Case S udies in The mal Enginee ing 32 (2022) 101869
A ailable online 15 Feb ua y 2022
2214-157X/© 2022 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY-NC-ND license
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
Expe imen al s udy on he he mal con ol o a oo - op collec i e
building an enna using a po ous ma ix illed wi h
Wa e -Coppe nano luid
F. Sas e
a
, A. Ma in-Ga in
b
, E. Ma in
c
,
*
, A. Velazquez
a
, A. Baï i
d
a
Uni e sidad Poli ´
ecnica de Mad id, Fluid Mechanics and Ae ospace P opulsion Depa men ., Plaza del Ca denal Cisne os 3, 28040, Mad id, Spain
b
Uni e sidad del País Vasco, UPV/EHU, ENEDI Resea ch G oup, Depa men o The mal Enginee ing, Plaza Eu opa 1, 20018, Donos ia-San
Sebas i´
an, Spain
c
Uni e sidade de Vigo, Depa amen o de Ingenie ía Mec´
anica, M´
aquinas y Mo o es T´
e micos y Fluidos, Escuela de Ingenie ía Indus ial, Campus
Ma cosende, 36310, Vigo, Spain
d
Uni e si ´
e de Pa is, Labo a oi e The mique In e aces En i onnemen (LTIE), EA 4415, 50 ue de S`
e es, F-92410, Pa is, F ance
HIGHLIGHTS
•The mal con ol o a collec i e building inned cylind ical an enna.
•Nano luid combined wi h po ous media.
•An enna empe a u e co ela ion o a ange o go e ning pa ame e s.
•Applica ion case o a sma building collec i e an enna.
ARTICLE INFO
Keywo ds:
Sma building
The mal egula ion
Elec onics enginee ing
Collec i e au oma ion an enna
Nano luid
Po ous media
ABSTRACT
This expe imen al wo k add essed he he mal con ol a oo - op collec i e building an enna
mean o con ol home equipmen in sma buildings. The an enna was placed inside a concen ic
quasi-cylind ical ca i y main ained a low empe a u e. Cooling was p o ided by a Wa e -Coppe
nano luid sa u a ed po ous ma ix placed be ween he an enna and he enclosu e. The a io o he
he mal conduc i i y o he po ous ma e ial o ha o he wa e a ied om 4 up o 41.2. The
nanopa icles olume ac ion a ied be ween 0% and 5%. The main esul was a new semi-
empi ical co ela ion ha allows o he de e mina ion o he an enna’s a e age su ace em-
pe a u e as a unc ion o he go e ning pa ame e s: a ios o nano luid o wa e and po ous media
o wa e he mal conduc i i ies, nanopa icles olume ac ion, and Rayleigh numbe . The
applicabili y o he co ela ion was illus a ed o a p ac ical applica ion case. I was ound ha ,
o some cases, he p oposed he mal con ol sys em imp o es powe dissipa ion by a ac o o
33% as compa ed wi h he case o pu e wa e (2 kW e sus 1.5 kW).
* Co esponding au ho .
E-mail add ess: [email p o ec ed] (E. Ma in).
Con en s lis s a ailable a ScienceDi ec
Case S udies in The mal Enginee ing
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h ps://doi.o g/10.1016/j.csi e.2022.101869
Recei ed 10 Decembe 2021; Recei ed in e ised o m 9 Feb ua y 2022; Accep ed 12 Feb ua y 2022
Case S udies in The mal Enginee ing 32 (2022) 101869
2
1. In oduc ion
The use o home au oma ion, ha applies o bo h indi idual apa men s and g oups o buildings, is becoming he endency in he
ield o mode n housing de elopmen . When conjoin ins alla ions a e planned, o en p e e ed because hey educe ope a ing and
main enance cos s, a collec i e an enna is no mally used. I s pu pose is o manage equipmen in eal and delayed ime, bo h on si e and
emo ely, and o se e o big da a ansmission in connec ion o building main enance and sa e y as desc ibed by Ma in-Ga in e al.
[1]. F om a he mal iewpoin , hese an ennae a e cha ac e ized by he s ong hea lux gene a ed by hei in e nal elec onic assembly
ha may lead o empe a u es so high ha hey could e en cause mal unc ions and shu down. The e o e, hei he mo egula ion is
c i ical o a eliable sys em ope a ion.
No mally, ai na u al con ec ion is p e e ed in hese ypes o he mal con ol sys ems gi en i s well-known ad an ages; among
Nomencla u e
a he mal di usi i y (m
2
s
−1
)
C speci ic hea a cons an p essu e (J.kg
−1
K
−1
)
g g a i y accele a ion (m.s
−2
)
H heigh o he cylinde (m)
I cu en in ensi y (A)
k(φ) unc ion in eq. (11.a), de ined in eq. (11.b)
L dis ance L=Re−Ri (m)
m(λ*) unc ion in eq. (11.a), de ined in eq. (11.c)
P powe (W)
q gene a ed hea lux (Wm
−2
)
Q measu ed global hea lux (Wm
−2
)
Re adius o he ca i y (m)
Ri adius o he ac i e cylinde (m)
RaL Rayleigh numbe (−)
S Con ac su ace wi h nano luid sa u a ed po ous medium (m
2
)
Si su ace o he i h elemen (m
2
)
Sh la e al su ace o he an enna (m
2
)
T empe a u e (K)
Tc a e age empe a u e o he ca i y (K)
Th a e age empe a u e o he an enna (K)
Ti empe a u e o he i h elemen (K)
U ol age (V)
G eek symbols
β olume ic expansion coe icien (K
−1
)
δ di e ence δ=ΔTc−ΔTm (K)
ΔY absolu e unce ain y o a gi en pa ame e Y (uni o Y)
ΔY/Y ela i e unce ain y o a gi en pa ame e Y (%)
ΔT di e ence empe a u e ΔT=Th-Tc (K)
ε
po osi y
φ Nanopa icles olume ac ion (%) in nano luid
λ he mal conduc i i y (Wm
−1
K
−1
)
λ* dimensionless he mal conduc i i y (−)
μ
dynamic iscosi y (Pa.s)
ρ
densi y (kg.m
−3
)
Subsc ip s
Cu solid Coppe nanopa icles
ca calcula ed
m measu ed
n nano luid
sp solid ma ix o he po ous medium
spm sa u a ed po ous medium
w base luid (pu e wa e )
F. Sas e e al.
Case S udies in The mal Enginee ing 32 (2022) 101869
3
o he s: simplici y o design and obus ness o ope a ion. This was he case in he s udy by Kim e al. [2] whe e he con ec i e hea
ans e coe icien was de e mined o a e ical su ace equipped wi h mic o- in a ays. Howe e , ai he mophysical p ope ies,
no wi hs anding hei well p o en applicabili y in enginee ing applica ions, a e some imes insu icien when a la ge olume ic hea
lux is gene a ed, as in he case discussed he e. In his si ua ion, o he wo king luids could be used such as wa e , oils, o alcohols wi h
di e en composi ions and he mophysical p ope ies (Bha i e al. [3], Baï i [4]). In his con ex , nano luids ha e ecen ly p o en o
be highly e ec i e in enhancing hea ans e , as shown, o ins ance, in he expe imen al wo k o Xu e al. [5] dealing wi h op o-
elec onic de ices. Fu he mo e, he imp o emen o nano luid’s manu ac u ing echniques, hei ease o use, and he educ ion in
hei cos p ice makes hem mo e and mo e a ac i e (Ali [6] and Sabou e al. [7]). Also, hei associa ion wi h po ous ma e ials leads
o u he enhanced hea ans e as highligh ed in se e al wo ks dealing wi h componen s and ca i ies o di e en shapes and a ious
nano luids: Abu-Nada and Oz op [8], Sheikholeslami e al. [9], Baï i and La aqi [10], She eme e al. [11], Alila [12], Pou azad e al.
[13], Hajipou e al. [14], Sha a e al. [15], Alila e al. [16], Salehi e al. [17], Gholinia e al. [18], El-Sho bagi e al. [19] and Li e al.
[20].
The he mophysical cha ac e is ics o he nano luid play an impo an ole in he hea exchange phenomena, as shown in Togh aie
e al. [21], Sol ani e al. [22], Baï i [23] and in he ecen c i ical e iew published by Zahma kesh e al. [24]. This e iew desc ibes,
among o he s, he e ec o nanopa icle shape, olume ac ion, and empe a u e on he cons i u i e cha ac e is ics o he nano luid,
whose e ec i e he mal conduc i i y and dynamic iscosi y a e he key he mophysical p ope ies o any modelling app oach. In his
con ex , se e al global models ha e been desc ibed in he specialized li e a u e, bu i should be no ed ha hey show a la ge
dispe sion. Acco dingly, esea ch is in ensi ying o classi y hose ha p esen he bes esul s alida ed by measu emen s, and o
associa e hem wi h p ecise ope a ing condi ions.
Basic models end o ollow, as a e e ence, he o mula ion p oposed by Maxwell [25] and B inkman [26] (sphe ical nano-
pa icles), and Hamil on and C osse [27] (non-sphe ical), o de e mine he e ec i e he mal conduc i i y and dynamic iscosi y o he
nano luid. A p ac ical implemen a ion o he Hamil on–C osse model [27] could be ound in he wo k by Abu-Nada e al. [28].
Ano he ele an model is he one p oposed by Koo and Kleins eue [29] ha decomposes he e ec i e he mal conduc i i y in o a
s a ic pa (Maxwell) and ano he pa aking accoun ing o he con ibu ion o B ownian mo ion. Recen applica ions o his model
could be ound in he wo ks o Sha a eldin e al. [30], and Babajani e al. [31]. An in e es ing conclusion o ano he model, his ime
p oposed by Timo ee a e al. [32], was ha he B ownian mo ion has less in luence on he e ec i e iscosi y han on he e ec i e
he mal conduc i i y. A ele an s udy, ha complemen s hose e iewed so a , is he one published by Co cione [33] in which he
p oposed cons i u i e co ela ions a e o an empi ical na u e. The s udy showed ha he ac ual nano luid e ec i e he mal conduc-
i i y ends o be ei he unde o o e -es ima ed by Maxwell’s model [25], and he e ec i e iscosi y o be unde -es ima ed by he
B inkman model [26].
The p esen wo k deals wi h he he mal con ol o a cylind ical an enna placed e ically. The en isaged applica ion is o i s use in
collec i e in eg a ed home au oma ion. The an enna is con ained in a concen ic quasi-cylind ical iso he mal enclosu e. Cooling is
p o ided by a po ous ma ix sa u a ed wi h H
2
O–Cu nano luid whose nanopa icles olume ac ion a ies in he ange om 0 (pu e
wa e ) o 5%. The a io be ween he mal conduc i i y o he po ous ma ix o ha o wa e spans om 4 o 41.2. A new co ela ion is
de eloped ha allows o de e mina ion o he mean empe a u e o he an enna as a unc ion o he go e ning pa ame e s. The esul s
ob ained could be used o op imize he global design o he an enna o a o i s ac ual implemen a ion.
2. Desc ip ion o he p oblem
The p oblem consis ed o he de e mina ion o he a e age empe a u e, Th , o a cylind ical, e ical, inned an enna du ing s eady
s a e ope a ion. A cons an hea lux q was gene a ed on he inside o he an enna whose la e al wall con ained eigh ec angula ins
Fig. 1(a). 3D ske ch o he conside ed con igu a ion.
F. Sas e e al.
Case S udies in The mal Enginee ing 32 (2022) 101869
4
a anged in he e ical di ec ion. This se up was enclosed in a quasi-cylind ical ca i y illed wi h a po ous ma ix sa u a ed wi h a
Wa e -Coppe based nano luid. The enclosing ca i y had a double ou e wall ha was kep iso he mal a empe a u e Tc by means o a
wa e -glycol dilu ion o ced o low in be ween he double wall. Two ske ches (3D and 2D) o he conside ed con igu a ion a e p e-
sen ed in Fig. 1(a) and Fig. 1(b), espec i ely.
Ri, Re, H, and Sh deno e inne and ex e nal adius o he sys em, heigh , and la e al su ace a ea espec i ely. The di e ence (Re−Ri)
was deno ed as L. The ho izon al base o he enclosu e (ca i y plus an enna) was insula ed and could be conside ed nea ly adiaba ic.
The nanopa icles olume ac ion in he Wa e -Coppe nano luid was a ied be ween 0 (pu e wa e ) up o 5%. The p ope ies ( he mal
conduc i i y, λ, densi y,
ρ
, speci ic hea , C, he mal expansion coe icien , β, and dynamic iscosi y,
μ
) o bo h wa e and coppe
nanopa icles (when applicable) a e de ailed in Table 1, see Abu-Nada e al. [28] and Rashidi e al. [34]. The a io be ween he he mal
conduc i i y o he po ous ma ix (po osi y
ε
=0.97) and he wa e was a ied in he ange om 4.0 up o 41.2.
2.1. Nano luid p ope ies
The Wa e -Coppe nano luid used in he expe imen s was egula ly eplaced o a oid p oblems like agglome a ion and sedi-
men a ion. Rega ding he de e mina ion o i s he mophysical p ope ies, he Maxwell [25] and B inkman [26] models we e used. The
expe imen al applicabili y o hese models has been es ed by Baï i [35] and Baï i e al. [36]. The e ec i e he mal conduc i i y, λn ,
dynamic iscosi y,
μ
n , densi y,
ρ
n , he mal expansion coe icien , βn , and speci ic hea , Cn , o he nano luid (subsc ip n ) we e
de ined as ollows:
Fig. 1(b). 2D iews o he conside ed con igu a ion.
Table 1
The mophysical cha ac e is ics o pu e wa e and Coppe nanopa icles.
Pu e wa e (w) Coppe nanopa icles (Cu)
λ (W m
−1
K
−1
) 0.613 401
ρ
(kg m
−3
) 997 8933
C (J kg
−1
K
−1
) 4180 395
β (K
−1
) 2.1 ×10
−4
1.67 ×10
−5
μ
(Pa s) 8.91 ×10
−4
F. Sas e e al.
Case S udies in The mal Enginee ing 32 (2022) 101869
5
λn =(λCu +2λw−2φ(λw−λCu)
λCu +2λw+φ(λw−λCu))λw(1)
μ
n =
μ
w
(1−φ)2.5(2)
ρ
n = (1−φ)
ρ
w+φ
ρ
Cu (3)
βn =(1−φ)(
ρ
β)w+φ(
ρ
β)Cu
ρ
n
(4)
Cn =(1−φ)(
ρ
C)w+φ(
ρ
C)Cu
ρ
n
(5)
whe e φ is he olume ac ion o he nano luid. P ope ies o he nano luid sa u a ed po ous media (subsc ip spm) we e calcula ed
using he po osi y
ε
.
λspm =
ε
λn + (1−
ε
)λsp (6)
(
ρ
C)spm =
ε
(
ρ
C)n + (1−
ε
)(
ρ
C)sp (7)
The a io be ween he mal conduc i i y o he po ous media’s solid ma ix (subsc ip sp) and ha o wa e , λ∗, was a go e ning
pa ame e oo.
λ∗=λsp
λw
(8)
2.2. Desc ip ion o he expe imen al se up
The main subsys ems (see he ske ch p esen ed in Fig. 2(a)) o he expe imen al ig we e:
•The he mo egula ed ba h ha gene a ed a glycol-wa e low a he desi ed empe a u e, Tc, in he ange be ween 278 and 300 K
(±0.1 K).
•The calib a ing he mome e needed o calib a e all he he mocouples o he assembly.
•The da a acquisi ion uni ha collec ed he he mal and elec ical da a o he es ed assembly.
•The s abilized powe supply enabling he an enna o be powe ed on. Cu en in ensi y I and ol age U we e measu ed wi h in-
eg a ed high p ecision mul ime e s (2 ×10
7
poin s, 6 digi s). Thei absolu e unce ain ies we e ΔI= ±0.1mA and ΔU= ± 0.01 V.
Rela i e unce ain y o he esul ing global powe P=UI was de e mined by means o he well-known pa ial de i a i e me hod as
ΔP/P=ΔU/U+ΔI/I. The unce ain y o he co esponding global measu ed hea lux Q=P/Sh was, hen, calcula ed by means o
he same de i a i e me hod ΔQ/Q=ΔP/P+ΔSh/Sh. A e accoun ing o he measu emen unce ain ies o he an enna’s di-
mensions, calcula ions lead o ΔQ/Q a ying in he span om 0.8 o 1.7%.
•The compu e needed o collec and p ocess he measu ed physical pa ame e s.
Fig. 2(a). Ske ch o he expe imen al ig.
F. Sas e e al.
Case S udies in The mal Enginee ing 32 (2022) 101869
6
Rega ding he es ed assembly, see Fig. 2(b) and Fig. 2(c), he an enna was ins alled on a ci cula base o insula ing ma e ial
(numbe 1 in Fig. 2(b)) placed on op o a hick ex uded polyu e hane oam pla e (numbe 2 in Fig. 2(b)). The a e age he mal
conduc i i y o his insula ing ma e ial in he ope a ing empe a u e ange (5–70 ◦C) was 0.036 ±0.002 Wm
−1
K
−1
measu ed by using
he T ansien Plane Sou ce (TPS) me hod [37,38]. The he mal conduc i i y o he po ous ma e ial, λsp, was also measu ed by means o
he same TPS me hod. Two ma e ials we e es ed in he expe imen al campaign wi h λ∗=λsp/λw=4.8 and 38.4 (±0.3) espec i ely. In
addi ion, he case wi hou po ous ma e ial (λ∗=0)was also es ed.
The po ous medium consis ed o sha ings o hese ma e ials made wi h a la he. Thei high po osi y
ε
=0.97 ±0.02 was measu ed
by using he Me cu y In usion Po osime y Me hod (IPM) [39]. The Wa e -Coppe nano luid was in oduced in o he ca i y by means
o wo pipes (numbe 3 o Fig. 2(b) and (c)). The pu pose o hese inle /ou le pipes was jus o ill and emp y he ca i y be o e and a e
he expe imen s espec i ely bu we e closed du ing he expe imen al campaigns. Two di e en concen a ions o Coppe nano-
pa icles we e conside ed in he es s. The co esponding olume ac ions we e φ=1.33 % and 4.81 % (±0.02). In addi ion, es s
we e also pe o med wi h pu e wa e (φ=0%). The mal conduc i i y o hese h ee luids ( wo Wa e -Coppe nano luids plus pu e
wa e ) was measu ed wi h he TPS me hod. The wa e -glycol mix u e gene a ed by he he mos a ic ba h lowed in o he double
hickness ou e wall o he ca i y h ough he ubes indica ed wi h numbe 4 in Fig. 2(b) and (c). This low kep he enclosu e’s wall a
Fig. 2(b). Pho og aph o he es ed assembly.
Fig. 2(c). Ske ch o he es ed assembly.
F. Sas e e al.
Case S udies in The mal Enginee ing 32 (2022) 101869
7
he desi ed cold empe a u e Tc wi hin ±0.1 K.
The an enna was powe ed by he wi ing (5) connec ed o he s abilized powe supply. The mocouple cables (numbe 6) we e
connec ed o he da a acquisi ion uni . The mal condi ions o he sys em we e measu ed by means o 0.05 mm diame e K- ype
he mocouples calib a ed a ±0.1 K by means o he calib a ing he mome e . Measu emen s we e pe o med a in e als o 0.05 s
un il he s eady s a e was eached. This s eady s a e was assumed o be hold when he empe a u e a ia ion was less han 0.5%. The
cold empe a u e Tc was measu ed a bo h he inle and ou le lines o he e ige an luid, as well as a ano he s a ion on he in e nal
su ace o he ca i y. Thei alues di e ed by less han 0.2 K when he luid low was adjus ed co ec ly.
The su ace empe a u e o he an enna was measu ed a wel e poin s e enly dis ibu ed on he cylinde ’s la e al a ea. The
empe a u e Th conside ed in his expe imen al app oach was he a i hme ic a e age o he measu ed su ace empe a u es. The
empe a u e o he ca i y’s base was measu ed a 4 poin s dis ibu ed uni o mly on he in e nal su ace o he base and a o he 4 poin s
loca ed on i s opposi e su ace ( he ex e nal ace). This allowed o he de e mina ion (using he linea ized Fou ie law) o he
Fig. 3. E olu ion o
ΔT=Th-Tc e sus RaL o some (λ∗,φ)combina ions.
F. Sas e e al.
Case S udies in The mal Enginee ing 32 (2022) 101869
8
conduc ion hea ans e passing h ough he insula ing pla e. Tes s based on he mos un a ou able con igu a ion ( he one wi h he
highes measu ed hea lux Q) showed ha he conduc i e losses associa ed o he insula ing pla e we e e y low, o abou 0.02Q. Since
he adia i e hea lux is ze o in his expe imen al se up, he hea lux gene a ed by he an enna q was conside ed o be 0.98Q,
measu ed wi h a ela i e unce ain y o Δq/q=2.5%.
The Rayleigh numbe was de ined as:
RaL=(β
ρ
μ
λa)w
gL4q(9)
whe e a is he he mal di usi i y. Wi h his de ini ion, he ela i e unce ain y o he Rayleigh numbe was de e mined using a
maximum ela i e unce ain y o 0.5% o all he mophysical cha ac e is ics o he base luid (wa e ) and o he dis ance L . The
Fig. 4. E olu ion o he di e ence δ=
ΔTca −ΔTm e sus RaL o some (λ∗,φ)combina ions.
F. Sas e e al.
Case S udies in The mal Enginee ing 32 (2022) 101869
9
maximum alue ha was ob ained was ΔRa/Ra ∼7%. Se e al alues o he Rayleigh numbe we e es ed in he ange om 3.43x105
up o 5.08x107 (co e ing abou wo o de s o magni ude).
3. Nume ical app oach
The nume ical app oach o he conside ed p oblem is based on he go e ning equa ions p esen ed in se e al documen s such as he
wo k by Pa anka [40] and Bejan [41]. The compu a ion p ocedu e adop ed he e is he same as ha o he ecen s udy by Ma in e al.
[42] dealing wi h he nume ical simula ion o a simila p oblem, whe e he eade could consul he de ails o he algo i hm. Then, o
ligh en his manusc ip , only he elemen s necessa y o i s unde s anding a e p esen ed he e.
The compu a ional domain was disc e ized wi h 553,523 elemen s o a ious shapes wi h a e inemen on he la e al su ace o he
an enna. This ope a ion allowed o he p ecise de e mina ion o he he mal g adien s ield a he in e ace o he an enna, and he
co esponding local empe a u es. Solu ion o he go e ning sys em was ob ained by means o a home-made so wa e based on he
con ol olume me hod associa ed wi h he SIMPLE algo i hm. The en i e compu a ional domain, including he an enna, was ini ially
assumed o be a he lowe Tc empe a u e. Du ing he nume ical con e gence p ocess, he he mal g adien was se o ze o a he
ho izon al base o he con igu a ion, while he ex e nal ca i y was main ained a iso he mal empe a u e Tc. A cons an hea lux q was
gene a ed by he an enna, and he no-slip condi ion was imposed on he ex e nal ace o he an enna and in e nal ace o he enclosu e.
The a e age empe a u e o he an enna Th was ob ained by weigh ing he local elemen s empe a u es Ti wi h he co esponding
su aces Si:
Th=1
Sh∮TidSi(10)
These calcula ions we e ca ied ou wi h he nano luid p ope ies ob ained a e he Maxwell and B inkman models, Eq. (1)–(5).
4. Expe imen al esul s and compa ison be ween he wo app oaches: expe imen al and nume ical
Resul s o he ΔT=Th−Tc measu emen s e sus RaL accompanied by hei unce ain y a e p esen ed in Fig. 3 o some ep e-
sen a i e (λ∗,φ)combina ions. The RaL unce ain ies a e, in en ionally, no ep esen ed gi en he chosen loga i hmic scale.
The nex s ep was, i s , o compu e a numbe o cases in he space o pa ame e s, and, second, o compa e he esul s ob ained o
hose ob ained expe imen ally. This compa ison was pe o med in e ms o he pa ame e δ=ΔTca −ΔTm , whe e ΔTca is he calcu-
la ed ΔT, and ΔTm is he measu ed ΔT. The esul s ob ained a e p esen ed in Fig. 4. In he case o cooling wi h pu e wa e and wi hou
po ous ma e ial (λ∗=0,φ=0), he measu ed alues ΔTm a e almos sys ema ically g ea e han he calcula ed ones ΔTca, wi h a
di e ence a ying be ween 0.5 and 3.5 K. Howe e , he measu emen s a e lowe o all he con igu a ions a ec ed by he nano luid
(φ∕= 0). The measu emen -calcula ion de ia ions a y be ween 1 and 5.5 K. The highes a e obse ed o he highes concen a ions o
nanopa icles (φ=4.81%), o all he alues o he λ∗ a io. These esul s a e accep able and p o ide and addi ional alida ion o he
nume ical model.
The p ac ical ou come o all hese esul s is he gene a ion o a co ela ion o ΔT as unc ion o he ou go e ning pa ame e s o he
p oblem; namely: λw/λn , φ, λ*, and RaL. The co ela ion is as ollows:
ΔT =3.532x10−4(λw
λn )[k(φ)m(λ*)]−1Ra0.683
L(11.a)
whe e:
k(φ) = ⎧
⎪
⎪
⎨
⎪
⎪
⎩
1 o φ =0(pu e wa e )
1.45φ0.03Ra−0.012
L o 1% ≤φ≤5%
(11.b)
m(λ*) = ⎧
⎪
⎪
⎨
⎪
⎪
⎩
1 o λ*=0
10.3(λ*)−0.31Ra−1+0.86(λ*)0.03
L o 4≤λ*≤41.2
(11.c)
λw
λn
=λCu +2λw+φ(λw−λCu)
λCu +2λw−2φ(λw−λCu)(11.d)
Co ela ion (11.a) o ΔT exhibi s dependence on ou pa ame e s: λw/λn , φ, λ*, andRaL. The i s wo pa ame e s, λw/λn and φ, a e
linked ia ela ions (11.b) and (11.d). Assuming ha he e m Ra−0.012
L adds up o he Ra0.683
L e m in co ela ion (11.a), changing φ om
0 o 0.01 and o 0.05 dec eases ΔT by a ac o o 23% and 35% espec i ely. On he o he hand, modi ying λ* om 0 o 4 and o 41.2
dec eases ΔT by a ac o o 20% and 40% espec i ely ( o a ypical RaL o 10
7
). This sugges s ha , om he physics poin o iew, he
in luence o he base luid (nano luid o no ), and he p esence (o no ) o he po ous ma ix ha e a simila in luence on he e iciency o
he p oposed he mal con ol sys em. In a sense, his could be expec ed because he p esence o bo h e ec s has he p ac ical ou come
F. Sas e e al.
