3. Mathematical model
3.5 Calculating Stribeck curves
3.5.2 Preliminary model results
When calculating the Stribeck curve, there are a number of parameters which can be varied. In the following sections the effect of the different parameters on the behaviour of the Stribeck curve are shown. When a specific param-eter is changed, the other model paramparam-eters are kept constant with values given in Table 3.2. Geometrical effects on the Stribeck curve are presented
Table 3.2: Operational conditions.
ηs β σs fc E0 B Do FN Pf
[m−2] [m] [m] [–] [Pa] [m] [m] [N] [–]
×1010 ×10−5 ×10−8 ×1011 ×10−3 ×10−3
1.45 5.59 6.1 0.25 4.4 6 82 240 0
in Sections 3.5.2.1 and 3.5.2.2 (macro-geometry), whilst the micro-geometrical effects on the Stribeck curve are given in Section 3.5.2.3. The effect of the operational parameters, load and temperature, i.e. viscosity, on the Stribeck curve are presented in Sections 3.5.2.5 and 3.5.2.6, respectively. The influence of the material property, E0, is given in Section 3.5.2.7. In Section 3.5.2.8 the effect of the hydrostatic sealed fluid pressure is shown.
3.5.2.1 Waviness amplitude, A
By changing the waviness amplitude, the hydrodynamic pressure generation is affected. The coefficient of friction in the boundary lubrication regime, fc is taken as 0.25, which is typical for silicon carbide/silicon carbide seal face combinations, see e.g. Summers-Smith (1988).
The different values, taken for the waviness amplitude, A, are presented in Table 3.3. In Figure 3.25 the coefficient of friction, f , and the separation, λ,
Table 3.3: Different values chosen for the waviness amplitude, A. The numbers 1–13 correspond with the numbers in Fig. 3.25.
1 2 3 4 5 6 7 8 9 10 11 12 13
A [µm] 0.5 0.75 1 2 5 10 15 20 25 30 35 40 50
are plotted as a function of the velocity, where λ is defined as the ratio of the
3.5 Calculating Stribeck curves 67
film thickness, h, and the standard deviation of the height distribution of the summits, σs. The numbers 1 to 13 in Fig. 3.25 correspond with the numbers in Table 3.3. It is shown that the waviness amplitude has a rather small effect on the transition from hydrodynamic lubrication (HL) to mixed lubrication (ML). Figure 3.25 (a) shows that the transition velocity, vt, acquires its lowest value for A = 2µm. For both lower and higher amplitudes the transition takes place at higher velocities. When the amplitude is further increased, see Fig. 3.25 (b), the transition from HL to ML shifts even further to the right.
For high speeds the separation, λ, no longer increases, but remains constant, as shown for A = 0.5µm (number 1 in the Figure).
3.5.2.2 Coning angle, a
In Fig. 3.26 a number of Stribeck curves (solid lines) are presented for different values of the coning angle, a. In Fig. 3.26 (a) the amplitude of the waviness is taken to be 1 µm, in Fig. 3.26 (b) the amplitude is taken to be 10 µm. In Table 3.4 the different values for the coning angle are shown, the numbers 1 to 9 correspond with the numbers in Fig. 3.26 (a) and (b). The coning angle has a
Table 3.4: Different values chosen for the coning angle, a. The numbers 1–9 correspond with the numbers in Fig. 3.26.
1 2 3 4 5 6 7 8 9
a [rad] × 10−5 0 1 2.5 5 7.5 10 15 20 25
rather large effect on the transition from HL to ML, especially in Fig. 3.26 (a), where a smaller waviness amplitude of 1µm is chosen. A larger coning angle is unfavourable with regard to hydrodynamic pressure generation, and therefore, as expected, the transition HL–ML shifts to the right for higher coning angles.
The separation, λ, becomes smaller for higher coning angles. As shown in Fig. 3.25 (a), an amplitude of A = 10µm can lead to higher separations than an amplitude of A = 1µm. As a consequence the transitions HL–ML will occur at lower speeds, as shown in Fig. 3.26 (b).
3.5.2.3 Roughness ηsβσs:
From the literature it is known that the product of ηsβσs is about 0.05. In Fig. 3.27 the effect of this product is presented. Three different values are
68 3. Mathematical model
chosen, ηsβσs =0.03, 0.05 and 0.07, respectively. In this product, only the ηs and the β are varied, both with the same factor. The operational conditions, including σs, are kept constant (see Table 3.2). Furthermore, two different values are chosen for the amplitude as well as for the coning. In Fig. 3.27 (a), A = 1µm and a = 0 rad and in Fig. 3.27 (b) A = 10µm and a = 1.5× 10−4rad.
It is shown that the product of ηsβσs hardly has any effect on the shape and the transitions of the Stribeck curve. In Fig. 3.27 (a) the transition moves a little to the right for a higher value of ηsβσ. Besides that, the separation, λ, is a little greater for a larger value of ηsβσs in the boundary and the mixed lubrication regime, as for a greater value of ηsβ, when both are increased by the same factor, the stiffness of the surface is greater. As a consequence, more hydrodynamic pressure has to be generated in order to obtain separation of the faces, which results in a small shift to the right for the HL–ML transition.
In Fig. 3.27 (b) the effect is even smaller. The transition shifts very little to the right with increasing values of ηsβσ.
σs:
The ratio of h to σs determines the lubrication regime. With an increasing value of λ the lubrication regime changes from boundary to mixed and from mixed to hydrodynamic lubrication. In this section the standard deviation of the height distribution of the summits, σs, is varied. The other parameters are kept constant, and are given in Table 3.2. The product of ηsβσ is kept constant at 0.05. When σs is multiplied by a factor x, ηs and β are both divided by a factor √
x.
In Fig. 3.28 two different combinations of the amplitude and the coning are chosen, in Fig. 3.28 (a) A = 1µm and a = 0, in Fig. 3.28 (b) A = 10µm and a = 1.5× 10−4rad. Six different values voor σs are chosen, which are given in Fig. 3.28.
As expected, for larger values of σs the transition from HL to ML shifts to the right. A larger film thickness h, and with that more hydrodynamic fluid pressure has to develop before full separation of the faces occurs.
3.5.2.4 Non-Gaussian height distribution; χ2n-height distribution In the friction model a Gaussian height distribution is assumed. However, when rough surfaces run in, the summits flatten, and a roughness distribution with a negative skewness remains, see e.g. Lubbinge (1994). Figure 3.29 shows a profile measurement of a run-in seal face with flattened summits. An exam-ple of a non-Gaussian height distribution function is the so-called M -inverted χ2n distribution function, see Adler and Firman (1981). This distribution func-tion is particularly suitable for studying the effect of negative skewness. The
3.5 Calculating Stribeck curves 69
probability density function of an M -inverted χ2n-distribution is defined as:
φχ2n(s) = exp −M −s2N
2n2Nn2Γ n2 (M − s)n2−1, (3.95) with:
M = n
√2n and:
N = 1
√2n. (3.96)
Figure 3.30 shows the effect of n on the χ2n-height distribution. For large values of n the Gaussian height distribution is approached, for smaller values of n the skewness becomes more negative and the kurtosis increases, see also de Rooij (1998).
In Fig. 3.31 some Stribeck curves are calculated for different shapes of the χ2n -height distribution, by varying n. The different values for n are indicated in the Figure. In the case of no coning, Fig. 3.31 (a), a lower value for n results in a lower transition velocity vt. As for lower values of n the χ2n-height distribution becomes steeper (right tail of curve 1 in Fig. 3.30), sooner full separation of the faces occurs, resulting in lower vt. In Fig. 3.31 (b), with coning, the same trend is found, however the effect of the different height distributions is small.
As expected from Fig. 3.30, the Stribeck curve for n = 100 is the same as Stribeck curve no. 3 in Fig. 3.25.
3.5.2.5 Axial load FN
In Fig. 3.32 the axial load FN is varied. In Table 3.5 the different values for FN are given. The numbers 1 to 5 correspond with the numbers in the figure. Two different combinations of the amplitude and the coning angle are taken, with the same values as in the previous sections. The other operational conditions are given in Table 3.2.
In both graphs (a) and (b) it is shown that for a greater load the Stribeck curves shift to the right. It is clear that when a greater load is present, more hydrodynamic pressure has to be generated, in order to separate the faces.
In Fig. 3.32 (b), where coning is present, the transitions take place at higher velocities compared to Fig. 3.32 (a), where a = 0; note the different scales.
Furthermore, it is shown that there is no effect of the amplitude and the coning on the relative displacement of the HL–ML transition. In both graphs, the velocity at which the transitions occur, changes almost with the same factor as the load, FN, does.
70 3. Mathematical model
Table 3.5: Different values for the axial load FN. The numbers 1–5 correspond with the numbers in Fig. 3.32.
1 2 3 4 5
FN [N] 100 500 1000 2000 5000
3.5.2.6 Viscosity
Three different values for the fluid viscosity, η, are taken, i.e. η = 1×10−3, 1× 10−2 and 5× 10−2Pa·s. The product of η and the velocity v determines the hydrodynamic pressure generation, as shown by Fig. 3.33. An increase of the viscosity results in lowering of the velocity at which the transition from HL to ML occurs. The effect of the viscosity is the same for both graphs, i.e. linear with the viscosity.
3.5.2.7 Reduced modulus of elasticity
In Fig. 3.34 three Stribeck curves are presented for three different values of E0. With an increase of E0 the transition HL–ML moves to the right. When coning is present (Fig. 3.34 (b)), the effect of E0 becomes smaller. For a stiffer material the separation in the ML regime is larger, and therefore a higher hydrodynamic pressure is required to enable separation of the faces. Furthermore, a lower E0 shows a higher coefficient of friction in the HL regime, due to an increase of the contact area. Practical values of E0 for mechanical face seals range between 2× 1010Pa for hard-soft seal face combinations and 4× 1011Pa for hard-hard seal face combinations.
3.5.2.8 Hydrostatic fluid pressure
In Section 3.4.6, three different film thickness equations have been derived for three different fluid pressure regions, i.e. Pf = 0, 0 < Pf ≤ 1 and 1 < Pf ≤ 1.75. The previous sections all concern calculated Stribeck curves with Pf = 0.
In the following Figures the effect of the sealed fluid pressure is demonstrated.
0 < Pf ≤ 1
In both graphs of Fig. 3.35, with waviness and coning values as in the for-mer sections, the transitions shift to the left with an increasing fluid pressure, Pf. When a sealed fluid pressure is present, the hydrostatic component be-comes active in the sealing interface. As a result, less hydrodynamic pressure