The application of Greenberg’s Model Modification for Estimating the Evacuation Time of People from Public Utility Buildings Zastosowanie modyfikacji modelu Greenberga do szacowania czasu ewakuacji

Objective: The article presents a proposition of a model for estimating people’s evacuation time from public utility buildings of category ZL III (not containing rooms designed for the simultaneous presence of more than 50 people who are not their regular users, not primarily intended for use by people with limited mobility). The model is based on the analogy between the theory of road traffic and the process of people’s movement during evacuation. Design and methods: In order to develop the model, a series of trial evacuations of people from public utility category ZL III buildings of varied geometry and number of users was conducted. A comparative analysis was performed concerning the evacuation times calculated with the use of models available in literature – a critical model of evacuation time, models designed by Togawa, Melenik and Booth, Galbreath, Pauls, methodology of the British Standard, and those derived from computer simulations performed with the use of the Pathfinder software. Based on the analysis of the conducted research and model considerations, an equation for the estimation of evacuation time was proposed based on a modified Greenberg’s equation derived from the road traffic theory. In the model modification, the concept of replacement length of evacuation route elements was applied, significantly slowing down people’s movement velocity, and a method for calculating them was proposed. Results: The evacuation times obtained in experimental research were compared to the model time values calculated from the models published in literature. A considerable dispersion of the achieved results was shown, ranging from –65.0% to +425.8% with respect to the evacuation times obtained experimentally. The performance of computer simulations brought evacuation times with a bias ranging from –54.4% to +26.0% with respect to the experiments conducted. Evacuation times calculated with the use of the proposed equation were in line with the experimental results with an error ranging from –12.3% to +13.8%. However, in comparison to the times obtained from additional computer simulations, representing the description of evacuation from buildings with highly varied geometry and various numbers of evacuees, the deviation of the calculated evacuation time from the proposed model was from –16.7% to +23.1%. In the vast majority of cases, the deviation of the result oscillated around ± 15% for a wide range of buildings’ geometry and the number of evacuees. Conclusions: The proposed model makes it possible to determine with sufficient accuracy the evacuation time of people from public utility buildings of category ZL III and can serve as a reliable source of comparative information.


Introduction
The basic fire safety requirement for buildings is to provide people staying in them with evacuation options [4]. Safe evacuation from a building in case of fire is the priority fire protection measure [5]. It is described on the basis of ASET (Available Safe Evacuation Time) and RSET (Required Safe Escape Time) [6][7][8]. Various tools can be applied in order to determine ASET and RSET, including empirical data (obtained, e.g., on the basis of real-scale tests, laboratory tests or practice evacuations), normative data specified in fire regulations and technical standards, computational models of evacuation time, computational models of temperature increase in a room as well as an increase in fogging, and simulation software based on the aforemen- Standard oraz modeli otrzymanych z symulacji komputerowych wykonanych za pomocą programu Pathfinder [1]. Uzyskane w eksperymentach czasy ewakuacji porównano z modelowymi wartościami czasów obliczonymi na podstawie opublikowanych w literaturze modeli. Wykazano duży rozrzut otrzymanych wyników wynoszący od -65,0% aż do +425,8% w stosunku do uzyskanych eksperymentalnie czasów ewakuacji. W symulacji komputerowej uzyskano czasy ewakuacji obarczone błędem od -54,4% do +26,0% w stosunku do przeprowadzonych eksperymentów [2].
3 determine ASET and RSET, including empirical data (obtained, e.g., on the basis tests, laboratory tests or practice evacuations), normative data specified in fire reg technical standards, computational models of evacuation time, computational temperature increase in a room as well as an increase in fogging, and simulation sof on the aforementioned models. Literature on the subject contains a considerable computational models of the transition time of people during evacuation. They complexity of computations and, primarily, in the accuracy of the obtained evacu [1].
In order to develop a proprietary model, a number of practice evacuations w from public utility buildings of category ZL III with varying geometries and numb A comparative analysis was performed of evacuation times calculated with the mod in literature: the critical evacuation time, the Togawa, Melenik and Booth, Galb models, the British Standard methodology and models obtained from computer performed with the Pathfinder software [1]. The evacuation times obtained in exper compared with the model time values calculated on the basis of models available A high dispersion of the results was identified from 65.0% to as many as comparison to evacuation times obtained in experiments. In the computer simulation ranging from 54.4% to +26.0% as compared to the experiments [2].
Due to the above, a proprietary equation was proposed for estimating the tra of evacuated people that would describe the analysed process with a higher accura

The traffic theory and the movement of evacuated people
The analysis of literature on traffic theory makes it possible to draw an analo the description of vehicles moving in traffic and the process of people's evac buildings [3]. The originator of the hydrodynamic theory, also referred to as th theory, was Greenberg [3]. He suggested equation (1), which is correct if the diff condition of intensity q and traffic density k is satisfied. The analysis of the traffic theory bears considerable resemblance to the evacuating people from buildings and can be applied for this purpose for the follow (1) gdzie:

INTRODUCTION
The basic fire safety requirement for buildings is to provide people staying in them with evacuation options [4]. Safe evacuation from a building in case of fire is the priority fire protection measure [5]. It is described on the basis of ASET (Available Safe Evacuation Time) and RSET (Required Safe Escape Time) [6][7][8]. Various tools can be applied in order to determine ASET and RSET, including empirical data (obtained, e.g., on the basis of real-scale tests, laboratory tests or practice evacuations), normative data specified in fire regulations and technical standards, computational models of evacuation time, computational models of temperature increase in a room as well as an increase in fogging, and simulation software based on the aforementioned models. Literature on the subject contains a considerable number of computational models of the transition time of people during evacuation. They differ in the complexity of computations and, primarily, in the accuracy of the obtained evacuation times [1].
In order to develop a proprietary model, a number of practice evacuations were arranged from public utility buildings of category ZL III with varying geometries and number of users. A comparative analysis was performed of evacuation times calculated with the models available in literature: the critical evacuation time, the Togawa, Melenik and Booth, Galbreath, Pauls models, the British Standard methodology and models obtained from computer simulations performed with the Pathfinder software [1]. The evacuation times obtained in experiments were compared with the model time values calculated on the basis of models available in literature. A high dispersion of the results was identified from 65.0% to as many as +425.8% in comparison to evacuation times obtained in experiments. In the computer simulation with a bias ranging from 54.4% to +26.0% as compared to the experiments [2].
Due to the above, a proprietary equation was proposed for estimating the transition time of evacuated people that would describe the analysed process with a higher accuracy.

The traffic theory and the movement of evacuated people
The analysis of literature on traffic theory makes it possible to draw an analogy between the description of vehicles moving in traffic and the process of people's evacuation from buildings [3]. The originator of the hydrodynamic theory, also referred to as the continuity theory, was Greenberg [3]. He suggested equation (1), which is correct if the differentiability condition of intensity q and traffic density k is satisfied. (1) where: opt v -optimum momentary velocity of traffic; kmax -traffic density in a traffic jam.
The analysis of the traffic theory bears considerable resemblance to the process of evacuating people from buildings and can be applied for this purpose for the following reasons: -optymalna prędkość chwilowa ruchu pojazdów; k max -gęstość ruchu w sytuacji korka drogowego.
In the computer simulation with a bias ranging from -54.4% to +26.0% as compared to the experiments [2].
Due to the above, a proprietary equation was proposed for estimating the transition time of evacuated people that would describe the analysed process with a higher accuracy.

The traffic theory and the movement of evacuated people
The analysis of literature on traffic theory makes it possible to draw an analogy between the description of vehicles moving in traffic and the process of people's evacuation from buildings [3].
The originator of the hydrodynamic theory, also referred to as the continuity theory, was Greenberg [3]. He suggested equation (1), which is correct if the differentiability condition of intensity q and traffic density k is satisfied.
3 ne ASET and RSET, including empirical data (obtained, e.g., on the basis of real-scale boratory tests or practice evacuations), normative data specified in fire regulations and al standards, computational models of evacuation time, computational models of ture increase in a room as well as an increase in fogging, and simulation software based aforementioned models. Literature on the subject contains a considerable number of ational models of the transition time of people during evacuation. They differ in the xity of computations and, primarily, in the accuracy of the obtained evacuation times In order to develop a proprietary model, a number of practice evacuations were arranged blic utility buildings of category ZL III with varying geometries and number of users. arative analysis was performed of evacuation times calculated with the models available ture: the critical evacuation time, the Togawa, Melenik and Booth, Galbreath, Pauls , the British Standard methodology and models obtained from computer simulations ed with the Pathfinder software [1]. The evacuation times obtained in experiments were ed with the model time values calculated on the basis of models available in literature.
dispersion of the results was identified from 65.0% to as many as +425.8% in ison to evacuation times obtained in experiments. In the computer simulation with a bias from 54.4% to +26.0% as compared to the experiments [2]. Due to the above, a proprietary equation was proposed for estimating the transition time uated people that would describe the analysed process with a higher accuracy.

e traffic theory and the movement of evacuated people
The analysis of literature on traffic theory makes it possible to draw an analogy between cription of vehicles moving in traffic and the process of people's evacuation from gs [3]. The originator of the hydrodynamic theory, also referred to as the continuity was Greenberg [3]. He suggested equation (1), which is correct if the differentiability n of intensity q and traffic density k is satisfied. (1) ptimum momentary velocity of traffic; raffic density in a traffic jam.
The analysis of the traffic theory bears considerable resemblance to the process of ing people from buildings and can be applied for this purpose for the following reasons: (1) where: 3 N ety requirement for buildings is to provide people staying in them with Safe evacuation from a building in case of fire is the priority fire t is described on the basis of ASET (Available Safe Evacuation Time) afe Escape Time) [6][7][8]. Various tools can be applied in order to ET, including empirical data (obtained, e.g., on the basis of real-scale practice evacuations), normative data specified in fire regulations and mputational models of evacuation time, computational models of room as well as an increase in fogging, and simulation software based odels. Literature on the subject contains a considerable number of f the transition time of people during evacuation. They differ in the ions and, primarily, in the accuracy of the obtained evacuation times p a proprietary model, a number of practice evacuations were arranged ings of category ZL III with varying geometries and number of users.
as performed of evacuation times calculated with the models available evacuation time, the Togawa, Melenik and Booth, Galbreath, Pauls dard methodology and models obtained from computer simulations inder software [1]. The evacuation times obtained in experiments were l time values calculated on the basis of models available in literature. e results was identified from 65.0% to as many as +425.8% in n times obtained in experiments. In the computer simulation with a bias +26.0% as compared to the experiments [2]. , a proprietary equation was proposed for estimating the transition time would describe the analysed process with a higher accuracy.
y and the movement of evacuated people terature on traffic theory makes it possible to draw an analogy between les moving in traffic and the process of people's evacuation from nator of the hydrodynamic theory, also referred to as the continuity 3]. He suggested equation (1), which is correct if the differentiability nd traffic density k is satisfied. 3) średnią prędkość chwilową 3 In order to develop a proprietary model, a number of practice evacuations from public utility buildings of category ZL III with varying geometries and nu A comparative analysis was performed of evacuation times calculated with the mo in literature: the critical evacuation time, the Togawa, Melenik and Booth, Ga models, the British Standard methodology and models obtained from comput performed with the Pathfinder software [1]. The evacuation times obtained in exp compared with the model time values calculated on the basis of models availabl A high dispersion of the results was identified from 65.0% to as many a comparison to evacuation times obtained in experiments. In the computer simulat ranging from 54.4% to +26.0% as compared to the experiments [2].
Due to the above, a proprietary equation was proposed for estimating the of evacuated people that would describe the analysed process with a higher accu

The traffic theory and the movement of evacuated people
The analysis of literature on traffic theory makes it possible to draw an an the description of vehicles moving in traffic and the process of people's ev buildings [3]. The originator of the hydrodynamic theory, also referred to as theory, was Greenberg [3]. He suggested equation (1), which is correct if the d condition of intensity q and traffic density k is satisfied.
kmax -traffic density in a traffic jam.
The analysis of the traffic theory bears considerable resemblance to t evacuating people from buildings and can be applied for this purpose for the follo  2) velocity of vehicles changes depending on their density, similarly to people in mo when their density increases, their velocity drops, and conversely, when the densit people on escape routes decreases, people move at a higher velocity; 3) similarly to vehicles in traffic, people move, collide with each other, pass obsta accelerate on straight road passages, when their density is low, and slow do especially when changing direction; 4) the trajectory of vehicles and evacuees is not straightforward. In both cases it dep on people's decisions and behaviour, which are difficult to predict; 5) moving vehicles and people evacuated from fire are influenced by a numbe variables, such as people's behaviour, road conditions, people's decisions such as choice of velocity, distance from other persons or obstacles, changes in road geom changes in direction of movement affecting velocity. Vehicles in traffic, similar as people during evacuation, constantly interact with other. Thus, in correspondence to the traffic theory, the state of the stream of moving pe can be described with three variables: 1) people's movement intensity Fd -the number of people passing through a spe section of the escape route in time, number of persons/m · s; 2) the density of evacuees D is the number of people in an area of a specific section o escape route, number of persons/sq. m; 3) average momentary velocity t v -the average velocity of people in motion, m/s.

Proposal of an equation for estimating the evacuation time of people from public utility buildings ZL III
In order to determine the evacuation time of people from a building, a model equa was derived (2), which, with an assumption that opt v corresponds to the velocity of evacuated person, specifies that people's evacuation time T is equal to:  1) the entire escape route was divided into horizontal (x) and vertical (y) sec which people's velocity is determined depending on their density according to (3) and where: Vs -velocity of a stream of people moving along a motion axis, m/s; D -density of people on the escape route's area, number of persons/sq. m; k -the coefficient of motion along the escape route, m/s; a -coefficient equal to 0.266 sq. m/persons.  In order to develop a proprietary model, a number of practice evacuations were arranged from public utility buildings of category ZL III with varying geometries and number of users. A comparative analysis was performed of evacuation times calculated with the models available in literature: the critical evacuation time, the Togawa, Melenik and Booth, Galbreath, Pauls models, the British Standard methodology and models obtained from computer simulations performed with the Pathfinder software [1]. The evacuation times obtained in experiments were compared with the model time values calculated on the basis of models available in literature. A high dispersion of the results was identified from 65.0% to as many as +425.8% in comparison to evacuation times obtained in experiments. In the computer simulation with a bias ranging from 54.4% to +26.0% as compared to the experiments [2].
Due to the above, a proprietary equation was proposed for estimating the transition time of evacuated people that would describe the analysed process with a higher accuracy.

The traffic theory and the movement of evacuated people
The analysis of literature on traffic theory makes it possible to draw an analogy between the description of vehicles moving in traffic and the process of people's evacuation from buildings [3]. The originator of the hydrodynamic theory, also referred to as the continuity theory, was Greenberg [3]. He suggested equation (1), which is correct if the differentiability condition of intensity q and traffic density k is satisfied. where: opt v -optimum momentary velocity of traffic; kmax -traffic density in a traffic jam.
The analysis of the traffic theory bears considerable resemblance to the process of evacuating people from buildings and can be applied for this purpose for the following reasons: -the average velocity of people in motion, m/s.

Proposal of an equation for estimating the evacuation time of people from public utility buildings ZL III
In order to determine the evacuation time of people from a building, a model equation was derived (2), which, with an assumption that corresponds to the velocity of the evacuated person, specifies that people's evacuation time T is equal to: 4 f vehicles changes depending on their density, similarly to people in motion: ir density increases, their velocity drops, and conversely, when the density of escape routes decreases, people move at a higher velocity; to vehicles in traffic, people move, collide with each other, pass obstacles, e on straight road passages, when their density is low, and slow down, when changing direction; tory of vehicles and evacuees is not straightforward. In both cases it depends 's decisions and behaviour, which are difficult to predict; vehicles and people evacuated from fire are influenced by a number of , such as people's behaviour, road conditions, people's decisions such as the velocity, distance from other persons or obstacles, changes in road geometry, n direction of movement affecting velocity. in traffic, similar as people during evacuation, constantly interact with each orrespondence to the traffic theory, the state of the stream of moving people with three variables: (2), forming a basis for the proposed proprietary computational model of was expanded with the following elements: 1) the entire escape route was divided into horizontal (x) and vertical (y) sections, for which people's velocity is determined depending on their density according to table 1 [7] and equation (3): entire escape route was divided into horizontal (x) and vertical (y) sections, for ich people's velocity is determined depending on their density according to equation ) and Table 1 where:      Na podstawie przeprowadzonych obserwacji długość zastępczą xd drogi przy zmianie ierunku ruchu na styku schodów i poziomej drogi ewakuacyjnej oraz na zakręcie drogi (4) where: x o -replacement length of the pinch point on the horizontal escape route, m; Zgodnie z przyjętym założeniem długość zastępczą drogi przy zmianie kierunku ruchu na poziomej drodze ewakuacyjnej oblicza się zgodnie z poniższym równaniem (5): (5) gdzie: In accordance with the adopted assumption, the replacement length of the route with a change in direction on the horizontal escape route is calculated in line with the following equation (5): (5) where: In line with the adopted principle, the replacement length of the route with a change of direction on the vertical escape route along which people move is calculated in line with the following equation (6): (6) where: In line with literature data [9] the assumption made in the    ion of vehicles' motion occurs regarding velocity values of e highest and lowest densities ed evacuation time models. uation time obtained from the pared to the actual stochastic hus, the modified form of the   e -liczba przewężeń występujących na l-tej pionowej drodze ewakuacyjnej, l=1,…,g, -; i c x -długość zastępcza przewężenia na l-tej pionowej drodze ewakuacyjnej, l=1,…,g, i=1,…,e, m.
Współczynnik korekcyjny A w wyniku analizy danych doświadczalnych przyjmuje następujące wartości dla poziomej drogi ewakuacyjnej:             y l -długość l-tej pionowej drogi ewakuacyjnych, l = 1,…,g, m; h -liczba zmian kierunków ruchu na l-tej pionowej drodze ewakuacyjnej, l = 1,…,g, -; 13 riginal Greenberg's model [3] the highest bias in the description of vehicles' motion occurs the highest and lowest traffic densities. In addition, there is a major controversy in the literature regarding velocity values of le moving along horizontal and vertical escape routes for the highest and lowest densities acuees, which has an impact on the accuracy of the computed evacuation time models.
In order to improve the accuracy of calculations of evacuation time obtained from the sed model with a number of simplifying assumptions as compared to the actual stochastic ation process, adjustment coefficient A was introduced. Thus, the modified form of the l takes the form of equation (8):   e -liczba przewężeń występujących na l-tej pionowej drodze ewakuacyjnej, l=1,…,g, -; i c x -długość zastępcza przewężenia na l-tej pionowej drodze ewakuacyjnej, l=1,…,g, i=1,…,e, m.
In a considerable majority of cases the result's deviation was around ±15% for a broad range of building geometries and various numbers of evacuees.