2.3. Uncertainty Analysis of Viscosity Measurement
There are a lot of uncertainties related with the viscosity measurement devices, dealing with processes, the mathematical expressions, the measurements, and so on. In this work, the described capillary tube viscometer was used for viscosity measurements. Regarding the process followed by this particular device, the following parameters are necessary for the uncertainty analysis:
Uncertainty in the measurement of pressure,
Uncertainty in the measurement of flow rate,
Uncertainty in the measurement of volume,
Uncertainty in the measurement of time,
Uncertainty from the radius of capillary tube,
Uncertainty from the length of capillary tube,
As all above input quantities are independent, the combined standard uncertainty is given by:
where
,
, and
are all the quantities that we took into consideration (
). The uncertainty value regarding viscosity using the values of
Table 1 is (
n) = ±0. 010304 Pa s with confidence 95% for each case.
The procedure of the uncertainty analysis is significant here in order to calculate through several measurements, with a higher accuracy, the viscosity values. All the tests were performed by an accredited device in a temperature controlled room, and the uncertainty quantities concern the device components.
2.4. CFD Quasi-Static Analysis of Piston Ring-Liner Conjunction
The numerical procedure reported here has been provided by Zavos and Nikolakopoulos [
12]. A one-dimensional analysis is used. The CFD model considers fully flooded inlet conditions. The side leakage of the lubricant is not taken into account around the periphery of the ring. The solution of the developed pressure between the ring-liner tribo-pair is produced using the Navier-Stokes equations [
24]. The set of equations is solved using the ANSYS Multiphysics code.
Figure 3 shows the boundary flow conditions in the ring-liner conjunction. The ring reciprocates along the liner following the piston sliding velocity [
25]:
where
rcr is the crank-pin radius,
ω is the rotational crankshaft speed,
ϕ is the crank angle, and
λCR is the control ratio. The ring twist is not accounted for. The in-cylinder pressure,
pc, and the outlet pressure,
pout (from crankcase), values are assumed as inputs, depending on the direction of the piston motion.
To obtain the minimum lubricant film thickness at any crank angle, a film shape expression is needed as follows:
where
is the minimum film thickness,
is the parabolic ring profile
, and
is the localised contact pressure-induced deflection. In reality, the ring profile has a rough axial asymmetric shape as reported in References [
13,
19]. However, an idealized parabolic ring shape is used in this investigation. In addition, the cylinder liner also has a perfect circular shape. The current study focuses on engine synthetic oil aging, which is suitable for the piston ring tribology undertaken here. The effect of the ring contact profile and the cylinder’s bore geometry is not the scope of this analysis. The latter parameter has a limited effect in the ring-liner contact for this analysis. According to the authors of [
12], this shows that localized ring deformation has a small effect on lubricant film distribution. More specifically, this factor has a critical impact when higher combustion pressures are expected.
Regarding in-plane ring behavior, two outward forces are applied between the piston and the back profile of the ring. The ring tension force is obtained as:
where the elastic pressure is taken as:
, while the ring cross section is defined as:
. Additionally, the back gas force is given as:
In the current analysis, the ring tension force
FT and the gas force
FG should be equal to the hydrodynamic reaction,
Wh, due to the lubricant film in the ring-liner conjunction, and the load, due to asperities,
Wc. At every crankshaft position, ring balance should take place according to the following expression:
Under mixed and hydrodynamic regimes of lubrication, the ring-liner conjunction is not covered by lubricant film only, but there is a part of the ring which suffers from the asperities. The hydrodynamic local capacity and the load carried by surface asperities are expressed as:
The load equilibrium is fulfilled when the relationship (10) is confirmed as:
When ring balance is not reached, then the minimum film thickness is recalculated through the iterative process. In this analysis, the value of the parameter
χ is taken as:
χ = 0.05.
To evaluate the load by the surface asperities, the Greenwood-Tripp contact model [
26] is used.
describes the root mean square (RMS) surface finish of the contact surfaces.
E’ is the effective elastic modulus of the ring-liner system and
A is the nominal contact area of the ring face-width. The terms
and
are also the roughness parameter and the asperity slope, respectively. Furthermore, the statistical function
F5/2(
λ) is expressed as a relation of the Stribeck oil film parameter
[
27]. The limits between various regimes of lubrication are used to indicate the changes at contact, and these regimes are depicted in the minimum film thickness curve.
In the piston ring-cylinder liner conjunction, the total friction is obtained as:
Under the hydrodynamic regime of lubrication, the viscous friction is given as:
where
τ is the viscous shear stress of the lubricant film:
When the film ratio is
, the load by surface asperities should be taken into account; therefore, the ring boundary friction can be described as [
28]:
where the non-Newtonian Eyring shear stress and the boundary shear strength of the conjugating surfaces are
MPa and
ς = 0.17, respectively. A coefficient of friction with a value of 0.17 was used for the formed ferrous surface oxide layer, as reported by Teodorescu et al. [
29]. The asperity contact area
Ac is also defined as [
26]:
where the statistical function
F2(
λ) is:
After grid sensitivity tests, 15 divisions are used along the lubricant film and 1000 divisions are considered along the ring face-width. The total number of cells is 15 × 10
3. The fluid region is meshed with finite volumes. The CFD procedure is specified by Zavos and Nikolakopoulos [
12] in more detail. The current analysis assumes an isothermal mixed lubrication model, where the thermal gradient in the contact is negligible. Therefore, the lubricant density and viscosity were evaluated with the pressure variation according to Roelands [
30], as follow:
where:
and the pressure–viscosity index is:
In the literature, there are a lot of pressure-viscosity models that were employed in the beginning of elasto-hydrodynamic line contact calculations. Barus [
31] and Dowson et al. [
32] are two well-known models. However, at high film pressures of the order of 1 GPa, encountered in many highly loaded elastohydrodynamic lubrication (EHL) contacts, the Barus equation is no longer applicable. The most popular one, in the open literature on EHL operations, is the Roelands pressure-viscosity equation. Bair [
33] refused the accuracy of the Roelands equation by showing that it is incorrect from about 0.5 GPa or higher, depending on the fluid considered, and proposed alternatives. However, the popularity of the Roelands equation shows that many of the film thickness approximations on the ring-liner contact have been based on it showing good agreement with experiments.
The pressure viscosity coefficient, in general, depends on the lubricant at hand and on the pressure, temperature, and shear rate in the contact areas. It is a very important factor, especially for heavily loaded elasto-hydrodynamically lubricated contacts, which requires the value of the pressure-viscosity coefficient to be known. The estimation of this value will easily lead to errors in pressure distribution and film thickness, for example. The pressure viscosity coefficient is generally recognized as temperature dependent, according to the Barus equation, and is usually considered constants in a wide range of temperatures, according to Roelands equation [
30,
31]. Nevertheless, a number of works have been made to combine in one equation the effects of temperature and pressure upon viscosity. Various researchers also report that their equations are accurate for a certain type of fluids, but either not so accurate or unusable for other types, highlighting pressure viscosity coefficient dependency to viscosity.
The nominal lubricant properties are obtained from the capillary tube viscometer described in
Section 2.1. For clarity, the overall methodology of the ring lubrication problem is summarized and shown in
Figure 4. The final solution was attained when both pressure and ring balance criteria were obtained simultaneously, and after the analysis was moved to the new crank angle location.