Equation of continuity for

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

It leads directly to the equation of continuity for the charge density in a closed volume, viz. 

If we substitute Eqs. (2.12) and (2.13) into Eq. (2.2) and perform the averaging operation indicated by the overbar, using the equation of continuity for an incompressible fluid,... 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

The equation of continuity for component A in a two-component system is according to Eq. (24) ... 

The basic expressions for the mass fluxes and the equations of continuity for multi-component mixtures are given in Sec. II,B. For a -component mixture of ideal gases in a system in which there is no pressure diffusion, forced diffusion, or thermal diffusion, the fluxes are given by... 

The conservation of mass in a continuous system is expressed by the equation of continuity for the density, p. 

In general, the local change of a physical quantity is due not only to the divergence of the current which is associated with it. but a source term has also to be taken into account. For instance, the equation of continuity for the density pr of a component y participating in a chemical reaction is... 

We will use the equation of continuity, assigning the subscript 1 to the barrel of the syringe and the subscript 2 to the needle. We are asked to calculate the speed of the medicine through the needle, which is represented by vr Our first step will be to solve the equation of continuity for v2 ... 

Subsequent to polymer manufacture, it is often necessary to remove dissolved volatiles, such as solvents, untreated monomer, moisture, and impurities from the product. Moreover, volatiles, water, and other components often need to be removed prior to the shaping step. For the dissolved volatiles to be removed, they must diffuse to some melt-vapor interface. This mass-transport operation, called devolatilization, constitutes an important elementary step in polymer processing, and is discussed in Chapter 8. For a detailed discussion of diffusion, the reader is referred to the many texts available on the subject here we will only present the equation of continuity for a binary system of constant density, where a low concentration of a minor component A diffuses through the major component ... 

The only nonvanishing velocity component is vg and it is a function only of the angular coordinates r and z. The other two velocity components vz and vr vanish, as implied from the equation of continuity for fully developed flow. The flow kinematics also imply that %rr tr xgg = xrz = 0 and the only nonvanishing stress components are xgz and xrg. 

The equation of continuity for an incompressible liquid, and with spherical symmetry, reduces to... 

Defining Pe as the pressure at the branching point of the two arterioles, the equation of continuity for the blood flow reads ... 

To obtain a relation between the volume moisture-content and the distance x for a given point, we may write the equation of continuity for one-dimensional flow as follows ... 

In one dimension tho equation of continuity for the particles within the thin layer is... 

Fick s second law defines the behavior of a diffusing chemical in space over time. Fick s second law is derived from Fick s first law and the equation of continuity for a solute. For simplicity, we derive Fick s second law in 1-D coordinates. This can readily be extended to multiple dimensions or to spherical coordinates [22]. 

In 1987 Valdes104 developed a model for composite deposition at a RDE taking into account the various ways in which a particle is transported to the cathode surface. As starting point an equation of continuity for the particle number concentration, C p, based on a differential mass balance was chosen, that is ... 

As we do for all mass transfer problems, we must satisfy the differential equation of continuity for each species as well as the differential momentum balance. Since we are dealing with a porous medium having a complex and normally unknown geometry, we choose to work in terms of the local volume averaged forms of these relations. Reviews of local volume averaging are available elsewhere (23-25). 

The equation of continuity for solid phase in a one-dimensional coordinate system can be written as... 

The equation of continuity for the liquid phase can be derived in a similar manner and is given by... 

Generally, it is necessary to carry out a simultaneous solution of the coupled equations of mass, momentum, and energy to account properly for the changes in T, q, and Uj and the effects of the changes in each of these variables on each other. In considering air pollution models, however, it is reasonable to assume that the presence of pollutants in the atmosphere does not affect the meteorology to any detectable extent thus, the equations of continuity for contaminant species can be solved independently of the coupled momentum and energy equations. Nevertheless, despite this simplification, solution of the coupled momentum... 

The equation of continuity for the liquid phase of the bubble column is obtained from a mass balance on tracer material for differential gas-liquid mixed phases. With the same procedure as for a homogeneous flow, the following equation is obtained when 4>(c) = 0 and the aspect ratio L/Dt 1 ... 

With a simplifying assumption that local adsorption equilibrium is instantaneously attained between gas and particles, the mass-transfer process is expressed by Fig. 58 for the cloud-overlap region. Here the influence of bubble wall curvature is neglected, since the region is very thin. When no catalyst is suspended in the bubble void, the equations of continuity for the reactant gas are as follows (M30) ... 

Morookact al. (M42) further solved the impulse response of a tracer gas for a fluidized catalyst bed according to the one-dimensional two-phase diffusion model (VI), where the influence of the particle capacitance effect was considered under the assumption of local adsorption equilibrium. The equations of continuity for the tracer gas are ... 

The problem consists of continuity equations and two types of condition equations. For the k-th collocation point, lying on the interface between the elements numbered m and m-1, the equation of continuity for W(z) is following ... 

Equation (1.13) must be satisfied for any macroscopic volume V, thus the expressions inside the volume integral must be equal to zero. This result is termed the equation of continuity for a mixture, and is given by ... 

In this section we present a development of the partial differential equation for the distribution function ip(R). This important equation is obtained by combining the equation of motion describing the relative motion of the beads, and an equation of continuity for the distribution function. Before discussing these, however, we need to make a few comments regarding the representation of the velocity field. 

Derivation of the Equations. The mass balance for oxygen in the tissue and capillaries is given by the equation of continuity for an element with volume V and surface F ... 

Simplify the equation of continuity for steady-state diffusion in one direction in rectangular coordinates and without chemical reaction. 

The equation of continuity for component A in a mixture describes the change of concentration of A with respect to time at a fixed point in space resulting from mass transfer of A and chemical reactions producing A. In vector symbolism, the resulting equation is (Bird et al., 1960) ... 

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