TENSILE STRENGTH OF MOIST AGGLOMERATES WOLFGANG
PIETSCH’,
EGON
HOFFMAN,
AND
HANS
RUMPF
Institut fur Mechanische Verfahrenstechnik, Uniuersitat ( T H ) Karlsruhe, Karlsruhe, West Germany A general theory for the tensile strength of moist granular materials was derived fer the whole moisture range. For the “pendular state” the equations of Pietsch and Rumpf were used. For the “capillary state” the tensile strength was taken to be equal to the capillary pressure in the porous body; two equations were used, neither of which is proved correct. A dimensionless capillary pressure characteristic was incorporated in the general theory. In the transition range a linearity proposal by Rumpf was used. A new experimental method was developed which enabled the tensile strength of moist agglomerates to be measured over a wide range of moisture contents and particle sizes. The results showed good agreement with the general theory.
IN an aggregate, a liquid can
have three characteristic distributions which depend on the pore structure and the amount of liquid present. I n the “pendular state” only liquid bridges exist between the individual particles, whereas in the extreme case of the “capillary state” all capillaries in the aggregate are completely filled with the liquid which forms concave menisci a t the pore ends. I n a transition range both liquid bridges and capillaries filled with liquid are present. Different liquid distributions influence the strength of moist agglomerates in different ways. Therefore, the moisture content, f, the relation between the amounts of liquid and dry solid material in the pellet, does not necessarily characterize the strength properties of a moist agglomerate correctly. T o be useful, a characteristic must consider the porosity of the pellet because a t a constant moisture content but differing porosity, different liquid distributions result. The relation between the pore volume occupied by the liquid and the total pore volume of the pellet, the so-called percentage liquid saturation, S, is such a characteristic. I t can be written as:
Theory
Based on the earlier work of Rumpf (1961) the use of the following set of equations was suggested to describe the tensile strength of moist agglomerates (Pietsch, 1968) :
Fz=
9 8
-
-.1 - t
FH = FZb for 0 5 S < S” (2a)
€
F Z = F Z S + s - S “ ( F P K - FZS ) ~
1
-s*
for
S”< S < 100
(2b)
with
and
F Z = FPK
for S = 100
(24
FZ and F P K are a dimensionless tensile strength characteristic and a dimensionless capillary pressure characteristic, respectively.
FZ = u t * X / a
I n the course of a research program on the strength of agglomerates, the influence of this percentage liquid saturation on the tensile strength of agglomerates bound only by a liquid was determined. The investigation, which used a new experimental method, led to a complete description of the strength of moist agglomerates in the whole moisture range.
’ Present address, Hutt G.m.b.H., 7101 Schluchtern-Heilbronn, West Germany 58
I & E C PRODUCT RESEARCH AND DEVELOPMENT
(34
F H = FH(p, 6, a/x) is a complicated function which depends on the angle, p (Figure l ) , the contact angle, 6, and the length ratio, a / x . I t is calculated for liquid bridges a t the coordination points of monosized spheres using the toroid approximation. The mathematic and graphic representation of this and other important functions was published earlier (Pietsch and Rumpf, 1967). With the assumption that the tensile strength increases linearly from a point SI-the critical percentage liquid saturation where liquid bridges in the aggregate begin
9-1,. 11 ,_ 1
5 ,
.
I-3 13 ~-
- -- -cm Scale
Figure 1 . Liquid bridge between t w o spherical particles of equal size (model)
I-t
cy e-
~
X O
t
*f(Q
10
Figure 2. Cross section of pressing device used
to touch each other--to S = loo'%, Equations 2 represent the most general solution of the problem. The characteristics, FZ and F P K , can incorporate any equation for the tensile strength, u t , and the capillary pressure, p . . I t is unfortunate that an absolutely correct, experimentally proved equation for the capillary pressure within a porous body is not yet available. Using the calculated diameter of the sphere with the same specific surface (surface-equivalent-diameter) as characteristic particle diameter, x,, and a mean pore radius, a conventional equation for the capillary pressure, p , in granular materials reads (Rumpf, 1961):
p c = const.
5
1
Courtes!, .jatuR
could be proved experimentally, they were used for comparison with the experimental results. Another uncertainty of the theory is the definition of the critical percentage liquid saturation, S*. However, if a uniform distribution of the liquid a t all coordination points between monosized spherical particles is assumed and the rules of the spherical trigonometry are used, the point a t which the liquid bridges touch each other and up to which Equation 2a is valid can be roughly estimated. Since, furthermore, the relationship between the coordination number, k , and the porosity, E , is estimated by e = a / k (Pietsch and Rumpf, 19671, a critical percentage liquid saturation can be calculated (Pietsch and Rumpf, 1967):
(4)
For a fluid wetting the solid entirely (6 = 0") the contact angle function f ( 6 ) is by definition f ( 6 = 0') = 1. T h e constant in Equation 4 must be found experimentally. According to present knowledge it has values between 6 and 8. The capillary pressure in a liquid bridge between two spherical particles of equal size can be written formally as (Pietsch and Rumpf, 1967):
FPC = FPC(p, 6, a / x ) is again a complicated function of the angle, p (Figure l ) , the contact angle, 6, and the length ratio, a / x (Pietsch and Rumpf, 1967). I n spite of the use of the toroid approximation, this function could be proved experimentally for a single bridge stretched between two monosized spheres (Schubert, 1968). If the breakthrough pressure of the pores in an aggregate is considered compa.rable with the capillary pressure of the porous body, Equation 5 can be employed to estimate the capillary pressure in an aggregate too. Although neither equation for the capillary pressure in porous bodies
Methods and Materials
T o be able to change the percentage liquid saturation and the particle size in wide limits and to eliminate some disadvantages of the existing experimental method (Rumpf, 1961), above all the uncontrolled drying of the pellet during preparation of the tensile strength specimen, an entirely new method was developed. Lsing a powder moistened with a calculated amount of water to produce a certain percentage liquid saturation, ring-shaped pellets were prepared by compacting the material a t about 70 kg. per. sq. cm. Then, the agglomerate which was still within the inner part of the die could be torn apart, thus measuring the tensile strength. Figure 2 shows a cross section of the pressing device used. The actual die consists of parts 4 and 5 . Parts 8, 9, and 10 produce the piston which is loaded concentrically by a sphere, 12; part 11 serves as a guide during the early stages of compaction. With shell 1 and nut 2, the system could be tightened against the ring ball bearing, 13, and the counterstand, 3. Previous experiments showed that if a cylindrical pellet was used, the measured tensile strength was lower than expected. This was caused by inhomogeneities in the pellet, VOL. 8 NO. 1 M A R C H 1 9 6 9
59
Scale 1
1
-
w
1
5
-
Scale -
-
cm
10
Figure 3. Specimen ready for tensile test
a low frictional resistance between pellet and die during the tensile test, and an excentric load producing a bending stress. Therefore, an insertion consisting of parts 6, 7, and 8 was employed. Pin 7 in the now ring-shaped agglomerate, 18, caused an additional guidance and an increased frictional resistance between the pellet and the walls during the tensile test. Simultaneously the inhomogeneous core of the pellet was eliminated. After removing all auxiliary parts of the device, pellet 18, still within the inner shells 4 and 5, was tested using a specially designed attachment (Figure 3). The stress introduction in the pellet was affected by friction between the shells, the pin, and the pellet. The failure always occurred a t the joint of the two die parts and was planar. Test experiments showed that the friction of part 2 in 6 and the adhesion a t the end faces of both parts and between die parts 4 and 5 were constant and so small that they could be neglected. This method can be used for liquid saturations down t o 2%. Below this point no pellets could be formed. The investigation of pellets with a higher moisture content turned out to be much more difficult. At about 50% liquid saturation a limit was reached; the air enclosed in the aggregate either created pressure domains which damaged the structure during unloading or carried liquid out of the pellet while escaping during compaction. T o produce liquid saturations up to 80%, pellets with 10% liquid saturation were prepared to which a calculated amount of liquid was added after compaction. For this reason the pellet had to be kept under a load with an auxiliary device (14, 15, 16, and 17 in Figure 4). The liquid was added to the pellet by way of sieve plate 6a (Figure 4). After about 6 to 10 minutes the liquid was found to be uniformly distributed in the pellet. Even with this method it was not possible to get liquid saturations higher than 80%, which was again due to air enclosures. 60
1
'
I & E C PRODUCT RESEARCH A N D DEVELOPMENT
I
5
-
-
cm
-
70
Figure 4. Auxiliary device to keep pellet under load
Immediately after fracture a sample was taken for moisture determination. With the known dimensions of the pellet and the properties of the material the porosity and the percentage liquid saturation could be determined. To obtain the tensile strength, the measured tensile force was referred to the fracture area. Eight limestone fractions were available for the experiments. Figure 5 shows the particle size distribution curves. The size distribution of the coarse fractions, 1 to 4, was determined by test sieving and that of the fine ones by pipet sedimentation. Each fraction was analyzed repeatedly; the deviation was always less than 5%. Whereas the size distributions 4 to 8 did not change during compression, the three coarsest materials showed a certain comminution effect. The dashed curves in Figure 5 show the initial size distribution curves for comparison. The surface-equivalent-diameter, x,, which was considered to be representative for the estimation of the tensile strength of moist agglomerates (Rumpf and Turba, 1964), was calculated from the size distributions:
6
xo= __ 0,
with 0, = 6 r z -
AD X
(7)
The shape factor, r, was assumed to be equal to unity and zero cumulative percentage undersize was correlated with a particle size of 0.5 micron arbitrarily. The representative particle size, x,, varied between 4.1 and 220.9 microns. Distilled water was used as the binding medium in all experiments. Experimental Results
Of the large number of results, some typical examples are presented in Figures 6 to 9. Each point is an average of eight individual measurements. After a small correction to constant porosity the arithmetic mean value and the confidence interval were calculated using the so-called
1
5
10
50 100 Particle Size x CpmJ
500
Figure 5. Particle size distribution curves of limestone fractions used in experiments xo. Surface-equivalent-diameter calculated from size distributions
student’s or “t-distribution” and a statistical certainty of 95%. The confidencle intervals of the tensile strength, characterized by vertical lines in the diagrams, lay between 2.23 and 27.5%0, but the deviation of 71% of all results was below 1076, so that reproducibility was assured. Entire wettability of the solid by the liquid (6 = 0”) was assumed in all cases for the theoretical calculations. Figure 6 shows the tensile strength, u t , obtained with fraction 4 pellets, plotted against the percentage liquid saturation, S. I n the middle moisture range the pellets were prepared using both premoistened material (open circles) and liquid addit ion after compaction (solid circles). The improvement of the pellet quality can be seen. I n the “pendular state,” below about S = 30%, the theoretical tensile strength curves for different dimensionless length ratios a / x are plotted. They reproduce the tendency of the experimental results well. Since the theoretical curves had easily distinguishable shapes a t different a / x values, the comparison with the experimental results could be used to estimate a mean particle distance. I n the case under consideration the estimation yielded an a / x value of 0.021. I n the range above the critical percentage liquid saturation, S”,the results are well reproduced by a linear increase up to p c , which was calculated using Equation 4. With the three coarsest fractions, measured strengths in the “pendular state” were always too high. Figure 7 , for example, shows the results obtained with fraction 2 pellets. An explanatilon of the deviation can be deduced from the comminution effect mentioned earlier. I n the pendular state the binding occurs a t the coordination points. At these points breakage will also take place, producing an increase of the contact area and a smaller mean particle distance. Both factors influence the strength in the pendular state. At higher percentages liquid saturation the macroscopic pore structure becomes more important for the binding mechanism and the experimental results follow the theoretical expectation again. Since the porosity of pellets obtained at lower compaction pressures is so high that the aiready low tensile strength of the coarse fraction agglomerates is even lower, the compaction pressure was kept constant in all experiments in spite of the comminution effect. For the mean dimensionless
80 I
Fraction 4
70
;
x, = 70.9pm
-A 0
Without With
Liquid Addition
50
6
I
&=0.45
;
Q=0021 X
I
0
0
10
I,
I
20
30
I
I
I
40 50 60 70 80 90 1 Percentage Liquid Saturation SL%J
Figure 6. Tensile strength of fraction 4 pellets
‘
2& = 0 4 0
I
I
ax= 0 0 0 3
----
a Fraction 2 , x, = 185 4 pm ,
-0
10
20
30
40 50 60 70 80 90 100 Percentage Liquid Saturation S C%3
Figure 7. Tensile strength of fraction 2 pellets VOL. 8 NO. 1 M A R C H 1 9 6 9
61
100
A
I
c
-
Fraction 4 , x, =709 p m Fraction5 , x, =352 (8
0
x, =130
Fraction 7
1'1
~
--4 /' u
__
-4
E 60 0
s
Y
P
,
$40 b "
L
20
0
0
10
40 50 60 70 80 90 Percentage Liquid Saturation S
20
9 shows, for example, the results of fraction 4 and 7 pellets in a dimensionless representation. The length ratio, a / x , was estimated in the pendular state and the half centriangle of the liquid bridges was rather arbitrarily chosen to be p = 28". I t seems that R u m p f s (1961) linearity proposal correctly describes the trend of the tensile strength of moist agglomerates in the transition range. Consequently, Equations 2a through 2d can be considered general relationships for the tensile strength of moist granular materials. The experiments indicate, however, that the correct capillary pressure equation has not yet been found. A theoretical and experimental study to solve this problem is under way a t the Institute for Mechanical Process Engineering, University of Karlsruhe, West Germany.
30
100
Figure 8. Tensile strength of fraction 4, 5 , a n d 7 pellets plotted as suggested by Rumpf ( 1 961)
Nomenclature a = distance between two particles a t coordination
point
D = percentage passing a given mesh size
f = moisture content, dry basis
FH = FH@, 6, a / x ) = function in Equations 2a and 2c Fraction 7
,
FPC FPK FZ
xo = 13.0 p m
;
= FPC(p, 6, a / x ) = function in Equation 5 = p.. x 01 = capillary pressure characteristic = o t . x 01 = tensile strength characteristic
k = mean coordination number of a particle in an aggregate M = mass 0, = specific surface per unit volume of powder pc = capillary pressure r = shape factor in Equation 7 percentage liquid saturation volume (without index: over-all pellet volume) particle size X surface-equivalent-diameter X, median of a particle size interval X surface tension of liquid 0 1 = P = angle (Figure 1) 6 = contact angle € = porosity o f agglomerate 3.1416 l r = P = density, specific gravity a = strength of agglomerates
s= v =
0
10
I 20
I
30
I
I
I
40 50 60 70 Bo 90 D O Percentage Liquid Saturation S f% J
Figure 9. Tensile strength of fraction 4 and 7 pellets plotted in modified form of Equation 2
SUBSCRIPTS
length ratio, a i x , a very low value of 0.003 was chosen in this case. From the critical percentage liquid saturation straight lines to both the capillary pressure, p c , obtained conventionally according to Equation 4 and the capillary pressure characteristic FPK (Equation 5 ) were plotted. I n this case, both are in good agreement with the experimental results. With increasing fineness of the powder particles, the experimentally obtained tensile strength values deviate more and more from the curves calculated according to Equation 4. Figure 8 shows the diagram proposed by Rumpf (1961) with the results of fractions 4, 5, and 7 as well as some theoretical tensile strength curves. Whereas the tensile strength values of the fraction 4 pellets still fit the diagram very well (compare Figure 6), the results obtained with the finest fraction 7 differ in the transition range. If the capillary pressure characteristic F P K calculated according to Equation 5 was employed, the tensile strength of all pellets examined could be well approximated. Figure 62
I & E C PRODUCT RESEARCH A N D DEVELOPMENT
b = bridge, pendular state L = liquid S = solid t = tensile strength * = critical value Literature Cited
Pietsch, W., Nature 217, 736 (1968). Pietsch, W., Rumpf, H., Chem.-lng.-Tech. 39, 885 (1967). Rumpf, H., International Symposium on Agglomeration, Philadelphia, Pa., 1961, W. A. Knepper, ed., pp. 379418, Interscience, New YorkiLondon, 1961. Rumpf, H., Turba, E., Ber. Deut.-Keram. Ges. 41, 78 (1964). Schubert, H., Chem.-1ng.-Tech. 40, 745 (1968). RECEIVED for review January 12, 1968 ACCEPTED November 8, 1968 Research made possible through funds by the Deutsche Forschungsgemeinschaft, Bad Godesberg, West Germany.
