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Modeling for Standard of Safety for Occupied Room
By:
Reuben D. Budiardja
Geoffrey S. Christanday
Reynard Hilman

(Won a "Honorable Mention" for Mathematical Contest of Modeling 1998 by MAA)

i. Table of Contents

Summary
Analysis of Problem
Assumptions
Model
    Variables Definition
    Assumptions of Values
    Design of the Model
Discussion
Appendix: Example Case
Bibliography
 

I. Summary

There are several factors involved in order to determine the maximum occupant load of a room. The effective method should include consideration on the number and size of egresses, the space area, and continuous rate of people through the way out. In addition to that we cannot ignore the factors that may create time delay during the evacuation process. This delay time will reduce the amount of time that is available for the evacuation.

We approach this problem by dividing the total evacuation time into two elements. The first element is the duration of the exiting process itself. The second element is the delay time, which includes other factors such as the response time after the emergency notice is issued and the time to pass the obstacles.

The following values need to be known before hand in order to find out the maximum occupancy load:

• Evacuation time
• Room area, shape, and configuration
• Number and size of egresses
• Comfortable area for each person

By knowing these facts and considering other general factors, i.e.: human velocity, time to respond to a danger, etc., we are able to make a model that is applicable to almost all situations.
 

II. Analysis of Problem

The problem requires us to come with a mathematical model that can be used to determine the number of people that is considerably safe to occupy a room. Hence, in the case of emergency, all the people could be evacuated from the room before the hazardous situation become uncontrollable. Consequently, we need to know the reasonable time before the hazardous situation become uncontrollable.

We agree that this reasonable evacuation time is applicable to all types of room regardless the differences between one room to another under the same hazardous situation. For an instance, in the case of earthquake or fire, the evacuation time for room A and room B have to be the same because after a certain amount of time then it will be too late. However, we also realize that the difficulties to reach the way out may differ from one room to another. Therefore, in our model we separate the total evacuation time into two elements. Those are the exiting time and the delay time. The exiting time is the time needed for people to go through the way out. The delay time is the time from when occupants first received the emergency notice until they pass through the exit door.
 

III. Assumptions

The following assumptions are established in order to generate a mathematical model of the problem:

• The people are panicked at the normal level.
• People want to reach the exit and get out as fast as they can.
• There is no obstacle outside the room, so that the outflow of people exiting the room is depending only to the factors inside the room.
• There is no evacuation priority given to different genders and ages.
• The average velocity of human is 1 m/s (V_p= 1m/s).
• The sizes of all the people are equal and normal.
• People are modeled as a square parallelepiped ().
• We assume that under all hazardous situations everyone should be able to get out within 180 seconds.
• For a small room it seems that there will always be enough time for the evacuation thus to determine the number of people we may use the maximum capacity where people feel comfortable.

IV. Model

IV.1 Variables Definition

In this mathematical model we decide to use the variables that we have defined as the following:

N  = The number of people inside the room.

T_total   = The total time required to evacuate everyone from inside the room to outside room. We measure T_total starting from the time when the emergency notice is issued until everyone is outside the room.

T_exit= The time required for everyone to pass the exit door assuming that everyone has been in the ready-to-exit position.

T_delay = The total delay which involves respond time, time to be ready, and others time that may affect.

T_ready = The average time required for everyone to move to the ready-to-exit position from their initial position.

T_respond= The average time required for everyone respond the emergency notice.

T_others= The time required for any other obstructions possibilities.

T_r = The average of response time to the emergency notice.

T_o = The average time to pass all the obstructions such as furniture, tables, seats, etc.

V_p = The reasonable velocity of human being in a panic situation.

q = The human speed reduction factor under the situation where there are obstructions that may reduce the speed, such as movable furniture, raisers, slope, etc.

S_e = The distance calculated from the exit to the point between the nearest and the farthest distances from the exit.

R = The rate of people exiting, i.e. the average number of people exiting for each second.

E = The number of egresses.

W_d = The average frontal width of the egresses.

W_p = The average frontal width of people.

t_passing = The average time required by one person to pass the exit door.

d = The density of the room, which is the number of people in one meter (people/m²) square where they still feel comfortable.

A_r = The area of the room

N = The number of people inside the room where they still feel comfortable.
 

IV.2 Assumption of Values

In order to obtain a mathematical model that is close to the real-world situation, we assume some reasonable value for our variables. We use these values to help us comparing and verifying our model with the real-world situation. However, these values are subject to changes when necessary.

V_p = 1 ^m/_s

W_p = 0.5 m

T_respond = 2 seconds

IV.3 Design of the Model

As we have mentioned earlier, our model separate the total evacuation time (T_total) into to the time to exit (T_exit) and the time to be in ready-to-exit position (T_delay).

T_{total =} T_{exit +} T_delay                                     (1)

Furthermore, we define the T_exit as the time required for everyone to pass the exit door. Hence T_exit should consider the number of the people to be transported (N) and the rate of the rate of people exiting (R).

                                                 (1.1)

The rate of people exiting (R) can be calculated with the consideration of the frontal width of the exit (W_d), the average frontal width of the people (W_p), the number of egresses (E), and the average time for each person to pass the exit (t_passing).

Dividing the W_d by the W_p will help us to determine the number of people can escape through one exit on the same event

In order to determine the amount of time required for each person to pass the door (t_passing) we divide the average distance traveled by each person when he or she crosses the exit (S_p) by the average velocity of human in a panic situation (V_p).

t_passing = S_p / V_p (1.1.1)

The average distance traveled by each person may be determined as the sum of the thickness of the door + the width of the human.

S_p = x + W_p (1.1.2.1)

Exiting or egress from a building is a prime concern for an emergency situation, since without rapid means of escape the congestion of people may lead to injury or death. The number of egress (E) is affecting the throughput rate of people exiting. Therefore, combining the equation (2), (3), and (4) we arrive to an equation to measure the rate of people exiting (R)

                         (1.1.2)

T_delay is defined as the average time required for everyone to move to the ready-to-exit position from his or her initial position. T_delay is very dependent on the room configuration, the room shape, and the location of people inside the room. Hence, the model should be general enough in order to cover the entire possible situation.

T_delay = T_ready + T_respond + T_others                           (1.2)

We expect that the minimum time ready, should be the time needed for the person nearest the exit to travel from his or her position to the exit door, and the maximum time ready is the time required by the person farthest from the exit to do the same action. However, as the farthest person is moving to exit, the nearest person has already exited. Based on those assumptions, the reasonable time ready should be the average of those two. We were thinking that we should be able to calculate this from the given area. Nonetheless, that is not always the case because the time ready is also dependent on the shape of the room. Thus, the best method to approximate the reasonable time ready is the time needed by a person to travel from the distance between the nearest and the farthest distances from the exit door (S_e). The nearest distance is, of course, at zero distance from the exit. The farthest distance need to consider the path traveled to pass the unmovable obstacles such as unmovable furniture, swimming pool, etc. We consider the unmovable obstacles as an additional distance traveled by the people because they need to walk around the unmovable obstacles. We also introduce another factor that may effect the velocity on the way to the exit door as q . This factor would be one unless under the situation where raisers, slope, slippery surface, etc. exist. The q is ranging from 0 to 1 with 1 meaning that there is no reduction factor. We consider the movable furniture as a velocity reduction factor because those do not change the path traveled by the people, but requires them to move them from their way. Thus, the T_ready can be formulated as the following:

T_respondis determined by the average time required by human to respond an emergency notice. In other words, it is the duration between the time emergency notice is issued and the time people start to react.

T_others is a variable where we can substitute with other matters that may affect the evacuation timing. One of the examples is the time required to open the emergency door.

The above equations may be used as the following to help determining the number of people allowed in a building (N):

1. Assume a reasonable total evacuation time (T_total), i.e. 180 seconds is a safe time to evacuate people out of a room
2. Calculate the T_delay using the equation 1.2
3. Substitute the T_total and T_delay to the equation 1 and then solve for T_exit
4. Calculate the R using the equation 1.1.3
5. Substitute the T_exit and R to the equation 1.1 and then solve for N

However, the maximum M number of people is only mathematically safe to occupy a room. The people inside the room may not be necessary comfortable. As a supplement to this equation we introduce another equation that consider the comfortable area (d) for each person.

M = d * A_r (2)

Thus to decide the maximum number of people we should use the M calculated from equation number 2 given room area and comfortable area per person as the limit of N. The logical representation may be as the following.

If N > M Then the maximum number of people is M

If N < M Then the maximum number of people is N

V. Discussion

Strengths

Our model considers many factors such as the number of egresses, the number of people that can go through the exit concurrently, the room configuration, the respond time, the obstacles which people may encounter while trying to reach the exit door. These are all the factors that we may encounter in the evacuation process. Thus, the involvement of these factors in our model has brought our model to be close enough to the actual emergency condition.

Weaknesses

The weakness of our model is we need to utilize to different essential equations in order to decide the maximum room occupancy. However, this is reasonable because for a small room it seems that there will always be enough time for everyone to escape. Similarly in the case of an open space area (a room with unlimited egresses) or a room with many egresses, the total evacuation time will be small. At least under these two different situations, we need to use the equation 2 that considers the density where people feel comfortable as the maximum limit.

Stability

The model is very stable because we may modify one or more the parameter’s values without affecting the integrity of this model. Furthermore, this model is very flexible since we can adjust the values of all the parameters as desired, with the consideration of different cases and different rooms. Even in extreme situation, such as the money storage in a bank that has many layers of security, which may require more time for the evacuation, we may include this consideration on the T_others.
 
 

VI. Appendix: Example Case

Assumed Values

V_p = 1 ^m/_s

W_p = 0.5 m

T_respond = 2 seconds

T_respond = 0 seconds

q = 1

Building Information

The room area is 1500 (A_r = 1500 m²)

The thickness of the door is 0.2m (x = 0.2 m)

The width of the door is 0.956 m (W_d = 0.956 m » 1 m)

Objective

To determine the maximum occupancy load that meets 180 seconds of evacuation time and the comfortable area of more than 0.9 m² each person.

T_total = 180 seconds with the comfortable area for each person.

d = 1/0.9m²
 
 

Solution

T_{total =} T_{exit +} T_delay (1)

First, we solve for T_exit

T_exit = N/R (1.1)

Because the throughput (R) is unknown, we need to find R

(1.1.2)

While t_passing = S_p / V_p

From the configuration of the room (see the picture), we can find S_p

S_{p =} x + W_p (1.1.2.1)

S_{p =} 0.2 m + 0.5 m = 0.7 m

t_passing = 0.7 m / 1 m/s = 0.7 seconds

Then, we get the throughput (R) :

(1.1.2)

R = 11.42 people/second » 11 people/second

After we get the throughput (R), we need to find the time delay (T_delay).

T_delay = T_ready + T_respond + T_others (1.2)

We need to find T_ready only to get T_delay because we already know T_respond and T_others.

T_ready = S_e / (q V_p) (1.2.1)

S_e = S_farthest / 2 = (7.5 + 50 + 7.5) / 2 see picture for the S_farthest

T_ready = 32.5 m / (1 * 1 m/s)

= 32.5 seconds

Thus,

T_delay = T_ready + T_respond + T_others

T_delay = 32.5 s + 2 s + 0 s

= 34.5 seconds

After that, we can solve for T_exit, and then solve for N, which will be the maximum occupant allowed based by this model.

T_exit = T_total – T_delay (1)

= 180 seconds – 34.5 seconds

= 145.5 seconds

Using the equation 1.1, we solve for N

T_exit = N/R à N = T_exit * R

N = 145.5 seconds * 11 people/seconds

N = 1600.5 » 1600 people

Comfortable area testing

M = d * A_r (2)

M = 1/0.9m² * 1500 m²

= 1666.66 » 1666 people

Since N or the number of maximum capacity where the room is still safe is less than M or the number of maximum capacity where people still feel comfortable, then the posted maximum capacity should be 1600 people. In the other words, 1666 is comfortable but does not meet the safety requirement.
 
 

VII. Bibliography

Mesterton, Michael., Gibbons. A Concrete Approach To Mathematical Modelling. New York: A Willey-Interscience Publication, 1995.
Dym, Clive L., Ivey, Elizabeth S. Principles of Mathematical Modeling. New York: Academic Press, Inc., 1980.
Uniform Building Code 1997 Vol. 1. New York: International Conference of Building Officials, 1997.
Hutching, Jonathan F. National Building Codes Handbook. New York: McGraw-Hill, 1998
 

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