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E Concepts of Probability Analysis
E.2
Reliability
E.2.1 Equipment Reliability
Reliability is an important aspect of equipment used within chemical processes.
A piece of equipment that does not operate as intended poses a significant safety
risk. More formally defined, equipment reliability is the probability that a piece
of process equipment will perform its intended function adequately for a specified
period (e.g., time or number of demands) and under specified process conditions.
If n identical equipment units operate without being replaced, after time t, a
certain number will have survived (n s (t)), and a certain number will have failed
(n f (t)). The probability of survival, or reliability R(t), and the probability of failure,
F (t) (note that F is used for ‘frequency’ in Sect. E.1, which is, of course, a different
quantity), is defined as:
R(t) =
n s (t)
n
= 1 − F (t) = 1 −
n f (t)
n
(E.4)
In order to convert a failure rate into a probability of failure, a continuous
probability distribution function can be developed as a function of the instantaneous
failure rate (λ):
λ(t) =
1
n − n f (t)
×
dn f (t)
dt
= −
1
R(t)
×
dR(t)
dt
(E.5)
This is also equal to −d ln(R(t))/dt. Through integration, this yields the
probability of reliability R(t) and probability of failure F (t):
R(t) = exp
−
t
0
λ(t
)dt
(E.6)
F (t) = 1 − R(t) = 1 − exp
−
t
0
λ(t
)dt
(E.7)
Figure E.1 shows the typical shape of an instantaneous failure rate curve (λ)
over time. Equipment reliability can be thought of similarly to the human body’s
susceptibility to developmental issues and disease. During infancy, the chance for
developmental abnormalities or having a weak immune system unable to fight off
diseases is high. After development is finished, the probability for health issues to
occur remains lower and more constant for some time. Then, as the body approaches
old age, the chance for health issues once again begins to increase. The same
thinking can be applied to equipment upon its installation and first calibration,
through its regular operating life, and then toward the end of its operating life.
E Concepts of Probability Analysis
E.2
Reliability
E.2.1 Equipment Reliability
Reliability is an important aspect of equipment used within chemical processes.
A piece of equipment that does not operate as intended poses a significant safety
risk. More formally defined, equipment reliability is the probability that a piece
of process equipment will perform its intended function adequately for a specified
period (e.g., time or number of demands) and under specified process conditions.
If n identical equipment units operate without being replaced, after time t, a
certain number will have survived (n s (t)), and a certain number will have failed
(n f (t)). The probability of survival, or reliability R(t), and the probability of failure,
F (t) (note that F is used for ‘frequency’ in Sect. E.1, which is, of course, a different
quantity), is defined as:
R(t) =
n s (t)
n
= 1 − F (t) = 1 −
n f (t)
n
(E.4)
In order to convert a failure rate into a probability of failure, a continuous
probability distribution function can be developed as a function of the instantaneous
failure rate (λ):
λ(t) =
1
n − n f (t)
×
dn f (t)
dt
= −
1
R(t)
×
dR(t)
dt
(E.5)
This is also equal to −d ln(R(t))/dt. Through integration, this yields the
probability of reliability R(t) and probability of failure F (t):
R(t) = exp
−
t
0
λ(t
)dt
(E.6)
F (t) = 1 − R(t) = 1 − exp
−
t
0
λ(t
)dt
(E.7)
Figure E.1 shows the typical shape of an instantaneous failure rate curve (λ)
over time. Equipment reliability can be thought of similarly to the human body’s
susceptibility to developmental issues and disease. During infancy, the chance for
developmental abnormalities or having a weak immune system unable to fight off
diseases is high. After development is finished, the probability for health issues to
occur remains lower and more constant for some time. Then, as the body approaches
old age, the chance for health issues once again begins to increase. The same
thinking can be applied to equipment upon its installation and first calibration,
through its regular operating life, and then toward the end of its operating life.