41
The pressure difference in this portion of the piston ring forces causes the fluid
velocity to lose
linearity thus requiring integration across the gap between the piston ring and the compressor
cylinder as:
𝑓 = 𝜇𝐴
𝑈
ℎ = 𝜇𝐴
𝑑𝑢
𝑑𝑦
Equation 39
Substituting the derivative of Equation 37 yields:
𝑓 = 𝜇𝐴 (
1
2𝜇
𝑑𝑃
𝑑𝑥
(2𝑦 − ℎ) +
𝑈
ℎ)
Equation 40
From here, we note that the area
(𝐴)
the friction force acts on is the bottom edge
of the piston
ring multiplied by the circumference of the piston ring. Since we are doing this for a one-
dimensional cross-
section, we remove the piston ring’s circumference to make the friction force
per unit length and evaluate the remaining equation at
(𝑦 = ℎ)
yielding:
𝑓
′
= (𝑥
3
− 𝑥
2
) (
ℎ
2
𝑑𝑃
𝑑𝑥 +
𝜇𝑈
ℎ )
Equation 41
Now we have the
frictional force acting on a majority of the piston ring’s area
(Equation 40) in
addition to the lubricant flowrate under the piston ring (Equation 36) and the
hydrodynamic
pressure generated under the piston ring (Equation 23, Equation 33, and Equation 35). This
provides a sufficient starting point with which to model a compressor piston ring as detailed in
Chapter 5
–
Modeling Compressor Lubrication. However, let us first make note of the
importance of viscosity in the equations above and its influence on compressor lubrication.
42
1.
Increasing the lubricant’s viscosity increases the hydrodynamic force built up under the
piston ring (see Equation 23 and Equation 33). This increases the separation gap
(ℎ)
between the piston ring and the compressor cylinder to prevent wear.
2.
Increasing the lubricant’s viscosity increases the
frictional force
acting against the piston
rings’ motion (see
Equation 41
).
3.
Increasing the lubricant’s viscosity
has counteracting effects on the lubricant flowrate
under the piston ring (see Equation 36
). Increasing the lubricant’s viscosity increased the
value in the denominator of Equation 36 but also increases the separation gap
(ℎ)
between the piston ring and the compressor cylinder.
Reviewing the equations in relation to an operating compressor, it is evident that the
compressor operator cannot vary the geometry of the piston rings and typically does not want to
vary the
compressor’s speed. Thus
, the lubricant viscosity and lube
rate are the only
parameters the operator has control over. Contemplating the equations governing the
hydrodynamic pressure (Equation 23, Equation 33, and Equation 35), one will note that once
the lubricant’s viscosity is too low, there is no amount of lubricant that can be supplied to keep
proper separation between the cylinder wall and the piston ring. Thus, higher viscosity lubricants
are typically suggested for harsher operating conditions as can be seen by investigating Table 4
and Table 5. There are many more sources besides the ones previously mentioned to aid an
operator in selecting the correct lubricant for a certain application. However, how can an
operator be sure these suggestions are correct for their specific application?
2.4 - Lubricant Viscosity and Gas Dilution Estimation
The lubricant viscosity and lubrication rate are the only controls an operator has to protect their
operating compressor. There are many suggestions for proper lubricants and lubrication rates
but how can the operator know these suggestions are correct for their specific application? Is
43
there a way to measure or calculate the viscosity of the lubricant in the compressor? The high
temperatures and pressures in a compressor preclude the use of many types of sensors that
could measure the lubricant’s viscosity in
-situ. This leaves calculations or laboratory
measurements. Addressing the first, the
lubricant’s
viscosity will depend on the compres
sor’s
temperature which the operator can estimate to be somewhere
between the compressor
’s
suction and discharge temperatures. However, the temperature will not provide the whole
picture as the high-pressure gases in the compressor are soluble in liquids (including lubricants)
which can cause the lubricant to be diluted with the gas in the cylinder. So, the operator needs a
way to estimate the amount of a gas that will dissolve into a lubricant and, using the amount of
dissolved gas, estimate the viscosity of the lubricant in the compressor. Unfortunately, there is
still one limitation to this method described nicely by Seeton (2009):
“
Given the nature and diversity of lubricants, there are no reliable prediction models
to estimate the solubility of liquefied gases in lubricants across the broad spectrum
of lubricant types and blends. Lubricating oils are generally made-up of a blend of
basestock fluids to reach a desired viscosity, viscosity index and lubricity for the final
product. This blending makes it possible to efficiently tailor products for different
applications; however, blending also makes it difficult to generalize the solubility of
gases into these blends within the same manufactures product line, and across
different manufactures of the same type classification. Therefore, solubility data must
be experimentally measured for specific combinations of interest to have accurate
information for the system of interest
”
(p 36).
So, it appears that the operator is stuck relying on recommendations from the lubricant
manufacturer or needs to experimentally measure the viscosity of multiple
lubricants when
subjected to their specific natural gas stream to see what is best for their specific application.
The focus of this thesis is on
the experimental determination of a lubricant’s viscosity when
44
diluted with natural gas components. Before addressing this though, let us first discuss the
solubility data that does exist for similar gas-lubricant systems and
ways to estimate how the
dissolved gas will affect the viscosity of the lubricant.
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