Search

Gas flow through layers of porous materials plays a crucial role in technical applications, geology, petrochemistry and space sciences (e.g. fuel cells, catalysis, shale gas production and outgassing of volatiles from comets). In many applications the Knudsen regime is predominant, where the pore size is small compared to the mean free path between intermolecular collisions. In this context common parameters to describe the gas percolation through layers of porous media are the probability of gas molecule transmission and the Knudsen diffusion coefficient of the medium. We show how probabilistic considerations on layer partitions lead to the analytical description of the permeability of a porous medium to gas flow as a function of layer thickness. The derivations are made on the preconditions that the reflection at pore surfaces is diffuse and that the pore structure is homogenous on a scale much larger than the pore size. By applying a bi-hemispherical Maxwell distribution, relations between the layer transmission probability, the half-transmission depth and the Knudsen diffusion coefficient are obtained. For packings of spheres, expressions of these parameters in terms of porosity and grain size are derived and compared with former standard models. A verification of the derived equations is given by means of numerical simulations, also giving evidence that our analytical model for sphere packing is more accurate than the former classical models.

In the previous years, the CoPhyLab* has established an experimental environment that makes it possible to simulate the activity of comets in the best way possible. Nevertheless, it is vital to know all parameters that have an impact on the outcome of such experiments, as well as parallelly developed models. Therefore, several smaller experiments provide important knowledge, especially about the used comet analogue, the CoPhyLab-dust. With the gas-flow experiment we conduct in Graz it is possible to learn about the permeability and effusion of gas through porous media in a dry environment. In this presentation we will present how that works and what challenges we needed to overcome to achieve good scientific outcomes. With the results of this experiment a special case of the dusty-gas model, established by Evans et al. 1961 [1], will be tested in high vacuum. A vacuum chamber with two compartments, which are connected via a tube that holds the tested sample, is used for the experiment (see Figure 1). In both compartments pressure sensors are mounted onto the chamber. For our measurements we use two different gas-flow controllers with different gas-flow ranges to cover a wide pressure range. As this is a continuing work, the theoretical scheme was established by Schweighart et al. 2021 [2]. According to Equation 10 from their work, knowledge of the upstream and downstream pressure as well as the mass flux flowing through the sample, allows us to measure the Knudsen diffusion Dk as well as the permeability B of the sample. Figure 1: Experiment Setup Because gas-flow must be known precisely, a leakage of gas into the chamber, which is always there could have an influence on the evaluation of the results. Therefore, a bleed-up test was performed. To work with porous media is a difficult task. Therefore, measurements with round glass beads, with a predictable packing behavior, were performed. We found that the results deviate from the regression using the dusty gas model at lower pressures or smaller gas flows (see Figure 2). There is a systematic decrease in pressure at smallest gas flows. We are currently investigating the possible causes for this behavior. On the one hand, we consider if the deviation could be explained by set up related properties, like a leak in the chamber or a systematic offset on the gas flow. On the other hand, an outgassing of volatiles or a temperature dependence are possibilities we are looking into. Nevertheless, we already gain high confidence results in the measured pressure regime. In future we will investigate how those results compare to theoretical models, which consider the porosity and the tortuosity of such porous samples (e.g., Asaeda et al. 1974 [3]). Figure 2: Experimental results for an asteroid analogue sample, with the apparent Diffusion Dapp depending on the mean pressure within the tested sample. Acknowledgements: This work is carried out in the framework of the CoPhyLab project funded by the D-A-CH programme (DFG GU 1620/3-1 and BL 298/26-1 / SNF 200021E 177964 / FWF I 3730-N36). References: [1] Evans III, R. B., G. M. Watson, and E. A. Mason. "Gaseous diffusion in porous media at uniform pressure." The journal of chemical physics 35.6 (1961): 2076-2083. [2] Schweighart, M., et al. "Viscous and Knudsen gas flow through dry porous cometary analogue material." Monthly Notices of the Royal Astronomical Society 504.4 (2021): 5513-5527. [3] Asaeda, Masashi, Shigeyuki Yoneda, and Ryozo Toei. "Flow of rarefied gases through packed beds of particles." Journal of Chemical Engineering of Japan 7.2 (1974): 93-98. *The CoPhyLab (Comet Physics Laboratory): https://www.cophylab.space/index.php?id=home

Introduction After the visit of comet 1P/Halley by the Giotto mission, “KOSI” (comet simulation) – the first large scale laboratory campaign for cometary research – investigated various physical phenomena occurring on comets by using analogue materials. The experiments were also accompanied by intensive modeling activity, which yielded several thermophysical models for the description of comets [1]. Following the revolutionary findings of the Rosetta mission, however, a new laboratory campaign became necessary to adapt existing models and to incorporate the new knowledge acquired. Thus, the “CoPhyLab – Comet Physics Laboratory” project was initiated in 2018 by an international consortium (www.cophylab.space). In the framework of this project so far, we have used several individual experiments to characterize various materials as potential ingredients for a new cometary analogue. In parallel, the construction of a large cryogenic vacuum chamber (“L-Chamber”) was completed, enabling the measurement of refractory-volatile mixtures by multiple instruments simultaneously. While cometary activity involves various physical processes, such as sublimation and gas diffusion in porous materials, understanding the comet’s thermal behavior is crucial. A rigorous thermophysical model (TPM) that considers the porous structure of the cometary material, as well as the volatile phases, and processes connected to them (sublimation, phase changes etc.), would significantly improve the understanding of the evolution of comets. Here, we present a strategy for the development of a new TPM that in the first phase is able to describe the laboratory experiments and ultimately can be scaled to cometary environments to assist the interpretation of data collected by observations and space missions. Strategy Previous modelling activity related to the “KOSI”-campaign was successful in the sense that the experiments could be well described by the models [2, 3]. However, a large variation of parameters between individual experiments impeded a proper understanding of phenomena that occur differently on comets than in the laboratory. Therefore, we pay special attention in the CoPhyLab campaign to repeating the most important baseline experiments. Moreover, we evolve the experiments from a simple form, where well characterized albeit idealized samples are used (e.g. glass beads), to more advanced iterations with more realistic analogue materials. On the one hand, this is to ensure that we understand the fundamental physical processes before considering complex interactions. On the other hand, the simple experiments provide us with the functional dependencies of thermophysical parameters (e.g. thermal conductivity) on the material properties, such as porosity, grain size distribution, grain shape and temperature. Numerical Model We approach the TPM analogously to the experiments, by starting with a very simplified macrophysical model using the finite element method (FEM). Thereby, the material-specific input parameters, which contain the microphysical relations, are taken from the characterization experiments. While most TPMs for comets are one-dimensional [2, 4], the consideration of special spatial features (e.g. sample boundaries, sensor hardware etc.) may require a 3D model [5]. To this end, we use the commercial FEM simulation software “COMSOL Multiphysics”, as it is ideally suited for our application. After the verification of our basic TPM with COMSOL, we will translate and evolve the model as a 1D code in python. This will provide a complementary, resource efficient method to simulate processes such as material ejection. First Phase In the first phase, we will establish the mathematical model that is required to describe the temperature evolution in a dry, well-defined sample (e.g. glass beads), illuminated in vacuum. A schematic of the corresponding experiment is shown in Figure 1. After satisfactory agreement between the model and experiment, more complex iterations will follow, for example by adding volatile phases and modelling their sublimation. These investigations lead to a better understanding of effects in the cometary analogue material during measurements, and in addition, of effects related to the measurement setup and its interaction with the sample. As an extra benefit, a higher accuracy of future laboratory experiments in this context can be achieved. Acknowledgements This work is carried out in the framework of the CoPhyLab project funded by the D-A-CH programme (DFG GU 1620/3-1 and BL 298/26-1 / SNF 200021E 177964 / FWF I 3730-N36). References [1] Sears D. W. G. et al.: Laboratory simulation of the physical processes occurring on and near thesurface of comet nuclei, Meteoritics & Planet. Sci., 34, pp. 497-525, 1999. [2] Spohn, T. and Benkhoff, J.: Thermal history models for KOSI sublimation experiments, Icarus Vol. 87, pp. 358-371. [3] Benkhoff, J. and Spohn, T.: Thermal histories of the KOSI samples, Geophys. Res. Letters Vol. 18, pp. 261-264, 1991. [4] Steiner, G. and Kömle, N. I.: A model of the thermal conductivity of porous water ice at low gas pressures, Planet. Space Sci. Vol. 39, pp. 507-513, 1991. [5] Macher, W et al.: 3D thermal modeling of two selected regions on comet 67P and comparison with Rosetta/MIRO measurements, Astron. Astrophys., Vol. 630, id. A12, 2019.

The success of the Rosetta mission to comet 67P/Churyumov–Gerasimenko has revolutionized our view of comets, while opening a plethora of new questions. In order to find the answers to them and harness the full potential of the new data, an international consortium named “Cophylab – Comet Physics Laboratory” (Cophylab.space) was launched in 2018. In this project several experiment campaigns were initialized to study cometary properties in a controlled environment. The idea was to isolate individual properties and processes in dedicated laboratory experiments. One of the experiment campaigns was designed to characterize gas flow properties of dry porous materials in a first step, with the aim of developing a model that improves our understanding of the outgassing of comets.Before the initial model is extended to consider the sublimation of volatile components, it needs to be validated by alternative methods, such as numerical simulations. For this purpose, we chose the finite element method, to test the combination of the Darcy and Knudsen flow model, which was used in the preceding study.Our approach was to use the results of the experiment as input in the simulations and compare the output with the measurements. This comparison confirmed the validity of the model and its assumptions. In particular, the sample is assumed to be homogenous and isotropic on a macroscopic scale, so that it can be described by a set of averaged parameters. While this description is relatively accurate for samples with well-defined grain shapes (e.g. spherical glass beads), significant discrepancies occur for inhomogeneous materials such as lunar, Asteroid or Martian analogues.We investigated various aspects that were initially neglected in the evaluation of the measurements, such as channel building in the sample, boundary effects and non-ideal geometry of the experimental setup, which will be complemented by inhomogeneities that occur naturally in random close packing or ballistic deposition samples. Furthermore, we assessed the models range of applicability through a thorough review of the different flow regimes encountered in the measurements. Our findings indicate that boundary effects, as well as non-ideal geometry have a significant influence particularly in samples with larger grains. For finer grained samples on the other hand, inhomogeneities are the most probable cause for discrepancies. The grain size also plays an important role regarding the flow regime and its corresponding parameters.The work for this study was performed in the framework of a master’s thesis, as part of the Cophylab project, which is funded by the D-A-CH program (DFG GU1620/3-1 and BL 298/26-1 / SNF 200021E 177964 / FWF I 3730-N36)

A profound understanding of gas flow in porous media is of great interest for various technological and scientific fields. Its investigation by laboratory measurements, however, poses several challenges. In particular, the determination of macroscopic flow parameters from pressure and gas flow measurements is prone to various error influences, some of which are very difficult to analyze experimentally. Computer simulations are a solution in this context as they facilitate modifications of the underlying geometry and boundary conditions in a flexible way. Here we present a simulation framework for the analysis of a recent experiment for determining the Knudsen diffusion coefficient and viscous permeability of various porous granular materials. By combining the finite element method with analytical models and other numerical methods, we were able to identify previously neglected physical effects that increase the uncertainty of the measurements. In particular, the porosity increase due to finite sample dimensions, in a layer of about a grain diameter thickness near the container wall, creates a deviation of the measured pressure gradient. This deviation amounts to ca. 5% for a sample width of about 100 grains and a porosity of 0.5, and is indirectly proportional to the porosity. The second most prominent error source, the sample support sieve, causes a slight constriction of the flow volume. Simulations of this effect show an error around 4%–7%, dependent on the grain size. Based on these findings we recommend an overall sample dimension of 100 grains or larger. As an example of failures of the sample homogeneity, we elaborate how channels through the sample influence the flow properties. Respective suggestions for keeping all discussed effects negligible are discussed in detail. Our methodology demonstrates how the combination of finite element computations with analytical representations of the involved macroscopic parameters can assess the validity and accuracy of laboratory experiments.