Evaporation as a Function of Insolation Citation Richardson, Omar Burt (1930) Evaporation as a Function of Insolation. Master's thesis, California Institute of Technology. doi:10.7907/668z-cj50. https://resolver.caltech.edu/CaltechTHESIS:08272024-172239326 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1930 Record Number: CaltechTHESIS:08272024-172239326 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:08272024-172239326 DOI: 10.7907/668z-cj50 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16674 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 29 Aug 2024 21:09 Last Modified: 29 Aug 2024 21:13 Thesis Files PDF - Final Version See Usage Policy. 7MB Repository Staff Only: item control page

Evaporation as a Function of Insolation, Thesis by Burt Richardson, In Partial Fulfilbnent of the Requirements for the Degree of Master of Science,California Institute of Teclmology ,Pasadena,California, 1930 . In the Western States water is scarce and valuable, and rates of evaporation are high. Heretofore, evaporation losses have been ascertained by direct measurements. The writer considers it possible to approach the problem by a study of the physical cause of evaporation. The principal cause is insolation which may be defined, briefly as ''exposure to the rays of the sun.'' This paper is concerned with the effect of insolation and with the measurement of the quantity of radiant energy imparted by the sun and sky to the water and the bottom of a reservoir. In a sense, this effect may be considered a measure of insolation. For example, it may be sai d that , in Pasadena,California, on a sunny day , the insolation is roughly 600 calories per sq. cm. per day. Introduction The quafutitative effect of insolation is variable and depends on the sun's elevation,the length of the day, cloudiness, smoke, etc. It may be determined by three independent methods: (1) By means of a Weather Bureau, recording , thermo-elect ric pyrheliometer. (2) By computing the q_uantity as a geometric problem from a knowledge of the sun's radiation, altitude, etc.,together with the latitude of the reservoir or lake. (3) By tracing the radiant heat energy which strikes the water surface of a heat-insulated tank. Having determined the value of insolation for a particular site it is possible to show what part of it goes into evaporation and to place definite limits on the quantity of evaporation . By using either Method (1) or Method (2) in combination with ]iiethod (3) the evaporation from a lake or proposed reservoir may be determined. Notation The notation adopted for use in this paper is as follows: solar radiation A = correction coefficient=-:---:--=-----total radiation a = the Stefan-Boltzmann radiation constant. Br= the radiation from water. b = a constant which permits the use of the air temperature above the body of water instead of the sky temperature. C = a factor representing the conduction of heat through the walls of the tank. C' = the percentage of cloudiness that occurred during a certain month, as given in Monthly Weather Review reports. 8 = the sun's declination for a certain day of the year. E = quantity of water evaporated. F = factor denoting the relative clearness of the sky above the -particular body of water in question. H = original vapor pressure of the air passing over the surface of the water. i = angle of incidence of energy striking water surface. I = the quantity of radiant heat, in calories per minute, that reaches a certain area on the surface of the earth. Iex= insolation incident upon the exterior of the earth's atrnosphere. K = the solar constant. k = the ratio of the incident to the reflected intensity of radiation. L = the latent heat of evaporation of water. n = index of refraction of water. p = atmospheric pressure. PW = original va:por pressure of the air in contact with the water surface. Q = the observed radiant energy that passes through local clouds. Q = the observed radiant energy that passes through a clear sky. s R = the ratio, convection to evaporation. r = the angle of refraction. s = sensible heat measured by the warming or cooling of the water. T = original temperature,centi grade, of the air passing over the a surface of the water. T = original temperature, centi grade, of the air in contact with w the water surface. t = time in minutes. 2TT w =----24 X 60 radians per min.= aI1c,"11lar velocity of the earth. 1 = latitude of the station or lake. p = the transmission coefficient of the earth's atmosphere. p' = the '' mean'' transmission coefficient of the atmosphere at a particular place. a. = the zenith angle of the sun. (1) Insolation Recorded by 'rhermo-Electric Pyrheliometers 'l'he Weather Bureau of the U.S.Depa r t ment of Agriculture (1) maintains six stations at which recording thermo-electric pyrheliometers are used and their records of insolation are published in the Monthly Weather Review. Due to the difference in latitude, cloudiness,etc., the value of insolation is, of course, not the sarne for all stations. For example, the average number of calories reaching each square centimeter of the earth's surface per day during the month of July,1928, at certain places is indicated in Table 1. Table 1. Values of Insolation at Seven Stations in the United States. Location. New York City,N.Y. Washington D.c. Chicago,Ill. l'-.'fadison, 'His. Lincoln, Nebr. r win Falls, Idaho Scripps Institution of Oceanography,I.a Jolla,Calif. Insolation in calories per square centimeters per day. 378.5 540.8 412.8 509.2 531.2 737.2 544.7 The insolation varies from day to day at any one station, as illustrated by data observed at the Scripps Institution of Oceanography at La Jolla, Calif. Vfuile this is not a U.S. Weather Bureau Station, it uses the u.s. Weather Bureau Pyrheliometer, No.17 which was taken by the writer from the California Institute of 'r echnology in June 1928. Since July 1928 the hourly values of insolation have been observed and computed by the writer and these values appear monthly in t he Monthly Weather Review. Table 2, which follows, gives the observations for each day of July,1 928. 'i:able 2. Insolation as Recorded by a U.S . Weather Bureau Pyrheliometer No.17, at the Scripps Institution of Oceanography,La Jolla, Calif. Dat~July,1928 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Calories per s quare centimeter per day. 662.2 627.1 669.l 679.l 620.6 614.0 610.3 591.3 665.9 605.1 586.5 471.9 482.9 633.6 571.4 15 Date,July,1928 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Calories per square centimeter per day. 600.6 604.1 579.6 632.3 481.9 534 .2 432.8 353.6 275.5 238.0 517.4 592.4 489.2 474.7 410.0 579.0 The total insolation f or the 31 days was 16,886.3 calories, which is the equivalent of 10.35 in. of evaporation, or an average of 544.7 calories per day. Consi dering that it requires 585.4 calories to evaporate 1 cc of water at 20° c., it is easily seen how much a given water surface . would drop each day due to evaporation, should all the insolation go into evaporation. Actually, the ratio • of observed evaporation to observed insolation is between 40 and 60 per cent. Table 3 gives all the records of insolation that have been observed at stations in the United States. It has been compiled from various sources. Items Nos.l to 12, inclusive, are records of ·six (2) stations of the U.S. Weathe~ Bureau where insolation is observed; Items Nos .13 t o 27., inclusive, are records reported from sundry other (3) sources; and the observations listed for the last three stations .• a'.9'e (4) those observed by the writer, using the pan method. Table 3. The Mean Monthl y Quantities of Observed Insolation. Also, the Mean Monthl y Insolation Computed by Equation (12) Incident upon the Exterior of the Earth's Atmos phere. Insolation,in Calories per Sq.Cm. per Day St ation New York City Nature Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec of data I 128 235 262 354 374 358 402 276 264 195 114 91 1 ex 373 560 650 793 964 999 999 888 700 518 377 297 Washington D.C . I Iex 178 242 338 419 472 489 4?3 420 360 2'79 188 143 377 577 678 820 994 999 998 894 705 543 410 326 Chicago,Ill. I Iex 98 1 51 223 328 432 335 390 380 336 167 87 78 317 547 643 785 96110191036 890 683 510 363 286 Madison, Wis . I I ex 165 245 315 414 450 519 505 434 334 221139 125 303 534 629 777 94710031085 891 684 524 357 279 Lincoln, Nebr. I I ex 208 300 374 451 475 556 570 486 392 290 203 171 324 561 650 794 965 9991020 889 690 515 379 298 Twin Falls, Ida. I Iex 162 261 250 345 380 614 717 '7 08 567 390 180 150 308 550 647 787 95410181024 887 688 504 354 280 Vancouver, B.C. I 1 ex 81 147 204 283 366 357 326 304 199 131 74 53 196 342 549 792 9431006 942 791 571 347 196 153 Victoria,B.C. I Ie::x 94 157 259 339 414 405 388 335 255 153 90 71 249 396 592 815 9571011 954 813 618 403 253 202 st .Johns N. B. I Iex 114 206 270 345 397 424 422 364 284 165 108 78 249 396 592 815 9571011 954 813 618 4L3 254 202 Boston Mass. I Iex 1 39 220 327 402 449 477 421 376 317 235 153 120 331 477 655 848 9641014 965 851 673 475 332 282 Eureka, Calif. I Iex 148 246 313 435 486 524 442 382 318 233 151 129 331 477 655 848 9641014 965 851 673 475 332 282 Lower Calif. ,1':iex . I Iex 306 364 415 482 476 482 449 437 420 341 301 281 494 615 761 898 973 997 973 898 774 618 496 453 Pasadena,Calif . I I ex 302 383 427 532 568 512 582 568 490 400 365 316 430 545 704 849 9601011 986 902 779 616 474 418 La Jolla, Calif. I 264 316 365 433 487 504 545 438 285 291 298 275 Fort Collins,Colo. I 370 San Francisco,Calif . I San Diego,Calif . I 564 662 (2). Insolation Computed as a Geometric Problem It is possible to determine the insolation value from a knowledge of the sun's radiation, its altitude, etc., and the latitude of the reservoir or lake. A formula may be set up for insolation,as: t I = K f ,_,<) sec a, cos a, p (1) dt tl Differentiating with respect to time to obtain the rate of insolation, dI sec a. (2) a:t"" = K cos a, p ( 5) The solar constant,K, as determined by Abbot, is 1.93 calories per square cm. per minute. To illustrate the use of Equation (2) consider the parallel of 40°N. latitude. Use Abbot's value for the solar constant, ( K = 1. 93 ) ; for simplicity assume a transmission coefficient of p = 0.'7, and insert proper values for a., the zenith angle of the sun, for each minute of each day of each month,successively. The values of I thus obtained are given in the following table. Table 4. Computed Values of I at 40° North Latitude. Calories per sq.cm. per day (average) 140.0 244.2 356.5 465.'7 5'73.'7 586.7 Month January February March April May June Month July August September October November December Calories per sq.cm. per day (average) 553.3 45'7.3 34'7.'7 240.0 136.0 100.1 At this point something might be said about p, the transmission ( 6) coefficient of the earth's atmosphere used in Equations (1) and (2). The only available values of this coefficient are as shown in 1~able 5. 'rhese are weighted values computed from the mean solar intensity and coefficient for a particular wavelength. Observed Values of p and A. Table 5. Mt. Wilson, Calif. Washington D.C. Mt. Whitney Calif. Madison, Wis. Lincoln, Nebr. 34°-13' 38°-56' Values of ,,.,, for Zenith Angles a oo 7 .5° 25° 30° 55° 60° p North latitude Station 0.841 0.689 0.882 40°-51' 43°-05' o.so o.so a.so 0.87 0.87 0.86 0.85 0.86 0.84 0.84 0.81 0.80 0.87 0.87 0.86 0.86 0.85 0.84 0.85 0.84 0.82 o.79 0.81 0.76 Values of the transmission coefficient for other places in the United States have not been computed so that research on this coefficient is highly desirable;, The v,ri ter suggests that this coefficient may be computed directly by means of Equation (2). The left side of this equation which represents the rate of insolation is known from the pyrheliometer record by direct reading for a particular minute of the (7) day, and the zenith angle corresponding to this minute is also known, leaving the one unknown, p. Total energy, including scattered sky radiation, is recorded by a pyrheliometer. Therefore, it is necessary to correct the pyrheliometer records by a coefficient, A, which represents the ratio of solar radiation to total radiation. Typical values of A for four ( 8) stations are given in Table 5. The rate of insolation from the sun ( not including sky radiation) equals: dI' dt sec a. = A £!_ = K cos a p dt ( 3) and, after making the necessary transpositions: log p = -1 log ( £!_) dt K cos a A sec a, ( 4) To illustrate this method by whi ch the transmission coefficient of the earth's atmosphere may be computed, the pyrheliometer records of fourteen days of 1926 for Pasadena, Calif., were used, and when the monthly values were plotted, a mean value of 0.751 for p was obtained. The value of insolation is not constant along any parallel of latitude, but changes at points of different longitude, due to variation in the transmi s sion coefficient, humidity, local clouds, smoke, dust, etc. Therefore, in order to use Equation (2) for points of di f f erent longitude, values of the transmission coefficient must be observed as was done by Abbot or computed by Equation (4). (3) Insolation by 'Pracing Radiant Heat Energy ( g) 'The third metho~: '. for measuring insolation is to trace the heat energy which comes to water contained in an open tank. In this case insolation (when corrected for reflection) will be equal to the summation of: (a)Evaporation; (b)convection losses, due to the circulation of air above the water; (c) sensible heat, measured by the warming of the water; (d) conduction, due to lo s s of heat through the walls of the tank; and (e) radiation, from water to colded air! This is simply a statement of the Law of Conservation of Energy . The ratio,k, of the incident to the reflected intensity of radiation is given by Fresnel's equation: 2 k 1 sin --2 2 sin 2 i - r i + r i tan 2 tan 2 i -r i •r in which, i and rare the angles of incidence and of refraction, sin i = respectively, and are connected by the relation n = sin r index of refraction. (5) Let n, for water, be eq_ual to 1.33; then r may be found for corresponding values of i and these, substituted in Equation (5) will give values of k for any angle of incidence. 'fhis is the correction factor for reflection. Insolation is a positive quantity during the day, but at night it is equal to zero. Evaporation is always positive, while the other terms - convection, sensible heat, conduction, and r adiation may be either positive or negative. In s ymbols, then: I IB + LER + S +C + Br = (6) in which, Lis the latent heat of evaporation ( 585.4 at 20°C.); Eis the nUJUber of cubic centimeters of. water evaporated; and R is the ratio of convection to evaporation. (10) Bowen has found that: R p 0.46 (7) )760 in which, Ta and Hare the original temperature and vapor pressure of the air passing over the surface of the water; Tw and Pw are the corresponding quantities for the layer of air in contact with the water surface; P is the atmospheric pressure; Sis the sensible heat measured by the warrning or cooling of the water; and C represents the conduction of heat through the walls of the tank. ( 4 -__) Br is the radiation from water and may be comput ed by the formula: 4 B = ( 0.906 a T r w (8) in which the decimal, 0.906, represents an important constant which takes care of the fact that water does not radiate as a perfectly black body ; a is the Stefan - Boltzinann constant, amounting to -5 5.7 x 10 ergs per sq. cm. per sec. per degree to the fourth power; while b permits the use of air temperature in the formula in place of temperature of the sky which receives the radiant heat energy. The average value of the term, b,(which remains q_uite constant for a particular placel in the radiation Equation (8) has been observed at four stations to be as follows: Station Mean value of b. Tank at Pasadena, California 0.815 Tank at Fort Collins,Colorado 0.760 Murray Lake, San Diego, California 0.794 Crystal Springs Lake, San Francisco,Calif. 0.757 Average (weighted) 0.790 When the temperature of the water and air is known, by using the mean value of 0.790 for b, radiation may be computed by Equation (8). However, when these temperatures were not known, it ·was possible, in thecase of California water surfaces, to estirnate the value of radiation from the data collected at other places. The radiation from t he water to the sky, in calories per square centimeter per day, has been found for five places for five places as shown in Table 6. 'I'able 6. Radiation from the Water to the Sky, in Calories per Square Centimeter per Day. Location. Radiation. Pasadena,California Fort Collins,Colorado Murray Lake, San Diego,California Crystal Lake,San Francisco,Calif'ornia La Jolla, California 117.0 132.0 128.0 148.0 130.3 Average value (weighted) 131.4 From Equation (6) the exact amount of radiation may be determined for the night, since during the night is zero and all the other quantities of terrns in the equation are measurable. Thus, the insolation may be determined by adding all the heat energy coming to the body of water. Tab le? indicates the order of magnitude of each te rm that enters into insolation when using the pan method. A similar tabulation for observations at Murr ay Lake , San Diego , California ( July? - July 27),gave corresponding average daily values, as follows: Factors Calories per s quare centimeter per day . 485.g 0.8 34 .4 128.2 13.0 Evaporati on Sensible h eat Convection Back radiation Conduction Observed insolation 662.3 Table ?. Items That 11a.ke Up Pan Insolation, As Observed At Crystal Springs Lake, San Francisco,Calif. August 10-30 1927. Day 1 2 3 4 5 Evaporation 389.l 232.5 438 . 9 403 . 0 351.8 494.2 355.5 326 . 4 395.3 412.2 393.4 392.5 327 . 4 311.5 381 . 8 370.2 315 . 0 243.8 383.4 379 . 2 6 ? 8 9 10 11 12 13 14 15 16 17 18 19 20 Total 20 days?,197.1 Average per day 359 . 9 Sensible ConHeat vection 0 103.2 40 .5 -1.9 51.1 - 28 . 0 6. 7 13.0 26.0 83 .? 9.4 101.4 94 .5 - 9.3 87.? - 3. 6 66 . 6 33 . 5 14.3 -11. 2 0 - 8.0 63.2 19.2 48 .4 10.5 22 . 6 - 5.5 15.3 -13.0 29.8 - 9.4 7.5 65 . 4 - 40 . 8 33 . 6 63.3 19.5 26 .1 24.6 Radiat ion 147.1 145.5 141.3 144 .5 148. 0 171.4 165. 0 162.5 158. 8 139.0 144.0 136.2 139.0 146.5 145.8 141.5 150.2 136.1 139.6 142.5 Conduction 19 . 3 17. 6 12.5 g. 6 14.4 30.6 27.6 26.2 20 . 2 10.5 2.8 10.9 11.1 13. l 13.1 11.2 15 .1 8 .7 8. 2 12.1 Pan Ob served Insol ation 658 .? 434.2 615.8 576.8 623 . g 707.0 633.3 5gg_ 2 674 . 4 564.8 532.2 622 . 0 536.4 488.2 543.0 543 . 3 553. 2 381 . 4 614 . 0 584.5 225.4 824 .5 2,944.5 294.8 11,486.3 11.3 41.2 147.2 14.? 574.3 As a test of this method f or measuring insolation, tanks of different areas and depths have been measured hourly over long periods of time with a resulting difference of less than 3 per cent. As one illustration, Table 8 will show a comparison of the results obtained from three tanks of different depths and areas and a Weather Bureau No.17 pyrheliometer over a period of five days, at the Scri pps Institution of Oceanography, La Jolla, California. '.f able 8. Comparison of Insolation Q,u anti ties Observed by Different Methods Method of observation Insolation, in calories per s quare centimeter per day. JulylO Julyll Aver. July9 Julyl3 Julyl2 Pyrheliometer •r ank No.l Tank No. 2 Tank No.3 662.5 653.6 629.0 638.3 614.2 617.7 590.3 626.7 583.0 594 .0 565.2 578 .9 468.2 463.4 445.9 474.5 560.8 565.0 542.2 552.3 476.2 496.3 480.7 443.4 Description of Method for Measuring Evaporation By transposing Equation (6) which is in metric units, and converting to inches, it may be written: I - S - C E Br ( 9) = 2.54 L ( 1 + R This expression, which gives evaporation as a function of insolation, water temperature, wet and dry bulb temperatures, and vapor and atmospheric pressures, is a true equation;but the tedious process of evaluating each term hourly ( as was done in the case of data mentioned herein) would defeat t h e purpose of this method for practical use. As a result of research, it is possible to assi gn limiting values to each term of Equation (9) and by considering certain variables constant, an equation may be obtained which is simple to handle. Thus, by considering sensible heat and conduction negli gible, by using 585 for the value of latent heat of evaporation, and adopting the value 0.22 for Bowen's ratio,R, Equation (9) becomes, in inches: E I Br 2.54 X 585 X 1.22 = I - Br 1814 (10) Thus far, no n1ention has been made of the effect of wind on evaporation, because in the expression for insolation given in Equation (6) this effect would be n1easured,calorie for calorie, in changes in sensible heat and in convection. Sensible heat as well as the value of conduction for a short period or over a yearly cycle, as deterrnined from a study of data collected in California, is negligible. Furthermore, no serious error will result from consi dering Ras a constant equal to 0.22. 'r his may be seen by referring to Table 9 which gives the value of R computed for dissimilar circumstances and over a period of time sufficiently long with hourly readings. Table 9. Comparison of Values of R Computed Under Dissimilar Circumstances. Location Date June 1926 Pasadena, California Sept .1926 Fort Collins,Colorado July 1927 Murray Lake , San Di ego , Calif. Crystal Spr.La.~e,San Francisco,Calif. Aug . 1927 July 1928 Scripps Institution, La Jolla,Calif. July 1928 Pacific Ocean, Santa Barbara, Calif'. Aug. 1928 u.s. Naval Air St. San Diego Bay I I T I I I T I Sept.1928 I I Oct. 1928 ' t '' 'T '! T I M ar. 1929 ' t '' Apr. 1929 '' '' '' '' Number days 10 10 20 20 5 12 31 30 30 3L 30 Ratio,R. 0.193 0.221 0.165 0 .280 0.1~0 0 .280 0.190 0.208 0.171 0 .170 0 .152 The maximun1 ueviation in t h ese values of R from the mean of 0.22 is 0. 06, and since evaporation equals on the average 50% of insolation, the resulting error in insolation caused by taking R eQual to a constant, 0.22, instead of t he largest or smallest value of R computed, would be 3%. It is interesting t o note t hat by using the insolation quantity as computed by Method No. 2 and by using the value of r a diation, ( Br), given for any one of f our places mentioned in Table 6 , t he evaporation compu t ed f or the f ifth pl a ce was within 5;b of the observed evaporat ion in each ca se. Bef ore taking up t he practical a pplicati on of Equat i on (10 ) it is well to consider Table 10, which contains good evaporation data for stations in the United States. Unfortunately these records were from tanks not of the same size. It is well also to bear in mind that the amount of evaporation from a small tank next to a lake is not the same as that from the larger body of water, but simply represents an index . The data in Table 10 were compiled from records presented in (11) various publications. Equation (10) may be solved by substituting observed values of I from sources such as 'rable 3 and by using values of Br determined by using Equation (8). Table 11 gives a comparison of the observed and computed evaporation near pyrheliometer stations. Table 11. Station Comparison of Observed and Computed Evaporation. Insolation by Radiation pyrheliometer in in inches inches per year per year divided by 1. 22 divided by Evaporation in inches per year. Computed Observed 42.7 33.l 48.7 32.2 39.11 31.76 1.22 Boston, Massachusetts. New York City, N.Y. Cincinnati,Ohio Great Lakes Birmingham,Ala. Mitchell, Nebraska Boise,Idaho Klamath,Oregon Lake Tahoe, California 60.0 50.4 66.0 4,9.5 Go.l 73. 7 78.0 62.7 62.7 17.3 17.3 17.3 17.3 15.0 17. 0 17.3 17.3 17.3 51.1 56.7 60 . 7 45.4 45.4 49.99 27 .96 51.30 65.67 77.43 54.40 42.20 Evaporation also has been computed for the stations of 'f able 10 with good agreement between the computed and observed values. 'fhe results of three other computations were as follows: (12) Hoger C.Wells of the U.S.Geological Survey, determined the evaporation of Chesapeake Bay to be 120.9 cm. per year, while the computed amount,using his fiata, ( 1~5) was 124 .3 cm. per 7Tear. McEwen by one method and Grunsky by another determined the evaporation of Lake Mendota (Madison,Wis,, a pyrheliometer station ) for a 5-month period, to be 41.45 and 45. 50 cm. respectively, while the computed amount was 39. 66 cm. ·rhe evaporation of Swiss Alps Table 10 - Observed Evaporation at Thirty-Two Stations in the United States, in Inches per Month. Station Jan Feb !Viar Apr May Jun Jul Aug Sep Oct Nov Dec Year Boston, Mass 0.90 1.20 1.80 3.10 4.61 5.86 6.78 5. 49 4.09 2.95 1.63 1.20 39.61 Rochester, N.Y.0.86 0.86 1.67 2.39 3.45 4.38 4.81 4.54 3.54 2.611.46 1.21 31.76 Cincinnatl 1. 00 1.50 2.50 4.12 5.07 6.21 7.20 7.26 5.63 3.00 1.50 1.00 45.99 Birmingham 1.50 1.50 2.25 4.45 5.91 7.28 7.36 7.34 6.00 4.00 2.25 1. 50 51.34 Great Lakes 0. 66 0.84 1. 03 2.12 2.76 3.67 4.38 3. 98 3.21 2.56 1.67 1.08 27.96 Mitchell Neb 1.75 1.75 3.00 4.50 6.25 8. 0510.95 9.39 7.44 5.59 4.00 3.00 65.67 Snake R.Ida. 2.25 2.50 4.00 7.00ll.2112.3115. 0013.5011.00 8.50 5.75 3.50 96.52 Boise Ida. 2.00 2.75 4.25 6.00 7.90 9.5710.5912.26 9.15 5 .42 5.52 2.00 77.43 Fallon Nev. 1.75 1.75 2.25 3.25 5.25 7.86 9 .86 8.70 5.13 3.35 2.50 2.00 53.65 Hermiston Ore 1.25 1.25 3.00 7.28 7.8 9 9.5412.0411.07 7.35 3.88 2. 00 1.50 68.05 Klamath Ore. 0.50 1.25 3.57 6.64 7.15 6.99 8.01 9.21 6.13 2.50 1.00 0.50 53. 45 N.Yacima Wn. 1.75 2.50 6.25 7.91 8.36 8.9010.74 9.41 5.51 3.15 2. 00 1.50 67.88 Lake Tahoe i.75 1.75 1.75 2.00 3.00 4.25 6.17 7.10 6.22 3.60 2.62 2.00 42.21 Salton Sea 3.41 5.09 5.95 8.7510.5013.0014.0312.1912.08 9.24 5.96 5.25105.45 Indio Ca lif. 3.18 5.08 7.5012.0515.8416.1116.3413.7812.37 8.91 5.17 3. 00119.33 Mecca Calif. 2. 92 5.00 8.0710.8712. 72H,-; 2315.2113.2210.29 8.17 4.13 2. 98107 .81 Brawley Calif.3.05 5.00 8.0010.7413.7913.6814.1411.2610.15 6. 99 4.09 2.66103.55 Mammoth Calif.4.24 5.67 8. 9912.0215.5216.7518. 0013.7312.16 9.49 5.26 3.70125.53 Phoenix Ariz 4.25 4.40 5.25 7.00 9.5012.0012.7512.5011.00 8.31 6.56 4.22 97.74 Lee's F .Ariz 1.74 3.52 5. 87 7.1611.7013.7013.7011.33 8.85 6.29 3.98 1.91 89.75 Roosevelt D. Ar2.29 3.10 5.33 7.3210.3712.6712.3410.56 8.60 5.78 3.52 2.13 84.02 MesaExp .Ariz 2.78 3.68 5.74 7.7810.2711.1710.55 8.29 6.39 4.73 3.36 2.62 77. 36 ~ ilcox Ariz 3.59 4 .73 7.2910.1011.2412.2010.64 9.03 8.10 6.70 4.67 3 .41 91.70 Yuma Ariz 3.09 3.89 5.72 7.23 8.24 8.9110.24 9.96 7.85 5.39 3.40 2.62 76.54 Carlsbad N. M. 4 .50 4.50 5.51 7.4510.1211.0%12. 8812.00 9. 50 7.00 5.75 4.50 94.76 N.Me:x: Agri C 2. 8 7 4.50 7.41 9.3711.1011.9111.15 9.7 9 8.09 5.93 3.63 2.52 88.27 ElButte N. IvI . 2.78 4.49 7.5610.4413.4314.4812.2210.83 9.07 7.76 4.32 3.05100.43 L. Avalon N. M. 2.34 3.26 5.49 7.49 7.87 8.7010.13 9.60 8. 57 5.51 3.17 2.42 75.55 Santa Fe N.M. 1.53 2.13 3.96 6.15 8.5110.14 8.93 8.06 6.55 4 .81 2.55 1.93 65.25 Spur Tex. 2.79 3.54 4.82 5.48 6.66 8.68 9 .33 7.88 5.59 4.48 3.42 2.34 65.01 Hill RanchTex 2. 47 3.43 5.29 6.01 6.81 7. 95 8. 97 8 .72 6 .36 4.96 3.18 2. 44 66.59 Beeville Tex 2.69 3.26 4.26 4.79 6.21 7.35 8.14 8.14 5.89 4.72 3.06 2.17 60.68 (14) Lakes amounted to 198.3 cm., while the computed a.mounted to 193.7 cm. When the value of insolation is not known from pyrheliometer records it is possible to compute the insolation, lex, incident upon the exterior of the earth's atmosphere for any latitude, and, later, to convert this into actual Insolation,I, by means of the two coefficients, p' and F, the mean transmission coefficient at a particular place and the clearness factor, respectively. The relation may be expressed by the integral: t I But e:x: = J o 2K cos u dt (11) cos u = sin 1 sin 6 + cos 1 cos integration o cos wt so Equation (11) upon becomes: Iex = ( 2K t sin 1 sin o + 2K (cos 1 cos 6 sin wt) (12) (J) in which, 1 = the latitude of the station; 8 = the sun's decUnation 2TT w = -2-4-x-6-0- - - = earth's angular for a certain day of the year; and velocity. The time, t, is the number of minutes of half-day, and K, the solar constant, has a value of 1.93 calories per sq. cm. per min. The values of declination, o, var-J from +23° on June 21 to -23° on December 21 of each year. The results of applying these values in Equation (12) to obtain quantitatively the insolation striking the exterior of the earth's atmosphere are given in Table 3 in ''Computed Insolation'! Table 12 gives additional values of exterior insolation fbr eleven different latitudes extending from Canada to Mexico, together with the eq_uivalent, in inches of evaporation. These values of evaporation would represent the physical maximum because they assume no clouds and a transmission coefficient of 1.00 Clearness Factors (15) By using Y..imball's fornrula we have a means of estimating the Table 12.- Evaporation as Determined by Computing the Insolation on the Exteri or of t he Earth's Atmosphere. Values of I Values of 1, in degrees No . Lat . Jan Feb 52 49 48 45 42 41 37 33 30 29 20 189 229 241 286 322 336 392 449 487 491 618 305 512 347 548 359 557 404 596 440 :_~623 450 634 505 678 556 717 585 735 600 746 702 714 Values of 1, in degrees No.Lat. 52 49 48 45 42 41 37 33 30 29 20 ex' in Calories per SQuare Centimeter per Day. Mar Apr May 743 765 770 796 809 815 844 864 871 875 904 Jun Jul Aug Sep Oct Nov Dec 916 993 929 1003 932 1000 945 1008 943 1004 951 1002 963 1032 963 1003 960 992 956 990 947 . 958 953 965 964 976 974 976 993 985 977 974 851 818 837 838 861 870 871 896 908 911 913 925 527 562 579 611 638 647 696 731 799 807 856 387 425 441 478 510 521 572 619 646 651 746 226 270 283 323 361 376 433 489 519 533 642 161 198 211 255 291 291 367 424 461 476 595 Values of Evaporation,E, in Inches of Evaporation from a Wat er Surface Corresponding to the Ab ove Values of I • ex 3.9 5.7 11.6 14.9 18.9 19.9 19.7 16.9 10.5 8.0 4.5 3.4 4 .7 6.5 11.3 15.3 19.2 20.l 19.9 17.3 11.2 8.8 5.4 4.1 5.0 6.7 11.5 15.4 19.3 20.1 19.9 17.3 11.6 9.1 5.7 4.4 5.9 7.5 12.3 15. 9 19.5 20.2 20.2 17.8 12.2 9 . 9 6.5 5.3 6.7 8.2 12.9 16.2 19.5 20.1 20.2 18.0 12.8 10.5 7.2 6.0 6.9 8.4 13.116.319.6 20.1 20.2 18.0 12.9 10.8 7.5 6.3 8.1 9.4 14.0 16.9 19.9 20.6 20.5 18.5 13.9 11.8 8 .7 7.6 9.3 10.4 14.8 17.3 19.9 20.i 20.4 18.8 14.6 12.8 9.8 8.8 10.1 10.9 15.2 17.4 19.8 19.8 20.2 18.9 16.0 13.3 10.4 9.5 10.3 11. 2 15.4 17.5 19.8 19.8 20.1 18.9 16.1 13.4 10.7 9 .8 12.8 13.l 14.8 18.l 19.6 19.2 19.6 19.l 17.115.412.8 12.3 rrable 13 . - The Rat i o of Observed Insolation and the Computed Insolation s·tri king a Horizontal Surface on the Ext erior of the Earth's Atmosphere for All Stati ons Where Insolation Records Have Been Kept. Station Values of t he Ratio Years obser ved ;r ~e:x: New York City N.Y.1924-2 9 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov De c . 34 .42 .40 .45 . 39 . 36 .40 .31 .38 .38 .30 . 31 "iVashington D. C. 24- 29 . 37 .42 . 50 .51 .48 .48 . 46 . 47 . 51 . 51 146 .44 Chicago Ill. 24-29 . 31 .28 . 35 .42 . 45 . 33 . 38 .43 . 49 .33 .24 . 27 Madison \'Tis . 24-29 . 55 . 46 .50 .53 .48 . 52 . 47 . 49 .49 .42 .39 . 45 Lincoln Nebr. 24-29 . 64 .53 . 58 .57 .49 .56 . 56 .55 . 57 . 56 . 54 .57 Twin Falls,Ida . 27- 28 .53 . 48 . 39 . 44 .40 . 60 .70 .so .82 .77 .51 .54 Vancouver B. C. 1924 . 41 . 4 r/, • 37 .36 .39 .35 .35 . 38 .35 .38 .38 .35 Vi ctoria B.C. '' . 38 . 40 . 44 . 42 . 43 . 40 . 41 . 41 .41 .38 .36 . 35 st.Johns N. B. f I .46 .52 . 46 . 42 .42 . 42 . 44 . 45 .46 . 41 .43 .39 Boston (Boat) t I .42 .47 .50 .47 . 47 .47 .44 .44 . 47 .50 . 46 . 43 1!.ureka Calif . I I .45 .52 .48 .51 .51 . 52 .46 . 45 . 47 .49 .46 .46 Florida (Boat) I I .43 . 40 . 48 . 47 . 47 .44 .44 . 44 . 42 . 42 .49 .43 Lower Calif .I,:ex. I f .62 .59 .55 .54 .49 .48 .46 .49 . 54 . 55 .61 .62 Santiago,Cuba. I I . 54 . 58 .54 .53 . 48 .52 .49 . 49 .51 .52 .52 .52 Pasadena,·Calif. 1926-27 Fort Colli ns;Colo. ~ .70 .70 . 61 .63 . 59 .51 • 59 .63 1926 La Jolla ,;,: 6 ._, . 57 San Francisco,Calif.1927 San Diego,Calif . . . 65 1927 .69 1928-29 .56 . 65 .7 6 .75 percentage of clear days: Q =Q,'s F = 0.22 = 0.78 ( 1.00 - C' ) (13) in which, Fis the clearness factor; Q is the observed radiant energy that passes through local clouds; Q8 , the observed radiant energy that passes through a clear sky; and C', the percentage of cloudiness that occurred during a certain month as given in Monthly Weather Review reports. The value of Fis useful inconverting the exterior insolation, Iex' into surface insolation, I, and then in-to evaporation. '.I'he following equation is useful and is sufficiently a ccurate to give good results: I = I ex p' F (14) Equation (14) demonstrates the combined effect of relative cloudiness and the atmospheric transmission coefficient and the product of the two terms, p'F, give the interesting ratio of the energy we receive to that which strikes the exterior or t he atmosphere. Table 13 includes values of this ratio. Conclusions It is possible to deterrnine the value of insolation by three distinct methods, and the results obtained by these different methods check experimentally within 5%. When it is not possible to get a ratio of evaporation to insolation from pyrheliometer records, it is necessary to compute the insolation o:fl. the exterior of the ea rth 's a tmosphere and to correct this quantity by a transmission coefficient f' or the earth 's atmosphere and by a clearness factor ( see Equation (14)). 1'he evaporation formula (Equation (9)) given in terms of insolation and r a diation checks experimental l y with obs erved evaporation in Calif ornia, a nd when applied to bodies of water outside California it gives satisfacto ry results. List of References Cited: (1) Kimball and Hobbs, Journal,Optical Society of .America,Vol.7 Sept.1923. (2) Monthly Weather Rev:i.ew,Vol.57,January 1929 . (3) Monthly 'H eather Review,October 1928,p.430. (4) Richardson and Montgomery, Nat. Res ,Council Bulletin 68,Apr.1929 Monthly Weather Review, July 1929 ,p.300 (5) and (6) Abbot,Annals,Astrophy.Observ.Smithsonian Inst. Vol.4, 1922 (7) Bowditch, ''American Practical Navigator''· (8) Monthly Weather Review,April 192 7, p .156. (9) Cummings and Richardson, Physical Revi ew, October 1927 p . 527. (10) Bowen, Physical Review, June 19 26,p.779. {11) Jl'reeman, ''Regulation 6£ the Great Lakes'' , October 1926. Monthly Weather Review,July 192?, p . 320; Eng.News 1910. {12) Wells, Journal Wash. Academy Sci. October 19, 1928 . Wells,U. S . Geol. Survey , Dep t .Int. Prof. Paper 156C . Mar.14,1 929 . (13 ) McEwen, 'I'ech.Series,Scripps Inst. u. of Cal. Vol.2, No.6 September 1929. (14) Monthly Weather Review, August 1925 , p.355 . {15) Monthly Weather Review, April 192'7 , p.156. Note: This thesis published Am . Soc .Civil Eng .l: > rocee dings l,:ay 1930 .