Page 119 - Air and gas Drilling Field Guide 3rd Edition

110 CHAPTER 5 Compressors and Nitrogen Generators

the initial state 1 of the gas, and area oabn measures the product P 1 v 1 , which is
the flow-work (ft-lb/lb or N-m/N) supplied in the passage of the fluid to the com-
pressor. The line bc represents the state change of the gas during compression.
The area mcbn measures
2
ð
Pdv (5-4)
1
or
ð 1
þ Pdv; (5-5)
2
which is the work ideally required for effecting the actual compression within the
compressor. Thus, aside from the work required for increasing the kinetic energy,
the net area abcd measures the net shaft work required for the induction, com-
pression, and delivery of the gas under conditions assumed to be ideal (absence
of fluid or mechanical friction, and mechanically reversible).
The total shaft work W s required for compression can be written as
V 2 2 V 1 2 ð 2
W s ¼ þ P 2 V 2 Pdv P 1 v 1 (5-6)
2g 2g 1
or

V 2 2 V 1 2 ð 1
W s ¼ þ P 2 v 2 þ Pdv P 1 v 1 ; (5-7)
2g 2g 2
where W s is the total shaft work (lb-ft/lb, N-m/N), V 1 is the velocity of the gas
entering the compressor (ft/sec, m/sec), V 2 is the velocity of the gas exiting the
2
compressor (ft/sec, m/sec), and g is the acceleration of gravity (32.2 ft/sec ,
2
9.81 m/sec ).
Equation (5-6) can be rewritten as
V 1 2 V 2 2 ð 2
W s ¼ þ P 1 v 1 P 2 v 2 ¼ Pdv: (5-8)
2g 2g 1
n
n
n
Assuming a polytropic process, where P 1 n ¼ P 2 n ¼ Pn ¼ constant, and the
1
2
simplest polytropic process where the exponent term n k, where k is the ratio
of specific heat for the gas involved in the process (e.g., for air, k ¼ 1.4), Equation
(5-8) can be rewritten as
ð 2 ð 2 dv
Pdv ¼ P 1 v k : (5-9)
1 k
1 1 v
The right-hand side of the aforementioned equation can be integrated to yield
" #
2
ð k 1
P 1 v 1 v 1
Pdv ¼ 1 : (5-10)
1 k 1 v 2