2611/2025
www.
inf
o
rmacjain
s
tal
.
c
o
m
.
pl
I
n
t
r
o
d
u
ct
i
on
The gr
ow
ing demand f
o
r energ
y
f
o
rce
s
u
s
t
o
create ne
w
techn
o
l
o
gie
s
and tr
y
t
o
impr
o
ve the e
x
i
s
ting
o
ne
s
.
The devel
o
pment
and implementati
o
n
o
f rene
w
able energ
y
ha
s
gained en
o
rm
o
u
s
imp
o
rtance
.
I
n recent
y
ear
s
, intere
s
t in
s
o
lar energ
y
ha
s
increa
s
ed,
and it i
s
currentl
y
c
o
n
s
idered
o
ne
o
f the main,
pr
o
mi
s
ing alternative energ
y
s
o
urce
s
[
2,6,7,10,12,13
].
The main element in
s
o
lar
s
y
s
tem
s
i
s
a
s
o
lar c
o
llect
o
r,
w
hich a
s
a heat
e
x
changer ab
s
o
rb
s
and intercept
s
incident
s
o
lar radiati
o
n
.
I
n the ne
x
t
s
tage, it c
o
nvert
s
it
int
o
heat and tran
s
fer
s
it t
o
the
wo
r
k
ing
medium (mainl
y
w
ater
o
r air) fl
ow
ing thr
o
ugh
the c
o
llect
o
r
.
The m
o
s
t p
o
pular and m
o
s
t fre-
quentl
y
te
s
ted and m
o
derni
z
ed t
y
pe
o
f
s
o
lar
c
o
llect
o
r
s
are flat plate c
o
llect
o
r
s
[
11
].
I
n hi
s
wo
r
k
,
A
lvare
z
[
1
]
carried
o
ut numerical
s
im-
ulati
o
n
s
o
f a flat c
o
llect
o
r and verified it
e
x
perimentall
y
in
o
rder t
o
determine the
thermal efficienc
y
, ma
x
imum
s
urface tem-
perature and temperature di
s
tributi
o
n ar
o
und
the c
o
llect
o
r
.
The auth
o
r
s
s
h
ow
ed that the
efficienc
y
o
f the flat plate c
o
llect
o
r i
s
influ-
enced b
y
s
uch parameter
s
a
s
o
perating
parameter
s
, envir
o
nmental parameter
s
, tem-
perature at the liquid inlet
o
r pipe
s
pacing
.
I
n
hi
s
wo
r
k
[
4
]
,
G
adi pre
s
ented numerical
m
o
del
s
ba
s
ed
o
n C
F
D
(C
o
mputati
o
nal
F
luid
Dy
namic
s
) m
o
del
s
, in
w
hich the numerical
re
s
ult
s
w
ere verified
w
ith a
s
o
lar
s
imulat
o
r
.
I
n
turn, the fl
ow
di
s
tributi
o
n
s
and temperature
pr
o
file
s
in the
s
o
lar c
o
llect
o
r
w
ere
s
tudied b
y
F
an et al
.
[
3
].
Numerical
s
tudie
s
o
f a flat
s
o
lar
c
o
llect
o
r
w
ere al
s
o
de
s
cribed b
y
Selmi et al
.
[
9
].
The
y
u
s
ed C
F
D
-
A
C
E
s
o
ft
w
are t
o
s
o
lve
the equati
o
n
s
o
f fluid fl
ow
, heat tran
s
fer and
radiati
o
n
.
I
t
w
a
s
s
h
ow
n that the predicted
temperature pr
o
file f
o
ll
ow
ed the
s
ame trend
a
s
the e
x
perimental pr
o
file
.
O
n the
o
ther
hand, the g
oo
d agreement
o
f the C
F
D
re
s
ult
s
w
ith the e
x
perimental data indicate
s
that
C
F
D
i
s
a g
oo
d t
oo
l f
o
r predicting the effi-
cienc
y
o
f
s
o
lar c
o
llect
o
r
s
.
While flat plate
s
o
lar c
o
llect
o
r
s
pr
o
duce l
ow
er temperature
s
,
the
y
have the advantage
o
f being
s
impler t
o
de
s
ign and have l
ow
er maintenance c
o
s
t
s
,
and are theref
o
re the m
o
s
t u
s
ed c
o
llect
o
r
s
in
s
o
lar
s
y
s
tem
s
in the re
s
idential and indu
s
trial
s
ect
o
r
s
.
A
t
y
pical flat c
o
llect
o
r c
o
n
s
i
s
t
s
o
f an
in
s
ulated metal b
o
x
w
ith a gla
ss
o
r pla
s
tic
c
o
ver and a dar
k
ab
s
o
rber plate
[
5
].
There
are man
y
s
tudie
s
in the literature dev
o
ted t
o
the
s
tud
y
o
f the impact
o
f pipe
s
hape
o
n the
o
perati
o
n
o
f a flat c
o
llect
o
r, but thi
s
t
o
pic i
s
Ź
r
ó
d
ł
a
c
i
ep
ł
a
i
e
n
e
r
g
ii
e
l
e
kt
r
yc
z
n
e
j
/
S
ou
rc
e
s
of
h
ea
t
a
n
d
e
l
e
ctr
i
c
i
ty
I
mpa
ct
of
t
h
e
ab
s
o
r
be
r
c
onf
i
g
u
r
a
t
i
on
on
t
h
e
s
o
l
a
r
t
h
e
r
ma
l
c
o
ll
e
ct
o
r
s
e
ff
i
c
i
e
n
cy
W
p
ływ k
onf
i
g
u
r
a
c
ji
ab
s
o
r
be
r
a
n
a
wy
da
j
no
ść k
o
l
e
kt
o
r
ó
w sł
on
e
c
z
n
yc
h
P
R
Z
EMY
SŁ
A
W S
Z
YM
A
N
E
K
DO
I 10
.
36119/15
.
2025
.
11
.
4
I
n the pre
s
ent paper, an anal
y
s
i
s
o
f the
o
perati
o
n c
o
nditi
o
n
s
o
f t
wo
flat plate
s
o
lar c
o
llect
o
r
s
that differ in the
ge
o
metr
y
o
f the tube line
s
y
s
tem (harp and meander) i
s
carried
o
ut
.
The c
o
llect
o
r
s
w
ere te
s
ted t
o
evaluate the
influence
o
f the t
y
pe
o
f
s
y
s
tem
o
n the thermal parameter
s
due t
o
the ge
o
metric difference
s
.
I
n b
o
th ca
s
e
s
, tube
s
w
ith
a thic
k
ne
ss
o
f 2 mm and an
o
uter diameter
o
f 30 mm
w
ere u
s
ed
.
The
wo
r
k
ing medium in the
s
y
s
tem
s
w
a
s
w
ater
w
ith
an inlet temperature
o
f 288K (15
o
C) and a vel
o
cit
y
o
f 1m/
s
.
Numerical inve
s
tigati
o
n
s
w
ere carried
o
ut
o
n full-
s
cale
s
imulati
o
n
s
u
s
ing the
A
n
s
y
s
F
luent
s
o
ft
w
are
.
The c
o
nducted
s
imulati
o
n
s
pr
o
vided inf
o
rmati
o
n
o
n the d
y
namic
s
o
f
change
s
in the
o
perating parameter
s
o
f the in
s
tallati
o
n
.
The anal
y
s
i
s
s
h
ow
ed that the efficienc
y
o
f the
s
y
s
tem can be
s
ignificantl
y
impr
o
ved b
y
in
s
talling an appr
o
priate ab
s
o
rber
s
y
s
tem
.
The re
s
ult
s
s
h
ow
that the greate
s
t heating
o
f the
circulating medium
o
ccur
s
in the elb
ow
s
and curve
s
o
f the meander c
o
llect
o
r
.
A
fter c
o
mparing the fl
ow
s
o
f the
wo
r
k
ing medium thr
o
ugh the meander and harp c
o
llect
o
r
s
, it
w
a
s
n
o
ticed that the meander c
o
llect
o
r i
s
a m
o
re
efficient
s
y
s
tem than the harp c
o
llect
o
r
.
Ke
ywo
rd
s
:
flat plate
s
o
lar c
o
llect
o
r, thermal efficienc
y
, c
o
mputati
o
nal fluid d
y
namic
s
W niniej
s
z
y
m art
yk
ule pr
z
epr
ow
ad
z
o
n
o
anali
z
ę
w
arun
ków
prac
y
d
wó
ch pła
s
k
ich
ko
le
k
t
o
r
ów
s
ł
o
nec
z
n
y
ch r
ó
ż
nią-
c
y
ch
s
ię ge
o
metrią u
k
ładu rur
ow
eg
o
(harf
ow
eg
o
i meandr
ow
eg
o
)
.
K
o
le
k
t
o
r
y
pr
z
ebadan
o
p
o
d
k
ątem
w
pł
yw
u
r
o
d
z
aju u
k
ładu na parametr
y
cieplne
wy
ni
k
ające
z
r
ó
ż
nic ge
o
metr
y
c
z
n
y
ch
.
W
o
bu pr
z
y
pad
k
ach
z
a
s
t
o
s
ow
an
o
rur
y
o
grub
o
ś
ci 2 mm i
ś
rednic
y
z
e
w
nętr
z
nej 30 mm
.
C
z
y
nni
k
iem r
o
b
o
c
z
y
m
w
u
k
ładach b
y
ła
wo
da
o
temperatur
z
e
w
l
o
t
ow
ej 288K (15
o
C) i pręd
ko
ś
ci 1 m/
s
.
B
adania numer
y
c
z
ne pr
z
epr
ow
ad
z
o
n
o
na p
o
d
s
ta
w
ie
s
y
mulacji pełn
o
-
s
k
al
owy
ch
z
wyko
r
z
y
s
taniem
o
pr
o
gram
ow
ania
A
n
s
y
s
F
luent
.
Pr
z
epr
ow
ad
z
o
ne
s
y
mulacje d
o
s
tarc
z
y
ł
y
inf
o
rmacji
o
d
y
namice
z
mian parametr
ów
prac
y
in
s
talacji
.
A
nali
z
a
wyk
a
z
ała,
ż
e
s
pra
w
n
o
ś
ć u
k
ładu m
o
ż
na
z
nac
z
ąc
o
p
o
pra-
w
ić p
o
pr
z
e
z
z
ain
s
tal
ow
anie
o
dp
ow
iednieg
o
u
k
ładu ab
s
o
rber
ów.
W
y
ni
k
i
w
s
k
a
z
ują,
ż
e naj
w
ię
k
s
z
e nagr
z
e
w
anie
s
ię
c
z
y
nni
k
a
o
bieg
ow
eg
o
wy
s
tępuje
w
ko
lan
k
ach i łu
k
ach
ko
le
k
t
o
ra meandr
ow
eg
o.
P
o
p
o
r
ów
naniu pr
z
epł
ywów
c
z
y
n-
ni
k
a r
o
b
o
c
z
eg
o
pr
z
e
z
ko
le
k
t
o
r
y
meandr
owy
i harf
owy
z
au
w
a
ż
o
n
o
,
ż
e
ko
le
k
t
o
r meandr
owy
je
s
t bard
z
iej
wy
daj-
n
y
m u
k
ładem ni
ż
ko
le
k
t
o
r harf
owy.
Sł
ow
a
k
luc
z
ow
e
:
pła
s
k
i
ko
le
k
t
o
r
s
ł
o
nec
z
n
y
,
s
pra
w
n
o
ś
ć cieplna,
o
blic
z
eni
ow
a mechani
k
a pł
y
n
ów
dr in
ż
.
Pr
z
em
y
s
ła
w
S
z
y
mane
k
http
s
:
//
o
rcid
.o
rg/0000-0001-5065-9311
‒
P
o
litechni
k
a C
z
ę
s
t
o
ch
ow
s
k
a, W
y
d
z
iał
I
nfra
s
tru
k
tur
y
i Śr
o
d
ow
i
s
k
a
A
dre
s
d
o
ko
re
s
p
o
ndencji/ C
o
rre
s
p
o
nding addre
ss
:
pr
z
em
y
s
la
w.
s
z
y
mane
k@
pc
z
.
pl
www.
inf
o
rmacjain
s
tal
.
c
o
m
.
pl11/202527
Ź
r
ó
d
ł
a
c
i
ep
ł
a
i
e
n
e
r
g
ii
e
l
e
kt
r
yc
z
n
e
j
n
o
t e
x
hau
s
ted and there are
s
till
s
o
me i
ss
ue
s
t
o
be anal
y
z
ed
.
I
n pre
s
ent paper
w
a
s
decided t
o
perf
o
rm
a numerical
s
tud
y
o
f t
wo
flat plate
s
o
lar c
o
l-
lect
o
r
s
and c
o
mpare the behavi
o
r
o
f the
wo
r
k
ing medium
.
The aim
o
f the
s
tud
y
w
a
s
t
o
anal
y
s
e the mechani
s
m
o
f heat tran
s
fer fl
ow
in a flat plate c
o
llect
o
r
.
Simulati
o
n
s
o
f flat
plate
s
o
lar c
o
llect
o
r
s
w
ith de
s
ign
s
differing
fr
o
m each
o
ther
w
ere carried
o
ut
.
The influ-
ence
o
f ge
o
metrical characteri
s
tic
s
w
a
s
c
o
n-
s
idered
.
The in
s
tallati
o
n
o
f the fir
s
t
s
o
lar c
o
l-
lect
o
r
w
a
s
characteri
s
ed b
y
a harp
s
tructure,
w
hile the in
s
tallati
o
n
o
f the
s
ec
o
nd c
o
llect
o
r
w
a
s
de
s
igned in a l
oo
p
s
hape (meander)
.
A
virtual m
o
del cl
o
s
e t
o
the actual perf
o
r-
mance characteri
s
tic
s
o
f the
s
o
lar
c
o
llect
o
r
s
w
a
s
m
o
delled
.
A
n
s
y
s
F
luent
c
o
mmercial
s
o
ft-
w
are
w
a
s
u
s
ed t
o
numericall
y
s
o
lve the fluid fl
ow
, heat tran
s
fer
and radiati
o
n equati
o
n
s
.
N
u
me
ri
c
a
l
m
o
de
li
n
g
i
s
485,200 f
o
r the harp c
o
llect
o
r and
537,680 f
o
r the meander c
o
llect
o
r
.
N
u
me
ri
c
a
l
me
t
ho
d
o
l
o
g
y
The three-dimen
s
i
o
nal m
o
del
s
o
f heat
and fluid fl
ow
in the
s
o
lar c
o
llect
o
r tube are
de
s
igned in
A
n
s
y
s
F
luent c
o
mputati
o
nal fluid
d
y
namic
s
s
o
ft
w
are
.
The fundamental equa-
ti
o
n
s
f
o
r
s
tead
y
vi
s
c
o
u
s
, inc
o
mpre
ss
ible lami-
nar fl
ow
in
s
ide tube
s
a
s
o
lar c
o
llect
o
r are
c
o
ntinuit
y
, m
o
mentum and energ
y
equati
o
n
s
,
w
hich ta
k
e the f
o
ll
ow
ing f
o
rm
.
F
o
r the ab
o
ve
a
ss
umpti
o
n
s
, the c
o
ntinuit
y
equati
o
n ma
y
be
w
ritten a
s
:
(1)
The m
o
mentum equati
o
n
s
in
x
,
y
and
z
directi
o
n
s
can be
w
ritten a
s
:
C
o
mp
u
t
a
t
i
on
a
l
c
onf
i
g
u
r
a
t
i
on
a
n
d
(2)
me
s
h
e
s
I
n thi
s
paper, t
wo
t
y
pe
s
o
f flat plate c
o
l-
lect
o
r
s
are anal
y
s
ed a
s
s
h
ow
n in
F
igure 1
.
The ge
o
metr
y
o
f a
s
traight tube and a
s
er-
pentine tube are c
o
n
s
idered
.
The m
o
del part
(3)
o
f the d
o
main
w
a
s
prepared u
s
ing
A
n
s
y
s
F
luent
s
o
ft
w
are
.
The harp
s
o
lar c
o
llect
o
r c
o
n-
s
i
s
ted
o
f three tube
s
250 mm apart at their
centre
s
.
The inner diameter
o
f each tube
w
a
s
26 mm, thic
k
ne
ss
w
a
s
2 mm and it
w
a
s
1 m
l
o
ng and c
o
ntained fl
ow
ing
w
ater
.
A
tube
o
f
(4)
the
s
ame diameter and thic
k
ne
ss
w
a
s
u
s
ed
(6)
w
hen de
s
igning the meander flat plate c
o
l-The energ
y
equati
o
n
s
f
o
r fluid and the
lect
o
r
.
frame
o
f c
o
llect
o
r are a
s
f
o
ll
ow
ing
:
F
o
r the ge
o
metr
y
d
o
main it i
s
clearl
y
imp
o
rtant t
o
build an accurate c
o
mputati
o
nal
grid in C
F
D
s
imulati
o
n
s
.
The accurac
y
o
f C
F
D
m
o
delling re
s
ult
s
depend
s
largel
y
o
n the
qualit
y
o
f the me
s
h
.
The finer me
s
he
s
tended
(5)
t
o
give m
o
re accurate
s
imulati
o
n re
s
ult
s
, the
l
o
nger the running time
.
The d
o
main di
s
creti-
s
ati
o
n
w
a
s
carried
o
ut c
o
n
s
idering a
s
truc-
tural me
s
h c
o
n
s
i
s
ting
o
f 0
.
8 mm tetrahedral
element
s
f
o
r the fluid part and 1 mm f
o
r the
s
o
lid part
.
The number
o
f regi
s
tered element
s
C
o
n
s
ervati
o
n equati
o
n
s
w
ere
s
o
lved in
in the grid f
o
r the c
o
mputati
o
nal d
o
main
o
rder t
o
y
ield the vel
o
cit
y
and temperature
c
o
n
s
i
s
ting
o
f
w
ater, tube
s
and ab
s
o
rber plate f
o
r the fluid (
w
ater) fl
ow
in the tube
s
.
I
n
o
rder
→
s
→
→
t
o
m
o
del the radiati
o
n, a di
s
crete-
o
rdinate
(
D
O
) radiati
o
n m
o
del
w
a
s
activated
.
The
di
s
crete
o
rdinate (
D
O
) radiati
o
n m
o
del
s
o
lve
s
the radiati
o
n tran
s
fer (
R
T
E
) equati
o
n
f
o
r a finite number
o
f di
s
crete
s
o
lid angle
s
,
each
o
f
w
hich i
s
related t
o
the vect
o
r direc-
ti
o
n determined in the gl
o
bal Carte
s
ian
s
y
s
-
tem (
x
,
y
,
z
)
.
D
O
m
o
del
s
o
lve
s
the radiative
tran
s
p
o
rt equati
o
n (
R
T
E
) f
o
r a finite number
o
f
directi
o
n
s
s
.
The
D
O
m
o
del treat
s
the radia-
ti
o
n tran
s
fer (
R
T
E
) equati
o
n in the directi
o
n
o
f
→
a
s
the field equati
o
n
.
R
T
E
o
f an ab
s
o
rbing,
emitting and
s
cattering medium can be
w
rit-
ten f
o
r the arbitrar
y
p
o
s
iti
o
n r in a d
o
main in
the directi
o
n
s
.
Thu
s
, equati
o
n i
s
w
ritten a
s
(7)
The efficienc
y
o
f the
s
o
lar c
o
llect
o
r i
s
determined b
y:
(8)
3
D
numerical
s
tudie
s
o
f flat plate
s
o
lar
c
o
llect
o
r
s
w
ere perf
o
rmed u
s
ing the C
F
D
technique and the g
o
verning equati
o
n
s
w
ere
s
o
lved u
s
ing the
A
n
s
y
s
F
luent
s
o
ft
w
are
.
B
oun
da
r
y c
on
d
i
t
i
on
s
a
n
d
o
pe
r
a
t
i
n
g
pa
r
ame
t
e
r
s
A
ppr
o
priate b
o
undar
y
c
o
nditi
o
n
s
are
imp
o
s
ed
o
n the c
o
mputati
o
nal d
o
main
.
The
inlet fl
ow
i
s
c
o
n
s
idered t
o
have a unif
o
rm
vel
o
cit
y
and varie
s
w
ith ma
ss
fl
ow
rate and
temperature
.
F
r
o
m the input fl
ow
rate, a fl
ow
vel
o
cit
y
o
f 1 m/
s
i
s
c
o
n
s
idered f
o
r the
s
imu-
lati
o
n and a fluid temperature
o
f 288 K
(15
o
C) i
s
s
elected f
o
r thi
s
s
tud
y.
The param-
eter
s
have been
s
elected in
s
uch a
w
a
y
a
s
t
o
o
btain the
o
ptimal fl
ow
, ta
k
ing int
o
acc
o
unt
the
s
mall diameter
s
o
f the tube
s
.
The appli-
cable equati
o
n
s
w
ere
s
o
lved u
s
ing the finite
v
o
lume technique
.
A
s
o
lver ba
s
ed
o
n
s
tead
y
-
s
tate pre
ss
ure
w
a
s
u
s
ed
w
ith a
s
ec
o
nd-
o
rder
s
cheme f
o
r the c
o
nvective c
o
nditi
o
n
s
in the
m
o
mentum and energ
y
equati
o
n
s
.
The
s
tan-
dard
s
cheme
w
a
s
u
s
ed t
o
di
s
creti
s
e the pre
s
-
s
ure,
w
hile the S
I
M
PL
E
alg
o
rithm
w
a
s
u
s
ed t
o
di
s
creti
s
e the pre
ss
ure-vel
o
cit
y
c
o
upling
.
The
“
S
y
mmetr
y”
c
o
nditi
o
n in the center
o
f the
ri
s
er pipe i
s
a
ss
umed
o
n the
w
all
.
The b
o
und-
ar
y
c
o
nditi
o
n
w
ith n
o
s
lip
w
a
s
imp
o
s
ed
o
n
the
w
all
s
.
The
“
s
hell c
o
nductivit
y”
o
pti
o
n
w
a
s
included f
o
r the ab
s
o
rber and ri
s
er f
o
r the
w
all heat c
o
nducti
o
n calculati
o
n
s
.
“
Semi-
tran
s
parent
”
w
a
s
turned
o
n f
o
r the gla
ss
s
hield t
o
tran
s
mit radiati
o
n t
o
the c
o
llect
o
r
area, and
“o
paque
”
w
a
s
turned
o
n f
o
r
o
ther
s
urface
s
.
a
)
b
)
F
i
g
u
r
e
1
.
S
c
h
ema
t
i
c
v
i
e
w
of
t
h
e
c
o
mp
u
t
a
t
i
on
a
l
d
o
ma
i
n
of
f
l
a
t
p
l
a
t
e
c
o
ll
e
ct
o
r
(
a
)
h
a
r
p
,
(
b
)
mea
n
de
r
Rys
un
e
k 1
.
S
c
h
ema
tyc
z
n
y w
i
d
o
k
d
o
me
n
y
o
b
li
c
z
e
n
i
o
w
e
j
k
o
l
e
kt
o
r
a
p
ł
a
sk
i
eg
o
(
a
)
h
a
r
f
a
,
(
b
)
mea
n
de
r
2811/2025
www.
inf
o
rmacjain
s
tal
.
c
o
m
.
pl
Ź
N
u
me
ri
c
a
l
r
e
s
u
l
ts
a
n
d
d
i
sc
u
ss
i
on
V
a
li
da
t
i
on
nu
me
ri
c
a
l
m
o
de
l
I
n thi
s
s
ecti
o
n,
w
e verif
y
h
ow
pr
o
p
o
s
ed
calculati
o
n meth
o
d tran
s
late
s
int
o
the accura-
c
y
o
f the re
s
ult
s
o
btained f
o
r the fl
ow
pr
o
b-
lem
s
including heat tran
s
fer
.
The te
s
t ca
s
e
w
ere
s
elected
s
uch a
s
t
o
reflect the fl
ow
phen
o
mena
o
ccurring in tube in
s
o
lar c
o
llec-
t
o
r i
.
e
.
, recirculati
o
n
z
o
ne
s
,
s
tagnati
o
n p
o
int
s
,
interacti
o
n
s
bet
w
een v
o
rtice
s
f
o
rmed ar
o
und
tube bend
s
.
A
ll thi
s
o
ccur
s
in a fl
ow
thr
o
ugh
s
elected
o
bject
s
.
The
s
implicit
y
o
f the ch
o
s
en
c
o
nfigurati
o
n
s
all
ow
ed f
o
r the preparati
o
n
o
f
b
o
d
y
-fitted me
s
he
s
w
hich
w
ere needed f
o
r
the c
o
mputati
o
n perf
o
rmed
w
ith the u
s
e
o
f
the
A
NS
Y
S
F
luent c
o
de
.
The c
o
mputati
o
n
s
carried
o
ut u
s
ing the
A
NS
Y
S
F
luent c
o
de
w
ere perf
o
rmed appl
y
ing the 2nd
o
rder di
s
-
creti
z
ati
o
n meth
o
d
o
n b
o
d
y
-fitted tetrahedral
me
s
he
s
e
x
actl
y
reflecting the
s
hape
s
o
f
o
bject
s
.
The
s
o
lar flat plate c
o
llect
o
r
w
hich i
s
c
o
n
s
idered f
o
r validati
o
n meth
o
d ha
s
the
dimen
s
i
o
n
s
o
f 0
.
915m l
o
ng, 0
.
810m
w
ide
and 0
.
095 m height
.
The ab
s
o
rber plate ha
s
0
.
84m
x
0
.
80m in length and
w
idth re
s
pec-
tivel
y
w
ith thic
k
ne
ss
o
f 0
.
00012m
.
The
w
ater
i
s
u
s
ed a
s
wo
r
k
ing fluid during the anal
y
s
i
s
.
Temperature
o
f the
wo
r
k
ing fluid at the
o
utlet
o
f the ri
s
er i
s
pre
s
ented in
F
igure 2
.
I
t
s
h
o
uld be n
o
ted that the re
s
ult
s
o
btained fr
o
m
the
A
NS
Y
S
F
luent c
o
de
w
ith pr
o
p
o
s
ed alcu-
lati
o
n meth
o
d and literature numerical re
s
ult
s
[
8
]
practicall
y
o
verlap,
w
hich mean
s
that the
accurac
y
i
s
ver
y
g
oo
d
.
T
empe
r
a
t
u
r
e
d
i
st
ri
b
u
t
i
on
I
n the ca
s
e
o
f the harp c
o
llect
o
r (
F
ig
.
3a),
blue and green c
o
nt
o
ur
s
(l
ow
value
s
o
f
w
ater
temperature) pred
o
minate
o
ver m
o
s
t
o
f the
tube length
.
A
s
the ‘height’ increa
s
e
s
,
y
ell
ow
and red
z
o
ne
s
f
o
rm,
w
hich inten
s
if
y
t
ow
ard
s
the upper regi
o
n
.
Clearl
y
, theref
o
re, increa
s
-
ing thermal c
o
nductivit
y
enc
o
urage
s
thermal
diffu
s
i
o
n and increa
s
e
s
heating
.
I
n the ca
s
e
o
f
the meander c
o
llect
o
r (
F
ig
.
3b) there i
s
al
s
o
a gradual heating fr
o
m the inlet
.
The blue
z
o
ne
s
are graduall
y
eliminated and the
green
z
o
ne
s
(higher value
s
o
f
w
ater temper-
ature) e
x
tend further until the red
z
o
ne
s
(ma
x
imum temperature
z
o
ne
s
) appear
.
The
curved
s
hape
o
f the c
o
llect
o
r tube
s
clearl
y
impr
o
ve
s
thermal c
o
nductivit
y
, and thi
s
enhance
s
thermal diffu
s
i
o
n and heat tran
s
fer
.
Cr
o
ss
-
s
ecti
o
nal vie
w
s
o
f the temperature di
s
-
tributi
o
n pr
o
vide clearer vi
s
uali
s
ati
o
n at vari-
o
u
s
l
o
cati
o
n
s
fr
o
m the ba
s
e
o
f the
s
o
lar c
o
l-
lect
o
r t
o
the upper end
.
When anal
y
s
ing the
change
s
in temperature value
s
, it can be
s
een that f
o
r the harp c
o
llect
o
r a variati
o
n
o
f
5 K
w
a
s
o
btained,
w
hile f
o
r the meander
c
o
llect
o
r the variati
o
n
w
a
s
21 K
.
A
dditi
o
nall
y
, figure
s
3a (right) and 3b
(right)
s
h
ow
the
w
ater temperature variati
o
n
s
in the tube
s
at the harp and meander c
o
llec-
t
o
r, re
s
pectivel
y
, f
o
ur and eleven different
tube
s
ecti
o
n
s
.
The
wo
r
k
ing fluid increa
s
e
s
it
s
temperature a
s
it circulate
s
thr
o
ugh the tube
.
I
n the
s
e t
wo
c
o
n
s
idered ca
s
e
s
, the
o
uter part
o
f each ge
o
metr
y
ha
s
a higher temperature
becau
s
e thi
s
part i
s
in direct c
o
ntact
w
ith the
inner part
o
f the tube, all
ow
ing m
o
re heat t
o
be ab
s
o
rbed
.
The fluid fed int
o
the tube in the
meander c
o
llect
o
r reached higher tempera-
ture value
s
o
f up t
o
319K (46
o
C),
w
hich i
s
ab
o
ut 16K m
o
re at the
o
utlet
o
f the tube
c
o
mpared t
o
the harp c
o
llect
o
r
.
V
e
l
o
c
i
ty
d
i
st
ri
b
u
t
i
on
I
n thi
s
s
ub
s
ecti
o
n, the di
s
tributi
o
n
o
f
vel
o
cit
y
in the di
s
cu
ss
ed c
o
llect
o
r
s
i
s
illu
s
trat-
ed
.
F
igure 4a,b pre
s
ent
s
the vel
o
cit
y
di
s
tribu-
ti
o
n
o
n the cr
o
ss
s
ecti
o
n f
o
r the di
s
cu
ss
ed
ca
s
e
s
.
F
igure 4 a, b (right)
s
h
ow
s
the change
s
in
w
ater vel
o
cit
y
at f
o
ur (harp) and eleven
(meander) different tube
s
ecti
o
n
s
.
The fluid
s
upplied t
o
the
s
ecti
o
n f
o
r the meander c
o
l-
lect
o
r
o
btained higher vel
o
cit
y
value
s
, reach-
ing value
s
up t
o
1
.
57 m/
s
at the
o
utlet
o
f the
tube
.
The harp t
y
pe
s
o
lar c
o
llect
o
r had the
m
o
s
t unif
o
rm vel
o
cit
y
di
s
tributi
o
n a
s
it
F
i
g
u
r
e
2
.
C
o
mpa
ri
s
on
of
t
h
e
r
e
s
u
l
ts
o
b
t
a
i
n
ed
u
s
i
n
g
t
h
e
An
sys F
l
u
e
n
t w
i
t
h
t
h
e
li
t
e
r
a
t
u
r
e
da
t
a
[8]
Rys
un
e
k 2
.
Po
r
ó
w
n
a
n
i
e
wy
n
i
k
ó
w
u
z
ysk
a
n
yc
h
p
r
z
y
u
ż
yc
i
u
o
p
r
o
g
r
am
o
w
a
n
i
a
An
sys F
l
u
e
n
t
z
da
n
y
m
i
li
t
e
r
a
t
u
r
o
wy
m
i
[8]
I
NL
E
T
I
NL
E
T
I
NL
E
T
I
NL
E
T
a
)
b
)
F
i
g
u
r
e
3
.
T
empe
r
a
t
u
r
e
c
on
t
ou
r
fo
r w
a
t
e
r
i
n
(
a
)
h
a
r
f
c
o
ll
e
ct
o
r
a
n
d
(
b
)
mea
n
de
r c
o
ll
e
ct
o
r
Rys
un
e
k 3
.
R
o
z
kł
ad
t
empe
r
a
t
u
ry w
o
d
y w
(
a
)
k
o
l
e
kt
o
r
z
e
h
a
r
fo
wy
m
i
(
b
)
k
o
l
e
kt
o
r
z
e
mea
n
d
r
o
wy
m
a
)
b
)
F
i
g
u
r
e
4
.
V
e
l
o
c
i
ty c
on
t
ou
r
fo
r w
a
t
e
r
i
n
(
a
)
h
a
r
f
c
o
ll
e
ct
o
r
a
n
d
(
b
)
mea
n
de
r c
o
ll
e
ct
o
r
Rys
un
e
k 4
.
R
o
z
kł
ad
p
r
ęd
k
o
śc
i
w
o
d
y w
(
a
)
k
o
l
e
kt
o
r
z
e
h
a
r
fo
wy
m
i
(
b
)
k
o
l
e
kt
o
r
z
e
mea
n
d
r
o
wy
m
29
www.
inf
o
rmacjain
s
tal
.
c
o
m
.
pl
11/2025
Ź
r
ó
d
ł
a
c
i
ep
ł
a
i
e
n
e
r
g
ii
e
l
e
kt
r
yc
z
n
e
j
o
btained a variati
o
n n
o
t e
x
ceeding 0
.
23
m/
s
,
w
hile f
o
r the harp/meander c
o
llect
o
r
the variati
o
n in vel
o
cit
y
value
s
w
a
s
0
.
57 m/
s
.
Clear difference
s
can be
o
b
s
erved in the
vel
o
cit
y
di
s
tributi
o
n
s
f
o
r b
o
th c
o
llect
o
r
s
.
F
o
r
the harp, a
s
traight tube, there i
s
a clear
increa
s
e in vel
o
cit
y
(up t
o
1
.
23 m/
s
)
o
b
s
erved in the middle part
o
f the tube
.
F
o
r
the meander,
o
n the
o
ther hand, the increa
s
e
in vel
o
cit
y
w
a
s
rec
o
rded in the end
s
ecti
o
n
s
o
f the
s
traight tube
s
(up t
o
1
.
22 m/
s
), and the
highe
s
t vel
o
cit
y
value
s
can be
o
b
s
erved near
the bend
s
(up t
o
1,57 m/
s
)
.
Thi
s
i
s
al
s
o
w
here
recirculati
o
n
z
o
ne
s
are created, a
s
s
h
ow
n in
F
igure 5
.
D
ue t
o
the m
o
re c
o
mple
x
ge
o
metr
y
,
larger difference
s
in pre
ss
ure value
s
w
ith
re
s
pect t
o
tube length
w
ere n
o
ted f
o
r the
meander c
o
llect
o
r
.
I
n general, there i
s
a
s
light
pre
ss
ure dr
o
p at the end
s
o
f the tube
s
(c
o
r-
re
s
p
o
nding t
o
the accelerati
o
n
o
f the fl
ow
)
.
The meander c
o
llect
o
r had higher perf
o
r-
mance b
y
dem
o
n
s
trating the l
ow
e
s
t vel
o
cit
y
in m
o
s
t part
s
o
f the tube,
w
hich reduced
pre
ss
ure dr
o
p and impr
o
ved heat tran
s
fer
fr
o
m the envir
o
nment t
o
the fluid
.
S
o
l
a
r
t
h
e
r
ma
l
c
o
ll
e
ct
o
r
s
e
ff
i
c
i
e
n
cy
The efficienc
y
o
f
s
o
lar c
o
llect
o
r
s
depend
s
o
n man
y
parameter
s
, including the tempera-
ture difference bet
w
een the
wo
r
k
ing fluid
and the envir
o
nment
.
The greater thi
s
differ-
ence, the l
ow
er the efficienc
y
, a
s
the
o
utlet
fluid temperature decrea
s
e
s
, limiting energ
y
tran
s
fer t
o
the receiver
.
A
flat c
o
llect
o
r
w
ith
a harf
s
y
s
tem, than
k
s
t
o
a
s
h
o
rt fl
ow
path
(ab
o
ut 1 meter), feature
s
l
ow
h
y
draulic re
s
i
s
-
tance and l
ow
energ
y
c
o
n
s
umpti
o
n
.
H
ow
-
ever, thi
s
s
y
s
tem e
x
hibit
s
uneven fl
ow
vel
o
cit
y
di
s
tributi
o
n
.
Thi
s
lead
s
t
o
uneven heating
o
f
the ab
s
o
rber
s
urface and a decrea
s
e in
s
y
s
-
tem efficienc
y.
I
n part
s
farthe
s
t fr
o
m the inlet
e
x
perience higher vel
o
citie
s
.
Thi
s
phen
o
me-
n
o
n increa
s
e
s
w
ith increa
s
ing temperature
o
f
the medium
.
I
n a meandered
s
y
s
tem,
w
ith
a l
o
nger fl
ow
pipe, h
y
draulic re
s
i
s
tance i
s
higher, but it all
ow
s
f
o
r better energ
y
utili
z
a-
ti
o
n and higher efficienc
y
(ar
o
und 49
%
c
o
mpared t
o
24
%
in the l
o
uvered
s
y
s
tem)
.
Clear pre
ss
ure dr
o
p
s
are n
o
ticeable in the
circuit
.
The l
o
nger fl
ow
path pr
o
m
o
te
s
turbu-
lent fl
ow
,
w
hich enhance
s
heat e
x
change
and en
s
ure
s
a m
o
re unif
o
rm vel
o
cit
y
di
s
tribu-
ti
o
n
.
The c
o
llect
o
r’
s
thermal efficienc
y
depend
s
o
n the unif
o
rmit
y
o
f the fluid fl
ow.
The m
o
re even the fl
ow
, the higher the effi-
cienc
y.
H
ow
ever, achieving perfect unif
o
r-
mit
y
i
s
challenging and require
s
further
re
s
earch int
o
fl
ow
d
y
namic
s
w
ithin the pipe
s
and ab
s
o
rber
.
E
ffect
o
f ab
s
o
rber thic
k
ne
ss
o
n efficienc
y.
Thi
s
s
ecti
o
n pre
s
ent
s
the influence
o
f the
ab
s
o
rber thic
k
ne
ss
o
n the c
o
llect
o
r efficienc
y
w
a
s
inve
s
tigated
.
A
s
s
h
ow
n in
F
igure 6, the
efficienc
y
o
f the
s
o
lar c
o
llect
o
r increa
s
e
s
w
ith
the thic
k
ne
ss
o
f the ab
s
o
rber plate
.
F
o
r
in
s
tance,
w
hen the ab
s
o
rber thic
k
ne
ss
ri
s
e
s
fr
o
m 0
.
1 mm t
o
0
.
6 mm, the
o
verall efficienc
y
impr
o
ve
s
b
y
m
o
re than 15
%
.
Thi
s
s
ignificant
enhancement can be mainl
y
attributed t
o
the
reducti
o
n in heat tran
s
fer re
s
i
s
tance bet
w
een
the ab
s
o
rber
s
urface and the
wo
r
k
ing fluid
.
A
thic
k
er ab
s
o
rber pr
o
vide
s
a larger cr
o
ss
-
s
ecti
o
nal area f
o
r heat c
o
nducti
o
n, all
ow
ing
ab
s
o
rbed
s
o
lar energ
y
t
o
be tran
s
ferred
m
o
re effectivel
y
and unif
o
rml
y
thr
o
ugh the
material
.
M
o
re
o
ver, increa
s
ing the thic
k
ne
ss
s
lightl
y
enhance
s
the ab
s
o
rber’
s
thermal
capacit
y.
Thi
s
higher heat capacit
y
help
s
t
o
buffer
s
h
o
rt-term temperature fluctuati
o
n
s
,
leading t
o
m
o
re
s
table thermal perf
o
rmance
under variable
s
o
lar radiati
o
n c
o
nditi
o
n
s
.
H
ow
ever, the influence
o
f heat capacit
y
i
s
relativel
y
min
o
r c
o
mpared t
o
the d
o
minant
effect
o
f impr
o
ved heat c
o
nducti
o
n
.
I
n prac-
tice, the
o
ptimal ab
s
o
rber thic
k
ne
ss
s
h
o
uld
balance higher efficienc
y
w
ith the increa
s
ed
material c
o
s
t and p
o
tential added
w
eight
o
f
the c
o
llect
o
r
.
E
ffect
o
f ab
s
o
rbing material
o
n efficienc
y.
A
b
s
o
rber i
s
o
ne
o
f the m
o
s
t imp
o
rtant
c
o
mp
o
nent
s
o
f a
s
o
lar c
o
llect
o
r, ab
s
o
rbing
s
o
lar radiati
o
n energ
y
and tran
s
ferring heat
ia
t
o
the tube
s
and fluid
s
.
Theref
o
re, the material
pr
o
pertie
s
o
f the ab
s
o
rber plate have a
s
ig-
nificant r
o
le in heat tran
s
fer fr
o
m the ab
s
o
rber
t
o
the tube
s
.
F
igure 7 pre
s
ent
s
the variati
o
n
s
o
f c
o
llect
o
r efficienc
y
ver
s
u
s
the reduced
temperature parameter
s
, (T
‒
T )/
G
, f
o
r
vari
o
u
s
ab
s
o
rber material
s
.
A
s
s
h
ow
n, c
o
l-
lect
o
r
s
w
ith c
o
pper ab
s
o
rber
s
achieve the
highe
s
t efficienc
y
, f
o
ll
ow
ed b
y
aluminum,
w
hile
s
teel e
x
hibit
s
the l
ow
e
s
t perf
o
rmance
.
Thi
s
trend i
s
c
o
n
s
i
s
tent f
o
r b
o
th fl
ow
t
y
pe
s
.
The
s
uperi
o
r thermal c
o
nductivit
y
o
f c
o
pper and
aluminum enhance
s
heat tran
s
fer fr
o
m the
ab
s
o
rber
s
urface t
o
the
wo
r
k
ing fluid, re
s
ult-
ing in higher efficienc
y
value
s
.
i
a
A
dditi
o
nall
y
, the difference bet
w
een the
meander (m) and harp (h) c
o
nfigurati
o
n
s
i
s
relativel
y
s
mall, but the meander t
y
pe c
o
llec-
t
o
r generall
y
s
h
ow
s
s
lightl
y
better perf
o
r-
mance due t
o
m
o
re unif
o
rm fluid di
s
tributi
o
n
and impr
o
ved heat e
x
change
.
The meander
c
o
llect
o
r en
s
ure
s
a m
o
re
s
table fl
ow
and
m
o
re c
o
mplete u
s
e
o
f heat
.
The
s
light
decrea
s
e in efficienc
y
w
ith increa
s
ing (T
‒
T )/
G
rati
o
indicate
s
higher thermal l
o
ss
e
s
at
larger temperature difference
s
,
w
hich i
s
t
y
pi-
cal behavi
o
r f
o
r flat-plate c
o
llect
o
r
s
.
S
u
mma
r
y
I
n thi
s
paper, C
F
D
numerical
s
imulati
o
n in
the
A
n
s
y
s
F
luent pac
k
age
w
a
s
u
s
ed t
o
inve
s
-
tigate the pr
o
ce
ss
e
s
in
s
ide the tube
s
o
f a harp
and meander flat plate c
o
llect
o
r
s
.
C
F
D
re
s
ult
s
o
f a flat plate
s
o
lar c
o
llect
o
r
s
w
ith different
cr
o
ss
s
ecti
o
n
s
,
s
imulated under the
s
ame
o
perating and
s
tead
y
s
tate c
o
nditi
o
n
s
,
s
h
ow
ed that the meander c
o
llect
o
r had the
highe
s
t perf
o
rmance
.
H
ow
ever, the harp c
o
l-
lect
o
r
w
a
s
the
o
ne that
s
h
ow
ed a m
o
re uni-
f
o
rm temperature di
s
tributi
o
n at the
o
utlet
o
f
the tube
.
I
n b
o
th ca
s
e
s
, it
w
a
s
n
o
ticed that the
liquid vel
o
cit
y
in
s
ide the tube
s
increa
s
e
s
w
hile
it decrea
s
e
s
at the
w
all
s
.
A
fter c
o
mparing
F
i
g
u
r
e
5
.
R
e
c
i
rc
u
l
a
t
i
on
z
on
e
i
n
t
h
e
a
r
ea
of
t
h
e
t
u
be
be
n
d
(
mea
n
de
r
)
Rys
un
e
k 5
.
S
t
r
e
f
a
r
e
cyrk
u
l
a
c
ji
w
o
b
s
z
a
r
z
e
z
a
kr
ę
t
u
r
u
r
y
(
mea
n
de
r
)
F
i
g
u
r
e
6
C
o
ll
e
ct
o
r
e
ff
i
c
i
e
n
cy
of
v
a
r
i
ou
s
ab
s
o
r
be
r t
h
i
ck
-
n
e
ss
e
s
(
mm
)
Rys
un
e
k 6
.
S
p
r
a
w
no
ść k
o
l
e
kt
o
r
a
d
l
a
r
ó
ż
n
yc
h
g
r
u
b
o
śc
i
ab
s
o
r
be
r
a
(
mm
)
F
i
g
u
r
e
7
C
o
ll
e
ct
o
r
e
ff
i
c
i
e
n
cy
of
v
a
r
i
ou
s
ab
s
o
r
be
r
ma
t
e
-
r
i
a
l
s
Rys
un
e
k 7
.
S
p
r
a
w
no
ść k
o
l
e
kt
o
r
a
d
l
a
r
ó
ż
n
yc
h
ma
t
e
r
i
a
ł
ó
w
ab
s
o
r
b
u
j
ą
cyc
h
Ź
the fl
ow
s
o
f the
wo
r
k
ing medium thr
o
ugh the
meander and harp c
o
llect
o
r
s
, it
w
a
s
n
o
ted
that the meander c
o
llect
o
r i
s
a m
o
re efficient
s
y
s
tem becau
s
e the
wo
r
k
ing medium travel
s
a greater di
s
tance and i
s
heated l
o
nger
.
The
greate
s
t heating
o
f the
w
ater
o
ccur
s
in the
elb
ow
s
and curve
s
o
f the meander c
o
llect
o
r,
w
here the temperature
o
f the
wo
r
k
ing medium
increa
s
e
s
s
ignificantl
y.
The fl
ow
o
f the medium
thr
o
ugh
s
everal elb
ow
s
further increa
s
e
s
thi
s
effect
.
E
lb
ow
s
al
s
o
affect the fl
ow
o
f the
wo
r
k
-
ing medium
.
Water m
o
ving in the curve
s
accelerate
s
rapidl
y
and then decelerate
s
,
h
o
ll
ow
s
pace
s
are f
o
rmed,
w
hich
w
a
s
n
o
t
o
b
s
erved in the ca
s
e
o
f the harp c
o
llect
o
r
.
I
n
the ca
s
e
o
f the harp c
o
llect
o
r, the fl
ow
pre
s
-
s
ure decrea
s
e
s
unif
o
rml
y
,
w
hile in the ca
s
e
o
f
the meander c
o
llect
o
r, large pre
ss
ure dr
o
p
s
w
ere
o
b
s
erved in the curve
s
and rapid pre
s
-
s
ure dr
o
p
s
in the re
s
ulting h
o
ll
ow
s
pace
s
.
N
o
me
n
c
l
a
t
u
r
e
c
p
z
→
→
s
a
–
ab
s
o
rpti
o
n c
o
efficient (dimen
s
i
o
nle
ss
),
A
–
c
o
llect
o
r ab
s
o
rpti
o
n area (m
2
),
c
–
heat capacit
y
[k
J/(
k
g·K)
]
,
g
–
c
o
mp
o
nent
s
o
f gravitati
o
nal accel-
erati
o
nin
z
directi
o
n
[
m/
s
2
]
,
G
–
s
o
lar irradiati
o
n
[k
W/m
2
]
,
I
–
radiati
o
n inten
s
it
y
(W/m
2
),
m
–
ma
ss
fl
ow
rate
[k
g/
s
]
,
n
–
refractive inde
x
(dimen
s
i
o
nle
ss
)
r
–
p
o
s
iti
o
n vect
o
r (m),
s
–
directi
o
n vect
o
r (m),
→
’
–
s
cattering directi
o
n vect
o
r (m),
i
o
o
u
s
T
–
l
o
cal temperature
[
K
]
,
T, T
–
fluid inlet,
o
utlet temperature
[
K
]
,
t
–
ambient temperature, in
s
tantane
o
u
s
and average
[
K
]
,
u, v,
w
–
vel
o
cit
y
in
x
,
y
,
z
directi
o
n
[
m/
s
]
,
Q
–
heat gain
[k
W/m
2
]
,
Q
–
rate
o
f u
s
eful energ
y
gained (W),
ρ
–
den
s
it
y
[k
g/m
3
)
]
,
s
–
Stefan-
B
o
lt
z
mann c
o
n
s
tant (5
.
669
* 10
-8
W/m
2
K
4
),
s
–
s
cattering c
o
efficient (dimen
s
i
o
n-
le
ss
),
μ
–
vi
s
c
o
s
it
y
[
Pa·
s
]
,
j
–
pha
s
e functi
o
n (dimen
s
i
o
nle
ss
),
Ω’
–
s
o
lid angle (dimen
s
i
o
nle
ss
)
A
ck
no
w
l
edgme
n
ts
R
e
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ub
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fr
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m the
F
acult
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o
f
I
nfra
s
tructure and
E
nvir
o
nment
o
f the C
z
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s
t
o
-
ch
ow
a Univer
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f Techn
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n
owy
m grunt
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m
wy
mien-
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k
u ciepła p
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o
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Heat Tran
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Valida-
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o
n
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f C
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D
s
imulati
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n f
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[
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Sła
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Nie
s
tacj
o
narna anali
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a
wy
mian
y
ciepła
w
grunt
owy
m
wy
mienni
k
u ciepła p
o
s
a-
d
ow
i
o
n
y
m na gruntach nie
wy
s
y
c
o
n
y
ch
.
I
n
s
tal
2019
,
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24-30
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D
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[
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Subiant
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A.
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.
A
nal
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del
s
f
o
r
the c
o
mputati
o
n and
o
ptimi
z
ati
o
n
o
f
s
ingle and
d
o
uble gla
z
ing flat plate
s
o
lar c
o
llect
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r
s
w
ith
n
o
rmal and
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[
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]
Ż
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s
k
i K
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E
fe
k
t
yw
n
o
ś
ć pr
z
y
g
o
t
ow
ania ciepłej
wo
d
y
w
w
ę
ź
le ciepln
y
m
w
s
p
ó
łpracując
y
m
z
in
s
talacją
ko
le
k
t
o
r
ów
s
ł
o
nec
z
n
y
ch
.
I
n
s
tal
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2
[
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Ż
ar
s
k
i K, Kr
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ż
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M
.
Warun
k
i efe
k
t
yw
neg
o
d
z
ia-
łania
s
ł
o
nec
z
n
y
ch u
k
ład
ów
pr
z
y
g
o
t
ow
ania
ciepłej
wo
d
y.
I
n
s
tal 2024, 7-8, 10-13
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:
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2024
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8
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1
n
