The Two Basic Models

In this section we want to illustrate the two basic ways in which diffusion can be described. To do this, we first imagine two large bulbs connected by a long thin capillary (Fig. 1.1-1). The bulbs are at constant temperature and pressure and are of equal volumes. However, one bulb contains carbon dioxide, and the other is filled with nitrogen. 

To find how fast these two gases will mix, we measure the concentration of carbon dioxide in the bulb that initially contains nitrogen. We make these measurements when only a trace of carbon dioxide has been transferred, and we find that the concentration of carbon dioxide varies linearly with time. From this, we know the amount transferred per unit time. 

We want to analyze this amount transferred to determine physical properties that will be applicable not only to this experiment but also in other experiments. To do this, we first define the flux  

In other words, if we double the cross-sectional area, we expect the amount transported to double. Defining the flux in this way is a first step in removing the influences of our 

The proportionality constant k is called a mass transfer coefficient. Its introduction signals one of the two basic models of diffusion. Alternatively, we can recognize 

---

Ionic and covalent bonding are two extreme models of the chemical bond. Most actual bonds lie somewhere between purely ionic and purely covalent. When we describe bonds between nonmetals, covalent bonding is a good model. When a metal and nonmetal are present in a simple compound, ionic bonding is a good model. However, the bonds in many compounds seem to have properties between the two extreme models of bonding. Can we describe these bonds more accurately by improving the two basic models ... 

The two basic models, as described previously (Section 3.4), have also been... 

Experimentally, it has been observed that many substances are transported across plasma membranes by more complicated mechanisms. Although no energy is expended by the cell and the net flux is still determined by the electrochemical potential, some substances are transported at a rate faster than predicted by their permeability coefficients. The transport of these substances is characterized by a saturable kinetic mechanism the rate of transport is not linearly proportional to the concentration gradient. A facilitated mechanism has been proposed for these systems. Substances interact and bind with cellular proteins, which facilitate transport across the membrane by forming a channel or carrier. The two basic models of facilitated diffusion, a charmel or a carrier, can be experimentally distinguished (1,2). 

The deformation behaviors have been interpreted in terms of the two basic models(11), (i) the deformed two-phase model in which the interparticle distances and the particles, initially giving rise to Debye s hard-sphere type scattering(1U), are affinely deformed under constant volumes (designated as "deformed hard-particles") and (ii) the deformed core-shell particle model in which a spherical core-shell particle is affinely deformed under constant volume into an ellipsoidal core-shell particle. 

FIGURE 17,1 1 Schematic representation of the two basic models used in bubble and fonm separalion technology. 

In all the preceding chapters the focus of attention was on the development of fundamental concepts and suitable models for describing the neutron population in elementary reactor geometries. The two basic models so far examined, the diffusion model and the Fermi age, were applied to bare reactors. In the present chapter our purpose is to show how these methods may be extended, with suitable modifications, to... 

The general expectations embodied in Equations 7.12, 7.16, and 7.19 are borne out to be valid as shown by experiments in dilute solutions of uncharged polymers. Depending on the experimental conditions, the value of the size exponent changes and this change is directly manifest in D, rj, and t in terms of their dependencies on the molecular weight of the polymer and solvent conditions. In order to obtain the numerical prefactors for the above scaling laws and to understand the internal dynamics of the polymer molecules, it is necessary to build polymer models that explicitly account for the chain connectivity. The two basic models of polymer dynamics are the Rouse and Zimm models (Rouse 1953, Kirkwood and Riseman 1948, Zimm 1956), which are discussed next. 

The experimental studies of a large number of low-temperature solid-phase reactions undertaken by many groups in 70s and 80s have confirmed the two basic consequences of the Goldanskii model, the existence of the low-temperature limit and the cross-over temperature. The aforementioned difference between quantum-chemical and classical reactions has also been established, namely, the values of k turned out to vary over many orders of magnitude even for reactions with similar values of Vq and hence with similar Arrhenius dependence. For illustration, fig. 1 presents a number of typical experimental examples of k T) dependence. 

Although the two quantum models are defined somewhat dilferently, both QCA-I and QCA-II start with the same basic premise, endowing the classical system with two characteristically quantum features. They both (1) replace each site variable with a quantum state containing all fc classical site-color possibilities, and (2) introduce a quantum transition operator "I , defining mixed color —> mixed a)lor transitions. Only in QCA-II, however, is also unitary see discussion below. 

What the three-step model really points out is that it is theoretically correct to carry out basic combustion calculations for a PBC system based on the mass flow and stoichiometry of the conversion gas from the conversion system and not based on the mass flow of solid fuel entering the conversion system. The two-step model approach applied on a PBC system, which is equivalent to assuming that the conversion efficiency is 100 %, is a functional engineering approach, because the conversion efficiency is in many cases very close to unity. However, there are cases where the two-step model approach results in a physical conflict, for example the mass flows in PBC sysfem of batch type cannot be theoretically analysed with a two-step model. 

Definition of site fractions. The multiple sublattice model is an extension of earlier treatments of the two-sublattice models of Hillert and Steffansson (1970), Harvig (1971) and Hillert and Waldenstrom (1977). It allows for the use of many sublattices and concentration dependent interaction terms on these sublattices. To woiic with sublattice models it is first necessary to define what are known as site fractions, y. These are basically the fiactional site occupation of each of the components on the various sublattices where... 

There are two basic models of CleanSoil machines. The standard 5-ton/hr unit is the Model CSI -200. The model CSI-1200 M has a capacity of up to 40 tons/hr. The vendor states that the CSI-200 typically costs 165,000, and the CSI-1200 costs approximately 430,000. According to the vendor, refurbished machines sell for under 100,000 (D18216L, p. 2 D22051Z, p. 1 D220520). 

The models developed for the low salt concentrations (0.0 and 0.1 M NaCl) were very different from those for the high salt concentration. The two basic Independent variables used were salt level and the absolute value of the pH minus 4.0. To avoid Implying that the behavior of each functional property on either side of pH 4.0 Is the mirror Image of Its behavior on the other side of pH 4.0, two variables were formed from the absolute value of pH minus 4.0. These were the absolute values of pH minus 4.0 for each pH above 4.0 (values of this variable for observations In which the pH was less than 4.0 were set equal to zero) and the absolute values of pH minus 4.0 for each pH below 4.0 (values of this variable for observations In which pH was greater than 4.0 were set equal to zero). As shown In the table, these basic variables were used In the estimated models along with their squares, their Interactions with salt level (0.0 and 0.1 M NaCl), and salt level to form the Independent variables In the final equations used. Other variables, such as cubic powers of pH minus 4.0, were tried and discarded due to lack of statistical significance In arriving at the final models. 

In order to study theoretically defect aggregation, several methods of physical and chemical kinetics were developed in recent years. Irrespective of the particular method used, the two basic approaches - a continuous and discrete-lattice ones - are used. In the former model intrinsic defect volume is ignored and thus a number of similar defects in any volume element is unlimited. In its turn, in the latter model any lattice site could be occupied by no more than a single particle (v or i) [15]. 

Based on the two-phase model of fluidization, the following basic relations hold ... 

Pair transfer interaction between the states of an electron system components can cause the gauge symmetry breaking realized in superconductivity. This circumstance forms the basis of the two-band model of superconductivity known already during a considerable time [1,2], The basic advantage of such approaches consists in the possibility to reach pairing by a repulsive interband interaction which operates in a considerable volume of the momentum space. An electronic energy scale is... 

The two physical models that can be used to describe an AOTF device depend as usual on optics, and on wave and particle descriptions of the interactions involved. The basic physical layout of the device will be described, and then its principle of operation will be discussed. 

---