Mathematical Modeling-Evacuation of Teaching Building-School Mathematical Modeling Lesson 2

mathematical modeling

Personnel evacuation

This topic was carefully prepared by me, my good friend Zhang Yong, the academic committee of our district team Xie Feifei and the instructor Shen Cong after several days and nights.

abstract

This paper analyzes the evacuation characteristics of large buildings, and taking the safe evacuation of teaching building 1 in our school as an example, preliminarily evaluates the evacuation design scheme in building fire, obtains the calculation method of evacuation time and the treatment method of bottleneck phenomenon in high crowd density building fire, and puts forward the methods of distance control evacuation process and bottleneck control evacuation process to analyze and calculate the evacuation of buildings.

key word

Distance-controlled evacuation process of personnel evacuation fluid model

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Prediction of evacuation time of teaching building

The teaching building of the school is a place where people are very concentrated, with large fire load and many fire factors. Once a fire breaks out, the fire and its smoke spread rapidly, easily causing serious casualties. For different types of buildings, the treatment methods of personnel evacuation are quite different. Combined with the structural form of the teaching building 1, this paper analyzes the typical fire scene of the teaching building, analyzes the present situation of the building's personnel evacuation design, and puts forward the basis of personnel evacuation, which provides useful opinions and suggestions for school leaders.

order

After a building fire, the safe evacuation of people is directly related to people's life safety, so it is of great significance to ensure that people in the building can be evacuated to a safe area in time. Whether people can be evacuated safely in a fire mainly depends on the length of time required to evacuate to a safe area. The safe evacuation of people in a fire refers to the action of evacuating all people in a building to a safe area before the fire smoke has reached a dangerous state for people. Evacuation time should not only consider environmental factors such as building structure and the distance between people and safe areas, but also consider people's natural conditions and psychology in case of fire emergency, which is a complex problem involving three basic factors: building structure, fire development process and personnel behavior.

With the development of performance-based safety evacuation design technology, countries all over the world have successively developed and studied evacuation safety assessment technology, and achieved certain results (models and procedures), such as CRISP, EXODUS, STEPS, Simulex in Britain, ELVAC, EVACNET4, EXIT89, HAZARDI in America, EGRESSPRO and FIREWIND in Australia, FIERA system in Canada, and EVACS in Japan. Research work has also been carried out in China's construction, fire research and teaching units, and related research has been included in the national "Ninth Five-Year Plan" and "Tenth Five-Year Plan" scientific and technological research topics.

Generally speaking, the evacuation assessment method consists of two parts: smoke characteristics prediction and evacuation prediction. The prediction of smoke characteristics is to predict the time when smoke will affect evacuees. Many fire cases show that the toxicity of fire smoke, hypoxia and suffocation and radiant heat are the main factors causing casualties.

Among them, smoke toxicity is the most important factor affecting safe evacuation and causing death in fire, and it is also the main factor causing fire danger. Studies have shown that people will die after being exposed to the concentration of 4X 10-3 carbon monoxide for 30 minutes.

In addition, hypoxia, suffocation and radiant heat are also the main factors leading to death. Research shows that the normal value of oxygen in the air is 2 1%. When the oxygen content drops to 12% ~ 15%, it will cause shortness of breath, headache, dizziness and drowsiness. When the oxygen content is as low as 6% ~ 8%, it will make people weak. The maximum radiant heat that human body can bear in a short time is 2.5 kW/m2 (the temperature of flue gas layer is about 200℃).

Figure 1 Influencing factors of evacuation

Predicting the influence of smoke on safe evacuation becomes a part of safe evacuation evaluation, which should consider the performance of smoke control equipment and the influence of walls and openings on smoke. By comparing the dangerous arrival time and the evacuation time, the rationality of evacuation design and the safety of evacuation are evaluated. If the evacuation time is less than the time when the danger comes, the evacuation is safe and the evacuation design scheme is feasible. On the contrary, evacuation is unsafe, and the evacuation design should be revised and re-evaluated.

Fig. 2 schematic diagram of the relationship between evacuation and smoke layer decline (two-layer regional model)

The time required for evacuation includes evacuation start time and evacuation action time. The evacuation start time is the time from fire to evacuation, which can be roughly divided into two stages: perception time (from fire to people's perception of fire) and evacuation preparation time (from perception of fire to evacuation). Generally speaking, the evacuation start time is related to fire detection system, alarm system, fire location, relative position of personnel, state and condition of evacuees, building shape and management status, evacuation guidance means and other factors.

Evacuation action time is the time from the start of evacuation to the end of evacuation, which consists of walking time (the time required to walk from the farthest evacuation point to the safety exit) and exit queuing time (the time required for all people in the area to pass through the exit). See Figure 3 for the parameters related to evacuation time prediction and their relationships.

Fig. 3 Parameters related to evacuation time prediction and their relationships.

Analysis and establishment of the model

We simulate the movement of people in the 1 teaching building as the flow of water in the pipeline, without considering the individual characteristics of people, but treat the evacuation of people as a whole movement, and make the following conservative assumptions about the evacuation process:

U Evacuees have the same characteristics and have enough physical conditions to evacuate to a safe place;

U The evacuees are awake and evacuated in an orderly manner at the initial stage of evacuation, and there will be no situation of returning halfway to choose other evacuation routes during the evacuation process;

U In the process of evacuation, the flow of people is directly proportional to the width of the evacuation passage, that is, the number of people evacuated from an exit is allocated according to the proportion of its width to the total width of the exit.

U Evacuate personnel from every available exit, and the evacuation speed of all personnel is the same and remains unchanged.

The above assumption is an ideal evacuation state, which may be different from the actual evacuation process. In order to make up for the influence of some uncertain factors in the evacuation process, a safety factor is usually conservatively considered when using this model to calculate evacuation, which is generally taken as 1.5 ~ 2, that is, the actual evacuation time is the calculated evacuation time multiplied by the safety factor.

1 floor plan of the teaching building

Simplification and calculation assumption of teaching building model

The teaching building 1 of our school is a building, which is divided into two buildings, A and B, with Block C connected in the middle (as shown above). Buildings A and B have five floors and Block C has two floors. There are several classrooms on each floor of Block A and Block B. Except for the fourth floor of Block A and the fifth floor of Block B, there are two large classrooms on each floor. The first floor of Block C is the hall, and the second floor of Block C is several offices. There are few people, so it is ignored and only used as a passage for people. In order to analyze the evacuation situation, the 10 small classroom (40 people), a middle classroom (100 people) and a large classroom (240 people) on each floor of Blocks A and B are simplified to six classrooms.

Fig. 4 Schematic diagram of the original classroom plan

In the corridor 1/2, classrooms 1, 2, 3, 4 and 5 are simplified as classrooms 13 and 14, and classrooms 6, 7, 8, 9 and 10 are simplified as classrooms10. At this time, the number of students in classrooms 13, 14, 15 and 16 is 100, and the exits of classrooms are14 and1/kloc-from both sides of the corridor respectively. We set up a large classroom with 100 people near the exit of the large classroom taking the stairs on the left, and the rest 140 people were evacuated from the stairs outside the large teaching building, thus using the exits of various passages. Because of the symmetry of building A and building B of the teaching building 1, the establishment of this schematic diagram can also be applied to any floor of building A and building B of the teaching building 1.

Fig. 5 Schematic diagram of classroom plan.

According to the survey, the total length of the corridor is 44m, the corridor width is 1.8m, the width of a single staircase is 0.3m, each staircase has 26 steps, the stairwell is 2.0m, and the area of each classroom is 125m2. Then 1/4 of the simplified corridor is the exit of the classroom, and the distance from the stairs should be 440.

Make the following assumptions about the fire scene:

U The fire broke out on the second floor/room KLOC-0/5;

The fire broke out because every classroom was full, so there were 600 people on this floor.

U teaching building is equipped with a centralized fire alarm system, but there is no emergency broadcast system;

U Failure to evacuate the fire floor within 10 minutes after the fire is considered as an escape failure;

Some simulation programs can be used to calculate the process of fire development and smoke spread in this scene, and the arrival time of dangerous situations in buildings can be determined accordingly. However, in order to highlight the key points, the calculation details are not discussed in detail here.

The whole evacuation time can be divided into three parts: the lag time before evacuation, the time after a certain distance during evacuation and the waiting time at some important exits. According to the structural characteristics of buildings, people's evacuation passages can be divided into several small sections. At some small exits, people may have to wait in line for a certain time when passing through. So the evacuation time ti of the ith person can be expressed as:

Among them, ti and ti, delay are the lag time before evacuation, including the time taken to detect and confirm the fire; Di, n is the length of the nth segment; Vi, n is the average walking speed of people in n section; Δ tm, queue is the waiting time at the exit of the nth segment. The total time for the last person to leave the teaching building is the evacuation time required for the evacuation of the teaching building.

Suppose the classroom on the second floor 15 is a fire room, and the people in it will evacuate immediately after they get the fire signs. The reaction lag time is 60s. Most of the teaching staff are students, and the fire information will spread quickly, so the staff in other classrooms on the same floor will get the warning from the staff in room 15 and start to decide to evacuate. Let this message spread 120s, that is, the total lag time of this group of people is120+60 =180s; Because of the left-right symmetry, here we calculate that people on the first, third, fourth and fifth floors will start to evacuate through the fire alarm system. They get fire information 60 seconds later than people in other classrooms on the second floor, so the total response delay is 240 seconds. Because the fire happened on the second floor, the danger to people on the first floor is relatively small, so the following discussion mainly focuses on the second, third and fourth floors.

In order to actually understand the walking situation of people in the teaching building, our group made many field observations and recorded the time when students passed through some typical sections. Referring to some other data [1, 2, 3], it is proposed that the main parameters of personnel evacuation can be shown in Figure 6. At the beginning of evacuation, the time someone stays in the classroom is regarded as the queuing time. People's walking speed should be selected according to different crowd densities. When the crowd density is greater than 1 person /m2, the evacuation speed is 0. 6m/ s, and the time required to pass through the corridor is 60s, and the time required to pass through the hall is12 s. When the crowd density is less than 1 person /m2, the evacuation speed is 1. 2m/ s, it takes 30s to pass through the corridor and 6s to pass through the hall.

Fig. 6 Some main parameters of personnel evacuation

Pauls[4] proposed that the flow f of people descending stairs is related to the effective width w of stairs and the number of people using stairs, and its calculation formula is:

Where the unit of flow f is person/second, and the unit of w is millimeter, and the applicable range of this formula is 0. 1

In this way, the evacuation time can be calculated by the flow and the number of people in the room. The effective width of the outlet is the actual width of the channel minus the boundary layer on both sides. Usually, the boundary layer on one side of the channel is set to150 mm.

3 Results and discussion

During the whole evacuation process, the following situations will occur:

(1) When the personnel in the fire fighting classroom started to evacuate, the crowd density was relatively small, and the evacuation space was relatively spacious compared with the personnel being evacuated. At this time, the key factor determining evacuation is the length of evacuation path. Now this evacuation process is defined as distance control evacuation process;

(2) People in other classrooms on the fire floor can get fire information quickly and decide to evacuate. Their whole evacuation process may be divided into two stages to calculate: when F enters the stairs on the second floor and exits the stairs on the second floor, the evacuation at this time belongs to the distance control evacuation process; When F enters the stairs on the second floor > F exits the stairs on the second floor, the width of the stairs on the second floor becomes the controlling factor in the evacuation process. Now this process is defined as the bottleneck control evacuation process;

(3) After the personnel on the third and fourth floors start to evacuate, the stairwell on the third floor and the stairwell on the second floor may become the bottleneck to control the evacuation process;

(4) When the classroom personnel on the first floor begin to evacuate, it may cause a bottleneck at the exit of the lobby on the first floor to control the evacuation process;

(5) In the later period of evacuation, people waiting for evacuation will meet the conditions of distance-controlled evacuation process, that is, the distance-controlled evacuation process will appear again.

The personnel density of the fire fighting classroom is 100/ 125 = 0.8 persons /m2. But there are many desks and chairs in the classroom, so it is not very convenient for people to move. According to the data given in table 1, the walking speed of indoor people is 1.1m/s. The width of classroom door is1. 80m。 In the process of evacuation, this width cannot be fully utilized, and its equivalent width is equal to this width minus 0. 30m。 The flow f0 of people coming out of the classroom is:

F0 = v0× s0× w0 =1.1× 0.8× 4.7 = 4.1(person/second) (3)

Where v0 and s0 are the walking speed and density of people in the classroom respectively, and w0 is the effective width of the classroom exit. At this rate, it will take 24.3 seconds for the people in the fire-fighting classroom to evacuate completely.

Personnel should enter the corridor at a flow rate of 4. 1 person/second. Since the density of people in the corridor is less than 1 person /m2, the speed is 1. The calculation adopts 2m/s. The time for available personnel to reach the stairs on the second floor is 9.2s, and the number of people who will use the stairs on the second floor at this stage is 100. At this time, p/w =100/1700 = 0.059.

After the fire broke out 120 seconds, the personnel of the other two classrooms on the fire floor (namely 1 1 and 13 classrooms) began to evacuate. Before entering the stairwell of this floor, the main parameters of evacuation are basically the same as those of the people in the fire fighting classroom. 129.2s, some of them reached the stairs on the second floor, and all the staff in the fire fighting classroom have been evacuated from the lobby on the second floor. Therefore, the number of people who will use the stairwell on the second floor p 1 is:

P 1 = 100 ×2 = 200 (person) (4)

At this time, F enters the stairs on the second floor > F flows out from the stairs on the second floor. From that moment on, the evacuation process shifted from distance control evacuation to the second floor stairwell bottleneck control evacuation stage. Since p/w = 200/1700 = 0.12, the evacuation flow f 1 can be calculated by Formula 2, namely:

/P & gt；

0.27

0.73

F 1 = (3400/ 8040) × 200 = 2.2 person/second) (5)

Where 3400 is the total effective width of the two stairs, and the unit is mm. People on the third and fourth floors did not start to evacuate until after the fire broke out 180s. 286.5s( 180+ 106.5) The people on the third floor went to the stairs on the second floor, and the people on the fourth floor went to the stairs on the fifth floor. At this time, the number of people waiting for evacuation in front of the stairs on the second floor is p' 1:

P'1= 200-(286.5–129.2) × 2.2 =-146.1(person)

So, everyone on the second floor went to the first floor.

After that, the number of people who need to use the stairwell on the second floor p2:

P2 = 100×3=300 (person) (7)

At this stage, the corresponding flow through the stairwell on the second floor is f 2:

0.27

0.73

F2 = (3400/8040) × 200 = 2.5 (person/second) (8)

T 1 evacuation time of building stairs:

t 1 = 300÷2.5 = 120(s)(9)

Because the third, fourth and fifth floors of the teaching building have the same structure, the time from the fifth floor to the fourth floor, the fourth floor to the third floor and the third floor to the second floor is equal, so there will be no bottleneck in the evacuation of people at the stairs.

Therefore, the total evacuation time through the stairs on the second floor is t:

t = 286.5+ 120×3 = 646.5(s)( 10)

Finally, according to the safety factor, the actual evacuation time is:

TActual = 646.5× (1.5 ~ 2) = 969.75 ~1293 (s) (11).

Fig. 7 Variation curve of two-story staircase flow with time.

A few supplementary explanations:

The above is our hypothetical analysis and calculation of the fire in Room 15 on the second floor of Block B. At this time, the evacuation is successful when people arrive at the first floor. Similarly, when there is a fire on the third floor, when people arrive at the second floor, evacuation is considered successful, as are the fourth and fifth floors. Because of the structural symmetry of Block A and Block B of the teaching building 1, other classrooms on this floor caught fire, which is also the same reason. Therefore, the above analysis and calculation in this paper are also applicable to two buildings, A and B. In addition, when there is a fire on the third floor or above, it will reflect the role of the second floor of Block C. When there is a fire on the third floor of Block B, the staff on the second floor of Block B must have reacted to the fire behind the staff on the third floor of Block B. Therefore, when the staff on the third floor evacuate to the second floor, the staff on the second floor will also start to evacuate, which will inevitably lead to Because the third, fourth and fifth floors of Block A and Block B are unconnected and independent structures, the fire will not directly threaten the safety of people on the third floor and other floors of Block A.. Therefore, in order to avoid the bottleneck of the stairs on the second floor, we will transfer all the people on the second floor to the second floor of Block A, so that the people on the burning floor can evacuate to a safe area faster. When the fourth and fifth floors of Block B catch fire, the personnel on the second floor will also move to the second floor of Block A, creating conditions for the evacuation of personnel above the second floor. Similarly, the same is true for block A. ..

When analyzing and calculating the fire hypothesis, we did not calculate the evacuation of the stairs at the back door of the big classroom. Because of the particularity of the 1 teaching building, there are no large classrooms on the fourth floor of Block A and the fifth floor of Block B, so the evacuation speed of the back door stairs of the large classroom is very fast, and there will be no bottleneck in the back door stairs of the large classroom.

Several exits of the teaching building 1;

There is a gate in hall u.

There is a door on the first floor of Block U near the main hall.

There is a door next to the big classroom in block u A.

The window near the main entrance of the classroom hall in block u B can be used as an emergency exit.

There are basements under Block A and Block B (when smoke spreads too fast to evacuate, it can be used as an escape destination when threatened by smoke).

Every big classroom in u A and B has a back door.

Total: 8 exits

A letter to the school leader.

Hello, school leaders.

According to the teaching building 1 of our school, our mathematical modeling team draws the following conclusions through actual measurement, modeling and model analysis: in the event of a fire in the teaching building 1, all personnel may not be able to evacuate safely.

The above analysis was carried out under ideal conditions without any modification. In fact, people's behavior in a fire is very complicated, especially those who have not received fire safety training may run blindly and walk backwards, which will also extend the total evacuation time.

This model is the basis of an evacuation analysis model at present, and it belongs to a theoretical model at present. The above calculation results are all calculated by hand or by Wenxing. The walking speed of people in the model is obtained by observing the walking speed of people after class in the teaching building for many times, referring to the walking speed of people during evacuation given by Fru2in, the walking speed of people given by NFPA and the general calculation speed in the current evacuation model, which has wide universality. The predicted evacuation time is obtained according to the structural characteristics of buildings and the walking speed of people. When calculating the evacuation time, except for the lag time (or pre-action time) before evacuation, the time obtained is reasonable. For the lag time of people before evacuation, refer to the experimental conclusions of T. J. Shields, etc. : 75% of the people started to evacuate after hearing the fire alarm in 15 ~ 40s, and the whole evacuation time was 646.5s In this case, the response lag time of the burning classroom was 60 s, counting from the beginning of the fire. Pre-action time has a great relationship with different types of buildings, the characteristics of people in buildings and the alarm system in buildings, and it is a very uncertain value. The pre-movement time used in this paper is less than 10% of the whole evacuation process. The curve of the flow rate of the second floor staircase with time is shown in Figure 7. As can be seen from the above, it takes 646.5 s for all people above the second floor to pass through the stairs on the second floor, which is longer than the previously set available safe evacuation time, and it is impossible to ensure the safe evacuation of all relevant personnel. The width of the stairs and the main entrance of the hall is obviously the bottleneck restricting evacuation. The basic reason for this situation is that the evacuation passage of the teaching building is improperly arranged and the width of the staircase passage is not enough, so the total width of the staircase can be appropriately increased. Or build another staircase on each branch of the teaching building, and the evacuation will be smoother; It is best to build a new exit similar to the main entrance in Block A and Block B, which will greatly relieve the pressure of evacuating people from the main entrance of the hall, and will not cause the crowd jam in the hall and affect the evacuation of people upstairs. On the other hand, schools should add more fire-fighting facilities, and every classroom should be equipped with fire extinguishers; Schools should also strengthen the cultivation and education of students' fire awareness, which can be diversified and innovative, such as giving lectures, taking practical classes, doing fire drills and so on. Let them know some common sense of fire escape and learn how to use some fire-fighting equipment, so that they can have a full understanding of the teaching building they use, so that they can know what evacuation methods should be taken in case of fire, so as to reach the safe area in the shortest time.

If the school has limited funds, it can also eliminate this fire hazard without spending a penny, that is, arrange classrooms reasonably to avoid all classrooms on each floor being used for classes. At least a few can be vacated on each floor, which will greatly alleviate the danger caused by unfavorable evacuation. But there are also shortcomings, that is, the use value of the classroom is not fully utilized and resources are wasted.