Balance Laws Of Energy Research Articles

Symmetries and conservation laws of the Euler equations in Lagrangian coordinates

We consider the Euler equations of incompressible inviscid fluid dynamics. We discuss a variational formulation of the governing equations in Lagrangian coordinates. We compute variational symmetries of the action functional and generate corresponding conservation laws in Lagrangian coordinates. We clarify and demonstrate relationships between symmetries and the classical balance laws of energy, linear momentum, center of mass, angular momentum, and the statement of vorticity advection. Using a newly obtained scaling symmetry, we obtain a new conservation law for the Euler equations in Lagrangian coordinates in n-dimensional space. The resulting integral balance relates the total kinetic energy to a new integral quantity defined in Lagrangian coordinates. This relationship implies an inequality which describes the radial deformation of the fluid, and shows the non-existence of time-periodic solutions with nonzero, finite energy.

On the thermodynamics of smooth muscle contraction

Cell function is based on many dynamically complex networks of interacting biochemical reactions. Enzymes may increase the rate of only those reactions that are thermodynamically consistent. In this paper we specifically treat the contraction of smooth muscle cells from the continuum thermodynamics point of view by considering them as an open system where matter passes through the cell membrane. We systematically set up a well-known four-state kinetic model for the cross-bridge interaction of actin and myosin in smooth muscle, where the transition between each state is driven by forward and reverse reactions. Chemical, mechanical and energy balance laws are provided in local forms, while energy balance is also formulated in the more convenient temperature form. We derive the local (non-negative) production of entropy from which we deduce the reduced entropy inequality and the constitutive equations for the first Piola–Kirchhoff stress tensor, the heat flux, the ion and molecular flux and the entropy. One example for smooth muscle contraction is analyzed in more detail in order to provide orientation within the established general thermodynamic framework. In particular the stress evolution, heat generation, muscle shorting rate and a condition for muscle cooling are derived.

Energy balance and second law analysis applied to buildings: an opportunity for Bejan’s theory

Energy balance and second law analysis applied to buildings: an opportunity for Bejan’s theory

Energy balance and second law analysis applied to buildings: an opportunity for Bejan’s theory

Energy balance and second law analysis applied to buildings: an opportunity for Bejan’s theory

Shape Memory Alloys Behaviour: A Review

Shape memory alloys are used in a variety of fields, such as medical or aeronautical. Other fields of knowledge have been researching these materials, attracted by their capacity to dissipate energy through high-strain hysteretic cycles without significant residual strains. Because of these interesting properties for seismic protection, an example of the possible beneficiaries of these materials are civil engineering structures. This paper reports a bibliographic review on the characteristics and uniaxial macroscale constitutive models for shape memory alloys, of interest for a significant number of applications, most often based on wires and bars. The constitutive model assessment focuses on mechanical and kinetic laws, as well as on the energy balance law, of relevance for dynamic loadings. Some characteristics of these materials are still not sufficiently well known, especially those related to ageing. With regard to behaviour prediction, the most frequently used uniaxial constitutive models result in similar responses.

Nonconventional thermodynamics, indeterminate couple stress elasticity and heat conduction

We present a phenomenological thermodynamic framework for continuum systems exhibiting responses which may be nonlocal in space and for which short time scales may be important. Nonlocality in space is engendered by state variables of gradient type, while nonlocalities over time can be modelled, e.g. by assuming the rate of the heat flux vector to enter into the heat conduction law. The central idea is to restate the energy budget of the system by postulating further balance laws of energy, besides the classical one. This allows for the proposed theory to deal with nonequilibrium state variables, which are excluded by the second law in conventional thermodynamics. The main features of our approach are explained by discussing micropolar indeterminate couple stress elasticity and heat conduction theories.

The stored energy of cold work, thermal annealing, and other thermodynamic issues in single crystal plasticity at small length scales

This paper develops a thermodynamically consistent gradient theory of single-crystal plasticity using the principle of virtual power as a paradigm to develop appropriate balance laws for forces and energy. The resulting theory leads to a system of microscopic force balances, one balance for each slip system, and to an energy balance that accounts for power expended during plastic flow via microscopic forces acting in concert with slip-rates and slip-rate gradients. Central to the theory are an internal energy and entropy, plastic in nature, dependent on densities that account for the accumulation of glide dislocations as well as geometrically necessary dislocations – and that, consequently, represent quantities associated with cold work. Our theory allows us to discuss – within the framework of a gradient theory – the fraction of plastic stress-power that goes into heating, as well as the reduction of the dislocation density in a cold-worked material upon subsequent (or concurrent) thermal annealing.

Transient Thermal Analysis of Wheel-Mounted Brake Disc of Gray Cast Iron

Determining how to accurately calculate the thermal loads and boundary conditions are the most critical issues in the thermal analysis of the brake disc. The heat flux acting on the disc surface was got by the uniform wear theory with consideration of the cover angle of pad. The result shows that: the heat flux determined by uniform wear is not only related to the braking time but also radial distance; the convective heat transfer coefficients will decrease by the raise of the time; the cover angle of pad increases at first and then decreases, while the heat partition coefficient is in the opposite situation; the results determined by uniform wear with the consideration of the cover angle of pad is more close to that calculated by energy balance law.

Condition Monitoring for Early Failure Detection. Frognerparken Pumping Station as Case Study

A case study has been carried out to investigate how process data from Frognerparken Pumping Station can be used more efficiently to get an early detection of degradation or failures of the pump systems. The technical condition is described by use of a single number from 0 to 100, denoted as the Technical Condition Index (TCI), where maximum value describes the design condition as new and the minimum value describes the state of total degradation. As presented in this paper, the pump station is broken down to subsystems where each subsystem consists of several nodes with its own TCI value. The overall pump station's TCI number is then computed by aggregating a weighted sum of all the subsystems’ TCIs.A method for early failure detection is also developed and is based on combining energy balance law together with the nonlinear extended Kalman filter to estimate model parameters for both the heating and the cooling phase of the pump. The model parameters are directly related to the temperature change and will give an indication of degradation during the transient phase even if the pump is working properly. During a visual inspection one of the pump had significantly higher temperature and vibration compared to the other pumps, and based on received timeseries for that specific day, we were able to validate the proposed method with realistic data.

On the Formulation of Mass, Momentum and Energy Conservation in the KdV Equation

The Korteweg-de Vries (KdV) equation is widely recognized as a simple model for unidirectional weakly nonlinear dispersive waves on the surface of a shallow body of fluid. While solutions of the KdV equation describe the shape of the free surface, information about the underlying fluid flow is encoded into the derivation of the equation, and the present article focuses on the formulation of mass, momentum and energy balance laws in the context of the KdV approximation. The densities and the associated fluxes appearing in these balance laws are given in terms of the principal unknown variable ? representing the deflection of the free surface from rest position. The formulae are validated by comparison with previous work on the steady KdV equation. In particular, the mass flux, total head and momentum flux in the current context are compared to the quantities Q, R and S used in the work of Benjamin and Lighthill (Proc. R. Soc. Lond. A 224:448---460, 1954) on cnoidal waves and undular bores.

Reviewing the roots of continuum formulations in molecular systems. Part II: Energy and angular momentum balance equations

This paper is the second part of a series dedicated to reviewing the fundamental link between discrete and continuum formulations, which is established by space averaging followed by probability density averaging. On obtaining the continuum balance laws of mass and linear momentum in the part I, here the balance laws of angular momentum and energy are re-derived from a discrete (atomistic/molecular) description. Different approaches (explicit and implicit) for the consideration of the potential energy are reviewed. Thereby for the explicit approach ambiguous possibilities for the localization of the potential energy are briefly discussed. Thereby we conclude that the explicit approach is preferable from the practical application point of view, however it becomes cumbersome when applied to multi-body interactions systems, whereas the implicit approach has no ambiguity in the localization of the potential energy to each particle and is easily applicable to any multi-body potential. Possible solutions for continuum fluxes (couple stresses, heat flux) are postponed until part III of the series.

Physics-Based Reasoning in Conceptual Design Using a Formal Representation of Function Structure Graphs

This paper validates that a previously published formal representation of function structure graphs actually supports the reasoning that motivated its development in the first place. In doing so, it presents the algorithms to perform those reasoning, provides justification for the reasoning, and presents a software implementation called Concept Modeler (ConMod) to demonstrate the reasoning. Specifically, the representation is shown to support constructing function structure graphs in a grammar-controlled manner so that logical and physics-based inconsistencies are prevented in real-time, thus ensuring logically consistent models. Further, it is demonstrated that the representation can support postmodeling reasoning to check the modeled concepts against two universal principles of physics: the balance laws of mass and energy, and the principle of irreversibility. The representation in question is recently published and its internal ontological and logical consistency has been already demonstrated. However, its ability to support the intended reasoning was not validated so far, which is accomplished in this paper.

Isogeometric divergence-conforming B-splines for the unsteady Navier–Stokes equations

Divergence-conforming B-splines are developed for application to the incompressible Navier–Stokes equations on geometrically mapped domains. These enable smooth, pointwise divergence-free solutions and thus satisfy mass conservation in the strongest possible sense. Semi-discrete methods based on divergence-conforming B-splines are shown to conserve linear and angular momentum and satisfy balance laws for energy, vorticity, enstrophy, and helicity. These are geometric structure-preserving quantities and numerical simulations that are sensitive to them are shown to be qualitatively correct and quantitatively accurate. The methods developed are anticipated to open new doors to the practical calculation of complex flows and to studies of their physical behavior.

Reviewing the roots of continuum formulations in molecular systems. Part I: Particle dynamics, statistical physics, mass and linear momentum balance equations

The link between atomistic quantities and continuum fields has been the subject of research for at least half a century. Nevertheless, there are still many open questions and misleading discussions in the literature. Therefore, based on the fundamental principles of classical mechanics and statistical physics we construct the basic framework for the link between the atomistic and continuum worlds. In doing so, considerable attention is paid to the central force decomposition and multi-body potentials, balance of angular momentum for the system of particles and its relationship to the extended third Newton axiom and the difference between the theorem of change of kinetic energy and the energy balance law. A number of general theorems related to the convolution properties of statistically averaged quantities, as well as their rates are also proven. These theorems make the derivation of balance equations far simpler when compared to the approaches used by others. Such theorems also make the link between space–time averaging and space–probability averaging more transparent. In this contribution the balance laws of mass and linear momentum are derived. The remaining balance laws of angular momentum and energy as well as the particular forms of fluxes, such as the stress, are discussed in the follow-up contributions of this series.

Numerical simulation for unsteady compressible Micropolar fluid flow

This paper reports the advancement of computational Micropolar fluid dynamics (CMFD) using the spectral difference (SD) method. The fundamentals, linear constitutive equations and generalized Stokes’ hypothesis for Micropolar fluid dynamics are briefly introduced. The additional degrees of freedom in MFD, gyration, can help to understand the coherent structures of vortices from the two-level energy transfer in the balance law of energy. The spectral difference (SD) method is proposed for solving unsteady compressible viscous Micropolar flow problems. The analytical and exact solution of compressible Micropolar Couette flow is solved and used to demonstrate the order of numerical accuracy of the SD method. As a numerical example, a 2D spectral difference solver is developed to simulate flow past a cylinder. The instantaneous gyration contours are plotted for the observation of vortex shedding behind the cylinder. A new physical phenomenon, the coupling effect, is discussed.

Ultrafast solid-liquid-vapor phase change of a thin gold film irradiated by femtosecond laser pulses and pulse trains

Effects of different parameters on the melting, vaporization and resolidification processes of thin gold film irradiated by femtosecond pulses and pulse train were systematically studied. The classical two-temperature model was adopted to depict the non-equilibrium heat transfer in electrons and lattice. The melting and resolidification processes, which was characterized by the solid-liquid interfacial velocity, as well as elevated melting temperature and depressed solidification temperature, was obtained by considering the interfacial energy balance and nucleation dynamics. Vaporization process which leads to ablation was described by tracking the location of liquid-vapor interface with an iterative procedure based on energy balance and gas kinetics law. The parameters in discussion included film thickness, laser fluence, pulse duration, pulse number, repetition rate, pulse train number, etc. Their effects on the maximum lattice temperature, melting depth and ablation depth were discussed based on the simulation results.

A comparative study of two macro-scale models of condensed-phase explosives

Two macro-scale models of high-energy, condensed-phase explosives are considered. The first, called ignition and growth, treats the heterogeneous explosive as a homogeneous mixture of reacting and product species. The second model treats the explosive as two distinct, but interacting, phases. Both models are hyperbolic systems of nonlinear partial differential equations involving balance laws of mass, momentum and energy, and involving assumed constitutive input and reaction kinetics. The two-phase model also involves stiff relaxation terms and nozzling terms that prevent the equations from being cast in conservation form. A high-resolution shock-capturing scheme on overlapping grids, with parallel adaptive mesh refinement, is used to solve the governing equations accurately. The purpose of the paper is to compare solutions of the two models for macro-scale problems involving detonation initiation and diffraction. This is done to illustrate strengths and weaknesses of the models, and to suggest modifications and extensions of the models for further study.

Formulation of Balance Laws as Mixture Theory for Nuclei and Matrix in Recrystallization

In the previous work, the authors formulated the balance laws of mass, momentum, angular momentum and energy of the lattice element used for recrystallization. These laws were summed up over a phase in a representative volume element (RVE) and averaged in the RVE so as to develop the discrete balance laws for single phase. Furthermore, the balance law of angular momentum was separated into a bulk and a lattice parts through the orderestimation with the representative lengths both in macroscopic and microscopic scales. In this paper, the RVE converges on a material point so that the laws are rewritten in the integration form. When the laws are summed up all over the phases and averaged in them, the balance laws of mass, momentum, angular momentum and energy for nuclei and matrix as mixture are formulated, using an useful theorem proposed for the mixing summation of unsteady terms. At this time, the macroscopic part of the balance law for angular momentum results in the usual equation of angular momentum, so that the stress tensor keeps symmetry even if the lattice rotation is considered. While, the microscopic one is localized as an equation of spin angular momentum for lattice, which is suggested to be equivalent to the evolution equation of crystal orientation in KWC type phase-field model. Moreover, the increase law of entropy for mixture is also formulated. During this process, the entropy flux is defined by use of relative mass flux and chemical potential of phase transformation.

Wave-crack interaction in finite elastic bodies

This paper continues studies in Lalegname et al. (Int J Fract 152:97–125, 2008) on crack propagation in a bounded linear elastic body under the influence of incident waves. In Lalegname et al. (2008) we have considered shear waves, whereas in this paper we discuss the influence of plane elastic waves to a running crack. Actually, the time dependent problem is formulated in a two-dimensional current cracked configuration by a system of linear elasto-dynamic equations. In order to describe the behaviour of the elastic fields near the straight crack tip, we transform these equations to a reference configuration and derive the dynamic stress singularities. Furthermore, we assume that an energy balance law is valid. Exploiting the knowledge on the singular behaviour of the crack fields, we derive from the energy balance law a dynamic energy release rate. Comparing this energy release rate with an experimentally given fracture toughness we get an ordinary differential equation for the crack tip motion. We present first numerical simulations for a Mode I crack propagation.

Transmission of longitudinal wave through micro-porous elastic solid interface

An investigation of reflection and transmission phenomena of plane longitudinal wave from a plane interface between two distinct micropolar porous elastic solid half-spaces in welded contact has been made. Using the method of potentials, the appropriate boundary conditions at the interface are solved to obtain the amplitudes of various reflected and transmitted waves. It has been noticed that these amplitudes are complex valued and their absolute values depend upon the angle of incidence, frequency of the incident wave and on the elastic characteristics of the medium. Numerical computations for a peculiar model are also performed. The variation of modulus of amplitudes of various reflected and transmitted waves against the angle of incidence are computed for obliquely incident longitudinal displacement wave traveling at high frequency as well as at low frequency. At each angle of incidence, the energy balance law has been verified at the interface.