesneither touch nor intersect the equilibriumcurve.The smaller the space between the equilibriumcurve and the balance lines, the more difcult isthe separation.

4.2.2. Prediction of Rectication Based onMaterial and Enthalpy Balances

The calculation of rectication based on mate-rial balance alone is possible only if the vaporand liquid streams ( G and L, respectively) in theindividual column sections are constant. If thisis not the case, the enthalpy balance must alsobe taken into consideration. The enthalpy con-centration (h x) diagram, shown in Figure 21for a binary mixture a b, is very convenient inthis respect.

Overall enthalpy balance I (see Fig. 18) forthe column gives the following relationship:

F hF+QR=D hD+B hB+QC (4.23)

where hF, hD, and hB are the specic enthalpiesof the feed, distillate, and bottom product, re-spectively; QR is the molar heat ow suppliedby the reboiler; and QC is the molar heat owremoved by the condenser. Thus,

F hF=D D+B B (4.24)

where D=hD+ QCD and B=hBQRB

Equation (4.24) represents a straight line inFigure 21 that extends from pointD to pointBand passes through F. The amounts of heat QRand QC can be determined easily as shown inthe diagram when the compositions of the feedand of the desired overhead or bottom productsare known (i.e., F, D, and B). Both the distillate( D) and the bottom product ( B) are assumed toleave the column in the boiling state.

Heat balance II in Figure 18 at the top of thecolumn (rectifying section) givesGhG=LhL+D D (4.25)

Here, hG and hL are the specic enthalpies ofthe vapor and liquid, respectively. This equationalso represents a straight line through the pointD on the enthalpy concentration diagram. If


the vapor in the column is always at the dewpoint and the liquid always at the boiling point(an assumption that is met closely), the compo-sitions of the vapor G and liquid L streams aswell as their concentrations y and x can be es-tablished as shown in Figure 21. Concentrationsy and x are points on the operating line on they x diagram (Fig. 19). Analogously, if the bal-ance lines in the stripping section are allowedto pass through B, the vapor and liquid statesare obtained from the points of intersection withthe dew point and the boiling point lines, respec-tively, as shown inFigure 21. Theoperating linesin the rectifying and stripping sections are ob-tained by transferring the vapor and liquid con-centrations to the y x diagram. The operatinglines are generally slightly curved. The quanti-ties of vapor and liquid can be determined byusing the lever arm rule:

Gl2=L (l1+l2) or LG=

l2

l1+l2(4.26)

where l1 is the distance between the boiling pointand dew point lines and l2 is the distance bet-ween the pole D and the dew point line.

In the special case in which the dew pointline runs parallel to the boiling point line onthe enthalpy concentration diagram, Equation(4.26) produces a constant value for L/ G. Thus,the criteria for constant vapor and liquid loadscan be formulated:

1) The enthalpies of vaporization of the twosubstances a and b must be equal.

2) No heat of mixing should occur.3) The boiling points of the substances should

be as close as possible.

The rst criterion is the most important. Equalenthalpies of vaporization can often be achievedby usingmolar enthalpies of vaporization,whichin turn require that the composition be given inmole fractions. Thus, rectication calculationsgenerally employ these units.

If the criteria for a constant liquid vapor ra-tio L/ G are not satised, the following approachis recommended:

1) The poles D and B are etablished on theenthalpy concentration diagram.

2) The vapor and liquid concentrations y andx are determined, in each case, by drawingstraight lines that radiate from these poles.

The values thus obtained are plotted on aMcCabe Thiele diagramand the curved op-erating line is obtained, point by point.

3) The number of equilibrium steps n or thenumber of transfer units NOG is determinedin the usual manner on the y x diagram.

The number of equilibrium steps can also bedetermined on the h x diagram [30], but thismethod is so complex that it has little practicaluse.

4.2.3. Rectication of Binary Mixtures atInnite Reux Ratio

If the internal streams G and L are much largerthan those of the feed ( F) and the products ( Dand B), the internal reux ratio L/ G in all partsof the column approaches 1. Thus, according toEquations (4.15) and (4.17),

R (4.27)

In this special case (i.e., at innite reux ratio),the energy input needed for separation is verylarge, but a minimum number of equilibriumstages n or transfer units NOG is required. Asshown in Figure 22, the operating line coincideswith the diagonal and is represented by the sim-ple equation

y=x (4.28)

The equilibrium state y is determined by start-ing with the liquid state xB. The new liquid con-centration x1 is now calculated from the vaporconcentration according to Equation (4.28). Thevapor state y1 in equilibrium with this liquidconcentration x1 is, in turn, determined and soon. This procedure is identical with the determi-nation of the distillation curve described in Sec-tion 2.2.3. Equation (2.25) thus applies to idealbinary mixtures with constant relative volatility:

xD=n xB

1+ (n 1) xB(4.29)

After rearrangement, the minimum number ofequilibrium steps nmin is obtained:

nmin=1ln

ln(xD

xB 1xB1xD

)(4.30)


Analogously, the minimum number of transferunits NOG,min is given by

NOG,min=1

1 ln[xD

xB(1xB1xD

)](4.31)

Figure 22.McCabe Thiele diagram of a binary mixture atinnite reux ratio

4.2.4. Rectication of Binary Mixtures atMinimum Reux Ratio

Operation with minimum column loads (i.e.,minimum reux ratio) has considerable practi-cal importance because it results in separation ofa mixture by using the smallest possible columndiameter and the lowest energy input.

Operation with minimal internal loads ischaracterized by intersection of the rectifyingand stripping operating lines at a single pointon the equilibrium curve. This point is called apinch point, and an innite number of equilib-rium stages n or transfer units NOG is requiredto reach this state. Figure 23 shows the y x di-agram for a binary system with minimal inter-nal loads. If the vapor and liquid ow rates ( Gand L, respectively) in the individual sections ofthe column are constant, the operating lines arestraight lines. The minimum slope of the recti-fying operating line for the more volatile com-ponent a can be calculated as follows:(L

G

)min

=xDyFxDxF

(4.32)

By use of the relationship L/ G =R/(R + 1), theminimum reux ratio becomes

Rmin=xDyFyFxF

(4.33)

Concentrations yF and xF are related by thephase equilibrium.By usingEquation (2.17), thefollowing relationship is obtained for ideal mix-tures:

Rmin=1

1

(xD

xF 1xD

1xF

)(4.34)

If the overhead product is required as pure sub-stance a (i.e., xD= 1),Equation (4.34) canbe sim-plied:

Rmin=1

( 1) xF(4.35)

The point of intersection of the feed line(Eq. 4.20) with the equilibrium line (Eq. 2.17)gives the concentration xF. A quadratic equa-tion is obtained which can be solved as follows:

xF= B

2+

B2

4+C (4.36)

where

B=1

(

1zF)

C=zF

( 1)zF is the mole fraction of substance a in a liq-uid vapor feed and is the thermal state of thefeed (caloric factor) dened in section 4.2.1.

Figure 23.McCabe Thiele diagram of a binary mixture atminimum reux ratio and minimum reboil ratioTheoperating lines intersect the equilibrium line at the pinchpoint.


If the feed is a boiling liquid with composi-tion xF, then

xF=xF (4.37)

If the feed is a saturated vapor with compositionyF, then

xF=yF/

1 ( 1) yF/(4.38)

Analogously, laws can be derived for the min-imum reboil ratio R = G/B. The slope of thestripping operating line can be calculated by us-ing Figure 23 as follows:(LG

)max

=yFxBxFxB

(4.39)

From the relationship L/ G =(R+1

)/R, the

minimum reboil ratio is

Rmin=xFxByFxF

(4.40)

The relationship yF = f (xF) can be formulatedeasily for ideal mixtures:

Rmin=

[1

1xB/xF(

1+ (1) xF1)]1

(4.41)

An important case exists when pure substanceb is to be separated as the bottom product (i.e.,xBa= 0):

Rmin=

[

1+ (1) xF1]1

(4.42)

The concentration xF can generally be calcu-lated from the composition zF and the thermalstate of the feed (Eq. 4.36). Again, the spe-cial cases in which the feed is a boiling liquid ora saturated vapor are very important. The rela-tionship between the minimum reux ratio andthe minimum reboil ratio can then be expressedby using Equations (4.21) and (4.22). Separa-tion using a rectication column with a mini-mum reux ratio requires an innite number ofequilibrium steps n or transfer units NOG. How-ever, even a slight increase in the reux ratio ofca. 5 10% can result in economical operation.Thus, the special case and laws discussed in thissection closely approximate actual recticationand have immediate practical importance.

4.2.5. Heat Requirements for Recticationof Binary Mixtures

In rectication columns, a quantity of heat QRis supplied in the reboiler at the bottom of thecolumn and a quantity QC is removed by thecondenser at the top. The following equationshold:

QR=r GorQC= r G (4.43)

where r is the molar latent heat of vaporizationfor the substance in question. Approximatelyequal molar heats of vaporization of the sub-stances involved are assumed.

Important relationships can be established forthe quantities of heat QR and QC. The heat re-quired b

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