Perfect-mixing models are commonly used for analyzing the performance of building ventilation systems. Recently, they have been used to estimate the strength of gas sources in buildings. However, buildings are generally partitioned in such a way that scalar first-order models are insufficient to completely describe the dynamic behavior of the gas transport in buildings. This paper addresses the question of when scalar, first-order differential equations are useful for describing the aggregate dynamical behavior of multi-variable perfect-mixing processes. Sufficient conditions for the input-output relation of a multi-variable perfect-mixing process to be a first-order differential equation are derived. The conditions are related to sensor location and system design. It is shown that the design condition is too restrictive to be widely applicable. Therefore, an alternative first-order relationship is derived by replacing the design condition with a leakage condition. The results enable the estimation of aggregate parameters of multi-zone ventilation systems from scalar, first-order differential equations, which substantially simplifies the estimation problem.

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