Continuity equation integral form

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

In step (2) of the solution of equation 9.1-17, we allow the core surface, fixed in step (1), to move, and integrate the continuity equation for B, using the first part of equation 9.1-6. For this purpose, we substitute both equations 9.1-23 and 9.1-6, the latter written in the form... 

As in the case of C(t) or E(t), the integral form in each equation is used for a continuous response, and the summation form for discrete response data. The result for / from equation 19.3-7 serves either as a second check on the accuracy of the tracer study, since / = V/q for constant density, or as a means of determining f, if the true value of V is unknown. 

Next one needs an expression for (xA — xA"). The difference in concentration between the two streams results from two effects thermal diffusion, which tends to increase the concentration difference, and convection, which tends to decrease it. Each of these effects is considered separately by obtaining an approximate integrated form of the steady state equation of continuity as applied to that particular process. If the only effect tending to produce a concentration difference were thermal diffusion, then according to Eq. (131) dxA/dx = — (kT/T)(d,T/dx) this expression may be written in difference form over the distance from x = — ( 4)a to x — + (M)° thus ... 

Ironically, since the mass-continuity equation was already used in the derivation of the substantial-derivative form of Eq. 3.2, it is not directly useful for deriving the continuity equation itself. Its application simply returns a trivial identity. Instead, we begin with the integral form as stated in Eq. 2.30 to yield... 

Based on the spherical control volume shown in Fig 3.15, derive the mass-continuity equation. Begin with the general statement of the Reynolds transport theorm in integral form (Eq. 2.19)... 

Using the integrals in the previous questions, derive a differential-equation form of the mass-continuity equation. For the differential control volume, explain how and why the volume integrals are eliminated from the analysis. 

The analytical solutions to Fick s continuity equation represent special cases for which the diflusion coefficient, D, is constant. In practice, this condition is met only when the concentration of diffusing dopants is below a certain level ( 1 x 1019 atoms/cm3). Above this doping density, D may depend on local dopant concentration levels through electric field effects, Fermi-level effects, strain, or the presence of other dopants. For these cases, equation 1 must be integrated with a computer. The form of equation 1 is essentially the same for a wide range of nonlinear diffusion effects. Thus, the research emphasis has been on understanding the complex behavior of the diffusion coefficient, D, which can be accomplished by studying diffusion at the atomic level. 

The integrated continuity equation is a weaker form of the full continuity equation. This is noticed in numerical solutions of mold filling problems, where continuity is never fully satisfied. However, this violation of continuity is insignificant and will not hinder the solution of practical mold filling problems. The integrated continuity equation reduces to... 

When the velocity at which the fluid approaches the collector becomes sufficiendy slow the iiuid may be considered to be essentially stagnant. The continuity equation for this situation takes on a particularly simple form which can be integrated to yield... 

This value of a is now introduced into the integrated form of the Clausius equation, and another equation is obtained which includes the variation of Q with T and which should apply to the whole vapour-pressure curve of liquid phosphorus on the assumption that it is continuous, i.e. that the liquid formed at lower pressures is really the same as that formed at higher pressures. The equations in question are... 

The above formula for Z, the NPT partition function, was first reported by Guggenheim [74], who wrote the expression down by analogy rather than based on a detailed derivation. While this form of the partition function is thought to be broadly valid and is widely applied (for example in molecular dynamics simulation [6]), it introduces the conceptual difficulty that the meaning of the discrete volumes Vi is not clear. Discrete energy states arise naturally from quantum statistics. Yet it is not necessarily obvious what discrete volumes to sum over in Equation (12.50). In fact for most applications it makes sense to replace the discrete sum with a continuous volume integral. Yet doing so results in a partition function that has units of volume, which is inappropriate for a partition function that formally should be unitless. 

Fig. 4. Analysis of the kinetic constants of the ferroxidase reaction catalyzed by soluble Fet3p. Fe(II) oxidation (A) and O2 consumption (B) were measured continuously and the residual substrate concentration was plotted with respect to time according to the integrated form of the Michealis-Menten equation as indicated in each panel. Fe(II) oxidation was followed by the appearance of Fe(III) at 315 nm while O2 consumption was determinedby the use of an O2 electrode. The [Fet3p] =0.2 fcM in 0.1 M MES buffer, pH 6.0, at 25°C. The curve in each panel is a linear least-squares fit of the data to...

The form assumed by the continuity equation, equation (1-19), can be derived formally by integration over the total cross-sectional area of the flow. The limits of the coordinates X2 and X3 [which appear in equation (1-13)] in such an integration must be independent of x because the boundaries of the cross section are streamlines and must therefore be parallel to the local X coordinate (that is, parallel to the local velocity vector). Thus, since the flow variables are independent of the coordinates X2 and X3, multiplication of equation (1-19) by 2 3 followed by integration over the cross-sectional area shows that... 

The continuity equation is re-written on the integral form, integrated in time and over a grid cell volume in the non-staggered grid for the scalar variables sketched in Fig C.2. The transient terms are discretized with the implicit Euler scheme. 

The analysis of this section is typical of all lubrication problems. First, the equations of motion are solved to obtain a profile for the tangential velocity component, which is always locally similar in form to the profile for unidirectional flow between parallel plane boundaries, but with the streamwise pressure gradient unknown. The continuity equation is then integrated to obtain the normal velocity component, but this requires only one of the two boundary conditions for the normal velocity. The second condition then yields a DE (known as the Reynolds equation) that can be used to determine the pressure distribution. 

The solution (6 126) with (6 127), is self-contained. Because of the integral constraint, (6 125), there is no need to consider w(,)) in order to obtain an equation for p(0>. Indeed, we see that u(()> is completely independent of x. This means that the flow in the core region is unidirectional at this leading order of approximation. The continuity equation, (6 120), is reduced to the form... 

From the integral form of the continuity equation and the velocity field within the current, derive the nonlinear diflusionlike equation for the transport of h(rj) ... 

Let us suppose a medium in which all electromagnetic field vectors F are continuous, bounded, single-valued functions of position and time F = F(x, y, z, t) and incorporate continuous derivatives as well. Then, the differential and integral form of Maxwell s equations can be expressed as... 

The principles of physics most useful in the applications of fluid mechanics are the mass-balance, or continuity, equations the linear- and angular-raomentum-balance equations and the mechanical-energy balance. They may be written in differential form, showing conditions at a point within a volume element of fluid, or in integrated form applicable to a finite volume or mass of fluid. Only the integral equations are discussed in this chapter. 

To complete the system of equations, we employ the integral form of the continuity equation, recognizing that the liquid surface layer set in motion by the surface tension force (Eq. 10.5.1) must be accompanied by a motion of the fluid in the opposite direction below the surface, as sketched in Fig. 10.5.1. With no net flux across any cross section, the continuity equation for the fully developed flow is... 

It is straightforward to rewrite the Maxwell equations and the continuity equation in an integral form. Specifically, integrating Eqs. (2.2) and (2.4) over a surface S bounded by a closed contour C (see Fig. 2.1) and applying the Stokes theorem. 

A type of continuous reactor with performance similar to a batch reactor is the plug-flow reactor, a tubular or pipeline reactor with continuous feed at one end and product removal at the other end. The conversion is a function of the residence time, which depends on the flow rate and the reactor volume. The data for plug-flow reactors are analyzed in the same way as for batch reactors. The conversion is compared with that predicted from an integrated form of an assumed rate expression. A trial-and-error procedure may be needed to determine the appropriate rate equations. 

As already mentioned, the form of the fundamental continuity equations is usually too complex to be conveniently solved for practical application to reactor design. If one or more terms are dropped from Eq. 7.2.a-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 the introduction. In these cases, it is often easier to apply Eq. 7.1.a-l directly to a volume element of the reactor. This will be done in the next chapters, dealing with basic or specific reactor types. In the present chapter, however, it will be shown how the simplified equations can be obtained from the fundamental ones. 

Such a situation can be dealt with in two ways. The first way is to analyze the data as such. The temperature dependence of the rate parameters is then directly included into the continuity equation and the resulting equation is numerically integrated along the tube with estimates for the parameters. If the gas temperature profile itself is not available or insufiBciently defined, the energy equation has to be coupled to the continuity equation. To determine both the form of the rate equation and the temperature dependence of the parameters directly from nonisothermal data would require excessive computations. 

Let Jji be the rate of mass transfer (evaporation, condensation, chemical reactions) from j-th to i-th phase, or the reverse. In the latter case, Jji < 0. Then the continuity equation for i-th phase in the integral form is... 

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