Length to Diameter Ratio of Extrudates in Catalyst Technology II. Bending Strength versus Impulsive Forces Jean W. L. Beeckman, Natalie A. Fassbender, and Theodore E. Datz Process Technology Dept., Catalyst Technology Div., ExxonMobil Research and Engineering Company, Annandale, NJ 08801 DOI 10.1002/aic.15231 Published online April 2, 2016 in Wiley Online Library (wileyonlinelibrary.com)
This article describes the reduction of the length to diameter ratio of extrudates, by breakage on collision with a surface. The approach links the rupture force of the extrudate by bending to the impulsive force the extrudate experiences due to collision. The bending or flexural strength of the extrudate is described by the Euler-Bernoulli modulus of rupture. The impulsive force the extrudate experiences is described by Newton’s second law. We apply the force balance at the asymptotic length to diameter ratio which is reached after many repeated impacts. This balance yields a dimensionless group as the ratio of the rupture force by bending to the impulsive force by collision. The analysis shows that the asymptotic length to diameter ratio is directly proportional to the square root of this group. This dimensionless group also allows one to define a severity of the collision via C 2016 The Authors AIChE Journal published by Wiley Periodithe impact velocity and the time of contact of the collision. V cals, Inc. on behalf of American Institute of Chemical Engineers AIChE J, 62: 2658–2669, 2016 Keywords: length to diameter ratio, aspect ratio, impulsive force, modulus of rupture, flexural strength Introduction For a comprehensive overview of typical catalyst strength properties, their measurement and their importance in applications, we refer to Le Page1, Woodcock2, and Bertolacini3. The authors point out that the use of catalyst strength tests is commonplace in the catalyst industry and that a large heuristic database exists. The interpretation of the results is often ad hoc and few common rules of thumb apply. Wu4 and Li5 give valuable insight into the intricacies of catalyst strength measurement. Herein, especially the single pellet crush strength test and the bulk crush strength test are described. The latter is also established as the ASTM D7084-04 test. The single pellet crush strength is measured on individual extrudates and the appropriate average force that crushes the extrudate is reported. In the bulk crush strength test, a vertical shallow cylinder is filled with a bed of catalyst extrudates and stress is applied downwards up to a certain value. Thereafter, the catalyst is unloaded and the fines produced by breakage are sieved out and measured. This fines make at different pressures is then used as a measure of the strength of the catalyst extrudates. More recently also the bending strength of the catalyst has drawn attention and for this we refer to Li6 and Staub7. The bending strength or flexural strength of the catalyst is characterized by the Euler-Bernoulli modulus of rupture and is typically obtained in a 3-point bending test. Correspondence concerning this article should be addressed to J.W.L. Beeckman at
[email protected]. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. C 2016 The Authors AIChE Journal published by Wiley Periodicals, V
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For the severity of the catalyst handling we refer to Papadopoulos8 who describes impact breakage of particulate solids. The author describes attrition and comminution of particulate solids that are typically of a more or less spherical shape and hence are less applicable to extruded catalysts. Salman9 and Subero-Couroyer10 describe the impact of particulates against a flat surface and the breakage of these granules. Literature data on the quality of the extrudates on handling from the perspective of their ultimate length to diameter ratio however is scarce. The authors of the present manuscript acknowledge that in this work they only model, correlate, and discuss the number average length to diameter ratio of a catalyst sample and refer to it as the aspect ratio of the catalyst. The aspect ratio of a catalyst sample is the sum total of the length to diameter ratios of the individual catalyst extrudates divided by the total number of catalyst extrudates in the sample. The current effort is a first order approximation to the change of the aspect ratio with regards to breakage due to impulsive forces generated during repeated impacts against a surface. We designate the aspect ratio by the Greek letter F sometimes followed by a subscript to clarify a specific meaning. The development of the finite difference modeling of the drop test has been described in detail by Beeckman11. The authors show that two asymptotic aspect ratios exist for collision against a surface. The first asymptotic aspect ratio, called F1 is reached when the impact against the surface is repeated over and over again. The second asymptotic aspect ratio, called Fa is reached after just one drop when the starting aspect ratio is sufficiently large. The authors show that extrudates reduce in aspect ratio according to a second order break law. The authors give the incentive for such modeling studies, and how it can benefit the overall operation of catalyst manufacturing plants by controlling the mechanical severity of the different unit operations. In this follow-up manuscript, we attempt to relate
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quantitatively, the parameters F1 and Fa to independently measurable properties of the catalyst extrudates, and also of course with the severity of the impact. For clarification, the authors drop the bolded L notation for aspect ratio used in Beeckman11, in favor of the letter F to avoid confusion with the notation L for the geometric length of the extrudate used herein.
Experimental The modulus of rupture is measured in a 3-point bending test. The instrument used for this is from Shimpo Instruments; Model FGS-50 PVH; serial # 061BE006. The force gauge is from Mecmesin; Model AFG 50N; serial # 06-0246-06. The anvil speed is about 0.25 m/min and kept constant. We have found that for this rate of speed, the catalyst is typically in a strain rate sensitive region, but that as long as the rate is kept constant the rupture force is reproducible. The drop test entails dropping a small quantity of catalyst from a given height onto a hard empty surface and measuring the aspect ratio as a function of the number of drops. The aspect ratio is determined by scanning the sample of about 100–300 catalyst extrudates with an Epson Perfection V700 Photo instrument. Thereafter, we apply the Alias version 3–4 software of Cascade Data Systems to analyze the digitized picture, and determine the aspect ratio. We draw attention to the fact that this scanner and software determines the diameter based on a top down view of the extrudates as they are lying flat on the scanner in their natural position. In the industry, the diameter of the extrudate is sometimes measured by hand with a caliper, and for some cross sectional shapes these values may be different than those obtained by optical scanner.
Fj 5ðcF0 1jF1 Þ=ðc1jÞ;c5ðFa 2F1 Þ=ðF0 2F1 Þ;j50;1;2;... Or, written as a finite difference Riccati equation: Fj11 5 Fa Fj 2F21 = Fa 22F1 1Fj ; j50; 1; 2; . . .
Modulus of rupture
During catalyst manufacturing, catalyst extrudates experience breakage by a number of operations that handle it in various ways and with different severities. For instance, catalyst transport is often accompanied by drops from one piece of equipment onto another, hence can cause breakage. Catalyst sieving used to separate “overs” and “fines,” is another example where breakage can occur. Comminution aka sizing is intended to reduce the aspect ratio on purpose because of customer demands. During the transport, handling, and comminution, catalyst extrudates break and the aspect ratio shows a more or less stepwise reduction. In essence, all operations in a plant or pilot/laboratory have the potential to break catalyst extrudates, and thereby reduce the aspect ratio. Hence, a question naturally arises whether this aspect ratio reduction continues indefinitely, or, whether the aspect ratio reaches an asymptotic value. As the catalyst shortens, the force required to break the extrudate into two or more pieces increases and is typically hyperbolic in nature from a torque argument perspective. Since a catalyst operation has an inherent severity, it is expected that at some point it is balanced with a certain aspect ratio and hence causes no further reduction thereof. Therefore, the aspect ratio reaches an asymptotic value. We anticipate that the asymptotic aspect ratio F1 is directly linked with the operational severity and the catalyst flexural strength. The asymptotic aspect ratio F1 can be reached for any arbitrary starting aspect ratio (but of course larger than the asymptotic aspect ratio) by repeatedly running the material through the device or the test until no further reduction in aspect ratio is observed. August 2016 Vol. 62, No. 8
Beeckman11 has shown that also a second asymptotic aspect ratio exists. It is observed after just one drop, when the starting aspect ratio is sufficiently large. This asymptotic aspect ratio is called Fa and just as F1 it is a function of the strength of the catalyst and the severity of the operation. Both parameters F1 and Fa have a physical-mechanical meaning, and can be obtained by non-linear regression using the following model equations for the case of a series of identical impacts against a hard, empty surface:
Where, F0 is the initial aspect ratio and Fj is the aspect ratio after j drops.
Theoretical Asymptotic aspect ratio concept
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Figure 1. Three-point bending of a catalyst extrudate by an external force F.
The modulus of rupture is a measure of the inherent bending strength of a catalyst. It can be obtained from the rupture force required to break a catalyst extrudate in a bending test. The theory on the bending and rupture of beams was originally championed by Leonhard Euler and Daniel Bernoulli back in the 1750s. Figure 1 shows a picture of a catalyst extrudate with the forces at hand in a 3-point bending test. Catalysts are brittle materials and the deformation of the extrudate as shown in Figure 1 is of course highly exaggerated. Typically, a catalyst bends when a small force F is applied and when the force is released, the body returns to its original shape. When the force is increased, the catalyst extrudate bends more until a breaking point is reached. The force applied at that breaking point is called the rupture force Fr , and it is used to determine the modulus of rupture. It is important to note that there are two regions in the catalyst extrudate where different material strength properties are at work due to the bending stress. The force in the extrudate at the top is compressive and it acts against the compressive strength of the extrudate. The stress at the very bottom of the extrudate (i.e., the area furthest away of the centroid aka the location of the extreme fiber) is the area of maximum tensile stress and it acts against the tensile strength of the extreme fiber. At the breaking point, the tensile stress at the extreme fiber has reached the rupture stress. The forces along the cross section of the extrudate go from compressive at the top of the extrudate to tensile at the bottom of the extrudate and hence change sign and therefore there is also a line of zero stress (aka the neutral stress line) along which the extrudate experiences no stress. Typically, for catalyst
Published on behalf of the AIChE
DOI 10.1002/aic
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Table 1. Moment of Inertia, Neutral axis and Shape Factors of the Various Cross-Sections Considered in this Study. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
a u
p/4
c/D I/D s w
1 ffiffi p 11 3=2 pffiffi p14 3ffi pffiffiffiffi ð21 3Þ2 pffiffi 11 p3ffiffiffi ð312 3Þ
–
4
1/2 p/64 8/p 2
pffiffiffi 1=ð11 2Þ 161p pffiffi 2
0
