As displayed in Fig. 1, the horizontal road tunnel was designed using the FDS, a modeling system that replicates the tunnel environment. The tunnel has measurements of 500 m in length, 5 m in width, and 9 m in height. Two smoke outlets were positioned in the model tunnel ceiling, each measuring 5 by 2 m. The smoke outlet’s short side lies parallel to the tunnel’s longitudinal axis. The distance from one smoke outlet to another was designated as D_e, which varied from 60 to 100 m at an interval of 20 m. The fire source is positioned along the tunnel’s longitudinal axis and is positioned between two smoke outlets. d was defined as the distance between the fire source and a single smoke outlet. More specifically, d₁ is the distance between the fire source and its upstream smoke outlet, and d₂ is the distance between the fire source and its downstream smoke outlet. When the fire source exists at the initial position, d₁ is 5 m. To study the impact of varying lengths from the fire to smoke outlets on smoke characteristics, the fire source was relocated from its initial position to the midpoint of the two smoke outlets in increments of 5 m until d₁ was equal to d₂. In the tunnel equipped with the centralized smoke extraction system, the top smoke duct is connected to external shafts at both ends. The fans in both shafts will switch on at the same time after the occurrence of fire. Thus, the exhaust volumetric flow rates upstream and downstream are the same in the smoke duct, as shown in Fig. 1.

By dictating a heat release rate per unit area over a 4 m² square solid surface which is parallel with the floor, the fire source boundary conditions have been established. The current temperature of the surroundings is 293 K, and the current pressure of the surroundings is 101 kPa. Temperature sensors were strategically set up 0.2 m below the tunnel ceiling centerline, spaced with a spacing of 1 m along the tunnel longitudinal axis. Nine measurement points for temperature and velocity were strategically positioned at the smoke outlet.

In sum, 147 test cases were established, as detailed in Table 1. Three smoke outlet spacing and 6~10 longitudinal fire source positions were considered. The heat release rates were set at 3, 5, and 10 MW, to simulate vehicle fire [27]. The exhaust volumetric flow rates were 60, 80, 100, and 120 m³/s.

The size of the grid is a vital factor that directly impacts the precision of the simulation outputs. Optimal grid size has significance for achieving accurate numerical simulation results and ensuring computational efficiency. In numerical simulations, the dimensionless parameter D∗/δx was employed to assess the grid quality across various fire scenarios. D∗ can be expressed as follows [28]: (1)D∗=(Q˙ρacpTag)52where Q˙ is the heat release rate (kW), ρ_a is the air density (kg/m³), c_p is the specific heat at constant pressure (kJ/(kg·K)), T_a is the ambient temperature (K), and g is the gravity acceleration (m/s²).

The recommended D∗/δx should be 4~16 [28]. Generally, a smaller grid size results in higher accuracy for numerical simulations; however, this also leads to longer simulation times. Grid quality verification was necessary to determine an appropriate grid size. The best grid size for Q˙ = 3 MW varies from 0.09 to 0.37 m, as determined by Eq. (1), and the recommended values of D∗/δx. Similarly, for Q˙ = 5 MW, the optimal grid size varies from 0.11 to 0.46 m, and for Q˙ = 10 MW, it varies from 0.15 to 0.6 m. Thus, this research picked the parameters Q˙ = 5 MW, De = 60 m, V = 60 m³/s, d1 = d2 = 30 m for the sensitivity analysis of the gird by using three different mesh sizes (0.2, 0.25, 0.35 m) to verify the independence of the mesh under three HRRs. As depicted in Fig. 2, the longitudinal temperature distribution is similar for 0.2 m grid size and 0.25 m grid size. However, the deviation in the simulation results for the 0.3 m grid size is markedly apparent. In order to achieve a compromise between simulation time and accuracy, the 0.25 m grid size was selected.

When smoke transitions from the point of origin to the ventilation outlet, a large volume of air is drawn in. This mostly happens during three stages: plume rise (Stage 1), one-dimensional longitudinal spreading (Stage 2), and smoke discharge from the smoke outlet (Stage 3) [29], as represented in Fig. 3. This work specifically examines the phenomenon of air entrainment in Stage 3.

The smoke mass flow rate emitted from the smoke outlet is determined by adding the smoke mass flow rate generated at the fire source to the fresh air mass flow rate that is drawn in, as indicated by the following equation: (2)m˙es=m˙fa+m˙sswhere m˙es is the mass flow rate expelled from the exhaust (kg/s), m˙fa is the fresh air mass flow rate (kg/s), m˙ss is the smoke mass flow rate (kg/s).

Depending on the principle of energy conservation, the following equations could be expressed: (3)Q˙es=Q˙ss (4)Q˙es=cpm˙esΔTes=cpρsvesSΔTes (5)Q˙ss=cpm˙ssΔTsswhere Q˙es is the heat expelled from the smoke outlet (kW), Q˙ss represents the heat of smoke produced by the fire burning (kW), ΔTes is the temperature increase at the smoke outlet (K), ΔTss is the temperature increase at the smoke outlet without centralized smoke exhaust (K), ρs is the smoke density at the smoke outlet (kg/m³), ves is the velocity at the smoke outlet (m/s), and S denotes the area of the smoke outlet (m²).

Combining Eqs. (4) and (5) with Eq. (2), the following equation can be expressed: (6)m˙fam˙es=1−ΔTesΔTss

The fresh air mass flow rate and smoke mass flow rate can be calculated using Eqs. (2) and (6). Fig. 4 demonstrates the trend of mass flow rate with longitudinal fire positions at the smoke outlet, considering various heat release rates. Fig. 4a demonstrates that m˙ss1 at the upstream smoke outlet exhibits a steady increase as d1 grows, while m˙fa1 at the upstream smoke outlet experiences a gradual drop with the increase of d1. Furthermore, m˙fa is oppositely related to HRR, while m˙ss is directly related to HRR, assuming the same positioning conditions. The closer the fire source is to the smoke outlet, the higher the temperature of the smoke below the smoke outlet. The greater the difference in density between the smoke and the ambient air, the greater the mass flow rate of air at the upstream smoke outlet. Thus, m˙fa1 increases as the density difference between the smoke and the ambient air becomes higher. Fig. 4b illustrates the variation in m˙fa2 and m˙ss2; both m˙fa2 and m˙ss2 exhibit minimal variations at the outlet when fire is at different positions. The distance between the fire and the downstream smoke outlet is greater compared to the upstream outlet, resulting in a negligible impact on the temperature difference beneath the smoke outlet due to varying fire locations. Accordingly, the fresh air mass flow rate and the smoke mass flow rate at the exhaust vent downstream changed a little.

Fig. 5 displays the changes in the m˙fa and m˙ss as the exhaust volumetric flow rates and longitudinal fire positions vary. When the smoke exhaust volumetric flow rate is held constant, m˙fa1 falls as the distance d1 increases, whereas m˙ss1 grows with d1. When d1 is constant, the fresh air and smoke mass flow rates both increase with V because of the enhanced extraction effect. From Fig. 5b, m˙fa2 and m˙ss2 are essentially unchanged with d2. When V varies, there is a substantial increase in m˙fa2, but a small increase in m˙ss2.

Fig. 6 illustrates the changes in m˙fa and m˙ss, influenced by varying intervals between two smoke outlets and different longitudinal fire positions. m˙fa1 falls in proportion to the ratio of d1 to De, while m˙ss1 increases in proportion to the same ratio, as depicted in Fig. 6a. However, m˙fa2 and m˙ss2 change with d2/De insignificantly from Fig. 6b. The ratio d/De indicates the relative location of the fire origin between the smoke outlets. Hence, the alterations in the mass flow rate are affected negligibly by alterations in the intervals between smoke outlets.

Given the unpredictability of fire locations, the length from the fire to its nearest upstream and downstream smoke outlets can vary. Consequently, this impacts the heat transfer both before and after the fire source. Define the heat exhaust coefficient as the ratio of the heat expelled from the exhaust outlet to Q˙, and it can be expressed by Eq. (7): (7)E=Q˙esQ˙=∑Ei=∑Q˙es,iQ˙where E is the heat exhaust coefficient of the centralized smoke extraction system, Ei is the heat exhaust coefficient of a single smoke outlet, and Q˙es,i is the heat released from a single exhaust outlet (kW).

When the fire source is equidistant from both sides of the smoke outlet, two outlets should discharge the same heat. In this scenario, a single smoke outlet’s heat exhaust coefficient can be denoted as E0, which is associated with Q˙, V, De, H, ρa, Ta, cp and g. As a result, E0 can be expressed by Eq. (8): (8)E0=f(Q˙,V,De,H,ρa,Ta,cp,g)

The fundamental magnitudes are selected as the primary quantities, representing the four basic physical quantities ρa, cp, Ta, and H. Eq. (9) can be obtained as follows: (9){π1=ρaα1cpβ1Taγ1Hε1Q˙=[ML−3]α1[L2T−2θ−1]β1θγ1Lε1[ML2T−3]π2=ρaα2cpβ2Taγ2Hε2V=[ML−3]α2[L2T−2θ−1]β2θγ2Lε2[L3T−1]π3=ρaα3cpβ3Taγ3Hε3De=[ML−3]α3[L2T−2θ−1]β3θγ3Lε3Lπ4=ρaα4cpβ4Taγ4Hε4g=[ML−3]α4[L2T−2θ−1]β4θγ4Lε4[LT−2]

In accordance with similarity theory, Eq. (9) should be substituted with the Eq. (10): (10)E0=f(Q˙ρacp3/2Ta3/2H2,Vcp1/2Ta1/2H2,DeH,HgcpTa)=f(Q˙ρacpTag1/2H5/2,VH5/2g1/2,DeH)=f(Q˙∗,V∗,De∗)

From Eq. (10), E0 is related to Q˙∗, V∗, and De∗. Fig. 7a illustrates the influence of varying HRRs on E0, demonstrating a decrease as the Q˙∗ increases for a constant exhaust volumetric flow rate. This phenomenon occurs because there is a direct connection between the increase in heat release rate and the corresponding increase in the production of smoke mass flow rate. Moreover, the mass flow rate of expelled smoke remains constant, resulting in a decrease in the heat exhaust coefficient. Fig. 7b depicts the relationship between different exhaust volumetric flow rates and E0. It shows that once the V∗ increases, E0 also increases. More smoke is exhausted and, subsequently, an elevation in the heat exhaust coefficient. Fig. 7c depicts the impact of various smoke outlet intervals on the heat exhaust coefficient E0; notably, E0 decreases as the dimensionless smoke outlet spacing increases. This phenomenon occurs because an increase in smoke outlet spacing lengthens the distance smoke travels within the tunnel. As a result, this lengthening increases the heat transfer between the tunnel walls and hot smoke, reducing the amount of heat that is released from the smoke outlets. The decrease in heat discharge leads to a fall in the heat exhaust coefficient.

To explore the relationship among E0, Q˙∗, V∗, and De∗, The Eq. (11) provided expresses the heat exhaust coefficient E0. (11)E0=χV∗λDe∗τQ˙∗σwhere χ, λ, τ, σ are the coefficients.

By analyzing the data we obtained the formula for E0 as shown in Eq. (12). As illustrated in Fig. 8, there is a good correlation between the heat exhaust coefficient E0 and the predicted values for different heat release rates, smoke extraction volumetric rates, and the spacing of smoke outlets. Its correlation coefficient is 0.95, indicating the reliability of the predictive model.

(12)E0=0.57V∗0.3De∗0.3Q˙∗0.06

Fig. 9 depicts the variation in Ei for a single smoke exhaust outlet as the longitudinal location of the fire source is altered. There is a noticeable pattern showing that Ei decreases as the length from the fire source to the smoke outlet rises. This phenomenon occurs because there is increased heat transmission between the tunnel walls and hot smoke. As a result, less heat is released from the smoke outlet, leading to a decrease in Ei. Therefore d has an impact on the heat transfer occurring at the smoke outlets. Furthermore, the tunnel heat exhaust coefficient is influenced by d1 and d2.

The previous investigation indicates a negative link between Ei and the distance separating the fire source from the smoke exhaust outlet. Therefore, it is implied that the correlation between Ei/E0 and d/De can be mathematically represented by the subsequent equation: (13)EiE0=f(dDe)

Fig. 10 demonstrates the functional relationship between Ei/E0 and d/De. It can be seen easily that there is a linear function between Ei/E0 and d/De. The constant term and slope of this function can be derived through fitting with data. It can be observed that for varying smoke outlet intervals, the slope (k1) of the fitted functional relationship decreases as the spacing of smoke outlets increases, according to Fig. 10b,d,e. Since the functional equation passes the point (0.5, 1), the slope can express the intercept of the equation. To unify the functional expression under different smoke outlet intervals, the relationship between smoke outlet intervals and k1 can be represented by the following formula:

(14)k1=−1200De−0.3

Bringing Eq. (14) into the relationship obtained from the fit, the following equation can be derived: (15)EiE0=1.15+1400De−(De200+0.3)dDe

By linking Eqs. (12) and (15), we can derive a prediction model Eq. (16) for the Ei, as displayed in Fig. 11. The calculated heat exhaust coefficient closely corresponds to the expected values, indicating a clear alignment between the two parameters. Hence, the prediction model can more effectively describe the heat exhaust coefficients associated with various longitudinal fire source locations within the tunnel.

(16)Ei=0.57V∗0.3De∗0.3Q∗0.06[1.15+1400De−(De200+0.3)dDe]

The length from the smoke exhaust outlet to the smoke front was defined as L_b which is called smoke spreading distance [30]. Fig. 12 illustrates the smoke movement under various fire source positions. The figure clearly demonstrates that while Q˙ = 5 MW, V = 80 m³/s, the distance at which the smoke spreads upstream decreases with d1 increases. Conversely, the downstream smoke spreading distance increases inversely with the decreased d2. When the distance between the fire source and the smoke outlet is shorter, the smoke is pulled more forcefully towards the outlet, resulting in an increased upstream movement of the smoke. Consequently, the distance that smoke spreads in the upstream direction is greater than the distance it spreads in the downstream direction.

Li et al. [13] gave a model for calculating the dimensionless smoke spreading distance combined with theoretical analysis, which can be represented by the subsequent equation: (17)LbH={18.5ln⁡(0.81Q˙∗1/3/v∗),Q˙∗≤0.1518.5ln⁡(0.43/v∗),Q˙∗>0.15 (18)Q˙∗=Q˙ρacpTag1/2H5/2 (19)v∗=vgHwhere v is the longitudinal velocity, m/s; v∗ is the dimensionless longitudinal velocity, m/s.

When d₁ = d₂, the heat produced by the fire source is discharged through the smoke outlets. Simultaneously, the remaining heat carried by the one-sided smoke in the tunnel due to the extraction effect Q˙e∗ could be written as: (20)Q˙e∗=12Q˙−cpm˙esΔTes=12Q˙−cpρsvesSΔTes

The dimensionless induced flow velocity can be defined as: (21)vin∗=m˙es/2AρagHwhere A is the area of the tunnel cross-section, m²; vin∗: is the dimensionless velocity of the induced air flow, m/s.

According to Eqs. (17), (20) and (21), in the scenario where the fire source exists at the midpoint of the tunnel, there would be a balance between the spread distance upstream and downstream. Therefore, only one side of the smoke spreading distance is considered, as displayed in Fig. 13. According to Fig. 13, the formulation can provide a more precise description of the smoke spreading distance with different HRRs, exhaust volumetric flow rates, and smoke outlet intervals. Since Q˙e∗ is less than 0.15 in this investigation, the model for prediction for the dimensionless spreading distance of smoke (Lb,mid) in the situation where the fire source is at the midpoint of the tunnel could be written as follows:

(22)Lb,midH=24.8ln⁡(0.81Q˙e∗1/3/vin∗)

The dimensionless quantity d∗=d/H is introduced. As depicted in Fig. 14, the change rate of the smoke spreading distance with d∗ varies depending on Q˙ and V. The change rate of the smoke spreading distance with d∗ is basically the same under different interval conditions.

Therefore, it is assumed that the relationship among the rate of change (k2), d∗, V∗, can be given by Eq. (23): (23)k2=ηQ˙∗φV∗ζwhere η, φ, ζ are the coefficients.

As shown in Fig. 15, to determine the relationship among k2, Q˙∗, and V∗, all simulated conditions were fitted with dimensionless Q˙∗ and dimensionless V∗ to obtain an empirical equation for k2:

(24)k2=−0.7(Q˙∗V∗3)0.5

The smoke spreading distance at other fire locations can be expressed by Eq. (25): (25)LbH=Lb,midH+k2(d∗−d0) (26)d0=De2/Hwhere d0 is the dimensionless parameter when the fire source is at the midpoint of two smoke outlets.

Therefore, by combining Eqs. (22), (24) and (25), a model for predicting the smoke spreading distance at different locations can be obtained: (27)LbH=24.8ln⁡(0.81Qe∗1/3/vin∗)−0.7(Q˙∗V∗3)0.5(De2H−d∗)

All the test results were compared to the predictions, as shown in Fig. 16. The prediction model can better predict the dimensionless smoke spreading distance with different HRRs, different intervals of smoke outlets, and different exhaust volumetric flow rates.