art of modelling in science and engineering

[...]... invoked in various engineering disciplines: conservation of charge in electrical engineering (Kirchhoff’s law) and conservation of moment, momentum and moment of momentum in mechanical and civil engineering Application of the laws we have chosen to a system or process under consideration leads to equations which are termed balances Thus, the law of conservation of mass leads to the mass balance of a species,... The rate of change of the mass in a tank being filled with water, for example, is given by the instantaneous rate of in ow and leads to a differential equation On the other hand, the actual mass of water in the tank at a given moment equals the cumulative amount introduced to that point and yields an algebraic TABLE 1.4 Categories of Balances and Resulting Equations Names and Model Types A Integral... application of these laws and relations? • What is the role of time, distance, and geometry in the formulation of the model? • Is there a relation between the type of physical process considered and the equations that result? • What type of information can be derived from their solution? These seemingly complex and sweeping questions have, in fact, well-defined and surprisingly simple answers The underlying... laws are those of mass, energy, and momentum Use of these laws is widespread and not confined to chemical engineering systems Fluid mechanics draws heavily on the law of conservation of mass (known there as the continuity equation) and the law of conservation of momentum which in its most general form leads to the celebrated Navier-Stokes equations In nuclear processes, conservation of mass is applied... momentum balances in each of the three coordinate directions Consideration of the independent variable is eased by the common occurrence, in all balances, of time t and the three coordinate directions as independent variables We have summarized these, as well as the relevant dependent variables in Table 1.3 1.2.2 INTEGRAL AND DIFFERENTIAL BALANCES: THE ROLE BALANCE SPACE AND GEOMETRY OF Spatial and geometrical... neutrons and includes diffusive transport as well as a form of reaction when these particles are produced by nuclear fission or absorbed in the reactor matrix The law of conservation of energy appears in various forms in the description of mechanical, metallurgical, nuclear, and other systems and in different areas of applied physics in general We note that conservation laws other than those mentioned are invoked... t Coordinate distances x, y, z Cartesian r, θ, z cylindrical r, θ, ϕ spherical Time t Coordinate distances x, y, z Cartesian r, θ, z cylindrical r, θ, ϕ spherical Time t Coordinate distances x, y, z Cartesian r, θ, z cylindrical r, θ, ϕ spherical is made between processes which are instantaneous in time, leading to differential equations, and those which are cumulative in time, usually yielding algebraic... uncertainties in the model Richard P Feynman (On the reliability of the Challenger space shuttle) Our opening remarks in this preamble are intended to acquaint the reader with some general features of the mathematical models we shall be encountering In particular, we wish to address the following questions: • What are the underlying laws and relations on which the model is based? • What type of equations... Homogeneous PDEs and BCs 7.1.3 PDEs with Variable Coefficients 7.1.4 Linear and Nonlinear PDEs: A New Category — Quasilinear PDEs 7.1.5 Another New Category: Elliptic, Parabolic, and Hyperbolic PDEs 7.1.6 Boundary and Initial Conditions Illustration 7.1.1 Classification of PDEs Illustration 7.1.2 Derivation of Boundary and Initial Condition 7.2 PDEs of Major Importance 7.2.1 First Order Partial Differential... distributions or “profiles” of the state variables in space, or in time and space Thus, a one-dimensional energy balance taken over a differential element of a tube -and- shell heat exchanger will, upon integration, yield the longitudinal temperature profiles in both the shell and the tubes When the balance is taken over a finite entity, we speak of “integral” or “macroscopic” balances, and the underlying models are . separation sciences, biomedical engineering, and applied mathematics with a keen interest in the craft of teaching. His current activities include writing, consulting, and performing science experi- ments. Congress Cataloging -in- Publication Data Basmadjian, Diran The art of modeling in science and engineering / Diran Basmadjian. p. cm. Includes bibliographical references and index. ISBN 1-58488-012-0 1 designed for undergraduate and graduate students, as well as prac- ticing professionals in the sciences and in engineering, with an interest in modeling based on mass, energy and momentum or force