This work presents a novel framework for numerically simulating the depressurization of tanks and pipelines containing carbon dioxide (CO2). The framework focuses on efficient solution strategies for the coupled system of fluid flow equations and thermodynamic constraints. A key contribution lies in proposing a new set of equations for phase equilibrium calculations which simplifies the traditional vapour–liquid equilibrium (VLE) calculations for two-phase CO2 mixtures. The first major novelty resides in the reduction of the conventional four-equation VLE system to a single equation, enabling efficient solution using a non-linear solver. This significantly reduces computational cost compared to traditional methods. Furthermore, a second novelty is introduced by deriving an ordinary differential equation (ODE) directly from the UV-Flash equation. This ODE can be integrated alongside the governing fluid flow equations, offering a computationally efficient approach for simulating depressurization processes.

Numerical simulations are conducted for the wave initiation, growth, and saturation at the oil-water interface in core-annular flow (CAF). The focus is on conditions with a turbulent water annulus, but the laminar water annulus is also considered. The simulation results are compared with lab measurements. The growth rate for the linear instability of different wavelengths in the case of a turbulent water annulus is obtained from two-dimensional (2D) axisymmetric Reynolds-averaged Navier-Stokes (RANS) simulations with the Launder-Sharma low-Reynolds number k-ϵ model. The latter simulation results provide the most unstable wavelength for the turbulent water annulus. Our study also shows the following. The maximum wave growth rate for a turbulent water annulus is significantly higher than for a laminar water annulus. The most unstable wavelength in the simulations is about 25% smaller than in the experiments. The wave amplitude for the different wavelengths in the simulations is typically 17% lower than in the experiments.

In this work we propose a novel method to ensure important entropy inequalities are satisfied semi-discretely when constructing reduced order models (ROMs) on nonlinear reduced manifolds. We are in particular interested in ROMs of systems of nonlinear hyperbolic conservation laws. The so-called entropy stability property endows the semi-discrete ROMs with physically admissible behaviour. The method generalizes earlier results on entropy-stable ROMs constructed on linear spaces. The ROM works by evaluating the projected system on a well-chosen approximation of the state that ensures entropy stability. To ensure accuracy of the ROM after this approximation we locally enrich the tangent space of the reduced manifold with important quantities. Using numerical experiments on some well-known equations (the inviscid Burgers equation, shallow water equations and compressible Euler equations) we show the improved structure-preserving properties of our ROM compared to standard approaches and that our approximations have minimal impact on the accuracy of the ROM. We additionally generalize the recently proposed polynomial reduced manifolds to rational polynomial manifolds and show that this leads to an increase in accuracy for our experiments.

In this paper we present a complete framework for the energy-stable simulation of stratified incompressible flow in channels, using the one-dimensional two-fluid model. Building on earlier energy-conserving work on the basic two-fluid model, our new framework includes diffusion, friction, and surface tension. We show that surface tension can be added in an energy-conserving manner, and that diffusion and friction have a strictly dissipative effect on the energy. We then propose spatial discretizations for these terms such that a semi-discrete model is obtained that has the same conservation properties as the continuous model. Additionally, we propose a new energy-stable advective flux scheme that is energy-conserving in smooth regions of the flow and strictly dissipative where sharp gradients appear. This is obtained by combining, using flux limiters, a previously developed energy-conserving advective flux with a novel first-order upwind scheme that is shown to be strictly dissipative. The complete framework, with diffusion, surface tension, and a bounded energy, is linearly stable to short wavelength perturbations, and exhibits nonlinear damping near shocks. The model yields smoothly converging numerical solutions, even under conditions for which the basic two-fluid model is ill-posed. With our explicit expressions for the dissipation rates, we are able to attribute the nonlinear damping to the different dissipation mechanisms, and compare their effects.

DNS and RANS simulations were carried out for core-annular flow in a horizontal pipe and results were compared with experiments carried out with water and oil in our lab. In contrast to most existing studies for core-annular flow available in the literature, the flow annulus is not laminar but turbulent. This makes the simulations more challenging. As DNS does not contain any closure correlations, this approach should give the best representation of the flow (provided a sufficiently accurate numerical mesh and numerical method is used). Various flow configurations were considered, such as without gravity (to enforce an on-average concentric oil core) and with gravity (to allow for eccentricity in the oil core location). Both single-phase and two-phase conditions were considered; single-phase flow refers to the water annulus with imposed wavy wall, whereas two-phase flow includes the determination of the wavy interface. Mesh refinement was carried out to assess the numerical accuracy of the simulation results.

The pressure-free two-fluid model (PFTFM) is a recent reformulation of the one-dimensional two-fluid model (TFM) for stratified incompressible flow in ducts (including pipes and channels), in which the pressure is eliminated through intricate use of the volume constraint. The disadvantage of the PFTFM was that the volumetric flow rate had to be specified as an input, even though it is an unknown quantity in case of periodic boundary conditions. In this work, we derive an expression for the volumetric flow rate that is based on the demand for energy (and momentum) conservation. This leads to PFTFM solutions that match those of the TFM, justifying the validity and necessity of the derived choice of volumetric flow rate. Furthermore, we extend an energy-conserving spatial discretization of the TFM, in the form of a finite volume scheme, to the PFTFM. We propose a discretization of the volumetric flow rate that yields discrete momentum and energy conservation. The discretization is extended with an energy-conserving discretization of the source terms related to gravity acting in the streamwise direction. Our numerical experiments confirm that the discrete energy is conserved for different problem settings, including sloshing in an inclined closed tank, and a traveling wave in a periodic domain. The PFTFM solutions and the volumetric flow rates match the TFM solutions, with reduced computation time, and with exact momentum and energy conservation.

The Reynolds-Averaged Navier Stokes (RANS) with the Launder & Sharma low-Reynolds number k−ε model was used to simulate core-annular flow in the same configuration with vertical upflow as considered by Kim & Choi (2018), who carried out Direct Numerical Simulations (DNS), and by Vanegas Prada (1999), who performed experiments. The DNS are numerically very accurate and can thus be used for benchmarking of the RANS turbulence model. There is a large ratio between the oil and water viscosities, and the density difference between the water and oil is only small. The frictional pressure drop was fixed and the water holdup fraction was varied. Differences between the RANS and DNS predictions, e.g. in the wave structure and in the Reynolds stresses, are discussed. Despite the shortcomings of the considered Launder & Sharma low-Reynolds number k−ε model in RANS, in comparison to DNS, the RANS approach properly describes the main flow structures for upward moving core-annular flow in a vertical pipe, like the travelling interfacial waves in combination with a turbulent water annulus. The Fanning friction factor with RANS is 18% lower than with DNS, and the holdup ratio with RANS is only slightly higher than with DNS (i.e. it has a slightly larger tendency to accumulate water in RANS than in DNS).

Interfacial waves in core-annular pipe flow are studied through two-phase numerical simulations. Here the water annulus is turbulent, whereas the oil core stays laminar. The low-Reynolds number Launder & Sharma k−ε model is applied. By extracting the moving wave shape from the two-phase results and imposing this as a solid boundary in a single-phase simulation for the water annulus gives single-phase results (for the pressure drop and holdup ratio) that are in close agreement with values obtained from the two-phase approach. The influence of wave amplitude and wave length on the pressure drop and hold up ratio is then studied by using the single-phase flow model. This gives insight in the appearance of core-annular flow, where the water-based Fanning wall-friction factor and the hold-up ratio are selected as the most important quantities. The effect of watercut and eccentricity on these quantities is also investigated.

1D, 2D and 3D numerical simulations were carried out with the Reynolds-Averaged Navier-Stokes equations (RANS) for horizontal oil-water core-annular flow in which the oil core stays laminar while the water layer is turbulent. The turbulence is described with the Launder-Sharma low-Reynolds number k−ϵ model. The simulation results are compared with experiments carried out in our lab in a 21 mm diameter pipe using oil and water with a viscosity ratio of 1150 and a density ratio of 0.91. The 1D results represent perfect turbulent CAF (i.e. no gravity, no interfacial waves), the 2D results represent axi-symmetric CAF (i.e. no gravity, with interfacial waves), and the 3D results represent eccentric CAF (i.e. with gravity, with interfacial waves). The simulation results typically show a turbulent water annulus in which the structure of the (high-Reynolds number) inertial sublayer can be recognized. The pressure drop reduction factor (which is the ratio between the pressure drop for CAF and the pressure drop for single phase viscous oil flow) for the 2D and 3D results is about the same, but its value is about 35% higher than in the experiment. The hold-up ratio in the 3D model is close to the experimental value, but the 2D prediction is slightly lower. The eccentricity predicted by the 3D simulations is much higher than in the experiment. Most likely, the observed differences between the simulations and the experiments are due to limitations of using a low-Reynolds number k−ϵ model. In particular the water layer at the top in the 3D results shows a relaminarization, which might be absent in the experiment.

A novel pressure-free two-fluid model formulation is proposed for the simulation of one-dimensional incompressible multiphase flow in pipelines and channels. The model is obtained by simultaneously eliminating the volume constraint and the pressure from the widely used two-fluid model (TFM). The resulting ‘pressure-free two-fluid model’ (PF-TFM) has a number of attractive features: (i) it features four evolution equations (without additional constraints) that can be solved very quickly with explicit time integration methods; (ii) it keeps the conservation properties of the original two-fluid model, and therefore the correct shock relations in case of discontinuities; (iii) its solutions satisfy the two TFM constraints exactly: the volume constraint and the volumetric flow constraint; (iv) it offers a convenient form to analytically analyse the equation system, since the constraint has been removed. A staggered-grid spatial discretization and an explicit Runge-Kutta time integration method are proposed, which keep the constraints exactly satisfied when numerically solving the PF-TFM. Furthermore, for the case of strongly imposed boundary conditions, a novel adapted Runge-Kutta formulation is proposed that keeps the volumetric flow exact in time while retaining high order accuracy. Several test cases confirm the theoretical properties and show the efficiency of the new pressure-free model.

Core-annular flow is an efficient way of transporting viscous oil through a pipeline. A sharp increase in the pressure drop will occur when the oil waves at the water-oil interface touch the pipe wall. Depending on the oil and pipe material physical properties, the oil may adhere to the wall leading to fouling. Therefore, a necessary requirement for the onset of oil fouling of the pipe wall is that the flow hydrodynamics allow the oil to reach and touch the wall. With respect to the problem statement, this study deals with finding the hydrodynamic conditions under which core-annular flow becomes unstable and the oil waves touch the pipe wall. The method that is followed is to resolve the first-principle set of equations that describe the hydrodynamics: the Reynolds-Averaged Navier-Stokes (RANS) equations are solved using Computational Fluid Dynamics (CFD) in the opensource package OpenFOAM. Simulations were carried out for the horizontal pipe with two diameters (10.5 and 21 mm), at a range of imposed pressure drops and water holdup fractions (giving the mixture velocity and watercut as output). Most simulations were carried out for an oil to water viscosity ratio of 1040 (but also a variation of this was considered). For each value of the pressure drop (or mixture velocity) there is a critical value of the watercut below which the oil reaches the pipe wall. This critical value of the watercut is lower for the larger pipe diameter of 21 mm, namely about 9.6%, than for the smaller pipe diameter of 10.5 mm, namely about 14% (for a viscosity ratio m = 1040). Wall touching occurs when the mixture velocity is too low, but this lower limit is significantly higher for the larger pipe diameter of 21 mm, namely about 1.1 m/s, than for the smaller pipe diameter, namely about 0.3 m/s (for a viscosity ratio m = 1040). The main conclusion is that a state-of-art CFD approach is capable of simulating the growth of waves at the oil-water interface until they touch the pipe wall, which is a necessary condition for the onset of fouling.

Hydrogen is one of the most popular alternatives for energy storage. Because of its low volumetric energy density, hydrogen should be compressed for practical storage and transportation purposes. Recently, electrochemical hydrogen compressors (EHCs) have been developed that are capable of compressing hydrogen up to P = 1000 bar, and have the potential of reducing compression costs to 3 kWh/kg. As EHC compressed hydrogen is saturated with water, the maximum water content in gaseous hydrogen should meet the fuel requirements issued by the International Organization for Standardization (ISO) when refuelling fuel cell electric vehicles. The ISO 14687-2:2012 standard has limited the water concentration in hydrogen gas to 5 μmol water per mol hydrogen fuel mixture. Knowledge on the vapor liquid equilibrium of H₂O-H₂ mixtures is crucial for designing a method to remove H₂O from compressed H₂. To the best of our knowledge, the only experimental high pressure data (P > 300 bar) for the H₂O-H₂ phase coexistence is from 1927 [J. Am. Chem. Soc., 1927, 49, 65-78]. In this paper, we have used molecular simulation and thermodynamic modeling to study the phase coexistence of the H₂O-H₂ system for temperatures between T = 283 K and T = 423 K and pressures between P = 10 bar and P = 1000 bar. It is shown that the Peng-Robinson equation of state and the Soave Redlich-Kwong equation of state with van der Waals mixing rules fail to accurately predict the equilibrium coexistence compositions of the liquid and gas phase, with or without fitted binary interaction parameters. We have shown that the solubility of water in compressed hydrogen is adequately predicted using force-field-based molecular simulations. The modeling of phase coexistence of H₂O-H₂ mixtures will be improved by using polarizable models for water. In the Supporting Information, we present a detailed overview of available experimental vapor-liquid equilibrium and solubility data for the H₂O-H₂ system at high pressures.

We present experimental and numerical results for by-pass pigging under low-pressure conditions which aided the design of a speed-controlled pig (Pipeline Inspection Gauge). Our study was carried out using air as working fluid at atmospheric pressure in a 52 mm diameter pipe of 62 m length. The experimental results have been used to validate simplified 1D models commonly used in the oil and gas industry to model transient pig behaviour. Due to the low pressure conditions oscillatory behavior is observed in the pig speed, which results in high pig velocity excursions. The oscillatory motion is described with a simplified model which is used to design a simple controller aimed at minimizing these oscillations. The controller relies on dynamically adjusting the by-pass area, which allows to release part of the excess pressure which builds up in the gas pocket upstream of the pig when the motion of the pig is arrested. Subsequently, the control algorithm is tested by a 1D transient numerical model and it was shown to successfully reduce the pig velocity excursions.

We have developed and applied an Eulerian-Lagrangian model for the transport, formation, break-up, deposition and re-entrainment of particle agglomerates. In this paper, we focus on agglomeration and break-up. Simulations were carried out to investigate what changes in the turbulent flow are inflicted by the presence of the agglomerates. Also, the dependence of the properties of the agglomerates on the Reynolds number of the flow and on the strength of the bonds between the primary particles is studied. The presence of the agglomerates attenuates the turbulence and thereby lowers the Reynolds stresses. As a result, the flow rate increases at constant pressure drop when agglomerates are formed (up to a certain dimension). If the agglomerates surpass this dimension, long-distance viscosity effects become dominant and a flow rate decrease occurs. The characteristics of the agglomerates are largely insensitive to the Reynolds number, provided the flow is turbulent. The agglomerates have an open and porous structure, and a fractal dimension of 1.8-2.3. Their mean mass scales exponentially with the strength of the internal bonds. Contrary to assumptions that are typically made in engineering models in the literature, agglomerates do not preferentially break into two fragments of similar size.

Based on the success of foam in subsurface applications it is of interest to investigate whether foam can also help overcome liquid management problems in surface flowlineriser systems. Therefore, flow experiments were carried out in the flow loop at the Shell Technology Centre Amsterdam. The facility consists of a 100 m long horizontal flowline (with 50.8 mm diameter) followed by a 16.8-m vertical riser (with 44 mm diameter). Air and water are the working fluids, and operation is at atmospheric outlet pressure. Foam is created by adding "DreftTM" (a dishwashing detergent) in various concentrations to the water/air flow. Experiments were taken both without and with foam. Various measurement techniques were used: differential pressure sensors, flow visualization, and Distributed Acoustic Sensors (DAS) (fibre optics). The focus is on (growing) slugs in the horizontal flowline, and severe slugging in the flowline-riser. It can be concluded from the small-scale lab experiments that adding a surfactant mitigates (growing) slugs in (nearly) horizontal flowlines, whereas the severe slugging cycle in a flowline-riser configuration cannot be broken.

Based on our earlier experimental work on the effect of surfactants on air-water flow in vertical pipes with internal diameters of 34 mm, 50 mm, and 80 mm, we create a mechanistic annular flow model for the pressure gradient. The major effect of the addition of surfactants is the formation of foam. We model the formation of foam and its impact on the flow. In the model we consider a gas core and a film at the wall, which consists of a layer of liquid at the wall and a layer of foam between the liquid layer and the gas core. We do not consider entrainment in the model. We developed four closure relations in order to solve the model: (i) for the density of the foam, (ii) for the viscosity of the foam, (iii) for the interfacial friction between the gas and the film, and (iv) for the thickness of the liquid layer at the wall. Subsequently, we solve for the film thickness that yields the imposed liquid flow rate. Comparing the experimental results for the pressure gradient to the results from the model, we observe that in most cases the model can predict the pressure gradient within 25%. Furthermore, the model is able to predict the onset of downwards flow in the film. Therefore, it can predict the transition between annular flow and churn flow. We show that the effect of five different surfactants on the flow is equal, apart from a scaling factor of the concentration, which means that the model can be applied for many different types of surfactants. The scaling factor is an input parameter to the model, which needs to be determined in a small scale experiment.

Stratified gas-liquid flow is a flow regime typically encountered in multiphase pipelines. The understanding and modeling of this regime is of engineering importance especially for the oil and gas industry. In this work, simulations have been conducted for stratified air-water flow in pipes. We solved the Reynolds-averaged Navier-Stokes (RANS) equations with the volume of fluid (VOF) method. The aim of this work was to evaluate the performance of the k-ω shear stress transport (SST) turbulence model with and without damping of the turbulence at the gas-liquid interface. Simulation results were compared with some of the latest experimental results found in the literature. A comparison between the simulated velocity and kinetic energy profiles and the experimental results obtained with the particle image velocimetry (PIV) technique was conducted. The characteristics of the interfacial waves were also extracted and compared with the experiments. It is shown that a proper damping of the turbulence close to the interface is needed to obtain agreement with the experimental pressure drop and liquid hold-up. In its current form, however, RANS with the k-ω turbulence model is still not able to give an accurate prediction of the velocity profiles and of the interface waves.