Page 150 - Chemical process engineering design and economics

Process Circuit Analysis 133

At this point in the analysis we do not know if the variables are over-
specified or under-specified. Table 3.5.3 gives the degrees of freedom for each
process unit. As usual prime the specified variables. Except for the splitter, the
analysis is straight forward. Since there is no composition change across the split-
ter, as stated by Equations 3.5.30 to 3.5.39, only the total mole balance is an inde-
pendent equation. Also, only the sum of the mole fractions for one of the three
streams is an independent equation. Table 3.5.3 shows that no process unit con-
tains zero degrees of freedom.
Before attempting to solve the equations in Table 3.5.3, calculate the de-
grees of freedom for the process. First, determine the number of unique variables
because some of the variables are repeated from process unit to process unit, as
shown in Table 3.5.3. The process variables are equal to the sum of all the unit
variables minus the repeated variables. To determine the repeated variables, ex-
amine the lines connecting the process units. Table 3.5.4 shows that the repeated
variables are mole fractions and molar flow rates. From Table 3.5.5, the total
number of variables for all units is 57, and the total number of repeated variables
is 23. Therefore, the number of unique process variables are 34, as shown in Table
3.5.5.
Next, determine the number of independent equations by again examining
each connecting line. The repeated equations are the mole fraction summations, as
shown in Table 3.5.4. To determine the number of independent equations for the
process, subtract the repeated equations from the sum of the equations for all the
process units. The total number of equations for all process units is 39, as shown in
Table 3.5.5. Although each process unit contains positive degrees of freedom, we
see that the process degrees of freedom equals minus two, which means that the
problem has been overspecified. Before unspecifying variables check if the num-
ber of equations are correct. By inspection - not an easy task - we find that Z y?.i
= 1, Equation 3.5.29 in Table 3.5.3, is not independent. It can be derived by sub-
stituting Equations 3.5.30 to 3.5.34 into Equation 3.5.6, Z yy- Therefor, the num-
ber of independent equations must be reduced by 1 - from 36 to 35 - and the de-
grees of freedom becomes minus one.
Once you are certain that all equations are independent and no equations are
missing, then unspecify one of the variables. For example, unspecify the nitrogen
concentration at the converter inlet, y 3;6. Because y 3;6 is now unspecified, correct
the degree of freedom analysis for both the mixer and converter. At the mixer and
converter the number of variables increases by one as shown in Table 3.5.6. Thus,
for the mixer F = 1 2 - 7 = 5 and for the converter F = 14-9 = 5. Because Equa-
tions 3.5.27 and 3.5.29 are not independent, the number of equations at the con-
denser-separator combination and the splitter are reduced by one, as shown in Ta-
ble 3.5.6. Finally, because Z y?,i is no longer valid, it is not a repeated equation.
Thus, the repeated equations in line 7 are now zero. The revised calculation
for the degrees of freedom in Table 3.5.6 shows that the process degrees of free-
dom is now zero.

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