# Modelling of the Process of Extrusion of Dry Ice through a Single-Hole Die Using the Smoothed Particle Hydrodynamics (SPH) Method

^{*}

## Abstract

**:**

_{2}. A ram-type extruder was considered in this analysis, in which dry ice was extruded through a single-hole die of varying geometry. The article presents the results of numerical analyses of the extrusion process, using a simulation method based on the Smoothed Particle Hydrodynamics (SPH) approach. The results from simulations were verified by the experimental data in terms of the maximum force required to complete the process, in order to assess the applicability of the proposed method in further research on dry ice compression.

## 1. Introduction

_{e}as a function of ram displacement s, subdivided into three main phases of the change of force applied to extrude dry ice pellets.

_{e}. This is because the displacement of ram reduces spaces between the particles. The exerted pressure does not induce elastic or plastic strains, due to ignorable contact between the particles. In Phase 2, the particles are packed closer together and the internal friction and elastic and plastic strain come into play as a result. The touching and relative displacement of particles increases the value of the exerted force F

_{e}, due to elastic and plastic strain, and internal friction. In addition, the compressed material interacts with the compression cavity walls, due to elastic strain. This results in friction between the compressed material and the compression cavity walls, increasing the compression resistance. The dissipation of energy during compression, in cylindrical chambers with tapered ends, has been extensively described in the literature [18,19].

_{e}increases until it equals the maximum resistance force of the process of compression F

_{L}. It has been demonstrated that the value of this force depends on the die parameters and the coefficient of friction between the extruded material and the side walls of the die [18]. The value of F

_{e}decreases as the process of extrusion proceeds and the density of the material ρ no longer increases as a result [20].

**Figure 2.**Phases of the compaction and extrusion process during one work cycle of the work system built for the purpose of this research: Phase 1—Initial compression, Phase 2—Final compression, Phase 3—Extrusion [21].

_{e}observed in Phase 3 marks the transition from compression to extrusion. The correlation coefficient defining the similarity of the F

_{e}vs. s curve, and a straight-line relationship is 0.9, which indicates a proportional decrease of force in relation to displacement. This is due to the linear change in the surface area of the side walls of the compression cavity (1 on Figure 1) that comes into contact with the material. This allows us to assume the constant values of E, ν, and μ, and use an elastic-plastic model to represent the process in numerical studies, as it has been done in previous studies reported in the literature [26,27,28].

_{j}—the mass of the j particle, ρ

_{j}—density of the j particle, and W

_{ij}—kernel function, depend on the smoothing length h

_{ij}and distance between particles r

_{ij}.

_{e}for different shapes of the extrusion die. The predicted values were compared with the experimental data to assess the accuracy of representation of the process by numerical methods, primarily SPH based. The primary goal was to develop a relatively simple technique for estimating the maximum resistance during compression of solid dry ice that could be used to maximise the process efficiency when designing new dies. To this end, it was required to find a numerical solution, for example with the application of SPH, to model the process of extrusion of dry ice without needing to use sophisticated material models, which require extensive testing to determine the material properties. With this approach it will be possible to expand the research also on other materials.

## 2. Materials and Methods

#### 2.1. Solid Carbon Dioxide

^{3}[30].

- −
- Young’s modulus E = 881 MPa;
- −
- Poisson’s ratio υ = 0.46;
- −
- Density ρ = 1625 kg/m
^{3}; - −
- Input yield stress σ
_{pl}= 3.5 MPa.

#### 2.2. Single-Hole Dies

- −
- Inlet diameter D = 28 mm;
- −
- Outlet diameter d = 16 mm;
- −
- Overall height H = 80 mm.

- −
- Conical-cylindrical (CS) with a truncated cone inlet section whose shape is defined by the wall taper angle α and conical section height h (Figure 3a);
- −
- Spherical-cylindrical (WK) with a convex frustum of a sphere defined by the sphere radius R
_{1}and height of the spherical part h_{1}(Figure 3b); - −
- Spherical-cylindrical (WP) with a concave frustum of a sphere defined by the sphere radius R
_{2}and height of the spherical part h_{2}(Figure 3c); - −
- Spherical-cylindrical (WKWP) with the inlet section made up of two spherical frustums arranged in a series, convex followed by concave. This shape is defined by the sphere radii R
_{1}and R_{2}and the heights of the spherical frustums h_{1}and h_{2}(Figure 3d).

- −
- for CS cavity:

- −
- for cavities WK and WP:

- −
- and for cavity WKWP, with an additional parameter H′ calculated as the sum of h
_{1}and h_{2}as follows:

_{1}and R

_{2}to the other geometric parameters of the cavity:

#### 2.3. Numerical Analysis

- −
- Reference point of the single-hole die (4), fixed on the die (1), and located in the bottom plane on the axis of symmetry;
- −
- Reference point of the ram (5), fixed on the ram (2), and located on its top surface and on the axis of symmetry.

- −
- Dynamic Explicit module analysis,
- −
- Duration of analysis: t = 16 s.
- −
- Mass Scaling feature was used to extend the step time to 0.0001 s. in order to improve the calculation efficiency.

- −
- The model of the die integrated with the feed barrel was treated as a non-deformable body with a pre-defined and invariable geometry;
- −
- The die was integrated with the cylindrical feed barrel of D = 28 mm in length and H
_{i}= 80 mm in height (Figure 5); - −
- R3D4 4-node, 3-D, quadrilateral, and infinitely rigid elements used were distributed symmetrically about the central axis of the symmetry of the die. The finite elements were distributed evenly throughout the die and had an averaged edge length of 2 mm.

- −
- Outside diameter of 27.5 mm and 2 mm thickness. This allowed the avoidance of friction between the ruled surface and the sides of the feed barrel. Furthermore, it was now possible to represent the process of extrusion with the ram moving inside a cylindrical chamber, on the specially built test bench.
- −
- FEM mesh of R3D4 4-node, 3-D, quadrilateral, and infinitely rigid elements of 1 mm averaged edge length, uniformly sized over the whole extrusion ram surface.

- −
- The compressed dry ice was modelled as a deformable elasto-plastic body;
- −
- The shape and size of this domain was assumed to be commensurate to the internal cavity of the integrated die (Figure 5). This shape was assumed to change in the process of extrusion, as is typical of a deformable body;
- −
- FEM mesh of C3D8R linear, 6-node, 3-D, reduced-integration elements. The FEM elements had 2 mm averaged edge length and were uniformly sized over the whole domain;
- −
- SPH based approach was applied to define the properties of the compressed dry ice to allow the use of the SPH method. This ensures undisturbed flow of the compressed material through the die, leaving out nodal interactions. Thus, the individual finite elements are modelled as non-interacting particles (Figure 7).

- −
- The die was modelled as a non-deformable body with a pre-determined, fixed geometry;
- −
- Full fixity boundary condition was assigned at the reference point (No. 4 in Figure 4) (constrained in all directions),
- −

- −
- Tangentially: μ = 0.1 friction, in the direction normal to the surface—no penetration;
- −
- With infinitely rigid walls of the integrated single-hole die (1) and ram (2) and “no penetration” boundary condition in place, the compressed dry ice (3) devoid of such conditions flew out through the opening in the bottom of the integrated die.

#### 2.4. Experimental Verification

- −
- After every three test measurements, the compression barrel and ram were cooled in a container with dry ice at 195 K in order to reduce sublimation, during the process of extrusion, which could increase due to the ambient temperature of the laboratory (ca. 293 K), which was higher than the temperature of dry ice (195 K);
- −
- Coaxial alignment of the ram, feed barrel, and compression cavity to ensure no contact between the ruled surface of the ram and the feed barrel opening. The ram diameter was 2% smaller than the feed barrel diameter to avoid the risk of the metallic surfaces coming in contact due to thermal expansion;
- −
- The die was filled up with dry ice pellets before the first extrusion cycle, and after each cooling of the compression assembly, in order to stabilise the distribution of stress (the value measured in the first cycle was always left out);
- −
- Constant speed of extrusion of v
_{e}= 5 mm/s.; - −
- 31 ± 1 g of dry ice snow was fed into the feed barrel each time;
- −
- Measurement of the applied force to an accuracy of 0.5% of the maximum measurement range of the test frame (0.5 accuracy class of the load cell fitted on the MTS test frame).

#### Statistical Analysis of the Test Data

## 3. Discussion

_{e}vs. displacement curves obtained with the above-mentioned test set-up are presented in Figure 11. Considering a relatively small scatter of data, these curves, based on their shape, can be sub-divided into phases, as was the case in previous studies (Figure 2), which attests to the accuracy of representation of the already used test method [21].

_{L}. The minimum value of p was 0.069 or more.

_{L}. This supports the hypothesis with statistically significant differences between the compared populations.

_{L_EMP}with the associated key statistical data.

_{e}vs. s

_{e}relationship. As it can be seen on the graph, two phases of the simulated process can be distinguished, which can be assigned to the relevant experimental data:

- −
- Phase 2, in which the value of F
_{e}decreases due to elastic strain of the material and its frictional interaction with the internal surfaces of the compression cavity. This increase continues up to the maximum compression force F_{L,}which marks the commencement of extrusion, - −
- Phase 3, in which the value F
_{e}decreases due to the plastic strain of the extruded material, whose value depends on the shape of the compression cavity.

_{L}is the most interesting parameter at this stage of research.

_{e}material force by the matrix, observed at the end of stage 2 and throughout Stage 3, while after exceeding the displacement s

_{e}about 30 mm, this oscillation significantly reduces its amplitude. This phenomenon is due to the stick-slip effect, which is associated with the elasto-plastic representation of the extruded material. In the area of pure occurrence of this phenomenon (approx. 8 mm < s

_{e}< approx. 30 mm), the extruded material does not extend from the matrix at total speed, equal to the speed of the piston movement v

_{e}. In this respect, s

_{e}displacement, due to the elastic deformation of its volume, the speed of the material flowing from the matrix is significantly lower. Considering the coincidence of this phenomenon, with friction of the material on the walls of the matrix and the internal friction of the deposit, are good conditions for the resulting effect of the stick-slip effect-i.e., periodic fluctuations in movement speed, as a result of the mutual impact of friction force and force responsible for pressing. The reduction of its intensity (decrease in the amplitude of the oscillation of force) is present after exceeding a certain displacement (s

_{e}= approx. 30 mm), when a portion of the compressed material (which is deformed in an elastic way), which is initially placed in front of the coincidence, begins to fill its volume. At the same time, the material originally present, leaves the matrix. Therefore, part of the material previously deformed in an elastic way (already in the range of plastic deformation) is less susceptible to this effect.

_{L_FEM}and the median of the experimental data F

_{L_EMP}. The results are compiled in Table 9 below.

_{L}, both F

_{L_FEM}(predicted) and F

_{L_EMP}(observed), and the relevant geometric parameters of the respective die shapes (Table 1, Table 2 and Table 3). A relatively high similarity of the obtained curves was observed in terms of monotonicity. Unfortunately, the values noticeably differ between the test points in all cases. This can be explained by a fixed value of yield stress σ

_{pl}used in all the numerical simulations, which was determined experimentally on compressed material. This implies inadequacy of the applied procedure as in the analysed system yield stress apparently depends on the type of die used in the process. However, this assumption appears correct as long as the above-mentioned value is taken as a process-specific parameter rather than an intrinsic property of the material.

_{L_FEM}and F

_{L_EMP}) mathematical functions were derived, describing the percent difference between the obtained values in relation to the observed extrusion resistance:

_{pl}used in the numerical elasto-plastic model from the following equation:

_{pl_COR}were obtained for the test points set on all the analysed dies (Figure 15). Relationships describing the variation of this parameter as a function of the key geometric parameters of the analysed dies, were derived in addition, considering the future application of this approach in optimisation of the geometric parameters of extrusion dies over the entire variation domain.

_{pl_COR}were used in the repeated numerical calculations performed with the use of the SPH method. Other settings remained unchanged. The results of the comparison between the corrected extrusion resistance force F

_{L_FEM_COR}and the observed values of this force F

_{L_EMP}are shown in Figure 16, together with the percent difference between these two ΔF

_{L_COR}. As it can be seen, this treatment improved the accuracy of representation of the process by numerical analysis, with regards to the obtaining of the maximum extrusion resistance force. After correction of the initial yield stress the percent difference did not exceed 10%, and the curves of F

_{L_FEM_COR}and F

_{L_EMP}became more similar in terms of monotonicity (as compared to the curves obtained for the constant value of σ

_{pl}).

## 4. Conclusions

_{pl}(determined by testing the compressed material) failed to accurately predict the maximum extrusion resistance force for the different die parameters. To cope with this problem, yield stress was treated as a process specific variable, depending on the geometric parameters of the die rather than a constant property of the extruded material. The corrected yield stress σ

_{pl_COR}variable in the domain of geometric parameters of the die was used, and the corrected FEM model reduced the difference between the predicted and observed values to less than 10%, thus solving the problem. Being a gross simplification of the actual process of extrusion, with the above-mentioned level of error, the accuracy of the proposed elasto-plastic model with the use of SPH should be considered satisfactory. This is especially true when the focus is on the maximum extrusion force rather than on the simulation of the entire process, as it is the case in this research. In addition, the tests performed on dry ice snow do not yield stable results, mainly due to technical constraints, with sublimation of this material being the main challenge. This is evidenced by the problem with reducing the probability of error to an acceptable level when comparing the subsequent data populations. This problem observed in laboratory experiments also applies to dry ice production on an industrial scale, i.e., under generally less controlled conditions.

## Author Contributions

## Funding

_{2}to reduce consumption of electricity and raw material”, number: “LIDER/3/0006/L-11/19/NCBR/2020” financed by the National Centre for Research and Development in Poland, https://www.gov.pl/web/ncbr (accessed on 15 December 2021).

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Main parts of the ram-type extruder: 1—compression cavity, 2—ram, 3—die, 4—loose dry ice, 5—compressed dry ice [17].

**Figure 3.**Shapes of the dies under analysis: (

**a**) conical-cylindrical (CS), (

**b**) with a convex spherical compression section (WK), (

**c**) with a concave spherical compression section (WP), (

**d**) with a convex section followed by concave section (WKWP).

**Figure 4.**Discretization by finite elements of the model used in the numerical analysis of the process of extrusion: 1—single-hole die integrated with feed barrel, 2—ram, 3—solid, compressed carbon ice snow, 4—reference point on the die, 5—reference point on the ram.

**Figure 5.**Shapes of integrated single-hole dies (examples): (

**a**) CS incl. cylindrical section, (

**b**) WKWP incl. cylindrical section, (

**c**) WP incl. cylindrical section; 1—feed barrel area, 2—compression cavity area; D—feed barrel diameter, H—feed barrel height, d—outlet diameter commensurate to the diameter of the produced pellets.

**Figure 6.**Shape and distribution of finite elements using an example of the compressed material model: (

**a**) dimensions, (

**b**) overview, (

**c**) distribution of finite elements over a plane parallel to the axis of symmetry, (

**d**) distribution of finite elements over a plane perpendicular to the axis of symmetry; 1—feed barrel area, 2—single-hole die area.

**Figure 7.**Distribution of particles in SPH analysis of the dry ice extrusion process: 1—integrated single-hole die, 2—ram, 3—particles of the extruded material.

**Figure 8.**Schematic of the extrusion process during numerical analysis: 1—single-hole die integrated with feed barrel, 2—ram, 3—compressed dry ice snow, v

_{e}—extrusion speed, D—inlet diameter, d—outlet diameter, H—overall height of the die, H

_{i}—initial height of the extruded material.

**Figure 9.**Test set-up used in this research for experimental verification of the compression and extrusion forces for the tested dies: 1—MTS Insight 50 kN test frame, 2—load cell, 3—grips of the moving crosshead, 4—ram mounting plate, 5—linear guides, 6—compression ram, 7—compression barrel assembly, 8—barrel base plate, 9—bottom support.

**Figure 11.**Relationships of the resistance force F

_{e}vs. ram displacement s

_{e}during extrusion of dry ice snow through WKWP70-50 single-hole die, showing division into phases of the process (different colours means the following samples).

**Figure 12.**Box plots relating the median of the maximum observed extrusion force F

_{L_EMP}with statistical data obtained for the dies under analysis, i.e., CS (

**a**), WK (

**b**), WP (

**c**) and WKWP (

**d**).

**Figure 13.**An example of the F

_{e}vs. s

_{e}relationship during extrusion of dry ice snow through a conical-cylindrical die (CS10), with marked division into phases.

**Figure 14.**Comparison of the curves representing the relationship of the maximum force during dry ice extrusion and the adopted variable geometric parameters of the analysed single-hole dies for: maximum predicted force (F

_{L_FEM}), maximum observed force (F

_{L_EMP}) and their percent difference ΔF

_{L}for the die groups CS (

**a**), WK (

**b**), WP (

**c**) and WKWP (

**d**).

**Figure 15.**Values of the corrected yield stress of compressed dry ice snow σ

_{pl_COR}depending on the geometric parameters of single-hole dies, and their comparison with the constant σ

_{pl}for dies of groups CS (

**a**), WK (

**b**), WP (

**c**), and WKWP (

**d**).

**Figure 16.**Comparison of the curves representing the relationship of the maximum force during dry ice extrusion and the adopted variable geometric parameters of the analysed single-hole dies for: corrected maximum predicted force (F

_{L_FEM_COR}), calculated with the corrected yield stress σ

_{pl_COR}, maximum observed force (F

_{L_EMP}), and their percent difference ΔF

_{L_COR}for the die groups CS (

**a**), WK (

**b**), WP (

**c**), and WKWP (

**d**).

Designation | Taper Angle α (°) | Length of Conical Section h (mm) |
---|---|---|

CS50 | 5 | 68.58 |

CS75 | 7.5 | 45.57 |

CS100 | 10 | 34.03 |

CS150 | 15 | 22.39 |

Designation | Rounding Radius R_{1} (mm) | Length of Spherical Section h_{1} (mm) |
---|---|---|

WK20/WP20 | 36.3 | 20 |

WK45/WP45 | 171.8 | 45 |

WK70/WP70 | 411.3 | 70 |

Length of the Convex/Concave Section H’, mm | Designation | Proportionality Coefficient a (−) | Rounding Radius R_{1}, mm | Length of Spherical Section, h_{1}, mm | Rounding Radius R_{2}, mm | Length of Spherical Section, h_{2}, mm |
---|---|---|---|---|---|---|

70 | WKWP70-25 | 0.250 | 102.8 | 17.5 | 308.5 | 52.5 |

WKWP70-50 | 0.500 | 205.7 | 35.0 | 205.7 | 35.0 | |

WKWP70-75 | 0.750 | 308.5 | 52.5 | 102.8 | 17.5 |

CS50 | CS75 | CS100 | CS150 | |
---|---|---|---|---|

CS50 | - | 0.000157 | 0.000164 | 0.000157 |

CS75 | 0.000157 | - | 0.000157 | 0.000157 |

CS100 | 0.000164 | 0.000157 | - | 0.000157 |

CS150 | 0.000157 | 0.000157 | 0.000157 | - |

WK20 | WK45 | WK70 | |
---|---|---|---|

WK20 | – | 0.000124 | 0.000124 |

WK45 | 0.000124 | – | 0.000124 |

WK70 | 0.000124 | 0.000124 | – |

WP20 | WP45 | WP70 | |
---|---|---|---|

WP20 | – | 0.000118 | 0.000118 |

WP45 | 0.000118 | – | 0.000333 |

WP70 | 0.000118 | 0.00333 | – |

WKWP70-25 | WKWP70-50 | WKWP70-75 | |
---|---|---|---|

WKWP70-25 | – | 0.000324 | 0.082838 |

WKWP70-50 | 0.000324 | – | 0.091569 |

WKWP70-75 | 0.082838 | 0.091569 | – |

F_{L}^{Min} (N) | F_{L}^{Q1} (N) | F_{L}^{Q2} (N) | F_{L}^{Q3} (N) | F_{L}^{Max} (N) | F_{L}^{AVG} (N) | |
---|---|---|---|---|---|---|

CS50 | 29,910 | 30,400 | 31,582 | 33,297 | 35,602 | 32,003 |

CS75 | 22,898 | 24,345 | 25,990 | 26,590 | 27,190 | 25,486 |

CS100 | 21,237 | 22,425 | 22,890 | 23,050 | 24,442 | 22,840 |

CS150 | 15,065 | 15,648 | 17,355 | 17,652 | 19,362 | 16,948 |

WK20 | 14,768 | 15,279 | 15,973 | 16,642 | 17,318 | 15,973 |

WK45 | 20,326 | 22,400 | 23,100 | 23,322 | 23,714 | 22,549 |

WK70 | 30,602 | 31,618 | 31,851 | 31,156 | 32,472 | 31,708 |

WP20 | 13,342 | 14,065 | 14,377 | 14,599 | 14,951 | 14,348 |

WP45 | 24,217 | 25,962 | 27,332 | 28,776 | 30,917 | 27,350 |

WP70 | 26,339 | 28,295 | 29,482 | 30,264 | 32,520 | 29,396 |

WKWP70-25 | 22,763 | 26,326 | 28,522 | 29,079 | 32,527 | 27,890 |

WKWP70-50 | 28,038 | 29,627 | 30,216 | 31,207 | 32,502 | 30,352 |

WKWP70-75 | 26,070 | 28,014 | 29,580 | 30,323 | 31,114 | 29,142 |

**Table 9.**Maximum predicted extrusion force F

_{L_FEM}and its average value calculated from the experimental data F

_{L_EMP}.

Die Designation | Predicted Maximum Extrusion Force F_{L_FEM} (N) | Average Maximum Extrusion Force Calculated from the Experimental Data F_{L_EMP} |
---|---|---|

CS50 | 25,934 | 31,582 |

CS75 | 19,137 | 25,990 |

CS100 | 14,943 | 22,890 |

CS150 | 11,637 | 17,355 |

WK20 | 10,091 | 15,973 |

WK45 | 18,062 | 23,100 |

WK70 | 25,861 | 31,851 |

WP20 | 17,358 | 14,377 |

WP45 | 23,802 | 27,332 |

WP70 | 25,604 | 29,482 |

WKWP70-25 | 27,533 | 28,522 |

WKWP70-50 | 24,853 | 30,216 |

WKWP70-75 | 25,871 | 29,580 |

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**MDPI and ACS Style**

Wałęsa, K.; Górecki, J.; Berdychowski, M.; Biszczanik, A.; Wojtkowiak, D.
Modelling of the Process of Extrusion of Dry Ice through a Single-Hole Die Using the Smoothed Particle Hydrodynamics (SPH) Method. *Materials* **2022**, *15*, 8242.
https://doi.org/10.3390/ma15228242

**AMA Style**

Wałęsa K, Górecki J, Berdychowski M, Biszczanik A, Wojtkowiak D.
Modelling of the Process of Extrusion of Dry Ice through a Single-Hole Die Using the Smoothed Particle Hydrodynamics (SPH) Method. *Materials*. 2022; 15(22):8242.
https://doi.org/10.3390/ma15228242

**Chicago/Turabian Style**

Wałęsa, Krzysztof, Jan Górecki, Maciej Berdychowski, Aleksandra Biszczanik, and Dominik Wojtkowiak.
2022. "Modelling of the Process of Extrusion of Dry Ice through a Single-Hole Die Using the Smoothed Particle Hydrodynamics (SPH) Method" *Materials* 15, no. 22: 8242.
https://doi.org/10.3390/ma15228242