Moving Boundary Approach Research Articles

Addressing Mass Conservation in Two-Dimensional Modeling of Lithium Metal Batteries with Electrochemically Plated/Stripped Interfaces

Several previous studies have aimed to develop mathematical models based on the moving boundary approach for predicting changes in lithium morphology during the cycling of lithium metal batteries under different operating conditions. These modeling frameworks play a crucial role in enhancing our fundamental understanding of the transport and reaction mechanisms and guide experimentalists in developing safer and more efficient lithium metal anodes. In this work, using a simple two-dimensional model for the lithium metal battery, we aim to bring attention to the bulk convection in the liquid electrolytes induced by the movement of the lithium metal surface modeled as a moving boundary. The back-and-forth motion of the lithium metal surface during the plating and stripping of lithium introduces a weak fluid motion in the liquid electrolyte that should be incorporated in the model equations and corresponding boundary conditions (Figure 1). The results for the electrochemical signatures and morphology evolution thus obtained by solving a coupled fluid model are compared with the case where the velocity distribution in the liquid electrolyte is ignored. This work extends our previously reported perspective on the convective flux correction to rectify the mass conservation failures at moving boundaries in one-dimensional models to two dimensions. This careful implementation of the correct boundary conditions ensures the mass conservation of lithium in two-dimensional simulations for predicting the morphological evolution of lithium metal electrodes over cycles. These revised battery models with accurate mass and charge conservation are crucial for predicting the performance of the battery over cycles. Additionally, these relative fluxes at the moving and fixed boundaries are sometimes ignored by assuming a bulk concentration condition at the far end, especially at the cathode/separator interface. While it may not affect overpotential signatures at the anode, it leads to mass conservation issues with implications for the accuracy of cycling simulations. Thus, with the help of a two-dimensional electro-convection model, this study addresses the role of bulk convection in liquid electrolytes and the importance of ensuring mass conservation in lithium metal battery models.

Navier-Stokes-Fourier fluids interacting with elastic shells

We study the motion of a compressible heat-conducting fluid in three dimensions interacting with a non-linear flexible shell. The fluid is described by the full Navier--Stokes--Fourier system. The shell constitutes an unknown part of the boundary of the physical domain of the fluid and is changing in time. The solid is described as an elastic non-linear shell of Koiter type; in particular it possesses a non-convex elastic energy. % Its deformation is modelled by a (non-linear) elastic Koiter shell. We show the existence of a weak solution to the corresponding system of PDEs which exists until the moving boundary approaches a self-intersection or the non-linear elastic energy of the shell degenerates. It is achieved by compactness results (in highest order spaces) for the solid-deformation and fluid-density. Our solutions comply with the first and second law of thermodynamics: the total energy is preserved and the entropy balance is understood as a variational inequality.

Modelling and analysis of the volume change behaviors of Li-ion batteries with silicon-graphene composite electrodes

Silicon-based composites are recognized as one of the most promising negative electrode materials owing to their high theoretical capacity. However, during (dis)charging, the silicon particles undergo significant volume changes, resulting in electrode thickness variation over the cycle. This maps to the cell scale as a change in cell thickness, which translates into outward pressure. To demonstrate the impact of swelling on the cell thickness variation, a model is developed using the moving boundary approach. The ratios of the composites are classified in detail to accurately analyse the contribution of two materials to the capacity. The impacts of the electrode design, such as the thickness of the active layer and the porosity of the negative electrode, as well as the operating conditions on the electrochemical performance and thickness variation of the cell are also investigated. The capacity contribution increases from 85% to 92% when the silicon content increases, and then changing the electrode structure. When the porosity raises from 40% to 60% and the thickness of the negative active layer increases from 55 μm to 85 μm, the capacity utilization varies from 58% to 79% and from 68% to 59%, respectively. In the rate test, the cell thickness variation reduces from 2.49 μm to 1.56 μm when the rate changes from 0.5 C to 2 C. The expansion/contraction behaviour of the active particles during the lithiation/delithiation process is utilized to investigate the variation of cell thickness under different conditions. The optimized simulation is applied to establish the stress variation model to provide insight into the stress evolution during cycling, which can be employed to more accurately assess the performance and potential risk.

Dynamic model and deep neural network-based surrogate model to predict dynamic behaviors and steady-state performance of solid propellant combustion

Understanding the dynamic combustion behavior of a solid propellant with reaction mechanisms is essential for designing rocket engines and safely disposing of expired propellants. In this study, a rigorous mathematical model was applied to predict the dynamic behavior of propellant combustion (HMX/GAP). Because accurate and fast prediction can contribute to saving design time and enhancing operational safety, a deep neural network surrogate model was also developed to predict propellant combustion characteristics, such as burning rate, gas phase temperature, and mole fraction. After the mathematical model with a moving boundary approach was validated with reported data, the DNN surrogate model was trained and tested with data generated through a stochastic simulation using the mathematical model. The prediction accuracies of the developed surrogate model were 96.46%, 99.72%, and 97.62% for the burning rate, gas phase temperature, and mole fraction, respectively. Compared with the dynamic simulation, the computational time of the surrogate model was 596 times faster for the prediction of the burning rate and gas phase temperature and 590 times faster for the mole fraction. The developed framework can be applied to the design of the optimal composition of solid propellants for the desired combustion behavior. It will also help in the safe incineration of expired propellants in real systems.

The Importance of a Moving Boundary Approach for Modeling the SEI Layer Growth to Predict Capacity Fade

One of the contributing factors to the aging of lithium-ion batteries is the growth of the solid-electrolyte interphase (SEI) layer. The growth of the SEI layer leads to the irreversible loss of lithium available for cycling and increases the resistance of the battery. Physics-based models in literature model the kinetically limited or solvent diffusion-limited growth. In such models, the interface resistance is a constant, and the contribution to the overpotential of the intercalation reaction from the SEI layer is considered to be ohmic. In this study, we propose a model that describes the growth of the SEI layer on the electrode surface as a moving interface. The transport of lithium ions and the solvent in the electrolyte are affected by this moving interface. The equations that govern the species transport and the potential drop across the SEI layer are derived from dilute solution theory and solved by transforming the coordinates of the moving boundary. The ion transport induces changes in the conductivity across the SEI layer, which affects the potential drop that arises due to its growth. The effects of this potential on capacity fade are studied over cycling the battery.

Directionality of Macrophages Movement in Tumour Invasion: A Multiscale Moving-Boundary Approach

Invasion of the surrounding tissue is one of the recognised hallmarks of cancer (Hanahan and Weinberg in Cell 100: 57–70, 2000. https://doi.org/10.1016/S0092-8674(00)81683-9), which is accomplished through a complex heterotypic multiscale dynamics involving tissue-scale random and directed movement of the population of both cancer cells and other accompanying cells (including here, the family of tumour-associated macrophages) as well as the emerging cell-scale activity of both the matrix-degrading enzymes and the rearrangement of the cell-scale constituents of the extracellular matrix (ECM) fibres. The involved processes include not only the presence of cell proliferation and cell adhesion (to other cells and to the extracellular matrix), but also the secretion of matrix-degrading enzymes. This is as a result of cancer cells as well as macrophages, which are one of the most abundant types of immune cells in the tumour micro-environment. In large tumours, these tumour-associated macrophages (TAMs) have a tumour-promoting phenotype, contributing to tumour proliferation and spread. In this paper, we extend a previous multiscale moving-boundary mathematical model for cancer invasion, by considering also the multiscale effects of TAMs, with special focus on the influence that their directional movement exerts on the overall tumour progression. Numerical investigation of this new model shows the importance of the interactions between pro-tumour TAMs and the fibrous ECM, highlighting the impact of the fibres on the spatial structure of solid tumour.

Comparison between moving-boundary and distributed models for predicting the time evolution of the solidification front in ice trays

The present paper is aimed at modelling the water solidification in a two-dimensional cavity that emulates an ice tray. A simplified moving-boundary approach is proposed based on the Stefan formulation to track the solidification front over time. Its prediction capabilities were compared with the results of a tailormade finite-volume model for different cavity aspect ratios and cooling conditions. It was found that the moving-boundary model predictions follow closely the time evolution of the solidification front obtained from the distributed model. A maximum deviation of 3.5% was observed for the freezing lead time, while the simplified approach demanded much lower computation time. A sensitivity analysis was also carried out. The results indicate that the ice production rates are ruled by the external air-side convection rather than the ice tray aspect ratio.

Moving boundary approach to forecast tight oil production

AbstractA novel semianalytical production predictive tool for tight reservoirs based on the application of material balance on a transient linear flow system is developed in this paper. This method considers two important regions during transient production of oil reservoirs: the saturated region where gas evolves and flows with oil, and the undersaturated region where only oil flows. A zonal moving boundary approach is used to evolve the two regions as the reservoir pressure gradually decreases. A semianalytical method is used to calculate pressures in the various regions and volumetric expansions. For both black oil and volatile oil scenarios, calculations from this analytical framework are able to match reservoir pressures, oil and gas rates, and cumulative gas–oil ratios determined using a reservoir simulator. The model was also applied to wells in tight reservoirs around the United States such as the Bakken (ND) and the Eagle Ford (TX) with reasonable success.

Optimal control analysis and Practical NMPC applied to refrigeration systems

This work is focused on optimal control of mechanical compression refrigeration systems. A reduced-order state-space model based on the moving boundary approach is proposed for the canonical cycle, which eases the controller design. The optimal cycle (that satisfying the cooling demand while maximizing efficiency) is defined by three variables, but only two inputs are available, therefore the controllability of the proposed model is studied. It is shown through optimization simulations how optimal cycles for a range of the cooling demand turn out not to be achieved by keeping the degree of superheating to a minimum. The Practical NMPC and a well-known feedback-plus-feedforward strategy from the literature are compared in simulation, both showing trouble in reaching the optimal cycle, which agrees with the controllability study.

Dynamic modeling and control strategies of organic Rankine cycle systems: Methods and challenges

Organic Rankine cycle systems are suitable technologies for utilization of low/medium-temperature heat sources, especially for small-scale systems. Waste heat from engines in the transportation sector, solar energy, and intermittent industrial waste heat are by nature transient heat sources, making it a challenging task to design and operate the organic Rankine cycle system safely and efficiently for these heat sources. Therefore, it is of crucial importance to investigate the dynamic behavior of the organic Rankine cycle system and develop suitable control strategies. This paper provides a comprehensive review of the previous studies in the area of dynamic modeling and control of the organic Rankine cycle system. The most common dynamic modeling approaches, typical issues during dynamic simulations, and different control strategies are discussed in detail. The most suitable dynamic modeling approaches of each component, solutions to common problems, and optimal control approaches are identified. Directions for future research are provided. The review indicates that the dynamics of the organic Rankine cycle system is mainly governed by the heat exchangers. Depending on the level of accuracy and computational effort, a moving boundary approach, a finite volume method or a two-volume simplification can be used for the modeling of the heat exchangers. From the control perspective, the model predictive controllers, especially improved model predictive controllers (e.g. the multiple model predictive control, switching model predictive control, and non-linear model predictive control approach), provide excellent control performance compared to conventional control strategies (e.g. proportional–integral controller, proportional–derivative controller, and proportional–integral–derivative controllers). We recommend that future research focuses on the integrated design and optimization, especially considering the design of the heat exchangers, the dynamic response of the system and its controllability.

Dynamic modeling and simulation of the combustion of aluminized solid propellant with HMX and GAP using moving boundary approach

This research describes the influence of nano-sized aluminum with varying contents (0–20 wt%) on the combustion behaviors of HMX/GAP/Al in aspects such as burning rate, surface temperature, gas phase temperature, mole fraction of species, and specific impulse. A rigorous mathematical model is developed for three phases (solid phase, condensed phase, and gas phase) with detailed kinetics. This model consists of 507 gas phase reactions and 4 condensed phase reactions of HMX/GAP/Al combustion. The model also emphasizes the phase transitions and reactions of aluminum in the gas phase. Based on this model, dynamic simulation is carried out at various propellant compositions and operating pressures by means of the moving boundary approach using gPROMS software package. The simulation results are in close agreement with the experimental data at slight marginal errors for HMX/Al combustion, predicting the combustion behaviors for the HMX/GAP/Al system. Accordingly, the model predicts the gas phase temperature of 3061 K for the 20 wt% Al and the specific impulse of 258.53 s for the 15 wt% Al of HMX/GAP/Al propellant under an operating pressure of 100 atm. The increase in burning rate and specific impulse by increasing the pressure is also indicated. According to this study, the addition of aluminum particles with a content range of 15–20 wt% considerably improves combustion behaviors. By dynamic modeling and simulation, a detailed framework for studying the multi-phase combustion of aluminized solid propellant is introduced.

Dynamic modeling of an R245fa ejector based refrigeration system

The dynamic behavior of the ejector refrigeration system (ERS) plays an important role in its performance prediction and control system design. This paper numerically and experimentally investigated ERS dynamic behavior by connecting generator, condenser and evaporator dynamic models into a single ERS dynamic model, as the heat exchangers dominate ERS thermal dynamics. The developed dynamic models of the heat exchangers were based on the moving boundary approach. A 1-D thermodynamic model of the ejector was also developed to evaluate the fluid behavior through the ejector. The pump and expansion valve were modeled using algebraic equations, as they were considered as steady state components in the system. The developed ERS dynamic model was validated experimentally by measuring the transient response to the step change of the pump speed. The validation studies revealed that the model is able to capture the refrigeration system's dynamics in terms of the pressures and outlet specific enthalpies of the heat exchangers, with relative errors of less than 3.78 and 0.5%, respectively.

A static moving boundary modelling approach for simulation of indirect evaporative free cooling systems

Evaporative Cooling and Free-Cooling technologies have gained a growing interest in air-conditioning systems and they are suitable in different air conditioning applications: commercial, industrial, residential, and data centers. The Evaporative Cooling technologies are environmentally friendly and they have a very low global warming impact. Moreover, under the right applications, conditions, and operations, these technologies can provide excellent cooling and ventilation with minimal energy consumption. Since computing experiment based on mathematical modelling enables conduction of research in a shorter time and at smaller costs, in this paper first we develop a First-Principle Data-Driven model for an Indirect Evaporative Cooling system with Free Cooling and then we accordingly design a Matlab-based simulation environment. In particular, we model the key component, i.e. the evaporative heat exchanger, by means of a static Moving Boundary approach that segments the heat exchanger depending on the physical phenomena that occur inside it, providing a good balance between model complexity and accuracy. Simulation examples show how the model mimics properly some fundamental thermal aspects of the indirect evaporative cooling process.

Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion

Recognised as one of the hallmarks of cancer, local cancer cell invasion is a complex multiscale process that combines the secretion of matrix-degrading enzymes with a series of altered key cell processes (such as abnormal cell proliferation and changes in cell–cell and cell–matrix adhesion leading to enhanced migration) to degrade important components of the surrounding extracellular matrix (ECM) and this way spread further in the human tissue. In order to gain a deeper understanding of the invasion process, we pay special attention to the interacting dynamics between the cancer cell population and various constituents of the surrounding tumour microenvironment. To that end, we consider the key role that ECM plays within the human body tissue, and in particular we focus on the special contribution of its fibrous proteins components, such as collagen and fibronectin, which play an important part in cell proliferation and migration. In this work, we consider the two-scale dynamic cross-talk between cancer cells and a two-component ECM (consisting of both a fibre and a non-fibre phase). To that end, we incorporate the interlinked two-scale dynamics of cell–ECM interactions within the tumour support that contributes simultaneously both to cell adhesion and to the dynamic rearrangement and restructuring of the ECM fibres. Furthermore, this is embedded within a multiscale moving boundary approach for the invading cancer cell population, in the presence of cell adhesion at the tissue scale and cell-scale fibre redistribution activity and leading edge matrix-degrading enzyme molecular proteolytic processes. The overall modelling framework will be accompanied by computational results that will explore the impact on cancer invasion patterns of different levels of cell adhesion in conjunction with the continuous ECM fibres rearrangement.

A Nonlinear Dynamic Switched-Mode Model of Twin-Roll Steel Strip Casting

During twin-roll steel strip casting, molten steel is poured onto the surface of two casting rolls where it solidifies to form a steel strip. The solidification process introduces a two-phase region of steel known as mushy steel which has a significant effect on the resulting quality of the manufactured steel strip. Therefore, an accurate model of the growth of mushy steel within the steel pool is imperative for ultimately improving strip quality. In this paper, we derive a reduced-order model of the twin-roll casting process that captures the dynamics of the mushy region of the steel pool and describes the effect that the casting roll speed and gap distance have on the solidification dynamics. We propose a switched-mode description that leverages a lumped parameter moving boundary approach, coupled with a thermal resistance network analogy, to model both the steel pool and roll dynamics. The integration of these models and simulation of the combined model are nontrivial and discussed in detail. The proposed reduced-order model accurately describes the dominant dynamics of the process while using approximately one-tenth of the number of states used in previously published models.

Capturing soil-water and groundwater interactions with an iterative feedback coupling scheme: new HYDRUS package for MODFLOW

Abstract. Accurately capturing the complex soil-water and groundwater interactions is vital for describing the coupling between subsurface–surface–atmospheric systems in regional-scale models. The nonlinearity of Richards' equation (RE) for water flow, however, introduces numerical complexity to large unsaturated–saturated modeling systems. An alternative is to use quasi-3-D methods with a feedback coupling scheme to practically join sub-models with different properties, such as governing equations, numerical scales, and dimensionalities. In this work, to reduce the nonlinearity in the coupling system, two different forms of RE are switched according to the soil-water content at each numerical node. A rigorous multi-scale water balance analysis is carried out at the phreatic interface to link the soil-water and groundwater models at separated spatial and temporal scales. For problems with dynamic groundwater flow, the nontrivial coupling errors introduced by the saturated lateral fluxes are minimized with a moving-boundary approach. It is shown that the developed iterative feedback coupling scheme results in significant error reduction and is numerically efficient for capturing drastic flow interactions at the water table, especially with dynamic local groundwater flow. The coupling scheme is developed into a new HYDRUS package for MODFLOW, which is applicable to regional-scale problems.

Dynamic simulation of ignition, combustion, and extinguishment processes of HMX/GAP solid propellant in rocket motor using moving boundary approach

Solid propellant burning rate, gas phase temperature, and condensed phase thickness depend on combustion chamber pressure, laser intensity, and propellant compositions. The ignition and combustion of solid propellants occur in three phases namely solid phase, condensed phase, and gas phase. In this study, moving boundary modeling was applied to each of the phases by coordinate transformation. This research includes modeling and dynamic simulation of the ignition and combustion of HMX/GAP, a high-energy material in the ratio of 8:2, with the gas phase of the combustion model consisting of 50 species and 234 reactions. The mathematical modeling used mass, energy, and momentum conservation equations, as well as constitutive equations for the moving boundary. Parametric studies were run under different operating conditions, with an initial temperature of 300 K, a pressure of 10–100 atm and a laser intensity of 100 W/cm2. A burning rate of 2.2 cm/s and a gas phase temperature of 2700 K were obtained under an operating pressure of 100 atm. Extinguishment of the solid propellant was rigorously analyzed in terms of dynamic simulation with various depressurization rates during combustion. This was carried out by depressurizing the solid propellant from 70 atm to 40 atm. Important factors of the extinguishment were discussed based on the mathematical model and various depressurization rates with parametric studies. At a depressurization rate of −8000 atm/s, the solid propellant was fully extinguished. From this study, one can identify the phenomenon for the extinguishment of HMX/GAP propellant using fast depressurization with rigorous mathematical model used for ignition and combustion.

A combined moving boundary and discretized approach for dynamic modeling and simulation of geothermal heat pump systems

The major goal of this study is to demonstrate the feasibility of the dynamic modeling and simulation of both conventional and direct exchange geothermal heat pump applications particularly with regard to performance evaluations. A direct exchange geothermal heat pump system is a geothermal heat pump system in which the refrigerant circulates through tubing placed in the ground. In this research, dynamic models for the simulation of geothermal heat pump systems with the working fluid propane for the application in building-scale energy systems have been developed based on the Moving Boundary approach and the challenges of dynamically evaluating these kind of energy systems have been hereby met. Additionally, the performance of both horizontal (slinky-coil) and vertical (borehole) geothermal heat exchangers, which have been modeled based on the Conventional Discretized approach, for such systems has been evaluated. Since these two modeling approaches are not compatible with one another, the coupling of the moving boundary model with the discretized model is therefore another challenge and of great importance. On the basis of a case study, the complete conventional and direct exchange geothermal heat pump systems with vertical and horizontal heat exchangers have been simulated and energetically and exergetically analyzed.

Lattice Boltzmann simulation of gas-solid heat transfer in random assemblies of spheres: The effect of solids volume fraction on the average Nusselt number for Re ≤ 100

A thermal lattice Boltzmann method has been applied to simulate gas-solid heat transfer in a random assembly of spheres using an immersed moving boundary approach. The numerical data was correlated to propose an expression for the Nusselt number as a function of the Reynolds number and solid volume fraction over a wide range of solid volume fractions (ϕ=[0,0.5]) for Reynolds numbers up to 100. It is hoped that the new Nusselt number correlation for gas-solid systems improves the accuracy of Euler-Euler and Euler-Lagrangian simulations of gas-solid heat transfer in packed and fluidized beds.

Compressible Fluids Interacting with a Linear-Elastic Shell

We study the Navier–Stokes equations governing the motion of an isentropic compressible fluid in three dimensions interacting with a flexible shell of Koiter type. The latter one constitutes a moving part of the boundary of the physical domain. Its deformation is modeled by a linearized version of Koiter’s elastic energy. We show the existence of weak solutions to the corresponding system of PDEs provided the adiabatic exponent satisfies {gamma > frac{12}{7}} ({gamma >1 } in two dimensions). The solution exists until the moving boundary approaches a self-intersection. This provides a compressible counterpart of the results in Lengeler and Růžičkaka (Arch Ration Mech Anal 211(1):205–255, 2014) on incompressible Navier–Stokes equations.