Mass-conserving

Level of enforcement of the incompressibility condition depends on the magnitude of the penalty parameter. If this parameter is chosen to be excessively large then the working equations of the scheme will be dominated by the incompressibility constraint and may become singular. On the other hand, if the selected penalty parameter is too small then the mass conservation will not be assured. In non-Newtonian flow problems, where shear-dependent viscosity varies locally, to enforce the continuity at the right level it is necessary to maintain a balance between the viscosity and the penalty parameter. To achieve this the penalty parameter should be related to the viscosity as A = Xorj (Nakazawa et al, 1982) where Ao is a large dimensionless parameter and tj is the local viscosity. The recommended value for Ao in typical polymer flow problems is about 10. ... 

FIG. 5-24 Flowchart iUnstrating problem solving approach using mass-transfer rate expressions in the context of mass conservation. 

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

Because we need to know how long the refined section of the bar is, it is important to describe the ramping up of the compositions in a quantitative way. We can do this by writing a differential equation which describes what happens as the zone moves from some general position x to a new position x + 8x (Fig. 4.4g). For a bar of unit cross-section we can write the mass conservation equation... 

For the ideal reactors considered, the design equations are based on the mass conservation equations. With this in mind, a suitable component is chosen (i.e., reactant or product). Consider an element of volume, 6V, and the changes occurring between time t and t + 6t (Figure 5-2) ... 

Considered are mass conservation of air and species (contaminants and humidity). Momentum equations are not considered on a global scale but have been used in some cases for the definition of the airflow-pressure relation of the individual links. Heat fluxes and thus energy conservation equations are not considered. 

The humidity and contaminant transport calculation is based on the previously calculated airflows, applying again the principle of mass conservation for the species under consideration. For each time step, the concentrations are calculated on the basis of the airflows, the source and sink strengths in the zones, and the concentration values at the previous time step. In contrast to the airflow calculation, which is a steady-state calculation at each time step, the contaminant transport calculation is dynamic. Therefore, the accuracy of the concentration results depends on the selected time-step interval. 

A numerical study of the MMEP kinetics, as described by the system of nonlinear differential equations (26), subject to mass conservation (Eq. (27)), has been carried out [64] for a total number of 1000 monomers and different initial MWDs. As expected, and in contrast to the case of wormlike micelles, it has been found that during relaxation to a new equilibrium state the temporal MWD does not preserve its exponential form. 

The numerical solution of these equations is not trivial, since for reasonably low viscosities the flow becomes turbulent. A popular method of treating these equations (together with the equations of energy and mass conservation) is the MAC method [156,157]. For the case of immiscible fluids or moving internal interface a phase-field-type approach seems to be successful [78,158,159]. Because of the enormous requirements of computing ressources the development in this field is still relatively slow. We expect, however, an impact from the more widespread availability of massively parallel computers in the near future. 

As a consequence of implicit mass conservation, the gas-dynamic conservation equations, expressed in Lagrangean form, can describe contact discontinuities. To prevent oscillating behavior in places where shock phenomena are resolved in the... 

Mass conservation requires that the production of one molecule of C result in the loss of one molecule of either A or AB, so... 

In fluid mechanics the principles of conservation of mass, conservation of momentum, the first and second laws of thermodynamics, and empirically developed correlations are used to predict the behavior of gases and liquids at rest or in motion. The field is generally divided into fluid statics and fluid dynamics and further subdivided on the basis of compressibility. Liquids can usually be considered as incompressible, while gases are usually assumed to be compressible. 

Flow through chokes and nozzles is a special case of fluid dynamics. For incompressible fluids the problem can be handled by mass conservation and Bernoulli s equation. Bernoulli s equation is solved for the pressure drop across the choke, assuming that the velocity of approach and the vertical displacement are negligible. The velocity term is replaced by the volumetric flow rate times the area at the choke throat to yield... 

Changes in free energy and the equilibrium constants for Reactions 1, 2, 3, and 4 are quite sensitive to temperature (Figures 2 and 3). These equilibrium constants were used to calculate the composition of the exit gas from the methanator by solving the coupled equilibrium relationships of Reactions 1 and 2 and mass conservation relationships by a Newton-Raphson technique it was assumed that carbon was not formed. Features of the computer program used were as follows (a) any pressure and temperature may be specified (b) an inert gas may be present (c) after... 

That the rates of reactant consumption and product growth are equal in the steady state is a consequence of setting d[Vjdt = 0. We can see this from the mass conservation relation,... 

Table 16-2 List of input components for the simplest case of the acid-base balance of unpolluted marine clouds. Also shown are the mass conservation statements, chemical equilibrium expressions and constants, and the requirement for charge balance...

Mass conservation and equilibrium fractionation together require that... 

Unsteady combustion is a strong source of acoustic noise. The emission of sound by gaseous combustion is governed by the classical set of conservation equations Mass conservation ... 

In PBPK models tissue blood perfusion and tissue composition can be characterized independently of the drug thus such a model can be created once and reused for many different drugs. Furthermore, because physical laws (mass conservation, diffusion, or facilitated transport mechanisms) are incor-... 

The Navier-Stokes equation defines a set of three relations for four unknown quantities, iq, Uj, M3 and p. Another equation is needed to close the set, which is the equation of mass conservation ... 

The Navier-Stokes equation [Eq. (1)] provides a framework for the description of both liquid and gas flows. Unlike gases, liquids are incompressible to a good approximation. For incompressible flow, i.e. a constant density p, the Navier-Stokes equation and the corresponding mass conservation equation simplify to... 

The situation is different for incompressible flow. In that case, no equation of motion for the pressure field exists and via the mass conservation equation Eq. (17) a dynamic constraint on the velocity field is defined. The pressure field entering the incompressible Navier-Stokes equation can be regarded as a parameter field to be adjusted such that the divergence of the velocity field vanishes. 

At that stage, the approximation obtained for the velocity field does generally not fulfil the mass conservation equation. In order to ensure mass conservation, corrections to the velocity and pressure field are introduced via... 

By demanding that the new velocity w field fulfils both the momentum and the mass conservation equation, the following equations for the velocity and pressure correction are derived ... 

Via Eq. (136) the kinematic condition Eq. (131) is fulfilled automatically. Furthermore, a conservative discretization of the transport equation such as achieved with the FVM method guarantees local mass conservation for the two phases separately. With a description based on the volume fraction fimction, the two fluids can be regarded as a single fluid with spatially varying density and viscosity, according to... 

The storage hold-up f/(r) is related to the input and output rates Fu(t) and Fd(0 by the mass conservation equation ... 

To describe the diffusion of solutes in the rhizosphere, where concentration gradients change with time, /, as well as space, mass conservation is invoked with the spatial geometry appropriate for the cylindrical root (8) ... 

Mass conservation at the rhizoplane means that the diffusive flux towards the root, Eq. (4), must equal the rate of extraction by the root, Eq. (9), leading to the boundary condition... 

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