A Mixed Convection Model for Estimating the Critical Velocity to Prevent Smoke Backlayering in Tunnels

Open Access
09-09-2024

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Authors: Michael Beyer; Conrad Stacey; Günter Brenn
Published in: Fire Technology | Issue 2/2025

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Abstract

This article introduces a mixed convection model for estimating the critical velocity needed to prevent smoke backlayering in tunnels during fires. The model is validated against data from the Memorial Tunnel fire tests and CFD simulations, showing consistent results across a range of heat release rates. The study also compares the model to small-scale test data, highlighting the importance of considering fire dimensions and intensity. A simplified equation for practical design purposes is provided, along with recommendations for design parameters. The article concludes by emphasizing the model's predictive capability and its potential application in tunnel designs, while acknowledging the need for further validation with full-scale data.

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Notes
Abbreviations

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$A$

Tunnel cross-sectional area at the fire site (m²)

$a$

Parameter used for non-dimensional trend lines (-)

${a}_{w}$

Empirical constant to offset the ‘plume blockage’ function. See Table 2 for pool fires, Equation (30) (-)

$b$

Maximum nondimensional critical velocity value for different non-dimensional trend lines (-)

${c}_{p}$

Specific isobaric heat capacity based on average temperature between approaching air and mean temperature just downstream of the fire, according to Equation (9). That equation needs to be adjusted if convective heat release rate is further reduced due to water mist systems or sprinklers. (J/kg K)

${d}_{b}$

Burner diameter (m)

${D}_{h}$, $\overline{H }$

Tunnel hydraulic diameter at the fire location (m)

${f}_{a}$

Empirical constant used in fire length function. See Table 2 for pool fires (-)

${f}_{g}$

Empirical constant to bias the ‘plume blockage’ function. See Table 2 for pool fires, Equation (30) (-)

${f}_{h}$

Empirical factor relating ${L}_{n}$ to the length of that part of the fire that influences backlayering propensity. See Table 2 for pool fires (-)

${f}_{o}$

Empirical constant used in fire length function. See Table 2 for pool fires (-)

${F}_{i}$

Inertial force (N)

${F}_{b}$

Buoyancy force (N)

$g$

Gravitational acceleration (m/s²)

${g}_{i}$

Component of the gravitational acceleration for the Cartesian coordinate direction $i$ (m/s²)

$h, H$

Tunnel height at fire location (from floor to highest point at ceiling) (m)

${H}_{DK}$

Height given different meanings in different references. Distance from the base of the fire source to the highest point in the ceiling in Danzinger & Kennedy [20] and in Annex D of NFPA 502 version 2023 [8]. Height of duct or tunnel at fire site in Annex D of NFPA 502 version 2014 [43] (m)

${H}_{scale}$

Tunnel height of a scaled tunnel (m)

${I}_{fire}$

Fire intensity (total heat release rate per unit area) (e.g. pool fire with oil fuel 2.25 MW/m² [9]) (W/m2)

$k$

Thermal conductivity (W/mK)

${K}_{F}$

Function for plume width influence on critical velocity (-)

${K}_{g}$

Grade factor, see Table 3 (-)

${K}_{g,DK}$

Grade factor as defined by [22] (-)

${K}_{L}$

Fire length function (-)

$L$

Characteristic length of the problem (m)

${L}_{fire}$

Total length of a continuous fire source (m)

${L}_{n}$

Characteristic length (height) for natural convection (for pool fires: tunnel height from the base of the fire to highest point at the ceiling) (m)

$m$

Spreading rate of rising plume relevant for interaction of ambient air with plume. See Table 2 for pool fires (-)

${p}^{*}$

Non-dimensional pressure defined as ${p}^{*}=p/\left({\rho }_{a}{U}^{2}\right)$ (-)

$\dot{Q}$

Total fire heat release rate (W)

${\dot{Q}}_{scale}$

Total fire heat release rate in a scaled tunnel (W)

${\dot{Q}}^{*}$

Nondimensional heat release rate (-)

$\dot{Q}_{kink}^{*}$

Non-dimensional total heat release rate where kink in non-dimensional trend lines happens (-)

$s$

Tunnel slope (ratio of height difference to length) (-)

${s}_{t}$

Transverse tunnel slope (ratio of height difference to transverse length) (-)

$T$

Temperature (K)

$\Delta T$

Temperature increase of a heat capacitance flow rate caused by a heat release rate $\dot{Q}$ (K)

$\Delta {T}_{p}$

Empirical temperature difference for describing the buoyancy force relevant for upstream smoke propagation (K)

${t}^{*}$

Non-dimensional time (-)

${T}_{a}$

Temperature of upstream air (K)

${T}_{p}, {T}_{f}$

Temperature of the hot gases (K)

$U$

Free stream velocity upstream of the fire or characteristic velocity (m/s)

${U}_{c}$

Critical velocity (m/s)

${U}_{c}^{*}$

Nondimensional critical velocity (-)

$U_{{c\_flat}}$

Critical velocity in a flat tunnel (m/s)

${U}_{conf}$

Confinement velocity (m/s)

${U}_{scale}$

Upstream air velocity in a scaled tunnel (m/s)

${u}_{i}^{*}, {u}_{j}^{*}$

Non-dimensional velocity in Cartesian coordinate direction $i$ and $j$ respectively (-)

${W}_{fire}$

Maximum width of the fire source (e.g. fire pans of Memorial Tunnel tests were 3.66 m wide) (m)

${x}_{i}^{* }, {x}_{j}^{*}$

Non-dimensional distance in Cartesian coordinate direction $i$ and $j$ respectively (-)

1 Introduction

“Critical velocity” is that speed of oncoming tunnel air which just prevents upstream backlayering of smoke in a tunnel fire. Achieving critical velocity as a minimum airspeed in fire response is often a design criterion for tunnel ventilation, particularly in the US, Australia and some countries in Asia. There is generally no practical upper limit in those jurisdictions. Other jurisdictions do not see this critical velocity approach as the best or recommended approach [1]. European approaches [2‐7] for example encourage more moderate air speeds, acknowledging that risks from over-ventilating are also significant. The US standard NFPA 502 historically relied on the critical velocity philosophy, but recently moved away from preventing backlayering to controlling backlayering [8], allowing some extent of smoke backlayering. However, even if absolute prevention of backlayering from a tunnel fire should not always be the first choice during the self-rescue phase, it can still be a useful design or benchmark criterion. Therefore, there is motivation to have a reliable tool for estimating critical velocity.

In the published literature, there have been three approaches to get to the answer: (i) conducting full-scale fire tests (expensive and hence rare), (ii) using a Computational Fluid Dynamics (CFD) model validated against full-scale tests, or (iii) calculating the critical velocity using accepted semi-empirical equations.

Over the last decades, many full-scale fire tests were carried out, but few of them were designed to accurately determine critical velocity in tunnels for large fires. The best-known full-scale data for critical velocity with a wide and relevant range of heat release rates (HRRs) are the Memorial Tunnel fire tests [9‐11]. These tests are well documented and thus well suited for validation. Several authors have compared different CFD models and approaches with the Memorial Tunnel full-scale tests previously (see Karki et al. [12], Rhodes [13], Galdo Vega [14] and McGrattan et al. [15]). For the purpose of the present study, a new CFD model validated by data from the Memorial Tunnel fire tests was developed and recently presented by Beyer and Stacey [16]. Conducting CFD simulations can be time consuming and requires knowledge about the software and the choice of physical models and boundary conditions but, when validated appropriately, is able to provide reasonably accurate answers for a specific tunnel project with tunnel geometry and fire scenario different from the known full-scale data. It also allows a “confinement velocity”, i.e., the air speed allowing for a designated length of hot smoke backlayering under the tunnel ceiling, to be determined.

The simplest and fastest way for estimating a critical velocity is the use of a well-validated semi-empirical equation. There are many semi-empirical approaches based on dimensional or non-dimensional models [17‐37]. The results and answers of those approaches can be very different. However, only a few of them have been adopted in versions of the US standard NFPA 502, which confers contractual relevance for real road tunnel projects. Therefore, more focus is laid on these calculation methods for critical velocity. Some of the references noted above [17‐37] are summarised and discussed in Sect. 2. For a more detailed explanation of the different methods, reference is made to other work [38‐42].

For many years, Annex D of NFPA 502 [43] proposed an equation introduced by Kennedy [22]. Even though the equation proposed by Kennedy was mainly based on the Memorial Tunnel fire test results, the equations in the Annex D of NFPA 502 were first amended [44] and then replaced [45] to better represent small-scale test results published by Li et al. [34‐37].

The work by Li et al. [34‐37] overpredicts the critical velocity [46‐49], for reasons established in several papers [46, 47, 49‐51]. That also explains why Kennedy [22] and Li et al. [34‐37] could both claim support from Memorial Tunnel data for very different critical velocity values. The representation of the Memorial Tunnel data by Kennedy [22] was correct.

As discussed above, there has been conflicting and confusing guidance from NFPA 502 for designers on estimating critical velocity of smoke backlayering in tunnel fires. The 2023 edition of NFPA 502 [8] did not achieve the required clarity. One recommendation to designers was to look at the original test data for values appropriate to a particular tunnel (Beyer et. al. [1]). However, even with real test data available, it is still helpful to have a physically based mathematical relation allowing the critical velocity to be estimated for tunnel geometries or cases not well represented by known tests.

The present work reports such a mathematical formulation for critical velocity and its validation. Section 2 reviews the current methodology. Section 3 presents the proposed physical model, and Sect. 4 provides relevant information on the CFD validation and a systematic parameter variation study to test the derived physical model. In Sect. 5, the model is validated against real test data. As the equation in Sect. 3 is perhaps too complex for design purposes, Sect. 6 provides a simpler, practical estimation of the critical velocity for design purposes. Conclusions of the work are summarized in Sect. 7. A summary of the CFD validation undertaken by Beyer and Stacey [16] is provided in the Appendix A.

2 Current Methodology

The present section reviews frequently discussed methods for calculating the critical velocity. Several articles that provide a comprehensive overview are already available [38‐42] and they are not repeated here. The methods are based on characteristic numbers of fluid mechanics, which are briefly reviewed. The models essentially yield the required critical velocity as a dimensional result.

2.1 Froude Number

The Froude number $Fr$ is a non-dimensional number representing the ratio of the inertial force to the gravity force;

$$Fr = \frac{U}{{\sqrt {gL} }}$$

(1)

where $g$ is the gravitational acceleration, $U$ a characteristic velocity and $L$ a characteristic length scale of the flow field. In fluid mechanics, the Froude number is significant for flows with free surface effects, like water waves in open channel flows, and for any isothermal fluid flow, which can be significantly influenced by the body force due to gravity. In the tunnel community and in fire engineering, a different form and definition of the Froude number, sometimes called the ‘critical Froude number’, can be found. The distinction is discussed in the following Sect. 2.2.

2.2 Dimensional Model

One of the first calculation models for the critical velocity was introduced by Thomas [17, 18]. He concluded that the upstream spread of hot smoke in tunnel fires can be described by the ratio between the inertial force (velocity head) and the buoyancy force (buoyancy head). It was assumed that, at the critical condition where backlayering of hot smoke is just prevented, the stagnation pressure (velocity) head will be of the same order of magnitude as the buoyancy head. The inertial force, ${F}_{i}$ and the buoyancy force, ${F}_{b}$, as defined by Thomas [17, 18] are:

$$F_{i} = \rho_{a} L^{2} U^{2}$$

(2)

$$F_{b} = \rho_{a} gL^{3} \beta \Delta T$$

(3)

where $L$ is a characteristic length scale in the problem, $\beta$ is the isobaric thermal expansion coefficient of the gas, and $\Delta T$ is the temperature difference relevant for the buoyancy force. The approach taken by Thomas determines $\Delta T$ in the buoyancy as the temperature increase of a heat capacitance flow rate ${c}_{p}{\rho }_{a}AU$ caused by the heat release rate (HRR) $\dot{Q}$, as it follows from the First Law of thermodynamics for an open system.

$$\Delta T = \frac{{\dot{Q}}}{{c_{p} \rho_{a} AU}}$$

(4)

Here, $A$ is the tunnel cross-sectional area and ${c}_{p}$ the specific isobaric heat capacity of the gas. By expressing the temperature rise as a function of air speed, the equation above connects the temperature increase, and hence the buoyancy force, to inertia effects. Substituting this equation into Equation (3) and forming the ratio of inertia force to buoyancy force (${F}_{i}/{F}_{b}$) gives the Froude number as defined by Thomas [17].

$$Fr_{c} = \frac{{\rho_{a} c_{p} AU^{3} }}{{gL\beta \dot{Q}}}$$

(5)

The number above is also named the ‘critical Froude number’, but in fluid dynamics, the number is a Richardson number $Ri$ (i.e., a ratio of Grashof number $Gr$ to Reynolds number squared ${Re}^{2}$) rather than a Froude number as given by Equation (1). In this approach there is only one characteristic length scale, other than that embodied in the tunnel cross-sectional area $A$. As a result, the method does not distinguish between tunnel height or hydraulic diameter, or the fire plume height. The length scale here is taken to be the tunnel height, $L=H$. Thomas then concluded that the critical Froude number is expected to be in the order of unity at the flow situation where backlayering of hot smoke is just prevented, arriving at a critical velocity formula of

$$U_{c} = \left( {\frac{{gH\dot{Q}}}{{\rho_{a} Ac_{p} T_{p} }}} \right)^{1/3}$$

(6)

where $\beta$ has been replaced by the inverse of the temperature of the hot gas, $1/{T}_{p}$, treating the gas as ideal.

A similar equation was proposed by Danziger and Kennedy [20, 22]. In contrast to Equation (6), the total HRR $\dot{Q}$ was replaced by the convective HRR, $\dot{Q}(1-\varepsilon )$, with the fractional convective heat reduction $\varepsilon$ due to radiation. A correction for the tunnel slope was also introduced. Furthermore, the temperature ${T}_{p}$ was replaced by the bulk temperature downstream of the fire, ${T}_{f}$. This led to the critical velocity

$$U_{c} = K_{g,DK} Fr_{c,DK}^{ - 1/3} \left( {\frac{{gH_{DK} \dot{Q}\left( {1 - \varepsilon } \right)}}{{\rho_{a} Ac_{p} T_{f} }}} \right)^{1/3}$$

(7)

with the grade correction factor for downgrade slopes $s$ given as

$$K_{g,DK} = 1 + 0.0374 \cdot \left[ {abs\left( s \right) \cdot 100} \right]^{0.8}$$

(8)

The proposed grade correction factor was derived from studies on the control of methane layers in coal mines. It was suggested that the grade correction should only be used for ventilating downgrade. For uphill grades, ${K}_{g,DK}$ is 1.0.

The temperature ${T}_{f}$ in Equation (7) is the bulk temperature downstream of the fire and is expressed as

$$T_{f} = \frac{{\dot{Q}\left( {1 - \varepsilon } \right)}}{{c_{p} \rho_{a} AU_{c} }} + T_{a}$$

(9)

where ${T}_{a}$ is the approaching ambient air temperature. For the critical Froude number, Kennedy [22] used a conservative value of 4.5 instead of unity as Thomas proposed. This conclusion resulted from the observations by Lee et al. [21].

Based on that value, Kennedy [22] validated the derived formula against the longitudinal ventilation test results of the Memorial Tunnel Fire Ventilation Test Program (MTFVTP) [9, 10] and concluded an under-prediction of the critical velocity for low fire heat release rates (< 20 MW) and an over-prediction for high heat release rates. The proposed formula was also adopted into Annex D of NFPA 502 [43]. Attention needs to be paid to the definition of ${H}_{DK}$ in the different quotations of this equation. The proposed formula by Kennedy [22] uses for ${H}_{DK}$ the distance from the base of the fire source to the highest point in the ceiling. The 2014 NFPA 502 Annex D uses the height of duct or tunnel at fire site (${H}$) for ${H}_{DK}$. From a design perspective, it makes sense to use the full tunnel height for ${H}_{DK}$ as the height of a design fire is usually not specified, and it is a more onerous case to assume that the base of the fire is at the roadway. The 2023 version of Annex D [8] also references the 2014 Annex D equation, but with ${H}_{DK}$ inexplicably redefined as being ‘height from the base of the fire to the tunnel ceiling at the fire site (not tunnel height ${H}$)’ without providing guidance on a typical fire height for a design fire.

2.3 Non-dimensional Froude Model

Non-dimensional Froude Models refer to the Froude number as defined by Equation (1) and not to the ‘critical Froude number’ or the Richardson number. The former neglects density variation and the latter two include density variation.

Non-dimensional Froude models are based on a non-dimensional critical velocity ${U}_{c}^{*}$ and HRR ${\dot{Q}}^{*}$, which are defined as

$$U_{c}^{*} { } = \frac{{U_{c} { }}}{{\sqrt {gH} }}$$

(10)

$$\dot{Q}^{*} = \frac{{\dot{Q}}}{{c_{p} \rho_{a} H^{2} \sqrt {gH} T_{a} }}$$

(11)

Both numbers use the reference velocity $\sqrt{gH}$ for the non-dimensionalisation in analogy to the definition of the Froude number in Equation (1). The reference heat release rate for Equation (11) is based on Equation (4) where in addition to the replacement of the velocity ${U}$ by $\sqrt{gH}$, the tunnel cross-sectional area $A$ is represented by ${H}^{2}$, and the temperature difference $\Delta T$ by the ambient temperature ${T}_{a}$. Relations between the non-dimensional critical velocity and fire heat release rate are exclusively investigated in model experiments at a scale as small as 1:20, as demonstrated by Oka and Atkinson [29], Wu and Bakar [31] and Li et al. [34]. In all these investigations, Froude similarity for scaling the velocity and fire HRR was assumed as follows

$$Fr = \frac{U}{{\sqrt {gH} }} = \frac{{U_{scale} }}{{\sqrt {gH_{scale} } }}$$

(12)

where the index _scale refers to the values in the scaled tunnel model. In the work of Oka and Atkinson [29], as well as in Li et al. [34], ${H}$ is the duct (tunnel) height, but in Wu and Bakar [31], tunnel height ${H}$ is replaced by the hydraulic diameter of the tunnel $\overline{H }$. The scaling as given by Equation (12) leads to the following relationship for velocity and heat release rate:

$$\frac{U}{{U_{scale} }} = \sqrt {\frac{H}{{H_{scale} }}}$$

(13)

$$\frac{{\dot{Q}}}{{\dot{Q}_{scale} }} = \left( {\frac{{H_{ } }}{{H_{scale} }}} \right)^{{\left( {2 + \frac{1}{2}} \right)}}$$

(14)

In the definition of the Froude number used by those authors, buoyancy is not included. Consequently, this approach inherently presumes that the temperature difference, and therefore the density deficit $\beta \Delta T$ between the scaled and the real tunnel, is the same. Moreover, it also assumes that the geometric proportions between area, hydraulic diameter and tunnel height are the same between the scaled and the real tunnel. For example, if the width to height ratio of the real tunnel is not the same as in the scaled duct, the mass flow rate, and therefore the gas temperature after the fire, are not scaled in the right manner. The same applies for the ratio between hydraulic diameter and tunnel height, which are relevant for the ratio of inertia to buoyancy force.

Beyond the velocity and the HRR, the fire intensity (heat release rate per unit area) also needs to be scaled in the right manner to avoid altering the plume density deficit between the real and the scaled tunnel. Failure to attend to the fire intensity would violate the assumption that allows Froude scaling to even be considered. This also includes the need to scale the flame length and plume volume in the right proportions.

The test setups of the different small-scale tests [29‐31, 34] are very similar. Wu and Bakar [31], as well as Li et al. [34], used a fixed burner size for all tests in the same test tunnel. The fuel flow rate was increased to increase the fire HRR, which automatically increased the heat release rate by unit area (fire intensity). Oka and Atkinson [29] also investigated different burner sizes and arrangement and derived a set of equations to account for the different results. The trendline for the critical velocity dependence on the HRR is different for the individual tests performed, but always has a similar characteristic as given below.

$$\begin{gathered} U_{c}^{*} = a \cdot \dot{Q}^{{*1/3}} \;for\;\dot{Q}^{*} \le \dot{Q}_{kink}^{*} \hfill \\ U_{c}^{*} = b\;for\;\dot{Q}^{*} > \dot{Q}_{kink}^{*} \hfill \\ \end{gathered}$$

(15)

The parameters $a$, $b$ and $\dot{Q}_{kink}^{*}$ for the different results are listed in Table 1. As discussed above, and detailed by Stacey and Beyer [46, 47], it is unlikely that all essential similarities are preserved in the scaled model, and hence unlikely that the trend lines provide relevant numbers when scaled to real tunnels.

Table 1

Parameters for Non-dimensional Trend Lines from Different Scale Model Tests

Authors	$a$	$b$	$\dot{Q}_{kink}^{*}$
Oka and Atkinson [29]	${{\left(0.22\,to\,0.354\right)}^{1}\cdot \left(0.12\right)}^{-1/3}$	${\left(0.22\,to\,0.354\right)}^{1}$	0.12
Wu and Bakar [31]	${\left(0.4\right)\cdot \left(0.2\right)}^{-1/3}$	0.4	0.2
Li et al. [34]	0.81	0.43	0.15

¹Value depends on the burner size and burner arrangement, see Oka and Atkinson [29]

Oka and Atkinson [29] and Wu and Bakar [31] used a water spray device to cool the tunnel wall near the fire source, which may have altered the heat loss from the smoke layer.

Any approach that does not include the density difference cannot be regarded as a physically reasonable approach for approximating density-driven flow behaviours with the confidence needed for design.

2.4 Practical Physical Problems with Prior Approaches

There are several observations to be made on the limitations of models previously applied.

The first is the dependence of the critical velocity on the heat release rate. That was a problem for Kennedy et al. [22] in matching the Memorial Tunnel data [9‐11, 52] (under- and overprediction of critical velocity as discussed in Sect. 2.2), and has been a problem since. Looking at the data relevant to road tunnel design, such a dependence could not be identified. Above a certain fire heat release rate, there is no further increase in critical velocity (see Figure 9).

Related to the problem of the heat release dependence is the second problem, which is the influence of the fire intensity. Fire intensity depends on the fuel type (heat of combustion) and the ability of the fuel to mix and burn before being advected downstream (burning rate). If the burning takes place while advecting downstream, the fire area has been increased and the intensity reduced. So, for a given fire type, an increase in the fire heat release rate involves an increase in the physical extent of the fire. For example, hydrocarbon pool fires that burn at around 2.25 MW/m² [9‐11, 52] do not obligingly burn at a higher rate (without changing the fuel type). To double the heat release rate, the pool area (area of the fuel source) also needs to be doubled. That has the effect of keeping the temperature in the hot plume relevant for the backlayering nearly constant with increasing HRR. A more intense fire (greater heat release rate per unit area) would result in a hotter plume relevant for the backlayering and greater propensity to backlayer. Experiments with a fixed burner area and varying gas flow are actually varying the intensity of the fire, introducing a variation that is not there for large real fires, yet the resulting formulae do not control for fire intensity.

The third problem is the extent of the fire, and of course, it is related to intensity. For a 20 m long fire (e.g. 20 m long truck on fire or 20 m long pool fire), the most upstream part of the fire will have the most chance of contributing to backlayering, with subsequent parts being progressively less influential. It seems, from CFD investigations, that heat released more than one tunnel height or so beyond the fire front has less additional influence on the backlayering propensity (see Figure 8). So, a 20 m long fire (e.g. truck or pool fire) uniformly burning at 100 MW would have a similar critical velocity value as a 10 m long fire burning at 50 MW, and probably not too much different to that for a 5 m long fire burning at 25 MW, when assuming that the width of the fire stays the same. Especially for wide fires, it seems that the downstream part of the fire after the first few metres does not contribute much to the required critical velocity value (as is seen in the Memorial Tunnel test results [9‐11, 52]). The heat release rate in those tests was increased from 10 MW to 100 MW by keeping the width of the fire the same but adding fire pans in the longitudinal direction. Consequently, the critical velocity value did not change very much from 25 MW to 100 MW (see Figure 9).

The fourth problem is the effect of tunnel aspect ratio. There are a couple of proposed corrections to formulae to account for the ratio of the tunnel width to the height if different from unity [28, 35, 36]. The reason that such corrections are required is clear. The recent (since about 1990) prior formulae all inherently use a temperature rise calculated by diluting the heat released by the fire into the entire flow along the tunnel (e.g. see Equations (6), (7) and (9)). It is obvious that for most tunnel fires, only a part of the tunnel flow enters the fire plume in the first few metres of the fire (where backlayering will be determined). As a tunnel gets wider, that fraction of the flow entering the plume goes down (see Sect. 4.2). Starting the formulation with an inappropriate temperature rise and then seeking to correct the answer empirically based on geometry does not seem the best approach. As part of this work, a more realistic basis for buoyancy, roughly modelling the plume area, is sought. Natural break points then enter the formulation when the tunnel is so narrow that the plume fills most of the cross section.

The next section seeks to develop a new, physically reasonable model, which addresses the above shortcomings.

3 The Present Mixed-Convection Model

All the transport processes in tunnel fire events are governed by the equations of motion of continuum fluid mechanics. These are the balance equations for mass, momentum and thermal energy. The fluid motion contributes convective transport of the conserved quantities. Tunnel fires inherently include both forced and natural convection, where the latter is due to Archimedian buoyancy caused by the temperature (or rather density) differences in the flow field. The occurrence of both types of convection at similar strengths produces what is called mixed convection. This is particularly the case when analysing the equilibrium where the buoyancy-driven convection upstream of the fire (natural convection) interacts with the inertial force of forced convection. The governing non-dimensional equations of motion are formulated in a Cartesian coordinate system, with x₁ (x) aligned with the tunnel axis in the flow direction, x₂ (y) horizontal, and x₃ (z) pointing in the direction of the tunnel height. For the present flow regime (steady state, Newtonian fluid and Mach number M < 0.3), the continuity equation, the momentum equation and the thermal energy equation for the incompressible fluid are given as

$$\frac{{\partial u_{i}^{*} }}{{\partial x_{i}^{*} }} = 0$$

(16)

$$u_{j}^{*} \frac{{\partial u_{i}^{*} }}{{\partial x_{j}^{*} }} = - \frac{{\partial p_{ }^{*} }}{{\partial x_{i}^{*} }} + \frac{1}{{Re^{ } }}\frac{{\partial^{2} u_{i}^{*} }}{{\partial x_{j}^{*2} }} - \frac{{Gr_{i} }}{{Re^{2} }}{\Theta }^{*}$$

(17)

$$u_{j}^{*} \frac{{\partial {\Theta }^{*} }}{{\partial x_{j}^{*} }} = Ec u_{j}^{*} \frac{{\partial p_{ }^{*} }}{{\partial x_{j}^{*} }} + \frac{1}{{RePr^{ } }}\frac{{\partial^{2} {\Theta }^{*} }}{{\partial x_{j}^{*} \partial x_{j}^{*} }} + \frac{Ec}{{Re}}{\Phi }^{*}$$

(18)

respectively. The non-dimensional variables and unknowns are defined with the length scale $L$, reference velocity $U$, the coordinates ${x}_{i}$ and velocities ${u}_{i}$. The non-dimensional pressure ${p}^{*}=p/\left({\rho }_{a}{U}^{2}\right)$, and the non-dimensional temperature is defined by ${\Theta }^{*}=\left(T-{T}_{a}\right)/\Delta {T}_{p}$. The thermal buoyancy caused by density differences owing to temperature gradients ${\rho }_{a}{g}_{i}\beta \Delta Tp$ is expressed via the body force in the momentum equation. This replaces the Froude number by the ratio ${Gr}_{i}/{Re}^{2}$. That is, the Froude number does not exist in the governing equations for mixed convection and hence should not have been central to scaling of such flow regimes as discussed in Sect. 2.3. The non-dimensional numbers $Re$, ${Gr}_{i}$, $Ec$ and $Pr$ are the Reynolds, Grashof, Eckert and Prandtl numbers, respectively, defined for this problem as

$$Re = \frac{UL}{{\nu_{a} }}$$

(19)

$$Gr_{i} = \frac{{g_{i} \beta L^{3} \Delta T_{p} }}{{\nu_{a}^{2} }}$$

(20)

$$Ec = \frac{{U^{2} }}{{c_{p} \Delta T_{p} }}$$

(21)

$$Pr = \frac{{\mu c_{p} }}{k}$$

(22)

The Reynolds number is the ratio of inertial to viscous forces, where $U$ is the characteristic velocity, $L$ the characteristic length of the flow field and ${\nu }_{a}$ the kinematic viscosity of ambient air. The Grashof number is the ratio of buoyancy to viscous forces. For an ideal gas, the isobaric thermal expansion coefficient $\beta =1/T$. The term ${g}_{i}$ represents the gravitational acceleration component in the Cartesian coordinate direction $i$. For tunnels with a moderate (typical) longitudinal slope $s$ and a transverse slope ${s}_{t}$, ${g}_{i}$ can be approximated as:

$$g_{1} = g_{x} \cong g \cdot s;\,g_{2} = g_{y} \cong g \cdot s_{t} ;\,g_{3} = g_{z} \cong g$$

(23)

That is, ${Gr}_{i}$ in the x and y directions becomes very small so that the body force in these coordinate directions can be neglected for the current problem (strength of forced convection dominates the strength of natural convection). In a further simplification, ${Gr}_{i}$ in the z direction will be declared just as $Gr$.

The Eckert number is the ratio of the kinetic energy to the enthalpy difference due to the temperature increase, and the Prandtl number represents the ratio of the diffusivities for momentum and thermal energy of the fluid. In flow with strong enthalpy differences as compared to kinetic energy, such as the present one, the Eckert number is not important for the physical analysis of the flow. Yet, the Prandtl number appears in a product with the Reynolds number in front of the heat conduction term in Equation (18). In flow dominated by convection, heat conduction is of minor importance, so that the Prandtl number does not appear as a quantity relevant for the flow modelling either.

When assuming that the Reynolds number is high (say Re > 10⁵), the extra stress term with $1/Re$ in front must be modelled for turbulent flow. The ratio of $Gr$ and $Re^2$ expresses the strength of natural convection relative to the strength of forced convection. It may be of the order of unity for mixed convection processes. The ratio $Gr/{Re}^{2}$ is also known as the Richardson number $Ri$ as per

$$Ri = \frac{Gr}{{Re^{2} }} = \frac{Strength\,of\,natural\,convection}{{Strength\,of\,forced\,convection^{ } }}$$

(24)

As noted above (Equation (5)) this has been referred to by prior authors as a “critical Froude number” (not to be confused with the Froude number that has been used in attempting to scale mixed convection problems), but is here referred to in standard fluid mechanics terminology.

The gas temperature $T$ in the relation $\beta =1/T$ can be expressed as the sum of the approaching air temperature and the temperature difference relevant for the buoyancy force (in z-direction towards the tunnel ceiling) caused by a tunnel fire, i.e.

$$T = T_{a} + \Delta T_{p}^{ }$$

(25)

Based on the relations above, the hydraulic diameter of the tunnel cross section is selected as the length scale L in the Reynolds number. The physical flow processes, however, are influenced by other length scales also, which enter via boundary conditions. For the modelling of the mixed convective flow effects of the present flow, therefore, the Grashof number for the z direction is modified by incorporating a length scale ratio and a grade factor for sloped tunnels. The Reynolds and modified Grashof numbers for the current problem are defined as:

$$Re = \frac{{UD_{h} }}{{\nu_{a} }}$$

(26)

$$\widetilde{Gr} = K_{g}^{2} \left( {\frac{{L_{n} }}{{D_{h} }}} \right)^{3} Gr = K_{g}^{2} \left( {\frac{{L_{n} }}{{D_{h} }}} \right)^{3} \frac{{gD_{h}^{3} }}{{\nu_{a}^{2} }}\frac{{\Delta T_{p}^{ } }}{{T_{a} + \Delta T_{p}^{ } }}$$

(27)

The grade factor for sloped tunnels ${K}_{g}$ is defined as being the ratio of the critical velocity in a sloped tunnel to the critical velocity in a flat tunnel, with the fire scenario and other geometrical tunnel parameters kept the same (see Table 3).

$$K_{g} = \frac{{U_{c} }}{{U_{{c\_flat}} }}$$

(28)

In this work, all transverse gradient (crossfall) has been taken as zero (${s}_{t}$ = 0 and ${g}_{2}$ = 0). Large crossfall may warrant an additional correction parameter.

Formulating the Richardson number with the Reynolds number and the modified Grashof number as defined by Equations (26) and (27) and solving for the velocity ${U}$ gives:

$$U_{ } = K_{g}^{ } \left( {\frac{{gL_{n}^{3} }}{{D_{h}^{2} }}\frac{1}{Ri}\frac{{\Delta T_{p} }}{{T_{a} + \Delta T_{p} }}} \right)^{1/2}$$

(29)

The velocity ${U}$ is related to smoke propagation, but is not critical velocity until $Ri$ and any empirical calibration is resolved for the equilibrium of the smoke layer at the front of the fire site.

As already discussed, it is presumed that, at the critical velocity, the strength of forced convection is of the same order as the strength of natural convection. According to Equation (24), this requires that the Richardson number is close to unity.

The frontal area of the rising plume represents an obstacle for the flow past the fire. If the plume frontal area is small compared to the tunnel area, a big fraction of the flow goes around the plume, so that the momentum acting onto the plume is lower than it would be for a plume frontal area that is nearly as big as the tunnel area (wide fires). The plume in the latter case becomes more deflected by the upstream flow, and more mixing into the plume occurs, so that the upstream velocity required to prevent backlayering goes down. The influence of the plume width on critical velocity can be considered by the factor ${K}_{F}$ defined here as:

$$K_{F} = f_{g} \cdot a_{w} \left[ {1 - {\text{min}}\left( {1,\frac{{W_{fire} L_{n} }}{A}} \right)} \right] + 1 - f_{g}$$

(30)

This correlation is a function of the plume frontal area ${W}_{fire}{L}_{n}$ in relation to the tunnel cross-sectional area $A$. The term ‘$\text{min}\left(1,{W}_{fire}{L}_{n} / A\right)$’ makes sure that the plume frontal area does not exceed the tunnel cross-sectional area. The parameters ${f}_{g}$ and ${a}_{w}$ are empirical constants that will be calibrated to measurements and a validated CFD model.

It is assumed that the effective temperature difference relevant for the buoyancy force (in z-direction towards the tunnel ceiling) $\Delta {{T}_{p}}$ is a function of the fire intensity (heat release rate by unit area), and the cooling due to the approaching air. The higher the approaching air mass flow and the bigger the plume frontal area, the more the hot plume gases are mixed and cooled down. The higher the fire intensity, the higher is the density deficit and thus the buoyancy force opposing the forced convection at the initial plume regime (fire front). It is also acknowledged that an increase in the heat release rate caused by extending the fire source in the longitudinal direction contributes less than the first part of the fire to the strength of natural convection at the front of the fire. This reasoning leads to the following representation of the effective temperature difference:

$$\Delta T_{p}^{ } = \frac{{{\text{min}}\left( {\dot{Q},I_{fire} W_{fire} L_{n} f_{h} K_{L} } \right)\left( {1 - \varepsilon } \right)\left( {1 - \eta } \right)}}{{{\text{min}}\left( {A,L_{n} \left( {W_{fire} + mL_{n} } \right)} \right)U_{ } \rho_{a} c_{p} }}$$

(31)

where $\dot{Q}$ is the total fire heat release rate, ${I}_{fire}$ is the fire intensity (total fire heat release rate per unit area), ${W}_{fire}$ the width of the fire, ${L}_{n}$ the tunnel height from the base of the fire to highest point of the ceiling, $\epsilon$ is the heat reduction due to imperfect combustion and/or radiation, $A$ is the tunnel cross-sectional area at the fire site, $U$ is the air velocity onto the fire, and ${f}_{h}{K}_{L}$ is describing the fraction of the length of the fire (as a proportion of ${L}_{n}$) that contributes to the natural convection at the front of the fire. The chosen function for the effective fire length ${K}_{L}$ is based on the ratio of the longitudinal extent of the fire ${L}_{fire}=\dot{Q}/\left( {W}_{fire}{I}_{fire}\right)$ (see Equation (34) below) to the tunnel height from the base of the fire ${L}_{n}$ and the empirical constants ${f}_{a}$, ${f}_{o}$ and ${\gamma }_{L}$ as follows:

$$K_{L} = \frac{{f_{a} + f_{{o_{ } }} }}{{f_{a} + e^{{ - \;\frac{{L_{fire} }}{{\gamma_{L} \cdot L_{n} }}}} }}$$

(32)

It was chosen simply as it has the desired form, and is able to be calibrated to observations.

Taking the minimum of $\dot{Q}$ and ${I}_{fire}$ times the effective fire area ${W}_{fire}{L}_{n}{f}_{h}{K}_{L}$ ensures that the modelled effective heat release rate does not exceed the actual total fire heat release rate if ${L}_{n}{f}_{h}{K}_{L}$ is larger than ${L}_{fire}$. The plume area interacting with the approaching air is taken as ${L}_{n}\left({W}_{fire}+m{L}_{n}\right)$. The basis for this is an assumption that plume shapes will be geometrically similar, growing in width with grade $m$ as the plume rises. As the plume width increases with height, the plume frontal area interacting with the ambient air will then vary with height from the base of the fire (conservatively tunnel height for design purposes). If a tunnel is wider than the mean plume width, the additional air passing the fire to the side of the fire does not dilute the effective temperature difference and therefore does not decrease the buoyancy force. However, the plume frontal area cannot be bigger than the tunnel area itself, which requires that the minimum of $A$ and ${L}_{n}\left({W}_{fire}+m{L}_{n}\right)$ be taken as representing the plume frontal area interacting with the ambient air. Deluge or water mist systems reduce the convective heat release rate and therefore the effective fire intensity, which can be considered via the parameter $\eta$.

With the definition of the fire intensity

$$I_{fire} = \frac{{\dot{Q}}}{{W_{fire} L_{fire} }}$$

(33)

the longitudinal extent of the fire can also be expressed as

$$L_{fire} = \frac{{\dot{Q}}}{{W_{fire} I_{fire} }}$$

(34)

where $\dot{Q}$ is the total fire heat release rate.

The definitions and assumptions above, finally lead, together with Equations (29) to (32), to the following system of implicit equations for the critical velocity

$$U_{c} = K_{F} K_{g} \left( {\frac{{gL_{n}^{3} }}{{D_{h}^{2} }}\frac{{\Delta T_{p} }}{{T_{a} + \Delta T_{p} }}} \right)^{1/2}$$

(35)

$$\Delta T_{p} = \frac{{{\text{min}}\left( {\dot{Q},I_{fire} W_{fire} L_{n}^{ } f_{h} K_{L} } \right)\left( {1 - \varepsilon } \right)\left( {1 - \eta } \right)}}{{{\text{min}}\left( {A,L_{n} \left( {W_{fire} + mL_{n} } \right)} \right)U_{c} \rho_{a} c_{p} }}$$

(36)

For flat tunnels, the grade factor ${K}_{g}$ is assumed to be unity. The characteristic length scale ${L}_{n}$ is defined as the height from the bottom of the heat source to the tunnel ceiling (conservatively the tunnel height for design purposes), ${D}_{h}$ is the tunnel hydraulic diameter, and ${W}_{fire}$ is the fire width (most likely between 2.0 m and 3.5 m for road tunnels). The fire intensity ${I}_{fire}$ depends on the type of the fire and the fire scenario. A typical value for a pool fire is 2.25 MW/m² [9] and is considered as a conservatively intense fire scenario. A higher fire intensity might be possible with different fuels or, in the experimental case, with burners introducing fuel at high rates in small areas. The empirical factors and constants in the semi-empirical correlation for calculating critical velocity (${f}_{h}, {f}_{a}, {f}_{o}, {\gamma }_{L}, {f}_{g}, {a}_{w}$ and ${m}$) might be variable, depending on the fire characteristics. For pool fires, it has been found that the values provided in Table 2 accurately represent the physical phenomena (see Sect. 5). These values were first selected as fitting the Memorial Tunnel test data and results from a validated CFD model (see Sect. 4), and were subsequently found to require no alteration to fit with the available small scale test data. The validity of the parameters listed in Table 2 is limited to the cases analysed or validated against (see Sects. 4 and 5). However, with no test data conflicting with the empirical results, they likely have wider applicability. How wide, is subject to further investigations. If water mist or deluge systems are not considered, $\eta = 0$. Values for $\varepsilon$ usually lie between 0.2 and 0.4 [53].

Table 2

Empirical Factors and Constants Used in the Derived Equations for Calculating Critical Velocity

${f}_{h}$	${f}_{a}$	${f}_{o}$	${\gamma }_{L}$	${f}_{g}$	${a}_{w}$	${m}$
0.176	0.95	0.88	2.20	0.58	1.82	0.35

Values are derived from validations against pool fires and results from a validated CFD model (See Sects. 4 and 5)

4 CFD Validation, Grade Factor and Parameter Variation

To properly calibrate Equations (35) and (36) together with (30) and (32) against the Memorial Tunnel data [9‐11, 52] it is required to understand the grade factor, as the Memorial Tunnel had a 3.2% downgrade. There are no critical velocity data from full size tunnels with varying grade, and scale models are not considered appropriate for the purpose. Further, the available data in the literature [22, 30] provide very different answers regarding the slope effect. It cannot be reliably said if one or none of them are correct. To reference the results back to a flat tunnel requires an analysis of the grade factor ${K}_{g}$ between a flat tunnel and the 3.2% downgrade. Clearly the same information on grade effects will also be useful in design of tunnel ventilation. For this, a CFD study using the simulation software ANSYS Fluent version 2021 R1 [54‐56] has been carried out. The first task was to carefully validate the CFD technique against Memorial Tunnel data, to be confident that simulations of slope effect would be meaningful. That validation has already been done and was documented in a separate paper by Beyer and Stacey [16]. A short summary of the different fire cases and results will be provided in the Appendix (A.1 and A.2). Table 11 in the Appendix A provides an overview of the simulation methodology and boundary conditions used. More comprehensive description of the Memorial Tunnel tests, the CFD model including all boundary conditions, and the methodology used, can be found in the work by Beyer & Stacey [16]. The CFD model validated for the Memorial Tunnel was then used to analyse the same model but with different tunnel slopes (Sect. 4.1). In a further step, a few parameters like width of fire pan, shape of tunnel profile and area, longitudinal extent of fire, and fire intensity were adapted and the results analysed to test the derived equations for a broader range of parameters (Sect. 4.2). Changes to the original validated CFD model, and the results of the additional CFD study, are documented here.

The Fire Dynamic Simulator (FDS) is widely used in the field of fire engineering. Therefore, the software was used in the work by Beyer and Stacey [16] where FDS results were compared to the Memorial Tunnel test data. That validation work concluded a drastic overprediction of the smoke backlayer, in combination with a weak prediction of the temperature field downstream of the fire. This has also been confirmed by McGrattan and Bilson [15] where results from FDS version 6.7.9 were compared to most of the relevant Memorial Tunnel test results, with the conclusion that FDS overpredicts critical velocity by 25% to 40%. With that knowledge, FDS was not useful or relevant for this work.

4.1 Grade Factor

The validated CFD-model described by Beyer & Stacey [16] and summarised in the Appendix A was used to identify the grade factor ${K}_{g}$ as defined in Sect. 3. For this purpose, the jet fans and all the instrument trees and other obstacles have been removed from the model, first to reduce the complexity and computational expense, and second, to simplify assessment of when critical velocity is achieved, which is made more difficult with the obstructions. As the models with different slopes are compared in a relative sense, these geometric modifications are not thought to be significant, and assessing critical velocity consistently across all tunnel slopes was seen as far more important. Grade factors have been analysed for a 50 MW fire (validation case 2 in the Appendix A), which is believed to be the most relevant fire scenario for road tunnels.

To accurately reproduce the point where upstream smoke propagation is just prevented, a simple feedback controller for adjusting the upstream velocity was implemented in the simulation model. Upstream velocity is automatically increased or decreased so that the tip of an upstream smoke layer is confined at the front edge of the fire pan. This was done by tracking the maximum occurring temperature at the tunnel cross sectional area in front of the fire front (‘tracking’ area) and adjusting the approaching air velocity in increments proportional to the temperature ratio (between maximum temperature at the ‘tracking’ area and the approaching air temperature). Once the temperature at the tracking area is equal to the ambient air temperature, the approaching air velocity was decreased by small, fixed increments. The inlet velocity stays constant once the temperature difference between the ‘tracking’ area and ambient air is > 10 K and < 40 K. By allowing sufficient computational time, the same criteria could be applied precisely to each slope case, eliminating any human judgement based on CFD images, and hence refining the relative grade effect. The outcome of that CFD study is summarised in Table 3.

Table 3

Grade Factor for Different Tunnel Slopes Based on a 50 MW Fire

Tunnel slope¹	0%	− 3.2%	− 6.0%
Heat release rate (MW)	54.3	54.3	54.3
Average upstream velocity (m/s)	3.006	3.037	3.050
Grade factor ${K}_{g}$ (–)	1.000	1.010	1.015

¹Negative value for downgrade

The conclusion from Table 3 is that, for most design purposes, the grade factor can be ignored. For the practical range of road tunnel longitudinal slopes, the effect of slope appears to be much smaller than tolerances in calculating the required ventilation to achieve critical velocity, or in measuring the result on commissioning. That is; for practical purposes, the factor ${K}_{g}$ may be deleted from Equation (35). Eliminating grade factor is a significant departure from prior practice, but the basis of the above assessment that grade effects are not significant for typical road tunnels is considered more robust than the prior trends based on very small models or tests with methane layering [22, 30]. It seems likely that no grade factor is required for transverse gradient (crossfall), but that has not been confirmed.

As the influence of grade is seen to be a second- or third-order effect, assessing it across the range of HRR was not a priority. Later work may look at it for higher HRR.

4.2 Parameter Variation

Based on the validated CFD model and the presented simulation methodology [16], geometric parameters like width of the fire, tunnel height, tunnel profile, longitudinal extent of the fire, as well as the fire intensity, were varied to better understand the effect of those parameters on the critical velocity. For most scenarios, the same velocity/backlayering controller described in Sect. 4.1 was used for the parameter variation study. The results in turn were used to verify the presented physical model (Equations (35) and (36)) and the empirical parameters as listed in Table 2.

4.2.1 Fire Width

The first parameter to be varied was the width of the fire pan. Fire width effects were studied based on the simplified Memorial Tunnel model that was used for the grade factor study. The tunnel slope of − 3.2% was kept the same for the different scenarios. The original 50 MW fire pan according to Figure 16 in the Appendix A was varied in width as illustrated in Figure 1, maintaining the fire pan length of 6.1 m and the fire intensity of 2.43 MW/m² according to validation case 2 (see Table 10 in the Appendix A). The upstream air temperature and therefore the wall temperature in the simulation was 6.13°C. Further parameters and results of the simulations are listed in Table 4.

Figure 1

Schematic of Memorial Tunnel profile used for fire pan width variation

Authors	\(a\)	\(b\)	\(\dot{Q}_{kink}^{*}\)
Oka and Atkinson [29]	\({{\left(0.22\,to\,0.354\right)}^{1}\cdot \left(0.12\right)}^{-1/3}\)	\({\left(0.22\,to\,0.354\right)}^{1}\)	0.12
Wu and Bakar [31]	\({\left(0.4\right)\cdot \left(0.2\right)}^{-1/3}\)	0.4	0.2
Li et al. [34]	0.81	0.43	0.15

Fire width	m	1.50	2.50	3.66	6.00
Area ratio plume/tunnel^#	–	0.82	0.70	0.57	0.29
Total HRR	MW	22.2	37.1	54.3	89.0
Fire intensity	MW/m²	2.43	2.43	2.43	2.43
Radiative fraction	–	0.27	0.27	0.27	0.27
Critical velocity (CFD)	m/s	3.20	3.19	3.03	2.50
Grade factor	–	1.01	1.01	1.01	1.01
Critical velocity obtained from proposed physical model	m/s	3.11	3.16	3.03	2.50

Tunnel height	m	3	6	9	12
Aspect ratio	–	1:4	1:2	3:4	1:1
Fire width	m	2.0	2.0	2.0	2.0
Total HRR	MW	50.0	50.0	50.0	50.0
Fire intensity	MW/m²	1.25	1.25	1.25	1.25
Radiative fraction	–	0.35	0.35	0.35	0.35
Critical velocity (CFD)	m/s	2.2	2.9	3.2	3.4
Grade factor	–	1.0	1.0	1.0	1.0
Critical velocity obtained from proposed physical model	m/s	2.0	2.5	2.8	3.2

Fire intensity	MW/m²	0.625	2.5	2.5
Total HRR	MW	50	50	200
Fire length	m	40	10	40
Fire width	m	2.0	2.0	2.0
Radiative fraction	–	0.35	0.35	0.35
Critical velocity (CFD)	m/s	2.20	3.10	3.28
Critical velocity obtained from proposed physical model (\({K}_{g}=1\))	m/s	2.14	2.94	3.26

Profile name	Description	Area (m²)	Hydraulic diameter (m)	Tunnel height (m)
TBM1	3-lane arched tunnel with false ceiling, see Figure 7	85.38	9.06	6.80
TBM2	3-lane TBM tunnel	85.60	9.28	8.07
TBM3	3-lane TBM tunnel, see Figure 5	106.0	10.7	9.82
TBM4	2-lane arched tunnel with false ceiling	40.00	5.99	5.00
TBM5	2-lane TBM tunnel	40.00	6.44	5.89
TBM6	2-lane arched tunnel, see Figure 1	59.95	8.13	7.86

Parameter	Unit	Location
1.476	-	First factor in Equation (37)
2.64	m	First factor in Equation (37)
294 (\({T}_{a}\))	°K	Second factor in Equation (37)
2.92⋅10^–4	s/(m²K)	First factor in Equation (38)
7.14	m	Equation (38)
0.95	-	Last factor in Equation (38)
8⋅10^–8	m/W	Exponent in exponential function of Equation (38)

Parameter	Case 1	Case 2	Case 3
Test number	606A	610	615B
Elapsed time used for averaging (sec.)	1651 to 1680.5	191.5 to 281	678.5 to 828.5
Measured total HRR¹(MW)	11.1	54.3	104.6
Activated jet fan/s	JF3	JF2, 5, 8, 11 & 14	JF1, 3, 4, 6, 7 & 9
Fire pan/s used, according to Figure 16	10 MW	50 MW	50, 30 & 20 MW
Flow rate—Loop 214 (m³/s)	143.2	148.3	139.8
Average velocity—Loop 305 (m/s)	2.92	3.00	2.85
Blockage area at Loop 305³ (m²) [9, 11, 52]	10.2	10.2	10.2
Free tunnel cross sectional area (m²)⁴	59.9	59.9	59.9
Average velocity unobstructed tunnel section (m/s)	2.44	2.51	2.38
Upstream temperature—Loop 214 (°C)	3.6	4.0	6.8
Upstream temperature—Loop 207 (°C)	8.5	7.3	12.0
Upstream temperature—Loop 307 (°C)	9.0	7.2	14.8
Measured bulk temperature after fire—Loop 302 (°C)	53	190	451
Measured bulk temperature after fire—Loop 303 (°C)	57	217	475
Upstream density—Loop 214 (kg/m³)	1.28	1.27	1.26
Specific isobaric heat capacity (kJ/kg K)	1.014	1.044	1.096
Theoretical bulk temperature after the fire based on total HRR(°C)	69	282	554
Radiative fraction² (-)	0.21	0.24	0.17

Solver	Pressure based, steady-state, Reynolds-averaged Navier–Stokes (RANS) with spatial discretisation method of 2nd order
Turbulence	Realizable k-ε model with scalable wall function
Thermal effects	Incompressible ideal gas law. Piecewise-polynomial function for the temperature dependent specific heat capacity for the individual species. Thermal dependency of the thermal conductivity and the dynamic viscosity for the individual species were estimated based on Sutherland’s law with three coefficients, in combination with a mass-weighted mixing law
Representation of fire	Eddy-dissipation combustion model
Thermal radiation	Not simulated. Instead, a radiative fraction approach as explained in the literature referenced [12, 16] has been adopted, capturing the loss of thermal energy in a gross sense
Fuel	Ideal combustion of fuel oil with a chemical formula of C₁₉H₃₀. Additional mass source term for ideal combustion and the radiative fraction approach (see [16]) are incorporated
Tunnel inlet*	Mass flow inlet with air temperature that was measured upstream of fire (~ Loop 207). Mass flow is based on measured flow rate and air density at Loop 214 (mass fraction of O₂ = 0.23, N₂ = 0.77 and 0 for all other species). See Table 10 and Beyer & Stacey [16] for further details
Tunnel outlet	Outflow
Tunnel wall	Wall temperature is ambient air temperature just upstream of fire (~ Loop 207), rough wall with roughness height = 0.0015 m. See Table 10 and Beyer & Stacey [16] for further details
Wall – instrument trees, obstacles, jet fans, fire pan frames	Adiabatic, rough wall with roughness height = 0.00025 m
Jet fans	Fan boundary to implement flow rate of 42.95 m³/s and discharge velocity of 34.2 m/s for activated jet fans according to Table 10
Fire pan	Mass flow inlet with ambient temperature (~ Loop 207) on surface located on top of fire pan. Mass fraction of O₂ and N₂ was 0, mass fraction of fuel and other species according to adopted fire HRR, radiative fraction and combustion efficiency. Combustion efficiency for extra mass source is assumed to be 95% [9, 10]. See Table 10 and Beyer & Stacey [16] for further details

Springer Professional

Abstract

Publisher's Note

1 Introduction

2 Current Methodology

2.1 Froude Number

2.2 Dimensional Model

2.3 Non-dimensional Froude Model

2.4 Practical Physical Problems with Prior Approaches

3 The Present Mixed-Convection Model

4 CFD Validation, Grade Factor and Parameter Variation

4.1 Grade Factor

4.2 Parameter Variation

4.2.1 Fire Width

4.2.2 Tunnel Height and Aspect Ratio

4.2.3 Variation of Fire Dimensions

4.2.4 Variation of Fire Intensity and Fire Length

5 Validation of Mixed-Convection Model Against Real Test Data

5.1 Memorial Tunnel Fire Ventilation Test Program

5.2 Small-Scale Test Data

6 Practical Estimation of Critical Velocity

6.1 Design Assumptions

6.2 Outcome of Design Assumptions for Typical Tunnel Profiles

6.3 Simplified Equation for Design Purposes

7 Conclusions

Acknowledgements

Declarations

Conflict of interest

Publisher's Note

A. Appendix—CFD Validation

A.1 Validation Cases

A.2 Validation Results and Discussion

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