# 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.

## 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

*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 (m

^{2})**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

^{−2}K^{−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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