University of Tennessee, Knoxville
Trace: Tennessee Research and Creative
Exchange
Doctoral Dissertations
Graduate School
8-1976
Lumped Parameter, State Variable Dynamic
Models for U-tube Recirculation Type Nuclear
Steam Generators
Mohamed Rabie Ahmed Ali
University of Tennessee - Knoxville
Recommended Citation
Ali, Mohamed Rabie Ahmed, "Lumped Parameter, State Variable Dynamic Models for U-tube Recirculation Type Nuclear Steam
Generators. " PhD diss., University of Tennessee, 1976.
http://trace.tennessee.edu/utk_graddiss/2548
This Dissertation is brought to you for free and open access by the Graduate School at Trace: Tennessee Research and Creative Exchange. It has been
accepted for inclusion in Doctoral Dissertations by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more
information, please contact [email protected].
To the Graduate Council:
I am submitting herewith a dissertation written by Mohamed Rabie Ahmed Ali entitled "Lumped
Parameter, State Variable Dynamic Models for U-tube Recirculation Type Nuclear Steam Generators." I
have examined the final electronic copy of this dissertation for form and content and recommend that it
be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a
major in Nuclear Engineering.
T. W. Kerlin, Major Professor
We have read this dissertation and recommend its acceptance:
P. F. Pasqua, J. C. Robinson, Kike Kiser, Jivan Thakkar
Accepted for the Council:
Dixie L. Thompson
Vice Provost and Dean of the Graduate School
(Original signatures are on file with official student records.)
July
9, 1976
To t h e G ra d u a t e C o u nc i l :
I am s u bm i t t i n g herewi t h a d i s s er ta t i o n wr i t t e n by Mo hamed
Ra b i e A hm e d Al i E n t i tl ed 11 L umped P a rameter , Sta t e V a r i a b l e Dynam i c
t·� od el s for U - t u be R e c i r c u l at i o n Typ e N uc l ea r S t eam G e n erators . "
I r ecomm e n d t ha t i t b e accep ted i n p a r t i a l ful f i l l me n t o f t h e
req u i remen t s fo r t h e d egree o f Doctor o f P h i l o s o p hy , w i t h a ma j o r
in Nucl ear E n g i neeri n g .
T . W. Kerl i n , Maj o r P rofe s s o r
We
rta t i o n
ta n c e:
Accep ted fo r t he Co u n c i l :
V i ce Chancel l or
G ra d u a t e S tu d i e s a n d R e s ea r c h
·'
.....
../
,.·� -
....,.:
.....
,_.:
L UM P E D PARAMETE R , STATE VARI A B L E DYNAMI C MO DELS
FOR U-TUBE RE C I RCULATION TYP E
N UC L EA R STEAt� GEN E RATORS
A D i s s e rta t i o n
P re se n t e d fo r t h e
Doc t o r o f P h i l o so p hy
De gree
The Un i ve r s i ty of Tenne s se e , Kno x v i l l e
Mo hame d Ra b i e Ahme d Al i
August
1976
1300528
ACKN OWL E D GE ME NTS
T h e a u t h o r wo u l d l i k e to t h a n k D r . T . W . Ke rl i n , t h e maj o r
) ro fe s s o r a n d t he re s e a rc h a dv i s o r , fo r h i s pat i en ce i n s mo o t h i n g
� h e ro u g h s po t s d u ri n g t he co u rs e o f t h i s wo rk .
H i s g u i d a n ce
i n d e n cou ra geme n t toge t h e r w i t h the val u a b l e s u gge s t i o n s o f t h e
lthe r membe r s o f t h e do cto ral commi t tee c o n t ri b uted t o t h e
; u c ce s s fu l c ompl e t i o n o f t h i s d i s s e rt at i on .
,f
H e wi s he s to e x p res s h i s g rat i t u de t o D r . P . F . P as q u a , h e a d
t h e N ucl e a r E n g i n e e ri n g Depa rtme n t fo r ma k i n g t he compl e t i o n o f
: h e res i dent req u i reme n t s fo r t h e P h . D . p rogram po s s i ble t h ro u gh a
r ra d u a t e a s s i s ta n ts h i p .
T he a s s i s t a n c e o f M s s rs . Mi ke K i s e r a n d J i van T h a k k a r i n
. h e c k i n g t h e s te a dy s t a t e c a l c u l a t i o n a l go ri thm a n d co upl i n g t h e
t e a m genera t o r a n d rea ct o r mode l s i s dee p l y a pp reci ated .
Many t ha n ks go to Mrs . J ua n i ta Rye fo r he r e ffo rt i n ty p i n g
h e ro u g h d r a ft s and t h e fi n a l fo rm o f t h e di s s e rtat i o n re p o rt .
T h e u s e o f The U n i vers i ty o f Ten n e s s ee Comput i n g C e n t e r
a c i l i t i es i s h i g h l y a c knowl e d ge d .
ii
ABSTRACT
A n umber of
1
umped paramet e r dynamic model s were d evelo ped for
a ver t i cal U-tube reci rcu l ati on steam g enerato r ( UTS G ) of the type
u s ed i n most pre s s u rized water reacto r n ucl ear steam supply systems.
.
T h e mode l s ran ged i n compl exity f rom a simplifie d model ( Model
A) wi th onl y three l umps to re present th e primary fl uid ( reactor
coo l ant ) , t u be me tal and the s econ dary f l uid to a d etail e d model
( Model D) with fo u rteen l umps an d a movin g boun dary between the
subcool ed and boil i n g s ections of the lleat exchange region.
The mode ls
are l in ear and are in the state variable fo rm which is con venie n t for
u sin g standard general purpo s e computer cod es for time or fre que ncy
domain anal y sis.
Th e adequacy of the mode l s was t ested in several ways.
The
cal c u l ated r e s pon s e f rom the models was s t u died fo r physical
pl au s i b i lity.
The adequacy of the anal ytical work was t e s t e d by
comparin g re s u l t s from pro gre s sivel y more detail ed models.
The
detail ed model re s pon s e was compared wit h the result s of other UTS G
dynamic mod els
Ro bin son (739
( 6,32) and with t e s t results obtain ed at the H . B .
MWe PWR ) pl an t . (3,4,S)
Al l of the ch eck s on model validity demon strated the adequacy
of the l umped parameter approach fo r the simul ation of the dynamic
res pon s e of a UTS G fo r norma l operatin g t ran sie n t s in volvin g small
deviation s from the steady state con ditio n s.
iii
TABLE OF CONTENTS
CHAPTER
I.
PAGE
I NTRODUCT I ON . . . . . .
Gen eral Con s i derat i on s . .
I I.
Descr i p t i on of the Phys i cal Sys t em
3
S tatemen t of the Problem . .
8
I m portance of the S tudy
8
Scope and Organ i zation of the Text .
9
REV I EW OF PERTINENT L I TERATURE .
. 11
I ntroduct i on .
11
Dynam i c Models for Heat Exchang er s and Fos s i l
Fueled Bo i lers
.
.
.
.
.
.
.
.
.
. 11
•
•
Dynami c Models for Bo i li ng Water Reactor ( BWR )
Sys terns . .
14
Dynam i c Models for Reci rculat i on Typ e Nuclear
S t eam Gen erators
. 18
Remarks
I I I.
DEVELOPI1ENT OF THE r1ATHEMAT I CAL MODELS
•
In troduct i on . . . .
•
24
•
25
•
Bas i c As sumpt i on s
•
S tructure of the Mathemat i cal Models .
.
25
25
26
S i mpli f i ed S team Ge n erat i ng Sys tem Model
( Model A )
.
.
•
•
.
F i rs t I n t ermed iate t�odel
iv
.
. 27
.
( Model B )
.
. . 30
v
:HAPTER
PAGE
S eco n d I n t e rmed i a te Mo del ( Mo d e l C )
30
The Oeta i l ed t1o d e 1 ( r1o de 1 D )
33
Go v ern i ng E q u a t i on s fo r Mo d e l A
35
P ri mary L ump ( PR L )
35
T u b e Met a l L um p ( MT L )
36
S econ da ry Fl u i d L u�p ( S FL )
37
S umma ry o f t h e t1o d e l E q ua t i on s
42
Go v ern i n g Equat i on s fo r Mo del B
43
P r i mary L um p ( P RLl ) .
43
P r i ma ry L um p ( PRL2 ) .
43
T u b e Meta l L ump ( MTL l )
44
Tube Meta l L ump ( MTL2 )
44
Secondary F l u i d L um p ( S FL )
44
S ummary of t he t�od el E q ua t i o n s
46
Go v ern i ng E qua t i o n s fo r t1o d e l C
48
Gen era 1 . . . . . .
48
P r i ma ry L ump ( P R L l )
48
P r i ma ry L um p ( P RL 2)
49
T u b e Meta 1 L ump ( MTL 1 )
49
T u b e Meta l L ump ( MTL 2 )
49
E ffec t i v e H ea t E x c ha n g e S econ d a ry F l u i d
L ump ( SFEH E L )
. . . . . . . . . . .
D ru m Equ i va l ent S e con dary F l u i d L ump (SFORL)
49
56
vi
CHAPTER
PAGE
Down comer Secondary Fl u i d L um p ( S FDCL )
.
Th e R ec i rc ul a t i on Loop E q ua t i on
•
. 65
S umma ry o f the Model E q ua t i o n s .
69
Go v ern i n g E q ua t i on s fo r Mod e l D
74
P r i ma ry S i d e E q ua t i on s
. 74
Tu b e Metal E quat i o n s .
. . 83
Sec o n d a ry S i d e E quat i on s .
87
The Re c i rcu l a t i o n Loop E q uat i o n
•
Summ a ry o f t h e Mo del Equa t i on s .
IV.
64
95
97
CALCULAT I ON P ROCEDURE AND RESULTS .
. 106
I n t ro d u ct i on
. 106
Cal c u l a t i on P rocedure fo r the P u re D i ffer e n ti a l
Formu l a t i o n
.
.
.
.
.
. .
.
107
Ca l cul a t i on P ro c ed u re fo r t h e Mi xed D i fferen t i a l
and A l g ebra i c Fo rmu l a t i on
Ti me R e s po n s e Cal c u l a t i o n
.111
. . . .
.
F r eq u e n cy R e s pon s e Ca l c u l a t i on . .
112
.114
Geomet r i ca l Cal c u l a t i on
P r i rna ry S i d e .
Tube t·1 eta 1 .
Sec o n d a ry S i d e
•
•
.
.
•
115
116
. .118
o
. . . 119
Res u l t s fo r H . B . Ro bi n so n S t eam Genera to r
Ca l c ul a tion o f H ea t Tra n s fer C o effi c i en ts .
F i l m H e a t Tra n s fer Co e ff i c i en t s
. . 121
.
•
•
1 22
. 1 23
vi i
riAP T E R
PAGE
Tube Metal Con ductance
.
.
•
.
.
•
.
•
1 25
O v e ra l l Heat T ran s fe r Coe ffi c i e n t s
1 25
Comb i ne d F i l m a n d T u be Me t a l Con d uc ta n ces
1 26
Res u l t s fo r H . B . Rob i nson Steam Gene rato r
.
•
S t e a dy S t a t e C a l cul at i ons
. 1 29
1 29
I n tro d u c t i o n .
1 29
Sol u t i o n U s i n g t he F i n i te D i ffe re nce A p p ro a c h
1 32
So l ut i o n U s i n g Lo ga ri thmi c Mean Tempe rat u re
D i ffe ren ce ( U1TO)
•
•
1 35
•
Res u l t s fo r H . B . Ro b i nson S team Ge n e rato r . . . 1 3 7
Gene ral Rema r k s fo r Dynami c Re s po n s e Cal cu l a t io n s
1 39
Dyn a m i c Res po n s e Res u l t s
1 41
. .
Re s po n s e o f �1ode 1 A
V.
.
.
1 41
Res pon s e o f Mode l B
1 43
Re s pon s e o f �1o d e l
1 46
c
Re s po n s e o f Mode 1 D
1 51
Comp a ri son o f t1ode 1 s Res po n s e s .
1 58
COfv1PARI SOrl O F DETA I LE D t10DE L PRE O I CT I OiJS \�I TH
160
E X P E R I M E NTAL RE SULTS
I nt roduc t i o n
•
.
.
160
•
De s c ri p t i o n of t hP. Sys tem .
160
P WR t•lo de 1 .
1 61
Co u p l i n g o f Reacto r a n d Stea� Ge n e rato r Mo de l s
167
Res u l ts
1 70
•
.
.
.
.
.
.
.
vi i i
HAPTER
VI .
PAGE
COMPARISON O F D E TA I L E D MODELS W I TH OTH E R UTSG
MODELS
.
.
.
1 83
.
I n t roduct i on
1 83
Comp a r i s o n w i t h C h r i sten sen 1 s UTSG Mo del
1 84
1 84
System De s c r i pt i on
Model D e sc r i pt i o n
1 34
•
1 86
Compa r i son o f Model Formu l a t i on
Compa r i son o f Dyn am i c Res po n s e s
•
•
.
.
Compa r i s o n w i t h A rwood1 s I E UTSG Model
VII .
1 88
1 88
Desc r i p t i on of t h e IEUTSG System
1 90
Compa r i son of Dyn a mi c Re s pon s e s . .
1 92
REt�ARKS AN D CGriCLUSIONS
•
.
•
.
.
.
1 95
E va l uat i on o f t h e Devel o pmen t P ro c edure
1 95
Po s s i bl e U s es of the C u r ren t Work
1 96
Re commenda t i o n s
1 97
fo r F u t u re Wo rk
1 93
Con c l u s io n s
I ST O F REFERENCES .
1 99
PPEIW I XES .
205
A.
NARROW RAtiGE LINEAR APPROX IMATIGrl OF SATURATION,
��AT E R
B.
C.
ANO
TH E CRITICAL FLO\� AS SUMPT I On .
BAS I C C Ofi SE RVAT I Oil E QU AT I O t l S FOR TWO PHASE FLO\J
SYSTEI� S
D.
STEA11 P ROPE RTI ES
.
.
•
.
. . . . .
RECIRCULATION LOOP EQUAT I ON
206
21 6
218
228
ix
PAGE
: HAPTE R
E.
AVERAGE DENS I TY I N TH E TWO - PHASE RE G I ON
F.
S U MMARY OF DY NArHC RESPONSE RESULTS
G.
I NSTRUC T I ONS FOR US I NG TH E DETA I LE D MO D E L DY NAM I C
RES PONSE COMPUT E R PROGRAM
V I TA
.
.
.
.
.
. 2 32
. .
. 2 36
. . 271
280
L I ST O F TAB L E S
PAGE
\BL E
[1 . 1
Sys tem Vari a b l e s fo r Mode l C
[1 . 2
Sys tem Vari a b l e s f o r the De tai l e d Mo de l (Model D )
[V. l
H. B. Rob i n s o n Steam Ge ne ra t o r De s i gn Data
[V
.
2
[V . 3
v. 1
V.2
•
•
•
.
•
•
.
Mod e l
c
.
98
1 17
. .
.
. .
.
. .
.
. .
.
.
.
.
.
.
.
.
•
1 47
0
.
0
.
.
1 52
.
?
.
.
. 1 63
Non z e ro E l eme n t s o f t h e t1at r i ce s Al and A2 fo r
Mo de l
0
0
0
0
.
0
0
.
.
De f i n i t i on o f t h e State V a r i ab l e s U s e d i n t h e
Reac t o r Mo de l .
0
0
. .
.
. .
0
.
0
.
.
.
.
Nonz e ro E l eme n t s o f t h e A Mat r i x fo r the Re a c t o r
0
0
. .
0
.
.
.
.
.
.
1 64
.
.
1 66
.
.
1 69
•
•
1 71
.
N on z e ro E l e me n t s o f t h e Fo rc i n g Vecto r fo r the Re a c t o r
�1o de l . .
0
. .
.
.
.
.
.
.
.
.
.
.
V.4
Sy s tem Vari a b l e s fo r the Ste am Ge ne rato r Mo de l
Vo 5
Fo r c i n g Funct i o n fo r S team V a l ve P e rt u rba t i o n
( + 1 0% Chan ge i n Ste am Fl ow Rate )
F. l
70
•
N on z e ro E l eme n t s o f t h e r�at ri ce s A l a n d A2 fo r
Mod e l .
V. 3
•
S umma ry o f Dyn amic Re s po n s e Re s u l t s
X
.
0
?
. 237
L I ST O F F I GURES
: I GURE
PAGE
1. 1 .
P re s s uri z e d W a te r Re a c t o r Coo l a n t Loop
1 . 2.
D e t ai l s o f a Ve rt i c a l U-T ube Re c i rc u l a t i o n Steam
Ge ne rator
1 . 3.
•
2
•
( UTSG )
4
S ch e ma t i c Di agram fo r a Ve rt i c a l U-T ube Re c i rcul a t i o n
( UTS G )
5
I I . l.
S i mp l i fi e d Steam Ge ne ra t o r System fo r t1o de l A
28
I ! . 2.
S c h e ma t i c D i a g ram for Model A
29
I I . 3.
S c hemat i c Di a g ram fo r Mode l
8
31
I I . 4.
S ch emat i c Di a g ram fo r Mo del
II.5.
S c h e ma t i c Di a gram fo r Mo de l D
II.6.
S e co n da ry L umps fo r t1o de l C
I I . 7.
V a r i a t i on o f T he rmo dyn ami c P ro pe rt i es o f t h e
Steam Ge n e rator
c
32
34
- 50
S e co n d a ry Fl u i d i n t h e E ffe ct i ve H e a t E x c h a n ge
53
L ump
II .8.
T h e D rum E q u i val e n t Secondary Fl u i d L ump
•
57
IV. l .
Comb i n i n g Fi l m a n d T ube Me tal Con d u ct an ce s .
. . . 128
I V . 2.
S c h emat i c o f t h e E ffe c t i ve H e a t E xc h a n g e Re g i o n
( Core )
IV. 3.
.
. . . . . .
.
.
•
. . . . .
.
•
.
Tempe rat ures a n d Ma s s Q u a l i ty P ro fi l e s Ca l c u l ated by
, 1 38
t h e Steady St ate P ro g ram
IV.4.
1 31
Ste p Re s po n s e o f f�ode l
V a l ve Coeffi c i e n t
xi
•
A
to + 1 0% Cha n ge i n Steam
•
.
•
1 42
xi i
;uRE
I. 5 .
PAGE
Ste p Res po n s e o f Mode l B to +1 0% C h a n g e i n Ste am
Va l ve Coe ff i c i e n t
I. 6.
{. 7 .
Va l ve Coe ffi c i e n t
•
1.2.
1.3.
1.4.
,.5.
Compa ri s o n o f T
.
. .
.
.
.
.
.
.
•
.
.
.
. . .
.
.
.
.
.
.
•
•
.
.
•
s
.
0
•
•
•
•
•
•
0
•
•
1 59
•
•
1 62
1 48
1 53
Re s po n s es i n t h e Fo ur
.
. . .
.
1 68
Lumpe d Struc t u re fo r a PWR Sys tem Mo d e l
Re acto r Powe r/ Rea c t i v i ty ( o P/ P / o p ) F reque n cy
0
Re s po n s e
•
•
•
•
•
•
•
•
•
.
•
•
•
•
•
.
•
.
1 72
.
•
•
•
•
•
1 74
•
•
•
0
176
•
0
•
•
Steam P re s s u re/ Rea c t i v i ty ( o P /o p ) F re q uency
s
Re s po n s e
•
•
•
•
•
•
Rea cto r Powe r/ Ste am Fl ow ( o P/ P / o W s ) Fre q ue n cy
0
0
Re s po n s e
•
•
.
Steam P re s s u re/Steam Fl ow ( oP /6W
s
.
.
.
.
. .
.
.
.
.
s0
) Fre q uency
. . .
Col d Le g Tempe rat u re/Steam Fl ow ( oT
CL
.
.
/ 6W
s0
.
•
178
)
1 80
F requency Re s po n s e
I 1
1 44
.
Schema t i c o f the H . B . Rob i n son P l a n t
Re s po n s e
V .l .
•
and P
Po
.
•
�.6.
.
Ste p Re s po n s e o f r�o de l D to + 1 0% C h a n g e i n Ste am
t�o de 1 s
I. 1
.
Ste p Re s po n s e o f Mode l C to +1 0% C h a n g e i n Ste am
Va l ve Co effi c i e n t
1.8.
•
Si mpl i fi e d Di a gram o f t h e Phys i ca l Sy s t e m
•
•
1 85
1 .2.
Compa ri s o n Betwee n Mo de l D and Ch ri s tens en•s Mode l
.
•
1 89
I .3.
Schemat i c Di a gram o f a n I E UTS G
1 .4 .
Compa ri s o n B e twe en UTSG a n d I E UTSG �1ode l s Us i n g t h e
•
•
Same Mode l i n g Appro a c h
1\ . 1 .
Vari a t i on o f T
1\ .2 .
Va ri a t i on o f V
sat
f
wi t h P re s s ure
w i t h Pre s s u re
•
1 91
•
.
•
•
1 93
•
208
2 09
xi i i
iURE
v9
3.
Va r i a t i o n o f
4.
Va ri a t i o n o f V
5
.
Va r i a t i o n o f h
.6.
Va ria t i o n o f h
•
7.
Va r i a t i o n o f h
•
1
�.
�.
.
•
2.
f
g
•
.
w i t h P res s u re
.
w i t h P re s s u re
.
.
.
. .
.
.
.
.
.
.
.
.
.
. . .
.
.
.
. 212
.
.
.
•
21 3
. .
.
.
.
.
.
.
.
.
.
.
.
21 9
. .
.
.
.
.
.
.
.
.
222
.
. . .
.
a
.
.
. .
.
. 226
Con t ra 1 Vo l ume fo r the Appl i cat i o n o f t h e
.
.
.
. .
.
.
Ca l cu l a t i o n o f t he D r i v i n g P re s s u re Head fo r t h e
Re c i r c u l a t i o n L o o p
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S t e p Re s po n s e fo r Ca s e A . l
2.
S t e p Res po n s e fo r C a s e A . 2
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S t e p Re s po n s e fo r C a s e A . 3
4.
S t e p Re s po n s e fo r C a s e B . 1
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S t e p Re s pon s e fo r Ca s e B . 2
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S te p Re s po n s e fo r C a s e 8 . 3
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S t e p Re s po n s e fo r Ca s e
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S t e p Re s po n s e fo r Ca s e C . 2
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S t e p Re s po n s e fo r C a s e C 3
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S t e p Re s po n s e fo r C a s e
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S t e p Re s po n s e fo r Ca s e 0 . 2
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S t e p Re s po n s e fo r C a s e 0 . 3
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Con t ro l Volume fo r t h e Appl i ca t i o n o f th e E n e rgy
Mome n t u m E q u a t i on
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w i th P re s s u re
fg
E q ua t i o n . .
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w i t h P re s s u re
wi th P re s s u re .
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Cont ro 1 Vo l ume fo r t h e Appl i ca t i on o f the Cont i n u i ty
Equat ion
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CHAPT E R I
INT RO DU CT I ON
I.l
Gen e r a l Co n s i de rat i on s
Al t ho u g h s t e am ge ne rato rs have been u s e d i n powe r ge n e rat i n g
pl a n t s fo r a l on g t i me , t he i n t e re s t i n t he i r dyn ami c beh a v i o r
ha s g rown o n l y s i n c e n u c l ea r ene rgy became a fe as i b l e s u b s t i t u te
fo r fo s s i l fuel s as a l oa d- fo l l ow i n g heat s o u rce fo r steam s u p p l y
sys tems.
l oo p .
F i g u re 1 . 1 s hows a p r e s s u r i zed wa ter rea c t o r ( PWR ) coo l a n t
I t i s c l e a r t ha t t h e s team gen e rato r i n t h i s l oo p acts a s
a t h e rmal c o u pl i n g betwe e n t he reacto r p r i ma ry coo l a n t sys tem a n d
t h e s e co n da ry s team cyc l e .
O n t h e p ri ma ry s i de , t h e re i s t h e
n u c l e a r powe r reacto r w h e re e ffi c i en t and s a fe h e a t remo val from
the reac to r c o re i s o f p ri ma ry co n ce rn to the re a c t o r de s i gn e r .
On tl1e s eco n d a ry s i de , t he re i s t h e e l e ct ri c a l e n e rgy c o n s ume r
who s e ma i n i n te re s t i s t h e re l i a bi l i ty o f s e rvi ce .
It i s
bec a u s e o f t h i s s t ra t e g i c po s i t i on o f the s t e am gene rato r between
t he s e two s i de s that i t s dyn ami c behavi o r bec ame of con s i de rabl e
i n te re s t fo r the s a fe des i g n a n d re l i a bl e o pe ra t i o n o f P\�R n u c l e a r
powe r p l a n t s .
T he wo r k reported i n t h i s d i s s ertat i o n was a pa rt o f a l a r ge r
tas k fo r the dyn ami c a n a l y s i s o f both s i n g l e a n d d u a l p u r po s e
( gen e rati ng e l e c t r i c i ty a n d des a l i na t i o n o f s e a w a t e r ) , n uc l e a r
powe r p l a n t s t ha t w a s c a r r i e d o ut a t t h e N u c l e a r E n g i n ee r i n g De pa rtme n t
1
2
r
Steam
Generator
Steam Valve
Secondary Steam
Cyc 1 e
,..._.___.._...,
Turbine
Generato
Pressurizer
UTSG
FWT 1--....,-'
Feedwater
Train
Hot Leg
Cold Leg
PWR
Pressurized
Water Reactor
F i g u re I . l
Primary
Systern
Pre s s ur i z e d W a t e r Reactor Coo l a n t Loo p .
Load
3
F
The U n i ver s i ty o f Ten n e s s ee wi t h the c o o pe ra t i on o f O a k Ri dge
1 t i o n a l Labo r a to ry and the Powe r Sys tems D i vi s i on o f Com b u s tion
1 g i nee ri n g , I n c .
Des c ri pt i o n o f t h e P hys i ca l Sys tem
2
T he s te a m gen e rator co n s i dered i n t h i s wo rk i s a ve rt i cal ,
t u be , rec i rc u l a t i on type s team gen e ra t o r
UTS GJ .
( h e re a fte r a b b re v i ated
I t i s t h e type u s e d i n mo s t o f t h e c u rre n t PWR n u c l ea r
earn s u p p l y sys tems
( NSSS ) .
The deta i l s o f s uc h a s team generator a re s hown i n F i g u re I . 2
d a s c hemat i c i s s hown i n F i g u re I . 3 .
Th e ho t reac to r cool a n t
r ry i n g t h e h e a t gene rat e d i n t h e co re e n t e rs a t t he bottom o f
e s team ge n e ra t o r t h ro u g h t he i n l et noz z l e t o a n i n l e t p l e n um ,
Jws t h ro u g h t he U- t u be s , t ran s fe rri n g h ea t to t h e s e co nda ry fl u i �
it flows o u t s i de t he t u b e s a n d t h e n e n t e rs a n o u t l e t p l en um
fo re l ea v i n g the s team gene rato r t h ro u g h the o u t l e t nozz l e ( s ) .
�
s e co n da ry feedwa te r to t h e s team ge n e ra t o r e n t e r s a t a l e ve l
;t a bo ve t h e U-tubes t h ro u g h a feedwa te r r i n g .
�
I t mi x e s wi t h
re c i rc u l a t e d water a n d t h e s l i g h t l y s u bcoo l ed mi x t u re pa s s e s
tnwa rd t h ro u g h t h e a n n u l a r re g i on betwee n t h e t u be wrappe r a n d
!
i
s h e l l b e fo re ente r i n g t h e U - t u be re g i o n .
H e a t i s t ra n s fe rre d
the seco n da ry fl u i d as i t fl ows upwa rd o u t s i de the U-t ub e s ,
a steam wa te r mixt u re i s fo rme d .
The steam wa ter mi x t u re then
ses t h ro u g h steam s epa rato rs and d rye rs t h a t l imi t t he moi s t u re
ten t of t he s t eam l e a v i n g t h e steam gene ra tor to l e s s t h a n
0.25%}2)
4
TEJ.t-4 OUTLET TO
MOISTURE SEPARATO�
URSINE GENERATOR
UPPH Si!EL.l.
SWIRL 'HMEI-IOISTURE
SEPARATOR
FEEOW.lTER
I MLE
----- AMTI-YIBRATIOII
BJ.RS
--
SHELL
TUBE SU�PORTS
TUBE BUNDLE
PUT I TION
TUBE PLATE
SUPPORT FOOT
CHA��EL HEAD
PRI�.lRY COOLAIIT
INLET
F i g u re I . 2
PRif.!,I.RY COOU!H
OUTLET
Deta i l s of a Ve rtical U - T u b e Rec i rc u l at i o n Steam
Gen e ra t o r (UTS G ) ( from Referen c e 1).
5
Steam O u t l e t
U p p er
Shel l
S e co n d a ry
S e pa ra to rs
Swi rl va n e
S e p a ra t o rs
-""
-
Fee dwa t e r
I nlet
U-Tube
' Bundl e
U- t u b e \� ra p pe r
L ower
She1 1
I n l et
P l e n um
Ou t l e t P l e n ur1
-
P r i ma ry
I nlet
Figure
I.3
P r i mary O ut l e t
Schematic Diag ram for a Ve rt i ca l U-t ube Reci rcu l a t ion
Steam Gene ra tor (UTSG).
6
The s e p a ra t e d wat e r ret u r n s t o mi x w i t h t he fee dwat e r fo r a n o t h e r
pa s s t h ro u g h t h e t u be b u n d l e re g i o n .
Refe r r i n g to F i gu re 1 . 3 , o n e c a n summa r i z e t h e d i fferen t
proces s e s e nco un t e re d i n t h e fun ct i o n i n g o f th i s UTS G a s fo l l ows:
A.
P ri ma ry S i de
1.
E n t ra n ce o f t he p r i ma ry wate r f rom t h e rea cto r hot
l eg to t h e i n l et p l en um .
2.
Fl ow o f t h e pri ma ry wate r t h ro u g h the U- t u be b u n dl e
a n d h e a t t ra n s fe r a l o n g i t s pat h to t h e s e co n da ry
fl u i d.
No t i ce t h a t fl ow o f t he p r i ma ry fl u i d i s
paral l e l to t he s e conda ry fl ow i n t h e i n l et b ra n c h
o f t h e U - t u bes a n d oppo s i te to i t i n t h e e x i t b r a n c h
o f t h e U- t u be s .
T h e re fo re , t h e U - t u b e s t eam gen e ra t o r
ca n b e co n s i dered a s compo s e d o f a p a r a l l e l fl ow
s team ge n e ra t o r a n d a counte r fl ow s team gene rato r
wi t h a common s e con da ry fl u i d .
3.
B.
O ut l e t o f t h e p r i ma ry water t h ro u g h t h e o ut l e t pl e n um .
T u be Meta l Wal l s
1.
The t u be met a l s e p a rates t he p r i ma ry water a n d the
s e c o n d a ry fl u i d .
The t u be con d uctan ce toge t h e r wi t h
t h e fi l m h e a t t ra n s fe r coe ffi c i e n t s determi n e t he
effec t i ve heat tra n s fer co effi c i e n t s r e l a t i ng the b u l k
mea n t empe rat u re s i n t h e p r i ma ry a n d s e con d a ry s i de s
a n d t h e t u be met a l tempera t u re .
7
C.
S e co n da ry S i de
1.
Fee dwate r e n t e rs t h ro u g h the fe e dwat e r r i n g i n to t h e
a n n u l us between t he U- t u be wra p pe r a n d t h e s h ell ,
refe r re d to a s t h e down come r s e ct i o n i n t h i s s t udy .
2.
Fee dwa t e r m i xes wi th s a t u ra t e d wat e r s e pa ra t e d
fro m t h e s team-wa ter mi x t u re i n t h e s team s e p a ra t e rs .
3.
Seconda ry wa t e r e n t e rs f ro m t he downcom e r i nto t h e
tube b u n dle re g i o n .
4.
Seco n da ry wa te r i s heated a s i t fl ows u pward a ro un d
t h e U - t u b e s a n d t h e tempe ra t u re i s ra i s e d t o t he bo i l i n g
po i n t .
5.
B o i l i n g o c c u rs i n t h e fl ow i n g s e co n d a ry fl u i d a n d s tea�
i s fo rme d as mo re h e a t i s t rans fe r red t h ro u g h t he U-tubes
fro m t he p r i ma ry f l ui d .
6.
T he s t eam-wa te r m i xtu re conti n u e s to fl ow u pwa rd
to t h e
s team s e p a ra t o r s where s team i s s eparated a n d col l ected
i n t h e upper pa rt o f the s team ge ne rato r s h e l l and
s eparated w a t e r i s mi xed wi t h feedwa t e r, t hen fl ows
downwa rd i n the downcomer .
7.
Steam e x i t s from the u ppe r pa rt o f t h e s tea� ge n e ra to r
at a rate dete rmi ned by t h e t u rb i n e s to p va l ve
o pe n i n g .
8
S ta temen t o f t h e P robl em
The purpo s e o f t he s t u dy a nd researc h wo rk re po rted in t h is
s e rt at io n was to inves t i gate the dynamic beha vio r o f a vert i cal
G u s i ng the s tate va ri a bl e, l umpe d pa ramete r techni q ue .
The
hemat i ca l mode l s devel o pe d were req u ired to be in a fo rm s uita bl e
us i ng e x i s t i ng state va riabl e dig i tal compu ter code s to obta i n
dyna mic res ponse in t he t ime and freq uen cy doma in s .
Al so ,
y we re req u i red to be in a fo rm s u i tabl e fo r co u pl i n g wi t h
e l s fo r othe r subsys tems s o t ha t t h e dynami c re sponse o f t he
e g ra te d P W R system can be obta ine d us i n g t h e same comput e r code s .
val i dity o f t he mode l wa s to be checked a ga in s t a va i l ab l e
eri menta l data from dynami c tests on t h e H . B . Robi nson Nuc l e ir
e r P l a nt . ( 3 , 4 , 5 )
Impo rtan c e o f the Study
The need fo r e ffi c i e nt a nd pract i ca l te chniq ues fo r a s ses smen t
t he dynami c beha v i o r o f n u c l e a r steam gene rators is g row i n g with
g rowth o f the n u c l ear po we r i ndust ry .
The de s ign o f a s team
era to r wit h in a n u c l ea r powe r pl ant dema n ds a deta i le d knowl edge
the t he rmo hyd raul ic p ro ce s s es a ffecti n g its perfo rma n ce , not o n l y
steady sta te , but a l so du r i ng t ra n s i en t s .
S in ce a d i gita l computer
1 be nece s s a ry to pred i ct the dynami c re s ponse of l a rge compl e x
terns, a n effi ci ent mat hema t i cal mo del must have the fo l l owin g
racterist i cs.
1.
The numbe r of equa t i o ns sho ul d be the min i mum t ha t
a deq uatel y si mul ates t h e dynamic be h a v io r o f the sys tem .
9
2.
T h e fo rm o f t he mo del sho u l d a l l ow t h e u s e o f s ta n d a rd
g e n e ra l p u rpo s e computer codes .
3.
It s ho u l d b e e a sy to co u p l e mo de l s fo r o t h e r s u b sy s t ems .
4.
T he s ame mo del s ho u l d be u s ea bl e fo r t r a n s i en t , fre q u e n cy
res pon s e o r s t ab i l i ty a n a l y s i s .
T he l a c k o f dynam i c mo del s w i t h t h e a bo ve c h a racte r i s t i c s
fo r n u c l e a r UTs G•s s t i mu l a t e d t he n ee d fo r the w o r k re ported i n t h i s
d i s s e rt a t i o n .
! . 5 Sco p e a n d O rgan i za tion of t he Text
Fol l ow i n g t h i s i n t ro d u c t o ry c h a pt e r , a re v i ew of p e rt i n e n t
l i t e ra t u re i s p re s e n t ed i n C h a p t e r II .
T he de velnpment o f t h e
ma t hema t i c a l mo de l s i s g i v e n i n Cha pte r II I .
I n Chapter I V, t he
c a l cula t i o n pro ce d u re i s de s c r i bed a n d t h e a p pl i c a t i on o f t h e s e
mo d e l s fo r dynami c re s po n s e s i mu l a t i o n o f a n i s o l ated U T S G i s
demo n s tra te d .
The re s po n s e o f a co u p l e d PWR/ s team ge n e ra t o r system
i s exami ned i n C ha p t e r V fo r t he p u rp o s e o f te s t i n g t h e v a l i d i ty
of t he mo de l s .
I n t h i s c h a p te r , t h e co rrel at i o n between t he
mat he mat i ca l mo del •s pred i c t i o n a n d t h e r e s u l t s obta i n ed from dyna� i c
te s t s o n H . B . Ro b i ns o n N uc l ea r Powe r P l a n t (
3
) i s d i s c u s s ed .
In
Chapte r VI , the d eta i l ed mod el r e s pon s e i s compared wi t h t h e resul t s
32
o bt ained f rom a f i n i te d i ffer ence mod e l . ( ) C h a p ter VI a l s o
con ta i n s t he com pa r i son b etween t h e d e ta i l ed U T S G mod e l a n d a
dynam i c mo d el fo r an Integral Economi z er U - t u b e Steam Ge nerato r
( I EUTSG ) w h i c h wa s de v e l o p ed u s i n g t h e same mo de l i n g a ppro a c h . (6)
In C ha pter V I I
a summary of t h e mo d e l eval u a t i o n i s p r e s en ted
to g e t h er wi t h some con c l u d i ng rema r k s .
10
The A p p end i xe s co n ta i n t h e ba s i c i n fo r ma t i on a n d deta i l ed
:a l c u l at i o n s t ha t wo u l d o t h erwi s e i n terfere wi t h t he smo o t h
:o n t i n u i ty o f t h e d i scu s s i on i n t he ma i n t ex t .
CHAPTER I I
RE V I EW O F P E RT I N E NT L I TE RATURE
II.1
I n tro d u c t i on
The pro b l em o f d evel o p i n g dyna m i c mod e l s to pred i c t t he dyn a m i c
beha v i or o f n u c l ea r s t eam g en era t o r s ha s r ec e i v e d con s i d e ra b l e
atten t i o n i n t h e p a s t d ec a de .
I n t h i s c ha p t er , a n umb er o f p u b l i ca­
t i o n s wi l l be re v i ewed fo r t he purpo s e of pre s en t i n g a sampl e of t he
ava i l a bl e l i t e ra t u r e p ert i n en t to the a rea o f dyn a m i c s i m ul a t i o n o f
two- p ha s e n a tu ra l c i rc u l a t i on sys tems i n g en eral a n d o f U - t u b e
rec i rcu l at i o n t y p e s t ea m gen erators i n pa r t i cu l a r .
rev i ew i s p r e s en t ed i n t hr ee part s .
The l i t era ture
In S ec t i on 1 1 . 2 , a r ev i ew o f
some dyn am i c mod e l i n g a p pro a c hes for h ea t exc han g er s a n d fo s s i l
fu el ed bo i l e r s a r e d i s c u s s ed .
I n S ec t i on I I . 3 , some o f t he dyn am i c
model s for bo i l i n g wa t e r r ea c t o r
( BWR ) sys t e m s a re d i scu s s ed wi th
the empha s i s o n t he s i m i l a r i ty between BWR and UTS G system s .
In
Sect i on I I . 4 , some o f t h e model i n g t e c hn i q u e s fo r n uc l e a r s t eam
gen era t o r sys t em s a r e re v i ewed w i t h the emp ha s i s on model i n g a p proa c he s
and t h e scarc i ty o f p u b l i s hed mode l de s c ri pti o n s t hut s t i mu l a t e d
t h e n eed fo r t h e st udy a n d r e s ea rch wo rk repo r t ed i n t h i s d i s s erta t i on .
Gen era l rema r k s a bo ut t he l i t erat ure re v i ew a re g i ve n i n S e ct i on I I . S .
1 1 .2
Dyn a mi c Mo d el s for Heat E x c hange r s a n d Fo s s i l Fuel ed B o i l ers
T he intere st in t h e dyn amic be havio r of h ea t excha n g e r s and
fo ssil fueled boilers ha s grown in t he pa st two decades and a n u mber
of publication s ha ve been r eported in t he o pe n l i t era ture .
11
12
Cl a r k et a l .
( ?)
p r e s en ted a n umbe r o f pa pers where t hey d er i ved
ana l yt i ca l s o l u t i on s fo r t h e dyn am i c re s pon s e of hea t excha n g e r s a s
a fun c t i o n o f po s i t i on and t i me to t hree d i ff e re n t pert u rba t i on s i n
t he h e a t g e n era t i on r a t e
( t he perturbat i on s were a s te p , a n a rb i t ra ry
fu n ct i o n , a n d a s i n u so i d ) .
C h i en et a l . (S) p r e s e n t ed the t ra n s i en t a n a l ys i s fo r a bo i l er
wi th a n i n t e rn a l r e c i rcu l a t i n g l oo p and a s u perheater b a n k i n t he
l oa d l oo p .
The model i n cl ude s a pre s s u re d r o p e q ua t i o n fo r t h e s u pe r -
heat e r a n d r ec i rcu l a t i n g l oo p toget her wi th a s t ea m d ru m wa t e r l evel
ca l cu l a t i on .
A n a l o g s o l ut i o n s o f t h e sys t em were obta i n ed for o pe n
l oo p a n d cl o s ed l oo p t ra n s i en t respo n s e s o f t h e d rum pr e s s u r e a n d
wat er l ev el t o c ha n g e s i n s t ea m f l ow , fuel s u p p l y a n d feedwa ter
fl ow .
Tha l - L a rson (
g)
e xa m i n ed t he dynami c b e ha v i o r o f hea t exchan g e r s
b y d er i v i n g t ra n s fer f u n ct i on s fo r freque n cy r e s po n s e an a l ys i s u s i n g
s i mpl e ma t hema t i ca l mode l s .
Impo rta n t sys t em t i m e co n s t a n t s were
i d en t i f i ed fo r ea ch mod e l p r e s en t ed .
A d e ta i l ed s t u dy by Ma s u buc h i (
l O)
t r ea t ed t h e dynam i c a na l ys i s
o f d i fferen t con f i g u ra t i on s o f h ea t excha n g e r s .
of
a
Tran s fer fu n ct i o n s
compl ex mod el were d e r i ved fo r fre qu ency r e s pon s e a na l ys i s .
a l s o presen t ed a s i m pl i f i e d model fo r s i mu l a t i n g t h e tra n s i en t
re s po n s e .
Enn s (
ll
) cl a s s i f i ed the many t h eo re t i ca l mode l s o f heat
excha n g er s i n t o d i s t r i b uted pa rameter a nd l umped pa ra meter mo de l s .
Tra n s fe r f u n ct i on s were d e r i ved w h i c h s how t he a de q u a cy o f l um�ed
He
13
pa rameter mo del s fo r f l ow r a t e o r heat fl u x i n d u c ed tran s i e n t s w h i l e
a d i s t r i b u t ed paramet er mod el i s recommended for t emperat ure i n d u c e d
tra n s i en t s .
D u s i n be rre
( 1 2)
a pproac hed t h e h ea t exc h a n g er dynam i c s prob l em
by pr e s en t i n g a n umer i ca l t ec hn i q ue fo r cal c ul a t i n g tran s i en t tempe r a t u re s .
H e i n d i ca t ed t h a t n umer i ca l met h o d s a re a n excel l en t way fo r
gen era t i n g a fami l y o f s o l u t i on s to a pa rt i c u l a r prob l em .
H e a l so
h i n t ed t ha t t here i s o ft en a great sa v i n g of t i me and effo rt i n
pro c e s s i n g o n a f i n i t e d i fferen c e ba s i s t hro u g ho u t w h en a n a na l yt i c a l
sol u t i o n i s not o bta i na b l e a n d when a d i g i ta l comp u t er i s a v a i l a b l e .
A s trom (
l3
) pres en t ed a s i mp l e n o n l i n ea r mo del fo r a drum b o i l er-
t u r b i n e u n i t where the d rum press ure i s t h e s t a t e var i a b l e a n d f u e l
fl ow , control va l ve s et t i n g a n d feedwa t e r f l ow are t h e con tro l
va r i a b l e s .
T h e model wa s s hown to a g re e wel l w i t h mea s u remen t s fo r
an a c t u a l bo i l er i n the ra n g e hal f power t o f u l l powe r .
T he mo del
ca n be deri v ed u s i n g phy s i c a l a rgume n t s and i s t h u s c h a r a c t er i zed by
several pa rameter s .
A der i v at i o n o f t h e s i mpl i f i e d mo del b a s ed o n
p hy s i c a l co n s i d era t i o n wa s g i ve n toget h er wi t h t he a s s um pt i o n s req u i red
to a rr i ve at the s i mp l i f i ed model .
A c r u d e met hod o f e s t i ma t i n g t he
mod el pa rameters wa s a l so g i v en .
Be fore l e a v i n g t h i s s ec t i o n , t h e a u t ho r wo u l d l i k e to c a l l
a t t en t i o n to a n u mber o f excel l en t papers o n t h e s ubj ec t o f b o i l er
mod e l i n g g i ven i n Refe r en c e 1 4 .
14
1.3
Dynamic Models for Boiling Water Reactor ( BWR ) Systems
An extensive research effort has been undertaken in the past
.wenty years in support of the boiling water reactor program with
�mphasis on hydrodynamic stability of natural circulation boiling
;ystems.
The heat exchange process in a BWR core is somewhat similar
:o that in the tube bundle region of a UTSG.
In both cases, heat
is added from a primary source of heat to a flowing secondary fluid
Nhich enters the heat exchange region as subcooled water and leaves
it as a two phase steam/water mixture.
Some of the references
dealing with the dynamic behavior of BWR systems are given below.
lS
Thie ( ) presented a theoretical analysis of the kinetic
behavior of a number of operating boiling water reactors based on
a simple model where steam void feedbacks dominate.
It was concluded
that it is possible within the framework of existing experimental
and theoretical BWR dynamic technology to design these reactors with
reduced instability limitations on the power.
Akcasu (
l6
) investigated the dynamic behavior of boiling water
reactors in a systematic way.
General expressions for the transfer
functions associated with the individual feedback mechanisms were
obtained for an arbitrary flux distribution, weighting function and
steam velocity distribution.
Specific forms of the transfer functions
were obtained anJ simplified by single time constant transfer functions.
The error involved in the lumped time constant approximation was
examined and was shown to be as large as 4 db in amplitude.
The
theoretical results were compared with experimental power-void transfer
15
1ct i o n s a n d t h e agreemen t i n some ca s e s wa s proven to b e b et t e r
tn 5 d b i n a m p 1 i tu d e a nd 1 0 deg r e e s i n p ha s e i n th e en t i r e
!qu ency ran g e f rom 0 . 0 1 to 1 00 rad/ s ec .
F l ec k
( l ?)
u se d t he l aw s o f co n s ervat i o n o f ma s s , energy , a n d
1en t um i n t h e i n t egral fo rm to der i ve a s e t o f o rd i na ry d i ffe ren t i a l
1at i o n s go vern i n g t h e hydro dyn ami c s o f n a t u r a l c i rc ul a t i o n bo i l i n g
;terns .
T h e eq u a t i o n s c a n b e s o l ved fo r t h e wa ter vel o c i t i e s a t
i n l e t a n d o u t l et o f t h e h eat ed s ect i o n , t h e o ut l et vo i d fra ct i o n
t he to tal fl u i d mome n t um .
T h e s e equat i o n s ca n b e a dded t o t h e
.cto r k i n et i c s equa t i on a n d t h e equat i o n s gov ern i n g t h e t r a n sfer
hea t from re a ctor fuel e l ement s to wat er a n d the rel a t i o n b e tween
tct i v i ty and t he a ve ra g e vo i d fra c t i o n to o b ta i n a mode l for t h e
1am i c s o f a b o i l i n g wa t e r re a c to r .
The cal cul ati ons u s i n g t h i s
lel i nd i c a ted t h a t u n d e r ce rta i n con d i t i on s , bo i l i n g \'late r re actors
1
o pera t e s ta b l y even fo r very s u b s ta n t i a l reac t i v i ty add i t i o n s .
a n a l ys i s a l s o revea l ed t h a t u n d e r p ro p e r c o n d i t i on s , pu re l y hyd r o -
1am i c i n s ta b i l i ty c a n occ u r i n de penden t l y o f a n y co u pl i n g b etwee n
id a n d wa ter .
The o n set o f t h i s u n sta b l e b eha v i o r woul d b e
lep en dent o f n uc l ea r a n d t h erma l parameters a l t ho u g h t he dyn am i c
1a v i o r o f t h e rea c t o r wo u l d depend o n t he m .
( l S)
Mu s c etto l a
pre sented a t h eo r et i ca l fo rmu l a t i o n fo r t h e
1am i c s o f a bo i l i n g wa ter reactor o f t h e p r e s s u re t u b e a n d fo rc ed
·cu l at i o n type .
The model treated t he n eu t ro n k i n e t i c s, fue l
�men t h ea t t ra n s fe r dynam i c s , t h e b o i l i n g c ha n n el , steam d rum ,
mcome r and r ec i rc u l a t i n g p ump .
T h e a ut ho r empha s i zed the n eed fo r
�ore t i c a l mode l s to s u ppo rt b o t h d e s i gn a n d e x p er i me n t a l p ro g rams .
16
M i da a n d S u da (
lg
) pre s en t ed s i mpl i f i ed ve r s i o n s o f t h e t ra n s fe r
func t i o n mode l s d er i ve d b y t h em i n an o t he r r e po rt .
T h e s i mpl i f i ed
mod e l s were u s ed to i n ve s t i ga t e t he dynam i c c ha ract e r i st i c s o f a BWR
syst em from d e s i gn pa rameter s .
Th e dyn am i c c ha ra c t e r i st i c s o f t h e
feedbac k t ra n s fe r funct i o n s a n d the sys tem s t a b i l i ty were a l so
i n ve s t i ga te d a n d some compa r i son w i t h o t h e r s t ud i e s were g i ven .
An d er s o n e t a l . (
20
) exam i n ed t h e tran s i en t r e s po n s e a nd sta b i l i ty
o f a two pha s e n a t ural c i rc u l a t i on sy stem for a ppl i ca t i o n to a bo i l i n g
wat e r reacto r .
The t r ea t ment i s ve ry s i m i l a r t o t h a t req u i red for
a n a t u ra l c i rc u l a t i on steam gen erato r .
The mod e l d e s c r i bed the
con s ervat i o n of ma s s , en ergy , and momen tum fo r a two - p ha s e l o o p .
T h e res u l t i n g equ a t i o n s were s o l ved s i mu l t a n eo u s l y w i t h a n a na l o g
:ompu t e r .
The mod el u s es a s l i p ra t i o co r rela t i on fo r det ermi n i n g
the two pha s e fl ow i n t h e bo i l i n g port i on o f t h e l oo p .
Naha va n d i a n d van llo l l en (
2l
) d i sc u s s ed t h e u s e o f s pa c e-dependent
jyn am i c a na l ys i s t o study the dynami c b e h a v i o r and s ta b i l i ty o f
Ja i l i n g water rea c t o r sys t ems wh ere d i sc re t e s pa ce node s a l o n g t h e
1xi s o f t he c o re w ere u s ed to d e t e rmi n e l o c a l vo i d fract i o n a nd
thermo hyd ra u l i c fe edba c k effec t s .
Neal and Zi vi (
22
) pre s en t ed a compa ra t i ve s t u dy o f a na l yt i ca l
1o del s a n d expe r i me n t a l da ta fo r the hydro dynami c sta b i l i ty o f
1a t u ra l c i rc u l a t i o n bo i l i n g sys t ems .
Th e a u t ho r s exa m i n ed t h e
;ta b i l i ty pro b l em i n n a t u ra l c i rc ula tio n b o i l i n g sys t ems a n d i n d i c ated
:ha t a l a rge n umb e r of paramet er s in bo t h the bo i l i n g a n d n o n - bo i l i n g
Jort i o n s o f s u c h a system j o i n t l y de term i n e i t s sta b i l i ty , h e n c e t h e
:ompl ex i ty o f t h e dyn am i c mo d e l s .
The s t u dy o f hyd rodyn am i c sta b i l i ty
17
pro b l em wa s pre s e n t ed i n t h re e part s .
I n t h e fi r s t part , t h e
mecha n i sms o f i n st a b i l i ty con s i de red by a n umb er o f a ut h o r s we re
d i s c u s s ed a n d the expe r i me n ta l l y o b s e rved effe c t s of va r i o u s sys t em
pa ramet e r s were s umma r i zed .
I n t he s ec o n d p a r t , t h e b a s i c c o n s erva-
t i o n eq u a t i on s were d e r i ved in general fo rm , a n d e a c h mod el wa s
desc r i bed r e l a t i ve to t he s e equat i on s .
F i n a l l y , t he mod e l predi c t i o n s
were compa r e d w i t h e x pe r i mental data , a n d t h e rel a t i ve p re d i c t i n g
power o f e a c h mod e l wa s p r e s en ted .
23
S p i gt ( ) p r e s en t ed t h e r e s u l t s o f a n e x pe r i mental and t h eo r e­
t i c a l s t u dy o n t h e hyd r a u l i c c h a racter i st i c s o f a s i n gl e c oo l ant
c ha n n el of s i mpl e a n n u l a r g eometry i n a bo i l i n g water reactor w i t h
ma i n empha s i s o n t h e sta b i l i ty c h a ra c teri s t i c s o f t h e fl ow pro c e s s
i n s u c h a c ha n ne l .
T h e re po rt i n cl uded a d e s c r i pt i o n o f t h e
pre s s u r i z ed a n d a tmo s p h e r i c bo i l i ng l oo p s wh i c h w e r e u s ed i n t h e
expe r i menta 1 pa r t o f t h e s t udy .
I n t h e t h eo r e t i c a 1 pa rt o f t h e
study , t h e g e n e ra l e q u at i o n s de s c ri bi n g t h e pe rfo rma n ce c h a racte r i s t i cs
of
a
bo i l i n g sys t em under steady sta t e a n d n o n - st ea dy state were
de r i ve d w i t h s pec i a l a tten t i o n t o the formul a t i o n o f bo unda ry con d i t i o n s
a n d the i n t roduct i o n o f p re s s u re e ffe c t s i nt o t h e equat i o n s .
The
dyn am i c equa t i on s w e re l i n ea r i z ed by a s s um i n g sma l l d i st u rb a n c e s from
ste ady state and t h en n umeri c a l l y i n t e g ra t ed by mea n s of a di g i ta l
computer.
Vassier (
24
) de ve l ope d a n a n a l o g mod e l to s t udy t h e dyn am i c
be ha v i o r of a bo i l i n g l oo p w i th n a t u ra l c i rc u l a t i o n .
T h e mod e l was
constructed for a l on ger t e s t l oop then t ho s e u s ed by Fl eck (
Zivi (
22
l?
) and
) a n d was k e p t as s i mpl e a s po ss i bl e by i so l a ting t h e mo st
18
i mpo rta n t facto r s a n d n egl ect i n g the o t h e r facto rs .
Th e mod e l s howed
ZJ
a ra t h e r g o o d a greeme n t w i t h t h e expe r i men t s of S p i g t ( ) rega rd i n g
t h e p red i c t i on o f t h e o n s e t o f i n s ta b i l i ty a s wel l a s t h e pow er/ i n l et
ve l o c i ty t ra n sfe r f u n c t i o n .
T h e mo del ha s been expanded to i n c l ude
the n eutro n k i n et i c s equat i on s , fuel el emen t a nd vo i d fe edba c k a n d
t h e c o n t ro l system .
T h e a u t ho r ind i cated t h a t w h en deal i n g w i th some
phenomen a i n ce rta i n fre q u e n cy ran g e s , i t i s not n e ce s sa ry to co n s i de r
other compo n e n t s o f t h e pl a nt w h i c h ha ve muc h l ower n a t u r a l freq u en c i e s .
1 1 .4
Dynami c Mode l s fo r Re c i rc u l at i on Type Nucl ear Steam Gen e ra tors
T h e probl em o f s i mu l a t i n g t h e dynami c beha v i o r of st eam
gen era t o r compo n e n t s of n uc l ea r power pl a n t s h a s recent l y rec e i ved
a co n s i de ra b l e a tten t i on bo t h from i n d u st ry and u t i l i ty pa rt i e s .
Z5
We s tmo re l a n d ( ) d e ve l o pe d a n a t u ra l c i rc u l a t i o n s t eam
generator mo de l w here he emph a s i zed r i s er l i qu i d - va por vo l ume fra c t i o n s
and c i rc u i t pre s s u re l o s s es i n desc r i b i n g a c l o s ed s eco n d a ry l oo p .
Th i s d i g i ta l compute r mo d e l o f a s h el l a n d t u b e type un i t u s ed a n
a n n u l a r two pha s e f l ow t r eatme n t i n c omput i n g t h e steady state a n d
tra n s i e n t s t eam fl ow , t h e r ec i rc u l a t i on rat i o a n d t he wa t er l e ve l .
26
Mc Gwan and Bodo i a ( ) h a v e perfo rmed e x pe r i mental s t u d i e s o f
t h e s ta b i l i ty o f a n a t u ra l c i rc u l a t i o n st eam g en erato r .
E x p e r i me n t a l
da ta from t h e t e s t l oo p w ere compared to pre d i c t i o n s from a d i g i t a l
computer mo del o f a s t eam gen erato r .
I t wa s fo u n d t h a t i n c re a s e d
power l e ve l and ri s e r resi stance have a d v e r s e effect s o n s t a bil ity
whil e additio nal downcomer re sistance i mpro v e d s t a b i lity .
2?
C l a r k ( ) p re sented a ma thema tical mo d el fo r t h e steady s t a t e
and tra n sien t ana l ys i s of a n at u ra l circ u l ation , s eparate d rum ,
19
hor i zo n t a l s team g en e ra t o r o f t h e type u s ed i n t h e S h i p p i n g po rt
N uc l ea r Pow e r P l a n t .
E q ua t i on s d e s c ri b i n g t he con s ervat i o n o f ma s s ,
e n e rgy , a n d mome n t um were w r i t t en fo r a seco nda ry s i de sys tem to
match a p r i ma ry s i de h e a t ba l a n c e a n d secon dary powe r d e l i vere d to
a t u rb i n e- g e n erator l oa d .
Maj o r a s pe c t s o f t h e mo del i n c l ude
s teady s t a t e and t ra n s i e n t rec i rc u l a t i o n fl ow , s i n gl e p h a s e de n s i ty
a n d p re s s u re d ro p ca l c u l a t i o n a n d a utoma t i c co n t ro l o f t h e s t eam
d r um wate r l e vel .
B o t h st eady state con d i t i o n s a n d t h e t r an s i e n t
re s po n se o f t h e s team gene rato r model w e re obta i n e d by mea n s o f a
d i git a l com p ut e r p rogram , u s i n g a f i n i t e d i ffe re n c i n g me t ho d .
The
mod e l res po n s e wa s compa re d w i t h e x pe r i men t a l da ta from t h e S h i p p i n g­
por t N uc l e a r Power P l an t .
Na h a vanJ i a nd B a t e n b u r g
( ZB )
p re s e n t ed a comb i n ed d i g i ta l - an a l o g
mathemat i ca l mo del fo r t h e dyn ami c a n a l ys i s o f ve rt i c a l U - t u b e
n a t u ral c i rc ul a t i on s t eam g e n e ra to r .
I n t h e f i rst pa rt , a s pa c e -
de pen d e n t d i g i t a l program s i mu l a t i o n wa s u s e d t o d e s c r i b e t h e dyn am i c
beha v i o r of t h e s t eam gen era t o r re c i rc u l a t i o n l oo p .
I n t h e s econd
pa r t, an a n a l o g s i mul a t o r con s i s t i n g o f f i ve d i s t i n c t b l o c k s was
used fo r feedwat e r con t ro l l er pa rameter o pt i mi z a t i o n s t u d i e s .
The
fi ve b l o c k s of t h e a n a l og s i mu l a to r were ( a ) feedwa t e r l i ne; ( b ) s team
drum , (c) steam ma i n to t u rb i n e; ( d ) water l e vel contro l l e r , a n d
( e ) rec i rc u l at i on l oo p mod e l .
I n t h e a n a l o g mo de l , t h e re c i rc u l a t i o n
loop was represented by six tra nsfer f u n ct i o n s obta i n e d by u s i n g
analog/digital techniques a n d the time res pon s e from t he d i g i t a l
program to specified inp�t perturba tions.
The o t h e r fo u r bloc k s
were represented by analytica l e xpre s s i on s u s i n g lum pe d parameter
20
o de l s .
T h e fundamen t a l e qu a t i o n s a n d method o f so l ut i o n fo r t h e
ec i rcul a t i on l oo p s pa c e- de pe n d e n t d i g i ta l mo de l ·were a l s o p re s e n t e d
y t h e auth o r s .
Ha rgro v (
2g
) pre s en t ed a pro gram to c a l cul ate mul t i l o o p d eta i l ed
ra n s i en t be h a v i o r o f a PWR sys tem.
T h e program s i mu l a t e d two
eactor cool a n t l oo p i nc l ud i n g two steam gene rators a n d a s so c i a ted
ystem s .
I t a l so s i mul ated reacto r ki n et i c s , re acto r con t ro l a n d
rotec t i on s y s t em s .
The t h e rma l a n d hyd raul i c c h a ra cte r i s t i c s o f
h e r ea c t o r c o o l a n t sys tem were d e s cr i b ed by t i me a n d s pa ce
i ffe r en t i a l e q ua t i on s , t h e n re duced to noda l fo rms, t h e so l ut i o n
f w h i c h a re t racta bl e by mean s o f fi n i te d i ffere n c e s .
E a c h st eam
en erato r wa s re p r e s en t ed by s i x node s , an i n l et h eade r , a n out l et
ea d e r and four p r i ma ry/ s econ da ry h ea t t ra n s fer node s .
T h e h ea t
ra n s fe r ra t e from each fl ow s ec t i o n to t he s econdary s i d e wa s c a l cu a t ed u s i n g t h e l o ga r i t hmi c mea n t empe rature di ffe re n ce ( L tiTD ) s o
h a t t h e powe r t ra n s fe r re d wa s comput ed co rrec t l y even a t v e ry l ow
The h ea t t ra n s fer co effi c i e n t a n d h e a t t ra n s fe r
l ow con d i t i on s .
rea we re t re a ted a s vari a b l e s a n d e va l uated a s fun c t i o n s o f t h e
epre s e n ta t i ve pa rameters.
Pe rturbat i o n s rel a t ed to t he s team
en era to r He re taken as p r i ma ry wate r i n l e t temre ra t u re ( re a c t o r h o t
e g t empera tu r e , steam fl ow ra t e, feedwa t e r fl ow r a t e anG feedw a t e r
n t ha l py .
)
Ten Wo l de (
30
) i n ve s t i ga t e d va r i ou s a s pe c t s o f t h e s i mul a t i o n
f t h e n o n l i n e a r t ra n s i e n t t h erma l beha v i o r o f s t eam gene rato r s .
h e e va po ra to r i n a sod i um co o l e d fa s t b re e d e r rea c to r pow e r pl a n t
Both
21
( LMFB R ) a n d t h e steam generator i n a p re s suri z ed water reacto r pl a nt
( PW R ) a re c o n s i dered i n t h e i n ve st i gat i o n .
Both a hyb r i d conti nuous
s pa c e d i s c rete t i me ( CS O T ) d i st r i buted pa ramet er mo del and a l umpe d
pa rameter mo del w ere con s i dered i n t h e stu dy.
Th e resu l t s from t h e s e
mode l s were compa red w i t h experi menta l r esul t s o bta i n ed from a 6 t·1W
PWR steam g enerato r .
Bauer et a l .
( Jl )
de s c r i bed a d i g i t a l mo del o f th e dynami c
pe rfo rma n c e o f ga s a n d pre s su r i z ed water reacto rs d e s i g n ed a n d
deve l o pe d by E l ectri c i te De F ra n ce fo r i t s equ i pment stud i es .
The
mod e l equat i o n s were pa rt i a l d i fferent i a l e quat i o n s a n d were
sol ved by t h e f i n i te d i ffere n c e tec h n i que.
The pa pe r s howed t h e mo d e l
equat i on s, t h e method o f s o l ut i o n , a n d t h e a p pl i cat i o n o f t h e mo del
fo r two c a se s .
a.
E x pl a n at i o n o f t h e pul sat i n g b e h a v i o r o f t h e h e at exc h a n g e r
o r i g i na l l y i nten d ed fo r t h e Bugey g ra p h i te g a s p l ant.
b.
An a l ys i s o f i n st a b i l ity of wat e r l evel in t he Chooz PWR
steam generato r u n d e r l ow out put .
( JZ )
C h r i s t en s en
deve l o pe d a o n e d i men s i o n a l mo del o f a UTSG
us i n g a m i xed d i st r i buted a n d l umped fo rmu l at i o n .
The gove rn i n g
e quat i o n s fo r t h e h eat t ra n s fe r zon e a n d t h e downcomer were de s c r i b ed
by pa rt i a l d i ffere nt i a l e quat i o n s wh i l e t h e ot h e r sect i o n s of t h e
steam generato r were d e s c r i bed by o rd i na ry d i ffe r ent i a l equat i o n s .
The part i a l diffe rent i a l e quat i o n s were s o l ved by sampl i n g i n t i me
a n d d i v i s i on i nto subs ect i on s i n s pa ce , thus tra n s fo rmi ng t h e
e quat i o n s i nto a l g e b ra i c equat i o ns .
T h e equat i on s c a n b e s o l ve d by
pu re d i g i t a l o r hyb r i d t�c h n i que s . T h e mo del ha s the adva nta ge o f
22
ca l cu l at i n g t h e steady state re s po n s e a s wel l a s t h e dynam i c re s po n s e
to s pec i fi ed pertu rbat i o n s .
The wo r k re ported i n th i s d i s s e rtat i on
wa s deve l o pe d i n d e pe n dent l y from the wo r k reported by C h r i sten s en .
A compa r i s o n between t he re s u l t s pre sented i n Re fe re n c e 32 a n d t h e
re s u l t s o bt a i ned f ro m t h e deta i l ed l umpe d pa rameter mo del i s g i ven
i n C ha pt e r V I .
Oe l e n e (
33
) deve l o pe d a d i g i tal s i m u l ato r fo r d u a l p u r po s e
d e sa l i n at i o n p l a n t s i n w h i c h m ul t i stage f l a sh ( MS F ) eva porat o rs
( PW R ) n uc l ear power p l ant
are cou p l e d to a P re s s u r i z e d Water Rea cto r
th ro u g h a b a c k p r e s s u re t u rb i n e .
T h i s s i mu l ator i nc l uded a state
va ri a b l e mo d e l fo r a U -t u be , drum type rec i rc ul at i n g b o i l e r .
The
p r i ma ry l i q u i d wa s d i v i de d i nto th ree s e ct i o n s , e a c h descr i b ed by
two o rd i n a ry d i f fe re nt i a l equat i o n s w i t h o ut l et a n d a v era ge temperat u re s of e a c h s ect i on as the state va r i a bl e s.
The s eco n d a ry s i de was
re pres ented by fo u r equat i o n s , two fo r t h e s u b cool ed
( n o n - bo i l i n g
s ect i o n ) , o n e fo r the down come r tem perat u re , a n d o n e fo r t h e ste a m
ma s s .
T h e a ut h o r state d t h at 11 s everal s i mp l i fy i n g a s s umpt i o n s a re
ma d e i n t h e steam gen e rato r mo del w h i c h may l i m i t the s i mu l ator• s
ab i l i ty t o pre d i ct t h e re s u l t s o f w i de r a n g i n g tran s i e n t s . 11
The
mo d e l wa s deve l o pe d o n l y fo r the p u r po s e o f a n o v e ra l l system
repre sentat i o n a n d h a s not been exp l i c i t l y t ested a ga i n st ot h er
models o r test result s .
Murrel
( 34 )
presented a descri pt i on o f t h e compo n ents compri s i n g
the s i mulat i on o f PWR nuclea r power plant s .
Th e reacto r wa s
represented by a s i ngle a v era ge channel def i ned as a fuel el ement
and the coolant a rea a s soci ated w i t h it .
Average den s ity i n t h e
23
pr i ma ry c i rc u i t wa s ca l cul at ed a s a fun c t i o n o f a ve ra ge tempe rature
a n d pre s su re .
T h e f l ow from t h e pri ma ry c i r cu i t i n to t h e p re s su r i ze r
wa s o bta i n ed from t he rat e o f c h a n g e o f p r i ma ry c i rcu i t den s i ty .
The re c i r cu l a t i o n fl ow i n t h e UTS G w a s c a l cu l a t e d from a p re s su re
b a l a n ce o f s ta t i c hea d d i fferen c e a g a i n s t fri ct i o n a l an d a c ce l era t i o n
head l o s s e s w h i c h w ere a s sume d pro po rt i on a l t o t h e s qua re o f t h e
fl ow .
Fre i e t a l . (
3S )
pre s e n t e d a de s c r i pt i on o f a compu t e r mo del fo r
a 660 MWe nuc l e a r power pl a n t .
The model i n c l uded t h e p re s su r i z ed
wa t e r reacto r , s t eam g e n e rato r , pre s su r i z er , s ec o n da ry s i de
power pl a n t s i de ) and con t ro l s .
( s team
The re sul t s obta i n e d f rom t he
dynami c mo d e l were com pa r e d a g a i n st t e s t re sul t s obta i n ed dur i n g
t h e comm i s s i o n i n g o f t h e pl a n t .
Ma tause k (
36 )
d e ve l o pe d a n o n l i ne a r l umped pa ramete r mo del
( S I NO D )
for a n a l y s i s o f two pha s e f l ow i n n a tural c i rcul a t i o n b o i l i n g w a t er
l oo ps .
T h e mod e l con s i s t s o f t he equa t i o n s for t he con s er va t i on o f
ma s s , momen tum , a n d e n ergy.
T h e resu l t i n g e qua t i o n s a re s o l ved w i t h
t i me a n d l en g t h a l on g t h e l oo p a s t he o n l y i n de pe n de n t va r i a b l es .
T h e s t ea dy state c a l cu l a t i o n s wa s pe rformed by s o l v i n g t h e same s et
o f e quat i on s w i th t h e t i me de r i va t i ve terms s et equa l to z e ro .
The
i n put d a t a to t h e mo d e l a re t h e pa ramet e r s s pe c i fyin g t h e geome try
of t h e sy s t em u n d e r co n s i de ra t i o n , p hy s i ca l p ro pe r t i e s o f t he fl u i d
and pre s sure dro p coeff i c i en t s a l o n g t h e l oo p .
O pt i o n a l corre l a t i o n s
a re pro v i ded for t he ca l cu l a t i o n of s l i p ra t i o a n d two p h a s e f r i ct i o n
mul t i pl i er s .
T h e mode l s h a ve a t h o roug h i n v e s t i g a t i o n o f t he i nt e r­
ac t i o n between ma s s vel o c i ty a n d vo i d f ra ct i o n w h i c h i s co n s i de r ed
24
the ba s i c mecha n i sm of hydrodyn ami c i n s t a b i l i ty .
When u s i n g t h e mo d e l ,
c a re mu s t b e t a ken to i n s u re that t h e a s s umpt i o n s b u i l t i n to i t a re
re a so n a bl e fo r t h e p hys i ca l sys t em under i n vest i ga t i o n .
11. 5
Rema r k s
From t h e prev i o u s d i scu s s i on , i t c a n b e s ee n t hat t h e pro b l em
o f a n a l yt i ca l a n d n umer i c a l s i mul a t i o n o f t h e dyna m i c b e h a v i o r o f
n uc l ea r s t e am g e n e ra to r s h a s recen t l y rece i ve d con s i derabl e a t t en t i o n
a s a tool for d e s i gn a n d t e st i n g o f PW R nucl e a r power p l an t s .
T h re e
mo d e l i n g a p proa c h e s a re n ow i n compet i t i o n .
1.
T ra n s f e r fun c t i o n a p proac h .
2.
N um e r i ca l
3.
State va r i a bl e l umpe d pa rame te r a pproa c h .
( fi n i t e d i fference ) a p proac h .
T h e f i r s t a pp roa c h i s ma i n l y a n a na l yti c a l a pp roa c h s u i t a b l e
fo r t h e so l ut i o n o n a na l o g compute r s .
The sys t em charact e r i s t i c s a re
h i dden i n t h e c ompl e x expre s s i on s o f t h e t ra n s fer f u n c t i o n s wh i ch
ma ke the i n ter preta t i o n o f r e su l t s ra t h er d i ff i c u l t .
Th e s econ d
a p proa c h i s a n umer i ca l a pp ro ac h wh ere t h e sy stem o f d i ffe re n t i a l
e qua t i o n s des c r i b i n g t h e system a re d i s c re t i z e d i n s pace a n d / o r
t i me t o o bta i n l o c a l e f fe c t s o f a g i ven perturbat i o n t o t he system .
The t h i rd a pp roac h comb i n es some o f t h e adva n tage s o f t he
anal yti c a l a p p roach and t h e numer i ca l a p proa c h .
A p hy s i cal i n s i g h t
i n to t h e system c a n b e ac h i eved by exami n i n g t h e dynami c b eha v i o r o f
the system state vari a b l es .
T h e wo rk re port ed i n t h i s d i s s e rtat i o n
i s an i n de pe n d en t con t r i bu t i on in t h e area of s i mu l a t i n g t he
dyn am i c behav i o r of PWR nuc l e a r power systems u s i n g th e l umped
paramete r state v a r i a b l e a p proa ch .
CHAPT E R I I I
DEVELOPMENT OF THE MATHOlA. T I CAL r�O DEL S
III.l
I n t ro d uc t i o n
An e s s en t i a l pa rt o f the dynami c a n al ys i s o f a n e n g i n e e r i n g
sys tem i s mod el i n g t h e phys i ca l p roc e s s ta k i n g pl ace w i th i n th e
sys t em d u r i n g a t ra n s i en t .
The p urpose o f th e mo de l i n g p ro ce s s
i s t o o b ta i n a ma t he ma t i ca l de s c r i pt i o n tha t can pred i c t
the re s po n s e o f the phys i ca l sys tem to a l l a n t i c i pated i n pu t s .
Fo r a t h e rma l hyd rau l i c sys tem l i ke a n uc l e a r s team gene rato r ,
the ma t h ema tica l mod e l co n s i s t s o f the equat i o n s go vern i n g th e
sto ra g e a n d t ra n s po rta t i o n o f ma s s , momen t um a n d energy ( ma i n l y
t h e rma l
) wi t h i n t h e sy s tem d u r i n g a t ra n s i en t .
I n th i s cha pt e r , a n umbe r o f ma thema t i cal mo de l s fo r a UTS G
a re devel o p e d u s i n g the s t a t e v a r i a b l e l umped parameter a pp ro a c h .
The mo de l s ra n g e from a t h ree - l ump mo del o f a s i mpl i fi ed s team
gen e ra t i n g sys tem to a fo u rteen - l um p deta i l e d mo de l of a UTS G
sys t em .
III.2
B a s i c Ass u mpt i o n s
I n t h e proce s s o f deve l o p i n g ma the�a t i cal mo de l s fo r a comp l e x
sys t em s u ch a s a UTS G , s e ve ra l a s s umpt i o n s h a ve to b e mad e .
The
Fol l ow i n g l i s t i n c l u d e s the a s s umpt i o n s used in th i s work :
1.
One d i men s i o n a l fl ow fo r both p r i ma ry a n d s e co n d a ry flu i d s .
This
me a n s that ra d i al var i a tio n s o f th erma l a n d hyd ra u l i c
propert i e s of t h e p rima ry a n d s e condary fl u i d s are n e g l e cted .
25
26
2.
Fo r t h e p re s su r i z e d p r i ma ry water a n d the s u bcoo l ed
s e co n da ry fl ui d , co n s ta nt de n s i ty and s pe c i fi c heat
a re a s sume d .
3.
T h e rma l c o n d u c t i vi ty o f t h e tube b u n d l e metal i s
a s sumed t o be co n s t a n t.
4.
H e a t t ra n s fe r coe ff i c i en t s a re a dj u s t e d to f i t t h e
h e a t t ra n s fe r ra tes a n d tempe ra t u re d i fferences reported
by t he s team gen erato r manufa c t u rer.
Heat t ra n s fe r
c o e ff i c i en t s a re a s s umed to b e co n s t a n t du r i n g t ra n s i en t s .
5.
T h e rmodyn ami c p ro p e rt i e s o f s a t u ra t e d wate r a n d
s atura t e d s team a re a s s ume d t o b e l i n ea r f u n ct i o n s o f
pre s su re o ve r a n a r row ran ge a b o u t t h e s t ea dy s tate
o pe rat i n g po i n t .
6.
T h e e n t ha l py a n d ma s s q u a l i ty o f t h e s te am wate r mi xt u re
i n t he s e condary fl u i d bo i l i n g reg i o n a re taken a s l i nea r
funct i o n s o f pos i t i on a l on g t h e h eat t ran s fe r p at h .
7.
N o heat t ra n s fe r t a ke s pl a c e between the t u be b u n d l e
re g i o n a n d t h e downcome r .
III . 3
S t ructu re o f t he Ma t h ema t i ca l Mo de l s
From t h e d i s cus s i o n i n C h a pter I , page
3 , i t c a n be s e e n t h a t
the UTS G i s cou p l e d t o t he rea c t o r coo l a n t sy s t em t h ro u g h t h e
pri ma ry cool a n t fl ow ra te a n d tempe ra t u re .
T h e s e p r i ma ry l o o p
co n d i t i o n s det e rm i n e t h e rate a t wh i c h t h e rma l e n e rgy i s i n put
to t h e s team ge n e ra t o r sys tem .
I n t h e deve l o pment t h a t f o l l ow s , t h e
pri ma ry cool a n t f l ow ra te i s a s s ume d co n s tan t .
T h e UTS G i s a l so
27
cou p l ed t o the s team cyc l e t h ro u g h t h e s team o u t l e t fl ow rate a n d
t h e feedw at e r i n put con d i t i on s .
The s te am o u t l et fl ow rate i s
d e t e rmi n e d by the steam dema n d s i g n a l to t h e t u rb i ne s to p va l ve .
T h e fee dwa t e r i n pu t c o n d i t i on s , ma i n l y fee dwater fl ow rate a n d
t h e fee dw a t e r tempe rat u re , a re determi ned by fe e dwater pump s pee d ,
feedwate r h e a te r s , a n d fe edwate r control syste m .
S i nce t h e ma i n
empha s i s i n t h i s wo rk w a s t h e devel o pme n t of UTS G mode l s , the fe e d ­
wate r c o n t ro l system i s n o t i n cl uded i n t h e mode l s a n d t h e fee dwa t e r
fl ow ra te w a s ma de e q u a l t o t h e steam fl ow ra te .
T h u s , t h e t u rb i ne
v a l ve c o e f f i c i ent a n d t h e fee dwater tempe ra t u re a re t h e o n l y
a l l owab l e s e conda ry s i de fo rc i n g funct i o n s .
The mode l i n g a pp roach u s ed i n the d e ve l o pme nt o f the ma thema t i c a l
mode l s do c ume n te d i n t h i s c h a pte r i s t h e s t e pwi s e l umped pa rame t e r
s t a t e v a r i a b l e a pproa c h .
T h e mo del s range from a t h ree l ump mo de l fo r
a s i mp l i fi e d steam genera t i n g sys tem ( t1o d e l A ) to a fo urteen l ump
det a i l ed mo d e l ( Mo del D) fo r a UTS G system w i th two mo de l s o f i n t e r­
med i a te com p l e x i ty ( B a n d C ) .
P ro gre s s i ve devel o pment o f i n crea s i n g l y
comp l e x mo de l s a n d compa r i son o f re s u l t s c re a t e d con fi dence i n t h e
va l i d i ty o f the fo rmu l a t i on o f the de ta i l e d mo de l .
T h e l umped s t ru c t u re
o f t h e s e mode l s a re de s c r i bed bel ow .
1 1 1 . 3.1
S i mp l i fi ed S team Gen e ra t i n g Sy s tem Mo del ( Mo del A)
Mode l A w a s de ve l o pe d a s a fi r s t s te p fo r the p u r po s e o f p ro v i d i n g
a 11 fe e l 11 for the dyn ami c re s po n s e o f t h e UT SG deta i l e d mo del .
I n th i s
mo de l � the steam gene ra tor i s re pre s e n t e d by t h e s i mpl i f i ed s t e a m
gen e ra t i n g system s hown i n F i gure I I I . l o
T h e mo del co n s i s t s o f t h e
fo l l ow i n g t h re e l umps a s s hown i n F i g u r e I I I , 2 .
28
St eam val ve
-----r:::>'C:o-J- ---;>o-
H
S t eam vol ume
sg
S a t u ra ted
wa ter poo 1
Hea t i n <;L__ _
S ec t i o n
P n0 ma r y
i n l et
F i gu re I I I . l
S team o u t l e t
•
P r i ma ry
outl et
'------
-<--- F e edi-Ja t er i n l e t
S i mp l i f i ed S team Gen e ra t o r System fo r Mode l A .
29
p
P r i ma ry
Lump
( P RL )
T
......___]_ __
Wp T P i
•
----->---·----
Fi gure I I I . 2
T
m
___,
_ ___
_
T u b e me ta l
l ump
( rn L )
S c h ema t i c D i a g ram fo r r 1o d e l A .
s
sat
S e con da ry
F l u i d L um p
( S FL )
30
l.
P r i ma ry fl u i d l um p ( P RL ) .
2.
H e a t c o n duct i ng tube metal l ump ( MT L ) .
3.
S e co n da ry fl u i d l ump ( S FL ) .
I 1 I . 3. 2
F i r s t I n te rme d i a te Model ( Model B ) _
I n Mod e l B, t h e p r i ma ry fl u i d l ump a n d t h e con duct i n g tube meta l
l um p o f Mo de l A a re eac h b ro k e n i nto two l um p s t o s i mul a t e t h e two
bra n c h e s o f t h e U-tube bun dl e .
b y o n e l um p .
The secondary f l u i d i s s t i l l repre s e n t e d
The mo de l con s i s t s o f t h e fo l l ow i n g f i ve l umps ( a l so
s e e n i n F i gure I I I . 3 ) :
1.
P a ra l l e l fl ow b ra n c h p r i ma ry fl u i d l ump ( P RLl )
2.
Counte r fl ow b ra n ch p r i ma ry fl u i d l um p ( P RL2 )
3.
P a ra l l e l fl ow b ra n ch conduct i n g tube met a l l ump ( MTL l )
4.
Cou n te r fl ow b ra n c h conduc t i ng tube me t a l l ump ( MTL2 )
5.
S e co n d a ry fl u i d l um p ( S FL ) .
II1.3.3
S e c o n d I nt e rme d i ate Mode l ( Model C )
I n t h i s mo de l , t h e s ec o n d a ry fl u i d l um p i n Mode l B i s b ro k e n i nt o
t h re e l umps a s s hown i n Fi gure I I I . 4 .
The mo de l t hu s c o n s i st s o f the
fo l l ow i n g s e ve n l um p s .
1.
P a ra l l e l fl ow b ra nc h p r i ma ry fl u i d l um p ( P RLl )
2.
Counte rfl ow b ra n ch p r i ma ry fl u i d l ump ( P RL2 )
3.
P a r a l l e l fl ow b r a n c h conduc t i n g tub e me ta l l um p ( tiT L l )
4.
Counte rfl ow b ra n ch conduct i n g t u be met a l l ump ( MTL2 )
5.
E ffe ct i ve h e a t e xcha n ge secon dary fl u i d l ump ( S FE H E L )
6.
Drum equ i va l e n t s e co n d a ry fl u i d l ump ( S F DRL )
7.
Downcome r secondary fl u i d l ump ( S FDC L ) .
31
,.
i
T
'RL 1 )
Pl
T
1-
sa t
,,
l
2
47
-·
TP
2
( P�L2 )
( S FL )
•
T ;
p
( rlfTL 1 )
( t�T L 2 )
...
gure
III.3
\�p
,.
S c h ema t i c D i a g ram for t1o de l B .
32
�� s
��
p
s
1 7
�I.J
---
--
/
--
/
--
/
I
T
pl
..,.., .
/
�� 2 '
H
x
P
U1TL l )
111.4
wl
- ---
__,.
w
�� F i
...
�� l
T
m2
T
'
T
T
' Fi
dw
p2
T
( PRL 2 )
( S F DC L )
d
�----- �
( MTL2 )
,T
Ld
-1
( S FEH E L )
Ll )
�- -
T
k-
- ->-
f.-- _�
w
r
' !<
p s , Ts , h s
ml
r--�
i gu r e
w g
s
t
--
)I\
o
-
d
S c h ema t i c Di a gram fo r Mo d e l C .
'
33
I I I . 3.4
T h e Det a i l ed t·1o del ( Mode l D )
I n t h e deta i l e d mo del , the e ffe ct i ve heat e xc h a n g e re g i o n ( tube
bun d l e re g i on ) i s d i v i de d i n to a subcoo l e d h ea t t ra n s fe r sect i on a n d
a bo i l i n g h e a t t ra n s fe r s ect i o n w i t h a dynami c boun da ry between t h e
two s e ct i o n s .
A l s o , t h e e ffect o f p r i ma ry coo l a n t i n l e t a n d outl e t
p l e nums i s exami n e d by a d d i n g a l um p fo r e a ch pl e num .
T h e deta i l ed mo del s t ru c t u re i s s hown i n F i gure I I I . 5 a n d i s
compo s e d o f t h e fol l ow i n g l umps .
a.
P r i ma ry s i de l um p s
1.
p r i ma ry i n l e t p l e num ( P R I N )
2.
p a ra l l e l fl ow b ra nc h subcoo l e d heat t ra n s fe r s e ct i o n
p r i ma ry l ump ( P RLl )
3.
pa ra l l e l fl ow b ra n ch bo i l i n g heat t ra n s fe r s e c t i o n
p r i ma ry l um p ( P RL2 )
4.
counte r fl ow branch bo i l i n g h eat t r a n s fe r s ec t i o n
pr i ma ry l um p ( P RL 3 )
5.
coun t e r fl ow b ra n ch subcoo l e d heat t ra n s fe r s ec t i o n
p r i ma ry l um p ( P RL4 )
6.
b.
p r i ma ry out l et p l enum ( P ROUT )
Conduct i n g tube metal l ump s .
7.
8.
pa ra l l e l fl ow b ra n c h subcoo l e d h ea t t ra n s fe r s e ct i on
tube meta l l ump ( ttfT L l )
paral l e l fl ow b ra n c h b o i l i n g heat t ra n s fe r s e ct i o n
tub e meta l l ump ( MTL 2 )
9.
counter fl ow b ra nc h b o i l i n g h eat t ra n s fe r s e ct i o n tube
me t a l l um p ( riT L 3 )
10.
cou n t e r fl ow b ra n c h subcoo l e d h ea t t ra n s fe r s ec t i o n
tube met a l l ump ( MTL4 )
\�
34
so
Ld
w
�----
L
}
( P RL 2 )
L - - - '[_
Tp
-t
( P RL 1 )
cr
m3
t
Tm
!----+-
��' T...sa c -
T
s
( SF SLl
t
w 1 , Td
r+-
( P RL 3 )
- _j_
-, -
Tm4
�
( �1TL 4 )
( PR I N )
pl
�
T sa
- - -- - - - -
( t1TL 1 )
-�
T
-
( S FBL )
t--· - �
1
( MTL 3 )
x,
m2
H
p,
F i gu r e
ll .
1
III.5
___,
.
___ _
( t1TL 2 )
T
H F i ' TF i
S c h ema t i c D i a g ram fo r r 1o d e l D .
-_t- - -
T [J
4
( P RL4 )
( S FD C LJ
J
r
Tp
( P RO UT )
o
fvJ
[J
,T
po
35
c.
I I I .4
S e c o n d a ry s i de l umps
11 .
s u bcoo l e d s econda ry fl u i d l um p ( S FSL )
12.
bo i l i n g s econd a ry fl u i d l um p ( S FB L )
13.
d rum eq u i va l en t s e c o n d a ry fl u i d l ump ( S FD R L )
14.
down come r s econda ry fl u i d l ump ( S FDC L ) .
Govern i ng Equ a t i o n s fo r Model A
I I I .4. 1
P r i ma ry L ump ( P R L )
An ene rgy bal a n ce o n t h e p r i ma ry wa te r l ump yi el d s t h e fol l ow i n g
equati on
( I I I .4.1 )
w h e re ,
t1
c
p
pl
T
p
w
p
T
pi
u
pm
mas s o f p r i ma ry water
=
s pe c i fi c h e a t o f p r i ma ry wate r
=
b u l k me a n tempe rat u re o f t h e p ri ma ry w a t e r l ump
=
ma s s fl ow ra t e o f pri ma ry wate r
=
i n l e t p r i ma ry water tempe ra t u re
=
e ffec t i ve h e a t t ra n s fe r coe ffi c i e n t between p r i ma ry
wa t e r a n d t u be me ta l l umps
S
pm
=
h e a t t ran s fe r a re a betwe e n t h e p r i ma ry wa t e r a n d t u b e
me t a l l umps
T
m
=
a ve ra ge tempe ra t u re o f t u be me ta l l ump .
S i nce t he den s i ty a n d s pe c i fi c h e a t o f t h e p r i ma ry wate r a re
a s sume d to be c o n s t an t , t h e n E q u a t i o n ( I I I . 4 . l ) c a n be wri tten i n
the form
( I I I .4.2)
36
· tP -
whe re ,
M
__£_ - res i den ce t i me of the pri ma ry wat e r i n the h e at i n g
wP -
sect i on .
I n t roduc i n g pe rturbat i on va r i a b l es , o n e obt a i n s :
( I I I . 4. 3)
whe re
6
mea n s the devi a t i o n o f the va ri a b l e from i t s val ue at
s te ady s t at e .
I I I .4.2
T u be Meta l L ump ( MTL )
An e n e rgy bal a nce o n t he tube metal l ump y i e l ds t h e fo l l owi n g
equat i o n
du 1 M C T )
dt m m m
�
=
where ,
MM
C
m
U
ms
U
- ) - U ms S ms ( T m- T s )
S
pm pm ( T p T m
( I I I .4 .4)
=
mas s o f t u be metal
=
s pec i fi c hea t o f t ube metal
=
e ffecti ve heat t ra n s fer coe ffi c i e n t between t h e tube
meta l a n d s econ d a ry fl u i d l umps
S
ms
=
heat t ra n s fe r a rea between t he tube met a l and s econdary
fl u i d l umps
T
s
=
b u l k mean tempe ra t u re i n the s e con da ry l ump ,
and other t e rms a re a s defi n e d be fo re .
Fo r co n s tant tube met a l den s i ty and s pe c i fi c heat , we h a ve
\· ( I I I .4.5)
37
T hu s , t h e
d cS T
m
pe rturba t i o n fo rm o f t he equa t i o n be come s
( U pm Spm ) O T
M C
_
dt -
m m
(
p
U
pm
S
+U S
pm ms ms eS T
)
+
M C
m
m m
u
s
ms ms �T . I I I . 4 . 6 )
u s (
M C
m m
I t i s a s sumed t h a t t h e f l u i d i n the s e co n d a ry l ump i s a s te am-wate r
mi xtu re a t t he rmo dyn ami c e qu i 1 i bri um .
Therefo re , a pre s sure c h a n g e
re sul t s i n a c h a n g e i n s a tu ra t i o n tempe ra tu re :
6
whe re
P
(
s
=
(
*
sat
) oP
aP
aT
( I I I . 4 . 7)
=
s team pres s u re
=
s l o pe o f t h e s t ra i ght l i ne a p proxi ma t i o n o f t he cu rve
aT
sat
)
aP
T
of T
s at
a ga i n s t P ( s ee Ap pe n d i x A ) .
T hu s Equa t i on ( I I I . 4 . 6 ) b ecome s
d6T
m
---eft
_
-
I I I . 4. 3
(
U
pm p m
S
M C
m m
) o Tp
_
(
Secon d a ry Fl u i d
U
S
S
pm pm+ U ms ms ) eS
T
M C
m
m m
+
(
U
S
ms ms
)
t1 C
m m
a Ts a t
aP
cS P
•
( I I I .4.8)
L ump ( S FL )
T he gove rn i n g equa t i o n fo r t h e s e co n d a ry fl u i d l ump i s
obta i n e d by a p p l y i n g ma s s b a l a n c e s fo r t he wate r a n d s te am
compo n e n ts ,
a vol ume ba l a n c e , a n d a n equa t i o n rel a t i n g t h e st eam
ge n e rat i o n rate W
sg
to the heat t ran s fe r rate to the s e c o n d a ry
fl u i d a n d t he degre e o f subcool i n g o f t h e feedwa te r e n te r i n g t h e
sy s tem ( he a t b a l a n ce ) .
*Al thoug h T s a t a n d o t h e r s a tu rat i o n p ro pe rt i e s a re co n s i d e re d a s
fun c t i o n s o f o n e va n a b l e ( P ) , t h e p a rt i a l de r i v a t i ve symbo l ( a/ ;:J P ) i s
used fo r c l a r i ty o f equat i o n s .
38
a.
Ma s s b a l a n ce
l.
W ater p h a s e
(III.4.9)
w h e re ,
r1
sw
w
sg
= mas s o f water i n t h e s e co n d a ry l ump
W Fi = i n l e t feedwa t e r fl ow ra te
2.
= s t eam gen e rat i n g ( e va po rat i o n ) ra te
S t eam ph a s e
( III .4. 10)
w h e re ,
M
= ma s s o f s team i n t he s e c o n d a ry l u mp
ss
W s o = o u t l e t s t e am fl ow ra te
b.
Vo l ume b a l a n ce
d v
sw
dt
--
+
d v
ss
dt
d v
=
dt
st = 0
( I I I . 4. l l )
w he re ,
V
= vo l u me o f wate r i n the s e co n da ry l ump
SW
v s s = vo l ume of s team i n t h e s e con da ry l ump
v s t = tota l vol ume o c c u p i ed by t h e seco n da ry fl u i d .
S u bs t i t u t i n g v s w = M
sw
and
vs s = Ms s
.
.
v
f
v '
g
w he re ,
v
v
f
g
=
s pec i fi c vo l ume o f s a t u rated wa ter
= s pe c i fi c vol ume o f s a t u rated s team
39
we get
v
d M
f
dt
sw
+
M
d v
d M
f
v
s w (ft + g
dt
ss
d v
�=
+ Ms s
0.
( III . 4 . 1 2 )
N e g l e ct i n g t he c h a n ge i n the s pe c i fi c vol u me o f s a t u ra t e d
wat e r d u ri n g a t ran s i e n t , o n e o b ta i ns :
v
d M
f
s-'w
__
dt
d v
u s i n g ____]_
dt
+
v
d
g
dt
�v
=
( __g_ )
ss
d v
M
__g_
+ s s dt
=
0
( III . 4 . 1 3 )
dP
dt
dP
we ge t
dt '
l
--a v
Ms s
•
;l P
an d so l v 1. n g f or
M
�
{
vf
d M sw
dt
+
v
d M
g
dt
ss }
( III . 4 . 1 4 )
a n d s u b s t i t u t i n g from E q ua t i o n s ( III . 4 . 9 ) a n d ( III . 4 . 1 0 )
i n to Eq u at i o n ( III . 4 . 1 4 ) a n d u s i n g the a s s umpt i o n
W
Fi
W
=
we h a v e
so '
( III . 4 . 1 5 )
whe re ,
- v .
f
9
U s i n g t he c r i ti c a l fl ow a s s umpt i o n ( see Appe n d i x
v
fg
v
have
w
so
=
c
L
. p
B ) , we
40
whe re ,
C L = s team v a l ve coe ffi ci e n t .
T h e r e fo r e ,
�� =
c.
d
dt
M
-1
ss
�
H e a t b a l ance
+
( M SW e f
3P
{v
- CL . P ) } .
(
fg W s g
( I I I . 4. 1 6)
M e ) = U S ( T - T ) + W F . c 2T F . - W h
ms ms m
s
1 P
1
so g
SS g
( I I I .4. 1 7)
w h e re ,
=
i n t e rn a 1 e n e rgy o f s at u ra t e d water
=
i nt e rn a l e n e rgy o f s a t u rated steam
C P2
=
s pec i fi c heat o f feedwate r
T Fi
=
i n l et tempe rat u re o f feedwat e r
e
e
f
g
a n d o t h e r t e rms a re a s defi ned be fo re .
e n ergy t e rms e
T he i nt e rn a l
a n d e 9 can b e re p l a c e d by t h e e n t h a l py
2
t e rms h a n d h 9 wi t hout s i gn i fi ca n t e rro r ( 2 ) d ue to
f
v
the s ma l l val u e of the fl ow wo rk te rm ( j ) ( s ee Appen di x
C).
f
S i n ce h f a n d h a re p res s u re de p e n dent , we h a ve
g
d M
3 hf
d
sw
_E.
+
h
M sw ap
. dt
f dt
+ h
d M
ss
d
t
g
=
U
+
M
h
3__g_
ss 3 p
•
d
_E.
dt
+
+ WF . c TF . - W h .
S (
ms ms T m - T s )
1 P2 1
s0 g
( I I I . 4. 1 8)
S u bs t i t ut i n g from E q u at i o n s ( I I I . 4 . 9 ) a n d ( I I I . 4 . 1 0 )
i nto Eq uat i on ( I I I . 4 . 1 8 ) a n d us i n g t h e a s s umpt i o n
41
Ms w
a
hf
dP
aP · d t
+
M
�
ss a P
+
�
•
dt
(W
- w
sg
so
) (h
g
- h )
f
=
( III.4.19)
F rom E q u a t i o n ( I I I . 4 . 1 5 ) we h a ve
( Ws g - Ws o )
av
M
a
�J
-
=
v
fg
dP
dt
-
·
P
T h u s , E q ua t i o n ( I I I . 4 . 1 9 ) be come s
( r�
sw
a hf
-
aP
+
r�
ah
_g_
ss a P
-
M
h
f .
...J:.9.
ss v
av
•
_
_g_) d P
aP
fg
dt
( I I I . 4 . 20 )
w h e re ,
h fg = h g
hf.
-
An d , the p er t u r ba t i o n equ a t i o n becomes
(M w
s
=
U
ah
-
f
+
-
aP
S ( OT
ms m s
m
M
ah
=
(M
sw
g
oT 5 =
ss v
av
_ ...9.) d o P
•
aP
fg
-
dt
_
-
a hf
aP
-
-
a T s at
aP
( I I I . 4 . 21 )
- C P 2T F } � W
so ·
i
+
and s u b s t i t u t e o W so
and
h
f
___!_9_
ah
o T s ) - W s o ( -a_g_P o P - C p 2 o T F 1. )
-
- (h
Let K
_g_ - M
ss a P
_
M
ss
=
ah
-
_9_
aP
-
t1
f
___I_9_
ss v
fg
C o P + PoC
L
L
o P t o o bta i n :
h
•
av
_ ..9.)
aP
42
I I I . 4. 4
S u mma ry o f t h e Mo de l Equat i o n s
T h e f i rs t s i mpl i fi e d mo del c a n be re p re s e n te d i n t h e fo rm o f
t h re e fi rs t o rder d i ffe re n t i a l e q u a t i o n s t h a t c a n b e w r i tte n i n the
ma t ri x fo rm
dx
A
dt _
X
-
oT
-
B u
( I I I . 4. 23)
rH 1
w h e re
oT
X +
-
Pi
P
-
u
m
Fi
OT
=
lo cl J
oP
E a c h o f t he ma t r i ces, A a n d � i s a 3 x 3 matr i x g i v i n g t h e coe ffi c i e n t s
o f t h e s t a t e v a r i a b l e ve cto r x and t h e fo rc i n g ve c t o r u .
T he n on z e ro e l eme n t s o f A and B a re l i s te d a s fo l l ows :
1
(Tp
u s
+ �m [2m )
A(l ,l )
=
A(l ,2)
=
u m s �m
2
A( 2 , l )
=
u�m s pm
Pl
MP C P l
Mm C m
u s
+ u s
m pm
ms ms )
( �
C
Mm m
aT
sat
A(2 , 2 )
A( 2 , 3)
3P
A(3,2)
=
A(3,3)
=
B(l ,l )
MP C
-
u
s
ms ms
K
l
- (U S
K
ms m s
1
T
-
p
aT
sat
3P
+
w so � + C L ( h g -C P 2 T F i ) )
ah
8 ( 3 ,2 )
8(3,3)
Go ve r n i ng Equat i o n s fo r t1o del 8
III.5
P r i ma ry L ump ( P RL l )
III .S. 1
Heat ba l an c e
�t
( r�
W C T
C T )
U (5 ) (
)
P l P l P l = P P l P i - pm pm 1 T Pl - T ml - W P C PT Pl
( III.5.1 )
whe re ,
M
ma s s o f p ri ma ry water i n P Rl
Pl =
T P l = bu l k mea n tempe ra t u re o f P Rl
(S
) = heat t ra n s fe r a re a between P Rl a n d MT L l
pm l
Tm = ave rage temp e ra t u re o f MT L 1 ,
1
and other te rms a re a s defi n e d befo re .
Fo l l owi n g the s ame p roce dure
as i n t he f i rs t s i mp l i fi ed model , we h a ve
where
:
M
l
= res i den ce t i me o f the p ri ma ry fl u i d i n P Rl .
'P l = w
III.5.2
P r i ma ry Lump { P RL2 )
The eq uati o n fo r p r i ma ry l ump P RL 2 can be obta i n ed f rom E q ua t i on
( I I I . 5 . 2 ) by re pl a c i n g T p ; by T , T
by T
and T
by T
Pl
Pl
P2
m2 as fo l l ows :
ml
dOT
P2
1
-.,..:....=. = - or
Pl
dt
' p2
44
I II . 5.3
T u be Met a 1 L ump U1TL 1 )
Hea t bal a n ce
where ,
M
ml
T
s
= mas s o f t ube met a l l ump t1TL 1
= bul k mean tempe ra t u re o f t he s e co n d a ry fl u i d l ump .
As s umi n g t hat T
s
i s equal to T
s at
' t he s a t u ra t i o n tempe ra t u re , and
us i n g t he same p ro cedure a s i n the case of the fi rst s i mpl i fi ed
model ( Mo d e l A ) , we get :
( I I I . 5 . 5)
III .5.4
Tu be Met a l L ump ( MT L 2 )
T h e e q ua t i o n fo r t ube metal l ump t.frL 2 c a n b e obta i n ed , by
s i mi l a ri ty , from Eq uat i o n ( I I I . 5 . 5 ) as fo l l ows :
d 6T m2
dt
aT
sat
aP
111.5.5
•
6P
•
( II I .5.6)
S econ dary Fl u i d Lump ( S F L )
The only d i ffe rence i n the fo rmu l a t i o n o f t h e s e co n d a ry l um p
eq u at i o n between t h i s mo d e l a n d t h e f i rs t one i s t h a t the heat i s
45
t ra n s fe rred to t he s e conda ry l ump from both me t a l l umps M 1 a n d M2 .
T h u s , t h e heat ba l a n ce e q ua t i on become s .
h ) - U ( S ) ( T -T
)
s!_
( h
ms ms 1
ml s a t
d t M sw f + M s s g
-
( I I I . 5 . 7)
That i s :
d M SW
dh
d MS S
d hf
M SW �
h
+
+ h f ---d�t- + M S S �
g dt
dt
( I I I . 5 . 8)
By us i n g t he re l a t i on s
a
dh
-at --a1=
and
h
dP
· dt
dM
dM
and s u bs t i tuti n g fo r d �w and d� s f rom E q u a t i on ( I I I . 4 . 9 ) a n d
Equa t i on ( I I I . 4 . 1 0 ) we get
dhf
dp
2
)
( M sw -aP + M s s �)
d P d t = U ms ( S ms ) 1 ( T m 1 +T m2 - T sa t
+ W Fi C P 2 T Fi - h g ( W s g - W s o )
- h f ( W Fi - W s g ) - w s o h g .
(III .5.9)
46
Us i n g t h e a s s umpt i on t h a t fee dwa te r i s a dj u s te d to mat c h s team fl ow
(W
Fi
=
W
s0
} , a n d s ubs t i t ut i n g fo r ( W
sg
- W
s0
} from E q u a t i o n ( I I I . 4 . 1 5 } ,
we get
(M
dh
sw
dh
h
f
f + M
..29.
_9. - M
ss v
s s dP
dP
fg
-
•
dv
dP
_9.) dP
dt
_
-
( III .S. lO}
I ntroduc i n g per tu r ba t i o n var i a bl es , we get :
( III.5. l l }
111 .5.6
S umma ry of t he Mod e l Equat i on s
t� d e l B c an be rep re sented i n the fo rm o f fi ve , f i rst o rde r
d i ffe re n t i a l eq u at i o n s t h a t can be wri tten i n t h e mat r i x fo rm g i ven
i n Equat i on ( ! ! 1 . 4 . 2 3 ) whe re
OTP l
6T
X =
-
oT
P2
ml
oTm
-
u
=
6T
Pi
6T
Fi
6C
L
2
oP
I n t h i s ca s e , A i s a 5 x 5 mat r i x and B i s a 5x3 ma tr i x g i v i n g
t he coe ffi c i en t s o f the s tate vari a b l e ve cto r i a n d the fo rc i n g
vecto r u , r e s p ec t i v el y .
47
T h e n o nz e ro e l eme n t s o f A a n d B a re l i s t e d a s fo l l ows :
A( l , l )
=
A( l , 3 )
=
A(2 ,1 )
A( 2 , 2 )
A( 2 , 4 )
A( 3 ,3 )
_
u
1
( -TPl
�m
(5
_
U
=
)
�m 1
1
- Tp l
T P2
=
(S )
p m pm )
MP l cP ,
MP C
l Pl
- -
=
U
+
(
,
)
U (S
m �m 2 )
+ Mp C
T P2
P2 P 1
--
(S )
p m pm 2
M P2 C P l
=
=
A( l , l )
A( l , 3)
U (S ) + U (5 )
ms ms l
l -,
m---pm�p-:-:
- ( --'-M C
ml m
__
_
_
A( 3,5)
U
m
A(4,4) - - ( p
A(4 , 5)
=
(S
) + U (S )
ms ms 2
pm 2
)
M m2 C m
A( 3,5)
A( 5,3)
A( 5 , 4 )
= A( 5 ,3)
=
A( 3 , 3)
B(l ,l
)
B ( 5 , 3)
111.6
Go v e rn i n g Equa t i on s fo r Model C
III.6.1
General
T he s t ru c t u re of t h i s seco n d i n te rme d i ate mo d e l was de s c ri bed
i n Sect i on 1 1 1 . 3 , page 2 6 , and s hown s c h ema t i c a l l y in F i g u re I I I . 4 , pa ge
32 .
The d i ffe re n ce between th i s mo del a n d the fi rst i n te rme d i ate mo de l
( Model B ) i s that a mo re deta i l e d l ump s t ru c t u re o f t h e secon dary
s i de i s u s e d .
The go vern i n g equat i ons fo r t h e p ri ma ry fl ui d a n d
t u be meta l l umps i n Mode l C a re s i mi l a r t o those i n Mo d e l B a n d a re
repea ted i n t h i s sect i on for conve n i en c e .
The govern i n g e q u a t i o ns
fo r t h e secondary fl u i d l umps a re deri ved i n de ta i l s i n ce t h ey wi l l
be u s e d i n the deve l o pmen t o f the det a i l e d model ( Model D ) .
111.6.2
P r i ma ry Lu mp
( PRLl )
The equat i o n fo r t he p r i ma ry l um p P RLl i n Mo de l C i s s i mi l a r
to E q u at i o n ( 1 1 1 . 5 . 2 ) .
That i s :
(III.6.1 )
49
I I I . 6. 3
P r i ma ry L ump ( P RL2 )
T h e e q u at i o n fo r t h e p ri ma ry l um p P RL 2 i n r1o de l C i s s i m i l a r
to E q u at i on ( I I I . 5 . 3 ) .
III.6.4
T hat i s :
T u be Met a l L ump ( MTL l )
T h e e q u at i o n fo r t h e t u be met a l l ump MTL l i n Mo de l C i s s i mi l a r
t o E q u at i on ( I I I . 5 . 5 ) e xc e pt t h a t T
s
i s con s i de red a s tate v a r i a b l e
( i n s t ea d o f p re s s u r e ) t o a c count for t h e c h a n ge i n s e co n d a ry
tempe rat u re between t h e s ub cool ed a n d bo i l i n g re g i o n s o f t h e e f fe c t i ve
h e a t exc h a n ge s e c o n d a ry fl u i d l ump .
d o Tml
u
�- =
dt
III.6.5
s
pm pml
M 1 cm
m
0
T
Pl
That i s :
U S
+U S
U S
( pm ml ms ms l ) oT , + ms ms l o T . ( I I I . 6 . 3 )
m
s
c
Mml c m
.
ml m
M
T u b e Met a l L um p ( MT L 2 )
The e q u a t i o n for t h e t u be met a l l ump MTL 2 i n r�o de l C i s s i m i l a r
to E q u at i o n ( 1 ! ! . 5 . 6 ) e xc e pt fo r t h e u s e o f T s a s a s tate v a r i a b l e
( s e e abo ve ) .
That i s :
d0T m2
dt
III.6.6
E f fect i ve H e a t E x c h a n ge S e c o n d a ry F l u i d L ump ( S FEH E L )
T h i s l ump re p re s e n t s the core o f t h e s te am gen e ra t o r s e co n d a ry
f l u i d whe re t h e ma i n h e a t e x c h a n ge p ro ce s s t a k e s p l ace as s hown i n
F i g u re I I I . 6 .
T he s e c o n d a ry fl u i d t h a t e n t e rs from the downcome r
50
[) rum
v a 1 en t
L um p
S F DR L )
1'-l!!!!!!-�a=i
w .h
2 xe
E ffec t i ve
Heat Exchan ge
Lump
( S FEH E L )
F i g ure I I I . 6 S econdary L umps fo r Mo del C .
Down comer
L um p
( S FD C L )
51
s e ct i o n re ce i ves h e a t f rom both bran c h e s o f t h e U - t u be b u n d l e a n d
i s co n ve r t e d from a s l i gh t l y s ubco o l e d l i q u i d i n to a two p h a s e
mi xt u re t h at r i s e s to e n t e r t h e d r um e q u i val e n t s e ct i o n .
A n ene rgy
b a l a n c e o n t h e con t ro l vol ume e n c l o s i n g t he e f fe c t i ve h e a t e xcha n ge
l ump y i e l d s t he fol l ow i n g e q u a t i o n
( III .6.5)
w h e re ,
�\
h
s
= ma s s o f s e c o n d a ry fl u i d
= a ve rage e n t ha l py o f s e co n da ry fl u i d
.
Q ms l = heat t ran s fe r rate from t h e p a ra l l e l fl ow t u be me t a l
b r a n c h t o t h e s e co n d a ry fl u i d = U
S ( T -T )
ms ms m 1 s
.
Q ms 2 - heat t ra n s fe r rate from t h e co u n t e r fl ow t u be me tal
-
w
b r a n c h t o t he s e conda ry fl u i d = U
1
S ( T -T )
ms ms m 2 s
= ma s s fl ow rate o f s e conda ry fl u i d from t h e down come r
to t h e ri s e r
c p 2 = s pe c i fi c h e a t o f the s u b coo l e d l i q u i d l e a v i n g t h e
down come r s e ct i o n
T
w
d
= temp e r a t u re o f t h e s u bcoo l e d l i q u i d l ea v i n g the
down come r s e ct i o n
2
= o u t l e t s e con da ry f l u i d ma s s fl ow rate
h xe = e n t h a l py of the out l e t s e c o n da ry fl u i d .
S i n c e t h e e ffe c t i ve heat exc h a n ge l ump i s b o u n de d a t t h e bottom
by t h e t u be s heet and at the top by a hypo t h et i cal p l a n e j u s t a bo ve
52
t he t u be b u n d l e , t he v o l ume o f t h e l ump i s con s t a n t a n d E q u a t i o n
( I I I . 6 . 5 ) become s
U
S (T
+ T 2 - 2T ) + w 1 cP Z T - W2 h
d
ms ms m l
m
s
e
(111 .6.6)
w h e re ,
V
s
=
vo l u me o f t h e s e c o n d a ry fl u i d i n t h e e ffe ct i ve h e a t
exc h a n ge l um p
ps
=
a v e ra ge den s i ty o f t h e s e co n d a ry f l u i d i n t h e e ffe c t i ve
h e a t e x c ha n ge l ump
T
s
=
ave ra ge tempe rat u re o f the s e c o n d a ry fl u i d i n the
e f fe c t i ve h e a t e xc h a n ge l ump .
T h e s e c o n d a ry f l u i d i n t he e ffect i ve heat e xc h a n ge l um p i s compo s e d
o f two reg i o n s , a s u bc o o l e d re g i o n whe re t h e b u l k me a n tempe rat u re
o f t he s u bcool e d wa te r i s ra i s e d from the down come r tempe rat u re to
t h e s a t u rat i o n tempe rat u re and a bo i l i n g re g i o n w h e re p a rt o f t h e
s at u rated wa t e r i s c o n ve rted i n to s te a m .
1 -- +-
1
L... C: l..o L.. s u b
b"
c
t h e l e n o,J t h
of
b e t h e l e n g t h o f t h e bo i l i n g re g i o n .
s u bcool e d re g i o n a n d L
bo i l
L
s ub
( t h e l e vel at w h i c h t h e bo i l i n g proce s s s t a rt s ) c a n b e dete rm i ned
from a c a l c u l a t i on o f t h e s e c o n d a ry tempe ra t u re p ro f i l e i n the
e ffec t i ve heat e x c h a n g e l ump a t s tea dy s ta te .
S i n c e the dynami c re s po n s e o f t h e s u bcoo l e d / bo i l i n g i n te rface
L b
s u ) i s a s s ume d to
L
(
a
n
d
h
e
n
c
e
i
n
t
h
i
s
model
,
not
c
o
n
s
i
d
e
re
d
is
L
s ub
bo i 1
the
rmo
dynami
c
p
ro
pe
rt
i
e
s
o f t h e s e co n d a ry
be c o n s t a n t . T he a ve ra g e
fl u i d i n the e ffect i ve h e at e x c h a n ge l ump a re de r i ve d from the
we i ghted a verage of t h e propert i e s i n t h e s u b co o l e d a n d b o i l i n g
re g i o n s .
Fi g u re I I I . ? s hows the v a r i a t i on o f t h e t h e rmodynami c
53
S u bcool ed
T
�--
Temper a t u r e
T
·­
r
I
d
B o i l i ng
sat
pd
De n s i ty
--
.. .... ...
E n t ha l py
hd
.
t�a s s Q u a l 1 ty
x *
d
�-·
-- - ------ --- - t1
·---
- - - -- ----- ----- -- _
_j
---- - --- - ----------1I
i
I
I
-------- - -- - - ·- - -�- - -
I
-
I
o . o ' - --��
- -- -
L
0
- --
'
sub
I
!
'------ --· --·- _
- _j_
Fi gure I I I . ?
--- - - - -
------ -- · - -----+ p xe
Lsl
- · ---
I
-- -- -- --�
-
-_
__ _ _
·
_ _ ... -
··
- -·
.-
:
I
I
_
U pwa rd Fl ow
-- -
_j
L
V a r i a t i on o f T hermo dyna m i c Pro pe r t i e s o f t he S ec o n d a ry
F l u i d i n t he E ffe c t i v e H eat E xc h a n g e L u m p .
x,
54
p ro p e rt i es o f the s econda ry fl u i d i n the s u bcoo l e d a n d bo i l i n g
re g i o n s o f t h e effe c t i v e h e a t e x c h a n g e l ump .
From t h i s f i g u re ,
t h e fol l owi n g eq u a t i o n s a re obta i ne d fo r t he a ve ra g e t h e rmo dynami c
p ro pe rt i es o f t h e s ec on d a ry fl u i d i n t h e effec t i v e h e a t exc ha n g e l ump .
T
(
s =
L
�+ T
s at
L
T +T
d s at
)
2
L
h
s
=
C
T +T
d sat
)
P2
2
(
•
sub +
L
L
( I I I .6 . 7)
( I I I . 6 . 8)
_
_
_
_
X
s ub
+ (h + � h )
L
f
2 fg
X
L
�+ �
2
L
L
.
bo i l
L
( III.6.9)
( II I .6.10)
I n t h e a bo ve e q u a t i o n s , t he s u b s c r i pt ( s ) i n d i cates a ve ra ge
p ro pe rt i es i n the e ffect i ve he a t e x c h a n ge re gi o n a n d t h e s u bs c r i pt
( d ) i n d i cates down come r e x i t p ro pe rt i e s .
S u b s t i t ut i n g from Eq uat i o n s
( I I I . 6 . 7 ) t o ( I I I . 6 . 1 0 ) i n to E qu a t i o n ( I I I . 6 . 6 ) a n d n e g l ec t i n g t he
c ha n g e s i n t h e s u bcool ed d en s i ty d u r i n g t he tran s i en t , o n e o b ta i n s
t h e fo l l owi ng equ a t i on :
T +T
d sa t
)
[C P ( 2
Ms �
dt
2
L
sub
+ (h +
L
f
-2e h fg )
d
-­
[--+ V h
x
s s dt
e
v f + 2 v fg
.
L
L
x
sub
.
+ T
s at
L
.
L
L
bo i l ]
L
boi l ]
L
bo i l
) ] + l·'� C T L
l P2 d
( III.6.ll )
55
P e r fo rmi n g t h e d i ffe ren t i a t i o n i n E q u a t i o n ( I I I . 6 . 1 1 ) a n d u s i n g t h e
a s s umpt i o n o f l i n ea r c ha n ge o f t h e t h e rmodyn ami c p ro pe rt i e s o f
s a t u rate d wate r a n d s team wi t h t h e s team gen e ra t o r p re s s u re ( s ee
Appen d i x A ) o n e can wr i te the eq ua t i o n de sc r i b i n g t h e h ea t ba l a n c e o f
the
effec t i v e h ea t exc ha n g e l ump i n t h e form :
L
dcP
bo i l ]
dt
L
(III.6.12)
of
o P and ox
e
i n the
de r i va t i ve
o f t h e a ve rage den s i ty i n the bo i l i n g re g i o n ( s e e A p pen d i x E ) .
F ro m
E q ua t i o n ( I I I . 6 . 7 ) we h a ve
0
T
L
s
=
s ub a T
2L
d
+
L
( s ub
2L
+
L
T h e rel a t i o n between t h e fl ow rates w
boi l )
L
1
aT
s at
aP
0
P
.
( I I I . 6 . 1 3)
a n d w 2 c a n be o b ta i n e d
from t h e ma s s ba l a n ce e q ua t i o n a s fo l l ows
( I I I . 6.14)
56
S u b s t i t u t i n g fo r
p
s
from E qu a t i on
( I I I . 6 . 8 ) and i ntrod u ci n g the
per turba t i o n va r i abl e s
V
Equat i on s
dox
L
do
e) J
b0 . 1
____.E. + c
1
(C
[
2b d t
s
L
1 b dt
=
- oWz .
ow ,
( I I I .6. 1 5)
( I I I . 6 . 1 2 ) , ( I I I . 6 . 1 3 ) , a n d ( I I I . 6 . 1 5 ) a re t h e de s c ri b i n g
equat i o n s fo r the e ffect i ve heat excha n ge l ump .
III.6.7
Drum Equ i va l en t Secon d a ry Fl u i d L umr
( S FD RL )
T he d rum eq u i va l en t s e conda ry fl u i d l um p i n t h i s mo de l i s bo u n de d
by t he hypothe t i cal pl ane abo ve the U-tube b u n d l e a n d t h e s team
generator s he l l i nn e r s u rface as s hown i n F i g u re I I I . S .
T h i s l ump
can be d i v i de d i nto t h ree parts
1.
Ri s e r/ s eparato r vol ume
2.
Drum wate r vol ume
3.
Drum s team vo l ume .
The ri s e r/ s e pa rato r vo l ume rep re s e n t s t he con t ro l vo l ume bo unde d
by t h e hy pothe t i c a l pl ane at t h e e x i t o f t h e e ffe c t i ve heat e xcha n ge
l ump . t he U-tube wrappe r and t he i n n e r s u rface o f the s team s eparators .
The d rum wate r vol ume re pre s e n t s the con t ro l vol ume bo unded by the
hy pot het i cal pl ane a t t he e x i t o f the effe c t i ve heat excha n ge l ump ,
the s team wate r i n te rfa ce , the hous i n g o f the s team s e p a rators and
the s team gen e rator s he l l i n n e r s u rface
( s ee F i g u re I I I . 8 ) . The d rum
s team vo l ume re p re s ents the contro l vol ume bounded by the s team wa t e r
i n te rface , t he s team s e p a rators a n d the s team gen e rator i n ne r s u rfa c e .
I n t h e ri s e r/ s e p a rator vo l ume , the two pha s e mi xture l e avi ng the
effe c t i ve heat exchan ge r l ump i s t ra n s po rte d to the steam s e p a rato rs
whe re t he s a t u rated s team part enters the d rum s team vol ume a n d t h e
s a t u rated wate r p a rt e n t e r s t he drum wa t e r vol ume .
I n the drum wate r
57
S t ea m O u t l e t
w
,h
g
so
D r um S t e a m --+----+­
Vo l u me
D r u mwater
V o l ume
� i ser/ S e p a r a t o r -- - ----1----�Vol ume
F i xed
- - - B o u n d a ry
H·-+--- U - t u b e Wra p p er
F i g u re I I I . 8
The Drum E q u i v a l e n t S e c o n da ry Fl u i d L ump .
58
vol ume , t he s a t u ra t e d w a t e r f rom t h e s team s e pa rat o r s i s mi xed wi t h
t he fee dw a te r e n t e r i n g t h ro u g h t h e feedwa te r ri n g a n d t h e s l i g h t l y
s u bcoo l e d l i qu i d t h u s fo rmed fl ows downwa rd t o e n t e r t h e down come r
l ump .
T he drum s team vo l ume a cts a s a s t e am s t o ra ge a n d del i ve ry
s e ct i on w h e re t h e s team from t h e s t eam s ep a rato rs e n t e rs a t a rate
p ro po rt i on a l to the e x i t q ua l i ty of the e va po ra t o r s ec t i o n a n d l ea ve s
a t a ra t e p ro po rt i o n a l t o t h e t u rb i n e l oa d dema n d .
T h e govern i n g
e q uat i on s fo r t h e a bo ve t h re e vo l ume s c o n s t i t ut i n g t h e d rum
e q u i va l en t l um p a re d e v el o ped bel o w .
Ri s e r/ S e p a r a t o r Vol ume
I I I . 6 . 7. 1
T h i s vo l ume i s ma i n l y a t ran s p o rt s e c t i o n b e tween t h e e ffe c t i ve
h e a t exc h a n ge l um p a n d t h e s team a n d wat e r vol umes o f t he d rum
e q u i v a l e n t l um p .
The t ra n s po rt t i me del ay a s s o c i ated w i t h t h i s
vo l ume can be a cc o u n te d fo r by cons i de r i n g a ma s s bal a n c e o n t h e
r i s e r/ s e p a rato r vo l u me a s fo l l ows :
( II I .6. 16)
w h e re ,
V
p
r
r
wz
w3
=
vo l ume o f t h e s e c o n d a ry fl u i d i n t h e ri s e r/ s e p a rato r vo l u me
=
den s i ty o f t h e s e con da ry fl u i d i n t h e r i s e r/ s e pa rato r vo l ume
=
ma s s fl ow rate o f t h e s e c o n d a ry fl u i d e n t e r i n g t h e ri s e r/
s e p a ra t o r vo l ume
=
ma s s fl ow rate of s e co n d a ry fl u i d l e a v i n g t h e r i s e r/
s e p a r a t o r v o 1 ume .
59
T h e den s i ty o f t h e s e c o n da ry fl u i d i n t h e ri s e r/ s e p a rato r vo l ume
c a n be e x p re s s e d by t h e fo l l owi n g e q uat i o n :
( I I I . 6 . 1 7)
w h e re
x
e
= ma s s q ua l i ty o f the two p h a s e mi xtu re l e a v i n g the
e f fe c t i ve h e a t e xc h a n ge l um p
v
v
f
g
=
s pe c i fi c vo l ume o f s a t u rated l i q u i d
= s pe c i f i c vo l ume o f s a t u ra t e d va po r .
F rom E q uat i o n s ( I I I . 6 . 1 9 ) and ( I I I . 6 . 2 0 ) , we h a ve
( I I I . 6 . 1 8)
S i n ce t h e r i s e r/ s e pa ra t o r vol ume i s bou n ded by t h e hy po t h e t i c a l
p l a n e a t t he e x i t o f t h e e conomi z e r - e va p o rato r s e c t i o n a n d t h e
U - t ube w ra p p e r t h en V
r
i s con stan t , a n d E q u a t i o n ( I I I . 6 . 1 8 ) becomes
( III .6.19)
D i f fe re n t i a t i n g and i n trod u c i n g t h e p e r tu rb a t i o n v a r i a b l e s , we g e t
Vr( c
d6 P
1 r dt
+
C
d6 x
e ) = 6W
W
2 - 6 3
2r �
whe re
and
c2
r
=
-
( v f+x v fg
e
)2
----­
[v
fg
] ( s e e Appen d i x E ) .
( I I I . 6 . 20 )
60
! ! ! . 6. 7.2
D rum W a t e r Vo l ume
T h e d rum wate r vol ume i s bounded at t h e bottom by t he
hypo t he t i ca l p l a n e s e pa rat i n g t h e down come r s e ct i o n from t h e
d rum s e ct i o n a n d i s bounded a t t h e t o p by t h e d rum s team/wa t e r
S u bcoo l e d wat e r from t h e fee dwa t e r t ra i n e n t e rs t h e
i n t e rfa c e .
d rum wa t e r vo l ume a n d mi xes wi t h t h e s a t u ra t e d wate r re c i rc u l a t e d
from t he s te am s e pa ra to rs .
The s l i gh t l y s ub coo l e d wate r re s u l t i n g
from t he mi xi n g p roces s l ea ve s t he d rum wat e r vol ume to t h e down come r s e c t i o n .
The ma s s a n d e n e rgy con s e rva t i o n e q ua t i o n s a p p l i e d
t o t h e d rum wate r vol ume s yi e l d :
( I I I . 6.21 )
and
d
C
C T
( M C T ) = WF C T
dw P 2 dw
i P 2 F i + W r P2 \ a t - Wdw P2 dw
dt
( I I I . 6 . 22 )
whe re
dw
=
ma s s o f d rum water
dw
=
d r um wate r t empe ra t u re
=
feedwa t e r fl ow rate
=
feedwa ter i n l et tempe ra t u re
r
=
re c i rcul a t e d wa t e r fl ow rate
sat
=
s a t u rat i o n tempe rat u re
W dw
=
M
T
WFi
T
T
Fi
w
d rum wate r vo l ume o ut l e t fl ow rat e
The re ci rcul ated wa t e r fl ow rate i s dete rmi n e d by t h e s e co n d a ry
fl u i d ma s s fl ow rate e n t e r i n g t h e s team s e p a ra t i o n s e ct i on a n d the
e v a p o ra to r exi t q ua l i ty t h u s
( I I I . 6.23)
61
where
x
and
e
w3
= e va po ra to r exi t q ua l i ty
= s e co n d a ry fl u i d ma s s fl ow ra t e e n te r i n g t h e s t eam
s e p a ra t i o n s e ct i on .
S ubs t i t u ti n g from Equat i o n ( I I I . 6 . 2 3 ) i n to Eq ua t i o n s ( I I I . 6 . 2 1 ) a n d
( I I I . 6 . 2 2 ) a n d n e g l ecti n g t he d i ffe re n c e i n s pe c i f i c h e a t between
s u bcoo l e d a n d s a t u rated wa t e r , we get :
( I I I . 6 . 24 )
and
( I I I . 6 . 25 )
Subst i tuti ng M
dw
= V
dwp dw
whe re
V
dw
=
vol ume o f d rum wa t e r
P dw = den s i ty o f d rum wate r
a n d n e g l e c t i n g a ny c ha n ge i n p
d u ri n g t h e t ran s i e n t s co n s i de re d i n
dw
t h i s a n a l ys i s , we get
( I I I . 6 . 26 )
and
As s umi n g t h a t t h e d r um wa te r vo l ume h a s a n e ffe c t i ve a rea , A dw
a n d t he l e ve l of t he d rum wa t e r i s L w ' t h e n
d
( I I I . 6 . 28 )
62
S u b s t i t u t i n g from E q u at i o n ( I I I . 6 . 2 8 ) i n to E q ua t i o n s ( 1 1 1 . 6 . 2 6 ) a n d
( I I I . 6 . 2 7 ) a n d i n troduc i n g perturba t i o n va r i a b l e s , we get
( I I I . 6 . 29 )
and
( II I . 6 . 3 0 )
E q ua t i o n s ( I I I . 6 . 29 ) a n d ( I I I . 6 . 30 ) a re t h e two go vern i n g e q u a t i o n s
fo r t he d ru m water va l u me .
III .6.7.3
Drum Steam Vol ume
The d r um s team vo l ume i s b o u n d e d by t h e s team g e n e r a t o r s ' s he l l
i n s i de s u rfa c e a n d t he s team/wate r i n t e rface i n the d rum e q u i v a l e n t
s e c t i on .
I t a c t s a s a s t o ra ge a n d de l i ve ry s e c t i o n w i t h t h e s team
gen e rato r pre s s u re a s t h e domi n a n t state va r i a b l e .
S t e am l e a v i n g
t h e s team s e p a ra to rs a c c umu l a t e s i n t he d r um s t e a m vo l ume wh i c h i n c l u d e s
the s t eam d ry e r s t h a t l i mi t t h e mo i s t u re c o n t e n t o f t h e s t e a m l e a v i n g
t h e s team gen e ra t o r to l es s t h a n 0 . 25% .
T h e rate o f steam l e a v i n g
t he s team gene rato r f rom t he s team o ut l et n o z z l e i s a fun c t i o n o f t h e
tu rb i ne s te am dema n d a n d i s determi n e d by t h e t u rb i ne val ve o pe n i n g
wh i c h i s adj u s ted a c c o rd i n g t o the e l e c t ri ca l l o a d deman d .
A ma s s bal a n c e o n t h e d rum s t eam vo l ume y i e l d s
( I I I . 6 . 31 )
63
w h e re ,
ma s s o f d rum s team
M s
d
X W
e 3
W
so
=
ma s s fl ow rate of steam from t he s team
=
ma s s f l ow rate o f s te am l e a v i n g t h e s t eam g e n e ra to r .
s e pa r a t o rs
As s umi n g t h a t t h e s team i n t he d rum s t eam vo l ume i s d ry a n d
s at u ra te d , t he n
( I I I . 6 . 32 )
where
p g = d en s i ty o f d ry s at u rated s team
V
ds
= vol ume of d rum s team.
The d ru m s te a m vo l ume can b e re l ated t o t h e d rum wa te r l e vel t h ro u g h
the equat i on :
( I I I . 6 . 33 )
where ,
V
A
dr
= vo l ume o f t h e e q u i va l e n t d rum
secti on
dw = e f fec t i ve a rea o f drum w a t e r vol ume
L
dw
= d rum wate r l e v e l .
S u b s t i t u t i n g f rom E q u a t i o n s ( I I I . 6 . 32 ) a n d ( I I I . 6 . 3 3 ) i n to E q ua t i o n
( I I I . 6 . 3 1 ) a n d d i ffe ren t i a t i n g we get
d
0 g dt ( V d r - A d J dw )
+
V
dp
s
-Tt
=
x e W 3 - W so
( I I I . 6. 34 )
E q ua t i on ( I I I . 6 . 34 ) i s t h e govern i n g e q ua t i on fo r t h e d rum s team
vol ume , howe ve r , befo re i t can be co u p l e d to t h e o t h e r e q uat i o n s
o f t h e mo de l ,
W50
may b e e i t h e r t reated a s a fo rc i � g e l eme n t o r
re l a ted t o o t he r s y s t e m s tate vari a b l e s a n d fo rc i ng e l ement s .
In
64
t h i s s t u dy , the c ri t i c a l fl ow equa t i o n i s u s e d to re l a te W
so
to the
s team gene rator p re s s u re ( P ) and the t u rb i ne val ue coe ffi ci ent ( C ) .
L
Thus
W
so
=
C
L
•
P.
( I I I . 6 . 35 )
S u bs t i t ut i n g fro m E q ua t i on ( ! 1 ! . 6 . 35 ) i n to Equa t i o n ( 1 ! ! . 6 . 34 )
and i n trodu c i n g per tu r ba t i o n v a r i a b l e s , we g et
( 1 1 1 . 6 . 36 )
Equat i on ( 1 1 1 . 6 . 36 ) i s t he go vern i n g eq ua t i on fo r the d rum s team
vol ume .
Oown come r Seco n d a ry F l u i d L ump ( S FDCL )
III .6.8
The down comer s e co n d a ry fl u i d l ump i n t h i s mo de l acts o n l y a s
a t i me del ay sect i on between t he exi t o f t he d rum wa ter vol ume t o
t h e i n l e t o f t he e ffe ct i ve h eat exchange l ump .
S i nce the down con� r
s ect i on has a fi xed vol ume and the c h a n ge i n s u bcoo l e d secon da ry
water den s i ty i s negl i g i bl e t hen a ma s s bal a n ce o n the downcome r
secon d a ry fl u i d l ump y i e l d s
w , = w dw "
( I I I . 6 . 37 )
A hea t bal ance o n t he down come r secondary fl u i d l ump re s ul t s i n
t he fol l owi n g :
( I I I . 6 . 38 )
where ,
M
d
cp2
= mas s o f t h e secondary fl u i d i n t h e down come r l ump
= s peci fi c heat of t he secon dary fl u i d i n the down come r l ump
65
T dw
=
tempe ra t u re o f t h e s u bcool e d fl u i d ente ri n g the downcomer
l ump
Td
=
tempe ra t u re of t h e s ubcoo l e d fl u i d l e avi ng t h e down come r
l ump .
S i n ce M
d
i s constan t , t hen
dT
d = w T
W T .
M -at
dw dw - l d
d
( I I I . 6 . 39 )
S u bs t i t u t i n g from E q uati on ( I I I . 6 . 3 7 ) i nt o E q uat i on ( I I I . 6 . 39 )
an d i ntrod u c i ng perturba t i on vari abl es , we get
( I I I . 6 . 40 )
E q ua t i o n ( I I I . 6 . 40 ) i s t h e de s c ri bi ng eq ua t i o n fo r t h e down come r
secon d a ry fl u i d l um p .
1!1.6.9
T he Rec i rcul at i o n Loop Equati on
In o rde r to s o l ve t he equa t i o ns fo r Model C, the va ri a b l e
w1
( t he fl ow f rom t he down comer i n to t he ri s e r ) h a s to be re l ated to
o t h e r sys tem s tate va ri a b l e s .
The d i re c t method fo r obta i n i n g t h i s
re l at i on s h i p i s t o a p pl y t he momentum theo ry o n t h e re c i rcul a t i on
secondary fl u i d l oo p between the down come r l ump a n d the e ffe ct i ve
hea t exchange l um p .
S i nce the rec i rcul a t i on ra t i o i s a s s ume d to
be known from the s t eam gen e ra to r des i gn data , t he mome n t um b a l ance
can be re p l aced by equat i ng t he dri vi ng pre s s u re d ue to t he s ta t i c
head d i ffe rence to the accel e rati on a nd fri cti on p re s s u re d ro ps a l ong
the l oo p .
S i n ce the dynami c p re s s u re d ro ps
( fri cti on a n d acce l e ra t i o n )
a re p ro po rt i on a l to t he s q u a re of t h e fl ow , then t h e mome n t um bal ance
e q u at i on can be repl a ced by t he fo l l ow i n g eq uati on
66
( I I I . 6 .41 )
w he re ,
�p
C
w
= d r i v i n g p re s s u re d u e t o s t at i c h e a d d i ffe re n c e
d
= e ffec t i ve dyn ami c p re s s u re coeffi c i e n t
d
= rec i rc u l a t i o n ma s s fl ow rat e .
1
From E q ua t i o n ( I I I . 6 .41 ) we h a ve :
1
W 1 = __
Let c
1
=
ICd
_1_ t he n ,
lfd
/�P
d
( I I I . 6 . 42 )
•
( I I I . 6 . 43 )
T he d ri v i n g s ta t i c h e a d p re s s u re � p
d
can be ca l c u l ated a s
fo l l ows :
( I I I . 6 . 44 )
w he re ,
( �P )
OT W
= p re s s u re h e a d of the re c i rc u l a t i o n l o o p o u t s i de
t he t u be w r a p p e r
( � p ) I TW = pres s u re h e a d of the re c i rc u l a t i o n l oo D i n s i de
t he t u be w ra p pe r .
T he s e con d a ry fl u i d o u t s i de t he t u be w ra p pe r c o n s i s t s o f two p a rt s
l.
t h e s u bcoo 1 e d l i q u i d i n t he d rum wa t e r vo l ume
2.
t he s u bcool e d l i q u i d i n t h e down come r l ump .
T h e re fo re , ( 6 P )
O TW
( �P )
OTW
c a n be ca l c u l a t e d from t he fo l l ow i n g e q u a t i o n
1
.
ps 1
=
l
( p dw dw
+
1 44
P d c l dc )
•
_g_
g
c
( I I I . 6 . 45 )
67
where
p dw = d en s i ty o f t he seco n d a ry s u bcoo l e d l i q u i d i n t h e drum
wat e r vo l ume
L dw = h e i g h t o f t he s e co n d a ry s u bcoo l e d l i q u i d i n t h e drum
wat e r vol ume above t he hy po t he t i c a l p l a n e a t t h e
e ffect i ve h e a t e x c ha n ge e x i t
p d c = den s i ty o f t h e s u bcoo l e d s e co n d a ry l i q u i d i n t h e
down come r
L
dc
1 ump
= h e i g ht o f t he down come r l ump me a s u re d from t h e t u be
s heet
2
g = g ra v i tat i o n a l a c ce l e rat i o n ( 32 . 2 ft/ sec )
l b m ft / s e c 2
).
g = con ve rs i o n fa cto r ( 32 . 2
lb
c
f
Si nce g and
g
c
are n ume ri c a l l y eq u a l i n t h e foo t - p o u n d - s e co n d ( fp s )
a re e q u a l , t h e n E q u at i o n ( I I I . 6 . 4 5 )
and p
u n i t sys tem a n d p
dw
dc
red u c e s to
( I I I . 6 . 46 )
w h e re
and
T h e s e conda ry fl u i d i n s i de t h e t ub e w ra p pe r cons i s t s o f two
pa rts
l.
t h e s econ d a ry fl ui d i n the e f fe ct i ve heat e xc h a n ge l ump
2.
t he s e conda ry f l u i d i n t h e r i s e r/ s e pa ra t o r v o l ume .
68
The re fo re , ( � P )
I TW
c a n be cal c u l a t e d f rom t he fo l l owi n g e q u a t i o n :
P L + p L
s s
r r . _9_
1 44
( I I I . 6 . 47)
whe re
Ps
=
a v e ra g e den s i ty o f s e co n da ry fl u i d i n t h e e ffe c t i ve
h e a t e x c ha n ge l um p
Ls
=
1 e n g t h o f t h e heat e x c ha n ge l ump
Pr
=
d en s i ty o f t h e s econ d a ry fl u i d i n t h e ri s e r/ s e pa ra to r
vo l ume
L
r
=
h e i g h t o f t h e s econ d a ry fl u i d i n t h e ri s e r/ s e p a ra t o r
vol ume .
S i n ce g a n d g
c
a re n ume ri c a l l y e q ua l i n t h e fps u n i t sy s te m ,
E q u at i o n ( I I I . 6 . 47 ) be come s
( �P )
nw\ .
=
pS l
p
L + p L
s s
r r
1 44
( I I I . 6 . 48)
S u b s t i t u t i n g from E q uat i o n s ( I I I . 6 .46 ) a n d ( I I I . 6 . 48 ) i n t o
Eq uati o n ( I I I . 6 . 44 ) a n d u s i n g t he re s u l ts i n E q ua t i o n ( I I I . 6 . 43 ) ,
we g e t
( I I I . 6 .49)
T h e v a l u e o f t h e re ci rc u l a t i o n fl ow coe ffi c i e n t
from t he s t e a dy s t ate c o n d i t i o n s .
c 1 can b e e s t i ma t e d
T h e de v i a t i o n o f
w1
from i t s
s teady s t ate v a l ue can be o bta i ne d by t a k i n g t h e deri vat i ve o f
Equat i o n ( I I I . 6 . 49 ) t h u s
oW 1
=
c
1
_
12
o(p l ) - o(p L + P L )
s s
r r
d d
2 /p d l d
-
( P s L s + P rL r )
( I I I . 6 . 50 )
1
1
69
W he n s ubs t i t ut i n g the e xpres s i on s fo r
(III.6. 8)
p5
a n d P r from E q u at i on s
a n d ( 1 1 1 . 6 . 1 7 ) a n d a s s umi n g t ha t the change i n v
f
and v
9
wi t h sys tem p re s s u re i s l i nea r , t hen E q u at i on ( 1 1 1 . 6 . 50 ) can be re duced
to t he fo l l owi n g e q u a t i on whi c h i s s u i ta b l e fo r coupl i n g w i t h other
model eq u at i o n s
( I I I . 6 . 5l )
w he re
ol
dw
oP
ox
e
=
d e v i a t i on o f d rum wat e r l e ve l from s te ady s tate
=
dev i a t i o n o f s team p re s s u re from s teady s tate
=
devi at i on of mas s q u a l i ty a t the o utl et of the
e ffec t i ve heat exchange l ump from s t e ady s t a te .
he e x p re s s i on s fo r t h e coeffi c i ents a , a , a n d a a re deri ve d
2
3
1
1
Appen d i x D .
1 .6 . 1 0
S ummary o f the r�odel Equa t i on s
T h e go ve rn i n g equat i on s o f t�de l C t h a t we re de ve l o pe d i n t h e
!ce e d i n g secti ons can be rep resented by t h e mat ri x e q ua t i on
·e
x
1d A
2
t
( d i fferen t i al + a l g ebra i c ) * vec to r
=
sys tem v a r i a b l e s
=
coeffi c i e nt mat ri ces
=
fo rc i n g vecto r .
A l i s t o f t he system v a r i abl e s i s g i ven i n Tabl e I I I . l a n d t he
·o
e l emen ts o f the ma t r i ces A
1
and A
2
a n d the fo rci n g ve c t o r t
ven bel ow .
1he defi n i t i on o f d i ffe ren t i al a n d a l gebra i c vari ab l e s wi l l be
ed i n Chapter I V .
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
70
TAB L E I I I . 1
SYSTEM VAR I AB L E S FOR MOD E L C
State V a r i a b l e
N umbe r
Symbol
De s c ri pt i o n
1
P a ra l l e l fl ow b ra n c h p ri ma ry wate r
t empe ra t u re
2
Co u n t e rfl ow b ran c h p r i ma ry wa t e r
temperat u re
3
Pa ra l l e l fl ow b ra n c h t ube me t a l
tempe ra t u re
4
Co u n te r fl ow b ra n ch t u be me tal
t e mp e ra t u re
5
p
S t e am p re s s u re i n t he d r um s t eam
vo l ume
6
Ma s s q ua l i ty at t h e e x i t o f the
e ffe ct i ve h eat e x c h a n ge l ump
7
Water t e m pe ra t u re i n the d rum wa te r
vol u me
8
9
D rum wate r l e vel
Down come r tempe ra t u re
*10
Avera ge tempe ra t u re i n t h e e ffe c t i ve
h e a t e x c h a n ge l um p
*11
Ma s s fl ow r a t e a t t h e i n l et o f t h e
e ffe c t i ve h e a t e xc h a n ge l um p
*12
Ma s s fl ow rate a t t h e e x i t o f t he
e ffe ct i ve h e a t exc h a n ge l um p
13
Ma s s fl ow ra te at t h e e x i t o f t h e
s e pa ra to r/ r i s e r vol ume
*Al g e b ra i c va r i a b l e s .
71
1.
T he A 1 Mat r i x
A ( 1 ,1 ) = 1 . 0
1
A ( 2 ,2 ) = 1 . 0
1
A (3,3) = 1 . 0
1
A (4,4) = 1 .0
1
L
C
. � . R
A ( 5 , 5 ) = [M
1
5
L
+ C 1 b Vs h s
A1 ( 7 ,5 ) =
v
A1 ( 7 ,6 ) =
v
L
•
s
L
•
s
v
2
L
•
boi l
L
boi l
L
. c
,r
A (8,6) = v . c
2r
r
1
A ( 9 , 8 ) = 0 dwA d
1
A1 ( 8 , 5 ) =
r
A ( 1 0,7) = M
1
dw
A 1 ( 1 0 ,8 ) = T 0 dwA
dw
dw
A ( 1 1 ,5) =
1
A ( 11 ,8) =
1
v
s
-
A ( 1 2 ,9 ) = M
1
d
�
3p
3P
P g A dw
•
•
bo i l
L
c,
b
cz
b
3T
s at
+ Ms
3P
72
2.
T he A
Mat ri x
2
1
A2 ( 1 , 1 ) = ( -'P 1
u
_
s
pm pml
r� 1 C )
P P1
- u pms pml
A2 ( 1 , 3 ) =
A2 ( 2 , 1
+
M
C
P l Pl
) = - _1_
T
Pl
s
pm pm1
M c
m1 m
u
A2 { 3 , 1 )
=
-
u
=
s
+ u s
pm pm 1
ms ms 1
M c
m1 m
u
s
ms ms
M 1c
m m
u
=
s
pm pm2
Mm 2 c
m
_
u
=
s
+ u s
ms ms 2
pm pm2
r� 2 c
m m
s
ms ms2
M 2c
m m
u
=
A2 ( 4 , 1 0 )
A (5 ,3) =
2
A2 ( s , 4 )
A2 ( 5 , 5 )
=
=
-
- U S
ms ms
- u ms s ms
W
3h
+
2 [�
,l P
x
<) h
__f£]
e aP
73
=
A (5,6)
2
A ( 5 ,9)
2
W 2 h fg
- w cp 2
1
=
A ( 5 , 1 0)
2
=
2 U
S
ms ms
A (5,12)
2
=
h
+ x h
e fg
aT
sat
JP
L
sub
- 2L
A (6,9)
2
_
A (6 . 1 o)
2
A (7,12)
2
A2 ( s , 1 2 )
A (8, 1 3)
2
A ( 9,6 )
2
=
- 1 .0
=
1 .0
=
- 1 .0
=
1 .0
w
=
)
A (9,1 3)
2
=
=
A ( 1 0,6)
2
=
A ( 10,7)
2
=
A ( 1 o,1 1 )
2
1
- ( 1 - xe ) w3
w
T
3
W dw
=
=
w
- ( 1 - xe )
=
A ( 1 0, 1 3)
2
=
3
1 .o
=
A2 ( 1 0 , 5 )
A ( 1 1 ,5 )
2
- 1 .o
=
A (7,11 )
2
A (g,1 1
2
Lsub
L i1
+ bo )
( 2
L
L
=
A (6,5)
2
f
T
JT
J
�
at
s at
=
W1
dw
(1
- xe ) \ a t
C L ( 0 . 0 i n c a s e o W s o i s u s e d a s fo rci n g )
74
A ( l l ,6)
2
=
A2 ( l 1 , 1 3 )
A2 ( 1 2 , 7 )
A2 ( 1 2 , 9 )
A2 ( 1 3 , 5 )
A ( 1 3,6)
3
w
-
- x
=
= - w
3
1
e
- w,
=
=
w,
- a2
-
- a
-
3
A2 ( 1 3 , 8 ) = - a 1
A ( 1 3,l l )
2
3.
=
1 .0
T h e Forc i n g V ec tor f
f( 1 )
1
T .
Tpl o Pl
= -
f ( 9 ) = oW
Fi
f( 1 1 )
III.?
P
o CL
( - cS W
so
i n case W
so
i s l e ft a s fo rc i n g ) .
Go ve rn i n g Equ a t i on s fo r Mo d e l D
III.?. 1
P r i ma ry S i de Equ a t i o n s
The p ri ma ry s i de eq ua t i o n s c o n s i s t o f two e q ua t i o n s fo r t h e
i n l e t a n d o u t l e t p l en ums a n d fo u r eq u a t i o n s fo r t h e p r i ma ry fl u i d
l umps i n the t u be b u n d l e re g i o n ,
I n de ve l o p i n g t h e eq u a t i on s fo r
t he p ri mary l umps i n t h e heat t ra n s fe r re g i o n , s pe c i a l co n s i de ra t i o n
i s g i ven to t h e dyn am i c bo unda ry betwe e n t h e pri ma ry l umps i n the
75
s u bcoo l e d r eg i o n a n d t ho s e i n t h e bo i l i n g r eg i o n .
B o t h co n t i n u i ty
( ma s s ba l an c e ) a n d en ergy ( hea t ba l an c e ) e q u a t i o n s a re con s i d e r ed
fo r d e sc r i b i n g t h e tran s i en t b e ha v i o r o f ea c h l ump .
T h e fo rmu l at i o n
t h a t r e s u l t s i s a s o - ca l l ed mo v i n g b o u n d a ry mod el .
I n thi s case, the
bo u n dary i s d e termi n ed by t he h ea t i n g l en g t h req u i red t o b r i n g t he
s u bcoo l ed wa t er u p to satu ra t i on t empe r a t u re .
T h i s d i s t a n c e a l so
f i xes t h e l en g t h of t h e met a l and pr i ma ry l umps t ha t are ad j a c e n t to
t h e s u bco o l ed s ec o n d a ry sect i o n .
I n l e t P l enum ( P R I N )
I I I .7.1 .1
T he i n l e t p l e n u m i s con s i d e r e d a s a wel l s t i rred tan k .
A heat
ba l an ce y i el d s the fo l l owi n g equa t i o n
( III.7.1 )
w he r e
M
T
Pi
Pi
e.
1
=
ma s s o f p r i ma ry wat e r i n t h e i n l et pl e n um
=
bu l k mea n t emperat u r e o f t h e i n l et pl en u m
=
p r i ma ry water i n l et t em p e ra t u r e
a n d o th e r terms a re a s d e f i n ed b e fo r e . .
S i n c e t h e d en s i ty a n d s p ec i f i c h ea t o f t h e p r i ma ry water are a s s umed
co n s tan t , t h e n
( I I I . 7. 2)
D i v i d i n g t hro u gh by M P C P w e g e t
i l
d Tp .
l
dt
-
w h e re
'P i
=
Mp .
W 1
-
Pi
=
l
T
p
-
i
o T P 1. (
e.
1
-
T p ,. )
( I I I . 7 . 3)
= r e s i d en ce t i me o f p r i ma ry wa t e r i n t h e i n l e t pl en u m .
A n d t he pertu r ba t i o n form o f t h e e q ua t i o n fo r t h e i n l et pl e n u m b ecome s
( I I I . 7.4)
76
l
- __ 6 T P
1
'p;
.
111.7.1 . 2
a.
+
_l_
'Pi
o e.
1
.
( III . 7.4)
P r i ma ry L ump ( PR L l )
Ma s s ba l an c e
( I I I . 7 . 5)
where
MP
= ma s s o f p r i ma ry water i n t h e l um p
l
W P i = ma s s fl ow rate a t t he e n t ra n c e o f t h e l ump
W P l = ma s s fl ow r a t e a t the e x i t o f the 1 ump .
But , M
= p A l
Pl
p p sl
where
p
A
p
p
( I I I . 7.6)
= d en s i ty o f p r i ma ry wa t e r
= fl ow a r ea o f p r i ma ry wa ter
L s l = l en g t h o f t h e l ump .
S i n c e t he fl ow a r ea a n d t h e d en s i ty o f t h e p r i ma ry wa ter
are con sta n t s , t hen E q u a t i on ( I I I . 7 . 5 ) becomes
b.
�t
p PA P
H e a t ba l a n c e
(M
C T
Pl Pl Pl )
where
T
Pl
Q pml
=
dL
d
1�
�
l
= Wp ; - WP l .
C T
P / P l TP i - WP l P l P l -
( I I I. 7 . 7)
Q pml
( I I I . 7 . 8)
= b u l k mean t emperat u re o f t he l um p
= h eat tran s fe r ra t e be tween ( P RL l ) and ( MT L l ) l um p s .
S u b s t i tu t i n g from E q ua t i on ( I I I . 7 . 6 ) i n to E q u a t i on ( I I I . 7 . 8 )
we get
dT
Pl
C
(
L
A
pp
dt
p Pl sl
+
T
dl
sl
---eft
) = W P i C P l T P i - �� P l C P l T P l - Q pml '
Pl
( II I . 7.9)
77
S u b s t i t ut i n g Q
pm l
=
U
p l (T
- T )
pm rl s l P l
Ml
whe re ,
P
h e a t t ra n s fe r a re a/ un i t l en gt h be t ween p ri ma ry
rl =
wat e r l ump ( P RLl ) a nd t ube me t a l l um p ( MT L l )
T
Ml
= a ve rage tempe ra t u re o f l ump ( MT L l )
a n d u s i n g E q u at i on ( I I I . 7 . 7 ) i n to E q u a t i o n ( I I I . 7 . 9 ) ,
we get
( I I I . 7. 1 0)
Re a r ran g i n g a n d d i vi d i n g t h ro u gh by M
C , then
Pl Pl
E q u a t i o n ( 1 1 1 . 7 . 1 0 ) red u ce s t o
( I I I . 7. 1 1 )
whe re ,
I I I . 7. 1 . 3
a.
P r i ma ry L ump ( P RL 2 )
Mas s b a l a n ce
( I I I . 7. 1 2 )
78
w h e re ,
M p 2 = mas s o f p ri ma ry wa te r i n t h e l um p
W P l , wp 2 = i n l et a n d o ut l e t ma s s fl ow ra t e s .
S u bs t i t ut i n g
( II I . 7. 1 3)
w h e re ,
L
s2
= l en gt h o f t h e l ump , t h e n
p P AP
Si nce L
51
dl 2
= WP l - WP 2 .
d
�
( II I .7. 14)
+ L 5 2 = con s t an t , t h e n
dl 2
s -
dl
sl
crt - - -at
( II I . 7. 1 5)
a n d E q u a t i on ( I I I . ? . 1 4 ) re duces to
dl
sl
W P 2 -- W P l + p A
P P -ar- .
( III .7. 16)
B ut from E q u a t i o n ( I I I . 7 . 7 ) , we h a ve
( I I I . 7. 1 7)
The re fo re , t h e ma s s b a l a n c e e q ua t i o n fo r t h e p r i ma ry
wate r l ump ( P RL 2 ) bec ome s
( I I I . 7 . 1 8)
b.
Heat bal ance
( I I I . 7. 1 9)
79
whe re ,
T
=
P2
b u l k mea n tempe rat u re o f the l um p
Q p mZ = heat t ran s fe r rate between ( P RL 2 ) and ( MT L2 )
1 umps .
S ub s t i t ut i n g fo r M
p 2 f rom E q u a t i o n ( 1 1 1 . 7 . 1 3 ) a n d u s i n g
E q u a t i o n s ( 1 1 ! . 7 . 1 7 ) a n d ( I I I . 7 . 1 8 ) , w e get
( 1 I I . 7 . 20 )
S ub st i t ut i n g Q Z = U p L ( T P Z - T )
pm r l s 2
M2
pm
( I I I . 7.21 )
where ,
T
M2
=
a ve ra ge t empe r a t u re o f l ump ( MT L 2 )
a n d d i v i d i n g t h ro u g h by M Z C , we ge t
P Pl
( I I I . 7 . 22 )
whi c h u po n i n t rod u c t i on o f p er t u rba t i o n va r i a b l e s becomes
( I I I . 7. 23)
80
whe re ,
1
5
I II . 7. 1 . 4
a.
MP 2
P2
-
Wp ;
= p
pm2
rl
•
L
s2
'
P ri ma ry L ump ( P RL 3 )
M a s s b a l a n ce
d
dt
( M P 3 ) = W P2 - W P 3 '
( I I I . 7 . 24)
= p A L
S i n ce M J = M
P
p2
p p s2
a n d wp = W ' t h e n
2
Pi
d
L ) = WP i - WP 3
(
dt P p A p s 2
( II I . 7.25)
u s i n g t h e a s s umpt i o n o f c o n s tant p ri ma ry wate r de n s i ty
a n d s o l v i n g fo r W P J ' w e get
W P 3 = Wp ;
- p A
P P
dL
d
�2
.
( I I I . 7 . 26 )
S ubs t i t ut i n g from E q ua t i o n ( I I I . 7 . 1 5 ) we get
WP 3 = WP i + p A
P P
b.
dl
sl
-ar- ·
( I II . 7. 27)
H e a t b a 1 a n ee
( I I I . 7 . 28)
U s i n g E q u a t i o n s ( 1 ! 1 . 7 . 1 8 ) a n d ( I I I . 7 . 26 ) to s ub s t i t ute
fo r w p and W ' we get
PJ
2
P pA p C P l
(T
dL
dT
__2_?_ + L s2 _12)
P 3 dt
dt
( I I I . 7 . 29 )
81
D i vi d i n g t h ro u g h by M
c
p3 Pl
=
p p A p L s 2 c P l , w e get
( I I I . 7 . 30 )
( I I I . 7 . 3l )
w h e re
T
and
+
a verage t empe ra t u re o f l ump ( MT L 3 ) ,
M3 =
i n trod u c in g pert u rba t i on va r i a b l e s , we g e t
u
s
p m pm3
M P lP l
6T
M3 =
( I I I . 7 . 32 )
w h e re ,
III.7. 1 .5
a.
P ri ma ry L ump ( P RL4)
Ma s s b a l a n ce
( I I I . 7 . 33 )
s u bs t i t u t i n g M
p PAP
A l
P4 = p p p s l , w e get
dl
d
�
1 =
W P 3 - W P4 '
( I I I . 7 . 34 )
From E q ua t i on ( I I I . 7 . 2 7 ) , we h a ve
dL
sl - (
A
---at
- WP i
Pp p
+
P p Ap
dL
sl )
--at
- W P4
( I I I . 7 . 35 )
82
from w h i c h we have
( I I I . 7 . 36 )
b.
Heat bal a n ce
( I I I . 3. 37)
S ub s t i t u t i n g fo r M
p4
= p A l
and us i ng Equati ons
p p sl
( ! ! 1 . 7 . 2 7 ) a n d ( I I I . 7 . 36 ) , w e g e t
p Ap C P l
p
dT
dl
T
( P4
s1 +
L
-en:-
sl
dl
P4
sl
�) = ( W P i + p P A P �) C P l T P 3
( I I I . 7 . 38 )
Di v i d i n g t h ro u g h by M
c
a n d re a r ra n g i n g , we h a v e
P4 P l
( I I I . 7 . 39 )
Q pm4
p
pm r l l s l
per t u r ba t i o n var i a b l e s , we get
S u bs t i t u t i n g fo r
w h e re ,
T P4 = T P l
s
pm4
= s
MP 4 = M
pml
P1 '
=
U
( T p4 -T �4 ) a n d i n trod u c i n g
t
83
111.7.1 .6
Outl et P l enum ( PROUT)
The equat i o n fo r the o u t l et pl enum i s obta i n ed i n the same
way a s fo r t h e i n l e t pl enum.
The equa t i on i s a s fo l l ows :
doT
1
Po
-:-:-_ = __ o T
dt
'
P4
Po
1
__ 0 T
'
Po
Po
_
( I I I . 7 . 41 )
where
= '
Pi
= pr i ma ry wat er o ut l et t emperature .
1 1 1 . 7. 2
Tu be Me t a l Equa t i o n s
As i n t h e ca se o f t h e pr i ma ry wa ter l umps i n t h e tube bundl e
reg i on , s pe c i a l con s i dera t i on i s neces sa ry for the trea tmen t o f t he
dynam i c bo undary between the su bcoo l ed and bo i l i n g sect i on s .
Howeve r , the t reatmen t of t he mov i n g bou nda ry i s d i ffe r en t from t h e
c a s e o f t h e p r i ma ry wa t e r l umps .
bo undary i s mo v i n g a t a rate dl
H e r e , i t wa s a s s umed that i f the
/ d t , i t add s (or s ub t ract s ) a
dl
l
quan t i ty of h ea t a t a ra t e p A C i
to the meta l l ump under
m m m m d
s1
con s i d e ra t i on
�
where
Pm
= den s i ty of t u b e metal
A m = cro s s sect i o n a l ar ea of tube met al l um p
C m = s pe c i f i c heat o f tube metal
im = a verage t empe ra ture a t t he bo undary .
Thi s a ppro a c h res u l ts i n ma i n ta i n i n g a dyna m i c hea t bal ance
for each l ump d u r i n g t h e tran s i en t .
84
1 1 1 . 7.2. 1
T ube Meta l Lump ( MT Ll )
Heat ba l ance
d (M
-- q·
(�
pml - 4ms l ) 1 + p mAmC mfm 1
ml C mT ml )
dt
T + T 2
S u bs t i t u t i n g Mml = p mAm l s l and T ml = m l 2 m t hen
dT
dl 1
ml
s
L
T
p mA mC m ( ml err-- + s l crt)
=
( I I I . 7 . 42 )
Q pml
T ml + T m2 d l s l
) crt
- { Q ms 1 ) l + pmAm C m (
2
.
( I I I . 7 . 43 )
D i v i di n g t h ro u gh by Mml c a n d col l ec t i n g te rms , then
m
( I I I . 7 . 44 )
- u ms 1 P r2 L s 1 ( T m 1 - r s 1 ) ] .
( I I I . 7 . 45 )
1
I n trodu c i ng pertu rba t i o n var i a bl e s a n d re pl a c i n g oT s l by � o T d + o T sa t ) ,
a T s at
Where .q sat =
a P oP the n t h e de scr i b i n g equat i o n for t he fi rst tube
u
me ta l l ump ( MT Ll ) becomes
85
u
s
pm pml eS T
Pl
Mm l C m
111. 7.2.2
_
(
+
u
S
. S
U pm
p ml + Ums l ms l ) cS T
ml
Mm l C m
s
ms l ms l
2 M C
ml m
aT
s a t cS P .
aP
( I I I . 7 . 46 )
T u be Metal L ump ( MTL 2 )
Hea t ba l an ce
�t
(M C T )
m2 m m2 = Q pm2
(Q
)
ms 2 2
T ml + T m2 d L s l
(
)
- 6 mAm C m
2
-ar-
( I I I . 7. 47)
U s i n g t h e s ame p ro cedure as i n l um p MTL l re s u l ts i n the fo l l ow i ng
eq u a t i on
u
s
pm pm2
M m 2 C m c5 T P2
_
U S
+U S
( pm pm2c ms 2 ms 2 ) c5 T m2
r� 2
m m
( I I I . 7 . 48 )
86
III.7.2.3
T u be Me t a l L ump ( MT L 3 )
H e a t b a l ance
( I I I . 7 . 49 )
Fo l l owi n g the s ame p rocedure as i n l ump MTL2 , we h a ve
d 0 Tm
3
dt
u
pms pm2
Mm 3 C m o T P 3
=
S
+ U ms 2 S ms 2
pm2
pm
) oT
(
m3
Mm 3c m
U
_
( I I I . 7 . 50)
I I I . 7. 2 .4
T ube Meta l L ump ( MTL 4 )
A ppl y i ng t he h e a t bal ance a n d us i n g t h e s ame p roce d u re , t h e n
the govern i n g equ at i o n for l ump MTL 4 c a n be w ri tten a s
do T m4
dt
u
=
+
+
u
S
s
pml o
TP 4
r� 4 c
m m
( pm pm 1
U
_
pm
+ U ms l S ms l
M m4 c m
) n m4
s
ms l ms l o
2 M m4 C m T d
S
U
:J T s a t
1
( 2m s m s l )
:J P
M m4 C m
,�
P.
( I I I . 7. 51 )
87
I I I . 7. 3
III.7.3.1
Secon d a ry S i de Equa t i on s
Gen e r a l C on s i de ra t i o n s
H e a t i s t ra n s ferred from t he hot p r i ma ry fl u i d ( rea cto r coo l ant )
to the s econdary f l u i d ma i nl y i n the t u b e b u n dl e re gi o n s omet i mes
referre d to as the 11 Co re . 11( 3 Z ) Part o f the hea t t ran s ferred i s u s ed
i n remo v i n g t h e s ubcool i n g o f t h e secondary wa ter e n t e ri n g the t ube
re gi on from t h e down come r .
The other part i s u s e d i n t h e bo i l i ng
p roce s s and con ve rt i n g part of the secondary wate r i nto s team .
T he
s team water m i x t u re t h u s formed con t i n ues to mo ve u pwards to the
s team s e p a ra to rs w he re the s te am i s s e pa rated and mo ves u pwa rds to
the steam s to ra ge and del i ve ry sect i on wh i l e the s e pa ra t e d water
moves downwa rds to mi x wi t h feedwat e r i n the downcomer .
The c i rc u l a t i o n
between t he downcome r a n d t he t u be reg i o n i s a ch i e ve d t h ro ugh the
n a t u ra l con vect i on p roces s due to the di f fe re n ce i n de n s i ty o f the
s e conda ry fl u i d i n these two re g i o n s .
Be ca u s e o f t h e two d i ffe re n t heat t ra n s fe r me c h a n i s ms between
the t u be me ta l and the secondary fl u i d , t he a c t i ve heat t ran s fer
re g i on i s d i v i ded i nto two s e c t i on s ; a s ub cool e d ( o r p reboi l i n g )
secti on a n d a bo i l i n g sect i on ( e va po rato r ) .
I n the s ubcool e d sect i on ,
t he tempe rat u re o f the secon da ry fl u i d ri s e s from T d a t the downcome r
o u t l et to T s a t at the s t a rt i n g o f t h e b o i l i n g s e c t i o n . I n t he
boi l i n g sect i on , t h e tempe rat u re stays a t T s a t wh i l e t he ma s s qua l i ty
( x ) o r vo i d fra ct i on ( ) i n crea ses a s t h e secondary fl u i d mo ves
a
u pwards and mo re heat i s t ran s fe rre d to i t .
A t the e n d of the heat
t ran s fe r reg i o n , the secondary fl u i d l ea ves the boi l i n g sect i o n w i t h
an ex i t q ua l i ty ( x e ) .
88
T h e l en g t h of the subcoo l e d sect i on i s de termi ned by the fracti on
o f the h e a t t rans fe rre d nece s s a ry to rai se the s econdary fl u i d
tempe ra t u re o f the s e conda ry fl u i d from T d to T s a t ·
T h e go vern i n g equat i on s fo r the boi l i n g sect i o n a re s i mpl i fi ed
by a s s um i n g that t he mass q ua l i ty o f t he two p h a s e mi x t u re c hanges
l i n ea rl y w i t h d i s t ance from the s ubcoo l e d/ bo i l i n g bo u n da ry .
I I I . 7 . 3 . 2 S u bcoo l ed Secondary F l u i d L ump
( SFS L )
S l i gh t l y s u bcool e d wate r from the downcome r enters t h e U -tube
re g i on a t a fl ow rate w 1 and tempe rat u re T d . Heat is t ra n s fe r red
to t he s e conda ry fl u i d fl owi n g u pwards ( by n a t u ra l re ci rc u l at i on )
from bot h bran ches o f t he U - t u bes .
A s a res u l t o f t h i s , the
tempe ratu re o f t he s e condary fl u i d ri s e s u n t i l i t re a c h es T s at , wh i c h
i s a fun ct i on o f t h e s eco nda ry system pre s s u re , a n d boi l i n g takes
T he l eve l at wh i ch boi l i n g s t a rt s i s a fun ct i o n of the
p l ace .
fol l owi n g .
1.
Fl ow rate a n d tempe ra t u re o f s e co n d a ry fl u i d from the
down comer
2.
Heat t ra n s fe r rate from tube met a l to the s ec o n d a ry fl u i d
T
•
sat
S i nce t he l en gt h o f the s ub coo l e d secondary l um p i s dete rmi ned
3.
Steam p re s s u re w h i c h dete rmi nes
by the s ta rt of t h e boi l i n g l evel , bot h ma s s ba l a n ce a n d heat bal ance
equ at i o n s a re req u i red to re p re s e n t the dynami c s o f t h e s u bcoo l ed l ump .
a.
Ma s s bal a n ce
( I I I . 7 . 52 )
89
d (
)
dt Psl A fs L s l
i .e. ,
( I I I . 7. 53)
=
where ,
Psl
A fs
Ls l
=
den s i ty o f wate r i n the s ub coo l e d 1 ump
=
s econdary fl u i d f l ow a rea
=
l en g t h o f s ubcoo l ed l ump .
Negl ecti n g the changes of s ubcool ed water den s i ty duri n g
t he t rans i en t , o n e obta i n s
( I I I . 7 . 54 )
whi c h yiel ds
do l s l
dt
b.
( I II . 7.55)
=
Heat ba l an ce
( I I I . 7. 56 )
where ,
Ms l
C P2
Ts 1
( Q ms 1 ) 1
( Q ms l ) 4
=
=
=
=
=
=
=
mas s o f wate r i n the s ubcool ed l ump
s pe c i f i c heat o f secondary water
avera ge tempe rat u re o f water i n t h e s ubcoo l ed 1 ump
hea t t ra n s fe r rate between
r�1
and s 1
U ms l p r2 L s 1 ( T m 1 - T s l )
heat tra n s fe r ra te between M4 a n d s 1
U ms 1 P r 2 L s l ( T m4 - T s l ) .
S i n c e t he d i ffe re n ce betwee n T
t hen we can a s s ume that
T
sl
=
Td + Ts at
2
d
and T
s at
i s us ua l l y s mal l ,
( I I I . 7. 57)
90
S u b s t i t u t i n g from E q ua t i o n ( I I I . 7 . 5 7 ) i nto E q uat i on ( I I I . 7 . 56 )
we get
T d+T s at
[M 1 C
(
) ] = Um s 1 P r 2 L s l ( T ml + T m4 - T d - T s a t )
2
s P2
df
d
( I I I . 7. 58)
S i nce Ms l = A fs p s l l s l , then
T d + T sat
d [
C
A
l
p
)]
fs s l s l P2 (
2
dt
=
U ms l p r 2 L s l ( T ml + T m 4 - T d - T s a t )
( ! ! ! . 7 . 59 )
U s i n g o T sat =
a T sat
o P , w e get
aP
rlT
s at o P ]
aP
( ! ! ! . 7 . 60 )
! 1 1 . 7. 3. 3
Bo i l i ng S econda ry Fl u i d L ump ( S FBL )
T he seconda ry boi l i n g s t a rt s when t he tempe ra t u re o f the
secon d a ry fl u i d rea ches T s a t a n d e n d s a t the e n d o f t h e acti ve heat
t r an s fe r re g i o n at a l e vel j us t above the t ube b u n d l e . A s the
seco n da ry fl u i d mo ves u pwards , the q ua l i ty o f t h e two - ph a s e mi x t u re
fo rmed by t he bo i l i n g proce s s i n creases from 0 to a n e x i t q ua l i ty x e .
91
A s s umi n g t he homo geneou s fl ow mo de l , we cons i de r the secondary fl u i d
i n t he b o i l i n g sect i o n as a homo geneou s s team wate r m i xt u re wi t h a n
a ve ra ge q u a l i ty x , dens i ty p b and entha l py h b where
( I I I . 7. 6l )
( I I I . 7 . 62 )
v + x v g
f
f
-
( II I. 7. 63)
and
S i nce t he s i ze and the den s i ty o f t he b o i l i n g l ump change d u ri n g
t he t ra n s i en t both ma s s bal ance and heat bal ance equat i o n s a re
nece s s a ry fo r t he de s cri pt i on o f t h e b o i l i n g l ump .
a.
Ma s s bal ance
( I I I . 7 . 64 )
S i n ce the fl ow a rea o f the s e co n d a ry fl i ud i s con stant ,
then
( I I I . 7 . 65 )
o r , i n term s of pertu rba t i on va r i abl es
( I I I . 7 . 66 )
S i n ce L s l + L s 2 con s t an t , then us i n g t h e e q ua t i on fo r
6 p b ( see A ppen d i x E ) , we get
=
A fs0 b
d6 Lsl
d <S P
l [
dt + A fs s2 C l b dT
+
d6 x
e = 6
C 2 b (ft"]
1-1 2
-
6 W3 .
( I II . 7·67)
92
b.
Heat ba l ance
( A4ms 2 ) 2 + ( A4ms 2 ) 3 + W 2 h f
-
W h
3 e
( 1 1 1 . 7. 68)
where ,
( Qms 2 ) 2
heat t ra n s fe r rate between M 2 and s 2
= U 2 P r2 L 2 ( T 2 - T
ms
. s
m
s at )
( 0ms 2 ) 3 heat t ra n s fe r rate between M 3 and s 2
Ums 2 P r2 L s 2 ( T m2 - T s a t )
h f ent ha l py o f s at u rated wate r e n te ri n g the
=
=
=
=
boi l i n g 1 ump
h
=
enthal py of secondary fl u i d l ea v i n g the
bo i l i n g l ump
( h x = h f + X e hf g )
e
where ,
h fg
xe
=
l atent heat o f vapo ri z a t i o n .
S u bs t i t ut i n g fo r h b ' ( Q ms 2 ) 2 a n d ( Q 2 ) 3 i n E qua ti on ( I I I . 7 . 6 8 ) ,
ms
we get
X
e
d [ A l
+
h
(
f 2 h fg ) ]
dt p b fs s 2
=
U ms 2 P r 2 L s 2 ( T m 2
Tsat )
( I I I . 7 . 69 )
w h i c h u pon l i neari z at i on a n d s ome al geb ra i c man i pul at i on
y i e l ds t he fol l owi n g equat i on
- U ms 2 P r2 [ Tm 2 + T m 3 - 2 T s at ] o l s l + h f o W 2 - h x 0W 3
e
ah
a hf
+ w 2 -af o P - w 3 x e
� oP
1 1 I . 7. 3. 4
( I I I . 7 . 70 )
- w 3 h fg o x e .
Drum Equ i va l en t Secon da ry Fl u i d L ump ( S F DRL )
T he d rum eq u i val e n t s econda ry fl u i d l ump i n t h i s mo de l
has t he s ame geome t ri cal bounda ri e s a s i n Mo del C .
I t can al s o
b e d i vi ded i nto the fo l l owi n g pa rt s :
1.
R i s e r/Separator vol ume
2.
D rum water vol ume
3.
Drum s team vol ume .
T he e q ua t i o n s des c ri b i n g thes e part s a re devel oped us i n g the s ame
pro cedure a s i n Model C a re repeated i n t hi s sect i o n fo r con ven i ence .
1.
R i se r/Sep a rator Vo l ume
The d e s cr i b i n g eq uat i o n fo r the ri s e r/ s e parator vol ume i s
the s a me a s Equati o n ( I I I . 6 . 2 0 ) de ri ved i n Se ct i o n I I I . 6 . 7 . 1 ,
page 58 , wi t h aw 3 a n d o W 4 repl a c i ng o W 2 a n d a w 3
re s pe cti ve l y , thus
Vr( Cl r
do xe
do P +
C2
dt
r Cl"t) =
( I I I . 7 . 71 )
94
1
whe re c r =
1
( see Appe n d i x E ) .
and
2.
Drum Wate r Vo l ume
T he des c ri b i n g equat i on s fo r t h e d rum wat e r vol ume a re
obtai ned from a ma s s bal a n ce equat i o n re l a t i n g the fl ows
i n , the fl ows o u t , a n d t h e water l e ve l a n d a heat bal ance
eq uat i on des c ri bi n g t h e mi x i n g between re c i rcul ated wate r
a n d fee dwater .
T h e d rum wate r l ump i s coupl e d to t h e
seco n da ry l umps i n t he e ffect i ve heat exchan ge re g i on
t h ro u g h t h e downcomer l ump a n d t h e re c i rcul a t i on l oop
equat i on s .
Fol l owi n g t he s ame p rocedure as i n Sect i on I I I . 6 . 7 . 2 ,
page 6 0 , we have t h e fo l l ow i n g e q ua t i on s
( I I I . 7 . 72 )
( I I I . 7. 73)
The a bo ve eq ua t i ons a re s i mi l a r to Eq ua t i on s ( I I I . 6 . 2 9 )
and ( 1 1 1 . 6 . 3 0 ) wi t h
w
3
rep l a ce d by
w
4
as t h e mas s fl ow
rate o f the two phase mi x t u re l ea vi n g t h e ri s e r/ s e pa rator
vo l ume .
95
3o
Drum Steam Vo l ume
The l i n e a ri zed e q ua t i o n fo r the drum s te am vol ume i s
s i mi l a r to E q u at i on ( 1 1 1 . 6 . 36 ) de ve l o pe d i n Sect i on
1 1 1 . 6 . 7 . 3 , pa ge 62,
wi t h w
4
re p l aci n g w
3
a s the ma s s
fl ow rate o f t h e two p h a s e mi xt u re l ea v i n g t h e r i s e r/
s e p a rator vol ume , t h u s
( 1 1 1 . 7 . 74 )
Down come r Secon dary Fl u i d L ump
1 1 1 . 7.3.5
( S FDCL )
As i n the case wi t h Model C , the down come r s econ d a ry fl ui d
l ump acts a s a tran s po rt t i me del ay between t he d rum water vol ume
a n d t he e ffe ct i ve h e a t exchange re gi o n .
The
equa t i o n for
t h e down comer l ump i s t h e same as Equa t i o n ( 1 1 1 . 6 . 4 0 ) and can
be wri tten i n the fo rm
dOT
d
1
-- = -
dt
whe re
r�
•
d
1 1 1 . 7.4
=
d
-w
l
=
'
d
( O T dw - OT d )
( I I I . 7 . 75 )
res i dence t i me o f the secondary fl u i d i n the
down come r 1 ump .
The Reci rcu l a t i o n Loop Equa t i on
T he reci rcul a t i o n l oo p e q u at i o n i s obta i ne d by a p pl y i n g t h e
momentum theory to t h e secondary fl u i d o u t s i de a n d i n s i de the t ube
wrappe r .
Fo l l owi n g the s ame p ro c e d u re de s c ri be d fo r Model C i n
S e ct i on I I I . 6 . 9 , page
6 5 , we noti ce that o ut s i de t h e t u be wrap pe r
t he p re s s u re head o f the re ci rcul a t i o n l oo p i s i de n t i c a l fo r Mode l
C a n d D , howe ver , the s econ dary fl u i d i n s i de the t ube wrappe r i n
Model D i s compo s e d o f t h ree pa rts .
96
1.
S ubcoo l ed re g i on o f l en gt h L s l , where t h e tempe ra t u re o f
t he s ubcoo l e d fl u i d i s ra i se d from T d a t t h e i n l e t to
T s at at t he outl et . T he den s i ty i s a s s ume d con s ta n t i n
t h i s reg i o n .
2.
N ucl eate boi l i n g re g i on o f l ength
LB
L - L s l , where
bo i l i n g o f t h e s at u rated water l ea vi n g the s ubcool e d re g i on
·,
=
takes p l ace a n d the mas s q ual i ty o f t h e two pha s e mi x t u re
i s a s s ume d to c hange l i ne ar l y from 0 a t t h e i n l et to x e
a t t he o ut l e t .
R i se r/ se p a rator reg i on o f l en gt h L R , whe re t h e two phase
mi x t u re l eavi n g the n u c l eate boi l i ng cont i n ues to mo ve
3.
upwa rds i n t h e ri s e r/ se pa rato r vol ume .
Therefo re ( � P ) OTW can be cal c u l ated i n t h e same way a s i n
E q u at i on ( 1 1 1 . 6 . 46 ) , t h u s
( I I I . 7 . 76 )
whe re ,
pd
Ld
=
=
P dw
=
P dc
L dw + L d c
=
den s i ty o f t he s econ dary fl u i d o ut s i de the
t ube w rappe r
=
hei g h t o f t h e s e conda ry f l u i d i n t h e d rum
wat e r vol ume and t h e downcome r t ra n s port
1 ump .
Howe ve r , ( � P ) I TW i s now cal cul ated f ro m the fo l l owi n g equat i on .
( �P )
1
I TW p s i
=
p
L
s sl
+
p
bL B
1 44
+ p L
r r
( 1 I I . 7. 77)
97
whe re ,
=
den s i ty i n t he s u bcoo l e d reg i on
=
l en gt h o f the s ubcool ed re g i on
P bs
=
a ve ra ge den s i ty i n t he n ucl eate bo i l i n g re g i on
L
=
l en gt h o f t h e boi l i n g re g i o n
Ps
L
sl
p
B
L
r
=
den s i ty i n t he ri s e r/ s e p a rator re g i o n
=
1 ength o f t he ri ser/ s eparator re gi o n .
r
Us i n g t h e s ame met ho d a s d i s cu s s e d o n page
68 , the e q u a t i o n fo r o W 1
can be re duced to t h e fo rm
( I I I . 7 . 78 )
T he expre s s i on s fo r t he coeff i ci e n t s a , a 2 , a , a n d a a re deri ved
1
3
4
i n Appen d i x D .
S umma ry o f the t1odel Equa t i o n s
III . 7.5
T he govern i ng equat i on s of Mo d e l
0
t h a t we re deve l o ped i n the
p receedi n g sect i o n s can be re presented by the ma t ri x e q u a t i o n
whe re
x
A
1
=
and A 2
r
=
system va r i abl e s ( di ffe ren t i a l + a l gebra i c ) * vector
=
coeffi c i ent matri ces
forc i n g vecto r .
A l i s t o f t he sys tem vari abl es i s g i ven i n Tabl e I I I . 2 a n d the nonzero
e l ements of t he mat r i ces A
1
a n d A 2 a n d the fo rc i ng ve cto r r a re g i ven
be l ow .
*T he de fi n i t i o n o f di ffe rent i a l and al ge b ra i c vari abl es wi l l be
di s c u s sed i n Chapter I V .
98
TABLE I I I . 2
S YSTEM VA R I AB LES FO R TH E DETA I L E D MODEL ( MODEL D )
l.
2.
3.
4.
5.
6.
7.
TP i
TP l
T P2
p ri ma ry wate r i n l et p l e n um tempe ra t u re
fi rst pri ma ry wate r l ump tempe ra t u re
second pri ma ry wate r l ump tempe rat u re
TP3
t h i rd pri ma ry wate r l ump tempe ra t u re
TPO
p r i ma ry water out l et pl e n um tempe ra t u re
TP 4
T ml
8.
T m2
fo urth pri ma ry wate r l ump tempe ra t u re
fi rs t tube met a l l ump tempe ra t u re
second tube met a l l ump tempe ra t u re
t h i rd tube meta l l ump tempe ra t u re
10.
Tm3
Trn4
fou rt h tube meta l l ump tempe rat u re
ll .
L dw
d rum wa ter l e ve l
9.
12.
13.
14.
1 5.
16.
*1 7 .
*1 8.
*1 9 .
*2 0 .
Lsl
nonbo i l i n g ( s u bcoo l ed ) l en gth
p
s team p re s s u re
xe
boi l i n g sect i on e x i t q ua l i ty
T dw
d rum wate r tempe rat u re
w1
f l ow rate to t he n o n bo i l i n g sect i on
Td
downc ome r out l e t tempe ra t u re
w2
fl ow rate to t h e b o i l i n g s ect i o n
w3
fl ow rate from t h e b o i l i n g sect i on
w4
fl ow rate from the r i s e r/ se pa rato r vol ume
*Al g e b ra i c v a ri a b l e s .
99
1.
The A , r1a t ri X
A1 ( l , l )
A1 ( 2 ,2 )
=
l .0
1 .0
=
A1 ( 3 , 3 ) = 1 . 0
T 1 -T 2
A1 ( 3 , 1 2 ) = ( PL P )
s2
A1 ( 4 , 4 )
=
1 .0
A1 ( 5 , 5 ) = 1 . 0
T 4-T 3
P P
Ls l
=
A1 ( 5 , 1 2 )
A1 ( 6 , 6 ) = 1 . 0
A1 ( 7 , 7 )
=
Al ( 7 , 1 2 )
/\ 1 ( 8 , 8 )
=
(
=
T ml - T m2
2L s l )
1 .0
T
=
A1 ( 8 , 1 2 )
A1 ( 9 , 9 )
l .0
=
-T
2L s 2
��
1 .0
1 .0
A1 ( 1 0 , 1 0 )
=
A1 ( 1 0 , 1 2 )
=
_
A1 ( 1 1 , 1 1 )
=
1 0
A1 ( 1 2 , 1 2 )
=
M
T m 3 - T m4
)
( 2
Ls l
•
C
s l P 3 T dt T s a t )
( L
2
sl
1 00
A1 ( 1 2 , 1 3 )
A1 ( 1 2 , 1 6 )
Ms 1 C P 2
2
_
aTs a t
aP
Ms 1 c P 2
2
=
A1 (1 3, 1 2 )
=
A1 (1 3 , 14)
=
A fs P b
-
A 1 ( 1 3 , 1 3 ) = A fs l s 2 c 1 b
A ( 1 4 ,1 2) =
1
A 1 ( 1 4 • 1 3)
=
A fs l s 2 c 2 b
-
A fs h b p b
ahf
A f s p bl s 2 (�
A 1 ( 1 4 , 1 4 ) = A fs p b l s 2 ·
A1 ( 1 5 , 1 3 )
=
v c1 r
r
A1 (l 5 ,14)
=
v rc 2 r
A1 ( 1 6 , l l ) = 1 . 0
A1 ( 1 7 , l l )
A1 ( 1 7 , 1 5 )
=
=
A1 ( 1 8 , 1 1 )
=
A1 ( 1 8 , 1 3 )
=
T dw
L dw
-
P
g
A dw
Cl p
v s _3
aP
A1 ( 1 9 , 1 6 ) = 1 . 0 .
2.
The A 2 M atri x
w P 1.
1
)
1
1
A2 ( ,
Mp ;
Tp;
hf
+
�
xe a h f
2
+
af!l)
+
A fs l s 2 h b C 1 b
A fs l s 2 h b C 2 b
1 01
1 + u pms pm 1
A2 ( 2 , 2 ) = (
M P1 C P 1 )
TP1
A2 ( 2 , 7 )
A2 ( 2 , 1 2 )
_
-
upm5 pm1
- M C
P1 P1
1 02
u
A2 ( 7 , 2 )
p
u
A2 ( 5 , 1 2 )
s
pm pm4
r1 P4 C P 1
_
_
pm r1 T
MP4 C P 1 ( P4 - T m4 )
=
u
(
=
s
p m pm1
M ml C m
_
=
u
+
u s s
m l ms 1
pms pm1
)
M m1 C m
u s s
m 1 ms 1
)
( 2M c
m1 m
u
_
_
s
ms 1 ms 1
2M C
m1 m
_
pm
u
s
pm2
M m2 c m
_
= (
u
+
u 2s s 2
ms m
pms pm 2
Mm 2 c m
u s 2s 2
ms
)
( Mm
m2 c m
A 2 ( 9 ,4 )
u
=
_
aTs
at
aP
s
pm pm2
��
m 3c m
aTs
at
aP
1 03
u
+ u
s
s
ms 2 ms 2 )
( pm pm2
r1 c
m3 m
=
dT
s
sat
aP
u
( ms 2 ms 2 )
r1 3c
m m
u
s
p m pm1
r� c
m4 m
_
_
+
ms 1 ms 1 )
( pm pmM 1
c
m4 m
u
A2 ( 1 0 , 1 0 )
=
A2 ( 1 0 , 1 3 )
-
-
A2 ( 1 0, 1 6 )
-
-
A2 ( 1 1 , 1 7 )
-
A2 ( 1 1 , 1 8 )
=
s
u
s
u
( ms 1 ms 1 )
2 t1m4 c
m
s
aT
sat
Cl P
u
-
ms l s r1s 1
2M 4 c
m m
1
A
p s fs
l
p s A fs
u
s
ms 1 ms 1
u
( ms �
s
s
�s l )
[ Tm l + Tm4 - T d - T s a t ]
+ W C )
= (U
2 P2
ms 1 5ms l
dT
at
a
�
1 04
A2 ( 1 3 , 1 8 )
A2 ( 1 3 , 1 9 )
A2 ( 1 4 , 8 )
-
=
1 .0
-
-
=
-
A2 ( 1 4 , 9 )
1 .0
-
ms 2 s m s 2
u
u ms 2 s
ms 2
s
u
ms 2 m s 2 ( T + T - 2 T
s at )
m2
m3
L5 2
ah
aT at
+
x
2
�
\·J 3 e
A 2 ( 1 4 , 1 3 ) = u ms 2 s ms 2 a
A2 ( 1 4 , 1 2 )
A2 ( 1 4 , 1 4 )
A2 ( 1 4 , 1 8 )
A2 ( 1 4 , 1 9 )
=
=
=
=
A2 ( 1 s , 1 9 )
=
A2 ( 1 6 , 1 4 )
=
A2 ( 1 6 , 1 7 )
=
�
W 3 h fg
- hf
hx e = h f
-
+
x e h fg
1 .0
A2 ( 1 s , z o ) = 1 . 0
A2 ( 1 6 , 2 0 )
A2 ( 1 7 , 1 3 )
-
-
A2 ( 1 7 , 1 4 )
=
A2 ( 1 7 , 1 6 )
=
w4
P dw Adw
1
P dwA dw
-
-
( 1 - xe )
P dwA dw
( 1 . - x e ) aT s a t
P dwA dw a P
w4
P dw f.\ dw
w1
P dwA dw
. T s at
1 05
A2 ( 1 7 , 1 7 )
=
Td
P dwAdw
( 1 - Xe )
P dw Adw
A2 ( 1 7 , 2 0 ) - A2 ( 1 8 , 1 3 )
=
cL
A2 ( 1 8 , 1 4 )
=
- w4
A2 ( 1 8 , 2 0 )
= X
Az ( 1 9 , 1 5 )
=
A2 ( 1 9 , 1 6 )
=
1
'd
=
a
=
a2
A2 ( 2 o , 1 1 )
A2 ( 2 0 , 1 2 )
- 1'd
3.
= -
1
e
=
w
- ,
Md
1
A2 ( 2 0 , 1 3 ) = a 3
A2 ( 2 0 , 1 4 ) = a 4
A2 ( zo , 1 7 )
·
1 .0.
The Fo rc i n g Vecto r f
wp ;
1
f ( l ) = - 6 8 . = u-1
"'p ;
'p;
f( 1 8)
=
- P
oCL .
oe
.
1
s at
C H APTER I V
C AL C ULATION P RO C E DURE AN D RESULTS
IV. 1
I n t roduc t i on
As wa s s een i n C h a pter I I I , pa ge 2 5 , t h e l umped pa rame ter s ta t e
va r i abl e a p proach r e su l t s i n two d i fferent type s of mod el formul a t i on s :
1.
Pure D i ffere n t i a l F o rmul a t i o n .
I n t h i s c a s e , a l l o f the
model equa t i on s a r e d i fferen t i a l .
Model s A a n d B bel o n g
t o t h i s g ro u p .
2.
M i x ed D i fferen t i a l and Al g eb ra i c Fo rmu l a t i o n .
I n t h i s ca s e ,
some o f t h e model equa t i o n s a r e d i fferen t i a l a n d some a re
a l g ebra i c .
T h e sys tem va r i a bl e s con s i st o f d i fferen t i a l
( o r s ta te ) va r i abl es a n d a l g eb ra i c ( o r i n termed i a t e )
va r i a bl es .
A d i fferen t i a l va r i a b l e i s def i n ed ( 3 7 ) a s a
var i abl e who s e der i va t i ve a ppea r s at l ea s t o n c e i n the
syst em equa t i o n s whi l e an a l g ebra i c var i a bl e i s d ef i n ed a s
a va r i abl e who se d er i va t i v e does n o t a ppear a nywhere i n t he
system equa t i o n s .
�1odel s C and D bel ong to t h i s g ro u p .
As
t he mod e l compl ex i ty i nc r ea s e s , t he m i x ed d i fferen t i a l and
a l gebra i c fo rmu l a t i on has the adva n ta g e o f red uc i n g t he
amount o f mat hema t i ca l man i pu l a t i on r equ i red by t he sys tem
anal yst a n d accord i n g l y r educ i ng t he po s s i b i l i ty o f
a l geb ra i c o r n umeri ca l erro r s i n t h e p ro c e s s o f dyn am i c
system a n a l ys i s .
T h e ca l c u l a t i on p rocedure fo r o b ta i n i ng
t h e system t i me respo n s e a n d frequ ency r e s po n s e fo r bo t h
l 06
1 07
formu l a t i on g ro u p s a n d t h e res u l t s o b ta i n ed from t h e
m a t h ema t i ca l mo d el s a r e d e sc r i b ed i n t h e fo l l ow i n g
section s .
IV . 2
C a l c u l a t i on P ro cedure fo r t h e P u re D i fferen t i a l Formu l a t i o n
A sys tem of c o u p l ed f i r s t o rder d i fferen t i a l equat i o n s can , i n
gen eral , b e wri tt en i n t h e fo rm
( IV . l )
where
-
x = s ta te va r i a b l e s v ec to r
f = fo rc i ng v ec t o r
C , D a n d E = coeff i c i en t ma tr i x e s .
( )
Pro v i ded tha t C i s a n on s i n g u l a r ma t r i x, 3B b o t h s i d e s of E q u a t i o n
( I V . l ) c a n b e pre-mul t i p l i ed b y C
-l
( i n ve r s e of C ) to g et t he
fo l l o wi n g e q u a t i o n
( IV . 2 )
w he r e
and
A
= C
B = C
-1
-l
D = s ta t e coeff i c i en t ma t r i x
E = fo rc i ng coeff i c i en t ma t r i x .
E q u a t i o n ( I V . 2 ) i s t h e " c l a s s i ca l '' form fo r the ma t hema t i c a l represen ta t i o n o f a system o f f i r s t ord e r d i fferen t i a l equa t i on s .
T h i s equa t i o n
( )
( )
fo rm s t h e ba s i s fo r t h e computer codes MATEX P 39 a n d S F R - 3 4 0 t ha t
a r e u s ed i n t h i s s t udy t o cal c u l a t e t h e t i me r e s po n s e a n d freq uency
r e s po n s e r e s pect i v el y .
1 08
T h e g en eral so l u t i on o f E q u a t i on ( I V . 2 ) i n t h e ti me doma i n i s
g i ven by (
41 )
A ( t- t )
0 x(
t )
0
x( t) = e
+
t A ( t- r )
B f( r ) dr
t e
0
( IV . 3)
where
=
x( t )
o
i n i t i a l co nd i t i o n s vector
and
""
{A( t-t ) }
0
L
( IV.4)
k!
k =O
T h e fo l l owi n g co n v en t i o n s a r e u s ed i n t h e ma t r i x ex pon en t i a l
i n Equa tion ( I V . 4 )
{ A ( t-t ) J
0
and
0
0!
= I = i d ent i ty ma t r i x
l.
=
F rom E q u a t i on ( IV . 3 ) , i t ca n b e s e e n t ha t g i ven the i n i t i a l c o n d i t i o n
vecto r x ( t 0 ) , t h e co eff i c i en t ma t r i c es A a n d B a nd t h e forc i n g v e c to r
f ( t ) , t h e s o l u t i o n a t a l a ter i n s ta n t t can be fo u nd .
T h e d eg r ee
of a c c u r a cy of t h e s o l u t i on d epend s on t h e number of term s u s ed i n
the c a l c u l a t i o n o f the ma t r i x expon e n t i a l i n E q u a t i on ( I V . 4 ) a n d the
me tho d of n umer i ca l i n t eg ra t i on u s ed to cal c u l a t e t h e i n tegral i n
E qu a t i o n ( I V . 3 ) .
T h e m e t hod for ha n d l i ng t h e i n t eg ra l t erm i n t h e
t�TE X P computer p ro g ram i s t o a s sume t ha t t h e fo rc i n g funct i o n i s
p i e c ewi s e c o n sta n t .
t
ft
o
e
A ( t - -r )
T h i s g i v es :
B f ( T ) d-r = [ I - e
A ( t-t )
o ]
A
-l
B f ( t0 ) .
( IV . 5)
1 09
T h e frequ ency r e s po n s e can b e o b t a i n ed by s t a rt i ng w i t h
E q u a t i o n ( I V . 2 ) a n d La p l a c e tran s fo rm i ng
S x( S )
=
(42)
bot h s i d e s , t h u s
A x(S) + B F(S)
( IV . 6 )
where
s
=
L a pl a c e t ra n s forma t ion v ar i a b l e
x(s )
=
L a pl a c e tra n s form of x ( t )
F(S)
=
L a pl a c e tran s fo rm o f f ( t )
=
co effi c i en t ma tr i c e s , a s s umed con s ta n t .
A ,B
Rearran g i ng E q u at i on ( I V . 5 ) a n d s o l v i ng fo r x ( S ) , we g e t
x(S)
=
[SI - A f
1
B F( S ) .
( IV . 7)
For a s i n gl e fo rc i n g e l ement f ( t ) i n t he fo rc i n g ve c tor f ( t ) ,
m
E q u a t i o n ( I V . 7 ) can b e wr i tt e n a s
x(S)
where
=
=
[SI
-
A]
-l
5
r1
F
m
(S)
( IV . S)
L a pl a c e tran sfo rm o f f ( t )
m
b
nm
T h e frequen cy r e s po n s e vecto r of t he sys t em s t a t e var i a b l es wi th r e s pe c t
to t h e fo rc i n g el ement f c a n be obta i n ed from E q u a t i o n ( I V . S ) by
m
r e p l ac i ng S by j w , t h u s
1 10
[jw
r
-
Ar 1
5
( IV . 9 )
m
F rom E qu a t i o n s ( I V . 3 ) and ( I V . 9 ) , t h e cal c u l a t i o n procedure
fo r o bt a i n i ng t h e dynam i c r e s po n s e when t h e mat h ema t i c a l mo del
co n s i s t s o f a sys tem o f fi r s t order d i f fe r en t i a l equa t i o n s ( Model s
A and B ) can b e s u mma r i zed a s fo l l o ws :
l.
Cal c u l a t e t h e g eomet r i cal pa rameters n eeded to est i mate
t h e v o l u m e s of t he d i fferen t l umps i n t h e mod el u s i n g
a va i l a b l e d e s i gn d a t a a n d / o r en g i n e e r i n g j u d g ement .
2.
Ca l c u l a t e t h e t he rmal a n d hyd ra u l i c pa ramet e r s re q u i red
for t he cal cul at i on of t h e co effi c i en t ma t r i c e s and
fo rc i n g vecto r s .
T h i s cal c ul a t i o n i n c l ud e s t h e ca l c u l a t i o n
o f t h e h ea t tra n s fer coeffi c i en t s and steady s t a t e
p a ramet e r s .
3.
Fo r t i me r e s po n s e an a l ys i s , i n p u t the coeffi c i e n t ma t r i x
a n d fo rc i n g func t i o n to t h e computer cod e r�ATEXP
( 39)
t o g e t h e r w i t h t h e prog ram con t ro l card s d e s c r i b ed i n
R efer en c e 3 9 .
4.
For frequ e n cy r e s po n s e a na l ys i s , i n p u t t h e c o e ffi c i en t
matr i x a n d t h e fo rc i n g funct i o n t o t h e computer code
S FR -3 ( 4 0) together w i t h t h e program c o n t r o l c a r d s desc r i b ed
i n Referen c e 4 0 .
111
C a l cu l a t i on P ro ced u r e fo r t he Mi xed D i ffere n t i a l a n d Al gebra i c
IV . 3
F o nnul a t i on
A s the dyn am i c model compl ex i ty i n crea s e s , t h e mathema t i ca l
man i pu l a t i o n effo r t c a n b e con s i d era bl y red u c ed whe n t he a l ge bra i c
e q uat i o n s r e l a t i ng sy stem i n termed i ate var i a bl e s to sys tem s ta t e
v a r i a b l e s a r e l ef t i n t h e mo del .
I n t h i s ca s e , t he ma t hema t i c a l
model o f t h e sys tem con s i s t s o f a set of f i r s t o rd e r e q ua t i on s ha v i n g
bot h d i ffer en t i a l a n d a l g ebra i c va r i a b l e s t h a t can be wr i tten i n
t h e form:
-
g u(t)
( IV . l O)
w h ere
z
-
-
g
=
vector o f sys tem va r i a b l e s ( d i fferen t i a l + a l gebrai c )
=
v ector of con sta n t mul t i pl i e rs o f t he t i me vary i n g
fo rc i n g fun c t i o n
u(t)
=
a r b i t r a ry fun c t i o n o f t i me represen t i n q sy s t em i n p u t
( fo rc i n g fun c t i on )
A
1
and A
2
=
coeff i c i en t ma tr i c e s .
I f the syst em va r i a bl e s a re compo sed o f n d i ffe re n t i a l ( o r st a t e )
v a r i a bl e s a n d r a l g e b ra i c ( o r i n t ermed i a t e ) va r i a b l e s , then ma tr i x
E q u at i on ( I V . l O ) ha s t h e order m wh i c h i s equal to ( n + r ) .
·
The met ho d o f o bta i n i n g t h e t i me r e s pon s e a n d t he frequ ency
r e s po n s e when t h e sy s tem dynami c model can b e r e p r e s e n t e d by
E quat i on ( I V . l O ) i s d e s c r i bed i n Referen c e 37 a n d i s s umma r i z e d
b e l ow fo r co n ve n i en c e .
112
T i me R espon s e C a l cu l a t i on
IV. 3 . 1
T h e s e t o f equ at i on s re presen ted by mat r i x E q ua t i on ( I V . l O )
i s f i r s t r eord ered i nt o m d i fferen t i a l equa t i on s fol l owed by r
a l g ebra i c equa t i o n s a n d then reduced to a system o f n d i ffe re n t i a l
(
e q u a t i on s u s i n g t he com p u t er code P U R E D I FF . J? )
s e t of p u r e d i fferen t i a l eq u a t i o n s i s s o l ved
I�TE X P . ( J g )
T hen t he redu ced
us i ng
t he co�p uter code
The method of remo v i n g the a l g eb ra i c var i a b l e s from the
equat i on s i s d e s c r i b ed bel ow.
After r eorder i n g E q ua t i o n ( I V . l O ) c a n be wr i tt e n i n the
pa rt i t i o n e d ma t r i x fo rm
( IV. l l )
wh ere
x-
y
d i fferen t i a l var i a b l e v ec t o r
=
a l g ebra i c va r i a b l e vecto r .
Equa t i on ( I V . l l ) can b e ex pa n d ed to o b ta i n t h e fo l l ow i ng equat i o n s
( IV . l 2 )
d
R 2 1 d tx + T 2 1 x- + T 22 y-
=
g2 u ( t )
( IV. l 3)
·
T h e a l gebra i c va r i a b l e v e c t o r y i s el i mi n a t ed by pre-mul t i p l y i n g
bot h s i d e s o f Equa t i o n ( I V . l 2 ) by t h e i n ve r s e o f T 1 2
and
pre-mu l t i pl yi n g bot h s i de s o f E q u a t i on ( I V . l 3 ) b y t h e i n v e r s e of r 2
2
a n d s u bt r a c t i n g t h e r e s u l t i n g e q ua t i on s to g et
113
- 1 g- - - 1 g
T
T2 2 2 J u ( t )
[ 12 1
( IV . 1 4)
pre -mul t i pl y i n g E qu a t i o n ( I V . 1 4 ) by r 1 2 we get
dx
- 1 -1
-1 _,
[ R l l = T l 2 T 22 R 2 1 J dt + c T l 1 - T l 2 T 22 T2 1 ] x
=
-1 -1
[ g l - T l 2 T 22 g 2 J u ( t ) .
( IV . 1 5 )
E qu at i on ( I V . 1 4 ) can b e wri t t en i n t h e con c i s e form
B
where
��
+
C x
=
5 u(t)
( IV . l 6)
-1
B = [R
l l - T 1 2 T 22 R 2 1 J
c
=
-1
[ T l l - T l 2 T 22 T2 1 J
6 = [g - T 2 T
l
z
l
1
g2]
rlu l t i 1J l yi n g bo t h s i d e s o f Equa t i on ( I V . l 6 ) by B - l g i ve s
��
+
B-1 C x
=
B-1 5 u ( t ) .
( IV . l 7)
Equa t i on ( I V . 1 7 ) i s i n t h e 11 C 1 a s s i ca l " fo rm s u i ta b l e for u s i n g t he
t i me r e s po n s e code MA T E X P . ( 3 9 )
An i mpor ta n t feat u re of t he a bo v e met hod i s t h a t i t i n vol v e s
mat r i x i n ver s i o n wh i c h may pro d u c e error i n t he re s u l t s w h e n any of
the ma tr i ce s i n vo l ved a re i l l - c o n d i t i o ned . ( 4 3 )
T h i s rro bl em �ay be
avo i ded by c ha n g i n g t h e o rder i n g o f t he sys t em equ a t i o n s .
114
IV . 3 . 2
F r equ ency R espo n s e Ca l c u l a t i o n
T he freq u e n cy r e s pon s e vec tor fo r t h e dynam i c sys t em re presented
by E quat i on ( I V . l O ) can b e obta i n ed by repl a c i n g S by jw i n the
tra n sf er f u n c t i on vecto r G ( S ) d e f i n ed a s
G(S ) =
�f�� .
( I V . l S)
The tran s fer f u n c t i on vector can b e o bta i n ed by two met ho d s .
1.
F o l l owi n g t he same proced u re a s i n t he t i me re s po n s e
c a l c u l a t i o n , o n e ca n red u c e t h e sys t em o f comb i n ed
d i ffe ren t i a l a n d al g ebra i c e q ua t i o n s to a set o f pure
d i ffere n t i a l eq uat i o n s o f t h e form
dx =
A x + -g u ( t ) .
dt
( IV . l 9)
The computer cod e , S F R - 3 , (4 0 ) c a n t hen b e u s ed to
obta i n t he freq u en cy respo n s e vector G ( j w ) u s i n g the
fol l ow i n g rel a t i o n
G ( j w ) = [j w I
2.
Th e
-
A]
-1 -
g.
( I V . 20)
sys tem equ a t i on s can b e k ept i n t he o r i g i na l m i x ed
mo d e ( d i ffe ren t i a l + a l g eb ra i c ) fo rmu l a t i on a n d a
mod i f i ed ver s i o n o f the S F R - 3 co de d evel o ped i n R eference
3 7 can b e u sed to obta i n the fre qu ency re s pon se vec tor
G ( j w ) u s i n g t h e fo l l owi ng rel a t i on
( IV . 21 )
115
From t h e a bo v e d i scu s s i on , i t can be seen t ha t t h e ca l c u l a t i on
proced ure fo r obta i n i n g t h e dyn am i c r e s po n s e when t h e ma t hema t i ca l
mo d el con s i s t s o f a m i xed s et o f d i fferen t i a l and a l geb ra i c e q u a t i on s
( Mo d e l s C a n d D ) can b e s umma ri z ed a s fo l l ows .
1.
C a l cu l a t e t he g eomet r i c p a ramet ers n eeded to e s t i ma t e t h e
vo l ume s o f t h e d i fferen t l umps i n t h e mo del u s i ng a va i l a b l e
d e s i g n d a t a a n d/ o r en g i n e e r i n g j u dgemen t .
2.
C a l c u l a t e t he t h e rma l a n d hyd ra u l i c para met e r s requ i red for
t h e c a l c u l a t i on of t h e coeff i c i e n t ma tr i c e s a n d forc i n g
vecto r s .
T h i s c a l c u l at i o n i n c l u d e s t he c a l c u l a t i o n o f t h e
h ea t t ran s fe r coeff i c i en t s a n d st eady sta t e pa rameter s .
3.
For t h e t i me r e s po n s e a na l y s i s , i n pu t t h e c o e f f i c i e n t ma t r i c e s
A1
and A
2
t o g et h e r wi t h t h e fo rc i n g fun c t i o n t o t he computer
code P U RE D I FF
fo rm .
( 3 ?)
to r ed u c e t h e system to p u re d i fferen t i a l
T h en u s e t h e compu t e r code 11ATE X P ( 3 g ) t o o b ta i n t h e
t i me res po n s e .
4.
For t h e freq u en cy r e s po n s e a n a l ys i s , e i t her u s e t h e mod i f i ed
vers i o n o f t h e S FR - 3 code ( 3 7 ) o r u s e t h e reduced syst em
mat r i x a n d fo rc i n g ve ctor obta i n ed from s t e p 3 a s i n pu t
to t he com p u t e r code S F R - 3 .
( 4 0)
V . 4 Geome t r i ca l C a l c u l a t i o n
W he n the d e s i gn d eta i l s of t h e steam g en e rator a re k n own to the
1a l ys t , t he ta s k o f c a l c u l a t i n g the l en g t h s , a r e a s a n d vol umes
?ce s sa ry fo r the dyn a m i c re s pon s e ca l c u l a t i o n red u c e s to a s i mpl e
t s k tha t ha s to be r e p ea t ed when t h e d i men s i o n s a n d / o r
'Omet r i cal con f i g u r a t i o n c han g e .
Ho weve r , s i n c e t he d e s i gn de ta i l s
116
of nucl ear s team g en erato r s a re often con s i d ered a s pro p r i e ta ry
i n fo rmat i o n , t he method o f ca l cu l a t i ng g eometri cal i n put data fo r
dynami c r e s po n s e c a l c u l a t i on from t he d es i gn deta i l s i s rul ed o u t
when thi s i n fo rma t i o n i s not a va i l a bl e .
I n thi s study , the on l y
i n fo rma t i o n a v a i l a bl e to t he a u t ho r i s th e s ee p i n g de s i gn data
u s u a l l y l i s ted i n t he sa fety a n a l ys i s re po r t o f a n uc l ear power
pl ant .
(2)
T he r e fo r e , en g i n e er i n g j ud gemen t and pro port i onal i ty
p r i n c i pl e s a re u sed to o bta i n t he geomet r i ca l pa ramete r s u s ed a s
i n pu t to t he dyn am i c res po n s e ca l c ul a t i on a l gor i thms .
I n thi s
sect i on , t he met hod o f ca l cul a t i n g t he geometr i ca l parameters i s
d e sc r i bed a n d t he n a pp l i ed to t h e H .
B.
Rob i n son Un i t 2 steam
g en erato r u s i ng t he steam g en erator des i gn data obta i n ed from
Refe ren ce 2 and s hown i n Tabl e I V . l .
P r i mary S i d e
IV.4.1
On t he p r i mary s i de , i t i s requ i red to dete rm i n e the vol ume of
the pr i ma ry water in t he i n l e t and o u t l et p l enums a n d fl ow a rea i n
t he effec t i ve heat exc ha n ge reg i on ( U -tube reg i o n ) .
T h e U-tu bes are
r e p l a ced by two stra i g ht - tube bra n c hes and t he average l en g t h of
ea c h bra n c h i s c a l cul a t ed as fo l l ows
( I V . 2l )
where
S
D
0
0
N
L
2
=
total heat t ra n s fer s u r fa c e , ft
=
U - t u be d i ameter , i n .
=
n umber o f U - tubes
=
l en g t h of s t ra i ght tube bran c h , ft .
117
TAB L E I V . l
H . B . RO B I NSON STEAt1 GErl E RA T OR
DES I GN DA T A*
H umbe r o f S team Gen e ra t o rs
3
De s i gn P re s s u re , Reacto r Coo l ant/Steam , ps i g
2485 / 1 085
Des i gn Tempera t u re , Rea cto r Coo l a nt/Steam , ° F 650/ 5 5 6
Rea ct o r Coo l an t F l ow , l b/ h r
33 . 9 3
To t a l Heat T ran s fe r S u r fa c e Area , ft
2
X
10
x
10
6
44 , 4 3 0
Steam Co n d i t i o n s a t F u l l Load , O ut l et N oz z l e :
S t e am F l ow , l b/ h r
3. 1 96
S t e am Tempe ra t u re , ° F
516
S t e am P re s s u re , p s i g
770
Fee dwate r Tempera t u re , ° F
4 35
O v e ra l l H e i ght , ft - i n .
63 - l . 6
S he l l O D , u pper/ l owe r , i n .
1 6 6/ 1 2 7 . 5
S he l l T h i c kn e s s , u ppe r/ l owe r , i n .
3 . 5/ 2 . 6 3
N umb e r o f U- t u be s
3260
U-tu be D i ame ter , i n .
0 . 8 75
Tube W a l l T h i c kn es s , ( a ve ra ge ) i n .
0 . 05 0
2 2 00 MWt
Reac to r Coo l a n t Wa te r Vo l ume , ft 3
Seco n d a ry S i de Wate r Vo l ume , ft
Secon d a ry S i d e S t e am Vo l ume , ft
3
3
*Ta b l e 4 . 1 -4 , Reference 2 .
6
Zero Powe r
928
928
1 5 36
3089
3203
1 64 0
118
The fl o w a r ea o f t h e pri mary wa ter i n t h e t u be re g i on i s de term i ned
by t he fol l owi n g equat i on
A
=
p
D.2
1T
�,-,1-.---
4x1 44
N
•
( IV . 23)
w here
A
p
D.
1
N
pr i ma ry fl ow a rea , ft
=
U - t u be i n s i d e d i ameter , i n
=
number o f t u be s .
i n s ide
The tube
2
=
d i amet er i s o bta i n ed f rom t he tube o ut s i d e d i ameter
T h e i n l e t a n d o utl et pl enums are eac h
and t he. tu b e wa l l t h i c ke n s .
a s s umed to ha ve t he same vo l ume wh i c h i s g i ven by
( I V . 24 )
where
V PL
vp
V
IV . 4 . 2
t
3
=
vo l ume of i n l et or o utl et pl enum , ft
=
steam g en erator p r i ma ry water vo l ume , ft
=
3
vo l ume o f pr i mary water i n s i d e t he U-tubes , ft .
3
Tube Meta l
The c r o s s sect i on area o f t h e tube metal ( A ) i s g i ven by
m
A
m
=
1T
( D 02
0�)
1
-.--.r....---'--
4xl 44
•
N.
( IV . 25 )
The surfac e area per un i t l en g t h i s u s ed i n the ca l cu l a t i on i n
the hea t t ran s fe r a r ea and can b e d etermi n ed a s fol l ows
p
ri =
D.
-,i1T
·
N
( I V . 26 )
119
n
=
D
0
12
N.
( IV. 27)
.
S ec o n d a r,t S i d e
IV .4 . 3
The s econdary s i de i s d i v i d ed i n to t hree pa r t s .
1 .
S econda ry fl u i d i n t h e effec t i ve heat exc han g e reg i o n .
2.
S ec o n d a ry fl u i d i n t he down comer reg i on .
3.
S e condary fl u i d i n t he d r um equ i va l en t r eg i on .
For t h e effec t i v e h eat exc ha n g e reg i on and t h e down comer r e g i o n ,
i t i s po s s i bl e to d e term i ne t h e fl ow area a n d t h erefo re t h e vol ume o f
a ny seco n d a ry fl u i d l um p i n t h e s e reg i on s from a k nowl ed g e o f t h e
l en g t h o f t he l um p .
However , fo r the d rum equ i va l ent reg i o n , i t i s
on l y po s s i bl e to e s t i ma t e l um ped vol umes d u e to t he compl ex
g eometr i ca l con f i gu ra t i o n i n t ha t reg i o n s i n c e i t i n c l u d e s t h e feed ­
wat er r i n g a n d t h e s t eam separa t ors a n d d r i er s .
T h e ca l c u l a t i on o f
t h e g eome tr i ca l parameters i n t he a b o v e reg i o n s i s d e s c r i b ed be l ow .
l.
E ffec t i ve H ea t Exchan g e Reg i o n .
T h e effec t i ve hea t
e x c ha n g e reg i on i s bou n de d by t he t u b e -wrapper , t he t u b e
p l a t e a n d a f i xed pl an e j u st a bo v e t he U - t u b e s .
The
s ec o n da ry fl u i d fl o w a rea i n t h i s reg i on can b e e s t i ma t ed
from t h e a rea i n s i d e t h e tu b e wra p pe r a n d t h e a r e a occu p i ed
by t he U -t u b e s , t hu s ,
A
s
=
( I V . 28 )
A - A
w
t
where
A
s
=
s econdary f l u i d fl ow a r ea , ft
2
1 20
A
A
w
t
2
=
a r e a i n s i d e t h e tu be wra p per , ft
=
2
a rea o cc u p i ed by t h e U - t u b e s , ft •
T he a rea i n s i d e the t u b e wra pper c a n b e e s t i ma t ed from t h e
t u bes a rra n gement , n umbe r a n d o u t s i de d i amete r .
T h e l e n gt h
o f t he e ffect i ve hea t exc han g e r eg i on i s a s sumed to b e
t h e same a s t he stra i g ht t u be b ra n c h l en gt h cal c u l a t ed by
E qu at i o n ( I V . 22 ) .
2.
Oown comer Reg i o n .
The down comer reg i o n i s de f i n ed by t he
a n n u l ar s p a c e b etween t he s t eam gen erator l ower s h el l
s ec t i on a n d t he tube wra p pe r .
T h e down come r fl ow a rea i s
d et erm i n ed by
( I V . 29 )
w here
A
A
d
LS
A
w
=
=
=
downcome r f l o w a rea , ft
2
i n s i d e a rea o f t h e l ower s he l l s e c t i on , ft
a rea o f t h e t u b e wra pper , ft
2
2
T h e l en g t h o f t he down comer s ect i on i s a s s umed to b e e q u a l
to t he same a s t he l en g t h o f t h e e ffect i ve h e a t e x c h a n g e
r eg i o n .
3.
Drum E q u i va l ent Reg i o n .
A s d e s c r i bed i n C ha pter I I I , pa g e 56 ,
t h e d ru m e qu i va l ent reg i o n i s compo s ed of t h ree sec t i on s .
a.
R i s er/ s epa rator vol ume
b.
Drum wa t er vol u�e
c.
Drum s t eam vol ume .
1 21
T he d i men s i o n s fo r t h e s e sec t i o n s were e s t i ma ted u s i ng
eng i ne er i ng j udgement and propo r t i onal i ty pr i n c i p l e s
u s i n g t he s t eam gen era tor d i men s i o n s and t he s econdary
s i d e wat e r and st eam vol ume s g i ven i n Ta b l e I V . l .
IV . 4 . 4
R e s u l t s for H .
B.
Rob i n son S team Gen era to r
T h e g eometr i ca l data cal cul ated fo r the H .
g en erat o r can be s umma r i zed a s fo l l ows .
A.
Ro b i n son st eam
P r i ma ry S i de
1
•
Number o f tube s = 326 0
3.
S tra i g h t tube branch l ength = 29 . 7 5 ft
2
P r i ma ry fl ow a rea = 1 0 . 68 ft
4.
Pri ma ry i n l et p l enum vol ume = 1 46 . 2 7 ft
5.
3
P r i mary o u t l et p l enum vol ume = 1 46 . 2 7 ft .
2.
B.
B.
T u be Meta l
2
1.
T u b e metal c ro s s s ect i o n = 2 . 934 ft
2.
P r i ma ry/meta l h eat tran s fer a rea p e r u n i t
l en g t h = 661 . 436 ft
3.
Meta l / s econdary h eat tran s fe r a rea per
u n i t l ength = 746 . 783 ft .
C.
3
S econdary S i de
1.
Effec t i ve exc hange reg i on
Fl ow a rea ( A ) = 5 2 . 9 3 ft
s
2
To tal l ength ( L ) = 2 9 . 7 5 ft .
t
1 22
2.
Down comer r eg i o n
F l ow a r ea ( A ) = 7 . 6 1 4 ft
d
2
L e n g t h ( L ) = 29 . 7 5
d
3.
Drum e q u i val en t reg i o n
Drum wa ter vol ume ( V
D r um wa ter a rea ( A
dw
dw
Drum wa ter l en g t h ( L
Drum s t eam vo l ume ( V
)
dw
ss
) = 9 7 3 . 88 ft
=
1 0 1 . 1 3 ft
=
2
) = 9 . 6 3 ft
3
) - 2966 . 38 ft
-
R i s er/ s e p a rato r vol ume ( V )
r
R i s er / s e parator a rea
3
=
408 . 5 f t
42 . 4 2 ft
3
2
R i s er/ s e parator l en g t h = 9 . 6 3 ft .
IV. 5
C a l c u l at i on o f Heat Tran s fer C o e ff i c i en t s
I n t h e effec t i v e heat excha n ge r eg i o n , h ea t i s t ra n s fe r red
from t he p r i mary s i d e to the s econdary s i d e thro u g h t he U - t u b e
wa l l s .
On t h e pr i ma ry s i d e , p re s s u r i z ed wa ter pa s s e s i n s i d e t h e
U-t u be s and tra n s fe r s h e a t to t h e t u b e met a l .
Ac ro s s t h e t u b e
t h i c k n es s , heat i s t ra n s ferred by rad i a l conduct i on .
O n t he
secon da ry s i de , s l i g h t l y s u bcool ed wa t er , the n two p h a s e s t e am/
wa t er m i x t u re fl ows by n a t u ra l c i rc u l a t i o n o u t s i d e t h e U - t u bes .
Me ta l -to - l i qu i d heat tran s fer o cc u r s i n t h e s u bcool ed s eco n d a ry
reg i o n whi l e n u c l ea t e bo i l i n g i s t h e p r i ma ry hea t tran s fer
mecha n i sm i n t he bo i l i n g s econda ry re g i o n .
1 23
I n t h i s s ec t i on , the me t ho d u s ed i n cal cul a t i n g t h e h ea t
tran sfer coeffi c i en ts i s desc r i b ed and the resu l t s u s ed a s i n p u t
for t h e d ynami c r e s pon s e cal c u l a t i o n s a r e reported .
F i l m H ea t Tra n s fer Co eff i c i en t s
IV. 5 . 1
Th e D i t tu s -B o e l ter Correl a t i on is t he mo st d e s i ra b l e corre­
2? )
l at i on
fo r c a l c u l at i n g f i l m h ea t tra n sfer coeffi c i en t s when
(
con v ec t i on i s the pr i ma ry heat tran sfer mec han i sm .
T h i s correl a t i on
can b e wri tten i n t he form
=
where
U FC
K
G
0
C
�
p
0 . 0 23
K
(0)
G D 0.8
(--�--)
�
C n
( �)
=
f i l m hea t tran sfer c o eff i c i en t , Btu/ hr ft
=
therma l conduct i v i ty , B t u/ hr f t o F
=
ma s s v el oc i ty , l bm/ h r ft
=
hyd ra u l i c d i ameter , ft
=
v i sco s i ty , l bm/ ft hr
( I V . 30 )
2
°F
2
= s p ec i f i c heat at c on s ta n t pres s ure , B t u/ l bm ° F .
T h e val u e o f t h e i nd ex n i s ta ken a s 0 . 3 for c a l c u l a t i n g t he
hea t tran sfer coeffi c i ent b etween the pr i ma ry wa ter a n d t h e i nn e r
s u r fa ce of the tubes and a s 0 . 4 fo r cal c u l a t i ng t h e h e a t tran sfer
co effi c i en t between t he ou t s i d e s u rface of the t u be s a nd the
s econd ary fl u i d i n the su bcool ed reg i on .
I n the bo i l i ng reg ion of
the s econdary fl u i d , the hea t tra n s fer i s en ha nced by t h e bo i l i n g
mec ha n i sm .
I n t h i s r eg i on , the fi l m heat t ra n s fer coeff i c i en t
c a n b e cal c u l a t i ng u s i n g a correl a t i o n pro po s ed by C he n .
( 44 )
In
1 24
t h i s co r r e l a ti on , the hea t tran sfer co eff i c i en t i n the s a tu ra ted
n u c l eate bo i l i ng reg i o n i s a s s umed to ha ve co n tr i bu t i o n from b o t h
n u c l ea te bo i l i ng a n d con v ec t i on , thu s
( I V . 31 )
where
U
U
TP
NCB
U
C
= heat tran s fer co effi c i en t i n the bo i l i ng reg i o n
= co ntr i b u t i o n d u e t o n u c l eat e bo i l i ng
= con t r i b u t i o n due to con v ect i o n .
I t wa s a s su me d tha t t h e con vect i o n compon en t , U ' coul d be
C
represented by a mod i fi ed D i tt u s -Boel ter typ e equa t i o n
U
-x) 0.8 � C 0 . 4 K f
( -) F
]
[.:._
K J__Q_
D
f
= 0 . 023 [ G ( l
C
�f
( IV . 32)
where F i s a parameter d ef i n ed s u c h that
( I V . 33 )
F i s a f un c t i o n of the Mart i nel l i fa ctor X
=
1
(2)
where
x
X
0. 9
(vf )
Vg
0. 5
( ,,.. f )
;g
tt
0.1
= ma s s qua 1 i ty
v f = s pec i f i c vol ume o f satura ted l i q u i d
vg = s pec i fi c vol ume o f saturated st eam
� f = v i sea s i ty o f saturated l i qu i d
� g = v i sco s i ty o f satura t ed steam .
defi n ed a s
( I V . 34 )
1 25
T h e n uc l eate bo i l i ng contr i bu t i o n h
i s g i ven by
N CB
0 . 79
0.45
0.49
K
C pf
pf
f
., 4 J
U N C B = 0 . 1 7 2 [-0.
=---==--=5_�
o ....
. 2o:-r
0 .�--=
24
0 . 29
h fg
�
P
o
g
f
5
•
�
T sat
o . 24
( 44 )
•
t,
P
sat
o 75
.
( I V . 35)
T h e reader i s referred t o R eference 4 4 fo r t h e d eta i l s o f c a l cul a t i ng
t h e hea i t r a n s fer co effi c i en t i n t h e bo i l i ng reg i o n u s i ng C hen ' s
correl a t i o n .
IV . 5 . 2
Tu be Metal Conductance
S i nc e the t h i c kn e s s o f the t u be wal l i s very sma l l rel a t i v e to
the tu be l eng t h , the tu be metal con d u c ta nc e can be e s t i ma ted u s i n g
t he fol l ow i n g equ a t i o n
( 45)
( I V . 36 )
where
�
b
w
= t hermal cond u ct i v i ty of tube meta l
= t h i c k n e s s o f tube wal l ( i n c h e s ) .
Effec t i v e area for hea t conduc t i on t hrou g h t he tube wal l i s g i ven by
A
m
I V . 5. 3
=
A
0
- A1·
A
l n ( o)
Ai
•
( I V . 37 )
Overal l H eat Tra n s fer Coeff i c i en ts
The o veral l hea t tran sfer r e s i sta n c e i s the sum o f the
res i s tances o f t h e pr i mary s i d e fi l m , the t u be wa l l , fo u l i ng a n d
the s econ dary s i d e fi l m.
B a s ed on t h e secon dary s i de heat tra n s fer
a rea we have
A
A
1
1
1
1 · o + f + o + = -U
u
U
U . A 1.
;a:;
1
m
o
( I V . 38 )
1 26
w here
U = o v e ra l l h ea t tra n sfer coeffi c i ent
u.
u
1
m
uo
= pr i ma ry s i d e f i l m heat tra n s fer coeffi c i ent
=
tu b e me tal condu c ta n c e
= s econd a ry s i d e f i l m he at t ran s fer coeff i c i en t
f = fo ul i ng fa c to r
A . = i n s i d e t u b e area
1
A
0
= o u t s i d e t u be area .
The secondary s i d e f i l m heat t ran s fer coeff i c i ent depends on whether
the secondary fl u i d i s s u bcoo l ed o r bo i l i n g .
Fou l i ng fac to r s may b e con s i d ered to repre sent sa fety fa c tors
t ha t i ncrea se t h e d e s i gn area o f the s team gen era to r to compen sate
for po s s i bl e s c a l e b u i l d up on t he tube meta l su rface s .
S i nce the
pr ima ry cool a n t i s ma i n taj n ed a t ext remel y h i g h pur i ty l evel ,
fou l i ng o n t he p r i ma ry s i d e i s o ften negl i g i bl e .
Secondary s i d e
sys tem pu r i ty i s l ower and some fo ul i ng a l l owa n c e mu s t b e con s i dered
i n cal c u l at i ng t he overa l l heat t ra n s fer co effi c i en ts .
sta t ed that an a l l owa n c e o f 0 . 0003 ( ° F ft
2
Fra a s e t a l . ( 4S )
ht/ B tu ) i s gen era l l y
con s i d ered rea sona bl e .
IV. 5.4
Comb i ned F i l m and Tube Meta l Cond u c tances
When t h e mea n t emperature of t h e t u b e metal l um p i s co n s i dered
a s a s tate var i a bl e i n the dynam i c mod e l , i t i s nece s s a ry to comb i n e
the fi l m hea t tran s fer con du c tance ( coeffi c i en t ) wi th the tu be meta l
conductance to o b ta i n an effec t i ve heat transfer coeff i c i ent between
the bu l k mean t emperatures o f the p r i ma ry and s econdary fl u i d s and
the tu be mean t empera ture .
The cal cul a t i o n fo r thi s effe c t i v e heat
1 27
tra n sfer co e f f i c i en t can b e deri ved from t h e a n a l ogy b e tween t h e fl o w
o f h ea t e n e r gy thro u g h t h ermal re s i s ta n c e t o t h e fl ow o f el e c t r i c
( 46 )
curren t i n a n e l ectr i ca l re s i s to r ,
thus
H ea t T ra n s f er R a t e
=
Tempera t u r e . D i fferen c e
T he rma l Re s 1 s ta n c e
Refer r i ng to F i g u re I V . l w e h a v e
Q pm
T
=
P
Tm
-
A 1
R . +R (�-)
m A.
1
=
U
A . ( T -T )
pm 1 p m
( I V . 39 )
1
and
( IV .40)
From t h e s e equ a t i on s we h a v e
A.
+ ( -,-)
U
A
i
ml
1
l
- = -
U
pm
a nd
A
(-0-)
- -
U
where
=
A -A .
m 1-'---;-A
1 n ( A� )
A -A
o m
A
o
l n(A )
m
rn s
•
A
A (
l)
i A� 1
------
=
1 n(
1
and
A
m2
=
A
A�
.
1
•
l
Urn
( I V . 41 )
( I V . 42 )
( IV . 43)
)
A
A ( 1 - �)
A
o
0
-�-A
o
1 n( )
A
m
( I V . 44 )
1 28
I
�� t- �-'
1---�
P
m
A
R.
1
R m (�)
A.
. R
1
P r i ma ry
S i de
Di
Om
Do
F i g u re I V . l
' -----
VV"-
A
(�
m A0 )
T
- _m
� T5
R
o
Secon d a ry
S i de
Tube .t1e t a 1
-�-...----�!
�
� ms
1
" . - ---- - -- - -------�
I
Comb i n i n g F i l m a n d T u b e Met a l C o n d uc ta n c e s .
1 29
where
=
p r i ma ry s i d e f i l m con d uc ta n c e
=
tube meta l cond uctance
=
s econ dary s i de f i l m conductance
A.
=
tube i n s i d e surface a r ea
A
=
tube mean s u rfa c e area
=
tu be o u t s i d e s u rfac e a rea .
u
u
i
m
u
o
1
m
A
IV. 5. 5
o
=
1 /R
=
1/R .
1
m
=
1 /R
0
Res u l t s fo r H . B . Rob i n s on Steam Gene rato r
T h e fo l l ow i n g va l ues o f the heat t ran s fe r coe ff i c i e n t s a re u s e d
a s i n pu t fo r t he refe rence c a se dyn am i c ca l cu l a t i o n .
P r i ma ry fi l m hea t t ran s fe r coe ffi c i e n t
Tube met a l conductance
=
=
2
4 500 B t u/ h r ft ° F
2
2 1 60 B t u / h r ft ° F
Tube ou t s i de heat t ra n s fe r a rea
=
44430 ft
2
Tube me ta l / s u bcoo l ed s e conda ry fi l m heat t ra n s fe r
coeffi c i ent
=
2
1 9 72 Btu/ h r ft ° F
Tu be metal / bo i l i n g s e condary fi l m heat t ran s fe r
2
coe ffi c i e n t = 6000 Btu/ h r ft ° F
IV.6
I V. 6 . 1
S teady State Cal cul a t i ons
I n t roduct i on
W hen t he mo v i n g boun d a ry a pp roach i s used i n t h e deve l o pme n t
of dynami c mo de l s fo r a UTS G , t h e cond i t i o n s a t stea dy state
a re pre requ i s i te s fo r t he dynam i c cal cul at i on a l gori t h m .
I n th i s
sect i on , i t i s a s s ume d t ha t t he rec i rc u l a t i o n rat i o o f the steam
generato r i s known a n d t h e pu rpo s e o f the s teady s tate cal cul a t i o n
i s t o dete rmi n e t h e tempe rat u re p ro f i l e s a l o n g t h e p r i ma ry a n d
1 30
s u bcoo l e d s econda ry fl ow pat h s a n d a l so t h e ma s s q u a l i ty profi l e a l o n g
t h e bo i l i n g s e c o n d a ry fl ow pa t h a fte r f i n d i n g t h e l e ve l a t w h i ch the
bo i l i n g proce s s i n t h e secondary fl u i d s t a rt s
( Ls 1 ) .
F i g u re I V . 2
s hows a s c hemat i c d i a g ram o f t h e e ffe c t i ve heat e x c h a n g e re g i o n ( co re )
o f a UTS G .
P r i ma ry water from t h e rea cto r h o t l e g e n t e r s t h e h ea t
e x c h a n g e re g i o n f rom t h e steam ge n e ra t o r i n l e t p l e n um a t tempe ra t u re
T
Pi
a n d a ft e r exc h a n g i n g hea t t o t h e s e c o n d a ry fl u i d pa s s i n g o ut s i de
t h e U - t u b e s , i t l ea v e s t h e heat e xc h a n g e re g i on t o t h e s team g e n e ra t o r
o ut l e t p l e n u m at t empe rat u re T
Po
"
The s e c o n da ry fl u i d e n t e r s the h e a t
exc hange r e g i o n f r o m t he downcomer s e c t i o n a t a tem pe r a t u r e T d wh i c h i s
l ower t h a n t he s a t u ra t i o n tempe rat u re c o r re s pon d i n g to t h e s team
gen erato r p re s s u re .
As t he s e c o n da ry fl u i d fl ow s u pwards i n t h e core
reg i on ( by n a t u r a l c i rc u l a t i on ) i t s tempe ra t u re i n c re a s e s u n t i l t h e
s at u ra t i o n t em pe rat u re T s a t i s re a c h e d a t a l e ve l
c o n s i d e re d a s t h e s ubcool ed/ bo i l i n g b o u n d a ry .
L sl w h i c h i s
T h e p r i ma ry t empe rat u re s
a t t h e s ub c o o l ed/ bo i l i n g bo u n d a ry a re de n o t ed by T
pa ra l l e l a n d co u n t e r fl ow bra n c h es re s pe c t i ve l y .
Pl
and T
Above L
P4
Sl
i n the
, t h e hea t
added to t h e s e c o n da ry fl u i d i s u s ed i n i n c rea s i n g the ma s s q u a l i ty
o f t h e steam/wa t er m i xt u re from xso a t t h e o n set o f b u l k n u c l e a t e
bo i l i n g t o x
=
x
e x c ha n g e re g i o n .
e a t t h e end o f t h e bo i l i n g s ec t i o n o f t h e h ea t
The p r i ma ry t em pe ra t u re s a t the e n d o f t h e b o i l i ng
s e c t i o n a re denoted by T Z a n d T p i n t h e p a ra l l e l and c o unte rfl ow
3
P
bra n c h e s re s pect i ve l y .
S i n ce t he p r i ma ry tempe rat u re a t t h e e x i t o f
t h e para l l e l fl ow b ra n c h must be e q u a l t o t h e i n l et tempe ra t u re t o
t h e c o unte rfl ow b ra n c h, T Z a n d T p s ho u l d b e e q u a l .
3
P
S i n ce t h e
1 31
!
I
t:: i
o'
-1-'
u
<l.!
(/')
C'l
t::
X.
T .+
J +1
rc J 1
- - - - - - - · T .
X.
J
___ _ - - - _P-SJ -
yj + l
y.
J
t
t
T
-f ·
. --- - -}
sat
yj + 1
yJ. _
_1_ .
l�
F i g u re I V . 2
t
p
,
T
pl
.
0
a::
--� \4
P
,
T
"0
<l.!
r-
0
0
u
..0
::l
(/')
t::
0
.,....
-1-'
u
<l.!
(/')
_)
_
po
S c hema t i c o f t h e E ffe c t i ve H e a t E xcha n ge Reg i o n ( Co re ) .
1 32
s econda ry i n l e t tempe ra t u re T
d
i s dependent o n t h e ma s s q u a l i ty at
the exi t of t he bo i l i ng sect i on , an i te ra t i o n tech n i q ue i s nece s sa ry
to sat i s fy an o v e ra l l heat ba l an ce between the p r i ma ry a n d s e condary
s i des. o f t h e s team gene rato r .
IV.6.2
Sol ut i on U s i ng the F i n i te Ui ffe ren ce A p p roach
I n t h i s met hod , t he pa t h a l o n g the heat exchange re g i o n i s
d i v i ded i n to a n umbe r o f f i n i te s ect i o n s .
T h e heat bal a nce
equat i on s fo r e a c h s ec t i on a re s o l ved fo r t h e e x i t con d i t i on s us i n g
t h e i nl e t con d i t i o n s a n d t he o t he r known sys tem pa rame te r s .
T he
s econdary t em pe ra t u re a t t he e x i t of the f i n i te sect i o n i s compa re d
wi t h the s a t u ra t i on t em pe rat u r�
There a re t h ree po s s i bl e con di ti on s :
1.
The seconda ry tempe rat u re i s l es s than T
2.
T h e s ec o n d a ry tempe ra t u re i s e q u a l to T
sat ·
sat
a bl e conve rgence a l l owance ) .
3.
The seconda ry tempe ra t u re i s g reater than T
( w i th i n a rea s o n sa t
•
I n t h e f i rs t c a s e , t he he i ght a l o n g t h e h e a t e x c h a n g e p a t h i s
i n c remen t ed and t he c a l cu l a t i o n s a re re peated fo r anothe r fi n i t e
sect i on u s i n g t h e e x i t cond i t i o n s o f t h e p re v i o u s s ect i o n s a s the
n ew i n l et cond i t i o n s .
I n t he second case , the s ubcool ed/bo i l i n g
boundary i s reached and new heat b a l a n ce equat i o n s a re s o l ved where
t he ma s s q u a l i ty i s the unknown va r i a b l e fo r the s e con dary fl u i d .
I n t he t h i rd case , t h e i n c remental h e i ght o f the f i n i te s ec t i o n i s
dec rea sed and t h e ca l c ul a t i on s a re re peated un t i l the secondary
t empera t u re a t the ex i t o f the ca l c u l a t i o n s ect i o n co n ve rge s to t h e
s a t u rat i on tempe ra t u re wi t h i n a re a s o na b l e convergence a l l owance .
1 33
I n t h e b o i l i n g sec t i o n , the fi n i te e l eme n t c a l c u l a t i o n c o n t i n u e s
u n t i l t h e t o t a l he i g h t o f t h e U - t u b e b ra n c h i s rea c h e d .
At t h i s
po i nt , t h e pri ma ry e x i t tempe ra t u re s , T p a n d T p a re te s te d fo r
2
3
e q u a l i ty i n c a s e t hey a re u n eq ua l , the p r i ma ry o u tl e t t empera t u r e i s
a d j u s t ed a n d t h e c a l c u l a t i on l o o p i s r e p eated .
T h i s a p pr o a c h ca u se s
a s l i g h t a dj u s tmen t o f t he p r i ma ry a v erag e t empe ra t u r e i n o rder to
a c h i ev e ba l a n c ed c on d i t i on s i n t he s t eam g en erato r .
T h e hea t ba l an c e e q u a t i on s fo r a f i n i t e el emen t i n t h e s u bcoo l e d
s ec t i on a r e g i ven b y ( s e e F i g u r e I V . 2 , pa g e 1 31 )
W C ( T . -T . )
P P 1 P PJ P PJ + 1
IJ
W C
p1
(T
. -P . )
PCJ + 1 pc J
=
u 1 A . . ( T ppm . -T Sr.l . )
lJ
J
J
( IV. 45)
u 1 A l. . ( T pcmj -T Sm .
J)
J
( I V . 46 )
)}
W C { ( T . -T . 1 ) + ( T . 1 -T
pc J
p CJ +
p pl
PPJ p p J +
.
=
w
-2
c 2 ( T S . + 1 -T S J. )
x
J
e p
( I V. 47)
w he r e ,
A. .
lJ
T
T
and
p pmj
.
pC fTIJ
T
n 00 N ( yj +l -yj )
=
T
=
T
=
2
+ T
e cj
TS .
J
Sfllj.
+ T
p ej
2
+
2
T
( I V . 48 )
!2J2j + 1
( I V . 49 )
J2C j + 1
( I V . 50 )
S J. + l
( I V.51 )
T h e i n i t i a l cond i t i o n s fo r t h e s u bcool ed sec t i o n ca l c ul a t i on
a re
=
=
and
=
T .
pl
T
po
T .
d
1 34
E q u a t i o n s ( I V . 4 5 ) t h ro u g h ( I V . 4 7 ) a re sol ved for t h e e x i t
con d i t i on s ( at y
j +l
) know i n g t h e i n l et con d i t i o n s ( a t y ) .
j
T h e h ea t ba l a n c e e q ua t i o n s fo r a f i n i te e l eme n t i n t he
bo i l i n g s ec t i o n a re g i ven by ( s ee F i g u re I V . 2 )
w
c
p p1
n
)
u 2 A 1. J. ( T p pmJ. - T sa t )
( I V . 52 )
)
W C ( T . -T . ) = u A . . ( T
2 l J pcmJ. -T s a t
p p l pc J + l PC J
( I V . 53)
W C ( T . -T .
p p l P PJ ppJ+ l
=
) + ( T . -T
.-T
)}
pC J + 1 p CJ
PP J p pJ + 1
.
.
=
w
-.2 ( X .
X
e
x.)
J +1 - J
( I V . 54 )
w h e r e t h e def i n i t i o n s i n E q ua t i o n s ( I V . 4 5 ) t hro u g h ( I V . S l ) a pp l y
wi t h t h e fo l l ow i n g i n i t i a l con d i t i o n s
Tppl = Tpl
T
pel
x
l
= T
p4
= 0.0.
E q u a t i o n s ( I V . 52 ) t h r o u g h ( I V . 54 ) a re s o l v ed fo r t h e ex i t c o n d i t i o n s
(at y
) know i n g t h e i n l et c o n d i t i on s ( a t y ) u n t i l y
i s equal
i
j +l
j +l
i s e q ua l to T
wi t h i n a
t o t h e t u b e b ra n c h l en g t h a n d T
p3
p2
rea son a b l e con ve r g en c e to l eran c e .
T h e a bo ve me t hod wa s prog rammed o n t h e I B t1/ 360 d i g i ta l computer
o f T he Un i vers i ty o f T en n e s s e e and i s u s ed a s a s u br o u t i n e wh i c h
pro v i d e s t h e s t ea dy s ta te bou n d a ry con d i t i on s n e ed ed for t h e
dyn am i c c a l c u l a t i o n a l go r i t hm fo r Mode l s C a n d D .
1 35
IV. 6 . 3
S o l u t i o n U s i n g Loga r i t hm i c Mean Tempera t u re D i ffere n c e ( L rnD )
I n t h i s me t ho d , t he e ffec t i ve h ea t e x c ha n g e reg i o n i s d i v i d ed
i n to two s e c t i o n s , a s u bcoo l ed s ec t i on a n d a bo i l i n g s ec t i on a s seen
E a ch sect i o n h a s a para l l e l fl ow b r a n c h and a
i n Fi gu re I V . 2 .
c o u n t e r f l o w b ra n c h w i t h a commo n s e conda ry fl u i d pa s s i n g o u t s i de the
t u bes .
The heat ba l a n c e equat i o n s fo r the s u bcoo l e d s e ct i on a re g i ven
by
( IV.55)
•
Q c s l = W pC p 1
( T p4 -T po ) = u 1 A s 1 ( LMTD ) c s l
•
Q p s l + �4 c s l
=
w
s
(h
f
-
h
si
)
( I V . 56 )
( IV.57)
whe r e t h e l o ga r i t h m i c mean tempe ra t u re d i �feren c e i n t h e pa ral l el
a n d cou n t er f l ow b ra n c he s o f t h e s ubco o l ed s ect i o n a r e g i v e n by
( LMT D )
=
psl
( T i -T ) - ( T - T t )
d
e
�1 sa
T . -T
d )
1 n ( e,
T p 1 - Ts a t
(4S)
( I V . 58 )
and
( L MTD )
cs1
=
( T�4 -T s a t ) - ( T�o -Td )
( IV . 59)
T
-T
e4 sat )
n
(
1 T -T
po d
I n t he bo i l i n g s e c t i o n , t he heat ba 1 a n c e equat i o n s a re g i ve n by
Q
ps 2
=
w c ( r -T )
P P 1 P 1 pm
=
u 2A 5 2 ( L MT D )
ps Z
( I V. 60)
1 36
Q•
cs2
=
W C ( T -T )
p pl pm p4
=
u 2A s 2 ( LMTD ) c s 2
( IV.6l )
( I V . 62 )
S i n c e t h e b u l k mean t empera t u re o f t h e secon d a ry fl u i d i n t he
bo i l i n g s ec t i on i s equa l to T
sa t '
t he l o ga r i t hm i c mea n t empera tu re
d i ffer en c e i n t he paral l el and co u n t e rfl ow b ra n c h e s o f the bo i l i n g
sec t i on a re g i ve n by
( L MTD )
ps2
( uno ) cs2
whe re T
pm
=
=
=
- T )
�m
�l
T -T
�m
at
)
1 n (T pm s a t
(T
(T
/
( I V.63)
)
�m - T �4
T -T
at
12m
).
1n (T p4 s a t
/
( I V . 64)
.
P3
T h e a bove eq ua t i o n s we re p ro g ra mmed o n t h e I B M/ 36 0 d i g i ta l
T
p2
=
T
com pute r o f T h e Un i ve r s i ty o f T e n n e s s ee .
E i t h e r the f i n i te d i ffe r -
e n c e o f t h e l o g me a n t empe ra t u re d i ffe r e n c e met h o d c a n b e u s e d t o
ca l c u l a t e t h e tempe rat ure a n d ma s s q u a l i ty profi l e s i n a UTS G .
Howeve r ,
s i n ce t h e f i n i te d i ffe rence met h o d c a n be e a s i l y i mp l emen ted i n t h e
dyn am i c ca l c u l at i o n a l g o r i t h � , i t wa s dec i de d to u s e i t i n obta i n i n g
t h e re s u l t s o f t h e dynami c res pon s e c a l c u l a t i on s re p o rted i n the
fo l l owi n g s ec t i o n .
A compa r i son betwee n t h e two met h o d s i s g i ven i n
t h e n ext sect i o n wh e re t h e stea dy state re s u l t s fo r H . B . Ro b i n so n
N u x l e a r Power P l a n t i s presen ted .
1 37
IV.?
Re s u l t s fo r H . B . Rob i n son S t eam Ge n er a t o r
T h e pr i ma ry a n d s u bcoo l e d
seco n d a ry t empe r a t u re v a r i a t i o n
a l ong t h e heat t ra n s fe r pa t h a s wel l a s t h e va r i a t i o n o f t h e ma s s
q u a l i ty o f t he s ec onda ry fl u i d i n t h e bo i l i n g re g i o n were c al c u l a te d
u s i n g t h e f i n i te e l emen t method a n d t h e re s u l t s a r e s hown i n
F i gure I V . 3 .
The c o n d i t i o n s d e s c r i b i n g t h e i n l e t a n d o ut l e t o f t h e
s u bcoo l ed a n d bo i l i n g s ect i on s a re a s fol l ows .
A.
P r i rna ry S i d e
Tp ;
T
T
T
T
B.
=
601 . 2 ° F
59 7 . 2 ° F
Pl
= 566 . 9 O F
Pm
P4
Po
=
548 . 1 O F
=
546 . 2 O F
S ec o n d a ry S i de
T
T
x
L
d
=
500 . 6 o F
= 51 7 °F
sat
e
sl
0 . 1 99 3
=
=
( 0 . 2 sta rt i n g v a l u e )
3 . 9 6 6 ft .
W hen the L MTD me t hod wa s u s ed to o b t a i n t h e s u bcoo l e d / bo i l i n g l um p s
bou nda ry con d i t i on s , t h e fo l l ow i n g re s u l t s were o bta i n ed .
A.
P r i ma ry S i d e
Tp ;
T
T
T
T
Pl
Pm
P4
Po
=
602 . 1 O F
= 597 . 3 3 O F
= 56 7 . 06 O F
=
548 . 2 O F
=
546 . 2 O F
1 38
-
-r
C o u n t e r fl 0 \1/
P a.r a l l e l fl ow
---r-·-- --
-- -- - - - - - - - --- - -
_ _
-
-
L_
I
I
P r i ma ry fl u i d
tempe ra t u re ( ° F ) I
I
�I
T pm
I
r.1a s s
•
-
..r
I
I
__.
q u a l i ty
I
I
i
.
. ·"
- - -- -- -�
- ---- --- - - �I
-
\_a_t__
-�
-<- - -----
--- -----�
I,
i I
S eco n da ry f l u i d I
tempera t u re ( ° F ) I \
'
I
I
I
I
�
' ---------- - - --- - - --�----'----�----
L
sl
L
2L-L51
F i g u re I V . 3 T empe r at u res a n d t1a s s Q u a l i ty P r o f i l e s C a l c u l ated by
t h e S t eady S t a t e P ro g ram .
L
Td
B.
1 39
S ec o n d a ry S i de
Td
500 . 6 ° F
=
51 7 O F
Tsat
=
x
0.2
=
e
L l
s
=
3 . 9 5 ft .
From t h e a bove r e s u l t s , we c a n s e e t h a t the res u l t s o b t a i n e d from
the fi n i te e l eme n t s method or the LMT D met h o d a re i n a g reeme n t
to wi t h i n 0 . 2% i n t h e boundary tempe rat u re ca l c u l a t i o n a n d w i t h i n
0 . 5% i n e s t i ma t i n g t he s u bcoo l ed l en gt h .
I t i s n o t i c e d t h a t t he
u se o f t h e fi n i te e l emen t met hod re s u l t ed i n a s l i g h t mod i f i c a t i o n
o f t h e p r i ma ry wat e r a ve ra g e t empe r a t u r e .
IV.8
Gen e r a l Rema r k s o n Dyn a m i c Respo n s e C a l c u l a t i o n s
T h e fo l l ow i n g rema r k s a re needed befo re pre s en t i n g t he re s u l t s
o f t he dyn a m i c res po n s e ca l c u l a t i on s o b t a i n ed from t he U T S G ma t h e ­
mat i ca l mod e l s d e ve l o pe d i n C ha pter I I I , pa g e 2 5 .
l.
W hen t h e c r i t i c a l f l ow a s s umpt i on wa s u s ed t o furn i s h
t he s team fl ow ra t e l ea v i n g t he s team generato r , t h e
t erm o W
so
wa s ex pre s s ed by t h e fo l l ow i n g e q ua t i on
( IV . 65 )
2.
T here a r e two ways to h a n d l e t h e feedwa t e r fl ow t e rm ,
o \4 Fi .
a.
U n co n t ro l l ed ca s e .
I n t h i s c a s e o \·J
Fi
i s a fo rc i n g
funct i o n ( i f th e pertu rba t i on o f i n t e re s t i s a fe e d ­
i s z e ro ( n o c ha n g e
wa t er fl ow p e r t u r b a t i o n ) , o r o W
Fi
i n feed wa t e r fl ow ) .
1 40
b.
I n t h i s c a s e , t he feedwa t e r
Perfe c t c o n t rol l er c a s e .
fl ow ra te i s c o n t i n uo u s l y s et equa l to t h e s t ea8 fl ow
rate ( oW .
Fl
=
oW
so
).
For the c a s e of t he c r i t i c al fl ow
a s s um p t i o n , oW F i = c W so
=
C
L
c P + P0 c C L .
t h a t t erms i n vo l v i n g c P n ow a p pea r i n t h e
T h i s mea n s
A
ma t r i x to
repre s en t t hi s con trol a s s umpt i on b ec a u s e o f t h e
C
3.
L
c P t erm.
I n obta i n i n g t h e dynami c re s po n s e from Mod e l C , i t wa s fo u n d
n e c es s a ry to remo v e t he a l gebra i c eq uat i o n e x p r e s s i n g c T
s
i n t erm s o f cT
d
and o P .
Al so , t h e equat i o n s a re reo rdered
i n order to a vo i d t h e i n v er s i on of i l l -con d i t i o n ed ma tr i c e s
3?
w h i c h may re s u l t i n erro n eo u s r e s u l t s from P URE D I F F . ( )
4.
F rom t h e e x p e r i en c e ga i n ed from Mo d el C , i t wa s fou n d t h a t
comb i n i n g t he d ru m wa t er v o l ume wi t h t he down c o mer l ump a n d
comb i n i n g t h e eva po rator/ r i s e r vol ume w i th t he bo i l i n g
s econdary fl u i d l um p i n Mod e l
0
red u c e s t h e o r d e r o f t he
sys t em equat i on s from 2 0 to 1 8 a n d s o l v e s t h e probl em o f
J?
t he i n v er s i o n o f i l l - co n d i t i o n ed ma t r i ce s i n P URE D I FF . ( )
5.
A compl ete s umma ry o f the dyn ami c re s po n se re s u l t s a s
o b ta i n e d ft·om t he mat hema t i ca l mo de l s i s g i ve n i n Appe n d i x
F.
I n t h e fo l l owi n g s e ct i o n , t h e s t e p re s po n s e re s u l t s o f
t he fo u r mo de l s , to a +1 0% c h a n ge i n steam val ve coeffi c i ent ,
a re p re s e n t e d fo r t h e p u rpo s e o f c h e c k i n g t h e p hys i c a l
p l a us i b i l i ty o f t he ma t he ma t i c a l de vel o pme n t a n d compa r i n g
t h e mo de l ' s c a pa b i l i t i es .
141
IV.9
IV. 9. 1
Dyn am i c R e s po n s e Re s u l t s
R e s po n s e o f Mod e l A
Th e n o n z ero el emen t s o f t h e A ma t r i x a nd B ma t r i x fo r t1odel A
( u s i n g i n pu t d a t a fo r H . B . Ro b i n so n s t eam generator ) a re g i ven
by :
)
=
- 0 . 9 7 83
A( l , 2 )
=
0 . 6486
A( 2 , 1 )
=
2 . 4 06
A(c , 2 )
=
- 5 . 40 0
A( 2 , 3 )
=
0 . 4 342
/\ ( 3 , 2 )
=
1 . 65 1
A( 3 ,3 )
=
- 0 . 2 864
B(l ,l
)
=
0 . 3296
8(3,2)
=
0 . 05 5 7 9
8(3,3)
=
A( l , l
- 33 . 2 8 .
The s t ep r e s po n s e o f Mo del A to a + 1 0% c ha n g e i n s t eam va l ve
coeff i c i en t i s g i v en i n F i g u r e I V . 4 .
I t s h ows a stabl e re s po n s e w i t h
a t i me con s tant o f a pprox i ma t e l y 1 5 s eco n d s .
I t can b e s ee n t h a t
t h e s t eam p r e s s u r e s hows t h e ea rl i e s t res pon s e t o t h e p e r t u rba t i on .
T h i s i s to b e ex pect ed s i n c e t h e s t eam va l ve pert u rba t i o n i s a
s econ d a ry s i d e pertu r ba t i o n .
T h e p r i ma ry o ut l et t empera t u r e res pon s e
s hows a s l i g ht d e l ay o f a bo u t l . 2 5 s econ d s .
Then i t s t a r t s t o fa l l
down fo l l owi n g t h e d ecrea s e i n t h e seconda ry s i de tempera t u r e c a u s ed
by the pres s u r e d ecay due to t h e st eam v a l ve per turba t i o n .
1 42
0
0
(Y"') �
>
(f)
s team p re s s u re P
0
\.D 0
.
s
� -�-----.:;:::====::;::=::===;:::=:; D. =
'
o . oo
30. oo
T I ME
9 0 . oo
60 . oo
-
1 20 . oo
38 . 4 1 8
(5ECJ
0
0
(\J 1
>
(f)
\.D 0
0
.
t ube me t a l temner
a t u re T
.
------
� �------�--�
'
o . oo
3 o . oo
T I ME
6o . oo
9o . oo
D. = - 4 . 3 8 4
1 2o . oo
( 5 E CJ
p r i ma ry te�pe ra t ure T
-.D o
0
m
p
� �------�====�==� �).=
1
0 . 00
30 . 00
T I ME
Fi g ure I V . 4
60 . 00
90 . 00
1 20
00
-2 . 9 0 6
(5ECJ
S t e p Re s po n s e o f �·1o del
Va l ve Coe ffi c i e n t .
A t o + 1 0% C h a n ge i n Steam
1 43
IV . 9 . 2
Re s po n s e o f Mo d el 8
The n o n z ero el emen t s o f t he coeff i c i en t ma t r i c e s A a n d B fo r
Mod el B
( ca l cu l a t e d u s i n g t h e H . B . Ro b i n so n s team g e n e rato r data )
a r e g i ve n by :
A( l , l
)
=
- 1 . 3 08
A( l ,3 )
=
0 . 6486
0 . 6593
A( 2 , l )
A(2 ,2 )
=
- 1 . 30 8
A( 2 ,4 )
=
0 . 6486
A(3 , 1 )
=
2 . 406
A( 3 , 3 )
=
- 5 . 400
A( 3 ,5 )
=
0 . 4342
A(4 , 2 )
=
A ( 4 ,4 )
=
A(4 , 5 )
=
A ( 5 , 3)
=
2 . 406
-5 . 400
0 . 4 342
0 . 8254
A( 5 ,4 )
.:::
0 . 3254
A(5,5)
=
- 0 . 2 364
B(l ,l )
=
0 . 6 59 3
B(5 ,2)
=
0 . 0558
8(5,3)
=
- 3 3 . 28
The r e s po n s e o f Mo de l B to +1 0% s t e p c h a n g e i n s t eam val ve co effi c i e n t
i s s hown i n F i g u re I V . 5 .
The g en era l trend o f t h e r e s po n s e i s
s i m i l a r t o Mod el A e x c e p t fo r t he mo r e d e l ayed pr i ma ry o ut l et temrera ­
t u re
( a bo u t
val u e s .
2
seco n d s ) a n d t h e s l i g h t c ha n g e i n t he n ew s t eady state
T h i s c ha n g e i s the r e s u l t o f u s i n g two l um p s to r e presen t
t he pr i ma ry water a n d t u be met a l i n s tead o f o n e l u mp a s u s ed i n Mo del
A.
1 44
0
0
(Y1 0'
>
<.n
t u be me t a l te�pe ra t u re T
...0 0
0
171
1
� �------�====�==� �=
•
a . oa
3 a . aa
T I ME
6a . ao
9a . oa
-
1 20 . oo
3
.
742
(SEC)
0
0
(\J a
>
(f)
...0 0
0
�
•
p r i �a ry te�pe ra t u re T
p2
-------
�------�--�
a . oa
3a. ao
T I ME
6a . oa
�=
-3 . 056
1 2 0 . 00
90 . 00
fSECJ
0
0
>
<.n
p r i ma ry tempe ra ture T
...0 0
C)
Pl
ru t-----�====�==� �=
•
o . oo
3a . oa
T I ME
F i g u re I V . 5
6a . oo
9o . oo
1 20 . 00
fSECJ
S t e p Re s po n s e o f Model B to +1 0% C h a n ge
S t e a� Va l ve Coe f fi c i e n t .
in
- 1 . 855
1 45
0
0
lf) 0"
>
(J1
s t e am pre s s u re P
0
..0 0
s
______ b =
� �------�--r--.--�
1 20 . o o
so. oo
go . oo
30. oo
'o . o o
0
T I ME
36 . 256
(5ECJ
0
0
:::r 1
>
(f)
..0 0
0
t u b e me tal tempe ra t u re T
_____
m2
b=
� �------,--.--�--�
'
o . oo
30 . 00
60 . 00
90 . 00
1 20 . 0 0
T I ME
F i g u re I V . 5
( Co n t i n ue d ) .
(SEC)
-4 . 277
1 46
IV.9.3
Re s po n s e o f Mod el C
S i n c e Mo del · c wa s deve l o pe d i n t h e m i x ed a l g e b ra i c pl u s
d i fferen t i a l fo rmu l a t i on , t here a r e two co effi c i e n t ma t r i c e s ,
A
1
( 37 )
a n d A , t o b e c a l c u l a t ed a s i n p u t t o P U RE D I FF .
2
el emen ts o f A
1
and A
2
a r e g i ven i n Ta b l e I V . 2 .
T h e n o n z e ro
Th e n o n z e ro e l emen t s
o f t h e fo rc i n g v e c to r f o r t h e ca se o f +1 0% s t ep c ha n g e i n t h e v a l ve
co effi c i e n t a n d a s s u m i n g that t h e feedwa ter fl ow a l ways ma tches t he
s t eam fl ow a re g i ven bel ow
f(6 )
=
-89 . 06
f(8)
=
89 . 06
f( 9)
=
7 . 96 7
X
l0
4
•
No t e t h a t t h e o u t p u t from P URE D I F F i s a 9 x9 red uc ed ma t r i x a n d a
9xl fo rc i n g v e c to r t h a t i s u s ed by r�TE X P to c a l c u l a t e t h e t i me
re s po n se .
T h i s s t e p i s do n e w i t h i n t h e dyna m i c re s po n s e c a l c u l a -
t i on a l gori thm w i t ho u t extern a l i n terferenc e .
The resul t s o f the
s t e p r e s po n s e cal c u l a t i o n s a re s hown i n F i g u r e I V . 6 .
I t i s not i c ed
t ha t t he re s po n s e o f t h e p r i ma ry a n d t u b e met a l l umps a r e a l mo s t
t he s a m e a s i n Mo d el B .
Th i s i s to b e e x p e c t ed s i nc e t h e d i ffe re n c e
b etween Mod e l s B a n d C i s o n l y i n t he treatme n t o f t h e s ec o n d a ry s i d e .
For t h e s ec o n d a ry s i de l umps , i t i s n o t i ced t h a t t h e d e v i a t i on
of t h e s team p re s s ur e i n t h e n ew s t eady s ta t e i s a b o u t 4 . 5 p s i l e s s
than fo r Mo del A a n d 2 . 4 p s i l e s s t h a n Model B .
by t h e fact t ha t i n Mod e l s A a n d
8,
T h i s can b e ex pl a i n ed
t he s econda ry s i d e tem p era t u r e
i s a s s umed t o b e the s a t u ra t i on t em p e ra t u re .
Wh i l e i n Model C , i t
i s t a k en a s a we i g hted a verage b etween the downcomer t empe ra t u re a n d
the
s a t u rat i on t empera t u r e .
TABLE I V . 2
NONZE RO ELE!�EIHS O F THE MATR I CE S
THE N O � Z E R n F L � M F NT S O F T H E
1
5
6
9
1
2
]
4
5
6
1
8
9
11
12
5
1
1
0
01
2
- O . l 777F 03
0 . 4675f 05
7
10
0. 1 000r
o . 2 3 7 7r o s
2
6
5
9
5
A 1
M l TR I X A RE :
3
3
- 5-- - - ·g
o o oi= o i
9
9
6
5
C)
12
10
11
7
0 . 1 30 8 F
01
-
oo
o3
•
•
3
-
s·
11
.
--- ---·
.
0. 1 OOOE 0 1 .
r·
·4
0 � l OO O E O f
o ; 4 l 5 E -0 5_ _ _____ 6- -- �- 5---- -- - - 0 � 7 6 6 E 0
7
11
.
- --
-
.
9
0 . 48 5 4 E 0 4
-0. 6558E 05
6
- 0 . 6 5 93 E
1
2
- 0 . 6 4 9 0E 0 0
3
1
3
3
1
3
0 2 4 06 E 0 1
4--- --z ---�- o �-t4-0 6F-01- -4--4
--- 4
- ---3
5
1
0 .
9
5
0 . 2 6 4 l f 04
- 0 . 6 4 9 0F 0 0
- o � 1 9 9 6 F" 'l o
- J . 1 9 96 r0 . 3 0 6 9 f7 0 7
6 0 . 1 132F 01
0 . 4 4 5 4 F 04
5
8
-o. 8007F 00
5
9
- - -- 9 ---- 1 2
o . 96 0 4 F
- 0 1 0 o o r· 0 l
12
-0 832 F 02
10
12
-
.
7
8
10
T H E N n � l I= R n f l F M F NT S 0 F T HE - A 2 '-1 Ar R IX 4. R E :
1
4
- ·-
.
A2 FOR r()OE l C
---- · ----· -----
----
0� 1
- �-0 � 64 5 6 E- - 0 8
0. 1 02 8E 0 5
0 . 1 7 5 8E 0 2
·
Al AND
· 2
2
5
3
4--5
00
01 .
0 . 5400E
0. 4 6 6 2 E
o. �0 57E
7
5
11
07
01
O l 3 08 E 0 1
• 0. 4 0 5 3 E 0 0
5 3 E o o · -- -- --- - - .
.
- o:-4o
o . s 75 3 E . o4· ·
---·
·
-·
·
·
s
s
·
··
::o . 1 0 9 5 F o s· · · · s · ·
- o . 1 a CJ s E · os " ·
o . 1 2 70 E 04 .
- 0 . 1 1 1 4 E 0 4 . . . 5 . i i --·-·
_
6
4
�
4
6 - --·-_- o 4 5 4F ··-·o 4--- -- -- '12 --- - - .;,;. o �- t993 E __o_cf .-�
_ 7-- · s �--� O � ft 4 5 4 E ' o - 3
.
0 1 0 00 E 0 1
10
8
-O . l l 2E 0 1
0 . 445 4 E 04
6
8
0 . 4 4 54E 04
3
9
0 . 43 5 1 E 07
6
9
- O . l 5 2 9 E 04
� o . 1e-2 o r:�o 3 ___ 1 0 ----ro - - -..: o :-.l' o o c;t-o i-1 o-u
o:r o o o E - o ! '- .
1 2 . . . . & - .. . : o . 2 6 8 9 r::· 0 4
. . 0 . 1 000F 0 1 . 1 2 5 .
0 . 72 3 9 E 00
.
-
.
o-:-s 4-ot>F.-61
.
.
.
.
·
·
-
.
0
•
1 0 00F
.
.
01
.
.
..
..
-
-
--
-
-
-
- -
..
-----
---
.
.
.
-
-
.
·
.
-
-
--
··--
.p.
""--1
..
.
.
-
.
-
----'---''--
-- --
---
- --
-
-
1 48
Q_
L
t-
0
0
0
.
t u be me t a l te�p e rature T
\0 0
0
�l
------- .D. =
.
� 4-------�---,---r--�
��
�
n r.
. u '""'
30 . 0 0
T I ME
60 . CC
( SEC)
-3 . 244
8Q . OQ
0
0
u
Q_
t-
� r i �a ry temp e ra t u re T
\.0 0
0
------ L'J. = - 2
.
� 4-------r---�--�
90 . oo
50 . oo
1 2 0 . oo
30 . oo
'a . o o
T I ME
CL
CL
t-
PZ
. 65 1
(SECl
0
0
0
.
\0 0
0
p r i na ry te�pe rat u re T
Pl
�------ L':,. =
N 4-------r---�--�
.
10 . 0 0
Fi g u re I V . 6
30 . 0 0
T I ME
60 . 00
(SECJ
90 . 00
-1
1 20 . 00
Step Re s po n s e o f r1o de l C t o +1 0% C h a n ge i n
Steam Va l ve Coe ffi c i en t .
.
609
1 49
co
o0
'
..--4
------- 6 =
•
0 . 009
( S FEH E L ) e x i t q u a l i ty
LLJ O
X � �------�--------�--------�------�
60 . 00
90 . 00
30 . 00
1 20 . 00
V) 9J . 0 0
T I ME fSECJ
len
(L
0
0
0
.
a
\,() 0
c
.
s tear1 p re s s u re P
s
_______ 6 =
-3 1 . 526
�· +------,---r--r---�
'o . o o
30 . oo
60 . oo
T I ME
lJ
2:
1-
go . oo
1 20 . o o
fSECl
0
0
a
.
t u b e �e ta l terJpe rat u re
1.0 0
: �----��====�==� 6=
'
o . oo
3o. oo
60 . oo
T I ME
F i g u re I V . 6
( Co n t i n ue d ) .
(SEC)
so . oo
T rJ2
1 2 0 . 00
-3 . 708
1 50
c.
0
0
t-
downcome r te�p e r a t u re T
..0 0
0
d
------ � =
� �------r---�---r--�
'
o . oo
3o . oo
6 Q , n i)
T I ME
<3 0 . c ':::
(5EC1
1 2 o , no
-4 . 362
a
0
::3:
D
t-
d r u� wate r te�pe rat u re T
1.0 a
0
.
dw
------ 6 =
� 4-------�--�--�
'o . oo
3:
D
_j
1.0
a
::r
.
0
3o . oo
6o . oo
T I ME
9 o . oo
( SECl
1 2 0 . 00
------
d r um wate r l ev e l L
0
0
dw
· �--------r--------,---------r------�
cu . 0 0
Fi g ure I V . 6
30 . 00
T I ME
( Co n t i n u e d ) .
60 . 00
(5EC1
6�
90 . 00
1 20 . 00
-4 . 363
0 . 352
1 51
Th e i n c rea s ed e x i t q u a l i ty a t t h e end o f t he effec t i ve h ea t
exc ha n ge l um p can b e expl a i ned b y
t h e re d u c t i o n
t empera t u r e d u e t o t he p re s s u r e decrea s e .
be a va i l a b l e fo r s team e va pora t i on .
Thi s
of
the sa turat i on
ca u s e s mo re
h e a t to
The i n c re a s e in t h e downc o me r
l e vel and t h e de c re a s e i n dawn come r tempe ra t u re a re due t o t he i n c re a s e
i n feedwa t e r fl ow i n to t h e s t e am g e n e r a t o r to ma t c h t h e i n c rea s e o f
s t eam fl ow r a t e d u e t o t he s t e p c h a n g e
in
t h e val ve o pen i n g .
The
downcomer temperat u r e l a g s b e h i n d t h e d r u m w a t e r t empera t u re by
a bo u t t hr e e seco n d s w h i c h i s t h e t ra n s po r t t i me i n t he downcome r
1 ump.
IV . 9 . 4
R e spon se of Model
0
The n o n z ero el emen t s of t h e c o e ff i c i en t mat r i ce s , A 1 a n d A 2 ,
fo r Mod e l D a re g i v en i n Tabl e I V . 3 . Th e n o n z ero el emen t s of t h e
fo rc i n g v ec t o r fo r a +1 0% s t e p chan g e i n t h e s team v a l v e coeffi c i en t
a r e g i ven by
f( l l )
=
f( l 2 )
=
f( 1 7 )
=
0 . 0667
59 . 6 7
-89 . 06 .
The s t e p r e s pon s e i s s hown i n F i g ure I V u 7 .
ha s a z e ro
r e s po n s e be c a u s e
it
acts
T h e p r i ma ry i n l e t pl en um
o n l y as a mi x i n g l ump
between t h e r ea ct o r hot l eg t empera t u re a n d the p r i ma ry i n l et
t empera t u r e .
The b e ha v i o r o f t h e pr i ma ry t empera t u r e i n t h e fi r s t
pr i ma ry l ump T P l l oo k s d i fferen t t ha n t he o ther p r i ma ry l umps ,
T h i s i s 'd u e to t h e mo v i n g boun d a ry t h a t
l en g t h o f t he l um p .
d e t e rm i n e s t h e
A l oo k a t t h e r e s po n s e o f t h e s u b coo l ed l en gt h
l f\jj L t
___ _ _ -- --
-
T HE
N n rH E fl 0
_
lV.J
_
HATRICES--Al AN D !\2 FO R�ll D E L D___ _ ___
fiON Z E R O - E L Et·1E NTS Or -THE
f:\,._f: m: NT S _ O E
JJAT � p: A R E :
_T_ ti� _ A l
_
1.
4
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_
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---- - -- - .. - - TJ �F N f11,J_l_�P O _ IL F.: M F. N_T_5_D_F_ _! H� __ A 2 �((1'Jil.X. A_I�J_� :___ _ .
_
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9
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5 845 E 0 .4 .... ... .1 5 . . J 2 _ .
Q.
1 6_1 3____ o .
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l _O O O E 0_ 1 _ __ __ :}
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1 58
s hows a s ha r p dro p t hen a g ra d u a l i n c r e a s e un t i l i t settl es a t a
l ower l ev e l a t t he n ew st eady s t a t e .
T h e down comer l evel a n d t h e
downc ome r t em p e ra tu r e b eha ve i n a s i mi l a r way a s i n Mo del C .
IV.9.5
Compa r i s o n o f Mode l s Re s po n s e s
S i n ce e a ch dyn am i c model h a s s ta te v a r i ab l e s t o re p re s e n t d i ffe r e n t
sect i o n s o f t h e sy stem , a one- to-one co rre s po n de n ce fo r co�ra r i n g t h e
re s ul t s o b t a i ned from t h e fo u r mo de l s i s n o t p os s i b l e .
T h e o n l y two
s ta t e v a r i a b l e s that a re common to a l l fo u r mo de l s a re the p r i �a ry
o ut l et te�pe rat u re
( T p 0 ) a n d t h e s team p re s s ure ( P s ) .
A com8a r i s o n
o f the s e two s t a te va r i a b l e s t o a + 1 0% c h a n ge i n t h e s team val ve
coeffi c i e n t i s s hown i n F i g u re I V . S .
I n f1o d e l
n,
the p r i ma ry o u t l e t
tempe ra t u re i s t a k e n a s the tempe ra t u re o f the l a st p r i ma ry l u� p
( P4 )
s i n ce t h e i n l e t a n d o u tl et p l e n um l umps we re n o t co n s i de re d i n t h e
o t he r mode l s .
Re fe r r i n g to F i g u re I V . 8 , i t c a n b e s e e n t h a t t h e re s po n s e o f
t h e p r i ma ry o ut l e t tempe ra t u re a n d t h e s team p re s s u re i s c o n s i s te n t
fo r a l l fo u r mo de l s .
The ma x i mum de v i at i o n i n th e n e w s te a dy s tate fo r
the p r i ma ry o ut l e t tempe rature i s l e s s t h a n 0 . 5 ° F a n d fo r t h e s tea�
pre s s u re i s l e s s than 8 p s i .
Th i s s hows the l umpe d n a t u re of t h e UTS G .
Th e d i ffe re n ce i n re s po n s e b e twee n t1o de 1 s A a n d B i s d u e to the u s e o f
the ba c kwa rd d i ffe re n c i n g w i th a s i n g l e p r i ma ry l ump fo r Mo del A a n d
two p r i ma ry l um p s fo r Mo del B .
C and
T h e d i ffe re nce i n re s po n s e betwee n �1o de l s
0 i s due t o t h e a s s umpt i o n o f a fi xed bou nda ry betwe en t h e s u b c oo l e d
a n d t h e b o i l i n g s e c t i o n s o f t h e e ffe c t i ve heat e xc h a n ge l ur1p i n �1ode l C
w h i l e a mo v i n g b o u n da ry i s con s i dered fo r t1od e l
D.
1 59
s t e a m pre s s u re ( P s )
0
0
0
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.
0
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.
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0
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Vl
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20 . 0
40 . 0
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60 . 0
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1 00 . 0
T i me ( s ec )
.
0
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p
0
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.
0
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1 20 . 0
T i me ( s e c )
F i g u re I V . 8
Compa r i s o n o f T p 0 a n d P s R e s po n se s i n t h e Fo u r Mode l s .
CHA P TE R V
C OM P A R I SON O F DETA I L E D MODEL P R E D I C T I ONS
W I TH EX P E R I ME NTAL RESULTS
V. I
I n t ro d u ct i o n
A ma t h ema t i c a l model o f a n e n g i n eer i n g syst em i s a s good a s i t s
c a p a b i l i ty o f p r ed i c t i n g t h e dyn am i c r es po n s e o f t h e a c t u a l system .
Therefo r e , t h e compa r i son betwee n t heoret i ca l mode l p r ed i c t i o n s a n d
ex per i men ta l r e s u l t s i s t h e b e s t method fo r test i n g t h e a d e q u acy o f
t h e dyn am i c s i mu l a t i o n mo de l s o f a n en g i n e er i n g system .
I n C h a p t e r I V , pa ge 1 06 , t h e ma t h ema t i ca l mo de l s devel o ped i n
C ha pt e r I I I , pa g e 2 5 , were a pp l i ed to s i mu l a te t h e s t e p r e s po n s e
a n d f requ ency r e s po n s e o f a UT SG w i t h a perfe c t feedwa ter con trol l er .
I n th i s c ha p te r , t h e end p ro d u c t UTSG mo del ( r1o d e l D ) i s c o u p l ed w i th
a mode l f o r a p re s s u r i zed wa ter r eac to r ,
(3 4)
a n d t h e frequen cy
•
r e s pon se o f t h e comb i ne d sys tem i s c ompa red wi t h t h e e x p e r i men tal
r e s u l ts o bta i n e d f rom dyn am i c t e s ts p e r fo rmed o n t he H . B . Ro b i n s on
n u c l ea r power p l a n t .
( 5)
T h e o bj e c t i ve of t h e compa r i so n i s to c h eck
the adequacy of t h e UTSG mo d e l .
V. 2
Desc r i pt i on o f the Sys t em
T h e Nuc l ea r S team S u p pl y Sys t em ( NS SS ) o f
Nu c l ea r Power P l a n t
(2)
the
H . B . Ro b i n son
i s a t h re e- l oo p 2 20 0 t·�.� t ( 7 3 9 t1\·le ) We s t i n g ho u s e
PW R sys tem own ed a n d o p e rated b y Ca ro l i na Power a n d L i g ht Compa n y .
E a c h l oo p i n c l u d e s a v e rt i ca l UTS G .
T he dyn am i c t e s t s were p e r fo rmed
d u r i n g fu l l power o p e ra t i on u s i n g n o rma l pl a n t eq u i pmen t to mea s u re
1 60
1 61
sys t em res po n s e s a t s evera l l o ca t i o n s o f t h e l oop a s s hown i n F i g u re
V. 1 .
T h e dyn ami c t e s t s i n vol ve d t h e p e r t u rb a t i on o f reac t i v i ty a n d
steam d ema n d u s i n g mu l t i fre quency b i n a ry s i g n a l s (
4)
a n d mon i to r i n g
the sys tem o u t p u t s a t t h e j unc t i on box whe re the s i gn a l s e n t e r t h e
pl a n t compu t e r .
V.3
PWR Mod e l
J
( )
The rea ctor power wa s mode l ed u s i n g t h e po i n t k i n e t i c s e q ua t i on s
wi th s i x g ro u ps o f d el ayed n e u t ro n s a n d reac t i v i ty fee d b a c k s d u e to
: ha n g e s i n fue l tempera t u r e , cool a n t/mode rato r tempe r a t u re and p r i ma ry
:oo l a n t system pre s s u r e .
Th e c o r e h eat t ran s fe r mod e l u s ed a noda l
i pp ro x i ma t i on fo r fuel a nd coo l a n t tempera t u r e .
E a c h a x i a l s ect i o n
i n c l u d e d a fu el tempe ra t u re node a n d two coo l a n t t empe ra t u re n ode s .
ih i s fo rmul a t i on wa s u se d to o bta i n a goo d a p prox i ma t i o n to t h e
tverage coo l a n t tempe ra t u re .
T h e r ea c t o r l ower a n d u pp e r p l en ums a n d
: h e p i p i n g b e tween t h e reactor a n d s t eam generator were mo de l ed t o
cc o u n t fo r t h e t ra n s po r t t i me d e l ays b e tween t h e re a c t o r c o re a n d
team gen era to r .
T h e reactor mo d e l u s ed fo r co u pl i n g w i t h the
eta i l ed steam gen erator model
( Mod e l D ) co n s i s t s of fo u r t ee n f i r st
rde r d i ffe r en t i a l equa t i on s tha t c a n b e ex pres sed i n the fo rm
dx
dt
1 ere
=
( v. 1 )
Ax + f' ( t )
x a n d r ( t ) a re t h e s t a t e va r i a b l e s a n d fo rc i n g v e c to r s r e s p e c -
i ve l y a n d
A i s the coeff i c i en t ma t r i x .
The def i n i t i o n s o f t he s t a t e
i r i a b l e s u s ed i n t h e rea ctor mo d e l a re g i ven i n Tabl e V . l .
Jn z e ro e l eme n t s o f t h e
A mat r i x a n d t he forc i n g vec to r
The
r a s o b ta i n ed
·orn Re fe re n c e 3 a re g i v en i n Ta b l e s V . 2 and V . 3 , r e s p ec t i ve l y .
F
p
I nput Si gnal
C o n t r o l Rod
T u rb i n e
P res s u r i z e r
S t e am
G e n e rat o r
T
0'>
N
Fe e dw a t e r H e a t e rs
N t----1
F
T l
F i g u re V . l
S c hemat i c o f t h e H . B . Ro b i n s o n P l a n t .
�
I n d i c a t e s Me a s u re me n t
N
T
P
F
L
=
=
=
=
=
n e u t ro n dens i
t e mp e ra t u re
p re s s u r e
fl ow
l e ve l
ty
1 63
TAB L E V . 1
DE F I N I T I ON O F TH E STATE VA R I ABL E S U S E D I N TH E REACTOR
State V a r i a bl e
N umb e r ---�S'->/.ycc.:m.:...:cb..::.
occ.:
.l
�10 D E L .
· t i on s
i n..:__1:._
e..:__
f ..:...;
�f)-=..:...:-=..:...:..::..
_______
______
o P/ P 0
o P = d e v i a t i on i n r e a c to r pow er
2
oC , f P 0
P 0 = s t eady state reactor power
3
oC z i P 0
4
6 C / P0
o C . = d ev i a t i on i n p r ec u r s o r co n c e n 1 trat i o n of g ro u p i , i = l , 2 ,
,6
5
o C 4/ P 0
6
7
o C5/ P 0
o C 6/ P0
3
o Tf
9
o TC l
10
11
12
13
14
6 TC 2
o TU
P.
6 \p
o THL *
o T **
cL
.
•
•
6 T f = d ev i a t i o� i n fu el t empera t u r e
o TC l = d ev i a t i on i n r ea c to r coo l a n t
temper a t u r e o f t h e f i r s t cool a n t
1 ump
o TC 2 = d e v i a t i on i n r ea c to r coo l an t
t em pera ture o f t h e second reactor
c oo l a n t l um p
o TU
P
= d e v i a t i on i n r ea c to r u p per pl e n um
temperat u r e
o Tf l
l
= d ev i a t i on i n reactor hot l eg
tempera tu re
oT
CL
oT
LP
=
d e v i a t i on i n r ea c tor col d l eg
t em p e ra t u r e
= de v i a t i on i n re a c to r l owe r pl en um
tempe rature
* To be c o u p l e d w i t h steam g e n erator pr i ma ry i n l et pl en u m .
**To b e co u pl ed w i t h st eam g en erator pri ma ry o u tl et pl e n u m .
1 64
TABLE V . 2
3
NONZERO ELEME NTS O F THE A MATR I X FOR TH E REACTO R f10 D E L ( )
Row
Number
C o l umn
N umber
1
1
1
0 . 01 2 5
3
0 . 0 3 05
4
0.1 1 1 0
5
0 . 30 1 0
6
1 . 1 40
7
3 . 01
8
-0 . 80 9 5
9
-6 . 2 2 7
10
-6 . 2 2 7
1 3 . 1 25
2
3
3
- 0 . 01 2 5
87 . 50
3
- 0 . 0 305
7 8 . 1 30
4
4
-400 . 0
2
2
2
E l emen t
Val ue
4
-0 . 1 1 1 0
1 58 . 1 0
5
5
5
- 0 . 301
6
1
46 . 2 5
6
6
-1 . 1 40
7
1
1 6 . 88
1 65
TAB L E V . 2 ( C ont i n u ed )
Row
Number
Co l umn
Nulilbe r
E l emen t
Val ue
7
7
-1 . 01
8
1 66 . 30
8
8
- 0 . 1 64 7
8
9
0 . 1 64 7
9
8
0 . 05707
9
9
- 2 . 440
9
12
2 . 383
10
8
0 . 05707
10
9
2 . 3 26
10
10
-2 . 333
11
10
0 . 3365
11
11
-0 . 3 36 5
12
12
-0 . 51 6
12
14
0.516
13
11
2 . 5 00
13
13
- 2 . 50 0
14
14
- 1 . 48
1 66
TAB L E V . 3
NONZERO E L E ME NTS O F THE FORC I NG V E CTOR
FOR TH E REAC TOR MODEL
f( 1 )
=
4 00 . 0
1 67
V.4
Cou p l i n g o f Rea cto r and S t eam Ge n e ra tor Mo de l s
Fi g u r e V . 2 s hows a sch ema t i c o f t he reacto r and s team g en er a t o r
l um p s u s ed i n t h e dynami c mod el s .
I t can b e seen t h a t t h e o u t p u t
from the r e a ct o r hot l eg l ump feed s i nto t h e s t eam g en�rato r p r i ma ry
i n l et pl en u m a n d t h e o u t pu t f rom t h e s t eam gen era to r p r i mary o u t l et
pl enum feed s i n to the reacto r c o l d l eg l um p .
T h i s con s t i t u t e s t h e
cou pl i n g me c ha n i sm betw een t h e r ea c to r mo del a n d t h e s t ea m gen era to r
mo d el .
The st eam gen era t o r mo d e l u s ed fo r c o u pl i n g w i t h t h e r e a c t o r
mod e l i s t he r ed u c ed v e r s i on o f Mode l D . I t was obta i n e d a s a n
J?
o u t pu t from t h e P U RE D I FF ( ) computer p rog ram w h en t h e m i xed
d i fferen t i a l and a l gebra i c equat i o n s d e sc r i b ed i n S e c t i on I I I . ? , pa g e
74 ,
a re u s ed a s i n put .
The mi x e d a l g ebra i c a n d d i ffer en t i a l fo rm
o f t h e mo de l con s i s t s o f e i g hteen d i ffe ren t i a l and a l g e b ra i c e qua t i o n s
i n v o l v i n g f i ftee n d i ffere n t i a l va r i a b l e s a n d three a l g e b ra i c v a r i a b l e s
a s s hown i n Ta b l e V . 4 .
Afte r e l i mi n a t i n g t h e a l gebra i c va r i a b l e s i n
PURE D I F F , t h e mod e l i s reduced t o f i fteen f i r s t o rd e r d i ffe r en t i a l
e q u a t i on s i n vo l v i n g t h e f i r s t f i fteen v a r i a b l e s i n Ta b l e V . 4 .
The
sta t e va r i a bl e s i n vo l ved i n t he c o u pl i n g w i t h t he r eac to r mo d e l a re
marked wi t h t h e s u persc r i pt
11 C 1 1 •
T h e comb i n ed rea c t o r - s t ea m
gen erator mo del c a n b e expr e s s ed i n t h e fo rm g i ven i n E q uat i on ( V . l ) ,
pa g e 1 6 1 , when i n t h i s ca s e x i s a
( 2 9 x l ) s t a t e vari a b l e v ec to r
w here t h e f i r s t 1 4 el ement s a re t h e reactor va r i a b l e s d e s c r i b ed i n
Tabl e V . l a n d t he n ext 1 5 e l emen t s a re the s t eam generator d i ffere n t i a l
va r i a b l e s g i ven i n Ta b l e V . 4 .
Heater
Hot LeCJ
Re acto r
Upper
P l en um
�
Tf
T
S t e m Generatot
ilL
.- - � -- I
'
T
UP
r-
l
Fue1
Reactor
Powe r
'
1- -
T
-
­
C2
3
T
Lower
P l en um
-
I
I
I
I
_J
I
I
'
el
I_
_±
\p
--
�·
f
I
I
I
CL
r---
S t ea m
Outl et
L ower
Pl e n um
P r i ma ry
S i de
L umps
I
Reacto r
Coo l a n t
T
t
U pper
P l enum
L - -j- - -
Co l d L e g
F i g u re V . 2
n pu t
L umped S t r u ct u re for a PWR Sy s t em r�del .
Tube
t1eta 1
L umps
-r - -� - - - -,
S eco nda ry
S i de
L umps
'
J
I
, - - -- - - - - - -
- - -'
j Feedwa t er
I n 1 et
m
co
1 69
TABL E V . 4
S YS TE M VAR I A B L ES FOR TH E STEAM GEN E RATO R
c
MO DEL
1.
T
2.
T
3.
T
4.
T
5.
T
6.
T
7.
T
ml
fi r s t t u b e meta l l ump t empera ture
8.
T
s ec o n d t u be met a l l ump t empera t u re
9.
T
1 0.
T
11.
L
1 2.
L 1
s
n o n bo i l i n g ( s ubcool e d ) l en gt h
1 3.
P
s t eam pre s s u re
Pi
Pl
P2
P3
P4
c
PO
m2
m3
m4
d
s
p r i ma ry wat e r i n l e t pl e n um tempe ra t u re
f i r s t p r i ma ry wat e r l ump t empera t u r e
s econ d p r i ma ry wa t e r l ump t empera tu re
t h i rd p r i ma ry wa t e r l ump tempera t u re
fo u r t h p r i ma ry wa ter i ump t empera t u re
p r i ma ry wa t e r o ut l et pl enum tempera t u re
t h i rd t u b e meta l l ump tempera t u re
fo u rt h t u b e meta l l ump t empe ra t u re
down come r 1 e ve 1
14.
bo i l i n g s ec t i o n exi t q u a l i ty
1 5.
down comer tempera t u re
*1 6 .
fl ow r a t e to t h e n o n b o i l i n g s ec t i o n
*1 7.
f l ow rate t o t h e b o i l i n g s ec t i o n
*18.
fl ow ra t e from t h e bo i l i n g s ect i on
*A l g ebra i c v a r i a b l e s .
1 70
V. 5
Re s u l t s
T h e f re q ue n cy re s po n s e of t h e comb i n e d rea cto r s team gen e ra t o r
mod e l wa s c a l c u l a t e d fo r two pe rt u rbat i on s , a re a c t o r s i de re a c t i v i ty
pe r t u rba t i o n a n d a s team gen e rato r s i de steam fl ow pe rt u rb a t i o n , u s i n g
the S F R - 3 c o d e .
( 40 )
T h e forc i n g vector fo r t h e rea ct i v i ty pe rt u r ba t i o n
ha s o n l y o ne n o n z e ro e l eme n t a s g i ve n i n T ab l e V . 3 .
Fo r t h e steam
fl ow pe rt u rba t i o n , t he fo rc i n g vector wa s o b ta i ne d fro� the t i me
re s po n se a n a l ys i s o f t h e s team gene rato r mo del a n d t h e n o nz e ro e l eme n t s
are g i ven i n Tab l e V . 5 .
F i g u re s V . 3 a n d V . 4 s how t h e fre q u e n cy re s po n s e o f rea ctor powe r
a n d steam ge n e ra to r pre s s u re fo r t h e reac t i v i ty pe rt u rb a t i o n a n d
F i g u re s V . 5 t h ro ug h V . 7 s how t h e freq u e n cy re s po n s e o f rea ctor power ,
c o l d l e g tempe ra t ure a n d steam pre s s u r e fo r t h e steam va l ve pert u rbat i on .
For t h e re a c t i v i ty pe rt u r ba t i on , i t wa s fo u n d t h a t t h e a greeme n t
between t h e t h e o re t i ca l mo del p re d i ct i o n s a n d t h e e x pe r i me n t a l re s u l t s
i n the l ower f re q ue n cy ra n ge depen d s o n t h e va l ue o f t h e mo de ra tor
coeff i c i en t of react i v i ty ( a ) .
c
The t h eo re t i c a l r e s po n s e wa s cal c u l a t e d
( S)
for two va l ue s o f a ' t h e re feren c e val u e ( a c r ) u s ed b y Ke rl i n e t a l .
c
and
6 0% of t h i s reference v a l ue .
I t c a n b e s e e n fro m F i g u r e V . 3
t ha t t h e a g reeme n t b e twee n t he t h eo re t i c a l mo d e l r e s po n s e a n d t h e
�xpe ri me n t a l re s u l t s i s b e t t e r f o r t h e s ma l l er a ' e s pe c i a l l y i n th e
c
ower fre q u e n cy ra n g e .
C l e a r l y , o n e wo u l d n e e d a b ette r va l ue o f a
c
h a n wa s a va i l ab l e fo r t h i s work t o pe rm i t conc l u s i ve compa r i son s o f
heory a n d e x pe r i ment a t l ow freq uen c i es .
T h e fol l ow i n g commen t s o n
h e compa r i s o n w i t h t e s t re s u l t s a r e i n o rde r .
1 71
TABL E V . 5
FOR C I NG FUN CTI O ri FOR STEM1 VAL V E P E RTURB AT I O N
( +1 0% Cha n ge i n S t e am F l ow Ra t e )
f( l 7)
=
2 . 69 79 9 9 E - O l
f( l 9)
=
f(21 )
=
4 . 732 00 0 E - O l
f ( 22 )
=
7 . 2 79 998E-02
f(23)
=
- 1 . 6 2 8000 E - 02
f ( 24 )
=
- 1 . 058000 E - O l
f(25)
=
1 . 68 1 000 E - O l
f( 26)
=
-2 . 298000 E - O l
f( 27)
=
- 6 . 072 000 E -O O
f ( 28 )
=
- 5 . 488999E - 0 3
f(29)
=
- 6 . 6 7 3998 E - 02
- 1 . 1 6 39 9 9 E - O l
. f.
1 - . i l�.,-r..,
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Part B .
Fi g u re V . ?
Phase
( Con t i n ue d ) .
1 82
l.
A compl ete s t at i s t i ca l a na l ys i s o f t h e e x pe r i men t a l data
(J
f rom the H . B . Rob i n son dynami c t e s t s ) h a s not been ma de .
Howe ve r , t h e standa rd d e v i a t i o n wa s c a l c u l ate d fo r some o f
t h e d a t a a n d t h e s e l ec ted res u l t s a r e s h own i n F i g u re s V . 3
t h ro ug h V . 7 .
2.
T h e compa r i s o n wi t h te s t re s u l t s pre s en t e d i n t h i s c ha pt e r
c a n o n l y s e r ve t h e p u r po s e o f c h e c k i n g fo r g ro s s i n a de q uac i e s
i n t h e steam gene rato r mod e l b ec a u s e o f t h e fo l l ow i n 9
fa c to r s .
a.
L i mi t e d a c c u ra cy o f t h e t e s t res u l t s d u e to e x p e r i men tal
n o i se i n some s i g n a l s .
b.
The e ffec t s o f t h e control systems were not i n c l uded i n
t h e t h eo ret i ca l mod e l re s u l t s , a n d cont rol system i n pu t s
h a ve a s i gn i f i ca n t e ffect o n steam generator re s po n se .
c.
Mo d e l s fo r ot h er compo n e n t s o f t h e system s uc h a s the
p re s s u r i z e r , rea c t o r cool a n t s , pumps a n d t u rb i n e
g e n e ra t o r were not i n c l uded i n t h e t he o re t i ca l mo de l
re s u l t s .
I t can b e con c l u d e d that n o g ro s s d i ffe re n c e s i n q ua l i ta t i ve
beha v i o r we re o b s e r ve d t h a t wou l d i nd i ca t e maj o r i na d e q u a c i e s i n t h e
d e ve l o pmen t o f t h e deta i l ed s team genera t o r mo del .
CHAPTER V I
COMPA R I SO N O F DETA I LE D MOD ELS W I TH OTH E R UTS G MODE LS
VI.l
I n t ro du c t i o n
I n C ha pter V , p a g e 1 6 0 , t h e frequen cy r e s po n s e o f t h e UTSG
d eta i l e d mo d e l ( Mo d e l D ) , c o u p l ed wi t h a PWR mo del wa s compa r ed
w i t h t h e experi men ta l r e s u l t s o b ta i n ed from dyn am i c t e s t s pe r fo rmed
on t h e H . B . Ro b i n so n N u c l ea r Power P l a n t . ( 5 )
I n th i s c ha p t e r ,
a n o t h er a pp roa c h fo r c he c k i n g t h e mat h ema t i ca l d e v e l o pmen t o f t h e
UTS G d eta i l ed mo d e l i s con s i d e red .
Th e st ep re s po n s e o f Mod el D
i s compa r ed wi t h t he s t e p r e s pon s e o b ta i n ed w i t h two o t her mod el s .
1.
A on e d i men s i o n a l fi n i t e d i fference mo d e l o f a We s t i n g hou s e
d e s i gn UTS G d ev e l o ped i nd ependen t l y by C h r i s te n s en .
2.
( JZ )
A s t a t e var i a b l e l umped parameter mo d e l o f a n ew Comb u s t i on
E n g i n e er i n g ( C E ) i n t egral econom i z e r U - t u b e st eam gen era to r
( I EUTS G ) d evel o ped a t T h e Un i ver s i ty o f Tenn e s see by
Arwood . ( 6 )
T he compa r i son wi t h t h e fi r s t mod e l i s to c h eck t h e l umped
pa r amet er , homo g e n eo u s fl ow , sta t e va r i a b l e a pproac h a g a i n s t a
f i n i te d i fferen c e , s l i p fl ow a p proac h .
T h e compa r i so n w i t h t h e
s econd mod el i s d o n e fo r t he pu rpo s e o f compa r i n g t h e b e ha v i o r o f
t h e convent i o n a l type ( UTS G ) wi t h t ha t o f t h e n ew d e s i gn ( I E U TS G ) .
I n each ca s e , a b r i e f d e s c r i p t i o n o f t h e sy s t em a n d t he
mat h emat i ca l mod e l i s i n t ro d u c e d fo l l owed by t he compa r i son o f t he
s t e p r e s pon s e of sel ect ed s t a t e va r i a b l e s .
1 83
1 84
VI.2
Compa r i son w i t h C hri sten s en ' s UTSG Model
System Oeser i pt i on
VI.2.1
T h e UTSG system con s i d ered by C hr i sten s en
d ev el o pmen t i s a West i n g ho u s e UTSG
i n Se c t i o n 1 . 2 , pag e 3 .
A
(l )
( 32 )
i n hi s mode l
s i m i l ar t o t ha t d e sc r i b ed
s i mpl i fi ed d i a g ram o f t he p hy s i c al
sys tem i s g i ven i n Reference 3 2 and i s re produced as F i g u re V I . l .
Mod el Desc r i pt i on
VI.2.2
( 32)
T h e s t eam gen era tor i s d i v i d ed i n to e i ght sect i on s a s shown
i n F i g u re V I . l .
T h e central " co re a n d the down comer s ect i o n s a re de­
s c r i bed by pa rt i a l d i fferent i a l e quat i on s , w h i l e the other sect i on s a re
de scr i bed by ordi nary d i fferen t i a l eq uat i on s .
The part i a l d i fferen t i a l
equa t i on s a re sol ved by sampl i n g i n t i me and d i v i s i on i n to s u b s ect i o n s
( f i n i te d i ffe ren c i ng ) i n s pac e , t h u s t ra n s fo rm i n g t h e d i fferen t i a l
e quat i o n s i nto a l ge bra i c equat i o n s .
T h e ma i n a p p ro x i ma t i o n s u s ed fo r t h e fo rmul a t i o n o f t h e
equat i on s are :
1.
Un i fo rm water and st eam ve l oc i ty i n t h e pr i ma ry and t he
s econda ry s i d e and con s tant s t eam to water vel oc i ty ra ti o
( s l i p fa c to r ) are u s ed .
2.
No su bcoo l ed bo i l i n g i s i n cl uded .
The wa ter i s hea te d
to satura t i o n and a ft erwa rd s a l l t he en ergy i s u s ed
for s t eam product i on .
3.
Therma l equ i l i br i um a t t he s a t u ra t i on po i n t i s a s s umed i n
the bo i l i n g part o f t h e " co re , " the r i s e r , the s team
vo l ume , and t he u p p er pa rt of the feedwa ter c hamber .
1 85
j
Steam outl et
I
_./
- - -- - -
..
·- S t ea m v o l ume
Ri ser
-ta te r
i n 1 et
-­
..-.. .
r - - - - - -.
. . __
__
F e e dwa t e r c ha m b e r
--�
- -
-t:
--
S e c o n d a ry s i d e
of " core"
_ _
i
I
..t..
·
I
-·ri
i
·
-
·
Oown c o m e r
U - tube
---- P r i ma ry s i d e
o f " co r e "
I
j
"+'
P r i ma ry i n 1 e t
chamber
·-
P r i ma ry
i g u re V I . l
�
P r i ma ry o u t l e t
c hamber
P r i mu ry o u t l e t
S i m p l i f i e d D i a g ra m o f t h e P h y s i c a l
Sy s t e � .
( J2 )
1 86
4.
Ho heat con ducti on a l ong t he t u be s ta k e s pl a c e .
T he
tubes a re d i v i ded i n two s h el l s , eac h wi t h h a l f of t he
heat ca pac i ty and a l l o f t he h ea t re s i sta nce a s s i g n ed
to t h e co u pl i ng between t h e s hel l s .
5.
No bo i l i n g i s a l l owed i n t he down comer and no h ea t
tra n sm i s s i on from t h e 11 core 11 to t h e down comer takes
pl a c e .
T h i s a s s umpt i on l i mi t s t he wo r k i n g ran ge of
the pre s su r e d e r i vat i ve to val u e s l es s t ha n 1 -2 bar/ s .
6.
Al l hea t exc ha n g e w i t h t he wal l and t h e st eel con struc­
t ion s in t h e steam vol ume i s n e g l ected .
The ba s i c equa t i o n s a re der i v ed by a ppl y i n g the ma s s , en ergy a nd/or
momen t um con s erva t i on pri n c i pl e s to t h e e i g ht s ec t i o n s of t h e
s t eamI g e n erator .
T h e equat i o n s fo r the 11 core 11 and downcomer s ec t i o n s
a r e pa rt i al d i fferen t i a l equa t i on s whi l e t ho s e fo r t h e o t h er sect i o n s
a r e ord i na ry d i ffe ren t i al equat i on s .
The d e ta i l s o f t h e model
equat i on s a re g i v en in Refe ren ce 3 2 .
Compa r i son of Model Formul a t i on
VI.2.3
The ma i n d i fferen ces between Ch r i s t e n s en ' s mod e l and Mo del D
can be summa r i zed a s fo l l ows .
1.
Mode l D u s e s a homo g enou s fl ow model where a ve rage d en s i ty
a n d a vera ge en t ha l py i n t he bo i l i n g reg i on a r e expres sed
i n t erms o f ma s s qual i ty ( x ) and t he s atura t i o n propert i es
wh i l e C h r i sten sen ' s mod e l ut i l i z e s a s l i p fl ow mod e l w i t h
con stant s l i p fa ctor ( S ) .
I n C h r i sten s en ' s wor k , t h e
av erage d en s i ty a n d a v e r a g e en tha l py i n t h e bo i l i n g reg ion
1 87
a re e x p r e s s e d i n t e r111 s o f t h e vo i d ra t i o
(n) .
Th e
re l at i o n between x , a , a n d S can be d e r i ved from the
b a s i c d e f i n i t i o n s of t h e s e q u a n t i t i e s and i s g i ven by
( 1-a )
a
2.
I n Mo d e l
0,
(VI . l )
the h e a t t r an s fe r c o e ffi c i en t s a re a s s umed
co n s t a n t d u r i n g t he t ra n s i en t .
I n C h r i s t en s en ' s mo del ,
t h ey a re ca l c u l a t ed a t e a c h t i me s t e p u s i n g i n s t a n t a n eo u s v a l u e s o f t empera t u re a n d pre s s u r e .
3.
T he rec i rc u l a t i o n fl ow i n Model D i s c a l c u l a t e d from a
q ua s i - s ta t i c moment um b a l a n c e a s s um i n g t ha t t h e
d i ffe r en t i a l g ra v i t y h ea d e q ua l s t he dyn a m i c p re s s u re
d r o p s i n t h e rec i rc u l a t i o n l oo p .
I n C h r i s t en s en ' s mo del ,
t h e dyna m i c pre s s u re d ro ps a re t r ea t ed e x pl i c i t l y w i t h
t he fl ow v e l oc i ty a s a va r i a b l e wh i c h s h o u l d e q u a l t he
to ta l ma s s f l ow d i v i d ed by t h e fl ow a re a a n d d e n s i ty .
4.
T h e me t hod o f so l u t i o n fo r Mode l D i s de sc r i b ed i n S e ct i o n
IV.3,
page 1 1 1 .
C hr i s ten s e n ' s mod e l may b e s o l ved e i t h e r
by a d i g i ta l pro gram o r b y hyb r i d s i m u l a t i o n .
Pa rt i a l
d i ffe ren t i a l equat i o n s a re s o l ved by d i v i s i o n o f s pa c e
i nt o s u b s ect i on s , 2 0 " co re " a n d downc omer s ect i on s , a n d
sampl i n g i n the t i me doma i n w i t h a sampl i n g ra t e o f 1 0- 2 0
p e r s econd .
T h e t i me de r i va t i ves a re repl a c ed b y f i r s t
o r d e r d i ffe re n c e s i n b o t h c a s es , wh i l e t h e s pa c e d e r i vat i ve s a re han dl ed i n d i ffe re n t ways .
Deta i l s o f both d i g i ta l
a n d hyb r i d sol ut i o n proc e d u r e s a re g i ven i n Re fere n c e 32 .
1 88
VI . 2 .4
Compa r i son o f Dynami c Re spon ses
In Refe renc e 3 2 , Chri sten s en pre s en ted t he s t e p re s po n s e of a
UTSG mo d e l to c ha n g e s i n t he pr i ma ry i n l et tem pe ra t u re a n d th e steam
val ve co effi c i en t .
The steam va l ve pe rtu rba t i o n re s u l ts a re u s ed
fo r t h e compa r i son o f t h e dynami c mod e l a s pred i cted by Model D and
C h r i ste n s en ' s mod e l .
I t i s not i c ed that t h e r e s u l t s gi ven i n
Re feren c e 32 a re ba s ed on s t eam generator de s i gn data that i s some­
what d i ffe rent from the H . B. Ro b i n s o n data ( C h r i s te n s en ' s system
i s l a rger and o perates a t h i gher pre s s ure ) .
c a n b e compa red o n l y on a qual i ta t i ve ba s i s .
There fore , t he r e s pon s e s
T h e sys t em var i ab l es
u s ed fo r t h e compa r i son a re pri ma ry o ut l e t temper a t u r e ( T p ) and
0
s t eam pre s s ure ( P ) .
s
T he res u l t s o f t h e compa r i son a re g i ven i n F i g u re V I . 2 .
It
c a n b e s een that t he resul t s obta i n ed from t h e deta i l ed l umped paramet er
model ( Model D) a gree fa i rl y wel l w i t h t he resul t s re po rted by
C h ri sten s en u s i n g t h e fi n i t e di fferenc e fo rmul a t i o n .
VI . 3
Compa r i son w i t h Arwood ' s I EUTS G Mod e l
( fi )
Arwood
d e ve l o pe d a ma t hema t i ca l model fo r an I n tegral
Econom i z er U-Tu b e S t ea m Gen erator ( I E UTS G ) u s i n g the state va r i a bl e
1 urnped pa ramet er a pproac h .
Arwood u s ed Mo del D o f t h i s s t udy a s a
s t a rt i n g po i n t a n d made t h e n eces s a ry c ha n g es to t h e l um p s t r ucture
to i n c l ude the i n tegra l econom i z er s ect i o n and to s i mu l a t e the new
steam gen erato r d e s i gn feature s .
I n th i s sect i o n , t he s t e p res pon s e
o f Mo d e l D w i l l b e compa red w i th t he re s u l t s re po rted b y Arwood fo r
the I EUTS G .
The ma i n o bj ec t i ve o f th i s compa r i son i s to h i ghl i g ht
t h e s i mi l a r i t i e s and d i fferences b etween t h e two UTS G d e s i gn s .
1 89
0
. ..
�
'
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-1 0
•
�-2 0
\.
.
'
..
'
.. .. ..
·,
V1
... ...
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... .. • ... ...
�- �
. ...... .
,..----- t1o d e 1
-
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�- � - - � - ---- - - - - -- - - - - ..c:::. __
v
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........:
VI . 2
- ·
-
·
50
-
....._
20
- - - - -- ­
- ·
- ·
- ·
-
60
· - ·
-
70
....
-
·
. ·- ·
·
� · - · - ·
·
- ·
·
- · -
- -- ..... .... _
� ·· /
·
30
0
· ---- C h r i s t e n s e n • s Mo d e l
�-
-
40
T i me
F i g u re
-· -·
( s ec )
-4
0
- · - · - ·
----� t1o de 1
' ., '
-3
- ·
40
T i me
0
·
30
20
0
D
..,./�
- - - - ...
-4 0
1
C h r i s t e n s e n • s t1o d e 1
_ __ __ _
50
- - - - - - ... - - - - - -
60
( sec )
C o m pa r i s o n S e tw e e n t1o de 1
0 a n d C h r i s t e n s c n • s t1o d e l .
70
1 90
V I . 3. 1
De s c r i pt i o n o f t h e I EUT S G Sys t en
T h e i n tegral eco n om i z er s t eam gen era t o r i s a n a d v a n c e d UTS G
de s i gn t ha t w i l l b e i n sta l l ed i n fu t u re PWR sys tems . (
i l l u s t ra t e d i n F i g ure V I . 3 .
4?)
T h e I E UTSG i s
A s te a m gen e ra t o r w i t h a n i n t e g r a l
economi z e r i s s i mi l a r i n mo st r e s pe c t s to t h e sta n d a rd U - t u b e
re c i rc u l a t i n g steam gen erator
( UTSG ) .
T h e ba s i c d i f ference i s t h a t
i n s t e a d o f i nt r o d uc i n g feedwat e r o n l y t h ro u g h a s pa rger r i n g to
m i x w i t h t h e rec i rc u l a t i n g wa t e r fl ow i n t he downcomer c h a n n el ,
fee dwate r i s a l so i n t ro d uc ed i n to a s eparate , b u t i n t eg r a l s ec t i o n
o f t h e s te a m g e n e ra to r .
A s emi - cyl i n d r i c a l s ect i on o f t h e t u be
bun d l e a t t he e x i t e n d o f t h e U - t u b e s i s s e pa rated from t he
rema i n der o f t h e t u b e b u n d l e by ve rt i c a l d i v i de r pl a t es a n d h o r i z o n t a l
ba ffl e p l a t e s .
Feedwat e r i s i n t ro d u c e d d i rect l y i n to t h i s s e c t i o n a n d
pre h ea t ed b efo re d i s c h a r g e i n to t he e va po ra t o r sec t i o n .
T h e economi z er
s e c t i o n i s d i v i de d i n to two hal ve s , i n t h e u p pe r o r co u n t erfl ow s ec t i o n
fee dwa ter f l ow s i n t h re e pa t h s a c r o s s t h e t u b e s a n d g e n e ra l l y c o u n t e r
c u r re n t to t h e d i rec t i o n o f p r i ma ry fl ow i n s i de t h e t u b e s .
In the
l ower o r pa ra l l e l fl ow s ec t i on , fe edwater a l so f l ow s i n t h ree pa t h s
ac ro s s t h e t u b e s a n d g en eral l y p a r a l l e l t o t he pr i ma ry fl ow i n s i de
the tubes .
T h i s s p l i t feed a r ra n g ement p re v en t s i n troduc t i o n o f
c o l d fe edwa ter o n t he t u b e s h ee t w i t h t h e a s s oci a t ed t h e rma l s t r e s s
pro b l ems a n d a t t h e s ame t i me reta i n s pa rt i a l l y t h e a d van t a g e o f
h i g he r l o g mea n tempera t u re d i f fe re n c e a s soc i ated w i t h t h e c o u n te rfl ow
a rra n gemen t .
The rema i n der of t h e s t eam gen era t o r i s l i t t l e
d i ffe ren t f rom t h e U TS G type exc ept t h a t t h e l ower po r t i o n o f t h e
1 91
Down comer Feedwat e r
U p pe r E c o n omi z e r Fe e dw a t e r
L owe r E c o n om i z e r Feedwa t e r
P r i r.1a ry I n l e t
P r i ma ry E x i t
F i g u re V I . 3
Sc h ema t i c D i a g ra m o f a n I E UTSG .
( 4l )
1 92
eva po ra to r s ec t i on a n d the down come r c ha n n e l oc c u p i e s on l y o n e h a l f
o f t he s t ea m g e n erator c ro s s s e ct i o n .
Compa r i s o n of Dyn ami c Re s po n s e s
VI . 3 . 2
F i g ur e V I . 4 s how t he compa ri son b etween t h e I E UTSG a n d UT S G
mod e l s s t e p r e s po n se t o +l 0 ° F c h a n ge i n p r i mary i n l et t em pera t u re .
A pr i ma ry s i de p ert u rb a t i o n wa s c ho s en so t h a t b o t h steam g enerator
mode l s c a n be compa red o n s i m i l a r ba s i s .
T h e s t a t e v a r i a b l e s c ho s en
fo r compa r i s o n p u r po s e s a re :
1.
P r i ma ry o u t l e t t empe ra t u r e
2.
S team gen erator p re s s u re
3.
Down comer l evel
4.
Down comer t empe ra t u re
( L D) .
( TPO ) .
( PS ) .
( TO ) .
I t c a n be seen t h a t t h e re s po n se o f t h e pr i ma ry TPO a n d PS
a re s i m i l a r i n bo t h s team gen erator mod e l s .
T h i s i s to b e expe c t ed
s i n ce t h e o n l y d i f fe re n c e b etween t h e I E UTSG a n d t h e UTS G i s i n
t he fee dwa t er pre h ea t i n g mec ha n i sm .
T h e d i ffere n ce i n t h e f i n a l
s t eady s t a t e val u e s i s d u e t o t h e d i ffere n c e i n t h e d e s i gn i n pu t
parame t er s u s ed fo r the dyn am i c re s po n s e c a l c u l a t i on .
T h e b a s i c de s i g n di ffere n c e b etween t he UTSG a n d t h e I E UTSG
can be s een from t h e re s po n s e o f the downcomer l e ve l a n d down comer
temperat ure as seen i n F i g u re V I . 4 .
S i n c e o n l y a s ma l l fra c t i o n o f
t h e fee dwat e r i s i n t ro d uced i n to t h e dow n come r s e c t i o n i n the I E UTS G ,
i t s hows a l a rg e r c h ange i n t h e downcome r l e ve l d u e t o t he decrea s e
o f t he rec i rc u l ated wat e r due t o t he i n c re a s e i n eva po ra t i on ra te
bec a u s e o f t he i n c re a s e i n h ea t i n p u t rate .
The down comer tempe ra t u re
1 93
I E UTSG
��--��--�� 6 =
0
0
0
::r
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_ _
ll_T_S_G
_ _ __
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l\
4 . 3777
=
3 . 408
- --
I-
Oowncomer tempe r a t u r e
50 . 00
25 . 00
T I ME
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UTS G
_ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1:1 =
0
0
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75 . 00
(SECJ
- 0 . 51 27
Down come r l ev e l
1 00 . 00
:::!' �----.---.--�
'.
25 . 00
00
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T I ME
(SECJ
--+_...__
--Jf--
-
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-+---4-
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UT S G
-
(f)
Q_
75 . 00
-
-
-
-
-
-
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-
-
-
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1'1
- - -
3 7 . 87
,
,
I
0
0
,
Steam pre s s u re
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0
0
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t-
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-
0-�------�--.--.r--.
2 5 00
50 0 0
75 0c
0 00
1 00
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Compa r i son B e twe en UTSG a n d I E U T S G r1o de l s Us i n g the Same
Mo de l i n g App roa ch .
1 94
i n t h e I E U TS G i s l a r gel y determi ned by t h e s a t u ra t i on tempera t u re
a n d t herefore i t s hows a l a rg e r i n c r ea s e t h a n i n t h e ca s e o f t h e
UTS G .
T h e a bo ve compa r i son pro v i de d a second c h e c k o n t h e
mo d e l i n g a p p ro a c h u s ed fo r t h e deta i l ed mod e l ( t�o d e l D ) deve l o pmen t .
CHAP TE R V I I
REt1ARKS A N D C mlCL US I O N S
E v a l ua t i o n o f t he De ve l o pment Procedure
VII .l
I n the p r ev i o u s c ha pte r s , a n a pproa c h wa s de s c r i b ed that
i n vo l ved s t e pw i s e d e v e l o pment of i n c r ea s i n g l y deta i l ed mo d e l s fo r
a UTS G .
T h i s a pproac h s t a r t s w i t h i d ent i fyi n g t h e p hys i c a l pro c e s s
ta k i n g p l a c e w i t h i n t he sys t em d u r i n g a n t i c i p ated trans i e n t . " T he
a p proac h proc eed s by d i v i d i n g t h e system i n to t h e mi n i mum n umber
of l u m p s that c a n d e s c r i be the re s po n s e to a n t i c i pa t e d i n p ut s .
Aft er a n a l yz i n g t he r e s pon s e o f t he sys t em a s o b t a i n ed from t h i s
fi r s t s t e p
( s i mp l E' s t ) mo del , fu rther s u b d i v i s i o n i n to mo re l u mps
c a n pro c e ed i n o r d e r to st u dy s pa t i a l e ffe c t s a n d / o r add n ew sys tem
va ri a bl e s tha t a re n e eded fo r d e s i g n p u r po s e s .
T h e a bo ve a pproa c h
ha s t he fo l l ow i n g a d va n ta g e s .
l.
An i d ea o f t h e re s po n se c ha ra c t er i s t i c s can b e o b ta i n ed
from a s i m p l e mod e l .
The s i mp l i c i ty ma k e s i t po s s i b l e
to c he c k t h e fo rmu l a t i o n i n deta i l to i n s u re t h at th ere
a r e no bl u n d e r s i n t he fo rm u l a t i o n or t h e a l gebra i c
man i pu l a t i on o f t he equa t i on s .
A s pro gre s s i ve l y mo re
deta i l e d mo de l s a re d e ve l o ped , t h ei r pe rfo rma n c e c a n be
compa red w i t h res u l t s from e a r i l er , s i m p l er mo d e l s .
Maj o r d i ffe re n c e s i n c o m pa ra b l e o utpu t s wo u l d i nd i c a t e
pro ba b l e errors i n t h e fo rmul a t i o n o f t he more de ta i l e d
IIIO d e l .
1 95
1 96
2.
T h i s pro gre s s i ve a pp ro a c h a l l ows a n a s s e s smen t o f mod e l
c o m p l ex i ty v s . f i d e l i ty .
Th i s i n fo rma t i o n c a n a s s i s t i n
s el e c t i n g t he mode l wi t h t h e a ppro p r i a t e deta i l a n d co st
fo r a pa rt i c u l a r a pp l i c a t i on .
VI I . 2
Po s s i b l e U s e s o f t h e C u r re n t Work
Po s s i bl e u se s o f the ma t h ema t i ca l mode l s d e ve l o ped i n t h i s
d i s se rta t i o n depend o n t he p urpo s e o f p e r fo rm i n g t h e dyn a m i c a n a l ys i s .
Mod e l s A a n d B a re t h e s i m pl e s t UTSG model s .
T h ey c a n b e u s e d
fo r see p i n g a n a l ys i s a n d when a s i mpl i fi ed mo d e l fo r t h e s t ea m
g e n erato r i s n eeded t o s i mul a t e a n o ve ra l l PWR sys tem .
T h e mod e l s
do n o t i nc l u d e t h e s t ea m gen erator l evel a s a s t a t e va r i a b l e ,
t h e refo re , t hey ca n no t be u s ed fo r a n a l ys i s i n vo l v i n g t h e dyna m i c
r e s po n s e o f t h i s q u an t i ty .
Mode l C comb i ne s t h e s i mpl i c i ty o f u s i n g t h e pr i ma ry a n d t u b e
me ta l s t r uc t u re o f r�ode l B w i t h t h e t reat men t o f wa t e r l ev e l s i n e e
t he s eco n da ry f l u i d l um p i s b ro k e n i n to t hree l um p s .
a.
E ffec t i ve h ea t exc h a n g e l um p .
b.
D rum eq u i va l en t l um p .
c.
Dow ncomer l um p .
T h e �od e l i s r ep r e se n t ed by 9 d i fferent i a l e q ua t i on s compared t o
fo r t1ode l D .
15
Therefo re , i t c a n b e u s ed fo r pa ramet r i c s team g en e ra tor
co n t ro l l er wo r k w i t ho u t i n t rod u c i n g t h e compl exi t i e s of t h e s u bcool ed/
bo i l i n g mo v i n g b o u n d a ry c on s i d e r ed i n Mo d e l D .
Mod e l D c a n b e con s i dered a s t h e en d p ro d u c t o f t h e wo r k
reported i n t h i s d i s s e rta t i o n .
I t i s n ow p ro g rammed s uc h t ha t i t
1 97
re cei ve s s t eam gen era to r de s i gn d a t a a n d produc e s t h e steady state
t em pera t u r e and ma s s q u a l i ty pro f i l e a l o n g t he h ea t exc ha n g e pa t h ,
t he n u s e s t h e s t eady state o u t p u t f o r t h e t ra n s i en t re s po n s e
ca l c u l a t i o n s .
T h e mo del re pres e n t s a fa i rl y deta i l ed l umped
pa ra�e t e r a pproa c h t h at can b e u s ed fo r the dynam i c a n a l ys i s o f UTS G
a s so c i a t e d w i t h PWR n uc l ear powe r p l a n t s .
VI I . 3
R e c ommen da t i o n s fo r F u t u r e Work
T he wo r k reported in t h i s d i s s er ta t i o n c a n b e e x panded i n
s everal d i r e c t i on s .
1.
Po s s i bl e f u t u r e w o r k may i nc l ude t h e fol l owi n g :
T h e s t ea dy s t a t e pro f i l e c a l c u l a t i on s c a n b e mod i f i ed to
i nc l u d e the momentum e q ua t i o n and ca n be u s ed to e st i ma t e
t h e rec i rc u l a t i on rat i o a t d i ffe rent l o a d c o n d i t i o n s .
2.
The s t ea� f l ow rate c a n b e re l at e d to t he t u rb i n e
e l ec t ri c a l l oad by mea n s o f a t u rb i n e mode l a n d t h u s th e
s t eam g e n e ra t o r no d e l ca n b e c o u pl ed t o t h e r ea c t o r o n
t he p r i ma ry s i de a n d t o t he t u r b i n e o n t h e s eco n da ry s i d e
t o s i mu l a t e l oa d fo l l ow i n g c a p a b i l i t i e s o f PWR syst em .
3.
A fee dwa t e r co n t ro l l er can b e co u pl ed t o t h e s t ea m
g en erato r mod el a n d t h e c l o s ed l oo p re s po n s e c a n b e u s ed
to d et ermi n e a n d o pt i m i z e t h e c on t ro l l e r para�e t e r s .
4.
The n on l i ne a r terms t h a t were e l i m i n a ted i n t h i s work c a n
b e i de n t i fi e d a n d re - i n t ro d u c e d i n to t h e mod e l .
5.
T h e e ffec t o f s u bcoo l e d bo i l i n g a n d t h e e ffect s h r i n k a n d
swe l l c a n b e added t o t h e mo d e l by i nt rod uc i n g t h e vo i d
f ra ct i on
( a ) i n stead o f t h e ma s s qua l i ty ( x ) a s a state
va r i a b l e i n the bo i l i n g secondary fl u i d l ump s .
1 98
VI I .4
Conc l u si o n s
A s e t o f l i n ea r , l umpe d parameter , s t a t e v a r i a b l e mo del s o f
va ry i n g compl ex i ty wa s devel o ped fo r s i mu l a t i o n o f U - t u b e steam
generato r s .
T h e mod e l s a ppear c a pa b l e o f p red i ct i n g t h e re s po n se
o f t h i s type o f sys t em .
T h e s u i ta b i l i ty o f t h e s e model s wa s
exami n e d by :
- a s se s s i n g t h e p hys i c a l p l a u s i b i l i ty o f t he t ra n s i en t
resul ts
- c o mpa r i s o n w i t h i nd e pen den t l y- devel o ped , mo r e d eta i l ed
mod e l r e s u l t s
- compa r i s o n wi t h a va i l a b l e t es t d a t a .
T h e mod e l s a re c a s t i n t h e s t a t e va r i a b l e fo rm , ma k i n g i t c o n ve n i e n t
t o c o u pl e t he m w i t h o t h er s u b sy s t em mo de l s fo r u se i n s t u dy i n g
o ve ra l l pl a n t performa n c e .
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36 .
Ma ta u s e k , M . R . , " S I NO D-A No n l i n ea r L umped Pa rame ter Mo d e l for
A na l y s i s of Two - P h a s e Fl o w i n N a t u ra l C i rcu l a t i on B o i l i n g
Wa t e r L oo p , " N u c l ea r S c i e n c e a n d E n g i n ee r i ng : 53 , 1 9 74 ,
p p . 440-4 5 7 .
37 .
Ha rri n gt o n , R . t� . • "An a l ys i s o f t h e Fo rt St . V ra i n n u c l e a r P l a n t
i n S u p po r t o f a Dynami c Te s t i n g P r o g ra m , " t1 . S . T h es i s , t l uc l ea r
E n g i n ee r i n g De partmen t , Th e Un i v e r s i ty o f Tenn e s s ee , Kn o x v i l l e ,
Ten n e s s e e ( A u g u st 1 9 74 ) .
38.
Kreys z i g , E . , Adva n c ed E n i n ee r i n Ma t h ema t i c s , J o h n W i l ey a n d
S on s , I n c . , New Y o r k , S e c o n d E d i t i o n , 1 9 6 7 .
39 .
Ha l l , S . J . a n d R . K . Adams , " MATE X P : A Gen eral P u r po s e Di g i t a l
Comp u t er P ro gram fo r S o l v i n g O rd i n a ry Di fferen t i a l E q u a t i o n s
by t he Ma t r i x E x po n en t i a l Me t ho d , 11 ORNL - TM - 1 9 3 3 , Oak R i dg e
Nat i o n a l Laboratory ( Au g u s t 1 96 7 ) .
40 .
Ker l i n , T . W . a n d J . L . L uc i u s , T h e S FR - 3 Code -A Fortran P rogram
fo r Ca l c u l a t i ng t he Frequ en cy Re spo n s e o f a M u l t i va r i a b l e
Sy s t em a n d i t s Sen s i t i v i t i e s to P a rame t er C hanges , USA E C
R e p o r t O RNL -TM- 1 5 7 5 ( 1 966 ) .
41 .
Wea ve r , L . E . , Reactor Dyn am i c s a n d C o n t ro l , Ame r i can E l s e v i e r
P u b l i s h i n g Compa ny , I n c . , New Y o r k ( 1 9 68 ) .
42 .
t�e l sa , J . L . a n d D . G . S h u l t z , L i n ea r Control Sys t ems , McGraw H i l l
Boo k Compa ny , I n c . , N ew Y o r k ( 1 96 9 ) .
43.
Ra l s t o n , A . a n d H . S . i·J i l f , ed . , Ma t hema t i c a l t�e t ho d s fo r
C ompu t er s , V o l . 2 , J o h n Wi l ey a n d S o n s , I n c . , N ew Y o r k
44 .
C o l l i e r , J o h n G . , C o n n ec t i v e B o i l i n g a n d Conde n sa t i o n , McGraw H i l l
Book Compa n y ( U K ) L i mi ted , L o n d o n ( 1 9 72 ) .
45.
Fraa s , A � P . a n d M . N eca t i o z i s i k , H e a t E xc h a n ger De s i gn , J o hn
W i l ey a n d S o n s , I n c . , New Y o r k , 1 96 5 .
46 .
Arpac i , Vedat S . , Co n du c t i on H e a t Tran s fer , Add i s o n -We s l ey
P u bl i s h i n g Compa n y , Rea d i n g , Ma s s ac h u s etts ( 1 9 6 6 ) .
204
47 .
" CE S SAR C E S ta n da rd S a fe ty Ana l y s i s Re port fo r System 80 NSSS ,
C E Power Systems , W i n d s o r , Co n n e c t i c ut , 1 9 74 .
48 .
E l -Wa k i l , M . M . , N uc l e a r H e a t T ra n s po rt , I n te rn a t i o n a l Textbo o k
C om pa ny , S c r a n to n , Pen n syl va n i a , 1 9 7 1 .
49 .
Kays , W . M . , C o n ve ct i ve H ea t a n d Ma s s T ra n s fe r , McGraw-H i l l B o o k
Compa ny , New Y o r k ( 1 9 6 6 ) .
A P P E N D I XE S
APPEND I X A
NARROW RAtJGE L I N EAR APPROX I MAT I O N OF S.IJ.T U RAT I O f'4
WAT E R AND STEAM PRO P E RTI ES
One o f t he p re re q u i s i te s for t h e sol ut i o n o f t h e dynam i c a n a l y s i s
mode l s fo r UTSG sys tems i s the e xpres s i on s fo r t h e e q ua t i o n o f s t a te fo r
sa t u ra t ed wa ter a n d s t eam .
The mo s t d i rect and a c c u rate me thod for
s a t i s fyi n g t h e a bo ve r e q u i reme n t i s to i n put t h e t h e rmodyn am i c
pro pe r t i e s o f s a t u ra t e d water a n d s t eam i n t a b u l a r form i n to t h e
d i g i ta l compu t er a n d u s e s ta n d a rd ta bl e l oo k u p a l go ri t h ms t o obta i n
t h e v a l u e o f a c e r ta i n p ro perty a t a g i ve n t h ermo dynami c s t a te .
Th i s
method i s s u i ta b l e fo r n umeri c a l s o l u t i on o f dyn ami c a n a l ys i s
mo de l s .
Ano t he r met hod t o o b ta i n t h e t h e rmodynami c p ro pert i e s i s to u s e
po l ynomi a l fi tt i n g o f t h e steam t a b l e s data .
Ob v i o u s l y , t h e h i g h e r
t h e o rde r o f the f i t t i n g po l ynomi a l , t h e mo re acc u r a te t h e sol u t i o n ( a t
t h e e x pe n se o f comp u t a t i o n t i me ) .
I n t h i s s t u dy , i n t u i t i o n a n d
en g i n e er i n g j ud geme n t w e r e u s ed to a r r i ve a t t h e l ea s t e x pen s i ve
met ho d for o bta i n i n g t h e s a tu ra t i o n pro pe rt i e s o f wate r a nd s team i n
t h e ra n g e o f i n t e re s t ( 600- 1 000 p s i a ) .
The method wa s a r r i ve d a t by fi r s t s t u dyi n g t h e b eh a v i o r o f t he
sa t u ra t i o n the rmodynami c pro p e rt i e s a s a fun c t i o n o f p re s s u re over a
w i d e r ran ge t h a n t h e ran g e o f i n t e r e s t fo r a pa rt i c ul a r a pp l i c a t i on .
Aft e r t h a t t h e w i de ran g e wa s d i v i d e d i n to a n umbe r o f n a rrow ra n g e s
a n d t h e beha v i o r o f t h e s a t u ra t i o n pro pe rt i e s wa s re exami n ed o v e r t h e s e
206
207
n a r row ra n g e s t o d e t e rmi n e t h e l ea st o rder pol yn om i a l t h a t can
a d e q ua t e l y re p r e s e n t t h e rel a t i o n b etween t h e s a t u ra t i on pro pe r t i e s
a n d the sys tem p re s s u re .
The a bo v e met hod was a p pl i e d t o o b t a i n a n
a pprox i ma t i o n fo r t h e eq uat i o n s o f s t a t e fo r s a t u rated wa t e r a n d
s team o v e r t h e pre s s u re ra n g e 600 - 1 000 p s i .
Thi s ran g e i s s u i ta b l e
fo r norma l o pe ra t i on t ran s i e n t s o f s te a m g en e ra t o r sy s t ems w h e r e t h e
st eady s t a te p re s s u re i s about 8 0 0 p s i .
T h e fo l l ow i n g s a t u ra t i o n
pro pert i e s were p l o tt ed a s a fu n c t i o n o f pre s s u r e u s i n g t he ASME
1 96 7 s t e a m t a b l e s .
=
s a t u ra t i o n tempe ra t u re ( 0 F )
l.
T
2.
vf
=
3
s pe c i f i c vol ume o f sa t urated wa te r ( ft / l brn)
3.
vg
=
3
s pe c i f i c vol ume o f sa t u ra t ed steam ( ft / l b m )
4.
v fg
5.
h
6.
h
7.
h fg
sat
f
g
=
vg - vf
=
e n t h a l py o f s a t u ra t e d wa t er ( B tu/ l bm)
=
e n t h a l py o f s a t u ra t ed s t eam ( B tu/ l bm)
=
h
h .
g - f
T h e res u l t s a r e s hown i n Fi g u r e s A . l t h ro u g h A . ? .
From t h e s e re s ul t s ,
i t c a n b e s ee n t h a t t h e c ha n g e o f t h e a b o v e pro per t i e s , o v e r t h e ra n g e
o f i n t e re s t i s fa i r l y l i n ea r .
T h e fo l l owi n g va l u es a r e re p re s en ta t i ve
o f t h e ra te o f c ha n g e o f t h e a b o ve s a t u ra t i o n p ro pe r t i e s w i t h re s pect
to pre s s u re fo r t h e H . B . Ro b i n so n steam g e n e ra to r .
l.
2.
a T�a t
aP
a vf
aP -
=
0 . 1 45
3 . 6x l 0 -
° F/ p s i a
6
3
ft I 1 bm/ p s i a
*A l t h o u g h T s a t a n d o t h e r s a t u ra t i o n p ro pe rt i es a re co n s i d e red a s
fun c t i o n s o f one va r i a b l e ( P ) , t h e pa rt i a l d e r i va t i ve symbo1 ( a / a P ) i s
u s ed fo r c l a r i ty o f eq ua t i on s .
208
0
0
0
'
0
0
0'\
\
0
0
OJ
0
0
r-..
\
0
0
1.0
\
\
\\
Vl
0..
0 ......
0
L!)
CJ
s...
::::l
Vl
Vl
CJ
s...
0 0..
0
0
0
M
\
\
"
.J
a
r.
0
1.0
0
t1j
. ,....
0
0
N
0
0
r-
�
--
"='
0
N
n
0
0
0
CJ
s...
::::l
(/')
(/')
CJ
s...
0..
..s::::
�
.......
209
0
0
0
l
0
0
m
0
0
aJ
0
0
1..0
ro
. ,...
V1
0..
0
0
LD
0
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<:::t
QJ
s...
:::::l
V1
V1
QJ
s...
a..
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QJ
s...
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V1
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QJ
s...
a..
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.jJ
3:
0
0
M
\
4>
40
c
0
0
0
N
+-'
ro
s...
ro
>
N
0
0
\
I
I
I
N
0
0
1\ 1
0
<:::(
QJ
s...
:::::l
01
LL
I
I
l
r-
0
0
I
I
I
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0
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21 0
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00
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lj
ro
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v
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s::
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0.
40
c
0
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ro
0
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N
/
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ro
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.
c::(
(V')
0
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QJ
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::I
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.
0
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0
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ro
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v
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<::t
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- '- -·
0
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(./l
c...
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s...
::::l
(./l
(./l
Q)
s...
0...
Q)
s...
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(./l
Q)
s...
0...
.,....
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3:
0
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c
0
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ro
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ro
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('Y')
N
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I
60 I
,-
50
40
I
.
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f
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t
1
..)�1
j
I
0
-
I
I
1 00
I
2 00
II
I
J
400
300
50 0
P res sure ( ps i a )
Fi g u re A . 5
V a r i a t i on o f h
--'
N
I
I
I
N
200
1 00
,
-·
I
----
----- ��
I
1/
30 I
I
I
f
w i th P re s s u r e .
I
I
6 00
j
700
I
800
I
_j
900
1 000
21 3
0
0
0
0
0
0"1
0
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co
0
0
r-..
0
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\..:>
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Vl
a..
0
c
1.!)
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oq-
<IJ
s...
:::::5
Vl
Vl
<IJ
s...
CL
<IJ
s...
:::::5
Vl
Vl
<IJ
s...
CL
..s::
-1--'
. ,....
3:
Ol
..s::
0
0
M
0
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N
40
c
0
. ,....
-1--'
"'
. ,....
s...
�
>
<..0
<
0
0
<IJ
s...
:::::5
Ol
1..1..
0
0
N
\
0
0
0
0
0
co
0
0
1..0
0
0
oq-
0
0
N
0
- ---�
1 200
1 200
1 000
800
I
I
1 00 0 .
"'---r------
----
800
600
6 00
4 00
400
200
200
0
0
1 00
2 00
0
3 00
4 00
500
P re s s u re ( p s i a )
F i g u re A . 7
Var i a t ion o f h
fg
w i t h Pres s u re .
600
7 00
800
900
1 000
--'
N
�
21 5
av
3.
_g_
4.
f
_2_9.
5.
6.
7.
aP
- 7 . 65x1 0
=
av
aP
ah
f
=
_
0. 1 75
=
- 2 . 7xl 0
----aP -
at
-7 . 7x1 0
ah
ah
f
7
=
- 0 . 203
-4
3
f t / 1 bm/ p s i a
-4
3
f t / 1 bm/ p s i a
B t u / 1 bm/ ps i a
_2
B t u/ 1 bm/ p s i a
B t u / 1 bm/ ps i a .
APPE @ I X B
THE C R I T I CAL FLOW ASSUMPT I O n
I n t h e a b s e n c e o f a t u rb i n e mo del t ha t c a n b e u s ed to r e l a t e
t h e steam f l ow ra t e f rom t h e s t eam g en erato r t o t h e t u rb i n e l o ad ,
some a s s um pt i o n s ha ve to be ma d e t o rel a t e t h e s t eam fl ow ra t e to
o t h e r s y s t em va r i a b l e s and/ o r fo rc i n g f u nc t i o n s .
The s i mpl e s t met ho d to a c h i eve t h i s rel a t i o n i s to l ea v e t h e
st eam f l o w r a t e i n t h e equat i o n s a s a fo rc i n g funct i o n .
T h i s method
i s s u i ta b l e wh en u s i n g t h e s t eam g en erator mo del fo r f eedwa t e r
c o n t ro l l er d e s i gn a n d o pt i m i za t i on .
A n o t h e r method i s t o rel a t e t h e
s t eam fl ow ra t e t o t he s t eam generato r pre s s ure a n d t h e tu rb i n e f i r s t
s t a g e pre s s u r e u s i n g t h e o r i f i c e fl ow eq ua t i o n
(B.l )
where
w
so
P
p
s
t
=
steam fl ow r a t e
=
steam gen era tor pre s s u r e
=
t u r b i n e f i r s t sta g e p re s s u r e .
When E q u a t i o n ( B . l ) i s l i n ea r i z ed , we g e t
(B.2)
wh er e C fl a n d c f 2 a re con sta n t c o e ff i c i e n t s t h at ca n b e cal c u l a t ed
from s t eady s t a t e con d i t i on s .
From E q ua t i o n ( 8 . 2 ) , i t can b e s een
that the steam f l ow rate term i n t h e go vern i n q equat i on s fo r t h e
steam gen era tor mod el i s repl a c ed b y two term s .
21 6
T h e fi r s t i n cl u d e s
21 7
t he st eam gen erato r p r e s s u re w h i c h i s on e o f t h e s t a t e va r i a b l e s and
p r e s s u r e w h i c h now
the o t h e r i n c l u d e s the tur b i n e fi r st stage
serve s
a s a fo r c i n g f u n c t i on .
T h e met hod u s ed i n t h e mat hema t i ca l d e v e l o pmen t s i n C ha pt er I I I ,
pag e
25,
i s to a s s ume t h a t t he s t eam fl ow rate i s d e t e rm i n ed o n l y by
t he u p s t r eam ( s team g en era to r ) pr e s s u r e a n d a ny d ro p i n t he down stream
( tu rb i n e ) pr e s s u re n o l o n g er r e s u l t s i n an i n c rea s e i n t h e s t eam fl ow
rat e from t he s team g en erato r .
fl ow" a s s umpt i o n .
T h i s i s u s u a l l y known a s t h e " c r i t i ca l
T h i s con d i t i on i s r eac hed when t h e rat i o b etwee n
t h e tu r b i n e p re s su r e t o t h e s t eam gen era t o r pre s s u re re a c h e s a c r i t i ca l
( 4S )
val ue.
For s a t u r a t ed s t eam , t h i s c r i t i ca l p re s s u re rat i o i s
a bo u t 0 . 54 5 .
( 43 )
W h en t h e c en t ral f l ow a s s umpt i o n i s u s ed , t he s t eam
f l ow ra t ed i s con s i dered to be a fun c t i o n of the u p s tream ( s team
g e n e ra t o r ) pre s s ur e o n l y , t h u s
(8.3)
E q u a t i o n ( B . 3 ) w hen l i n ea r i z ed yi e l d s t h e fo l l owi n g e q ua t i o n .
oW
so
=
C
L
oP
s
+ P
s
oC
L
(B .4)
T hu s , t h e steam fl ow term i s re p l a c e d by a t e rm i n c l u d i n g t h e
s t eam p r e s s u r e P
co effi c i ent C
L
s
a n d another t erm i n v o l v i n g t h e fl ow ( o r v a l v e )
w h i c h i s now t a k e n a s t h e forc i n g f u n c t i o n .
A P P E ND I X C
BAS I C C O N S E RVAT I ON E QUAT I O N S FO R TWO PHASE FLOW S Y S T EMS
T h e ba s i c con s erva t i on e q u a t i o n s are the b u i l d i n g b l o c k s i n
t he d e vel o pmen t o f dyn ami c mod e l s for t he rma l hyd ra u l i c sys tems .
T hey expre s s t he s t o r a g e a n d t ra n s po r t o f ma s s , e n e rgy , a nd
momen tum w h i c h a r e t he maj o r fa c t o r s govern i n g t h e dyn a m i c b e ha v i or
o f t h e sy s t em .
I n t h i s a ppend i x , t h e general sta temen t o f t he
ba s i c co n s e rva t i on e q ua t i on s i s g i ven a n d a p pl i ed to d er i v e t h e
ma t h ema t i c a l expre s s i on s fo r t h e con s er va t i o n equa t i o n s i n a o n e
d i men s i o n a l two p ha s e fl ow sys t em .
C.l
P r i n c i pl e o f C o n s erva t i o n o f Ma s s ( Con t i n u i ty Equa t i o n )
The p r i nc i pl e o f c o n s erva t i on o f ma s s e qu a t i on c a n b e s t a t ed
a s fol l ows
( Ra te o f i nc r e a s e o f ma s s s t o ra g e ) = ( fl ow i n ) - ( fl ow o u t ) . ( C . l )
Con s i d er t h e c o n tro l vo l u me bou nd ed by t h e ho r i z o n t a l l ev e l s
( z ) a n d ( z+ L ) i n t he v e r t i c a l f l ow sy stem s hown i n F i g u re C . l ,
a pp l i ca t i o n o f E q ua t i on ( C . l ) t o t h i s c o n t ro l vo l ume y i e l d s t h e
fo l l ow i n g equat i o n
d
dt
e-P AL ] =
where
p
= a vera g e d en s i ty o f t h� fl u i d i n t h e co n t ro l vol ume
A = cro s s s ec t i o n a rea
21 8
219
[W ] z
+L
z +L
o
AL
-+-------,�--� z
[W ] z
Fi gure C . l
C o n tro l V o l u me fo r t h e A p p l i c a t i o n o f t h e C o n t i n u i ty
Equa t i o n .
220
=
l en g t h of t he control vo l ume
=
fl ow i n
=
fl ow o u t .
L
W
z
W +
z L
Fo r con s tan t c ro s s sect i on a rea , we can d i v i d e b o t h s i d e s o f
Equa t i on ( C . 2 ) b y A t o get
(c. 3)
where
G
G
2
z +L
=
i n l et ma s s vel oc i ty
=
o u t l et ma s s vel oc i ty .
Fo r a homo g e n eo u s fl ow mod e l , we have
p
=
1
---­
v +x v fg
f
(C.4)
(C.S)
(C.6)
where
v
v
=
s pec i f i c vol ume o f s a t u ra ted wa t e r
=
s pec i f i c vol ume of s a t u rated s team
=
X
( v9 - vf)
=
a ve ra g e ma s s qual i ty
u
=
fl ow ve l oc i ty
=
1 oca 1 ma s s q u a l i ty.
f
g
v fg
-
X
221
Fo r a sl i p fl ow mod el , we ha ve
(c. 7)
G
G
z
=
z+L
=
where
p
a
pf
pg
uf
u
C.2
g
[ p f u f ( 1 -a ) + p g u g a ] z
(C.8)
[ p f u f ( l - a ) + P g u g a ] z+L
(C.9)
=
a vera g e d en s i ty
=
a v erag e vo i d f r i c t i on
=
d en s i ty o f satu rated wa ter
=
d en s i ty o f saturated s team
=
vel o c i ty o f wa ter pha s e
=
v el oc i ty o f s team pha se .
Pr i n c i pl e o f Con s erva t i on o f E n ergy ( En e rgy Equa t i o n )
T he pr i nc i pl e o f conservat i o n o f en ergy can be stated a s
fol l ows
Ra te o f Crea t i on o f E n ergy
=
0
( C . 1 0)
g
where the term crea t i on i s de f i n ed a s ( 4 )
C re a t i on
( o u t f l ow ) - ( i n f l ow) + ( i n crea se o f s torage ) . ( C . l l )
=
Th e appl i ca ti on o f E q u at i on ( C . l O ) to the con tro l vol ume s hown
i n F i gure C . 2 , y i el d s the fol l owi ng equ a t i on
�t
[ p AL
•
e]
=
[W
•
(e +
p
�) ] 2
- [W
•
(e +
p
� ) ] z+L +
Q
(C.l 2)
222
-+-----+---J- z + L
z
[W · h ] z
F i gure C . 2
C o n trol Vol ume for th e A p p l i ca t i o n o f t he E n ergy E q u a t i o n .
223
where
-
e
a ve ra g e i n te rn a l t h e rma l en ergy
=
e
=
l o c a l i n t ern a l therma l ene rgy
Q
=
hea t i n pu t ra t e
=
fl ow wor k ( i n t herma l e n ergy u n i t s ) .
EJ
EJ i s
T h e c omb i n a t i on e +
k nown a s t h e en t h a l py a n d i s d e not ed
by h , t h u s , E qu a t i o n ( C l 2 ) can be wri tten i n t h e fo rm
.
d
df
[ p-
AL
•
e]
=
[W
h]
•
2
-
[14
•
h]
z+L
+ "ll ·
(C.l 3)
T h e fl ow wor k t e rm c a n b e s hown to b e n e g l i g i b l e compa red to t h e
en t h a l py t e rm i n typ i ca l
U TSG o perat i o n .
Therefo re , t h e e n t ha l py
h c a n b e u s ed i n bo t h s i d es o f E q u a t i o n ( C . l 3 ) , t h u s
d
dt
[p
AL
•
h]
=
[W
h]
•
z
-
[W
•
h]
z+ L
+
Q
•
(C.l4)
T h e r e l a t i ve i mpo r ta n c e o f t he fl ow wo r k t e rm fo r a ppl i c a t i on s
i n vo l v i n g l ow ma s s q u a l i ty two p h a s e m i xt u res c a n b e ex ami n ed from
the fo l l ow i n g exampl e .
Fo r P
h
h
v
v
t he n
800 p s i a , we ha v e ,
=
pv
f
509 . 7 B t u/ l bm
g
=
1 1 99 . 3 B t u/ l bm
f
=
3
0 . 02087 ft / l bm
=
3
0 . 5691 ft / l bm
g
f
---
J
=
=
800
X
1 44 X 0 . 0 2 08 7
778. 1 6
=
3 . 09 B tu / l bm
2 24
�
J
=
BOO x 1 44 x 0. 5 6 9 1
7 78 . 1 6
=
84 . 26 B t u / 1 bm
and
fo r a ma s s q u a l i ty X
e
=
0 . 2 , t h e we i g h t ed error i nv o l ved i n u s i n g
en t h a l py i n s t ead o f i n tern a l en ergy can b e e st i ma t ed a s fo l l ows
E
e
=
0 . 8 ( 0 . 6 0 6 ) + 0 . 2 ( 7 . 02 6 )
a n d fo r an a verage ma s s q u a l i ty
�
=
x
=
=
1 . 89%
0 . 1 , we have
0 . 9 ( 0 . 606 ) + 0 . 1 ( 7 . 026 )
=
1 . 2 5% .
F rom t h e a bo ve exampl e , i t c a n be s ee n t h a t the e r ro r i n vo l ved i n
u s i n g t h e e n t ha l py i n s t ead o f t he i n ternal en ergy t e rm fo r UTSG
a pp l i c a t i on s i s l e s s t han 2% .
D i v i d i n g bo t h s i d e s o f E q u a t i o n ( C . l 4 ) by A , we g e t
[L
Q_
dt
•
p n]
=
[G h ] z - ( G · h ]
z+L
.
( C . l 5)
For the homo g en eo u s fl ow mod el , we h a v e
(C.l6)
( C . l 7)
( C . l 8)
225
F o r the s l i p f l ow model
(C.19)
( C . 20 )
(c.21 )
T h e Momen tum P r i n c i pl e ( Momen t u n Equ a t i o n )
C.3
T h e mom e n t u m equa t i o n c a n g e n e ra l l y b e stated a s fol l ows .
" Ra t e o f momen t um a c c umu l a t i o n i n a c o n trol vol ume i s equ a l
t o t he n e t ra t e a t w h i c h momentum fl ows i n to t h e c o n t r o l v o l ume
p l u s t h e sum o f the s urface and body fo r c e s a c t i n g u po n t h e
c o n trol v o l ume . 11
W h e n t h e a bo v e s ta teme n t i s a p p l i ed to the c o n t r o l v o l ume s hown
i n F i g u r e C . 3 , we o b ta i n the fo l l owi n g mat h ema t i ca l e x p r e s s i o n s fo r
t h e momen tum equa t i o n
d
A dt [ L p u ]
-­
A [p u
=
-
T
w
p
2
yw
+ P]
2
L - g
- A[p u
2
+ P]
z+L
p AL
where
-
=
a v e r a g e v el oc i ty o f t h e f l u i d i n t he c o n tr o l vol ume
(l)
=
wa l l s he er s t r e s s
yw
=
wa l l wet ted pa ramet e r
=
a c ce l era t i o n d u e t o gra v i ty
u
T
p
g
a n d o t h er terms a re a s d e f i ned b e fo r e .
( C . 22 )
226
z+L
(p AL ) · u
T
w
J
T
[P]
�
w
z
z
[W · u ] 2
F i gure C . 3
C o n trol Vol ume fo r t h e A p p l i c a t i o n o f t h e Mome n t u� E � u a t i o n .
227
I n t h e c u rrent a pp l i ca t i o n , t h e mome n t um eq ua t i o n i s u s e d i n
i t s stea dy s t a t e fo rm ( n o t i me de ri va t i ve ) t o ca l c u l ate t h e s e conda ry
fl u i d ma s s fl ow ra t e a t the e n t ra n c e o f t h e t ube b un d l e re g i on a s
de sc ri b e d i n A p pe n d i x D .
AP PE N D I X D
R E C I RCULAT I O N LOOP E QUAT I ON
T h e ma s s fl ow r a t e of t h e s econdary fl u i d e n t e r i n g t h e t u b e
b u n d l e r e g i on f ro m t h e down comer s ec t i o n ( W 1 ) i s co n s i d er e d a s a n
i n l et c o n d i t i o n fo r t h e so l u t i on o f t h e ma s s a n d ene rgy ba l a nc e
equ a t i on s i n t h e effe c t i ve heat exc ha n g e re g i on .
b e tween w
1
The rel at ion
a n d o t h e r state va r i a b l e s i n t h e UTSG mo del i s d e r i ved
h e r e u s i n g t h e a s s u mp t i o n that the n et p r es s u r e dro p a l on g t h e
r ec i rc u l a t i o n l oo p i s equal t o z ero du r i n g t he tra n s i en t .
Thi s
a s s umpt i o n mea n s t hat t h e d r i v i n g s ta t i c head due to t h e d i ffe rence
i n d en s i ty b e tween t h e down comer sect i on and the t u b e b u n d l e reg i o n
i s equal t o t h e to ta l dyna mi c head l o s s a l o n g t he re c i rc u l a t i o n
l oo p .
Tha t i s
(D.l )
w h er e ,
6P
6P
6P
= s ta t i c pre s s u re d ro p a l o n g t h e l oo p
d
f = fr i c t i o n a l p r e s s ure d ro p a l o n g t h e l oo p
= a c c e l erat i o n pr e s s u r e drop a l o n g the l oo p .
a
S i nc e the fr i c t i o n a n d a ccel era t i o n pres s u r e d ro p s a re pro po r t i o n a l
to t h e s q u a r e o f t h e ma s s fl ow rate , t hen E q uat i o n ( D . l ) c a n b e
wr i tten a s
( D . 2)
w here
C
d
= effec t i v e dyn am i c h ead l o s s coeffi c i en t .
228
229
Sol v i n g E q u a t i o n ( 0 . 2 ) fo r w , w e g e t
1
w
L et c 1 =
_1_
v"cd
1
= __ ltJJ: .
d
;c:d
1
( 0. 3)
t hen
w1 = c1 16 P
d
•
(0.4)
Referr i n g to F i g u re 0 . 1 , we h a ve
(0 . 5)
where
p d = d en s i ty i n t h e down comer s ec t i on
L
dw
L
d
= drum wa ter l evel
= down comer l en g t h
P s l = d en s i ty o f t h e s eco n d a ry fl u i d i n t he s u bcoo l ed r eg i on
L51
= l en g t h o f t h e s u bcoo l ed r eg i o n
p b = d en s i ty i n t h e s ec o n d a ry fl u i d i n t h e b o i l i n g r e g i o n
Lb
= l en g t h o f t he bo i l i n g reg i on
L
= effec t i ve l en g t h o f t he r i s er/ s ep a rator vol ume
r
p f = d en s i ty of the s econd a ry fl u i d i n t he r i s e r / s e pa r a t o r
vol ume .
S i n c e t he d en s i ty o f the s u bcoo l ed l i q u i d i n t h e down c omer reg i o n
i s n ea r l y equal t o t he d en s i ty i n t h e s ubcool ed r eg i o n , t hen
E qu a t i on ( 0 . 5 ) can b e w r i t ten a s fo l l ow s
=
p d ( L dw+L d - L S l ) - L b p b - L r p r
1 44
(0.6)
2 30
f
S team o u t l et
\.J
sg
r
-
L
,.._,. ._ ._
..p.. - .... ....
L dw
r
F eedwa t er
i n l et
I
•
I
L
b
( SFBL )
I
I
I
I
'
l
I
I
I
L
d
'
( S FSL )
Fi g u re
D. l
I
-,
I
I
P r i ma ry o u t l et
C a l c u l a t i on o f t h e D r i v i n g P r e s s u r e Head fo r t h e
Re c i rcu l at i on Loo p .
2 31
S u b s t i t u t i n g from E q u a t i o n ( 0 . 6 ) i n to E q u a t i o n ( 0 . 4 ) , we g e t
The d e v i a t i o n o f w
1
f rom s teady s ta t e can b e o bt a i ned from E q u a t i o n
( D . 7 ) a s fo l l ows
c,
o Wl
24 R
�
{ p ( o l -o L ) - ( p l +L o p ) - L o p }
Sl
b b b b
d
r r
dw
( 0 . 8)
w h er e
T h e d e v i a t i o n i n t he two - p ha s e d en s i ty i n t he bo i l i ng re g i o n a n d i n
t h e r i s e r/ s e p a ra to r vo l ume can b e e x p r e s s e d a s ( s ee A p p e n d i x E )
( 0. 1 0)
and
( 0. 1 1 )
W h e n E q ua t i o n s ( 0 . 1 0 ) a n d ( 0 . 1 1 ) a re s u b s t i t u ted i n to E q u a t i o n
( 0.8) , we get
where
a
a
a
a
c
l
2
3
4
�
-
-
-
-
-
-
, pd
24R
c , ( p -p b )
d
24 R
c ( c L b +c 1 rL r )
1 1b
24R
c, (c
l +C l )
2b b 2 r r
24R
APPE N D I X E
AVERAGE DE N S I TY I N THE TWO-PHASE REG I O N
A n expres s i o n for the d en s i ty i n the two - p ha s e reg i o n , i n terms
of sys t em p re s s u r e and bubb l e popu l a ti o n i n the two-pha s e m i x ture , i s
nec e s s a ry fo r the so l u t i on of the d e s cr i b i n g equ a t i o n s i n the two
pha s e s ec o nda ry fl u i d l umps .
I n the s l i p fl ow mo d el , t h e bubbl e popul a t i on i s repre sented
by the vo i d fra c t i o n a def i ned by the equa t i on
A
=
.....9.
a
A
(E.1 )
where
a
A
= vo i d fra c t i o n
= area occupi ed by the vapor pha s e
9
A = tota l fl ow area .
The den s i ty o f t h e two -pha s e mi xture i s g i ven by
p
< 44 )
= a P + ( 1 -a ) P f
g
( E . 2)
where
p
p
P
f
g
= two - p ha s e mi xture den s i ty
= d e n s i ty o f the l i qu i d p ha s e ( wa te r )
= d e n s i ty of the vapor p ha se ( s team) .
A s s umi n g that P f a nd P are l i near func t i o n s of system pre s s ure P
g
s
( se e Append i x A , page 206 ) d i ffe ren t i a t i ng E q ua t i o n ( E . 2 ) a n d
l i near i z i ng g i v e s
2 32
2 33
d op
_
30
f do P
dt - aP df -
30
[a( --a-Pf -
�)
�P
o P + ( p f-p g )
do
].
d�
( E . 3)
Negl ec t i ng t h e c ha ng e i n l i q u i d p ha s e d en s i ty w i t h p re s s u re , we
have
( E .4 )
T h e per t u r ba ti o n equa t i on fo r t h e two pha s e m i x t u r e d e n s i ty
can be w r i tten a s
(E.5)
E q u a t i o n ( E . 5 ) r e l a t e s t h e p e r t u r ba t i o n s i n
i n P a nd
a.
P
w i t h t h e pertu rba t i o n
( g)
I n t he homo g e n eo u s fl ow mod el , we h a v e 4
(E.6)
where
p
v
v
f
= d en s i ty o f t he two p ha s e m i x t u r e
= s p ec i f i c vo l ume o f l i q u i d p ha s e
=
s pec i f i c vol ume o f v a p o r p h a s e
9
x = ma s s q u a l i ty = ma s s fl ow n o t e o f vapor pha s e
tota l ma s s fl ow r a t e
D i fferen t i a t i n g E q u a t i o n ( E . 6 ) g i ve s
(E.7)
2 34
where
I f we n e g l e c t t h e c ha n g e i n v
f
wi t h p re s s u r e , t h e p e r t u rba t i on
equa t i on for the two p ha s e m i x t u r e d en s i ty c a n b e wri tten a s
op
( E .B)
=
I n t h e bo i l i n g s ec o nda ry fl u i d l ump , we h a v e x
=
x
=
X
e
2 and
E q u a t i on ( E . 8 ) becomes
(E.9)
E qu a t i on ( E . 9 ) c a n b e w r i tten i n t h e s i mpl e form
( E . l O)
where
c
lb
=
3V
-
-X
( v +x v )
fg
f
fg
3P
2
\1
a nd
Cz
b
=
-vf
g
2 ( v +x v )
f
f9
2
s
s
I n the r i s e r/ separa to r 1 ump x
=
x e a n d E q u a t i o n ( E . S ) be come s
(Eal l )
2 35
E q u a t i on ( E . l l ) can b e wr i t ten i n the s i mp l e fo rm
op
r
where
c
-x
lr
=
a nd
c
( v +x v )
f e fg
-v
2r
=
e
fg
( v +x e v g )
f
f
=
c
1Y
a vf
2
2
oP + c2
r
aP
ox
e
g
ss
ss
W h en u s i n g t he a bo v e ex pres s i o n s , i t i s u n d e r s to o d t h a t t h e
u n perturbed pa ramet e r s a re ca l c u l a t ed a t t h e steady s t a t e cond i t i o n s
corre s po n d i n g t o t h e system pres s ure , e v e n when t h e s u b s c r i pt ( s s . )
i s n o t s hown i n t h e expre s s i o n fo r t h e s e p a rame ters .
A P P E ND I X F
S UMMARY O F DYNAM I C RE S PONSE RESULTS
I n C ha pter I V , p a g e 1 4 1 , t h e s t e p r e s po n s e o f t h e fo u r U TSG
dyn am i c m o d e l s to a s t eam va l ve perturba t i o n wa s p r e s e n ted for
mod e l s compa r i son .
I n t h i s a pp e n d i x , a s umma ry o f t h e dynami c
r e s po n s e o f the fou r mod e l s to a l l a n t i c i pa t ed pertu rba t i on s i s
pre s e n t ed fo r f u t u r e referen c e .
T h e fo u r mo del s a re d e s i g n a t ed a s
fo l l ows :
Mod e l A :
S i mpl i fi ed mod e l
Mo d e l B :
F i r s t i n term ed i a t e mo d e l
Mod e l C :
S e c o n d i n termed i a te mod e l
Mod e l D :
D e ta i l ed mo del .
T h e i n p u t p e r tu rba t i o n s a re
l.
P r i ma ry i n l e t tempera t u r e pertu rba t i o n
2,
Fe edwa ter tempe ra t u r e pertu rba t i o n
3.
S t e am va l ve p e r t u rba t i o n .
Tab l e F . l g i ve s t h e c a s e i de n t i fi cat i on s a n d F i g u re s F . l t h ro u g h F . l 2
show t h e r e s u l t s o f t h e dynami c re s po n s e fo r these c a s e s .
2 36
237
TAB L E F . I
S UMMARY O F DYNAM I C RESPONSE RESULTS
Ca s e No .
Mode l
P e rt u r ba t i o n s
F i gu r e
A-1
A
1
F- 1
A-2
A
2
F 2
A- 3
A
3
F-3
B-1
B
B-2
B
2
F- 5
B-3
B
3
F -6
C-1
c
C-2
c
2
F-8
C-3
c
3
F-9
D- 1
0
1
F-1 0
0-2
0
2
F-1 1
D-3
0
3
F-1 2
-
F -4
F- 7
238
0
0
---
·--
>
(f)
\C)
S t e am p r e s s ure P
0
6=
35 . 884
s
0
0· �------r--.--�---.
0 . 00
30 . 00
60 . 00
90 . 00
1 20
C
(S
.T_
I_
M_
E_
_E
__J
___
_
_
0
0
. 00
6=
6 . 226
ru ::l'·
>
(f)
\C)
0
0
T u b e meta l te�pe ra t u r e T
4---------�------�--------�------�
en . o o
'
3 o . oo
60 . oo
T I ME ( 5ECJ
9o . oo
1 20 . oo
·------
>
(f)
\C)
0
0
6=
7 . 498
P r i ma ry t empe r a t u r e T
p
�--------�-------r--------�-------.
� . 00
·
m
Fi g u r e F . l
30 . 00
60 . 00
T I ME (SEC)
90 . 00
S t e p Re s pon s e fo r C a s e A . l .
1 20 . 00
239
0
0
en :::1'·
��v
S t e am · pre s s ure P
I
9J . 0 0
T I ME
o..O
0
0
·�
��
----
0
:::1'
.
..-. a
o..O
I
1 20 . 00
(5ECJ
�:::. = 0 . 6 s 0
T ube metal tempera t u re T
9J . 0 0
>
(J)
90 . 00
------:----
0
If)
ru 0·
>
(f)
I
I
60 . 00
30 . 0 0
s
�--
----
30 . 00
T I ME
----�------�
--
60 . 00
1 20 . 00
(5ECJ
�:::. = 0 . 4 3 1
P r i ma ry tempe ra t u re T p
0
0
T I ME
Fi g u r e F . 2
90 . 00
------
30 . 00
m
60 . 00
(5ECJ
S t e p Re s pon s e fo r Ca s e A . 2 .
90 . 00
1 20 . 00
2 40
0
0
CD 0.
>
en
0
...0 0
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-0 . 38 1
1 20 . 00
I
(SECJ
-----
a
('f)
>a
w
_J
I
downcomer l eve l L
.6 =
0 . 400
d
.
0 �-------.--�--.
• o . oo
3o . oo
..0
9o. oo
1 2o . oo
CSECJ
------ .6 =
0
(X)
::r
:L
1-
6o . oo
T I ME
.
0
t ub e met a l t empe ra t u re T
0 . 994
m4
a
0
· �------�--------�------�------�
� . 00
30 . 0 0
60 . 0 0
T I ME
Fi g u re F . l l
( Co n t i n ue d ) .
(SECJ
90 . 00
1 20 . 0 0
265
------- D=
0
0
z: •
2 . 693
W C\J
0
..0
downcomer t emperat ure
0
0
· �----�-,--------.--------.--------.
� . 00
30 . 00
60. 00
T I ME
90. 00
1 20 . 00
£SECl
______
a:::l N
wo
b=
bo i l i n g s e ct i on e x i t q ua l i ty X
>< a �---�--�
30 . oo
60 . oo
so. oo
1 2 0 . oo
\.() ' a . o o
T I ME
t­
(f)
Q_
..0
.
::r
steam p re s s u re P
s
0
0
· ��----�--------�------�------�
� . 00
30 . 00
60 . 00
T IME
F i g u re F . l l
e
CSECJ
------ D=
0
0
-0 . 00 1
( Con t i n u ed ) .
(SEC)
90 . 00
1 20 . 00
5 . 556
266
0
0
C\.1 0'
0:
o._
11.0
p r i ma ry te mpera t u re T
0
c:::
P2
�------ �=
- 1 848
1 20 . 0 0
N 4-------r--r--�--�
'
o . oo
3 o . oo
T I ME
6o. oo
s o . oo
(SEC)
pr i ma ry tempe rat ure T
1.0
0
0
Pl
------- � =
' 4---------r--------r--------�-------,
� . 00
3 0 . 00
T I ME
.
60 . 00
90 . 00
0 . 075
1 20 . 0 0
( S E C)
0
0
�_J
(L
t1.0
0
i n l et p l en um temp e ra t u re T
p;
c::: -+-----.....--..- b,. = 0 0 0 0
30 . 00
6 0 . 00
9 0 . 00
1 2 0 . 00
�. 00
•
T IME
F i g u re F . l 2
(SECJ
S t e p Re s po n se fo r C a s e 0 . 3 .
26 7
0
0
o o·
_J
Q_
t-
o u t l e t p l en um tempe rat u re T
\.0 0
0
Po
-3 . 033
.
� 4-------�--�--.--.
1 2o . o o
so . oo
6o . oo
3 o . oo
'o . oo
------- b. =
T I ME
(5ECJ
pr i ma ry temperat u r e T p
4
------- b. = - 3
.
� 4-------�--�--.---�
1 20 . oo
90 . oo
3 0 . oo
60 . oo
•o . o o
\.0 0
0
T I ME
..0 0
0
(5ECJ
pr i ma ry tempe rat u re T p
3
.
� �------�--�
'
30 . 00
o . oo
6 0 . 00
1 20 . 00
90 . 00
------ b. =
T I ME
Fi g u re F . l 2
( Co n t i n u ed ) .
. 033
(5ECJ
-3 . 075
268
0
0
(l'") l
-
tube met a l t empe ra t u r e T
m
..0 0
0
.
m �------.--r--�--�
•o . o o
3
------ b=
3 0 . 00
6 0 . 00
T I ME (SECJ
90 . 00
-4 . 286
1 20 . 0 0
0
0
..o o
o
t ube me t a l t empe rature T
. m2
;�----�====�==� b=
•o . oo
6o . oo
3o . o o
T I ME (SEC)
so . oo
-3 . 740
1 2o . o o
0
0
.
..--i
:L
1-
0
t ub e met a l tempe ra t u re T
..0 0
0
.
ml
-------
N �------��--�---.
•a . oo
F i g u re F . l 2
60 . oo
3 0 . oo
T I ME (SEC)
( Con t i n ue d ) .
90. oo
�=
1 20 . oo
-1 .813
269
cn
0
C\.1
.
o
::::> '
�---- � =
�-�-�--------,-------�--------,-------�
-0 . 266
s ubcool e d l en g t h L 1
5
1 01. o o
"' · - �
w
_J
0
3o. oo
T I ME
sb . oo
(SEC)
90. oo
1 2 o . oo
1 . 292
------- � =
0
0
> ....:
..0
0
0
down comer l ev e l L
d
" 4-------�--------�------�------�
93 . 00
::f4
L
t-
3 0 . 00
T I ME
60 . 00
C S ECl
90 . 00
1 20 . 00
0
0
.
0
t ub e met a l t empera t u re T
..0 0
0
m4
��------��====�==� � =
'o . oo
3 o . oo
6o . oo
so . oo
1 2o . oo
Fi gure F . l 2
-
T IME
( Con t i n ue d ) .
(SECJ
3 745
.
2 70
0
0
� _....
w '
t­
o
downcomer tempera t u re
� 0
0
.
- 4. . 5 7 0
------- 6 =
� �------�--�
'o . o o
30. 00
60 . 00
T IME
9 0 . 00
1 20 . 00
(SEC)
0 . 004
------- 6 =
bo i l i n g s e ct i o n e x i t q u a l i ty X
co co
wo
>< o ;,i:J
ten
o._
I0
. 00
30 . 00
T IME
60 . 00
90 . 00
(5ECJ
----r--r--�--�
1 2Q
•
e
00
0
0
.
0
0
"" b
s t eam p re s s u r e P
s
6= - 3 6
� ;-------�.---�--�--�
30 . oo
so. oo
'o . o o
so. oo
1 20 . ao
0
T I ME
Fi gure F . l 2
( C o n t i n ued ) .
(5ECJ
. 2 7 4.
AP P E N D I X G
I NS TRUC T I O N S FOR U S I NG TH E D E TA I LED MO D E L DY NAM I C
RES PONSE CQt.1 PUTE R PROGRAt1
W h en Mod e l A or B i s u sed fo r t h e dynami c r e s po n s e c a l c u l a t i o n
fo r a U TS G , t he on l y computer i n pu t s n ee d ed a re t h e c o e ff i c i en t
matr i c e s a n d fo rc i ng vec to r s o b ta i n ed from steam gen erato r d e s i gn
a n d i n pu t data .
O n c e t h e coeff i c i en t mat r i c e s a re c a l c u l a ted , t h ey
ca n b e u s ed to pro v i d e t h e i n p u t t o t h e comp uter codes MATEXP (
o r S FR- 3 (
40
39 )
) to ca l c u l a t e t h e t i me o r frequ en cy r e s po n s e r e s p ec t i vel y .
I n c a s e o f Mo de l s C a n d D , t h e c a l c u l a t i on proced u r e i s mo re
com p l i ca ted d u e to t h e u s e o f the m i x ed
( d i fferen t i a l + a l gebra i c )
fo rmu l a t i o n , t he c omp uter program t ha t wa s prepa red to c a l c u l a t e t h e
t i me r e s po n s e fo r Mo d e l 0 i s o u t l i n ed a n d t h e i n s t r u c t i on s fo r u s i n g
i t a r e g i ven i n t h i s a p pend i x .
The computer program i s a FORTRA N I V program prepa red to b e
i m p l emen t ed on t h e I BM 3 6 0 Mo del 65 compu t e r .
T h e prog ram con s i st s
o f a ma i n rou t i n e a n d s ev er a l s u bro u t i n e s , t h e fu n c t i o n s o f w h i c h
a r e d e s c r i bed bel o w .
l.
The �� r�A I W ro u t i n e s r e a d i n i n pu t data a nd c a l c u l a t e
t he
pa rameters n eeded fo r t h e st eady state c a l c u l at i on .
Th ere
are some bu i l t - i n data t ha t c a n be col l ec t ed i n d a ta
sta temen t s fo r i mpro v i n g t he reada b i l i ty o f t he pro g ra m .
2 71
2.
2 72
T h e s ub r o u t i n e " S TlJYST" i s c a l l ed from " �1/\ I W to c a l c u l a t e
t h e t em p era t u re s a n d ma s s q u a l i ty prof i l e s a l o n g t he heat
tran s fe r pat h in the effec t i ve heat exc ha n g e re g i on
( core) .
The s u bcoo l ed l e n g t h a n d b o u n d a ry temper a t u r e s a re pa s s ed
from " S TDYST" to '' MA I N " so t ha t i t can be u s ed to ca l c u l ate
the c o e f f i c i e nt mat r i c e s AONE a n d ATWO .
3.
After AO N E a n d ATWO a re c a l c u l a t ed , t h e progr ar.1 proc eeds
by r e ad i n g i n t he con t ro l i n pu t data u s ed for s u b ro u t i ne
" PART" w h i c h ha ndl e s t h e p a rt i t i on i n g o f AO NE a n d ATI-JO to
red uc e the sy stem of a l g e b ra i c pl u s d i ffere n t i a l equat i o n s
i n to a s et o f p u r e d i ffere n t i a l equat i on s .
After t hat ,
t h e fo rc i n g vecto r fo r " PU RE D I F F " i s read i n .
4.
Aft er t he p u r e d i ffe ren t i a l sys t em i s obt a i n ed , s u b rou t i n e
" MATE X P " i s c a l l ed to c a l c u l a t e a n d p l o t t h e t i me r e s pon s e
u s i n g t h e r e d u ced A mat r i x a n d fo rc i n g vecto r .
5.
T h e same procedure can b e u s ed t o ca l c u l a t e t h e f re q u ency
r e s po n s e by ca l l i n g t he " S FR - 3 " pro g ra m i n s t ead o f
" 11ATE X P " .
G. 1
I n put Data and Fo rma t
T h e fo l l ow i n g a re t he i n p u t d a ta con ta i n ed i n ea c h d a t a c a rd
tog e ther w i t h t h e r ea d i n g format u s ed .
2 73
Card :lo . l ( Geometr i ca l pa rame t e r )
C o l ur,1 11
l -5
Fo rma t
I5
I n put
NT
NT
H1T
TO O
t
I
1 6 -25
1
1
7 ( El 0 . 4 )
TMT
I
USHO
l
1
36-45
26-35
USHT
I
46-55
LSHO
\
56 - 6 5
1
66-75
l L S H T I OUHT
n umber of U - t u b e s
=
TO O
6-1 5
t u be o u t s i d e d i ameter ( i n c h e s )
=
t u b e meta l t h i c k n e s s ( i n c he s )
=
USH D
=
s team gen erator u p p e r s he l l d i ameter ( i n c h e s )
USHT
=
s t eam g en erato r u p p er s he l l t h i c k n e s s ( i nc he s )
LSH D
=
s t eam g en e rator l ower s he l l d i amet er ( i nc h e s )
LSHT
=
OVHT
=
s t eam g e n e ra to r l ower s h e l l t h i c k n e s s ( i n c he s )
s team generator o v eral l h e i g ht* ( ft ) .
C a rd No .
2.
( P r i ma ry s i d e p a ramet er s )
l -1 0
C o l umn
Fo rma t
\iP
In put
/
I
l l -2 0
VP
l
2 l - 30
l
3l -40
B( E l 0 . 4 )
I CPl I
TP I
1
4l -50
,
J TPO I
5l -60
pp
/
Gl -70
1
7 1 -80
I TO P I PHC
pr i ma ry wa ter ( r eactor coo l a n t ) ma s s fl ow ra t e i n to t he
WP
s t eam g en era to r ( l b/ h r )
steam gen erator pr i ma ry wa t er v o l ume ( i nc l ud i n g i n l e t
3
a n d o u t l e t pl en um s ) ( ft )
VP
CPl
=
s p ec i f i c hea t at con sta n t pre s s u r e o f t he pr i ma ry wa t e r
( l3 t u / l b " F )
u sed
* Th i s i n pu t i s re d u n d a n t . I t wa s read i n beca u s e i t may b e
t h e fu t u r e to ca l c u l a t e wa t e r a n d s t eam vol ur:1es .
in
2 74
TPI
=
p r i ma ry wa t e r i n l et t em p e r a t u r e ( ° F )
TPO
=
pr i ma ry wat e r o u t l et t empera t u re ( ° F )
PP
p r i rna ry l oo p a verage pre s s u re ( ps i a )
=
ROP
=
3
a v e r a g e d en s i ty o f pr i ma ry wa t e r ( l b / f t )
PHC
=
pr i �a ry s i de h ea t co n te n t* ( B tu ) .
C a rd No . 3 ( Se co n da ry s i de parameter s )
Col umn
w
P
T
T
V
50
=
STG
s team fl o w ra te ( l b/ hr )
=
s t eam g e n erato r pr e s s u r e ( p s i g )
=
s a t u ra t i on t empera t u r e a t P
=
f e edwa t e r i n l et t emper a t u r e ( ° F )
SAT
FW I
SW
1 -1 0
STG
( °F)
3
v o l ume o f s econdary wa t e r ( ft ) i n t h e s t eam g en era to r
=
( i n c l u d i n g downcomer )
v 55
R
05 1
CP
2
3
v o l u me o f s econdary s team i n t he d rum st eam v o l ume ( ft )
=
=
=
3
s u bcool ed s ec o n da ry wat er a v era g e d en s i ty ( l b / ft )
s pec i f i c heat o f s econ d a ry s i de s ubcool ed wa t e r ( B t u / l b ° F )
C a r d No . 4 ( He a t tran s fer coeffi c i en t s )
C o l umn
1 -1 o
Fo rma t
I n pu t
H TA
1
I
1 1 -20
1
2 1 -30
5 ( El 0 . 4 )
HP
1
u r�
1
I
3 l -4o
Hs
1
j
4 1 - 5o
I HS 2
*T h i s i n p u t i s red u n da n t . I t c a n b e repl a c ed by 0 . 0 wi t hout
a ffect i n g t he r e s u l t s ; it wa s i n c l u ded for po s s i b l e f u t u re u s e .
2 75
2
H TA = o v e r a l l heat tran s fer a rea o f t h e s t eam gen era t o r U - t u b e s ( ft )
2
H P = pr i ma ry s i de f i l m h ea t t ra n s fer co eff i c i en t ( B t u / h r n o F )
2
U M = t u b e meta l co n d u ct a n c e ( B t u / h r ft ° F )
llS
HS
1
2
2
= s u b coo l ed s econ dary f i l m h ea t tran s f e r coeffi c i en t ( B t u / h r ft ° F )
2
= bo i l i n g s e co n d a ry fi l m h ea t tra n s fer coeffi c i e n t ( B t u / hr ft ° F ) .
C a rd No . 5 ( S t ea dy s t a t e th ermodyn am i c pro pert i e s )
C o l u mn
1 -1 0
Forma t
I n put
1
1 1 1 -20
1
2 1 -30
6 ( El 0.4)
I H FG / H G
HF
1
31 - 4 0
I
!vF
4 1 -50
V FG
1
51 -60
I VG
H F = en thal py of s a t u ra t ed wa t e r ( Bt u / l b )
H FG = l a t e n t hea t o f v a po r i za t i o n ( B tu/ l bm)
H G = en thal py· o f s a t u ra ted s team ( Bt u/ l b )
3
V F = s pe c i f i c heat o f saturated wa ter ( ft / l b )
V FG = d i ffe r en c e b etween s pe c i f i c vo l umes fo r s a t u ra ted s t ea m and
3
wa t er ( ft / l b )
3
V G = s p ec i fi c heat o f sa turated s t ea m ( ft / l b )
C a r d No . 6 ( The rmo dyn a m i c p ro pe rty g rad i en t s )
C o l umn
1 -1 0
Fo rma t
1 1 -2 0
DTSAT l oH F
I n pu t
DT
1
I
= aT
SAT
1
l
2 1 -30
DHFT
sa t / a P ( ° F / p s i a )
1
l
D H F = a h / a P ( B t u/ l b/ ps i a )
f
DHFG
=
ah
fg
/ a P ( B t u / l b/ p s i a )
31 -4 0
1
41 -50
8 ( E1 0 . 4 )
DHG
l
DV F
1
I
5 1 -60
DVFG
1
I
6 1 -70
DVG
1
1
7 1 -80
DROG
2 76
DHG = a h / a P ( B t u/ l b/ ps i a )
9
3
DV F = a v / a P ( ft / l b/ p s i a )
f
3
DVFG = a v / a P ( ft / l b/ p s i a )
fg
3
DVG = a v / a P ( ft / l b / p s i a )
9
3
DROG = a p / a P ( l b/ ft / pS i a )
g
(p
g
= d en s i ty o f s a t u ra t e d s t eam) .
C a r d s 7 throu g h 1 1 ( 5 t i tl e card s )
Fo rma t ( 1 8 A . 4 ) eac h .
Card No . 1 2
C o l umn
1 -5
6-1 0
Format
15
15
I n pu t
rmE D
NALG
U se the fo l l ow i n g i n pu t v a l u es :
NRE D = 1 5 = order o f reduced ( pu r e d i fferen t i a l sys t em )
NAL G = 3 = n umber o f a l g ebra i c v a r i a b l e s i n t he m i xed mod e formul a t i on .
C a rd r�o . 1 3
Col umn
1 -5
6 -1 0
1 1 -1 5
Format
I5
I5
I5
I n pu t
LALG ( l )
LJ.\LG ( 2 )
LALG ( 3 )
LALG ( I ) , I = 1 , 2 ,
•
.
.
,
NAL G i s t h e l i s t o f NAL G a l g eb ra i c v a r i a bl e s .
T h e al gebra i c va r i a bl e s i n t h e c u rr en t prog ram a r e 1 6 , 1 7 , a n d 1 8 .
2 77
C a rd N o . 1 4 ( No n z ero fo rc i n g fu n c t i on el emen t s )
Co l umn
l -5
6-1 5
Forma t
I5
E l 0-4
I n pu t
IU(J )
UX ( J )
Re peat 4 p e r c a rd
I U ( J ) = row n umber o f ea c h n o n z ero compo n e n t o f t h e forc i n g
fu n c t i o n vector
UX ( J ) = v a l u e o f eac h n o n z ero compo n en t o f the fo r c i n g fun c t i o n
vecto r . *
Card No . 1 5
B l a n k ca rd to end the i n pu t o f t he non zero fo rc i n g fun ct i on
el emen t s .
Ca rd No . 1 6
C o l umn
1 -2
Fo rma t
I2
I n pu t
riE
( MATE X P con tro l pa ramet er s )
6-7
I2
3X
3X
LL
l l -2 0
2 1 - 30
3 1 -4 0
4 1 -50
5 1 - GO
6 1 -6 2
Fl 0 . 0
Fl O . O
Fl 0 . 0
Fl 0 . 0
Fl O. O
I2
p
TZERO
T
TnAX
P L T I N C MATYES
t�TEXP co n trol paramet e r s ( co n t i n ue d )
C o l umn
Fo r:-1i1 �
I n pu t
6 3-64
I2
I CSS
65-66
I2
J FLAG
67 -69
70
I3
Il
12
I H1AX
LASTCC
IlZ
7 1 - 72
73 - 74
75-80
12
F6 . 0
r c o rnR VAR
N E = n u mber o f e q u a t i on s = 1 5
L L = co effi c i e n t mat r i x tag n umber, u se L L = l
P = prec i s i o n o f C a n d H P - r ecommen d 1 0 -
6
or l e ss
TZE RO = z ero t i me ( u s ual l y 0 . 0 )
*See S ec t i o n I I I . 7 . 5 , pa g e 1 0 5 , fo r the c a l c u l a t i o n of the
fo rc i n g vecto r el emen t s .
2 78
� i me
T = compu ta t i on
i nterval
TMAX = ma x i mum t i me
P LT I NC = p r i n t i n g t i me i n t e r v a l
MATY E S = c o e ffi c i e n t ma tr i x
( A)
c o n t ro l
fl a g
= u s e p re c i o u s A a n d T
2
= re a d. n ew c o e ff i c i e n t s to a l t e r A
3
= r e a d e n t i re n ew A
( n o n z e ro
val ues )
4 = D I STRB t o c a l c u l a t e e n t i re n ew A ,
( Use
ICSS = i n i ti a l
MAT Y E S =
3
for t h i s a p pl i cat i o n
c o n d i t i o n v e c to r
( XIC )
)
fl ag
= re a d i n a l l n ew n o n z ero v a l u e s
2
= r ea d n ew v a l u e s to a l t e r p r ev i o u s vecto r
3
= u se p r ev i o u s ve cto r
4 = ve c to r =
5
0
= u s e l a s t v a l ue o f X v e c t o r from p r ev i o u s run
( Current
pro gram u s e s I C S S = 4 . )
J FLAG = forc i n g fu n c t i on
( Z)
fl a g
l
t h r u 4 = same a s fo r I C S S for c o n s t a n t
5
= cal l
Z
D I STRB a t e a c h t i me ste p fo r va r i a b l e
( c ur r e n t
program u s e s J FLAG =
3.)
I H1AX = ma x i mum n umb er of t e rm s i n s e r i e s a p pro x i ma t i on o f e x p
( r ecomm e n d
I TMAX =
10)
LAS TCC = n o n z e ro for l a s t ca s e
I l Z = row o f
Z
i f o n l y o n e n o n z ero , o t herw i s e =
0
( AT )
279
I CO NTR
for i n ternal contro l o pt i o n s
=
0
-1
=
read n ew contro l c a r d f o r n ext c a s e
=
go to 2 1 2 c a l l D I STRB fo r n ew
=
go to 2 1 5 cal l D I STRB fo r n ew i n i t i a l c o n d i t i on s
( C u rren t p ro gram u s e s I C O NT R
VAR
=
=
A
or T
0. )
ma x i mum a l l owa b l e va l ue o f l a r g e s t c o e f f i c i en t �� t r i x e l emen t
*T ( Re c ommen d VAR
=
1 .0) .
C a rd N o . 1 7
B l a n k c a r d to end t he i n pu t data .
V ITA
Mo hame d Ra b i e Ahme d Al i w a s born i n Mata i , E l - i1i nya , E gypt on
t·1ay 9 , 1 940 .
Aft e r rece i v i n g t he B a c h e l o r o f S c i e n ce de g ree i n
E l ectri c a l En g i ne e r i n g ( Powe r S e ct i o n ) i n 1 9 62 , he worke d i n
t e a c h i n g E l e ct r i c a l E n g i neeri n g a n d s pe c i a l t e c h n o l o gy o f e l e ct r i c a l
powe r e q u i pme n t a t t he Abba s s i a E l e c t r i c a n d E l e ct ron i c T e c h n i ca l
T r a i n i n g Cen t e r i n Ca i ro , E gypt from No vembe r 1 96 2 to Feb rua ry 1 96 5 .
S i nce f1a rch , 1 96 5 , he h a s wo rke d fo r t h e E gy pt i a n Atomi c E n e rgy
Comm i s s i on N u c l e a r Powe r P roj e c t De pa rtme n t .
I n 1 96 5 , h e mat r i cul a t e d a t Ca i ro Un i ve rs i ty a n d o b ta i ne d t h e re
t h e Di pl oma o f H i gh e r S t ud i e s ( Eq u i va l en t t o a non - t h es i s
M.
S.
de g ree ) i n Powe r S t a t i on s a n d E l e c t r i c a l Netwo rk s i n 1 96 7 .
H e c a me t o t h e Un i ted S t a t e s i n J u ne 1 96 9 wh e re h e wo r k e d wi t h
t h e A g r o I n d u s t r i a l Compl e x S t u dy T e a m at O a k R i dge N a t i o n a l L abo rato ry
from J un e t o De cemb e r o f 1 96 9 .
I n J a n ua ry 1 9 70 , h e s ta rt e d h i s s t u dy
a t t h e G ra duate S c ho o l of T h e U n i vers i ty o f Tenn e s s ee o n a n I AEA
Fel l ow s h i p s po n so re d by the U . S . Nat i o na l Aca demy of S c i e n ces fo r
a pe r i o d o f f i fte e n mon t h s .
S t a rt i n g Apr i l 1 , 1 9 71 , he h a s b e e n
awarded a re s e a r c h a s s i s tan t s h i p from t he N ucl e a r E n g i ne e r i n g De pa rtme n t
o f T he Un i v e r s i ty o f Tennes s e e .
I n De cemb e r 1 9 7 1 , h e obta i ne d t h e
M . S . degre e i n N uc l ea r E n g i n ee ri n g .
I n J ul y 1 9 74 , h e wa s emp l oyed
by t he Powe r Sys tems G ro u p of Comb u s t i o n E n g i ne e r i n g , I n c .
He h e l d
t h e po s i t i o n o f a s en i o r N S S S E n g i n e e r i n the Re actor Dynami c s S e c t i on
un t i l May 1 9 76 w h e n he ret u rn e d to T h e Un i vers i ty o f Te n n e s s ee a n d
obta i ne d t he Ph . D . de gree i n N uc l e a r E n g i ne e r i n g i n A u g u s t 1 9 76 .
2 80
281
Ra b i e i s m a rr i e d to t h e fo rme r Mi s s Sohe i r Moh amed 11o s t a fa Kame l
o f C a i ro , E gypt a n d h a s a d a u ghter n a me d Ama n i a n d a s o n n amed Y a s e r .