# Solution of Orifice Hollow Cathode Plasma Model Equations by Means of Particle Swarm Optimization
## 4 Results
Figures 4-6 report the best ten solutions obtained for the operative point at 1A and 2A in the whole mass flow rate span. These ten solutions have been selected from the PSO algorithmooking at the cost function. In fact, the cost function \(\dagger\) is defined as the sum of the residuals \(\epsilon\) evaluated for each equation, as reported in Section 3.2. Thus, with this assumption, the value of \(\dagger\) can be used as a direct and global index of the solution quality and hence is presented in Figure 7.  
Figure 4: (a) Plasma density; (b) neutral density; (c) electronic temperature; (d) emitter temperature as function of mass flow rate. All the figures refer to physical properties evaluated in the orifice region in the case of operative points at 1A and 2A. Figure 5: (**a**) Plasma density; (**b**) neutral density; (**c**) electronic temperature as function of mass flow rate. All the figures refer to physical properties evaluated in the keeper region in the case of operative points at 1A and 2A. Figure 6: (**a**) Plasma density; (**b**) neutral density; (**c**) electronic temperature; (**d**) emitter temperature as function of mass flow rate. All the figures refer to physical properties evaluated in the emitter region in the case of operative points at 1A and 2A. Figure 7 presents the errors \(\epsilon\) evaluated as the mean value of the ten solutions at 2A, for example, with respect to the mass flow rates. A similar figure would be drawn at 1A and 3A. The ten solutions presented at 1A and 2A, Figures 4-6, are characterised by a J value that typically remains under the 5% from the minimum mass flow rate up to 0.6 mg/s. Beyond this value, and up to 1 mg/s, the residuals, \(\epsilon\), tend to increase as obviously do the cost function J. This circumstance is also evidenced by the greater dispersion of the solutions while moving toward the higher mass flow rate values. Also in this region of greater uncertainties, the J values areimited to 20% and 40%, respectively, for the cases at 2A and 1A. Moreover, in the worst case, the mean value of a single residual is not greater than 10% as in Figure 7, where the greater error is due to the calculation of the pressure in the orifice region. All the trends of plasma properties (Figures 4-6) reflect those reported in the openiterature [5; 6]: the neutral densities increase because of the internal pressure, particularly the trend isinear in the orifice and keeper regions (Equations (3) and (6)) and with 0.57 power in the emitter (Equation (16)); the plasma densities trends have also an increasing trend because more neutrals means more collisions, and thus ionization. The electron temperatures decrease with the mass flow rate in all regions because it decreases with the internal pressure as demonstrated in ref. [1]. The orifice temperature slightly increases because of the higher heat flux to the wall, whereas no particular trend can be assumed for the emitter temperature where, as for the orifice region, an increasing trend is expected, too. The electron temperatures (Figures 4c, 5c and 6c), and the plasma (Figures 4a, 5a and 6a), and neutral (Figure 4b, 5b and 6b) densities found by PSO are very close to each other, generating a confidence region that seems to be very tight. The higher dispersion is that of emitter plasma density (i.e., \(\pm\)20% of the average value). The wall temperatures of the emitter and orificeay withiness than 100 K bound. Nevertheless, it must be remarked that the PSO has been able to obtain the order of magnitude of neutral and plasma densities by searching within a range of three orders of magnitude. Figure 8 shows the PSO targeting the experimental measurements [26] of the discharge voltage as a function of mass flow rate at 2A (i.e., PSO using Equation (36)). It has been found that the results perfectly match the results calculated without the assumption made with Equation (36) The only difference regards the emitter wall temperature. Figure 9 highlights that with this approach, the expected increasing trend of the temperature can be detected.