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34
= the thermal diffusivity; The convective heat flux (in W m-2) is calculated as:
= k/C where k is the thermal conductivity;
= the density; and Qc = hc (Ts - Ta ) (18)
C = the pavement heat capacity.
where hc is the heat transfer coefficient from the empirical
Together with this equation, a flux boundary condition at the equation (33).
pavement surface and a second flux condition at 3 m below
the surface are considered. T +T
0.3
0.00144 abs s a U d
hc = 698.24 a
i i 2 (19)
+ 0.00097 ( abs (Ts + Ta ))
0.3
The Surface Boundary Condition
Considering a differential element of the pavement surface, where U is the hourly wind speed (i m s-1) and a and d are
its thermal energy (temperature) will change to the extent two-dimensionless empirical parameters.
that the fluxes from above and from below do not balance. The The heat flux within the pavement at the surface is
various fluxes shown in Figure 31 lead to the following sur- expressed by Fourier's equation:
face condition:
Ts
Qf = k (20)
x Ts x
C = Qs - i Qs + Qa - Qr - Qc - Q f (15)
2 t
where Ts is pavement surface temperature, and k is thermal
where conductivity of asphalt concrete (in W m-1 k-1).
Combining these results, the following equation serves as
C = volumetric heat capacity of the pavement; the surface boundary condition:
T5 = pavement surface temperature;
x = the depth below the pavement surface; x Ts
C = Qs - i Qs + a T a 4 - T 4
x 2 t
s
= the (differential) pavement thickness for the energy
T
2 balance; - hc (Ts - Ta ) + k s (21)
Qs = heat flux due to solar radiation; x
~ = albedo of pavement surface, the fraction of reflected
solar radiation; The Bottom Boundary Condition
Qa = down-welling long-wave radiation heat flux from
the atmosphere; Commonly, a constant-temperature boundary condition
Qr = outgoing long-wave radiation heat flux from the at some distance below the surface is reported. For example,
pavement surface; Hermansson (19) used the annual mean temperature 5 m
Qc = the convective heat flux between the surface and the below the surface as a bottom boundary condition and Gui
air; and (20) used a measured temperature of 33.5°C at a depth of
Qf = the heat flux within the pavement at the pavement 3 m as the boundary condition. In the EICM model, temper-
surface. atures were measured from water wells across the country at
a depth of 10 to 18 m, from which an isothermal map was
The incoming and outgoing long-wave radiation (in W m-2) constructed. Such a constant-temperature boundary condi-
are calculated by: tion has the advantage of simplicity.
In this project, a different approach was used. Measured data
Qa = a T a
4 (16) in the LTPP database showed that temperatures at a depth
beyond 2 m tend to vary approximately linearly with depth.
Qr = T s4 (17) Therefore, a flux boundary condition was used at a depth of
3 m. That is
where
T
a = absorption coefficient of pavement; 3m = independent of depth (22)
x
= emission coefficient of pavement;
Ts = pavement surface temperature, k; This boundary condition is independent of location and
Ta = air temperature, k; and does not require a specific value. In addition, it is easy to
= 5.68 × 10-8W m-2K-4 is Stefan-Boltzman constant implement this boundary condition in the finite difference
