Transition of Component States N F Component fails Component is repaired Failed state continues Normal state continues

The Repair-to-Failure Process

Definition of Reliability The reliability of an item is the probability that it will adequately perform its specified purpose for a specified period of time under specified environmental conditions.

REPAIR -TO-FAILURE PROCESS MORTALITY DATA t=age in years ; L(t) =number of living at age t t L(t) t L(t) t L(t) t L(t) 0 1,023, , , , ,000, , , , , , , , , , , , , , , , , , , ,765 After Bompas-Smith. J.H. Mechanical Survival : The Use of Reliability Data, McGraw-Hill Book Company, New York, 1971.

HUMAN RELIABILITY t L(t), Number Living at Age in Years Age t R(t)=L(t)/N F(t)=1-R(t) 0 1,023, ,000, , , , , , , , , , , , , , , , , , , , , , repair= birth failure = death Meaning of R(t): (1) Prob. Of Survival (0.87) of an individual of an individual to age t (40) (2) Proportion of a population that is expected to Survive to a given age t.

Reliability, R(t) = probability of survival to (inclusive) age t = the number of surviving at t divided by the total sample Unreliability, F(t) = probability of death to age t (t is not included) =the total number of death before age t divided by the total population Probability of Survival R(t) and Death F(t) Figure 4.3 Survival and failure distributions.

P Survival distribution Failure distribution Probability of Survival R(t) and Death F(t)

FALURE DENSITY FUNCTION f(t) Age in Years No. of Failures (death) ,102 5,770 4,116 3,347 2,950 12,013 9,543 10,787 12,286 14,588 18,055 23,212 30,788 41,654 56,709 99, , , , ,554 56,634 18,566 2,

Age in Years (t) Figure 4.4 Histogram and smooth curve Numbre of Deaths (thousands) Failure Density f(t)

Number of Deaths (thousands) Age in Years (t)

Age in Years (t) Failure Density f (t)

CALCULATION OF FAILURE RATE r(t) Age in Years No. of Failures (death) r(t)= Age in Years No. of Failures (death) r(t)= ,102 5,770 4,116 3,347 2,950 12,013 9,534 10,787 12,286 14,588 18,055 23, ,788 41,654 56,709 76,420 99, , , , ,554 56,634 18,566 2,

Random failures Bathtub Curve t,years Figure 4.6 Failure rate r (t) versus t Failure rate f (t) Wearout failures

Early failures Random failures Wearout failures Failure Rate r(t) Failure rate r(t) versus t.

Reliability - R(t) The probability that the component experiences no failure during the the time interval (0,t). Example: exponential distribution

Unreliability - F(t) The probability that the component experiences the first failure during (0,t). Example: exponential distribution

Failure Density - f(t) (exponential distribution)

Failure Rate - r(t) The probability that the component fails per unit time at time t, given that the component has survived to time t. Example: The component with a constant failure rate is considered as good as new, if it is functioning.

Mean Time to Failure - MTTF

Failure RateFailure DensityUnreliabilityReliability t (a) t (b) t (c) t (d) f (t)F (t)R (t) Area = 11 - F (t) Figure 11-1 Typical plots of (a) the failure rate (b) the failure density f (t), (c) the unreliability F(t), and (d) the reliability R (t).

Period of Approximately Constant failure rate Infant MortalityOld Age Time Failure Rate, (faults/time) Figure 11-2 A typical “bathtub” failure rate curve for process hardware. The failure rate is approximately constant over the mid-life of the component.

TABLE 11-1: FAILURE RATE DATA FOR VARIOUS SELECTED PROCESS COMPONENTS 1 Instrument Fault/year Controller 0.29 Control valve 0.60 Flow measurement (fluids) 1.14 Flow measurement (solids) 3.75 Flow switch 1.12 Gas - liquid chromatograph 30.6 Hand valve 0.13 Indicator lamp Level measurement (liquids) 1.70 Level measurement (solids) 6.86 Oxygen analyzer 5.65 pH meter 5.88 Pressure measurement 1.41 Pressure relief valve Pressure switch 0.14 Solenoid valve 0.42 Stepper motor Strip chart recorder 0.22 Thermocouple temperature measurement 0.52 Thermometer temperature measurement Valve positioner Selected from Frank P. Lees, Loss Prevention in the Process Industries (London: Butterworths, 1986), p. 343.

A System with n Components in Parallel Unreliability Reliability

A System with n Components in Series Reliability Unreliability

Upper Bound of Unreliability for Systems with n Components in Series

Reactor PIA PIC Alarm at P > P A Pressure Switch Pressure Feed Solenoid Valve Figure 11-5 A chemical reactor with an alarm and inlet feed solenoid. The alarm and feed shutdown systems are linked in parallel.

Alarm System The components are in series Faults/year years

Shutdown System The components are also in series:

The Overall Reactor System The alarm and shutdown systems are in parallel:

The Failure-to-Repair Process

Repair Probability - G(t) The probability that repair is completed before time t, given that the component failed at time zero. If the component is non-repairable

Repair Density - g(t)

Repair Rate - m(t) The probability that the component is repaired per unit time at time t, given that the component failed at time zero and has been failed to time t. If the component is non-repairable

Mean Time to Repair - MTTR

The Whole Process

Availability - A(t) The probability that the component is normal at time t. For non-repairable components For repairable components

Unavailability - Q(t) The probability that the component fails at time t. For non-repairable components For repairable components

Unconditional Repair Density, w(t) The probability that a component fails per unit time at time t, given that it jumped into the normal state at time zero. Note, for non-repairable components.

Unconditional Repair Density, v(t) The probability that the component is repaired per unit time at time t, give that it jumped into the normal state at time zero.

Conditional Failure Intensity, λ(t) The probability that the component fails per unit time, given that it is in the normal state at time zero and normal at time t. In general, λ(t)≠r(t). For non-repairable components, λ(t) = r(t). However, if the failure rate is constant (λ), then λ(t) = r(t) = λ for both repairable and non- repairable components.

Conditional Repair Intensity, µ(t) The probability that a component is repaired per unit time at time t, given that it is jumped into the normal state at time zero and is failed at time t, For non-repairable component, µ(t)=m(t)=0. For constant repair rate m, µ(t)=m.

ENF over an interval, W(t 1,t 2 ) Expected number of failures during (t1,t2) given that the component jumped into the normal state at time zero. For non-repairable components

SHORT-CUT CALCULATION METHODS Information Required Approximation of Event Unavailability When time is long compared with MTTR and, the following approximation can be made, Where, is the MTTR of component j.

Z AND X Y IF X and Y are Independent

Z OR X Y

COMPUTATION OF ACROSS LOGIC GATES 2 INPUTS 3 INPUTS n INPUTS AND GATES OR GATES

CUT SET IMPORTANCE The importance of a cut set K is defined as Where, is the probability of the top event. may be interpreted as the conditional probability that the cut set occurs given that the top event has occurred. PRIMAL EVENT IMPORTANCE The importance of a primal event is defined as or Where, the sum is taken over all cut sets which contain primal event.

[ EXAMPLE ] As an example, consider the tree used in the section on cut sets. The cut sets for this tree are (1), (2), (6), (3,4),(3,5). The following data are given from which we compute the unavailabilities for each event E-5 (.125) 2.4E E-5 (.125) 3.0E E-4 (6) 9.8E E-4 (1) 3.3E E-4 (1) 5.5E E-5 (.5) 2.75E-5 Now, compute the probability of occurrence for each cut set and top event probability. (1) 2.4E-6 (2) 3.0E-6 (6) 2.75E-5 (3,4) 3.23E-6 (3,5) 5.39E E-5