## 2 Models and Analyses
### Analysis of Model Calculation Results  
A highly sensitive type of magnetic particle product, 14 A fluorescent magnetic particles, was prepared to locate very fine discontinuities in critical parts and applications. The main material of the magnetic powder was ferric oxide, as per the information provided by the manufacturer. The diameter of the magnetic particles was 8 \(\upmu\)m, the particle density was 4800 kg/m\({}^{3}\), the volume magnetic susceptibility of the particles was \(8.64\times 10^{3}\), and the density of the magnetic suspension carrier was 1000 kg/m\({}^{3}\). BH curve is a curve that represents the relationship between the magnetic induction intensity B and the magnetic field intensity H in the magnetization process of ferromagnetic materials. The BH curve of the workpiece is displayed in Figure 4, where 9 CrWMn steel was used, with the 4 th grade of grain size, hardness of 90 HRB, surface roughness of 3.2 m, and size of 500 mm (L) \(\times\) 150 mm (W) \(\times\) 70 mm (D). The internal magnetic field strength and saturation magnetic  
Figure 3: The schematic diagram of the multi-particle model. induction strength of the workpiece were 2800 A/m and 1.3 T, respectively, the crack width and height were 30 \(\upmu\)m and 60 \(\upmu\)m, respectively, the static friction coefficient between workpiece surface and the particles was \(2\times 10^{-7}\). The number density of magnetic particles in the suspension was converted by the magnetic particle concentration (Table 1). A pear-shape concentration measurement tube is used for measurement, the error is \(\pm\)5%. The influences of the magnetic field strength on the critical distance for four different magnetic particle densities (0.1 1/L, 1 1/L, 10 1/L, and \(100\times 10^{7}\) 1/L) are depicted in Figure 5.  
The critical distance first increased and then remained unchanged with the increase of the magnetic field strength. As the effect of the magnetic force is inversely proportional to the square of the distance, the effect of the magnetic field strength will gradually decrease with the increase of the distance. With the increase of the magnetic particle concentration, the influence of multiple particles on each other increased, and the critical distance became smaller. As the magnetic field strength increased, the critical distance first increased and then remained unchanged. When the magnetic particle concentration was low, single particles and multiple particles had similar critical distance, whereas when the magnetic particle concentration increased, the result of the multi-particle situation was consistent with the actual situation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Magnetic Particle Concentration** & **Magnetic Particle Number** & **Interval Between Particles (\(\upmu\)m)** \\
**(mL/L)** & **Density (10\({}^{7}\)/L)** & **Interval Between Particles (\(\upmu\)m)** \\ \hline
0.50 & 18.5 & 175 \\
1.00 & 37 & 139 \\
2.00 & 74 & 111 \\
3.00 & 111 & 97 \\
4.00 & 148 & 88 \\
6.00 & 222 & 77 \\
10.00 & 370 & 65 \\
20.00 & 740 & 51 \\
30.00 & 1110 & 45 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Concentration and number density of magnetic particles. Figure 4: BH curve of the workpiece. Figure 6 shows the effects of magnetic particle concentrations on the \"critical distance\" for magnetic particle accumulation. With the increase of the magnetic particle concentration, the critical distance dropped sharply and then decreased slowly. The reason for this phenomenon is that after the magnetic particles are magnetized, the magnetic field direction of the particles themselves does not completely match the leakage magnetic field formed by the simple defects on the particles around them, consequently they may be partially neutralized by each other. Therefore, as the particle concentration increases, the adsorption limit distance will decrease.