ABSTRACT. The material behavior of dry, moist and wet spherical elastic-plastic granules subjected to multiple stressing conditions have been experimentally ...
Material Behavior of Spherical Elastic-Plastic Granules under Diametrical Compression Alexander Russell1, Peter Müller1 and Jürgen Tomas1 1
Mechanical Process Engineering, Otto von Guericke University, D-39106 Magdeburg, Germany
ABSTRACT The material behavior of dry, moist and wet spherical elastic-plastic granules subjected to multiple stressing conditions have been experimentally studied using diametrical compression tests. The force-displacement curves were approximated using the Hertz model for non-linear non-adhesive elastic contact behavior, the StrongeAntonyuk model for non-linear non-adhesive elastic contact behavior with viscous effects and the Tomas-Antonyuk model for non-linear non-adhesive elastic-plastic contact behavior. The energy characteristics for single particle stressing by diametrical compression were analyzed and a corresponding energetic restitution coefficient was determined. Furthermore, the influences of particle size and moisture content on the material behavior and the energetic restitution coefficient have been discussed. 1.
INTRODUCTION
In general, particulate solids such as granules and agglomerates are stressed repeatedly by mechanical forces at random locations and intensities during almost every industrial mechanical operation such as processing, transportation, storage and handling. Several quality deteriorative processes advance during prolonged exposure to such forces. The nature, characteristics, intensities, velocities and frequencies of the stressing forces determine the rate and the magnitude of the deformation and the breakage behavior of the stressed material. Extensive theoretical, experimental and simulative efforts are necessary to realize a good understanding of the mechanisms and characteristics of the material behavior of particles under mechanical stressing conditions. Since several decades, quasi-static compression and dynamic impact have been the most popular single particle stressing techniques that have been used worldwide in the experimental approaches. In literature [1–5] one can find a lot of interesting research works striving to achieve an in-depth understanding of the material behavior of granular and particulate materials under different mechanical stressing conditions. In spite of such notable developments, there still exists the inability to characterize the material, comprehensively understand the interactions within the material in the micro level and above all manufacture designer materials with optimum properties for appropriate applications. Thus, the material behavior of granular and particulate materials continues to remain as an area requiring intense investigation. This work aims to accomplish a deeper understanding of the material behavior of spherical elastic-plastic granules under multiple stressing conditions as observed during fluidized bed granulation, drying, pneumatic conveying, mixing, etc., and strives to arrive at conclusive dependencies of granular compression behavior on particle size and moisture content. Further evaluations concerning contact deformation and energy characteristics are also presented in this paper. 2.
EXPERIMENTAL
2.1.
Material Sample
Industrially manufactured synthetic hygroscopic spherical elastic-plastic zeolite 4A granules were chosen as test samples (Figure 1). They are produced by Chemiewerk Bad Köstritz, Germany and are marketed under the commercial name ‘Köstrolith 4A’. Their chemical formula is Na2O·Al2O3·2SiO2·nH2O. They exhibit a uniform pore size of diameter 0.38 nm as well as possess a well-defined pore and channel system [6]. This well-defined pore and channel system give them the remarkable ability to selectively retain molecules of unique sizes and chemical properties within their porous network. For this reason, they are widely known as molecular sieves. They find their prime application in the drying of air, noble gases, aliphatic and aromatic liquids. Furthermore, they are used in the cleaning of technical gases and petrochemicals.
Figure 1 Digital image of the examined granules [7] In order to investigate the influence of particle size on the material behavior under quasi-static compression, three different size ranged granule samples were chosen. Due to the observed high value of sphericity (see Table 1), the granules have been assumed as perfectly homogeneous isotropic spheres and therefore the principles of continuum mechanics and contact mechanics for spheres have been comprehensively applied to arrive at the results described in this study. Similarly, to investigate the influence of moisture content, the granules were moistened using wet air in an air-conditioned climatic chamber at a temperature of 45 °C and a relative humidity of 95 % to achieve a certain moist condition (where the pore saturation degree S ranges between zero and unity) and by temporarily immersing the granules in a water bath to achieve a totally wet condition (where the pore saturation degree S equals unity).
Mean diameter d50 in mm 1.745 2.033 3.267 2.2.
Table 1 Physical properties of the examined granules Moisture content Pore saturation degree XW in kgH20 kgDS-1 S Sphericity Porosity ψ ε in % Wet air Water bath Wet air Water bath 0.980 0.984 0.984
50.8 50.1 53.6
0.30 0.30 0.30
0.45 0.46 0.49
0.67 0.65 0.62
1 1 1
Measurement Technique
A strength tester produced by Etewe Testing and Automation Systems, Germany was used to carry out the experiments described in this study. It is widely used by the departments of research and development in several chemical, pharmaceutical, food, ceramic and pigment industries to characterize granular products [7]. Particles of sizes ranging from 0.05 mm to 5 mm can be diametrically compressed with forces up to 2 kN at a wide range of loading velocities. Figure 2 depicts a schematic representation of the principle of quasi-static diametrical compression tests. Force levels with increasing intensities were chosen on the basis of their particle sizes and moisture contents such that the final force level was approximately equal to 90% of the average breakage force. The granules were subjected to diametrical loading and subsequent unloading at a unique contact position with a constant loading velocity of vL = 0.02 mm s-1. In order to achieve representative results, each of the granule samples were tested thirty times for each of the selected compressive force levels.
Figure 2 Principle of quasi-static diametrical compression tests [8]
3.
RESULTS AND DISCUSSION
3.1.
Force-Displacement Behavior
On diametrical loading, the granule initially undergoes elastic deformation until the yield point as described by the Hertzian theory. Hertz described the non-adhesive quasi-static collision between two perfectly elastic isotropic solids mutually exerting a certain finite pressure on each other by means of an elliptical distribution of the contact pressure within the contact area [9]. In case of quasi-static diametrical compression of a spherical granule between two rigid flat walls, the normal elastic non-adhesive contact force in analogy with the Hertzian theory is given by FN,el = 0.33 E (1- ν2)-1 d0.5 sN1.5
(1)
where FN,el is the normal elastic non-adhesive contact force, sN the normal displacement, E the modulus of elasticity of the spherical granule, d its diameter and ν its Poisson’s ratio. Figure 3 shows a good fit of the Hertz model for the observed force-displacement behavior within the yield limit.
Figure 3 Observed force-displacement behavior and the suitable approximations applied to it On achieving the yield criterion at the yield point, plastic deformation adds on to the existing elastic deformation constituting the elastic-plastic deformation. Tomas defined the dimensionless contact area coefficient κA for an adhesive contact of fine particle which analytically represents the ratio of the plastic deformation area A pl to the total deformation area Atotal including a certain constant normal elastic displacement according to [10] κA = 0.67 + 0.33 Apl Atotal-1 = 1 - 0.33 sN,y0.33 sN-0.33
(2)
where sN,y is the normal contact displacement at the yield point (Figure 3). In case of quasi-static diametrical compression of a spherical granule between two rigid flat walls, the normal elastic-plastic non-adhesive contact force in analogy with the Tomas-Antonyuk model is given by [8, 11, 12] FN,el-pl = 0.5 λel-pl E (1-ν2)-1 d0.5 sN,y0.5 (1 – 0.33 sN,y0.33 sN-0.33) sN
(3)
where FN,el-pl is the normal elastic-plastic non-adhesive contact force and λel-pl the elastic-plastic fit parameter. However, Eqn. (3) represents the resultant normal elastic-plastic non-adhesive contact force comprising of a certain linear and a slightly non-linear normal elastic-plastic contact. The respective linear and non-linear normal elasticplastic non-adhesive contact force contributions namely, FN,lin,el-pl and FN,non-lin,el-pl to the overall normal elasticplastic non-adhesive contact force is given by FN,lin,el-pl = 0.5 λel-pl E (1-ν2)-1 d0.5 sN,y0.5 sN
(4)
FN,non-lin,el-pl = - 0.17 λel-pl E (1-ν2)-1 d0.5 sN,y0.83 sN0.67
(5)
Following a slight pause on reaching the maximum normal displacement sN,max (Figure 3) at the maximum normal force, diametrical unloading at the same constant velocity of vL = 0.02 mm s-1 was carried out. A sudden deviation from the characteristic elastic recovery of the unloading curve was observed just before the end of its course resulting in a faulty value of sN,U which is the normal displacement at unloading corresponding to zero normal force (Figure 3). This observation is due to the adhesion occurring at the contacts of the granule and the piston plates of the strength tester. This limitation was overcome by approximating the unloading curve based on the StrongeAntonyuk model according to [12, 13] FN,U,non-lin,el = 0.33 E (1-ν2)-1 µD0.5 d0.5 (2 sN,max sN,y-1 - 1)0.25 (sN – sN,U)1.5
(6)
where FN,U,non-lin,el is the normal elastic non-adhesive unloading force and µD the coefficient that considers additional displacements due to viscous effects. 3.1.
Energetic Restitution Coefficient
The restitution coefficient e is a characteristic material parameter which is governed by several interdependent dissipative mechanisms [7]. It gives a vivid picture of the material behavior of a body at the collision event of a certain dynamic stressing condition. The restitution coefficient can be mathematically expressed as e =│vrebound││vimpact│-1
(7)
where vrebound and vimpact are the respective velocities of the body at the rebound and impact phases of a dynamic collision event. However, due to the dependency of the restitution coefficient on the energy characteristics occurring during the collision event, an energetic restitution coefficient eE is used in most cases to describe the material behavior during quasi-static collision events such as diametrical compression. The energetic restitution coefficient is given by [13] eE = Wnon-lin,el,R0.5 Wtotal,C-0.5
(8)
where Wnon-lin,el,R is the elastic energy released at restitution and W total,C the total energy absorbed at compression which can be determined according to Wtotal,C = Wnon-lin,el + Wlin,el-pl + Wnon-lin,el-pl
(9)
Wnon-lin,el = 0.4 FN,el sN,y
(10)
Wlin,el-pl = 0.5 FN,lin,el-pl (sN,max2 - sN,y2) sN,max-1
(11)
Wnon-lin,el-pl = 0.6 FN,non-lin,el-pl (sN,max - sN,y1.67 sN,max-0.67)
(12)
Wnon-lin,el,R = 0.4 FN,U,non-lin,el (sN,max - sN,U)
(13)
where Wnon-lin,el is the energy absorbed at the non-linear elastic contact, Wlin,el-pl the energy absorbed at the linear elastic-plastic contact and Wnon-lin,el-pl the energy absorbed at the non-linear elastic-plastic contact. No trending dependency of the energetic restitution coefficient is observed on the investigated particle size ranges (Figure 4). However, from the results furnished in this study it is seen that the energetic restitution coefficient remains approximately constant within the investigated range of particle sizes. With increasing moisture content the energetic restitution coefficient decreases significantly as shown in Figure 4. The deformation of pore geometry, swelling up of solid bridges, weakening of bridge bonds and many more such undesirable effects caused by the moisture present within the pores are responsible for this observation.
Figure 4 Energetic restitution coefficient expressed as a function of particle size and moisture content 4.
CONCLUSIONS
The material behavior of spherical elastic-plastic zeolite 4A granules of different particle sizes and moisture contents under compressive loading had been experimentally studied using single particle diametrical compression tests. Valuable information regarding their processing, transportation, storage and handling have been achieved. The experimental force-displacement behavior was approximated using the Hertz model for non-linear non-adhesive elastic contact behavior, the Stronge-Antonyuk model for non-linear non-adhesive elastic contact behavior with viscous effects and the Tomas-Antonyuk model for non-linear non-adhesive elastic-plastic contact behavior. The rate independent energy characteristics at compression and subsequent restitution was studied. An energetic restitution coefficient characterizing the material behavior under diametrical compression was determined. With increasing particle size, the energetic restitution coefficient approximately remains to be constant within the investigated range of the particle sizes used in this study. With increasing moisture content, the energetic restitution coefficient decreases significantly. This observed behavior is due to the absorption and release of inner stresses during wetting and subsequent drying processes where several localized dislocations, propagation of micro-cracks, deformation of pore network, swelling up of solid bridges, weakening of bridge bonds, etc., occur. As a consequence, the inelastic strain energy contributions to the total strain energy absorption during compression increases resulting in significant irreversible deformations and energy loss. ACKNOWLEDGEMENTS The authors express their sincere gratitude to the Ministry of Science and Economy, Saxony Anhalt, Germany, for the financial support of this study through the ‘Graduate Scholarship of the Land’. 5. Apl Atotal d d50 E e eE FN,el FN,el-pl
NOMENCLATURE : plastic deformation area [m2] : total deformation area [m2] : particle diameter [m] : mean particle diameter [mm] : modulus of elasticity [N m-2] : restitution coefficient : energetic restitution coefficient : normal non-adhesive elastic contact force [N] : normal non-adhesive elastic-plastic contact force [N]
FN,lin,el-pl FN,non-lin,el FN,non-lin,el-pl FN,U,non-lin,el S sN sN,max sN,U sN,y vimpact vrebound vL Wlin,el-pl Wnon-lin,el Wnon-lin,el-pl Wnon-lin,el,R Wtotal,C XW ε κA λel-pl μD ν ψ 6. [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13]
: normal loading force at non-adhesive linear elastic-plastic contact [N] : normal loading force at non-adhesive non-linear elastic contact [N] : normal loading force at non-adhesive non-linear elastic-plastic contact [N] : normal unloading force at non-adhesive non-linear elastic contact [N] : pore saturation degree : normal contact displacement [m] : maximum normal contact displacement [m] : normal contact displacement at unloading corresponding to zero unloading force [m] : normal contact displacement at yield point [m] : impact velocity [m s-1] : rebound velocity [m s-1] : loading velocity [mm s-1] : energy absorbed due to linear elastic-plastic contact [N m] : energy absorbed due to non-linear elastic contact [N m] : energy absorbed due to non-linear elastic-plastic contact [N m] : elastic energy restored at restitution [N m] : total energy absorbed at compression [N m] : moisture content [kgH2O kgDS-1] : porosity [%] : dimensionless contact area coefficient : elastic-plastic fit parameter : coefficient for viscous effects : Poisson’s ratio : sphericity
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