Electronic Spray Analyzer for Electrically Conducting Particles J A C O B M . GEIST', J . LOUIS YORK,
AND
GEO. G R A N G E R B R O W N
UNIVERSITY.OF MICHIGAN, ANN ARBOR. MICH.
*
*
OST instruments for determination of particle size distribution and
M
Analyses of sprays and other suspensions frequently involve sampling w i t h microscope, slides, cells, or other relatively large devices, followed by tedious counting procedures. T h i s paper presents some calculations t o emphasize t h e advantage of small samplers and describes preliminary work in t h e developm e n t of a n electronic analyzer which utilizes a small sampler t o measure and t o count t h e particles. Metal spheres, w i t h diameters f r o m 500 t o 6340 microns, and drops of water, alcohol, and acetone, w i t h diameters f r o m 2500 t o 4500 microns, provide data t o show t h a t t h e electrical pulses created upon interception of t h e particles by t h e probe wire are proportional t o t h e 1.6 power of t h e particle diameter. The effects of probe geometry and potential are shown, and t h e underlying mechanism is discussed. W i t h f u r t h e r development of t h e geometry of t h e probe, t h e electronic spray analyzer may offer an extremely rapid method for determining t h e drop size and size distribution in t h e spray of an aperating nozzle, w i t h a minimum sampling error.
number of particles in a suspension rely upon physical sampling and a study of the sample. Usually the sampling involves capture and withdrawal of the particles from suspension; this forces the sampler to be many times as large as the largest pmticle in suspension. Many investigators have shown that a sampling bias is inevitable with physical sampling, and that the bias may be very large for large samplers and small particles. This bias is reduced to manageable proportions if the size of the sampler - is reduced to the order of magnitude of the small particles, but such a sampler is incapable of capturing and removing the particles for leisurely study. This paper presents some calculations to emphaaize the advantages of the smaller size of sampler, and describes preliminary work performed in the hope of developing an analyzer which can utilise a small sampler to measure and to count the particles without withdrawing them permanently from the suspension. VARIATION OF SAMPLING BIAS
A bias in physical sampling results from deflection of flow lines around the sampler. If all suspended particles maintained their poaition in the flow lines during this deflection, none would be intercepted by a sampler which removed none of the fluid, If the particles all had sufficient inertia to leave their flow lines and to
continue in a straight path, the sampler would intercept all whose paths project into it. Actual operation lies between these extremes; larger particles are intercepted and smaller particles tend to follow the flow lines around the sampler. The tendency to follow the flow lines increases with decreasing size of particles. Flow lines which would pass through the center of the sampler, if undeflected, have a large proportion of all sizes of their suspended particles intercepted; those near the edges of the sampler carry all except the largest particles around the sampler. The effectiveness of sampling is defined as the ratio of the number of particles intercepted by the sampler to the number that would be intercepted if none were deflected around the sampler. For particles of uniform size and shape, this may be restated aa the ratio of the distance between streamlines within which all particles are intercepted t o the distance across the sampler. Calculations made by A l b r e c h t ( I ) , S e l l (8), Landahl and Herrmann (6), and Langmuir and Blodgett (6) agree that higher velocities, smaller samplers, a n d larger particle8 give higher v a l u e s of effectiveness. Although there ia disagreement in the quantitative results of their calculations, there is sufficient qualitative agreement to estimate some limitations of spray sampling devices. 1 Present address, Department of Chemical Engineering, Massachusetts Instituts of Technology, Cambridge Mass.
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INDUSTRIAL AND ENGINEERING CHEMISTRY
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Vol. 43, No. 6
1.0
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Figure 2.
DIAMETER OF WATER DROPS MICRONS Collection Effectiveness of 1-Inch Microscope Slide
DIAMETER
Figure 3.
OF WATER DROP- MICRONS
Collection Effectiveness of 16-Gage Wire 1295-micron diameter
Each of them calculated the trajectories of small particles in the field of flow around an infinite cylinder or ribbon, and defined effectiveness by the ratio of distances given above. Their results, recalculated to a common basis, are given in Figure 1. The ordinate is the calculated effectiveness of sampling. The abscissa is a dimensionless number, D2vpD/18p*D,, defined as the ratio of the maximum horizontal distance traveled by the particle, if injected horizontally into the nonflowing suspension medium with a velocity equal t o that of the suspension relative to the sampler, to the distance across the sampler. This abscissa, calculated from Stokes’ laFv, wavs common t o all investigators. The parameter is Reynolds number, defined as D v ~ A / P . Figure 1 shows that Sell agrees with Langmuir and Blodgett for ribbon-shaped samplers except at small values of the abscissa, corresponding to small particles. Albrecht and sell agree closely for cylindrical samplers except for small values of the abscissa, and are w-ithin 25% of data of Langmuir and Blodgett. Lan-
dah1 and Hermann agree with Langmuir and Blodgett (for Re = 10) within 20%. Based on the method of Langmuir and Blodgett ( 6 ) , the effectivenees of a 1-inch ribbon (microscope slide), a 16-gage wire, and a 24-gage wire were calculated for xater drops moving in air at 70” F. and 740 mm. (mercury) pressure. The results are presented in Figures 2, 3, and 4 . For a particle velocity of 32 feet per second and an effectiveness of a t least 90%, a 1-inch slide is an effective interceptor for particles larger than 62 microns, a 16-gage wire (1295 microns) is effective for particles larger than 20 microns, and a 24-gage wire (511 microns) is effective for particles larger than 11.5 microns. A t a velocity of 32 feet per second, a 1-inch slide will intercept no particles smaller than 7 microns, a 16-inch wire, no particles smaller than 1.5 microns, and a 24-gage wire, no particles smaller than 1 micron.
INDUSTRIAL A N D ENGINEERING CHEMISTRY
June 1951
1373
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$ 0.6 IA. 0
3w z w
.
0.4
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.
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2
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20
Figure 4.
200
60 80 100
40
DIAMETER OF WATER DROPS
- MICRONS
400
600 800 1000
Collection Effectiveness of 24-Gage Wire 511-micron diameter
These considerations indicate that data on size distribution obtained with large samplers are to be questioned for particle sizes smaller than 50 microns. Techniques using samplers as small as the wires of Figures 3 and 4 would seem more reliable because of the reduced sampling bias, but probably must rely on some property, related to the particle size, which can be measured and transmitted electrically. ELECTRONIC SPRAY ANALYZER
U
Guyton (4) reported on an analyzer based upon electrical pulses created upon interception of particles by a wire. The pulses were amplified, classified by size, and counted. His apparatus was intended for analysis of slow-moving clouds, a sample of the suspension being aspirated through a tube and emitted a t high velocity through a jet onto a wire. Apparently, the particles became charged electrostatically as they moved through the tube, the charge being proportional to the square of the particle diameter. The wire was grounded except for a few inconclusive determinations, and greatest success was obtained with nonconducting particles.
tricity results in negative electrical pulses a t the interceptor, the pulses varying with the particle size. These pulses are preamplified by a directly connected cathode follower to permit transmission by shielded cable to a high-gain amplifier. The amplified pulses are sent to a discriminator which passes only pulses greater than a set voltage. The pulses passed on are counted by a scaler. After calibration, the data obtained by varying the set voltage a t the discriminator and counting the pulses to the scaler provide cumulative size-distribution curves of the intercepted particles. The difference between successive counts is the number of intercepted particles in the size range corresponding to successive set voltages. The interceptor is a 4-inch piece of 18-gage copper wire, fixed directly to the grid of a Type 6AK5 pentode, which is connected as a cathode follower. The electronic tube and component parts are shielded completely within a grounded brass cylinder, 2 inches in diameter. The brass cylinder and 2 inches of the interceptor are enclosed within Plexiglas tubing to minimize surface-film conduction between the charged probe wire and the grounded shield. A battery of dry cells was used for a power supply to the probe and a 6-volt wet cell was used to supply the filaments of the tube in the probe. The gain of the cathode-follower stage was TO 600 VOLT B A T T E R Y
.
FOLLOWER
' AMPLIFIER
OISCRIMINATOR
TI piiF-1
V I
OSCILLOSCOPE
Figure 5.
TO AMPLIFIER
Block Diagram of Electronic Spray Analyzer
The analyzer described here consists of a charged wire, inserted into a moving suspension or moving through the suspension, connected to electronic circuits which amplify, classify, and count the electrical pulses created upon interception of the particles by the probe. Figure 5 is a block diagram showing the principal circuit components. The wire interceptor is maintained a t a high positive electric potential. Impingement of solid or liquid particles which conduct elec-
-Figure 6.
Wiring Diagram of Probe
Ail condensers in microfarads. All resistors in ohms. K = thousand. M = mllllon. 6-volt battery used for filament
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
1374 201
t
L
I
SYMBOL
;
8
A
D
0 D
NCHES MATERIAL FROM TIF METAL 0 METAL I14 METAL 112 ME TAL I METAL 1-1/2 WATER WATER WAER WATER
D 0
0
2
Vol. 43, No. 6
0 1/4 i/ 2
I
0.2
__
--
0.0 8 0.06
INCHES FROM TIP
0.0 4
0.0 2
-
IO
Figure 7.
w
0 114 ---+-///
c
I - 1/2-
20
40
60 80 100
200 433 DIAMETER- MiCRONS
I
IC
1
2000
I
I
I l l l l
4000
IC )OO
Relationship between Pulse Size and Particle Sire for Probe Potential of 430 Volts
measured and found to be approximately unity over the frequency range from 100 t o 1,000,000 cycles per second a t a grid potential of 430 volts. This potential is substantially that of the intercep tor and was used for most of the data presented here. Figure 6 shows the wiring diagram of the probe. The signal from the probe is fed to the pulse amplifier through a low-impedance attenuator which controls the amplitude of the input voltsge to the amplifier. The output of the first tube of sthe amplifier section is selected by means of a switch from either the plate or the cathode, so that only positive pulses are fed t o the discriminator. The last tube of the amplifier is a cathode follower which allows the amplifier to be physically separated from the discriminator without signal distortion in the connecting cable. The gain of the amplifier is variable in ten steps up to a maximum of 82.0, measured as the ratio of the output voltage to the input voltage a t frequencies from 100 to 1,000,000 cycles per second. The gain was found to be uniform within 10.3 decibel from 2000 to 800,000 cycles per second, and within *l decibel from 200 to 1,000,000 cycles per second. The discriminator was built on principles outlined by Lewis (7), so that pulses below a predetermined voltage are not passed on, but pulses exceeding this voltage are passed on as large uniform pulses. The operation of the discriminator was checked with uniform positive pulses. The scaler measures the time required to register a predetermined number of pulses with an electric timer having a smallest division of 0.01 second. A resolution time of less than 5 microseconds is necessary.
A detailed description of the electronic equipment and its operation is given by Geist ( 8 ) .
CALI BRATION
The relationship between the particle size and the size of the pulse obtained at the input to the amplifier was determined for metal spheres with diameters from 500 to 6340 microns and for liquid drops with diameters from 2590 t o 4550 microns. The data for metal spheres and water drops, which followed the same curves, are plotted in Figure 7 for a probe potential of 430 volts. Each metal ball was cemented to the end of a small rod of polystyrene rotated through a circle at about 60 r.p.m.? adjusted so that the metal sphere made one glancing contact with the probe wire during each revolution. A grounded strip of copper waO placed to make electrical contact with the metal ball just before each contact of the ball with the probe wire. The liquid drops were allowed to fall singlyfrom a buret or hypodermic needle maintained a t ground potential. The volumes of the drops were obtained by counting the number of drops delivered from a measured volume of liquid, and the diameters calculated by assuming the drops to be uniform spheres. The ranges of pulse sines for repeated contacts with metal spheres of various sizes at various contact points on the probe wire are given in Table I. Similar data for successive liquid drops are given in Table 11. The discriminator settings were recorded at the voltage above which no pulses were being counted by the scaler, and again a t the voltage below which tbe scaler counted
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1951
Table 1.
Diameter, Microna
Material Steel Steel Steel Steel Stael Steel Steel Solder Solder
At ti of proge 13.8 -14.7 10.7 -11.6 8.5 8.8 6.33 6.7 4.63 - 4.92 2.91 3.15 1.46 - 1.57 0.511- 0.524 0.207- 0.219
--
1375
Ranges of Pulse Sizes Obtained from Metal Sph'eres (Probe potential, 430 volts) Pulse Size, Volts Inehea from Tip
0.26 6.63 -6.93 5.15 -5.45 4.00 -4.12 2.76 -2.86 1.82 -1.88 1.14 -1.18 0.719-0.743 0.207-0.231
.......
0.5 5.75 -6.14 4.95 -5.25 3.15 -3.33 2.33 -2.42 1.63 -1.72 0.98 -1.02 0.646-0.670 0.195-0.207
...
0.76 5.26 -6.55 4.91 -5.09 3.03 -3.15 2.24 -2.36 1.51 -1.60 0.90 -0.96 0.673-0.598 0.183-0.189
. ... ...
1 4.61 -4.79 3.94 -4.12 2.79 -2.97 2.00 -2.12 1.44 -1.54 0.80 -0.84 0.536-0.561 0.183-0.195 0.073-0.070
1.25 4.18 -4.36 3.58 -3.82 2.61 -a.79 1.94 -2.06 1.32 -1.40 0.79 -0.82 0.511-0.523 0.159-0.171
.......
1.6 4.06 -4.36 3.34 -3.58 2.31 -2.55 1.88 -2.00 1.37 -1.41 0.80 -0.83 0.451-0.486 0.146-0.169
... ....
Table II. Pulse Sizes Obtained from Drops of Tap Water, Acetone, and 95% Alcohol (Probe potential, 430 volts) Pulse Size, Volts Diameter, Miorons 4660°
Material Tap water Tap water Tap water Aaetone Alcohol 0
8060 asgo
33~0 3330
Inohes from Tip
At tip of probe 7.17-8.02 3.96-4.06 2.87-2.97 1.98 1.98
0.26 3.07-8.86 1.44 1.18 0.96 1.04
0.5 2.78-8,02 1.29 1.14 0.81 0.89
0.76 2.78-a.27 1.29 1.12 0.77 0.87
1 2.87-2.97 1.29 1.09 0.76 0.84
1.25 2.72-2.97 1.34 1.07 0.67 0.87
-
1.75 4.20 -4.35 3.40 -3.68 2.26 -2.67 1.91 -2.03 1.39 -1.47 0.79 -0.86 0.476-0.510 0.169-0.171
-_
1.5 2.33-2.72 1.29 1.02 0.67 0.87
1.75 2.13-2 67 1.m 0.97 0.67 0.87
Ranges of pulse sire8 obtained from three series of measurements.
500,000 OHMS
1
(-> f
.
SPHERE
T 7pcpf
Figure 8.
Equivalent Circuit of Probe
all pulses. The range of pulse size for a given particle size waa calculated by dividing the above discriminator settings by the gain of the amplifier. At the same attenuator setting no measurable pulses were obtained when the metal spheres were not restored t o ground potential between successive contact. No pulses were obtained when the polystyrene rod, without a metal sphere, was placed gently in
contact with the probe wire, but pulses of less than 0.1 volt were produced in bursts when the probe wire was struck a sharp blow with the bare polystyrene rod. The pulse resulting from the mechanical impact of the polystyrene rod was apparently a microphonic effect, and was of the same order of magnitude as the pulse obtained from a metal sphere 500 microns in diameter. Visual observation of the pulses on an oscilloscope indicated that the pulses obtained from contact of the particles with the charged probe were negative. The time required for each pulse to rise t o a maximum was less than 100 microseconds, and the total duration of the pulse was less than 1 millisecond. No measurable pulses were obtained when drops of carbon tetrachloride, benzene, toluene, or kerosene impinged on the charged probe. The pulses obtained with drops of acetone and 95% ethyl alcohol were smaller than the pulses obtained from water drops of the same diameter, and are presented with them in Table 11.
0 -
4-
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w
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100
Figure g.
200
-
So0
PROBE POTENTIAL VOLTS Variation of Pulse Size w i t h Potential of Probe 0.126-inch:ball
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INDUSTRIAL AND ENGINEERING CHEMISTRY
The probe was inserted in a Fater spray to test its performance with mixtures of particles of various sizes. It functioned in a satisfactory manner, giving reproducible counts of particles over the size range for which it had been calibrated, although the greater proportion of the spray was smaller than these sizes.
Vol. 43, No. 6
and for a small change in capacitance
AV
=
-I7( AC/C)
or
AV/V = -AC/C
(4)
Equation 4 indicates that for a known initial potential and capacitance the pulse size might be estimated from the increase in capacitance of the probe upon addition of the spherical conductor. Equation 4 also indicates that a negative pulse should appear on the probe when the capacitance is increased; this was confirmed by experiment. The experimental relationships between the pulse size and the particle diameter a t several points of contact along the probe, as given in Figure 7, may be expressed empirically for this system as
l
NOTE: NUMBERS ARE DIAMETERS OF STEEL BALLS -INCHES
6
AV = 4
2
0 0
I
3/4
114
DISTANCE FROM TIP
Figure I O .
-
1-1/4
1-112
1-3/4
INCHES
Variation of Pulse Size with Position of Impact on Probe
SOURCE
OF
T H E PULSES
When an uncharged conductor is placed in contact with a charged conductor, the distribution of the charge is changed. The capacitance of the system is increased above the capacitance of the original charged conductor, and the potential of the system is decreased. The magnitude of the increase in capacitance or the decrease in potential is related to the sizes of the conductors, the potential of the charged conductor, and the original intensity of the charge at the point of contact. In this system these correspond to the particle diameter, the probe size, the probe potential, and the point of contact along the probe. If the probe is considered as a condenser connected to a battery, its equivalent circuit is given in Figure 8. After the capacitance of a condenser is suddenly changed, a finite time is required to restore the condenser to its original potential. The time constant of a circuit is the time required to ,,each the final condition if the initial rate of current flow is maintained. It is equal to the product of the resistance in the circuit in ohms and the capacitance in farads ( 2 ) . The capacitance of the probe, measured to be 7 micromicrofarads %-hencold, and the capacitance of a 10,000-micron sphere in space may be used to estimate the relative time constants. The capacitance in farads of a sphere in space is given ( 2 ) by C = 2aeD
(1)
where E = the dielectric constant = 8.854 X farad per meter. For a 10,000-micron sphere, the capacitance is calculated to be approximately 0.5 X 10-l2 farad. Assuming a contact resistance of 10 ohms between sphere and probe, the time constant for charging a 10,000-micron sphere is 5 X 10-f2second as compared with the time constant for charging the probe through a 500,000-ohm resistor of approximately 4 X second. Thus, even for a sphere as large as 10,000microns, the probe may be considered a condenser with a fixed charge. For a condenser,
Q = CV
(2)
dV = (-&/C2)dC
(3)
By differentiation
This variation of pulse size with the 1.6 povier of the diameter differs from the 2.0 power reported by Guyton ( 4 ) . Variation of the pulse size with the potential of the probe is illustrated in Figure 9 for a 0.125-inch steel ball a t three positions of contact along the probe. Similar curves were obtained from the data for other diameters. The straight lines passing through the origin of the coordinates indicate that the pulse size and probe potential are in a direct ratio, as is indicated by Equation 4. Both Figures 7 and 9 show an effect with position of contact along the probes. Some data are cross-plotted in Figure 10 to show the effect more clearly. The sharp rise in pulse size a t the tip of the probe may be qualitatively explained n-ith the aid of a sketch (Figure 11) of an estimated electric field surrounding the probe. The egg-shaped lines are equipotential lines and the hyperbolic-shaped lines represent flux lines. The variation of the concentration of electrical charge along the probe is indicated by the proximity of the flux lines and of the equipotential lines: The closer the lines, the greater the charge density. The charge density is greatest a t the tip of the probe; therefore, the pulse size should be greatest at the tip.
TO QRlD OF
I
PLEXIGLAS
INSULATION
.
INTERCEPTOR
PREAMPLIFIER
A------ ------------ j Figure 11.
M a p of Electrostatic Field around Probe
I n the work reported here, the size of the probe was not varipd; therefore, its effect is not considered. CONCLUSIONS
The possibility of determining the size of particulate material in suspension by inserting a charged probe into the flowing material has been demonstrated. I n order to make the device generally useful, it must be improved to apply to the smaller sizes of particles and to eliminate or control the effect of geometry of the probe. Partial success has already been attained. With further work, it may be possible to accomplish these objectives and t o develop equipment of better stability or reproducibility. ACKNOWLEDGMENT
This work was conducted with the aid of a financial grant for equipment from the DeVilbiss Co., Toledo, Ohio, and with the aid of fellowship grants from E. I. du Pont de Semours & Co., Inc., Wilmington, Del., and from the Engineering Research Institute of the University of Michigan.
June 1951
INDUSTRIAL AND ENGINEERING CHEMISTRY NOMENCLATURE
A C D
(I
*
a function of point of contact along the probe electrical capacitance, farads diameter of particle or drop, cm. D, controlling distance across sampler, cm. Q = electrical charge. coulombs R e = Reynolds number V = electrical potential, volts AT7 = electrical pulse, volts v = velocity of carrier gas stream, cm. per second = viscosity of carrier gas stream, grams per cm.-second P A = density of carrier gas stream, grams per cc. p~ = density of particle or drop, grams per cc. = = = =
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LITERATURE CITED
(1) Albrecht, F., Physilz. Z.,32, 48 (1931). (2) Attwood, S. S., “Electric and Magnetic Fields,” 3rd ed., New York, John Wiley & Sons, 1949. (3) Geist, J. M., Ph.D. dissertation, University of Michigan, 1950. (4) Guyton, A. C., J . Ind. Hyg. Tozicol., 28, 133 (1946). (5) Landahl, H. O., and Herrmann, R. G., J . Colloid Sei., 4, 103 (1949). (6) Langmuir, I., and Blodgett, K. B., “Mathematical Investigation of Water Droplet Trajectories,” U. S. Army Air Force, Tech. Rept. 5418 (1946). (7) Lewis, W. B., “Electrical Counting,” Cambridge, University Press. 1942. (8) Sell, W.’, Forschungsheft, 347, Ausgabe B,Band 2 (August 1931). RECEIVED January 3, 1951.
Fitting Bimodal Particle Size Distribution Curves COMPARISON OF METHODS J. M.DALLAVALLE, C . ORR, JR.,AND H. 0 . BLOCKER GEORGIA INSTITUTE O F TECHNOLOGY, ATLANTA, GA.
B
cs
IMODAL size frequency distribuIncreasing interest in the technology of particles and droplets has made it tions are often encountered in miimperative t h a t mathematical procedures for describing bimodal size districromeritics. Thus far, they have rebutions be considered. c e i v e d l i t t l e attention. With the Methods for describing two bimodal distribution curves which occurred in development of interest in particle size studies of aerosol aggregation phenomena are presented. distributions over wider and wider ranges A selected-ordinate method i s developed 2nd is shown t o describe these of size, greater consideration must inbimodal distributions better than t h e more complicated methods of moments evitably be given to the fitting of and least squares. bimodal and even multimodal distributions. This paper discusses procedures for analyzing bimodal distributions in particular, although the can also be applied to unimodal distributions. As will be shown, procedure can be expanded to multimodal forms. There are as the form of equation used and the method of evaluating its conyet no simple parameters for characterizing differences in various stants yield the important properties of a bimodal distribution bimodal distributions other than the location of the modes and and serve to characterize its salient features. the minimal values lying between. It is hoped that this report The bimodal form was recognized as a possibility in statistical will stimulate further interest in the subject. distribution by Kapteyn ( d ) and later by Fagerholt (1). FagerMultimodal probability distributions are not uncommon. holt’s general equation is rather difficult to deal with and is based They occur in quantum theory in connection with the distribuon Kapteyn’s theory of “reaction functions” as the causative tion of electrons about the nucleus of an atom ( 6 ) . Recently factor in determining distribution types. Karl Pearson’s ( 7 ) imSchoenholz and Kimball (8)’ studying emulsions, and Lipscomb, portant contributions to the analysis of skew distributions in Rubin, and Sturdivant (S),investigating smokes, encountered terms of normal curves could undoubtedly be applied to the bimodal distributions. The bimodal distributions to be discussed analysis of bimodal distributions. However, the computation of here occurred in a study of aerosol aggregation phenomena. five moments and the solution for the roots of a n equation of the Bimodal distributions in which the value of one mode is small ninth order are involved for a single skew distribution. in comparison with the other are sometimes represented by a THEORY normal curve. The equation of a normal curve is of the form Three methods for obtaining the fit of a bimodal distribution are discussed here; all are based on the general equation y = exp [ - ( a4x4 a3x3 a2x2 a12 ao) ] (3) where y = number of particles having a stated size x, N = numwhich was shown by O’Toole ( 4 ) to represent (under certain conber of particles measured,’M = arithmetic mean of the distribuditions) a bimodal distribution. For simplicity g(z) will be tion, and u = standard deviation of the distribution. written for the exponential polynomial, or With values computed from the data of Table I of M = 2.76, u = 0.36 (or 1.80 class intervals), and N = 268, the normal equag ( x ) = a424 a323 a& alx a0 (4) tion becomes Equation 3 then takes the form y = 59.4 exp [ - (x - 2.76)2/0.26] (2) -In y = (g)z (5) As shown in Figure 1, this unimodal form does not represent the The general polynomial of the fourth degree can be simplified data, and i t is erroneous t o assume that it does. It must be by methods outlined in texts on advanced algebra. Thus, by recognized that many forms of bimodal distributions can be multiplying and dividing through by a2 a4 broken down into two normal or skewed distributions. However, g(x) = aZ(x4 a:xa a:zs f aiz a;) (6) there are some distributions for which this cannot be done witha u t introducing errors of considerable magnitude. The bimodal where ai = a3/a2, pi = az/aa, a; = al/a2, and ai = ao/az. If x is expression as developed in this paper is of such generality that i t replaced by x - a3/4, after simplification Equation 6 becomes
+
+
+
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+
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+ +
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