Internal coordinate velocity

External coordinate velocity (m/s) r-th moment average velocity in SQMOM Internal coordinate velocity Particle drift velocity (m/s)... 

The vibrational kinetic energy can also be expressed in terms of the velocities in internal coordinates by taking the partial derivatives of Eq. (49). Thus, S = GP and, as G is square and nonsingular, P G lS and its transpose... 

The resulting conditional average is implicitly a function (A)p = (A)p( ) of the soft coordinates. Here and in what follows, )5 is used to indicate a conditional average with respect to fluctuations in the state of the surrounding solvent, for fixed values of the system s internal coordinates q and momenta p. This average over solvent degrees of freedom is unnecessary in Eq. (2.79) if A = A q,p) is a quantity (such as a bead velocity) that depends only on the system s coordinates and momenta, but is necessary if A is a quantity (such as the total force on a bead) that depends explicitly on the forces exerted on the system by surrounding solvent molecules. 

In this equation the angular velocity vector co is referred to the center of mass frame coordinate system and the s are the time derivatives of the internal coordinates. Moreover, the kinetic energy matrix coefficients may be expressed as... 

Assumptions Same as microdistributed balance. Only internal coordinate particle velocities are considered. 

Bji matrix elements relating Cartesian coordinates Xj to the internal coordinate rj c velocity of light... 

Defining a single distribution function, p(x, r, c, t)dxdrdc, as the probable number of particles with internal coordinates in the range dx. about x, located in the spatial range dr about the position r, with a velocity range dc about c. 

The single distribution function /(x, r, t)dxdr thus denotes the probable number of particles within the internal coordinate space in the range dx about x, in the external (spatial) range dr about r at time t. is the mean velocity of all particles of properties x at a location r at time t. The velocity independent birth and death terms are defined by ... 

The growth terms need some further attention. Considering that for fluid particles the mass change rate (or an imaginary velocity in the internal coordinates) of the particle, is negative for the case of condensation or particle dissolution, and is positive in the case of evaporation or mass diffusion into the particle the distribution function was discretized using an upwinding approach. 

The disperse phase is constituted by discrete elements. One of the main assumptions of our analysis is that the characteristic length scales of the elements are smaller than the characteristic length scale of the variation of properties of interest (i.e. chemical species concentration, temperature, continuous phase velocities). If this hypothesis holds, the particulate system can be described by a continuum or mean-field theory. Each element of the disperse phase is generally identified by a number of properties known as coordinates. Two elements are identical if they have identical values for their coordinates, otherwise elements are indistinguishable. Usually coordinates are classified as internal and external. External coordinates are spatial coordinates in fact, the position of the elements in physical space is not an internal property of the elements. Internal coordinates refer to more intimate properties of the elements such as their momenta (or velocities), their enthalpy... 

A special case of considerable interest occurs when the internal-coordinate vector is the particle-velocity vector, which we will denote by the phase-space variable v. In fact, particle velocity is a special internal coordinate since it is related to particle position (i.e. external coordinates) through Newton s law, and therefore a special treatment is necessary. We will come back to this aspect later, but for the time being let us imagine that otherwise identical particles are moving with velocities that may be different from particle to particle (and different from the surrounding fluid velocity). It is therefore possible to define a velocity-based NDF nv(t, x, v) that is parameterized by the velocity components V = (vi, V2, V3). In order to obtain the total number concentration (i.e. number of particles per unit volume) it is sufficient to integrate over all possible values of particle velocity Oy ... 

Note that, although we treat x and f in the same manner, they are in fact different types of vectors. The vectors x and v are the standard vectors for position and velocity used in continuum mechanics. The internal-coordinate vector on the other hand, is a generalized vector of length N in the sense of linear algebra. 

Equation (2.14) must be coupled with initial conditions given for the starting time and with boundaries conditions in physical space O and in phase space O. Analytical solutions to Eq. (2.14) are available for a few special cases and only under conditions specified by some very simple hypotheses. However, numerical methods can be used to solve this equation and will be presented in Chapters 7 and 8. The numerical solution of Eq. (2.14) provides knowledge of the NDE for each time instant and at every physical point in the computational domain, as well as at every point in phase space. As has already been mentioned, sometimes the population of particles is described by just one internal coordinate, for example particle length (i.e. f = L), and the PBE is said to be univariate. When two internal coordinates are needed, for example particle volume and surface area (i.e. = (v, a)), the PBE is said to be bivariate. More generally, higher-dimensional cases are referred to as multivariate PBEs. Another important case occurs when part of the internal-coordinate vector is equal to the particle-velocity vector (i.e. when the particles are characterized not by a unique velocity field but by their own velocity distribution). In that case, the PBE becomes the GPBE, as described next. 

By using a very similar approach to the one outlined above for the PBE, it is possible to derive a GPBE for an NDF that includes particle velocity as an internal variable. We will denote this general NDF as n(t,x,, ) (i.e. without subscripts on n). The simplest GPBE (i.e. velocity without other internal coordinates) is known as the Boltzmann kinetic equation and was first derived in the context of gas theory (Chapman Cowling, 1961). The final form of the GPBE is... 

When developing models for polydisperse multiphase flows, it is often useful to resort to conditioning on particle size. For example, in gas-solid flows the momentum-exchange terms between the gas phase and a solid particle will depend on the particle size. Thus, the conditional particle velocity given that the particle has internal-coordinate vector will... 

But mi is usually not zero when the internal coordinate represents particle mass, surface area, size, etc. In these cases the PD algorithm can be safely used. The case of null mi occurs more often when the internal coordinate is a particle velocity that, ranging from negative to positive real values, can result in distributions with zero mean velocity. Another frequent case in which the mean is null is when central moments (moments translated with respect to the mean of the distribution) are used to build the quadrature approximation. These cases will be discussed later on, when describing the algorithms for building multivariate quadratures. 

Since for velocity distributions (in three spatial dimensions) three internal coordinates (M = 3) are needed, we discuss in the following example the construction of a quadrature approximation for a trivariate tensor-product QMOM. 

We could use the particle momentum instead of the velocity. However, since the particle mass can be one of the internal coordinates, there is no loss of generality by using die velocity. 

Recall that the random variables in this equation are the complete set of particle positions, velocities, and internal coordinates. The state-space variables x, v, and are fixed. 

Note that left-hand side of this expression is, in fact, a continuity equation for which states that the multi-particle joint PDF is constant along trajectories in phase space. The term on the right-hand side of Fq. (4.32) has a contribution due to the Alp-particle collision operator, which generates discontinuous changes in particle velocities Up" and internal coordinates p", and to particle nucleation or evaporation. The first term on the left-hand side is accumulation of The remaining terms on the left-hand side represent... 

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