Modeling of Heat Transfer in Two-Phase Flow Using the Level-Set Method

The implementation of a two-phase flow model, developed at SINTEF Energy Research and relying on the level-set method, was extended through the discretization and implementation of an advection-diffusion equation for temperature and a Boussinesq coupling between the temperature and velocity fields. In two-phase flow, both the continuum surface force method and the ghost-fluid method was employed for handling jumps at the interface. Results from simulated cases indicated that the implementation for both single- and two-phase flow with the ghost-fluid method was correct, with second- and first-order convergence, respectively.Also, a model for phase transition was implemented to allow for vaporization and condensation mass transport between the phases. Results from simulated one-dimensional cases indicated that the implementation of this model was correct in one dimension, with first-order convergence. These results from one dimension and the qualitatively correct results from two dimensions gave reason to believe that the implementation was correct also in two dimensions.Through the introduction of heat-transport physics in the implementation of the two-phase flow model, this implementation has been developed in direction of performing more detailed simulations that are relevant for natural gas liquefaction processes

Modelling of two-phase equilibrium, stability and steady-state flow in porous media

This thesis concerns fundamental aspects of coexistence and flow of two fluid phases within porous media. Specifically, the focus is on thermodynamic stability and equilibrium on the scale of a single pore and on macroscopic steady-state properties of immiscible two-phase flow. In this work, macroscopic steady-state properties refers to time-averages of time dependent quantities that describe flow through a volume element consisting of many pores. Capillary models are derived for free and adsorbed droplets and bubbles, and thick films in a pore. The thermodynamic stability of these structures in a specific pore geometry is mapped out and the effect of pore size and the pore being open or closed w.r.t. exchange of particles with the surroundings is explored. Equilibrium structures are found. When structures are unstable, they are found to be unstable against perturbations that can be classified as either translation or as condensation/evaporation. Furthermore, stability of thin films and droplets on a flat solid surface is considered. It is found that the often-cited criterion for stability of a flat film, which states that the flat films are stable when the derivative of disjoining pressure w.r.t. film thickness is negative, applies in open systems. In closed systems, however, stability is governed by mechanical instabilities that require a large enough substrate size to render the film unstable. Unstable droplets in both the open and closed containers are found to represent saddle points and activation barriers in their respective energy landscapes. Numerical methods are presented that enable stable and fast time integration of a pore network model. These eliminate previous problems with numerical instabilities observed at low capillary numbers. The new methods extend the range of capillary numbers for which the pore network model is a tractable alternative and enables e.g. future studies of Haines jumps in the low capillary number regime. By pore network modelling and lattice-Boltzmann simulations in the high capillary number limit, it is found that the total flow rate follows a Darcy-type equation where the fluid viscosity is replaced by an effective viscosity. This effective viscosity can be modelled by the Lichtenecker-Rother equation. Results from more than 6000 steady-state simulations using the pore network model are presented that range from the high capillary number limit and down to ~10-4. Variation in dimensionless output from the model is shown to be explainable by three dimensionless variables: the wetting fluid saturation, the viscosity ratio and a dimensionless pressure gradient. In the high capillary number limit, the model gives relative permeabilities that do not form straight lines when the viscosity ratio is different from one. This thesis also addresses the computational challenges associated with calculating the thermodynamic stability limits of multi-component mixtures and the identification of extrema as minima, maxima or saddle points in variational calculus

Thermodynamically consistent modeling of gas flow and adsorption in porous media

In modeling of gas flow through porous media width adsorption, the thermodynamic properties of the adsorbed phase are usually approximated by those of the bulk liquid. Using non-isothermal, gaseous transport of moist air through a porous insulation material as example, we show that this leads to violation of the second law of thermodynamics and a negative entropy production. To resolve this violation, we use information about the adsorption and thermodynamic properties of bulk fluids to derive consistent thermodynamic properties of the adsorbed phase, such as the chemical potential, enthalpy and entropy. The resulting chemical potential of the adsorbed phase is a starting point for rate-based models for adsorption based on non-equilibrium thermodynamics. Incorporating the consistent thermodynamic description into the energy, entropy and momentum balances restores agreement with the second law of thermodynamics. We show that the temperature evolution in the porous medium from the consistent description differs from the standard formulation only if the adsorption depends explicitly on temperature. This highlights the importance of characterizing the temperature dependence of the adsorption with experiments or molecular simulations for accurate non-isothermal modeling of porous media.publishedVersio

Fluid Meniscus Algorithms for Dynamic Pore-Network Modeling of Immiscible Two-Phase Flow in Porous Media

We present in detail a set of algorithms for a dynamic pore-network model of immiscible two-phase flow in porous media to carry out fluid displacements in pores. The algorithms are universal for regular and irregular pore networks in two or three dimensions and can be applied to simulate both drainage displacements and steady-state flow. They execute the mixing of incoming fluids at the network nodes, then distribute them to the outgoing links and perform the coalescence of bubbles. Implementing these algorithms in a dynamic pore-network model, we reproduce some of the fundamental results of transient and steady-state two-phase flow in porous media. For drainage displacements, we show that the model can reproduce the flow patterns corresponding to viscous fingering, capillary fingering and stable displacement by varying the capillary number and viscosity ratio. For steady-state flow, we verify non-linear rheological properties and transition to linear Darcy behavior while increasing the flow rate. Finally we verify the relations between seepage velocities of two-phase flow in porous media considering both disordered regular networks and irregular networks reconstructed from real samples.publishedVersionCopyright © 2021 Sinha, Gjennestad, Vassvik and Hansen. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms

A volume-averaged model for acoustic streaming induced by focused ultrasound in soft porous media

Equations describing acoustic streaming in soft, porous media driven by focused ultrasound are derived based on the assumption that acoustic waves pass through the porous material as if it were homogeneous. From these equations, a model that predicts the time-averaged flow on the macroscopic scale, as well as the advective transport of the trace components, is created. The model is used to perform simulations for different shapes of the focused ultrasound beam. For a given shape, and using the paraxial approximation for the ultrasound, the acoustic streaming is found to be linearly proportional to the applied ultrasound intensity, to the permeability of the porous material and to the attenuation coefficient, and inversely proportional to the liquid viscosity. Results from simulations are compared to a simplified expression stating that the dimensionless volumetric liquid flux is equal to the dimensionless acoustic radiation force. This approximation for the acoustic streaming is found to be reasonable near the beam axis for focused ultrasound beam shapes that are long in the axial direction, compared to their width. Finally, a comparison is made between the model and experimental results on acoustic streaming in a gel, and good agreement is found.acceptedVersio

The influence of thermal diffusion on water migration through a porous insulation material

Excess water on pipes and equipment under porous insulation materials can lead to undesired corrosion. This work aims to clarify to what extent thermal diffusion affects the migration of water inside insulation materials subject to large temperature gradients. Since no experimental data is available on the thermal diffusion coefficients of humid air, revised Enskog theory for Mie fluids is used to estimate transport properties. Comparison to experimental data from literature shows that the theory reproduces the diffusion coefficient, viscosity and thermal conductivity of humid air within 8.7%, 5.0% and 3.5% respectively. The small discrepancies suggest that the theory can also provide reliable estimates of the thermal diffusion coefficients. In the investigated composition and temperature range, the theory predicts the Soret coefficient of water to be approximately [Formula presented], while the Soret coefficient of oxygen varies from [Formula presented] to +[Formula presented]. A case study with heating of glass wool insulation containing humid air, encapsulating a cylindrical pipe is investigated. Non-equilibrium thermodynamics is used to consistently incorporate the Soret coefficients into the flux equations in a dynamic, non-isothermal model that includes diffusion, convection, thermal conduction and water sorption in the porous medium. With 50 K temperature difference across 5 cm of insulation, we find that at steady-state, thermal diffusion leads to a mole fraction of water in the gas phase that is about 1.5% higher at the hot location than if thermal diffusion is neglected. © 2024 The Author(s)The influence of thermal diffusion on water migration through a porous insulation materialpublishedVersio

Thermal performance estimation for cryogenic storage tanks: Application to liquid hydrogen

The design of cryogenic liquid storage solutions requires accurate methods for estimating heat ingress, from the material level to the tank level. For insulation materials, thermal performance is usually measured using ambient conditions and liquid nitrogen at 77 K as boundary temperatures. A key question is how much heat ingress increases when storing liquid hydrogen (LH2) at 20 K. We address this by introducing the Concavity Hypothesis, namely that heat ingress is a concave function of the cold boundary temperature, and show that the increase in heat ingress is below 26 %. Additionally, we demonstrate that heat ingress is much more sensitive to the warm boundary temperature than the cold boundary temperature. At the tank level, we compare two methods for assessing the steady-state thermal performance of cryogenic tanks: thermal network models and the heat equation solved with the finite element method. The latter offers high accuracy and adaptability for complex geometries, while thermal network models benefit from simplicity, speed and robustness. We apply both approaches to a self-supported 40 000 m LH2 tank concept for maritime transport that operates at constant pressure, and analyze sensitivity to structural support thickness, warm boundary temperature, and choice of insulation material. The thermal network model can estimate heat ingress with -1% error and the cold-spot temperature with error less than 1 K.Thermal performance estimation for cryogenic storage tanks: Application to liquid hydrogenpublishedVersio

The effect of temperature constraints on the treatment of tumors using focused ultrasound-induced acoustic streaming

The transport of drugs into tumor cells near the center of the tumor is known to be severely hindered due to the high interstitial pressure and poor vascularization. The aim of this work is to investigate the possibility to induce acoustic streaming in a tumor. Two tumor cases (breast and abdomen) are simulated to find the acoustic streaming and temperature rise, while varying the focused ultrasound transducer radius, frequency, and power for a constant duty cycle (1%). In the absence of perfusion, the simulated rise in temperature, despite the low duty cycle, never reaches a steady state and is fitted to a logarithmic equation, enabling predictions of the temperature for long treatment times. Higher frequencies and larger probe radii are found to result in shorter treatment times relative to the temperature rise, at the cost of a smaller treated area. Results from the simulations indicate that it may be possible to achieve reasonable acoustic streaming values in tumor without the temperature exceeding 50 °C. Treatment times for streaming a distance of 50 μm in the breast case are shown to range from less than one and a half hour to 93 h, depending on the probe settings. © The Author(s) 2024.publishedVersio

Analysis of thermal stratification and global heat and mass balance in cryogenic storage tanks

A modelling approach is presented that couples a detailed model of the heat flow through the insulation and walls of a cryogenic storage tank to a description of the fluid that accounts isobaric boil-off and for thermal stratification of the gas phase. We apply this approach to simulate the boil-off of both liquefied nitrogen (LN2) and liquefied hydrogen (LH2) at different fill ratios in an example case. The simulation results show that experimentally measured boil-off rates are captured well by the model. Thermal stratification in the gas phase resulted in heat flow from the gas side of the tank to the liquid side through the inner tank wall. This heat flow was most important for the LN2 case, but was also of some significance for the LH2 case, and lumped-phase tank models may therefore benefit from accounting for it. An equation for this heat flow was fitted and found to describe the variation with fill level accurately.acceptedVersio