Fluid Leakage in Static Rubber Seals

The discussion in Sect. 3 is valid as long as the fluid pressure \(p_\text{a}\) on the high pressure side is much smaller than the maximum contact pressure. When this is not the case the fluid will increase the separation between the surfaces and result in an increased leakage rate. In particular, for large enough fluid pressure lift-off occur resulting in a catastrophic failure of the seal. In this section we will discuss the influence of the fluid pressure on the leakage rate.

The fluid pressure has two effects on a seal: it will elastically deform the seal and change its macroscopic shape (see Fig. 7) which will affect the contact pressure distribution. In addition, the fluid pressure will affect the surface separation in the nominal contact region. The first effect can be studied using standard Finite Element Method (FEM) calculations. In some applications, like in most O-ring applications (see Fig. 7), the nominal contact pressure increases nearly proportional to the fluid pressure while the fluid pressure at the contacting interface increases slower (it decreases from \(p_\text{a}\) at the inlet to \(p_\text{b}\) on the exit side and is hence smaller than \(p_\text{a}\) in the sealing region where the contact pressure is highest). Hence an O-ring seal may leak when the fluid pressure \(p_\text{a}\) is small while it is tight when the fluid pressure is high. This effect was observed recently in a syringe application (with a steel plunger road with a rubber O-ring seal) [14]. However, in some other sealing configurations the fluid pressure may have a negligible influence on the nominal squeezing pressure, and in these cases lift-off occur if the fluid pressure is high enough, as observed in Ref. [6].

Since the effect of the fluid pressure on the nominal contact pressure is well understood, and in order to have a clean situation, we will not considered this effect here, but we assume that the contact pressure distribution \(p_0(x)\) is known and of Hertz-like form for simplicity.

We will study the influence of the fluid pressure on the leakrate for two cases, where, in the absence of the fluid, the contact area does not percolate or does percolate. In the former case the nominal contact pressure is below the percolation pressure everywhere in the nominal contact region, while in the second case the contact area percolate in a rectangular strip \(-b<x<b\) at the center of the contact area (see Fig. 8).

4.1

Theory

We will calculate the fluid leakage using the effective medium theory developed in Ref. [6] (see also Ref. [15,16,17,18,19,20]). The ensemble averaged fluid flow current

$$\begin{aligned} \mathbf{J} = -\sigma _\text{eff} \nabla p_\text{fluid} \end{aligned}$$

(11)

where \(p_\text{fluid}\) is the ensemble averaged fluid pressure. The effective conductivity \(\sigma _\text{eff}\) is calculated using the Bruggeman effective medium theory. In this theory enters the probability distribution of interfacial separations which is calculated using the Persson contact mechanics theory. For the cylinder geometry which interest us here the fluid pressure \(p_\text{fluid} (x)\) and the (asperity) contact pressure (also ensemble averaged) \(p_\text{cont} (x)\) depends only on the coordinate x orthogonal to the cylinder axis. If we denote

$$\begin{aligned} S(x) = \int _0^{x} dx' \sigma _\text{eff}^{-1} (p_\text{cont} (x')) \end{aligned}$$

(12)

then the leakrate (volume per unit time) is given by

$$\begin{aligned} \dot{Q} = {L_y (p_\text{a}-p_\text{b}) \over S(L_x)} \end{aligned}$$

(13)

where \(L_x=2a\) is the width of the nominal contact region in the fluid flow direction, orthogonal to the cylinder axis. The fluid pressure

$$\begin{aligned} p_\text{fluid} (x) = p_\text{a} -(p_\text{a}-p_\text{b}){S(x)\over S(L_x)} \end{aligned}$$

(14)

The (ensemble averaged) pressure acting on the rubber surface is

$$\begin{aligned} p_0(x)=p_\text{cont} (x)+p_\text{fluid}(x) \end{aligned}$$

(15)

In the absence of the fluid pressure the contact pressure is assumed to be Hertz-like:

$$\begin{aligned} p_0(x) = p_0 \left( 1-\left( {x\over a}\right) ^2\right) ^{1/2} \end{aligned}$$

(16)

This pressure result from the macroscopic deformations of the rubber. Thus a small change in the interfacial surface separation induced by the fluid pressure will have a negligible influence on the (ensemble averaged) pressure acting on the rubber surface except in the entrance region (\(x \approx -a\)) where \(p_0(x)\) is very small [\(p_0(x)\rightarrow 0\) as \(x \rightarrow -a\)]. In this region lift-off will occur and the pressure acting on the rubber will equal the fluid pressure \(p_\text{a}\). Thus we take \(p_0(x)\) equal to \(p_\text{a}\) in the lift-off region and equal to (16) in the remaining part of the contact region.

The logarithm of the leakrate as a function of the fluid pressure \(p_\text{a}\) for the \(H=1\) and \(h_\text{rms}=6 \ {\mu} \text{m}\) and for the compression \(\delta /R = 0.15\). The seal is \(L_y = 1 \ \text{m}\) long. The contact area does not percolate for any fluid pressure

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The same as in Fig. 9 but with the pressure scale logarithmic. The strait line has the slope 1 showing that for fluid pressures \(p_\text{a} < 0.1 \ \text{MPa}\) the leakrate is proportional to the fluid pressure

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The dependency of the fluid pressure \(p_\text{fluid}\) (green line), the contact pressure \(p_\text{cont}\) (blue line) and the total pressure \(p_\text{tot} = p_\text{cont}+p_\text{fluid}\) (red line) on the coordinate x in the nominal rubber-countersurface contact area. The contact pressure \(p_\text{cont}\) is below the percolation pressure \(p_\text{perc}\approx 7.6 \ \text{MPa}\) everywhere. For \(H=1\), \(\delta /R =0.15\) and \(h_\text{rms} = 6 \ {\mu} \text{m}\) and the fluid pressure on the high pressure side \(p_\text{a} = 2.39 \ \text{MPa}\) (Color figure online)

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4.2

Contact Area Does not Percolate

In this case open fluid flow channels occur at the interface for all applied fluid pressures \(p_\text{a}\). As an illustration consider the \(H=1\) surface but with a roughness scaled up with a factor of 20 so that \(h_\text{rms}=6 \ {\mu} \text{m}\). For this case the rms-slope will be 20 times bigger than for the \(H=1\) surface with \(h_\text{rms}=0.3 \ {\mu} \text{m}\). Thus \(\xi \approx 20 \times 0.103 \approx 2\) and the contact area at the center of the contact region will percolate for the compression \(\delta /R > 0.22 \xi ^2 \approx 0.88\). Hence, for the compression 0.15 and 0.3 the contact area will not percolate and the seal will leak already for arbitrary small fluid pressures \(p_\text{a}\). To illustrate this case Fig. 9 shows the logarithm of the leakrate as a function of the fluid pressure \(p_\text{a}\) for the compression \(\delta /R = 0.15\). Figure 10 shows the same results but now as a function of the logarithm of the fluid pressure \(p_\text{a}\). The strait line has the slope 1 showing that for fluid pressures \(p_\text{a} < 0.1 \ \text{MPa}\) the leakrate is proportional to the fluid pressure as expected when the separation between the surfaces is unchanged [15, 16].

Figure 11 shows the dependency of the fluid pressure \(p_\text{fluid}\) (green line), the contact pressure \(p_\text{cont}\) (blue line) and the total pressure \(p_\text{tot} = p_\text{cont}+p_\text{fluid}\) (red line) on the coordinate x in the nominal rubber-countersurface contact area. The fluid pressure on the high pressure side \(p_\text{a} = 2.39 \ \text{MPa}\). Note that the fluid pressure has separated the surfaces on the high pressure side so that no real contact occur between the rubber and the countersurface for \(0<x<1 \ \text{mm}\).

The logarithm of the leakrate as a function of the fluid pressure \(p_\text{a}\) for the \(H=1\) (green lines) and \(H=0.8\) (blue lines). The seal is \(L_y = 1 \ \text{m}\) long. The two curves at the lowest pressure are for the compression \(\delta /R = 0.15\) and the two upper ones for \(\delta /R = 0.3\). For low water pressure the contact area percolate and the leakage is negligible. Higher compression result in higher contact pressure and a higher water pressure is needed to separate the surfaces enough in order for the contact area not to percolate. For large enough water pressure the surface separation becomes so large that the surface roughness has negligible influence on the leakage rate. This is the reason the blue (\(H=0.8\) surface) and green (\(H=1\) surface) lines for each compression merge at the highest water pressures (Color figure online)

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The dependency of the fluid pressure \(p_\text{fluid}\) (green line), \(p_\text{cont}\) (blue line) and the total pressure \(p_\text{tot} = p_\text{cont}+p_\text{fluid}\) (red line) on the coordinate x in the nominal rubber-countersurface contact area. The fluid pressure is so high that the contact pressure \(p_\text{cont}\) is always below the percolation pressure \(p_\text{perc} \approx 1.64 \ \text{MPa}\) (see Fig. 6). For \(H=0.8\), \(\delta /R =0.3\) and the fluid pressure on the high pressure side \(p_\text{a} = 3.39 \ \text{MPa}\) (Color figure online)

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The dependency of the fluid pressure \(p_\text{fluid}\) (green line) and the total pressure \(p_\text{tot} = p_\text{cont}+p_\text{fluid}\) (red line) on the coordinate x in the nominal rubber-countersurface contact area. The contact pressure \(p_\text{tot} -p_\text{fluid}\) is above the percolation pressure \(p_\text{perc}\) in a region \(1.80 \ \text{mm}< x < 5.25 \ \text{mm} \). In this region the fluid pressure is decreasing linearly with x as a result of assuming a contact fluid conductivity in this region (see text). For \(H=0.8\), \(\delta /R =0.3\) and the fluid pressure on the high pressure side \(p_\text{a} = 2.19 \ \text{MPa}\) (Color figure online)

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The logarithm of the average surface separation as a function the coordinate x in the nominal rubber-countersurface contact area. We show results for several fluid pressures indicated in the figure (\(p_\text{a} = 0.73\), 1.46, 2.19, 3.29 and \(3.65 \ \text{MPa}\))

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4.3

Contact Area Percolate

We consider now the case where, when the fluid pressure vanish, the contact area percolate in a strip at the center of the contact. The percolated area will act as a barrier towards fluid leakage (see Fig. 8). In this case it is necessary to discuss how the seal was mounted. If the seal was mounted in the dry state regions with compressed air may form in the area where the contact area percolate. If the seal was mounted in the fluid, fluid filled (and pressurized) regions (lakes) may form in the percolated area. In order to have a well-defined system in this case we will assume that there is a very small fluid leak current through the contact region where the contact area percolate. This may be due to diffusion of water molecules through the rubber matrix. In this case, after a long enough contact time, a well-defined stationary state will be formed, where some (extremely small) leakage occur even when the contact area percolate. We will describe this very small fluid flow by a constant (extremely small) flow conductivity \(\sigma ^*_\text{eff}\) in the region where the contact area percolate. The results presented below does not depend on the magnitude of \(\sigma ^*_\text{eff}\) as long as it is extremely small.

To illustrate the case when the contact area percolate for small fluid pressures we consider the two systems studied in Sect. 3 where \(\xi = 0.103\) (surface \(H=1\)) and \(\xi =0.445\) (surface \(H=0.8\)). For these two cases the contact area percolate for the compression \(\delta /R \approx 0.22 \xi ^2 = 0.0023\) and 0.0436, respectively.

Figure 12 shows the logarithm of the leakrate as a function of the fluid pressure \(p_\text{a}\) for the \(H=1\) (green lines) and \(H=0.8\) (blue lines) surfaces. The two curves at the lowest pressure are for the compression \(\delta /R = 0.15\), and the two upper ones for \(\delta /R = 0.3\). For low water pressure the contact area percolate and the leakage is negligible. Higher compression result in higher contact pressure and a higher water pressure is needed to separate the surfaces enough in order for the contact area not to percolate. For large enough water pressure the surface separation becomes so large that only the longest wavelength roughness (with the largest amplitude) matter. This is the reason the blue (\(H=0.8\) surface) and green (\(H=1\) surface) lines for each compression merge at the highest water pressures. At even higher water pressure the surface roughness has a negligible influence on the leakage rate.

Figure 13 shows the dependency of the fluid pressure \(p_\text{fluid}\) (green line), the contact pressure \(p_\text{cont}\) (blue line), and the total pressure \(p_\text{tot} = p_\text{cont}+p_\text{fluid}\) (red line) on the coordinate x in the nominal rubber-countersurface contact area for \(H=0.8\) and \(\delta /R =0.3\). The fluid pressure \(p_\text{a} = 3.39 \ \text{MPa}\) is so high that the contact pressure \(p_\text{cont}\) is always below the percolation pressure \(p_\text{perc} = 1.64 \ \text{MPa}\). For the same system, in Fig. 14 we show the dependency of the fluid pressure and the total pressure on the coordinate x when the fluid pressure on the high pressure side \(p_\text{a} = 2.19 \ \text{MPa}\). In this case the contact pressure \(p_\text{cont}= p_\text{tot} -p_\text{fluid}\) is above the percolation pressure \(p_\text{perc}\) in a region \(1.80 \ \text{mm}< x < 5.25 \ \text{mm}\). In this region the fluid pressure is decreasing linearly with x as a result of the assumption of a constant fluid conductivity in this region. The result in Fig. 14 is of no practical importance since the leakage rate is negligible small (and determined by \(\sigma ^*_\text{eff}\)) but the figure shows how deep into the contact the fluid penetrate by lift-off (to \(x\approx 0.5 \ \text{mm}\)) and by infiltration into the net-yet percolated contact region (to \(x\approx 1.8 \ \text{mm}\)). Note also that the fluid pressure vanish in a narrow x-region (from \(x\approx 5.25\) to \(5.4 \ \text{mm}\)) at the exit of the contact region, where the contact area is not percolated. Thus in this x-region the resistance to fluid flow is negligible because of the extremly small leak correct which result from the finite but extremly small flow conductivity \(\sigma ^*_\text{eff}\) in the percolated region.

Figure 15 shows the logarithm of the average surface separation as a function the coordinate x in the nominal rubber-countersurface contact area. We show results for the fluid pressures \(p_\text{a} = 0.73\), 1.46, 2.19, 3.29 and \(3.65 \ \text{MPa}\).