Sunny Laddha

To validate the physical model used to describe gas flow through porous, granular media in recent experiments [1], we conducted a finite element analysis (FEA). The main parameters of a sample, such as gas permeability, Knudsen diffusion coefficient and porosity were determined from the experiment and used as an input for the simulations, with the aim of replicating the pressure evolution (measured quantity) depending on external gas flow. While the results matched well for samples with large and spherical grains, the deviations were significant for materials with small and irregularly shaped grains. An improvement of the model could be achieved with complementary simulation methods and a more detailed analysis of the samples.1. IntroductionIn the past decades, multiple cometary space missions (e.g. Deep Impact and Rosetta) have provided us with better data and understanding of the solar system’s primordial objects. The most recent project is a cooperation of six European institutions, called “Cometary Physics Laboratory (CoPhyLab)” and has already produced a plethora of results (see webpage https://cophylab.space/index.php?id=27). In this work we investigated gas flow through porous media with a simulation approach, in order to compare it with the experiment and validate its physical model.2. Experiment The basis and motivation of this work was the experiment performed by [1] (see Figure 1), where the gas flux through a set of artificial and natural porous materials was analysed. To describe the flux J [mol∙m-2∙s-1] of dry air in the given pressure range, a combination of viscous flow and free molecular diffusion was chosen as the physical model:The gas permeability B­0 (viscous flow) and Knudsen diffusion coefficient DK (molecular diffusion) are the main parameters which describe a sample’s ability to allow gas to flow through it. They were determined by measuring the steady state pressure difference across the inlet and outlet of the sample for different gas flux levels. The porosity of the materials was measured through packing experiments.3. SimulationSince the physical model describes the gas flow on a macroscopic level, with the main parameters being determined as average values across the whole sample, a FEA is the ideal method. The simulations were computed with the software “COMSOL Multiphysics®”, in which the sample was modelled as a cylinder (see Figure 2). The built-in physics module “Darcy’s Law” was selected to provide the governing equations and all input parameters were taken from the measurements (sample dimensions, B0, DK, porosity, viscosity etc.). In stationary studies, the pressure at the sample outlet and the inward mass flux of the measurements were defined as boundary conditions for the simulation, with the pressure in the rest of the sample being the free variable. For transient studies, the empty volumes of the vacuum chamber, as well as the pressure dependent performance characteristic of the pump were taken into consideration to be able to compare the dynamic process. Furthermore, variation studies of the main parameters were carried out to investigate their individual influence on the pressure evolution. Boundary effects such as the facilitation of gas flow due to lower porosity at the container wall or opening of channels in the sample, or impediment due to the holding sieve, were also analysed by adapting the material parameters for the respective boundary layers.4. ResultsFor the comparison of stationary simulation results with those of the experiment, the measured pressure at the inlet face of the sample was taken as a boundary condition and for transient simulations also the outlet face pressure (see Figure 3 for an exemplary comparison of results). The deviations were relatively small for synthetic material with large grain sizes (i.e. glass beads with a diameter above 200 µm). However, for smaller grain sizes and natural materials the discrepancies were quite large (up to 35%) and featured no apparent correlation regarding the amount of gas flow.Regarding parameter variation studies, it is important to note that the creation of physical samples where only one parameter varies is very difficult, because many of the parameters are directly or indirectly coupled with each other. Nevertheless, the results indicated that in regimes of low B0 and DK, as is the case for small grain size samples, a variation of either parameter has a large influence on the pressure evolution. This is particularly important when considering measurement and regression errors.When boundary layers were added for the analysis of boundary effects (higher permeability and diffusion at the cylinder wall or in a channel), most of the gas flowed through it and the pressure difference across the sample was minimal. Deviations between simulations with and without the holding sieve, manifested themselves as temporary spikes when the gas flow was increased, with steady state differences below 2%.5. Summary and OutlookThe results indicate that the applied model is well consistent with the measurements of homogenous samples (e.g. synthetic materials with large, spherical grains). However, in the case of small and irregularly shaped grains, the material’s anisotropy could explain the discrepancies. To improve the existing model, it is necessary to analyse the physical samples on a smaller scale and better understand the interdependencies of the governing parameters. This can be achieved by creating virtual samples and using alternative simulation methods (i.e. Monte Carlo), which work on a microscopic scale.6. AcknowledgementsThis work was carried out in the framework of the CoPhyLab project funded by the D-A-CH programme (GU 1620/3-1 and BL298/26-1/SNF 200021E 177964/FWF I 3730-N36).References[1] M. Schweighart, W. Macher, G. Kargl, B. Gundlach, H. L. Capelo, Viscous and Knudsen gas flow through dry...

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 molecule 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 thickness, 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 providing evidence that our analytical model for sphere packing is more accurate than the former classical models.

The diffusion of gas through porous material is important to understand the physical processes underlying cometary activity. We study the diffusion of a rarefied gas (Knudsen regime) through a packed bed of monodisperse spheres via experiments and numerical modelling, providing an absolute value of the diffusion coefficient and compare it to published analytical models. The experiments are designed to be directly comparable to numerical simulations, by using precision steel beads, simple geometries, and a trade-off of the sample size between small boundary effects and efficient computation. For direct comparison, the diffusion coefficient is determined in Direct Simulation Monte Carlo (DSMC) simulations, yielding a good match with experiments. This model is further-on used on a microscopic scale, which cannot be studied in experiments, to determine the mean path of gas molecules and its distribution, and compare it against an analytical model. Scaling with sample properties (particle size, porosity) and gas properties (molecular mass, temperature) is consistent with analytical models. As predicted by these, results are very sensitive on sample porosity and we find that a tortuosity q(ϵ) depending linearly on the porosity ϵ can well reconcile the analytical model with experiments and simulations. Mean paths of molecules are close to those described in the literature, but their distribution deviates from the expectation for small path lengths. The provided diffusion coefficients and scaling laws are directly applicable to thermophysical models of idealised cometary material.

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.

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.

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

The CoPhyLab (Cometary Physics Laboratory) project is designed to study the physics of comets through a series of earth-based experiments. For these experiments, a dust analogue was created with physical properties comparable to those of the non-volatile dust found on comets. This ‘CoPhyLab dust’ is planned to be mixed with water and CO₂ ice and placed under cometary conditions in vacuum chambers to study the physical processes taking place on the nuclei of comets. In order to develop this dust analogue, we mixed two components representative for the non-volatile materials present in cometary nuclei. We chose silica dust as a representative for the mineral phase and charcoal for the organic phase, which also acts as a darkening agent. In this paper, we provide an overview of known cometary analogues before presenting measurements of eight physical properties of different mixtures of the two materials and a comparison of these measurements with known cometary values. The physical properties of interest are particle size, density, gas permeability, spectrophotometry, and mechanical, thermal, and electrical properties. We found that the analogue dust that matches the highest number of physical properties of cometary materials consists of a mixture of either 60 per cent/40 per cent or 70 per cent/30 per cent of silica dust/charcoal by mass. These best-fit dust analogue will be used in future CoPhyLab experiments.

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)