Nonlinearity of Dynamic Processes in a Drum Boiler Circulation Loop

( :ojl\rlglll

co

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NONLINEARITY OF DYNAMIC PROCESSES IN A DRUM BOILER CIRCULATION LOOP V. Kecman

.\bstrrtct. r:': l C :)[lpe r

(JeA1s \\"it: } rIlnt11emniicf11

moc1ellin;;: flnd nllmericflJ si-

InuIAt .i.on of t! ,e dynAmics of flow FIll cl lient processes insi{le tl,e circulAtioll loo:) or '" pOh'er sten:n c:enerntor. Tile processes in tile evaporAtor, drlil'l alHi downcomers arc mode 11 ed selwratc ly. Tile eVAporator is treated flS fl system witi; distributed !lArnmeters and is descrihed hy three nonlineAr, pArtial differentiAl equations for the mass flow, entropy and !)ressure. ,',n ei r>.: envfllue analysis of eVAporAtor system matrix!! l10s heen mAde. LineRr nnd nonlinear models of circulAtion loop describe the pllenomen? of sllrillk and s\.;el I of drum wflter level. The comparative ana.lysis of nonlincAr llIodel respotlses to the inputs of the same maf!;nitude ohtfline(' at t)1re different load levels, clearly presents the nonlinearity of t il e circulAtion loop processes. :: eyworcls. ,jo ilers; modeJlinp;; nonlineAr silllulntion; linear analysis; p,wtiCll {]ifferential equations; discretisation

part of the riser is the hasic condition for the vnlidity of the model of entire circulntiOIl [lElth. Therefore, a specilll concern is given to the modelling of this part of the circulatjon loop. In contrast to most pnpers dealing with drum boilers the processes in the eVllporl1tor, hoiler's drum and dOMlcomers are modelled sepnratly. The riser is treated as A system witll distrihuted parameters and the processes oft he fl 01.; and lient exchClnl(e in it are descrihed hy pArtiAl differential e(]uatiol1s (I'JlE) in terms of the mllss fl m.; (m), entro:ly (s) and pressure (T'). TIle ann! ysis of the ei(~envAllIes of the discretised a nd lineariserl set of equations ohtAined from these 1'01': presents a F;ood insif!;llt into dynnmic cllarActeristics of the evaporlltor. The transients of pressure and water level in the drum are described IVith two nonlinear, ordinary differential equations (ODE). The downcomers' dynamics is neglected.

:'; uccessful npplicntion of modern cOlltrol metllOds depends lI1ost]y upon validity of mat ilemClticCll models which form the !)f1.se for [l control AI!';oritlllllllS synthesis. These models, l)esides they should he ahle to describe \,'e I I the process dynl1mics, olJ(':ht to he ill tile form of I inenr dynnmic efllwtiolls of stl1te spnce (i.e. in tIle form of mfltrix differential eqllntions), hecallse the llIodern control theory uses Just this mfltllematical tool. Tllere is always the same question flbollt the lineArisation - what is tile rAn~e of clwnp:e of state variahles inside which the ohtAined linear model can still reprodllce n process dynamics very well. It is the fluestion about the nonlinearity of modelled processes. In tllis pflper, the flnnlysis of nonlineAr charflcter of the processes ,.hich tnke part inside the circulntion pAth of power plnnt boiler with forced cjrculntion is p:iven.

A necessary tool for fulfil ling this ~oal is an appropriate mr tllernatica I mode 1 nb] e to give n true representntion of the process dynnmics on the steam - water side. The precise description of the evaporatinl!;

The mathematical model IVns applied to the circnJation PAtl: of the ;\0 . .. unit of Cromhy stntiOI1, l'hiladel phia, US.~ . All necessary geometric data were foulld at Clelland (197G). 21

22

V. Ke cman

dm n

dt :-i"'lre : ill !Ist:-Ates sllem[\ticnl~' Ft circ'.llntiOI . 11)0;1 for wldc:) 11 ;lroper nlflt! ' emntic 3 1 model sl ,0'l1<1 lle estC:'llisl\ed. T:le whole 1 0 0 :1 is cli.vi(l('n into tl1rc l' nrts - evopOl'ntor (riser), drnm, downcomers, nnd N1C!' :1:11' t '"-i1!" heen mane I I eo se~)~rf1t 1y. Tl:e "lo(lcl is developed on t he fJasis o f n ' first f' ri nci:ll () " nnnlysis . t i!nt is, s;; overnin r: equnti ons "re eOllntions of mass, ~omentllr nnd cner ~ y fJalance. The <1cvelopmer.t of these erl'Hltions h'i 11 he not given (i t ca!! he found i 11 an~' .:roo(l hook on f 1 uid dynRmics) nno just their finAl forms will he presented bere. I\evertheless, fl curiotls render co1l1(1 find 01Jt tl.at it hFls heen tryed to be in nccordance with }:IVI1tny, ~ ic : )onal d, ~ pnre (I <)71), Konopacki (1 <) 7 9 ), EeCI1lDll (l 98C,l98"), Lermak, r'eter1.<\, ,:flvorl(a (! <)7 , ) • Sv:t ~, )orator

':';le riser section hns !-:leen treated as t he system with distrihuted [larameters and t:le model of dynnmic processes in it hns t'le fo 1 J owi nr; form,

(1)

os

ut

u: ~ ut

v(~ £.!l - !!!,\ ~ " I uZ uz .,. 1

[ " <>m v

~T~

o!\

m .•.": L'.

1

:' I,.}

(~)

m os v(l ~ mv ·, ~ TI uZ - 'X uZ + '1'rCl I re

j'

-,\ n

-f'

n-1 _

"5t

us

m-m n n-l

/\.

~z

mn

-T' (4)

(5)

di ' n-l

---ar-

"

Y~-J -.,--

o;"n_l

n

~,

-,\-

I\'E +

mn-m n _ I l:J.z

0nn_lvll_

-

(6)

1

r-.ow, this system of cquations can he used for the numerical simulation of dynamic behaviour of the evaporator. ~Inthematica­ ly spealdnp;, from the thre PDE, we derive a system of :\.1\1': ODE in time, where i\E is the numher of elements. Ilefore simul ation, the boundary conditions should he stated. In the riser, which is thc closed circulation loop, the physical houndary conditions are inlet flow m, inlet ent ha 1 JlY . hd c and out 1 et pressure I' nn = I' d with the heat flux dq/dz as the drivin/!: force over the heated length. In this paper, internal variahle is the movinr-; sl!hcoolinf',' lenp;th Ls "'hicll can he calculated from tI le following; equations, ()q / oh' dp (:L c>zLs-mdc(h-hdc)- ,?"Ls(up or s

AP'

dt

Ani (11'

-rr-

'--y---'

QV

v

~m

~

ClZ

- LdC

)

-

,

)~').h

/

(7)

(3 )

()t

Tlli 5 set of t lIree non 1 ineFlr H)S is not convenient for computer simulation and it should be previously rewriten in a proper form. The most common means for solvin~ this system of equrttions is the discretisation method. EssentiRlIy, this means dividing the evaporating part of the riser into a numher of elements and any obtained section spFltinl derivative replace with back-ward difference. (Clearly, the greater tile nnmber of sections, the hetter the approximation, hut the lRrger the model the more expensive is the simulation). Applying this hack-ward method to the riser's nonlinear model the following set of equations is ohtflined:

(At Konopacki (1 979) and ~ ecman (1980 , 198~) L is constflllt). s 111 order to get hetter insight into dynamic characteristic of the evaporfltor part, a lineflrisation of the nonlinear model has 1Jeen made and the analysis of the eigenvaJ ue of tIle system matrix ,\ bas been done. Discretised and linearised set of equations for flow rate, entropy and pressure is given below, uL'.m

n

~

(-

4iii 'In AL'.z

23

S

(-

-5

IT:

v

.;. \ - "'~") c.~\),

62

•

.'

Ti; v

(

vt

0

\1 .

: '-1

I ) Cc/-nil). '"~ V • J ~ [ i) "r1 IV

. cC.Z,IS . , _

11

() '. -' ,

+

() I

.(). 11

,1 "]

.(V-Vw)h +-+-(V-V )· '1 - V 0; , h' J

'!l-I

Cl

Input v.9r i nh .ies are:

feech'l1ter flow me'

stenrn flow

circ ~ llation

Itl(l'

rn,

f.lOh'

riser olltJet flow and entll
(-

l? :In -

1

Line{\r

fllH1Jysi~

~ives

tl!nt the c1rtlm,

lysed sepflrfltly, s11o,,'s inte .2 ;ral

R,; n_1

nnrt-

Ch[ll'Clcter

for \,ater level Hand Ilrop
clw.rflc-

ter (T"':;u seconds) for drllm pressure t'd. i.' i."· .

~

ill ,1strHtcs t:,e st;;te v ;\ r::.a' lle

f'or ' rl ,f or

t ~' c

C[1se \L=t! ~~'J11l ~

is (1i -J idcd into Lndicntes

(cvA.porntin ,~:

c~er 'lents).

in fl ne!lce of

t!IC

elements ( \\:.:) , ," It

of

'L,

t~;e

pnrt

;' i s;'

~'

The do\vncomers' (lynarnics is ne!!"] ected and of

itil!11 i )Cr

on the

;O~(~,

l)oWI1comers

merely t' le

nlixin~:

of two strenms is f: iven

(for the calCllinti.on of the enthl11pv lldc) be.low,

t': e

t he \,L, tl-: e wider

f~ rent(~r

of ei.~'eTivf\ 1 nes re I~ rO(jllccd.

noted t:!(1t rec

t 1", e

0

r

~

ci r~e nv Ll l

a n (~ t

i .~ i 11

townros the

~re?

i.s decrcnsinr.·.

1"1 e

r

i .r:r: ~;

t

t~ \ e

rnn ,?.: e

It s! -· oul

(~

;n

he

t!cs move t o\\'flrd h':-t [)

nf instn1)i

fl

~ j

pl

t1 11 c,

:i. e •

ty, w!:e n

\ftC?'r n 1l1l('lbcr of

;\=

c;-tlcn~n-

iI e e +

Cm

-

m ):/

e

m

.\ 1 I!;ehrAie e'llll1tion shows tl ' flt the Clcumu 111tion of I1IIlE'S or energ:y inside the (lowncomel's is not tl1kel1 into the I1cconnt. But this er,llation is importrmt ',ecausc tI,e suhcoolin ['; len:z;tl1 Ls is determined "'Y hnc'

entropy (I'>s) per eJeme 'l t onJ~'

criteriOl ~)

(and tl11t is the

is r:renter th11r1 C .n:;l:>

kJ/kp;ol'; ' Tl:is criterion I':ives the sm1111est ;\c,

wilici ') ensures the st111'le model. For

Cromhy :':0 smallest

G.

~E

t'nit at lClI equals

of lond the

and the dynnmics of the riser depends upon L

s

•

FinAlly, the nonlineRr find lineRr model of the entire circulation loop is ohtained hy coupling the approprillte e(]llRtions of

0

a]l tLree subsystems.

Drum ;, dynamics of the processes in the drum is descrihed \-dth tllc following two ODE,

The division of the riser has n dOlll)] e impact on the size of the model Ilnd cost of the simulation of tIle boiler's dynamics.

"t") ( P'h' -J'l

Firstly, the smaller the NE, the smaller ( 11 )

the numher of differential equations describing the evaporation processes. Secondly, the smaller the 1\1<;, the smaller

24

the re al

a nd illwp;innry p Arts or t:le

eigenvnl ucs (Fi r!: . :;) i. e. tI le l11r,! !:er t !le and hecnuse the step of integr a tion is prollOrtioTlfll to the time lAr g er the

ste~)

.\ dru m :'oi ler cirCll 'clt ion :0 0,'

proces~,

time constllnt s descrihing tile

l;t

con~tant,

the

of inte g ration 6t, con-

senquently, the cheaper the simulation. Table I shows the

inte ~ ration

steps 6t

\,ltich, for given !\E, still ensure t,;e o and stahle simulation rllr1S. ( F or ;\E l;t

= l ; . 0(,n ;j

mics tRke " :1

G"u second" of

fir~t

S

'C)'

of U:\ I V,iC

j 1(; ( ,

n~'na­

CO'TiT lter

near model

oht<1inec1 at 7"

;; :~

on

trnnsients of wRter level :: find pressure I ' fit

,)U ,:, of

load i l: t he case of slldclen

increase of heat rate for

It.

, J .

The swell

phenomenon! is more pronounced in the case of rOll f!her discretis<:1tion , l)ut even more interestin~

is t hat the pressure dynamics

does not depencl u po n division of the eVllporAtin :::: pflrt. To

~ et

: (If

, J. -

CAll

1011(1

ratin [o!: part o f t: ;e : oop is tr cnte(1 RS " distrihuted process, !'i mulroti<)!l r lln" n re ver~ '

pretty cost ly heC£\llSe of ~:rAtion

!'mf1 11

inte-

step. Tlle most reco illlile nded fllrt ';(> l'

cont i nuntion of t h is wor k 1\'0 1:1 11 he t o a I"o re np:lro priHte (faster) i l' te ;:T il to

ilS

;; !ln ~ys e

",,-

ssihle reductions of ti le evo;' orntor 'lIod ,o" ne c; lectin g momentum e(]untion s . Tl e d i v i, sion of riser has not all i m!,!1c t re dynnrrd cs Rnr1 t ile phen ;) men R

0

on

'lI'eSS II-

r s:n' j Il k

and sweJ lore marc pronounced in the cnsc of ron;rh di,scretisotion. The r e is n :im i t in the smo 11 est num l)er of el ements o ',t[l ined b y t he division, And tl ,is ljlr.i t determined :JY tI le entropy c lla rwe

tl .e answer to the question n'lOut

1 ,

Rccejltnhle re sponsp.s. I f t ' )e eV flpo -

~ ive

tion procesure n s \Vel] ;"i r; . ·1 indicl1tes tlte influence of

nO Il I 1.-

C~,,"":~ es,

the \vllO l e up ;l er ::<:11f o f lone]

a ppJ y

ti me usin ~ " 7 ~ of memory).

~ ' flS

near dynllmic c !lR r ncte ri s tics. T t'OI I'c " , Co,,

is

t Le

0 1 Oll !:'

discrete riser clement.

the nOlllinearity of processes which tflke p lace inside the circ1l1ation loop, !'imulation rllns IIRve h een made with the d is turh:'nces of the same mar,ni tuc1e n t

three

different 10l'l<) levels. l,' i /2: . :; and G illustrate the dependence of' dyn ami c helHlviour of circlllation loop in the cnse of increflse of heat rate for C . :' 8;; kJ/ms (J'i [l: .S) Illld

in t ile Cflse of increase of feedwnter

rAte for g iq!;/s (F'i!<.G). iJotl' fi p; ures intlic n te tile difference in dynamic behAviollr at different level s and show t1 1nt tile linea r model ohtained at one .load level cllilnot descri!)e the dynllmics in the who] e area of load chanp:es. At

the same time

simulation runs show t ha t del ohtnined at

7~

~

the linear mo-

of load could repre-

sent the dynamics in area from 50 '" to fill 1 lORd, sntisfRctorily.

Clellnnd, !'.J. (197G). ,' mvel' j' l.'1rlt ;, nt H Base, I ·~\.:ntny,

P:~Cl'

lI. G.,

Cromhy :\0. ~lcDo n :110.,

,J . ~ .,

boiler-turbine~enerator

~~pa re

. .1. ::

.

svs t ems,

!'art 11 Deve I o!lIllent , J:',CC of t ' ,e " .',V:::, PAper

]'\ 0.

:>-])!';.

Konopacki, '" . ,\ .(1 9 7 '3) . The i;ynomics oncl Stahi 1 i ty of

i~vn]lorat

lon in

S te R~

',e-

nerators, i'h. D. Thesis. lj r exel l.nivers,:ity, i 'hiladelphin. };ecman, ', .

(198 0) . S im1.11ntio n

,·; n"I~' s i s

tIle Dynamics of a Drum :10 ] I er tion Loop,

i(eSearC~ 1

80 - 8 4 ::; - (> 1. [)rexel

of

~ ]rC'l: i l -

::cpnr t :>'0 .

t'n ivcr s itv., 1 )' jl ,: -

delphi a . Ke cma n. V.

(19 8") . '!odelirn n,je i

cion a ann] i

been ohtained hy simulat ing: ,j ust the dy-

nom kotll1 (in

namics of circullltion loop, and it is

l:ni t, Volu me

(1971). ,\ nonline n r llloele1 for l'e LC'H t

In this paper, the results presented have

rather hard to compRre them with values

n

Thesis

1

Zil

si :':l'.l l ? -

dinami.cld h proces<1 ~ er"ocroatian ).

!)

:' ~.

nR r;~ .

liniversitv of / ,i1 ,!cre".

Cermak , 1., f.- et erkn, V. , /.:1 vor :·~Ft, .J .

(~:) 7 ('1 " .

me asu red on the wllole plant. Such experi-

:J inamika re g:uJ iruemil ; sis tem v te ;o lo-

mentally ohtained values for Cromby No.

ener ge tike i

2 Unit a re presented in Thompson (1967),

1ated from Czech) ~lir

and Fi g: . 7 shows the comparison of results

himii (in Russi an, tr n ns1

c:osco"'.

Thompson, F' . T. (J 9(7). ,\ Dy namic

~:ode 1

of

at 90 '/0 of 10Rc] and for the chan p; e of he-

a Drum-Type Boiler Sys tem, ISEE Trans.

atin g: of :\ 'io. The short dashed curve is

on \-''\S, Vo1. J',\S - 86, !'; o. ;),

the Cromby P lant response, while the solid

S. G"5-G3!'i.

curve g:ives t h e model response.

25

::on l inea r i ty o f Dy n a mi c Process e s

X X X

X

"'2

X X

S2

X X

X

'2 P,

X

X

X X

X

P,

X X X

0)3

X X X

X X X X X

X X

X X X

'3 P2

X

54

P , 3

x

X

X

X

X

X

X

x

X X

X

's

X X

P,

Vi g . ,

circulat io n l oop.

X

"

X

P3

';'5

Y.

X X

55

X X

X X X

P,

X

X

A

1. The h oil er

~:

m,

X X X

X X

rhs

X

X

P2 X X X

I Ps" '1

.

B

riser model.

"~ 3~

18

"

I

a

..

a

200

o a o

-- - - r -- --:s

I

a a

100

.,

NE.6

real c i g Ql1\'n 111Its-_·~.I_--.....1- NE.2

Vi K. 3 . I nfluence of NE on the ei Kenvalues spectrum - 10 0% load.

TA BLE ) Dependence of the Greatest Integra tion Step 6t

NE

.l

61....,..[S)

~=~ . = ~ = ~

I

1------- -

MCE -B

0,02

---

2

0.004

3

0.001

5

0.00075

7

0.0005

10

o,OOCI

U

State variable form of linear

300

o

max

upon

~E

1

I

I ~Q /v:

mJ 'J

X

X X

r:"

r,'i ~ .

r rn,

"'2

26

0,2

Ll.P [bar]

0,6

NE

.~

--------.--------

0,4

2,3,5

.~------

0,2 -

_ _ -+-------+----_-+-- ____' +---+-- _ 4 - _ _ -+- -

50

100

.. - -t--------t--j..-----
300

250

200

150

t (s]

"i v . 1\. Jnflllcnce of riser rljvision « \ 2) on loop's trClnsients, ~' ;. "

1oCld,

1G,J

j

ncrCflSC ill heat rate.

Ml Cm]

0,. of load

/ ... ........................., .......

100 50 10

,

i: 0)

..... , ....... _........................

!

..... ~ ................

!

...........

t (5]

Z\.

P [bar)

0/. of load 100 50 10

0))

0,4

,"" ..... "

-- -

---- --

..................... . . . . . . . . . . . . . . . . .. .. _

-_ .... _---------------

•••• ••••• • • • • • • • • • • •• •••• •••••••••• •• • • • • • • • • • • •• • • • • • • • • •••• •

T---~--~-~ · -·~I----rO

50

ICO

__- 4_ _- +__- 4_ _ _ _

150

200

eo • • • • • • • • • • • • • • • • •

~--.~--+_--_ _~_

250

300

Fig. 5. Dependence of transients upon load level, increase of heat rate for O.~85 kJ/ms.

t (s]

~:on

linearity of l'ynamic Proc esses

27

AH[m]

°/. of load - - - iOO 50

0,2

............ 10

I

200 ~",p

250

300

t[ s1

[bar]

0,1

-0,1

~

;' ... _. ~.:::-.:-::.::-.~. ~ .:::::.:::'::::.:::.::-:.::: ::: ::: .7:".::-:.::-..::'.::::.:::.:::-. :-:: :-:7.::-:. ~.:::'.::-: :-:7.7:

V. ~

'1. of load

--100

I

- - - 50 ........ - ... 10 Fig. 6. Dependence of loop's transients u non load level. increase of feecl\vater rate for (l kg;/s.

0,02 300

400

t [5J

- 0,02

fie: . 7. ;·lode I and plant res ponses. 3 ~.)

decrease in heat rate.

9() ~ "

load.