of the grid are the probabilities that the particle \vi11 finally leave the cascade after zero, one, two, etc., backtracks. and are equal to the Lt’coefficients. T h e polynomials ai’e grrirrated by starting at the upper left and multiplying thr 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, ,\ and y can have any 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 on 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 numbrr 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 thr It”s. Y for the last vrjsrl \vns assuinrd to be the same as for the Jther vc-asels [except the first vtxel). 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, and 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’here are two approximations in this method. ‘l‘here ir no backflow into the last vessel, and there is no backflow oiit of the first vessel, so that Z for these two end vessels is really rqual t o
F.”/ 4 is:
1
0
~-
1
7
1 0' ~+j 3 4
1
0
f44
--
0 0
.
,
,
,
,
0
.
.
.
0
7
7
0
0
0
1 3 4 --
7
1
I
1 J33
a condition which ib similar to but more conservative than the necrssary stability criterion developed by \'an Heerden ( 2 ) and Aris and Arnundson ( 7 ) in single-stage linear analysrs:
co
1
j 5 6 f 6 6 ..
-
bgi
7
7 4
.
.
.
.
i
o
If. as is the case for most common kinetics. (hi. bo) equality 8 reduces t o '
7
1
0
--
0
0
> 0.
In-
7
-
0 0 0 0
or. if its elements are grouped into second-order submatrices, it may bc written as:
r
P,
1
-1
1
(0
0
0
0
0
0
0
0
0
1
7
1 1
,
~
192
7
1
0
1
--I 7
1
--I
F,
1
--I
7
7
. ~'
0
.
'
1
~0 L
0
0
0
1
0
--
I
--I
1
7
1
P,
7
1
J
where each
2
->++)I 1
(&
ai.
\Vhile this criterion is the only nebv contribution for the fifth stage and beyond. the earlier resrrictions on the conditions of the fourth and lower stages still apply. I n establishing the sign-definite character of P, through its leading principal minors. the sign of each term of the diagonal i n Equation 5 is fixed:
< 0; i
F,
= 1:
2 , . . ..\
(11)
But this restriction \vas showm in the previous work cited to be sufficient to ensure the as)-mptotic stability of a single, uncoupled reactor stage. Hence, the sufficient conditions arising from this analysis are also sufficient to establish the asymptotic stability of each stage. uncoupled. Anorher corollarbarises from the observation that the sign definite restriction on the first 2i leading principal minors of a '.\--order F, matrix are identically the terms that would arise from a truncated cascade consisting of stages 1 to i. Consequentlk. this analysis establishes also the asymptotic stability of all such subcascades. T h e problrm of geometrical analysis for many stages becomes increasingly difficult as higher order spaces are required. For .\-stages. a 2.\--Jimensional state space is required to illustrate both the I i a p u n o v function and the confining separatrices. The increased complexity of the higher order problems suggests the use of a high speed computer. In spite of this. it is still possible to dran. some qualitative conclusions lvith regard to general cascade stability. and in certain cases. as illustrated by the following example. even to predict thr quality of a control mode xvith regard to stability.
1 7
2-1.2.
In general. the negative definiteness of the F, matrix may be established by 2.\- conditions. each condition corresponding to a leading principal minor. If each leading principal minor of -@, is positive. then bv Krasovskii's theorem. the cascade is asymptotically stable From an examination of the successive leading principal minors of 8, it may be found that the addition of an ith stage to a n (i - 1)-stage subcascade contribute< two new restrictions (for sufficiency); ho\vever. if i > 4. the many zero entries in 8 , reducp this to a single additional restriction on the ne\v F,, determinant :
Two-Stage Cascade
Consider rhe case of t\vo stirred reactors in series. normalized statr equations are
'I he
'
(12)
1965
91
2\
VOL. 4
NO. 1
FEBRUARY
T h e - 8, matrix is:
- _1 T
i
0
L For a tLvo-stage reactor there are tbvo possible choices of a control temperature: TI or TP. If Tu is a function of T1 only, the effect of the feedback control appears only in the 8, submatrix of 811. T h e use of T I as a control variable was arialyzrd in the previous study ( 3 ) ,where it was shown that a sufficient condition for improved srability is :
T h e sufficient criteria for stability are: (i)
>0 - fl?
(ii)
fllf~
(iii)
fllf~f33
-
>0 1
- - f ~ - f122f33 T2
>0
T h e alternative control requires that Tube a function of I n this case, the normalized state equations are:
Inequalities 14 i , ii and 15 may be written more concisely as:
