Simplifying assumptions

The primary assumptions for modeling granularity in the transverse plane of a rotating cylinder are as follows  

The bed material consists of cohesionless particles that possess a relatively high coefficient of restitution. This assumption places important emphasis on the role of interparticle collision on momentum transfer and permits the use of the equations of Lun et al. (1984). 

Particles are spherical, rigid, and slightly inelastic, such as the polyethylene pellets used in the experimental work. 

The bed motion is in the rolling mode and the active layer is considered to be thin relative to the bed depth (the ratio of the active layer depth at mid-chord to the chord length was less than 0.04 in the experiments). The granular temperature in the active layer is assumed constant in the radial direction at each bed surface position. 

The motion is essentially two-dimensional in the transverse plane since the transverse velocity is several orders of magnitude greater than the axial velocity. Also particle flux into the active layer at the right quadrant is assumed to equal the particle flux into the plug flow 

2 Average length of a molecule between cross-links 

Consider a freely jointed chain with n links, each of length /. The length of a 

For a large number of chains q), or one chain considered at many different times, the mean length 

We follow the procedure used, for example, with sinusoidally varying quantities such as alternating current and voltage, where the mean value is zero, and calculate the mean square chain length 

The root mean square chain length is therefore 

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We make the simplifying assumption that both and are functions only of temperature, not of pressure and composition. For a condensable component it follows that at the same tempera-ture, . ... 

Witn the advent of electronic computers, it is no longer necessary to make drastic simplifying assumptions to reduce the... 

Table 3 shows results obtained from a five-component, isothermal flash calculation. In this system there are two condensable components (acetone and benzene) and three noncondensable components (hydrogen, carbon monoxide, and methane). Henry s constants for each of the noncondensables were obtained from Equations (18-22) the simplifying assumption for dilute solutions [Equation (17)] was also used for each of the noncondensables. Activity coefficients for both condensable components were calculated with the UNIQUAC equation. For that calculation, all liquid-phase composition variables are on a solute-free basis the only required binary parameters are those for the acetone-benzene system. While no experimental data are available for comparison, the calculated results are probably reliable because all simplifying assumptions are reasonable the... 

Since we make the simplifying assumption that the partial molar volumes are functions only of temperature, we assume that, for our purposes, pressure has no effect on liquid-liquid equilibria. Therefore, in Equation (23), pressure is not a variable. The activity coefficients depend only on temperature and composition. As for vapor-liquid equilibria, the activity coefficients used here are given by the UNIQUAC equation. Equation (15). ... 

The maximum temperature cross which can be tolerated is normally set by rules of thumb, e.g., FrSQ,75 °. It is important to ensure that Ft > 0.75, since any violation of the simplifying assumptions used in the approach tends to have a particularly significant effect in areas of the Ft chart where slopes are particularly steep. Any uncertainties or inaccuracies in design data also have a more significant effect when slopes are steep. Consequently, to be confident in a design, those parts of the Ft chart where slopes are steep should be avoided, irrespective of Ft 0.75. 

There are different possibilities to address the above set of equations which can be solved provided 2in > 3i, and provided the measured ToF information varies between measurement points. For the purpose of the present work we have taken two simplifying assumptions (a) one virtual source predominates at each measurement point, m and (b) each virtual source predominates at more than one measurement point. Note that assumption (b) ensures the condition 2m > 3i that is necessary to obtain solutions for Equations (2) and (3). These assumptions are justified by considering the defect surface as an acoustic secondary field source. At each measurement point the transducer predominantly receives signals from an... 

This is a third set of equations that essentially allows us to locate a virtual source for every two adjacent measurements. Intuitively this can be simply described as finding the point of intersection of two radii of lengths and Therefore the simplifying assumptions... 

The microscopic contour of a meniscus or a drop is a matter that presents some mathematical problems even with the simplifying assumption of a uniform, rigid solid. Since bulk liquid is present, the system must be in equilibrium with the local vapor pressure so that an equilibrium adsorbed film must also be present. The likely picture for the case of a nonwetting drop on a flat surface is... 

Rate laws have also been observed that correspond to there being two kinds of surface, one adsorbing reactant A and the other reactant B and with the rate proportional to 5a x 5b- For traditional discussions of Langmuir-Hinshelwood rate laws, see Refs. 240-242. Many catalytic systems involve a series of intermediates, and the simplifying assumption of steady-state equilibrium is usually made. See Boudart and co-workers [243-245] for a contemporary discussion of such complexities. 

Reaction (5) proceeds mostly heterogeneously, reaction (6) mostly homogeneously. This mechanism can be integrated with simplifying assumptions to demonstrate the main features of gas-phase explosion kinetics [8]... 

X- ) quantities. In practice, the complexity of the general relationship between and X- means that progress requires the introduction of certain simplifying assumptions. These usually follow from symmetry... 

The standard chemical potentials are approximately tire same if tire surfactant in each aggregate sees nearly tire same interaction witli tire solvent. This simplifying assumption tlien gives... 

With these simplifying assumptions equations (12 13) and (12.14) reduce to... 

A number of simulation methods based on Equation (7.115) have been described. Thess differ in the assumptions that are made about the nature of frictional and random forces A common simplifying assumption is that the collision frequency 7 is independent o time and position. The random force R(f) is often assumed to be uncorrelated with th particle velocities, positions and the forces acting on them, and to obey a Gaussiar distribution with zero mean. The force F, is assumed to be constant over the time step o the integration. 

In the independent particle approximation, the simplifying assumption is made that V i) is an average potential due to a core that consists of the nuclei and all elections other than elechon i... 

If we do not make any simplifying assumptions, we must calculate the matrix... 

Langmuir referred to the possibility that the evaporation-condensation mechanism could also apply to second and higher molecular layers, but the equation he derived for the isotherm was complex and has been little used. By adopting the Langmuir mechanism but introducing a number of simplifying assumptions Brunauer, Emmett and Teller in 1938 were able to arrive at their well known equation for multilayer adsorption, which has enjoyed widespread use ever since. 

Now in principle each layer will have its own values of a, q, and v, and consequently the summation of Equation (2.11) cannot be carried out unless simplifying assumptions are made. Brunauer, Emmett and Teller made three such assumptions (a) that in all layers except the first the heat of adsorption is equal to the molar heat of condensation q, (b) that in all layers except the first the evaporation-condensation conditions are identical, i.e. that... 

Dyna.micPerforma.nce, Most models do not attempt to separate the equiUbrium behavior from the mass-transfer behavior. Rather they treat adsorption as one dynamic process with an overall dynamic response of the adsorbent bed to the feed stream. Although numerical solutions can be attempted for the rigorous partial differential equations, simplifying assumptions are often made to yield more manageable calculating techniques. 

In order to calculate the distribution function must be obtained in terms of local gas properties, electric and magnetic fields, etc, by direct solution of the Boltzmann equation. One such Boltzmann equation exists for each species in the gas, resulting in the need to solve many Boltzmann equations with as many unknowns. This is not possible in practice. Instead, a number of expressions are derived, using different simplifying assumptions and with varying degrees of vaUdity. A more complete discussion can be found in Reference 34. 

The McCabe-Thiele method employs the simplifying assumption that the molal overflows in the stripping and the rectification sections are constant. This assumption reduces the rectifying and stripping operating line equations to ... 

The kinetics of the ethylene hydration reaction have been investigated for a tungstic oxide—siHca gel catalyst, and the energy of activation for the reaction deterrnined to be 125 kJ/mol (- 30 kcal/mol) (106,120). The kinetics over a phosphoric acid-siHca gel catalyst have been examined (121). By making some simplifying assumptions to Taft s mechanism, a rate equation was derived ... 

The physical properties of the liquid, rather than those of the vapor, are used For determining the film coefficient for condensation. Nus-selt [2. Ver. Dt.sch. Ing., 60, 541, 569 (1916)] derived theoretical relationships for predicting the film coefficient of heat transfer for condensation of a pure saturated vapor. A number of simplifying assumptions were used in the derivation. 

The foregoing designs were discussed as ring expansion joints by Kopp and Sayre, Expansion Joints for Heat Exchangers (ASME Misc. Pap., vol. 6, no. 211). All are statically indeterminate but are subjected to analysis by introducing various simplifying assumptions. Some joints in current industrial use are of lighter wall constniction than is indicated by the method of this paper. 

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