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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

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.

Keywords

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

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

nit

number of iterations

nsb

number of sub-surfaces

k

number of subdivisions of a vertical surface

r

number of grid nodes

rse

sun–earth distance factor

z

zenith angle (rad)

A

surface area (m2)

A, A1, A2

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, E1

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

Jday

Julian day

I0

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)

TL

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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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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