Figure 1.
Apparatus for determining pressure drop in a moving bed
H. L. TOORl and S. D. EAGLETON Monsanto Chemicals, Ltd., Fulmer Hall, Fulmer Bucks, England
Plug Flow and Lubrication of Polymer Particles I N FABRICATION of high polymers, a bed of polymer particles frequently is forced through a conduit under fairly high pressures. I n the injection molding process, for example, particles are forced by a piston from a cold cylinder into a heating chamber and the molten polymer which is displaced from the heating chamber enters a mold where it is formed into the finished product. The pressure necessary to force the molten polymer from the chamber into the mold must be transmitted through the particle bed; thus any pressure loss in the particles is unavailable a t the mold. A knowledge of the pressure transmitted through the bed and the variables which affect it are important both in studying the operation of the process and in the del Present address, Carnegie Institute of Technology, Pittsburgh, Pa.
sign of polymer formulations for the process. I n the design of polymer formulations, one objective is to reduce the pressure drop to a minimum and a common method is to add lubricants to the surface of the particles. Little information is available on the flow of a particle bed. Spencer and coworkers (3) have examined the process in which a bed of particles contained in a tube is compressed by a piston. They restricted their investigation to a static bed in which the porosity, although it varies with distance along the bed, is not a function of time. They assumed that there is no motion of particles with respect to each other, that contact forces between the particlesare the same in all directions a t each point, and that the number of particles per unit area contacting the pistons, or in contact a t any cross section, is
constant. With these assumptions they were able to equate the frictional force at the wall for a differential length of bed to the change in the axial force. Their final equation in our terminology is :
This equation was checked by Spencer and coworkers who applied force to a movable piston in contact with one end of a cylinder of granules and measured the force transmitted to a fixed piston in contact with the other end of the bed. Their results confirmed the above equation and from the experimental data they calculated a coefficient of friction for polystyrene on steel of 0.262 which compares favorably with the directly measured value of 0.284 ( 3 ) . Both of VOL. 48, NO. 10
OCTOBER 1956
1825
0.28
0.26
no \
n. 0.24
0.22
0.20 2
0
4
6 pa. p.8.i. x
IO-^
8
IO
12
Figure 2. Typical variation of pressure ratio with applied pressure for unlubricated particles; I = 1.57 inches; granule size, between No. 12 and 22 screen
SCREEN S1ZE.B.S 410 5
0.0 0.00 0.02
404
0.06 a08 0.10 0.12 0.14 APERTURE OPENING, INCHES
0.16
Apparatus
The apparatus shown in Figure 1 was used to measure the applied force and transmitted force simultaneously: while the bed was in motion. The cylinder was constructed of carbon steel and had 1 826
1.5
2.0
2.5
3.0
INCHES
Figure 3. Effect of bed length on pressure ratio for various applied pressures; granule size between No. 10 and 22 screen
an inside diameter of 0.749 inch. The apparatus rested on the lower platen of a Baldwin-Tate-Emory testing machine which could apply a maximum force of 5000 pounds. This machine was equipped for the automatic recording of stress-strain curves. When force was applied to the upper piston by the upper platen, the transmitted force caused the springs to compress and the entire bed moved downward. The motion of the l o ~ e piston r was picked up by the extensometer which gave a plot of applied force against deflection of the springs on the machine recorder. By using the spring constants and the cylinder area, it was possible to convert the recorded curve into a curve of applied pressure versus transmitted pressure. These pressures could be estimated to within 25 lb./sq. in. I n addition to allowing the entire bed to move, the apparatus made the collection of results very rapid, for one curve covered the entire range of pressures. The pistons fitted smoothly in the cylinder and would drop out of the cyl-
INDUSTRIAL AND ENGINEERING CHEMISTRY
1.o L,,
0.18
Figure 4. Effect of particle size on pressure ratioI = 1.57inches
these results are lower than the value of 0.3 measured directly by Shooter and Thomas (2). Spencer and coworkers ( 3 ) also presented some data which indicated that the addition of lubricants to granules or the tube wall increased the transmitted force by decreasing the coefficient of friction between the granules and the cylinder wall. The purpose of this work is to determine the pressure transmitted through a bed of particles which is moving through a tube and to study the effect of some of the variables which are met in practice, especially the effect of surface lubrication of the polymer particles. Much of the work is exploratory, and no attempt has been made to investigate the effect of all the variables in detail.
0.5
inder under their own weight at the end of the runs. This indicated that there was no binding of the pistons due to fine particles which might have been trapped between the pistons and cylinder wall, even during runs with fine particles. Commercial unlubricated polystyrene (Monsanto Lustrex) was used in all the runs which Tvere made in a testing room kept at 72' F. and 50% relative humidity. The polymer was dried and allowed to remain in the testing room for a number of days before use. For runs with lubricated polymer, the dried polymer was thoroughly mixed with dried lubricant and then left in the testing room. This procedure was necessary to obtain reproducible results. Care was taken that no extraneous lubricant films were on the metal surfaces. Granulated polymer with particle sizes in a narrow range were obtained by' screening them out of a large batch using British Standard 410 test sieves. These particles were of irregular shapes. The polymer pellets used were roughly
cylinders '/a inch in diameter and l/s inch long with smooth surfaces, obtained by chopping extruded rods. The cylinder was tapped while the particles were being charged and the tapping was continued until no further settling of the bed was observed.
Unlubricated Particles Effect of Applied Pressure. Equation l indicates that P,/Pa should be independent of the applied pressure for any particular bed of particles. This was not completely true in this work as may be seen from Figure 2 which is a typical plot of P,/P, vs. Pa plotted on an expanded scale, The deviation decreases with increasing pressure and becomes very small a t higher pressures. This systematic deviation occurred in all the systems examined, but for practical purposes it is not considered serious. Equation 1 is plotted on Figure 2 for various coefficients of friction. Frequently, the particles tended to weld together after being compressed, especially in the upper part of the cylinder where the pressure was greatest. At times it took a severe impact to dislodge them. Effect of Initial Length of Plug. Figure 3 shows the results obtained when the initial length of bed was varied. The log of the ratio, P,/P, is plotted against Lo. The particles used are those which passed a No. 10 screen but were held on a No. 12 screen. The equation holds fairly well for any fixed value of the applied pressure, although there is a deviation for small values of Lo. At any one value of the initial length, the points spread because of the deviation discussed before. The curves for coefficients of friction, 0.15 and 0.18 calculated from Equation 1, are shown in the figure. Effect of Particle Size, Shape a n d Size Distribution. Not enough work has been done on this complicated subject to allow definite statements to be made. If all conditions but particle size are kept constant, the transmitted pressure increases linearly as the particle size increases. This is shown in Figure 4 where the data are at a constant applied pressure and initial length. The aperture corresponds to the screen in which the particles were retained and is only a rough measure of the true particle size in the batch. When a mixture of particles is used, the transmitted pressure falls somewhere between the extremes of particle sizes making up the batch. Pelleted polymer gives results lower than those for granules of approximately the same size; the results are about the same as small granules. Effect of Upper Piston Velocity. All the work presented in this paper was done at a piston velocityof0.5 in./minute.
Weight 7 . Lubricant Figure 5. Effect of lubrication on pressure ratio-L Ib./sq. inch
Negligible variations in the results were obtained when the velocity was increased to 4.0 in./minute. Discussion. Equation 1 correlates the data for any one type and size of particle if the applied pressure and original bed length are above certain values; for small enough particles, the results are independent of particle size. However, in all cases the apparent coefficient of friction is lower than the reported values which range from 0.25 ( 3 ) to 0.3 ( 2 ) . In Figure 2, for example, p ranges from about 0.15 to 0.19. It might be expected that deviations from Equation 1 are due to motion of the bed. Attempts were made to check this by stopping the testing machine during a run and then restarting. Any difference between the static and dynamic case would be expected to cause a break in the curve, but this was not observed and a continuous line resulted. I t seems then, that the difference between the coefficients of static and dynamic friction is negligible, and that the effect of changing porosity and frictional temperature rise a t the wall is small, a t least u p to a velocity of 4.0 in./minute. A second possible explanation for the low apparent coefficient of friction is that a lubricating film existed. Films were observed on the wall a t times,
= 1.57 inches; P, = 1 1,400
but care was taken to remove them with solvents and a t times by heating the wall to high temperatures. By heating it was always possible to reproduce data obtained on a freshly polished surface. There is still the possibility that the supposedly unlubricated polymer did contain small amounts of lubricant, but as will be shown subsequently, small amounts of lubricant added to the surface of the polymer decreased the transmitted pressure, giving an apparent increase in the coefficient of friction. However, it is not clear what the effect of small amounts of lubricants incorporated in the polymer might be. A third possibility is that the contact forces between particles are not the same in all directions. If the particles are entangled, so that the bed possesses structural rigidity, the force a t any point would be larger in the axial direction than in the direction normal to the wall. Under these conditions, normal forces on the wall would be lower than would be the case if the contact forces were all equal, as assumed in the derivation of Equation 1, and the apparent coefficient of friction would be less than the true value. If (Y is defined as the ratio of the normal force to the axial force on any particle in the bed and if it is assumed to be constant, then, VOL. 48,
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0
OCTOBER 1956
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value. The cause of this discrepancy between the two sets of data is not known.
Lubricated Particles
00
0.08 012 016 Weight % Lubricant
004
020
024
Figure 6. Effect of lubrication on apparent coefficient of fiiction; I = 1.57 inches; P, = 1 1,400 Ib./sq. inch
by referring to Spencer and coworkers (3), it is easy to show that Equation 1 is replaced by:
So that
c y = -
I*
P’
(3)
Figure 2 indicates that p decreases with increasing applied pressure, or in terms of Equation 3, a decreases from 0.76 to 0.60 if the true coefficient of friction is taken as 0.25. This might be explained as an increase in the structural rigidity of the bed as it is compacted and the particles are welded together. The assumptions made in the derivation of Equation 1 cannot be expected to hold when the size of the particles becomes large relative to the cylinder diameter. The decreasing apparent coefficient of friction with increasing particle size (Figure 4) could be due to a decrease in a, although it is likely that the other assumptions are also in error here. The deviation from Equation 1 a t small values of the bed length in Figure 3 is most probably due to the somewhat ambiguous nature of Lo, for the original length varied slightly with the method of charging the particles. Because of the scatter in their data, it is not possible to determine whether deviations from the functional form of Equation 1 appear in the results of Spencer and coworkers. However, their average apparent coefficient of friction is larger than found in this work and is very close to the directly measured 1828
One of the major objects of this work was to obtain information concerning the action of lubricants on the pressure transmitted through a particle bed. For the static case, Spencer and coworkers found that the addition of lubricating oil to the tube wall decreased the coefficient of friction to 0.08, and adding an unnamed solid lubricant to the surface of the particles caused an increase in the transmitted force approximately proportional to the weight per cent of lubricant added. No experimental details were given. Experimental Procedure. All of this work was done with zinc stearate because it is one of the more common lubricants used. In order to eliminate as many variables as possible, runs were made with a fixed particle size and initial length under varying conditions of lubrication. Values of the transmitted pressure for each run were taken from the measured curve at one value of the applied pressure. The value of the applied pressure used did not significantly affect the results, although it was considered preferable to work with the highest pressures where p ’ is relatively independent of pressure. Two types of particles were used. The first type was granules which passed a No. 12 screen and were retained on a No. 22 screen, averaging roughly 0.04 inch on a side; the second was ‘/*-inch pellets. The two types of particles had shown similar results when unlubricated. The initial bed length was 1.57 inches. I t was found that the results were strongly affected by the history of the cylinder wall as well as by the amount of lubricant used. The following technique gave reproducible results: Starting with a freshly cleaned cylinder wall, runs were repeated a t a fixed lubricant concentration until the transmitted pressure approached a constant value. Between each run the apparatus was dismantled, the polymer was removed, and the cylinder wall was wiped with a cloth to remove any loose material. At the end of each series of runs at a constant lubricant concentration, one or more runs were made with unlubricated polymer using the dismantling procedure given above, The wall was then thoroughly cleaned and another run was made with unlubricated polymer, the criterion for a clean wall being that the unlubricated polymer gives the same results as would be obtained on a freshly polished wall. The procedure was then repeated with another lubricant concentration. Results. In Figure 5 the ratio of the
INDUSTRIAL AND ENGINEERING CHEMISTRY
transmitted pressure to the applied pressure is plotted against the weight per cent of lubricant with the applied pressure constant a t 11,400 Ib./sq. in. Curves A and B pertain t o No. 22 granules and curves C and D are for I/g-inch pellets. Curves A and C are the first runs a t each concentration starting with a clean wall and curves B and D are the last runs. The corresponding apparent coefficients of friction, calculated from Equation 1, are presented in the same manner in Figure
6. The maximum value of the pressure ratio which was measured in these experiments was obtained by heavily dusting the cylinder wall with lubricant before making a run. The results for pellets were almost the same whether or not additional lubricant was added to the particles, and these results are represented by the straight line labeled “coated wall” in Figure 5. This line corresponds to a minimum apparent coefficient of friction of 0.047 and is also shown in Figure 6. Figure 5 shows the surprising fact that addition of small amounts of lubricant to the polymer decreases the transmitted force in all four cases. It can be seen that the pressure ratio for the initial granule runs, curve A , drops from 0.266 for no lubricant to 0.128 at O . O l ~ o lubricant and does not reach the unlubricated value again until a concentration of almost 0.270 is used. As runs were repeated at one concentration, the pressure ratio increased with the number of runs and eventually reached a maximum value which is given by curve B. As the lubricant concentration increases, the maximum value gets farther and farther from the initial value.
The results for pellets are qualitatively similar to those for granules. Curve C (Figure 5) for the initial points also shows a decrease in low concentrations which is not as pronounced as the decrease found with granules, although this could be caused by insufficient data a t low concentrations. After the initial dip, the curve rises rapidly and passes the original value at 0.02y0 lubricant. T h e pressure ratio for the final points, shown by curve D (Figure 5 ) , rises rapidly, although it still shows a tendency t o dip at low concentrations. Detailed results for the pellets are shown in Figure 7 where the pressure ratio is plotted against the number of runs starting from a clean wall, with weight per cent lubricant as a parameter. The corresponding apparent coefficients of friction are given in Figure 8. The maximum pressure ratio, obtained as mentioned above, by directly coating the cylinder wall with lubricant, is shown in Figure 7. The line is labeled “coated wall” as in Figure 5. The corresponding minimum apparent coefficient of friction of 0.047 is also shown in Figure 8. These lines are given for comparison. I t can be seen from Figure 7 that the curves for concentrations from 0.02% to 0.04% have not reached their maximum values and it is quite possible that the pressure ratio a t the latter concentration would have approached the value shown for a directly coated wall if enough runs had been made. The same results as those given for
0
2
4 6 8 NUMBER OF RUNS
IO
12
Figure 8. Effect of repeated runs -on apparent coefficient of friction for -various amounts of lubricant (weight per cent); L = 1.57 inches; Pa = 11,400 Ib./sq. inch; pellets, ‘/8 inch
No. 22 granules were obtained with granules with a wide range of particle sizes. I n all cases the addition of lubricant eliminated the tendency for the particles to weld together during a run. The results of the runs made without lubrication and without cleaning which followed the runs with lubricated particles, fell into two well-defined classes. If the pressure ratio obtained in the last of the lubricated runs was less than the pressure ratio corresponding to unlubricated particles on a freshly cleaned wall, then the ratio was equal to the latter value and was independent of the number of unlubricated runs. If the pressure ratio in the last lubricant run was greater than the unlubricated value with a clean wall then the ratio obtained was slightly less than the last lubricant run, and it dropped slowly with subsequent unlubricated runs. Discussion. Although more experimental work is needed for a complete explanation of the results, some tentative theories may be considered. In the first part of this article it was postulated that the low apparent coefficient of polystyrene on steel was caused by unequal contact forces in the bed. Since small amounts of lubricant eliminate any visible agglomeration or rigidity in the bed, it is reasonable to suppose that this addition tends to equalize the contact forces so that a increases as the lubricant concentration increases. This would give an apparent increase in the coefficient of friction and from curve A (Figure 6) it can be seen that the apparent coefficient of friction rises from 0.16 with no lubricant to almost 0.25 a t 0.01% lubricant. This is quite interesting since 0.248 is one value reported for the coefficient of friction for polystyrene on steel ( 3 ) . The close correspondence between the apparent and true coefficients indicates that a t O . O l ~ o lubricant, a equals 1, so that the contact forces in the bed of granules are equalized. Thus the small amount of lubricant present has a negligible effect on the coefficient of friction between the particles and the wall, for if either of the above two assumptions were incorrect, the apparent coefficient of friction would be expected to be less than the true value. (If the coefficient of friction of polystyrene on steel is taken as 0.3 instead of 0.25 then it is preferable to qualify the first assumption to that of partial equalization of contact forces.) The increase in pressure ratio, or decrease in apparent coefficient of friction, with the number of runs at one lubricant concentration is caused by deposition of lubricant on the cylinder wall. The small difference between curves A and B in Figures 5 and 6 , for lubricant concentrations of 0.1% or less, indicate that in this region the
amount of lubricant deposited on the wall is too small to change significantly the coefficient of friction between the particles and wall, a conclusion reached previously. As the lubricant concentration on the particles increases, the pressure ratio rises and the intial and final curves diverge. This effect, also caused by a build-up of lubricant on the wall, is much more marked with pellets than with granules; it is discussed subsequently. Another set of results explained by this picture was that obtained with unlubricated granules after the lubricated runs. When the lubricant concentration on the granules used immediately before the runs was 0.1% or less, the pressure ratio was independent of the previous history of the wall and was the same as that obtained with no lubricant and a clean wall. This is what would be expected, since the lubricant on the wall is negligible and a is a t its low, unlubricated value. With pellets as with granules, the rise in the normal force a t low lubricant concentrations causes an increase in the apparent coefficient of friction. With pellets, however, this rise is counteracted, even a t low lubricant concentrations, by a decrease in the true coefficient of friction caused by a lubricant film on the wall. The lubricant film is evidenced by the difference between curves C and D. I t does not take a great stretch of imagination to extrapolate curves C and D to a coefficient of friction of 0.25 a t zero lubricant concentration as shown in Figure 6, which leads to the conclusion that a has risen close to its maximum value of 1 in the region from 0 to 0.01% lubricant. When a series of runs is made at one lubricant concentration, the resulting curves presented in Figures 7 and 8 have no inflection point, for there is no change in the amount of lubricant in the bed; the only possible change is a t the wall where some type of lubricating film must build up. The presence of the film is readily observed when a run is made with unlubricated particles, for it is impossible to reproduce the data obtained with a clean wall until the lubricant is removed. All the curves in Figures 7 and 8 also can be extrapolated to a coefficient of about 0.25 at 0% lubricant, because the contact forces have already been equalized at the lowest lubricant concentration. These results show that the decrease of curve C (Figure 6) with increasing lubricant concentration is caused by the lubricant deposited during the initial runs, and the decrease in curve D is caused by the lubricant deposited by all of the preceding runs in the series. This explains why the apparent coefficient of friction for pellets does not VOL. 48,
NO. 10
OCTOBER 1956
1829
reach 0.25 as it does with granules; with pellets a significant amount of lubricant is deposited on the wall during the first run a t O.OlyGlubricant. The decreasing coefficients of friction are the cause of the rise in curves C and D (Figure 5). Curves A and B in Figure 5 also increase and diverge with increasing lubricant concentration for the same reasons. In a particle bed only a fraction of the area of the wall is in contact with the particles. During a run, each particle moves along the wall leaving a track behind it, and after a number of runs, it is likely that the entire wall has been in contact with a particle a t least once. The effect of stearic acid films o n the friction between two metal surfaces indicates that a monomolecular film reduces the friction as much as a thick film, but it is worn away more rapidly ( 7 ) . If we assume that polymer metal lubrication is similar to metal-metal lubrication, then from Figure 7 it is obvious that lubrication is not due simply to a deposition of lubricant on the wall; if this were the case the maximum values of P,/P, should be the same for all lubricant concentrations. The increase in the maximum values of the pressure ratio with lubricant concentration can be explained by considering the lubricant to be continuously deposited on the wall by polymer particles and a t the same time to be continuously worn off. A crude picture of the process can be obtained by assuming the rate of deposition and removal of lubricant to be first order. The amount of lubricant deposited on the wall per run is taken as proportional to the initial concentration of lubricant on the particles, and the amount removed is taken as proportional to the amount of lubricant left on the wall a t the end of the run. Thus, using units of mass per unit area,
which gives
and
The choice of variables is for convenience only, since the details of each individual run are being ignored. It is difficult to relate the above equations to measured results in a quantitative manner because the relationship between concentration and coefficient of friction is not known. However, the curves of Figure 7 can be empirically correlated quite well by Equation 5 if concentration is replaced by P,/P,. and kz is allowed to vary with initial lubricant concentration.
1 830
It would be expected that the results for pellets would not be the same as for granules because the projected surface area per unit mass of the granules is about twice that of the pellets. The area ratio is somewhat greater than this because of the roughness of the granule surfaces. Therefore, at an equal weight per cent lubricant, the lubricant concentration on an area basis is at least twice as great in pellets as in granules. However, the two sets of curves in Figure 5 differ by considerably more than a scale factor of 2. Visual observation in a glass tube indicated that in a bed of pellets, the wall is in contact over a large part of its area with the smooth pellet surfaces, while with granules, the contact area is much less, for contact only takes place at the peaks of the rough particle surfaces. This difference in area would cause the rate of deposition and removal of lubricant from the wall to be less for granules than for pellets, even at the same surface concentration of lubricant on both types of particles. If the rates of deposition and removal decreased in the same proportion, Equations 5 and 6 indicate that the maximum lubricant concentration would be unchanged, although it would take more runs to approach the maximum value. As mentioned before, the maximum lubrication with granules is much less than with pellets, evtn when the initial lubricant concentration is made the same on a n area basis for both types of particles. It appears, therefore, that with granules the rate of removal of the lubricant from the wall is relatively high, possibly because of the high pressures a t the small contact areas. As mentioned earlier, the pressure ratio obtained by directly lubricating the wall, shown in Figures 5 and 7 by the lines labeled “coated wall,” was almost the same with and without additional particle lubrication. The film buildup on the wall during runs with lubricated pellets behaved in a manner similar to that which was applied directly. I n both cases, when runs were made with unlubricated pellets without removing the film. the first run gave a pressure ratio slightly lower than the figure obtained in the last run of lubricated material. As the runs \\-ere continued, the pressure ratio decreased slowly, indicating that the film was being worn off. The closeness of the results with and Lvithout lubricaticn is inconsistent with the idea of variation in contact forces. W-hen a film is present on the wall, the unlubricated particles would be expected to give an even lower apparent coefficient of friction than the lubricated particles, since the value of o( is assumed lower with the unlubricated particles. For example. from Figure 6, the value of p for unlubricated pellets is 0.161
INDUSTRIAL A N D ENGINEERING CHEMISTRY
and with p’ taken as 0.25. a! is 0.644. Our earlier results indicated that with this figure for p ’ , the contact forces are equalized with lubricated pellets so that the apparent coefficient is equal to the true coefficient. The minimum apparent coefficient found with 0.047, lubricant was 0.055 and this is presumed to be the true coefficient. By means of Equation 2 it can then be seen that the pressure ratio with unlubricated pellets should be 18% higher than with pellets containing 0.0470 lubricant. The failure to find a significant difference between the lubricated and unlubricated particles when a film is present on the wall could be due to the removal of part of the film during the first unlubricated run. However, the subsequent unlubricated runs showed only a slight wear of the film so that this explanation would only be correct if the first run were much more effective in removing the lubricant than the following ones. This is thought to be rather unlikely and it is possible that the concept of a variation in contact forces is incorrect. Although this concept is quite useful in explaining a number of results. in the light of the above inconsistency more data are needed to decide on its validity.
Nomenclature Go = Original lubricant concentration on particles, mass/unit area.
C, = Maximum lubricant concentration deposited on wall a t any value of C,, mass/unit area. C, = Lubricant concentration on wall: mass/unit area. D = Cylinder diameter, inches. kl = Rate constant for deposition of 1 lubricant on wall, n’ ka = Rate constant for removal of 1 n’
lubricant from wall. -
Lo = Original bed length, inches. n = No. of runs starting with a clean wall. Pa = Average axial pressure applied to bed. lb. isq. in. P t = Average axial pressure transmitted through bed, lb./sq. in. p = Apparent coefficient of friction as defined by Equation 1. p’ = True coefficient of friction brtween polymer and wall. literature Cited (1) Bowden, F. P., Tabor, D., “The Friction and Lubrication of Solids,” p. 184, Oxford University Press, hTew York. 1950. \-,
-~~
( 3 ) Spence
RECEIVED for review October 24, 1955 ACCEPTED April 2, 1956
