Page 155 - Plant design and economics for chemical engineers

COMPUTER-AIDED DESIGN 129

must have this number of equations, but most models have more, because the
changes occurring in the operation are complex and more equations are needed
to represent these changes. Distillation is an operation for which many equa-
tions must be solved within the module in order to determine the output stream
conditions. The model equations introduce new unknowns as well. These new
unknowns are parameters of the operations that are necessary to relate the
outputs to the inputs. A simple example involves splitting one stream into two
streams with no changes in temperature, pressure, or composition: the split
between the two output streams is a variable in the overall mass-balance
equation, and must be specified, e.g., by fixing the fraction of the input that goes
into one of the outputs.
The total number of stream variables, the number of equations, and the
number of equipment parameters can be summed. and the total degrees of
freedom (unknowns minus equations) then determined. A unique solution to a
problem exists only when the numbers of unknowns and equations are equal.
Therefore, a number of variables equal to the number of degrees of freedom
must be given values so that there will be a unique solution.
Example 3 Degrees of freedom for styrene process. Determine the degrees of
freedom for the styrene process using the process flow diagram, Fig. 4-5.
Solution. There are 8 components (= C’) and 17 streams in the process. Thus
there are
(8 + 2) * 17 = 170 total stream variables
The process has 9 units and there are 15 output streams from these units:
17 - 2 = 15
streams external feeds outputs from
process units
Therefore, there are
15 * 10 = 150
outputs c’+2 mass balance
equations
which results in
170 - 150 = 20
variables equations more stream variables
than equations
Twenty stream variables need to be specified in order for a unique solution to
exist. In principle, any 20 stream variables could be supplied; however, the usual
solution strategy requires that the process feed streams be specified. Specifying the
flow rate of each of the eight components, plus the temperature and pressure of
the two feed streams (ethylbenzene and steam) reduces the number of variables to
150. Hence, a unique solution is available.

Equation Solution
The complete model of a chemical process can consist of hundreds, even
thousands, of equations. The very simple process represented by the styrene