Howmuch work can be extracted from a radiation reservoir?Q1 Viorel Badescu ∗
Candida Oancea Institute, Polytechnic University of Bucharest, Spl. Independentei 313, Bucharest 060042, Romania
h i g h l i g h t s
• Radiation reservoirs are more complex than heat reservoirs.• Reversible and endoreversible work extraction from radiation reservoirs is analyzed.• The upper bound for reversible work extraction is not Carnot efficiency.• All upper bound efficiencies depend on the geometric factor of the radiation reservoir.
a r t i c l e i n f o
Article history:Received 17 January 2014Received in revised form 7 April 2014Available online xxxx
Radiation reservoirs are more complex than heat reservoirs. They depend on the intensivethermodynamic parameters (such as temperature, pressure, and chemical potential) aswell as on other state parameters (such as the geometric factors). The paper refers to workextraction from a high temperature radiation reservoir (the pump), the sink being a heatreservoir, respectively a radiation reservoir. The simplest case of radiation reservoir (i.e.blackbody isotropic radiation) is considered. Reversible and endoreversible conversion isanalyzed. The upper bound for reversiblework extraction is not Carnot efficiency. All upperbound efficiencies obtained here depend on the pump geometric factor.
the endoreversibility assumption, i.e. irreversibility occurs in the interaction between conversion systems. When work ex-1
traction from heat reservoirs is considered, this procedure yields the well-known Chambadal–Novikov–Curzon–Ahlborn2
efficiency [10]. In case of work extraction from radiation reservoirs, the procedure has been used in Refs. [11,12] and the3
results obtained there are less known.4
The present paper systematically treats the work extraction from a high temperature radiation reservoir, the sink being5
a heat or a radiation reservoir. Both reversible and endoreversible conversion are considered. The case of endoreversible6
conversion and sink heat reservoir has been considered in Refs. [1,11–13]. The case of endoreversible conversion and sink7
radiation reservoir is treated here for the first time. More accurate upper bound efficiencies are obtained in case of endore-8
versible conversion, for sink reservoirs consisting of heat or radiation. Finally, the accuracy of reversible and endoreversible9
upper bound efficiencies is emphasized by inter-comparison.10
2. Generalities11
We consider work extractors operating between a radiation reservoir (the pump) and a heat or radiation reservoir (the12
sink). The next assumption is that the reservoirs are in thermodynamic equilibrium. The temperatures of the pump and sink13
are denoted as TH and TL, respectively. The work extractor may be a thermal engine, providing mechanical work, or other14
device providing electrical work (such as a photovoltaic cell; see Refs. [1,14] for early works) or chemical work. Heat fluxes15
are denoted Q and radiation energy density fluxes are denoted ϕ. Entropy fluxes associated with heat transfer are denoted S16
and radiation entropy density fluxes are denotedψ . Work rate is denoted W while the entropy generation rate during work17
extraction is denoted Sgen(≥ 0). Steady state operation is considered here.18
The geometric factor of a radiation reservoir is denoted f (≤ 1). The solid angleΩ of a spherical source is given by:19
Ω = 2π(1 − cos δ) (1)20
where δ is the half-angle of the cone subtending the sphere when viewed from the observer [15,16]. When the sphere’s21
center has the zenith angle θ0 [15,16]:22
f =Ω
π
1 −
Ω
4π
cos θ0. (2)23
The simplest case is considered here, i.e. isotropic blackbody radiation reservoirs, which are characterized by two24
parameters only, i.e. their temperature T and geometrical factor f . The energy and entropy density fluxes are, respectively:25
ϕ = f σT 4, (3a)26
ψ =43f σT 3. (3b)27
Two cases are treated next: (i) the sink is a heat reservoir and (ii) the sink is a hemispherical radiation reservoir (δ = π/2,28
f = 1).29
Work extraction from radiation reservoirs involves radiation absorbers and/or emitters. They may be materials with/30
without band-gap energy or bodies with selective radiative properties. The simplest case is considered here: the absorber31
(emitter) is a (non-selective) plane Lambertian blackbody of surface area Aa(Ae) receiving (and emitting) radiation over the32
whole hemisphere. Therefore, their geometric factors are fe = fa = 1. Local thermal equilibrium is assumed and the temper-33
ature of the absorber (emitter) is denoted Ta(Te).34
The following notation is used:35
a ≡TLTH, (4a)36
x ≡TaTH, (4b)37
y ≡TeTH, (4c)38
r ≡Ae
Aa. (4d)39
Lambertian blackbody absorbers imply full thermalization of radiation energy (i.e. the radiation energy is entirely40
transformed into internal energy of the absorbing body). Therefore, specific work extractors such as thermal engines (for41
early works see Refs. [11,17]) or thermophotovoltaic devices should be considered.42
Two ideal operation regimes are treated next. First, reversible operation is considered. This is associated with vanishing43
entropy generation rate inside the work extractor and on the borders of the work extractor (these borders are the radiation44
emitter and the radiation absorber). Second, endoreversible operation is considered. In this case the entropy generation45
rate inside the work extractor is vanishing but no hypothesis is made about the entropy generation rate at work extractor46
V. Badescu / Physica A xx (xxxx) xxx–xxx 3
Fig. 1. Work extractor in contact with a radiation reservoir at high temperature TH and a heat reservoir at low temperature TL .
borders. In both cases the conversion efficiency constitutes upper bound for the efficiency of real work extractors. However, 1
taking into account Gouy–Stodola theorem [2,18,19], the second ideal operation regime provides smaller work rate than the 2
first ideal operation regime. Therefore, the second ideal operation regime provides more accurate (i.e., lower) upper bound 3
efficiency for the efficiency of real work extractors than the first ideal operation regime. 4
3. High temperature radiation reservoir and low temperature heat reservoir 5
Work extractors from solar radiation operating on the Earth’s surface are particular cases of the general approach con- 6
sidered here. 7
3.1. Reversible operation 8
The absorber receives radiation from the pump and emits radiation (Fig. 1). The first and second laws for work extractor 9
yield, respectively: 10
AaϕH − Aaϕa − QL = W (5) 11
AaψH − Aaψa − SL + Sgen = 0. (6) 12
Note that Sgen includes the entropy generation inside the absorber. The work extractor efficiency ηrh is defined (subscript rh 13
stands for radiation-heat): 14
ηrh ≡W
AaϕH= 1 −
Aaϕ + QL
AaϕH. (7) 15
Here Eq. (5) has been used. The relationship between the heat and entropy fluxes in the linear theory is: 16
SL =QL
TL. (8) 17
Usage of Eqs. (6)–(8) yields: 18
ηrh = 1 −ϕa + TL(ψH − ψa + Sgen/Aa)
ϕH. (9) 19
The energy and entropy density fluxes received by the Lambertian absorber from the radiation reservoir of high temperature 20
and emitted hemispherically by the Lambertian absorber, respectively, are shown in Table 1. 21
Eqs. (5), (6) and Table 1 state that equilibrium occurs for TL = Ta = TH and fH = 1. In the general, non-equilibrium case, 22
the following constraints apply: TL ≤ Ta ≤ TH . Also, the work rate W is positive under the following constraints: 23
ϕH ≥ ϕa, (10a) 24
QL ≥ 0. (10b) 25
Gouy–Stodola theorem states that the maximum work rate is generated at reversible operation [2,18,19]. Thus, the as- 26
sumption Sgen = 0 is adopted and usage of Eqs. (4a), (4b), (9) and Table 1 yields the maximum efficiency ηrh,max: 27
ηrh ≤ ηrh,max = 1 −x4revfH
−43a1 −
x3revfH
(11) 28
4 V. Badescu / Physica A xx (xxxx) xxx–xxx
Table 1Energy and entropy density fluxes received by the Lambertian absorber from the pump radiation reservoir (at temperature TH ) and by the Lambertianemitter from the sink hemispherical radiation reservoir (at temperature TL). The energy and entropy density fluxes emitted by the absorber(temperature Ta) and emitter (temperature Te) are also shown.
Case Energy density flux(W/m2)
Entropy density flux(W/(m2K))
Radiation received by absorber from pump radiation reservoir ϕH = fHσT 4H ψH =
43 fHσT
3H
Radiation received by emitter from the sink hemispherical radiation reservoir ϕL = σT 4L ψL =
43σT
3L
Radiation emitted hemispherically by the absorber ϕa = σT 4a ψa =
43σT
3a
Radiation emitted hemispherically by the emitter ϕe = σT 4e ψe =
43σT
3e
where xrev is reduced absorber temperature at hypothetic reversible operation. Usage of Eqs. (10a), (4b) and Table 1 gives1
xrev ≤ f 1/4H . However, usage of Eqs. (10b), (4b), (6), (8) and Table 1 gives a more restrictive constraint:2
xrev ≤ f 1/3H
≤ f 1/4H
(12)3
ηrh,max monotonously decreases by increasing xrev and a. It reaches a maximum value for xrev = a, i.e. for absorber operat-4
ing at the low temperature TL of the sink heat reservoir. Then, usage of Eq. (11) gives the maximum maximorum efficiency5
ηrh,max,max:6
ηrh,max,max = 1 −43a +
13a4
fH(13)7
ηrh,max .max increases by decreasing fH . It ranges between a maximum value 1 (for a → 0) and 1 − f 1/3H (for a = f 1/3H ).8
Petela–Landsberg–Press efficiency [5–7] is a particular case of the upper bound efficiency Eq. (13) (it applies for hemispher-9
ical pump radiation reservoirs, i.e. for fH = 1).10
ηrh,max,max Eq. (13) does not depend on the parameters of thework extractor. It depends on the parameters of the radiation11
and heat reservoirs only. It constitutes the upper bound efficiency for reversible operation of work extractors in contact with12
a pump radiation reservoir and a sink heat reservoir. Further comments and discussions may be found in Ref. [9].13
3.2. Endoreversible operation14
The energy balance for the absorber gives15
Aa(ϕH − ϕa)− Qa = 0 (14)16
where Qa is the unbalanced flux of thermal energy, which is partially converted into the work rate:17
Qa − QL = W . (15)18
The entropy balance associated with the heat fluxes Qa and QL is:19
Qa
Ta+ S ′
gen =QL
TL(16)20
where the entropy generation rate S ′gen differs from Sgen of Eq. (6) since the two quantities are associatedwith different series21
of irreversible processes. For instance, S ′gen does not refer to the absorption or emission process at the boundaries of thework22
extractor.23
The work extractor efficiency η′
rh is defined as in Eq. (7) and:24
η′
rh ≡W
AaϕH=
1 −
TLTa
1 −
ϕa
ϕH
−
TLS ′gen
AaϕH. (17)25
Here (14)–(16) have been used. The endoreversibility assumption S ′gen = 0 is now adopted and the maximum efficiency26
η′
rh,max is defined:27
η′
rh ≤ η′
rh,max =
1 −
axrev′
1 −
x4rev′fH
. (18)28
Here Eqs. (17), (4a), (4b) and Table 1 have been used while rev′ shows that the assumption of endoreversibility refers to29
those processes associated with S ′gen. The work extractor generates positive work rate when:30
0 ≤ a ≤ xrev′ ≤ f 1/4H ≤ 1 (19)31
V. Badescu / Physica A xx (xxxx) xxx–xxx 5
η′
rh,max Eq. (18) may be used to obtain upper bound efficiencies for the endoreversible operation of work extractors in 1
contact with a pump radiation reservoir and a sink heat reservoir. Two approaches are considered in the next Sections 3.2.1 2
and 3.2.2, respectively. 3
3.2.1. Analytical upper bound efficiency 4
The approach first presented in Ref. [12] is shown here. Eq. (18) may be written as: 5
η′
rh,max = 1 −43
a
f 1/4H
+
13
a
f 1/4H
4
− S(a, xrev′ , fH) (20) 6
where: 7
S (a, xrev′ , fH) ≡1
3axrev′ f3/4H
a
f 1/4H
x4rev′ (xrev′ − a)
3 −
axrev′
−a2
x2rev′−
a3
x3rev′
8
+ f 3/4H af 1/4H − xrev′
3 −
xrev′
f 1/4H
−x2rev′
f 2/4H
−x3rev′
f 3/4H
. (21) 9
The function S(a, xrev′ , fH) is non-negative under the constraints Eq. (19). Thus, 10
η′
rh ≤ η′
rh,max ≤ η′≡ 1 −
43
a
f 1/4H
+
13
a
f 1/4H
4
. (22) 11
Therefore, η′ given by Eq. (22) constitutes an (analytical) upper bound efficiency for the endoreversible operation of work 12
extractors in contact with a pump radiation reservoir and a sink heat reservoir. 13
3.2.2. Numerical upper bound efficiency 14
Eq. (18) is maximized numerically under the constraints Eq. (19). The optimum value xrev′,opt is found and the resulted 15
quantity η′
rh,max,max is the upper bound efficiency for the endoreversible operation of work extractors in contact with a pump 16
radiation reservoir and a sink heat reservoir. 17
3.3. Discussion 18
The reversible upper bound efficiency ηrh,max,max Eq. (13) and the endoreversible upper bound efficiency η′
rh,max,max 19
obtained by the constrained maximization of Eq. (18) (by using the software developed in Refs. [20,21]) are shown in Fig. 2, 20
together with the analytical endoreversible upper bound efficiency η′ Eq. (22), for two values of the pump geometric factor 21
fH . These efficiencies are positive for a limited range of variation of a, depending on the value of fH : ηrh,max,max is positive 22
for a < f 1/3H while η′
rh,max,max and η′ are positive for a < f 1/4H . All these upper bound efficiencies decrease by increasing the 23
ratio a ≡ TL/TH . 24
The Carnot efficiency ηC = 1−a is numerically larger than themaximumefficiencies predicted by Eqs. (13), (18) and (22), 25
respectively (see Fig. 2a,b). However, ηrh,max,max (or η′
rh,max,max, or η′) is an upper bound efficiency, since efficiency values 26
between ηrh,max,max (or η′
rh,max,max, or η′) and ηC are not allowed. Note that ηC is positive for a ≥ f 1/4H , when in fact the work 27
extractor does not provide positive work rate. 28
Themost accurate (i.e., lower) upper bound efficiency is the quantity η′
rh,max,max obtained numerically (Fig. 2). The simple 29
analytical formula η′ provides the second best accurate upper bound efficiency. The reversible upper bound ηrh,max,max is less 30
accurate, as expected. PLP efficiency is a particular case of ηrh,max,max; it applies only for fH = 1. Using PLP efficiency in case 31
of fH = 1 may inappropriately predict positive efficiency for a ≥ f 1/4H (when the work rate is in fact negative). 32
4. High temperature radiation reservoir and low temperature radiation reservoir 33
Work extractors from solar radiation operating in space are particular cases of the general approach considered here. 34
4.1. Reversible operation 35
The first and second laws for work extractor yield, respectively (Fig. 3): 36
AaϕH + AeϕL − Aaϕa − Aeϕe = W (23) 37
Aaψa + Aeψe − AaψH − AeψL − Sgen = 0. (24) 38
The work extractor efficiency ηrr is defined (subscript rr stands for radiation–radiation): 39
ηrr ≡W
AaϕH= 1 −
Aaϕa + Aeϕe − AeϕL
AaϕH. (25) 40
6 V. Badescu / Physica A xx (xxxx) xxx–xxx
Fig. 2. Upper bound conversion efficiencies as a function of a ≡ TL/TH in case of a pump radiation reservoir and a sink heat reservoir. (a) pump geometricfactor fH = 0.001; (b) fH = 0.3. The Carnot efficiency 1 − a is also shown.
Fig. 3. Work extractor in contact with two radiation reservoirs of temperature TH and TL , respectively.
Here Eq. (23) has been used. The energy and entropy flux densities emitted by the emitter and received by it from the sink1
are shown in Table 1. Eqs. (4a)–(4d), (24) and Table 1 yield:2
y =a3 + fH/r − x3/r + (3/4)Sgen/(σAeT 3
H)1/3
. (26)3
Eqs. (4a)–(4d), (25), (26) and Table 1 yield:4
ηrr = 1 + (ra4 − x4 − ry4)/fH . (27)5
The work extractor (Fig. 3) generates positive work rate under several constraints. First, AaϕH ≥ Aaϕa, i.e. x ≤ f 1/4H (we6
used Eq. (4b) and Table 1). Second, Aeϕe ≥ AeϕL, i.e. y ≥ a (we used Eqs. (4a), (4c) and Table 1). Since y ≥ a and Sgen ≥ 0,7
the upper limit of x is found:8
x ≤ (x)max = f 1/3H
≤ f 1/4H
. (28)9
Therefore, a ≤ y ≤ x ≤ f 1/3H .10
V. Badescu / Physica A xx (xxxx) xxx–xxx 7
The reversibility assumption Sgen = 0 is now adopted. Then, from Eq. (26) one sees that: 1
yrev =
a3 +
fHr
−x3revr
1/3
≤ y (29) 2
and the work extraction efficiency is a maximum, ηrr,max: 3
ηrr ≤ ηrr,max = 1 −x4revfH
+ra4
fH−
rfH
a3 +
fHr
−x3revr
4/3
. (30) 4
Here Eqs. (27) and (29) have been used. The equation ∂ηrr,max/∂xrev = 0 is solved by using Eq. (30) and the optimum reduced 5
absorber temperature xrev,opt is found: 6
xrev,opt =
fH + ra3
1 + r
1/3
(31a) 7
( = yrev,opt). (31b) 8
Here Eq. (29) has been also used, giving the optimum reduced emitter temperature yrev,opt . Usage of Eqs. (30) and (31a) 9
yields themaximum maximorum efficiency: 10
ηrr,max,max = 1 −1 + rfH
fH + ra3
1 + r
4/3
+ra4
fH(32) 11
ηrr,max,maxdepends on the design of the work extractor, specified by the parameter r . In the extreme case r → ∞, Eq. (32) 12
yields: 13
ηrr,max,max,max = 1 −43a +
13a4
fH
≥ ηrr,max,max
(33) 14
while Eqs. (31a) and (31b) become: 15
xrev,opt = yrev,opt (34a) 16
= a (34b) 17
ηrr,max,max,max Eq. (33) depends on the properties of the two reservoirs only, ηrr,max,max,max is the upper bound efficiency for 18
reversible operation of work extractors in contact with a pump radiation reservoir and a sink radiation reservoir. Further 19
comments and discussions may be found in Ref. [9]. 20
4.2. Endoreversible operation 21
The energy balance for the absorber is given by Eq. (14), where the unbalanced flux of thermal energy Qa is partially 22
converted into the work rate: 23
Qa − Qe = W (35) 24
where Qe is the heat flux transferred to the emitter. The energy balance for the emitter yields: 25
Qe = Ae(ϕe − ϕL). (36) 26
The entropy balance associated with the heat fluxes Qa and Qe is: 27
Qa
Ta+ S ′
gen =Qe
Te(37) 28
where the entropy generation rate S ′gen takes into account irreversible processes but not those associated with absorption 29
or emission of radiation at work extractor boundaries. 30
The work extractor efficiency η′rr is defined as in Eq. (7) and: 31
η′
rr ≡W
AaϕH=
1 −
TeTa
1 −
ϕa
ϕH
−
TLS ′gen
AaϕH. (38) 32
Here Eqs. (14), (35) and (37) have been used. The endoreversibility assumption S ′gen = 0 is now adopted: 33
η′
rr ≤ η′
rr,max ≡
1 −
yrev′xrev′
1 −
x4rev′fH
. (39) 34
8 V. Badescu / Physica A xx (xxxx) xxx–xxx
Here Eqs. (38), (4b), (4c) and Table 1 have been used while rev′ shows that the assumption of endoreversibility refers to1
those processes associated with S ′gen. The work extractor generates positive work rate when:2
0 ≤ a ≤ yrev′ ≤ xrev′ ≤ f 1/4H ≤ 1. (40)3
When S ′gen = 0, Eq. (37) becomes:4
yrev′
1 −
x4rev′fH
− xrev′
rfH
y4rev′ − a4
= 0 (41)5
where Eqs. (4), (15), (36) and Table 1 have been used.6
η′rr,max Eq. (39) and constraint equation (41) may be used to obtain upper bound efficiencies for endoreversible operation7
of work extractors in contact with a pump radiation reservoir and a sink radiation reservoir. Two approaches are considered8
in the next Sections 4.2.1 and 4.2.2, respectively.9
4.2.1. Analytical upper bound efficiency10
Eq. (40) states that yrev′ ≥ a. Therefore, from Eq. (39) one sees that11
η′
rr ≤ η′
rr,max ≤ η′
rh,max ≡
1 −
axrev′
1 −
x4rev′fH
(42)12
where η′
rh,max has been already defined in Eq. (18). But η′
rh,max ≤ η′ (see Eq. (22)). Thus:13
η′
rr ≤ η′
rr,max ≤ η′
rh,max ≤ η′≡ 1 −
43
a
f 1/4H
+
13
a
f 1/4H
4
. (43)14
Therefore, η′ Eq. (43), earlier obtained as Eq. (22), is an (analytical) upper bound efficiency for the endoreversible operation15
of work extractors in contact with a pump radiation reservoir and a sink radiation reservoir.16
4.2.2. Numerical upper bound efficiency17
Eq. (39) is maximized numerically under the constraints Eqs. (40) and (41). The optimum value xrev′,opt and yrev′,opt are18
found and the resulted quantity η′rr,max,max constitutes the upper bound efficiency for endoreversible operation of work19
extractors in contact with a pump radiation reservoir and a sink radiation reservoir.20
4.3. Discussion21
The endoreversible upper bound efficiencies η′rr,max,max (obtained by the constrainedmaximization of Eq. (39)) are shown22
in Figs. 4 and 5, together with the simple analytical endoreversible upper bound efficiency η′ Eq. (43) and the reversible23
upper bound ηrr,max,max Eq. (32). These efficiencies are positive for a limited range of variation of a, depending on efficiency24
and value of fH . All these efficiencies decrease by increasing the ratio a ≡ TL/TH (Fig. 4). The endoreversible upper bound25
efficiencies η′rr,max,max and η
′ generally increase by increasing the pump geometric factor fH (Fig. 5a,b). The reversible upper26
bound efficiency ηrr,max,max decreases by increasing fH .27
The Carnot efficiency ηC = 1−a is numerically larger than themaximumefficiencies predicted by Eqs. (39), (43) and (32),28
respectively (see Figs. 4 and 5). However, Eqs. (39), (43) and (32), give the upper bound efficiencies, as explained when Fig. 229
has been discussed. The most accurate (i.e., lower) upper bound efficiency is the quantity η′rr,max,max obtained numerically.30
The simple analytical formula η′ provides the second best accurate upper bound efficiency. The reversible upper bound31
ηrr,max,max is less accurate, as expected.32
5. Conclusions33
Radiation reservoirs are more complex than heat reservoirs. Thermodynamic equilibrium between two bodies emitting34
and receiving radiation requires equality of their intensive thermodynamic parameters (such as temperature, pressure, and35
chemical potential) as well as equality of the other state parameters (such as the geometric factors). Here we considered the36
simplest case of radiation reservoir (i.e. blackbody isotropic radiation) and the simplest case of absorber/emitter (i.e. plane37
Lambertian blackbody).38
The main conclusions of this paper are:39
1. The upper bound for reversible work extraction from radiation reservoirs is not Carnot efficiency.40
2. The upper bound for reversible work extraction from radiation reservoirs is ηrh,max .max Eq. (13), when the sink is a heat41
reservoir, and ηrr,max,max,max Eq. (33), when the sink is a radiation reservoir. Note that the right hand side member of Eq.42
(13), and Eq. (33), respectively, is identical.43
V. Badescu / Physica A xx (xxxx) xxx–xxx 9
Fig. 4. Upper bound conversion efficiencies as a function of a ≡ TL/TH in case of a pump radiation reservoir and a sink radiation reservoir. Pump geometricfactor fH = 0.01; (a) r ≡ Ae/Aa = 1; (b) r = 10. The Carnot efficiency 1 − a is also shown.
Fig. 5. Upper bound conversion efficiencies as a function of the pump geometric factor fH in case of a pump radiation reservoir and a sink radiationreservoir. (a) r ≡ Ae/Aa = 10 and a ≡ TL/TH = 0.02; (b) r = 10 and a = 0.16. The Carnot efficiency 1 − a is also shown.
3. When endoreversible operation of work extractors in contact with a pump radiation reservoir and a sink heat reservoir 1
is considered, the most accurate upper bound efficiency is obtained by constraint numerical maximization of η′
rh,max 2
Eq. (18). This upper bound depends on the pump geometric factor fH . The results reported here are new. 3
4. When endoreversible operation of work extractors in contact with a pump radiation reservoir and a sink radiation 4
reservoir is considered, the maximum efficiency is obtained by constraint numerical maximization of η′rr,max Eq. (39). 5
It depends on the pump geometric factor fH and the design of the work extractor specified by the ratio r ≡ Ae/Aa. 6
The most accurate upper bound efficiency depending only on the properties of the radiation reservoirs is obtained by 7
maximization of Eq. (39) in the extreme case r → ∞. The results reported here are new. 8
10 V. Badescu / Physica A xx (xxxx) xxx–xxx
5. The simple analytical formula η′ given by Eq. (22) (or Eq. (43)) may be used as an upper bound efficiency for work1
extraction from radiation reservoirs when the sink is a heat reservoir (this result has been reported in Ref. [12]) and2
when the sink is a radiation reservoir (new result obtained here). It does not depend on the design of the work extractor3
but only on the properties of the two radiation reservoirs.4
Acknowledgments5
The author thanks the reviewers for useful comments and suggestions.6
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