Continuity equations, defined

Referring to one fiber alone, the scheme of the reacting system is similar to those examined so far. However, the reaction does not occur in either regions 1, 2 or 3. The set of differential mass transfer and continuity equations defining substrate and product concentration in these regions are equal to those previously examined. The description of mass transport in the shell-side region is somewhat more complicated. Differences in the environment surrounding each fiber, the position of fibers in the bundle, and the ultrafiltration fluxes make both the analytical and the numerical approach quite difficult. 

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... 

Analytical solutions for the closure problem in particular unit cells made of two concentric circles have been developed by Chang [68,69] and extended by Hadden et al. [145], In order to use the solution of the potential equation in the determination of the effective transport parameters for the species continuity equation, the deviations of the potential in the unit cell, defined by... 

In reality, the slip velocity may not be neglected (except perhaps in a microgravity environment). A drift flux model has therefore been introduced (Zuber and Findlay, 1965) which is an improvement of the homogeneous model. In the drift flux model for one-dimensional two-phase flow, equations of continuity, momentum, and energy are written for the mixture (in three equations). In addition, another continuity equation for one phase is also written, usually for the gas phase. To allow a slip velocity to take place between the two phases, a drift velocity, uGJ, or a diffusion velocity, uGM (gas velocity relative to the velocity of center of mass), is defined as... 

Note that 7Zu = 0 due to the continuity equation. Thus, the pressure-rate-of-strain tensor s role in a turbulent flow is to redistribute turbulent kinetic energy among the various components of the Reynolds stress tensor. The pressure-diffusion term T is defined... 

Applying this expression in (3.115), and using the continuity equation for the mean velocity, yields Sf = 0. Thus, in high-Reynolds-number flows,29 Sf will be negligible. The mean-scalar-gradient term Gf is defined by... 

Based on the quantum mechanical equation of motion, the continuity equation actually defines the relation between the charge density p(r, t) and the current density j(r,t) at a given time,... 

In addition to the vorticity transport equation, a relationship between vorticity and stream function can be developed for two-dimensional steady-state problems. Continuing to use the r-6 plane as an example, the stream function is defined to satisfy the continuity equation exactly (Section 3.1.3),... 

In a true binary system, the transport problem, which includes the boundary morphology, is completely defined by 1) the continuity equation (11.2) at the moving... 

The continuity equations are supplemented by the constitutive relations involving the current of number, momentum and energy. The local velocity field u(r, t) is defined via the following relation,... 

For general three-dimensional impinging streams it is difficult to define a stream function satisfying the continuity equation. However, if the velocity vector is defined by... 

If we take the schematic of a differential fiber element presented in Fig. 6.19, we can define the fiber geometry by the function R(x) and the unit normal vector n. The continuity equation tells us that the volumetric flow rate through any cross-section along the rr-direction must be Q... 

Because they do not contain the pressure as a variable. Eqs. (2.76) and (2.77) have been used quite extensively in solving problems for which the boundary layer equations (see later) cannot be used. For this purpose, instead of solving the Navier-Stokes and energy, simultaneously with the continuity equation, it is convenient to introduce the stream function, ip, which is defined such that... 

As shown in Chapter 2, the stream function as so defined satisfies the continuity equation. 

By definition, with the area-averaged velocity defined as above, the continuity equation for flow in a porous medium w ill have the same form as that for the flow of a pure fluid, i.e., the continuity equation for flow through a porous medium is, if density variations are negligible ... 

These two equations (4.2) and (4.3) together with (4.4) (continuity equation for incompressible fluids) and with the boundary conditions of the particular reactor define the convective mass transport in electrochemical cells. It is important to take in mind that this exhaustive description is frequently used in electrochemical engineering, especially in cases such as the electroplating processes where the current distribution becomes a key factor in the performance of the process. 

Here, tj is called the similarity variable, and g(r/) is the function we seek as a solution. In accordance with the continuity equation, a stream function i/ may be defined so that... 

When packed beds or fluidized-expanded beds of defined bed expansion are involved, further consideration must come into play. Thus, by assuming that axial diffusion can be neglected, the continuity equation linking concentration, axial distance and time for nonporous particle HPLC sorbents takes the modified form of Eq. (120), namely411,412... 

Let us reiterate that diffusion defined by (102) can be derived using the Brownian motion approach and a continuity equation. Moreover, one can imply... 

While the reaction rate is usually defined by the (normalized) concentration increase of species k, i.e., by dCk/St, the flux density jk (concentration times velocity) can be introduced by the continuity equation... 

FORTRAN computer program that predicts the species, temperature, and velocity profiles in two-dimensional (planar or axisymmetric) channels. The model uses the boundary layer approximations for the fluid flow equations, coupled to gas-phase and surface species continuity equations. The program runs in conjunction with CHEMKIN preprocessors (CHEMKIN, SURFACE CHEMKIN, and TRAN-FIT) for the gas-phase and surface chemical reaction mechanisms and transport properties. The finite difference representation of the defining equations forms a set of differential algebraic equations which are solved using the computer program DASSL (dassal.f, L. R. Petzold, Sandia National Laboratories Report, SAND 82-8637, 1982). 

SO that the continuity equation (Eq. 6-39) is automatically satisfied and thus eliminated (this can be verified easily by direct substitution). Next he defined a fiinction /(7)) as the dependent variable as... 

The ensuing step is implementation of the mass continuity principle, which results in a material balance for both the solid and liquid phases within a defined control volume in the carbon bed. In its general form, the mass continuity equation is ... 

Hulburt and Katz (HI7) developed a framework for the analysis of particulate systems with the population balance equation for a multivariate particle number density. This number density is defined over phase space which is characterized by a vector of the least number of independent coordinates attached to a particle distribution that allow complete description of the properties of the distribution. Phase space is composed of three external particle coordinates x and m internal particle coordinates Xj. The former (Xei, x 2, A es) refer to the spatial distribution of particles. The latter coordinate properties Ocu,Xa,. . , Xt ) give a quantitative description of the state of an individual particle, such as its mass, concentration, temperature, age, etc. In the case of a homogeneous dispersion such as in a well-mixed vessel the external coordinates are unnecessary whereas for a nonideal stirred vessel or tubular configuration they may be needed. Thus (x t)d represents the number of particles per unit volume of dispersion at time t in the incremental range x, x -I- d, where x represents both coordinate sets. The number density continuity equation in particle phase space is shown to be (HI 8, R6)... 

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