System Of Balance Equations Research Articles

A two-stage preconditioner for multiphase poromechanics in reservoir simulation

Many applications involving porous media–notably reservoir engineering and geologic applications–involve tight coupling between multiphase fluid flow, transport, and poromechanical deformation. While numerical models for these processes have become commonplace in research and industry, the poor scalability of existing solution algorithms has limited the size and resolution of models that may be practically solved. In this work, we propose a two-stage Newton–Krylov solution algorithm to address this shortfall. The proposed solver exhibits rapid convergence, good parallel scalability, and is robust in the presence of highly heterogeneous material properties. The key to success of the solver is a block-preconditioning strategy that breaks the fully-coupled system of mass and momentum balance equations into simpler sub-problems that may be readily addressed using targeted algebraic methods. Numerical results are presented to illustrate the performance of the solver on challenging benchmark problems.

Hyperbolic thermoelasticity in gas medium

A numerical approach is used to investigate waves propagation due to laser radiation. A thin layer of gas medium is considered, and gas is modeled based on ideal gas concept. The aim of the research is to examine time relaxation constant’s influence on the coupled thermoelastic system. Heat flux relaxation according to Maxwell–Cattaneo law is studied. The spatial description is used in order to describe continuum fields. Density, temperature, heat flux and velocity are evaluated through the system of balance equations: mass, energy and momentum balances are taken in integral form. The high-speed thermal impact is modeled by defining the distribution of heat sources in the volume for the semitransparent medium with help of Bouguer law. The power of the laser pulse depends on time as the Dirac delta function or as the Heaviside function do. A number of problems with different time relaxation constant are considered, and the interaction between thermal and acoustic waves is examined. For numerical calculation, an explicit technique is applied.

Ion-Exchange Inverse Problem for Low-Salinity Coreflooding

The data input for reservoir simulation of low-salinity waterflood in sandstones includes adsorption isotherms, or equilibrium constants for all cation-exchange reactions, and cation-exchange capacity (CEC). We develop a novel method for determination of those functions from the laboratory data of low-salinity coreflooding; the inverse problem is solved; rather, matching is applied. Changing independent time variable to Lagrangian coordinate transforms the governing system for two-phase multicomponent flow into a single-phase system of mass balance equations for each ion and the volumetric balance equation for water. It allows determining the adsorption isotherms for all ions, or equilibrium constants and CEC, from the breakthrough ion concentrations, using the exact solution of ion-exchange inverse problem. Further, the fractional flow and relative permeability can be determined from the effluent water cut and pressure drop histories, using the Welge’s and JBN methods; here each data point corresponds to the breakthrough values of ion concentrations. We show that for four-ion waterflooding, only two constants from two equilibrium constants and CEC can be determined. Treatment of five coreflood data sets shows a close agreement between the laboratory data and the results of inverse problem solution.

A comparison of the finite‐difference and finite‐volume methods for a numerical solution of a hyperbolic thermoelasticity problem utilizing the implicit and explicit schemes

AbstractA numerical solution for a coupled hyperbolic thermoelasticity problem of the Lord‐Shulman type is presented. An infinite layer with free boundaries that are kept at a constant temperature is irradiated by a short laser impulse. In this case the problem can be studied in the one‐dimensional formulation. An impact of the laser impulse on a medium is modeled by an internal heat source. The laser impulse intensity decays with a distance from an irradiated surface according to the Bouguer law. The layer is in an unperturbed state at the initial moment of time. The material properties of copper are taken to be the thermo‐mechanical properties of the layer.Two approaches for the boundary value problem are considered: the finite‐difference method (FDM) and the finite‐volume method (FVM). In the first case the system of two second‐order PDE is investigated, in the second case a system of integral balance equations and an integral thermal conductivity equation are composed. To integrate these equations both the explicit and implicit schemes are used. The obtained results are compared with each other and with an analytical solution. A numerical error is estimated for different time and space step sizes. The features inherent in various ways of integration of the coupled hyperbolic thermoelastic problem with the thermal excitation are found.

Theoretical description and numerical modelling of dehydration of tert-butanol via reactive distillation at concurrent flow of liquid and vapor phases

An original simplified mathematical model of the process of tert-butyl alcohol dehydration in a catalytic zone of the reactive distillation unit with the downward concurrent flow of liquid and vapor phases is developed. The system of heat and mass balance equations includes relationships that allow determining changes in temperature as well as gas and liquid compositions caused by a chemical reaction along the catalyst bed height. Results of calculation of the temperature profile along the catalyst bed height, carried out using the developed model, are in good agreement with data from our previous laboratory studies as well as with data obtained during trial operation of an industrial reactor.

Geometric spatial reduction for port-Hamiltonian systems

A geometric spatial reduction method is presented in this paper. It applies to port Hamiltonian models for open systems of balance equations. It is based on projections which make use of the spatial symmetries in the model and preserve the “natural” power pairing. Reductions from 3D to 2D and 1D domains are illustrated via two examples. The first one is a vibro-acoustic system with cylindrical symmetry where 3D–2D reduction is applied. The second one is the system of two coupled parabolic equations describing the poloidal magnetic flux diffusion and heat radial transport in tokamak reactors. In this latter example the toroidal symmetry allows to perform a 3D–1D reduction. Obtained reduced models are compared with the common control models found in the literature for these two examples.

Рекомендуемые способы предотвращения попадания речных наносов и шуги в водоводы деривационных гидроэлектростанций

The article discusses methods for preventing river drift from getting into the diversion canals of hydroelectric power plants (HPPs). The considered methods imply the use of a curvilinear intake channel, within which longitudinal-transverse circulation occurs, during the high-water period, and the use of drift protection thresholds for setting up artificial circulation during the period of low water level. The river flow hydromorphometric characteristics in the river bed curvilinear part were analyzed, as a result of which empirical dependencies for determining the water intake location and for calculating the intake stream canal straight section length have been obtained. A system of discharge balance equations in the section of the river hydroelectric power plant’s dike structure is given. Based on experimental investigation results, three water intake layout arrangements for mountain rivers are recommended. A method for protecting the diversion canal from sludge ice is suggested. According to this method, in the winter season water is taken from the stream lower layers, and sludge and ice are dumped into the hydroelectric complex tail race by means of special sludge dumping devices. The problem of modeling spatial turbulent fluid flow within the water intake structure head race for diversion power plants (DPP WIS) is solved using the FLOW 3D software package. The velocity distribution patterns along the flushing path axis for favorable and unfavorable sludge passage conditions are given. The diagram of water intake structure flushing path regimes for diversion power plants has been drawn. The obtained investigation results can be used in the design, construction, and operation of the water intake structures of diversion power plants in mountain and piedmont areas.

Feedback control of bilinear distributed parameter system by input-output linearisation

In this paper, a control law that enforces the tracking of a boundary controlled output for a bilinear distributed parameter system is developed in the framework of geometric control. The dynamic behaviour of the system is described by two weakly coupled linear hyperbolic partial differential equations. The stability of the resulting closed-loop system is investigated based on eigenvalues of the spatial operator of a weakly coupled system of balance equations. It is shown that, under some reasonable assumptions, the stability condition is related to the choice of the tuning parameter of the control law. The performance of the developed control law is demonstrated, through numerical simulation, in the case of a co-current heat exchanger. The control objective is to control the outlet cold fluid temperature by manipulating its velocity. Both tracking and disturbance rejection problems are considered.

A new model of thermodynamics. Extension of chemical thermodynamics to complex materials of cellular structure

This paper presents a structural model of complex materials that are partly or entirely of cellular structure, and a new model of thermodynamics, which can be applied to the processes taking place in complex materials. Since the complex materials always contain cellular fraction, the supposition of cellular equilibrium is real. The complex materials are characterized by polyagent behaviour. Therefore, new concepts such as principal variables, redundancy, macroscopic/microscopic uncertainty are defined, moreover, the application of chemical thermodynamics is limited to micro processes only, and the free energy (F) and the free enthalpy (G) function cannot be generally applied to calculations concerning the bulk amount. Temperature as only a local intensive variable can be discussed. The first law of thermodynamics is expressed in the traditional way. The second law applied to the processes taking place both in open and in adiabatically closed systems is formulated as in words as an equation of stability, completed by the relations to first and second differential of entropy. In connection with the second law the Damköhler balance equation system, the Rabinowitsch–Mooney equation, and the application of dimensional analysis are presented. Only the apparent thermal capacities can be used for complex materials in general. The data of specific thermal capacity of complex materials in the proximity of absolute zero temperature are not sufficient for drawing conclusions on their entropy. The concept ‘shelf-life’ is essential in food science and practice, a Monte-Carlo method is presented for its calculation.

Numerical simulation of fire integrity resistance of full-scale gypsum-faced cross-laminated timber wall

Fire design requires three fire resistance criteria, namely integrity, insulation and load-bearing. In particular, repartition wall systems used in residential and commercial buildings are required to provide certification of insulation and integrity fire resistance and have to be analysed in the context of the characteristic time scales of a fire. This paper presents a numerical methodology to simulate the fire integrity resistance of full-scale timber-based wall systems exposed to fire. The main contribution of this paper, with regard to the existing literature, is the presence of cross-laminated timber (CLT) panel in the central part of the wall system, where the pyrolysis of timber was modelled explicitly in the energy balance equation system. For this end, a user-subroutine, called UMATHT, is developed and successfully implemented in the Abaqus finite element code for thermal analysis. The falling off of the gypsum boards under fire was considered implicitly in the FE model based on experimental observations. The obtained results show clearly the need to take into account explicitly the cracks and falling off of the gypsum boards for an appropriate prediction of the integrity fire resistance in order to reduce expensive experimental procedures.

On the Formulation of the Problem of Optimal Control of Production Parameters using a Two-Level Model of the Production Process

Using a statistical approach, widespread in natural sciences, a two-level model to control the parameters of the flow line production system has been built. The state of the system is given by the amounts of sets of the objects of labour. The state of the subject of labour is given by a point in the phase space. The function of the distribution of objects of labour by the state is introduced and the kinetic equation for the distribution function is written. Now we have closed system of dynamical equations for parameters of flow production line. The null and the first moments of the distribution function of labour objects in terms of the state characterize the magnitude of inter-operational stocks and the rate of processing of labour objects from operations of the technological route and are the main parameters of the management of the production line. The limiting transition from the kinetic description of the state of objects of labour to the stream description of the processing of objects of labour is accomplished. Integration of the kinetic equation by the states of the objects of labour made it possible to construct a closed system of balance equations for the parameters of the production line. The task of optimal control of the flow parameters of the production line has been set. The balance equations for the moments of the distribution function of objects of labour by states determine the constraint equations in the control problem

Implicit solution of harmonic balance equation system using the LU-SGS method and one-step Jacobi/Gauss-Seidel iteration

ABSTRACTThis paper presents efficient alternative numerical methods for an implicit solution of the harmonic balance equation system for analysing temporal periodic unsteady flows. The proposed method employs approximate factorisation to decouple the common residual term and the time spectral source term of a harmonic balance equation system when it is discretised implicitly. With this approximate factorisation, the complexity of implicit solution of the discrete system is greatly reduced. The common residual term can be dealt with using a lower-upper symmetric-Gauss-Seidel (LU-SGS) method and the time spectral source term is integrated using a Jacobi iteration (JI) or one step Gauss-Seidel (GS) iteration, leading to the LU-SGS/JI method or LU-SGS/GS method. The NASA stage 35 compressor and the 1.5 stage Aachen turbine were used to demonstrate the effectiveness of the proposed methods in stabilising solution and its advantages in comparison with the existing lower-upper symmetric-Gauss-Seidel/block Jacobi (LU-SGS/BJ) method. The LU-SGS/GS method and the LU-SGS/JI method are more robust than the LU-SGS/BJ method in stabilising solution. The LU-SGS/GS method also has faster and tighter convergence and lower memory consumption in comparison with the LU-SGS/BJ method.

Extended thermodynamics of dense polyatomic gases: modeling of molecular energy exchange

We revisit the extended thermodynamic (ET) theory of dense polyatomic gases developed in the paper (Arima et al. in Phys Rev Fluids 2:013401, 2017) from the viewpoint of the energy exchange between two subsystems: the subsystem with the kinetic and potential energies and the subsystem of the internal modes such as molecular rotation and vibration. We confirm that the system of balance equations derived from the viewpoint is completely the same as the previous one, but thereby we can obtain its complementary physical implications. We also point out a possible alternative procedure in the modeling method based on ET.

Modelling of strain softening materials based on equivalent damage force

The main aim of the work presented in this paper was treatment of damage and deformation localisation observed in the finite element method (FEM) analysis of strain softening materials combined with local constitutive models where damage is represented using continuum damage mechanics (CDM). The CDM/FEM approach typically suffers from a number of shortcomings, including mathematical (change of the type of partial differential equations leading to ill-posed boundary value problem), numerical (pronounced mesh dependency) and physical (infinitely small softening zone with the zero dissipated energy). The approach proposed here is still based on the local constitutive model including damage, but introduces an alternative representation of damage effects in the system of linear momentum balance equations. The damage effects are included through equivalent damage force (EDF), which contributes to the right-hand side of the momentum balance equations. The main advantages of this approach are that the problem remains well posed, as the type of partial differential equations remains unchanged when the material enters softening; numerical stability is preserved without a need for regularisation measures; and significantly reduced mesh dependency. In addition, the EDF approach can be used in combination with existing local CDM damage models and does not violate symmetry of the material stiffness tensor.The EDF approach is applicable to modelling of strain softening typically observed in damaged quasi brittle materials such as fibre reinforced composites and concrete.The EDF model was implemented in the in-house developed coupled FEM-SPH code, where an explicit FEM code is coupled with a stable Total-Lagrange form of SPH. Its performance is demonstrated in the analysis of a dynamic one dimensional (1D) stress wave propagation problem, which was analytically solved by Bazant and Belytschko in 1985. For a range of loading rates that correspond to the material softening regime, the numerical results shown nonlocal character with a finite size of the damaged zone, controlled with the damage characteristic length, which can be experimentally determined and is an input parameter independent of the discretisation density.

Analysis of photoinduced reverse changes in bacterial reaction centers

Background: The membrane protein-pigment complexes of photosynthetic isolated reaction centers (RC) Rhodobacter Sphaeroides are macromolecular systems for studying the physical mechanisms of electron and proton transport in biological structures, the role of molecular dynamics. The experimental kinetics of cyclic electron transfer in molecular complexes has a multiexponential character with negative values of decrements. For their description, a system of balance equations is used. Objective of the work is to determine the features of the kinetics of cyclic electron transfer in the RC using two models of electron transfer and the connection of such features with space-time motions in the RC. Materials and methods: Measurement of the absorption kinetics was performed at 865 nm using a two-channel diode spectrometer. The experimental kinetics of RC absorption (the main reaction of the system) was represented by the fitting method in the form of a sum of three exponential functions. In the first model with time-variable rate constants of the balance equations, the wavelet transform method of the logarithmic derivative of the electron transfer kinetics was used. In the second model, the equation of state and three differential equations with constant coefficients were used as the algebraic sum of the rate constants. To determine the values of the rate constants in the balance equation, an optimization problem was solved. The solution of the system of balance equations by the matrix method made it possible to determine the features of the kinetics of the population of substates of the RC. Results of calculations showed that the features of the wavelet spectrum of the logarithmic derivative of the electron transfer kinetics in the first model coincided with the features of the population kinetics of substates of the RC of the second RC model. These features were in the bands 1 s, 3 s, 60 s from the moment of switching on (off) the light and depend on the photoexcitation parameters. Conclusions: The features of the kinetics of the populations of substates in the RC both at the stage of illumination and at the relaxation stage are determined by changes in the structure of the RC in the form of effects of hidden parameters of the structural self-regulation of the RC (feedback through the RC structure).

About the Stability of Flow Parameters of a Production Line

The model of the technological process for enterprises with the production method of production is investigated. The dynamics of macroparameters of the production process is considered: inter-operational stocks and the rate of processing of subjects of labour on technological operations along the technological route. Balance equations are presented for macroparameters of the technological process, which express differential dependencies between macroparameters. The linearization of the system of balance equations is performed and the equations of the perturbed state for macroparameters of the closed system of balance equations are written down. Criteria of stability for macroparameters of the production system relative to the normative unperturbed state are determined. Conditions of the stability of the production system functioning are presented. The interrelation between the magnitude of the tempo and the magnitude of the inter-operational reserve of basic products is shown, which ensures the stable functioning of the production process. A system of equations for the perturbed state of a production system for a special case of stability theory is considered (the case of one zero root of the characteristic equation of the system).

Homogeneous Hamiltonian Control Systems Part II: Application to thermodynamic systems

In this paper, we give several examples of the formulation of the dynamics of open physical systems in terms of homogeneous Hamiltonian control systems. Therefore we have systematically used Gibbs-Massieu equations in the entropy form for the definition of the Lagrangian submanifold associated with the thermodynamical properties of the system. Accordingly we have used the total entropy function of the system to define the generating function of this Lagrange submanifold and have used the canonical lifting of systems of balance equations of the remaining independent extensive variables. In the first part we consider two examples of thermodynamic systems: two cells exchanging a heat flow and the Continuous Stirred Tank Reactor (CSTR). In the second part, we consider an elementary dissipative mechanical, the mass-spring-damper system and then embed it into the gas-piston system.

On the Formulation of the Problem of Optimal Control of Production Parameters Using a Two-Level Model of the Production Process

Using a statistical approach, widespread in natural sciences, a two-level model to control the parameters of the flow line production system has been built. The state of the system is given by the amounts of sets of the objects of labour. The state of the subject of labour is given by a point in the phase space. The function of distribution of objects of labour by state is introduced and the kinetic equation for the distribution function is written. Now we have closed system of dynamical equations for parameters of flow production line. The null and the first moments of the distribution function of labor objects in terms of the state characterize the magnitude of interoperational stocks and the rate of processing of labor objects from operations of the technological route and are the main parameters of the management of the production line. The limiting transition from the kinetic description of the state of objects of labor to the stream description of the processing of objects of labor is accomplished. Integration of the kinetic equation by the states of the objects of labor made it possible to construct a closed system of balance equations for the parameters of the production line. The task of optimal control of the flow parameters of the production line has been set. The balance equations for the moments of the distribution function of objects of labor by states determine the constraint equations in the control problem.

Estimation of the signal gain in a linear-cavity ytterbium laser

Based on the exponential representation of signal amplification along an active fibre, we construct an analytical approximation of the solution to a system of balance equations describing the dynamics of an average signal power and pump inside a linear cavity. The output power of the signal at the ends of the linear cavity is estimated. The output power is optimised for a linear-cavity ytterbium fibre laser.

New Non-Stationary Gradient Model of Heat-Mass-Electric Charge Transfer in Thin Porous Media

The well-known complicated system of non-equilibrium balance equations for a continuous fluid (f) medium needs the new non-Gibbsian model of f-phase to be applicable for description of the heterogeneous porous media (PMs). It should be supplemented by the respective coupled thermal and caloric equations of state (EOS) developed specially for PMs to become adequate and solvable for the irreversible transport f-processes. The set of standard assumptions adopted by the linear (or quasi-linear) non-equilibrium thermodynamics are based on the empirical gradient-caused correlations between flows and forces. It leads, in particular, to the oversimplified stationary solutions for PMs. The most questionable but typical modeling suppositions of the stationary gradient (SG) theory are: 1) the assumption of incompressibility accepted, as a rule, for f-flows; 2) the ignorance of distinctions between the hydrophilic and hydrophobic influence of a porous matrix on the properties; 3) the omission of effects arising due to the concomitant phase intra-porous transitions between the neighboring f-fragments with the sharp differences in densities; 4) the use of exclusively Gibbsian (i.e. homogeneous and everywhere differentiable) description of any f-phase in PM; 5) the very restrictive reduction of the mechanical velocity field to its specific potential form in the balance equation of f-motion as well as of the heat velocity field in the balance equation of internal energy; 6) the neglect of the new specific peculiarities arising due to the study of any non-equilibrium PM in the meso- and nano-scales of a finite-size macroscopic (N,V)-system of discrete particles. This work is an attempt to develop the alternative non-stationary gradient (NSG) model of real irreversible processes in PM. Another aim is to apply it without the above restrictions 1)-6) to the description of f-flows through the obviously non-Gibbsian thin porous medium (TPM). We will suppose that it is composed by two inter-penetrable fractal sf-structures of f-phase (formed by the “mixture” of g- and l-phases termed, in total, interphase) and solid (s) porous matrix termed below s-phase. The permanent influence of humidity and the respective increase of the moisture content in TPM including the unavoidable phenomenon of capillary condensation are the main factors to occur the non-stationary transport f-flows through its texture.