Page 141 - Plant design and economics for chemical engineers

116 PLANT DESIGN AND ECONOMICS FOR CHEMICAL ENGINEERS

Clearly the values have changed since the previous calculation; convergence has
not been attained. The user requests another calculation with the result
b C
6 153.3 65.7
7 153.3 0.0
8 0.0 65.7

Again the values have changed, so the calculations are repeated. Calculations are
continued until the user is satisfied that no further “significant” change has
occurred (the definition of significant is up to the user; a maximum change in
values between iterations of 0.01 percent is a reasonable criterion).
The final results (to 0.01 percent) for this example are
b c
6 233.31 99.99
7 333.31 0.0
8 0.0 99.99

The exact values for this problem (by algebraic solution) are 233$, 100, and 3334,
respectively.
If, after several manual trials, the calculations appear to be converging, it is
reasonable to switch from manual to automatic iteration. It is strongly
recommended that the calculations always be checked by a simple hand calculation.
In this example, for instance, the user may add the feed and recycle carbon
monoxide flow rates, multiply the sum by the conversion, and compare the result
with the tabulated value for the methanol product rate c8.
A “scratch pad” can be very useful for a spread-sheet user. A scratch pad is
an area (e.g., columns to the right of the main spread sheet) set aside for doing
side calculations of parameters or constants for the main spread sheet, writing
comments, locating macros, and other uses. A scratch pad would be useful in this
example if the user needed mass balances for a particular product flow rate rather
than a specified feed rate as just illustrated. The problem has been solved for a
carbon dioxide feed rate of 100 mol/h as shown above, but suppose it is desired to
know the feed rate required to produce 880 mol/h of methanol product. The
problem might intentionally be solved first for a specified feed rate, because it is
often more convenient to write the mass-balance equations for known process feed
rates rather than known product rates.
The preceding results may be converted to the basis of the new product flow
rate. Mass-balance equations are linear, so that if the product flow rate is doubled,
for example, all flow rates in the process are doubled. In this example then, the
new flow rates are obtained by multiplying the previously calculated values by the
ratio of the new-to-previous product-stream flow rates (880/99.99) = 8.801. This
ratio can be calculated in the scratch pad (say location e3), the new CO feed rate,
880.1, entered into a3, and the iterative calculations repeated until convergence is
attained. The iteration could be avoided, however, if each mass balance had been
entered as shown above, but multiplied by a coefficient e3 (the scratch-pad
location). Initially, e3 would have been set equal to 1 and the calculations would
have proceeded just as shown above. The user then enters 88O/c8 for location e3,
repeats the calculation, and obtains the final flow rates without iteration.