Ion mobility spectrum for Gerdien tubes by integral equation method
Ion mobility is an important parameter used in many ion-related research areas, such as atmospheric electricity, ion flow field around high voltage direct current (HVDC) transmission lines and so on. The spectrum of ion mobility is an elaborate tool that can be utilized to study the impacts of ambient conditions, such as temperature and humidity, on the atmospheric ion chemistry and motion of ions and hence help understand the underlying physics. This paper proposes an integral equation method for the recovery of ion mobility spectrum for Gerdien tubes widely used in ion mobility measurements. In view of the experimental observation of the multiplicity of ion species in the gas ionization products, it is believed that the multiplicity is associated with the difference between the ideal and real current-voltage (I-U) characteristics of the tubes and a qualitative explanation of this association is given. With the introduction of ion mobility spectrum density function, the qualitative explanation is formulated as an integral description which renders us to obtain the ion mobility spectrum by solving a Fredholm integral equation of the first kind. Tikhonov regularization method is utilized to solve this integral equation. To demonstrate the effectiveness of the integral equation method, numerical simulations and practical case studies are conducted. By simulations, it is found that the ion mobility spectrum recovered by the integral equation method is powerful enough for our practical use. Further, the integral equation method is applied to several sets of I-U characteristic data of air corona discharge ions and SF6 ions due to natural radiation recorded by Gerdien tubes. The calculated mean ion mobilities and total ion densities are in consistency with that by the commonly-used graphical method. The peaks in the recovered ion mobility spectrum show that there are several dominant species of ions in the ionization products. This helps us in understanding phenomena that are not comprehensible before. This integral equation method maybe very promising in the measurements of mobility spectrum of the ions under conductors with DC corona discharges.