Enemy Vol. 5. pp. 69&707 Pergamon Press Ltd., 1980
Printed in Great Britain
SECOND LAW AND RADIATION ROBERTH. EDGERTON Oakland University School of Engineering, Rochester, MI 48063,U.S.A. Abstract-The available energy of thermal radiation and solar radiation is examined in this paper. The extension of the available energy concept to the evaluation of the potential energy conversion in solar converters is outlined. The fundamental question discussed is: How much of a given solar radiation flux is convertible to thermodynamic work? The basic relations for evaluating the available energy in radiation processes are developed. The effects of both the spectral and spatial distribution of the radiation on the available energy are discussed. Atmospheric effects are examined, using NASA standard atmosphere solar spectral distributions. The available energy of the spectral characteristics is compared with the available energy of thermal equilibrium radiation at the same solar flux. This is used to illustrate the available energy losses in thermal energy converters. The technical evaluation of solar energy converters are discussed, based on the available energy of the input energy. A method for evaluating spectral sharing solar conversion devices and solar energy simulators is outlined.
INTRODUCTION
The objective of this paper is to develop a method for evaluating the available energy of solar radiation. The basic question posed is: In a given radiation distribution, what portion of the energy is convertible to thermodynamic work? The solar energy incident on the earth surface is modified in several ways by the atmosphere. The two main processes are absorption and scattering. These processes alter the spectral distribution, intensity, and spatial distribution of the radiation. Their effect on the available energy of solar radiation are of special concern in this paper. In the development of new solar energy converters, it will be important to determine the available energy at specific geographic locations so that selective devices can be utilized for maximum conversion efficiency. The nature of radiation, and how electromagnetic theory and thermodynamics relate, was consolidated and formulated by Planck in his classical treatise “The Theory of Heat Radiation”.’ The present work extends this to the consideration of solar energy. We will begin by outlining and defining radiation processes in geometrical and thermodynamic terms. The effects of spectral and spatial distributions on the available energy of radiation will then be developed. These will include specification of temperature, entropy, and available energy. BLACK BODY RADIATION
Black body radiation has been thermodynamically examined by many researchers.‘-’ They begin with the assumption of quantized photon energies ei = hvi, in which h is Planck’s constant and vi is the frequency of the photon’s electromagnetic wave. The energy and number of photons in a given volume are next calculated at equilibrium by statistical thermodynamic techniques. The expected number of photons in an equilibrium volume V at temperature T can then be shown to be (n)=0.488
(F >3v,
where k is Boltzmann’s constant. The expected energy (or internal energy) is (E) = 0.658(F)lkTV
(2)
and the thermodynamic pressure is p
J-9 -.
3
693
(31
R.H.E~CERTON
694
From these, the following thermodynamic properties are easily found 4(E) Entropy: s = 3~ Pressure-Volume: p V = @ 3
(51
Helmholts Free Energy: F = (E) - 7’s = -y
(6)
Gibbs Free Energy: G = H - TS = (E) + PV - TS = 0
(71
AvailableEnergy:
A=H-T$=$(l-$‘)(E)=(l-$)H
(8)
To is the datum state temperature, and H is the enthalpy. The Gibbs Free Energy is proposed by some as a measure of the quality of energy, Thermal radiation in this context has no value for conversion to work forms. We follow the more acceptable procedure of using available energy as the measure of the value of energy. In this case, radiation, as shown in Eq. (8), has the expected available energy-enthalpy Camot factor relationship. The relationship of thermodynamic quantities requires an interpretation in terms other than energy (E). The reason for this is that radiation processes are usually described in terms of energy fluxes through or on surfaces. The relationship between energy density and every flux can be simply determined.’ If each particle has average energy z, then the energy impinging on the surface per unit area is
0 44s _ c(E)
f+A=4V-4V’
(91
The “radiant flux” per unit area can then be written as
i-oc_?q$Lu~4 4v
1% c
’
(10)
where u is the Stefan-Boltzmann constant (5.67 x IO-” W/cm2 . K4). The standard relationships between the radiation parameters of intensity, energy density, and energy flux per unit area are easily derived from geometrical relations.’ The may be summarized for axisymmetric geometries as 2e
Energy density: u =
u=- In
C
Energyflux:
e=ZR
(11)
(12)
(131
where I is the intensity and R is the solid angle of the radiation Equations (10) and (12) can be combined to relate energy density to temperature for any black body radiation distribution in a solid angle fi as u
d-l ~-7’~ CT
.
Secondlaw and radiation
695
This is sufficient for cases where the angular distribution of the radiation is known as well as
the temperature of the radiation. If the radiation process is done reversibly (as on perfect reflection), the temperature remains constant. In observable terms, the “color” of the radiation is determined by the frequency distribution which is specified by the temperature. Intensity remains constant in reversible optical processes. The energy flux relationship to angular distribution and temperature can be found by using Eq. (13) as e
+(I+).
(15)
For the special case of uniform radiation in all directions, the energy flux impinging on a surface is equivalent to n = 2~ to give e = uT4. The net flux across a transparent surface is of course zero for fi = 41r. In both of these expressions, e is the net flux per unit area in one direction across an area. Inverting Eq. (15) to solve for T4 gives
or, from Eq. (12),
In optical problems, a constant temperature of radiation is indicated if the intensity remains the same in the process. The temperature may be determined by measurement of the energy flux per unit area and the solid angle of the radiation. The “temperature” of solar radiation may then be calculated for e, = 1.353W/m2 and 0 = 6.78 x 10m5,the solid angle of direct radiation from the solar disks as seen from the mean earth distance, to yield lr(1.353)
T4=
6.78 x 1O-5C 1 -6’78 ;b”-‘) ( or T = 5775°K. This temperature closely represents the energy distribution of radiation from the sun as shown in Fig. 1. The calculated temperature is the temperature which would be observed if the sun emitted radiation as a black body. Another way to consider it is that this is the temperature of radiation if the space were filled with radiation moving in all directions, with the same intensity of the sunlight per unit angle in all directions. Another interesting “temperature” is the temperature which would be obtained if this radiation were absorbed in a black cavity. The cavity would come to equilibrium, with the flux leaving equal to the fhrx entering. The excess flux leaving would be distributed over the angle 0 = 2~. Using Eq. (13) and Eq. (14), we get
e
=?0(1-g),
6% 0.25
0.20
Irradiation Curve Outside Atmosphere Irradiation Curve at Sea Level 0.15
Cll?Xe
for Blackbody at 5900'K
03
rl 0'4
,H20
: 0.10 z I w
I _
0.05
I
I/
I
I
0 0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
WAVELENGTH (u)
Fig. 1. Spectral distribution curves for solar energy with atmospheric absorption.
and solving for T gives T = 393°K. This is a significant temperature, as it represents the maximum temperature attainable by a black surface outside the earth’s atmosphere at the earth-sun distance. If the earth and its atmosphere acted like a black body, this would represent the temperature 0: the earth as observed from space. The next thermodynamic variable of concern is entropy. Equation (4) indicates that the entropy of the radiation is determined by the energy and the temperature S _ 4(E) 3 T’ The entropy per unit volume for uniform radiation at equilibrium is found by using Eq. (10) as
The entropy intensity Sr is then s =Sc_4aT3 I R 37T’ The entropy flux S, follows from Eq. (14) as
697
Second law and radiation
For uniform fhrx in all directions fi = 2a, the entropy flux across a unit area is
The entropy flux may be written in terms of the energy flux by combining with Eq. (15) to give
s, =$?,-[~(1-~)]“4.
(19)
For uniform radiation in all directions
These relations become impor@nt when examining the energy in radiation which can be converted to thermodynamic work. The entropy flux for sunlight in a solid angle R = 6.78 x lo-’ can now be calculated from Eq. (19). Using the energy flux e = 1353W/m’, gives Se = 0*313 W/m*“K. The entropy fly from a hohlraum, if the entering energy flux is solar energy of 1353W/m’, will return through the opening with a solid angle 0 = 2~. This entropy flux out will be S =-e4 eB 3
314
_ crl/4 -4.59 W/m*“K.
The rate of entropy generation in the hohlraum is the difference between the entropy leaving and that entering, or !g
=
se,
-
se
= ;
1 -e
e3/4(g/4 [
1 A
2>1’”
= 4.28 W/m*“K. The entropy generation at any adiabatic surface may therefore be calculated from knowledge of the energy flux to the surface and the solid angle distribution of radiation to the surface and that flux leaving the surface. The available energy per unit volume can be found from Eqs. (8) and (14) as
The available energy flux per unit area is then found by using Eq. (13) as b=c
(1-2 > a.
(21)
In terms of the solid angle and temperature, this can be written as b
2(1-$y$l-o-$).
(22)
R. H.
6%
EDGERTON
The Camot Factor (1 -(TO/T)) is applicable, and the factor of (4/3) accounts for the radiation pressure which is also available for work processes. This is analogous in steady flow systems to applying the Carnot Factor to the enthalpy, not the internal energy. For radiation problems, in classical heat transfer theory, the momentum flux is assumed as zero, thus neglecting the radiation pressure effects. In fact, this energy is assumed as not available because of the technological diiculty of utilizing it. Boltunarm originally hypothesized a radiation expansion engine in his consideration of the thermodynamics of black body radiation. Utilizing this effect in a throughflow machine which includes the flow of gases would not be feasible due to the extremely low radiation pressure relative to practical gas pressures. In space applications, where the density of particules is extremely low, the expansion energy utilization might prove useful. The available energy flux can be found in terms of the energy flux by substituting the temperature from Eq. (19) to give
b=$ 1 l1
TO
l/4
.
(23)
( an( 1 T&4?r)) Ji
The rate of available energy loss when sunlight is uniformly scattered from a surface can then be calculated knowing the initial solid angle of the light and the final solid angle of 2~. Any other device which is adiabatic, and alters the solid angular distribution of the radiation, also changes the available energy. The available energy loss when solar energy is scattered or when absorbed by a black collector is then
boss = h=6.nx,o-5
-
bn=z,.
For e = 1353W/m2,
or
Thus, 76% of the available energy of sunlight is lost in the absorption process. Designs of energy conversion systems which either rely on black absorber surfaces or do not utilize the solid angle of solar radiation give up 76% of the available energy as a penalty. A focusing collector utilizes the narrow angular distribution of solar radiation to advantage. The energy flux per unit area in a perfect focusing collector can increase the energy flux per unit area in the ratio
This corresponds to a temperature of T = 5775”K, and the available energy loss is zero.t SPECTRAL
FACTORS OF SOLAR RADIATION
In order to deal with the spectral characteristics of solar radiation, we must consider the radiant energy between the frequency of v and v + dv as d(E)=Fdv. Weveral short notes on the available energy of radiation have appeared in the literature.“‘0 They should be consulted regarding other approaches to radiation thermodynamics.
Second law and radiation
699
Now the expected energy per unit frequency can be written in statistical mechanics for black body radiation’ as a(E) 87rV e-Bhu 3
-=7hv l3V
1 _
eMvs
(25)
In the equilibrium situation, the distribution of energy over the spectrum is determined by the temperature. If the radiation is in a state which is not in equilibrium, the specification of the energy distribution over the spectrum requires measurement of the energy at each wavelength or frequency. Equation (25) indicates that measurement of the energy over a given frequency u, = L~E/cYv can be used to specify a /3, or “temperature,” for that specific wavelength. Inverting this expression to solve for p gives B=kln(l+F),
Y
(24)
or
If the energy flux e, is uniform in all directions, then f?,
=z.
j3 can then be written in measurable terms as
The following relationships then follow, in which the “temperature” or /3, does not appear: -The pV function:
a(pV) -=
( _&) 8lrVh~~~~ ’ +2Irhv’
8V
-The
(30)
entropy function: (31)
-The
available energy function is then:
A =z_4Ve, ” au --+--7c
8?rVhv3
ln(I+$) 1*(1+?$)
(32) --The available energy flux b, = (~A/c?v)(c/4 V) is then:
700
-The relation:
R. H. E~GERTON
available energy in terms of wavelength can also be obtained by the derivative
a c a -=--iah A au’ The available energy then becomes
The available energy flux per unit area for uniform radiation becomes
SPACIAL
DISTRIBUTION
CONSIDERATIONS
A general expression relating energy flux, intensity, and energy density is needed for the situation where the intensity is variable with spatial coordinates. In order to deal with evaluation of the available energy in nonuniform spatial distributions, the choice between knowing the energy flux from each spatial element or the intensity in each direction must be made. From a measurement perspective, the intensity is a secondary or derived measurement. The basic measurement device is a pyroheliometer, which measures the energy flux in a known solid angle determined by the aperture of the viewing tube. This energy flux ‘fir,, as measured, is perpendicular to the measurement surface of the pyroheliometer. If the pyroheliometer surface is at angle 4 to the surface of interest, then the actual surface flux is en = enp cos 4.
From the practical point of view, the choice of determining the total flux can be done by adding up all the en for each special segment, and then combining the measurement enP and the given angle, making sure the full space is sampled. The other possibility is to compute In = e&l from the aperture solid angle R and energy flux, and then integrate over all the space de
e,=dw=bcos~=~=InCOSd=
2
cos
c$.
From Eq. (ll), the energy density is then computed as du
de -=-=-_ 1
U”=dw=dwcosf#l
e, ccosc#J
IA _& c
ncv
(36)
For thermal black body radiation from a solid angle w IA=?, and Eq. (12) then yields UT4 uw=z
(37)
Second law and radiation
701
The energy flux per unit area is then
e, =
$
cos 4,
(38)
and e=
I
4
$cosddw=
5
cos 4 sin 4 d4 d0.
(39)
If the temperature of the space is a function of spherical coordinates 4 and 8, then T(4,O)
will be required to determine the energy flux e. The entropy intensity is, from Eq. (17),
The entropy flux is then 4UT4
Se, = 37
cos4 = de, j_~,
WV
and S?=;/p$
cos4 sin 4 d4 de,
(411
The available energy intensity is then 4 alw =- 3
(
I-$
qcos4, >
(42)
and b=I(i(l-$‘)$cos+sin4d+dB.
(431
Now, we must get T from the energy flux 4 e,=$cosf$
Since
T4 = %
acos4’
substitution gives
This expression can be used to determine the available energy flux, knowing the energy flux per unit area eoP on surfaces perpendicular to the sky area as measured with a pyrheliometer b=f_/i
[l-(~,,;,)%.os4sin+d4d& \
R. ii. EDGERTUN
702
If enP is uniform over the hemisphere above the area, then b =;(l-$)e.
The next problem is to determine the available energy flux in the case where the spectral radiation distribution is a function of the spherical coordinates 8 and (b-This is characteristic of solar radiation as it is scattered and absorbed by the atmosphere. The available energy of this radiation is of principal concern in evaluating the site specific work potentiai of soIar energy. We will start from the energy density relationship as a function of frequency v and solid angle w u =@=2w lJ av
hv3 7
eeBhv 1 -
(47)
e-BlrY*
This equation indicates the energy density of space as a function of the total solid angle w and frequency v. Next uw=--
au, _ 2hv3 ebBhv aw c3 1 - epBhv’
Then, using the flux to density relation 2hv3 e-flhv e VW= cu,, cos (b = TI-e-8h”
cos 0.
(49)
Using the measured variable obtained with a spectral pyrheliometer
Then, p or (l/KT) can be evaluated from inversion of this relationship
The entropy flux per unit frequency per unit solid angle is then found as
$,,=qcos4In(l+&) +&cos~$
In l+ [ ( &-J-G%)]
The total entropy flux through the area is then
S=
S,, sin #J d4 de dv,
(51)
Second
703
law and radiation
and the available energy flux b,, can be found as ln(l+&)
3
ln(l I 2h;h;)
bAw=eAw +*:: i
cos9 I
-T,il[(~)cos~ln(*+&)+~(l*(l+$&j)-In(&))].
(52)
Using
these expressions can be written in terms of wavelength instead of frequency
I
cos I$
This form presents numerical computational problems with negative logrithmic relations. The frequency relations are thus preferred for actual computations.
PRELIMINARY
RESULTS
OF ATMOSPHERIC
INFLUENCE
ON AVAILABLE
ENERGY
The
NASA solar spectral data of Thekaekara”-‘3 were utilized to determine the available energy of sunlight with different air mass conditions and turbidities. The general results are illustrated in Fig. 2. In these calculations the intensity was assumed to be distributed uniformly over the full hemisphere w = 27~. Complete scattering of the atmosphere was assumed in these calculations. The spectral data 0.6,
0.5..
"z -
fill /-
0.4..
0.3..
! 2
o.2
--s
--o-
,
3 s G B g
-
______----_-
__ 8
0.1..
I
_a__-
Ul=l, m=l, m=lO, m=lO,
CX=1.3, 8=0.02 c%=0.66, 5=0.17 rpl.3 0=0.02 a=0.66, 4=0.17
-o----u--
--a-
Lightest of Data
Darkest of Data
-__n____fJ
/ xp
I
WAVELENGTH (MICRON) Fig. 2. Available EGY Vo.5,Nos 819-D
energy
variation
with cut-off
wavelength
for sunlight
through
different
atmospheres
R. H. ECCERTON
704
did not include intensity variation with angular position. The results therefore apply to the available energy flux from a surface which scatters radiation uniformly in all directions (or a perfect light diffuser). The turbidity has significantly more effect at high air mass than at low air mass. This would imply that turbid atmospheres will not affect photoconversion devices when the sun is near the zenith nearly as much as at large zenith angles during the early morning or late afternoon. The available energy plot vs cutoff wavelength (Fig. 2) shows an interesting characteristic. The available energy reaches a maximum between 2.5 and 3.5 CL.This indicates the trade-off between narrow bandwith and total energy in the determination of the temperature of the radiation. Beyond the peak wavelength, low energy photons reduce the available energy of the high energy photons, and thus reduce the energy convertible to work. This would indicate the importance of a cutoff filter in this range (2.5-3.5 II) for photo conversion processes. Photons beyond this range are detrimental, in that they provide heating with little quality energy. In thermal devices, a cutoff filter can be used to increase this available energy input. This effect is very small and may not be of practical significance. The ratio of available energy to total energy is shown across the spectrum in Fig. 3. For the spectra1 distributions examined, the total available energy to total energy ratio ranges from 0.5 to 0.6. This is considerably higher than the available energy to total energy ratio for radiation at equilibrium as shown in Table 1. This difference is the most important consideration in the development of better solar energy conversion systems. The use of spectrally selective surfaces for thermal converters utilizes this difference to advantage to increase the temperature of a collector. Optimization of solar thermal converters in the future should be based on variation of the selective cutoff wavelength with atmospheric conditions and time of day and season. Present selective surfaces are based on black body radiation distribution calculations, which do not optimize for the varying spectral distributions. Also important is the prediction of the performance of solar converters in different geographical locations. This will include consideration of the variable optical characteristics of the atmosphere, including altitude, cloud effects, and atmospheric constituent gases and particles. Loferski14 shows that the efficiency of a solar cell can be increased by water vapor absorption of solar energy. He illustrates a situation where the electrical energy output of a silicon solar cell is increased 4% with water absorption relative to the response to radiation without absorption. At present, the comparison of solar heating collectors is based on performance under standard conditions at one location. The estimation of performance at other locations of thermal and photo converters, however, requires more information on the spectral distribution and the percentage of diffuse radiation. These will require more detailed atmospheric measurements and have hardly been explored in present research. r
WAVELENGTH (MICRON)
Fig. 3. The ratio of available energy to total energy at the earth’s surface through various atmospheres with different cut-off wavelengths.
0.54
0.02
0.04
0.085
0.17
0.02
0.04
0.085
0.17
1.3
1.3
0.66
0.66
1.3
1.3
0.66
0.66
1.3
1.3
0.66
0.66
1.3
1.3
0.66
0.66
1
1
1
1 4 4
4
4
7
7
7
7
10
10
10
10
0.04
0.02
0.17
0.085
0.04
0.02
0.17
0.085
0.56
%
0.06
0.15
0.08
0.03
0.23
0.12
0.29
0.16
322 309
0.002
332
0.02 0.01
339
0.03
319
0.01
0.13
334
0.03
0.25
343
350
0.04
0.06
0.40
340
355
362
0.33
0.04
0.07
0.30
0.06
0.13
0.18
0.22
0.16
0.24
0.44
0.09
0.29
0.52
0.29
368
0.10
0.58
385
0.17
0.34
0.31
392
0.21
0.88 0.79
394
0.025
0.065
0.10
0.13
0.05
0.10
0.14
0.16
0.12
0.17
0.20
0.24
0.32
396
0.22
0.33
Available Energy Loss in Thermal Conversion (Kw/m')
0.23
Thenal Equilibrium Temperature ("k)
0.91
0.94
Available Energy Flux at Thermal Equilibrium
0.46
0.52
Available Energy Flux (Kw/m')
Air Mass
=
Table 1. Solar spectral energy data.
706
R. H. E~GERTON
An instructive way to look at the solar radiation is that the total energy is nearly all available outside the atmosphere, since it is a stream of particles from a source at about 5800°K. The atmosphere reduces this availability by dispersion and absorption processes. The available energy of the resultant radiation is dependent upon the devices we apply for the conversion of this energy to useful effects. Conversion to thermal energy at a black surface is the least efficient thermodynamic process, as it does not utilize the non-equilibrium energy distribution effect nor the directional characteristics. The temperature calculated across the spectrum and the resultant available energy are measures of the usefulness of the radiation for devices which are designed to utilize the whole solar spectrum within the atmosphere. If spectral selective devices or directional collectors are proposed, then the available energy and temperature of that portion of the spectrum can be similarly calculated. This will then provide a measure of the potential of spectral selective conversion devices. A solar cell with sensitivity out to a cutoff wavelength can be evaluated relative to the available energy by this procedure. Mallinson and Landsberg” have attempted this to a limited extent in trying to optimize photodevices for use in Great Britain, where the solar energy character is significantly different from that in bright sky areas where most photodevice research has been conducted. Available energy comparison of photodevices based on effectiveness measures would help to clarify the many discussions regarding different advantages of devices with different spectral sensitivities. Another factor which must be included in the discussion of the output of photo conversion devices is the effect of the energy outside the spectral response wavelengths or frequencies. This is typically dealt with in terms of the heating effects. With spectral sharing systems, the systematic evaluation of this remaining energy becomes more important. Recently experiments with thermo-photovoltaic cells for solar to electric conversion are being carried out.16 The results appear promising, and an optimization with thermodynamic analysis of the conversions where spectral availability is traded off against “spatial availability” could be useful. REFERENCES 1. M. Planck, The Theory of Hear Radiation. Dover, New York (1957) 2. L. Page, Theoretical Physics. Van Nostrand, Amsterdam (1928). 3. M. Planck, Ann. Physik 4 (1901). 4. S. N. Rose, Z. Physik 26, 178(1924). 5. A. Einstein, Berl. Ber. 261 (1924). 6. L. Bolzmamt, Wiedmannsche Anna/en der Physik 22,291 (1884). 7. N. A. Leontovich, Sou. Phys. Usp. 17, 963 (1975). 8. R. Petela, J. Heat Transfer 187-192,May (1964). 9. W. H. Press, Nature 264,734 (1976). 10. J. U. Thoma, RM-78-14,Int. Inst. Appi. Syst. Anal., Laxemburg, Austria (1978). 11. M. P. Thekaekara, “Data on Incident Solar Energy Institute of Environmental Sciences Meeting,” 20th Supplement 2, 21 (1974). 12. Anonymous, ASTM Standard, E 4!&73a, 1974Annual Book of ASTM, Philadelphia, Pemtsylivania (1974). 13. Anonymous, “Solar Electromagnetic Radiation”, NASA SP 8005, Washington, DC, May (1971). 14. J. J. Loferski, J. Appl. Phy. 27, 777 (1956). IS. J. R. Mallinson and P. T. Landsberg, Proc. Roy. Sot. (London) 377, 115(1977). 16. R. M. Swanson, Proc. IEEE 67,446 (1979).
DISCUSSION U.S.A. In considering a single threshold process, such as a photovoltaic device, isn’t it true that more work can be produced at a lower equivalent temperature and still increase the total stored energy? Author’s reply. The answer is yes, if we are talking about a thermophotovoltaic type device where the solar energy is used to heat an element that radiates to the photo device. This is because of device limitations, and not because of radiation thermodynamics. Available energy is lost in the conversion to thermal equilibrium and in the spatial broadening of the radiation. In these thermophotovoltaic devices, the available energy is lost in the element “heating” process. One of the objectives of my paper was to point out that the availability of solar energy should be determined independently of the device contemplated for the conversion process. Once a device has been selected, the fraction of available energy of the sunlight that can ideally be converted by that device can be calculated. More importantly, the available energy of the radiation not utilizable by that device can be determined, to evaluate whether it is useful for conversion in a different device. This is the spectral sharing idea. Himanshu B. Vakil, General Electric Corporation,
John Ahem, Aerojet Hectrosystems Company, U.S.A. In rejecting heat in spacecraft to deep space, the black body radiation is about 4°K. Why, then, did you state that 3.93”Kis not a good reference temperature for your system? Author’s reply. For the calculations of interest, the effect of the earth’s atmosphere on the radiation is of main concern. The earth’s surface temperature is then a better datum temperature. For energy conversion in space, certainly 4°K would be a better datum, but the atmospheric effects would certainly be of smaller interest.
Second law and radiation
707
Robert Euons, Georgia Institute of Technology, U.S.A. How do you explain the fact that it is impossible to focus ultraviolet radiation in such a way that a stagnation temperature equal to the sun’s surface temperature may be achieved? Author’s reply. If the high energy ultraviolet radiation is of solar origin, then focusing the radiation simply increases its solid angle. However, bringing this radiation to equilibrium means that the total energy is distributed over the full equilibrium spectrum. This equilibrium spectrum will have a lower temperature than that of the initial beam. Remembering that the temperature is related to intensity (or radiant energy per unit solid angle per unit area), the area is decreased on focusing but the solid angle of the beam is increased. Richard Lyon, Oak Ridge National Laboratory, U.S.A. Have you calculated a quantity of entropy generation associated with a quantity of radiant energy in a spatial distribution? Author’s reply. Yes, an illustration is given in the paper where solar radiation in a solid angle of 0 = 6.78X !O-’ with energy flux e = 1353W/m* and entropy flux of 0.313W/m’K (representation of solar energy outside the atmosphere) is spread out over a solid angle of 2n in a hohlranm with no energy loss. The entropy flux out is then 4.59W/m*K. 76% of the available energy is lost in this process. Other calculations are easily made, indicating the large impact of atmospheric scattering on the available energy of solar radiation at the earth’s surface.