# A Study on Liquid Leak Rates in Packing Seals
## 2 Theoretical Model  
The fluid flow in the packing rings was assumed to be in the stem axial direction. To simulate the flow through the porous material, a model consisting of several straight circular capillaries with a one-dimensional flow in each was considered--due to the axisymmetric flow assumption (as shown in Figure 1) for a packing seal.  
The material was assumed to be made of N capillaries of radius R through which the fluid passed. The capillary r dependency of the flow is neglected because of the simplified consideration of the momentum correction factor. Therefore, the only dependency of the flow is the capillary axial z direction.  
The correlation between velocity and pressure for a fully developed and isothermal flow in a capillary was obtained by the conservation of momentum equation:  
\[\frac{1}{r}\frac{d}{dr}\left(r\frac{du}{dr}\right)=\frac{1}{\mu}\frac{dP}{dz} \tag{1}\]  
The maximum velocity was at the middle of tube such that:  
\[\left.\frac{du(r)}{dr}\right|_{r=0}=0 \tag{2}\]  
The velocity at the wall, based on a first-order slip model of liquids, was  
\[u(r,z)=\left.-L_{\mathrm{s}}\frac{du}{dr}\right|_{r=R} \tag{3}\]  
where \(L_{\mathrm{s}}\) is the slip length with a liquid in a circular capillary and is given by [30]:  
\[L_{\mathrm{s}}=0.059\times\gamma^{0.485}\mathrm{\;and\;}\gamma=\frac{\Delta PR }{2\mu\ell} \tag{4}\]  
Integrating Equation (1) twice and applying the boundary conditions of Equations (2) and (3) gives the liquid velocity profile as a function of the pressure gradient in the \(z\) direction:  
\[u(r,z)=\left.\frac{1}{\mu}\frac{dP}{dz}\right[\frac{r^{2}}{4}-\frac{R^{2}}{4} -L_{\mathrm{s}}\frac{R}{2}\right] \tag{5}\]  
Figure 1: Capillary tubes in a simulated packing seal.  
Integrating the velocity through the area of \(N\) capillaries gives the leak rate as:  
\[L=N\int\limits_{0}^{R}\rho 2\pi r\cdot u(r,z)\ dr \tag{6}\]  
Substituting for \(u\) from Equation (5) into Equation (6) gives  
\[L=-\frac{NR^{4}\pi}{8\ \mu}\rho\frac{dP}{dz}\left[1+\frac{4L_{s}}{R}\right] \tag{7}\]  
The total leak rate through the packing rings is independent of the length and therefore  
\[\int_{0}^{\ell}Ldz=\int_{P_{i}}^{P_{o}}-\frac{NR^{4}\pi}{8\ \mu}\rho\left[1+ \frac{4L_{s}}{R}\right]dP \tag{8}\]  
This gives the total liquid leak rate for \(N\) capillaries  
\[L=\frac{NR^{4}\pi\rho(P_{i}-P_{o})}{8\ \mu\ell}\left[1+\frac{4L_{s}}{R}\right] \tag{9}\]  
The porosity parameters \(N\) and \(R\) were obtained from a curve fitting of the measured data, obtained experimentally with helium gas. \(A\) similar theoretical approach for the flow through capillaries was applied--but for gases. Reference [15] gives the details on how to get the expression of the total leak rate for gas flow in packing rings given hereafter:  
\[L_{gas}=\frac{NR^{4}\pi P_{O}^{2}(\Pi^{2}-1)}{16\Re T\mu\ell}\left[1+\frac{2- \sigma}{\sigma}\frac{16 Kn}{(\Pi+1)}\right] \tag{10}\]  
where \(Kn\) is the Knudsen number defined as:  
\[Kn=\frac{\lambda}{2\ R} \tag{11}\]  
\(\Pi\) is the pressure ratio given by:  
\[\Pi=\frac{P_{i}}{P_{o}} \tag{12}\]  
and \(\lambda\) is the mean free path given by:  
\[\lambda=\frac{16}{5\ p_{o}}\sqrt{\frac{\Re T}{2\ \pi}} \tag{13}\]  
The equation of gas leak rate can be linearized in terms of the reciprocal pressure to give:  
\[A=NR^{4}\left[1+B\frac{1}{(\Pi+1)}\right] \tag{14}\]  
where  
\[A=\frac{16\mu\delta\Re TL_{gas}}{\pi P_{o}^{2}(\Pi^{2}-1)} \tag{15}\]  
and  
\[B=16\frac{2-\sigma}{\sigma}Kn \tag{16}\]  
\(\sigma\) is the tangential momentum accommodation coefficient, and is taken as 1. The porosity parameters \(N\) and \(R\) can be obtained from the slope \(A\) and intercept \(B\) from the curve fitted lines of the helium gas leak data as a function of the pressure ratio \(1/(\Pi+1)\).