Rt P

where the ratio P2/P1 is the compression ratio (r).

B. isentropic Compression

For an ideal gas under adiabatic frictionless (i.e., isentropic) conditions, P c R

—r = constant , where k = —; cp = cv H----(8-19)

pr cv M

The specific heat ratio k is approximately 1.4 for diatomic gases (O2, N2, etc.) and 1.3 for triatomic and higher gases (NH3, H2O, CO2, etc.). The corresponding expression for isothermal conditions follows from Eq. (8-17):

Note that the isothermal condition can be considered a special case of the isentropic condition for k = 1. The "constant" in Eq. (8-19) or (8-20) can be evaluated from known conditions at some point in the system (e.g., P1 and T1). Using Eq. (8-19) to eliminate the density from Eq. (8-15) and evaluating the integral leads to

RT1k

Although it is not obvious by inspection, setting k = 1 in Eq. (8-21) reduces that equation to Eq. (8-18) (this follows by application of l'Hospital's rule).

If we compare the work required to compress a given gas to a given compression ratio by isothermal and isentropic processes, we see that the isothermal work is always less than the isentropic work. That is, less energy would be required if compressors could be made to operate under isothermal conditions. However, in most cases a compressor operates under more nearly adiabatic conditions (isentropic, if frictionless) because of the relatively short residence time of the gas in the compressor, which allows very little time for heat generated by compression to be transferred away. The temperature rise during an isentropic compression is determined by eliminating p from Eqs. (8-17) and (8-19):

In reality, most compressor conditions are neither purely isothermal nor purely isentropic but somewhere in between. This can be accounted for in calculating the compression work by using the isentropic equation [Eq. (8-21)], but replacing the specific heat ratio k by a "polytropic" constant, y, where 1 < y < k. The value of y is a function of the compressor design as well as the properties of the gas.

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