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1-3 MECHANISTIC AND EMPIRICAL MODELS 11

actually fall below the lower control limit. This is a very strong signal that corrective action is
required in this process. If we can find and eliminate the underlying cause of this upset, we can
improve process performance considerably.
Control charts are a very important application of statistics for monitoring, controlling,
and improving a process. The branch of statistics that makes use of control charts is called sta-
tistical process control, or SPC. We will discuss SPC and control charts in Chapter 16.

1-3 MECHANISTIC AND EMPIRICAL MODELS

Models play an important role in the analysis of nearly all engineering problems. Much of the
formal education of engineers involves learning about the models relevant to specific fields
and the techniques for applying these models in problem formulation and solution. As a sim-
ple example, suppose we are measuring the flow of current in a thin copper wire. Our model
for this phenomenon might be Ohm’s law:

Current voltage resistance

or

I E R (1-2)

We call this type of model a mechanistic model because it is built from our underlying
knowledge of the basic physical mechanism that relates these variables. However, if we
performed this measurement process more than once, perhaps at different times, or even on
different days, the observed current could differ slightly because of small changes or varia-
tions in factors that are not completely controlled, such as changes in ambient temperature,
fluctuations in performance of the gauge, small impurities present at different locations in the
wire, and drifts in the voltage source. Consequently, a more realistic model of the observed
current might be

I E R (1-3)

where is a term added to the model to account for the fact that the observed values of
current flow do not perfectly conform to the mechanistic model. We can think of as a
term that includes the effects of all of the unmodeled sources of variability that affect this
system.
Sometimes engineers work with problems for which there is no simple or well-
understood mechanistic model that explains the phenomenon. For instance, suppose we are
interested in the number average molecular weight (M n ) of a polymer. Now we know that M n
is related to the viscosity of the material (V), and it also depends on the amount of catalyst (C)
and the temperature (T) in the polymerization reactor when the material is manufactured. The
relationship between M n and these variables is

M f 1V, C, T 2 (1-4)
n
say, where the form of the function f is unknown. Perhaps a working model could be devel-
oped from a first-order Taylor series expansion, which would produce a model of the form

V C T (1-5)
M n 0 1 2 3