Page 247 - Design of Simple and Robust Process Plants

6.2 Basic Theory of Reliability 233
When only the process performance is of interest, the components of the supplies
can be eliminated, or the reliability of the supplies can be designated as a one. All
components in the RBD have their specific reliability data included in the model.
The reliability model needs to have the capability of calculating the overall process
reliability and availability, as well as identifying the individual, or groups of compo-
nents, that contribute to the failure(s). The number of outages over the mission
time are calculated, as well as the related down-times, and the contribution of the
individual components are ranked. There are two techniques available for analyzing
these problems: one is analytical, and the other is based on Monte Carlo simula-
tions.

6.2.4.1 The analytical technique
The analytical technique is based on the product and summation rules. The product or
multiplication rule is applicable when we have components in series whose failures
are statistically independent. Such a system can only function if all components
function.

The following mathematical nomenclature used is to define operations:
Operation of union
S
Operation of intersection
T
C Operation of complementation
In other words; C = Component C is down, C = Component C is up
The probability (P) that the system is up is the product of the probability each
component is up. In formulae, this is expressed as:
Q n
P (system is up) = P(C 1 ) . P(C 2 ) . P(C 3 ) ¼¼.P(C n )= P C 1 † (7)
iˆ1
The system will not perform the function or the system is down is described as:
P (system is down) = 1 ± P (system is up) (8)
This can be written in two ways:
Q n
P (system is down) = 1± P C 1 †
iˆ1
or Q n
=1± 1 P C 1 †† (9)
iˆ1
The possibility that two components fail at the same time is very low, when the
probability of failure is P C† < 0.1.
In that case, the probability that the system is down is reduced to the following
approximation:
n
P
P (system is down) P C 1 † (10)
iˆ1
This last reduction is called the rare event approximation.