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SIMULATION OF A SOLAR HEATED HOUSE
USING THE BOND GRAPH MODELING APPROACH
AND THE DYMOLA MODELING SOFTWARE
by
Spyr os Andr eou
A Thesi s Subm t t ed t o t he Facul t y of t he
DEPARTMENT OF ELECTRI CAL AND COMPUTER ENGI NEERI NG
I n Par t i al Ful f i l l ment of t he Requi r ement sFor t he Degr ee of
MASTER OF SCI ENCEWTH A MAJ OR I N ELECTRI CAL ENGI NEERI NG
I n t he Gr aduat e Col l ege
THE UNI VERSI TY OF ARI ZONA
99
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STATEMENT BY AUTHOR
Thi s t hesi s has been subm t t ed i n par t i al f ul f i l l mentof r equi r ement s f or an advanced degr ee at The Uni ver si t y ofAr i zona and i s deposi t ed i n t he Uni ver si t y Li br ar y t o be madeavai l abl e t o bor rower s under r ul es of t he Li br ar y.
Br i ef quot at i ons f rom t hi s t hesi s ar e al l owabl ew t hout speci al per m ssi on, pr ovi ded t hat accur at e acknow -edgment of sour ce i s made. Request s f or perm ssi on f orext ended quot at i on f r om or r epr oduct i on of t hi s manuscr i pt i nwhol e or i n par t may be gr ant ed by t he head of t he maj or
depar tment or t he Dean of t he Gr aduat e Col l ege when i n hi s orher j udgment t he pr oposed use of t he mat er i al i s i n t hei nt er est s of schol ar shi p. I n al l ot her i nst ances, however ,perm ssi on must be obt ai ned f r om t he aut hor .
SI GNED:
APPROVAL BY THESI S DI RECTOR
Thi s t hesi s has been appr oved on t he dat e shown bel ow:
~~~rt ~Associ at e Prof essor
El ect r i cal and Comput er Engi neer i ng
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ACKNOWLEDGEMENTS
I wi sh t o expr ess my gr at ef ul appr eci at i on t o my
parent s , Savvas and Andromachi , f or t hei r cont i nuous suppor t
t hr oughout my academ c car eer .
Fur ther mor e, I woul d l i ke t o ext end my deepest
appr eci at i on t o my academc advi sor Dr . Fr ancoi s Cel l i er f or
hi s val uabl e assi s t ance t o t hi s pr oj ect . He was al ways
wi l l i ng and avai l abl e f or hel p whenever I r equest ed i t . I
al so thank the other two members of t he exam ni ng comm tt ee,
Dr . Mal ur Sundareshan and Dr . Hal Tharp.
Speci al t hanks t o Qi ngsu Wang f or her assi st ance i n
t he DYMOLA sof t war e. Al so, t o Dr . Gr ani no Kor n f or maki ng
avai l abl e hi s comput er t o obt ai n t he DESI RE out put s. And,
f i nal l y t o al l my f r i ends who cont r i but ed t o t hi s pr oj ect t o
become r eal i t y.
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TABLE OF CONTENTS
LIST OF FIGURES •• • • • • •• ••••• • ••• • 7
ABS TRACT ••• •••••••••••••••••••••••••••••••••••••••••• 11
CHAPTER 1 INTRODUCTION •••••• ••••••••• • 12
CHAPTER 2 BOND GRAPHS ••••••••••••••••••••••••••••••• 16
2 1 Overview 16
2. 2 Basi c Def i ni t i ons . ••. . •••••. . . •. . •. . ••. ••. ••••. 17
2. 2. 1 Mul t i por t El ement s, Por t s, and Bonds •. . . 17
2. 2. 2 Bond Graphs •••. . •. •. . •••. •••••. •. ••. ••. • 19
2. 2. 3 Por t Var i abl es . . . . . . . •. . . . . ••. ••. . . •. . . . 19
2. 2. 4 Basi c Mul t i por t El ements •. . . . . . . ••. . . . . . 20
2 . 2. 5 Ext ended Def i ni t i ons •. •••••••••. •••. •. •• 21
2. 2. 6 General i zat i on t o Basi c Physi calTypes of Syst ems . . . •. . . . . . . . . •. •••. . . . . • 21
2. 3 The Concept of Causal i t y. . . . . . . . . . . . . . . . . . . . . . . 24
2. 4 Pseudo Bond Gr aphs and Thermal Syst ems . •. •. ••. . 27
CHAPTER 3 DYMOLA 30
3 1 Overview 3
3. 2 Speci al Proper t i es of DYMOLA ModelDescriptions 31
3 . 2. 1 Some Pr oper t i es ••. . •. . ••••••. ••. ••. •. . •• 31
3. 2. 2 The Cut Concept . . . . . . •. •••. ••. ••. ••. . . 34
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TABLE OF CONTENTS ( cont i nued)
3. 2. 3 The Submodel Concept and nodes i n DYMOl A ••••••••••••••••••••••. 38
3. 2. 4 Hi er ar chi cal Model st r uct ur e i n DYMOl A. . 41
3. 3 Gener at i on of DESI RE Model s . 44
3. 3. 1 Cr eat i on of a DESI RE Si mul at i onl ?~( ) ~~Cl I n•••••••••••••••••••••••••••••••• 47
3. 3. 1. 1 Descr i pt i on of t he Si mul at i onCont r ol Model . . . . . . . . . . . . . . . . . . 47
3. 3. 1. 2 Obtai ni ng Executabl e DESI REPrograms • • 48
3. 4 Some Unsol ved Probl ems •. . . . •••. ••. ••. •. . •. . •••. 49
CHAPTER 4 CONSTRUCTI ON OF BOND GRAPHS AND THEI RTRANSFORMATI ON I NTO DYMOl A ••••. • . . •. . . • . • . 52
4 1 Overview 52
4. 2 Some Basi c Rul es f or Const r uct i ng BondDi agr ams f or El ect r i cal Networ ks •. •. . •. . •••••. . 53
4. 3 Const ruct i on of a Bond Di agr am of a Si mpl eEl ect r i cal Network • . . •. . . . . . . • . . . •. ••. ••. . . . . . . 54
4. 3. 1 The St ep by St ep Pr ocedur e • . . • . . • . •. . . . . 54
4. 4 Tr ansf ormat i on of Bond Gr aphs i nt oDYMOLA Code ••••••••••••••••••••••••••••••••• •• 60
CHAPTER 5 CASE STUDY: MODELI NG- SI MUl ATI NGA SOLAR- HEATED HOUSE . •. •••. •••••••••••. •. • 71
5 1 Overview 71
5.2 Solar Heating.................................. 72
5. 3 Basi c Thermodynam c and General Concepts •. . . . . . 76
5. 4 Fl at Pl at e Sol ar - Col l ect or Model i ng . . . . . . •. . . . . 79
5. 5 Heat st or age Tank Model i ng 97
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TABLE OF CONTENTS ( cont i nued)
5. 6 Water Loop Model i ng 100
5. 7 Habi tabl e Space Model i ng •••••••••••••••••••. ••• 104
5. 8 The Tot al Sol ar - Heat ed House ••••••••••••••••••• 113
5. 9 Choosi ng Appr opr i at e Par ameter s f or Anal yzi ngt he Ef f ect i veness of Our Syst em ••••••••••••. . •. 114
CHAPTER 6 CONCLUSION •• ••• •••••• •••••• •••••• • 118
APPENDI X RESULTS •••••••••••••••••••• ••••••••••••• 120
REFERENCES 121
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Fi gur e 2. 1a
Fi gur e 2. 1b
Fi gur e 2. 1c
Fi gur e 2. 2a
Fi gur e 2. 2b
Fi gur e 2. 3
Fi gur e 2. 4
Fi gur e 2. 5
Fi gur e 2. 6
Fi gur e 2. 7
Fi gur e 2. 8a
Fi gur e 2. 8b
Fi gur e 3. 1
Fi gur e 3. 2
Fi gur e 3. 3
Fi gur e 3. 4
Fi gur e 3. 5
Fi gur e 3. 6
7
LIST OF FIGURES
Mul t i por t el ement s 18
The el ement s and t hei r por t s •••••••••••• 18
Format i on of a bond ••••••••••••••••••••• 18
Bond graph 18
A bond 18
The bond gr aph w t h power s di r ect edand bonds l abel ed . . . . . . . . . . . . . . . . •. . •. •. 19
Def i ni t i ons of t he basi c mul t i por tel ements 22
Pr esent at i on of a summar y of t hef our gener i c var i abl es bei ng usedi n some common physi cal syst ems . . . •••. •. 23
Meani ng of causal st rokes . •••••. . . . . . . . . 25
Desi r ed causal f orms and r el at i onsof t he basi c ni ne mul t i por t el ement s 26
Thermal r esi s t or and i - j unct i on •. •. . •. . . 28
Ther mal capaci t or and O- j unct i on . ••. •. •. 28
Model of a conduct ance usi ng i nputout put decl ar at i on •••••. •••. ••. •. •••. ••. 35
Model of a conduct ance usi ng cutdeclaration 35
Thr ee submodel s connect ed at por t A 37
Exampl es of at om c model s i n DYMOLA 39
Exampl es of a coupl ed model pr c 40
Coupl ed model s i n DYMOLA •••. . . . . . . . . . ••. 42
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Fi gur e 3. 7
Fi gur e 3. 8
Fi gur e 3. 9
Fi gur e 4. 1
Fi gur e 4. 2
Fi gur e 4. 3
Fi gur e 4. 4
Fi gur e 4. 5
Fi gur e 4. 6
Fi gur e 4. 7
Fi gur e 4. 8
Fi gur e 4. 9
Fi gur e 4. 10
Fi gur e 4. 11
Fi gur e 4. 12
Fi gur e 4. 13
Fi gur e 4. 14
8
LIST OF FIGURES ( cont i nued)
A hi er ar chi cal l y st r uct ur ed syst em 43
Descr i pt i on of t he hi er ar chi calst r uct ur e of a syst em i n DYMOLA 45
Model speci f i cat i on f or pr c usi ngmodel type ............... 46
An el ect r i cal net wor k w t h nodesl abel l ed ( r
r ef er ence) ••••••. . •. •••••• 55
Layout of vol t age j unct i ons( a- j unct i ons) . ••••. . •••. . . . . . . . •. . . . . . . . 55
The assembl y of component s and sour ce 56
The cancel l at i on of r ef er ence nodeand associ at ed bonds •••••. . . . •. . •. . . . . . • 56
The condensat i on of bonds . . . . . . •. . . . . . •. 58
The r educed gr aph •. . •••••. •. . . . . •. . . . . . • 58
The bond graph 58
The compl et ed bond gr aph w t hi t s causal i t i es •••••••••. •••. . •••. . . ••. • 59
DYMOLA expanded bond graph w th eachnode i ndi cat ed . . ••. . •••. ••. . •. •••. •••. •. 61
DYMOLA code of the bond graph shownon Figure 4 9 63
The var i ous basi c DYMOLA model t ypes . . . . 64
Exper i ment used f or t he net wor k •. •. . •. . • 65
Gener at ed DESI RE pr ogr am •••••. ••. •. . •. . • 67
St at e- s pace r epr esent at i on of t henetwork 68
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Fi gur e 4. 15
Fi gur e 5. 1
Fi gur e 5. 2
Fi gur e 5. 3a
Fi gur e 5. 3b
Fi gur e 5. 3c
Fi gur e 5. 3d
Fi gur e 5. 3e
Fi gur e 5. 3f
Fi gur e 5. 3g
Fi gur e 5. 3h
Fi gur e 5. 3i
Fi gur e 5. 3j
Fi gur e 5. 3k
Fi gur e 5. 31
Fi gur e 5. 3m
Fi gur e 5. 3n
Fi gur e 5. 30
Fi gur e 5. 3p
Fi gur e 5. 4a
Fi gur e 5. 4b
Fi gur e 5. 4c
LIST OF FIGURES ( cont i nued)
DESI RE out put
A sol ar heat ed house ••••••••••••••••••••
Model of a f l at - pl at e col l ect or •••••••••
Bond di agr am of a one- di mensi onal cel l
DYMOLA model t ype of a one- di mensi onal
9
69
74
8
83
cell 83
84odul ated conduct i ve sour ce ••. ••••. •. •. •
Bond di agr am of a heat exchanger .
DYMOLA model t ype of a heat exchanger . . .
Bond di agr am of a wat er spi r al •. . . •. ••••
DYMOLA model t ype of a wat er spi r al •. •. .
Bond di agr am of t her m c l oss
DYMOLA model t ype of therm c loss .
Bond gr aph of t he col l ect or . ••. •. ••. . . •.
DYMOLA model t ype of t he col l ect or . •. . •.
DYMOLA model t ype of mG •. ••. . . . •. ••. •. •.
DYMOLA model t ype of mGS
DYMOLA model t ype of mRS
DYMOLA model type of mC
DYMOLA model t ype of RS
85
85
87
88
89
89
91
91
95
95
96
96
96
The st or age t ank w t h t he col l ect orwat er l oop and heat er wat er l oop 99
Bond gr aph of t he st or age t ank . . •. . . . •. . 99
DYMOLA model t ype of t he st or age t ank . . . 1 1
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Fi gur e 5. 5a
Fi gur e 5. 5b
Fi gur e 5. 6a
Fi gur e 5. 6b
Fi gur e 5. 6c
Fi gur e 5. 6d
Fi gur e 5. 7
Fi gur e 5. 8
LIST OF FIGURES ( cont i nued)
Bond di agr am f or wat er l oop
DYMOLA model t ype f or wat er l oop
Thr ee- di mensi onal di f f usi on cel l
DYMOLA model t ype of a t hree- di mens i onal
10
102
102
107
cell 108
The house r oom r epr esent ed as a10X10X10 cube •. ••••. . •••. ••••. . •. •. . •. . . 109
DYMOLA model t ype of t he SPACE ( house)
Aggr egat ed bond gr aph of t he over al l
110
Tabl e of some r esul t s
system • • • • 113
116
Si mul at i on r esul t s at var i ous nodes . . . . • 120
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ABSTRACT
Thi s t hesi s di scusse~ t he appl i cat i on of t he bond
gr aph model i ng t echni que di r ect l y coded i nt o t he Dynam c
Model i ng Language ( DYMOLA) f or si mul at i ng a sol ar - heat ed
house. Sci ent i st s t hr oughout t he year s have i nvest i gat ed t he
expl oi t at i on of sol ar r adi at i on f or space heat i ng. I n t hi s
t hesi s , t he physi cal behavi or of such a syst em i s model ed
and si mul at ed i n a conveni ent , r obust and f ast manner . The
bond gr aph model i ng met hodol ogy has f ound w despr ead use i n
a w de r ange of syst ems. DYMOLA i s a model i ng l anguage wel l
sui t ed t o r epr esent bond gr aphs. DYMOLA i s a pr ogr am gener a-
t or t hat can map a t opol ogi cal syst em descr i pt i on, such as a
bond gr aph, i nt o a st at e- space descr i pt i on expr essed i n t he
f orm of a DESI RE si mul at i on pr ogr am
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CHAPTER 1
INTRODUCTION
Engi neer s ar e t r ai ned i n i nvent i ng new means whi ch
w l l event ual l y l ead t o easi er sol ut i ons of t hei r pr obl ems.
One of t hem i s t o model and t hen t o si mul at e a cer t ai n number
of physi cal syst ems encount er ed i n t hei r ever yday l i f e w t h
t he mai n t ar get bei ng t o f i r st pr edi ct and secondl y t o st udy
t hei r physi cal behavi or .
The goal of t hi s t hesi s i s t o pr esent a moder n and
advanced model i ng- s i mul at i on t echni que appl i ed t o a sol ar -
heat ed house. The bond gr aph model i ng t echni que as wel l as
t he Dynamc Model i ng Language (DYMOLA) w l l be used.
Ther e exi st a number of bond gr aph model i ng t ool s on
t he . mar ket . The best est abl i shed t ool i s ENPORT- 7 (Rosencode
Associ at es I nc. , 1989) , a SPI CE- l i ke bond gr aph l anguage w t h
a gr aphi cal f r ont end. Ot her t ool s ar e TUTSI M ( van Di xhoor n,
1982) and CAMP (Granda, 1982) . However , none of these systems
i s abl e t o handl e t rul y hi er ar chi cal bond gr aphs as t hey w l l
be essent i al f or our endeavor . DYMOLA ( El mqvi st , 1978) i s t he
onl y model i ng l anguage avai l abl e whi ch can handl e t rul y
hi er ar chi cal nonl i near bond gr aphs i n a compl et el y gener al
f ashi on.
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I t i s at t r act i ve t o many engi neer s t o st udy t he
possi bi l i t y of expl oi t i ng t he f r eel y avai l abl e sol ar r adi a-
t i on f or heat i ng a house. For . a successf ul desi gn of such a
f aci l i t y, i t i s essent i al t hat t he syst em behavi or can be
si mul at ed so t hat var i ous al t er nat i ves can be t est ed pr i or t o
i mpl ement at i on. DYMOLA t oget her wi t h t he bond gr aph appr oach
t o physi cal syst em model i ng i s expect ed t o be t he qui ckest
and most accur at e method compar ed wi t h ot her s used i n t he
past t o descr i be such a syst em Bond gr aphs wer e i nvent ed i n
1960 by Henry Paynter , an M T prof essor ( Paynter , 1961) , and
DYMOLA was desi gned at t he Lund I nst i t ut e of Technol ogy i n
1979 by Hi l di ng El mqvi st i n hi s Ph. D. di sser t at i on (El mqvi st ,
1978) . However , t he appl i cat i on of DYMOLA to expr ess bond
graphs i s new and has never been done bef ore.
Bond gr aphs f i nd many appl i cat i ons i n var i ous
engi neer i ng di sci pl i nes because t hey make model i ng more
syst emat i c, because t hey make i t easi er t o deal wi t h
i nt er f aces bet ween subsyst ems of di f f er ent t ypes ( e. g. ,
el ect r o- mechani cal coupl er s ) , and because they si mpl i f y the
ver i f i cat i on of a cor r ect ener gy f l ow acr oss such i nt er f aces
and wi thi n the sUbsys tems. They are abl e to provi de a common
model i ng met hodol ogy not onl y f or el ect r i cal , mechani cal and
ot her f r equent l y si mul at ed syst ems, but al so f or l ess
commonl y si mul ated systems such as chem cal , ecol ogi cal or
bi omedi cal syst ems. They of f er a mor e gener al gr aphi cal
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r epr esent at i on t han ei t her bl ock di agr ams or si gnal f l ow
gr aphs si nce t hey pr eser ve bot h t he comput at i onal and t he
topol ogi cal st r ucture of al l t he systems ment i oned above. As
t he wor d i ndi cat es, a bond gr aph i s a col l ect i on of el ement s
bonded t oget her . More i nf ormat i on about t hi s uni que model i ng
techni que i s provi ded i n the second chapter .
Af t er model i ng our sol ar - heat ed house i nt o bond
gr aphs, t he pr oduced di agr ams ar e di r ect l y coded i nt o DYMOLA,
a modul ar hi er ar chi cal cont i nuous- system model i ng l anguage.
I t s mai n advant age i s t hat i t can deal wi t h l ar ge- scal e
syst ems i n a modul ar and hi er ar chi cal manner . Mor eover , i t
i s ver y wel l sui t ed t o i mpl ement t he bond gr aph model i ng
methodol ogy, and i s abl e to map bond graphs i nto s tate- space
descr i pt i ons of t he t ype ~ = i ( ~, y, t ) . speci al f eat ur es of
DYMOLA are f ound i n the thi r d chapter .
The mai n subj ect of t he f our t h chapt er i s a
demonst rat i on of t he way i n whi ch DYMOLA can be used t o sol ve
t he pr esent ed pr obl em The t r ansi t i on f r om the bond di agr am
t o DYMOLA code i s a st r ai ght f or war d pr ocedur e r equi r i ng
sever al si mpl e r ul es bei ng pr esent ed i n a concr et e and
succi nct manner . DYMOLA i s so power f ul t hat i t can aut o-
mat i cal l y eval uat e t he causal i t y of t he bond gr aph, pr oduce
a st at e- space descr i pt i on f or t he syst em as wel l as gener at e
a si mul at i on program coded i n ei ther DESI RE (Korn, 1989b) or
SI MNON (El mqvi st , 1975) , two di r ect execut i ng cont i nuous-
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system si mul at i on l anguages. Moreover , a si mpl e el ect r i cal
networ k i s i ncl uded, t r ansf ormed f i r st i nt o i t s bond di agr am
and then i nto DYMOLA code, wi th the hope that the reader wi l l
f ol l ow and comprehend al l t he present ed steps i n a conveni ent
manner .
As ment i oned bef ore, t he case study present ed i n thi s
t hesi s i s a sol ar - heat ed house, a r el at i vel y compl i cat ed
syst em i nvol vi ng var i ous subsyst ems and var i ous t ypes of
ener gy. The conf i gur at i on under st udy consi st s of a f l at -
pl at e sol ar col l ect or , one sol i d body st or age t ank, wat er
l oops, a heat exchanger , and t he habi t abl e space. Each par t
i s gover ned by a set of f i r st or der di f f er ent i al equat i ons
i l l ust r at i ng t he ener gy f l ow t hr ough t he subsyst em Each
subsyst em i s di r ect l y t r ansf ormed i nt o a bond gr aph
r epr esent at i on. The var i ous par amet er s used f or t he
si mul at i on wer e t aken f r oman ol der st udy of a si m l ar sol ar -
heat ed house per f ormed i n t he l at e 70 s ( Kass, 1978) , f r om
other sour ces i n t he l i t er at ur e (Def f i e and Bechman, 1980)
and f romusi ng our physi cal i nt ui t i on and common sense.
I t i s hoped t o have t he oppor t uni t y t o appl y bot h t he
bond gr aph model i ng t echni que and t he dynamc model i ng
l anguage i n i ndust r y obser vi ng t he physi cal pr oper t i es of
var i ous syst ems. Bei ng abl e t o t r ansl at e t hem i nt o bond
di agrams and then code t hemdi rect l y i nt o DYMOLA i s , i ndeed,
an exci t i ng exper i ence.
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CHAPTER 2
BOND GRAPHS
I n t hi s chapt er t he Bond Gr aph met hodol ogy i s
di scussed ext ensi vel y. I t st ar t s w t h an over vi ew of t hi s
uni que model i ng t echni que, t hen i t gi ves some basi c def i ni -
t i ons w t h i l l ust r at i ons and i t di scusses t he concept of
causal i t y. Fur t hermor e, a r ef er ence t o Pseudo Bond Gr aphs and
Thermal Systems i s gi ven.
2 1 ov er v i ew
Engi neer s needed t o f i nd a mor e gener al gr aphi cal
( symbol i c) r epr esent at i on whi ch at t empt s t o pr eser ve bot h t he
comput at i onal and t opol ogi cal st r uct ur e of any ki nd of
physi cal syst em They f ound out t hat bl ock di agr ams and
si gnal f l ow gr aphs onl y pr eser ve t he comput at i onal but not
t he t opol ogi cal st r uct ur e. Thus, a r el at i vel y new and
power f ul r epr esent at i on i s t hat of Bond Gr aphs whi ch has been
i nt roduced by Henr y Paynt er i n t he ear l y si xt i es ( Paynt er ,
1961) . Many t ypes of physi cal syst ems have been st udi ed
usi ng bond gr aphs i ncl udi ng el ect r i cal networ ks, mechani cal
r i gi d bodi es, hydr aul i c, t her mal and ener gy t ransduct i on
phenomena. Some r esear cher s r ef er t o Bond Gr aphs al so as Bond
Di agr ams. We shal l use bot h t ermnol ogi es i nt er changeabl y.
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I t i s t r ue, however , t hat f or t he begi nner t hi s
model i ng l anguage i s qui t e abst ract . Bl ock di agr ams and
si gnal f l ow di agr ams can be mor e easi l y compr ehended.
Never t hel ess, f or t he case of model i ng t he sol ar - house, a
r el at i vel y compl i cat ed syst em i nvol vi ng many di f f er ent t ypes
of ener gy f l ow bet ween i t s i nt er connect ed par t s, i t appear s
t hat t he bond gr aph pr ocedur e i s mor e appeal i ng due t o i t s
ease of appl i cat i on and gr eat er i nf ormat i on cont ent .
Model i ng a physi cal syst em i s a si mpl i f i ed abst ract
const ruct i on used t o pr edi ct i t s physi cal behavi or . That i s
exact l y what t he bond gr aph model i ng met hodol ogy i s
per f orm ng.
The pur pose of t hi s chapt er i s t o i nt r oduce t he
r eader t o t hi s abst r act model i ng met hodol ogy and t o pr ovi de
enough i nf ormat i on so t hat he/ she can easi l y compr ehend i t .
2. 2 Basi c Def i ni t i ons
2. 2. 1 Mu1t i por t El ement s, Por t s, and Bonds
The nodes of t he gr aph ar e cal l ed Mul t i por t El ement s
desi gnat ed by al pha- numer i c char act er s such as 1 and R, as
shown i n Fi gur e 2. 1( a) . The pl aces wher e a mul t i por t el ement
can i nt er act w t h i t s envi r onment ar e cal l ed Por t s desi gnat ed
by l i ne segment s i nci dent on t he el ement at one end. Fi gur e
2. 1 ( b) shows t he 1 el ement havi ng t hr ee por t s and t he R
el ement havi ng one por t . When pai r s of por t s ar e combi ned
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1 R R
I
I C
R
a b )
Fi gur e 2. 1 ( a) Mul t i por t el ement s( b) The el ement s and t hei r por t s
( c) For mat i on of a bond
R e
SE OTF 1 ~
I If
C I a) b )
Fi g~r e 2. 2 (a) Bond graph( b) A bond
134SE~O~TF~
2~
C
R
~6
I
Fi gur e 2. 3 The Bond Gr aph w t h power s di r ect edand bonds l abel ed
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t oget her , bonds ar e f ormed. Thus, bonds ar e connect i ons
between pai r s of mul t i por t el ement s. For exampl e, Fi gur e
2. 1( c) shows a f ormat i on of t he bond between 1 and R.
2. 2. 2 Bond Graphs
A bond gr aph i s a col l ect i on of mul t i por t el ement s
bonded t oget her . I n a more gener al per spect i ve i t i s a l i near
gr aph wi t h nodes bei ng t he mul t i por t el ement s and wi t h
branches bei ng t he bonds. An exampl e of a bond graph i s shown
i n Fi gur e 2. 2( a) havi ng seven mul t i por t el ement s and si x
bonds.
Anot her def i ni t i on:
A bond, r epr esent ed by a bol d hal f ar r ow, i s not hi ng
but a connector that si mul taneousl y connects two var i abl es,
one acr oss var i abl e, i n bond gr aph t er mnol ogy usual l y
r ef er red t o as t he ef f or t e, and one t hr ough var i abl e,
cal l ed t he f l ow f ( Cel l i er , 1990a) . Ref er t o Fi gur e 2. 2( b)
as wel l as t o t he next sUbsect i on f or mor e i nf ormat i on.
2. 2. 3 Por t var i abl es
Ther e ar e t hr ee di r ect and t hr ee i nt egr al quant i t i es
associ at ed wi t h a gi ven por t .
The f i r st t wo di r ect quant i t i es ar e cal l ed Ef f or t ,
e( t ) , one acr oss var i abl e and Fl ow, f ( t ) , one t hr ough
var i abl e, assumed t o be scal ar f unct i ons of an i ndependent
var i abl e ( t ) .
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J unct i ons . The t wo sources , t he t wo storages ( capaci t ance and
i ner tance) and t he di ssi pat i on ( resi st ance) ar e I - por t
el ement s wher eas t wo of t he j unct i ons ( t r ansf or mer and
gyr at or ) ar e 2- por t ones and t he ot her t wo ( 0 and 1) ar e at
l east 3- por t el ement s. The f ol l owi ng Fi gur e 2. 4 shows t he
symboI , def i ni t i on and name of t he ni ne basi c mul t i por t
el ement s. I n t he f i gur e, ~ st ands f or a gener al f unct i on
rel at i ng two var i abl es.
2. 2. 5 Extended Def i ni t i ons
Al though the f ol l owi ng f eatures are beyond the scope
of t hi s t hesi s t hey ar e wor t h ment i oni ng. The t erm Fi el d i s
al so used i n bond gr aph t erm nol ogy. Thus, t her e ar e
C- f i el ds, I - f i el ds and R- f i el ds whi ch ar e mul t i por t
gener al i zat i ons of - C, - I and - R r espect i vel y. Mor eover ,
t her e ar e t he Modul at ed Tr ansf ormer (MTF) and Modul at ed
Gyr at or ( MGY) .
Lat er i n t he f i f t h chapt er , when t he bond gr aph of a
t hr ee- di mensi onal cel l i s const r uct ed t he R- f i el d i s used
( t hr ee r esi st or s ar e connect ed i n x, y, z di r ect i ons, see
Fi gure 5. 6a) .
2. 2. 6 Gener al i zat i on t o Basi c Physi cal Types of Syst ems
We have al r eady seen f our gener i c var i abl es ef f or t ,
f l ow, momentum and di spl acement . The f ol l owi ng Fi gure 2. 5
demonstr at es a pr esent at i on summar i z i ng t he above f our
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SYMBOL DEFINITION N ME
S Ee ,
e = e t) s ou rc e o f e ffo rt
S Ff
f = f t) s o u rc e o f flo w
Ce e = < 1>q )
d
c a p a c i t a n c ef q t) = q to ) J fd t
e f = < 1 > p )d
i n e r t a n c ef p t) = p to ) +J e d t
Re
<1>e ,f) = 0d
r e s i s t a n c ef
1 2 e 1= m e 2T F
t r a n s f o r m e rf2 = m f 1m
1 2 e 1 = r f 2
G Y
g y r a t o re 2 rf 1r
1 3 e 1 = e 2 = e 3 c om m o n e ffo rt 0
f 1 + f 2 -f3 = 0 j u n c tio n
t 21 3 f1 = f2 = f3 c o m m o n flo w
e 1 e 2 - e 3 = 0 j u n c t i o n
t 2Fi gur e 2. 4 Def i ni t i ons of t he basi c mul t i por t el ement s
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Efforte Flowf
Generalized Generalized
Momentum Displacementp q
E l e c t r i c a l v O l t ~ e c u r re n t flu x c h a r g eu [ ] i[ A ] c l > [ V s ] q [ A s ]
T r a n s l a t i o n a lf o r c e v e l o c i t y m o m e n t u m d i s p l a c e m e n t
F [ N ] . u [ m s ] I [N s ] x [m ]
t o r q u ea n g u l a r
t w i s t a n g l eR o t a t i o n a l v e l o c i t yT [ N m ]
Q[ ra d s · 1 ] t[ N m s ] e [ra d ]
p r e s s u r e v o lu m e p re s s u r ev o l u m e
H y d r a u l i c P [ N m -2] f lo w m o m e n t u mv [m 3]c l> v[ m 3 s 1 ] IlN m -2s ]
c h e m i c a l m o la r flo w m o la r m a ssC h e m i c a l p o t e n t i a l Q N [ m o le s - 1 ] N [m o l]
u ] J e m o l - 1 ] d t
T h e r m o - t e m p e r a t u r e e n tro p y flo w e n t r o p y
d y n a m i c a l T [ O K ] ~ [W O K 1 ] S [ J e O K 1 ]d t
Fi gur e 2. 5 Pr esent at i on of a summar y of t he f our gener i cvar i abl es bei ng used i n some common physi cal syst ems
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gener i c var i abl es bei ng used t o t he most common physi cal
system t ypes.
2. 3 The Concept of causal i t y
Bond gr aphs have t he pr oper t y of pr eser vi ng t he
t opol ogi cal as wel l as t he comput at i onal st ruct ur e of a
syst em When, f or exampl e, a gi ven el ect r i cal syst em i s
t r ansf ormed i nt o bond gr aphs i t s t opol ogi cal st r uct ur e i s
qui te evi dent to the reader . Never t hel ess, i t s computat i onal
st r uct ur e cannot be seen easi l y. Thus, t he i nt r oduct i on of
bond graph causal i t y comes i nt o account .
We say t hat i n bond gr aphs i nput s and out put s ar e
speci f i ed by means of t he causal st r oke. I t i s a shor t
per pendi cul ar l i ne made at one end of a bond or por t l i ne. I t
i ndi cat es t he di r ect i on i n whi ch t he ef f or t si gnal i s
di r ect ed, i mpl yi ng t hat t he ot her end whi ch does not have a
causal st r oke i s t he one t hat t he f l ow si gnal ar row poi nt s.
Fi gur es 2. 6( a) and 2. 6( b) i l l ust r at e succi nct l y t he meani ng
of causal i t y ( causal st r oke) .
The f ol l owi ng Fi gur e 2. 7 shows t he ni ne mul t i par t
el ement s wi t h t hei r desi r ed causal f orms and r el at i ons. I t
i s wor t hwhi l e sayi ng t hat f or r esi st ance bot h causal f orms
( as shown) ar e physi cal l y and comput at i onal l y possi bl e.
However , f or capaci t ances and i nert ances we woul d rat her pi ck
t he causal i t i es t hat numer i cal l y i nt egr at e over al l st at e
var i abl es.
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ELEMENT
E ff o rt S ou rc e
F lo w S ou rc e
R e s i s t a n c e
C a p a c it a n c e
In e r t a n c e
T r a n s f o r m e r
G y r a t o r
0 - J u n c tio n
1 - J un ctio n
US LFORM
S E > I
S F I >
- - - - > • . . .1 R
R I >
->-C
> ~ I
I1 > T F I 2 >
1 > J T F 2 > I
I 1 > G Y 2 > I
1 > I G Y I 2 >
1~ I
3
26
US LREL T I ON
e t) = E t}
f t) = F t}
f = C l> ~ e )
e = C l> R Q
e = C l > d J f d t )
1 = C l > ; U e d t }
e 1= m e 2 , f2 = m f1
f1 = f 2 /m e 2 = e 1 1 m
e 1= rf2 e 2 = rf111= e 2 / r 12 = e 1 / r
e 2 = e l , e 3 = e 1
1 1 = - f 2 + f 3 }
12 = 11 . 13 = 11
e l = - e 2 + e 3 )
Fi gur e 2. 7 Desi r ed causal f or ms and r el at i ons of t he basi cni ne mul t i por t el ement s
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2 4 Pseudo Bond Graphs and Thermal Systems
Because of t he f act t hat t he sol ar - heat ed house i s a
t her mal syst em i t i s t i me t o i nt r oduce some bond- gr aph
representat i ons f or such a thermal system Thermal systems
have been pr esent ed as anal ogous t o el ect r i cal syst ems,
usual l y wi th temperature anal ogous to vol t age and heat f l ow
anal ogous t o cur rent . wi t h t hi s anal ogy i n m nd we have
sources anal ogous to vol t age and cur rent sour ces, thermal
resi stor s and capaci t or s, and a and 1 j unct i ons. However ,
there are no thermal i ner t i as ( i ner t ances) .
Ther e i s one maj or obst acl e. The pr oduct of
t emperat ure and heat f l ow i sn t power . Heat f l ow i s by i t sel f
a power . Engi neer s, t hen, deci ded to name such a bond graph
i n whi ch t he pr oduct of ef f or t and f l ow i sn t power a pseudo
bond gr aph. As l ong as the basi c el ement s i n t he pseudo bond
gr aph ar e cor r ect l y r el at ed t o t he e, f , p, and q var i abl es,
t he r ul es f or t he r egul ar bond gr aph t echni que can be
usef ul l y appl i ed. The t r ue bond gr aph r esul t s ( see Fi gur e
2. 8) , i f t emperatur e and ent r opy f l ow are used as ef f or t and
f l ow var i abl es r espect i vel y. I ndeed, t he pr oduct of
t emper at ure and ent ropy f l ow i s power .
The f ol l owi ng Fi gure 2. 8 shows a thermal resi stor and
I - j unct i on as wel l as a t her mal capaci t or and a- j unct i on
whi ch ar e goi ng t o be used i n t he f i f t h chapt er dur i ng t he
model i ng procedure of t he sol ar house.
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Despi t e t he f act t hat i n t he l i t er at ur e pseudo- bond gr aphs ar e
mor e popul ar t han t he t r ue- bond gr aphs i n model i ng t her mal
syst ems, i t may be ar gued t hat usi ng t he l at t er ones w l l be
mor e appr opr i at e f or model i ng t he sol ar house. Tr ue- bond gr aphs
ar e bet t er sui t ed t o r epr esent t he ener gy f l ow acr oss a
j unct i on t o and f r om ot her t ypes of ener gy, such as mechani cal ,
el ect r i cal , hydr aul i c, or pneumat i c. Thus, as shown i n t he
pr evi ous f i gur e, t emper at ur e ( T) woul d be t he ef f or t var i abl e
and ent ropy f l ow ( S ) woul d be t he f l ow var i abl e.
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Present l y, DYMOLA suppor t s DESI RE, SI MNON and FORTRAN and i t
woul d not be di f f i cul t t o enhance i t t o suppor t ot her
l anguages, such as ACSL, as w~ l .
DYMOLA uses t wo concept s: t he submodel concept as
wel l as t he cut concept . These w l l be cl ar i f i ed l at er i n
thi s chapter .
Ther e exi st cur r ent l y two di f f er ent i mpl ement at i ons
of DYMOLA, one coded i n PASCAL and t he ot her coded i n SI MULA.
The f i r st one r uns on VAX/ VMS and on PC compat i bl es, whi l e
t he l at t er r uns on UNI VAC comput er s.
3. 2 Speci al Pr oper t i es of DYMOLA Model Descr i pt i ons
3. 2. 1 Some Pr oper t i es
The f ol l ow ng ar e pr oper t i es of a DYMOLA model . Some
ar e quot ed di r ect l y f rom Cel l i er s book ( Cel l i er , 1990a) ,
others are paraphrased:
( 1) DYMOLA var i abl es can be of t wo t ypes: t he
t erm nal t ype and t he l ocal t ype. I f t hey ar e
connect ed t o somet hi ng out si de t he model , t hey
w l l be of t he t erm nal t ype; ot herw se, t hey
w l l be of t he l ocal t ype ( connect ed i nsi de t he
model ) .
( 2) Term nal s m ght be ei t her i nput s or out put s,
f r equent l y dependi ng on t he sur r oundi ngs t o
whi ch t hey ar e connect ed. The user has t he r i ght
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t o decl ar e t hem t he way he want s t hem t o be by
expl i ci t l y speci f yi ng i nput or out put .
( 3) DYMOLA const ant s can be of t he par amet er t ype i f
t he user w shes t o do so. Par amet er val ues can
be assi gned f r om out si de t he model , but t hey can
al t er nat i vel y al so assume def aul t val ues.
( 4) Ter mnal s can have def aul t val ues. I n t hi s way,
t hey don t need t o be ext er nal l y connect ed
(Cel l i er , 1990a) .
( 5) The f i r st t i me der i vat i ve of st at e var i abl e x
can be expr essed i n t wo ways, ei t her t hr ough
der ( x) or t hr ough x . Second der i vat i ves can be
wr i t t en as ei t her der 2( x) or x .
( 6) The user cannot set i ni t i al condi t i ons f or t he
i nt egr at or s i nsi de a model , show ng cl ear l y a
f l aw of DYMOLA.
( 7) The synt ax expr essi on = expr essi on i s used i n
DYMOLA equat i ons, bei ng sol ved f or t he pr oper
var i abl e dur i ng t he pr ocess of a model
expansi on. DYMOLA accept s t he f act t hat t he l ef t
hand si de of an equat i on can have der
( t emperature) , whi l e t emperature appear s on the
l ef t hand si de of anot her .
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( 8) When mul t i pl yi ng t erms by a zer o par amet er , t hey
ar e aut omat i cal l y el i m nat ed dur i ng a model
expansi on. For exampl e, i f we have
La = 0. 0
and the model equat i on
La * der ( i a) ua - ui - Ra * i a
t hen t he above i s r epl aced by
0. 0 ua - ui - Ra * i a
3.1)
(3.2)
r esul t i ng i n t he f ol l ow ng t hr ee
s i mul at i on equat i ons:
( a) ua ui Ra * i a ( 3. 4)
( b) ui ua - Ra * i a ( 3. 5)
( c) i a = ( uo - ui ) / Ra ( 3. 6)
dependi ng on t he envi r onment i n whi ch t he model
i s used.
3.3)
possi bl e
I f La ~ 0. 0, t hen t he model equat i on i s al ways
t r ansf ormed i nto
der ( i a) = ( ua - ui - Ra * i a) / La ( 3. 7)
( 9) The above r ul e i ndi cat es t hat par amet er s w t h
val ue 0. 0 ar e t r eat ed i n a compl et el y di f f er ent
manner t han al l ot her par amet er s ( Cel l i er ,
1990a) . Par amet er s whi ch ar e not equal t o zer o
ar e mai nt ai ned i n t he gener at ed si mul at i on code,
wher eas t he ones w th 0. 0 val ue ar e not
r epr esent ed i n t he si mul at i on code.
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( 10) DYMOLA model s ar e modul ar because t he equat i ons
can aut omat i cal l y be sol ved dur i ng model
expansi on.
3 2 2 The Cut Concept
When advanci ng to hi gher l evel s of t he hi erarchy, t he
number of t he par amet er s wi l l be gr owi ng. Si m l ar t o r eal
syst ems wher e wi r es ar e gr ouped i nt o cabl es and cabl es ar e
grouped i nto t r unks, t he concept of cut has been i nt r oduced
i n DYMOLA t o gr oup var i abl es t oget her . Cut s cor r espond t o
compl ex connect i on mechani sms of physi cal syst ems l i ke
el ect r i cal wi r es, pi pes and shaf t s . A more pr eci se def i ni t i on
i s t he f ol l owi ng: Cut s ar e hi er ar chi cal dat a st r uct ur es t hat
enabl e t he user t o gr oup i ndi vi dual wi r es i nt o buses or
cabl es and cabl es i nt o t r unks. A cut i s l i ke a pl ug or a
socket . I t def i nes an i nt er f ace t o t he out si de wor l d
( Cel l i er , 1990b) .
The f ol l owi ng two f i gur es, 3. 1 and 3. 2, show a model
of a conduct ance ( i nver se of r esi st or ) i l l ust rat i ng t he
di f f er ent model descr i pt i ons bef or e and af t er usi ng t he
concept of cut . Thi s exampl e demonst r ates how a cont i nuous
model achi eves modul ar i t y.
wi t h cut decl ar at i ons the i nput and output var i abl es
do not change t he model descr i pt i on by swi t chi ng t hem
Never t hel ess, t he mai n advant age of cut i s t he two t ypes of
var i abl es, t he across and through var i abl es wi t h whi ch there
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m o de l n am e : c on du cta nc einpu t: Iou tpu t: Vp ara m ete r: Ge qu atio ns : V I /G
o r m ode l nam e: conduc tanceinpu t: Vou tpu t: Ip ara m ete r: G
e qua tio ns : I = V • G
Fi gur e 3. 1 Model of a conduct ance usi ng i nput out putdecl ar at i on
m o de l n am e : c on du cta ncecu t: A Va I I 8 V b I I
lo ca l: Vp ara m ete r: Ge qu atio ns : V = V a -V b
V = I I G
Fi gur e 3. 2 Model of a conduct ance usi ng cut decl ar at i on
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i s associ at ed as i n t he r eal physi cal wor l d a connect i on
mechani sm The equat i ons whi ch descr i be t he phys i cal l aws at
t he connect i on mechani sm are aut omat i cal l y generat ed by t he
decl arat i on of cut and the connect i on statements .
Consi der t he f ol l owi ng exampl e: Thr ee submodel s
def i ned as GH Gz and G 3 have A and B as t hei r cut
var i abl es. V« and I ar e t he acr oss var i abl e and t he t hr ough
var i abl e associ at i ng wi th cut A, respect i vel y, bei ng decl ared
as ( see Fi gur e 3. 3)
cut A (V«/I)
Usi ng t he connect statement
connect G1:A at Gz:A at G3:A,
t he f ol l owi ng equat i ons are aut omat i cal l y generat ed:
GpV« Gz.V« (3.8)
Gz•V« G3• V« (3 • 9 )
Gp I + Gz• I + G3• I = 0 (3. 10)
The above equat i ons descr i be what exact l y happens at t he
boundar y of t he subsyst em wher e two or mor e el ement s ar e
connect ed. Ther eby, al l acr oss var i abl es ( t o t he l ef t of t he
sl ash separ at or ) ar e set equal , and al l t he t hr ough var i abl es
( t o t he r i ght of t he sl ash oper at or ) ar e summed up t o zer o
( Cel l i er , 1990b) .
When sever al cut s ar e gr ouped t oget her , a
hi er ar chi cal cut i s f or med i n t he same way as i ndi vi dual
wi r es ar e gr ouped t oget her i nt o a cabl e.
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V a G 1 V b i t B
B•
I
V a G 2 V b i t B
I
V a G 3 V b i t B B B
•I
a b
Fi gur e 3. 3 Thr ee submodel s connect ed at por t A
( a) Thr ee submodel s( b) Connect ed at A
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Usi ng t he concept of cut , t he f ol l ow ng concl usi ons
can be der i ved:
( 1) When var yi ng t he I / O var i abl es, a model i n a
cont i nuous syst em can avoi d a change i n i t s
model descr i pt i on.
(2) I t separ at es t he physi cal l aws whi ch descr i be
t he st at i c and dynam c pr oper t i es of t he model
f r om t he physi cal l aws whi ch domnat e at sever al
subsyst ems at t hei r connect i ng poi nt s.
(3) Model s i n DYMOLA ar e sai d t o be i n pr oper
modul ar f or m so t hat t he user can bui l d t hem i n
a hi erarchi cal modul ar manner .
The above concept can be ext ended t o ot her syst ems
such as mechani cal , hydr aul i c and t her mal syst ems. Bei ng
i nt er est ed i n t he l ast ones, i t i s wor t hwhi l e ment i oni ng t hat
t emper at ur e and pr essur e ar e acr oss var i abl es, wher eas heat
f l ow i s a t hr ough var i abl e.
3. 2. 3 The Submode1 Concept and nodes i n DYMOLA
A submodel m ght be an at om c model , i . e. , a model
w t hout coupl i ng, or a coupl ed model .
Fi gur e 3. 4 i l l ust rat es at om c model s , wher eas t he
f ol l ow ng one ( Fi gur e 3. 5) i l l ust rat es a coupl ed model . I n
t he l at t er one, t he submodel s of a r esi s t or and a capaci t or
ar e depi ct ed whi ch ar e i n modul ar f orm The r esi st or s onl y
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m o d e l p r c
s u b m o d e l r e s i s t o r 3 0
s u b m o d e l c a p a c i t o r 2 0c u t A V A i l
c u t B V A / - Ic o n n e c t r e s i s t o r : B a t c a p a c i t o r : B a t B
c o n n e c t r e s i s t o r : A a t c a p a c i t o r : A a t A
e n d
Fi gur e 3. 5 Exampl e of a coupl ed model pr c
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par amet er R i s 30 ( i ndi cat ed by t he ( 30) i n t he submodel
statement ) , whereas the capaci tor s onl y parameter C i s 20.
By coupl i ng t he t wo at om c model s t oget her , a coupl ed
model prc i s produced. prc stands f or paral l el connect ed
r esi st or and capaci t or . The coupl ed model i s i n pr oper
modul ar f orm and can be used t o const r uct l ar ger syst ems.
Thi s concept of coupl ed model s i n DYMOLA i s shown i n
Fi gure 3. 6.
The node st at ement wi l l be seen ver y of t en i n a
DYMOLA progr am Nodes are conveni ent ways t o make several
connect i ons act i ng l i ke t he power di st r i but or . We pl ug
several appl i ances i nto one di st r i butor . For exampl e, we can
have
node n
connect x: A at n
connect y: B at n
whi ch i s equi val ent to the si ngl e statement
connect x: A at y: B
3. 2. 4 Hi er ar chi cal Model st r uct ur e i n DYMOLA
Fi gur e 3. 7 depi cts a syst emnamed S decomposed i nt o
sever al sUbsystems: Sl , S2 and S3 . S2 i s decomposed
i nto S21 and the l ast subsystem i s f ur t her decomposed i nto
S31 and S32 showi ng an overal l hi erarchi cal st r ucture.
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RR i I e B PRC
AT T
B
C
A II • BC
a
cut 1 •
cu t 1 cu t 2• .• 1Ll i J ·
u t 2
cu
Zc
t 1
ZA : cut- ZB :
2
cu~ ZA I •• ~2
b
Fi gur e 3. 6 Coupl ed model s i n DYMOLA
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s
Fi gur e 3. 7 A hi er ar chi cal l y st r uct ur ed syst em
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Fi gur e 3. 8 depi ct s one way t o descr i be t he
hi erarchi cal s t ruct ure of t he system ( S) i n DYMOLA. However ,
t hi s t echni que has a ser i ous f l aw. For exampl e, i f syst em
S21 and S32 are the same, the model speci f i cat i on must be
r epeat ed. I n or der t o avoi d dupl i cat i ng subsyst ems wi t h t he
same model s, DYMOLA i nt roduces a t erm cal l ed model t ype.
A model speci f i ed as model t ype r epr esent s a
gener i c model of a gener al cl ass of obj ect s. Thi s model
t ype can be used t o generat e several model s wi t h a submodel
stat ement so t hat dupl i cat i on wi l l be avoi ded ( Wang, 1989) .
For i nst ance, t he model r esi st or and t he model
capaci t or i n the model speci f i cat i on can now be def i ned as
model t ype r esi st or and model t ype capaci t or . The
f ol l owi ng Fi gur e 3. 9 demonst r at es t he same model
speci f i cat i on as bef ore but now usi ng model t ypes.
Af t er cr eat i ng model t ypes of any syst em i t
nat ur al l y comes t o t he user t o decl ar e l i br ar i es of model s.
Thi s l i br ar y i s set up f i r st when a syst em i s model ed and
then the hi er ar chy can be speci f i ed.
3. 3 Gener at i on of DESI RE Model s
As ment i oned bef ore, DYMOLA i s used to generate not
onl y SI MNON and FORTRAN model but al so DESI RE model s. The
f ol l owi ng command i s used f or thi s purpose:
out put desi r e model
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M o d e l S
m o d e l S
en d
m o d e l S
m o d e l 8
en d
en d
m o d e l S3
m o d e l S3
en d .
m o d e l S
en d
e n d
en d
Fi gur e 3. 8 Descr i pt i on of t he hi er ar chi cal st ruct ur eof a syst em i n DYMOLA
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m o d e l t y p e P R es u b m o d e l r e s i s t o r r t w o 3 0
s u b m o d e l c a p a c i t o r c o n e 2 0
c u t A V A / 1
c u t B V B / - I
c o n n e c t r t w o : A a t c o n e : A a t A
c o n n e c t r t w o : B a t c o n e : B a t B
e n d
Fi gur e 3. 9 Model speci f i cat i on f or pr eusi ng model t ype
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However , bef or e pr oceedi ng, t he user has t o i ssue t he
command:
par t i t i on
whi ch mani pul at es al l t he equat i ons emanat ed by t he model
descr i pt i on and connect i on mechani sm Thi s wor ks i n t he
f ol l ow ng way: Fi r st , t he comput er w l l det erm ne i f a
var i abl e i s pr esent i n an equat i on or not . Secondl y, i t f i nds
out f or whi ch var i abl e each equat i on must be sol ved. Thi r dl y,
i t par t i t i ons t he equat i ons i nt o smal l er syst ems of equat i ons
whi ch must be sol ved at t he same t i me. At t he ver y end, i t
sor t s t he equat i ons i nt o t he cor r ect comput at i onal or der .
3. 3. 1 cr eat i on of a DESI RE Si mul at i on Pr ogr am
To cr eat e a DESI RE Si mul at i on Pr ogr am a cont rol
por t i on of t he DYMOLA pr ogr am i s added. I n or der t o r un t he
si mul at i on of a cont i nuous syst em t he basi c i nf or mat i on f or
si mul at i on cont r ol such as si mul at i on st ep, communi cat i on
poi nt s and si mul at i on t i me ar e r equi r ed.
3. 3. 1. 1 Descr i pt i on of t he Si mul at i on Cont r ol Model
I t s synt ax i s cmodel and i t must be st or ed i nt o a
f i l e w t h t he same f i l ename as t hat of t he cont rol l ed syst em
I t i s i ndi cat ed by t he f i l e ext ensi on ct l and i s compr i sed
of t hr ee par t s:
( 1) basi c par t
( 2) r un cont r ol bl ock
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(3) out put bl ock
I n t he basi c par t t he f ol l owi ng i nf ormat i on i s st or ed:
1 si mul at i on t i me
(2) s i mul at i on step si ze
(3) number of communi cat i on poi nt s
(4) i nput s ( opt i onal )
The r eader shoul d consul t Wang s t hesi s (Wang, 1989
concer ni ng t he f ormat of t he basi c par t .
The r un cont r ol bl ock i nvol ves t he r un cont r ol
st at ement s whi ch can appear i n t he r un- t i me cont r ol par t of
a DESI RE progr am
The output bl ock must contai n the si mul at i on output
r equi r ement s. Ther e ar e f our out put st at ement s whi ch ar e
di spt , di spxy , t ype , and stash. Wang s thesi s gi ves
ext ens i ve det ai l s concerni ng t hei r synt act i c st ruct ures whi ch
ar e beyond t he scope of t hi s t hesi s . For t hi s t hesi s , we
requi r e si mul at i on graphs, so the di spt st atement i s goi ng
t o be used.
3. 3. 1. 2 Obt ai ni ng Execut abl e DESI RE Programs
The command
out put desi re program
wi l l cr eat e execut abl e DESI RE pr ogr ams. Fi r st , t he pr ogr am
ver i f i es i f t he si mul at i on cont r ol model associ at ed wi t h t he
syst em exi st s. Secondl y, i f t he above i s t r ue, t hen an
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execut abl e DESI RE program i s generat ed; ot herwi se, an er r or
message i s di spl ayed.
The pr ocedur e of gener at i ng DESI RE model s wi l l be
shown wi t h exampl es i n t he next chapt er wher e a di r ect
pr ocedur e of t r ansf ormng bond gr aphs i nt o DYMOLA code i s
devel oped.
3 4 Some Unsolved Problems
Cur r ent l y, DYMOLA i s st i l l i n a devel opi ng st age. A
f ai r amount of r esear ch i s needed t o make DYMOLA a mor e
pr oducti onal code. Ther e ar e, i ndeed, some unsol ved pr obl ems
whi ch ar e l i st ed bel ow. They ar e good r esear ch t opi cs f or
DYMOLA s f ut ur e enhancement and advancement .
( 1) DYMOLA i s cur r ent l y abl e t o el i mnat e var i abl es
f r om equat i ons of t ype Q = f 3 . However , i t i s
unabl e to el i mnate var i abl es f r omequat i ons of
t ype ± f 3 O.
( 2) DYMOLA must be abl e t o f i nd out dupl i cat e
equat i ons and t o get r i d of one of t hese
aut omat i cal l y. Thi s i s ver y i mpor t ant f or
hi er ar chi cal l y connected submodel s.
( 3) DYMOLA shoul d be abl e t o handl e super f l uous
connect i ons , i . e. , i f we speci f y t hat w2 = - w1,
i t i s obvi ousl y t r ue t hat al so b2 = - b1
(Cel l i er , 1990a) . (w i s the angul ar vel oci t y and
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b i s i t s cor r espondi ng angl e. ) Cur r ent l y,
DYMOLA cannot l et t he user speci f y t hi saddi t i onal connect i on and el i m nat e superf l uous
connect i ons dur i ng t he model expansi on.
( 4) DYMOLA must be capabl e of r ecogni zi ng t hat
connect i ons of out put s of i nt egrat ors can al ways
be t r ansf ormed i nt o connect i ons of i nput s of
such i ntegrator s . For exampl e, havi ng i a3 i a2,
i t i s obvi ousl y t r ue t hat i adot 3 i adot 2• Thi s
r ef ormul at i on can hel p el i m nat e st r uct ur al
s i ngul ar i t i es.
( 5) Gr oups of l i near al gebr ai c equat i ons ar e
cur r ent l y gr ouped t oget her and pr i nt ed out by
DYMOLA wi t hout bei ng sol ved. DYMOLA shoul d be
abl e t o r ewr i t e t he syst em of equat i ons i nt o a
mat r i x f or m si nce DESI RE can handl e mat r i x
expr essi ons ef f i ci ent l y and f ut ur e ver si ons of
DESI RE wi l l i ncl ude ef f i ci ent al gor i t hms f or
i nver t i ng mat r i ces (Wang, 1989) .
( 6) I f , f or exampl e, t he f ol l owi ng expr essi on i s
wr i t t en
x2 + z
2 + 2 * Y - 10 0
and i t i s desi r ed t o be sol ved f or x or z, t hen
pr obl ems wi l l ar i se. DYMOLA cannot sol ve f or
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second or hi gher or der equat i ons. I t can sol ve
f or Y, however .
( 7) DYMOLA can handl e onl y cont i nuous- t i me systems.
I t st i l l cannot handl e di scr et e t i me syst ems
al t hough DESI RE can handl e t hem
The af or ement i oned unsol ved pr obl ems ar e t he most
not i ceabl e ones. For mor e i nf or mat i on, t he r eader can r ef er
t o Cel l i er s book ( Cel l i er , 1990a) and Wang s t hesi s ( Wang,
1989) .
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CHAPTER 4
CONSTRUCTI ON OF BOND GRAPHS
AND THEI R TRANSFORMATI ON I NTO DYMOLA
Af t er di scussi ng bot h t he bond gr aph met hodol ogy and
t he Dynamc Model i ng Language i n t he pr evi ous t wo chapt er s,
t hi s chapt er f ocuses on t he way t o combi ne t hese t wo t ool s
f or model i ng and si mul at i ng. A demonst r at i on f or const r uct i ng
a bond gr aph f or a si mpl e el ect r i cal net wor k i s gi ven and
t hen i t s gr aph i s t r ansf ormed i nt o DYMOLA code. I t i s a
si mpl e, di r ect pr ocedur e as w l l be seen.
4. 1 Over vi ew
A det ai l ed pr ocedur e f or const r uct i ng t he bond gr aph
i s pr ovi ded. The sampl e syst em i s goi ng t o be a si mpl e
el ect r i cal network. Several di agrams are drawn demonst rat i ng
t he st ep by st ep pr ocedur e so t hat t he r eader can f ol l ow i t
w t hout any di f f i cul t y.
Once t he bond gr aph f or t he gi ven syst em has been
const r uct ed, i t can be di r ect l y coded i nt o DYMOLA. Ther e ar e,
however , sever al r ul es f or t hi s pr ocedur e t hat shoul d be
obser ved. They ar e st ressed i n t he subsequent sect i ons of
t hi s chapt er . The basi c bond gr aph model i ng el ement s of R,
C, L, TF, GY and bond can be descr i bed once and f or al l and
st or ed away i n a DYMOLA model l i br ar y cal l ed bond. l i b.
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At t he ver y end, t he DYMOLA coded pr ogr am i s r un on
t hepc
I t i s goi ng t o be seen t hat DYMOLA i s so power f ul
t hat i t can aut omat i cal l y eval uat e t he causal i t y of a bond
gr aph, gener at e a st at e- space descr i pt i on f or t he syst em and
f i nal l y gener at e a si mul at i on pr ogr am i n cur r ent l y ei t her
DESI RE or SI MNON, t wo f l at di r ect execut i ng cont i nuous-
syst em si mul at i on l anguages. DESI RE i s goi ng t o be used f or
t hi s purpose.
4. 2 Some Basi c Rul es f or const r uct i ng Bond Di agr amsf or El ect r i cal Networ ks
Bef or e pr oceedi ng t o our const r uct i on of a bond
di agr am f or a si mpl e el ect r i cal net wor k, we need t o meet some
r egul at i ons gi ven i n t hi s sect i on.
( 1) I n t he a- j unct i on, al l ef f or t var i abl es ar e
equal , wher eas al l f l ow var i abl es add up t o
zero.
( 2) I n t he I - j unct i on, al l f l ow var i abl es ar e equal ,
wher eas al l ef f or t var i abl es add up t o zer o.
Ther ef or e, f or an el ect r i c ci r cui t di agr am t he a- j unct i on i s
equi val ent t o a node, or a node i n a DYMOLA pr ogr am
( El mqvi st , 1978). Moreover , t he a- j unct i on r epr esent s
Ki r chhof f s cur rent l aw, wher eas t he I - j unct i on r epr esent s
Ki r chhof f s vol t age l aw. I f t wo j unct i ons ar e connect ed w t h
a bond, one i s al ways of t he a- j unct i on t ype whi l e t he ot her
i s al ways of t he I - j unct i on t ype. I t can be sai d t hat
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a- j unct i ons and 1- j unct i ons al ways t oggl e. Nei ghbor i ng
j unct i ons of the same type can be amal gamated i nt o one.
4 3 Cons t r uc t i on of a Bond Diagram
of a Simple Electrical Network
Because of my f am l i ar i t y t o el ect r i cal net wor ks, I
have chosen a si mpl e el ect r i cal networ k t o demonst r at e t he
st ep by st ep pr ocedur e f or const r uct i ng i t s bond gr aph.
The net wor k i s shown on Fi gur e 4. 1, wi t h i t s node
vol t ages l abel l ed a, b, c and r .
4 3 1 The Step by Step Procedure
The f ol l owi ng st eps must be f ol l owed f or const r uct i ng
i t s bond graph:
( 1) I t i s bet t er t o use vol t ages t han cur r ent s, so
Fi gur e 4. 2 shows t hr ee a- j unct i ons ( vol t age
j unct i ons) bei ng l ai d out wi t h subscr i pt s
cor r espondi ng to the nodes. The ref erence node
i s not r epr esent ed by a a- j unct i on.
( 2) Then, we r epr esent each br anch of t he ci r cui t
di agr am by a pai r of bonds r epr esent i ng t wo
a- j unct i ons wi t h a 1- j unct i on i n between t hem
( l - j unct i on = cur r ent - j unct i on) . Thi s i s
di spl ayed i n Fi gur e 4. 3
( 3) Set t i ng Vr t o zer o, we can r emove t he bonds
connect i ng t he r est of t he ci r cui t ( see
Fi gure 4. 4) .
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a Rb c
r
Fi gur e 4. 1 An el ect r i cal net wor k w t h nodes l abel l ed( r r ef er ence)
V a R c V c
Fi gur e 4. 2 Layout of vol t age j unct i ons ( O- j unct i ons)
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R
I 1
c
I 1
S E 1
Fi gur e 4. 3 The assembl y of component s and sour ce
R C
0 1 0 1
I I I1 1 1
I S E L 1 L 2
Fi gur e 4. 4 The cancel l at i on of r ef er ence nodeand associ at ed bonds
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1
Fi gur e 4. 5 The condensat i on of bonds
R C SE 0 L 2
L 1
Fi gur e 4. 6 The reduced graph
R C
SE
> -
> - 0 > - > L 2
~
L 1
Fi gur e 4. 7 The bond gr aph
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R : R 1
= 2 0 0 0
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.v e c
S E : u U o v 1 v 1 v L 2> I : L 2
> 1 > 0 > 1= 2 0 V
i e
.= 1 m H
1 0 1 0
i L VIe
I : L 1= 1 . 5 m H
Fi gur e 4. 8 The compl et ed bond gr aph w t h i t s causal i t i es
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4. 4 Transformat i on of Bond Graphs i nt o DYMOLA Code
Af t er const r uct i ng t he bond gr aph f or t he sel ect ed
si mpl e el ect r i cal networ k, we ar e r eady t o t r ansf orm i t i nt o
DYMOLA code whi ch i s a st r ai ght f or war d pr ocedur e. The
f ol l owi ng rul es must be observed, however :
( 1) The O- j unct i ons ar e equi val ent t o DYMOLA s
nodes.
( 2) Ther e i s no DYMOLA equi val ent f or 1- j unct i ons;
however , i f t he ef f or t and f l ow var i abl es ar e
i nt er changed, t hen t hey ar e t he same as
a- j uncti ons.
( 3) Havi ng t he above i n m nd, a model t ype bond
whi ch si mpl y exchanges t he ef f or t and f l ow
var i abl es can be cr eat ed and i nst al l ed i n
DYMOLA s l i brary. Bes i des , t he el ement s R, C, L,
TF and GY whi ch descr i be t he basi c bond gr aph
mul t i por t el ement s ar e i nst al l ed once and f or
al l i n DYMOLA s l i br ar y. They ar e i l l ust r at ed i n
Fi gur e 4. 11.
( 4) I n DYMOLA, al l el ement s shoul d be at t ached t o
O- j unct i ons onl y. I f we want t o at t ach an
el ement t o a 1- j unct i on, t hen we need t o pl ace
a bond i n bet ween ( Cel l i er , 1990a) . The
expanded bond graph i s shown on Fi gure 4. 9.
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R:R1 C :C 1
1 1 dR 1 dC 1
1 V1 1SE : U
> 0 > 1),
0 > 1 , 0 >
v a.
dL2 I:L20
~
Ie
I:L 1
Fi gur e 4. 9 DYMOLA expanded bond graphw t h each node i ndi cat ed
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(5) Nei ghbor i ng j unct i ons are al ways of t he opposi t e
sex, i . e. , O- j unct i ons and I - j unct i ons al ways
t oggl e.
( 6) For t unat el y, we do not need t o wor r y about
causal i t i es. DYMOLA i s per f ect l y capabl e of
handl i ng t he causal i t i es as i s seen dur i ng t he
execut i on of t he al gor i t hm assi gni ng t hem
However , as we saw ear l i er , we wer e per f ect l y
capabl e of assi gni ng causal i t i es. Thi s i s not
t rue ever y t i me. For exampl e, a non- causal
syst em r esul t s when we t r y t o connect t wo
sources of di f f erent val ues.
Now we are r eady t o t r ans l at e t he expanded bond graph
i nt o DYMOLA code as i ndi cat ed by Fi gur e 4. 10. The code i s
sel f - expl anat or y as we use t he st at ement s submodel ,
connect , and node whi ch had been anal yzed i n t he pr evi ous
chapt er . Fur t hermor e, t he var i ous DYMOLA model t ypes as wel l
as t he Exper i ment used f or si mul at i ng the network are shown
i n the next two f i gures. Exper i ment i s t he si mul at i on cont r ol
model as descr i bed i n chapter t hree.
DYMOLA can f ur t hermor e be used f or obt ai ni ng var i ous
r esul t s such as causal i t y, el i m nat i on of r edundant
equat i ons, der i vat i on of a st at e- space represent at i on, and
generat i on of a si mul at i on program f or DESI RE. The ul t i mate
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{bond gr aph model f or a s i mpl e RLC net wor k}
@r l c . r
@r l c . c
@r l c . i
@r l c . s e
@r l c . bnd
model RL C
s ubmodel ( SE) UO
s ubmodel ( R) RI ( R=200. 0)
s ubmodel ( 1 ) L l ( 1 =1. 5E- 3) , L 2( 1=I . OE- 3)s ubmodel ( C) CI ( C=O. l E- 6)
s ubmodel ( b ond) BI , B2, B3, B4, B5, B6
node vO, i O, v l , dRI , i c , d L2, dCl
out put y l
c onnec t UO at vOconnec t BI f r om vO t o i O
connec t B2 f r om i O t o dRI
connec t RI at dRI
c onnect B3 f r om i O t o v I
connec t L I at vl
connect B4 f r om vI t o i c
connect B5 f r om i c t o dCIconnec t CI at dCI
connec t B6 f r om i c t o dL 2
connec t L2 at dL2
UO. EO=20. 0
y l =L2. e
end
Fi gur e 4. 10 DYMOLA code of t he bond gr aphshown on Fi gur e 4. 9
63
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model t y pe Rcut A ( e / f )par amet er R=l . OR* f = e
end
model t ype GY
cut A( el / f l ) B( e2/ - f 2)mai n cut C[ A B)
mai n pat h P<A - B>par amet e r r =l . Oel =r * f 2
e2=r * f 1end
model t y pe I
cut A e / f
par amet er 1=1. 0I * der ( f ) = e
end
model t ype TF
cut A( el / f 1) B( e2/ - f 2)mai n cut C[ A B)
mai n pat h P<A - B>par ame te r m=l . O
e1=m* e2f 2=m* f 1
end
model t ype C
cut A e / f)
pa r amet er C=l . 0
C* der ( e) = f
end
model t ype SFcut A( . / - f )
t er m nal FO
FO=f
end
model t ype SE
cut A e / .)
t er m nal EOEO = e
end
model t ype bond
cut A x / y) B ( y / - x)
mai n cut C [ A B)
mai n pat h P <A - B>
end
Fi gur e 4. 11 The var i ous basi c DYMOLA model t ypes
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cmodel
s i mut i me 50. 0E- 6s t ep 50. 0E- 9
commupoi nt s
c t b l oc k
s cal e = 1
XCCC = 1l abel TRY
drunr I i f XCCC<O t hen XCCC = - XCCC I s c al e = 2* s c al e I go t o TRYel s e pr oceed
c t end
out b l o ckOUT
yl =L2Se
di s pt y lout end
end
Fi gur e 4. 12 Exper i ment used f or t he net wor k
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goal i s t he gener at i on of an execut abl e DESI RE pr ogr am usi ng
the f ol l ow ng commands:
dymol a
> ent er model
- @r l c. dym
> ent er exper i ment
- @r l c. ct l
> out f i l e r l c. des
> par t i t i on el i m nat e
> out put desi r e pr ogr am
> stop
cl ar i f yi ng t he l ast por t i ons of t he l ast chapt er .
Then, we can r un DESI RE usi ng t he f ol l ow ng commands
desi r e
> l oad ' r l c. des'
> r un
> bye
The gener at ed DESI RE pr ogr am as wel l as t he st at e- space
r epr esent at i on ar e shown i n t he f ol l ow ng two f i gur es
r espect i vel y ( see Fi gur e 4. 13 and Fi gur e 4. 14) . The
st at ement s above t he DYNAM C decl ar at i on of t he gener at ed
DESI RE program descr i be t he exper i ment t o be per f ormed on t he
model , and t he ot her st at ement s descr i be t he dynamc model .
The t i me of t he whol e compi l at i on i s l ess t han a t ent h of a
second. Fi nal l y, t he DESI RE out put of our net wor k i s shown i n
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- - CONTI NUOUS SYSTEM RLC
STATE C1Se LI Sf L2 f
DER dC1Se dLl f dL2 f
OUTPUT yl
PARAMETERS and CONSTANTS:
R=200. 0
C=0. l E- 6
L1SI = . 5E- 3
L2SI = . OE- 3
- - I NI TI AL VALUES OF STATES:
CI Se=O
LI Sf =O
L2Sf =0
TMAX=50. 0E- 6 DT=50. 0E- 9 NN= OI
scal e = I
XCCC = 1
l abel TRY
drunr I i f XCCC<O then XCCC = - XCCC I scal e = 2*scal e I go t o TRY
el se proceed
DYNAM C
- - Submodel : RLC
B3Sx = L2Sf + LI Sf
- - Submodel : RI
RI Se = R*B3Sx
- - Submodel : CI
d/ dt CI Se = L2Sf / C
- - Submodel : RLC
BI Sx = 20. 0B4Sx = BI x - RI e
Fi gur e 4. 13 Gener at ed DESI RE Pr ogram(cont i nued on next page)
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- - Submodel : L l
d/ dt L l f = B4 x / L l l- - Submodel : RLC
L2 e = B4 x - Cl e
- - Submodel : L2
d/ dt L2 f = L2 e/ L2 l
OUTyl =L2 edi s pt y l
/ P I C r l c . PRC
Fi gur e 4. 13 Generat ed DESI RE program ( cont i nued)
RLC B3 . x = L2. f L l . fRl e = R*B3. xCl de re = L 2 . f / CRLC Bl . x = 20. 0
B4 . x = Bl . x - Rl . eL l de r f = B4. x/ lRLC L 2. e = B4. x Cl . eL2 der f = e/ l
Fi gur e 4. 14 St at e- space r epr esent at i on of t he net wor k
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+
f
I····················································· .
8
e B e e B e
scale = 3 2 B e B l
2 .58e-85
y l u t
S.88e-85
Fi gur e 4. 15 DESI RE out put
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CHAPTER 5
CASE STUDY
MODELING SIMULATING A SOLAR HEATED HOUSE
The mai n goal of t hi s t hesi s i s pr esent ed i n t hi s
chapt er . Havi ng st udi ed t he bond gr aph methodol ogy, DYMOLA,
and seen how t hese t wo t ool s can be combi ned t oget her , we ar e
r eady t o model and t hen t o si mul at e our sol ar - heat ed house.Bei ng a r el at i vel y compl i cat ed syst em i t i s appr opr i at e f or
model i ng pur poses t o di vi de i t i nt o sever al par t s, t hat i s ,
i nt o a hi er ar chi cal l y descr i bed st r uct ur e. Each par t i s
pr esent ed by i t s bond gr aph conver t ed i nt o i t s DYMOLA code as
wel l . Fi nal l y, al l t he par t s ar e combi ned t oget her r esul t i ng
i n t he whol e model of t he sol ar - heat ed house.
5 1 Overview
sci ent i st s t hr oughout t he year s have i nvest i gat ed t he
expl oi t at i on of sol ar r adi at i on f or space heat i ng. A sol ar
heat i ng syst em l i ke t he i nvest i gat ed one i s any col l ect i on of
equi pment desi gned pr i mar i l y t o use t he sun' s ener gy f or
heat i ng pur poses.
The above syst em i s a r el at i vel y compl i cat ed one
i nvol vi ng many di f f er ent t ypes of ener gy. Var i ous met hods
wer e used t hr oughout t he year s f or model i ng and si mul at i ng
such a syst em w t h m xed r esul t s. I t i s expect ed t hat usi ng
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t he met hod descr i bed i n t hi s t hesi s, t hat i s, t he bond gr aph
model i ng met hodol ogy as wel l as DYMOLA f or gener at i ng a
si mul at i ng program f or DESI RE, the physi cal behavi or of such
syst ems can be model ed, si mul at ed and eval uat ed i n a
conveni ent , robust , and f ast manner .
The i nvest i gat ed conf i gur at i on consi st s of a f l at -
pl at e sol ar col l ect or , a sol i d body st or age t ank and t he
habi t abl e space. They ar e connect ed wi t h wat er l oops ci r cu-
l at i ng wat er t hr ough pi pes. Each par t i s t hor oughl y st udi ed
and anal yzed i l l ust r at i ng t he energy f l ow t hr ough each sub-
syst emand across t he bar r i er bet ween sUbsyst ems. Each one i s
t r ansf ormed i nt o a bond gr aph r epr esent at i on and i s t hen
di rect l y coded i nt o DYMOLA whi ch not onl y generates a DESI RE
pr ogr am but can al so pr ovi de us wi t h a set of f i r st or der
di f f er ent i al equat i ons ( a st at e- space represent at i on) . The
var i ous par amet er s used f or t he si mul at i on wer e t aken f r om
var i ous sources as ment i oned bef ore ( see l ast por t i on of t hi s
chapt er ) .
5. 2 Sol ar Heat i ng
A popul ar concept i on of sol ar heat i ng i s t o use t he
sol ar r adi at i on mor e or l ess di r ect l y wi t hout any nat ur al
i nt er medi at e st eps such as phot osynt hesi s. Thi s can be
pr i mar i l y accompl i shed by col l ect or s whi ch ar e devi ces
col l ect i ng sol ar r adi at i on ar r i vi ng f rom t he sun and
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conver t i ng t hi s r adi ant ener gy t o a mor e desi r abl e one such
as heat . Thi s conver ted ener gy can be t ransf er red by a f l ui d
( usual l y hot wat er ) and ei t her ut i l i zed i mmedi at el y or st or ed
f or l at er use. Thi s heat can be used f or a si mpl e space
heat i ng. A gener al sol ar heat i ng syst em i s shown i n
Fi gur e 5. 1.
Let us descr i be i n gener al t erms t he col l ect or s and
t he st or age t ank as wel l as t he habi t abl e space.
Col l ect or s ar e t he hear t of any sol ar heat i ng syst em
col l ect i ng and t hen conver t i ng t he sol ar r adi at i on. The
si mpl est and cheapest one ( see Fi gur e 5. 2 i n sect i on 5. 4) i s
cal l ed t he f l at pl at e col l ect or . I t i s a f l at sheet of dar k
sur f aced met al possessi ng one or mor e l ayer s of gl ass above
and a l ayer of common i nsul at i on bel ow. The met al sheet i s
heat ed by sunl i ght whi ch comes t hr ough t he gl ass. The amount
of heat t hat can escape and di ssi pat e can be r educed by t he
gl ass and i nsul at i on; t her ef or e, t he met al sheet becomes ver y
hot . I n or der t o obt ai n t hi s heat f or ut i l i z i ng i t , t her e ar e
two ways t o do i t . Ei t her ai r can be passed above t he met al
or a f l ui d can be passed t hr ough t ubes bonded t o t he met al .
Ther ef or e, t he sunl i ght heat s ei t her ai r or wat er whi ch ar e
t r ansf er red t o ot her conveni ent l ocat i ons f or use.
When t he col l ect or suppl i es t he heat ed ai r or wat er ,
one of t wo t hi ngs must be done- - ei t her i t can be used at once
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0- ~~:~
Col,.ctor :.
0 . • • •
E\tctric;ty
Fi gur e 5. 1 A sol ar heat ed house
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or i t can be st or ed f or l at er use. Par t i cul ar l y, t he hot
wat er can be st or ed i n t ank syst ems desi gned i n such a way
t hat cool er wat er f r om t he bot tom can be sent t hr ough t he
col l ect or f or heat i ng and t hen r et ur ned t o t he upper par t of
t he t ank. I t i s not pr act i cal t o st or e hot ai r . The st or age
t anks ar e heavy and usual l y ar e set bel ow gr ound.
Havi ng heat ei t her f r om col l ect or s or t he st or age
t ank, we have t o use i t ; f or exampl e, f or space heat i ng
( habi t abl e space) . Hot wat er passes t hr ough a heat pump
( mght be cool i ng or heat i ng devi ce) and a heat exchanger i n
whi ch t he ai r bl ows ar ound t he hot wat er coi l s f r om t he heat
st or age t ank. Ther eby, t he habi t abl e space i s heat ed\
Above, t he pr ocedur e has been descr i bed i n whi ch
sol ar r adi at i on i s conver ted t o a f or m of ener gy f or heat i ng
a house. We ar e r eady now t o model t he basi c par t s, t hat i s
t he col l ect or , t he st or age t ank, t he wat er l oops ( col l ect or
and heat er wat er l oop) and t he habi t abl e space. The col l ect or
wat er l oop ( CWL) i s connect ed bet ween t he col l ect or and t he
st or age t ank; wher eas, t he ot her one ( heat er wat er l oop:
HWL) i s connect ed bet ween t he st or age t ank and t he heat er
( heat exchanger ) . I n model i ng our sol ar house, we have been
car ef ul t o make our syst em causal , t hat i s, t o sat i sf y al l
causal i t y condi t i ons. To st ar t w t h, we have t o know some
basi c t hermodynamc concept s whi ch ar e pr esent ed i n t he next
secti on.
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5. 3 Basi c Thermodynam c n Gener al Concept s
The f i r st t hi ng needed i s t o def i ne ent r opy and t he
f i r st l aw of t hermodynam cs. So, ent ropy ( S) i s def i ned as
S - Q- T 5.1
wher e Q i s heat ( i n J oul es) and T i s t emper at ur e ( i n Kel vi n) .
The f i r st l aw of t hermodynam cs st at es t hat t he t ot al
energy Et, bei ng a const ant , equal s t o t he sum of f r ee ener gy
E and t he t hermal ener gy Q.
( 5. 2)
Al so ent ropy f l ow can be def i ned as
dS _ .1 QQ
dt - T dt ( 5. 3)
and when mul t i pl i ed by t he t emper at ur e T gi ves heat f l ow
whi ch i s power needed t o const r uct t he bond di agr ams.
Moreover , t he heat equat i on
5.4)
descr i bes bot h t he t hermal conduct i ve and convect i ve f l ow of
heat .
I n t hermodynam cs, we need t o f am l i ar i ze our sel ves
w th three separate physi cal phenomena provi di ng mechani sms
f or heat t r ansf er or heat f l ow. They ar e conduct i on, convec-
t i on, and r adi at i on.
I n heat conduct i on, t hermal ener gy i s t r anspor t ed by
t he i nt er act i ons of i t s mol ecul es i n spi t e of t he f act t hat
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mol ecul es do not move t hemsel ves . For i nst ance, when one end
of a r od i s heat ed, t he l at t i ce at oms i n t he heat ed end
vi br at e wi t h gr eat er ener gy t han t hose at t he cool er end so
t hat t hi s ener gy i s t ransf er red al ong t he r od. I n t he case
of a met al r od, t he t r anspor t of t hermal ener gy i s ai ded by
f ree el ect rons whi ch are movi ng t hroughout the met al and they
col l i de wi t h t he l at t i ce at oms.
I n convect i on, heat i s t r anspor t ed by a di r ect mass
t r ansf er . For i nst ance, warm ai r near t he f l oor expands and
r i ses because i t possesses l ower densi t y. Thermal energy i n
t hi s war m ai r i s t ransf er red f r om t he f l oor t o t he cei l i ng
al ong wi t h t he mass of warm ai r .
The l ast mechani sm of heat t r ansf er i s t hr ough
thermal radi at i on i n whi ch energy i s emt ted and absorbed by
al l bodi es i n t he f or m of el ect romagnet i c r adi at i on. I f a
body i s i n t hermal equi l i br i umwi t h t he envi r onment , i t em t s
and absor bs ener gy at t he same r at e. However , when i t i s
warmed t o a hi gher t emper at ur e t han i t s envi r onment , i t
r adi at es away mor e ener gy t han i t absor bs so t hat i t cool s
down as t he sur r oundi ngs get warmer . As a r esul t , i n Ar i zona
peopl e avoi d havi ng dar k pai nt ed car s because t hey
emt / absorb l i ght much more st r ongl y than l i ght ones.
As we saw i n t he t hi r d chapt er , DYMOLA pr ovi des a
modul ar i zed hi er ar chi cal l y st r uct ur ed model descr i pt i on.
Thus, t he ent i r e sol ar house has been di vi ded i nt o f i ve maj or
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hi erar chi cal st r uctures bei ng the ones as ment i oned bef ore.
Each of t hese consi st s of smal l er hi er ar chi cal st r uct ur es( submodel s) . For i nst ance, t he sol ar col l ect or consi st s of
t he l oss and t he spi r al submodel s. Fur t hermor e, t he spi r al
compr i ses of t wo ot her smal l er submodel s bei ng t he heat
exchanger and t he one- di mensi onal cel l . Thi s hi er ar chy
cont i nues even f ur ther wi t h t he one- di mensi onal cel l
consi st i ng of two other submodel s, the modul ated conduct i ve
source (mGS) and the modul at ed capaci t ance (mC) . Al l t hese
ar e descr i bed i n det ai l l at er i n t hi s chapt er .
Al l t hese hi er ar chi cal st r uct ur es pr ovi de t he
researcher a conveni ent way to study the physi cal behavi or of
each par t i cul ar par t of t he sol ar house i n gr eat er det ai l .
I n t he l ast por t i ons of t he pr evi ous sect i on,
t he t ermcausal was ment i oned, i . e. , t o sat i sf y al l causal i t y
condi t i ons. To achi eve t hi s, we have t o avoi d t he so- cal l ed
al gebr ai c l oops and t he st r uct ur al si ngul ar i t i es. Thi s has
been done by choosi ng t he pr oper el ement s i n assi gni ng
causal i t i es and not t o have any f r ee choi ce as depi ct ed i n
Fi gur e 2. 7 and by not over speci f yi ng i n t he descr i pt i on of
each par t i cul ar model . Thus, our sol ar - heat ed house wi l l be
a causal syst em and wi l l possess a uni quel y det er mned
causal i t y.
Now we ar e r eady t o pr oceed wi t h t he model i ng
procedure st ar t i ng i n the next chapter .
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5 4 Flat Plate Solar Collector Modeling
We shal l st ar t model i ng t he ent i r e sol ar house by
model i ng i t s f l at pl at e col l ect or . A si mpl er descr i pt i on t han
t he pr evi ous gi ven one f or t he sol ar col l ect or i s t o i magi ne
i t as a bl ack body accumul at i ng sol ar heat t hr ough r adi at i on
so t hat t he t emper at ur e r ai ses i nsi de. The col l ect or s ( may be
one or sever al ones) ar e usual l y f i l l ed by ai r possessi ng a
l ar ge heat capaci t y. I nsi de t hem a w ndi ng wat er pi pe goes
back and f or th bet ween t he t wo ends i n or der t o maxi m ze t he
pi pe sur f ace. Let ' s cal l t hi s a wat er spi r al . The heat i ng of
t he wat er i n t he pi pe occur s when a most l y conduct i ve heat
exchange t akes pl ace bet ween t he col l ect or chamber and t he
wat er pi pe. We shal l descr i be t he col l ect or wat er l oop as a
pump whi ch ci r cul at es t he wat er f r om t he col l ect or s t o t he
st or age t ank. As a r esul t , t he heat t r ansf er occur s i n a
most l y convect i ve manner . The wat er spi r al s can be connect ed
ei t her i n par al l el or i n ser i es and t he pump i s dr i ven by a
sol ar panel . The sol ar l i ght i s conver t ed i nt o el ect r i ci t y
i nsi de t he panel . As a r esul t , t he pump ci r cul at es t he wat er
onl y on a sunny day, whi ch i s meani ngf ul . Fur ther mor e, t he
wat er pi pe i s pr ot ect ed by a f r eeze pr ot ect i on devi ce whi ch
al so sw t ches t he pump on when t he t emper at ur e f al l s bel ow
5°C out s i de.
A model depi ct i ng such a f l at pl at e sol ar col l ect or
i s shown i n Fi gur e 5. 2. The ef f i c i ency of t he sol ar col l ect or
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Sol¥ ~
R.o~hon \, :. .
Fi gur e 5. 2 Model of a f l at - pl at e col l ect or
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depends upon sever al f act or s such as cl i mat i c condi t i ons
( ambi ent t emper at ur e, wi nd) , number of cover s and t hei r
r adi at i ve pr oper t i es, i nci dent sol ar angl e, r adi at i ve
proper t i es of absorber pl ate, spaci ng of covers and absorber ,
f l ui d t ype and i nsul at i on of col l ect or encl osur e. The
f ol l owi ng assumpt i ons wer e made bef or e model i ng t he
col l ect or :
( a) The heat f l ow i nt o t he col l ect or i s basi cal l y a
r adi at i ve heat f l ow, model ed by a f l ow sour ce
whi ch i s dependent on t hr ee f act or s, day of t he
year , t i me of t he day and weat her .
( b) Ther e i s l oss f r om t he col l ect or t o t he
surr oundi ngs whi ch has conduct i ve, convect i ve
and r adi at i ve el ement s wi t h t he f i r st t wo mor e
domnant . I t i s basi cal l y model ed as a t empera-
t ur e sour ce and as a modul at ed conduct ance
char act er i zed by t he absor ber and envi r onment .
( c) There exi st s conduct i ve heat exchange between
t he col l ect or space and t he hydr aul i c spi r al .
And f i nal l y,
( d) Ther e i s convect i ve heat t r anspor t i n t he
spi ral .
The wat er spi r al i nt roduced pr evi ousl y wi l l be
r epr esent ed as a ser i es of one- di mensi onal cel l s. The bond
di agram and the DYMOLA model type of such a cel l are depi cted
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i n Fi gur es S. 3a and S. 3b, r espect i vel y, w t h t he causal i t i es
cor r ect l y mar ked. The mGS el ement i s a modul at ed conduct i ve
sour ce modul at ed w t h t emper at ur e and, f ur ther mor e, i t i s
modul at ed w t h t he wat er vel oci t y i n t he pi pe as shown i n
Fi gur e S. 3c. Thi s el ement ( one- di mensi onal cel l ) has been
model ed t hr ough i t s conduct ance r at her t han t hr ough i t s
r esi st ance because t he conduct ance changes l i near l y w t h t he
water vel oci t y.
The bond gr aph of t he one- di mensi onal cel l f avor s
heat f l ow f r om t he l ef t t o t he r i ght ; t her ef or e, i t i s not
symmet r i cal . Our deci si on t o r epr esent t he heat ( ent ropy)
f l ow i n such a way i s j ust an appr oxi mat i on. I f , f or exampl e,
t he mGS el ement i s spl i t i nt o t wo equal par t s, one t ur ni ng
l ef t and t he ot her r i ght , t hi s choi ce i s not desi rabl e
because of t he i nt roduct i on of al gebr ai c l oops i dent i f i ed
w t h t he choi ces of causal i t i es ( Cel l i er , 1990a) .
The next st ep i s t o devel op t he heat exchanger model
bei ng used t o descr i be t he exchange of heat acr oss t he bor der
of t wo medi a. I n t hi s par t i cul ar case, t he heat exchanger i s
used t o model t he heat t r anspor t f r om t he col l ect or chamber
t o t he wat er spi r al . I t s bond di agr am as wel l as i t s DYMOLA
model t ype ar e shown i n t he f ol l ow ng t wo f i gur es ( S. 3d and
S. 3e) .
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~ 11D 1
Fi gur e 5. 3a Bond di agr am of a one- di mensi onal cel l
{bond gr aph f or one di mens i o nal c el l }
model t ype oneD
submode l ( MGS) Gc el l ( a=1. 5, b=O. 72)submode l ( MC) Cce l l ( gamma=72. 0 )
s ubmodel ( b ond) Bl , B2, B3
node nl , n2
t er m nal vwat er
cut Cx( ex/ f x) , Ci ( ei / - f i )
mai n pat h P<Cx - Ci >
connec t B1 f r om Cx t o n1
connec t B2 f r om n1 t o n2
connec t Gcel l f r om n2 t o Ci
connec t B3 f r om n1 t o Ci
connec t Ccel l at Ci
Gce l l . vel =vwat er
end
Fi gur e 5. 3b DYMOLA model t ype of a one- di mensi onal cel l
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G
eonwetiw.-::~ --. ;a;;;. . . . ._ ht t t r 6n sport
eondvct t-t..t trlf lsportv
Fi gur e 5. 3c Modul at ed conduct i ve source
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rmRS~
o
-tmRS ./
Fi gur e 5. 3d Bond di agr am of a heat exchanger
{bond gr aph f o r Heat er ( heat exchanger ) }
model t ype HE
submode l ( MRS) RI H( t het a=8 . 0E+2) , R2H( t het a=8 . 0E+2)cut A( el / f l ) , B( e2/ - f 2)mai n cut C[ A B]mai n pat h P<A - B>
connec t RI H f r om A t o Bconnec t R2H f r om B t o Aend
Fi gur e 5. 3e DYMOLA model t ype of a heat exchanger
85
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Af t er model i ng t he one- di mensi onal cel l model and t he
heat er model , we ar e r eady t o pr oceed wi t h t he model i ng of
t he wat er spi r al whi ch i s a di st r i but ed paramet er syst em We
have deci ded t o r epr esent t he wat er spi r al wi t h t hr ee one-
di mensi onal cel l s connect ed i n ser i es and heat exchanger s
at t ached i n between. Obvi ousl y, our deci si on i s an approxi -
mat i on of a pr ocess wi t h di st r i but ed par amet er s. The bond
gr aph of a wat er spi r al whi ch i s a 3- por t el ement i s depi ct ed
next ( see Fi gur e 5. 3f ) . Fur t hermore, i t s cor r espondi ng DYMOLA
model t ype i s shown i n Fi gur e 5. 3g.
The f i nal el ement t o be devel oped f or t he compl et e
col l ect or model i s t he l oss el ement f r om t he col l ect or
chamber t o t he sur r oundi ngs. Thi s l oss i s par t l y conduct i ve
and par t l y convect i ve and i t s bond di agr am i s shown i n
Fi gur e S. 3h. I t i s a I - por t el ement . I t s DYMOLA model t ype
i s al so shown ( Fi gur e 5. 3i ) . The ef f or t sour ce denot es t he
out si de envi r onment , wher eas t he mG el ement denot es t he heat
di ssi pat i on t o t he envi r onment . The di ssi pat ed heat i s
pr opor t i onal t o t he di f f er ence i n t emperat ur es between t he
i nsi de and t he out si de. The mG el ement i s a modul at ed
conduct ance whi ch i s ver y si m l ar t o t he mGS el ement f ound
ear l i er . I t i s model ed wi t h t he t emper at ur e as wel l , but t hi s
t i me, t he modul at i on i s wi t h r espect t o t he wi nd vel oci t y
i nst ead of t he wat er vel oci t y.
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000 0
111 1
O~lD~O~lD~O~lD~O
o
1O~Sp1~O
Fi gur e 5. 3f Bond di agr am of a wat er spi r al
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{bond gr aph Spi r al }
model t ype Spi
5ubmodel ( HE) HEI , HE2, HE3, HE45ubmode l ( oneD) oneDl , oneD2 , oneD3
node nl , n2
t er m nal vwat er
cut i nwat er l ( el / f l ) , o ut wat er l ( e2/ - f 2 ) , C( e3/ f 3)
mai n cut D[ i nwat er l out wat er l ]
mai n pat h P<i nwat er l - out wat er l >
connect HEI f r om C t o i nwat er l
connect oneDl f r om i nwat er l t o nl
connect HE2 f r om C t o nlconnect oneD2 f r om nl t o n2
connec t HE3 f r om C t o n2
connec t oneD3 f r om n2 t o out wat er lconnec t HE4 f r om C t o out wat er l
oneDI . vwat er =vwat er
oneD2. vwat er =vwat er
oneD3. vwat er =vwat er
end
Fi gur e 5. 3g DYMOLA model t ype of a wat er spi r al
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mG
f ....O ~ l D S BO~Ik-SE
Fi gur e 5. 3h Bond di agr am of t her mc l oss
{bond gr aph f or LOs s }
model t ype Los s
s ubmodel ( SE) out t emp
s ubmodel ( MG) Gl os s( a=1. 5, b=O. 72)
s ubmodel ( bond) Bl , B2, B3
node n, nG, nS
mai n cut A( e/ f )
t er m nal Tout , vwi nd
connec t Bl f r om A t o n
connec t B2 f r om n t o nG
connec t Gl o s s at nG
connec t B3 f r om n t o nS
connec t out t emp at nS
out t emp. EO = Tout
Gl os s . vel = vwi nd
Gl os s . Tout = Tout
end
Fi gur e 5. 3i DYMOLA model t ype of t her mc l oss
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For t he one- di mensi onal cel l , t he mGS el ement i s used
because t he ener gy i s not l ost . The ener gy i s si mpl y t rans-
por t ed r i ght away t o t he next node. Thi s i s t he r eason t hat
a new bond gr aph el ement cal l ed a r esi st i ve sour ce ( RS) has
been i nt r oduced ( Thoma, 1975) . Obvi ousl y t he GS el ement i s
l / RS. On t he ot her hand, t he mG el ement i s used i n t he l oss
because t he behavi or i s l i ke t he el ect r i cal case wher e t he
r esi st ances ( conduct ances) di ssi pat e heat and l ose ener gy.
Not i ce t hat i n t her modynam cs, t he RS ( R) and C el ement s ar e
nonl i near . I n DYMOLA, t hey ar e model ed by t wo new bond gr aph
el ements , mRS and mC, respect i vel y.
The over al l bond di agr am f or t he col l ect or can now be
dr awn as shown i n t he Fi gur e 5. 3j . The mC el ement whi ch i s
modul at ed w t h t emper at ur e i s t he heat capaci t ance of t he
col l ect or chamber . The SF el ement i s t he heat i nput f rom
sol ar r adi at i on whi ch must be model ed separ at el y.
We can use t he hi er ar chi cal cut concept of DYMOLA t o
combi ne t he two cut s i nt o an aggr egat ed bond gr aph r epr esen-
t at i on pi ct or i al l y r epr esent ed by a doubl e bond. The t wo cut s
can be named as i nwat er and outwat er and t he hi er ar chi cal cut
can be named as wat er . The di sadvant age of t he doubl e bond
r epr esent at i on i s t hat causal i t i es cannot be shown any
l onger .
Fi nal l y, t he DYMOLA model t ype of t he col l ect or can
be devel oped as depi ct ed i n t he next f i gur e ( Fi gur e 5. 3k) .
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- O~COl1~O
COll~O
Fi gur e 5. 3j Bond gr aph of t he col l ect or
{bond gr aph f o r Col l ec t or }
model t y pe COL L
s ubmodel ( MC) Cc ol l
s ubmodel ( SF ) SDOT
s ubmodel ( S pi ) Col l Spi r a l
s ubmodel ( L os s ) Col l L os s
t erm nal SO, Tout , v wi nd, v wat er
cut i nwat er ( el / f l ) , out wat er ( e2/ - f 2)
mai n cut wat er [ i nwat er out wat e r l
connec t SDOT at Col l Spi r al : C
connec t Ccol l at Col l Spi r al : C
connec t Col l Los s at Col l Spi r al : C
SDOT. FO=SO
Col l L os s . Tout = Tout
Col l L os s . vwi nd = vwi nd
Col l Spi r a l . vwat e r = vwat er
end
Fi gur e 5. 3k DYMOLA model t ype of t he col l ect or
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I n usi ng t he f our el ement s, i . e. , mC, mG, mGS and
mRS, t her e ar e some physi cal concept s whi ch must be men-
t i oned. Cel l i er pr ovi des a ver y compr ehensi ve anal ysi s f or
t hese physi cal concept s and, t her ef or e, i t i s used i n t hi s
t hesi s ( Cel l i er , 1990a) .
We can wr i t e t he capaci t y of a body t r anspor t i ng heat
i n a di ssi pat i ve manner as
llT=
8
QQ=dt ( 8 . T) dSdt ( 5. 5)
wher e
~T = t emper at ure di f f er ence
8 = t hermal res i s tance
S = ent r opy
Q = heat
The above equat i on l ooks l i ke Ohm s l aw and i t can be
wr i t t en al so as
dS~T = R dt R = 8 . T ( 5. 6)
The t hermal r esi st ance can now be wr i t t en
8 = ~) . ~) ( 5. 7)
wher e
= speci f i c t hermal conduct ance of t he mat er i al
A = ar ea of cr oss sect i on
= l engt h
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The three el ements mG, mGS and mRS, are model ed based
on t he above concept s and t hei r DYMOLA model t ypes ar e
i l l ust r at ed at t he end of t hi s sect i on al ong w t h t he me
el ement .
NOw, t he capaci t y of t he body t o st or e heat can be
wr i t t en as
dT~ = y
dt( 5. 8)
wher e y = t hermal capaci t ance.
The pr evi ous equat i on can al so be wr i t t en as:
~S = C dTdt
( 5. 9)
wher e C t
T
The t erms t hermal r esi st ance and t hermal
capaci t ance ar e i nt r oduced because of t he t r adi t i onalrel at i onshi p between t emperature and heat al though t hroughout
t hi s t hesi s ent r opy i s used ext ensi vel y.
Cont i nui ng, t he t hermal capaci t ance of a body can be
descr i bed as
-y = c . m 5. 10
wher e
m = mass of t he body
c = speci f i c t hermal capaci t ance of t he mat er i al
Mass can f ur ther be wr i t t en as
m = V 5. 11
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wher e
= densi t y
v = vol ume
and
v = A . dx
Now f r om ( 5. 10) we have
c = ~ = ~c~ ~e~~A~_ ~n=xT T
( 5. 12)
The t i me const ant can now be det ermned:
T = R • C = 8 • Y =c .
( 5. 13)
The l ast equat i on pr ovi des us w t h t he capabi l i t y of det er -
m ni ng t he di mensi ons of bot h t he r esi s t i ve and capaci t i ve
el ement s i n our bond gr aph.
Let us i l l ust r at e t he model i ng by means of t he i - t h
comput at i onal cel l . The equat i ons descr i bi ng such a cel l wer e
devel oped t o be:
dTi 1nSf ( 5. 14)
dt C
nTi = Ti-1 - Ti ( 5. 15)
Sf -1=
nTi ( 5. 16)R
S~x = Sf -1nTi ( 5. 17)Ti
nSf = S f-1 S~x - S~ ( 5. 18)
where Ti i s t he t emper at ur e of t he comput at i onal cel l t o t he
l ef t and Sf i s t he ent ropy f l ow t o t he comput at i onal cel l t o
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{bond gr a ph modul at e d c onduc t anc e}
model t ype MG
mai n cut A e/ f )
t er m nal vel , T out
par amet er a=l . O, b=l . O
l oc al G, Gl
GI = a* vel b
G = Gl / Tout
G* e = f
end
Fi gur e 5. 31 DYMOLA model t ype of mG
(modul at ed w t h T and Vw nd)
{bond gr aph conduc t ance s our ce f o r one di mens i onal cel l }
model t ype MGS
cut A el / f l ) , B e2/ - f 2 )
mai n cut e[ A B)mai n pat h P<A - B>
t er m nal vel
par amet e r a=l . O, b=l . Ol ocal G, GI
GI =a*vel +b
G=f 2* GI
G* e l =f l
f l * el =f 2* e2
end
Fi gur e 5. 3m DYMOLA model t ype of mGS
(modul at ed w th T and Vwat er )
95
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{ Bond Gr aph of a heat modul a t ed r es i s t i ve s our c e )
model t ype MRS
cut A el / f l ) , B e2/ - f 2)
mai n cut C[ A B]mai n pat h P<A - B>
par amet e r t het a =l . O
l oc al R
R = e2* t het a
R* f l = el
el * f l = · e 2* f 2
end
Fi gur e 5. 3n DYMOLA model t ype of mRS
(modul at ed w t h T and 8)
{ Bond Gr aph modul at e d c apac i t o r / c ompl i anc e}
model t ype MC
cut A e / f
pa r ame t e r gamma=I . O
l oc al C
C=gamma/ e
C* der e) = f
end
Fi gur e 5. 30 DYMOLA model t ype of mC(modul at ed w t h T and 7)
{bond gr aph f or a r es i s t i v e s our ce}
model t ype RS
cut A el / f l ) B e2/ - f 2)
mai n cut C[ A B]
mai n pat h P<A - B>
par amet e r R=l . 0
R* f l = el
el * f l = e2* f 2
end
Fi gur e 5. 3p DYMOLA model t ype of RS( el ect r i cal pr i mar y si de)
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t he r i ght ( pl ease see t he bond di agr am on Fi gur e 5. 3a t o
f ol l ow t he above equat i ons) .
One cl ar i f i cat i on shoul d be made whi ch i s t he
f ol l owi ng: The RS el ement s may have both si des, pr i mary and
secondary, as thermal ones and they are model ed as shown i n
Fi gur e 5. 3n. On t he ot her hand, i f t hei r pr i mar y si de i s
el ect r i cal t hen t hei r DYMOLA model t ype i s di f f er ent and i s
shown i n Fi gur e 5. 3p. We ar e goi ng t o meet t hi s case when
desi gni ng t he el ect r i cal backup devi ce f or t he st or age t ank
( see next sect i on) .
5 5 Heat storage Tank Modeling
Af t er t he col l ect or model was made avai l abl e, t he
i mmedi at e next st ep i s t o model t he heat st or age t ank.
Fr equent l y, t he st or age t ank i s r eal i zed as a l ar geand wel l i nsul at ed wat er heat er . However , i n order t o model
a wat er heat er cor r ect l y, we must t ake t he mxi ng t hermo-
dynamcs i nt o account . Thi s makes t he model i ng pr ocedur e
di f f i cul t . Ther ef or e, a sol i d body st or age t ank was used
t oget her wi t h anot her wat er spi r al so t hat t he wat er f r omthe
col l ect or l oop and f r omthe heat er l oop never mx. One wat er
spi r al deposi t s heat i n t he st or age t ank, whi l e t he ot her
pi cks i t up agai n.
I nsi de t he st or age t ank t her e i s a second wat er
spi r al whi ch r epresent s t he heat er wat er l oop pi cki ng up t he
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heat f r om t he st or age t ank. Whenever t he st or age t ank
t emper at ur e f al l s bel ow a cr i t i cal val ue, an i nst al l ed
el ect r i cal heat er heat s t he st or age t ank el ect r i cal l y up t o
t he m ni mum mai nt enance t emper at ur e.
Anot her pump dr i ves t he heat er wat er l oop and t hi s
pump i s swi t ched on whenever t he r oomt emper at ur e f al l s bel ow
20°C dur i ng t he day or 18°C dur i ng t he ni ght . I t i s swi t ched
of f whenever t he room t emperat ure rai ses beyond 22°C dur i ng
t he day or 20°C dur i ng t he ni ght .
Summar i zi ng, t he st or age t ank cont ai ns two wat er
spi r al s, one bel ongi ng t o t he col l ect or wat er l oop and t he
ot her one bel ongi ng t o t he heat er wat er l oop. Thi s i s
depi ct ed i n Fi gur e 5. 4a. Mor eover , an el ect r i cal backup
devi ce i s i nst al l ed and i t i s t ur ned on onl y i f t he t emper a-
t ur e i n t he st or age t ank f al l s bel ow i t s cr i t i cal val ue. I t
i s used onl y dur i ng eveni ng hour s when t he pr i ce of el ec-
t r i ci t y i s l ower . The over al l bond di agr am f or t he st or age
t ank i s shown i n Fi gur e 5. 4b. The mC el ement r epr esent s t he
heat capaci t y of t he st or age t ank, wher eas t he f l ow sour ce
t oget her wi t h t he RS el ement denot e t he backup devi ce. The
pr i mar y si de of t hi s r esi st i ve sour ce i s el ect r i cal whi l e t he
secondar y si de i s t herm c.
As we not i ce, t he st or age t ank i s a 4- por t el ement .
When wr i t i ng i t s DYMOLA model t ype, we combi ne t he cut
i nwat er l wi t h t he cut outwat er l t o t he hi er ar chi cal cut
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S p i r a l s
e a t e ro l l e c t o r
Fi gur e 5. 4a The st or age t ank w t h t he col l ect or wat er l oopand heat er wat er l oop
o 0
1 Sp1k-O~Spl
1 o 0
SF~ RS~O
1
Fi gur e 5. 4b Bond gr aph of t he st or age t ank
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- .
100
i nwat er , wher eas we combi ne t he cut out wat er 2 w t h t he cut
i nwat er 2 t o t he hi er ar chi cal cut out wat er . By decl ar i ng a
mai n pat h wat er , a l ogi cal br i dge has been cr eat ed f r om t he
hi er ar chi cal cut i nwat er t o t he hi er ar chi cal cut out wat er .
Thi s i s depi ct ed i n t he f ol l ow ng f i gur e ( Fi gur e 5. 4c) .
5 6 water Loop Modeling
An exampl e of a convect i ve mechani sm i n our sol ar
house i s t he t ranspor t of t he heat f r om t he sol ar col l ect or
t o t he wat er heat er connect ed by a pi pe cont ai ni ng wat er . The
wat er i s ci r cul at ed f r om t he wat er heat er ( st or age t ank) t o
t he col l ect or and back by a pump.
We have al r eady seen t wo wat er l oops, t he col l ect or
wat er l oop and t he heat er wat er l oop, and bot h ar e model ed
exact l y i n t he same way.
Each of t he pi pes i s model ed by t hr ee one-di mensi onal
cel l s connect ed i n ser i es as i l l ust rat ed i n t he bond di agr am
( see Fi gur e 5. 5a) . The one- di mensi onal cel l has been
devel oped pr evi ousl y. We shal l assume t hat t he pi pes ar e
t herm cal l y wel l i nsul at ed, t hat i s, t her e i s not any l ost
heat t o t he sur roundi ngs on t he way. As shown, i t i s anot her
4- por t el ement . I n devel opi ng i t s DYMOLA model t ype, we shal l
combi ne t he cut i nwat er 1 w t h t he cut out wat er 2 t o t he
hi er ar chi cal cut i nwat er . I n addi t i on, we shal l combi ne t he
cut out wat er l w t h t he cut i nwat er 2 t o t he hi er ar chi cal cut
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{bond gr aph s t or age t ank}
model t ype ST
s ubmodel SF ) SOOTs ubmodel RS) Rl R=l O. O)
submodel MC) Ct ank gamma=9. 0E+4)
s ubmodel Spi ) Spi t a nk l , Spi t a nk2
t e r m nal SO, vwat e r
cut i nwat er l e l / f l ) , out wat er l e2/ - f 2)
cut i nwat er 2 e3/ f 3) , out wat er 2 e4/ - f 4 )
mai n cut i nwat er [ i nwat er l out wat er l ]mai n cut out wat er [ o ut wat e r 2 i nwat e r 2]
mai n pat h wat er <i nwat er - out wat er >
connec t Spi t ank l : C at Spi t ank2: C
connec t Ct ank at Spi t ank l : C
connec t Rl f r om SOOT t o Spi t ank l : C
SDOT. FO=SO
Spi t ankl . vwat er = vwat e r
Spi t a nk2. vwat e r = vwat e r
end
Fi gur e 5. 4c DYMOLA model t ype of t he st or age t ank
101
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Ok 1Dk Ok 1Dk Ok 1DJ c O
I I I
Fi gur e 5. 5a Bond di agr am f or wat er l oop
{bond gr aph w t r l oop heat e r +col l ec t or ) }
model t ype WL
5ubmodel oneD) oneDl , oneD2, oneD3 , oneD4 , oneDS, oneD6
node nl , n2, n3, n4
t er m nal vwat er
cut i nwat er l el / f l ) , out wat er l e2/ - f 2)
cut i nwat er 2 e3/ f 3 ) , out wat er 2 e4/ - f 4)
mai n cut i nwat er [ i nwat er l out wat er 2]
mai n cut out wat er [ i nwat e r 2 out wat er l ]
mai n pat h wat er <i nwat er - out wat er >
connec t oneDl f r om i nwat er l t o nlc onnect oneD2 f r om nl t o n2
connect oneD3 f r om n2 t o out wat e r l
connec t oneD4 f r om i nwat e r 2 t o n3connec t oneD5 f r om n3 t o n4
connect oneD6 f r om n4 t o out wat er 2
oneDl . vwat er =vwat er
oneD2. vwat er =vwat er
oneD3. vwat er =vwat er
oneD4. vwat er =vwat er
oneDS. vwat er =vwat er
oneD6. vwat er =vwat er
end
Fi gure 5. 5b DYMOLA model t ype f or water l oop
102
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outwater . Moreover , we decl are a mai n path water creat i ng a
l ogi cal br i dge f r om t he hi er ar chi cal cut i nwat er t o t he
hi erar chi cal cut outwater ( see Fi gure 5. 5b) .
Assum ng t hat t her e i s no ai r i n t he pi pe and t he
wat er i n i t i s t ot al l y i ncompr essi bl e, sever al i mpl i ci t
physi cal si mpl i f i cat i ons can be t aken i nt o consi der at i on.
Thi s l eads t o t he concl usi on t hat t he wat er f l ow vi a t he
whol e pi pe has a const ant vel oci t y Vw•
The hydr aul i c f l ow i sexpressed i n ms- 1 denot ed by ~v and t he vol ume of wat er i n a
one- di mensi onal cel l i s V = A . fiX. Thus, t he amount of
ent ropy l eavi ng t he i - t h cel l per t i me uni t t o t he r i ght i s
gi ven by
~v8 i out = fl8i V 5. 19
whi ch can al so be wr i t t en as
8 f out = c ~v Ti 5.20)
Dur i ng the same t i me, a si m l ar amount of heat i s t r anspor t ed
i nt o t he cel l f r om i t s l ef t nei ghbor bei ng
8 f i n = C . ~ v ) T i 1 5. 21
Combi ng t he pr evi ous t wo equat i ons, we concl ude t hat
8 f conv = G conv flTi
and
G conv= C ~v
V 5.22)
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Consequent l y, convect i on i s si mpl y a second convect i ve
r esi st ance bei ng connect ed i n par al l el w t h t he conduct i ve
r esi s t ance. Ther ef or e, convect i on augment s t he t her mal
conducti vi t y.
As ment i oned bef or e, sever al si mpl i f i cat i ons wer e
made. I n r eal i t y, t her e i s f r i ct i on bet ween t he l i qui d and
t he wal l , and f r i ct i on w t hi n t he l i qui d. We woul d not i ce
t hat i n t hi s case t he l i qui d f l ows f ast er at t he cent er of
t he pi pe and sl ower near t he wal l . The hydr aul i c f r i ct i on i s
a di ssi pat i ve pr ocess pr oduci ng mor e heat and t her eby mor e
ent r opy sour ces shoul d be appl i ed t o t he t hermal uni t .
Fur t hermor e, i f t he assumpt i on of i ncompr essi bi l i t y
wer e not made, t hen t he whol e si t uat i on woul d be much mor e
compl i cat ed. I n t hi s case, we woul d need t o t ake i nt o con-
si der at i on t he pneumat i c pr ocess besi des t he t hermal pr ocess.
The pneumat i c process generates a t i me- and space- dependent
f l ow r at e ~v( t , x} whi ch can be used t o modul at e t he convect i ve
r esi st ance of t he t her mal uni t . Over al l , t he whol e si t uat i on
becomes ver y i nvol ved.
5. 7 Habi t abl e Space Model i ng
I t was deci ded t hat t he habi t abl e space ( house) woul d
be a cube w t h di mensi ons 10m x 10m x 10m ( see Fi gur e 5. 6c) ,
mai nl y f or r easons of si mpl i ci t y.
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10S
However , bef or e st ar t i ng t o model t he house, a t hr ee-
di mensi onal cel l as wel l as a t wo- di mensi onal cel l have been
devel oped. The concept of a one- di mensi onal cel l whi ch has
been devel oped pr evi ousl y can be ext ended t o t he t wo- and
t hr ee- di mensi onal case. So, l et us assume t hat each t hr ee-
di mensi onal cel l consi st s of one capaci t or and t hr ee
r esi st or s, one t o i t s l ef t , one t o i t s f ront and one bel ow as
depi ct ed i n Fi gur e S. 6a t oget her w t h i t s bond di agr am I t
can al so be seen f r om t he f i gur e t hat t he necessi t y t o at tach
ever y el ement t o a- j unct i ons f or DYMOLA model i ng has been
t aken i nt o consi der at i on. The DYMOLA model t ype CEL
descr i bi ng t he t hr ee- di mensi onal cel l has been devel oped as
wel l and i t i s depi ct ed i n Fi gur e S. 6b. The t wo- di mensi onal
case consi st s of t wo r esi st or s ( xy, yz, xz di r ect i ons) and a
capaci t or .
We shal l assume t hat t he ent i r e house consi st s of one
r oom onl y and t hat a si ngl e l ar ge r adi at or i s used f or
heat i ng pur poses. The r adi at or exchanges heat w t h t he r oom
i n a par t l y conduct i ve and par t l y convect i ve manner . I t i s
at t ached t o t he l ef t wal l of t he house somewher e cl ose t o t he
f l oor so t hat t he heat i nput t akes pl ace at t he l ef t l ow
out s i de center t hree- di mensi onal cel l . Because the di mensi ons
of t he r adi at or ar e much smal l er t han t hose of t he house, a
deci si on was made not t o model t he r adi at or . Ther ef or e, t he
out wat er l of t he heat er wat er l oop has been si mpl y connect ed
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w t h t he i nwat er 2 of t he heat er wat er l oop. Mor eover , anot her
heat exchanger has been at t ached at t hi s node w t h t he
r esponsi bi l i t y of exchangi ng t he heat bet ween t he heat er
wat er l oop and t he house.
Fr om Fi gur e S. 6c, i t can be seen t hat t he house has
27 nodes. At t hese nodes one- , t wo- and t hr ee- di mensi onal
cel l s ar e pl aced w t h t he except i on of t he f i r st node. So,
nodes 2 and 3 ar e one- di mensi onal i n x di r ect i on, 4 and 7 ar e
one-di mensi onal i n y di r ect i on, 10 and 19 are one-di mensi onal
i n z di r ect i on. The t wo- di mensi onal cel l s ar e l ocat ed at
nodes 5, 6, 8, 9 ( xy di rect i on) , 11, 12, 20, 21 ( xz di r ec-
t i on) and 13, 16, 22, 25 ( yz di r ect i on) . Fi nal l y t hr ee-
di mensi onal cel l s ar e pl aced at t he r emai ni ng nodes, i . e. ,
14, 15, 17, 18, 23, 24, 26, 27. These nodes ar e connect ed
t hr ough t hei r pat hs accor di ngl y as depi ct ed i n Fi gur e 5. 6d.
We need al so t o pl ace our heat sour ce and one addi t i onal
capaci t or at t he f i r st node whi ch i s act ual l y t he ex poi nt of
t he second node. The house l oses heat t hr ough t he f our wal l s
and t hr ough t he r oof , but not t hr ough t he f l oor so t hat at
nodes 5 and 14 t her e ar e no l osses. The pr evi ousl y devel oped
l oss el ement s ar e at t ached t o each of t he a- j unct i ons as
appr opr i at e. I n t he case of a cel l adj acent t o t wo or t hr ee
out si de wal l s, we at tach one combi ned l oss el ement t o t he
cor respondi ng node onl y because of t he emergence of al gebrai c
l oops. Ther ef or e, nodes 2, 4, 6, 8, 11, 13, 15, 17, 23 have
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Fi gur e 5. 6a Thr ee- di mensi onal di f f usi on cel l(RS el ements are actual l y mRS el ements )
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bond gr aph f or a t hr ee di mens i o nal c el l )
model t ype CEL
s ubmo del MRS) Rx t he t a =O. 5) , Ry t he t a =O. 5) , - >
Rz t heta=O. 5 )
s ubmode l MC) C gamr na=l 5 2310. 0 )s ubmodel bond) Bx l , Bx 2, Bxa, By l , By 2, Bya, Bz l , Bz 2, Bz anode Nx , Nx a, Ny , Ny a, Nz , Nz a
cut Cx ex/ f x) , Cy ey/ f y) , Cz ez / f z ) , Ci e i / - f i )pat h Px<Cx - Ci >, Py<Cy - Ci >, Pz<Cz - Ci >
connec t Bxl f r om Cx t o Nx
connect By l f r o m Cy t o Ny
connect Bz l f r o m Cz t o Nzconnec t Bx2 f r om Nx t o Ci
connec t By2 f r om Ny t o Ci
connect Bz 2 f r o m Nz t o Ci
c onnec t Bxa f r o m Nx t o Nxaconnec t Bya f r om Ny t o Nya
connec t Bz a f r om Nz t o Nz a
connec t Rx f r om Nxa t o Ci
connec t Ry f r om Nya t o Ci
connec t Rz f r om Nz a t o Ciconnec t C at Ci
end
Fi gur e 5. 6b DYMOLA model t ype of a t hr ee- di mensi onal cel l
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19
1810
1
---_ 5 > < :. • .. • . . • .. • . . • .. • .. • . . • .
- - 9
3
Fi gur e 5. 6c The house r oom repr esent ed as a 10x10x10 cube
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model SPACE
submodel S F) Ts p
submodel MC) C gamma=152310. 0 )
submode 1 L S1) 12, 14, 16, 18, 111, 113, 115 , 117, 123
submode 1 L S2) 11, 13, 17, 19, 110, 112, 116 , 118, 120, 122, 124, 126
s ubmodel L S3) 119 , 121 , 125 , 127
s ubmodel CXD) d2, d3s ubmode1 CYD) d4, d7
s ubmode1 CZD) dl 0, d19submode1 XYC) d5, d6, d8, d9
s ubmode 1 XZC) d11, d12, d20, d21
submode 1 YZC) d13, d16 , d22, d25
s ubmode1 CE L) d14, d15 , d17, d18, d23, d24, d26, d27
i nput Tout , vwi nd, SO
out put yl
c onnec t Px) d2- d3, d5- d6, d8- d 9, - >
dl l - d12, dI 4- d15, d17- d18, - >d20- d21 , d23- d24 , d26- d27
connec t Py) d4- d7, d5- d8, d6- d9, - >dI 3- dI 6, d14- dI 7, dI 5- dI 8, - >
d22- d25, d23 - d26, d24 - d27
connec t pz ) dl 0- dI 9, d11- d20, dI 2- d21, - >
d13- d 22, d14 - d 23 , dI 5 - d 24, - >d I 6 - d25 , d17 - d26 , d18 - d27
connect 11 at d2: Cx
c onnec t 12 at d2: Ci
connec t 13 at d3: Ci
c onnec t 14 at d4: Ci
c onnec t 16 at d6: Ci
connec t 17 at d7: Ci
c onnec t 18 at d8: Ci
c onnec t 19 at d9: Ci
c onnec t 110 at dl O: Ci
connec t 111 at dl l : Ci
c onnec t 112 at d12: Ciconnec t 113 at dl 3: Ci
connect 115 at dl 5: Ci
connect 116 at d16: Ci
connect 117 at d17: Ci
conne c t 118 at d18: Ci
Fi gur e 5. 6d DYMOLA model t ype of t he SPACE ( house)
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connec t 119 at d19: Ci
c onnec t 120 at d20: Ci
c onnec t 121 at d21: Ci
c onnec t 122 at d22: Ci
c onnec t 123 at d23: Ci
c onnec t 124 at d24: Ci
c onnec t 125 at d25: Ci
c onnec t 126 at d26: Ci
c onnec t 127 at d27: Ci
c onnec t Ts p at d2: Cx
connec t C at d2: Cx
connec t d5: Cx at d4: Ci
c onnec t dB: Cx at d7: Ci
c onnec t dl l : Cx at dl 0 : Cic onnec t dl 4: Cx at d13: Ci
c onnec t d17: Cx at dl 6 : Cic onnec t d20: Cx at dl 9: Ci
c onnec t d23: Cx at d22: Ci
c onnec t d26: Cx at d25: Ci
c onnec t d4: Cy at d2: Cx
connec t d5: Cy at d2: Ci
connec t d6: Cy at d3: Ci
c onnec t dl 3: Cy at dI O: Ci
c onnec t dl 4 : Cy at dl l : Ci
c onnec t dl 5 : Cy at dl 2 : Ci
c onnec t d22: Cy at dl 9: Ci
c onnec t d23: Cy at d20: Cic onnec t d24: Cy at d21: Ci
c onnec t dI O: Cz at d2: Cx
connec t dl 1 : Cz at d2: Ci
c onnec t dl 2 : Cz at d3: Ci
c onnec t dl 3 : Cz at d4: Ci
c onnec t dl 4: Cz at d5: Ci
c onnec t dl 5: Cz at d6: Cic onnec t d16: Cz at d7: Ci
connec t d17: Cz at dB: Ci
c onnec t dl B: Cz at d9: Ci
I I . Tout = Tout
I 2. T out = ToutI 3. T out = Tout
14. Tout = Tout
16. Tout = Tout
17. Tout = Tout
I B. T out = Tout
Fi gur e 5. 6d ( Cont i nued)
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19. Tout = Tout110. Tout = Tout
I 1I . Tout = Tout
112 . Tout = Tout
113 . Tout = Tout115. Tout = Tout
116. Tout = Tout
117. Tout = Tout
118. Tout = Tout
119. Tout = Tout
120. Tout = Tout
121. To ut = Tout
122 . Tout = Tout
123. Tout = Tout
124 . Tout = Tout
12S. Tout = Tout
126. Tout = Tout
127. T out = Tout
11 . vwi nd = vwi nd 117. vwi nd = vwi nd12 . vwi nd = vwi nd 118. vwi nd = vwi nd13. vwi nd = vwi nd 119 . vwi nd = vwi nd14. vwi nd = vwi nd 120 . vwi nd = vwi nd16. vwi nd = vwi nd 121. vwi nd = vwi nd17. vwi nd = vwi nd 122 . vwi nd = vwi nd18 . vwi nd = vwi nd 123 . vwi nd = vwi nd19. vwi nd = vwi nd 124 . vwi nd = vwi nd110. vwi nd = vwi nd 125. vwi nd = vwi nd111. vwi nd = vwi nd 126. vwi nd = vwi nd112 . vwi nd = vwi nd 127 . vwi nd = vwi nd
113 . vwi nd = vwi nd115. vwi nd = vwi nd Tsp. FO = SO
116. vwi nd = vwi nd yl = d ei
end
Fi gur e 5. 6d ( Cont i nued)
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a l *MG l oss el ement , nodes 1, 3, 7, 9, 10, 12, 16, 18, 20,
22, 24, 26 have a 2*MG l oss el ement and nodes 19, 21, 25, 27
have a 3*MG l oss el ement . ( The r eader m ght go back t o Fi gur e
5. 3h and see t he l oss el ement . ) The cor respondi ng DYMOLA
model t ype f or t he house has been named SPACE and i t i s
depi ct ed i n Fi gur e 5. 6d.
Thi s concl udes t he model i ng of al l t he par t s of t he
sol ar house.
5 8 The Total Solar Heated House
The over al l syst em i s a ser i es connect i on of t he
previ ousl y pr esent ed aggregat ed bond graph el ement s , t hat i s,
t he col l ect or , t he col l ect or wat er l oop, t he st or age t ank,
t he heat er wat er l oop, t he heat exchanger and t he house. Thi s
i s depi ct ed i n Fi gur e 5. 7.
COll..: ::aWL~ST~ WL~HE l.House
Fi gur e 5. 7 Aggr egat ed bond gr aph of t he over al l syst em
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5. 9 Choosi ng Appr opr i at e Par amet er s f or Anal yzi ngt he Ef f ect i veness of Our Syst em
unt i l now, t he model i ng of t he t ot al sol ar house has
been di scussed. Now we ar e r eady t o t est t he ver sat i l i t y of
DYMOLA to descr i be such a compl ex physi cal syst em af t er i t
has been model ed by t he bond gr aph met hodol ogy.
The best appr oach t o si mul at e i s t o st ar t f r om t he
habi t abl e space ( house i t sel f ) as i ndi cat ed i n t he t ot al
aggr egat ed f i gur e. We shal l i magi ne t hat t her e i s an
ar bi t r ar y heat sour ce heat i ng t he house, assumng t hat t he
i ni t i al t emper at ur e i nsi de t he house i s l SDC ( 2SSDK) . Our
goal i s t o det ermne t he t i me t hat i t t akes f or t he t emper a-
t ur e t o r each i t s st eady- st at e val ue i n var i ous l ocat i ons
i ns i de t he house.
The DYMOLA model t ype SPACE (model SPACE i n t hi s case
because t he house i s our mai n syst em now) has been used t o
gener at e t he DESI RE pr ogr am The same pr ocedur e as descr i bed
i n Chapt er 4 has been used. However , t he PC was unabl e t o
gener at e t he DESI RE pr ogr am because i t was exceedi ng i t s
memor y ( heap) capabi l i t y.
Theref ore, a deci si on was made to use the VAX. Usi ng
t he VAX, we wer e abl e t o gener at e a SI MNON pr ogr am ( cur -
r ent 1y, t he VAX ver si on does not pr ovi de a DESI RE pr ogr am
generat i on capabi l i t y yet ) . However , t he conver si on of t he
SI MNON pr ogr am i nt o a DESI RE pr ogr am i s not a di f f i cul t
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pr ocedur e. Af t er doi ng t hat , we ar e r eady t o si mul at e t he
house ( habi t abl e space) .
The f ol l owi ng el ement s must be cal cul at ed bef or e
pr oceedi ng: t he modul at ed r esi st i ve sour ce ( t het a) , t he
modul at ed capaci t ance ( gamma) bei ng i nsi de t he one- , t wo- and
t hr ee- di mensi onal cel l s. These val ues wi l l be t he same
ever ywher e i n t he house. Mor eover , t he modul at ed conduct ance
i n t he l oss el ement s ( a and b) must be eval uat ed. Pl ease
r ef er t o Fi gur es 5. 3n, 5. 30 and 5. 31 wher e t he par amet er
val ues f or t hese el ement s are speci f i ed i n parentheses.
A l ogi cal and econom cal heat source ( ent ropy source)
i s f ound t o be 20 J / K. Our i nt ui t i on was based on an aver age
mont hl y ut i l i t y bi l l t hat peopl e spend f or heat i ng t hei r
house dur i ng wi nt er t i me.
I t i s f ound t hat t he a, b and t het a par amet er val ues
af f ect t he over al l heat i ng of t he house. Fi gur e 5. 3c hel ps
us t o f i nd a, b. The angl e must be kept smal l ar ound 25° and
b i s appr oxi mat el y one- hal f of t he t angent of t hat angl e.
These a and b val ues det erm ne how wel l t he house i s i nsu-
l at ed. Moreover , t he val ue of t het a depends on t he ai r i nsi de
t he house. Formul a 5. 7 det ermnes t he val ue of t het a but t he
physi cal constant f or ai r i s not r el i abl e. We have used
some f l exi bi l i t y i n deci di ng t he val ue of t het a whi ch i s
about 0. 5 sec. K/ J .
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.
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Fi nal l y, t he val ue of gamma can be f ound usi ngf ormul a 5. 12 and i t i s f ound t o be 152310 J j K ( not t o be
conf used wi t h ent r opy) .
Havi ng f ound al l t he necessar y par amet er s, t he
s i mul at i on can now be per f ormed. We s i mul ated t he house, and
di spl ayed the temperature i n the vi ci ni t y of t he heat source,
and al so at f ar t her away nodes. The f ar t hest one, node 27,
caused t he most pr obl ems and di d not gi ve sat i s f act or y
r esul t s.
Var i ous r esul t s ar e i n t he f ol l owi ng t abl e: ( see
Appendi x A f or gr aphs wi t h y1, y2, y3, y4 and y5 cor r espond-
i ng t o nodes 16, 20, 22, 9 and 26 r espect i vel y. Not i ce t hat
y2 i s t he same as y3 because of symmet r y of t he nodes,
t her ef or e onl y one of t hei r gr aphs i s shown) .
Node Tsteady state 0C) Ti me ( see)
3 77 1. 40E+59 24 2. 00E+5
14 63 1. 50E+516 3S 1. 50E+517 29 1. SOE+520 32 1. 50E+5
22 321. 50E+5
26 15 1. SOE+527 10 1. 70E+5
Fi gur e 5. S Tabl e of some r esul t s
The out s i de t emper at ure was assumed t o be O· C.
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From the above resul t s , we can observe that the temperature
i n most of t he nodes of t he house i s not consi st ent . I n t he
vi ci ni t y of t he heat sour ce, t he t emperat ur es ar e ver y hi gh,
whereas i n the f ar t hes t nodes of the house, the temperatures
ar e l ow. Thi s makes us bel i eve t hat t he heat di ssi pat i on
t hr ough t he house ( e.g. by means of convect i on) i s not
model ed cor r ect l y.
The t emper at ur e at ever y node r eaches i t s st eady-st at e val ue i n a l i t t l e over a day. Thi s makes sense. I t
t akes a l ong t i me t o heat t he house t o i t s st eady- st at e
t emper at ur e wi t h an economcal heat er such as t he one we
used.
The next st ep was t o add t he heat er wat er l oop and
constant temperature sour ce at the storage tank to produce
t he 20 J / K ent ropy ( heat ) source. Never t hel ess, combi ni ng al l
t he DYMOLA model t ypes t oget her , t he whol e pr ogr am wi l l
become ver y l ar ge so we deci ded t o st op t he si mul at i on
anal ysi s. I t i s t r ue, however , t hat by havi ng a computer wi t h
enough heap (memor y) t hat can handl e such l ar ge pr ogr ams, t he
whol e s i mul at i on anal ys i s can be per f ormed unt i l we reach t he
col l ect or . At t he end, we wi l l have a ver y l ar ge DYMOLApr ogr am wi t h al l t he hi er ar chi cal st r uct ur es of t he sol ar
house connect ed t oget her .
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.
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CHAPTER 6
CONCLUSION
Thi s t hesi s t ouches on a moder n and advanced
model i ng- s i mul at i on t echni que appl i ed t o a l arge and compl ex
physi cal system - the sol ar heated house.
The bond gr aph model i ng met hodol ogy has been st udi ed
extensi vel y as wel l as a sof tware tool cal l ed DYMOLA desi gned
t o i mpl ement bond graphs. How wel l t hey wor k t oget her was
demonstr at ed i n Chapt er 4.
Bond gr aphs wer e successf ul i n pr ovi di ng us wi t h a
compl et e and easi l y comprehens i bl e model of the sol ar house,
a rel at i vel y compl i cat ed syst em Furt hermore, DYMOLA proved
t o be a sui t abl e sof twar e t ool f or i mpl ement i ng t he
hi er ar chi cal bond gr aphs encount er ed i n t he syst em Bot h
t ool s, l i ke SPI CE, can be combi ned t oget her f or st udyi ng t he
behavi or of several l ess compl ex syst ems such as el ect r i cal
and mechani cal ones.
On t he negat i ve aspect , bond gr aphs as t hey ar e
devel oped t oday, ar e not sui t abl e f or model i ng di st r i but ed
parameter syst ems i n several space di mensi ons. As al l ot her
gr aphi cal t echni ques, bond gr aphs become cl umsy when appl i ed
t o di st r i but ed par amet er pr obl ems i n mor e t han one space
di mensi on.
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Ther e ar e many oppor t uni t i es f or r esear ch. Bot h bond
gr aphs and DYMOLA can be f ur t her devel oped so t hat t he st udy
of compl ex syst ems can become mor e f easi bl e and at t r act i ve t o
t he r esear cher.
Thi s t hesi s has pr ovi ded new i nsi ght i nt o t he pr ocess
of model i ng compl ex physi cal syst ems. For t he f i r st t i me, t he
bond gr aph model i ng t echni que was expanded t o hi er ar chi cal
model descr i pt i ons. I t has been shown t hat t he gener al
pur pose cont i nuous- s yst em model i ng l anguage DYMOLA can be
ef f ect i vel y used t o descr i be hi er ar chi cal nonl i near bond
graphs.
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APPENDIX
GRAPHS
+ ~ : : .
~ : :: . .~ :
· .· . y1
~ ~ ~ j ~
· .
. .e : : 8 8 ~ · ~ B e · · · · · · · · · · · · · · · · · · · · ·. · : · · ·. . · · · · · . . · i : 5 · ~ B 5 ·. . · . . · · . . · · · · i~ · ~B5
scale = 3 . 2 8 e B l 111II111 ~ 1 ~ ~ 3 ~ ~ us. t
Si mul at i on r esul t s at var i ous nodes
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REFERENCES
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