UNIVERSITY OF WARMIA AND MAZURY IN OLSZTYN FACULTY OF TECHNICAL SCIENCES
POLISH SOCIETY OF THEORETICAL AND APPLIED MECHANICS
2nd Workshop on Porous Media
BOOK OF ABSTRACTS
OLSZTYN - POLAND 28-30 JUNE 2018
2 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8
SCIENTIFIC COMMITEE Wojciech Sobieski (chair of committee) Janusz Badur Mariusz Kaczmarek Adam Lipiński Teodor Skiepko Mieczysław Cieszko Piotr Srokosz Adam Szymkiewicz Anna Trykozko Maciej Marek Maciej Matyka Joanna Wiącek
ORGANIZING COMMITEE Wojciech Sobieski (chair of committee) Seweryn Lipiński Dariusz Grygo Tomasz Tabaka Aneta Molenda
Book of abstracts published from materials provided by Authors Edited by Seweryn Lipiński 3 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8
CONTENTS
Dariusz Asendrych, Paweł Niegodajew 5 Interfacial heat transfer modelling in packed beds Janusz Badur, Marcin Lemański, Tomasz Kowalczyk, Paweł Ziółkowski, Bartosz Kraszewski 6 Porous media as an important element of fuel cell in high efficiency energy production Włodzimierz Bielski, Ryszard Wojnar 7 Plane flow through the porous medium with chessboard-like distribution of permeability Renata Cicha-Szot, Piotr Such, Grzegorz Leśniak 8 A comparative study of nanoscale pore structures in geomaterials Mieczysław Cieszko 9 Minkowski Metric, Dirichlet Energy and Pore Tortuosity Tomasz Czerwiński, Mieczysław Cieszko, Chaplya Yevhen 10 Influence of pore structure parameters on saturation of a ball with liquid during intrusion-extrusion process Waldemar Dudda, Wojciech Sobieski 11 Experimental investigations of pressure drops in chosen granular beds Marek Gawor 12 Experimental determination of the kinetics of sorption and the kinetics of gas filtration in coal Karolina Grabowska, Jarosław Krzywański, Karol Sztekler, Wojciech Kalawa, Wojciech Nowak 13 Fuzzy logic approach in the analysis of heat transfer in a porous sorbent bed of the adsorption chiller Paweł Guzik, Krzysztof Mudryk 14 A concept of a two-stage process of non-pressure granulation of mineral materials Daniel Janecki 15 Mathematical model of the course of process of catalytic wet air oxidation of phenol (CWAO) in trickle bed reactors (TBR) Łukasz Jasiński, Marcin Dąbrowski 16 Depth-averaged model for flow in a propped fracture Dariusz Kardaś, Sylwia Polesek-Karczewska, Paweł Ziółkowski, Janusz Badur 17 A unified thermodynamic approach to description of whole phenomena during a coke production Marcin Kempiński, Mieczysław Cieszko, Marcin Burzyński, Zbigniew Szczepański 18 Determination of pore size distribution in sintered glass bead samples based on mercury porosimetry and microtomographic image analysis Wojciech Kiński, Wojciech Sobieski 19 Geometry extraction from GCODE files destined for 3D printing Wojciech Kiński, Wojciech Sobieski 20 The application of 3D printing technology in the investigations of porous media Jarosław Krzywański 21 Selected artificial intelligence methods in modeling of energy devices and systems Anna Kułakowska 22 Evaluation of plasticity criterion applicability in the porous materials research Sylwia Kwiatkowska-Marks, Justyna Miłek, Ilona Trawczyńska 23 Diffusion of Cd(II), Pb(II) and Zn(II) on calcium alginate beads Grzegorz Leśniak 24 Effect of microfracture on ultratight matrix permeability Grzegorz Leśniak, Renata Cicha-Szot, Krzysztof Labus 25 Multi-scale core to pore imaging and modelling of the heterogonous rocks Grzegorz Leśniak, Karol Spunda, Renata Cicha-Szot 26 Pore scale modelling of fluid transport using FIB-SEM images Seweryn Lipiński, Wojciech Sobieski 27 Reliability of the tortuosity value obtained on the basis of other parameters of the porous bed Seweryn Lipiński, Zenon Syroka 28 Relations between the various probability density functions describing the distribution of particles in a granular bed Wojciech Ludwig 29 Effect of friction coefficient on results of particles velocity calculation using Euler-Lagrange model in spout-fluid bed apparatus 4 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Janusz Łukowski, Mieczysław Cieszko 30 Analysis of Influence of Model Parameters on the Capillary Imbibition of Porous Materials with Liquid Marcin Majkrzak 31 Possibilities of using glass microspheres to build models and simulation of fluid flow through geological strata Ewelina Małek, Danuta Miedzińska, Wiesław Szymczyk, Arkadiusz Popławski 32 Application of 3D printing technology to random porous structures study Marcin Małek, Wojciech Życiński, Mateusz Jackowski, Marcin Wachowski, Waldemar Łasica 33 Effect of polypropylene fiber addition on mechanical properties of concrete based on portland cement Maciej Marek 34 Numerical generation of a random packed bed of saddles Kinga Michalak, Janusz A. Szpaczyński 35 Sludge dewatering by thin-film freezing in the north-east of Poland Danuta Miedzińska 37 Numerical modeling of porous ceramics microstructure Justyna Miłek, Sylwia Kwiatkowska-Marks, Ilona Trawczyńska 38 Application of silica gel in the process of trypsin immobilization Agnieszka Niedźwiedzka 39 Numerical modeling of cavitation phenomenon in a small-sized converging-diverging nozzle Anna Pajdak 40 The influence of comminution degree on structural properties of rocks of various porous structure Norbert Skoczylas, Mateusz Kudasik 41 A multiparameter description of the rock-gas system by the use of original methods Karolina Słomka-Polonis, Jakub Fitas, Jakub Styks, Bogusława Kordon-Łapczyńska 42 Measurement of porosity of comminuted Salix Viminalis L. in aspects of uncertainty of measurement Wojciech Sobieski 43 Numerical investigations of tortuosity in randomly generated pore structures Wojciech Sobieski 44 Path Searching Algorithm Aleksander Sulkowski 45 Application of the contour erosion function in shape analysis of a solid particle Zbigniew Szczepański, Mieczysław Cieszko 46 Influence of Microscopic Pore Geometry on the Parameter of Pore Tortuosity Janusz A. Szpaczyński 47 The use of natural soil as a porous medium for the treatment of secondary effluent in the northern climate Adriana Szydłowska, Jerzy Hapanowicz 48 The stand for testing flow of two-phase system through the metal foams Piotr Szymczak, Filip Dutka, Florian Osselin 49 Dissolution of porous media studied in a simple microfluidic setup Adam Szymkiewicz, Witold Tisler, Wioletta Gorczewska-Langner, 50 Rafał Ossowski, Danuta Leśniewska, Stanisław Maciejewski Numerical simulations of laboratory-scale dike overflowing using two phase flow model Ilona Trawczyńska, Justyna Miłek, Sylwia Kwiatkowska-Marks 51 Effect of temperature, concentration of alcohols and time on baker's yeast permeabilization process Anna Trykozko 52 Pore scale simulations of flows in weakly permeable porous media Grzegorz Wałowski, Gabriel Filipczak 53 Using of the SEM image method to evaluate the porosity of materials with varied internal structure Joanna Wiącek, Marek Molenda, Mateusz Stasiak 54 Numerical analysis of compression mechanics of granular packings with various number of particle size fractions Eliza Wolak, Elżbieta Vogt, Jakub Szczurowski 55 Determination of the heat of wetting of selected liquids on modified activated carbons
5 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Interfacial heat transfer modelling in packed beds Dariusz Asendrych, Paweł Niegodajew Institute of Thermal Machinery, Częstochowa University of Technology [email protected]
Keywords: packed bed, multiphase flow, trickling flow regime, interfacial heat transfer
Packed bed columns are commonly used in chemical and process industries, just to mention absorption, distillation or rectification processes as examples. The packing material ensures large contact area between phases as well as sufficient liquid holdup required to reach expected process efficiency. Modelling of 2-phase flows in packed beds is a challenging task [1] even if isothermal conditions are assumed. However, most of processes include heat transfer phenomena, thus the changes in material properties have to be taken into account. During recent years a lot of effort has been done in the field of packed beds operated in trickling flow regime, mostly dedicated to flow hydrodynamics. Prediction of pressure drop, liquid holdup, liquid spreading or wetting efficiency has become much more accurate and reliable. Much less works, however, is related to heat transfer phenomena and in principle they are devoted to overall heat transfer in packed beds. None of the papers deals with interfacial (gas-liquid) heat transfer except for the experimental work of Heidari and Hashemabadi [2]. The weak point of this work is, however, a simplified model of a packed bed as composed of large spheres filling the column of the same diameter. Practical lack of reliable data does not allow to model nonisothermal multiphase flows in packed beds. That is why it was decided to undertake experimental studies to investigate the interfacial gas- liquid heat transfer. The experiment was conducted with the use of a small laboratory test rig equipped with a 100mm column filled in with 6mm glass Raschig rings. Air and water were used as working media. Hot water was circulated in a closed loop, it was supplied to the top of the packing section and flowed down. The ambient air was sucked at the bottom of the column and it flowed countercurrently to the liquid phase. Water and air flowrates as well as water inlet temperature were controlled independently. A system of flowmeters, thermocouples and humidity meters allowed to measure heat fluxes and thus to determine the interfacial gas-liquid heat transfer coefficient and the corresponding Nusselt number. More details about the experiment and the methodology can be found in [3].
Fig.1. Nusselt number dependence on gas and liquid loads and liquid inlet temperature.
The results of measurements are shown in Fig. 1 as the Nusselt number dependence on superficial gas and liquid velocities for 3 different inlet water temperatures. As can be seen Nu is strongly influenced by the gas load which intensifies the heat transfer. Fig. 1 indicates also a slight Nu dependence on liquid velocity which becomes especially evident for the lowest loads. Moreover temperature difference between gas and liquid phases influences the Nusselt number. The Nu results were then used to develop a correlation describing the interfacial heat transfer for the trickling flow regime with the use of various group numbers to account for such influences as gravity, inertia, surface tension, viscosity, buoyancy and thermal diffusion. As a results of detailed regression analysis the following correlation was found as the best fit to the experimental data 1.17 −0.84 0.72 푁푢 = 푅푒퐺 ∙ 퐺푎퐺 ∙ 퐸표 (1) where ReG and GaG are the gas Reynolds and gas Galileo numbers and Eo is the Eotvos number. The value of correlation coefficient R=0.992 indicates nearly perfect fitting proving its relevance. The correlation (1) describing the interfacial gas-liquid heat transfer coefficient for the trickling flow regime is a step ahead in modelling complex thermal phenomena in multiphase flow systems in packed beds. It should be remarked that the correlation was developed for limited gas and liquid loads and for particular type and size of packing elements. Thus there is a need for further research including wider (closer to industrial application) ranges of loads as well as different catalyst types.
References: [1] Niegodajew P., Asendrych D.: Amine based CO2 capture - CFD simulation of absorber performance, Appl. Math. Model. 40: 10222-10237, 2016 [2] Heidari A., Hashemabadi S.H.: Numerical evaluation of the gas–liquid interfacial heat transfer in the trickle flow regime of packed beds at the micro and meso-scale, Chem. Eng. Sci. 104: 674-689, 2013 [3] Niegodajew P., Asendrych D.: An interfacial heat transfer in a countercurrent gas-liquid flow in a Trickle Bed Reactor, Int. J. of Heat & Mass Transfer 108A: 703-711, 2017
6 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Porous media as an important element of fuel cell in high efficiency energy production Janusz Badur, Marcin Lemański, Tomasz Kowalczyk, Paweł Ziółkowski, Bartosz Kraszewski Energy Conversion Department, Institute of Fluid Flow Machinery PAS-ci, Gdańsk [email protected]; [email protected]
Keywords: porosity, SOFC, poro-thermo-chemo-mechanical interactions, continua with microstructure
In the future, hydrogen energy and solid oxide fuel cells (SOFCs) may be a competitive technology for large power units [1]. Promising electrical efficiency and emission level, respectively ca. 60% and 0.30 kg CO2/ MWh for hybrid fuel cell systems (SOFC/GT), cause large energy companies to become interested in this technology. Fuel cells are profitable modern devices being the best examples of a useful machinery where complex conversion of energy at nanoscale with porous media takes place. Especially, we observe such conversion at the high temperature solid oxide fuel cell (SOFC) that is built from ceramic nanomaterials. Anode supported fuel cell consist mainly of two nanoporous electrodes (cermets, lanthanum strontium manginite) separated, as was presented in Fig.1a, by a thin, very dense solid electrolyte (yttria-stabilized zirconia or perovskite-type material). Finding of a mathematical model of an acting SOFC at temperatures as high as 1000oC is a serious challenge as well for nanomechanics as for nanothermo-chemistry. In our works [1-3], a further development of the authors model of thermo-chemical flow of fuel, air, oxygen, steam water, species, ionic and electron currents within nanochannels and nanostructures of novel divides is presented. Integrated geometrical characteristics of working fluids and canal materials such as: porosity, tortuosity and mean radii are finally involved into a macroscopic continuum model, and implemented into the standard CFD code. The mean pore radii has been taken to be equal 300 nm. In Fig. 1b) dependence of generated voltage on current at various porosities are presented. When lower porosities are assumed the concentration polarization is higher [4].
a) b) Fig.1. a) 3D Computational domain for a SOFC tubular with electrolyte as a porous media, b) Influence of porosity nanoparameter on SOFC performance [4].
Different transport enhancement models should be taken into account –among them the most important are: the velocity slip connected with complex external friction, the Darcy mobility and the Reynolds transpiration. Increasing gas path to the triple- phase-boundary (TPB) enhances mass and electricity fluxes due to the concentration jump and the electron resistivity jump.
References: [1] J. Badur, M. Karcz, M. Lemański, L. Nastałek. Enhancement transport phenomena in the Navier-Stokes shell like slip layer. Computer Modeling In Engineering and Science, 73:299–310, 2011. [2] P. Ziółkowski, J. Badur: A theoretical, numerical and experimental verification of the Reynolds thermal transpiration law. International Journal of Numerical Methods for Heat & Fluid Flow , vol:28, iss:1; pp. 64-80 . [3] J. Badur, P. Ziółkowski, S. Kornet, T. Kowalczyk, K. Banaś, M. Bryk, P.J. Ziółkowski, M. Stajnke: Enhanced energy conversion as a result of fluid-solid interaction in micro and nanoscale. Journal of Theoretical and Applied Mechanics, 56, 1, pp. 329-332, Warsaw 2018, DOI: 10.15632/jtam-pl.56.1.329 . [4] M. Karcz, S. Kowalczyk, and J. Badur. On the influence of geometric nanostructural properties of porous materials on the modelling of a tubular fuel cell. Chem. Proc. Eng., 31:489–503, 2010.
7 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Plane flow through the porous medium with chessboard-like distribution of permeability Włodzimierz Bielski1, Ryszard Wojnar2 1 Institute of Geophysics, Polish Academy of Sciences 2 Institute of Fundamental Technological Research, PAS [email protected]; [email protected]
Keywords: geometrical mean, effective permeability, Darcy’s law, Brinkman’s flow, Keller - Dykhne formula
Theoretical trials to determine the effective properties of heterogeneous porous media are widely performed. In particular, the methods of asymptotical homogenisation are applied, cf. [1, 2]. In this contribution we study a special case of a chessboard-like distribution of permeability coefficients, in which simple algebraic arguments are applied. First, Darcy’s flow is considered, described by the equation. 퐾 풗 = − ∇(푝 + 푈) 휂 Here the vector 풗 is the flow velocity, K – permeability, 휂 - viscosity, p – the pressure, and U – the external potential. We assume the flow being incompressible, it is ∇ ∙ 풗 = 0. It is known, from the papers by Keller [3] and Dykhne [4] that the symmetry of equations describing stationary potential flows in the planar systems composed of two constituents, combined with geometrical symmetry of both constituents of the considered medium permits to find the effective conductivity of such a body. This elegant non-perturbative result is known as Keller - Dykhne's formula. There are three main assumptions needed: (i) considered fields are two-dimensional; (ii) the flow is stationary and has a potential; (iii) statistical symmetry and isotropy of the composite is assumed; both components are equivalent in statistical meaning: they have the areas of the same dimension, and can be mutually changed without the change of the whole composite, cf. Figure 1.
If one constituent has the permeability K1, and the second K2, then the effective permeability is √ 퐾1 퐾2, in agreement with experiments and simulations of Warren and Price [5].
. Fig.1. Chessboard-like distributions of the permeability. Examples of different tessellations of more dense (dark) and more rare (light) areas of a rock cross-section, which are in agreement with Keller-Dykhne’s geometrical assumption.
Similar result we obtain for the problem in which the flow is described by Brinkman’s equation, cf. [6], 퐾 풗 = [− ∇(푝 + 푈) + 휂′ ∆ 풗 ] 휂 Here the coefficient 휂′ means Brinkman’s effective viscosity. This equation permits to satisfy in full the boundary conditions at the interfaces of regions with different permeabilities, what, as it is known, is impossible when Darcy’s equation is used. But now, the flow 풗 is not potential, any more. Let us substitute 풖 = 풗 − (퐾 휂′/휂) ∆ 풗. Then, Brinkman’s equation has the form of Darcy’s equation 풖 = −(퐾/휂)∇(푝 + 푈) . Such vector 풖 is the potential and divergence-free one, as the vector 풗 is divergence-free. Thus, it satisfies all Keller - Dykhne's formula assumptions. Hence, in the case Brinkman’s flow the effective permeability is √ 퐾1 퐾2 also.
References: [1] V. V. Jikov, S. M. Kozlov, O. A. Oleinik, Homogenization of differential operators and integral functionals, Springer-Verlag Berlin Heidelberg 1994. [2] G. Allaire, One-phase newtonian flow, in: Homogenization and porous media, Ed. U. Hornung, Springer Verlag, New York - Berlin - Heidelberg 1997, pp. 45-76. [3] J. B. Keller, A theorem on the conductivity of a composite medium, Journal of Mathematical Physics 5 (4) 548-549, 1964. [4] A. M. Dykhne, Conductivity of a two-dimensional two-phase system, Soviet Physics JETP, 32(1) 63-65, 1971; in Russian: Zh. Eksp. Teor. Fiz. 59(1) 110-115, 1970. [5] J.E. Warren, H.S. Price, Flow in heterogeneous porous media, Society of Petroleum Engineers Journal 1 (03), 1961. [6] H.C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Applied Scientific Research 1(1), 27–34, 1949.
8 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 A comparative study of nanoscale pore structures in geomaterials Renata Cicha-Szot, Piotr Such, Grzegorz Leśniak Oil and Gas Institute – National Research Institute, 31-503 Kraków, Lubicz 25A Str., Poland [email protected]
Keywords: nanoscale pores, shales, gas slippage
Nanoscale pore structures are common in unconventional hydrocarbon reservoirs. The most diverse pore structures might be found in shale rocks, which are investigated all over the world due to its hydrocarbon potential. One of the challenges of economically exploiting shale gas reservoirs is proper estimation of gas flow rate. To achieve this goal, better understanding of pore structure and flow paths in fine grained rocks is needed. The nano to micrometer scales of pore systems requires more complex analysis and application of different techniques to understand pore changes in shale systems. With the exception of pore size analysis using scanning electron microscopy, in this study mercury intrusion capillary pressure (MICP), nitrogen adsorption and Klinkenberg gas slippage analysis are taken into account for evaluating pore size. Moreover, the latter technique characterizes only the part of the pore system that is responsible for fluid flow. Very few studies have been published that include gas slippage measurement on stressed shale samples [1],[2]; however, these studies were performed on mudstones or siltstones, whereas Central European shale formations are mainly claystones, which are more prone to deformation of pore space with effective stress. In this study, slippage was measured on plugs oriented parallel to bedding. All samples were pre-stressed at reservoir conditions (σeff~33 MPa). Due to the high heterogeneity of the analyzed shale samples, Klinkenberg permeability, SEM, MICP and nitrogen adsorption analysis were performed on the same plug in order to obtain the most adequate and coherent pore size results. Theoretical calculations of average pore diameter derived from slippage data collected in this study for samples with high clay content and low carbonate content yielded lower diameters than estimates from MICP data. Pore size evaluated with MICP and Klinkenberg gave similar results for samples with existing microfractures. Although similar pore diameters might be distinguished on SEM images, detailed pore size analysis using FIB-SEM showed very different results when evaluated with gas slippage.
Table 1 Comparison of slit-shaped pore size and tube-shaped pore size from SEM and Klinkenberg
Pore width Pore diameter Pore width SEM Pore diameter FIBSEM (median) Sample ID Klinkenberg Klinkenberg
[m] [m] [m] [m] 8 4.05·10-8 11.98 ·10-8 1.61·10-8 1.00·10-7 32 2.35 ·10-8 7.21 ·10-8 9.39·10-9 2.50·10-8 42 3.31·10-8 7.58 ·10-8 1.32·10-8 1.53·10-8
In order to provide more accurate estimation of clay rich shale permeability, additional pore characterization using Klinkenberg slippage is needed on an existing well-characterized plug. Pore size distribution obtained by conventional analysis performed in unstressed conditions may lead to significant errors. However, as has been shown in the paper, microfractures may reduce the effect of stress in rocks with specific mineral composition and pore space structure; thus, dominant pore diameter might be included in estimations of permeability. Initially, it seemed that SEM image analysis produced similar pore diameter measurements as Klinkenberg slippage. However, more detailed analysis showed a large discrepancy between the results. Moreover, for production simulation, correction related to reservoir temperature needs to be taken into account (as with other reservoir conditions, temperature is naturally included in permeability Pulse Decay measurements).
The research leading to these results were performed within the project: Methodology for sweet spots determination based on geochemical, petrophysical and geomechanical properties based on the correlation between laboratory investigations and geophysical measurements and a 3D generation model, co-funded by the National Centre for Research and Development as part of the programme BLUE GAS – POLISH SHALE GAS. Contract No. BG1/MWSSSG/13.
References: [1] Cui A., Wust R., Nassichuk B., Glover K., Brezovski R., Twemlow C.: A nearly complete characterization of permeability to hydrocarbon gas and liquid for unconventional reservoirs: a challenge to conventional thinking, SPE 168730, Unconventional Resources Technology Conference, 12-14 August, Denver, USA, 2013 [2] Heller R., Vermylen J., Zoback M.: Experimental investigation of matric permeability of gas shales AAPG Bulletin, 98, 975-995, 2014
9 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Minkowski Metric, Dirichlet Energy and Pore Tortuosity Mieczysław Cieszko Institute of Mechanics and Applied Computer Science Kazimierz Wielki University, Bydgoszcz [email protected]
Keywords: anisotropic pore space, Minkowski metric space, Dirichleta energy, pore tortuosity and its microscopic representation
Parameter of the pore tortuosity together with the volume porosity and permeability form a set of basic parameters characterizing macroscopic pore space structure of permeable porous materials. This parameter plays important role in all transport processes taking place in porous materials. This concerns among others the flow of fluids, electrical current, and also diffusion and heat conduction. The importance of general description of the pore space structure is determined by the fact that engineering of transport processes in porous media is directly related with engineering of pore structure. In spite of the fundamental character of the tortuosity parameter and great number of publications devoted to its definition, analysis of the physical and geometrical meaning and the methods of determination (e.g. [1-3]), there is still no commonly accepted, general definition of this macroscopic notion and its relation with microscopic pore structure. This problem becomes even more complicated in materials with anisotropic pore space structure. The aim of the paper is to present the general solution of the problem of macroscopic description of the anisotropic pore space structure, which allows precise and consistent formulation of definitions of macroscopic parameters of pore space structure: pore tortuosity and surface porosity, and also their natural introduction into macroscopic description of processes occurring in porous materials. The general character of these definitions is also a necessary condition for formulation of general representation of these parameters by quantities characterizing microscopic pore structure. Considerations have been based on the model assumptions presented in papers [4] and [5]. It was assumed that at the macroscopic point of view interconnected pores in permeable porous materials form anisotropic space the structure of which is determined by its metric and this space is modelled as Minkowski metric space. Such approach to this problem raises a number of consequences: a) modelling of the pore space structure is a primary problem in comparison to the modelling of processes occurring in the pore space; b) parameters of the pore space structure are defined by the metric of the space; c) pore structure parameters codetermine the course of each process occurring in the pore space and are independent of them. Application of the concept of Minkowski metric space as a model of anisotropic pore space enables precise and consistent definition of macroscopic measures of distance, surface and volume in this space, and as a consequence, also definition of macroscopic parameters of pore space structure: pore tortuosity and surface porosity, directly related to these measures. It was shown that these parameters and their tensor characteristics are directly defined by the metric tensor of the pore space. This means that character of these parameters is purely geometrical. Definitions of the pore structure parameters formulated based on the concept of Minkowski metric space are also the basis for precise determination of their relation with quantities characterising microscopic pore structure. General form of such relation for surface porosity and pore tortuosity have been obtained requiring the full representation of macroscopic density of fluid kinetic energy in the potential flow, by microscopic velocity field. The microscopic representation of tortuosity for porous media with isotropic pore space structure has the form
1 1 푟 2 2 = ∫ (퐧 ∙ 퐧) 푑푉 (1) 훿 푉푝 Ω푝 where 퐧 i 퐧푟 are unit vectors defining directions of gradient of macroscopic potential inducing fluid flow and of microscopic potential in the pore space, respectively. It was shown that such approach is directly related with the variational problem of minimization of scalar field inhomogeneity defined in the pore region the measure of which is the integral of square of gradient of this scalar field, called Dirichlet integral or Dirichlet energy. Euler equation for this problem takes form of the Laplace equation that is the basic equation describing various types of potential transport. This equation do not contain any material characteristics, and due to the pure geometrical character of the variational problem, its solutions are contingent also upon geometry of the region on which it is defined. In the case of potential flow of fluid, the variational problem means minimization of kinetic energy of fluid in the considered pore region, and with regard to the microscopic representation of the pore tortuosity, given by expression (1), it means minimization of the value of integral present there. References: [1] Carman P., Fluid Flow through Granular Beds, Trans. Inst. Chem. Engng, 15, 150-166, 1937. [2] Lorentz P. B., Tortuosity in Porous Media, Nature, 189, 386-387, 1961. [3] Clennel M., Tortuosity: A Guide through the Maze, in Developments in Petrophysics, eds Lovell M.A. & Harvey P.K., Geological Society London Special Publication No 122: 299-344, 1997. [4] Cieszko M., Fluid Mechanics in Anisotropic Pore Space of Permeable Materials. Application of Minkowski Metric Space (in polish), Wyd. Uniwersytetu Kazimierza Wielkiego, Bydgoszcz 2001. [5] Cieszko M., Description of Anisotropic Pore Space Structure of Permeable Materials Based on Minkowski Metric Space, Archives of Mechanics, 61, 6 (425-444), 2009.
10 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Influence of pore structure parameters on saturation of a ball with liquid during intrusion-extrusion process Tomasz Czerwiński1, Mieczysław Cieszko2, Chaplya Yevhen3 Institute of Mechanics and Applied Computer Science Kazimierz Wielki University, Bydgoszcz [email protected], [email protected], [email protected]
Keywords: macroscopic description, non-wetting liquid intrusion, porous ball, intrusion-extrusion hysteresis
The problem of macroscopic description of non-wetting liquid (mercury) intrusion and extrusion from a ball of a porous material is presented in the paper and the influence of pore structure parameters on saturation of the ball with liquid is analyzed. Modeling of these processes play an important role in the interpretation of mercury intrusion curves used for determination of pore size distribution and other pore structure parameters. For interpretation of such curves the simplified capillary models of pore structure [1] or computer network models [2] are commonly used. The use of macroscopic description of mercury intrusion- extrusion process is a new approach to this problem. The analysis is based on the new continuum model of the capillary transport of liquid and gas in unsaturated porous material presented in the paper [3], [4]. The key assumptions of this model are: division of liquid in the pore space into two macroscopic constituents called mobile liquid and capillary liquid; description of menisci motion by an additional macroscopic velocity field; parametrization of saturation changes by a macroscopic pressure-like quantity that for quasi static and stationary processes is equal to the capillary pressure. This model significantly changes the image of processes of liquid and gas capillary transport in an unsaturated porous material. It shows that these processes are running in a five dimensional pressure-time-space continuum, and quasi static processes of intrusion and extrusion of non-wetting liquid are non-stationary processes running in pressure-space continuum. The intrusion-extrusion processes in a porous ball are described by equation
2 휕푠푚 푑푠푐 휕푠푚 (1 + ) − 푐푚(푠푚, 푝) ( ) = 푐푠(푠푚, 푝) (1) 휕푝 푑푠푚 휕푟 where 푠푚(푟, 푝) and 푠푐(푟, 푝) are local parameters describing saturation of a ball with the mobile and capillary liquid, respectively, while 푝 represent pressure in the mobile liquid. Quantities 푐푚(푠푚, 푝) and 푐푠(푠푚, 푝) are coefficients characterizing gradient and volume diffusive transport of the menisci in the pore space. Equation (1) is an extension of the model proposed in the paper [4]. In the case when the process of liquid extrusion is starting from full saturation of the sample the whole process will be described by the reduced form of equation (2),
휕푠푚 푑푠푐 (1 + ) = 푐푠(푠푚, 푝). (2) 휕푝 푑푠푚 Such form of equation (2) results from the fact that in the state of complete saturation it is homogeneous, and during the extrusion process there are no mechanisms inducing inhomogeneity of this distribution. In addition to the change of liquid wetting coefficient in the intrusion-extrusion processes, this is the main mechanism generating hysteresis of the capillary potential curve. The problem liquid intrusion into a ball has been solved analytically using the method of characteristics. The expressions describing liquid distribution in the ball and the curves of its capillary potential were determined. This made it possible to analyze the dependence of these curves on the pore structure parameters and to determine the saturation curves with mercury during the extrusion process.
References: [1] Winslow D.N.: Advances in Experimental Techniques for Mercury Intrusion Porosimetry, Surface and Colloid Science 13: 259-282, 1984. [2] Xiong Q., Baychev T.G., Jivkov A.P.: Review of pore network modelling of porous media: Experimental characterisations, network constructions and applications to reactive transport. J. Contam. Hydrol. 192, 101–117, 2016. [3] Cieszko M.: Macroscopic Description of Capillary Transport of Liquid and Gas in Unsaturated Porous Materials, Meccanica 22: 1-22, 2016. [4] Cieszko M., Czapla E., Kempiński M.: Continuum Description of Quasi Static Intrusion of non-Wetting Liquid into a Porous Body. Continuum Mechanics and Thermodynamics 27(1): 133-144, 2015.
11 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Experimental investigations of pressure drops in chosen granular beds Waldemar Dudda, Wojciech Sobieski University of Warmia and Mazury, Olsztyn, Poland [email protected]
Keywords: granular porous media, permeability, Forchheimer law.
Investigations of pressure drop in porous media filled by water and/or air are very important in many areas of science and technique. The nature of such flows is very complicated and dependent on many factors. In particular, very interesting is the question whether the pressure drops obtained for the air flow correlate with the pressure drops received for the water flow (for the same porous column). Such investigations were performed in the past [1,2,3], however the results are not unambiguous. To continue these investigations, a new laboratory stand was designed and built (Fig. 1, left), and appropriate experiments were performed. The experiment was performed for three diameters of glass marbles (4, 6, and 8 [mm], see Fig. 1, right), two fluids (air and water) and over 20 various filtration velocities. Every measurement was repeated 10 times. Important is that in both sets of experiments (with air and water), the spatial arrangement of glass marbles was the same.
Fig. 1. Schema of the used laboratory stand (left) and a sample of glass marbles used in the experiment (right)
Fig. 2. Comparison of results for air (left) and water (right) flow through porous column (d = 6 [mm])
In Fig. 2, exemplary results of measurements for air (left) and water (right) flow for one chosen size of glass marbles are shown. All results were compared with the Forchheimer, Andreasen-Poulsen and Kozeny-Carman mathematical formulas. To obtain the Forchheimer coefficient, the so-called Forchheimer Plot Method was applied [4]. The data of glass marbles were taken from the monograph [4]. The presented results will serve as data for a comparative study of air and water flows through granular porous beds.
References: 1. Andreasen, R.R., Canga, E., Kjaergaard, C. et al. Water Air Soil Pollut (2013) 224: 1469. doi:10.1007/s11270-013- 1469-5. 2. Loll P., Moldrup P., Schjønning P., Rile H.: Predicting saturated hydraulic conductivity from air permeability: Application in stochastic water infiltration modelling. Water Resouces Research, Vol. 35(8), 1999, pp. 2387-2400. 3. Pugliese L.; Poulsen T.G.: Linking Gas and Liquid Pressure Loss to Particle Size Distribution and Particle Shape in Granular Filter Materials. Water Air Soil Pollut (2014) 225:1811. DOI 10.1007/s11270-013-1811-y. 4. Sobieski, W., Lipiński, S., Dudda, W., Trykozko, A., Marek, M., Wiacek, J., Matyka, M., Gołembiewski, J.: Granular porous media (in Polish). University of Warmia and Mazury in Olsztyn, Olsztyn, p. 180 (2016).
12 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Experimental determination of the kinetics of sorption and the kinetics of gas filtration in coal Marek Gawor The Strata Mechanics Research Institute of the Polish Academy of Sciences [email protected]
Keywords: sorption, filtration, sorption kinetics, filtration kinetics
The paper presents tests rigs for experiments on sorption kinetics and gas filtration kinetics in a porous medium. It was observed that two phenomena occur in these processes: transportation of gas into the porous solid and settling of gas molecules on the walls of the solid or in its volume. The adsorption kinetics is understood as the gas molecule transition rate from the free state to the bound state. This sorption process takes place when a molecule of the sorbed gas is already on the surface of the solid and there is a reaction of sorption forces between that particle and the atoms of the solid. Filtration is the gas transportation process into the solid that takes place as a result of the gas pressure gradient. Both sorption and filtration occur at a certain rate. Sorption involves heat release (adsorption) or heat transfer, which is connected with the change of temperature. The paper presents the results of three experiments. In the first one, a thin resistance thermometer was quickly taken out of an argon stream and placed in carbon dioxide or the other way round (Fig.1). The measurement made it possible to determine the sorption time constant. It was demonstrated that the sorption rate is much higher than the filtration rate. Thus filtration is the process describing the rate at which gas molecules penetrating the porous substance are adsorbed or desorbed. The sorption time constant is not higher than 50 msec. In the second experiment the authors determined the rate at which gas is liberated from coal grains. The measurement method was based on the measurement of the pressure of desorbing gas in constant volume (Fig.2). The experiment involved a measurement of the pressure of the gas liberated from the coal grains in a closed chamber. The kinetics curves obtained in this way were used to determine the carbon dioxide filtration coefficient in coal grains. During the experiment a particular focus was put on the initial stage of gas liberation (up to 0.4 sec) Gas transportation in a porous structure of coal briquettes is a slower process. In the third experiment the variety of the boundary conditions allowed for a fuller verification of the assumed theoretical model and if necessary, a more precise specification of filtration parameters. The test rig that was built for that purpose (Fig.3) allowed for the measurement of the pressure and the temperature in the edge of a briquette.
Fig. 1 The measurement principle of the sorption time Fig. 2 The scheme of the rig for experiments on gas filtration constant. coefficient in coal grains.
Fig. 3 The scheme of the measurement rig for determination of gas filtration coefficient in coal briquettes.
13 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Fuzzy logic approach in the analysis of heat transfer in a porous sorbent bed of the adsorption chiller Karolina Grabowska1, Jarosław Krzywański1, Karol Sztekler2, Wojciech Kalawa2, Wojciech Nowak2 1Jan Dlugosz University, al. 13/15 Armii Krajowej; PL42-200 Czestochowa, Poland 2AGH University of Science and Technology, ul. Czarnowiejska 30; PL30-059 Cracow, Poland [email protected]
Keywords: adsorption chiller, porous media, fuzzy logic, thermal conductivity, coated adsorption bed
Thermal conductivity in boundary layer of heat exchange surface is the crucial parameter of adsorption processes efficiency which are carried out in the adsorption bed. In order to improve of heat transfer conditions in the adsorption chiller the novel constructions of adsorption beds are currently investigated. The porous structure of sorbent layer is low thermal conductivity of the whole adsorption bed. One of the methods is the modification of porous media bed structure with glue which is characterized by higher thermal conductivity. The optimum parameters of sorbents and glues to build the novel coated construction in terms of improving the COP (Coefficient of Performance) of chiller were defined in [1]. The paper implemented the fuzzy logic approach to predict the thermal conductivity of modified porous media layer. The analogic approach was applied in [2] for the prediction of local heat transfer coefficient in the combustion chamber of a circulating fluidized bed combustor. The developed model present changes of sorbent layer thermal conductivity depending on the change of input parameters defining the heat transfer layer. The data from empirical research were used to build up the model by the fuzzy logic techniques. The sample geometry used in the experiment is shown in the figure 1.
Fig.1. Coated sorbent layer
References: [1] Grabowska K., Krzywański J., Nowak W., Wesołowska M., Budowa innowacyjnej konfiguracji złoża sorbentu w adsorpcyjnym agregacie chłodniczym – Kryterium doboru optymalnej pary sorbent – klej, XXIII Zjazd Termodynamików 2017. Beskid Śląski; 2017. [2] Krzywanski J., Nowak W. Modeling of bed-to-wall heat transfer coefficient in a large-scale CFBC by fuzzy logic approach, International Journal of Heat and Mass Transfer, 94, 327-334, 2016.
14 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 A concept of a two-stage process of non-pressure granulation of mineral materials. Paweł Guzik1, Krzysztof Mudryk1 1 1University of Agriculture in Krakow, Faculty of Production and Power Engineering, Department of Mechanical Engineering and Agrophysics Krakow, ul. Balicka 120, 30-149 Kraków, Poland [email protected]
Keywords: Slow-release fertilizers, non-pressurized agglomeration, Compressive strength, disc fertilizer granulator
Mechanism of granules formation in the disk granulation process is based on the adhesive properties of drops of the liquid used. Substance is putting with the liquid into the fluidized bed, is deposited partly on the surface of the granules in the layer and forms separate particles. The growth of granules is possible when the strength of liquid drop bonds with solid particles are large. In the fluidized bed, apart from the formation of new granules, there is an increase in already existing granules. A characteristic feature of the fluidized bed is the classification of particles in different zones of the layer. Granulation process in the apparatus with the fluidized bed, as a rule, proceeds with the simultaneous separation of particles in terms of their size. This phenomenon significantly affects the nature of the granulation process and should be taken into account when calculating the granulometric composition of the layer. In order to obtain a homogeneous granulate, a selective discharge of granules from the apparatus and the return of fine particles to the sprinkling zone is used. At this point, there is a chance to use an additional device (surrounding drum) cooperating with the disk granulator. As the amount of mass in the granulator increases, the granule bed will exceed the edge of the granulator, pouring out of the granulator. Receiving pouring granules in this way and directing them to further relaxing surrounding may allow obtaining granules with greater mechanical strength. It has been proved that the surrounding time has a positive effect in correlation with the obtained higher compressive strength in strength tests. In addition to the main concept of influencing the hardening of the obtained granules, a rotating drum could fulfill the task of sifting sieve (through the applied perforation) too small granule fraction and shredding too large. Only a part of the raw material would be recycled to re-granulation. In my earlier research on the granulation process, I showed the dependence of the mechanical strength of granules on the angle of the granulator's disc and the rotational speed.
Fig.1. . Comparison of strength from the angle of inclination of the granulator's plate.
Graph above shows the dependence of the strength of the granules on the rotational speed, the highest result is characterized by the target fraction obtained at the highest angle of inclination of the granulator's disc of 55 ° at any rotational speed. It is difficult to indicate unequivocally what effect of the parameters results in a lower compressive strength of the obtained raw material, but it is clearly visible that the average highest results are on the cattle speed of 20 rpm in all angles of inclination of the granulator's plate. Visually, the lowest compressive strength parameters below 4 N are characterized by samples obtained at 45 ° and 50 ° and the lowest rotational speed of 17.5 RPM. The obtained results are the starting point for further agglomeration tests. Results above in the necessity of further searching for parameters allowing to control the process of non-pressure granulation. From sources of literature there are indications that prolong the time of surrounding granules. However, leaving granules too long in the fluidized bed causes their excessive growth, which is undesirable in the case of the selected range of expected granule size for fertilizing purposes. The homogeneous particle size allows precise application of the fertilizer dose. The aforementioned idea is the concept of a two-stage granulation allowing to continue the process of hardening the granules outside the main bed, at the same time allowing testing in a continuous granulation process and not as in the intermittent form after full granulator platter has been obtained. The presented concept of a two-stage, non-pressure granulation is subject to further research in my field.
References: [1] Pietsch W.: Aninterdisciplinary approach to size enlargement by agglomeration, 2003 [2] Domoradzki M.: The kinetics of dust granulation in a disk granulator, 1978 [3] P. W. Kłassien, I. G. Griszajew.: Basics of granulation techniques, 1989. 15 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Mathematical model of the course of process of catalytic wet air oxidation of phenol (CWAO) in trickle bed reactors (TBR) Daniel Janecki Department of Process Engineering, University of Opole, ul. Dmowskiego 7-9, 45-365 Opole, Poland [email protected]
Keywords: trickle-bed reactors; multiphase flow (CFD); catalytic phenol wet oxidation; active carbon as catalyst
Despite progress achieved in recent years in the use of Computational Fluid Dynamics (CFD) in modelling work of trickle bed reactor (TBR), still predicting hydrodynamic and kinetic parameters compliant to an experiment encounters major difficulties. It is connected among other things with lack of reliable reports describing interactions between phases for multiphase systems, as well as limited number of experimental data used at verification of estimated spatial and temporal changes of the discussed parameters. The aim of the present study is modelling of process of catalytic wet air oxidation of phenol (CWAO) in TBR. This process was chosen because phenol and its derivatives are the most frequent ingredients of pollution present in industrial waste and their toxicity and ability of bioaccumulation forces to establish more and more effective methods of removing phenol from the polluted water. One of the possibilities is the use of catalytic process of wet air oxidation of phenol, while in the experiments instead of commonly used metal oxides (the most frequent being copper oxide) active carbon was used as a catalyst of the process (granules of active carbon of 1.5 mm diameter piled in a layer 0.70 m high). The reactor was working at the pressure of 1.85 MPa (Janecki et al. (2016)). In the discussed process one could differentiate stages: transport of oxygen from gas to liquid phase and directly from gas to external surface of the catalyst, transport of oxygen and phenol from the core of liquid to external surface of the catalyst, transport of oxygen and phenol inside the pores of the catalyst and reaction on the internal surface of contact. Making mass balance of reacting substances it was assumed that the reactor works in the established state in isothermal and isobaric conditions at stable activity of the catalyst. It was assumed also, that external surface of the catalyst is partially moistened with liquid which makes contact of the oxygen contained in the air with the surface of the catalyst possible, while the pores of the catalyst, sprinkled intensively before each experiment, are filled with liquid. Speed of reaction and factors of transport of mass from gas to liquid and solid phase were obtained from available literature data. Balance equations were adjusted to the shape which allows to use them in Fluent program. Source elements which appear in the equations of this model have been introduced to this program with the use of user function UDF. To recreate the area taken by the bed and liquids flowing through it a structural net was generated with the use of GAMBIT preprocessor. As initial conditions experimental values of gas and liquid velocity, average volume share of liquid phase and mass shares of elements in gas and liquid at the inlet to the reactor were adopted. Calculations were performed in two stages. In order to obtain the distribution of velocity of liquid and gas and distribution of volume share of gas and liquid phase the Eulerian model was used which includes the balance of mass and momentum for each phase. For calculations in which the change of porosity of the bed along the radius of the apparatus is considered, with the use of Martin correlations (1978) (the ratio of the diameter of the reactor to the diameter of the catalyst was equal 13.3), it is best to use Ergun constant values estimated with the use of neuron networks by Iliuta et al. (1998) (Janecki et al. (2014)). For the reactor and catalyst used in the research these constants equal E1 = 155; E2 = 2.07. In the next stage a simulation of the field of concentration of elements in the whole reactor at the assumption of invariability of the field of flows and volume shares of all phases was conducted, which was fully justified due to small interphase streams of mass. As result of the calculations local values of concentration of elements were obtained as well as their values at the outlet from the reactor. For comparison a single-dimensional isothermal model of catalytic process of oxidation of phenol in trickle reactor with piston flow of both phases was worked out with the use of which a change of concentration of elements along the reactor and their values at the outlet from the apparatus was calculated. The equations of the model were solved numerically with the use of Runge-Kutta method of the fourth order. While analysing the obtained results one may state that the results of numerical simulations obtained with the use of CFD model are closer to experimental values. Average relative error of estimation of phenol concentration at the outlet of the reactor obtained from CFD model equals 11.6%, while average standard deviation is 5.2%; for values obtained from single-dimension model it is respectively 16.9% and 12.2%. Taking into account the number of parameters the value of which had to be calculated from empirical correlations (equation of reaction velocity, mass-transfer coefficients) one may notice that a satisfying compliance of measured values and those calculated from the model was achieved.
References: [1] Iliuta I., Larachi F., Grandjean B.P.A.: Pressure drop and liquid holdup in trickle flow reactors: improved Ergun constants and slip correlations for the slit model. Ind. Eng. Chem. Res., 37: 4542-4550, 1998 [2] Janecki D., Burghardt A., Bartelmus G.: Influence of the porosity profile and sets of Ergun constants on the main hydrodynamic parameters in the trickle-bed reactors; Chem. Eng. J., 237: 176-188, 2014 [3] Janecki D., Szczotka A., Burghardt A., Bartelmus G: Modelling of wet-air oxidation of phenol in a trickle-bed reactor using active carbon as a catalyst; J. of Chem. Techn. & Biotech., 91: 596-607, 2016 [4] Martin H.: Low Peclet number particle to fluid heat and mass transfer in packed beds, Chem. Eng. Sci. 33: 913-919, 1978
16 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Depth-averaged model for flow in a propped fracture Łukasz Jasiński1, Marcin Dąbrowski1,2 1Computational Geology Laboratory (CGL), Polish Geological Institute – National Research Institute, Wrocław, Poland 2Physics of Geological Processes, University of Oslo, Norway [email protected]; [email protected]
Keywords: fracture flow, Brinkman flow model, numerical simulation
Propped fractures largely contribute in overall flow through low matrix permeability formations, such as shales or crystalline rocks. Therefore, they may play an important role during natural processes (i.e. hydrocarbon migration. hydrothermal circulation, metamorphism), as well as, in industrial applications (i.e. gas storage in caverns, radioactive waste repositories, oil and gas extraction from reservoirs). Moreover, similar geometries may also be found in manufactured system, for example: microfluidic devices for microbiological filtration and separation systems, high performance liquid chromatography, dielectrophoretic platforms or heat spreading and dissipation systems for microelectronic.
Fig. 4. A geometric model of a plane-walled fracture with cylindrical circular obstacles. b denotes the fracture aperture, and R and d are the cylindrical obstacle radius and diameter respectively.
In our model, the fracture walls are planar and proppant grains are approximated by circular cylindrical obstacles (see Fig. 1). We study the local and the effective flow properties in a propped fracture, and how they are influenced by two geometric parameters: the obstacle fraction f and the ratio between the fracture aperture b and the obstacle diameter d. Due to the symmetry of geometry along the fracture aperture, the problem is reduced to two dimensions, and to fulfill the no-slip boundary condition at the rims of the obstacles, the Brinkman flow model [1] is adopted to the fracture flow problem [2]:
푏 −푏∇푝 + 휇∇2푱 − 푱 + 푏풇 = 0 (1) 퐾 ∇ ∙ 푱 = 0 (2) where p is fluid pressure, µ denotes fluid shear viscosity, f is the body force, J is the local flow rate (i.e., the depth-integrated in- plane components of the velocity vector), and K is the permeability of an empty plane-walled channel of an aperture b, 퐾 = 휇 1/12푏2. The additional Brinkman term 푱 in equation (1) corresponds to the viscous drag due to the presence of the fracture 퐾 walls. The Brinkman flow model is carefully validated against the analytical solution for a single obstacle, and compared to fully resolved three-dimensional Stokes model for the case with many obstacles. Described flow models are solved numerically using Finite Element Method, based on the modified version of MILAMIN codes [3]. Finally, the results of systematic calculations are analysed in terms of the velocity fields probability distributions and effective fracture transmissivity.
References: [1] Brinkman H. C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Flow Turbulence and Combustion, 1(1), 27, 1949 [2] Jasinski L., Dabrowski M.: The Effective Transmissivity of a Plane-Walled Fracture With Circular Cylindrical Obstacles, Journal of Geophysical Research: Solid Earth 123(1): 242–263, 2018 [3] Dabrowski M., Krotkiewski M., Schmid D. W.: MILAMIN: MATLAB-based finite element method solver for large problems, Geochemistry, Geophysics Geosystems, 9(4): Q04030, 2008
17 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 A unified thermodynamic approach to description of whole phenomena during a coke production Dariusz Kardaś, Sylwia Polesek-Karczewska, Paweł Ziółkowski, Janusz Badur Institute of Fluid Flow Machinery PAS-ci, Gdańsk [email protected], [email protected]
Keywords: coke, poro-mechanics, poro-thermo-chemo-mechanical interactions
Coke industrial production still is based on a mysterious and hidden technology procedures coming from the beginning of XIX- century. These procedures, speaking directly, have a roofs from the renascence alchemy and have nothing common with modern science. Many parameters, that play a key role in the coking process, such as a rate of heating, the length of thermal plateau, etc., are taken from an accidental knowledge of a producer. We have found that due to lack of scientific recognize, a coking process is, generally, at engineering level – therefore, it is a chance for science to improve its efficiency and to reduce its cost.
In our best opinion, a proposed unified thermodynamic approach should take into account the following phenomena such as: in the wall zone: surface radiation, turbulent heat transfer, depletion of flue gases;
in the semi-coke sorption/desorption, swelling, cracking, zone: thermal creep, thermal and sorption stresses, radiation, heat diffusion, porous flow of gases;
in the closed chemical reaction, porosity creation, porosity zone: effective stresses, heat conductivity, vaporization; internal pressure, gas confinement;
in the plastic enhancement heat transport, layer zone: devolution, thermoplasticity;
in the coal zone: heat and mass conductivity, water/steam adsorption, swelling and hydration behavior of coal, gas sorption/desorption in porous media.
It is evident that spatial and temporal monitoring of the progress of coke would also aid in characterizing the local variations in pore size and tortuosity, of broad relevance to all unconventionals and transport processes. In this study, starting from a common thermodynamic approach, and from our previous works on fluid flow in such systems dominated by various mechanisms (such as Darcy flow, slip flow, desorption, Knudsen diffusion, Reynolds transpiration, thermal creep, multiscale radiation) we propose a thermodynamically justified FSI model for modeling whole coke production process. A particular data of several processes are taken from the literature and form our own experiments – the process from the beginning should be treated to be nonstationary (whole about 3 hours). Modeling has also been used to describe fluid transfer from coal matrix to fractures, incorporating the significant Navier diffusion in nanopores, Graham sorption gas, mechanism of thermal pyrolysis, and so on. The coal pores, which occur in the process, could be classified into micropores (pore size d=6-10 nm), mesopores (10 nm
18 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Determination of pore size distribution in sintered glass bead samples based on mercury porosimetry and microtomographic image analysis Marcin Kempiński1, Mieczysław Cieszko, Marcin Burzyński2, Zbigniew Szczepański Institute of Mechanics and Applied Computer Science, Kazimierz Wielki University, Kopernika 1, 85-074 Bydgoszcz, Poland 1 [email protected], 2 [email protected]
Keywords: pore size distribution, sintered glass beads, MIP, CT
In the paper the capillary and random chain models of pore architecture, presented in the paper [1], are applied for determining the limit pore size distributions in sintered glass bead samples based on mercury intrusion curves, [2]. They estimate the range of pore sizes in the investigated material. It is proved that capillary model, commonly used in mercury porosimetry, and the chain model are two limit cases of net models of pore architecture for a given pore size distribution. Both distributions determine the range in which the pore diameter distribution of the investigated material occurs and defines the degree of inaccuracy of the method based on the mercury intrusion data caused by the indeterminacy of the sample shape and its pore space structure. The capillary and chain model with constant length has been used as a basis for the procedure of determining the limit pore size distributions in sintered glass beads samples. The limit distributions are compared with distributions determined from microscopic image analysis of samples obtained by the micro computed tomography (CT) method. These distributions are considered as actual distributions of the pore space and are determined by prescription to each point (voxel) of the pore space of porous material, the diameter of the maximal sphere that contains this point and is completely inside the pore space, [3]. The models of the pore space of porous materials are analysed in which the individual pores are cylindrical links of random length and diameter distributions. Two independent factors determinate the pore space structure of such media: the link size distributions and the way of their connections. The second factor is called the pore space architecture. The pore architecture causes that even for the same pore diameter distribution in the model material its pore space structure can be different. Regarding the pore architecture, in the paper we will distinguish three kinds of models of the pore space structures: capillary, chain and network. In the capillary model the links of equal diameter are joined in series and form the long capillaries of constant diameters crossing the whole material. In the chain model the links are joined at random in series and form the capillaries of step-wise changing cross- section. In the network model a random connected links from the space network. Fife special cases of the capillary potential curves of porous layer are presented in Figure 1. Four cases concern the chain pore architecture of the layer with the same link diameter distribution and different distributions of link length: capillary model (CM), periodic (PM) and random periodic (RPM) models with constant link length and model with random link length (RM). The fifth case concerns the porous ball with isotropic chain pore architecture. It was assumed in this case that the mean length of the capillary chords in the ball is equal to the thickness of the layer.
Figure 1: Capillary potential curves of the porous layer with capillary (CM), periodic (PM), random-periodic (RPM), random (RM) pore space architecture and porous ball with isotropic chain pore structure (BM).
From this figure results that link length distribution in the layer and capillary chord distribution in the ball does not influence significantly the capillary potential curves. Therefore, description of this curve for porous material of chain pore architecture can be effectively represented by the model with constant link length. Then, the thickness of the layer can be interpreted as the mean length of capillary chains in sample of any shape. It was shown that capillary potential curves for capillary and periodic models are limit curves for network models of the same pore size distribution. This means that both models can be used for estimation of the range of pore diameter distribution determined basing on the mercury intrusion data. Both models have been applied for determination of limit pore size distributions of sintered bead samples. The obtained results were compared with distributions determined from micro tomographic images using the methods of morphological image analysis.
References: [1] Cieszko M., Kempiński M., Determination of Limit Pore Size Distributions of Porous Materials from Mercury Intrusion Curves. Engng. Trans. 54, 2, pp. 143-158, 2006. [2] Webb P.A., Orr C., Analitical Methods in Fine Particle Technology. Micrometitics Instrument Corporation. Norcross. GA USA, 1997. [3] Hildebrand T., Ruegsegger P., A new Method for the Model-Independent Assessment of Thickness in Three-Dimensional Images, Journal of Microscopy, 185, 1, pp. 67-75, 1997. 19 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Geometry extraction from GCODE files destined for 3D printing Wojciech Kiński, Wojciech Sobieski University of Warmia and Mazury, Olsztyn, Poland [email protected]
Keywords: 3D printing, GCODE, reverse engineering.
Nowadays the 3D printers are very popular and their importance grows every year. They are used in many areas of engineering, technique, medicine and even food industry. To obtain any printed object, first its geometry must be defined, with taking into account all limitations of the 3D printing technology. To reach this aim, any software destined to preparing 3D geometry (in the STL file format) may be used. At this stage assumptions related to the used material must be accept, too. Very important is that the virtual geometry cannot be printed directly on a 3D printer. The geometry must to be converted to a set of instructions, which will be sequentially read and execute by the printer. In particular, the movement of the print head and the material feeding speed must be specified. Here, a specific programming language, the so-called GCODE is used. It is a standardized command language that it uses to operate numerically controlled devices (CNC). The practical experience in 3D printing shows that sometimes the original geometry data are no longer available. That means that having only a GCODE file there is no possibility to see the geometry prepared to printing. This problem as well as the lack of appropriate software, was the main motivation to performing investigations described in the paper. The aim is to develop an algorithm, which convert GCODE files to a format, which may be easy visualised.
Fig. 1. Examples of geometry prepared in the MatterControl [2] software and saved to STL files
The applied methodology cover three steps: 1) preparing exemplary geometries in the STL format; 2) converting STL files to GCODE files; 3) decoding GCODE files to reconstruct the geometry. It should be stressed, that the fully reconstruction of STL files is very difficult (and perhaps not possible at all). In the investigations was assumed, that geometry representation in the form of a point cloud would be sufficient.
Fig. 2. Examples of decoding chosen GCODE files
In Fig. 1 three examples of geometry are shown: thin-walled object (left), filled object (centre), and object with a thread (right). In turn, in Fig. 2 the decoded geometry of the same objects in a form of point clouds are visible. As it can be seen, the obtained shapes correspond well to the original geometry. Such result is enough to identify the geometry saved in GCODE files. It should be stressed, that the developed algorithm is destined only for files prepared for 3d printers and don’t have an universal character. The data converter is written in the Fortran language [1]. Visualization of point clouds is made in the ParaView software [3] with the use of VTK file format [4]. The main observations and final conclusions are as follows: it is possible to reconstruct the geometry saved in a GCODE file, and the most convenient form of its visualization is a point cloud; the movement of the print head must be interpolated to reconstruct filled structures; duplicates of points in a cloud should be detected and removed; the decoding process is hindered by the fact, that from one geometry many differing GCODE files may be obtained (it depends on settings of the algorithm converting the STL files to the GCODE files).The last problem causes sometimes appearing points located outside the original geometry. It is difficult to develop a universal algorithm to filter such points for all possible variants of GCODE files.
References: 1. GNU Fortran Home Page [on-line]. URL:https://gcc.gnu.org/fortran/ (available at February 6, 2018) 2. MatterControl Home Page [on-line]. URL: https://www.matterhackers.com/ (available at February 6, 2018) 3. ParaView Home Page [on-line]. URL: https://www.paraview.org/ (available at February 6, 2018) 4. VTK - The Visualization Toolkit[on-line]. URL: https://www.vtk.org/ (available at February 6, 2018)
20 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 The application of 3D printing technology in the investigations of porous media Wojciech Kiński, Wojciech Sobieski University of Warmia and Mazury, Olsztyn, Poland [email protected]
Keywords: 3D printing, porous media, controlled geometry.
The porous media found in the nature as well as in the modern technique are very diverse [1]. Usually the spatial structure of porous media cannot be freely modified and depends on many factors, including random ones. However, to obtain specific and desirable features, it would be good to control the porosity, tortuosity, specific surface and the other parameters characterizing the geometry of the porous body. The question is whether the 3D printing technology may be applicable to creating porous media with such controlled geometry. The main aim of the presented study is to recognize possibilities in this area. Currently two possibilities are taken into account: 1) generation of porous media based on filling patterns; 2) generation of porous media based on full 3D geometry. In Fig. 1, left, the most popular filling patterns are visible. They are applied to save the material and to shorting the printing time. Usually the same patterns are repeated in every layer due the fact, that every next layer must be supported from below. In Fig. 1, right, other filling pattern is shown. In this case, the holes have an regular three-dimensional shape.
Fig. 1. Exemplary filling of models
In Fig. 2 application of the second mentioned possibility is shown. First, a 3D model with specific porosity and tortuosity was created, which next was printed. Important is, that many geometrical information from a 3D model may be obtained. Some of them are assumed, the other may be calculated (Tab. 1). The ranges of geometrical parameters, which are possible in the praxis is the key question in such investigations. It can mention here the maximum number of channels, its diameter, the minimum porosity and the maximum tortuosity as well as correlations between these parameters.
Fig. 2. Stages of creating an porous object with specific porosity and tortuosity
Table 1. Geometrical parameters of the porous body. number of channels height [mm] volume [mm3] porosity [-] tortuosity [i] straight channels 16 100 334 880 0.898 1 curved channels 16 106.8 345 800 0.87 1.068
References: 1. Thingiverse Home Page [on-line]. URL:https://thingiverse.com/ (available at February 20, 2018)
21 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Selected artificial intelligence methods in modeling of energy devices and systems Jarosław Krzywański Akademia im. Jana Długosza w Częstochowie, Wydział Matematyczno-Przyrodniczy j.krzywanski @ajd.czest.pl
Keywords: artificial intelligence, neural networks, fuzzy logic, energy systems, modelling.
The work deals with the artificial intelligence methods in modeling of energy systems. Selected artificial intelligence techniques, i.e. artificial neural networks, fuzzy logic and genetic algorithms are applied in the study. A wide range of applications are considered. Heat transfer in a large-scale circulating fluidized bed boiler, hydrogen production via CaO sorption enhanced gasification of sawdust in a fluidized bed unit, NOx emissions from calcium looping in fluidized bed systems and the CaSO4 decomposition during coal pyrolysis are discussed in the work [1-3]. The developed models constitute a series of easily-applicable and powerful tools allowing to describe the complex energy systems. Original models developed in the book have a high value from practical point of view and are consistent with the latest scientific achievements in the discussed subject.
References: [1] Krzywanski J., Nowak W.: Modeling of bed-to-wall heat transfer coefficient in a large-scale CFBC by fuzzy logic approach, International Journal of Heat and Mass Transfer, 2016;94:327– 34. [2] Krzywanski J., Grabowska K., Herman F., Pyrka P., Sosnowski M., Prauzner T., Nowak W.: Optimization of a three-bed adsorption chiller by genetic algorithms and neural networks, Energy Conversion and Management, 153:2017, 313–322. [3] Krzywański J., Nowak W.: Neurocomputing approach for the prediction of NOx emissions from CFBC in air-fired and oxygen- enriched atmospheres, Journal of Power Technologies, 97 (2) 2017:75-84.
22 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Evaluation of plasticity criterion applicability in the porous materials research Anna Kułakowska Institute of Technical Education & Safety Systems, Jan Długosz University in Częstochowa, Al. Armii Krajowej 13/15, 42-218 Częstochowa [email protected]
Keywords: plasticity criterion, shear deformation, destruction
At the moment, conventional bulk materials are being increasingly replaced by sintered materials with the same chemical composition. These materials could be characterised by a high value of porosity that quite often deteriorates their properties in comparison with bulk materials. One of the possible ways to improve the properties of such materials is to deform sintered billets, what may lead to additional strengthening of surface layers. In this case, it is very important to determine whether these materials are able to undergo deformation. The article describes an approach for determination the yield criterion according to the Kolmogorov model. he plasticity criterion is the simplest and most reliable way for determination strain capacity of materials [1, 2]. According to the Kolmogorov model, the use of plasticity at any time point from the period t can be determined via equation [1, 3]:
푡 퐻푑휏 휓 = ∫ (1) 0 훬푝 where: H – shear deformation rate intensity; τ – shear stress; Λp - shear deformation to destruction (range plasticity): when t=0-ψ=0; at the moment of destruction tp-ψ=1, and at any time 0 훬푝 = 훬푝(푘휎, 휇휎, 퐻, 푇, 퐵(휏), 푋푖) (3) where: kσ=σśr/τi – stress status indicator; τi – intensity of shear stress; μσ – Lode indicator; B(τ) – an index of deformation non-monotonicity; H – intensity of shear deformation rate; T – temperature; Xi – physicochemical and structural parameters of the deformed material. The ability to plastic deformation without cracking is limited and therefore plasticity (deformability) is one of the main features of material that determines its sensitivity to plastic forming. References: [1] Kolmogorov V.L.: Napriazhenija. Dieformacja. Plastichnost. Mietalurgija. 1970. [2] Kolmogorov V.L.: Plasticznost i raz ruszenie. Mashynostrojenije1977. [3] Dyja H., Gałkin A., Knapiński M.: Reologia metali odkształcanych plastycznie. Wyd. Politechniki Częstochowskiej. 2010. 23 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Diffusion of Cd(II), Pb(II) and Zn(II) on calcium alginate beads Sylwia Kwiatkowska-Marks, Justyna Miłek, Ilona Trawczyńska Faculty of Chemical Technology and Engineering, University of Technology and Life Sciences, Bydgoszcz [email protected] Keywords: diffusion, cadmium, zinc, lead, alginate beads Heavy metals have a proven harmful effect on many forms of life. Lead and cadmium are known to be especially harmful to man and the environment [1]. Wastewater that contains zinc is harmful for both irrigational and industrial applications [2]. Biosorption on materials of natural origin seems to provide the most prospective results: in addition to being highly efficient, it enables elimination of the entire content of metal ions, even if they are present at very low concentrations in the liquid waste. Alginates are linear copolymers of β-D-mannuronate (M) and α-L-guluronate (G) residues in (1→4)-linkage, arranged in a block- wise pattern along the linear chain [3]. Alginates are biopolymers with high sorption capacity for heavy metals, even at low concentrations of the metals in solutions [4]. The sorption of metal ions on alginates takes place at a very fast rate and is only limited by diffusion phenomena. Therefore, according to the commonly accepted belief, the rate of sorption with this type of sorbent is limited by internal diffusion. In order to use the quantitative approach to the diffusive-mass movement within the porous beads having a complicated geometrical structure, the notion of effective diffusion coefficient, De, has been introduced. Since the rate of sorption on alginate beads is determined by the rate of diffusion in the sorbent pores, it is essential to know the effective diffusion coefficient to design the equipment. In the case of alginate gels, it is convenient to use the diffusion retardation coefficient, φ: D (1) e Daq where: De - effective coefficient of sorbate diffusion in sorbent pores, Daq - diffusion coefficients for a highly dilute aqueous solution (in this work were calculated using the Nernst equation) The main objectives of this research work are to determine by the conductometric method the effective diffusion coefficient for different heavy-metal salts: Cd, Zn and Pb in calcium alginate beads, and to determine the effect of the metal type, anion from the metal salt and the alginate content in the beads on the De value. In the conductometric method, determination of the effective diffusion coefficient is based on measurements of conductivity of the solution into which the sorbate diffuses; therefore, assuming that dependence of conductivity on concentration is linear, the following equation is obtained (by transforming the non-stationary diffusion equation): P 6 1 D n 2 2t t 1 exp e (2) P 2 2 2 n1 n R where: Pt – conductivity of the solution after the time t, P - conductivity of the solution after the time . A typical dependence Pt/P on the process duration is shown in Fig. 1. 1,2 1 0,8 0,6 /P t P 0,4 0,2 0 0 10 20 30 40 50 60 t [min] Fig.1. Dependence of Pt/P on the process duration for diffusion of CdSO4 from alginate beads with a dry weight of 1.5%. The experimental results clearly indicate a decrease in the values of De, caused by an increase in the alginate content in the sorbent beads. This is in agreement with the mechanism of the diffusion process taking place in porous carriers. Good agreement between the experimental data and the mathematical model was obtained, as shown by the high values of correlation coefficients. The value of the effective diffusion coefficient is affected by the metal salt anion, therefore, it should also be taken into account in the calculations. All the values of De, obtained by the conductometric method, are lower than the calculated diffusion coefficients in highly dilute aqueous solution of the given this salt, Daq. More often than not, the condition is not satisfied in literature reports, especially in calculations by conventional methods (SCM, LAM). The conductometric method is simple and it provides good results in calculating the effective diffusion coefficients for heavy metals in alginate sorbents. References: [1] Meena AK, Kadirvelu K, Mishra GK, Rajagopal C, Nagar PN: Adsorption of Pb(II) and Cd(II) metal ions from aqueous solutions by mustard husk. J. Hazard. Mater. 150:619-625, 2008 [2] Lai Y.L., Annadurai G., Huang F.C., Lee J.F.: Biosorption of Zn(II) on the different Ca-alginate beads from aqueous solution. Bioresource Technology 99: 6480-6487, 2008 [3] Davis T.A., Volesky B., Mucci A.: A review of the biochemistry of heavy metal biosorption by brown algae. Water Res 37:4311– 4330, 2003 [4] Papageorgiou S.K., Katsaros F.K., Kouvelos E.P., Nolan J.W., Le Deit H., Kanellopoulos N.K.: Heavy metal sorption by calcium alginate beads from Laminaria Digitata. Journal Of Hazardous Materials B137:1765-1772, 2006 24 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Effect of microfracture on ultratight matrix permeability Grzegorz Leśniak Oil and Gas Institute – NRI Cracow [email protected] Keywords: anomalous permeability, shale rocks, microfracture width. Measurements of reservoir rock permeability have been executed in the oil industry for several decades. Depending on the type of permeability, oil and gas reservoirs can be divided into those with pore reservoirs, pore – fracture reservoirs and fracture reservoirs. In practice, pore and fracture permeability is observed in all unconventional reservoirs with varying proportions of either type. Natural microfracture systems increasing the permeability of the rock matrix are also recorded in shale rocks. In this paper anomalous results of permeability for rock samples from shale formations have been analysed. Observations with the use of SEM and petrographic microscope allow us to distinguish microfractures generated as a result of decompression of rocks (change of stress) and natural ones. The fractures generated as a result of core decompression are usually associated with very fine laminations with a material of different grain size composition comparing with rock matrix (smaller or larger grains), or with clay laminations within mudstones (author's microscopic observations). It has been concluded that the microfracture systems present in the examined rocks are the reason for anomalous values of permeability measured by the Pulse-Decay method. Dependences of overburden pressure on fracture permeability have been analysed. Simulative research performed for plug-type core samples allowed us to obtain permeability values in a function of the microfractures width. Finally, dependence of reservoir conditions on fracture width as well as on porosity was examined. The problem encountered during the selection of shale samples for permeability measurements is that the sample is cut in the form of a cylinder of 2.54 cm in diameter and 4 cm in length, which is often unfeasible. This is caused by typically shale cleavage and the presence of microfractures. Observations in the petrographic microscope allows us to divide the microfractures resulting from the expansion of the rock after pulling to the surface (change of stress) and the system of natural microfractures. The fracture resulting from the expansion of the core are associated with very fine laminations with different granulometric material from the rock matrix (smaller or larger grains) or clay laminates within the mudstones. The research procedure was as follows: • preparation of plug samples, • permeability measurement - Pulse Decay (PDP-250) , • 3D imaging - X-ray microtomography (CT) – resolution 10-12 m, • 2D X-ray imaging (RTG) • selection of samples in which we deal with the microfracture permeability (based on permeability measurements and 3D and 2D imaging), • calculation of width of microfracutures based on permeability measurements 48 samples of the Baltic basin shale cores were selected for the study. The range of obtained permeability researches cover the range from 3.8 mD to 0.4 nD (that is, six orders of magnitude). Samples with permeability above 1 mD were recognized as the samples with fractures formed during their preparation. For the rest of samples the base question is: are the obtained permeability values only connected with rock matrix or also with microfratures? Assuming that the value of 0.4 nD is the lowest permeability of the rock matrix, and taking into account permeability of the rock matrix (according to available publications) [1,2,3] can be up to 1.3 D, we should accept that all measurements of permeability values greater than 0.0013 mD (1.3 D) should be connected with the existence of the microfracture system present in the samples. The obtained results of the width of microfractures range from 1.035 to 15.201 μm. This width can be compared with the average pore size in classic sandstone reservoirs. For the four microfratures in the sample, these values range from 0.652 to 9.576 μm. In the case of the samples with permeability of less than 600 nD for one microfracture, the width is from 0.291 to 0.652 μm, and for 4 microfractures from 0.183 to 0.411 μm. It should be noted that the change in the number of microfractures in a sample does not alter the calculated of width of microfractures in the same way. Also, the differences of the width of microfractures – if we ignore the permeability values of the rock matrix - do not change drastically. The variation in the width of microfractures is approximately 40%. Studies have shown that the "anomalous" values of permeability in shale rocks correspond to microfracture systems. Therefore, it is important to put more emphasis on the correct determination of permeability of the rock matrix (without microfractures). The impact of microfracture parameters on the permeability of shale under reservoir conditions was analyzed. The width existing microfractures was calculated. The permeability value over which microfracture samples (matrix permeability) should be expected in the specimen samples of shale rock was estimated. ACKNOWLEDGEMENTS This article is the result of research conducted as part of the project Methodology of determining sweet spots on the basis of geochemical, petrophysical, and geomechanical properties, based on the correlation of the results of laboratory examinations with the geophysical measurements and the 3D generation model, co-funded by the National Centre for Research and Development as part of the BLUE GAS – POLISH SHALE GAS program. Contract no. BG1/MWSSSG/13 References: 1. Apaydin O., 2012. New coupling considerations between matrix and multiscale natural fractures in unconventional resource reservoirs. PhD thesis, Colorado School of Mines 2. Cho Y., Apaydin O.G., Ozkan E., 2013. Pressure-dependent natural fracture permeability in shale and its effect on shale-gas well production. SPE Reservoir Evaluation & Engineering (216-228) 3. Subrata Roy, Reni Raju, Chuang H.F., Cruden B.A., Meyyappan M. 2003: Modeling gas flow through microchannels and nanopores", Journ of Applied Physics, vol 93, no8 (4870 - 4879) 25 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Multi-scale core to pore imaging and modelling of the heterogonous rocks Grzegorz Leśniak1, Renata Cicha-Szot1, Krzysztof Labus2 Oil and Gas Institute – National Research Institute, 31-503 Kraków, Lubicz 25A St., Poland Silesian University of Technology, Faculty of Mining and Geology, 44-100 Gliwice, 2 Akademicka St., Poland [email protected] Keywords: fractal approach, computer analysis of images, pore space, net model, numerical modelling Geologic systems are heterogenous on all length scales [1]. Reservoir engineering captures such heterogeneity by classifying the formation into multiple lithological facies, constructing reservoir models and sampling these for further examination using core analysis. This process, however, only captures heterogeneity down to the scale of the 1 – 1.5” “core plug”, which is then upscaled back to the reservoir model. Rotliegend sandstone is one of the most perspective reservoir rock in Europe. It is considered as sediments which show a wide variety of geological characteristics, from homogenous rocks dominated by more or less uniform quartz cementation, to heterogeneous samples with complex distributions of authigenic clay minerals and quartz, carbonate and anhydrite cementation [2],[3]. Due to sedimentary and diagenesis history[3], this kind of rocks have complicated pore structures which impedes estimation of filtration parameters with the use of conventional analysis. Experimental quantification of filtration parameters requires regularly shaped samples. These fragments (plugs) are cut from cores extracted from wells. Moreover, coring is expensive in general and, arguably, impossible where new drilling technologies (e.g., coiled tubing) are employed. Consequently, the most important application of this study will be for estimating permeability on cuttings and irregularly shaped sidewall core samples. Recent developments in imaging technology solved that problem and can be used to bridge the gap between pore and reservoir- core scale descriptions. In this paper, we present the results of laboratory study where we compare permeability estimates obtained from several mercury injection capillary pressure based models, Klinkenberg corrected permeability in tight sands and prepared for analyzed samples net model of pore space obtained by complex validation of the obtained parameters of rocks [4] to calculations based on DRP. Moreover, the paper will show estimation of applicability limits for classical evaluation of reservoir rocks heterogeneity degree in comparison with microscope 3D images of pore space. Benchmarking of fluid flow parameters show that corrections of Mercury Injection Capillary Pressure (MICP) experiments are absolutely necessary, particularly conformance correction at low capillary pressures. On the other hand, provided in this paper Digital Rock Physics calculations and results of flow modelling shed light on the benefits of this techniques in order to explain discrepancies due to heterogeneity and tremendous influence of compaction and cementation which are the cause of very complex flow paths in tight gas or low permeability sandstones. Moreover, this paper presents a novel methodology for classifying, sampling and imaging a 1” core-plug such that essential heterogeneity is maintained throughout the workflow. This is applied to a subsurface sample from the Rotleigend sandstone, a Central European reservoir showing strong lamination on the mm-cm scale. First the entire core-plug sample was imaged with a resolution of around 19µm. While the pore structure was not resolvable, macroscopic bedding related heterogeneity was, along with a prominent fracture. This image was classified, creating a label image where each voxel was either labelled as a “high porosity”, “low porosity” or fracture. Locations for high resolution (1-3µm) non-destructive interior tomographies were defined using this macroscopic lithological map. As mechanical sample extraction was not required, sample sites could be much more accurately identified, and the association between litho-type and microscopic structure maintained. Permeability tensors were calculated for each interior image. A second model was then constructed using the macroscopically classified image, so each voxel from the high porosity and low porosity lithologies were populated with the associated average permeability tensor derived from the high resolution images. A simulation was then performed, computing stokes flow through the macroscopically visible fracture and Darcy flow through the microscopically sampled lithologies. The resulting permeability tensor was highly anisotropic, with a high permeability in directions parallel to layering and low permeabilities perpendicular to it. This result was only possible because the cm scale heterogeneity associated with primary sedimentary layering was maintained, through the macroscopic model. The research leading to these results has received funding from the Polish-Norwegian Research Programme operated by the National Centre for Research and Development under the Norwegian Financial Mechanism 2009-2014 in the frame of Project Contract No. Pol- Nor/196923/49/2013. In addition, we wish to thank the Carl Zeiss Microscopy and Math2Market who developed presented methodology with Oil and Gas Institute – National Research Institute. . References: [1]. Ringrose P.S., Martinius W. Alvestad, J.: Multiscale geological reservoir modelling in practice. Geol. Soc. London, Spec. Publ. 309, 123–134, 2008 [2]. Such P, Maliszewska A, Leśniak G.: Właściwości filtracyjnego utworów górnego czerwonego spągowca a jego wykształcenie facjalne, Prace IGNiG 104, ISSN 0209 0724,1999 [3]. Such P., Leśniak G., Słota M.: Quantitative porosity and permeability characterization of potential Rotliegend tight gas reservoirs, . Przegląd Geologiczny, Vol. 58, 4, p. 347 351, 2010 [4]. Leśniak G., Such P.: Fractal approach, Analysis of images and diagenesis in pore space evaluation. Natural Resources Research, 14, 4, p. 317 324 ISNN: 1520 7439, 2005 26 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Pore scale modelling of fluid transport using FIB-SEM images Grzegorz Leśniak, Karol Spunda, Renata Cicha-Szot Oil and Gas Institute – National Research Institute, 31-503 Kraków, Lubicz 25A Str., Poland [email protected] Keywords: pore scale, network modelling, FIB-SEM Pore scale imaging and modelling using focused ion beam scanning electron microscope (FIB-SEM) images develops into the most robust technology which allows to characterize porous rock not only from the mineralogical, geochemical or petrographic point of view but also quantify petrophysical parameters such as porosity (effective and total) and permeability. This paper presents the workflow developed for examination and estimation of basic parameters such us porosity and permeability of one of the most complicated porous system which is heterogeneous shale gas rocks using FIB-SEM images. Based on the high resolution maps (HRI) and SEM images which were take after ion beam polishing of sample surface spots for more detailed 3D reconstruction are chosen. Using ion beam mounted on the Helios NanoLab 450HP further analysis of sample by etching 20nm slices of rocks using Auto Slice &View automated serial sectioning and imaging through a defined volume of a specimen was performed. The sequence of images captured by Auto Slice &View was used as input for 3-dimensional reconstruction of the sliced volume. One of the crucial step is parametrization of obtained results which was performed in this work using Avizo 3D software (Table 1). Table 1. Example of results obtained from the pore space 3D reconstruction Median of pore Maximum Median of Maximum pore Sample No. No. of pores volume pore volume pore surface surface [nm3] [nm3] [m2] [nm3] A-1 715 5.25·105 1.56·109 4.64·104 4.49·107 B-1 2617 1.60·104 1.62·108 3.19·103 7.42·106 Obtained results allow us to perform evaluation based on the coupling of the Navier-Stokes and Darcy equations for modeling the porous media flows of selected region with interconnected pore structures (w kierunkach XYZ) Fig.1. Model of fluid flow in the X-XY axis. The lines on the left illustrate gas inflow in the pore space, the lines on the right of the sample gas outflow [1] Results obtained from simulation and modelling were compared to standard porosity and permeability measurements. The paper concludes by discussing limitations and challenges including finding representative samples, imaging and simulating flow and transport in pore space over several orders of magnitude in size, and interpretation obtained values to reservoir conditions. We conclude that pore scale modeling gives satisfactory results which might be applied to developed petrophysical models, however it needs to be used with caution due to sample size and different stress conditions adopt during FIB-SEM imaging. The research leading to these results were performed within the project: Methodology for sweet spots determination based on geochemical, petrophysical and geomechanical properties based on the correlation between laboratory investigations and geophysical measurements and a 3D generation model, co-funded by the National Centre for Research and Development as part of the programme BLUE GAS – POLISH SHALE GAS. Contract No. BG1/MWSSSG/13. References: [1] Leśniak G., Such P., Mroczkowska–Szerszeń M., Dudek L., Cicha-Szot R., Spunda K.: Metodyka analizy przestrzeni porowej skał łupkowych, Intytut nafty i Gazu – Państwowy Instytut Badawczy, 2017 ISSN 2353-2718 27 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Reliability of the tortuosity value obtained on the basis of other parameters of the porous bed Seweryn Lipiński1, Wojciech Sobieski2 1 Department of Electric and Power Engineering, Electronics and Automation 2 Department of Mechanics and Basics of Machine Construction Faculty of Technical Sciences, University of Warmia and Mazury in Olsztyn [email protected] Keywords: granular beds, tortuosity, porosity Tortuosity τ is one of the most important parameters describing porous media. It is defined as the ratio of the length of the actual path through pore channels to the thickness of the considered medium, along the chosen axis [1]. Tortuosity may be understood as a geometrical quantity or as a flow property. In the first case it is referenced as geometric tortuosity while in the second case, as hydraulic or diffusive tortuosity [2]. The paper considers the first understanding of the term. In general, geometric tortuosity can be obtained in many different ways, i.e. experimentally, numerically or indirectly - through relationship with other parameter(s) of the analysed porous bed [3]. The last approach is the most common one, as it is the simplest, but there appears a problem of reliability of the tortuosity value when obtained on the basis of other parameters of the porous bed. This particular issue is being considered in this paper. A set of virtual porous beds consisting of spherical particles was analysed. The set was created with the use of Discrete Element Method and it consists of 75 beds with common mean value of particles diameters (6.072 mm) and 25 values of standard deviation (each bed was created three times, hence 75 beds). Particles diameters were normally distributed. For each virtual porous bed, we calculated a set of parameters describing it, including i.a. porosity and tortuosity (using the so- called Path Tracking Method [4]). Then we obtained approximation functions linking these parameters with standard deviation of the particles diameters. Optimal approximation functions for tortuosity τ and porosity ø were as follows: 휏 = 1.208 + 0006∙ 푒1.54휎, (1) ø = 0.4124 + 0.0005σ - 0.0013 𝜎2. (2) Above equations were used for comparison of obtained values of tortuosity with values obtained based on relationships between tortuosity and porosity [3]. We used 8 functions linking porosity with tortuosity as well as Eq. 1. The achieved results, given as a function of standard deviation, are shown in Fig. 1. 1,7 1,6 1,5 ] - 1,4 1,3 1,2 Tortuosity Tortuosity [ 1,1 1 0,9 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 Standard deviation of particles diameter [mm] Eq. (1) Maxwell (1981) Weissberg (1963) Kim et al. (1987) Comiti & Renaud (1989) Iversen & Jorgensen (1993) Boudreau (1996) Lanfrey et al. (2010) Liu &Kitanidis (2013) Fig.1. Tortuosity vs. standard deviation of particles diameters obtained using various analytical formulas and using Eq. 1. As it can be seen, obtained curves differ significantly but their common feature is the increase of tortuosity value with the increase of standard deviation. This increase is the most noticeable for the Eq. 1 what suggests that literature relationships between tortuosity and porosity underestimate the influence of standard deviation on the parameters of porous beds. Both these observation lead to the conclusion that the use of literature functions linking porosity with tortuosity is not advisable approach. References: [1] Latour L.L., Kleinberg R.L., Mitra P.P., Sotak C.H: Pore-size distributions and tortuosity in heterogeneous porous media, Journal of Magnetic Resonance A 112(1), 83-91, 1995 [2] Clennell M. B.: Tortuosity: a guide through the maze. Geological Society, London, Special Publications 122(1), 299-344, 1997 [3] Sobieski W., Lipiński S.: The analysis of relations between porosity and tortuosity in granular beds, Technical Sciences 20(1): 75-85, 2017 [4] Sobieski W.: The use of Path Tracking Method for determining the tortuosity field in a porous bed. Granular Matter 18(3), 72, 2016. 28 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Relations between the various probability density functions describing the distribution of particles in a granular bed Seweryn Lipiński, Zenon Syroka Department of Electric and Power Engineering, Electronics and Automation, Faculty of Technical Sciences, University of Warmia and Mazury in Olsztyn [email protected] Keywords: particle size distribution, probability density functions, granular beds One of the most important topics in the field of granular beds research is the analysis of the particle size distribution (PSD) - regardless of whether it is the bed of natural or artificial origin [1]. It is very important in understanding its physical and chemical properties as it affects inter alia the strength and load-bearing properties of rocks and soils, as well as the reactivity of solids; that is why PSD needs to be strictly controlled in many industrial products [2-6]. There is many probability distribution functions, which are being used for the purpose of description of particle size distribution. The most commonly used are the log-normal (Galton) distribution, the Weibull distribution (also known as Rosin–Rammler distribution), the log-hyperbolic distribution and the skew log-Laplace distribution, but there are many other distributions being used for that purpose; many of them was developed strictly for the purpose of granular media description. Particular attention should be paid to the following types of distributions: Anderson, Fréchet, Fredlund, Gates–Gaudin–Schuhmann, Gompertz, Gumbel, Jaky, Johnson Sb, Lifshitz–Slyozov–Wagner, Nukiyama–Tanasawa and Skaggs [7-11]. Thera are also modifications of basic distributions that are being utilized for the purpose of granular beds description [8,12]. Such big number of various distributions being used to describe the granular beds raises a question about the relationships between them and that is the main subject of our research. An attempt was made to find mathematical relationships between most important probability distributions. A network of dependencies between the most commonly used distribution types has been built, including inter alia distributions of Gompertz, Gumbel, Fréchet and Weibull. Knowledge on this subject makes it possible to analyse and compare granular beds described with the use of different models. References: [1] Lipiński S.: Pozyskiwanie informacji o typie rozkładu złoża granularnego oraz generacja rozkładów wirtualnych. Granularne ośrodki porowate, University of Warmia and Mazury in Olsztyn Publishing House, 57-70, 2016 [2] Grace J.R., Sun, G.: Influence of particle size distribution on the performance of fluidized bed reactors, The Canadian Journal of Chemical Engineering 69(5), 1126-1134, 1991 [3] Meddah M.S., Zitouni S., Belâabes S.: Effect of content and particle size distribution of coarse aggregate on the compressive strength of concrete, Construction and Building Materials 24(4), 505-512, 2010 [4] Nan C. W., Clarke D.R.: The influence of particle size and particle fracture on the elastic/plastic deformation of metal matrix composites, Acta Materialia 44(9), 3801-3811, 1996 [5] Servais C., Jones R., Roberts I.: The influence of particle size distribution on the processing of food. Journal of Food Engineering 51(3), 201-208, 2002 [6] Verbeek C.J.R.: The influence of interfacial adhesion, particle size and size distribution on the predicted mechanical properties of particulate thermoplastic composites. Materials Letters 57(13-14), 1919-1924, 2003 [7] González-Tello P., Camacho F., Vicaria J.M., González P.A.: A modified Nukiyama–Tanasawa distribution function and a Rosin–Rammler model for the particle-size-distribution analysis, Powder Technology 186(3), 278-281, 2008 [8] Hwang S.I., Lee K.P., Lee D.S., Powers S. E.: Models for estimating soil particle-size distributions, Soil Science Society of America Journal 66(4), 1143-1150, 2002 [9] Macıas-Garcıa A., Cuerda-Correa E.M., Dıaz-Dıez M.A.: Application of the Rosin–Rammler and Gates–Gaudin–Schuhmann models to the particle size distribution analysis of agglomerated cork, Materials Characterization 52(2), 159-164, 2004 [10] Nabizadeh E., Harchegani H.B.: Performance of Eight Mathematical Models in Describing Particle Size Distribution of Some Soils from Charmahal-va-Bakhtiari Province, Journal of Water and Soil Science 15(57), 63-75, 2011 [11] Taşdemir A., Taşdemir T.: A comparative study on PSD models for chromite ores comminuted by different devices, Particle and Particle Systems Characterization 26(1‐2), 69-79, 2009 [12] Hwang S.I.: Effect of texture on the performance of soil particle-size distribution models. Geoderma 123(3), 363-371, 2004 29 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Effect of friction coefficient on results of particles velocity calculation using Euler-Lagrange model in spout-fluid bed apparatus Wojciech Ludwig1 1Wroclaw University of Technology, Faculty of Chemistry, Department of Chemical Engineering [email protected] Keywords: friction coefficient, tangential restitution coefficient, particles velocity The presented paper is a continuation of research concerning gas-solid flow modelling using the Euler-Lagrange approach in a spout-fluid bed apparatus with a circulating dilute bed [1]. The greatest problem in this case was to determine the friction coefficient for particles hitting against the walls of the apparatus. On the basis of the properties of similar materials the value of this quantity was estimated at 0.2. Therefore, it proved useful to check the model’s sensitivity to the value of this parameter. The study investigated the effect of friction coefficient in the range of 0.1-0.4 on calculated values of particles velocity in the draft tube and the annular zone of the device for various masses of the circulating bed (from 100 to 390 g). The flow of the dispersed phase elements (microcrystalline cellulose particles Cellets® 1000) was modelled by solving the motion equation individually for each particle. Due to the low value of the solid volume fraction, interactions between particles were neglected and the main focus was on the description of particle collisions with the walls of the device. The hard-sphere method was applied, assuming that all particle-wall interactions are binary and immediate (the contact time is infinitely small) and contact forces are impulsive. Particles have a spherical shape and their shape is preserved after collision. During the point contact between particles and the wall the particles undergo normal and tangential deformations resulting from the elastic forces. The tangential and normal components of the velocity after collision are calculated as the product of their pre-impact values multiplied by the corresponding restitution coefficients. Particle impact collisions with the wall were divided into elastic and elastoplastic ones, which were diagnosed on the basis of the yield velocity Vy depending only on the physical properties of the material (microcrystalline cellulose) and the wall (glass and aluminium) such as Young’s modulus, Poisson’s ratio, plasticizing pressure. The normal restitution coefficient en, depending only on the normal component of the incident velocity Vni, was based on the Thorton model [2, 3]: 0.5 0.73 푉3/5 (1 − 푛푖 ) , 푉 < 푉 100 푛푖 푦 1 푒푛 = 1 −1 (1) 2 2 4 6√3 1 푉푦 2 푉푦 푉푦 푉푦 {( ) [1 − ( ) ]} {( ) [( ) + 2√1.2 − 0.2 ( ) ] } , 푉푛푖 > 푉푦 { 5 6 푉푛푖 푉푛푖 푉푛푖 푉푛푖 In order to calculate the tangential restitution coefficient, a kinematic model of Wu et al. [4] was used. It makes the value of the tangential coefficient of restitution et dependent on the angle of incidence, normal restitution coefficient and friction coefficient for the particle colliding with the wall [4]: 2 1 − [푐 + 푐 푡푎푛ℎ(푐 + 푐 훩)], 훩 < 훩 훩 1 2 3 4 푐푟푖푡 푒푡 = { 2 (2) 1 − , 훩 ≥ 훩 훩 푐푟푖푡 2 훩 = 푡푎푛(90° − 휃) (3) (1+푒푛)휇 where: c1,c2,c3,c4 - equation constants, 훩푐푟푖푡 - dimensionless, critical angle of incidence, 훩 - dimensionless angle of incidence, 휃 - angle of incidence, en - normal coefficient of restitution, µ - friction coefficient. In the course of calculations, a relatively small influence of friction coefficient on particles velocity was observed in the tested zones of the apparatus. Its increase by 300 % from 0.1 to 0.4 caused an increase in particles velocity in the draft tube by maximum 16 % (on average 14 %), while its decrease in the annular zone by maximum 25 % (on average 18 %). The changes were most visible for large masses of the bed, which was connected with an increase in the number of collisions of particles with the walls. The different dependency of particle velocity on the friction coefficient in the draft tube and the annular zone results from the properties of equation (2). According to it, in the area of low test angles (below approximately 700, depending on the normal restitution coefficient), an increase in the value of the coefficient of friction causes a decrease in the tangential restitution coefficient. For high incidence angles this relationship is reversed. A relatively small number of collisions with a high incidence angle occur in the draft tube, especially in the bottom part of the apparatus, where the bed is intensively circulating. In the annular zone, on the other hand, the collisions are more frequent and the particles collide with the walls at low angles. On the basis of the conducted tests, it can be concluded that a possible error in estimating the value of the friction coefficient for particles colliding with the walls of the apparatus has little influence on the accuracy of calculation of the dispersed phase velocity by means of the applied model. Acknowledgments The studies were funded by the Polish National Science Centre within the framework of the research grant UMO- 2013/09/B/ST8/00157. References: [1] Ludwig W., Płuszka P.: Euler-Lagrange model of particles circulation in a spout-fluid bed apparatus for dry coating, Powder Technol. 328: 375-388, 2018. [2] Wu C.Y., Li L.Y., Thornton C.: Energy dissipation during normal impact of elastic and elastic-plastic spheres, Int. J. Impact. Eng. 32: 593-604, 2005. [3] Thornton C., Coefficient of restitution for collinear collisions of elastic-perfectly plastic spheres, J. Appl. Mech. 64: 383-386, 1997. [4] Wu C.Y., Thornton C., Li L.Y.: A semi-analytical model for oblique impacts of elastoplastic spheres, Proc. R. Soc. A. 465: 937–960, 2009. 30 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Analysis of Influence of Model Parameters on the Capillary Imbibition of Porous Materials with Liquid Janusz Łukowski1, Mieczysław Cieszko2 Institute of Mechanics and Applied Computer Science Kazimierz Wielki University, Bydgoszcz [email protected], [email protected] Keywords: unsaturated porous materials; macroscopic process of fluid imbibition; numerical analysis Numerical analysis of influence of model parameters on the capillary imbibition of porous materials with liquid is performed. A new macroscopic description of fluid imbibition is proposed, based on the continuum model of the capillary transport of liquid and gas in unsaturated porous materials of homogeneous and isotropic pore space structure, proposed in the paper [1]. The key assumptions of this model are: division of liquid in the pore space into two macroscopic constituents called mobile liquid and capillary liquid; description of menisci motion by an additional macroscopic velocity field; parametrization of saturation changes by a macroscopic pressure-like quantity that for quasi static and stationary processes is equal to the capillary pressure. This model significantly changes the image of processes of liquid and gas capillary transport in an unsaturated porous material. It shows that these processes are running in a five dimensional pressure-time-space continuum, and quasi static processes of intrusion and extrusion of non-wetting liquid are non-stationary processes running in pressure-space continuum. A one dimensional system is considered, composed of two half spaces (Fig.1). The half space z > 0 is occupied by undeformable porous material filled with gas (air), whereas the half space z < 0 is occupied by incompressible wetting fluid (water). We analyze a nonstationary process of liquid imbibition of the porous material caused by the capillary forces and impeded by the gravity and viscous interactions of liquid with the skeleton. It is assumed that gas is inert and its pressure is constant. The inertial forces in the mobile liquid are omitted and at the beginning of the process liquid is present at the contact surface z = 0. Fig.1. Scheme of the analyzed system. The system of equations describing nonstationary imbibition process is composed of: balance equations for mass and for linear momentum of the mobile liquid; evolution equation for saturation; constitutive equations for the stress tensor and viscous interaction force of the mobile liquid with the skeleton, for velocity of diffusive transport of menisci, and for relation between saturations with the mobile and capillary liquids. For the limit case of static distribution of the mobile liquid attained at the end of the imbibition process the system of equations takes the form 2 s ds s s m c m m 1 Cm (sm, p) mg Cs (sm, p) , (1) p dsm dz p dp g (2) dz m where sm , sc , m , g, p are saturations with the mobile and capillary liquid, mass density, gravity acceleration and pressure, respectively. Quantities Cm(sm, p), Cs (sm, p) have the form 1 푝 푚 푝 −푛 퐶푚(푠푚, 푝) = ( ) , 퐶푠(푠푚, 푝) = 푠푚 ( ) (3) 푠푚 푝표 푝표 and are coefficients characterizing menisci transport in the pore space. They are directly related with the pore space structure of the porous material. Equation (1) is strongly nonlinear differential equation of the first order and is solved using the method of characteristics. Analysis of influence of model parameters on the imbibition process has been performed numerically. References: [1] Cieszko M.: Macroscopic Description of Capillary Transport of Liquid and Gas in Unsaturated Porous Materials, Meccanica 22: 1-22, 2016 31 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Possibilities of using glass microspheres to build models and simulation of fluid flow through geological strata Marcin Majkrzak Department of Petroleum Engineering, Oil and Gas Institute – National Research Institute, Cracow [email protected] Keywords: glass microspheres, glass bed packs, fluid flow simulation This paper presents the results of evaluation potential possibilities of using glass microspheres to build glass bed packs and simulation of fluid flow through created porous models. Although, using original core samples enable initiate real petrophysical parameters of the porous media, it has also some limitation. One of the most important limit is the fact that core samples are very valuable, therefore it is impossible to carry out this type of research on a large scale. Moreover, in many cases they can be used just one time (destructive measurements). The alternative is models of porous media made of sands or glass microspheres with known granulation. Projects that were performed by many researches [1], [2], [3] indicate satisfactory results of using homogenous glass beds models which opens up a new perspective area for flow simulation of multiphase systems. Fig.1.Measurement equipment (TEMCO C.O) with model of glass bed pack. For the need of the task several models (glass microspheres with known granulation) with various structures and layers layout were built and characterized in terms of their basic petrophysical and filtration properties – pore volume, porosity, absolute permeability for gas and brine, relative permeability for oil. The final part of work was to define recovery factor (the recoverable amount of hydrocarbon initially in place) for each of model. Additional, comparison of permeability results obtained from measurements performed on glass bed pack to values for theoretical model of parallel and vertical layers were carried out. References: [1] Mai A., Kantzas A.: Heavy Oil Water Flooding: Effects of Flow Rate and Oil Viscosity. Journal of Canadian Petroleum Technology, 48, 42-51, 2009 [2] Mai A., Kantzas A.: Improved Heavy Oil Recovery by Low Rate Waterflooding. SPE International Thermal Operations and Heavy Oil Symposium, Calgary, 2008 [3] Metin C. O., Bonnecaze R. T., Nguyen Q. P.: The Viscosity of Silica Nanoparticle Dispersions in Permeable Media. SPE International Oilfield Nanotechnology Conference, Noordwijk, 2012 32 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Application of 3D printing technology to random porous structures study Ewelina Małek, Danuta Miedzińska, Wiesław Szymczyk, Arkadiusz Popławski Military University of Technology, Faculty of Mechanical Engineering, Urbanowicza 2 St, 00-908 Warsaw, Poland [email protected] Keywords: Rapid Prototyping, SLA (stereolitography), Fractal Models In the field of numerical research there are various approaches and methods for modeling structures of porous materials. Therefore various material models are used for numerical analyzes of materials such as, among others, ceramics, metallic foams, ceramic or metal composites. In the case of modeling of natural porous structures, for example rocks, their complex structure prevents the identification of properties and finding of material data. As a consequence, there is a problem with modeling the structure of these materials. The solution is the use of fractal models to modeling of porous structure [1,2]. An example of geometry of fractal model, built on the basis of a spatial raster is Menger's sponge (Fig.1). On each of the walls of the sponge, the Sierpinski carpet pattern is visible [3]. Fig.1. Fractal models - Sierpinski carpet (2D structure) and Menger's sponge (3D structure) [3] In the case of modeling the geometry of natural (random) materials, there is a problem of compatibility of the FEM geometry and real geometry. This is a source of differences between the results of calculations and experimental results. Application of 3D printing technology will allow to receive a real structure in a controlled manner, which exactly reflects the designed structure and is consistent with the geometry of the numerical model. The 3D printing technique used is the stereolithography method, which consists in selectively fixing the layer-applied resin with UV laser rays (Fig.2) [4]. Fig.2. Schematic of the upside-down SLA system [4] An experimental research on the standard samples made of photopolymer resin using 3D printing technique is presented in the paper. The aim of the research was to determine the base material properties and, consequently, to select the constitutive model necessary to carry out numerical analyses. Variants of own geometry of models simulating the structure of porous materials based on the basic fractal model were also presented. Acknowledgements: The paper supported by a grant No BG2/DIOX4SHELL/14 financed in the years 2014-2018 by The National Centre for Research and Development, Poland. References: [1] Banhart J., Manufacture, characterization and application of cellular metals and metal foams. Progress in Materials Science 46, 559–632, 2001. [2] Mills N. J., The high strain mechanical response of the wet Kelvin model for open-cell foams. International Journal of Solids and Structure, 44, 1, 51-65, 2007. [3] Menger K. Classics on fractals. Studies in Nonlinearity, Westview Press, 2004. [4] The Ultimate Guide to Stereolithography (SLA) 3D Printing, Marzec 2017, formlabs.com 33 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Effect of polypropylene fiber addition on mechanical properties of concrete based on portland cement Marcin Małek1, Wojciech Życiński1, Mateusz Jackowski1, Marcin Wachowski2 and Waldemar Łasica1 1Military University of Technology, Faculty of Civil Engineering and Geodesy, Warsaw, Poland 2Military University of Technology, Faculty of Mechanical Engineering, Warsaw, Poland [email protected] Keywords: polypropylene fiber, concrete, mechanical properties Negative features of traditional concrete can be eliminated by using dispersed fibers in a concrete mixtures. These fibers, after mixing process, form composite materials called fibre concrete, whose tensile strength and resistance to fatigue is higher than in traditional concrete materials [1]. On the one hand in fiber concrete material, concrete provides high compressive strength, stiffness and provides fiber protection against corrosion but on the other hand, the fibers provide tensile strength and reduce shrinkage causes cracks in the concrete. Both elements are complement each other [2]. Due to the application and properties fibres can be divided into three groups of materials: polymer fibers, steel fibers and carbon fibers [3]. The smallest and lightest of the whole group are polypropylene fibers. Most often, they are added in an amount of about 0.6 kg / m3 of concrete mixtures. Their dispersion strengthens the structure of concrete in all directions. Their main task is to reduce the contraction of concrete, especially during its first binding phase. The shrinkage cracks are small, irregular and appear most often within 24 hours of laying the concrete mixtures. These cracks are caused by plastic shrinkage or by the drying process of concrete [4]. In this work results of polypropylene fibers addition into concrete mixture based on portland cement was characterized. The main purpose of this research was to identify directly influence fibers addition on concrete mechanical strength. The recipe of concrete was prepared using three types of aggregates: 0.125 – 0.250; 0.250 – 0.500 and 0.500 – 1.000 mm. To identify structures of researched samples and fibres SEM and LM observation were widely done. Basic properties of concrete mixture were defined by: chemical composition, sieve curve, slump cone test and setting time. Mechanical properties such as compressive strength and bending test after 1, 7, 14 and 28 days were characterized. Obtained results was compare with mixtures without fibers modifications. Study was proven that all chosen modifiers revealed increase effect on final mechanical properties and are very perspective for future application in concrete technology. References: [1 Shankar, G.R.V., Sundarraja, M.C., ; Kim, Y.Y., ; Prabhu, GG.: Using carbon-fibre-reinforced polymer to strengthen concrete- filled steel tubular columns, Proceedings of the Institution of Civil Engineers-Structures and Buildings, Volume: 170, Issue: 12, Pages: 917-927, 2017 [2] Wattick, J.A., ; Chen, A.: Development of a prototype fiber Reinforced Polymer - Concrete Filled wall panel, Engineering Structures, Volume: 147, Pages: 297-308, 2017 [3] Mansour, R., El Abidine, R.Z., Brahim, B.: Performance of polymer concrete incorporating waste marble and alfa fibers, Advances in Concrete Construction, Volume: 5, Issue: 4, Pages: 331-343, 2017 [4] Li, W., Huang, Z., Hu, G., Duan, W. H., Shah, S.P.: Early-age shrinkage development of ultra-high-performance concrete under heat curing treatment, Construction and Building Materials, Volume 131, Pages 767-774, 2017. 34 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Numerical generation of a random packed bed of saddles Maciej Marek Faculty of Mechanical Engineering and Computer Science, Częstochowa University of Technology, Częstochowa, Poland [email protected] Keywords: random packed bed, porosity, saddles, DEM Random packed beds have a large variety of applications. They can be used e.g. in chemical engineering systems as a means for increasing the contact surface between reactants delivered as fluids either in counter- or co-current fashion. Numerical simulations of flows in such beds performed at the scale of individual particles require the details of the bed geometry to be represented in the model. The geometry can be obtained with the use of complex and expensive experimental techniques (e.g. computer tomography) or in numerical simulations – much cheaper and promising alternative. The present work is devoted to numerical generation of a geometry of a random packed of saddles being a simplified variant of Intalox saddles, quite popular elements in packed columns. The concept of the algorithm is an extension of the method presented in [1] for cylinders and in [2,3] for rings packed in a cylindrical container. Generation of the bed is treated as the simulation of the filling process with simplified mechanics. The particles are added to the structure in a sequential manner, they move due to gravity and reaction forces from other elements and the container’s wall until they reach the state of mechanical equilibrium. The reaction force resulting from the contact between the active particle and the bed depends on the overlap between the particles. In order to facilitate the calculation of the overlap, the active particle is covered with a uniform grid of markers. Packing density is increased by additional artificial force acting in the direction from the container’s axis to its walls. Fig.1. Subsequent steps in generation of the random packed bed. The final outcome of the work, which will be shown during the presentation, includes: sample geometries of beds of saddles (Fig. 1), characteristics of such beds (global porosity, distribution of local porosity, orientation of particles) and their dependence on the particular dimensions of individual particles. Acknowledgements This study was performed within the framework of the contract: UMO-2014/15/B/ST8/04762 funded by National Science Centre. References: [1] Marek M.: Numerical generation of a fixed bed structure, Chemical and Process Engineering 34(3): 347-359, 2013 [2] Marek M.: Numerical simulation of a gas flow in a real geometry of random packed of Raschig rings, Chemical Engineering Science 161: 382-393, 2017 [3] Niegodajew P., Marek M.: Analysis of orientation distribution in numerically generated random packings of Raschig rings in a cylindrical container, Powder Technology 297:193-201 , 2016 35 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Sludge dewatering by thin-film freezing in the north-east of Poland Kinga Michalak, Janusz A. Szpaczyński Wrocław University of Science and Technology, Faculty of Mechanical and Power Engineering, [email protected] Keywords: sludge dewatering, freeze/thaw. Sludge dewatering and disposal is still one of the most pressing research needs in recent years. Scientists and technologists are constantly looking for new, more effective methods of sludge dewatering. In northern climate particular interest is currently placed on the use of natural freezing. The freezing process uses the fact that the crystallographic structure of the ice makes it difficult to absorb contaminants in the form of a solid or dissolved phase. The solid particles are pushed by the ice crystals that incorporate only pure water. Frozen water in the form of ice creates frozen channels in sludge. After melting, free spaces are created, which are used by melt water to flow out. Freeze/thaw process can replace the use of chemical conditioners, improve efficiency of dewatering, filterability [1] and can significantly reduce O&M costs. In a cold climate, the freezing process can be carried out without significant energy inputs using negative ambient temperatures in the winter. In the paper, several sets of samples of clay sludge were frozen and after that lyophilized to reveal the pore structure (Fig. 1, 2.). Tests with melting and drying of samples showed deformation of pores. Only lyophilisation of samples allowed examining the structure of sludge after freezing. Fig. 1. Sludge before freezing Fig. 2. Sludge after freezing and lyophilisation Although most of application of freeze/thaw for sludge dewatering are reported from Canada and USA it is possible to use this method in Poland too. Based on the weather data from the last ten years, obtained from the Institute of Meteorology and Water Management for the Suwałki region, a simulation of thin-layer freezing in a settling tank was carried out. Sludge was loaded by 8 cm high layers. To simulate the time of freezing a model developed by Martel [2] was used. The results are shown in Fig. 3. Fig. 3. Calculated depth of frozen sludge for Suwałki region. The coldest winter was in 2009 and 2010. It was possible to freeze almost 3 m of sludge. It should be noted that freezing can be used interchangeably with mechanical dewatering. On cold nights the sludge can be frozen. In the case of positive temperatures for an extended period of time the filtration press or solid bowl centrifuge can be used for sludge dewatering. References: [1] Szpaczynski J.: Poprawa własności filtracyjnych i sedymentacyjnych zawiesin poprzez naturalny proces zamrażania, Przemysł Chemiczny, 96, 9, 2017 [2] Martel C. J..,: Development and design of sludge freezing beds, CRREL Report 88-20, 1988 36 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Numerical modeling of porous ceramics microstructure Danuta Miedzińska Military University of Technology, Urbanowicza 2 St, 00-908 Warsaw [email protected] Keywords: porous ceramics, finite element method, microstructure Porous ceramics is a group of new and very interesting materials. It can be used for thermal insulation, filters, bio-scaffolds for tissue engineering, and preforms for composite fabrication [1]. One of the most interesting applications is usage of those materials for production of proppants for hydraulic fracturing of shale rocks. Porous structure of a ceramic can be prepared through many processing techniques. One technique is to simply sinter coarse powders or partially sinter a green ceramic to hinder full densification [1]. Other traditional methods of fabricating porous ceramics can be divided into three basic processing techniques: replica; sacrificial template; and direct foaming as seen in Fig. 1 [2]. The development process influences the microstructure of the material, what is presented in Fig. 2. Fig.1. Typical processing methods for the production of macroporous ceramics: (a) replica technique; (b) sacrificial template technique and (c) direct foaming technique [2]. a) b) Fig.2. Porous ceramics microstructure: a) grain structure made by sintering [3], structure made by replication [4]. In the paper the main interest is directed to the grain porous ceramics. In the presented research the numerical modelling of idealized microstructure of such structure was presented to study the influence of grains distribution on the porous ceramics mechanical behaviour. Some examples of the developed model are shown in Fig. 3. 37 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Fig.3. Geometry of numerical models of grain porous ceramics idealized structure. The analyses were conducted with the use of finite element method. Acknowledgements: The paper supported by a grant No BG2/DIOX4SHELL/14 financed in the years 2014-2018 by The National Centre for Research and Development, Poland. References: [1] Hammel E.C., Ighodaro O.L.-R., Okoli O.I..: Processing and properties of advanced porous ceramics: An application based review, Ceramics International, 40(10): 15351-15370, 2014 [2] Studart A.R., Gonzenbach U.T., Tervoort E., Gauckler L.J.: Processing routes to macroporous ceramics: a review, Journal of the American Ceramic Society 89: 1771-1789, 2006 [3] http://www.tech-ceramics.co.uk (access 25.02.2018) [4] Walsh D., Boanini E., Tanaka J., Mann S.: Synthesis of tri-calcium phosphate sponges by interfacial deposition and thermal transformation of self-supporting calcium phosphate films, Journal of Materials Chemistry 15: 1043-1048, 2005 38 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Application of silica gel in the process of trypsin immobilization Justyna Miłek, Sylwia Kwiatkowska-Marks, Ilona Trawczyńska Faculty of Chemical Technology and Engineering, University of Technology and Life Sciences, Seminaryjna 3, 85-326 Bydgoszcz, Poland e-mail:[email protected] Keywords: trypsin, immobilization, silica gel Trypsin (EC 3.4.21.4) is hydrolysis of proteins. It causes certain peptide bonds to break, decreasing the number of allergenic proteins in hypoallergenic food production [1]. Enzyme immobilization on carriers may significantly contribute to cost reduction and improvement of the process, due to the reusability of the same portion [2]. Process immobilization using involves several stages. Firstly, activation of the carrier is conducted by attaching a reactive group, then the enzyme attaches [3]. Cross-linking agents can be glutaraldehyde and 3- aminopropyltriethoxysilane [4]. SHEN et al. [4] presented that surface modification of silica nanoparticle by aminopropyl groups (3-aminopropyltriethoxysilane) causes an increase in adsorption ability of bovine serum albumin (BSA), in comparison to unmodified silica nanoparticles. YANG et al. [3] proved that activity of immobilized lipase on aminosilica gel activated by glutaraldehyde is greater compared to immobilized lipase without being activated by glutaraldehyde. Figure 1 illustrates the process of trypsin immobilization on silica gel. OH Si OH Si Si OC2H5 O C2H5-Si-CH2-CH2-CH2NH2 O O O O OC2H5 Si Si OH Si O Si NH2 50° C (15-120 min) O 25° C (15-120 min) O O Si Si OH Si OH (1) Si OH Si OH O trypsin O O H O trypsin Si O Si N NH O Si O Si N O O O O Si OH Si OH (2) (3) Fig.1. The process scheme of trypsin immobilization with the use of 3-APTES and glutaraldehyde The aim of the study was to produce immobilized trypsin from bovine pancreas on modified silica gel in optimal conditions. Modification of silica gel by 3-aminopropyltriethoxysilane (3-APTES) and glutaraldehyde in immobilization process was applied for the first time. Activity of native and immobilized enzyme was measured using Kunitz’s method [5, 6]. Effects of temperature within the range of 35-75℃ and pH between 3 and 9 on activity of native and immobilized trypsin on modified silica gel were determined. Reusability of the biocatalyst was also tested. Optimal conditions for production of immobilized trypsin on modified silica gel are as follows: activation time of 10% 3- aminopropyltriethoxysilane solution, 2.5% glutaraldehyde solution and trypsin solution is 120 minutes for each of the solutions. Optimal temperature and pH for immobilized trypsin is 55℃ and 7.6 respectively. Immobilized trypsin on modified silica gel stored for 14 days at 4℃ exhibits 75% of its activity. After being reused for eight cycles immobilized trypsin on modified silica gel exhibits 60% of its activity. References: [1] Mohamad N. R., Marzuki, N. H. C., Buang, N. A., Huyop, F., Waha, R. A. Review. Agriculture and environmental biotechnology, Biotechnology and Biotechnological Equipment 29 (2): 205-220, 2015 [2] Acton, Q. A. PhD (ed.) Endopeptidases: Advances in Research and Application: 2011 (pp. 159-160). A Scholarly Editions eBook, Atlanta, Georgia, USA, 2012 [3] Yang G., Wu J., Xu G., Yang L. Comparative study of properties of immobilized lipase onto glutaraldehyde-activated amino-silica gel via different methods, Colloids and Surfaces B: Biointerfaces 78: 351–356, 2010 [4] Shen S.-C., Ng W.K., Chia L., Dong Y.-C., Tan R.B.H. Sonochemical synthesis of (3-aminopropyl)triethoxysilane-modified monodispersed silica nanoparticles for protein immobilization. Materials Research Bulletin 46:1665–1669, 2011 [5] Salara S., Mehrnejada F., Sajedib R. H., Arough J. M. Chitosan nanoparticles-trypsin interactions: Bio-physicochemical andmolecular dynamics simulation studies, International Journal of Biological Macromolecules 103: 902–909, 2017 [6] Zhou C., Wu X., Jiang B., Shen S. Immobilization strategy of accessible transmission for trypsin to catalyze synthesis of dipeptide in mesoporous suport, Korean Journal of Chemical Engineering 28(12): 2300-2305, 2011 39 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Numerical modeling of cavitation phenomenon in a small-sized converging-diverging nozzle Agnieszka Niedźwiedzka Department of Mechanics and Basics of Machine Construction, Faculty of Technical Sciences, University of Warmia and Mazury in Poland [email protected] Keywords: cavitation, converging-diverging nozzle, homogeneous approach Cavitation is a phenomenon of liquid evaporation in the areas, where the static fluid pressure drops below the saturated vapour pressure at the given temperature. Until today appeared many methods of numerical modeling of this phenomenon. These methods include a one-fluid method with the homogeneous approach. In this approach, apart from the balance equations of mass, momentum and energy, an additional transport equation is solved. Niedźwiedzka et al. [1] summarized and compared the most important homogeneous cavitation models. Although cavitation is an undesirable phenomenon, because it causes noise and erosion, there are a small number of devices, where cavitation is required. These group of devices include small-sized converging-diverging nozzles. The main role of converging-diverging nozzles is the control of mass flow rate. Small sized converging-diverging nozzles deliver constantly a very small liquid flow rate and for that reason, they find application in e.g. lab scale monopropellants or hybrid rocket motors. The main aim of numerical simulations is to predict the area of cavitation and based on these data, to modify the construction of the chosen devices to minimalize the negative consequences of this phenomenon. In the case of the small-sized converging- diverging nozzles, the high-pressure cavitating flow is observed. Numerical modeling of cavitating flow is very difficult, because the phenomenon is until today not enough physically explained and mathematically described. The factor, which additionally makes the simulations more difficult, are the high-pressure conditions. The main aim of the presented research is to show results of numerical calculations of cavitating flow in an example of small- sized converging-diverging nozzle using the homogeneous cavitation model – the Schnerr and Sauer model. The motivation to the research is lack of any material about simulations of the high-speed cavitating flow in the small-sized converging-diverging nozzles with the small throat ratio in the literature. Some experimental measurements considering these devices have been already published [2]. The source terms of the Schnerr and Sauer model for the evaporation and condensation are formulated respectively as follows: (1) + 𝜌푣𝜌푙 3 2 (푝 − 푝푠푎푡) 푚̇ = 훼푣(1 − 훼푣) √ 𝜌푚 푅 3 𝜌푙 (2) + 𝜌푣𝜌푙 3 2 (푝 − 푝푠푎푡) 푚̇ = − 훼푣(1 − 훼푣) √ 𝜌푚 푅 3 𝜌푙 The transient simulations were performed using Fluent software. The time step was 0.00003 s. The boundary conditions are defined as follow: velocity inlet – 0.5 m/s, pressure outlet – 101325 Pa, turbulence model – k-ε model. The achieved contours of vapour volume fraction are presented in Fig. 1. Fig.1. The contours of vapour volume fraction in the analysed converging-diverging nozzle. The main aim of the research was achieved. In the numerical calculations using Fluent software were obtained the contours of the vapour volume fraction. It was established that the biggest intensity of the cavitation phenomenon was observed in the throat. In the downstream region appeared the cavitation cloud. The area and the intensity of the cavitation cloud changes in the time. Also, the periodicity of this phenomenon is visible. To have a comprehensive picture of the high-pressure cavitating flow in the analysed small-sized converging-diverging nozzle with a small throat ratio is necessary to conduct more simulations considering other homogeneous cavitation models. References: [1] Niedźwiedzka A., Schnerr G. H., Sobieski W. Review of numerical models of cavitating flows with the use of the homogeneous approach. Archives of thermodynamics, 37 (2): 71-88, 2016. [2] Niedźwiedzka A. Sobieski W. Analytical analysis of cavitating flow in Venturi tube on the basis of experimental data, Technical Sciences, 19(3): 215-229, 2016. [3] Schnerr G. H., Sauer J. Physical and numerical modeling of unsteady cavitation dynamics. In: Proceedings of the Fourth International Conference on Multiphase Flow (ICMF’01), New Orleans, USA, 2001. [4] Singhal A. K., Athavale M. M., Li H., Jiang Y. Mathematical basis and validation of the full cavitation model. Journal of Fluids Engineering. 124: 617–624, 2002. [5] Zwart P. J., Gerber A. G., Belamri T. A two-phase flow model for prediction cavitation dynamics. In: Proceedings of the Fifth International Conference on Multiphase Flow (ICMF 2004), Yokohama, Japan, 2004. 40 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 The influence of comminution degree on structural properties of rocks of various porous structure Anna Pajdak The Strata Mechanics Research Institute of the Polish Academy of Sciences [email protected] Keywords: rock porosity, coal structure, dolomite structure The knowledge of structural parameters of rocks such as: surface area, the size and the distribution of pores, is important from the point of view of the extraction, enrichment and technological use of rocks. This paper presents the results of structural analyses of the two types of rocks that differ considerably in terms of their porous structure: coal and dolomite of different grain fractions. Coal has a developed system of pores, of which 90 % are micropores and submicropores [1], [2], [3]. Submicropores are flexible in relation to sorbed particles of gas and the diffusion into them requires a certain activation energy. Their share in the deposition of sorbent particles is the biggest. The porous structure of dolomite is dominated by mesopores and macropores (according to UIPAC classification [4]). Those pores are of various sizes and shapes. They can function as transporters of gas in the rock strata or they can be spaces that have no contact with the outside [5], [6]. The author analyzed three coal samples from different mines and three samples of dolomite from O/ZG Polkowice-Sieroszowice copper mine. Each sample was crushed to a few grain fractions and was subjected to surface structural analysis using volumetric low-pressure gas adsorption method. For each grain fraction the author determined the sorption capacity in relation to nitrogen (77K) and carbon dioxide (273K) in the absolute pressure range from 0 bar to 1 bar. On the basis of the adsorption measurements and using the Langmuir, BET and BJH (Barrett, Joyner and Halenda) models as well as classical pore distribution models, the author described the porous texture of those materials. The author determined the surface area, mean pore size and mean pore volume, which varied depending of the type of the sample and its grain size. The analysis revealed structural diversity of the analyzed rocks, which was determined by the type of porosity and degree of comminution. On the basis of the degree of the diversity, the author correlated the structural parameters and the grain size. References: [1] Kreiner K., Żyła M.: Binary character of surface of coal, Górnictwo i Geoinżynieria 30 2, 2006 [2] Pajdak A., Skoczylas N.: Comparison of surface area and pore size distribution of coal use of sorption methods at different temperatures, Prace IMG PAN 16 3-4, 2014 [3] Pajdak A.: Theoretical models of the surface area and the distribution of the pores as a tool of analysing the equilibrium data of a low-pressure CO2 adsorption on carbon adsorbents, Prace IMG PAN 16 3-4, 2015 [4] IUPAC: Physical chemistry division commission on colloid and surface chemistry subcommittee on characterization of porous solids. Recommendations for the characterization of porous solids, Technical Report, Pure and Applied Chemistry 66 8, 1994 [5] Pajdak A., Godyń K., Kudasik M., Murzyn T.: The use of selected research methods to describe the pore space of dolomite from copper ore mine, Poland, Environmental Earth Sciences 76:389, 2017 [6] Pajdak A., Kudasik M.: Structural and textural characteristic of selected copper-bearing rocks as one of the elements aiding in the assessment of gasogeodynamic hazard, Studia Geotechnica et Mechanica 39 2 51-59, 2017 41 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 A multiparameter description of the rock-gas system by the use of original methods Norbert Skoczylas, Mateusz Kudasik The Strata Mechanics Research Institute of the Polish Academy of Sciences [email protected], [email protected] Keywords: rock-gas system, exchange sorption under confining pressure One of the main research topics that preoccupies the researches from The Strata Mechanics Research Institute of the Polish Academy of Sciences concerns the rock-gas system. We carry out basic research concerning both the presence of gas in rock and its transport in porous structure of rock. Also, we determine parameters related to the presence of gas in rock for mining industry. On the basis of laboratory experiments and experiments in-situ, we developed innovative and inventive research methods of the rock-gas system analysis. Designed for the use in copper mines was a device that identifies the amount and the quality of gas in the porous structure of dolomites and anhydrites and especially in their closed pores. Scientific basis of the proposed method were presented. The fundamental measurement is based on the volumetric analysis supported by simultaneous analysis of pressure change. The paper presents the physical principles of the processes describing the rock-gas system on which the metrological concepts of the presented methods are based as well as the way to evaluate the sorption parameters of coal. We described a digital methane emission reader – a metrological system that allows for the evaluation of desorbed amount of methane in coal and an effective methane diffusion coefficient in coal under in-situ conditions. The paper also presents a unique in the world and innovative experimental rig together with the measurement methodology. It was developed for the purpose of thorough identification of the CO2/CH4 exchange sorption in coal under confining pressure. The equipment allows for the analysis of the following phenomena: changes of the sorption capacity of a sample resulting from the changes of the confining pressure, changes of the sorption capacity of a sample under various confining pressures for particular points of sorption equilibrium, the course of the exchange sorption process under various confining pressure and various pressure gradient, changes in the coal structure, changes of the parameters describing filtration of gas through porous media under changeable confining pressure and during the exchange sorption process. The multitude of dependencies that can be observed and evaluated in terms of quantity during the sorption experiments under confining pressure is a missing link of the broadly understood analysis of the coal-gas system. 42 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Measurement of porosity of comminuted Salix Viminalis L. in aspects of uncertainty of measurement Karolina Słomka-Polonis1, Jakub Fitas1, Jakub Styks1, Bogusława Kordon-Łapczyńska1 1 University of Agriculture in Krakow, Department of Mechanical Engineering and Agrophysics [email protected] Keywords: Porosimetry, Measurement, Density, Salix Viminalis L. The knowledge of physical or mechanical properties of materials, including biomass for energy production, is of definite importance for the identification of thermophysical properties, modelling, optimalisation, and design of their thermal and mechanical processing. The density and porosity of biomass are among the more important properties, widely used in description and design of biomass processing. Different methods of measuring these properties differ in regards to their accuracy. Therefore, the aim of this paper was to analyse the influence of the method of measuring the porosity of comminuted willow Salix Viminalis L. on the value of measurement uncertainties. Dried wood for dried substance was comminuted by a knife mill using three sieves 15, 8, 4 mm. For porosity calculation, in the first stage true and volumetric densities were assessed. For each fraction the densities were assessed by different devices: AccuPyc II 1342 and GeoPyc 1365 pycnometers. Using thus discerned density porosity was indirectly measured for each fraction. True density of willow was measured by AccuPyc II 1342 as 1604,39 kgm-3. Meanwhile, volumetric density was measured by the hydrostatic method, where it was found equal to 618,14 kgm-3 for the 15 mm fraction, 635,56 kgm-3 for 8 mm, and 636,84 kgm-3 for 4 mm (Tab 1). Basing on these measurements porosity was calculated for each fraction as respectively: 0,6147, 0,6038, and 0,6031. In the course of research it was discovered that volumetric density measurement by GeoPyc 1365 is not adequate, therefore the resulting data were discarded. Table 1. Average physical parameters of the willow in dependence of measurement method and particle size. Particle size Parameter Unit 15mm 8 mm 4 mm -3 Absolute density, r kgm 1604,39 1604,39 1604,39 -3 Envelope density, f – by the pycnometry kgm 618,14 635,56 636,84 Porosity, P – with the use of the pycnometry - 0,6147 0,6038 0,6031 -3 Envelope density, f – by the quasi-fluid pycnometry kgm 503,38 486,80 330,5 Porosity, P - with the use of the quasi-fluid pycnometry - 0,6862 0,6966 0,7940 The data analysis in respect to each employed method was conducted according to the prescriptions of GUM 1995. Relative uncertainties calculated for the two techniques were respectively 2,1 % for the hydrostatic method, and 1,1% for the quasi-liquid method. For porosity, the respective absolute uncertainties were 1,4 % and 0,4 %. The results of the measurements and calculation are presented in the Table 2. Table 2. Average uncertainty (U) and standard deviations (SD) of measurements in dependence of measurement method and particle size. Parameter, x Unit 푈푎푏푠(푥) U(x) SD -3 * Absolute density, U(r) kgm 0,2 0,32 87,21 -3 Envelope density, U(f) – by the pycnometry kgm 2,1 13,27 21,30 Porosity, U(P) - with the use of the pycnometry - 1,4 0,0083 0,0133 -3 * Envelope density, U(f) – by the quasi-fluid pycnometry kgm 1,1 9,25 11,56 Porosity, U(f) - with the use of the quasi-fluid pycnometry - 0,4 0,0030 0,0060 Based on the results of the research, the following conclusions were formulated: - the smallest uncertainty for the assessment of the porosity of comminuted common osier was observed for the quasi-liquid method using GeoPyc 1365; - uncertainty fell as the degree of comminution increased; - with the increase in size of particles the porosity of samples grew. 43 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Numerical investigations of tortuosity in randomly generated pore structures Wojciech Sobieski University of Warmia and Mazury, Faculty of Technical Sciences, Olsztyn, Poland [email protected] Keywords: porosity, tortuosity, Lattice Boltzmann Method, Path Searching Algorithm In the study, the tortuosity in randomly generated 2D pore structures with various porosity and size of a structural element forming the solid body is investigated. Two different numerical methods are applied: the Lattice Boltzmann Method (LBM) and the self-developed algorithm, the so-called Path Searching Algorithm (PSA). The first method serves to calculate the hydraulic tortuosity based on the velocity field. The second method is used to detect free passages in the porous medium in the chosen space direction. In the study, 15000 different pore structures are investigated with 5 different resolutions of the numerical grid. The main result of the investigations is a new empirical function destined to estimate the tortuosity in cases where the porosity and the size of the structural element forming the solid body are known. Such a functions are not known in the literature. In Fig. 1 all the obtained values of the LBM tortuosity for the highest grid are shown together with the results of the PSA analysis. The small, filled points represent the values of the LBM tortuosity. Circles mean that in the current case at least one free path exists in the direction of the main fluid flow. Squares are shown in cases in which the LBM tortuosity has a negative value. Fig.2. Comparison of the data obtained in LBM calculations and PSA analysis The proposed function has a following form 푎 푑 휏(휙, |푠|) = + , (1) 휙푏∙|푠|푐 휙푒∙|푠|푓 where fit parameters a, b, c, d, e, and f are equal to 1.1, 0.33, -0.01, 5.3e-08, 6.8 and 2.86, respectively. The other symbols means: 휏 – the tortuosity, 휙 – the porosity, |푠| – the normalized size of the structural element forming the solid body. In Fig. 2 it can be seen that the new function matches the extreme values of the normalized size of the structural element especially well. Fig.2. Comparison of LBM data for grid 150(·4)× 150(·4) to the new proposed function and functions by Koponen [1] and Matyka [2] In the investigations of the tortuosity based on a velocity field, the data should be filtered and only these pore structures in which at least one passage in the main space direction exists should be taken into account. This issue is not obvious because sometimes, when porosity is relatively high, the flow in a chosen direction is impossible. For this reason, additional methods destined to analyse free passages in the pore space are needed. The filtration process allows eliminating almost all cases in which the tortuosity value is non-physical (less than 1). To eliminate the rest of the incorrect data, it is enough to reject the extremely high values of tortuosity (however, after rejecting all pore structures without at least one passage, such cases are very rare). References: [1] Koponen A., Kataja M., Timonen J.: Permeability and effective porosity of porous media. Phys. Rev. E 56, 3319 (1997). [2] Matyka M., Khalili A., Koza Z.: Tortuosity-porosity relation in porous media flow. Phys. Rev. E 78, 026306 (2008). 44 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Path Searching Algorithm Wojciech Sobieski University of Warmia and Mazury, Faculty of Technical Sciences, Olsztyn, Poland [email protected] Keywords: porosity, tortuosity, Path Searching Algorithm Path Searching Algorithm it is a iteration algorithm destined to conduct searches of continuous pore channels in a selected space direction in the earlier generated 2D pore structures. The path lengths as well as the length of the porous body (퐿0) are counted in the grid nodes. An example of the acting of the developed algorithm is visible in Fig. 1. The red colour denotes the nodes belonging to the current path. The blue nodes represent locations to which the path returned when no further movement was possible (e.g. due to the existence of a cavity). All points omitted in this process are in green. As it can be seen in the drawing, in some places the path is very tortuous and passes through many neighbouring nodes (red areas). Consequently, the length of the path is overestimated. In this Figure, the acting of the periodic boundary condition is additionally visible. The symbols 휙 and 푠 mean the porosity and the size of the structural element forming the porous body, respectively. Fig. 1. An example of a path obtained for grid 150×150, 휙 = 0.4 and 푠 = 10 (repetition 1) In the investigations the so-called path density indicator was introduced in the following form 푛푝 (1) 𝜌푝 = , 푐푚푎푥 where: 푛푝 – the number of free paths (passages) in the pore structure, 푐푚푎푥 – the maximum number of times the current grid node is assigned to a free path. It was noted, that the values of the LBM tortuosity (calculated with the methodology described in [1] or [2]) are high in areas in which the path density indicator tends to be zero (Fig. 2). Fig. 2. Correlation between the path density indicator and the LBM tortuosity for grid 150(·4)×150(·4) (all repetitions) The developed Path Searching Algorithm is particularly useful to reject these pore structures in which there are no free channels in the main flow direction. In such cases, calculating the tortuosity does not make much sense. This algorithm allows to calculate 푃푆퐴 푛푝 a parameter similar to tortuosity (휏 = , where 푛푝 is the number of “effective” nodes in the current path and 푛푥 is the number 푛푥 of nodes in the X-derection), which may serve to estimate the complexity of the pore space. However, the obtained values are overestimated in the relation to the classical tortuosity. Therefore, determining the practical usefulness of this parameter requires further research and perhaps an improvement of the developed algorithm. References: [1] Koponen A., Kataja M., Timonen J.: Permeability and effective porosity of porous media. Phys. Rev. E 56, 3319 (1997). [2] Matyka M., Khalili A., Koza Z.: Tortuosity-porosity relation in porous media flow. Phys. Rev. E 78, 026306 (2008). 45 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Application of the contour erosion function in shape analysis of a solid particle Aleksander Sulkowski Wrocław University of Technology, Faculty of Mechanical and Power Engineering, Department of Cryogenic, Aeronautic and Process Engineering [email protected] Keywords: solid particle, projective image, convex shell, erosion, convexity deficit Particle shape is one of the most essential parameters affecting structure of granular material and determining its properties and technological behaviour [1], [2]. Thus proper geometrical description of solid particle shape becomes one of the most important problems in powder technology [1]. In the paper a solid particle was defined to be some material object represented by its geometrical model C, assumed to be a subset of three dimensional real space, satisfying the following conditions: C is compact (closed and bounded) and connected set, being the closure of its connected interior Int(C) , ( C Int(C) ) , [2]. It is assumed that a geometric description of a solid particle may be based on the information provided by the analysis of geometric features of its two-dimensional projective image C. In this work a method of description of the shape of a particle projective image has been proposed. The theoretical basis of the presented method is the concept of analysis of the deviation of the geometric structure of the particle image C from its convex shell Cˆ Conv(C) , which is the smallest convex set containing the considered image [3], as shown in Fig.1. Considerations concern the so called convexity deficit zone QC of particle projective image C, which is defined as the ˆ ˆ difference between the convex shell C and the image being considered, ( QC C C ), as shown in Fig.1. The convexity deficit of particle image itself, marked with the symbol (C) , is defined by the formula (C) (C) / (Cˆ ) , (1) where (C) is the Lebesgue measure [4] (surface area) of the set C in two dimensional real space R 2 . For the purpose of the description of convexity deficit distribution within the QC zone, the so-called contour erosion function C (t) was proposed. It required introducing the concept of the erosion layer LC () with the depth , which was defined as the common part of the convex shell and the set of all spheres with the radius , whose centres are the boundary points of the convex shell , as shown in Fig.1. For further considerations, two specific depth values of the erosion layer LC () are important – critical depth X and threshold depth T . Critical depth is the smallest value of parameter , for which the erosion layer absorbs the convex shell . The threshold depth T is the smallest value of the parameter , for which the difference between the projective image of the particle C and the erosion layer LC () becomes a convex set. The aforementioned contour erosion function C() is given by the formula ˆ C (t) ((C LC (t X )) ((C LC (t X )) (2) It can be shown that the contour erosion function C (t) has the following properties: 1.) C :[0,1] [0,1] , 2.) C (t) is a continuous, non decreasing function, 3.) C(0) (C) , 4.) C(t) 1 for t (T X ) . Thus the contour erosion function may be applied for description of the geometric structure of a solid particle. In particular it can be used to describe the distribution of the convexity deficit in the erosion layer of the particle projective image. In Fig.2 an exemplary projective image of a real particle with its convex shell and erosion layer with a threshold depth, has been presented. A graph of the contour erosion function corresponding to the image of the real particle considered is shown in Fig.3. C L C Conv(C) Fig.1. Convex shell and erosion layer Fig.2. Convex shell and kernel Fig.3. Contour erosion function of a particle projective image C. of a real particle image of a real particle image. In addition, contour erosion itself, is some morphological operation [5], which can be used for the purpose of modelling the process of evolution of the solid particle shape in case it is subject to destructive external factors such as abrasion or grinding. References: [1] Wanibe Y., Itoh T.: New Quantitative Approach to Powder Technology, John Wiley & Sons Ltd., N.Y., 1998 [2] Ohser J., Mucklich F.: Statistical Analysis of Microstructures in Materials Science, John Wiley & Sons, New York, 2000. [3] Rudin W.: Functional Analysis, Mc. Graw-Hill Book Company, N.Y., 1973 [4] Fremlin D.H.: Measure theory vol. 1,2, Torres Fremlin, Colchester, 2000 [5] Serra J.: Image Analysis and Mathematical Morphology, Academic Press, London, 1982. 46 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Influence of Microscopic Pore Geometry on the Parameter of Pore Tortuosity Zbigniew Szczepański, Mieczysław Cieszko Institute of Mechanics and Applied Computer Science Kazimierz Wielki University, Bydgoszcz e-mail: [email protected], [email protected] Keywords: pore tortuosity, microscopic representation, influence of pore geometry The aim of the paper is to analyse the influence of microscopic pore geometry on the parameter of pore tortuosity of porous materials with isotropic pore space structure. Considerations are based on the general relation of this macroscopic parameter with quantities characterising microscopic pore structure. The general form of this relation have been obtained in the paper [1] requiring the full representation of macroscopic density of kinetic energy of fluid in the potential flow, by microscopic velocity field. The microscopic representation of tortuosity has the form 1 1 . 푟 2 2 = ∫ (퐧 ∙ 퐧) 푑푉 (1) 푇 푉푝 Ω푝 where 퐧 i 퐧푟 are unit vectors defining directions of gradient of macroscopic potential inducing fluid flow and of microscopic potential in the pore space, respectively, and quantity 푉푝 denotes the volume of the representative region of poresΩ푝in porous material. Considerations presented in the paper [1] have been based on the model assumptions presented in papers [2] and [3]. It was assumed that at the macroscopic point of view interconnected pores in permeable porous materials form anisotropic space the structure of which is determined by its metric and this space is modelled as Minkowski metric space. Such approach enabled precise and consistent definition of macroscopic parameters of pore space structure: pore tortuosity and surface porosity, directly determined by the metric tensor of the pore space. Analysis of the influence of microscopic pore geometry on the parameter of pore tortuosity has been performed both analytically and numerically for two types of pore geometry models: capillary and two dimensional network. In the first case, the influence of distribution of pore diameter, length and cross-section were analysed. It was shown among others that tortuosity of porous layer with capillary pores of two different length is equal to the geometric mean of the arithmetic and harmonic means of tortuosities produced by both length. Fig.1. Tetragonal system of pores. Two examples of the two dimensional network models analysed in the paper have been presented in Fig.1, and the obtained values of tortuosity were presented in Fig.2. It was shown that both types of pore structure are isotopic independently of the values of their volume porosities. Fig.2. Dependence of pore tortuosity of tetragonal pore space structure on the volume porosity. References: [1] Cieszko M., Minkowski Metric, Dirichlet Energy and Pore Tortuosity, INT J ENG SCI (in preparation for publication). [2] Cieszko M., Fluid Mechanics in Anisotropic Pore Space of Permeable Materials. Application of Minkowski Metric Space (in polish), Wyd. Uniwersytetu Kazimierza Wielkiego, Bydgoszcz 2001. [3] Cieszko M., Description of Anisotropic Pore Space Structure of Permeable MaterialsBased on Minkowski Metric Space, Archives of Mechanics, 61, 6 (425-444), 2009. 47 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 The use of natural soil as a porous medium for the treatment of secondary effluent in the northern climate Janusz A. Szpaczyński Wrocław University of Science and Technology, Faculty of Mechanical and Power Engineering, [email protected] Keywords: freeze crystallization, land application, wastewater treatment, spray irrigation. In northern climate, cold temperature can significantly impact the efficiency of the biological wastewater treatment process. Systems with lagoons and spray irrigation are particularly vulnerable. Unfortunately such systems are often located on the northern rural areas because they are well suited for small communities where land is usually available and where it is important to keep costs as low as possible. In addition, in cold climate, during winter months, discharge of an effluent to local lakes or tributaries is usually prohibited. Long term storage is required over winter months. Therefore, the development of each village or municipality must be preceded by an increase in the capacity of the lagoons. The idea of converting lagoon effluent to man-made snow during winter months can help to solve the problem [1]. The concept has a number of advantages that have already been described in the professional literature [2]. In the spring, melting water soaks into the soil as it does during summer irrigation. The system uses soil properties as a porous medium for physical and biochemical filtration. Moreover, the process of phytoremediation takes place. Movement of an effluent of melt water in the pores of the soil produces its purification. The soil matrix in this system acts as a deep-bed bioreactor and has a tremendous potential to treat a secondary effluent. In the upper soil layer, essentially all suspended solids and biodegradable materials can be restrained and decomposed. In the paper, the results of the studies of the groundwater quality from two different sites of full scale snow storage and treatment are presented. Process factors, such as the concentration of nitrogen, conductivity and phosphorus have been adopted as parameters of concern and are used to assess the effectiveness of the atomizing freeze crystallization process and its effect on the quality of groundwater at the snow deposit site. Significant reduction of phosphorus was achieved by its sorption to the fine soil particles and precipitation. The rest of dissolved phosphorus, as well as other nutrients, are up taken by plants. Usually, It happens before they reach the ground water table (Fig 1, 2). Fig. 1.Total phosphorus in the soil at different depths. Fig. 2. Sodium bicarbonate extractable phosphorus at different depths. The analyses of soil showed that there was an evident adsorption of phosphorus in the soil profile. Sodium bicarbonate extractable phosphorus was detected only at the location of the main snowpack, where the highest load of effluent took place. However, it decreases significantly with the depth of the soil profile. This fact suggests the conclusion that extractable P was up taken by vegetation, immobilized by soil microorganisms and fixed. Nitrate is broken down through a process of denitrification in anaerobic conditions present in the soil. Vegetation at the snow deposit site and buffer zone removes most of the nitrogen as nitrates compounds and other nutrients. Monitoring ground water quality confirms that atomizing freeze crystallization process and the treatment of secondary effluent in the soil profile have effectively removed most of contaminants. It was found that nitrates at the property boundary of the snow deposit site are not detectable or are below drinking water standards. This system is a well performing option for final polishing of wastewater effluent. References: [1] Huber D., Palmateer G., Snowfluent - A join experimental project between the Ministry of the Environment and Delta Engineering in the storage and renovation of sewage effluent by conversion to snow. MOE Report. Ontario, Canada. [2] White J.A., Lefebvre P., 1997. Snowfluent® - the use of atomizing freeze crystallization on municipal, agricultural and hog manure wastes. Conference, World-Wise’97, Selkirk, Manitoba. [3] Szpaczynski J.A., White J.A., 2000a. Experimental studies on the application of natural process of snow metamorphism for concentration and purification of liquid wastes. WEF & Purdue University, Industrial Wastes Technical Conference. May 21, 2000, St. Louis, Missouri, USA. 48 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 The stand for testing flow of two-phase system through the metal foams Adriana Szydłowska, Jerzy Hapanowicz Department of Process Engineering, Faculty of Mechanical Engineering, Opole University of Technology [email protected] Keywords: metal foam, two-phase flow, gas-liquid system, hydrodynamic Flow of multiphase systems occurs in a number of industrial technologies, mainly in the chemical and petrochemical sectors, but also in the pharmaceutical, food and energy sectors. Processing of such substances almost always involves momentum, heat and mas transfer processes what spells necessity of having an unequivocal description of accompanying their flow phenomena. Heat transfer processes are a in an area of interest of a number of researchers which mostly aspire to improvement in the heat exchange effectiveness. Metal foam with open cells may be one of the suggested solutions in this range. The metal foam as a packing of flow pipe constitutes a specific “fin” exchanging the heat with flow substance. The metal foams which are produced for flow apparatuses packing are manufactured of materials which are good heat conductors e.g. aluminium (Fig. 1.) or copper alloys. They are distinguished by continuity of frame, a well-developed specific surface area (a few thousands m2/m3) and high porosity usually exceeding 90%. Such properties of this group of materials make them an interesting substitute for granular beds or wire-net packings. Fig. 1. Metal foam Within several last years a number of research on flow single-phase and two-phase gas-Newtonian liquid systems through the metal foam hydrodynamic have been shown up [1]. Knowledge of description of phenomena accompanying that flow is necessary for a successful design and operation of apparatuses and technological systems. There are few references concerning two-phase systems: gas-liquid and liquid-liquid flow through the metal foams. Here it is noteworthy that a correct description of the heat transfer in a given substance often requires taking into consideration the phenomena resulting from the flow hydrodynamic of this substance, particularly when it is a two-phase system. Therefore, there is a justified need to conduct some research in this regard, however it demands a properly constructed and equipped measuring position. The subject of this paper is to present conception of the stand, its component parts, measuring possibilities, established methodology of carrying out the research and assessment of results of the test studies. The stand enables measurement resistances of flow of two-phase mixtures through sections of horizontal packed as well as unpacked with the foam pipe and also assessment of influence of presence of this foam on the flow pattern of the two-phase system and its rheological behaviour. A diameter of the flow pipe is equal to 10 mm and it constitutes a replaceable part of the stand which can be packed with the foam with different parameters. The length of the measuring pipe is divided into five segments. The length of each of them corresponds to relation L/d=15. Three middle segments are packed with the metal foam permanently while the first and the last one function as pipe rheometers. The set of supply the systems with operating fluids enables to select such fluids fluxes that formation of the flow patterns typical of the given two-phase system flow will be possible. In order to form researched mixtures air, water, oil and non-Newtonian liquid may be used. The measurement of these components fluxes and resistances of two-phase mixtures flow through successive measuring sections is carried out with using analog-to-digital signal converters working with a computer which has proper software. Photographic recording of observed flow patterns in transparent fragments of flow pipes placed at the beginning and at the end of the system supports visual identification of these patterns. As a result of the research it was found that the constructed measuring position gives opportunity of reliable assessment of resistances of two-phase systems flow through pipe packed with the metal foam with definite parameters. However, the impact of the metal foam on a possible change of flow pattern or rheological properties of the two-phase mixture will be feasible after carrying out experiments in significantly larger scale. References: [1] Dyga R., Wymiana ciepła i hydrodynamika przepływu przez piany metalowe, Oficyna Wydawnicza Politechniki Opolskiej, 2015 49 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Dissolution of porous media studied in a simple microfluidic setup Piotr Szymczak1, Filip Dutka1, Florian Osselin2 1University of Warsaw 2University of Calgary [email protected] Keywords: reactive transport, dissolution, porous media Dissolution of fractured and porous media introduces a positive feedback between fluid transport and chemical reactions at mineral surfaces, which can lead to self-focusing of the flow in pronounced wormhole-like channels [1,2]. We study the flow-induced dissolution in a simple microfluidic setup, with a gypsum block inserted in between two polycarbonate plates, which is the simplest model of a fracture [3]. This gives us a unique opportunity to observe the evolution of the dissolution patterns in-situ and in real-time. By changing the flow rate and the aperture of the fracture we can scan a relatively wide range of Péclet and Damköhler numbers, characterizing the relative magnitude of advection, diffusion and reaction in the system. Additionally, as the aperture is increased, a transition is observed between the fractal and regular dissolution patterns. For small gaps, the patterns are ramified fractals. For larger gaps, the dissolution fingers are found to have regular forms of two different kinds: either linear (for high flow rates) or parabolic (for lower flow rates). The experiments are supplemented with numerical simulations and analytical modeling which allow for a better understanding of evolving flow patterns. In particular, we find the shapes and propagation velocities of dominant fingers for different widths of the system, flow rates and reaction rates. Finally, we comment on the link between the experimentally observed patterns and the natural karst systems - both cave conduits and epikarst solution pipes. Fig.1. Microfluidic setup (left panel) and the examples of dissolution patterns formed spontaneously in this system References: [1] Hoefner, M. L. and Fogler, H. S. Pore evolution and channel formation during flow and reaction in porous media. AIChE J. 34, 45–54, 1988 [2] P. Szymczak, A. J. C. Ladd, Wormhole formation in dissolving fractures, J. Geophys. Res., 114, B06203, 2009 [3] F. Osselin, P. Kondratiuk, A Budek, O. Cybulski, P. Garstecki, P. Szymczak Microfluidic observation of the onset of reactive infiltration instability in an analog fracture, Geophys. Res. Lett., 43, 6907-6915, 2016 50 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Numerical simulations of laboratory-scale dike overflowing using two phase flow model Adam Szymkiewicz1, Witold Tisler1, Wioletta Gorczewska-Langner1, Rafał Ossowski1, Danuta Leśniewska2, Stanisław Maciejewski2 1Gdańsk University of Technology, Faculty of Civil and Environmental Engineering 2Koszalin University of Technology, Faculty of Civil Engineering, Environmental and Geodetic Sciences [email protected] Keywords: flood dikes, unsaturated flow, two phase flow, infiltration, air trapping Water flow in the shallow subsurface, natural and artificial slopes and earth structures occurs in partially saturated conditions, i.e. part of the pore space is occupied by water and the other part by air. In typical engineering analyses it is commonly assumed that the air in pores is connected to the atmosphere and much more mobile than water. As a consequence the air pressure can be considered constant and only the water flow is considered. In such a case the water flow can be described by the Richards equation. However, the assumptions underlying the Richards equation are not always valid, especially in the case of heterogeneous soils or in the presence of barriers impermeable to air flow. Air can be trapped in the soil pores when large areas of land are ponded with surface water or when water flows over the crest of a flood dike, earth dam or similar structure [1]. This contribution presents the results of laboratory experiments carried out on a model dike made of fine sand [2]. During the overflowing experiment a significant amount of air was trapped in the core of the dike. Increase of the air pressure led to expulsion of air in form of bursts, leading to damages of the soil structure and creation of cracks. The observations are compared to the results of numerical simulations based on two phase flow model, which accounts for both water and air phases. Simulations were carried out using finite volume discretization method implemented in an in-house code developed by the first author. While the numerical model does not include deformation, the results for the early stage of infiltration process (before cracks appear) are in a good agreement with the visual observations of the water saturation field and with air pressure measurements (Fig. 1) [2]. Fig.1. Calculated and observed distribution of water saturation in the model dike [2]. References: [1] Bogacz P., Kaczmarek J., Leśniewska D. (2008), Influence of air entrapment on flood embankment failure mechanics - model tests, Technol. Sci. 11, 188-201. [2] Tisler W., Gorczewska-Langner W., Leśniewska D., Maciejewski S., Ossowski R., Szymkiewicz A.: Simulations of air and water flow in a model dike during overflow experiments, submitted to Computational Geosciences, 2018. 51 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Effect of temperature, concentration of alcohols and time on baker's yeast permeabilization process Ilona Trawczyńska, Justyna Miłek, Sylwia Kwiatkowska-Marks Department of Chemical and Biochemical Engineering, Jan and Jędrzej Śniadecki University of Technology and Life Sciences in Bydgoszcz, Seminaryjna 3, 85-326 Bydgoszcz, Poland [email protected] Keywords: permeabilization, baker’s yeast, biocatalyst, response surface methodology Many scientists have proven that the use of biocatalysts in the form of whole cells of microorganisms is often more effective than the use of purified enzymes [1, 2]. However, low permeability of cell wall and membrane contributes to the slow rate of reactions catalyzed by whole cells enzymes. Such difficulties can be overcome by the application of permeabilization technique which is to improve the permeability of cell wall and membrane of microorganisms for facilitating the diffusion of reaction reagents, while also maintaining the cell’s properties, including enzymatic activity and structure [3, 4]. In this paper, the influence of physical and chemical parameters on the effectiveness of permeabilization of baker's yeast cells using alcohols was analysed. For this purpose the response surface methodology (RSM) has been applied in the study. The effectiveness of the process has been represented by measuring intracellular catalase activity. The research program was designed in such manner, that it was possible to obtain the necessary information performing the fewest number of analysis possible. Therefore, the research was conducted according to the points of the compositional design, plan which in the form of coded variables is given in table 1. The number of necessary experiments to conduct was 20 for each of the permeabilization processes, i.e. with the use of ethanol, 1-propanol, 2-propanol. Table 1. Central composite design matrix nr temperature concentration time enzyme activity [U∙g-1] ethanol 1-propanol 2-propanol 1 – 1 – 1 – 1 190 140 230 2 – 1 1 – 1 2540 3410 4840 3 – 1 – 1 1 260 235 1990 4 – 1 1 1 3085 2560 5050 5 1 – 1 – 1 620 340 2690 6 1 1 – 1 2040 2580 4990 7 1 – 1 1 2120 1620 3620 8 1 1 1 1780 1935 3600 9 1.682 0 0 140 1370 2570 10 – 1.682 0 0 210 1135 360 11 0 1.682 0 1005 2210 4980 12 0 – 1.682 0 300 140 120 13 0 0 1.682 4340 3590 4760 14 0 0 – 1.682 1690 2570 3090 15 0 0 0 4390 4100 5900 16 0 0 0 5070 4220 5750 17 0 0 0 5150 4070 5880 18 0 0 0 5050 3970 5780 19 0 0 0 4760 4170 5800 20 0 0 0 5190 4145 5765 Response surface plots were created based upon experiments conducted in the points of compositional plans. Plots present an effect of two process variables on activity of catalase, assuming that the value of the third variable is constant. Based upon the research, conclusion can be drawn, that alcohol of lower concentration provides better results, along with the increase of temperature in permeabilization process of baker’s yeast cells. Applying solvents of too high concentrations contribute to the decrease of permeabilization effectiveness. At high temperatures of approx. 30℃, better results can be achieved by applying lower concentrations of alcohol, along with the increased duration of cells shaking. Treatment time of the process proved to be a parameter bearing little influence on the effectiveness of the process. References: [1] Sekhar S., Bhat N., Bhat S.G.: Preparation of detergent permeabilized Bakers’ yeast whole cell catalase, Process Biochemistry 34(4): 349-354, 1999 [2] Xu P., Zheng G.-W., Du P.-X., Zong M.-H., Lou W.-Y.: Whole-Cell Biocatalytic Processes with Ionic Liquids, ACS Sustainable Chemistry & Engineering 4 (2): 371-386, 2016 [3] Kumari S., Panesar P.S., Bera, M.B., Singh B.: Permeabilization of yeast cells for beta–galactosidase activity using mixture of organic solvents: A response surface methodology approach. Asian Journal of Biotechnology 3(4): 406-414, 2011 [4] Panesar P.S., Panesar R., Singh R.S., Bera M.B.: Permeabilization of yeast yells with organic solvents for β–galactosidase activity. Research Journal of Microbiology 2(1): 34-41, 2007 52 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Pore scale simulations of flows in weakly permeable porous media Anna Trykozko ICM - Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw, ul. Pawińskiego 5a, 02-106 Warszawa [email protected] Keywords: pore scale, micro-CT imaging, resolution, simulations of flow 1. Computational modelling of flows through porous media performed at pore scale has become a standard procedure in the last couple of years. It is based on solving Navier-Stokes equations defined in a domain available for flow (pores). In a standard approach only pores are taken into account in computations and a percolation through a sample is required in order to simulate flow. Moreover, all pore spaces across a sample should be connected and therefore dead-end pores are removed from a domain of flow. Simulation results obtained at pore scale, which are pressure and velocity fields, can be ‘translated’ by upscaling to larger scales which are more relevant in practical engineering applications. The main steps of the computational procedure we use are described in [1] and [2]. Complex shapes of flow domains result in large sizes of computational problems what remains a challenge in spite of rapidly growing available computer resources. 2. X-ray computed tomography (micro-CT) has become the more and more accessible source of realistic data describing pore structures. As a result of a segmentation of a micro-CT image, a map of voids and impervious ‘solid’ voxels can be obtained to serve as data for defining flow domain in simulations. A voxel size depends on a resolution of a microimage. Our study was performed based on micro-CT images of a set of samples of soils. The samples were manufactured as mixtures of sand and clay containing from 100% to 20% of sand. There was no percolation detected at a micro-CT resolution in many samples and therefore the standard computational procedure was not applicable. 3. On the other hand there were significant differences in porosities evaluated at micro-CT resolution as compared to the values obtained in laboratory measurements. In order to take into account this subscale porosity we proposed an extended procedure in which voxels classified as 'solids' during segmentation were modelled as porous media characterized by a small permeability. This permeability was determined based on data provided by a Scanning Electron Microscopy which is another high-resolution microimaging technique. 4. Details of the extended computational procedure and results of simulations performed over various weakly permeable samples will be presented. Acknowledgements: Samples used in this study were manufactured and imaged for the needs of the project nr Pol- Nor/209820/14/2013 supported by the Polish-Norwegian Programme operated by the National Centre for Research and Development under the Norwegian Financial Mechanism 2009-2014. References: [1] Peszynska M., Trykozko A.: Pore-to-core simulations of flow with large velocities using continuum models and imaging data, Computational Geosciences, 4(17): 623-645, 2013 [2] Trykozko A., Peszynska M., Dohnalik M.: Modeling non-Darcy flows in realistic pore-scale proppant geometries, Computers and Geotechnics, (71): 352-360, 2016 53 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Using of the SEM image method to evaluate the porosity of materials with varied internal structure Grzegorz Wałowski1, Gabriel Filipczak2 1Institute of Technology and Life Sciences, Renewable Energy Department - Poznan Branch, 2Opole University of Technology, Faculty of Mechanical Engineering, Department of Process Engineering, [email protected] (or) [email protected] Keywords: porous material, SEM image, anisotropy An important feature of porous materials, resulting directly from their structure, is porosity. In quantitative terms, this parameter results directly from the pores volume [1], [2], as the free volume space for fluid flow. When the porous material comprises a rigid skeletal structure (Fig. 1), a part of the pores and channels volume can be closed or blinded, and the flow of fluid through such material is limited only to the pores space (or channels) connected to each other’s, with a free space less than this as results from the real scale of the porosity of material. This greatly impedes assessment of the hydrodynamics of fluid flow through skeletal porous media, especially the linking of the permeability of these materials with their properties. The problem becomes even more complicated in case of anisotropic structure, when there is a diverse spatial structure (Fig. 1). In own investigation, to porosity evaluation the image analysis method based on scanning surface topography (SEM) was used – Fig. 2. Due to the anisotropic structure of tested materials, this porosity was determined in three selected directions (X, Y, Z), and next the average porosity value as resulting from the configuration of surface pores was determining. - X direction - Y direction - Z direction Fig.1. Image of the surface of a porous material (coal char) with a differentiated spatial structure. - X direction - Y direction - Z direction Fig.2. SEM images of porous material (coke) with varied elementary structure. In the technical analysis, a specialized computer program for graphic image analysis was used for the quantitative evaluation of porous structure – Fig. 3a. This program (IRIS) makes it possible to calculate geometrical parameters of structures, including linear dimensions, pores and channels surfaces, with the possibility of eliminating closed structures, as it exemplified in Fig. 3b. The results obtained during the laboratory tests showed that based on SEM images method the assessment of porosity can be successfully used in description of hydrodynamics of gas flow through porous materials with skeletal structure [3]. a) b) Fig.3. Calculation proceedings for determination of porosity: a) program procedure, b) detail of SEM scanning area. References: [1] Aksielrud G.A., Altszuler M.A.: Ruch masy w ciałach porowatych. WNT, Warszawa, 1987 [2] Strzelecki T., Kostecki S., Żak S.: Modelowanie przepływów przez ośrodki porowate. Dolnośląskie Wydawnictwo Edukacyjne, Wrocław, 2008 [3] Wałowski G.: Hydrodynamika przepływu gazu przez złoże porowate, Praca doktorska, Politechnika Opolska, 2008 54 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Numerical analysis of compression mechanics of granular packings with various number of particle size fractions Joanna Wiącek, Marek Molenda, Mateusz Stasiak Institute of Agrophysics Polish Academy of Sciences Doswiadczalna 4, 20-290 Lublin 27, Poland [email protected] Keywords: polydisperse granular packings, granulometric fractions, structure, mechanical properties Polydisperse granular materials exhibit complex and undefined behaviors, posing considerable challenges in the design and operation of processing plants. Granular packings may be composed of one, two, three or more components different from one another in size, shape, material properties et al.. Size polydispersity is an inevitable feature of granular materials which determines structural and mechanical properties of particulate systems [1-3]. Although a number of studies has been so far conducted for polydisperse granular packings, no study on effect of number of granulometric fractions on structure and mechanical characteristics of grain assemblies with uniform particle size distribution has been so far undertaken. Thus, the objective of the presented project was to analyze the effect of number of particle size fractions on structural and mechanical properties of granular mixtures composed of spheres with size distribution uniform by number of grains. A study was conducted for mixtures with various ratios between diameter of the largest and the smallest particle to verify whether value of particle size ratio has an influence on relationship between number of fractions and structure and micromechanics of granular packings. In this study, simple granular packings composed of three, five and seven granulometric fractions were examined, which are the simplest polydisperse particulate systems, after binary ones. Spherical particles with random initial coordinates were generated inside the rectangular box and settled down onto the bottom of the test chamber under gravity (see Fig. 1). The rigid and frictional walls did not deform under the applied load. Next, spheres were compressed through the top cover of the chamber that moved vertically downwards at a constant velocity of until a maximum vertical pressure on the uppermost particles reached 100 kPa. Fig.1. Initial configuration of mixture. The effect of number of granulometric fractions on structural and micromechanical properties of frictional packings of spheres with various particle size dispersity has been investigated by means of the discrete element simulations of confined uniaxial compression. Three-dimensional simulations were conducted using the EDEM software [4], based on the Discrete Element Method [5]. The study has shown an influence of both, number of particle size fractions and degree of particle size heterogeneity on packing density of assembly. Results have shown a presence of the certain value of ratio between diameter of the largest and the smallest particle in multicomponent granular system below which packing density decreases significantly with increasing number of granulometric fractions; however, estimation of that value requires further investigations. The largest average coordination numbers were obtained in ternary samples with the smallest ratio between diameter of the largest and the smallest particle. Coordination number decreased with increasing number of granulometric fractions. In more heterogeneous and disordered packings with higher particle size dispersity, small spheres filled the pores between larger particles only partially, resulting in smaller number of interparticle contacts. It was found that corrected coordination number, including particles with more than three contacts, was larger in samples with higher ratio of the diameter between the largest and the smallest particle, which was related to larger number of contacts between mechanically stable large spheres and surrounding them small particles. Significant influence of the particle size ratio and number of particle size fractions on distribution of contact forces in granular packings was observed, which determined global stress and energy dissipation in assemblies. The dissipation of energy was found to increase with increasing vertical pressure and increasing grain size heterogeneity. Results presented in this paper indicate that structural and micromechanical properties of granular packings with uniform discrete particle size distribution are determined by both, degree of particle size dispersity and number of grain size fractions in mixture. Knowledge of these geometric and statistical factors, describing composition of granular mixtures, thus plays a crucial role in predicting and interpreting effects observed in particulate systems. References: [1] McGeary R.K: Mechanical Packing of Spherical Particles. J. Am. Ceram. Soc., 44, 513-523, 1961 [2] Wiącek J., Molenda, M.: Effect of particle size distribution on micro- and macromechanical response of granular packings under compression. Int. J. Solids Struct., 51, 4189-4195, 2014 [3] Wiącek J., Stasiak M., Parafiniuk P.: Effective elastic properties and pressure distribution in bidisperse granular packings: DEM simulations and experiment, Arch. Civ. Mech. Eng. 17, 271-280, 2017 [4] EDEM Software, 2016. Retrieved from www.dem-solutions.com/software/edem-software [5] Cundall P.A., Strack O.D.: A discrete element model for granular assemblies. Géotechnique, 29, 47-65, 1979 55 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 Determination of the heat of wetting of selected liquids on modified activated carbons Eliza Wolak, Elżbieta Vogt, Jakub Szczurowski AGH University of Science and Technology, Faculty of Energy and Fuels, Cracow, Poland [email protected] Keywords: activated carbon, hydrophobization, heat of wetting Active carbons are intensively used materials in both basic research and industrial processes, mainly in the processes of purification, separation and catalysis. They are one of the most popular adsorbents that are used for gas cleaning and separation, solvent recovery, gas storage, water treatment and many other applications. Thanks to their numerous advantages, microporous carbon adsorbents are also used in unconventional mass and energy storage processes, e.g. in adsorption cooling or air conditioning systems [1], [2]. Their effectiveness in these kind of processes primarily depends on structural properties and chemical properties of the surface. The most widely used treatment-enhancing properties of carbon sorbents are appropriate modifications, which result in the changing of their structural and surface parameters [3], [4]. The change of the hydrophobic properties of the carbon materials, which is applied during production filters for removing oil and organic pollutants [5] or in the purification processes used frying oils, is also possible [6]. Techniques to modify and characterize the surface chemical properties of activated carbons constituted the subject of interest for many scientists [7]. In this paper commercially available activated WD-extra carbon (Gryfskand) which is applied for water treatment was used. Activted carbon was modified by the following chemical agents: H2O2, HNO3 and HCl. The textural characteristics of the samples were determined by the analysis of physical adsorption isotherms of nitrogen vapor at 77 K. The Boehm titration general procedure was used to determine the distribution of the surface functional groups. Chemical modifications significantly affect the chemical, structural and surface properties of activated carbons. Hydrophobization with ethereal stearic acid was performed on the raw material and samples after chemical modification. Hydrophobic properties of the samples were specified. The relationship of the chemical modification agents with hydrophobization degree was indicated. The heat effect of adsorption was measured in accordance with Polish Standard PN-90/C-97554 and using a home-designed apparatus. Gryfskand WG-12 active carbon was used as a reference material. The enthalpy of wetting by wetting liquids was calculated. The author gratefully acknowledgements the financial support of this work by the Polish Ministry of Science and Information Society Technologies under AGH statutory project No. 11.11.210.374. References: [1] Anyanwu E.E.: Energy Conversion and Management 45(7-8): 1279-1295, 2004 [2] Wolak E., Buczek B.: Inżynieria i Aparatura Chemiczna 6(1): 11-1, 2005 [3] Repelewicz M., Jedynak K., Choma J.: Ochrona środowska 31(3): 45-50, 2009 [4] Buczek B., Wolak E.: Adsorption 14(2-3): 283-287, 2008` [5] Lee C.H., Johnson N., Drelich J., Yap Y.K.: Carbon 49(2): 669-676, 2011 [6] Buczek B., Chwiałkowski W.: Żywność. Nauka. Technologia. Jakość 4(45): 85-99, 2005 [7] Menendez J.A., Phillips J., Xia B., Radovic L.R.: Langmuir 12(18): 4404-4410, 1996 56 | 2nd Workshop on Porous Media, Olsztyn, 28 - 3 0 J u n e 2 0 1 8 NOTES