COM MU N ICATl ON
D I S T R I B U T I O N OF R E S I D E N C E T I M E S I N A CASCADE OF M I X E D V E S S E L S W I T H BACKMIXING When there i s no backmixing, the probability density for any residence time is given b y a single term, but in the case of backmixing, this density i s the summation of an infinite series. A particle passing through the cascade can backtrack to the previous vessel any number of times from zero to infinity, and there i s a term in the series for each number of backtracks. Described i s a computer calculation for generating the coefficients for the terms in the series.
for the distribution without backmixing was published over 40 years ago, a n d is \vel1 knoivn. H e r e the equation is extended to include backmixing. LVhen a stream of particles (or molecules) fl0u.s through a cascade of perfectly mixing vessels. such as in Figure 1, the probability that a pdrticle \vi11 remain in the cascade between t and t dt units of time is ( 7 . 2) :
A
This equation applies only \\hen there is no backmixingthat is, \
N EQuA'riox
+
+
+
where
+
/7 =
Qt ~
~
(flow rate) (time) ~~~
~~~~~
~~
pI
ez
~~
~
V volume of a single vessel
...
(n
+
+ I ) ! ' (n + 3 ) ! ~~~
~~~
~~
~
+
Zn13
ez
Zntd
~~~~~~
~~~~~
'
(n
+ 5 ) ! etc.: ,
First Vessel
4 X
=/ X
Leave f i f t h vessel with no back track. x
After one backtrack.
t Figure 1 , 88
+
t
+
x44
Cascade of n mixed vessels with backmixing, and part of the probability grid
l&EC FUNDAMENTALS
+ 4x2y5 + 8x2y6
After two backtracks. for n = 5
Table 1.
W Coefficients Calculated from the Probability Grid in Figure 1, with a Digital Computer n = 10 Vessels 71 = 10 VLssels _ _ ~ ~. ~~~~
x
0 1 2 3 4 3
(3
0 65610
0 40960
0 242’6
0 0 0 0 0
0 0’2’0 0 02056
0 005’3 0 00158 0 00041
0 0
8 9
0
0 0
10
11 12 13 14 15 16 118 19 20 21 22 23 24 25 26 228 29 30 31 32 33 34 35 36 3’ 38 39 40
0 0
0 12960 0 14515 0 13022 0 10962 0 0902’ 0 07374 0 06006 0 04886 0 03973 0 03231 0 0262’ 0 02135 0 0136 0 01411 0 0114’ 0 00933 0 0058 0 00610 00501 0 0040’ 0 00331 0 00269 0 00219 0 001’8 0 00145 0 00118 0 00096 0 00078 0 00061 0 00051 0 00042 0 00034 0 00028 0 00022 0 00018 0 00015 0 00012 0 00010 0 00008 0 00006 0 00005
2’853 15270 0’888 03996 02010 01009 00506 00254 00127 00064 00032 00016
0 0 0
0 0 0 0 0
0
~~~~~~
0.1 38742 31768 17019 0586 03062 01164 00426 00152 00053
0
0 0 0 0 0 0 0 0
0 0
0 0 0 0 0 0 0 0 0
a n d the probability density for a cascade of n actual vessels will be some composite oE these individual densities, Lvith each lveighted in proportion to the probabi1ir) of its occurrence:
Ll’,,+ ,e (I1
+
Z n+3
+ 3) !
~~
p
. . .
0.2 13422 19864 19177 15384 11162 07621 05002 0319’ 02005 01241 00761 00463 00280 00169 00101 00061 00036 00022 00013 00008
0.4 0 01008 0 02338 0 03528 0 04413 0 04981 0 05281 0 0535 0 05318 0 05158 0 04931 0 04663 0 04374 0 04077 0 0382 0 03495 0 03220 0 02961 0 02710 02489 0 02278 0 02083 0 01903 0 01’38 0 01586 0 01447 0 01320 0 01203 0 0109’ 0 01000 0 C0911 0 00830 0 00’5’ 0 00689 0 00628 0 00572 0 00521 0 00475 0 00433 0 00394 0 00359 0 00327
x =
x =
=
also Z
=
n.
x =
02
x =
04
0 00000
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 00104 0 00352 0 00850 0 01640 0 02695 0 03914 0 05150 0625 0 07146 0 0698 0 0’906 0 0’92 0 0’410 0 06828 0 06118 0.05348 0 04572 0 03832 0 03154 0.02555 0 02038 0 01605 0.01248 0 00959 0 00730 0 00550 0 00410 0 00304 0 00223 0 00163 0 00118 0 00085 0 00061 0 00043 0 00031 0 00022 0 00015 0 00011 0 00007
0 00000 0 00000 0 u0000 0 00000 0 00000 0 00001 0 00001 0 00002 0 00004 0 00007 0 00010 0 00015 0 00022 0 00030 0 00041 0 00054 0 000’0 0 00089 0 00111 0 00135 0 00163 0 00194 0 00228 0 00264 0 00103 0 00345 0 00389 0 00435 0 00482 0 00531 0 00581 0 00611 0 00682 0 00’33 0 00783 0 00833 0 00882 0 00910 0 009’’ 0 01022
05781 10942 1493 16011 14753 12020 08882 06062 038’3 02341 01349 00’46 00398 00206 00104 00051 00024 0 00011

\$’hen n is as large as a b o u t 10: we c a n use Stir
1b.71+
(n
x =
0 00017
ling’s approximation for n ! finally reduces to:
(2)
01
0 01642
1 p3n3
~
+ 3 ) (n + 2 ) ( n + 1)
d2%(n
+
e)n,
so that Equation 3
l~t’71+6 pjn5
(n
+ 5)
~~~~~
..
(n
+ 1) + ~
.
,] (4)
+ 3 ) + 2 ) + 1) . ~~~
(n
(7~
(ii
+
~~~
(n
+
It 7,L+6 Zj 5) . . . (n
+ 1) (3)
T h e I17’s are the weighting coefficients a n d are equal to the probability that a particle \vi11 follo\v a path including zero (11’,1), one (I17,12)! etc:.. backtracks. F r o m this it follo\vs that = unity. F o r backmixing Z becomes equal to (Q 2q)t l’instead of Qt ‘ V . T h e curve of x f ( Z )cs. Zfrom Equation 3 passes through its m a x i m u m in the neighborhood of Z = n . Since this is the region of greatest interest: it is natural to substitute p n for Z so that p \vould become the only variable in Equation 3. Of course. p c a n takc on all values from zero to infinity. but the peak o n the curve \vi11 lie in the neighborhood of p = 1 , \\.here
XII
+
Even this equation ib not simple to use for large values of n. because the 11 coefficients have to be calculated individually, as described belo\v. ‘To calculate the I I ’ coefficients. \ve define .t = y ‘(2 2q as the probabilit). that a parcicle \vi11 backtrack instead of joining the for\\ard Q stream leaving a n y veisel. a n d 1 = 1  t as the probability that it \\ill join the forxvard stream. ‘l‘he lo\ver part of Figure 1 may be called a probabilit) grid and i, the locus of all possible paths through a cascade of five vessrl,. ‘The five junctions along a n ) horizontal line in the grid rrpresent the five vessels. .AL each junction there i, the probabilit>. I . that a particle \vi11 niove horizontall) to the nrxt the probabilit).. I , that it will move do\vn\vard or backtrack. Each polynomial \\.ritten along a connrcting line of thr grid equals the probability that the stated line \vi11 lie in the path of the particle. 7’he pol>nornials along thr righthand boundary
+
VOL. 4
NO. 1
FEBRUARY
1965
89
of the grid are the probabilities that the particle \vi11 finally leave the cascade after zero, one, two, etc., backtracks. a n d are equal to the Lt’coefficients. T h e polynomials ai’e grrirrated by starting at the upper left a n d multiplying t h r input to every junction by 1 to get the horizontal output and by x to get the downward output (backtrack). As far as the grid is concerned, ,\ a n d y can have a n y pair of values srich that .\ y = 1, but for a real cascade .I cannot exceed ‘,‘q, which corresponds to the limiting case where Q = 0. 1 he T17’s i n ‘l’able I \\.ere generated o n a computer, using the grid calc,ul;ition i n Figure 1 . T h e tabulated valiies show that bvtien n and Y are both small, coefficient It’ decreases continuolie!; with thc number of backtracks, biit increasing either n or x causes I I ’ to go through a niaxinirim n i t h the niimber of backtracks. Eventually 11’ will approach zero, of coiirse, biit the n u m b r r of coefficients that have to br calculatrd beforr 11’ + zero can exceed n. ‘l‘his makes Equation 4 uinvirldy for large values of
+
+
+
(Q q)t/lzvinsteadof (Q 2q)t: L’as used in Cqiiations 2 to 4. Also, in calculating t h r It”s. Y for the last vrjsrl \vns assuinrd to be the same as for the Jther vcasels [except the first v t x e l ) . whilr the correct value is q Q q inrtrad of q Q tZ q . ‘1 he method given here could be used t o rebolve the oritpiit signal from a cascade to find the valiie of . A . ‘I he calciilation is made by plotting a family of dktribution ciirvrj for varioiis valiirs of 1, a n d then comparing these with the euprrirnental CurVe.
+
\V. R . RE‘IA121,1C:Ki
Corisolidution Coal Co Librurj: Pa. literature Cited
(1) Hani. A , . ( h e , H. S.,Chrrrr. . k f p t . ?,‘ri,q. 19, 6 6 3 (1918). (2) hfachfiilliii. K. R., \Veht>r. hl., Trcins. Am. T r i i / . Chrrrr. 29, 409 (1935).
n.
l ’ h e r e are two approximations in this method. ‘l‘here ir no backflow into the last vessel, a n d there is no backflow oiit of the first vessel, so that Z for these two end vessels is really rqual t o
F.”/
KECFJVFDfor rrlirv, M a y 13. 1963 ACC:EPTLI) .Uo\clntirr. 3 . 1964 1 Prrsriit address. Houriry Process and Chrmical Cn., M a r r u s IIook, Pa.
C O M M U N I C A T l ON
STABILITY OF CASCADED R E A C T O R S Previous work on the stability of a singlestage stirred reactor i s extended to an Nstage coscade.
The
conditions which are shown to be sufficient for a region of asymptotic stability include stage interactions for the first four stages, but do not require such restrictions for subsequent stages. An example is included which compares alternatives for a twostage feedback control application.
N PREL’IOLTS work
(1)the stability of a single stirred reactor
I was analyzed, a n d sufficient conditions wei’r established to determine a region of asymptotic stability (RAS) for that reactor. This work is concerned with the possible extensions of Liapunov’s direct method to a system of staged reactors.
theorem, if it is first transformed into normalized form a\ previously shown ( 2 ) .
dt
co
a
T
General Staged Reactor
Considering a cascadr of ,Y stirred reactors in series, a n y stage can be debcribed by simultaneous heat a n d material balances For the zth stage these a r e : dTZ
pT’C, 
=
dt
AHI’ri
l&EC
U(%,
..
.9.v. Ql
.
.
Qs)
=
I

[ ’ A ( T i 7,)  pqC, ( T ,  Ti,) 1
1
‘ l h u s , for a n A\\’stagrcascade, a system of 2,V differential rquations \ \ i l l br reqiiired to describe the cascade completely. .l.his s\ Stein of rquations may be analyzed by Krasovskii’s 90
?’he Liapunov function for this system of equations is
FUNDAMENTALS
provided that suficirnt conditions can be established to assure that ‘c’ < 0. O n e \vay to do this is through conditions ~ v h i c h establish the negarive definitrnrai of F,lthe sum of the Jarobian matrix for Eqiiation 2 and its transpose. T h e grneral k,) matrix \vi11 br the 2.\’th order square matrix :