1
Flexible dynamic model o PHEX o ansien simula ions in
Ma lab/Simulink using ini e con ol olume me hod
E ik Salaza -He an, Koldobika Ma in-Escude o, Luis A. del Po illo-Valdes, I an
Flo es-Abascal, Naia a Rome o-An on
ENEDI Resea ch g oup, Depa men o The mal Enginee ing, Uni e si y o he Basque
Coun y (UPV/EHU), Plaza To es Que edo 1, 48013 Bilbao, Spain
Email: e ik.salaza @ehu.eus
Telephone numbe : + (34) 94 601 7322
ABSTRACT
In o de o imp o e he ene gy e iciency and con ol o hea pump sys ems, i is
necessa y o de elop dynamic models ha accu a ely simula e hei eal pe o mance.
In addi ion, hese models will help o ca y ou u u e wo ks o esea ch, such as new low
ca bon e ige an es ing.
Physics-based models ollow a se o physics laws ha cha ac e ize he model as he
mos accu a e, e sa ile and obus o simula e di e en hea pump sys ems. Taking in o
accoun he ac ha he dynamics o he elemen s ha egula e mass low (comp esso s
and al es) a e much as e han he dynamics o he componen s ha egula e hea
ans e (hea exchange s), he model complexi y usually esides in he la e .
This pape p o ides a de ailed explana ion o he physics-based dynamic model in
Ma lab/Simulink using he ini e-con ol olume app oach applied o a e ige an - o-liquid
pla e hea exchange . Dynamic expe imen al es s we e de eloped o alida e he model
unde ou possible si ua ions: condense and e apo a o hea exchange s wo king in
bo h coun e - and pa allel- low. In addi ion, an app oxima ion o he numbe o ini e
con ol olumes equi ed o each a good accu acy, while main aining a easonable
simula ion ime is p esen ed.
Simula ion esul s show g ea accu acy when compa ed o expe imen al es s. I was
p o ed by calcula ing he No malized Residual E o , which is be ween 1.1 E-04 and 1.0
E-03 in all cases. I was also concluded ha using wen y ini e con ol olumes, he e is
good ag eemen be ween he accu acy o he esul s and he compu a ional ime.
Keywo ds: Pla e hea exchange s; Dynamic modeling; Expe imen al alida ion; Fini e
con ol olume app oach; Re e sible hea pump sys ems; Ma lab/Simulink.
This is he accep ed manusc ip o he a icle ha appea ed in inal o m in In e na ional Jou nal o Re ige a ion 110 : 83-94
(2020) , which has been published in inal o m a h ps://doi.o g/10.1016/j.ij e ig.2019.11.003. © 2019 Else ie L d and IIR.
unde CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
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Nomencla u e
Symbols
A a ea [m2]
c speci ic hea [kJ kg-1 K-1]]
h en halpy [kJ kg-1]
ℎ󰇗 en halpy ime de i a i e [kJ kg-1 s-1]
L wid h o he pla es [m]
m mass [kg]
𝑚󰇗 mass low a e [kg s-1]
N numbe o ini e olumes [-]
P p essu e [kPa]
𝑃󰇗 p essu e ime de i a i e [kPa s-1]
𝑞󰇗 hea ans e a e [kW s-1]
ime [s]
T empe a u e [K]
V olume [m3]
x leng h o he pla es in he low di ec ion [m]
G eek le e s
𝛼 con ec ion hea ans e coe icien [kW m-2 K-1]
𝜕 pa ial de i a i e
𝜌 densi y [kg m-3]
𝜌󰇗 densi y ime de i a i e [kg m-3 s-1]
Subsc ip s
a e a e age p ope y
c c oss-sec ional a ea
h a cons an en halpy
i con ol olume index
in inle
ou ou le
p a cons an p essu e
R e ige an
s hea ans e a ea
S seconda y luid
W wall/pla e
Ac onyms
COP coe icien o pe o mance [-]
FCV Fini e Con ol Volume
GSHP G ound-Sou ce Hea Pump sys ems
HEX Hea Exchange s
HTC Hea T ans e Coe icien [kW m-2 K-1]
MB Mo ing Bounda y
NRE No malized Residual E o
PHEX Pla e Hea Exchange
RTF Real Time Fac o
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1. In oduc ion
Ene gy sus ainabili y is a global challenge ha needs o be achie ed in he coming yea s.
Di e en s eps ha e been aken owa ds his goal o e he las ew yea s, such as he
Pa is Ag eemen [1] and he Ho izon 2020 p og am [2]. Wi hin his challenge, Hea Pump
sys ems a e des ined o be one o he sys ems ha will help o achie e hese
in e na ionally p oposed goals. In ac , since 2009 in he Eu opean Union, a pa o he
ene gy p oduced by hese sys ems can be conside ed enewable [3].
G ound sou ce hea pump sys ems (GSHP) a e a ype o apo comp ession sys em
ha , d i en by elec ic ene gy, can hea o cool wa e o ai h oughou he yea by
ans e ing hea om o o a sou ce, such as he subsoil o a wa e ese oi [4]. These
sys ems ha e been p o ed o ha e be e ene gy pe o mance han ai sou ce hea pump
sys ems [5].
Imp o ing he pe o mance and con ol o hese sys ems is a key issue in making GSHP
an inc easingly compe i i e echnology as compa ed o con en ional hea ing and cooling
sys ems, as well as o eaching he abo e-men ioned ene ge ic sus ainabili y. In his
issue, hea pump manu ac u e s play a e y impo an ole in de eloping inno a i e
GSHP wi h upg aded p edic i e con ol sys ems and be e pe o mance. Ne e heless,
hea pump p o o ypes and expe imen al es s wi h new equipmen equi e a high
in es men and his is ime-consuming. Dynamic simula ion models and so wa e, which
could educe he cos s o new p oduc s and imp o e hei pe o mance [6], would be
bene icial all o e he wo ld, bu such models need o be accu a e, e sa ile and obus .
In his line, physics-based models a e an app op ia e solu ion. These kinds o model use
physical laws and pa ial de i a i es o simula e he dynamic beha io o he sys em,
p o iding g ea accu acy and he oppo uni y o simula e di e en componen s and
con igu a ions. Unde wood [7] e iewed GSHP simula ions, compa ing models based on
ans e unc ions and physics-based models, emphasizing he accu acy o he la e .
Simila ly, Rasmussen [8] compa ed physics-based and da a-based models, highligh ing
he e sa ili y o he o me .
The challenge when modeling a dynamic sys em usually lies in modeling he
componen s wi h as dynamics. In he case o hea pump sys ems, he hea exchange s
ha e much slowe dynamics han he componen s ha egula e he mass low a e, such
as comp esso s o al es.
Fo modeling hea exchange s (HEX) wi h physics-based models, di e en app oaches
and esolu ion me hods can be ound. Among o he s, wo o he mos commonly used
app oaches a e he mo ing-bounda y (MB) [9] and ini e-con ol olume (FCV) [10].
While FCV di ides he o al leng h o he HEX in o a ini e numbe o olumes o equal
size; he MB app oach di ides i in o a educed numbe o unequal and a iable size
ini e olumes. The sizes o hose ini e olumes a ies along he simula ion and depends
on a chosen p ope y, such as he mean oid ac ion [11].
Some compa isons be ween bo h app oaches ha e been p esen ed o e he las ew
yea s. Bendapudi e al. [12] compa ed hem in a shell-and- ube hea exchange o a
cen i ugal chille sys em. The esul s we e compa ed wi h expe imen al da a and i was
concluded ha , du ing s eady-s a es, MB was as e eaching equal accu acy.
Ne e heless, du ing ansien -s a es, he FCV o mula ion was ound o be mo e obus .
He schel e al. [13] made a compa ison be ween modeling an ai - o-liquid HEX wi h bo h
FVC and MB app oaches, concluding ha he compu a ional ime o MB was lowe han
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ha o FCV, main aining a good o de o accu acy in bo h cases. Deside i e al. [14]
compa ed MB and FCV app oaches o simula e a pla e hea exchange (PHEX)
concluding he simila good g ade o accu acy o bo h model and highligh ing he lowe
compu a ional ime o he MB app oach. Simila conclusions we e eached by Bonilla e
al. [15] simula ing an e apo a o o a di ec sola s eam acili y. Rod iguez e al. [16]
compa ed ou di e en dynamic modeling pa adigms o an ai - o- e ige an e apo a o .
I was compa ed FCV app oach wi h h ee MB me hods: en halpy, oid ac ion and
densi y based MB. They highligh ed he ad an ages o he FCV me hod due o he no
need o swi ching igge s, h esholds o ole ances o each an accu a e esul s,
al hough i is done a he expenses o a highe compu a ional ime.
The MB app oach has been widely used in ai - o- e ige an HEX dynamic modeling, as
desc ibed in [8,17,18]. Fo ins ance, Ib ahim e al. [19] used i o p edic he ene gy
pe o mance o an ai sou ce hea pump wa e hea e . Simila ly, MB app oach has been
used o simula e liquid- o- e ige an HEX, such as [14,20]. Bell e al. [20] de eloped an
MB model o coun e low HEX aking in o accoun he luid phase change, bu was only
able o p edic s a iona y s a es. Chu e al. [21] we e able o model a coun e low PHEX
by using a MB-FCV coupling algo i hm.
Rega ding FCV app oach, Ozana e al. [22] implemen ed a dynamic model o a s eam
supe hea e HEX o an indus ial boile using he FCV app oach. I was alida ed o bo h
pa allel and coun e low connec ions. Ne e heless, as o a supe hea ed HEX, i was
no necessa y o model he phase change o he luids. Bendapudi e al. [23] de eloped
a cen i ugal chille sys em model using he FCV app oach o he shell-and- ube hea
exchange , ocusing on such aspec s as mesh dependence, in eg a o o de and s ep-
size. I was concluded ha an inc ease in he numbe o ini e con ol olumes inc eases
he simula ion accu acy up o a limi . S iha i e al. [24] analyzed he e ec o a bad
dis ibu ion o he low in o he pla e hea exchange (PHEX) channels o single-phase
luids using an FCV model. The ansien pe o mance o he ai side hea exchange o
an ai -wa e hea pump sys em unde condi ions o os ing was analyzed using a
dynamic FCV model by Gao e al. [25].
Two o he mos common modeling en i onmen s o physic-based models o HEX a e
Modelica [26] and Ma lab/Simulink [27]. In Modelica many esea ches ha e been ca ied
ou , bo h wi h MB and FCV app oach [14,15,28,29]. Mo eo e , some d awbacks o FCV
such as cha e ing [30] ha e been s udied concluding ha ha phenomenon would be
educed by using MB app oach [31].
Rega ding Ma lab/Simulink, among o he s [32,33], one o he mos known he mal
sys ems simula ion models a e in he oolbox The mosys [34]. I allows he use o
simula e di e en e ige a ion sys ems wi h di e en componen s such as ai - o-
e ige an condense s and e apo a o s o liquid- o- e ige an condense s and
e apo a o s. Howe e , his las ones only in pa allel low con igu a ion. These
componen s a e de eloped by using MB app oach, which make di icul o de elop a
liquid- o- e ige an HEX wi h coun e - low con igu a ion.
Taking all his in o conside a ion and o he bes o au ho s knowledge, a e ige an - o-
liquid HEX dynamic model ha includes phase-change phenomena (bo h condensa ion
and e apo a ion) and ha can be used indis inc ly o bo h coun e - low and pa allel- low
con igu a ions in Ma lab/Simulink en i onmen canno be ound in he li e a u e. This kind
o model allows calcula ions o be ca ied ou in GSHP, ha ing he possibili y o simula e
swi ching mode ope a ion. In addi ion, he ex ensi e use o Ma lab/Simulink so wa e
5
mus be aken in o accoun in he wo ld o he mal acili ies manu ac u e s, especially o
he de elopmen o con ol loops.
The main goal o his documen is o p esen a high accu acy dynamic model o simula e
ansi o y changes in PHEX ha can be connec ed in bo h pa allel and coun e - low.
Addi ionally, he model allows he same PHEX o wo k as a condense o e apo a o .
The go e ning equa ions o he model and i s ma ix o m a e de eloped in de ail.
Mo eo e , an example o he o mula ion o a h ee ini e con ol olumes PHEX o bo h
coun e and pa allel- low is p esen ed. Finally, he alida ion o he model unde di e en
expe imen al si ua ions is ca ied ou and a pa ame ic analysis o assess he in luence
o he numbe o FVC in he simula ion speed is done.
2. Model de elopmen
The model p esen ed is a dynamic model based on physical equa ions using a
dis ibu ed-pa ame e ini e olume modeling me hod de eloped in a Ma lab/Simulink
en i onmen . As we ha e al eady said, he de eloped model is able o simula e he
beha io o a e ige an - o-liquid PHEX unde di e en si ua ions. All hese si ua ions
a e g ouped unde he cases shown in Table 1. The go e ning equa ions p esen ed and
deal wi h he e a e equal o all o he said cases. Ne e heless, when he equa ions a e
disc e ized in o di e en ini e con ol olumes and b ough oge he in a ma ix o m,
depending on whe he he PHEX connec ions a e in coun e - low o pa allel- low, he
esul ing ma ices a e di e en . As he di ec ion o he e ige an is in e ed, wha was
he inle po o he e ige an becomes he ou le po and ice e sa, changing he sign
o some a iables o he ma ix. Addi ionally, he co ela ions o calcula ing he hea
ans e coe icien (HTC) o ob ain he hea ans e a es a e subo dina e o he wo king
mode o he PHEX, and hese a e di e en o he condense o e apo a o wo king
modes.
Table 1. Cases analyzed o PHEX wo king modes and connec ions
Case numbe
PHEX connec ions and wo king mode
Case 1
Coun e - low condense
Case 2
Pa allel- low condense
Case 3
Coun e - low e apo a o
Case 4
Pa allel- low e apo a o
2.1. Model equa ions
The equa ions used o desc ibe he dynamic, physical beha io o he luids inside he
PHEX a e now p esen ed. The ollowing assump ions ha e been aken in o accoun o
modeling he PHEX.
• P essu e d ops h ough he PHEX a e negligible.
• The maldis ibu ion o he luids is neglec ed.
• Axial hea conduc ion is negligible, so he p ope ies inside each ini e olume a e
conside ed o be uni o m.
• The PHEX is supposed o be pe ec ly insula ed om he su ounding
en i onmen .
• The luid low is modeled as one-dimensional in he longi udinal di ec ion.

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The FCV app oach is used o ca y ou he simula ions. This model can cap u e he
beha io and dynamics inside hea exchange s in g ea de ail. The he mo-physical
g adien s and dis ibu ed pa ame e s o each small disc e iza ion o he hea exchange
can be ob ained.
2.1.1. Re ige an equa ions
The e ige an luid unde goes a phase change when i ci cula es h ough he PHEX. I
can be condensed, going om supe hea ed gas o sub-cooled liquid, o e apo a ed,
going om wo-phase luid o supe hea ed gas. Addi ionally, i can lea e he PHEX as a
wo-phase luid i he hea ans e ed be ween luids is insu icien o conclude he
condensa ion o e apo a ion p ocesses.
Du ing ansien s a es, he p ope ies o he e ige an luid change o e ime. In o de
o model he changes in he luid p ope ies, ene gy and mass conse a ion equa ions
a e applied o he e ige an luid. The momen um equa ion is no applied because, as
said be o e, he p essu e d op h ough he PHEX is negligible.
Eq. (1) and (2) p esen he mass and ene gy conse a ion equa ion applied o he
e ige an luid espec i ely.
The ollowing s eps a e ollowed o ans o m bo h equa ions:
• Apply 𝑐𝑅𝑇𝑅=ℎ𝑅 equali y o calcula e he en halpy a ia ion ins ead o empe a u e
a ia ion.
• In eg a e o e he leng h o he con ol olume, which in his case is he o al leng h o
he hea exchange .
• Apply he Leibniz in eg al ule [36].
• Assume a e age alues o luid and pla e p ope ies inside he PHEX.
• Sol e ime de i a i es.
• Take he en halpy and he p essu e as he independen a iables o calcula e he
densi y ime de i a i e.
Now, he mass conse a ion equa ion and ene gy conse a ion equa ion applied o he
e ige an luid has been modi ied in o de o be implemen ed in a simula ion code.
whe e |𝜕𝜌
𝜕𝑃|ℎ ep esen s he e ige an densi y a ia ion wi h espec o he p essu e
a ia ion a cons an en halpy and |𝜕𝜌
𝜕ℎ|𝑃 ep esen s he e ige an densi y a ia ion wi h
espec o he en halpy a ia ion a cons an p essu e [37].
𝜕(𝜌𝑅𝐴𝑐)
𝜕𝑡 +𝜕(𝑚󰇗𝑅)
𝜕𝑥 =0
(1)
𝜕(𝜌𝑅𝑐𝑅𝐴𝑐𝑇𝑅)
𝜕𝑡 −𝜕(𝐴𝑐𝑃𝑅)
𝜕𝑡 +𝜕(𝑚󰇗𝑅𝑐𝑅𝑇𝑅)
𝜕𝑥 =𝛼𝑅𝐿(𝑇𝑅−𝑇𝑊)
(2)
(|𝜕𝜌
𝜕𝑃|ℎ𝑉)𝑃󰇗𝑅+(|𝜕𝜌
𝜕ℎ|𝑃𝑉)ℎ󰇗𝑅+𝑚󰇗𝑅,𝑜𝑢𝑡−𝑚󰇗𝑅,𝑖𝑛=0
(3)
(|𝜕𝜌
𝜕𝑃|ℎℎ𝑅,𝑎𝑣𝑒−1)𝑉𝑃󰇗𝑅+(|𝜕𝜌
𝜕ℎ|𝑃ℎ𝑅,𝑎𝑣𝑒+𝜌𝑅,𝑎𝑣𝑒)𝑉ℎ󰇗𝑅+𝑚󰇗𝑅,𝑜𝑢𝑡ℎ𝑅,𝑜𝑢𝑡−𝑚󰇗𝑅,𝑖𝑛ℎ𝑅,𝑖𝑛
= 𝛼𝑅𝐴𝑠(𝑇𝑅−𝑇𝑊)
(4)
7
Implemen ing Eq. (3) and Eq. (4) oge he in a simula ion code, he a ia ion o he
e ige an p ope ies du ing ansien s a es can be p edic ed.
2.1.2. Seconda y luid equa ions
The eloci y and he densi y o he seconda y luid, gene ally wa e , a e assumed o be
cons an all along he hea exchange . This means he equa ion o he mass conse a ion
does no need o be applied o he seconda y luid.
Eq. (5) p esen s he ene gy conse a ion equa ion o he seconda y luid. Unlike he
e ige an luid, he seconda y luid will emain in a liquid phase h oughou he PHEX.
The e o e, he empe a u e a ia ion can be di ec ly calcula ed ins ead o en halpy
a ia ion.
Since he seconda y luid is assumed o be an incomp essible luid, he a ia ion in he
p essu e is neglec ed. The e o e, he pa ial de i a i e o he p essu e in e ms o he
ime does no appea in Eq. (5).
𝜕(𝜌𝑆𝐴𝑐𝑐𝑆𝑇𝑆)
𝜕𝑡 +𝜕(𝑚󰇗𝑆𝑐𝑆𝑇𝑆)
𝜕𝑥 =𝛼𝑆𝐿(𝑇𝑊−𝑇𝑆)
(5)
In eg a ing each e m o he equa ion o e he leng h o he PHEX, assuming a e age
p ope ies inside he PHEX and sol ing he ime de i a i es, Eq (6) is ob ained.
𝑉𝜌𝑆,𝑎𝑣𝑒𝑐𝑆,𝑎𝑣𝑒𝑇󰇗𝑆+𝑚󰇗𝑆𝑐𝑆,𝑎𝑣𝑒(𝑇𝑆,𝑜𝑢𝑡−𝑇𝑆,𝑖𝑛)= 𝛼𝑆𝐴𝑠(𝑇𝑊−𝑇𝑆)
(6)
By implemen ing Eq. (6) in a simula ion code, he seconda y luid empe a u e a ia ion
in a HEX can be calcula ed wi h espec o he ime.
2.1.3. Hea exchange pla e equa ions
The empe a u e a ia ion o he hea exchange pla e is desc ibed by applying he
ene gy conse a ion equa ion. I is p esen ed in Eq. (7) and no ans o ma ions a e
needed.
𝑐𝑊𝑚𝑊𝑇󰇗𝑊=𝛼𝑅𝐴𝑠(𝑇𝑅−𝑇𝑊)−𝛼𝑆𝐴𝑠(𝑇𝑆−𝑇𝑊)
(7)
The empe a u e o he pla e depends on he mass, he he mal p ope ies o he pla e
ma e ial and he hea ans e ed o o om bo h e ige an and seconda y luids.
2.2. Model implemen a ion
Once he necessa y equa ions ha e been ans o med in o empo al de i a i e
equa ions, hey a e hen disc e ized o use wi h he FCV app oach. Fo an 𝑁 ini e
numbe o con ol olumes, Eqs. (3), (4), (6) and (7) a e ans o med in o Eqs. (8) - (11).
𝑉𝑐𝑣[(|𝜕𝜌
𝜕𝑃|ℎ)𝑖ℎ𝑅,𝑖−1]𝑃󰇗𝑅+𝑉𝑐𝑣[(|𝜕𝜌
𝜕ℎ|𝑃)𝑖ℎ𝑅,𝑖+𝜌𝑅,𝑖]ℎ󰇗𝑅,𝑖+𝑚󰇗𝑅,𝑜𝑢𝑡,𝑖ℎ𝑅,𝑜𝑢𝑡,𝑖
−𝑚󰇗𝑅,𝑖𝑛,𝑖ℎ𝑅,𝑖𝑛,𝑖= 𝛼𝑅,𝑖𝐴𝑠,𝑐𝑣(𝑇𝑅,𝑖−𝑇𝑊,𝑖)
(8)
𝑉𝑐𝑣(|𝜕𝜌
𝜕𝑃|ℎ)𝑖𝑃󰇗𝑅+𝑉𝑐𝑣(|𝜕𝜌
𝜕ℎ|𝑃)𝑖ℎ󰇗𝑅,𝑖+𝑚󰇗𝑅,𝑜𝑢𝑡,𝑖−𝑚󰇗𝑅,𝑖𝑛,𝑖=0
(9)
𝑉𝑐𝑣𝜌𝑆,𝑖𝑐𝑆,𝑖𝑇󰇗𝑆,𝑖+𝑚󰇗𝑆,𝑖𝑐𝑆,𝑖(𝑇𝑆,𝑜𝑢𝑡,𝑖−𝑇𝑆,𝑖𝑛,𝑖)= 𝛼𝑆,𝑖𝐴𝑠,𝑐𝑣(𝑇𝑊,𝑖−𝑇𝑆,𝑖)
(10)
8
𝑐𝑊,𝑐𝑣𝑚𝑊,𝑐𝑣𝑇󰇗𝑊,𝑖=𝛼𝑅,𝑖𝐴𝑠,𝑐𝑣(𝑇𝑅,𝑖−𝑇𝑊,𝑖)−𝛼𝑆,𝑖𝐴𝑠,𝑐𝑣(𝑇𝑆,𝑖−𝑇𝑊,𝑖)
(11)
Then, in o de o implemen hem in a Ma lab code, hey a e eo ganized in o a ma ix
equa ion.
[
𝑉𝑐𝑣[(|𝜕𝜌
𝜕𝑃|ℎ)𝑖ℎ𝑅,𝑖−1]𝑉𝑐𝑣[(|𝜕𝜌
𝜕ℎ|𝑃)𝑖ℎ𝑅,𝑖+𝜌𝑅,𝑖]0 0
𝑉𝑐𝑣(|𝜕𝜌
𝜕𝑃|ℎ)𝑖𝑉𝑐𝑣(|𝜕𝜌
𝜕ℎ|𝑃)𝑖0 0
0 0 𝑉𝑐𝑣𝜌𝑆,𝑖𝑐𝑝,𝑆,𝑖 0
00 0 𝑐𝑊𝑚𝑊,𝑐𝑣
]
×
[
𝑃󰇗𝑅
ℎ󰇗𝑅,𝑖
𝑇󰇗𝑆,𝑖
𝑇󰇗𝑊,𝑖
]
=
[
𝑚󰇗𝑅,𝑖𝑛,𝑖ℎ𝑅,𝑖𝑛,𝑖−𝑚󰇗𝑅,𝑜𝑢𝑡,𝑖ℎ𝑅,𝑜𝑢𝑡,𝑖+2𝑞󰇗𝑅,𝑖
𝑚󰇗𝑅,𝑖𝑛,𝑖−𝑚󰇗𝑅,𝑜𝑢𝑡,𝑖
−𝑚󰇗𝑆,𝑖𝑐𝑆,𝑖(𝑇𝑆,𝑜𝑢𝑡,𝑖−𝑇𝑆,𝑖𝑛,𝑖)+2𝑞󰇗𝑆,𝑖
𝑞󰇗𝑅,𝑖−𝑞󰇗𝑆,𝑖
]
(12)
Bo h hea ans e a es (𝑞󰇗𝑅 and 𝑞󰇗𝑆) a e mul iplied by wo because, in he ac ual PHEX,
e e y e ige an o seconda y luid a e in con ac wi h wo in e media e pla es, excep
o he bounda y seconda y luid channels. Empi ical co ela ions a e used in he
calcula ion o he hea ans e a es. Fo he one-phase con ec ion HTC, he Di us-
Boel e co ela ion is used [38]. In addi ion, he co ela ions p esen ed in Han e al.
[39,40] a e used o he wo-phase con ec ion HTC.
In o de o sol e Eq. (12), he inle and ou le mass low a es o he e ige an luid mus
be known. When a s eady-s a e is calcula ed, bo h a e equal. Ne e heless, du ing a
ansien -s a e, expansion and comp ession de ices egula e he e ige an mass low
and he one can be di e en om he o he . On he o he hand, he inle en halpy o he
e ige an luid and he inle empe a u e and mass low a e o he seconda y luid mus
also be known.
As will be shown la e , when Eq. (12) is ex ended o an a bi a y numbe o con ol
olumes and he e ige an inle and ou le mass low a es a e known, he in e media e
mass low a es mus be calcula ed a each ime s ep. These in e media e e ige an
mass low a es will be mo ed o he second e m o he ma ix equa ion, o ming ano he
s a e ec o .
In he agg ega e, o each ime s ep, 4𝑁 de i a i e s a es and in eg a ions mus be
sol ed. Ne e heless, only he e ige an ene gy and mass equa ions ha e o be sol ed
oge he . The ene gy equa ion o pla es and seconda y luid can be emo ed om he
ma ix and sol ed sepa a ely, hus educing he compu a ional ime.
On he o he hand, a he ime o applying he p esen ed o mula ion o a ixed numbe
o FCV, coun e - low and pa allel- low connec ions mus be di e en ia ed. Figu e 1 and
Figu e 2 show a g aph desc ip ion o a PHEX disc e ized in N numbe o FCV o coun e -
low and pa allel- low connec ions, espec i ely.
9
Figu e 1. Fini e con ol olume coun e - low PHEX model.
Figu e 2. Fini e con ol olume pa allel- low PHEX model.
As can be seen, he ou le s and inle s o he e ige an luid a e in e ed when he
connec ion o he PHEX is in coun e - low o pa allel- low. The in e sion o he e ige an
luid allows he model o be used in e e sible hea pump sys ems which can hea and
cool-down wa e , depending on he use demand.
2.3. Th ee FCV example cases
Eq. (12) will now be ex ended o a h ee con ol olume o mula ion o bo h a coun e -
low and a pa allel- low connec ion. The di e ences in he o mula ion be ween bo h
connec ions a e shown below.
Eqs. (13) and (14) show he ma ix equa ions ha sol e he e ige an ene gy and mass
equa ions o coun e - low and pa allel- low PHEX, espec i ely. As he ene gy equa ion
o seconda y luid and pla es can be sol ed sepa a ely and a e equal ega dless o he
connec ion o he PHEX, hey ha e been emo ed om he main ma ix. Eq. (15) shows
he seconda y luid ene gy ma ix equa ion and Eq. (16) shows he in e media e pla e
ene gy ma ix equa ion o he h ee con ol olume example.
Seconda y
luid
Re ige an
luid
Pla e
Con ol
olumen 1 Con ol
olumen 2 Con ol
olumen n-1 Con ol
olumen n
. . .
𝑚󰇗𝑆𝑚󰇗𝑆
𝑚󰇗𝑅,𝑜𝑢𝑡 𝑚󰇗𝑅,𝑖𝑛
𝑞󰇗𝑅,
𝑞󰇗𝑆, 𝑇𝑆,𝑜𝑢𝑡
ℎ𝑅,𝑜𝑢𝑡
𝑇𝑆,𝑖𝑛
ℎ𝑅,𝑖𝑛
𝑇𝑆, 𝑇𝑆,𝑛 𝑛
𝑇𝑆, 𝑇𝑆, 𝑇𝑆,𝑛 𝑇𝑆,𝑛
ℎ𝑅, ℎ𝑅,𝑛 𝑛
ℎ𝑅, ℎ𝑅, ℎ𝑅,𝑛 ℎ󰇗𝑅,𝑛
𝑞󰇗𝑅, 𝑞󰇗𝑆, 𝑞󰇗𝑅,𝑛
𝑞󰇗𝑆,𝑛 𝑞󰇗𝑅,𝑛
𝑞󰇗𝑆,𝑛
𝑞󰇗𝑆, 𝑞󰇗𝑆, 𝑞󰇗𝑆,𝑛 𝑞󰇗𝑆,𝑛
𝑞󰇗𝑅, 𝑞󰇗𝑅, 𝑞󰇗𝑅,𝑛 𝑞󰇗𝑅,𝑛
Pla e
Pla e
𝑚󰇗𝑅, 𝑚󰇗𝑅,𝑛 𝑛
Seconda y
luid
Re ige an
luid
Pla e
Con ol
olumen 1 Con ol
olumen 2 Con ol
olumen n-1 Con ol
olumen n
. . .
𝑚󰇗𝑆𝑚󰇗𝑆
𝑚󰇗𝑅,𝑜𝑢𝑡
𝑚󰇗𝑅,𝑖𝑛 𝑞󰇗𝑅,
𝑞󰇗𝑆, 𝑇𝑆,𝑜𝑢𝑡
ℎ𝑅,𝑜𝑢𝑡
𝑇𝑆,𝑖𝑛
ℎ𝑅,𝑖𝑛
𝑇𝑆, 𝑇𝑆,𝑛 𝑛
𝑇𝑆, 𝑇𝑆, 𝑇𝑆,𝑛 𝑇𝑆,𝑛
ℎ𝑅, ℎ𝑅,𝑛 𝑛
ℎ𝑅, ℎ𝑅, ℎ𝑅,𝑛 ℎ𝑅,𝑛
𝑞󰇗𝑅, 𝑞󰇗𝑆, 𝑞󰇗𝑅,𝑛
𝑞󰇗𝑆,𝑛 𝑞󰇗𝑅,𝑛
𝑞󰇗𝑆,𝑛
𝑞󰇗𝑆, 𝑞󰇗𝑆, 𝑞󰇗𝑆,𝑛 𝑞󰇗𝑆,𝑛
𝑞󰇗𝑅, 𝑞󰇗𝑅, 𝑞󰇗𝑅,𝑛 𝑞󰇗𝑅,𝑛
Pla e
Pla e
𝑚󰇗𝑅, 𝑚󰇗𝑅,𝑛 𝑛
16
(a)
(b)
(c)
Figu e 8. Pa allel- low condense (Case 2) esul s o e ige an p essu e (a), wa e
empe a u es (b) and empe a u e dis ibu ion a ime 800 seconds (c).
Figu e 8(c) depic s he empe a u e dis ibu ion in he pa allel- low condense . As can be
seen, he e ige an luid jus eaches he sub-cooled sa u a ion line in he inal pa o
he PHEX.
On he o he hand, Figu e 9(c) and Figu e 10(c) ep esen he empe a u e dis ibu ion
when he PHEX wo ks as an e apo a o o he coun e - low and he pa allel- low,
espec i ely. As can be seen, he e ige an luid goes in o he PHEX as a wo-phase
luid. Inside he PHEX, i is e apo a ed and supe hea ed.
(a)
(b)
2400
2500
2600
2700
2800
2900
0 200 400 600 800 1000
Condensa ion p essu e [kPa]
Time [s]
Tes da a
Model esul s
25
30
35
40
45
0 200 400 600 800 1000
Wa e empe a u e [ºC]
Time [s]
Ou le (model)
Ou le ( es )
Inle
30
40
50
60
70
80
010 20 30 40 50
Tempe a u e [ºC]
PHEX leng h [cm]
Re ige an
In e media e pla e
Wa e
700
800
900
1000
1100
0 200 400 600 800 1000
E apo a ion p essu e [kPa]
Time [s]
Tes da a
Model esul s
0
5
10
15
20
0 200 400 600 800 1000
Wa e empe a u e [ºC]
Time [s]
Ou le (model)
Ou le ( es )
Inle

17
(c)
Figu e 9. Coun e - low e apo a o (Case 3) esul s o e ige an p essu e (a), wa e
empe a u es (b) and empe a u e dis ibu ion a ime 800 seconds (c).
(a)
(b)
(c)
Figu e 10. Pa allel- low e apo a o (Case 4) esul s o e ige an p essu e (a), wa e
empe a u es (b) and empe a u e dis ibu ion a ime 800 seconds (c).
Rega ding he PHEX connec ions, in bo h Case 2 and Case 3, he wa e inside he PHEX
is cooled down by a ound 5 ºC. Ne e heless, in Case 2, he e ige an luid goes ou
om he PHEX wi h a supe hea ing o a ound 5ºC, while in Case 3, he supe hea ing is
o app oxima ely 2 ºC. Such a di e ence is ela ed o he PHEX connec ion, he hea
ans e being highe in a coun e - low connec ion han in a pa allel- low connec ion.
In Table 6, he maximum and mean nume ical di e ences be ween he model esul s
and he da a om he es s o e ige an p essu e and wa e ou le empe a u e a e
p esen ed.
Table 6. Maximum and mean nume ical di e ences be ween model esul s and es s da a o
he ou s udied cases.
Re ige an p essu e [kPa]
Wa e ou le empe a u e [ºC]
5
7
9
11
13
15
17
010 20 30 40 50
Tempe a u e [ºC]
PHEX leng h [cm]
Re ige an In e media e pla e Wa e
900
950
1000
1050
1100
1150
0 200 400 600 800 1000
E apo a ion p essu e [kPa]
Time [s]
Tes da a
Model esul s
0
5
10
15
20
0 200 400 600 800 1000
Wa e empe a u e [ºC]
Time [s]
Ou le (model)
Ou le ( es )
Inle
4
6
8
10
12
14
010 20 30 40 50
Tempe a u e [ºC]
PHEX leng h [cm]
Re ige an
In e media e pla e
Wa e
18
Maximum
Mean
Maximum
Mean
Case 1
51.5
20.3
1.1
0.3
Case 2
52.6
30.4
1.3
0.6
Case 3
46.9
12.4
0.5
0.1
Case 4
51.9
21.8
0.8
0.2
The esul s show ha he maximum nume ical di e ence o he e ige an p essu e is
a ound 50 kPa o any s udied case. Rega ding he compa ison o he wa e ou le
empe a u e, he g ea es di e ence is ound in Case 2. He e, he maximum di e ence
inc eases up o 1.3 ºC.
Addi ionally, he NRE o he e ige an p essu es and wa e ou le empe a u es o he
ou cases a e shown in Table 7. As can be seen, in all he cases s udied, he NRE is
smalle han he accep ed alue o 0.05.
Table 7. No malized Residual E o o he ou s udied cases.
Re ige an
p essu e
Wa e ou le
empe a u e
Case 1
1.1 E-04
1.2 E-04
Case 2
1.6 E-04
2.5 E-04
Case 3
2.2 E-04
2.5 E-04
Case 4
5.8 E-04
1.0 E-03
4.3. FVC numbe s simula ion speed
Finally, he simula ion speed and accu acy ha e been s udied by inc easing he numbe
o FCV used o simula e he desc ibed model om i e o one hund ed. The coun e - low
condense (Case 1) has been simula ed o 20 minu es and he Real Time Fac o (RTF)
o he simula ion has been calcula ed, di iding he compu a ional ime by he simula ed
ime.
Figu e 11 (a) shows he di e ences in he condensing p essu e calcula ion using a
di e en numbe o ini e olumes. Al hough he simula ion ime was 1200 seconds, in
he image only he pe iod whe e he ansien occu s has been shown, so he di e ences
can be be e pe cei ed. Figu e 11 (b) shows he RTF o di e en numbe s o FCV.
As can be app ecia ed in Figu e 11 (a), he shape o he sys em dynamic is well ollowed,
e en wi h a low numbe o FCV. Ne e heless, he nume ical esul s do no accu a ely i
wi h a low numbe o FCV. Wi h en FCV, he esul s can be imp o ed. The nume ical
esul s a e accu a e enough once he numbe o FCV is equal o o highe han wen y.
Rega ding he compu a ional ime, he RTF emains low up o 20 FCV and inc eases
g ea ly wi h 100 FCV. These esul s ag ee wi h he conclusions gi en in [42], whe e i is
s a ed ha a leas 15 FCV a e equi ed o achie e good accu acy.
19
(a)
(b)
Figu e 11. Condensing p essu e wi h di e en numbe s o ini e con ol olumes (a) and Real
Time Fac o o di e en numbe s o ini e con ol olumes (b).
5. Conclusions
In o de o imp o e he con ol sys ems o e e sible hea pump sys ems, models ha
can p edic he dynamic beha io and ansien s a es o hese sys ems a e needed. In
his ask, physics-based models a e he mos accu a e ones.
In his pape , a physics-based model using he ini e olume con ol app oach o simula e
a PHEX is p esen ed. The model has been de eloped in Ma lab/Simulink en i onmen .
This kind o hea exchange is commonly used in esiden ial e e sible g ound sou ce
hea pump sys ems. As i is based on physical equa ions, his model can be used o
model di e en ypes o PHEX, achie ing a good le el o accu acy. The model has been
alida ed wi h expe imen al dynamic es s o a PHEX wo king as condense and as
e apo a o . Mo eo e , o each wo king mode, i has been simula ed unde coun e - low
and pa allel- low a angemen s.
The accu acy o he model has been p o ed by calcula ing he NRE o he simula ed
esul s wi h espec o he es da a o he e ige an p essu e and he wa e ou le
empe a u e. The minimum alue o he NRE was 1.1 E-04, while he maximum was 1.0
E-03, which is lowe han he accep ed alue o his kind o model. I has also been
demons a ed ha disc e izing he model in wen y ini e olumes allows a high deg ee
o accu acy o be eached, while main aining he compu a ional ime low.
This pape p o ides a use ul ool o hea pump sys ems and PHEX design as i allows
he pe o mance o a PHEX o be compa ed, a ying he wo king mode o he PHEX, he
luid connec ions, he PHEX size, he numbe o PHEX pla es, he wo king luids, e c.
Addi ionally, as he model can simula e a PHEX wo king as a condense and as an
e apo a o , i can be used o simula e he beha io o e e sible hea pump sys ems.
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