Expansion ratio definition

The edges of this dodecahedron sized a - 8.5 cm. When the volume of the rubber balloon at inflation became bigger than the volume of the sphere inscribed in the dodecahedron, the balloon was deformed by the dedecahedron faces and took a shape close to the respective shape of a bubble in a monodisperse dodecahedral foam with a definite expansion ratio. The expansion ratio of the foam was determined by the volume of liquid (surfactant solution or black ink in the presence of sodium dodecylsulphate) poured into the dodecahedron. An electric bulb fixed in the centre of the balloon was used to take pictures of the model of the foam cell obtained. The film shape and the projection of the borders and vertexes on the dodecahedron face are clearly seen in Fig. 1.10. 

By definition at an expansion ratio variable by height, the total foam resistance in Eq. (4.32), appears to be its harmonic mean value. That is why the experimental expansion ratio (Eq. 4.32) at its equilibrium distribution along the height of the cells is compared to the calculated harmonic mean ng value A2Al... 

The initial expansion ratio and dispersity of polyhedral foams are related through the quantitative dependence, given by Eq. (4.9). There at Ap > 103 Pa the content of the liquid phase in the films can be neglected. Thus, the connection of the structure parameters n, a and r can be expressed by the simple relation in Eq. (4.10). It follows from it that under given foaming conditions a definite expansion ratio can be reached by changing the border pressure, foam dispersity and surface tension of the foaming solution. 

Figure 8.4 Dependency of the expansion ratio on the level of reduction, y, for various kinds of electrolytes in polyaniline films at pH = 0. The definition of y is shown by the inset, and y = 0 is taken as the ES state, /q is the length of film at y = 0. BSA is benzene sulfonic acid.

The crucial property of the integrand in Eq. (10.16), which facilitates calculation, is that the denominator admits expansion in the small parameter /i prior to momentum integration. This is true due to the inequality j 2 2 2 which is valid according to the definitions of the functions a and b. In this way, we may easily reproduce the nonrecoil skeleton integral in (9.9), and obtain once again the nonrecoil corrections induced by the radiative insertions in the electron line [32, 33, 34]. This approach admits also an analytic calculation of the radiative-recoil corrections of the first order in the mass ratio. 

Solution It is apparent from the units of b] that solute concentration has been expressed in g/cm3. Dividing this concentration by the density of the unsolvated protein converts the concentration to dry volume fraction units. Since the concentration appears as a reciprocal in the definition of [17], we must multiply bl by p2 to obtain (lAA Mb/r/o) - 1]. For this protein the latter is given by (3.36)(1.34) = 4.50. If the particles were unsolvated, this quantity would equal 2.5 since the molecules are stated to be spherical. Hence the ratio 4.50/2.50 = 1.80 gives the volume expansion factor, which equals [1 + (mhb/m2)(p2/p )]. Therefore (m, tblm2) = 0.80(1.00/ 1.34) = 0.6O. The intrinsic viscosity reveals the solvation of these particles to be 0.60 g HaO per gram of protein. ... 

Equation (54) represents the ratio of real volume of holes and expansion volume to the total free-volume, as distinct from its usual definition as the ratio of free-volume to total volume, the latter being determined by... 

The situation is more complicated if expansion or contraction of a volume element does occur and the volumetric flowrate is not constant throughout the reactor. The ratio VJv, where v is the volume flow into the reactor, no longer gives the true residence time or contact time. However, the ratio VJv may still be quoted but is called the space time and its reciprocal v/V, the space velocity. The space velocity is not in fact a velocity at all it has dimensions of (time) 1 and is therefore really a reactor volume displacement frequency. When a space velocity is quoted in the literature, its definition needs to be examined carefully sometimes a ratio Vi/ V, is used, where V/ is a liquid volume rate of flow of a reactant which is metered as a liquid but subsequently vaporised before feeding to the reactor. 

To summarize, strict e-expansion a priori seems to yield unambiguous results. Closer inspection, however, reveals that in low order calculations considerable ambiguity is hidden in the definition of the physical observables used as variables or chosen to calculate. What is worse, the e-expansion does not incorporate relevant physical ideas predicting the behavior outside the small momentum range or beyond the dilute limit. In particular, it does not give a reasonable form for crossover scaling functions. On the other hand, it can be used to calculate well-defined critical ratios, which are a function of dimensionality only, Even then, however, the precise definition of the ratio matters,... 

The second way of establishing the parameter relations focusses on three points (encompassing two boxes), Xr, t,Xn,Xn+1 obtained from (B.50). If they also obey the box expansion formula, then a definite expression for /J should be obtained from the ratio of the two box lengths,... 

By definition, the compression ratio is r = Vc/ Vol in addition the expansion rati... 

X 10" K" and ap(1000 K) = 10.4 x 10 K". As can be seen in Table 4, these are in much better agreement with the experimental data than are the fluctuation formula results. Particularly striking is the fact that the ratios of both the calculated and the experimental values at the two temperatures are now nearly the same, 1.4 (calc.) vs 1.36 (exp.). (With the fluctuation formula, this ratio is 2.2.) It should also be mentioned that when the coefficient of thermal expansion is obtained from its definition, which involves the volume and not its fluctuation, the convergence is much more rapid than with the fluctuation method. Figure 12 shows that the final volume is reached within a few picoseconds. 

The characteristic velocity is determined by the ratio of the characteristic tangential (Marangoni) stress, 0(PAT/L), which drives this motion to the viscous forces ()(p,uc/d) that derive from this motion. The definition (6 212) also allows us to return to the condition for neglect of buoyancy forces compared with Marangoni forces as a potential source of fluid motion in the thin cavity. To do this, we introduce the thermal expansion coefficient, which we denote as a, so that the characteristic density difference Ap = O(paAT). Then the condition (Apge2t2/puc) 1 can be expressed in the form... 

Note that the nozzle inlet enthalpy, ho, is taken as the actual value, not the stagnation value in this definition. The expansion over the blades will, of course, be associated with a drop in pressure, and we may deduce its value by expressing the enthalpy drops in terms of temperature ratios and then pressure ratios as follows ... 

Despite the apparent success, the above considerations are flawed because one of the possible stereochemieal choices has been overlooked - polyhedral expansion has not been considered could not Ru3(CO)i2 have an icosahedral ligand envelope with looser carbonyl contacts and a larger internal cavity The a priori dismissal of this possibility (which follows from radius ratio considerations) implies the hidden assumption that COs behave like sticky rigid spheres, which can neither interpenetrate nor become detached, i.e. that polyhedral interconversions (between polyhedra with equal edges) are always energetically favored relative to polyhedral expansions of any size. Such a rigid assumption is definitely unjustified when one further considers that ... 

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