Collector Energy Balance
M.Fossa, Sust. En. Solar & Geo 1, UniGe - Pag. 19 / 95
Collector Energy Balance (I)
G”u = G”a - G”disp
G”inc = G”direct + G”diff
G”u = G”a – K” (Tp, ave – Ta)
G”a = G”inc (τα)ave
(Tp, ave – Tfluid, ave) = G”u / h fluid
Tp, ave = G”u / h fluid + Tfluid, ave
G”inc
G”u (1 + K”/ h fluid ) = G”a – K” (Tfluid, ave – Ta)
G”disp
G”u = F’[G”a – K” (Tfluid, ave – Ta)]
Ta
F’ = h fluid / (K” + h fluid ) Efficiency factor
Glass cover
G”abs
Tp
G”useful
Tf
Flow rate m”
Thermal fluid
Insulation
M.Fossa, Sust. En. Solar & Geo 1, UniGe - Pag. 20 / 95
Collector Energy Balance (II)
G”u = F’[G”a – K” (Tfluid, ave – Ta)]
G”u = m”c (Tfi – Tfu)]
A=WL dA = Wdx
G”u,x dA = G”u,x (W dx) = (m” A) c dTf,x
m”c L dTf,x / dx = G”u = F’[G”a – K” (Tf, x – Ta)]
being d[G”a – K” (Tf, x – Ta)] = -K” dTf
dTf = -1/K” d[G”a – K” (Tf, x – Ta)]
∫dx=L
F’/(m”c L) dx = (-1/K”) d[G”a – K” (Tf, x – Ta)] / [G”a – K” (Tf, x – Ta)]
(G”a – K” (Tf, u – Ta)) / (G”a – K” (Tf, i – Ta)) = exp [-F’K”/(m”c)]
G”inc
G”disp
Ta
L
W
Glass cover
y
x
G”abs
G”useful
TFI
Tp
TFU
TF
Insulation
m” [kg/(s m2)]
Therm. fluid
Mass flow rate m”
(per unit surface)
Δx
M.Fossa, Sust. En. Solar & Geo 1, UniGe - Pag. 21 / 95
Collector Energy Balance (IIIa)
(heat removal factor) Fattore di rimozione termica FR
G”u = F’[G”a – K” (Tfluid, ave – Ta)]
FR= G”u / [G”a – K” (Tfluid, in – Ta)]
= [m”c (Tout – Tin)] / [G”a – K” (Tin – Ta)]
FR= (1/K”)m”c [K”(Tout – Tin] / [G”a – K” (Tin – Ta)] =
(1/K”)m”c [K”(Tout – Ta)-K”(Tin – Ta)] / [G”a – K” (Tin – Ta)] =
(1/K”)m”c [ [G”a -K”(Tin – Ta)] - [G”a - K”(Tout – Ta)] ] / [G”a – K” (Tin – Ta)] =
(1- [ [G”a - K”(Tout – Ta)] / [G”a -K”(Tin – Ta)] ]) (1/K”)m”c =
(1(1-exp [-F’K”/(m”c)]) (1/K”)m”c
FR= (1 – exp[-F’K”/(m”c)])(m”c)/K”
M.Fossa, Sust. En. Solar & Geo 1, UniGe - Pag. 22 / 95
Collector Energy Balance (IIIb)
(heat removal factor) Fattore di rimozione termica FR
G”u = F’[G”a – K” (Tfluid, ave – Ta)]
FR= G”u / [G”a – K” (Tfluid, in – Ta)]
FR is close to unity for high flow rates, high h, low K”. It can be
demonstrated that:
FR= [1 – exp(-F’K”/m”c)](m”c)/K”
Hence:
G”u = FR[G”a – K” (Tfluid, in – Ta)]
(Bliss Equation)
G”inc
G”disp
Ta
Copertura vetrata
G”ass
Tp
TFU
TFI
Tf
G”utile
Fluido vettore
Isolante Termico
Portata fluido m”
(per unità di
superficie)
M.Fossa, Sust. En. Solar & Geo 1, UniGe - Pag. 23 / 95
Collector Energy Balance (IV)
G”u = FR[G”a – K” (Tfluid, in – Ta)]
(Bliss equation)
G”a = G”inc (τα)
Ea = Einc, ave (τα)ave
[J/m2 day]
(on a daily basis, monthly ave)
η
Collector efficiency η is defined as:
η
= G”u /G”inc
= FR [G”a – K” (Tfluid, in – Ta)] / G”inc
ηave = FR [Ea – K” (Tfluid, in – Ta, ave)Δτ] / Einc
(on a daily basis, monthly ave)
Ea = Einc (τα) ave
G”inc
G”disp
Ta
Einc = ET
G”ass
TFI
Tp
Tf
G”utile
ET = Overall insolation on
tilted surface
TFU
m”
(per unit
surface
M.Fossa, Sust. En. Solar & Geo 1, UniGe - Pag. 24 / 95
Collector Efficiency (I)
ηave = FR (τα) – FRK” [(Tfluid, in – Ta)]/G”inc
2
2
2
2
2
2
M.Fossa, Sust. En. Solar & Geo 1, UniGe - Pag. 25 / 95
Collector Efficiency (II)
η= FR (τα) – FRK” [(Tfluid, in – Ta)] / G”inc
η=a + bx
Notice:
1) Sometimes FR (τα) is written as η0
2) Sometimes η= a + b1x + b2x2
η
1
Iso 98069806-1
Solar Energy – Methods of
Test for the Thermal
Performance of Liquid
Heating Collectors
FR τα
Atan (K”FR )
EN 1297512975-2:2006 Thermal
solar systems and
components – Solar
collectors – Part 2: Test
methods
[(Tfluid, in – Ta)] / G”inc ]
M.Fossa, Sust. En. Solar & Geo 1, UniGe - Pag. 26 / 95
Collector Efficiency (III)
EN 12975
Indoor Test
The lamps capable of
producing a mean irradiance
over the collector aperture of
at least 700 W/m2. Values in
the range 300 to 1000 W/m2
can be used for special tests.
The irradiance at a point on
the collector aperture shall
not differ from the mean
irradiance over the aperture
by more than ±15 %
The spectral distribution of
the simulated solar radiation
shall be approximately
equivalent to that of the solar
spectrum at optical air mass
1.5
M.Fossa, Sust. En. Solar & Geo 1, UniGe - Pag. 27 / 95
Collector Efficiency (III)
M.Fossa, Sust. En. Solar & Geo 1, UniGe - Pag. 28 / 95
Collector Efficiency (IV)
Coeff. In efficiency expression, as measured at CSTB (France)
MARQUES de CAPTEURS
Avis Technique
CSTB
Coef. τ α
CSTB
Coef. K“
W/m2.°C
BUDERUS - Logasol SKS
14-00/577
0,79
4,89
AT 14 + 5/03839
0,73
4,26
4-00/576
0,68
3,82
GASOKOL - Enersol GKAN
et GKAQ
14/02-716
0,77
3,86
GIORDANO - C8 HI
14/02-747
0,72
4,36
14+5/02-756
0,72
4,80
SOLAHART - Solahart Ko
14/01-672
0,79
4,76
SONNENKRAFT - SK500
(Solar Connexion)
14-00/575
0,76
3,78
SUNMASTER - SK20 LM
(New Point Products)
14/01 – 650
0,77
4,17
VIESSMANN - Vitosol 100
S1,7
14/00-584
0,76
4,34
WAGNER - EURO C20 AR
14/03-844
0,85
3,34
WEISHAUP - WTS-F
14+5/03-793
0,77
2,75
ZENIT - Thermic
14+5/01-609
0,77
CLIPSOL - TGD Y1200
DE DIETRICH - Sol 1
PHÖNIX - Infinity 21
3,62
M.Fossa, Sust. En. Solar & Geo 1, UniGe - Pag. 29 / 95
Collector Efficiency (V)
Incidence angle effects on daily efficiency
ηave= Cθw, ave FR (ταN) – FRK” [(Tfluid, in – Ta)] / G”inc
Cθw=C(IAM(θw))
IAM(θw)
w
M.Fossa, Sust. En. Solar & Geo 1, UniGe - Pag. 30 / 95
Collector Efficiency(VI)
Reference Area
Very important is to know Efficiency to which Area is referred to.
Agross, Aaperture, Aabsorber
M.Fossa, Sust. En. Solar & Geo 1, UniGe - Pag. 31 / 95
Collector
Efficiency(VIB)
Reference Area
M.Fossa, Sust. En. Solar & Geo 1, UniGe - Pag. 32 / 95