Keywords

9.1 Introduction

Relatively large fires are needed in order for flames to extend along a tunnel ceiling. This corresponds to HRRs that are usually over 20 MWs for most tunnels. Due to the confined tunnel space, the ceiling flame length is normally longer than that in an open fire or a room fire. This horizontal extension results in higher risk for fire spread to the neighbouring vehicles. As the risk of fire spread increases with flame length, it is identified as a key parameter that needs to be thoroughly investigated.

The visible flame is a product of combustion which emits visible radiation. The flames indicate regions with high gas temperatures. Note that, the room or enclosure fires are always related to vertical buoyancy driven turbulent diffusion flames. In some special scenarios the fires could be related to turbulent jet flames, for example, a liquefied gas tank ejects burning gases from small orifices to several hundred times the orifice diameter. Therefore, the focus will be on turbulent diffusion flames as they are more likely in large tunnel fires. The existence of the turbulent diffusion flame along the tunnel ceiling is mainly due to insufficient air entrainment.

The combustion of the fuels can be categorized into two parts. In the vertical flame zone, part of the combustible gas burns and flows upward until it impinges on the tunnel ceiling. In the horizontal flame zone, another part of combustible gas burns along the ceiling to a certain distance downstream of the fire source. The flame length, L f , discussed in this chapter is defined as the horizontal distance from the impingement point of the fire source center to the flame tip.

In a tunnel fire with natural ventilation or a low ventilation rate, the horizontal flame could exist both upstream and downstream of the fire source, as shown in Fig. 9.1. Note that generally, the flames on the two sides are not symmetrically distributed since, even under natural ventilation there is a dominating wind direction. Here, the downstream side refers to the region with a slightly longer flame length.

Fig. 9.1
figure 1

Flame length in a large tunnel fire under low ventilation rate or natural ventilation

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In a tunnel fire with a high ventilation rate, the ceiling flame only exists downstream of the fire source, see Fig. 9.2. Compared to the fire under a low ventilation rate, more heat is blown to the downstream side and the smoke layer height is lower. The distinction condition between the low ventilation rate and high ventilation rate will be discussed later.

Fig. 9.2
figure 2

Flame length in a large tunnel fire under high ventilation rate

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9.2 Overview of Flame Length in Open and Enclosure Fires

For buoyancy driven turbulent diffusion flames in a quiescent environment (no wind), the flame length or flame height has been found to be well correlated with the Froude number defined using the parameters of the fuel gas and the flame length, L f , can be estimated using [1]:

$$ {L_f} = 0.235{\dot{Q}^{2/5}}-1.02{D_f} $$
(9.1)

where \(\dot{Q}\) is the HRR (kW), D f is the equivalent diameter of the fire source (m). Note that, this equation is only suitable for turbulent diffusion flames. It is widely used in fire safety engineering.

The work on flame lengths under unconfined ceilings gives us some valuable information about the flame length in a large tunnel fire. When the flame impinges on an unconfined ceiling, the unburnt gases spread radialy and a circular disc flame can be observed under the ceiling. You and Faeth [2] carried out small-scale experiments and the results show that the ceiling flame length is about half the difference between the free flame height in an open fire and the ceiling height. However, this estimation is very rough since the tested HRRs were very small. Heskestad and Hamada [3] carried out fire tests using larger HRRs ranging from 93 to 760 kW and the tests results show that the flame length below the ceiling is about the same as the difference between the free flame height and the ceiling height. In other words, the total flame length, that is, sum of the ceiling height and the ceiling flame length in an enclosure fire equals the free flame height in an open fire.

The flame length in a corridor fire is more similar to a large tunnel fire. Hinkley [4] carried out an experimental investigation of flames spreading under an incombustible ceiling in a corridor with one closed end. The fire source was a town gas burner placed at one end of the simulated corridor and covered the whole corridor width. Hinkley [4] considered the flame length in a corridor as the distance between the flame tip and a virtual origin. The horizontal distance between the virtual origin of the horizontal flames and the fire source appeared to be twice the ceiling height above the fire source, however, more test data are required to support this. Based on a dimensional analysis and investigation of the test results, Hinkley [4] presented a correlation for the flame length in a corridor with one closed end and an open bottom (no floor). For a dimensionless parameter m * < 0.024 with either no front end screen or a 150 mm deep screen fitted to the end of a 7.3 m long corridor, and for m * < 0.018 with a 230 mm deep front end screen, the flame length in a corridor with one closed end can be expressed as:

$$ \frac{{{{{L}'}_f}}}{H} = 220{m^*}^{2/3} $$
(9.2)

where the dimensionless parameter, m *, is defined as:

$$ {m^*} = \frac{{{m}'}}{{{\rho_o}{g^{1/2}}{H^{3/2}}}}$$

In the above equation, m ′ is the fuel flow rate per cm corridor width (kgcm−1 s−1), H is the ceiling height above the fire source (m), L f ′, is the horizontal flame length from the virtual origin estimated based on an empirical equation to account for the vertical flame part (m). The results showed that when m * > 0.024 the flame lengths were underestimated. Note that according to the above equation, the horizontal flame length is actually independent of the smoke thickness or corridor height. The flame length in the above equation was estimated based on a virtual origin which was not well defined. Further, the fire source was attached to the closed end of the corridor and covered the whole corridor width, and the corridor had an open bottom (no floor). Moreover, the tests were carried out under no ventilation. Therefore, the scenario is different to a fire in a tunnel.

Based on Alpert’s work [5] on infinite line plume impinging on a ceiling and flowing equally in both directions of a corridor, Babrauskas [6] presented a method to estimate the horizontal flame length, assuming that the total entrained mass flow rate required for ceiling combustion is the same as that required for vertical combustion flames in the open. Results from Hinkley et al.’s [4] and Atallah’s tests [7] were used in the analysis. However, this assumption of entrainment cannot be reasonable for the tunnel/corridor case since the air flow that can be entrained is much smaller in a tunnel fire compared to in an open fire.

9.3 Overview of Flame Length in Tunnel Fires

Limited research has been carried out on the flame length in tunnel fires. The focus has been on flame lengths on the downstream side. Rew and Deaves [8] presented an equation for flame length in tunnels, which included HRR and longitudinal velocity but not the tunnel width or height. Much of their work was based on the investigation of the Channel Tunnel Fire in 1996 and test data from the HGV-EUREKA 499 fire test [9] and the Memorial Tests [10]. They defined the horizontal flame length, L f , as the distance of the 600 ºC contour from the center of the HGV or the pool, or from the rear of the HGV. The flame length from the rear of the HGV was represented by the following equation:

$$ {L_f} = 0.02\left({\frac{{\dot{Q}}}{{120}}}\right){\left({\frac{{{u_o}}}{{10}}}\right)^{-0.4}}$$
(9.3)

where u o is the longitudinal air velocity (m/s). Note that in the above equation, the HRR \(\dot{Q}\) is expressed in MW. This equation is a conservative fit to a limited data obtained from the HGV-EUREKA 499 test. The weakness of Eq. (9.3) is that no geometrical parameter has been taken into account, which makes it impossible to predict the flame length for other tunnels due to different geometries of tunnel and fire source.

Lönnermark and Ingason [11] investigated the flame lengths of the Runehamar tests. Alpert’s equation for ceiling jet temperatures [12] were used to estimate the form of the equation for the flame length, and the uncertain coefficients were determined by regression analysis which gave a best fit for an exponent of 0.8 for the HRR. Data from the Runehamar tests and some data from the Memorial tests were used in the analysis. The proposed equation was expressed as follows:

$$ {L_f} = \frac{{1370{{\dot{Q}}^{0.8}}{u_o}^{-0.4}}}{{{{({T_f}-{T_o})}^{3/2}}{H^{3/2}}}}$$
(9.4)

where \(\dot{Q}\) is the HRR (kW), L f is the horizontal flame length (m), T f is the flame tip temperature (K) and H is the tunnel height (m). Given, that the ceiling jet equation used in the analysis is only suitable for ceiling jets under unconfined ceilings, the generic use of the equation is limited.

Both equations above imply that the effect of longitudinal ventilation velocity on flame length is not as important as the HRR. This phenomenon was observed by Ingason and Li [13] in their model-scale tests that flame length is insensitive to the longitudinal ventilation velocity.

9.4 Flame Lengths in Tunnel Fires

9.4.1 Transition Between Low and High Ventilation Rate

If the longitudinal ventilation velocity is much lower than the critical velocity, there exist two parts of horizontal flame regions, that is, upstream region (L f, ds ) and downstream region (L f, us ). For high ventilation velocities, the flames only exist downstream of the fire. The transition point is therefore, defined as the longitudinal velocity under which no ceiling flame exists upstream of the fire source. Accordingly, the “high ventilation” for the flame length is defined as the case with the ventilation velocity larger than the transition point, and the “low ventilation” corresponds to the ventilation velocity less than the transition point.

Based on Li et al.’s study [14], the ratio of back-layering length to the tunnel height is related to the ratio of the ventilation velocity to the critical velocity at a given HRR. For high HRRs, the back-layering length is only dependent on the ventilation velocity, regardless of the HRR. Note that the upstream flame length is part of the back-layering length, and the fires with ceiling flames only correspond to high HRRs. Therefore, similar to the critical velocity, a dimensionless ventilation velocity at the transition point is defined:

$$ u_{tp}^* = \frac{{{u_{o,tp}}}}{{\sqrt {gH} }}$$
(9.5)

A dimensionless HRR was defined according to the following equation:

$$ {Q^*} = \frac{{\dot{Q}}}{{{\rho_o}{c_p}{T_o}{g^{1/2}}{H^{5/2}}}}$$
(9.6)

where u o is the longitudinal velocity (m/s), g is the gravitational acceleration (m/s2), H is the tunnel height (m), \(\dot{Q}\) is the HRR (kW), ρ o is an ambient air density (kg/m3), c p is the heat of capacity (kJ/kg K), T o is an ambient air temperature (K). Subscript tp indicates transition point.

Figure 9.3 shows a plot of data with and without upstream flames. Data from longitudinal tunnel fire tests conducted at SP [13], point extraction tests also conducted at SP [15], the Memorial tunnel tests [10] and the Runehamar tests [16] are used in the analysis. For further information about large- and model-scale tests see Chap. 3. The solid data points represent a situation when the flames existed on the upstream side in the tests, and the hollow data points indicate when no flames were obtained on the upstream side for different longitudinal velocities. The data show that there is a clear transition line that exists between the solid and hollow data points. This line can be expressed as:

Fig. 9.3
figure 3

Transition line between low ventilation rate and high ventilation rate

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$$ u_{tp}^* = 0.3 $$
(9.7)

Given that the dimensionless critical velocity approaches 0.43 for large fires [14], the results shown in Fig. 9.3 indicate that the transition point corresponds to a longitudinal velocity of approximately 70 % of the critical velocity.

Example 9.1

Estimate the location of the ceiling flames for a 150 MW fire in a tunnel with dimensions of 6 m high and 6 m wide at a longitudinal velocity of 1 and 3 m/s, respectively. What are the scenarios if the tunnel instead is 6 m high and 12 m wide.

Solution:Calculate the dimensionless longitudinal velocity using Eq. (9.5), u *  = 0.13 <0.3 for the velocity of 1 m/s and u *  = 0.39 > 0.3 for the velocity of 3 m/s. Therefore, we know that both upstream and downstream flames exist for 1 m/s and only downstream flame exists for 3 m/s. If the tunnel width is 12 m instead, it means that the scenarios are the same as the tunnel width does not influence the results.

9.4.2 Model of Flame Length in Tunnel Fires

Under low ventilation rate conditions, two parts of the flame exist: upstream and downstream of the fire, respectively.

In the ceiling flame zone, relatively good stratification probably exists, that is, there appears to be a clear layer interface between the fire and the fresh air. The total mass flow rate at the flame tip in the horizontal combustion region can be estimated as follows:

$$ {\dot{m}_{hr}}= \int_0^{{L_f}}{{\rho_o}vWdx}$$
(9.8)

where \({\dot{m}_{hr}}\)is the mass flow rate of the entrained air from the lower layer by the horizontal flame (kg/s), ρ o is the density of the entrained air (kg/m3), v is the entrainment velocity (m/s), W is the tunnel width (m), x is the distance from the fire, and L f is the horizontal flame length (m).

The combustion in the horizontal flame is mainly dependent on the entrainment of the air flow and the mixing at the interface. The entrainment velocity for the mixing layer is given in the same form as that for the vertical plume. More information can be found in Chap. 12. The upstream and downstream entrainment velocities, can be expressed as:

$$ {v_{t,us}}= \beta \left|{{u_{us}}+ {u_o}}\right|,\quad {v_{t,ds}}= \beta \left|{{u_{ds}}-{u_o}}\right| $$
(9.9)

where subscripts us and ds represent upstream and downstream, respectively, “+” indicates opposite directions between longitudinal flow and smoke, and vice versa.

Note that, the entrainment coefficient in a vertical plume is assumed to be a constant. For simplicity, we also assume the average entrainment coefficient along the flame β is a constant.

Thus, the mass flow rate of total entrained air downstream,\({\dot{m}_{ds}}\)(kg/s), and upstream of the fire,\({\dot{m}_{us}}\)(kg/s), can be respectively expressed as:

$$ {\dot{m}_{ds}}= {\rho_o}{v_{ds}}W{L_{f,ds}},\quad {\dot{m}_{us}}= {\rho_o}{v_{us}}W{L_{f,}}_{us} $$
(9.10)

We also know from the research on open fires that there is a relationship between the HRR and the entrained air, that is, the HRR should be intimately related to the mass of entrained air flows. Here, we assume that the ratio of air flows involved in reaction and total entrained air flows is k. Therefore, the energy equation can be expressed as:

$$ \dot{Q} = {\dot{Q}_{vt}}+ ({\xi_{ds}}{\dot{m}_{ds}}{Y_{{O_2},ds}}+ {\xi_{us}}{\dot{m}_{us}}{Y_{{O_2},us}})\Delta {H_{{O_2}}}$$
(9.11)

where \({\dot{Q}_{vt}}\) is the heat released in the vertical flame region (kW), \(\xi \)is the ratio of the oxygen involved in the combustion to the oxygen entrained (or the ratio of air flow involved in the combustion to the total entrained air flow), ΔH O2 is the heat released while consuming 1 kg of oxygen (kJ/kg), Y O 2 is the mass concentration of oxygen in the air flow at the lower layer. The second term on the right-hand side means the heat released in the horizontal flame regions.

Inserting Eq. (9.10) into Eq. (9.11) gives:

$$ \dot{Q} = {\dot{Q}_{vt}}+ ({\xi_{ds}}{\rho_o}W{L_{ds}}\beta {Y_{{O_2},ds}}\left|{{u_{ds}}-{u_o}}\right| + {\xi_{us}}{\rho_o}W{L_{us}}\beta {Y_{{O_2},us}}\left|{{u_{us}}+ {u_o}}\right|)\Delta {H_{{O_2}}}$$
(9.12)

For upstream and downstream ceiling flames, the ratio k should be approximately the same, that is \({\xi_{ds}}= {\xi_{us}}= \xi \). Therefore, we have

$$ {L_{f,ds}}{Y_{{O_2},ds}}({u_{ds}}-{u_o}) + {L_{f,us}}{Y_{{O_2},us}}({u_{us}}+ {u_o}) = \frac{{\dot{Q}-{{\dot{Q}}_{vt}}}}{{\xi {\rho_o}\beta W\Delta {H_{{O_2}}}}}$$
(9.13)

The mass flow rate in the vertical flame region under different ventilation conditions has not yet been explored thoroughly. Li et al. [17, 18] carried out a theoretical analysis of the maximum gas temperature beneath a tunnel ceiling and the mass flow rate of the fire plume in a ventilated flow based on a plume theory. The HRR, longitudinal ventilation velocity and tunnel geometry were taken into account. Test data were also used to verify the theoretical model and good agreement was obtained between the theory and test data. However, the entrainment of the flame zone is very different compared to the plume zone. Due to lack of information, we may assume that the entrainment inside the flame region in tunnel flows is similar to that in open fires. Delichatois [19] proposed simple correlations for the mass flow rate inside the flame:

$$ \dot{m}(z)\propto D_F^2{z^{1/2}}\quad {\text{for }}z/{D_F}<1 $$
(9.14)

where z is the height above the fire source (m), \(\dot{m}(z)\) is the mass flow rate inside the flame at height z (kg/s), and D F is the diameter of the fire source (m). Note that, the equation for the mass flow at height z can also be expressed as:

$$ \dot{m}(z)\propto \rho uD{(z)^2} $$
(9.15)

where u is the vertical gas velocity (m/s) and D(z) is the diameter of the plume at height z (m). It can be expected that the fire plume diameter is proportional to the diameter of the fire source, that is, D(z)∝D F , and the temperature inside the continuous flame zone can be reasonably considered as constant. Combining the above two equations suggests that the maximum vertical gas velocity, u max,v , can be expressed as:

$$ {u_{\max,v}}\propto H_{ef}^{1/2} $$
(9.16)

where u max,v , is the maximum velocity of the vertical flame (m/s), H ef is the effective tunnel height (m), that is, the tunnel height above the fire source. For vehicle fires or solid fuel fires, the effective tunnel height is the vertical distance between the bottom of the fire source and the tunnel ceiling. The above relationship correlates well with Thomas’ equation for gas velocities in the flame zone [20].

The velocity of the fire plume after impingement on the ceiling slightly decreases, however, it can be assumed to be proportional to the maximum velocity in the vertical plume, as proved in the research on ceiling jets [12]. This indicates the maximum horizontal gas velocity, u max,h , could also be expressed as:

$$ {u_{\max,h}}\propto H_{ef}^{1/2} $$
(9.17)

where u max,h , is the maximum velocity of the horizontal flame (m/s). The horizontal maximum gas velocity could be considered as a characteristic velocity for the ceiling flames.

9.4.3 Flame Length with High Ventilation Rate

According to the previous definition, the high ventilation here corresponds to the ventilation velocity under which no ceiling flame exists upstream of the fire source, that is, a dimensionless ventilation velocity greater than 0.3u *. In such cases, the ceiling flames only exist downstream of the fire, see Fig. 9.4.

Fig. 9.4
figure 4

A schematic diagram of entrainment under high ventilation

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Note that, generally the longitudinal ventilation velocity is around 3 m/s for a tunnel with longitudinal ventilation during a fire and even less for transverse and semi-transverse ventilation. For a large tunnel fire with a relatively long ceiling flame length, the smoke velocity right above the fire could range from 6 to 12 m/s or even higher. Apparently, this velocity is relatively low compared to the gas velocity in the ceiling flame region. Therefore, we may use the gas velocity under natural ventilation to approximately express the velocity difference between the ceiling flame layer and lower layer, that is

$$ {u_{ds}}-{u_o}\propto H_{ef}^{1/2} $$
(9.18)

For a HGV fire, the horizontal flame could be very long, that is several times or even over ten times the tunnel height. The combustion in the vertical flame region could be limited due to the confinement of the tunnel configuration. Moreover, the flame was deflected and thus, the vertical flame region also contributes to the flame length. Therefore, as a first attempt, the combustion in the vertical flame region is ignored. Given that the air entrained from the lower layer is generally not highly vitiated, the oxygen concentration for the entrained air is considered to be constant, that is close to ambient. Thus from Eq. (9.13) one gets:

$$ {L_{f,ds}}= \frac{{\dot{Q}}}{{\xi {\rho_o}\beta \Delta {H_{{O_2}}}{Y_{{O_2}}}WH_{ef}^{1/2}}}\propto \frac{{\dot{Q}}}{{WH_{ef}^{1/2}}}$$
(9.19)

For a rectangular tunnel, A = WH. For other shapes, the tunnel width could vary with the flame layer height above the floor. For simplicity, let us estimate the tunnel width using W = A/H. The above equation can therefore, be written as:

$$ {L_{f,}}_{ds}\propto \frac{{\dot{Q}H}}{{AH_{ef}^{1/2}}}$$
(9.20)

To normalize the results, two dimensionless parameters are defined here. The dimensionless flame length is defined as:

$$ L_{_f}^* = \frac{{{L_f}}}{H} $$
(9.21)

The dimensionless heat release rate is defined as:

$$ Q_f^* = \frac{{\dot{Q}}}{{{\rho_o}{c_p}{T_o}{g^{1/2}}AH_{ef}^{1/2}}}$$
(9.22)

Therefore, we expect that the dimensionless flame length is proportional to the dimensionless HRR:

$$ L_{f,ds}^* = {C_f}Q_f^* $$
(9.23)

where C f is a coefficient which will be determined by experimental data. It can be seen that the flame length is independent of the ventilation velocity, under the above assumptions. This conclusion is in accordance with the previous findings presented in Sect. 9.3. Figure 9.5 shows the dimensionless flame lengths under high ventilation as a function of the dimensionless HRR. Test data from longitudinal tunnel fire tests conducted at SP [13], point extraction tests also conducted at SP [15], EUREKA 499 programme [9], Memorial tunnel tests [10], and Runehamar tests [16] were plotted. Clearly, the proposed equation correlates well with the test data. The correlation can be expressed as:

Fig. 9.5
figure 5

Flame lengths under high ventilation rate

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$$ L_{f,ds}^* = 5.5Q_f^* $$
(9.24)

It can be concluded that under high ventilation, the flame length in a tunnel fire is mainly dependent on the HRR, tunnel width and the effective tunnel height, and insensitive to the ventilation velocity.

Example 9.2

Calculate the flame length for a 150 MW fire in a tunnel with dimensions of 6 m high and 6 m wide (H × W) at a longitudinal velocity of 3 m/s, respectively. The bottom of the fire sources is 1 m (H ef = 5 m) above the tunnel floor. What is the flame length (L f, ds ) if the tunnel instead is 6 m high and 12 m wide?

Solution: From Example 9.1, we know that for the tunnel with a width of either 6 or 12 m, only downstream flame exists at a longitudinal velocity of 3 m/s. The dimensionless HRR can be estimated using Eq.  (9.22):Q f *  = (150 × 1000)/(1.2 × 1 × 293  × 9.8 0.5  × 6 × 6 × 5 0.5 ) = 1.69. For 3 m/s, the downstream flame length can be calculated using Eq. (9.21) and Eq. (9.24), that is, L f, ds  = 5.5 × 1.69 × 6 = 56  m. If the tunnel width is 12 m instead, the flame lengths calculated will reduce by 50 %, since the width is in the denominator in Eq. (9.22), that is, L f, ds  = 28 m.

9.4.4 Flame Length Under Low Ventilation Rate

Under low ventilation rate conditions, two parts of horizontal flame regions exist, that is the upstream and downstream regions. The relative velocity can be estimated using:

$$ \left|{{u_{ds}}-{u_o}}\right|\propto H_{ef}^{1/2},\quad \left|{{u_{us}}+ {u_o}}\right|\propto H_{ef}^{1/2} $$
(9.25)

The combustion in the vertical flame region is also considered as being limited and thus, ignored as a first approximation.

For a slightly higher ventilation velocity, for example, 1.5 m/s, the length of the upstream flame and the back-layering is short, and therefore, the entrained air in the upstream flame region will only be slightly vitiated (inerted). However, the air could be highly vitiated in the downstream flame region. If no dominating ventilation direction exists, the fresh air will be entrained from both sides of the tunnel by thermal pressure created by the hot smoke. The air entrained into the flame region on both upstream and downstream sides could be highly vitiated, see Fig. 9.6. This phenomenon has not been clearly understood and needs to be further investigated. As a first approximation, the oxygen concentration will be implicitly accounted for in the following analysis. Therefore, we have:

Fig. 9.6
figure 6

A schematic diagram of the vitiated gas entrained into the ceiling flame region under natural ventilation or low ventilation

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$$ {L_{f,tot}}= {L_{f,ds}}+ {L_{f,us}}= \frac{{\dot{Q}-{{\dot{Q}}_v}}}{{\xi \beta {\rho_o}W{{\bar{Y}}_{{O_2}}}\Delta {H_{O2}}H_{ef}^{1/2}}}$$
(9.26)

where \({\bar{Y}_{{O_2}}}\)is the average oxygen concentration of the entrained air, L f,tot is the total flame length (m).

Therefore, the dimensionless total flame length could be expressed as:

$$ L_{f,tot}^* = {{C}'_f}Q_f^* $$
(9.27)

where C f ′ is a coefficient which will be determined by experimental data.

Figure 9.7 shows the dimensionless total flame lengths under low ventilation conditions as a function of the dimensionless HRR. Test data from EUREKA 499 programme [9], Memorial tunnel tests [10], and Hinkley’s tests [4] were used. Note that in Hinkley’s tests [4] the fire sources were attached to one closed end, and thus the scenario could be considered as being symmetrical, that is both the flame lengths and HRRs are doubled while plotting in the figure. It is clearly shown that the proposed equation correlate very well with the test data. The correlation can be expressed as:

Fig. 9.7
figure 7

Total flame length under low ventilation rate

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$$ L_{f,tot}^* = 10.2Q_f^* $$
(9.28)

We may also want to know how the downstream flame length varies with the longitudinal ventilation velocity. Figure 9.8 shows the dimensionless downstream flame lengths under high ventilation as a function of the dimensionless HRR.

Fig. 9.8
figure 8

Downstream flame length under low ventilation

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Test data from EUREKA 499 programme [9], Memorial tunnel tests [10], and Hinkley’s tests [4] were used, see Chap. 3. For test data from Hinkley’s tests [4], the HRRs are doubled in the figure. Under natural ventilation or very low ventilation, the air entrained into the ceiling flame region is vitiated, as shown in Fig. 9.6. Especially in the vicinity of the fire the air quality gets worse. This suggests that more gas needs to be entrained into the flame region for combustion, that is, the flame length could be longer than with high ventilation, which will be discussed later.

The downstream flame length under low ventilation rate can be expressed as:

$$ L_{f,ds}^* = 5.2Q_f^* $$
(9.29)

Comparing the above equation to the equation for flame length under high ventilation shows a very small difference. This suggests that the downstream flame length is insensitive to the ventilation velocity. In other words, the flame length under high ventilation is approximately equal to the downstream flame length under low ventilation. For simplicity, the downstream flame length can be estimated using Eq. (9.24), regardless of the ventilation conditions.

Thus, as the ventilation velocity decreases, the total flame length increases, although, the downstream flame length is approximately invariant. In other words, the increase of total flame length due to a lower ventilation velocity is due to the existence of the upstream flame.

Example 9.3

Calculate the flame length in the same scenarios as described in Example 9.2 but at a longitudinal velocity of 1 m/s.

Solution: From Example 9.1, we know that both upstream and downstream flames exist for the velocity of 1 m/s. The dimensionless heat release rate is Q f *  = (150 × 1000)/(1.2 × 1 × 293  × 9.8 0.5  × 6 × 6 × 5 0.5 ) = 1.69. For 1 m/s, the maximum total flame length can be calculated using Eq. (9.28), that is L f,tot  = 10.2 × 1.69 × 6 = 103  m, and the downstream flame length can be calculated using Eq. (9.21) and Eq. (9.24) although the results could tend to be conservative, that is, L f,ds  = 5.2 × 1.69 × 6 = 53  m. The upstream flame length can be estimated as: L f,us  = 103−53 = 50 m. In reality, the upstream flame length could be slightly shorter than this value for a velocity of 1 m/s. If the tunnel width is 12 m instead, the flame lengths calculated will reduce by 50 % since the width is in the denominator in Eq. (9.22).

9.5 Summary

In a large tunnel fire, the flame impinges on the ceiling and then extends along the tunnel ceiling for a certain distance. Under high ventilation rate conditions only downstream flame exists, and under low ventilation rate conditions, both upstream and downstream flames exist. The horizontal distance between the fire source and the flame tip is called the flame length in a tunnel fire. A simple theoretical model for the flame length in a large tunnel fire was proposed. A large amount of experimental data relevant to the flame length was used to verify the model.

The downstream flame length was found to be directly related to the HRR, tunnel width and effective tunnel height, and insensitive to the ventilation velocity. The downstream flame length can be estimated using Eq. (9.24), regardless of the ventilation conditions.

Under low ventilation, that is, u * < 0.3, the total flame length increases with the decreasing ventilation velocity, despite that the downstream flame length is approximately invariant. This indicates that increase in the total flame length due to a lower ventilation velocity is only due to the existence of the upstream flame. The maximum total flame length is obtained when there is no ventilation in the tunnel, and it is approximately twice the downstream flame length in tunnel fires under high ventilation. The maximum total flame length under low ventilation can be estimated using Eq. (9.28).