Transcript Reliability

Transition of Component States
Normal
state continues
Component fails
N
F
Component is repaired
Failed
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)
0 1,023,102
1 1,000,000
2
994,230
3
990,114
4
986,767
5
983,817
10 971,804
t
L(t)
t
L(t)
t
L(t)
15
20
25
30
35
40
45
962,270
951,483
939,197
924,609
906,554
883,342
852,554
50
55
60
65
70
75
80
810,900
754,191
677,771
577,822
454,548
315,982
181,765
85
90
95
95
78,221
21,577
3,011
125
After Bompas-Smith. J.H. Mechanical Survival : The Use of Reliability
Data, McGraw-Hill Book Company, New York , 1971.
HUMAN RELIABILITY
t
Age in Years
0
1
2
3
4
5
10
15
20
25
30
40
45
50
55
60
65
70
75
80
85
90
95
99
100
L(t), Number Living at
Age t
R(t)=L(t)/N F(t)=1-R(t)
1,023,102
1,000,000
994,230
986,767
983,817
983,817
971,804
962,270
951,483
939,197
924,609
883,342
852,554
810,900
754,191
677,771
577,882
454,548
315,982
181,765
78,221
21,577
3,011
125
0
1.
0.9774
0.9718
0.9645
0.9616
0.9616
0.9499
0.9405
0.9300
0.9180
0.9037
0.8634
0.8333
0.7926
0.7372
0.6625
0.5648
0.4443
0.3088
0.1777
0.0765
0.0211
0.0029
0.0001
0.
0.
0.0226
0.0282
0.0322
0.0355
0.0384
0.0501
0.0595
0.0700
0.0820
0.0963
0.1139
0.1667
0.2074
0.2628
0.3375
0.4352
0.5557
0.6912
0.8223
0.9235
0.9789
0.9971
0.9999
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.
= 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)
Reliability, R(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.
1.0
P
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
FALURE DENSITY FUNCTION f(t)
n(t  )  n(t )
f (t ) 
Age in Years
0
1
2
3
4
5
10
15
20
25
30
35
40
45
50
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
41,654
56,709
99,889
123,334
138,566
134,217
103,554
56,634
18,566
2,886
125
0
n(t   )  n(t )
N
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
0.00814
0.01110
0.01500
0.01950
0.02410
0.02710
0.02620
0.02020
0.01110
0.00363
0.00071
000012
f (t ) 
dF (t )
dt
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
-
0.14
120
0.12
Failure Density f(t)
Numbre of Deaths (thousands)
140
100
80
60
40
0.10
0.8
0.6
0.4
20
0.2
0.0
0
20
40
60
80
Age in Years (t)
Figure 4.4 Histogram and smooth curve
100
0
20
40
60
80
100
140
Number of Deaths (thousands)
120
100
80
60
40
20
20
40
60
80
100
Age in Years (t)
0.14
0.12
Failure Density f (t)
0.10
0.8
0.6
0.4
0.2
20
40
60
80
100
Age in Years (t)
number of deathsduring [t , t  ) f (t )
r (t ) 

number of survivalsat age t
R(t )
CALCULATION OF FAILURE RATE r(t)
Age in Years
0
1
2
3
4
5
10
15
20
25
30
35
No. of Failures r(t)= f (t )
1  F (t )
(death)
23,102
5,770
4,116
3,347
2,950
12,013
9,534
10,787
12,286
14,588
18,055
23,212
0.02260
0.00570
0.00414
0.00338
0.00299
0.00244
0.00196
0.00224
0.00258
0.00311
0.00391
0.0512
Age in Years
No. of Failures
r(t)=
f (t )
1  F (t )
(death)
40
45
50
55
60
65
70
75
80
85
90
95
99
30,788
41,654
56,709
76,420
99,889
123,334
138,566
134,217
103,554
56,634
18,566
2,886
125
0.00697
0.00977
0.01400
0.02030
0.02950
0.04270
0.06100
0.08500
0.11400
0.14480
0.17200
0.24000
1.20000
Random failures
0.2
Wearout failures
Failure rate f (t)
0.15
0.1
0.05
0
20
40
60
80
t,years
Figure 4.6 Failure rate r (t) versus t.
100
Bathtub Curve
Random failures
Early failures
Wearout failures
0.2
0.15
0.1
0.05
20
40
60
80
Failure rate r(t) versus t.
100
Reliability - R(t)
• The probability that the component
experiences no failure during the the time
interval (0,t).
lim R(t )  1
t 0
lim R(t )  0
t 
• Example: exponential distribution
R(t )  e
 t
Unreliability - F(t)
• The probability that the component
experiences the first failure during (0,t).
R(t )  F (t )  1
lim F (t )  0
t 0
lim F (t )  1
t 
• Example: exponential distribution
F (t ) 1  e
 t
Failure Density - f(t)
dF (t )
 t
f (t ) 
 e
dt
t
F (t )   f (u )du
0

R(t )   f (u )du
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.
f (t )
f (t )
r (t ) 

R(t ) 1  F (t )
• Example: r (t )  
The component with a constant failure rate is considered
as good as new, if it is functioning.
Mean Time to Failure - MTTF

1
0

MTTF   tf (t )dt 
Failure Rate
Failure Density
Unreliability
1
f (t) Area = 1

F (t)
1
 f t dt
t
0
t
(a)
t
(b)
0
Reliability
t
(c)
R (t)
0
1 - F (t)
t
(d)
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).
Failure Rate,

(faults/time)
Period of Approximately Constant
failure rate
Infant Mortality
Old Age
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
Control valve
Flow measurement (fluids)
0.29
0.60
1.14
Flow measurement (solids)
Flow switch
Gas - liquid chromatograph
3.75
1.12
30.6
Hand valve
Indicator lamp
Level measurement (liquids)
0.13
0.044
1.70
Level measurement (solids)
Oxygen analyzer
pH meter
6.86
5.65
5.88
Pressure measurement
Pressure relief valve
Pressure switch
1.41
0.022
0.14
Solenoid valve
Stepper motor
Strip chart recorder
0.42
0.044
0.22
Thermocouple temperature measurement
Thermometer temperature measurement
Valve positioner
0.52
0.027
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
F   Fi
i 1
n
R  1  F  1   (1  Ri )
i 1
A System with n Components
in Series
• Reliability
• Unreliability
n
R   Ri
i 1
n
F  1  R  1   (1  Fi )
i 1
Upper Bound of Unreliability
for Systems with n Components
in Series
n
n
i 1
F   Fi   Fi Fj    (1)
i 1
n
  Fi
i 1
i  2 j 1
n 1
n
F
l
l 1
Pressure
Switch
Alarm
at
P > PA
PIA
PIC
Pressure
Feed
Solenoid
Valve
Reactor
Figure 11-5 A chemical reactor with an alarm and inlet feed solenoid. The alarm and feed shutdown systems
are linked in parallel.
Component
Pressure Switch #1
Alarm Indicator
Pressure Switch #2
Solenoid Valve
Failure Rate
(Faults/yr)

0.14
0.044
0.14
0.42
Reliability
Unreliability
F=1-R
0.87
0.96
0.87
0.66
0.13
0.04
0.13
0.34
R (t )  e
 t
Alarm System
• The components are in series
2
R   Ri  (0.87)(0.96)  0.835
i 1
F  1  R  1  0.835  0.165
   ln R  0.180 Faults/year
MTTF 
1

 5.56
years
Shutdown System
• The components are also in series:
2
R   Ri  (0.87)(0.66)  0.574
i 1
F  1  R  1  0.574  0.426
   ln R  0.555
MTTF 
1

 1.80
The Overall Reactor System
• The alarm and shutdown systems are in
parallel: 2
F   F j  (0.165)(0.426)  0.070
j 1
R  1  F  1  0.070  0.930
   ln R  0.073
MTTF 
1

 13.7
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
G(t )  0
Repair Density - g(t)
dG(t )
g (t ) 
dt
t
G (t )   g (u )du
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.
g (t )
m(t ) 
1  G(t )
• If the component is non-repairable
m(t )  0
Mean Time to Repair - MTTR

MTTR   tg (t )dt
0
The Whole Process
Availability - A(t)
• The probability that the component is
normal at time t.
• For non-repairable components
A(t )  R(t )
A()  0
• For repairable components
A(t )  R(t )
A()  0
Unavailability - Q(t)
• The probability that the component fails at
time t.
A(t )  Q(t )  1
• For non-repairable components
Q(t )  F (t )
Q ()  1
• For repairable components
Q(t )  F (t )
Q ()  1
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, w(t )  f (t )
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 nonrepairable components.
w(t )
 (t ) 
1  Q(t )
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.
t2
W (t1 , t2 )   w(t )dt
t1
For non-repairable components
W (0, t )  F (t )
SHORT-CUT CALCULATION METHODS
Information Required
(1) j  failure rate  cons tan t
(2) j  repair rate  cons tan t
(3) min imum cut sets
Approximation of Event Unavailability
Qj 
1 e
j
j   j
When time is long compared with MTTR and
can be made,
j
Where,
j
(
j
j
(  j   j ) t

, the following approximation
 j  0.1
j)
j
lim Q j 


  j j
t 

 j   j ( j ) 1  j
j
is the MTTR of component j.
Z
AND
X
IF X and Y are
Y
Independent
Z   X Y ( X   Y )
 X Y
Z 
 X Y
Z
OR
X
Y
z  x   y
( x x   y y )
z 
( x   y )
COMPUTATION OF  ,  ACROSS
GATES
2 INPUTS

AND
GATES
OR
GATES

3 INPUTS
n INPUTS

12 (1   2 )    (        ) 
1
1 2 3
 1 2
1   2

1  2

11  2 2
1  2
2 3
1 3
2
3
2
1
1
2
1

2
1
2
1
2
2
3
3
3

1
2
3
1

3
n 1
2
1



2
1

n
   
1
3
      
  
1
n
1
  
1
 (    
   )
1
3
3
2
1 2
 
     
1
LOGIC
2
n
       
    
1
1
2
1
2
2
n
3
n
n
n
CUT SET IMPORTANCE
The importance of a cut set K is defined as
Q
I 
Q
k
k
s
Where, QS is the probability of the top event. I K may be interpreted
as the conditional probability that the cut set K occurs given that the
top event has occurred.
PRIMAL EVENT IMPORTANCE
The importance of a primal event X is defined as
1
i 
Q
X
or
S
Q
M
K
K 1
i  I
M
X
K
K 1
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.
Event
1
2
3
4
5
6
  yr
1
.16
.2
1.4
30
5
.5
  yr (hr )
qi
1.5E-5 (.125)
1.5E-5 (.125)
7E-4 (6)
1.1E-4 (1)
1.1E-4 (1)
5.5E-5 (.5)
2.4E-6
3.0E-6
9.8E-4
3.3E-3
5.5E-4
2.75E-5
Now, compute the probability of occurrence for each cut set and top event
probability.
Q
Cut Set
K
(1)
2.4E-6
(2)
3.0E-6
(6)
2.75E-5
(3,4)
3.23E-6
(3,5)
5.39E-7
Q 
S
3.67E-5