Number balance

A dynamic ordinary differential equation was written for the number concentration of particles in the reactor. In the development of EPM, we have assumed that the size dependence of the coagulation rate coefficients can be ignored above a certain maximum size, which should be chosen sufficiently large so as not to affect the final result. If the particle size distribution is desired, the particle number balance would have to be a partial differential equation in volume and time as shown by other investigators ( ). 

Figure 34. Control volume for performing a number balance on granules. (From Litster and Ennis, 1994.)...

In some areas, e.g. aerosol physics and crystallisation, population balance models are used in situations when a number balance equation is required as well as conventional mass and energy balances. Randolph and Larson review this theory as it applies specifically to particulate systems [15], whilst Froment and Bischoff [16] present population balance equations in the context of an extension of classical RTD theory. 

Keeping an eye on electrons by using oxidation numbers Balancing redox reactions in the presence of acid Balancing redox reactions in the presence of base... 

This equation shows that transmutes to the two elements written to the right of the arrow. When this transmutation happens, energy is released, partly in the form of gamma radiation and partly in the form of kinetic energy in the alpha particle (2He) and the thorium atom. In this and all other nuclear equations, the mass numbers balance (238 = 234 + 4) and the atomic numbers also balance (92 = 90 + 2). 

A chemical equation has reactants on the left side and products on the right side. The reactants and products are separated by a yields sign (—>). Numbers called coefficients are placed in front of the reactant and product symbols these numbers balance the equation. What is being balanced in a chemical equation Incorporate the law of conservation of matter into your answer. 

Again, we assume that the spatial variation of H+ is described by Eq. (106). A number balance on the functional groups gives... 

Note that if the atomic number of the beta particle is considered —1 and its atomic mass number 0, both atomic numbers and atomic mass numbers balance in the above equation, as is the case with equation (1). The net effect of the transformation is the conversion of a neutron in the nucleus to a proton and an electron, the latter being ejected Although the net reaction in such cases is generally clear, the path by which such a conversion occurs still causes consternation among theoretical physicists. 

Notice that the sum of the mass numbers on the right (222 + 4) equals the mass on the left (226). As well, the atomic numbers balance (88 = 86 + 2). Thus, this nuclear equation is balanced. 

Mass numbers balance 90 = 90 + 0 Atomic numbers balance 38 = 39 + (-1)... 

There are two ways to solve this problem. The first is the use of a classical unsaturation number that can be obtained from the empirical formula, as one is taught in freshman chemistry courses. Comparison of the atomic composition of starting materials and products yields the unsaturation number balance that one needs to assess oxidation level changes as a whole. Frequently, however, it is also useful to learn the way by which parts of the molecule undergo oxidation level changes in order to concentrate one s attention on that particular portion of the compound. This leads us to discuss the second way of solving the oxidation level problem. 

The model derivation given above using the Liouville theorem is in many ways equivalent to the Lagrangian balance formulation [83]. Of course, a consistent Eulerian balance formulation would give the same result, but includes some more manipulations of the terms in the number balance. However, the Eulerian formulation is of special interest as we have adopted this framework in the preceding discussion of the governing equations of classical fluid dynamics, chap 1. 

In this section the population balance modeling approach established by Randolph [95], Randolph and Larson [96], Himmelblau and Bischoff [35], and Ramkrishna [93, 94] is outlined. The population balance model is considered a concept for describing the evolution of populations of countable entities like bubble, drops and particles. In particular, in multiphase reactive flow the dispersed phase is treated as a population of particles distributed not only in physical space (i.e., in the ambient continuous phase) but also in an abstract property space [37, 95]. In the terminology of Hulburt and Katz [37], one refers to the spatial coordinates as external coordinates and the property coordinates as internal coordinates. The joint space of internal and external coordinates is referred to as the particle phase space. In this case the quantity of basic interest is a density function like the average number of particles per unit volume of the particle state space. The population balance may thus be considered an equation for the number density and regarded as a number balance for particles of a particular state. 

This approach makes the velocities in the population balance different from the mass- or phase weighted average velocities obtained solving the two-fluid model. This discrepancy is an argument for the formulation of a mass density population balance instead of a number balance to achieve a consistently integrated population balance within the two-fluid modeling framework. 

The PBE is a simple continuity statement written in terms of the NDE. It can be derived as a balance for particles in some fixed subregion of phase and physical space (Ramkrishna, 2000). Let us consider a finite control volume in physical space O and in phase space with boundaries defined as dO. and dO., respectively. In the PBE, the advection velocity V is assumed to be known (e.g. equal to the local fluid velocity in the continuous phase or directly derivable from this variable). The particle-number-balance equation can be written as... 

When a nucleus of uranium-238 undergoes alpha decay, a nucleus of thorium-234 and a nucleus of helium-4 are produced. The atomic numbers balance on the two sides of the equation, as do the mass numbers. 

Is this equation balanced Check to see whether the sum of the mass numbers on the right side of the equation is 238, which is the mass number of the uranium isotope on the left side. It is, because the mass number of the thorium atom is 234 and that of the alpha particle is 4. Also make sure that the sums of the atomic numbers on both sides of the equation are the same. Both sums equal 92, so the atomic number balances, and the nuclear equation as a whole is balanced. Note that a balanced nuclear equation differs from balanced chemical equations because the kinds of atoms involved need not remain the same. 

Equation (145) is the number balance (accounting) on cells of group t Eq. (147) is the number balance on cells of group 0. Equation (146) is a material balance on the /th chemical species in the medium Eq. (132) is its steady-state form. In addition a balance on biomass yields the generalization of Eq. (131) ... 

The boundary condition which solutions of Eq. (170) must satisfy could be obtained by making a number balance on cells of all sizes present. However, this can be reduced to the boundary condition given, Eq. (172). This is done by integrating Eq. (170) over all m and comparing the result with the number balance on cells of all sizes. In this process, Eq. (167b) must be used. 

The change of number concentration of drops is described by the drop-number-balance equation... 

Determining oxidation numbers Balancing redox reactions Purifying elements... 

The mathematical description of a crystal size distribution and of its change in space and time makes use of the conservative character of the number of particles in space and state (i.e., particle size L). In the respective number balance, the particle size distribution is represented by the number density, see Hulburt and Katz (1964) and Randolph and Larson (1988),... 

Base approach on mass and energy balances, population or number balances. Follow the population density of number versus size. Must know the type of crystals and the mode of operation. 

Mass balance Heat balance Energy consumption Particle number balance... 

Development of a system of ganglia number balance equations governing the dynamics of oil ganglia populations. 

Mere inspection of these results shows that deviation from linearity actually occurs, at least in certain cases. As discussed earlier, such deviations are due to elongation. They can be treated quantitatively as follows. Consider the conceptual slug experiment. Simple v-ganglia number balances give... 

Nuclear reactions can be represented by balanced equations in which the focus is on mass number and atomic number balance on each side. Symbols are used in nuclear equations that make it convenient to balance mass and atomic numbers. 

As before, the preceding equation must be supplemented with initial and boundary conditions. Equation (2.7.3) continues to serve as the boundary condition. In view of the total number balance... 

From the population balance (2.11.22), and the boundary conditions, show that the total number balance is given by... 

Ramkrishna (35) provided a brief explanation on the population balance equation The population balance equation is an equation in the foregoing number density and may be regarded as representing a number balance on particles of a particular state. The equation is often coupled with conservation equation for entities in the particles environmental (or continuous) phase. ... 

---