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The physics of confined flow and its application to water leaks, water permeation and water nanoflows: a review
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2016 Rep. Prog. Phys. 79 025901 (http://iopscience.iop.org/0034-4885/79/2/025901) View the table of contents for this issue, or go to the journal homepage for more
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Reports on Progress in Physics Rep. Prog. Phys. 79 (2016) 025901 (45pp)
doi:10.1088/0034-4885/79/2/025901
Review
The physics of confined flow and its application to water leaks, water permeation and water nanoflows: a review Wenwen Lei, Michelle K Rigozzi and David R McKenzie School of Physics, University of Sydney, NSW, 2006, Australia E-mail:
[email protected] Received 5 May 2015, revised 10 September 2015 Accepted for publication 3 November 2015 Published 2 February 2016 Abstract
This review assesses the current state of understanding of the calculation of the rate of flow of gases, vapours and liquids confined in channels, in porous media and in permeable materials with an emphasis on the flow of water and its vapour. One motivation is to investigate the relation between the permeation rate of moisture and that of a noncondensable test gas such as helium, another is to assist in unifying theory and experiment across disparate fields. Available theories of single component ideal gas flows in channels of defined geometry (cylindrical, rectangular and elliptical) are described and their predictions compared with measurement over a wide range of conditions defined by the Knudsen number. Theory for two phase flows is assembled in order to understand the behaviour of four standard water leak configurations: vapour, slug, Washburn and liquid flow, distinguished by the number and location of phase boundaries (menisci). Air may or may not be present as a background gas. Slip length is an important parameter that greatly affects leak rates. Measurements of water vapour flows confirm that water vapour shows ideal gas behaviour. Results on carbon nanotubes show that smooth walls may lead to anomalously high slip lengths arising from the properties of ‘confined’ water. In porous media, behaviour can be matched to the four standard leaks. Traditional membrane permeation models consider that the permeant dissolves, diffuses and evaporates at the outlet side, ideas we align with those from channel flow. Recent results on graphite oxide membranes show examples where helium which does not permeate while at the same time moisture is almost unimpeded, again a result of confined water. We conclude that while there is no a priori relation between a noncondensable gas flow and a moisture flow, measurements using helium will give results within two orders of magnitude of the moisture flow rate, except in the case where there is anomalous slip or confined water, when moisture specific measurements are essential. Keywords: water leaks, moisture permeation, micro- and nano-fluidics (Some figures may appear in colour only in the online journal)
1. Introduction
conductive pathways or flow channels. Moisture permeation is an important problem in many fields that require the long term isolation of packaged items. Moisture permeation involves the transport of water molecules in the liquid phase,
This review covers the prediction and measurement of the rate of moisture permeation through barriers that contain 0034-4885/16/025901+45$33.00
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© 2016 IOP Publishing Ltd Printed in the UK
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vapour phase and as adsorbed surface layers. Packaging of food is a well known application area where effort has been directed towards improved barrier performance, while even more demanding applications are the isolation of components, especially electronic components, in adverse environments. For example, solar cells are required to operate with a long service life while exposed to the weather so that the energy and capital invested in the installation can be recovered. Organic light emitting diodes are sensitive to moisture, but even more stringent requirements are placed on the moisture hermeticity of encapsulations for implantable medical devices where electronic components must operate with a long service life in an environment surrounded by body fluids. Microfluidic devices are used for measuring and processing of fluids by confining them in small channels. The design of microfluidic devices also requires an understanding of flow in small channels and is an active field of research. An emerging field that relies on the permeation of fluids is that of drug delivery by means of particles, especially nanoparticles. These drug delivery particles are required to control the drug release to provide local administration of the drug [1]. The subject of water flows and permeation has received a resurgence of interest as part of a trend in which phenomena traditionally studied by observation on the macroscopic scale are now being viewed on the nanometre scale where new and often unexpected findings are being made. The availability of atomistic and molecular simulations has assisted this trend towards an examination of the small scale and has delivered valuable insights. As the dimensions of the channels are reduced to achieve miniaturization of the devices, the flow becomes highly dependent on the boundary conditions at the walls and on the permeability of the walls (see for example [2]). One of the motivations for this review is to present available theories of liquid water and water vapour flow in small channels and in permeable media and test them against experiment where possible over all distance scales. A considerable literature exists on permeation in membranes and porous media, subjects pioneered by Graham, Fick, Darcy and others. Another large body of literature exists devoted to the study of the flow of fluids through a single channel of specified geometry, a subject pioneered by Poiseuille, Stokes, Navier, Knudsen and others. The evolution of these related areas of research has been largely separate. A motiv ation of this review is to move towards a unification of these fields, to enable the lessons learnt from each to be utilized to increase understanding of gas and condensable vapour flows, with an emphasis on water and water vapour. Recent unexpected findings have challenged our understanding of flows in small smooth walled channels while our understanding of the flow of condensable vapours in membranes has also been challenged in membranes with layered microstructures (see for example the work of Nair et al [3] in which the flow of water is observed to be up to ten orders of magnitude higher than the flow of helium). Such findings have outlined the need for deeper understanding in these fields and have increased interest in what seemed only a few years ago to be a mature research field with no surprises. While recent findings seem unexpected at first sight, there are allusions to
their origins in much older literature. For example, the fact that high flow rates of gases exceeding Knudsen diffusion could be found in smooth walled channels was predicted by Smoluchowski in 1910 [4]. The importance of slip length in capillary filling was recognized by Washburn [5]. The importance of two phase flows in porous media was understood by the 1960s [6]. The subject of fluid flow in a single uniform channel of circular or rectangular cross section is relatively well developed and has been addressed in works such as the book by George Karniadakis et al [7] and in the review of Sharipov [8]. The book by Karniadakis focuses on simulations for gas flow through two parallel plates for pressure driven flow (flow driven by pressure difference) and for liquid transport driven by surface tension and electrokinetic forces, but does not place emphasis on two component flow and the determination of mass flow rates in channels. The review of Sharipov provides numerical and analytical models on mass flow rate and heat flux but does not compare with experiment. The translation of these results to the study of the hermeticity of encapsulations has led to an emphasis on ideal gas behaviour [9, 10], even for water permeation [11]. The flow of condensable gases and vapours such as carbon dioxide and organic vapours which may show strong departures from ideal gas behaviour in porous media has been reviewed by Choi et al [12] but a thorough discussion of water vapour flows is lacking. Reviews of the literature covering specifically moisture permeation rates have been limited to summaries of measurements of permeabilities rather than attempts to relate moisture permeability to theories based on microstructural properties. Water is a strongly polar molecule with strong intermolecular interactions in the form of hydrogen bonding and dipole–dipole coupling. These interactions impart an ability to wet many surfaces, so that potentially, liquid and vapour states can both contribute to barrier penetration. The selectivity of a barrier or leak for water relative to a non-condensable gas is clearly important in barrier testing and leak testing with gases other than water. While it is much simpler to test an encapsulation for the penetration of a noncondensable gas such as helium, and such tests are routinely used for hermeticity testing, there is presently no agreed scientific basis for converting a measurement of a helium transmission rate into one for water. A motivation of this review is to provide a basis for the development of sensitive, reliable and convenient tests for predicting the performance of barriers against moisture penetration by examining the principles that determine whether a correlation between a helium permeation test and a water permeation test can be made. We address flow and permeation of noncondensable gases as a necessary precursor to the understanding of the behaviour of water and water vapour. The verification that a barrier is suitable for its application is usually a test for establishing that the total permeation rate through the pathways present in the barrier is lower than a chosen threshold. This review is concerned mainly with flows under nominal isothermal conditions, where there are no imposed temperature gradients across the material or flow channel under study. Measurement of isothermal water vapour permeation rates has been carried out by gravimetric testing in 2
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area of the conducting aperture. Some authors have tended to emphasize mass flow rate [19] as we do in this review, while others use volume [20], particle number or molar flow rates. The mass flux is defined by m˙ /A, where A is the cross sectional area of the aperture defining the boundaries of the flow. The particle number flow rate and molar flow rate are often convenient for determining the effects on the contents of packages or encapsulations and are useful when comparing two different gases or vapours. The mass flow rate is readily converted to particle number flow rate using the conversion:
which the mass of accumulated water is measured by weighing a test cell containing a water absorbing material such as a desiccant before and after a known exposure time to water vapour [13–15]. Such tests are time consuming and not readily compatible with routine testing for performance compliance during manufacture. For this reason, testing for water penetration under accelerated conditions may be carried out in equipment such as that produced by MOCON [16] or Vacutran [17]. In the MOCON equipment, water molecules are detected by their effect on electrical conductivity using a coulometric cell. While this method is more convenient than the gravimetric test, there remain significant time delays associated with the response time of the sensor and the time for the water background to come into equilibrium with and be cleared from the interior of the equipment. A variant of this method uses an electrical sensor of metallic calcium that responds upon reaction with water [13–15]. Mass spectrometry is a method of detecting and quantifying the partial pressures of gases present in a chamber that has been used for measuring water vapour permeation rates [18]. Mass spectrometry has an excellent sensitivity and offers some advantages in response time and sensitivity. However, it is still subject to the time delays in the system that result from the need for the flow rates to come to equilibrium and the need for residual or background water to be cleared from the system. The use of isotopically modified water with mass 20 such as H2O18 [18] and heavy water D2O16 are of considerable value in minimizing the background of adventitious water vapour. This review is organized so that there is a progression in complexity from single component flows in individual channels with well defined geometry through flows in porous media where the channel geometry is highly complex and finally to flows in membrane materials. Within each of these areas there is a discussion of single component flows and binary component flows. The discussion of single component flows is further divided into non-condensable gases for which there is a considerable literature and into condensable vapours, especially water, which have been less well studied. Binary component flows are relevant to the flow of moisture through a barrier with air as a background gas at a pressure much higher than that of the vapour pressure of water. The case where water is one component and air is the other component is a typical one for an encapsulation exposed to the environment of the body. The problem of penetration of an encapsulation under these conditions is then one in which the flow of water is driven by a partial pressure gradient while the total pressure gradient is small or zero. We place an emphasis on quantitative prediction of flow rates. Formulas of particular interest for the prediction of mass flow rates that have been tested against experiment are highlighted in the text by enclosing them in a box.
n˙ = m˙ /m m, (1)
where n˙ is the number of particles flowing per second and m m is the molecular mass. The particle flow rate is converted to a molar flow rate by dividing by Avogadro’s number. The volume flow rate is related to the mass flow rate by: V˙ = m˙ /ρ, (2)
where ρ is the density of the fluid. The flow velocity averaged over the cross section is given by: V˙ v= . (3) A
In cases where there are two species present, labelled 1 and 2, the composition of the fluid under flow is described by the contributions of each to the total particle concentration, the particle number per unit volume: n n n c = = 1 + 2 = c1 + c2, (4) V V V In the case where species are gases, it is usual to write the total pressure as the sum of the two partial pressures: PT = P1 + P2. (5)
For ideal gases the equation of state applies: PV = nkBT , (6)
where V is the volume, n is the number of particles, kB is the Boltzmann constant and T is temperature. Equations (5) and (6) can be used to relate the particle concentrations to partial pressure for ideal gases: P P c1 = 1 , c2 = 2 . (7) kBT kBT 1.1.2. Permeability, permeance and water vapour transmission rate. Here we follow the ASTM standard definitions
of permeability, permeance and water vapour transmission rate, of a medium, but will express them in SI units. The lateral boundaries of the medium under consideration are normally considered distant and to cause no perturbation of the flow, as in a thin membrane covering an aperture. We will use axes in which the flow is directed along x. The particle number flow rate under steady state conditions through a medium with a defined aperture A is frequently observed to be proportional to the particle concentration gradient of permeant across the medium, a result originally arrived at by Fick from a consideration of diffusive processes and named Fick’s law:
1.1. Basic definitions and regimes of flow 1.1.1. Mass, volume and molar flow rates and fluxes. Quanti-
ties that are basic to the quantification of a flow are the flow rate and the flux. The flow rate is the quantity of flow per unit time and a flux is defined as the flow rate divided by the 3
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where L is the capillary length. The Poiseuille formula can also be applied to gas flows where the gas is dense and behaves as a viscous medium. For an ideal gas, µ has the value [25]:
n˙ ∂c (8) = −D F . A ∂x For a gas diffusing in a volume where the boundaries are distant, the Fick’s law diffusion coefficient D F is derivable from kinetic theory:
i ⎛ k Tm ⎞1/2 µ = 2 ⎜ B m⎟ , (14) dm ⎝ π ⎠
1 (9) D F = λu m , 3 where um is the mean molecular speed and λ is the mean free path which are given for an ideal gas by [21]:
where i is a numerical factor of order unity. For a hard sphere model i has the value 5/16 [25] which is approximately 1/π. This view of permeation as a flow through interconnected channels is also referred to as the pore-flow model [26]. The measured dynamic viscosity of water vapour at 25 °C at satur ation pressure is about 9.87 × 10−6 kg m−1 s−1 (see the database of National Institute of Standards and Technology (NIST) [27]), while from equation (14) it is 2.58 × 10−5 kg m−1 s−1. The discrepancy arises since water is a polar molecule and it is not well approximated using an elastic hard sphere assumption for equation (14). To describe the permeation process in a material membrane, the solution-diffusion model, originating in the work of Graham in 1866 [28], proposes that the flow through a membrane is established in three steps. First, the permeant dissolves in the surface of the material to give an internal concentration that is dependent on the pressure immediately outside the input surface (assuming, for now, the linear relation known as Henry’s Law which does not always apply):
8kBT um = (10) πm m
and kBT λ = , (11) 2 πPd 2m
where dm is the molecular diameter. Wroblewski [22] further developed Fick’s law and related the mass flux to a pressure gradient: m˙ ∆P (12) = Pmass , A l where Pmass is the mass permeability (kg m−1 Pa−1 s−1) of the medium to the permeant under study, l is the thickness of the medium and ∆P = Pi − Po is the pressure difference across the medium defined as the difference in the pressures of the permeant outside the medium on the inlet (Pi) and the outlet side (Po). In the special case where the medium under permeation is another ideal gas, the concentration of the permeant is continuous across the boundaries, but in general for other media there is a discontinuity in the concentration on passing from the outside to the inside of the medium. It is possible to assign volume, particle number and molar permeabilities of a membrane by changing the mass flux to volume, particle number and molar fluxes respectively. For membranes the non-SI unit of Barrer for volume permeability is sometimes used. A measurement in Barrer is converted to the SI unit of mass permeability by dividing by 2.9882 × 1023/M where M is the molecular weight of the permeant [23]. Mass permeance in units of kg Pa−1 m−2 s−1 is the mass permeability divided by the thickness of the test piece in the direction of flow. In the special case of water permeation through a membrane, a quantity termed water vapour transmission rate (WVTR) is defined as the mass transfer rate of water vapour per unit area of membrane (kg m−2 s−1) under known conditions of water vapour pressure on each side of the membrane. Following this definition, WVTR is the product of the permeance of the material to water vapour multiplied by the difference in water vapour pressure just outside the inlet and outlet surfaces of the test piece. Liquid flows were studied by Poiseuille [24] who developed a formula expressing the flow rate of a liquid in a capillary of radius rc in terms of the fluid dynamic viscosity µ (Pa s). For the mass flow rate of the liquid:
ci = PS (15) i H,
where SH is the Henry law solubility, Pi is the pressure on the inlet side of the membrane and ci is the internal concentration in the inlet side of the membrane. Second, the concentration gradient of the permeant inside the membrane is established by diffusion over time to a steady state level also determined by the conditions at the outlet side, in turn determined by the external pressure of the permeant just outside the outlet surface of the membrane, Po. Finally, the evaporation of the permeant from the outlet surface of the membrane, where again the external pressure, is related to the internal concentration at the outlet side of the membrane by the Henry’s law solubility. In this view of permeation, it is clear that the internal concentration gradient is related to the external pressure gradient by multiplication by the Henry’s law solubility of the permeant in the material. From equations (8) and (15) we write the mass flux as: m˙ ∂c P − Po = −D F = D FSH i . (16) A ∂x l
So that from equation (12): Pmass = SHD F. (17)
In some of the literature on membranes, the term permeation coefficient is used instead of permeability [29]. In polymers, the Fick’s law diffusion coefficient is often referred to simply as the diffusivity. An analogy can be made with the electrical conductivity of a medium where the mass flux is analogous to the current density and the pressure is analogous to electrical potential. The subject of permeation will be discussed further in section 8.
πrc 4∆Pρ , m˙ = (13) 8µL 4
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1.1.3. Dimensionless constants of flow in a channel. The
In the case where Henry’s law applies, equation (17) may be used to obtain any of three variables in it when the other two are known. The solubility can be determined separately by the total permeant uptake when the medium is immersed in permeant. In the case of water, a water absorbance is often measured in this way as a mass of water per unit mass of medium. For the purpose of this review, a porous medium is one having interconnected void spaces that can be filled with liquid or gas. The geometry of the pores may be complex, leading to flow channels with both constrictions and tortuosity. The nature of the interconnectivity may also be complex, with divergent and convergent connections. The simplest case is for ideal gas flows through a porous medium, in which case the concentration of the gas is continuous on entering the medium (corresponding to a Henry’s law solubility of unity). For condensable gas flows there will be a concentration increase on entering the medium when condensation occurs on the surface of the pores. The proportionality of the flow rate to the pressure difference ∆P across the medium often applies in viscous flows through porous media where it is referred to as Darcy’s law. When written as an expression for the volume flow rate [30] Darcy’s law is:
Reynolds number is useful in determining whether a viscous flow is turbulent or laminar and for a flow having a fluid velocity v is defined by: ρvdh Re = . (20) µ
Here dh is the hydraulic diameter of the channel defined by: 4A dh = , (21) pw
where pw is the wetted perimeter of the channel. Laminar flows are described by streamlines representing the vector direction of the flow. The Knudsen number is useful in defining flow regimes and is defined as: kBT λ Kn = = , (22) l0 4 2 πrc 2Pl 0
where l 0 is a characteristic dimension of the channel for which it is common practice to use the diameter 2rc for a cylindrical channel or the smallest separation of the walls h for a rectangular channel. The flow regimes are defined by the following ranges of Kn:
∆P k A ∆P V˙ = −Pvolume A =− , (18) l µ l
where Pvolume is the volume permeability (m2 s−1 Pa−1) and k is the intrinsic Darcy’s law permeability of the porous medium (m2). Note that when equation (18) applies, the intrinsic Darcy’s law permeability is a property of the porous medium and not of the permeant since the properties of the permeant enter through the viscosity. In this respect, the intrinsic Darcy’s law permeability differs from the mass and volume permeability definitions given above. The permeation process in a porous medium is frequently more complex than Darcy’s law suggests, especially for condensable permeants such as water vapour. We will discuss this in detail in section 7. It is therefore important to distinguish between gases that do not condense on the surface of a flow channel and vapours that condense to form a liquid layer. We divide flows into cases where there is a single species and cases where there are multiple species. For two comp onent flows, Landau and Lifshitz [31] have given a useful breakdown of the flux of one species in a two component flow. If species 1 is considered as the permeant of interest then its particle flux is given by:
• For 0 < Kn ⩽ 0.01, we define the continuum flow regime (also called hydrodynamic flow or viscous flow). The classical Navier–Stokes equations with no-slip boundary conditions are usually valid for this flow regime. • For 0.01 ⩽ Kn ⩽ 0.1, we define the slip flow regime. In this flow regime, a sublayer, known as the Knudsen layer, exists between the bulk of the fluid and the surface which has a thickness of the order of the mean free path. The majority of the flow is governed by the classical Navier– Stokes equations, while the flow in the Knudsen layer, which usually covers much less than 10% of the channel width or diameter, can be neglected. It is more accurate to use Maxwell’s velocity slip boundary conditions (first order boundary conditions). • For 0.1 ⩽ Kn ⩽ 10, we define the transition flow regime. In this flow regime, the classical Navier–Stokes equations for continuum require higher-order corrections to the boundary conditions. A simplified form of the Boltzmann transport equation known as the Bhatnagar–Gross–Krook (BGK) equation is one approach for modelling flow in this flow regime. • Kn ⩾ 10, we define the free molecular flow regime. In this flow regime, intermolecular collisions are assumed to be negligible in determining the gas dynamics. The BGK equation reduces to the Knudsen equation in this flow regime.
⎡ c ∂c ⎤ ∂T ∂c J1 = − ⎢D12 1 + D Tc1 + DF 1 T ⎥ , (19) ⎣ c T ∂x ⎦ ∂x ∂x
where D12 is the mutual Fick’s law interdiffusion coefficient of species 1 in species 2 describing the ‘interdiffusive’ component of the flow. D T is the ‘thermodiffusion’ coefficient describing temperature driven flow and D F is the Fick’s law diffusion coefficient describing the flow of both species together. Equation (19) is a statement that the fluxes are independent and additive. For example, the interdiffusive flux is carried with the total flux of the mixed gases. In a two comp onent flow, it is entirely interdiffusive if there are only partial pressure gradients without a total pressure gradient.
1.2. Single component gas flows in a channel: Knudsen diffusion and the Knudsen equation
The diffusion of a single component gas in a conduction channel was first studied quantitatively by Knudsen [32] who derived a value for the diffusion coefficient under molecular 5
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flow conditions in which collisions with the walls are dominant over intermolecular collisions, a process termed Knudsen diffusion. Knudsen’s expression for the Knudsen diffusion coefficient DK is analogous to that for the Fick’s law diffusion coefficient (equation (9)) where the mean free path is replaced by a path length determined by the channel dimensions:
The Boltzmann equation describes the evolution of the probability distribution function of the particles in the fluid, f (r, v, t ), describing the probability of there being a particle in a given phase space volume d 3rd 3v at time t, where r is the position of a particle and v its velocity. The particles are assumed identical and uncorrelated (molecular chaos) with no internal degrees of freedom. The Boltzmann equation is written as:
1 DK = umH , (23) 3
∂f ∂f ∂f +v⋅ +F⋅ = Q. (28) ∂t ∂r ∂v
where H is a shape-dependent length equal to the diameter 2rc for a cylindrical channel. The Knudsen diffusion coefficient is readily related to the mass flow rate in the channel by invoking the ideal gas law (equation (6)), so that the particle flux is related to the pressure gradient along the diffusion direction x by:
The first term gives the rate of change of the number of molecules at r having velocity v in the phase space; the second term describes molecules passing through a fluid volume at r with velocity v; the third term describes molecules at r passing through velocity space under the influence of an external force F. The collision term Q is a function that describes the nature of the collisions between particles. Q is expressed by the int egral giving the operator that gives the change in the number of particles with velocity v as a result of a collision between two particles of velocity v and v*:
n˙ n˙ D dP = 2 = K . (24) A kBT dx πr c
The mass flow rate is: π r 2m mDK dP . m˙ = m m n˙ = c (25) dx kBT
Using equations (10) and (23), we obtain the Knudsen equation for the mass flow rate in the molecular flow regime:
Q=
∫R ∫0 3
4π
v − v* I ( v − v* , Ω)
×[ f (r, v*)f (r, v) − f (r, v*′ )f (r, v ′ )] dΩdv*,
(29)
where I ( v − v* , Ω) is the differential cross section of the col lision, where the relative velocity changes from the incident direction to the element of the solid angle dΩ as a result of col lision. R3 is the domain of integration over all velocities. The term Q considers two particles colliding, starting with velocities v and v* and leaving the collision elastically with velocities v ′ and v*′ . Only binary collisions are considered, which applies provided the gas is sufficiently dilute. The terms subtracted from one another in the square brackets have the form ‘the number of particles scattered in, minus the number of particles scattered out’; some particles that had velocity v will be scattered through collisions to have new velocities, while other particles will join the group of particles with velocity v after colliding. The Boltzmann equation has been solved numerically for f, using approximations for Q, as the nature of Q as described in equation (29) makes the equation (28) impossible to solve analytically. Macroscopic variables describing the flow can then be obtained from the distribution function:
4 2π r 3c m m dP . m˙ = (26) 3 kBT dx
The coefficient that relates the mass flow rate to the pres sure gradient in a channel is sometimes referred to as the mass conductance of the channel. The Knudsen equation works well for the low pressure flow of molecules in channels and predicts that the mass conductance is independent of the pres sure (and therefore the Knudsen number for a given channel through equation (22)). There is an implicit assumption that the collisions with the walls of a channel are similar to col lisions between molecules in the sense that the momentum of a molecule in the direction of flow is not preserved after the collision. This corresponds to an assumption of diffuse reflection at the walls, to be discussed later. Soon after Knudsen’s paper appeared, Smoluchowski [4] provided a generalization of the Knudsen result for the case where a fraction α of the collisions with the walls are diffuse and the remainder specular. We will discuss the fraction α in more detail in section 2.1 to which Maxwell gave the name of (tangential momentum) accommodation coefficient, often abbreviated as TMAC. Smoluchowski’s result is:
∫
Particle number density : n = f (r, v, t )dv (30)
∫
4 2π r 3c m m dP 2 − α m˙ = . (27) 3 kBT dx α
Density : ρ(r, t ) = m m f (r, v, t )dv (31)
∫
1.3. The Boltzmann and related equations
1 Bulk velocity : u = vf (r, v, t )dv. (32) n
A starting point for the understanding of flows under a wide variety of conditions is the Boltzmann transport equation, derived by Ludwig Boltzmann in 1872. This approach is applicable in principle to all flow regimes.
The quantity of interest here is the mass flow rate of the fluid, which can be calculated by integrating the product of density ρ and fluid velocity u over an area (e.g.: the crosssectional area of a conduction channel): 6
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Table 1. Governing equations for steady flows in approximate
Governing equation Collision term
simplifying the Boltzmann equation is the Chapman–Enskog method for which the phase space distribution function f takes the form of a perturbation expansion including terms up to Kn2:
BGK equation
f = f 0 + f 1 + f 2 = f 0 (1 + aKn + bKn2 ), (37)
forms of the Boltzmann equation and the corresponding collision term.
Linearized Boltzmann equation (LBE)
1 ( floc − f ) τ QLBE = 3 + f 0 (v*)(h′ + h*′ − h − h*)dn dv* QBGK =
where a and b are functions of fluid density, temperature and velocity.
∫R ∫S
1.4. Navier–Stokes equations
The Navier–Stokes equations were first derived by the French mathematician L M H Navier in 1822 and then further developed by the English mechanician Sir G G Stokes in 1845. These equations apply to the single phase viscous flow of fluids including liquids as well as vapours and gases in the continuum flow regime. While some researchers have focused on the derivation of the Navier–Stokes equation from the Boltzmann equation [42–44], the Navier–Stokes equations are more simply derived from Cauchy’s equation of motion, which is based on Newton’s second law:
∫ ∫cross section
m˙ = ρ( y, z )u( y, z )dy dz. (33)
When considering flow in a channel, collisions with the boundary must be included in the collision term. For more details of the derivation of the equations in this section the reader is referred to the book of Karniadakis et al [7]. Various approximation schemes have been made to evaluate the collision term Q, depending on the flow regime. In the molecular flow and transition regimes, approximate forms of the Boltzmann equation have been applied such as the linearized Boltzmann equation (LBE) [33, 34] and the Bhatnagar– Gross–Krook (BGK) equation [8, 20, 35, 36] which may also be linearized. In the LBE and BGK equations, linearizing is carried out by writing the velocity distribution function as a perturbation expansion using a perturbation distribution function h(r, v, t ) [37]:
⎛ ∂u ⎞ ⎜g − ⎟ρ + ∇ ⋅ σ = 0, (38) ⎝ ∂t ⎠
This equation is valid for both compressible and incompressible fluids in the presence of the gravitational acceleration g. By expressing equation (38) in Cartesian coordinates and evaluating the contributions to the total stress tensor σ from pressure gradients and shear stresses and then relating the shear stresses to the rate of strain tensor (using the fluid constitutive relations), we obtain the compressible Navier– Stokes equation [45]. For which the x-component is:
f = f 0 [1 + h(r, v, t )] , (34)
where f 0 is the absolute Maxwellian velocity distribution function: ⎛ m m ⎞3/2 ⎛ m ⎞ f 0 = n 0⎜ ⎟ exp⎜− m v2⎟, (35) ⎝ 2πkBT ⎠ ⎝ 2kBT ⎠
where n 0 is the equilibrium number density. The above approaches differ in the form of the collision term in the Boltzmann equation, as shown in table 1 for the case of steady flows. In this table, floc is the local Maxwellian distribution function:
µ ∂ux ∂u ∂u ∂u 1 ∂P + ux x + u y x + uz x = gx − + ∂t ∂x ∂y ∂z ρ ρ ∂x 2 2 ⎛ ∂ 2ux ⎞ ∂Dv ∂ ux ∂ ux + + ⋅⎜ 2 + ⎟, (39) ∂x ⎠ ∂z 2 ∂y 2 ⎝ ∂x
with corresponding equations for the y and z-components. As in previous sections, the x direction is the direction of flow. Dv is a volumetric dilation and in the case of incompressible flow, Dv = 0. In vector notation for the incompressible case the equations are:
⎡ m (v − u)2 ⎤ ⎛ m m ⎞3/2 floc = n⎜ ⎟ exp ⎢− m (36) ⎥, ⎝ 2πkBT ⎠ ⎣ ⎦ 2kBT
where n = n(r, t ) is the local number density. It has been estimated that the difference in the mass flow rate between parallel plates calculated by solving the BGK equation and that obtained by solving the full Boltzmann equation for isothermal plane Poiseuille flow is only about 2% [8]. Radtke and Hadjiconstantinou [38] have devised a particle simulation technique based on the BGK approach that offers simulation of near-equilibrium conditions with reduced statistical uncertainty. Gladkov [39] and Kang et al [40] have shown that the Boltzmann equation can be reduced to a non-linear form of Darcy’s Law that linearizes at small flow velocities. Note that while the original Boltzmann equation is only valid for dilute gases, it has been developed for non-ideal dense gases and liquids by Enskog [41]. For dense fluids, a popular approach to
µ ∂u 1 + (u ⋅ ∇)u = g − ∇P + ∇2 u. (40) ∂t ρ ρ I. II. III. IV. V.
In the case where the flow velocity depends only on a single spatial variable, the flow is termed 1D. The terms numbered in equation (40) with Roman numerals have the meanings shown in table 2. The meanings of steady flow and uniform flow are also defined in table 2. In parallel flow the velocity is in one direction at all positions and times. Table 3 gives the special conditions for incompressible flows that lead to related governing equations that are special cases of equation (40), well known in fluid mechanics. The Navier–Stokes equations have been extended to cover all flow regimes by ensuring that the expressions approach 7
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Table 2. The meaning of the terms in the incompressible Navier–Stokes equations, equation (40).
Term No.
Meaning of the terms
I.
The temporal change in velocity at a fixed control volume in the flow. For a steady flow, in which the velocities at each position in the flow are independent of time, this term is zero. All steady flows are laminar. The convective acceleration of the fluid, which describes the acceleration of a control volume in the flow arising from the change in velocity with position. Uniform flow, as for example in a channel with dimensions independent of position, has no convective acceleration. The body forces acting on the fluid. The example used is gravity. The force arising from pressure gradients. The acceleration (deceleration) due to the frictional resistance (viscosity) of the fluid.
II.
III. IV. V.
Table 3. Special conditions applying to the Navier–Stokes equation, equation (40), which lead to the other principal equations of fluid flow,
with the applications listed. Name
Formula
Euler’s equation
1 ∂u + (u ⋅ ∇)u = g − ∇ P ∂t ρ ρu2 + P + ρgz = constant 2 µ ∂u 1 = − ∇ P + ∇2 u ∂t ρ ρ µ 2 1 ∇ P = g + ∇ u ρ ρ
Bernoulli’s equation Parallel flow equation Stokes equation
Special conditions on the terms
Application
V. = 0
Inviscid and incompressible flow.
I. = V. = 0
1D, steady, inviscid and incompressible flow.
II. = III. = 0
Uniform, incompressible flow with velocity in one direction in a horizontal channel or in zero gravity conditions.
I. = II. = 0
Steady, parallel flow. The inertial forces are small compared with viscous forces.
2. Ideal gas flow in a channel
the Knudsen flow (equation (26)) at large Kn. An empirically determined parameter is usually needed to achieve a smooth dependence on Kn for intermediate values of Kn [7, 46]. In one approach, the matching process is formalized by adding an additional diffusive mass flow rate term to the Navier– Stokes equations based on Fick’s law [47–49]. The resulting extended Navier–Stokes equations provide agreement with experimental data over a wide Knudsen number range covering the continuum to free molecular flow regimes for many gases such as helium, argon, nitrogen, and carbon dioxide (see section 2.1.2). For water vapour, flow measurements in the literature are rare as experimental difficulties in maintaining the constant pressure conditions required in conventional measurement techniques in the presence of large and temperature sensitive adsorption and desorption rates on the surfaces of experimental apparatus. The extended equations also provide good agreement with the results obtained using a numerical simulation method known as the Direct Simulation Monte Carlo (DSMC) method. The DSMC method was developed by Bird [50] and is a powerful computational method suitable for a wide range of Knudsen numbers, including the flow of rarefied gases at high Knudsen numbers. The efficiency of the method is partly due to the way in which the simulation cell size is chosen to be proportional to the mean free path and the time step is chosen to be smaller than, but of the same order as, the mean collision time [7]. The available theories for describing single non-condensing gas or vapour flow without special conditions have been summarized diagrammatically by Bird [50] as a function of Kn. An updated summary of Bird’s diagram is shown in figure 1.
We consider here ideal gas flows as a good approximation for the flow of the gases helium and air of relevance in leak assessment as well as describing an important limiting case of water vapour flows. At sufficiently low pressures, where there is no condensation of liquid water, water vapour can be considered to behave as an ideal gas. 2.1. Gas flow in a uniform cylindrical channel
The systematic study of flow in a cylindrical channel began in the nineteenth century. Jean Louis Marie Poiseuille, a French anatomist, first studied the pressure driven flow of viscous fluids through capillaries at low Knudsen numbers, publishing his work in 1841. He devised the well-known Poiseuille law for the flow rate in a cylindrical channel with an assumption of the flow velocity being zero at the walls [24]. When this no-slip boundary condition applies, the dominant forces are the viscous drag forces and the forces due to the pressure gradient in the tube. The Poiseuille law then follows. However, at lower pressures when the mean free path becomes comparable with the dimensions of the tube, friction forces operating at the walls of the tube become more important than the viscous drag in the fluid. The Poiseuille law fails to predict the flow rate under these conditions. Many researchers have worked to extend the range of the Poiseuille law to lower pressures [54]. By 1909, Knudsen had completed a theoretical and experimental investigation of flows in cylindrical channels at low pressures in which he derived the equation that bears his name for both cylindrical and rectangular channels. Knudsen assumed that 8
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Figure 1. The Knudsen number Kn ranges defining flow regimes. Also shown is a selection of equations for predicting flow rates (adapted with permission from [50], copyright Oxford University Press) with their regimes of applicability. There are three types of theory covering the values of Kn between the slip flow regime and the molecular flow regime, those using higher order slip boundary conditions (red box) [51, 52], those requiring an empirical parameter (pink box) [46, 49] and those arising from an approximate solution of the Boltzmann equation (blue box) [20, 33, 36, 53].
molecules incident on the walls move in a random direction after the collision, which applies in most cases except in special cases when the walls are exceptionally smooth. Knudsen’s important paper describing the equation that bears his name was published in 1909. In the following year, Smoluchowski [4] extended Knudsen’s equation by using Maxwell’s definition for the tangential momentum accommodation coefficient (TMAC) which describes the fraction of molecules which suffer fully diffuse reflections at the walls in which tangential momentum is lost, with the remainder having fully specular reflections in which tangential momentum is preserved. Up to the present time, the goal of obtaining increasingly accurate and convenient methods for predicting gas flow in cylindrical channels is still being pursued [20, 46, 53, 55, 56]. Here we outline the Poiseuille law for the continuum flow regime [21], the modified Poiseuille law for the slip flow regime [56] and introduce the extended Navier–Stokes equation (theory of Cha and McCoy [46]) for the transition and molecular flow regimes. We consider 2D isothermal flow through a cylindrical channel in the form of a tube of length L, radius rc, as shown in figure 2. We consider the case of long tubes where L/rc ≫ 1, which allows us to neglect end effects near the inlet and the outlet of the channel.
Figure 2. A cylindrical channel showing relevant variables. The flow direction is along x for which the flow velocity u is positive.
πrc 4PO2 m m 2 πr 4∆PPmm m m˙ = c = (Π − 1), (42) 8µkBTL 16µkBTL
where Pm is the mean pressure across the channel, and we define Π = Pi /Po as the pressure ratio between inlet and outlet of the channel of length L. It is useful to plot the mass flow rate against pressure ratio Π when the outlet pressure is constant. When fluid enters the channel or exits the channel by means of a larger diameter flow channel, turbulence may affect the pressure difference across the channel. In reality, the velocity of flow at the walls is not zero, but has a non-zero slip velocity, given by u w. Reviews of the subject of slip velocity are given in Neto et al [57] and Cao et al [58]. Maxwell considered two limiting cases to take into account the collisions between the flowing molecules and the surface of the walls, each described by a value of TMAC. If the surface of the wall is smooth on the molecular size scale, the reflection is perfectly specular. In this case, there is no momentum transferred from the molecules in the direction tangent to the surface of the walls, a case defined by α = 0 where α is the average value of the ratio of the tangential momentum transferred to the surface divided by the initial momentum. It is easy to show that this average is also equal to
2.1.1. The continuum and slip flow regimes. The flow is sustained by a pressure drop (∆P = Pi – Po) from the inlet (i) to the outlet (o) of the channel. For a cylindrical channel of uniform diameter, the Stokes equation applies (see table 3). In cylindrical coordinates, the Stokes equation is:
µ ⎛ ∂u ∂ 2u ⎞ ∂P + r 2 ⎟ = 0. + ⎜ − (41) r ⎝ ∂r ∂r ⎠ ∂x
When the no-slip boundary condition is applied at the wall, the equation is solved to give the velocity as a function of r. The mass flow rate in the continuum is then given by the well known Poiseuille law for an ideal gas: 9
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Kn(x ) by Knm, we obtain a result convenient for calculation of mass flow rates in the slip flow regime:
the fraction of molecules that undergo diffuse reflection, if it is assumed that specular reflection and diffuse reflection are the only two options available to a molecule. If the surface of the wall is rough on the molecular scale, diffuse reflection occurs. A complete diffuse reflection transfers, on average, all of the tangential momentum of a molecule to the wall, a case defined by α = 1. In general, the tangential momentum accommodation coefficient is in the range of 0 ⩽ α ⩽ 1. Maxwell obtained the following equation for the slip velocity of flow at walls as a boundary condition involving only the first order derivative of the velocity:
πr 4∆pPmm m m˙ = c (1 + 8A1Knm ). (47) 8µkBTL
This mass flow rate equation applies to the slip flow regime. The first order boundary condition introduced by Maxwell can be extended by including in it a series of derivatives of the velocity of increasing order. The highest order of derivative specifies the order of the boundary condition. Where second-order and higher-order boundary conditions are used, a straightforward solution of the Navier–Stokes equations is no longer possible. The Burnett equations [7] are then applicable but are much more difficult to solve. As an approximate method, a separate analysis of the Navier–Stokes equation for the interior region and the wall region has been used with higher order boundary conditions at the walls [61]. Maurer et al [61] compared the use of first and second order boundary conditions with experimental data ranging from the slip flow to the transition flow regimes. In order to illustrate the effects of boundary conditions on flow rate more clearly, it is useful to define a nondimensional normalized flow rate S defined as the ratio of mass flow rates under the slip boundary conditions to the mass flow rate under no-slip boundary conditions (the continuum flow regime):
2 − α ∂u uw = ± λ . (43) α ∂r
Here the mean free path λ is not the kinetic theory result of equation (11), but an expression derived using the hard-sphere model [25]: 1/2 µ ⎛ πk T ⎞ λ= ⎜ B ⎟ . (44) P ⎝ 2m m ⎠
This equation gives the previously quoted result, equation (11), for the mean free path in an ideal gas when the viscosity µ of an ideal gas is calculated from a hard sphere model, equation (14). Since the Knudsen number changes with pressure, its value in the channel will depend on the distance x along the flow direction. Kn(x ) is the local Knudsen number, Kn(x ) = λ(x )/d where λ(x ) is the local mean free path, defined as λ(x ) =
( )
µ πkBT 1/2 P(x ) 2m m
S = 1 + 8A1Knm + 16A2 ln Π K n2m, (48)
where A2 is a dimensionless constant dependent on the molecular interaction model. S is useful for comparing the order of the boundary conditions as it shows only the departure from the no-slip predictions. For the hard sphere model,
and P(x ) is the local
pressure. Using an approach in which the presence of a slip velocity is used to correct the Stokes equation, a mass flow rate equation for slip flow conditions can be obtained by integrating equation (41) over the cross section to give the flow at location x in the channel [55]:
σ2p Π − 1 , A2 = 2 (49) kλ Π + 1
πr 4m dP m˙ = − c m [1 + 8A1Kn(x )], (45) 8µkBT dx σ
where σ2p is the second velocity slip coefficient obtained from fitting to experimental data. For the variable hard sphere model [59]:
2−α
where A1 = kp , σp = α is known as the first velocity slip λ coefficient and kλ is a coefficient dependent on the assumed
⎛ σ2p 1 ⎞ Π+1 A2 = ⎜ 2 + . ⎟ (50) ⎝ kλ 2kλ 2 ⎠ Π−1
π
molecular interaction model (for example, kλ = 2 for a hard sphere model [25], and kλ = 0.731 for nitrogen for a ‘variable hard sphere’ model [59]). An improved expression for the first velocity slip coefficient is obtained by Albertoni et al [60]:
Although the first and second order boundary conditions give somewhat different theoretical predictions, the exper imental precision has so far been insufficient to discriminate between the predictions based on the different orders of boundary condition. In this work we will use the equation (47) for the slip flow regime, in the range of Knudsen number from 0.01 to 0.1. In this flow regime the boundary condition order makes no significant difference to the prediction since the separation of the predicted curves begin for values of Knm above 0.1 (see figure 3).
2−α σp = (σp(1) − 0.1211(1 − α)), (46) α
where σp(1) = 1.016 is the first velocity slip coefficient when α = 1. In the case where the flow conditions do not change substantially in a channel, it is convenient to use a mean Knudsen number Knm to replace Kn(x ). A mean Knudsen number can be obtained from the relation Knm = λ /d = 1
2.1.2. The transition and molecular flow regimes. Cha and
( )
µ πkBT 1/2 , Pm 2m m
McCoy [46] developed equations for ideal gas flows based on the standard assumptions of Poiseuille flow, that the flow is laminar, independent of time, isothermal, that the end effects are negligible, the pressure is uniform in the plane normal to
where the mean pressure is Pm = 2 (Pi + Po ). By integrating equation (45) with respect to x and replacing the integral of 10
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Figure 3. (a) The points show the experimental measurements of mass flow rate for helium in a cylindrical channel of diameter 25 μm from Ewart et al [59] through the continuum regime, slip flow regime and part of the transition flow regime for a pressure ratio of 3, as a function of mean Knudsen number Knm. The first order slip boundary conditions (lower curve, blue) and the second order slip boundary conditions (upper curve, red) are calculated from equations (47) and (48). (b) shows the normalized nondimensional flow rate S defined in equation (48) as a function of Knm for the same two cases as in the theoretical curves in (a) to illustrate more clearly the difference caused by the choice of boundary conditions.
the flow and that the rate of pressure drop along the flow is constant. The approach of their work was to solve the Cauchy equation (equation (38)) by expanding the stress tensor in terms up to third order in velocity gradient and applying the Chapman–Enskog theory to the Krook equation to obtain the expansion coefficient. They required the resulting flow to approach Smoluchowski’s extension of the Knudsen equation (equation (27)) at large Kn and the Poiseuille law (equation (13)) at small Kn. They found a single adjustable numerical constant c0 was required to match experimental data in the transition flow regime, giving the equation: πrc 3∆P ⎛ m m ⎞ 2 ⎡ 8 2 − α π 1 ⎤ + ⎜ ⎟ ⎢ ⎥ 0 L ⎝ 2kBT ⎠ ⎣ 3 π α 8 Kn ⎦ ⎧ ⎫ ⎪ ⎪ ⎡ ⎛ 1 ⎞⎤ 1 ⎬. ×⎨1 − c0Kn tan hKn ⎢1 − 2NJ1⎜ ⎟⎥ − 2 − α 64 ⎝ ⎠ ⎣ ⎦ N ⎪ 2 1 + α 3π Kn ⎪ ⎩ ⎭
m˙ =
∫
rc
8 2−α G= . (53) 3 π α
For TMAC of unity this limit is 1.505. Cha and McCoy’s result for G as a function of the mean Knudsen number for three different choices of c0 is shown in figure 4 with a comparison with experimental data for 5 gases [32, 62, 63] and the Knudsen equation [32]. The theory of Beskok and Karniadakis [64] also applies across all flow regimes but is not used in this work as it requires more than one empirical parameter and has been tested only against the results of one author [65], see figure 4(b). Theories without any empirical parameters based on a solution of the Boltzmann equation by Cercignani and Daneri [20], and Loyalka [33], that apply in the transition and molecular flow regimes are also shown in figure 4 and will be discussed further for elliptical and rectangular channels in sections 2.2 and 2.3. The theory of Cha and McCoy illustrates the Knudsen paradox, first enunciated by Knudsen in 1909 [32] which states that there is a minimum value of the mass flow rate per unit pressure difference when plotted as a function of Knudsen number. The occurrence of this minimum value happens when the diffusive flow begins to be dominant where it relates to the physical properties of the gas or vapour and the pressure ratio. In the Cha and McCoy theory, this value applies when the Kn is around 1.43 for a cylindrical channel [46].
1
2πrρu dr =
(
)
(51)
This equation has the correct asymptotic behaviour at small and large Knudsen numbers as required. A second nondimensional reduced flow rate G is useful to assist comparison between theories: 1/2 L ⎛ 2kBT ⎞ m˙ . G= ⎜ ⎟ (52) πrc 3∆P ⎝ m m ⎠
G is analogous, except for the factor
( )
2kBT 1/2 , mm
to the
2.1.3. Theory and experiment across all flow regimes. The
concept of conductivity in electrical property measurements, as it normalizes for the length and the radius of the channel. Note that the other non-dimensional flow rate S introduced earlier is useful mainly in the continuum and slip flow regimes, whereas G is useful across all flow regimes. In the limit of large Knudsen numbers, the asymptotic value of G from equation (52) depends only on the value of the Maxwell TMAC, α:
flow rate of nitrogen has been studied experimentally by Lei and McKenzie [69] across all flow regimes in silica channels of 25 μm diameter of various lengths prepared from the same capillary tubing. The results for the mass flow rate are shown in figure 5. The theory of Cha and McCoy provided a good fit with c0 of 6.5 ± 1.5 to all of the data. By fitting the Navier– Stokes equation with first order Maxwell boundary conditions to the data in the slip flow regime, a value of the Maxwell 11
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Figure 5. Experimental nitrogen mass flow rates as a function of mean Knm. The solid line is the fit with Cha and McCoy (C&M) [46] theory (c0 = 6.5 ± 1.5). The data from Lei and McKenzie [69] are for a pressure ratio of 3 were measured using some 1 cm channels and some 20 cm channels depending on the flow rates relative to the background outgassing rate. The mass flow rate quoted is converted to the equivalent flow for a channel of 20 cm length. Reproduced with permission from Lei and McKenzie [69]. Copyright 2014 Elsevier.
specular and define an ensemble average of the diffuse fraction as the TMAC to which we give the symbol f. Arya et al showed by simulation that the use of the Maxwell TMAC definition overestimates flow in the Knudsen regime. Therefore, the values of the Maxwell TMAC α in the slip flow regime and the molecular flow regime will not be the same, as observed. 2.1.4. Ideal gas flow in cylindrical nanometre dimensioned channels. All theories discussed so far converge to the
Figure 4. The reduced flow rate G of equation (52) as a function of mean Knudsen number Knm calculated for a cylindrical channel with d = 25 μm. (a) shows results of theoretical predictions and experimental results for the transition flow regime. The result of Cercignani is a solution of the BGK equation [20].The result of Knudsen is obtained with an assumption of α = 1 [32]. The effect of varying the empirically determined number co in the theory of Cha and McCoy [46] is shown. Experimental data shown for comparison are: Smith [66] and Tomothée for N2 Knudsen [59] and Dong [67] for CO2 and Dong for H2, He and air [68]. (b) shows results for all flow regimes. The theory of Cha and McCoy is shown for four different values of co. The theory of Beskok and Karniadakis [64] is also shown together with experimental results for N2 of Lei and McKenzie [69], Ewart et al [59], Otani et al [66] and Tison [65].
Knudsen result at high values of Kn. An interesting test of the theories is to explore the limit of small channel diameter, where the strong constraint provided by the walls may lead to new phenomena. The availability of perfectly formed smooth walled carbon nanotubes has provided an interesting test case for the theory at the smallest diameter well formed channels so far investigated. Until results on carbon nanotubes became available, it appeared that even for the highest Knudsen numbers, the Knudsen assumption of diffuse reflection at the walls appeared to hold. The conduction of air in arrays of carbon nanotubes has been found to exceed the prediction on the basis of the Knudsen equation by large factors, of between 16 and 120 [72]. This excess flow rate may be the result of the very smooth walls of the nanotubes that allow specular reflection of molecules at the walls. If this is the case, the equation (51) can be used to derive an equation that allows molecular flow at high Knudsen numbers for the case of smooth walls, with values of α of less than 1. From equation (51) we obtain the result of Smoluchowski (equation (27)) for large Kn [4]:
TMAC of α = 0.91 ± 0.051 was found which agrees with the value found by Maurer et al [61] in silicon wafer microtubes (0.87 ± 0.03), by Ewart et al [59] in fused silica mircotubes (0.933 ± 0.037) and by Porodnov et al [70] in glass capillaries (0.925 ± 0.014). The value of the Maxwell TMAC α found by fitting the theory of Smoluchowski in the molecular flow regime is larger, and very close to unity. The origin of this discrepancy was proposed in [69] to lie in the use of Maxwell’s definition of TMAC for the molecular flow regime. Arya et al [71] discussed the fact that Maxwell’s assumption that reflections are either fully diffuse or fully specular, embodied in his definition of α, is not correct and propose instead the assumption that all reflections are partially diffuse and partially
πr 3∆P ⎛ m m ⎞ 2 ⎡ 8 2 − α ⎤ m˙ = c ⎜ ⎟ ⎢ ⎥ L ⎝ 2kBT ⎠ ⎣ 3 π α ⎦ 1
=
12
4 2π 3 m m dP 2 − α . rc 3 kBT dx α
(54)
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This equation could be used to explain the high conductances observed for nanotubes. Using the result of Holt et al [72] that air flow in a small polycarbonate tube with an internal diameter of 15 nm has an enhancement over the Knudsen result of 2.1, the value of α would be 0.65. In an extreme case, for a double walled nanotube, the value of α would be 0.05. This large increase in flow was attributed by Holt et al to the smoothness of the nanotube walls. We conclude that in such cases, the value of α would be as low as 0.05. It is not immediately clear that the nanotube case is an exception in having very small values of α. In order to test whether such accelerated flow phenomena occur in fabricated silicon channels, Gruener and Huber [73] have constructed arrays of conduction channels in silicon has extended observations in microchannels up to very large Knudsen numbers, claimed to be the highest Knudsen number for which observations have been made. No evidence of enhanced flows were found, with the scaling with temperature and molecular mass according to Knudsen’s diffuse reflection assumption (α = 1) holding true over the whole range, up to Kn of 107. An obvious difference between the case of a fabricated channel and a nanotube is the difference in effective roughness of the wall that may lead to differences in the amount of specular as opposed to diffuse reflection. The effect of adsorbed water on a silica surface on the value of α for nitrogen molecules incident on the surface was studied by Seo et al [74] by measuring the damping of a spherical particle which was separated from a flat plate by a distance ranging from 10 nm to 10 μm. The value of the TMAC α was found to be a complex function of the humidity. The value of α was 0.4 for the dry hydrophilic silica surface, and increased to 1.0 as the surface became covered with a very thin layer of adsorbed water, then decreased to 0.4 again as the layer became thicker, finally approaching 1.0 for very thick layers of adsorbed water. This unexpectedly complex behaviour of α with the humidity is not completely understood at present (see more on this in section 2.2.1 where water vapour flows are discussed).
Figure 6. Cross section of an elliptical channel showing variables. The flow direction is along x.
The cross section of an elliptical channel is shown in figure 6. For the continuum and slip flow regimes, we begin with the Stokes equation shown in table 3. For the continuum flow regime, no-slip boundary conditions are applied on the walls so that u(z, y ) E = 0, where the boundary E is defined by the elliptical equation z 2d 2 + y 2c 2 = 1. Then the Stokes equation can be integrated over the cross sectional area to give the Poiseuille law for an elliptical channel: π c 3d 3 m m Pm∆P . m˙ = (55) 8 (c 2 + d 2 ) kBT µL
This equation reduces to the result for a cylindrical channel (equation (42)) in the case where c = d. For the slip flow regime, the velocity needs to be corrected using the slip boundary conditions. Following a derivation procedure similar to that of section 2.1, the mass flow rate with first order boundary conditions in the slip flow regime is expressed as: π c3d 3 m m Pm∆P (1 + 8A1Knm ). m˙ = (56) 8 (c 2 + d 2 ) kBT µL
This result reduces to that of a cylindrical channel, equation (47), when c = d. For the transition and molecular flow regimes, the mass flow rate for an elliptical channel has been derived from the BGK equation following the approach of Sharipov [75]. The BGK equation applies to a steady isothermal flow:
2.2. Gas flow in an elliptical channel
A solution of the BGK equation developed by Sharipov et al [75] has been applied to channels with an elliptical shape, cylindrical being a special case. The treatment here is covered separately for each flow regime. In this section, we introduce the theory of Sharipov, based on a solution of the BGK equation for a general elliptical channel, and compare it with the theories developed specifically for cylindrical channels, such as the theory of Lo and Loyalka [35], a solution of BGK equation by the integro-moment method [35], the theory of Loyalka and Hamoodi [33], a solution of the LBE by the neutron transport method [33], the theory of Valougeorgis and Thomas [36], a solution of the BGK equation by the neutron transport method [36] and the theory of Cha and McCoy [46] discussed previously. These theories except the theory of Cha and McCoy in this section are complete theories in the sense that they do not have an empirical parameter, however, in the continuum regime these theories are not well adapted and do not always converge well to the Navier–Stokes behaviour at small Kn.
∂f 1 v⋅ = ( floc − f ), (57) ∂r τ
where r is the position vector, τ (= µ /P) is the relaxation time and the local Maxwellian distribution floc is given in equation (36). The BGK equation may be linearized to gain further computational advantage. The velocity distribution functions floc and f are given by equations (35) and (36). Using the substitutions: π d π Rarefaction parameter : ϕe = = (58) 2 λ 2Kn ⎛ m ⎞1/2 Scaled velocity : c = ⎜ m ⎟ v. (59) ⎝ 2kBT ⎠ 13
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Figure 7. Comparison of experiment with theories for the nitrogen reduced mass flow rate G in cylindrical channels across the full range of Knudsen number. The work by Graur and Sharipov [75] derived for elliptical channels agrees well in the cylindrical limit with theories developed for cylindrical channels and the experimental results of Otani et al [66] and Lei and McKenzie [69] for cylindrical channels. Cylindrical theories are Loyalka and Hamoodi [33], Lo and Loyalka [35] and Valougeorgis and Thomas [36], based on a solution of the linearized Boltzmann equation or the BGK equation. The semiemprical theory of Cha and McCoy [46] and the result for the Navier–Stokes equation are also shown. Figure 8. (a) The mass conductance of 10 μm (pink hollow triangles), 25 μm (squares) and 50 μm (circles) silica capillaries of 1 cm length for water vapour at 295.5 K as a function of the mean Knudsen number Knm. The channels are new or have received heat treatment as shown in the legend. The results for 10 and 50 μm channels are converted to the equivalent flow for a 25 μm capillary. The blue line is the theory of Cha and McCoy [46] for water vapour. The dashed part of the line is for water vapour flows that cannot be accessed at the test temperature (295.5 K) because of the formation of liquid water. The dashed curves through the data points are guides to the eye. (b) The same water vapour flow measurements and theory as in (a) are here shown as a function of Pm /Ps. The stars refer to a t test comparing theory and experiment (★★★ p
