Volume of spheres

Total volume of spheres f 16/3)Trr3 16tt Total volume of cube 83 2r3 3 X 83 2... 

Initial temperature, T0 = 20 °C. Specific heat, cp = 460J/kgK Density, p = 7850 kg/m3 Radius, R = 1 cm Volume of sphere = (4/3)7tR3 Suface area of sphere = 4nR2... 

Here, the subscript Total Solv. stands for all the solvent (of volume VTo,ai = NV, i.e., N times the solvent cells of volume V each). Of course, the volume fraction remains unchanged in Equation (36) since both the volume of spheres and the volume of solvent cells are multiplied by the amount N. Thus, one has from Equation (36)... 

A coke-oven pitch fraction which had been partly carbonized and contained about 30% by volume of spheres was reduced with lithium in ethylene-... 

Rate of reaction per unit volume of reactor Rates of reaction of D and G materials Rate of substrate reaction Local rate of reaction of substrate per unit volume of sphere Rate of biomass reaction Rate of reaction per unit area of biocatalyst Maximal rate of reaction based on unit area of biocatalyst... 

This equation has also a simple physical meaning. Actually jtR3 is the volume of sphere around the acceptor such that if a donor-particle gets into it, this will result in its decay during time t with a probability practically equal to unity. The term exp ( —f jtRfN) represents the probability for the donor particle not to get into this sphere around an arbitrary acceptor particle, i.e. the probability that a particle of the donor will not decay by time t as a result of electron tunneling. 

Volume of spheres added Total solid volume of spheres 0.524d 0.037d 0.012d8 0.021d3 0.0064d3... 

Consider now the mean oxygen density conditional on a specific alkyl configuration. Since that conditional mean oxygen density is less traditionally analyzed than the density profile shown in Fig. 1.9, we exploit another characterization tool, the proximal radial distribution (Ashbaugh and Paulaitis, 2001). Consider the volume that is the union of the volumes of spheres of radius r centered on each carbon atom see Fig. 1.10. The surface of that volume that is closer to atom i than to any other carbon atom has area fl, (r) with 0 < ff, (r) < 4tt. The proximal radial distribution function ( ) is defined as... 

This model is frequently used considering mono-atomic uncharged molecules. However, this model gives a very crude representation of the actual physics (e.g., repulsive forces and volume of sphere), since molecules in fact are complicated electronic structures, and can by no means resemble rigid spheres. 

This program can easily create geometric models of hypothetical blends. For example, consider two pools of spheres available for placement in a three-dimensional space. Each pool can represent a distinct size distribution, and the pools of spheres are kept separate in files available to the program for sphere placement. Figure 3 shows the results of three runs of the program. In the first run, 6.0% discrete-phase volume of spheres from one pool (2.0 jxm diameter, monodispersed) and zero-discrete-phase volume from the other pool (3.0 xm diameter, monodispersed) were selected. The second model shows the placement of only the larger spheres at the 6.0% phase-volume level. Placement of... 

Fkj. 10-8. Critical mass and volume of spheres and circular cylinders containing solutions of and surrounded by water. Straight lines show mass of U ... 

Fj = the volume of sphere G and capillary tube K (measured from the junction at R), Fj = the volume of the compressed gas in capillary tube K. 

Volume of sphere and required amount of mercury in commercial McLeod gauges... 

Table 9.8 shows the values of V and of the radius of the sphere of action calculated at different oxygen concentration. We can notice that the values differ from one concentration to another. At the highest oxygen concentrations, the values are smaller. This clearly indicates that upon increasing oxygen concentration, fluorescence lifetime decreases and the studied protein area is more and more small. This result is in complete opposition to that obtained when the asymptote was drawn at low oxygen concentration. In this case, at all oxygen concentrations, one can determine very close volume of sphere of action. 

All of the above considerations were concerned with the radius of gyration of unbranched chain molecules at infinite dilution. With increasing concentration the coils fill the available space more and more. The loose coils tend to become compressed above a certain critical concentration. This critical concentration can be roughly approximated on the basis of hexagonal close packing (about 75% of the total volume) of spheres of radius r ... 

The volume inerement AV of an atom under eonsideration is ealeulated as volume of sphere of the atom minus volumes of spherieal segments, whieh are eut off on this sphere by the adjaeent eovalently-bound atoms ... 

Figure 4.1.9. Simulated impedance and modulus spectra for a two-phase microstructure, based on the effective medium model. Values of the input parameters are given in Table 4.1.1. (a, b) Spectra for a matrix of phase 1 containing 25% by volume of spheres of phase 2. Resolution is achieved in the modulus spectrum (b) but not the impedance spectrum (a), (c, d) Spectra for a spherical grain of phase 2 surrounded by a grain boundary shell of phase 1. The ratio of shell thickness to sphere radius is 10" Resolution is achieved in the impedance spectrum (c) but not the modulus spectrum (d).

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