Heat Transfer Model

The basic geometrical configurations for concrete pavements and bridge decks considered in this study are shown in Figure 1. For a pavement, the concrete thickness is considered to be 0.2 m, with a 0.2 m thick layer of soil (subbase) immediately beneath it. For the bridge deck, both surfaces of the concrete are assumed to be exposed to the environment, which is generally the case when temporary wooden forms are used [6,7]. While often the bottom surface is protected from the environment by steel forms which remain in place [6,7], the thickness of these forms (20 gauge or 0.9 mm) is such that their contribution to heat transfer should be minimal. For the bridge decks, it is further assumed that the convection coefficient for heat transfer is the same for both the top and bottom surfaces, and that no heat transfer by radiation (incoming sunlight or emitted radiation) occurs at the bottom surface. For the pavements, it is further assumed that at a depth of 0.2 m, the soil temperature is constant at a value of 13 ºC. (Variation of this soil temperature did not significantly influence the model results in this study).

  

Figure 1: Basic configurations of one-dimensional heat transfer model for concrete pavements and bridge decks.

Both systems shown in Figure 1 are modelled using a one-dimensional finite difference grid with a spatial resolution of 20 mm (e.g., ten nodes within the concrete). At the top surface, four modes of heat transfer are considered: conduction into the concrete, convection, solar absorption, and grey-body irradiation to the surroundings. Specifically, the heat flow contribution (in $\frac{W}{m^2}$) due to conduction is given by:

$$Q_{cond} = k_{conc} \times \frac{(T_0 - T_1)}{\Delta x}$$ (1)

where k_conc is the thermal conductivity of the concrete in $\frac{W}{m~^\circ C}$, T₀ and T₁ are the surface temperature and internal temperature at the first internal node, respectively, and Δx is the node spacing (20 mm). For convection, the heat flow is given by:

$$Q_{conv} = h_{conv} \times (T_0 - T_{ambient})$$ (2)

where T_ambient is the ambient temperature and h_conv is the convection coefficient in $\frac{W}{m^2~^\circ C}$. While several empirical relationships to estimate convection coefficients are available in the literature [2], for this study, the convection coefficient was calculated based on the wind speed available in the weather data files and the following equations used in the commercially available FEMMASSE system [8]:

$$h_{conv} = 5.6 + 4.0 v_{wind} \hspace{2.0cm} \mbox{for}~v_{wind}~\le~5~m/s$$ (3)

$h_{conv} = 7.2 \times v_{wind}^{0.78} \hspace{2.0cm} \mbox{for}~v_{wind}~>~5~m/s$

where v_wind is the measured wind speed in m/s.

For radiative heat transfer at the top concrete surface, two contributions are considered. The first is the radiation absorbed from the incoming sunlight. A simple equation for the incoming heat flow due to this source is given by [2]:

$$Q_{sun} = \gamma_{abs} * Q_{inc}$$ (4)

where Q_inc is the incident solar radiation (W/m² ) and γ _abs is the solar absorptivity of the concrete. For this study, the incident solar radiation was taken directly from the weather data files [5], while a value of 0.65 was used for the solar absorptivity [8]. Second, one must consider the emission of radiation from the "warm" concrete to the night sky. This heat flow can be estimated as:

$$Q_{sky} = \sigma \epsilon \times (T_{0K}^4 - T_{sky}^4)$$ (5)

where σ is the Stefan-Boltzmann constant (5.669 x 10^-8 W/(m² ºC⁴)), ε is the emissivity of the concrete, T_0K is the concrete surface temperature (in K), and T_sky is the calculated sky temperature also in K. For this study, a value of 0.9 was used for the concrete emissivity [2,8,9]. The sky temperature was estimated based on an algorithm presented by Walton [10] using the following series of equations:

$$T_{sky} = {\epsilon_s}^{\frac{1}{4}} \times T_{ambient} \hspace{0.5cm} (T~in~K)$$ (6)

where the sky emissivity, ε_s is given by:

$$\epsilon_s = 0.787 + 0.764 \times ln (\frac{T_{dew}}{273.}) \times F_{cloud}$$ (7)

where T_dew is the dewpoint temperature in K and with the cloud cover factor, F_cloud, as:

 

F_cloud = 1.0 + 0.024 N - 0.0035 N ² + 0.00028 N ³ (8)

where N is the "tenths cloud cover", taking values between 0.0 and 1.0. Because the above equations for surface irradiation are significantly different from approximations used elsewhere [2], it is worth noting that the equations provided here have been successfully used in a variety of predictive models of relevance to the construction community [10,11].

The above equations are employed in a general finite difference solution to one-dimensional heat transfer [4]. The time step in the finite difference scheme is established based on the layer thickness (20 mm) and the thermal properties of the materials, to ensure numerical convergence of the solution [4]. The ambient environmental conditions at any specific time are calculated by linear interpolation of the hourly values available from the weather files.