State-space methods for calculating concentration dynamics in multizone buildings

This work explores analytic solutions to contaminant transport in multizone systems as a means of solving concentration dynamics more rapidly and providing insight into system behaviour. A general formulation is developed that is consistent with state-space theory. This is used to provide general and specific solutions for the concentration time series. In particular, an analytical expression is presented for the case of constant system parameters and constant input conditions using the eigenvalues and eigenvectors for the case of a diagonalisable state matrix for the multizone system. A method for using this expression to solve concentrations over a series of varying building and input states is developed, with potential for use as a rapid means of calculating concentrations at any time. In principle, this method could be used as a complementary method for solving concentration dynamics within multizone software. Analytical expressions for cumulative exposure for the same case are also presented. The characteristic behaviour of the solutions and their dependence on the eigenvalues and eigenvectors of the state matrix are explored for the steady state case and the decay solution. In particular, the behaviour in the late decay phase is shown to be characterised by a single decay rate, given by the smallest magnitude eigenvalue, and a fixed concentration ratio, given by the associated eigenvector. The independence of the concentration ratio in this phase to the initial conditions is also demonstrated. Two example cases are used to illustrate these and additional features.