Energy Engineering - RELIABILITY, SAFETY AND RISK ANALYSIS C

Chapter 10 - Exercises collection

Divided by topic

Chapter 10 Markov chains 10.1 Nuclear steam supply system A nuclear steam supply system has two turbo -generator units; unit l operates and unit 2 is in standby whenever both are good. The units have a constant MTTF of λi-1=l and 2, during active operation while during standby unit 2 has a MTTF of (λ2*)-1. The repair of a unit is assumed to begin instantaneously after it fails, but its duration is random so that the instantaneous repair rates will be μ1 and μ2 , respectively. The repairs can be done on only one unit at a time and any unit under repair will remain so until the task is completed. l. Draw the system diagram. 2. Write the Markov equations. 10.2 Alarm system An alarm system is subject to both unrevealed (u) and revealed (r) faults each of which have time to occurrence which are exponentially distributed with mean values of 200 h and 100 h respectively. lf a revealed fault occurs, then the complete system is restored to the time -zero condition by a repair process which has exponentially distributed times to completion with a mean value of 10 h. lf an unrevealed fault occurs, then it remains in existence until a revealed fau1t occurs when it is repaired along with the revealed fault . l. What is the asymptotic unavai1ability of the alarm system? 2. What is the mean number of system failures in a total time of 1,000 h? 10.3 Two unit standby system Consider a two -unit standby system. The failure rate of the operating unit is λ1 and that of the standby unit is λ2 . Both components are operative at time 0. l. Find the reliability of the system. 2. lf, in addition, the system is exposed to a hazard, characterized by a Poisson process with parameter λ12, that is detrimental to both components, develop an expression for the reliability of the system. 3. Find the M TTF in (l) and (2). 4. Assume now that λ1=λ, λ2=λ 12=0 (cold standby, no common cause failures), and that there is one repairman (repair rate μ). Develop expressions for the steady -state availability, the reliability function, the failure intensity and the mean time to first system failure. 10.4 Item with partial repair after first failure A new item starts operating on line. When it fails (failure rate λ1) a partial repair is performed (repair rate μP) which enables the item to continue operation, but with a new failure rate λ2> λ1 . When it fails for the second time, a thorough repair (repair rate μT< μP) restores the item to the as -good -as-new state and the cycle is repeated. l. Find the asymptotic unavailability. 2. How can you get the familiar expression for a single item under exponential failure and repair? 3. What is the asymptotic failure intensity? 10.5 Two unit standby system Consider a two unit standby system, with failure rate λa and λb during active operation and λb+ in the standby mode in which there is a switching failure probability p. l. Draw the transition diagram. 2. Write the Markov equations. 3. Solve for the system reliability. 4. Reduce the reliability to the situation in which the units are identica l, λa = λb = λb+ = λ 10.6 Water chlorination system The water chlorination system of a small town has two separate pipelines, each with a pump that supplies chlorine to the water at prescribed rates. The two pumps are denoted A and B respectively. During normal operation both pump s are functioning and thus are sharing the load. In this case each pump is operated on approximately 60% of its capacity (cap% = 0.60). When one of the pumps fails, the corresponding pipeline is closed down, and the other pump has to supply chlor ine at a higher rate. In this case the single pump is operated at full capacity (cap% = 1.00). We assume that the pumps have the following constant failure rates: λcap% =cap%·6.3 y-1 Assume that the probability of common cause failures is negligible. Repair is initiated as soon as one of the pumps fails. The mean time to repair a pump has been estimated to be eight hours, and the pump is put into operatio n again as soon as the repair is co mpleted. Repairs are carried out independent of each other (i.e., maintenance crew is not a li miting factor). If both pu mps are in a failed state, unchlorinated water will be supplied to the custo mers. Both pu mps are assu med to be functioning at ti me t=0. l. Define the possible syste m states, and establish a state space (Markov) diagra m for the syste m. 2. Write down the corresponding state equations on matrix for mat. 3. Determine the steady state probabilities for each of the syste m states. 4. Determine the mean nu mber of pu mp repairs during a period of 3 years. 5. Determine the percentage of ti me exactly when one of the pu mps is in a failed state. 6. Determine the mean ti me to the first syste m failure (i.e., the mean ti me until unchlorinated water is supplied to the custo mers for the first ti me after ti me t= 0). 7. Determine the percentage of ti me unchlorinated water is supplied to the custo mers. 10.7 Two identical pumps in paral lel logic Two identica l pumps are working in parallel logic. During normal operation both pumps are functioning. When one pump fails, the other has to do the whole job alone, with a higher load. The pumps are assumed to have exponentially distributed failure times: λH = 1.5·10 -4 h-1 when the pumps are bearing 'half load'. λF = 3.5·10 -4 h-1 when one of the pumps bears the 'full load'. Both pu mps may fail at the same time due to some external stresses. The failure rate with respect to this common cause failure has been estimated to be λC =3.0·10 -5 h-1. This type of external stress affects the system irrespective of how many of its units are functioning. Repair is initiated as soon as one of t he pumps fails. T he mean time to repair a pump, μ-1, is 15 bours. W hen bot h pumps are in t he failed state, t he w hole system bas to be s hut down. In t his case, t he system will not be put into operation again until bot h pumps have been repaired. T he mean downtime, μB-1 , w hen bot h pump s are failed, bas been estimated to be 25 bours. l. Establis h a state -space diagram for t he system. 2. Write down t he state equation in matrix format. 3. Determine t he steady states probabilities. 4. Determine t he percentage of time w hen: i. Bot h pumps are functioning. ii. Only one of t he pumps is functioning. iii. Bot h pumps are in t he failed state. 5. Determine t he mean number of pump repairs t hat are needed during a period of 5 years. 6. How many times we may expect to have a total pump failure (i.e. bot h pumps in a failed state at t he same time) during a period of 5 years?