*3.2. Maximum Excess Temperature Beneath the Ceiling*

In this paper, the maximum excess temperature beneath the ceiling is defined as the difference between the maximum temperature beneath the ceiling and the ambient temperature. The maximum excess temperature at different bridge deck spacings varies with time, as shown in Figure 4. It can be clearly seen that the maximum excess temperature changes with time and can be divided into three stages: slow growth stage, rapid growth stage, and relatively stable stage. In the slow growth stage, the maximum excess temperature increases slowly with time, which occurs between approximately 0 and 300 s. In the rapid growth stage, the maximum excess temperature increases rapidly with time, which occurs between approximately 300 and 600 s. In the relatively stable stage, the maximum excess temperature increases rapidly with time, which occurs between approximately 600 and 4200 s. The maximum excess temperature does not change significantly with time and fluctuates around a certain value. With the increase in bridge deck spacing, the overall excess temperature of the bridge ceiling decreases gradually. The reason Is that as the bridge deck spacing increases, the entrainment route of the air beneath the ceiling decreases, which increases heat flux loss.

**Figure 3.** Influence range on the ceiling at different bridge deck spacings: (**a**) H = 9.8; (**b**) H = 10.6; (**c**) H = 11.4; (**d**) H = 12.2; (**e**) H = 13.0; (**f**) H = 13.8. **Figure 3.** Influence range on the ceiling at different bridge deck spacings: (**a**) H = 9.8; (**b**) H = 10.6; (**c**) H = 11.4; (**d**) H = 12.2; (**e**) H = 13.0; (**f**) H = 13.8. *Fire* **2022**, *5*, x FOR PEER REVIEW 7 of 14

**Figure 4.** The maximum excess temperature versus time beneath the ceiling. **Figure 4.** The maximum excess temperature versus time beneath the ceiling.

most cases by many scholars, which can be expressed as [20]:

<sup>−</sup>

A lot of research has been performed on the formula of maximum excess tunnel ceiling temperature. Li proposed the formula of the maximum excess ceiling temperature un-

1/3

<sup>−</sup> ∆ = <sup>&</sup>gt; (8)

*<sup>w</sup>* <sup>=</sup> (9)

<sup>=</sup> (10)

*p o ef C g <sup>Q</sup> <sup>u</sup> C T uD h*

2.68 (1 ) <sup>T</sup> , ' 0.19 ( )

\* ' *<sup>u</sup> u*

\* 1/3

*Q g <sup>w</sup>*

where *w*\* is the characteristic plume velocity, and *Qc* is the convective heat release rate.

0 ( ) *<sup>c</sup>*

To verify whether the formula of the maximum excess temperature of the tunnel ceiling is suitable for a bridge under natural ventilation, Li's formula is used to fit the numerical simulation data. To reduce the error of numerical simulation results, the average value of the stable stage is selected as the maximum excess temperature beneath the ceiling. Figure 5 shows the fitting of the formula and simulation data. It is obvious that the formula of the maximum excess tunnel ceiling temperature also applies to the bridge. The simulation data of maximum excess temperature beneath the top floor can be correlated

max c 1/3 5/3 T 0.397

Compared with Li's empirical formula, the coefficient is relatively small. The reason is as follows: firstly, the maximum excess temperature here is the average of the stable

where ΔTmax-*<sup>c</sup>* is the maximum excess temperature beneath the ceiling.

*ef*

∆ = <sup>−</sup> (11)

*Q*

*uD h*

*D CT* ρ

*p o*

max 1/3 1/3 5/3

χ

where Δ*Tmax-c* is the maximum excess ceiling temperature, *Q* is the total heat release rate, *To* is the ambient temperature, *Cp* is the thermal capacity of air, *CT* is coefficient, *χ<sup>r</sup>* is the fraction of radiative heat release rate, *g* is the gravitational acceleration, *D* is the radius of the fire source, *u* is the ventilation velocity, and *u*′ is the dimensionless ventilation velocity.

*T r*

0

ρ

*c*

*U*′ can be expressed as [21]:

well with Equation (11):

A lot of research has been performed on the formula of maximum excess tunnel ceiling temperature. Li proposed the formula of the maximum excess ceiling temperature under longitudinal ventilation velocity. The formula has been proved to be applicable to most cases by many scholars, which can be expressed as [20]:

$$
\Delta T\_{\text{max}-\varepsilon} = \frac{2.68 \mathcal{C}\_T (1 - \chi\_r) g^{1/3}}{\left(\rho\_0 \mathcal{C}\_p T\_o\right)^{1/3}} \frac{Q}{u D^{1/3} h\_{ef}^{5/3}}, \mu' > 0.19\tag{8}
$$

where ∆*Tmax-c* is the maximum excess ceiling temperature, *Q* is the total heat release rate, *T<sup>o</sup>* is the ambient temperature, *C<sup>p</sup>* is the thermal capacity of air, *C<sup>T</sup>* is coefficient, *χ<sup>r</sup>* is the fraction of radiative heat release rate, *g* is the gravitational acceleration, *D* is the radius of the fire source, *u* is the ventilation velocity, and *u* 0 is the dimensionless ventilation velocity.

*u* 0 can be expressed as [21]:

$$u' = \frac{u}{w^\*}\tag{9}$$

$$w^\* = \left(\frac{Q\_c g}{D\rho\_0 C\_p T\_o}\right)^{1/3} \tag{10}$$

where *w*\* is the characteristic plume velocity, and *Q<sup>c</sup>* is the convective heat release rate.

To verify whether the formula of the maximum excess temperature of the tunnel ceiling is suitable for a bridge under natural ventilation, Li's formula is used to fit the numerical simulation data. To reduce the error of numerical simulation results, the average value of the stable stage is selected as the maximum excess temperature beneath the ceiling. Figure 5 shows the fitting of the formula and simulation data. It is obvious that the formula of the maximum excess tunnel ceiling temperature also applies to the bridge. The simulation data of maximum excess temperature beneath the top floor can be correlated well with Equation (11): *Fire* **2022**, *5*, x FOR PEER REVIEW 8 of 14

$$
\Delta T\_{\text{max}-c} = 0.397 \frac{Q}{\text{uD}^{1/3} h\_{ef}^{5/3}} \tag{11}
$$

where ∆*T*max-*<sup>c</sup>* is the maximum excess temperature beneath the ceiling. to limit heat dissipation.

**Figure 5.** Fitting of Li's formula and simulation data. **Figure 5.** Fitting of Li's formula and simulation data.

*3.3. Temperature Distribution of Truss*

m, which leads to the flame combustion reaction increasing.

maximum excess temperature of the truss at different bridge deck spacings varies with time and is shown in Figure 6. The maximum excess temperature varies with time and can also be divided into three stages: slow growth stage, rapid growth stage, and relatively stable stage. With the increase in the bridge deck spacing, the overall excess temperature gradually decreases. This is because as the bridge deck spacing increases, air entrainment increases and more cold air flows through the truss surface, which leads to the temperature decreasing. However, the excess temperature of the 12.2 m bridge deck spacing is bigger than that of the 11.4 m bridge deck spacing. This may be caused by the oxygen supply from outside of the 12.2 m bridge deck spacing becoming better than that of 11.4

Compared with Li's empirical formula, the coefficient is relatively small. The reason is as follows: firstly, the maximum excess temperature here is the average of the stable stage. Secondly, compared with tunnels, the double-deck bridge does not have sidewalls to limit heat dissipation.
