Page 244 - Design of Simple and Robust Process Plants

230 Chapter 6 Process Design Based on Reliability
6.2.2.3 The exponential distribution
This distribution is frequently applicable to repairable systems, and is frequently
used in reliability and safety studies. It implies that the failure rates are constant
and independent, and so the mean time between consecutive failures is constant.
Thus, the MTTF is equal to the MTBF. To illustrate this, when the bearings of a
mechanical device fail and are repaired, the mean time to the next failure remains
the same.
The exponential expression is R(t) = e ±k.t or F(t) = 1 ± R(t) = 1± e ±k.t ,
where k is a constant failure rate.
Then, the expected number of failures in operating time period t is k t.

6.2.2.4 The normal distribution
The normal distribution is in relation to a stochastic variable which can have values
between 1 and ±1. The probability density function is:
1 1 t l 2

f(t) = p  exp ± 1 <t< 1
r 2 2 r
where the variable is t,
2
The mean value is l, and the variance Var(t) = r .

6.2.2.5 The log normal distribution
This distribution is often applicable to reliability studies. The variable t is said to be
log normal distributed if ln(t) is normally distributed. The probability density func-
tion is now
1 ln t† l† 2
f(t) = p  exp 2
tr 2 2r
2

x mean = exp (l + 0.5r ) Var(t) = exp 2l ‡ r 2 exp r 2 1

6.2.2.6 The Weibull distribution
The Weibull distribution has a specific characteristic in that the distribution does
not have a specific shape, but can have different shapes depending on the selection
of a set of parameters. The distribution is commonly used in reliability analysis as it
can describe decreasing, as well as increasing, failure rates. Earlier, we referred to
the aging of components that are often covered in Weibull distributions.
The Weibull distribution for two parameters is defined as:
b 1 " b #
b t t
F(t) = exp ,0 £ t< 1
g g g
Where b is called the shape parameter (dimensionless), and g is called the scale pa-
rameter (dimension t). Increasing and decreasing failure rates are realized by the
choice of the shape parameter b, where: