Simple Drum-Boiler Models Åström, Karl Johan; Bell, Rodney D. 1988 Document Version: Publisher's PDF, also known as Version of record Link to publication Citation for published version (APA): Åström, K. J., & Bell, R. D. (1988). Simple Drum-Boiler Models. (Technical Reports TFRT-7402). Department of Automatic Control, Lund Institute of Technology (LTH). General rights Unless other specific re-use rights are stated the following general rights apply: Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Read more about Creative commons licenses: https://creativecommons.org/licenses/ Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.
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LUND UNIVERSITY
PO Box 117221 00 Lund+46 46-222 00 00
Simple Drum-Boiler Models
Åström, Karl Johan; Bell, Rodney D.
1988
Document Version:Publisher's PDF, also known as Version of record
Link to publication
Citation for published version (APA):Åström, K. J., & Bell, R. D. (1988). Simple Drum-Boiler Models. (Technical Reports TFRT-7402). Department ofAutomatic Control, Lund Institute of Technology (LTH).
General rightsUnless other specific re-use rights are stated the following general rights apply:Copyright and moral rights for the publications made accessible in the public portal are retained by the authorsand/or other copyright owners and it is a condition of accessing publications that users recognise and abide by thelegal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private studyor research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal
Read more about Creative commons licenses: https://creativecommons.org/licenses/Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will removeaccess to the work immediately and investigate your claim.
This paper describes a simple nonlinear models for a drum-boiler. The modelg are derived from fi.rst principles.They can be characterized by a few physical parameters thaü are easily obtained from construction data. Themodels also require steam tables for a limited operating range, which can be approximated by polynomials.The models have been validated against experimental data. A complete simulation program is provided.
Key words
Classification systctn and/or index tcrms (íî any)
Supplemcntary bìblíographical ínîormation
I5,9N and key title ISBN
Language
EnglishNu¡nbe¡ ofpages
34Recípient's notcs
S ccurity classífrcat ío n
The rcpott may bc o¡de¡ed îrom the Department of Automatic Cont¡ol or bortowed. tårough the tlnìvercíty Library 2, Box 7010,5-227 03 Lund, Sweden, Telex: 33248 lubbìs lund.
Paper presented onIFAC Symposi-um on Power Systems Modelling and ControlApplications, Brussels, Belgium, September 5-8, 1988.
Simple Drum-Boiler Models
K J ÄströmDepartment of -A,utomatic Cont¡ol
Lurd rnrtitute of Tech,ology5-221 00 Lu¡d
Sweden
R D BellSchool of M¡ther¡¡atics and physics
Macquarie UnìversityNorth Ryde New South Wales 2213
Aust¡alia
1. IntroductionThe¡e are -onJ¡ mod.els of drum-boile¡s i¡ the lite¡atu¡e. Seethe ¡efe¡eace ljst. The models desc¡ibed in thi¡ paper are d.e-¡ived f¡om first prhciples. They are cha¡acte¡ized by u fu. pu-¡amete¡s only which caq be obtained f¡om fust p.in.ipler, Chumodels a¡e validated by comparison with exte¡slve plant data.
.4, key feature of s d¡um-boiler is that there is a very eñcieute1e-rg¿ md mss transfq betweu all pute th¿t ¡¡e in cont¿ctwith the steam. The mechani¡m ,*poiriUtu ioiì¡!-¡"s¡ ¿¡o;"_fs¡ is þeiling and conden¡atiou. Â consequence of tl¡.is i¡ th¿tit is a very good approrirution to assume th¿t all w¡ter ste&ma¡d metal i¡ in thermat equilibrium- This me¿¡s that the total_Ae¡$y can be represented by a global energ¡r br'l¡¡ce. The v¿-lial,ity of thi¡ approrimatiou h¿¡ beea show-a-by -,.y mod.eling.' exe¡cises.
The paper is org¡.izsd as follows. Â fi¡st order mod.el is'. presented in Sectioa 2. This model j¡ obtained from a giobaleuerg¡r balance for the total plant. The model has one stste. va¡ìible which is chosen as the d¡um p¡essure. This model hasthe same structu¡e ¿s the model presented in Àst¡õm and Ek_luad (1972). The par¡meters are, however, obtained ùom fi¡stprhciples. To model the d¡r¡ar w¿te¡ level it is necessary to
. account fo¡ the ¡hri¡.k and swell phenomena. This is d.one inSection 3. ¡, thi¡d orde¡ model is obtained. This model hasd¡unr p¡es6rue, w¿te¡ volume and ¡teo- quality in the ¡iser¡-; as state va¡iables. The model "=hibits a compler beh¿viour ini spite of being of low o¡d.er. gimnt¿fie¡ of ,i"p ,urf o*u, ,"u' preseoted in Section 4.
:j
,{bsú¡actThis paper desc¡ibe¡ a rimple no¡lincar models for a d.rum-boi-le¡. The models ¡¡e de¡ived f¡om first principles. They canbe. ch¡¡¿cterized by a few physical par^-etls that are easily ob-tained f¡om construction data. The model¡ also require steamtables. for a li'nited operating range, which can be apiroximatedby polynomids. The models h¿ve been y¿lidated ugui*t .-pur-imental data. Ä complete ¡imr¡-lation p¡ogra¡n is pr-ovideil.
2. Ä First Order Model '
Because of the efEcient he¿t a¡d m¡s¡ t¡ansfe¡ duc ¡s þ6iting andconde¡.sation all parts of the rystcn which a¡e il contact withthe ste"m will be i¡ the¡m¿l equilibria. ft is the¡efo¡e natu¡al todesc¡ihe tbe plaut with global -^.s and eaergy balaoces ès wasdone ir .Ast¡öm and El¡lu¡d (1g22). The global energr b^loncecan be w¡itten æ
d.¡¡l,e,h'V,t* p-hnV**mcrT)= P { q¡-h¡- - q,h, (1)
whce g denotes specific denrit¡ ¡ u¡¡h"llh¡ 7 volume and. gmas¡ 0.ow. The iaöces r, u and /u refere to steam, water and.feedwater respectively. The total mass of the metal tubes is m,the specific heat is ç and the average metal temperatu¡e is ?.
The iaput power from the fuel is denoted by p. The total steamvolume i¡ given by
Vt=Vdttø-V-+ a^V (2)
where I/¿*- is the dru::o volme, I/- the volu:ne of w¿te¡ i¡the drum, I/, the riser volum.e md c- the average stem-watsvolume ratio. The total w¿ter voh:me is
Vu = V- * V¿ + (1 - o^)V, (3)
The right hand side of equation (1) represeuts the energy flowto the system from fuel and feedwate¡ and the energy llow fromthe system via the steam- Siace all parts ¿¡s in t[s¡ñ:l squiliþ-ria the state of the system can be represented by one ,r,a,¡iable
which we choose as the steam p¡essu¡e. Using steam tables thevariables Qt, Q-, h, and À. can then be expressed. as fu¡ctioroof steam p¡essue. ginil¡rly ? can be expressed as a functionof presaure by assu_m.iag that ? is equal to the satuation tem-perature of ¡te¡m which correrponds to p.
This model rep¡esents the dynamics clue to input powerwell. When the feedwater f.ow or the steam flow is changed itis, however, ¡ì.ecessa¡y to also take i¡to account that the wate¡¡nd ¡te¿m m¿s¡e¡ a¡e also changing, This can be ¿ccou¡ted forwith a giobal massbalance.
d.lîIe,v,' + p-Vø) = q¡- - Ç, (4)
The dyn"-i6¡ which desc¡ibe how the dnm ptessure is infu-enced by input power, feedwate¡ flow and steam flow is rvellcaptured by equations (t) al'd (a).
The de¡ivative of the total water volme (dØ_t/dt) can be
-eliminated between equations (t) and (+). I{jtiplicatián of (+)
The conderoation enthalpy h" = h, - It- has also been int¡o_d-uced. lfthe boiler is provided with a good level control systemthe total water volu¡ne (7-,) "¡d the total steam volu.me (7,r)do not change much. Ecluation (b) can then be simplifed to
dn
"rrà = P - q¡-(h-- h¡-)- q,h" (6)
where
"n = h"v,,ff + e,rr,dfr + p-v-d* + *",ff.A.part from steao table data it is thus suficient to klow totalsteam and wate¡ volumes aad total metal mass. The model (6)is identical to the model in Àsrröm and Eklund (1972). ño:tice howeve¡ th¡t in this cæe the pa¡ameters are obtained f¡omcomtruction data. Also notice that the term
c.= -*¡n,",,*+ e-v-,** *",#l
can be interpreted æ the total condensation f.ow. ft is observedthat the teløs ilh,ldp artd dh_ldp are key quantitier in pre-dictiag the energ¡r and m¿ss transfe¡ between steam and w¿te¡.These terms also appeared in the d¡um-boile¡ model of Mo¡tonand Price (1977).
&m a(.,()d( = : I"' a(c,()dþ,oî1
J"1
Ef 1""'a(z)dz ( 11)
(15)
Fo¡ somp control t¿skc e.g. drum level cont¡ol it is necessaryto model the dy¡"miçs of the drum level. This is mo¡e difficultbecause of the sh¡i¡l ¡nd ¡well efect. To describe this ii isrecessuy to accor¡lxt fo¡ the d.istribution of ¡te¿m and wate¡and the t¡anqfer ofm¿ss and energy between steam and w¿ter.
The steam-mter d.ist¡ibution varies along the ¡iser¡. parti&ldiffe¡ential equstioD¡ a¡e needecl to desc¡ibe this properly. Tokeep a finite áimsrrsiqnrl model we will assume that ihe shapeof the di¡tribution i¡ l¡own, The ¡¡sumed shape ir ba¡ed oneolving the partial dillereutial equrtionr in the rterdy rtato.This gives a line¿r di¡tribution of the ¡team-w¿te¡ mas¡ ratioalong the ¡i¡e¡s, lVe will the¡efo¡e assune th¿t the ¡atio r¡¿ries
c({)=a,{ 0SdS1 (?)
where f is a normalized lengt¡ s66¡¡linate along the risers and ¡iis the stese-w¿te¡ m¿ss ¡atio at the ¡iser outlet. The t¡aqsfe¡ ofm¿se and euergr between ste¿m and water by condeas¿tion andevaporation is a key elemeut þ t[g ¡û6de[ing. I{len 66dallingstem and water reparately the transfer mut be accounted. forexplicitly. This ca¡. be avoided by writing joiat balance equa_tio¡s fo¡ wate¡ and steÂE- The global -o"s balance fo¡ the ¡ise¡section is
d. d.. dt(e,o^W) + ¿(e-(t - e^)W) = Qac- j¡ (8)
' whe¡e ç is the-total m¿ss flow out of the risera. The globai. **gX bala¡ce for the rise¡ ¡ection i¡
. The flow out of the risus (g,) can be eliminated by multiplyingequation (8) by -(Ir-1c,lr") and ¿d¡ting to equation (S). Hencã
d, ,1
îî(p,h'o-V) - (lr- + æ,h.) fr(p,a^U,)¿
+ dtþ-h-(l - d-)V,) - (/r- + z,/¿")
d,;t(P.(1 -"-)%) - P-x',h"q¿"
.:::' This ca¡r be simplified to
,r; h.(1- ù*fu,o^V) + n-(t - "^)Vd!¡î,1
,. - z,h"frk-O - o^)V)* n,o^Voþ|" " -,,0"0*
(10)
Ì:
Drurn Level
To calculate ¡[g d¡r,E level it is necessary to know the avcagesteam-w¿te¡ volune ratio i¡ the ¡ise¡s (a-). We have
z = --3:c.-p,a¡ p_(l - z)
Solving this equation for a we get
c=a(a)= ,rQ-' "e'*(P-- e,)¿
Assr¡:o.e that the steam-wôte¡ ûås¡ ratio is linear along the ¡ise¡as erp¡essed by equatioa (?). The Bve¡age Eteaqr-water volu¡¡e¡atio i¡ the ¡ise¡s is
= e- l, - , n' , ¿n(t + e- - e,z-llo. - o. L- (p- - p.)o, ''' \^ ' B. "/l
'We can now obtain the followiag eqution fo¡ the d¡um level
,.=r-*;^n (12)
whe¡e ,{ ie the v¡et surf¿ce of the d¡r::n. This equation tellr thatthe d¡um level is composed of two terrru, the total amourt ofwatc in the drum, aad the d-isplacement due to changes of thesteam-wate¡ ratio in the ¡isers. The model has the same basicform es the water level model i¡ Bell and :4.¡t¡öm (1g7g). Thismodel was, however, developed heuristically and not from fi¡stprinciplee.
Downcomer Flow
The flow through the downcome¡s (g¿) can be obtained f¡oma momeú.tum' balance, In natu¡al ci¡cul¿tion boile¡s the flowis d¡iver by the difereuce between the dencities of wate¡ andsteam. Â momentum balance gives
a^V(p-- ù=lrxqre (13)
whe¡e È is a f¡iction coeflìcient. The ¡iser flow g, can be calcu_lated from equation (8). We get
d. de¡ = e¿¿- î¡\p,a^v,) - d¿
(p-(1 - "^)V.) (14)
3. Shrink and Swell
4. Simulations
whe¡e
The equations derived in Section 3 will now be sumrnrized.The state equations a¡e given by (f), (l) and (8). The stateva¡iables a¡e chosen as d¡um plessu¡e p, wate¡ vol,,me in d¡u:nV- and average steam quality ¿t rise¡ outlet 2,. Equation (1),(a) and (8) can thm be w¡itten æ
To execute the si¡nr¡lation equation (15) hae to be solved for thede¡iv-¿tives of the state v¿¡iables. The right hand side of (15)contain ilput y¿riables P, 91- and g,, aad fu¡ctions of the statevariables. Notice th¿t dowucome¡ flow g¿ is given by equation(13). À detailed description of the si:¡ulation is given ia. thecode in the Appendir.
Parametere
The model is cha¡sctetized by the rariablesV¿* drum volume
V ¡iser volume
V¿. downcomer volumem total metal m¿sgcp specific heat of metalå friction coefficient
¿nd the fi¡nctions e,(p), e-(p), Ir,(p), lr-(p), f,(p), h¡-(n)Í'hich a¡e obtai¡ed f¡om ste"- tables. Quadratic approxirna-tions to the gteam tablee are given in the program \isting ilr theA.ppendir.
Dquilibriurn Conditions
Equiïbrium conditio¡s a¡e obtained from (15). Hence
q!ú = lt, (17)
P = q,(h, - h¡-) (18)
P = q¿.,,h¿ (19)
The equilibrium l,"¿lue of the drum plessure caa be determinedfrom equation (fB) since h, arldþ¡- depend oa the pressu-re.
Dyna:øric Response
Responses to steps in fuel flow ¿rd steam ff.ow a¡e given in. Figures 1 and Figure 2. The simul¡tions illustrate the d¡.namic
:.i featu¡es th¿t a¡e captured by the model. Figure 1 shows the.r ¡espoDse to a etep çïange in fuel flow. The pressure respondsI Iile a pu.re integrator. The total amount of w¿ter in the dru.m
Drum pressure
7.6
02040Water volume
60 80
14.5
13.5
0204060Steam fract¡on mass
80
0.1
0.096
0.092
inc¡eæes because ste¿m is generated in the ¡ise¡s. The totalôrnor¡!.t of steam ir the risers i¡.c¡eases because of the inc¡easedsteam generation. The steam quality in volume ratios i¡creucsiaitially but it will later dec¡eæe because of the compressionelïect.
The drum level inc¡eases rapidly at fust but the ¡¡te ofi:rc¡e¿¡e dec¡e¡ses. The downcomcr fÌow matdres thc stcalnfractiou volume ratio. Thue i¡ an i¡stantaneous inc¡ease ofthe rige¡ flow at the beginning of the step. The ¡ise¡ llow willthen dec¡ease at the same rate ¿s the dow¡.come¡ llow. Figure 2
shows the response to I step change i:r steam fl.ow. The globaieffect is that the pressu¡e and the volume will respond ìike in-tegrato¡s, There will, however, be a swell effect because of theinitial enporation of steam.
5. Conclusions
This paper has presmted si:nple models for a d¡rm boiler sys-tem, The models capture the major dynamical behaviour. Theya¡e derived f¡om fi¡st principles and require only a few physicalparametere that a¡e easily obtained f¡om construction data andsteam t¿bles. The behaviou¡ h¿s been shown by simulating steplesportses to fuel ¿nd steam flow changes. Reasonable ¡esults a¡eobtained even fo¡ the difficult problems of predictiag ci¡culationBow and il¡u:¡. wate¡ level sh¡int and ¡well, The model can etr-ily be augmented by equationr for tu¡bi¡e ud electrical outputgiven in .Ä,st¡ðm a¡d Eklurd (1972, 19?5) o¡ Bell. and Àst¡öm(1979) to produce a simple model fo¡ a compiete boiler-tu¡birealternator system. A strong feature is that the model capturethe essence of the steam genuation in a heated pipe. It has alsobeen used successfully to model eteam generation in a nuclea¡piant. It can also be adapted to model once-through boilers.
1.54 Drum level
1.5
r.460201060
Downcomer & riser flow
EO
1540
1500
1460
0204060Steam fraction vo¡ume
80
0.432
0.428
0.424
'àiir; ,
i
020406080 020406080
Figure 1. Rcrpouo to À ¡tcP i! fud fow.
7.5
7.1
Drum pressure
volume
0204060
020406080
level
20 40 60
Downcomer & riser fow
20 10 60
Steam fraction volume
1.46
t.42
1500
0.43s
0.425
80 0
0
0204060
0.093
0.091
mass
i 60 80
80
80
80020
Figure 2. Rcrponrc lo c ctcp il ¡tcu flow,
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Appendix
6. References
:;.ì¡ti
COITTII{UOUS SYSTEM DBI'Hrrl{o¡Linear thlld otder Eode]. for dro-d.ovaconer-¡ise¡rr.tutho¡ K J ÀBtroE 87OBo5
INPUT pos qly tfv q6OUTPUI dl' ãESTÅTE p Ve xrDER dp dVv dxr
rrPropertiea of stean s,¡d eato! Ln ¡atuletod stato
