17370 Reliability

8/12/2019 17370 Reliability

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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 1,023,102 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.


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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 toage t (t is not included)

=the total number of

death before age t

divided by the total

population

ProbabilityofSurvivalR(t)andDeathF(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

P

FALURE DENSITY FUNCTION f(t)

Age in Years

+=

N

)()()(

tntntf

dt

tdFtf

)()( =

0

1

2

3

4

5

10

15

20

25

30

35

40

4550

60

65

70

75

80

85

90

95

99

100

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

41,65456,709

99,889

123,334

138,566

134,217

103,554

56,634

18,566

2,886

125

0

0.02260

0.00564

0.00402

0.00327

0.00288

0.00235

0.00186

0.00211

0.00240

0.00285

0.00353

0.00454

0.00602

0.008140.01110

0.01500

0.01950

0.02410

0.02710

0.02620

0.02020

0.01110

0.00363

0.00071

000012

0.00540

0.00454

0.00284

0.00330

0.00287

0.00192

0.00198

0.00224

0.00259

0.00364

0.00393

0.00436

0.00637

0.00962

0.01367

0.01800

0.02200

0.02490

0.02610

0.02460

0.01950

0.00970

0.00210

-

-

-

)()( tntn +


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Age in Years (t)

Figure 4.4 Histogram and smooth curve

NumbreofDeaths(thousands)

140

120

100

80

60

40

20

0 20 40 60 80 1000 20 40 60 80 100

0.14

0.12

0.10

0.8

0.6

0.4

0.2

0.0

FailureDensityf(t)

20 40 60

20

40

60

80

80

100

120

140

100

NumberofDeaths(thousands)

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)

FailureDensityf(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

45

10

15

20

25

30

35

23,102

5,770

4,116

3,347

2,95012,013

9,534

10,787

12,286

14,588

18,055

23,212

0.02260

0.00570

0.00414

0.00338

0.002990.00244

0.00196

0.00224

0.00258

0.00311

0.00391

0.0512

40

45

50

55

6065

70

75

80

85

90

95

99

30,788

41,654

56,709

76,420

99,889123,334

138,566

134,217

103,554

56,634

18,566

2,886

125

0.00697

0.00977

0.01400

0.02030

0.029500.04270

0.06100

0.08500

0.11400

0.14480

0.17200

0.24000

1.20000

)(

)(

ageatsurvivalsofnumber

),[duringdeathsofnumber)(

tR

tf

t

tttr

=

+=


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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

Failureratef(t)

Wearout failures

20 40 60 80 100

0.05

0.1

0.15

0.2

Early failures

Random failures

Wearout failures

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)()( =+ tFtR 1)(lim

0)(lim0

=

=

tF

tF

t

t

tetF = 1)(


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Failure Density - f(t)

=

=

==

t

t

t

duuftR

duuftF

edt

tdFtf

)()(

)()(

)()(

0

(exponentialdistribution)

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

tf

tr ==

=)(trThe component with a constant failure rateis 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 reliabilityR(t).


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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 COMP ONENTS1

Instrument Fault/year

Controller 0.29Control valve 0.60Flow measurement (fluids) 1.14

Flow measurement (solids) 3.75Flow switch 1.12Gas - liquid chromatograp h 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 temperatur e 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

i

iFF1

===

n

ii

RFR1

)1(11

A System with n Components

in Series

Reliability

Unreliability

=

=n

i

iRR1

=

==n

i

iFRF1

)1(11


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Upper Bound of Unreliability

for Systems with n Components

in Series

=

=

==

++=n

l

l

n

j

n

i

i

j

i

n

i

i FFFFF1

1

2

1

11

)1(L

=

n

i

iF1

Reactor

PIA PIC

Alarmat

P > 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.

Component

Failure Rate

(Faults/yr)

ReliabilitytetR =)(

Unreliability

F=1-R

Pressure Switch #1 0.14 0.87 0.13

Alarm Indicator 0.044 0.96 0.04

Pressure Switch #2 0.14 0.87 0.13

Solenoid Valve 0.42 0.66 0.34

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


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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


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Repair Density - g(t)

=

=

t

duugtG

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


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Availability - A(t)


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 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)


repaired per unit time at time t, give

that it jumped into the normal state at

time zero.


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Conditional Failure Intensity, (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 , (t)r(t).

For non-repairable components, (t) = r(t).

However, if the failure rate is constant () ,

then (t) = r(t) = for both repairable and non-

repairable components.

)(1)()(tQ

twt

=

Conditional Repair Intensity, (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,(t)=m(t)=0. For constant repair rate

m, (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.


)(),0( tFtW =

= 2

1 )(),( 21

t

t dttwttW

SHORT-CUT CALCULATION METHODS

Information Requiredj

j

(1 ) f a i lu r e r a t e c o n s t a n t

( 2 ) r e p a i r r a te c o n s t a n t

(3) m in im u m cu t s e t s

= =

= =

Approximation of Event Unavailability

When time is long compared with MTTRand , the following approximat ion

can be made,

Where, is the MTTR of component j.

[ ]t

jj

j

j

jjeQ )(1

+

+=

1.0


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Z

AND

X Y

IF X and Y are Indepe ndent

YX

YXZ

YXYXZ

+=

+= )(

Z

OR

X Y

( )

( )

z x y

x x y y

z

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

+

+ +

L L L

L L

1 2

1

1 1 1

n

+ + +L

1 2 n + + +L

1 1 2 2

1 2 3

n n

n

+ + +

+ + +

L

L

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 probabilit y of the top event. may be interpreted

as the conditional probab ility that the cut set occurs given that the

top event has occurred.

SQ

KI

K

PRIMAL EVENT IMPORTANCE

The importance of a primal event is defined asX

1

1

1 M

X K

KS

M

X K

K

i QQ

i I

=

=

=

=

or

Where, the sum is taken over all cut sets which contain p rimal

event .X


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[ EXAMPLE]

As an example , consider the tr ee 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 unavailabiliti es 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 eac h cut set and top event

probability.Cut Set KQ

(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

17370 Reliability

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