Momentum Balance Equations Research Articles

Part-load performance analysis of a modular biomass boiler with a combined heat and power industrial Rankine cycle and supplementary sCO2 Brayton cycle

Abstract The addition of a supplementary high efficiency cycle integrated with an existing steam power cycle may increase energy efficiency and net generation. In this paper, part load performance and operation of a modular biomass boiler with an existing industrial Rankine steam heat and power cycle and supplementary supercritical CO2 (sCO2) Brayton cycle is analysed. The aim is to leverage the high efficiency of the sCO2 cycle by retrofitting sCO2 heaters in the existing biomass boiler, increasing net power output and thermal efficiency. With nominal load performance previously investigated, understanding part-load performance and operation is vital to determine cycle feasibility. A quasi-steady-state one dimensional thermofluid network model was used to simulate the integrated cycle performance for loads ranging from 100% to 60%. The model solves the mass, energy, momentum, and species balance equations, capturing detailed component characteristics. Two control methodologies are explored for the sCO2 Brayton cycle, namely inventory control and inventory control combined with throttling valve control. Inventory control is selected as the better performing control strategy for load following, maintaining high thermal efficiency across partial loads. At 60% load, the sCO2 compressor operates near the pseudo-critical point, leading to a sharp decrease in sCO2 cycle capacity, which requires careful management of inventory control. Two sCO2 heater configurations are investigated, namely a single convective-dominant heater, and a dual heater configuration with a radiative and a convective heater. The single heater configuration is preferred to minimise adverse impacts on the Rankine cycle superheaters.

Phase Field Coupled Finite Deformation Plasticity Formulation of Ductile Fracture With Nonlinear Kinematic Hardening and Modified Energy Release Function

ABSTRACTA ductile damage theory is presented by coupling the covariant formulation of finite deformation plasticity with the phase field modeling of fracture, including kinematic hardening for the ductile response of the materials. A phase field coupled nonlinear kinematic hardening equation is proposed in the reference configuration having equivalent representation through the Lie derivative of the kinematic hardening tensor by push‐forward operation in the spatial configuration, thus ensuring the satisfaction of frame invariance. To capture the correct physical response of the material by the phase field evolution equation, the fracture driving function, that is, the difference between the sum of elastic and plastic energies and threshold energy, after the damage initiation is modified by an energy release controlling function, which is an empirical relation of equivalent plastic strain. In defining the energy release controlling function, well‐defined points with physical significance in the experimental load versus displacement curve are used. To simulate the response of the material in relatively large time steps, a modified staggered scheme is presented, evaluating the fracture driving and energy release controlling functions from the previous converged step and using the updated phase field variable in the weak form of the momentum balance equation. To quantify different material parameters from available experimental results in the literature, the developed phase field coupled elasto‐plastic model uses a neural network optimization procedure consisting of neural network training together with optimization in MATLAB and finite element model evaluation in Abaqus user element subroutine UEL. Model capabilities are demonstrated by simulating the crack propagation in complex 3D geometries such as the second and third Sandia Fracture Challenges.

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.

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.

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.

A free-energy based multiple-distribution-function lattice Boltzmann method for multi-component and multi-phase flows

This study presents the development of a multiple-distribution-function lattice Boltzmann model (MDF-LBM) for the accurate simulation of multi-component and multi-phase flow. The model is based on the diffuse interface theory and free energy model, which enable the derivation of hydrodynamic equations for the system. These equations comprise a Cahn-Hilliard (CH) type mass balance equation, which accounts for cross diffusion terms for each species, and a momentum balance equation. By establishing a relationship between the total chemical potential and the general pressure, the momentum balance equation is reformulated in a potential form. This potential form, together with the CH type mass balance equation, is then utilized to construct the MDF-LBM as a coupled convection–diffusion system. Numerical simulations demonstrate that the proposed MDF-LBM accurately captures phase behavior and ensures mass conservation. Additionally, the calculated interface tension exhibits good agreement with experimental data obtained from laboratory studies.

On the comparison between phenomenological and kinetic theories of gas mixtures with applications to flocking

We study the comparison between the phenomenological and kinetic models for a mixture of gases from the viewpoint of collective dynamics. In the case in which constituents are Eulerian gases, balance equations for mass, momentum, and energy are the same in the main differential part, but production terms due to the interchanges between constituents are different. They coincide only when the thermal and mechanical diffusion are sufficiently small. In this paper, we first verify that both models satisfy the universal requirements of conservation laws of total mass, momentum, and energy, Galilean invariance and entropy principle. Following the work of Ha and Ruggeri (ARMA 2017), we consider spatially homogeneous models which correspond to the generalizations of the Cucker Smale model with thermal effect. In these circumstances, we provide analytical results for the comparison between two resulting models and also present several numerical simulations to complement analytical results.

Turbulent Exchange in Unsteady Air-Sea Interaction at Small and Submesoscales

An adequate description of the interaction between the atmosphere and ocean remains one of the most important problems of modern oceanology and climatology. An extremely wide variety of physical processes occurring in the coupled layers, a large range of scales, a moving boundary, all this factors significantly complicates the creation of models that would allow calculating the physical characteristics in both media with the necessary accuracy. In this paper the temporal variability of dynamic parameters in the driving layer of the atmosphere and in the near-surface layer of the sea on small and sub-mesoscales from one to several tens of hours is considered. The collected experimental data show a very high correlation between the dynamic wind speed and turbulence intensity in the upper sea layer on all scales recorded. An important distinguishing feature of all measured physical quantities in both media is the presence of quasi-periodic oscillations with different periods. For a more accurate description of momentum and energy fluxes from the atmosphere a non-stationary model of turbulent exchange in the near-surface layer of the sea is proposed. The model takes into account quasi-periodicity in the intensity of dynamic interaction between the atmosphere and the sea at these scales. In the model we use the equations of momentum and turbulent energy balance, the system of equations is solved numerically, the calculation results are compared with other models and with experimental data. It is shown that taking into account the non-stationarity of the wind strain improves the correspondence between the calculations and the experimental data. It is noted that in the nonstationary case, the energy and momentum flux from the atmosphere and the turbulence intensity increases compared to the action of a constant average wind of the same duration. Therefore, the strong averaging often used in global models may markedly underestimate the intensity of the dynamic interaction between the atmosphere and ocean.

CFD study of heat transfer in power‐law fluids over multiple corrugated circular cylinders in a heat exchanger

AbstractThe heat transfer in power‐law fluids across three corrugated circular cylinders placed in a triangular pitch arrangement is studied computationally in a confined channel. Continuity, momentum, and energy balance equations were solved using ANSYS FLUENT (Version 18.0). The flow is assumed to be steady, incompressible, two‐dimensional, and laminar. A square domain of side 300Dh is selected after a detailed domain study. An optimized grid with 98,187 cells is used in the study. The convergence criteria of 10−7 for the continuity, x‐momentum, and y‐momentum balances and 10−12 for the energy equation were used. Constant density and non‐Newtonian power‐law viscosity modules were used. The diffusive term is discretized using a central difference scheme. Convective terms are discretized using the Second‐Order Upwind scheme. Pressure–velocity coupling between continuity and momentum equations was implemented using the semi‐implicit method for pressure‐linked equation scheme. Streamlines show wake development behind the cylinders, which is very dominant at large ReN and n. Isotherm contours are cramped at higher values of ReN and PrN, implying higher heat transfer. Global parameters, like, Cd and Nu, are computed for the wide ranges of controlling dimensionless parameters, such as power‐law index (0.3 ≤ n ≤ 1.5), Reynolds (0.1 ≤ ReN ≤ 40), and Prandtl (0.72 ≤ PrN ≤ 500) numbers. The NuLocal plot attains a pitch near the corrugation of the surface due to abrupt changes in velocity and temperature gradients. Nu increases with ReN and/or PrN and decreases with n under ot herwise identical situations. Nu is correlated with pertinent parameters, namely, ReN, PrN, and n.

The Modified Guyer-Krumhansl Equations Derived From the Linear Boltzmann Transport Equation

Abstract Phonon hydrodynamics originated from the macroscopic energy and momentum balance equations (called Guyer-Krumhansl equations) proposed by Guyer and Krumhansl by solving the linearized Boltzmann transport equation for studying second sound in the normal-process collision dominated phonon transports in an isotropic nonmetallic crystal with a dispersionless frequency spectrum. However, the low-dimensional dielectric materials and semiconductors are anisotropic, and the different branches in their phonon frequency spectrum usually have different group velocities. For such materials, we derive the macroscopic energy and momentum balance equations from the linear Boltzmann transport equation to describe the phonon hydrodynamic transport, and solve the longstanding debate about whether the energy balance equation contains the second-order spatial derivatives of temperature. Finally, by solving the modified Guyer–Krumhansl equations, we find the minimum and maximum values of the length required by the occurrence of second sound in suspended single-layer graphene with the rectangular shape.

Wing-to-Wall Distance Effect On The Large-Scale Turbulent Structures Interacting With A Wall

We experimentally investigate the effect of the wing-to-wall distance, also called ground clearance, on the three-dimensional flow generated by a wing at a 4° angle of attack and a chord Reynolds number of 1400. We perform 3D velocity measurements using Particle tracking velocimetry to characterize the wake downstream of the wing, evaluate the evolution of the intensity and the size of the wingtip and secondary counter-rotating vortices, and finally compute the aerodynamic coefficients using momentum balance equations. We extended the control volume method for drag and lift calculation to the case where the transverse plane downstream of the wing does not contain the whole wake. The secondary flow's vorticity isocontours and velocity vectors show a decrease in the downwash and vortex size with the ground clearance ratio (i.e., as we approach the ground). The computation of the circulation shows that this latter decays rapidly streamwise as we get closer to the wall, indicating a dissipation of the kinetic energy contained in the trailing vortices. The aerodynamic coefficients' results based on the control volume method show that the tested wing manifests a two-force regime evolution of the drag coefficient with the ground clearance ratio, as it first decreases when moving far from the ground before increasing as we move farther from the ground. A noteworthy high lift is generated when too close to the ground. Further application of the extended control volume method on DNS data is needed to validate the method.

Multi-dimensional CFD-Mask R-CNN and CFD-watershed segmentation approach for multiphase non-catalytic gas-solid reactions: A case study for hydrogen reduction of porous iron oxide pellets

The direct reduction of iron oxide (DRI) using hydrogen is a promising method for clean steelmaking. Previous studies used one-dimensional models to explore this non-catalytic gas–solid reaction, but they lacked accuracy in assessing the impact of pellet shape. To tackle this issue, a multi-dimensional computational fluid dynamic (CFD) method has been employed, integrating a novel porous three-interface unreacted shrinking core model (USCM) to explore the effects of pellet shape through multiphase reactions of Fe2O3 → Fe3O4 → FeO → Fe by pure hydrogen. The model incorporates Knudsen, molecular, and effective diffusion mechanisms, alongside a continuous porosity function. Validation against previous experiments within a temperature range of 800–1000 °C and varying porosity is performed by solving four partial differential equations (PDEs) of mass and momentum balance in a porous media using the finite element method (FEM). Subsequently, a dataset comprising 754 images of pellets from the industrial DRI unit is collected and applied for the mask region-based convolutional neural network (Mask R-CNN) algorithm coupled with CFD, to draw geometries and simulate DRI to study indentations and protrusions of pellets. Furthermore, watershed segmentation is applied to investigate the shapes and real sizes of pellets from their images. The Mask R-CNN achieves intersection over union (IoU), precision, and recall percentages of 86.1 %, 87.3 %, and 87.2 % on average, respectively. Pellet shape investigations reveal notable discrepancies in wustite phase profiles and porosity. A nuanced impact is discerned regarding the DRI in 2D shape deviation, while 3D model errors in solid conversion time range from 3 % to 12.8 %.

Optimization and heat transfer analysis for MHD radiative natural convective flow of hybrid nanofluid within a partially heated cavity

The current study consists of optimization and heat transfer analysis for magnetohydrodynamic (MHD) radiative-natural convective flow of hybrid nanofluid (mixture of silver and copper nanoparticles with ethylene glycol (base fluid)) saturated inside the partially heated isosceles triangular cavity. Additionally, the current analysis investigates the distraction of available energy during thermal mixing and flow friction through local entropy generation. The conservation laws of total mass, momentum, and energy balance equations have been utilized to govern the physical flow problem. After eradicating the pressure terms in the governing partial differential equations (PDEs) via the penalty technique, the Galerkin finite element method (FEM) with the Newton-Raphson technique has been utilized to solve it. The outcomes established that the concentration of silver nanoparticles, as compared to copper nanoparticles, gives better energy transport. An increase in the concentration of nano-particles from ( ϕ 1 = ϕ 2 = 0 ) to ( ϕ 1 = ϕ 2 = 0.1 ) increased the intensity of streamlines from ψ max = 1.5 to ψ max = 4 and energy transmit rate increases by 57.4 % . The influence of thermal radiation decreased the fluid flow friction or local entropy via fluid flow.

Improvements on the discretisation of boundary conditions to the momentum balance for glacial ice

Abstract The flow of glacial ice is typically approximated as a nonNewtonian viscous fluid, with the momentum balance described by (an approximation to) the Stokes equations, and the nonlinear rheology described by a flow law. The most commonly used rheological law for glacial ice, Glen's flow law, yields infinite viscosity in the case of zero deformation, which can be the case at the ice surface. This poses a problem when solving the momentum balance numerically. We show that two commonly-used discretisation schemes for the boundary conditions at the ice surface and base, which yield proper numerical convergence when applied to simpler problems, produce poor numerical convergence and large errors, when used to solve the momentum balance with Glen's flow law. We show that a discretisation scheme based on the concept of ghost nodes, which substitutes the boundary conditions directly into the momentum balance equations, yields second-order numerical convergence and errors that can be up to four orders of magnitude smaller. These results are robust across different momentum balance approximations. We show that the improved boundary conditions are particularly useful for solving the 3-D higher-order Blatter-Pattyn Approximation (BPA). In general, this work underlines the importance of thoroughly verifying the numerical solvers used in ice-sheet models, before applying them to future projections of ice-sheet mass loss.

Impact of rotational direction on wake replenishment mechanisms of counter-rotating paired vertical-axis wind turbines

Recent studies have shown that the placement of paired vertical-axis wind turbines (VAWTs) in close proximity is a promising approach to design efficient VAWT farm layouts, mainly due to the increased power output. This study aims to experimentally investigate the impact of the rotational direction of paired counter-rotating VAWTs on their wake characteristics and replenishment mechanisms through a quantitative analysis using the momentum balance equation. The results of this study enhance our understanding of the effect of rotational direction on wake dynamics of counter-rotating paired VAWTs and offer valuable insights for optimising the design of VAWT farm configurations.

Analytical study on the liquid–particle mass transfer coefficient for multiparticle systems

This work focuses on liquid–particle mass transfer coefficient correlations for multiparticle systems. After discussing the mass transfer coefficient definitions, we review the available theoretical correlations for isolated particles and multiparticle systems. The latter are based on assumptions that are not rigorous. To estimate the mass transfer coefficient and derive a correlation for it, we apply scaling and order-of-magnitude analysis to the mass and linear momentum balance equations. The analysis features an undetermined constant. For isolated particles, we determine the value of this constant by matching the predictions of our correlation with those of other correlations available in the literature, obtaining a good fit. For multiparticle systems, we estimate the value of the constant via regression of experimental data; the predictions of our correlation align well with the experimental data, the relative percent error being less than 30% for most of the data.

Matrix methods and teaching the fundamentals of fluid and gas mechanics in higher education institutions

Problem. The instruction of physical and mathematical disciplines in higher education institutions employing cutting-edge methods and universal approaches, which seamlessly integrate scientific and engineering activities of future specialists, remains one of the pertinent and priority tasks. The utilization of multidimensional or even infinite-dimensional spaces has become an effective and nearly indispensable tool in mathematical modeling of physical phenomena. This circumstance is directly linked to the increasing level of abstraction and the refinement of mathematical methods. The use of geometric methods inherent in vector algebra and vector analysis as the foundation for studying mechanics, despite their illustrative nature, is losing its relevance. Methods built upon matrix formalism are evolving as substitutes. The matrix framework enables the exploitation of phase space advantages to derive canonical equations of motion for continuous media and electromagnetic field equations in covariant forms. The electromagnetic potential acquires mechanical significance, allowing the utilization of electromechanical analogies at a fundamental level rather than on a merely formal basis, as done in classical electrodynamics. Goal. The aim of this research is to study and justify the advantages of matrix methods for deriving equations of motion for fluids and gases from the canonical equations of mechanics by utilizing a continuum model in phase and physical space. The research is directed towards demonstrating the connection between the internal microscopic and macroscopic motion of fluids and gases, as well as highlighting the potential of matrix formalism both in terms of modeling and the utilization of computational tools. Methodology. The methodological basis for selecting matrix methods lies in the application of tensor analysis and its generalizations for modeling the motion of fluids and gases in physical and phase space. Results. It has been demonstrated that the matrix method, previously applied to classical and relativistic mechanics, allows for the consideration of the molecular structure of the environment by folding the multidimensional phase space of the mechanical system, represented by a particle of the medium, and thereby deriving equations of motion for fluids and gases from the canonical equations of classical mechanics and the equations of momentum balance of the medium. Originality. The combination of canonical equations as a result of applying the conservation law of matter in the form of balance equations in phase space and momentum balance equations in physical space allows for elucidating the relationship between macroscopic and microscopic motions of the medium, as well as the mathematical structure of the equations of motion for fluids and gases. Practical value. The proposed method allows, while remaining within the standard mathematical training offered by technical educational institutions, to effectively formalize the equations of motion for fluids and gases and provide them with an invariant form. By guiding oneself through invariance, both fundamental laws (such as the law of universal gravitation) and partial laws (the generalized Newton's law) can be obtained.

Integrated Performance of a Modular Biomass Boiler With a Combined Heat and Power Industrial Rankine Cycle and Supplementary sCO2 Brayton Cycle

Abstract In this paper, the integrated performance of a modular biomass boiler with an existing industrial Rankine steam heat and power cycle and a supplementary supercritical-carbon dioxide (sCO2) Brayton cycle is analyzed. The aim is to leverage the high efficiency supplementary sCO2 cycle to increase net generation and energy efficiency from the existing biomass boiler. Two sCO2 heater configurations situated within the flue gas flow path are investigated, namely a single convective-dominant heater, and a dual heater configuration with a radiative and a convective heater. A quasi-steady-state 1D model was developed to simulate the integrated cycle, including detailed component characteristics for the Rankine and Brayton cycles. The model solves the mass, energy, momentum, and species balance equations. The system is analyzed for three cases: (i) the existing Rankine cycle without the sCO2 integration, (ii) with the single convective-dominant sCO2 heater configuration, and (iii) the dual sCO2 heater configuration. The results show the required rate of overfiring for the sCO2 configurations, with a 15.3% increase in fuel flowrate resulting in an additional 21.2% in net power output. The model quantifies the impact of the sCO2 heaters, with reduced heat uptakes for downstream boiler heat exchangers. Furnace water wall heat uptake increased due to overfiring, offsetting the reduced heat uptakes at downstream evaporative heat exchangers. The dual configuration has more impact on Rankine cycle operation due to the radiative sCO2 heater placement in front of the second superheater, absorbing some of the direct radiation from the furnace.

Finite element modeling of thermal‐hydro‐mechanical coupled processes in unsaturated freezing soils considering air‐water capillary pressure and cryosuction

AbstractThis paper presents a comprehensive computational model for analyzing thermo‐hydro‐mechanical coupled processes in unsaturated porous media under frost actions. The model employs the finite element method to simulate multiphase fluid flows, heat transfer, phase change, and deformation behaviors. A new soil freezing characteristic curve model is proposed to consider the suctions from air‐water capillary pressure and water‐ice cryosuction. A total pore pressure with components from liquid water pressure, air pressure, and ice pressure is used in the effective stress law. Vapor and dry air are considered miscible gases, utilizing the ideal gas law and Dalton's law. The governing equations encompass the linear momentum balance equation, the energy balance equation, and mass conservation equations for water species (ice, liquid, and vapor) and dry air. Weak forms are formulated based on primary variables of displacement, water pressure, air pressure, and temperature. The spatial discretization is achieved through the finite element method, while temporal discretization employs the fully implicit finite difference method, resulting in a system of fully coupled nonlinear equations. To verify the proposed computational model, a numerical implementation is developed and validated against a set of experimental data from the literature. The successful verification demonstrates the robustness of the model. A detailed discussion of the contributions from phase change strain and different sources of pore pressure is also addressed.

Annular unidirectional slip Couette flow with heat transfer and variable viscosity by Laplace transform and homotopy perturbation method

This article investigates the thermal, variable viscosity axial Couette flow between two con- centric circular cylinders, taking into account both the Navier and the dynamic slip boundary conditions. In order to put in evidence the effect of dynamic slip on the behavior of the solution, we have kept the flowing fluid to a simple Newtonian viscous fluid. The nonlinear governing equations for momentum and energy balance are solved under startup condition using the Laplace transform (LT) technique, in conjunction with the Homotopy Perturbation Method (HPM). The first two orders of approximation for temperature and velocity are obtained, as well as the entropy generation in the space occupied by the flow and the volumetric flux. Numerical results and graphical representations of the solution are provided and discussed to depict the effects of the slip parameters on the velocity and the temperature profiles, on the entropy generation rate and on the volumetric flux.