A METHOD FOR COMPARING T H E T I M E S OF MIXING OF TJ’iO T R A S S P A R E S T LIQCIDS I S D I F F E R E S T PROPORTIONSl AKD SOJIE E S P E R I M E S T A L RESCLTS, P1RTICULA4RLY W I T H GXSOLISE AXD CARBOS RISULPHIDE‘ BY L. E. DODD
Introduction The purpose of this article is to describe an optical method for determining the “time of mixing” of two transparent liquids, not too readily miscible, and to record some experimental results. The degree of reproducibility attained appears to justify further work with the method, which is a particular adaptation of the more general “mcthod of striae.” The “Schlierenmethode” of Toepler,? the aberrational effect on the rays of light passing through a glass lens8 because of the striae in it, particularly of a certain general type,4 that in some cases results in a blurring of the image, and the unsteadiness, or “boiling,” of a telescopic image due to turbulence of the atmosphere, which consists of motion of atmospheric striae that in turn are due to varying density in the air between the object and the telescope 5are all well known. In general, striae are physical inhomogeneities of a given medium, which must be transparent for their study by an optical method such as that described here, and they are accompanied by corresponding variations in the optical density, so that the refractive index changes from point to point. Their presence can be detected by the scattering effect upon a beam of light transmitted through the medium. The more pronounced striae are easily seen by simply looking into the transparent’ medium when it is illuminated in a suitable direction by ordinary daylight. -1common example of striae in air is that of t’he visible so-called “heat” rising from a hot surface. The mass of striae in this case is formed of air heated unevenly. Often the index variation is narrowly localized, as in much optical glass that is still usable. I n such cases the striae appear as threads or sheets of transparent material, imbedded in a medium that otherwise has a uniform refractive index. Striae are rendered visible in any one of three different ways: ( I ) the niedium in which they occur may be viewed directly, as mentioned above, the direction of view being generally toward the source of illumination when the latter is not too bright; ( 2 ) a r e a l image of the medium, or, more exactly, of a l h i s article includes the material of a paper presented before the .Imerican Physical Society, Pacific Coast Section, at the Pasadena meeting, September 1 8 f1923), abstract of whirh appeared in the Physical Review, 22,527 ( 1 9 2 3 ) . * Toepler: Pogg. Ann., 1 3 1 , 3 3 ( 1 8 6 7 ) ;see also 11. W. Kood: “Physical Optics.” L-. 9 . Bureau of Standards Scientific Paper, 373 :1920). “Characteristics of Striae in Optical Glass,” by Smith, Bennett, and hlerritt. C.9. Bureau of Standards Scientific Paper, 333 ( 1 9 1 9 ) . “Striae in Optical Class,” b y .I..I. hlichelson. See discussion of scintillation in JVood’s “Physical Optics.”
1;62
L. E. DODD
definite sheet, or plane, of it perpendicular to the axis of the rays, may be projected to a screen, where the striae appear in the image; or, (3) by simple means,‘ “refraction shadows” of the striae may be made to appear on a suitable screen, preferably translucent, held in the path of the light. For highest visibility, all three methods employ a source of light as concentrated as possible, in order that the aberrations, caused by the striae, in numerous pencils of rays of light from the many different points in an extended source, may not overlap and produce a field of quite uniform intensity, in spite of the presence of the striae, in this way masking them. The type of striae occurring where one liquid has been added to another, and which persists until the “mixing,” aa the term is defined in this article, has been completed, is common. The particular method to be described makes possible a particular definition of “mixing” that is stated in fairly specific experimental terms. Two liquids are here regarded as “mixed” when the striae have disappeared, as judged by the clearing up of a real image formed by rays that have traversed the liquid mixture. The precision of such a method is raised by the use of a long focus lens. The relatively rapid final clearing up of the image gives approximately the time at which the mixing is completed according to the definition. The “time of mixing,” that is, the elapsed time which under the given experimental conditions is necessary for the clearing up of the image, depends among other factors upon the method of determining the “instant of start” of the mixing, or time zero. This requires a separate definition, one that is based likewise on the experimental conditions, and is likewise fairly precise. It will be stated later in the article, following a description of the apparatus. The “time of mixing” depends on a number of factors, including the particular liquids used, and the “rate of stir,” which in turn is affected by the particular design of the stirrer, its size, shape, and amplitude of motion relative to the shape and size of the vessel and the total volume of liquid in it, the location of the stirrer in the vessel, and the nature and rate of its motion. A temperature effect is to be expected, but none was established in the present work, which was all done at the approximately constant temperature of the room. Apparatus and Method of Stir Optical System. Fig. I , with legend, shows the scheme of the optical system as viewed from above. The length of the path of the rays from the objective, e, to the translucent screen, g, was approximately 15 ft. Tank. The mixing vessel, d, was a commercial rectangular battery jar, of outside dimensions 2 X 2 1 X 4 inches. The inside dimensions of the horizontal section were about 4 X 6 cms. The two parallel walls of larger area, used as windows of the tank, were ground and polished plane on the outside, being left from 3 to 3.5 mms. thick. No attempt was made to modify the inside surfaces, which were not precisely flat, although smooth, as the vessel was blown in a mold during manufacture. However the aberrational effect ‘L.E.Dodd: J . A m CeramicSoc.,2,97; (1919).
A METHOD FOR COMPARING THE T I M E S O F MIXING
I763
upon the transmitted light, due to departure from flatness on the interior, was lessened because of the relatively low index between the glass and the contained liquid. No greater flatness could be claimed for the tank floor than for the inside walls. None of the inside corners were sharply angular. For the present use the vessel was provided with a ground plate-glass cover fitting tightly on the upper, open end, whose edges were likewise ground flat. The cover had a 4-mm. hole drilled in it to permit the handle of the stirrer to project upward out of the tank.
Y FIG.I Optical System for determining the Time of Mixing of Transparent Liquids by Method of Striae. a, 100 w. Mas& b, condenser lens c, opaque screen with 2 mm. aperture d, g l e mixing tank e, objective, f = 30 inches f, plane, plate glass mirror g, translucent screen to receive real image of aperture a t c.
Stirrer. A metal meshwork stirrer combined simplicity of construction and operation with thoroughness in its stirring effect. It consisted of a rectangular loop of iron wire, about 2 . 5 mms. diameter, bent sharply at t h e corners and soldered so as to form a rigid frame. The loop was 3 X 5 cms. in size, and had a sheet of galvanized wire cloth (ordinary fly screen) fastened rigidly to it below by local soldering, to give it the motion of the loop itself. The sheet of metal screen was about the same size as the loop; its edges did not project much beyond the loop toward the glass wall. There were fourteen squares of the mesh-work in each linear inch of the screen, including the wires, which were about 0.3 mm. diam. The vertical handle, also of the same size of iron wire, was attached rigidly by solder a t one corner of the loop, and extended perpendicularly to its plane, passing out through the tank cover as already mentioned. Thus the plane of the loop, covered with its metal network, re. mained approximately horizontal. The free end of the handle was filed to a hook shape, to receive a fine piano wire suspension. Motion of the Stirrer. The suspension wire was in turn hung from the end of a hardwood lever about 2 . 5 ft. long, mounted on a horizontal axle so that it formed a walking beam. The outer end of the lever was jointed to the
1764
L. E. DODD
iron rod of a reciprocating system operated by a small D. C. motor. The vertical motion of the stirrer suspension could be regarded as very nearly simple harmonic. The plane of the meshwork itself had very little angular motion. Adjustments were such that when the fine wire suspension was at the upper end of the lever stroke, the meshwork was lifted clear of the free surface of the liquid by about $ inch, and with the suspension at its lower position, the meshwork rested moment~arilyon the bottom of the tank with slack in the suspension wire. This slack amounted to about t inch also, so that the motion of the stirrer in the liquid was symmetrical in character.
___ /uo , / a 7__ \
FIG.2 Approximate Motion of the Stirrer
Fig. 2 gives a cosine curve of velocity that holds for true simple harmonic motion. It is taken to represent approximately the actual motion of the stirrer. The parts of 6he curve, each including a crest or a trough, drawn in as continuous lines, correspond to time intervals when the stirrer is in motion in the liquid, and the parts drawn in as dashed lines correspond to the intervals when the stirrer is, in effect, at rest relative to the liquid, that is, when it is either out of the liquid, or at rest, on the tank floor. At the points .A and E: the stirrer emerges from the liquid into the air. At point R it reenters the liquid, at C it stops abruptly on the tank floor, and at D it starts abruptly upward t,hrough the liquid. I n the present work the period of the stirrer could be kept approximately constant at about 1.7 secs., as tested wit,h stopclock, by maintaining constant voltage on the motor. The performance of t,he stirrer is shown in Table I. The presence of the liquid in the tank had little or no effect on the period. The constant total volume of liquid used, 60 cc., gave an approximate depth of liquid in the tank of z ; nims. The double aniplitude of the end of the wooden lever at the point where the suspension wire was attached, was 38 mms. Akssuniingthat thp meshwork was precisely at the center of thc liquid vertically when the suspension was at its oscillation center, the velocity of the meshwork at, that instant was 7.0 cms./'sec., with the amplitude of the suspension t,aken at 19 mms. and the period at I.; secs. The actual
h METHOD FOR COMPARING T H E TIMES O F MIXING
1765
period may be regarded as within 5% of this figure, which will serve the purpose of discussion. The harmonic velocity increased from 5 . 2 a t the tank Hoor to 7.0 cms./sec. at the center of the liquid, and then decreased to 5 . 2 cms. 'sec. upon emergence from the liquid. The same succession of velocity values at the same places was repeated during the down stroke.. Thus the
TABLE I Performance of the Stirrer Date
Time
S o . of Cycles 40
67 j
I
40
67 o
1.68
40
66 8
1.67
40
66.8
1.67
Period of suspension with stirrer attached and moving in 60 cc. gasoline in tank.
40
66.8
1.6;
Same, but stirrer moving in 60 cc. CS2 in tank.
40
66.8
1.67
Stirrer moving in air.
40
66.5
1.66
10
16 4
I
64
End of series 6.
IO
16.2
1.62
End of Series 7 .
Yept. .I
IO
16.6
1.66
Sept.
IO
17
4
I
74
20
35 1
1
7i
I
81
20
36 z 35 6 35 4 35.6
20
35 4
20
35.0
1.75
(1923)
.-lug.
- 1
( 1
,!
.lug.
Ij
>,
*, ,1
16
Sppt. 3 .,
..
j
(secs)
T 69
Remarks
Period of stirrer suspension without stirrer. Period of suspension with stirrer attached and moving in air.
,,
20 20 20
20
Sept. I 2
I
78
1 I
7i 78
1
ii
\
>,
,*
,)
Series I O started this date. Voltmeter and rheostat control on motor The seven readings were taken successively.
During Series
IO.
1766
L. E. DODD
stirrer was alternately in mot.ian in the liquid and t h q i n e f f e c t , a t rest relativa to it. The manner of corning to rest on the tank floor WBS different to he s w e from the effective coming to rest as the stirrer broke through the surface and was lift.& clear of tho liquid. The “nst” intervals wspo slightly longer than the motion intervals, being 0.45 and 0.40 soes. respectively. During the periodic ”mst” intervals of t.he stirrer the inertia of the currents set up by its passage through the liquid maintained in eonaiderahle degree the process of stirring, 80 that it was in a so= continuous. The preeisc quantitative effect on t,he mixing, of the periodic pause in the motion of the meshwork stirrer, would be interesting to know. That it lengthens the mixing time is probable, and t.his circumsta~iceshould contribute to relatively low errors and to consistcncy in results by increming the magnitudes to he measurrvl. The . . . . ..... ..... -..
.................
__ .........
-. .
..... ._ -.
.
l h3 I’rriuriic Shadowernphs of the Striae during miring, shewing the (‘haraatri of the Hechanical Agitation
distribution of the turbulence was quit,e uniform throuKhout t.he whole body of liquid, as indicated by the snapshot “shadowgrephs” (“rzfraction shadows” by short range projection) of Fig. 3. These six snapshots, taken successively at q u a l and relatively long time intervals, were made in some preliminary work ahere measuring by photography the time of mixing, was being considered. In each of the six snapshots tho meshwork stirrer is seen cdgewisr at an instantaneous position near the floor of the tank. The striac in the g h s floor, whcrc the rays of light traversed it edgewise, and reached the film, are seen at the bottom of t,he prints. Particularly in nos. j, 4, 5 , 6 , refraction shadows of a different character are noted in the vicinity of the free surface. Those below the surface are probably due to a temperature effect caused poasibly by slight evaporation at the surface, or by the entrance of the meshwork into the liquid, because of B slightly lowered tnnperat.ure of the metal due to cvaporation from it while in the air, even with thc ground glass C O Y P ~in
A METHOD FOR COMPARING THE TIMES O F hlIXISG
1767
position on the tank. Temperature change may have contributed to, or even been chiefly responsible for, the refraction shadows above the surface, although the mere presence of some liquid flowing down the glass because of waves, might account for them. However, for the results presented in this article a greater volume of liquid was used, and these effects in the neighborhood of the surface did not themselves disturb the beam of light.
Start of Stir. Because of the relatively high speed of the motor, since reduction gearing was employed, and its quick response to the current, the “instant of start” was taken as that of closing the switch, with the stirrer, attached to the suspension wire, resting initially on the tank floor. An alternative method was to have the motor already running, but the loop of the suspension wire unhooked from the stirrer handle, and then suddenly to engage the hook with the loop, at, the same time starting the stopclock. The method of closing the switch was chosen as being the simpler. In accepting this definition of the “time of start” the effect of the “initial stir” (see discussion below) is neglected. Initial Stir. By this term is meant the stirring that results from adding the second liquid to that already in the tank without any motion of the stirrer itself. The amount of this initial agitation can be lessened, even to a negligible degree, &s is evident from the present results, at any rate up to fully jo% gasoline, by permitting the liquid being added to flow into the tank slowly and gently. This lengt,hens the time before the stirrer is set into moLion, giving diffusion or possible ‘‘chemical” action a longer time in which t o act, but there seemed to be no marked effect attributable to these, at any rate up to j o % on the axis of abscissae. Furt’her reference to the initial stir will be made in the discussion of results. Appearance of Image. The presence of the striae, even when their structure had become fine through stirring, was indirectly known, either from lorn “definition” in the image, or from “boiling” of the image, or from both effects. When the image became “clear” and steady the striae were regarded to have disappeared, to the degree of precision attained in the present experimental setup. Due to the aberrational (prismatic) effects in the tank, caused by lack of planeness, and absence of strict parallelism, of the inside wall surfaces, the image of the illuminated circular aperture serving as object was not itself circular, but usually roughly triangular in shape. In size it was an inch or more across. For the reason that the mean index of the mixture varied with the proportions of the two liquids, the dimensions of the image changed soniewhat. In the region above 907’ gasoline the shape of the image had materially changed to that of an oblong field, the colors not being so pronounced. This circumstance may have contributed to a somewhat erratic tendency shown by curve I O , Fig. 6, in this region, because of possibly less precise determination of the completion of the mixing. The image may be described ronghly as a color field with a darker central portion. Rut the color and the edges of the image were well defined. Any
1768
L. E. DODD
“boiling” of the image, even local, due to moving striae in the tank, was plainl!. seen, perhaps as easily as if the image had been perfect. Throwing the image slightly out of focus appeared to accenhate a little the boiling effect, due possibly to the striae themselves approaching a focus. Procedure After the first of the two liquids had been poured into the tank, the stirrer was immersed so that it rrsted on the tank floor, with the ground glass cover in position, but rotated sufficiently in its own plane about the stirrer handle to leave an opening for the adding of the second liquid, aft,er which the cover mas moved to the closed position. The hook on the stirrer handle was nest engaged in the loop of the suspension wire, and the stirrer started by closing the motor switch, as already mentioned. The stopclock was tripped when the real image cleared up with considerable abruptness. Room temperature, recorded in the d a t a , was read from a standardized thermometer kept within a foot of the tank and on a level with it. There were no appreciable air currents about the apparatus. Experimental Results Ten series of readings altogether (Tables I1 to X I ) were taken in August and September, and curves for five of them, viz., I , z 1 3, 9, I O , are presented. Series 3 was an incidental t,est with distilled water and concentrated common salt solution as the two liquids mixed. The remaining series, 4, j, 6 , 7 , 8, were t.ests for reproducibility. Series 4) 5 , 6 , were taken successively on the same day, series 8 and 9 successively on the following day. The continuous time intervals during which the readings of the ten series were obtainrd, and other inforination of a general nature, are given in Table XII.’ For series IO, however, the actual time consumed by the readings \vas I()& hours in four separate continuous time intervals as follows: 1st day, 2 hrs.; z i d , 5 4 ; 3rd, 44; 4th day, i t hrs. (September). Each of the total of 2 4 8 readings recorded here, required an experimental “run” lasting anywhere from18 secs. to 61 rnins. I n the nine series, excepting series 3, the two liquids were gasoline and carbon bisulphide, both commercial. The gasoline used, while not all from the same purchase, was obtained from the same filling station, the gravity being given at 6 2 . The
