## 2 Materials and Methods
### Separation of Variable Gas-Ejection  
In-line separation is nearly as important as identification. For lighter materials, gas-ejection separation is most popular. As mentioned above, because the shading of low-energy and thickness calibration can be used to calculate the mass and center-of-gravity coordinates of materials, with which combines the variable gas-ejection, it is possible to achieve the in-line automatic sorting of multicategory scrap in one single measure, which would be a great solution to the sorting efficiency described in Section 1.4.5. In order to improve the traditional rejected - accepted method and realize identification and separation  
Figure 5: Flow-process diagram of non-ferrous metal identification by DE-XRT.  
of multicategory materials in one detection, the variable gas-ejection separation method is used to control the falling point range of multicategory materials. The schematic diagram is shown in Figure 6.  
Position synchronization and variable action on a range of materials during gas-ejection were also expounded comprehensively in our previous research work [39]. Here, variable gas pressure and/or variable action time were used to change the action impulse of gas-ejection, by which the falling points of various materials could be well controlled and one-time separation of various materials also could be realized. The operation principles are as follows: (i) gas pressure can be changed by means of analog signal to control the electro-pneumatic proportional valve. (ii) Action time can be changed by means of changing the ejection range of materials mentioned in Reference [39] because the action range of gas-ejection is directly proportional to the action time at constant line speed. For the shredded bulk materials, in general, the lower the density, the lighter the weight. Therefore, the falling point by gas-ejection of lower density materials should be usually controlled a little further. The action direction of gas-ejection at the ejection point was the normal direction of its trajectory, and the elevation angle of gas-ejection was \(\theta\).  
If gas-ejection impulse \(I_{e}\) = \(F_{eM}\)\(\times\)\(\triangle t\), \(F_{eM}\) is the mean action force of gas-ejection and \(\triangle t\) is the action time of gas-ejection, the gas-ejection impulse \(I_{e}\) under the condition of preset separation distance \(y_{n}\) is determined by the following equation:  
\[I_{e}=\frac{-b-\sqrt{b^{2}-4 ac}}{2a} \tag{13}\]  
where,  
\(D=\sqrt{g^{2}t_{e}^{2}+v^{2}}\)  
\(a=2 Dmg(t_{e}^{2}v^{2}-t_{e}vy_{n}-gt_{e}^{2}H_{b2l})\)  
\(b=2D^{2}m^{2}(g^{2}t_{e}^{2}y_{n}+t_{e}v^{3}-g^{2}t_{e}^{3}v-v^{2}y_{n}-2 gt_{e} H_{b2l})\)  
\(c=D^{3}m^{3}(Sy_{n}^{2}-gt_{e}^{2}v^{2}-2v^{2}H_{b2l})+gt_{e}^{2}\)  
where \(v\) is line speed of belt conveyor, \(t_{e}\) is time from material off belt to ejection point, \(\mathrm{P_{e}}\) (\(y_{e},z_{e}\)) is ejection point and its coordinates, \(\mathrm{P_{0}}\) (\(y_{o},H_{b2l}\)) is free parabolic falling point and its coordinates, \(\mathrm{P_{n}}\) (\(y_{n},H_{b2l}\)) is preset falling point and its coordinates, \(g\) is acceleration of gravity, \(H_{b2l}\) is vertical height of belt to falling point and \(m\) is mass of material, and the following equation holds:  
\[t_{e}=\frac{v}{g\tan\theta} \tag{14}\]  
\[y_{0}=v\sqrt{\frac{2H_{b2l}}{g}} \tag{15}\]  
Figure 6: Schematic diagram of falling point control by separation of variable gas-ejection.  
The mathematical derivation of the above conclusions is detailed in Appendix B.