Continuity equation derivation

The continuity equation is a statement of mass conservation. As presented in Section 3.1, however, no distinction is made as to the chemical identity of individual species in the flow. Mass of any sort flowing into or out of a differential element contributes to the net rate of change of mass in the element. Thus the overall continuity equation does not need to explicitly demonstrate the fact that the flow may be composed of different chemical constituents. Of course, the equation of state that relates the mass density to other state variables does indirectly bring the chemical composition of the flow into the continuity equation. Also, as presented, the continuity-equation derivation does not include diffusive flux of mass across the differential element s surfaces. Moreover there is no provision for mass to be created or destroyed within the differential element s volume. 

An example of the continuity equation derived from the stationary fluid element and thus written in the conservation form with the vector notation appropriate for any coordinate system is... 

The generalized form of the transport theorem, Eq. (10), provides a clear route for developing conservation equations valid at phase interfaces as well as in bulk phases. For modelling the growth of SEI layers, we are primarily interested in continuity equations derived from the law of conservation of mass. 

This equation is analogous to the continuity equation derived for the conservation of mass in a continuum... 

The variational derivative of this with respect to ((> yields the continuity equation... 

When one takes its vaiiational derivative with respect to the phases < >, one obtains the continuity equation in the form... 

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... 

When electrons are injected as minority carriers into a -type semiconductor they may diffuse, drift, or disappear. That is, their electrical behavior is determined by diffusion in concentration gradients, drift in electric fields (potential gradients), or disappearance through recombination with majority carrier holes. Thus, the transport behavior of minority carriers can be described by a continuity equation. To derive the p—n junction equation, steady-state is assumed, so that = 0, and a neutral region outside the depletion region is assumed, so that the electric field is zero. Under these circumstances,... 

The substantial derivative, also called the material derivative, is the rate of change in a Lagrangian reference frame, that is, following a material particle. In vector notation the continuity equation may oe expressed as... 

The method is apphcable for unflocculated pulps or those in which the ionic characteristics of the solution produce a flocculent structure. If polymeric flocculants are used, the floccule size will be highly dependent on the feed concentration, and an approach based on the Kynch theoiy is preferred. In this method, the test is carried out at the expec ted feed solids concentration and is continued until underflow concentration is achieved in the cyhnder. To determine the unit area, Talmage and Fitch (op. cit.) proposed an equation derived from a relationship equivalent to that shown in Eq. (18-45) ... 

This model v/as used by Atwood et al (1989) to compare the performance of 12 m and 1.2 m long tubular reactors using the UCKRON test problem. Although it was obvious that axial conduction of matter and heat can be expected in the short tube and not in the long tube, the second derivative conduction terms were included in the model so that no difference can be blamed on differences in the models. The continuity equations for the compounds was presented as ... 

We first derive the so-called continuity equation, which is a direct consequence of the conservation of mass. If p is the density, or mass per unit volume, then the total mass of a fluid contained in F is equal to M = fj p dF. Letting dS — fi dS be an element of the surface, with n a unit vector perpendicular to the surface, the mass flow per unit time through the surface element is pv dS. The total fluid flow out of the volume F is then given by... 

In this section we show how the fundamental equations of hydrodynamics — namely, the continuity equation (equation 9.3), Euler s equation (equation 9.7) and the Navier-Stokes equation (equation 9.16) - can all be recovered from the Boltzman equation by exploiting the fact that in any microscopic collision there are dynamical quantities that are always conserved namely (for spinless particles), mass, momentum and energy. The derivations in this section follow mostly [huangk63]. 

Recall that equations 9.86 and 9.100 have been both derived using only the first-order terms in the Taylor series expansion of our basic kinetic equation (equation 9.77). It is easy to show that if instead all terms through second-order in 6x and 6t are retained, the continuity equation ( 9.86) remains invariant but the momentum equation ( 9.100) requires correction terms [wolf86c]. The LHS of equation 9.100, to second order in (ia (5 << 1, is given by... 

In the preceding section, we have established the importance of the power series q x) r(x), 5(x), t x) in combinatorics. Here we examine their analytical properties radius of convergence, singularities on the circle of convergence, analytic continuation. We derive these characteristics from the functional equations whose solutions these series present. I start with a summary of the equations and some notations. 

Studying the steady motion of a single medium-size bubble rising in a liquid medium under the influence of gravity, Levich (L3, L4) solved the continuity equation simultaneously with the equations of motion by introducing the concept of a boundary layer for the case of a bubble. This boundary layer accounts for the zero, or extremely low, shear stress at the interface. Despite some errors in deriving the equations, his result was later confirmed with minor improvements (A4, M3, M10). 

Multiphase reactors can be batch, fed-batch, or continuous. Most of the design equations derived in this chapter are general and apply to any of the operating modes. Unsteady operation of nominally continuous processes is treated in Chapter 14. 

The objective of most of the theories of transport in porous media is to derive analytical or numerical functions for the effective diffusion coefficient to use in the preceed-ing averaged species continuity equations based on the structure of the media and, more recently, the structure of the solute. 

Trinh et al. [399] derived a number of similar expressions for mobility and diffusion coefficients in a similar unit cell. The cases considered by Trinh et al. were (1) electrophoretic transport with the same uniform electric field in the large pore and in the constriction, (2) hindered electrophoretic transport in the pore with uniform electric fields, (3) hydrodynamic flow in the pore, where the velocity in the second pore was related to the velocity in the first pore by the overall mass continuity equation, and (4) hindered hydrodynamic flow. All of these four cases were investigated with two different boundary condi-... 

The problem of controlling the outcome of photodissociation processes has been considered by many authors [63, 79-87]. The basic theory is derived in detail in Appendix B. Our set objective in this application is to maximize the flux of dissociation products in a chosen exit channel or final quantum state. The theory differs from that set out in Appendix A in that the final state is a continuum or dissociative state and that there is a continuous range of possible energies (i.e., quantum states) available to the system. The equations derived for this case are... 

The estimation of f from Stokes law when the bead is similar in size to a solvent molecule represents a dubious application of a classical equation derived for a continuous medium to a molecular phenomenon. The value used for f above could be considerably in error. Hence the real test of whether or not it is justifiable to neglect the second term in Eq. (19) is to be sought in experiment. It should be remarked also that the Kirkwood-Riseman theory, including their theory of viscosity to be discussed below, has been developed on the assumption that the hydrodynamics of the molecule, like its thermodynamic interactions, are equivalent to those of a cloud distribution of independent beads. A better approximation to the actual molecule would consist of a cylinder of roughly uniform cross section bent irregularly into a random, tortuous configuration. The accuracy with which the cloud model represents the behavior of the real polymer chain can be decided at present only from analysis of experimental data. 

The conservation of mass can be applied to an arbitrarily small fluid element to derive the microscopic continuity equation, which must be satisfied at all points within any continuous fluid. This can be done by considering an arbitrary (cubical) differential element of dimensions dx, dy, dz, with mass... 

Let us now, as we normally do, continue to derive the expressions for absorbance noise again referring to our previous chapter [8], we can start with equation 52-29 ... 

In this Eq. (Js)n is the Jacobi matrix for the solid phase, which contains the derivatives of the mass residuals for the particulate phase to the solid volume fraction. Explicit expressions for the elements of the Jacobi matrix can be obtained from the continuity for the solid phase and the momentum equations. For example for the central element, the following expression is obtained from the solid phase continuity equation, in which the convective terms are evaluated with central finite difference expressions ... 

Since LB describes the NS equations in the incompressible limit, the local pressures can directly be obtained from the local densities and the speed of sound. Hence, a distinct step for calculating the pressures via a Poisson equation (derived from the continuity equation) as required in incompressible FV schemes, is absent in LB. 

To obtain an expression for tj, we first derive the continuity equation governing steady-state diffusion of A through the pores of the particle. This is based on a material balance for A across the control volume consisting of the thin strip of width dx shown in Figure 8.10(a). We then solve the resulting differential equation to obtain the concentration profile for A through the particle (shown in Figure 8.10(b)), and, finally, use this result to obtain an expression for tj in terms of particle, reaction, and diffusion characteristics. 

Nonisothermal spherical particle. The energy equation describing the profile for T through the particle, equivalent to the continuity equation 9.1-5 describing the profile for cA, may be derived in a similar manner from an energy (enthalpy) balance around the thin shell in Figure 9.1(b). The result is... 

The basis for the analysis using the SCM is illustrated in Figure 9.3. The gas film, outer product (ash) layer, and unreacted core of B are three distinct regions. We derive the continuity equation for A by means of a material balance across a thin spherical shell in the ash layer at radial position r and with a thickness dr. The procedure is the same as that leading up to equation 9.1-5, except that there is no reaction term involving (- rA), since no reaction occurs in the ash layer. The result corresponding to equation 9.1-5 is... 

This diffusive flow must be taken into account in the derivation of the material-balance or continuity equation in terms of A. The result is the axial dispersion or dispersed plug flow (DPF) model for nonideal flow. It is a single-parameter model, the parameter being DL or its equivalent as a dimensionless parameter. It was originally developed to describe relatively small departures from PF in pipes and packed beds, that is, for relatively small amounts of backmixing, but, in principle, can be used for any degree of backmixing. 

The derivation of the material-balance or continuity equation for reactant A is similar to that of equations 19.4-48 and -49 for nonreacting tracer A, except that steady state replaces unsteady state (cA at a point is not a function of t), and a reaction term must be added. Thus, using the control volume in Figure 19.15, we obtain the equivalent of equation 19.4-48 as ... 

Some of the simplifications that may be possible are illustrated by the case of steady, fully-developed, laminar, incompressible flow of a Newtonian fluid in a horizontal pipe. The flow is assumed to be axisymmetric with no swirl component of velocity so that derivatives wrt 6 vanish and vg = 0. For fully-developed flow, derivatives wrt z are zero. With these simplifications and noting that the flow is incompressible, the continuity equation (equation A. 11) reduces to... 

The change in the order of differentiation to give the last term in equation A.29 is permissible because the velocity field will satisfy the sufficient conditions, namely that the two mixed partial derivatives are continuous. Equation A.27 can now be written as... 

We continue with deriving the next set of components by maximizing the initial problem (Equation 4.67). This maximum is searched in a direction orthogonal to tx, and searching in the orthogonal complement is conveniently done by deflation of X. The deflated matrix X is... 

The key problem in using Eq. (3.1) is the specification of p. We ask whether we can derive an expression for p. The velocity components u, v, and w, although random, are related through conservation of mass and momentum for the flow, that is, they are governed by the stochastic Navier-Stokes and continuity equations. In general, as we have noted, an exact solution for u, v, and w is unobtainable. We can, however, consider an idealized situation in which the statistical properties of u, v, and w are specified a priori. Then, in so doing, we wish to see if we can obatin an exact solution of Eq. (2.4) from which p can be obtained through Eq. (2.6). 

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