Sphere in continuum

Henderson 575 presented a set of new correlations for drag coefficient of a single sphere in continuum and rarefied flows (Table 5.1). These correlations simplify in the limit to certain equations derived from theory and offer significantly improved agreement with experimental data. The flow regimes covered include continuum, slip, transition, and molecular flows at Mach numbers up to 6 and at Reynolds numbers up to the laminar-turbulent transition. The effect on drag of temperature difference between a sphere and gas is also incorporated. 

Table 5.1. Correlations for Drag Coefficient of a Single Sphere in Continuum and Rarefied Flows1575 ...

Fuoss and Accascina [49] have shown, however, that for ionic concentrations as low as 10 M, Ostwald s equation is not exact because of its neglect of long range interionic attraction upon the conductance and activities of ions. Maintaining the concept of the sphere in continuum model, in which ions are regarded as hard spheres immersed in a continuous medium, Fuoss corrected the equation from first principles and derived the relationship... 

It is a serious drawback that it is not possible to determine the transfer activity coefficient of the proton (or of any other single-ion species) directly by thermodynamic methods, because only the values for both the proton and its counterion are obtained. Therefore, approximation methods are used to separate the medium effect on the proton. One is based on the simple sphere-in-continuum model of Born, calculating the electrostatic contribution of the Gibb s free energy of transfer. This approach is clearly too weak, because it does not consider solvation effects. Different ex-trathermodynamic approximation methods, unfortunately, lead not only to different values of the medium effect but also to different signs in some cases. Some examples are given in the following log yH+ for methanol -1-1.7 (standard deviation 0.4) ethanol -1-2.5 (1.8), n-butanol -t-2.3 (2.0), dimethyl sulfoxide -3.6 (2.0), acetonitrile -1-4.3 (1.5), formic acid -1-7.9 (1.7), NH3 -16. From these data, it can be seen that methanol has about the same basicity as water the other alcohols are less basic, as is acetonitrile. Di-... 

A first approach to take into account the solvent s effect on the absolute mobility of an ion was made by Walden. It is based on the Stokes law of frictional resistance. Walden s rule states that the product of absolute mobility and solvent viscosity is constant. It is clear that the serious limitation of this model is that it does not consider specific solvation effects, because it is based on the sphere-in-continuum model. However, it delivers an appropriate explanation for the fact that, within a given solvent, the mobility depends on temperature to the same extent as the viscosity (in water, for example, the mobility increases by about 2.5% per degree Kelvin). The mobilities do not deviate too... 

Non-hydrogen-bonded liquids of high dielectric constant are often referred to as dipolar aprotic solvents. Nitrobenzene, V,V-dimethyl-formamide, acetonitrile, and nitromethane have dielectric constants in the range 35 to 38, close to that of methanol. The dielectric constants of sulpholane, dimethylsulphoxide and propylene carbonate are higher. Any peculiarities of acid-base behaviour in dipolar aprotic solvents as compared with methanol cannot be accounted for on the basis of the Born sphere-in-continuum model. 

This type of study, still in its infancy, is important if it provides new insight into the short-range forces in electrolyte solutions because classical measurements can only give limited information. A new picture of the contact ion pair emerges and emphasis is placed on the role of solvent structure in interionic interactions an emphasis which is being recognised as essential if models are to be extended beyond the limited sphere in continuum picture of the very dilute solution. ... 

All results obtained are consistent with a simple theoretical treatment of the ionic association-dissociation processes based on the sphere-in-continuum model. 

In order to utilise our colloids as near hard spheres in terms of the thermodynamics we need to account for the presence of the medium and the species it contains. If the ions and molecules intervening between a pair of colloidal particles are small relative to the colloidal species we can treat the medium as a continuum. The role of the molecules and ions can be allowed for by the use of pair potentials between particles. These can be determined so as to include the role of the solution species as an energy of interaction with distance. The limit of the medium forms the boundary of the system and so determines its volume. We can consider the thermodynamic properties of the colloidal system as those in excess of the solvent. The pressure exerted by the colloidal species is now that in excess of the solvent, and is the osmotic pressure II of the colloid. These ideas form the basis of pseudo one-component thermodynamics. This allows us to calculate an elastic rheological property. Let us consider some important thermodynamic quantities for the system. We may apply the first law of thermodynamics to the system. The work done in an osmotic pressure and volume experiment on the colloidal system is related to the excess heat adsorbed d Q and the internal energy change d E ... 

For small particles, subject to noncontinuum effects but not to compressibility, Re is very low see Eq. (10-52). In this case, nonradiative heat transfer occurs purely by conduction. This situation has been examined theoretically in the near-free-molecule limit (SI4) and in the near-continuum limit (T8). The following equation interpolates between these limits for a sphere in a motionless gas ... 

As a first approximation to the motion of two spheres in a solvent (which can be regarded as a continuum), the spheres can be presumed to move about the solvent sufficiently slowly that the very much simplified Navier—Stokes equation of fluid flow is applicable. The application of a pressure gradient VP(r) in the fluid develops velocity gradients within the fluid, Vv(r). If another force F(r) is included in the fluid, this can generate a pressure gradient and further affect the velocity gradients. The Navier— Stokes equations [476] becomes... 

Association Phenomena According to the theoretical model of spheres in a dielectric continuum the ions are represented as rigid, charged spheres that do not interact with solvent, which is considered to be a medium without any kind of structure. The only interaction is that which occurs between the ions, and the formation of ion pairs is controlled only by electrostatic forces. On these bases, the association constant may be expressed by the Fuoss equation (29) ... 

Koelman and Hoogerbrugge (1993) have developed a particle-based method that combines features from molecular dynamics (MD) and lattice-gas automata (LGA) to simulate the dynamics of hard sphere suspensions. A similar approach has been followed by Ge and Li (1996) who used a pseudo-particle approach to study the hydrodynamics of gas-solid two-phase flow. In both studies, instead of the Navier-Stokes equations, fictitious gas particles were used to represent and model the flow behavior of the interstial fluid while collisional particle-particle interactions were also accounted for. The power of these approaches is given by the fact that both particle-particle interactions (i.e., collisions) and hydrodynamic interactions in the particle assembly are taken into account. Moreover, these modeling approaches do not require the specification of closure laws for the interphase momentum transfer between the particles and the interstitial fluid. Although these types of models cannot yet be applied to macroscopic systems of interest to the chemical engineer they can provide detailed information which can subsequently be used in (continuum) models which are suited for simulation of macroscopic systems. In this context improved rheological models and boundary condition descriptions can be mentioned as examples. 

In continuum percolation (see Section 1.2.1(g)), we suppose that the defects are introduced in a solid sample as randomly placed insulating holes with the shape of a circle (in two dimensions) or a sphere (in three dimensions) and we include the possibility of overlap of the defects (Swiss cheese model). This last possibility gives near Pc an infinite cluster with the the links having different cross-sectional width 6. This property is essentially responsible for the differences between lattice and continuum percolations. 

The generic model for a ionic fluid is the primitive model ( or RPM ) of charged hard spheres in a dielectric continuum with the dielectric constant e. Thus, the potential is just the Coulomb potential between the ions labelled i and k, which is cut-off at the collision diameter [Pg.150]

The potential profile is the least reliable feature of the GC model. Certainly, Monte Carlo calculations in which the ions are represented as charged hard spheres in a dielectric continuum show that the GC potential profile is seriously in error at high electrolyte concentrations. However, it is sometimes used at very low concentrations to obtain an approximate idea of potential variation in the diflhse layer. 

The rate of heterogeneous condensation depends on the exchange of matter and heat between a particle and the continuous phase. The extreme cases of a particle much larger or much smaller than the mean free path of the suspending gas are easy to analyze. In the continuum range (dp ip), diffusion theory can be used to calculate the transport rate. For a single sphere in an infinite medium, the steady-state equation of diffusion in spherical coordinates takes the form... 

One of the simplest realistic molecular models of an ionic solution is the restricted primitive model. This consists of an equimolar mixture of oppositely charged but equal-sized hard spheres in a dielectric continuum. 

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