Geografisk Tidsskrift, Bind 75 (1976)THE ENERGY OF RAINDROPSK. Høgh-Schmidt and S. Brogaard Side 24
Høgh-Schmidt, K.
& Brogaard, S.: The energy of raindrops,
K. Høgh-Schmidt, cand. mag., Physics Laboratory, Royal Veterinary and Agricultural University, Copenhagen, Denmark. S. Brogaard, cand. mag., Physics Laboratory, Royal Veterinary and Agricultural University, Copenhagen. Introduction.In the autumn 1974 the total energy of precipitation as a main factor in soil erosion was discussed in a colloquim at The Geographical Institute. All the papers presented were based upon the idea that the precipitation energy - with identical intensity - was a function of the terminal velocities of the drops. It seemed strange that the wind should have no effect, since increasing wind should increase the horizontal velocities of the drops, and hence the kinetic energy of the precipitation. It was therefore decided to investigate this in greater details, and the present paper presents a simple mathematical model allowing an approximate calculation of velocities as a function of (a) the wind velocity, and (b) different shapes of the wind profile. Mathematical treatment.The calculations are based upon a model assuming that all drops, situated above a given height H, have the same horizontal speed as the atmosphere. When the drops have descended the level H, horizontal speed of the drops is presumed to deviate from the horizontal speed of the atmosphere. Finally, it is assumed that no coalescense occurs below the height H. Over an adequately
thick layer, the wind profiles obey ![]() where u is the horizontal wind speed at the height z, UQ the horizontal wind speed at a reference level ZQ, and the power n is in particular a function of the stability of the atmosphere, and to a lesser degree of the roughness of the surface. At a drop's
relative motion through atmosphere, the ![]() where yis some
coefficient of resistance, pL the density If the horizontal speed of the drop is v, its relative speed in horizontal direction will be v - u. The vertical speed of the drop being w, the speed wa relative to the air is given by ![]() Fig. 1. The frictional resistance is directly opposite the relative velocity wa, determined by the vertical velocity w and the relative horizontal velocity v-u of the drop. The horizontal component of the frictional resistance is called Fx, the vertikal component Fz. Fig. L Modstanden mod en dråbes bevægelse er modsat rettet den relative hastighed wa, bestemt ved vertikal faldhastighed wog relativ horisontal hastighed v-u. Gnidningsmodstandens horisontale og vertikale komposanter er henholdsvis Fx og Fz. ![]() Side 25
If a is the angle between w and wa (see Fig. 1), the following two equations are obtained for the determination of the acceleration of the drop, Ah and Av, in the horizontal and the vertical direction, respectively ![]() (1) ![]() (2) where pw is the
density of the drop, g the acceleration (3) ![]() The second part of
equation (2) represents gravity Using a first order approximation Av is zero, hence the left side of equation (2) is (nearly) constant. During the fall of the drop wa increases, but a must decrease in such a manner that wa 2 cos a is (nearly) constant: ![]() (4) At the starting
level H the equation can be written ![]() (5) where yo is the
coefficient of resistance, WQ the vertical Since ![]() (6) the equations (1),
(3), (4), (5), and (6) may be combined ![]() (7) The coefficient of resistance, yO, may be calculated by means of equation (5) knowing the terminal velocities of waterdrops in still air. Using the value PL/PW = 1.196 • 10~3 some calculations of p0 and DQ have been made, and the results are given in Table 1. Numerical solutions.The horizontal
speed of a drop is determined by equation Assuming u to be a function of z given by the power law, the value of the drop velocity v for decreasing levels can be calculated successively. Down to 20 m over the plane surface the calculation is made for every meter, from 20 to 5 for every half meter, and from 5 and down to the surface for every 10 cm. Calculations are made for drop diameters from 1 to 5 mm for each value of the speed UQ (assumed to be determined at the reference level ZQ = 10 m) and for each value of the power n. ![]() Table 1. Terminal velocities wO, drag coefficient X0 and factor D0 (see the text) for water drops falling in still air. Tabe! l. Terminalhastighed wO, modstandskoefficient YO og proportionalitetsfaktor D0 for forskellige dråbediametre ved fald i stillestående In solving equation (7) the initial condition is that v- u = 0 at the starting level H. In practice this assumption is not valid since a drop moves faster than the air. It is shown that the selected initial condition has no influence on the terminal horizontal velocity of the drop at the surface by comparing the calculated values of the terminal velocity obtained for starting levels H equal to 200 m and 100 m, respectively. ![]() Table 2. The distance, a drop has to move-according to the modelbefore obtaining the same velocity (difference 5) as a free falling drop of the same size. Tabel 2. Indsvingningsdistance for forskellige dråbediametre som funktion af de valgte krav om hastighedsoverensstemmelse (6). The vertical
distance a drop starting at the initial level Side 26
initial level H =
200 m (apart from a small difference 0) The main result of these calculations is that the assumption v - u = 0 at the starting level H has no influence on the resultant terminal velocity of the drop, as long as the starting level is at least 100 m above the surface. In other words, as long as the cloud base is 100 m ore more aloft, drops of a given size will reach a terminal horizontal velocity that depends on drop size and wind profile but not on the starting level of the drop. The horizontal velocity of the raindrops.In making the numerical calculations of the terminal velocities (i. e. the velocities at the surface of the earth) a starting height of H = 200 m is chosen. Table 3 shows the results of the calculations using a value of n = 0.30 (approximately the average stability). ![]() Tabel 3. Dråbers horisontale terminalhastighed Vht (n = 0.30). Table 3. The horizontal terminal velocities of raindrops (n = 0.30). From the table it is seen, that for a given drop diameter the horizontal terminal velocity of the drops is proportional to the wind speed at a height of 10 m (cf. Fig. 3). From the table it is also seen, that for a given wind speed the horizontal velocity of the drops increase with increasing drop diameter, but the relationship is far from linear, since the increase in horizontal velocity per mm increase in drop diameter diminishes as the drop diameter grows bigger (Fig. 4). ![]() Fig. 4. Horizontal terminal velocity vs. diameter of the drops. Fig. 4. Horisontal terminalhastighed som funktion af dråbediameter (n = 0.30). Fig. 3. Horisontal terminalhastighed som funktion af vindhastigheden u0 i referencehøjden z0 = 10 m(n = 0.30). Fig. 3. Horizontal terminal velocity of the drops at different wind speeds at reference level 10 m. ![]() Table 4. The horizontal terminal velocities of rain drops for n = 0.20, 0.25,0.35 and 0.40. Tabel 4. Dråbers horisontale terminalhastighed vht for n = 0.20, 0.25, 0.35 og 0.40. The results of the numerical calculations of the horizontal terminal velocity made for various values of the exponent n are shown in Table 4. From the table it is seen, that for fixed values of n and drop diameter, the horizontal
terminal velocity vht of the drop is proportional ![]() Tabel 5. Konstanterne fO, a og b i ligningen f =f0 + a exp(-b • dgs), hvorf=vh,(n)/vht(o.3o). Table 5. The factors fo,aand b from the equation f = f0 + aexp(-b • dgs), wheref=vht(n)/vht(o.3o). The horizontal terminal velocity as a function of n is illustrated by plotting ß = vht/u0 against n (Fig. 5). It is seen that in the case of small drops there is a distinct decrease in ß, and consequently in vht, for increasing values of n, while ß for larger drops decreases very little with increasing n. From Fig. 6, which shows ß as a function of drop diameter for various values of n, it appears that the smaller the n value (i. e. the less stable Side 27
the atmosphere)
the greater the horizontal terminal velocity Method used in calculating the energy of precipitation.![]() Fig. 5. ß = Vht/Uo som funktion af n for d = l, 3 og 5 mm. Fig. 6. ß = Vht/uo vs. diameter for n= 0.20, 0.30 and 0.40 Fig. 6. ß = Vht/Uo som funktion af dråbediameter for n= 0.20, 0.30 og 0.40. Fig. 5. ß = vht/uo vs. n for d = l, 3 and 5 mm. ![]() Table 6. The size distribution (after Laws and Parsons) at an intensity of 0.5 in h~'. Tabel6. Dråbefordeling (efter Laws og Parsons) veden nedbørsintensitet på ca. 12.5 mmh~\ When the distribution of the drops in the precipitation (rain) and the terminal velocities of the drops both in horizontal and vertical direction are known, it is possible to calculate the total kinetic energy of a shower. The calculations
are based on the size distribution found by Laws and Parsons at an intensity of 0.5 in h 1 (cf. Table 6) and the calculated horizontal component of the terminal velocities. The approach used in making the calculations of the energy of the precipitation is as follows: a) For every
interval in drop size, the drop diameter dgs, matching the
average mass of drops in the interval, b) For every value
of d , the vertical terminal velocity minal velocity as
a function of drop diameter. c) For every value
of d , the horizontal terminal velocity based upon the
calculated horizontal velocities for The values of the horizontal terminal velocities for dgs> smm are obtained by extrapolation. This might seem doubtful, but vht increases only slowly for increasing d and in this case even relatively large errors would only affect the energy of the total shower to a small extent, since large drops are relatively rare. d) For other
values of n the horizontal terminal velocity ![]() where vht(o.3o) is the horizontal terminal velocity of the drop concerned for n = 0.30 at the given speed of wind. The factor f is obtained from the empirically based relation ![]() where e is the
base of the natural logarithms, and The terminal
velocities for drop diameters of 1, 3, e) The
precipitational energy is expressed per mm precipitation
Laws and Parson's
precipitational distribution factor Side 28
In order to obtain the total energy of precipitation (per m2), the value found for the precipitational energy is multiplied by the total amount of precipitation (per m2). 0 For a given
value of uO, the total energy per mm ![]() where ijjj is the
amount of precipitation in drop size ![]() ![]() where E0 = 1/2 I qji wi2 is the total energy of the drops in calm weather, and where un= % I Vi ßi2 is a quantity dependant on the stability index n. It is seen that the total energy is a parabolic function of wind speed, when the size distribution and wind profile are kept constant. ![]() Fig. 7. The factor n vs. n. Fig. 7. Proportionalitetsfaktoren \in i ligningen for nedbørenergien E=EO+ • Ho2 som funktionn. Results from calculation of the energy of prepicitation.![]() \in af Fig. 8. The horizontal energy per mm rain at an intensity of 0.5 in h~l. Fig. 8. Nedbørens horisontalenergi pr. mm regn ved en nedbørintensitet på ca. 12.5 mm h~\ The resulting E0 and un values obtained from the selected size distribution and wind profile are shown in Table 7. The graph of unversus n is shown in Fig. 7. un appears to be a non-linear decreasing function of n. The calculations show that for a given wind speed, at the reference level z0 = 10 m, the total energy of precipitation is larger in unstable conditions of the atmosphere (n small) than in stable conditions (n large). ![]() Fig. 10. The total energy per mm rain at an intensity of 0.5 in h~1 vs. wind speed at reference level 4 m. Fig. 10. Nedbørenergi pr. mm regn ved en nedbørintensitet på ca. 12.5 mm h~} som funktion af vindhastigheden i referencehøjden z0 = 4 m. ![]() Fig. 9. The distribution of the energy per mm rain at an intensity of 0.5 in h~\ The area beneath the curve represents the total energy of the rainfall per mm rain. Fig.9. Nedbørens energifordeling pr. mm regn ved en intensitet på ca. 12.5 mm h~^ (u0 = 12 m s~\ n = 0.30). Arealet under kurven repræsenterer den samlede energi pr. mm regn. ![]() Tabel 7. Konstanten E0 og proportionalitetsfaktor \j„ (ligningen for nedbørenergi:E =E0 + p/? u0 2)forforskellige værdier af n. Table 7. The constant energy E0 and the factor |jn, determining the energy of precipitation: E = E0 + Mn u0 2. Side 29
Fig. 8 depicts the »horizontal« energy (pn • u0 2) per mm rain at the precipitation intensity considered (about 12.5 mm • h~1) as a function of wind speed at the 10 m level for 3 different values of the exponent n. It is noted that the »horizontal« energy of the rain is a significant part of the total energy at wind velocities greater than 6 m • s~1. The graph depicting precipitational energy per mm rain at an intensity of 0.5 in-bf1 as a function of drop size is shown in Fig. 9 for a wind velocity of 12 m-s~1 at the 10 m level and n = 0.30. It is seen that for drop sizes less than 1 mm and larger than 5 mm the energy is quite insignificant. Finally, Fig. 10 shows that by selecting an appropriate reference level at which to measure the wind speed uO, it is possible to obtain a relation between precipitational energy and wind speed at reference level, which is independant of the stability index n. For the rain intensity selected and the related size distribution, it is found that the appropriate reference level is z0 =4m.ln this case the total energy of the precipitation is given by ![]() whereEo =
21.27Jm~2mm~I,u0=0.412Js2m~4mnT1, RESUMEVed anvendelse af en simpel matematisk model for vanddråbers fald gennem luften kan dråbernes horisontalhastighed approksimativt bestemmes ved numerisk integration af en differentialligning. Ved løsningen af ligningen forudsættes det 1) at
vindhastighed og horisontal dråbehastighed er 2) at dråben kan
betragtes som en kugle, 3) at der ikke
optræder koalescens i atmosfæren i 4) at
vindprofilen kan beskrives ved en potenslov. Beregningerne viser, at den første antagelse ikke medfører fejl ved beregningen af dråbernes horisontale terminalhastighed, når blot udgangshøjden H er større end 100 m overjordoverfladen. De horisontale terminalhastigheder viser sig ved beregningerne at være en funktion af vindhastigheden u0 i referenceniveauet (z0 = 10 m) og af eksponenten ni den potenslov, der gengiver vindprofilen (fig. 3-6). For nedbørenergien vises det, at for samme dråbefordeling i nedbøren og for samme værdi af eksponenten i potensloven er den totale energi en parabolsk funktion af vindhastigheden i referenceniveauet. På grundlag af den dråbefordeling, der er fundet af Laws og Parsons, er nedbørens energi i horisontal retning beregnet som funktion af vindhastigheden i referenceniveauet 10 m og for 3 forskellige værdier af eksponenten (u= 0.20, 0.30 og 0.40), fig 7-8. Ved passende valg af referenceniveau z0 for vindhastigheden u0 kan man opnå en praktisk talt entydig sammenhæng mellem nedbørenergien og vindhastigheden uafhængigt af eksponenten n. For den angivne dråbefordeling finder man, at referenceniveauet skal være z0 = 4m, fig 10. LITTERATUR.Norman Hudson
(191 \)\ Soil Conservation, The Anchor Press G. J. Mason
(1971): The Physics of Clouds, 2. ed., Oxford
/. O. Laws and D.
A. Parsons (1943): The relationship of W. H.
Wischmeier andDwightD. Smith (1958): Rainfall Energy
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