Keywords:Sensor placement Markov matrix Set cover problem Indoor air quality Transmission of infectious diseases Chemical and biological warfare a b s t r a c t Air quality has been an important issue in public health for many years. Sensing the level and distributions of impurities help in the control of building systems and mitigate long term health risks. Rapid detection of infectious diseases in large public areas like airports and train stations may help limit exposure and aid in reducing the spread of the disease. Complete coverage by sensors to account for any release scenario of chemical or biological warfare agents may provide the opportunity to develop isolation and evacuation plans that mitigate the impact of the attack. All these scenarios involve strategic placement of sensors to promptly detect and rapidly respond.This paper presents a data driven sensor placement algorithm based on a dynamical systems approach. The approach utilizes the finite dimensional Perron-Frobenius (PF) concept. The PF operator (or the Markov matrix) is used to construct an observability gramian that naturally incorporates sensor accuracy, location constraints, and sensing constraints. The algorithm determines the response times, sensor coverage maps, and the number of sensors needed. The utility of the procedure is illustrated using four examples: a literature example of the flow field inside an aircraft cabin and three air flow fields in different geometries. The effect of the constraints on the response times for different sensor placement scenarios is investigated. Knowledge of the response time and coverage of the multiple sensors aides in the design of mechanical systems and response mechanisms. The methodology provides a simple process for place sensors in a building, analyze the sensor coverage maps and response time necessary during extreme events, as well as evaluate indoor air quality. The theory established in this paper also allows for future work in topics related to construction of classical estimator problems for the sensors, real-time contaminant transport, and development of agent dispersion, contaminant isolation/removal, and evacuation strategies.

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Predicting the movement of contaminants in the indoor environment has applications in tracking airborne infectious disease, ventilation of gaseous contaminants, and the isolation of spaces during biological attacks. Markov matrices provide a convenient way to perform contaminant transport analysis. However no standardized method exists for calculating these matrices. A methodology based on set theory is developed for calculating contaminant transport in real-time utilizing Markov matrices from CFD flow data (or discrete flow field data). The methodology provides a rigorous yet simple strategy for determining the number and size of the Markov states, the time step associated with the Markov matrix, and calculation of individual entries of the Markov matrix. The procedure is benchmarked against scalar transport of validated airflow fields in enclosed and ventilated spaces. The approach can be applied to any general airflow field, and is shown to calculate contaminant transport over 3,000 times faster than solving the corresponding scalar transport partial differential equation. This near real-time methodology allows for the development of more robust sensing and control procedures of critical care environments (clean rooms and hospital wards), small enclosed spaces (like airplane cabins) and high traffic public areas (train stations and airports).

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Volatile organic compounds, particulate matter, airborne infectious disease, and harmful chemical or biological agents are examples of gaseous and particulate contaminants affecting human health in indoor environments. Fast and accurate methods are needed for detection, predictive transport, and contaminant source identification. Markov matrices have shown promise for these applications. However, current (Lagrangian and flux based) Markov methods are limited to small time steps and steady-flow fields. We extend the application of Markov matrices by developing a methodology based on Eulerian approaches. This allows construction of Markov matrices with time steps corresponding to very large Courant numbers. We generalize this framework for steady and transient flow fields with constant and time varying contaminant sources. We illustrate this methodology using three published flow fields. The Markov methods show excellent agreement with conventional PDE methods and are up to 100 times faster than the PDE methods. These methods show promise for developing real-time evacuation and containment strategies, demand response control and estimation of contaminant fields of potential harmful particulate or gaseous contaminants in the indoor environment.