Page 22 - Modelling in Transport Phenomena A Conceptual Approach

1.2. DEFINITIONS 3

engineering volume, the result is the macroscopic level engineering Bernoulli equa-
tion.
Constitutive equations, when combined with the equations of change, may or
may not comprise a determinate mathematical system. For a determinate math-
ematical system, Le., number of unknowns = number of independent equations,
the solutions of the equations of change together with the constitutive equations
result in the velocity, temperature, pressure, and concentration profiles within the
system of interest. These profiles are called theoretical (or, analytical) solutions. A
theoretical solution enables one to design and operate a process without resorting
to experiments or scaleup. Unfortunately, the number of such theoretical solutions
is small relative to the number of engineering problems which must be solved.
If the required number of constitutive equations is not available, i.e., number of
unknowns > number of independent equations, then the mathematical description
at the microscopic level is indeterminate. In this case, the design procedure appeals
to an experimental information called process correlation to replace the theoretical
solution. All process correlations are limited to a specific geometry, equipment
configuration, boundary conditions, and substance.

1.2 DEFINITIONS

The functional notation
cp = cp (t, 2, Y, (1.2-1)
indicates that there are three independent space variables, x, y, z, and one inde-
pendent time variable, t. The cp on the right side of Eq. (1.2-1) represents the
functional form, and the cp on the left side represents the value of the dependent
variable, cp.

1.2.1 S teady-S tat e

The term steady-state means that at a particular location in space, the dependent
variable does not change as a function of time. If the dependent variable is cp, then

(1.2-2)

The partial derivative notation indicates that the dependent variable is a func-
tion of more than one independent variable. In this particular case, the independent
variables are (z, y, z) and t. The specified location in space is indicated by the
subscripts (2, y, z) and Eq. (1.2-2) implies that cp is not a function of time, t.
When an ordinary derivative is used, Le., dpldt = 0, then this implies that cp is a
constant. It is important to distinguish between partial and ordinary derivatives
because the conclusions are very different.