ene gies
A icle
Pe o man and Simple Nume ical Modeling o
Dis ic Hea ing Pipes wi h Hea Accumula ion
Libo Kudela * , Radomi Chylek and Ji i Pospisil
Ene gy Ins i u e, Facul y o Mechanical Enginee ing, B no Uni e si y o Technology—VUT B no,
Technicka 2896/2, 61669 B no, Czech Republic; Radomi [email p o ec ed] (R.C.); ji i.pospisil@ u b .cz (J.P.)
*Co espondence: Libo .Kudela@ u b .cz; Tel.: +420-54-114-2579
Recei ed: 24 Janua y 2019; Accep ed: 13 Feb ua y 2019; Published: 16 Feb ua y 2019
Abs ac :
This pape compa es app oaches o accu a e nume ical modeling o ansien s in he pipe
elemen o dis ic hea ing sys ems. The dis ibu ion g id i sel a ec s he hea low dynamics o a
dis ic hea ing ne wo k, which subsequen ly go e ns he hea delays and en i e e iciency o he
dis ibu ion. Fo an e icien con ol o he ne wo k, a con ol sys em mus be able o p edic how
“ empe a u e wa es” mo e h ough he ne wo k. This p edic ion mus be su icien ly accu a e o
eal- ime compu a ions o ope a ional pa ame e s. Fu u e con ol sys ems may also bene i om
he accumula ion capabili ies o pipes. In his a icle, he key physical phenomena a ec ing he
ansien s in pipes we e iden i ied, and an e icien nume ical model o abo eg ound dis ic hea ing
pipe wi h hea accumula ion was de eloped. The model used analy ical me hods o he e alua ion
o sou ce e ms. Physics o hea ans e in he pipe shells was cap u ed by one-dimensional ini e
elemen me hod ha is based on he s eady-s a e solu ion. Simple ad ec ion scheme was used o
disc e iza ion o he luid egion. Me hod o lines and ime in eg a ion was used o ma ching. The
complexi y o simula ed physical phenomena was highly lexible and allowed o ade accu acy
o compu a ional ime. In compa ison wi h he e y inely disc e ized model, highly compa able
ansien s we e ob ained e en o he hick accumula ion wall.
Keywo ds:
dis ic hea ing; hea accumula ion; pipe; nume ical model; Modelica language; Julia
language; pe o mance
1. In oduc ion
In he nex ew decades, he dis ic hea ing (DH) will unde go subs an ial changes. The majo
ope a ional changes associa ed wi h he 4 h gene a ion o DH a e lowe empe a u es in dis ibu ion
g ids ( he hea -ca ie will be liquid wa e ), enhancemen o accumula ion, and use o hea pumps o
o he empe a u e boos e s [
1
,
2
]. U iliza ion o enewable sou ces and was e hea (bo h o en luc ua e)
aim a he educ ion o ca bon dioxide emissions and ossil uel consump ion [
3
]. Consume s, supplie s,
and ne wo k p o ide s will become membe s o he mul ile el hea ma ke . The p oduc ion and
consump ion o hea o cold can be shi ed be ween seasons using la ge long- e m he mal s o age,
which has a lowe ela i e p ice han smalle ones [
4
]. In such a complica ed a angemen , he ne wo k
mus be su icien ly lexible, p edic i e, and in elligen .
Ad anced and holis ic ope a ional op imiza ion o he con ol s a egies will become c ucial.
Al hough he ope a ional op imiza ion is usually being pe o med on al eady exis ing g ids wi h gi en
opologies, he simul aneous op imiza ion o bo h g id opology and ope a ion may be mo e bene icial
han a sepa a e app oach. In o he wo ds, some g id opologies can allow be e con ol s a egies
while o he s do no ha e o. The ideal con ol sys em should ake long- e m consequences in o accoun
(delays a e mos o en a ec ed by he mal ine ia) [
5
]. The con olle has o ind he con ol ac ions by
minimizing he sum o all he penal ies (ene gy losses, demand misma ches, empo a y p ice, e c.)
Ene gies 2019,12, 633; doi:10.3390/en12040633 www.mdpi.com/jou nal/ene gies
Ene gies 2019,12, 633 2 o 23
de ined inside he models o he indi idual componen s. This is called ope a ional op imiza ions.
I is c ucial o i as many simula ions as possible o he con ol ho izon (se e al minu es) o ind he
op imal alues o he disc e e con ol a iables o he p edic ion ho izon (se e al hou s, days, weeks
o e en mon hs). This depends on he models being as and accu a e. Rega ding he pipes hemsel es,
he co ec p edic ion o s ep change p opaga ion is mos impo an as i de e mines he delay in
empe a u e deli e y. These delays a e no only caused by pu e ad ec ion h ough he sys em.
To ensu e he su icien pe o mance o he model, he ela i e simula ion ime ( a io o physical
ime and compu a ional ime) is o be educed. This migh be achie ed by a combina ion o
se e al p inciples, such as lowe ing he numbe o used spa ial de i a i es (e.g., educing o he
one-dimensional desc ip ion) o a oiding some o he disc e iza ions al oge he by using he analy ic
desc ip ion o hea ans e [
6
] and ic ion [
7
] o all egimes in he pipes. When i comes o he
hyd aulic calcula ions, which a e necessa y o e alua e mass lows o hea ca ie in he b anches, i is
usually conside ed sa is ac o y o use he s a ic model, since he p essu e wa es sp ead signi ican ly
as e han he mal wa es. I is also help ul o de i e speci ic solu ions ha ake ad an age o
known ac s a he han hose ha co e a wide ange o possible op ions (e.g., he p esump ion o
incomp essible luid simpli ies he dealing wi h he mass conse a ion laws signi ican ly).
In p e ious wo k, a ew main app oaches ha deal wi h simula ions o DH g ids a e u ilized.
The i s g oup is based on he p esump ion ha he g id ope a es mos ly in s eady-s a e. Ad anced
echniques based on his can be ound a [
8
], whe e au ho s de elop a model and a me hod o
pa ame e calib a ion ha allows o be e i he measu ed da a o simula ion.
Ano he g ea example o s eady-s a e es ima ion o DH g id is [
9
]. In his pape , he au ho s use
au oma ed cus ome me e sys em o e alua e mass lows, empe a u es and consequen ly hea losses
in pipes o a DH ne wo k. The op imiza ion algo i hm is u ilized o es ima e empe a u es in he pipes,
so simula ed empe a u es in he cus ome 's loca ion i he measu ed da a.
The second g oup o models, which include dynamic e ec s o some ex en , is based on highly
simpli ied models ha di ec ly link inle empe a u e signal o he ou le signal. In he so-called node
me hod [
10
], nodes ha a e connec ed by pipes o known geome ical and he mal p ope ies ep esen
he ne wo k. Based on mass lows in he pipes and he empe a u es o wa e in indi idual nodes, he
me hod s eps h ough he ime o e alua e empe a u e in all he nodes o all he ime s eps. I u ilizes
known ime delays caused by ad ec ion and he mal ine ias o he pipes.
Au ho s o [
11
] de elop new so-called unc ion me hod o he e alua ion o he mal ansien s
in DH g ids. The me hod is based on he inse ion o a Fou ie se ies in o he simpli ied Pa ial
Di e en ial Equa ion (PDE) o he luid and solid egions o he ci cula pipe o de i e he ela ionship
be ween inle and ou le empe a u e. Time delay and ela i e a enua ions a e inco po a ed in o he
model. The new me hod is epo ed o be abou 37% as e han he s anda d node me hod. Model is
alida ed in eal DH sys em. No axial u bulen di usion is conside ed.
Pu e ad ec ion o inle ime o he wa e o he pipe is used in [
12
]. This alue is hen sub ac ed
om he ime when he liquid lea es he pipe. The esul an alue ep esen s he ime he liquid
spends inside he pipe which is hen used o es ima ing he empe a u e d op due o hea loss. To
inco po a e he mal ine ia, equi alen mixing olume is placed a he ou le . This esul s in a so-called
plug- low model. The model accu a ely simula ed 168 h o ope a ion o dis ic hea ing g id in Pongau
(Aus ia) in less han 2 s. No axial di usion is inco po a ed.
Au ho s o [
13
] ocus on ansien s in pipes wi h s able mass lows. They use s eady-s a e
axial empe a u e dis ibu ions and con olu ion wi h he axial di usion ke nel o a i e a an explici
ma hema ical unc ion ha desc ibes he empo a y ansien beha io . Thei solu ion is hen compa ed
wi h he nume ical solu ion o he app op ia e Pa ial Di e en ial Equa ion (PDE) wi h posi i e esul s.
No dynamic e-accumula ion in he pipe walls is included.
A s eady-s a e hea ans e model is combined wi h a a iable anspo delay model in [
14
] o
ob ain a as model capable o simula ions wi h a iable lows. The model esul ed in app oxima ely
4000 imes less in ensi e compu a ion in compa ison wi h one-dimensional PDE app oach (bo h
Ene gies 2019,12, 633 3 o 23
models we e de eloped in Ma lab/Simulink). This speedup ac o ises wi h he u iliza ion o la ge
ime s eps bu a he cos o highe e o . Axial di usion and he mal ine ia o he wall is conside ed.
Compa ison o he node me hod and pseudo ansien app oach wi h expe imen al da a is
conduc ed in [
15
]. Au ho s o he a icle conclude ha he mal di usion caused by u bulen beha io
o he luid inside he pipe has a signi ican e ec on he he mal wa e p opaga ion. The e ec o
u bulen di usion is mo e signi ican wi h sha pe inle empe a u e changes and ha he p ope
modeling should conside his. Fu he , hey conclude ha he pseudo ansien app oach p edic s he
he mal p opaga ion mo e accu a ely han he node me hod.
The hi d g oup o models is based on he disc e iza ion o he PDE desc ibing he pipe in one
o mo e dimensions (nume ical/physical models). Me hod o cha ac e is ics wi h he hi d-o de
nume ical scheme is used in [
16
] o sol e he ene gy Equa ion. I seems o p ese e he shapes and
ampli udes o he hea wa e, which indica es ha he e is li le o none nume ical di usion in he
solu ion. Axial di usion and he mal dynamics o he wall is no inco po a ed.
Imp o ed hi d o de nume ical solu ion wi h o al a ia ion diminishing p ope y ( educes
oscilla ions nea sudden empe a u e changes) is used in [
17
]. Simpli ied dynamic accumula ion o
hea in o he pipe’s wall is conside ed. An indica o ha desc ibes he in luence o he he mal ine ia
o he wall on o he wa e p opaga ion is de eloped. The p oposed model is alida ed in a eal sys em.
Axial u bulen di usion is no inco po a ed in o he model.
Al hough nume ical models a e usually ega ded as slow, hey ha e one signi ican ad an age.
They can cap u e mos o he dynamic phenomena a ec ing he a eling he mal wa e in conside able
de ail. Acco ding o he e iew o ele an li e a u e gi en abo e, any o he s a es o he a models
(o any g oup) does no inco po a e all he men ioned e ec s a once. To speed up nume ical models, i
is desi able o limi he numbe o ope a ions as well as he numbe o memo y alloca ions (sa ing
ewe da a and c ea ing ewe empo a ies). To ob ain a smoo h solu ion, he e is a need o choose
mino s eps unless an app op ia e nonlinea in e pola ion is possible.
Usually, p oblems, such as ad ec ion, a e nume ically sol ed by ini e olume me hods (FVM).
Ei he explici o implici schemes a e used. The PDE is o en disc e ized along all dimensions,
including he ime. The quad a ic ups eam in e pola ion wi h es ima ion e ms (QUICKEST) is
o en conside ed o be he mos e icien scheme [
18
–
21
]. A good FVM scheme should p ese e
sha p empe a u e changes while no in oducing unphysical oscilla ion. Non-oscilla o y schemes a e
nonlinea and usually in ol e a lux limi e , which makes hem mo e expensi e du ing un- ime. The
only linea non-oscilla o y FVM is i s o de upwind scheme, bu i su e s om se e e unphysical
(nume ical) di usion. Luckily, when eal di usion is in ol ed, he oscilla ion o highe -o de schemes
ades ou . An example o his is p esen ed in he ollowing sec ion.
Hea ans e is e y o en sol ed by ini e elemen me hod (FEM), which is based on a weak
o mula ion o he PDEs. Nodal alues o he s encil ampli y local non-ze o no malized ield unc ions
(shape unc ions). The global empe a u e ield unc ion is hen a supe posi ion o he magni ied local
ields. The e o e, a good FEM scheme is p oduced by shape unc ions ha closely esemble he ue
local empe a u e dis ibu ion. In he ollowing sec ion, he e is a de ailed desc ip ion o an app oach
ha esul s in he s encil coe icien s o he adial model o a solid wall. S eady-s a e biased shape
unc ions a e used o cap u e he less ansien pe iods e y accu a ely e en h ough oughes meshes.
In o de o acili a e he a ious models o indi idual componen s o a DH sys em, all componen s
should be sol ed simul aneously o include hei mu ual e ec s. Me hod o lines can be u ilized he e.
In his me hod, he PDEs a e app oxima ed by se s o o dina y di e en ial equa ions (ODE), whe e
only he spa ial de i a i es a e disc e ized. Such models a e easy o combine using an ODE sol e .
Open-sou ce en i onmen s wi h powe ul ODE sol e s can be used o sol e hese sys ems. Two
in e es ing ones a e he OpenModelica and Julia languages. The i s uses a componen -o ien ed,
Equa ion-based language called Modelica. I specializes in he modeling o complex sys ems dealing
wi h a ious physics backg ounds [
22
]. The e a e many lib a ies con aining p ede ined models o
componen s which can be easily combined oge he o c ea e a complex sys em. The OpenModelica
Ene gies 2019,12, 633 4 o 23
compile compiles code ha sol es e icien ly ime-dependen a iables based on gi en pa ame e s. I
uses sol e s o Di e en ial algeb aic Equa ions (DAE) o sol e implici sys ems, which allows he
use o speci y he ela ionship be ween a iables wi hou he need o being explici . Communica ion
wi h MATLAB can be also es ablished h ough sc ip ing and ile exchange [
23
] ( his is ad an ageous
because MATLAB’s de aul ODE sol e s a e e y slow).
Julia is a gene al-pu pose, high-le el, dynamic language ha app oaches he speeds o
s a ically-compiled languages like C [
24
]. The e a e se e al ools o benchma king and p o iling o
w i en algo i hms, which helps o iden i y hei po en ial speedups. The e a e many well-de eloped
packages al eady, amongs which is a e y pe o man (meaning ha i is wo king well enough
o be conside ed unc ional bu may no ye be ully op imized) package dealing wi h Di e en ial
Equa ions [
25
]. This package con ains many sol e s and can e en in e ace o ad anced C lib a ies,
such as Sundials [
26
]. The esul s use in e pola ion wi h p ede ined ole ance o e u n alues a an
a bi a y ime. This oge he makes he language highly sui able o scien i ic compu ing.
The aim o his pape was o de i e and es specialized and as nume ical model o a simple DH
pipe ha inco po a es sha p in e ace p ese a ion, u bulen di usion, and de ailed adial he mal
ine ia e alua ion caused by he wall.
2. Ma e ials and Me hods
This chap e con ains a e y in-dep h desc ip ion o he ma hema ics behind he model. In
o de o ensu e ha he esul an model is as , a one-dimensional PDE is used o a oid wo spa ial
de i a i es. He e, he incomp essibili y o he luid is assumed. Indi idual a iables depend on he
posi ion and ime as deno ed by a iables xand in he ollowing equa ion:
dT(x, )
d =−u( )dT(x, )
dx +Dd2T(x, )
dx2+S(x, ), (1)
whe e Tis he local c oss-sec ional empe a u e o he luid, uis he axial luid eloci y, Dis he axial
di usion coe icien , and Sis a sou ce e m (dealing wi h hea low om inne wall and ic ion).
Because he me hod o lines is used he e, only he spa ial de i a i es mus be disc e ized. The pipe is
di ided in o n
a
cylind ical olumes, whe e spa ially dependen a iables a e localized (see Figu e 1).
Localized empe a u es ep esen he olume-a e aged empe a u e and, localized sou ces ep esen
he incoming hea om he solid shell di ided by he hea capaci y o one olume elemen .
Ene gies 2019, 12, x FOR PEER REVIEW 4 o 23
compile compiles code ha sol es e icien ly ime-dependen a iables based on gi en pa ame e s.
I uses sol e s o Di e en ial algeb aic Equa ions (DAE) o sol e implici sys ems, which allows he
use o speci y he ela ionship be ween a iables wi hou he need o being explici .
Communica ion wi h MATLAB can be also es ablished h ough sc ip ing and ile exchange [23] ( his
is ad an ageous because MATLAB’s de aul ODE sol e s a e e y slow).
Julia is a gene al-pu pose, high-le el, dynamic language ha app oaches he speeds o
s a ically-compiled languages like C [24]. The e a e se e al ools o benchma king and p o iling o
w i en algo i hms, which helps o iden i y hei po en ial speedups. The e a e many
well-de eloped packages al eady, amongs which is a e y pe o man (meaning ha i is wo king
well enough o be conside ed unc ional bu may no ye be ully op imized) package dealing wi h
Di e en ial Equa ions [25]. This package con ains many sol e s and can e en in e ace o ad anced
C lib a ies, such as Sundials [26]. The esul s use in e pola ion wi h p ede ined ole ance o e u n
alues a an a bi a y ime. This oge he makes he language highly sui able o scien i ic
compu ing.
The aim o his pape was o de i e and es specialized and as nume ical model o a simple
DH pipe ha inco po a es sha p in e ace p ese a ion, u bulen di usion, and de ailed adial
he mal ine ia e alua ion caused by he wall.
2. Ma e ials and Me hods
This chap e con ains a e y in-dep h desc ip ion o he ma hema ics behind he model. In o de
o ensu e ha he esul an model is as , a one-dimensional PDE is used o a oid wo spa ial
de i a i es. He e, he incomp essibili y o he luid is assumed. Indi idual a iables depend on he
posi ion and ime as deno ed by a iables x and in he ollowing equa ion:
2
2
, , ,
,
dT x dT x d T x
u D S x
d dx dx
, (1)
whe e T is he local c oss-sec ional empe a u e o he luid, u is he axial luid eloci y, D is he axial
di usion coe icien , and S is a sou ce e m (dealing wi h hea low om inne wall and ic ion).
Because he me hod o lines is used he e, only he spa ial de i a i es mus be disc e ized. The pipe is
di ided in o na cylind ical olumes, whe e spa ially dependen a iables a e localized (see Figu e 1).
Localized empe a u es ep esen he olume-a e aged empe a u e and, localized sou ces ep esen
he incoming hea om he solid shell di ided by he hea capaci y o one olume elemen .
Figu e 1. Di ision o pipe’s luid egion in o olumes, wi h localized spa ial dependen a iables.
Fini e di e ences a e used o disc e iza ion. The e is a simple ela ionship be ween he ini e
di e ence schemes o any de i a ion o de (including ze o) and gi en n s encil poin s [27]:
1
1
m
c Z
x
, (2)
whe e c is a ec o o he s encil coe icien s, m is he de i a ion o de , Z is an n×n ma ix such as:
1
0
,
j
i
i j
x x
Zx
, (3)
Figu e 1. Di ision o pipe’s luid egion in o olumes, wi h localized spa ial dependen a iables.
Fini e di e ences a e used o disc e iza ion. The e is a simple ela ionship be ween he ini e
di e ence schemes o any de i a ion o de (including ze o) and gi en ns encil poin s [27]:
c=1
∆xmZ−1δ, (2)
whe e cis a ec o o he s encil coe icien s, mis he de i a ion o de , Zis an n×nma ix such as:
Zi,j=xi−x0
∆xj−1
, (3)
Ene gies 2019,12, 633 5 o 23
in which x
i
is he axial posi ion o a s encil poin , x
0
is he axial posi ion o he poin in which he
de i a i e is demanded, and δ ep esen s a ec o such as:
δi=(0i6=m+1
m!i=m+1. (4)
The i s e m on he igh -hand side o Equa ion (1) dealing wi h ad ec ion is disc e ized in
wo ways. The i s way u ilizes Equa ion (2) di ec ly, which esul s in he ini e di e ence me hod
(FDM). The second way is based on FVM, whe e Equa ion (2) is used o he e alua ion o he alues
a elemen aces (ze o o de de i a i es). The disc e iza ion o he ad ec ion pa should be good
in shock cap u ing; he e o e, 660 es s a e conduc ed o compa e his abili y among he di e en
schemes. By illing ows wi h he s encil coe icien s, a sc ip w i en in Julia cons uc s he ma ix
A ha app oxima es he de i a i e ope a o . Among he inpu pa ame e s, he e is a numbe o
used upwind and downwind s encil poin s. While illing he i s ew ows o ma ix A, bo h hese
pa ame e s a e au oma ically dec eased. When illing he las ew ows, only he downwind pa ame e
is dec eased because o he missing nodes. Fo example, he maximum numbe o a ailable upwind
nodes a he inle is one ( he bounda y empe a u e). No oscilla ion is caused by cen al schemes nea
he bounda ies (unless a cen al scheme is used explici ly) because he local upwind bias is always
p ese ed ( i s ows) o inc eased (las ows). The bounda y ec o A
in
is cons uc ed as well. Bo h
a e used o e alua e he pu e ad ec ion p oblem in he ollowing o m:
dT
d =−u(AT +Tin Ain), (5)
whe e Tis he ec o o localized luid olume empe a u es and T
in
is he ime-dependen scala
ep esen ing he inle (bounda y) empe a u e signal. P oblems a e sol ed using no malized a iables,
and he beha io o he schemes is p esen ed in a no malized a iable diag am (NVD). Sundials lib a y
is used o sol ing he p oblem in ime, namely he CVODE algo i hm wi h backwa d di e encing and
wi h ela i e and absolu e ole ances o 10
−6
. I has an adap i e ime s ep and p oduces a con inuous
empe a u e signal a he ou le by using He mi e in e pola ion. Examples o NVDs a e shown in
Figu e 2. The diag ams con ain hin s abou he schemes (Uand Da e he numbe o upwind and
downwind nodes, espec i ely, and nis a numbe o olume elemen s).
Ene gies 2019, 12, x FOR PEER REVIEW 5 o 23
in which xi is he axial posi ion o a s encil poin , x0 is he axial posi ion o he poin in which he
de i a i e is demanded, and δ ep esen s a ec o such as:
0 1
! 1
i
i m
m i m
. (4)
The i s e m on he igh -hand side o Equa ion (1) dealing wi h ad ec ion is disc e ized in wo
ways. The i s way u ilizes Equa ion (2) di ec ly, which esul s in he ini e di e ence me hod
(FDM). The second way is based on FVM, whe e Equa ion (2) is used o he e alua ion o he alues
a elemen aces (ze o o de de i a i es). The disc e iza ion o he ad ec ion pa should be good in
shock cap u ing; he e o e, 660 es s a e conduc ed o compa e his abili y among he di e en
schemes. By illing ows wi h he s encil coe icien s, a sc ip w i en in Julia cons uc s he ma ix A
ha app oxima es he de i a i e ope a o . Among he inpu pa ame e s, he e is a numbe o used
upwind and downwind s encil poin s. While illing he i s ew ows o ma ix A, bo h hese
pa ame e s a e au oma ically dec eased. When illing he las ew ows, only he downwind
pa ame e is dec eased because o he missing nodes. Fo example, he maximum numbe o
a ailable upwind nodes a he inle is one ( he bounda y empe a u e). No oscilla ion is caused by
cen al schemes nea he bounda ies (unless a cen al scheme is used explici ly) because he local
upwind bias is always p ese ed ( i s ows) o inc eased (las ows). The bounda y ec o Ain is
cons uc ed as well. Bo h a e used o e alua e he pu e ad ec ion p oblem in he ollowing o m:
in in
dT
u AT T A
d , (5)
whe e T is he ec o o localized luid olume empe a u es and Tin is he ime-dependen scala
ep esen ing he inle (bounda y) empe a u e signal. P oblems a e sol ed using no malized
a iables, and he beha io o he schemes is p esen ed in a no malized a iable diag am (NVD).
Sundials lib a y is used o sol ing he p oblem in ime, namely he CVODE algo i hm wi h
backwa d di e encing and wi h ela i e and absolu e ole ances o 10−6. I has an adap i e ime s ep
and p oduces a con inuous empe a u e signal a he ou le by using He mi e in e pola ion.
Examples o NVDs a e shown in Figu e 2. The diag ams con ain hin s abou he schemes (U and D
a e he numbe o upwind and downwind nodes, espec i ely, and n is a numbe o olume
elemen s).
(a) (b)
Figu e 2. Examples o bad-beha ing ad ec ion schemes. (a): Low o de upwind biased scheme
causes a lo o nume ical di usion; (b): Low o de cen al scheme is being oo oscilla o y.
I is obse ed ha cen al schemes ha e a s onge endency o oscilla e (Figu e 2b) and a e
mo e compu a ionally expensi e han upwind-biased schemes (see Figu e 3). O e all, highe o de
upwind schemes (see Figu e 4) seem o show a good comp omise be ween he deg ee o hei
oscilla ion, hei abili y o shock cap u ing, and compu a ional expensi eness.
Figu e 2.
Examples o bad-beha ing ad ec ion schemes. (
a
): Low o de upwind biased scheme causes
a lo o nume ical di usion; (b): Low o de cen al scheme is being oo oscilla o y.
I is obse ed ha cen al schemes ha e a s onge endency o oscilla e (Figu e 2b) and a e mo e
compu a ionally expensi e han upwind-biased schemes (see Figu e 3). O e all, highe o de upwind
schemes (see Figu e 4) seem o show a good comp omise be ween he deg ee o hei oscilla ion, hei
abili y o shock cap u ing, and compu a ional expensi eness.
Ene gies 2019,12, 633 6 o 23
Ene gies 2019, 12, x FOR PEER REVIEW 6 o 23
Figu e 3. Compa ison o compu a ional ime o he es s be ween upwind-biased and cen al
schemes o FDM. Simila esul s can be obse ed o FVM. The alues in b acke s ep esen ounded
ow s encil coe icien s in he ypical ow o he de i a i e ope a o A (when Δx = 1).
(a) (b)
Figu e 4. Examples o well-beha ing ad ec ion schemes. (a): Highe o de upwind biased FVM
scheme wi h mild nume ical di usion and mild oscilla ions; (b): Highe o de upwind biased FDM
scheme wi h mild nume ical di usion and e y mild oscilla ions.
The i s o de cen al scheme is used o he cons uc ion o B, which is he ma ix ha
app oxima es he second de i a i e ope a o dealing wi h axial di usion in Equa ion (1). Bounda y
ec o Bin allows he calcula ion o he second de i a i e a he inle node as well. Equa ion (5) is
ex ended and esul s in Equa ion (6):
in in in in
dT
u AT T A D BT T B
d , (6)
Figu e 3.
Compa ison o compu a ional ime o he es s be ween upwind-biased and cen al schemes
o FDM. Simila esul s can be obse ed o FVM. The alues in b acke s ep esen ounded ow
s encil coe icien s in he ypical ow o he de i a i e ope a o A(when ∆x= 1).
Ene gies 2019, 12, x FOR PEER REVIEW 6 o 23
Figu e 3. Compa ison o compu a ional ime o he es s be ween upwind-biased and cen al
schemes o FDM. Simila esul s can be obse ed o FVM. The alues in b acke s ep esen ounded
ow s encil coe icien s in he ypical ow o he de i a i e ope a o A (when Δx = 1).
(a) (b)
Figu e 4. Examples o well-beha ing ad ec ion schemes. (a): Highe o de upwind biased FVM
scheme wi h mild nume ical di usion and mild oscilla ions; (b): Highe o de upwind biased FDM
scheme wi h mild nume ical di usion and e y mild oscilla ions.
The i s o de cen al scheme is used o he cons uc ion o B, which is he ma ix ha
app oxima es he second de i a i e ope a o dealing wi h axial di usion in Equa ion (1). Bounda y
ec o Bin allows he calcula ion o he second de i a i e a he inle node as well. Equa ion (5) is
ex ended and esul s in Equa ion (6):
in in in in
dT
u AT T A D BT T B
d , (6)
Figu e 4.
Examples o well-beha ing ad ec ion schemes. (
a
): Highe o de upwind biased FVM
scheme wi h mild nume ical di usion and mild oscilla ions; (
b
): Highe o de upwind biased FDM
scheme wi h mild nume ical di usion and e y mild oscilla ions.
The i s o de cen al scheme is used o he cons uc ion o B, which is he ma ix ha
app oxima es he second de i a i e ope a o dealing wi h axial di usion in Equa ion (1). Bounda y
ec o B
in
allows he calcula ion o he second de i a i e a he inle node as well. Equa ion (5) is
ex ended and esul s in Equa ion (6):
dT
d =−u(AT +Tin Ain)+D(BT +TinBin), (6)
Ene gies 2019,12, 633 7 o 23
di usion coe icien Dmay ei he be lumped o localized in o he o m:
Di=w1,i·u·d1+w2,i, (7)
The i s e m in Equa ion (7) deals wi h u bulen di usion. The second e m, which is s ic ly
non-nega i e, co esponds o eloci y-independen e ec s, such as conduc i e di usion (which is
shown la e o be negligible) o sp eading due o e ec s o g a i y. I he second e m in Equa ion (7) is
neglec ed, he Equa ion (6) can be simpli ied:
dT
d =u[(d1B−A)T+(d1Bin −Ain)Tin]=u(MT +MinTin), (8)
whe e Mand M
in
a e he cons an ma ix ope a o and bounda y ec o o he luid egion, espec i ely.
This s ep should imp o e he pe o mance o he model. I also makes he Peclé numbe independen
o low eloci y. This means ha when p ope axial disc e iza ion is chosen o ensu e non-oscilla o y
beha io o he me hod, i is p ese ed o all eloci ies o he luid. Au oma ic axial disc e iza ion,
based on alues w
1,i
, is hen possible. The weigh s in Equa ion (7) allow he uning o he pipe o
mimic a eal ins alla ion mo e accu a ely (simila weigh ing is possible o he sou ce e m as well).
Using he analy ic solu ion o s ep inpu in he NVD, he e ec o D
i
on oscilla ion o Equa ion
(6) is s udied. Any alue o he nume ical solu ion ha is ou side o he uni in e al is conside ed o
be an o e sho caused by oscilla ion. This is a e y simple way o es ima e he esidual oscilla ion o
Equa ion (6). FDM is chosen o he model because o i s lowe oscilla ions (see Figu e 5).
Ene gies 2019, 12, x FOR PEER REVIEW 7 o 23
di usion coe icien D may ei he be lumped o localized in o he o m:
1, 1 2,i i i
D w u d w
, (7)
The i s e m in Equa ion (7) deals wi h u bulen di usion. The second e m, which is s ic ly
non-nega i e, co esponds o eloci y-independen e ec s, such as conduc i e di usion (which is
shown la e o be negligible) o sp eading due o e ec s o g a i y. I he second e m in Equa ion (7)
is neglec ed, he Equa ion (6) can be simpli ied:
1 1
in in in in in
dT
u d B A T d B A T u MT M T
d
, (8)
whe e M and Min a e he cons an ma ix ope a o and bounda y ec o o he luid egion,
espec i ely. This s ep should imp o e he pe o mance o he model. I also makes he Peclé
numbe independen o low eloci y. This means ha when p ope axial disc e iza ion is chosen o
ensu e non-oscilla o y beha io o he me hod, i is p ese ed o all eloci ies o he luid.
Au oma ic axial disc e iza ion, based on alues w1,i, is hen possible. The weigh s in Equa ion (7)
allow he uning o he pipe o mimic a eal ins alla ion mo e accu a ely (simila weigh ing is
possible o he sou ce e m as well).
Using he analy ic solu ion o s ep inpu in he NVD, he e ec o Di on oscilla ion o Equa ion
(6) is s udied. Any alue o he nume ical solu ion ha is ou side o he uni in e al is conside ed o
be an o e sho caused by oscilla ion. This is a e y simple way o es ima e he esidual oscilla ion o
Equa ion (6). FDM is chosen o he model because o i s lowe oscilla ions (see Figu e 5).
Figu e 5. E ec o physical di usion on unphysical oscilla ion on sha p in e aces o na = 100 ( he c i ical
Peclé numbe , which makes he esidual oscilla ion less han 10−6. I is app oxima ely 6.47 o he
U2D1FVM and 8.51 o he U2D1FDM).
The las e m o Equa ion (1) is mo e complica ed since i mus deal wi h hea dynamics in he
solid shells su ounding he pipe. F om he luid iewpoin , he only impo an alue is he inne
wall empe a u e o he i s shell. Using publica ion [6], he Nussel numbe o all low egimes is:
Figu e 5.
E ec o physical di usion on unphysical oscilla ion on sha p in e aces o n
a
= 100 ( he
c i ical Peclé numbe , which makes he esidual oscilla ion less han 10
−6
. I is app oxima ely 6.47 o
he U2D1FVM and 8.51 o he U2D1FDM).
The las e m o Equa ion (1) is mo e complica ed since i mus deal wi h hea dynamics in he
solid shells su ounding he pipe. F om he luid iewpoin , he only impo an alue is he inne wall
empe a u e o he i s shell. Using publica ion [6], he Nussel numbe o all low egimes is:
Ene gies 2019,12, 633 8 o 23
Nu =
3.66 i Re ≤2300
3.52 ·U4−45.148 ·U3+212.13 ·U2−427.45 ·U2+316.08 i 2300 ≤Re ≤3100
8·(Re−1000)·P
1+12.7·
81/2·(P 2/3−1)i Re <3000 , (9)
whe e Re is he Reynolds numbe , U=Re/1000,
is he Da cy-Weisbach ic ion ac o , and P is he
P and l Numbe . The ic ion ac o migh be e alua ed using he Chu chill Equa ion p esen ed in [
7
]
o all low egimes:
=8· 8
Re 12
+1
(θ1+θ2)3/2 !1/12
, (10)
whe e
θ1=−2.457 ·log7
Re 0.9 +0.27 ·ε
d116
θ2=37530
Re 16 , (11)
in which
ε
is he oughness o he inne wall and d
1
is he inne diame e . I is a simple s ep o e alua e
he local sou ce e m Sin he o m o a ec o :
Si=4·Nu ·λ
d2
1·ρ·cp
·(Tsn,i,1 −Ti). (12)
whe e T
sn,i,1
ep esen s he inne wall (la e i s node o he solid egion) empe a u es o he solid
shell, c
p
is he speci ic hea capaci y o he luid,
ρ
is he mass densi y o he luid, and
λ
is he hea
conduc i i y o he luid.
The ime-dependen dis ibu ion o empe a u e in he solid shells a ec s he axial ansien s
o he hea wa es (cha ging and discha ging o he accumula ion laye s). Tangen ial and axial hea
conduc ions in he shells a e igno ed because e en when he low eloci y is small, he a e in which
he hea is being anspo ed by ad ec ion is o en much highe han ha by conduc ion. This migh
be exp essed in he o m o he Pécle numbe o conduc ion in bo h he luid:
Pe =d1·u·ρ·cp
λ≈d1·u·6,29 ·1061, (13)
and he solid (e.g., s eel):
Pes=d1·u·ρs·cp,s
λs
≈d1·u·8·1041. (14)
As shown, he inne diame e s o he pipes o he low eloci ies would ha e o be e y small
o he conduc i e axial di usion o be compa able wi h ad ec ion. This is no common wi h eal
applica ions in he DH sys ems which also educes he numbe o necessa y spa ial de i a i es used in
he model. The luid olumes can be ma hema ically coupled wi h one-dimensional models o pu ely
adial hea ans e (la e solid segmen s) desc ibed by PDE in he o m:
dTs,i
d ·ρs·cps =1
d
d λs· ·dTs,i
d , (15)
whe e is he adial coo dina e, T
s,i
is he con inuous empe a u e in he solid segmen ,
ρs
is he
mass densi y, c
ps
is he speci ic hea capaci y, and
λs
is he hea conduc i i y. Equa ion (15) migh be
ew i en in o he weak o m using he shape unc ion V(no ice ha in eg a ed e m in he b acke
a ec s he sys em only a he bounda ies).
2·π·∆x·
2
R
1
·V·dTs,i
d ·ρs·cps ·d =2·π·∆x·"−
2
R
1
·dV
d ·dTs,i
d ·λs·d +h ·V·λs·dTs,i
d i 2
1#.(16)
Ene gies 2019,12, 633 9 o 23
The s eady s a e empe a u e dis ibu ion is used as he base o he nodal shape unc ions:
Tse,i,j( )=ln( /Rj)·(Tsn,i,j+1−Tsn,i,j)
ln(Rj+1/Rj)+Tsn,i,j=Tsn,i,j+1·ln( /Rj)
ln(Rj+1/Rj)+Tsn,i,j1−ln( /Rj)
ln(Rj+1/Rj),(17)
whe e T
se,i,j
a e he empe a u e dis ibu ions in he cylind ical elemen s, T
sn,i,j
a e nodal empe a u es,
and R
j
a e adial coo dina es o he nodes. Equa ion (17) is he solu ion o he ollowing PDE wi h
gi en bounda y condi ions (see also Figu e 6):
1
d
d λs,j· ·dTse,i,j
d =0
Tse,i,jRj+1=Tsn,i,j+1
Tse,i,jRj=Tsn,i,j
, (18)
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2
2
2
1
1
1
,
,,
2 π 2 π
s i
s i s i
s
s ps s
dT
dT dT
dV
d
x V c d x V
d d
d d
. (16)
The s eady s a e empe a u e dis ibu ion is used as he base o he nodal shape unc ions:
, , 1 , ,
, , , , , , 1 , ,
1 1 1
ln ln ln
1
ln ln ln
j sn i j sn i j j j
se i j sn i j sn i j sn i j
j j j j j j
R T T R R
T T T T
R R R R R R
, (17
)
whe e Tse,i,j a e he empe a u e dis ibu ions in he cylind ical elemen s, Tsn,i,j a e nodal
empe a u es, and Rj a e adial coo dina es o he nodes. Equa ion (17) is he solu ion o he
ollowing PDE wi h gi en bounda y condi ions (see also Figu e 6):
, ,
,
, , 1 , , 1
, , , ,
1
0
se i j
s j
se i j j sn i j
se i j j sn i j
dT
d
d d
T R T
T R T
, (18)
`
Figu e 6. G aphical ep esen a ion o he a iables.
Nodal shape unc ions a e de i ed om Equa ion (17) by conside ing he e ec o one node o
he global empe a u e dis ibu ion. The e o e, hey ake he ollowing piecewise o m (no ice ha
bounda y nodes a e neighbo ing one elemen only, so he shape unc ions ha e only he app op ia e
hal ):
1
1
1
1
1
1
,
1
1
1 1
1
1
1
1
ln
1 , , 1
ln
0,
ln ,
ln
( ) , 2 1
ln
1 ,
ln
0,
ln , ,
ln
0,
j
j j
j j
j j
j
j j
j j
sn j j
j j
j j
j j
j
j j
j j
j j
R R R j
R R
R R
R R R
R R
V j n
R R R
R R
R R
R R R
R R
R R
j n
, (19)
When mass densi ies and speci ic hea capaci ies o he elemen s a e cons an , he ollowing
in eg a ion (acco ding o he le -hand side o Equa ion (16)) esul s in disc e ized hea capaci y:
Figu e 6. G aphical ep esen a ion o he a iables.
Nodal shape unc ions a e de i ed om Equa ion (17) by conside ing he e ec o one node o
he global empe a u e dis ibu ion. The e o e, hey ake he ollowing piecewise o m (no ice ha
bounda y nodes a e neighbo ing one elemen only, so he shape unc ions ha e only he app op ia e
hal ):
Vsn,j( ) =
1−ln( /Rj)
ln(Rj+1/Rj),Rj≤ ≤Rj+1
0, Rj> >Rj+1
,j=1
ln( /Rj−1)
ln(Rj/Rj−1),Rj−1≤ <Rj
1−ln( /Rj)
ln(Rj+1/Rj),Rj≤ ≤Rj+1
0, Rj−1> >Rj+1
, 2 ≤j≤n−1
ln( /Rj−1)
ln(Rj/Rj−1),Rj−1≤ <Rj
0, Rj−1> >Rj
,j=n
, (19)
When mass densi ies and speci ic hea capaci ies o he elemen s a e cons an , he ollowing
in eg a ion (acco ding o he le -hand side o Equa ion (16)) esul s in disc e ized hea capaci y:
Csn,j=2·π·∆x·
ρs,j−1·cps,j−1·
Rj
Z
Rj−1
·Vsn,j·d +ρs,j·cps,j·
Rj+1
Z
Rj
·Vsn,j·d
, (20)
whe e C
sn,j
a e he nodal hea capaci ies,
ρse,i,j
a e he mass densi ies o he elemen s, and c
ps,j
is he
speci ic hea capaci y o he elemen s. Fo inside nodes, he in eg a ion o (20) esul s in:
Csn,j=π·∆x·ρs,j−1·cps,j−1·R2
j−1−R2
j
2·ln(Rj/Rj−1)+R2
j−ρs,j·cps,j·R2
j−R2
j+1
2·ln(Rj+1/Rj)+R2
j,(21)
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Figu e 17. Es ima ions o maximal ela i e e o s a he ou le o he pipe (Case 1), whe e he ed
a ow signi ies he ideal di ec ion o disc e iza ion.
Figu e 18. Compu a ion imes o he composi e model es s (Case 1).
Simila es s o he second case clea ly demons a e he independence on he adial numbe o
subdi isions (see Figu es 19 and 20). The e is no hing o be e ined in he adial di ec ion. Fu he
subdi isions only ake mo e compu a ional ime (see Figu e 21). The highe compu a ional ime in
he igh op co ne o Figu e 21 is caused by smalle ime s eps o ensu e nume ical s abili y.
Figu e 18. Compu a ion imes o he composi e model es s (Case 1).
Ene gies 2019, 12, x FOR PEER REVIEW 16 o 23
Figu e 17. Es ima ions o maximal ela i e e o s a he ou le o he pipe (Case 1), whe e he ed
a ow signi ies he ideal di ec ion o disc e iza ion.
Figu e 18. Compu a ion imes o he composi e model es s (Case 1).
Simila es s o he second case clea ly demons a e he independence on he adial numbe o
subdi isions (see Figu es 19 and 20). The e is no hing o be e ined in he adial di ec ion. Fu he
subdi isions only ake mo e compu a ional ime (see Figu e 21). The highe compu a ional ime in
he igh op co ne o Figu e 21 is caused by smalle ime s eps o ensu e nume ical s abili y.
Figu e 19.
Es ima ions o maximal absolu e e o s a he ou le o he pipe (Case 2), whe e he ed
a ow signi ies he ideal di ec ion o disc e iza ion.
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Figu e 19. Es ima ions o maximal absolu e e o s a he ou le o he pipe (Case 2), whe e he ed
a ow signi ies he ideal di ec ion o disc e iza ion.
Figu e 20. Es ima ions o maximal ela i e e o s a he ou le o he pipe (Case 2), whe e he ed
a ow signi ies he ideal di ec ion o disc e iza ion.
Figu e 21. Compu a ion imes o he composi e model es s (Case 2).
The Figu e 22 demons a es he e ec o he s ong accumula ion using he ou le empe a u e
pe spec i e. The abo e esul s deal wi h he model alone. Un o una ely, eal da a we e no
a ailable o he au ho s o alida e he model p ope ly. The model is a leas compa ed (pa ially
alida ed) o a e y ine model de eloped in Fluen (axisymme ic 2D in Figu es 23 and 24) and
STAR-CCM+ ( ully 3D in Figu es 25 and 26). The ole ance o he new model was lowe ed o 10−5 and
samples we e sa ed e e y 10 s o demons a e he e ec i eness o he new model. The ime span o
he simula ion was 16,000 s.
Figu e 20.
Es ima ions o maximal ela i e e o s a he ou le o he pipe (Case 2), whe e he ed a ow
signi ies he ideal di ec ion o disc e iza ion.
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Figu e 19. Es ima ions o maximal absolu e e o s a he ou le o he pipe (Case 2), whe e he ed
a ow signi ies he ideal di ec ion o disc e iza ion.
Figu e 20. Es ima ions o maximal ela i e e o s a he ou le o he pipe (Case 2), whe e he ed
a ow signi ies he ideal di ec ion o disc e iza ion.
Figu e 21. Compu a ion imes o he composi e model es s (Case 2).
The Figu e 22 demons a es he e ec o he s ong accumula ion using he ou le empe a u e
pe spec i e. The abo e esul s deal wi h he model alone. Un o una ely, eal da a we e no
a ailable o he au ho s o alida e he model p ope ly. The model is a leas compa ed (pa ially
alida ed) o a e y ine model de eloped in Fluen (axisymme ic 2D in Figu es 23 and 24) and
STAR-CCM+ ( ully 3D in Figu es 25 and 26). The ole ance o he new model was lowe ed o 10−5 and
samples we e sa ed e e y 10 s o demons a e he e ec i eness o he new model. The ime span o
he simula ion was 16,000 s.
Figu e 21. Compu a ion imes o he composi e model es s (Case 2).
The Figu e 22 demons a es he e ec o he s ong accumula ion using he ou le empe a u e
pe spec i e. The abo e esul s deal wi h he model alone. Un o una ely, eal da a we e no a ailable
o he au ho s o alida e he model p ope ly. The model is a leas compa ed (pa ially alida ed)
o a e y ine model de eloped in Fluen (axisymme ic 2D in Figu es 23 and 24) and STAR-CCM+
( ully 3D in Figu es 25 and 26). The ole ance o he new model was lowe ed o 10
−5
and samples we e
sa ed e e y 10 s o demons a e he e ec i eness o he new model. The ime span o he simula ion
was 16,000 s.
Ene gies 2019, 12, x FOR PEER REVIEW 18 o 23
Figu e 22. Compa ison o Case 1 and Case 2 (wi h cons an mass low o 282.74 kg/s).
Figu e 23. Compa ison wi h a model om Fluen ( he solid ma e ial p ope y di e sligh ly
he e—mass densi y is 7850 kg/m3, hea conduc i i y is 50 W/(mK), and speci ic hea capaci y is 500
J/(kgK)).
Figu e 24. De ail o he di e ence in empe a u e on compa ison wi h a model om Fluen .
Figu e 22. Compa ison o Case 1 and Case 2 (wi h cons an mass low o 282.74 kg/s).
Ene gies 2019,12, 633 18 o 23
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Figu e 22. Compa ison o Case 1 and Case 2 (wi h cons an mass low o 282.74 kg/s).
Figu e 23. Compa ison wi h a model om Fluen ( he solid ma e ial p ope y di e sligh ly
he e—mass densi y is 7850 kg/m3, hea conduc i i y is 50 W/(mK), and speci ic hea capaci y is 500
J/(kgK)).
Figu e 24. De ail o he di e ence in empe a u e on compa ison wi h a model om Fluen .
Figu e 23.
Compa ison wi h a model om Fluen ( he solid ma e ial p ope y di e sligh ly he e—mass
densi y is 7850 kg/m3, hea conduc i i y is 50 W/(mK), and speci ic hea capaci y is 500 J/(kgK)).
Ene gies 2019, 12, x FOR PEER REVIEW 18 o 23
Figu e 22. Compa ison o Case 1 and Case 2 (wi h cons an mass low o 282.74 kg/s).
Figu e 23. Compa ison wi h a model om Fluen ( he solid ma e ial p ope y di e sligh ly
he e—mass densi y is 7850 kg/m3, hea conduc i i y is 50 W/(mK), and speci ic hea capaci y is 500
J/(kgK)).
Figu e 24. De ail o he di e ence in empe a u e on compa ison wi h a model om Fluen .
Figu e 24. De ail o he di e ence in empe a u e on compa ison wi h a model om Fluen .
Ene gies 2019, 12, x FOR PEER REVIEW 19 o 23
Figu e 25. Compa ison wi h a model om STAR-CCM+.
Figu e 26. De ail o he di e ence in empe a u e on compa ison wi h a model om STAR-CCM+.
Al hough he maximal empe a u e di e ence is highe han he model om STAR-CCM+, he
empe a u e di e ence alls close o ze o quicke han he model om Fluen . The e ec o g a i y
on wa e p opaga ion is such ha i beha es as addi ional axial di usion, which can be uned up by
adjus ing weigh s in Equa ion (7). Figu e 26 shows his ac by he mo e ime-sp ead ise in
empe a u e a he ou le . Figu es 27 and 28 show how he g a i y sp eads he empe a u e on in
he STAR-CCM+ model 7 s a e bo h ho and cold wa es en e ed he pipe.
Figu e 27. E ec o g a i y on he empe a u e dis ibu ion o a ho wa e ( he uppe pa shows he
empe a u e on wi hou e ec s o g a i y, he lowe pa shows he con a y).
Figu e 25. Compa ison wi h a model om STAR-CCM+.
Ene gies 2019,12, 633 19 o 23
Ene gies 2019, 12, x FOR PEER REVIEW 19 o 23
Figu e 25. Compa ison wi h a model om STAR-CCM+.
Figu e 26. De ail o he di e ence in empe a u e on compa ison wi h a model om STAR-CCM+.
Al hough he maximal empe a u e di e ence is highe han he model om STAR-CCM+, he
empe a u e di e ence alls close o ze o quicke han he model om Fluen . The e ec o g a i y
on wa e p opaga ion is such ha i beha es as addi ional axial di usion, which can be uned up by
adjus ing weigh s in Equa ion (7). Figu e 26 shows his ac by he mo e ime-sp ead ise in
empe a u e a he ou le . Figu es 27 and 28 show how he g a i y sp eads he empe a u e on in
he STAR-CCM+ model 7 s a e bo h ho and cold wa es en e ed he pipe.
Figu e 27. E ec o g a i y on he empe a u e dis ibu ion o a ho wa e ( he uppe pa shows he
empe a u e on wi hou e ec s o g a i y, he lowe pa shows he con a y).
Figu e 26. De ail o he di e ence in empe a u e on compa ison wi h a model om STAR-CCM+.
Al hough he maximal empe a u e di e ence is highe han he model om STAR-CCM+, he
empe a u e di e ence alls close o ze o quicke han he model om Fluen . The e ec o g a i y
on wa e p opaga ion is such ha i beha es as addi ional axial di usion, which can be uned up by
adjus ing weigh s in Equa ion (7). Figu e 26 shows his ac by he mo e ime-sp ead ise in empe a u e
a he ou le . Figu es 27 and 28 show how he g a i y sp eads he empe a u e on in he STAR-CCM+
model 7 s a e bo h ho and cold wa es en e ed he pipe.
Ene gies 2019, 12, x FOR PEER REVIEW 19 o 23
Figu e 25. Compa ison wi h a model om STAR-CCM+.
Figu e 26. De ail o he di e ence in empe a u e on compa ison wi h a model om STAR-CCM+.
Al hough he maximal empe a u e di e ence is highe han he model om STAR-CCM+, he
empe a u e di e ence alls close o ze o quicke han he model om Fluen . The e ec o g a i y
on wa e p opaga ion is such ha i beha es as addi ional axial di usion, which can be uned up by
adjus ing weigh s in Equa ion (7). Figu e 26 shows his ac by he mo e ime-sp ead ise in
empe a u e a he ou le . Figu es 27 and 28 show how he g a i y sp eads he empe a u e on in
he STAR-CCM+ model 7 s a e bo h ho and cold wa es en e ed he pipe.
Figu e 27. E ec o g a i y on he empe a u e dis ibu ion o a ho wa e ( he uppe pa shows he
empe a u e on wi hou e ec s o g a i y, he lowe pa shows he con a y).
Figu e 27.
E ec o g a i y on he empe a u e dis ibu ion o a ho wa e ( he uppe pa shows he
empe a u e on wi hou e ec s o g a i y, he lowe pa shows he con a y).
Ene gies 2019, 12, x FOR PEER REVIEW 20 o 23
Figu e 28. E ec o g a i y on he empe a u e dis ibu ion o cold wa e ( he uppe pic u e shows
he empe a u e on wi hou e ec s o g a i y, he lowe pic u e shows he con a y).
The compa ison o he new model wi h he e y ine nume ical coun e pa s shows ha he
new model o e es ima es he wa e o each he ou le sligh ly la e (asymme ic peaks in
empe a u e di e ence in he Figu es 23–26). This migh be caused by he ac ha he empe a u e
and eloci y p o iles in he new model a e conside ed o be ully de eloped igh om he
beginning, while wi h he e y ine coun e pa s hey a e no .
4. Summa y o he Resul s
The shape unc ion used in he adial model as well as he use o analy ic desc ip ion o hea
exchange be ween he luid and he inne wall su ace ensu es ha he solu ion o he new
model ag ee wi h he analy ic solu ion comple ely e en o he oughes adial disc e iza ion
(also isible h ough as s abiliza ion in Figu e 22 a e he sudden empe a u e change).
The new model allows simple modeling o ci cula pipes wi h an a bi a y numbe o shells and
hei ma e ial p ope ies (e.g., a s eel wall ollowed by insula ion and a plas ic shell). All shells
hen conside adial dynamic e ec s based on he newly de i ed adial model, and
OpenModelica’s ma ching algo i hm au oma ically adap s he ime s ep o ensu e i s nume ical
s abili y.
Al hough elemen s g owing in size in he adial di ec ion be e cap u e he sudden changes o
empe a u e, he necessi y o smalle ime s eps o ensu e he s abili y o he ma ching
algo i hm comple ely nega es his ad an age (compa e he CPU ime in Figu e 7) and makes
he uni o m mesh mo e p e e able.
Wi h he p e e ed use o he uni o m mesh in he adial model, he hick wall has o be
modeled wi h highe numbe o elemen s, while he hin wall is cap u ed well wi h e y ew o
e en single elemen (compa e e o es ima ions o g ow ac o 1 in Figu es 8–15 o he e o
maps in Figu es 16 and 19).
The use o he me hod o lines and app op ia e algo i hm wi h adap able ime s ep makes he
p ecision easily adable o compu a ion ime, wi hou he necessi y o ocus on he nume ical
s abili y condi ions o he composi e model ( ading he cap u ing o he ansien s o CPU
ime, he s eady-s a es will be cap u ed accu a ely e en wi h ough adial meshes).
The e a e egions in which a sole imp o emen o he mesh in one di ec ion will no esul in
supe io solu ion (e.g., see he egion o 100 axial and 20 adial elemen s in Figu e 16, whe e sole
imp o emen o adial disc e iza ion does no lowe e o s). The e is, howe e , an ideal
di ec ion o he disc e iza ion in each case.
Figu e 22 shows how does s ong he mal ine ia o he wall a ec he ou le empe a u e in
idealized case ( he e ec las s app oxima ely o hal an hou o he hick s eel wall).
Maximal de ia ion in ou le empe a u e in compa ison wi h Fluen (2D) is abou 3 K.
Maximal de ia ion in ou le empe a u e in compa ison wi h STAR-CCM+ (3D) is abou 6 K o
he case wi hou g a i y and 4 K o he case ha includes g a i y.
Du ing all simula ions, posi i e weigh s in he de ini ion o axial di usion a e deployed, which
shows ha u bulen axial mixing plays a ole in he mal wa e p opaga ion.
Figu e 28.
E ec o g a i y on he empe a u e dis ibu ion o cold wa e ( he uppe pic u e shows he
empe a u e on wi hou e ec s o g a i y, he lowe pic u e shows he con a y).
The compa ison o he new model wi h he e y ine nume ical coun e pa s shows ha he new
model o e es ima es he wa e o each he ou le sligh ly la e (asymme ic peaks in empe a u e
di e ence in he Figu es 23–26). This migh be caused by he ac ha he empe a u e and eloci y
Ene gies 2019,12, 633 20 o 23
p o iles in he new model a e conside ed o be ully de eloped igh om he beginning, while wi h
he e y ine coun e pa s hey a e no .
4. Summa y o he Resul s
•
The shape unc ion used in he adial model as well as he use o analy ic desc ip ion o hea
exchange be ween he luid and he inne wall su ace ensu es ha he solu ion o he new model
ag ee wi h he analy ic solu ion comple ely e en o he oughes adial disc e iza ion (also isible
h ough as s abiliza ion in Figu e 22 a e he sudden empe a u e change).
•
The new model allows simple modeling o ci cula pipes wi h an a bi a y numbe o shells and
hei ma e ial p ope ies (e.g., a s eel wall ollowed by insula ion and a plas ic shell). All shells hen
conside adial dynamic e ec s based on he newly de i ed adial model, and OpenModelica’s
ma ching algo i hm au oma ically adap s he ime s ep o ensu e i s nume ical s abili y.
•
Al hough elemen s g owing in size in he adial di ec ion be e cap u e he sudden changes o
empe a u e, he necessi y o smalle ime s eps o ensu e he s abili y o he ma ching algo i hm
comple ely nega es his ad an age (compa e he CPU ime in Figu e 7) and makes he uni o m
mesh mo e p e e able.
•
Wi h he p e e ed use o he uni o m mesh in he adial model, he hick wall has o be modeled
wi h highe numbe o elemen s, while he hin wall is cap u ed well wi h e y ew o e en
single elemen (compa e e o es ima ions o g ow ac o 1 in Figu es 8–15 o he e o maps in
Figu es 16 and 19).
•
The use o he me hod o lines and app op ia e algo i hm wi h adap able ime s ep makes he
p ecision easily adable o compu a ion ime, wi hou he necessi y o ocus on he nume ical
s abili y condi ions o he composi e model ( ading he cap u ing o he ansien s o CPU ime,
he s eady-s a es will be cap u ed accu a ely e en wi h ough adial meshes).
•
The e a e egions in which a sole imp o emen o he mesh in one di ec ion will no esul in
supe io solu ion (e.g., see he egion o 100 axial and 20 adial elemen s in Figu e 16, whe e sole
imp o emen o adial disc e iza ion does no lowe e o s). The e is, howe e , an ideal di ec ion
o he disc e iza ion in each case.
•
Figu e 22 shows how does s ong he mal ine ia o he wall a ec he ou le empe a u e in
idealized case ( he e ec las s app oxima ely o hal an hou o he hick s eel wall).
•Maximal de ia ion in ou le empe a u e in compa ison wi h Fluen (2D) is abou 3 K.
•
Maximal de ia ion in ou le empe a u e in compa ison wi h STAR-CCM+ (3D) is abou 6 K o
he case wi hou g a i y and 4 K o he case ha includes g a i y.
•
Du ing all simula ions, posi i e weigh s in he de ini ion o axial di usion a e deployed, which
shows ha u bulen axial mixing plays a ole in he mal wa e p opaga ion.
•
P esence o g a i y in ully ho izon al pipes ac s as addi ional axial di usion om he iewpoin
o c oss-sec ional empe a u e a e age (compa ison o he wo cu es om STAR-CCM+ in
Figu e 26).
•
Rela i e simula ion ime o he new model o a single 100 m long pipe wi h s ong accumula ion
is app oxima ely in o de o 3.75 ×10−5.
•
A simple quasi-s a ic e alua ion o he p essu e losses ( he ic ion coe icien is al eady being
e alua ed du ing he mal simula ion) would allow he use o he new model o simula ion o
whole g ids wi h au oma ic mass lows e alua ion in each b anch (e.g., OpenModelica connec o
de ini ion includes Ki chho laws by de aul ).
5. Conclusions
This pape deals wi h ansien s in ci cula pipes, whe e he accumula ion o hea in o su ounding
ma e ial has been conside ed in de ail. Model is a combina ion o he analy ic desc ip ion o a hea
ans e be ween he luid and he solid wall, simple ad ec ion scheme, u bulen di usion, and a
Ene gies 2019,12, 633 21 o 23
ini e elemen scheme dealing wi h he ci cula wall in which he shape unc ions a e based on he
s eady-s a e solu ion. Me hod o lines is used o ime in eg a ion. The new model ep esen s an
e ec i e nume ical solu ion o he he mal wa e p opaga ion p oblem, which includes ad anced
e alua ion o he mal ine ia o he pipe’s wall and he axial di usion ha is caused by u bulence.
The di usion coe icien e alua ion is mainly p opo ional o low eloci y.
O e all, i can be said ha s ong accumula ion a ec s he he mal wa e p opaga ion o
a s e ched pe iod o ime in compa ison wi h he non-accumula ion case. The e ec o he
e-accumula ion is no only dependen on he pu e amoun o e-accumula ed hea , bu also on
he pa icula way his e en happens. In o he wo ds, o cap u e he e en accu a ely is o ocus on
he p ope e alua ion o he inne mos su ace empe a u e o he pipe’s wall because i di ec ly a ec s
he a e in which he empe a u e o he luid changes. The disc e iza ion o bo h axial and adial
di ec ion mus go hand in hand o imp o e he solu ion. Du ing compa ison wi h he ine models
om Fluen and STAR-CCM+, deploymen o posi i e weigh s in he di usion coe icien e alua ion
a e always necessa y o ma ch he simula ed da a. This means ha u bulen di usion plays a ole
in empe a u e wa e p opaga ion. P esence o physical di usion also educes unphysical oscilla ion
on sha p in e aces. G a i y is shown o esemble he beha io o an addi ional axial di usion as
well. The new model esembles he ansien beha io o he accumula ing pipe wall wi h highly
compa able esul s o he equi alen , e y ine models om Fluen and STAR-CCM+, which should
ep esen he ideal physics-based modeling. The alida ion wi h a b anched eal sys em needs o be
done in he u u e.
The composi e model had un as e in OpenModelica han in Julia possibly because o a
non-op imized jacobian e alua ion. I migh be possible o speed up he Julia execu ion using unc ion
de ini ion mac os om Di e en ial Equa ions package, which could be able o de i e symbolic jacobian
au oma ically. The model can be ex ended by p essu e loss e alua ion, which would allow simula ion
o whole DH g ids based on Ki chho laws. Regula iza ion o mass lows would be necessa y he e o
a oid singula i ies (di ision by ze o) in he nodes.
The u u e wo k should aim a a sel -iden i ica ion algo i hm using inle /ou le empe a u e
signals om eal ins alla ions o calib a e he app op ia e weigh s in he model. I may be possible o
gene alize he app oach o mo e complex con igu a ions, such as win-pipes, bu ied pipes, and so
on. An op imiza ion algo i hm can be used o ind he ma ix coe icien desc ibing he adial hea
dynamics. A backg ound om machine lea ning may be u ilized o such an endea o . The scaling o
he new model, when used o simula ion o whole g ids wi h au oma ic mass low e alua ion in each
b anch based on he p essu e d ops be ween indi idual nodes, is also pa o he u u e wo k.
Au ho Con ibu ions:
Concep ualiza ion, L.K.; Da a cu a ion, L.K. and R.C.; Fo mal analysis, L.K.; Funding
acquisi ion, J.P.; In es iga ion, L.K.; Me hodology, L.K. and R.C.; P ojec adminis a ion, J.P.; So wa e, L.K. and
R.C.; Supe ision, J.P.; Valida ion, L.K. and R.C.; Visualiza ion, L.K., R.C., and J.P.; W i ing-o iginal d a , L.K. and
J.P.; W i ing- e iew & edi ing, L.K.
Funding:
This pape has been suppo ed by he p ojec “Compu e Simula ions o E ec i e Low-Emission
Ene gy” unded as p ojec No. CZ.02.1.01/0.0/0.0/16_026/0008392 by Ope a ional P og amme Resea ch,
De elopmen and Educa ion, P io i y axis 1: S eng hening capaci y o high-quali y esea ch.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
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2019 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access
a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion
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