A Software Tool for Domino Effect Risk Assessment in Industrial Plants

by Antonio C. Caputo, Ph.D., L'Aquila, Italy

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Probability of domino effect triggered by fragment projection:
a) Determination of damage area radius
The previous model adopted the Clancey correlation [Ref. 16] linking the equivalent mass of explosive W_TNT to the maximum distance (D_max) reached by projectiles resulting from unit i explosion, which is assumed to be the boundary of the damage area (r_A = D_max). In this revised model, the more precise model of Baker [Refs. 10, 17] is adopted. That model utilizes empirical correlations to compute a scaled initial fragment velocity from the scaled internal pressure of the vessel, and then a scaled fragment range from the scaled initial velocity. For the sake of simplicity in this work, the following parameter values were assumed: lift-to-drag ratio of fragments set at zero which represents a null lift coefficient of fragment as typical of large vessel fragments, a fragment mass of 480 kg, a diameter of 1.6 m and a drag coefficient C_D = 0.47.

b) Determination of damage probability P_ij,FP
The probability P_ij,FP of actual damage to unit j is computed by Equation 9:

(9)

where P_pr is the probability that fragments are actually projected following an explosion of a process vessel (P_pr = 0.8 for pressurized vessels and 0.4 for atmospheric vessels), P_imp is the fragment impact probability computed from Equation 10, and DF is the Damage Factor.

(10)

In Equation 10, F_D is a directional factor considering the relative position of the target vessel, with respect to the projectile's trajectory and the position of the source vessel, as it is more likely that fragments will be projected from the two ends of the vessel, rather than the sides. If the target vessel is located within an area covered by arcs 30° to either side of the vessel's front and rear axial directions F_D = 1.5, while if it is not located in those two 60° sectors, F_D = 0.67 [Refs. 5, 18]. The number of fragments is instead denoted by n. As suggested by Holden and Reeves [Ref. 19], n = 4 for cylindrical vessels, n=8 for spherical vessels with volume lower than 1200 m³ and n=16 for greater volume.

The probability of fragment impact is computed as P_{imp_frag} = /2 (0.5 - PI), according to the method of Gubinelli et al. [Ref. 20], where is the angular width of the target as seen from the source vessel and PI is a trajectory-dependent probability factor, which may be evaluated from Equation 11:

(11)

being D_min is the minimum distance of the target from the source and D_max is the previously determined maximum distance reached by the projectile. To compute PI, the assumption of a conservative value of the initial fragment velocity of 200 m/s is made which, with the above described fragment characteristics, according to the Gubinelli et al. procedure, yields z₁ = 0.125, z₂ = 0.21, z₃ = 0.24.

Finally, the Damage Factor DF is computed using Equation 12:

(12)

F_M is a fragment mass factor function of the thickness of the primary unit walls, as a higher fragment's mass can give rise to a higher damage to the target. It is assumed that F_M = 1 for thick walls (thickness > 20 mm) and 0.7 for thin walls. F_S is a damage susceptibility factor (F_S = 0.7 for thick walls and 1 for thin walls), meaning that the probability that the target unit is actually damaged by the impact of a fragment of given kinetic velocity is an inverse function of the thickness of its walls. Finally, p is a factor representing the kinetic energy of the fragments and is computed as the probability that fragments will travel a distance further than the source-target distance D. On the assumption that the distance traveled by fragments has a Gaussian probability distribution with maximum value D_max and minimum value 0, p can be computed by referring to the standard normal distribution and the corresponding standardized variable Z = (D – ) / (i.e., the distribution has = D_max/2 and = D_max/6).

Probability of domino effect triggered by pool fire:
a) Determination of damage area radius
The damage area is limited by the radius at which the threshold value of thermal radiation flow I_S = 12.5 kW/m² is obtained. This can be computed, assuming a point source model, by Equation 13, which relates the radiation intensity to the distance and pool fire characteristics [Refs. 21, 22]:

(13)

where r is the radius, F_S is the percent irradiated energy (usually ranging between 0.13 and 0.4), m is the combustion velocity per unit surface area of the pool, = 2.02 (p_w r)^-0.08 is the atmospheric transmissivity coefficient, with p_w being the water saturation pressure at ambient temperature, H_c being the heat value of the burning substance and D being the pool diameter.

b) Determination of damage probability P_ij,PF
The damage probability is computed as

(14)

In Equation 14, is the view factor, with being the angular width of the target as seen from the source vessel and _osc being the angular width, within , of any obstacle shielding the target from the thermal radiation. F_RP is a factor accounting for the possibility of cooling the target vessel walls, ranging from 0.75 to 1. This depends on the availability of a vessel cooling system and a rapid deployment firefighting squad. F_RT is instead a wall thermal resistance coefficient ranging from 0.55 to 1 and accounting for the thickness of vessel walls and the presence of an insulating layer. Finally, DF is the damage factor given by Equation 15 as a function of the absorbed dose of thermal radiation DTR [Ref. 22].

(15)

The dose of thermal radiation is computed as , where I is the actual thermal radiation computed from Equation 13, I_S is the threshold value of thermal radiation and t_B is the exposure time (i.e., the pool burning time).

The exposure time is computed as , where V is the pool volume, A is the pool surface area and v is the burning velocity

(16)

where k₁ is a constant depending on the flammable liquid [Ref. 21], and v is the combustion velocity of a pool with infinite diameter.

17)

with k₂ = 0.0076 cm/min, and H_V being the latent heat of vaporization.

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