A Matrix Approach Coupled with Monte Carlo Techniques for Solving the Net Radiative Balance of the Urban Block

A new method is developed for solving the shortwave and longwave net radiative balance of a three-dimensional urban structure, represented by parallelepiped blocks uniformly distributed in each direction. The method is based on a novel approach to determine the shape factors among surfaces, which are estimated by Monte Carlo techniques due to the complex geometry associated with the three-dimensional urban structure. Then, a set of linear equations is solved to quantify the radiative balance, in order to obtain their exact solution, considering all the inter-reflections among surfaces. The comparison between the new and the ray-tracing tracking methods resulted in a Pearson correlation coefficient of 0.996. However, by integrating the linear equations’ exact solution with Monte Carlo techniques, the new method reduces by a factor of 36 the central processing unit (CPU) time used to perform the calculations of the ray-tracing tracking method. The use of the model for a sensitivity study allows us to verify the effective absorptance and emittance increases with the canyon aspect ratio of the urban layout. An urban structure formed by square cross-sectional blocks absorbs more solar radiation than an urban structure formed by rectangular cross-sectional blocks. The approximation of a specific geometry for an equivalent bi-dimensional infinite street can be applied for rectangular cross-sectional blocks, where the width is 11 times or more greater than the depth dimension.