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Boundary conditions

By convention, upward fluxes are positive. At the top surface, the energy balance (W m⁻²) is given by:

(6)

where −Q*_A is the net radiation (net shortwave + net longwave); Q_GA is the heat flux through the top surface of the concrete; Q_HA is the convective sensible heat flux; Q_EA is the evaporation heat flux; and Q_RA is the runoff water heat flux (Fig. 1).

The total top surface energy flux is explicitly given by:

The second term on the right hand side of Eq. (7) is the evaporative heat flux, the third is the sensible heat flux, and the fourth is the runoff water heat flux. M is the volume flow rate of spray water running off the bridge (m³ s⁻¹); A_top is the area of the bridge's top surface (m²); q_a and q_s are the air and surface specific humidities (g_w g_a⁻¹); T_a and T_s are the air and surface temperatures (ºC); r _w and r are the water and air densities (kg m⁻³); c_w and c_p are the water and air specific heat capacities (J kg⁻¹ K⁻¹); L_v is the latent heat of evaporation (J kg⁻¹); U is the wind speed (m s⁻¹); C is the dimensionless exchange coefficient; z is the vertical dimension (m); and T_wi and T_wf are the initial (when hitting the top surface) and final (when running off the bridge) spray water temperatures (ºC).

For conditions other than free convection (free convection is assumed when the atmospheric stability indicator, the bulk Richardson number, Rb, < −1 (14) and U < 1 m s⁻¹), the exchange coefficients were determined by Wojcik and Fitzjarrald (6). Free convection exchange speeds (equal to CU in Eq. (7)) are given by Kondo and Ishida (15) for a smooth surface.

The net radiation is computed by determining the incoming and outgoing longwave and shortwave radiation components with schemes given by Stull (14), Freedman et al. (16), Prata (17), and Brutsaert (18).

The spray water drops will adjust to the environmental conditions as they rise and fall after being ejected from the irrigation hoses. Because the drops do not remain airborne for more than a few seconds, their temperatures generally do not approach the air wet-bulb temperature. A model by Pruppacher and Klett (19) is used to calculate the temperature of the spray water as it hits the top surface (T_wi).

At the sheet metal form, the heat flux at the bridge deck bottom, Q_GB (W m⁻²), is given by:

where the second term on the right-hand side accounts for convective heat loss; T_cf is the temperature of the form; Q*_B is the net radiation at the bottom surface; and C_b is the exchange coefficient for the form. To model the heat transfer through the steel support beams (Fig. 1), a formulation of heat flow through an infinite steel fin (20) is used.

Next: Boundary condition sensitivity Up: Analytical Investigations Previous: Analytical Investigations