Page 201 - Compression Machinery for Oil and Gas

190 SECTION II Types of Equipment

2 3

R
6 sinθcosθ 7
L
6 7
6 7 (5.9)
_ x ¼ Rω sinθ + s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6 7
R
2
4 2 5
1 sin θ
L
π
2
V¼ D _ x (5.10)
4
Pressure within the cylinder (P) changes with an inverse relationship to the
cylinder volume, which causes mass flow to be exchanged through the suction
and discharge valves according to the suction pressure (P s ) and discharge pres-
sure (P d ) and the valve dynamics. The physical principles governing these
aspects of the reciprocating compressor operation are discussed in the subse-
quent sections.
The following assumptions are considered for the simulation of the system:
l The suction and discharge volumes are sufficiently large such that the
respective pressures are considered constant, and the gas velocities are
low enough that dynamic pressure drop can be ignored.
l The operation of the piston is slow enough that the cylinder thermodynamic
properties (i.e., pressures, temperatures, etc.) may be treated as quasi steady
and without any spatial variation.
l The respective motions of the piston and valves are one-dimensional—that
is, no misalignments and angular or orthogonal motion—and frictionless.
Recall Eq. (5.1) represented the first law of thermodynamics for the cylinder.
As a simplification, heat transfer effects are ignored and piston seal
_
leakage is neglected. With this, and also substituting E ¼ _ mu + m _ u and
_ m ¼ _ m s _ m d _ m l , Eq. (5.1) is rewritten as Eq. (5.11). Other forms of the
first law equation can be derived using ideal gas assumptions to relate
pressure and temperature. However, depending on the gas species and condi-
tions, this may not be appropriate. Eq. (5.11) is the form used to generate exam-
ple results.

PV + _ m s h s _ m d h d ¼ _ m s _ m d Þu + m _ u (5.11)
ð

Example Calculations
Simulations of a reciprocating compressor (Fig. 5.8) are presented in this sec-
tion to demonstrate some aspects of reciprocating compressor performance.
Baseline simulation parameters are listed in Table 5.1. Time integration is car-
ried out with a simple Euler method and a time step providing just under 20
steps per degree of crank angle. The simulation is transient but is executed long
enough to achieve the steady-state cycle. The steady cycles are displayed by
truncating the transient portions.