Damage Function and Its Application

The types of damage functions for objects in the event of various emergency situations fires, accidents, emergencies, etc. are considered. The methods of constructing damage functions using expert methods and taking into account the actions of emergency services to eliminate emergency situations and their consequences are presented. The possibility of comparing the effectiveness of various methods of emergency response using the damage function is shown. An example of assessing damage reduction while increasing the efficiency of fire extinguishing by automatic installations is provided.


Introduction
In the event of emergency situations (fires, accidents, emergencies, etc.) at industrial [1] and other facilities, they are usually damaged. At the same time, the amount of damage depends both on the object itself and on the type of emergency situation and measures to minimize it -on the effectiveness of fire extinguishing, emergency rescue operations (ERO) [2], etc. When developing planning documents [3][4][5] and risk assessment [6,7] is of interest to determine the dynamics of possible damage, for which it was proposed [8,9] to use a special curve U(t), called the «damage function».

Materials and methods
The initial damage function U(t) is a curve that monotonically increases from U=0 (at time t=0 of the onset of an emergency situation, there is still no damage) to U=1, when the damage reaches its greatest value (Fig. 1). Mathematical expressions for the function U(t) that meet the requirements for it can have different forms (for example, [8,9]): where k, n, A, m, B, λ, μ, a, b, с, ν, ξ -coefficients that are to be determined. The representation of the damage function is carried out using expert methods [10] by 2 ( fig. 1, a) or 3 ( fig. 1, b) points. It's easy to notice that by 2 points: U(t1)=U1 и U(t2)=U2 the following coefficients may be obtained : (k, n), (A, m) и (В, λ) of the equations (1)-(3) respectively. Coefficients (μ, a, b) and (с, ν, ξ) of equations (4) and (5) -by 3 points: fig. 1, b).
Coefficients (k, n), (A, m) и (В, λ) can be determined from solutions of systems of two equations: Coefficients (μ, a, b) can be determined from solutions of systems of three equations: Coefficients с, ν, ξ can be determined similarly.

Results and discussion
Using the damage function, it is possible to solve the problem of assessing the effectiveness of eliminating an emergency situation. In this case, two situations should be distinguishedwhen at the moment tl the damage reaches its maximum Umax and remains at this level for some time (for example, a fire) and when after reaching Umax the damage is completely eliminated after the time tк, i.e. U(tк) = 0, which may correspond to the neutralization of the release of an emergency chemical hazardous substance. Graphically similar situations are shown in Fig. 2.
If we assume that the elimination of an abnormal situation will begin in time tp after its occurrence and will occur at a rate of kl (can be determined by expert methods [10]), then the dynamics of damage is shown in Fig. 3, provided that equation (12) is valid.

Fig. 3. An example of a damage function during actions to localize and eliminate an emergency situation
Taking into account the actions to localize and eliminate the contingency situation, the dynamics of damage U(t) can be described in equation: (16) where U0 -is an initial damage function determined by (1)-(5); T=tl and Uк=Umax, when, for example, fire extinguishing takes place; T=tк and Uк=0, when, for example, there is a neutralization of the release of hazardous chemicals.
Time tl can be found from the solution of the transcendental equation: If the damage function is described by expression (1), then equation (17) takes the form [9]: (18) where Е=ехр (-k(tltp)). Maximal damage Umax=U(tl) at the initial damage function (1) can be determined from the equation (16): Equations (18) and (19) can be solved numerically [11]. Time tк can be found from the solution of the transcendental equation: (20) As an example, Fig. 3 shows function (12) and damage reduction functions at different speeds kl from 0.01 to 0.05 min -1 .
With the help of the damage function, it becomes possible to solve the problem of comparing various methods of neutralizing an abnormal situation at an object, for example, a fire in a room when using automatic fire extinguishing installations (AFEI) [12]. In Fig. 4, using the damage function, it is clearly shown that with an increase in the operational efficiency of the AFEI by the value of Δt, the direct damage from the fire can be reduced by the value of ΔU. Specific values of Δt and ΔU can be obtained using the damage function for specific objects and specific types of AFEI.

Conclusion
Thus, if the facility has a risk of various emergency situations (for example, fires), the above methods for constructing the damage function make it possible to objectively assess the