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A Computational Design Method for Multicomponent

Condensation

Eine Methode zur Berechnung der Kondensation von Vielstoffgemischen

ANDRZEJ BURGHARDT and MAREK BEREZOWSKI

Polish Academy of Sciences, I nstitute of Chemical Engineeri ng, Bdtycka 5, 44-100 Gliwice (Poland)

(Received June 30, 1988)

Abstract

A mathematical model has been developed which describes quantitatively the phenomena occurring during the

condensation of multicomponent mixtures in which all constituents condense to form a homogeneous liquid

phase. As a result of the analysis of the model equations two specific modes of the vapour mixture condensation

have been defined and described: equilibrium condensation and total local condensation.

The derived model of the process and the results of the analysis of the model equations have served as a basis

for the proposed method of condenser design as well as for the relevant algorithm and numerical calculation

procedure. The proposed numerical algorithm for the condenser design was tested for three- and tive-compo-

nent mixtures. By changing the composition and the inlet temperature of the vapour, as well as the temperature

of the cooling medium, a wide range of condensation intensities was covered, and thus the stability and

convergence of the numerical method were tested.

The results of the theoretical analysis as well as those of the numerical calculations led to the development

of a simplified method of calculation which is particularly useful for mixtures with more than three components,

and has the advantage of being almost five times faster.

Kurzfassung

Ein mathematisches Model1 wurde entwickelt, das den Vorgang der Kondensation eines Vielstoffgemisches,

dessen Komponenten alle kondensierbar und dessen Kondensat eine homogene fliissige Phase bildet, quantitativ

beschreibt. Als Ergebnis der Analyse der Modellgleichungen lie&n sich zwei Grenzfalle der Kondensation eines

Vielstoffgemisches definieren und beschreiben: die Gleichgewichtskondensation und die lokale Totalkondensa-

tion.

Dieses Model1 bildet die Grundlage fiir den Vorschlag zu einer Berechnungsmethode fur die Auslegung von

Kondensatoren fur Vielstoffgemische. Beispielhaft wird der Berechnungsalgorithmus an einem Drei- und einem

Fiinfstoffgemisch erhiutert. Die Parameter Eintrittszusammensetzung,

Temperatur des Dampfes und

Eintrittstemperatur des Kiihlmittels wurden in breiter Variation untersucht, urn damit von der Gleichgewichts-

kondensation bis zur lokalen Totalkondensation die Stabilitlit und die Konvergenz des Berechnungsverfahrens

zu priifen.

Aufgrund der theoretischen Analyse und der numerischen Berechnungen konnte ein einfaches Ngherungsver-

fahren entwickelt werden, da13 besonders ntitzlich bei Vielstoffgemischen mit mehr als drei Komponenten ist.

Die Rechenzeiten des Naherungsverfahrens sind nur ein funftel derer der vollstandigen Methoden.

Wnopse

Ein mathematisches ModeN wurde entwickelt, das

die Kondensation eines Vielstoff gemisches quantitati v

beschr eibt, desser t Komponenten all e kondensierbar

sind und dessen K ondensat eine homogene Jl ii ssige

Phase bildet. Di eses Modell ist die Grundlage der

vorgeschlagenen Methode zur Auslegung entsprechen -

der Kondensatoren.

Die Verii nderung der Stofl stri ime beider Phasen,

deren Zusammensetzung und Temperatu r im Verlauf

der Kondensation wir d durch ein System von dif feren-

tiellen Massen- und Enthat piebilanzen wiedergegeben.

Urn die in diesen Bil anzen auftr etenden Stoff -

stromdichten der von der Dampf - in die Fbissigkeits-

phase ii bergehenden Komponenten und die Wii rme-

stromdichte durch den Kondensatfi lm berechnen zu

kii nnen, ist die Kenntn is der an der Phasengrenze

0255-2701/88/%3.50

Chem. Eng. Process.. 24 ( 1988) 189-202

0 Elsevier Sequoia/Printed in The Netherlands

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herrschenden Bedingungen erf orderl ich. Di e Zu-

standsvari ablen (Zusammensetzung der Dampf- und

der F li issigkeitsphase, Temperatur an der Phasen-

grenze) lassen sich mit den Gln . (I )+) aufgrund eines

Systems algebra& her Verkni ipf ungen berechnen. Di e

Stoff stromdichten der Komponenten von Vielstoff -

gemischen sind durch die Gln . (6)-o-() defi niert.

Der an der Phasengrenze angenommene Gleich-

gewichtszustand wir d durch die Matri xrelati onen (13)

und (14) beschr ieben. Nach Ei nf tih rung oben genannter

Abhii ngigkeiten in die Gl. (I ) erhalten wir die Matrix -

formel (15), aufgrund derer die Molenbrti che der Kom-

ponenten an der Phasengrenze bestimmt werden

konnen.

Di e Matri zen der Stoflti bergangskoefizienten und

der Kor rekturkoefi zienten fu r Vielstoffgemische sind

durch die Gln . (16)-(23) gegeben. Bei Einsetzen der

Gln . (16)-(18) in der Beziehung (I S) gelangt man nach

Transformationen zur Gl. (28). Aus dieser Gl eichung

geht hervor , da@ die Zusammensetzung der fr i issigen

Phase an der Phasengrenze stark von der Kondensa-

tionsintensittit abhiingt, die durch die globale Molen-

stromdichte N charakteri siert ist, welche in den

Matri zen 8 und 4 vorkommt. Es wurde die Ver-

ii nderung von x, als Funkti on von 4 analysiert, und es

wurden zwei Grenzfti ll e des Kondensationsmechanis-

mus fi r 4 -B 0 und 4 --t co herausgestellt. Aus der Gl.

(28) geht hervor, da@ im Fall e sehr gr oper Molen-

stromdichten des kondensierenden Dampfes N + co

die Zusammensetzung der Fhi ssigkeit an der Phasen-

grenze xi , gleich der Dampfzusammensetzung yi ist. I n

diesem Fall ist die Temperatur der Phasengrenze, die

aus den Gleichgewichtsbeziehungen hervorgeht, gleich

der Siedetemperatur der F li issigkeit von der Zusam-

mensetzung x,, = yi. Den KondensationsprozeB bei

groaen Molenstromdichten N bezeichnet man als art-

lithe Totalkondensation.

Im zweiten Grenzfall , das he@ fu r N + 0, gelten

die Gln. (33) und (34), die die sog. Gleichgewichtskon-

densation de@ieren. Fur die Werte von N, die von

Nul l und Unendli ch abweichen, ist das I nterval1 der

Phasengrenztemperaturen durch die Ungleichung (35)

bestimmt.

Die Abhtingigkeit (28) kann erheblich vereinfacht

we& en durch Einf ii hrung der Annahmen (36) und (37),

die den Stofltr ansport im Kondensatji lm charakte-

ri sieren. Di e Anwendung der Filmtheorie fur die Be-

stimmung a& Matrix der Korr ekturkoefi zienten (20)

fi ih rt zum System der algebrai schen Gleichungen (39)

(41), aufgr und deren die Zustandsvari ablen an der

Phasengrenze und die globale Molenstromdichte

berechnet werden ki innen.

Di e quanti tative Formuli erung des Modell s, die die

Berechnung der Veranderungen der Zustandrvariablen

in der Dampfphase und im Kondensat er laubt, ist

durch a& System der Dtzerentialgleichungen (43)-

(46) gegeben. In detail li er ten Berechnungen ist es ni itz-

li ch, den Kondensationsgrad (47) einzuftihren und das

oben genannte System der Dtr erentialgleichungen in

die Gln . (48)-(50) zu transformieren. Di e Stofl str iime

der beiden Phasen und die Zusammensetzung

er piis-

sigen Phase kii nnen aus den Gln. (51)-(53) berechnet

wet& n,

Urn die Wiirmeaustauschflache aufgrund der Gl.

(54) bestimmen zu ki innen, miissen die Dtr erential -

gleichungen (48H50) im Bereich des Kondensations-

grades & I fi ir gegebene Randbedingungen in tegri ert

wetden. Di eses Dtrerentialgl eichungensystem wur de

mit H ii fe der modtj?zierten Methode von Eul er i nte-

gri er t. Di e erhaltenen Rekursionsgleichungen (58x65)

& fi nieren direkt die Zustandsvariablen (yi, xi, T,,

T,, GG, G3 am Ende jedes l ntegrationsschr ittes. Di e

Genauigkeit dieser Methode ist vergleichbar mit der

der Runge-Kutta-Gil l Methode vierter Ordnung, die

ln tegrati onszeit ist aber vie1 ki ir zer.

Die Zustandrvariablen an der Phasengrenze, die bei

jedem l ntegrationsschr itt bendtigt wet-den, wurden aus

den Gln . (39x41) mit H il fe der Newton-Raphson

Methode berechnet. D ie Eigenwerte der Matri x der

D usionskoefl zienten (68) und (69), die zur Berech-

nung der Matrix funktion exp( -4& notig sind, wur-

den mit Hi I fe der Transformationsmethode, die in Li t.

I 4 bearbeitet wurde, berechnet.

Der vorgeschlagene Berechnungsalgori thmus zum

Entwur f von Kondensatoren wurde anhand von

Berechnungen eines Dreistoff- und eines F ii nf-

stoff gemisches erli iu tert. D ie Eintr ittszusammenset-

zungen und Temperatur en des Dr eistoflgembches

Benzol-Toluol-Xylol wie such die Ein tri ttstempera-

tur en des Ki ih lmi ttels sind in der Tabell e I angegeben.

GroJe Anderungen der oben genannten Parameter

wurden vorgenommen, urn einen breiten Bereich & r

Kondensationsin tensit ii ten zu analysieren (von der

Gleichgewichtskondensation bis zur Sirt li chen Total-

komiensation) und urn auf diese Weise die Stabil it ii t

und Konvergenz der Methode zu pri if en.

Es hat sich gezeigt, da4 in alien in & r Tabelle 1

zusammengestell ten Berechnungsbeispielen die Berech-

nungsprozedur konvergent und stabil war, sogar fti r

Kondensationsstromdichten, die gegen Null gehen

(N + 0). I n den B il dern 2-6 sind die Zusammenset-

zungs- und TemperaturproJl e als Funk tionen des Kon-

densationsgrades fi ir zwei gewii hl te Temperaturen des

Kiihlmittels dargestellt.

Die Ergebnisse der Berechnungen fii r ah F& f-

stofl gemisch, dessen Eintr ittszusammensetzung in & r

Tabelle 2 gegeben ist, wurden i n den Bildern 7-9

dargestellt.

Di e wachsende Zahl der Komponenten ver li in gert

erhebli ch die Zeit der numerischen Berechnungen. Das

wir d dadurch verur sacht, a?$ in jedem I ntegrations-

schr itt die Eigenwerte der Di ff usionskoefi zientenmatri x

berechnet werden mtissen. Di es ist die zeitraubendste

numerische Prozedur des ganzen Al gori thmus. Deshalb

wurde eine verein fachte Methode zur Auslegung von

Kondensatoren fi ir Vielstofigemische vorgeschlagen.

Da fti r die Mehrzahl der Vielstoflgemische die Dt ii -

sionskoeflzientenmatrix eine diagonal-dominierend

Matrix ist, wurden die nichtdiagonalen Elemente

dieser Matri x vernachli issigt. Auf diese Weise werden

die Kreuzefl ekte des Stofl transportes vernachll issigt,

und die Eigenwerte mu& en nicht berechnet werden.

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Bei dieser Methode wir d die Berechnungszeit ungeftihr

auf ein Funftel reduziert.

Ein Vergleich der Ergebnisse, die mit H il fe dieser

Methode erhalten wurden, mit den Ergebnissen der

genauen Berechnungen ist in

den

Bil dern IO und 11

dargestelit. Dar in zeigt sich, daJ3 diese Methode zur

ni iherungsweisen Auslegung von Kondensatoren f ii r

Vielstofiemische empfohlen werden kann.

1. Introduction

The methods currently used in the design of heat

exchangers in which multicomponent condensation

takes place fall into two groups: the equilibrium

methods employing the works of Kern [ 11, Silver [2]

and Bell and Chaly [3], and those based on the

original paper by Colbum and Drew [4]. To describe

the phenomena occurring during the condensation,

the latter group makes use of the relations defining

the mass and energy fluxes both in the vapour phase

and in the layer of condensate. The discussion car-

ried out in ref. 5 shows that the use of the equi-

librium methods in the design of condensers may

lead to substantial errors incurred in determining the

heat fluxes and, consequently, to incorrect dimen-

sions of these apparatuses.

Although there are a number of papers dealing

with the condensation of multicomponent mixtures

containing inert components [6-lo], the condensa-

tion of mixtures in which all constituent species

condense has not been properly analysed. The exten-

sive study presented in ref. 5 is restricted to binary

mixtures.

The literature survey carried out in ref. 5 reveals

that previous research, both experimental and theo-

retical, was concerned exclusively with binary mix-

tures. Moreover, the authors were interested mainly

in determining the overall coefficients of heat transfer

from the condensing vapour to the wall, based on the

measured heat fluxes and assuming various empirical

values of the driving force. Thus, a real mechanism

of the process, which also includes the diffusion of

the condensing species, was not considered.

The reason for the apparent lack of interest in the

process discussed lies in the considerable mathemati-

cal difficulties concerned with the design procedures

for such condensers. While, by employing certain

simplifying assumptions, there is always a possibility

of obtaining explicit relations defining the individual

mass fluxes for mixtures containing inert components

[ 11, 121, the fluxes for mixtures in which all species

condense are given by implicit formulae.

It should also be noted that the evaluation of the

state variables at the vapour-liquid interface along

the condenser becomes much more complicated than

in the case of mixtures with some inert components.

This has also been pointed out in ref. 5, where a

specially developed procedure for determining the

interfacial state variables was presented.

The main objective of the present paper is to

derive a mathematical model describing quantita-

tively the phenomena occurring during the condensa-

tion of multicomponent mixtures in which all con-

stituents condense, employing both the conclusions

given in ref. 5 and the existing experimental results

for binary mixtures.

The model will serve as a basis for developing an

appropriate numerical design procedure.

2. The model and its analysis

The basic assumptions of the model of condensa-

tion of vapour mixtures are as follows.

(1) The layer of condensate flows down the wall

as a laminar film, and the flow itself is determined by

the equilibrium between the gravitational and vis-

cous forces. The possible effect of turbulence of the

layer of condensate and that of the vapour velocity

on the heat transfer resistance may be allowed for

using relations valid for the condensation of pure

substances.

(2) The flow of the vapour phase is a plug flow

with perfect radial mixing.

(3) According to the film theory, the diffusional

and heat resistances are located in suitably thin

layers adjacent to the interface and are defined by the

mass and heat transfer coefficients.

(4) Thermodynamic equilibrium prevails at the

interface.

(5) The wall is cooled from the outside by an

appropriate coolant which takes away the heat of

condensation.

(6) The possibility of the formation of mist in the

vapour phase is not allowed for. It is also assumed

that the temperature of the vapour is determined

solely by the equilibrium relations from the moment

when it reaches the dew point temperature.

Typical concentration and temperature profiles

are given in Fig. 1, together with the positive direc-

tions of the mass and heat fluxes.

The changes of the flow rates, composition and

temperatures of both phases during the condensation

are described by the appropriate differential mass

and energy balances.

Basic quantities appearing in these balances are

the mass fluxes of the condensing species and the flux

of energy transferred between the phases. Unequivo-

cal definition of the fluxes requires knowledge of the

interfacial state variables, which determine the kinet-

ics of the condensation. To evaluate these variables

the relations employed are those which hold at

steady state for any cross-section perpendicular to

the interface:

-equality of the mass fluxes at the interface

Nio = N,L

i=l,2,...,n-1

-

energy balance at the interface

i Nj AHj + a,(7, - Z-,) =

k(T, - 7,-)

j I

(1)

(2)

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CONDENSAT

I

1

GC

G,+dG, G,+dG.

T, + dT,

T,+fl, Ta+flo

5, + dX, v, + dv,

Fig. 1. Schematic diagram of concentration and temperature

profiles in a cross-sectional area of the condenser.

where

1

-z~+~+$

k

W

-equilibrium relations at the interface

y,=.L(x,,, -5, . . . 2 %I,,T*, PI

i=l,2,...,n

(3)

(4)

i=

I

If we assume that the pressure P, the vapour

temperature TG, the complete composition of both

phases, yi and xi, and the coolant temperature Tc

are all given for a certain cross-section of the con-

denser, then 2n + 2 unknown variables may be calcu-

lated using eqns. (l)-(S), namely n mole fractions

y2(i=l,2,...,n),

n mole fractions ya (i =

. . ,

n),

the interfacial temperature

T I

and the

t&~l.flux of the condensing vapour N.

The mass flux of the ith component in the vapour

phase is given by

Nit = 4.a + Y, f Nj = AGI + Y,N

j=l

i-1,2,...

,n-1 (6)

where the diffusional flux is

n-1

LX = c B?gAYj Yj,)

(7)

j=l

For the liquid phase a similar relation is valid:

n-1

N,L = c &X&1 - Xi) + XiIN

i=l,2,...,n-1

j= I

(8)

If the flow rates, composition and temperatures of

both phases are known in a given cross-section, then

the matrices of the mass transfer coefficients, &

and &, may be determined using the procedure

described in refs. 11 and 12.

By introducing the relations defining the mass

fluxes, eqns. (6)-(g), into eqn. (1) we obtain

n--l

n--l

jc,

(S - xi) + -TIN=

jc,

B~(rj -

y,d +

Y,~N

i=l,2,...

*n--l (9)

which can be written in a concise matrix form:

fMx, - x) + x,N = B%Y - ~1) + YIN

(10)

where E and p; are the (n - 1) x (n - 1) matrices of

mass transfer coefficients, and y, x, y, and x1 are the

(n - 1) x 1 column matrices of mole fractions in the

bulk vapour and liquid phases and at the interface,

respectively.

According to the assumptions of the model the

mole fractions x,, and y, are related by the equi-

librium formulae (3), which may be written as fol-

lows:

yi1 = Ki (x,, T,, P)

i =

1,2,. . . ,n

(11)

where, as was pointed out, the equilibrium constants

will, in general, depend on the composition of the

liquid phase, the interfacial temperature and the

pressure.

For an ideal mixture, the equilibrium constants

will be functions of the temperature and pressure

only:

K, = Pn(Td

,

P

(2)

Upon introducing the diagonal matrix K, the

equilibrium relation ( 11) may be expressed in the

matrix form

that is,

YI = Kx,

(13)

(14)

Inserting eqn. (14) into ( 10) and solving for x1 we

get

x Y + mi)-VW

(1%

Each of the matrices of mass transfer coefficients

is a function of the total mass flux of the condensing

vapour N through the matrix of correction factors 8

[ 131, that is,

Bt; = B&G

(16)

BIL= &@L

(17)

where the matrices & and 8, are functions of the

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following matrices:

&=NBo-

(18)

&= NBC

(19)

The form of these functions depends on the mass

transport model employed.

Assuming that the positive directions of the fluxes

pertain to species leaving the vapour phase, the

following functions defining the correction matrices

are obtained [ 131:

-according to the film theory

@G

= @o[K - exp( -W-

0, = h_[exp~~d - El - I

-according

to the penetration theory

O,=exp(- z)k-erf( )]-

O,=exp(- Z)[E+erf(- 5)]-

(20)

(21)

(22)

(23)

It has also been shown in ref. 13 that the limiting

values of the correction matrices are independent of

the mass transfer model used. Thus, for the gas

phase we have

Oor o~co

for

&i+cc (24)

Oo=E

for

&=O

(25)

interfacial temperature for the vapour composition

considered, namely T, y) . Th is mode of condensa-

tion is termed the local total condensation.

The second limiting case is obtained for very small

fluxes of the condensing vapour: N 40, that is,

@+O.

Making use of eqns. (25) and (27) we get for this

case the limiting value of the liquid-phase composi-

tion at the interface:

~~~ox,= Bo-~~+K)- ~+PG-l~~~)

(30)

*L+O

Equation (30) was obtained by assuming a value

of zero for the total flux N, without discussing

further the mechanism of such a process. Owing to

the fact that in the condensation process the mass

transport is unidirectional, the value N = KY_ r NJ

may be equal to zero only when all fluxes of the

condensing species also approach zero (Ni +O,

i =l ,2 ,..., n ).

Therefore, in order to determine correctly the

concentrations at the interface, the interfacial energy

balance should be taken into account, with the indi-

vidual mass fluxes given by eqn. (6).

Hence, after some transformations we obtain

N = k T , - T c ) - a ; TG T , ) - i I s I AH j

A&

j-1

II

31)

while for the liquid phase

@L=o

for

cbI_+co

26)

AH;, = 2 Y7AHj

j=- I

&=E

for

(PI_=0

(27)

Upon introducing eqns. ( 16)-( 19) into eqn. ( 15),

which defines the mole fractions at the interface, we

may transform it into the following form:

x1 = [@,--&&,-OL + K +

O,-e&E - K)] --I

x (Y + -4k+L-@LX)

(28)

It follows from the above equation that the liquid

composition at the interface indeed depends on the

intensity of condensation, characterized by the mag-

nitude of the tlux of the condensing vapour, N,

which appears in the matrices 0 and 4. Let us

examine more closely the change of x, as a function

of 8 and 9.

Using eqns. (24) and (26) it may be proved that

d_@_

Xl =

Y

TB XI = Y) < TI < T,(Y) (35)

When

TG > T,, y),

that is, when the vapour is

Thus, we reach an interesting conclusion: for very superheated, some temperature difference exists be-

large fluxes of the condensing vapour the liquid

tween the bulk vapour and the interface, but there is

composition at the interface becomes equal to that in

no further driving force to take away the superheat

the bulk vapour. Since, according to the assumptions

Tc = T , , y) = T I ) . Th us,

cooling of the superheated

of the model, there exists thermodynamic equi-

vapour may occur with simultaneous evaporation of

librium at the interface, the interfacial temperature

the condensate. Generally, however, in such in-

T I must be equal to the boiling point of the liquid

stances (N + 0) the superheated vapour already un-

whose composition is equal to the bulk vapour com-

dergoes cooling down to the dew point temperature

position (y,). This is the lowest possible value of the

in the inlet part of the heat exchanger.

32)

It may be seen from eqn. (31) that if N is to be

zero the following relations must hold:

Tc = T I = TG

33)

&ir=O*Y,=Y,

i=l,2,...,n

(34)

Thus, no concentration gradients appear in the

vapour, and its interfacial composition is equal to

that of the bulk phase. Since thermodynamic equi-

librium prevails at the interface, Y, ( =yJ is in equi-

librium with x,, and, consequently,

T I

must be equal

to the dew point temperature of the vapour of

composition y, namely,

T, y).

This mode of the

process is called the equilibrium condensation.

For finite values of N from the interval (0, co) the

interfacial temperature falls within the range

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194

In the quantitative description of the condensa-

tion certain simplifying assumptions are usually em-

ployed concerning the mass transfer in the liquid

phase.

As has already been mentioned, for large values

of N the correction matrix e,_ tends to zero and,

consequently, the matrix of transfer coefficients B:

also takes very small values. Moreover, the flow of

the condensate is usually laminar, which also leads to

very small mass transfer coefficients.

Hence, the following model of mass transport in

the liquid phase may be assumed:

Pi-0

and hence I

IL0 (36)

that is,

N, z x,~N

(37)

With such an assumption the calculations become

considerably simpler as the diffusional fluxes in the

liquid phase need no longer be determined. Equation

(28) may thus be reduced to

x, =

[K + 8,-+,(E -K)] -y

(38)

satisfying the limiting cases for Q. + co and &. + 0

derived using the general formula (28).

Finally, making use of the film theory we obtain

the following set of algebraic equations defining the

interfacial parameters and the total flux of the con-

densing vapour:

x, =

[E - exp( -&)(E - K)]-*y

(39)

N = [k(T, - TG) -

az T, -

T,)]/AH,

(41)

where

NCmx

aE 1 - exp( -NC,,,/c&

(42)

3. Concentratioo and temperature profdes

along the condense r

The interfacial state variables, determined using

the relations derived in the previous Section, un-

equivocally define all the mass and energy fluxes.

These fluxes may now be employed in the appropri-

ate mass and heat balances characterizing the con-

denser analysed.

For the parallel flow of the vapour, condensate

and cooling medium the mass and heat balances are

given by the following set of differential equations:

(43)

d.vi

1

dF= +QVi-yy,N)

i=l ,2,...,n-I

(44)

dT,

dF +

c

NC

wm,

-=

G&&y

(45)

d

Tc

-=+

w-c- TX)

dF

- G&c

(46)

where the negative sign pertains to the vapour and

the coolant flowing cocurrently with the condensate

(which, in turn, flows down the condenser under the

action of gravity).

In the detailed design calculations it is convenient

to introduce a parameter describing the degree of

condensation,

u = (Gco - GG)/GGO

which varies within the range [0, 11.

(47)

Inserting o into eqns. (43))(46) and using the

assumptions introduced earlier, finally we obtain the

following differential equations:

dYi

Yi - X~I

_-

du-

l-v

i=l,2,...,n-1

dTo

---Z

dv

(49)

dTc W, - Tc) Go

-=

dv G&c N

(50)

The cocurrent flow of the vapour, condensate and

cooling medium is assumed in these equations.

Moreover, in each cross-sectional area of the con-

denser defined by the degree of condensation v, the

following relations must hold which determine the

flow rates of both phases together with the average

composition of the liquid:

Go=G&l-t))

(51)

G,_ = Go,,v

(52)

xi=Ye-Ylv -4

i=l,2,...,n-1

V

(53)

The heat transfer area is calculated using the

equation

F= Goo

dv/N

It shozld be noted

that the intensity of condensa-

tion has a substantial effect on the state variables of

the condensing mixture. Applying the conclusions

given in the previous Section, for the local total

condensation we have

(54)

dri

-0

do-

i=l,2,...,n-1

and

dTo/dv = 0

(56)

that is, for large fluxes of the condensing vapour its

composition and temperature do not change as the

condensation proceeds.

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4. Numerical solution of tbe model equations

The boundary conditions associated with these

In order to determine from eqn. (54) the heat

transfer area required for the condensation of a given

equations are defined by the inlet composition and

multicomponent mixture, it is necessary to integrate

eqns. (48)-(50) over the whole range of degrees of

temperature of the vapour mixture:

condensation (V = O-l).

To = Too,

Yi =y,

i=l,2,...,n-I

for v = 0

(57)

Depending on the direction of flow of the cooling

medium we thus have either an initial value problem

(cocurrent flow:

Tc = T ,,

for v = 0) or a boundary

value problem (countercurrent flow: T, = T,, for

v = 1).

The equations analysed were integrated numeri-

cally using the modified Euler method. Employing

certain special features of the right-hand sides of

eqns. (48)-(50), their partial analytical integration

was carried out within a single integration step. The

recurrent formulae which explicitly define the values

of yi,

T ,

and

T ,

at the end of each step were thus

obtained.

The use of such an algorithm substantially

reduces the calculation time with respect to the

fourth-order Runge-Kutta-Gill method, without di-

minishing the overall accuracy.

Thus, the numerical integration was performed on

the basis of the following recurrent formulae defining

the values of the parameter at the end of each

integration step:

vk+,=vk+Av

(58)

GGO

Fk+,=Fk+-Au

N/C

(59)

1 -v,

Y i , k+ l =X i l , k + Y i , k -X i * , k ) 1 _v

& + I

Tcc ,k +, = T,,k +(Tak - T1.k ) ( ; I> (61)

where

T

C.k+ l = T,,k +(TC,k --T , ,k )

=P

-k, Go Av

--

GzCpc Nk

>

G

G,k+I =GG~(l-vk+ l )

G

,_,,~+,=GGoQ+,

(62)

(63)

(W

x i , k + I =

YzU-Yi ,k + l( l -vk+ l )

v k + l

The values of the interfacial state variables (x,,

T,)

required at each integration step were calculated

iteratively from the algebraic equations (39) and

195

(40), employing the Newton-Raphson procedure for

an assumed value of the total flux of condensation N.

The value of N was subsequently verified using the

energy balance (41). If the error was found to be

larger than an arbitrarily selected value E, N was

modified according to the method of interpolation

and the interfacial state variables were recalculated.

The matrix function exp( -I&) appearing in eqn.

(39) was evaluated using Sylvesters theorem:

n-1

exp( -+o) = C Zo(&) exp( -*o,)

r-1

(66)

where the matrix coefficients Z,(A,) are defined by the

following formulae:

n-1

n (D- -SE)

j=

Z,(4) = ,_

jg, (2, 4)

j#V

(67)

in which D-r = A represents the inverse of the multi-

component diffusion coefficients matrix and ;lj are

the eigenvalues of this matrix.

The elements of the inverse matrix D- = A are

given by

A..=J L+

2

22

Din ]=I D,

iti

(68)

The necessary eigenvalues Iii of D-i = A were

computed by employing two separate methods: the

characteristic polynomial for a ternary mixture, and

the transformation method of Burghardt and War-

muzit%ki [ 141 for mixtures containing more than

three components.

The mass transfer coefficients goi appearing in

&ii.

doi = NIBoi (70)

were calculated from the known correlations for

binary systems in which the binary diffusion co-

efficients have been replaced by the eigenvalues

of the multicomponent diffusion matrix, Li. So the

Sherwood and Schmidt numbers were evaluated as

follows:

s ii =& j

SCoi=*

8 Pd 1

U

(71)

The physical properties of the pure species in both

phases were taken from the literature [ 15, 161, while

those for

of the formulae in ref.

At each integration step the physicochemical

parameters the vapour and liquid were

TG and TL ) and compo-

sition of the phases. The vapour-liquid equilibrium

data were taken from refs. 18 and 19.

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196

5. Numerical results ad conclusions

In order to test the numerical procedures and

algorithms proposed, the calculations were carried

out for two different systems, a ternary mixture and

a five-component system.

As a ternary mixture the system benzene-toluene-

xylene, condensing inside vertical tubes of diameter

d = 0.021 m, was selected.

The local vapour mass and heat transfer co-

efficients Boi and cto

were calculated for turbulent

flow with the aid of the following correlations:

Nu, = 0.026 Re,O.* Pr,13

(72)

Sh,, = 0.026 Reo.s Scoi3

(73)

For laminar flow the equations which approxi-

mate the theoretical Graetz solution were used.

The heat transfer coefficient for the condensate

film was evaluated from the formula [ 151

Nu, = ( Nu,~ + Nu,~) 4

(74)

where

Nu, = 0.797 Re,-1

(75)

Nu, = 8.66 x lo- Re,0.382 Prr,o,5689

(76)

The inlet vapour and coolant flow rates and the

coefficient of heat transfer into the cooling medium

were kept constant in all calculations:

Go0 = 1.89 x lO-5 kmol s-

Gco = 0.248 kg s-

t(c = 1745 W m-* K-

However, the initial composition and temperature

of the condensing mixture, as well as the inlet tem-

perature of the cooling medium, were varied appre-

ciably (the initial conditions for which the

computations were performed are summarized in

Table 1). By making substantial changes in the initial

composition and, especially, in the temperature of

the coolant, the stability and convergence of the

numerical method were tested over a wide range of

condensation intensities. As the choice of the initial

temperature of the coolant, Too, affects the magni-

tude of the total flux of the condensing vapour, N, it

is possible to verify the method over the whole

region between the local total condensation and the

equilibrium condensation.

TABLE I. Initial conditions for the mxture benzene-toluene-xylene

In Table 1 the values of the final degree of

condensation, v,, above which condensation could

no longer occur, are given for certain temperatures

of the cooling medium. In a cross-section of the

condenser characterized by this particular value, vk,

the temperature difference between the interface and

the coolant dropped to a few tenths of a degree

Kelvin and the flux N tended to zero.

In all the calculations shown in Table 1, that is,

for both very large and very small intensities of

condensation, the method proposed here was conver-

gent and stable.

Figures 2-6 illustrate the concentration and tem-

perature profiles in the condenser as functions of the

degree of condensation for two values of the temper-

ature of the cooling medium,

T ,-, =

293 K and

Fig. 2. Bulk and interface. concentration profiles in the vapour

and liquid phases as functions of the degree of condensation for

the mxture benzenetohtene-xylene (y,, = yss = 0.330).

T [Kl

Tm-293 K

39l

lb

375.

fi -

Tr

I I

I I I

I I

I I

0

92 a4 06

96 v

l .0

Fig. 3. Profiles of the dew point, boiling point and interface

temperatures as functions of the degree of condensation for the

mxture benzene-toluette-xyleneene(y,, = yrn = 0.330).

Initial vapour Temperature of cooling

composition

medium, T,, (K)

YIO Y20 293 323 343 348 352 353 315 382 383 385 387 389 410

0.33 0.33 x x x , = 0.75 I& 0.4 k = 0.3 k o.z * 0.09

0.99 0.001 x x x 0, = 0.98

0.001 0.99 x 0, = 0.98

0.001 0.001 x ut O.%

x ,

numerical calculations performed; I+, maximal degree of condensation.

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Q37

Q36_

Q35-

Q32-

0

a2 a4

a6

aa

v

1.0

Fig. 4. Vapour-phase and interface concentration profiles as func-

tions of the degree of condensation for the mixture benzene-

toluene-xylene (y10 = y,, = 0.330).

a33

a3(

1

1 I

I

I

I I I

I I

I I

a2 a4

0.6

a8

1.0 v

Fig. 5. Liquid-phase concentration profiles as functions of the

degree of condensation for the mixture benzen~toluene-xylene

(Y10 = y20 = 0.330).

____

Tkl,

389

t

l-b

387 -

385 -

383-

38f -

379-

To-353 K

ii;

0

0 1

d.2 - a3

1

a4

a5

a6

a7

as

1.

Fig. 6. Profrles of the dew point, boiling point and interface temperatures as functions of the degree of condensation for the mxture

benzene-toluene-xylene ( y10 = yzO= 0.330).

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198

353 K, and thus for widely different condensation

intensities.

For To, = 293 K the process fully corresponds to

the local total condensation. This is demonstrated by

the fact that the vapour and liquid composition

remain unchanged along the condenser and that the

liquid composition at the interface is equal to that of

the bulk vapour (Fig. 2). The interfacial tempera-

ture, also constant throughout the condenser, is

equal to the boiling point of the liquid whose compo-

sition is yi (Fig. 3).

In Figs. 46, showing the course of condensation

at T ,-, =

353 K, the differences between the inter-

facial liquid composition and that of the bulk vapour

are clearly visible; also, the interfacial temperature

differs from the boiling point of the liquid of the

same composition as the vapour phase. Even in that

case, however, the course of the process, characteris-

tic of the final stages of condensation (v -+ l), be-

comes evident.

For v = 0.8, the liquid composition at the inter-

face and that of the bulk vapour become equal; the

interfacial temperature T , also becomes equal to the

boiling point of the liquid phase, Ts. This is clear

evidence of the fact that, for such degrees of conden-

sation, local total condensation occurs in the system:

the matrix tends to infinity as the matrix of mass

transfer coefficients BG becomes equal to zero, which,

in turn, results from the vapour flow rate approach-

ing zero. The analysis of this phenomenon is much

the same as that for binary mixtures [5].

Another example is a design procedure for a

condenser-evaporator in which a five-component

mixture condenses in the intertubular space, while

the cooling medium, R-l 14, evaporates inside the

copper tubes of

d ,Jd i 20 /16

mm and pitch

t = 1.5

do.

The inlet vapour flow rate, temperature

and pressure are GGO 40 kg SC, TGo 448.2 K and

P =

0.1013 MPa, respectively. The evaporation of

R-l 14 occurs at constant pressure; it is also assumed

that the coefficient of heat transfer into the coolant is

constant and equal to 7500 W rn- K-.

Because of the nature of the process, the highest

evaporation temperature of R-l 14 at which the total

condensation of the mixture was still possible had to

be determined. The composition of the mixture is

given in Table 2.

The calculations were performed for various boil-

ing points of R-l 14. At the boiling points equal to

293 K and 298 K and for the corresponding pres-

sures P, = 0.187 and 0.221 MPa, the mixture con-

densed completely. For the boiling point of 303 K

TABLE 2. Composition of the five-component mixture

the condensation is no longer complete, since for

L)~= 0.99 the interfacial temperature

T ,

reaches

303.3 K and the condensation ceases.

The concentration and temperature profiles calcu-

lated for Tco = 293, 298 and 303 K are shown in

Figs. 7-9.

Appreciable variation of the liquid and vapour

composition along the condenser, together with large

differences between the interfacial liquid composition

and that of the bulk vapour, which may be seen in

Figs. 7 and 8, suggest that the process analysed

differs from the local total condensation.

It is also evident from Fig. 9, which shows the

interfacial temperature profiles

T,,

that the process

does not approach that of the equilibrium condensa-

tion, as T , is markedly different from the saturation

Fig. 7. Vapour-phase and interface concentration profiles as func-

tions of the degree of condensation for the five-component mix-

ture Table 2).

Component

Initial composition y, Boiling temperature (K)

Molecular mass

2-Methylbutane

0.15

301.0

72.15

Ethylbenzene

0.25

409.3

106.17

Methylcyclohexane

0.20

374.1

98.19

Butylcyclopentane

0.25

429.9

126.24

n-Hexylcyclopentane

0.15

476.3

154.30

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199

and becomes identical with that for binary mixtures:

Tco=293 K

Fig. 8. Liquid-phase concentration profiles as functions of the

degree of condensation for the five-component mixture Table 2).

temperature of the mixture,

T,,.

Even in this case,

however, the local total condensation may be ob-

served during the final stages of the process (v + 1),

thus confirming theoretical conclusions reached in

the previous Sections. The numerical example pre-

sented here is rather special from two points of view.

Firstly, the difference between the boiling points of

the individual components is large (about 175 K);

secondly, at the final stage of condensation the differ-

ence between the interfacial temperature and that of

the coolant does not exceed more than a few degrees

(for TcO = 303 K it falls to zero). Consequently, the

condensation intensities in the process studied cover

a relatively wide range of values.

As the number of components increases, the cal-

culations become progressively more laborious and

time-consuming. This results mainly from having to

evaluate the matrix function exp( -4o) at each inte-

gration step, and thus determining n - 1 eigenvalues

of the matrix of diffusion coefficients (n = number of

components). As this is the most time-consuming

operation in the whole algorithm, a simplified calcu-

lational procedure has been proposed: assuming that

for the majority of multicomponent mixtures en-

countered in practice the matrices of diffusion co-

efficients

are diagonally dominant [ 141, the

non-diagonal elements (and, consequently, the diffu-

sional cross-effects) may be neglected.

The matrix relation (39) is thus markedly simpler

Yi

Xi=l+(Ki-l)exp(-~oi)

(77)

where

+oi = NIBoi (78)

leading to a fivefold reduction of the calculation

time.

In order to compare the simplified method with

the exact results, the error involved in evaluating the

total flux of the condensing vapour was determined

as a function of the degree of condensation; the

results are presented in Fig. 10. It may be seen from

this Figure that the error becomes appreciable only

during the terminal stage of the process, when the

flux of the condensing vapour decreases as a result of

diminishing temperature differences.

Figure 11 illustrates the error resulting from the

use of the simplified method in determining the

length of the tubes of the condenser. This error is

large only for the cases where the temperature of the

cooling medium approaches that of the interface.

Taking into account the fact that the example of

the design calculations for the condenser-evaporator

presented in this work is rather special and that the

temperature differences in industrial condensers sel-

dom reach such low values, the simplified method

seems to be fully applicable to determining the area

of heat transfer during multicomponent condensation.

Nomenclature

C

CP

D,

D

d

E

F

G

g

AH

4

K

k

M

Ni

N

P

P&l3

T

V

W

xi

Yi

molar density of fluid mixture, kmol me3

specific heat, kJ mol- K-

binary diffusion coefficient, m2 s-i

multicomponent diffusion coefficients matrix,

m* s-

tube diameter, m

unit matrix

heat transfer area, m*

molar flow rate, kmol s-

acceleration due to gravity, m s-*

molar enthalpy of evaporation, kJ kmoll

diffusional flux of component i , kmol m-* s-i

= Y,~/x,, equilibrium constant

overall heat transfer coefficient, W m-* K-i

molar mass, kg kmoll

molar flux of component

i ,

kmol m-* s-i

=X7=, Nj, total molar flux of mixture,

kmol rnd2 ss

pressure, Pa

vapour pressure of component i , Pa

temperature, K

degree of condensation

velocity of fluid, m s-l

mole fraction in liquid phase

mole fraction in vapour phase

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200

428-

0

a2

Q3 Q4

a5

a7

a9 v

T [Kl r

r

1s

333 1

303

, I

0

02 a4 06 08 v

0

Fig. 9. Profiles of the dew point, boiling point and interface temperatures as functions of the degree of condensation for the five-component

mixture Table 2).

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201

18

46

4

2

4

~ 298 K

I

I

Ta-293 K

I

I

Fig. 10. Error in the total condensation flux calculated using the

simplified method as a function of the degree of condensation for

the five-component mixture (Table 2).

CL

B

u

Nur_

Pr

Re

ReL

SCGi

Sk

heat

t ra nsfe r co efficient, W m - K -

mass transfer coefficients matrix,

kmol m-* s-

mass transfer coefficient, kmol m - 2 s-

= G,M,/dx, wetting rate, kg m s-

thickness, m

dynamic viscosity, kg m-

s

correction factor matrix

the rm a l conductivi ty , W m - K -

eigenvalue of matrix

D-

me2

s

density, kg m--3

=

ad/l,

Nusselt number

= a,(q,*/p,g) 3/,IL, Nusselt number for

condensate

= CJ /A., Prandtl number

= wdplq, Reynolds number

=r _, Reynolds number for condensate

= v/( I/ ), calculational Schmidt number for

component i

= /& d/C( I/ ), calculational Sherwood

number for component i

Subscripts

:

refers to boiling point

refers to coolant

D refers to dew point

I

I

.FoIK]

293 29s

3 :

4 _

-2 -

-3 -

-4 -

-5 -

b=D_L

I

,M)%

-6 _

9

1

il

Fig. II. Error in the length of the condenser tubes calculated

using the simplified method as a function of the cooling medium

temperature for the five-component mixture (Table 2).

refers to vapour

I

refers to interface

i

denotes component

L

refers to condensate film

M

refers to mixture

W

refers to wall

0 refers to inlet conditions

Superscript

denotes corrected heat and mass transfer

coefficients

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