The tensile strength of agglomerates with localized bondings can be approximated by the following equation (Rumpf, 1962):
in which a, is the mean tensile strength per unit section area, e is the void fraction of the agglomerate, d is the diameter of the particles, and H is the tensile strength of a single bond. The equation was derived by considering a particle assembly of monosized spheres.
An approximate solution of equation 1 for agglomerates in the pendular state was given by Pietsch (1969):
The cohesive strength H of a pendular bonding depends on liquid surface tension y, volume of liquid in the bridge, diameter d of the solid particles and the distance between the particles. Pietsch showed that H can be approximated by 2yd when the particles are in close contact and when the contact angle of the liquid to the solid is zero. Equation 2 predicts that the tensile strength of an agglomerate in the pendular liquid state is constant and, hence, independent of the liquid saturation. The validity of the equation was demonstrated experimentally.
The tensile strength of agglomerates in the capillary state is controlled entirely by the pressure deficiency P in the liquid. P can be calculated from the Laplace equation for a circular capillary:
where r is the radius of the capillary and 6 the contact angle.
Rumpf (1962) showed that the radius of the capillary may be related to the properties of the particle assembly by a hydraulic radius derived from the specific surface of the particles and the porosity of the assembly. Thus, the tensile strength due to the maximum pressure deficiency in an agglomerate of uniform spheres is:
Equations 2 and 4 show that the tensile strength of agglomerates in the pendular state is about one-third of the maximum strength in the capillary state. The tensile strength of agglomerates in intermediate states is usually approximated by the value of equation 4 multiplied by the liquid saturation S. Hence, the following equation describes the tensile strength of the agglomerate in the funicular and capillary liquid states:
C is a constant that takes the value 6 when the particles are uniform spheres. For irregular sand particles, values of C between 6.5 and 8 have been reported (Capes, 1980).
Figure 2 shows the theoretical tensile strength in the funicular and capillary state of an agglomerate consisting of 20 /tm spheres wetted with a binder liquid with a surface tension of 68 mN m_1, e.g. an aqueous povidone (PVP) solution. The graph demonstrates that, for example, an agglomerate with 20% porosity and a liquid content close to saturation has tensile strength of about 8Ncm-2 which is close to 1 kg cm-2.
The equations for the liquid bonding strength apply to idealized agglomerates consisting of uniform spheres. When the particle assembly is polydisperse, an approximate value of the tensile strength, resulting from mobile liquid bondings, can be obtained by substituting the volume-surface diameter dvs of the particle system for d in the equations (Rumpf and Turba, 1964).
Equation 5 predicts that the tensile strength is proportional to porosity function (1 — e)/e. There is, however, experimental evidence for a larger variation of tensile strength with changes in porosity than that predicted by the equation. According to Cheng (1968), the major factors determining the tensile strength are the particle size distribution and the interparticle forces which are strongly dependent on the surface separation between
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