## 1

When power is expressed in hp, pressure in psia, and capacity in cfm, power can be expressed, based on Equation 30-4, as:

process by multiplying

Fig. 30-22 Theoretical Pressure-Volume Diagram of Helical Screw-Type Compressor. Source: Compressed Air and Gas Institute

Fig. 30-22 Theoretical Pressure-Volume Diagram of Helical Screw-Type Compressor. Source: Compressed Air and Gas Institute

For a positive displacement compressor, at any given rate of flow, the ideal power is affected by the inlet pressure (pj), the ratio of specific heats (k), and the compression ratio (r). The actual power requirement is increased as a result of losses through the intake and discharge valves (or ports). Additional losses result from turbulence in and leakage from the compression chamber and preheating of the inlet gas.

Figure 30-22 is a theoretical pressure-volume diagram of the helical screw-type compressor. The compression process approaches isothermal compression because of the cooling effect of the oil. The power and capacity loss can be represented by the shaded area, although leaking also occurs along the compression line.

Isentropic equations are often used so that isentropic work can be determined from thermodynamic reference charts or tables. However, isentropic, or reversible adia-batic, compression is ideal compression in which there are no internal energy losses due to friction, windage, or throttling and no heat transfer takes place. When internal compressor losses due to friction, windage, and throttling are considered, the process is known as irreversible adia-batic. This represents the actual work required to compress the gas. When thermal losses through the compressor casing are accounted for, the process is known as polytropic. Polytropic compression, therefore, represents the actual compression process.

For ideal gases with constant specific heats, the relationship of isentropic efficiency to polytropic efficiency can be expressed as:

With the volume capacity (V) expressed in m3/s and the pressure (p) in Pascal, Equation 30-4 yields actual power in watts.

Where:

nc = Adiabatic efficiency np = Polytropic efficiency

This relationship applies with reasonable accuracy to real gases that do not deviate greatly from ideal gases because the real-gas errors cancel.

Adiabatic equations can be applied to a polytropic process by multiplying k - 1

This may be expressed as

, where n represents the polytropic coefficient, which can be determined if inlet and discharge temperature and pressures are known.

For positive displacement compressors in which velocities, turbulence, and slip are relatively low, power requirements can be reasonably approximated on an isentropic basis. However, dynamic compressors are more typically evaluated on a polytropic basis. Power, often referred to as gas power, on a volumetric flow basis is expressed as:

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