Ind. Eng. Chem. Res. 1999, 38, 1911-1916
1911
A Model for the Bulk Crushing Strength of Spherical Catalysts Yongdan Li,*,† Dongfang Wu,† Liu Chang,† Yahua Shi,‡ Dihua Wu,§ and Zhiping Fang§ Department of Catalysis Science and Technology and State Key Lab on C1 Chemical Technology, School of Chemical Engineering, Tianjin University, Tianjin 300072, China, Research Institute of Petroleum Processing, Beijing 100083, China, and SINOPEC Technology Company, China Petrochemical Corporation, A-6 Huixin Dongjie, Chaoyang District, Beijing 100029, China
Statistics and experiments reveal that the mechanical failure of a spherical catalyst under singleparticle crushing strength and bulk crushing strength tests are both brittle fracture and the fact that the single particle strength data follow a Weibull distribution. A model is proposed for the packing of spherical particles, which assumes tetrahedral contact and force transmission between the spheres. A deduction from this packing is that the broken percentage of spheres during bulk strength measurement follows the relationship of the Weibull equation. Experimental results confirm this conclusion, which relates the single particle property and the bulk behavior of the packed bed of catalytic materials. 1. Introduction The mechanical strength of solid catalyst is one of the most important factors for the reliable and efficient performance of a fixed-bed reactor.1,2 Its measurement was discussed in a small committee of AIChE in 19743-6 and was written as one of the most important characterization techniques in the textbooks of catalysis.7,8 Single-particle crushing strength (SPCS) and bulk crushing strength (BCS) of a small bed have been proposed as the two most important methods. The SPCS method has been accepted as the standard in many countries,9-11 while the BCS method and data are often found in the literature.7 Nevertheless, up to now, the scientific basis for both the SPCS and BCS measurements and the relationships between the SPCS and BCS data of one single sample are still not clear. A probabilistic method has been used in publications from this group in the description of the scattering property of the SPCS data of cylindrical tablets.12-17 It has been proposed that the solid catalysts are typical brittle materials and their strength failure is due to brittle fracture.12 The Weibull equation has been proved to be successful in the description of the scattering properties of the SPCS data of tablets.12-17 Solid catalysts have a wide spectrum of shapes. They are normally in the state of mixed oxides, though in less common cases a prereduced form is available, which is a mixed oxides supported highly dispersed metal; in these cases the matrix which offers the strength is mixed oxides. These are all brittle materials. Their behaviors before strength failure can be assumed to obey the laws of elasticity. Among their different shapes, spherical and cylindrical shapes are the ones with which the analytical solution of elastic equations is possible. The BCS data of solid catalysts have been recommended to be more reliable than those of SPCS. BCS has a closer meaning to packed beds and is supposed to be applicable to catalysts of all shapes.7 The literature * Corresponding author. E-mail:
[email protected]. † Tianjin University. ‡ Research Institute of Petroleum Processing. § China Petrochemical Corp.
presents BCS data as the curves of the percentage of fines and displacement versus the pressure applied.4,7 A major difficulty encountered in the modeling and correlation of the data to the particle properties exists mainly due to the unclearness of the packing and contact laws of various shaped particles in a packed bed. Recently, a number of papers studied the packing density and coordination number by computer simulation;18-22 some others presented idealized mechanical models of heaped granular materials.23,24 Kanda et al.25 investigated the energy consumption of compressive crushing of large particles to small ones. Ouwerkert26 simulated the bulk crushing strength of spherical particles by a distinct element method. However, there is still a lack of literature on the relationship between the percentage of failure and applied force and the dependence of BCS on the properties of the particles. In the present work, the relationship between BCS and SPCS of ideal shaped spherical catalysts is discussed. Simplified models for the packing and the correlation between BCS and SPCS are proposed. A quite good fit of the experimental data is obtained. 2. Theoretical Section 2.1. Weibull Statistics for SPCS of Spherical Catalysts. Because of the brittleness of catalytic materials and the tensile fracture nature of its strength failure,12 the SPCS data of catalysts scatter in rather large ranges. In our previous work, the Weibull distribution has been used in the calculation of the mechanical reliability of tablets. Here again, the SPCS data are described by the Weibull distribution27
F(σ) ) 1 - exp(-β′σm)
(1)
Hiramatsu and Oka28 gave an analytical solution of the elastic equations for the two-point pressing of an elastic sphere. They found that the tensile stress induced has a maximum value at around the axis passing the two contact points. They proposed that this stress leads to the fracture and has a relation to the
10.1021/ie980360j CCC: $18.00 © 1999 American Chemical Society Published on Web 03/30/1999
1912 Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999
Figure 1. Tetrahedral packing of spheres (top view).
Figure 2. Force transmission system of four spheres with three contacts.
force as follows:
F σ ) 2.8 πdp2
(2)
If the failure initiates on the surface, the tensile stress leading to fracture should be calculated by29,30
F σ ) 1.6 πdp2
(3)
It is reasonable to assume that the tensile stress leading to fracture is proportional to the total load for spheres with an identical diameter, no matter how the fracture initiates. Combining eqs 1-3, we get
F(F) ) 1 - exp(-βFm)
(4)
2.2. An Idealized Three-Dimensional Crushing Model. In the BCS test, particles of catalyst are randomly packed in a cylindrical container. An axial pressure P, applied on catalyst particles via a piston, is transmitted from one particle to another through contacts. The effect of the weight of the particles in this case is ignored. A three-dimensional ideal pack of equal spheres has been presented by Li and Bagster23 and can be shown as in Figure 1. It has tetrahedral packing, which can be approached by setting a distance between the particles within each layer of spheres of hexagonal dense packing. Set the distance between the surfaces of any two adjacent spheres in the same layer as 2δ; then each sphere receives three supports and three contacts on its top. In this case the coordination number of any interior sphere is 6. A tetrahedral force transmission system of the packing can be shown as in Figure 2. When a pressure P is applied on the top layer of the packing, each sphere of the top layer is subjected to force F1.
F1 )
(
)
x3 2δ Pdp2 1 + 2 dp
2
F1
x6
(
1-
)
2δ2 2δ dp2 dp
F(σ) ) 1 - exp(-β0σm0)
1/2
(6)
(7)
In eq 7, the tensile stress σ leading to fracture within each sphere is supposed to be proportional to F2, i.e.
σ ) φF2
(8)
Substituting eqs 5, 6, and 8 into 7, we obtain
F(P) ) 1 - exp(-BPM)
(9)
M ) m0
(10)
where
[
(5)
The force F1 from each of the three adjacent top layer spheres, in turn, applies force F2 at three contacts of a second layer sphere.
F2 )
For the interior sphere, the resultant force of three forces F2 equals force F1. Therefore, every layer of spheres has the same forced condition as the top layer ones; that is, in the tetrahedral packing, all of the interior spheres are each subjected to six contact forces given by eq 6. In the case of δ ) 0, the tetrahedral packing becomes dense rhombohedral packing with a coordination number of 12. Any interior sphere contacts with another six spheres in the same layer. The spheres inside are subject to six contact forces as in the above model and another six contact forces to the other spheres in the same layer, which results from the limitation effect of these spheres to the deformation of itself. However, the dense packing cannot be reached; the real packing of the spheres is between the model proposed above and the dense packing. It can be convincing to do the analysis on the cases of δ > 0. As mentioned by Ouwerkert,26 the force and stress relation for the SPCS needs to be generalized to the compression of a particle with a number of contacts. According to eqs 2 and 3, the stress may be expressed as proportional to F/dp2 for multipoint contacts. It is reasonable to assume that the crushing strength of spherical catalysts under six contact forces F2 also follows a Weibull distribution.
B ) β0 f
( )]
x2 δ φdp2 f 4 dp
( ) ( )( 2δ δ ) 1+ dp dp
2
1-
m0
(11)
)
2δ2 2δ dp2 dp
1/2
(12)
Equation 9 gives the relationship between the applied pressure and fracture probability of each sphere in the packing. Consequently, the broken percentage (i.e.,
Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 1913 Table 1. Geometric Size and Textural Properties of the Catalyst Samples
sample
diameter (mm)
skeletal density (g/cm3)
A B C
4.43 ( 0.14 6.02 ( 0.14 5.75 ( 0.17
2.594 2.149 2.041
particle density (g/cm3)
porosity (mL/g)
surface area (m2/g)
mean pore diameter (µm)
1.190 1.371 1.364
0.4548 0.2640 0.2434
168.3 24.35 25.69
0.0119 0.1971 0.1870
ratio of pore volume in a specific diameter range (%) >50 nm >20 nm 36.9 75.2 75.4
42.5 87.1 83.9
BCS Measurement. About 125 mL of spheres, selected beforehand to eliminate broken and defected ones, was charged. An increasing piston pressure was applied to the top of the packed bed by a hydraulic system oiled at a uniform rate. After pressure application, the bed was removed and the broken spheres including fines were chosen by hand. The spheres broken or cracked with visible defects were taken as failed ones. The weight percentage of broken spheres and the ultimate piston pressure were recorded. It should be mentioned that before all tests the samples were heated in air at 200 °C for 4 h. 3.4. Data Treating. SPCS. Equation 4 can be put into the form
(
ln ln
)
1 ) m ln F + ln β 1 - F(F)
(14)
The maximum load F can be measured; F(F) can be estimated by listing the SPCS data from the minimum to the maximum value, i.e.
F1 e F2 e ... e Fi e ... e Fn-1 e Fn Figure 3. Apparatus for the bulk crushing strength measurement.
Then the probability is estimated by31-33
relative amount of breakage) in the packing can be obtained
dm/m ) 1 - exp(-BPM)
(13)
where the broken percentage dm/m is the mass ratio of broken spheres. 3. Experimental Section 3.1. Samples Used. Three spherical samples were used in this work. One is γ-alumina (denoted as sample A); the other two are 3A zeolite with different physical properties (denoted as samples B and C). Their geometric size and texture properties are listed in Table 1. Their texture properties and the pore size distribution were obtained by an Autopore 9220 II porosimeter. 3.2. Apparatus. A scale permitting one to measure two decimals after millimeters was used to get the diameters of the spheres. SPCS measurements were carried out with a ZQJ-II strength tester made in Dalian, China. A balance used for the determination of the weight has one decimal of precision after grams. The BCS tests were performed using a typical device designed by Beaver.3,4 The schematic diagram of the apparatus is shown in Figure 3. The cylindrical sample container is 54.26 mm in diameter and about 100 mm in depth. The piston diameter is nearly equal to the inside diameter of the container. The piston and container are both made of stainless steel. 3.3. Experimental Method. SPCS Measurement. The particle was loaded diametrically until failure between the two anvils of the strength tester. The load at which fracture of the sample occurred was recorded.
Fi(F) )
i - 0.5 n
(15)
The Weibull parameters m and β can be obtained by linear least-squares regression between ln F and ln ln(1/(1 - F(F))). BCS. Equation 13 can be written as
1 (1 - dm/m ) ) M ln P + ln B
ln ln
(16)
The total pressure P and percentage broken dm/m can be measured. The parameters M and B can also be obtained by linear least-squares regression. 4. Results and Discussion 4.1. SPCS. In the SPCS test, brittle fracture happened suddenly and completely, with the sphere breaking into several fragments. In some cases, the fragments were nearly hemispherical, while fracturing to many segment-shaped pieces was more frequent. The surface of the section was not smooth but rather irregular and has the features of typical brittle fracture, similar to that of cylindrical tablets after the same measurement.12,17 Table 2 lists the results of the SPCS tests on the three samples. It can be seen that the SPCS data have a large scattering range, which is the inevitable result of the brittle fracture. For catalysts, they are porous and full of defects, dislocations, and discontinuations in the bulk phase; these are in the same range of size and nature as the microcracks defined by fracture mechanics, which are the origins of the stress concentration and lead to the scattering of the data.
1914 Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 Table 2. Results of the SPCS Tests of the Three Samples SPCS (N) sample
sample size
mean value
standard deviation
minimum value
maximum value
Weibull modulus
Weibull size parameter
A B C
53 88 86
56.2 36.3 43.9
19.6 11.1 17.9
22 11 10
99 66 90
3.02 3.56 2.87
3.83 × 10-6 1.81 × 10-6 1.40 × 10-5
Figure 4. Correlation of single-particle crushing strength with a Weibull distribution funciton: ], sample A; 4, sample B; O, sample C.
Figure 4 gives the quality of the fit of the SPCS data with eq 4. These data are fit quite well. This is an indirect evidence for the brittle fracture nature of the strength failure and provides a possibility of the calculation of the probability of strength failure under specific loading or stress conditions for these catalysts, which
in turn provides the possibility of reliability prediction. The description of catalyst strength failure by probability should be more advanced than the method described in the national standards.9-11 4.2. Bulk Crushing Strength. Many differently shaped fragments were found after the BCS test. Figure 5 shows photographs of several typical fragments. These cases can be divided into two categories. One is as illustrated by parts a-c; these have the same features as that after the SPCS test. The fragments are either approximately hemispherical (parts a and b), or segment-shaped (part c). The other one does not appear in the SPCS test. Part d shows that fracture happens at the same time on two sides of the sample, while in part e a local peeling feature is seen on a large part of the fractured sphere. These sections all present the typical features of brittle fracture. This is evidence for the tensile stress to be the reason of fracture. It should be mentioned that an amount of fines was also generated when the applied pressure was high. In Figure 6, the broken percentage in the packed bed is plotted as a function of piston pressure, where the curves are drawn by eq 13 and the points are the measured values. The parameters in the equation are
Figure 5. Photographs of the typical fragments after the BCS measurements.
Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 1915
Figure 6. Correlation of the percentage of particles broken after the BCS measurement with the total pressure applied: ], sample A; 4, sample B; O, sample C. Table 3. BCS Distribution Parameters of the Samples sample
M
B
A B C
3.02 3.18 2.80
0.0266 0.0918 0.119
given in Table 3. It is noticed that the measured and calculated values deviate when the pressure applied exceeds some critical value. In Figure 6, the broken percentages of samples B and C for point P ) 1.8 MPa are less than the calculated values. One explanation is that when the pressure is sufficiently high, a large amount of fines is generated and fills the voids between the particles, which scatters the contact force and reduces the fracture probability. Another less important reason might be that the wall effect for relatively large particles is enhanced when pressure exceeds a critical value. Therefore, it is likely that the breakage curve obeys the Weibull equation only in the low loading range. However, the model is of practical significance because the breakage nature and the broken percentage in the low loading range are the most interesting for the industrial practice. The amount of sample loaded has an effect on the experimental result because of the statistical nature of the fracture. In this work, we adopted the recommendation of Dart:3,5 the initial bed height L and the diameter D is in a ratio of about 1:1. The results show it is fairly good. The wall of the container also influences the geometrical arrangements of the particles in space. However, it can be neglected if the diameter of the particles is much smaller than the inside diameter of the container. The applicability of a Weibull equation to the description of the relationship between the percentage of broken particles in a packed bed and the total force applied along the axial direction of the bed is reasonable. The BCS data represent the failure of spheres in the packed bed as group behavior. This correlation provides a possibility for the reliability prediction of a fixed-bed converter. There are also possibilities of doing experiments at high pressure and temperature with the configuration of the present apparatus, and there is strong reasoning for the applicability of eq 13 to the data of high pressure and temperature cases. 4.3. Relationship between SPCS and BCS. From eq 10, M is predicted to be equal to m0, while m and m0 should be identical for the same sample because they relate only to the defect distribution. The results presented in Tables 2 and 3 show that these values are
not exactly the same; however, the sequence of M in Table 3 for different samples is the same as that of m in Table 2. This deviation may come from two origins. First, it can come from the statistical nature of the SPCS and BCS data and the comparatively small number of points measured in the experiments. Second, it is noticed that the deviation between the two values increases with an increase of the diameter of the spheres, i.e., the smaller D/dp is, the larger the deviation is. For the SPCS data, the Weibull modulus describes the scattering property of the data. The larger m is, the smaller the scattering range is. For BCS, the larger M is, the higher the slope of rising of the curve is. It seems very reasonable that the curves of SPCS and BCS should have some similarities, as the features of both have nearly the same physical meaning. Another deduction is that when m is same, a higher mean value of SPCS leads to a shift of the BCS curve to higher range. SPCS is simple and informative if a probabilistic method is introduced to the interpretation of the data, while the BCS method is closer to the meaning of the fixed bed. The two methods give an identical Weibull parameter for the same sample, in the case that the sample size is large enough; however, in the SPCS case the modulus has a meaning different from that of the parameter in BCS. BCS gives a more direct feeling of the strength reliability. For instance, in Figure 4, it is difficult to determine which of samples B and C has a higher strength, while in Figure 6, it is very clear. However, BCS measurement is much more complex than that of SPCS; it needs more complex equipment and patience. 5. Concluding Remarks The strength failure of a spherical catalyst under both SPCS and BCS testing conditions is the case of brittle fracture. As a consequence, the SPCS data scatter in a rather large range and can be fit quite well by the Weibull distribution. This provides a method for the calculation of the probability of strength failure under specific loading conditions. The real packing of spheres is between the one extreme as described by the model and the other which is the dense packing; i.e., the number of contact points is between 6 and 12. Under a multipoint contact situation, the origin of failure is still tensile stress brittle fracture. This is consistent with the elastic theory. A deduction is that the crushing strength of individual particles within the bed follows the Weibull distribution. This results directly in the conclusion that the percentage of breakage and the total pressure applied in BCS measurement can be related by a Weibull equation, which now does not represent a concept of data distribution but a definite law for the behavior of a packed bed. It is assumed that the Weibull modulus m of the SPCS data and the factor M in the BCS correlation have an identical value. This was found to be really true for one of the samples, while for the samples with larger diameters, the deviation between the two values increases with an increase of the sphere diameter. This indicates the size effect of the cylinder in the BCS equipment. The model presented here has several theoretical gaps. First, it needs a more clear description of the packing of the spheres in a bed. Second, it needs an
1916 Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999
exact solution of the elastic equations for the multipoint pressed configuration of a sphere. The experimental results confirm the final relationship between the percentage of failure and the total pressure applied and the deduced conclusions from the model. This model has a rather important practical significance, as it relates directly the percentage of failed particles in a bed under a specific loading condition, which has a close meaning as the reliability of a fixed-bed converter. The model also relates the property of a single particle to the bulk performance of catalytic materials. Acknowledgment The financial support for the catalyst mechanical strength issues from the NSF of China, Ministry of Education, and SINOPEC Technology Co. is gratefully acknowledged. Nomenclature B ) parameter in the model BCS ) bulk crushing strength D ) sample container diameter, mm dm/m ) percentage of broken particles, wt % dp ) spherical particle diameter, mm F ) total load at fracture in the SPCS test, N Fi ) individual crushing strength data of a sample F1 ) force applied on each sphere of the top layer, N F2 ) contact force between particles, N F(σ), F(F), F(P) ) fracture probability i ) sequential number of the SPCS data L ) initial bed height, mm m, m0 ) Weibull modulus M ) parameter in the model n ) number of SPCS data measured for a sample P ) axial piston pressure, MPa SPCS ) single-particle crushing strength Greek Letters σ ) maximum tensile stress leading to fracture, MPa β, β′, β0 ) Weibull size parameter φ ) constant in eq 8, mm-2 δ ) distance between the surfaces of any two adjacent spheres in the same layer, mm
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Received for review June 8, 1998 Revised manuscript received January 22, 1999 Accepted February 3, 1999 IE980360J
