Solar Energy Vol. 46, No. 4, pp. 231-236, 1991 0038~)92X/91 $3.00 + .00 Printed in the U.S.A. Copyright © 1991 Pergamon Press pie

A NEW LOOK AT THE MAXIMUM CONVERSION EFFICIENCY OF BLACK-BODY RADIATION

STEPHAN KABELAC Institute of Thermodynamics, University of Hannover, 3 Hannover,

Callinstr. 36, Germany

Abstract--A new derivation of the maximum conversion efficiency for a continuous, steady-flow radiation conversion process is given. Using basic thermodynamic considerations only, this theoretical efficiency is shown to be the Carnot efficiency. Other existing conversion efficieneies give smaller values, because they suffer from constraints which are discussed in detail.

1. INTRODUCTION

Economic use of solar energy must be based on an understanding of the maximum work-producing po- tential that is associated with thermal radiation. This potential, the exergy of thermal radiation, has been the subject of intense research activity within the last 25 years. The resulting literature may be divided into two groups, which differ by the initial question posed. One group of analyses considers a "closed system," which is a certain volume that is initially filled with equilib- rium black-body radiation of a given temperature, and asks for the work which could ideally be extracted as the system reaches the dead state, defined by a specified ambient. The second group of analyses considers a "steady flow process," where a steady flux of black- body radiation of a given temperature is incident on an ideal conversion device. The question asked here is how much mechanical power (representing the high grade rate of energy) may be continuously drawn from the converter.

In each group exist competing results. The contro- versy that accompanies the ideal conversion of enclosed black-body radiation is thoroughly discussed by Be- jan [ 1 ]. This article is concerned with the derivation of the efficiency formulae for the second case.* For the theoretical upper limit of the elficiency of the contin- uous conversion of a black-body radiative energy flux ,I% having the temperature T~,, there exist three com- peting expressions in the literature, namely

P"a~- 1 4 To 1 To 4

o r

~ in -- T~. 1~ 1 -

with

0 = 4 " TSopt- 3" To" T4pt - T~." To

o r

Pmax TO - 1 - - ( 3 )

,I, in T~°"

To is the temperature of the ambient. Equation (1) was derived independently by Landsberg and Mallin- son[2], see also Landsberg[3], by Press[4] and by Bognjakovid[5], while the second equation is favored by Castafis [ 6 ], Haught [ 7 ], De Vos and Pauwels [ 8 ], and Bejan[1]. Jeter[9] obtains the Carnot efficiency, eqn (3), as the limiting value for a steady flow con- version process, but, as Bejan[l] pointed out, his der- ivation actually belongs to the group of closed system processes. There has been some confusion about which of these expressions ( 1 )-( 3 ) to use, since they all claim to answer the same question. Equations ( 1 ) and (2) result in values less than those of eqn (3), therefore some constraints must exist which have not been clearly stated in the literature. Based on a new approach, these constraints will be clarified in this article.

The following derivation is restricted to the Planck formulation of the black-body radiation energy flux <I, and entropy flux ~I'[10]

d~ = A a T 4 ; ff~ = 4 A a T 3 "

( 1 ) This restriction has the advantage that the entropy flux accompanying the radiation energy flux is known ex- actly. Real radiation, as for example the solar radiation incident on earth, calls for a d d i t i o n a l models enabling one to calculate the entropy flux. Such a model, for example the Landsberg model of diluted black-body

(2) radiation[3], complicates the derivation and will be treated in a later article.

* Because this case (representing the "energy transfer view" ) is conceptually different from the case mentioned first (which deals with a system of classical equilibrium thermo- dynamics)[l], the term "radiative energy flux" will be used instead of the inexact term "radiation."

2. THE M O D I F I E D CARNOT ENGINE

The derivation of the exergy of a black-body radia- tion flux will make use of a device closely related to a Carnot engine. A conventional Carnot engine (Fig.

231

232 S. KABELAC

I

0,oi --y,

I Ti,~ I

system boundary . /

i

- t F (a)

system boundary /

T.,. A.,

7-"

I

I - F - J \ (b)

, / J

~2

J

¢'0

-....

Fig. 1. The conventional Carnot engine (a) and the modified Carnot engine (b). emitting a black-body radiation flux if2

instead of releasing a heat flux Oo~t.

with the conventional Carnot engine). This, however, results in a conflict with the second law of thermo- dynamics, as can be seen from Fig. 2. This graph shows the quantity P + TjSi, (eqn (6)), which results from eqns (4) and (5), as a function of 7"2, the temperature of the emitting surface.

P + T , S ~ = ~ 0 1 - ~ 0 0 - ~ 2 1 - ~ . (6)

The three curves account for three different input tem- peratures T~. If Sir~ is set to zero, the ordinate gives the power output P. The modified Carnot engine could then continuously perform work even though the out- put temperature T2 is higher than the input tempera- ture T, (see for example the curve T, = 650 K, which gives positive quantities up to 7"2 ~ 830 K). This par- adox was pointed out by Wiirfel[l 1] and clarified by De Vos[12]. The conversion of a heat flux into a ra- diation flux (and vice versa) is inherently irreversible, if the temperatures T2 and To are not equal. This ir- reversibility Sirr,Q can be quantified by eqn (7), an expression which originates from Planck [ 10 ]:

Sirr, Q = ~ A 2 o ' T 3 + A2aTo~ ' -~2 -

l (a)) will be understood as an ideal device, which converts a given heat flux (~, into mechanical power P. The third energy flux coupled to the engine is the "waste" heat flux Qou~, which takes care of the entropy flux transported into the engine via (~io. In order to facilitate the understanding of the radiation conversion device introduced in section 3, a modified Carnot en- gine (Fig. l (b) ) will be discussed as an intermediate model. This modified engine rejects the heat flux not by means of conduction/convection as in the conven- tional case, but by means of a radiation flux from area A2 into the surrounding half space with an ambient temperature To. Thus during this intermediate step the conversion of a heat flux into a radiation flux is considered• Given the rate of heat (~io delivered by conduction or convection at T,, it is asked for the maximum power output P. This is found by applying the energy conservation law (eqn (4)) and the second law of thermodynamics (eqn ( 5 )), introducing the en- tropy generation rate Si~.

- - A2°'( T2 - To )2 ( T 2 + 3 T 2 + 2 T z T o ) . (7) 3T2

The quantity S~=,o is always positive except for the case of equal temperatures T2 = To . This case, however, is not very helpful, since with 7"2 = To the entropy pro- duction rate in eqn (5) becomes negative unless 0in is set to zero, as well.

Thus it seems, as if there is no possible operational mode of the modified Carnot engine without entropy generation. This is contrary to the conventional Carnot engine, which gives rise to the Carnot efficiency in the reversible case. But until now the rrle of the emitting area A2 has not been taken into account properly. This area, which does not appear in the corresponding equations for the conventional Carnot engine, may be chosen arbitrarily, since it is a "machine-specific" value. In order to enable a reversible mode of operation, this area A2 must increase in a way that a certain finite input heat flux is reached as 7"2 approaches To. Taking A2 as a function of temperature, eqn (5) reads

P = O_i. + '~o - ¢b2 = Oi . - A z a ( T ~ - T g) (4)

4 ~P2 4 t~ 0 0in 3 T 2 3To T1

4 3 Qin = 3 A2~(7"2 - T ~ ) - T--S (5)

These two equations do not suffice to calculate P ex- plicitly, since there are three unknowns, i.e. P, Si~, and 7"2. In a first step we set Si~ = 0, as we are ex- amining the reversible case (analogous to the foregoing

lim ] aA2( T)[ T2 3 - T g] = 4 a C T 2 "- Qin r2~ To T ~

( 8 )

Here the limiting rule ofl 'Hrpital has been applied for the first part of the equation using the following tem- perature dependent function A = A (T)*

C A ( T ) - - - (9)

T2- To"

* The foregoing may easily be extended to functions A(T) = C / ( T ~ - T g ) , n > O .

Maximum conversion etficiency of black-body radiation 233

+

20

kW

15

A z = 1 m e T o = 3 0 0 K

I I I I I I I I

- T , ~ " ~ ' ~ 2 ~ , ~ "

1 0 - . ~

P

5 -

, \ % ~ \ ",,), \ - 1 0 I I I t I ~ 1 \ I I

0 100 2 0 0 3 0 0 4 0 0 500 600 700 800 9 0 0 K 1000

' T e To

Fig. 2. The plot of eqn (6) against the emitter temperature T2 for the modified Carnot engine showing the violation of the second law in case of Si~ = 0.

This type of function for A seems uncomfortable at first, because the area A2 grows to infinity as T2 ap- proaches To. But a comparison with the conventional Carnot engine reveals the same phenomenon there. Nevertheless the conventional Carnot engine is well accepted as the ideal conversion device which gives the maximum power Pm,x from a given heat flux (~i,. Looking at the conventional Carnot engine, let the outgoing heat flux Qo,t = 0~n - P be transported to the ambient via convective heat transfer described by

Oout = ~A2( T2 - To) .

Solving for the heat transfer area A2 with a fixed value 0o,t and a given heat transfer coefficient h, one obtains an equation similar to eqn (9). When a reversible heat transfer interaction is required, the temperatures ?'2 and To have to become equal, so that the (reversible) Carnot engine has infinite large heat transfer areas, as well.

Returning to the modified Carnot engine, the con- stant C in eqn (9) is determined via the input heat flux 0i , . Inserting eqn (9) into eqn (8) and applying the rule of l 'Hrpital results in

C = Qi. 1 Yi. 4 ~ T o ~

The energy conservation law, eqn (4), then yields

Pmax = lim (~i, - - C o - = Oin 1 To r:~7~ T2 - To ]

With this new look at A 2 , the same maximum power Pm~x results as in the case of the conventional Carnot engine case. It must be kept in mind that a reversible operation of both types of Carnot engines requires in- finite large heat transfer areas. Summarizing this sec- tion, it was shown that the modified Carnot engine (Fig. I (b ) ) is equivalent to the conventional Carnot engine. A (theoretical) reversible operational mode is possible in both cases.

3. THE RADIATION CONVERSION ENGINE

The conversion of radiation is considered by means of the conversion device shown in Fig. 3. It receives a black-body radiation flux ~i,, characterized by the temperature T~n, from the surrounding half space and releases a heat flux 0oul, power P and a black-body radiation flux ~ . As was shown above, it does not make any difference if the entropy (i.e., the "'waste" heat 0o.,) is rejected conventionally by conduct ion/ convection or by means of radiation. The device shown in Fig. 3 is not necessarily a heat engine. It might be e.g., a photovoltaic cell as well, as long as continuous energy conversion is realized.

The first and second law of thermodynamics for this case are

P : A,~(T?n - T~) - 0oo, (lO)

5"irr - Qo,t 4 To 3A,a(T3,- T~). (11)

234 S. KABELAC

\

J

1 t i 11. A I

"-5 -7 0o~,

Fig. 3. A device for the continuous conversion of a black- body radiation flux.

The question which arises is if one is allowed to set S~r = 0 in eqn (11). In general, if the conversion of a radiation flux into a heat flux is allowed, Si~ must be taken into account according to eqn (7). But con- verting a radiation flux into a heat flux is not the only possibility• One could also think of a (direct) conver- sion into rates of electrical or chemical energy, Because of this uncertainty, different cases will be studied and compared•

explicitly. Thus, S~ is not set to zero, but equal to girt.Q, eqn ( 7 )

(1 T4~ 4 ) '~irr = Sirr,Q = mlO ~ T~ Jr- ~-1 - 3 T3n ' (13)

This is a severe restriction, since if the radiation flux is not converted into a heat, but into an electrical or chemical energy flux instead, the above eqn ( 13 ) is not correct. By inserting eqn (13) into eqn ( 11 ), one can eliminate Qo,t, which results in

( T , ) ,41aTo " '" - T 3 (14) P = A'tr( Ti4"- T 4 ) - T, "

Computing the maximum power output with re- spect to the unknown temperature T~ by setting the derivation OP/OTj equal to zero, one obtains

0 = 4 T ~ - 3 T o T 4 - T4, To. (15)

Solving this equation gives the optimal operational temperature T~ = Too,. Thus, eqn (14) reads

3.1 Case a." (Si~, " 0; A, = const. ) In this first case we take Sj , to be definitely zero

and leave A~ as a constant. Eliminating Qo~, in eqn (10) and ( 11 ) results in

( 4To) ( 4To) P = A , a T 4 . 1 - ~ " ~ i ~ - AI~rT ~ 1 - ' ~ ~ .

The temperature TI of the radiation intercepting area A~ is unknown. To determine the optimal mode of operation, the power P is maximized with respect to T~ (i.e,, OP/OT~ - 0), which gives T~ = To, and thus

Pmax= ~in l - ~ ' ~ i ~ ] + ~ A l a T 4 or

P . . . . (1 4 To 1 T g \

• , ° _

This result is the same as eqn ( 1 ), representing the first of the three efficiency formulae for a black-body ra- diation flux found in the literature. Recognizing that T~ 4= T~,, this result is not valid, when the incoming radiation flux is converted into a heat flux, because then Si~ would definitely not be zero• Thus, eqn (12), which results when S~, is forced to be zero while A~ is left as a constant, is not applicable to a heat engine device, but would hold true, e.g., for a photovoltaic cell. Nevertheless, in most solar energy applications solar radiative energy fluxes are converted into heat fluxes•

3.2 Case b: (S~r = Sift, O; AI = const . ) Here we consider a case, where just the opposite of

case a) is true. The incident radiation flux must be converted into a heat flux, because this is the only case for which the entropy production rate '~irr is known

Topt / (16)

This is the second well-known relation for the maxi- mum conversion efficiency of a black-body radiation flux, which has already been stated in eqn (2). This version holds true only if the radiation flux is definitely converted into a heat flux within the intercepting area A~. Certainly this will be true for many (solar) systems, but searching for the theoretical limit, no restrictions should be made (see for example the dispute between Castafis and Jeter in [ 13 ], where Castafis contends that eqn (16) is universal, whereas Jeter accepts it only as one possibility), 'I',e, in eqn (16) is the difference be- tween the incoming radiation energy flux ff~n and the re-emitted energy rate 'Ih, ~,e, = ~I,~, - ~I,j, i.e., the net-energy rate transferred from the absorber into the inner part of the conversion device. Because of the intrinsic entropy production in this case (T, = Top,

Ti.), eqn (16) may not be termed exergy. (It must be noted that this upper limit, eqn ( 16 ), can be slightly raised when an infinite set of spectral selective con- verters is considered[7,8]. The resulting expression gives values below the Carnot efficiency.)

3.3 Case c: ( S i , -+ 0; A~ variable) This case represents the transfer of a radiation en-

ergy flux into a heat flux without entropy production. This can be realized by letting T~ approach Tin. Finite rates of energy in this case (see eqns (10) and (11)) will result only ifA~ = A ~ ( T ) ~ oo. In analogy to section 2 this is achieved with the function A l (T) = C~ ( T~, - T~ ). The constant C is determined by a given

Maximum conversion efficiency of black-body radiation

finite net energy flux ~ne~, which actually reaches the inside of the converter. Letting T~ approach Ti. with the requirement

B

lim A ~ ( T ) a ( T 4 , n - T 4 ) - ~ .... (17) TI~ Tin

which yields

C : ~nel

4a T ~n "

Insertion into eqn ( 11 ) gives

4 T ~ . - T ~ ] dPnet 0out lim Ca - - + Si~.e . . . . . .

T~7"i, -~ T i , - TI Tin To

Sirr,Q approaches zero as T~ approaches Ti., so that the second part of this equation, together with eqn (10) and (17), gives the maximum power Pm,x

Thus, the theoretical maximum efficiency for the con- tinuous conversion of a black-body radiation flux is described by the Carnot efficiency. This was already stated by Jeter in 1981 [9 ], but because of his disputed derivation this result has not been established in the literature. Two arguments may be raised against the spirit o f e q n (18). The first one (which may, for ex- ample, emerge from a comment made by Bejan[l , p. 502]) could reject the "totally unrealistic" case A ~ ~c. This surely is a fictitious machine, but what we are looking for is the theoretical maximum con- version efficiency. Admitting for instance a temperature difference of 2 K between T~, and T~, one is close to the reversible case while the area A~ is still finite. As has been shown with the help of the modified Carnot engine in section 2, this derivation parallels the con- ventional Carnot engine, which is well accepted as the theoretical upper limit of the conversion of a given heat flux. The effect of a limited heat transfer rate on the efficiency of a conventional Carnot engine has been studied by Curzon and Ahlborn [ 14].

The second drawback of eqn (18) (which is ad- dressed in a comment by De Vos[15, p. 82]) is that the correct definition of the conversion efficiency is disputable, i.e., which denominator is to be used for normalizing Pma~- Concerning solar application De Vos may be right in saying that "we are not paying for keeping the sun hot," so consequently ~i, should be used. But there is no way to eliminate the re-emission of the energy rate (Ih, which is intrinsically coupled to the absorption of a radiation energy flux. (One inter- esting idea, however, which is not pursued here, is the delay of the re-emission using an optical circula- tor[16].) Thus looking for the basic derivation con- sistent with analogous definitions in thermodynamics, e.g., the Carnot efficiency, the proper denominator is

235

':I'.~t = (I,~, - ~ j , because this corresponds to the energy transfer to the device just as 0~, does in conventional Carnot engines. Note that 0 and ~n~t incorporate a temperature difference, but ~ , is a function of one temperature only. Even though many parallels have been drawn between the Carnot engine and the radia- tion conversion engine, these considerations are con- strained to blackbody radiation fluxes, which represent a very special case, These results hopefully will not be used to strengthen the idea of radiation being "heat" (see, e.g., Jeter's comment in [ 15 ]). A black-body ra- diation flux may be somewhat similar (as the funda- mental relation 0~/0qt = Tholds) , but this idea would be a great handicap when other types of radiation are considered (the laser being one example).

4. CONCLUSION

From the thermodynamic viewpoint it has been shown that, using a derivation closely related to the Carnot engine, the reversible emission and absorption of a black-body radiation flux with finite energy rates is possible. Thus, the maximum efficiency of the con- tinuous conversion of a black-body radiation flux is the Carnot efficiency. Other efficiency formulae pub- lished in the literature suffer from restrictions, which will certainly become important when looking at solar application, but which should not be confused with a theoretical upper limit for the conversion of a black- body radiation flux.

NOMENCLATURE

A area, m 2 ( ' constant, m 2 K

energy flux, W h heat transfer coefficient, W/m 2 K P power, W 0 heat flux, W ko entropy flux, W/K

• ~rr entropy production rate, W/K • ~i,.# entropy production rate due to absorption or emission.

W/K 7" temperature, K

To~, optimum operation temperature (see. 3.2), K a Stefan-Boltzmann constant, a - 5.67.10 s W/m z K 4

Subscripts 0 ambient 1 energy input port of conversion device 2 energy output port of conversion device

in energy source

REFERENCES

1. A. Bejan, Advanced engineering thermoclvnamkw. Wiley, New York (1988).

2. P. T. Landsberg and J. R. Mallinson, Thermodynamic constraints, effective temperatures and solar cells. In: Int. Colloquium on Solar Electricity, CNES, Toulouse, pp. 27-46 (1976).

3. P. T. Landsberg, An introduction to nonequilibrium problems involving electromagnetic radiation. In: J. Casas- Vfizquez et al.. (eds.), Recent developments in nonequi- [ibrium thermodynamics. Springer-Verlag, Berlin, pp. 224-267 ( 1986 ).

236 S. KABELAC

4. W. H. Press, Theoretical maximum for energy from direct and diffuse sunlight, Nature (London) 264, 734-735 (1976).

5. F. Bosnjakovi6 and K. F. Knoche, Technische Thermo- dynarnik, Teil 1. 7th ed.. Steinkopff-Verlag, Darmstadt ( 1988 ).

6. M. Castafis, Bases fisicas del aprovechamiento de la ener- gia solar, Rev. Geofis. 35, 227-239 (1976).

7. A. F. Haught, Physics considerations of solar energy con- version. In: Int. Conf. on Photochemical Conversion and Storage of Solar Energy, Boulder, CO (August 1980).

8. A. De Vos and H. Pauwels, On the thermodynamic limit of photovoltaic energy conversion, Appl. Phj,s. 25, 119- 125(1981).

9. S. J. Jeter, Maximum conversion efficiency for the utili- zation of direct solar radiation, Solar Energy 26, 231- 236 (1981).

10. M. Planck, Theorie der Wdrmestrahlung. 6th ed., Barth Verlag, Leipzig ( 1966 ).

I 1. P. Wfirfel, The chemical potential of radiation, J. Phys. C: Solid State Phys. 15, 3967-3985 (1982).

12. A. De Vos and H. Pauwels, Comment on a thermody- namic paradox presented by P. Wfirfel, J. Phys. C. Solid State Phys. 16, 6897-6909 (1983).

13. M. Castafis and S. M. Jeter, Comments on "Maximum conversion efficiency for the utilization of direct solar ra- diation," Solar Energy 30, 293 ( 1983 ).

14. F. L. Curzon and B. Ahlborn, Efficiency ofa Carnot engine at maximum power output, Am. Z Phys. 43, 22-24 (1975).

15. A. De Vos and H. Pauwels, Discussion of"The second law efficiency of solar energy conversion," J. Solar Energy Eng. 108, 80-83 (1986).

16. H. Ries, Konzentration diffuser Strahlung, Ph.D. Thesis, University of Munich ( 1985 ).