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The solution of multicomponent distillation columns
Chemrcal Eng~neenng SCKWX,19115, vol 4, pp 222 to 227
The solution
Pcgamon
Prew Ltd
of multicomponent
distillation
columns
GERHARD KLEIN* and D. N. HANSON Departmentof Chemistryand Chemmal Engineering, University of California, Berkeley (Recemd
7 Februmy
1955)
Summary-The equations of A J V UNDERWOODfor solutron of drstillation columns are arranged to grve a solutron m the form of a sum of difference products For many drstdlatron problems the present solutron IS much less tedrous than the strarghtforward solutron of UNDE~WOOD’S equatrons or the determmantal solutron proposed by ALDER and HANSON [l]. The solutron 1s dlustrated by an example employmg a Ave-component system R&arm&-Pour la solutron des &matrons de A J V UNDERWOODpermettant le calcul des colonnes ir drstdler, les auteurs consedlent l’emplor d’une somme de prodmts d’une &ff&ence Dans de nombreux probkmes de dlstdlatlon, cette solution est beaucoup moms phble que la solution duwte des 6quatlons ~‘UNDERWOODou que la solutron sous forme de d&.ermmenta proposh par ALDEBet HANSON La solution est. rlhrstr& par un exemple sur un systeme B canq composants.
THE solution of multlcomponent dlstlllatlon problems IS, m general, time-consummg and tedious work The methods whrch are avadable for such solutions employ various basrc assumptions and varrous techmques, and the partrcular method whrch is best suited to the problem at hand depends largely on the accuracy of the answer desired and on the physical make-up of the system to be dlstlled. As an example, calculation procedures are plate-by-plate drstmctly useful for some systems, for example systems m which all of the components of the feed appear m one or the other of the products rn well-definable amounts and m which the drfference IQ volatrlity between the key components is large. For such systems, calculatron plate-byplate from one end of the column ~111 provrde the solutron wrthm a few plates. If, on the other hand, the key components he Intermediate m volatlhty wlthm a large number of components, and a large number of plates is to be expected, plate-by-plate calculatrons become difficult, and procedures which allow the calculation of the number of rectlfymg and strlppmg plates as complete sectrons become most useful. Such “group ” procedures have only recently been l
Present
address
-
developed, and to the authors’ knowledge, four basrc solutrons presently exrst. The relative merits of these are as yet undetermmed but one which shows promrse of consrderable usefulness IS the set of equatrons derived by A. J. V. UNDERWOOD
[S,
4,
51.
The equations provide exact solutions of drstlllatlon columns under the basm assumptions of constant relatrve volatrhtres and constant sectional flows. The correctness of these assumptions depends upon the system, and although they are not m general exact, distlllatron equatrons mvolvmg then use have found wide apphcatron. The flows are usually set by the internal reflux at the top of the column and the thermal condrtron of the feed. Varrous procedures are m use for estlmatmg average relatrve volatihtres. Experience has shown that satisfactory values may be obtamed usmg an average of the top plate and reboller temperatures werghted by the molar amounts of top and bottom products or, alternatrvely, that temperature at whmh the product of the equilibrium constants of the hght and heavy key 1s equal to umty. The solution of UNDERWOOD’Sequations grves directly the number of rectrfymg and strrpping
%~~ita~ Engmeenng PNBJ~&, Unlvemlty of California, Richmond, &lif~rn&. 229
GERHARD
and D. N. HNWON
KLEIN
plates. The systems ordmarlly encountered m multlcomponent dlstlllatron contam components both lighter than the light key and heavier than the heavy key. The trial and error solution followmg the lines proposed by USDEHWOOD [5] then becomes prohrbltrvely time consummg. ALDER and HANSON [l] have proposed a solution of the equations m determmantal form, bypassing the calculation of feed plate concentrations proposed by UNDERWOOD. However, since the order of the determinants which must be solved m the method of ALDER and HANSONIS equal to the number of components m the feed, the dlficulty of the solution mounts rapidly with an increase m the number of components, and a solution which would avoid this difficulty IS considerably more useful. The present paper proposes a solution whrch accomplishes this by developing the determinants mto a sum of term\, each of which contains a quotient of products of differences. For the convenience of the reader, and to mtroduce the notation to be used, a brief review of the developments which led to the solutron 1s presented. To avoid excessive duphcatron of previously published material, the review and the development of the solution will be presented for a specific case. The use of the solution m solving other cases can be obtained from a brief study of the previous paper by ALDER and HANSON. For a system of three components A, B and C, then, the followmg equations may be written :
a, = relative volatility of component i. All relative volatlhtles to be based on the same component. + = a rectifying section parameter 4’ = a stripping section parameter USDEHWOOD employs equations (1) and (2) to define the rectifying and strippmg section parameters, + and 4’. It is apparent that for this three component system there are three values of C#which will satisfy eq. (1) such that if these are numbered m decreasing order of magnitude : aA > 41 > ag aB > 9b > h2 ac > 48 There are also three values of 4’ which will satisfy equatron (2) and if these are again listed m decreasing order of magnitude :
aA
aE >
aA ___bA)f
+
aB-h
aA -++ bA)f aA-
aB @B)f
&A)/
a.4 -
+
aB -
4
ac - 4
(2)
where V = rectifying section vapour flow, moles v’ = stripping section vapour flow, moles
d (a~,)~= amount of any component i in the top product, moles b (;L& = amount of any component i m the bottom product, moles 280
aA ___ @A)f aA-
NC+
aB-+!2
+
aA_h
aAbb~)b + aBbbB)b + acbbdb
ac
aB (%)f + aC b%)f ~
aA-h
-aA -Vv’
+‘a >
aA aB
UNDERWOODhas shown that two independent equations may be wrrtten relating the three values of +, the concentrations of A, B and C in the liquid on the feed plate, and the number of rectifying plates. These equations are .
and
+
+‘I > > +‘a>
+
$1
=
aB h?), + % ~ - (%)f w-+2 w--h -aB @B)f aB-48
+
(8)
aC (Gf ac+S
aC (%)j
Al ” =
0&
(8’)
QC-$8
where (a+ = mole fraction of component i in the liquid on the feed plate. R = rectifying plates, mcludmg the feed plate. The authors, to simphfy the wrltmg of such equations have adopted a shorter nomenclature, where
The solution of multicomponent clistillatlon columns aA
-= a.4
-_!k_
A,,
-
+l
wi
-
=
ALDER and HANSON have suggested that the solution of these equations be accomphshed by solution of the thud order determinant which can be set up as follows :
B,, etc.
+l
and
0 +2t* &
Rewriting become
2’
equations
(3) and
(S’),
4
these then
-42
(8.4)~
+
J32 @B)f
+
c2
kk)c)/
A2
@A)f
+
B2
@B),
+
c2
h),
A2
@A)1
+
B2
@B)f
+
c2
@c)/
(3)
a’
and =
+
(3’) 82
Slmllarly, two independent equations can be written for the strlppmg section * A,
@A)f
+
J32’
A,’
(*A),
+
B,‘@B)~
A,’ -
@A)f
+
B,’
hj),
+
‘&’
6%),
&A)/
+
B2’
(zB)f
+
c2’
(Q)/
where A,’
A2
A;--
AI @A),+ B, bB)f + CI (@ = +
-42’
-
@B,f
+
‘7,’
@d,
+
cl’(~)/
=
+’ 21
=
+’
-,
etc.
82
(4) (4’)
: =
rA a.4
-
B,’ ~62”
=
A-aB
-
+A’
and
where m = stripping plates, mcludmg the reboller but not the feed plate. If components B and C are the key components and their separation 1s set, then the amount\ of these components m the top and bottom As an approximation products are known. for the first solution, It will be assumed that component A goes completely mto the top product. This approxlmatlon may be corrected to any desired degree of accuracy later and does not represent a loss m accuracy of the calculation. If, then, the reflux 1s set, the quantities V and v’ are known and the + and 4’ values may be calculated from equations (1) and (2). It will be noted that a consequence of the approximation of no component A m the bottom product 1s or that dl’ = aA. This means that A,’ aA/(aA - &‘) is equal to infinity and as a result eq. (4) becomes useless. Equations (3), (a’), and (4’) can be written out however and solved for n and m.
$21
-42--2h32
A;&
Bl -
332 421
Cl
-
C2
$21
B2---~,+a2
C2--C39%,2
B;-
C;--C;I,&
B&A
=o
(5)
The unknowns n and m are contamed as exponents m the # and I,Y terms. If a value of 12 1s set, the correspondmg value of m may be calculated from eq. (5), and m this way the picture of all possible combmatlons of rectifying and strlppmg plates which ~111 perform the dlstdlatlon can be found. The solution of eq. (5) IS not difficult. However, if, for example, a five component system were to be dlstllled, eq. (5) would become a fifth order determinant and the solution would be considerably more tedious. Since the number of components m the feed 14 equal to the order of the determinant to be solved, ab the number of components increases, the labour of solution by further However, becomes prohlbltlve development, the solution can be reduced from determmantal form to a difference product form which 1s much easier to solve. Kq. (5) may be wrltten : J~,Y%-A,#,
W/5--2+2
C,#l-C2+2
-42$2--A,&
B292-%#2
C2#2-G&
A&;-A;&
B&;-B;&
C;t&C;la,b;
=o
N6)
where & = #Q*, #2 = #Jo*,etc. & = (V&P,
(Ii = (V&P,
etc.
It is apparent that the term aA aB ac can be cancelled out of the determinant, eq. (6). The term
A, t,bl, for example,
IS then ~ aA
1 -
#I
*1*
Rewritmg eq. (6) in a new nomenclature to show the droppmg of all a values m the numerators of the mdlvldual terms 1s not worthwhile, however, and It can merely be understood that for the rest of the development,
231
A, =
GEEENUI
and
-N
etc. This lapse m preciseness of nomenclature here leaves no ambiguity m the final solution. Eq. (6) can also be factored mto six third order determmants to give :
D. N. HANSON
and
If these factors are dropped, then the first term in equation (7) becomes : (9192
Y4J P
(4142
6,
UA UB UC
and if this IS written out it is :
*s43 [Ml -
-($1
[taA
If the first determinant m eq. (7) is written out, it is :
Such a determmant is a double alternant and can also be written [2] as
-
92)(+I - 42 (A4- &,I 41)taA - +a)taA - hi)]
Eta,- 41)taB
’ 4%)ta - &]
x
’
It will be noted that the liberty of writmg the difference product, [f (4, +a +i) with all terms reversed has been taken. It IS more natural to calculate the difference product this way, subtracting + and 4’ values from $ or +’ values precedmg, and it yields the same results m the summation of terms in eq. (7). For illustration, the second term m eq. (7) 1S
where k = the order of the determmant. 5f = difference product obtained by taking each term and subtractmg from it m turn each term precedmg, then multiplymg all of these differences together. 214
UB
=
=
or written out -
difference product obtained by multiplying together all terms (aA - 4,) where +r is a general value of 4 which will take all values of 4 used in the determmant. a difference product exactly hke uA except that aB is used in place of aR. etc.
Before proceedmg, some simpbfication can be effected. Two factors enter mto each determinant m eq. (7) and may be cancelled. Thus we may drop the terms (4
(- 1)
p-1'
($1A4vuWl - 44)(41- 6, (42- 9ul [taA 41)caA - d2)taA - +d]
[caB- ‘$1)taB
t_ h?) taB -
hi)]
x
’
The other terms m eq. (7) are constructed in the same way. At first acquamtance, the terms m the difference product solution of eq. (7) seem quite long and mvolved, and m the extension of the method to systems of several components, it appears at first that even the work of keepmg track of the malor terms m equations of the type of eq. (7) ~111 be difficult. Since this is one of the chief
283
The solutionof multicomponent dbtillation columns
difficulties m the difference it to set up a standard procedure a set of as automatic as possible. of equation be written as : ~
(_
l)b+P+P+
r+a+;+) ,
u, up uq . . . 24,
I
ut . . .
24,
(404p4q* * * 4; 4: 4;. - -1 ,
*
s
u, up uq . . . u, u, ut . . .
lf (4, $34q* * * 4; 4: 4;) u,upuq...
u; u;24;. . .
1 J =
()
4.
All of the $ values are written down first, followed by the #’ values.
5.
The combmations of # and 11’ values used are all possible combmations under rules (l), (2), (8) and (4) which do not contain any repeating 1,4or #’ values. Thus, the number of combinations possible and also the number of major terms m the summation is equal to the number of useful + values times the number of useful 6 values.
6.
The difference product 5’ (4. $P 9, . . . . . +I +; +; * * * -1 is obtained by subtracting each + or 4’ value from all Q or 4’ values preceding it in the order of listing of the 7% and $’ values and multiplying these differences together. Thus I* (4, &, 4s . . .
(7)
In general this equation is the solution of a determmant of the type of eq. (6). The order of the determmant is equal to the number of components m the feed, and m a case where the separation of the two key components has been set and the reflux has been set, one such determinant can be written [l]. This determinant will m general employ all of the 4 values which are useful and all of the 4’ values which are useful, thus excluding 4 and 4’ values which are equal to relative rolatihties. The total number of useful ~3and +’ values will equal the number of components plus two : the number of + values and the number of #’ values ~111 vary depending on where the key components he m the hst of components. The rules which may then be set up for the solution of equation (7) are as follows : 1.
order of magnitude of #). The #’ values are written down in ascending order of magnitude of subscripts (descendmg order of magnitude of 4’).
* -4; 4; 6 . * 4 = [bib -
2.
The total number of 16 and 4’ values written down m each term 1s equal to the number of components.
8.
The $ values are written down in ascending order of magnitude of subscripts (descending
(&- 3;) (&-- +;j * .I [W, - 42 ($I - 9%) .I* - * * [C41 - 4,) (4; - 4;)- **.]
[(&
7.
$J
. . .] . . . . . .
The number of u values m each term 1s equal to the number of $ and +’ values used m the term. For convenience these have been grouped and characterized by a 4 or 4’ value rather than by component. Thus u 1 = (a~ - 41) (an - 74) (ac - 41) (an - &) . . . . . . . , & = (aA - 6) (as -
8.
-
6) (ac -
6) (an -
6) . . . . etc.
Each term in the summation is multiplied bY - 1 or + 1 depending on the sum of the subscripts of all 4 and 4’ values used m the term. Smce the choice is arbitrary, the convention has been adopted that if the sum of the subscripts is even, the term is multiplied by + 1 and if the sum of the subscripts rs odd, the term 1s multiplied by -1.
The rules set down here completely 288
9,) * ’ *
....
Each major term m the summation will be characterized by one less than the number of useful Q values and one less than the Thus the number of useful 4’ values. number of 7%values written down m each term is one less than the number of useful + values, and the number of ~5’ values written down m each term is one less than the number of useful 4’ values.
$JJ MO -
detlne
GNENAEIIKLEIN and D. N. the solution of the determinant and enable calculation of the column. Some study of them is necessary, however, before much facility m such solutions is gamed, and to aid in this understandmg of the method an example of a distillation column separating a five component system has been solved for presentation here. Consider a five component system of A, B, C, D, E. The feed to the column is saturated liquid of the composition shown in the table below, and the relative volatilities based on the heavy key, D, are as shown :
&hNI3ON
the minimum m the curve of total plates verses feed plate location 1s very flat, and a wide choice of feed plate location can be made with httle effect on the total plate requirements. The fist step in solution of the problem is the calculation of the + and 4’ values. This calculation is the solution of the following equations : For values of 4 : 16= 5(005)+ 5- +
25(0 25)+ 22(088) + l(OO2) 2.5 - 4 2.2 - $6
For values of #’ : XF
A B c
D E
0 05
6”
0.25 0.40 0 20
25 22 1.0 06
0 10
_
2.2 - lp'
Components C and D are the key components. Specifications on their separation are set so that 95% of C goes mto the top product and 90% of D goes into the bottom product. It is assumed as a first approximation that the light dluents, A and B, go completely into the top product and that the heavy dluent, E, goes Thus completely into the bottom product. the products, based on one mole of feed, are : d tx,)d
A B c
D E
005 0.25 088 0 02
-
~(j=ww+
b tx,)b
-002 018 0 10
VW+
06(010)
I - 4'
0.6 - $b'
(9)
Equations (8) and (9) may be solved by any of the iteration methods available for solution of cubic and higher order equations. Actually, trial-and-error solution is quite rapld and has the advantage that m the solution for 4 or +‘, the quantities aA - +i, etc., are calculated. Since these drfferences are of use later it 1s convement to have them tabulated as a result of this first calculation. Solving eq. (8),then, the following values of I# are found : +1 = 4885 Qa = 2.898 Al = 1882 G=;y
The mimmum rectrfying section vapour flow for this separation IS calculated to be 1 206 moles/mole of feed. An operating rectifying section vapour flow of 1.6 moles/mole of feed, or approximately l8Oo/oof mmimum, was chosen for the calculation. With these variables set, there is an infinite number of rectifymg and stripping plate combmatrons which will perform the distillation, and the answer desired is the combmatron of rectlfymg plates and strippmg plates which ~111 constitute a mmimum of total plates, or will give the optimum feed plate location. Generally
It will be noted that a value of 4s IS listed. If the assumption 1s made that d (~x)~ = 0, then $6 = aE and may be bsted as such although It 1s of no use m solving the column. Simdarly, from eq. (9) the followmg values of 4’ are found : $b; = 5 &=25 +; = 22811 +; = 1118 9% = 0.6284 Here again values are listed for 4; and 4;. Under the assumptions of b (z,& = 0 and b (z~)~ = 0, the values of +; and 4; are respectively equal to aA and ag. These values of $; and & are of no
284
The solutionof multicomponentdi&.llationcolumns use in solving the column, but are listed in order to clarify the system for subscript numbering of the +’ values. Three sigmficant figures are being carried m the calculations at this point, although it is apparent that this necessitates having more than three sigmficant figures m the + and 4’ values. The + values to be used m the solution are 419 429 4*9 and +4. The 4’ values to be used are +;, &, and 4;. Generally, the next step m calculation is to form a table listing the characterizing 7%and #’ combinations. In the table may be listed, then, the other terms which accompany the characterizmg 9 and #’ combmation m equation (7). These are - m the numerator, the term (- 1) (O+P+*+ * “+“+“+ 1, which will be called simply the sign ; m the numerator the value of the difference product, [f (4. . . . .) ; and m the denominator the term U, up. . . Such a table has been constructed for the example and is shown in Table 1. In the first column are listed all of the characterizmg I/ and #’ combmations which are possible under the rules. As an illustration, the various items m the first row of the table will be calculated out here :
The various values of a - $l are available from the calculation of 4l and it is convenient following calculation of the 4 and 4’ values to calculate all values of u and tabulate them for use. These terms are entered in the table, and the rest of the table is calculated, whereupon eq. (7) may be written :
If, now, a value of n, the number of rectifying plates, is set, from eq. (7) the correspondmg value of m may be calculated. In this example the followmg values have been calculated :
The sign by which the term is to be multiplied is determined by (- 1) @ subsrripts On$ ti v valuea)which, for the first combmation, ls (_ 1)(1+2+5+4+5)= (-_ 1~15~ _ 1. The sign is thus minus. Next the difference written out :
~lW](1.22)
product
(177)]
is,
when
[(a264 ( 74
= + 81-O Lastly the product of u values = ul t(s us eC;u; where ul = (a.4 - 41) (ag - ~4) (ac (ag - M (ax - 41) = 12 8 etc. Then ul U, us ui u; =
+
0787.
$1)
n 6 5 45
m 45 52 60
n+m 10 5 10 2 10 5
It is apparent that the mmimum in the curve of total plates is flat, and that n = 5 and m = 5 2 essentially represents the optimum point. The assumption, made at the beginning of the example, that components A and B, which are lighter than the light key, go completely mto the top product, and that component E, heavier than the heavy key, goes completely mto the bottom product, should be checked. The actual amounts of these components m the products will vary to some extent with the number of rectifymg and strippmg plates. Since the minimum point in total plates is of most interest, the amounts of the dduents m the products should be calculated at this point, i.e., with rectifying plates equal to 5 and stripping plates equal to 5 2. The calculation of the amount of each dduent m the products requires the solution of one additional determinant. This is identical to the solution for plates, except that the 4
285
GEltliARD
%.ElN
and 4’ values used are different. The correct values of + and 4’ to be used for the checkmg of each dlluent is drscussed m the previous paper by ALDER and HANSON [l]. Followmg the calculation of the amounts of the dduents m the products, the question of whether or not the calculation for the number of plates should be repeated with more correct 4 and 4’ values is a matter of judgment and depends on the accuracy desired m the dnal answer. The calculation of this example, while it may appear so at first glance, was not excessively time consuming. Actually the most labour m such a calculation occurs m calculation of the + and 4’ values. Famlhanty with the rest of the calculations leads to many short cuts which save considerable time. For example, It should be apparent that if eq. (7) is multlphed through by the product of all u values, there remam m each major term only two u values m the numerator, the values of u corresponding to the one + value and the one 4’ value which do not appear m the characterlzmg 9 term. Along with this slmphficatlon, others can be effected, and convement tables can be set up for calculatmg the difference product terms. The amount of time which will be spent on the calculation of a column by this method 1s a definite function of the amount of experience with the method which the calculator possesses. Whether the method
and
D. N.
-ON
1s efficient m comparison with plate-by-plate procedure is mostly dependent on the system to be fractronated, and again It must be left to the judgment of the calculator to determine whether it is desirable to employ tlus method. Certainly there are many cases where it will prove of considerable value. NOTATION i in the bottom product, moles/mole of feed. d (z,)~ = amount of component f in the top product, moles/mole of feed k = order of a determmant. n = rectrfymg plates, mcludmg the feed plate. WC= strippmg plates, mcludmg the reboder u_,f = product of all terms (aA - &), where &, is any value of 0 or & occurrmg m the altemant bemg solved. uB = solar product of all terms (a~ - &), etc. ur = product of all terms (CC+- #r), where. 9 rs the relatrve volatihty of a general component. The product contams a term for each component. U; = product of all terms (q - 4;). etc. (z,), = mole fractron of component t m the hqmd on the feed plate A, B, C, etc , refer to components ; m general A being the hghtest, B next hghtest, etc. b (z&, =
A,
A;
=
=
amountof component
abbrevlatlon abbrevratron
a,.4 for ad-41 a4
for ---L-
OLA-h
Table I.
Sample calculation
SW + + + + + +
Cl(4 * * ..I
uou*....
+ 810 - 22.2 - 454 - 18.2
+007WY
- 80.4 + 064 - 0464 + 284 -
890 Om717
+ 0.0768 -000955
-
0~0108
+ 0.0269
-000958 + 0.00125 - 0+9828 -0.0705 +090922 -0a241 - 0090787 + OaOO101 -0000288
; R, = -
OlB
, etc.
aB-6l
, ; B, = 7
aB
; etc
aB-44 flow, moles/mole
V = rectrfymg section vapour of feed. V’ = stnppmg sectron vapour flow, moles/mole of feed. (X,)F = mole fraction of component a m the tota feed. Greek Letlera a, = relative volatrhty of component a. All relatrve volatrbtres are based on the same component. 4 = rectifymg sectron parameter # = stnppmg sectron parameter. +r = abbrevratron
for 4rn, #s -
4; = abbrevuatron
for l/(4;)“,
+221=abbreviatronfor
(:r,
I&
= abbreviation
&“s ek #; = 1/(4a)m~ etc.
+a = ($)‘,
for ($r,4;2=($i,etc.
et&
The 8olut.h of multieomponent distihtion eolumna
ft = dlffexenoe product obtained by subtmethg e8ehtermin8n8ltem8ntfrome8ehpreceding term and multiplying these dWereneea together.
General Submzriph o,p,q...r’,d,t’reiaton~eu~riptson9Md~ v8h.w. ~refer8to8gener8leomponeut.
REFEBENCES [I]
ALDER, B. J., and -ON,
[a]
MUIR, T., A Treatsue on fhe T-of
D. N. ; CAcm. Eng. Pro& la50 46 48.
[8]
thDEEWOOD,
lkWmananta, New York, 1930.
A. J. V. ; J. h?f. hfrdeum,
lW5 31 111.
[4]
thDEEWOOD,
A. J. V. ; J. Inst. Pdfolcum, 1948 32 5%
[5]
UNDEXWOOD,
k
J. V. ; Chc9n.ET@!.Prog., 1948 44 003.
The solution
Pcgamon
Prew Ltd
of multicomponent
distillation
columns
GERHARD KLEIN* and D. N. HANSON Departmentof Chemistryand Chemmal Engineering, University of California, Berkeley (Recemd
7 Februmy
1955)
Summary-The equations of A J V UNDERWOODfor solutron of drstillation columns are arranged to grve a solutron m the form of a sum of difference products For many drstdlatron problems the present solutron IS much less tedrous than the strarghtforward solutron of UNDE~WOOD’S equatrons or the determmantal solutron proposed by ALDER and HANSON [l]. The solutron 1s dlustrated by an example employmg a Ave-component system R&arm&-Pour la solutron des &matrons de A J V UNDERWOODpermettant le calcul des colonnes ir drstdler, les auteurs consedlent l’emplor d’une somme de prodmts d’une &ff&ence Dans de nombreux probkmes de dlstdlatlon, cette solution est beaucoup moms phble que la solution duwte des 6quatlons ~‘UNDERWOODou que la solutron sous forme de d&.ermmenta proposh par ALDEBet HANSON La solution est. rlhrstr& par un exemple sur un systeme B canq composants.
THE solution of multlcomponent dlstlllatlon problems IS, m general, time-consummg and tedious work The methods whrch are avadable for such solutions employ various basrc assumptions and varrous techmques, and the partrcular method whrch is best suited to the problem at hand depends largely on the accuracy of the answer desired and on the physical make-up of the system to be dlstlled. As an example, calculation procedures are plate-by-plate drstmctly useful for some systems, for example systems m which all of the components of the feed appear m one or the other of the products rn well-definable amounts and m which the drfference IQ volatrlity between the key components is large. For such systems, calculatron plate-byplate from one end of the column ~111 provrde the solutron wrthm a few plates. If, on the other hand, the key components he Intermediate m volatlhty wlthm a large number of components, and a large number of plates is to be expected, plate-by-plate calculatrons become difficult, and procedures which allow the calculation of the number of rectlfymg and strlppmg plates as complete sectrons become most useful. Such “group ” procedures have only recently been l
Present
address
-
developed, and to the authors’ knowledge, four basrc solutrons presently exrst. The relative merits of these are as yet undetermmed but one which shows promrse of consrderable usefulness IS the set of equatrons derived by A. J. V. UNDERWOOD
[S,
4,
51.
The equations provide exact solutions of drstlllatlon columns under the basm assumptions of constant relatrve volatrhtres and constant sectional flows. The correctness of these assumptions depends upon the system, and although they are not m general exact, distlllatron equatrons mvolvmg then use have found wide apphcatron. The flows are usually set by the internal reflux at the top of the column and the thermal condrtron of the feed. Varrous procedures are m use for estlmatmg average relatrve volatihtres. Experience has shown that satisfactory values may be obtamed usmg an average of the top plate and reboller temperatures werghted by the molar amounts of top and bottom products or, alternatrvely, that temperature at whmh the product of the equilibrium constants of the hght and heavy key 1s equal to umty. The solution of UNDERWOOD’Sequations grves directly the number of rectrfymg and strrpping
%~~ita~ Engmeenng PNBJ~&, Unlvemlty of California, Richmond, &lif~rn&. 229
GERHARD
and D. N. HNWON
KLEIN
plates. The systems ordmarlly encountered m multlcomponent dlstlllatron contam components both lighter than the light key and heavier than the heavy key. The trial and error solution followmg the lines proposed by USDEHWOOD [5] then becomes prohrbltrvely time consummg. ALDER and HANSON [l] have proposed a solution of the equations m determmantal form, bypassing the calculation of feed plate concentrations proposed by UNDERWOOD. However, since the order of the determinants which must be solved m the method of ALDER and HANSONIS equal to the number of components m the feed, the dlficulty of the solution mounts rapidly with an increase m the number of components, and a solution which would avoid this difficulty IS considerably more useful. The present paper proposes a solution whrch accomplishes this by developing the determinants mto a sum of term\, each of which contains a quotient of products of differences. For the convenience of the reader, and to mtroduce the notation to be used, a brief review of the developments which led to the solutron 1s presented. To avoid excessive duphcatron of previously published material, the review and the development of the solution will be presented for a specific case. The use of the solution m solving other cases can be obtained from a brief study of the previous paper by ALDER and HANSON. For a system of three components A, B and C, then, the followmg equations may be written :
a, = relative volatility of component i. All relative volatlhtles to be based on the same component. + = a rectifying section parameter 4’ = a stripping section parameter USDEHWOOD employs equations (1) and (2) to define the rectifying and strippmg section parameters, + and 4’. It is apparent that for this three component system there are three values of C#which will satisfy eq. (1) such that if these are numbered m decreasing order of magnitude : aA > 41 > ag aB > 9b > h2 ac > 48 There are also three values of 4’ which will satisfy equatron (2) and if these are again listed m decreasing order of magnitude :
aA
aE >
aA ___bA)f
+
aB-h
aA -++ bA)f aA-
aB @B)f
&A)/
a.4 -
+
aB -
4
ac - 4
(2)
where V = rectifying section vapour flow, moles v’ = stripping section vapour flow, moles
d (a~,)~= amount of any component i in the top product, moles b (;L& = amount of any component i m the bottom product, moles 280
aA ___ @A)f aA-
NC+
aB-+!2
+
aA_h
aAbb~)b + aBbbB)b + acbbdb
ac
aB (%)f + aC b%)f ~
aA-h
-aA -Vv’
+‘a >
aA aB
UNDERWOODhas shown that two independent equations may be wrrtten relating the three values of +, the concentrations of A, B and C in the liquid on the feed plate, and the number of rectifying plates. These equations are .
and
+
+‘I > > +‘a>
+
$1
=
aB h?), + % ~ - (%)f w-+2 w--h -aB @B)f aB-48
+
(8)
aC (Gf ac+S
aC (%)j
Al ” =
0&
(8’)
QC-$8
where (a+ = mole fraction of component i in the liquid on the feed plate. R = rectifying plates, mcludmg the feed plate. The authors, to simphfy the wrltmg of such equations have adopted a shorter nomenclature, where
The solution of multicomponent clistillatlon columns aA
-= a.4
-_!k_
A,,
-
+l
wi
-
=
ALDER and HANSON have suggested that the solution of these equations be accomphshed by solution of the thud order determinant which can be set up as follows :
B,, etc.
+l
and
0 +2t* &
Rewriting become
2’
equations
(3) and
(S’),
4
these then
-42
(8.4)~
+
J32 @B)f
+
c2
kk)c)/
A2
@A)f
+
B2
@B),
+
c2
h),
A2
@A)1
+
B2
@B)f
+
c2
@c)/
(3)
a’
and =
+
(3’) 82
Slmllarly, two independent equations can be written for the strlppmg section * A,
@A)f
+
J32’
A,’
(*A),
+
B,‘@B)~
A,’ -
@A)f
+
B,’
hj),
+
‘&’
6%),
&A)/
+
B2’
(zB)f
+
c2’
(Q)/
where A,’
A2
A;--
AI @A),+ B, bB)f + CI (@ = +
-42’
-
@B,f
+
‘7,’
@d,
+
cl’(~)/
=
+’ 21
=
+’
-,
etc.
82
(4) (4’)
: =
rA a.4
-
B,’ ~62”
=
A-aB
-
+A’
and
where m = stripping plates, mcludmg the reboller but not the feed plate. If components B and C are the key components and their separation 1s set, then the amount\ of these components m the top and bottom As an approximation products are known. for the first solution, It will be assumed that component A goes completely mto the top product. This approxlmatlon may be corrected to any desired degree of accuracy later and does not represent a loss m accuracy of the calculation. If, then, the reflux 1s set, the quantities V and v’ are known and the + and 4’ values may be calculated from equations (1) and (2). It will be noted that a consequence of the approximation of no component A m the bottom product 1s or that dl’ = aA. This means that A,’ aA/(aA - &‘) is equal to infinity and as a result eq. (4) becomes useless. Equations (3), (a’), and (4’) can be written out however and solved for n and m.
$21
-42--2h32
A;&
Bl -
332 421
Cl
-
C2
$21
B2---~,+a2
C2--C39%,2
B;-
C;--C;I,&
B&A
=o
(5)
The unknowns n and m are contamed as exponents m the # and I,Y terms. If a value of 12 1s set, the correspondmg value of m may be calculated from eq. (5), and m this way the picture of all possible combmatlons of rectifying and strlppmg plates which ~111 perform the dlstdlatlon can be found. The solution of eq. (5) IS not difficult. However, if, for example, a five component system were to be dlstllled, eq. (5) would become a fifth order determinant and the solution would be considerably more tedious. Since the number of components m the feed 14 equal to the order of the determinant to be solved, ab the number of components increases, the labour of solution by further However, becomes prohlbltlve development, the solution can be reduced from determmantal form to a difference product form which 1s much easier to solve. Kq. (5) may be wrltten : J~,Y%-A,#,
W/5--2+2
C,#l-C2+2
-42$2--A,&
B292-%#2
C2#2-G&
A&;-A;&
B&;-B;&
C;t&C;la,b;
=o
N6)
where & = #Q*, #2 = #Jo*,etc. & = (V&P,
(Ii = (V&P,
etc.
It is apparent that the term aA aB ac can be cancelled out of the determinant, eq. (6). The term
A, t,bl, for example,
IS then ~ aA
1 -
#I
*1*
Rewritmg eq. (6) in a new nomenclature to show the droppmg of all a values m the numerators of the mdlvldual terms 1s not worthwhile, however, and It can merely be understood that for the rest of the development,
231
A, =
GEEENUI
and
-N
etc. This lapse m preciseness of nomenclature here leaves no ambiguity m the final solution. Eq. (6) can also be factored mto six third order determmants to give :
D. N. HANSON
and
If these factors are dropped, then the first term in equation (7) becomes : (9192
Y4J P
(4142
6,
UA UB UC
and if this IS written out it is :
*s43 [Ml -
-($1
[taA
If the first determinant m eq. (7) is written out, it is :
Such a determmant is a double alternant and can also be written [2] as
-
92)(+I - 42 (A4- &,I 41)taA - +a)taA - hi)]
Eta,- 41)taB
’ 4%)ta - &]
x
’
It will be noted that the liberty of writmg the difference product, [f (4, +a +i) with all terms reversed has been taken. It IS more natural to calculate the difference product this way, subtracting + and 4’ values from $ or +’ values precedmg, and it yields the same results m the summation of terms in eq. (7). For illustration, the second term m eq. (7) 1S
where k = the order of the determmant. 5f = difference product obtained by taking each term and subtractmg from it m turn each term precedmg, then multiplymg all of these differences together. 214
UB
=
=
or written out -
difference product obtained by multiplying together all terms (aA - 4,) where +r is a general value of 4 which will take all values of 4 used in the determmant. a difference product exactly hke uA except that aB is used in place of aR. etc.
Before proceedmg, some simpbfication can be effected. Two factors enter mto each determinant m eq. (7) and may be cancelled. Thus we may drop the terms (4
(- 1)
p-1'
($1A4vuWl - 44)(41- 6, (42- 9ul [taA 41)caA - d2)taA - +d]
[caB- ‘$1)taB
t_ h?) taB -
hi)]
x
’
The other terms m eq. (7) are constructed in the same way. At first acquamtance, the terms m the difference product solution of eq. (7) seem quite long and mvolved, and m the extension of the method to systems of several components, it appears at first that even the work of keepmg track of the malor terms m equations of the type of eq. (7) ~111 be difficult. Since this is one of the chief
283
The solutionof multicomponent dbtillation columns
difficulties m the difference it to set up a standard procedure a set of as automatic as possible. of equation be written as : ~
(_
l)b+P+P+
r+a+;+) ,
u, up uq . . . 24,
I
ut . . .
24,
(404p4q* * * 4; 4: 4;. - -1 ,
*
s
u, up uq . . . u, u, ut . . .
lf (4, $34q* * * 4; 4: 4;) u,upuq...
u; u;24;. . .
1 J =
()
4.
All of the $ values are written down first, followed by the #’ values.
5.
The combmations of # and 11’ values used are all possible combmations under rules (l), (2), (8) and (4) which do not contain any repeating 1,4or #’ values. Thus, the number of combinations possible and also the number of major terms m the summation is equal to the number of useful + values times the number of useful 6 values.
6.
The difference product 5’ (4. $P 9, . . . . . +I +; +; * * * -1 is obtained by subtracting each + or 4’ value from all Q or 4’ values preceding it in the order of listing of the 7% and $’ values and multiplying these differences together. Thus I* (4, &, 4s . . .
(7)
In general this equation is the solution of a determmant of the type of eq. (6). The order of the determmant is equal to the number of components m the feed, and m a case where the separation of the two key components has been set and the reflux has been set, one such determinant can be written [l]. This determinant will m general employ all of the 4 values which are useful and all of the 4’ values which are useful, thus excluding 4 and 4’ values which are equal to relative rolatihties. The total number of useful ~3and +’ values will equal the number of components plus two : the number of + values and the number of #’ values ~111 vary depending on where the key components he m the hst of components. The rules which may then be set up for the solution of equation (7) are as follows : 1.
order of magnitude of #). The #’ values are written down in ascending order of magnitude of subscripts (descendmg order of magnitude of 4’).
* -4; 4; 6 . * 4 = [bib -
2.
The total number of 16 and 4’ values written down m each term 1s equal to the number of components.
8.
The $ values are written down in ascending order of magnitude of subscripts (descending
(&- 3;) (&-- +;j * .I [W, - 42 ($I - 9%) .I* - * * [C41 - 4,) (4; - 4;)- **.]
[(&
7.
$J
. . .] . . . . . .
The number of u values m each term 1s equal to the number of $ and +’ values used m the term. For convenience these have been grouped and characterized by a 4 or 4’ value rather than by component. Thus u 1 = (a~ - 41) (an - 74) (ac - 41) (an - &) . . . . . . . , & = (aA - 6) (as -
8.
-
6) (ac -
6) (an -
6) . . . . etc.
Each term in the summation is multiplied bY - 1 or + 1 depending on the sum of the subscripts of all 4 and 4’ values used m the term. Smce the choice is arbitrary, the convention has been adopted that if the sum of the subscripts is even, the term is multiplied by + 1 and if the sum of the subscripts rs odd, the term 1s multiplied by -1.
The rules set down here completely 288
9,) * ’ *
....
Each major term m the summation will be characterized by one less than the number of useful Q values and one less than the Thus the number of useful 4’ values. number of 7%values written down m each term is one less than the number of useful + values, and the number of ~5’ values written down m each term is one less than the number of useful 4’ values.
$JJ MO -
detlne
GNENAEIIKLEIN and D. N. the solution of the determinant and enable calculation of the column. Some study of them is necessary, however, before much facility m such solutions is gamed, and to aid in this understandmg of the method an example of a distillation column separating a five component system has been solved for presentation here. Consider a five component system of A, B, C, D, E. The feed to the column is saturated liquid of the composition shown in the table below, and the relative volatilities based on the heavy key, D, are as shown :
&hNI3ON
the minimum m the curve of total plates verses feed plate location 1s very flat, and a wide choice of feed plate location can be made with httle effect on the total plate requirements. The fist step in solution of the problem is the calculation of the + and 4’ values. This calculation is the solution of the following equations : For values of 4 : 16= 5(005)+ 5- +
25(0 25)+ 22(088) + l(OO2) 2.5 - 4 2.2 - $6
For values of #’ : XF
A B c
D E
0 05
6”
0.25 0.40 0 20
25 22 1.0 06
0 10
_
2.2 - lp'
Components C and D are the key components. Specifications on their separation are set so that 95% of C goes mto the top product and 90% of D goes into the bottom product. It is assumed as a first approximation that the light dluents, A and B, go completely into the top product and that the heavy dluent, E, goes Thus completely into the bottom product. the products, based on one mole of feed, are : d tx,)d
A B c
D E
005 0.25 088 0 02
-
~(j=ww+
b tx,)b
-002 018 0 10
VW+
06(010)
I - 4'
0.6 - $b'
(9)
Equations (8) and (9) may be solved by any of the iteration methods available for solution of cubic and higher order equations. Actually, trial-and-error solution is quite rapld and has the advantage that m the solution for 4 or +‘, the quantities aA - +i, etc., are calculated. Since these drfferences are of use later it 1s convement to have them tabulated as a result of this first calculation. Solving eq. (8),then, the following values of I# are found : +1 = 4885 Qa = 2.898 Al = 1882 G=;y
The mimmum rectrfying section vapour flow for this separation IS calculated to be 1 206 moles/mole of feed. An operating rectifying section vapour flow of 1.6 moles/mole of feed, or approximately l8Oo/oof mmimum, was chosen for the calculation. With these variables set, there is an infinite number of rectifymg and stripping plate combmatrons which will perform the distillation, and the answer desired is the combmatron of rectlfymg plates and strippmg plates which ~111 constitute a mmimum of total plates, or will give the optimum feed plate location. Generally
It will be noted that a value of 4s IS listed. If the assumption 1s made that d (~x)~ = 0, then $6 = aE and may be bsted as such although It 1s of no use m solving the column. Simdarly, from eq. (9) the followmg values of 4’ are found : $b; = 5 &=25 +; = 22811 +; = 1118 9% = 0.6284 Here again values are listed for 4; and 4;. Under the assumptions of b (z,& = 0 and b (z~)~ = 0, the values of +; and 4; are respectively equal to aA and ag. These values of $; and & are of no
284
The solutionof multicomponentdi&.llationcolumns use in solving the column, but are listed in order to clarify the system for subscript numbering of the +’ values. Three sigmficant figures are being carried m the calculations at this point, although it is apparent that this necessitates having more than three sigmficant figures m the + and 4’ values. The + values to be used m the solution are 419 429 4*9 and +4. The 4’ values to be used are +;, &, and 4;. Generally, the next step m calculation is to form a table listing the characterizing 7%and #’ combinations. In the table may be listed, then, the other terms which accompany the characterizmg 9 and #’ combmation m equation (7). These are - m the numerator, the term (- 1) (O+P+*+ * “+“+“+ 1, which will be called simply the sign ; m the numerator the value of the difference product, [f (4. . . . .) ; and m the denominator the term U, up. . . Such a table has been constructed for the example and is shown in Table 1. In the first column are listed all of the characterizmg I/ and #’ combmations which are possible under the rules. As an illustration, the various items m the first row of the table will be calculated out here :
The various values of a - $l are available from the calculation of 4l and it is convenient following calculation of the 4 and 4’ values to calculate all values of u and tabulate them for use. These terms are entered in the table, and the rest of the table is calculated, whereupon eq. (7) may be written :
If, now, a value of n, the number of rectifying plates, is set, from eq. (7) the correspondmg value of m may be calculated. In this example the followmg values have been calculated :
The sign by which the term is to be multiplied is determined by (- 1) @ subsrripts On$ ti v valuea)which, for the first combmation, ls (_ 1)(1+2+5+4+5)= (-_ 1~15~ _ 1. The sign is thus minus. Next the difference written out :
~lW](1.22)
product
(177)]
is,
when
[(a264 ( 74
= + 81-O Lastly the product of u values = ul t(s us eC;u; where ul = (a.4 - 41) (ag - ~4) (ac (ag - M (ax - 41) = 12 8 etc. Then ul U, us ui u; =
+
0787.
$1)
n 6 5 45
m 45 52 60
n+m 10 5 10 2 10 5
It is apparent that the mmimum in the curve of total plates is flat, and that n = 5 and m = 5 2 essentially represents the optimum point. The assumption, made at the beginning of the example, that components A and B, which are lighter than the light key, go completely mto the top product, and that component E, heavier than the heavy key, goes completely mto the bottom product, should be checked. The actual amounts of these components m the products will vary to some extent with the number of rectifymg and strippmg plates. Since the minimum point in total plates is of most interest, the amounts of the dduents m the products should be calculated at this point, i.e., with rectifying plates equal to 5 and stripping plates equal to 5 2. The calculation of the amount of each dduent m the products requires the solution of one additional determinant. This is identical to the solution for plates, except that the 4
285
GEltliARD
%.ElN
and 4’ values used are different. The correct values of + and 4’ to be used for the checkmg of each dlluent is drscussed m the previous paper by ALDER and HANSON [l]. Followmg the calculation of the amounts of the dduents m the products, the question of whether or not the calculation for the number of plates should be repeated with more correct 4 and 4’ values is a matter of judgment and depends on the accuracy desired m the dnal answer. The calculation of this example, while it may appear so at first glance, was not excessively time consuming. Actually the most labour m such a calculation occurs m calculation of the + and 4’ values. Famlhanty with the rest of the calculations leads to many short cuts which save considerable time. For example, It should be apparent that if eq. (7) is multlphed through by the product of all u values, there remam m each major term only two u values m the numerator, the values of u corresponding to the one + value and the one 4’ value which do not appear m the characterlzmg 9 term. Along with this slmphficatlon, others can be effected, and convement tables can be set up for calculatmg the difference product terms. The amount of time which will be spent on the calculation of a column by this method 1s a definite function of the amount of experience with the method which the calculator possesses. Whether the method
and
D. N.
-ON
1s efficient m comparison with plate-by-plate procedure is mostly dependent on the system to be fractronated, and again It must be left to the judgment of the calculator to determine whether it is desirable to employ tlus method. Certainly there are many cases where it will prove of considerable value. NOTATION i in the bottom product, moles/mole of feed. d (z,)~ = amount of component f in the top product, moles/mole of feed k = order of a determmant. n = rectrfymg plates, mcludmg the feed plate. WC= strippmg plates, mcludmg the reboder u_,f = product of all terms (aA - &), where &, is any value of 0 or & occurrmg m the altemant bemg solved. uB = solar product of all terms (a~ - &), etc. ur = product of all terms (CC+- #r), where. 9 rs the relatrve volatihty of a general component. The product contams a term for each component. U; = product of all terms (q - 4;). etc. (z,), = mole fractron of component t m the hqmd on the feed plate A, B, C, etc , refer to components ; m general A being the hghtest, B next hghtest, etc. b (z&, =
A,
A;
=
=
amountof component
abbrevlatlon abbrevratron
a,.4 for ad-41 a4
for ---L-
OLA-h
Table I.
Sample calculation
SW + + + + + +
Cl(4 * * ..I
uou*....
+ 810 - 22.2 - 454 - 18.2
+007WY
- 80.4 + 064 - 0464 + 284 -
890 Om717
+ 0.0768 -000955
-
0~0108
+ 0.0269
-000958 + 0.00125 - 0+9828 -0.0705 +090922 -0a241 - 0090787 + OaOO101 -0000288
; R, = -
OlB
, etc.
aB-6l
, ; B, = 7
aB
; etc
aB-44 flow, moles/mole
V = rectrfymg section vapour of feed. V’ = stnppmg sectron vapour flow, moles/mole of feed. (X,)F = mole fraction of component a m the tota feed. Greek Letlera a, = relative volatrhty of component a. All relatrve volatrbtres are based on the same component. 4 = rectifymg sectron parameter # = stnppmg sectron parameter. +r = abbrevratron
for 4rn, #s -
4; = abbrevuatron
for l/(4;)“,
+221=abbreviatronfor
(:r,
I&
= abbreviation
&“s ek #; = 1/(4a)m~ etc.
+a = ($)‘,
for ($r,4;2=($i,etc.
et&
The 8olut.h of multieomponent distihtion eolumna
ft = dlffexenoe product obtained by subtmethg e8ehtermin8n8ltem8ntfrome8ehpreceding term and multiplying these dWereneea together.
General Submzriph o,p,q...r’,d,t’reiaton~eu~riptson9Md~ v8h.w. ~refer8to8gener8leomponeut.
REFEBENCES [I]
ALDER, B. J., and -ON,
[a]
MUIR, T., A Treatsue on fhe T-of
D. N. ; CAcm. Eng. Pro& la50 46 48.
[8]
thDEEWOOD,
lkWmananta, New York, 1930.
A. J. V. ; J. h?f. hfrdeum,
lW5 31 111.
[4]
thDEEWOOD,
A. J. V. ; J. Inst. Pdfolcum, 1948 32 5%
[5]
UNDEXWOOD,
k
J. V. ; Chc9n.ET@!.Prog., 1948 44 003.