HYDROLOGICAL PROCESSES
Hydrol. Process. 14, 37±49 (2000)
Measuring drop size distribution and kinetic energy of
rainfall using a force transducer
A. W. Jayawardena* and Rezaur R. B.
Department of Civil Engineering, The University of Hong Kong, Hong Kong
Abstract:
The relatively high cost of commercially available raindrop spectrometers and disdrometers has inhibited
detailed and intensive research on drop size distribution, kinetic energy and momentum of rainfall which are
important for understanding and modelling soil erosion caused by raindrop detachment. In this study, an
approach to ®nd the drop size distribution, momentum and kinetic energy of rainfall using a relatively
inexpensive device that uses a piezoelectric force transducer for sensing raindrop impact response is introduced.
The instrument continuously and automatically records, on a time-scale, the amplitude of electrical pulses
produced by the impact of raindrops on the surface of the transducer. The size distribution of the raindrops and
their respective kinetic energy are calculated by analysing the number and amplitude of pulses recorded, and
from the measured volume of total rainfall using a calibration curve. Simultaneous measurements of the
instrument, a rain gauge and a dye-stain method were used to assess the performance of the instrument. Test
results from natural and simulated rainfalls are presented. Copyright # 2000 John Wiley & Sons, Ltd.
KEY WORDS
drop size distribution; kinetic energy; momentum; piezoelectric transducer; soil erosion; rainfall
INTRODUCTION
Since erosion starts with the process of soil detachment by raindrop impact, the basic unit of raindrop
erosivity can be represented by the stress, momentum or kinetic energy of a single raindrop (Sharma, 1996),
which are all functions of the drop size, drop shape and the terminal velocity. Of these, the kinetic energy of a
single drop is the most commonly used unit of raindrop erosivity (Hudson, 1995). Frequent and routine
measurements of drop size distribution, momentum, kinetic energy or impact forces of rainfall are often
needed to better understand the mechanics of soil detachment by raindrop impact, and data on the kinetic
energy load of rainstorms are basic in order to develop and verify physically based models of soil detachment
by raindrop impact in interrill erosion processes (Jayawardena and Rezaur, 1999). There are several studies
in which the response of the soil surface to single water drop impact force (Ghadiri and Payne, 1977, 1981;
Nearing and Bradford, 1985), kinetic energy (Al-Durrah and Bradford, 1981; Sharma and Gupta, 1989;
Sharma et al., 1991) and momentum (Riezebos and Epema, 1985) have been examined. All these models
require speci®c information about the size and velocity of the impacting drops.
The total kinetic energy of rainfall is calculated by summing up the individual kinetic energies of raindrops
with the aid of drop size distribution and raindrop terminal velocity information for a rainstorm (Sharma
et al., 1993, 1995). The commonly used methods of measuring the size distribution of natural or arti®cial
rainfall are the dye-stain method, ¯our pellet method, high speed photographic method and oil immersion
method (Eigel and Moore, 1983; Coutinho and Tomas, 1995; Cerda, 1997). These methods are laborious,
* Correspondence to: A. W. Jayawardena, Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong
Kong. E-mail: hrecjaw@hkucc.hku.hk
CCC 0885±6087/2000/010037±13$17.50
Copyright # 2000 John Wiley & Sons, Ltd.
Received 7 September 1998
Accepted 27 April 1999
38
A. W. JAYAWARDENA AND REZAUR R. B.
time consuming and incapable of providing a continuous record. Since measurement of drop size
distribution is cumbersome and subject to storm-type variability, such methods do not lend themselves
conveniently to frequent and routine use. Even if detailed and accurate information on drop size distribution
has been collected, there still remains the problem of combining such data with the terminal velocity of
simulated or natural rainfall into momentum, kinetic energy or some similar function (Hudson, 1981).
Automatic sampling devices designed to give a continuous record of the drop size distribution such as the
raindrop spectrometer (Mason and Ramanadham, 1953), the photoelectric raindrop size spectrometer
(Dingle and Schulte, 1962), balloon-borne instruments and instruments using the wind tunnel principle
(described in Mason, 1971) have the disadvantage of being costly, complicated or having moving components that are inconvenient for routine use. The acoustic method based on the conversion into an electrical
pulse of the sound made by the drop upon impact on the diaphragm of a microphone (Kinnell, 1972;
De Wulf and Gabriels, 1980) has limitations for use as a disdrometer because of the duration of the decaying
waveform. The pulses produced by successive impacts of raindrops tend to interfere with each other and the
device can only be used at low rainfall intensity. A piezoelectric sensor used with unsophisticated sorting and
storage equipment (Kowal et al., 1973) have given important results but the sensor, not being sensitive
enough, was unable to detect smaller drops. Hudson (1981, 1995) recognized these diculties, and emphasized the need for an inexpensive, simple and robust instrument for widespread use in soil erosion studies.
The most recent and advanced techniques, the optical disdrometer (Grossklaus et al., 1998), the 2D-video
disdrometer (Schonhuber et al., 1994, 1995) and the Joss±Waldvogel rainfall disdrometer (Kinnell, 1976) use
highly sophisticated electronic instruments capable of providing a completely automatic record of the drop
size distribution. In common with all other electronic methods these instruments were primarily designed for
meteorological studies of cloud physics and are too sophisticated and, at the present time, too expensive
for routine studies of soil erosion. Recognizing that research in erosivity or the study of the potential of
raindrops to cause soil erosion is impeded by a lack of simple instrumentation for routine use and of
techniques for measuring the size distribution and kinetic energy or momentum of raindrops during
individual rainfall event or soil erosion experiments, this study attempts to design and fabricate a simple and
relatively inexpensive instrument by using commercially available materials for routine use in measuring the
drop size distribution and kinetic energy load of rainstorms applicable to studies of water and soil
conservation.
INSTRUMENT DESIGN
The design of the instrument centres around the choice of a sensing device which must be large enough to
record a representative sample of drops during rainfall, but not too large that coincident counts will be too
numerous. It must be sensitive enough so that a drop of given size will produce the same response within
small limits of error at all points in the sensitive area, except at the very edges. The sensor must be fast
enough to respond to the dynamic impact of raindrops and it should be capable of being served by a wide
variety of electronic readout devices capable of recording the data as a function of time, either directly
without analysis or with varying degrees of analysis as desired. Obviously, a facility for remotely controlled
measurements through a computer is desirable. Of the various possibilities that exist, each has distinct
advantages and disadvantages.
The conclusion reached in the present case was that a piezoelectric force transducer would most nearly and
most ¯exibly meet the needs detailed above and the needs for soil erosion studies. In the past, piezoelectric
transducers have not been utilized to their full potential because transducers were expensive and had slow
response time (Dowd and Williams, 1989). The high frequency response, fast rise time, high sensitivity and
good response to dynamic and short duration static loading (Nearing et al., 1986; Taylor, 1997) made
piezoelectric force transducers a logical choice for measuring transient water drop impact response for this
study. The instrument essentially consists of two components; a piezoelectric force transducer and a
computer data acquisition system.
Copyright # 2000 John Wiley & Sons, Ltd.
Hydrol. Process., Vol. 14, 37±49 (2000)
RAINFALL DROP SIZE AND KINETIC ENERGY
39
Figure 1. Schematic diagram of system components
Operation and type of piezoelectric transducer
Piezoelectric force transducers usually work on the piezoelectric e€ect of crystalline quartz discs which
produce electric charges proportional to the force applied (Taylor, 1997). Although the charge in a crystalline
dielectric is formed at the location of an acting dynamic force, metal electrodes equalize the charges along the
surface making the transducer sensing area not selectively sensitive (Fraden, 1997). To measure the response
due to rainfall impact, a force version of a Bruel & Kjaer* type 8200 surface mounting force transducer
(Figure 1) was chosen. The transducer had a rugged, all-welded, hermetically sealed construction with a
ceramic insulated microplug connector sealed with modulated glass, and a stainless steel sensing area, 16 mm
in diameter, allowing it to be used under very severe environmental conditions. The connections between the
transducer and the cables were waterproof sealed by RTV (room temperature vulcanizing) for enhanced
survivability and to prevent ¯uid from entering and short circuiting or corroding the internal electronics. The
overall thickness of the transducer was 13 mm. The transducer had a resonant frequency, fr , of 35 000 Hz,
and a rise time of about 0.25/fr 7 ms (Bruel and Kjaer, 1998).
It is dicult to obtain a uniform response over a large sensing area because the noise level increases
rapidly with the size of the sensing area. Therefore, the detection of small drops or multiple drops
impacting the transducer diaphragm simultaneously is prejudiced (Mason, 1971). The relatively small
sensing area of the transducer ensured uniform response and reduced the chance of a number of drops
hitting simultaneously on the sensing area. The issue of collecting responses from a representative sample
of raindrops can be achieved by exposing the transducer to rainfall for a relatively long period of time.
Since piezoelectric force transducers have no static response (Fraden, 1997), the accumulation of water on
the transducer surface does not produce any response. The possibility of a damping e€ect due to
accumulation of water on the surface of the transducer was eliminated by coating the transducer
diaphragm with RTV and mounting it slightly inclined to facilitate removal of water by gravity ¯ow. By
having the transducer surface slightly inclined (less than 108, to facilitate gravity drainage) the sensitivity of
the transducer to drop impact is not a€ected.
Data acquisition system
The data acquisition system is based around a commercially available charge ampli®er (Bruel & Kjaer,
Type 2636), an analogue to digital (A/D) board (Data Translation, model 2801-A) and an IBM AT
compatible 16-bit personal computer (Figure 1). The transducer output terminals were interfaced with the
computer through the charge ampli®er and the A/D board. This enables fast datalogging through the use of
* Trade names and company names, included for the bene®t of the reader, do not imply endorsement or preferential treatment of the
product by the authors.
Copyright # 2000 John Wiley & Sons, Ltd.
Hydrol. Process., Vol. 14, 37±49 (2000)
40
A. W. JAYAWARDENA AND REZAUR R. B.
data acquisition programs, resulting in considerably greater control of the measurement regime. The charge
ampli®er converts the output charge (coulomb) of the transducer to a voltage. It also allows the use of long
or varying lengths of input cables without disturbing the sensitivity of the transducer. The A/D board scans
the transducer at a frequency of 2500 Hz, converts the analogue voltage pulses to a digital record as a
function of time and sends the data to the computer for analysis and ®nal storage. The high frequency
scanning provides adequate mapping of the amplitude of the voltage pulses produced by the transducer and
resolves the dynamic process considerations imposed on the transducer by the raindrop impact. The A/D
board also serves to record signals from multiple transducers simultaneously and provides user-speci®ed
controlled measurements as circumstances dictate. The computer was used to store the data and to
programme the A/D converter to monitor the transducer. The computer also served as a remote control.
Because of the fast scanning rate of the A/D board, there is an in¯ux of a high number of data. A data
acquisition program was written in HP VEE visual programming to control the A/D board and to extract the
peak amplitude of the voltage pulses and the time of occurrence of the drop before they were ®nally stored.
This arrangement reduced the volume of stored data. A schematic diagram of the system components of the
instrumentation is shown in Figure 1.
CALIBRATION
The transducer response (voltage output) was calibrated for momentum, kinetic energy and equivalent drop
mass with water drops of known size and known fall velocity. An inexpensive set-up (Figure 2) was developed
which uses visible laser beams and photocells to measure the fall velocity of water drops. Within the
open space between two stairs of a staircase, a variation of fall height between 0.5 and 14 m was accomplished
by placing a burette provided with a capillary at di€erent elevations. Di€erent capillaries produced drops with
diameters between 1.50 and 5.25 mm. The time interval between the formation of consecutive drops was 1±2
seconds. At the lower end of their fall trajectory, and before they hit the transducer, the drops were made to
pass through a 10 cm wide and 50 cm long plexiglas casing. Within the casing, two horizontal laser beams at a
vertical distance of 50 cm were interrupted by the falling drops. From the time interval between the interruption of the higher beam and the lower one, measured by a timer counter, the fall velocity of the drops was
calculated.
Figure 2. Schematic diagram of set-up for calibrating transducer. A: Drop former; B1, B2: laser beam emitter (laser diode module);
C1, C2: photodetector; D: force transducer; E: feed for laser unit; F: timer counter; G: transducer response recorder
Copyright # 2000 John Wiley & Sons, Ltd.
Hydrol. Process., Vol. 14, 37±49 (2000)
41
RAINFALL DROP SIZE AND KINETIC ENERGY
Table I. Water drop characteristics
Drop diameter (mm)
Velocity (m s ÿ1)
Drop mass (mg)
m
s
m
s
Measured
From Epema and Riezebos*
1.51
2.45
2.80
3.56
3.76
4.73
5.25
0.022
0.012
0.015
0.021
0.023
0.030
0.028
1.76
7.73
11.50
24.00
28.00
55.28
75.60
0.021
0.022
0.029
0.021
0.032
0.089
0.078
5.30
7.27
7.80
8.32
8.57
9.01
9.12
5.27
7.30
7.76
8.40
8.61
8.98
9.11
m ˆ mean.
s ˆ standard deviation.
Sample size ˆ 30 for each drop size and velocity measurements.
*From terminal velocity data of Epema and Riezebos (1983) by interpolation.
Transducer responses vs. momentum were measured on 1.51, 2.45, 2.80, 3.56, 3.76, 4.73 and 5.25 mm
diameter drops. The mean and the variation of the drop sizes were determined by weighing 100 drops to the
nearest 0.1 mg, repeated 30 times for each drop size. Between 30 and 35 impacts on the transducer were
recorded for each drop size. For each drop size the mean and the variation of peak amplitude of voltage pulse
were determined. The regression equations in the calibration curve were calculated from the complete data
set but for clarity the data points for the mean values only are shown.
The characteristics of the drops are given in Table I. Many of the drops which impacted the transducer did
not fall wholly on the sensing surface. The drops that fell partially on the edge of the sensing area could be
determined from the splash pattern on the block on which the transducers were mounted. The drops that fell
completely on the sensing area formed a single ring centred around the transducer after impact. Erroneous
data due to drop impingement on the transducer edge were eliminated in this way during data collection for
calibration.
The mean transducer response vs. drop size, momentum and kinetic energy curves and their best ®t
regression equations are shown in Figure 3. The transducer response vs. drop size and drop momentum
relationships were linear. The transducer response vs. drop kinetic energy relationship was analysed by ®tting
a linear model and a non-linear model. The linear regression model did not ®t well to the data and was found
to overestimate the kinetic energy for smaller drops and underestimate for larger drops. The non-linear
model provided good ®t of the data. This is perhaps because of the presence of the velocity squared term
(0.5 mv2) in the kinetic energy equation which did not allow a good ®t for a linear model. However, although
the initial slope of the relationship between transducer response vs. drop kinetic energy for drop volumes
smaller than 4 mm3 (1.96 mm diameter) is curvilinear, this is taken into account during data analysis for
rainfall by grouping the drop sizes with volumes less than 4 mm3 into one size class. Since the kinetic energy
of drops in the range 0.5±2.0 mm diameter is very small compared with the kinetic energy of drops larger
than 2 mm diameter [about 57, 2.0 mm diameter drops or 37 771 drops of 0.5 mm diameter have the same
kinetic energy as one 6 mm diameter drop, considered to be the largest stable drop by Lal (1990)], the total
kinetic energy load of the rainfall is not greatly a€ected by grouping drop sizes of 0.0±2.11 mm diameter into
one size class.
CALCULATION OF DROP SIZE DISTRIBUTION AND KINETIC ENERGY OF RAINFALL
The drop size distribution and kinetic energy of a rainfall event were calculated by analysing the voltage
pulses stored, where each pulse represents a drop and the magnitude of the pulse corresponds to drop size,
momentum and kinetic energy of the drop. The linear response between the transducer output and drop
Copyright # 2000 John Wiley & Sons, Ltd.
Hydrol. Process., Vol. 14, 37±49 (2000)
42
A. W. JAYAWARDENA AND REZAUR R. B.
Figure 3. Relationship between transducer peak output voltage, momentum, kinetic energy and drop size for drops falling at terminal
velocity. (Each data point in the graph is the mean of 30 samples)
momentum allows one to write an equation relating drop volume intercepted per unit area of the transducer
to its output signal in the form (Kowal et al., 1973)
Vi ˆ bli ni
1†
where Vi is the volume intercepted per unit area of transducer (mm3 cm ÿ2) in a given amplitude size class i, b
is a coecient (mm3 mV ÿ1) (slope), li is the mean transducer output amplitude (mV) in class i, and ni is the
number of drops intercepted per unit area (cm ÿ2) of the transducer in amplitude class i.
Since the total volume of rainfall must be equal to the summation of the volume of drops in di€erent size
classes, the volume of rainfall, TV, measured by a rain gauge (mm3 cm ÿ2) can be expressed as
TV ˆ
N
X
iˆ1
Vi ˆ b
N
X
li ni †
2†
iˆ1
where N is the number of amplitude classes. The average volume (mm3) of each drop V0 i within a given
amplitude size class i is obtained as
0
Vi ˆ Vi =ni
3†
and the average diameter di of each drop (mm) within a given amplitude size class i is obtained as
di ˆ
6V0i
p
1=3
4†
Since TV is known from rain gauge measurements, and li , ni are known from transducer output, Equation (2)
is solved for b. Substitution of b in Equation (1) yields the volume of drops in amplitude size class i.
Equations (3) and (4) are then subsequently used to ®nd the average volume and diameter of drops within
each size class. The kinetic energy of each drop is obtained from the calibration curve using the information
on transducer output amplitude. The steps for calculating the size distribution and kinetic energy of rainfall
for a rainfall event recorded in Hong Kong on 22 May 1998 are shown in Table II. The ®rst step in preparing
Copyright # 2000 John Wiley & Sons, Ltd.
Hydrol. Process., Vol. 14, 37±49 (2000)
Transducer response
Voltage class
(mV)
Mid value
(mV)
(li)
0.00±2.44
1.22
2.45±4.88
3.67
4.89±7.32
6.11
7.33±9.76
8.55
9.77±12.21
10.99
12.22±14.65
13.43
14.66±17.09
15.87
17.10±19.53
18.31
Total
Total KE (J mÿ2 mm ÿ1)
No. of
pulses
No. of drops
per cm2
(ni)
108
54
14
7
6
5
4
4
201
54.0
27.0
7.0
3.5
3.0
2.5
2.0
2.0
(lini)
65.88
99.09
42.77
29.93
32.97
33.58
31.74
36.62
372.58
Volume in
each class
(mm3 cm ÿ2)
(Vi ˆ blini)
264.18
397.35
171.51
120.02
132.21
134.21
127.28
146.85
Drop size
Kinetic energy
Volume (mm3) Diameter (mm) mJ cm ÿ2 drop ÿ1
(V0 i ˆ Vi/ni)
(di)
*
{
4.89
14.72
24.50
34.29
44.07
53.68
63.64
73.43
2.11
3.04
3.60
4.03
4.38
4.68
4.95
5.20
0.054
0.239
0.435
0.658
0.873
1.087
1.315
1.527
0.058
0.221
0.412
0.621
0.844
1.077
1.321
1.572
5.83
12.91
6.09
4.61
5.24
5.44
5.26
6.11
51.48
34.43
6.26
11.93
5.77
4.35
5.06
5.39
5.28
6.29
50.33
33.69
43
Hydrol. Process., Vol. 14, 37±49 (2000)
TR ˆ 14.94 mm ˆ 1494 mm3 cmÿ2, b ˆ TR/S(lini) ˆ 4.010.
The odd fractions in the voltage class are due to dividing the minimum and maximum transducer response into eight class intervals of equal increment.
*Using terminal velocity data from Epema and Riezebos (1983).
{Using calibration curve of Figure 3.
mJ cm ÿ2
*
{
RAINFALL DROP SIZE AND KINETIC ENERGY
Copyright # 2000 John Wiley & Sons, Ltd.
Table II. Drop size distribution and kinetic energy of raindrops of a 14.94 mm rainstorm recorded for 15 minutes by the instrument at the University of
Hong Kong on 22 May 1998
44
A. W. JAYAWARDENA AND REZAUR R. B.
the Table is to ®ll in column 3 showing the number of pulses corresponding to each amplitude class. The
second step is to include in the Table the number of drops per cm2 corresponding to each amplitude or drop
size, by dividing the number of pulses by the e€ective area of the transducer (201 mm2). Then, from the total
value of lini and rainfall measurement TR, the value of b is obtained from Equation (2). The drop size
distribution is then calculated from Equations (1), (3) and (4).
RESULTS AND DISCUSSION
The instrument provides a convenient, fast and relatively simple means of assessing the drop size distribution
of natural or arti®cial rainfall, from which the kinetic energy or momentum can be deduced and used to
assess the erosivity of rainfall and to complement studies on soil erosion. Since the instrument records pulses
in a time sequence, measurement of the rainfall can be made for any particular time during the storm. The
reliability of the instrument and the technique was assessed by comparing the cumulative rainfall volume
measured with a rain gauge with that obtained with the force transducer. The cumulative volume of rainfall
from the force transducer was obtained by calculating the size (volume) of each rain drop using the
calibration curve (Figure 3) and the cumulative summation of drop volume for that particular time. For all
events, comparisons of rainfall rates measured by the force transducer and by a rain gauge reveal a good
match. Figure 4 shows one such comparison for a simulated rainfall ( from a nozzle and spinning disk-type
Arm®eld FEL3 rainfall simulator) of 60 mm hr ÿ1 intensity and 15 minute duration. The highest magnitude
of absolute deviation in cumulative rainfall volume measurement by the instrument, for each successive one
minute interval, was 10% of that measured by the rain gauge, and the mean deviation was 3%. This clearly
demonstrates the reliability of the instrument. Figure 5 shows the drop size distribution obtained by the
instrument for the simulated rainstorm.
A further check for the reliability of the instrument is provided by comparing the drop size distribution
with that obtained simultaneously by the dye-stain method. Figure 6 shows such a comparison for short time
intervals during experimentation with simulated rainfall for various intensities and duration. Each point on
the graph represents the number of drops of a particular size determined by the two methods. The regression
Figure 4. Comparison of cumulative rainfall measured by a rain gauge and the instrument for a simulated rainfall of 60 mm hr ÿ1
intensity
Copyright # 2000 John Wiley & Sons, Ltd.
Hydrol. Process., Vol. 14, 37±49 (2000)
45
RAINFALL DROP SIZE AND KINETIC ENERGY
Figure 5. Drop size distribution measured by the instrument for a 14.94 mm rainfall recorded for 15 minute duration
Figure 6. Comparison of number of drops of di€erent sizes recorded by the instrument and the dye-stain technique for simulated
rainfall of various intensities
line ®tted to the data set gave a coecient of determination of 0.895 indicating good performance of the
instrument.
The Marshall±Palmer distribution function has been found to be a reasonable general predictor of drop
size distribution for drop sizes greater than about 1.5 mm (Mason, 1971; Brandt, 1990). The function takes
the form
N…d† ˆ No e
ÿfd
f ˆ 41R
Copyright # 2000 John Wiley & Sons, Ltd.
ÿ021
5†
Hydrol. Process., Vol. 14, 37±49 (2000)
46
A. W. JAYAWARDENA AND REZAUR R. B.
Figure 7. Comparison of results on drop size distribution obtained with the instrument and the law given by Marshall±Palmer
where No ˆ 8000 (mm ÿ1 m ÿ3), d is the drop diameter (mm), R is the rainfall intensity (mm hr ÿ1) and N(d) is
the number of drops per millimetre diameter interval and per cubic metre of air (mm ÿ1 m ÿ3). In order to
obtain the number of drops in each size class in 1 m2 on the ground, it is necessary to multiply N(d) by the
terminal velocity of a drop of a diameter in the middle of the class interval (Brandt, 1990). Figure 7 shows a
comparison of the drop size distribution obtained from the instrument for a 14.94 mm rainfall recorded for
15 minutes with the Marshall±Palmer model. The results from the instrument conform quite well to the
distribution law of Marshall±Palmer which is shown in Figure 7. Assuming the numbers of drops in di€erent
size class are distributed according to the Poisson distribution, the small probability of two or more drops
intercepting the transducer sensing area at the same time and producing a single pulse can be shown using
the Marshall±Palmer distribution function. The assumption of the Poisson distribution was con®rmed by
theoretical and experimental investigations carried out by Sasyo (1965). According to Equation (5),
assuming the mean value n~ i of number of drops between diameters d and d ‡ dd (Joss and Waldvogel, 1969)
intercepting the transducer surface in unit time (s ÿ1) to be
ÿfd
n~ i ˆ vi No e i ddA
6†
where A is the transducer surface area (m2), dd is the increment in drop size (mm) between diameter d and
d ‡ dd and vi is the terminal velocity (m s ÿ1) of drops in size class i. The probability p(xi) of ®nding xi drops
with diameters between d and d ‡ dd on the sensor area in one second is
xi …n~ i †
~
7†
e…ÿni †
p…xi † ˆ
xi †!
To simplify the calculations, a constant rainfall rate R ˆ 60 mm hr ÿ1 was chosen (as it would produce a
greater concentration of drops in air space). The probabilities of 2, 3 or 4 drops intercepting the sensor at the
same time, for di€erent drop sizes, were calculated using Equations (6) and (7). The terminal velocity vi for
the drops were taken from Epema and Riezebos (1983) and a dd of 0.25 mm was used. The results are plotted
in Figure 8 which illustrates the small probability of having coincident impacts from drops of di€erent size
intercepting the transducer surface at the same time. In common with other mechanically de®ned sampling
devices, errors may arise due to interception of drops at the edge of the transducer. Using the same analogy
Copyright # 2000 John Wiley & Sons, Ltd.
Hydrol. Process., Vol. 14, 37±49 (2000)
RAINFALL DROP SIZE AND KINETIC ENERGY
47
Figure 8. Probability of multiple impacts and edge e€ects from drops of di€erent size intercepting the transducer sensing area
simultaneously or a drop intercepting the edge of the transducer
as in Equations (6) and (7), the small probability of a drop interception at the edge of the transducer (area
within 1 mm from the edge) is calculated and is shown in Figure 8.
Table II lists the kinetic energy for each drop and the kinetic energy load of the storm as calculated using
terminal velocity data from Epema and Riezebos (1983) and using the calibration curve in Figure 3. The
highest magnitude of absolute deviation in kinetic energy (mJ cm ÿ2 drop ÿ1) measurements using the
instrument was about 8% of that calculated using the fall velocity data from Epema and Riezebos (1983), and
the mean deviation was 2%. Calculation of rainfall momentum followed the same procedure and, therefore, is
not shown.
The results are calculated from a record of the number and amplitude of the pulses recorded and from
the measured volume of rainfall. Changes in the sensitivity of the instrument are re¯ected in the value of
the regression coecient b, which accounts for all pertinent factors relating to the rainfall characteristics and
the sensitivity of the instrument. If drops are falling at their terminal velocities, their momentum and kinetic
energy can be calculated by making use of published data (e.g. Laws, 1941; Epema and Riezebos, 1983) on
terminal velocity. However, most soil erosion and hydrological experiments, either in the laboratory or in
®eld plots, use rainfall simulators to produce rainfall to speed-up data acquisition and to control rainfall
conditions. The calibration of the force transducer responses to drop momentum or kinetic energy for
known drop mass and fall velocity enables direct calculation of the kinetic energy or momentum from
transducer response information and the calibration curve. This can be useful in cases of simulated rainfall
where drop size distribution can be measured, but fall velocities are unknown or are very dicult to measure.
The performance and accuracy of the instrument and of the technique were found to be satisfactory to
complement studies of soil erosion. The technique, however, is based on a few assumptions that should be
taken into account when interpreting the results. Errors in counting and classi®cation of raindrops may arise
from the fact that drops falling at the extreme edge of the transducer produce smaller responses than those
falling near the centre, and by two or more drops falling on the transducer at the same time to produce only
one pulse. However, since metal electrodes within the piezoelectric transducer equalize the charges along the
surface, making the transducer sensing surface not selectively sensitive (Fraden, 1997), the possibility of edge
errors is greatly reduced. Observation of rain shows that, even in the heaviest showers, drops of 1 mm and
Copyright # 2000 John Wiley & Sons, Ltd.
Hydrol. Process., Vol. 14, 37±49 (2000)
48
A. W. JAYAWARDENA AND REZAUR R. B.
Figure 9. A typical trace of direct output of signals from the instrument showing transducer response to drop impact
larger occur only about once in every 103 cm3 of air space (Dingle and Schulte, 1962), and rainfall intensity in
tropical areas seldom exceeds 3000 drops m ÿ2 s ÿ1 (Kowal et al., 1973). The sensitive area of 201 mm2 and the
rise time 7 ms of the instrument is such that coincidence in this size range is therefore unlikely.
It is considered that splashing of the intercepted drops on the surface of the transducer due to raindrop
impact has no e€ect on the results since the momentum of the minute drops produced in splash is too small
to produce a signi®cant transducer response. This was veri®ed by observing the amplitude spectrum of the
voltage pulses from a typical trace of direct output signals of the transducer during a rainfall event (Figure 9).
The large distinct pulses (Figure 9), each with a positive peak followed by a negative peak, represent a drop.
The small pulses ¯uctuating around zero are the noise and are not considered as data. Positive peaks are due
to compression produced by drop impact and negative peaks are due to the tensile force produced by the
drop rebound and collapse on the force transducer.
CONCLUSIONS
The simple and relatively inexpensive instrument described and veri®ed in this study, though lacking the
superior capability of optical raindrop spectrometers, resolves the frequently encountered problem in soil
erosion studies and has the advantage that it can be constructed from readily available materials and
assembled in most laboratories without diculty. This makes assembly a simple task for modestly equipped
electronic laboratories in research and teaching organizations and eases on-site maintenance. Its simple and
¯exible design allows the relatively straightforward incorporation of multiple sensors as circumstances
dictate. It has the potential to establish within realistic budgets, relatively dense measurement networks for
detailed spatiotemporal analysis of rainfall erosivity. It is lightweight, robust and can be monitored continuously and automatically through PCs. The sensor occupies only a nominal space and, therefore, can be
used in small erosion plots with simulated or natural rainfall to develop and verify models of soil detachment
rate to kinetic energy of rainfall in interrill erosion processes.
ACKNOWLEDGEMENTS
The authors gratefully acknowledge the assistance of the laboratory sta€ of the Civil Engineering Department of The University of Hong Kong, during the design, fabrication, experimental set-up, troubleshooting
and data collection for this study.
Copyright # 2000 John Wiley & Sons, Ltd.
Hydrol. Process., Vol. 14, 37±49 (2000)
RAINFALL DROP SIZE AND KINETIC ENERGY
49
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