General continuity equation

Boris, J. P. 1976. Flux-Corrected Transport modules for solving generalized continuity equations. NRL Memorandum report 3237. Naval Research Laboratory, Washington, D.C. 

Besides the resuspension of particles, the perfect sink model also neglects the effect of deposited particles on incoming particles. To overcome these limitations, recent models [72, 97-99] assume that particles accumulate within a thin adsorption layer adjacent to the collector surface, and replace the perfect sink conditions with the boundary condition that particles cannot penetrate the collector. General continuity equations are formulated both for the mobile phase and for the immobilized particles in which the immobilization reaction term is decomposed in an accumulation and a removal term, respectively. Through such equations, one can keep track of the particles which arrive at the primary minimum distance and account for their normal and tangential motion. These equations were solved both approximately, and by numerical integration of the governing non-stationary transport equations. 

For example, the general continuity equation for groundwater (Equation 1.12 5AM/5t = 5np 5t) written in terms of material derivatives in a deforming coordinate system following the motion of the solids, becomes (e.g. Palciauskas and Domenico, 1989 Shi and Wang, 1986)... 

In this paragraph we consider the translational terms in the context of a generalized continuity equation [100, 83, 85],... 

Averaging the pore scale transport process over the REV and assigning the average properties to the centroid of the REV results in continuous functions in space of the hydrodynamic properties and state variables. As for the flow equation (1), differential calculus can be applied to establish mass and momentum balance equations for infinitesimal small soil volume and time increments. For the case of inert solute transport in a macroscopic homogeneous soil, the general continuity equation applies ... 

A general continuity equation (mass balance) for a component in a layer, where diffusion and chemical reactions take place simultaneously, is written for an infinitesimal volume element... 

By letting Ax, Ay, and Az approach zero we have shrunk the system to a single point. Thus Eq. 3.36 is the mass balance for any point in space it is often called the general continuity equation Equation 3.37 is the mass balance for any point in space which contains a constant-derisity fluid. 

Technically, Liouville s equation refers to the continuity equation in the setting of a volume preserving flow. Here we use the term liouviUian to refer to the operator whose action on a density gives the right hand side of the general continuity equation. 

Rietema s theory does not take into account the radial fluid flow, it neglects any effects of inertia, it takes no account of hindered settling at higher concentrations and it assumes any influence of turbulence to be negligible. A more recent version of the residence-time theory, the so-called bulk model due to HoUand-Batt °, does take into account the radial fluid flow. He simply used the hold-up time of the liquid in the cyclone (flow rate per cyclone volume) as the residence time, average radial fluid velocity (flow rate per wall area of the cyclone) and a general continuity equation for two-dimensional flow to derive an expression for the cut size. 

Equation 8.3) is derived from the general continuity equation, into which the densities of components fluxes at interdiffusion (Equation 8.2) are substituted. 

The form of the general continuity equations is usually too complex to be conveniently solved for practical application for reactor design or simulation. If one or more terms are dropped from (7.3.1.1-6) and/or integral averages over the spatial directions are considered, the continuity equation for each component reduces to that of an ideal, basic reactor type, as outlined in Section 7.2. In these cases it is often easier to apply (7.2-1) directly to a volume element of the reactor. This will be done in the next chapters, dealing with basic or specific reactor types, but in the present chapter, it will be shown how the simphfied equations can be obtained from the fundamental ones. 

The mass of the cell body does not change thus, the general continuity equation (Equation 12.36) can be ignored. The energy equation relates the heat transferred by convecfion to the anode, cathode, and environment the heat generated during the chemical reaction and the electrical power generated. This can be characterized as... 

For a general dimension d, the cluster size distribution fiinction n(R, x) is defined such that n(R, x)dR equals the number of clusters per unit volume with a radius between andi + dR. Assuming no nucleation of new clusters and no coalescence, n(R, x) satisfies a continuity equation... 

In Equation (4.12) the discretization of velocity and pressure is based on different shape functions (i.e. NjJ = l,n and Mil= l,m where, in general, mweight function used in the continuity equation is selected as -Mi to retain the symmetry of the discretized equations. After application of Green s theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and the pressure terms (to maintain the consistency of the formulation) and algebraic manipulations the working equations of the U-V-P scheme are obtained as... 

Dimensional Analysis. Dimensional analysis can be helpful in analyzing reactor performance and developing scale-up criteria. Seven dimensionless groups used in generalized rate equations for continuous flow reaction systems are Hsted in Table 4. Other dimensionless groups apply in specific situations (58—61). Compromising assumptions are often necessary, and their vaHdation must be estabHshed experimentally or by analogy to previously studied systems. 

A drawback of the Lagrangean artificial-viscosity method is that, if sufficient artificial viscosity is added to produce an oscillation-free distribution, the solution becomes fairly inaccurate because wave amplitudes are damped, and sharp discontinuities are smeared over an increasing number of grid points during computation. To overcome these deficiencies a variety of new methods have been developed since 1970. Flux-corrected transport (FCT) is a popular exponent in this area of development in computational fluid dynamics. FCT is generally applicable to finite difference schemes to solve continuity equations, and, according to Boris and Book (1976), its principles may be represented as follows. 

A generalized Darcy equation and equation of continuity for each fluid phase is used to describe the flow of multiple immiscible fluid phases ... 

As an alternative to film models, McNamara and Amidon [6] included convection, or mass transfer via fluid flow, into the general solid dissolution and reaction modeling scheme. The idea was to recognize that diffusion was not the only process by which mass could be transferred from the solid surface through the boundary layer [7], McNamara and Amidon constructed a set of steady-state convective diffusion continuity equations such as... 

Show how the Hagen-Poiseuille equation for the steady laminar flow of a Newtonian fluid in a uniform cylindrical tube can be derived starting from the general microscopic equations of motion (e.g., the continuity and momentum equations). 

For an incompressible fluid, the term in parentheses is zero as a result of the conservation of mass (e.g., the microscopic continuity equation). Equation (13-25) can be generalized to three dimensions as... 

As the continuity equation, the NS equations, and the transport equations for the turbulent variables are highly nonlinear, any CFD-calculation is essentially iterative. Generally, the convergence rate of simulations depends on the number of grid points and on the number of equations to be solved. 

The general equation used for conservation of mass (the continuity equation) may be written as follows ... 

The use of the excess ligand condition, equation (57), spares the need to consider the continuity equation (52) for the ligand. Then, two limiting cases of kinetic behaviour are particularly simple the inert case and the fully labile case. As we will see, these cases can be treated with the expressions (for transient and steady-state biouptake) developed in Section 2, and they provide clear boundaries for the general kinetic case, which will be considered in Section 3.4. 

There is no general solution of the Navier-Stokes equations, which is due in part to the non-linear inertial terms. Analytical solutions are possible in cases when several of the terms vanish or are negligible. The skill in obtaining analytical solutions of the Navier-Stokes equations lies in recognizing simplifications that can be made for the particular flow being analysed. Use of the continuity equation is usually essential. 

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