Appendix A GHS—Globally Harmonized System of Classication and Labelling of Chemicals The Globally Harmonized System of Classification and Labelling of Chemicals (GHS) was published by the UN in 2003 [A-1] with the objective to harmonize the differing approaches of classifying and labeling chemicals in different countries. The GHS was introduced in the European Community by [A-2]. It came into force on January 20th, 2009. The regulation comprises to a large extent the provisions of [A-1] and is also known as the CLP regulation (Regulation on Classification, Labelling and Packaging of Substances and Mixtures). The purpose of the regulation is described in [A-2] as follows: ‘‘This Regulation should ensure a high level of protection of human health and the environment as well as the free movement of chemical substances, mixtures and certain specific articles, while enhancing competitiveness and innovation’’. In order to achieve this, materials are assigned to hazard classes which describe the physical hazard, the hazards for human health or the environment. The classes are divided into hazard categories in order to characterize the severity of a hazard. In addition pictograms and signal words are introduced. Pictograms are intended to graphically convey specific information on the hazard concerned. A ‘signal word’ means a word that indicates the relative level of severity of hazards to alert the reader to a potential hazard. For example, the word ‘danger’ indicates the more severe hazard categories, whilst ‘warning’ signals the less severe hazard categories. In Annex I of [A-2] the general principles for classification and labelling are treated in part 1. Part 2 deals with the physical hazards and uses the classes listed in Table A.1. The subject of part 3 of Annex I are health hazards. Table A.2 lists the classes of health hazards. Further to that part 4 of Annex I deals with substances, which constitute hazards for the aquatic environment, and part 5 with substances which are hazardous to the ozone layer. Ó Springer-Verlag Berlin Heidelberg 2015 U. Hauptmanns, Process and Plant Safety, DOI 10.1007/978-3-642-40954-7 625
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Appendix AGHS—Globally Harmonized Systemof Classication and Labelling of Chemicals
The Globally Harmonized System of Classification and Labelling of Chemicals(GHS) was published by the UN in 2003 [A-1] with the objective to harmonize thediffering approaches of classifying and labeling chemicals in different countries.The GHS was introduced in the European Community by [A-2]. It came into forceon January 20th, 2009. The regulation comprises to a large extent the provisions of[A-1] and is also known as the CLP regulation (Regulation on Classification,Labelling and Packaging of Substances and Mixtures).
The purpose of the regulation is described in [A-2] as follows: ‘‘This Regulationshould ensure a high level of protection of human health and the environment aswell as the free movement of chemical substances, mixtures and certain specificarticles, while enhancing competitiveness and innovation’’.
In order to achieve this, materials are assigned to hazard classes which describethe physical hazard, the hazards for human health or the environment. The classesare divided into hazard categories in order to characterize the severity of a hazard.In addition pictograms and signal words are introduced. Pictograms are intended tographically convey specific information on the hazard concerned. A ‘signal word’means a word that indicates the relative level of severity of hazards to alert thereader to a potential hazard. For example, the word ‘danger’ indicates the moresevere hazard categories, whilst ‘warning’ signals the less severe hazardcategories.
In Annex I of [A-2] the general principles for classification and labelling aretreated in part 1. Part 2 deals with the physical hazards and uses the classes listedin Table A.1.
The subject of part 3 of Annex I are health hazards. Table A.2 lists the classesof health hazards.
Further to that part 4 of Annex I deals with substances, which constitute hazardsfor the aquatic environment, and part 5 with substances which are hazardous to theozone layer.
� Springer-Verlag Berlin Heidelberg 2015U. Hauptmanns, Process and Plant Safety,DOI 10.1007/978-3-642-40954-7
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Table A.1 Physical hazards [A-2]
Numberingaccording to Annex I,part 2
Description of the class
2.1 Explosives, substances and mixtures as well as articles withexplosives
2.2 Flammable gases
2.3 Flammable aerosols
2.4 Oxidizing gases
2.5 Gases under pressure
2.6 Flammable liquids
2.7 Flammable solids
2.8 Self-reactive substances and mixtures
2.9 Pyrophoric liquids
2.10 Pyrophoric solids
2.11 Self-heating substances and mixtures
2.12 Substances and mixtures which in contact with water emitflammable gases
2.13 Oxidizing liquids
2.14 Oxidizing solids
2.15 Organic peroxides
2.16 Corrosive to metals
Table A.2 Health hazards [A-2]
Numbering according toAnnex I, part 3
Description of the class Differentiationaccording to
3.1 Acute toxicity Acute oral toxicity
Acute dermaltoxicity
Acute inhalationtoxicity
3.2 Skin corrosion/irritation
3.3 Serious eye damage/eye irritation
3.4 Respiratory or skin sensitization
3.5 Germ cell mutagenicity
3.6 Carcinogenicity
3.7 Reproductive toxicity
3.8 Specific target organ toxicity—singleexposure
3.9 Specific target organ toxicity—repeated exposure
3.10 Aspiration hazard
626 Appendix A: GHS—Globally Harmonized System…
References
[A-1] United Nations (2003) Globally harmonized system of classification and labelling ofchemicals (GHS), ST/SG/AC. 10/30, New York and Geneva
[A-2] Regulation (EC) No 1272/2008 of the European parliament and of the council of 16December 2008 on classification, labelling and packaging of substances and mixtures,amending and repealing Directives 67/548/EEC and 1999/45/EC, and amendingRegulation (EC) No 1907/2006. Official J Eur Union L 353/1, 31.12.2008
Appendix A: GHS—Globally Harmonized System… 627
Appendix BProbit Relations, Reference and Limit Values
B.1 Probit Relations
B.1.1 Fatal Toxic Effects for Selected Materials [B-1]–[B-3]
Acrolein
Y ¼ �9:931þ 2:049 � ln C � tð Þ ðB:1Þ
Acrylonitrile
Y ¼ �29:42þ 3:008 � ln C1:43 � t� �
ðB:2Þ
Ammonia
Y ¼ �30:75þ 1:385 � lnZ t
0
C t0ð Þ2:75�dt0
0
@
1
A ðB:3aÞ
Y ¼ �28:33þ 2:27 � lnZ t
0
C t0ð Þ1:36�dt0
0
@
1
A ðB:3bÞ
Y ¼ �35:9þ 1:85 � ln C2 � t� �
ðB:3cÞ
Benzene
Y ¼ �109:78þ 5:3 � ln C2 � t� �
ðB:4Þ
Hydrogen cyanide
Y ¼ �29:42þ 3:008 � ln C1:43 � t� �
ðB:5Þ
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Bromine
Y ¼ �9:04þ 0:92 � ln C2 � t� �
ðB:6Þ
Chlorine
Y ¼ �17:1þ 1:69 � lnZ t
0
C t0ð Þ2:75�dt0
0
@
1
A ðB:7aÞ
Y ¼ �36:45þ 3:13 � lnZ t
0
C t0ð Þ2:64�dt0
0
@
1
A ðB:7bÞ
Y ¼ �11:4þ 0:82 � lnZ t
0
C t0ð Þ2:75�dt0
0
@
1
A ðB:7cÞ
Y ¼ �5:04þ 0:5 � lnZ t
0
C t0ð Þ2:75�dt0
0
@
1
A ðB:7dÞ
Hydrogen chloride
Y ¼ �16:85þ 2:0 � ln C � tð Þ ðB:8Þ
Ethylene oxide
Y ¼ �6:8þ ln C � tð Þ� ðB:9Þ
Fluorine
Y ¼ �8:56þ 1:08 � ln C1:85 � t� �� ðB:10Þ
Hydrogen fluoride
Y ¼ �48:33þ 4:853 � ln C � tð Þ ðB:11aÞ
Y ¼ �26:36þ 2:854 � ln C � tð Þ ðB:11bÞ
Y ¼ �35:87þ 3:354 � ln C � tð Þ ðB:11cÞ
Y ¼ �25:87þ 3:354 � ln C � tð Þ ðB:11dÞ
Formaldehyde
Y ¼ �12:24þ 1:3 � ln C2 � t� �
ðB:12Þ
Carbon disulphide
Y ¼ �46:56þ 4:2 � ln C � tð Þ ðB:13Þ
630 Appendix B: Probit Relations, Reference and Limit Values
Carbon monoxide
Y ¼ �37:98þ 3:7 � ln C � tð Þ ðB:14Þ
Methanol
Y ¼ �6:34734þ 0:66358 � ln C � tð Þ ðB:15Þ
Fuming sulphuric acid (oleum)
Y ¼ �14:2þ 1:6 � ln C1:8 � t� �� ðB:16Þ
Phosgene
Y ¼ �27:2þ 5:1 � ln C � tð Þ ðB:17aÞ
Y ¼ �19:27þ 3:686 � ln C � tð Þ ðB:17bÞ
Phosphine
Y ¼ �2:25þ ln C � tð Þ ðB:18Þ
Sulphur dioxide
Y ¼ �15:67þ 2:1 � ln C � tð Þ ðB:19Þ
Hydrogen sulphide
Y ¼ �11:15þ ln C1:9 � t� �
ðB:20Þ
Toluene
Y ¼ �6:794þ 0:408 � ln C2:5 � t� �
ðB:21Þ
where C(t) is the time-dependent concentration in ppm and the time is in minutes(exception: * in mg/m3 and minutes).
B.1.2 Pressure and Heat Radiation Exposures [B-1, B-4]
Death from lung haemorrhage due to a blast wave
Y ¼ �77:1þ 6:91 � ln ps ðB:22Þ
Eardrum rupture due to a blast wave
Y ¼ �15:6þ 1:93 � ln ps ðB:23aÞ
Y ¼ �12:6þ 1:524 � ln ps ðB:23bÞ
Death following body translation due to impulse
Y ¼ �46:1þ 4:82 � ln J ðB:24Þ
Appendix B: Probit Relations, Reference and Limit Values 631
Injuries from impact
Y ¼ �39:1þ 4:45 � ln J ðB:25Þ
Serious injuries from flying fragments (particularly glass)
Y ¼ �27:1þ 4:26 � ln J ðB:26Þ
Structural damage
Y ¼ �23:8þ 2:92 � ln ps ðB:27Þ
Glass breakage
Y ¼ �18:1þ 2:79 � ln ps ðB:28Þ
Death due to thermal radiation
Y ¼ �14:9þ 2:56 � ln te � q004=3 � 10�4� �
ðB:29Þ
Death due to thermal radiation (unprotected by clothing)
Y ¼ �36:38þ 2:65 � ln te � q004=3� �
ðB:30Þ
Death due to thermal radiation (protected by clothing)
Y ¼ �37:23þ 2:56 � ln te � q004=3� �
ðB:31Þ
First degree burns
Y ¼ �39:83þ 3:02 � ln te � q004=3� �
ðB:32Þ
Second degree burns
Y ¼ �43:14þ 3:02 � ln te � q004=3� �
ðB:33Þ
The symbols have the following meaning:
ps peak side-on overpressure in N/m2;J impulse in Ns/m2;te duration of exposure in s;q00 radiation intensity (heat flux) in W/m2
632 Appendix B: Probit Relations, Reference and Limit Values
B.2 Reference Values for Damage to Health, Property,and Buildings
Tables B.1, B.2, B.3 and B.4.
Appendix B: Probit Relations, Reference and Limit Values 633
Table B.1 Reference values for health damage from thermal radiation [B-1]
Thermal dose in kJ/m2 Effect
375 Third degree burns
250 Second degree burns
125 First degree burns
65 Threshold of pain, no reddening or blistering of skin
Table B.2 Reference values for property damage from thermal radiation [B-1, B-4]
Thermal radiationintensity limit in kW/m2
Effect
37.5 Damage to process plant equipment
35 Spontaneous ignition of wood (without ignition source)
35 Textiles ignite (without ignition source)
18–20 Cable insulation degrades
12 Plastic melts
Table B.3 Reference values for damage from thermal radiation with durations of exposure[30min [B-4]
Material Thermal radiation intensity limit in kW/m2
Damage level 1a Damage level 2b
Wood 15 2
Synthetic material 15 2
Glass 4 –
Steel 100 25a Damage level 1 catching of fire by surfaces of materials exposed to heat radiation as well as therupture or other type of failure (collapse) of structural elementsb Damage level 2 damage caused by serious discoloration of the surface of materials, peeling-offof paint and/or substantial deformation of structural elements
B.3 Limit Values in Germany and Other EuropeanCountries for Damage Causing Loads (After [B-5])
Tables B.5, B.6, B.7, B.8 and B.9.
Table B.4 Reference values for building damage caused by blast waves (after [B-1])
Damage type Peak side-onoverpressure in Pa
Shattering of glass windows large and small, occasional framedamage
3447.4–6894.8
Blowing in of wood siding panels 6894.8–13789.6
Shattering of concrete or cinder-block wall panels, 20 or 30 cmthick, not reinforced
10342.2–37921.4
Nearly complete destruction of houses 34500.0–48300.0
Rupture of oil storage tanks 20684.4–27579.2
Table B.5 Reference values for impacts on people of different forms of energy (bold print limitvalues proposed in [B-5])
Damage causingfactor
Limit value Valuation according to the MajorAccident Ordinance (StörfallV)
Peak side-onoverpressure
1.85 bar (lung haemorrhage) §2 no. 4a StörfallV
Threat to the life of humans
Thermalradiation
10.5 kW/m2 (lethal burns in40 s) Grave health damage (irreversible
damage)
- of concern even if only one personis affected
Peak side-onoverpressure
0.175 bar (eardrum rupture) - small;
Thermalradiation
2.9 kW/m2 (threshold of painreached after 30 s)
- number of affected people large
Peak side-onoverpressure
0.1 bar (destruction of brickwalls)
§2 no. 4b StörfallV
Thermalradiation
1.6 kW/m2 (adverse effect)Health impairment of a large number ofpeople (reversible damage)
Peak side-onoverpressure
0.003 bar (loud bang) Harassment
Thermalradiation
1.3 kW/m2 (maximum ofsolar radiation)
634 Appendix B: Probit Relations, Reference and Limit Values
Table B.6 Limit values in Belgium
Thermal radiation in kW/m2 Explosion pressure in mbar Missile flight
Safety zonea – – –
Risk zoneb 2.5 during 30 s 20 –a Zone, where reversible effects are observedb Zone, where specific measures must be taken for limiting accident consequences with dueconsideration of the duration of exposure
Table B.7 Limit values in France
Thermal radiationb inkW/m
Explosion pressure inmbar
Missileflight
Irreversibleconsequences
3 50 —
Lethal consequences 5 140 —
Risk of a Dominoeffecta
8 for unprotectedstructures12 for protectedstructures
200 for significantdamage350 for grave damage500 for very gravedamage
—
a these threshold values are used by INERIS Institut National des Risques, but are not officialb if exposure is longer than 60 s
Appendix B: Probit Relations, Reference and Limit Values 635
Table B.8 Limit values in Italya
Thermal radiation in kW/m2
Explosion pressure inmbar
Missileflight
Reversibleconsequences
3 30 –
Irreversibleconsequences
5 70 –
Start of lethality 7 140 –
High risk of lethality 12.5 300 –
Risk of a domino effect 12.5 300 –a In Italy the following threshold values are used as well for non-stationary thermal radiation (incase of a fireball): 125 kJ/m2 for reversible effects, 200 kJ/m2 for irreversible effects, 350 kJ/m2
for the threshold to lethality, radius of the fireball for high lethality: 200–800 m, Domino effects.For instantaneous thermal radiation of short duration (in case of a flash fire): � � LFL (start oflethality) and LFL (high lethality)
References
[B-1] Mannan S (ed) (2005) Lees’ loss prevention in the process industries, hazard identification,assessment and control, 3rd edn. Elsevier, Amsterdam
[B-2] Louvar JF, Louvar BD (1998) Health and environmental risk analysis: fundamentals withapplications, vol 2. Prentice Hall, Upper Saddle River
[B-3] PHAST Version 6.51 (2006)
[B-4] The Director-General of Labour (1989) Methods for the determination of possible damageto people and objects resulting from the release of hazardous materials. Green Book,Voorburg, December 1989
[B-5] Kommission für Anlagensicherheit beim Bundesminister für Umwelt, Naturschutz undReaktorsicherheit, Leitfaden ,,Empfehlung für Abstände zwischen Betriebsbereichen nachder Störfall-Verordnung und schutzbedürftigen Gebieten im Rahmen der Bauleitplanung-Umsetzung §50 BImSchG, 2. Überarbeitete Fassung, KAS-18, November 2010Short version of Guidance KAS-18 (2014) Recommendations for separation distancesbetween establishments covered by the major accidents ordinance (Störfall-Verordnung)and areas worthy of protection within the framework of land-use planning implementationof Article 50 of the Federal Immission Control Act (Bundes-Immissionsschutzgesetz,BImSchG). http://www.kas-u.de/publikationen/pub_gb.htm. Last visited on 13 May 2014
Table B.9 Limit values in Spain
Thermal radiation in kW/m2
Explosion pressure inmbar
Missile flight
Alarm zonea 3 50 99.9 % of the rangeof the missile flight
Interventionzoneb
5 125 95 % of the range ofthe missile flight
Dominoeffect zone
12 for unprotectedstructural elements insidethe plant37 for protected elementsinside the plant
100 for buildings160 for equipmentunder atmosphericpressure350 for equipmentunder overpressure
100 % of the rangeof the missile flight
a the consequences of the accident can be perceived by the population, but do not justify anintervention except with critical groups of peopleb the consequences of the accident are so grave that an immediate intervention is justified
636 Appendix B: Probit Relations, Reference and Limit Values
In what follows an overview of selected results of probability calculations is given;the presentation draws upon [C-1].
C.1 Events and Random Experiments
Probability calculations deal with random events and phenomena. The underlyingprocesses are either random like, for example, the disintegration of radioactiveisotopes, or they are so complex that we are either not willing or incapable todescribe them exactly in quantitative terms. For example, we could, on the basis ofinfluenza cases of the year 2012, estimate an expected number of cases for the year2013, although they might be counted in the year 2013. Yet this can only be doneafter the end of 2013. This tells us that a probability can be assigned to eventswhich may possibly occur in the future. In retrospect we are then certain; eitherone or none of the prospectively considered possible events has become true.
If we throw a die, we carry out an experiment which takes place according toknown physical laws. Yet its outcome cannot be predicted with certainty. Such anexperiment is called a random experiment. It can be identified on the basis of thefollowing prescriptions [C-2]
1. A prescription exists for carrying out the experiment (hence it takes placeaccording to strict rules).
2. The experiment can be repeated as often as desired.3. At least two outcomes are possible.4. The outcome is not predictable.
The set of possible outcomes of a random experiment forms the so-called eventspace or sample space, which generally is denoted by X. For a die X = {1, 2, 3, 4,
� Springer-Verlag Berlin Heidelberg 2015U. Hauptmanns, Process and Plant Safety,DOI 10.1007/978-3-642-40954-7
637
5, 6} applies. With random events we may be interested not only in a particularevent, but also in a combination of several events, for example the occurrence of 3or 4 pips on throwing a die. This is illustrated by set operations such as
• union: A[B; at least one of the two events A or B occurs• intersection: A\B; both A and B occur• difference: A - B; A, but not B occurs• complement: A = X - A; A does not occur, A is the event complementary to A
The relationships are illustrated by Fig. C.1.The universal set X, which contains all conceivable events, is called the certain
event, its complement �X the impossible event. Two events for which A \ B ¼ ; istrue, are called incompatible or disjunct, where ; denotes the empty set.
Example C.1 Quality of screws [C-2]In a production of screws we wish to check, if the required length, which is to lie
between 1.9 and 2.1 cm, is satisfied. For this purpose a screw is selected at randomand its length is measured (random experiment). Let A be the event that the screw isshorter than 1.9 cm and B the event that it is longer than 2.1 cm. Then A[B meansthat the screw does not have the required length and A [ B means that it satisfies thelength requirement. If C were the event that the screw is at least 2.0 cm long, then C\ A [ B is the event that the length of the screw is between 2.0 and 2.1 cm. h
638 Appendix C: Basics of Probability Calculations
Fig C.1 Set operations represented by set or Venn diagrams
C.2 Probabilities
One cannot predict the outcome of a random experiment, but it is possible toindicate a probability for a particular outcome. Thus it is known that 5 pips showup with a probability of 1/6 when throwing an ideal die. If this event is denoted byC we write
P Cð Þ ¼ 16
ðC:1Þ
Since it is mathematically inexact to base areas of knowledge on experimentswith ideal—but in reality non-existent—objects, Kolmogoroff established axioms.These axioms, however, comprise the results which would intuitively be expectedif the experiment were repeated an infinite number of times. The axioms are
1.
P Að Þ� 0 for any event A � X ðpositivityÞ
2.
P Xð Þ ¼ 1 ðunitarityÞ ðC:2Þ
3.
P[1
i¼1
Ai
!
¼X1
i¼1
P Aið Þ ðr� additivityÞ
The third property of course implies the finite additivity
P[n
i¼1
Ai
!
¼Xn
i¼1
P Aið Þ ðC:3Þ
If there are just two disjunct (mutually exclusive) events, A and B, we have
P A [ Bð Þ ¼ P Að Þ þ P Bð Þ ðC:4Þ
All calculation rules for probabilities can be derived from the above properties,e.g.
P A [ Bð Þ ¼ P Að Þ þ P Bð Þ � P A \ Bð Þ for any arbitrary A and B
P �Að Þ ¼ 1� P Að Þ ðC:5Þ
P A� Bð Þ ¼ P Að Þ � P Bð Þ; if B � A
Appendix C: Basics of Probability Calculations 639
Example C.2 Game of DiceWe are looking for the probability that when throwing a die two or four pips
appear. This event is described by the set {2, 4}. According to Eq. (C.4) we have
P 2; 4f gð Þ ¼ P 2f gð Þ þ P 4f gð Þ
¼ 16þ 1
6¼ 1
3
Another way of solving the problem consists in subtracting from the certain eventall events which we are not looking for, i.e.
P 2; 4f gð Þ ¼ P 1; 2; 3; 4; 5; 6f gð Þ � P 1; 3; 5; 6f gð Þ¼ 1� P 1f gð Þ � P 3f gð Þ � P 5f gð Þ � P 6f gð Þ
¼ 1� 16� 1
6� 1
6� 1
6¼ 1
3:
h
C.3 Conditional Probabilities and Independence
Often we are interested in the probability of the occurrence of an event A under thecondition that a particular event B has already occurred. For example, the failureof a pump in a process plant under the condition that the plant has been flooded.Such a probability is called conditional probability. It is explained below usingexamples from [C-2].
Example C.3 Relative riskThose who are exposed to a particular risk factor are called exposed persons
and those who are not, unexposed or control persons (members of the controlgroup). The probability of falling ill of disease K, if the risk factor R prevails isdenoted by P(K|R). Then we obtain the possibilities and probabilities of falling illor not listed in Table C.1.
The parameter d ¼ P K Rjð Þ � P K �Rjð Þ is called the risk which can be attributedto the risk factor R. h
Example C.4 Probability of survivalThe probability for a male newborn baby to reach his 70th birthday and to
survive until his 71st is P(A) = 0.95. The probability of living until the 72nd
640 Appendix C: Basics of Probability Calculations
Table C.1 Possibilities and probabilities for exposed and unexposed persons to fall ill or not
K �K
R P(K|R) P �KjRð Þ P(R)�R P Kj�Rð Þ P �Kj�Rð Þ P �Rð Þ
P Kð Þ P �Kð Þ 1
birthday after having reached the 71st is P(B|A) = 0.945. Hence, we obtain theprobability of reaching the 72nd birthday after having lived until 70 years as
P A \ Bð Þ ¼ P Að Þ � P B Ajð Þ ¼ 0:950 � 0:945 ¼ 0:898
h
The conditional probability for B to occur under the condition that A hasoccurred is understood to be
P B Ajð Þ ¼ P A \ Bð ÞP Að Þ ðC:6Þ
where P(A) 6¼ 0 has to hold. In this way we obtain the rule for multiplication, i.e.
P A \ Bð Þ ¼ P B Ajð Þ � P Að Þ ¼ P A Bjð Þ � P Bð Þ ¼ P B \ Að Þ ðC:7Þ
Equation (C.7) can be extended analogously to more than two events. Eventsare stochastically independent, if
P A \ Bð Þ ¼ P Bð Þ � P Að Þ ¼ P Að Þ � P Bð Þ ¼ P B \ Að Þ ðC:8Þ
holds. Stochastic dependence has to be distinguished from causal dependence. Thelatter is directed, i.e. the cause produces the consequence. Stochastic dependence,on the other hand, is symmetric. Two quantities depend on each other. Causaldependence implies stochastic dependence. However, the inverse argument is nottrue.
C.4 Total Probability and Bayes’ Theorem
If K denotes a particular disease, F a woman and M a man, then we obtain asprobability for a randomly chosen person of being ill
P Kð Þ ¼ P Fð Þ � P K Fjð Þ þ P Mð Þ � P K Mjð Þ ðC:9Þ
Using Eqs. (C.7) and (C.9) is written as follows
P Kð Þ ¼ P F \ Kð Þ þ P M \ Kð Þ ðC:10Þ
or generalized
P Kð Þ ¼X
i
P Ai \ Kð Þ ðC:11Þ
Equation (C.11) is known as the total probability of event K.Combining Eqs. (C.9) and (C.10) in such a way that we can answer the question
whether a person suffering from disease K is a man, we obtain the probability
P M Kjð Þ ¼ P M \ Kð ÞP Kð Þ ðC:12Þ
Appendix C: Basics of Probability Calculations 641
In Eq. (C.12) we ask for a particular circumstance related to an event. In thepresent context the question is if a person affected by the disease K (event) is aman (circumstance).
Inserting Eq. (C.10) in Eq. (C.12) and using Eq. (C.9), one obtains
P M Kjð Þ ¼ P K Mjð Þ � P Mð ÞP Fð Þ � P K Fjð Þ þ P Mð Þ � P K Mjð Þ ðC:13Þ
In this way we obtain Bayes’ theorem, which in generalized form reads
P Ak Kjð Þ ¼ P Akð Þ � P K Akjð ÞPn
i¼1 P Aið Þ � P K Aijð Þ ðC:14Þ
The following example from [C-2] shows an application of Bayes’ theorem.
Example C.5 Terrorism and air trafficAs a precaution all passengers in an airport are controlled. A terrorist is
detained with a conditional probability of P F Tjð Þ ¼ 0:98, a non-terrorist withprobability P F �Tjð Þ ¼ 0:001. Every one hundred thousandth tourist is assumed tobe a terrorist, i.e. P(T) = 0.00001. What is the probability that a detained personreally is a terrorist? The solution is
P T Fjð Þ ¼ P F Tjð Þ � P Tð ÞP F Tjð Þ � P Tð Þ þ P F �Tjð Þ � P �Tð Þ ¼
0:98 � 0:000010:98 � 0:00001þ 0:001 � 0:99999
¼ 0:0097
Despite the quality (reliability) of the controls (probability of success: 0.98) thedetention of 99.03 % of the passengers is unjustified, they are not terrorists. h
C.5 Random Variables and Distributions
Variables which adopt a particular value with a certain probability are calledrandom variables. They may result, for example, from an experiment. Thus theprobability of having six pips when throwing a die is 1/6. In general such a processcan be described as follows. An experiment was carried out in which a randomvariable X adopted a value x; x is called a realization of X. The universal set is theset of all possible realizations of X (here: x = 1, 2, 3, 4, 5, 6). A sample isunderstood to be the n-fold realization of X.
In case of a die the random variable is discrete. It can at most adopt countablymany values xi. A probability P(X = xi) is assigned to each of these values, the sumof all of them is equal to 1.
If we are dealing with a continuous variable, for example the weights offragments after the explosion of a vessel, we use a distribution function for itsdescription. This function indicates the probability for X B x. Hence we have
642 Appendix C: Basics of Probability Calculations
F xð Þ ¼ P X� xð Þ ðC:15Þ
F(x) is thus defined for all real numbers. F(x) is also called the cumulativedistribution function. If F(x) is differentiable, which normally is the case, weobtain its probability density function (pdf)
f tð Þ ¼ P t�X� tþ dtð Þ ðC:16Þ
Equation (C.16) is the probability for X lying between t and t + dt.By combining Eqs. (C.15) and (C.16) we obtain
F xð Þ ¼Zx
�1
f tð Þdt withZ1
�1
f tð Þdt ¼ 1 ðC:17Þ
Probability distributions are characterised by so-called moments. The firstmoment is the expected value. In case of discrete variables we have
EðXÞ ¼Xn
i¼1
xi � P X ¼ xið Þ ðC:18Þ
and for continuous variables
EðXÞ ¼Z1
�1
t � f tð Þdt ðC:19Þ
Furthermore the variance is used. It is obtained from
V Xð Þ ¼ E X� E Xð Þð Þ2h i
ðC:20Þ
Using Steiner’s theorem Eq. (C.20) becomes
V Xð Þ ¼ E X2� �
� E Xð Þ2 ðC:21Þ
where E X2� �
is the second moment. The square root of the variance is calledstandard deviation, i.e.
S Xð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiV Xð Þ
pðC:22Þ
Example C.6 Expected value and varianceThe expected values and the variance for throws of an ideal die and for an
exponential distribution with parameter k = 1/6 are to be calculated.Note: the probability density function of the exponential distribution is
f tð Þ ¼ k � exp �ktð Þ k; t� 0
Appendix C: Basics of Probability Calculations 643
Solution
Die
• Expected value according to Eq. (C.18)
E Xð Þ ¼X6
i¼1
i � 16¼ 3:5
• Second moment in analogy with Eq. (C.18)
E X2� �
¼X6
i¼1
i2 � 16¼ 15:1667
• Variance according to Eq. (C.21)
V Xð Þ ¼ E X2� �
� E Xð Þ2¼ 15:1667� 3:52 ¼ 2:9167
Exponential distribution
• Expected value according to Eq. (C.19)
EðXÞ ¼Z1
0
t � k � e�kt dt ¼ 1k¼ 6
• Second moment in analogy with Eq. (C.19)
EðX2Þ ¼Z1
0
t2 � k � e�kt dt ¼ 2
k2 ¼ 72
• Variance according to Eq. (C.21)
V Xð Þ ¼ E X2� �
� E Xð Þ2¼ 2
k2 �1
k2 ¼1
k2 ¼ 36
h
In addition to expected value and variance the distribution percentiles are usedto characterize a distribution. The percentiles are values below which a certainfraction of the distribution lies. In use are the 5th, 50th (median) and 95thpercentiles. Using Eq. (C.17) we obtain for continuous random variables
F x�ð Þ ¼Zx�
�1
f tð Þ dt ¼ 1� c2
ðC:23Þ
Equation (C.23) gives for c = 0.9 the 5th respectively the 95-th percentiles andfor c = 0 the median.
644 Appendix C: Basics of Probability Calculations
C.6 Selected Types of Distributions
The exponential distribution was presented in the preceding Section. Thisdistribution is a one-parameter distribution (k). Mathematical statistics uses a largenumber of distributions, which may serve, for example, to describe empirical dataor random processes. Below the probability density functions of several two-parameter distributions are listed, some of which also exist in versions with threeparameters. Details are found in [C-1–C-5].
• Normal distribution
fXðx) ¼ 1
rx
ffiffiffiffiffiffi2pp exp � 1
2x� �xx
rx
� �2" #
�1\x\1 ðC:24Þ
• Truncated normal distribution
fXðx) ¼ 1
r � rx
ffiffiffiffiffiffi2pp exp � 1
2x� �xx
rx
� �2" #
0\x\1 ðC:25Þ
with
r ¼ 1� / � �xx
rx
� �
and / denoting the standard normal distribution• Inverse Gaussian distribution
fXðx) ¼ a2 � p � x3
� �12� exp
�a � x� sð Þ2
2 � s2 � x
!
0� x�1 ðC:26Þ
• Logarithmic normal (lognormal) distribution
fXðx) ¼ 1
xsx
ffiffiffiffiffiffi2pp exp � 1
2ln x� lx
sx
� �2" #
0\x\1 ðC:27Þ
• Gamma distribution
fXðx) ¼ gb
C bð Þ � xb�1 � exp �g � xð Þ x, b, g [ 0 ðC:28Þ
• Inverse gamma distribution
fXðx) ¼ gb
C bð Þ �1x
� �bþ1
� exp �gx
� �x, b, g [ 0 ðC:29Þ
Appendix C: Basics of Probability Calculations 645
• Weibull distribution
fXðx) ¼ g � b � g � xð Þb�1� exp �g � xð Þb x, b, g [ 0 ðC:30Þ
• Log-logistic distribution
fXðx) ¼ d � e�c � x�d�1
1þ e�c � x�dð Þ2x, d[ 0 ðC:31Þ
• Beta distribution
fXðx) ¼ C aþ bð ÞC að Þ � C bð Þ x
a�1 � 1� xð Þb�1 a [ 0; b [ 0; xe 0; 1½ ðC:32Þ
• Rectangular distribution (constant probability density function)
fxðxÞ ¼1
b�a if b� x� a
0 otherwise
ðC:33Þ
• Right-sided triangular distribution
fxðxÞ ¼2�b
b�að Þ2 �2�x
b�að Þ2 if b� x� a
0 if b� x� a
(
ðC:34Þ
• Bivariate lognormal distribution
fX;Y x,yð Þ ¼exp � 1
2� 1�q2ð Þ �ln x�l1
s1
� �2�2 � q ln x�l1
s1
� �� ln y�l2
s2
� �þ ln y�l2
s2
� �2 � �
2 � p � s1 � s2 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2ð Þ
p� x � y
0� x; y\1; s1; s2 [ 0; qj j\1
ðC:35Þ
C.7 Estimation of Parameters
Let the sequence of observations x1, x2, …, xn of a random sample be realizationsof n independent random variables X1, X2,…, Xn, all of which possess the samedistribution; n is called the sample size. The expected value of the distribution isEðXÞ ¼ l. E(X) is estimated by the mean or average value
�x ¼ 1n
Xn
i¼1
xi ðC:36Þ
and the variance V(X) by
646 Appendix C: Basics of Probability Calculations
r2 ¼ 1n� 1
Xn
1¼1
x2i � n�x2
!
ðC:37Þ
Equations (C.36) and (C.37) result from applying the maximum-likelihoodestimation (MLE) to normally distributed variables. The estimation of theparameters of other distributions leads to more complicated systems of equations.Details are found, for example in [C-1, C-3]. An application is given in the nextExample.
Example C.7 Estimation of the parameters of a discrete and a continuousdistribution
In a die game the following numbers of pips appeared:
Calculate the mean value and the variance and compare them with thetheoretical results of Example C.6.
According to Eq. (C.36) the mean value is
�x ¼ 1n
Xn
i¼1
xi ¼1
20� 67 ¼ 3:35
The variance results from Eq. (C.37)
r2 ¼ 1n� 1
Xn
1¼1
x2i � n�x2
!
¼ 2:6605
The corresponding theoretical values are 3.5 and 2.9167. The standarddeviation is r = 1.6311. The circumflex above �x and r2 indicates that we aredealing with an empirical estimator. These estimators take the places in therelationships of the corresponding true but unknown parameters.
When observing the lifetimes of gas vessels the following values were found:
t1 ¼ 800,000 h; t2 ¼ 1,000,000 h; t3 ¼ 650,000 h and t4 ¼ 1,200,000 h
Calculate the failure rate assuming exponentially distributed lifetimes.The failure rate is determined using the maximum-likelihood method, whichrequires the probability density function
f tð Þ ¼ k � e�kt k; t� 0
The likelihood function then is
L ¼ f t1ð Þ � f t2ð Þ � f t3ð Þ � f t4ð Þ
Usually the logarithm of function L is formed and derived with respect to theparameter, k in this case. If the result is set equal to zero, we have the necessary
Appendix C: Basics of Probability Calculations 647
condition for the maximum of the function, from which k is determined.
d ln Ldk¼ 4
k� t1 þ t2 þ t3 þ t4ð Þ
where from
k ¼ 4t1 þ t2 þ t3 þ t4
¼ 1:1 � 10�6 h�1
results. h
C.8 Probability Trees
Based on the methods described above probability calculations for sequences ofevents can be performed, as shown in the following example from [C-2].
Example C.8 Engine damage of a jet planeA rickety jet aeroplane has three engines (A, B, C), which would survive an
overseas flight with the probabilities of P(A) = 0.95, P(B) = 0.96 and P(C) = 0.97.For being capable of flying, the plane needs at least two functioning engines(‘success criterion’). What is the probability that the aeroplane survives theoverseas flight? The corresponding tree structure is shown in Fig. C.2.
+
+
+ -
-
+ -
Root
0.95
0.96
0.97
0.04
0.030,970.03
0.05
-
+
+ -
-
+ -
0.970.030.97 0.03
0.96 0.04
Node
Final node
1st engine
2nd engine
3rd engine
Overseas flight succesful
0.95·0.96·0.97=0.88464
0.95·0.96·0.03=0.02736
0.95·0.04·0.97=0.03648
0.05·0.96·0.97=0.04656
P(success) =0.99542
Crash
0.95·0.04·0.03=0.00114
0.05·0.96·0.03=0.00144
0.05·0.04·0.97=0.00194
0.05·0.04·0.03=0.00006
P(crash) =0.00458
Flight successful
Crash
Fig C.2 Tree structure for treating engine failures of an aeroplane with probabilities (after [C-2])
648 Appendix C: Basics of Probability Calculations
The flight is successful if any one of the following situations occurs:
• engines A and B survive, C fails
PðA \ B \ CÞ ¼ P Að Þ � P Bð Þ � 1� P Cð Þð Þ ¼ 0:02736
• engines B and C survive, A fails
PðB \ C \ AÞ ¼ P Bð Þ � P Cð Þ � 1� P Að Þð Þ ¼ 0:04656
• engines A and C survive, B fails
PðA \ C \ BÞ ¼ P Að Þ � P Cð Þ � 1� P Bð Þð Þ ¼ 0:03686
• all engines survive
PðA \ C \ BÞ ¼ P Að Þ � P Bð Þ � P Cð Þ ¼ 0:88464
Since we are dealing with mutually exclusive events the total probability of asuccessful flight is calculated according to Eq. (C.4), which gives
P successful flightð Þ ¼ 0:99542 and hence P crashð Þ ¼ 0:00458:
h
References
[C-1] Hartung J (1991) Statistik: Lehr- und Handbuch der angewandten Statistik. R. OldenbourgVerlag, München
[C-2] Sachs L (1999) Angewandte Statistik—Anwendung statistischer Methoden. Springer,Heidelberg
[C-3] Härtler G (1983) Statistische Modelle für die Zuverlässigkeitsanalyse. VEB VerlagTechnik, Berlin
[C-4] Abramowitz M, Stegun IA (eds) (1972) Handbook of mathematical functions withformulas, graphs, and mathematical tables. Department of Commerce, Washington
[C-5] Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, vol 2.Wiley, New York
Appendix C: Basics of Probability Calculations 649
Appendix DCoefficients for the TNO Multienergy Modeland the BST Model
Tables D.1 and D.2.
� Springer-Verlag Berlin Heidelberg 2015U. Hauptmanns, Process and Plant Safety,DOI 10.1007/978-3-642-40954-7
651
652 Appendix D: Coefficients for the TNO Multienergy Model and the BST Model
Table D.1 Coefficients for the TNO multienergy model Eq. (10.163) [D-1–D-3]
[D-1] Arizal R (2012) Development of methodology for treating pressure waves from explosionsaccounting for modelling and data uncertainties. Dissertation, Fakultät für Verfahrens- undSystemtechnik, Otto-von-Guericke-Universität Magdeburg
[D-2] Alonso FD, Ferradas EG, Perez JFS, Aznar AM, Gimeno JR, Alonso JM (2006)Characteristic overpressure-impulse-distance curves for the detonation of explosives,pyrotechnics or unstable substances. J Loss Prev Process Ind 19:724–728
[D-3] Assael MJ, Kakosimos KE (2010) Fires, explosions, and toxic gas dispersions: effectcalculation and risk analysis. CRC Press Taylor & Francis Group, New York
[D-4] Det Norske Veritas (DNV) London, PHAST software version 6.7
654 Appendix D: Coefficients for the TNO Multienergy Model and the BST Model
of gas (vapour), 33–34of dust, 47–49, 297of an explosive, 49–57Explosion effectsfuel gas and explosive, 533–550, 602–609physical (BLEVE), 550–559dust, 559–561
Explosion energy, 51–54, 108Explosion limits (LEL and UEL)
accidents to be prevented, 119batch, 71–81, 129–137continuous stirred tank reactor, 80–82,
120–129, 223–228cooling (HAZOP analysis), 304–306cooling control (LOPA analysis), 313–315emergency discharge, 115–117, 428-437failure of stirrer and cooling control (fault
tree analysis), 414–427hazard potential after the Dow Index,
299–301reduction of inventory for reducing the
hazard potential, 537–538upgrading (retrofit) for satisfying SIL
requirements, 594–595
662 Index
Reactor (cont.)trip system of an injector reactor (fault treeanalysis), 406–410tubular flow, 82–85
Boolean representation, 345–356definition, 286dependent failures, 378–387failure mode and effect analysis, 306–309fault tree analysis, 316, 323–325Hazard and operability study (HAZOP),
301–306increase of availability, 356–378Layer of protection analysis (LOPA),