Balance Equations Research Articles

Coating flow of a liquid film with colloidal particles on a vertical fiber

A thin liquid film of colloidal suspension is considered which is spreading down on a vertical cylinder under the effect of gravity. A precursor film model is employed at the three-phase contact line to relieve the stress singularity. The curvature pressure leads to the formation of a capillary ridge at the contact-line. Bulk and surface colloids are assumed to be present in the liquid film. The interfacial pattern is governed by the film evolution equation, obtained by simplifying mass and momentum balance equations within the lubrication assumption, while rapid vertical diffusion is assumed for the advection–diffusion equation of the bulk concentration. Fluid viscosity and diffusivity are considered to be functions of the particle volume fraction. The effects of the bulk and surface colloids on the spreading film dynamics are systematically studied. The presence of bulk colloids leads to the thinning of the capillary ridge for smaller inlet bulk colloid concentrations. On the other hand, a larger inlet bulk concentration leads to the formation of a secondary advancing front behind the capillary ridge in the film profile. A reduced contact point velocity is observed with an increase in the upstream bulk concentration. Surface concentration is found to result in a hump-like structure behind the capillary ridge in the upstream direction due to solutal Marangoni stress. The Marangoni stress also hinders the surface-tension-driven spreading, resulting in a smaller contact line velocity. Reducing the Bond number results in unstable film profiles, which lead to a wave-like structure in the region with no bulk concentration gradients.

Performance analysis and accuracy of numerical methods for solving simultaneous nonlinear particulate processes with multi-dimensional extension

The simultaneous population balance equation (PBE) incorporating diverse particulate process aggregation, breakage, growth, nucleation and source is a long-standing and significant problem in the field of particulate sciences such as pharmaceutical industry, chemical engineering, astrophysics and biology. Owing to the complex and nonlinear nature of the governing equations, finding an analytical solution is quite challenging for empirical kernels. In this work, simultaneous PBEs are solved using sectional and semi-analytical techniques. The performance, adaptability and accuracy of both methods for solving simultaneous PBEs are discussed in detail. The iterative semi-analytical scheme for solving simultaneous PBE is introduced, and a detailed convergence analysis using Banach fixed point theory is also presented. Both the schemes are further extended to solve simultaneous PBE in multi-dimensions. Performance analysis is also executed for multi-dimensional models. This article is a comprehensive study on various solution techniques for solving simultaneous PBEs. Therefore, it will be beneficial for the researchers working in industry and computational fluid dynamics modelling to identify the best method according to their requirements.

Development of dynamic simulation model for 20 kWe micro heat pipe fission battery with dual power conversion system

Recent advancements in politics, technology, and economics have sparked interest in small-scale nuclear power generation. Heat-pipe micro-reactors, a type of small modular reactor (SMR), are ideal for remote and specialized applications like deep-sea and deep-space exploration. Traditionally, these reactors have used thermoelectric generators (TEGs) or Stirling engines for power conversion. However, TEGs suffer from low efficiency, and while Stirling engines have improved, they still have lower long-term stability compared to TEGs. Furthermore, comprehensive studies on system-level transients, dynamic behavior, and thermal responses are lacking. This study designs and models a dual power conversion system integrating TEGs and Free-Piston Stirling Generators (FPSGs) for micro heat pipe reactors to enhance efficiency, stability, and reliability. By integrating the two systems in parallel, the limitations of each technology are addressed, offering a robust solution for small, portable nuclear power generation. A dynamic simulation model incorporating the reactor core, heat pipes, and power conversion system enabled a comprehensive study of system transients and interactions. The core was modeled using thermal energy balance equations, and a thermal resistance approach was applied to the heat pipe. The TEG model included the Thomson and Seebeck effects, while neutronics feedback used a point kinetics model. The nonlinear analysis model was applied to simulate the FPSG. The system reached a stable operation after about 1000 s, generating 20kWe with 29.39% efficiency. It maintained stability without external control during transients within a specific range, demonstrating its feasibility. The TEG and FPSG exhibited independent yet complementary behaviors, ensuring reliable performance across scenarios.

Application of dynamic material balance to evaluate oil wells in Buzurgan oil field

The Material Balance Equation is a crucial tool utilized in reservoir studies to evaluate fluids and rock properties at static pressures. The Flowing and Dynamic Material Balance methods offer a significant advantage by avoiding the requirement to shut down wells, as they use flowing pressure instead of static pressure under constant or variable flow rates. The concept of "Dynamic Material Balance" involves converting the bottom hole flowing pressure at any point at any given time to the average reservoir pressure at that point. This allows for the use of classical material balance calculations and the development of classical material balance plots. In this study, the Dynamic Material Balance and Agrawal Type Curve techniques were used to estimate average reservoir pressures, initial hydrocarbon in place, and ultimate oil recovery for a well in the Mishrif reservoir, the main reservoir in the Buzurgan oil field. Many wells in this field experience problems such as high-pressure decline or continuous water production, necessitating ongoing evaluation. While the Dynamic Material Balance method focuses on boundary-dominated flow data, the Agrawal-type curve technique analyzes data from both transient and boundary flow periods. Agarwal decline curves were constructed using relationships of pseudo pressure normalized production, material balance pseudo time, and dimensionless variables in well-test analysis. The results from both methods showed comparable results with an absolute percentage error of (0.738) %, (3.07) %, and 5.7% for oil-in-place, drainage area, and average reservoir pressure, respectively. This strong correlation between the Dynamic Material Balance and Type Curve results indicates their accuracy and reliability.

Analytical model for flexoelectric sensing of structural response considering bonding compliance

Flexoelectricity has generated huge interest as an alternative to piezoelectricity for developing electromechanical systems such as actuators, sensors, and energy harvesters. This article presents a generic theoretical framework for the sensing mechanism of a flexoelectric sensor bonded to a host beam through an adhesive layer. The model incorporates piezoelectric and flexoelectric effects and considers both shear-lag and peel stresses at the sensor-beam interface. The formulation also includes the electric field gradient terms that are often overlooked. Consistent one-dimensional constitutive relations and governing equations of equilibrium are derived from the electric Gibb’s energy density function and extended Hamilton’s principle. The sensor is assumed to follow the Euler–Bernoulli beam-type membrane and bending deformation behaviour. Closed-form solutions are obtained for the interfacial stresses by analytically solving a seventh-order non-homogeneous ordinary differential equation, satisfying the stress-free boundary conditions at the sensor edges. The induced electric potential at the sensor top is derived by solving a fourth-order differential equation obtained from the charge balance equation, satisfying the electric boundary conditions. For validation, the sensor output is compared with the results of the existing non-rigid bonding piezoelectric sensor model. Numerical results show a significant impact of non-rigid bonding and the electric field gradient terms on the induced electric potential. Further, the importance of bonding compliance on the interfacial stress distributions is illustrated. Finally, the effects of adhesive and transducer thicknesses on the peak amplitudes of interfacial stresses and sensory potential are presented.

A generalized kinetic theory of Ostwald ripening in porous media

Partially miscible bubbles (e.g., CO2) trapped inside a porous medium and surrounded by a wetting phase (e.g., water) occur in a number of applications including underground hydrogen storage, geologic carbon sequestration, and the operation of electrochemcial devices such as fuel cells and electrolyzers. Such bubbles evolve due to a process called Ostwald ripening that is driven by differences in their interfacial curvature. For spherical bubbles, small bubbles shrink and vanish while feeding into larger ones, resulting in one large bubble at equilibrium. Within the confinement of a porous medium, however, bubbles can attain a distribution of sizes at equilibrium that have identical curvature. This work concerns itself with the formulation of a kinetic theory that predicts the statistical evolution of bubble states, defined as the sizes of the pores within which bubbles are trapped and the extent to which those pores are saturated with bubbles. The theory consists of a population balance equation and appropriate closure approximations. Systematic comparisons against a previously published pore network model (PNM) are conducted to validate the theory. Our theory generalizes existing variants in the literature limited to spherical bubbles trapped in homogeneous media to non-spherical (deformed) bubbles inside microstructures with arbitrary heterogeneity and spatial correlation in pore/throat sizes. We discuss the applicability, limitations, and implications of the theory towards future extensions.

Water Resources Management for Multi-Source Ecological Restoration Goals in an Oasis: A Case Study of Bohu County Irrigation Area in Xinjiang, China

Oases in arid regions consist of river–lake–groundwater systems characterized by complex hydrological cycles and fragile ecosystems. Sustainable water resource management, aimed at multi-source ecological restoration, is crucial for oasis ecological protection and represents a current research challenge. This study focuses on the Bohu irrigation area, using ecological water levels, the MIKE-SHE hydrological model, and the water balance equation to propose a multi-objective groundwater and surface water regulation scheme that meets both the ecological safety requirements of the irrigation area and the ecological water demands of the Small Lake. Key findings include the following: (1) The regional ecological water level ranges from 1.69 m to 4 m, with about 74% of the area exceeding this range, threatening local ecology. (2) The proposed regulation method adjusts 91.25% of areas exceeding the ecological water level to within the acceptable range. (3) Under various planting scenarios, the minimum water distribution from the west branch of the BLSM water diversion hub should be 824.632 × 106 m3/a to meet Small Lake ecological demands. When this volume exceeds 831.902 × 106 m3/a, both groundwater regulation and Small Lake ecological demands are satisfied. This paper quantifies the water cycle mechanisms in complex hydrological interaction areas, providing specific solutions to regional ecological problems, which holds significant practical relevance.

Theoretical study of particle and energy balance equations in locally bounded plasmas

Abstract In this study, new particle and energy balance equations have been developed to predict the electron temperature and density in locally bounded plasmas. Classical particle and energy balance equations assume that all plasma within a reactor is completely confined only by the reactor walls. However, in industrial plasma reactors for semiconductor manufacturing, the plasma is partially confined by internal reactor structures. We predict the effect of the open boundary area (A_L,eff') and ion escape velocity (u_i) on electron temperature and density by developing new particle and energy balance equations. Theoretically, we found a low ion escape velocity (u_i/u_B≈0.2) and high open boundary area (A_L,eff'/A_T,eff)≈0.6) to result in an approximately 38% increase in electron density and a 8% decrease in electron temperature compared to values in a fully bounded reactor. Additionally, we suggest that the velocity of ions passing through the open boundary should exceed ω_pi *λ_De under the condition (E_0)^2≪(Φ/λ_De)^2.

On the Energy Cost for Nitrogen Fixation in DC Glow Discharges in Atmospheric‐Pressure Air

ABSTRACTModeling of the production of nitrogen oxide molecules in DC discharges in laminar longitudinal flows of atmospheric‐pressure air is performed. Two‐dimensional nonequilibrium discharge model includes the system of gas dynamic equations and balance equations for various plasma components. Simulations are performed for the discharge currents 50–600 mA and gas flow velocities 1–10 m/s. In considered conditions, the gas temperature in the discharge core varies in the range of 3000–6000 K. Spatial profiles of the densities and fluxes of produced species are obtained, both inside the discharge and in the afterglow region. The energy cost for the production of nitrogen oxides is calculated, depending on the discharge current, gap length, and gas flow velocity. The results of the simulation are compared with available experimental data.

Non-linear biphasic mixture model: Existence and uniqueness results

Abstract This paper is concerned with the development and analysis of a mathematical model that is motivated by interstitial hydrodynamics and tissue deformation mechanics (poro-elasto-hydrodynamics) within an in-vitro solid tumour. The classical mixture theory is adopted for mass and momentum balance equations for a two-phase system. A main contribution of this study is we treat the physiological transport parameter (i.e., hydraulic resistivity) as anisotropic and heterogeneous, thus the governing system is strongly coupled and non-linear. We derived a weak formulation and then formulated the equivalent fixed-point problem. This enabled us to use the Galerkin method, and the classical results on monotone operators combined with the well-known Schauder and Banach fixed-point theorems to prove the existence and uniqueness of results.

Modelling the ejection of primary aerosols during the fast pyrolysis of biomass anisotropic particles

A model for the fast pyrolysis of anisotropic biomass particles is presented which considers bubbling dynamics within the liquid intermediate phase (metaplast) and aerosol ejection from this phase. The model employs the population balance equation and the method of moments to estimate the production rate and resultant size distribution of aerosol ejections, incorporating a detailed CRECK reaction mechanism, and considers the effect of anisotropic biomass microstructure on the intraparticle transport of mass and energy. This study investigates the impact of particle size, heating rate (heat transfer coefficient), and lignocellulosic composition on aerosol ejection. The model predicts that, at high heating rates (convective heat transfer coefficient of 359 W/m2.K), aerosols can contribute over 20% to the heavy fraction yield in bio-oil for small particles (1 mm diameter, 4 mm length). The model can predict aerosol size distribution and surface area, indicating an average size of 20 μm for bubbles and 5 μm for aerosols during increased bubble production and aerosol ejection rates. These findings are consistent with prior experimental results and provide essential information for future modeling of extra-particle reactions of the aerosols as they progress through the reactor.

Comparative research on monitoring methods for nitrate nitrogen leaching in tea plantation soils

Great concern has long been raised about nitrate leaching in cropland due to its possible environmental side effects in ground water contamination. Here we employed two common techniques to measure nitrate leaching in tea plantation soils in subtropical China. Using drainage lysimeter as a reference method, the adaptability of estimating drainage and nitrate leaching by combining the water balance equation with the suction cup technique was investigated. Results showed that the final cumulative leachate volume for the calculated and measured method was 721.43 mm and 729.92 mm respectively during the study period. However, nitrate concentration exerted great influence in the estimation of nitrate leaching from the suction cup-based method. The cumulative nitrate leaching loss from the lysimeter and suction cup-based method was 47.45 kg ha−1 and 43.58 kg ha−1 under lysimeter nitrate concentrations ranging from 7 mg L−1 to 13 mg L−1, 156.28 kg ha−1 and 79.95 kg ha−1 under lysimeter nitrate concentrations exceeding 13 mg L−1. Therefore, the suction cup-based method could be an alternative way of monitoring nitrate leaching loss within a range of 7–13 mg L−1 of nitrate concentrations in leachate. Besides, lower results occurred in suction cup samplers due to lack of representative samples which mainly leached via preferential flow when in strong leaching events. Thus, it is advisable to increase sampling frequency under such special conditions. The results of this experiment can serve as a reference and guidance for the application of ceramic cups in monitoring nitrogen and other nutrient-ion leaching in tea plantation soils.

Dynamic prediction of overhead transmission line ampacity based on the BP neural network using Bayesian optimization

Traditionally, the ampacity of an overhead transmission line (OHTL) is a static value obtained based on adverse weather conditions, which constrains the transmission capacity. With the continuous growth of power system load, it is increasingly necessary to dynamically adjust the ampacity based on weather conditions. To this end, this paper models the heat balance relationship of the OHTL based on a BP neural network using Bayesian optimization (BO-BP). On this basis, an OHTL ampacity prediction method considering the model error is proposed. First, a two-stage current-stepping ampacity prediction model is established to obtain the initial ampacity prediction results. Then, the risk control strategy of ampacity prediction considering the model error is proposed to correct the ampacity based on the quartile of the model error to reduce the risk of the conductor overheating caused by the model error. Finally, a simulation is carried out based on the operation data of a 220-kV transmission line. The simulation results show that the accuracy of the BO-BP model is improved by more than 20% compared with the traditional heat balance equation. The proposed ampacity prediction method can improve the transmission capacity by more than 150% compared with the original static ampacity.

Consistent solutions for simultaneous dynamic water, solute, stable isotope and radon balances to estimate average exchange fluxes between surface water bodies and groundwater

Many papers show that water, conservative solute, stable isotope and radon balances can be applied to the study of surface water – groundwater interaction. When undertaken simultaneously, such analysis provides an even more powerful method for estimating groundwater inflows and outflows, i.e. the exchange fluxes that for most people are hardest to estimate. The overarching objective of this paper is to provide practitioners with a consistent conceptual and simulation modelling framework with which to achieve a rapid initial understanding of surface water – groundwater interaction, as well as to guide the efficient and targeted acquisition of field data. This paper makes several important contributions. First, after an introduction that explains the complexities of groundwater flow patterns near surface water bodies, a set of general balance equations is developed using consistent terminology and notation. Second, after focusing on a simple conceptual model with steady groundwater inflow I, groundwater outflow O, precipitation P and ever-present evaporation E, analytical solutions are developed for transient and steady state conditions: one water balance solution is found to apply in all situations; solutions for the solute balance equation are developed for 23 different subcases; a transient balance equation for stable isotopes written in terms of 1+δ is found to apply in all situations; and a new transient solution for radon is found in terms of the upper incomplete gamma function. On the path towards developing the isotope balance equation, we present a rigorous approach, based on coupled nonlinear balance equations for abundant and rare isotope pairs (1H and 2H, then 16O and 18O), using isotope ratios and the fractions of each isotope. Third, we describe a Lake Water Balance Calculator (LWBC) written using GoldSim simulation software, a tool to help others to conduct the simultaneous analyses that we promote; we use GoldSim’s numerical methods to solve the general balance equations, we implement the analytical solutions to demonstrate that both GoldSim’s numerical methods and the analytical solutions are correct, and we also show that the 1+δ approximation is equivalent to the rigorous coupled equations, at least for heavy stable isotopes with very low abundance. Fourth, we describe field studies: we provide three examples of applications of LWBC to short-term studies of Perry Lakes East, Lake Jasper and Georgetown Billabong in Australia; we then describe WSIBal, a simulation model based on the same balance equations and describe its application to Lake Jasper over a 19-year period; finally, we describe R-WSIBal, which evolved from WSIBal, and explain its application to 2500 km of the Murray River in Australia.

An improved phase-field model for oil–water two-phase flow and mixed-mode fracture propagation in hydraulic fracturing

An improved novel phase-field model is proposed for describing fracture propagation, effectively characterizing the interactions between oil–water two-phase fluids and solids as well as mixed-mode fracturing. This model encompasses equations for two-phase flow stress balance, fluid flow, and a complex mode fracture phase-field specific to two-phase flow. Within the fluid flow equations, the capillary pressure caused by the oil–water interface during the fluid loss process in the matrix is accounted for, and the anisotropic relative permeability during fluid flow is linked to normalized saturation. The driving forces for fracture propagation in the phase-field are categorized into Mode I and Mode II forces, with the contribution from the oil–water two-phase fluid attributed to the Mode I (tensile) driving force. The model uses finite element numerical discretization and the Newton-Raphson iterative method to establish a numerical iteration format, employing an implicit-explicit staggered solution scheme. Additionally, the model is used to investigate several key factors. It assesses the impact of different intersection angles on fracture propagation in naturally porous media. It also studies the effect of different natural fracture permeabilities on hydraulic fracture propagation. Finally, the model analyzes the propagation patterns of hydraulic fractures under different perforation phase angles.

Study on fluid dynamic lubrication numerical analysis and surface topography design of slipper pair of valve distribution piston pump

In order to improve the fluid dynamic lubrication performance of slipper pair of valve distribution (digital distribution) axial piston pump, this paper proposes a theory analytical model of fluid dynamic lubrication for slipper pairs with different surface structures of valve distribution pump and its tribological optimization mechanism of surface microstructure topography. In the model, based on the finite volume method, the least square method combined with the trust region dogleg was proposed to solve the nonlinear system of equations of slipper dynamic balance. The fluid lubrication mechanism of slipper pairs with different surface structures was analyzed, the probability of overturning and eccentric wear of slipper pairs with complex surface structure and the viscous friction torque were quantitatively analyzed, and experimental research was conducted. The results show that the surface microstructure topography has great influence on the fluid lubrication performance of slipper pairs. Compared with before optimization, the friction torque of slipper with single-support belt, double-support belt, and four-support belt is decreased by 8.58%, 13.65%, and 17.07% at most, respectively, but the probability of overturning and eccentric wear of slipper with double-support belt and four-support belt is increased and the oil film becomes thicker. Reducing the friction loss and enhancing the anti-overturning ability of the slipper pair are often difficult to be compatible. The research results lay a theoretical and technical foundation for the optimization of lubrication performance of slipper pair of valve distribution pump.

Modeling, parameter estimation and optimization of fluidized bed-based alternative ironmaking process for CO2 emission reduction

This study presents the development of a comprehensive process model for simulating and optimizing the FINEX process consists of the multi-stage fluidized beds, the melter-gasifier unit, and a gas recycling system. The model is developed using Pyomo, a Python-based open-source software package that provides extensive capabilities for formulating, solving, and analyzing optimization models. It incorporates heat and mass balance equations and captures a wide range of chemical reactions, including the reduction of iron ore by hydrogen and carbon monoxide, the calcination of carbonate materials, the water–gas shift reaction, and coal gasification and combustion. Unknown parameters in the model, such as heat loss in each reactor, the extent of the calcination reaction, and the outlet gas temperature from the melter-gasifier, were estimated to calibrate the model. These parameters were estimated by solving an optimization problem that minimizes the gap between the model and real plant data. The optimized model was employed to investigate various scenarios for minimizing CO2 emissions in the FINEX process. This included assessing the impact of HBI charging rates, iron ore quality, and the integration of a CCUS unit. For each scenario, an optimization problem was formulated to minimize production costs across a range of CO2 tax levels. The optimal solutions revealed the relationships between process economics and CO2 emissions for each variable.

Neural force functional for non-equilibrium many-body colloidal systems

Abstract We combine power functional theory and machine learning to study non-equilibrium overdamped many-body systems of colloidal particles at the level of one-body fields. We first sample in steady state the one-body fields relevant for the dynamics from computer simulations of Brownian particles under the influence of randomly generated external fields. A neural network is then trained with this data to represent locally in space the formally exact functional mapping from the one-body density and velocity profiles to the one-body internal force field. The trained network is used to analyse the non-equilibrium superadiabatic force field and the transport coefficients such as shear and bulk viscosities. Due to the local learning approach, the network can be applied to systems much larger than the original simulation box in which the one-body fields are sampled. Complemented with the exact non-equilibrium one-body force balance equation and a continuity equation, the network yields viable predictions of the dynamics in time-dependent situations. Even though training is based on steady states only, the predicted dynamics is in good agreement with simulation results. A neural dynamical density functional theory can be straightforwardly implemented as a limiting case in which the internal force field is that of an equilibrium system. The framework is general and directly applicable to other many-body systems of interacting particles following Brownian dynamics.

Numerical assessment of transient flow and energy dissipation in a Pelton turbine during startup

The Pelton turbine, known for its high application water head, wide efficient operating range, and rapid start-stop capability, is ideal for addressing intermittent and stochastic load issues. This study numerically analyzes the transient two-phase flow and energy dissipation during the startup of a Pelton turbine. Dynamic mesh technology controlled nozzle opening changes, and momentum balance equations managed runner rotation. Findings showed that the runner speed initially increased rapidly and then more slowly, and flow rate matched the nozzle opening variations. Runner torque first rose linearly, then decreased, with the fastest decline during nozzle closing. Hydraulic efficiency peaked early in nozzle reduction but then dropped sharply. Strong vortices formed due to upstream inflow and downstream backflow impact in the distributor pipe. The jet needle and guide vane improved flow in the converging section of nozzle, but flow began to diffuse with increased stroke. Initially, the jet spread fully on the bucket surface, but later only affected the bucket tips. Pressure fluctuations in the water supply mechanism were primarily due to jet needle motion, with higher amplitude during movement and lower when stationary. These fluctuations propagated upstream, weakening over distance. Reynolds stress work and turbulent kinetic energy generation, respectively, dominated energy transmission and energy dissipation, with their maximum contribution exceeding 96% and 70%. High-energy clusters corresponded to jet impact positions, highlighting jet-bucket interference as crucial for energy transport. This study established a performance evaluation method for Pelton turbine startups, supporting further investigation into characteristic parameters, flow evolution, and energy dissipation patterns.

Aggregating nanoparticle transport with nonlinear attachment: Modeling and experimental validation

A conceptual mathematical model was developed to simulate the transport of migrating nanoparticles in homogeneous, water saturated, 1-dimensional porous media. The model assumes that nanoparticles can collide with each other and aggregate. Nanoparticles can be found attached reversibly and/or irreversibly onto the solid matrix of the aquifer or suspended in aqueous phase. Attached particles may either contribute to the acceleration of subsequent particle deposition or hinder it, leading to the ripening or blocking process, respectively. The aggregation process was simulated based on the Smoluchowski Population Balance Equation (PBE) and was coupled with the advection-dispersion-attachment equation (ADA) to form a family of partial differential equations that govern the migration of nanoparticles in porous media. For the solution of the PBE, an efficient finite volume solver was employed that significantly accelerated computation times, by reducing the number of participating equations, while maintaining the required accuracy. The developed model was applied to nanoparticle transport experimental data available in literature. The model successfully matched the breakthrough concentration curves, and estimated the corresponding nanoparticle diameter, proving its ability to capture the physical processes participating in nanoparticle transport.