# Breakage Strength of Wood Sawdust Pellets: Measurements and Modelling

^{*}

## Abstract

**:**

## 1. Introduction

^{−}

^{3}, which is the typical density of the pellets [5,6]. The bulk density of pellets is significantly higher than that of natural wood block (400–700 kg m

^{−}

^{3}); however, it is lower than the solid-phase density of sawdust particles (1450 kg m

^{−}

^{3}) [7]. Hydrogen binding at the surfaces of lignocellulosic particles of sawdust provides the main type of binding force used in manufacturing of pellets [1]. With increasing compaction pressure, the contact area between the particles increases and the pellets become denser and more durable. Binding forces are higher when a contact zone between particles covers a larger area; that is, the forces increase with an increase in compaction pressure and a decrease in particle size [8]. To increase binding forces, steam explosion pretreatment is applied; this causes lignin melting and a reduction in particle size [3]. The moisture content (MC) markedly affects the binding forces. It has both antagonistic (with water molecules replacing wood polymer bonds) and protagonistic (decreasing the melting temperature of lignin) actions on pellet durability [2]; therefore, an optimum MC should be maintained during the pelletisation process [9]. Pressure (contact between particles), temperature (plastic deformation of lignin), particle size, and optimum MC are key factors in particle binding [1,3]. Therefore, properly pretreated materials have better physical quality than untreated materials [2,6].

## 2. Experiments

#### 2.1. Materials and Methods

#### 2.1.1. Materials

#### 2.1.2. Pellets Preparation

^{−1}. To remove the pellet, the base of the mould was removed and the pellet was pressed out [42]. Twelve combinations of pellets were prepared utilising three types of wood, two levels of moisture content, and two levels of compaction pressure. Ten specimens of each variant of pellets were prepared.

#### 2.1.3. Determination of Solid-Phase and Bulk Densities

_{s}, and of the intact wood (40 mm diameter and 70 mm height blocks), ρ

_{w}, was measured using helium pycnometry (Ultrapyc 1200e, Quantachrome Instruments, Boynton Beach, FL, USA) in five replicates.

_{w}, of the pellets prepared at both levels of compaction pressure, BD

_{p}, and the initial bulk density of the sawdust (just before densification), BD

_{s}, were determined from the mass and volume of the specimens. The volume of the pellets was determined from their diameter and height measured just before the diametral compression test using an electronic calliper with an accuracy of 0.01 mm. The volume of the sawdust was determined from the mould diameter and initial height of the bulk of the sawdust placed in the mould.

#### 2.1.4. Estimation of Porosity and Pore Size Distribution

_{x}of the studied materials was estimated from their bulk density BD

_{x}as

_{s}is the solid density of the sawdust, the index x equals w for the intact wood and p for pellets, and s is the initial bulk of the sawdust. In all cases, the density of the solid phase was assumed to be equal to the density of the sawdust.

^{3}kg

^{−1}), intruded at a given pressure P (in pascals) gave the pore volume that could be accessed. The intrusion pressure was translated on the equivalent pore radius R (in metres) using the Washburn equation

_{m}is the mercury surface tension, α

_{m}is the mercury–solid contact angle (taken as 141.3° for all studied materials), and A is a shape factor (equal to 2 for the assumed capillary pores).

#### 2.1.5. Scanning Electron Microscopy

#### 2.1.6. Diametral Compression Test

^{−}

^{1}. The load was measured with an accuracy of ±0.02 N.

_{x}and σ

_{y}are the principal stresses σ

_{1}and σ

_{2}, respectively [14]. The tensile strength at failure, σ

_{f}, is identified by the maximum tensile stress σ

_{1,max}in the direction perpendicular to the load in the centre of the object [13]:

_{f}is the failure load, R

_{p}is the radius of the pellet, and h

_{p}is its height.

#### 2.1.7. Statistics

#### 2.2. Experimental Results

^{−3}[44]) is close to the densities of all the types of sawdust studied, whereas the density of the wood with its closed pores inside is markedly lower. This may indicate that the vast majority of closed wood pores were damaged during sawdust preparation, with only a small portion eventually remaining. Therefore, the density of the sawdust particles was considered as the true density of the wood solid phase. This was the reason for applying its value for bulk density calculations in Equation (1). The lower solid-phase density of the intact wood indicates that its structure contained closed pores that were not available for gas penetration. The volumetric percentages of these pores, calculated using data from Table 1, were 26.8 ± 0.3 for oak, 2.9 ± 0.2 for pine and 6.5 ± 0.1 for birch.

_{z}versus Δh) for pelletisation are presented in Figure 5a. All dependencies were similar. The noticeable increase in the compaction pressure started at a piston displacement of >10 mm (i.e., >50% of the total displacement). For a higher MC, the maximum compaction pressure was reached at a considerably lower piston displacement (except with oak) and the unloading process lasted longer (Figure 5a inset). This resulted in a lower bulk density of pellets produced from sawdust with a higher MC.

_{1}on ΔL/D were smooth and round without any sudden drops, which is typical for the ductile breakage mode.

_{f}and the deformation at failure, ΔL

_{f}/D, indicate the degree of variability of the experimental relationships resulting from the natural differentiation of pellets (Table 3). Three groups of pellets can be distinguished with respect to the tensile strength: σ

_{f}> 1 MPa (oak, σ

_{z}= 120 MPa, and MC = 8%); σ

_{f}~ 0.5 MPa (oak, σ

_{z}= 60 MPa, and MC = 8%; oak, σ

_{z}= 120 MPa, and MC = 20%; pine, σ

_{z}= 120 MPa, and MC = 8%); and σ

_{f}< 0.3 MPa (all others).

_{z}resulted in a decrease in the tensile strength of the pellets. The highest tensile strength was observed for oak pellets (MC = 8% and σ

_{z}= 120 MPa) and the lowest for birch pellets (MC = 20% and σ

_{z}= 60 MPa). An increase in MC and decrease in the compaction pressure resulted in a fourfold decrease in the tensile strength of oak and birch pellets and an eightfold decrease for pine pellets. The range of the determined values of σ

_{f}is consistent with the findings of other researchers for pellets produced from untreated wood materials [4,6,22]. The range of the deformation at failure, ΔL

_{f}/D, found in the present study (0.56–0.82) is very similar to that obtained by Gilvari et al. [40] in the diametral compression tests of biomass pellets (0.05–0.07).

## 3. Modelling

#### 3.1. DEM Setup

#### 3.1.1. Contact Model

_{1}is the loading (plastic) stiffness, k

_{2}is the unloading (elastic) stiffness, k

_{c}is the adhesive stiffness, δ

_{n}is the overlap in the normal direction, and δ

_{n,}

_{0}is the residual overlap during unloading when the force ${f}_{n}^{}$ changes from repulsive to attractive interaction. Adhesive interactions keep the overlap δ

_{n}among the contacts within the pellet removed from the mould at a level very close to the residual overlap δ

_{n,}

_{0}.

_{1}is related to the yield strength p

_{y}of a particle [47],

_{n,y}is the yielding overlap, ${r}^{*}={r}_{i}{r}_{j}/({r}_{i}+{r}_{j})$ is the equivalent radius of the contacting particles, and p

_{y,i}and p

_{y,j}are the yield strengths of particles i and j, respectively.

_{2}for unloading and reloading is related to k

_{1}through the restitution coefficient e as follows [48]:

_{t}and δ

_{t}are the stiffness and overlap in the tangential direction, respectively, and μ

_{p-p}is the particle–particle friction coefficient.

_{t}, was assumed to be equal to the stiffness in the normal direction. The velocity-dependent dissipative component ${f}_{t}^{d}$ of the tangential force f

_{t}is defined as [48,49]:

_{t}is the relative velocity in the tangential direction, and e is the same restitution coefficient used for the energy dissipation in the normal direction.

_{i}associated with rolling friction m

_{r}was introduced as

_{i}is the unit angular velocity vector of particle i at the contact point.

^{−8}J and the overlap was very close to the residual overlap, δ~δ

_{n}

_{,0}(Figure 7).

_{n}and v

_{t}are the relative velocities in the normal and tangential directions, respectively; ${k}_{n}^{b}$ and ${k}_{t}^{b}$ are the stiffnesses in the normal and tangential directions, respectively; $A=\pi {r}_{b}^{2}$ and $J=\frac{\pi {r}_{b}^{4}}{2}$ are the area and moment of inertia of the bond cross-section, respectively; r

_{b}is the radius of the bond; and Δt is the time increment.

_{c}or the maximum tangent stress ${\tau}_{\mathrm{max}}^{b}$ exceeds the shear strength τ

_{c}:

_{c}is

#### 3.1.2. DEM Modelling Parameters

^{−3}as the uniform value for the solid density of all types of sawdust particles. This value was slightly higher than the bulk density of intact oak wood and higher than those of pine and birch. We decided to set the same particle densities for each type of sawdust because we surmised that the structure of the sawdust of all types of wood would be uniform during sawing. Almost identical values of solid-phase density of each type of sawdust (see Table 1), along with almost identical pore size distributions of all pellets (see Figure 4), confirmed this assumption. The assumed high sawdust particle density was a direct consequence of the high solid-phase density. By setting the bulk density of the intact wood as the solid density of particles, in the first stage of compaction, the free spaces between particles were reduced to zero, and, with further compaction, an increase in the density of the pellet above the density of the intact wood reproduced the closing of internal micro- and macro-pores inside sawdust particles.

_{p-p}, and particle–wall friction, μ

_{p-w}, to 0.5 and 0.15, respectively, in accordance with similar studies on biomass pellets [40] and pinewood chips [38]. The effect of the rolling friction coefficient m

_{r}on the behaviour of pellets was not considered because, in reality, the rolling of elongated dust particles seems to be absent. Therefore, a very low default value of m

_{r}= 0.01 was used in the simulations, as recommended by the EDEM software package [48]. A Poisson ratio of ν = 0.35 and a coefficient of restitution of e = 0.5 were adopted for the DEM simulations, as typical values for wood materials [7,30].

_{y}and the resulting plastic stiffness k

_{1}were determined as the values that best fit the σ

_{z}(Δh/h

_{0}) relationship during the compaction process. The bond radius r

_{b}evaluated from the SEM images of the cross-sections of the pellets was used as baseline input data for the DEM simulations. The elastic properties of the bonds were treated as adjustable parameters for the calibration process. The bond elasticity modulus E

^{b}, the strength σ

_{c}, and the final value of the bond radius r

_{b}were determined as the values providing the best fit of the σ

_{1}(ΔL/D) relationship during diametral compression.

^{4}. To keep the gravitational force unchanged, the gravitational acceleration was reduced by a factor of 10

^{4}. As also shown in [29], scaling the density by a factor of 10

^{6}did not change the shape of the σ

_{1}(ΔL/D) characteristics and reduced the tensile strength σ

_{f}by only 1.2% of the strength of the sample without density scaling. Therefore, scaling density by a factor of 10

^{4}was considered to introduce an error of <1.2%. Time integration was performed with steps of 2 × 10

^{−6}s, that is, 4% of the Rayleigh timestep [50].

#### 3.1.3. Stages of DEM Simulations

^{−5}m s

^{−1}up to the assumed value of the compaction pressure σ

_{z}of 60 or 120 MPa. The unloading process was performed in two stages: (1) unloading in the z direction by piston removal with the same velocity as during loading and (2) mould removal by increasing the mould radius. Next, after the assembly was relaxed, the BPM was initiated and the adhesive interactions were gradually reduced to zero. The last stage was the diametral compression between parallel plates with the same displacement rate as in the previous stages. The colours in Figure 8 represent the average compressive force exerted on the particles. The force scale ranges illustrate the compressive force differences in the simulated objects: the maximum compressive force in the relaxed pellet was five orders of magnitude lower than that in the compacted pellet (compare the scales “c” and “b” in Figure 8) and the maximum compressive force during diametral compression at σ

_{1}= 0.7 MPa (~0.5σ

_{f}) was one order of magnitude lower than that in the compacted pellet (scales “d” and “b” in Figure 8).

#### 3.1.4. Fit Quality

_{1}(ΔL/D) obtained from the simulation was evaluated using the relative root mean-square error (RRMSE):

_{1}(ΔL/D) relationship.

#### 3.1.5. Calibration of Material Parameters for DEM Modelling

_{y}, the coordination number CN, the bond radius r

_{b}, the bond elasticity modulus E

^{b}, and the bond tension and shear strength σ

_{c}= τ

_{c}. The values of p

_{y}and CN were calibrated against the experimental data of sawdust compaction and pellet relaxation. The bond radius was evaluated based on SEM images of broken pellets and a preliminary modelling of the diametral compression test fitting to the range of experimental values of tensile strength and deformation at failure. The elastic modulus and strength of the bonds were determined as a set of two independent parameters that provided the best fit to the experimental diametral compression stress–deformation relationship.

#### 3.2. DEM Run

#### 3.2.1. Compaction of Sawdust: Plastic and Elastic Stiffness of Particles

_{y}was determined as the value providing the minimum RRMSE of the DEM approximation of the experimental stress–deformation relationship of compaction process. As presented in Figure 9, the experimental relationships between σ

_{z}and Δh/h

_{0}, where h

_{0}is the initial height of the sawdust in the mould, for compaction of sawdust were very similar for the three wood materials. Therefore, a common value of p

_{y}was applied for all three materials. The DEM simulation performed for p

_{y}= 100 MPa fitted all the experimental relationships very well for Δh/h

_{0}< 0.75 (RRMSE < 0.15). The ratio of the yielding overlap to the particle radius, δ

_{n,y}/r, was 9.63 × 10

^{−3}. The plastic stiffness of particles, k

_{1}, corresponding to p

_{y}= 100 MPa, was 1 × 10

^{5}N m

^{−1}, and the elastic stiffness, k

_{2}, was 4 × 10

^{5}N m

^{−1}; consequently, the assumed value of the restitution coefficient was 0.5. The lower quality of approximation of the experimental relationship for the highest level of compaction (Δh/h

_{0}> 0.75) resulted from the difference in curvature between the experimental and simulated lines. It is possible that a nonlinear contact model could fit the experimental data better.

#### 3.2.2. Coordination Number

_{p}/BD

_{w}, during compaction. The coordination number CN increased nonlinearly with increasing BD

_{p}/BD

_{w}, being faster and nonlinear for BD

_{p}/BD

_{w}< 1 (as free spaces closed between particles) and slower and almost linear for BD

_{p}/BD

_{w}> 1 (as there remained no free spaces between particles, the surface pores on the sawdust particles and intraparticle pores being closed). In this figure, three values of CN of relaxed pellets (12.9, 12.2, and 10.5) were obtained in the process of unloading and relaxation of the bulk of particles compacted to σ

_{z}= 120, 60, and 48 MPa, respectively. The values of the BD

_{p}/BD

_{w}ratio, corresponding to the indicated values of CN, respectively, covered the entire range of variability of the experimental values of the BD

_{p}/BD

_{w}ratio of 1.62 ± 0.19, 1.32 ± 0.14, and 1.13 ± 0.08, determined as the averaged values for the three wood materials (Table 2) and for three cases: (1) MC = 8%, σ

_{z}= 120 MPa; (2) MC = 8%, σ

_{z}= 60 MPa and MC = 20%, σ

_{z}= 120 MPa; and (3) MC = 20%, σ

_{z}=120 MPa.

_{f}increased slightly faster than linearly as the coordination number increased (Figure 11b). However, Gilvari et al. [40] found that the tensile strength of pellets increased linearly with the coordination number.

#### 3.2.3. Bond Radius Estimation

^{b}= 120 MPa and σ

_{c}= 36 MPa) illustrate the strong dependence of the stress–deformation relationships on the bond radius (Figure 12a). However, simulations performed for the bond parameters fulfilling the conditions ${E}^{b}/{r}_{b}^{2}=\mathrm{constant}$ and ${E}^{b}/{\sigma}_{c}=\mathrm{constant}$ had almost identical stress–deformation relationships (Figure 12b). This similarity means that the model can be easily recalibrated to different values of micro-parameters to simulate other stages of compaction (coordination number), other bond radii, or other materials (elastic parameters). This indicates an important need for precise experimental verification of the micro-parameters of bonds to obtain reliable modelling results.

^{b}and σ

_{c}and variable bond radii indicated that the breakage strength σ

_{f}(Figure 13a) and the deformation at failure, ΔL

_{f}/D (Figure 13b), increased almost linearly with the bond cross-sectional area A. Experimental values of σ

_{f}and ΔL

_{f}/D for oak pellets (MC = 8% and 20% and σ

_{z}= 60, 120 MPa) presented in this figure served as reference data to find the best-fitting values of E

^{b}and σ

_{c}for the selected value of r

_{b}of 20 μm and the particular values of MC and σ

_{z}. By taking into account the fact that the value of the bond radius approximated from SEM images provides a similarity between the simulated and experimental stress–deformation relationships and the breakage profiles, a bond radius of 20 μm was set as the fixed material parameter for all further DEM simulations.

#### 3.2.4. Bond Elasticity and Strength

^{b}and strength σ

_{c}were determined as values providing the minimum RRMSE of the DEM approximation of the experimental stress–deformation relationship. Details of the procedure for searching for the best-fitting parameters are exemplarily illustrated for oak pellets with MC = 8% compacted to σ

_{z}= 120 MPa (Figure 14). Generally, 8–10 simulations, divided into two or three groups with sequentially changed E

^{b}(Figure 14a) and σ

_{c}(Figure 14b) as the constant and variable parameters, were sufficient to find the set of E

^{b}and σ

_{c}, bringing the RRMSE relatively close to its global minimum. This procedure was applied separately to each variant of the experimental relationship. Eighty simulations were performed to find the best fit for all experimental cases.

## 4. Simulation Results

_{1}(ΔL/D) fitted by the DEM approximations. The DEM simulations approximated very well the individual experimental stress–deformation relationships that were the closest to the mean tensile strength (σ

_{f}, ΔL

_{f}/D).

_{p}/BD

_{w}ratio. This can be clearly seen for the oak pellets. The bond elasticity modulus E

^{b}and its strength σ

_{c}decreased as MC increased and σ

_{z}decreased. The range of variability of the bond elasticity modulus applied in the DEM simulations (Table 5) from 9.2 MPa (birch, MC = 20%, and σ

_{z}= 60 MPa) to 120 MPa (oak, MC = 8%, and σ

_{z}= 120 MPa) was consistent with a range of values (3–146 MPa) determined experimentally for wood pellets in the puncture test [51,52,53]. The approximately twofold decrease in the bond elasticity modulus with an increase in MC from 8% to 20% applied in our simulations fitted very well to the rate of decrease in pellet elasticity determined experimentally by Gallego et al. [51]. The order of magnitude of the bond elasticity modulus (10

^{7}–10

^{8}Pa) and the corresponding range of the normal stiffness coefficient of the bond (10

^{10}–10

^{11}N m

^{−3}) applied in our study were similar to the values of the bond elasticity and stiffness applied in the DEM modelling of loading of pinewood chip briquettes by Xia et al. [38], the durability of wood pellets found by Mahajan et al. [39], and the breakage of biomass pellets studied by Gilvari et al. [40]. The maximum bond tensile strength σ

_{c}of 36 MPa applied in our study for oak (MC = 8% and σ

_{z}= 120 MPa) corresponded to that applied by Gilvari et al. [40] (σ

_{c}= 35 MPa) and Mahajan et al. [39] (σ

_{c}= 40 MPa) to model pellet behaviour. The ~30% decrease in the bond strength with an increase in MC from 8% to 20% was consistent with the results of Whittaker and Shield [2] and Li and Liu [9], who indicated that an excessively high MC reduces the binding forces between particles.

_{1}(ΔL/D) relationship typical for the ductile breakage mode. As shown by Horabik et al. [54], σ

_{c}/E

^{b}was <0.1 for the semi-brittle breakage mode and >0.15 for ductile breakage. The σ

_{c}/E

^{b}values found in this study ranged from 0.25 to 0.5, which is typical for the ductile breakage mode.

## 5. General Remarks

_{n}/r~0.33). The coordination number of relaxed pellets in the range of 10.5–12.9 was consistent with the values obtained by Nordström et al. [27] in their DEM modelling of the formation of microcrystalline cellulose tablets under pressure in the range of 100–300 MPa.

## 6. Conclusions

- The stress–deformation relationship during pelletisation of wood sawdust comprising a large range of particle deformations was successfully modelled using a DEM equipped with a linear hysteretic contact model. Good agreement between the simulated range of change in the bulk density during pelletisation with experimental data was obtained because of the application density of the intact wood as the input density of particles in the DEM simulations;
- The highest tensile strength was obtained for oak and the lowest for birch pellets. For all materials, the tensile strength was the highest for MC = 8% of sawdust compacted under a pressure of 120 MPa and the lowest for MC = 20% of sawdust compacted under a pressure of 60 MPa;
- The breakage processes of pellets of all tested materials were successfully simulated using the DEM with the BPM;
- All pellets exhibited a ductile breakage mode characterised by a smooth and round stress–deformation relationship without any sudden drops. Cracks were initiated in locations close to the centre of the pellet, and progressive deformation developed in the direction of loading and toward the interior of the pellet;
- Applying values of the bond elasticity modulus E
^{b}and the tensile strength σ_{c}fulfilling the condition σ_{c}/E^{b}> 0.25 in the DEM simulations allowed the stress–deformation relationship and crack formation to be reproduced well for all studied pellets.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Dependencies of the pore volume on the pore radius for (

**a**) the studied wood and (

**b**) pellets. Average curves from two replicates are presented.

**Figure 3.**Pore size distribution functions for (

**a**) the studied wood and (

**b**) pellets. Average curves from two replicates are presented.

**Figure 4.**Representative SEM images (at 1000× magnification) of the surfaces of cross-sections of the wood blocks, sawdust particles, and pellets. White ellipses surround examples of potential particle–particle bonds. The 40 µm scale bar shown in the oak pellet image is valid for all other materials.

**Figure 5.**Stress–deformation dependencies during (

**a**) the compaction–unloading cycle of sawdust and (

**b**) the diametral compression of pellets.

**Figure 8.**Stages of the DEM simulations: (

**a**) filling, (

**b**) compaction, (

**c**) relaxation, and (

**d**) diametral compression.

**Figure 10.**Comparison of the coordination number CN versus the bulk density ratio BD

_{p}/BD

_{w}of sawdust during compaction and for relaxed pellets.

**Figure 11.**Impact of the coordination number CN on (

**a**) the shape of the stress–deformation relationship during diametral compression and (

**b**) the tensile strength for r

_{b}= 20 μm, E

^{b}= 120 MPa, and σ

_{c}= 36 MPa.

**Figure 12.**Stress–deformation relationship during diametral compression: (

**a**) impact of the bond radius for E

^{b}= 120 MPa, σ

_{c}= 36 MPa, and CN = 12.9; (

**b**) impact of the bond radius compensated by the constant value of the following proportions: E

^{b}⋅r

_{b}

^{2}= constant and E

^{b}/σ

_{c}= constant.

**Figure 13.**DEM simulations fitting the range of experimental values for oak as influenced by the bond cross-sectional area A for CN = 12.9: (

**a**) the tensile strength, σ

_{f}; (

**b**) deformation at failure, ΔL

_{f}/D.

**Figure 14.**RRMSE of fitting the experimental stress–deformation relationship during diametral compression of oak pellets (MC = 8% and σ

_{z}= 120 MPa) performed for CN = 12.9 and r

_{b}= 20 μm as influenced by (

**a**) the bond elasticity modulus E

^{b}and (

**b**) the bond strength σ

_{c}.

**Figure 15.**Fitting the σ

_{1}(ΔL/D) relationships of the diametral compression for birch, oak, and pine. The bars indicate the SD of the mean value.

**Table 1.**Density of the solid phase of the intact wood and of the sawdust measured by helium pycnometry.

Material | Wood Solid Phase (Including Closed Pores) | Sawdust |
---|---|---|

Density ρ_{w} (kg m^{−3}) | Density ρ_{s} (kg m^{−3}) | |

Birch | 1369.8 ± 0.3 ^{a} | 1465.3 ± 0.8 ^{a} |

Oak | 1068.4 ± 1.6 ^{b} | 1459.9 ± 1.1 ^{b} |

Pine | 1426.6 ± 0.3 ^{c} | 1468.7 ± 0.6 ^{c} |

**Table 2.**Bulk density and porosity of the studied intact wood, sawdust, and pellets at various values of MC.

MC (%) | Intact Wood | Sawdust (Initial) | Pellets | ||||||
---|---|---|---|---|---|---|---|---|---|

BD_{w}(kg m ^{−3}) | p_{w}(%) | BD_{s}(kg m ^{−3}) | p_{s}(%) | Compacted to 60 MPa | Compacted to 120 MPa | ||||

BD_{p}(kg m ^{−3}) | p_{p} (%) | BD_{p}(kg m ^{−3}) | p_{p} (%) | ||||||

Birch | 8 | 550 ± 2 ^{a} | 62.5 ± 0.2 ^{a} | 301.7 ± 6.4 ^{a} | 79.4 ± 1.7 ^{a} | 701.2 ± 9.9 ^{a} | 52.1 ± 0.7 ^{a} | 843.9 ± 4.8 ^{a} | 42.4 ± 0.2 ^{a} |

Birch | 20 | 272.1 ± 9.7 ^{b} | 81.4 ± 2.9 ^{b} | 634.6 ± 6.7 ^{b} | 56.7 ± 0.6 ^{b} | 781.9 ± 22 ^{b} | 46.6 ± 1.3 ^{b} | ||

Oak | 8 | 675 ± 3 ^{b} | 53.8 ± 0.2 ^{b} | 312.6 ± 5.4 ^{a} | 78.6 ± 1.4 ^{a} | 824.5 ± 10.4 ^{c} | 43.5 ± 0.5 ^{c} | 1005.8 ± 11.8 ^{c} | 31.1 ± 0.4 ^{c} |

Oak | 20 | 267.8 ± 6.3 ^{b} | 81.6 ± 1.9 ^{b} | 700.3 ± 25.5 ^{a} | 52.0 ± 1.9 ^{a} | 826.4 ± 35.5 ^{a} | 43.3 ± 1.9 ^{a} | ||

Pine | 8 | 510 ± 2 ^{c} | 65.2 ± 0.2 ^{c} | 267.9 ± 6.7 ^{b} | 81.8 ± 2.0 ^{b} | 779.6 ± 12.9 ^{d} | 46.9 ± 0.8 ^{d} | 937.8 ± 17.8 ^{d} | 36.2 ± 0.7 ^{d} |

Pine | 20 | 250.9 ± 3.4 ^{c} | 82.9 ± 1.1 ^{c} | 610.9 ± 44.2 ^{ab} | 58.4 ± 4.2 ^{ab} | 673.1 ± 33.3 ^{b} | 54.2 ± 2.7 ^{e} |

Material | σ_{z} (MPa) | MC (%) | ΔL_{f}/D | σ_{f} (MPa) |
---|---|---|---|---|

Birch | 60 | 8 | 0.056 ± 0.005 ^{a} | 0.112 ± 0.012 ^{a} |

Birch | 120 | 8 | 0.056 ± 0.003 ^{a} | 0.307 ± 0.009 ^{b} |

Birch | 60 | 20 | 0.057 ± 0.005 ^{a} | 0.081 ± 0.008 ^{a} |

Birch | 120 | 20 | 0.061 ± 0.008 ^{b} | 0.183 ± 0.018 ^{c} |

Oak | 60 | 8 | 0.056 ± 0.001 ^{a} | 0.511 ± 0.069 ^{d} |

Oak | 120 | 8 | 0.061 ± 0.003 ^{b} | 1.346 ± 0.068 ^{e} |

Oak | 60 | 20 | 0.059 ± 0.004 ^{c} | 0.277 ± 0.039 ^{b} |

Oak | 120 | 20 | 0.058 ± 0.002 ^{c} | 0.683 ± 0.067 ^{d} |

Pine | 60 | 8 | 0.064 ± 0.008 ^{b} | 0.262 ± 0.019 ^{b} |

Pine | 120 | 8 | 0.057 ± 0.003 ^{a} | 0.608 ± 0.034 ^{d} |

Pine | 60 | 20 | 0.068 ± 0.006 ^{d} | 0.085 ± 0.025 ^{a} |

Pine | 120 | 20 | 0.082 ± 0.002 ^{e} | 0.182 ± 0.071 ^{c} |

Parameter | Symbol | Value |
---|---|---|

Container: | ||

Diameter (mm) | D_{c} | 10 |

Height (mm) | H_{c} | 25 |

Solid density (kg m^{–3}) | ρ | 7800 |

Young’s modulus (MPa) | E | 1.561 × 10^{6} |

Poisson’s ratio | ν | 0.3 |

Particles: | ||

Number | 40,000 | |

Mean particle radius (mm) | r | 0.2 |

SD of particle radius (mm) | r_{sd} | 0.02 |

Particle radius range (mm) | 0.14–0.26 | |

Particle solid density (kg m^{–3}) | ρ | 680 |

Young’s modulus (MPa) | E | 1.57 × 10^{4} |

Poisson’s ratio | ν | 0.35 |

Yield strength (MPa) | p_{y} | 100; 150 |

Mean loading (plastic) stiffness (N m^{–1}) | k_{1} | 1 × 10^{5} |

Mean unloading (elastic) stiffness (N m^{–1}) | k_{2} | 4 × 10^{5} |

Mean adhesion stiffness (N m^{–1}) | k_{c} | 0; 600 |

Restitution coefficient | e | 0.5 |

Particle–particle friction coefficient | μ_{p-p} | 0.5 |

Particle–wall friction coefficient | μ_{p-w} | 0.15 |

Rolling friction coefficient | m_{r} | 0.01 |

Bond radius (μm) | r_{b} | 10; 20; 30; 40 |

Bond tension strength (MPa) | σ_{c} | 3–40 |

Bond shear strength (MPa) | τ_{c} | 3–40 |

Bond Young’s modulus (MPa) | E^{b} | 5–120 |

Material | σ_{z} (MPa) | MC (%) | CN | E^{b} (MPa) | σ_{c} (MPa) | RRMSE |
---|---|---|---|---|---|---|

Birch | 60 | 8 | 12.2 | 12.8 | 4.6 | 0.027 |

Birch | 120 | 8 | 12.2 | 36.6 | 10.1 | 0.039 |

Birch | 60 | 20 | 12.2 | 9.2 | 4 | 0.037 |

Birch | 120 | 20 | 12.2 | 22.4 | 8.05 | 0.039 |

Oak | 60 | 8 | 12.2 | 60 | 20 | 0.027 |

Oak | 120 | 8 | 12.9 | 120 | 36 | 0.111 |

Oak | 60 | 20 | 10.5 | 52.8 | 13 | 0.089 |

Oak | 120 | 20 | 12.2 | 73.2 | 24 | 0.116 |

Pine | 60 | 8 | 12.2 | 27.2 | 13.5 | 0.052 |

Pine | 120 | 8 | 12.9 | 50.4 | 13.5 | 0.028 |

Pine | 60 | 20 | 12.2 | 8.6 | 4.3 | 0.053 |

Pine | 120 | 20 | 12.2 | 19.7 | 9.9 | 0.049 |

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**MDPI and ACS Style**

Horabik, J.; Bańda, M.; Józefaciuk, G.; Adamczuk, A.; Polakowski, C.; Stasiak, M.; Parafiniuk, P.; Wiącek, J.; Kobyłka, R.; Molenda, M.
Breakage Strength of Wood Sawdust Pellets: Measurements and Modelling. *Materials* **2021**, *14*, 3273.
https://doi.org/10.3390/ma14123273

**AMA Style**

Horabik J, Bańda M, Józefaciuk G, Adamczuk A, Polakowski C, Stasiak M, Parafiniuk P, Wiącek J, Kobyłka R, Molenda M.
Breakage Strength of Wood Sawdust Pellets: Measurements and Modelling. *Materials*. 2021; 14(12):3273.
https://doi.org/10.3390/ma14123273

**Chicago/Turabian Style**

Horabik, Józef, Maciej Bańda, Grzegorz Józefaciuk, Agnieszka Adamczuk, Cezary Polakowski, Mateusz Stasiak, Piotr Parafiniuk, Joanna Wiącek, Rafał Kobyłka, and Marek Molenda.
2021. "Breakage Strength of Wood Sawdust Pellets: Measurements and Modelling" *Materials* 14, no. 12: 3273.
https://doi.org/10.3390/ma14123273