Mass integration definition

In many applications, the particles will be composed of multiple chemical species. In such cases, it is necessary to introduce a vector of internal coordinates p whose components are the mass of each chemical species. Obviously, the sum of these internal coordinates is equal to the particle mass. By definition, if pa, is the mass of component a, then integration over phase space leads to a component disperse-phase mass density ... 

As was pointed out earlier. Equation (1.6) allows us to find the attraction field everywhere, but it requires a volume integration, that in general is a rather cumbersome procedure. Fortunately, in many cases the calculation of the field g(p) can be greatly simplified. First, consider an elementary mass with density 6 q), located in the volume AV. Now let us start to increase the density and decrease the volume in such a way that the mass remains the same. By definition, these changes do not make a noticeable influence on the field because the observation point p is far away. In the limit, when... 

By definition, any plane 0 — constant is a plane of symmetry. In other words, there are always two elementary masses, which are equal to each other, and located at opposite sides of this plane but at the same distance. As is seen from Fig. 1.5d, the sum of 0-components, caused by both masses is equal to zero. Representing the total mass as a sum of such pairs we conclude that the 0-component, gg, due to the spherical mass is absent at every point outside and inside the body. In the same manner we can prove that — 0. Of course, volume integration, Equation (1.6), can prove this fact, but this procedure is much more complicated. Thus, the attraction field has only a radial component, g, and the field is directed toward the origin, 0. In order to determine this component we will proceed from Equation (1.26) and consider a spherical surface with radius R, as is shown in Fig. 1.5c. Inasmuch as dS — dSiR and the scalar component g is constant at points of the spherical surface, we have for the flux ... 

By definition of the center of mass the last two integrals are zero and we obtain... 

Now the function displays the number fraction of molecules with a certain molecular mass. Its integral is 1 by definition. Nevertheless, we still call it the number molecular weight distribution because the factor /N (A/) dM is nothing but a constant. 

As shown above in (6.162), the Lagrangian fluid-particle PDF can be related to the Eulerian velocity, composition PDF by integrating over all initial conditions. As shown below in (6.168), for the Lagrangian notional-particle PDF, the same transformation introduces a weighting factor which involves the PDF of the initial positions y) and the PDF of the current position /x.(x t). If we let V denote a closed volume containing a fixed mass of fluid, then, by definition, x, y e V. The first condition needed to reproduce the Eulerian PDF is that the initial locations be uniform ... 

Vacuum systems are integral parts of any mass spectrometer, but vacuum technology definitely is a field of its own. [251-255] Thus, the discussion of mass spectrometer vacuum systems will be restricted to the very basics. 

If now the definitions of the average gas volume-fraction and of the mass-flow rates, Wl and Wq, are written, and Eq. (65) and (66) are substituted in them together with the definition of the quality, x = + Wo), then integration gives... 

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. 

As stated in Eq. 4.156, it might appear that a the solution (i.e., u(r)) would exist for any value of the parameter Re/. However, the velocity profile must be constrained to require that the net mass flow rate is consistent with m = pU Ac, where U is the mean velocity used in the Reynolds number definition. Based on the integral-constraint relationship,... 

This expression states that the product of the mean mass fraction and the overall mass flow rate must equal the integral over the channel width of the local mass flow rate of species ft. An analogous definition for the energy flow was used to define a mean temperature in the Graetz problem (Section 4.10). In nondimensional terms,... 

The RDF between the center-of-mass of the solute and the solvent molecules, Gcm-cm(t) is another natural possibility of describing the solvation shells around the /3-carotene. In figure 4a, the RDF between the center-of-mass of /3-carotene and acetone molecules is shown as an example of the liquid structure. This Gcm-cm t) presents a clear definition of four peaks that characterize the solvation shells around the center-of-mass of the /3-carotene. The number of solvent molecules in each shell was obtained by integrating the peaks. In the case presented in figure 4a, 7 acetone molecules were found in the first shell (integrating until 6.35 A), 30 in the second shell (from 6.35 to 10.65 A), 46 in the third shell (from 10.65 to 13.85 A) and finally 108 in the fourth shell (from 13.85 to 18A). Figure 4b will be discussed soon below. 

Energy of Zero-Point-Motion is calculated for each nucleus rather than estimated or ignored. (3) Elaborate shape definitions are replaced by a matching procedure where the fragment interaction has the correct asymptotic form. (4) Microscopically calculated mass paramater functions are employed in two-dimensional action integrals. Mass asymmetry as well as charge asymmetry are fully taken into account. 

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