Residue Curve Maps

1. ON THE DYNAMICS OF DISTILLATION PROCESSES-III
THE TOPOLOGICAL STRUCTURE OF TERNARY
RESIDUE CURVE MAPS
M F DOHERTY
Department of Chemical Engmeenng. Goessmann Laboratory. Umverslty of Massachusetts, Amherst, MA 01003,
USA
and
J D PERKINS
Department of Chenucal Engmeenng and Chemical Technology, Impenal College, South Kensmgton, London SW7
2BY. England
(Received for pubhcatwn 6 Apnl 1979)
Abstract-The dtierentlal equations descnkung the sunple dlstdlatlon of azeotropic ternary mixtures place a
physrcally meanmgful structure (tangent vector field) on ternary phase &grams By recogmzmg that such
structures are subJect ot the PorncarLHopf mdex theorem It has been possible to obtam a topologrcal relatlonshrp
between the azeotropes and pure components occumng m a ternary mature Thus relahonshlp gives useful
mformatlon about the dlstdlatlon behavior of ternary rruxtures and also predicts sltuahons m which ternary
azeotropes cannot occur
INTRODUCTION
One of the most tnterestlng problems m the ther-
modynanucs of azeotrop~c rmxtures IS the theoretical
predIction of the hkehhood of the appearance of ternary
azeotropes usmg only a knowledge of the propeties of
the three constituent bmary mrxtures If a detailed
knowledge of the bmary mixtures IS avadable (as
expressed by knowmg the parameters m a hqrud soluUon
model for each of the bmary paus) then It 1s possible to
determme the presence or absence of a ternary azeo-
trope by numerically solvmg the azeotroplc equlhbnum
equations
Px, -P:(T)x,yi(A,x)=O I = 1,2,3 (1)
In wntmg these equations we have assumed that the
vapour LSan adeal mixture and that the ternary actwlty
coefficients can be determmed from a knowledge of the
binary mteraction parameters
Unless we are mterested m the detied behavlour of a
spectic system this approach requires too much m-
formation and provides httle InsIght as to why and when
ternary azeotropes occur Furthermore, there 1s no
guarantee that the model even gives quahtahvely correct
results (I e It may predict the presence of a ternary
azeotrope which does not exist m reality) It IS known
that hquad solution models sometunes do predict quahta-
tively incorrect phase behavlour [1, 51
The estlmatlon of the concentration regon to which a
thllxtures showmg the behavior gwen m Figs 1, 10 and II are
hsted m Table 1
ternary azeotrope must be confined has been the subject
of several arttcles tn the Russian hterature[2-41 (other
Refs are gwen tn those works) These articles state
necessary condltlons for the occurrence of ternary azeo-
tropes As Maruuchev[3] points out, “it IS possible with
the aid of the Margules-Wohl method, without calculat-
mg hqmd-vapour eqmhbnum data, to estunate the con-
centration re@on of the ternary azeotrope from data on
the binary systems and pure components The questlon
of the existence of the ternary azeotrope at a gven
temperature remams open”
It IS possible to obtam a few general results on this
question without recourse to solution models In obtam-
mg these results we also obtam useful mformatlon about
the distlllatlon characteristics of the ternary mixture
This will become apparent as the methodology IS des-
cnbed
Instead of dealing with solution models, we formulate
the pair of ordinary ddferentlal equations
dxa
dS=X’-Y’
dx2
dS =x2- y2
and mvestlgate the properties of the smgular pomts
(steady state solutions) Clearly, this 1s an enmely
equivalent problem but appears to be more amenable to
analysts than other formulations
EquaUon (2) defines the locus of the residue hquld
compos%tion as It changes with time m a simple dls-
tdlation process The tra]ectone.s of (2) are usually called
residue curves and typical residue curve diagrams are
shown 111Fig 1 t It can be proved[6-81 that eqn (2) has
the followmg properties
1401

2. 1402 M F DOHERTY and J D PERKINS
B
(al
C B
(bl
C
AL.I!!kLAhB
lc)
C 0
Cd)
C
Rg I Examples of resbduecurve maps See Table 1 for example
mixtures
P(l) The solutions are confined to he either on the
boundary of or interior to the nght mangle of umt side
P(u) The Independent vmable, 5, IS a dlmenslontess
measure of time It 1s a nonhnear, stictly monotomcally
mcreasmg function of tune and IS defined on the mterval
[O,+ 001
P(m) The singular points of eqn (2) occur at all pure
component vertices, bmary azeotropes and ternary
azeotropes These points are always isolated
P(W) When the equatkon IS Imeamed about azeotro-
plc smgular pomts the elgenvalues are datmct and real
It follows that there can be at most one zero elgenvalue
P(v) When the equation 1s hneanzed about pure
component vertices the elgenvalues are real but not
necessanly dlstmct One zero eigenvalue occurs for
every dIrectIon, k, In which
PX =
0
the denvatwe hemg evaluated at the vertex of Interest
Therefore, one zero elgenvalue occurs for every binary
nurture which exhibits a tangential azeotrope at the
vertex of Interest It IShighly unhkely that more than one
tangential azeotrope would occur at a vertex at any one
time so we will assume that in general, there will be at
most one zero elgenvalue More details are gwen m
Appendix I
P(vi) Limit cycles cannot occur
(vu) The temperature surface IS a naturally occurring
Llapounov function for this equation The movement of
the hqmd composltlon, x(t), IS always m a direction
which makes the temperature increase
Having estabhshed the general propertles of the solu-
tions of the dlfferentml equation we are now m a position
to ask the questlon, what ISthe relabonshlp between the
number and the type of singular points which eqn (2) can
display7
Since the tune of Porncar& a large body of mathema-
tlcal literature has been devoted to answenng Just such a
question for abstract dtierentmi equations The relevant
theory is calIed degree theory and by its apphcation we
are able to answer the question we have posed regardmg
eqn (2)
In order that the non-specmhst reader may follow our
reasoning, we include a short summary of the pnnclpal
concepts of degree theory
Degree
The followmg IS based on the excelIent monograph by
Temme [9]
Let fi be an open set 111R”, its boundary ISdenoted by
Xl and Its closure by fi The degree of a smooth
function f fi+ R” Hrlth respect to the regon a at the
pomt p E R” -f(&k) -f(Z) IS denoted by deg (f, fi, p)
and defined by
deg (fi a, P) = 7 sgn I;(K) (3)
where xi E f-‘(p) n n (I e it IS a solution of the equa-
tion f(x) = p) and the sum in eqn (3) IS taken over all
such solutions contamed m n The symbol J,(x*) sIgndies
the determinant of the Jacobian matnx of first partial
denvatlves of f evaluated at the point x, The symbol
f(z) represents the image of the set of all points in fl for
which J,(x) = 0
It can be seen at the outset that degree theory
concerns itself vvlth the propertIes of the solutions of an
algebrac equation of the form f(x) = p If we have a set
of dlfferentlal equations x = f(x) then degree theory can
help us determme the propeties of the singular points d
we mvestlgate the degree at p = 0, deg cf. a, 0)
If &(x,) = 0 then the degree of f at p IS defined by
deg ti a, PI = da (f, Q, 4) (4)
where q IS any point near p at which Jf(x,) # 0 for all x,
which satisfy f(x) = q The degree deg (f, il. q) is then
found by the use of eqn (3) It can be proved that the
degree at all pomts 4 near p IS the same and so eqn (4) IS
well defined
For two dlmenslonal systems there are two
(equivalent) geometnc ways of finding the degree These
are,
C(i) The degree ISthe number of hmes that the Image
f(an) encircles p where anticlockwlse rotation 1s coun-
ted posltlvely and clockwIse rotation negatively Exam-
ples are aven m Fu 2
C(u)
deg (f, Q. PI = g (3
where At? IS the change m angle that f makes with
respect to some fixed duecuon as x traverses an in the
positwe direction (antlctockwse)

3. On the dynamics of dlshIlafion processes-III 1403
f(ad
Q-f-&a 609-0
1 n 1_._qY-&L-J
I I
d”
F@ 2 Examples of degree
A necessary (but not sticlent) wn&Uon for the
degree to change 1s that p moves across the unage of ail
(Ie across f(8l-i)) It IS because of thus fact that the
degree of a pomt p m the unage of ail (1e m f(an)) IS
not defined If such is the case then much of the mate&
described m this section ceases to be apphcable We wll
take up this pomt agam later
Degree has many useful propertres, some of them are
hsted below
D(1) If f-‘(p) IS empty then deg (f. ik p) = 0
D(2) If deg (f, a, p) # 0 then f(x) = p has at least one
solution m R
D(3) Let f,, and f1 be smooth functions from fi+R”
Define for 0 5 t 5 1, ft = rfl + (I- t)fo Suppose that
p +Zfi(ail) for all t E 10.11 then deg(ft,il,p) IS m-
dependent of t In partuzular, deg (f, , fk P) =
deg(f,,n,p) Thus 1s a resticted version of the homo-
topy mvarumce character of degree and IS a very useful
property
D(4) Let p E f(Xk) and K be a closed subset of fi
such that P G f(K) Then deg ti Q, P) =
deg (f, i-i - K, p) This 1s called the exclsslon property
We can use degree to define the mdex of an isolated
solution of f(x) = p
Index
Let f be a smooth function from ai-, R” If x E n IS
an Isolated pomt of f-‘(p) then we define the mdex of f
relative to p at the pomt x by,
md Cf,x.P) = deg (f, &(x), P) (6)
where B,(x) IS an open ball of radius I around x such
that B,(x) C l2 and contams a smgle solution of the
equation f(x) = p Property D(4) ensures that this
defimtlon does not depend on r In fact D(4) allows us to
replace B,(x) m the above defimtion by any open set
around x provided the set hes wholly 111a and contams
only one solution of f(x) = p
It follows from eqn (3) that
md(f,x,p)=sguJ,(x)=sgn G A~
i-1
Q
where A, are the elgenvalues of the Jacobian matnx of f
at x (multiple elgenvalues being counted multiple times)
Therefore, when J,(x) # 0 , md (f, x, p) = + 1 or - 1 If
Jr(x) = 0 the right hand side of (6) must be replaced by
deg(f, B,(x), q) and now the possibdlty exists that
f(x) = q has multiple solutions mslde B,(x) In this case
deg cf. E,(x). q) may be dtierent from + 1 or - 1. con-
sequently md (f, x, p) may be ddferent from + 1 or - 1
By apphcatlon of D(4) we have the useful relatlonshlp,
deg Cf.Cl,P) = z md(f..G, P)
1
x,Ef’t.P)nn (8)
Consider now the pau of ddferentlal equations,
Assume (9) has k isolated smgular points, xt, xll, xX
Then the mdex of the rth smgular pomt, md (f, xp, 0), IS
snnply the sign of &(xF) Eqtnvalently, we could draw a
small c&e around x;” and note the change 111angle, A@,
which the trajectones crossrng the cucle make relative to
some fixed drrection as the circle IStraversed once m the
antlclockwlse duectton, then
md (f, xl+, 0) = g (10)
This procedure uses the fact that the tralectory through
any point, x, pomts m the same direction as f at that
same point
If If(x:) = 0 then the md (f, x,*, 0) cannot be equated
with sgn Jf(x:) but the index can stdl be calculated usmg
eqn (10)
The geometnc method of calculatmg index 1s therefore
apphcable to both elementary (I e J”(x) # 0) and non-
elementary (.&(x) = 0) singular pomts
Bentison ([lo] Chapter X) has summarized this
geometnc method very precisely m the followmg
theorem
Theorem (Ben&son)
Let (0,O) be an Isolated smgular pomt of the dIfferenti
eqn (9) (thus can always be achieved by means of a linear
transformation) Either (0,O) IS
(I) a centre
or (n) a focus
or (m) the open nelghbourhood of the 0~ can be
drvlded mto a fimte number of sectors such that each
sector IS either a fan, a hy-perbohc sector or an elhptic
sector (see FQg 3)
Furthermore, the mdex of the singular pomt at the oqgn
ISgven by
md(f,O,O)= l+v (11)

5. On the dynamics of dlstlllatcon processes-III 1405
lemm and Pollack[l4] Chapter 3 and Mrlnor[lS] Chap-
ters 5 and 6
Havmg estabhshed the mathematrcal framework we
now return to our problem of studytng the smguhu pomts
of eqn (2) subject to the facts P(t)-P(vn) Let us call the
mtertor of the rtght trtangle wtth umt stde @, its boun-
dary a@ and tts closure & The natural startmg pomt ISto
see what f(4) looks hke The fun&tons fr = x1- yr and
f2 = x2- y2 are constramed to he m the unrt square about
the ortgm The Image of &D ts gtven by,
And so we tind the usage space as shown m Ftg 5 We
cannot apply eqn (8) for two reasons, smgular pomts of
the drtTerentta1 equatrons he on a@ (pure component
verttces and bmary axeotropes) and the pomt p = 0 hes
on f(M) Rather than attemptmg to find stumble exten-
sions of f we well dtscard the classrcal approach m
favour of our earlier suggeshon of “piecmg together”
eight tnangles to form a closed, hollow 2-dunenslonal
polyhedron rn R’ (Frg 6) whtch IShomeomorphtc to the
2-sphere We then tnvoke the Pomcar6-Hopf Theorem m
the form,
7md (f, x,,0) = 2 (14)
All that remams to be done 1s estabhsh the mdex of the
various kmds of smgular pomts of eqn (2). multrply the
number of each type occurrmg by its index and sum
them to obtam the left hand side of (14) The resuhmg
expression 1san exphctt relattonshtp between the number
of each type of smgular pomt occurrmg and every three
component axeotroptc mtxture must sattsfy this relatton
The remamder of the paper ts split mto two parts In
Part A we assume that all the singular pomts are ele-
mentary and in Part B we allow for the posstbthty of
Fig 6 Construction of the polyhedron by “puxmg together”
eight compos&on tnangles The polyhedron IShomeomorphlc to
the 2-sphere
non-elementary smguhu pomts tn accordance with pro-
perttes P(w) and P(v)
PART A. ELEMENTARY SINGULAR WINTS
For this case there are two kmds of mtenor smguIar
pomts shown m Frg 7(a),(b) (ternary axeotropes), two
kmds of vertex smgular pomts shown m Ftg 7(c),(d)
(pure components) and two kmds of edge smguku pomts
shown m Frg 7(e). (f) (bmary azeotropes)
There ISa need to discuss the shape of the trajectones
m the vlcrruty of a pure component vertex m more detzul
m order to ~ustiy that Figs 7(c) and (d) are the only
elementary shapes This 1s done m Appendtces I and II
Let us call Fw 7(d) a pure component quarter saddle
and Ftg 7(e) a bmary half saddle Define,
N3 = number of ternary nodes 111the tnangle
S, = number of ternary saddles m the trtangle
A%= number of bmary nodes m the trtangle
S2= number of bmary half saddles m the &tangle
N1 = number of pure component nodes m the b-tangle
S1= number of pure component quarter saddles m the
trtangle
Frg 5 Compostnon tnangle sad rts image under f

6. 1406 MF
Then from eqn (14).
IN3(8N3)+ r~8s3)+~N2(4N2)+~9(4s2)
+IN,(2N, ) + Is,(2Sl) = 2
Do~m-r and J D PERKINS
A (101-c)
(15)
where lN3 IS the mdex associated with a single ternary
node on the surface of the sphere, etc The reason for
the terms 8N3 and 8S3 m (15) IS that each ternary
smgular pomt m the truingle gves nse to e&t unages of
Itself on the surface of the sphere Sumlarly, each bmary
smgular pomt m the triangle gves nse to four unages of
Itself on the surface of the sphere and each pure com-
ponent smgular pomt in the mangle gves nse to two
unages of Itself
Equation (15) must be coupled w&h one constramt
equation,
N,+S,=3 (16)
expressmg the fact that there are exactly three pure
components present
The mdlces m (15) take the values,
IN, = + 1 lg = - 1
I,=+1 r*=-1 (17)
IN‘= + 1 b, = - 1
The values for 1~3, I-, 1~~and lNI need no explanation
The values for I~ and lsl are both - 1 because these
types of singular point m the mangle give nse to genutne
four-hyperbohc-sector saddles on the surface of the
sphere
Between eqns (U&(17) we obtam the desved refa-
tlonshlp.
2N,-2S3+N2-Sz+N,=2 (18)
All the diagrams grven m FQ 1 can be seen to conform
with thts equation One of the most comphcated ternary
IO) (bl
IO) (I)
Fig 7 Elementary smgutar pomts In the composition tnangle
(7s 9-a
(eo 2%) (60 I-c)
Fa 8 Residue curve map for the system methyl
cyclohexane(A) + hexafluorobenzene(E) + benzene(C) at 740 mm
of Hg
diagrams known IS shown m FQ 8 (see Wade and
TayIor [16]) and this too conforms vvlth the equation
Equation (18) can be used to enumerate the possible
residue curve maps ansmg from any pven set of azeo-
tropic data as demonstrated tn the followtng examples
Example 1
Thermodynanucists have often asked whether It IS
possible for a ternary mixture to exhlbtt a ternary azeo-
trope m the absence of any binary azeotropes bemg
present 1171 The problem IS specfied m Fig 9 It can be
seen that vertex 2 ISa quarter saddle and vertices 1 and 3
are nodes, hence, N1 = 2, SZ= 0, N2 = 0 Equation (18)
reduces to,
N3-S,=O (19)
which IS satisfied by N3 = S3 = 0,l. 2 but IS not
satisfied by a smgle ternary azeotrope
We conclude that when all the smgular pomts are
elementary it is not posstble for a si@e ternary azeo-
T, > Ta > T,
F@ 9 Boundary data for ternary mtxture wth no bmary azeo-
tropes

7. On the dynamics of dlstdlation processes-III 1407
trope to occur in a nuxture which displays no bmary
az.eotropes
Example 2
Consider a ternary nuxture with two binary maxunum
bo&ng azeotropes and one bmary muumum boihng
azeotrope as shown m Fu lo(a) Vetices A and B are
quarter saddles and vertex C ISa node, hence N, = 1 and
S, = 2 Also, for this example,
N*+s2=3 (20)
so eqn (18) becomes
Ns-&=2- N2 (21)
Smce N2 IS restncted to take one of the values 0, 1,2,3
we find the followmg structural relationshIps
(I) N2 = 0 implies
Ns-S3=2 (22)
(a)
A 1, (100)
F@ 10 (a) Boundary date for a ternary mlxtve w&h three
maximum bodmtt bmary azeotropes aad one rmmmum botbng
hnary azeotrope (b)-(g) Elementary residue carve maps See
Table 1 for example mutures
The sunplest solution, N, = 2, S, = 0 1s shown m Fig
lo(b)
(II) N2 = 1 unphes
N,-S,= 1 (23)
The sunplest solution, N, = 1, S, = 0 is shown m Fig
10(c) (Na = nummum borlmg) and Fig lo(d) (N3 = maxl-
mum boding)
(m) N2 = 2 Implies
N3-S,=O (24)
The solution N, = S, = 0 is shown in Fig 10(e)
(IV) N2 = 3 implies
N3=Ss-1+N3-Sa=-1 (25)
The simplest solution, N3 = 0, S, = 1 1s shown m Fig
10(f) It 1s mterestmg to note that m Figs 10(b)-(e) there
ISonly one dlstdlation repon From Fig 10(f) we get the
adddlonal information that the bollmg temperature of the
ternary azeotrope must be m the interval T4< TAE < T3
A ternary azeotrope cannot exist m the temperature
range Ts<TH<T4
There are many ways of organumg the sequence of
temperatures m Fx 10 wNe preservmg the specdied
boundary condlttons of two bmary maxunum boding
azeotropes and one bmary nummum boding azeotrope
For example, putting T3equal to 105 leaves the boundary
conditions unchanged but gwes the azeotrope at D a
higher bothng pomt than pure A The temperature
sequence becomes Tl > Ts > T2> T. > TS> Ta It 1s
natural to speculate whether the residue curve maps are
&ected by rearrangements m the temperature sequenc-
mg of this sort Unfortunately, the answer IS yes but the
aBect ISnot so serious as to destroy the ortgmal structure
altogether
Equations (21)-(25) are mdependent of the detads of
the temperature sequence, they are based only on the
boundary condltlon that there are two maxunum bollmg
binary azeotropes and one binary nununum These
equations are mvarmnt under boundary condltlon
preserving temperature sequencmg rearrangements- This
IS sticlent to guarantee preservation of the global fea-
tures of the residue curve maps but not sufficient to
always guarantee local preservation For example, m the
case N3 = S, = 0, we can be sure that N2= 2 for all
boundary condition preservmg temperature sequencmg
rearrangements (so the global feature IS preserved) but
we cannot guarantee that the same two binary azeo-
tropes will remam as nodes after the temperature rear-
rangement has occurred This can be demonstrated by
lettmg T3= 103, T5= 101and T6= 95 thus preserving the
boundary con&tion The new temperature sequence 1s
Tt > T3> T5> Tz > T6> T4 Figures lo(b), (c), (d), (f) are
preserved both globally and locally under this rear-
rangement whde Ftg lo(e) changes to lo(g) Comparmg
Figs 10(e) and (g) we see that N2 = 2 for both of them
but azeotrope E changes its character, becommg a
saddle and azeotrope D becomes a node This change

8. 1408 M F DOHERTYandJ D PERKINS
gves nse to two dlstmct dlstdlation regons m FIN 10(g)
whrch contrasts wrth the smgle dlstdlation region m Fig
10(e) So, the temperature rearrangement leads to no
change m four of the diagrams but a very dramatic
change m the fifth
It ISworthwhrle pomtmg out that the first temperature
rearrangement we suggested (puthng T3 = 105) Ieaves aIl
the dmgrams unchanged
Thrs kmd of pattern recurs throughout all the examples
we have considered which suggests that there IS some
simple undertymg rule which governs whether a tem-
perature rearrangement alters the residue curve map or
not Work IScontmumg m an effort to make this obser-
vation quantitatively precise m the hope that qmck
screenmg procedures can be devised for the selection of
extractrve and azeotroplc dlstiation agents
By reversing all the arrows (Ie reversmg tnne) m Figs
IO(a)-(g) we obtam the set of diagrams generated by the
boundary condltron of two munmum boding bmary
azeotropes and one maximum bollmg bmary azeotrope
Consequently, we need not consider this as a separate
case
Example 3
Consider a ternary murture with three binary maxi-
mum boding azeotropes as shown m Fig 11(a) AU three
vertices are nodes so eqn (18) becomes,
N,-S,= l-N* (26)
A T. CIDO)
Fig I1 (a) Boundary data for a ternary muture with three
maxlmum bodmg bmary azeotropes (b)-(e) Elementary residue
curve maps See Table 1 for example mixtures
Figure Number
Table 1 Example mixtures for figures shown in text
Component A Component B Reference
(Boiling Polnt ‘C) (Boiling Point ‘C)
Caaponent C
(Boiling Polnt lC)
1(a) Acetone (56 4)
1(b) Benzene (80 1)
1 (cl
1(d)
1(d)
lo(b)
10(c)
10(d)
10(e)
10(f)
10(g)
11(b)
11(c)
Aniline (184 4)
Ethylene (-104)
Dichloronethane
(41 5)
Iso~ro~l Ether
Ethyl Fotmate
(54 1)
Cycl ohexane
(80 75)
Paraffin (110)
Acetone (56 15)
Dichlorolnethane
(41 5)
Acetone (56.4)
Water (100)
Acetylene (-84)
Methanol (64.7)
No example known
See
See
See
coarsen t be1 ow***
carment belowC”*
Acetone (56 4)
P-Bmmopropane
(59 4)
c-n t be1 ow***
Methanol (64 7)
Methanol (64 7)
Carbon Disulphide
(46 25)
11 (d)
11 (el
See conenznt below***
No example known
Chloroform (61.2)
Chloroform (61 2)
Hydrazine (113 5)
Ethane (-88)
Acetone (56.4)
Chloroform (61.2)
Chlorofonn (61 2)
Acetone (56.4)
Toluene (110.7)
Methanol (64 7)
Ewe11 and Welch*(20)
Ewe11 and We1ch*(20)
Reinders and BeHinjer
(21.22.23)
Wilson, et al *(24)
Rowlinson(25) (p 184)
Ewe11 and Welch*(20)
Ewe11 and Welch*(PO)
Horsley(26) System
numbers 16301. 1450.
5485, 1447
Horsley”(26) System
numbers 16327. 5378.
1963. 2079
Benedict and Rubin (27)
Horsley)*(26) System
nunbers 16267, 1194.
1175. 1963.
*These papers are concerned with batch drshllatron but contam enough mformatmn to construct sunple dlsUllation resulue curve maps
**These systems are topolo@cally slmdar to the dlagmms but ~tb the arrows reversed
***There ISnothmg topolog~callyunusual about these dmgrams although example systems are not readily avmlable m the bterature

9. On the dynamics of dutdl8hon processes-111 1409
Substitutmg the values 0, 1, 2, 3 for N2 gwes the cases,
(1) N2 = 0 unphes
N,-s,= 1 (n)
The sunplest solution, Na = 1, S3= 0 1s shown m FU
11(b) for the case of a ternary maxunum boll~ng azeo-
trope It is clearly ImposslMe for a smgle ternary muu-
mum bodmg azeotrope to occur because this reqmres
that azeotrope E be a stable node (highest bodmg pomt
on the dmgram) which 1s contrary to the statement that
Nz=O
(u) N2 = 1 lmpbes
N,-&=O m
The sunplest soluaon, N, = Sa = 0 ISshown m Fig 1l(c)
(m) N2 = 2 Imphes
N,-&=-I (29)
The solution Ns = 0, SX= I 1s shown IIIFg 1l(d) The
ternary azeotrope must lie in the temperature range
100<TU<105
(IV) N2 = 3 unphes
N,-s,=-2 m
The solution N3 = 0, & = 2 1s shown m Fig 1l(e)
These !igures appear to be mvammt Consider, for
example, the temperature rearrangement T.= 115 with
all other temperatures bemg kept the same Figures 1l(b)
and (e) remam exactly the same Fiiures 1l(c) and (d) are
changed m order to accommodate the fact that azeotrope
D 1s now the west b&ng pomt but a simple relabel-
hng of the axes enables us to recover the form of the
o~lgmal diagrams exactly
By reversmg the arrows m these figures we obtam the
set of dmgrams generated by the boundary con&Uon of
three mmunum bow bmary azeotropes Consequently
we need not consider this as a separate case Apply-
mg the same reasonmg as m case (1)above we conclude
that a smgle ternary maxunum bolllng azeotrope cannot
occur when each of the three bmary faces exlublts a
mmunum bodmg azeotrope
We could consider examples of ternary nuxtures
which exhrbrt fewer than three bmary azeotropes but it 1s
not our purpose to present such an exhaustive analysis at
ttis pomt
Theorem (Ben-son)
The local phase potit of an isolated smgular pomt
with a smgle zero elgenvalue Is of one of the followmg
three types node, saddle pomt (four separatnces), two
hyperbolic sectors and a fan (three separatices, see Frg
4(d)) The correspondmg m&ces are 1, - 1,O which serve
to distmgmsh the three types
This theorem guarantees that the elementary singular
points shown m Fig 7 need only be augmented by a few
addltlonal types
The zero eigenvalue mtroduces one new type of ter-
nary smgular pomt shown m Fig 12(a), two new types of
pure component singular pomts shown m Figs 12(b) and
(c) and one new type of binary smgular pomt shown m
Fig 12(d) The ten dmgrams shown m FQS 7 and 12
exhaust all the possible singular pomts which can occur
in a ternary nuxture
We can now denve the more general version of eqn
(18) Let us call Fe 12(a) a ternary node-saddle, FQ
12(b) a pure component node-saddle, Fe 12(c) a pure
component half saddle and Fig 12(d) a bmary node
saddle Define,
NS, = number of ternary node-saddles m the mangle
N& = number of bmary node-saddles m the tmu&e
N.9, = number of pure component node-saddles m the
tnangle
Then, from eqn (14)
S? = number of pure component half saddles m the
*angle
IN,(SN,) + 1&3S,) + lMJ8NS3)
+ 1&4Nz) + 1*(4&) + #~9(4NS2)
+I,,(~N,)+I~,(~SI)+IN~,(~NSI)+~~~(~S~)=~ 131)
with the constramt,
Nl+S,+NS,+S:=3 (32)
PART B NON-- ARYBtNGInARPo~
In accordance ~rlth propeNes P(lv) and P(v) of eqn
(2) we will now allow for the case of a smgle zero
elgenvalue of the Jacobtan matnx
In general, nonelementary smgular pomts are capable
of assunung the most mhrcate patterns m the phase
plane as a browse through Sansone and ConttC191 WrIl
tesbfy However, we are fortunate msofar as we can ICl td)
prove that there 1s at most one zero eIgenvahUZfor which pj 12 N
case Bendtxson ([lo] p 230) has proved
on-elementary smgular potnts 111the cornposItiontn-

10. 1410 M F DOHERTY andJ D PERKINS
The rndlces take on the values grven by eqn (17) and
usmg eqn (11) we obtam.
lNsj=o INS,= - 1
lN*=o 1&q= -3
Between eqns (17) and (31)-(33) we obtam
(33)
2N,-2&+Nz-Sz+N,-S:=2 04)
Puttmg S? = 0 we obtam the spectal case of eqn (18)
We ~111bnefly review the conclusions given m exam-
ples one, two and three m the l&t of eqn (34)
Example I
Our earher conclusion must be mod&d to accom-
modate FM 13(a) which exphcltly demonstrates the way
m which a non-elementary pure component smgular
point at T, allows for the appearance of a maxnnum
bothng (or mmtmum bollmg) ternary azeotrope Ftgure
13(a) conforms to eqn (34) However, it ISnnposslble for
a smgle ternary saddle point to occur as can be proved
by contradiction
Smce vertices of this sort have not been reported
expenmentally we will not pursue this classticafion any
further
Example 3
AH we will say here 1s that the conclusion reached
from eqn (27) regardmg the unposslblllty of a ternary
muumum botlmg azeotrope ceases to be true when we
allow for non-elementary singular pomts This is shown
111Fui 13(b)
Our analysis of non-elementary singular points has
unphcabons about the shape of the boding temperature
surface in the viclIuty of ternary azeotropes This 1s
discussed m the foliowmg section
Bodmg temperature surface
In a previous article (7) we have proved that In a
c-component mrxture, maximum bollmg c-component
azeotropes gwe nse to stable modes and mmlmum boding
c-component azeotropes @ve nse to unstable nodes
The correspondence between the type of stationary
pomt in the bothng temperature surface and the type of
stablltty of the assocmted smgular point can be analyzed
further by use of the followmg equation (see Doherty
and Perkms[7]) for the determinant of the Jacobian
matix at a multicomponent azeotrope of composition 2,
(37)
Assume a single ternary saddle does occur, then SX=
1, N, = 0, N2 = S2= 0 hence eqn (34) becomes,
N,-S:=4 (35)
The left hand side of (35) has a rnaxlmum possible value
of three So, we can state the general proposlfion
A ternary saddle point azeotrope cannot occur in a
ternary mixture whtch exhtblts no bmary azeotropes
PhysIcally this IS a very reasonable result because a
saddle pomt has four separatrrces, each of which
requires a smgular pomt to tend towards (as g+ + COor
-m) In a ternary mixture with no bmary azeotropes
there are only three singular points avdable (the ver-
tices)
Example 2
Equation (34) becomes,
2N,-2&+2N2+N,-S:=S (361
Unhke the elementary case we cannot determme the
nature of a pure component smgular point purely from
the due&on of the arrows on the bmary faces Th~3
makes eqn (36) more unwieldy than Its counterpart, eqn
(21)
Fwures IO(b)-(g) can be augmented by many more
diagrams m which we allow for non-elementary vetices
where B 1s a positive number
For a ternary nuxture this reduces to,
det J(z) = 8* det
where 8* ISa posttlve number
Equation (38) tells us that saddle pomts m the bollmg
surface gave flse to smgular pomts which are saddle
pomts and vice versa This equation also tells us that
semldefinlte stationary pomts m the boding surface gwe
nse to non-elementary smgular pomts and vice-versa
Because eqn (38) cannot mve any mformatton about
the possible types of seml-defimte stationary pomts we
were not able to answer the question of Just how “bad”
these semtdefintte stationary points can be In general,
semldefimte stationary pomts can be as comphcated as
we choose, wtth multiple valleys and ndges convergmg
on the stationary pomt
This question no longer remams unanswered We have
proved that the only new smgular pomt Introduced by a
zero elgenvalue LSthe node-saddle shown m Fig 12(a)
Thus smgular pomt must @ve nse to an armchau hke
stationary pomt m the bollmg temperature surface
because the temperature always rises along a residue
curve (see (7) Theorem Four) This proves

11. TI E
1110)
On the dynamtcs of dtstdlatlon processes-III 1411
B
D T+ = 5
To (4s) (40)
I*01
tbl
Rg 13 Examples of non-elementary resrdue curve maps
trope to occur m a rmxture which exhlblts a bmary
muumum bodmg azeotrope on each of the bmary faces
(IV) If a ternary saddle pomt azeotrope occurs, Its
bollmg temperature can be bracketed Thus enables us to
determme temperature mtervals m whch a ternary
azeotrope cannot occur Thus goes some way towards
answermg the problem described by Maruuchev[31
quoted in the mtroductlon, that, “the question of the
exrstence of the ternary azeotrope at a gven tempera-
ture remams open”
(v) Allowmg for non-elementary smgular pomts, It Is
not possible for a smgle saddle point azeotrope to occur
m a mixture which exhlblts no bmary azeotropes
(VI) The bollmg pomt surface can exhlblt only four
types of stationary pomt maxlma, muuma, saddles and
chatrs
Acknowledgement-One of us (M F D ) wshes to thank the
Nattonal Science Foundatton for provldmgfinancialsupport m the
form of a Research Imtlatlon Grant (Grant Number ENG 7%
05565)
NOTATION
A vector of parameters m eqn (2)
Theorem
The only kmds of ternary azeotropes which can occur
are maxlmum bodmg, mmlmum bollmg. saddle pomt and
armchau-hke azeotropes
The bolhng surface cannot be any more comphcated
B,(x) ball, centre x, radius r
c number of components m a multicomponent
mrxture
e, unit vector m the rth orthogonal dlrectlon
e number of elhptlc sectors
f vector functions
than the restictions Imposed by this theorem
It IS worth notmg the tnvlal pomt that m bmary
nurtures the only kmd of seml-defimte stationary pomt
which can occur IS an mflexlon
CONCLUSION
The dlstdlatlon eqn (2) place a physIcally meanmgful
structure (I e tangent vector field) on ternary phase
diagrams By recogmzmg that such structures are sublect
to some very general mathematical laws It has been
passable to obtam structural relationshIps between the
azeotropes and pure components occumng tn ternary
mixtures
These relationships gwe useful mformatlon about the
dlstlllatlon behavlour of ternary mixtures and also allow
us to obtam some general results concemmg the nature
of ternary azeotropes These may be summarued as
follows
(I) When all the smgular pomts are elementary rt IS not
possible for a smgIe ternary azeotrope to occur m a
mixture which exhlblts no bmary azeotropes
(u) When all the smgular pomts are elementary It 1s
not possible for a smgle ternary nummum bolng azeo-
trope to occur III a murture which exhibits a bmary
maxlmum boding azeotrope on each of the bmary faces
(m) When all the smgular pomts are elementary it Is
not possible for a smgle ternary maxnnum bofimg azeo-
N.%
mverse of f
Image of an
image of the set of cntical pomts Z
genus (eqn 13)
number of hyperbohc sectors
mdex of smgular pomt
mdex of a smgular pomt of type a
Jacoblan matnx of first partA denvatlves of f
compact boundaryless mamfold
number of nodes m the mangle of type Q
number of node-saddles m the mangle of type a
P pressure
P,O saturated vapour pressure of pure component I
R gas constant
R” space of dlmenslon n
S, number of saddles m the mangle of type a
St number of pure component half saddles m the
-de
T temperature
TaF bodmg temperature of pure component c
x (1) abstract state vector, (2) vector of mole frac-
tions m the lrqmd phase
y vector of mole fractions in the vapour phase
2 azeotrop~c composition
Z set of cntical pomts (1e pomts such that J,(x) =
0)

12. 1412 M F DonmtnandJ D PERKINS
Greek symbols
Yi achvity coefficient of component I
Ai elgenvalue
@ open rtght angled trtangle with unit side (the open
composition h-tangle)
13@ the boundary of @
6 the closure of @
R open set m R”
an the boundary of n
fi the closure of Q
2‘ dlmensroniess tie
9
19* posrttve numbers
x Euler number
Mathemattcal symbols
[ ] closed interval
E contamed m
!z not contamed m
sgn sign
Superscripts
m mfirute dllutron
0 pure component
AZ azeotroprc value
Subscripts
I,J dummy subscripts
x’ keep the vanables xl, x2
stant
q-t, xj+b xc-l con-
PI
r21
f31
r41
r51
[61
r71
181
191
[toI
[ttl
WI
u31
1141
WI
WI
[I71
REFERENCES
Herdemann R A and Mandhane J M , Chem Engng SCI
1975 38 425
Marmrchev A N , J Appl Chem USSR (Eng Tmns ) 1971
44 2399
Maruuchev A N , J Appi Chem USSR (Eng Tmns ) 1972
43 2602
Susarev M P and Totia A M , Russ J Phys Chem (Eng
Trans) 1974 48 1584
Mamuchev A N and Vmtchenko 1 G , J ADD/ Chem__
USSR (Eng Tmns ) I%9 42 525
Doherty M F and Perkms J D. Chem Ennnp Scr 1977 32
1112 -
-_
Doherty M F and Perkms J D, Chem Engng Scr 1978 33
281
Doherty M F and Perkms J D, Chem Engng Scr 1978 33
569
Temme N h4 (Ed ), Noniuteur Analysrs, Vol 1 Mathema-
ttsch Centrum, Amsterdam 1976
Lefschetz S , D$etzntIal Equatwns Geometnc Theory, 2nd
Edn Intersctence. New York 1957
Gavalas G R , No&near Dtffenznturl Equutrons of Chem-
uxlly Reucturg Systems Sormger-Verlag, New York l%g
Courant R and Robbms H , In The World of Mathematrcs
(Edged by Newman J R 1. DD 581-599 Stmon and Schus-
ter. New -York.1956 .- __
DoCarmo M P , hfferentuzl Geometry of Curves and Sur-
faces Prentrce-Hall;Englewood Cl& Nkw Jersey 1976
Gudlemm V and Pollack A. LWerentral Tooolo~v Pren-
tree-Hall. Englewood Cldfs, New jersey 1974 _ -_
Mdnor, J W , Topology fmm the D#erentmble Kewpornt
The Unrverstty Press of Vugmta, Charlottesvdle 1976
Wade J C and Taylor Z L , I Chem Engng aOtu 1973 18
424
Zemrke J . Chemrcai Phase Theory Kluwer, Antwerp, 1956
[W
1191
1201
Et1
[221
1231
I241
WI
WI
1271
Horsley L H , Azeottvprc Data III Amerrcan Chemical
Socrety, Wasbmgton. D C 1973
Sansone G and Conb R , Nonlmear Dtff-erenttal Equotrons
Pergamon Press, Oxford 1%4
Ewe11 R H and Welch L M , Znd &gng Chem 1945 37
1224
Remders W and DeMmler C H , Ret Trau Chrm 1940 59
207
Remders W and DeMmJer C H , Ret True Chrm 1940 59
369
Remders W and DeMinJer C H , Ret Tmw Chum 1940 59
392
Wilson R Q , Mink W H , Munger H P and Clegg J W,
AIChEJ.19552220
Rowlmson J S , Lrqurds and bqurd Mxtures, 2nd Edn
Buttenvorths, London 1%9
Horsley L H , Azeotroprc Data [II Advances m Chemistry
Senes 116 Amencan Chemical Soctety. Washmgton, D C
1973
Benedtct M and Rubm L C , Natwnal Petroleum News
1945 37 R729
AFP-I
The drsttllatton equattons for a c-component mtxture are,
dx
dz=x-y
Wdhout loss of generabty we can study the properttes of these
equattons at any vertex by studymg them at the ortgm (see
Doherty and Perkms[8], Appendtx I)
Lmeanzmg (Al) about the ongm gtves.
dx I-Y,,
Clf=
1 -Y22
0
1
0
- Yc-
1
X
(A21
-1 r--l J
where Y,,= (~Y&WP ==O The off-dtagonal terms are zero
because y, = 0 when x, = 0 for all values of x,, I# I hence
w,ia+ x=a= 0 for all J# I (see Frg Al and eqn (32) m Ref
[81)
The Jacobtan matnx m (A2) ISdiagonal so Its normalized ergen
vectors are the orthonormal duecttons e,, ez. e3 e,_, It ISwell
known that the soluttons of a system of lmear ordmary ddferen-
teal equatrons exhibit the properttes,
(I) If the ortgm ISa node, the tralectones approach tangenttalty
along an ergendtrectton (see Appendtx II)
(II) If the ortgm ISa saddle, the traJectorres become asymptottc
to the etgenduecttons
This lustrfles Figs 7(c) and (d) and proves that these are the only
elementary pure component smgular pomts
Of course, the above reasonmg fads tf y,, = 1 for any I
Phystcally, the term y,, represents the mrttal slope of the yi vs
xl diagram for the bmary submrxture of components l and c (see
Fig Al)
TJus slope can be wrttten m terms of other thermodynamtc
quantrttes as follows Assume an eqmhbnum relatton of the
form,
PYr = PPx,r, (A3)
DdIerentratmg with respect to x, and letting x, = x2 = X,_, = 0
gtves
( )ay, =rPC’( Tic)
ax, P I -0 P
(A4)
where yi” ISthe mtimte ddutron acttvity coe&tent of component
I and P,‘(Tb,) IS the saturated vapour pressure of pure I at the

13. OF I All0 3 When a tangentml azeotrope behveen components k and r
(rf c) occurs at vertex cI then
(3)
( >app
a-% PII
*3
gwes
Fig Al Vapour-lrqtud composrtlon surface, y, vs x, and xt for a
a
non-azeotroplc ternary mixture Pco(ekh (ek)-pro(ek)Yr(ek) r= I
c-l
bodmg temperature of pure component c The r&t hand side of
(A4) IS equal to umty when the binary mixture of components L
(A13)
and c exhlbrt a tangenhal azeotrope at the orlgm, m whmh case
We need to establish the condltlons under which the binary
mixture of k and r exhlblts a tangentml azeotrope at vertex ek
cl W) The basic condltlon which we must work from IS
We can show this correspondence as follows Usmg the same 0 (A14)
procedure described m Appendix I of Doherty and Perkms 181we
obtam
where z denotes a movement m the k - r face, I e such that
dT
( >ax,
=p-P3 L ) Y,“(O)
++c)
W) q + x, = I (AIS)
P x-0
From (A6) we see
Usmg the chain rule on (A14) gives
dT
( > *, ~
a*, p x a~ ( >
ax,=,
ax, p I az (Al6)
On the dynanucs of dlstdlatlon processes-III 1413
PP(e&P(ek) = l
P
W 1)
and hence by (A9) we get A. = 0
We prove statement (Al 1) below Summmg eqn (A3) over all I
gwes
C-l
PCOYA1 -11 -x2 --XC_,)+ c P#%,y, = P (A13
,-I
DtlTerentlatrng (Al2) with respect to x, r = l c - 1, taking the
hnut as xk* 1 and remembermg,
(1)
(2)
Yk(ek) = 1
r,-.L ax, #?.+O
lim a
I > r=l
c-l
Between (AlS) and (Al6) we find the requued condltlon,
This corresoondence extends to all vertmes but ts more
dltlicult to ex&act for those other than the oruun (Al7)
The Jacobian matnx for the vertex elr IS gwen by (see Ref
WI)
1
Between (A13) and (A17) we find,
0 P%k )%*(e&) = Pk%k )Yk(ek) (‘418)
J(ek I=
The rrght hand side of (A18) ISequal to the total pressure, P. and
so we obtam
l_J%
py:-det)
I
P,“(e,)vr%t) = ,
P
(Al9)
(A81
Equation (Al9) IS the condttlon for a tangential azeotrope to
The elgenvalues are @ven by occur between components k and r at vertex et This completes
what we set out to prove
A = , _ P%kh%k)
P
1=1 C-l I# k (A91
In summary, we have shown that one zero ergenvalue IS
Introduced for every tangentml azeotrope occumng at a given
vertex
Ak= -~~(~~)(~)p (AlO)
.%-en APPENDIXU
When a tangentml azeotrope between components k and c Shape of the rexdue cumes in the vwmty of pure component
occurs at vertex ek then (AIO) shows that Ak = 0 venkces

14. 1414 M F L&WFXY and J D &SKINS
From eqns (A2) and (A4) we find that the ergenvalues of the Kk=(er ) = rc”(ek )PP( TM)
bnearlzed equations about the 0~ are,
P
A, = 1 - K,_(O) 1= 1 C-l (A20) We wdl wnte eqns (A20) and (A22) as,
where K,-(O) IS the mfimte dllutlon K-value of component I
evaluated at the ongm, A,=]-KI”
(‘424)
(A=)
Y,-(o)pP(Tbc)_ aY1
K-(O) = p -z( ) I=1 c-l It bemg understood that the Kp”‘sare evaluated In the proper way
P r=0 at the vertex of Interest
(A=) In accordance with the theory of lmear differential equations
we have
From eqns (A9). (AlO) and (Al3) we find that the elgenvalues of (a) The vertex IS a stable node d K,“C 1, I = 1 c-l The
the lmeanzed equahons about vertex q are, residue curves are tangent to the hne connectmg the vertex about
A, = 1 - ZCp”(ek) (A221
which we have lmeamed with vertex V, such that A. = max {A,}
(I e A. has the smallest absolute value)
c-l
where K;“(et) IS the mfnute dllutlon K-value of component I
(b) The vertex IS an unstable node d K,- > 1. d= 1
evaluated at the vertex ek,
The restdue curves are tangent to the bne connectmg the vertex
about which we have bneanzed w&h vertex 0. such that A. =
mm i&l
s=l c-l r#k (c) The vertex IS a saddle under any other cvcumstances
(A=)
(except the particular case Kjm= 1 for some J This case has
already been dlscussed m Appeadlx I)