Page 236 - Statistics for Environmental Engineers

L1592_frame_C27.fm Page 239 Tuesday, December 18, 2001 2:47 PM

27

Factorial Experimental Designs

KEY WORDS additivity, cube plot, density, design matrix, effect, factor, fly ash, factorial design,
interaction, main effect, model matrix, normal order scores, normal plot, orthogonal, permeability,
randomization, rankits, two-level design.

Experiments are performed to (1) screen a set of factors (independent variables) and learn which produce
an effect, (2) estimate the magnitude of effects produced by changing the experimental factors, (3)
develop an empirical model, and (4) develop a mechanistic model. Factorial experimental designs are
efficient tools for meeting the first two objectives. Many times, they are also excellent for objective three
and, at times, they can provide a useful strategy for building mechanistic models.
Factorial designs allow a large number of variables to be investigated in few experimental runs. They
have the additional advantage that no complicated calculations are needed to analyze the data produced.
In fact, important effects are sometimes apparent without any calculations. The efficiency stems from
using settings of the independent variables that are completely uncorrelated with each other. In mathe-
matical terms, the experimental designs are orthogonal. The consequence of the orthogonal design is
that the main effect of each experimental factor, and also the interactions between factors, can be
estimated independent of the other effects.

Case Study: Compaction of Fly Ash

There was a proposal to use pozzolanic fly ash from a large coal-fired electric generating plant to build
impermeable liners for storage lagoons and landfills. Pozzolanic fly ash reacts with water and sets into
a rock-like material. With proper compaction this material can be made very impermeable. A typical
−7
criterion is that the liner must have a permeability of no more than 10 cm/sec. This is easily achieved
using small quantities of fly ash in the laboratory, but in the field there are difficulties because the rapid
pozzolanic chemical reaction can start to set the fly ash mixture before it is properly compacted. If this
−7
happens, the permeability will probably exceed the target of 10 cm/sec.
As a first step it was decided to study the importance of water content (%), compaction effort (psi),
and reaction time (min) before compaction. These three factors were each investigated at two levels. This
is a two-level, three-factor experimental design. Three factors at two levels gives a total of eight experi-
mental conditions. The eight conditions are given in Table 27.1, where W denotes water content (4% or
10%), C denotes compaction effort (60 psi or 260 psi), and T denotes reaction time (5 or 20 min). Also
3
given are the measured densities, in lb/ft . The permeability of each test specimen was also measured.
The data are not presented, but permeability was inversely proportional to density. The eight test specimens
were made at the same time and the eight permeability tests started simultaneously (Edil et al., 1987).
The results of the experiment are presented as a cube plot in Figure 27.1. Each corner of the cube
represents one experimental condition. The plus (+) and minus (−) signs indicate the levels of the factors.
The top of the cube represents the four tests at high compression, whereas the bottom represents the
four tests at low pressure. The front of the cube shows the four tests at low reaction time, while the back
shows long reaction time.
It is apparent without any calculations that each of the three factors has some effect on density. Of
the investigated conditions, the best is run 4 with high water content, high compaction effort, and short

© 2002 By CRC Press LLC