Page 21 - Modelling in Transport Phenomena A Conceptual Approach

2 CHAPTER 1. INTRODUCTION

The entropy inequality is also a basic concept but it only indicates the feasibility
of a process and, as such, is not expressed as an inventory rate equation.
A rate equation based on the conservation of the value of money can also be
considered as a basic concept, i.e., economics. Economics, however, is outside the
scope of this text.

1.1.1 Characteristics of the Basic Concepts

The basic concepts have certain characteristics that are always taken for granted
but seldom stated explicitly. The basic concepts are

Independent of the level of application,
Independent of the coordinate system to which they are applied,

Independent of the substance to which they are applied.
The basic concepts are applied both at the microscopic and the macroscopic
levels as shown in Table 1.1.

Table 1.1 Levels of application of the basic concepts.
Level Theory Experiment
Microscopic Equations of Change Constitutive Equations
Macroscopic Design Equations Process Correlations

At the microscopic level, the basic concepts appear as partial differential equa
tions in three independent space variables and time. Basic concepts at the mitre
scopic level are called the equations of change, i.e., conservation of chemical species,
mass, momentum and energy.
Any mathematical description of the response of a material to spatial gradients
is called a constitutive equation. Just as the reaction of different people to the same
joke may vary, the response of materials to the variable condition in a process
differs. Constitutive equations are postulated and cannot be derived from the
fundamental principles1. The coefficients appearing in the constitutive equations
are obtained from experiments.
Integration of the equations of change over an arbitrary engineering volume
which exchanges mass and energy with the surroundings gives the basic concepts
at the macroscopic level. The resulting equations appear as ordinary differential
equations with time as the only independent variable. The basic concepts at this
level are called the design equations or macroscopic balances. For example, when
the microscopic level mechanical energy balance is integrated over an arbitrary
'The mathematical form of a constitutive equation is constrained by the second law of ther-
modynamics so as to yield a positive entropy generation.