striking the black end cap, then almost all of the solar radiation will be absorbed, a≈ 1; and the hot gas emerging from the exit pipe of the cooling jacket will be approximately at the same temperature as the closed end of the inner tube, which will be filled with essentially blackbody radiation characteristic of this temperature, designated T. In practical use a tube length to diameter ratio of 15 is adequate.

Even with much lower reradiation losses from the volume near the entrance due to the lower temperatures with coolant flow and the strong T temperature law for radiation, much of the entering solar power would escape again as thermal radiation if it were not for the wavelength selective properties of the mirror tube walls. Only a small fraction of the intense black body radiation filling the end of the tube remote from the entrance can escape directly, and much of that directly escaping flux would be reflected back by the shadowed or central section of the concentrating mirror. For most of the radiation generated by walls at temperature TL or less, the glass walls are opaque, and consequently the intense wall radiation at the far end of the tube cannot see the silver film and mirror its way out like the solar radiation did on its way in. (Glass is opaque to infrared radiation because it is an absorber not because it reflects it.3) In this respect the tubular device resem

B-2

bles a set of nested greenhouses except that solar reflection at the glass greenhouse surfaces reduces the transmission to the innermost greenhouse and thus limits the number of greenhouses that can be usefully nested to three, whereas solar reflection by the glass surface in the tubular device actually increases the transmission of solar energy to the end remote from the entrance. In the interest of higher Carnot efficiency (higher T) quartz may replace glass at the hot end of the mirror tube, but it should grade into glass or be doped with an infrared absorber near the cooler entrance end to preserve this selective mirror action. Thus while thin glass walls are desirable if heat is transferred through them, they should nonetheless be thick enough to be opaque to the wall radiation.

This tubular absorber can be significantly improved by refinements suggested in Sec. II.B, but then the analysis required to predict its performance is made much more difficult and will not be presented here, as even this inferior version will be shown to be quite attractive. Environmental radiation entering along with the sunlight will be neglected in the analysis, but it also aids the system.

(1) All the energy added to the gas flowing in the coolant jacket must pass through the walls of the mirror tube, and even if they are thin, some inefficiency results. Thus if the coolant gas is transparent to sunlight it is both more economical and more efficient to eliminate the cooling jacket annulus and simply connect the coolant flow exit tube through a very porous black end cap at the remote end of the mirror tube. In order to do this on earth for a gas other than air, one can simply enclose both the tubular device and the paraboloid mirror in a thin, pressure supported, transparent plastic bubble. Even for air this may be economically attractive to protect the concentrating mirror and to remove wind loads from the heliotrope. In space applications the entrance sealing bubble need not enclose the concentrating mirror but is simply a pressure window remote enough from the entrance to tolerate the pressure load with the partially concentrated solar flux incident, perhaps including a fluxtrap as part of its support structure as space mirrors are of low quality. Thus far we have tacitly assumed that the gas inside the mirror tube is transparent for both the solar and heat radiation; however, many molecular gases, i.e., water vapor laden air have strong absorption bands in the infrared. Consequently much of the wall radiation that would otherwise escape can be internally reabsorbed in the coolant flow. Not only does this effectively prevent wall radiation from escaping, but at high temperatures it significantly aids in the radial transport of energy in the cooling gas and reduces radial temperature gradients. Unfortunately the radiation from the hot gas partially replaces the wall radiation it prevents. But at no wavelength can this hot gas radiation exceed the essentially blackbody ra

October 1974/ Vol. 13, No. 10 / APPLIED OPTICS 2431

diation it replaces; thus the net effect is beneficial. Because the hot gas radiates strong lines of bands that the cold gas does not absorb well, this calculation of radiative transport inside the tube is very complex. Therefore in our analysis we have assumed that there is a vacuum in the interior of the mirror tube and that coolant flow in the annulus is employed.

(2) With uniform solar reflectivity the thermal flux on the mirror tube walls decreases with distance from the entrance. Because of this and the fact that the coolant temperature is steadily increasing, the heat flux into the coolant decreases with distance more rapidly than the thermal flux on the walls. Economic considerations related to the insulation cost, may make an intentional reduction in the solar reflectivity with distance from the entrance desirable even though the reradiation losses would increase. Then shorter tube lengths with the same net solar absorption would be possible.

(3) Also the cross section of the mirror tube should increase with distance from the entrance in the initial section of the tube so that solar radiation collected by the surface of the paraboloid mirror near the rim is converted into more nearly paraxial rays after its first reflection inside the tube and thus travels much further into the tube before being absorbed. The paraboloidotoroidal phocon5 is useful for this purpose. For example, if the rim angle of the concentrating mirror is 60° the length of this initial phocon section required to convert all rays to 30° or less inclination to the tube axis is only 1.15 maintube diameters. The entrance area for this example would be only 28% of the maintube cross section and consequently much of the shortest wavelength wall radiation or hot gas radiation not blocked by the selective mirror action of the walls would fail to escape from this convergent phocon exit. Both the increased distance between reflections for the solar radiation propagating into the tube and the relatively larger main tube diameter also act to make the thermal wall flux load very much smaller than the entrance flux. However, this paraboloidotoroidal phocon entrance section increases the complexity of the analysis and like the other improvements will be omitted.

C. Analysis

Since we assume that the absorber tube is perfectly insulated, the total energy transport rate across each cross section is a constant. This net power absorbed is

P1 = C(x) - K(x) − ƒ„(x) + F,(x) + ƒ,(x), (1) where x is the distance from the entrance, C(x) is the convective energy rate, K(x) is the power transported back toward the entrance by thermal conduction, fn(x) is the net wall radiation traveling back toward the entrance, F,(x) and f,(x) are, respectively, the short wavelength radiation from the sun and the long wavelength radiation, escaping from the tube but reflected back into it, which reach x without prior ab

and RT is the temperature that a black disk covering the entrance would attain in the cold radiation field of space. Now R, e, p, and To are independent parameters, but Ea is dependent on them and S, i.e., Ea is a monotonically increasing function of Cm. Unfortunately, Ea(R, e, p, To, S) is not known a prio

ri. When Cm is large (S small), the temperature achieved deep in the tube, TL, (L » D) is low. Hence, even though E, is large when Cm is large, the over-all efficiency

B-4

functions of Cm (or S) assuming that the parameters R, e, p, and To are fixed.

In order to find the monotonic function, E.(Cm), and consequently TL(Cm) or TL(S) via Eqs. (14) and then E(Cm) or E(S) via Eqs. (15), we first characterize the system under study by a set of values for the parameters of Eq. (11), and also for both p, the wall reflectivity for the incident short wavelength flux, and f(0), its angular distribution, so that 7,(x) can be calculated as outlined in Ref. 7. Then we select a particular E, of interest, calculate G, and guess a test value of Cm that we shall designate C, If C, > Cm, then as Eq. (11) is solved for T(x) at increasing x, using the initial conditions, the derivative T'(x) will remain too large for x ≥ 0 and in fact TL →. Likewise, if C, < Cm, then TL → ∞ as x → ∞. Consequently, for each physically realistic E. there is a unique value of C, for which T remains finite as x →

and the relationship between E, and Cm or S is found from this condition.

But, for a simplified but practical example of this procedure, we assume

E≤ E.E.

(12)

B-5

Often the true value of Cm (or S) differs from the equality value given by Eqs. (18) by only a few percent. We will avoid numerical analysis here and continue by using the largest value of Cm permitted by Eqs. (18), i.e., the equality value. This approximation is conservative, as the value of T calculated from Eqs. (14) for fixed values of all the parameters on the right side of Eqs. (18), including E., is a few percent too low. Thus the Carnot efficiency and consequently the over-all efficiency E calculated with Eqs. (15) is underestimated by a few percent if Cm is taken at the largest value permitted by Eq. (18). Now with reference to Fig. 2 of Ref. 7, we assume

A conventional black absorber-radiator in the same solar flux would have much greater reradiation losses, lower E., and reach temperatures far less than TL. For example, a conventional black absorber in an R = 2 flux has a peak temperature even with no useful power output of only 2To, thus its Carnot efficiency would not exceed 50% even if the over-all efficiency fell to zero. Thus this tubular device is three to four times more efficient and functions well in poorly concentrated sunlight.

We can define an effective emissivity, e, for the entrance surface in terms of the exhaust temperature achieved and the reradiation losses, i.e.,

E.

Discussion

This analysis has assumed that the absorber was Bedeployed in the cold radiation field of space. cause of its high efficiency, relatively little waste heat must be rejected. Thus, in space, with a relatively small waste heat radiator located in the shadow of

1.0

€ = 0.9

and assume

P = 0.5

105

as a conservative estimate of the fraction of wall radiation that can be reflected back into the tube with a good optical design-phocon or external mirrors. Next, for a particular R, i.e., R = 2, we assume various values for E, and with Eq. (18) calculate the corresponding value of S and then for this S and E, calculate E from Eq. (15b). The results of this R = 2 case are shown in Fig. 2, and we see that a maximum over-all efficiency of 73% results when E. = 90%. More rapid coolant flow would produce higher absorber efficiency but lowers the exit temperature excessively. Likewise less rapid coolant flow can result in better Carnot efficiency, but reradiation losses reduce E. excessively. The efficiency at optimum coolant flow rate for other values of R is shown in Fig. 3.

2434 APPLIED OPTICS / Vol. 13, No. 10 October 1974

8 MERIT INDEX ale

103

10.7

CONVERSION EFFICIENCY, E

Fig. 3. Conversion efficiency, E, and effective a/e merit index at optimum coolant flow vs solar flux index, R, ratio of a one-sided black disk absorber temperature to coolant entrance temperature.

the concentrating mirror, To « 300 K would be possible. Then rather large values of TL/To could be tolerated without damage to a quartz mirror tube. In addition to higher predicted efficiency, for units producing several kilowatts or more, this absorber and advanced turbine generators4,8 developed for space applications offer significantly lower cost and liftoff weights compared with either solar cells or isotopic power supplies.

On earth, To ≥ 300 K, and material problems limit the TL/To ratio to about 5. Thus for most of the day, even in northern latitudes, the flow rate would be automatically controlled to maintain the highest possible Carnot efficiency permitted by the construction materials and the absorber would operate at greater than optimum efficiency. A linear slot absorber over a parabolic mirror is also attractive and may also be able to produce a materials limited exhaust tempera

ture.

In spite of the fact that E, is nearly unity, economic considerations would result in a significant part of the sunlight collected by the concentrator being poorly focused and wasted. However, unlike the uncollected waste heat that goes up the chimney of a conventional power plant and adds to the local environmental heat load, this concentrated sunlight can be reflected so as to escape from the earth at negligible incremental cost. Thus there need be no change in the local albedo. Since this simple economic absorber is not dangerous, does not require intense sunlight, and need not have any net local ecological impact, it should be located in urban power demand areas to avoid transmission losses. It can also make

B-6

fuel by producing hot steam for a chemical cracking process.9

III. Summary

A high temperature solar absorber utilizing new operating principles to reduce reradiation losses has been described. One, less than optimum, example has been analyzed, neglecting both thermal conduction losses and environmental radiation gains, using expressions for 7, and 7 that are simplified mathematically but physically quite plausible. High efficiency is predicted even with poorly concentrated sunlight such as would be available from a flexible mirror deployed in space or from concentration optics economically deployed on earth. Ironically a/e <<< 1 for the mirror-tube walls permits fantastically high effective a/e ratios for the tubular absorber.

References

1. A. B. Meinel and M. P. Meinel, Phys. Today 25, (2), 44 (1972). 2. B. O. Seraphin, Appl. Opt. 12, 349 (1973).

3. R. Gordon, J. Am. Ceram. Soc. 44, 305 (1961).

4. K. E. Nichols, Progress in Astronautics and Aeronautics (Academic Press, New York, 1963), Vol. 11, p. 891.

5. V. K. Baranov, Geliotekh. 2, (3), 11 (1966).

6. W. R. Powell, Appl. Opt. 13, 593 (1974).

7. W. R. Powell, Appl. Opt. 13, 952 (1974).

8. D. A. Perz, 7th Intersociety Energy Conversion Engineering Conference (American Chemical Society, Washington, D.C., 1972), p. 346.

9. Euratom CCR-ISPRA, Report 1, "Hydrogen Production from Water Using Nuclear Heat" (1972) Available from NTIS (EUR4776e).

24-215 O- 78 - 58