Energy Engineering - RELIABILITY, SAFETY AND RISK ANALYSIS C

Chapter 05 - Exercises collection

Divided by topic

Chapter 5 Reliability of simple systems 5.1 Satellite with two transmitters Consider a satellite with two transmitters, one of which is in cold standby. Loss of transmission can occur if either both transmitters have failed or solar disturbances permanently interfere with transmission. lf the rate of failure of the on -line transmitter is λ and the rate of solar disturbances is λcm , find: l. The reliability of transmission. 2. The mean time to transmission failure. 5.2 Simple system Consider a system of two independent components with exponentially distributed failure times. The failure rates are λ1 and λ2, respectively. Determine the probability that component l fails before component 2. 5.3 Parallel system Suppose that in the system shown below, the two components have the same cost, and their reliabilities are R1 = 0.7, R2 = 0.95, respectively. lf it is perr missible to add two components to the system, would it be preferable to a) replace component l by three components in parallel or b) to replace components l and 2 each by simple parallel systems? 5.5 Common mode failure Suppose that a unit has a design -life reliability of 0.95. Assume an exponential distribution for the failure times and the rare -event approximation for the reliability, R(t) = e xp( -λt) ≈ 1- λt. l. Estimate the reliability if two of these units are put in active paral lel. 2. Consider now the possibility of common -mode failures (shocks which simultaneously fail both components). In this case, the failure rate λ has two contributions from independent ( I) and common -mode (C) failures λ = λ I + λC = (1 -β)λ + βλ where β=λ C/λ Estimate the maximum fraction β of common failures that is acceptable if the parallel units in part l are to retain a system reliability of at least 0.99. 1 2 5.6 Active parallel system In an active parallel system each unit has a failure rate of 0.002 hr -1. l. What is the MITF of the system if there is no load sharing? 2. What is the MITF of the system if the failure rate increases by 20% as a result of increased load? 3. What is the MITF of the system if one simply (and conservati vely) increased both unit f ailure rates by 20%? 5.7 Shared load parallel system In a "shared load parallel system," the partial components equally share the load, and, as a component fails, the surviving components must sustain an increased load. Thus, as successive components fail, the failure rates of the surviving components increa se. An example of a shared parallel load configuration would be when bolts are used to hold a machine member; if one bolt breaks, the remainders must support the load. Consider such a system with two components whose constant failure rates are defined as follows: λh= half -load failure rate λf = full -load failure rate Find the time -dependent reliability of the system. 5.8 Temperature sensor A temperature sensor with failure rate λ is to have a design -life reliability of no less than 0.98. Since a single sensor is known to have a reliability of only 0.90, the design engineer decides to put two of them in parallel. The reliability should then be 0.99. Upon reliability testing, howev er, the reliability is estimated to be only 0.97. The engineer f irst deduces that the degradation is due to common -mode failures and then considers two options: (l) putting a third sensor in parallel, and (2) reducing the probability of common -mode failures λC. l. Assuming that the sensors have constant failure rates, find the value of β = λC/λ that characterizes the common -mode failures. 2. Will adding a third sensor in parallel meet the reliability criterion if nothing is done about common - mode failures? 3. By how much must β be reduced if the two sensors in parallel are to meet the criterion? 5.9 A two engine p lane Consider a two engine piane in a one -out -of-two logic configuration. When both engines A and B are fully energized they share the total load and the failure time densities are f A(t) and f B(t). lf either one fails, the survivor must carry the full load and its failure density becomes g A(t) or g B(t). l. Derive an expression for the reliability of the system R(t). 2. Find the reliability if fA(t)=f B(t)= λexp (-λt) gA(t)=g B(t)=k λexp( -kλt) k>1 5.10 One -out -of-two system Consider two components A and B in a one -out -of-two logic configuration. When both A and B are fully energized they share the total load and the failure densities are f A(t) and fB(t). lf either one fails, the survivor must carry the full load and its failure density becomes g A(t) or g B(t). (A simple example would be a two -engine piane which, if one engine fails, can stili keep flying, but the surviving engine now has to carry the fullload. ) Find the reliability R(t) of the system if fA(t)=f B(t)= λexp (-λt) gA(t)=g B(t)=k λexp( -kλt) k>1 5.11 Pressure vessel A pressure vessel is equipped with six relief valves. Pressure transients can be controlled successfully by any three of these valves. lf the probability that any one of these valves will fail to operate on demand is 0.04, what is the probability on demand that the relief valve system will fail to control a pressure transient? Assume that the failures are independent. 5.12 r -out -of-N detection system You are to design an r -out -of-N detection system. The number of components, N, must be as small as possible to minimize cost. The fail -to-danger (the component is requested to detect an actually present danger but fails to do so) and the fail -safe (the sys tem gives a false alarm in absence of danger) probabilities for the identica l components are q d = 10 -2 and q s=l0-2. Your design must meet the following criteria: l. Probability of system fail -to-danger < 10 -4. 2. Probability of system fail -safe < l 0-2 3. What values of r and N should be used? 5.13 Cold standby system of two units Consider a "cold" standby system of two units. The on -line unit has an MTTF of 2 years. When it fails, the standby unit comes on line and its MTTF is 3 years. Assume that each component has an exponential failure times distribution. l. What is the probability density function of the system failure time? 2. What is the MTTF ofthe system? 3. Repeat l and 2, assuming that the two components are in parallel in a one -out -of-two configuration. 5.14 Temperature sensing elements Three nominally identica l temperature sensing elements are connected to nominally the same point on a process plant. An alarm is designed to be given if any two or more of these temperature sensors record a temperature above a certain prescribed level. The times to f ailure of each element are exponentially distributed with a mean value of 5 000 h. What is: l. The probability of the alarm system not working, if an excessive plant temperature rise takes piace at 500 h, or secondly at 2,000 h? 2. The mean time to complete failure of the alarm system? 3. The average unavailability over a period of 500 h?