argonne bational xaboratorganl-6560 mathematics and computers (tid-4500, 19th ed.) aec research and...

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





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



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

î*,' ^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

î*/î\ û/îi kij /k i i

0 0

'û/'îi

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

http://3i29.Si2.Sii

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


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<


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


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

argonne bational xaboratorganl-6560 mathematics and computers (tid-4500, 19th ed.) aec research and...

Documents