Mathematical models for the dynamic response of continuous particulate air monitors

Mathematical models are developed for the dynamic, time-dependent count rate response of fixed-filter (FF), rectangular-window moving filter (RW), and circular-window moving filter (CW) continuous particulate air monitors. These models relate the monitor input, a time-dependent concentration, to the monitor output, its dynamic countrate. The models take the form of a single integral for the FF case, the sum of two double integrals for the RW case, and the sum of three triple integrals for the CW case. The models apply for a single isotope of any half-life, and for sampled concentrations of any time dependence, at any time during a concentration transient. The models are solved for selected concentration behaviors, providing analytical expressions for the prediction of monitor response, and plots comparing these responses of the three monitor types are presented. A numerical simulation of monitor response has also been developed, which starts from first principles, and does not use the analytical solutions. It is shown that this simulation agrees with the analytical solution.