Transition of Component States

N F

Component fails

Component is repaired

Failedstatecontinues

Normalstate 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 1,023,102 15 962,270 50 810,900 85 78,221

1 1,000,000 20 951,483 55 754,191 90 21,577

2 994,230 25 939,197 60 677,771 95 3,011

3 990,114 30 924,609 65 577,822 95 125

4 986,767 35 906,554 70 454,548

5 983,817 40 883,342 75 315,982

10 971,804 45 852,554 80 181,765

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 1,023,102 1. 0. 1 1,000,000 0.9774 0.0226 2 994,230 0.9718 0.0282 3 986,767 0.9645 0.0322 4 983,817 0.9616 0.0355 5 983,817 0.9616 0.0384 10 971,804 0.9499 0.0501 15 962,270 0.9405 0.0595 20 951,483 0.9300 0.0700 25 939,197 0.9180 0.0820 30 924,609 0.9037 0.0963 40 883,342 0.8634 0.1139 45 852,554 0.8333 0.1667 50 810,900 0.7926 0.2074 55 754,191 0.7372 0.2628 60 677,771 0.6625 0.3375 65 577,882 0.5648 0.4352 70 454,548 0.4443 0.5557 75 315,982 0.3088 0.6912 80 181,765 0.1777 0.8223 85 78,221 0.0765 0.9235 90 21,577 0.0211 0.9789 95 3,011 0.0029 0.9971 99 125 0.0001 0.9999 100 0 0. 1.

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

Prob

ability of S

urvival R

(t) and

Death

F(t)

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 10 20 30 40 50 60 70 80 90 100

Figure 4.3 Survival and failure distributions.

0 10 20 30 40 50 60 70 80 90 100

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

PSurvival distribution

Failur

e dist

ributio

n

Pro

babi

lity

of

Sur

viva

l R(t

) an

d D

eath

F(t

)

FALURE DENSITY FUNCTION f(t)

Age in Years No. of Failures (death)

N

)()()(

tntntf

dt

tdFtf

)()(

012345101520253035404550606570758085909599100

23,102 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 99,889123,334138,566134,217103,554 56,634 18,566 2,886 125 0

0.022600.005640.004020.003270.002880.002350.001860.002110.002400.002850.003530.004540.006020.008140.011100.015000.019500.024100.027100.026200.020200.011100.003630.00071000012

0.005400.004540.002840.003300.002870.001920.001980.002240.002590.003640.003930.004360.006370.009620.013670.018000.022000.024900.026100.024600.019500.009700.00210 - - -

)()( tntn

Age in Years (t)

Figure 4.4 Histogram and smooth curve

Nu

mb

re of Death

s (thou

sand

s)

140

120

100

80

60

40

200 20 40 60 80 100

0 20 40 60 80 100

0.14

0.12

0.10

0.8

0.6

0.4

0.2

0.0

Failu

re Den

sity f(t)

20 40 60

20

40

60

80

80

100

120

140

100

Num

ber

of D

eath

s (t

hous

ands

)

Age in Years (t)

20 40 60

0.2

0.4

0.6

80

0.8

0.10

0.12

0.14

100 Age in Years (t)

Fai

lure

Den

sity

f (

t)

CALCULATION OF FAILURE RATE r(t)

Age in Years No. of Failures

(death)

r(t)=)(1

)(

tF

tf

Age in Years No. of Failures

(death)

r(t)=)(1

)(

tF

tf

0 1 2 3 4 5101520253035

23,102 5,770 4,116 3,347 2,95012,013 9,53410,78712,28614,58818,05523,212

0.022600.005700.004140.003380.002990.002440.001960.002240.002580.003110.003910.0512

40455055606570758085909599

30,788 41,654 56,709 76,420 99,889123,334138,566134,217103,554 56,634 18,566 2,886 125

0.006970.009770.014000.020300.029500.042700.061000.085000.114000.144800.172000.240001.20000

)(

)(

ageat survivals ofnumber

),[ during deaths ofnumber )(

tR

tf

t

tttr

Random failures

0 20 40 60 80 100

Bathtub Curvet,years

Figure 4.6 Failure rate r (t) versus t.

0.2

0.15

0.1

0.05

Failu

re rate f (t)

Wearout failures

20 40 60 80 100

0.05

0.1

0.15

0.2Early failures

Random failures

Wearout failuresF

ailu

re R

ate

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

tetR )(

0)(lim

1)(lim0

tR

tR

t

t

Unreliability - F(t)

• The probability that the component experiences the first failure during (0,t).

• Example: exponential distribution

1)()( tFtR1)(lim

0)(lim0

tF

tF

t

t

tetF 1)(

Failure Density - f(t)

t

t

t

duuftR

duuftF

edt

tdFtf

)()(

)()(

)()(

0

(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:

)(1

)(

)(

)()(

tF

tf

tR

tftr

)(trThe component with a constant failure rate is considered as good as new, if it is functioning.

Mean Time to Failure - MTTF

1

)(0

dtttfMTTF

Failure Rate Failure Density Unreliability Reliability

t(a)

t(b)

t(c)

t(d)

f (t) F (t) R (t)

1 1

0 0

Area = 1

t

0dttf

1 - 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 Mortality Old 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 COMPONENTS1

Instrument Fault/year

Controller 0.29Control valve 0.60Flow measurement (fluids) 1.14

Flow measurement (solids) 3.75Flow switch 1.12Gas - liquid chromatograph 30.6

Hand valve 0.13Indicator lamp 0.044Level measurement (liquids) 1.70

Level measurement (solids) 6.86Oxygen analyzer 5.65pH meter 5.88

Pressure measurement 1.41Pressure relief valve 0.022Pressure switch 0.14

Solenoid valve 0.42Stepper motor 0.044Strip chart recorder 0.22

Thermocouple temperature measurement 0.52Thermometer temperature measurement 0.027Valve positioner 0.44

1Selected from Frank P. Lees, Loss Prevention in the Process Industries (London: Butterworths, 1986), p. 343.

A System with n Components in Parallel

• Unreliability

• Reliability

n

iiFF

1

n

iiRFR

1

)1(11

A System with n Components in Series

• Reliability

• Unreliability

n

iiRR

1

n

iiFRF

1

)1(11

Upper Bound of Unreliability for Systems with n Components

in Series

n

ll

nj

n

i

i

ji

n

ii FFFFF

1

1

2

1

11

)1(

n

iiF

1

Reactor

PIA PICAlarm

atP > PA

PressureSwitch

PressureFeed

SolenoidValve

Figure 11-5 A chemical reactor with an alarm and inlet feed solenoid. The alarm and feed shutdown systems are linked in parallel.

C o m p o n e n t

F a i l u r e R a t e( F a u l t s / y r )

R e l i a b i l i t y

tetR )(U n r e l i a b i l i t y

F = 1 - R

P r e s s u r e S w i t c h # 1 0 . 1 4 0 . 8 7 0 . 1 3A l a r m I n d i c a t o r 0 . 0 4 4 0 . 9 6 0 . 0 4P r e s s u r e S w i t c h # 2 0 . 1 4 0 . 8 7 0 . 1 3S o l e n o i d V a l v e 0 . 4 2 0 . 6 6 0 . 3 4

Alarm System

• The components are in series

56.51

180.0ln

165.0835.011

835.0)96.0)(87.0(2

1

MTTF

R

RF

RRi

i

Faults/year

years

Shutdown System

• The components are also in series:

80.11

555.0ln

426.0574.011

574.0)66.0)(87.0(2

1

MTTF

R

RF

RRi

i

The Overall Reactor System

• The alarm and shutdown systems are in parallel:

7.131

073.0ln

930.0070.011

070.0)426.0)(165.0(2

1

MTTF

R

FR

FFj

j

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

0)( tG

Repair Density - g(t)

tduugtG

dt

tdGtg

0)()(

)()(

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)(1

)()(

tG

tgtm

0)( tm

Mean Time to Repair - MTTR

0

)( dtttgMTTR

The Whole Process

Availability - A(t)

• The probability that the component is normal at time t.

• For non-repairable components

• For repairable components0)(

)()(

A

tRtA

0)(

)()(

A

tRtA

Unavailability - Q(t)

• The probability that the component fails at time t.

• For non-repairable components

• For repairable components

1)()( tQtA

1)(

)()(

Q

tFtQ

1)(

)()(

Q

tFtQ

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.

)()( tftw

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.

)(1

)()(

tQ

twt

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(t1,t2 )

Expected number of failures during (t1,t2) given that the component jumped into the normal state at time zero.

For non-repairable components

)(),0( tFtW

2

1

)(),( 21

t

tdttwttW

SHORT-CUT CALCULATION METHODS

Information Requiredj

j

(1) failure rate cons tan t

(2) repair rate cons tan t

(3) min imum cut sets

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.

t

jj

jj

jjeQ )(1

1.0j

j

jjj

j

j

j

j

j

jj

jj

tQ

1)(

)(lim

j

Z

AND

X Y

IF X and Y are Independent

YX

YXZ

YXYXZ

)(

Z

OR

X Y

( )

( )

z x y

x x y yz

x y

COMPUTATION OF ACROSS LOGIC GATES

,

2 INPUTS 3 INPUTS n INPUTS

1 2 1 2( )

1 2

1 2

1 2

1 1 2 2

1 2

1 2 3 2 3 1 3 1 2( )

1 2 3

2 3 1 3 1 2

1 2 3

1 1 2 2 3 3

1 2 3

1 2 2 3 1 3

1 2 1

(

)n n

n

1 2

11 1 1

n

1 2 n

1 1 2 2

1 2 3

n n

n

AND

GATES

OR

GATES

CUT SET IMPORTANCE

The importance of a cut set K is defined as

k

k

s

QI

Q

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.

SQ

KI

K

PRIMAL EVENT IMPORTANCEThe importance of a primal event is defined asX

1

1

1 M

X KK

S

M

X KK

i QQ

i I

or

Where, the sum is taken over all cut sets which contain primal event .X

[ 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.

1yr Event ( )yr hr

1 .16 1.5E-5 (.125) 2.4E-6

2 .2 1.5E-5 (.125) 3.0E-6

3 1.4 7E-4 (6) 9.8E-4

4 30 1.1E-4 (1) 3.3E-3

5 5 1.1E-4 (1) 5.5E-4

6 .5 5.5E-5 (.5) 2.75E-5

Now, compute the probability of occurrence for each cut set and top event

probability. Cut Set K

Q (1) 2.4E-6

(2) 3.0E-6

(6) 2.75E-5

(3,4) 3.23E-6

(3,5) 5.39E-7

SQ 3.67E-5

iq