Final dimensions of the regenerator

4.5. Designing the regenerator 139

AMR). Additionally, three different kinds of fluid flow profile were implemented: that of the rotary valve system (bedtype 0) as shown in Fig. 64-a, and those for bedtype 1 and bedtype 2 as shown in Fig. 64-b. In the model, the fluid flow profile is multiplied by the fluid flow rate.

A parametric analysis was carried out to determine the most suitable numerical mesh for the actual operating conditions. The analysis is based on the number of spatial steps, 𝑁x, the number of time steps, 𝑁t and the convergence tolerance, 𝑡𝑜𝑙. The latter determines when the convergence (periodic steady state) is achieved (Engelbrecht, 2008):

⃒

⃒

⃒

⃒

Δ𝐸_s+ Δ𝐸_f 𝐸_max−𝐸_min

⃒

⃒

⃒

⃒

< 𝑡𝑜𝑙 (4.9) whereΔ𝐸_s and Δ𝐸_f corresponds to the total energy change of the bed solid and fluid entering in the bed in one cycle, respectively.

While𝐸max and𝐸minare the maximum and minimum values of the energy in the regenerator during a cycle, respectively.

The accumulated cooling capacity difference as a function of the numerical mesh parameters is shown in Fig. 68. By balancing the accuracy in the prediction of the cooling capacity and the time required for numerical convergence (computational time), the op- timal numerical mesh for the UFSC device consisted of 80 spatial steps, 1000 time steps and a convergence tolerance of 0.0005.

0.0%

0.1%

0.2%

0.3%

0.4%

0 100 200 300

Acc. Cool. Cap. difference (%)

Nx (-) Nt = 800 tol = 0.0005

(a)

0.0%

3.0%

6.0%

9.0%

12.0%

15.0%

0 1000 2000 3000

Acc. Cool. Cap. difference (%)

Nt (-) Nx = 80 tol = 0.0005

(b)

0.0%

0.4%

0.8%

1.2%

1.6%

0 0.0005 0.001 0.0015 0.002

Acc. Cool. Cap. difference (%)

tol (-) Nx = 80 Nt = 800

(c)

Figure 68 – Parametric analysis of the (a) spatial and (b) time mesh size, and (c) the convergence tolerance for the 1D model developed by (Engelbrechtet al., 2006) and modified for the conditions of the UFSC device.

Regarding the width of the beds, it should be small in order to reduceN^bed_D . However, the device grows in complexity as the number of beds increases. On the other hand, magnetic forces decrease by having a more continuous regenerator ring, such as wider beds with less spacing in between. Eventually,𝑛bedand𝑤bedwere determined by the perimeter of the magnetic gap, Pergap = 628.3 mm (360º) and by the opening of the high magnetic field generated by the magnetic circuit (Fig. 57). Since the flow distribution system was designed to have an 8-port rotary valve, the regenerator ring was, initially, divided into 8 beds with an opening angle in the magnetic gap of 36º, which results in𝑤_bed∼63.4 mm. The spacing left at the regenerator ring is used by the screws fixing the channeling rings.

It has been found later that having wide regenerator beds

4.5. Designing the regenerator 141

could generate large stresses in the thin casing wall during opera- tion at high pressures. Moreover, wider beds are susceptible to flow maldistribution. Therefore, a structural analysis of the casing was carried out using the SolidWorks Simulation^® software to include an strengthening rib into the regenerator beds. Application of the von Mises criterion resulted in a longitudinal rib with a 4 mm thick- ness neck that divides each original bed into two beds that receive simultaneously the fluid flow from the rotary valves bifurcated at the hot end flow distributor.

Circular ribs were machined on the outer surface of the rege- nerator ring to create air cavities to suppress heat convection and mechanical deformation of the casing. The numerical simulation in- dicated that when the beds are pressurized at 10 bar the inner wall could have a maximum displacement (URES) of 0.17 mm and a maximum von Mises stress of 18.4 MPa (POM yield stress is about 63 MPa). Fig. 69 shows the resultant displacement with a 50 times distortion scale. Thus, effectively, the regenerator was split into 16 beds, each with an opening of 16.8º, i.e.,𝑤_bed∼29.7 mm. The beds were fabricated in a milling cutter and the edges were machined rounded with radius of 4.5 mm. The cross-section view of a pair of regenerator beds is shown in Fig. 70(a) and a transversal cutaway view (A-A) of the regenerator bed is presented in Fig. 70(b).

Magnetic circuit Regenerator

Cold end flow distributor

Hot end flow distributor Longitudinal rib

Circular ribs

Figure 69 – Resultant of the displacement (URES) for a bed pair when pres- surized at 10 bar. The distortion scale is enlarged 50 times. The image shows the second lengthwise half of the beds.

The resulting cross-sectional area per bed is 𝐴_c,bed ∼ 279.7 mm². Calculations showed that a regenerator with a porosity 𝜀 ∼ 0.36, a sphere diameter 𝑑_p ∼ 0.4 mm and 𝐿_bed ∼ 80 mm resulted

142 Chapter 4. Designing a novel rotary magnetic refrigerator

Magnetic circuit Regenerator

Cold end flow distributor

Cold heat exchanger Cold end flow

distributor

Hot end flow distributor Flow

bypass

Flow meter T2

T3

Cartridge heater

P1

P4 P2

Connection tools

T2 T1

T4

80 mm

(a) _(a)(b) ^Magnetic _circuit

Regenerator

Cold end flow distributor

Cold heat exchanger Cold end flow

distributor

Hot end flow distributor Flow

bypass

Flow meter

P3

T2

T3

Cartridge heater

P1

P4 P2 Connection tools

T2 T3

T1

T4

80 mm

(a) (b)

(b)

Figure 70 – (a) Cross-section view of a pair of regenerator beds and (b) a transversal cutaway view (A-A) of the regenerator bed.

in moderate pressure drops in the range of 0.2 to 2.5 bar for 0.1 <

𝜑< 1.0 and𝑓 =1 Hz. The final regenerator design shown in Fig. 71 can hold approximately 1.8 kg of packed Gd spheres. Depending on the pressure drop in the other device components (valves, heat exchangers, etc.) and on local pressure changes at the inlet and out- let of the regenerator, the pressure drop associated with the porous bed is such that the device can operate with moderate to high vol- umetric flow rates in the range of 50 to400^L/h. A detailed analysis of the experimental regenerator pressure drop will be presented in Chapter 5.

The regenerator was made from polyoxymethylene (POM) for its good mechanical properties, machinability, low water absorption, very low magnetic permeability, and good dimensional stability. The regenerator beds are coupled with a channeling ring at each end, also manufactured in POM, as shown in Fig. 72(a) and Fig. 72(b). Each regenerator bed has four channels (i.e., two at the inlet and two at oulet, on each end), totalling 64 channels. Each channel is composed of: (i) a stainless steel mesh screen (Mesh 100, 0.15 mm) held in place by (ii) a POM stopper to hold the cyclical loads, (iii) an acetal check valve and (iv) a Buna-N o-ring. An exploded view of the regenerator ring with all of its components is shown in Fig. 72(c). The sealing between the beds and the channeling rings is made by nitrile gaskets.

Stainless steel anchors were installed to avoid polyacetal wear.

The stainless steel wire screens were cut into 5.4-mm diam- eter circles to fit into the 6-mm diameter channels, as shown in Fig. 73(a). A special die cutting tool, shown in Fig. 73(b), was spe- cially designed and fabricated for this purpose.

The fluid flow is unidirectional in each channel. The parallel

4.5. Designing the regenerator 143

Figure 71 – Regenerator ring.

channels on each side are not interconnected to avoid dead volumes and losses due to fluid (blow) mixing (Jacobs, 2009). The entrances of the channels are tapered out to improve the flow distribution, although it is known that this increases the inlet pressure drop.

The inlet and outlet channels are diagonally opposed to each other across the regenerator beds to improve mixing.