Page 33 - Coulson Richardson's Chemical Engineering Vol.6 Chemical Engineering Design 4th Edition

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Input
Unit
streams CHEMICAL ENGINEERING Output
streams
Input Calculation Output
information method information

Figure 1.6. The “design unit”

representing the unit are shown diagrammatically in Figure 1.6. In the “design unit” the
flow of material is replaced by a flow of information into the unit and a flow of derived
information from the unit.
The information flows are the values of the variables which are involved in the design;
such as, stream compositions, temperatures, pressure, stream flow-rates, and stream
enthalpies. Composition, temperature and pressure are intensive variables: independent of
the quantity of material (flow-rate). The constraints on the design will place restrictions on
the possible values that these variables can take. The values of some of the variables will
be fixed directly by process specifications. The values of other variables will be determined
by “design relationships” arising from constraints. Some of the design relationships will
be in the form of explicit mathematical equations (design equations); such as those
arising from material and energy balances, thermodynamic relationships, and equipment
performance parameters. Other relationships will be less precise; such as those arising
from the use of standards and preferred sizes, and safety considerations.
The difference between the number of variables involved in a design and the number
of design relationships has been called the number of “degrees of freedom”; similar to the
use of the term in the phase rule. The number of variables in the system is analogous to the
number of variables in a set of simultaneous equations, and the number of relationships
analogous to the number of equations. The difference between the number of variables
and equations is called the variance of the set of equations.
If N v is the number of possible variables in a design problem and N r the number of
design relationships, then the “degrees of freedom” N d is given by:

N d D N v N r 1.1
N d represents the freedom that the designer has to manipulate the variables to find the
best design.
If N v D N r ,N d D 0 and there is only one, unique, solution to the problem. The problem
is not a true design problem, no optimisation is possible.
If N v <N r ,N d < 0, and the problem is over defined; only a trivial solution is possible.
If N v >N r ,N d > 0, and there is an infinite number of possible solutions. However,
for a practical problem there will be only a limited number of feasible solutions. The
value of N d is the number of variables which the designer must assign values to solve
the problem.
How the number of process variables, design relationships, and design variables defines
a system can be best illustrated by considering the simplest system; a single-phase, process
stream.