*3.5. Porosity*

Porosity is defined as the ratio of the volume of empty space to the total regenerator volume. By increasing the porosity, the free space of regenerator increases, therefore more fluid can pass through the regenerator bed and more heat is absorbed from the solid refrigerant, leading to an increase in the refrigeration capacity and the coe fficient of performance. On the other hand, an increase in porosity reduces the amount of magnetocaloric material and the magnetocaloric e ffect, thereby reducing the refrigeration capacity and coe fficient of performance. In this case, a more intense magnetic field can be used to increase the refrigeration capacity and coe fficient of performance. Furthermore, it should be noted that in a very-low-porosity condition, the viscous dissipation is increased, so that a lower refrigeration capacity and coe fficient of performance can be expected (Figure 12).

#### *3.6. Pump Power*

One of the most important parameters influencing the performance of the magnetic refrigeration system is the pump power, that is, the viscous dissipation. Viscous dissipation in the fluid is the irreversible process which causes mechanical energy to transform into heat and may increase the heat losses. The impact of the viscous dissipation is included in the AMR model via a friction factor, as shown in Equation (3). The e ffect of viscosity loss at high frequencies will increase and, in some cases, become significant in the models of compact AMRs, because the small geometries require higher fluid flow to maintain the same cooling capacity at a large scale. Excessive pressure drops (viscous dissipation) increase the work required to pump the fluid through the AMR.

In this study, it was assumed that there is no leakage in the system, and the mechanical parts of the system such as the piping system were not considered in the numerical model. The impact of these parameters on the performance of the AMR could be considered as a correction factor to the pump power.

An important parameter in the viscous dissipation is the spherical particle diameter. As shown in Figure 13, the work of the pump was increased due to the increased pressure drop by reducing the diameter of the spherical particles and increasing the mass flow rate of the fluid.

#### *3.7. Design Analysis*

Figures 9–13 show the e ffect of di fferent parameters on the performance of the magnetic refrigeration system. The performance of the AMR considerably depends on the operational parameters. Figure 9 shows that low or high mass flow rate is not desirable for the AMR and by increasing the frequency cooling power will increase. The results which are presented in Figure 10 show that there is a linear dependency of COP and refrigeration capacity on the temperature span. Figure 11 shows the refrigeration capacity and COP as a function of sphere diameters. It is evident that there is an optimum point for sphere diameter for each mass flow rates. The e ffect of diameter of spherical particles and mass flow rate simultaneously on AMR performance in are presented in Figure 13.

**Figure 12.** Chart of (**a**) the coefficient of performance and (**b**) refrigeration capacity based on porosity for a temperature range of 1 K, frequency of 4 Hz, and a sphere diameter of 0.5 mm at different volumetric flow rates.

(**a**) 

**Figure 13.** *Cont*.

**Figure 13.** The power pump based on the of the sphere particle diameter at a different volumetric flow rate in the temperature span of 1 K: (**a**) frequency of 4 Hz, (**b**) frequency of 2 Hz, and (**c**) frequency of 1 Hz.

The designer of a new magnetic refrigeration system can select the parameters that are appropriate for the working conditions by using the diagrams presented in this study. According to the parameters reported in Table 3, it is possible to predict the third parameter by having two parameters. For example, as shown in Figure 9, the refrigeration capacity and the coefficient of performance can be calculated by using the flow rate and operating frequency. Furthermore, using the frequency and the refrigeration capacity, it can be predicted how much refrigeration capacity is being met at a specific rate of mass fluid flow. In the same way, like the other parameters, the porosity, temperature range, and diameter of the spherical particles can be calculated. Design charts can be categorized into different groups: The design diagram based on the operational parameters, the price of the magnetocaloric material, the dimensions, and geometry of the regenerator. Design charts can be considered as a tool for developing a magnetic refrigeration system without performing mathematical calculations that leading to time-saving. The designer of the cooling system must be aware of the application of the system and understand what the purpose and operating conditions related to the system are and consider all the aspects of their designs, including the limitations, designers must also offer all possible options for the client's system requirements. Some of the losses in the magnetic refrigeration system that affect the AMR performance are the insufficient heat transfer between the heat transfer fluid and the magnetocaloric material, the magnetic hysteresis, insufficient heat transfer in heat exchangers, and the pressure drops in the piping and heat exchanger. Another important point that should be considered in the system design is the desired economics of the refrigeration system, that is, having the lowest cost and the highest efficiency. The basic information generally required in order to design an active-reactive magnetic refrigeration system is shown in Table 5.

**Table 5.** Parameters required for the design of a magnetic refrigeration system.

