# A Study on Liquid Leak Rates in Packing Seals
## 4 Results and Discussion  
The porosity parameters were determined using tests with a reference gas; in this case, helium. Helium was selected because it is the smallest safe particle, but nitrogen could also have been used. Figure 4a,b shows the results of the gas leak tests for the flexible graphite and die ring seals, respectively. It is to be noted that the relationship between leak and pressure for the different gland stresses was rather linear in a log-log scale, because the laminar flow was predominant under the testing conditions with helium. Indeed, this is confirmed by the first term of Equation (10). Figure 3: Liquid leak measuring device for packing rig. Figure 5a,b shows the relationship between the porosity parameter A, given by Equation (14), and the reciprocal pressure \(\Pi\) for the braided flexible graphite and graphite die rings, respectively. From the linear behavior observed in the curves obtained at the different gland stress levels, the two porosity parameters (the number and diameter of the capillary, \(N\) and \(R\)) were determined using Equations (15) and (16). The intercept of line \(A\) gives NR\({}^{4}\), as per Equation (12), whereas the slope gives BNR\({}^{4}\), and hence B was obtained. The Knudsen number was then obtained from Equation (16) for \(\sigma\) equal to 1, from which the hydraulic diameter D = 2R was deduced for the specific gas used for the characterization test. Finally, the number of capillaries N was obtained. Figures 6a and 7a show the variations of the porosity parameter NR\({}^{4}\) and Knudsen number with the gland stress. The porosity parameter was exponentially decaying with stress and the Knudsen number was below 2.5, which indicates a transitional flow regime. Figures 6b and 7b show the variation of the radius and number of capillaries as a function of the gland stress. Using these curves, the capillary size and number under any specific packing stress were obtained, in order to predict the leak rate of the liquid under consideration using Equation (8). While the relationship between NR\({}^{4}\) and stress was exponential,  
Figure 4: Helium leak tests (**a**) Flexible graphite, (**b**) die ring. Figure 5: Graphs to obtain porosity parameters. (**a**) Flexible graphite, (**b**) die ring. the relationship became linear when the two parameters, N and R, were taken individually. The linear relationships were obtained by a curve fit for both materials. It is to be noted that both parameters decreased with a gland stress increase. The graphite die rings had a comparatively lower radius and a lower number of capillaries. The relationships for the braided flexible graphite packing were:  
\[R=-5.05\times Sg+161\] \[N=-7.61\times 10^{8}Sg+1.92\times 10^{10} \tag{17}\]  
Likewise, for the graphite die rings:  
\[R=-2.93\times Sg+150\] \[N=-3.62\times 10^{8}Sg+9.33\times 10^{9} \tag{18}\]  
From this relationship, the extrapolated lines gave a radius of 161 nm and 150 nm, respectively, for the braided flexible graphite packing and graphite die rings, at no stress or as received. Capillaries numbered at 1.92 \(\times\) 10\({}^{10}\) and 9.33 \(\times\) 10\({}^{9}\) for these two materials, respectively. Equations (17) and (18) were used--in conjunction with kerosene and water properties and Equation (14)--to predict leak rates through these two materials. Parallel experimental tests were conducted with these two liquids on both materials under the  
Figure 6: Variation of porosity parameters and Knudsen number for of Flexible graphite. Figure 7: Variation of porosity parameters and Knudsen number for die rings. -gland stresses and pressures given above. The comparisons are shown in Figures 8 and 9. The predictions for both materials were within acceptable limits at low stress. However, the model was less accurate, showing a considerable difference from the experimental measurements, especially for high gland stresses. The difference depended on the applied liquid pressure and the level of the gland stress.