1047 pág.

# Fenômentos de Transporte

Pré-visualização50 páginas

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