Steady-state Analytical Solutions Research Articles

Part III: Dynamic evolution of the particle size distribution in batch and continuous particulate processes: A Galerkin on finite elements approach

The present study provides a comprehensive investigation on the numerical solution of the dynamic population balance equation (PBE) in batch and continuous flow particulate processes. The general PBE was numerically solved using the Galerkin on finite elements method (GFEM) for particulate processes undergoing simultaneous particle aggregation, growth and nucleation. The performance of the GFEM in terms of accuracy and stability was assessed by a direct comparison of the calculated particle size distributions (PSD) and/or their corresponding moments to available analytical solutions. Numerical simulations were carried out over a wide range of variation of particle aggregation and growth rates. Simulations of batch particulate processes revealed that the GFEM was generally more accurate than the standard fixed-grid sectional methods and more robust than the orthogonal collocation on finite elements method. In the case of continuous flow processes, numerical simulations were carried out till the dynamic solution reached its final steady-state. It was found that the calculated PSDs converged exactly to available analytical steady-state solutions of the PBE. The transient PSDs were often characterized by steep moving fronts that required careful discretization of the particle volume domain and the inclusion of an artificial diffusion term in order to eliminate numerical oscillations in the solution. Finally, for continuous particulate processes, the effect of various size-dependent particle growth models on the calculated steady-state PSDs was investigated.

Comparison of analytical solutions for cmsmpr crystallizer with QMOM population balance modeling in fluent

Fluent version 6.2 computational fluid dynamics environment has been enhanced with a population balance capability that operates in conjunction with its multiphase calculations to predict the particle size distribution within the flow field. The population balance is solved by the quadrature method of moments (QMOM). Fluent's prediction capabilities are tested by using a 2-dimensional analogy of a constantly stirred tank reactor with a fluid flow compartment that mixes the fluid quickly and efficiently using wall movement and has a feed stream and a product stream. The results of these Fluent simulations using QMOM population balance solver are compared to steady state analytical solutions for the population balance in a stirred tank where 1) growth, 2) aggregation, and 3) breakage, take place separately and 4) combined nucleation and growth and 5) combined nucleation, growth and aggregation take place. The results of these comparisons show that the moments of the population balance are accurately predicted for nucleation, growth, aggregation and breakage when the flow field is turbulent. With laminar flow the mixing is not ideal and as a result the steady state well mixed solutions are not accurately simulated.

Unsteady analytical solutions of the spherical shallow water equations

A new class of unsteady analytical solutions of the spherical shallow water equations (SSWE) is presented. Analytical solutions of the SSWE are fundamental for the validation of barotropic atmospheric models. To date, only steady-state analytical solutions are known from the literature. The unsteady analytical solutions of the SSWE are derived by applying the transformation method to the transition from a fixed cartesian to a rotating coordinate system. Fundamental examples of the new unsteady analytical solutions are presented for specific wind profiles. With the presented unsteady analytical solutions one can provide a measure of the numerical convergence in the case of a temporally evolving system. An application to the atmospheric model PLASMA shows the benefit of unsteady analytical solutions for the quantification of convergence properties.

Dynamic evolution of PSD in continuous flow processes: A comparative study of fixed and moving grid numerical techniques

The present study provides a comprehensive investigation on the numerical solution of the dynamic population balance equation (PBE) in continuous flow processes. Specifically, continuous particulate processes undergoing particle aggregation and/or growth are examined. The dynamic PBE is numerically solved in both the continuous and its equivalent discrete form using the Galerkin on finite elements method (GFEM) and the moving grid technique (MGT) of Kumar and Ramkrishna [1997. Chemical Engineering Science 52, 4659–4679], respectively. Numerical simulations are carried out over a wide range of variation of particle aggregation and growth rates till the dynamic solution has reached its final steady-state value. The performance of the two numerical methods is assessed by a direct comparison of the calculated particle size distributions and/or their moments to available steady-state analytical solutions.

Model for Wind-Driven Vertical Mixing in a Shallow Lake with Submersed Macrophytes

An energy flux model to simulate vertical mixing in a temperature stratified shallow lake with submersed macrophytes has been formulated. The model is based on a one-dimensional equation for transport and dissipation of kinetic energy produced by wind, coupled to heating at the water surface. Analytic solutions of the energy flux equation characterize the steady-state mixed layer depth for constant wind shear and constant heat input at the water surface. It is found that in water columns with sparse macrophytes, the mixed layer depth is controlled by the buoyancy length scale, which is proportional to the heat flux. In dense macrophyte beds, the mixed layer depth is controlled primarily by the macrophyte length scale, which is a function of the plant surface area. The analytic steady-state solutions give a lower bound for daytime mixed layer depth, which is controlled by simultaneous heating and wind mixing from the water surface.

Extensions of the LuGre tyre friction model related to variable slip speed along the contact patch length

The recently developed LuGre model describes the three-dimensional tyre friction dynamics given in the brush representation. The model has a three-state lumped-parameter form and an analytical solution for tyre static curves. It is simplified using the assumption of a constant slip speed along the contact patch length. This assumption is abandoned in this paper in order to extend the model with several secondary but potentially important effects such as camber, carcass compliance, conicity, ply steer and rolling resistance. Two approaches to the introduction of a variable-slip-speed effect are considered. The first is based on stepwise approximation of the slip speed along the contact patch length, and the second includes a specific, spatially distributed additive deflection term. Both approaches preserve the three-state lumped-parameter model form and an analytical steady-state solution. Qualitative and quantitative validation results point to consistency and benefits of the proposed model extensions.

Handling the water content discontinuity at the interface between layered soils within a numerical scheme

In general, the water content (θ) form of Richards’ equation is not used when modeling water flow through layered soil since θ is discontinuous across soil layers. Within the literature, there have been some examples of models developed for layered soils using the θ-form of Richards’ equation. However, these models usually rely on an approximation of the discontinuity at the soil layer interface. For the first time, we will develop an iterative scheme based on Newton’s method, to explicitly solve for θ at the interface between 2 soils within a numerical scheme. The numerical scheme used here is the Method of Lines (MoL); however, the principles of the iterative solution could be used in other numerical techniques. It will be shown that the iterative scheme is highly effective, converging within 1 to 2 iterations. To ensure the convergence behaviour holds, the numerical scheme will be tested on a fine-over-coarse and a coarse-over-fine soil with highly contrasting soil properties. For each case, the contrast between the soil types will be controlled artificially to extend and decrease the extent of the θ discontinuity. In addition, the numerical solution will be compared against a steady-state analytical solution and a numerical solution from the literature.

Exact Steady State Reconnection Solutions in Weakly Collisional Plasmas

We consider the problem of reconnection in weakly collisional plasmas in the strong guide field limit. In this regime the standard resistive Ohm’s law is modified to include electron compressibility and electron inertia effects. Despite the increased complexity of the governing equations, we show that analytic steady state solutions, like those discovered by Craig and Henton for the purely resistive case, can be developed for this new system. The resulting solutions are somewhat richer than those of Craig and Henton, and there are various different regimes in parameter space to consider that exhibit multiple length scales, boundary layer effects, and other features. We also examine the dynamical behavior of these new solutions by solving the time-dependent problem numerically. Subject headingg MHD — plasmas

Transition waves in bistable structures. II. Analytical solution: wave speed and energy dissipation

We consider dynamics of chains of rigid masses connected by links described by irreversible, piecewise linear constitutive relation: the force–elongation diagram consists of two stable branches with a jump discontinuity at the transition point. The transition from one stable state to the other propagates along the chain and excites a complex system of waves. In the first part of the paper ( Cherkaev et al., 2004, Transition waves in bistable structures. I. Delocalization of damage), the branches could be separated by a gap where the tensile force is zero, the transition wave was treated as a wave of partial damage. Here we assume that there is no zero-force gap between the branches. This allows us to obtain steady-state analytical solutions for a general piecewise linear trimeric diagram with parallel and nonparallel branches and an arbitrary jump at the transition. We derive necessary conditions for the existence of the transition waves and compute the speed of the wave. We also determine the energy of dissipation which can be significantly increased in a structure characterized by a nonlinear discontinuous constitutive relation. The considered chain model reveals some phenomena typical for waves of failure or crushing in constructions and materials under collision, waves in a structure specially designed as a dynamic energy absorber and waves of phase transitions in artificial and natural passive and active systems.

Simulation of diffusion–convection processes in microfluidic channels equipped with double band microelectrode assemblies: approach through quasi-conformal mapping

Two different applications of conformal mapping approaches are proposed for the treatment of double band assemblies operating in a microfluidic channel: (i) a quasi-conformal map for the numerical simulation of diffusion–convection processes occurring when the channel is submitted to a microfluidic flow, and (ii) a conformal map specifically designed for obtaining the steady-state analytical solution for the pure diffusional regime (i.e., without flow) within the same channel. The latter applies also when such assemblies perform within a thin film of electrolyte. The results are presented for a generator–collector operation of the assemblies but the methods apply also to other situations (e.g., generator–generator mode).

A Simulation Study on the Role of the Vibrationally Excited Nitrogen in the Ionosphere

AbstractVibrationally excited molecular nitrogen plays an important role in the F region ionosphere. It increases the loss rate of the dominant ion O+ and reduces the electron concentration in the F region of the ionosphere. In this paper, we use a theoretical numerical ionospheric model to study the effect of on the ionosphere during four time periods, each including both geomagnetically quiet and storm disturbed periods, with considering and without considering the role of in the numerical ionospheric model. The four periods simulated are (1) May 14–20, 1997; (2) May 1–10, 1998; (3) June 11–15, 1990 and (4) April 6–10, 2000. By Comparing the simulated results with the observational ones, it is shown that has a significant effect on the ionosphere during high solar activity years. While for the low solar activity years, the role played by is very limited, much less than that in high solar activity years. It is also shown that for the high solar activity years, when constructing an ionospheric physical model, one must take into account not only for geomagnetically disturbed period but also for quiet period. Moreover, it is found that the effect of on the ionospheric electron density distribution depends on the vibrational temperature Tv adopted. When taking the value of Tn as Tv(where Tn is the background neutral temperature) the simulated result, compared with the observation one, is not as good as that when Tv is calculated by the steady state analytical solution. Moreover, our simulation results show that, for high solar activity years, the effect of on the distribution of ionospheric electron concentration is to reduce the electron density for the altitudes above 150km, while its effect on the electron density below 150km is very small. The F2 layer peak height hmF2 is not affected very much by .

Sructure of steady-state Debye layers in a low-density plasma near a dielectric surface

Exact steady-state analytic solutions describing kinetic processes in a low-density plasma layer near a dielectric surface are found in a time-dependent one-dimensional model with allowance for secondary electron emission. It is shown that, at low electron temperatures, both the electric potential and electron density monotonically decrease toward the dielectric surface (Debye layer). As the electron temperature increases, an anti-Debye layer first forms, in which the potential monotonically increases toward the wall, and regimes with a nonmonotonic potential profile then arise.

AN EXPLICIT SUBPARAMETRIC SPECTRAL ELEMENT METHOD OF LINES APPLIED TO A TUMOUR ANGIOGENESIS SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS

In this paper we consider a numerical solution to Anderson and Chaplain's tumour angiogenesis model1 over two-dimensional complex geometry. The numerical solution of the governing system of non-linear evolutionary partial differential equations is obtained using the method of lines: after a spatial semi-discretisation based on the subparametric Legendre spectral element method is performed, the original system of partial differential equations is replaced by an augmented system of stiff ordinary differential equations in autonomous form, which is then advanced forward in time using an explicit time integrator based on the fourth-order Chebyshev polynomial. Numerical simulations show the convergence of the steady state numerical solution towards the linearly stable steady state analytical solution.

Diffusion Approach to Long Distance Charge Migration in DNA: Time-Dependent and Steady-State Analytical Solutions for the Product Yields

In this study we report analytical solutions for both time-dependent and steady-state problems of unbiased charge transfer through a regular DNA sequence via a hopping mechanism. The phenomenon is treated as a diffusion of charge in a one-dimensional array of equally spaced and energetically equivalent temporary trapping sites. The solutions take into account the rates of charge hopping (k), side reactions (k(r)), and charge transfer to a terminal charge acceptor (k(t)). A detailed analysis of the time-dependent problem is performed for the diffusion-controlled regime, i.e., under the assumption that k(t) >> k, which is also equivalent to the fast relaxation limit of charge trapping. The analysis shows that the kinetics of charge hopping through DNA is always multiexponential, but under certain circumstances it can be asymptotically approximated by a single-exponential term. In that case, the efficiency of charge transfer can be characterized by a single rate constant k(CT) = 1.23kN(-2) + k(r), where N is the DNA length expressed in terms of the number of equidistant trapping sites and k(r) is the rate of competing chemical processes. The absolute yield of charge transfer under steady-state conditions in general is obtained as Y(infinity) = omega [alpha sinh(alphaN) + omega cosh(alphaN)](-1), where alpha = (2k(r)/k)(1/2) and omega = 2k(t)/k. For the diffusion-controlled regime and small N, in particular, it turns into the known "algebraic" dependence Y(infinity) = [1 + (k(r)/k)N(2)](-1). At large N the solution is asymptotically exponential with the parameter alpha mimicking the tunneling parameter beta in agreement with earlier predictions. Similar equations and distance dependencies have also been obtained for the damage ratios at the intermediate and terminal trapping sites in DNA. The nonlinear least-squares fit of one of these equations to experimental yields of guanine oxidation available from the literature returns kinetic parameters that are in reasonable agreement with those obtained by Bixon et al. [Proc. Natl. Acad. Sci. U.S.A.1999, 96, 11713-11716] by numerical simulations, suggesting that these two approaches are physically equivalent.

Semi-analytical solutions for solute transport in fractured porous media using a strip source of finite width

Transient and steady-state analytical solutions are derived to investigate solute transport in a fractured porous medium consisting of evenly spaced, parallel discrete fractures. The solutions incorporate a finite width strip source, longitudinal and transverse dispersion in the fractures, source decay, aqueous phase decay, one-dimensional diffusion into the matrix, sorption to fracture walls, and sorption within the matrix. The solutions are derived using Laplace and Fourier transforms, and inverted by interchanging the order of integration and utilizing a numerical Laplace inversion algorithm. The solutions are verified for simplified cases by comparison to solutions derived by Batu [Batu V. A generalized two-dimensional analytical solution for hydrodynamic dispersion in bounded media with the first-type condition at the source. Wat Resour Res 1989;25(6):1125] and Sudicky and Frind [Sudicky EA, Frind EO. Contaminant transport in fractured porous media: analytical solutions for a system of parallel fractures. Wat Resour Res 1982;18(6):1634]. The application of the solutions to a fractured sandstone demonstrates that narrower source widths and larger values of transverse dispersivity both lead to lower downstream concentrations in the fractures and shorter steady-state plumes. The incorporation of aqueous phase decay and source concentration decay both lead to lower concentrations and shorter plumes, with even moderate amounts of decay significantly shortening the persistence of contamination.

Theory of transient and steady-state ECL generation at double-hemicylinder assemblies using conformal mapping and simulations

Rigorous analytical expressions for steady-state currents and electrogenerated chemiluminescence (ECL) intensities are derived for two-hemicylinder assemblies operating in the ECL generation mode based on a specific conformal mapping transform. This transform is also used for fast and accurate simulations of transient currents and ECL intensities leading to an excellent agreement at long times with the analytical steady state solutions. These results are compared to those relative to two-band assemblies to investigate the effect of hemicylindrical electrode protuberance on ECL efficiencies. This evidence that electrode protuberance results in a drastic increase of the ECL generation and shortening of commutation times when the gap is not too large as compared to the electrode radii.

Determination of the Thermal Dispersion Coefficient During Radial Filling of a Porous Medium

Resin Transfer Molding is one of the Liquid Composite Molding processes in which a thermoset resin is infiltrated into a fibrous porous media in a closed mold. To reduce the curing time of the resin, the mold may be heated, influencing other filling parameters such as the resin viscosity. Analysis of the non-isothermal effects during filling will help to understand the manufacturing process. One of the issues of non-isothermal filling in porous media is the variation of the velocity profile at the micro scale level, which as it is averaged, cannot be included in the convective term. To account for it, the thermal conductivity tensor is modified and a thermal dispersion coefficient Kd is introduced to model the micro convection effects. In this paper, we explore the temperature profile under non-isothermal conditions for radial injection during Resin Transfer Molding in order to determine the thermal dispersion coefficient. An approximate solution is derived from the series solution and validated with a numerical method. Experiments using carbon fibers and polyester resin were conducted. The thermal dispersion coefficient is determined by comparing experimental results with the steady state analytical solution. The comparison between radial and linear injection results shows that the same degree of dispersion is present in isotropic fibrous porous media.

Steady-state analytical solutions for the lattice boltzmann equation

Abstract A general class of analytical solutions of the lattice Boltzmann equation is derived for two-dimensional, steady-state unidirectional flows. A subset of the solutions that verifies the corresponding Navier-Stokes equations is given. It is pointed out that this class includes, e.g., the Couette and the Poiseuille flow but not, e.g., the basic Kolmogorov flow. For steady-state non-unidirectional flows, first and second order solutions of the lattice Boltzmann equation are derived. Practical consequences of the analysis are mentioned. Differences between the technique applied here and those used in some earlier works are emphasized.

Analytical solution for arbitrarily located multiwells in an anisotropic homogeneous confined aquifer

Analytical solutions have been suggested to delineate the capture zone created by multiwells for the remediation design of contaminated groundwater. These previous analytical solutions are often restricted to pumping wells only, specific well locations, a limited number of wells, and isotropic aquifer. An alternative analytical steady state solution was suggested to deal with arbitrarily located multiinjection‐pumping wells in an anisotropic homogeneous aquifer. The solution can be an alternative solution for easily and conveniently calculating the hydraulic head and gradient in a complex flow field caused by multiinjection‐pumping wells at different rates, and will consequently be useful for pump‐and‐treat design.

On the diffusion from a slender bubblein an extensional flow

Mass transfer between a slender bubble and a Newtonian liquid in a simple extensional and creeping flow has been theoretically studied. The analytical steady-state solution when the Peclet number is zero has been obtained using the works of Szegö and Payne, who studied the electrostatic capacity of a spindle surface in a bispherical coordinate system. The result shows, as expected, that the modified Sherwood number increases as the capillary number increases since the surface area of the bubble also increases.