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Endoreversible terrestrial solar energy conversion
ARTICLE IN PRESS Renewable Energy 33 (2008) 2631– 2636
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Renewable Energy journal homepage: www.elsevier.com/locate/renene
Endoreversible terrestrial solar energy conversion S.E. Wright University of Colorado at Denver and Health Sciences Center (UCDHSC), Campus Box 112, P.O. Box 173364, Denver, CO 80217-3364, USA
a r t i c l e in fo
abstract
Article history: Received 31 October 2006 Accepted 21 February 2008 Available online 18 April 2008
The analysis of reversible radiation conversion is insightful in process optimization and provides the upper limit to performance of any conceptual device. However, in practice, irreversibilities are unavoidable and play an important role in performance optimization. The finite absorption of a radiative source flux, and the simultaneous emission of radiation, is an inherently irreversible process. Likewise, heat rejection from the conversion device is unavoidable and dependent on local environmental conditions and resources. The endoreversible model treats these irreversibilities as inherent but external to the conversion process. In this paper, the effect of these irreversibilities on performance is investigated for a model with irreversible radiative absorption combined with irreversible conductive or convective heat rejection. Previous models either do not include the magnitude of entropy rejection required by the second law or do not otherwise accurately represent terrestrial solar conversion. Analysis of the model provides a guide for optimal operating conditions. For relatively poor heat rejection, as can occur in arid climates, analysis of the model reveals regions of operating parameters that will result in zero or low theoretical work output, and/or high operating temperatures and increased risk of failure. This is of particular concern for photovoltaic systems with low maximum temperature limits. Analytical expressions for maximum ideal work are provided, given the specific radiative source flux and heat rejection conditions. A fair evaluation of system performance is thereby obtained by comparing actual work production to the ideal, given the specific external operating conditions or restrictions. Published by Elsevier Ltd.
Keywords: Endoreversible Solar engineering Energy conversion Entropy Second law Radiation
1. Introduction Efficiency evaluation and optimization play a vital role in developing cost-effective solar energy conversion devices. In the field of solar engineering, performance evaluation is based almost entirely on energy (first law) principles. However, a first law approach to performance evaluation and optimization is of limited effectiveness and at times misleading. For example, energy-based efficiencies can be misleading as they compare actual work output to an upper limit that is not even theoretically obtainable, the energy flux of the incident solar radiation. The second law of thermodynamics is vital as it determines the absolute upper limit to the performance of solar energy conversion devices, a limit that can be far lower than the energy of the source flux [1]. For practical operation of a conversion device, irreversibility rate calculations provide the rate of entropy production or exergy destruction caused by various losses in the conversion process. The purpose of the present analysis is to investigate the
Abbreviation: BR, blackbody radiation; DBR, diluted blackbody radiation; GR, graybody radiation; HE, heat engine; SR, solar radiation; TR, thermal radiation. Tel.: +1 303 556 8348; fax: +1 303 556 6371. E-mail address: [email protected] 0960-1481/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.renene.2008.02.021
irreversibilities associated with endoreversible (or internally reversible) solar energy conversion. That is, to consider the external irreversibilities associated with the non-ideal interaction of a solar energy conversion device with its surroundings. Energy interactions for solar energy conversion include the absorption of source radiation, simultaneously emission of thermal radiation, and heat rejection to a low temperature sink. Analysis of endoreversible operation of heat engines (introduced by Curzorn and Ahlborn [2]) has led to a number of important results and provides an illustrative example of the benefits of endoreversible analysis. The second law imposes a theoretical upper limit on the efficiency of any heat engine (Carnot efficiency). However, under practical considerations, the Carnot engine is viewed as a zero power device because irreversibilities in a real heat engine result in an efficiency less than the Carnot efficiency when the work output is non-zero. For both thermal and quantum solar energy conversion processes the entropy of the absorbed radiation source flux, in addition to any entropy increase due to entropy production, cannot be destroyed and must be rejected to the low-temperature heat/entropy sink. Inherent radiation emission by the conversion device carries entropy from the device but it is not of sufficient magnitude [3]. As a results heat/entropy rejection to the lowtemperature sink by conduction or convection is required.
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Nomenclature A g r I˙ JS k Q˙ q R T ˙ W w x
area (m2) thermal conductance (W/K or W/K4 for radiative transfer) constant in (22) irreversibility rate (W) entropy flux (W/m2 K) thermal conductivity (W/m K) heat transfer rate (W) heat flux (W/m2) thermal resistance (K/W or K4/W for radiative transfer temperature (K) work transfer rate (W) work output rate per unit area of source radiation absorption (W/m2) dimensional quantity (m)
In general, solar energy conversion models considered in the literature either do not have sufficient entropy rejection required by the second law of thermodynamics or do not accurately represent terrestrial solar energy conversion. The present analysis considers endoreversible solar energy conversion with irreversible radiative transfer from the source and irreversible heat conduction or convection to the low-temperature sink. The effect of these external irreversibilities on maximum power optimization is investigated.
graybody radiation emittance energetic efficiency entropy production rate (W/K) entropy production rate per unit volume (W/m3 K) Stefan–Boltzmann constant ¼ (5.67)108 W/m2 K4
e Z _ P p_ s
Subscripts A B CA H L o opt
Absorber Heat rejection temperature Curzorn–Ahlborn High-temperature source Low-temperature sink Environment or atmosphere conditions Optimal
_ temperature (To) and the entropy production rate (P), _I ¼ T o P _
(2)
For the endoreversible heat engine the entropy production rates are _ H ¼ Q_ H 1 1 and P _ L ¼ Q_ L 1 1 P (3) TA TH TL TB and the irreversibility rates, using (1), are given by 2
2
_IH ¼ T o ðT H T A Þ and _IL ¼ T o ðT B T L Þ RH T A T H RL T B T L
2. Background In the endoreversible model the heat engine is internally reversible, so all irreversibilities are due to external heat transfer from the high-temperature source (TH) and heat rejection to the low temperature sink (TL). The external irreversibilities are modeled by thermal resistances to heat transfer as depicted in Fig. 1. The thermal resistances are defined by the relations TH TA TB TL and Q_ L ¼ Q_ H ¼ RH RL
(1)
where the units of thermal resistance for heat conduction and convection are K/W. The non-zero thermal resistances result in entropy production (irreversibility). The irreversibility rate ð_IÞ is determined using the Guoy–Stodola theorem (see for example, Bejan [4]), which shows that the irreversibility is simply equal to the product of the environment
TH
QH
RH Internally Reversible Conversion
TA
WOut
TB RL
QL
TL Fig. 1. Endoreversible heat engine.
(4)
For a real heat engine with non-zero thermal resistances, maximum efficiency operation occurs when all the energy flows are infinitesimal and operation is reversible. This maximum efficiency limit is given by the Carnot efficiency expression ZC ¼
_ Out TL W ¼1 TH Q_
(5)
H
The reversible Carnot efficiency for ideal operation is less than unity because the entropy flow from the source flux (Q˙H/TH) cannot be destroyed and is diverted to the low temperature sink such that Q˙H/TH ¼ Q˙L/TL. The rejected energy flow Q˙L is thus the product of Q˙H/TH and TL, that is, TL(Q˙H/TH). In contrast, the efficiency for maximum power operation, with RH ¼ RL, is given by Curzorn-Ahlborn [2]: 1=2 TL ZCA ¼ 1 (6) TH where the subscript CA denotes Curzorn–Ahlborn. For fixed and equivalent thermal resistances (RH and RL), as the heat flux qH increases the efficiency (Z) decreases, and as a result there is an optimum operating point that delivers maximum power. For maximum power operation the Curzorn–Ahlborn efficiency (6) is independent of the magnitude of the thermal resistances (RH ¼ RL), and is less than the maximum possible theoretical efficiency (5). In the present analysis of the solar energy conversion model, we are concerned with the specific character and role of the thermal resistances to heat transfer to and from the conversion devices. If the external heat flux is due to heat conduction we have the general form (Fourier’s law for one dimensional heat conduction) q ¼ k
dT dx
(7)
ARTICLE IN PRESS S.E. Wright / Renewable Energy 33 (2008) 2631–2636
where k is the thermal conductivity. The associated entropy flux is given by the relation q k dT JS ¼ ¼ T T dx
(8)
where T is the local material temperature. For steady-state operation (fixed q), the local entropy production rate per unit _ using (7) and (8) is volume (p) 2 d q dT k dT p_ ¼ JS ¼ 2 ¼ 2 dx dx dx T T
then the thermal resistances are simply 1 RH ¼ g 1 ; H ¼ ðsH AH Þ
1 RL ¼ g 1 L ¼ ðsL AL Þ
(12)
where AH and AL are the heat transfer areas at the high- and lowtemperature sides of the conversion device.
3. Endoreversible model for terrestrial solar radiation conversion
(9)
The entropy production rate (9) is only zero when the temperature gradient is infinitesimal or zero. Although this is shown for one-dimensional heat conduction, it is true for any type of heat conduction or convection. Consequently, for a finite heat flux the thermal conductivity must be infinite to have a zero temperature gradient and zero entropy production. This can be seen by re-stating the entropy production rate (9), using (7), as k q2 q2 1 p_ ¼ þ 2 (10) ¼þ 2 k k T T In contrast, for a finite heat transfer rate, finite thermal conductivity results in a thermal resistance and entropy production. De Vos [5] compares a number of solar energy conversion models with and without thermal resistance to heat transfer (external irreversibilities). These models include the solar cell model, the endoreversible Stefan–Boltzmann engine, and the Muser engine [6]. The solar cell model includes radiation absorption and emission without external irreversibilities. The Muser engine includes irreversible radiation transfer from the source and reversible heat conduction to the low temperature sink, whereas the endoreversible Stefan–Boltzmann engine models irreversible radiation transfer from the source as well as to the sink. The Stefan–Boltzmann engine is representative of solar energy conversion for extraterrestrial applications where heat conduction or convection to a low-temperature sink is not possible. The endoreversible Stefan–Boltzmann engine is depicted in Fig. 2. The radiative heat transfer rates are specified in the general form Q_ H ¼ g H ðT 4H T 4A Þ;
2633
Q_ L ¼ g L ðT 4B T 4L Þ
(11)
where the thermal conductances gH and gL are the inverse of the thermal resistances for the high- and low-temperature sides of the conversion device, respectively. Assuming the emissivity of the heat transfer surfaces at TA and TB are eH and eL, respectively, that the source radiation from TH is isotropic blackbody radiation (BR), and that the sink at temperature TL behaves as a blackbody,
For both thermal and quantum conversion processes there must be heat/entropy rejected to the environment (Q˙L) even for ideal reversible operation because entropy is absorbed from the source flux. For non-ideal conversion, the required heat/entropy rejection is increased due to entropy production. The radiation emission by the absorbing material carries entropy from the device but it is not of sufficient magnitude. Consequently, there must be heat rejection by conduction or convection to the lowtemperature heat sink for both thermal and quantum conversion devices even when they are ideal. Heat rejection by conduction and convection in practice is an irreversible process. In this analysis we consider the case where the engine has irreversible radiative transfer from the source and irreversible heat conduction or convection to the low-temperature sink. Fig. 3 depicts the conversion model considered in this analysis. The heat transfer rate to the low temperature sink may be expressed as Q_ L ¼ g L ðT B T L Þ
(13)
where gL is the heat rejection conductance by convection or conduction. For example, gL ¼ hAL for heat rejection by convection. The power flux is the product of the energy flux from the source (qH) and the efficiency (Z), that is, w ¼ ZqH. Applying the first law of thermodynamics, the work output rate is given by _ Out ¼ g ðT 4 T 4 Þð1 T B =T A Þ W H H A
(14)
Application of the second law of thermodynamics to the reversible portion of the conversion device allows the temperature TB to be expressed as " TB ¼ TL 1
g H ðT 4H T 4A Þ TA gL
#1 (15)
For the endoreversible solar energy conversion model the emissivity eH of the absorber is a primary variable that affects the work output rate. However, analogous to the Curzorn–Ahlborn analysis, the temperature of the absorber will also be taken in this analysis as an independent variable. TH
TH
Radiative: Radiative: Internally
QH = gH (TH4 − TA4 ) TA
QH = gH (TH4 − TA4 ) TA
Reversible
Reversible
WOut
Conversion Device
Internally
Device TB
WOut
Conversion TB
Heat Rejection:
Radiative:
QL = gL (TB − TL )
QL = gL (TB 4− TL 4) TL Fig. 2. Stefan–Boltzmann engine.
TL Fig. 3. Endoreversible conversion with irreversible heat rejection by conduction or convection.
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Assuming the absorbing material behaves as a graybody with emissivity eH equal to absorptivity aH, the absorbed heat flow from the incident, concentrated, diluted blackbody radiation (DBR) flux is given by (16)
where TH is an effective source temperature (T 4H ¼ E_ Inc =s). The heat transfer rate to the low-temperature sink is given by (13). Using (15), the efficiency1 can be expressed as 1 TB g ¼ 1 T L T A H ðT 4H T 4A Þ (17) Z¼1 TA gL
0.6 Efficiency
Q_ H ¼ AH qH ¼ sH AH ðT 4H T 4A Þ
0.8
0.4 0.2 0 1
This efficiency provides a limit for both quantum and thermal conversion processes. For quantum conversion, the temperature TA represents the emission temperature and the temperature TB is the heat or entropy rejection (lattice) temperature.
900 0.5
600
Emissivity 0
4. Analysis and results
(18)
must be specified from practical considerations of heat transfer parameters, material properties and geometric considerations. The range of the conductance constant considered in this analysis is from r ¼ 5E9 K3 for poor heat rejection conditions, to r ¼ 1E11 K3 for good heat rejection conditions. The r-value of 5E9 K3 represents, for example, an overall convective heat transfer coefficient of 140 W/m2 K with an area ratio AL/AH of 2. The r-value of 1E11 K3 for good heat rejection represents, for example, an overall thermal conductivity of 300 W/m K with a Dx of 0.26 m and an area ratio AL/AH of 5. Using the definition of the conductance constant (18), the efficiency (17) may be expressed as Z ¼ 1 T L ½T A H ðT 4H T 4A Þ=r1
(19)
The work output rate for this model is the product of the heat flux (16) and the efficiency (19). The work output rate w from the endoreversible model, per unit absorption area AH, is " # ðT A T L Þ H ðT 4H T 4A Þ=r w ¼ sH ðT 4H T 4A Þ (20) T A H ðT 4H T 4A Þ=r Figs. 4 and 5 depict the variation of the efficiency, and the work output rate per unit absorption area, with the emissivity eH and the absorber temperature TA for r ¼ 5E9 K3. Note, that this value of the constant r represents the case of relatively poor heat rejection (ex. h ¼ 140 W/m2 K, AL/AH ¼ 2). The efficiency clearly increases with increasing absorber temperature and/or decreasing emissivity, as seen in Fig. 4. In both instances, this occurs because the energy fluxes through the conversion device decrease, allowing in turn a farther spread between the absorber temperature TA and the heat rejection temperature TB. In contrast to the efficiency, the heat flux through 1 This efficiency represents the part of the conversion process operating between TA and TB. For example, this efficiency applies to the thermal power plant performance not including the optical concentration process.
15000 Work Output (W/m2)
gL g H =H
Absorber Temperature (K)
Fig. 4. The efficiency Z (19) versus emissivity (eH) and absorption temperature (TA) with r ¼ 5E9 K3, TH ¼ 900 K and TL ¼ 300 K.
To obtain numeric plots we must specify values for a number of parameters. The effective source temperature will be taken as TH ¼ 900 K. This is representative of medium-level concentration of typical, unconcentrated, terrestrial solar fluxes. For example, this effective temperature represents an unconcentrated flux of 800 W/m2 with a concentration factor of 45. The ambient heat rejection temperature is taken as TL ¼ 300 K. The numeric value of the conductance constant (r), defined as, r¼
300
T > 508.5 K for region III
10000
5000 III
II
0 1
900 I
0.5
Emissivity
600 0
300
Absorber Temperature (K)
Fig. 5. The work output w per unit absorption area versus emissivity and the absorber temperature (TA) with r ¼ 5E9 K3, TH ¼ 900 K and TL ¼ 300 K.
the conversion device is maximized for low absorber temperatures and/or high emissivities/absorptivities. This competition results in three distinct regions with respect to work optimization, as shown in Fig. 5:
Region I: W˙ ¼ 0 and Z ¼ 0 for all aH (note: aH ¼ eH). Region II: W˙max occurs at aHo1. Region III: W˙max occurs at aH ¼ 1. In region I, the energy flows through the conversion device are such that there is no distinction between the absorber temperature TA and the required heat rejection temperature TB (15), and consequently the result is zero efficiency and work output. As thermal conductance to the low-temperature sink decreases, the heat rejection temperature TB increases, thus reducing the efficiency of the conversion process. In region II, there is some distinction between the absorber temperature and the heat rejection temperature. Note that in Fig. 5 the boundary between region II and III is given by TA ¼ 508.5 K (235.2 1C). Positive work production is possible in region II, but maximum work occurs with absorption of only a fraction of the incident SR flux (aHo1). Higher absorption rates increase the heat flux while decreasing the efficiency to such an extent that the work production decreases.
ARTICLE IN PRESS S.E. Wright / Renewable Energy 33 (2008) 2631–2636
_ max W
pffiffiffiffiffiffi pffiffiffiffiffi2 TA TL ¼ gL
5000
0 1 900 0.5
(21)
and the efficiency concisely is sffiffiffiffiffiffi TL Z¼1 TA
(22)
(23)
pffiffiffiffiffiffi pffiffiffiffiffi2 _ max W ¼ sr TA TL AH
(24)
For a fixed absorber emissivity, on the other hand, maximum work always occurs at some optimum value of the absorber temperature, as can be seen in Fig. 5. From the comparison of Figs. 4 and 5, one may also conclude that efficiency and power optimizations always refer to distinctly different operating conditions. Fig. 6 depicts the variation of the work output rate per unit absorption area, for cases of good heat rejection. In this plot, it can be seen that maximum work generally occurs at eH ¼ 1 (region III). The small or absent region II indicates that, for systems with strict temperature limitations, lower operating temperatures may be realized with positive work production. The magnitude of the irreversibility on the cold side of the conversion device, due to heat rejection, can be compared to the irreversibility due to radiation absorption and emission by considering the ratio
Rad
0
300
Absorber Temperature (K)
occupy the same hemisphere of directions. Treating the incident radiation as isotropic GR with an emission temperature TS, and a non-uniform spectral emissivity en, the incident entropy flux is given by Planck’s entropy formula [7] Z 1 4 45 S_ Inc ¼ sT 3S AH xS ½ð1 þ f I Þ lnð1 þ f I Þ f I lnðf I Þ dxS (27) 3 4p4 0 where f I ¼ u =ðexS 1Þ
(28)
xS ¼ hn/kTS, and n is frequency. The outgoing entropy flux due to emission and reflection is similarly, Z 1 4 45 S_ Ref ¼ sT 3A AH xA ½ð1 þ f RE Þ lnð1 þ f RE Þ f RE lnðf RE Þ dxA 4 3 4p þEmi 0 (29)
As can be seen in Fig. 5, for case III (TA4508.5 K) maximum work occurs at eH ¼ 1. The magnitude of maximum work in this case is " # TL (25) w ¼ sðT 4H T 4A Þ 1 T A ðT 4H T 4A Þ=r
_ICS p_ CS Q_ L =T L Q_ H =T A ¼ ¼ _IHS p_ HS Q_ =T A þ S_ Emi; S_ Inc; H
600
Fig. 6. The work output w per unit absorption area versus emissivity and the absorber temperature (TA) with r ¼ 1E11 K3, TH ¼ 900 K and TL ¼ 300 K.
Notably, the efficiency (23) does not depend on the thermal conductances from the source, or to the low-temperature sink, even if these conductances are unequal in magnitude. The maximum work output (21) can also be expressed on a per unit absorber area basis as
RI
10000
Emissivity
where the optimum emissivity eH is given by pffiffiffiffiffiffiffiffiffiffiffi T TATL H;opt ¼ A 4 ðT H T 4A Þ=r
wmax ¼
15000 Work Output (W/m2)
These observations regarding regions I and II reveal the importance of good heat rejection for viable system designs, as well as the fair evaluation of actual performance of a device with respect to what can be theoretically obtained by any system given the specific heat rejection conditions. Poor heat rejection is particularly a problem when water is in short supply so that heat must be rejected to atmospheric air, particularly if water resources are insufficient to the point that evaporative cooling is not possible. This is especially a problem in designs for arid or desert regions. Poor heat rejection conditions can result in zero or low work output, and/or higher operating temperatures and risk of failure. This is of particular concern to photovoltaic systems that have low maximum temperature limits. In case II, for accurate performance evaluation the maximum possible work output can be expressed by the analytical result
2635
(26)
where f RE ¼ ð1 n Þ=ðexS 1Þ þ =ðexA 1Þ
and where xA ¼ hn/kTA. However, the magnitude of the entropy fluxes of the incoming and outgoing radiation cannot be generally determined using (27) and (29) as they depend on the spectral and directional distribution of the terrestrial source radiation, and on the concentration process. The entropy-to-energy ratio of DBR is not the same as BR with the same emission temperature. However, the entropy of the incident flux can be approximated as that of the entropy of BR with the same energy flux, but lower emission temperature [8]. For typical terrestrial SR, this approximation generally results in less than 10% overestimate of the incident entropy flux. With this approximation, and limiting the analysis to region III so that the emissivity is unity, the net incident entropy flux can be simply estimated as 4 S_ Inc S_ Emi ¼ sAH ðT 3H T 3A Þ 3
(31)
where TH is the effective source temperature of the concentrated DBR source radiation. By defining y as the ratio of the absorber temperature to that of the effective temperature of the incident source radiation, y ¼ TA/TH, using Eqs. (11) and (25), and after lengthy algebraic manipulation, the irreversibility ratio (26) can be expressed in the following simplified form:
Rad
The entropies of the reflected and emitted fluxes are not independent because their energy spectra overlap and they
(30)
RI ¼
ð1 y4 Þ2 h i ½1 43y þ 13y4 ry=T 3H ð1 y4 Þ
(32)
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It is worthwhile to note that minimizing the entropy production rate in the conversion device does not result in maximum work production. Fig. 8 shows a plot of the non-dimensionalized ˙ /sAHT4H) and the total entropy production rate work .output (W (p_ total sAH T 3H ) as functions of the temperature ratio y, with r ¼ 1E11 K3. The entropy production rate decreases with increasing y, and the minimum occurs at y ¼ 1, while maximum work occurs at y ¼ 0.6748 (TA ¼ 607.3 K for TH ¼ 900 K).
1.4
The Irreversibility ratio RI
1.2 r = 5E9 K3 1 0.8 gL increasing
0.6 0.4
r = 2.5E10 K3
4. Conclusions
0.2 r = 1E11 K3 0 0.35
0.45
0.55 0.65 0.75 The temperature ratio (θ θ=TA/TH)
0.85
0.95
Fig. 7. The irreversibility ratio RI in (28) versus the temperature ratio y ¼ TA/TH, for maximum work output, TH ¼ 900 K and TL ¼ 300 K.
1.80 1.60
Total Entropy Production Rate
Non-Dimensional
1.40 1.20 1.00 0.80 0.60
Work Output Rate
0.40 0.20 0 0.3
0.4
0.5 0.6 0.7 0.8 The temperature ratio θ = TA/TH
0.9
1
The finite absorption of a radiative source flux, and the simultaneous emission of radiation, is an inherently irreversible process. Likewise, heat rejection from the process is unavoidable and depends partly on local environmental conditions and resources. The endoreversible model treats these irreversibilities as inherent but external to the conversion process. The endoreversible model considered in this analysis represents typical solar radiation conversion and incorporates the necessary level of entropy rejection required by the second law of thermodynamics. The model provides a guide for optimal absorber temperature and heat fluxes for maximizing work production. For relatively poor heat rejection such as in arid climates, analysis of the model reveals regions of operating parameters that will result in zero or low theoretical work output, and/or high operating temperatures and risk of failure. This is of particular concern to photovoltaic systems that have low maximum temperature limits. This model also provides analytical expressions and numeric plots of the maximum ideal work limits for a conversion device operating with specific radiative source flux and heat rejection conditions. A fair evaluation of system performance is thereby obtained by comparing actual work production in comparison to the ideal, given the specific operating conditions or restrictions.
Fig. 8. The non-dimensionalized work output and total entropy production rate versus the temperature ratio y, for TH ¼ 900 K and r ¼ 1E11 K3.
References
Fig. 7 depicts the irreversibility ratio RI, for maximum work output, as a function of the temperature ratio y, for three values of the heat conductance ratio r. The plot shows that, for good heat rejection (high r-value), the irreversibility rate due to thermal radiation (TR) absorption dominates the calculation of the total irreversibility rate. However, for relatively poor heat rejection conductance, r ¼ 5E9 K3 (ex. h ¼ 140 W/m2 K, AL/AH ¼ 2), the irreversibility due to heat rejection can be greater than that due to radiation absorption. Under these conditions the work output of the device can be very low or zero, and the operating temperature of the device can be high enough to cause failure, particularly with photovoltaic systems that have strict temperature limitations.
[1] Wright SE, Rosen MA. Exergetic efficiencies and the exergy content of terrestrial solar radiation. ASME J Solar Energy Eng 2004;126:673–6. [2] Curzon F, Ahlborn B. Efficiency of a carnot engine at maximum power output. Am J Phys 1975;43:22–4. [3] Wright SE, Rosen MA, Scott DS, Haddow JB. The exergy flux of radiative heat transfer for the special case of blackbody radiation. Exergy Int J 2002;2(1): 24–33. [4] Bejan A. Advanced engineering thermodynamics. 3rd ed. New Jersey: Wiley; 2006. [5] De Vos A. Endoreversible thermodynamics of solar energy conversion. 1st ed. New York: Oxford University Press; 1992. [6] Muser H. Behandlung von Elektronenprozessen in Halbleiter Randschichten. Zeitschr Phys 1957;148:380–90. [7] Planck M. Translation by Morton Mausius, The theory of heat radiation. New York: Dover Publications; 1914. [8] Wright SE, Scott DS, Rosen MA, Haddow JB. On the entropy of radiative heat transfer in engineering thermodynamics. Int J Eng Sci 2001;39:1691–706.
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Renewable Energy journal homepage: www.elsevier.com/locate/renene
Endoreversible terrestrial solar energy conversion S.E. Wright University of Colorado at Denver and Health Sciences Center (UCDHSC), Campus Box 112, P.O. Box 173364, Denver, CO 80217-3364, USA
a r t i c l e in fo
abstract
Article history: Received 31 October 2006 Accepted 21 February 2008 Available online 18 April 2008
The analysis of reversible radiation conversion is insightful in process optimization and provides the upper limit to performance of any conceptual device. However, in practice, irreversibilities are unavoidable and play an important role in performance optimization. The finite absorption of a radiative source flux, and the simultaneous emission of radiation, is an inherently irreversible process. Likewise, heat rejection from the conversion device is unavoidable and dependent on local environmental conditions and resources. The endoreversible model treats these irreversibilities as inherent but external to the conversion process. In this paper, the effect of these irreversibilities on performance is investigated for a model with irreversible radiative absorption combined with irreversible conductive or convective heat rejection. Previous models either do not include the magnitude of entropy rejection required by the second law or do not otherwise accurately represent terrestrial solar conversion. Analysis of the model provides a guide for optimal operating conditions. For relatively poor heat rejection, as can occur in arid climates, analysis of the model reveals regions of operating parameters that will result in zero or low theoretical work output, and/or high operating temperatures and increased risk of failure. This is of particular concern for photovoltaic systems with low maximum temperature limits. Analytical expressions for maximum ideal work are provided, given the specific radiative source flux and heat rejection conditions. A fair evaluation of system performance is thereby obtained by comparing actual work production to the ideal, given the specific external operating conditions or restrictions. Published by Elsevier Ltd.
Keywords: Endoreversible Solar engineering Energy conversion Entropy Second law Radiation
1. Introduction Efficiency evaluation and optimization play a vital role in developing cost-effective solar energy conversion devices. In the field of solar engineering, performance evaluation is based almost entirely on energy (first law) principles. However, a first law approach to performance evaluation and optimization is of limited effectiveness and at times misleading. For example, energy-based efficiencies can be misleading as they compare actual work output to an upper limit that is not even theoretically obtainable, the energy flux of the incident solar radiation. The second law of thermodynamics is vital as it determines the absolute upper limit to the performance of solar energy conversion devices, a limit that can be far lower than the energy of the source flux [1]. For practical operation of a conversion device, irreversibility rate calculations provide the rate of entropy production or exergy destruction caused by various losses in the conversion process. The purpose of the present analysis is to investigate the
Abbreviation: BR, blackbody radiation; DBR, diluted blackbody radiation; GR, graybody radiation; HE, heat engine; SR, solar radiation; TR, thermal radiation. Tel.: +1 303 556 8348; fax: +1 303 556 6371. E-mail address: [email protected] 0960-1481/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.renene.2008.02.021
irreversibilities associated with endoreversible (or internally reversible) solar energy conversion. That is, to consider the external irreversibilities associated with the non-ideal interaction of a solar energy conversion device with its surroundings. Energy interactions for solar energy conversion include the absorption of source radiation, simultaneously emission of thermal radiation, and heat rejection to a low temperature sink. Analysis of endoreversible operation of heat engines (introduced by Curzorn and Ahlborn [2]) has led to a number of important results and provides an illustrative example of the benefits of endoreversible analysis. The second law imposes a theoretical upper limit on the efficiency of any heat engine (Carnot efficiency). However, under practical considerations, the Carnot engine is viewed as a zero power device because irreversibilities in a real heat engine result in an efficiency less than the Carnot efficiency when the work output is non-zero. For both thermal and quantum solar energy conversion processes the entropy of the absorbed radiation source flux, in addition to any entropy increase due to entropy production, cannot be destroyed and must be rejected to the low-temperature heat/entropy sink. Inherent radiation emission by the conversion device carries entropy from the device but it is not of sufficient magnitude [3]. As a results heat/entropy rejection to the lowtemperature sink by conduction or convection is required.
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S.E. Wright / Renewable Energy 33 (2008) 2631–2636
Nomenclature A g r I˙ JS k Q˙ q R T ˙ W w x
area (m2) thermal conductance (W/K or W/K4 for radiative transfer) constant in (22) irreversibility rate (W) entropy flux (W/m2 K) thermal conductivity (W/m K) heat transfer rate (W) heat flux (W/m2) thermal resistance (K/W or K4/W for radiative transfer temperature (K) work transfer rate (W) work output rate per unit area of source radiation absorption (W/m2) dimensional quantity (m)
In general, solar energy conversion models considered in the literature either do not have sufficient entropy rejection required by the second law of thermodynamics or do not accurately represent terrestrial solar energy conversion. The present analysis considers endoreversible solar energy conversion with irreversible radiative transfer from the source and irreversible heat conduction or convection to the low-temperature sink. The effect of these external irreversibilities on maximum power optimization is investigated.
graybody radiation emittance energetic efficiency entropy production rate (W/K) entropy production rate per unit volume (W/m3 K) Stefan–Boltzmann constant ¼ (5.67)108 W/m2 K4
e Z _ P p_ s
Subscripts A B CA H L o opt
Absorber Heat rejection temperature Curzorn–Ahlborn High-temperature source Low-temperature sink Environment or atmosphere conditions Optimal
_ temperature (To) and the entropy production rate (P), _I ¼ T o P _
(2)
For the endoreversible heat engine the entropy production rates are _ H ¼ Q_ H 1 1 and P _ L ¼ Q_ L 1 1 P (3) TA TH TL TB and the irreversibility rates, using (1), are given by 2
2
_IH ¼ T o ðT H T A Þ and _IL ¼ T o ðT B T L Þ RH T A T H RL T B T L
2. Background In the endoreversible model the heat engine is internally reversible, so all irreversibilities are due to external heat transfer from the high-temperature source (TH) and heat rejection to the low temperature sink (TL). The external irreversibilities are modeled by thermal resistances to heat transfer as depicted in Fig. 1. The thermal resistances are defined by the relations TH TA TB TL and Q_ L ¼ Q_ H ¼ RH RL
(1)
where the units of thermal resistance for heat conduction and convection are K/W. The non-zero thermal resistances result in entropy production (irreversibility). The irreversibility rate ð_IÞ is determined using the Guoy–Stodola theorem (see for example, Bejan [4]), which shows that the irreversibility is simply equal to the product of the environment
TH
QH
RH Internally Reversible Conversion
TA
WOut
TB RL
QL
TL Fig. 1. Endoreversible heat engine.
(4)
For a real heat engine with non-zero thermal resistances, maximum efficiency operation occurs when all the energy flows are infinitesimal and operation is reversible. This maximum efficiency limit is given by the Carnot efficiency expression ZC ¼
_ Out TL W ¼1 TH Q_
(5)
H
The reversible Carnot efficiency for ideal operation is less than unity because the entropy flow from the source flux (Q˙H/TH) cannot be destroyed and is diverted to the low temperature sink such that Q˙H/TH ¼ Q˙L/TL. The rejected energy flow Q˙L is thus the product of Q˙H/TH and TL, that is, TL(Q˙H/TH). In contrast, the efficiency for maximum power operation, with RH ¼ RL, is given by Curzorn-Ahlborn [2]: 1=2 TL ZCA ¼ 1 (6) TH where the subscript CA denotes Curzorn–Ahlborn. For fixed and equivalent thermal resistances (RH and RL), as the heat flux qH increases the efficiency (Z) decreases, and as a result there is an optimum operating point that delivers maximum power. For maximum power operation the Curzorn–Ahlborn efficiency (6) is independent of the magnitude of the thermal resistances (RH ¼ RL), and is less than the maximum possible theoretical efficiency (5). In the present analysis of the solar energy conversion model, we are concerned with the specific character and role of the thermal resistances to heat transfer to and from the conversion devices. If the external heat flux is due to heat conduction we have the general form (Fourier’s law for one dimensional heat conduction) q ¼ k
dT dx
(7)
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where k is the thermal conductivity. The associated entropy flux is given by the relation q k dT JS ¼ ¼ T T dx
(8)
where T is the local material temperature. For steady-state operation (fixed q), the local entropy production rate per unit _ using (7) and (8) is volume (p) 2 d q dT k dT p_ ¼ JS ¼ 2 ¼ 2 dx dx dx T T
then the thermal resistances are simply 1 RH ¼ g 1 ; H ¼ ðsH AH Þ
1 RL ¼ g 1 L ¼ ðsL AL Þ
(12)
where AH and AL are the heat transfer areas at the high- and lowtemperature sides of the conversion device.
3. Endoreversible model for terrestrial solar radiation conversion
(9)
The entropy production rate (9) is only zero when the temperature gradient is infinitesimal or zero. Although this is shown for one-dimensional heat conduction, it is true for any type of heat conduction or convection. Consequently, for a finite heat flux the thermal conductivity must be infinite to have a zero temperature gradient and zero entropy production. This can be seen by re-stating the entropy production rate (9), using (7), as k q2 q2 1 p_ ¼ þ 2 (10) ¼þ 2 k k T T In contrast, for a finite heat transfer rate, finite thermal conductivity results in a thermal resistance and entropy production. De Vos [5] compares a number of solar energy conversion models with and without thermal resistance to heat transfer (external irreversibilities). These models include the solar cell model, the endoreversible Stefan–Boltzmann engine, and the Muser engine [6]. The solar cell model includes radiation absorption and emission without external irreversibilities. The Muser engine includes irreversible radiation transfer from the source and reversible heat conduction to the low temperature sink, whereas the endoreversible Stefan–Boltzmann engine models irreversible radiation transfer from the source as well as to the sink. The Stefan–Boltzmann engine is representative of solar energy conversion for extraterrestrial applications where heat conduction or convection to a low-temperature sink is not possible. The endoreversible Stefan–Boltzmann engine is depicted in Fig. 2. The radiative heat transfer rates are specified in the general form Q_ H ¼ g H ðT 4H T 4A Þ;
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Q_ L ¼ g L ðT 4B T 4L Þ
(11)
where the thermal conductances gH and gL are the inverse of the thermal resistances for the high- and low-temperature sides of the conversion device, respectively. Assuming the emissivity of the heat transfer surfaces at TA and TB are eH and eL, respectively, that the source radiation from TH is isotropic blackbody radiation (BR), and that the sink at temperature TL behaves as a blackbody,
For both thermal and quantum conversion processes there must be heat/entropy rejected to the environment (Q˙L) even for ideal reversible operation because entropy is absorbed from the source flux. For non-ideal conversion, the required heat/entropy rejection is increased due to entropy production. The radiation emission by the absorbing material carries entropy from the device but it is not of sufficient magnitude. Consequently, there must be heat rejection by conduction or convection to the lowtemperature heat sink for both thermal and quantum conversion devices even when they are ideal. Heat rejection by conduction and convection in practice is an irreversible process. In this analysis we consider the case where the engine has irreversible radiative transfer from the source and irreversible heat conduction or convection to the low-temperature sink. Fig. 3 depicts the conversion model considered in this analysis. The heat transfer rate to the low temperature sink may be expressed as Q_ L ¼ g L ðT B T L Þ
(13)
where gL is the heat rejection conductance by convection or conduction. For example, gL ¼ hAL for heat rejection by convection. The power flux is the product of the energy flux from the source (qH) and the efficiency (Z), that is, w ¼ ZqH. Applying the first law of thermodynamics, the work output rate is given by _ Out ¼ g ðT 4 T 4 Þð1 T B =T A Þ W H H A
(14)
Application of the second law of thermodynamics to the reversible portion of the conversion device allows the temperature TB to be expressed as " TB ¼ TL 1
g H ðT 4H T 4A Þ TA gL
#1 (15)
For the endoreversible solar energy conversion model the emissivity eH of the absorber is a primary variable that affects the work output rate. However, analogous to the Curzorn–Ahlborn analysis, the temperature of the absorber will also be taken in this analysis as an independent variable. TH
TH
Radiative: Radiative: Internally
QH = gH (TH4 − TA4 ) TA
QH = gH (TH4 − TA4 ) TA
Reversible
Reversible
WOut
Conversion Device
Internally
Device TB
WOut
Conversion TB
Heat Rejection:
Radiative:
QL = gL (TB − TL )
QL = gL (TB 4− TL 4) TL Fig. 2. Stefan–Boltzmann engine.
TL Fig. 3. Endoreversible conversion with irreversible heat rejection by conduction or convection.
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Assuming the absorbing material behaves as a graybody with emissivity eH equal to absorptivity aH, the absorbed heat flow from the incident, concentrated, diluted blackbody radiation (DBR) flux is given by (16)
where TH is an effective source temperature (T 4H ¼ E_ Inc =s). The heat transfer rate to the low-temperature sink is given by (13). Using (15), the efficiency1 can be expressed as 1 TB g ¼ 1 T L T A H ðT 4H T 4A Þ (17) Z¼1 TA gL
0.6 Efficiency
Q_ H ¼ AH qH ¼ sH AH ðT 4H T 4A Þ
0.8
0.4 0.2 0 1
This efficiency provides a limit for both quantum and thermal conversion processes. For quantum conversion, the temperature TA represents the emission temperature and the temperature TB is the heat or entropy rejection (lattice) temperature.
900 0.5
600
Emissivity 0
4. Analysis and results
(18)
must be specified from practical considerations of heat transfer parameters, material properties and geometric considerations. The range of the conductance constant considered in this analysis is from r ¼ 5E9 K3 for poor heat rejection conditions, to r ¼ 1E11 K3 for good heat rejection conditions. The r-value of 5E9 K3 represents, for example, an overall convective heat transfer coefficient of 140 W/m2 K with an area ratio AL/AH of 2. The r-value of 1E11 K3 for good heat rejection represents, for example, an overall thermal conductivity of 300 W/m K with a Dx of 0.26 m and an area ratio AL/AH of 5. Using the definition of the conductance constant (18), the efficiency (17) may be expressed as Z ¼ 1 T L ½T A H ðT 4H T 4A Þ=r1
(19)
The work output rate for this model is the product of the heat flux (16) and the efficiency (19). The work output rate w from the endoreversible model, per unit absorption area AH, is " # ðT A T L Þ H ðT 4H T 4A Þ=r w ¼ sH ðT 4H T 4A Þ (20) T A H ðT 4H T 4A Þ=r Figs. 4 and 5 depict the variation of the efficiency, and the work output rate per unit absorption area, with the emissivity eH and the absorber temperature TA for r ¼ 5E9 K3. Note, that this value of the constant r represents the case of relatively poor heat rejection (ex. h ¼ 140 W/m2 K, AL/AH ¼ 2). The efficiency clearly increases with increasing absorber temperature and/or decreasing emissivity, as seen in Fig. 4. In both instances, this occurs because the energy fluxes through the conversion device decrease, allowing in turn a farther spread between the absorber temperature TA and the heat rejection temperature TB. In contrast to the efficiency, the heat flux through 1 This efficiency represents the part of the conversion process operating between TA and TB. For example, this efficiency applies to the thermal power plant performance not including the optical concentration process.
15000 Work Output (W/m2)
gL g H =H
Absorber Temperature (K)
Fig. 4. The efficiency Z (19) versus emissivity (eH) and absorption temperature (TA) with r ¼ 5E9 K3, TH ¼ 900 K and TL ¼ 300 K.
To obtain numeric plots we must specify values for a number of parameters. The effective source temperature will be taken as TH ¼ 900 K. This is representative of medium-level concentration of typical, unconcentrated, terrestrial solar fluxes. For example, this effective temperature represents an unconcentrated flux of 800 W/m2 with a concentration factor of 45. The ambient heat rejection temperature is taken as TL ¼ 300 K. The numeric value of the conductance constant (r), defined as, r¼
300
T > 508.5 K for region III
10000
5000 III
II
0 1
900 I
0.5
Emissivity
600 0
300
Absorber Temperature (K)
Fig. 5. The work output w per unit absorption area versus emissivity and the absorber temperature (TA) with r ¼ 5E9 K3, TH ¼ 900 K and TL ¼ 300 K.
the conversion device is maximized for low absorber temperatures and/or high emissivities/absorptivities. This competition results in three distinct regions with respect to work optimization, as shown in Fig. 5:
Region I: W˙ ¼ 0 and Z ¼ 0 for all aH (note: aH ¼ eH). Region II: W˙max occurs at aHo1. Region III: W˙max occurs at aH ¼ 1. In region I, the energy flows through the conversion device are such that there is no distinction between the absorber temperature TA and the required heat rejection temperature TB (15), and consequently the result is zero efficiency and work output. As thermal conductance to the low-temperature sink decreases, the heat rejection temperature TB increases, thus reducing the efficiency of the conversion process. In region II, there is some distinction between the absorber temperature and the heat rejection temperature. Note that in Fig. 5 the boundary between region II and III is given by TA ¼ 508.5 K (235.2 1C). Positive work production is possible in region II, but maximum work occurs with absorption of only a fraction of the incident SR flux (aHo1). Higher absorption rates increase the heat flux while decreasing the efficiency to such an extent that the work production decreases.
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_ max W
pffiffiffiffiffiffi pffiffiffiffiffi2 TA TL ¼ gL
5000
0 1 900 0.5
(21)
and the efficiency concisely is sffiffiffiffiffiffi TL Z¼1 TA
(22)
(23)
pffiffiffiffiffiffi pffiffiffiffiffi2 _ max W ¼ sr TA TL AH
(24)
For a fixed absorber emissivity, on the other hand, maximum work always occurs at some optimum value of the absorber temperature, as can be seen in Fig. 5. From the comparison of Figs. 4 and 5, one may also conclude that efficiency and power optimizations always refer to distinctly different operating conditions. Fig. 6 depicts the variation of the work output rate per unit absorption area, for cases of good heat rejection. In this plot, it can be seen that maximum work generally occurs at eH ¼ 1 (region III). The small or absent region II indicates that, for systems with strict temperature limitations, lower operating temperatures may be realized with positive work production. The magnitude of the irreversibility on the cold side of the conversion device, due to heat rejection, can be compared to the irreversibility due to radiation absorption and emission by considering the ratio
Rad
0
300
Absorber Temperature (K)
occupy the same hemisphere of directions. Treating the incident radiation as isotropic GR with an emission temperature TS, and a non-uniform spectral emissivity en, the incident entropy flux is given by Planck’s entropy formula [7] Z 1 4 45 S_ Inc ¼ sT 3S AH xS ½ð1 þ f I Þ lnð1 þ f I Þ f I lnðf I Þ dxS (27) 3 4p4 0 where f I ¼ u =ðexS 1Þ
(28)
xS ¼ hn/kTS, and n is frequency. The outgoing entropy flux due to emission and reflection is similarly, Z 1 4 45 S_ Ref ¼ sT 3A AH xA ½ð1 þ f RE Þ lnð1 þ f RE Þ f RE lnðf RE Þ dxA 4 3 4p þEmi 0 (29)
As can be seen in Fig. 5, for case III (TA4508.5 K) maximum work occurs at eH ¼ 1. The magnitude of maximum work in this case is " # TL (25) w ¼ sðT 4H T 4A Þ 1 T A ðT 4H T 4A Þ=r
_ICS p_ CS Q_ L =T L Q_ H =T A ¼ ¼ _IHS p_ HS Q_ =T A þ S_ Emi; S_ Inc; H
600
Fig. 6. The work output w per unit absorption area versus emissivity and the absorber temperature (TA) with r ¼ 1E11 K3, TH ¼ 900 K and TL ¼ 300 K.
Notably, the efficiency (23) does not depend on the thermal conductances from the source, or to the low-temperature sink, even if these conductances are unequal in magnitude. The maximum work output (21) can also be expressed on a per unit absorber area basis as
RI
10000
Emissivity
where the optimum emissivity eH is given by pffiffiffiffiffiffiffiffiffiffiffi T TATL H;opt ¼ A 4 ðT H T 4A Þ=r
wmax ¼
15000 Work Output (W/m2)
These observations regarding regions I and II reveal the importance of good heat rejection for viable system designs, as well as the fair evaluation of actual performance of a device with respect to what can be theoretically obtained by any system given the specific heat rejection conditions. Poor heat rejection is particularly a problem when water is in short supply so that heat must be rejected to atmospheric air, particularly if water resources are insufficient to the point that evaporative cooling is not possible. This is especially a problem in designs for arid or desert regions. Poor heat rejection conditions can result in zero or low work output, and/or higher operating temperatures and risk of failure. This is of particular concern to photovoltaic systems that have low maximum temperature limits. In case II, for accurate performance evaluation the maximum possible work output can be expressed by the analytical result
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(26)
where f RE ¼ ð1 n Þ=ðexS 1Þ þ =ðexA 1Þ
and where xA ¼ hn/kTA. However, the magnitude of the entropy fluxes of the incoming and outgoing radiation cannot be generally determined using (27) and (29) as they depend on the spectral and directional distribution of the terrestrial source radiation, and on the concentration process. The entropy-to-energy ratio of DBR is not the same as BR with the same emission temperature. However, the entropy of the incident flux can be approximated as that of the entropy of BR with the same energy flux, but lower emission temperature [8]. For typical terrestrial SR, this approximation generally results in less than 10% overestimate of the incident entropy flux. With this approximation, and limiting the analysis to region III so that the emissivity is unity, the net incident entropy flux can be simply estimated as 4 S_ Inc S_ Emi ¼ sAH ðT 3H T 3A Þ 3
(31)
where TH is the effective source temperature of the concentrated DBR source radiation. By defining y as the ratio of the absorber temperature to that of the effective temperature of the incident source radiation, y ¼ TA/TH, using Eqs. (11) and (25), and after lengthy algebraic manipulation, the irreversibility ratio (26) can be expressed in the following simplified form:
Rad
The entropies of the reflected and emitted fluxes are not independent because their energy spectra overlap and they
(30)
RI ¼
ð1 y4 Þ2 h i ½1 43y þ 13y4 ry=T 3H ð1 y4 Þ
(32)
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It is worthwhile to note that minimizing the entropy production rate in the conversion device does not result in maximum work production. Fig. 8 shows a plot of the non-dimensionalized ˙ /sAHT4H) and the total entropy production rate work .output (W (p_ total sAH T 3H ) as functions of the temperature ratio y, with r ¼ 1E11 K3. The entropy production rate decreases with increasing y, and the minimum occurs at y ¼ 1, while maximum work occurs at y ¼ 0.6748 (TA ¼ 607.3 K for TH ¼ 900 K).
1.4
The Irreversibility ratio RI
1.2 r = 5E9 K3 1 0.8 gL increasing
0.6 0.4
r = 2.5E10 K3
4. Conclusions
0.2 r = 1E11 K3 0 0.35
0.45
0.55 0.65 0.75 The temperature ratio (θ θ=TA/TH)
0.85
0.95
Fig. 7. The irreversibility ratio RI in (28) versus the temperature ratio y ¼ TA/TH, for maximum work output, TH ¼ 900 K and TL ¼ 300 K.
1.80 1.60
Total Entropy Production Rate
Non-Dimensional
1.40 1.20 1.00 0.80 0.60
Work Output Rate
0.40 0.20 0 0.3
0.4
0.5 0.6 0.7 0.8 The temperature ratio θ = TA/TH
0.9
1
The finite absorption of a radiative source flux, and the simultaneous emission of radiation, is an inherently irreversible process. Likewise, heat rejection from the process is unavoidable and depends partly on local environmental conditions and resources. The endoreversible model treats these irreversibilities as inherent but external to the conversion process. The endoreversible model considered in this analysis represents typical solar radiation conversion and incorporates the necessary level of entropy rejection required by the second law of thermodynamics. The model provides a guide for optimal absorber temperature and heat fluxes for maximizing work production. For relatively poor heat rejection such as in arid climates, analysis of the model reveals regions of operating parameters that will result in zero or low theoretical work output, and/or high operating temperatures and risk of failure. This is of particular concern to photovoltaic systems that have low maximum temperature limits. This model also provides analytical expressions and numeric plots of the maximum ideal work limits for a conversion device operating with specific radiative source flux and heat rejection conditions. A fair evaluation of system performance is thereby obtained by comparing actual work production in comparison to the ideal, given the specific operating conditions or restrictions.
Fig. 8. The non-dimensionalized work output and total entropy production rate versus the temperature ratio y, for TH ¼ 900 K and r ¼ 1E11 K3.
References
Fig. 7 depicts the irreversibility ratio RI, for maximum work output, as a function of the temperature ratio y, for three values of the heat conductance ratio r. The plot shows that, for good heat rejection (high r-value), the irreversibility rate due to thermal radiation (TR) absorption dominates the calculation of the total irreversibility rate. However, for relatively poor heat rejection conductance, r ¼ 5E9 K3 (ex. h ¼ 140 W/m2 K, AL/AH ¼ 2), the irreversibility due to heat rejection can be greater than that due to radiation absorption. Under these conditions the work output of the device can be very low or zero, and the operating temperature of the device can be high enough to cause failure, particularly with photovoltaic systems that have strict temperature limitations.
[1] Wright SE, Rosen MA. Exergetic efficiencies and the exergy content of terrestrial solar radiation. ASME J Solar Energy Eng 2004;126:673–6. [2] Curzon F, Ahlborn B. Efficiency of a carnot engine at maximum power output. Am J Phys 1975;43:22–4. [3] Wright SE, Rosen MA, Scott DS, Haddow JB. The exergy flux of radiative heat transfer for the special case of blackbody radiation. Exergy Int J 2002;2(1): 24–33. [4] Bejan A. Advanced engineering thermodynamics. 3rd ed. New Jersey: Wiley; 2006. [5] De Vos A. Endoreversible thermodynamics of solar energy conversion. 1st ed. New York: Oxford University Press; 1992. [6] Muser H. Behandlung von Elektronenprozessen in Halbleiter Randschichten. Zeitschr Phys 1957;148:380–90. [7] Planck M. Translation by Morton Mausius, The theory of heat radiation. New York: Dover Publications; 1914. [8] Wright SE, Scott DS, Rosen MA, Haddow JB. On the entropy of radiative heat transfer in engineering thermodynamics. Int J Eng Sci 2001;39:1691–706.