DISSIPATION OF ENERGY IN SINGLE PARTICLE CRUSHING R I C H A R D A.
Z E L E N Y ' A N D
E D G A R L . P l R E T *
University of Minnesota, Minneapolis, Minn. Single particles o f gloss and quartz w e r e fractured within a small colorimeter by impact crushing with a dual pendulum
device.
written about the sample and calorimeter. referring to the sample and calorimeter W , = Qt
The energy input and heat p r o d u c e d w e r e colcu-
lated from hammer height and temperature measurements. The energy input t o a particle per unit area formed was constant ot 77,000 ergs per sq. cm. o v e r a 24-fold energy range f o r vorious sample geometries.
M o s t o f the energy
received by the sample was dissipated t o heat.
From z e r o to
o v e r 50% of the energy input t o the calorimeter was lost t o plastic deformation o f the hordened steel crusher surfaces. Nonequilibrium surface energies o f gloss and quartz w e r e found to be less than 5000 ergs per sq. cm.
HE ultimate distribution of the energy input required to Tfracture single particles of quartz and annealed glass was determined from work and calorimetric measurements. Cylinders. spheres, and rectangular prisms were crushed by impact \vithin a small calorimeter which also served as the crushing device. A dual pendulum apparatus was used to introduce and measure the work input to the calorimetercrusher and surface areas were determined by the B.E.T. gas adsorption technique. Of particular interest were the order of magnitude of the surface energies of glars and quartz as produced in a crushing process, the heat evolvled in the particle, and the energy loss to plastic deformation of the crushing device.
Energy Balances for Crushing Process
In the process considered. the initial and final temperatures of the particles and external pressures were the same Sound. electric, and other minor energy terms were considered negligible Then, bv the first law of thermodynamics. and referring to the samp!e ti',
=
Qa
+ F~ AA + ADs
(11
Since the fracture process occurs under nonequilibrium conditions, 8, is the average surface energy for the newly formed nonequilibrium surface Materials such as g l a s and quartz may be considered to be perfectly elastic (72) so that AD, equals zero and Equation 1 becomes : W8 = Q a
+ F~ AA
(2)
The surface energy, 8, and the heat, Qs. were calculated from experimental measurements by use of energy and heat balances 1 Present address, Michigan State University, East Lansing, hlich. 2 Present address. Scientific Attache, American Embassy, Paris, France.
- hAA
By the first law
+ asAA + A D ,
(3)
I n Equation 3, the work input W awas evaluated by an energy balance about the pendulum device. The heat: Qt: was determined from temperature measurements and specific heat data. An estimate of 600 ergs per sq. cm. was used for the heat of adsorption, h, and was based on Harkins' measurement of adsorption of water on glass (7). This approximation is sufficiently accurate since hAA was less than 1% PV,. Several initial areas and all final areas were measured by the BrunauerEmmett-Teller method using ethane as the adsorbed gas (2, 9 ) . The term AD,,, accounts for the change in internal energy of the calorimeter metal due to permanent plastic deformation, and the surface energy change of the metal due to the formation of pits and dents was considered negligible. Maximum values of the surface energy of glass were determined b y letting AD, equal zero in Equation 3. a s m S x A A= Wa -
(Qi
- hAA)
(4)
I n many experiments 8, maz, A A was within the experimental error of the heat and work terms. Therefore, only the maximum order of magnitude of 8, could be determined from Equation 4. Based on these results, a value of CS was chosen for the calculation of the work, W,, and the deformation energy, AD,. T h e heat removed from the sample, Qs, was computed from a heat balance about the sample and calorimeter-Le.
Q.
- hAA
= Qt
-
(5)
Qm
T h e heat produced within the calorimeter metal, Q,, due to elastic vibrations was estimated by calculations based on elasticity theory and from experiments in which the sample did not break on impact. Hence, Qs was calculated from Equation 5, the magnitude of 8, was determined from Equation 4, the work W , was then calculated from Equation 2: and the deformation energy loss, ADTn,was found from Equation 3. Samples
Borosilicate glass spheres and cylinders (fabricated by J. R. Kilburn Glass Co., Chartley, Mass.) were annealed by heating above 560" C. for 3 hours and cooled a t less than 33" C. per minute. Examination under polarized light showed the absence of internal strains. The cylinders, 0.499 + 0.001 inch long and 0.50 f 0.01 inch in diameter, had faces flat and parallel within +0.0005 inch, and were round with 1 0 . 0 0 5 inch. T h e spheres were 0.4993 i 0,0005inch in diameter and were spherical within 10.0001 inch. Natural quartz crystals in the form of rectangular parallelepipeds measuring 7 X 8 X 11 mm. were cut and polished by a lapidary. They were not annealed. Two were crushed simultaneously by placing one crystal above the other in the calorimeter. Figure 1 shows the samples and crushed product. VOL.
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Description of A p p a r a t u s
Figure 1.
Figure 2.
Samples and crushed praducfs
Calorimeter-Crusher. Two hardened tool steel conccntric cylinders separated by an annular rubber gasket served as the calorimeter, Figure 2. Thin hard end plates screwed to one end of each cylinder served as the crushing surfaces. The sample was inserted in the inner shell, using two small pieces of Tygon far centering and the calorimeter was assembled to make good contact with the sample. Two steel springs located outside the calorimeter held the shells together and were also used to attach the calorimeter to one of the pendulum hammers, Figure 3. Two copper-constantan thermocouples were imbedded in the metal of the shells, and a third inserted within the calarimeter. A Speedomax potentiometer connected to an amplifier recorded temperature changes on a 0.5 to 2.0' C. full scale chart. Dual Pendulum. Two 3-pound Star Zenith tool steel cylindrical hammers suspended by piano wire rolled on ball bearings against a 6-foot radius vertical semicircular masonite hoard after being simultaneously released by two electromagnets which could be set at any desired height, Figure 4. The rebound distance of the hammer was marked by a flow of sparks which punctured a horizontal coated strip of paper laid on the spark wire shown in Figure 4. A switch, tripped on impact of the hammers, started the spark coil. A third suspension wire leading from the middle of the hammer to the pendulum pivot served as the ground. Two hardened tool steel stops, screwed to one hammer face and slightly shorter than the calorimeter length when containing the unbroken sample, protected the calorimeter from damage following fracture of the particle. At the instant of impact, the calorimeter was automatically released from the hammer and swung free.
Calorimeter and 0.5-inch glass cylindei
Work l n p u l l o Calorimeter-Sample
Assembly,
W,
The energy balance for the pendulums is P.E. = W,
+ R f F + Vo f
V,
(6)
Permanent deformations of the hammers and stops were not observed. The terms in Equation 6 were evaluated as follows: ~. ....... . ..... . ..~
SPRING
Figure 3.
Calorimeter and pendulum h a m m e r
P.E. = The initial potential energy of the hammers was calculated from height and weight measurements. R = The rebound energy of one hammer was calculated from rebound distance, and the rebound of the second hammer was calculated from conservation of momentum. Vh = Vibration energy losses in the hammer due to impact within the calorimeter were approximated by two methods, An estimate of less than 4.5% P.E. was calculated for an elastic collision with the sample by applying conservation of momentum and using a coefficient of restitution of 0.98. Estimates of 8.3, 9.7, and 11.47, P.E. were obtained from experiments in which the sample did not break. Similar experiments in a drop weight device indicated the losses were smaller when sample fracture occurred. O n the above bases, a value o f 6 & 397, P.E. was used for Vh. Vs = When the calorimeter was removed, the rebound was from direct contact of the hammers with the sto s, and the rebound energy, R, was found to be 6& P.E. Thus V, was 40% P.E. and V,/R equals 2/3. This relation was used whenever the hammer struck the stops. The error in R V, was estimated at 10% R V,. = Friction energy losses of less than 1% P.E. were F determined from attenuation measurements of a free swinging hammer.
+
Figure 38
4. Dual
pendulum device
16EC PROCESS D E S I G N A N D DEVELOPMENT
+
The maximum error in work input. tl,, found from Equation 6 was only 3Yc when R C; was zero. However, the error CIA increased-e.g.. when R V, increased rapidly as R was 507, P.E.. the error in LITa \\as 177,.
+ +
Heat Removed from Calorimeter-Sample
X
+
INNER S H E L L OUTER S H E L L
1.5
Assembly, Q,
The calorimeter was calibrated by supply electrical energy from 5 to 20 seconds. in amounts about equal to the heat produced in a crushing experiment. to a coiled resistance wire inserted within a broken sample in the calorimeter. T h e ensuing temperature changes of the calorimeter suspended in air were measured antd are illustrated in Figure 5. It was assumed the temperatures of the sample and gasket were equal to the inner shell temptrature during the cooling portion of the temperature-time plot. and the temperature of each shell \cas uniform throughout the shell. From several calibration experiments. the specific heat of the steel of the calorimeter was calculated to be 0.111 calorie per gram ' C. This is in close agreement Lvith reported values of 0.116 to 0.109 a t 25' C. (3. 7 0 ) : thus supporting the calorimetric technique used in this Lvork. The error in total heat? Q t . is estimated to be 47, when Q t \cas greater than 1 calorie. Heat Produced within Calorimeter Metal, Q,
.in estimate of Qnr was obtained by setting QIrlequal to the calculated elastic energ'( stored in the end plates of the calorimeter immediately prior to fracture. For samples of uniform cross section. the elastic energy stored in the same cross section of the plates. as a perctntage of the total energy in the sample and plates, \cas calculated from the relation for static loadingi.e.
For a ?-inch long glass cylinder and 82-inch thick steel plates, 3,6Ycof the total energy \vas calculated as being stored in the plates. The total energy in the sample and plates approximately equals the pendulum energy before impact-i.e., P.E. - F . In four experiments in which the sample did not break: the total heat .removed, Qt, was 6.6, 6.6, 8.9, 8.9% P.E. - F and represents an upper limit for Q m since Q 1 = Qs Q?,( in this case. O n the above bases, the intermediate value of 6 i 3 7 , P.E. -. F was chosen to estimate Q m in all the experiments. I n the above analysis any heat resulting from plastic deformation of the metal was assumed small compared with the heat produced uithin the sample, Qs.
+
I
0.0-*
2
4
TIME
6
10
8
MINUTES
Figure 5. Temperature-time plot of the calorimeter and sample during cooling
specific heat, and heat of sublimation. 'Theoretical surface energies of other materials such as sodium chloride, silver oxide, and metals are of the same order of magnitude as that of quartz. .4surface energy for quartz of 106.000 ergs per sq. cm. was determined from hall-mill-calorimeter experiments (7.1). The neglect of the deformation energy change of the mill could account for the high surface energy value. I n drop iceight experiments. values of the work input per unit area of 100.000 ergs per sq. cm. \.\-eredetermined ( 6 , 9 ) . Since the drop Lveight crusher is similar in action to pendulum device reported here, most of the work input \vould appear as heat and only a small percentage as surface energy. Experimental surface energy applied to the nonequilibrium surface formed under rapid fracture conditions. Agreement between the theoretical values for equilibrium surfaces and experimental values show that the fracture conditions d o not affect the magnitude of the surface energy. Theoretical values for quartz are believed to be closer to the actual value of aT for glass than the experimental estimate of 2500 ergs per sq. cm. Therefore, based on the range of theoretical surface energies, a value of 1000 ergs per sq. cm. was chosen to calculate Ll', and ADm in all experiments.
Surface Energy of Glass and Quartz
XLlaximum values of the suiface energy of glass were determined from Equation 4 using the data from six experiments in Lvhich the error in 8, mnl L 4 was less than 8% 116 and 8 , lllRX was smaller than values from other experiments. T h e average value of a\ n,H* was 250Cl ergs per sq. cm., and the mean deviation was 2400 ergs per sq. cm. Thus 5000 ergs per sq. cm. would certainly represent a n upper limit of the surface energy,
aq. L*sing all the data on quartz. the average value of maL plus the mean deviation \vas 3700 ergs per sq. cm., indicating the low order of magnitude of the surface energy of quartz. These results are in accord with theoretical surface energies of 920, 510, 995, and 2300 ergs per sq. cm. (4, 5, 7 7 , 76) mhich were based on van der Lt'aal's equation. density, coefficient of expansion, breaking, strength. distance between planes,
Y
I
I CYLINDERS ON F L A T S
I
CYLINDERS SPHERES QUARTZ ON ROUNDS
Figure 6. Percentage of the work input to the calorimeter assembly appearing as deformation energy Vertical lines represent the maximum expected error
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Deformation Energy Change, AD,,,
The deformation energy change, ADm, was expressed as a percentage of the work input to the sample and calorimeter, and these percentages are shown in Figure 6 where the vertical lines indicate the estimated maximum errors. T h e maximum error in AD, depended largely upon errors in the measurement of the rebound energy which in turn depended upon the energy absorbed by the sample. Since the energy absorbed by each sample was a highly variable quantity which could not be controlled, the observed variations of the error in ADm were expected. Since spheres and cylinders crushed on the round surface cause nonuniform stresses across the sample-metal interface prior to fracture, it was expected that the per cent of W, appearing as deformation energy would be greater for these samples than for uniformly loaded cylinders, Figure 6. The spheres tended to dent the center of the calorimeter faces while the cylinders produced pit marks and scratches. I n multiple particle drop weight crushing, the sharp corners of the particles produced score marks on the crushing surface, and it was estimated that 50% of the energy input could have been consumed in forming the deformation (75). The consistent results obtained in multiple particle crushing (74:75) were expected since the deformation energy change represents a n average of the individual changes associated with particles near the surface.
Heat Produced within Sample, Q.
Figure 7 shows the linear relation found between the heat Qs and the new surface area AA for the various sample geometries tested. A 24-fold energy range was covered and the heatnew area proportionality constant was 76,000 ergs per sq. cm. Since the surface energy is approximately 1000 ergs per sq. cm., the heat Qs is approximately equal to the work Ws (see Equation 2) ; hence, Figure 7 also represents the work inputnew area relation. In calculating Qs, it was assumed that any heat due to plastic deformation of the metal surfaces was negligible. This seems to be in accordance with the fact that the points representing experiments in which large amounts of plastic deformation occurred also fit the correlation of Figure 7 .
Work Required for Fracture, W, The work input to the sample, W/s, was a variable quantity for a given group of samples of the same size and method of loading-e.g., glass cylinders crushed on the flat surface. These variations were probably caused by variations in orientation, size, and number of flaws or weak spots present in the samples. Furthermore, the samples were not identically loaded because of problems in centering the sample and in the alignment of the pendulums a t the time of impact. Other investigators using drop weight or slow compression methods (7,8) have, of course, encountered similar variations in the energy required for fracture or in the breaking strength of glass and quartz. The work per unit area, LVS;AA, was approximately constant a t 77,000 ergs per sq. cm. for all the samples tested. In these experiments, the rate of fracture was rapid and could not be controlled. Roesler (73) controlled the rate of fracture of glass sample by varying the force applied to a conical indenter which rested on the sample. H e calculated the work input per unit of geometric area and obtained values of 4000 ergs per sq. cm. to 40
I&EC
PROCESS DESIGN AND DEVELOPMENT
X
CYLINDERS ON ROUNDS
100
-0
H E A T WITHIN
S A M P L E , 0s
200 KG. CM.
-
Figure 7. Plot of heat produced in the sample vs. the new area formed
20,000 ergs per sq. cm., the large value corresponding to the more rapid rate of fracture. These values are approximately equivalent to 2000 and 10,000 ergs per sq. cm. of gas adsorption area [l sq. cm. of geometric area equals 2 sq. cm. of gas adsorption area ( 9 ) ] . Thus, it appears from these data and the above value of 77,000 ergs per sq. cm. that the net work input per unit area formed can be changed by altering the rate of fracture. And at small fracture rates, the work input is utilized more efficiently for forming surface than at larger fracture rates. Also, Roesler’s W,/AA value of 2000 ergs per sq. cm. represents an upper limit to the surface energy in his experiments. Nomenclature
AA = surface area change ADm = deformation energy change (defined as the change in the energy of the bulk material) of the calorimeter metal ADs = deformation energy change of the sample F = friction energy P.E. = initial hammer potential energy Qm = heat produced in the calorimeter metal (positive as defined) = heat produced in the sample (positive as defined) Qs = total heat removed from sample and calorimeter Qt (positive as defined) R = final hammer potential energy = energy loss in the hammers due to impact with the v h calorimeter = energy loss in the stops and hammer due to impact VS with the hammer = work input to the sample and calorimeter = work input to the sample = thickness of the calorimeter end plates = thickness of the sample = Young’s modulus of the steel end plates = Young’s modulus of the glass sample = average surface energ! per unit area of sample References
(1) Axelson, J. W., Piret, E. L., IND.ENG.CHEM. 42,665 (1950). (21 Brunauer. S.. Emmett, R. H., Teller, E., J . A m . Chem. Sod. ‘ ’60, 390 (1 938).’ (3) Chemical Engineers’ Handbook (J. H. Perry, editor), 3rd ed ., p. 221, McGraw-Hill, New York, 1950. (4) Edser, E., Brit. Assoc. Advance. Sci.,Rept. 4, 281 (1922). (5) Fahrenwald, A. W., Hammer, 0. IV., Lee, H. E., Staley, W. W., Trans. A m . Inst. Minin,
