Conclusions
The difference method is of interest in a wide variety of problems involving moving or unknown boundaries and in several areas of application. Apparently, the simple potential floiv model used here gives a useful approximation to the physical phenomenon. Prediction of the rippling effect is a particularly interesting result of the work. Literature Cited
Banks, R. B., Chandraselthara, D. V., J . Fluid M e c h . 15, 13 (1963). Collins, R. D., Lubanska, H., Brit. J . Appl. Phys. 5 , 2 2 (1954).
Greenspan, D., J . Franklin Znst. 266, 39 (1958). Harlow, F. H., kVelch, J. E., Phys. Fluids 8, 2182 (1965). Milne-Thornson, L. M., “Theoretical Hydrodynamics,” p. 291, Macniillan, London, 1960. Olmstead, \Y. E., private communication, 1966. Olmstead, \V. E., Raynor, S., J . Fluid M e c h . 19, 561 (1964). Young, David, Ehrlich, Louis, “Boundary Problems in Differential Equations,” R. E. Langer, Ed., p. 143, Cniversity of \$-isconsin Press, Madison, 1960. RECEIVED for reLiew January 6, 1967 ACCEPTED June 23, 1967 Research work carried out under the support of the National Science Foundation, Grant GP-661. .4cknowledgment is due to the Rice University Computer Project for having made available its computer facilities.
EXPERIMENTAL TECHNIQUE
ON-LINE PARTICLE SIZE ANALYZER Y . N A K A J I M A , K . GOTOH, A N D T. T A N A K A Department of Chemical Process Engineerinp, Hokkaido Unicersity, Sapporo, J a p a n Two new instruments for on-line measurement of particle fineness are described. Experimental results show that average particle size can be detected continuously within about 10 to 15 seconds and the cumulative oversize distribution curve obtained in less than 1 minute. observed.
n powder technology. it is important to know the fineness or Many types of devices have been developed for this purpose, but not all of them can be used for continuous measurement. Therefore, a n on-line particle size analyzer is required for the automatic control of powder handling processes. This paper discusses tlvo kinds of new instruments-one for the on-line measurement of average size of particles and another for cumulative size distribution.
I size distribution of particles.
Basic Principle
A stream of solid particles having a range of sizes is fed into a classifier and separated continuously into two streams at a certain size \\ hich is called the cutoff size. If a clean cut of particle sizes is attained in the classifier, the cumulative oversize fraction corresponding to the cutoff size, x,, in the feed materials-Le.. the cumulative mass fraction of particles in the feed stream nhich are larger than the cutoff size-is described as follows:
M here F,, and F,,, are the mass flow rates of feed stream and of tailings, respectively. By using tlvo particle flowmeters, therefore, the cumulative oversize fraction corresponding to the cutoff size can be detected continuously. Alternatively. the cutoff size generally can be represented by some function of the manipulated variable, J . of the classifier-e.g., air velocity in an elutriator-as follo\vs:
xc = g ( 2 ?
(2)
Hence, the cutoff size is determined from Equation 2 if the function is knolvn. Consequently, if the cumulative oversize fraction, R(x,) = Fout/’FID. corresponding to the cutoff size is controlled so as to keep a preset \.slue, R,, by automatic adjustment of the ma-
Only particles from 100 to 500 microns are
nipulated variable of the classifier, J , in response to the flow r a t e signals of F,,, and Fout, then the manipulated variable, J , gives the cutoff size, Jvhich is regarded as an average or representative particle size of the feed mixture. For instance, ivhen R, is set at SO%, the 50% average particle size of the feed mixture can be obtained continuously. I n contrast to the system mentioned above, if the manipulated variable, 3 . is changed continuously by manual operation or programmer by the use of a continuous divider and a n analog computer for solving Equations 1 and 2, the particle size distribution curve can be drawn automatically on an X-Y recorder. The test apparatus for those two systems will be shown schematically in Figures 1 and 6. respectively. Type 1.
On-Line Measurement of Average Particle Size
Experimental Apparatus. Figure 1 shows the on-line measuring equipment of average particle sizes. The classifier should satisfy five requirements. T h e cutoff size of particles can be calculated from the manipulated variable of the classifier. T h e manipulated variable of the classifier can be changed easily and continuously. T h e particles can be fed into and discharged from the classifier continuously. At least one of the flow rates from the classifier can be measured immediately. T h e particles are separated in a relatively short time. For simplicity, a n elutriator of the suction type shown in Figure 2 was used as the classifier. T h e particles are fed a t the middle point of the tube and separated continuously by a n uplvard stream of air. T h e fine particles are carried over from the top of the tube and collected at a cyclone, Lvhile the coarse ones (tailings) flow out from the bottom of the tube, and the floiv rate is measured immediately. T h e relationship
vot.
6
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N O V E M B E R1 9 6 7
587
ORIFICE
r-----
STORAGE
I
I
Fin
SIGNAL
I
RS
j
'LIJ
70
I
Figure 1,
CONTROL VALVE .A
COARSE PARTICLES (TAILINGS)
AB
1
7 1
g@R
-
CE.dt .
Schematic diagram of apparatus for detecting average particle size 1)
2) 3) 4) 5)
6)
Particle flowmeter 1 Particle flowmeter 2 Elutriation tube Variable transformer Integrator (master controller) Orifice air flowmeter
between cutoff size, xe, and manipulated variable of the classifier-Le., air velocity, u, in the elutriator-is given by the following equations :
Re, 10
< 10;
u = (p,
< Re, < 500;
- p)gx>/18p
u = {4(pp
Stokes' equation
- ~ ) ~ g ~ / 2p p2 )51 / 3. x c Allen's equation
Re,
> 500;
u = {3(p,
PARTICLE INLET
(3)
(4)
INSTALLING TAPS FOR PITOT TUBE
139
STATIC PRESSURE TAP
- p ) g / p ) l / z . x,1/2 Newton's equation
(5)
59 GLASS TUBE
where Re, = upxJp I n the present experiment, fineness of sample powder was selected within Allen's range for convenience. Then, Equation 2 corresponds to Equation 4. A linear relationship bet\veen the cutoff size and the manipulated variable as in Equation 4 is not always necessary in the basic principle; hence, any other form of Equation 2: even if it is represented graphically, is usable according to the range of cutoff size required. The use of Newton's, Allen's, or Stokes' equation is convenient and will not result in serious error because the purpose of this apparatus is to measure average particle size, Lvhich is expected to fluctuate in a comparatively narrow range. T h e range of cutoff size need not be very wide. In Figure 1, particulate materials from a process line, a crusher for example, are fed continuously into the elutriation tube, 3. T h e feed rate of particles, Fin,is measured by the particle flo\vrneter, 1, which consists of a beam balance and a differential transformer, and is converted into an electrical signal, Ein. The input voltage to the differential transformer of the flowmeter, 1, is dropped doxvn by R,( in Figure 1: the control system is made optimal experimentally, so that the setting time may become shortest with a reasonable stability. Some noises are included in the error signal and, therefore, t h e derivative action of the controller \vas omitted. The optimal setting of the controller remains unsolved here because the performance function and u becomes nonlinear and hard to obtain. relating Fout Figure 5 shows a transient response curve obtained for a step change of 50% average particle size from 0.34 to 0.26 m m . ; the settling time is about 10 to 15 seconds. The classifier used in the present experiments has large time lags and, therefore, the settling time can be shortened by improving the classifier. Type II.
Cumulative Size Distribution Autorecording
Experimental Apparatus. Figure 6 is a schematic diagram of the on-line measuring equipment for cumulative size distributions. The classifier should satisfy the same requirements as stated above. .4n elutriator as shown in Figure 7 \vas used. T h e feed materials are fed without vertical component of the initial velocity in the elutriation tube by using a particle distributer: so that the relative velocity of the particles, ivhose sizes are in the vicinity of the cutoff size with respect to the surrounding air, becomes immediately equal to the terminal velocity. T h e special purpose analog computer made for the calculation of Equation 2 consists of a n amplifier having a varistor diode and a resistor in the feed-back circuit. Equation 2 was calculated numerically using drag coefficients from Lapple and Shepherd (1940) and Perry (1950) and the relationship between input and output signal of the computer graphically approximated this result. .4s in the average size measurement, it may be possible to use Equation 3, 4, or 5 in place of Equation 2. I n this apparatus. however, the range of cutoff size is generally very wide and cannot be covered u i t h only Stokes’, Allen’s, or Xewton‘s equation, because for this apparatus the cutoff size must be changed over the full size range of feed materials. The divider in Figure 6 consists of a direct current amplifier and a servomechanism; it has some time lag for the computation, especially for changes of Fin. This is not desirable, as the particle feed rate must be kept as constant as possible. I n this experiment, therefore, a powder orifice was used to keep the feed rate constant. The flo\v rate of tailings, F,,,, sometimes varied, even Jvhen the feed rate of particles, the air velocity in the elutriation tube, and the particle size distribution of feed materials remained
W
W
GLASS SPUERES
b 0.4
E
5
3
W
2 0.2
u 3
I I I
0.0 0.0
0.1
0.2
I
0.3 0.4 PARTICLE SIZE, m m
I
0.5
Figure 4. Curnulative oversize distribution curve for silica sand (Type I)
I
0
5
10
15
20
TIME, SECONDS
Figure 5. Transient response curve (Type I) VOL. 6
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589
=BLOWER
ORIFICE
AIR
*
--.___
VARIABLE TRANSFORMER
U fi
y
F,"
EQUATION 2
I
PARTICLE FLOW ELUTRIATOR
VARSTER DIODE
FILTER -
$
i
CIRCUIT h
h
l
r
~
-
PARTICLES
FOR EQUATION 1 X-Y RECORDER
Figure 6. tion curve
Schematic diagram of apparatus for obtaining cumulative oversize distribu-
PARTICLE DISTRIBUTER
J --
'-loo+ I
Figure 7.
k-2094 UNIT
: mm.
Modified elutriation tube
unchanged. Especially at a larger feed rate, the pulsations became very large and the flow rate could not be measured. T h e fluctuation of troidage in the elutriator seemed to influence this phenomenon, but the origin of such a problem is still unknown. T h e feed rate, F,,, \vas adjusted to be sufficiently small to avoid the pulsations, and a filter circuit having a large time constant was provided to remove the hunting components from the flow rate signal of tailings. The continuous change of xc \vas carried out manually by adjustment of the blower speed using a variable transformer. 590
l&EC FUNDAMENTALS
Experimental Results. Glass spheres (0.1 to 0.3 mm.) were used in the experiment. As fluctuation in particle feed rate should be avoided, by using a po\vder orifice, the feed rate was kept constant. For this apparatus, it was found experimentally that the feed rate should be less than 3 grams per second, because of the variations in the flow rate of tailings, to obtain a clean-cut separation in the elutriator. The measuring system sholvn in Figure 6 has many components which cause time lags. Hence, the sweeping speed with which the cutoff size is changed must be slow enough to make the dynamic error negligible. Suppose that the total time lag for an R(.Y,) signal is T~ seconds and that for an x,signal is T~ seconds, and that it requires T seconds to move x , with a constant speed from the smallest particle size, x,in, to the largest one, xmaX. Since the sweeping action begins from x, = 0, the x , signal will be delayed by T~ seconds when x c approaches x,,,. O n the other hand, the R(x,) signal will be delayed by the amount of 0 to r R seconds. Figure 8 illustrates such a positional relationship on the chart of an X-Y recorder. The range of dynamic error in the ordinate direction is
where --cy is the slope of the cumulative oversize distribution curve when the interval from xminto xmaXis unity. If the value of a: can be regarded as approximately constant in the range of xmin < x < x,,,, the dynamic error may be nearly equal to a ( r R - T,)/T except in the vicinity of x = xmin and xmax. Therefore, if the time constants can be adjusted to make ( r R - T,) = 0, the error in the major part of the cumulative oversize distribution curve obtained \vi11 be tolerably small and the maximum error becomes less than --a . T J T . I n this apparatus, r R r Z Y 2 seconds and -a < 2 ; about 40 seconds are necessary to obtain the curve lvith an accuracy within 10%. The experiments \+ereconducted ui,der the condition of feed rate, Fin= 2 . 4 grams per second, and sveeping time T = 60 seconds. The nveeping speed \vas not necessarily uniform because of the manual operation. Figure 9 shows a typical experimental result for glass spheres. Agreement with the results of sieve screening tests is satisfactory.
above, the second and third terms of Equation 8 were neglected, because their absolute values become comparable with each other for a n appropriate design of the classifier. Considering the accuracy of the fractional recovery curve, it is not always necessary to compute the integral from xmin to x, in Equation 8 ; then the fractional recovery curve can be reasonably truncated as shown in Figure 10:
TRUE CUMULATIVE OVERSIZE CURVE
I
RECORDED POINT IF THERE WERE NO TIME LAG
w
< xc - Ax1; x > xc + AX^; x
=
q(x,,x)
0
?(x,,x) = 1
where
-t
0
?(xc,x) =
FULL SPAN = 1
X I ,
?(xc,x) = 0.95
PARTICLE SIZE
Figure 8.
0.05
a t x = xc - Ax1
Xn-
Positional relationship of the error
a t x = xc
+ Ax2
Then Equation 8 can be approximated by
s;,
zc+ Ax2
SWEEPINIG TIME + 60 SECONDS
\
o S I E V ESCREENING
f(x)dx
(9)
If Axl and Ax2 are small, and the frequency distribution curve is relatively flat in comparison with the fractional recovery curve,
\
f(Xc
- hxJ
N- f ( x c ) y _ f ( X c
+ Ax21
(10)
Then Equation 9 becomes
PARTICLE SIZE, mm.
R(xc) + f ( x c ) ( s i
Figure 9. Cumulative oversize distribution curve for glass spheres (Type II)
- S2)
(11)
where S1 and S2 are the areas of the hatched regions in Figure 10. If SI SZ and Equation 10 hold, therefore, any kind of classifier can be used for precise measurement of the cumulative oversize fraction even if clean-cut separation cannot be expected. Some improvements on the classifier will make the measuring time shorter and the range much wider. Conclusions
PARTICLE SIZE
Figure 10.
Results of experimental confirmation in the static characteristics of both types are satisfactory. T h e average particle size can be detected continuously within about 10 to 15 seconds, and the cumulative oversize distribution curve obtained automatically in less than 1 minute for the particles larger than 100 microns. Measuring procedures of both instruments are simple.
xc
Fractional recovery curve
Effect of Incomplete Separation of Particles. T h e separation of particles by the dassifier is not always complete. T h e sharpness of separation is given by the fractional recovery curve, q ( x c , x ) , as illustrated in Figure 10. I n this case, the relationship between Finand F,,, becomes
Nomenclature = output signal of particle flowmeter
Eout
=
AE Fin Fou t
f
J xmin J
~ m i n
La’
f ( x ) 11
-
tl ( x d ) dx
(8)
in place of Equation 1, where f ( x ) is the particle size frequency distribution function 0.F feed materials. I n the discussion
1 (inlet), volts output signal of particle flowmeter 2 (outlet), volts = Ei, - Eout= error signal, volts = particle feed rate, grams/sec. = particle flow rate of tailings, gramdsec. = particle size frequency distribution function, -dR/dx = acceleration of gravity, cm./sec.2 = a function of y, cm. = sensitivity constant of particle flowmeter, volts sec./gram = cumulative oversize fraction, dimensionless = cumulative oversize fraction corresponding to xc, dimensionless = set point or setting parameter of R(x,), dimensionless
Ein
VOL. 6
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NOVEMBER 1 9 6 7
591
= Reynolds’ number around a particle, dimensionless = electric resistance of varister diode, ohms = areas of hatched regions in Figure 10, cm. (dimen-
Re,
R,
Si, S:! 7’ XC
= = = =
x,,,
=
U X
x,in = AXI, Ax? = Y =
sionless ordinate) sweeping time, sec. air velocity in elutriation tube, cm./sec. particle size, cm. cutoff size of classifier, cm. the largest particle size in feed materials, cm. the smallest particle size in feed materials, cm. size differences represented in Figure 10, cm. manipulated variable of classifier
GREEKLETTERS = slope of cumulative oversize fraction curve (see
-CY
Figure 8), dimensionless
E
7 (x,,x)
cc
Pa
P PP
rZ, r R
X-Y recorder, dimensionless = fractional recovery function, dimensionless = viscosity of fluid, gramsjcm. sec. = gain of amplifier, dimensionless = density of fluid, grams/cc. = density of particle, grams/cc. = total lags and dead times of x, signal and R ( x , ) signal, respectively, sec. = error in ordinate direction of
literature Cited
Chem. Engr’s. Handbook, J. H. Perry, Ed., 3rd. ed., p. 1018, McGraw-Hill, New York, 1950. Lapple, C. E., Shepherd, C. B., Ind. Eng. Chem., Anal. Ed., 32, 605 (1940). RECEIVED for review January 24, 1967 ACCEPTED July 17, 1967
HIGH SPEED CINEPHQTOMICROGRAPHY D . F. S H E R O N Y , C. B. T A N , A N D R . C. K I N T N E R Illinois Institute of Technology, Chicago, Ill.
Techniques of high speed photomicrography are presented. Attention is focused on the equipment and practices for taking motion pictures a t framing rates up to 4000 p.p.s. Consideration is given to data accumulation from films and the error involved in velocity determinations.
BSERVATIOT
of some phenomena may be impossible be-
0 cause of the rapidity and minuteness of the occurrence.
Such was the case in examining the role of the fibers when a fibrous bed was used to coalesce a secondary emulsion. T h e solution lay in photographing the phenomena through a microscope (1OX objective and 1OX ocular) with a high speed motion picture camera operating at about 500 p.p.s. T h e films served as a source of data and as a permanent record of the qualitative observations made during the investigation. The optical system for taking pictures through a microscope has been described by various investigators (7-3, 5). T h e following is a summary of some current practices with high speed photomicrography. Figure 1 illustrates the assembled apparatus, which must be carefully aligned to prevent aberration and nonuniform illumination. I t is evident that back lighting was used, but
H A n
B
Figure 1.
C
D _E
F G -
Apparatus for high speed cinephotomicrography
Bausch & tomb illuminator, 6-volt, 1 8 - a m p e r e lamp Rack for heat absorber and light filters C. Condensing lens system D. Substage condenser and iris diaphragm E. Flow cell containing fibers F. Microscope optical system G. tight t r a p H. Fastax WF3 camera
A. B.
592
l&EC FUNDAMENTALS
reflected light could be employed in limited cases. The light source could be a 6-volt, 18-ampere ribbon filament or a 110volt coiled filament lamp, the only constraints being that sufficient light be available for photography and that the light density be uniform over the optical field of view. A slide projector can be used in some applications. T h e Bausch & Lomb illuminator employed here also provides a mount for heat and light filters. The condensing lens system consists of two plano-convex lenses arranged a t a position along the optical axis, so that an image of the lamp filament is formed on the iris diaphragm of the microscope substage condenser. T h e condenser lenses should be of sufficient diameter to collect as much light from the lamp as practical. TYhen in use the iris diaphragm should be fully open, allowing the substage condenser to collect all available light and any attenuation of the beam should be done Lvith the iris diaphragm a t the light source or with a voltage control in the current supply to the illuminator. For observing fibrous bed coalescence, a rectangular transparent viewing cell was constructed of Plexiglas. The emulsion flowed through a fibrous mat which was packed into the cell from one end. T h e type and construction of the viewing cell would be determined by the phenomena involved. If fluid motion is under investigation, the cell must be so made that the particular local flow can be brought into view and focused. The observer must bear in mind the particular point that he is photographing in the fluid field. The distance between this point and the microscope objective is the sum of two quantities: the working distance of the objective and the field depth. T h e working distance is constant and is furnished by the lens manufacturer. I n general it is inversely proportional to the magnification. For a 1OX objective, it could be as high as 10 mm. For a 60X objective, it lies in the range of 0.2 mm. Special lenses are available with longer working distances. The field depth of the lens ranges from about 10 to about 50 microns. The fine adjustment of the microscope is usually calibrated in microns and refers to the field depth, so that a reasonable approximation of position can be made. For these short optical distances many observations of the flowing field would ordinarily be in the boundary layer behind the viewing surface of the cell, even if the main flow of fluid were in turbulent motion. If the interest is in the turbulent regime, the optical system and cell must be so designed that the focal plane of the microrxope objective is in such a zone.
