Boundary-Layer Meteorology

, Volume 122, Issue 1, pp 217–241 | Cite as

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

  • Marta J. N. Oliveira Panão
  • Helder J. P. Gonçalves
  • Paulo M. C. Ferrão
Original Article


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.


Monte Carlo Multiple reflections Radiative balance Three-dimensional geometry Urban longwave matrix Urban shortwave matrix 

List of Symbols

a, b, c, d

wall building surfaces


fraction of rays which intersect the surface


altitude above sea (km)


proportion between block width and depth


total number of surfaces


number of neighbour urban units


number of iterations


number of sub-surfaces


number of subdivisions of a vertical surface


number of grid nodes


sun–earth distance factor


zenith angle (rad)


surface area (m2)

A, A1, A2

absorptivity matrices


total outgoing radiative flux density (W m−2)


total outgoing radiative flux density vector (W m−2)


horizontal sky diffuse radiation flux density (W m−2)

E, E1

emissivity matrices


shape factor between surfaces


shape factor matrix


global radiation flux density (W m−2)


building block height (m)


identity matrix


Julian day


solar constant (W m−2)


direct surface irradiation flux density (W m−2)


normal direct radiation flux density (W m−2)


sky downward longwave radiative flux density (W m−2)


building block width (m)


air mass (kg)


space between blocks (m)


absolute temperature (K)


Linke turbidity factor



\(\mathbf{ \alpha}\)

absorptivity vector


layout azimuth (deg)



\(\mathbf{\varepsilon} \)

emissivity vector

\(\mathbf{\kappa} \)

Ψ weighting area vector


Pearson correlation coefficient


Stephan–Boltzman constant (W m−2K−4)

\(\mathbf{\omega} \)

\(\Omega \) normalized vector


surface net radiative flux density (W m−2)

\(\mathbf{\Phi} \)

net radiative flux vector (W m−2)


transformation matrix


total incoming radiative flux density (W m−2)

\({\bf \Omega}_{\rm L}\)

black surface emitted radiation vector (W m−2)

\({\bf \Omega}_{\rm S}\)

shortwave irradiation vector (W m−2)


urban matrix



i, j

general surfaces indexes


ground surface


roof surface


generic surface


urban block


wall surface


walls and ground surfaces

x, y

x- and y-axis






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Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  • Marta J. N. Oliveira Panão
    • 1
  • Helder J. P. Gonçalves
    • 1
  • Paulo M. C. Ferrão
    • 2
  1. 1.Renewable Energy Department, National Institute of EngineeringTechnology and Innovation (INETI)LisbonPortugal
  2. 2.IN+, Center for InnovationTechnology and Policy Research Instituto Superior TécnicoLisbonPortugal

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