Energy Dissipation Law Research Articles

A new approach for evaluating photosynthetic bio-hydrogen production: The dissipation rate method

A dissipative rate model is established by using the non-equilibrium thermodynamics method to evaluate photosynthetic bio-hydrogen production process. The mass transfer dissipation rate, chemical reaction dissipation rate and total dissipation rate for the bio-hydrogen production processes with glucose and acetic acid are calculated and analyzed respectively by using the proposed dissipation rate model. When substrate concentration is taken as variable (15 g/L,20 g/L and25 g/L), the total dissipation rate and mass transfer phenomenological coefficient of hydrogen production process with 20 g/L are the lowest, and the maximum dissipation rate is 7.7288J/h. The phenomenological coefficient of chemical reaction is larger at high concentration. When the temperature is variable (298.15 K,303.15Kand308.15 K), the dissipation rate is lowest at 303.15 K and the chemical reaction coefficient is largest and the maximum dissipation rate is 6.0596J/h. In addition, the mass transfer phenomenological coefficient is larger at high temperature. With the initial pH as variable (6.5,7.0,7.5), the maximum dissipation rate of 11.2271J/h is obtained at pH 7.5. At high pH, the mass transfer phenomenological coefficient is large and the chemical reaction coefficient is exactly the opposite. This model can predict the energy dissipation law in the process of photosynthetic biohydrogen production, and provide a theoretical basis for the subsequent process optimization.

An energy-stable and conservative numerical method for multicomponent Maxwell–Stefan model with rock compressibility

Numerical simulation of gas flow in porous media is becoming increasingly attractive due to its importance in shale and natural gas production and carbon dioxide sequestration. In this paper, taking molar densities as the primary unknowns rather than the pressure and molar fractions, we propose an alternative formulation of multicomponent Maxwell–Stefan (MS) model with rock compressibility. Benefiting from the definitions of gas and solid free energies, this MS formulation has a distinct feature that it follows an energy dissipation law, and namely, it is consistent with the second law of thermodynamics. Additionally, the formulation obeys the famous Onsager's reciprocal principle. An efficient energy-stable numerical scheme is constructed using the stabilized energy factorization approach for the Helmholtz free energy density and certain carefully designed formulations involving explicit and implicit mixed treatments for the coupling between molar densities, pressure, and porosity. We rigorously prove that the scheme inherits the energy dissipation law at the discrete level. The fully discrete scheme has the ability to ensure the mass conservation law for each component as well as preserve the Onsager's reciprocal principle. Numerical tests are conducted to verify our theories, and in particular, to demonstrate the good performance of the proposed scheme in energy stability and mass conservation as expected from our theories.

Gradient-descent-like scheme for the Allen–Cahn equation

The phase-field equations have many attractive characteristics. First, phase separation can be induced by the phase-field equations. It transforms from a single homogeneous mixture to two distinct phases in a nascent state. Second, the solution of the phase-field equations is bounded by a finite value. It is beneficial to ensure numerical stability. Third, the motion of the interface can be described by geometric features. It is helpful for expressing natural phenomena in mathematical terms. Fourth, the phase-field equations possess the energy dissipation law. This law is about degeneration and decay. It tells us in thermodynamics that all occurrences are irreversible processes. In this paper, we would like to investigate the numerical implementation of the Allen–Cahn (AC) equation, which is the classical one of the phase-field equations. In phase field modeling, the binary phase system is described using a continuous variable called the order parameter. The order parameter can be categorized into two forms: conserved, which represents the physical property such as concentration or mass, and non-conserved, which does not have the conserved physical property. We consider both the non-conservative and conservative AC equations. Our interest is more precisely to scrutinize the utilization of the discrete Laplacian operator in the AC equation by considering the conservative and non-conservative order parameter ϕ. Constructing linearly implicit methods for solving the AC equation, we formulate a gradient-descent-like scheme. Therefore, reinterpreting the implicit scheme for the AC equation, we propose a novel numerical scheme in which solutions are bounded by 1 for all t > 0. Together with the conservative Allen–Cahn equation, our proposed scheme is consistent when mass is conserved as well. From a numerical point of view, a linear, unconditionally energy stable splitting scheme is transformed into a gradient-descent-like scheme. Various numerical simulations are illustrated to demonstrate the validity of the proposed scheme. We also make distinctions between the proposed one and existing numerical schemes.

Energy consumption model of coal grinding based on energy dissipation fractal dimension

Based on particle size distribution characteristics, this paper explored the energy dissipation law in the coal ultrafine grinding process. By studying the fractal dimension of coal particle size distribution under the multi-domain fractal rule, the truncated particle size was introduced to characterize the energy consumption changes of coarse, intermediate, and fine particles. Combined with macro-energy consumption laws and three-piecewise fractal dimensions, a micro-energy consumption theory considering lattice fracture and agglomeration of ultrafine particles was proposed. On this basis, the energy consumption fractal dimension DE was proposed, and its theoretical calculation value was 2.021. The energy consumption formula of coal ultrafine grinding was established, in which energy consumption was characterized by the DE th power of the added surface area of coal particles. The DE obtained by the specific surface area and the adsorption equilibrium constant of pores were 2.285 and 2.564, respectively, which confirmed the reliability of the above theoretical analysis.

L 2 norm error estimates of BDF methods up to fifth-order for the phase field crystal model

Abstract The well-known backward difference formula (BDF) of the third, the fourth and the fifth orders are investigated for time integration of the phase field crystal model. By building up novel discrete gradient structures of the BDF-$\textrm{k}$ ($\textrm{k}=3,4,5$) formulas, we establish energy dissipation laws at the discrete levels and then obtain a priori solution estimates for the associated numerical schemes; however, we can not build any discrete energy dissipation law for the corresponding BDF-6 scheme because the BDF-6 formula itself does not have any discrete gradient structures. With the help of the discrete orthogonal convolution kernels and Young-type convolution inequalities, some concise $L^{2}$ norm error estimates (with respect to the starting data in the $L^{2}$ norm) are established via the discrete energy technique. To the best of our knowledge, this is the first time such type $L^{2}$ norm error estimates of non-A-stable BDF schemes are obtained for nonlinear parabolic equations. Numerical examples are presented to verify and support the theoretical analysis.

Linear energy storage and dissipation laws and damage evolution characteristics of rock under triaxial cyclic compression with different confining pressures

Triaxial cyclic compression tests with different confining pressures were conducted to explore the energy distribution characteristics of rock and their responses to confining pressure. The pre-peak input strain energy, pre-peak elastic strain energy, and pre-peak dissipated strain energy at different stress levels were computed based on the resultant stress−strain curves. The results indicated that under different confining pressures, both the pre-peak elastic strain energy and pre-peak dissipated strain energy were linearly related to the pre-peak input strain energy, and thus the linear energy storage and dissipation laws in triaxial cyclic compression were confirmed. The energy storage coefficient in triaxial compression of the rock was positively correlated with the confining pressure, whereas the corresponding energy dissipation coefficient exhibited an opposite correlation. Using the linear energy storage law, the peak elastic strain energy and peak dissipated strain energy of rock in triaxial compression were deduced. Furthermore, the damage evolution characteristics of rock under different confining pressures were analysed by considering the linear energy dissipation law.

Two-Phase Flows with Bulk–Surface Interaction: Thermodynamically Consistent Navier–Stokes–Cahn–Hilliard Models with Dynamic Boundary Conditions

We derive a novel thermodynamically consistent Navier–Stokes–Cahn–Hilliard system with dynamic boundary conditions. This model describes the motion of viscous incompressible binary fluids with different densities. In contrast to previous models in the literature, our new model allows for surface diffusion, a variable contact angle between the diffuse interface and the boundary, and mass transfer between bulk and surface. In particular, this transfer of material is subject to a mass conservation law including both a bulk and a surface contribution. The derivation is carried out by means of local energy dissipation laws and the Lagrange multiplier approach. Next, in the case of fluids with matched densities, we show the existence of global weak solutions in two and three dimensions as well as the uniqueness of weak solutions in two dimensions.

Phase-field modeling and consistent energy-stable simulation of binary creeping flows in contact with solid

In this work, we present an efficient and practical model for describing two-phase creeping fluid flows in contact with a solid substrate. In the framework of phase-field method, a ternary Cahn–Hilliard model is modified by adding a term reflecting the wetting condition of liquid phase on the liquid–solid interface. The contact angle dynamics can be implicitly achieved by solving the phase-field equations and the explicit treatment on liquid–solid boundary is absent. Therefore, various discretization methods in space can be naturally adopted. To update the creeping flows in arbitrary domains, we herein consider the incompressible Darcy equations with a penalty term. The coupled binary fluid system theoretically satisfies the energy dissipation law with respect to a total energy functional. An energy dissipation-preserving time-marching scheme is constructed based on the auxiliary variable approach. Furthermore, a simple correction technique is utilized to improve the consistency between original and numerical values. We analytically prove that the proposed scheme still satisfies the energy law in its discrete version. Extensive numerical experiments are performed to validate the accuracy, consistent stability, and capability of our method.

Influence of water on rockburst proneness of sandstone: Insights from relative and absolute energy storage

To fully understand the influence of water on rockburst proneness from energy storage viewpoint of rock and to check the performance of energy-related rockburst proneness indexes, several groups of uniaxial compression tests at different stress levels are performed on three types of red sandstone specimens (i.e., the saturated, natural, and oven-dried specimens). The method of combining strain energy calculations and experimental observations is used to assess the rockburst proneness. The progressive failure of the three types of specimens is recorded by a high-speed (H-S) camera, and the rockburst proneness is comprehensively assessed by considering the failure features of the rock specimens (including the fracture sound, fragment ejection, fragment distribution, and far-field ejection mass ratio). The laboratory results reveal that the rockburst proneness of the natural and oven-dried sandstone specimens is similar. However, compared with the natural and oven-dried specimens, the water-saturated specimens exhibit a visible weaker rockburst proneness. It is also evident that the linear energy storage and dissipation laws are applicable to the saturated, natural, and oven-dried specimens. Most interestingly, the energy storage and energy dissipation coefficients are nearly immune to the presence of water in the red sandstone. Based on this, the definitions of absolute energy storage capacity and relative energy storage capacity are adopted to describe the influence of water on the energy storage performance of rocks. It is found that water dramatically weakens the absolute energy storage capacity, but has little and negligible effect on the relative energy storage capacity of sandstone. Considering the direct and indirect indexes, the accuracy of several representative energy-related rockburst proneness indexes is comprehensively compared in terms of the far-field ejection mass ratio, and their performance is discussed from the relative and absolute energy storage aspects. According to their performance, it is strongly recommended to use the direct index for determination of rockburst proneness when energy-related indexes are used.

Passivity Fractional-Order Sliding-Mode Control of Grid-Connected Converter With LCL Filter

Grid-connected converter with LCL-filter (LCL-GCC) plays an important role in renewable power generation systems. However, existing control techniques face several challenges in practice (e.g., difficulty in parameter design, slow dynamic response, and poor robustness under slowly time-varying filter parameter, dead-time of power devices and external disturbances). To solve these difficulties, a passivity-based fractional-order sliding-mode control (PBC-FOSMC) hybrid controller, which combines the merits of PBC and FOSMC, is proposed, where the inputs of the passive controller are the outputs of the FOSMC controller. First, a passive current controller is designed based on the Euler-Lagrange (EL) model established by damping injection according to the passivity-based control (PBC) theory, which can make the system automatically converge to meet the energy dissipation law. Secondly, a FOSMC controller is designed to further enhance the system robustness to counter disturbances and compensate the reference current accuracy of PBC. The introduced fractional order can successfully suppress the undesired chattering due to the switching behaviors in conventional SMC. Thirdly, the hybrid PBC-FOSMC controller is derived and system stability is analyzed. Finally, experimental results under 10-kW prototype validate the excellent performance of the proposed method in terms of robustness, dynamic performance, strong perturbation rejection ability, etc., and the desired control targets are achieved.

Acoustic emission parameters and energy dissipation law of cyclic load and unload damage of dihydrate gypsum under different loading rate

In order to explore the effect of loading rate on physical and mechanical properties of dihydrate gypsum, cyclic loading and unloading mechanical tests were carried out at different loading rates. Test results were analyzed from the aspects of stress-strain curve, energy distribution mode, damage law and failure mode of specimen. The main research results obtained in the thesis are as follows: with the increase of the loading rate, the peak value of specimen damage first increases rapidly, and then in-creases slowly, and there is a damage threshold. In the early stage of loading, the dissipated energy of the specimen accounts for about 70% of the total energy, most of the total energy input is converted into dissipated energy. The elastic energy density shows an increasing trend with the increase of the loading rate. The elastic energy density is the highest when the loading rate is 400 ​N/s, and more elastic energy can be stored. The ratio of elastic energy ue/u increases with the in-crease of loading rate and tends to be stable. The acoustic emission data show that the acoustic emission signals present a certain agglomeration phenomenon at the unloading point, and there is a “blank period” between the unloading point and the emergence of the next acoustic emission activity. In the early stage of specimen loading, friction-type acoustic emission is mainly generated. The cumulative ringing count when the load reaches the peak failure stress at low loading rate is more, indicating that low loading rate will produce more acoustic emission activities. With the increase of loading rate, the cumulative ringing number per unit time increases, indicating that the increase of loading rate accelerates the damage and failure of dihydrate gypsum near the peak value. The failure mode of gypsum specimens is shear failure, and the increase of loading rate of shear failure angle shows an increasing trend. The larger the loading rate is, the higher the strength of the specimen is. The more energy the press inputs during the loading process, the higher the energy absorbed by the unit volume specimen, which aggravates the development, expansion and penetration of the internal cracks of the specimen, resulting in the larger shear angle of the specimen. The test results provide a more comprehensive theoretical basis for the study of damage characteristics of dihydrate gypsum during cyclic loading and unloading.

A fully decoupled numerical method for Cahn–Hilliard–Navier–Stokes–Darcy equations based on auxiliary variable approaches

A fully decoupled, linearized, and unconditionally stable finite element method is developed to solve the Cahn–Hilliard–Navier–Stokes–Darcy model in the coupled free fluid region and porous medium region. By introducing two auxiliary energy variables, we derive the equivalent system that is consistent with the original system. The energy dissipation law of the proposed equivalent model is proven. To lay a solid foundation, we first present a coupled linearized time-stepping method for the reformulated system, and prove its unconditionally energy stability. In order to further improve the computational efficiency, special treatment for the interface conditions and the artificial compression approach are utilized to decouple the two subdomains and the Navier–Stokes equation. Therefore, with the discretization techniques of two existing auxiliary variable approaches, a fully decoupled and linearized numerical scheme can be developed, under the framework of a semi-implicit semi-explicit scheme for temporal discretization and Galerkin finite element method for spatial discretization. The grad-div stabilization is also employed to further improve the stability of auxiliary variable algorithm. The full discretization obeys the desired energy dissipation law without any temporal restriction. Moreover, the implementation process is discussed, including the adaptive mesh strategy to accurately capture the diffuse interface. Ample numerical experiments are performed to validate the typical features of developed numerical schemes, such as the accuracy, energy stability without restriction for time step size, and adaptive mesh refinement in space. Furthermore, we apply the proposed numerical method to simulate the shape relaxation and the Buoyancy-driven flows, which demonstrate the applicability of the proposed method.

Linear, second-order, unconditionally energy stable scheme for an electrohydrodynamic model with variable density and conductivity

The motion of ions in the complex fluids widely appears in biofluids, hydrodynamics, geodynamics and geophysics. Performing efficient and accurate computation for electrohydrodynamic flows is challenging due to the multiphysics and complex nature of the problem. In this paper, we describe a new second-order linear energy stable time-stepping scheme for a variable density electrohydrodynamic power-law model, which satisfies an energy dissipation law. Firstly, a square root transformation for the fluid density and a logarithmic transformation for the electrical conductivity are used to enforce the positivity-preserving property for these two physical parameters. Secondly, by using the zero energy contribution property of the nonlinear coupling terms in the model and introducing a nonlocal auxiliary variable, an equivalent continuous formulation of the model is obtained. Finally, a two-step backward differentiation scheme is proposed for the resulting equivalent model to construct the time-stepping scheme, and the finite element method is used for spatial discretization. The scheme is proved to be unconditionally energy stable and a nonlocal splitting technique can be utilized to yield several linear subsystems, which can be solved more easily. Numerical simulations are carried out to demonstrate the accuracy and stability of the scheme. The dispersion of electrical conductivity and the effect of electrical force on the Rayleigh–Taylor instability are studied for both shear-thinning and shear-thicking fluids.

Theoretical shear damage characterization of intact rock under compressive-shear stress considering energy dissipation

Investigation into rock damage is of great significance for evaluating and predicting the stability of underground rock applications, such as deep mining or tunneling structures. Considering the energy dissipation properties during rock deformation, this paper proposes a novel theoretical characterization of the damage induced by compressive-shear stress and its evolution in intact rocks. The linear energy dissipation (LED) law is derived from shear stress and deformation data of rocks resulting from the preset angle shear experiment. Based on the LED law, two damage variables are separately constructed from the theoretical and experimental aspects. Several sets of experimental data are subsequently utilized to validate the two constructed damage variables. Results show that both damage variables grow first slowly and then rapidly with shear displacement or shear stress in nonlinear relations. By comparison, however, it is found that the theoretical damage variable outperforms the experimental damage variable, which can accurately reflect the stress and deformation data during progressive rock damage with favorable continuity. This study contributes to a novel theoretical approach to quantifying the pre-peak damage in intact rocks subject to compressive-shear stress.

Unconditional energy dissipation law and optimal error estimate of fast L1 schemes for a time-fractional Cahn–Hilliard problem

In this paper, we will investigate a fully discrete semi-implicit stabilized scheme for the time-fractional Cahn–Hilliard equation, which adopts the nonuniform fast L1 scheme in time, the mixed finite element method in space, and the stabilization technique for the derivative of the double-well potential. By using the temporal–spatial splitting technique, the boundedness of L∞-norm of the computed solution Uhn is obtained. Combining this boundedness, the unconditional optimal error estimate is given without certain temporal restrictions dependent on the spatial mesh size. Moreover, with the help of the boundedness of ‖Uhn‖∞ and the positive definiteness of L1 kernels, it is shown that the proposed scheme preserves the modified discrete energy dissipation property. Finally, numerical experiments are provided to further verify our theoretical convergent result.

Modified diffuse interface fluid model and its consistent energy-stable computation in arbitrary domains

Binary fluid flows in irregular domains are common phenomena in natural and industrial fields. To describe these physical processes, we herein present a novel fluid flow-coupled two-material model reflecting different wetting conditions on an arbitrary fluid-solid interface. A modified ternary Cahn–Hilliard (CH) diffuse interface model is adopted to capture the fluid-fluid interface. One of three components is fixed for all time to represent the profile of complex domains. By considering the contact angle condition on solid-fluid interface and equilibrium interface assumption, an extra wetting term is derived. To treat the fluid flow in domains with solid obstacles, we consider the entire system as a porous medium with variable permeability and add a large penalty term into the incompressible Navier–Stokes (NS) equations. It can be proven that the proposed modified Cahn–Hilliard–Navier–Stokes (CHNS) system satisfies the energy dissipation law (energy stability). Based on the scalar auxiliary variable (SAV) approach with appropriate correction techniques, the linear and consistent energy-stable scheme is developed. We introduce the numerical implementation and estimate the discrete version of energy stability in detail. The proposed method can be implemented on regular Cartesian grids with the absence of explicit boundary treatment. Moreover, the calculations are totally decoupled in each time step. Extensive numerical experiments not only indicate the expected accuracy and stability but also show the superior performance in arbitrary domains with different wetting conditions.

Analysis of the Dynamic Stability of Tailing Dams: An Experimental Study on the Dynamic Characteristics of Tailing Silt

With the improvement in tailing mining-grade requirements and in mineral processing technology, tailing materials tend to be fine-grained. Under the action of earthquakes, a tailing dam is prone to liquefaction, which endangers the safety and stability of the dam. To further explore the dynamic properties of tailing silt under cyclic stress, through a series of dynamic triaxial experiments, we investigated the growth of the hysteresis curve, the development of pore pressure, and the energy dissipation law of tailing silt. The experimental findings indicated that increasing the density of the sample significantly improves its liquefaction resistance and the pore pressure development curve can be fitted using the BiDoseResp function. At the same cyclic stress ratio, the sample’s anti-liquefaction strength did not rise monotonically with increasing confining pressure but changed variably at values near a specified low confining pressure; when the sample density rose under the same settings, the specific confining pressure reduced. We also further discussed the evolution law of the stress–strain curves of tailing silt. The results further explored the dynamic characteristics of tailing silt, which can provide some reference for the seismic design and reinforcement measures of many fine-grained tailing dams.

Study on the Mechanical and Energy Evolution Characteristics of Oxidized Coal

ABSTRACT To study the variation laws of the mechanical properties of coal with different oxidation degrees, coal samples were subjected to constant temperature oxidation at 80°C, 120°C, 160°C and 200°C by using a coal oxidation spontaneous combustion test system. Through a uniaxial compression experiment, the variation laws of mechanical parameters, failure characteristics and energy dissipation law of coals with different oxidation degrees were analyzed and obtained. The study shows that the stress – strain curves of coal with different oxidation degrees vary in different stages. The peak stress, elastic modulus, secant modulus and Poisson’s ratio appear to decrease to different degrees with the deepening of oxidation, and the peak strain increases. The cracks on the surface of the coal samples increase and become more with the deepening of oxidation, the exothermic characteristics of the coal samples decrease, and the failure characteristics of the coal samples change from single shear-type failure characteristic to conjugate shear-type characteristic and then to shatter-type failure characteristic. With an increasing degree of oxidation, the total energy required for coal destruction gradually decreases, the conversion rate of total energy into elastic energy decreases, the dissipation energy gradually increases, and the release rate of elastic energy and the increase rate of dissipation energy gradually slow down after the peak strain.

Linearly implicit methods for Allen-Cahn equation

It is well known that the Allen-Cahn equation satisfies a nonlinear stability property, i.e., the free-energy functional decreases in time. Linearly implicit integrators have been developed for energy-preserving methods for conservative systems with polynomial Hamiltonians, which are based on the concept of polarization. In this paper, we construct linearly implicit methods for gradient flows preserving the energy dissipation by polarizing the free-energy functional. Two-step linearly implicit methods are derived for the Allen-Cahn equation inheriting energy dissipation law. Numerical experiments for one-, two-, and three-dimensional Allen-Cahn equations demonstrate the energy dissipation and the accuracy of the linearly implicit methods.

Sharp L2 Norm Convergence of Variable-Step BDF2 Implicit Scheme for the Extended Fisher–Kolmogorov Equation

A variable-step BDF2 time-stepping method is investigated for simulating the extended Fisher-Kolmogorov equation. The time-stepping scheme is shown to preserve a discrete energy dissipation law if the adjacent time-step ratios r n ≔ Τ n / Τ n − 1 < 3 + 17 / 2 ≈ 3.561 . With the aid of discrete orthogonal convolution kernels, concise L 2 norm error estimates are proved, for the first time, under the mild step ratios constraint 0 < r n < 3.561 . Our error estimates are almost independent of the step ratios r n so that the proposed numerical scheme is robust with respect to the variations of time steps. An adaptive time-stepping strategy based on solution accuracy is then applied to update the computational efficiency. Numerical examples are included to illustrate our theoretical results.