Compressors polytropic compression

To start, convert the flow to values estimated to be the compressor inlet conditions. Initially, the polytropic head equation (Equation 2.73) will be used with n as the polytropic compression exponent. If prior knowledge of the gas indicates a substantial nonlinear tendency, the real gas compression exponent (Equation 2.76) should be substituted. As discussed m Chapter 2, an approximation may be made by using the linear average ut the inlet and outlet k values as the exponent or for the determination of the polytropic exponent. If only the inlet value of k is known, don t be too concerned. The calculations will be repeated several times as knowledge of the process for the compression cycle is developed. After selecting the k value, u,se Equation 2.71 and an estimated stage efficiency of 15 / to de clop the polytropic compression exponent n. 

For centrifugal and axial compressors, the compression path is usually based on polytropic compression. The power required for a polytropic compression can be expressed as... 

To size a conq)ressor requires calculating the power needed for compression. This can be done by assuming an isentropic compression and then correcting the result by dividing by an isentropic efficiency. The power can also be calculated by assuming a polytropic compression, and then correcting the result by dividing by a polytropic efficiency. Both methods will be considered. The isentropic method is also used for blowers and vacuum pumps, but the polytropic method could also be used if data were available. First, we need to derive relationships to calculate the compressor power. 

Pcp compressor power for a polytropic compression Pe electric motor power Pp fan power Pp pump power Pj turbine power R gas constant s entropy T absolute temperature... 

Polytropic compression. In large compressors the path of the fluid is neither isothermal nor adiabatic. The process may still be assumed to be frictionless, however. It is customary to assume that the relation between pressure and density is given by the equation... 

Distillation column with top vapour recompression heat pump The flow diagram of the top vapour recompression scheme is shown in Fig. 2. The top column outlet stream is compressed with compressor to raise its temperature and promoting its energy content to be more usable. When the top column pressure is 21.6 bar, the temperature is increased from 81 to 139.9 C and also the pressure is increased from 21.6 to 52.6 bar. The compressor polytropic efficiency was assumed to be 70%. After the compressor, the heat exchanger allows transfer of the energy of this stream to boil up the bottom column outlet stream. With the same top column pressure and reflux ratio as before, the compressor outlet stream is condensed and cooled to 109.2 C, while the bottom column outlet stream is partially boiled. 

Polytropic system. Figures 12-37D and 12-37E, gets nearer to the conditions of a practical system, with polytropic process having n = 1.2. The gray area Wi compresses air from 1 to 2 while the air rejects heat to atmosphere through compressor cylinder walls. equals area 1-2-y-z in 4e. In the aftercooler-receiver, the air rejects heat Q i equal to gray area 2-3-w-y. 

The gas compression in practically all commercial machines is polytropic. That is, it is not adiabatic or isothermal, but some form peculiar to the gas properties and the hydraulic design of the compressor. Actual machines may be rated on adiabatic performance and then related to polytropic conditions by the polytropic efficiency. Other performance rating procedures handle the calculations as polytropic. For reference, both methods are presented. 

The usual centrifugal compressor is uncooled internally, and hence, operates with polytropic characteristics having n greater than k however, if the unit is internally cooled, then n will be greater than 1.0 but may be less than k. The inefficiencies caused by internal losses (friction, etc.) keep the operation from being truly adiabatic however, some compressions are close to this condition and may be used for approximations. 

Calculate the compression ratio, Pd/Ps = P2/P1 = Pc-From Table 12-9B for the compressor frame selected, select polytropic efficiency, Cp, and using Figure 12-65A, determine adiabatic efficiency, e d. 

In industrial compressors or expanders the compression or expansion path will be polytropic , approximated by the expression ... 

A poly tropic compression is neither adiabatic nor isothermal, but specific to the physical properties of the gas and the design of the compressor. The polytropic coefficient n must therefore be determined experimentally. If the initial and final conditions for a given compression process are known, then n can be determined from a rearrangement of Equation B.23 ... 

Whereas reciprocating compressors are normally designed on the basis of adiabatic work (together with an isentropic efficiency), centrifugal compressors are usually designed on the basis of polytropic work. By analogy with Equation B.15, the work required for a poly tropic compression is given by ... 

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. 

The thermodynamic state of liquids is a changing polytropic one (close to isentropic) during quick compression (see Fig.4.1-1, for C02) and the corresponding temperature rise is low for liquids and large for gases (see AB in Fig. 4.1-1). The final compression temperature limits the compression ratio per stage with gas compressors. 

For reciprocating positive pumps and compressors the fluid displacement per stroke depends on the geometry, the fluid compressibility, the polytropic coefficient, and the internal leakages and volumetric losses, but basically not on the speed of machine. 

The centrifugal compressor, unless it is dirty or mechanically defective, has to operate on its curve. As the compressor discharge pressure increases, then Hp, the feet of polytropic head required, must also increase. Also, as can be seen from the compressor curve, the volume of gas compressed (ACFM) must decrease. When the volume of gas drops below a critical flow, the compressor will be backed up to its surge point. 

Figure 7.27. Efficiencies of centrifugal and reciprocating compressors, (a) Polytropic efficiencies of centrifugal compressors as a function of suction volume and compression ratio (Clark Brothers Co.), (b) Relation between isentropic and polytropic efficiencies, Eqs. (7.22) (7.23). (c) Isentropic efficiencies of reciprocating compressors (De Laval Handbook, McGraw-Hill, New York, 1970). Multiply by 0.95 for motor drive. Gas engines require 7000-8000 Btu/HP.

Polytropic efficiencies are obtained from measurements of power consumption of test equipment. They are essentially independent of the nature of the gas. As the data of Figure 7.27 indicate, however, they are somewhat dependent on the suction volumetric rate, particularly at low values, and on the compression ratio. Polytropic efficiencies of some large centrifugal compressors are listed in Table 7.6. These data are used in Example 7.9 in the selection of a machine for a specified duty. 

The exponent n depends on the amount of cooling, mechanical friction, and fluid friction dining compression. It is determined experimentally for any particular compressor. By integrating Equation 5.8 over a polytropic path, using Equation 5.16, it can be shown that the polytropic work of compression for an ideal gas. 

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