# The Need to Accurately Define and Measure the Properties of Particles

^{1}

^{2}

^{3}

^{*}

*Standards*)

## Abstract

**:**

## 1. Introduction

## 2. Particle Size and Its Size Distribution (PSD)

#### 2.1. The Average Particle Size

^{3}(e.g., the density of water) which settles in still air at the same velocity as the particle in question [13,14,15]. The latter classifies charged particles according to their mobility in an electric field, followed by a particle counter to count particles of a specific mobility [16].

#### 2.2. The Particle Size Distribution

_{N}(d) and F

_{N}(d) for the distribution by numbers; f

_{L}(d) and F

_{L}(d) for the distribution by length; f

_{s}(d) and F

_{S}(d) for the distribution by surface area; and f

_{M}(d) and F

_{M}(d) for a mass distribution. In these definitions, d is the relevant particle size (see Table 1). The distribution by length is seldom used in practice, but is given for reasons of completeness.

_{i}, an average size d

_{i}and a value of the distribution f(d

_{i}). In Equations (4)–(7), the definitions of the discrete distribution densities and the cumulative fraction are given with j as the index of the j-th part of the discrete distribution that corresponds with the value of d

_{i}:

_{i}, l

_{i}, s

_{i}, m

_{i}being the number, length, surface and mass in a size range i and N, L, S and M the total number, length surface and mass.

_{1}, k

_{2}and k

_{3}:

_{2}is the following:

_{m}(the size with maximum distribution density) is used as a parameter instead of the mean.

_{i}as the average size in an increment range of size ∆d

_{i}.

_{m}) is, as already stated, the most commonly found size in the distribution. As opposed to the mean and the modus, the median (d

_{50%}) is most easily identified using the cumulative fraction, F(d), where it corresponds to the 50 %-value.

_{50%,}with the following definitions:

_{84%}and d

_{16%}correspond with the particle size with a cumulative fraction F(d) equal to 84 % and 16 %, respectively. Some particle size analyzers (e.g., laser diffraction) use a slightly different definition with the spread evaluated between d

_{90%}and d

_{10%}.

#### 2.3. Particle Size and Size Distribution Measurements

#### 2.3.1. Common Instrumental Techniques

#### 2.3.2. Parameters Affecting the Instrumental Particle Size Measurement

#### 2.3.3. Comparing the Common Particle Size Analyzers

_{2}O

_{3}(3 130 kg/m

^{3}) and SiC (3 960 kg/m

^{3}), with additional properties given in Table 7 [25]. Ten measurements were carried out. The results and coefficients of variation are given in Table 8.

_{50}are generally less than 10%, except for photo-sedimentation with a ~20% relative accuracy.

## 3. Particle Shape

_{v}) and the surface to volume diameter (d

_{sv}) have to be determined experimentally.

_{,}the particle density (kg/m

^{3}), which will be discussed in detail in the following section.

## 4. Particle Density and Bed Voidage

^{3}), bed bulk density, ${\rho}_{\mathrm{B}}$ (kg/m

^{3}), density of the fluid occupying the intra- and interparticle empty space, ${\rho}_{\mathrm{g}}$, and bed voidage, ε (−), are correlated by:

^{3}), of the material out of which the particle is composed. Due to a possible internal porous structure of the particle, the particle density will often be lower than the absolute density. Sometimes the particle density is denoted with other names such as hydrodynamic, apparent, envelope, effective or piece density [38,39].

#### 4.1. Particle Density

- (1)
- Caking end-point measurements are sometimes performed in the petrochemical industry for rapid and cheap estimations of pore volume. In these measurements, the investigated powder is put in a vibrating flask and a liquid with low viscosity or volatility (for example, water) is added incrementally. As long as the liquid is absorbed into the microscopic pores, the powder remains free-flowing. If the pores are completely filled, any surplus of liquid will coat the surface of the particles and cause the formation of liquid bridges, i.e., caking. This surplus depends on pore size and the surface tension. A complete filling-up of the pores is often impossible due to surface tension constraints. As a result, the caking end-point measurement method tends to overestimate the particle density. If the pore volume is determined, the particle density can be calculated with:$${\rho}_{\mathrm{p}}=\frac{1}{x+1/{\rho}_{\mathrm{abs}}}$$
^{3}/kg).

- (2)
- In a porosimeter, mercury under high pressure is forced into to the pores of the particles. Eventually, the pore size can be determined. As in the caking end-point method, the particle density is determined from Equation (30). A major setback of this measuring method is its high cost.
- (3)
- The particle density can also be determined in the comparative method by examining the tapped bulk density, ${\rho}_{\mathrm{BT}}$, of both the sample and a control powder. Then, applying Equation (31) yields the particle density of the investigated sample powder.$${\rho}_{\mathrm{pX}}=k\frac{{\rho}_{\mathrm{BTX}}}{{\rho}_{\mathrm{BTC}}}{\rho}_{\mathrm{pc}}$$

- k = 1 for identically shaped sample particles and control particles
- k ≈ 0.82 for rounded or spherical sample particles and angular control particles
- k ≈ 1/0.82 for angular sample particles and spherical or rounded control particles

- (4)
- In the adapted gas flow technique of Ergun, the particle density is determined by comparing the pressure drop over a bed with minimum voidage to a bed with a maximum voidage. Maximum voidage can be achieved by fluidizing the sample and letting it gently settle. The resulting bed can next be tapped for a sufficient length of time to reach the state of minimal voidage. In both situations, the bed height L
_{A}(aerated) and L_{T}(tapped) is measured. Additionally, the pressure drop, ∆p, is recorded for at least four different gas velocities. Next, the pressure drop is plotted against the superficial velocity (v) and the slope within the laminar flow regime (Re < 2) is measured. With these values, the particle density can be calculated using a rearranged form of Equation (28), i.e., the Ergun equation in the laminar flow regime, with L, the bed length (m), ε the bed voidage (-) and µ the gas viscosity (Pa s). - (5)
- Rearranged, the slopes of the graphs S
_{A}and S_{B}for the two beds are$${S}_{\mathrm{A}}={\left[\frac{\mathsf{\Delta}p}{v}\right]}_{\mathrm{T}}=\frac{150\mu}{{d}_{\mathrm{sv}}^{2}}{L}_{\mathrm{A}}\frac{{\left(1-{\epsilon}_{\mathrm{A}}\right)}^{2}}{{\epsilon}_{\mathrm{A}}^{3}}$$$${S}_{\mathrm{T}}={\left[\frac{\mathsf{\Delta}p}{v}\right]}_{\mathrm{T}}=\frac{150\mu}{{d}_{\mathrm{sv}}^{2}}{L}_{\mathrm{T}}\frac{{\left(1-{\epsilon}_{\mathrm{T}}\right)}^{2}}{{\epsilon}_{\mathrm{T}}^{3}}$$$$\frac{{L}_{\mathrm{A}}}{{L}_{\mathrm{T}}}=\frac{{\rho}_{\mathrm{BT}}}{{\rho}_{\mathrm{BA}}}$$$$1-{\epsilon}_{\mathrm{A}}=\frac{{\rho}_{\mathrm{BA}}}{{\rho}_{\mathrm{P}}}$$$$1-{\epsilon}_{\mathrm{T}}=\frac{{\rho}_{\mathrm{BT}}}{{\rho}_{\mathrm{P}}}$$

_{A}, L

_{T}, ε

_{A}and ε

_{T}, yields:

- (6)
- In the powder displacement method, the particle density is measured by comparing the tapped bulk density of a control powder with a mixture of the control and the sample powder. This technique is specific because the fine powder is used as pycnometric fluid to fill the open pores in the investigated particles. As such, the pycnometric powder must be free-flowing, non-porous and sufficiently smaller than the sample particles. If the latter condition is not fulfilled, the comparison between the control powder and the mixture of control and sample powder will give erroneous results. This test can be performed in the apparatus illustrated in Figure 10b. If the control tapped bulk density is ${\rho}_{\mathrm{BTC}}$, up to 20 wt% of the larger unknown porous particles is mixed with the control powder and tapped in the cup of Figure 10b.

_{c}, is determined. With this, the particle density of the unknown particles, ${\rho}_{\mathrm{pX}}$, is determined with Equation (40):

_{x}being the mass of sample in the mixture and V the volume of the cup.

- (7)
- Additionally, the minimum fluidization velocity is related to the particle density in the Ergun equation:$${\rho}_{\mathrm{p}}=\left[\frac{1.75}{{\epsilon}_{\mathrm{MF}}^{3}}{\left[\frac{{d}_{\mathrm{sv}}{\mathsf{\nu}}_{\mathrm{MF}}{\rho}_{\mathrm{g}}}{\mu}\right]}^{2}+\frac{150\left(1-{\epsilon}_{\mathrm{MF}}\right)}{{\epsilon}_{\mathrm{MF}}^{3}}\left[\frac{{d}_{\mathrm{sv}}{\mathsf{\nu}}_{\mathrm{MF}}{\rho}_{\mathrm{g}}}{\mu}\right]\right]\frac{{\mu}^{2}}{{d}_{\mathrm{sv}}^{3}{\rho}_{\mathrm{g}}\mathrm{g}}+{\rho}_{\mathrm{g}}$$

_{MF}itself is dependent on the particle density as follows:

_{sv}are required. The latter is not troublesome for spherical particles since the sieve size d

_{a}in that case equals the surface-to-volume ratio, d

_{sv}.

- (8)

#### 4.2. Bulk Density

_{H}), powders can be classified in different categories:

- R
_{H}< 1.25: Group A, B, or D - R
_{H}> 1.4: Group C - 1.25 < R
_{H}< 1.4: Transition group AC

#### 4.3. Bed Voidage

- The compaction state: Obviously, a tapped bed will have a smaller voidage than an aerated bed. Two extreme conditions, assuming random packing, are used as a reference: ‘loose’ packing with the maximum voidage and ‘dense’ packing with minimum voidage.
- The particle shape: The voidage increases with decreasing sphericity. This is illustrated in Figure 11.

- The particle size: For loosely packed beds, the voidage decreases with increasing particle size. The densely packed bed voidage, on the other hand, is quite insensitive to size. This is illustrated in Figure 12.
- The particle size distribution: The voidage decreases with increasing spread.
- The particle and wall roughness: The voidage increases with increasing surface roughness.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Distribution f(d

_{i}) of the sieved sand with indications of the different mean values, with the modus, d

_{m}; and $\overline{{d}_{\mathrm{a}}},\text{}\overline{{d}_{\mathrm{q}}},\text{}\overline{{d}_{\mathrm{c}}},\text{}\overline{{d}_{\mathrm{h}}},\text{}\overline{{d}_{\mathrm{g}}}$ as arithmetic, quadratic, cubic, harmonic and geometric means.

**Figure 6.**Cumulative fraction with indications of the median d

_{50%}and the fractions d

_{16%}and d

_{84%}. The spread σ is 197 µm while the relative spread σ/d

_{50%}equals 0.79.

**Figure 11.**Voidage versus sphericity for loosely and densely packed beds of uniformly sized particles larger than about 500 µm [44].

**Figure 12.**Variation of packed bed voidage vs. particle size for spherical particles and sand. The particle size distribution is narrow in both cases.

**Table 1.**Definitions of the particle diameter (adapted from [17]).

Symbol | Diameter Definition | Equivalent Sphere Diameters |
---|---|---|

d_{A} | Sieve | Largest sphere diameter that can pass through the square aperture of the sieve. |

d_{v} | Volume | Sphere diameter when particle and sphere volumes are equal. |

d_{s} | Surface | Sphere diameter when particle and sphere surfaces are equal. |

d_{SV} | Surface to Volume | Sphere diameter when the surface area to volume ratio of the sphere and the particle are equal. |

Sieve Size (mm) | Average Size (mm) | Sieve Mass (g) | Mass Fraction |
---|---|---|---|

0.04–0.06 | 0.05 | 0.1 | 0.03 |

0.06–0.10 | 0.08 | 0.4 | 0.11 |

0.010–0.18 | 0.14 | 0.7 | 0.19 |

0.18–0.30 | 0.24 | 0.9 | 0.25 |

0.30–0.42 | 0.36 | 0.7 | 0.19 |

0.42–0.59 | 0.5 | 0.5 | 0.14 |

0.59–0.83 | 0.71 | 0.2 | 0.06 |

0.83–1.00 | 0.92 | 0.1 | 0.03 |

Total | 3.6 | 1 |

Mean | g(d) | Formula |
---|---|---|

Arithmetic | d | ${\overline{d}}_{\mathrm{a}}={\displaystyle {\displaystyle \sum}_{i}^{\infty}}{d}_{i}f\left({d}_{i}\right)\mathsf{\Delta}{d}_{i}$ |

Quadratic | d^{2} | ${\overline{d}}_{\mathrm{q}}=\sqrt{{\displaystyle {\displaystyle \sum}_{i}^{\infty}}{d}_{i}^{\text{}2}f\left({d}_{i}\right)\mathsf{\Delta}{d}_{i}}$ |

Cubic | d^{3} | ${\overline{d}}_{\mathrm{c}}=\sqrt[3]{{\displaystyle {\displaystyle \sum}_{i}^{\infty}}{d}_{i}^{\text{}3}f\left({d}_{i}\right)\mathsf{\Delta}{d}_{i}}$ |

Geometric | Log(d) | ${\overline{d}}_{\mathrm{g}}={10}^{{\displaystyle {{\displaystyle \sum}}_{i}^{\infty}}\mathrm{log}({d}_{\mathrm{i}})f\left({d}_{i}\right)\mathsf{\Delta}{d}_{i}}$ |

Harmonic | d^{−1} | ${\overline{d}}_{\mathrm{h}}={\left({\displaystyle {\displaystyle \sum}_{i}^{\infty}}{d}_{i}^{-1}f\left({d}_{i}\right)\mathsf{\Delta}{d}_{i}\right)}^{-1}$ |

Sieve Aperture (µm) | Size d_{A} (µm) | Weight% in Range ∆d_{i} |
---|---|---|

600–500 | 550 | 0.5 |

500–420 | 460 | 11.6 |

420–350 | 385 | 11.25 |

350–300 | 325 | 14.45 |

300–250 | 275 | 20.8 |

250–210 | 230 | 13.85 |

210–180 | 195 | 12.5 |

180–150 | 165 | 11.9 |

150–125 | 137 | 3.15 |

Method | Approx. Size (µm) | Size Type | Basis of the Size Distribution |
---|---|---|---|

Sieving (wet/dry) | 25–4000 5–120 | d_{A} | Mass |

Woven mesh | |||

Electro-formed mesh | |||

Microscopy | 0.8–150 0.001–5 | d_{z,} d_{F,} d_{M}d _{SH,}d_{CH} | Number |

Optical | |||

Electron | |||

Gravity sedimentation | 2–100 | d_{St}, d_{f} | Mass |

Centrifugal sedimentation | 0.01–10 | d_{St}, d_{f} | Mass |

Elutriation (dry) | 5–100 | d_{St}, d_{f} | Mass |

Centrifugal elutriation (dry) | 2–50 | Mass | |

Impactors (dry) | 0.3–50 | Mass or number | |

Coulter Counter (electrical resistance) | 0.8–200 | d_{v} | Number |

Fraunhofer diffraction (laser) | 1–2000 | Specific diameter | Volume |

Mie light scattering (laser) | 0.1–40 | Specific diameter | Volume |

Photon correlation spectroscopy | 0.003–3 | Specific diameter | Number |

Doppler phase shift (laser) | 1–10^{4} | Specific diameter | Mean only |

**Table 6.**Parameters affecting the instrumental particle size measurement (adapted from [25]).

Parameters | |
---|---|

Powder sample | Particle density, particle refractive index and weight of the sample |

Solvent | Type, density, refractive index and viscosity |

Dispersant | Organic/anorganic, concentration |

Dispersion | Ultrasonication bath or tip (position, size, material), suspension volume, power, frequency and ultrasonication duration |

SiC | Al_{2}O_{3} | |
---|---|---|

refractive index (−) | 2.65 | 1.76 |

dispersant and concentration (wt%) | tri-sodium phosphate 0.025 | sodium hexametaphosphate 0.05 |

ζ-potential (mV) | −64 | −97.5 |

Al_{2}O_{3} | X-ray Sedimentation | Photo-Sedimentation | Light Obscuration | Electrical Sensing Zone | Laser Diffraction |
---|---|---|---|---|---|

d10 (µm) | (0.95) | (0.95) | 1.16 | 1.16 | 0.71 |

CV (%) | 2.80 | 14.20 | 5.80 | 8.30 | 35.9 |

d50 (µm) | 1.81 | 1.69 | 2.88 | 2.16 | 2.10 |

CV (%) | 3.00 | 12.60 | 7.20 | 4.80 | 12.70 |

d90 (µm) | 3.68 | 4.13 | 4.89 | 4.07 | 4.69 |

CV (%) | 5.20 | 41.80 | 3.20 | 4.60 | 9.60 |

SiC | X-ray Sedimentation | Photo-Sedimentation | Light Obscuration | Electrical Sensing Zone | Laser Diffraction |

d10 (µm) | (0.11) | (0.16) | 0.63 | (0.20) | (0.24) |

CV (%) | (15.20) | (27.20) | 3.50 | (21.30) | 34.50 |

d50 (µm) | 0.47 | 0.47 | 1.02 | 0.68 | 0.64 |

CV (%) | 21.70 | 39.40 | 6.90 | 10.40 | 18.00 |

d90 (µm) | 1.92 | 1.60 | 3.12 | 2.71 | 1.96 |

CV (%) | 10.80 | 34.70 | 17.70 | 14.50 | 31.20 |

Shape | Relative Proportions | ψ |
---|---|---|

Spheroid | 1 : 1 : 2 | 0.93 |

1 : 2 : 2 | 0.92 | |

1 : 1 : 4 | 0.78 | |

1 : 4 : 4 | 0.70 | |

1 : 2 : 4 | 0.79 | |

Cylinder | Height = 0.5 × diameter | 0.83 |

Height = 0.25 × diameter | 0.69 | |

Cube | - | 0.81 |

Material | ψ |
---|---|

Crushed coal | 0.75 |

Crushed sandstone | 0.8–0.9 |

Sand (average) | 0.75 |

Round sand | 0.83 |

Flint sand, jagged | 0.65 |

Crushed glass | 0.65 |

Common salt | 0.84 |

Most crushed materials | 0.6–0.8 |

Method | Relative Equipment Cost | Suitable Types of Powder in Rank Order According to Geldart’s Classification |
---|---|---|

Caking end-point | Negligible | A |

Mercury porosimeters | Very high | D, B, A |

Comparative | Low | B, A |

Gas flow | Low | A, B |

Powder displacement | Low | D, B |

Minimum fluidization velocity | Low | D, B, A spherical |

Photographic | High | B, D, A |

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**MDPI and ACS Style**

Deng, Y.; Dewil, R.; Appels, L.; Zhang, H.; Li, S.; Baeyens, J. The Need to Accurately Define and Measure the Properties of Particles. *Standards* **2021**, *1*, 19-38.
https://doi.org/10.3390/standards1010004

**AMA Style**

Deng Y, Dewil R, Appels L, Zhang H, Li S, Baeyens J. The Need to Accurately Define and Measure the Properties of Particles. *Standards*. 2021; 1(1):19-38.
https://doi.org/10.3390/standards1010004

**Chicago/Turabian Style**

Deng, Yimin, Raf Dewil, Lise Appels, Huili Zhang, Shuo Li, and Jan Baeyens. 2021. "The Need to Accurately Define and Measure the Properties of Particles" *Standards* 1, no. 1: 19-38.
https://doi.org/10.3390/standards1010004