Space Solar Power Review Vol 3 Num 2 1982

SPACE SOLAR POWER REVIEW Volume 3, Number 2, 1982

SPACE SOLAR POWER REVIEW Published under the auspices of the SUNSAT Energy Council Editor-in-Chief Dr. John W. Freeman Space Solar Power Research Program Rice University, P.O. Box 1892 Houston, TX 77001, USA Associate Editors Dr. Eleanor A. Blakely Lawrence Berkeley Laboratory Colonel Gerald P. Carr Bovay Engineers, Inc. Dr. M. Claverie Centre National de la Recherche Scientifique Dr. David Criswell California Space Institute Mr. Leonard David PRC Energy Analysis Company Mr. Hubert P. Davis Eagle Engineering Professor Alex J. Dessler Rice University Mr. Gerald W. Driggers L-5 Society Mr. Arthur M. Dula Attorney; Houston, Texas Professor Arthur A. Few Rice University Mr. I.V. Franklin British Aerospace, Dynamics Group Dr. Owen K. Garriott National Aeronautics and Space Administration Professor Norman E. Gary University of California, Davis Dr. Peter E. Glaser Arthur D. Little Inc. Professor Chad Gordon Rice University Dean William E. Gordon Rice University Dr. Arthur Kanfrowitz Dartmouth College Mr. Richard L. Kline Grumman Aerospace Corporation Dr. Harold Liemohn Boeing Aerospace Company Dr. James W. Moyer Southern California Edison Company Professor Gerard K. O'Neill Princeton University Dr. Eckehard F. Schmidt AEG—Telefunken Dr. Klaus Schroeder Rockwell International Professor George L. Siscoe University of California, Los Angeles Professor Harlan J. Smith University of Texas Mr. Gordon R. Woodcock Boeing Aerospace Company Dr. John Zinn Los Alamos Scientific Laboratories Editorial Assistant: Edith R. Mahone Editorial Office: John W. Freeman, Editor-in-Chief, Space Solar Power Research Program, Rice University, P.O. Box 1892, Houston, TX 77001, USA.

0191 -9067/82/020097-01$03.00/0 Copyright ® 1982 SUNSAT Energy Council EDITORIAL Space Solar Power Review invites papers dealing with any area of space utilization or space industrialization. Papers are welcome on any subject pertaining to the use of space for the benefit of mankind. This includes technical papers on scientific or engineering aspects as well as social, environmental, or economic areas. Papers may be full length papers not to exceed 25 typewritten pages or short reports or technical notes not to exceed 5 pages. Manuscripts should be submitted to me with an original and two copies; all manuscripts will be refereed by members of the Board of Associate Editors. John W. Freeman Editor-in-Chief

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0191 -9067/82/020099-21 $03.00/0 Copyright ® 1982 SUNS AT Energy Council TRANSMISSION OF MICROWAVE BEAMED-POWER FROM AN ORBITING SPACE STATION TO THE GROUND R. M. WELCH* Institute of Meteorology Johannes Gutenberg University 6500 Mainz, West Germany J, M. DAVIS and S. K. COX Department of Atmospheric Science Colorado State University Fort Collins, Colorado 80523, USA Abstract — Transmission efficiencies and surface power densities are calculated from the interaction of a 10 GW microwave beam with rain clouds. Computations are made as a function of [a] frequency (2.45 to 10 GHz); [b] beam nadir angle; [c] raindrop size distribution; and [d] cloud shape. Scattered surface power densities outside of the receiving rectenna do not exceed 10 gW/cm2 for frequencies of 2.45 and 3.3 GHz, even for extremely heavy rainfall rates. At higher frequencies exposure levels outside of the rectenna may reach 100 q.W/cm2, or two orders of magnitude less than the U.S. safety standard. From the standpoint of public health and safety, the scattering of microwaves by rain clouds is not a serious problem, with scattered fluxes outside of the rectenna much smaller than sidelobe fluxes. Beam losses due to absorption in rain clouds are significant in some cases, with absorption losses far more important than scattering losses. The amount of scattering increases with increasing microwave frequency, increasing drop size and drop concentration and increasing nadir angle of the beam. I. INTRODUCTION The concept of a geosynchronous solar power satellite (SPS) has been given increased attention as an alternative energy source. In spite of the costs associated with construction in orbit, the power satellite would be competitive with nuclear energy production. The proposed system would provide energy all but 72 min during the day and would be immune to various ground-based limitations such as dust and clouds. Studies of the SPS concept have been reported by many authors (1-8). The current SPS proposal centers around a 21.3 x 5.3 km2 silicon-solar-cell array, oriented towards the sun and providing 17 GW of continuous power to one or two 1 km-in-diameter slotted waveguide transmitters. The transmitter(s) consists of approximately 104 klystron or amplitron oscillators. The surface rectenna consists of large arrays of dipole antennas with a solid-state rectifier at each dipole to collect the microwave energy and convert it to d.c. The microwave d.c. power then is collected, converted to a.c. and distributed over conventional power grids. A review of the various efficiencies and design criteria has been given (9). An extensive review of the *Present address: Department of Geophysical Sciences, Old Dominion University, Norfolk, VA 23508.

various considerations involved in the SPS concept reaching maturity has been presented by Glaser (10). The American Institute of Aeronautics and Astronautics (AIAA) has recently completed a detailed position paper on space power system options (11). They report that “The major questions regarding these transmission schemes center on their environmental impact. Much more needs to be known about the interaction of such beams with the ionosphere, and there is considerable divergence of opinion about safe intensity levels for birds, other wildlife, vegetation, or humans living in the area surrounding a receiving antenna.” One of the prime reasons for the selection of a microwave transmission frequency is the fact that microwaves interact only weakly with cloud drops and aerosol particles. However, scattering and absorption of microwaves by rain drops increases rapidly with increasing size of the drops and with increasing beam frequency. It is the interaction of the microwave beam with rain drops which constitutes the main topic of this paper. Surface microwave power densities are calculated for various rain drop size distributions and at frequencies ranging from 2.45 to 10 GHz. II. COMPUTATIONAL PROCEDURE Even though absorption dominates scattering in the microwave region, it is important to determine the spatial distribution of power scattered out of the beam in order to determine probable system efficiencies and design criteria, as well as to evaluate health and ecosystem impacts. The Monte Carlo radiative transfer technique is ideally suited for the study of both radiance and irradiance patterns in finite clouds, or for beamed energy propagating through clouds (12,13). The present model consists of gaseous and droplet absorption superimposed on the basic Monte Carlo scattering procedure (14). Absorption by water vapor and oxygen is calculated from the top of the atmosphere to the point of entry into the scattering zone. The scattering routine considers the path of photons undergoing pure scattering by rain drops. Within the scattering zone, absorption along photon paths is determined both from drop and gaseous contributions. In order to complete the absorption profile, photon trajectories and absorptions are computed from the point of cloud exit either to the Earth's surface or to the top of the Earth's atmosphere. The scattering model closely resembles that developed by McKee and Cox (12). A rectangular parallelepiped cloud region is subdivided by mutually perpendicular intersecting planes into smaller parallelepiped boxes. By assigning absorption and scattering parameters within each box, inhomogeneous clouds with the desired attenuation characteristics are constructed. Photon entry points on the top of the cloud may take on the desired spatial distribution, from uniform to Gaussian. A photon enters the cloud region with a specified nadir and azimuth angle and travels a distance s, at which point scattering occurs. Transmittance T is the probability that a photon travels a distance 5 between scattering events, where r is the optical depth, and ^3, is the volume extinction coefficient for the rth box. A random number is chosen for T, and 5 is determined by summing along the photon path until the right-hand-side of Eq. 1 equals the given random number. At

this point, scattering is assumed, and a new direction of travel is determined by selecting values for the Eulerian angles a and A relative to the old direction. Angle A represents a counterclockwise rotation of the original xy coordinate axes about the z axis. The value for A is selected from a uniform angular distribution ranging from 0 to 2tt. The direction of propagation along the new z axis is obtained by a counterclockwise rotation of the old z and new y axes about the new x axis through the angle a. The probability of a photon being scattered between 0 and the angle a is given by the normalized integral over solid angle of the scattering phase function P(a). A

random number RN between 0 and 1 is assigned to this probability, and the angle a is then determined from the solution of Water vapor and molecular oxygen absorption coefficients are taken from Ref. 15. Oxygen absorption is based upon Van Vleck's theory (16) with line parameters determined by Carteret al. (17). Calculations of water vapor absorption are based on parameters derived from the data of Becker and Autler (18). Absorption from other trace constituents such as NH;i and O3 is relatively small (19). The mid-latitude summer and winter atmospheres given by McClatchey et al. (20) are used for the vertical water vapor distribution. Oxygen and water vapor extinction coefficients (km-1) are given as a function of height and frequency in Fig. 1 for a clear atmosphere. Absorption by both oxygen and water vapor increases strongly with decreasing altitude. At a height of 10 km, absorption by oxygen is significantly larger than absorption by water vapor, except in the 20-25 GHz region. However, absorption by water vapor increases rapidly with decreasing altitude. At a height of 1 km, absorption by water vapor is less than that of oxygen only for frequencies less than 10 GHz. The indices of refraction for ice and water are strongly dependent upon temperature as well as wavelength within the microwave region. The complex permittivity e = (e' - ie") is determined as a function of temperature, from which the complex index of refraction n = (nr - in^ is then calculated (21). The indices of refraction of

ice and water as a function of temperature and frequency are given in Table 1. These values closely agree with values reported by other authors (22,23). III. MICROPHYSICAL PARAMETERS A number of droplet size distribution and rainfall rate models are used to examine the attenuation of microwave radiation by both scattering and absorption processes. The Rain L and Rain M size distribution functions for light and moderate rainfall conditions have been proposed by Deirmendjian (24). Rain L has a liquid water (wj content of 0.117 g/m3, with drop mode radius (rm) of 0.07 mm, while Rain M has wL = 0.495 g/m3 and rm = 0.05 mm. For moderate and heavy rainfall conditions, Deirmendjian has added the Rain 10 and Rain 50 rain drop distribution functions, respectively. Rain 10 represents drop spectra below the cloud base for a precipitation rate of 10 mm/h at the ground with rm = 0.33 mm and wL = 0.509 g/m3. Rain 50 represents heavy rain during rainfall rates of 50 mm/h at the ground, with rm = 0.60 mm and wL = 2.11 g/m3. Table 2 shows the volume extinction (/3e) and absorption (/3„) coefficients (km-1) for selected microwave frequencies as a function of temperature for the four Deirmendjian drop distribution functions mentioned above. Rain 50 represents rainfall rates of 50 mm/h, while measurements indicate that rates as high as 150-250 mm/h are not uncommon (25). The Marshall-Palmer (26) drop size distribution n(r) has been found to provide realistic fits for most continental mid-latitude situations, expressed with a negative exponential form: n(r) = noe~kr , (3) where n0 = 1.6 x 104 m-3 mm-1 for all rain (27) and r is the drop radius. The slope parameter X depends only upon rainfall rate according to X(R) = 8.2R-0-21, (4) with X in mm-1 and R in mm/h. These values are appropriate for moderate widespread rain. However, n0 is related to the intensity of convective activity and the process responsible for precipitation; therefore, n0 is not a constant (28). Applying the Marshall-Palmer (MP) distribution function for the values given above, Table 3 shows attenuation rates as a function of temperature for rainfall rates between 25 and 150 mm/h. The MP distribution is not appropriate for heavy rainfall. Nevertheless, it is utilized here in order to provide at least qualitative estimates. Underestimation of the number density of large drops using the MP distribution leads to an underestimation of the scattering characteristics. Comparison of Tables 2 and 3 shows that for a rainfall rate of 50 mm/h there are noticeable differences between the attenuation rates for the assumed drop size distributions. In general, the extinction and absorption coefficients are larger for the MP distribution than for the Deirmendjian modified gamma size distribution. While the liquid water content is slightly larger for the Marshall-Palmer Rain 50 (MP Rain 50) distribution than for the Deirmendjian Rain 50 (D Rain 50) distribution, this difference cannot account for the large variation in attenuation coefficients. The MP distribution for Rain 50 has a larger drop mode radius and a larger concentration of large drops. Attenuation parameters, in general, increase in value with decreasing tempera-

ture. This variation is far greater at low microwave frequencies than at high frequencies. For most of the distribution functions at a frequency of 3.3 GHz, a decrease in cloud temperature from 20 to -20 °C may lead to an increase in the attenuation parameters by up to a factor of three. However, while both extinction and absorption coefficients are highly temperature dependent, the scattering coefficient (/?, = fie - /3a) is almost temperature independent. In the consideration of the size of precipitating elements, there are two scales. Thunderstorms generally consist of very tall clouds (up to 10 km or higher) with widths approximately equal to heights (29). In such cases the cloud width may be approximately equal to the microwave beam diameter. On the other hand, widespread nimbostratus formations generally are of smaller height (up to about 3 km thick), but are much wider than the beam diameter. In both cloud types, temperature, drop size distributions, and rainfall distributions show wide vertical variability (30-37). Microphysical processes are at present so insufficiently known, and measurements of liquid water contents and drop size distributions so imprecise, that no general framework for detailed calculations is available. For these reasons the large number of cases found in Tables 2 and 3, spanning a broad range of possible conditions, are studied in order to adequately represent the variability of drop attenuation parameters found in clouds. For the purposes of the present investigation, scattering and absorption properties are assumed uniform throughout the rain cloud. Attenuation parameters corresponding to T = 0 °C are used unless otherwise noted. While it is recognized that real clouds have drop size distributions and temperature variations which are strongly height dependent, inclusion of these complexities is beyond the scope of the present study. IV. BEAM GEOMETRY The choice of beam geometry for the transmitted microwave power has not been defined. The two most severe restrictions on beam shape are maximum power density at beam center and maximum power density outside of the receiving rectenna. It is generally assumed that the beam will take a Gaussian shape with power densities P defined in terms of the distance r from beam center, as where r0 defines the beam shape. Figure 2 shows beam power density shapes for several possible configurations. Limiting the value of P (i.e., Px in Fig. 2) to 10 mW/cm2 at the edge of the rectenna (r = 5 km), leads to values of r0 = 5.48 km and Er = 21.7 GW. For such a beam shape, the power densities outside of the rectenna decrease slowly with increasing distance, having values of 6.93, 4.50, 2.73, 1.55, and 0.82 mW/cm2 at distances of 1, 2, 3, 4, and 5 km, respectively, from the edge of the rectenna (6, 7, 8, 9, and 10 km from the center of the beam). The Soviet microwave exposure standard is attained 10 km from the edge of the rectenna. Since biological studies have shown that these exposure levels are dangerous, a

more restrictive beam geometry is required. However, the safe level of exposure is currently unknown. The dashed lines in Fig. 2 show beam power densities at 2.45 GHz for assumed maximum power densities limited at the edge of the receiver to be 1, 0.1, and 0.01 mW/cm2 (P2, P^, and P4, respectively), with values of r0 = 2.82 km and Er = 5.7 GW, r0 = 2.14 km and ET = 3.3 GW, and r0 = 1.8 km and ET = 2.3 GW, respectively. Decreasing the maximum value of power density at the receiver edge also rapidly decreases the exposure levels in surrounding areas outside of the rectenna, but also rapidly decreases the total raidated beam power. The curve designated at P^ with r0 = 3.72 km radiates 10 GW of total power, with 3.9 mW/cm2 at the edge of the rectenna. V. RESULTS It is assumed that 10 GW of continuous microwave power, with a beam radius of 5

km, is incident at the top of the atmosphere (15 km in the present investigation). Ionospheric interactions with the beam are neglected, being outside of the scope of the present study. Photon paths are traced through the cloud until they are absorbed or scattered either back out of the atmosphere or to the surface. The total number of photon paths is chosen so that about 25,000 photons are scattered to the surface, allowing adequate statistics of surface power density levels. A uniform distribution of photons is assumed in most cases with a uniform power level across the beam of 12.7 mW/cm2, Actual beam configurations are dependent upon the required sidelobe suppression and beam frequency. In many cases it may be expected that up to 5% of the total beam power will fall outside of the 5 km-in-diameter rectenna. The results in this section examine several important microwave beamed power parameters as well as beam interaction with various rain cloud shape and attenuation parameters. First, the transmission efficiencies and surface power densities are computed as a function of beam frequency. Second, the effect of cloud temperature is discussed. Third, calculations as a function of beam nadir angle are presented. Fourth, the effect of cloud size on surface power densities is examined. Last, the effect of various drop size distributions upon transmission losses and surface power densities are compared. The drop size distribution is assumed as a first approximation to be vertically and horizontally homogeneous. Unless otherwise noted, beam nadir angle has been selected to be 0°, and rain cloud dimensions are 7 km in vertical extent and 24 km on a side. Clouds of this arbitrary size structure are used as a basis for later comparison with other cloud shapes. Cumulonimbus clouds are generally smaller than 24 km in diameter, while nimbostratus clouds are generally less than 7 km in vertical extent. A. Beam Frequency The Rain M drop size distribution (Table 2) arbitrarily has been chosen for the preliminary set of calculations presented in Table 4a. The effect of various beam frequencies from 2.45 to 10 GHz is examined at beam nadir angle of 0°. At 2.45 GHz, all but 84 MW of the total 10 GW of beamed-power propagates directly through the rain cloud without absorption or scattering. Of this loss to the beam, about 8 MW is scattered to the surface, 0.7 MW is scattered back to space, and 75 MW is absorbed by the cloud. The greatest loss to the beam power at all frequencies is absorption by drops within the cloud. The attenuation of the beamed- power increases rapidly with increasing frequency. Doubling the frequency from 2.45 to 5 GHz at this particular cloud optical depth increases beam attenuation to approximately 1.0 GW (or 10% of the total beamed-power); at 7 GHz the loss has increased to 2.2 GW, while at 10 GHz the loss is 4.1 GW (or 41% of the total beamed-power). It is generally assumed that beam transmission efficiencies of about 85% are acceptable for the SPS concept to be feasible, so that total attenuation losses of about 15% to the beam power are tolerable. Greater losses would make the proposed system efficiencies too low to be economically competitive with alternative energy approaches. Therefore, the previous discussion indicates that beam frequencies of up to about 5 GHz may be tolerable for conditions of medium-to-heavy rainfall rates. Much larger attenuation losses to the microwave beam may be experienced for very heavy rainfall conditions and at higher frequencies. However, cloud bodies with very large rainfall rates are generally fast moving and short lived.

Figure 3 shows scattered radiation power densities at the surface as a function of distance from the beam center for frequencies ranging between 2.45 and 10 GHz. At 2.45 GHz almost all of the scattered radiation still lies within the receiving antenna. However, the intensity of scattered radiation increases rapidly with frequency. At the edge of the rectenna (r = 5 km), the power density levels are about 0.4, 2.0, 10, 50, and 100 gW/cm2 at the frequencies of 2.45, 3.3, 5, 7, and 10 GHz, respectively. These are the maximum values of scattered radiation which would be found at the surface near the edge of the rectenna. At a distance of 5 km from the receiving site (10 km from the center of the beam), power densities at the surface decrease to values of 0.03, 0.15, 1.3, 4, and 10 p.W/cm2, respectively. Therefore, the Soviet safety standard of 10 ^.W/cm2 occurs about 1 km beyond the edge of the rectenna at 5 GHz, 2 km from the edge of the rectenna at 7 GHz, and 5 km from the edge of the rectenna at 10 GHz. At frequencies of 2.45 and 3.3 GHz the Soviet microwave standard is never exceeded outside of the rectenna. Superimposed on the curves shown in Fig. 3 are the Gaussian beam power densities given in Fig. 2. At frequencies less than about 5 GHz, scattered radiation outside of the rectenna exceeds that due to the beam geometry only far from the edge of the rectenna. At higher frequencies (greater than about 5 GHz) surface power density contributions from scattering in some cases may be larger than those contributions by the beam geometry outside of the rectenna. Note that even at a frequency of 10 GHz scattered surface power densities outside of the rectenna may only reach 100 /xW/cm2, or two orders of magnitude smaller than the current U.S. microwave exposure standard. Therefore, radiation power density outside of the rectenna is determined primarily by beam geometry (and sidelobe suppression) rather than by scattered radiation. Total beam efficiencies may, however, be critically influenced by the beam frequency and cloud raindrop size spectrum. Heating rates of approximately 0.05 °K/h would result if the entire microwave beam were absorbed by this 7 km thick rain cloud. For all practical purposes, cloud heating rates due to microwave beam heating may be neglected. B. Cloud Temperature Table 4b shows the effect of temperature upon cloud attenuation characteristics for a frequency of 5 GHz. The amount of radiation reaching the rectenna unattenuated remains relatively unchanged, but the amount of scattered radiation reaching the ground increases from 54.7 MW for a cloud uniformly at a temperature of -20 °C to 63.9 MW for a warm cloud at a temperature of +20 °C. Warmer clouds scatter more microwave radiation while absorbing less.

C. Beam Nadir Angle Table 4c shows calculations as a function of beam nadir angle at a frequency of 5 GHz. The effect of increasing beam nadir angle in a horizontally homogeneous cloud which is wider than the beam is to increase total attenuation of the beam and to increase scattering at the ground. Comparison of Tables 4b and 4c shows that increase of beam nadir angle from 0° to 30° has about the same effect on surface scattering as a change in cloud temperature from 0 to +20 °C. However, while absorption within the cloud increases slowly with increasing nadir angle, the total surface scattering increases rapidly, by nearly an order of magnitude as beam nadir angle increases from 0° to 60°. D. The Effect of Cloud Width Figure 4 shows surface power densities for a frequency of 5 GHz as a function of horizontal cloud width and beam nadir angle. The cloud is assumed to be 7 km thick in all cases, and the Rain M drop size distribution is used. The center of the microwave beam is assumed to intercept the center of the cloud (Fig. 5). For a small cloud 4 km in diameter, a large fraction of the beam misses the cloud entirely at nadir angle of 0°, resulting in very small scattered power densities at the surface. However, surface power densities increase rapidly with increasing cloud diameter. Note that increasing cloud diameter beyond about 10 km (the beam diameter) adds very little to surface power density levels at 0°. However, at larger nadir angles surface power density levels are more strongly affected by the cloud shape. For a beam nadir angle of 30°, the maximum scattered power density for a cloud 4 km in diameter becomes very large within the rectenna, but decreases more rapidly outside of the rectenna than at an angle of 0°. With increasing beam nadir angle, a portion of the beam increases its optical path length through the cloud; at the same time, other portions of the beam traverse decreased path lengths through the cloud (Fig. 5). Therefore, the average path length of the beam through the cloud may be less than or greater than the path length at 0°, depending upon cloud shape. Likewise, surface power densities may be larger or smaller than those given for nadir angles of 0°. As the cloud increases in diameter, the peak in maximum surface power density decreases in magnitude, but shifts to the outer edges of the rectenna (Fig. 4). For a beam nadir angle of 30° and cloud diameter of 24 km, surface power densities increase only slightly with further widening of the cloud. Much more dramatic changes occur at large nadir angles. For nadir angle of 60°, the peak in maximum surface power density steadily decreases with increasing cloud diameter, while continuing to shift to larger distances from beam center. In general, surface power density levels outside of the rectenna increase with increasing beam nadir angle and also with increasing cloud width. Comparison of Fig. 4 with Fig. 2 shows that at 60° scattered surface power density levels outside of the rectenna may equal or exceed those density levels of some of the beam geometries at environmentally significant levels (i.e., for surface power densities > 10 g.W/cm2). E. Cloud Vertical Dimensions Table 5 shows calculations of surface power densities outside of the rectenna as a function of distance from beam center for several variations in cloud height and

cloud base. Beam nadir angle is 0°, cloud width is 24 km, and the Rain M drop size distribution is assumed. Decreasing rain cloud thickness from 7 km to 5 km at a beam frequency of 5 GHz leads to decreased surface power density levels, as well as decreased absorption of the beam within the cloud. Total surface scattered power is decreased in this case from 57.5 to 40.7 MW, while rain absorption within the cloud decreases from 907 to 658 MW. A further decrease in rain cloud thickness to 3 km decreases total surface scattered power to 22 MW and rain absorption to 403 MW. Decreasing rain cloud thickness, while keeping cloud base height fixed at the surface, decreases the surface power density levels at all surface position. At the edge of the rectenna (r = 5 km), maximum surface power densities are 10.4, 6.6, and 4.4 /zW/cm2 for cloud thicknesses of 7, 5, and 3 km, respectively. At r = 10 km (5 km from the edge of the rectenna), corresponding power densities are 1.3, 0.6, and 0.25 /xW/cm2, respectively, while at r = 20 km corresponding values drop to 0.16, 0.07, and 0.03 /zW/cm2.

However, surface power densities are a function not only of rain cloud thickness, but also cloud base height. Normally large drops present in clouds may be assumed to reach the ground as precipitation. However, in dry climatic regions, virga conditions may prevail in which drops evaporate before reaching the surface. In addition, large drops may form in developing cloud cores long before precipitation is initiated. These large drops are often found near the cloud tops. In order to simulate such situations, the following cases were studied with variable cloud base height. Increasing the cloud base height to 2 km for the 5 km thick cloud decreases surface power density at the edge of the receiver to aboout 4.2 gW/cm2, but significantly increases surface power density levels at larger distances from the rectenna. In fact, the surface power density levels at distances of r = 10 km to r = 20 km from the beam center for the 5 km thick cloud with base height of 2 km are nearly identical to those power density levels produced by the 7 km thick cloud with base height at the surface. Similar results hold for the 3 km thick cloud. Increasing cloud base height from the surface to 2 and 4 km steadily decreases the power density levels at the edge of the rectenna, but strongly increases surface power density levels at greater distances from the rectenna. These results demonstrate that the vertical distribution of large drops within a cloud may be an important consideration when considering environmental impact. This may be particularly important in strongly developing clouds in which the maximum drop sizes and the largest liquid water contents are found in the upper regions of the cloud. Due to the lack of detailed microphysical results, more detailed calculations involving cloud vertical structure are beyond the scope of the present investigation. F. Drop Size Distribution Surface scattering is very small for frequencies less than about 5 GHz for light rain (using the Deirmendjian Rain L drop size distribution). For the 7 km thick rain cloud at a beam nadir angle of 0°, surface power density is only 0.2 gW/cm2 at the edge of

the rectenna at a beam frequency of 5 GHz; at a frequency of 7 GHz surface power density increases only to 1 /zW/cm2 at r = 5 km, and to 7 /zW/cm2 at 10 GHz. For moderate rain using the Diermendjian Rain 10 drop size distribution function, surface power density levels are obtained which are very different from those obtained using the Rain M drop size distribution function. This is due to the fact that the Rain M distribution has much larger scattering coefficients, due to larger drop sizes, and is more similar to the MP drop distribution functions. Note, however, that at the larger rainfall rates even the MP drop distribution functions may underestimate the number density of large drops, and, therefore, scattering. Using the Rain 10 drop distribution, surface power densities were calculated to be 0.07, 1, 8, and 20 /zW/cm2 at the edge of the rectenna (r = 5 km) at beam frequencies of 2.45, 5, 7, and 10 GHz, respectively. For heavy rain of 50 mm/h, the Deirmendjian D-50 drop spectrum produces surface scattering patterns similar to, although slightly larger than, those obtained using the Rain M drop distribution function. At the edge of the rectenna, surface power densities are 0.8 gW/cm2 at beam frequency of 2.45 GHz, and 15 /zW/cm2 at 5 GHz. The surface scattering pattern at beam nadir angle of 60° is also similar to that obtained using the Rain M distribution. However, the Rain 50 drop distribution absorbs more of the microwave beam within the cloud, absorbing 0.24, 0.51, 1.50, 3.57, and 7.24 GW at beam frequencies of 2.45, 3.3, 5, 7, and 10 GHz, respectively (compare with Table 4a). Increasing rainfall rates (drop sizes and drop concentrations) has a large impact upon the microwave optical properties and upon the resulting surface power density levels. The surface power densities at beam frequency of 2.45 GHz for heavy rain are larger than those at 5 GHz for moderate rain. This behavior can be seen more clearly by assuming a progression of MP drop size distributions, MP-25, MP-50, MP-100, and MP-150, representing rainfall rates of 25, 50, 100, and 150 mm/h, respectively. For the same rainfall rate the MP drop size distribution provides larger scattering coefficients G8S) and larger single scattering albedos (w0) than do the corresponding Deirmendjian drop distribution functions. In any case, these variations support observations that drop size distributions are sufficiently variable to account for a factor of three in attenuation (at 16 GHz) for the same observed rainfall rate. Figure 6 shows surface power density levels as a function of distance from beam center for a cloud 7 km thick and 24 km in diameter, for the MP-25, MP-50, MP-100, and MP-150 drop size distribution functions, for beam nadir angle of 0°, and for beam frequencies of 2.45, 3.3, 5, and 7 GHz. At beam frequency of 2.45 GHz, maximum surface power densities at the edge of the rectenna (r = 5 km) are 0.3, 1.0, 3.0, and 6.0 /zW/cm2 for rainfall rates of 25, 50, 100, and 150 mm/h, respectively. These values are well below the Soviet microwave exposure safety standard. At a beam frequency of 3.3 GHz, corresponding surface power densities at r = 5 km are 2, 5, 9, and 20 /zW/cm2, respectively. Only at very large rainfall rates (150 mm/h) do the surface power densities exceed the Soviet safety standard. At 5 GHz, corresponding levels are 8, 20, 50, and 100 /zW/cm2. These values are at or above the Soviet safety standard, but two orders of magnitude lower than the allowable exposure levels under the U.S. microwave safety standards. It would appear that even at a beam frequency of 5 GHz, scattered radiation power density levels do not exceed the Soviet safety standards at distances greater than 5 km from the edge of the rectenna even under extremely heavy rainfall rates. Comparison with Fig. 2 shows that even for heavy rainfall rates surface power densities due to beam shape (and sidelobes) dominate those densities due to scatter-

ing by raindrops within the biologically sensitive levels (3= 10 ju.W/cm2). For lower levels of exposure below the Soviet safety standard, power density levels due to scattering outside of the rectenna may exceed those due to the beam geometry. At a beam frequency of 7 GHz, surface power densities within the rectenna region show an inversion of the normal situation, with the lower rainfall rate producing the largest surface scattered power densities. This is due to the fact that multiple scattering within the cloud becomes more significant with increasing drop size and increasing beam frequency. Increasing photon path lengths, along with extremely strong drop absorption, causes this behavior. However, at larger distances from the rectenna, the largest rainfall rates produce the largest surface power density levels. Similar behavior is also found at larger frequencies (10 GHz). VI. CONCLUSIONS The present investigation calculates surface power density levels, for a microwave beam propagated through various sized clouds, (a) at frequencies ranging from 2.45 to 10 GHz, (b) for beam nadir angles ranging from 0° to 60°, and (c) for a wide range of raindrop size distributions ranging from light rain to extremely heavy rainfall rates of 150 mm/h. Scattered surface power density levels outside of the rectenna remain below the Soviet exposure standard (10 /u.W/cm2) at frequencies of 2.45 and 3.3 GHz, even for extremely heavy rainfall rates. However, exposure levels outside of the rectenna may reach 100 gW/cm2 at higher frequencies for heavy rainfall rates. Therefore, from the point of view of public health and safety, the scattering of microwaves by rain clouds is not a serious problem even at larger frequencies. Exposure levels due to scattering are often smaller than those due to antenna sidelobe characteristics. Cloud shape (and orientation with respect to beam position) becomes increasingly important at large beam nadir angles. For increasing cloud width, the peak surface power density decreases, while shifting away from beam center. The effect of increased beam nadir angle is to lower transmission efficiency of the beam while increasing surface scattering. The position of large drop concentrations within the cloud may also significantly affect surface power density levels outside of the collecting rectenna. Increasing the level of scattering within the cloud decreases power density levels near the rectenna, but strongly increases those levels at greater distances from the receiver. While there appear to be no severe environmental restrictions due to scattering of the microwave beam, the total transmission efficiency may be strongly reduced by higher beam frequencies, by large beam nadir angles and by large raindrops. In order for the microwave beamed power concept to be feasible, total system efficiencies of about 60% are required. Of this total, it is estimated that transmission losses no larger than about 10%^15% can be tolerated. Beam losses due to scattering are much smaller than beam losses due to raindrop absorption. Doubling beam transmission frequency from 2.45 to 5 GHz at moderate rainfall and at a beam nadir angle of 0° in a 7 km thick cloud increases transmission loss by about an order of magnitude, from about 1% to about 10%. Increasing beam nadir angle to 45° at 5 GHz increases beam loss to about 14%. Much larger losses occur for higher frequencies, for larger beam nadir angles, and for larger rainfall rates. Acknowledgements — The authors extend their thanks to S. Wunch who typed the manuscript, Judy Sorbie who drafted the figures, and to P. Martin who proofread the manuscript and made many helpful suggestions.

This research was sponsored jointly by the GARP Project of the National Science Foundation and the GATE Project Office of NOAA under grants OCD 74-21678 and ATM 77-15369. Computer calculations were made at the National Center for Atmospheric Research, which is sponsored by the National Science Foundation. The Editor wishes to thank John Zinn and W. E. Gordon for reviewing this paper. REFERENCES 1. P.E. Glaser, The Future Power from the Sun, IECEC Record, pp. 98-103. IEEE Publication 68C21- Energy, 1968. 2. P.E. Glaser, Power from the Sun: Its Future, Science 162, 857-886, 1968. 3. P.E. Glaser, Solar Power via Satellite. Astronaut. Aeronaut. 11, 60-68, 1973. 4. P.E. Glaser, Space Shuttle Payloads: Hearing before the Committee on Aeronautical and Space Sciences, U.S. Senate, 93rd Congress, 1st Session on Candidate Mission for the Space Shuttle. Government Printing Office, Washington, DC, Part 2, 31 October 1973, pp. 11-62, 1973. 5. P.E. Glaser, Solar Power from Satellites, Physics Today 30, 30-38, 1977. 6. P.E. Glaser, Replies to Letters, Physics Today 30, 66-69, 1977. 7. G.R. Woodcock, Solar Satellites: Space Key to Our Power Future, Astronaut. Aeronaut. 14, 30-43, 1977. 8. B.M. Elson, Space-Based Solar Power Study Near Completion, Aviat. Week Space Technol. 107, 58-69, 1977. 9. W.C. Brown, A Profile of Power Transmission by Microwaves, Astronaut. Aeronaut. 17, 50-55, 1979. 10. P.E. Glaser, The Potential of Satellite Solar Power, Proc. IEEE 65, 1162-1176, 1977. 11. AIAA, Solar Power Satellites: An AIAA Position Paper, Astronaut. Aeronaut. 17, 14-17, 1979. 12. T.B. McKee and S.K. Cox, Scattering of Visible Radiation by Finite Clouds, J. Atmos. Sci. 31, 1885-1892, 1974. 13. T.B. McKee and S.K. Cox, Simulated Radiance Patterns for Finite Cubic Clouds. J. Atmos. Sci. 33, 2014-2020, 1976. 14. J.M. Davis, S.K. Cox, and T.B. McKee, Total Shortwave Characteristics of Absorbing Finite Clouds, J. Atmos. Sci. 36, 508-518, 1979. 15. E.R. Westwater, Microwave Emission from Clouds, NOAA Technical Report ERK 219-WPL 18, 1972. 16. J.H. Van Vleck, The Absorption of Microwaves by Oxygen, Physics Rev. 71, 413-424, 1947. 17. C.J. Carter, R.L. Mitchell, and E.E. Reber, Oxygen Absorption Measurements in the Lower Atmosphere, J. Geophys. Res. 73, 3123-3127, 1968. 18. G.E. Becker and S.H. Autler, Water Vapor Absorption of Electromagnetic Radiation in the Centimeter Wavelength Range, Physics Rev. 70, 300-307, 1946. 19. W.M. Caton, W.J. Welch, and S. Silver, Absorption and Emission in the 8 mm Region by Ozone in the Upper Atmosphere, J. Geophys. Res. 72, 6137-6148, 1967. 20. R.A. McClatchey, R.W. Fenn, J.E.A. Selby, F.E. Volz, and J.S. Garing, Optical Properties of the Atmosphere, revised, Air Force Cambridge Res. Lab., AFCRL-71-0279, Environmental Research Paper No. 354, 1971. 21. P.S. Ray, Broadband Complex Refractive Indices of Ice and Water, Appl. Optics 11, 1836-1844, 1972. 22. D. Deirmendjian, Far-Infrared and Submillimeter Wave Attenuation by Clouds and Rain, J. Appl. Meteorol. 14, 1584-1593, 1975. 23. R.G. Medhurst, Rainfall Attenuation of Centimeter Waves: Comparison with Theory and Experiment. IEEE Trans. Antenas Propagat. AP-13, 550-563, 1965. 24. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions, Elsevier Press, 1969. 25. R.A. Semplak, The Influence of Heavy Rainfall on Attenuation at 18.5 and 30.9 GHz, IEEE Trans. Antennas Propagat. AP-18, 507-511, 1970. 26. J.S. Marshall and W. Mck. Palmer, The Distribution of Raindrops with Size, J. Meteorol. 5, 165-166, 1948. 27. R.R. Rogers, Statiatical Rainstorm Models: Their Theoretical and Physical Foundation, IEEE Trans. Antennas Propagat. AP-31, 547-566, 1976. 28. A. Waldvogel, The n„ Jump of Raindrop Spectra, J. Atmos. Sci. 31, 1067-1078, 1974. 29. V.G. Plank, The Size Distribution of Cumulus Clouds in Representative Florida Populations, J. Appl. Meteorol. 8, 46-67, 1969. 30. P. Squires, The Microstructure and Colloidal Stability of Warm Clouds. Part I. The Relation Between Structure and Stability, Tellus 10, 256-261, 1958. 31. P. Squires, The Spatial Variation of Liquid Water and Droplet Concentration in Cumuli, Tellus 10, 372-380, 1958.

32. F. Singleton and D.J. Smith, Some Observations of Drop-Size Distributions in Low Layer Clouds, Q. J. Roy. Meteorol. Soc. 86, 454-467, 1960. 33. S. Twomey, The Effects of Fluctuations of Liquid Water Content on the Evolution of Large Drops by Coalescence, J. Atmos. Sci. 33, 720-723, 1976. 34. A.M. Borovikov, I.P. Mazin. and A.N. Nevzorov, Some Observations of the Distribution of Large Particles in Various Clouds, IZV. Atmos. Oceanic Phys. 1, 291-301, 1965. 35. T. Takahashi, Precipitation Mechanisms in a Shallow Convective Cloud Model, J. Atmos. Sci. 35, 277-283, 1978. 36. R.A. Houze, Jr., P.V. Hobbs, P.H. Herzegh, and D.B. Parsons, Size Distributions of Precipitation Particles in Frontal Clouds, J. Atmos. Sci. 36, 156-162, 1979. 37. D.K. Lilly, The Dynamic Structure and Evolution of Thumderstorms and Squall Lines. Ann. Rev. Earth Planet Sci. 7, 117-161, 1979.

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0191-9067/82/020121-11$03.00/0 Copyright ® 1982 SUNSAT Energy Council LUNAR RESOURCE BENEFICIATION BY MAGNETIC SEPARATION DAVID R. KELLAND Francis Bitter National Magnet Laboratory Massachusetts Institute of Technology Cambridge, Massachusetts 02139, USA The use of lunar material for construction on the Moon and in space has been proposed for such projects as solar power satellites. Glasses and metals would be produced with the advantage of a great reduction in cost over shipping terrestial resources for construction of support structures. As with terrestrial materials, the collection, transportation, and beneficiation of lunar raw materials will be required to make suitable feed stocks available for refining and manufacturing. One beneficiation method which may have distinct advantages in a space environment and has been the subject of recent extensive technological improvement is magnetic separation. Its features and the possibilities for application in solar power satellite construction were outlined at the Glass and Ceramics Workshop at the Lunar and Planetary Institute in 1979. Recent research has enhanced the applicability of magnetic separation to lunar resources. MAGNETIC SEPARATION The retention or separation of solid particulates through the use of magnetic separators has been a well-established art for most of this century. Until the late 1960's, most of the commercial applications were confined to the removal of ferromagnetic materials with fairly large particle sizes. The equations for the forces on magnetic dipoles were formulated during the previous century but until this time there had not been any major theoretical consideration of the overall magnetic separation process. A breakthrough in this field occurred when high gradient magnetic separation (HGMS) was introduced about ten years ago. Reviews of the more recent developments can be found in the IEEE Transactions on Magnetics since 1973. The theoretical basis for HGMS can be found in a paper on electrostatic separation by Zebel (1). In electrostatic separation and in HGMS, high gradients (and hence, large forces) are obtained by placing small dielectric or ferromagnetic fibers in a uniform electric or magnetic field. In most practical HGMS devices, a matrix of ferromagnetic steel wool with a packing fraction of the order of 5% is placed in a magnetic field produced by an iron-clad electromagnet or by a superconducting coil. The substances to be separated are passed through the matrix in a gaseous or, more commonly, a liquid carrier. It is possible to remove weakly paramagnetic (magnetic Supported by the National Science Foundation.

properties orders of magnitude smaller than ferromagnetic iron) micron size particles from a fluid slurry in HGMS devices which are available commercially. A wide range of magnetic properties are encountered in particulate systems. At one end of the scale, the collection of large amounts of ferromagnetic solids from dense slurries has been accomplished (2). At the other end of the scale, the selective capture of diamagnetic (again, magnetic properties small but negative, i.e., repelled by a field that would attract paramagnets) particles has been observed in a system in which capture forces on these particles have been enhanced to a level comparable to that achieved in paramagnetic separations (3). The collection of magnetite in a high gradient magnetic separator was first observed at MIT during research on the beneficiation of oxidized taconite iron ore (4). A Mesabi (Minnesota) ore containing about 10% of magnetite by weight was observed to form dendrites along the magnetic field lines surrounding a 50 /xm stainless steel wool strand such as those used as a matrix in HGMS. Figure 1 illustrates the general principles of magnetic separation. A magnetic field gradient (d/7/dr) is required for a force to be exerted on magnetizable material. That force is given in one dimension as

where x is the magnetic susceptibility relative to the fluid carrier, defined as the ratio of the magnetization to the magnetizing field, V is the material volume and dH/dx is the gradient of the field over the volume of the material. In the example of Fig. 1, unsymmetrical poles produce a magnetic field which varies in space. Magnetic particles, if free to move, do so toward the pointed pole under the influence of the magnetic force. (A number of separators have been used over the years that have shaped poles.) Figure 1 illustrates a slurry of particles passing through such a separator. The magnetic particles migrate transversely in the slurry stream; the stream is then split and the materials are separated. Unfortunately, the gradient, which normally needs to be large in order to produce a large magnetic force, is not large between widely separated pole pieces. A high gradient magnetic separator, in contrast, uses materials such as steel wool or expanded metal to fill the space occupied by the field. Ferromagnetic steel wool strands 50 to 100 /xm in diameter cause large, local distortions in the field. In the case of HGMS, the gradients extend over micrometers. Therefore, the local gradients are much larger than those attainable with conventional methods. Very large field gradients can be obtained when high field strengths are applied. Hence the force, which depends directly on the gradient as well as susceptiblity and particle size, is very large, and very small, weakly magnetic particles can be trapped. The creation of high magnetic fields is the specialty of the Francis Bitter National Magnet Laboratory.

Magnets designed for HGMS make efficient use of iron to concentrate the field and have coils inside the iron return path. This design, shown in Fig. 2, allows a large material throughput. The steel wool used as a matrix in HGMS to produce high gradients is self-supporting and fills the volume of the separator, taking up 3% to 5% of the volume. A stream of material which contains 40% solids or more, as is often the case in mining, passes through the separator with its 95% or so open space, almost as if it were an open pipe. Figure 3 illustrates the principle of particle capture in HGMS. For a particle traveling toward a wire magnetized transverse to the wire axis, there is a certain capture radius. A particle passing within the capture radius is trapped by the magnetic force — if outside it, it is not. There are a number of wires throughout the volume giving a statistical probability of capturing a particle on its way through the separator. If one calculates the magnetic force in cylindrical coordinates for a paramagnetic particle, there is a maximum angle 0 for which the radial force component is positive. Outside that angle it is negative. Inside 6, a particle will be drawn in along a trajectory and captured on the wire. If it happens to pass outside the capture radius, it passes into the zone where the force is outward and will be rejected, possibly to be captured by another wire. The same is true for a diamagnetic (y = - ) particle except it will be captured where the gradient dH/dx is negative. Several approaches for removing collected particles from the matrix wires are discussed in a

following section. The field can be approximated by the superposition of a dipole field on the uniform background field. Industrial magnetic separation is performed most often on ferromagnetic materials such as iron ores or magnetite used in coal cleaning but kaolin clay is cleaned of paramagnetic impurities commercially and other HGMS applications await only economic improvement. To indicate that very small particles can be trapped. Fig. 4 shows the recovery of hematite with a size range from 5 to 10 p,m. Hematite is actually antiferromagnetic with a small susceptibility. The curve represents a model done in part by Clarkson (4). Notice that we have to go to very high fields in order to magnetize this material, but it can be collected. Recently, matrices have been developed which collect diamagnetic mineral components such as those labeled in Fig. 5 and magnetite shown collected on a screen matrix in Fig. 6. MAGNET SYSTEMS There are a number of possible candidates for magnet systems for lunar surface use. The three important categories are superconducting magnets operated at cryogenic temperatures, electromagnets with iron return paths and low voltage coils to be compatible with solar cell power supplies, and permanent magnets. The features and advantages of each would have to be considered in light of a specific application. The collection of 100 p.m iron-nickel particles from soil requires a very different approach than separating diamagnetic aluminum ores from crushed rock. Superconducting magnets generally are considered to be perhaps the only suitable means for high magnetic field applications. Because of economics and the reluctance of some industries to install sophisticated technological equipment to do dirty jobs, there has been limited terrestrial application of superconducting magnets. For lunar

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