Abstract. Many modelling approaches can be used to assess evacuation time of buildings in fire situations. We present in this paper a new model, implemented in a collaborative work between LEMTA and CNPP, and based on a macroscopic continuous approach. First, the assumptions and algorithm of pedestrian movement model are detailed. Secondly, our model is confronted to results from full-scale tests carried out without any fire constraints and to other egress models on typical test cases extracted from literature. Finally, the complete model (including fire stresses) is compared with FDS+Evac on a simple fire scenario.
Assessment of fire safety performance of buildings requires an estimation of Required Safe Egress Time (RSET) in fire situation. Many egress models have therefore been developed during the last decades, using various mechanisms. First evacuation models were based on empiric relations between people density, walking speed and geometric characteristics of the building (Togawa [1], Pauls [2]). Then, advancement of numerical simulation and progress in human behaviour modelling led to finer analysis of pedestrian motion, particularly microscopic approaches. For instance, Lagrangian-type models (Evac [3], Simulex) consider people as particles, whose respective motion is governed by different attractive or repulsive forces and characteristic velocities. Another kind of physical model, such as cellular automata (Blue and Adler [4], Klüpfel [5], Tissera [6]), considers a space-and-time discretization with people who jump from cell to neighboring cell following given transition laws. The main part of current evacuation computational tools are based on microscopic approaches. However, these kinds of models provide output results which are significant for the only simulated evacuation scenario and a particular initial people distribution. So, these microscopic descriptions need a statistical treatment of numerous simulations for a particular configuration to provide average data (required evacuation time, flowrates through openings in terms of persons per time unit) which can be considered representative of all scenarios likely to occur.
A macroscopic continuous egress model was developed in the present work, as an alternative to this limitation. Instead of considering people as individual entities with particular characteristics, our model tracks the evolution of people density in space and time and takes into consideration average input parameters to describe the population. The model is based on a people density balance solved in a discretized two-dimensional space, with additional constraints, such as critical flows through doors or congestion constraints, which make the problem more complex. The complete model is also able to take fire stresses into account, in terms of temperature, heat flux and visibility.
This paper details the algorithm used to solve the problem and our perspectives in order to integrate fire effects into the pedestrian motion model. Real-scale tests were carried out to check that our basis pedestrian motion model is able to reproduce experimental data (without any fire constraint). At this stage, validation of the complete model including fire stresses seems very difficult, due to the lack of experimental data. Nevertheless, a comparison between our model and FDS+Evac considering a simple test case extracted from literature leads to promising results.
The model presented here relies on a macroscopic approach, in which people are assembled to their people density. These types of models, based on partial differential equations solved in a continuous medium, have been studied since the 1950's with the emergence of first vehicular traffic models [7]. The application of these models inspired from fluid mechanics to people motion modelling is more recent and was initiated with Hughes [8]. In these studies, motion of people is described with the mass continuity equation, with some additional hypothesis in order to integrate human specificities and simple features of human behavior. Hughes' model was the source of further developments and mathematical studies. We can notice that macroscopic models are generally used in crowd motion with high densities. In fact, microscopic approaches are not optimal to study large crowds, due to the complex interactions between persons and the consequent computation time. This limitation is overcame with macroscopic approaches, in which calculation time depends on the size of computational domain instead of the number of persons. We can also notice that macroscopic egress models have been studied from a mathematical point of view, but have not been implemented in a numerical tool devoted to evacuation in fire situation.
The people motion model follows the temporal and spatial evolution of a macroscopic quantity, which is the people density ρ (person per square meter). It is based on three fundamental assumptions:
Mathematically, a balance is written for the time- and space-varying people density in a pre-calculated velocity field:
where is the effective transport velocity of people density and is the preferred walking speed of people. The operator P refers to the projection of preferred speeds on the permissible velocity space in a least square sense. So, real speed is necessarily lower than preferred walking speed due to the presence of other persons.
To complete the evacuation model, we have to introduce four key parameters:
The implementation of our model and the algorithm used in order to solve it numerically (which is similar to the one adopted by Roudneff-Chupin [9]) are presented in figure 1. Two distinct parts are involved in a sequential process: subroutines PAULO (Pathfinding Algorithm Using Length Optimization) and MARCOE (Macroscopic Analysis of Rescue Configuration for Optimal Evacuation).
The computational domain is discretized with a mesh of square cells, which can be classified into three categories:
The scenario (initial spatial distribution of persons) may be defined in a realistic way taking into account the nature of the building. Nevertheless, in order to study a mean configuration representative for all potential scenarios, the initial distribution of people density in the calculation area can be considered uniform and fairly distributed on free cells.
The preliminary step for the resolution of the density balance equation is the computation of the velocity field in which the people density is transported. This calculation is based on a simple observation: people tend to minimize their travel time to their goal [10]. So it is assumed in this study that people have a perfect knowledge of their environment and of the configuration of the building. For each free cell, the velocity is directed along the shortest path to the nearest exit cell. The shortest path between a current cell and its nearest exit cell is computed thanks to the PAULO algorithm (Pathfinding Algorithm Using Length Optimization) inspired from Dijkstra and able to take into consideration the specificities of the geometry (particularly blocked cells). At this stage, it can be noticed that the norm of the velocity is the preferred walking speed .
The space- and time-varying evolution of people density in the pre-calculated velocity field is governed by the equation (1). The equation is solved numerically using a finite volume method with an explicit temporal scheme. The advective part of the equation is treated thanks to a QUICK scheme, which consists in a second order interpolation in space and represents a good compromise between accuracy and computational time. At this step, exit cells are treated as particular cases and their outgoing flow is limited by the critical flux . Moreover, influence of surrounding people density on walking speed is not considered at this stage. This transport step is the first part of the MARCOE algorithm (Macroscopic Analysis of Rescue Configuration for Optimal Evacuation).
The previous calculation step (transport at preferred velocity) can lead to non-physical situations where the people density can locally exceed the critical density . So a corrective part extracted from literature [11] is introduced in order to respect congestion constraint for each cell. This corrective part consists in a stochastic projection of local excess density:
It should be noticed that this corrective part is mathematically equivalent to Equation (2) and consists in its algorithmic implementation. This step is the second part of the MARCOE algorithm.
The present model allows the inclusion of fire influence on egress conditions.
Fire effects on humans are generally classified into three categories:
In this study, it was decided not to deal with the question of toxic effects, because we can consider that toxic effects are correlated with optical and thermal effects. Moreover, reliability of gas concentration estimation sub-models (especially those embedded in Fire Dynamics Simulator) is questionable.
Three different ways are considered to represent fire impact on evacuation circumstances:
Burning cells are treated as blocked cells (physical obstacles)
The cells with thermal constraints (temperature, heat flux) and optical constraints (optical density) above threshold values introduced in table 1 are also considered as blocked cells. In fact, beyond these threshold values, it can be generally considered that people are not able to ensure their own evacuation. The critical value chosen for extinction coefficient corresponds to a visibility distance of 10 m according to Jin's correlation for a light-reflecting object.
Extinction coefficient of the smoke locally reduces the walking speed of humans according to the empirical formula given by Frantzich and Nilsson [12]:
where is the effective speed, is the preferred speed and is the extinction coefficient (). Fire-related data are extracted from Fire Dynamics Simulator 6 (developed by the NIST) and are updated at each time step.
Fire effect | Threshold value |
---|---|
Temperature | 60°C |
Thermal flux | 2,5 |
Extinction coefficient | 0,3 |
To the authors' knowledge, macroscopic models based on people density conservation have not been validated rightfully against real data yet. The objective in this section is to validate the evacuation model at a small scale with repeated tests and without fire constraints.
The egress model was confronted against experience on a very simple configuration: a 10 m² room (4 m long and 2,5 m wide) with a single 90 centimeter-wide exit (figures 2 and 3). A GoPro camera is set at the door in order to take the evacuation time of each person and to determine at every moment the proportion of initial population which is out of the room. Initial positions and orientations of the 10 persons are randomly chosen and the start for evacuation is given by a sound signal.
The aim of this preliminary step is to determine, for this particular experimental configuration, the average free walking speed and reaction time of the 10 members of the population. 100 evacuation tests were carried out with a single person to obtain exit rate (percentage of the population who left the room) over time (figure 4).