Optimizing Product Quality Performance ◾  59
7.2 Frequency of Failure
The quality performance of a stream from a distillation column can be evaluated by the average impurity concentration
in the stream and the distance from the nearest specification
limit (DNS) compared to the standard deviation (Equation 7.1).
The value of DNS is three times the process capability index,
C pk . The DNS value is easy to communicate in the number of
sigma units:
DNS = (USL – X a )/σ
(7.1)
where:
DNS = distance from nearest specification limit
USL = upper specification limit
X a = average key impurity concentration
σ = standard deviation (sigma)
When the concentration of impurity follows a normal distribution in samples, the frequency of failure rate can be
calculated (Table 7.1). These values are taken from a single
side of the normal Gaussian error distribution, which can
be found in statistics books such as the one by Montgomery
and Runger. 1
When samples of distillate and bottoms are taken once
every 8-hour shift, there are 21 samples per week and 1,095
samples per year. If the samples follow a normal distribution
and the average impurity concentration is 2σ, that is, DNS =
two standard deviations, below the upper specification limit
(USL) there would be one shift sample out of spec about every
2 weeks. If the average were 3σ below the upper specification
limit, there would be one shift sample out of spec about every
9 months from common cause variability. If the average were
4σ below the upper specification limit, there would be one
shift sample out of spec about every 27 years.