On modeling evaporation

We develop a theoretical model for evaporation of a pure liquid drop on a thermally conductive solid substrate. We discuss a variety of effects regarding evaporation regime, the state of the liquid/gas interface and the content of gas phase. Then, we further consider two models: the one resulting from the one-sided non-equilibrium assumption and the other that assumes diffusion-limited regime and equilibrium at the liquid/gas interface. A single governing equation for the evolution of drop thickness is derived for both models. We show that although the model predicts qualitatively different temperature along liquid/gas and liquid/solid interface, the dynamics of the drops is almost the same.      

Dynamical behavior in a stage-structured differential-algebraic prey-predator model with discrete time delay and harvesting

A differential-algebraic model system which considers a prey-predator system with stage structure for prey and harvest effort on predator is proposed. By using the differential-algebraic system theory and bifurcation theory, dynamic behavior of the proposed model system with and without discrete time delay is investigated. Local stability analysis of the model system without discrete time delay reveals that there is a phenomenon of singularity induced bifurcation due to variation of the economic interest of harvesting, and a state feedback controller is designed to stabilize the proposed model system at the interior equilibrium; Furthermore, local stability of the model system with discrete time delay is studied. It reveals that the discrete time delay has a destabilizing effect in the population dynamics, and a phenomenon of Hopf bifurcation occurs as the discrete time delay increases through a certain threshold. Finally, numerical simulations are carried out to show the consistency with theoretical analysis obtained in this paper.     

Development and tending to steady state of subsonic gas condensation on a plane condensed phase

The kinetic equation for a monatomic gas with a model collision operator (S-model) is used to study the development and tending to steady state of one-dimensional unsteady half-space gas condensation on a plane condensed phase. Initially, the gas is at rest and in equilibrium with the body’s surface and, then, the body temperature suddenly drops to a constant value. The problem is solved using an implicit second-order accurate quasi-monotone scheme. The process of reaching a steady flow regime is of primary interest. The effect of the evaporation (condensation) coefficient on the flow pattern is analyzed.     

Two-fluid and population balance models for subcooled boiling flow

Population balance equations combined with a three-dimensional two-fluid model are employed to predict subcooled boiling flow at low pressure in a vertical annular channel. The MUSIG (MUltiple-SIze-Group) model implemented in the computer code CFX4.4 is further developed to accommodate the wall nucleation at the heated wall and condensation in the subcooled boiling regime. Comparison of model predictions against local measurements is made for the void fraction, bubble Sauter mean diameter and gas and liquid velocities covering a range of different mass and heat fluxes and inlet subcooling temperatures. Additional comparison using empirical relationships for the active nucelation site density and local bubble diameter is also investigated. Good agreement is achieved with the local radial void fraction, bubble Sauter diameter and liquid velocity profiles against measurements. However, significant weakness of the model is evidenced in the prediction of the vapour velocity. Work is in progress to circumvent the deficiency through the consideration of additional momentum equations or developing an algebraic slip model to account for bubble separation.     

On the self-similar solution of navier-stokes equations with volume sources and sinks of mass : PMM vol. 40, n 6, 1976, pp. 1121–1124

Unlike the investigations in [1, 2] of the motion of fluid with surface sources and sinks of mass (injection and suction), the flow is considered here in the presence of uniformly distributed mobile volume sources and sinks in flat and round channels. It is shown that far away from the inlet a self-similar solution of the system of equations of motion can be obtained. The results are applicable, for instance, to two-phase (vapor-liquid) streams with condensation or evaporation for small volume concentrations of the discrete phase and absence of phase slip.     

The modeling and numerical simulation of gas flow networks

Summary. A network formulation is introduced for the modeling and numerical simulation of complex gas transmission systems like a multi-cylinder internal combustion engine. Several simulation levels are discussed which result in different network representations of a specific system. Basic elements of a network are chambers of finite volume, straight pipes and connections like valves or nozzles. The pipe flow is modeled by the unsteady, one-dimensional Euler equations of gas dynamics. Semi-empirical approaches for the chambers and the connections yield differential-algebraic equations (DAEs) in time. The numerical solution is based on a TVD scheme for the pipe equations and a predictor-corrector method for the DAE-system. Simulation results for an internal combustion engine demonstrate the practical interest of the new approach. Received May 12, 1994 / Revised version received August 26, 1994     

Simulating the motion of the leech: A biomechanical application of DAEs

In this paper a mathematical model is developed for the dynamical behaviour of a hydrostatic skeleton. The basic configuration is taken from the worm-like shape of the medicinal leech. It consists of a sequence of hexahedra with damped elastic springs as edges to model the various parts of the musculature. The system is stabilized by the constraint of constant volume either in the whole body or in prescribed compartments. We set up Lagrange's equations of motion with the Lagrange multipliers being the pressure values in the compartments. The equations of motion lead to a large differential-algebraic system which is solved by an application of semi-explicit numerical methods. Though the model has not yet been adapted to experimental data, first simulations with a simplified set of parameters show that it is capable of generating basic movements of the leech such as crawling and swimming. This revised version was published online in June 2006 with corrections to the Cover Date.     

Convergence analysis of a partial differential algebraic system from coupling a semiconductor model to a circuit model

We are interested in circuit simulation including distributed semiconductor models. The circuit itself is modeled by the modified nodal analysis. The stationary drift diffusion equations are used to describe the semiconductors. The complete system is then a partial differential-algebraic system. We discretize it first in space with finite elements and the Scharfetter-Gummel discretization. The resulting semi-discrete system can be analyzed as a differential-algebraic equation with properly stated leading term. We present topological index one criteria. They coincide with previous results for the non-discretized partial differential-algebraic equation. For the time discretization we use standard BDF methods (implicit Gear formulas). Finally we derive a convergence estimate for the whole partial differential-algebraic system close to equilibrium.     

Stationary solutions of the kinetic Broadwell model

We consider a stationary discrete model of the Boltzmann equation for four velocities (the Broadwell model). We obtain new exact automodel solutions of the model corresponding to an incompressible and a compressible gas. We show that one class of solutions satisfies the problem of gas evaporation and condensation on the boundary of a disk and external space. The system turns out to be strongly nonequilibrium, and continuous medium equations are not applicable to it.     

Modeling of three dimensional thermocapillary flows with evaporation at the interface based on the solutions of a special type of the convection equations

Theoretical and numerical study of the convection processes, which are accompanied by evaporation/condensation, in the framework of new non-standard problem is largely motivated by new physical experiments. One of the principal questions is to understand the character and to evaluate the degree of influence of particular factors or their combined action on the structure of the joint flows of liquid and gas-vapor mixture. The flow topology is determined by four main mechanisms: natural and thermocapillary convection, tangential stresses and mass transfer due to evaporation at the interface. The mathematical modeling of the fluid flows in an infinite channel with a rectangular cross section is carried out on the basis of the solution of a special type of the convection equations. The effects of thermodiffusion and diffusive thermal conductivity in the gas phase and evaporation at the thermocapillary interface are taken into consideration. Numerical investigations are performed for the liquid – gas (ethanol – nitrogen) system under normal and low gravity. The fluid flows are characterized as translational and progressively rotational motions and can be realized in various forms.     

具有不完全信息的内部交易动态博弈

在Kyle模型中的线性均衡假设进行了修正的基础上,针对内部交易者只具有资产价值不完全信息情况,建立两期风险厌恶型内部交易均衡模型,并求得该模型的子博弈纳什均衡解.由此发现资产价值不完信息中噪音对市场干扰程度愈小(波动程度愈小),就愈有利于内部交易者的收益;内部交易者的交易就愈活跃;交易均衡价格包含资产价值信息就愈多.     

The stability of constant equilibrium states in relaxation models

We consider various first-order systems of PDEs with partial dissipation, as well as partial conservation. This class includes relaxation models, for instance the one designed by S. Jin and Z. Xin, as well as discrete velocity models for gases, as the Broadwell system. As we showed in a recent paper, the Jin-Xin model admits a convex compact positively invariant region, whenever the equilibrium system does. As a by-product, we obtained the existence of global weak solutions for the Cauchy problem with large data. For more general systems, the global existence of a uniformly bounded entropy solution will be a basic assumption in this work. We consider one-dimensional data which are either space periodic or square integrable. We prove that the (expected globally bounded) entropy solution relaxes to the equilibrium state; the latter is either zero or is determined by the mean value of the conserved components. We emphasize that we do not need any assumption about the nonlinearity of the underlying equilibrium system. We give two different proofs of the stabilization, which apply in different contexts. The first one uses compensated compactness and has a rather broad efficiency. For instance, it applies to several quasi-linear models. But the convergence result does not provide any decay rate in the periodic setting. The other one uses a dispersion estimate for the principal part of the model. It applies to periodic data and needs the strong assumption of semi-linearity, but yields an exponential decay in theL ²-norm. We expect that it could extend to multi-dimensional contexts.     

Instability of the Phase Transition Front during Water Injection into High-Temperature Rock

Water injection into a high-temperature geothermal reservoir saturated with superheated vapor is investigated. A solution to the one-dimensional problem in the form of a traveling wave is found. It is shown that there exist two types of solutions which correspond to the boiling of water and the condensation of vapor. In the condensation regime with high initial pressure, vapor ahead of the phase transition front is shown to be in a supercooled state. For moderate or law initial pressure, solutions with condensation and boiling are thermodynamically consistent. Linear stability of the phase transition surface between the water and vapor regions is analyzed. It is shown that the phase transition front moving at constant velocity is always unstable.     

A new coupled model for alloy solidification

A new coupled model in the binary alloy solidification has been developed. The model is based on the cellular automaton (CA) technique to calculate the evolution of the interface governed by temperature, solute diffusion and Gibbs-Thomson effect. The diffusion equation of temperature with the release of latent heat on the solid/liquid (S/L) interface is valid in the entire domain. The temperature diffusion without the release of latent heat and solute diffusion are solved in the entire domain. In the interface cells, the energy and solute conservation, thermodynamic and chemical potential equilibrium are adopted to calculate the temperature, solid concentration, liquid concentration and the increment of solid fraction. Compared with other models where the release of latent heat is solved in implicit or explicit form according to the solid/liquid (S/L) interface velocity, the energy diffusion and the release of latent heat in this model are solved at different scales, i.e. the macro-scale and micro-scale. The variation of solid fraction in this model is solved using several algebraic relations coming from the chemical potential equilibrium and thermodynamic equilibrium which can be cheaply solved instead of the calculation of S/L interface velocity. With the assumption of the solute conservation and energy conservation, the solid fraction can be directly obtained according to the thermodynamic data. This model is natural to be applied to multiple (< 2) spatial dimension case and multiple (< 2) component alloy. The morphologies of equiaxed dendrite are obtained in numerical experiments.     

A new coupled model for alloy solidification

A new coupled model in the binary alloy solidification has been developed. The model is based on the cellular automaton (CA) technique to calculate the evolution of the interface governed by temperature, solute diffusion and Gibbs-Thomson effect. The diffusion equation of temperature with the release of latent heat on the solid/liquid (S/L) interface is valid in the entire domain. The temperature diffusion without the release of latent heat and solute diffusion are solved in the entire domain. In the interface cells, the     

微分代数系统的渐近性

从动力系统的角度研究微分代数系统,利用单调流理论中的结果和方法讨论微分代数系统渐近性态.首先,我们把所考察的系统嵌入到一族相关的系统,引进使得系统族中的每个系统生成单调流的相应偏序和条件.然后给出了若干关于解收敛于平衡点的一般性结果,并对一类微分代数系统的渐近性作了较为精细的讨论.我们的结果是Hirch等关于常微分方程的相关结果的推广和改进.     

Numerical modelling of bubbly flows with and without heat and mass transfer

In this paper, modelling gas–liquid bubbly flows is achieved by the introduction of a population balance equation combined with the three-dimensional two-fluid model. For gas–liquid bubbly flows without heat and mass transfer, an average bubble number density transport equation has been incorporated in the commercial code CFX5.7 to better describe the temporal and spatial evolution of the geometrical structure of the gas bubbles. The coalescence and breakage effects of the gas bubbles are modelled according to the coalescence by the random collisions driven by turbulence and wake entrainment while for bubble breakage by the impact of turbulent eddies. Local radial distributions of the void fraction, interfacial area concentration, bubble Sauter mean diameter, and gas and liquid velocities, are compared against experimental data in a vertical pipe flow. Satisfactory agreements for the local distributions are achieved between the predictions and measurements. For gas–liquid bubbly flows with heat and mass transfer, boiling flows at subcooled conditions are considered. Based on the formulation of the MUSIG (multiple-size-group) boiling model and a model considering the forces acting on departing bubbles at the heated surface implemented in the computer code CFX4.4, comparison of model predictions against local measurements is made for the void fraction, bubble Sauter mean diameter, interfacial area concentration, and gas and liquid velocities covering a range of different mass and heat fluxes and inlet subcooling temperatures. Good agreement is achieved with the local radial void fraction, bubble Sauter mean diameter, interfacial area concentration and liquid velocity profiles against measurements. However, significant weakness of the model is evidenced in the prediction of the vapour velocity. Work is in progress through the consideration of additional momentum equations or developing an algebraic slip model to account for the effects of bubble separation.     

A Semenov approach to the modelling of thermal runaway of damp combustible material

Semenov theory for the self-heating of a reactive slab is extendedto take account of the presence of water vapour. In this paper,mass changes due to evaporation/condensation are neglected butheat exchange is retained in the energy equation. By doing this,a simple easily solvable set of equations can be set up to representthe thermal behaviour of the slab. No account is taken of possiblewet exothermic reactions in this paper. The aim is simply tounderstand the effects of evaporation/condensation on the overallthermal history. Using a simple model which treats the masschanges within the material as negligible, the competitive effectsof condensation and evaporation are shown to produce a two-timesituation which depends crucially on the surface mass transfer/heattransfer ratio h_m. Either self-heating occurs at a lower ratethan that due to dry oxidation, or else a maximum temperatureis reached before a lower equilibrium steady-state temperatureis achieved. Thus, compared to the dry case, in general terms,evaporation certainly encourages stability. However, the finalstrictly subcritical steady state will not always be achieveddue to the competitive process between recondensation and evaporationloss at the surface at medium timescales. A set of quasi-steadystates is identified which yield plots of a more restrictivecritical value of temperature against the Frank-Kamenetskiiparameter (proportional to the thickness of the slab and itsreactivity). If the value of h_m is such that the maximum temperaturereaches this critical value, then thermal runaway can stilltake place even though the starting value of temperature wasstrictly below the true (damp) final steady-state critical value.     

On the hyperbolicity of the two-fluid model for gas–liquid bubbly flows

The hyperbolicity condition of the system of partial differential equations (PDEs) of the incompressible two-fluid model, applied to gas–liquid flows, is investigated. It is shown that the addition of a dispersion term, which depends on the drag coefficient and the gradient of the gas volume fraction, ensures the hyperbolicity of the PDEs, and prevents the nonphysical onset of instabilities in the predicted multiphase flows upon grid refinement. A constraint to be satisfied by the coefficient of the dispersion term to ensure hyperbolicity is obtained. The effect of the dispersion term on the numerical solution and on its grid convergence is then illustrated with numerical experiments in a one-dimensional shock tube, in a column with a falling fluid, and in a two-dimensional bubble column.