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Optimization of Multicolor Photothermal Power Plants in the Solar System: A Finite - Time Thermodynamic
Approach
V. Bãdescu, C. Dinu
To cite this version:
V. Bãdescu, C. Dinu. Optimization of Multicolor Photothermal Power Plants in the Solar System:
A Finite - Time Thermodynamic Approach. Journal de Physique III, EDP Sciences, 1996, 6 (1), pp.143-163. �10.1051/jp3:1996118�. �jpa-00249444�
Optimization of Multicolor Photothermal Power Plants in the
Solar System: A Finite Time Thermodynamic Approach
V. Bidescu and C. Dinu
Group of Solar Energy and Applications, Corp Facultate Mecanici, Sala CG 129, Polytechnic University of Bucharest, Spl Independentei 313, Bucharest 79590, Romania
(Received 4 January1995, revised 24 July 1995, accepted 16 October 1995)
PACS.05.70-a Thermodynamics PACS.44.40+a Radiative heat transfer
Abstract. This paper deals with the performance of photothermal multicolor and omnicolor
converters in the Solar system. Both interplanetary power stations and power systems placed
on the surface of different planets were analyzed. The power station consists of a multicolor
converter endoreversible thermal engine combination. In the case of ground based planetary
stations the thermal engine is assumed to be of the Curzon Ahlborn type. The interplanetary stations, which are characterized by a nonlinear heat transfer between the engine and the ambi- ent, are treated by using a simple model developed in the paper. The influence of the radiation
concentration on the system performance is outlined. The effect of the Sun's zenith angle is also discussed. Spectral distributions of the collector and radiator optimum temperatures are shown.
1. Introduction
Presently, there are two major types of solar power systems used in space missions. The first type uses photovoltaic cells to convert the solar radiation directly into electrical energy, in combination with electrochemical storage. The second type is the so called "dynamic" system.
In this, solar concentrators reflect the flux of solar radiation toward a collector where a working
fluid is heated up to drive conventional thermal engines. Electrical energy may be generated by
alternators coupled to these engines. This "dynamic" system will be considered in this paper.
Various authors have proposed different ways to convert solar energy into useful work effi-
ciently. In case of photovoltaic conversion Shockley and Queisser ill treated a single cell (I.e. a single gap system). More recently it has been realized that the use of a system involving more
than one energy gap should enable a higher efficiency to be attained. It is believed that the
highest efficiency would be realized by an infinite stack of p-n junctions with smoothly varying bandgaps from cc to zero, such that there is a single junction adapted to each frequency in the solar spectrum. In the system proposed by De Vos [2] each of the individual cells is a selective black-body such that photons of frequency v of the solar spectrum are completely absorbed by the cell with bandgap Eg = hv. Such a system is termed as a "fully selective" or
an "omnicolor" photovoltaic converter.
Useful approximations for the mathematical treatment were proposed subsequently by Gros- jean and De Vos [3]. A minor deficiency of the model was corrected in references [4,5] where
a simple relation between the Carnot efficiency of thermodynamic engines and photovoltaic
@ Les #ditions de Physique 1996
energy conversion was derived. A further improvement of the model was performed in refer-
ence [6] where De Vos and Vyncke considered the influence of the ambient radiation and proved that both photovoltaic and photothermal omnicolor converters have the same maximum effi- ciency. The influence of the radiation concentration on the efficiency of omnicolor converters
was studied by Haught [7]. Note that the geometric factor affecting the ambient radiation in- cident on the collector have to be corrected in that paper. However, this error has insignificant
influence on the results. Some of the above papers were reviewed in references [8,9].
All the previously published results refer to the omnicolor converters placed on the Earth's surface. The use of the highly (or even fully) selective collectors for cosmic applications is more attractive. Indeed, the risks and the limited opportunity to correct possible damage during
the space missions, require the use of the most sophisticated human technologies. From this
point of view the multigap systems are to be seriously considered [10,11].
This paper deals with the performance of the multicolor converters in the solar system.
We first approached this subject in [12] where both photothermal and photovoltaic converters were envisaged. In the quoted paper the photothermal conversion was analyzed by assuming
that the omnicolor converter was coupled with an ideal Carnot thermal engine. It is well
known, however, that this hypothesis usually leads to too high upper bounds for the converter
efficiency. Here, ~ve will analyse the more realistic case of an endoreversible thermal engine.
So, when thermal power stations placed on the planetary surface are considered, a combination omnicolor solar converter + a Curzon Ahlborn thermal engine [15-18] is considered. Indeed, in this case a linear coupling between the engine and both thermal reservoirs can be accepted.
The situation is different for interplanetary vehicles, where the coupling between the thermal engine and the cold reservoir is nonlinear. In this case a simple original model has been
developed.
Our analysis refers, on one hand to the power systems placed on the surface of different planets, and, on the other hand to the interplanetary space power stations. In the latter case, the maximum performance of both omnicolor and multicolor solar converters is discussed. The influence of the radiation concentration on the system performance ~vill be outlined for some
cases of practical interest. The effect of the radiation incidence angle will be also discussed.
Spectral distribution for various optimum temperatures and efficiencies are presented.
2. Surface Planetary Power Plants
2.I. MODEL. The maximum conversion efficiency with a photothermal system is obtained in the limit of an infinite collector array, as shown in Figure 1. Each radiation splitter selects from the (concentrated) radiation spectrum the photons from a narrow band of width dv
around a given frequency (say v) These photons are used to heat a collector which absorbs and emits around the frequency v. This collector has a temperature Tc(v) and its absorptance o(v) is assumed to be given by:
1 for the frequency range iv du/2, v + dv/2]
~~
° ~
0 for all other frequencies
In order to draw up the energy balance for the collector at temperature Tc(v) we have to remind readers of some basic facts. The spectral irradiance 4l from a source of black-body
radiation at temperature T is given by:
4l(u,T)
=
~Ip(u,T)hu (2)
~°°~ ~°~'~~~°~
Solar radiation Ambient raciation Emitted radiation Heat flux and [on<entrator
"~~~~~~~~ ~"~~
Ambient
~
thermal thermal
Splitters collectors engines
' ' )+dJ
/ '
'
~ Ri'~' Jl
1,ldl 1~
_
~
' '~~
'
~- dJ
~
'~ ii)-dJi
Fig. 1. Omnicolor photothermal converter. For details see the text.
~~~~~
~P~~ ~~ ~"~ je% lj~~ (~)
~2
is Planck's distribution (see e-g. [13]). Here c, h and k are the light speed, the Planck and the Boltzmann constants, respectively, and B is a geometric factor.
The net thermal flux Qi(u)du supplied by the collector at temperature Tc(u) towards its
accompanying heat engine is assumed to be given by:
Qi(u)du
i j~~(u,T~) + ~~(u,T~) ~~ju,T~(u)jj du (4)
where the indices s, a and c refer to the Sun, ambient and collector, respectively. The first two terms from the right side of equation (4) represent the incident solar and ambient radiation, respectively, and the last term is the flux of radiation emitted by the collector.
In the first approximation, the Sun can be considered as a source of isotropic radiation. In this case, Ts m 5769.8 K and the following equation applies:
Bs(Qs, 8)
= 7r~t~(Qs) cos 8 (5)
~~~~~
~(a~)
=
~S i )) ~~~
= sin b (6)
~r ~
Here as is the solid angle subtended by the Sun when viewed from the receiving surface and 8 is the angle between the normal of the receiving surface and the direction to the center of the solar disc (I.e. the Sun's zenith angle). Also, b is the half angle of the cone subtended by
the Sun. The equation (5) can be used only when the following condition is fulfilled:
e + ~ ~ yr/2 (7)
I-e- when the Sun is completely visible.
When unconcentrated direct solar radiation is considered, the solid angle as subtended by
the Sun can be computed by equation (6), by taking into account the simple geometrical relationship b
= tg~~(lls Id) where lls(m 6.9599 x 105 km) is the Sun's radius. In the case of concentrated radiation, first we observe that in the presence of the concentrator, the Sun is viewed from the collector surface as having an enlarged solid angle (say Qc). The concentration ratio C is naturally defined as:
~ ~~ ~ ~~
~ i(Q)) 8
= 0) ~~~
i-e- the ratio between the geometric factors of the concentrated and non-concentrated radiation, respectively, both evaluated at normal incidence (8
= 0). By using equations (8), (5) and (6)
one obtains:
onsequently, for given distance to the Sun d, and oncentration ratio C,
equation(9) can be
used to ompute the nlarged
by means of quation (2).
The eometric
B~=7r-Bs (10)
Note that equation
(10) is igorous in the case of concentrated solar radiation, hen the
con-
centrator'smirror covers a part of the
celestial vault (17]. However, this
equation is
very good in the case of the
unconcentrated
radiation, because of the
value which Bs has in this case (at the Earth-Sun mean Bs m 6.83 x 0~5). Equa-
tion (10) was accepted by several uthors[6,13]. Haught assumed B~ = 7r in his
reference [14] we proved that each of the two hypotheses is
valid under
special
circumstances.
this paper we use equation
(10), ecause
to significant rror for
large values of C.
The ollector is assumed to emitdiation
toward the hole (~ = 27r). Conse-
uently, its geometric factor is Be (Q = 2~, 8 = 0) = 7r, as equation
By using equations 2),(3) and (10)
thermal
engine
working at equency u:
=
c
(11)
If we
take into account equation (I), we see that this engine uses the energy
only in a narrow range around frequency u. Of course, neighboring enginesuse
from other infinitesimal equencyntervals. Our analysis assumes that
I-e-
consists of the three parts [15]:
I) a eversible part (I.e. a Carnot part)
workingetween two
heat reservoirs one at high
say ti, and one at low temperature, say t2i usually
it) t~vo irreversible parts containing temperature drops (I. e. the temperature falls Tc ti
and t2 Ta).
When a surface planetary station is considered, both temperature drops are linear in tem- peratures, because the heat transfer between the working fluid and the hot and cold reservoirs
(the converter and the atmosphere or soil of the planet, respectively) is mainly convective or
conductive and, consequently, the Newton or Fourier laws apply. As a result, a special kind of endoreversible converter can be used, namely the Curzon Ahlborn engine. This engine is
characterized by a linear relationship between the heat flows and the temperature gradients.
Details on the endoreversible and Curzon-Ahlborn engines can be found in references [15-18].
The power lktot supplied by the whole array of monochromatic convertei~s (I.e. by the omnicolor converter) is given by:
lj~ ~~~
~ (12)
co
a u
/°~ pr(~)du / Qi(")
1
T (u) Wtot
~ 0 ~
where the square brackets contain the efficiency of the spectral Curzon Ahlborn engine. The maximum power supply lktot,max can be obtained by optimizing the collector temperature Tc(u)
for conversion of the radiation of frequency u. Consequently, we have to solve the equation:
~~°~
= 0 (13)
0Tc(V)
and then to replace its root (sa» Tc,opt(v)) in equation (12). The maximum overall efficiency of the omnicolor converter is simply given by:
j~~~~~~~ j~~~~~~~
~~~~ ~
4~i(V)dV
~ ~~~~~ ~~~~
The following notation will be used:
The overall maximum efficiency of the omnicolor converter can be computed by using equations (11-15). The result is:
~~'~~
i~ ~~~
~~~~~~~~ ~~ xp(~~a) -1 ~ ~p(~~~~ xp(~tx~pt) ll~~~~ ~~~~
Here the optimum value xopt(~t) is obtained by solving the equation:
exp(u/pt ~ ~~~~~~ ~~~~~(e~~~~~~~l)2
~~ xp(~t~a) 1 ~ ~ xp(i~) ~ ~~~~
By using (15), we see that once xopt(~t) is known, the optimum dependence of Tc,opt(~t) is given by:
Tc,opt(u) =
~
j~~ (18)
opt
If we take into account that:
~ i~~a
~~~~
equation (18) can be used to obtain the optimum dependence ofTc on the photon wavelength 1.
Table I. Data abo~tt planets.
Mean distance Planet surface
to the Sun temperature
x lo6 Km
57.9 623
lo8 2 753
149 300
227.9 296
775 133
1430 123
2870 lo3
4505 53
5900 43
7400 43
o.7 iooo
900 0.6
80D
.( ~'~ ~~~
-.- Max. en. CA.
1 0.4 ~~~ i
-- Max en Carnot
( 500 il
g ~ ~ ?
-- Pmax C A.
E 400 (
f ~ ~- PmaX Carnot
'I 0 2 ~°° .j
~
200 ~
~'~
l 00
o o
0 20 40 60 80
zenith angle degrees
Fig. 2. Maximum efficiency and maximum power provided by an omnicolor photothermal converter
in combination with a Camot and a Curzon-Ahlbom (CA) thermal engine, respectively, placed on the Earth surface, as a function of the solar zenith angle 8. Unconcentrated radiation (C
= 1) was
considered.
2.2. RESULTS AND DIscussIoNs. First, let us analyze the case of a surface planetary po~ver
station which uses unconcentrated solar radiation. Data about the planets whose atmosphere
is transparent for direct solar radiation are given in Table I. Of course, it seems unrealistic to
speak about the use of solar energy on the surface of Jupiter or Saturn, but it could be possible
on the surface of
one of their satellites, such as Titan, where the atmospheric temperature is around 100 K [19].
There is a significant difference between the results obtained by modeling the power station
as a Carnot engine [12] and a Curzon-Ahlborn thermal engine, respectively. This can be seen from Figure 2 which corresponds to a power station placed on the Earth surface. As we see, the maximum efficiency is about 20 percent lower in the case of the more realistic (Curzon- Ahlborn) thermal engine. The dependence of the maximum efficiency on the zenith angle 8 is
300U 25U0
Eaih-C.A. 2°°°
S
g 2000 -- Eaih-Carnot I
g i~ ~~~g w
( e -.-- Mars-CA- j/
j ~~~
-~- Mars-Carnot .I )
f f 1000 g f
j ~~~ j
500 ~°°
o o
0 2 DA 0.6 0.8 1.2 4 1.6 1.8 2
Wavelenath tmicronsl
Fig. 3. The optimum temperature of an omnicolor photothermal converter in combination with
a Camot and a Curzon Ahlbom (CA) thermal engine, respectively, placed on the Earth and Mars
surfaces, as a function of the ~vavelength ~. Unconcentrated radiation (C
= 1) at normal incidence
(8 = 0) was considered.
1-educed. Ho~vever, at larger values of the zenith angle 8 the difference between the two sorts of engines is lesser.
The spectral variation of the optimum temperature Tc,opt(I) is shown in Figure 3 for two
planets to a similar distance fi.om the Sun (Earth and Mars). The differences between them
are relatively small. The spectral temperature is slightly influenced by the irreversibilities of the thermal engine. Indeed, the temperature is always bigger in case of the Curzon Ahlborn
engine, especially at the small wavelengths.
Now, let us look at some results at large scale. The maximum efficiency of a Curzon-Ahlborn power station using omnicolor converters is higher in case of the outer planets (Fig. 4). Of
course, this is not a direct consequence of the larger distance to the Sun, but of the lower
ambient temperature (see Tab. I.), whose influence is stronger than the effect of the geometric factor B which decreases with distance. Note, however, that the power supplied by the station is strongly decreasing by increasing the distance to the Sun, because of the smaller value of the
solar radiation flux incoming on the converter (Tab. II.). The zenith angle 8 of the Sun has a little influence on ilm~x. The dependence of the optimum temperature Tc,opt of the omnicolor
photothermal converters on the photon wavelength I is shown in Figure 5 for some planets.
Tc,opt is strongly influenced by the ambient temperature, being higher in case of the planets
close to the Sun. The variation in Tc,opt(I) for the outer planets is insignificant. Note that at
higher wavelengths the optimum temperature becomes practically a non spectral quantity.
Let us consider no~v the influence of solar radiation concentration on the performance of omnicolor photothermal converters. Figure 6 shows the dependence of the maximum efficiency
ilm~x on the concentration ratio C. As we see, increasing C leads to a significantly higher efficiency. However, the performance improvement diminishes at large values of the concentra-
tion ratio and at large distance from the Sun. The dependence of the maximum mechanical power lktot,max on the concentration ratio is shown in Table III. lktot,max is practically a linear function of C in the case of po~ver stations placed on the surface of the outer planets. This rule fails for the planets close to the Sun.
o 7~
0.6 ° ~°~~
o 74 Lefi ams
.~~ -- Mercuri
f ~
0.73 Mars
( ~'~
~~ ~ n
Saturn fi
~ -- Uranus
) 0A
0.71 Rlghtams
j ~°~~ NeP,Une
.( ~'~~ 0.7
--o- Pluto(Penhelion)
~ ° ~
-.- Pluto(Aphelion)
0.25 ~'~~
02 0.68
0 20 40 60 80
zenith angle degrees
Fig. 4. The dependence of the maximum efficiency ~ma~ of
a photothermal omnicolor converter on the Sun's zenith angle B for a Curzon-Ahlbom power station situated
on the ground of different
planets. For the ambient temperature and distance to the Sun, see Table 1.
Table II. The dependence of the maxim~tm mechanical po~er s~tppty llltot,max on the solar zenith angle 8 in case of ~tnconcentrated radiation. A photothermat omnicotor converter placed
on a s~trface planetary po~er station ~as considered. For the distance to the S~tn and ambient temperat~tre, see Table1.
Zenith 8
0 20 40 60 80
1876 366
2 1. 0
o
o o o o
0.31 0 0
The dependence of the optimum spectral temperature Tc,opt(I) of an omnicolor converter on the concentration ratio C is shown in Figure 7 in the case of a power station placed on the
Mars surface. The optimum temperature increases when the concentration augments. For a
given C, the largest variation ofTc,opt occurs in the range of the visible and near IR spectrum.
Figure 8 shows the spectral dependence of Tc,opt(I) for a concentration ratio C
= 1000, which is quite accessible to the solar paraboloidal concentrators. Significant differences exist between the values obtained for the inner planets. Note that the maximum concentration factor (which
can be obtained by using equations (5) and (9)) is 6,922; 45,833 and 107,223 for Mercury, Earth and Mar8, respectively. Consequently. the value C
= 1000 used in the computations is
high in the case of Mercury and relatively important, too for Earth and Mars. However, small differences exist between the outer planets, whose maximum concentration ratio are very high.
(Cmax = 4.221 x 10~ and 7.186 x 10~ for Saturn and Pluto, respectively).
