15

7 i ..(^ ANL-7374 ANL-7374

V s ^

0 0

argonnc Bational Caboratorg

A HYBRID-COMPUTER PROGRAM FOR TfiAMSiENOEMEEfiATURE CALCULAJJONS ON TREAT FAST REACTOR SAFETY EXPERIMENTS

by

Lawrence T. Bryant, Lawrence W. Amiot, Charles E. Dickerman, and William P. Stephany

DfSTumunoN OF THIS DOCUMENT IS UNIIMITEO

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.

The faci l i t ies of Argonne National Labora tory a r e owned by the United States Govern­ment . Under the t e r m s of a cont rac t (W-31 -109-Eng-38) between the U. S. Atomic Energy Commiss ion , Argonne Univers i t ies Associa t ion and The Universi ty of Chicago, the Univers i ty employs the staff and opera tes the Labora tory m accordance with pol icies and p r o g r a m s fornnu-lated, approved and reviewed by the Associat ion,

MEMBERS OF ARGONNE UNIVERSITIES ASSOCIATION

The Universi ty of Arizona Carnegie Institute of Technology Case Institute of Technology The Universi ty of Chicago Univers i ty of Cincinnati Illinois Insti tute of Technology Universi ty of Illinois Indiana Univers i ty Iowa State Universi ty

The Univers i ty of Iowa Kansas State Univers i ty The Universi ty of Kansas Loyola Universi ty Marquet te Univers i ty Michigan State Univers i ty The Universi ty of Michigan Universi ty of Minnesota Universi ty of Missour i

Nor thwestern Universi ty Universi ty of Notre Dame The Ohio State Universi ty Purdue Universi ty Saint Louis Univers i ty Washington Universi ty Wayne State Universi ty The Universi ty of Wisconsin

LEGAL NOTICE

This r e p o r t was p repa red as an account of Government sponsored work. Nei ther the United States, nor the Connmission, nor any person acting on behalf of the Commiss ion

A. Makes any war ran ty or represen ta t ion , exp res sed or implied, with r e ­spect to the accuracy , comple teness , or usefulness of the information contained m this r epo r t , or that the use of any information, appa ra tus , method, or p r o c e s s d isc losed m this r epo r t may not infringe pr iva te ly owned r ights , or

B. Assumes any l iabi l i t ies with r e spec t to the use of, or for damages r e ­sulting f rom the use of any information, appara tus , nnethod, or p r o c e s s d isc losed m this r epor t .

As used m the above, "person acting on behalf of the Commiss ion" in­cludes any employee or contrac tor of the Commiss ion , or employee of such con t rac to r , to the extent that such employee or cont rac tor of the Commission, or employee of such cont rac tor p r e p a r e s , d i s semina t e s , or provides acces s to, any information pursuant to his employment or contract with the Commiss ion, or his employment with such cont rac tor .

Pr in ted m the United States of Amer ica Available from

Clear inghouse for Fede ra l Scientific and Technical Information National Bureau of Standards , U S Depar tment of Commerce

Springfield, Virginia 22151 P r i c e Pr in ted Copy $3.00, Microfiche $0 65

ANL-7374 Mathematics and

Computers (TID-4500) AEC R e s e a r c h and

Development Report

ARGONNE NATIONAL LABORATORY 9700 South Cass Avenue Argonne, Illinois 60439

A HYBRID-COMPUTER PROGRAM FOR TRANSIENT TEMPERATURE CALCULATIONS ON TREAT FAST REACTOR SAFETY EXPERIMENTS

by

Lawrence T. Bryant and Lawrence W. Amiot

Applied Mathemat ics Division

and

Char les E. Dickerman and William P. Stephany*

Reactor Phys ics Division

September 1967

*Now at The Universi ty of Michigan.

L E G A L N O T I C E ThiB report was prepared as an account of Government 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 in 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 em­

ployee 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.

tKTRIBUTION OF THIS DOCUMENT IS UNLfMITTO

to

r .

TABLE OF CONTENTS

P a g

NOMENCLATURE 6

SUMMARY 7

I. MATHEMATICAL ANALYSIS 8

II. DESCRIPTION OF THE HYBRID SIMULATION 13

III. SYSTEM MONITOR 20

IV. RESULTS 20

V. DISCUSSION 22

APPENDIXES

A. Detai ls of Operat ion of the Digi ta l -computer P r o g r a m . . . . 24

1. Introduction 24 2. Detai ls of Operat ion 35

B. Special F e a t u r e s Allowed by ADP 37

REFERENCES

LIST OF FIGURES

Title Page

Radial Segment of Annulus 9

Radial Subdivision of an Axial Section 12

Model of a Typical Fue l Cell 12

Thermal -conduct iv i ty Curve 13

Analog Circui t Diagram 16

Digital Flow Char t 17

S teady-s ta te T e m p e r a t u r e Calculations (Example 1) 21

Normal ized Power Trans ient (Example 2) 22

P o w e r - t r a n s i e n t Response of Innermost Node 22

P o w e r - t r a n s i e n t Response of Edge of Fuel 22

Response with Variable T h e r m a l Conductivity 23

LIST OF TABLES

Title Page

Source Language List ing of Mainline P r o g r a m 25

Valid P a r a m e t e r Names 35

N O M E N C L A T U R E

6

NOMENCLATURE

Symbol

A

Cp

G

h

K,K(T)

K^i,i + l

q

^i

t

T

U

V

V

'^"•fusion

Ar^

Ar

Ar

C O

f

Az

P

Subscr ip ts

c

CO

f

i

Supe r sc r ip t s for T9

i

i - i

C r o s s - s e c t i o n a l flow a r e a

Heat capaci ty

Mass flow ra te = PQQVA

Fi lm coefficient used to calcula te heat flux from outer surface of cladding to coolant

T h e r m a l conductivity

Linear average of the t h e r m a l conductivity between the ith and (i+l)th nodes

Power densi ty

r -component of the posit ion vector to the ith node point

T ime

T e m p e r a t u r e

Gap conductivity used to calculate the heat flux from the outer surface of the fuel to the inner surface of the cladding

Average speed of coolant

Volume

Heat of fusion

Cladding th ickness

Width of coolant annulus

Dis tance between node points of fuel

Height of a rad ia l sect ion

Density

Cladding

Coolant

Fue l

Node point, i = 1, . . . , 9

cm

j / c c deg C

g/sec

w/cm^ deg C

W/cm deg C

W/cm deg C

w/cc

sec

°C

w/cm^ deg C

c m / s e c

cc

j / c c

cm

cm

cnn

cm

g/cc

Outlet of the ith sect ion

Inlet for the ith sect ion

A HYBRID-COMPUTER PROGRAM FOR TRANSIENT TEMPERATURE CALCULATIONS ON TREAT FAST REACTOR SAFETY EXPERIMENTS

by

La-wrence T. Bryant, Lawrence W. Amiot, Char les E. Dickerman, and William P . Stephany

SUMMARY

This repor t gives a detailed descr ipt ion of a hybr id-computer p r o ­g r a m for calculating t e m p e r a t u r e s in a mult i region, ax i symmet r i c , cylin­d r i ca l configuration consist ing of solid m a t e r i a l s bounded by flowing coolant. Included is an explanation of the mathemat ica l methods , together with a d i s ­cussion of special fea tures , input-output descr ip t ions , and severa l sample p r o b l e m s .

The number of concentr ic regions allowed by the p r o g r a m is l imited by the amount of analog computer equipment avai lable. Each region naay contain e i ther s ta t ionary or turbulent ly flowing m a t e r i a l with t e m p e r a t u r e -dependent p r o p e r t i e s . The t e m p e r a t u r e s a r e calculated at node points equally spaced within a region. The sys tem descr ibed here uses nine rad ia l node points and ten axial i n c r e m e n t s .

The var ia t ion of conductivity and t empe ra tu r e along the radius is a s sumed l inear between adjacent node points over a region. Thermal con­ductivity is a s sumed to be a continuous finite function of t empe ra tu r e . Phase change in the fuel is considered, with heats of t ransformat ion added to the m a t e r i a l at the t r ans format ion t e m p e r a t u r e s .

The hybrid computer consis ts of a s t o r e d - p r o g r a m digital computer , with an 8K core m e m o r y , connected through a link to an analog computer . The analog computer solves the o rd inary differential equations ar i s ing from a f ini te-difference approximation of the hea t - t r ans f e r equations.

The duties of the digital computer a re : (l) p a r a m e t e r input, (2) a r i t h ­met ic computation of the coefficients of the sys tem, (3) analog potent iometer sett ing, (4) sys t em check, (5) control , and (6) s torage and me mory .

The physical p a r a m e t e r s , ini t ial conditions, and scale factors a r e the input p a r a m e t e r s to the digital computer . The digital computer calculates the potent iometer set t ings of the analog computer , se ts all analog potentiom­e t e r s , r e s e t s the analog computer , pe r fo rms a sys tem check, and makes a

decision to "go" or "not go." The rese t , hold, and operate analog modes a re "slaved" to those of the link, so that the analog is under the control of the digital computer . The analog components a r e miultiplexed, and grea t savings of analog ha rdware a re real ized.

The digital computer sets the init ial conditions of the p rob lem and gives the "opera te" o rde r to the link, which ini t iates the first i tera t ion.

The exit coolant t e m p e r a t u r e of an increment (A £)i, where £ is the length of the fuel pin (or cell), is computed by the analog, sampled in t ime, and s tored in the meraory of the digital computer . This becomes the inlet t empe ra tu r e of the next axial inc rement (Ai)i4-i, 3- < s ° o > until

n y (Ai)j^ = £ (axial conduction of heat is neglected).

i = i

Thus, by multiplexing, only one set of t empe ra tu r e equations is p r o g r a m m e d on the analog computer to obtain the radia l and axial t empe ra tu r e prof i les .

The analog and digital p r o g r a m s a r e descr ibed in Appendix A, and sample p rob lems a r e included in the body of the text. Special features a r e d iscussed in Appendix B.

I. MATHEMATICAL ANALYSIS

Calculation of t r ans ien t t e m p e r a t u r e s in various exper imenta l con­figurations is pa r t of the Fas t Reac tor Safety P r o g r a m . The configuration usually consis ts of fuel cells in the form of a right cylinder with axially flowing coolant. A typical fuel cell consis ts of a cyl indrical rod of f ission­able m a t e r i a l with concentr ic bonding ma te r i a l , cladding, and coolant.

The equations for calculating t e m p e r a t u r e s in concentr ic cyl indrical configurations a r e derived below. The regions may contain ei ther s ta t ionary or flowing m a t e r i a l , with heat generat ion permi t ted in the s ta t ionary region. Axial conduction of heat is neglected. Therefore , the t empe ra tu r e equations a r e coupled in the direct ion of flow only through the var ia t ion of the coolant t e m p e r a t u r e along the z - ax i s . This var iat ion in t e m p e r a t u r e is dictated by energy t r a n s p o r t due to coolant flow. Thus the equation for radia l heat flow is the same (in form) at any axial location.

The genera l diffusion equation descr ibing the t h e r m a l behavior of s ta t ionary (nonconvective) m a t e r i a l s i s '

pCp 5- = V • (KVT) + q(r , t) . (1)

A s s u m i n g t h a t T i s a f u n c t i o n of r a n d t o n l y , E q . 1, i n c y l i n d r i c a l c o ­

o r d i n a t e s , r e d u c e s t o

^ B T 1 1 LAl Sr V Sr

+ q ( r , z , t ) . (2)

M o s t c o d e s u s e a f i n i t e - d i f f e r e n c e f o r m of E q . 2 a t t a i n e d b y a s s u m i n g c o n d u c t i v i t y K = c o n s t a n t , y i e l d i n g t h e e q u a t i o n

p C p t - •<( ^^2 • r d r b^T 1 a T \ / .

(3)

E q u a t i o n 3 n e g l e c t s ( ^ K / S T ) ( S T / B r ) ^ i m p l i c i t l y c o n t a i n e d in E q . 2 . T h i s , i n g e n e r a l , i s n o t c o r r e c t .

R a t h e r t h a n s o l v i n g E q . 3 , w e c a n p u t E q . 2 i n t o f i n i t e - d i f f e r e n c e f o r m , a n d t h e r e s u l t i s e q u i v a l e n t t o a p p l y i n g t h e p r i n c i p l e of c o n s e r v a t i o n of e n e r g y

d i r e c t l y t o t h e s y s t e m . T h i s i s t h e m e t h o d u s e d i n t h e d e v e l o p m e n t of t h e p r o b l e m f o r t h e h y b r i d s y s t e m .

C o n s i d e r a h o m o g e n e o u s m a t e r i a l w i t h a n i n ­t e r n a l h e a t s o u r c e . A t y p i c a l g e o m e t r i c r e p r e s e n t a ­t i o n f o r t h e i ^ " i n t e r i o r n o d e p o i n t i s s h o w n i n F i g . 1. E q u a t i o n 2 c a n b e pu t d i r e c t l y i n t o f i n i t e - d i f f e r e n c e f o r m , g i v i n g r e s u l t s i d e n t i c a l t o t h o s e of t h e m e t h o d p r e s e n t e d h e r e ( t h e i n t e g r a l f o r n n u l a t i o n b e i n g m o r e e x p l i c i t ) . I n t e g r a t i n g b o t h m e m b e r s of E q . 2 o v e r t h e v o l u m e V j a s s o c i a t e d w i t h t h e i t h n o d e , w e g e t

145-921

Fig. 1. Radial Segment of Annulus

X. pCp I f dV -~ijH-?>-ir' (4)

w i t h d V = r d r dQ d z .

F o r xonit h e i g h t , a n a n g u l a r s e g m e n t of o n e r a d i a n , a n d e x c l u d i n g 9 d e p e n d e n c e , d V = 27rr d r . T h e r e f o r e ,

/ - r i + A r / z ^ ^

/ ( p C p ) i - ^ r d r ' r ^ - A r / z

r i + A r / 2

J , a-rr^S7)^^M , ^i^^^' > ^ r ^ - A r / 2 ^ ' - ' r ^ - A r / z r ^ - A r / z

(5)

o r

^ T i ^ T ( p C p ) i - ^ A r r i = r K ^

r j^+Ar/z

r j ^ -Ar / 2 + q ^ A r r ^ , (6)

w h e r e pCpi and qj a r e a s s u m e d c o n s t a n t o v e r Vj, and ^ T / S t for Vi i s a s ­s u m e d to be BTx/^t . E v a l u a t i n g t h e f i r s t t e r m on t h e r i gh t s ide of Eq . 6 g ives

r K ST rj^+Ar/ z

r^- A r / z

Ar , ^1 + — ) Ki , i+i

^i--r)^i' i- i

Ti+i - T j Ar

T j - T j . t

Ar

T h e v a r i a t i o n of K and T w i th r i s a s s u m e d to be l i n e a r o v e r t h e r e g i o n . T h e r e f o r e ,

Ki , i+i = K(Ti ) + K(Ti+, )

w h e r e K ( T ) is a s s u m e d to be a con t inuous , f in i te funct ion of t e m p e r a t u r e . A s i m i l a r def in i t ion ho lds for Ki^i_i .

We h a v e t h u s d e r i v e d t h e " w o r k i n g " d i f f e ren t i a l equa t ion ,

d T i 1

d t (pCp)i rAr2 _

-Ku-.fi4')

Ki,i+ifri + ^ ^ ( T i - T i + i )

( T i - i - T i ) q i

(pCp)i-

Apply ing the l aws of c o n s e r v a t i o n of e n e r g y and a s s u m i n g TQ we ob ta in t h e fo l lowing s e t of s i m u l t a n e o u s d i f f e r en t i a l e q u a t i o n s :

d T i

dt

dTz

"dT

1 / ^^f\/ V q , ^ . — z K i , 2 r i + — - ( T i - T z ) +

•i(pCp)f Arf ' \ 2 /

r2(pCp)f 2^1 _

2

Ar ,

(pCp)f'

(7)

(8)

^3 ( r z + - ^ ) K 2 , 3 ( r 2 + - ^ ) ( T 2 - T 3 )

K . . z ( r z - ^ ) ( T 1 - T 2 ) + (pCp)f' (9)

d T ,

"dT 1

- K

r3(pCp)f Arf

Arf

K , ,4(r3 + ^ ) (T3 - T4)

= , 3 ( r 3 - ^ ) (Tz - T3) (pCp)f'

(10)

dT4 dt r4(pCp)f ^ f

Arf

K4,5(r4+^yT4-T5)

K,

d T j

"dT

3 , 4 ^ ^ 4 2 "

1

r5(pCp)f A7f

Arf

( T 3 - T 4 ) (pCp)f'

K 4 , 5 l r 5 - 2

dt fr6-^ypCp)f Arf .

K 5 , 6 ( r 6 - - ^ ) ( T 5 - T 6 )

U r 6 A r f ( T 6 - T 7 )

dTy

dt ( r 7 + - ^ j ( p C p ) c Ar^ _

+

K,

(pCp)f'

( T 7 - T 8 )

- U r 7 A r c ( T 6 - T 7 )

dTg

dt r 8 - ^ ) ( p C p ) ^ Ar2L

K c ( r 8 - - ^ ) ( T v - T 8 )

h r 8 A r c ( T 8 - T 9 )

d T ,

"dT h r 8 ( T 8 - T 9 ) G f ( t ) ( T ^ T ^ - ^ )

A r , A r c o ( p C p ) c o ( r 8 + - ^ ] ^TiAzpco Ar^of ^g + — ^ J

(11)

(12)

(13)

(14)

(15)

(16)

w h e r e Tg = (T , " + T^) /2 , and for the f i r s t ax i a l s ec t i on , Tg"^ == cons tan t in le t coo lan t t e m p e r a t u r e .

T h i s o r i g i n a l f o r m u l a t i o n w h i c h c o n s i d e r s the fuel a s a hol low cy l ­i n d e r , r e d u c e s to the so l id c y l i n d r i c a l c a s e by s e t t i ng r j = A r f / 2 in Eq . 8. F i g u r e 2 shows t h e r a d i a l s u b d i v i s i o n s for a t y p i c a l a x i a l s e c t i o n .

t

CLAD COOLANT

145-920

Fig. 2. Radial Subdivision of an Axial Section

145-919 Rev. 1

T h e f u e l c e l l i s d i v i d e d i n t o " s m a l l " i n c r e m e n t s ,

A z i , w h e r e t h e l e n g t h of t h e c e l l i s

n

1 '

a s s h o w n i n F i g . 3 . T h e r a d i a l t e m p e r a t u r e d i s t r i b u t i o n in t h e f u e l , b o n d , a n d c l a d d i n g i s c o n s i d e r e d t o b e i n v a r i ­a n t o v e r t h e d i s t a n c e A z . T h e c o o l a n t t e m p e r a t u r e i s a s s u m e d t o v a r y l i n e a r l y a l o n g Az. T h e c o n d u c t i v i t y a t t h e b o u n d a r y b e t w e e n t h e r a d i a l s e c t i o n i a n d i + i of t h e f u e l , d e n o t e d b y K^ ^.^.i, i s e v a l u a t e d a t t h e boxonda ry t e m ­p e r a t u r e T i^ i+ i = ( T i + T i - | - i ) / 2 . C o n d u c t i v i t y a c r o s s t h e f u e l b o u n d a r y i s g i v e n b y a q u a d r a t i c f u n c t i o n of t e m ­p e r a t u r e . F o r t h e p r o b l e m g i v e n h e r e , t h e fxinct ion i s

Ki,i+i = 4.756 X 10"^ - 3.7165 x 10"^ ^^M^ + 8.8405 x 10"^ Ti_i+,

i n W / c m d e g C f o r 0 < T i i + j < 2 2 0 0 ° C . T h i s e q u a t i o n Fig. 3. ModelofaTyp- w a s o b t a i n e d b y a p o l y n o m i a l a p p r o x i m a t i o n of t h e c o n ­

ical Fuel Cell d u c t i v i t y c u r v e s h o w n in F i g . 4 . T h i s c u r v e w a s o b ­t a i n e d b y u s i n g t h e d a t a of H e d g e a n d F i e l d h o u s e , ^

K i n g e r y e t a l . , ^ a n d R e i s w i g , t o g e t h e r w i t h a p p r o p r i a t e c o r r e c t i o n f a c t o r s t o c o r r e c t f o r d e v i a t i o n f r o m t h e o r e t i c a l d e n s i t y a n d t o c o m p e n s a t e f o r p l u t o -n i u m c o n t e n t . [In o u r p r o b l e m , (pCp) f w a s 3 .52 j / c c d e g C , w h i c h i s 9 5 % of t h e h e a t c a p a c i t y of f u l l y d e n s e UO2. T h e c o r r e c t i o n f a c t o r u s e d f o r p l u t o -n i u m c o n t e n t w a s 0 . 7 . 1

The melting point of the fuel is assumed to be 2600°C and AHfusion = 2990 j / c c . Using a semiempi r ica l relat ionship, the AHfusion °^ fully dense UO2 was est imated to be 3150 j / c c . ^ This value will be used for the (U + Pu)02. Since the mixed oxide is 95% of theore t ica l density,

^Hfusion = °-^^ "^ ^^^° '^/'^^ " ^^^° '^Z^'^-

The gap conductivity is taken as a constant. The value used is 0.5 w/cm^ deg C and is based on previous UO2 exper iments .

The cladding and coolant a re s ta inless steel and sodium, r e spec ­tively. The t h e r m a l p roper t i e s of the cladding a re the same as those of Type 304 s ta inless steel;^ the coolant p roper t i es a re presented in Ref. 8.

The equations developed thus far define the mathemat ica l model. With this model, the phenomena a r e descr ibed quantitatively. This sys tem can be simulated on purely analog or digital equipment; but simulation on a hybrid computer has an outstanding advantage in t e r m s of programming effort and computer equipment.

II. DESCRIPTION OF THE HYBRID SIMULATION

The Argonne Hybrid Computer System consists of a s to red -p rog ram PDP-7 digital computer connected through a control and interfacing unit to the analog-computer consoles. ' '

The interfacing e lectronics handles the control between computers , the conversion of digital information to analog voltages, the conversion of analog voltages to digital information, and special functions, such as auto­mat ic potent iometer sett ings on the analog consoles. The conversion equipment consis ts of 10 channels in each direction: analog-to-digi tal (A-to-D) and digi tal- to-analog (D-to-A).

Analog computers a r e efficient in handling ordinary differential equations of the type presented h e r e . However, the analog computing equipment required to effect a solution is enormous. The nature of the equations suggests multiplexing the analog equipment. This can be done since the equations for the radial t empera tu re profile a re the same at any axial increment .

o 4 + e

E o

* 34

ESTIMATE OF THERMAL CONDUCTIVITY - v S S ' / . THEORETICAL DENSITY

500 1000 1500 2000 TEMPERATURE, °C

2500 3000

145-918

Fig. 4. Thermal-conductivity Curve

Since a x i a l conduc t ion of hea t i s n e g l e c t e d , t h e t h e r m a l coupl ing b e t w e e n s u c c e s s i v e a x i a l i n c r e m e n t s (Az) is only t h r o u g h the f lowing c o o l ­an t . The in le t coo lan t t e m p e r a t u r e to an a x i a l s e g m e n t is t h e ex i t coo lan t t e m p e r a t u r e f r o m t h e s e g m e n t i m m e d i a t e l y p r e c e d i n g i t . T h i s t e m p e r a t u r e m u s t be " r e m e m b e r e d " f r o m rxon to run ( s ec t i on to s e c t i o n ) to be u s e d a s input da t a to t h e s u c c e e d i n g s e c t i o n . It is in t h i s " r e m e m b e r i n g " tha t t h e d ig i t a l s y s t e m p l a y s a m o s t v i t a l r o l e . T h u s , only t h e s i m u l a t i o n of one se t of E q s . 8 -16 i s n e c e s s a r y . ^ °

T h e t w o - d i m e n s i o n a l t e m p e r a t u r e p ro f i l e in a fuel c e l l i s d e t e r m i n e d in the fol lowing m a n n e r : T h e flow r a t e and in l e t coo lan t t e m p e r a t u r e s a r e known func t ions of t i m e . The r a d i a l t e m p e r a t u r e p r o f i l e of the f i r s t a x i a l s e g m e n t i s c o m p u t e d on t h e ana log c o m p u t e r . The exi t coo lan t t e m p e r a t u r e TQO, f^on^ t h i s s e g m e n t i s sanapled in t i m e a t a r a p i d r a t e and s t o r e d a s s a m p l e d b i n a r y n u m b e r s in the m e m o r y of t h e d ig i t a l c o m p u t e r . The t e m ­p e r a t u r e p r o f i l e i s p l o t t e d on X - Y r e c o r d e r s d u r i n g e a c h i t e r a t i o n . T h i s w a s n e c e s s a r y b e c a u s e of l a c k of s t o r a g e s p a c e on t h e d ig i t a l c o m p u t e r . T h u s , t h e d ig i t a l c o m p u t e r n e e d only r e t a i n t h e e x i t - c o o l a n t t e m p e r a t u r e funct ion (T^o^) f r o m t h e p r e v i o u s i t e r a t i o n . By naaking T ^ Q , a v a i l a b l e a s t h e in l e t t e m p e r a t u r e to t h e nex t a x i a l s e c t i o n , and u s i n g t h e s a m e ana log e q u i p m e n t aga in , we c o m p u t e the r a d i a l t e m p e r a t u r e p r o f i l e of t h i s nex t s e g m e n t . T h i s p r o c e s s i s t hen r e p e a t e d for a l l a x i a l d i v i s i o n s of the fuel ce l l , t h u s enab l ing t h e e n t i r e r a d i a l and a x i a l t e m p e r a t u r e p r o f i l e s of the ce l l to be c a l c u l a t e d .

C h a n g e of s t a t e (me l t i ng ) of the fuel i s hand l ed by a s y s t e m of t r a n ­s i s t o r i z e d s w i t c h e s and e n e r g y s t o r a g e m e c h a n i s m s on the a n a l o g . As a fuel n o d e r e a c h e s i t s m e l t i n g point , t h e node t e m p e r a t u r e i s h e l d c o n s t a n t wh i l e t h e e n e r g y n e e d e d to m e l t t h e node is s t o r e d by the s y s t e m . When t h e s t o r a g e m e c h a n i s m d e t e r m i n e s t h a t t h e p r e s c r i b e d a m o u n t of e n e r g y h a s b e e n s t o r e d , t h e n o d e t e m p e r a t u r e i s r e l e a s e d for con t inued t e m p e r a ­t u r e i n c r e a s e . The e n e r g y can be r e l e a s e d in t h e so l id i fy ing of t h e m o l t e n m a t e r i a l . O t h e r fuel c e l l s m a y be t r e a t e d s i m u l t a n e o u s l y , if enough h a r d ­w a r e i s a v a i l a b l e , o r s e q u e n t i a l l y in the s a m e m a n n e r a s d e s c r i b e d .

T h e P D P - 7 d i g i t a l c o m p u t e r i s not only i m p o r t a n t in t h e b a s i c m e t h o d of so lu t ion of t h e p r o b l e m , bu t i s a l s o invo lved in t h e o v e r a l l con­t r o l of t h e h y b r i d s y s t e m . In add i t ion to p r o v i d i n g for o p e r a t o r i n t e r v e n ­t ion d u r i n g t h e so lu t i on p r o c e s s , t h e d ig i t a l s y s t e m a l s o p r o v i d e s for i n i t i a l s e t u p of the p r o b l e m , a u t o m a t i c p o t e n t i o m e t e r s e t t i n g on t h e a n a l o g c o m ­p u t e r , c h e c k o u t of the s y s t e m , input of the p r o b l e m p a r a m e t e r s , c o n t r o l of the o p e r a t i o n of the a n a l o g c o m p u t e r s , and g e n e r a l c o n t r o l of t h e p r o b l e m so lu t ion . The d ig i t a l naach ine i s c o n t r o l l e d by s t o r e d d i g i t a l p r o g r a m w r i t t e n s p e c i f i c a l l y for t h i s p r o b l e m . In add i t ion to t h i s m a i n l i n e p r o g r a m , s e v e r a l r o u t i n e s r e s i d e n t in the c o m p u t e r m e m o r y can be e n t e r e d to a l low for s p e c i a l f e a t u r e s . A m o n g t h e s e a r e r o u t i n e s to p r i n t out p o t e n t i o m e t e r s e t t i n g s and to debug t h e p r o g r a m .

The digital p rog rams a r e wri t ten in assembly language, the a s s e m ­b le r for the P D P - 7 being run on the CDC-3600. ' ' The language for the a s s e m b l e r is essent ia l ly that used by the Digital Equipnaent Corp. a s s e m ­b le r for the P D P - 7 , with cer ta in mnemonics , macro ins t ruc t ions , and sub­rout ines added to handle the hybrid sys tem. The output of the CDC-3600 for the assembled object code is on magnetic tape suitable for input to the P D P - 7 at run t ime . These p r o g r a m s a r e d iscussed in Appendix A.

To ensure that the Eqs . 8-16 do not exceed the l imits of the machine, the equations must be scaled. The following relat ionships define the scale factors of the var ious var iab les :

at = f, bTi = T| , cq = q',

where the pr imed var iab les a r e analog machine voltages and analog machine t ime , and a, b, and c a r e p rede te rmined constants which ensure that our analog var iab les do not exceed the computer limit over the period of in te res t . Substitution of these "scaled" var iab les yields the so-cal led machine equa­t ions . F igure 5 shows the analog circui t d iagram for these equations.

A flow d iagram of the digital p r o g r a m used in the problem solution is shown in F ig . 6. The p r o g r a m shown is n e c e s s a r y to p rocess the i t e r a ­t ions for a t e m p e r a t u r e profile for one set of input p a r a m e t e r s .

The input p a r a m e t e r s to the digital computer a r e the init ial condi­t ions, d imensions , and t h e r m a l p rope r t i e s of the sys tem, together with all analog scale fac tors . This enables the m a t e r i a l s , d imensions, or scale to be changed readi ly so that other m a t e r i a l s can be investigated. The follow­ing is a list of the input data to the digital p rogram:

A. Dimensions of the sys tem and scale factors

Az

^1

^2

^3

1-4

c m

c m

c m

c m

c m

^5

^6

1-7

^8

Arf

c m

c m

c m

c m

c m

••^co

a

b

c

c m

v/sec

v/deg C

V cc/w

B. T h e r m a l p rope r t i e s

Fuel

K,

(pCp)f

W/cra deg C

J / c c deg C

x-7

r-^

1

-

*« 1

Qio

-M

1

avi

3 1

1 o>

-li o

1

V I

o

o

> 3 !U

O

OQ

17

^ BEGIN j

I NIT PROGRAM INIT.HYBRID

RESET ANALOG

READ PARA. NAME FROM

TYPEWRITER

CALC POT

VALUES

SET POTS

TYPE FLOATING

POINT PARAMETER

TYPE OUT

f HALT j

INIT. PROGRAM RESET ANALOG

SET SWITCH 'FIRST T I M E

T H R U "

NOT OK .^ . . ' ^YSTEMXS.

^ \ C H E C K ^ ^

OK

I N I T . DTA,CHANNEL ZERO TO ZERO

CONVERT ATD.CHANNEL

0

CONVERT ATD

CHANNEL2

INIT.PROGRAM INIT HYBRID

RESET ANALOG

MNTERRUPTJ

GET CONVERTED ATD INPUT

SET ATD CLOCK PERIOD

CONVERT ATD, CHANNEL I

UNDER CLOCK CONTROL

ENABLE INTERRUPT

SYSTEM

OPERATE ANALOG

LOOP WAIT FOR

INTERRUPTS

NOT JKAXIMUM

HOLD ANALOG

SET SWITCH "NOT FIRST T IME THRU"

YES

\ v T H R U , / ' ^

|N0 •

GET DTA OUTPUT

FROM LIST

CONVERT DTA

CHANNEL 0

^ w

STORE ATD INPUT IN

LIST

INIT PROGRAM INIT HYBRID

NOT MAXIMUM

RESTART

HALT

145-916

Fig. 6. Digital Flow Chart

2. Cladding

(pCp)c

3. Coolant

P

W/cm deg C

j / c c deg C

g /cc

j / g deg C CO

CPco

U w / c m ^ deg C

Inlet coolant t e m p e r a t u r e °C.

G g / sec

h w / c m ^ deg C

Initial t e m p e r a t u r e s of the sys tem in °C.

The power density is input data to the analog computer .

The f i r s t portion of the p r o g r a m allows for init ial ization of the nna-chine, p rog ram, and assoc ia ted ha rdware . P rob lem p a r a m e t e r s a re entered froin the console keyboard. The p a r a m e t e r s descr ibe the physical constants of the p rob lem to be run as indicated above. The p a r a m e t e r s a r e inputed as floating-point nximbers into the magne t i c - co re s torage . Once p a r a m e t e r s a r e entered, a special routine may be used to punch them onto paper tape. This paper tape can then be used in place of the typewr i te r on subsequent r e runs of the p rob lem. Once the paper tape has been entered, any p a r a m ­e te r s may be changed by the typewr i te r .

After the problem p a r a m e t e r s a r e co r rec t ly entered, GO is typed and the digital computer calculates potent iometer values via p rog rammed a lgor i thms , using the inputed p a r a m e t e r s . These values a r e s tored in digi­ta l form in a table in core s to rage . The potent iometer values can be p r o ­duced by:

1. Di rec t typing of coefficients from the typewr i te r .

2. Calculation from typed p a r a m e t e r s .

3. Stored coefficients p rog rammed into the problem.

Once al l values have been calculated, they a r e sequential ly converted to analog voltages by D-to-A conversion o r d e r s , and the corresponding poten­t i o m e t e r s a r e set to these values under control of the digital machine .

A sys t em stat ic check is made on the analog computer , and the status of the sys t em is indicated. Two manual switches on the analog console a re in te r rogated . The first indicates whether the opera tor wishes the computer to proceed with the p rob lem solution or to continue to loop and r e in t e r roga te the swi tches . The second indicates whether the p r o g r a m should re ini t ia l ize

itself or continue i te ra t ions along the axis upon signal from the f irs t switch. The switches have plus or minus voltages on the poles to control the com­pu te r ' s in ter rogat ion .

When the p r o g r a m is allowed to proceed with the problem solution, it sets the clock sampling ra te and places the analog consoles in the operate mode. The exit t e m p e r a t u r e for the p resen t i terat ion on segment Azj is sampled, converted A- to -D, and put into a list in core s torage . At the same t ime , a sample is taken from the l is t . This sample re la tes to the exit t em­p e r a t u r e of the previous section from the previous i te ra t ion . Samples a r e repet i t ively taken at a r a t e corresponding to the clock until the list is filled with s ample s . The number of samples taken, divided by the sample ra te , then gives the total t ime for one i tera t ion. This can be var ied by changing a location in core m e m o r y which controls the clock sample ra te .

This i t e ra t ive p rocedure is c a r r i e d on in this fashion over each axia l segment , Az^, until

n ^ Azi - H. i = i

The resul t ing t e m p e r a t u r e dis t r ibut ions of each segment a r e plotted on X-Y r e c o r d e r s , resul t ing in the two-dimensional t e m p e r a t u r e distr ibution of the cell .

The inlet t e m p e r a t u r e for the f i rs t section is constant (supplied by analog). The digital coraputer must then supply a zero inlet value during the f i rs t i te ra t ion . This is done by init ial ly converting A- to-D with ze ro and then bypassing further A- to-D convers ions on the f irs t i tera t ion.

To conserve core s to rage , we keep only one list for sampled data, and it is shared by both inlet and outlet t e m p e r a t u r e data. In concept, it ac ts as a push-down list , inlet t e m p e r a t u r e data being pushed in one end and outlet t e m p e r a t u r e data emptying out the other . Actually, as we move down the fixed l ist , inlet t e m p e r a t u r e data a r e f i rs t moved from the l ist and converted A- to -D, and then the D-to-A converted outlet data a re s tored in the i r p lace . As we move down the l ist , the front par t will contain outlet t e m p e r a t u r e data and the las t par t will contain inlet t e m p e r a t u r e data. The outlet data will become the inlet data for the next i tera t ion.

Nine radia l and ten axial segments gave good r e su l t s .

III. SYSTEM MONITOR

The s y s t e m l i b r a r y con t a i n s an A u t o m a t i c Debug P r o g r a m (ADP), w h i c h w a s w r i t t e n to a l low for o p e r a t o r c o n t r o l o v e r the p r o g r a m in t h e i n i t i a l p h a s e s of debugg ing . T h i s p r o g r a m , h o w e v e r , a l s o p r o v i d e s a con­v e n i e n t and power fu l m e a n s for the e x p e r i m e n t e r to c o n t r o l t he p r o b l e m d u r i n g t h e a c t u a l h y b r i d p r o b l e m so lu t ion .

A D P p r o v i d e s t h e fol lowing f a c i l i t i e s :

1. The m e a n s to i n t e r r o g a t e ind iv idua l l o c a t i o n s o r l i s t of con­s e c u t i v e l o c a t i o n s in m e m o r y and p r i n t t h e m on the c o n s o l e t y p e w r i t e r .

2. The m e a n s to i n t e r r o g a t e l o c a t i o n s and, if d e s i r e d , change t h e s e l o c a t i o n s w i t h the t y p e w r i t e r .

3. The m e a n s to punch c o n s e c u t i v e l o c a t i o n s onto p a p e r t a p e .

4 . The m e a n s to put b r e a k p o i n t s ( p r o g r a m m e d j u m p s b a c k in to A D P ) at any d e s i r e d poin t in t h e p r o g r a m (up to e igh t ) .

5. The m e a n s to con t inue t h e p r o g r a m at the b r e a k p o i n t in the p r o g r a m .

Al though p r o b l e m p a r a m e t e r s a r e inpu ted t h r o u g h t h e n o r n a a l p r o -grana in f l oa t i ng -po in t f o r m , A D P a l l o w s s e l d o m - c h a n g e d p a r a m e t e r s to be e a s i l y m o d i f i e d . We t h u s u s e A D P to modi fy the s a m p l i n g r a t e , t h e n u m b e r of s a m p l e s p e r i t e r a t i o n , o r t h e n u m b e r of s e g m e n t s to be a n a l y z e d . The s w i t c h e s on the ana log c o n s o l e t ha t c o n t r o l r e s t a r t and con t inue a long wi th the b r e a k p o i n t s t h a t can be added at r a n d o m p r o v i d e for p a u s e s in the p r o ­g r a m , at w h i c h t i m e v a l u e s can be i n t e r r o g a t e d , r e s u l t s s tud ied , e t c .

IV. R E S U L T S

The e x a m p l e s t h a t follow r e p r e s e n t t y p i c a l p r o b l e m s tha t w e r e so lved u s i n g t h e h y b r i d s y s t e m . In s o m e c a s e s , d ig i t a l o r a n a l y t i c r e s u l t s a r e c o m p a r e d wi th t h e h y b r i d so lu t ion .

A. E x a m p l e 1

Da ta for t h e f i r s t p r o b l e m to be u s e d a s a m a c h i n e check a r e a s fo l lows:

Input Data

Az = 3.76 c m

r i = 0.092 c m

V2 - 0 .1236 c m

r j = 0.1552 c m

T4 = 0.1868 c m

r s = 0.2184 c m

r^ = 0.25 c m

r-j = 0.25 c m

rg = 0.2921 c m

Pco = 0.833 g / c c

Cpco = 1-2636 j / g deg C

G = 93.4 g / s e c

a = 100

b •= 0.02

c = 0.01

Arf = 0.0316 c m

Arc = 0.0421 c m

Arco = 0.0859 c m

Kf = 4.756 X 10-^ w / c m deg C

K, 0.228 w / c m deg C

(pCp)f = 3.52 j / c c deg C

(pCp)c = 4.7 j / c c deg C

U = 0.5 w /cm^ deg C

h = 18.3 w /cm^ deg C

I n i t i a l t e m p e r a t u r e s = 400°C; p o w e r dens i t y q = 2000 W / c c .

Us ing the above da t a , we c a l c u l a t e d s t e a d y - s t a t e t e m p e r a t u r e s for the f i r s t a x i a l s e c t i o n of the s y s t e m us ing bo th the a n a l y t i c f o r m of the

diffusion equa t ion and the equa ­t i ons p r o g r a m m e d on the h y b r i d conaputer . F u e l conduct iv i ty w a s held cons t an t in th i s e x a m p l e . F i g u r e 7 c o m p a r e s the hyb r id c o m p u t e r so lu t ion wi th the c a l ­cu la t ed r e s u l t s , for the f i r s t s e g m e n t .

B. E x a m p l e 2

E x a m p l e 2 is p r e s e n t e d a s a check of t h e hyb r id r e s p o n s e to a n o r m a l i z e d p o w e r t r a n s i e n t of the f o r m given in F i g . 8. As in E x a m p l e 1, t he conduc t iv i ty of the fuel w a s held cons t an t . Th i s w a s done to check our r e ­su l t s wi th the r e s u l t s ob ta ined

u s i n g the ARGUS p r o g r a m . The ARGUS p r o g r a m u s e s only five ax ia l in ­c r e m e n t s for t h i s c a l c u l a t i o n . T h e r e s u l t s for the i n n e r m o s t node and the edge of t h e fuel a r e shown in F i g s . 9 and 10, r e s p e c t i v e l y . Note tha t the a x i a l r a t h e r t han the r a d i a l t e m p e r a t u r e p r o f i l e s of the v a r i o u s s e g m e n t s (Az) of t h e fuel a r e shown.

1400

1300

1200

HOC

1000

900

800

700

600

500

400

nodes-

°C

hollow-

- 1 2

—analytic • hybrid

— assumed

3 4 5 *

NL 1 i i N v i i I N [ 1 j \

lq=2000 w/cc IKJ= 4.756 10" W/Cm-

iT;,g(0)=400°C

• l U C I

i

"c

^ ^

•—clod —*

J

••coolant - ^

145-915

Fig. 7. Steady-state Temperature Calculations (Example 1)

145-914 Fig. 8. Normalized Power Transient

(Example 2)

1750

1500

1250

1000

750

500 -

250-

°C

y

/ /

Jx ^ ^

- " ^ ^ ^

t sees

^ ,—- „5 ,6

X^'^ ^ ^ - ^ — « l . 2

0 10 20 30 4 0 50 60 70 SO 90

145-913

Fig. 9. Power-transient Response of Innermost Node

t sees

0 10 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0

145-912

Fig. 10. Power-transient Response of Edge of Fuel

C. E x a m p l e 3

T h i s e x a m p l e shows t h e r e s p o n s e of the s y s t e m when the fuel con­duc t iv i ty i s a con t inuous function of fuel t e m p e r a t u r e . F i g u r e 11 c o m p a r e s the r a d i a l t e m p e r a t u r e p r o f i l e wi th an ana ly t i c so lu t ion of the p a r t i a l dif­f e r e n t i a l e q u a t i o n s .

V. DISCUSSION

The ou t s t and ing f e a t u r e of t h i s m e t h o d for c a l c u l a t i n g the t r a n s i e n t t e m p e r a t u r e r e s p o n s e of m u l t i r e g i o n , a x i s y m m e t r i c , c y l i n d r i c a l conf igura­t i ons i s t he g r e a t sav ing of c o m p u t e r equ ipmen t . Th i s i s a c c o i n p l i s h e d by

3200

2800

2400

2000

1600

1200

800

400

m u l t i p l e x i n g the ana log eqxiipment of t h e h y b r i d c o m p u t e r , a l lowing g r e a t s av ings of ana log c i r c u i t r y . Without t h i s e c o n o m y , the t e n s e t s of n ine s i m u l t a n e o u s d i f f e ren ­t i a l equa t i ons t h a t r e p r e s e n t e d ou r m o d e l could not have been hand led .

o HYBRID - ANALYTIC P h a s e change (mel t ing) of t h e fuel

i s h a n d l e d by a s y s t e m of t r a n s i s t o r i z e d s w i t c h e s and e n e r g y - s t o r i n g m e c h a n i s m s ( i n t e g r a t o r s ) , a s shown in the ana log c i r ­cui t d i a g r a m (F ig . 5). When a node of the fuel r e a c h e s i t s m e l t i n g point (2600°C), t he d e r i v a t i v e of the d i f fe ren t i a l equat ion r e p r e s e n t i n g the node is m a d e z e r o , ho ld­ing the node t e m p e r a t u r e and the node conduc t iv i ty c o n s t a n t . The e n e r g y - s t o r i n g i n t e g r a t o r then c a l c u l a t e s the amount of

e n e r g y n e c e s s a r y to m e l t t he node , and, when the p r o p e r annount of e n e r g y h a s b e e n s t o r e d , t he t e m p e r a t u r e of the node is a l lowed to i n c r e a s e .

02 06 10 .14 18 RADIUS,CM

22 26 30

145-911

Fig. 11. Response with Variable Thermal Conductivity

W e u s e d a s i m p l e e x p e r i m e n t t o a r r i v e a t t h e s a m p l i n g r a t e u s e d i n t h e s y s t e m . W e o b s e r v e d t h e t r u e a n a l o g r e s p o n s e to a p o w e r fxinct ion in e x c e s s of a n y p o w e r w e w o u l d u s e i n o u r c a l c u l a t i o n s . W e s a m p l e d t h i s r e s p o n s e o v e r t h e i n t e r v a l a t a r a t e t h a t e x h a u s t e d t h e r e s t of t h e d i g i t a l m e m o r y . W e r e c o n v e r t e d t h e s a m p l e d r e s p o n s e D - A a n d c o m p a r e d i t w i t h t h e o b s e r v e d a n a l o g r e s p o n s e . We t h e n d e c r e a s e d t h e s a m p l i n g r a t e a n d r e p e a t e d t h e p r o c e s s u n t i l w e f o u n d a s a m p l i n g r a t e t h a t c a u s e d t h e s a m p l e d -r e c o n v e r t e d D - A c u r v e t o d i f f e r f r o m t h e t r u e a n a l o g c u r v e . S i n c e a s a m ­p l i n g r a t e t h a t e x h a u s t e d t h e m e m o r y o v e r a n y i n t e r v a l of i n t e r e s t w a s m o r e t h a n a d e q u a t e , w e c h o s e t h a t r a t e in a l l c a s e s .

T h u s , t h e t i m e i n t e r v a l i s l i m i t e d b y t h e a m o u n t of m e m o r y a v a i l ­a b l e , a n d t h i s w i l l d e t e r m i n e t h e s a m p l i n g r a t e . If a l o n g e r i n t e r v a l i s t o b e c o n s i d e r e d , t h e i n t e r v a l c a n b e s u b d i v i d e d i n t o s m a l l e r i n t e r v a l s ( w h o s e l e n g t h s a r e c o m p a t i b l e w i t h t h e m a c h i n e l i m i t a t i o n s ) a n d t h e r e s p o n s e s c o m p u t e d s e q u e n t i a l l y .

I n t h i s s i m u l a t i o n , t h e s p e e d of t h e d i g i t a l c o m p u t e r i s n o t a f a c t o r . T h i s i s t r u e s i n c e no a r i t h m e t i c c a l c u l a t i o n s a r e m a d e d u r i n g s a m p l i n g i n ­t e r v a l s o r i t e r a t i v e r u n s .

A P P E N D I X A

D e t a i l s of O p e r a t i o n of the D i g i t a l - c o m p u t e r P r o g r a m

1. I n t r o d u c t i o n

T h i s a p p e n d i x g i v e s a d e t a i l e d d e s c r i p t i o n of the d ig i t a l p o r t i o n of the h y b r i d c o m p u t e r p r o g r a m . Since the d ig i t a l c o m p u t e r c o n t r o l s the h y b r i d s y s t e m , the c o n t r o l f e a t u r e s of the p r o g r a m a r e e m p h a s i z e d . The h y b r i d p r o g r a m w a s d e s i g n e d to r u n a s an a u t o m a t i c p r o d u c t i o n - t y p e a n a l ­y s i s p r o g r a m . We r e c o g n i z e d tha t an i n t e r a c t i o n b e t w e e n the e x p e r i m e n t e r and the s i m u l a t i o n p r o c e s s w a s n e e d e d , and p r o g r a m m e d the d ig i t a l c o n t r o l so tha t a "hands on" f e a t u r e w a s a v a i l a b l e .

T a b l e I g i v e s a l i s t i n g of the a s s e m b l y l a n g u a g e for the m a i n l i n e p r o g r a m . L i b r a r y s u b r o u t i n e c a l l s a r e e x e c u t e d by m i c r o and m a c r o i n ­s t r u c t i o n s . The r o u t i n e s a r e l oaded in to the p r o g r a m a t a s s e m b l y t i m e by the a s s e m b l e r fronn a l i b r a r y m a g n e t i c t ape and thus do not a p p e a r in the s o u r c e l a n g u a g e l i s t i n g .

The a r i t h m e t i c s u b r o u t i n e s a r e s i n g l e - p r e c i s i o n f l oa t i ng -po in t r o u ­t i n e s . C o m m u n i c a t i o n of p a r a r a e t e r s v ia the t e l e t y p e i s in f l oa t ing -po in t n o t a t i o n . The f l oa t i ng -po in t n u m b e r s a r e "f ixed" b e f o r e c o n v e r s i o n (D-to-A) or " f loa ted" a f t e r c o n v e r s i o n ( A - t o - D ) .

The d i g i t a l p r o g r a m c a n be g r o u p e d into t h r e e c a t e g o r i e s :

1. A g e n e r a l p r o g r a m , which a l l ows p a r a m e t e r input , c a l c u l a t i o n of coe f f i c i en t s and in i t i a l cond i t i ons ( p o t e n t i o m e t e r s e t t i n g s ) , and the c o n t r o l p r o g r a m for i n i t i a t i ng and runn ing the m u l t i ­p l e x e d h y b r i d s i m u l a t i o n .

2. A se t of s u b p r o g r a m s , wh ich a l low the e x p e r i m e n t e r to p e r f o r m s p e c i a l funct ions s u c h a s , "hands on . "

3. A D P , wh ich s e r v e s a s a l i m i t e d m o n i t o r c o n t r o l p r o g r a m . (See Append ix B. )

TABLE I. Source Language List ing of Mainline P r o g r a m

FRED

PAST

GO

TEST

TESTB

SIG

START

KEYS

KEY

RES LAC DAC CLAD JMP LAC» DAC CIDAO ION LAC DAC* ISZ ISZ JMP LAC DAC HRTC LAC DAC DZM« LAC DAC ICG2 LAC DAC ClADO JKP CLAD ION LAC SMA JMP LAC DAC C1AD2 JMP CLAD ION LAC SPA JMP JMP LAC DAC RRTC ICG2 SCKl CIADIC URTC JMP lOF ICRTC RRTC LAM STA TCR RWORD LAC DAC LAC SAD* JMP

1 = ATD = BINA

PAST =LISTA =DTA = DTA

= BINA =LISTA =LISTA = CT1 • =7400006 FRED

= BINB =LISTA =LISTA SAMPCT =CT1

OROA 00 1 = ATD •

= ATD

SIG URDD 001 = ATD •

= ATD

GO-2 CONT ORDB UOl

10 = ATD

«

-33 = KEYfc

= KEY KtYrz = KEYA =KEY = KEYA KEYM

INTERRUPT LOCATION

SET UP TABLE COUNI

CARRIAGE RETURN LINE FEED GET FIRST 3 CHARACTERS

S£T UP ADDRESS OF TABLE

COMPARE TYPED WORD TO TABLF

26

TABLE I (Contd.)

KEYM

KEYTZ CI C2 C3 C4 C5 C6 07

ce C9 CIO Cll FIEMPl FTEMP? FIEMPS FIEMP4 POTT

C

B

G

LAC ACD STA ISZ JI'P JMP ISZ XCT» CAL FCEC FCEC FDEC FCEC FDEC FCEC FCEC FCEC FCEC FDEC FDEC RES RES RtS RtS KES RES DEC DEC DEC RES RtS KES DEC DEC DEC RES RES RFS DEC DEC DEC RES RES RFS DEC DEC DEC RES RCS RES DEC DEC DEC RES RFS RES LAW STA JMP LAW JMP LAW

^KEYA -7 =KEYA = KEYe KEY KEYS = KEYA = KEyA KEYI lE-1 37165E-2 8e405t-4 •SEO ILO 2EI -25E3 25E0 -5E0 159E-1 2t0 2 2 2 2 9 6 ZQOOOjb 2COC03B 171564B 1 1 2 20000.B 2G00GL'B

l7156''tB I I 2 20000^8 20000.B I71'564B I 1 2 200G0'.H 2000G'.B 171564B 1 1 2 20000. B 200000B 171S64B 1 I 9 CZ = ADDZ INPUT BZ C+I GZ

INCREMENT ADDRESS UF TABLE

END OF TABLE TPY NEXT WORD NO MATCH IN TABLE N'ATCH FOUND

I N I T I A L POT PUT PUT POT PUT PUT Pl;T POT PCT PCT PUT P u l PUT PUT PCT PUT P(1T PUT PCT PUT PUT PLT Pf iT PL,T POT PUT

Pf;T PCT PCT PUT PUT SET

9 -lb 16 17 18 19 20 22 23 24 25 26 21 29 3u 31 32 33 34 36 37 38 39 40 41, 43 44 45 46 47 48 UP

CONDITION 14 = .5 = .5

21

POTS

,28

35

42

- 56 PARAMETERS

27

TABLE I (Contd.)

u

H

LA

Rl

R2

R3

R4

R5

R6

R7

R8

BA

DZ

PF

KC

DHF

DRF

PCO

DRC

PCP

INPUT

JMP LAW JMP LAW JMP LAW STA JMS JMP LAW JMP LAW JMP LAW JMP LAW JMP LAW JMP LAW JMP LAW JMP LAW JMP LAW JMP LAW JMP LAW JMP LAW JMP LAW STA JMS JMS JMP LAW JMP LAW JMP JMS SAD JMP LAW JMP LAW JMP JMS SAD JMP LAW JMP JMS SAD JMP LAW JMP LAW JMP JMS

C+I UZ C + 1 HZ C + 1 LAZ = ADOZ RCHAR INPUT RIZ LA + 1 R2Z LA+l R3Z LA-t-l R4Z LA + 1 R5Z LA+I R6Z LA + l R7Z LA+l R8Z LA+1 BAZ LA + l DZZ LA + 1 PFZ LA + 1 KCZ LA + 1 DhFZ = ADDZ RCHAR RCHAR INPUT DRFZ DHF + 1 PCOZ DHF + 1 RCHAR =0002^58 • •3 DRCOZ DHF + 1 ORCZ LA + 1 RCHAR =0003030 * + 3 PCPFZ DHF + 1 RCHAK =0U02^5B * + 3 PCPCOZ DHF + 1 PCPCZ LA+1 RFDEC

COMPARE TO =

CUMPARE TO C

CUMPARE rU =

TYPE FLOATING NUMBER INTO F. AC

28

TABLE I (Contd.)

ICO

ICl

IC2

IC3

ICA

IC5

IC6

IC7

IC8

KEYT

LAC STA» ISZ LAC STA* JMP LAW JMP LAW JMP LAW JMP LAW JMP LAW JMP LAW JMP LAW JMP LAW JMP LAW JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP

= FACt = ADDZ = AODZ = FACF = AOOZ KEYS ICOZ DHF + 1 ICIZ OHF + 1 £C2Z DHF + 1 IC3Z DhF + 1 IC4Z DHF + 1 IC5Z DHF + 1 IC6Z DHF + 1 IC7Z DHF + 1 IC8Z DHF + 1 C37540 C 02 7540 B 07754U G 257540 U 10754u H 140175 LA 226175 Rl 226275 R2 226375 R3 226475 R4 226575 R5 226675 R6 226775 R7 227075 R8 020175 BA 043275 DZ 200675 PF 130375 KC 041006 OHF

GET NEXT PARAMETER

29

TABLE I (Contd.)

cz BZ LAZ RIZ R2Z R3Z R4Z R5Z R6Z R7Z RfcZ DRFZ DRCZ DRCOZ BAZ GZ DZZ PFZ PCPFZ UZ PCPCZ KCZ HZ PCOZ PCPCOZ DhFZ ICOZ ICIZ IC2Z IC3Z IC4Z IC5Z IC6Z IC7Z

OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP OCT JMP RES RES RES RES RES RES RES RES RES RES RES RES RES RES RFS RES RES RES RES RES RES RFS RES RES RES RES RES RES RES RES RES RES RES RES

042206 DRF 042203 DRC 2C0360 PCO 20032? PCP 071717 GOO 110360 ICO 110361 ICl 110362 IC2 110363 IC3 110364 IC4 110365 IC5 110366 IC6 110367 IC7 110370 IC8 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

30

TABLE I (Contd.)

RES FLAC FMPY FSTA FMPY UNFLTF FLAC FMPY UNFLTF FLAC FMPY UNFLTF FLAC FMPY UNFLTF FLAC FMPY UNFLTF FLAC FMPY UNFLTF FLAC FMPY UNFLTF FLAC FMPY UNFLTF FLAC FMPY UNFLTF FLAC FMPY FSTA FMPY FSTA FLAC FOIV UNFLTF LAC DAC DAC DAC DAC DAC FLAC FOIV UNFLTF LAC DAC DAC DAC DAC FLAC FDIV FDIV UNFLTF LAC DAC DAC DAC DAC FLAC

2 CI BZ FTEMPl ICOZ POTT FTEMPl ICIZ POTT+1 FTEMPl IC2Z POTT+2 FTEMPl IC3Z POTT+3 FTEMPl IC4Z POTT+4 FTEMPl IC5Z POTT+5 FTEMPl IC6Z POTT+6 FTEMPl IC7Z POTT+7 FTEMPl 1C8Z PQTT+8 LAZ PCPFZ FTEMPl CZ FTEMP2 BZ FTEMP2 POTT+9 POTT+9 POTT+IO PCTT+ll POTT+12 POTT+13 PUTT+14 C2 BZ POTT+18 POTT+IS POTT+25 POTT+32 POTT+39 POTT+46 C3 BZ BZ POTT+19 POTT+19 POTr+26 POTT+33 POTT+40 POTT+47 FTEMPl

TABLE I (Contd.)

FMPY FMPY FSTA FMPY FSTA FLAC FMPY FSTA FADD FOIV FMPY UNFLTF FLAC FSUB FMPY FDIV FDIV UNFLTF FLAC FADD FDIV FDIV FMPY UNFLTF FLAC FSUB FMPY FDIV FDIV UNFLTF FLAC FADD FMPY FDIV FDIV UNFLTF FLAC FSUB FMPY FCIV FDIV UNFLTF FLAC FADD FMPY FCIV FCIV UNFLTF FLAC FSUB FMPY FDIV FCIV UNFLTF FLAC FADD FMPY FDIV FCIV UNFLTF FLAC FMPY

DRFZ ORFZ FTEMP2 RIZ FTEMP3 ORFZ C4 FTEMP4 RIZ FTEMP3 C5 P0TT+20 R2Z FTEMP4 C5 FTEMP2 R2Z POTT+21 R2Z FTEMP4 FTEMP2 R2Z C5 POTT+27 R3Z FTEMP4 C5 FTEMP2 R3Z POTT+28 R3Z FTEMP4 C5 FTEMP2 R3Z PQTT+34 R4Z FTEMP4 C5 FTEMP2 R4Z POTT+35 R4Z FTEMP4 C5 FTEMP2 R4Z POTT+41 R5Z FTEMP4 C5 FTEMP2 R5Z POTT+42 R5Z FTEMP4 C5 FTEMP2 R5Z POTT+48 DRFZ C7

TABLE I (Contd.)

FADD FSTA FLAC FSUB FMPY FCIV FDIV UNFLTF FLAC FMPY FMPY FMPY FDIV FDIV UNFLTF FLAC FMPY FADD FSTA FLAC FDIV FCIV FDIV FDIV FSTA FDIV FSTA FMPY FMPY FMPY UNFLTF FLAC FMPY FADD FMPY FMPY UNFLTF FLAC FMPY FADD FSTA FLAC FDIV FSTA FLAC FMPY FADD FMPY FMPY UNFLTF FLAC FMPY FMPY FMPY FMPY UNFLTF FLAC FMPY FADD FSTA FLAC FMPY

R6Z FrEMP3 R6Z FTEMP4 Cll FTEMP2 FTEMP3 POTT+49 C6 R6Z DRFZ UZ FTfcMP2 FTEMP3 PCTT+50 DRCZ C6 R7Z FTEMP3 C6 LAZ PCPCZ DRCZ DRCZ FTfcMP2 FTEMP3 FTEMP3 UZ R7Z DRCZ POTT+51 DRCZ C4 R7Z KCZ FTEMP3 POTT+52 DRCZ C7 R8Z FTEMP3 FTEMP2 FTEMP3 FTEMP3 DRCZ C9 R8Z KCZ FTEMP3 POrT+53 HZ R8Z DRCZ FTEMP3 C5 POTT+54 ORCCZ C4 R8Z FTEMP3 HZ C5

33

TABLE I (Contd.)

FMPY FDIV FDIV FDIV FCIV UNFLTF FLAC FDIV FMPY FDIV FDIV FDIV FDIV UNFLTF LAM STA LAW STA OZM

NXTPOT LAC* STA LAC STA JMS

CALLS OCT RtS ISZ JMP JMP ISZ ISZ LAC AND SAD JMP JMP

BCD LAC AND ADD DAC JMP

CCiNT DZM ICRTC RRTC CIDAC LAC DAC LAC DAC LAC DAC LAW AND DAC DAC ION JMP

ORDA JMP URDB LAC ORDC JMP OROC JMP

JMS

R8Z FTEMP3 LAZ PCPCOZ DRCOZ POTT+55 GZ FTEMPJ CIO LAZ DZZ PCOZ DRCOZ POTT+56 -5/ = KEYe POTT =POTA0 =POTNU =POFAD CALLS+1 =POTNU CALLS+2 POTSX 777377B 2 = KbY£ • + 2 CONT =PUTAD =POTNU =0 000178 =POTNU = 10 BCD NXTPOT =POTNU =7777608 =0000208 =POTNU NXTPOT = DTA

= DTA ORDA 00 1 URDC FRED SAMPCT =CT1 LISTA =C17777B =LISTA = BINB

GO TEST = ATD PAST TESTB ADP

SFT UP COUNT OF POTS TO Bfc SEI

SLT UP ADDRESS OF POT VALUE SET UP POT NUMBER

GET VALUE

GET POT NUMBER ENTER PGT SET ROUTINE CONSOLE A CALLING SEQUENCE SFT ALL POTS -

INCREMENT POT VALUE ADDRESS INCREMENT POT NUMBER

34

TABLE I (Contd.)

PCTSX

P0TS4

ORD POTSl PCTS2 PLTS3 TYPE

TYPEE

RES lOF LAC* AND STA ISZ LAC* STA C1DA4 LAC STA ISZ JMP CLDA ISZ LAC* XOR RTCO LAC STA LAC STA LAC STA ION ISZ JMP ISZ JMP CLAP lUF LAC RTCO LAC STA JMS LAC* JMS ISZ JMP LAC DAC ISZ JMP CLAP ISZ JMP* JMP RES RES RES LAM DAC LAW DAC JMS DZM LAC JMS LAC JMS LAC*

1

*-2 =2004008 P0TS3 *-5 *-6 POTSl POTSl =7777u0B P0TS2 P0TS2 *-l

POTSX POTSX P0TS3

=4000008 P0TS2 =7777408 POTSl ORD 001

P0TS2 •-1 POTSl •-3

=2010008

=4000006 P0TS2 TCR PCTSX TDEC P0TS2 *-l =4000006 P0TS2 PCTS2 • -1

POTSX POTSX PGTS4 1 I 1 -57 =TYPEC POTT =TYPEA TCR =TYPEP =TYPEP TDEC =4040406 TWORD =TYPEA

PUT SET SUBROUTINE INTERRUPT OFF GET CONSOLE A OR B CODE

GET VALUE TO BE SET TEMP STCRE

CLEAR INTERRUPT

GET POT NUMBER

PUT SET

INTR ON

INTERRUPT OFF

CLEAR PCT ORDER

T A B L E I (Contd.)

SAMPCr

L ISTA

JMS JMS ISZ ISZ ISZ JMP JMP DEC SUBR L ITLS RES JMP END

2. D e t a i l s o f Op

IFRAC TCR =TYPEP =TYPEA =TYPEC TYPEE ADP 76422JB

1 ADP

e r a t i o n

Once the p r o g r a m h a s b e e n l o a d e d , c o n t r o l i s t r a n s f e r r e d to A D P . While the m a i n p r o g r a n n i s u n d e r c o n t r o l of A D P , we type 105G to e n t e r th i s p r o g r a m a t s y m b o l i c l o c a t i o n S T A R T . Th i s s e t s up the d ig i ta l p a r t of the hyb r id c o m p u t e r p r o g r a m and effects a p a u s e with the keyboa rd r e q u e s t a c t i v e . Val id p a r a m e t e r n a m e s (see T a b l e II) and t h e i r s igned f loa t ing-po in t v a l u e s can now be e n t e r e d t h r o u g h the k e y b o a r d . The p r o g r a m s t a y s in t h i s loop unt i l the e x p e r i m e n t e r w i s h e s to p r o c e e d with the s i m u l a t i o n . The m n e m o n i c GOO is e n t e r e d t h r o u g h the k e y b o a r d , and the p r o g r a m p r o c e e d s to s y m b o l i c l o c a t i o n GOO. P o t e n t i o m e t e r v a l u e s a r e c a l cu l a t ed us ing the p r o g r a m m e d a l g o r i t h m s and the p a r a m e t e r s tha t w e r e e n t e r e d by the e x ­p e r i m e n t e r . T h e s e v a l u e s a r e s t o r e d a s f ixed-poin t b i n a r y n u m b e r s in a t ab l e in the p r o g r a n n s t a r t i n g a t s y m b o l i c loca t ion P O T T .

TABLE II. Valid P a r a m e t e r Names

System P a r a m e t e r Typed P a r a m e t e r Scale F a c t o r s Typed P a r a m e t e r s

G U h r i

TE

^3

1-4

rs ^6 r? rs A Az

Pi Kc '^ ^fusion Arf

Pco A r , pCp

(pCp)c (pCp)co

G U H R l

R2 R3 R 4 R 5 R 6 R 7 R 8 B A D Z P F K C D H F D R F P C O D R C P C P DRCO P C P C PCPCO

a b c

Initial Condition

F i r s t fuel segment Second fuel segment Thi rd fuel segment Four th fuel segment Fifth fuel segnnent Final edge Homogenized bond and

cladding Edge of cladding Coolant t empe ra tu re

L A B C

ICO I C l IC2 I C 3 I C 4 I C 5

IC6 I C 7 I C 8

The automatic po ten t iometer - se t t ing routine is entered, and the s e r v o - s e t po ten t iometers of the analog computer a r e set, via the hybrid o r d e r s , to the values s tored in the POTT table. If a potent iometer fails to set , an e r r o r message is indicated for it, descr ibing the potent iometer number and the value to which it should be set.

The p r o g r a m proceeds to symbolic location CONT, and the hybrid sys tem is r e s t a r t e d . A D- to-A conversion is made on channel 0, which r e p r e s e n t s the initial inlet t e m p e r a t u r e . The p r o g r a m then proceeds to synnbolic location GO, where two checks a r e made on the analog console:

1. Should the p r o g r a m proceed , or should it continue to loop and check on the two condit ions.

2. Should the p r o g r a m r e s t a r t at location CONT, or should it con­tinue to loop and check on the two conditions.

The two conditions a r e controlled by two switches , on the analog console, which a re under manual control of the opera tor .

If the p r o g r a m is to proceed , it b ranches to location SIG. An i t e r a ­tion is per formed using a p rede te rmined c lock-sampl ing ra te and total sample count. On each "beat" of the clock, an A-to-D conversion is made on the change in outlet coolant t e m p e r a t u r e , and the converted data a r e s tored in core in a l is t . Simultaneously, a D- to-A conversion is made on the inlet-coolant- tennperature change, which was stored in the outlet-coolant t empera tu re change l i s t during the previous run. The f i rs t i tera t ion is made using zero as the in l e t - coo lan t - t empera tu re change.

At the end of an i tera t ion, the p r o g r a m re tu rns to GO, where the continue and r e s t a r t switches a r e tes ted. I tera t ions a r e continued along the axial length until the maximum number of axial segments have been i te ra ted a n d / o r a r e s t a r t is signaled. As noted before, t e m p e r a t u r e s of in te res t to the exper imente r a r e plotted by X-Y r e c o r d e r s during each i tera t ion. Thus we achieve the radia l and axial t empera tu re profi les of our fuel cell .

APPENDIX B

Special F e a t u r e s Allowed by ADP

The use of ADP in conjunction with the p r o g r a m allows cer ta in special functions to be per formed. ADP may be entered at any time by manually s tar t ing the computer at 1637 (octal).

ADP allows the inser t ion of up to eight breakpoints at any des i red point in the p r o g r a m . They a r e inser ted by typing ZZZBX, where X is the breakpoint number and 0 ^ X ^ 7, while the sys tem is under control of ADP. ZZZ (octal) is the m e m o r y add re s s where the breakpoint is to be inser ted . When the main p r o g r a m is executed and proceeds to ZZZ, a branch back to ADP will be effected. Any ADP function can now be p e r ­formed, including enter ing a C via the typewri ter keyboard; this causes the main p r o g r a m to continue execution at the point where the breakpoint occur red .

Frequent ly it is des i rab le to know the value of the potentiometer set t ings. While the sys tem is under ADP control , a routine at location TYPE may be entered by typing 1614G. The potent iometer numbers and corresponding values will be typed out in decimal form. Upon completion, control is re turned to ADP.

To re l ieve the opera tor of the task of entering the p a r a m e t e r s every t ime the p r o g r a m is to be reloaded for subsequent runs , the p a r a m e t e r s may be punched out from core m e m o r y onto paper tape. The paper tape can then be loaded after the main p r o g r a m is loaded. The paper tape is punched out under control of ADP by typing 522P106P. The paper tape is read in under computer manual load control s tar t ing at address 522 (octal). Any or all p a r a m e t e r s may then be modified through the typewri ter keyboard.

The clock, which controls the sampling rate of the conversion equip­ment , is p r o g r a m m e d to sample every 10 m s e c . Six thousand samples a r e taken per i tera t ion. The clock t ime can be var ied in mill isecond increments by storing the octal b inary number in mil l i seconds in location 73 octal. This is a lso facilitated by using ADP. For example, 73/xXXXXX000017 will change the value of XXXXXX to 15 m s e c

REFERENCES

Dusinberre, G. M., Heat Transfer Calculation by Finite Differences^ International Text Book Co., Scranton (1961).

Hedge, J. C , and Fieldhouse, J. B., Measurement of Thermal Conduc­tivity of Uranium Dioxide^ AECU-3381 (1956)„

Kingery, W. D., et al. ^ The Measurement of Thermal Conductivity of Refractory Materials^ NYO-3647 (1953).

Reiswig, R. D., Thermal Conductivity of Uranium Dioxide to 2100°C^ J. Am. Ceram. Sec. Ze (1961), p. 48,

Chikalla, T. D., The Liquids for the System UO2-PUO2, HW-69832 (1961).

Dickerman, C. E., Robinson, L. E., Sowa, E. S., and Skladzien, S„ B., Studies of Fast Reactor Fuel Element Behavior under Transient Heating to Failurei II. In-pile Experiments on UO2 Samples in the Absence of Coolant^ ANL-6845 (Jan 1965), pp. 34-36.

Engineering Materials Handbook^ Table 5-4 (1958 ed,), pp. 5-12.

See, e.g.. Dunning, E. L., The Thermodynamic and Transport Properties of Sodium and Sodium Vapor^ ANL-6246 (Oct 1960).

Amiot, L. W., et al., The Argonne Hybrid Computer Maintenance Manual^ Applied Mathematics Division Technical Memorandum No. 115 (1965) (unpublished).

Sanathanan, C. K., et al.j The Application of a Hybrid Computer to the Analysis of Transient Phenomena in a Fast Reactor Core^ Trans. Am. Nuc. Soc. 9,1 (June 1966), p. 272 (abstract).

Clark, R. K., and Hodges, D., An Assembly Program for the PDP-7 Computer Executed on a CDC-SdOO^ Applied Mathematics Division Technical Memorandum No. 102 (Jan 1966) (unpublished).