6.12 A flow rate of 32,000 cfm of gas at a temperature 120°F and a pressure 25 psia is to be compressed to 2500 psia. Under these conditions, the gas behaves according to the ideal gas law. The choice of compressor type is influenced by the fact that centrifugal compressors can handle high-volume flow rates but develop only low-pressure ratios per stage. A reciprocating compressor, on the other hand, is suited to low-volume flow rates but can develop high-pressure ratios. To combine the advantages of each, the compression will be done using a low-stage centrifugal compressor and a high-stage reciprocating compressor. Between the compressor stages, an intercooler is used to return the gas temperature to 120°F.

A sketch of this dual-stage compressor system is shown in Figure P6.12.

The first cost of each compressor, expressed in terms of volume flow rates and pressure ratios are given by the following equations:
= a V + a P P ICc 1 1 2 (centrifugal compressor)
2 1 = b V + b P P ICr 1 3 2 (reciprocating compressor)
4 3 In the abovementioned equations, a1 = $0.03/cfm, a2 = $1600, b1 = $0.09/cfm, and b2 = $800. Using the Lagrange Multiplier Theorem, determine the following:

a. The minimum first cost of the compressors

b. The pressure ratios across each compressor to achieve this optimum condition

c. Determine the parameters in parts

a. and

b. using a numerical optimization algorithm. How does the numerical solution compare to the solution using the Lagrange Multiplier Theorem?