Page 248 - Design of Simple and Robust Process Plants

234 Chapter 6 Process Design Based on Reliability

C 1

C 2 A
X
C 3 B
Y
C 4 C

C
5
One out of five Two out of three One out of two

Fig. 6.8. Parallel components for different configurations.

The summation rule is applicable to systems with parallel components, and failures
of these components are mutually exclusive. In a parallel system, the overall system
fails if all the subsystems fail. A parallel system is shown in Figure 6.8 for different
configurations: one out of five; or one out of two only fails if all parallel components
fail. This is expressed mathematically as:
P(system is down) = P (C 1 down T C 2 T C 3 .... T C n )

= P C 1 P C 2 P C 3 .... P C n
Q n
= P C† (11)
iˆ1
For the system two out of three, Eq. (11) becomes:
n o
P(2/3 system fails) = P A T B† S A T C† S B T C

C +P B
=P A T
T
T
T T
B +P A

B
C 2.P A
C (12)
The results of serial and parallel systems are tabulated in Table 6.1. The probabil-
ity of failures for the different configurations have been calculated for two probabil-
ities of component failure 0.1 and 0.05. The failures are assumed to be mutually
exclusive. The configurations which have been calculated are:
. Serial components, one up to five with Eq. (7) and approximated with Eq. (8)
. Parallel components one up to five with Eq. (11)
. Parallel components two out of three with Eq. (12)
The results of the table clearly lead to the following conclusions:
1. Decrease in probability of component failures give higher system reliability
2. More components in series goes at the expense of system reliability
3. The approximation equation for serial systems is accurate for small failure
probabilities and a limited number of components in series