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7.6. THEORY AND  CALCULATIONS  OF  GAS  COMPRESSiON  159

                                                               In terms  of  a known isentropic efficiency the  final temperature  T,
                                                               then is found by trial from
                                                                            T2
                                                                  (AH)Jqs = 1 C; dT + AH; - AH;.           (7.49)
                                                                           Ti
                                                                  In these equations the heat capacity C;  is that of  the ideal gas
                                                               state or that of  the real gas near zero or atmospheric pressure. The
                                                               residual properties AS;  and AH; are evaluated at (Pl, Tl)  and AS;
                                                               and AH;  at (P2, TJ. Figure 7.28 gives them as functions of  reduced
                                                               temperature  T/T, and  reduced  pressure  P/P,. More  accurate
                                                               methods and charts for finding residual properties from appropriate
                                                               equations of  state  are presented  in  the cited books of  Reid et al.
                                                               (1977) and Walas (1985).
                                                                  For  mixtures,  pseudocritical  properties  are  used  for  the
                                                               evaluation  of  the  reduced  properties.  For  use  with  Figure  7.28,
                                                               Kay’s rules are applicable, namely,
                                                                                                           (7.50)
                                                                                                           (7.51)

                                                               but many equations of  state employ particular combining rules.
                                                                  Example  7.8  compares  a  solution  by  this  method  with  the
                                                               assumption of  ideal behavior.
                                                               EFFICIENCY
                                                               The  efficiencies of  fluid  handling  equipment  such  as  fans  and
                                                               compressors are empirically derived quantities. Each manufacturer
                                                               will supply either an efficiency or a statement of  power requirement
                                                               for a specified performance. Some general rules have been devised
                                                               for  ranges in which  efficiencies of  some classes equipment usually
                                                               fall. Figure 7.27 gives such estimates for reciprocating compressors.
                                                               Fan  efficiencies can  be  deduced  from  the  power-head  curves  of
                                                               Figure  7.24.  Power  consumption  or  efficiencies  of  rotary  and
                                                               reciprocating machines are shown in Tables 7.7, 7.8, and 7.9.
                                                                  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  0f  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  most  nearly  correct  methods  of  Section  7.6.4  require
                                                               knowledge of  isentropic efficiencies which  are obtainable from the
                                                               polytropic  values.  For  a  given  polytropic  efficiency,  which  is
                                                               independent  of  the  nature  of  the  gas,  the  isentropic  value  is
                                                              obtained with Eq.  (7.39) or Figure 7.27(b). Since the heat capacity
                                                              is involved in this transformation, the isentropic efficiency depends
                                                               on  the  nature  of  the  substance  and  to  some  extent  on  the
                                                               temperature also.

                                                              TEMPERATURE RISE, COMPRESSION RATIO,
                                                              VOLUMETRIC EFFICIENCY
                                Reduced pressure.pr
                                                              The isentropic temperature  in terms  of  compression ratio  is given
                                  tb)                         for ideal gases by

           Figure 7.28. Residual entropy and enthalpy as functions of  reduced   (T&  = TpyP1)‘k-””.      (7.52)
           properties.  (a) Residual entropy. (b) Residual enthalpy. Drawn  by
           Smith  and  Van  Ness  (Introduction  to  Chemical  Engineering   For  polytropic compression the final temperature  is given  directly
           Thermodynamics,  McGraw-Hill,  New  York,  1959) from  data  of   bY
           Lydersen et al.  For illustrative purposes primarily; see text for other
           sources.]                                              T, = T&/Pl)‘”-l””                       (7.53)
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