Thermodynamics

368 | ThermodynamicsPP 22Work savedTP 2P xP 1Polytropic2P xIntercoolingT 11IsothermalIntercoolingFIGURE 7–46P-v and T-s diagrams for a two-stagesteady-flow compression process.P 11vsThe size of the colored area (the saved work input) varies with the valueof the intermediate pressure P x , and it is of practical interest to determinethe conditions under which this area is maximized. The total work input fora two-stage compressor is the sum of the work inputs for each stage of compression,as determined from Eq. 7–57b:w comp,in w comp I,in w comp II,in(7–58) nRT 1n12>n1n 1 caP xb 1 d nRT 1n12>n1P 1 n 1 caP 2b 1 dP xThe only variable in this equation is P x . The P x value that minimizes thetotal work is determined by differentiating this expression with respect to P xand setting the resulting expression equal to zero. It yieldsP x 1P 1 P 2 2 1>2 or P xP 1 P 2P x(7–59)That is, to minimize compression work during two-stage compression, thepressure ratio across each stage of the compressor must be the same. Whenthis condition is satisfied, the compression work at each stage becomesidentical, that is, w comp I,in w comp II,in .EXAMPLE 7–13Work Input for Various Compression ProcessesAir is compressed steadily by a reversible compressor from an inlet state of100 kPa and 300 K to an exit pressure of 900 kPa. Determine the compressorwork per unit mass for (a) isentropic compression with k 1.4, (b) polytropiccompression with n 1.3, (c) isothermal compression, and (d) ideal twostagecompression with intercooling with a polytropic exponent of 1.3.Solution Air is compressed reversibly from a specified state to a specifiedpressure. The compressor work is to be determined for the cases ofisentropic, polytropic, isothermal, and two-stage compression.