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International Journal on Mechanical Engineering and Robotics (IJMER) ______

Benefit-Function of Two- Identical Cold Standby Nuclear Reactors System subject to failure due to radioactivity or Overheating, , fire, and meltdown

Ashok Kumar Saini BLJS COLLEGE, TOSHAM (BHIWANI) HARYANA INDIA Email ID [email protected]

The accident killed 30 people directly and damaged Abstract- A nuclear and radiation accident is defined by approximately $7 billion of property. A study published in the International Atomic Energy Agency as "an event that 2005 estimates that there will eventually be up to 4,000 has led to significant consequences to people, the additional deaths related to the accident among environment or the facility." Examples include lethal those exposed to significant radiation levels. Radioactive effects to individuals, large radioactivity release to fallout from the accident was concentrated in areas of the environment, or reactor core melt." The prime Belarus, Ukraine and Russia. Approximately 350,000 example of a "major nuclear accident" is one in which people were forcibly resettled away from these areas soon a reactor core is damaged and significant amounts after the accident. of radioactivity are released, such as in the in 1986. Benjamin K. Sovacool has reported that worldwide there have been 99 accidents at plants from 1952 The impact of nuclear accidents has been a topic of debate to 2009 (defined as incidents that either resulted in the loss practically since the first nuclear reactors were constructed of human life or more than US$50,000 of property damage, in 1954. It has also been a key factor in public concern the amount the US federal government uses to define about nuclear facilities. Some technical measures to reduce major that must be reported), totaling the risk of accidents or to minimize the amount of US$20.5 billion in property damages. Fifty-seven accidents radioactivity released to the environment have been have occurred since the Chernobyl disaster, and almost adopted. Despite the use of such measures, human two-thirds (56 out of 99) of all nuclear-related accidents error remains, and "there have been many accidents with have occurred in the US. There have been comparatively varying impacts as well near misses and incidents". few fatalities associated with Worldwide there have been 99 accidents at nuclear power accidents. plants. Fifty-seven accidents have occurred since the In this paper we have taken failure due to radioactivity or Chernobyl disaster, and 57% (56 out of 99) of all nuclear- Overheating, steam explosion, fire, and meltdown. When related accidents have occurred in the USA. the main unit fails due to Overheating, steam explosion, Serious nuclear power plant accidents include fire, and meltdown then cold standby system becomes the Fukushima Daiichi nuclear disaster (2011), Chernobyl operative. Overheating, steam explosion, fire, and disaster (1986), Three Mile Island accident (1979), and meltdown cannot occur simultaneously in both the units the SL-1 accident (1961). Nuclear advocate Stuart Arm and after failure the unit undergoes very costly repair maintains that, "apart from Chernobyl, no nuclear facility immediately. Applying the regenerative point workers or members of the public have ever died as a technique with renewal process theory the various result of exposure to radiation due to a commercial nuclear reliability parameters MTSF, Availability, Busy period, reactor incident." Benefit-Function analysis have been evaluated. Nuclear-powered submarine core meltdown and other Keywords: Cold Standby, Overheating, steam explosion, mishaps include the K-19 (1961), K-11 (1965), K- fire, and meltdown, first come first serve, MTSF, 27 (1968), K-140 (1968), K-429 (1970), K-222 (1980), K- Availability, Busy period, Benefit -Function. 314 (1985), and K-431 (1985). Serious radiation accidents include the disaster, , radiotherapy INTRODUCTION accident in Costa Rica, radiotherapy accident in Serious radiation and other accidents and incidents Zaragoza, radiation accident in Morocco, Goiania include: 1940s accident, radiation accident in Mexico City, radiotherapy unit accident in Thailand, and the Mayapuri radiological  May 1945: Albert Stevens was the subject of accident in India. a human radiation experiment, and was injected The International Atomic Energy Agency maintains a with without his knowledge or informed website reporting recent accidents. consent. Although Stevens was the person who One of the worst nuclear accidents to date was received the highest dose of radiation during the the Chernobyl disaster which occurred in 1986 in Ukraine. plutonium experiments, he was neither the first nor ______ISSN (Print) : 2321-5747, Volume-2, Issue-6,2014 26 International Journal on Mechanical Engineering and Robotics (IJMER) ______

the last subject to be studied. Eighteen people aged a conservative estimate; 270,000 people were 4 to 69 were injected with plutonium. Subjects who exposed to dangerous radiation levels. Over thirty were chosen for the experiment had been small communities were removed from Soviet diagnosed with a terminal disease. They lived from maps between 1958 and 1991. (INES level 6) 6 days up to 44 years past the time of their October 1957: Windscale fire, UK. Fire ignites injection. Eight of the 18 died within 2 years of the plutonium piles and contaminates surrounding injection. All died from their preexisting terminal dairy farms. An estimated 33 cancer deaths. illness, or cardiac illnesses. None died from the plutonium itself. Patients from Rochester, Chicago,  1957-1964: Rocketdyne located at the Santa and Oak Ridge were also injected with plutonium Susanna Field Lab, 30 miles north of Los Angeles, in the Manhattan Project human experiments. California operated ten experimental nuclear reactors. Numerous accidents occurred including a  6–9 August 1945: On the orders of President Harry core meltdown. Experimental reactors of that era S. Truman, a -gun design bomb, Little were not required to have the same type of Boy, was used against the city of , containment structures that shield modern nuclear Japan. Fat Man, a plutonium implosion-design reactors. During the Cold War time in which the bomb was used against the city of Nagasaki. The accidents that occurred at Rockedyne, these events two weapons killed approximately 120,000 to were not publicly reported by the Department of 140,000 civilians and military personnel instantly Energy. 10/02/1958 - , Former Soviet and thousands more have died over the years Union. accident in SCR plant. from radiation sickness and related . Conducted experiments to determine the of in a cylindrical container  August 1945: at US Los with different concentrations of uranium in Alamos National Laboratory. Harry K. Daghlian, solution. Staff broke the rules and instructions for Jr., dies. May 1946: Criticality accident at Los working with YADM (nuclear ). Alamos National Laboratory. Louis Slotindies. When SCR personnel received doses from 7600 to 1950s 13,000 rem. Three people died, one man got  February 13, 1950 : a Convair B-36B crashed in radiation sickness and went blind. northern British Columbia after jettisoning a Mark  December 30, 1958: Cecil Kelley criticality IV atomic bomb. This was the first such nuclear accident at Los Alamos National Laboratory. weapon loss in history.  March 1959: Santa Susana Field Laboratory, Los  December 12, 1952: NRX AECL Chalk River Angeles, California. Fire in a fuel processing Laboratories, Chalk River, Ontario, Canada. Partial facility. meltdown, about 10,000 Curies released. Approximately 1202 people were  July 1959: Santa Susana Field Laboratory, Los involved in the two year cleanup. President Jimmy Angeles, California. Partial meltdown. 1960s Carter was one of the many people that helped clean up the accident. 15/03/1953 – Mayak, Former  7 June 1960: the 1960 Fort Dix IM-99 . Criticality accident. Contamination accident destroyed a CIM-10 Bomarc nuclear of plant personnel occurred. missile and shelter and contaminated the BOMARC Missile Accident Site in New  1954: The 15 Mt Castle Bravo shot of 1954 which Jersey. spread considerable on many Pacific islands, including several which were  24 January 1961: the 1961 Goldsboro B-52 inhabited, and some that had not been evacuated. crash occurred near Goldsboro, North Carolina. A B-52 Stratofortress carrying two Mark  March 1, 1954: Daigo Fukuryū Maru, 1 fatality. 39 nuclear bombs broke up in mid-air, dropping its nuclear payload in the process.  September 1957: a plutonium fire occurred at the Rocky Flats Plant, which resulted in  accident. Eight fatalities and more than 30 people the contamination of Building 71 and the release of were over-exposed to radiation. plutonium into the atmosphere, causing US $818,600 in damage.  March, 21 -August 1962: radiation accident in Mexico City, four fatalities.  21/04/1957 - Mayak, Former Soviet Union. Criticality accident in the factory number 20 in the  May 1962: The was a 13-day collection oxalate decantate after filtering sediment confrontation in October 1962 between the Soviet oxalate enriched uranium. Six people received Union and Cuba on one side and the United doses of 300 to 1,000 rem (four women and two States on the other side. The crisis is generally men), one woman died. September 1957: Kyshtym regarded as the moment in which the Cold disaster: Nuclear waste storage tank explosion at Warcame closest to turning into a nuclear Chelyabinsk, Russia. 200+ fatalities, believed to be conflict and is also the first documented instance ______ISSN (Print) : 2321-5747, Volume-2, Issue-6,2014 27 International Journal on Mechanical Engineering and Robotics (IJMER) ______

of mutual assured destruction (MAD) being  10/12/1968 - Mayak, Former Soviet Union. discussed as a determining factor in a major Criticality accident. Plutonium solution was poured international arms agreement. into a cylindrical container with dangerous geometry. One person died, another took a high  1964, 1969: Santa Susana Field Laboratory, Los dose of radiation and radiation sickness, after Angeles, California. Partial meltdowns. which he had two legs and his right arm amputated.  1965 Philippine Sea A-4 crash, where  : reactor in a Skyhawk attack aircraft with a undergoes partial core meltdown leading to fell into the sea. The pilot, the aircraft, and the B43 massive radioactive contamination of a cavern. nuclear bomb were never recovered. It was not 1970s until the 1980s that the Pentagon revealed the loss of the one-megaton bomb.  1974–1976: Columbus radiotherapy accident, 10 fatalities, 88 injuries from Cobalt-60 source. July  October 1965: US CIA-led expedition abandons a 1978: Anatoli Bugorski was working on U-70, the nuclear-powered telemetry relay listening device largest Soviet particle accelerator, when he on Nanda Devi accidentally exposed his head directly to the  January 17, 1966: the 1966 Palomares B-52 beam. He survived, despite suffering some long- crash occurred when a B-52G bomber of term damage. the USAF collided with a KC-135  July 1979: Church Rock Uranium Mill tanker during mid-air refuelling off the coast Spill in New Mexico, USA, when United Nuclear of Spain. The KC-135 was completely destroyed Corporation's uranium mill tailings disposal pond when its fuel load ignited, killing all four crew breached its dam. Over 1,000 tons of radioactive members. The B-52G broke apart, killing three of mill waste and millions of gallons of mine effluent the seven crew members aboard. Of the flowed into the Puerco River, and contaminants fourMk28 type hydrogen bombs the B-52G traveled downstream. 1980s carried, three were found on land near Almería, Spain. The non-nuclear explosives in two of the  1980: Houston radiotherapy accident, 7 fatalities. weapons detonated upon impact with the ground, resulting in the contamination of a 2-square-  October 5, 1982: Lost radiation source, Baku, kilometer (490-acre) (0.78 square mile) area Azerbaijan, USSR. 5 fatalities, 13 injuries. by radioactive plutonium. The fourth, which fell  March 1984: Radiation accident in Morocco, eight into the Mediterranean Sea, was recovered intact fatalities from overexposure to radiation from a after a 2½-month-long search. January 21, 1968: lost iridium-192source. the 1968 Thule Air Base B-52 crash involved a United States Air Force (USAF) B-52 bomber.  1984: Fernald Feed Materials Production The aircraft was carrying four hydrogen Center gained notoriety when it was learned that bombs when a cabin fire forced the crew to the plant was releasing millions of pounds of abandon the aircraft. Six crew members ejected uranium dust into the atmosphere, causing major safely, but one who did not have an ejection radioactive contamination of the surrounding areas. seat was killed while trying to bail out. The bomber That same year, employee Dave Bocks, a 39 year crashed onto sea ice in Greenland, causing the old pipefitter, disappeared during the facility's nuclear payload to rupture and disperse, which graveyard shift and was later reported missing. resulted in widespread radioactive contamination. Eventually, his remains were discovered inside a uranium processing furnace located in Plant 6.  May 1968: Soviet submarine K-27 reactor near meltdown. 9 people died, 83 people were  August 1985: Soviet submarine K-431 accident. injured. In August 1968, the Project 667 A - Ten fatalities and 49 other people suffered Yankee class nuclear submarine K-140 was in the radiation injuries. naval yard at Severodvinsk for repairs. On August  October 1986: Soviet submarine K-219 reactor 27, an uncontrolled increase of the reactor's power almost had a meltdown. Sergei Preminin died after occurred following work to upgrade the vessel. he manually lowered the control rods, and stopped One of the reactors started up automatically when the explosion. The submarine sank three days later. the control rods were raised to a higher position. Power increased to 18 times its normal amount,  September 1987: Goiania accident. Four fatalities, while pressure and temperature levels in the reactor and following radiological screening of more than increased to four times the normal amount. The 100,000 people, it was ascertained that 249 people automatic start-up of the reactor was caused by the received serious radiation contamination from incorrect installation of the electrical exposure to Cesium-137. In the cleanup cables and by operator error. Radiation levels operation, topsoil had to be removed from several aboard the vessel deteriorated. sites, and several houses were demolished. All the

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objects from within those houses were removed  August 2000 – March 2001: Instituto Oncologico and examined. Time magazine has identified the Nacional of Panama, 17 fatalities. Patients accident as one of the world's "worstnuclear receiving treatment for prostate cancer and cancer disasters" and the International Atomic Energy of the cervix receive lethal doses of radiation. Agency called it "one of the world's worst radiological incidents".  August 9, 2004: Mihama Nuclear Power Plant accident, 4 fatalities. Hot water and steam  1989: San Salvador, El Salvador; one fatality due leaked from a broken pipe (not actually a radiation to violation of safety rules at Cobalt-60 irradiation accident). facility. 1990s  9 May 2005: it was announced that Thermal Oxide  1990: Soreq, Israel; one fatality due to violation of Reprocessing Plant in the UK suffered a large leak safety rules at Cobalt-60 irradiation facility. of a highly radioactive solution, which first started December 16 - 1990: radiotherapy accident in in July 2004. Zaragoza. Eleven fatalities and 27 other patients were injured.  April 2010: Mayapuri radiological accident, India, one fatality after a cobalt-60 research irradiator was  1991: Neswizh, Belarus; one fatality due to sold to a scrap metal dealer and dismantled. violation of safety rules at Cobalt-60 irradiation facility.  2010s  1992: Jilin, China; three fatalities at Cobalt-  March 2011: Fukushima I nuclear accidents, Japan 60 irradiation facility. and the radioactive discharge at the Fukushima Daiichi Power Station.  1992: USA; one fatality.  January 17, 2014: At the Rössing Uranium Mine,  April 1993: accident at the Tomsk-7 Reprocessing Namibia, a catastrophic structural failure of a leach Complex, when a tank exploded while being tank resulted in a major spill. The France-based cleaned with nitric acid. The explosion released a laboratory, CRIIAD, reported elevated levels of cloud of radioactive gas. (INES level 4). radioactive materials in the area surrounding the mine. Workers were not informed of the dangers of  1994: Tammiku, Estonia; one fatality from working with radioactive materials and the health disposed caesium-137 source. effects thereof.  August — December 1996: Radiotherapy accident  February 1, 2014: Designed to last tens thousand in Costa Rica. Thirteen fatalities and 114 other years, the Waste Isolation Pilot Plant (WIPP) site patients received an overdose of radiation. had its first leak of airborne radioactive  1996: an accident at Pelindaba research facility in materials. 140 employees working underground at South Africa results in the exposure of workers to the time were sheltered indoors. 13 of these tested radiation. Harold Daniels and several others die positive for internal radioactive contamination. from cancers and radiation burns related to the Internal exposure to radioactive isotopes is more exposure. serious than external exposure, as these particles lodge in the body for decades, irradiating the  June 1997: Sarov, Russia; one fatality due to surrounding tissues, thus increasing the risk of violation of safety rules. future cancers and other health effects. A second  May 1998: The Acerinox accident was an incident leak at the plant occurred shortly after the first, of radioactive contamination in Southern Spain. releasing plutonium and other radiotoxins, causing A caesium-137 source managed to pass through the concern for communities living near the repository. monitoring equipment in an Acerinox scrap Stochastic behavior of systems operating under metal reprocessing plant. When melted, the changing environments has widely been studied. . caesium-137 caused the release of a radioactive Dhillon , B.S. and Natesan, J. (1983) studied an outdoor cloud. power systems in fluctuating environment . Kan Cheng  September 1999: two fatalities at criticality (1985) has studied reliability analysis of a system in a accident at Tokaimura nuclear accident (Japan) randomly changing environment. Jinhua Cao (1989) has 2000s studied a man machine system operating under changing environment subject to a Markov process with two  January–February 2000: Samut Prakan radiation states. The change in operating conditions viz. accident: three deaths and ten injuries resulted fluctuations of voltage, corrosive atmosphere, very low in Samut Prakarn when acobalt-60 radiation- gravity etc. may make a system completely inoperative. therapy unit was dismantled. Severe environmental conditions can make the actual mission duration longer than the ideal mission duration.  May 2000: Meet Halfa, Egypt; two fatalities due In this paper we have taken failure due to radioactivity to radiography accident. or Overheating, steam explosion, fire, and meltdown. ______ISSN (Print) : 2321-5747, Volume-2, Issue-6,2014 29 International Journal on Mechanical Engineering and Robotics (IJMER) ______

When the main operative unit fails then cold standby Symbols for states of the System system becomes operative. Overheating, steam Superscripts O, CS, RAF, OSFMF explosion, fire, and meltdown failure cannot occur simultaneously in both the units and after failure the unit Operative, Cold Standby, failure due to radioactivity or undergoes repair facility of very high cost in case of Failure due to Overheating, steam explosion, fire, and Overheating, steam explosion, fire, and meltdown meltdown respectively immediately. The repair is done on the basis of first fail Subscripts nosfm, osfm, raf, ur, wr, uR first repaired. Assumptions No Overheating, steam explosion, fire, and meltdown, Overheating, steam explosion, fire, and meltdown, 1. 1 , 2, are constant failure rates for failure due to failure due to radioactivity, under repair, waiting for radioactivity or Overheating, steam explosion, repair, under repair continued from previous state fire, and meltdown respectively. The CDF of respectively repair time distribution of Type I and Type II are Up states – 0, 1, 2, 7, 8 ; G1(t) and G2(t). Down states – 3, 4, 5, 6 2. The failure due to Overheating, steam explosion, fire, and meltdown is non-instantaneous and it regeneration point – 0,1,2, 7, 8 cannot come simultaneously in both the units. States of the System 3. The repair starts immediately after the failure due 0(O , CS ) to radioactivity or Overheating, steam explosion, nosfm nosfm fire, and meltdown works on the principle of first One unit is operative and the other unit is cold standby fail first repaired basis. and there is no failure due to Overheating, steam explosion, fire, and meltdown in both the units. 4. The repair facility does no damage to the units

and after repair units are as good as new. 1(RAFraf, ur , Onosfm) 5. The switches are perfect and instantaneous. The operating unit fails due to radioactivity and is under repair immediately of very costly Type- I and 6. All random variables are mutually independent. standby unit starts operating with no failure due to 7. When both the units fail, we give priority to Overheating, steam explosion, fire, and meltdown. operative unit for repair. 2(OSFMF osfm, ur , Onosfm) 8. Repairs are perfect and failure of a unit is The operative unit fails due to OSFMF resulting from detected immediately and perfectly. failure due to Overheating, steam explosion, fire, and 9. The system is down when both the units are non- meltdown and undergoes repair of type II and the operative. standby unit becomes operative with no failure due to Overheating, steam explosion, fire, and meltdown. Notations 3(RAFraf,uR , OSFMF osfm,wr) 1 , 2 are the failure rates due to radioactivity or Overheating, steam explosion, fire, and meltdown The first unit fails due to radioactivity and under very respectively. G1(t), G2(t) – repair time distribution Type costly Type-1repair is continued from state 1 and the -I, Type-II due to radioactivity or Overheating, steam other unit fails due to OSFMF resulting from Failure due explosion, fire, and meltdown respectively. to Overheating, steam explosion, fire, and meltdown and is waiting for repair of Type -II. p, q - probability of failure due to radioactivity ; Overheating, steam explosion, fire, and meltdown 4(RAF raf,uR , RAF raf,wr) respectively such that p+ q=1 The repair of the unit is failed due to RAF resulting from Mi(t) System having started from state i is up at time t failure due to radioactivity is continued from state 1and without visiting any other regenerative state the other unit failed due to RAF resulting from failure due to radioactivity is waiting for repair of Type-I. Ai (t) state is up state as instant t 5(OSFMF osfm, uR , OSFMF osfm, wr) Ri (t) System having started from state i is busy for repair at time t without visiting any other regenerative The operating unit fails due to failure due to state. Overheating, steam explosion, fire, and meltdown (OSFMF mode) and under repair of Type - II continues B (t) the server is busy for repair at time t. i from the state 2 and the other unit fails also due to

Hi(t) Expected number of visits by the server for failure due to Overheating, steam explosion, fire, and repairing given that the system initially starts from meltdown is waiting for repair of Type- II. regenerative state i 6(OSFMF osfm,uR , RAF raf,wr)

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The operative unit fails due to OSFMF resulting from Making use of relations (1) & (2) it can be shown that * failure due to Overheating, steam explosion, fire, and ø0 (0) =1 , which implies that ø0 (t) is a proper meltdown and under repair continues from state 2 of distribution. Type –II and the other unit is failed due to RAF resulting from failure due to radioactivity and under very costly Type-1 MTSF = E[T] = (s) 7(O nosfm , RAF raf,ur) s=0 The repair of the unit failed due to RAF resulting from = (D ’(0) - N ’(0)) / D (0) failure due to radioactivity failure is completed and 1 1 1 there is no failure due to Overheating, steam explosion, = ( +p01 + p02 ) / (1 - p01 p10 - p02 p20 ) fire, and meltdown and the other unit is failed due to RAF resulting from failure due to radioactivity is under where repair of very costly Type-1 휇0 = 휇01+ 휇02 ,

8(O nosfm , OSFMF osfm,ur) (4) (3) 휇1 = 휇01 + 휇17 + 휇18 , The repair of the unit failed due to RAF resulting from (6) (5) 휇2 = 휇02+휇27 + 휇28 failure due to radioactivity failure is completed and there is no failure due to Overheating, steam explosion, fire, Availability analysis and meltdown and the other unit is failed due to OSFMF Let Mi(t) be the probability of the system having started resulting from failure due to Overheating, steam from state i is up at time t without making any other explosion, fire, and meltdown is under repair of Type-II. regenerative state. By probabilistic arguments, we have Transition Probabilities − t − t M0(t) = 푒 1 푒 2 , Simple probabilistic considerations yield the following M (t) =p G (t) e - (  +  ) = M (t) expressions: 1 1 1 2 7 - (   ) M2(t) =q G2(t) e 1+ 2 = M8(t) p01 = p, p02 = q, The point wise availability A (t) have the following p = pG *(  )+q G *(  )= p i 10 1 1 1 2 70 , recursive relations * * p20 = pG2 ( 1)+q G2 ( 2)= p80 , A0(t) = M0(t) + q01(t)[c]A1(t) + q02(t)[c]A2(t) (3) * (4) (5) * p11 = p(1- G1 ( 1))= p14 = p71 p28 = q(1- G2 ( 2))= (5) A1(t) = M1(t) + q10(t)[c]A0(t) + p25 = p82 (1) q (3)(t)[c]A (t)+ q (4)(t)[c]A (t) We can easily verify that 18 8 17 7 (4) (3) A2(t) = M2(t) + q20(t)[c]A0(t) + p01 + p02 = 1, p10 + p17 (= p14) + p18 (=p13 ) = 1, (5) (6) (5) (6) [q28 (t)[c] A8(t) + q27 (t)] [c]A7(t) p80 + p82 + p87 = 1 (2) A (t) = M (t) + q (t)[c]A (t) + And mean sojourn time is 7 7 70 0 (4) (3) [q71 (t)[c] A1(t) + q78 (t)] [c]A8(t) A8(t) = M8(t) + µ0 = E(T) = q 80 (t)[c]A 0 (t) +[q (5)(t)[c] A (t) + q (6)(t)] [c]A (t) (7-11) Mean Time To System Failure 82 2 87 7 Taking Laplace Transform of eq. (7-11) and solving for Ø0(t) = Q01(t)[s] Ø1(t) + Q02(t)[s] Ø2(t)

Ø1(t) = Q10 (t)[s] Ø0(t) + Q13(t) + Q14(t) = N (s) / D (s) (12) Ø2(t) = Q20 (t)[s] Ø0(t) + Q25(t) + Q26(t) (3-5) 2 2 We can regard the failed state as absorbing where

(3) (6) (5) Taking Laplace-Stiljes transform of eq. (3-5) and N2(s) = 0 (1 - 78 - 87 )- 82 solving for (6) (3) (5) (4) * ( 27 78 + 28 - 71 ø0 (s) = N1(s) / D1(s) (6) (4) (6) (3) (4) (5) where ( 17 + 87 18 )+ 71 82 * * * * * (4) (6) (3) N1(s) = Q01 [ Q13 (s) + Q14 (s) ] + Q02 [ Q25 (s) + Q26 ( 17 - 27 18 )]+ 01[ 1(1 – * (s) ] (3) (6) (4) (3) * * * * 78 87 ) + 71 ( 7 + 78 D1(s) = 1 - Q01 Q10 - Q02 Q20 (3) (6) 8)+ 18 ( 7 87 - 8)-

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(5) (6) (3) (5) 82 ( 1( 27 78 + 28 )+ So that (15)

(4) (3) (5) 17 (- 2( 78 + 7 28 )- (3) (6) (3) 18 ( 2+ 7 27 )}] 02[ 2(1 – 78 The expected busy period of the server when there is (6)) + (6)( OSFMF - Failure due to Overheating, steam explosion, 87 27 fire, and meltdown or RAF- failure due to Failure due (3) (5) (6) (4) 7 + 78 8)+ 28 ( 7 87 + 8) - 71 ( to radioactivity in (0,t] (6) (5) 1(- 27 - 28 + R0(t) = q01(t)[c]R1(t) + q02(t)[c]R 2(t) (6) (4) (5) (3) (6) R (t) = S (t) + q (t)[c]R (t) + 87 )+ 17 ( 2+ 28 8)- 18 (- 2 87 + 1 1 10 0 (6) (3) (4) 8 27 )}] q18 (t)[c] R8 (t) + q17 (t)[c]R7(t) (5) (3) (6) R2(t) = S2(t) + q20(t)[c]R0(t) + q28 (t) 18 ( 2+ 7 27 )}] (6) (3) (6) (5) R8(t) +q27 (t)][c]R7(t) D2(s) = (1 - 78 - 87 ) - 82 ( (4) R7(t) = S7(t) + q70(t)[c]R0(t) + Q71 (t) (6) (3) (5) (4) 27 78 + 28 )- 71 (3) R1(t) +q78 (t)][c]R8(t) ( (4)+ (6) (3))+ (4) (5) ( (4) (5)- (5) 17 87 18 71 82 17 28 R8(t) = S8(t) + q80(t)[c]R0(t) + Q82 (t) (3) 18 )]+ 01[- 10 (1 – (6) R2(t) +q87 (t)][c]R7(t) (16-20) (3) (6) (4) (3) 78 87 ) - 71 ( 70+ 78 Taking Laplace Transform of eq. (16-20) and solving for

(3) (6) 80)- 18 ( 70 87 - 80 )-

(5) (6) (3) (5) = N3(s) / D2(s) (21) 82 ( - 10( 27 78 + 28 )+

(4) (3) (5) where 17 ( 20 ( 78 - 70 28 )+ (3) (6) (3) (6) (3) (6) N 3(s) = 01[ 푆1(1 – 78 87 ) + 18 ( 20+ 70 27 )}] 02[- 20(1 – 78 87 ) (6) (3) (5) (6) (4) (3) (3) - 27 ( 70 + 78 80 ) - 28 ( 70 87 + 80 ) - 71 ( 푆7 + 78 푆8)+ 18 ( 푆7 (4) 71 ( (6) (5) (6) (3) (5) 87 - 푆 8)]- 01 82 ( 푆 1 27 78 + 28 )+ (6) (5) (6) (4) (4) (3) (5) 10 ( 27 + 28 87 )- 17 ( 20- 17 (푆2 78 + 푆7 28 )-

(5) (3) (6) (3) (6) 28 80 )- 18 ( 20 87 + 80 18 ( 푆2+ 푆7 27 )]+ 02[푆2(1 –

(6) (3) (6) (6) (3) (5) 27 )}] 78 87 ) + 27 ( 푆7 + 78 푆8)+ 28 ( 푆7 (6)+ 푆 ) - (4)( 푆 (- (Omitting the arguments s for brevity) 87 8 02 71 1 (6) (5) (6) (4) (5) (3) The steady state availability 27 - 28 87 17 (푆 2+ 28 푆 8)- 18 (-푆 2 (6) (6) 87 + 푆 8 27 )] A0 = and = = D 2(s) is already defined. Using L’ Hospitals rule, we get (Omitting the arguments s for brevity)

In the long run, R = (22) A0 = = (13) 0

The expected up time of the system in (0,t] is The expected period of the system under OSFMF - failure due to Overheating, steam explosion, fire, and (t) = meltdown or RAF- Failure due to radioactivity in (0,t] is So that (14) (t) = So that The expected down time of the system in (0,t] is The expected number of visits by the repairman for (t) = t- (t) repairing the identical units in (0,t]

H0(t) = Q01(t)[s][1+ H1(t)] + ______ISSN (Print) : 2321-5747, Volume-2, Issue-6,2014 32 International Journal on Mechanical Engineering and Robotics (IJMER) ______

Q02(t)[s][1+ H2(t)] CONCLUSION (3) H1(t) = Q10(t)[s]H0(t)] + Q18 (t)[s] After studying the system , we have analyzed (4) graphically that when the failure rate due to radioactivity H8(t) + Q17 (t)] [s]H7(t) , or Failure due to Overheating, steam explosion, fire, and (5) H2(t) = Q20(t)[s]H0(t) + Q28 (t) [s] meltdown increases, the MTSF and steady state H (t) +Q (6)(t)] [c]H (t) availability decreases and the Benefit-function decreased 8 27 7 as the failure increases. (4) H7(t) = Q70(t)[s]H0(t) + Q71 (t) [s] (3) REFERENCES H1(t) +Q78 (t)] [c]H8(t) (5) [1] Dhillon, B.S. and Natesen, J, Stochastic Anaysis H8(t) = Q80(t)[s]H0(t) + Q82 (t) [s] of outdoor Power Systems in fluctuating (6) H2(t) +Q87 (t)] [c]H7(t) (23-27) environment, Microelectron. Reliab. ,1983; 23, 867-881. Taking Laplace Transform of eq. (23-27) and solving for [2] Kan, Cheng, Reliability analysis of a system in a

randomly changing environment, Acta Math. Appl. Sin. 1985, 2, pp.219-228. = N4(s) / D3(s) (28) In the long run , H = N (0) / D ’(0) (29) [3] Cao, Jinhua, Stochatic Behaviour of a Man 0 4 3 Machine System operating under changing Benefit- Function Analysis environment subject to a Markov Process with The Benefit-Function analysis of the system considering two states, Microelectron. Reliab. ,1989; 28, pp. mean up-time, expected busy period of the system under 373-378. failure due to radioactivity or Failure due to [4 Barlow, R.E. and Proschan, F., Mathematical Overheating, steam explosion, fire, and meltdown, theory of Reliability, 1965; John Wiley, New expected number of visits by the repairman for unit York. failure. [5] Gnedanke, B.V., Belyayar, Yu.K. and Soloyer , The expected total Benefit-Function incurred in (0,t] is A.D. , Mathematical Methods of Relability C(t) = Expected total revenue in (0,t] Theory, 1969 ; Academic Press, New York. - expected busy period of the system under failure due to radioactivity or failure due to Overheating, steam explosion, fire, and meltdown for repairing the units in (0,t ] - expected number of visits by the repairman for repairing of identical the units in (0,t] The expected total cost per unit time in steady state is

C = =

= K1A0 - K 2R0 - K 3H0 where

K1 - revenue per unit up-time,

K2 - cost per unit time for which the system is under repair of type- I or type- II

K3 - cost per visit by the repairman for units repair. Fig. The State Space Diagram Up state down state  Regeration point

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