Energy balance

Figure 1shows the energy sources that are considered when estimating the energy balance at the pavement surface. The energy that enters (or leaves) the ground comes from different sources such as the sun, the objects that surround the pavement and the air. This energy balance can be expressed as (Dempsey, 1985):
Q_g = Q_s + Q_l + Q_c                                                                                                                                            (1)
where
Qg is the energy absorbed by the ground [W m-2]
Qs is the net short wave radiation [W m-2]
Ql is the net long wave radiation [W m-2]
Qc is the convective heat transfer [W m-2]

Figure 1: Energy exchange between pavement surface and environment
It has to be noted that this approach does not include the effects of transpiration, condensation, evaporation and sublimation.
Q_s – net short wave radiation
The short wave radiation can be identified with the high frequency energy radiated by the sun that reaches the pavement surface and part of which is visible to the human eye. It is, therefore, a function of the extraterrestrial radiation incident on the atmosphere at a particular latitude, day and time and of the filtering and reflecting effect of the clouds that may be present at that location (ME-PDG, 2004).
                                                                                                                        (2)
where
a_s is the surface short wave absorptivity (0.8 – 0.98)
R is the potential extraterrestrial radiation [W m^-2]
S_c is the percentage of sunshine
A and B are constants that account for diffuse scattering and absorption by the atmosphere. The ME-PDG suggests values of 0.202 and 0.539 respectively but they will change with location.
The potential extraterrestrial radiation R is a function of latitude Ф [rad], declination δ [rad] and time of day. Its average value during one day, R_a, can be calculated by:
                              (3)
where
Ф is the latitude expressed in radians
δ is the declination expressed in radians and calculated by:
                                                                                              (4)
where day is the day of the year expressed as an integer.
The average value of R_a does not serve the purpose of estimating temperatures on an hourly basis. Nonetheless, assuming a parabolic distribution of R during daytime that starts from sunrise and finishes at sunset (Dempsey, 1985), it is possible to use R_a to estimate R at any given time. The total energy radiated during daytime must be equal to R_a multiplied by 24h, i.e. the areas of Figure 2(a) and (b) must be the same.
Figure 2: Calculating R from R_a (a) (b)
The time of sunrise and sunset for one particular day and latitude can be calculated as follows (University of Oregon, 2009):
Sunrise_angle = [rad]                                                                               (5)
Sunset_angle = [rad]                                                                        (6)
Sunrise_time = [s]                                                              (7)
Sunset_time = [s]                                                               (8)
From what given above, the parabola’s coefficients can be calculated:
                                                                                       (9)
                                                                                         (10)
[s]                                                                                    (11)
for                   (12)
R=0 for  or (13)
Q_l – net long wave radiation
Any object that is at a temperature higher than the absolute zero radiates energy. Since we are dealing with objects at low temperatures (certainly not comparable to the sun’s temperature), they radiate at low frequencies / long wave lengths, well outside the visible spectrum.
The general formula to calculate this energy is given by:
[W m^-2]                                                                                                                         (14)
where:
ε is the object’s emissivity
σ is the Stefan-Boltzman constant [W m^-2 K^-4]
T is the object’s temperature [K]
The net long wave radiation that contributes to the energy balance at the pavement’s surface is calculated considering the incoming (positive) energy radiated by the air and the outgoing (negative) energy radiated by the pavement itself. Therefore:
[W m^-2]                                                                                                          (15)
where:
R_ldc is the clear sky downward radiation:
                                                                                                                                         (16)
where the air emissivity εa can be calculated as a function of vapour pressure and Ta is the air temperature (Jacobs, 2004).
Q_x is the outgoing radiation:
                                                                                                                                           (17)
where εs is the surface emissivity and Ts is the surface temperature.
C_cc is the cloud cover correction factor:
                                                                                                                                      (18)
where N is the cloud base factor (0.9-0.8 depending on height) and W is the percentage of clouds.
Q_c – convective heat transfer
The pavement’s surface exchanges heat with the surrounding air by convection. This energy transfer is function of air and surface temperature and wind speed. The ME-PDG suggests the following equation:
                                                                                    (19)
where:
T_m = average of T_a and T_s [K]
U = average daily wind speed [m s^-1]

Posted in: model, temperature