Continuum and discrete modelling of isothermal water and air flow in porous media

Transcription

1 Continuum and discrete modelling of isothermal water and air flow in porous media Filip Descamps, Building physics laboratory, KU Leuven. Anyone who has ever watered the roses or given a baby its bottle, will have witnessed the phenomenon of air bubbling induced by liquid transfer.

2 Jury: prof. dr. ir. J. Delrue, voorzitter prof. dr. ir. H. Hens, promotor prof. dr. ir. G. De Roeck prof. dr. ir. J. De Baerdemaeker prof. dr. ir. P. Moldenaars prof. dr. ir. J. Carmeliet prof. dr. ir. J. Claesson (Lund University) prof. dr. ir. S. Bories (IMFT Toulouse) UDC D/1996/7515/50 ISBN

3 Continuum and discrete modelling of water and air transfer in porous media ir. Filip Descamps key words: two-phase flow, imbibition, calcium silicate brick, rain, percolation, building material ABSTRACT Water imbibition into porous building materials is a two-phase flow process in which the imbibing water displaces the air that is initially present in the pore space. Accurate non-destructive measurements of transient moisture content profiles during imbibition into calcium silicate brick indicate that the air outflow boundary condition has a predominant influence on the water imbibition process. Significant differences are observed in the water sorption coefficients and the inflow surface moisture contents between water imbibition experiments with free and bounded air outflow boundary conditions. The time scale differences between the water and the air transfer process and the limited space resolution of the non-destructive moisture content measurement preclude the simultaneous and accurate measurement of both transfer potentials. A direct calculation of the water and air transfer coefficients from an imbibition experiment is thus impossible. Discrete modelling of the pore space is used as an indirect way to calculate these coefficients. Six parameters are identified that characterise the geometrical and topological properties of the pore space. The values of these parameters for calcium silicate brick are calculated so as to conserve the crucial water transfer properties of the porous medium: (i) the critical moisture content, (ii) the capillary moisture content and (iii) the moisture content dependence of the water diffusivity. Percolation concepts are used to incorporate these macroscopic water transfer properties into the discrete network model. The one-phase mass transfer coefficients for water and air are then calculated from the discrete network model using a stationary approach. The residual air saturation for different boundary conditions is calculated from a dynamic immiscible displacement approach. Continuum modelling of the imbibition process using a two-phase flow approach and the calculated mass transfer coefficients confirms the experimental observations. (283 words)

4 List of symbols δp vapour permeability (s) δpa vapour permeability of air (s) ϕ relative humidity (-) λ γ radioisotope decay constant (-) λ Boltzmann variable (ms -0.5 ) µ vapour resistance factor (-) µ γ mass attenuation coefficient (m 2 /kg) ηi dynamic viscosity of phase i (Pa s) ρ bulk density (kg/m 3 ) σ surface tension (interfacial potential energy) (N/m, J/m 2 ) θ contact angle ψ0 open porosity (m 3 /m 3 ) ψi volumetric content of phase i (m 3 /m 3 ) A water sorption coefficient (kg/m 2 s 0.5 ) B intrinsic permeability (m 2 ) Dl liquid diffusivity (m 2 /s) E γ photon energy (J) f Darcy-Weisbach friction factor (-) Ia absolute intensity (-) K permeability (s) L thickness, length (m) &M mass flow rate (kg/s) &m density of mass flow rate (kg/m 2 s) n cluster size (-) N number of events (-) &N event rate (cps) o occupation probability (-) otr percolation treshold (-) p pressure (Pa) PR pore radius distribution (m) PV pore volume distribution (m 3 ) R number of counts (-) r radius (m) rref reference radius for pore radius and pore volume distribution (= 1 m) &R count rate (cps) Rg perfect gas constant (J/kgK) s network size (-) T temperature (K) t time (s) u log(r/rref) where rref = 1 m

5 v velocity (m/s) w moisture content by volume (kg/m 3 ) xi distance from the surface to measuring point i (m) z coordination number Subscripts a air b bubbling BC boundary condition D driving rain i interface l liquid M maximum m minimum v vapour f fluid (gas or liquid) g gas s solid c capillary

6 Table of contents 1 Introduction 1.1 Aim of the work 1.2 Method 2 Isothermal mass transfer in homogeneous porous media 2.1 Continuum modelling The representative elementary volume The phenomenological transport equations. Diffusion transfer Bulk fluid transfer Capillary liquid transfer The balance equations for moisture transfer 2.2 Discrete modelling Isothermal laminar flow in a tube Incompressible and compressible steady state flow in a tube Laminar incompressible liquid flow in a tube Laminar compressible gas flow in a tube Simultaneous water and air flow in a tube D pore space models Parallel tube model Series-parallel model (cutting-and-random-rejoining model) Capillary string model Network modelling and percolation concepts Network modelling: a topological approach of the pore space Percolation theory: universal characteristics of flow in random networks Percolation quantities and concepts Capillary fluid transfer applications of percolation concepts Invasion percolation and immiscible displacement models 3 Experimental determination of the pore structure and mass transfer properties of calcium silicate brick 3.1 Basic hygric properties Open porosity and bulk density Sorption coefficient and capillary moisture content

7 3.2 Characterisation of the pore space Water retention curve Pressure plate technique Mercury intrution technique Pore volume distribution 3.3 Experimental determination of the liquid diffusivity γ-ray attenuation technique Characterisation of a γ-ray source The interaction of γ-rays with matter Experimental equipment Calibration procedure Moisture content profile measurement and accuracy analysis Transient water transfer experiments Free imbibition experiment Horizontal drying experiment Bounded imbibition experiment The inverse water transfer problem Boltzmann transform method Flow-gradient method Optimisation method Discussion 3.4 Experimental determination of the air permeability Air permeability of the dry material Steady state method Transient method Air permeability as a function of moisture content 4 Discrete modelling of isothermal water and air transfer 4.1 Design of a discrete model Pore radius distribution and pore geometry Network topology 4.2 Percolation models Physical principles and numerical algorithm Topological and geometrical network parameters of calcium silicate brick Pore space topology Pore space geometry Backbone and dead-end fractions

8 4.3 Immiscible displacement models Characterisation of two-phase mass transfer The viscosity ratio κ The capillary number Ca Two-phase flow in a discrete pore model Basic principles and algorithms Handling of constrained interfaces Residual air saturation for the water imbibition process Discrete network parameters Acceptance profile and residual air saturation 4.4 Discussion 5 Continuum modelling of isothermal water and air transfer 5.1 Immiscible mass transfer equations Immiscible water and air transfer in a rigid porous medium Numerical implementation of the immiscible transfer equations One-phase water transfer in a rigid porous medium 5.2 Simulation of the horizontal water imbibition process The diffusivity equation approach The conductivity equation approach The immiscible transfer approach Free imbibition process Bounded imbibition process Pressure infiltration process 5.3 Discussion The influence of air on water imbibition The concept of the capillary moisture content Practical relevance of bounded profile air outflow boundary conditions 6 Conclusions and perspectives 6.1 The one-phase inverse problem 6.2 Topological and geometrical characteristics of the pore space 6.3 The influence of air on water imbibition 6.4 Water-vapour-air modelling of non-isothermal water imbibition Appendix. Simplified modelling of capillary rainwater uptake A.1 Simplified modelling of the water transfer coefficients A.2 Simplified analytical model of the capillary rainwater uptake

9 1 Introduction 1.1 Aim of the work In this thesis we present an analysis of the phenomena that can be observed during water imbibition into rigid and homogeneous porous media. Water imbibition is defined as the process of entry into a porous medium of liquid water that is available at its surface. We limit to horizontal and isothermal systems in which the effects of gravity and of temperature gradients may be neglected. Water imbibition into a dry porous medium is modelled as a displacement process of two immiscible fluids: the air that is initially present in the pore space is displaced by the imbibing water. The displacement process is governed by the viscous forces within each phase and by the capillary forces at the phase interface. Our purpose is to make clear how water and air interact during water imbibition. The influence of the air phase on water imbibition is twofold: an air pressure head is built up ahead of the wetting front, and air is trapped within the pore structure of the medium. The quantitative analysis of these phenomena will result in a better definition of the water sorption coefficient and the boundary conditions at the water inflow surface. It may improve future experimental determination of the water transfer properties of porous media. 1.2 Method [Philip 1969] introduced the non-linear diffusion approach to the horizontal isothermal water transfer process: w = ( D ( w) w) [1.1] t l The transfer coefficient D l is called the liquid diffusivity. The strongly non-linear dependence of the liquid diffusivity D l on the moisture content w is governed by the microscopic topology and geometry of the

10 2 Chapter 2 pore space. It is calculated from the one-phase mass transfer analysis of transient moisture content profiles or from a discrete model of the pore space. This continuum modelling approach is discussed in detail in chapter 2. A state-of-the-art review of different discrete models of the pore space is given. The use of these discrete models for calculating the water transfer coefficient D l is indicated. Chapter 3 is devoted to the experimental determination of the pore space characteristics and mass transfer properties of calcium silicate brick. The γ-ray attenuation technique for the non-destructive measurement of transient moisture content profiles during water imbibition is discussed. The calibration and the accuracy analysis of this technique are crucial to the solution of the inverse transfer problem. An in-depth analysis of the numerical solutions of the one-phase inverse problem reveals the predominant influence of the air outflow boundary condition. Significant differences are observed in the water sorption coefficients and the inflow surface moisture contents of water imbibition experiments with free and bounded air outflow boundary conditions. The time scale differences between the water and the air transfer process and the limited time and place resolution of the γ-ray attenuation technique preclude the simultaneous and accurate measurement of both transfer potentials. A two-phase mass transfer analysis of the imbibition experiments is thus impossible. This is why a discrete network model is used in chapter 4 to calculate the mass transfer coefficients of calcium silicate brick. In a first step, six parameters are identified that characterise the geometrical and topological properties of the pore space. The values of these parameters for calcium silicate brick are calculated so as to conserve the crucial water transfer properties of the porous medium: (i) the critical moisture content, (ii) the capillary moisture content and (iii) the moisture content dependence of the water diffusivity. Percolation concepts are used to incorporate these macroscopic water transfer properties into the discrete network model. In a second step, the water and air transfer coefficients are calculated from the discrete network model using a stationary mass transfer approach. Finally, the residual air saturation for different boundary conditions is obtained from a dynamic immiscible displacement approach.

11 Method 3 In chapter 5 continuum modelling of the imbibition process using a twophase mass transfer approach and the calculated mass transfer coefficients confirms the experimental observations. The assumptions that allow to reduce the two-phase mass transfer process to the one-phase mass transfer process [1.1] are identified. Chapter 6 summarises the achievements and conclusions of this work. The conclusions give rise to new questions that are presented in a more global framework for suggested research in the field of mass transfer in porous building materials. In the appendix, a set of guidelines is compiled to determine the water transfer properties of capillary building materials in a simplified way. Building physicists can use these guidelines to calculate the rain water intake by capillary exterior wall leafs or the initial phase of the vertical water uptake process in the case of rising damp.

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13 2 Isothermal mass transfer in homogeneous porous media A porous medium is a multi-phase system consisting of a solid phase and one or more fluid phases that occupy the pore space. It is microscopically quantified by the geometrical and topological characteristics of the pore space and by a set of relevant state variables for each phase. We limit to a rigid and homogeneous porous medium consisting of three phases: a rigid solid matrix phase s, a liquid water phase l and a gas phase g (figure 2.1). The gas phase g consists of water vapour and air. It is assumed that there is no chemical interaction between the solid and the fluid phases. Mass transfer phenomena in a porous medium may be described at the microscopic level, at which we focus on what happens at a mathematical point within a given phase. This microscopic approach of mass transfer phenomena is usually intractable because of the inability to describe the complex geometry of the phase interfaces. A discrete model is a conceptual model of the pore space that incorporates only its basic features of geometry and topology. Discrete models are especially useful in facilitating the understanding of the macroscopically observed mass transfer phenomena. Figure 2.1 : Porous medium. The continuum approach to mass transfer phenomena is a common way to avoid the necessity of modelling the microsopic pore space. It

14 6 Chapter 2 represents the classical engineering approach to the modelling of media with a complex microscopic geometry. Continuum modelling constitutes a phenomenological approach as the mass transfer coefficients that appear in the macroscopic balances have to be determined experimentally. In this introductory chapter we review the modelling of mass transfer phenomena in homogeneous porous media. The first section of this chapter is devoted to continuum modelling of mass transfer phenomena, the second to discrete modelling of the pore space. 2.1 Continuum modelling Continuum modelling of mass transfer phenomena in porous media rests upon three basic assumptions concerning the mathematical description of microscopically heterogeneous systems: the physical properties of the porous medium can be associated with mathematical field variables whose spatial domain is a set of centroids of representative elementary volumes; the microscopic (molecular) mass transfer phenomena can be modelled with phenomenological mass transfer equations in terms of these mathematical field variables; the time and space dependence of the field variables can be represented in the form of differential balance equations for mass, momentum and energy. In this section we discuss these three assumptions The representative elementary volume The concept of the averaging elementary volume is introduced to relate the microscopic geometrical and physical properties of the real porous medium to the macroscopic (or averaging) properties of the continuum model. With every point of the spatial domain we associate an elementary volume V that is used to evaluate the macroscopic properties of the porous medium (figure 2.1). [Withaker 1969] uses a sphere as the elementary volume, but states that any shape will suffice provided that the dimensions are constant and the orientation relative to some inertial frame remains unchanged. A microscopic property of the real porous

15 Continuum modelling 7 medium is denoted by P if it is defined in all points of the spatial domain (i.e. in each phase) or by P i if it is defined only over the spatial domain of phase i. In order to define the macroscopic field variables in any mathematical point of the spatial domain, the microscopic phase properties P i are set to zero for all phases other than i. The three-phase real porous medium is thus conceptualised as a fictitious model of overlapping continua s, l and g in which each phase fills up the entire spatial domain [Slattery 1972]. [Withaker 1979] defines three types of macroscopic properties: the spatial average <P>, the phase average <P i > and the intrinsic phase average <P i > i. The spatial average of a microscopic property P in a point m is defined as the average of the microscopic property P over an elementary averaging volume V with centroid m: 1 < P >= V PdV V [2.1] Usually one is not interested in the volume average over all phases, but rather in the volume average over a single phase. The phase average of a microscopic property P i is defined as 1 < P >= = = V P dv 1 V P dv i s l g i i i,,, [2.2] V Vi According to the concept of overlapping continua, <P i > is defined over all points of the spatial domain, including those points where P i was set to zero. This definition implies that, for a constant microscopic property P i, the value of the phase average <P i > is different from the value of the microscopic property P i. This inconvenience is removed by defining the intrinsic phase average as 1 < P > = V P dv i i i i V i [2.3] where < P >= ψ < P > [2.4] i i i i The macroscopic value of a property must be independent of the size, the shape and the orientation of the elementary volume V that is used for its

16 8 Chapter 2 evaluation. The smallest elementary volume that satisfies this condition is, by definition, the representative elementary volume (REV) [Bear 1975] (figure 2.2): < P >= V VREV lim 1 V PdV [2.5] V [Whitaker 1969] indicated that the averaging process will lead to meaningful results if d << l << L [2.6] where d is a microscopic characteristic length over which significant variations in the mathematical point properties occur, l is a characteristic length of the representative elementary volume V REV, and L is a macroscopic characteristic length. Typically l = m in sands [Prévost 1980]. The first part of the inequality [2.6] states that the characteristic length of the REV must be sufficiently larger than the size of a single pore so that the REV includes a sufficient number of pores. The spatial average <P> can then be considered as an adequate estimate of the value of the microscopic property P at the centroid of the REV. The second part of the inequality [2.6] states that the characteristic length of the REV must be sufficiently smaller than the length over which significant macroscopic variations in the point properties occur (figure 2.2). We only consider macroscopically homogeneous porous media so that this inequality is always satisfied. Different properties may involve different characteristic lengths of the representative elementary volume. The continuum modelling of a process is valid only if a common characteristic length of the REV for each property can be found. An exact mathematical treatment is thus limited to the case where all macroscopic properties present a closely similar functional dependence on the length scale of the elementary volume [Bear 1975].

17 Continuum modelling 9 Value of the macroscopic property <P(V)> microscopic approach continuum approach macroscopically inhomogeneous medium macroscopically homogeneous medium V REV <P(V REV )> Size of the averaging volume V Figure 2.2 : Definition of the representative elementary volume. The passage from the microscopic level of description to the macroscopic one can then be viewed as an operation of smoothing out the microscopic heterogeneity of the relevant properties. The averaging procedure is used to obtain a field of macroscopic properties for each phase. For a rigid and homogeneous porous media, the geometrical and topological macroscopic properties of the pore space are single-valued functions of space and time. These media are then characterised by welldefined and unique geometrical pore space properties such as porosity and water retention curve, and by size-independent state variables and mass transfer properties [Sahimi 1993] The phenomenological mass transfer equations Mass transfer phenomena are observed through the macroscopic manifestations that are the result of the involved microscopic (molecular) mass transfer processes. The mass transfer equations relate the macroscopically observed flow to the relevant macroscopic state variable. The simplest mass transfer equations assume a unique relationship between the flow and the gradient of the macroscopic state variable. In this case the proportionality factor is called a mass transfer coefficient. It is assumed that all non-linearities of the mass transfer process can be accomodated by the non-linearity of the mass transfer coefficient.

18 10 Chapter 2 The mass transfer coefficients are obtained from experiments or from a conceptual discrete model calculation. The use of a discrete model may lead to a better understanding of the dependence of the transfer coefficient on the microscopic geometrical parameters of the pore space. But even in this case, there always remain unknown numerical constants that must be determined from experiments. Because of their dependence on observations and experimental evidence, the mass transfer equations are often referred to as phenomenological equations. They model the assumed behaviour of ideal continua. We now have a closer look at the mass transfer equations for three major mass transfer processes: vapour diffusion transfer, bulk fluid transfer and capillary liquid transfer. An elaborate introduction to mass transfer processes is given in [Krischer 1963]. Recent overviews can be found in [Garrecht 1992] and [Künzel 1994] Vapour diffusion transfer All molecular mass transfer processes in which molecules move under a concentration gradient or a partial pressure gradient are called diffusion processes. This definition excludes the presence of a total pressure gradient which induces bulk mass transfer. In this paragraph we focus on the binary diffusion process of water vapour in air within the pore structure of a porous medium. The gas diffusion process in an infinite volume is governed by the interaction between the gas molecules. This interaction is characterised by the mean free path length λ of the gas molecules. This property depends on the gas type, the temperature and the mean pressure [Carman 1956]: λ ηg πrgt = g = v, a [2.7] p 2 g For vapour and air at atmospheric pressure and room temperature, the mean free path length λ is of the order magnitude of m. For the gas diffusion process within the pore structure of a porous material, the interaction between the gas molecules and the pore walls should also be considered. For pores with a radius r >> λ, however, the latter interaction is negligible and the mass transfer process reduces to molecular diffusion of vapour in air, which is called pure diffusion.

19 Continuum modelling 11 [Krischer 1963] derived an expression for the density of the vapour flow rate in pure diffusion: &m v Dva = R T p v [2.8] v where D va is the binary diffusion coefficient of a vapour-air mixture, for which [Schirmer 1938] suggested the empirical relation D va 5 patm T 1 81 = ( ). [2.9] p + p 273 a v For pore radii r << λ, the interaction of the gas molecules with the pore walls is far more important than the interaction among the gas molecules. The mass transfer process is then called Knudsen diffusion or effusion. For a cylindrical pore with radius r the density of vapour flow rate in effusion is given by [Carman 1956, Krisher 1963]: m & v 8 = 2r 9πR T p v v [2.10] In a transition range where r = O( λ ), an intermediate diffusion process takes place, which [Gertis 1976] conveniently called mixed transfer. In capillary porous media, where pore radii range from to m, the macroscopically observed vapour flow is assumed to be the result of these three microscopic diffusional mass transfer processes. The expressions [2.8] for pure diffusion and [2.10] for effusion motivate the equation for diffusion transfer: < & >= ( ) δ pa( T) mv δ p T < pv >= < pv > µ [2.11] where δ p is the vapour permeability of the medium, δ pa is the vapour permeability of air and µ is the vapour resistance factor [Klopfer 1974]. The transfer coefficient vapour permeability δ p and the vapour resistance factor µ account (i) for the reduced cross section that is available for vapour flow, (ii) for the tortuosity of the flow path [Carman 1956] and (iii) for any other microscopic vapour transfer process that contributes to the macroscopic vapour flow. The vapour permeability δ p and the vapour resistance factor µ are measurable properties. Laboratory measurements were recently compiled

20 12 Chapter 2 within the frame of [IEA Annex 24]. [Quenard 1989] used a conceptual discrete network model to calculate the vapour permeability δ p Bulk fluid transfer In bulk fluid transfer processes, the fluid is transported under a total pressure gradient. The type of the fluid flow depends on the pressure range and the average pressure. In this paragraph we limit to the laminar flow of a Newtonian fluid in a rigid porous medium under such conditions that inertial effects can be neglected. We assume that the pressure changes are small so that the fluid may be assumed to be incompressible. If the velocity at the fluid-solid interface is zero (i.e. the no-slip condition) then the density of steady fluid flow rate in a single cylindrical pore with radius r is obtained from the Hagen-Poiseuille equation: 2 & r mf = ρ f p f, f = l, a 8η f [2.12] In the case of air flow, the no-slip condition is no longer satisfied if the mean free path length λ of the gas molecules is of the same order of magnitude as the pore radius r. An expression for the density of slip air flow rate in a single cylindrical pore with radius r is given by [Carman 1956] (figure 2.3): m & a 2 r π = ( ρa + r ) p 8η 8R T a a a [2.13]

21 Continuum modelling 13 Density of air flow rate m a under unit pressure gradient (kg/m 2 s) Poiseuille flow 10-9 slip flow λ= O( m) p a = p atm Pore radius r (m) Figure 2.3 : Bulk fluid transfer: slip flow and Poiseuille viscous flow. The Darcy mass transfer equation relates the density of the macroscopic bulk fluid flow rate to the averaged fluid pressure gradient: < & ρ f mf >= K f < p f >= B < p f >, f = l, a η f [2.14] where K is the permeability and B is the intrinsic permeability of the porous medium. Measurements of saturated water permeabilities of building materials other than concrete are very scarce. [Garrecht 1992] performed measurements on some natural sandstones. Air permeabilities were compiled by [Kronvall 1980] and [Bumbaru 1988]. Air permeabilities of some insulation materials were measured by [Lecompte 1989]. [Bamforth 1987] showed that slip flow results in gas permeabilities that are significantly higher than liquid permeabilities for fine-porous materials such as concrete Capillary liquid transfer Capillary phenomena arise as a result of forces that are related to the interfaces between immiscible phases. We first formulate capillary phenomena in terms of the mechanical properties of the system. The subsequent thermodynamic approach allows to formulate the Kelvin

22 14 Chapter 2 interface equilibrium equation. An excellent introduction to capillary phenomena is given in [Dullien 1979]. A mechanical approach to capillarity We first consider a two-phase immiscible fluid-fluid equilibrium system: a spherical drop of water in air. Young and Laplace [cited in Boucher 1980] first stated that, from a mechanical point of view, this system behaves as if it consisted of two homogeneous phases seperated by a tension surface of infinitesimal thickness. Let us consider the equilibrium of the cap of a spherical drop of water (figure 2.4). Figure 2.4 : Mechanical equilibrium of the water cap of a spherical drop of water. The water pressure p l acts on the plane surface A l while the gas pressure p g acts on the curved cap surface A g. In the absence of gravitational effects, the pressure p g and p l are uniform throughout their respective phases. The surface tension force F c acts along the cap edge. Equilibrium along the y-axis yields Laplace s equation: 2 2 2πrσ cos( γ ) + p πr = p πr [2.15] or p l lg g l pg = 2 σ = 2 lg σ lg cos ( γ ) [2.16] r R where r is the radius of the plane surface of the cap and R is the drop radius (and thus the radius of curvature of the liquid-gas interface). The surface tension σ in the curved surface A g creates a pressure difference p l - p g between the water at the concave side of the surface and the gas. This pressure jump at the interface is called the capillary pressure p c : p = p ( concave side) p ( convex side) = p p [2.17] c f f l g In the case of a drop of water, the water pressure p l exceeds the gas pressure p g.

23 Continuum modelling 15 We now turn to a three-phase solid-liquid-gas system that is relevant for the description of capillary phenomena in porous media. In analogy to the surface tension at a fluid-fluid interface, there also exists a surface tension at a solid-fluid interface. At a line of contact between the solid and the two fluid phases, equilibrium is established between the surface tension forces at the solid-liquid interface σ sl, the solid-gas interface σ sg and the liquid-gas interface σ lg. Since both the solid-liquid and the solid-gas interfaces are rigid, the only way to establish equilibrium along the x-axis is by adjusting the curvature (and by this the direction of the interface tension force) of the mobile liquid-gas interface (figure 2.5). This curvature is defined by the contact angle θ which is the angle between the solid-liquid interface and the liquid-gas interface, taken through the denser phase [Boucher 1980]. Equilibrium along the x-axis then yields: σ sg 2πr = σ sl 2πr + σ lg cos ( θ) 2πr [2.18] or cos( θ) = σ sg σ σ lg sl 1 [2.19] This equation is known as Young s formula. If σ sl < σ sg and σ sg - σ sl < σ lg, the contact angle θ is smaller than 90 and the liquid is said to preferentially wet the solid. gas phase σ lg θ liquid phase 2r σ lg σ sg σ sl solid phase x σ sl σ sg Figure 2.5 : Equilibrium at the three-phase line contact. It is interesting to illustrate that the preferential wettability of water in a solid-water-gas system induces water transfer in a capillary without any external pressure gradient.

24 16 Chapter 2 p g = p l = p atm A σ sl σ sg B σ lg p g = p atm L 2r Figure 2.6a : Water-gas interface in free imbibition: initial position p g = p l = p atm p g = p atm A B p l = p g - p c θ 2r L Figure 2.6b : Water-gas interface in free imbibition: intermediate position. Consider a preferentially water-wet horizontal capillary with radius r which is initially filled with gas at atmospheric pressure p g = p atm. At one side of the capillary, water is applied at atmospheric pressure p l = p atm (figure 2.6a). The initial water-gas interface is the plane surface AB. This surface is in equilibrium except at the contact line with the solid phase where a positive net force F c along the x-axis causes the interface to enter the capillary: F = 2πr ( σ σ ) > 0 [2.20] c sg sl The liquid-gas interface quickly adopts a curved shape as enforced by the contact angle θ (figure 2.6b). The curvature of the interface creates a capillary pressure p c over the interface: 2πrσ lgcos( θ) 2σ lg 2σ lg pc = pg pl = = cos ( θ) = [2.21] 2 πr r R

25 Continuum modelling 17 where r is the radius of the capillary and R is the radius of curvature of the water-gas interface. Expression [2.21] is called the Washburn equation. For building materials it is usually assumed that the contact angle θ is equal to zero. In this geometric configuration, water is at the convex side of the interface so that the water pressure p l is lower than the gas pressure p g. The capillary pressure p c creates a pressure gradient over the water phase which causes the water to flow and which sustains the motion initially created by unbalanced surface tension forces (figure 2.6b). The definition of the capillary pressure in equation [2.16] does obviously not require the presence of a capillary. However, the presence of the solid wall of a capillary determines the curvature of the liquid-liquid interface and thus the magnitude of the capillary pressure in equation [2.21]. A thermodynamic approach to capillarity Consider again a two-phase closed system of a spherical drop of water with radius R in equilibrium with a gas phase consisting of water vapour and air. We denote the interface area between the liquid and the gas phase by A lg. For a reversible isothermal compression process, the fundamental Gibbs equation for the Helmholtz free energy function F equals the work done on the system [Rowlinson 1982]: df = ( p dv p dv + σ da ) [2.22] l l g g lg lg T This equation defines the surface tension σ lg as the rate of change of the surface free energy with the interfacial area [Morrow 1970]: σ lg = ( df ), da [2.23] V T lg The condition of mechanical equilibrium of the system is then written as the derivative of the free energy F to the drop radius R [Landau and Lifshitz 1980]: df lg ( ) p dv lg dr dr p dv da l g T = l g + σ = 0 where [2.24] dr dr dvl dr 2 = 4π R, da dr lg = 2π R and dv g dr dvl =. dr

26 18 Chapter 2 The condition of mechanical equilibrium [2.24] results in the expression for the capillary pressure p c that is identical to the above derived equation [2.16]. In the case of a three-phase solid-water-gas system we are especially interested in the equilibrium partial vapour pressure p v = p g - p a at the curved interface in a capillary. The equilibrium condition at the interface is written in terms of the chemical potential µ [Defay 1966, Hall 1977, Landau and Lifshitz 1980]: µ ( p, T) = µ ( p, T) [2.25] l l v v The saturated vapour pressure p vs at a plane water surface at atmospheric pressure p l = p g = p atm is given by the analogous equation [Kreider 1994]: µ ( p, T) = µ ( p, T) [2.26] l atm v vs where p vs = p Tc K T c10 1 ( ) [2.27] p c = MPa T c = K K T T T = ( ) ( ) ( ) Subtracting [2.26] from [2.25] and using the expressions for the chemical potential of an incompressible liquid and an ideal gas yields: 1 ( p p ) = R T(ln( p ) ln( p )) or [2.28] ρ w l g v v vs pv pc 2σ cos( θ) ln( ) = ln( ϕ) = = p ρ R T rρ R T vs l v l v [2.29] This expression for the equilibrium vapour pressure p v at a curved surface is called the Kelvin equation [Skinner 1972]. The mass transfer equation for capillary liquid transfer The Darcy equation for unsaturated liquid transfer relates the density of the macroscopic capillary liquid flow rate to the averaged capillary

27 Continuum modelling 19 pressure gradient or moisture content gradient [Childs 1950, Philip 1969, Withaker 1979]: < &m l >= Kl < pc >= Dl w [2.30] l where < pc >=< pg > g < pl > and w l l =< ρ l >. K l is called the (unsaturated) permeability and D l the liquid diffusivity of the porous medium. Equation [2.30] is only valid under the assumption of a continuous liquid phase [Withaker 1979]. Both the permeability and the diffusivity are strongly non-linear functions of their respective transfer potentials. Moisture content is not a potential stricto sensu since it is not unique at material interfaces. It is often used though as a pseudo-potential since it is a measurable property. The measurement of the unsaturated water permeability K l requires the measurement of the capillary pressure gradient. For bulk materials such as soil, tensiometers can be used for this purpose [Vachaud 1972]. For rigid building materials, one reverts to the measurement of transient moisture content profiles to calculate the water diffusivity D l (see chapter 3). We mention some recent experimental studies on mortar [Daï an 1986, Garrecht 1992], fired clay brick [Sommer 1971, Nielsen 1976, Kießl 1983, Garrecht 1992, Pel 1995, Krus 1995], natural sandstone [Garrecht 1992, Krus 1995], wood [Descamps 1991b, 1992], cellular concrete [Sommer 1971, Van der Kooi 1972, Nielsen 1976, Kießl 1983, Krus 1995], calcium silicate brick [Sommer 1971, Kießl 1983, Krus 1995], concrete [Kießl 1983] and mineral wool insulation [Hokoi 1991] The balance equations for moisture transfer In this paragraph we set up the averaged balance equations that describe the water and vapour transfer in a porous medium. [Philip and De Vries 1957] and [De Vries 1958] first formulated the equations that describe vapour transfer by molecular diffusion and water transfer by capillary action in rigid homogeneous porous media. A comprehensive treatment of their theory is given in [van der Kooi 1971]. An overview of its applications is given in [De Vries 1987]. A more rigorous approach was developed by [Withaker 1979] and [Kohonen 1984]. Here we limit to the case of isothermal moisture transfer. We start from the microscopic mass balance equation for the water phase l:

28 20 Chapter 2 ρl ρl t + ( v ) = l 0 [2.31] For a representative elementary volume V with surface area A, the macroscopic mass balance equation for the water phase l can be written in terms of the microscopic quantities (figure 2.1): 1 ρl 1 0 V t dv + ( ρlv l ) dv = [2.32] V Vl Vl To rewrite this macroscopic mass balance equation in terms of averaged (or macroscopic) quantities we use the general transport theorem and the spatial averaging theorem of [Slattery 1972]. The general transport theorem relates the time derivative of an averaged quantity to the average of the time derivative of the quantity: 1 Pi 1 V t dv Pi Pi = < > = < > Pi ( v ij nij ) da [2.33] t t V j Vi This expression allows to rewrite the first term in equation [2.32] as 1 ρl 1 1 V t dv < ρl > = ρl ( v n ) da ρl da t V V ( v n ) Vl Alg Aij lg lg ls ls [2.34] Als where the last term vanishes since the liquid-solid interface A ls is immobile. The spatial averaging theorem relates the divergence of an averaged quantity to the average of the divergence of the quantity: 1 < P >= < P > + P n da V j i i i ij Aij [2.35] The second term in the macroscopic mass balance equation [2.32] is then rewritten as ( ρlv ) dv = < ρlv > + ρl( v n ) da + l da V V V ρ ( v n ) Vl l l l lg l ls [2.36] Alg Als The last term vanishes since the liquid velocity v l is zero at the liquidsolid interface A ls (i.e. no-slip condition). If we assume the liquid water phase to be incompressible and we substitute ρ l v l by &m l and <ρ l > by w l, we obtain

29 Continuum modelling 21 w t l + < m& 1 l > + ρl(v l v lg ) n lg da = 0 [2.37] V Alg The last term may be considered the density of phase change rate from the liquid phase to the vapour phase [Kohonen 1984]: 1 ρ l(v l v lg ) n lgda = E V Alg lv [2.38] By introducing the mass transfer equation for capillary liquid flow [2.30] we obtain the macroscopic mass balance for the liquid water phase [De Vries 1958]: w t l = ( K < p > ) E [2.39] l c lv If the gas pressure is uniform and constant, no bulk mass transfer takes place and a similar expression for the macroscopic mass balance for the vapour constituent of the gas phase g can be written [De Vries 1962]: w t v = ( δ < p > ) + E [2.40] p v lv Summing [2.39] and [2.40] yields the macroscopic moisture balance (omitting the <...> notation in macroscopic balances for brevity): w t wv wl = + = ( δ p pv ) + ( Kl pc ) [2.41] t t We now rewrite this equation in terms of a single driving potential, which requires two additional definitions. The relation between the liquid water content w l and the capillary pressure p c is called the water retention curve w l (p c ), while the relation between the liquid water content w l and the relative humidity ϕ = p v /p vs at constant temperature is called the sorption isotherm w l (ϕ). The relation between the vapour content w v and the liquid water content w l is given by w v wl = ( ψ o ) v ρ ρ [2.42] l

30 22 Chapter 2 These expressions and the Kelvin equation [2.29] allow to rewrite the macroscopic moisture balance [2.41] in terms of the liquid moisture content w l as ρ 1 + [ ] = v wl ψ ρ ρ ρ ϕ wl ϕ ( δ + o ) vs( ) T ppvs( ) T wl Dl wl [2.43] l l wl t wl or in terms of the capillary pressure p c as w ρ ρ [ ] ρ ρ ψ δ ρ l v v wl 1 pc v ( 1 ) + ( o ) p pc Kl pc pc l l ρl RvT t = ρ + [2.44] l The Kelvin equation, the water retention curve w l (p c ) and the sorption isotherm w l (ϕ) constitute equilibrium relations that are not strictly valid under dynamic conditions. Therefore the equations [ ] constitute a quasi-equilibrium theory. A consistent distinction has been made between changes of moisture content in the liquid and the vapour phase. In most simplified models it is assumed that changes of moisture content in the vapour phase are negligible so that w w l [2.45] This simplifies the macroscopic moisture balance equation [2.44] to w p c pc t δ ρ v = ( p Kl ) pc ρ [2.46] l 2.2 Discrete modelling Discrete modelling starts from a conceptual model of the pore space. The basic strategy is to disassemble the pore space into discrete elements and to reassemble these elements into a model that preserves the basic geometrical and topological features of the pore space. This strategy is guided by two important considerations: (i) the flow through the basic discrete element can be derived from fundamental physical laws; (ii) the resulting model should be tractable from a calculational point of view.

31 Discrete modelling 23 The model is not expected to give a complete picture of the complex microstructure of the porous medium. It has to incorporate though, its basic features so that it is useful in facilitating the understanding of observed macroscopic phenomena. Discrete models can play an important role in interpreting the measured transfer coefficients of a porous medium. The discrete modelling approach can serve as an alternative to experiments for obtaining the transfer coefficients of the porous medium. We begin our discussion of these models by setting up the equations for isothermal and laminar flow in a cylindrical tube. This basic element is common to most discrete models. In the next section we review three historically important one-dimensional discrete models. They serve as a good introduction to the discussion of the two-dimensional percolation and immiscible displacement models in the last section of this chapter. An extensive review of discrete modelling is given in [Van Brakel 1975] Isothermal laminar flow in a tube In this paragraph we review the modelling of incompressible liquid flow and compressible gas flow in a tube. The basic equations for these onephase flows are then combined to describe the simultaneous flow of water and air in a tube. Incompressible and compressible steady state flow in a tube The modelling of the flow of a fluid in a long, straight, cylindrical tube with constant cross section results from the application of the momentum equation, the Darcy-Weisbach friction equation and the equation of continuity. The momentum equation is based on Newton s second law of motion. It states that the change in momentum as a fluid element of infinitesimal length flows through a tube is equal to all external forces acting on its surface (figure 2.7).

32 24 Chapter 2 τ o p p + dp 2r τ o dx Figure 2.7 : External forces on a fluid element. The external forces include the external pressure and the fluid friction along the tube walls: dp τ v = = v v o ( ρ ) ρ + v dx r t x 2 2 ρ x [2.47] For laminar incompressible flow the wall frictional shear stress τo is calculated from the Darcy-Weisbach friction factor f (head loss by established cylindrical tube friction): τ o = fρv 8 2 where f 64 32η 32η = = = Re ρvr mr & (Re < 2000 [ASHRAE 1993]) [2.48] In isothermal steady state flow the Reynolds number Re and the friction factor f are constant: both the viscosity η, which only depends on temperature, and the density of mass flow rate &m are constant. To secure isothermal flow in a tube the heat transferred out of the fluid through the tube walls and the energy converted into heat by the friction process must be adjusted so that the fluid temperature remains constant. This flow condition is fulfilled in tubes where velocities are low and where temperature inside and outside the tube are of the same order. [Shapiro 1953] showed that the friction factor values f for incompressible flow are also applicable to compressible flow as long as cp v < c RT = m/s at room temperature [2.49] v In the case of compressible air flow this condition is more restrictive than the condition for laminar flow (Re < 2000) for r < m. It is always fulfilled in building physics applications.

33 Discrete modelling 25 Combining [2.47] and [2.48] yields the differential equation for isothermal steady state laminar flow in a cylindrical tube: 2 p fρv ρ ρ v v v = 0 where f is a constant [2.50] x 4r x x Dividing this equation by ρv 2 and substituting &m for ρv yields an equivalent flow differential equation: ρ p f 1 v 1 ρ = 0 where &m is constant [2.51] 2 &m x 4r v x ρ x This equation is now applied to both incompressible liquid and compressible gas flow. Laminar incompressible liquid flow in a tube The further analysis is based on the finite control volume concept [Vennard 1982]. The control volume has a finite length L (figure 2.8). The control surfaces A are perpendicular to the tube walls. The fluid properties ρ and v are averaged over the control surfaces A. ρ 1 v 1 A 1 2 ρ 2 v 2 A L Figure 2.8 : Finite control volume concept. In the case of steady state flow of an incompressible liquid, the last term at the left hand side of the flow equation [2.51] vanishes and the equation of continuity for the finite control volume 1-2 (figure 2.8) is &m v v v l = ρl 1 = ρl 2 = ρl or v = constan t [2.52] Integrating the differential flow equation [2.51] over the finite control volume 1-2 yields

34 26 Chapter 2 2 ρ l l m dp f & 2 l r dx v dv = 0 [2.53] 1 1 The last term on the left hand side vanishes according to the continuity equation [2.52]. The density of mass flow rate &m in laminar incompressible liquid flow is then given by 2 & ρ r p p l l2 l1 ml = ( ) 8η L l [2.54] The expression for the mass flow rate & M l is named after Hagen and Poiseuille, who first established it experimentally: & M 4 r pl p = ρ Av = ρ π 8 η L l l l l 2 l1 [2.55] An alternative approach to incompressible liquid flow in a tube is based on the Bernoulli energy equation between two points 1 and 2 on a horizontal streamline: ρ 2 lv v + + ρ t dx F l + = 1 pl 1 where 2 8 lvl F = η for a circular cross section [2.56] 2 1 r For steady state flow in a tube with constant cross section the second and the third term vanish and [2.54] is obtained again. Laminar compressible gas flow in a tube We limit the analysis to conditions under which the gas acts as a perfect gas. The equation of state for a perfect gas is p = ρ R T [2.57] g g g In the case of steady state flow of a compressible gas the equation of continuity for the finite control volume 1-2 is &m g = ρg1v1 = ρg2v where & 2 mg is constant or p p g1 g2 v2 = [2.58] v Thus, the differential flow equation [2.51] can be rewritten as 1

35 Discrete modelling &m R T p g g g p g x f 1 v 1 pg = 0 where &m is constant [2.59] 4r v x p x g Integrating [2.59] over the finite control volume 1-2 yields m& R T p dp f( ρg ) dx 4r v dv p dp g g g = g g g [2.60] or 1 & 2 m R T g g 2 2 pg2 pg1 f( ρ) L v p 2 g2 + + ln( ) = 0 2 4r v p 1 g1 The last term on the left hand side vanishes according to the continuity equation [2.58]. The density of mass flow rate in laminar compressible gas flow is then given by m& r pg2 pg1 ρgjr pg 2 pg1 g = ( ) = ( ) 16η R T L 16η p L g g g gj where j = 1 or 2 [2.61] For small pressure differences this expression reduces to equation [2.54] for incompressible fluid flow. Simultaneous water and air flow in a tube Now we combine the expressions for steady state incompressible water and compressible air flow to model the imbibition process in a tube. The transient water imbibition and air outflow process in a tube is modelled as a quasi-steady state phenomenon. Inertia effects are limited to the very beginning of the imbibition process [Sommer 1971] and are neglected here. Consider a cylindrical tube of length L and radius r that is initially filled with air at atmospheric pressure p atm. When water is supplied to one side of the tube at a pressure p l,bc, a capillary water-air interface is built. Let s denote the position of the meniscus by x F with x F,t=0 = 0. We assume that the only effect of this capillary meniscus is to provide a pressure jump localised over an interfacial region of infinitesimal thickness. Away from the meniscus the flow is assumed to be isothermal and laminar. When the water interface pressure is denoted by p li, and the air interface pressure by p ai, then the capillary pressure p c is obtained from equation [2.21]: