Page 29 - Applied statistics and probability for engineers

Section 1-2/Collecting Engineering Data 7

5"#-& t 1-1 The Designed Experiment (Factorial Design) for the Distillation Column
Reboil Temp. Condensate Temp. Re ux Rate
–1 –1 –1
+1 –1 –1
–1 +1 –1
+1 +1 –1
–1 –1 +1
+1 –1 +1
–1 +1 +1
+1 +1 +1

speciic inferences about the effect of these factors. Such an approach allows us to proactively
study a population or process.
Interaction can be a An important advantage of factorial experiments is that they allow one to detect an interac-
Key Element in tion between factors. Consider only the two temperature factors in the distillation experiment.
Problem Solving Suppose that the response concentration is poor when the reboil temperature is low, regardless
of the condensate temperature. That is, the condensate temperature has no effect when the reboil
temperature is low. However, when the reboil temperature is high, a high condensate tempera-
ture generates a good response, but a low condensate temperature generates a poor response.
That is, the condensate temperature changes the response when the reboil temperature is high.
The effect of condensate temperature depends on the setting of the reboil temperature, and these
two factors are said to interact in this case. If the four combinations of high and low reboil and
condensate temperatures were not tested, such an interaction would not be detected.
We can easily extend the factorial strategy to more factors. Suppose that the engineer wants
to consider a fourth factor, type of distillation column. There are two types: the standard one
and a newer design. Figure 1-6 illustrates how all four factors—reboil temperature, conden-

sate temperature, relux rate, and column design—could be investigated in a factorial design.
Because all four factors are still at two levels, the experimental design can still be represented
geometrically as a cube (actually, it’s a hypercube). Notice that as in any factorial design, all
possible combinations of the four factors are tested. The experiment requires 16 trials.
Generally, if there are k factors and each has two levels, a factorial experimental design will
4
k
require 2 runs. For example, with k = 4, the 2 design in Fig. 1-6 requires 16 tests. Clearly, as the
number of factors increases, the number of trials required in a factorial experiment increases rap-
idly; for instance, eight factors each at two levels would require 256 trials. This quickly becomes
unfeasible from the viewpoint of time and other resources. Fortunately, with four to i ve or more
factors, it is usually unnecessary to test all possible combinations of factor levels. A fractional
factorial experiment is a variation of the basic factorial arrangement in which only a subset of the
factor combinations is actually tested. Figure 1-7 shows a fractional factorial experimental design
for the four-factor version of the distillation experiment. The circled test combinations in this i gure
are the only test combinations that need to be run. This experimental design requires only 8 runs
instead of the original 16; consequently it would be called a one-half fraction. This is an excellent
experimental design in which to study all four factors. It will provide good information about the
individual effects of the four factors and some information about how these factors interact.

Reflux rate +1 +1
temperature
FIGURE 1-5 The Condensate
factorial design –1 –1
for the distillation –1 +1
column. Reboil temperature