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as the model output the content of the 
label wbem the dart lands. 
Figure 1.1-2 A poor model of the weather 
Several things are immediately obvious about this model. First, it does not 
include anywhere near all possible weather patterns (outputs) - as just one 
example, there is no label for hail on a windy, cool day. Second, it does not 
include all (input) parameters. For example, it incorporates effects of humidity, 
barometric pressure, prevailing winds, etc., only insofar as they might affect the 
Probably the worst model ever used is the frontier method (hopefully apocryphal) 
for weighing hogs. First, they would laboriously tie the squirming hog to the end of a 
long sturdy log, then place the log across a stump for a fulcrum. Next they would 
search for a rock which, when tied to the other end of the log, balanced the weight of 
the hog. Then they would guess the weight of the rock. 
Chapter I : Essentials 3 
person throwing the dart (and hdshe is probably indoors). In fact, the output of 
the model bears no systematic relationship to the inputs. Third, it is not very 
precise in its outputs - it predicts only "windy" or "calm" without giving a wind 
velocity. 
It will, however, give the correct prediction occasionally, if only by chance. 
One should always be alert to the possibility that bad models can 
fortuitously give correct results, even though the probability of this 
occurring is usually vanishingly small. For this reason, statistical methods 
should be used to test models, even though we will not have room here to 
explore this subject thoroughly. 
At the other end of the spectrum are weather prediction models resident on 
large computers, containing hundreds of variables and large systems of partial 
differential equations. If we use such a very sophisticated model, we still need to 
look inside the black box and ask the same questions that we ask of the dart 
board model. (For example, some of the results of chaos theory pose the 
question of whether accurate long-term modeling of the weather is even 
possible.) 
Although engineers and scientists model many of the same physical 
situations, the scientist is interested in the model that gives maximum accuracy 
and best conformity with physical reality, while the engineer is interested in the 
model which offers the precision and accuracy sufficient for an adequate solution 
of the immediate problem without unnecessary expenditure of time and money. 
It should be noted that the best model of a system is the system itself. One 
would like to put the system to be modeled within the black box and use the 
system itself to generate the outputs. One would then obtain the most 
representative outputs.2 
These outputs would be the solution of the true differential equations that 
describe the system behavior, even though from the outside of the box one 
would not know the form of these equations. The system, through its response 
to the inputs, "solves" the equations and presents the solution as the outputs. 
(We will see later, in Chapter 5 , that through dimensional analysis of the inputs 
to the system it is possible to obtain the coefficients in the equations describing 
the system, even absent knowledge of the form of the equations themselves.) 
Even if we conduct measurements on the actual system under the same input 
conditions we still must face questions of reproducibility and experimental error. 
4 Chapter I : Essentials 
If we cannot make measurements on the system itself" under the desired 
inputs, we must replace the system with a more convenient model. One of two 
avenues is usually chosen: either a scaled laboratory model or a mathematical 
model. The choice of an appropriately scaled laboratory model and conditions can 
be accomplished with the techniques presented in Chapter 5 . Using a 
mathematical model presents the problem of including all the important effects 
while retaining sufficient simplicity to permit the model to be solved. 
Originally, engineering mathematical models could be formulated solely in 
terms of intuitive quantities such as size, shape, number, etc. For example, it 
was intuitively obvious to the designer of a battering ram that if he made it 
twice as heavy andor had twice as many men swing it, the gates to the city 
would yield somewhat sooner. This happy but unsophisticated state of affairs 
has evolved into one aesthetically more pleasing (more elegant and more 
accurate) but intuitively less satisfying. 
Nature, though simple in detail, has proved to be very complex in 
aggregate. Describing the effect of a change in crude oil composition on the 
product distribution of a refinery is a far cry from deducing the effect of adding 
another man to the battering ram. This leaves us with a choice - we may model 
the processes of nature either by complex manipulations of simple relationships, 
or we may avoid complexity in the manipulations by making the relationships 
more abstract. 
For example, in designing a pipeline to carry a given flow rate of a liquid, 
we might wish to know the pressure drop required to force the liquid through the 
pipeline at a given rate. One approach to this problem would be to take a series 
of pipes of varying roughness, diameter and length, apply various pressure drops, 
and measure the resulting flow rates using various liquids. The information 
could be collected in an encyclopedic set of tables or graphs, and we could design 
pipeline flow by looking up the exact set of conditions for each design. The 
work involved would be extreme and the information would be hard to store and 
retrieve, but one would need only such intuitively satisfying concepts as 
diameter, length and flow rate.4 If, however, we are willing to give up these 
elementary concepts in favor of two abstractions called the Reynolds number 
and the friction factor, we can gather all the information needed in twenty or 
For example, the system may be inaccessible, as in the case of a reservoir of crude 
oil far beneath the ground; a hostile environment, as in the case of the interior of a 
nuclear reactor; or the act of measurement may destroy the system, as in explosives 
research. 
To run all combinations of only 10 different pipe diameters, 10 pressure drops and 
100 fluids would require 10 x 10 x 100 = 10,OOO experiments - and we still would have 
completely neglected important variables such as pipe roughness. 
Chapter I : Essentials 5 
thirty experiments and summarize it on a single page. (We will learn how this is 
done in Chapter 6.) 
1.1.1 Mathematical models and the real world 
As the quantities we manipulate grow more abstract, our rules for 
manipulating these quantities grow more formal; that is, more prescribed by that 
system of logic called mathematics, and less by that accumulation of experience 
and application of analogy which we call physical intuition. We are thus driven 
to methods whereby we replace the system by a mathematical model, operate on 
the model with minimal use of physical intuition, and then apply the result of 
the model to the system. Obviously, the better the model incorporates the desired 
features of the system, the better our conclusions will be, and conversely. We 
must always be aware of the fact that the model is not the real world, a fact all 
too easy to ignore (to the detriment of our conclusions and recommendations) as 
the model becomes more and more abstract. 
This procedure is exemplified by the way one balances a checkbook. The 
symbols in the checkbook do not come from observing the physical flow of 
money to and from the bank, but, if the manipulations are performed correctly, 
the checkbook balance (the output of the model) will agree with the amount in 
the bank at the end of the month. In this case the mathematical model represents 
one aspect of the real world exactly