3.2.2. Temperature Decay Model

For the downstream smoke under ceilings far away from fire source: Mass equation:

$$\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}(\rho u A) = \rho\_a \mathcal{W}u\_\varepsilon \tag{2}$$

Energy equation:

$$\frac{d}{dx}(\rho A u c\_p T) = \rho\_a \mathcal{W} u\_e c\_p T\_a - h\_t w\_p (T - T\_a) \tag{3}$$

The entrainment velocity *u<sup>e</sup>* can be expressed as:

$$
\mu\_{\ell} = \beta(\mu - \mu\_o) \tag{4}
$$

where *ρ* and *u* are the density (kg/m<sup>3</sup> ) and velocity (m/s) of smoke; *A* is the smoke flow section area (m<sup>2</sup> ); *W* is the width of the tunnel (m); *T* is the temperature of the smoke (K); *ue* is the entrainment velocity of the smoke (m/s); *w<sup>p</sup>* is the wet perimeter of the smoke flow (m); *h<sup>t</sup>* is the total net heat transfer coefficient on the tunnel walls (kW/m<sup>2</sup> ·K); *β* is a coefficient measured in the test; and *u<sup>o</sup>* is the velocity of the air (m/s).

In tunnel fires, smoke entrainment velocity is very small [41]. In order to simplify the energy equation, we usually ignore the influence of smoke entrainment on the energy equation. The tunnel wet perimeter can be calculated according to the tunnel design parameters, so the tunnel wet perimeter can be considered a constant. It is assumed that *h<sup>t</sup>* is also a constant. Based on Equations (2)–(4), the temperature attenuation downstream is obtained as:

$$\frac{\Delta T(\mathbf{x})}{\Delta T\_{\text{max}}} = \exp\left(-\frac{h\_l w\_p + \rho\_d \mu\_\varepsilon \mathcal{W} c\_p}{\rho u A c\_p} \mathbf{x}\right) \tag{5}$$

Because the coefficient *β* (=0.00015) is very small [42], the horizontal entrainment of smoke in the tunnel can be ignored:

$$\frac{\Delta T(\mathbf{x})}{\Delta T\_{\text{max}}} \approx \exp\left(-\frac{h\_{\text{f}} w\_p}{\rho u A c\_p} \mathbf{x}\right) \tag{6}$$

For a tunnel with a rectangular section, the temperature decay downstream can be expressed as:

$$\frac{\Delta T(\mathbf{x})}{\Delta T\_{\text{max}}} \approx \exp\left(-(\frac{2h}{W} + 1) \cdot \frac{h\_l}{\rho u c\_p} \cdot \frac{\mathbf{x}}{h}\right) = \exp\left(-\mathfrak{f} \cdot \frac{\mathbf{x}}{h}\right) \tag{7}$$

$$\xi = (\frac{2h}{W} + 1) \cdot \frac{h\_t}{\rho u c\_p} \approx \frac{h\_t}{\rho u c\_p} \tag{8}$$

Although the coefficient *h<sup>t</sup>* is different in different locations and different fire development stages, when the fire tends to be stable and far enough from fire, the difference of *ht* in different locations is very small. In addition, the height of the smoke layer can also be replaced by tunnel height. In view of this, the temperature decay can be approximated using an exponential function [20,40]. Li et al. [43] found that the form of the sum of two exponential functions, that is <sup>∆</sup>*T*(*x*)/∆*T*max<sup>=</sup> *<sup>a</sup>* <sup>×</sup> exp *<sup>b</sup>*<sup>×</sup> *<sup>x</sup> H* <sup>+</sup>*<sup>c</sup>* <sup>×</sup> exp(*d*<sup>×</sup> *<sup>x</sup> H* , can well describe the decay of temperature under the tunnel ceiling.

Taking the location of the maximum temperature as a reference point, this paper studies the temperature decay from the reference point to the first vent downstream. The smoke control effect is the best in Case "4", and the decay rate of the temperature is like that of other smoke vent opening schemes. Therefore, this paper only presents the fitting results for the temperature attenuation of Case "4".

When four smoke vents are opened, the temperature decay rate is the same under different smoke exhaust rates (H is the height of the tunnel, m), as shown in Figure 8. The results of numerical calculation are in good agreement with the fitting curve (*R* <sup>2</sup> = 0.992). The constants a, b, c, and d are 0.444, −0.156, 0.598, and −1.913, respectively. The temperature decays the slowest when the layout of the exhaust vent is Case "5B" and decays the fastest when the layout of exhaust vent is Case "3B". The correlation coefficient (*R* 2 ) of the fitting curve of the temperature attenuation under different smoke vent opening schemes was greater than 0.976. Except for the smoke vent opening schemes of Case "5B" and Case

"3A", the temperature decays rapidly and the decay rates are similar. Furthermore, the fitting curves of the scheme with fast decay rate are close to the results of temperature for Case "4". Therefore, we only need to focus on the temperature attenuation of Case "4". "3A", the temperature decays rapidly and the decay rates are similar. Furthermore, the fitting curves of the scheme with fast decay rate are close to the results of temperature for Case "4". Therefore, we only need to focus on the temperature attenuation of Case "4".

*Fire* **2022**, *5*, x FOR PEER REVIEW 10 of 17

describe the decay of temperature under the tunnel ceiling.

results for the temperature attenuation of Case "4".

of two exponential functions, that is ∆

ξ

2

T

( 1) *t t*

ρ

Although the coefficient *<sup>t</sup> h* is different in different locations and different fire development stages, when the fire tends to be stable and far enough from fire, the difference of *<sup>t</sup> h* in different locations is very small. In addition, the height of the smoke layer can also be replaced by tunnel height. In view of this, the temperature decay can be approximated using an exponential function [20,40]. Li et al. [43] found that the form of the sum

Taking the location of the maximum temperature as a reference point, this paper studies the temperature decay from the reference point to the first vent downstream. The smoke control effect is the best in Case "4", and the decay rate of the temperature is like that of other smoke vent opening schemes. Therefore, this paper only presents the fitting

When four smoke vents are opened, the temperature decay rate is the same under different smoke exhaust rates (H is the height of the tunnel, m), as shown in Figure 8. The results of numerical calculation are in good agreement with the fitting curve (*R*2 = 0.992). The constants a, b, c, and d are 0.444, −0.156, 0.598, and −1.913, respectively. The temperature decays the slowest when the layout of the exhaust vent is Case "5B" and decays the fastest when the layout of exhaust vent is Case "3B". The correlation coefficient (*R*2) of the fitting curve of the temperature attenuation under different smoke vent opening schemes was greater than 0.976. Except for the smoke vent opening schemes of Case "5B" and Case

*h h h W uc uc*

*p p*

=+ ∝ (8)

ு<sup>ቁ</sup> +c×exp(d<sup>×</sup> <sup>௫</sup>

ு ), can well

 ρ

(*x*)/∆*Tmax*=*a*×exp <sup>ቀ</sup>b<sup>×</sup> <sup>௫</sup>

**Figure 8.** Temperature attenuation beneath tunnel ceiling of Case "4". **Figure 8.** Temperature attenuation beneath tunnel ceiling of Case "4".

In order to make the empirical formula for the temperature attenuation more representative, the temperature distributions under different HRRs and exhaust volumes are calculated for Case "4". The detailed calculation conditions are shown in Table 4. Under different HRRs, the exhaust volume has little effect on the temperature attenuation, as shown in Figure 9. The influence of the HRR on the attenuation rate is very small. When the HRR of the fire source is 10–30 MW, the temperature decay rate is similar. Taking the average value of each coefficient, the empirical formula of temperature attenuation under the tunnel ceiling with four smoke vents open can be obtained as follows:

$$
\Delta T(x) / \Delta T\_{\text{max}} = 0.40e^{-0.147(\frac{x - x\_{\text{max}}}{H})} + 0.60e^{-2.17(\frac{x - x\_{\text{max}}}{H})} \tag{9}
$$

**Table 4.** Calculation condition of 4 smoke vents.


To highlight the difference between the single-side multi-point exhaust tunnel and the previous research results, we compared the temperature attenuation model in this paper with some existing temperature attenuation models. Ji et al. [37] studied the effect of pressure and HRR on the temperature decay beneath the ceiling and found that the impact of pressure on temperature decay is very small, and the temperature attenuation conforms to the sum of two exponential attenuations:

$$
\Delta T\_{\text{x}} / \Delta T\_{\text{r}} = 0.33e^{-0.59\frac{(\mathbf{x} - \mathbf{x}\_{\text{r}})}{H}} + 0.67e^{-0.048\frac{(\mathbf{x} - \mathbf{x}\_{\text{r}})}{H}} \tag{10}
$$

Ingason and Li [43] concluded that the temperature attenuation beneath the ceiling conforms to the sum of two exponential attenuations: shown in Figure 9. The influence of the HRR on the attenuation rate is very small. When the HRR of the fire source is 10–30 MW, the temperature decay rate is similar. Taking the

In order to make the empirical formula for the temperature attenuation more representative, the temperature distributions under different HRRs and exhaust volumes are calculated for Case "4". The detailed calculation conditions are shown in Table 4. Under different HRRs, the exhaust volume has little effect on the temperature attenuation, as

*Fire* **2022**, *5*, x FOR PEER REVIEW 11 of 17

$$
\Delta T\_{\text{X}} / \Delta T\_{\text{I}} = 0.57e^{-0.13\frac{(\mathbf{r} - \mathbf{r}\_{\text{T}})}{H}} + 0.43e^{-0.021\frac{(\mathbf{r} - \mathbf{r}\_{\text{T}})}{H}} \tag{11}
$$

The numerical results are in good agreement with the prediction model proposed in this paper, and the error between the numerical results and the prediction model is basically within 15%, as shown in Figure 10. The closer to the smoke vent, the greater the error between the numerical results and the prediction model, with the numerical calculation results slightly higher than the prediction model. This is because the smoke vent will inhibit the movement of high-temperature smoke and reduce the temperature attenuation rate. Because there are four smoke vents in the tunnel that are continuously discharging high-temperature smoke, the rate of the temperature attenuation model in this paper is faster than that of natural ventilation or longitudinal ventilation tunnel, and the choice of reference points may also be one of the reasons for the large differences. <sup>∆</sup>*T*(*x*)/∆*Tmax* <sup>=</sup> 0.40eି0.147( *x*ି*x*ೌೣ ு ) + 0.60eି2.17( *x*ି*x*ೌೣ ு ) (9) **Table 4.** Calculation condition of 4 smoke vents. **Test No. Smoke Vent Number HRR (MW) Smoke Exhaust Rate (m3/s)**  31–35 4 10 100, 120, 140, 160, 180 36–40 20 160, 180, 200, 220, 240

**Figure 9.** Temperature attenuation with different HRRs of Case "4". (**a**) 10 MW, (**b**) 20 MW, (**c**) 30 MW. **Figure 9.** Temperature attenuation with different HRRs of Case "4". (**a**) 10 MW, (**b**) 20 MW, (**c**) 30 MW.

To highlight the difference between the single-side multi-point exhaust tunnel and the previous research results, we compared the temperature attenuation model in this paper with some existing temperature attenuation models. Ji et al. [37] studied the effect of

reference points may also be one of the reasons for the large differences.

pressure and HRR on the temperature decay beneath the ceiling and found that the impact of pressure on temperature decay is very small, and the temperature attenuation conforms

Ingason and Li [43] concluded that the temperature attenuation beneath the ceiling

The numerical results are in good agreement with the prediction model proposed in this paper, and the error between the numerical results and the prediction model is basically within 15%, as shown in Figure 10. The closer to the smoke vent, the greater the error between the numerical results and the prediction model, with the numerical calculation results slightly higher than the prediction model. This is because the smoke vent will inhibit the movement of high-temperature smoke and reduce the temperature attenuation rate. Because there are four smoke vents in the tunnel that are continuously discharging high-temperature smoke, the rate of the temperature attenuation model in this paper is faster than that of natural ventilation or longitudinal ventilation tunnel, and the choice of

*<sup>H</sup>* <sup>+</sup> 0.67eି0.048(*x*ି௫)

*<sup>H</sup>* <sup>+</sup> 0.43eି0.021(*x*ି௫)

*<sup>H</sup>* (10)

*<sup>H</sup>* (11)

<sup>∆</sup>*Tx/*∆*Tr <sup>=</sup>* 0.33eି0.59(*x*ି௫)

<sup>∆</sup>*Tx/*∆*Tr <sup>=</sup>* 0.57eି0.13(*x*ି௫)

to the sum of two exponential attenuations:

conforms to the sum of two exponential attenuations:

**Figure 10.** Comparison between existing temperature decay models and the model proposed in this paper. **Figure 10.** Comparison between existing temperature decay models and the model proposed in this paper.
