Fuzzy approach and possibility to solve uncertainty weaknesses in conventional quantitative risk assessment

Abstract

The disadvantage of the methods used in dependability and especially the QRA (Quantitative Risk Assessment) method is that of data imperfection and the lack of robustness in the results (subjective estimation of frequencies and consequences effects of accident scenarios). This subjectivity results from several factors, in particular, the high number of components in the system studies, connections and structural interactions, and functional dependencies between the components including operating conditions. Also, physical factors and environmental influence the state and functioning of the system. All these factors alternate the credibility and perfection of data as well as security analysis results. Although the QRA method is effective, its application is very costly in terms of time and effort, which requires competent specialists. It is right that several aspects are not well supported by conventional QRA. Paving the way for further improvements and suggesting soft computing approaches using fuzzy sets and logic although various other approaches can be adopted such as fuzzy probability theory approach. A good review of the QRA approach in various fields is also appreciated as the availability and quality of the data restrict the objectivity and effectiveness of QRA and secondly the results of QRA either the steps taken are qualitative or quantitative are associated with uncertainty or impreciseness and compel to adopt soft computing approach. Four types of uncertainties are covered: Uncertainty associated with input parameters, stochastic/ontological uncertainty, Assessment and reduction in risk and modeling of dangerous effects. The uncertainties inherent in QRA approach are dealt with using tools from the theory of fuzzy sets and that of possibility and helpful in flexible solutions allowing for adaptive risk management.

Appendices

Appendix

To treat the uncertainty modeling of thermal effects like pool fire and fireball, the fuzzy QRA model and particularly in the calculating of hazardous effects consequences. We used the calculating equations of conventional QRA, but will place it in fuzzy term (Eqs. 5, 6) which are illustrated as follows:

Fuzzy fireball

2.1 The fuzzy maximum radius of a fireball and total time duration:

The fuzzy maximum radius $\widetilde{{r}_{b}}$ (m), and the fuzzy total time duration $\widetilde{{t}_{b}}$ (s), of a fireball must be calculated from the following empirical expressions (Crossthwaite et al. 1988):

$$ \tilde{r}_{b} = 2.9\tilde{M}^{0.333} ;\tilde{t}_{b} = \left\{ \begin{gathered} 0.45\tilde{M}^{0.333} ;\tilde{M} < 37000 \hfill \\ 2.59\tilde{M}^{0.167} ;\tilde{M} \ge 37000 \hfill \\ \end{gathered} \right. $$

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where $\widetilde{M}$(kg) is the fuzzy Mass of LPG (Butane).

2.2 The Fuzzy lift-off height of a fireball

The fuzzy lift-off height $\widetilde{{\mathrm{H}}_{\mathrm{b}}}$(m) of the fireball is defined as the height from the center of the fireball to the ground under the fireball. Based on the HSE fireball model (Crossthwaite et al. 1988) can be calculated:

$$ \tilde{H}_{b} = \tilde{r}_{b} $$

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2.3 The fuzzy surface emissive power of the fireball

The Rebert’s correlation (Roberts 1981) for the Fuzzy surface emissive power $\widetilde{\mathrm{E}}$ $\left(\mathrm{Kw}/{\mathrm{m}}^{2}\right)$ is employed in the HSE model, it is given by:

$$ \tilde{E} = \frac{{f_{S} \times \Delta H_{C} \times \tilde{M}}}{{4\pi \times \tilde{t}_{b} \times \tilde{r}_{b} }};f_{S} = 0.27 \times \left( {\frac{{P_{sat} }}{{10^{6} }}} \right)^{0.32} $$

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where

• ${f}_{s}(-)$ is the fraction of total available heat energy radiated by the flame;

• $\Delta {H}_{c}\left(J/kg\right)$ is the combustion heat of LPG(Butane) and

• ${P}_{sat }\left(\mathrm{N}/{\mathrm{m}}^{2}\right)$ is the burst pressure.

2.4 Fuzzy view factor

The Fuzzy view factor ${\widetilde{F}}_{view}(-)$ expresses the fraction of the emitted radiation that reaches the receptor per unit area (the receptor can be human or any object). In this case, the shape of the fire is considered as a perfect sphere. Therefore, the view factor, that the receptor is (Van Den Bosh and Weterings.,1997):

$$ \tilde{F}_{view} = \left( {\frac{{\tilde{r}_{b} }}{{\tilde{X}}}} \right)^{2} ;\tilde{X} = \sqrt {\left( {x^{2} + \tilde{r}_{b}^{2} } \right)} $$

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where $x$(m) is the distance from the fireball center.

2.5 Fuzzy atmospheric transmissivity

For the calculation of fuzzy atmospheric transmissivity ${\widetilde{\tau }}_{a}(-)$, the following empirical expression (Wayne 1991) can be employed:

$$ \begin{aligned} \tilde{\tau }_{a} = & \left\{ \begin{gathered} 1.006 - 0.01171 \times \log_{10} \tilde{X}\left( {H_{2} O} \right) - 0.02368 \times \left( {\log_{10} \tilde{X}\left( {H_{2} O} \right)} \right)^{2}+ \hfill \\ - 0.03188 \times \log_{10} \tilde{X}\left( {CO_{2} } \right) + 0.001164 \times \left( {\log_{10} \tilde{X}\left( {CO_{2} } \right)} \right)^{2} \hfill \\ \end{gathered} \right. \end{aligned} $$

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$$ \begin{aligned} \left\{ \begin{gathered} \tilde{X}\left( {CO_{2} } \right) = \frac{{2.165 \times \tilde{P}_{W} \times \tilde{X}}}{{\tilde{T}_{a} }};\tilde{X}\left( {CO_{2} } \right) = \frac{{273 \times \tilde{X}}}{{\tilde{T}_{a} }} \hfill \\ \tilde{P}_{W} = 99.98 \times \tilde{R}_{H} \times \exp \left( {21.66 - \frac{5431.3}{{\tilde{T}_{a} }}} \right) \hfill \\ \end{gathered} \right.\\ \end{aligned} $$

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where

• $\widetilde{PW}$is the fuzzy partial water vapor pressure in the air

• $\widetilde{RH}$ (%) is the fuzzy relative Humidity

• $\widetilde{{T}_{a}}$(K) is the fuzzy Ambient temperature

• $X\widetilde{{(H_{2} o)}}\left( {-} \right)$ is the fuzzy absorption coefficients for water vapor and

• $X\widetilde{{(Co_{2} )}}\left( {-} \right)$is the fuzzy absorption coefficients for carbon dioxide.

Fuzzy pool fire

In the heat radiation calculations for pool fire, the following steps can be distinguished:

3.1 Calculation of the liquid pool fire fuzzy diameter $\widetilde{{{\varvec{D}}}_{{\varvec{P}}}}({\varvec{m}})$ (Van Den Bosh and Weterings 1997):

$$ \tilde{D}_{P} = \sqrt {\frac{{4 \times \tilde{V}}}{\pi \times 0.02}} $$

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where $\widetilde{V}$ (${m}^{3}$) is the fuzzy volume of n-octane.

• a) Calculation of the fuzzy burning rate $\widetilde{{{\varvec{m}}}_{{\varvec{B}}}}\left({\varvec{k}}{\varvec{g}}/{{\varvec{m}}}^{2}\cdot {\varvec{s}}\right)$:

  $$ \tilde{m}_{B} = m_{\max } \times \left( {1 - e^{{{{\_\tilde{D}_{p} } \mathord{\left/ {\vphantom {{\_\tilde{D}_{p} } {L_{S}^{ - 1} }}} \right. \kern-0pt} {L_{S}^{ - 1} }}}} } \right) $$

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  ${\mathrm{m}}_{\mathrm{max}}\left(\mathrm{n}-\mathrm{octane}\right)=0.146\mathrm{ and }{\mathrm{L}}_{\mathrm{s}}^{-1}\left(\mathrm{n}-\mathrm{octane}\right)=0.7$(Rew et al. 1997).

Where:

• ${m}_{max}$ $\left(\mathrm{kg}/{\mathrm{m}}^{2}\cdot \mathrm{s}\right)$ is the maximum burn rate fuel and

• ${L}_{s}$ (m) is the characteristic burn length.

3.2 Calculation of the flame dimensions of a pool fire:

Determine the fuzzy Pool fire flame length ${\widetilde{L}}_{P}(m)$ (Thomas. 1963):

$$ \tilde{L}_{P} = 42 \times \tilde{D}_{P} \times \left( {\frac{{\tilde{m}_{B} }}{{\tilde{p}_{air} \times \left( {g \times \tilde{D}_{P} } \right)^{0.61} }}} \right) $$

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In which:

$$ \tilde{p}_{air} = \left( {\frac{{101325 \times W_{air} }}{{R_{C} \times \tilde{T}_{a} }}} \right) $$

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where

• $\widetilde{{\rho }_{air}}$ (kg/m³) is the fuzzy density of ambient air;

• $g$ $\left(m/{s}^{2}\right)$ is the gravitational acceleration 9.81;

• ${\mathrm{W}}_{\mathrm{air}}\left(\mathrm{kg}/\mathrm{mol}\right)$ is the molecular weight air and

• ${R}_{c}(J/mol. K)$ is the Gas constant 8.314.

Determine the fuzzy flame tilt angle $\widetilde{\uptheta }$, using the fuzzy Froude ($\widetilde{\mathrm{Fr})}$ and fuzzy Reynolds numbers $\left(\widetilde{\mathrm{Re}}\right)$:

$$ \tilde{\theta } = \arcsin \left( {\frac{{1 + \sqrt {1 + 4\tilde{t}^{2} } }}{{2 \times \tilde{t}}}} \right);\tilde{t} = 0.7 \times \left( {\tilde{F}_{r}^{0.428} \times \tilde{R}_{e}^{0.109} } \right) $$

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In which:

$$ \tilde{F}_{r} = \frac{{\tilde{U}_{W}^{2} }}{{g \times \tilde{D}_{P} }};\tilde{R}_{e} = \frac{{\tilde{U}_{w} \times \tilde{D}_{P} }}{{\tilde{v}}} $$

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$$ \tilde{v} = - 1.1555 \times 10^{ - 14} \times \tilde{T}_{a}^{3} + 9.5728 \times 10^{ - 11} \times \tilde{T}_{a}^{2} + 3.7604 \times \, 10^{ - 8} \times \tilde{T}_{a} - 3.4484 \times 10^{ - 6} $$

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where

• $\widetilde{\nu }$ $\left({m}^{2}/s\right)$ is the fuzzy kinematic viscosity of air and

• ${U}_{w}\left(\mathrm{m}/\mathrm{s}\right)$ is the fuzzy wind velocity at a height of 10 m.

3.3 The fuzzy surface emissive power of the pool fire $\widetilde{{\varvec{E}}}\left(\mathbf{K}\mathbf{w}/{\mathbf{m}}^{2}\right)$ (DNV 2022):

$$ \tilde{E} = \frac{{\tilde{m}_{B} \times \Delta H_{C} \times F_{S} }}{{\left( {1 + \frac{{4 \times \tilde{L}_{P} }}{{\tilde{D}_{P} }}} \right)}}_{{}} $$

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In which: ${F}_{s}=0.4$ (Mudan and Crose 1995).

Where

• $\Delta {H}_{C}$ (J/Kg) is the net heat of combustion of fuel at boiling temperature and

• ${F}_{s}\left(-\right)$is the fraction of the generated heat radiated from the flame surface.

3.4 Fuzzy atmospheric transmissivity

$$ \tilde{\tau }_{a} = \left\{ {\begin{array}{*{20}l} 1; \hfill & {x < \frac{{\tilde{D}_{p} }}{2}} \hfill \\ {1.006 - 0.01171 \times \log_{10} \tilde{X}\left( {H_{2} O} \right) - 0.02368 \times \left( {\log_{10} \tilde{X}\left( {H_{2} O} \right)} \right)^{2} + }; \hfill & {x \ge \frac{{\tilde{D}_{p} }}{2}} \hfill \\ { - 0.03188 \times \log_{10} \tilde{X}\left( {CO_{2} } \right) + 0.001164 \times \left( {\log_{10} \tilde{X}\left( {CO_{2} } \right)} \right)^{2} } \hfill & {} \hfill \\ \end{array} } \right. $$

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$$ \left\{ \begin{gathered} \tilde{X}\left( {H_{2} O} \right) = \frac{{2.165 \times \tilde{P}_{W} \times \tilde{X}}}{{\tilde{T}_{a} }};\tilde{X}\left( {CO_{2} } \right) = \frac{{273 \times \tilde{X}}}{{\tilde{T}_{a} }}; \hfill \\ \tilde{P}_{W} = 99.98 \times \tilde{R}_{H} \times \exp \left( {21.66 - \frac{5431.3}{{\tilde{T}_{a} }}} \right);\tilde{X} = x - \frac{{\tilde{D}_{P} }}{2} \hfill \\ \end{gathered} \right. $$

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3.5 Fuzzy view factor

Consider a cylindrical approximation to the listed flame. The fuzzy vertical $\left(\widetilde{{F}_{v}}\right)$ and horizontal $\left(\widetilde{{F}_{h}}\right)$ area view factors are following expressions (Atallah and Shal 1990):

$$ \begin{aligned} \pi \tilde{F}_{V} = & - \tilde{E}\tan^{ - 1} \tilde{D} + \tilde{E} \times \left( {\frac{{\tilde{\alpha }^{2} + \left( {\tilde{\beta } + 1} \right)^{2} - 2\tilde{\beta }\left( {1 + \tilde{\alpha }\sin \left( {\tilde{\theta }} \right)} \right)}}{{\tilde{A} \times \tilde{B}}}} \right)\\ & \times \tan^{ - 1} \left( {\frac{{\tilde{A} \times \tilde{D}}}{{\tilde{B}}}} \right) + \left( {\frac{{\cos \left( {\tilde{\theta }} \right)}}{{\tilde{C}}}} \right) \\& + {\kern 1pt} \left( {\frac{{\tilde{\alpha } \times \tilde{B} - \tilde{F}^{ - 2} \sin \left( {\tilde{\theta }} \right)}}{{\tilde{F} \times \tilde{C}}}} \right) + \tan^{ - 1} \left( {\frac{{\tilde{F} \times \sin \left( {\tilde{\theta }} \right)}}{{\tilde{C}}}} \right);and \\ \end{aligned} $$

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$$ \begin{aligned} \pi \tilde{F}_{H} = & \tan ^{{ - 1}} \left( {\frac{1}{{\tilde{D}}}} \right) \\ & + \frac{{\sin \left( {\tilde{\theta }} \right)}}{{\tilde{C}}} \times \left( {\tan ^{{ - 1}} \left( {\frac{{\tilde{\alpha } \times \tilde{B} - \tilde{F}^{{ - 2}} \sin \left( {\tilde{\theta }} \right)}}{{\tilde{F} \times \tilde{C}}}} \right) + \tan ^{{ - 1}} \left( {\frac{{\tilde{F} \times \sin \left( {\tilde{\theta }} \right)}}{{\tilde{C}}}} \right)} \right) \\ & - \left( {\frac{{\tilde{\alpha }^{2} + \left( {\tilde{\beta } + 1} \right)^{2} - 2\left( {\tilde{\beta } + 1 + \tilde{\alpha } \times \tilde{B} \times \sin \left( {\tilde{\theta }} \right)} \right)}}{{\tilde{A} \times \tilde{B}}}} \right) \times \tan ^{{ - 1}} \left( {\frac{{\tilde{A} \times \tilde{D}}}{{\tilde{B}}}} \right) \\ \end{aligned} $$

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In which:

$$ \tilde{\alpha } = \frac{{\tilde{L}_{P} }}{{\left( {\frac{{\tilde{D}_{P} }}{2}} \right)}};\tilde{B} = \frac{x}{{\left( {\frac{{\tilde{D}_{P} }}{2}} \right)}};\tilde{A} = \sqrt {\tilde{\alpha }^{2} + \left( {\tilde{\beta } + 1} \right)^{2} - 2\tilde{\alpha }\left( {\tilde{\beta } + 1} \right)\sin \left( {\tilde{\theta }} \right)} ;\tilde{B} = \sqrt {\tilde{\alpha }^{2} + \left( {\tilde{\beta } + 1} \right)^{2} - 2\tilde{\alpha }\left( {\tilde{\beta } - 1} \right)\sin \left( {\tilde{\theta }} \right)} $$

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$$ \tilde{C} = \sqrt {1 + \left( {\tilde{\beta }^{2} - 1} \right) - \cos \left( {\tilde{\theta }} \right)} ;\tilde{D} = \sqrt {\frac{{\left( {\tilde{\beta } - 1} \right)}}{{\left( {\tilde{\beta } + 1} \right)}}} ;\tilde{E} = \frac{{\tilde{\alpha }\cos \left( {\tilde{\theta }} \right)}}{{\tilde{\beta } - \tilde{\alpha }\sin \left( {\tilde{\theta }} \right)}};\tilde{F} = \sqrt {\left( {\tilde{\beta }^{2} - 1} \right)} $$

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The fuzzy maximum view factor ${\widetilde{F}}_{view}(-)$ is estimated from the vector sum of $\widetilde{{F}_{v}}$ and $\widetilde{{F}_{h}}$:

$$ \tilde{F}_{view} = \sqrt {\tilde{F}_{V}^{2} + \tilde{F}_{H}^{2} } $$

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