Computational aspects of nodal multizone airflow systems

The multizone approach to steady-state airflow problems models a building as a network of discrete mass flow paths. A nodal formulation of the problem writes the governing equations in terms of the unknown pressures at the points where the flow paths connect. This paper proves conditions under which the nodal equations yield symmetric positive-definite matrices, guaranteeing a unique solution to the flow network. It also establishes relaxed conditions under which a nodal airflow system yields asymmetric matrices with positive eigenvalues, guaranteeing at least one solution. Properly exploiting the system properties greatly reduces the cost of numerical solution. Thus, multizone airflow programs such as C and C depend on symmetric positive-definite systems. However, the background literature neglects or simplifies the underlying assumptions, does not assert existence and uniqueness, and even contains factual errors. This paper corrects those errors, states the implicit assumptions made in the programs, and discusses implications for modelers and programmers.