0
Research Papers

# Predicting Leak Rate Through Valve Stem Packing in Nuclear ApplicationsPUBLIC ACCESS

[+] Author and Article Information
Ali Salah Omar Aweimer

Ecole de Technologie Superieure,
1100 Notre-Dame Ouest,
e-mail: ali-salah-omar.aweimer.1@ens.etsmtl.ca

Abdel-Hakim Bouzid

Professor
Fellow ASME
Ecole de Technologie Superieure,
1100 Notre-Dame Ouest,
e-mail: hakim.bouzid@etsmtl.ca

Mehdi Kazeminia

Ecole de Technologie Superieure,
1100 Notre-Dame Ouest,
e-mail: mehdi.kazeminia.1@ens.etsmtl.ca

1Corresponding author.

Manuscript received September 8, 2017; final manuscript received May 30, 2018; published online January 24, 2019. Assoc. Editor: Robert Stakenborghs.

ASME J of Nuclear Rad Sci 5(1), 011009 (Jan 24, 2019) (7 pages) Paper No: NERS-17-1113; doi: 10.1115/1.4040493 History: Received September 08, 2017; Revised May 30, 2018

## Abstract

Leaking valves have forced shutdown in many nuclear power plants. The myth of zero leakage or adequate sealing must give way to more realistic maximum leak rate criterion in design of nuclear bolted flange joints and valve packed stuffing boxes. It is well established that the predicting leakage in these pressure vessel components is a major engineering challenge to designers. This is particularly true in nuclear valves due to different working conditions and material variations. The prediction of the leak rate through packing rings is not a straightforward task to achieve. This work presents a study on the ability of microchannel flow models to predict leak rates through packing rings made of flexible graphite. A methodology based on experimental characterization of packing material porosity parameters is developed to predict leak rates at different compression stress levels. Three different models are compared to predict leakage; the diffusive and second-order flow models are derived from Naiver–Stokes equations and incorporate the boundary conditions of an intermediate flow regime to cover the wide range of leak rate levels and the lattice model is based on porous media of packing rings as packing bed ($Dp$). The flow porosity parameters ($N, R$) of the microchannels assumed to simulate the leak paths present in the packing are obtained experimentally. The predicted leak rates from different gases ($He, N2, and Ar$) are compared to those measured experimentally in which the set of packing rings is mainly subjected to different gland stresses and pressures.

<>

## Introduction

Further to the global concern of the amount of hazardous pollutants that is released to the atmosphere and the consequence on the environment, health, and safety, stricter regulations on fugitive emissions are legislated. To tune the quality of industrial equipment performance with these regulations, in recent years, research programs that cover experimental, numerical, and theoretical studies on gasketed joints and valve stem packing have been launched in North America, Europe, and Japan recently. The characterization of gas leak rate through valve packing seals at room and high temperatures is the latest. The development of a model to predict the actual behavior of packed stuffing boxes at room and high temperatures represents a major contribution from this stand point. Theoretical and applied research in flow, heat, and mass transfer in porous media has drawn a lot of attention during the past two decades. This is due to the importance of this research area in many engineering applications including valve stem sealing. It is important that the mechanism of a packed stuffing box maintains a threshold amount of contact pressure during the operation of a valve. In order to improve the sealing performance of packed stuffing boxes, it is essential to study their leakage behavior, the factors which contribute to leak, and the influence level of the affecting parameters. The purpose of this work is to use theoretical and simulation models to investigate the leakage behavior of packing seals under different operating conditions, such as pressure, stress, and type of fluid. Most valves use packing seals to prevent stem leakage. Valve packing must be properly compressed to prevent fluid escape and stem damage. If a packing seal is too loose, leak occurs between the stem and the packing rings, which is a safety hazard; if it is too tight, it will impair the movement and possibly damage the stem. Furthermore, 60% of fugitive emissions of pressure vessel equipment requiring sealing compliance come from valve leakage. Therefore, it is important to predict the long-term leakage behavior and ensure a long working lifetime and reduce maintenance work of valves. The flow in packing seals can take place through cracks, serrations, and material pores. These types of leak are very difficult to measure or predict separately. Therefore, they are very difficult to model using simple analytical approaches. Up until now, there have been few analytical models to simulate gas flow through packing seals expressed as continuous flow through capillaries or tortuous flow through porous media. There are different flow regimes that could possibly be used to model the flow in packing rings. The suitability of a model that covers a wide range of leaks due to the various applications is a difficult task to achieve. The type of flow regime is dictated by Knudsen number [1]; for values greater than 1, the flow is considered to be in the free molecular regime; for a range between 0.01 and 1, the transitional flow regime is present; and for values less than 0.01, continuum flow regime is predominant. Ewart et al. [2] measured the mass flow rate for a gas flow in slip regime and isotherm steady conditions in cylindrical micro tubes. The measured values were compared to the ones obtained by analytic models based on gas dynamics-continuum equations. They conclude that in a slip regime, there is a significant $Kn$ second order effect. In another study, Kazeminia and Bouzid [3] used different analytical approaches based on first-order slip model and modified Darcy model to predict the leakage through packing seals. Their results show that these flow models have nearly the same accuracy for the level of leak rate above 10−3 mg/s for a 28.575 mm (1 1/8 in) stem diameter with 9.525 mm (3/8 in) square packing rings. The effect of inlet pressure has been studied by Arkilic et al. [4], where they investigated filtration velocity with slip boundary conditions applied to Navier–Stokes equations (NSE). They conclude that the slip boundary conditions are not an accurate assumption with a larger Knudsen number. Adanhounmè et al. [5] found better agreement with their experiments results when using the domain decomposition method instead of the nonlinear functional NSE.

Numerous studies have been conducted on rarefied gas flow in porous media. Hong et al. [6] developed a methodology to estimate gaseous leak flow rates in a narrow crack for a wide range of flow conditions using Darcy's friction factor and Reynolds number. They validated the predicted leak rates by comparing them to both numerical results and available experimental data. Diany and Bouzid [7] determined the leak rate for two packing types, where they found that Teflon required smaller compression stresses to seal than flexible graphite. Agrawal and Dongari [8] estimated the mass flow rate using NSE with a Knudsen number equal to four. They found an agreement with experimental measurements. In another study conducted by same authors [9], the microscale gas flow was characterized by NSE with second-order slip boundary condition. They employed a generalized form of the second-order slip model, earlier developed by Sreekanth [10], and solved the integral form of the NSE retaining the inertia terms. The solution obtained for long microchannel gave satisfactory results up to a Knudsen number value of 2. Singh et al. [11] investigated the leakage behavior in low-speed isotherm microscale gas conditions and obtained good results up to a Knudsen Number of 2.2. Amyx et al. [12] measured the permeability phenomenon by investigating the capacity of fluid flow through rock samples. In the macroscale regime, the diffusion model based on wall adsorption prevails. Dongari et al. [13] developed a theoretical treatment for mass flow rate through a microchannel using NSE and including the diffusion effect caused by density and temperature gradients. The theoretical predictions using the modified NSE are found to be in good agreement with the available experimental data, while ignoring wall-slip boundary conditions.

In parallel, a lot of research has been dedicated to the other approach based of fuid flow through porous media. The use of Brinkman–Forchheimer-extended Darcy or the more generalized model is suitable for an extensive variety of engineering applications, including flows through packed beds, packing seals, and gaskets. The porosity in terms of the size and shape of pores has a significance effect on the flow properties using Ergun equation; Handley and Heggs [14] investigated the effect of cylindrical and spherical pore shapes. Macdonald et al. [15] studied impact on flow of the porosity range between 0.36 <$ε$< 0.92. Zou and Yu [16] proposed an empirical equation to quantify the relationship between the porosity and the sphericity of cylindrical particles in dense random packing using an experimental approach. They found that the formulated relations obtained demonstrate to be useful in characterizing porosity prediction of nonspherical particle mixtures. Parkhouse and Kelly [17] investigated the relationship between the length to diameter ratio of the bed and the distribution of the pores in the stacks using a statistical approach. The model agrees with experimental results. Liu et al. [18] focused on the fundamentals of flow through porous materials and attempted to include a factor to account for the wall effect. They discovered that the total length of the leak paths varies with the void ratio. The materials used in their study were spherical glass beads of varying diameters, with a porosity ranging from 36 to 44%. The Ergun equation was only valid for Reynolds number less than 1000. With a modification to the Ergun equation, Jones and Krier [19] extended its validity up to a Reynolds number of 76,000.

The aim of this paper is to shed light on the underlying prediction of packing seals leakage by examining different analytical models to describe leakage behavior in packing seals. Experiment validation is conduct to verify the suitability of the analytical models. A set of four flexible graphite-based packing rings is tested under three types of gases, namely nitrogen, argon, and helium at different gland stress levels and internal pressures.

## Physical Flow Models

###### Capillary Model With Second-Order Slip Condition.

The analytical solution of the NSE with the second-order velocity slip condition in circular channels is employed to evaluate the mass leak rate through a set of porous packing rings. The model implemented in the present work is already exploited in microchannels within known dimensions for different types of geometry such as circular [20], rectangular [21], and triangular and trapezoidal [22]. The leak paths that fluid particles follow through porous packing materials such as flexible graphite used in this study are modeled as circular channels oriented in the axial direction. A set of $Nc$ parallel capillaries of uniform radius $Rc$ with the second-order slip condition is shown in Fig. 1.

The equation of conservation of momentum in cylindrical coordinates for an ideal gas, without taking into account the effect of inertia at low Reynolds number and large $l/2Rc$, reduces to Ref. [23]: Display Formula

(1)$1rddr(rdudr)=1μdPdz$

In the case of isothermal flow with slip condition at the wall of a microtube of circular section, the first and second boundary conditions are Display Formula

(2)$dudr|r=0=0$
Display Formula
(3)$u|r=R=−2−σσλdudr|r=R+A2λ2d2udr2|r=R$

σ is the tangential momentum accommodation coefficient taken as 1 for simplicity and λ is the mean free path. Integrating Eq. (1) twice and applying the above boundary conditions gives the solution for the velocity in the z direction as a function of the radial position such that Display Formula

(4)$u(r,z)=1μdPdz[r24−R44−2−σσλR2+A2λ212]$

The leak rate for $N$ capillaries can be obtained by integrating the velocity through the area as Display Formula

(5)$L=Nρ∫0R2πru(r,z)dr$

Substituting for the velocity and using the ideal gas law give the leak rate as Display Formula

(6)$L=NπR4Po2(Π2−1)16μℜTℓ[1+162−σσKn(Π+1)−32A2Kn2In(Π)(Π2−1)]$

where Kn is the Knudsen number defined as Display Formula

(7)$Kn=λ2R$

$Π$ is the pressure ratio such that: Display Formula

(8)$Π=PiPo$

and $λ$ is the mean free path given by: Display Formula

(9)$λ=16μ5PoℜT2π$

## Diffusive Flow Model

The analytic solution of the NSE with the diffusive conditions in circular channels is employed to evaluate the mass leak rate [13] in the simple case of an isotherm flow and with slip condition at the wall. The first- and second-order boundary conditions are Display Formula

(10)$dudr|r=0=0$

(11)$u|r=R=−μρPdPdz$

Neglecting the temperature gradient in the z direction Soret's term has no contribution to the diffusion velocity and only Fick's term contributes by following Eq. (11). The expression for the velocity is therefore Display Formula

(12)$u(r,z)=−12μdPdz(R4−r2)−(μρPdPdz)$

Integrating the velocity over the area using Eq. (5) gives the mass flow rate as Display Formula

(13)$L=NπR4Po2(Π2−1)16μℜTℓ(1+16μ2ℜTR2Po2In(Π)(Π2−1))$

## Ergun Model for a Porous Media

The Ergun model is often used to quantify the flow through porous materials [24]. Based on the Forchheimer's Law, the flow through a packed bed of spheres is given by Ergun equation such that Display Formula

(14)$fp=150Rep+1.75$

where $fp$ is the friction factor for packed beds and Rep is the Reynolds number for packed beds, (150,1.75) are values found experimentally [24]. $fp$ and Rep can be expressed as Display Formula

(15)$fp=ΔPℓDpρu2ε3(1−ε)$
Display Formula
(16)$Rep=Dpuρ(1−ε)μ$

The pressure drop $∇P=Pi−Po$ through a randomly packed bed of spheres is therefore obtained by combining Eqs. (14)(16) such that Display Formula

(17)$∇P=150μuℓφ2Dp2(1−ε)2ε3+1.75ρu2ℓφDp(1−ε)ε3$

This model can be applied to packing seals simulated as a material made with particles of diameter $Dp$ and having a porosity, $ε$. The porosity $ε$ of the material is defined using the bulk density $ρb$ and the solid density $ρs$, such that Display Formula

(18)$ε=1−ρbρs$

The porosity is an indication of the closeness of the packing, with a smaller void ratio indicating a tighter packing. For the flexible graphite packing material tested, the values of the porosity at the different gland stresses are presented in Table 1. For the case of nonspherical particles, a correction or shape factor known as sphericity $φ$ and defined in Ref. [25] is frequently used. It is the ratio of the surface area of spheres with the same volume the particles over the surface area of particles. This is used to obtain the equivalent sphere diameter, where in the case of sphere particles a sphericity of 1 applies, while in our case the equivalent diameter is estimated to 0.9.

## Experimental Setup

The experimental leak measurement tests are conducted on the stuffing box test bench shown in Fig. 2. This test bench has three main systems: a hydraulic tensioner system with a manual pump and accumulator, a pressurization system composed of different gases, and a leak detection system. Every system has its own instrumentation and controls connected to a data acquisition system using LabVIEW software as an interface program between the data logger and the computer. The housing of a total length of 104.8 mm (4.125 in) in height can accommodate up to five packing rings of 9.5 mm (3/8 in) square section. A set of four packing rings was used for the purpose of the current test study. The stuffing box housing has an outside diameter of 79.4 mm (3.125 in) and an inside diameter of 47.6 mm (1.875 in). The stem is made of steel with a diameter of 28.6 mm (1.125 in). Depending on the leak rate, four different leak measurement techniques can be used: flow meter, pressure decay, pressurized rise, and spectrometry, with the latter being able to detect down to 10−10 ml/s. For this study, the flowmeter and the pressure rise methods were used to measure the leak rates. A gland stress ranging from 7 to 41 MPa (1000 to 6000 psi) is applied by using the hydraulic tensioner. The gland stress is deduced from the measurement of the load through a Wheatstone bridge strain gage placed on the stem. For every gland stress level, pressures ranging from 0.34 to 2.76 MPa (50 to 400 psi) in steps with 0.34 MPa are applied to the packed stuffing box. Three gases, Nitrogen, Argon, and Helium, were used in the experiment. The physical properties of gases used in the experimental investigation of this study are presented in Table 2. Using the pressurized method to measure leak and taking into consideration that the uncertainties of pressure, temperature, volume, and time as 7 × 10−4, 3.36 × 10−3, 3.18 × 10−6, and 6.67 × 10−4, respectively, the uncertainties of helium, argon, and nitrogen leak rate are shown in Table 3.

## Porosity Parameters Evaluation

###### Capillary Model With Second Slip Velocity.

The mass flow rate equation can be written in the following form: Display Formula

(19)$Ac=NR4[1+Bc1(Π+1)−DcIn(Π)(Π2−1)]$

where Display Formula

(20a)$Ac=16LμℜTℓπP02(Π2−1)$
Display Formula
(20b)$Bc=162−σσKn$
Display Formula
(20c)$Dc=32A2Kn2$

The porosity parameters $N$ and $R$ can be obtained from a curve fit of the data obtained with tests using helium as a reference gas and Eq. (19) that expresses Ac as a function of the pressure ratio $Π$.

###### Capillary Model With Diffusive Velocity.

The mass flow rate equation can be written as follows: Display Formula

(21)$AD=NR4[1+BDln(Π)(Π2−1)]$

where Display Formula

(22a)$AD=16LμℜTℓπP02(Π2−1)$
Display Formula
(22b)$BD=16μ2ℜTR2Po2$

Following the same procedure as with the second-order model, the porosity parameters $N$ and $R$ of the diffusivity models are obtained.

###### Parameters of Porous Media Condition.

The leak rate and pressure drop being known for each stress level during the tests with helium, the values of Dp can be obtained and used to predict the leak rate for other gases such as nitrogen and argon.

## Results and Discussion

Helium is used as the reference gas to determine the porosity parameters embedded in each of the analytical models. Figure 3 shows the leak rate measurements versus helium pressure at four different gland stresses. Using the data from these tests and a curve fitting of a plot of $Ac$ versus the pressure ratio $Π$, the porosity parameters $N$ and $R$ are obtained for the second-order and diffusion models. Figure 4 gives a plot of $NR4$ as a function of the gland stress. The two curves given by the second-order slip and diffusion models are very similar. The porosity parameter values from the two models are close to each other at the low and intermediate stresses. However, for the high stress, the second-order slip model predicts smaller values of the porosity parameter. The porosity and the Ergun model solid phase sphere size $Dp$ are plotted against the gland stress in Fig. 5. As the stress increases, both parameters decrease. In addition, the sphere size decreases with a pressure increase although the effect is comparatively small as shown in Fig. 6. The porosity parameters and the other constants obtained from data curve fitting have been used to predict the leak rate for other types of gases namely nitrogen and argon. For the second-order and diffusivity models, the number and radius of capillary obtained from Helium tests were incorporated in Eqs. (6) and (13), respectively, while the diameter of particles is used in Eq. (17) to predict the leak rate for other gases. In parallel, tests were conducted with these two gases in order to compare the results with all three models. The measured leak rates are conducted at four stress levels varying from 7 to 41 MPa and eight different gas pressures ranging from 0.34 to 2.7 MPa. In general, the theoretical leak predictions are slightly higher than the experimental leak measurements depending on the gas pressure and the level of stress on the packing rings. Figures 7 and 8 present the Nitrogen gas results for 7, 14, 28, and 41 MPa gland stresses, respectively. The second-order slip model is shown to predict leak better than the diffusivity and Ergun models with the latter showing less accuracy. In addition, as the gland stress is increased, the leak becomes smaller and the diffusivity and Ergun model curves separate from the measured curve indicating that the flow does not obey the porous media and diffusive flow regimes. The capillary flow with the second-order slip model predicts better results in this case. Figures 9 and 10 present the Argon gas results for 7, 14, 28, and 41 MPa gland stresses, respectively. Although all models predict leak rates with acceptable error margin, the second-order slip model gives a better agreement with experimental data for all pressure values. The gap between the experimental curves and the diffusion and Ergun models is relatively large for higher gland stress values indicating a larger percentage difference even in a log scale. Therefore, the second-order slip model simulates better the leakage behavior of flexible graphite packing seals over a wide range of flow from 0.0001 to 1 mg/s of helium. Tighter packing seals are the subject of future studies. It is suspected that the second-order slip model could still be used to predict even leak rates up to 10−6 mg/s where the molecular flow prevails.

## Conclusion

This study investigates the suitability of capillary and porous media models to predict gaseous leaks in flexible graphite packing rings of valve stuffing boxes. The packing rings are a homogeneously distributed porous media with disordered porosity. In the slip and diffusive flow models, the porous media was hypothetically constructed with microtubes in the direction of the fluid flow. While in the Ergun model, the porous media was assumed to be a packed bed made of spheres. Helium was used as the reference gas to determine the porosity parameters according to different models. These parameters were then used to predict the leak rate for other types of gases. In general, the results showed a good agreement between the predicted leak rates and those measured experimentally. However, the second-order slip model showed a better agreement over the four decade range of leak rates from 0.0001 to 1 mg/s with helium. This model is currently being tested for other tighter packings such as those made of polytetrafluoroethylene and other materials.

## Nomenclature

• $Ac,Bc,Dc$ =

parameters for capillary second-order slip model

• $AD,BD$ =

porosity parameters for diffusivity model

• A2 =

constant for capillary second-order slip model

• $Dp$ =

diameter of spherical particles, m

• $L$ =

leak rate, kg/s

• $ℓ$ =

length of packing, m

• N =

number of capillaries

• $NR4$ =

porosity parameters for capillary and diffusivity models, m4

• $P$ =

pressure, Pa

• $R$ =

• $T$ =

temperature, K

• $u$ =

axial velocity, m/s

• $z$ =

axial direction, m

Greek Symbols
• $∇P$ =

pressure difference between upstream and downstream, Pa

• $ε$ =

porosity

• $φ$ =

sphericity, m

• $λ$ =

mean free path, m

• $μ$ =

dynamic viscosity, Pa s

• $ρ$ =

density, kg/m3

• $σ$ =

tangential momentum accommodation coefficient = 1

• $ℜ$ =

specific gas constant, J/kg K

Subscripts or Superscripts
• $b$ =

bulk

• $c$ =

capillary second-order model

• $D$ =

capillary diffusive model

• $i$ =

inlet or upstream

• $o$ =

outlet or downstream

• $p$ =

packed beds model

• $s$ =

sphere

Acronyms
• NSE =

Navier–Stokes equations

Nondimensional Number
• Kn =

Knudsen number $(λ/2R)$

• Re =

Reynolds number $(Dpuρ/(1−ε)μ)$

## References

Knudsen, M. , 1909, “The Law of the Molecular Flow and Viscosity of Gases Moving Through Tubes,” J. Ann. Phys., 28(1), pp. 75–130.
Ewart, T. , Perrier, P. , Graur, I. , and Meolans, J. G. , 2006, “Mass Flow Rate Measurements in Gas Micro Flows,” J. Exp. Fluids, 41(3), pp. 487–498.
Kazeminia, M. , and Bouzid, A. , 2016, “Predicting Leakage in Packed Stuffing Boxes,” 23rd International Conference on Fluid Sealing, Manchester, UK, Mar. 2–3, pp. 45–59.
Arkilic, E. B. , Schmidt, M. A. , and Breuer, K. S. , 1994, “Slip Flow in Microchannels,” 19th International Symposium on Rarefied Gas Dynamics, Oxford, UK, July 25–29, pp. 57–65.
Adanhounmè, V. , Codo, F. D. P. , and Adomou, A. , 2012, “Solving the Navier Stokes Flow Equations of Micro-Polar Fluids by Adomian Decomposition Method,” J. Bull. Math. Sci. Appl., 2, pp. 30–37.
Hong, C. , Asako, Y. , and Lee, J.-H. , 2009, “Estimation of Leak Flow Rates Through Narrow Cracks,” ASME J. Pressure Vessel Technol., 131(5), p. 051405.
Diany, M. , and Bouzid, A.-H. , 2011, “An Experimental-Numerical Procedure for Stuffing Box Packing Characterization and Leak Tests,” ASME J. Tribol., 133(1), p. 012201.
Agrawal, A. , and Dongari, N. , 2012, “Application of Navier-Stokes Equations to High Knudsen Number Flow in a Fine Capillary,” Int. J. Microscale Nanoscale Therm. Fluid Transp. Phenom., 3(2), pp. 125–130.
Dongari, N. , Agrawal, A. , and Agrawal, A. , 2007, “Analytical Solution of Gaseous Slip Flow in Long Microchannels,” Int. J. Heat Mass Transfer, 50(17–18), pp. 3411–3421.
Sreekanth, A. , 1969, “Slip Flow Through Long Circular Tubes,” Sixth International Symposium on Rarefied Gas Dynamics, Cambridge, MA, July 22–26, pp. 667–680.
Singh, N. , Dongari, N. , and Agrawal, A. , 2014, “Analytical Solution of Plane Poiseuille Flow Within Burnett Hydrodynamics,” J. Microfluid. Nanofluid., 16(1–2), pp. 403–412.
Amyx, J. , Bass, D. , and Whiting, R. , 1960, “Petroleum Reservoir Engineering: Physical Properties,” Properties of Porous Media Containing Multiple Fluid Saturations, McGraw-Hill, New York, pp. 147–211.
Dongari, N. , Sharma, A. , and Durst, F. , 2009, “Pressure-Driven Diffusive Gas Flows in Micro-Channels: From the Knudsen to the Continuum Regimes,” J. Microfluid. Nanofluid., 6(5), pp. 679–692.
Handley, D. , and Heggs, P. , 1968, “Momentum and Heat Transfer Mechanisms in Regular Shaped Packings,” J. Trans. Inst. Chem. Eng., 46(9), pp. 251–264.
Macdonald, I. , El-Sayed, M. , Mow, K. , and Dullien, F. , 1979, “Flow Through Porous Media—The Ergun Equation Revisited,” J. Ind. Eng. Chem. Fundamentals, 18(3), pp. 199–208.
Zou, R. , and Yu, A. , 1996, “Evaluation of the Packing Characteristics of Mono-Sized Non-Spherical Particles,” J. Powder Technol., 88(1), pp. 71–79.
Parkhouse, J. , and Kelly, A. , 1995, “The Random Packing of Fibres in Three Dimensions,” Proc. R. Soc. London, (Math. Phys. Sci.), 451(1943), pp. 737–746.
Liu, S. , Afacan, A. , and Masliyah, J. , 1994, “Steady Incompressible Laminar Flow in Porous Media,” J. Chem. Eng. Sci., 49(21), pp. 3565–3586.
Jones, D. P. , and Krier, H. , 1983, “Gas Flow Resistance Measurements Through Packed Beds at High Reynolds Numbers,” ASME J. Fluids Eng., 105(2), pp. 168–172.
Tison, S. , 1993, “Experimental Data and Theoretical Modeling of Gas Flows Through Metal Capillary Leaks,” Seventh International School on Vacuum Conference, Plovdiv, Bulgaria, Sept. 23–27, pp. 1171–1175.
Maurer, J. , Tabeling, P. , Joseph, P. , and Willaime, H. , 2003, “Second-Order Slip Laws in Microchannels for Helium and Nitrogen,” J. Phys. Fluids, 15(9), pp. 2613–2621.
Araki, T. , Kim, M. S. , Iwai, H. , and Suzuki, K. , 2002, “An Experimental Investigation of Gaseous Flow Characteristics in Microchannels,” J. Microscale Thermophys. Eng., 6(2), pp. 117–130.
Grine, L. , and Bouzid, A.-H. , 2011, “Liquid Leak Predictions in Micro and Nanoporous Gaskets,” ASME J. Pressure Vessel Technol., 133(5), p. 051402.
Ergun, S. , 1952, “Fluid Flow Through Packed Columns,” Chem. Eng. Prog., 48(2), pp. 89–94.
Carman, P. C. , 1956, Flow of Gases Through Porous Media, Academic press, New York.
View article in PDF format.

## References

Knudsen, M. , 1909, “The Law of the Molecular Flow and Viscosity of Gases Moving Through Tubes,” J. Ann. Phys., 28(1), pp. 75–130.
Ewart, T. , Perrier, P. , Graur, I. , and Meolans, J. G. , 2006, “Mass Flow Rate Measurements in Gas Micro Flows,” J. Exp. Fluids, 41(3), pp. 487–498.
Kazeminia, M. , and Bouzid, A. , 2016, “Predicting Leakage in Packed Stuffing Boxes,” 23rd International Conference on Fluid Sealing, Manchester, UK, Mar. 2–3, pp. 45–59.
Arkilic, E. B. , Schmidt, M. A. , and Breuer, K. S. , 1994, “Slip Flow in Microchannels,” 19th International Symposium on Rarefied Gas Dynamics, Oxford, UK, July 25–29, pp. 57–65.
Adanhounmè, V. , Codo, F. D. P. , and Adomou, A. , 2012, “Solving the Navier Stokes Flow Equations of Micro-Polar Fluids by Adomian Decomposition Method,” J. Bull. Math. Sci. Appl., 2, pp. 30–37.
Hong, C. , Asako, Y. , and Lee, J.-H. , 2009, “Estimation of Leak Flow Rates Through Narrow Cracks,” ASME J. Pressure Vessel Technol., 131(5), p. 051405.
Diany, M. , and Bouzid, A.-H. , 2011, “An Experimental-Numerical Procedure for Stuffing Box Packing Characterization and Leak Tests,” ASME J. Tribol., 133(1), p. 012201.
Agrawal, A. , and Dongari, N. , 2012, “Application of Navier-Stokes Equations to High Knudsen Number Flow in a Fine Capillary,” Int. J. Microscale Nanoscale Therm. Fluid Transp. Phenom., 3(2), pp. 125–130.
Dongari, N. , Agrawal, A. , and Agrawal, A. , 2007, “Analytical Solution of Gaseous Slip Flow in Long Microchannels,” Int. J. Heat Mass Transfer, 50(17–18), pp. 3411–3421.
Sreekanth, A. , 1969, “Slip Flow Through Long Circular Tubes,” Sixth International Symposium on Rarefied Gas Dynamics, Cambridge, MA, July 22–26, pp. 667–680.
Singh, N. , Dongari, N. , and Agrawal, A. , 2014, “Analytical Solution of Plane Poiseuille Flow Within Burnett Hydrodynamics,” J. Microfluid. Nanofluid., 16(1–2), pp. 403–412.
Amyx, J. , Bass, D. , and Whiting, R. , 1960, “Petroleum Reservoir Engineering: Physical Properties,” Properties of Porous Media Containing Multiple Fluid Saturations, McGraw-Hill, New York, pp. 147–211.
Dongari, N. , Sharma, A. , and Durst, F. , 2009, “Pressure-Driven Diffusive Gas Flows in Micro-Channels: From the Knudsen to the Continuum Regimes,” J. Microfluid. Nanofluid., 6(5), pp. 679–692.
Handley, D. , and Heggs, P. , 1968, “Momentum and Heat Transfer Mechanisms in Regular Shaped Packings,” J. Trans. Inst. Chem. Eng., 46(9), pp. 251–264.
Macdonald, I. , El-Sayed, M. , Mow, K. , and Dullien, F. , 1979, “Flow Through Porous Media—The Ergun Equation Revisited,” J. Ind. Eng. Chem. Fundamentals, 18(3), pp. 199–208.
Zou, R. , and Yu, A. , 1996, “Evaluation of the Packing Characteristics of Mono-Sized Non-Spherical Particles,” J. Powder Technol., 88(1), pp. 71–79.
Parkhouse, J. , and Kelly, A. , 1995, “The Random Packing of Fibres in Three Dimensions,” Proc. R. Soc. London, (Math. Phys. Sci.), 451(1943), pp. 737–746.
Liu, S. , Afacan, A. , and Masliyah, J. , 1994, “Steady Incompressible Laminar Flow in Porous Media,” J. Chem. Eng. Sci., 49(21), pp. 3565–3586.
Jones, D. P. , and Krier, H. , 1983, “Gas Flow Resistance Measurements Through Packed Beds at High Reynolds Numbers,” ASME J. Fluids Eng., 105(2), pp. 168–172.
Tison, S. , 1993, “Experimental Data and Theoretical Modeling of Gas Flows Through Metal Capillary Leaks,” Seventh International School on Vacuum Conference, Plovdiv, Bulgaria, Sept. 23–27, pp. 1171–1175.
Maurer, J. , Tabeling, P. , Joseph, P. , and Willaime, H. , 2003, “Second-Order Slip Laws in Microchannels for Helium and Nitrogen,” J. Phys. Fluids, 15(9), pp. 2613–2621.
Araki, T. , Kim, M. S. , Iwai, H. , and Suzuki, K. , 2002, “An Experimental Investigation of Gaseous Flow Characteristics in Microchannels,” J. Microscale Thermophys. Eng., 6(2), pp. 117–130.
Grine, L. , and Bouzid, A.-H. , 2011, “Liquid Leak Predictions in Micro and Nanoporous Gaskets,” ASME J. Pressure Vessel Technol., 133(5), p. 051402.
Ergun, S. , 1952, “Fluid Flow Through Packed Columns,” Chem. Eng. Prog., 48(2), pp. 89–94.
Carman, P. C. , 1956, Flow of Gases Through Porous Media, Academic press, New York.

## Figures

Fig. 1

Capillary model with second-order slip flow

Fig. 2

General configuration of the test bench for leak detection test

Fig. 3

Measured leak rates using helium

Fig. 4

Porosity parameter NR4 for second-order and diffusivity models

Fig. 5

Average sphere diameter Dp and porosity ε

Fig. 6

Ergun pore size versus pressure at different stress levels

Fig. 7

Comparison of leak rates of Nitrogen at 7 and 14 MPa gland stress

Fig. 8

Comparison of leak rates of Nitrogen at 28 and 41 MPa gland stress

Fig. 9

Comparison of leak rates of Argon at 7 and 14 MPa gland stress

Fig. 10

Comparison of leak rates for Argon at 28 and 41 MPa gland stress

## Tables

Table 1 Flexible graphite packing porosity
Table 2 Physical properties of the gases used in experimental investigations
Table 3 Uncertainties for lowest and highest mass leak rates

## Errata

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles