Enskog Dense Gas Theory Research Articles

Closed-loop task allocation in robot swarms using inter-robot encounters

In swarm robotics systems, coordinated behaviors emerge via local interactions among the robots as well as between robots and the environment. For a swarm of robots performing a set of pre-defined tasks in an enclosed region, this paper develops a decentralized mechanism to allocate tasks to each robot by leveraging the spatial interactions occurring among the robots as they move around the domain. With the aim of achieving a desired percentage of the swarm performing each task, the developed strategy allows individual robots to switch between different tasks with a certain probability when they encounter other robots in the region. We develop an analytical model to describe the inter-robot encounters occurring in a densely packed swarm of robots using ideas from the Enskog theory of dense gases and illustrate how the swarm can leverage this model to achieve the desired allocation levels. Furthermore, the inter-robot encounters enable the robots to measure the current allocation of the swarm, which is then used to regulate the rate at which they switch between tasks. This allows the swarm to speed up or slow down the rate of transitions between tasks depending on how far the current allocation is from the desired values, ultimately facilitating uninterrupted task execution by the robots. The methods introduced in this paper illustrate how naturally occurring encounters among robots in a swarm can be used to allocate tasks to the robots in a closed-loop manner. The developed algorithm is completely decentralized and can be deployed on minimalistic robots without the need for communication or a central coordinator. The performance of the algorithm is demonstrated on a swarm of real robots.

Theoretical scheme for the shear viscosity of Lennard-Jones fluids

We use the shear viscosity expression from the Enskog theory of dense gases in a perturbative scheme for the Lennard-Jones (LJ) fluid. This perturbative scheme is formulated by combining the analytic rational function approximation method of Bravo Yuste and Santos [Phys. Rev. A 43, 5418 (1991)] for the radial distribution function of hard-sphere fluids and the well known Mansoori-Canfield/Rasaiah-Stell perturbation theory to determine an effective diameter for the LJ fluid. The scheme is reliable on a wide range of temperatures and densities, and is very accurate around the critical point. Using this information, we build an accurate empirical formula for the shear viscosity in the liquid phase, which fits the recent data [K. Meier et al., J. Chem. Phys. 121, 3671 (2004)] in the whole simulation range.

One-dimensional surface diffusion: Density dependence in a smooth potential

A kinetic treatment of the transport of mobile molecules in nanometer channels is presented, that explores ideas of the Enskog theory of dense gases combined with molecular relaxation to the substrate. Transport diffusivity is shown to increase strongly with density, as an effect of effective transfer of mass in the nonlocal intermolecular collisions. The effect is partially compensated by the additional collisional relaxation, that would be absent in the truly one-dimensional system of hard rods, but is shown to occur in the realistic cases of quasi-one-dimensional channels and/or “soft” particles. A resulting behavior of diffusivity as a function of density close to 1/(1−θ), where θ is the degree of occupancy, is expected. This suggests an explanation for the puzzling universal behavior observed earlier in zeolites.

A theory for granular materials exhibiting normal stress effects based on Enskog's dense gas theory

Granular materials exhibit phenomena such as normal stress differences, which are typical of materials whose response is non-linear. For example, when a non-linearly elastic slab is sheared, its motion is not determined by the shear force but by the normal forces that manifest themselves due to the shearing (Poynting effect). Another example is a non-linear fluid which exhibits normal stress differences that lead to phenomena like “die-swell” or “rod-climbing,” which is again a manifestation of the stresses that develop orthogonal to planes of shear. In this paper, an expression for the stress tensor of a granular material that can exhibit normal-stress effects due to a solids fraction gradient is derived from both continuum and kinetic models. The continuum model motivates and develops the form of the stress tensor, but introduces undetermined coefficients. The kinetic model evaluates those coefficients using Enskog's dense gas theory. The dependence of the granular stress tensor on the solids fraction gradient arises by requiring that the correlating factor that links the two-particle distribution function to the two single-particle distribution functions be the contact value for the radial distribution function of a non-homogeneous, hard-sphere fluid. A representation for that contact value is found by developing the generalized van der Waals theory expression for a stress tensor element of a nonhomogeneous fluid (a fluid that exhibits a density gradient) in equilibrium, and comparing it to the exact expression. That representation of the contact value is introduced into the two-particle distribution function, and its contribution to the stress tensor is found. The resulting stress tensor expression is applied to a simple shear flow problem in which a linear, solids-fraction profile is transverse to the flow. The resulting normal-stress effects increase with the solids-fraction and its gradient.

Modeling of solid particles in fluidized beds

A theory is proposed for the modeling of solid particles in fluidized beds based on ideas in the theory of interacting continua. The theory is capable of predicting the behavior of the mixture in the two extremes. That is, in one limit, as the concentration of solid particles becomes small, the theory reduces to the classical theory of incompressible viscous fluid. In the other limit, as the effect of the interstitial fluid becomes negligible, the theory reduces to the flow of incompressible granular materials. The mixture is considered to be made up of a linearly viscous fluid and granular materials. The stress tensor for the solid particles is derived using a modified version of Enskog's dense gas theory. The normal-stress effects that are observed experimentally in granular materials are predicted in this theory due to a solids fraction gradient. The dependence of the granular stress tensor on the solids fraction gradient arises by requiring that the correlating function that links the two-particle distribution function to the two single-particle distribution functions be the contact value for the radial distribution function of a non-homogeneous, hard sphere fluid. The interaction force is assumed to have contributions from density gradients, relative velocity (drag force), and relative acceleration (virtual mass force).

The inverse operator representation of thermal diffusion factors in dense gas mixtures of rigid spheres

It is shown that, expressed in terms of the inverse of the Enskog collision operator, the Stefan–Maxwell equations in a multicomponent dense gas of rigid spheres take a form quite similar to that obtained in gases at moderate pressure. Two thermal diffusion factors are defined. The first is a ‘‘kinetic’’ thermal diffusion factor formally identical with the factors found by kinetic theory at moderate pressures. The second is a ‘‘phenomenological’’ thermal diffusion factor which is rather more pressure dependent. A brief comparison with experiment shows that the Enskog dense gas theory works best in systems rich in helium at elevated temperature.

Generalized corresponding states correlation for self-diffusion coefficients of fluids

The simple principle of corresponding states may be generalized to include substances that depart from strict conformality by introducing state dependent shape factors. This approach is applied to self-diffusion coefficient using a modified Enskog dense gas theory. Results are obtained for both saturated liquids and supercritical fluids and are generally within experimental error.

Quantum statistical theory of chemical reactions

Starting from the exact rate equations for chemical reactions derived in a previous paper [1], approximate rate equations are derived by considering a convenient class of diagrams containing both two-body dynamics and quasistatic two-body correlations. Thus, in the markovian limit, the rate constants are formulated in terms of the T-matrix elements of two-molecule reactive scattering theory and of pair distribution functions describing the correlations of the molecules of the different species in thermal, but not chemical, equilibrium. The pair correlation functions depend in a complicated way on the concentrations of the different species, which leads to a concentration dependence of the rate constants. Physically, our expressions for the rate constants are closely related to the usual expressions for the transport coefficients of the Enskog dense-gas theory of non-reactive fluids.

Quantum statistical theory of chemical reactions

Starting from the exact rate equations for chemical reactions derived in a previous paper [1], approximate rate equations are derived by considering a convenient class of diagrams containing both two-body dynamics and quasistatic two-body correlations. Thus, in the markovian limit, the rate constants are formulated in terms of the T-matrix elements of two-molecule reactive scattering theory and of pair distribution functions describing the correlations of the molecules of the different species in thermal, but not chemical, equilibrium. The pair correlation functions depend in a complicated way on the concentrations of the different species, which leads to a concentration dependence of the rate constants. Physically, our expressions for the rate constants are closely related to the usual expressions for the transport coefficients of the Enskog dense-gas theory of non-reactive fluids.

Corresponding states correlation for thermal conductivity of dense fluids

The simple principle of corresponding states may be generalized to include substances that depart from strict conformality by introducing state dependent shape factors. This approach is applied to thermal conductivity, and a general correlation is presented for the shape factors. Mixing rules derived from statistical mechanics, based on Enskog dense gas theory for thermal conductivity, provide a means for predicting thermal conductivities of dense gas and liquid mixtures of nonpolar substances. Results obtained are generally within experimental error.

Conformal solution theory for viscosity and thermal conductivity of mixtures

The first-order terms are derived in a conformal solution theory expansion for the viscosity and thermal conductivity of a fluid mixture. Provided that pairwise additivity of the intermolecular potentials is assumed, these terms can be made to vanish exactly by a suitable choice of the potential parameters and molecular mass for the reference system. A generalization of Enskog's dense gas theory to molecules with a general intermolecular potential law is used to suggest suitable combinations of potential parameters and reduced masses as expansion variables. Several sets of mixing rules are thus suggested. These are first tested for dilute gases by comparison with the Chapman-Enskog results, and the best mixing rules for gases are determined. This is followed by a comparison with experimental data for simple liquid mixtures. The general behaviour of the excess viscosity and thermal conductivity for binary mixtures as a function of the potential and mass parameters is investigated using the theory. It is shown that whereas the excess viscosity is nearly always negative or zero, the excess thermal conductivity may be positive or negative.

The binary diffusion coefficient of moderately dense gas mixtures

A model of the three-body encounters which occur in a dense gas is used to evaluate their effect on the binary diffusion coefficient of a moderately dense gas mixture. The behaviour predicted for the diffusion coefficient as a function of density is in direct opposition to that resulting from the Enskog theory of dense gases in which three-body collisions are neglected. When the entire effect of increasing gas density on the diffusion coefficient is considered an expression is obtained which leads to substantial agreement with available experimental data.

Kinetic Theory of Moderately Dense Gases

A kinetic equation approach to the kinetic theory of moderately dense gases is developed by asymptotic expansions from a viewpoint independent of Bogoliubov chaos assumptions. The theory is applied to the rigid sphere model where the contributions which determine the Enskog dense gas theory are identified. The existence of divergences in the time integrations for triple collision events in the case of rigid discs and for quadruple collision events in the case of rigid spheres is confirmed by means of variable transformations. A scheme for renormalizing the divergences into an asymptotic density expansion containing nonanalytic logarithmic terms is proposed, illustrated for the case of rigid discs, and employed to deduce the fact that the first density corrections for rigid spheres, being convergent, remain intact as the lead term of the asymptotic density expansion. Numerical studies of the first density corrections to the thermal conductivity and shear viscosity of rigid spheres are performed in order to assess the relative importance of Enskog dense gas contributions and dynamical correlation effects. The results are in substantial agreement with preliminary Monte Carlo calculations of Sengers based upon Bogoliubov theory.

Second-Order Kinetic Theory of Stress for a Dense Gas of Rigid Spheres

The Enskog dense gas theory is generalized to arbitrary order in gradients of macroscopic field variables. A constitutive relation for the pressure tensor involving products of velocity gradients is presented in an irreducible tensor formulation. The second-order transport coefficients are evaluated for smooth rigid spheres, and the theory is applied to certain viscometric flows. Expressions are derived for normal stress effects.

NMR Measurements of Self-Diffusion in Normal Hydrogen Gas from 55° to 90°K

An absolute measurement of the self-diffusion coefficient D11 in hydrogen gas has been made at 78°K and 117 Amagats. Relative measurements have been made of D11 as a function of gas density at 55°, 78°, and 90°K, up to a maximum density of 650 Amagats. The experimental results are compared with viscosity data and with the theories of Buckingham and Mason. The observed density dependence of D11 is in marked disagreement with the Enskog dense-gas theory.

Self-Diffusion in Argon to 300 Atmospheres

The coefficient of self-diffusion for argon is measured from 68 to 291 atmos at 49.4°C. A radioactive tracer technique employing A37 is used and coefficients are measured in a cell in which the path of diffusion is through a plug of porous bronze. Results agree reasonably well with the Enskog dense gas theory calculated from p-v-T data.

Diffusion in CO2–CH4 Mixtures to 225 Atmospheres Pressure

Diffusion coefficients have been measured in two mixtures of CO2 and CH4. (1) 50 mole percent CO2—50 percent CH4; (2) 75 mole percent CO2—25 mole percent CH4. C14 tagged CO2 was used, so that no gradient in chemical composition was necessary. For the first mixture the results agreed with Enskog's dense gas theory at all temperatures and pressures. For the 75 percent CO2 mixture at 25°C there was a positive deviation from theory, corresponding to the increased importance of glancing collisions for CO2 near saturation. The results are consistent with our previous data for self-diffusion in the two pure gases.

Diffusion in the System CH4—TCH3 to 300 Atmospheres Pressure

Diffusion coefficients have been measured in the system CH4—TCH3 to 300 atmospheres pressure. In general the results are consistent with the predictions of Enskog's dense gas theory. At the highest densities, the diffusion coefficients are lower than the theory predicts. This is expected as, at these densities, unevaluated terms in the dense gas correction become significant.