Abstract
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.
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Abbreviations
- a, b, c, d :
-
wall building surfaces
- f :
-
fraction of rays which intersect the surface
- h :
-
altitude above sea (km)
- l :
-
proportion between block width and depth
- m :
-
total number of surfaces
- n :
-
number of neighbour urban units
- n it :
-
number of iterations
- n sb :
-
number of sub-surfaces
- k :
-
number of subdivisions of a vertical surface
- r :
-
number of grid nodes
- r se :
-
sun–earth distance factor
- z :
-
zenith angle (rad)
- A :
-
surface area (m2)
- A, A 1, A 2 :
-
absorptivity matrices
- B :
-
total outgoing radiative flux density (W m−2)
- B :
-
total outgoing radiative flux density vector (W m−2)
- D :
-
horizontal sky diffuse radiation flux density (W m−2)
- E, E 1 :
-
emissivity matrices
- F :
-
shape factor between surfaces
- F :
-
shape factor matrix
- G :
-
global radiation flux density (W m−2)
- H :
-
building block height (m)
- I :
-
identity matrix
- J day :
-
Julian day
- I 0 :
-
solar constant (W m−2)
- K :
-
direct surface irradiation flux density (W m−2)
- K ⊥ :
-
normal direct radiation flux density (W m−2)
- L ↓ :
-
sky downward longwave radiative flux density (W m−2)
- L :
-
building block width (m)
- M :
-
air mass (kg)
- W :
-
space between blocks (m)
- T :
-
absolute temperature (K)
- T L :
-
Linke turbidity factor
- α:
-
absorptance
- \(\mathbf{ \alpha}\) :
-
absorptivity vector
- δ:
-
layout azimuth (deg)
- ɛ:
-
emittance
- \(\mathbf{\varepsilon} \) :
-
emissivity vector
- \(\mathbf{\kappa} \) :
-
Ψ weighting area vector
- ρ:
-
Pearson correlation coefficient
- σ:
-
Stephan–Boltzman constant (W m−2K−4)
- \(\mathbf{\omega} \) :
-
\(\Omega \) normalized vector
- Φ:
-
surface net radiative flux density (W m−2)
- \(\mathbf{\Phi} \) :
-
net radiative flux vector (W m−2)
- Γ:
-
transformation matrix
- Λ:
-
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)
- Ψ:
-
urban matrix
- Subscripts:
-
- i, j :
-
general surfaces indexes
- g:
-
ground surface
- rf:
-
roof surface
- sf:
-
generic surface
- ub:
-
urban block
- w:
-
wall surface
- wg:
-
walls and ground surfaces
- x, y :
-
x- and y-axis
- S:
-
shortwave
- L:
-
longwave
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Panão, M.J.N.O., Gonçalves, H.J.P. & Ferrão, P.M.C. A Matrix Approach Coupled with Monte Carlo Techniques for Solving the Net Radiative Balance of the Urban Block. Boundary-Layer Meteorol 122, 217–241 (2007). https://doi.org/10.1007/s10546-006-9088-y
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DOI: https://doi.org/10.1007/s10546-006-9088-y