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1. Education and Culture Lifelong Learning Programme ERASMUS Assist.Prof.Dr. Lucija Hanžič LECTURE 1 Transport phenomena in porous materials www.fg.uni-mb.si/lucija/presentations/2010covilha-lecture1.ppt CONTENTS 1. Introduction 2. Structure of solids and transport routes 3. Diffusion 4. Capillary absorption 10 – 14 May 2010 Host institution University of Beira Interior Department of Civil Engineering and Architecture Covilha, Portugal Home institution University of Maribor Faculty of Civil Engineering Maribor, Slovenia

2. Assis.Prof.Dr. Lucija Hanžič Transport phenomena in porous materials Transport phenomena in porous materials MASS ENERGY Diffusion Capillary absorption Suction Conduction Convection NANO SCALE MICRO SCALE NANO SCALE MICRO SCALE Movement of atoms or molecules Movement of fluid, where fluid is considered as CONTINUUM Atomic vibrations and interactions Convective current of fluid in voids 2/8 University of Maribor, Faculty of Civil Engineering Introduction Diffusion Suction Conduction

3. Assis.Prof.Dr. Lucija Hanžič Transport phenomena in porous materials Permeable Open Ink-bottle Pores Closed 3/8 University of Maribor, Faculty of Civil Engineering Structure of solids and transport routes GRAIN BOUNDARY DIFFUSION SURFACE DIFFUSION CAPILLARY ABSORPTION SUCTION VOLUME DIFFUSION Active figure 1. Routes of mass transport in solid materials.

4. Assis.Prof.Dr. Lucija Hanžič Transport phenomena in porous materials Figure 1. Steady-state diffusion D.R. Askeland, The sceience and Engineering of Materials, SI ed., Stanley Thornes (Publishers) Ltd, Cheltenham, 1998, p.117. 4/8 University of Maribor, Faculty of Civil Engineering Diffusion Diffusion is the movement of discrete particles (atoms, molecules) within a material. Particles move due to concentration differences in order to remove them. Thus, particles move from the region of higher concentration towards the region of lower concentration. The rate of diffusion or diffusion flux is expressed as a number of atoms - N (1) that pass through a unit cross-sectional area - A (m2) of solid per unit of time - t (s): (1). Steady-state diffusion: Fick’s first law When the concentration difference - Δc (1/m3) over distance - Δx (m) does not change with time, a steady-state condition exist and diffusion flux is described by Fick’s first law: (2) where D is diffusion coefficient (m2/s) calculated according to Arrhenius equation (3) where D0is diffusion constant (m2/s), Q is activation energy (J/mol), R is gas constant (8.31 J/mol K) and T is temperature (K).

5. Assis.Prof.Dr. Lucija Hanžič Transport phenomena in porous materials 5/8 University of Maribor, Faculty of Civil Engineering Dynamic diffusion: Fick’s second law Under nonsteady state or dynamic conditions partial differential equation, known as Fick’s second law, is used: Its solution depends on boundary conditions. One important solution is for semi-infinite solid. (4) (5) where cs is the surface concentration (at. %), c0 initial concentration in material, cx concentration at depth x (m) after elapsed time t (s) and erf is “Gaussian error function”. Figure 2. Semi-infinite solid boundary conditions. (a) (b) Figure 3. Concentration cx (a) as function of depth x at time t and (b) as function of time t at depth x.

6. Assis.Prof.Dr. Lucija Hanžič Transport phenomena in porous materials Fg– Gravity force Fk– Capillary force Fp – Capillary pressure force Fa – Atmospheric pressure force Fv – Viscosity force (6) 6/8 University of Maribor, Faculty of Civil Engineering Capillary absorption Particles of the same kind exert cohesive forces (K) on each other, while particles of different kind act on each other by adhesive forces (A). Figure 4. Shape of a droplet on the solid surface and shape of the meniscus in a capillary tube when (a) cohesive forces (K) are smaller then adhesive forces (A) and contact angle (Θ)is smaller then 90° and(b) when cohesive forces are larger than adhesive forces hence, contact angle is larger then 90°. (a) (b) Interactions between particles on the solid-liquid-gas interface result in surface tensionγ(N/m) and contact angle - Θ(°). In the case of narrow tubes forces arising from surface tension are of the same magnitude as gravitational forces acting on the liquid column. Figure 5. Forces acting on the liquid column in a capillary tube.

7. Assis.Prof.Dr. Lucija Hanžič Transport phenomena in porous materials (7) (9) (8) 7/8 University of Maribor, Faculty of Civil Engineering By employing some simplifications and generalization derivation of Eq. (6) yields Lucas-Washburn equation, which describes liquid movement in porous material due to capillary absorption. Hight of liquid front is determined as where kis capillary coefficient (m s-1/2) defined as: where γis surface tension (N/m), ris capillary radius (m), Θ is contact angle (°), and η is dynamic viscosity of the liquid (Pa s). Figure 6. Computer model of porous material. M. Borovinšek, Computational modelling of irregular cellular structures, PhD, University of maribor, 2009. By means of gravimetric method, sorptivity(S) is often determined instead of capillarity: where iis volume of liquid absorbed per unit cross-sectional area (m3/m2). Figure 8. Graphical presentation of Lucas-Washburn equation. Figure 7. Measurement of liquid front on neutron radiograph.

8. Assis.Prof.Dr. Lucija Hanžič Transport phenomena in porous materials 8/8 University of Maribor, Faculty of Civil Engineering Figure 9. Transport of water from fresh mortar into a brick.