*3.2. Frequency*

As shown in Figure 9, the refrigeration capacity and coe fficient of performance increased with decreasing frequency at a low flow rate because the heat transfer time between the fluid and the magnetocaloric material increased. At high volumetric flow rates, by increasing the frequency, the number of cycles completed at the time step increased, which increased the refrigeration capacity and the coe fficient of performance. As it observed, higher frequencies are achievable in higher mass flow rates and, therefore, it would result in higher viscous dissipation and consequently, higher input pump work was required that reduced the overall performance.

#### *3.3. Temperature Span (Hot- and Cold-Source Temperature Di*ff*erence)*

The temperature span is another important design parameter. In this study, the e ffects of the temperature span variation on the refrigeration capacity and coe fficient of performance were evaluated and the results are shown in Figure 10. According to Figure 10, it can be concluded that the refrigeration capacity and coe fficient of performance were inversely proportional to the temperature range. In the low-temperature span, due to the fact that less power and energy are needed to transfer heat from a cold source to a hot supplier, it was expected that the refrigeration capacity and coe fficient of performance would be higher than in the higher temperature span. Hence, the refrigeration capacity and the coe fficient of performance decreased by increasing the temperature span. As the temperature span increased due to the axial heat conductivity from the hot end to the cold end of the regenerator, heat loss is increased, and it would result in reducing the overall e fficiency.

#### *3.4. Spherical Particle Diameter*

One of the key parameters of the packed sphere bed regenerator is the size of the spherical particle diameter, as shown in Figure 11. By increasing the diameter of the spherical particles, the refrigeration capacity and the coe fficient of performance first increase and then decreased meaning that there will be an optimal diameter for the spherical particles. Below the optimal diameter of spherical particles, increasing it reduces the viscous dissipation, thereby increasing the refrigeration capacity. On the other hand, beyond the optimal diameter of spherical particles, increasing it results in a decrease in the coe fficient of the heat transfer and, therefore, the coe fficient of performance and the refrigeration capacity decrease.

**Figure 10.** The effects of the span temperature variation on the coefficient of performance (**a**) and refrigeration capacity (**b**).

(**b**) 

**Figure 11.** (**a**) the coefficient of performance and (**b**) refrigeration capacity based on the sphere particle diameter for a temperature span of 1 K and a frequency of 4 Hz at different volumetric flow rates.
