Page 24 - Modelling in Transport Phenomena A Conceptual Approach

1.3. MATHEMATICAL FORMULATION OF THE BASIC CONCEPTS 5

1.2.2 Uniform

The term uniform means that at a particular instant in time, the dependent vari-
able is not a function of position. This requires that all three of the partial deriva-
tives with respect to position be zero, i.e.,

(1.2-3)

The variation of a physical quantity with respect to position is called gradient.
Therefore, the gradient of a quantity must be zero for a uniform condition to exist
with respect to that quantity.

1.2.3 Equilibrium
A system is in equilibrium if both steady-state and uniform conditions are met si-
multaneously. An equilibrium system does not exhibit any variations with respect
to position or time. The state of an equilibrium system is specified completely by
the non-Euclidean coordinates2 (P, V, T). The response of a material under equi-
librium conditions is called property correlation. The ideal gas law is an example
of a thermodynamic property correlation that is called an equation of state.

1.2.4 Flux
The flux of a certain quantity is defined by

Flow of a qusntity/Time - Flow rate
Flux = - (1.2-4)
Area Area
where area is normal to the direction of flow. The units of momentum, energy,
mass and molar fluxes are Pa ( N/ m2, or kg/ m. s2), W/ m2 ( J/ m2. s), kg/m2. s,
and kmol/ m2. s, respectively.

1.3 MATHEMATICAL FORMULATION OF
THE BASIC CONCEPTS

In order to obtain the mathematical description of a process, the general inventory
rate equation given by Eq. (1.1-1) should be translated into mathematical terms.

1.3.1 Inlet and Outlet Terms

A quantity may enter or leave the system by two means: (i) by inlet and/or outlet
streams, (ii) by exchange of a particular quantity between the system and its
2A Euclidean coordinate system is one in which length can be defined. The coordinate system
(P, V, T) is non-Euclidean.