Numerical Discretization

Vd satisfying PdVd = Vd, which means that Vd is an approximation of an invariant measure. For an invariant measure, any numerical discretization may be interpreted as a stochastic perturbation of the original problem. 

Discretization of the governing equations. In this step, the exact partial differential equations to be solved are replaced by approximate algebraic equations written in terms of the nodal values of the dependent variables. Among the numerous discretization methods, finite difference, finite volume, and finite element methods are the most common. Tlxe finite difference method estimates spatial derivatives in terms of the nodal values and spacing between nodes. The governing equations are then written in terms of... 

Hounslow etal. (1988), Hounslow (1990a), Hostomsky and Jones (1991), Lister etal. (1995), Hill and Ng (1995) and Kumar and Ramkrishna (1996a,b) present numerical discretization schemes for solution of the population balance and compute correction factors in order to preserve total mass and number whilst Wojcik and Jones (1998a) evaluated various methods. 

The excitation of oscillations with a quasi-natural system frequency and numerous discrete stationary amplitudes, depending only on the initial conditions (i.e. discretization of the processes of absorption by the system of energy, coming from the high-frequency source). A new in principle property is the possibility for excitation of oscillations with the system s natural frequency under the influence of an external high-frequency force on unperturbed linear and conservative linear and non-linear oscillating systems. 

As we formulate and solve specific problems, we will spend more time discussing characteristics and their effects on numerical discretization and solution algorithms. 

This discretization method obeys a conservation property, and therefore is called conservative. With the exception of the first element and the last element, every element face is a part of two elements. The areas of the coincident faces and the forces on them are computed in exactly the same way (except possibly for sign). Note that the sign conventions for the directions of the positive stresses is important in this regard. The force on the left face of some element is equal and opposite to the force on the right face of its leftward neighbor. Therefore, when the net forces are summed across all the elements, there is exact cancellation except for the first and last elements. For this reason no spurious forces can enter the system through the numerical discretization itself. The net force on the system of elements must be the net force caused by the boundary conditions on the left face of the first element and the right face of the last element. 

Textbooks include Fletcher (Computational Techniques for Fluid Dynamics, vol. 1 Fundamental and General Techniques, and vol. 2 Specific Techniques for Different Flow Categories, Springer-Verlag, Berlin, 1988), Hirscft (Numerical Computation of Internal and External Flows, vol. 1 Fundamentals of Numerical Discretization, and vol. 2 Computational Methods for Inviscid and Viscous Flows, Wiley New York, 1988), Peyret and Taylor (Computational Methods for Fluid... 

Fig. 2. Influence of the porosity on the overall Sherwood number derived both analytically (compact line) and numerically (discrete points)...

Numerical discretization of the three-dimensional integral on the r.h.s. of eq. (9) (p and i indices run through abscissas of the radial and angular quadratures, respectively)... 

The parameterization of the particle collision densities was obviously performed emplo3ung elementary concepts from the kinetic theory of gases, thus the derivation of the source term closures at the microscopic level have been followed by some kind of averaging and numerical discretization by a discrete numerical scheme [f6, 92, 118]. 

The more recent coalescence closures that were formulated directly within the macroscopic framework [77, 113, 92, 73] are frequently transformed and expressed in terms of probability densities and used within the average microscopic balance framework without further considerations emplo3dng a discrete numerical discretization scheme. 

Over the years numerous discretization methods for the convection /advection terms have been proposed, some of them are stable for steady-state simulations solely, others were designed for transient simulations solely, but many techniques can be used for both t3q)es of problems. 

In this section, attention is given to the numerical discretization of the finite volume calculation domain and the choice of variable arrangement on the grid. 

This part of the chapter is devoted to a few of the popular numerical discretization schemes used to solve the population balance equation for the (fluid) particle size distribution. In this section we discuss the method of moments, the quadrature method of moments (QMOM), the direct quadrature method of moments (DQMOM), the discrete method, the chzss method, the multi-group method, and the least squares method. 

For the continuous microscopic formulations, the closures and the numerical discretizations are split so an optimal numerical solution method has to be found after the closure laws are derived. The numerical solution methods are in general problem dependent and optimized for particular applications. For the discrete macroscopic formulations, the closures and the numerical discretizations are not split thus the numerical solution method is determining a basis for the closure laws that are derived. The numerical solution method chosen is thus a part of the model closure and cannot be optimized for particular applications. 

Hirsch C (1988) Numerical Computation of Internal and External Flows. Volume I Fundamentals of Numerical Discretization. John Wiley Sons, Chichester... 

As the energy is increased (the wavelength decreased), vibrational transitions occur in addition to the rotational transitions, with different combinations of vibrational-rotational transitions. Each rotational level of the lowest vibrational level can be excited to different rotational levels of the excited vibrational level B in Figure 16.3). In addition, there may be several different excited vibrational levels, each with a number of rotational levels. This leads to numerous discrete transitions. The result is a spectrum of peaks or envelopes of unresolved fine stracture. The wavelengths at which these peaks occur can be related to vibrational modes within the molecule. These occur in the mid- and far-infrared regions. Some typical infrared spectra are shown in Figure 16.4. 

During condensation, the liquid collects in one of two ways, depending on whether it wets the cold surface or not. If the liquid condensate wets the surface, a continuous film will collect, and this is referred to as filmwise condensation. If the liquid does not wet the surface, it will form into numerous discrete droplets, referred to as dropwise condensation. All surface condensers today are designed to operate in the filmwise mode, since long-term dropwise conditions have not been successfully sustained. 

In practice, the friction function y(t) appearing in the STGLE is not periodic but a decaying function. However, one may use it to construct the periodic function y(t r) = x "y(/ — m)6(t — nT)0[(n + 1)t — r] where 0(x) is the unit step function. The continuum limit is obtained when the period t goes to o°. In any numerical discretization of the STGLE care must always be taken not to extend the dynamics beyond the chosen value of the period t, as beyond this time one is following the dynamics of a system which is considerably different from the continuum STGLE. 

The genesis of the terms plate height" and "number of thcoreiical plates is a pioneering theoretical study of Marlin and Synge in which they treated a chromatographic column as if it were similar to a distillation column made up of numerous discrete but contiguous narrow layers called iheoretical plates. At each plate, equilibration of the solute between the mobile and stationary phases was assumed to take place. Movement of the solute down the column was then treated as a stepwise transfer of equilibrated mobile phase from one plate to I he next. 

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