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On the regime boundaries of moisture in granular materials
Powder Technology, 66 (1991) 191-194
191
On the regime boundaries
of moisture
in granular
materials
C. L. Flemmer Department
of Mechanical
and Aerospace
Engineen’ng,
West Virginia University, Morgantown,
WV 26506
(U.S.A.)
(Received April 20, 1990; in revised form August 28, 1990)
Abstract
In modelling pellet strength, researchers have presented analyses for the three regimes of pellet strength (the pendular, funicular and capillary regimes) without defining quantitatively the boundaries of the regimes. The problem reduces to a determination of the upper limit of moisture content of a pellet in the pendular state. For a pellet comprised of randomly packed, equal-sized spherical particles, this upper limit of moisture content is computed at 13.6 vol.% corresponding to a saturation of 34.1 vol.%.
Introduction
and literature
review
In considering pellet strength, Newitt and ConwayJones [l] define three states of water content in a pellet, namely the pendular, funicular and capillary states. These states are described with reference to Fig. 1 as follows: The pendular state (Fig. 1A) refers to low moisture content in the granule, the water being held as discrete lenticular rings at the points of contact of the particles. At a higher moisture content, the rings coalesce forming a continuous network of liquid interspersed with air (Fig. 1B) and this is called the funicular state. Further increase in the water content leads progressively to the capillary state (Fig. 1C) in which all the pore spaces in the granule are completely filled. A further increase in moisture content would result in the liquid completely enveloping the solid and the concave liquid surfaces would be replaced by the convex surface of the liquid droplet. While the phenomenon of a totally saturated pellet is important A
B
’
A B L
C
Peidular State F~rmcular- State Capillary State
Fig. 1. Three states of water content for an assembly of spherical particles.
0032-5910/91/%3.50
during granule growth, it is not of practical importance for pellet strength. Newitt and Conway-Jones [l] presented the first analysis of pellet strength for a pellet in each of these three states but did not specify quantitatively the actual ranges of moisture content of each state. Following on from this work, Rumpf [2] provided theoretical derivations of pellet strength of pellets of mono-sized spherical particles in both the pendular and capillary states. Rumpf did not specify the moisture contents corresponding to the different states. However, he presents a table of values of S, the fraction of void filled with moisture, for a pendular state matrix and has a maximum value of S of 0.4. Other researchers in the field of pellet strength [3-71 present theories of pellet strength for specific moisture content regimes, but do not state the values of moisture content for which their analysis holds. While it is relatively easy to distinguish between these three states in a qualitative manner, a quantitative description of the states is not available. The capillary state is defined by a moisture content of 100%. The pendular state covers a range of moisture content from zero to some upper limit where the lenses of water at adjacent points of contact are on the point of coalescence. This upper limit of the pendular state corresponds to the lower limit of the funicular state, which then extends to a moisture content of just below 100%. Thus, in order to define the three states of a pellet we need only compute the upper limit of moisture content for the pendular state.
0 Elsevier Sequoia/Printed in The Netherlands
192 Assumptions
can be seen by inspection
(i) The pellet consists of randomly
packed equal-
sized spherical particles of radius r. (ii) The mean co-ordination number of a particle is 9.96. This was derived from the work of Mason and Clark [8]. (iii) On average, because the packing is isotropic, the points of contact of three neighbouring particles upon any reference particle are at the vertices of an equilateral triangle. (iv) The surface of the liquid lens between two particles has constant curvature (i.e., the lens shape is assumed to be toroidal). In reality, the lens surface is a catenoid rather than a toroid, but numerical analysis shows that the two shapes are almost identical for liquid lenses. (iv) There is perfect wetting, i.e., the contact angle between the liquid lens and the particle surface is zero.
Analysis
Figure 2 shows a lens of water between two equalsized spherical particles. The upper limit of the moisture content of the pendular regime corresponds to the maximum lens diameter d such that adjoining lenses do not coalesce. Consider a figure defined in three space by N points such that lines joining the points (nodes) define triangular faces which form the surface of the figure. In other words, consider a three-dimensional graph [9] where all nearest neighbour points are connected by edges and no edges exist between other points. By assumption (iii), all edges are of equal length. Physically, each edge of the graph is a line connecting points of contact of adjacent spheres upon a reference sphere. The equilateral triangles bounded by the edges approximate the surface of the reference sphere which they cover. As the number of points of contact increases, so the approximation improves. If there are N such nodes (or points of contact), they form a three-dimensional figure which has F faces, and the relationship between F and N lens
of water
F=4+(N-4)x2
to be (1)
The relationship expressed in eqn. (1) has been checked for ~
Equation (1) is derived for integral N and the figure of 15.92 represents an average of many integers. These average faces are equilateral triangles of side d. Thus, 9.96 nearest-neighbour spheres touch the reference sphere, covering its surface with 15.92 faces which are, on average, equilateral triangles of side d. The area A of such a triangle is
A4$ Setting 15.92 times this area equal to the surface area of the reference sphere, we find
Therefore, ,
(
\ liz
n-
d=4r15.926
)
Equation (2) defines the upper limit of the lens diameter d. This may be seen by noting that the limit of lens growth in the pendular regime is reached when two adjacent lenses just touch. This will occur (on average) when the lens diameter equals the spacing between the points of contact between neighbouring spheres, i.e., when the lens diameter equals the distance d. It should be noted that although d is a function of particle radius r, the moisture content and sat-
(4 Fig. 2. Computation
of maximum lens diameter
d.
Fig. 3. Relationship between nodes and faces. (a), 4-Point graph, N=4, F=4; (b), S-point graph, N=5, F=6.
193
uration apply to any particle size since all the volumes (solid, liquid and gas) in the unit cell are proportional to the cube of the radius (i.e., there is geometric similarity). A particle of unit radius was therefore selected for the computation of moisture content and saturation. For small particles, the effect of gravity is small in comparison with surface tension forces and is neglected. The application of geometric scaling requires that we ignore the effect of gravity for the unit radius sphere. For a sphere of unit radius, eqn. (2) yields d = 1.350 15
which, from eqn. (5), is
Equations (6) and (7) are solved, using eqn. (3) for V, and computing V, as the volume of a sphere of unit radius, to yield a moisture content of 13.6 vol.% and a saturation of 34.1 vol.%, corresponding to the upper limit of the pendular regime.
Conclusion
Referring to Fig. 2, the volume of liquid V in the half-lens between points A and B, based on assumption (iv), is V= 9.558 x 1O-2 Each spherical particle has 9.96 lenses associated with it, but each lens is shared by two particles. Therefore, a single reference particle has a volume of liquid V, associated with it, where
For a pellet comprised of randomly packed, equalsized, spherical particles, the upper limit of the pendular regime moisture content has been determined to be 13.6 vol.%. The boundaries of the three moisture content regimes can thus be stated to be: Pendular
regime: 0
Funicular
regime: 13.6 < M (%) < 100
Capillary regime: M (%) = 100 Consider a unit cell within the pellet consisting of a single solid particle, 9.96 half-lenses of water and the air associated with the arrangement. A randomly packed set of spherical particles of equal size has voidage E of 0.4 [lo]. Thus,
(4) Rearranging
v,=
d F
where V, is the volume of the solid spherical particle and V, is the volume of air in the unit cell. The moisture content M of a pellet is defined as
v
v,+v,+v,
Substituting
List of symbols
A
eqn. (4),
$(K-K)
M(%)=
The analysis is based on the assumption of perfect wetting (contact angle = O”). Contact angles greater than 0” would be associated with a higher moisture content boundary for the lower limit of the funicular state, since each lens would have a larger volume than its counterpart in the case of perfect wetting.
x 100
for VP from eqn. (5),
M N i V
v, v, K E
area of equilateral triangle of side d [L’] maximum lens diameter corresponding to upper limit of pendular regime, defined in Fig. 2 [L] number of faces percentage moisture content (by volume) number of nodes radius of particle [L] percentage saturation (by volume) volume of liquid in half-lens [L3] volume of air in unit cell [L3] volume of liquid in unit cell [L3] volume of solid particle [L3] fractional voidage of particulate
References
The saturation S of the pellet is defined as the volume per cent of the voids which is occupied by water, i.e.
s (%)=
&1
x 100 _e
D. M. Newitt and J. M. Conway-Jones, Trans. Inst. Chem. Eng., 36 (1958) 422. H. Rumpf, in W. A. Knepper (ed.), Aggbmerafion, Interscience Publishers, New York, 1962, pp. 379-418. E. Cohen and W. K. Ng, Symp. on Advances in Ektractive MetaNqy, London (1967) 127.
194 4 W. B. Haines, J. Agric. Sci., 17 (1927) 164. 5 P. M. Heertjes and N. W. F. Kossen, Chem. Eng. Sci., 20 (1955) 593. 6 S. N. Omenyi and C. E. Capes, Powder TechnoL, 33 (1982) 167. 7 H. Schubert, Powder Technol., II (1975) 107.
8 G. Mason and W. Clark, Nature, 207 (1965) 512. 9 D. E. Johnson and J. R. Johnson, Graph Theory with EngineeringApplications, Roland Press, New York, 1972. 10 J. M. Coulson and J. F. Richardson, Chemical Engineering, Vol. 2, Pergamon International Library of Science, Technology, Engineering and Social Studies, Oxford, 3rd edn. (SI Units), 1977, p. 3.
191
On the regime boundaries
of moisture
in granular
materials
C. L. Flemmer Department
of Mechanical
and Aerospace
Engineen’ng,
West Virginia University, Morgantown,
WV 26506
(U.S.A.)
(Received April 20, 1990; in revised form August 28, 1990)
Abstract
In modelling pellet strength, researchers have presented analyses for the three regimes of pellet strength (the pendular, funicular and capillary regimes) without defining quantitatively the boundaries of the regimes. The problem reduces to a determination of the upper limit of moisture content of a pellet in the pendular state. For a pellet comprised of randomly packed, equal-sized spherical particles, this upper limit of moisture content is computed at 13.6 vol.% corresponding to a saturation of 34.1 vol.%.
Introduction
and literature
review
In considering pellet strength, Newitt and ConwayJones [l] define three states of water content in a pellet, namely the pendular, funicular and capillary states. These states are described with reference to Fig. 1 as follows: The pendular state (Fig. 1A) refers to low moisture content in the granule, the water being held as discrete lenticular rings at the points of contact of the particles. At a higher moisture content, the rings coalesce forming a continuous network of liquid interspersed with air (Fig. 1B) and this is called the funicular state. Further increase in the water content leads progressively to the capillary state (Fig. 1C) in which all the pore spaces in the granule are completely filled. A further increase in moisture content would result in the liquid completely enveloping the solid and the concave liquid surfaces would be replaced by the convex surface of the liquid droplet. While the phenomenon of a totally saturated pellet is important A
B
’
A B L
C
Peidular State F~rmcular- State Capillary State
Fig. 1. Three states of water content for an assembly of spherical particles.
0032-5910/91/%3.50
during granule growth, it is not of practical importance for pellet strength. Newitt and Conway-Jones [l] presented the first analysis of pellet strength for a pellet in each of these three states but did not specify quantitatively the actual ranges of moisture content of each state. Following on from this work, Rumpf [2] provided theoretical derivations of pellet strength of pellets of mono-sized spherical particles in both the pendular and capillary states. Rumpf did not specify the moisture contents corresponding to the different states. However, he presents a table of values of S, the fraction of void filled with moisture, for a pendular state matrix and has a maximum value of S of 0.4. Other researchers in the field of pellet strength [3-71 present theories of pellet strength for specific moisture content regimes, but do not state the values of moisture content for which their analysis holds. While it is relatively easy to distinguish between these three states in a qualitative manner, a quantitative description of the states is not available. The capillary state is defined by a moisture content of 100%. The pendular state covers a range of moisture content from zero to some upper limit where the lenses of water at adjacent points of contact are on the point of coalescence. This upper limit of the pendular state corresponds to the lower limit of the funicular state, which then extends to a moisture content of just below 100%. Thus, in order to define the three states of a pellet we need only compute the upper limit of moisture content for the pendular state.
0 Elsevier Sequoia/Printed in The Netherlands
192 Assumptions
can be seen by inspection
(i) The pellet consists of randomly
packed equal-
sized spherical particles of radius r. (ii) The mean co-ordination number of a particle is 9.96. This was derived from the work of Mason and Clark [8]. (iii) On average, because the packing is isotropic, the points of contact of three neighbouring particles upon any reference particle are at the vertices of an equilateral triangle. (iv) The surface of the liquid lens between two particles has constant curvature (i.e., the lens shape is assumed to be toroidal). In reality, the lens surface is a catenoid rather than a toroid, but numerical analysis shows that the two shapes are almost identical for liquid lenses. (iv) There is perfect wetting, i.e., the contact angle between the liquid lens and the particle surface is zero.
Analysis
Figure 2 shows a lens of water between two equalsized spherical particles. The upper limit of the moisture content of the pendular regime corresponds to the maximum lens diameter d such that adjoining lenses do not coalesce. Consider a figure defined in three space by N points such that lines joining the points (nodes) define triangular faces which form the surface of the figure. In other words, consider a three-dimensional graph [9] where all nearest neighbour points are connected by edges and no edges exist between other points. By assumption (iii), all edges are of equal length. Physically, each edge of the graph is a line connecting points of contact of adjacent spheres upon a reference sphere. The equilateral triangles bounded by the edges approximate the surface of the reference sphere which they cover. As the number of points of contact increases, so the approximation improves. If there are N such nodes (or points of contact), they form a three-dimensional figure which has F faces, and the relationship between F and N lens
of water
F=4+(N-4)x2
to be (1)
The relationship expressed in eqn. (1) has been checked for ~
Equation (1) is derived for integral N and the figure of 15.92 represents an average of many integers. These average faces are equilateral triangles of side d. Thus, 9.96 nearest-neighbour spheres touch the reference sphere, covering its surface with 15.92 faces which are, on average, equilateral triangles of side d. The area A of such a triangle is
A4$ Setting 15.92 times this area equal to the surface area of the reference sphere, we find
Therefore, ,
(
\ liz
n-
d=4r15.926
)
Equation (2) defines the upper limit of the lens diameter d. This may be seen by noting that the limit of lens growth in the pendular regime is reached when two adjacent lenses just touch. This will occur (on average) when the lens diameter equals the spacing between the points of contact between neighbouring spheres, i.e., when the lens diameter equals the distance d. It should be noted that although d is a function of particle radius r, the moisture content and sat-
(4 Fig. 2. Computation
of maximum lens diameter
d.
Fig. 3. Relationship between nodes and faces. (a), 4-Point graph, N=4, F=4; (b), S-point graph, N=5, F=6.
193
uration apply to any particle size since all the volumes (solid, liquid and gas) in the unit cell are proportional to the cube of the radius (i.e., there is geometric similarity). A particle of unit radius was therefore selected for the computation of moisture content and saturation. For small particles, the effect of gravity is small in comparison with surface tension forces and is neglected. The application of geometric scaling requires that we ignore the effect of gravity for the unit radius sphere. For a sphere of unit radius, eqn. (2) yields d = 1.350 15
which, from eqn. (5), is
Equations (6) and (7) are solved, using eqn. (3) for V, and computing V, as the volume of a sphere of unit radius, to yield a moisture content of 13.6 vol.% and a saturation of 34.1 vol.%, corresponding to the upper limit of the pendular regime.
Conclusion
Referring to Fig. 2, the volume of liquid V in the half-lens between points A and B, based on assumption (iv), is V= 9.558 x 1O-2 Each spherical particle has 9.96 lenses associated with it, but each lens is shared by two particles. Therefore, a single reference particle has a volume of liquid V, associated with it, where
For a pellet comprised of randomly packed, equalsized, spherical particles, the upper limit of the pendular regime moisture content has been determined to be 13.6 vol.%. The boundaries of the three moisture content regimes can thus be stated to be: Pendular
regime: 0
Funicular
regime: 13.6 < M (%) < 100
Capillary regime: M (%) = 100 Consider a unit cell within the pellet consisting of a single solid particle, 9.96 half-lenses of water and the air associated with the arrangement. A randomly packed set of spherical particles of equal size has voidage E of 0.4 [lo]. Thus,
(4) Rearranging
v,=
d F
where V, is the volume of the solid spherical particle and V, is the volume of air in the unit cell. The moisture content M of a pellet is defined as
v
v,+v,+v,
Substituting
List of symbols
A
eqn. (4),
$(K-K)
M(%)=
The analysis is based on the assumption of perfect wetting (contact angle = O”). Contact angles greater than 0” would be associated with a higher moisture content boundary for the lower limit of the funicular state, since each lens would have a larger volume than its counterpart in the case of perfect wetting.
x 100
for VP from eqn. (5),
M N i V
v, v, K E
area of equilateral triangle of side d [L’] maximum lens diameter corresponding to upper limit of pendular regime, defined in Fig. 2 [L] number of faces percentage moisture content (by volume) number of nodes radius of particle [L] percentage saturation (by volume) volume of liquid in half-lens [L3] volume of air in unit cell [L3] volume of liquid in unit cell [L3] volume of solid particle [L3] fractional voidage of particulate
References
The saturation S of the pellet is defined as the volume per cent of the voids which is occupied by water, i.e.
s (%)=
&1
x 100 _e
D. M. Newitt and J. M. Conway-Jones, Trans. Inst. Chem. Eng., 36 (1958) 422. H. Rumpf, in W. A. Knepper (ed.), Aggbmerafion, Interscience Publishers, New York, 1962, pp. 379-418. E. Cohen and W. K. Ng, Symp. on Advances in Ektractive MetaNqy, London (1967) 127.
194 4 W. B. Haines, J. Agric. Sci., 17 (1927) 164. 5 P. M. Heertjes and N. W. F. Kossen, Chem. Eng. Sci., 20 (1955) 593. 6 S. N. Omenyi and C. E. Capes, Powder TechnoL, 33 (1982) 167. 7 H. Schubert, Powder Technol., II (1975) 107.
8 G. Mason and W. Clark, Nature, 207 (1965) 512. 9 D. E. Johnson and J. R. Johnson, Graph Theory with EngineeringApplications, Roland Press, New York, 1972. 10 J. M. Coulson and J. F. Richardson, Chemical Engineering, Vol. 2, Pergamon International Library of Science, Technology, Engineering and Social Studies, Oxford, 3rd edn. (SI Units), 1977, p. 3.