Chemical Equilibrium Code Research Articles

Depletion-driven thermochemistry of molten salt reactors: review, method, and analysis

Molten salt reactors (MSRs) are innovative advanced nuclear reactors that utilize nuclear fuel by dissolving it in a high-temperature liquid salt. This unique feature differentiates MSRs from other types of reactors and allows for enhanced safety and economic performance. The liquid fuel also entails several multiphysics effects that can complicate reactor design and operation. One primary effect termed here as depletion-driven thermochemistry is a driving force in altering the multiphysics behavior of the reactor. Essentially, depletion-driven thermochemistry is the effect that fuel depletion has on changing the chemical redox potential of the fuel salt over time. As the fuel is consumed, the redox potential shifts toward a more oxidizing state. Without active control, the changing chemistry due to depletion increases corrosion thereby limiting reactor component lifetimes. Additionally, the changing redox potential of the fuel salt alters the vapor pressures of chemical species dissolved in the fuel salt. Changing vapor pressures of species in the fuel salt is an important parameter to understand when off-gassing volatile species during normal reactor operation, and for source term characterization during accident scenario transients. The present work represents a fundamental step toward modeling and coupling the driving physics (i.e., neutronics and chemistry) involved in altering the redox potential in an MSR. Here, the neutronic code Griffin models the depletion of the fuel-salt system, while the chemical equilibrium code Thermochimica calculates the thermochemical state of the isotopic inventory, using the Molten Salt Thermodynamic Database - Thermochemical (MSTDB-TC). These two codes are tightly coupled to predict the impact of fuel depletion in altering the chemistry in MSR systems. Redox potential control methods are discussed and can be modeled using this multiphysics approach. The vapor pressures of chemical species that could be extracted to an off-gas system, as determined by the reactor’s thermochemical state, are examined. The neutronics-chemistry coupling developed in this work is expected to have potential application for analyzing corrosion, source term evolution, and material safeguards in MSR systems. Lastly, suggestions for areas of further improvements of the models to expand these capabilities by incorporating other coupled physics effects is provided.

Spreading Effect of Enhanced Blast in Explosives and Propellants

We performed tests on one of our propellants to determine its TNT equivalence. In such tests we detonate a charge of explosive at one end of a propellant cylindrical sample. We use two types of diagnostics: 1) to determine the propellant energy release rate, we wrap it with a thin aluminum shell, and measure the radial velocity of the shell along the propellant sample with velocity interferometers (PDV); and 2) we use commercial blast gauges at a distance of about 10m from the test sample, to determine the blast effect of the propellant (after deducting the blast effect of the driving explosive). We expected the two types of diagnostics to agree with one another, but this has not been the case. From the velocity gauges we see that only the part of the propellant which is near the explosive has fully reacted, and that the extent of this part depends on the size and configuration of the driving explosive. But from the blast diagnostics it seems, that as if the whole propellant has reacted. We’ve seen some sporadic indications in the literature to such an enhanced blast effect from propellants [1-5]. We refer here to such an enhanced blast effect as a spreading effect. We claim, that when a reacting propellant (or explosive) is spread out in space (without changing its total energy release upon reacting), the blast effect at a given distance is enhanced substantially compared to the same energy yield without spreading. In our tests, the relative amount of reacted propellant decreases substantially with distance from the initiating explosive, but because it does not react in a continuous fashion, there is a spreading effect, that enhances the level of air blast substantially. To demonstrate the action of the spreading effect, we make 1D simulations of outgoing detonations of an HB (H=HMX, B=Binder) explosive in spherical symmetry. For different runs we dilute the explosive with the binder to various extents, from W=1 (pure explosive) to W=0.01 (1% explosive). For each such formulation we compute the detonation parameters using our in-house chemical equilibrium code (not presented here). For each formulation we adjust the explosive radius so that the total detonation energy stays the same. For all runs we monitor the rigid boundary reflected pressure (pr) of the created blast wave in air, at a radial distance of 12m. We get that as W decreases below 0.5, the total amount of HMX decreases below 6% of the amount for pure HMX, but pr increases by about 12% relative to that of pure HMX. We attribute this enhanced blast effect to the spreading of the explosive, while keeping the amount of energy released unchanged (spreading effect).

Theoretical Performance and Application Cases of Water Electrolysis Propulsion

Water Electrolysis Propulsion offers a clean alternative to current toxic and detonable spacecraft propellants. Beyond the potential replacement of hydrazine as propellant, water based propulsion can be highly attractive for future exploration missions, since the availability of water on many potential landing sites offers the opportunity of refueling. The paper discusses components and state of the art of water propulsion and lists possible variants. A performance analysis is made by use of the chemical equilibrium code CEA. Trade-offs and comparisons are performed with conventional chemical and electric propulsion systems.

Effect of Argon Concentration on Laminar Burning Velocity and Flame Speed of Hydrogen Mixtures in a Constant Volume Combustion Chamber

Abstract Hydrogen combustion, coupled with the use of argon as a working fluid, is a promising approach to delivering clean and efficient energy from internal combustion (IC) engines. The use of hydrogen-oxygen-argon (H2/O2/Ar) mixtures in combustion aids in mitigating harmful environmental pollutants and enables a highly efficient energy conversion process. The use of argon as a working fluid decreases the NOx emissions and increases the thermal efficiency of internal combustion engines due to the high specific heat ratio of noble gases. In this study, premixed hydrogen combustion was investigated with the purpose of examining the effect of the full or partial substitution of argon for nitrogen in air on laminar burning velocity (LBV), flame speed, flame morphology, and instability. The experimental approach uses an optically accessible constant volume combustion chamber (CVCC) with central ignition; the spherical flame development was studied using a high-speed Z-type Schlieren visualization system. Moreover, a numerical model was developed to convert the experimental dynamic pressure rise data to laminar burning velocity. Coupling the model to a chemical equilibrium code aids in determining the burned gas properties. Additionally, an image processing technique has been suggested to compute the flame propagation speed. The experimental and numerical investigations indicate that increasing the concentration of argon as the working fluid in the mixture increases the laminar burning velocity and flame speed while extending the lean flammability limit.

Dust sputtering within the inner heliosphere: a modelling study

Abstract. The aim of this study is to investigate through modelling how sputtering by impacting solar wind ions influences the lifetime of dust particles in the inner heliosphere near the Sun. We consider three typical dust materials, silicate, Fe0.4Mg0.6O, and carbon, and describe their sputtering yields based on atomic yields given by the Stopping and Range of Ions in Matter (SRIM) package. The influence of the solar wind is characterized by plasma density, solar wind speed, and solar wind composition, and we assume for these parameter values that are typical for fast solar wind, slow solar wind, and coronal mass ejection (CME) conditions to calculate the sputtering lifetimes of dust. To compare the sputtering lifetimes to typical sublimation lifetimes, we use temperature estimates based on Mie calculations and material vapour pressure derived with the MAGMA chemical equilibrium code. We also compare the sputtering lifetimes to the Poynting–Robertson lifetime and to the collision lifetime. We present a set of sputtering rates and lifetimes that can be used for estimating dust destruction in the fast and slow solar wind and during CME conditions. Our results can be applied to solid particles of a few nanometres and larger. The sputtering lifetimes increase linearly with the size of particles. We show that sputtering rates increase during CME conditions, primarily because of the high number densities of heavy ions in the CME plasma. The shortest sputtering lifetimes we find are for silicate, followed by Fe0.4Mg0.6O and carbon. In a comparison between sputtering and sublimation lifetimes we concentrate on the nanodust population. The comparison shows that sublimation is the faster destruction process within 0.1 AU for Fe0.4Mg0.6O, within 0.05 AU for carbon dust, and within 0.07 AU for silicate dust. The destruction by sputtering can play a role in the vicinity of the Sun. We discuss our findings in the context of recent F-corona intensity measurements onboard Parker Solar Probe.

Internal Ballistics of a Boron-Containing Solid Fuel Ramjet

ABSTRACT The solid fuel ramjet is one of the simplest air-breathing engines, characterized by increased specific impulse. This study focuses on a parametric investigation of the internal ballistics of a boron-containing solid fuel ramjet. A 2-D axisymmetric model of a ramjet combustor flying at Mach 2.5 was developed and solved numerically utilizing a commercial computational fluid dynamics code modified to address the specific issues of the problem. The simulation results were compared to the theoretical performance calculated by the chemical equilibrium code. The solid fuel regression rate was found to depend on the port diameter and thus decreases with time. Thrust regulation of ramjet engine was demonstrated using bypass air. The specific impulse was found to decrease with boron content, probably because of low combustion efficiency caused by either insufficient oxygen supply or short residence time of the boron particles.

Jones-Wilkins-Lee Unreacted and Reaction Product Equations of State for Overdriven Detonations in Octogen- and Triaminotrinitrobenzene-Based Plastic-Bonded Explosives.

The Jones-Wilkins-Lee (JWL) equation of state (EOS) is used to calculate expansion of detonation reaction products from the chemical equilibrium Chapman-Jouguet (C-J) state to large volumes. Overdriven detonation waves with shock pressures higher than C-J are created by high-velocity impacts or converging detonation waves. Reflection from high-impedance materials, multiple shock impacts, and Mach stem wave interactions creates similar pressures. When overdriven states were first measured experimentally, the original reaction product JWL EOSs predicted excess compression. This problem was resolved by modifying the JWL EOS to produce less compression at high pressures while still correctly calculating expansion from the C-J state. Zeldovich-von Neumann-Doring (ZND) reactive flow models, which include the measured reaction zone momentum, explained experimental observations that lower C-J pressures are required to smoothly connect the C-J state to overdriven states on the product Hugoniot curve. Experimental data on overdriven detonation waves for two octogen (HMX)-based plastic-bonded explosives (PBXs), PBX 9501 and PBX 9404, and for two triaminotrinitrobenzene (TATB)-based PBXs, LX-17 and PBX 9502, are compared to various JWL reaction product EOSs, including ones generated by the CHEETAH chemical equilibrium code. Excellent agreement is obtained using JWL EOSs for overdriven shock pressures and densities up to 130 GPa and 3.8 g/cm3 for both HMX- and TATB-based PBXs.

Equilibrium chemistry down to 100 K

We have introduced a fast and versatile computer code, GGCHEM, to determine the chemical composition of gases in thermo-chemical equilibrium down to 100 K, with or without equilibrium condensation. We have reviewed the data for molecular equilibrium constants, kp(T), from several sources and discussed which functional fits are most suitable for low temperatures. We benchmarked our results against another chemical equilibrium code. We collected Gibbs free energies, ΔGf⊖, for about 200 solid and liquid species from the NIST-JANAF database and the geophysical database SUPCRTBL. We discussed the condensation sequence of the elements with solar abundances in phase equilibrium down to 100 K. Once the major magnesium silicates Mg2SiO4[s] and MgSiO3[s] have formed, the dust to gas mass ratio jumps to a value of about 0.0045 which is significantly lower than the often assumed value of 0.01. Silicate condensation is found to increase the carbon to oxygen ratio (C/O) in the gas from its solar value of ~0.55 up to ~0.71, and, by the additional intake of water and hydroxyl into the solid matrix, the formation of phyllosilicates at temperatures below ~400 K increases the gaseous C/O further to about 0.83. Metallic tungsten (W) is the first condensate found to become thermodynamically stable around 1600–2200 K (depending on pressure), several hundreds of Kelvin before subsequent materials such as zirconium dioxide (ZrO2) or corundum (Al2O3) can condense. We briefly discuss whether tungsten, despite its low abundance of ~2 × 10−7 times the silicon abundance, could provide the first seed particles for astrophysical dust formation.

High density thermite mixture for shaped charge ordnance disposal

Abstract The effect of thermite mixture based on aluminum and ferric oxides for ammunition neutralization has been studied and tested. Thermochemical calculations have been carried out for different percentage of Al using Chemical Equilibrium Code to expect the highest performance thermite mixture used for shaped charge ordnance disposal. Densities and enthalpy of different formulations have been calculated and demonstrated. The optimized thermite formulation has been prepared experimentally using cold iso-static pressing technique, which exhibited relatively high density and high burning rate thermite mixture. The produced green product compacted powder mixture was tested against small caliber shaped charge bomblet for neutralization. Theoretical and experimental results showed that the prepared thermite mixture containing 33% of aluminum as a fuel with ferric oxide can be successfully used for shaped charge ordnance disposal.

Modeling of combustion and flow in a single element GH2/GO2 combustor

A single element gaseous H2/O2 combustion test case set up by the Pennsylvania State University is investigated using the new Reynolds Averaged Navier–Stokes (RANS) code, Rocflam3. There are considerable numbers of publications on this test case. Some of the publishing authors assume that it is not possible to achieve good results for this test case with simple RANS methods whereas others demonstrate the opposite. This is a reason for discussion which continues in this work. To obtain a reasonable solution for the test case, different simulation settings have been examined. This includes a grid study, a comparison of various $$k{\text{-}} \epsilon$$ type turbulence models and the variation of the turbulent Prandtl and Schmidt numbers. Turbulent combustion is treated via an equilibrium-based, presumed probability density function approach with a mixture fraction formulation. The 2D steady state simulations show good agreement with the experimentally measured heat flux. However, a discrepancy between the computed chamber pressure and the measured one is observed and is discussed incorporating considerations about combustion efficiency. This discussion shows that the experimental data are inconsistent. Computations performed with the chemical equilibrium code by Gordon and McBride are used to support the statements.

Laser ignition of hypersonic air–hydrogen flow

An experimental investigation of the behaviour of laser-induced ignition in a hypersonic air–hydrogen flow is presented. A compression-ramp model with port-hole injection, fuelled with hydrogen gas, is used in the study. The experiments were conducted in the T-ADFA shock tunnel using a flow condition with a specific total enthalpy of 2.5 MJ/kg and a freestream velocity of 2 km/s. This study is the first comprehensive laser spark study in a hypersonic flow and demonstrates that laser-induced ignition at the fuel-injection site can be effective in terms of hydroxyl production. A semi-empirical method to estimate the conditions in the laser-heated gas kernel is presented in the paper. This method uses blast-wave theory together with an expansion-wave model to estimate the laser-heated gas conditions. The spatially averaged conditions found with this approach are matched to enthalpy curves generated using a standard chemical equilibrium code (NASA CEA). This allows us to account for differences that are introduced due to the idealised description of the blast wave, the isentropic expansion wave as well as thermochemical effects.

Unlike-pair interactions of detonation products at high pressure and high temperature

In this paper, we present a method that unlike-pair interactions are determinated by explosive detonation data. The unlike-pair interaction parameters kN2-H2O= 1.03, kN2-CO2= 1.035, kH 2O-CO2= 0.96 are deduced from the calculated results of explosive RX-23-AB,HNB and PETN with our statistical mechanical chemical equilibrium code CHEQ-like. At the same time, the theoretical Hugoniots of explosive PBX9404 are calculated with these parameters. The results in better agreement with the experimental data show that the parameters and method are reliable.

The effect of the regular solution model in the condensation of protoplanetary dust

We utilize a chemical equilibrium code in order to study the condensation process which occurs in protoplanetary discs during the formation of the first solids. The model specifically focuses on the thermodynamic behaviour on the solid species assuming the regular solution model. For each solution, we establish the relationship between the activity of the species, the composition and the temperature using experimental data from the literature. We then apply the Gibbs free energy minimization method and study the resulting condensation sequence for a range of temperatures and pressures within a protoplanetary disc. Our results using the regular solution model show that grains condense over a large temperature range and therefore throughout a large portion of the disc. In the high-temperature region (T≥ 1400 K) hibonite and gehlenite dominate, and we find that the formation of corundum is sensitive to the pressure. The mid-temperature region is dominated by Fe(s) and silicates such as Mg2SiO4 and MgSiO3. The chemistry of forsterite and that of enstatite are strictly related, and our simulations show a sequence of forsterite–enstatite–forsterite with decreasing temperature and the abundance of the first high-temperature peak of forsterite is also pressure sensitive. In the low-temperature regions (T≤ 600 K), a range of iron compounds (FeS, Fe2SiO3, FeAl2O3) form. We find that all the condensation sequences move towards lower temperature as the pressure decreases. We also run simulations using the ideal solution model and see clear differences in the resulting condensation sequences with changing solution model. In particular, we find that the turning point in which forsterite replaces enstatite in the low-temperature region is sensitive to the solution model. In this same temperature region, fayalite is the most stable compound for the regular solution, while magnetite replaces fayalite in the ideal solution model at the lowest values of temperature. Our results show that the ideal solution model is often a poor approximation to experimental data at most temperatures important in protoplanetary discs. We find some important differences in the resulting condensation sequences when using the regular solution model and suggest that this model should provide a more realistic condensation sequence.

Experimental correction of combustion gas properties of AN-based composite solid propellants used for turbo-pump starter

Solid propellant gas generators play a role as a turbo-pump starter in liquid propellant propulsion systems by supplying pressurized gas to power turbines for engine start. Among the required combustion gas properties provided by solid propellant gas generators, the combustion gas temperature should not exceed a certain temperature which may damage the turbine blades. For such purposes, phase stabilized ammonium nitrate (AN)-based propellants have been widely used with a low combustion temperature. However, gas generator propellants with ammonium nitrate have historically exhibited incomplete combustion resulting in increased flame temperatures differing significantly from equilibrium values. In consideration of design requirements, an engineering model of solid propellant gas generator was manufactured using the combustion gas properties calculated by a chemical equilibrium code and then hot-fire tests were performed. Procedures for the correction of T 0 , k and M w of the combustion gas from the experimental results are introduced and the following effects on the design of the solid propellant gas generator are presented. From the experimental correction of the combustion gas properties, it is found that the amount of the propellant could be reduced while providing the same amount of available power to the turbines and consequently, the size of the gas generator could also be decreased.

Chemical Energy Release in Several Recently Discovered Detonation and Deflagration Flows

Several recent experiments on complex detonation and deflagration flows are analyzed in terms of the chemical energy release required to sustain these flows. The observed double cellular structures in detonating gaseous nitromethane–oxygen and NO2–fuel (H2, CH4, and C2H6) mixtures are explained by the amplification of two distinct pressure wave frequencies by two exothermic reactions, the faster reaction forming vibrationally excited NO* and the slower reaction forming highly vibrationally excited N2**. The establishment of a Chapman-Jouguet (C-J) deflagration behind a weak shock wave, the C-J detonation established after a head-on collision with a shock front, and the C-J detonation conditions established in reactive supersonic flows are quantitatively calculated using the chemical energy release of a H2 + Cl2 mixture. For these three reactive flows, these calculations illustrate that different fractions of the exothermic chemical energy are used to sustain steady-state propagation. C-J detonation calculations on the various initial states using the CHEETAH chemical equilibrium code are shown to be in good agreement with experimental detonation velocity measurements for the head-on collision and supersonic flow detonations.

Performance analysis of the pulse detonation rocket engine based on constant volume cycle model

As the propagation time of detonation is very short in the actual operation process of a pulse detonation rocket engine (PDRE), the gases expelled slightly from the detonation chamber in this process is negligible. Therefore, a mathematical model of constant volume cycle (CVC) can be used to approximately depict the operation process of PDRE. Based on CVC model, a systematic analysis of PDRE’s performance is conducted in this paper. Combined CVC model with Chemical Equilibrium and Applications (CEA) code, the specific impulse of PDRE using different propellants could be obtained. Utilizing this model, the specific impulse of kerosene/oxygen PDRE in vacuum and at finite backpressures is estimated. When the backpressure exists, there is an optimum nozzle area expansion ratio in the nozzle design of PDRE, similar to conventional steady-state rocket. The performance of PDRE and conventional steady rocket is compared respectively, under the two conditions of the same supply system and the same maximum material stress. When PDRE and conventional steady rocket have a comparable performance, the filling pressure of PDRE is much lower than that of steady rocket.

エンジン風洞における超音速流の閉塞

Breakdown of diffuser flow was often observed in our scramjet engine tests. This facility operation may damage the engine wind-tunnel and should be prevented. An one-dimensional analysis was applied to the diffuser flow to identify the causes of the flow breakdown. All the losses and gains by engine and friction loss in the diffuser were represented by point-sources of mass, momentum and energy. The thermal choking condition was calculated by uses of a chemical equilibrium code. The fuel rates causing the flow-choking successfully reproduced the limit fuel rates observed in our tests. Inlet-unstart of engine lost the ejector-pumping effect in the diffuser system to trigger the flow choking. The choking was also promoted by the drag of the gas sampling rakes. The choking in diffuser flow and the engine unstart may couple each other to cause hysteresis in the diffuser breakdown, which was also experienced in our tests. A rocket-based, combined-cycle (RBCC) engine will be tested under the Mach 4 condition. The engine easily causes the choking of diffuser because of the large propellant supply rates and the relatively-low specific impulse. Operation of the wind-tunnel was discussed to control the flow choking in the tests.

연료 과농 가스발생기의 연소 가스 물성치에 관한 연구

액체 로켓 엔진용 가스발생기 개발을 위해서는 추진제 O/F ratio에 따른 연소 가스의 열역학적 물성치 예측이 필수적이다. 본 연구에서는 액체산소/Jet A-1 조합의 연료 과농 가스발생기의 실 추진제 연소 시험을 통해 O/F ratio 변화에 따른 연소 생성 가스의 온도를 계측하였다. 또한 연소실 내 동압 섭동 측정 및 정압 측정 결과를 이용하여 비열비, 가스 상수, 정압 비열과 같은 연소 가스의 열역학적 물성치를 간접적으로 산출해내었다. 본 실험값은 화학평형코드 결과를 통해 구한 보간 계수를 이용한 예측 결과와 비교해보았을 때 동일한 경향 및 유사한 값을 가지는 것으로 밝혀졌으며 이는 보간 계수 예측 방법이 설계 도구로 충분히 적용 가능하다는 결과를 확인하였다. It is essential to predict thermodynamic properties of combustion gas with respect to a propellant mixture ratio for the development of a gas generator for a liquid rocket engine. The present study shows the temperature measurement of exit combustion gas as a function of a mixture ratio through the series of combustion tests of a fuel-rich gas generator with liquid oxygen and Jet A-1. The measurements of dynamic and static pressures, and combustion gas temperatures allowed the estimation of thermodynamic properties like a specific heat ratio, a gas constant, and a constant pressure specific heat of the combustion gas. The comparison of the experimental results with predictions made by interpolation parameters obtained from the modification of the chemical equilibrium code indicates that the interpolation method calibrated using the temperature measurements can be utilized as an effective tool for the initial design of a fuel-rich gas generator.

A thermodynamic model of high temperature lava vaporization on Io

We modified the MAGMA chemical equilibrium code developed by Fegley and Cameron (1987, Earth Planet. Sci. Lett. 82, 207–222) and used it to model vaporization of high temperature silicate lavas on Io. The MAGMA code computes chemical equilibria in a melt, between melt and its equilibrium vapor, and in the gas phase. The good agreement of MAGMA code results with experimental data and with other computer codes is demonstrated. The temperature-dependent pressure and composition of vapor in equilibrium with lava is calculated from 1700 to 2400 K for 109 different silicate lavas in the ONaKFeSiMgCaAlTi system. Results for five lavas (tholeiitic basalt, alkali basalt, Barberton komatiite, dunite, and a molten type B1 Ca, Al-rich inclusion) are discussed in detail. The effects of continuous fractional vaporization on chemistry of these lavas and their equilibrium vapor are presented. The predicted abundances (relative to Na) of K, Fe, Si, Al, Ca, and Ti in the vapor equilibrated with lavas at 1900 K are lower than published upper limits for Io's atmosphere (which do not include Mg). We predict evaporative loss of alkalis, Fe, and Si during volcanic eruptions. Sodium is more volatile than K, and the Na/K ratio in the gas is decreased by fractional vaporization. This process can match Io's atmospheric Na/K ratio of 10±3 reported by Brown (2001, Icarus 151, 190–195). Silicon monoxide is an abundant species in the vapor above lavas. Spectroscopic searches are recommended for SiO at IR and mm wavelengths. Reactions of metallic vapors with S- and Cl-bearing volcanic gases may form other unusual gases including MgCl 2, MgS, MgCl, FeCl 2, FeS, FeCl, and SiS.

Modeling the behavior of unsaturated cement systems exposed to aggressive chemical environments

The main features of a numerical model that predicts the mechanisms of ionic transport in unsaturated cement systems are described. The model, called STADIUM, is divided into four parts: ionic diffusion, moisture transport, chemical reactions, and chemical damage. The diffusion of all ions present in the system is modeled by solving the extended Nernst-Planck/Poisson set of equations. The model takes into account the electrical coupling between the various ionic species in solution and chemical activity effects. The transport of water by capillary suction is described by a diffusion equation. In the model, the transport phenomena and the chemical reactions are uncoupled. Phase dissolution and precipitation phenomena are modeled through a separate chemical equilibrium code. The model considers the influence of chemically induced microstructural alterations on the transport properties of the material. A brief description of the model is presented in the first sections of this paper. Typical applications of the model are given in the last section.