*3.3. Temperature Distribution of Truss*

In this paper, the maximum excess temperature of the truss is defined as the difference between the maximum temperature of the truss and the ambient temperature. The 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 m, which leads to the flame combustion reaction increasing. *Fire* **2022**, *5*, x FOR PEER REVIEW 9 of 14

**Figure 6.** Maximum excess temperature of the truss versus time. **Figure 6.** Maximum excess temperature of the truss versus time.

A great number of correlations for maximum excess temperature have been presented in the literature, even though there is some inconsistency with the different forms of them. An analysis of the maximum excess temperature in ventilation velocity has established the dependence of the dimensionless excess temperature (ΔT/ΔTmax-c) on the dimensionless HRR denoted as *Q*\* ( = ρ \* 1/2 2/5 0 0 / *Q Q CTg h <sup>p</sup> ef* [22]) and the Froude number denoted as *Fr* ( = <sup>2</sup> / *e f Fr u gh* [22]). In addition, some scholars find that the HRR is closely related to the equivalent diameter of the fire source [9]. To quantify the maximum excess truss temperature, a dimensional analysis method is used. Based on the analysis above, the main factors affecting the maximum excess truss temperature are effective bridge deck spacing (*hef*), equivalent diameter of the fire source (*D*), flame heat release rate (*Q*), ambient A great number of correlations for maximum excess temperature have been presented in the literature, even though there is some inconsistency with the different forms of them. An analysis of the maximum excess temperature in ventilation velocity has established the dependence of the dimensionless excess temperature (∆*T*/∆*T*max-c) on the dimensionless HRR denoted as *Q*\* (*Q*<sup>∗</sup> = *Q*/*ρ*0*CpT*0*g* 1/2*he f* 2/5 [22]) and the Froude number denoted as *Fr* (*Fr* = *u* <sup>2</sup>/*gh<sup>e</sup> <sup>f</sup>* [22]). In addition, some scholars find that the HRR is closely related to the equivalent diameter of the fire source [9]. To quantify the maximum excess truss temperature, a dimensional analysis method is used. Based on the analysis above, the main factors affecting the maximum excess truss temperature are effective bridge deck spacing (*hef*), equivalent diameter of the fire source (*D*), flame heat release rate (*Q*), ambient air density (*ρ0*), ambient temperature (*T0*), specific heat capacity at constant pressure

air density (*ρ0*), ambient temperature (*T0*), specific heat capacity at constant pressure (*Cp*), gravitational acceleration (g), and ventilation velocity (*u*). The maximum excess tempera-

Based on the principle of dimensional consistency, the formula can be simplified as:

( ,, ) *p ef <sup>t</sup> <sup>T</sup> Q u CT h T Dg D D g D g*

73 11

( ) () *<sup>t</sup> p ef*

<sup>∆</sup> <sup>−</sup> <sup>=</sup> , max <sup>0</sup>

<sup>−</sup> <sup>∆</sup> <sup>=</sup> <sup>=</sup> max '

<sup>0</sup> 0 max ∆ = <sup>−</sup> ( , , , , ,,, , ) 0 *ef <sup>p</sup> <sup>t</sup> f h DQ C guT T* (12)

 ϕ

*<sup>T</sup> <sup>Q</sup> <sup>Q</sup> T DC T uh* (14)

(13)

ρ

ϕ

ρ

ϕ

0 22 22 0

ρ

0 00

where ΔTmax-*<sup>t</sup>* is the maximum excess temperature of the truss.

The right four terms of the formula can be combined:

where *Q*′ is the modified dimensionless heat release rate.

(*Cp*), gravitational acceleration (g), and ventilation velocity (*u*). The maximum excess temperature of the truss can be expressed by the following formula:

$$f(h\_{\varepsilon f}, D\_\prime Q\_\prime \rho\_0, \mathbb{C}\_{p\prime} \mathbb{g}\_\prime \mu\_\prime T\_{0\prime} \Delta T\_{\max-t}) = 0\tag{12}$$

where ∆*T*max-*<sup>t</sup>* is the maximum excess temperature of the truss.

Based on the principle of dimensional consistency, the formula can be simplified as:

$$\frac{\Delta T\_{\text{max}-t}}{T\_0} = \varphi(\frac{Q}{\rho\_0 D^{\frac{\gamma}{2}} \mathcal{S}^{\frac{3}{2}}}, \frac{u}{D^{\frac{1}{2}} \mathcal{S}^{\frac{1}{2}}}, \frac{\mathcal{C}\_p T\_0}{D \mathcal{S}}, \frac{h\_{ef}}{D}) \tag{13}$$

The right four terms of the formula can be combined: *Fire* **2022**, *5*, x FOR PEER REVIEW 10 of 14

$$\frac{\Delta T\_{\text{max}-t}}{T\_0} = \varphi(\frac{\mathcal{Q}}{\rho\_0 D \mathcal{C}\_p T\_0 \imath h t\_{\ell f}}) = \varphi(\mathcal{Q}') \tag{14}$$

(15)

where *Q*0 is the modified dimensionless heat release rate. ble stage is selected as the maximum excess temperature of the truss. Figure 7 shows that

To reduce the error of the numerical simulation results, the average value of the stable stage is selected as the maximum excess temperature of the truss. Figure 7 shows that the simulation data of maximum excess temperature can be plotted as a function of *Q*/*ρ*0*DCpT*0*uhe f* in this region. The simulation data of maximum excess temperature of the truss can be correlated well with Equation (15): the simulation data of maximum excess temperature can be plotted as a function of ρ0 0 / *Q DC T uh p ef* in this region. The simulation data of maximum excess temperature of the truss can be correlated well with Equation (15): <sup>−</sup> <sup>∆</sup> <sup>=</sup> max 1.257 0.183( ) *<sup>t</sup> T Q*

**Figure 7.** Dimensionless maximum excess temperature of the truss versus modified dimensionless heat release rate. **Figure 7.** Dimensionless maximum excess temperature of the truss versus modified dimensionless heat release rate.

From one-dimensional analysis, Yang [23] estimated the heat loss intensity and established the energy equation of the gas beneath the ceiling. However, the excess temperature of the truss is mainly affected by the flame plume. Therefore, the energy diffusion of the micro length *dh* of the truss to the surrounding can be approximately expressed as: From one-dimensional analysis, Yang [23] estimated the heat loss intensity and established the energy equation of the gas beneath the ceiling. However, the excess temperature of the truss is mainly affected by the flame plume. Therefore, the energy diffusion of the micro length *dh* of the truss to the surrounding can be approximately expressed as:

where *C* is the perimeter of the contact surface between the element length and the surrounding environment, *q*″ is the heat flux to truss, and *m* is the mass flow rate of the

''

 = ρ

where *we* is horizontal entrainment velocity, which can be expressed as:

*m* can be approximated as the mass rate of the micro side enters the flame plume:

$$q'''\mathbb{C}dh = -c\_p \dot{m} d(\Delta T) \tag{16}$$

= ∆*<sup>c</sup> q hT* (17)

*m w hC* <sup>0</sup> *<sup>e</sup>* (18)

where *hc* is the lumped heat transfer coefficient.

truss.

where *C* is the perimeter of the contact surface between the element length and the surrounding environment, *q* <sup>00</sup> is the heat flux to truss, and . *m* is the mass flow rate of the truss. *q* 00 can be expressed by the following formula:

*q*

.

00 = *hc*∆*T* (17)

where *h<sup>c</sup>* is the lumped heat transfer coefficient. .

*m* can be approximated as the mass rate of the micro side enters the flame plume:

$$
\dot{m} = \rho\_0 w\_\epsilon \hbar \mathbf{C} \tag{18}
$$

where *w<sup>e</sup>* is horizontal entrainment velocity, which can be expressed as:

$$w\_{\varepsilon} = 1.94a \left(\frac{g}{\pi \text{C} \, p \, T\_0 \rho\_0}\right)^{1/3} \text{Q}^{1/3} \text{h}^{-1/3} \tag{19}$$

Incorporating Equations (17)–(19) into Equations (16) and (19) can be obtained as follows:

$$\frac{d(\Delta T)}{\Delta T} = -\frac{h\_c}{1.94 \alpha \rho\_0 \left(\frac{\mathcal{S}}{\pi C \rho T\_0 \rho\_0}\right)^{1/3} Q^{1/3}} \frac{dh}{h^{2/3}}\tag{20}$$

The boundary conditions are given by the following equation:

$$\begin{cases} \text{ } h = h\_0\\ \Delta T = \Delta T\_{\text{max}-t} \end{cases} \tag{21}$$

where *h<sup>0</sup>* is the location of maximum excess temperature of the truss.

Incorporating Equations (20) and (21) can obtain the attenuation formula of excess temperature of the truss:

$$f(h) = \frac{\Delta T}{\Delta T\_{\text{max}-t}} = \mathbf{e}^{-A(h^{1/3} - h\_0^{1/3})}, h > h\_0 \tag{22}$$

where *A* =

3*hc* 1.94*αρ*0( *g πCpT*0 *ρ*0 ) 1/3*Q*1/3 .

Assuming that the excess temperature at the same distance from the maximum excess temperature position is almost same, the formula of the increasing excess temperature of the truss can be obtained as follows:

$$f(h) = f(\mathfrak{A}h\_0 - h) \tag{23}$$

By substituting Equation (23) into Equation (22), Equation (24) can be expressed as:

$$f(h) = \frac{\Delta T}{\Delta T\_{\text{max}-t}} = \mathbf{e}^{-A\prime(\left(2h\_0 - h\right)^{1/3} - h\_0^{1/3})}, h \le h\_0 \tag{24}$$

To obtain the dimensionless temperature as a function of the dimensionless distance, *hef* is used as the characteristic scale to the dimensionless distance. Incorporating Equations (22) and (24), the dimensionless excess temperature above and below the position of the maximum temperature can be expressed as:

$$\frac{\Delta T}{\Delta T\_{\text{max}-t}} = \begin{cases} e^{-k\left(\left(\frac{h}{h\varepsilon f}\right)^{1/3} - \left(\frac{h\_0}{h\varepsilon f}\right)^{1/3}\right)}, h > h\_0 \\\ e^{-k'\left(\left(\frac{2h\_0 - h}{h\varepsilon f}\right)^{1/3} - \left(\frac{h\_0}{h\varepsilon f}\right)^{1/3}\right)}, h \le h\_0 \end{cases} \tag{25}$$

where *k* and *k* 0 are coefficients.

In this work, the maximum excess temperature of the truss is located at the top of the truss, namely *h<sup>0</sup> = hef*. So, the excess temperature of the truss can be expressed as:

$$\frac{\Delta T}{\Delta T\_{\text{max}-t}} = e^{-k'\left(\left(2-\frac{h}{k\varepsilon f}\right)^{1/3}-1\right)}, h \le h\_{\varepsilon f} \tag{26}$$

Figure 8 shows that the simulation data fit well with the formula. The values of *k* 0 under different working conditions are shown in Table 3. When H ≤ 10.6, the value of *k* 0 is close to 1. However, when H > 11.4, the value of *k* 0 is larger than 1. This means that when the bridge deck spacing is less than 11.2, the excess temperature varies slightly along the truss. When the bridge deck spacing continues to increase, the change of excess temperature is great. Figure 8 shows that the simulation data fit well with the formula. The values of *k*′ under different working conditions are shown in Table 3. When H ≤ 10.6, the value of *k*′ is close to 1. However, when H > 11.4, the value of *k*′ is larger than 1. This means that when the bridge deck spacing is less than 11.2, the excess temperature varies slightly along the truss. When the bridge deck spacing continues to increase, the change of excess temperature is great.

**Figure 8.** Excess temperature along the truss versus normalized effective height: (**a**) H = 9.8; (**b**) H = 10.6; (**c**) H = 11.4; (**d**) H = 12.2; (**e**) H = 13.0; (**f**) H = 13.8.


**Table 3.** Values of *k* 0 under different bridge deck spacings.
