Influence of the face surface shape of the piston on the limit value of compaction stress in the process of dry ice agglomeration

This paper presents the results of research on improving the efficiency of the mechanical agglomeration through describing the influence of the shape of the piston pressure face on the limit force value. The focus of this study is the geometrical parameters which are describing the shape of the piston pressure face. Where these parameters influence on the value of the limit stress in the process of mechanical agglomeration of crystallized carbon dioxide. The first part of the paper proposes a model describing the influence of the geometric parameters of the piston pressure face on the value of axial force during the subsequent compression phase of the process. The research part of the paper presents the empirical research results to verify the proposed model. The results of the work will be used to determine the influence of the described geometrical parameters of the dry ice residue on the axial force value. The derived mathematical model will be used for defining the design requirements as the starting point for the design and building of dry ice compression and granulation machines.


Introduction
In modern economy, a very important part of the economical balance is the utilization of waste material created by the production processes. Such by-products often find many interested recipients [1]. Such materials can include crystallized carbon dioxide which is the waste product in the manufacturing of ammonia [2,3]. The material is compressed and delivered to interested recipients in liquid form. As a result of sudden expansion of the liquid carbon dioxide, it crystallizes [4]. The end product of this process is available in broken down form and exhibits peculiar properties due to its low temperature of approx. -78.5°C and the ability of sublimation in normal conditions [5][6][7][8][9]. The indicated properties make for a broad industrial application for such material, e.g. in refrigeration processes, when transporting thermolabile materials, disinfection and cleaning of surfaces [8][9][10][11][12][13][14][15][16][17][18].
The efficiency of refrigeration processes utilizing dry ice depends on sublimation time [2,8,11]. Therefore, the broken down material is agglomerated under elevated pressure which causes a decrease in the area of the phase transition, consequently increasing the indicated time of transition. Commercially available machines include equipment for agglomeration of dry ice. Literature research [19,20] indicates that in most cases the piston-based method is utilized for this purpose. The method is carried out using special working units fitted with single-channel and multi-channel dies (Fig. 1). However, regardless of the type of die employed, the process remains the same from the standpoint of its general description. The agglomeration of dry ice is achieved as a result of displacement of the compacting piston (1) inside the compacting chamber (2). The process was divided into 3 sequentially recurring stages. During the first stage, the material is compacted until the moment in which the force applied to the piston FT is balanced by the resistance force FOP. In the subsequent step, the material is pressed through the channels forming the multi-channel die (4). After the piston achieves the boundary position, it is retracted to the initial position and the system is ready to begin another cycle. The change of force value on the piston as a function of its displacement was described in subject literature for an example multichannel die [1, 11, 19,].
The employed technique is classified as one of the binder-free methods of agglomeration, characterized by a high degree of energy consumption [19,21]. Therefore, it is recognized that works undertaken to determine the influence of the geometric parameters of the working system components of the machines employed to this end and their influence on the energy required to carry out the process are sound. Such assertion is justified in subject literature, which indicates that the yield stress value of the process is directly related to the geometrical parameters of the working system components [22,23]. Piston-type extrusion assembly 1 -piston, 2 -extrusion tube, 3 -multi-channel die, 4 -spacing ring, 5 -dry ice snow, 6 -compressed dry ice snow [1] In available subject literature it was pointed out that the process characteristics of dry ice agglomeration is progressive (Fig. 2). Based on the above, it was determined that the agglomeration process is similar to mineral materials such as e.g. salt [19,21,24]. The information provided in available subject literature indicate that contribution of tensile stress is omitted for the compaction process [5]. This served to simplify the formulated algebraic models by not accounting for friction between the agglomerated material and the pressure face surface of the piston. Additionally, one needs to consider the negligible factor of external friction, not exceeding 0.02 [19].

Fig. 2.
Dry ice compaction characteristics in the working system with a cylindrical chamber [19,24] The present paper describes the results of study aiming to describe the characteristic of change in the value of yield stress of the agglomeration of crystallized carbon dioxide as a function of geometric parameters describing the shape of the pressure face of the compaction piston. This stems from the recognized connection between the value of compressing stress σZ and the surface area of the pressure face of the piston ST, which, after taking into account the above described simplification, can be described as below.

=
(1) Whereas the surface area of the pressure face of the piston ST can be described in the cylindrical system using the following formula where f(r) stands for the two-side limited function describing the path curve of the surface ST around the axis z (Fig. 3) of the piston. The limit of the path curve is related to piston diameter (DT = 2RT), which is approximately equal to the diameter of the compaction chamber DK. No information was found in available subject literature regarding the analysis of the influence of geometric parameters of the piston pressure face on the value of compaction yield stress σZ in the process of agglomeration of dry ice. However, based on the study of the available subject literature it was noticed that the geometric parameters of the indicated surface may affect the limit value of force FT necessary to carry out the agglomeration process. The following chapters of this article present the results of empirical and numerical study carried out to formulate the relationship describing the change of force limit value FT as a function of the surface area of the pressure face ST.
During the review of the available literature, it was noticed that published results of FEM analysis of various processes are often verified or performed on the basis of empirical research [25][26][27]. In this paper has decided to use a similar approach to numerical analysis.

Methodology of Research
In order to determine the characteristics of change of the required force value on the piston FT, as a function of the piston pressure face area ST, a study was carried out according to the methodology described in subject literature [11,19,28,29]. To this end, a durometer by MTS model Insight 50 kN was employed to register the force value and displacement of the measuring head with frequency 10 Hz.
To carry out the study, a special compaction head (Fig. 4) was used. In order to minimize the measuring error resulting from the possibility of off-center mounting of the compaction head, the durometer jaws were fitted with a jig to ensure the rightangle arrangement of the compacting force and the sample cross-section (5).
Before carrying out the test in the piston system (Fig. 5a), a special end with set shape of pressure face was mounted (Fig. 5b, 5c). In the case of piston end type 1, the pressure surface was flat and its area can be described with the following formula Whereas for the piston end of the second type, the pressure face was spherical with height h = 5 mm and radius RS = 25 mm. The indicated surface can be described with the following formula, At the beginning of the study, the compaction chamber ( Fig. 4 item 1) was filled with broken down dry ice with weight approx. 41.5 g. Next, the piston (3) was introduced into the compacting chamber (2). At the final stage, the head was fastened to the durometer jaws (6). After taring the machine, the experiment was carried out in which the piston system (3) was moved downwards at a constant velocity of 5 mm/s. After the force FT has reached the equal value to the force of resistance depending on the utilized multi-channel die, (4), the material was pressed through its forming dies. The examination was carried out in 10 repetitions, the results were averaged and presented as the compaction characteristics describing the change in force at piston FT as a function of relative strain of the agglomerated material along axis Z (Fig. 6). Additionally, during testing, measurement was taken for the weight of the tested material before m0 and after m1 the examination. Results are presented in Table 1. The determined average value of compacting stress σZ for compacting the material using piston with pressure face type 1 is 11.83 MPa. Whereas for face type 2 it was 11.16 MPa.

FEM Analysis
Different shapes of the piston face do not exhibit a considerable difference in the compaction force necessary to carry out the compaction process. However, they might unequivocally affect the density distribution of the compacted material, in particular in the immediate vicinity of the piston pressure face. In order to examine this phenomenon, the process of compacting of crystallized carbon dioxide was modeled in the Abaqus software. The compacted material was modeled using Drucker-Prager/Cap solution with the following parameters: flexural modulus -3.2 GPa, Poisson's coefficient -0.46, cohesion coefficient -3.4 MPa, Drucker-Prager friction angle -24°, α = 0.01. Fig. 7 presents the graph of the change of plastic strain pb as a function on volumetric plastic strain εpl. The details regarding the employed parameters in the material model can be found in paper [20]. The MES analysis was carried out for four shapes of piston pressure face: flat, concave, dual-sin concave and quadro-sin concave. Their external diameter DT was equal to 36 mm. Fig. 8 shows the construction drawings of the pistons with examined shapes. Whereas Fig. 9 presents the results of the MES analysis which was to determine the distribution of density of the material during its compaction.

Conclusions
The results of the carried out empirical and analytical studies allow to formulate the following conclusions: 1. No dependency between the piston pressure face area and the value of force on the piston wad determined. The force value is associated with the boundary value of compacting stress. 2. Numerical analyses indicate a relationship between the distribution of density of the agglomerated deposit in the compaction chamber and the shape of the compaction piston pressure face. 3. Numerical analyses indicate, that the distribution of density of the agglomerated deposit in the course of the compaction process using the examined shapes of the piston pressure surface is the most even when the process is carried out with flat shaped piston face. 4. Regarding the agglomeration process of dry ice utilizing multi-channel dies, the material is subsequently compacted in the convergent section of the channels. Therefore, the shape of examined compaction piston face does not directly affect the final density of agglomerated dry ice. The performed examinations allow to suspect that the varied shape of the compaction piston may be relevant for the agglomeration process of dry ice utilizing single-channel dies. The conducted program of the study did not include the process of compaction utilizing such dies, therefore no results for the indicated case were presented in this paper.