ANL-6560 ANL-6560
argonne Bational Xaboratorg RESULTS OBTAINED FROM THE FRICKE
DIFFUSION KINETICS CODE
by
Erwin H. Bareiss, Cynthia Chamot,
and Hugo Fricke
RETURN TO REFERENCE FILE TtCllNlCAL PiJ3L!GAT13NS
DEPARTMENT
LEGAL NOTICE
This report was prepared as an account of Govemment sponsored work. Neither the United States, nor the Commission, nor any person acting on behalf of the Commission:
A Makes any warranty or representation, expressed or implied, with respect to the accuracy, completeness, or usefulness of the information contained m this report, or that the use of any information, apparatus, method, or process disclosed in this report may not infringe privately owned rights; or
B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of any information, apparatus, method, or process disclosed in this report.
As used in the above, "person acting on behalf of the Commission" includes any employee or contractor of the Commission, or employee of such contractor, to the extent that such employee or contractor of the Commission, or employee of such contractor prepares, disseminates, or provides access to, any information pursuant to his employment or contract with the Commission, or his employment with such contractor.
ANL-6560 Mathematics and Computers (TID-4500, 19th Ed.) AEC R e s e a r c h and Development Report
ARGONNE NATIONAL LABORATORY 9700 South Cass Avenue
Argonne, Il l inois
RESULTS OBTAINED FROM THE FRICKE DIFFUSION KINETICS CODE
by
Erwin H. B a r e i s s , Cynthia Chamot,
Applied Mathematics Divis ion
and
Hugo Fricke
Chemistry Divis ion
September 1962
Operated by The Univers i ty of Chicago under
Contract W - 3 1 - 1 0 9 - e n g - 3 8 with the
U. S. Atomic Energy C o m m i s s i o n
TABLE OF CONTENTS
Page
INTRODUCTION AND PHYSICAL INTERPRETATIONS OF SYMBOLS 9
[Table I ] .
SECTION ONE: FINAL RESULTS FOR a-RAY CASES. For s e v eral values of E. there are Tables of Final Resu l t s , Graphs of Final Fjj v s . B3 and B4 , and graphs of normal ized F, , v s . Bj^^ . 11
[Tables Il-XI and F igures 1-22).
SECTION TWO: FINAL RESULTS FOR 7-RAY CASES. For var ious combinat ions of p a r a m e t e r s , there are Tables of Final Resul ts for c a s e s with Two Radicals (Code A), followed by Tables of Resul t s for various c a s e s with Four Components (Code B) and the a s soc ia t ed graphs of final FJ; VS. B J and B4 23
[Tables XII-XV and F igures 2 3 - 3 5 ] .
SECTION THREE: INFLUENCE OF PARAMETERS ON RESULTS FOR a-RAY AND -,-RAY CASES. Final resu l t s are grouped in Tables and Graphs to show the effects of varying different paranneters 31
[Tables XVl-XIX and F igures 36 -45 ] .
SECTION FOUR: FIGURES TRACING THE REACTIONS OVER TIME AND SPACE FOR > R A Y CASES. Concentrat ions of the var ious reactants and react ion products as functions of t ime and space are shown 38
[Tables XX-XXIIl and F igures 46-103) .
SECTION FIVE: RESULTS OBTAINED FROM CODE C, FOR THREE PRIMARY AND ONE SECONDARY REACTION 57
[Tables XXIV-XXV).
ACKNOWLEDGEMENT 58
REFERENCE.S 58
LIST O F FIGURES
No. Tit le Page
1-4 Dependence of N o r m a l i z e d R e c o m b i n a t i o n F r a c t i o n s on
Bi (E = 5.305) 17
5-8 Dependence of N o r m a l i z e d R e c o m b i n a t i o n F r a c t i o n s on Bi (E = 2 .48) . 18
9-10 D e p e n d e n c e of N o r m a l i z e d R e c o m b i n a t i o n F r a c t i o n s on Bi (E= 1.29) 19
11-14 Dependence of N o r m a l i z e d R e c o m b i n a t i o n F r a c t i o n s on Bi (E = .675) 20
15-18 D e p e n d e n c e of N o r m a l i z e d R e c o m b i n a t i o n F r a c t i o n s on Bi (E= .3535) 21
19-22 D e p e n d e n c e of N o r m a l i z e d R e c o m b i n a t i o n F r a c t i o n s on
B ; (E= .1875) 22
.13-25 D e p e n d e n c e of F i n a l R e c o m b i n a t i o n F r a c t i o n s on Bi 27
2 6 - 2 9 Dependence of F i n a l R e c o m b i n a t i o n F r a c t i o n s on B^ 28
30-33 Dependence of F i n a l R e c o m b i n a t i o n F r a c t i o n s on Bi 29
34-35 D e p e n d e n c e of F i n a l R e c o m b i n a t i o n F r a c t i o n s on Bi 30
36 Inf luence of Ini t ia l F r e e R a d i c a l C o n c e n t r a t i o n s on F - n ( B ) / F „ ( 0 ) 33
37 Inf luence of Ini t ia l F r e e Rad ica l C o n c e n t r a t i o n s on F Z 2 ( B ) / F „ ( 0 ) 33
38 Inf luence of Diffusion Coef f i c i en t s on F J 2 ( B ) / F 2 2 ( 0 ) 34
39 Inf luence of R e a c t i o n R a t e s on F 2 2 ( B ) / F 2 2 ( 0 ) 34
40 Inf luence of R e a c t i o n R a t e s on F n 3b
41 Inf luence of Diffusion R a t e s on F n 36
42 Inf luence of R e a c t i o n Rate on Fi j for B4 10 36
43 Inf luence of Diffusion Rate on Fjj for B4 10 56
• LIST O F FIGURES
No. 11^ - ^ ^
44 Influence of R e a c t i o n R a t e on F22
45 Influence of Diffusion Ra te on F22
46 G e n e r a l Shapes of Solut ions Uj (R, T) 38
47-51 T ime Dependence of R e c o m b i n a t i o n F r a c t i o n s 39
52-54 T ime Dependence of R e c o m b i n a t i o n F r a c t i o n s 40
55-57 T ime Dependence of R e c o m b i n a t i o n F r a c t i o n s 40
58-59 T ime Dependence of R e c o m b i n a t i o n F r a c t i o n s 41
60-61 T ime Dependence of R e c o m b i n a t i o n F r a c t i o n s 41
62-65 Dependence of N o r m a l i z e d C o n c e n t r a t i o n s on P o s i t i o n s in
Space and T ime 46
66-69 Dependence of N o r m a l i z e d C o n c e n t r a t i o n s on P o s i t i o n s in Space and T ime 47
70-73 Dependence of N o r m a l i z e d C o n c e n t r a t i o n s on P o s i t i o n s in Space and T ime 48
74-77 Dependence of N o r m a l i z e d C o n c e n t r a t i o n s on P o s i t i o n s in Space and T ime 49
78-81 Dependence of Norma l i zed C o n c e n t r a t i o n s on P o s i t i o n s in Space and Time SO
82-85 Dependence of Norma l i zed C o n c e n t r a t i o n s on P o s i t i o n s m Space and T ime - ,
86-89 Dependence of Norma l i zed C o n c e n t r a t i o n s on P o s i t i o n s in Space and T ime
90-91 Dependence of N o r m a l i z e d C o n c e n t r a t i o n s on P o s i t i o n s in Space and T ime
92-93 T ime Dependence of Recombina t ion F r a c t i o n s
52
53
54
94-95 T ime Dependence of Recombina t ion F r a c t i o n s 54
LIST OF FIGURES
No. Tit le Page
9 6 - 9 7 Time Dependence of Recombination Fract ions 55
9 8 - 9 9 T i m e Dependence of Recombination Fract ions 55
100-101 Time Dependence of Recombination Fract ions 56
102-103 Time Dependence of R e c o m b i n a t i o n F r a c t i o n s 56
No.
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
XI.
XII.
xm.
XIV.
XV.
XVI.
XVII.
XVIII.
XIX.
XX.
XXI.
LIST O F T A B L E S
T i t l e _ P a g e _
Symbols and T h e i r P h y s i c a l M e a n i n g s 1°
Values of P a r a m e t e r s for S t a n d a r d a - r a y C a s e s 11
P a r a m e t e r s for P r e l i m i n a r y a - r a y (e = l) C a s e s wi th Two Rad ica l s Only (Code A) ^^
P a r a m e t e r s and F i n a l R e s u l t s of Some F o u r - c o m p o n e n t C a s e s B a s e d on S t a n d a r d I wi th E = 1.97 and e = 1 (for
\ . . 12 a r ays ) •
Case N u m b e r s of a - r a y C a s e s wi th S t a n d a r d II P a r a m e t e r s and Six Values of E for Which Bj and B4 V a r y 12
Resu l t s for S tandard II and Ep^ = 5.305 ( E j ^ = 10.61) 13
Resu l t s for S tandard II and Ej^ = 2.48 ( E ^ = 4.96) 14
Resu l t s for S tandard II and Ej^ = 1.29 ( E M = 2.58) 14
Resu l t s for S tandard II and E R = 0.675 ( E j ^ = 1-35) 15
Resu l t s for S tandard II and E R = 0.3535 ( E ^ = 0.707) . . . . 15
Resu l t s for S tandard II and Ej^ = 0.1875 (E]y[ = 0.375) . . . . 16
Values of P a r a m e t e r s for S t a n d a r d 7 - r a y C a s e s . . . . . . . 23
P a r a m e t e r s and Resu l t s for C a s e s wi th Only One R a d i c a l
(with and without a solute) 23
Spher ica l C a s e s with Two Componen t s Only 24
Resu l t s for 4 -componen t -y-ray C a s e s . . . . . . . . . . . . . . 25 & 26
Resu l t s of Cy l indr ica l C a s e s for Dif ferent Va lues of E to
P lo t N o r m a l i z e d F ' s v s . B 32
Resu l t s of S tandard C a s e s with S p h e r i c a l S y m m e t r y 34
Resu l t s when k i j / k n and D3/D1 Vary for 7 - r a y C a s e s . . . . 35
Resu l t s when kj^ /kn and D ^ / D J Vary for 7 - r a y C a s e s . . . . 37 Concen t ra t ions of Reac t an t s at V a r i o u s T i m e s and P o s i t ions in Space for Case 257: S p h e r i c a l G e o m e t r y ; S tandard II; B3 = B,j = 0. . . . . . . Concen t ra t ions of Reac t an t s at V a r i o u s T i m e s and P o s i t ions in Space for C a s e 305: S p h e r i c a l G e o m e t r y ; S tandard II; B, = 10"^, B . = 10
LIST OF TABLES
No. Tit le Page
XXII. Concentrat ions of Reactants at Various T i m e s and P o s i t ions in Space for Case 303: Spherical Geometry; Standard II; B j = 10"*, B4 ^ O.l 44
XXIII. Concentrat ions of Reactants at Various T i m e s and P o s i t i ons in Space for Case 301: Spherical Geometry ,
Standard II, Bj = B4 = 10"' 45
XXIV. P a r a m e t e r s for S t anda rd II C a s e s Run with Code C 57
XXV. Resul t s of C a s e s D e s c r i b e d in Tab le XXIV 57
RESULTS OBTAINED FROM THE FRICKE DIFFUSION KINETICS CODE
by
Erwin H. B a r e i s s , Cynthia Chamot, and Hugo Fricke
ABSTRACT
The Diffusion Kinetics codes descr ibed by the authors in ANL-6556 were used to study hypothetical react ions of free radicals produced by pass ing a rays and 7 rays through aqueous so lut ions . Tables and graphs of resu l t s are given for about 200 c a s e s run on the IBM 704 with various parameter combinat ions . Intermediate as well as final concentrat ions are given for s o m e 7-ray c a s e s .
INTRODUCTION
The IBM 704 Diffusion Kinetics c o d e s ( l ) were used for over 200 c a s e s to study from two to four primary react ions involving free ' . idicals , and a l so a secondary react ion. C a s e s were run to determine alues of various parameters , giving resu l t s , Rj and Rj. cons is tent with
those obtained exper imenta l ly for water and for water:iodine:iodide s y s t e m s . Other parameters were varied to study their e f fects . In this report, resu l t s are tabulated and a large number of graphs are included to i l lustrate various e f fec t s . Also , some results of the previous Avidac codes(^) are included for compar i son .
Complete output was normally obtained; so react ions of interes t may be traced over all space up to the t ime they were removed from the machine . Although most a-ray c a s e s (c - 1 for cyl indrical symmetry ) were run until the reaction was complete , some of the first c a s e s were not al lowed to finish, but those react ions can be followed part way to T = 0 .001 . Complete output is avai lable for s evera l Y-ray c a s e s (e = 2 for spher ica l symmetry ) which were run to complet ion . However, it was profitable to stop many of these c a s e s at T - O.l, s ince final concentrat ions of reaction products could be obtained by extrapolation.
The actual react ions being considered were: H • H - H2; H * OH = H2O; and OH • OH - H2O2 m all three codes ; and H t 1 - H* * I" and OH + I" OH' + 1 in code B only; and a secondary react ion, OH ^ HiOi - HjO * HOj in code C only.
10
For the machine codes, H = Xi, OH = Xj, I = X3, I" = X4, and HjOj = ^4 • The amounts of reaction products formed are Fn for H2, F12 for H2O, F22 i °^ H2O2, Fi3 for H + ,F24 for OH", and Fft for HO2.
The physical interpretations of the symbols are given in Table I. K should be noted that these are all dimensionless pa rame te r s . For further information, see Ref. 1.
T a b l e I
SYMBOLS AND T H E I R P H Y S I C A L M E A N I N G S
Syrnbol
Input P a r a m e t e r s
D e f i n i t i o n
e i = D i / D i
E = E R
e
A m e a s u r e of the i n i t i a l s c a t t e r of Xi r e l a t i v e to Xg.
The diffusion r a t e for X^ r e l a t i v e to t h a t of X^.
A m e a s u r e of t he i n i t i a l c o n c e n t r a t i o n of X j . It s h o u l d b e n o t e d t ha t Ef^ r e f e r r e d to in r e f e r e n c e s 2 and 3 wou ld be 2 E R .
T h i s i s 1 for c y l i n d r i c a l s y m m e t r y {a r a y s ) and 2 for s p h e r i c a l
s y m m e t r y ( - y r a y s ) .
Ra t io of the i n i t i a l c o n c e n t r a t i o n s of t he s o l u t e s to t he c o n c e n
t r a t i o n of Xi at t he o r i g i n .
The r a t e of r e a c t i o n for Xj c o m b i n i n g w i t h X; r e l a t i v e to t h a t of Xi with X j .
P r o b l e m P a r a m e t e r s
Symbol
R
T
Ui(R.T)
A m e a s u r e of the i n i t i a l c o n c e n t r a t i o n s of t he s o l u t e s , X3 a n d X4, i n c o r p o r a t e d into 7^ for input d a t a .
T r a n s f o r m e d f r a c t i o n a l a m o u n t s of r e a c t i o n p r o d u c t s f o r m e d (or of f r e e r a d i c a l s r e m a i n i n g if j - 0 ) .
P o s i t i o n in t r a n s f o r m e d s p a c e , go ing f r o m 1 at t he c e n t e r of the i n i t i a l f r e e r a d i c a l c o n c e n t r a t i o n s , to 0 f a r a w a y .
T r a n s f o r m e d t i m e , going f r o m 1 at t i m e of i n i t i a l f r e e r a d i c a l f o r m a t i o n , to 0 a s r e a l t i m e t i n c r e a s e s to co.
T r a n s f o r m e d c o n c e n t r a t i o n s of r e a c t a n t s Xi at v a r i o u s p o i n t s in s p a c e at t i m e T .
E x p e r i m e n t a l l y D e t e r m i n e d Q u a n t i t i e s (at T = 0)
Symbol
Ri
Rz
F z z / F i i - 1.62 for 7 r a y s wi th no s o l u t e p r e s e n t .
(Fio + E2o}/(Fii + Fiz) = 2.40 for 7 r a y s w i t h no s o l u t e p r e s e n t .
Note : The s u b s c r i p t s i and j r e f e r to t he r e a c t a n t s Xi , X^, X3 c o m b i n a t i o n s a r e l i m i t e d to 11 , 12, 13 , 22, and 24 .
o r X4. T h e ij
11
S E C T I O N O N E : F I N A L R E S U L T S F O R a - R A Y C A S E S ( C Y L I N D R I C A L G E O M E T R Y )
T a b l e II g i v e s c e r t a i n s t a n d a r d p a r a m e t e r v a l u e s w h i c h w e r e c o m -i n o n t o s e v e r a l c a s e s .
T a b l e III g i v e s s o m e p a r a m e t e r c o m b i n a t i o n s f o r w h i c h the c o m p u t a t i o n s w e r e not c o m p l e t e d , s o that o n l y i n t e r m e d i a t e r e s u l t s a r e a v a i l a b l e .
T a b l e IV g i v e s r e s u l t s f o r s o m e c a s e s b a s e d on S t a n d a r d I .
T a b l e V g i v e s p a r a m e t e r c o m b i n a t i o n s for c a s e s b a s e d on Stamd-a r d II, w i t h s i x d i f f e r e n t v a l u e s of £ . It s h o u l d be n o t e d that f o r the s m a l l e r v a l u e s of E , s m a l l e r v a l u e s of B m u s t b e u s e d t o f a c i l i t a t e t h e n o r m a l i z a t i o n s of c u r v e s t o B = 0 .
T a b l e s VI to XI g i v e the f i n a l r e s u l t s for the c a s e s in T a b l e V.
F i g u r e s l - i i a r e g r a p h s of f ina l F j j v s . B and n o r n n a l i z e d F j i v s . Bj.^^ f o r e a c h v a l u e of E in T a b l e V.
Tabic II
VALUES OF PARAMETERS FOR STANDARD a-RAY CASES
B , and B4 v a r y .
STA.VDARD I STA.NDARD 11
c
I 1
a
1 1
D j / D ,
0 4 5 4 5 0 .25
D ^ D , D J / D |
0 15 0 .15
k i i / k n
2
*^lj/'*Il
2«
l^i i /ki i
1 0 . 5 5
*lt should be noted that for 2-radical cases only, k i^kn and kj^/kn are •et equal to zero
Table III
PARAMETERS FOR PRELIMINARY u-RAY (t = 1) CASES WITH TWO RADICALS ONLY (CODE A)
C a s e N o .
I - I 1-2 1- 1 1-4
1-5 1-6 1-7 1 -8
26
P
1
E
1.8 1.8 1.8 1.8
2 .5 2 5 2 5 2 .5
1.97
D J / D ,
0 333 0 50 0 667 1.0
0 . 3 3 3 0 , 5 0 0 667 1,0
0 . 4 5 4 5
•^li/^^ll
2 2
i i.
I
^u/^\ 1
Standard 1
Thete case* were not computed all thr way. Cases No. 1-1 to 1-8 were stopped at T = 0.001, when F|o and Fjo were still near O.OS, and No. lf> was run until T - 10"'*, at whirh timr F,o still exceeded 0,01,
12
T a b l e IV
PARAMETERS AND FINAL R E S U L T S O F SOME F O U R - C O M P O N E N T CASES BASED ON STANDARD I WITH E = 1.97 AND £ = 1 (FOR a RAYS)
Case No.
126 127
128 129 130
126 132
133 134 135
B3
10-5
1 0 - '
1 0 - ' 1
10
10-5 10-5 10-5 10-5 10-5
B4
10-5
10-5 10 -5 10 -5 10 -5
10 -5
1 0 - ' 1 0 - '
1
10
F„
0.4565 0,4332 0.3312 0.1788 0,0404
0.4565 0.4540 0.4240 0.3789 0.3426
F12
0.4711 0.4567 0.3759 0.2372 0.0697
0,4711 0.4566 0.3778 0.2387 0,0699
F l 3
0,0724 0,1101 0.2929 0.5840 0.8899
0.0724 0.0894 0.1962 0.3824 0.5875
F22
0.5070 0.5034 0,4653 0.4086 0.3621
0.5070 0.4896 0 .3778 0.1951 0.0413
F24
0.0219 0.0399 0 ,1588 0,3542 0,5682
0.0219 0 .0538 0.2443 0.5662 0 .8888
Table Y
CASE NUMBERS OF a -RAY CASES WITH STANDARD U PARAMETERS AND SIX VALUES OF E FOR WHICH B3 AND B4 VARY
E • 5.305
Bi
10 1
0.1
10-J
10-5 10-' 10-11)
10
580
iW
1
581
521
0.1
582
522
B4
10-5
583
523
w-i
525
526 527
528
524
10-' 10-10 E . 0.675
Bi
10 1
0,1
10-3 10-5 10-10
10-11
10
585
550
1
586
551
0.1
587
557
10-3
588 553
B4
10-5
555
556
557
558
554
517
10-' 10-10
516
515
10-15
515A
E • 2,48
»3
10 1
0.1 10-3
10-i 10-'
10-10
h 10
505
5'30
1
506
511
0.1
507
53?
10-3
508 511
10-5
535
536
517 518
514
1 0 - '
509
10-11 E • 0,3535
Bi
10 1
0.1
10-i
10-5
10-10 10-15
10
500
560
1
501
561
0.1
502
562
10-3
503
561
B4
10-i
565
566
567 568
564 511)
1 0 - ' 10-10
518
514
lO-li
529
B3
10 1
0.1 10-3 10-5
10-'
10-10
B4
10
540
1
541
0.1
542
10-3
543
10-5
544
10-'
539
10-10
539A
E • 0,1875
B3
10
1
0,1 10-3
10-5
10-10 10-15
10
590
570
1
591
571
0.1
59P
57?
10-3
593
573
B4
10-5
575
576
577
578 574
599
10-'
595
10-10
579
10-15
597
512
T a b l e VI
13
RESULTS FOR STANDARD II AND E L 5.305 ( E M ^ 10.61)
C a s e No.
BJ = 1 0 - ' B« =
F„ Fu Fu fu Fi«
520
10
0.35475 0 .14213 0.5031 0 .05295 0 .8049
521
1
0 .38508 0.36699 0.2479 0.21260 0.4204
522
0.1
0 .40787 0.48859 0.1035 0.36252 0.1489
523
1 0 - '
0.41951 0.53667 0 .0438 0 .43966 0.0237
524
1 0 "
0.42131 0 54404 0.0347 0 .45048 0.0055
C a s e No.
B« = 10-* B J ^
F u F u F . j
fl*
525
10
0.08592 0 ,14327 0 .7708 0.28252 0.5742
526
1
0.26192 0.36144 0.3766 0.35229 0.2863
527
1 0 - '
0.371 17 0 .48488 0.1440 0.41697 0 0982
528
1 0 - '
0 .41371 0 .53603 0 0503 0.44714 0 .0168
524
10-*
0.42131 0.54404 0.0347 0 .45048 0.0055
Case No.
B, = 84 =
F„ F u F . j fu fzA
5 80
10
0 .08785 0 .09246 0 ,8197
0 .05369 0 .8539
581
1
0 .26150 0 .30574 0 ,4328 0 20501 0 .4893
582
1 0 - '
0 .36887 0 46292 0 .1682 0 .35512 0 .1820
583
1 0 - '
0 .41355 0 .53402 0 .0524 0 .43907 0 .0269
524
10 - *
0 42131 0 .54404 0 .0347 0 .45048 0 .0055
14
RESULTS F O R STANDARD 11 AND E R = 2 ,48 ( E M = 4.96)
C a s e No.
B3 = 10"* B4 =
F u F u
F24
530
0.34104 0.08548 0,5735 0.02886 0.8B57
531
0.36493 0.28650 0.3486 0.14926 0.5642
532
0.39316 0.44167 0.1652 0.31906 0.2393
533
1 0 - '
0 .41169 0.51852 0,0698 0 ,43293 0 ,0486
534
10-*
0 .41284
0 .53033 0 ,0568 0 ,44942 0 .0203
Case No.
B4 = 10"* B3 =
F u F12 Fi3 F22 F24
535
10
0,04871 0,08620 0.8651 0.27090 0.6429
536
1
0,19602 0,28473 0.5193 0,33118 0.3841
537
0,1
0.32785 0.43688 0.2353 0,40221 0,1609
538
1 0 " '
0 ,39924 0,51771 0,0831 0.44568 0.0366
534
10"*
0 .41284 0 ,53033 0 ,0558 0 ,44942 0 ,0203
C a s e No.
B3 = B , =
F n F,2 Fl3 F22 F24
505
10
0.04862 0.05036 0.9010 0,02874 0.9209
506
1
0.19410 0.22234 0.5835 0.14500 0.6327
507
0.1
0,32542 0.40597 0.2686 0.31140 0.2826
508
1 0 - '
0 .39883 0.51379 0,0874 0,43202 0,0542
534
10-*
0 .41284 0 .53033 0 .0568 0 ,44942 0 .0203
509
1 0 " '
0 .41849 0 .53559 0 .0458 0 ,45400 0 .0103
RESULTS FOR STANDARD II AND E R = 1.29 ( E M = 2.58
Case No.
B3 = 10-* B4 =
F u F u Fi3 F22 F24
540
10
0.32624 0.05056 0.6232 0.01601 0.9334
541
1
0.34420 0,21258 0.4432 0.10080 0.5866
542
0,1
0,37382 0.38402 0.2422 0,26972 0.3463
543
1 0 " '
0 ,39781 0 ,49168 0,1105 0 ,42124 0,0871
544
10"*
0 .39950 0 .51141
0,0891 0 .44738 0.0412
539
1 0 " '
0 .39971 0 .51345 0 .0858 0 .45388 0 ,0327
539A
10" ' °
0-39971 0 .51347 0 .0868 0 .4555 0 0310
15
MM IX
•tSUti 'W SIANOMO a AND (R • at» XM • \»
Cmik
1 , . lO-t ( , .
'II >a 'IJ 'a 1*
MO
M
a)otit aoati otta aooM a«tn
Ml
1
aBin a i e i ) ai)4o aotBi a m i
Ml
ai
0)410) a ) i s i a)4(B ajiao a4n)
») 10-'
OUBI 04ym a i m a)n4} a 14*
M4
10'
asm a4ii« ai4a a4«ii2 aon<
wt 10'
>u M-IO
a ooo d
C > H N L
h • w ^ l l •
'II 'u riJ ra ff
w 10
a o i n a a s i OfW a^i atu)
Ht
1
aOM4 ai44t2 anti a«jM aMt
MI
ai
antti ami ) 04IK OltlH 0.119
Mi
10'
anti) a44«it 02111 04)1)2 a i m
»4
10»
a m i a4im 01422 044012
aom
m
10'
ill
ujio Illll ddddd
C«>ll>
• ) - « 4 -
'11 »u »u f a
se
u aoi» aois42 a«« aooM atm
w 1
aoioi a o v ) a i m aotie a «
w
10-1
021)40 aiMU auD 0201)0 a}2H
w I0>
a))it) 04401) aau OWt) Oltlt
M4
io-»
amn a4>m 01422 044012 o o n t
M4
10-'
m 10-10
ddddd
SUA
lO-lJ
atiMs OSItJt ootu OttSB aoiu
lakX
CntNt
1 , . iiri t,.
l l fu f u fa
)tO
10
a2Ma aouM attn 000414 o t n i
KSUIIS 'OR SIWOADO n MO la • 0 B B IM • 0 70I>
Ml
1
a2«at OOtlS) OtlM OOVit 01112
HI
01
0)11)0 a2im 045a oiun OtII)
W)
U-)
a)B40
02112 0)t l t> 02410
M4
10-5
ddddd
i04
10-'
Ml
»•'» anm a4(Bi 021O 044tB alios
CMNt
u-tr* •!• '11 fu fu fB It
Ml
U
OOBUO
tinsn tffH LMUt o iv t
Mt
1
aeiu ootut OMI a a i e OtM)
M)
01
auiv 022tll ot in OtBtt 04)41
Ml
lo-J
0200 o)tin 0BI4 04104) Oltl»
M4
io-»
Hnm ami ona 042in OINI
Mt
ur'
Ht
10-10
a.wM OICM o i m 0420 aiMi
C M ML
U-ti-
fll 'u f l l fB fM
MO
U
OlODIIO OOOBI
t m OLtOM O.WI
Ml
1
ddSSd
M2
01
OlUlt OltOt) ai>i4 OMtlt OMtl
)01
10-'
021701 anm OUR OBtIt amo
M4
10-5
o a ddd
>IJ
lo-i
M4
10-10
ddddd
Sit
io-» iiiii ddddd
16
RESULTS FOR STANDARD II AND E R = 0,1875 ( E M = 0.375)
Case No.
B3 = 10"* B4 =
F u F,2 Fl3 F22 F24
5 7 0
10
0.25275 0.00853 0.7387 0.00249 0.9890
571
1
0.25595 0.05458 0.6895 0.02118 0.9242
5 7 2
0 ,1
0,26637 0.16093 0.5727 0.10130 0.7378
5 7 3
10" '
0.28365 0.32126 0.3951 0.30567 0.3731
5 7 4
10-*
0.28590 0.37912 0.3350 0.38629 0.2346
5 9 5
1 0 - '
0.28597 0.38558 0.3285 0.4156 0.1988
5 9 6
10"'°
5 9 7
10"'*
0,28605 0.38579 0.3282 0.43026 0.1840
Case No.
B4 = 10"* B3 =
F u F , 2
F l 3
F22 F24
5 7 5
10
0.00436 0.00854 0.9871 0.23493 0,7565
5 7 6
1
0.03055 0.05447 0,9150 0.25395 0.5916
5 7 7
0 . 1
0.10035 0,15836 0,7413 0.30141 0.5402
5 7 8
10" '
0.22601 0.31895 0,4550 0,37313 0,3079
5 7 4
10"*
0,28590 0,37912 0.3350 0.38629 0,2346
5 9 4
1 0 " '
5 9 8
10" '°
5 9 9
10"'*
0,36987 0.38638 0.2438 0.38672 0.2269
Case No,
B3 = B4 =
F u F12 Fl3 F22 F24
590
10
0.00436 0,00444 0.9912 0.00249 0.9931
591
1
0.03053 0,03377 0.9357 0.02110 0.9451
5 9 2
0 . 1
0.10010 0,12582 0.7741 0,09993 0.7743
5 9 3
10" '
0,22572 0.30462 0.4697 0.30421 0.3912
5 74
10"*
0.28590 0.37912 0.3350 0,38629 0,2346
5 7 9
10" '°
0.3497 0.4432 0,2071 0,4437 0,1131
5 1 2
10"'*
0,3779 0,4655 0.1566 0.4574 0,0771
17
ar
at
as
' • I
0 4
O j
O t
O l
0
/ /
/
CVUNOfBCAL KoyfTiiT«>i CAtit uo.eii.sti.ftss.»i«
fTAHOANO D Cu* 9 90«
0|/0t-S9 0|/0i* •5*D4'0i
i
—.-—.'»
o'" r,'ht
;•*. F i g s . 1-4. Dependence of N o r m a l i z e d R e c o m b i n a t i o n
Fract ions on Bi (E = 5.305)
CYLINDRir-Al. SEOMETRVfl'l
E R = 2 4 6
O-Fz2(e)/F22l0) B3rlO-=.B4VARIES X-F||(e)/Fn(0> 84=10-5,33 VARIES a-Ft|(B)/F|| (0) 85 = 84-VARY a-Fjj[Q)/F22(0) 83 = 84 -VARY V-F(B)/F(0) aVIDAC 6 VARIES
CYLIfjPRICflL GEOMETRY fl=l CASES 505,506,507.508,
534, 509 STANDARD H ER = 2 46 Kl2 = 2 = K,i K22 = 55 K24 = 2 D2/0|= 25 D3/D| = I5 = D4/D(
F i g s . 5- 8. Dependence of N o r m a l i z e d R e c o m b i n a t i o n F r a c t i o n s on B^ (E = 2.48)
19
(•••n O-^HlOI/f^IO). tt'iO-*. ^ « M ( t
Fig». 9-10. Dependence of Normalized Recombination Fractions on Bi (E = 1.^9)
20
TYi INDRICAL GEOMETRY fl = l
ER = 6 7 5 . 0 -Fz2 lB ) /F22 (0 ) ;B j= ro -5B4 VARIES K-F| |{B)/FntOI.B4:IO"^B3 VARIES a-F||(B)/F| |(0);B3 = S4-VARY i-F2g(B)/F22lO);B3 = B^-vaRY
I
\ \
~ \ \
r1 -
-1 / i / i 1
CYLINDRICAL GEOMETRY fl=l
CASES 550, 551,552,553, 554,569,516
STANDARD K ER = . 6 7 5
D2/D,=.2S 03/D|= I5 = D4/D|
^^^' / ^
J/^ . r V
V,. 1 1
O F
! 10 1 0 1
Figs. 11-14. Dependence of Normalized Recombination Fractions on Bi (E = 0.675)
21
O l Oil • ? * tO^ io^* KT*
CTUWOWICAL ftgOMgTWT * • !
CUKS MO.MI.M2.90S.9M.
STMO&MD 0 Ca* US9
0t/t>i*2» D)/Di>t9>D«/0i
F i g s . 15-18. Dependence of Normal i zed Recombination Fract ion* on Bj (E = 0.3S35)
22
rvl lNnRICAL GEOMETRY ff=l
E R = . I 8 7 5
O-F22(B)/F22(0);S3= 10-5,64 VARIES X-Fii ( B ) / F | | ( 0 ) , 64= 10-5, B J VARIES D-F | | (B) /F | | tO) ,B3-B4-VARY i -F22(B) /F2£(0) ;93 = B4-VARr
CYLINDRICAL GEOMETRY 3=1
CASES 570.571, 572, 573,
574, 595 ,596 ,597
STANDARD H E R = . I B 7 5
K-|2= K\i = 2 = 'C24 ^22 =.55
D2/0|=.25 D3/D|=.I5= D4/D(
CYLINDRICAL GEOMETRY fl=l
CASES 575. 576, 577, 578,
574, 594, 598, 599
STANDARD H Efi=.l875
l<-|2 = K|3 = 2 = K24 K-22=,55
Da/Di =.25 03/01=15 = 04/01
CYLmORICAL GEOMETRY g= I
CASES 590, 591, 592, 593,
574,579,512
STANDARD I ER=1875
K\z'Kis= 2 = /C24K-22=.55
D2/D|=,25 D3/D1 =.I5 = D4/D|
F i g s . 19-22. Dependence of N o r m a l i z e d Recombina t i on F r a c t i o n s on Bj (E = 0.1875)
23
SECTION TWO: FINAL RESULTS FOR 7-RAY CASES (SPHERICAL GEOMETRY)
Table XII g ive s the standard values for five s e t s of parameter combinat ions which were invest igated most thoroughly.
Table XIII g ive s va lues of p a r a m e t e r s used for some pre l iminary spher ica l c a s e s l imi ted to one radical and one solute (Code B). Compar i sons w e r e made with previous runs by F landers (Ref. 2) and Phi l l ips (Ref. 3) on the Avidac and George , with codes that could accommodate only two reac t ions .
Table XIV g i v e s p a r a m e t e r s and resu l t s for c a s e s with two radi ca l s only (Code A).
Table XV g ive s re su l t s for the spher ica l c a s e s run with four c o m ponents (Code B), m o s t of which are based on the five standard se t s of va lues
F i g u r e s 23 -35 are graphs of final F n vs B for the c a s e s in Table XV.
Table XII
VALUES OF PARAMETERS FOR STANDARD ->-RAY CASES
Standard
I II
III IV V
•-
2 2 2 2 2
P E
0.87 0.87 0.91 0.87 0.87
D j / D ,
0.4545 0.25 0.25 0.25 0.125
D ^ D ,
0.15 0,15 0.15 0.15 0 15
k ,z /kn
2 2 4 1 2
^i*,' ^11 l^n/kii
2 2 2 2 2
kia/kii
1 0.55 0.60 0.55 0.30
Table XIII
PARAMETERS AND RESULTS FOR CASES WITH ONLY ONE RADICAL (WITH AND WITHOUT A SOLUTE)
Case
D P DAF
t
2 2
/5
I I
E
2.5 2.5
D«/D,
0 0
D , / D ,
1 0
^i*/^i\ ^u/^ii kij /k i i
0 0
'^u/'^ii
1 u
Tf
U.17 0.05
f l l
0.1505 0.430
fIJ or f | 0
0.8495 0.570
24
^ Case No.
1 2-1
1 1-1
1 2-3
1 2-4
1 2-5
1 Z-6
1 2-7 1 2-8
1 2-9
1 2-10
1 2 -U
1 2-12
1
} . 1 4 1 5
1 6 1 7
1 8 1 10
1 11 1 251 2 252
1 253 2 254
1 255
1 256 1 257
1 258 1 259
1 260
1 261 1 262 1 263 1 264
1 265
1 266 1 267
1 268 1 269
1 270 1 271 1 272
1 273 1 274
1 275 1 276
1 279 1 280
0.5 281
2 282 1 283
0.5 284 2 285
E
1,8 1.8
1.8
1.8
2.5
2,5
2,5
2.5
3,2 3,2 3,2 3.2 1,25 1,25
1,25 1.25
0.6 0.6 0,6 0.6 0.8 0,8 0.87 0,87
0.87 0,87
0.87
0.87 0,87
1,10 0,75
0,8 0,8 0,8 0,89 0,87
0,87
0-435 0,87
0,87
0.87 1,74 0,91
0.87 0,435
0,87 0,87
1.74
0.75 0.87
0.87 0.87
0.87
0.87 0,87
D2/D1
0,3333
0,5 0,6667
0,3333
0.5
0.6667
0,3333
0,5 0,6667
0.3333
0,5 0,6667 1 0,333
0,5 0,6667 1 0,45
0,5 0,4545 0,4545
0,4545
0,25
0.4545
0,25 0,25
0,25 0,25
0,25 0,25 0-25
0,25 0,125
0,25
0,25 0,25
0,25
0,5 0-25
0.25 0,125 0,25
0,25
0,25 0.25
0.125 0.125
0,25 0,25
0,25
0,25 0,25
SPHERICAL CASES WITH TWO COMPONENT
ki2 /k l l
2
J
2 2
2 2 2
2 2 2 2 2 2 2 2 2 2 1,5 2 2 1,3 2 1,75
2 1,25 3,125
1 2 4 2 2 1 2 2 4 2 2 4 1 1 0,5 1 1 2 2 2 2 2 2 2
k22/kll
0,5
0,3 1 0,75 0.55
0.25 1 0,5 0,5 0,5 0,53 0,55
0,55
0,55
0-275 0.55
0-55
0-55
0.6 0-55 0-55
0-55
0-275 0-55
0-275
0-30 0-275 0-275 1-10
0-55
0-55
Fu
0-2477
0-2503 0-2539
0-2615
0-2810
0-2846
0-2886
0-2972
0-3053 0-3089
0-3131
0-3224
0-2094
0-2114
0-2144
0-2201
0-1365
0-1381
0-1398 0-1423
0-1651
0-1658
0-1812 0-1659
0-1737 0.1746
0.1708
0.1776
0.1670
0.2102
0-1388
0-1814 0-1589
0-1282
0-1690 0-1651
0-1918
0-1073 0-1649
0-1337
0-1711
0-2340
0-1370 0-1879
0-1175 0-2075
0-1902 0-2827
0-1496 0-1629
0-1531 0-1712
0-1586
F12
0-2999 0-2939
0-2858
0-2615
0-3317
0-3279
0-3208
0-2972 0-3543
0-3513
0-3448 0-3224
0-2604
0-2535
0-2441
0-2201
0-1803
0-1714
0-1623
0-1423
0-2053
0-2026 0-1854
0-2060 0,2145
0-1860
0-2236
0-2007
0-2412
0-2081 0-2871
0-1344
0-2331
0-3697
0-2450 0-2434
0-1397
0-1602
0-2558
0-3802
0-2265
0-3264 0-3844
0-1416
0-0883 0-0759
0-1499
0-1990 0-2414
0,2592
0,2622 0,2199
0,2335
i ONLY
F22
0,4238 0,3697
0,3302
0,2615
0,4485
0,4016 0,3649
0,2972 0,4641
0,4221 0,3879
0-3224
0-3861
0-3285
0-2875
0-2201
0-2948
0-2334
0-1948
0-1423
0-2857
0-2701
0-1943
0-4384
0-S51
0-2280
0-3904
0-2511
0-2735 0-2004
0-3289
0-2874
0-2476
0-1938
0-2265
0-3264 0-3844
0-1416
0-0883
0-0759
0-1843
0-3426 0-2293
0-4381 0-2154
0-3473
0-1968 0-4192 0-2341
0-2633
0-2808 0-4096
0-4157
FlO
0-4525 0-4558
0-4603
0-4771
0-3873
0-3876 0-3907
0-4056 0-3404
0-3398
0-3420
0-3552
0-5302
0-5352
0-5410
0-5599
0-6832
0-6905
0-6979
0-7154
0-6296
0-6317
0-6335
0-6281
0-6118
0-6394
0-6056
0-6217
0-5918
0-5817
0-5741
0-6842
0-6080
0-5021
0-5860 0-5915
0-6685
0-7325
0-5793
0-4861
0-6024
0-4397
0-4787
0-6706
0-7942
0-7166
0-6599 0,5183
0.6090
0.5780
0,5847
0,6089
0.6079
F20
0.2764
0,3365 0,3840
0,4771
0,2198
0,2705
0.3143
0,4056 0.1816
0.2267
0.2672
0.3552
0.3535
0,4180
0,4678
0.5599
0,5249
0.5952
0.6429
0-7154
0-5090
0-5273
0-6204
0-3556
0-4904
0-5859
0-3860
0.5482
0.4853
0.5915
0.3840
0.5782
0.5193
0.4365
0.4856
0.3810
0-5417
0-6456 0-5784
0-4107
0-5893
0-3310
0-3864
0-4203
0-6963
0-5768
0-6533
0-3818
0.5245
0.4775
0-4571
0-3705
0-3508
Rl
1-711
1-477
1-300
1-000
1.596
1.411
1.265
1-000 1-520
1-366
1-239
1-000
1.844
1.554
1.341
1.000
2.160
1.690
1-393
1-000
1,731
1-629
1-072
2-642
1-699
1-306
2-286
1-413
1-637
0-953
2.370
1.584
1-558
1-512
1-594
2-274
1-661
1-SlO
1-006
1-564
1.077
1.465 1.674
2-332
1-834 1-674
1-035
1-483
1-565
1-616
1-834
2-392
2-621
R2
1-086
1-278
1-445
1-825
0-832
0-959
1-079
1-365 0.678
0.775
0.869
1.102
1.484
1-766
2-010
2-544
2-801
3-461 4-007
5-027 2-526
2.659
3.340
1.628
2.351
3.043
1.767
2.729
2.445
2.857
2.048
2.693
2.773
2.915
2.444
1.798
2.371
4.902
3.501
2.615
3.353
1.337
2.362
1.743
4.477
2.331
3.394
1.282
2.954
2.476
2.401
1-686
1.669
25
C M Ik
M m m m Ml
I d 19 m 110
IM I I I m m uo
w l a I S IM I B
M Ml UJ U l
i n 19 I B IM i »
li li
a i n i l U 7U 215
n t
O l
R I
TO 2M
O M M o M B SunaN I
1
«.(
a i
0
D| iO|-ai
Df B| • 1.0
CawUnIi
l< • 10 »
i 4 t 10-'
• , i i i r >
• j u r '
S lananlC lu
1 • •J2
Dfia
K
" U - W
«„-ir'
«„-ir»
K ) i - a B
x , , ! « a - i « j , - 2
c
» j -1» ->
0
OyDl -aiM}^
IVO) • o e e -
oyo, -o
04ID, • o n
V H M M I
l< • 1 0 ' la> 10-1 1
10
0 ) t i r » t 10 ' t i f f ' e 1 CIO
£ l f f ' l l f f l t 1 t i o
t | I Iff-' t i i r ' t i f f i t 1 CIO
I 4 - 1 0 - * I f f ' Iff" 1
10
I ) • I ff ' Var ia i t l l
I f f ' I f f ' I f f l 1
10
I f f ' Iff" 10
I f f ' Iff" 10
I f f ' I f f ' Iff" 1
10
10
I f f ' I f f ' Iff" 1
10
I f f ' I f f ' Iff l 1
10
U c u U M ConunlrUWit t Carncut
'11
0I62« ami a.tfm o o n s
0.0062
0.1120 01612 01032 oom OOOM
o i m 0 1 m 017» OI62i
01720 0.I7II o i«n 0.1610 01602
'12
02062 o w ; 01)12 00126 00120
021)4 0.20H o i4 i ; 0.06)4 0.0121
ana 02061 01421 ootn 00121
h2
iiii
i
02»)0 02«6 02106 0.2S67 02414
017« 01701 0 1056 oont 00064
02»» 027U 01)07 O0470 00066
'IS
0-6404 0 6447 0 762?
o a m o w n
0 6146 062W 0 755)
o n 7 2 o m 5
06412 0-6117 0 7011 07745
eeo
op
illll
' a
04KB OW77 0 5711 06750 0 7421
0 « I 6 04M5 0 5771 0 67O 07464
0 6412 06557 07711 01025 o w n
0 « I 6 0 5171 07065 0-mi 0.«12
OI220 o i r z o o i m 01612 0160)
021)5 O2065 01452 0.0661 O012V
02«51 02776 0 1547 oora 0 0070
06145 0-6216 0615) 07MO 01262
04115 05160 O7001 OB34 OMO
OI I I I 01720 O I W O.IW 0.21»
01720 oirai 0.166)
01720 o i t n OIKD
01220 01222 o i m o m ) o.a«2
01220 ai22r ffinn 01612 OltOI
0.1220 01220 OIIM 01641 01601
0 21)2 O206) OI470 00611 001)0
0 21)5 01454 00 I2«
021)5 01441 00121
02 in 0!«6 01471 O0(B2 001))
001)2
021)5 0 206)-0 145) 00661 0 0121
0 2 in 0 206) 0 14)1 0 0654 0 0121
0214; 02272 0 1547 0 0M7 0 0070
02»51 OIS42 O0070
02»il 0IS4I oooro
02151 0 2776 01541 oom 0 0022
0 0022
0 2»)l 0 276(r 0I5M ooaw OOD'O
02151 027ra 01507 0 0415
oooro
06150 06217 062)1 073)1 0 7685
0 6145 Otl)7 01207
06145 06tS6 0 0 7
06145 06211 0623) 02)25 atm
0 6145 0620-0 6(51 0 76W 01267
0 6145 06217 0 6175 0 7704 01261
04421 05164 0««4 0022 0«00
04114 OWW OWO
0 4114 O70II o w n
04114 0 5151 0 M » 0020 o i m
0« I5 0517^ oam o n ) ) 01W0
0MI5 0 5167 0 7061 0061 OWB
26
Case No.
f236 237
•^238 239
[240
246 247 248 249 250
SIIII
3025 303 304 305
Std- IE
312
315
Std- m
316 317 3175 318 319 320
Std-¥ 306 307 3075 308 309 310
Std- n-
301
400 401 402
403 404
405
409
271A
271C
•55 po nts inst
TableZ5Z(Cont'd-l
Deviation from Standard I
K
/<13 • 0-1
D2/D1
0-25
0-25
0-25
0-125
0-25
E
'<12 '<22
2 0-55
4 0-60
1 0-55
2 0-30
2 0-55
K
ki3/k,i.0,l
*13'*11 • »,5
D
Dj/Dj . 0-015
E
0-87
0-91
0-87
0-87
0-87
0
03/0, • 0-015
Deviation from Standard IE
adof ttie usual 3-
BJ • 10-5 Variable B4
10-5 10-3 10-1
10
10-5 10-3
10-1
10
B
B3- 10-5 8 4 . 1 0 - 5
10-3 10-2
0-1 1
10
B3 • 10-5 B4 • 10-5
10-3
10-2 0-1 1
10
B3 . 10-5
BJ • 10-10-3 10-2
0-1 1
10
B3 • 10-5 B, • 10-5
10-3 10-2
0-1 1
10
0 3 - 6 4 BJ . 10-5
10-3
10-2
0-1 1
10
8 3 - 10-5
Variable B4
10 1 0-1
10-3
10 1 0-1
10-3
1 10 0-1
Fll
0-1720 0-1720 0-1690
0-1633 0-1591
0-1720 0-1722
0-1789 0-1969 0-2149
Calculated Concentrations & Corrected
F12
0-2135 0-2064 0-1450 0-0667 0-0129
0-2135 0-2066
0-1470 0-0681
0-0130
F22
0-2951 0-2776 0-1547 0-0498 0-0070
0-2951 0-2776
0-1548 0-0498 0-0070
Fl3
0-6145 0-6216 0-6859 0-7699 0-8280
0-6145 0-6211
0-6740 0-7350
0-7720
Extrapolated Concentrations and By oiffere
0-1667
0-1666 0-1663 0-1650 0-1620 0-1600
0-1368 0-1374 0-1404
0-1500
0-1912 0-1899 0-1841
0-1711
0-1625 0-1625 0-1624
0-1616 0-1603 0-1597
0-1667 0-1610
0-1435 0-1032 0-0412 0-0068
0-2405
0-2356 0-2141 0-1617
0-0712 0-0132
0-3769
0-3480 0-2743 0-1324
0-1361 0-1225 0-0902 0-0377
0-2585 0-2539
0-2313 0-1740
0-0743 0-0133
0-2405 0-2313
0-2018 0-1361 0,0472 0,0070
0-2731 0-2575 0-2092
0-1196 0-0312 0-0040
0-2170 0-1799 0-1080 0-0317
0-2993
0-2406 0-1345
0-0331
0-2657 0-2470 0-1862
0-0902 0-0187 0-0022
0-2731 0-2571
0-2081 0-1180
0-0310 0-0040
0-5928 0-5978 0-6196
0-6733 0-7668 0-8268
0-4864 0-5146
0-5853 0-7176
0-6727 0-6876 0-7357 0-7912
0-5790 0-5836 0-6063
0-6645 0-7655
0-8270
0-5928 0-6074 0-6547
0-7607 0-9117
0-9863
Calculated Concentrations & Corrected
0-2160 0-1981
0-1800 0-1730
0-2010 0-1893 0-1777 0-1730
0-1798 0-2170
0-1474
0-0132 0-0683
0-1473 0-209
0-0132 0-0680 0-1469 0-2068
0-1351 0-0269
0-2778
0-0072 0-0500 0-1551
0-280
0-0072 0-0500 0-1550 0-2778
0-0317 O-0O44
0-1080
0-0771 0-734
0-673 0-618
0-786 0-743 0-675 0-620
0-685 0-756
0-575
F24
0-5160
0-7003 0-8835 0-9800
0-4914
0-5158
0-6981 0-8820 0.9799
nee
0-4864
0-5169
0-5767 0-7187
0-8976 0-9829
0-4061 0-4721 0-6177 0-8359
0-5646 0-6365 0-7753 0-9292
0-4759 0-4991
0-5825 0-7359
0-9070 0-9875
0-4864
0-5116 0-5901 0-7459
0-9218 0-9891
0-0980 0-882
0.698 0.511
0.980 0,882 0,698 0,515
0,833 0.969 0.614
27
O t
1
O l
• 1/
^ ^ * " l l
9T*NO*nO 1
< • z
*• • c • oar
D|/D, • 0 » 0 ^ 0 , • 0 15
.«> «,» ^
• 019 > 2
• 2
• 2 . I0'»
CAMS 211.212. 21) 2 i« . 2 i»
Figa 2 3 - 2 5 . Dependence of Final Recombination Fract ions on Bj.
28
CASES 236,237, 236, 239, 240
STANDARD I*(D3/D| = .OI5)
/ ^
/ / CASE 231, 232. 233, P/ 234,235
/ d STANDARD I (»C24=.02)
Fgg
"^1^12
1
10 I 10 '
Figs . 26-29. Dependence of F ina l R e c o m b i n a t i o n F r a c t i o n s on B; .
C*« Ji4.S<4,liS. 3i29.Si2.Sii
-h
0 1
o t
\
0 1
/
'J. .-^
1 1 . * • ! - , - »
CAM >»}.>4.S<6 Sin.Si7.54
9TMOAM0 n
1
-••
' t t
' i i
»«
-<— ^ n
CASC 30e.>0'.SO0.30« S09.Si0.2*0
STATiOM Z $TU«AI»0 S (•)-B4)
Figs. 30-33. Dependence of Final Reconnbination Fractions on B^.
NO
30
CASE 4 0 0 - 4 0 4
STANDARD l'lK,^= l . D s / O r 015)
Figs . 34-35. Dependence of F i n a l R e c o m b i n a t i o n F r a c t i o n s on B^.
31
SECTION THREE: INFLUENCE OF PARAMETERS ON RESULTS FOR a-RAY AND 7-RAY CASES
Tabic XVI g ive s resu l t s of cy l indrical c a s e s grouped to faci l i tate malting c o m p a r i s o n s for different va lues of E.
F i g u r e s 36 and 37 are plots of normal ized F j , v s . 8;+.^ for the different va lues of E in Table XVI.
Table XVU g ives r e s u l t s of the Standard Spherical c a s e s , grouped to faci l i tate making c o m p a r i s o n s for different va lues of K:\I and D^/Di.
F i g u r e s 38 and 39 are plots of normal ized Fjj v s . B4 for the different va lues of PC^ and D J / D J in Table XVII.
Table XVIII g ives re su l t s of spherical c a s e s grouped to show the ef fects of varying «:i3 and D3/D1.
F i g u r e s 4 0 - 4 3 are graphs of F^j v s . B4, <i j , or D J / D I for cer ta in va lues of B4 a s given in Table XVIII.
Table XIX g ives resu l t s of spherical c a s e s grouped to show the ef fects of varying icn and D4/D1.
F i g u r e s 44-45 are graphs of F^j vs . B^, 1C24, or D4/D, for cer ta in a lues of B4 a s given in Table XIX.
32
Case No.
520.,
530-,
540,.
550'•
560'•
570,
Case No,
525,,
535,-
545-,
555,,
565,,
575,.
Table B Z I
RESUITS OF CYLINDRICAL CASES FOR DIFFERENT VALUES
E
5,305
110,61)
2,48
I4,%l
1.29
12.581
0.675
11.35)
0.3535
10.707)
0.1875
10,3751
10,07071
E
5,305
2,48
1.29
0,675
0-3535
0-1875
63 • 10-5, 84 •
F22 F22IFIO) 1-Avidac
F22 f22;FI0l l-Avidac
F22 F22/FIO) 1-Avidac
F22
F22/FIO)
1-Avidac
F22
F22;F(0)
1-Avidac
F22
F22/FI0I
1-Avidac
1-Avidac
84-10-5 , B 3 .
F l l Fll/FIOI
Fu Fll/FlO)
F l l Fii/FlO)
F l l Fll/FIOI
f l l fll/FlO)
F l l Fll/FIOI
10
0-0530
0-115
0-190
0,0289
0,063
0,101
0,0160
0,035
0,0087
0,019
0-0046
0,010
0,020
0,0025
0,005
10
0,0859
0,202
0-0487
0-115
0-0151
0.036
0.0081
0.019
0.0044
0.010
1
0-2126
0-462
0-547
0-1493
0-325
0-411
0-1008
0-219
0-0632
0-137
0-0372
0-081
0-110
0-0212
0-046
0-012
1
0-2619
0-616
0-1960
0-462
0.0886
0.208
0.0532
0.116
0.0306
0.072
0-1
0-3625
0-788
0-824
0-3191
0-694
0-715
0.2697
0.586
0-2122
0-462
0-1528
0-332
0-320
0-1013
0-220
0-1
0-3712
0-873
0-3279
0.772
0-2146
0-505
0-1528
0.360
0.1004
0.236
OF E TO PL
10-3
0-4397
0-956
0-957
0-4329
0-941
0-923
0-4212
0-916
0-3995
0-869
0-3616
0-787
0-643
0-3057
0-665
10-3
0-4137
0.973
0.3992
0.940
0-3391
0-798
0-2875
0-676
0-2260
0-532
OT NORMAL
10-5
0-4505
0-979
0-4494
0-977
0-4474
0-973
0-4401
0-957
0-4219
0-918
0-760
0-3863
0-840
10-5
0.4213
0.991
0.4128
0.972
0.3758
0.884
0.3379
0.795
0,2859
0,673
ZEO F'S V
10- '
0,4539
0.987
0,833
10-10
0,4156
0,903
0,337
10-7
S. B
10-10
0,4555
0,990
0,4541
0,988
0,4463
0,971
10-15
0.4303
0.935
10-10
0.4049
0.953
0.3825
0.900
10-15
0.3699
0.870
0-
0,46
1
1
0.46
1
1
0.46
1
1
0.46
1
1
0,46
1
1
0,46 1
1
0-
0,425
1
0,425 1
0.425 1
0.425 1
0,425 1
° Rough exlrapolated values.
33
Fig . 36. Influence of Ini t ia l F r e e Rad ica l Concentrat ions on F I I ( B ) / F I I ( 0 )
CTLiNoaiCAi. oeoueTn CASES STANDARD 0 WITH Bs'iO'^
O t • 5 505
X E • 2 AO
• E- 1 W
o E -o»n
L l- 0SM9
V E • aiare
10 •>
Fig. 37 Influence of Initial Frrc K.ulu al Concentrations on Fji(U), Fji(il)
34
SId, I
B3 • 10-5
Sid, n
BJ • 10-5
SId. m
B3 • 10-5
Std, n
BJ • 10-5
Sld,Y
BJ • 10-5
Std, I *
83-64
64
Case No,
F22(B)
F I6 | ;F I0)
Case NO-
F22IB)
FIBI/FIO)
Case No,
F22IB)
FIBI/FlO)
Case No,
F22I6I
FI6)/FI0)
Case No,
F22IB)
flBUFlO)
Case No,
Fi i lBI
f l l lBl/FIOI
F22IBI
F22(6)/F(0)
RESULTS OF STANDARD CASES WITH SPHERICAL SYMMETRY
10
145
0,0070
0,024
305
0,0039
0-014
315
0-0044
0-020
320
0-0040
0-029
310
0-0022
0-008
325
0-0068
0-041
0-0040
0-014
1
144
0-0498
0-169
304
0-0312
0-114
314
0-0317
0-138
319
0-0331
0-104
309
0,0187
0,071
324
0,0412
0,247
0,0310
0,113
0,1
143
0,1547
0,524
303
0,1195
0,437
313
0-1080
0-471
318
0-1345
0-422
308
0-CB02
0-343
323
0-1032
0-618
0-1180
0-431
10-2
3025
0-2092
0-765
3125
0-1799
0-785
3175
0-2409
0-756
3075
0-1862
0-707
322
0-1435
0-859
0-2081
0-761
10-3
142
0-2776
0-941
302
0-2575
0-941
312
0-2170
0-946
317
0-2993
0-939
307
0-2470
0-938
321
0-1610
0-964
0-2571
0-940
10-5
141
0-2951
1-000
301
0-2713
0-992
311
316
306
0-2657
1
301
0-1667
0-998
0-2731
0-999
0
253
0-2951
1
257
0-2735
1
271
0-2293
1
265
0-3186
1
280
0-2633
1
257
0-1670
1
0-2735
1
SPHERICAL CASES WITH B ,
O STANDARD I (Da /D , = 0.4545)
X STANDARD n ( D 2 / D | : 0 , 2 5 )
i STANDARD IT ( D j / D , ^ 0.125)
F i g . 38
Influence of Diffusion Coef f i c i en t s on
F 2 2 ( B ) / F 2 2 { 0 )
Fig . 39
Influence of React ion Rates on
F 2 2 ( B ) / F , 2 ( 0 ) SPHERICAL GEOMETRY CASES WITH B3
0 STANDARDU C ' l j / l ' i i = 4)
X STANDARD n ( k | £ / l i | | = 2 )
A STANDARD W ( k i g / k , , ' 1)
10-5
35
Table XVIII
RESULTS WHEN k i i A i i AND D J / D , VARY FOR 7-RAY CASES
Case No.
141-145 405-409 246-250 151-155 211-215
203-201 141-145 2J6-240
Deviat ion f r o m Std. I
< „ Dv^D,
2.0
0.5 0 .1
10- ' lo - "
0.4545 0.15 0.015
Std. I
Std. I
10 - '
0.1720
0.1720 0 1718 0.1720
0.1720 0.1720 0.1720
Fina
1 0 - '
0.1720 0.1730 0.1722 0.1720 0.1720
. 0.1720 0 1720
Values Q i& B4
10- '
0.1695 0.1777 0 1789 0.1792 0.1795
0.1709 0.1695 0.1690
f F „
1
0 1642 0 1893 0 1969 0.1987 0.1993
_ 0.1642 0 1633
10
0.1603 0 2010 0.2149 0.2185 0.2197
0.1663 0.1603 0.1591
Case No.
145 405 250 155 215
201 145 240
Deviat ion f r o m Std. I
< „ D J / D ,
2.0 0.5 0.1 10 - ' 10-*
0.4545 0.15 0.015
Std. 1
Std. I
Fina l Values for B , = 10
F , ,
0.1603 0 2010 0.2149 0.2185 0.2197
0.1663 0.1603 0.1591
F i :
0.0129 0.0132 0.0130 0.0130 0.0133
0.0129 0.0129 0.0129
fu
0.0070 0.0072 0.0070 0.0070 0.0072
0.0070 0.0070 0 0070
F l ,
0.8267 0.786 0.7720 0.7685 0.7670
0.8207 0.8267 0.8280
F «
0.9800 0.980 0.9799 0.9800 0.9796
0.9800 0.9800 0.9800
36
SPHERICAL GEOMETRY
X
o £i
V
STANDARD
STANDARD
STANDARD
STANDARD
STANDARD
I
I *
I "
I *
I "
WITH B J ^10-5
'f|3
Na "^13
l K | j
' 13
2,0)
0.5)
0,1)
10-2)
lO-"*)
SPHERICAL GEOMETRV WITH I
- STANDARD I " { D S / D , = 0,4545)
X STANDARD I ( D J / D , ^0.15)
O STANDARD I * ( D j / D , = 0,015)
Fig . 40. Influence of Reac t ion Ra tes on F l l
F ig . 4 1 . Influence of Diffusion R a t e s on F n
SPHERICAL GEOMETRY WITH B3 ^ 10"^
STANDARD I AND I * CASES (VARY/Cij)
I I I I I 2.0 1.0 0.5 10"' 10-2
dl
Fig . 42. Influence of Reac t ion Rate on Fj j for B4 = 10
^
-
-
SPHERICAL GEOMETRY WITH B j = 10"^
STANDARD I AND I * CASES (VARY Da/D, )
i
/ / -F22
i 1 1 ^ ' 1 0-2 0-3
D 5 / D ,
Fig . 43 . Inf luence of Diffusion Ra te on F j j for B4 = 10
37
i«h zn
Msuin wKiikitni, AND 04101 v«av ton fHiti CASES
CaiiMi.
M-M
M-M
m-m
•nMlanlnaSH.
"M
U OK
DKIOl
OLB
aa
SM.
SM
nMlVMuHe(F]|«l«
10^
ami ami
ur>
amo arro
a2n6 amo
nr'
auo au«
au4i auo aun
1
aom aoos
11
10
aOOTD a0072
00070 aoo70 aoo70
Ornm.
M
a
m M IB
OHWHB I W SM.
m u O K
BtttI
a«e-»i3 aa
SM
SM
Find Uliai lor 14 • 10
fu
tlMS
aian auoi auo) awa
fu
aoin aom
fa
aooTO aooro
00070 00070 00070
f|J
aaw aa2u
0»7 aaa O L I W
f24
a«oo o w e
OMO OWO
Fig . 44
Influence of Reaction Rate
on F i j
Fig. 45
Influence of Diffusion Rate
on F22
0>
OJ
01
^ 1 1
WWnCM.MOMtTKT »ITH Oi-IO-*
> ITWOUO 0 ITMOMD 0 ITW.OAMO
1
I* IO, /0, 1 IO, /0,
1" "t/o,
0 « M « 0151 0091
1
38
SECTION FOUR: FIGURES TRACING THE REACTIONS OVER TIME AND SPACE FOR 7-RAY CASES
As indicated previously, . . . . . f o r m e d time T goes from 1 to 0, while real time goes from 0 to ~; - ^ transformed spac^ R goes ^ ^ ^ 1 to 0. so P goes from some integer at the origm down to 0. wh le the real space goes from 0 to 00. All but Figure 46 are based on actual cases described in Table XV.
Figure 46: General shapes of solutions.
Figures 47-57: Graphs of the amounts of Xi (i=l,2,3,4) at the origin (center of initial concentration of free radicals) as a function of the r ea l time plotted on a log scale. Figures 47-51 are for Standard I at different values of B4. whereas Figures 52-57 are all for B4 = 1 for different parameter combinations.
Figures 58-61: Similar plots for either X] and X3 or X^ and X4 for B4 = 10 only, with different cases grouped so each graph shows the influence of some parameter for cases similar to Standard I. The pa rameters being varied are D3/D1, /c 13, D / D ] , and ^34; in each case the concentration of Xi at the origin is plotted against the log of the rea l t ime.
Tables XX to XXIII: Values of Ui ( i = -j"!^) to be used in plotting
U- vs. T at certain values of R (or P) and to plot Uj vs. R at certain values of T for Standard II cases.
Figures 62-89: Plots based on the information in Tables XX-XXIII for the cases described in Table XV.
Figures 90 and 91 are similar plots for Standard V Cases.
Figures 92-103: Plots of F vs. log t for Standard I cases and others with selected values of B4.
i j
Fig. 46
General Shapes of Solutions Ui(R,T)
39
use •«& tT*ND*l*0 t I •,••01
Figs. 47-51 . Time Deprndrncr of Recombination Fract ions
40
CASE 225 STANDARD I (D,/D, = .05)
32,000
28.000
24,000
20,000
16.000
12,000
8,000
4,000
0 0
Figs. 5Z-57. Time Dependence of Recombination Fractions
41
•ONOMOI Oi'lO >,a,.io
onomi/o,.oow O M t IO|/D|.Oi«l _ * 101 ID|/e,.0«»4«)
u.ooo
KAOO
- | M , 0 0 0
- KVOOO
- 19.000 I
—{•1.000
•.000
4.000
H 0
I t ,000
OZ»{Gta/0, '00»l * t49 mt /D, -0 i9 l O H » ( 0 4 / 0 f 0 « M S
10
a M
0*0
070
~ o « o
k 5 a so
0 40
0 3 0
010
a i o
,S
.±2
MfeBOFO^
B,<io 9aa*io o z » (x,«*ao2t O 149 (x^> Z)
'k 1 1 1 J
F i g s . 58 -61 . Time Dependence of Recombination Fract ions
42
T a b l e XX
C O N C E N T R A T I O N S O F R E A C T A N T S A T VARIOUS T I M E S AND P O S I T I O N S IN S P A C E FOR CASE 257: S P H E R I C A L G E O M E T R Y ; S T A N D A R D II; B3 - B4 - 0
0.2727
0.2727
0.1009
0.1033
0.2611''
0.201r'
0.2822-2
0.9436-^
0.1233"'
0.5152
0.5152
0.1780
0.2472
0.4527-'
0.7421''
0.5012-2
0.7008-2
0.2328-'
0.7576
0.7576
0.2467
0.4083
0.6167-'
0.1575
0.G947-2
0.2258-'
0.3422-'
1.0
1.0
0.3084
0.5748
0.75G9-'
0.2614
0.8612-2
0.5067-'
0.4516-'
0.1460"'° 0.2865"°
P
17
43
Tabl* XXI
CONCENTRATIONS OF REACTANTS AT VARIOUS TIMES AND POSITIONS IN SPACE
FOR CASE JOS: SPHERICAL GEOMETRY; STANDARD II; B, = 10 ' ' , B, = 10
l.l 1.0
1.0
1.0
0 [l£_ 0 1.0
0.JJJ7 10 0.5IS2 1.0 1.0
0.2171
o.iooa 056M'-*' 09955
0.2591'""
0.2 750'•"
0 1772
0 1466'""
0.4435'""
09993
0.4«06'">'
0 7576
0.2443
0.26261""
3521.97 0.5920'"'
0.9965
23(53 0.6461'""
20495.45 0.3037
0.9795
0.9906
0 9 2 1 7 ' ' * ' 10 0 1 7 4 1 ' " ' " 1.0 0.2560'"'" 1.0
0.9905
10 1.0 1.0
0.4032 ' " "
30363.72
0.9696
11788.33
0.9812
1469.55 0,753»'""
0.3378'"'" 1.0002
1.0
5495.13
0.9945
P
M
44
T a b l e XXII
C O N C E N T R A T I O N S O F R E A C T A N T S A T VARIOUS T I M E S A N D P O S I T I O N S IN S P A C E
F O R CASE 3 0 3 : S P H E R I C A L G E O M E T R Y ; S T A N D A R D II; B , = 10 , B4 "•
0.1009
0.95 85<-"
0.2609'-"
0.1514'-"
0.2808'"2'
0.2240'""
0.6029'"" I0.9995
0 1.0
0.4517'-"
0.5685'""
0.4963'"2>
0.1793'"2>
0.1132'"'' 1.0181
0.6139'""
0.1234
1.0 1.0
0.3082
0.7512'""
0.2104
0.8283'"2>
0.1676'""
1873.71
0.8127
3171.24
0.6830
2598.76
0.7402
0.9998
P
17
45
T«blr XX I I l
CONCENTRATIONS OF REACTANTS AT VARIOUS TIMES AND POSITIONS IN SPACE FOR CASE tOl -iPHKRICAL GEOMETRY; STANDARD II. 8 , 8 , = lO"'
10
10
02727 1.0 0.5152 1.0 0.272)
0.1009
0.1033
0.0261 0.9902
0.0201
02822'
0.9000'""
0.1157'" '" 0.999J
1.0 0.5152
0.9992 0.I7I0
1.0008
0.0453
1.0098
0.9(85 0.5012'""
0.7008'""
O.I8»2'"'" 0.9999
0.757(
1.0122 0.2467
0.9(78
0.9840 0.1575
0 9953 0.6946'"
1.0047 0.2258'""
1.0
0.9191
1.0656
0.1I73'""I 10 01576'"'" 1.0 0 .74 (3 ' " ' " 0 9999
1.0
1.0551
0.9448
0 . (611'"
0.5067'""
1.1749
0.8251
I.24I3
0.75(6
0.2738""'" 0,9999 0.3614'""" 1.0
0,2307''" 0,9999
P
17
0.9
0.8
0.7
0.6
1 -
5 0.5
0.4
0,3
0.2
O.l
--
CASE
U, vs
257
R y / j - \ 0
^ ' T= 0.75
T - 0 . 5 0
1
0,9
0,8
0.7
0 6
p £ 0 , 5
3
0,4
0.3
0,2
0.1
^
---—
CASE 257
R
—\—— T
>^= I .O
^ ^ J--01'b
. y ^ T ^ 0 . 5 i D ,
- " - " ' ' ' ' ^ T = 0,25
0.9
0.8
0 .7
0 .6
p i O . 5 3
0.4
0.3
0,2
0 1
--
—
CASE 257
U, vs T
^ - • - ° ° ° ^ ^ ^
R - 3 3 - 1 , 0 /
/ /
/Ay ^^^^^^^^^^^-" '^
- \
0.9
0,8
0,7
0,6
£ 0 . 5 3
0.4
0 ,3
0.2
0, 1
-
_ -
-
--
CASE
U,vs
257
T
R
" ^ ^ ^ 1
33 33 'io/\
R - ^ " " 3 3
" " 3 3
R-.2.
Figs. 6Z-65. Dependence of Normalized Concentrations on Positions in Space and Time.
<»»(- c u t MO
Fies. 66-69. Dependence of Normalized Concentrations on Positions in Space and Time.
1,0
0.75
0,50
0,25
-
-
CaSE 305
U, vs R
^ _ 4 \ 1
T=I.O
T= 0,75
T = 0,50
i 1 1 1 5/33 10/33 15/33 20/33 25/33 30/33 33/33 = 1.0
1.0
0.75
0,50
0,25
-
-
CASE 305
U, vs R
1 1 1
^ T =
1 1
1,0
5/33 9/33
32,000
28,000
24.000
20.000
16,000
12,000
8,000
4,000
^C
--
-
CASE 305
Uj vs R
9/33 " " l
17/33 25/33
T = 0,75
T = 0.50
T:0,25
1 33/33=1,0
1.0
0,99
0.98
0,97
-
CASE 305
U, vs R
•—_..___^ T = 0 25
^*'*v,s,„^ ^ \ ^ T ' 0 50
^ \ ^ = 0 75
Figs. 70-73. Dependence of Normalized Concentrations on Positions in Space and Time.
0 «
OffO
p.oas K
oao
ora
aro
c
-c*u u« M
SOS
T
! V 0 29 " * » * — ^ 0 9 0
-—^^^^-^"'y^ '']/// ' 'v /
/ - , - , • •
1 0 T 3 l (
Figs. 74-77. Dependence of Normalized Concentrations on Positions in Space and Time
1 0
0.90
0,80
0.70
P 0 . 6 0
£ -0 .50
0,40
0.30
0.20
0.10
---
CASE
U, vs
303
R
1 1
^ T = I.O
_ — T=0,75
T= O hn
1 1
1,0
0 90
0.80
0.70
K 0.60
^ 0,50
0.40
0.30
0.20
O l O
-
-----
CASE 303
U; vs R
- - - - - ' 4 f ^
jf T= l .0
^ T- 0.75
^ _ _ „ . — • T=0.50
1 1
3000
2000
1000
-
-
CASE 303
U, vs R V T=0.50
y ^ , T=0,25
^ ^ y / ^ ^ 1^0.75
1 1
0,95
0.90
P 0,85
-^0,80
0,75
0.70
0 6 5
0,60
-— ~ --
CASE 303 ^ ^
U, vs R
1 1
^ S ^ ^ \ - — T^O.75
\ ^ ^ T = 0 2 5
^ T=0.50
1 1
Figs. 78-81. Dependence of Normalized Concentrations on Positions in Space and Time
l O
OffO
oao
on)
C oao 6 ^oao
0 4 0
0 9 0
oxo
O l O
-
--
_ -
C A U M i
0 *- \
OffO
aas
oao
- \
-
y on
-K 1
^ - " ^ y ^
X-H-.o
/ C*K »Oi
1 1 0.90 ors i<
Figs. 82-85. Dependence of Normalized Concentrations on Positions in Space and Time
1.0
0 ,90
0 . 8 0
0 .70
0 . 6 0
0 , 5 0
0 . 4 0
0 ,30
0 . 2 0
0 .10
^
--
CASE
U, vs.
301
R
9 / 3 3 1 1
17/33 2 5 / 3 3
T= 1,0,..
T : 0 . 7 5
T=0,50
I 3 3 / 3 3 = 1.0
1,0
0,90
0 60
0.70
0.60
0.50
0.40
0.30
0.20
0.10
-
----
CASE
U,vs
301
R
— J ' i
y ^ T=0,75
"-""'^ T = 0.50
' T = 0.25
1.30
Pl.25 (E
^ 1 . 2 0
1.15
I.IO
1.05
1.00
CASE 301
Uj vs R
1 9 / 3 3
1 17/33
1 2 5 / 3 3
T = 0 , 2 5
^ ^ T = 0 , 5 0
T = 0 . 7 5
1 3 3 / 3 3 = 1 . 0
0,95
K 0.90
0,85
0.80
-
"
-
CASE 301
U4 VE R
1 1
^S. ^ \ '""~"-----^ ' ° ^
Y \ T = 0.50
T = 0 2 5 \
1 1
Figs. 86-89. Dependence of Normalized Concentrations on Positions in Space and Time
53
oio
;: OM E
0 « 0
OJO
0 2 0
OlO
0, OIS
F i g s . 9 0 - 9 1 . Dependence of Normal ized Concentrations on Pos i t ions in Space and Time
54
1.00
0.95
0.90
0 8 5
0 60
0 7 5
0 70
0.65
0,60
0 5 5
0,50
0.45
0.40
0,35
0,30
0 2 5
0.20
OIS
0.10
0O5
-
/
VIC
• ^ 2 0 ^ , ^
^ u-1/^ / /
1 1 1 J
CASE 141
IB.^IO"'
/ \ I V
~M,
— F24
r ^
— F|2
1
1.00
0 95
0 90
0.85
0 80
0.75
0 70
0.65
0 60
0 55
Fjj 0 50
0 45
0 40
0 35
0 30
0 25
O20
0 15
0 10
0 05
- \ -----_ ~ -
VlO
•^20^ \ ^
1 j / \ ^
CASE 142
I B . = 10-'
r \ 1 1 V l
F|3
^ 2 4
F22
F|2
' '
1
0 9 5
0 .85
0.80
0.75
0.70
0.65
0 6 0
0.55
0.50
0 4 5
0 4 0
0 35
0.30
0 2 5
0.20
0 . 1 5
0 1 0
0 0 5
OO
~\ \ F
FZOI
-- ' -Y ---- \ : /
:j
10
'X \ 1
CASE
( B , =
1 1
144
)
1 1
^ - ^ 1 3
F||
Figs. 92-95. Time Dependence of Reconnbination Fractions
55
A "-9 -4 -1 -«
OffO
OffO
C i »
oao
0 ' 9
o n
O M
N/« - Y
oao4|
O M
r . ,o»
0 4 9
0 4 0
0 9 9
0 9 0
0 2 9
0 « 0
0 1 9
O ' O
0 0 9
C*St i i " la.ooi
•'t*
• ^ l
- •
,
. 0 0
OM
OfO
0 » » -
O I D -
OtO
OW
UM Ol i»,.io-»i
•II Ht -.0 - t
Figs. 96-99. Time Dependence of Recombination Fractions
56
1.0
0.95
0 90
0 65
0 80
0.75
0.70
0 6 5
0 60
0 55
0.50
0.45
0.40
0 3 5
0.30
0.25
0.20
0.15
0.10
0.05
1 \'°
- V t
IF 30
UL ^
CASE 215
J>' \ \ \
' F i 3
= ( f Z 2 l
0.9b
0.90
0 8 5
0.80
0 75
0 70
0 65
0 6 0
0.55
0.50
045
0 4 0
0.35
O30
025
0.20
0 10
0.05
0 0
/VlO
• \
^
1
"
- \ K - •< tM 1 1
CASE 235
Fl2 \ I I I I '
^^\^
F|l
P4F22I
0.95
0.90
0.85
o.eo
0 75
0 70
0 65
0.60
0 55
F|,0.50
0 45
0 4 0
0.35
0.30
0 2 5
0.20
0.15
0.10
0.05
_ -
Y 1
CASE 2 5 3
I l l l l 1 1
Figs. 100-103. Time Dependence of Recombination Fractions
57
SECTION FIVE: RESULTS OBTAINED FROM CODE C. FOR THREE PRIMARY AND ONE SECONDARY REACTION. THUS, D I / D | , <C|J. AND B , ARE ALL ZERO.
Table XXIV
PARAMETERS FOR STANDARD II CASES RUN WITH CODE C
Case No.
1000
1001
1002
1003
1004
c
^
z y
2
1
P E
0.87
0.87
0.87
0.87
1.60
Di/D,
0.25
0.25
0.25
0.25
0.25
Dj/D,
0
1.0
0.5
0.125
0.125
<u
2
2
2
_?
2
"•u
0.55
0.55
0.55
0.55
0.55
-:?«
0.1
0.1
0.1
0.1
0.1
B:
0.02175
0.02175
0.02175
0.02175
0.08
Table XXV
R E S U L T S O F CASES DESCRIBED IN T A B L E XXIV
Case No.
1000
1001
1002
1003
1004
F, ,
0 .1670
0.1670
0.1670
0.475
F u
0.2410
0.2410
0.2410
0.525
fu F?4 = HOz Hj - j F i i
No Resu l t (uns tab le )
0.2731
0.2724
0.2720
0.437
0.0014
0.0023
0.0042
0.038
0.0835
0.0835
0.0835
0.238
HOj= \Fu-f**
0.1351
0.1339
0.1318
0.181
58
ACKNOWLEDGEMENTS
Sandra Janoucek and Joanne Griffin of Argonne National Labora tory ' s Applied Mathematics Division assisted with the calculations and graph plotting.
REFERENCES
1. E. H. Bareiss , C. Chamot, and H. Fr icke, Mathematical Formulation, Analysis, and Description of the Fricke Diffusion Kinetics Code, ANL-6556 {May 1962).
2. D. A. Flanders and H. Fr icke, Applications of a High-speed Electronic Computer in Diffusion Kinetics, J. Chem. Phys. , 28, 126-9 (1958).
3. H. Fricke and D. L. Phillips, High Speed Computations in Diffusion Kinetics III: Solute Depletion in the Cylindrical One Radical-One Solute Model, J. Chem. Phys. , ^2. 1183 (I960).
•I