Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.

On the Forcing of the Radial-vertical Circulation within Cyclones-Part 1: Concepts and Equations

お? Following the theoretical result of Eliassen, the Sawyer-Eliassen equation for frontal circulations and the equation for forcing the meridional circulation within a circumpolar vortex are extended in isentropic coordinates to describe the forcing of the azimuthally averaged mass-weighted radial-vertical circulation within translating extratropical and tropical cyclones. Several physical processes which are not evident in studies employing isobaric coordinates are isolated in this isentropic study. These processes include the effects of pressure torque, inertial torque and storm translation that are associated with the asymmetric structure in isentropic coordinates. This isentropic study also includes the effects of eddy angular momentum transport, diabatic heating and frictional torque that are common in both isentropic and isobaric studies. All of the processes are modulated by static, inertial and baroclinic stabilities. Consistent with the theoretical result of Eliassen, the numerical solution from this isentropic study shows that the roles of torque, diabatic heating and hydrodynamic stability in forcing the radial-vertical circulation within stable vortices are that 1) positive (negative) torque which results in the counterclockwise (clockwise) rotation of vortices also forces the outflow (inflow) branch of the radial-vertical circulation, 2) diabatic heating (cooling) forces the ascent (descent) branch of the radial-vertical circulation and 3) for given forcing, the weaker hydrodynamic stability results in a stronger radial-vertical circulation. It is the net inflow or convergence (net outflow or divergence), vertical motions and the associated redistribution of properties that favor the evolution of vortices with colorful weather events. Numerical solutions of this isentropic study are given in companion articles. The relatively important contribution of various physical processes to the forcing of the azimuthally-averaged mass-weighted radial-vertical circulation within different translating cyclones and in their different stages of development will be investigated.

Multi-head neural networks for simulating particle breakage dynamics

The breakage of brittle particulate materials into smaller particles under compressive or impact loads can be modelled as an instantiation of the population balance integro-differential equation.In this paper,the emerging computational science paradigm of physics-informed neural networks is studied for the first time for solving both linear and nonlinear variants of the governing dynamics.Unlike conventional methods,the proposed neural network provides rapid simulations of arbitrarily high resolution in particle size,predicting values on arbitrarily fine grids without the need for model retraining.The network is assigned a simple multi-head architecture tailored to uphold monotonicity of the modelled cumulative distribution function over particle sizes.The method is theoretically analyzed and validated against analytical results before being applied to real-world data of a batch grinding mill.The agreement between laboratory data and numerical simulation encourages the use of physics-informed neural nets for optimal planning and control of industrial comminution processes.

A Survey of Unbalanced Flow Diagnostics and Their Application

This paper presents an extensive survey of the most commonly used tools for diagnosing unbalanced flow in the atmosphere, namely the Lagrangian Rossby number, Psi vector, divergence equation, nonlinear balance equation, generalized omega-equation, and departure from fields obtained by potential vorticity (PV) inversion. The basic thoery, assumptions as well as implementation and limitations for each of the tools are all discussed. These tools are applied to high—resolution mesoscale model data to assess the role of unbalanced dynamics in the generation of a mesoscale gravity wave event over the East Coast of the United States. Comparison of these tools in this case study shows that these various methods agree to a large extent with each other though they differ in details. Key words Unbalanced flow - Geostrophic adjustment - Gravity waves - Nonlinear balance equation - Potential vorticity inversion - Omega equations - Rossby number This research was conducted under support from NSF grant ATM-9700626 of the United States. The numerical computations described herein were performed on the Cray T90 at the North Carolina Supercomputing Center and the Cray supercomputer at the NCAR Scientific Computing Division, which also provided the initialization fields for the MM5. Thanks are extended to Mark Stoelinga at University of Washington for the RIP post-processing package.

Modelling Wave Transmission and Overtopping Based on Energy Balance Equation

Wave transmission and overtopping around nearshore breakwaters can have significant influence on the transmitted wave parameters,which affects wave conditions and sediment transportation and becomes the focus of design in engineering.The objective of this paper is to present a simplified model to estimate these important wave parameters.This paper describes the incorporation of wave transmission and overtopping module into a wave model for multi-directional random wave transformation based on energy balance equation with the consideration of wave shoaling,refraction,diffraction,reflection and breaking.Wen's frequency spectrum and non-linear dispersion relation are also included in this model.The influence of wave parameters of transmitted waves through a smooth submerged breakwater has been considered in this model with an improved description of the transmitted wave spectrum of van der Meer et al.(2000) by Carevic et al.(2013).This improved wave model has been validated through available laboratory experiments.Then the verified model is applied to investigate the effect of wave transmission and overtopping on wave heights behind low-crested breakwaters in a project for nearshore area.Numerical calculations are carried out with and without consideration of the wave transmission and overtopping,and comparison of them indicates that there is a considerable difference in wave height and thus it is important to include wave transmission and overtopping in modelling nearshore wave field with the presence of low-crested breakwaters.Therefore,this model can provide a general estimate of the desired wave field parameters,which is adequate for engineers at the preliminary design stage of low-crested breakwaters.

MULTI-MONTE-CARLO METHOD FOR GENERAL DYNAMIC EQUATION CONSIDERING PARTICLE COAGULATION

Monte-Carlo （MC） method is widely adopted to take into account general dynamic equation （GDE） for particle coagulation, however popular MC method has high computation cost and statistical fatigue. A new Multi-Monte-Carlo （MMC） method, which has characteristics of time-driven MC method, constant number method and constant volume method, was promoted to solve GDE for coagulation. Firstly MMC method was described in details, including the introduction of weighted fictitious particle, the scheme of MMC method, the setting of time step, the judgment of the occurrence of coagulation event, the choice of coagulation partner and the consequential treatment of coagulation event. Secondly MMC method was validated by five special coagulation cases in which analytical solutions exist. The good agreement between the simulation results of MMC method and analytical solutions shows MMC method conserves high computation precision and has low computation cost. Lastly the different influence of different kinds of coagulation kernel on the process of coagulation was analyzed： constant coagulation kernel and Brownian coagulation kernel in continuum regime affect small particles much more than linear and quadratic coagulation kernel,whereas affect big particles much less than linear and quadratic coagulation kernel.

The (G'/G, 1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations

The （G＇/G, 1/G）-expansion method for finding exact travelling wave solutions of nonlinear evolution equations, which can be thought of as an extension of the （G＇/G）-expansion method proposed recently, is presented. By using this method abundant travelling wave so- lutions with arbitrary parameters of the Zakharov equations are successfully obtained. When the parameters are replaced by special values, the well-known solitary wave solutions of the equations are rediscovered from the travelling waves.

The generalized method for estimating reserves of shale gas and coalbed methane reservoirs based on material balance equation

As the main unconventional natural gas reservoirs,shale gas reservoirs and coalbed methane(CBM)reservoirs belong to adsorptive gas reservoirs,i.e.,gas reservoirs containing adsorbed gas.Shale gas and CBM reservoirs usually have the characteristics of rich adsorbed gas and obvious dynamic changes of porosity and permeability.A generalized material balance equation and the corresponding reserve evaluation method considering all the mechanisms for both shale gas reservoirs and CBM reservoirs are necessary.In this work,a generalized material balance equation(GMBE)considering the effects of critical desorption pressure,stress sensitivity,matrix shrinkage,water production,water influx,and solubility of natural gas in water is established.Then,by converting the GMBE to a linear relationship between two parameter groups related with known formation/fluid properties and dynamic performance data,the straight-line reserve evaluation method is proposed.By using the slope and the y-intercept of this straight line,the original adsorbed gas in place(OAGIP),original free gas in place(OFGIP),original dissolved gas in place(ODGIP),and the original gas in place(OGIP)can be quickly calculated.Third,two validation cases for shale gas reservoir and CBM reservoir are conducted using commercial reservoir simulator and the coalbed methane dynamic performance analysis software,respectively.Finally,two field studies in the Fuling shale gas field and the Baode CBM field are presented.Results show that the GMBE and the corresponding straight-line reserve evaluation method are rational,accurate,and effective for both shale gas reservoirs and CBM reservoirs.More detailed information about reserves of shale gas and CBM reservoirs can be clarified,and only the straight-line fitting approach is used to determine all kinds of reserves without iteration,proving that the proposed method has great advantages compared with other current methods.

A Note on the Derivation of Wave Action Balance Equation in Frequency Space

In this paper the wave action balance equation in terms of frequency-direction spectrum is derived. A theoretical formulation is presented to generate an invariant frequency space to replace the varying wavenumber space through a Jacobian transformation in the wave action balance equation. The physical properties of the Jacobian incorporating the effects of water depths are discussed. The results provide a theoretical basis of wave action balance equations and ensure that the wave balance equations used in the SWAN or other numerical models are correct. It should be noted that the Jacobian is omitted in the wave action balance equations which are identical to a conventional action balance equation.

Spin-dependent balance equations in spintronics

It is commonly known that the hydrodynamic equations can be derived from the Boltzmann equation. In this paper, we derive similar spin-dependent balance equations based on the spinor Boltzmann equation. Besides the usual charge current, heat current, and pressure tensor, we also explore the characteristic spin accumulation and spin current as well as the spin-dependent pressure tensor and heat current in spintronics. The numerical results of these physical quantities are demonstrated using an example of spin-polarized transport through a mesoscopic ferromagnet.

The Explicitly Covariant Stress-Energy-Momentum Balance Equation for Electromagnetism in Material Media

Beginning with the explicitly covariant Maxwell equations in media, we deduce an explicitly covariant stress-energy-momentum balance equation in material media. Proceeding in this way we avoid mixing external fields and self fields, as occurs if one begins with Lorentz＇s law, the most usual approach appearing in textbooks. Indeed our deduction implies a generalized force density in which the total fields appear. As an application of the present deduction, we discuss briefly the Abraham-Minkowski controversy, showing its relation to open or closed electromagnetic systems. This approach will be interesting for scholars as well as graduate students interested in conceptual problems of relativistic electromagnetism.

Balance the Equation

On the morning of July 2, at the Sandton Convention Center in Johannesburg, Kobus van der Wath was addressing dozens of South Afri -- o can suppliers and buyers about the changes of China＇s role in the global supply chain. Wath is a South African who has worked in Asia for 17 yeaxs. His company The Beijing Axis specializes in providing international bush^ess consultation and procurement services to foreign companies coming to China and to Chinese companies going overseas.

Bubble size modeling approach for the simulation of bubble columns

The constant bubble size modeling approach(CBSM)and variable bubble size modeling approach(VBSM)are frequently employed in Eulerian–Eulerian simulation of bubble columns.However,the accuracy of CBSM is limited while the computational efficiency of VBSM needs to be improved.This work aims to develop method for bubble size modeling which has high computational efficiency and accuracy in the simulation of bubble columns.The distribution of bubble sizes is represented by a series of discrete points,and the percentage of bubbles with various sizes at gas inlet is determined by the results of computational fluid dynamics(CFD)–population balance model(PBM)simulations,whereas the influence of bubble coalescence and breakup is neglected.The simulated results of a 0.15 m diameter bubble column suggest that the developed method has high computational speed and can achieve similar accuracy as CFD–PBM modeling.Furthermore,the convergence issues caused by solving population balance equations are addressed.

Simulation of gas-solid adsorption process considering particle-size distribution

The particle-size distribution of adsorbents usually plays an important role on the adsorption performance. In this study, population balance equation(PBE) is utilized in the simulation of an adsorption process to model the time-dependent adsorption amount distribution on adsorbent particles of a certain size distribution. Different adsorption kinetics model can be used to build the adsorption rate function in PBE according to specific adsorption processes. Two adsorption processes, including formaldehyde on activated carbon and CO_(2)/N_(2)/CH_(4) mixture on 4A zeolite are simulated as case studies, and the effect of particle-size distribution of adsorbent is analyzed. The simulation results proved that the influence of particle-size distribution is significant. The proposed model can help consider the influence of particlesize distribution of adsorbents on adsorption processes to improve the prediction accuracy of the performance of adsorbents.

Seasonal Change in CO2 Production Rate along Depth in a Grassland Field

Soil is a large terrestrial carbon pool so that the evaluation and prediction of soil respiration is important for understanding and managing carbon cycling between the pedosphere and the atmosphere. For better understanding about characteristics and mechanisms of soil respiration, this study monitored seasonal behaviors of soil gaseous CO2 concentration profile with relevant soil physical conditions in a meadow field, and numerically analyzed the monitored data sets to inversely determine time-series of depth distributions of CO2 production rate in the field by assuming optimum ranges of depth and moisture condition for aerobic respiration of soil fauna and flora. The results of the inverse analyses showed that the depth range of intense CO2 production resided in top soil layers during summer and moved down into subsoil layers in winter, implying that the depth range of main CO2 sources can change dynamically with seasons. The surface CO2 emission rates derived from the inverse analyses fell in the range typically found in the same kind of land use. The evaluated mean residence time of gaseous CO2 in the study field was around half a day. These findings suggested that the modelling assumptions about soil respiration in this study are effective to probe spatial and temporal behavior of respiratory activity in a soil layer, and it is still important to integrate facts about in-situ CO2 concentration profiles with soil physical parameters for quantitatively predicting possible behaviors of soil respiration in response to hypothetical changes in atmospheric and soil climates.

Heat Transfer Analysis on Wire Icing and the Current Preventing from Icing

This study concerns the heat transfer processes during ice accretion on wires. The steady state heat balance equation assumed to describe the thermodynamics at the surface of a current heated wire subjected to icing is obtained by analyzing and computing each terms of heat flux. The surface temperature of wire is derived from the heat balance equation, which gives out a proposed estimation of the current intensity to prevent the wire icing

Wave Numerical Model for Shallow Water

The history of forecasting wind waves by wave energy conservation equation Is briefly described. Several currently used wave numerical models for shallow water based on different wave theories are discussed. Wave energy conservation models for the simulation of shallow water waves are introduced, with emphasis placed on the SWAN model, which takes use of the most advanced wave research achievements and has been applied to several theoretical and field conditions. The characteristics and applicability of the model, the finite difference numerical scheme of the action balance equation and its source terms computing methods are described in detail. The model has been verified with the propagation refraction numerical experiments for waves propagating in following and opposing currents; finally, the model is applied to the Haian Gulf area to simulate the wave height and wave period field there, and the results are compared with observed data.

Estimation of water balance in the source region of the Yellow River based on GRACE satellite data

Water storage has important significance for understanding water cycles of global and local domains and for monitoring climate and environmental changes. As a key variable in hydrology, water storage change represents the sum of precipitation, evaporation, surface runoff, soil water and groundwater exchanges. Water storage change data during the period of 2003-2008 for the source region of the Yellow River were collected from Gravity Recovery and Climate Experiment （GRACE） satellite data. The monthly actual evaporation was estimated according to the water balance equation. The simulated actual evaporation was significantly consistent and correlative with not only the observed pan （20 cm） data, but also the simulated results of the version 2 of Simple Biosphere model. The average annual evaporation of the Tangnaihai Basin was 506.4 mm, where evaporation in spring, summer, autumn and winter was 130.9 mm, 275.2 mm, 74.3 mm and 26.1 mm, and accounted for 25.8%, 54.3%, 14.7% and 5.2% of the average annual evaporation, respectively, The precipitation increased slightly and the actual evaporation showed an obvious decrease. The water storage change of the source region of the Yellow River displayed an increase of 0.51 mm per month from 2003 to 2008, which indicated that the storage capacity has significantly increased, probably caused by the degradation of permafrost and the increase of the thickness of active layers. The decline of actual evaporation and the increase of water storage capacity resulted in the increase of river runoff.

CFD-PBE simulation of a bubble column in OpenFOAM

A general CFD-PBE(computational fluid dynamics-population balance equation) solver for gas–liquid poly-dispersed flows of both low and high gas volume fractions is developed in OpenFOAM(open-source field operation and manipulation) in this work. Implementation of this solver in OpenFOAM is illustrated in detail. The PBE is solved with the cell average technique. The coupling between pressure and velocity is dealt with the transient PIMPLE algorithm, which is a merged PISO-SIMPLE(pressure implicit split operator-semi-implicit method for pressure-linked equations) algorithm. Results show generally good agreement with the published experimental data, whereas the modeling precision could be improved further with more sophisticated closure models for interfacial forces, the models for the bubble-induced turbulence and those for bubble coalescence and breakage.The results also indicate that the PBE could be solved out the PIMPLE loop to save much computation time while still preserving the time information on variables. This is important for CFD-PBE modeling of many actual gas–liquid problems, which are commonly high-turbulent flows with intrinsic transient and 3 D characteristics.

The Effects of Climate on Development of Ecosystem in Oasis

When vegetation and bare soil coexist, in consideration of some ecological conditions of plant, the total evapotranspiration rate of the oasis and the temperature of vegetation and soil in different climatic and ecological conditions are calculated by using the thermal energy balance equations of vegetation and soil The evapotranspiration rate depends on climatic and ecological conditions, in some conditions, quasi-bifurcation and multi-equilibrium state appear in the solutions of evapotranspiration rate in the areas covered by small part of vegetation.