Page 121 - Air and gas Drilling Field Guide 3rd Edition

112 CHAPTER 5 Compressors and Nitrogen Generators

Substituting Equation (5-20) into Equation (5-8) gives
V 2 2 V 1 2
W s þ þ P 1 v 1 P 2 v 2 ¼ R g ð T 2 T 1 Þ: (5-21)
2g 2g k 1
Equation (5-17) defined Pv at state 1. Similarly, the definition of Pv at state 2 is
P 2 n 2 ¼ R g T 2 : (5-22)
Substituting Equations (5-17) and (5-22) into Equation (5-21) and rearranging
gives
V 2 2 V 1 2 R g
W s ¼ þ R g ðT 2 T 1 Þþ ðT 2 T 1 Þ: (5-23)
2g 2g k 1
The aforementioned can be rearranged to give
2 2
k T 2 V V 1
2
W s ¼ R g T 1 1 þ : (5-24)
k 1 T 1 2g
It is more useful to have Equation (5-24) written in terms of pressure instead
of temperature. This can be accomplished by substituting Equations (5-17) and
(5-19) into Equation (5-24). This gives
k 1
2 3
2 2
k P 2 k V V
6 7 2 1
W s ¼ P 1 n 14 15 þ : (5-25)
k 1 P 1 2g
The last term (the kinetic energy) in Equation (5-25) can be shown in practical
applications to be quite small relative to the first term. Thus, this latter term is
usually neglected. Therefore, the shaft work for compression of gas in a compres-
sor is reduced to
k 1
2 3

k P 2 k
6 7
W s ¼ P 1 n 14 15: (5-26)
k 1 P 1
Compressors can actually be considered steady-state flow mechanical devices
(even intermittent flow machines). If the weight rate of flow through the
compressor is _ w (lb/sec, N/sec), then the time rate of shaft work done _ W s
(lb-ft/sec, N-m/sec) to compress gas in a compressor can be obtained by multiply-
ing Equation (5-26) by _ w. This gives
k 1
2 3

k P 2 k
6 7
_ W s ¼ P 1 _ w n 14 15: (5-27)
k 1 P 1
The term _ W s (lb-ft/sec, N-m/sec) is the theoretical power required by the com-
pressor to compress the gas. Using the general relationship between specific vol-
ume and specific weight given in Equation (5-15), the volumetric flow rate at
state 1 (entering the compressor) Q 1 is