Closure Equations Research Articles

Extended thermodynamics for dense gases up to whatever order

The 14 moments model for dense gases, introduced in the last years by Arima, Taniguchi Ruggeri, Sugiyama, is here considered. They have found the closure of the balance equations up to second order with respect to equilibrium; here the closure is found up to whatever order with respect to equilibrium, but for a more constrained system where more symmetry conditions are imposed and this in agreement with the suggestion of the kinetic theory. The results, when restricted at second order with respect to equilibrium, are the same of the previously cited model but under the further restriction of full symmetries.

Predictive model of solute transport with reversible adsorption in spatially periodic hierarchical porous media

Solute transport in hierarchical porous media with reversible adsorption is predicted by the volume averaging method. A transient macroscopic advection-diffusion equation is derived to describe the multiscale solute transport problem. The theoretical expression of the dispersion tensor is obtained which is the function of pore-scale velocity profile. Steady closure equations are derived to calculate the dispersion tensor. With the aid of pore-scale simulation in unit cells of the hierarchical porous media, the dispersion tensor can be calculated. The model is verified by comparing its predictions and obtaining favorable agreement with results of direct numerical simulations and with experimental data for columns comprised of ordered, porous pillars. It is straightforward to use the model to predict the solute transport behavior in fixed beds packed with particles.

Analysis of the single toggle jaw crusher kinematics

Purpose – This paper aims to obtain equations that can be used to describe the motion of any given point in the swing jaw of a single toggle jaw crusher. Design/methodology/approach – The swing jaw drive mechanism of a single toggle jaw crusher is modelled as a planar crank and rocker mechanism with the swing jaw as the coupler link. Starting with the vector loop closure equation for the mechanism, equations of the position, velocity and acceleration of any given point in the swing jaw are obtained. Findings – Application of the kinematical equations that were obtained is demonstrated using the dimensional data of a practical single toggle jaw crusher. Thus, a description of the kinematics of any given point in the swing jaw of a single toggle jaw crusher is realized. Originality/value – The model of the single toggle jaw crusher mechanism as a planar crank and rocker mechanism is a realistic one. The equations obtained in this paper should be useful in further studies on the mechanics and design of the single toggle jaw crusher.

Modal time correlation functions for homogeneous isotropic turbulence in a projection operator method

Applying a projection operator method to homogeneous isotropic turbulence in the Lagrangian and Eulerian specifications of the flow field, we obtain a closure equation, which is called the similarity approximation (SA) equation, for the time correlation function. We compare its solutions with the time correlation function obtained from direct numerical simulation in the Eulerian specification and with that obtained from direct interaction approximation in the Lagrangian specification. Results of the comparison show that the SA equation can express time correlation for homogeneous isotropic turbulence in the Lagrangian and Eulerian specifications.

Computing mobility condition using Groebner basis

A mechanism is a set of rigid links interconnected together with ideal joints and featuring at least one degree of freedom. An overconstrained mechanism is mobile provided the links' dimensions fulfill a certain relation named the “mobility condition”. The dimensions are not independent from each other and the goal is to obtain this mobility condition. Firstly, parameters are divided into two categories: dimensional parameters and positional parameters. Dimensional parameters represent links' sizes and positional parameters represent the relative positions between the links. The closure equation models the geometric problem by capturing the relationships between the two types of parameters. The principle of the paper is to generate the mobility condition by applying Groebner basis computation to the closure equation. Three methods are presented and investigated. The first one is called the univariate polynomial method (UNIPOL); the second method is called multi-order derivation method (MOD) and the third one is called the finitely separated configurations (FISECO) method. Practical implementation of these various techniques is explained by using a standard computer algebra system. The three methods are applied on a 2D overconstrained mechanism.

A New Co-Located Pressure-Based Discretization Method for the Numerical Solution of Incompressible Navier-Stokes Equations

There are two commonly used numerical strategies for the solution of incompressible Navier-Stokes equations in the context of the finite-volume method. These equations are either discretized using geometric interpolations and solved on staggered grids, or solved on co-located grids using geometric and momentum-based interpolations for convected and convecting velocities, respectively. This article presents an alternative finite-volume method for the solution of incompressible Navier-Stokes equations on co-located grids without resorting to two different interpolation formulas for convected and convecting velocities. The key idea for achieving physical coupling between pressure and velocity fields in the numerical model is the employment of proper closure equations during the discretization. These closure equations are used to convert the finite-volume balance equations to proper computational molecules at nodal points. A number of steady two-dimensional test problems are solved which show the applicability and excellent performance of the proposed method. The method does not have any inherent limitation and is extendable to three-dimensional flows as well.

Evaluation of solvent effect on the amine-based CO2 absorbents by theory and experiment

The three-dimensional reference interaction site model (3D-RISM) integral equation is applied to predict the hydration free-energies and solution enthalpies at infinite dilution for amines and alkanolamines. In particular, the closure equation for 3D-RISM is integrated with the bridge functions derived from minimizing the hard-sphere free-energy functional for different interactions. The density distributions of water molecules around different solute molecules are calculated to intuitively describe the solvent effects. The solution enthalpies and hydration free-energies are then determined by the density distributions. Meanwhile, the solution enthalpies for some alkanolamines with low volatility are measured, and the values at infinite dilution are thereby obtained through extrapolation. It is shown that the theoretical predictions are generally in agreement with the experimental results, suggesting that the current theoretical approach is reasonable to quantitatively evaluate the solvent effect of various amine-based CO2 absorbents.

Conservative conservation equations: Numerical approach and code-to-code benchmarks

One of the principle features of RELAP5-based system thermal hydraulic codes is the use of a two-fluid, non-equilibrium, non-homogeneous, hydrodynamic model for the transient simulation of the two-phase system behavior. This model includes six governing equations to describe the mass, energy, and momentum of the two fluids. The current version of RELAP-5 is not a fully conservative code because it uses both non-conservative and conservative numerical approximation forms of conservation equations. The current version of RELAP5 versions have mass and energy errors during time advancements, either resulting in (a) automatic reduction of time steps used in the advancement of the equations and increased run times or (b) the growth of unacceptably large errors in the transient results. Therefore, fully conservative conservation equations and closure equations have recently been developed to address this problem. This article demonstrates the numerical approach to implement the developed fully conservative conservation equations into RELAP5 and the results of RELAP5 including developed conservative form of conservation equations. RELAP5 versions including conservative and non-conservative conservation equations are compared for various tests from a single pipe to a whole Pressurized Water Reactor (PWR) model.

Kinematics and Simulation of a Hybrid Mechanism: MATLAB/SimMechanics

Kinematic analysis and simulation of hybrid drive system are addressed in this study. A seven link mechanism with two degrees of freedom is selected as the configuration of the system. Kinematic analysis is performed by loop closure equations and required inputs of servo motor are given to get desired ram motion scenario. MATLAB/SimMechanics platform is used to model the hybrid driven mechanical system mechanism characteristics. The simulation results are presented herein.

Conditional statistics in a planar liquid jet with a second-order chemical reaction

Conditional statistics, conditioned on mixture fraction, are experimentally investigated in a planar liquid jet with a second-order isothermal chemical reaction A+B→R. Reactants A and B are contained in the jet and ambient flows, respectively, and are supplied into the test section under the non-premixed condition. Streamwise velocity, mixture fraction and concentrations of all reactive species are simultaneously measured by using an I-type hot-film probe and an optical fibre probe based on light absorption spectrometry. The cross-streamwise profiles of conditional mean concentrations and conditional mean reaction rate show that the conditional mean concentrations and the conditional mean reaction rate near the jet exit change with the cross-streamwise position, whereas they are independent of the cross-streamwise position in the downstream regions. On the jet centreline, the conditional mean reaction rate has a peak value on the condition that the conditional mean concentration of product R has a peak value. A comparison between the conditional mean reaction rate and the covariance of conditional concentration fluctuations shows that the covariance of conditional concentration fluctuations is so small that a first moment closure model for the conditional mean reaction rate is valid. The conditional mean streamwise velocity is almost linear to the mixture fraction for small mixture fraction fluctuations. The conditional scalar dissipation rate is calculated from the budget of the conditional moment closure equation. The results show that the conditional scalar dissipation rate has a single peak value in the upstream region, whereas in the downstream region, it has two peaks for the large and small mixture fraction values. The conventional mean scalar dissipation rate is calculated from the probability density function of the mixture fraction and the conditional scalar dissipation rate. The conventional mean scalar dissipation rate on the jet centreline decreases in the downstream direction as (x∗)-2.9 (where x∗ is the distance from the virtual origin), which is almost the same as that expected from scaling arguments.

Forecasting of global horizontal irradiance by exponential smoothing, using decompositions

Time series methods are frequently used in solar irradiance forecasting when two dimensional cloud information provided by satellite or sky camera is unavailable. ETS (exponential smoothing) has received extensive attention in the recent years since the invention of its state space formulation. In this work, we combine these models with knowledge based heuristic time series decomposition methods to improve the forecasting accuracy and computational efficiency.In particular, three decomposition methods are proposed. The first method implements an additive seasonal-trend decomposition as a preprocessing technique prior to ETS. This can reduce the state space thus improve the computational efficiency. The second method decomposes the GHI (global horizontal irradiance) time series into a direct component and a diffuse component. These two components are used as forecasting model inputs separately; and their corresponding results are recombined via the closure equation to obtain the GHI forecasts. In the third method, the time series of the cloud cover index is considered. ETS is applied to the cloud cover time series to obtain the cloud cover forecast thus the forecast GHI through polynomial regressions. The results show that the third method performs the best among three methods and all proposed methods outperform the persistence models.

A Unified Algorithm for Analysis and Simulation of Planar Four-Bar Motions Defined With R- and P-Joints

This paper presents a single, unified, and efficient algorithm for animating the coupler motions of all four-bar mechanisms formed with revolute (R) and prismatic (P) joints. This is achieved without having to formulate and solve the loop closure equation for each type of four-bar linkages separately. Recently, we developed a unified algorithm for synthesizing various four-bar linkages by mapping planar displacements from Cartesian space to the image space using planar quaternions. Given a set of image points that represent planar displacements, the problem of synthesizing a planar four-bar linkage is reduced to finding a pencil of generalized manifolds (or G-manifolds) that best fit the image points in the least squares sense. In this paper, we show that the same unified formulation for linkage synthesis leads to a unified algorithm for linkage analysis and simulation as well. Both the unified synthesis and analysis algorithms have been implemented on Apple's iOS platform.

An 18 Moments Model for Dense Gases: Entropy and Galilean Relativity Principles without Expansions

The 14 moments model for dense gases, introduced in the last few years by Arima, Taniguchi, Ruggeri and Sugiyama, is here extended up to 18 moments. They have found the closure of the balance equations up to a finite order with respect to equilibrium; it is also possible to impose for that model the entropy and Galilean relativity principles up to whatever order with respect to equilibrium, but by using Taylor’s expansion. Here, the exact solution is found, without expansions, but a bigger number of moments has to be considered and reasons will be shown suggesting that this number is at least 18.

Closure of the single fluid magnetohydrodynamic equations in presence of electron cyclotron current drive

In the presence of electron cyclotron current drive (ECCD), the Ohm’s law of single fluid magnetohydrodynamics (MHD) is modified as E + v × B = η(J – JECCD). This paper presents a new closure relation for the EC driven current density appearing in this modified Ohm’s law. The new relation faithfully represents the nonlocal character of the EC driven current and its main origin in the Fisch-Boozer effect. The closure relation is validated on both an analytical solution of an approximated Fokker-Planck equation as well as on full bounce-averaged, quasi-linear Fokker-Planck code simulations of ECCD inside rotating magnetic islands.

A two-phase shallow debris flow model with energy balance

This paper proposes a thin layer depth-averaged two-phase model provided by a dissipative energy balance to describe avalanches of solid-fluid mixtures. This model is derived from a 3D two-phase model based on the equations proposed by Jackson [R. Jackson, The Dynamics of Fluidized Particles, 2000] which takes into account the force of buoyancy and the forces of interaction between the solid and fluid phases. Jackson's model is based on mass and momentum conservation within the two phases, i.e. two vector and two scalar equations. This system has five unknowns: the solid volume fraction, the solid and fluid pressures and the solid and fluid velocities, i.e. three scalars and two vectors. As a result, an additional equation is necessary to close the system. Surprisingly, this issue is inadequately accounted for in the models that have been developed on the basis of Jackson's work. In particular, Pitman and Le [E.B. Pitman, L. Le, Phil. Trans. R. Soc. A, 2005] replaced this closure simply by imposing an extra boundary condition. If the pressure is assumed to be hydrostatic, this condition can be considered as a closure condition. However, the corresponding model cannot account for a dissipative energy balance. We propose here a closure equation to complete Jackson's model, imposing incompressibility of the solid phase. We prove that the resulting whole 3D model is compatible with a dissipative energy balance. From this model, we deduce a 2D depth-averaged model and we also prove that the energy balance associated with this model is dissipative. Finally, we propose a numerical scheme to approximate the depth-averaged model. We present several numerical tests for the 1D case that are compared to the results of the model proposed by Pitman and Le.

A model for annular displacements of wellbore completion fluids involving casing movement

Abstract This paper presents a mathematical and a numerical model for solving the flow and displacement of completion fluids in the annular space formed by the gap between the outer wall of the casing and the rock face. Such flows occur during mud circulation and cementing operations and may involve casing rotation and reciprocation. Most completion fluids have a shear-dependent apparent viscosity. Additionally, muds and cement slurries often exhibit a yield stress and a gel strength. The displacement patterns and the final placement of the cement depend on injection rate and casing movement histories, rheology contrasts, density contrasts as well as the actual shape and orientation of the annular space which may vary along the wellbore axis. All the above listed phenomena are included in the model. The mathematical model is derived using the lubrication approach and the narrow-slot approximation for the momentum balance equations. These methods provide a (2+1)D-averaged model where the radial dimension is not neglected but averaged across the gap. The numerical model is developed in the goal of minimizing computational time. It takes advantage of multiprocessor architectures, first to pre-compute the closure equations linking local flow velocity to local pressure gradient, prior to running the displacement simulation, and second, to solve the non-linear pressure equation by sampling multiple choices of relaxation parameters.

Modeling of liquid–liquid equilibrium of systems relevant for biodiesel production using Backtracking Search Optimization

In this paper, the liquid–liquid equilibrium of thirty ternary systems relevant for biodiesel production was modeled using the NRTL and UNIQUAC equations and the Backtracking Search Optimization algorithm (BSOA). Performance of this new optimization method for liquid–liquid equilibrium (LLE) parameter estimation, with and without the application of the closure equations, has been analyzed and compared with those obtained using the genetic algorithm (GA) and harmony search algorithm (HAS). Results on the estimation of the interaction parameters of local composition models showed that BSOA with the closure equations can provide promising results in tested biofuel systems. The use of closure equations improves the performance of BSOA for LLE parameter estimation in biofuels mixtures. However, the selection in the parameter elimination using closure equation is relevant for obtaining the best results in LLE data modeling. Finally, NRTL offers the best performance for LLE data modeling of tested ternary systems and this model is attractive for process system engineering in biodiesel production and purification.

Closure relations for mobile bed debris flows in a wide range of slopes and concentrations

Debris flows are flows of water and sediment driven by gravity that initiate in the upper part of a stream, where the slope is very steep, allowing high values of solid concentration (hyperconcentrated flows), and that stop in the lower part of the basin, which is characterized by much lower slopes and reduced speeds and concentrations. Modelling these flows requires mathematical and numerical tools capable of simulating the behavior of a fluid in a wide range of concentrations of the solid phase, spanning from hyperconcentrated flows to flows in the fluvial regime. According to a two-phase approach, the depth integrated equations of mass and momentum conservation for water and sediments, under the shallow water hypothesis, are employed to solve field problems related to debris flows. These equations require suitable closure relations that in this case should be valid in a very wide range of slopes. In the hypothesis of absence of cohesive material, we derived these closure relations properly combining the relative relations valid separately in the fluvial and in the hyperconcentrated regimes. In the intermediate regime, the shear stress is due to the combined effect of the deformation of the liquid phase (grain roughness turbulence) and of inter-particle collisions. Therefore, an approach based on the sum of the effects of the two causes has been proposed, combining the Darcy–Weisbach equation and the Bagnoldian grain-inertia theory.A similar treatment has been made for the transport capacity relations, combing the Bagnold expression of the collisional regime with a transport capacity monomial formula valid in the fluvial regime.The closure relations are expressed in non-dimensional form as a function of the Froude number, of the solid concentration, of the relative submergence, and of the slope.In order to test the closure relation, a set of experiments with mixtures of non-cohesive sediments and water have been carried out in a laboratory flume under steady uniform flow conditions, with different solid and liquid discharges and different grain size distributions. The closure equations are satisfactorily tested against experimental investigation.

A direct expansion method of moments for Brownian coagulation

We propose a new method of moments for solving a coagulation equation based on the Taylor-expansion method of moments (TEMOM). In this model, the closure of the moment equations is achieved by using the Taylor expansion technique, and the fractional-order moments can be used to improve the accuracy of the results. The newly proposed method has no prior requirement for the shape of the particle size distribution. By comparing it with other known methods both in free molecular and continuum regimes, the new approach is found to be the most accurate method. The results show that the predictions of the lower-order moments are not significantly influenced by the higher-order ones, and the higher-order moments are not necessary if one intends to increase the accuracy of the numerical predictions.

Chemical master equation closure for computer-aided synthetic biology.

With inexpensive DNA synthesis technologies, we can now construct biological systems by quickly piecing together DNA sequences. Synthetic biology is the promising discipline that focuses on the construction of these new biological systems. Synthetic biology is an engineering discipline, and as such, it can benefit from mathematical modeling. This chapter focuses on mathematical models of biological systems. These models take the form of chemical reaction networks. The importance of stochasticity is discussed and methods to simulate stochastic reaction networks are reviewed. A closure scheme solution is also presented for the master equation of chemical reaction networks. The master equation is a complete model of randomly evolving molecular populations. Because of its ambitious character, the master equation remained unsolved for all but the simplest of molecular interaction networks for over 70 years. With the first complete solution of chemical master equations, a wide range of experimental observations of biomolecular interactions may be mathematically conceptualized. We anticipate that models based on the closure scheme described herein may assist in rationally designing synthetic biological systems.