< ~ H
von
KARMAN INSTITUTE
FOR PLUID DYNAMICS
TECHNICAL MEMORANDUM
AN AERODYNAMIC PENDULUM OF STANDARD DESIGN AND SIMPLE CONSTRUCTION FOR VERIFICATION OF ANEMOMETER CALIBRATIONS AT LOW VELOCITIES
by
P. L. CLEMENS
RHODE-SAINT-GENESE, BELGIUM
•
von KARMAN INSTITUTE FOR FLUID DYNAMICS
TECHNICAL MEMORANDUM 22
AN AERODYNAMIC PENDULUM OF STANDARD DESIGN
AND SIMPLE CONSTRUCTION FOR VERIFICATION
OF ANEMOMETER CALIBRATIONS AT LOW VELOCITIES
by
P. L. CLEMENS·
MAY 1971
Visiting Professor, von Karman Institute for F1uid Dynamics, and Assistant Manager, Aerospace Instrumentation Branch,
von Karman Gas Dynamics Faci1ity, ARO, Inc., Arno1d Engineering Development Center.
CONTENTS
ACKNOWLEDGEMENTS • • • • • • • . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i ABSTRACT • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ~ ~ INTRODUCTION -- A Need tor an Anemometer Calibration
Standard . . . • . . . 1 PRINCIPLE OF THE PENDULUM ANEMOMETER .•••.•••••.•••••.• 3 DESIGN OF A STANDARD PENDULUM •...•.••.•••••.••••••.•
6
WIND-TUNNEL CALIBRATIONS ANp RESULTS ••••••..•••••••••. 8 SOURCES OF ERROR •...•••••••••.•..•.•••••••.•••.•••••. 13 NOTES OF ADVICE ON USE • • . • . • . . . • • . . • . . . . • • • . • • • • . . 17 CONCLUSIONS •••••••••••••••••••••••••••.•••••••••••••• '8 REFERENCES • • • • • • . • . • • • • • • • • • • • • • • • . • • • • • • • • • • • • • • • . • • 20 APPENDIX -- The Beaufort Anemometric Scale . . . • • • . . . 21TABLES
TABLE I -- ITTF Table-Tennis BalI Characteristics
TABLE 11 -- Pendulum Anemometers for Wind-tunnel Test. TABLE 111 - - General Calibration for Spherical
33
34
Pendulum Anemometer . . . • . . . • • • . • • . . . . • • • • TABLE IV -- Error SOurces and Magnitudes • . . • • . . . • . •
35 - 36 37 FIG.l-a FIG.1-b FIG.l-c FIG.1-d FIG.2 FIG.3 FIGURES
Tapered-Tube (Variable-Aperture) Anemometer 38 Commercial Pendulum Anemometer . . • . • . . . . • . . 39 Pendulum Anemometer (Internal Flow) •..•••.
40
Cup Anemometer . . . • 41 Pendulum Anemometer Equilibrium ForceDiagram . . . • . . . '. . . .. 42 BalI Cross Section Showing Cord
FIG.4 FIG.5
FIG.6
FIG.8
Table-Tennis Ball Anemometer ••.••••••.•.•. ~~ Anemometer Test in VKI Low Speed Wind'
Tunnel L-1 . . . 45 Wind-Tunnel Test Results from Pendulum
Anemometers . . . • . . . . 46 General Calibration for Spherical Pendulum An emomet er • . . • . • . . . • . . • . . . • • . . . • 471 Pendulum Anemometer Standard in Field Use • 48
- 1
-ACKNOWLEDGEMENTS
The author is indebted to Professor Melvin H. Snyder of Wichita State University, and Visiting Professor at the von Karman Institute, with whom valuable discussions on
low-speed anemometry were held in the course of this work. Thanks are also due to Mr. Carbonaro of the faculty of the von Karman Institute Low Speed Department who assisted 1n conducting the wind tunnel tests and who provided much useful guidance. The design of the anemometer standard could not have evolved as reported here without the generous assistance
of Dr. J. Rufford Harrison, Chairman of the Equipment Committee of the International Table Tennis Federation, who supplied
invaluable information concerning the variability of the
ABSTRACT
In a large variety of activities~ there is need for the accurate measurement of surface wind velocities. The calibrations of the anemometers with which these
measurements are made of ten fall subject to doubt, and means are rarely available to verify the calibrations in the
field. A pendulum anemometer has been designed which can be constructed easily and inexpensively to close performance tolerances from readily available components. Wind-tunnel tests have provided both calibration of the anemometer and demonstration that this calibration is transferable to other units of like construction. The device may be used either as a field calibration standard for other anemometers, or directly as an anemometer itself. Details of design and construct ion
are presented, and calibration data are provided in (1) equation, (2) tabular, and (3) graphical form. Sources of error are
explored analytically. Velocity errors of the order ot six percent are typical over the velocity range to
11.5 meters/sec. An Appendix traces the historical development of the Beaufort Scale used in anemometry.
INTRODUCTION - A Need for an Anemometer Calibration Standard
Ground level winds exert an important influence on the play of many outdoors sports. Examples are found in
the classical track and field sports : the discus, the javelin, running and jumping. More modern examples are found in the
piloting of sailplanes and gliders and in parachuting. In these latter cases, the effects of ground winds must be reckoned
with not only because of their influence upon performance, but also out of consideration for the safety of those who participate. In such sports as yacht racing, even very feeble surface winds are of quite understandable concern. So sensitive is sailboat performance to the effects of small changes in
wind velocity that an elaborate statistically-based handicapping system has been developed (Refs.1 through 3) in which handicap coefficients assigned to the craft of anyone design are
adjusted as functions of wind velocity in order to enable the racing of yachts of mixed class designs on an equitable basis over a range of wind velocities.
Various methods of gag1ng the velocity of surface winds by subjective observation have arisen in attempts to
safety the need for accurate anemometry. Among these subjective methods, the one constructed around the Beaufort Scale is
unquestionably the best known and most widly employed. The Appendix seeks to trace the origins, the evolution, and the shortcomings of the Beaufort Scale. As the Appendix indicates, serious errors may arise in the gaging of wind velocity
subjectively by such methods as the observation of wind effects upon the form assumed by waves on a body of water. Objective measurement is much to be preferred, and yachtsmen, athletic coaches and others find themselves considering the anemometers which are commercially available. These operate on a variety of principles. Figure 1 presents a sampling of common types recently calibrated at the von Karman Institute for Fluid Dynamics.
A~ter some period of use, there inevitably arises the question whether the commercial anemometer has retained the accuracy claimed ~or it by its manu~acturer. One
questions whether rough handling or exposure to ~oul weather and salt spray might have altered the instrument's calibration. In the case o~ the very inexpensive instrument, one may be
properly justi~ied in questioning its accuracy even be~ore
exposing it to normal wear and tear. Veri~ication o~ calibration tends toward two extremes : the simple though inaccurate and the accurate but expensive. Wind tunnel calibrations generally ~all in the latter category o~ great expense (Ref.4). In the former, one ~inds such attempts at anemometer calibration as are realized by supporting the instrument outside a moving automobile, a practice certainly to be discouraged if only because of the increased risk of accident with a car whose driver may be distracted by the peculiar demands of the operation. These hazardous automotive calibrations of
anemometers are resorted to with such frequency and such lack of success that it is worth mentioning their additional
disadvantages. Rarely are automobile speedometers reliably
accurate, and this is especially true at the low veloeities which are of chie~ interest in anemometry for sports. Furthermore,
the anemometer which is being calibrated is itself immersed in a field of aerodynamic flow which is greatly disturbed by the transporting vehicle, and the local air velocity which the
anemometer experiences may be quite different from the gross velocity of the vehicle.
Thus there is a need for a method whereby the
calibrations of anemometers can be verified, with reasonable accuracy, with relative ease, at low cost, and in safety. A readily constructed anemometry field standard, to be described here, satisfies this need.
- 3
-PRINCIPLE OF THE PENDULUM ANEMOMETER
•
The pendulum anemometer employs, as a pendulum bob, an object whose geometrie form and surface area give it
resistance to the flow of air around i t . This object is
hinged or suspended from an arm or a cord allowing it freedom to swing in the plane containing the wind velocity vector. Provision is made to observe the angle through which it is
deflected when acted upon by the aerodynamic forces of drag and lift which result from the flow. If friction in the suspension system is neglected, then the hinge reaction (or cord tension), the weight of the pendulum, and the aerodynamic forces arrive at equilibrium as indicated in Fig.2, from which :
D
cot a
=
(w - L) (Eq. 1 )If the aerodynamic coefficients of drag (CD) and l i f t (CL) for the pendulum bob are known, then
D
=
q C AlD (Eq.2)
and
where Al and A2 are the effective surface areas acted upon by the dynamic pressure, q. Thus pendulum deflection becomes a measure of the dynamic pressure developed by air of given density, p, moving at a given velocity, u, according to the relationship
•
q = 1 pu2
2 (Eq.4)
Pendulum anemometers are not at all new. Pannell, for
example, notes in Ref.5 that a revised edition of Dr. Rooke's "nirections to Seamen" published in 1667 contained a
des-cription of a pendulum anemometer which consisted of a small flat plate hanging from an arm.
From Eqs.1 through 3, the pendulum suspension angle a resulting for a given value of dynamic pressure can be expressed as
a = co t - l
The aerodynamic force coefficients C
n
and CL will vary with angular deflection for many pendulum configurations. However, if the pendulurn bob is a sphere, then CL will vanish and the Cn
term will no longer be a function of angulardeflection. Thus from Eq.5, for the spherical pendulum bob
*
(Eq.6)
Here the term CnAl/W is the reciprocal of the familiar ballistic coefficient, representing the ratio of aerodynamic forces to inertial forces. It is apparent that the angular deflection of the pendulum in response to a given dynamic pressure is determined by this term. Thus the ballistic coefficient establishes the sensitivity of the spherical pendulum anemometer.
The t erms Wand Al are readily measured, and the coefficient C
n
has been established for spheres by manyinvestigators over a broad range of velocities (Ref.6). So it is possible to compute the ballistic coefficient for a given sphere and, by using Eq.6, to arrive at the values of pendulum deflection angle which might be expected over a given range of dynamic pressures. Then with dynamic pressure related to velocity (Eq.4), the spherical pendulum device would be expected to serve as an anemometer calibration standard. However, the values of C
n
determined by previousinvestigators as reported extensively in Ref.6 apply generally to unrestrained spheres. The pendulum anemometer, on the other
•
In the work related here, the area term A,for the spherical pendulum anemometer is without exception the projected'!rd 2
5
-hand, makes use of what might be termed a tethered sphere. Aerodynamic interference introduced by the tethering
arrangement (i.e., the attached suspension cord) might be
expected to affect the behaviour of the sphere. The stationary support mayalso exert an influence, especially at low values of a where the sphere may enter the wake of the tethering
support. It is preferabIe, then, tomeasure the drag coefficient for a spherical pendulum anemometer of a particular design
in the laboratory and to demonstrate either that i t is constant or that it varies 1n a predictabIe way over the velocity
range of interest.
Were such a spherical pendulum device to be of
simple design, and w~re i t demonstrabIe that its construction could be readily and faithfully reproduced, then its laboratory calibration might be considered transferable to other, identi-cally constructed units, and it might serve not only as an
anemometer itself but perhaps even more usefully as a convenient sort of field standard for use in verifying the calibrations of more elegant devices. Such is the purpose of the design described in the Section which follows.
Various sources of error will merit attent ion in considering the construction and in evaluating the performance of the practical spherical pendulum anemometer : the effect of variations in the diameter and length of the suspension cord are examples. A source of "error" of another kind is apparent from an examination of Eqs.
4
and6.
The deflection angle of a pendulum anemometer is determined not alone by velocity; it is a function of aerodynamic density as weIl. While this may be counted among the disadvantages of the pendulum anemometer, i t is one for which compensation can easily be made by measurement of atmospheric pressure and temperature and the computation of a corresponding correct ion for any change in density.required to produce a given pendulum deflection can be written :
Introducing compensation for variations in the density of air as a result of changes in atmospheric pressure and temperature
o s T amb P atm T amb P atm
density of air -- standard conditions absolute ambient air temperature
atmospheric pressure
These forms are convenient for later use.
DESIGN OF A STANDARD PENDULUM
(Eq.8)
As has been demonstrated,the sensitivity of the spherical pendulum anemometer is a function of the ballistic coefficient of the bob. Therefore the first consideration in the design of such an anemometer is the select ion of a sphere capable of giving the device reasonable deflection angles over the velocity range of its intended use. It is also desirable that it be a sphere whose diameter and weight can be accurately reproduced fr om unit to unit, if the calibration
is to be transferable.
In the work described here, it was found that table-tennis balls satisfy these requirements quite well. Their manufacturersare obliged to impose close control on ball
diameter and weight in order to satisfy standards established by the International Table Tennis Federation. Abstracted
from the Laws of Table Tennis (Refs.7 and 8), Table I
7
-specified by the ITTF.
In the selection of the suspension cord with which to support the pendulum bob, attention must be given to those properties of the cord which might influence the performance of the anemometer. It is desirable that the cord be of
negligible weight and that its diameter be as small as possible. In the initial work reported here, use was made of monofilament Nylon cord having a diameter of 0.08 mm o
and weighing 0.011 gm per meter*. Subsequently, monofilament cord of much greater diameter (0.2 mm) has been used with
no noticeable difference in the performance of the anemometer.
Sturdy attachment of the cord to the ball is necessary lest the ball be swept away by strong winds or torn free by rough handling. The attachment must add a minimum of weight and introduce a minimum of aerodynamic
interference. The method sketched in Fig.3 has been used with success in the work reported here. The balI is pierced from the outside at two diametrically opposite points by forcing a sewing needIe through its surface, producing slight
indentations or "dimples" at the points of perforation, as shown. The cord, threaded through the needle eye, is then passed through the ball. A bit of plastic mOdeling cement anchors the free end of the cord. The surplus of this free cord beyond the glue joint is trimmed away, as close to the balI surface as possible, af ter the cement has hardened.
(Care must be taken to select a cement which will not chemically attack the balI material). Weighings made
beflore and af ter gluing have verified that the weight added
•
This cord is available in short, inexpensive lengths from sporting goods shops which sell it for use in tying fishing flies. Breaking strength is customarily about 0.4 Kg.Monofilament Nylon thread in suitable diameters is also available from many sewing shops.
by this method, performed with reasonable care, is negligible (less than 0.001 gm). Platic adhesive tape is useful for
retaining the cord in position during the gluing.
An ordinary plastic protractor (Fig.3) serves well as ascale against which to make readings of deflection
angle, which is defined here as the angle separating the suspension cord from the horizontal. Used in this way, the term may seem a misnomer. It is a convenient definition, however,in consideration of the usual scale markings found on commercially made protractors. As shown in Fig.4, the suspension cord may be passed through the index hole at the center of the protractor, and an ordinary spirit level may be cemented near the base line of the protractor to ensure that the base line can be held level during use. The
assembly may then be attached to a short baton to facilitate its being held by hand at a suitable distance from the user so as to avoid the flow disturbances near the body. A baton length of 40 cm has been found quite suitable,and a suspension cord length near 30 cm has proved near optimum. (See Notes of Advice on Use).
WIND-TUNNEL CALIBRATIONS AND RESULTS
A number of pendulum anemometers constructed to the general design described in the preceding Section have been tested in the open test section of the
low-speed wind tunnel L-1 of the von Karman Institute for Fluid Dynamics. A description of this tunnel is found in Ref.9, and its aerodynamic characteristics are presented in Ref.10.
The results of a particular set of tests, to be discussed, were obtained using a battery of four rake-mounted anemometers, as seen in Fig.5, and a single hand-held unit of the kind picturedin Fig.4. Weights and diameters of the
- 9
-five balls used were weIl within ITTF-specified tolerances. Table II presents the pertinent characteristics of each of the anemometers.
Dynamic pressure measurements were made during the tests using a probe, permanently instalIed in the wind tunnel. A manometer manufactured by E.SCHILTKNECHT, Ing.
(Zurich) and filled with distilled water was used to provide readout. An error tolerance of ± 0.1 mm applies to this
instrument, according to the manufacturer.
Deflection angles of the anemometers tested were read by an observer standing on a platform in the wind tunnel test section. Distortion of the flow field surrounding the anemometers as a result of the approach of the observer was seen to have negligible effect on their readings, provided that the observer's body did not move upstream of the unit being read and provided that his body remained three of its own diameters to one side. (With the hand-held unit, this is easily accomplished by use of the short baton described in the Section on Design of a Standard Pendulum).
Values of anemometer deflection angle as a function of dynamic pressure realized in these wind tunnel tests
appear plotted in Fig.6. Except as indicated by the vertical
errOr bars at the three pressure values of 7.0, 8.86 and 9.78 mm, scatter among the deflection data for the five units tested
did not exceed one degree. The markedly increased scatter
seen in the data at these three pressure values is attributable to difficulty which was realized in obtaining readings from rake-mounted anemometers of two particular suspension lengths
excur-•
• •
s~ons at these levels of dynam~c pressure . No attempt
was made to reduce the amplitude of these oscillations (e.g. by varying suspension length; see Section on Notes of Advice on Use). One of the oscillating balls was swept away at a
dynamic pressure just aboye
7.0
mm and the other at a pressure of ab out 10.0 mmo (This explains the apparent disappearance and recurrence of scatter with increasing pressure). Thus data beyond a dynamic pressure of7.0
mm represent the performance of four units rather than five, and the number drops to three units beyond a pressure of 10.0 mmo Data from the hand-held model are included throughout. In no casewas the deflection angle of the hand-held unit responsible for producing data at either the maximum or the minimum
extreme indicated by the error bars which describe data scatter.
Also plotted in Fig.6 is a curve of theoretical deflection as a function of dynamic pressure. Equation 6 was used in producing this curve. Median values of ITTF-approved balI veight and diameter (TabIe I) were taken, together with a coefficient of drag of
0.47
for untethered spheres (Ref.6), in arriving at the value of ballistic coefficient used in plotting this curve. From Eq.6, the corresponding equation isa
=
cot- 1 {0.222 q) { q : mm H20Again using Eq.6 and ITTF median values of balI weight and diamete~values of drag coefficient were computed for the average deflection angles which were realized at
each of the values of dynamic pressure for which the wind-tunnel
•
At these same dynamic pressures, the other anemometers could also be made to oscillate by adjusting their suspensionlengths to these same values. It is supposed that the oscillatory behaviour arises from pendular resonance with the frequency at which vortices are shed from the spheres.
- 11
-test data were taken. The f0110wing equation (from Eq.6) was used :
2. 12 t
=
coq (l ave.
The individua1 va1ues of drag coefficient computed with this equation for the average def1ection ang1es rea1ized at
each pressure level during testing are a1so p10tted in Fig.6.
The range of Reyno1ds Numbers (based on sphere diameter) encompassed in the testing extended from about
103 to 3.5x104
• (Reyno1ds Number va1ues are indicated in Fig.6). Over this range of Reyno1ds Numbers, a re1ative1y constant drag coefficient wou1d have been expected for smooth untethered spheres (Ref.6). Instead, a wide spread of values for the drag coefficient was found for ~ tethered anemometer spheres, and some explanation for this is necessary.
The three data points of Fig.6 indicating values of drag coefficient 1ess than 0.4 resu1ted from measurements of dynamic pressure at levels of 0.6 mm or lower. At these low pressure levels, a consistent smal 1 error in dynamic pressure measurement, but within the 1-mm error tolerance of the
manometer used, cou1d easi1y produce the deviations seen 1n drag coefficient. At the deflection ang1es of less than some 30 degrees, there is evidence that the anemometer sphere enters an area of flow which has been disturbed by the pro-tractor and anemometer support. This is thought to account for the abrupt apparent rise in drag coefficient seen for values of dynamic pressure above 9.0 mmo Support for this supposition comes from the behaviour of the anemometer having the greatest suspension 1ength (i.e., 60 cm). The def1ection ang1es which it produced correspond consistently to the data va1ues at the 10wer ends of the error bars (i.e., higher
angles). Because of the greater suspension length for this anemometer, its sphere wou1d be expended to enter the flow
disturbed by the protractor and its support later (i.e., at a lower deflection angle) than would those of the other units.
An average value of drag coefficient was computed from the wind-tunnel test data plotted in Fig.6. In computing this average value, the data below and above mea~ured values of dynamic pressure of 0.6 and 9.0 mm, respectively, were discounted on the arguments outlined above. The value of drag coefficient thus produced is 0.442. This is six percent lower than the accepted value of 0.47 found in Ref.6 for
untethered spheres. The tethering is thought to be responsible for the difference. It yould be expected that flow in the
boundary layer of the sphere would become locally turbulent upon passing around the tether. A lower value of CD is the expected result of this tripping of the boundary layer as is discussed in detail in Ref.6.
Using the value of drag coefficient of 0.442 thus determined, and taking the ITTF median values for balI weight
and diameter, a value of ballistic coefficient for the anemometer was computed. Fromthis ballistic coefficient and the value of aerodynamic density for standard conditions (760 mm Hg and
15°C), Eq.7 becomes the general calibration equation for this spherical pendulum anemometer
u = 8.76 ,I cot a (meters/sec) (Eq.7')
Similarly Eq.8, which provides compensation for variation in aerodynamic density caused by variations from the standard values of atmospheric temperature and pressure, becomes :
u
=
14.2~7~+T
cot a (meters/sec) {~
Deg.C mm Hg (Eq.8')
- 13
-The IBM 1130 computer of the von Karman Institute has been used to prepare Table 111 from Eq.7'. This is a
general calibration table, expressing wind velocity in various systems of units for one-degree increments in anemometer
deflection angle. The values of dynamic pressure which are
also tabulated were computed from Eq.6. They apply irrespective of temperature and pressure, of course.
The general velocity calihration data from Table 111 appear plotted in Fig.7. Equation 7', the plot of Fig.7, and the data of Table 111 will be useful in interpreting
deflection readings given by anemometers constructed to the design described here. If velocity readings which are
compensated for variations in atmospheric pressure and temperature are desired, then Eq.B' will be of use.
SOURCE13 OF ERROR
Various factors which might contribute to errors in the measurement of wind velocity with the anemometer
described here are summarized in Table IV. Changes in length and diameter of the suspension cord between the values shown have had no discernible effect on deflection readings in the test work, so long. as the oscillatory phenomenon thought to be attributable to pendular resonance with vortex shedding
is avoided. Greater ranges of cord length and diameter than those shown have not been examined. Thus it is probable that those shown do not constitute limits, and that still
greater variations are permissible. Also indicated in Table IV is the limit for observer-body approach to the anemometer, as determined experimentally.
Use of a large protractor is desireable for accuracy in reading the scale divisions. Increasing the protractor radius for a given length of the suspension cord increases the deflection angle at which the ball enters flow
disturbed by the protractor, thereby faulting accuracy at higher velocities. Thus protractor size (hence facility of reading) must be compromised against the maximum velocity to which an anemometer of given cord length can be used. In the work reported here, it was found that a velocity
near 11.5 meters/sec., producing a deflection angle of about 30 degrees (measured from the horizontal), is a maximum
for accurate work when using a suspension length of 30 cm and a protractor having a 6 cm radius. Lengthening the suspens~on cord increases the velocity range for a given protractor radius.
The error in velocity measurement resulting from an error in reading the anemometer deflection angle is
variable with the angle itself. Velocity errors attributable to one-degree reading errors at several deflection values have been computed from the calibration data of Table l I l . and are shown in Table IV. Experience with the anemometer has indicated that,with reasonable care, reading errors exceeding one degree are rather easily avoided when using a protractor of 6-cm radius or larger. One-degree scatter has been typical in the taking of data such as those plotted in Fig.6. and greater scatter has been rare except in the oscillatory cases already discussed.
The influence of the remaining factors listed in Table IV on the accuracy of the anemometer is best examined analytically.
Equation 8 may be written as
T - (K amb U - I P atm W 1/2 C A cot a)" D 1 (Eq.8")
Consider, first. the influence of errors in sphere weight. Let
Ó
w
=
numerical deviation of sphere weight, W, from- 15
-6
=
numerical deviation in velocity, correspondingu
to a deviation in sphere weight by an amount Ö
w
Then, 6
I
'
u
expresses the fractional deviation in velocityu
corresponding to a fractional deviation of Ö
Iw
in spherew
weight. The numerical deviation in u corresponding to a given numerical change in W may now be expressed as :
6
u
d'
- u Ö
- dW 'w for Öw ~ 0
Treating the other parameters as a lumped constant (K2)
so as to evaluate only the intluence of variations in W for given values of angular deflection, the original 1unction
(Eq.8") may be differencti~ted with respect to W
(Eq.10)
Substituting Eq.10 into Eq.9
(Eq.11)
Since Eq.8" has been taken as exact, Eq. 11 may be normalized.
by dividing by Eq.8"
Hence,
~
_ 1 ö\!u - 2 W (Eq.12)
Equation 12, then, gives the fractional uncertainty in velocity resulting from a given fractional uncertainty in sphere weight.
This derivation holds only for small perturbations, inasmuch as it has been made implicit in the statement of
Eq. 9 that 6. must approach zero. Similar derivations for
v
the influence the other parameters give, for temperature
for for 11 1 6T -!! = u 2 T pressure 11 1 <Sp -!! = u 2 p drag coefficient 11 u
and for sphere diameter (d
=~)
~.riationsof appropriate magnitude in each of these potential contributors to velocity error have been assumed. The variations, and the velocity errors each would produce, appear in Table IV. BalI weight and diameter variations assumed are those allowed by the ITTF specifications.
Variations assumed in atmospheric temperature and pressure
are those recorded ~n the Rhode-Saint-Genèse area over the past
several years. The drag coefficient variations which have been assumed are those which were experienced in the wind tunnel
testingamong d~ used in computing the average value of
Cn.
Also shown in Table IV are values for the probable total error which might be anticipated both with and without
- 17
-In arriving at these estimates of overall error, it has been assumed that the several components of error will not simply sum; instead, the square root of the sum of their squares has been taken as characteristic of the most
probable error. Overall velocity measurement errors of thè order of six percent are to be expected over the velocity
range from about
3.5
to 11.5 meters/sec., based on the use ofEq.8' which compensates tor variations in temperature and
pressure. Errors in velocity measurement of some seven percent would appear realistic over this same range of veloeities
for readings taken directly from Table 111 or arrived at
by using the general calibration equation (Eq.7') and
presuming the very large excursions in pressure and temperature indicated in Table IV.
NOTES OF ADVICE ON USE
In use, the pendulum anemometer constructed as
shown in Fig.4 ~s best held with the baton gripped in the
right hand and extended more-or-Iess at arm's length. The observer may then stand with his right shoulder toward
the wind (Fig.8) and, taking care to maintain the instrument level, he may observe the deflection readings directly from
the protractor scale. Should difficulty arise ~n viewing
the fine suspension cord, it may be found useful to cement a small bit of cotton thread to it at the point where it intersects the protractor arc.
As with any anemometer, when using this instrument the observer should take up a position in an open area, free of obstructions to windward. Velocity gradients and turbulent eddies can be expected in the flow which surrounds and
continues downwind of such obstacles as buildings, trees,
shrubber~ fences, and the observer himself. As an example, the pendulum anemometer shown in use in Fig.8 is held too near to the gunwale of the boat for accurate measurement. Wetting of the balI by rain or spray will increase its weight and fault accuracy.
The pendulum responds quickly to gusts and lulls, and so the reading may change quickly. With a bit of patient practice under such conditions, it will be found that
accurate average readings are readily arrived at by noting the maximum and minimum extremes of the angular excursions made by the suspension cord. Even in winds of relatively constant velocity, it may sometimes be noted that the pendulum will oscillate through a small angle ·.bout ita average reading. If this effect becomes troublesome, it may be relieved by altering the suspension cord length to vary the pendular frequency. (An exhaustive and interesting treatment of the stability of such an aerodynamic system will be found in Ref."). Surplus suspension cord can be wound conveniently around the baton, ready for use should a change in length be desired.
Finally, it may be found useful to overlay the
protractor with a calibration scale constructed from Table 111 so as to enable velocity readings to be made directly.
CONCLUSIONS
The pendulum anemometer the design of which has been described ~s easily and inexpensively constructed from readily available components. The wind tunnel tests which have provided its calibration have also demonstrated that this
calibration is transferable to other units of like construction.
In using the calibration data which are presented here in tabular (Table 111), equation (Eqs.7'and 8'), and graphical (Fig.7) form, the particular conditions which apply to their development must be taken into account :
1. The calibration data apply, with their greatest accuracy, to a pendulum anemometer which has been constructed using a table-tennis ball whose
19
-weight and diameter equal the median values specified by the ITTF.
2. The calibration data apply under standard atmospheric conditions i.e.,
15°C(59°F)
and
760
mm Hg. Although accuracy depreciates with departure from these conditions, it can ~e restored by use of Eq.8'.3. Although the general calibration data have
been tabulated and plotted to deflection angles as small as 20 degrees, their accuracy at angles smaller than 30 degrees should be considered suspect because of the entry of the sphere at lower angles into the reg10n of flow which has been disturbed by the protractor and its support.
Over the velocity range extending to
11.5
meters/s., probable errors are about seven percent for the anemometer used without compensation for variations in atmospheric pressure and temperature. If such co~pensation is made, the probable error diminishes to about six percent.While this instrument may be used directly as an anemometer itself, it is considered that its chief utility is as a field standard for the verification of calibrations of other anemometers.
1. "At the Masthead" (Editorial) , One Desi~n and Offshore Yachtsman, Vol.7, No.1, p.4, January 19 8.
2. Richards, Nolan,"Adapting the Portsmouth System", Ibid, p.41.
3. "At the Masthead" (Editorial), One Design and Offshore Yachtsman, Vol.9, No.4, p.4, April 1970.
4. Pike, Julian M., "A Calibration Facility for Meteorological Instruments", Instrument Society of America Paper
No. 718-70, October 1970.
5. Pannel, J.R., "Fluid Velocity and Pressure" (pub. :Arnold), p.45, 1924.
6. Hoerner, S.F., "Fluid-Dynamic Drag" (pub. : Author), 1958.
7. Harrison, J. Rufford, Chairman : Equipment Committee of the International Table Tennis Federation (Private
Communication)
8. "Official Rules of Sports and Games -- 1966-67" (pub.: Corgi), p.401.
9.
"Installations Experimentales de l'I.V.K.", VKI TM 11, Edition 2, 1967.10. Colin, P.E. and N. Collard, "Calibrage de la Soufflerie à Faible Vitesse
a
Veine Libre de 3 m. de Diam~tre",Ministère des Communications.Administration de l'Aeronauti-que, Rapport No. L.A.06-52, Decembre 1952.
11. Delaurier, J.D.,"A First Order Theory for Predicting the Stability of Cable Towed and Tethered Bodies where the Cable has a General Curvature and Tension VariabIe", VKI TN 68, December 1970.
- 21
-APPENDIX -- The Beaufort Anemometric Scale
The Beaufort Scale is not without predecessors. In 1735, Jan Noppen began a systematic recording of weather and wind effects in the community of HALFWEG, midway between AMSTERDAM and HAARLEM, where he was a superintendent of dikes and locks. Havinga (Ref.A-1) provides an excellent study of Noppen's empirical development of a well-defined wind intensity scale of 17 steps, each of which he associated with particular subjective observations of the noises of the wind and its
effects upon trees, reeds, and the surface of a nearby lake which was being drained.
During the years 177~ and 1773, a crude anemometer
was instalIed on the roof of the building occupied by Noppen, and he supplemented nis meticulous logging of subjective
observations with notations of the readings given by the anemometer. In his own work nearly two centuries later, Havinga (Ref.A-1) reconstructed the Noppen anemometer and fitted it to a model of the building on which the original had been mounted. Using Noppen's records, he was then able to assign modern units of absolute velocity to the steps of the scale which Noppen had developed. Havinga's work thus has
made it possible to co~pare Noppen's early scale with other
subjective ones developed more recently, including that of Beaufort, and such comparisons have been drawn by Verploeg
(Ref.A-2) .
It 15 interesting that af ter some years of its use,
the 17-step Noppen Scale came to be regarded as too demanding of precision in the making of the subjective
observations. Accordingly, the odd-numbered steps feIl into disuse, and the even-numbered steps continued to be employed long af ter Noppen's death.
Seeking to satisfy a serious need for a standardized system of noting and reporting the strength of surface winds at sea, Francis Beaufort (1774-1857; eventually Sir Francis) drew up the first scale which came to bear his name. The references are generally in accord on the fact that Beaufort entered the British Royal Navy in 1787 at the age of 13,
but beyond that there arises much disagreement as to the year in which Beaufort first constructed his scale and as to
year in which Beaufort first constructed his scale and as to the naval rank he held at the time : Reference A-3 notes that he was alieutenant, Ref.A-4 that he was a commande~
and others (e.g., Refs.A-S, A-6, A-7) assert either that he was or later became either an admiral or rear admiral.
Reference A-3 places the date of his achievement at 180S, as does Ref.A-8. Many others (e.g., Refs.A-S, A-6, A-7) fix 1806 as the date, and Ref.A-4 under its entry headed Wind marks the date as 1806, while indicating under a separate entry headed Beaufort that the year was 1805.
Whatever the year, there is no doubt that the original Beaufort Scale used a series of numbers from zero through twelve to identify winds of increasing strength. As had been the case with the Noppen Scale, the Beaufort
scale was based on subjective observations and made no mention of wind velocities. Beaufort identified each number of his scale with a description of the effects of winds upon
"a well-conditioned man-of-war" (Ref.A-4). Thus Force ~ of the original Beaufort Scale became a "strong breeze in which the
.•• ship could just carry, close hauled, single reefs and top-gallant sails" (Ref.A-4), and Force 12 was ominously described as "that which no canvas could withstand"(Ref.A-6).
Beaufort's Scale was officially recognized for use on the open sea by the British Admiralty in either 1834
(Ref.A-3) or 1838 (Ref.A-6), and in 1874 it was adopted by the International Meteorological Committee at the First Conference
- 23
-of Maritime Meteorology (London) for use ln international weather messages (Hefs A-3 and A-6).
The numbers of the Beaufort Scale had meanwhile come to be identified not with descriptions of wind effects upon sailing vessels but with equally subjective descriptions of the appearance of the sea surface. These "sea criteria", as they are of ten called, may be found with some variations ln many modern references, and their use continues to be widespread. Reference A-c, for example, carries a listing of sea criteria in one of the several forms of ten seen today. Two of these are quoted in part here, to exemplify :
Beaufort Number 4 6 Sea Criterion
Small vaves, becoming longer : fairly frequent white horses
Large ~ves begin to form; .•. white foam crests are ... extensive everywhere.(Probably some spray) .
Probable Wave Height - Feet
Similar tables have been constructed relating the numbers of the Beaufort Scale to phenomena subjectively observable on land: the rustling of leaves, behaviour of smoke from chimneys, uprooting of trees, etc. As examples, two of these are quoted here as they appear in Ref. A-10:
Beaufort Number
4
6
Description
Raises dust and loose paper; small branches are moved.
Large branches in motion; telegraph wires whistle; umbrellas used with difficulty.
During the early part of the 20th century, with the Beaufort Scale in one or another of its subjective
forms being used by almost all of the sea-going world, attempts were intensified to standardize a correlation of its force
numbers with objective anemometric measurements of absolute ~ind velocity. In a meeting of the International Meteorological Committee held in Berlin during 1939 (Ref.A-6), an International Scale of correspondence between ranges of velocity and Beaufort Force Numbers was adopted by many nations. Among those nations which did not accept this International Scale were the
United States and Great Britain, both electing instead to continue the use of a somewhat earlier and somewhat different scale. Both of these scales of Beaufort Numbers appear 1n Table A-1 as they are found 1n several references, and both
continue in use. The two sc~les differ not only in the ranges
of velocity assigned to particular Beaufort Numbers, but also as to the elevation at which velocity measurements are to be made. (See notes, Table A-1.) The specification of the elevation at which the measurements are to be made is a matter of no
small importance. Substantial vertical velocity gradients are found in the shear layers of surface winds.
Confusion and ~isunderstanding abound in the
attempts by reputable popular references to present the
velocity equivalence of the force numbers of the Beaufort
Scale, the elevation at which anemometric measurements are to be made, and the descriptions of subjectively observable criteria. The fact of there being two scales (TabIe A-1) both having official sanction and vide recognition and each having
• •
•
evolved from several s1m1lar predecessors must surely
contribute to that confusion. The references rarely signal
whether it is the International or the U.S.-U.K. scale they
present (e.g., Refs. A-4, A-5, A-i, A-i, A-9, A-10, A-13, A-14).
*
Reference A-12 notes that some 25 separate and differingBeaufort Scales have emerged, each carrying its own schedule of velocity equivalence.
- 25
-Although the
u.s.
Weather Bureau added Beau~ort Force Numbers 13 through 17 to the scale in 1955, the references rarely note numbers beyond 12. (Reference A-6 presents veloeities corresponding to these additional steps, and Ref.A-10 takes note of them.) Reference A-13, without comment as to which of the seales it is presenting, tabulates one which conforms neither to the International nor to the U.S.-U.K. versions listed by others.Curiously, of a large number ofpopular referenees consulted, only Ref.A-14 cites an equation for velocity (V) in terms of Beaufort Number
(B)
which produees approximate median values of the veloeity steps (beyond B=
0) making*
up the U.S.-U.K. seale :
v
=
1.87lJ31
(miles/hour) Ref.A-14 E quat10ns of the general form . V=
C B K . do roughly approx1mate many of the Beaufort Scales, old and new (Ref.A-12). When such an approximation is applied, i t is ~enerally found for. 68*~
most seales that the value of Y. l1es between 0.97 and 1. . Verploegh has shown (Re~.h-2) that while the assumption of
such an exponential law is sometimes a convenient simplifieation, i t is not necessarily a good one. Referenee A-12 notes,
however, that it was just sueh an equation (with K
=
3/2) whieh served as the basis for a vers ion of the Beaufort Scale whieh was adopted by the Meteorological Office of London in1906, and another (again with K
=
3/2) whieh 1S used to the present time in the Code 1100 Beaufort Scale of l'Organisation Météorologique Mondiale.IJ
Referenee A-14 fails to mention, however, which of the scales this equation is intended to represent.
Furthermore, there is a tendeney among many of the Beaufort Seales to differ least from one another in the region
A new schedule of equivalence between the Force Numbers (through 12) of the Beaufort Scale and absolute velocities has recently been proposed to l'Organisation .. Meteorologique Mondiale (Geneva) by the Commission de
Meteorologie Maritime. This new Beaufort Scale, known as New Proposed Code 1100, alters both the velocity correspondence and the subjectively observable sea criteria of previous scales. The Executive Committee of l'Organisation Meteorolo-gique Mondiale has now adopted the position (Ref.A-12) of
recommending the use of the schedule of velocity correspondence of the new scale, but has decided to devote further study to the descriptive terms making up the sea criteria. This new Beaufort Scale appears here aS Table A-II.
Using Havinga's work at reconstruction (Ref.A-1), i t is interesting to compare a portion of this new Beaufort Scale with corresponding even-numbered steps of the scale which Noppen first conceived in 1735. The correlation is altogether striking :
Noppen New Proposed
Scale Code 1100.
Force Velocity Velocity Beaufort
No. m/sec m/sec No.
2 3.8 2.8- 4.5 2 4 6.2 4.6- 6.6 3 6 8.3 6.7- 8.9 4 8 10. 3 9 • 0- 1 1 . 3 5 10 12.0 11.4-13.8 6 12 14.8 13.9-16.4 7
With all of the more common versions of the Beaufort Scale, including those of Tables A-I and A-II, there are dis-quieting aspects which diminish their appeal to the metrologist-meteorologist. Voids appear between the adjacent steps of the scales, and these voids introduce ambiguities in expressing
- 27
-velocities. For example, upon measuring a wind velocity of
3.5 miles per hour at whatever elevation might be appropriate, one can only then wonder whether to interpret this as a
Beaufort Number of 1 or of 2 in either the U.S.-U.K. or the International Scale of Table A-I.
The imprecision of modern adaptations of the Beaufort Scale is more profound than would be indicated
simply by these ambiguous velocity voids, by the annoyance of disagreement between competing scales, and by the vast
confusion extant among the references. In none of the current versions of the scale (Tables A-I and A-II) do Beaufort
Numbers progress either arithmetically with velocity or with its square. Thus these scales of Beaufort Number are
flawed in that they convey neither a linear measure of velocity itself nor do they relate nirectly to the aerodynamic forces developed. The magnitude of the individual steps of the scales mayalso be disputed on aerodynamic gro~nds : For each of the
first six steps of the scale, the dynamic pressure (hence the aerodynamic forces) represented by the upper velocity limit exceeds that at the lower limit by a factor near or exceeding two. These aspects reduce the worth of the scales to many potential users. Consider, for example, the skipper of the small sailing dinghy who finds that the force exerted by the wind on the sails of his craft does not increase in an orderly manner with increases in Beaufort Number, and that these
forces may vary by a factor of two or more within the range of a single Beaufort Force Number.
Detracting s t i l l further from trust in the use of the Beaufort Scale is the still-persistent practice of gaging the scale by totally subjective observations such as, for example, the form assumed by waves on a body of water. Large errors are inevitable. The form and scale of waves
(i.e., wave length, crest-to-trough height, etc.) have been shown to be related in quite profound ways to other factors than wind velocity alone. (References A-15 and A-16 treat this
subject in great depth.) The more important of these other
paramet~rs a r e : exposure duration, water depth, and fetch
(i.e., unobstructed distance to windward). So it is,then,
that waves of relatively similar form observed on two different bodies of water my each have been produced there by quite
different wind velocities. Moreover, similar waves seen at different times at a given point on a single body of water may also result from differing wind velocities, if the prior
histories of their development differ (the duration factor) or if wind directions differ. (Fetch may vary as directional changes bring differing obstructions into prominence.)
Particularly are these factors evident on protected inland bodies of water (Ref.A-16) where there is great justification
for disdain of subjective observation in favour of objective measurement.
It is indeed lamentable that the Beaufort Scale,
born of such nObility of purpose, h~s been allowed by modern
technology to arrive at so chaotic a turn.The mistreatment which the scale has suffered has done disservice to the man whose name continues in association with it.
- 29 -APPENDIX REFERENCES A-1 A-2 A-3 A-4 A-5 A-6 A-7 A-8 A-9 A-10 A- 11 A-12 A-13
Havinga, A., "\-lindwaarnemingen in Holland in de 18 eeuw"; Verh. Bataafsch Genootschap Proefond; Wijsbegeerte, Rotterdam; 1948
Verploegh, G., KNMI 102, Meded.en Verh.89; 1967
"Sir Francis Beaufort Biography"; J.Inst.Nav.XI,No.3
Entries :"Wind" and "Beaufort", Encyclopedia International (pub. Grolier, Limitée), 1965
Entry "Beaufort (échelle de)", Grand Larousse Encyclopédique (pub. : Librairie Larousse), 1960 Entry
(pub. Entry (pub. Entry
"Beaufort Scale", Encyclopedia Britannica William Benton), 1964
"l'Air en Mouve~ent", La Science pour Tous Grolier, Limitée), 1964
"Beaufort Scale", Webster's New \-Torld
Dictionary of the American Language -- Encyclopedic Edition (pub. : The VTorld Publishing Co.), 1951
Entry :"Beaufort Wind Scale", Reed's Nautical Almanac (pub. Thomas Reed Publications Limited), 1965
Entry "Beaufort Scale", Webster's Seventh New Collegiate Dictionary (pub. : G. and C. Merriam Company), 1963
Entry :"Beaufort Edition of Funlt.
(pub. Funk and
Scale", Britannica World Language and Wagnalls Standard Dictionary Wagnalls Company), 1964
Anon., "1'Echelle Beaufort de Force du Vent (Aspects techniques- et opérationnels)", Rapport No.3,
Organisation Météorologique Mondiale, 1970.
Entry : "Winds and Air Circulation", Cyclopedia of Aviation Terms (pub. McGraw-Hill Book Co., Inc.),
APPENDIX REFERENCES (eoneluded) A-14
A-15
A-16
Entry : "Beaufort Seale", Chambers's Teehnieal Dietionary (pub. : W. and R. Chambers, Ltd.), 1953 Briseoe, M.G., "The Surf"aee of the Oeean '- lts Dynamies and Deseription", VKI LS 19, January 1970 Kitaigorodski, S .A., "Applieations of the Theory of Similarity to the Analysis of Wind-Generated
Wave Motion as a Stoehastie proeess", Izvestia, Akad. Nauk SSSR, Ser. Geofiz., Vol.1, 1961.
- 31
-TABLE A-I -- Forms of the Beaufort Scale from Popular References
BEAUFORT NUMBER WIND VELOCITY International
*
ScaleMILES PER HOUR British and
**
U.S. Scale ~...
0 0- 0-2- 3 1- 3 2 4- 7 4- 7 3 8-11 8-12 4 12-16 13-18 5 17-21 19-24 6 22-27 25-31 7 28-33 32-38 8 34-40 39-46 9 41-48 47-54 10 49-56 55-63 1 1 57-65 64-75·· .. 12 a.bove 65 above 75**.-.
From Ref.A-6; anemometer at 6 meters elevation.
Adopted Berlin, 1939, by Interna.tional Meteorolo~ical
Committee.
From Ref.A-4; a.nemometer at 10 meters elevation. Sa.me
sca.le also a.ppea.rsin Refs.A-7, A-l1. Reference A-6 gives
same scale but notes elevation to be 11 meters.
' Beaufort No.11 corresponds to 64-72 miles/hour, Bccording
to Refs. A-6 a.nd A-10.
Beaufort No.12 corresponds to 73-82 miles/hour, a.ccording to Ref.A-6, which then continues scale for individual Beaufort Numbers through 17. Reference A-10 groups
Beaufort Numbers 12-17, corresponding to 73-136 miles/hour,
•
TABLE A-II -- New Proposed Code 1100 B~aufort ScaleWIND VELOCITY'"
BEAUFORT Average Range Range
NUMBER (meters/sec) (meters/sec) (Knots)
•
**
0 0.8 0- 1.3 0- 2 2.0 1. 4- 2.7 3- 5 2 3.6 2.8- 4.5 6-· 8 3 5.6 4.6- 6.6 9-12 4 7.9 6.7- 8.9 13-16 5 10.2 9.0-11.3 17-21 6 12.6 1 1 .4- 13.8 :!2-26 7 15. 1 13.9-16.4 27-31 8 17.8 16.5-19.2 32-37 9 20.8 19.3-22.4 38-43 10 24.2 22.5-26.0 44-50 1 1 28.0 21.1-30 51-5712 32.2 . (no upper limit applies)
As taken from Ref.A-12 and recommended by the Executive Committee of l'Organisation ~~t~orologiaue Mondiale
(Geneva) .
33
-TABLE I -- ITTF Table-Tennis Ball Characteristics
Diameter Weight
..
..
M~n~mum 37.2 mm (1.46 in) 2.40 gm (37 grains)
.
..
Max~mum 38.2 mm (1.50 in) 2.53 gm (39 grains) Median 37.7 mm (1.48 in) 2.465 gm (38 grains)
Percent Variability ± 1.3% ± 2.56%
..
Data from Refs. 7 and 8 Laws of Table TennisNOTE Only table-tennis balls which bear the approval of the International Table Tennis Federation can be reasonably assured of conforming to these limits. Many balls are marketed which are not ITTF-~pproved
and whose weights and diameters fall well outside these limits.
TABLE 11 - - Pendulum Anemometers for Wind-Tunnel Test
Rake Mounted (Fig.5) Unit Characteristic A B C D Suspension Length - cm 14 20 26 60 Suspension Cord Diameter - mm 0.08 0.08 0.08 0.08 Ball Weight
-
gm 2.458 2.474 2.474 2.476 Deviation from ITTFMedian Weight
-
gm -0.007 +0.009 +0.009 +0.011 Ball Diameter•
- mm 37.75 37.90 37.52 37.46 Deviation from ITTFMedian Diam.
-
mm +0.05 +0.20 -0. 18 -0.24 Protractor Radius-
cm 6 6 6 6 Hand Held"lJ E 30 0.2 2.478 +0.013 37.60 -0.10 7.5•
Mean values are given; typical diameter variability for a single ball is 0.1 mmo- 35
-TABLE 111 -- General Calibration for Spherical Pendulum Anemometer
STD. AIR AT 760. MM HG AND 15 DEG. C
C AVE. EXPER.- 0..442
D
ALPHA
U· U U U Q ...DEG
M./SEC.
KNOTS
ST.MI./HR.
KM./HR.
~~MH20
20.
14.5
28.2
32.5
52.3
13.2
121
14.1
27.5
31.6
50..9
12.5
22
13.8
26.8
.
30..8
49.6
.
11.9
23
13.5
26.1
30..1
48.4
11.3
.
24
13.1
25.5
29.4
.
47.3
10..8
.
25
12.8
24.9
28.7
46.2
10..3
.
26
12.5
24.4
28.1
45.2
.
9.84
27
12.3
23.9
.
27.5
44.2
9.4
.
2
28
12.
Q.23.4
26.9
43.3
9.0.3
29
11.8
22.9
26.3
42.4
8.66
30.
11.5
22.4
25.8
.
41.5
8.31
1
31
11.3
22.0.
25.3
40.1
7.9
.
9
32
11.1
21.6
24.8
39.9
7.68
33
10.9
21.1
24.3
39.1
7.3
.
9
34
10.7
20.7.
23.9
.
38.4
7.12
1
35
10..5
20..4
23.4
37.7
6.8
.
6
36
10.3
20.0
23.
o..37.0
6.6
.
1
137
10.1
19.6
.
22.6
36.3
.
6.37
38
9.9.1
19.3
22.2
35.7
6.14
39
9. 7
.
4
18.
g.21.8
35.1
5.93
40
9.5
.
7
18.6
21.4
34.4
5. 7
.
2
41
9.4
.
0.
18.3
21.0
33.8
5.52
42
9.24
18.0
20.7
33.2
5.3
.
3
43
9.0.8
17.6
20.3
32.7
5.15
44
8.92
17.3
19.9
.
32.1
4.97
45
8.7
.
6
17.
Q.19.6
.
31.5
4.80
46
8.61
16.7
19.3
.
31.0
4.64
47
8.46
16.5
.
18.9
30.5
.
4.48
48
8.32
16.2
18.6
29.9
.
4.3
.
2
49
8.17
15.9
18.3
29.4
4.17
50.
8.0.
.
3
15.6
18.0
28.9
4.0.3
51
7.89
15.3
17.6
28.4
3.89
52
7.75
15.1
17.3
27.9
3.75
53
7.6
.
1
14.8
17. O
.
27.4
3.6
.
2
54
7.47
14.5
16.7
26.9
3.49
55
7.33
14.3
16.4
.
26.4
3.3
.
6
(continued, next pap.:e)
