Thermodynamics

Chapter 7 | 367Isentropic (Pv k constant):w comp,in kR 1T 2 T 1 2k 1 kRT 1k12>k1k 1 caP 2b 1 dP 1(7–57a)Polytropic (Pv n constant):w comp,in nR 1T 2 T 1 2n 1 nRT 1n12>n1n 1 caP 2b 1 dP 1(7–57b)Isothermal (Pv constant):PP 2w comp,in RT lnP 1(7–57c)The three processes are plotted on a P-v diagram in Fig. 7–45 for thesame inlet state and exit pressure. On a P-v diagram, the area to the left ofthe process curve is the integral of v dP. Thus it is a measure of the steadyflowcompression work. It is interesting to observe from this diagram that ofthe three internally reversible cases considered, the adiabatic compression(Pv k constant) requires the maximum work and the isothermal compression(T constant or Pv constant) requires the minimum. The workinput requirement for the polytropic case (Pv n constant) is between thesetwo and decreases as the polytropic exponent n is decreased, by increasingthe heat rejection during the compression process. If sufficient heat isremoved, the value of n approaches unity and the process becomes isothermal.One common way of cooling the gas during compression is to usecooling jackets around the casing of the compressors.P 2Isentropic (n = k)Polytropic (1 < n < k)Isothermal (n = 1)P 1FIGURE 7–45P-v diagrams of isentropic, polytropic,and isothermal compression processesbetween the same pressure limits.1vMultistage Compression with IntercoolingIt is clear from these arguments that cooling a gas as it is compressed is desirablesince this reduces the required work input to the compressor. However,often it is not possible to have adequate cooling through the casing of thecompressor, and it becomes necessary to use other techniques to achieveeffective cooling. One such technique is multistage compression with intercooling,where the gas is compressed in stages and cooled between each stageby passing it through a heat exchanger called an intercooler. Ideally, the coolingprocess takes place at constant pressure, and the gas is cooled to the initialtemperature T 1 at each intercooler. Multistage compression with intercoolingis especially attractive when a gas is to be compressed to very high pressures.The effect of intercooling on compressor work is graphically illustrated onP-v and T-s diagrams in Fig. 7–46 for a two-stage compressor. The gas iscompressed in the first stage from P 1 to an intermediate pressure P x , cooled atconstant pressure to the initial temperature T 1 , and compressed in the secondstage to the final pressure P 2 . The compression processes, in general, can bemodeled as polytropic (Pv n constant) where the value of n varies betweenk and 1. The colored area on the P-v diagram represents the work saved as aresult of two-stage compression with intercooling. The process paths for singlestageisothermal and polytropic processes are also shown for comparison.