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

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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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Authors and Affiliations

  1. Renewable Energy Department, National Institute of Engineering, Technology and Innovation (INETI), Estrada do Paço do Lumiar 22, 1649-038, Lisbon, Portugal

    Marta J. N. Oliveira Panão & Helder J. P. Gonçalves

  2. IN+, Center for Innovation, Technology and Policy Research Instituto Superior Técnico, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Paulo M. C. Ferrão

Authors
  1. Marta J. N. Oliveira Panão
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  2. Helder J. P. Gonçalves
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  3. Paulo M. C. Ferrão
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Correspondence to Marta J. N. Oliveira Panão.

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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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