The design and application of bridge deicing systems require an understanding of heat-transfer mechanisms of the bridge. One of these systems is geothermal foundations that support structural loads and utilize heat exchange with the surrounding soil. To design such a bridge deicing system and ensure ice-free surfaces during winter, accurate prediction of the temperature of bridge decks is vital. Heat-transfer mechanisms of the bridge deck include many factors such as convection between air and deck, solar radiation, and longwave radiation. Despite considerable research in this area, traffic vehicle effects on the heat balance of the bridge have not been fully investigated. In this paper, a two-dimensional finite-element analysis that focuses on natural factors is proposed and validated using measured data. After the validation, the model was extended to include the vehicle effects for conditions of both light and heavy traffic, considering tire frictional heat and vehicle-induced convection and radiation. The results show that the vehicular traffic increased the bridge surface temperature by up to 2°C with light traffic and by up to 4°C with heavy traffic, thus providing an advantage for deicing. Traffic effects can cool the bridge surface temperature down by up to 2°C, mainly during the summertime. Therefore, the traffic effects can be optionally considered during the design of bridge deicing systems.
Road infrastructure systems support safe, rapid, and reliable transportation of people, goods, and services in a wide range of weather conditions, including ice and snow conditions. The most common method for pavement deicing during winter utilizes chemical salts that lower the freezing point of water on the pavement or bridge-deck surfaces. However, deicing chemicals are corrosive and reduce the longevity of bridge infrastructure making it difficult to achieve the national goal of a 100-yr or more bridge service life (Koch et al. 2002; Azizinamini et al. 2014). Furthermore, pavement or bridge temperature can be lower than the working range of these chemicals in extreme cold weather (Minsk 1999; Lund 1999; Joerger and Martinez 2006). Alternative approaches to mitigate the corrosion damage caused by the deicing chemicals have been attempted since the 1950s. For instance, the pavement of the bridge deck can be heated by electric cables, conductive concrete, or thermal pipes circulating a heated fluid or by using geothermal heat exchange between foundations supporting the bridge structure and surrounding soils connected to pipes embedded in the bridge deck (Zenewitz 1977; Lee et al. 1984; Cress 1995; Xie et al. 1996; Xiao et al. 2013). Design of bridge deicing methods requires an understanding of heat-transfer mechanisms of the bridge and accurate prediction of bridge-deck temperature. In addition, the thermal energy demand for different geographic locations and weather conditions is needed to design efficient bridge heating systems considering the variations of heat-transfer boundaries at the bridge surface.
The bridge temperature depends on local weather and environmental conditions (i.e., air temperature, wind speed, precipitation, longwave or infrared radiation, and solar or shortwave radiation). One of the important factors that balances the bridge surface temperature is the incoming and outgoing longwave radiation. However, incoming longwave radiation is the most poorly quantified from observations (Trenberth et al. 2009; Trenberth and Fasullo 2012), which will be discussed in this paper.
The passage of vehicles also influences the temperature of the pavement and bridge surfaces (Parmenter and Thornes 1986). The temperature of the pavement surface decreases with increasing vehicle speed and increases with vehicle body temperature and traffic volume (Ishikawa et al. 1999). Limited numerical modeling simulated the effects of vehicles (e.g., vehicle-induced convection and radiation) on the road surface temperature to investigate the contributions of those vehicle factors (Prusa et al. 2002; Fujimoto et al. 2008, 2012; Khalifa et al. 2016, 2018). Moreover, there is no model combining the effects of natural wind and vehicle-induced wind on the convection boundary on the bridge-deck surface, which is included in the analysis presented in this paper. To improve the design of winter deicing systems for bridges, a predictive approach is proposed that can accurately predict the temperature of the bridge deck and assess the effects of friction heat of tires and vehicle-induced radiation and convection.
a. Natural factors
The pavement heat-transfer mechanisms have been studied by a number of researchers who contributed to the design guidelines of deicing systems (e.g., Chapman 1952; Leal and Miller 1972; Schnurr and Falk 1973; Ramsey et al. 1999; Chiasson et al. 2000; Rees et al. 2002; Qin and Hiller 2013). From these studies, the heat fluxes due to natural factors are summarized in Fig. 1a. These include 1) conduction within the bridge slab due to temperature gradients between the top and bottom surfaces; 2) convection (or sensible heat exchange) between the ambient air and the bridge; 3) radiation to the bridge (i.e., solar or shortwave radiation and incoming longwave radiation) and from the bridge (i.e., outgoing longwave radiation to ambient air, water, and ground, etc.); and 4) precipitation, which includes both sensible (associated with rainfall and snowfall on the bridge deck) and latent heat due to evaporation and melting (heat of fusion).
Two modeling approaches are used to simulate the temperature of the bridge with natural factors; one is defined as the “Improved model,” and the other one is defined as the “American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) model.” The ASHRAE model is based on the method suggested by the ASHRAE Handbook (ASHRAE 2011). In addition to the factors considered in the ASHRAE-based model, in the Improved model 1) the atmosphere is considered as a graybody with an effective emissivity for the calculation of the incoming longwave radiation, and 2) natural convection is considered. The third model used in this paper is identified as the “Vehicle model,” which combines the Improved model (natural factors) and vehicle effects.
b. Vehicle effects
The model shown in Fig. 1a considers the natural factors in the prediction of the bridge surface temperature. However, the passing vehicle also alters the heat-transfer boundary at the pavement surface for different traffic conditions. Using the thermal mapping technique, heavy traffic flow was found to increase the temperature of the pavement surface by approximately 1.5°C during rush hours (Gustavsson and Bogren 1991). Chapman and Thornes (2005) reported 1.5°C temperature increase of the road surface due to the increase of traffic on a multilane highway.
Numerical results of Parmenter and Thornes (1986) indicated that the increased volume of vehicles at a low speed increased the road surface temperature (RST) up to 2°C. Ishikawa et al. (1999) conducted a boundary simulation of road surface with snow cover to evaluate the contribution of infrared radiation from the vehicles under different traffic volumes. The total incoming longwave radiation of the road surface with traffic increased by 50% in comparison with no traffic for 30 min. Chapman et al. (2001) assumed an increase of 2 m s−1 of wind speeds induced by vehicles in the theoretical simulation to simplify the vehicle effect on the convection. The results of the Town Energy Balance model for urban areas proposed by Khalifa et al. (2016, 2018) showed that traffic increased the RST by 2°–3°C depending on the traffic rate and vehicle speed. Fujimoto et al. (2010) reported that vehicles stopped at traffic signals increased RST by 3°–4°C. Fujimoto et al. (2008, 2012) also conducted numerical simulations considering the vehicle effect on pavement condition and presented the relationship between the vehicle speed and vehicle-induced wind; the simulation showed that the RST at the vehicle-passage area is ~3°C lower than non-vehicle-passage area as a result of the vehicle-induced convection. However, the work by Fujimoto et al. (2008, 2012) and Khalifa et al. (2016, 2018) did not consider the interaction between the natural wind and vehicle-induced wind, which is addressed in this paper.
As shown in Fig. 1b, the effects of vehicles on the heat transfer of the bridge-deck surface include 1) longwave radiation from the bottom of the vehicle , 2) corrected convection heat induced by the vehicle and natural wind , 3) frictional heat from tires , and 4) the effect of the presence of vehicles on the incoming longwave from the atmosphere and incoming shortwave radiation .
To estimate the bridge surface temperature, the ASHRAE model was initially utilized. An improved boundary model (Improved model) of the bridge-deck surface was then developed that considers natural factors and models proposed by ASHRAE (2011), Ramsey et al. (1999), Chiasson et al. (2000), and Xiao et al. (2013). A two-dimensional (2D) simulation of the Improved model was validated using results of measured temperatures of the Jamestown Verrazzano Bridge (Rhode Island) that were reported by Tsiatas et al. (2002). Using the 2D model with a series of meteorological data from the local weather station (e.g., air temperature, solar radiation, and hourly average wind speed), the temperatures of the bridge-deck surface were calculated and compared with measured temperatures. After validation, the Improved model was extended to incorporate the effects of vehicles, which is referred to as the Vehicle model in this paper. The effective wind velocity was introduced using the concept of the effective pollutant advection velocity proposed by Rao et al. (2002) to combine the natural wind and vehicle-induced wind. The daily and seasonal vehicle effects were also investigated. Furthermore, the heat flux contribution of the vehicles on the bridge surface is also presented in clear and snowing weather conditions, separately.
3. Numerical models
a. Governing equations
All of the equations that are mentioned in this paper are given in Table 1, and the main notations and subscripts used in the paper are shown in Table 2. In the Improved model, the finite-element method was used to solve the heat conduction equation that describes the temperature distribution of the bridge slab. The governing transient 2D heat conduction equation can be expressed as Eq. (1), where α is the thermal diffusivity of the bridge-slab material that is calculated using Eq. (2).
b. Natural boundaries
The heat fluxes at the top surface of the bridge deck arising from all natural factors shown in Fig. 1a are presented in a partial differential equation [Eq. (3)], which includes conductive heat flux , net radiation (longwave and shortwave radiation: and ), convective heat flux , sensible heat flux due to falling rain , latent heat flux due to snow melting , and evaporation heat flux of water or melted snow . As discussed in the following sections, and in Eq. (3) can be obtained using Eqs. (4)–(12). Determining the heat fluxes related to precipitation (, , and ) is based on the method suggested by ASHRAE (2011). The bottom surface of the bridge slab is subjected to longwave radiation and convection , which are given by Eqs. (13)–(15).
1) Net radiation
(i) Shortwave radiation
The shortwave radiation absorbed by the top surface of the bridge can be estimated by Eq. (5), where a is albedo (reflection coefficient) of the bridge surface, which ranges from 0.25 to 0.3 for weathered concrete, and an average value of 0.275 was used in the models when the surface is dry (Bretz et al. 1998). The albedo of the bridge surface was assumed to be 0.45 during snow and 0.14 during rain (Dingman 1994; Levinson and Akbari 2002).
(ii) Longwave radiation
Longwave radiation at the top surface of the bridge includes incoming longwave radiation and outgoing longwave radiation . The incoming longwave radiation is defined as the total irradiance within the infrared part of the electromagnetic spectrum (4–100 μm). There are two common methods of representing from the atmosphere on the basis of the Stefan–Boltzmann law. The first method treats the atmosphere as a blackbody with a uniform temperature Tsky. Another method treats the atmosphere as an isothermal graybody with temperature same as the air, and an effective emissivity of the atmosphere is employed (Ramsey et al. 1982). The first method was used in the ASHRAE model (ASHRAE 2011; Ramsey et al. 1999; Chiasson et al. 2000; Liu et al. 2007; Xiao et al. 2013) for the heat-transfer simulation of pavement or bridge systems, and the calculated Tsky is 10°–40°C lower than air temperature (Ramsey et al. 1982). However, extensive studies showed that the longwave radiation received by the ground surface corresponds essentially to radiation emitted by the lowest 100 m of the atmosphere, where the temperature is close to air temperature Ta (Zhao et al. 1994; Brown and Dunn 1998; Trigo et al. 2010; Moene and van Dam 2014). Therefore, the second method was employed to calculate in the Improved model presented in this study, where Ta was considered as the radiant temperature, and the effective emissivity of the atmosphere was used.
Following the procedures of Brock and Arnold (2000) and Brown and Dunn (1998), the longwave radiation at the top bridge surface was computed using Eqs. (6)–(8). The incoming longwave radiation is estimated by Eq. (7), where ε* is the effective emissivity of the atmosphere, which ranges from 0.6 to 1 depending on the cloud cover and air temperature (Unsworth and Monteith 1975; Prata 1996; Brock and Arnold 2000; Herrero and Polo 2012), and σ is the Stefan–Boltzmann constant (5.67 × 10−8 W m2 K−4). According to Ryu et al. (2008), the outgoing longwave radiation can be estimated by Eq. (8), where εs is the bridge surface emissivity, which is equal to 0.9 for concrete according to Bergman et al. (2011) and Levinson and Akbari (2002).
Radiation at the bottom surface only includes longwave radiation , which is estimated using Eq. (14). The incoming longwave at the bottom originates from ambient environment (ground or water surface). The air temperature is assumed to be equal to the ambient environment temperature (Chiasson et al. 2000), and the emissivity of the ambient environment beneath the bridge εe is assumed to be equal to 0.96 according to Bergman et al. (2011).
The convective heat flux at the bridge top and bottom surfaces was calculated using Eqs. (9) and (15), and the convective heat-transfer coefficient h at the bridge surface for the forced convection is defined in Eq. (10). Nusselt number and Reynolds number ReL were calculated using Eqs. (11) and (12) according to ASHRAE (2011). ASHRAE (2011) recommends using the width of the bridge slab as the characteristic length Lch for the calculation of h in Eqs. (10) and (12). The convective heat-transfer coefficient for natural convection was estimated using the method suggested by Walton (1981) and Xiao (2002).
3) Precipitation (+ + )
The method of estimating sensible heat flux due to falling rain , latent heat flux due to snow melting , and evaporation heat flux of water or melted snow recommended by ASHRAE (2011) was used in this study. Deicing salts are commonly used on the bridge surface during the snowing period, and both physical effects (e.g., hygroscopicity) of the salts and snow accumulation on the bridge surface were neglected in the model. For the heated bridge deck, the snow accumulation can be also neglected.
The original snow-melting model proposed in ASHRAE (2011) neglects the effects of solar radiation during snow periods. However, the model referred to in this paper as the ASHRAE model does account for the effects of solar radiation to allow for applying it to a year-round simulation. The ASHRAE model is compared with the Improved model without vehicle effects to examine the contributions of longwave radiation and natural convection as described earlier in this paper.
c. Vehicle boundaries
As shown in Fig. 1b, the vehicle effects include convective heat flux combining natural wind and vehicle-induced wind, longwave radiation from vehicle bottom , corrected incoming longwave and shortwave radiation affected by the vehicles, and tire frictional heat . The effects of vehicle exhaust gases on the temperature and humidity of the surrounding air were neglected in the model, since bridges are usually constructed in an open space, and the diffusion of the exhaust gases to the surrounding air is much faster than the urban area. The heat fluxes at the top surface of the bridge deck due to both natural factors (Fig. 1a) and vehicle effects shown in Figs. 1b and 1c can be represented by the partial differential equation [Eq. (16)], which will be described in this section.
1) Vehicle-induced wind
The vehicle-induced wind speed (VIWS) was calculated following the method proposed by Fujimoto et al. (2008). Figure 2 describes the VIWS when a vehicle passes a point (point O in the figure). The VIWS increases linearly to a peak speed at time tvmax and decreases exponentially after the peak velocity until time tviw. The tviw is the lifetime of vehicle-induced wind that alters the convection boundary of the bridge-deck surface. During the passing time of the vehicle tpv, both the sensible heat and longwave radiation induced by the vehicle will influence the temperature of the bridge-deck top surface. In the postpassing time tpp, only the convection or sensible heat is affected by the vehicle. The equations proposed by Fujimoto et al. (2008, 2012) were used to calculate tvmax, tviw, and tpv.
Fujimoto et al. (2008, 2012) presented the relationship of vehicle speed |Vυ| and induced wind speed |Vviw| under calm conditions; however, the model does not combine the effects of the natural wind in various directions and vehicle-induced wind. Therefore, the effective wind velocity was introduced, which uses the concept of the effective pollutant advection velocity proposed by Rao et al. (2002). The effective pollutant advection velocity components can be calculated by adding the natural wind velocity and the total wake velocity deficit components induced by the vehicle in the horizontal plane. The wake velocity deficit is a function of the relative velocity Vrd between the vehicle Vυ and natural wind Vnw in the driving direction, which was studied by many researchers using turbulent dispersion modeling to estimate the carbon monoxide concentration in the vehicle wake of the highways (e.g., Eskridge and Hunt 1979; Eskridge and Thompson 1982; Hider et al. 1997). For the heat transfer of the bridge, the related region only includes the turbulent wake of the vehicle close to or above the traffic lane. The wake region in the pollutant dispersion models is much larger than the road or pavement area. Therefore, the concept considering the wind direction in the pollutant dispersion model was combined with the vehicle-induced wind model proposed by Fujimoto et al. (2008, 2012). The relative velocity Vrd of the vehicle and natural wind in the driving direction calculated using Eq. (17) will be used to obtain the |Vviw| in the driving direction, which is different from the model proposed by Fujimoto et al. (2008).
The |Vviw| is integrated and equalized to uniform vehicle-induced wind speed |Veviw| in the period of tviw as represented by the dotted line in Fig. 2. Once the |Veviw| by one vehicle is computed, the hourly traffic volume (vehicles per hour) is used to calculate the hourly average vehicle-induced wind speed |Vaviw| using Eq. (18). The |Vaviw| is combined with the natural wind speed |Vnw| using Eq. (19) to obtain the effective wind speed |Vew| shown in Fig. 3. In addition, |Vnw| was replaced by |Vew| to calculate the Reynolds number in Eq. (12).
From the observation of Fujimoto et al. (2012) on the distribution of the vehicle-induced wind speed in the transversal direction of roads, the vehicle-induced wind speed on the ambient lane is ~7% of the vehicle-induced wind speed on the lane where the vehicles are passing. Therefore, the effect of traffic on the convection from other lanes is neglected. In the model, uniform traffic flow rate is assumed, and the length of all of the vehicles Lυ is 4.7 m.
2) Vehicle-induced radiation effect
The temperature of the vehicle bottom is 4°–44°C higher than the ambient air depending on the part of vehicle and the running time (Ishikawa et al. 1999; Fujimoto et al. 2008). The average temperature at the bottom of the vehicle Tvb was assumed as 26.2°C higher than the air temperature as recommended by Fujimoto et al. (2008). The vehicle-induced infrared radiation is given by Eq. (20), where ευ is the emissivity of the vehicle bottom, with value equal to 0.8 according to Fujimoto et al. (2008), and OV is the occupancy of vehicle (i.e., percent of time for a point on the road to be occupied by vehicles) as evaluated by Eq. (21). In consideration that the pavement is screened by running vehicles, the incoming shortwave and longwave radiation are corrected by Eqs. (22) and (23) so as to decrease with increasing traffic flow rate. The longwave radiation from the bottom of the vehicle, however, increases with traffic flow rate. The effect zone of the vehicle for radiation is the width of the vehicle Wυ, which is 1.8 m in the model as shown in Fig. 1c. The radiation on the road surface outside the passing vehicle is the same as the natural boundary. The width of the lane Wla is assumed as 3.6 m. The effects and contribution of the vehicle on radiation boundary are shown in Eq. (16).
3) Frictional heat of tires
The friction between moving surfaces in contact converts kinetic energy into heat (Juga et al. 2013). Most of the rolling friction heat of tires is dissipated away into the ambient air and only a fraction is deposited into the road surface. The tire frictional heat generated by vehicles Ef can be estimated by Eq. (24), and the contribution of the frictional heat to the whole lane of the bridge is calculated using Eq. (25) according to Prusa et al. (2002). The vehicle mass mυ is assumed as 1600 kg. The rolling friction coefficient μ and fraction of the frictional heat directly affecting the road surface βf are shown in Table 3 for different conditions including dry surface, rain, and snow. The light, moderate, and heavy snow rates were classified by Rasmussen et al. (1999).
4. Model validation
a. Description of the bridge
The Jamestown Verrazzano Bridge spans the West Passage of Narragansett Bay and links North Kingstown and Jamestown, Rhode Island. The bridge, with a width of 22.5 m, has four travel lanes separated by a concrete Jersey barrier. The total length of the bridge is 2240 m, with a main center span of 183 m and two side spans of 83 m. The bridge is a symmetric double-cell (south and north cells), posttensioned concrete box girder bridge. Figure 4a shows the dimension of the bridge girder of the north cell. The thicknesses of the top slab and the bottom slab are 0.31 and 0.20 m, respectively, and the depth of the girder is 3.05 m. Tsiatas et al. (2002) monitored the strains and temperatures at different locations of the bridge between 17 September 1997 and 23 September 1998 (approximately 1 yr) and summarized results in a research report submitted to the Rhode Island Department of Transportation. The locations of the sensors in the bridge are shown in Fig. 4b. The temperatures of the top and bottom slabs were measured at 3.63 and 3.33 m from the centerline of the bridge girder, respectively (Fig. 4a). The temperatures of the bridge top slab were monitored at depth increments of 50.8 mm (i.e., 50.8, 101.6, 152.4, and 203.2 mm). Temperatures of the bottom slab were measured at depths 25.4, 50.8, 101.6, 152.4, and 177.6 mm from the bottom surface as shown in Fig. 4b.
To validate the model, monitored data summarized by Tsiatas et al. (2002) were compared with the computed results. In addition to the whole-year data, Tsiatas et al. (2002) also presented data over a period of 10 days for different weather conditions during the monitoring period for five different periods. The weather conditions of these 10-day periods are described in Table 4. Both the whole-year data and the 10-day data were used for the validation of the model.
b. Finite-element models
A cross section of the bridge slab was modeled using a two-dimensional model in the ANSYS software package to simulate the heat transfer of the bridge (ANSYS, Inc., 2013). The “PLANE55” 2D thermal solid four-node rectangular element was used with heat flux of solar radiation , infrared radiation , and sensible and latent heat fluxes of precipitation (, , and ) applied on the top-surface nodes. As shown in Fig. 1a, however, the bridge surface transfers heat through both heat flux and convection boundaries, which can override each other on the surface nodes. To model the heat convection, the “SURF151” element was used at the bridge–air interface at which the convection between ambient air and concrete was applied. Because of the complicated boundaries at the bridge surface, the finite-element mesh of the model was refined near the bridge-slab surface, resulting in approximately 0.05 m × 0.05 m elements to yield accurate results (see Fig. 4).
c. Material properties
The thermal properties of concrete and air used in the model are summarized in Table 5. Since the heat transfer of the bridge slab was modeled for a full year, the thermal properties of air at average temperature of 10°C were used as based on the properties reported by Bergman et al. (2011).
d. Input weather data
The hourly weather data inputs were acquired from Weather Underground (2012) except the cloud cover and solar radiation, which were available at the National Solar Radiation Database (National Renewable Energy Laboratory 2010). To evaluate the difference between the input weather data from Weather Underground (2012) and local weather at the Jamestown Verrazzano Bridge, the weather data from two local weather stations were compared. One weather station (station identifier: KUUU) belongs to the Automated Surface Observing Systems Program of the National Weather Service in Newport, Rhode Island, which is around 6 mi (10 km) from the bridge. The other weather station is in North Kingstown (station identifier: KOQU), which is around 4 mi (6.5 km) from the bridge. The bridge is approximately located between those two weather stations. The hourly weather data from the North Kingstown weather station are only available from 0800 to 2200 local time (LT) each day during the simulation period. Therefore, the weather data from KUUU were used in the simulation. The weather data differences between those two weather stations for the simulation period are as follow: 1) air temperature and dewpoint differ by less than 2°C, 2) relative humidity differences are less than 10%, and 3) wind speed differences are less than 25%. Therefore, the input weather data may have an error of 1°C air temperature, 5% relative humidity, and 15% wind speed differences between the local weather of the bridge and the input weather data. A sensitivity analysis of wind speed showed that 15% wind speed differences can lead to ~0.5°C temperature difference of the bridge deck.
The daily weather data used for the simulation period from Weather Underground (for KUUU) were also compared with those of the National Oceanic and Atmospheric Administration (station identifier: USW00014787) to validate the input data for the models (Weather Underground 2012; NOAA 2018). The weather data of the two sources have slight deviations. For example, the root-mean-square deviation (RMSD) for the comparison of air temperatures is less than 0.5°C, and the RMSD for the comparison of the precipitation is 2.3 mm day−1. However, hourly weather data are not available from NOAA (2018) except for precipitation. Therefore, Weather Underground data were utilized.
The wind speed obtained from the weather station is measured at 10 m above the ground surface; therefore, the wind speed at the bridge surface was corrected using Eq. (26) (Petersen et al. 1998a,b) because the bridge is at an elevation of 53.3 m above sea level. It is assumed that the wind speed at sea level below the bridge is the same as that at the ground surface of the weather station. The corrected input wind speed was 1.3 times that of the local weather station. The temperature and relative humidity of air at the bridge location were assumed to be the same as at the weather station. In the simulations, the time step is 10 min, and the inputs of each time step were calculated from hourly weather data using a linear interpolation method.
Since the Jamestown Verrazzano Bridge is separated by Jersey barriers, the characteristic length of the top slab is 11.4 m, which is one-half of the top slab width. The characteristic length for the bottom surface is the width of the bottom slab of 12 m. The character length for free convection can be obtained using the ratio of deck surface area and perimeter considering the Jersey barriers. During the period of simulation of the Jamestown Verrazzano Bridge, the maximum snow precipitation at the site was 2 mm h−1 (0.08 in. h−1), and deicing salt was usually used during snowfall; therefore, the snow accumulation effect on heat transfer was neglected in the validation analysis.
e. Results and discussion
1) Validation of models
The temperatures of the Jamestown Verrazzano Bridge were calculated using both the ASHRAE model and the Improved model. Figure 5 shows the comparison between the measured and calculated temperatures at 51 mm below the top surface of the bridge slab (point P1 in Fig. 4b) and 51 mm from the bottom of the bridge (point P2 in Fig. 4b). These results show that the ASHRAE and Improved models yield a good estimation of the bridge temperatures especially during wintertime. Results calculated using the ASHRAE model and the Improved model are shown in Fig. 6 and compared to the measured temperature for specific days.
Figures 6a and 6b show the comparison of specific days in January and May at P1 of the top bridge slab. The Improved model presents better prediction than the ASHRAE model, and the average temperature at P1 predicted by the Improved model in those days is 1.0°C higher than that of ASHRAE, which could be attributed to the underestimation of incoming radiation in the ASHRAE model since this was the major difference between the ASHRAE model and the Improved model. In the ASHRAE model, the atmosphere was treated as a blackbody with a uniform sky temperature (mean radiant temperature), which was 15°–30°C lower than the air temperature. The longwave radiation from the atmosphere may be underestimated, which leads to the average predicted bridge temperatures being 2°C lower than the measured data (Figs. 6a,b). As shown in Fig. 6a, the predicted temperatures from ASHRAE and Improved models after raining were 1°–3°C lower than the measured data, which could be attributed to overestimation of the heat flux due to evaporation in the models.
Figure 6c shows the comparison of 10 days at location P1, with the Improved model having an RMSD of 1.7°C as compared with 2.6°C for the ASHRAE model. For the wintertime, the RMSD of the Improved model is 1.4°C as compared with 2.1°C for the ASHRAE model, and the RMSD of the Improved model is 1.8°C as compared with 3.0°C for the ASHRAE model during the summer. The predicted temperatures at the bottom of the bridge for the two models are almost the same; thus, only the results from the Improved model are shown in Fig. 6d; the RMSD is 1.8°C.
Figures 6e and 6f show a comparison of the temperatures at the top of the bridge slab at different depths for the Improved model and measured results on specific days in January and May. In general, the difference between the calculated and measured temperatures is less than 2°C. The temperature distribution along the depth of the bridge slab is influenced by the boundary, which also depends on the thermal properties of the concrete used in the simulation. The good match of the temperature along the depth of the top slab indicates that the thermal properties used in the simulation are reasonable. The measured results show that the temperature changes along the depth of the bridge slab on sunny days (Fig. 6f) are almost 2 times those on cloudy and rainy days (Fig. 6e).
From the comparison, a 0.7°C difference of the RMSD between the Improved and ASHARE models seems to be a minor improvement during wintertime in the Improved model; however, it can affect phase change of the water on the bridge surface when the temperature of the bridge is close to the freezing point of the water. Thus, it is better to consider the recommendations made in the Improved model that include the effect of longwave radiation and natural convection to predict the bridge temperature.
2) Discussion of convection
Figures 5 and 6 indicate that the calculated results match the measured results very well during the wintertime; however, the deviation of temperatures during the summertime is over 3°C. Some of the deviations may be due to differences of the weather between the location of the bridge and the weather station. Another reason may be that the model underestimates the convective heat-transfer coefficient (CHTC) at the lower wind speeds, which reduces the convective heat flux contribution in the heat transfer of the bridge based on Eq. (10). Figure 7 shows the relationships between CHTC and wind speed based on different models. Qin and Hiller (2013) investigated the influences of wind speed and CHTC on the pavement temperature prediction. The calculated results from different models for CHTC are compared with one day’s measured pavement temperature. From the comparisons of Qin and Hiller (2013), the CHTC calculated using the Priestley and Thurston (1979) and Bentz (2000) methods would be overestimated, whereas the modified Blasius model (Qin and Hiller 2013) and the model proposed by Sharples and Charlesworth (1998) have a better match with the measured data. The CHTC obtained from Eqs. (10)–(12) (ASHRAE 2011) is also plotted in Fig. 7, which increases almost linearly with wind speed. However, the CHTC is smaller than that of other models when the wind speed is lower than 8 m s−1 and may be underestimated by the equations that were used in the simulation.
The effect of underestimation for CHTC on temperature prediction will be significant when the temperature difference between the air and the bridge surface is large based on Eq. (9). As shown in Fig. 8, the solar radiation and deck–air temperature differences Tst − Ta show similar seasonal variations. The daily average solar radiation during summertime (from June to August) is over twice that of the wintertime (from December to February). Meanwhile, the temperature of the bridge-deck surface is higher than surrounding air temperature in most of the summertime. If CHTC is underestimated in the model, the thermal energy stored in the bridge deck is unable to be transferred to the air through convection, which leads to larger predicted temperature of the bridge deck. Additionally, wind speed may influence the pavement surface temperature predication in a more pronounced manner if the solar radiation is higher (Qin and Hiller 2013). As shown in Fig. 8, the daily average wind speed measured by the weather station during the summer and winter are 4.4 and 5.7 m s−1, respectively, and the instantaneous wind speed is less than 8 m s−1 during most of the summertime. For this wind speed range, the CHTC may be underestimated using the method in ASHRAE (2011) shown in Fig. 7. Therefore, the variation of CHTC has a significant influence on convective heat flux during summertime based on Eq. (9). In other words, if the CHTC is underestimated, the predicted bridge surface temperature could be larger than the measured data during summer as shown in Fig. 5. This may be one of the main reasons that the Improved model produces the predicted temperature with larger deviation during summer. However, the main interest here is that the model produces very good results during the wintertime.
5. Vehicle effects
After validating the 2D Improved model for heat-transfer analysis, the analysis was extended to evaluate vehicle effects on the bridge temperature. As discussed above, the vehicle effects considered in the model includes the convective heat due to airflow induced by the passage of the vehicle, longwave radiation from vehicle bottom, longwave/shortwave radiation blocked by the vehicles, and frictional heat of tires.
To investigate the effects of different traffic flow rates on the energy balance of the bridge, both low and high traffic rates were simulated using the same local weather conditions of Jamestown Verrazzano Bridge. The average daily traffic (ADT) of the Jamestown Verrazzano Bridge was 28 400 in 1998, which was reported by Rhode Island Geographic Information System (RIGIS 2018). Therefore, the ADT from September 1997 to September 1998 can be assumed as 28 400 passenger vehicles per day (7100 passenger vehicles per day per lane), which is defined as light traffic conditions. For the analysis, the traffic condition similar to the Brooklyn Bridge in 2010 was used as an example of heavy traffic input in the model.
The model also accounts for commuting-time effects. Laffont et al. (1999) present six typical daily traffic patterns for a weekday. The traffic volume between 2200 and 0600 LT is usually less than 10% of the total daily traffic, and the hourly traffic flow rate in the commuting hour is 1~2 times that of the normal time, which is consistent with the results of 1-yr traffic flow data reported by Chapman and Thornes (2005). Traffic breakdown may occur with some probability at various flow levels, and the probability of breakdown increases dramatically, with traffic flow rate above 1500 passenger vehicles per hour per lane (hereinafter pv/h/ln) according to observations by Dong and Mahmassani (2012) and Chen et al. (2015). Therefore, in this study, a traffic flow rate of less than 1500 pv/h/ln was considered to be light traffic. A traffic flow rate of larger than 1500 pv/h/ln was considered to be heavy traffic condition. The following assumptions were made:
The traffic flow during commuting time (0600–0900 and 1600–1900 LT) is 2 times the traffic during other times (0900–1600 and 1900–2200 LT), and the traffic volume in the period from 2200 to 0600 LT was assumed to be zero to simplify the model of traffic flow, on the basis of the traffic data reported by Laffont et al. (1999) and Chapman and Thornes (2005).
The vehicle speed is 18 m s−1 (speed limit of the Jamestown Verrazzano Bridge) when the traffic flow rate is less than 1500 pv/h/ln and is 9 m s−1 when the traffic flow rate is larger than 1500 pv/h/ln.
The traffic flow rate and average vehicle speed are shown in Fig. 9 during the heavy and light traffic conditions. For the heavy traffic rate condition, the traffic flow rate is 1800 pv/h/ln during the commuting time and 900 pv/h/ln during the period of normal traffic time. The total daily traffic is 19 800 vehicles per lane, which is close to the 20 600 vehicles per lane of the Brooklyn Bridge (New York City Department of Transportation 2012). The total daily traffic of Jamestown Verrazzano Bridge is 7100 passenger vehicles per lane (RIGIS 2018), and the traffic rate is 645 and 323 pv/h/ln during the commuting and normal traffic hours.
a. Results with light traffic
Heat-transfer analysis of the bridge was conducted for the Jamestown Verrazzano Bridge under light traffic conditions. Measured and predicted temperatures at location P1 (close to the top surface of the bridge) were compared in Fig. 10. The RMSD of the temperature comparisons is 1.3°C for the Vehicle model and 1.7°C for the Improved model (no vehicle effects). For the wintertime, the RMSD of the Vehicle model is 1.1°C as compared with 1.4°C for the Improved model. The comparison between the Improved and Vehicle models is also shown in Figs. 6a and 6b for specific days, and the Vehicle model shows better results than both the ASHRAE and Improved models.
The predicted temperature differences of the bridge surface between the Improved and Vehicle models considering effects of the light traffic are compared in Fig. 11 in polar coordinates; Tsv is the bridge surface temperature obtained by the Vehicle model, and Ts is the bridge surface temperature obtained by the Improved model. The figure shows the traffic effects in 24 h of a day, which presents that the bridge surface temperature was increased by the vehicles (up to 4°C). The heating effect of the vehicle is mainly attributed to the longwave radiation from the bottom of the vehicle and frictional heat of tires. The bridge surface was also cooled down by 1°C, which mainly occurs around noontime (Fig. 11). This is because the vehicle-induced wind enhances the convection between the bridge deck and air in the vicinity.
Figure 12 represents 1 year of results for the top-surface temperature differences between the Improved and Vehicle models. Figure 12a summarizes the results of the analysis for a light traffic condition, which shows that the predicted surface temperature of the model with vehicle effects (Vehicle model) is 0°–2°C higher than that without traffic (Improved model) during most of the year.
b. Effects of heavy traffic
To estimate the effects of heavy traffic, traffic data similar to those of the Brooklyn Bridge were used as traffic input in the Vehicle model coupled with the weather data at the Jamestown Verrazzano Bridge. Figure 12b shows the results for heavy traffic conditions and indicates that traffic mostly increases the bridge surface temperature by 0°–4°C. Figure 12 also shows that vehicles can cool down the bridge surface temperature by up to 1° and 2°C as a result of vehicle-induced convection with light and heavy traffic conditions, respectively. Note that the cooling-down effects mainly occur during the summertime as shown in Fig. 12. The cooling-down effect is consistent with the results of numerical simulation performed by Fujimoto et al. (2012) and Khalifa et al. (2016). RST differences of up to 0.5°C between the models with and without traffic were predicted by Khalifa et al. (2016), and vehicle-induced convection caused up to 3°C temperature differences in Fujimoto et al. (2012).
Figure 13 shows the results of the bridge surface temperature for two specific days with different weather conditions in January 1998. Figure 13a shows the bridge surface temperature of different traffic conditions on a clear day, and Fig. 13b summarizes the results for a day with precipitation. When comparing the results of different traffic conditions, the bridge surface can be ~3°C warmer during heavy traffic conditions than during light traffic during the commuting time, which is similar to the conclusions by Parmenter and Thornes (1986) and Fujimoto et al. (2010). Thus, heavier traffic could lead to more significant effects under different weather conditions. As shown in Fig. 13, the surface temperature differences between the Improved and Vehicle models for light traffic are less than 2°C for the analyzed weather conditions.
From the simulations discussed in Fig. 13, the heat fluxes at two selected times (0700–0800 and 1300–1400 LT) during the day are presented in Fig. 14 with different weather conditions. In early afternoon (~1300 LT), the vehicle affects the heat fluxes caused by natural factors (e.g., , , and ) by less than 35 W m−2 (see Fig. 14a), and the temperature of the bridge surface with heavy traffic is 2.5°C higher than with light traffic and 4°C higher than with no traffic, results which are attributed to the radiation and tire frictional heat boundaries (see Fig. 13a). The heat fluxes of tire friction were 18 W m−2 for light traffic and 49 W m−2 for heavy traffic, and the heat fluxes of longwave radiation from vehicles were 11 W m−2 for light traffic and 31 W m−2 for heavy traffic.
During a snowy morning commute (~0700 LT), heavy traffic more significantly affects the longwave radiation heat fluxes (see Fig. 14b). For example, the heat flux of longwave radiation (infrared radiation) received from vehicles could reach 114 W m−2 for the heavy traffic condition. The heat fluxes of tire friction were 35 W m−2 for light traffic and 49 W m−2 for heavy traffic. At the same time, the heat loss through longwave radiation also increased to approximately 95 W m−2. The changes of heat fluxes due to other factors (e.g., , , and ) are less than 35 W m−2. Therefore, the effect of traffic in increasing bridge surface temperature during the morning commute is largely caused by infrared radiation from the vehicle and tire frictional heat, which cause the bridge surface to release more outgoing longwave radiation.
6. Summary and conclusions
The development of heat-transfer models to predict the temperature and heat flux of the bridge-deck surface has been described considering both natural and vehicle-induced factors. The model proposed by the ASHRAE Handbook (ASHRAE model) was modified by replacing the longwave radiation formula proposed by Brock and Arnold (2000) and Brown and Dunn (1998) and by incorporating the effects of solar radiation and natural convection. Furthermore, the Improved model incorporated the effective emissivity of the atmosphere in the longwave radiation model considering the cloud cover effect. The Improved model with the effects of natural factors was validated by the comparison of 1 year’s measured results. The comparisons indicate that the Improved model may underestimate the convective heat-transfer coefficient (CHTC) during summer yet produces very good results for the whole year.
In the ASHRAE model, the atmosphere was treated as a blackbody with a uniform sky temperature (mean radiant temperature), which was 15°–30°C lower than the air temperature. The longwave radiation from the atmosphere may be underestimated using this method, which leads to a predicated temperature of the bridge that is 2°C lower than the measured data.
After validating the Improved model, the analysis was modified to include vehicle effects (Vehicle model). The effective wind velocity was used to investigate the vehicle-induced sensible heat on the bridge surface. Moreover, the longwave radiation from vehicles and the tire frictional heat were considered in the Vehicle model. Including vehicle effects improved the accuracy of the predicted temperature as shown by a lower RMSD.
The bridge surface temperature was calculated for different traffic rates with the same 1 year of weather data. The results show that vehicular traffic increased the bridge surface temperature by up to 2°C for the light traffic condition and by up to 4°C for the heavy traffic condition during the wintertime, thus providing an advantage for deicing. The heating effects of the traffic were mainly caused by the longwave radiation from vehicles and the tire frictional heat. The effect of traffic on heat flux for each factor (e.g., convection, shortwave radiation, longwave radiation) was less than 35 W m−2. However, the heat flux of longwave radiation (infrared radiation) received from vehicles could reach 114 W m−2 for the heavy traffic condition during commuting time. In theory, with the heating effects of traffic, vehicles could run on only one lane of a multiple-lane bridge in the light traffic condition to reduce the icing probability during snows; however, it is practically difficult to achieve.
Traffic effects can cool the bridge surface temperature down by up to 2°C as a result of vehicle-induced convection, which mainly occurs during the summertime. Because the winter effects of traffic almost exclusively result in warming, they can be carefully considered as beneficial factors for the bridge deicing. A conservative design of the heating system for bridge deicing can neglect the vehicle effects on bridge temperature.
From the validation and results presented in this paper, it is recommended that natural factors (mainly convection and longwave radiation) and vehicle effects be included when modeling bridge deicing. The Vehicle model can be used in the planning stage for different heating technologies (e.g., electric cables, geothermal energy, and boiler) to perform life-cycle analysis for decision-making. If those heating technologies for a specific location are more expensive than the conventional deicing method (snow removal and using deicing salts) considering corrosion effects of deicing salts on the bridge service life, the conventional deicing method could be used. The research team is currently using the models discussed in this paper to investigate the efficiency of geothermal foundations in different weather conditions.