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A microscale three-dimensional urban energy balance model for studying surface temperatures

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Abstract

A microscale three-dimensional (3-D) urban energy balance model, Temperatures of Urban Facets in 3-D (TUF-3D), is developed to predict urban surface temperatures for a variety of surface geometries and properties, weather conditions, and solar angles. The surface is composed of plane-parallel facets: roofs, walls, and streets, which are further sub-divided into identical square patches, resulting in a 3-D raster-type model geometry. The model code is structured into radiation, conduction and convection sub-models. The radiation sub-model uses the radiosity approach and accounts for multiple reflections and shading of direct solar radiation. Conduction is solved by finite differencing of the heat conduction equation, and convection is modelled by empirically relating patch heat transfer coefficients to the momentum forcing and the building morphology. The radiation and conduction sub-models are tested individually against measurements, and the complete model is tested against full-scale urban surface temperature and energy balance observations. Modelled surface temperatures perform well at both the facet-average and the sub-facet scales given the precision of the observations and the uncertainties in the model inputs. The model has several potential applications, such as the calculation of radiative loads, and the investigation of effective thermal anisotropy (when combined with a sensor-view model).

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Abbreviations

cair,H:

average heat capacity per unit plan area of air below z H , J m−2 K−1

C :

volumetric heat capacities, J m−3 K−1

ea :

water vapour pressure at z ref, hPa

G :

conductive heat flux density, W m−2

H/W:

mean building-height-to-street-width ratio of canyon or regular cube array

H/W3D:

mean wall-to-street area ratio of regular cube arrays

H can :

summed canopy patch convective sensible heat fluxes per canopy plan area, W m−2

h :

convective heat transfer coefficient (patch), W m−2 K−1

H :

convective sensible heat flux density (patch), W m−2

h top :

convective heat transfer coefficient between canopy air and boundary layer, W m−2 K−1

H top :

convective sensible heat flux density between canopy air and boundary layer, W m−2

i, j:

patch index/number

k :

thermal conductivity, W m−1 K−1

K↑:

upward shortwave radiative flux density, W m−2

K↓dif:

incident diffuse shortwave flux density, W m−2

K↓dir:

incident direct shortwave flux density, W m−2

K↓ i :

incident shortwave radiative flux density at patch i after multiple reflections, W m−2

K↑ i :

reflected shortwave radiative flux density at patch i after multiple reflections, W m−2

L h :

forcing height for patch convection, m

l p :

length of a patch side, m

L R :

average roof length, m

L↑:

upward longwave radiative flux density, W m−2

L↓:

downward longwave flux density from the sky, W m−2

L↓ i :

incident longwave flux density at patch i after multiple reflections, W m−2

m :

timestep index

n :

total number of patches

p :

number of layers in a patch (conduction)

P a :

atmospheric pressure at z ref, hPa

Q*:

net radiation flux density, W m−2

Q h :

convective sensible heat flux density (volume), W m−2

q :

reflection number

R q :

reflected radiative flux density (solar or longwave) at reflection q, W m−2

r w :

wall roughness coefficient

S d :

refers to the sub-domain

T(z):

air temperature at height z, K

T a :

air temperature at z ref, K

T app :

apparent surface temperature, K

T b (b = 1,...,p):

substrate layer temperatures, K

T can :

canopy air temperature (z <  z H ), K

T D :

deep-soil temperature, K

T INT :

building internal temperature, K

Tlog(z):

air temperature profile above z H , K

T R :

mean roof surface temperature, K

T r :

mean road surface temperature, K

T sfc :

surface temperature, K

T W :

mean wall surface temperature, K

U(z):

wind speed at height z, m s−1

U a :

wind speed at z ref, m s−1

Ueff(z):

effective wind speed at height z, m s−1

x, y, z:

location in space

x L :

building width, m

x W :

canyon (street) width, m

z 0h :

roughness length for heat, m

z 0m :

roughness length for momentum (patch), m

z 0town :

roughness length for momentum (domain), m

z d :

displacement height, m

z H :

mean building height, m

z horz :

height of patch forcing U(z) and T(z) above street level, m

z ref :

reference height for forcing data, m

α :

shortwave albedo

α r :

shortwave albedo of streets

α R :

shortwave albedo of roofs

α W :

shortwave albedo of walls

δ:

solar azimuth angle, °

Δα D :

sub-domain albedo change threshold

ΔQ s :

storage heat flux density (volume), W m−2

Δt :

timestep size, s

ΔT crit :

surface temperature change threshold, K

Δx :

layer thickness, m

\({\varepsilon }\) :

longwave emissivity

\({\varepsilon_{r}}\) :

longwave emissivity of streets

\({\varepsilon_{r}}\) :

longwave emissivity of roofs

\({\varepsilon_{W}}\) :

longwave emissivity of walls

\({\phi}\) :

solar zenith angle, °

γ:

degree of implicitness (conduction)

\({\eta}\) :

domain rotation from north (clockwise), °

λ c :

complete-to-plan area ratio

λ f :

frontal-area-to-plan area ratio

λ p :

building-to-plan area ratio

λ pH :

building-to-plan area ratio at z H

σ:

Stefan–Boltzmann constant, W m−2 K−1

\({\varOmega_{\rm INT}}\) :

building internal resistance, W m−1 K−1

\({\psi_{i,{\rm external}}}\) :

view factor from a patch i to external surfaces

\({\psi_{i,j}}\) :

view factor from a patch i to a patch j

\({\psi_{i,{\rm sky}}}\) :

view factor from a patch i to the sky

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Authors and Affiliations

  1. Department of Geography, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    E. Scott Krayenhoff

  2. Department of Geography, University of Western Ontario, London, ON, N6A 5C2, Canada

    James A. Voogt

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  1. E. Scott Krayenhoff
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  2. James A. Voogt
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Correspondence to E. Scott Krayenhoff.

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Krayenhoff, E.S., Voogt, J.A. A microscale three-dimensional urban energy balance model for studying surface temperatures. Boundary-Layer Meteorol 123, 433–461 (2007). https://doi.org/10.1007/s10546-006-9153-6

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