*5.1. Validation of the Mathematical Model*

The flow characteristics of the single and dual XMFCDs are both simulated by the proposed mathematical model. The boundary conditions of the simulation are the same as the working conditions in an electric propulsion system. The upstream xenon pressure of the XMFCD is set to 0.1 to 0.2 MPa and the downstream pressure of the XMFCD is set to 0. The temperature is set to 20 ◦C. The structure parameters of the proposed XMFCD and the physical properties of the xenon are both listed in Table 1.


**Table 1.** The simulation parameters of the XMFCD.

The flows at 11 pressure points between 0.1 to 0.2 MPa are calculated by the proposed mathematical model. In order to validate the mathematical model, the flows at the same pressures are tested. The measured and simulated mass flows of the single and dual XMFCDs are both listed in Table 2. Moreover, its errors are also calculated, which is shown in Table 2. The experimental and simulated flows of the single and dual XMFCDs are both plotted in Figure 7a,b, which show that the simulation flow results are in good agreement with the experiment results. According to Table 2, the error's absolute value of the measured and simulated mass flows of the single XMFCD is 0.00256 to 0.49 mg/s, which is 0.2% to

5.5% of the experimental results. Moreover, the error's absolute value of the measured and simulated mass flows of the dual XMFCD is 0.00318 to 0.03589 mg/s, which is 0.3% to 5.7% of the experimental results. Therefore, the mathematical model proposed in this paper is proven to be very effective for predicting the flow characteristics of the proposed XMFCD. Besides, the experimental results show that the mass flow of the proposed XMFCD is linearly proportional to the inlet pressure.


**Figure 7.** Comparison of the experimental and simulated mass flow of the proposed XMFCD. (**a**) Single XMFCD; (**b**) Dual XMFCD.

#### *5.2. Influence of Structural Parameters*

In order to realize the optimization design of the proposed XMFCD, the influence of structural parameters on flow characteristics should be investigated. According to Figure 3, the diameter of orifice *D*o, and the width of groove *w*<sup>g</sup> are the parameters that are easy to adjust under volume limitation. So the influence of these two parameters on the flow characteristic is focused on analysis using the mathematical model validated above. In the following simulations, the upstream xenon pressure of the XMFCDs is set to 0.15 MPa.

Firstly, the mass flow of the single and dual XMFCDs with different *D*o (40, 50, 60, 70, 80, 90, and 100 μm) and other parameters the same as shown in Table 1 are simulated. Moreover, the simulated mass flow results are shown in Table 3 and Figure 8a. According to the data analysis, two conclusions can be obtained as follows: (1) for XMFCDs with the same structural parameters, the mass flow *Q*<sup>1</sup> of the single XMFCD can be approximately described as 1.4 times of the mass flow *Q*<sup>2</sup> of the dual XMFCD, that proves the flow of N series orifices is 1/ √ N of a single orifice's flow, and (2) for the proposed XMFCDs, the mass flow is proportional to the square of the orifice diameter.


**Table 3.** The simulated mass flow of the XMFCD with different diameters of orifice (*D*o) and widths of groove (*w*g).

**Figure 8.** The simulated flow of the XMFCD with different structural parameters. (**a**) Simulated flow with different *D*o; (**b**) Simulated flow with different *w*g.

Then, the mass flow of the single and dual XMFCDs with different *w*<sup>g</sup> (30, 50, 80, 100, 130, 150, 180, and 200 μm) and other parameters the same as shown in Table 1 are simulated. The simulated mass flow results are listed in Table 3 and drawn in Figure 8b. Referring to the simulated data, we can get conclusions as follows: (1) when *w*<sup>g</sup> ≥ 80 μm, the flow of the XMFCD is mainly determined by the orifices and the effect of the grooves on the flow is not obvious, and the mass flow in this case conforms

to the above flow formula of the series orifices; and (2) when *w*<sup>g</sup> < 80 μm, the flow of the XMFCD decreases sharply as *w*<sup>g</sup> decreases because the throttling effect of the grooves becomes increasingly obvious. It is worth noting that when *w*<sup>g</sup> = 80 μm, its equivalent diameter of the groove is about 100 μm which is as same as the diameter of the orifices. Therefore, in other words, only when the equivalent diameter of the groove is smaller than the diameter of the orifices, the impact of the groove on the XMFCD's flow will become significant.
