Flux material volume

In a moving medium, velocity therefore represents the flux of volume. The flux of material J associated with the movement is the amount of matter carried by the flux... 

The volume integral on the RHS is defined over a control volume V fixed in space, which coincides with the moving material volume V(t) at the considered instant, t, in time. Similarly, the fixed control surface A coincides at time t with the closed surface A(t) bounding the material volume V(t). In the surface integral, n denotes the unit outward normal to the surface A t) at time t, and V is the material velocity of points of the boundary A t). The first term on the RHS of (1.8) is the partial time derivative of the volume integral. The boundary integral represents the flux of the scalar quantity / across the fixed boundary of the control volume V. 

Although the derivation of the continuity equation by use of a fixed control volume is perfectly satisfactory, it is less obvious how to apply Newton s laws of mechanics in this framework. The familiar use of these principles from coursework in classical mechanics is that they are applied to describe the motion of a specific body subject to various forces or torques. To apply these same laws to a fluid (i.e., a liquid or a gas), we introduce the concepts of material points and a material volume (or material control volume) that we denote as Vm(t). Now a material point is a continuum point that moves with the local continuum velocity of the fluid. A material volume Vm (t), is a macroscopic control volume whose shape at some initial instant, / = 0, is arbitrary, that contains a fixed set of material points. Because the material volume contains a fixed set of such points, it must move with the local continuum velocity of the fluid at every point. Hence, as illustrated in Fig. 2-3, it must deform and change volume in such a way that the local flux of mass through all points on its surface is identically zero for all time (though, of course, there may still be exchange of molecules due to random molecular motion). Because mass is neither created nor destroyed according to the principle of mass conservation, the total mass contained... 

Because the initial choice of Vm(t) is arbitrary, we obtain the same differential form for the continuity equation, (2-5), that we derived earlier by using a fixed control volume. Of course, the fact that we obtain the same form for the continuity equation is not surprising. The two derivations are entirely equivalent. In the first, conservation of mass is imposed by the requirement that the time rate of change of mass in a fixed control volume be exactly balanced by a net imbalance in the influx and efflux of mass through the surface. In particular, no mass is created or destroyed. In the second approach, we define the material volume element so that the mass flux through its surface is everywhere equal to zero. In this case, the condition that mass is conserved means that the total mass in the material volume element is constant. The differential form (2-5) of the statement of mass conservation, which we have called the continuity equation, is the main result of this section. 

Region Material Volume, cm Volume fraction u Mass m, kg Molecular weight M Density P, g/cm Cone. TV, molecules/cm (Xl0- ) Relative thermal flux ... 

Region Material Volume, cm /cm Density, g/cm Relative thermal flux... 

Accordingly, consider a solid body B occupying a material volume V bounded by a surface A. Let the solid, of mass density p, absorb vapor through its boundary and let m denote the vapor-mass per unit volume of the solid. Also, and let f, q, and V denote fluxes of vapor-mass and of heat, and the velocity of the solid particles, respectively. 

The volume flux is the rate at which material volume sweeps across the left or right face of the slab, per unit area. 

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

Approximately 5% of the U.S. consumption of is in agriculture. Boron is a necessary trace nutrient for plants and is added in small quantities to a number of fertilizers. Borates are also used in crop sprays for fast rehef of boron deficiency. Borates, when apphed at relatively high concentration, act as nonselective herbicides. Small quantities of borates are used in the manufacture of alloys and refractories (qv). Molten borates readily dissolve other metal oxides usage as a flux in metallurgy is an important apphcation. Other important small volume apphcations for borates are in fire retardants for both plastics and ceUulosic materials, in hydrocarbon fuels for fungus control, and in automotive antifreeze for corrosion control (see Corrosion and corrosion inhibitors). Borates are used as neutron absorbers in nuclear reactors. Several borates, which are registered with the Environmental Protection Agency (EPA) can be used for insecticidal purposes, eg, TIM-BOR. 

In membrane separations, the product S Dj is referred to as the per-meabihty, p (kmoL/m s Pa). The rate of passage of material through a membrane is referred to as flux, with symbol J . Jj is equal to Ni in the equations given above. Generally Jj has the dimensions of velocity, m/s (more conveniently, Im/s), or conventionally as /m hr, gal/ft day, or ftVft day. For most apphcations, throughput is expressed in volumes instead of moles or mass. 

Initial plume volume flux for dense gas dispersion, voliime/time Continuous release rate of material, mass/time Instantaneous release of material, mass Release duration, time T Absolute temperature, K... 

Lagranglan codes are characterized by moving the mesh with the material motion, u = y, in (9.1)-(9.4), [24]. The convection terms drop out of (9.1)-(9.4) simplifying all the equations. The convection terms are the first terms on the right-hand side of the conservation equations that give rise to fluxes between the elements. Equations (9.1)-(9.2) are satisfied automatically, since the computational mesh moves with the material and, hence, no volume or mass flux occurs across element boundaries. Momentum and energy still flow through the mesh and, therefore, (9.3)-(9.4) must be solved. 

The major losses within any core material are the hysteresis loss and eddy current loss. These losses are typically lumped together by the core manufacturer and given in a graph of watts lost per unit volume V5. the peak operational flux density (5max) and frequency of operation. Hysteresis loss is given as... 

In the early days of the commercial development of PVC, emulsion polymers were preferred for general purpose applications. This was because these materials exist in the form of the fine primary particles of diameter of the order of 0.1-1.0 p,m, which in the case of some commercial grades aggregate into hollow secondary particles or cenospheres with diameters of 30-100 p,m. These emulsion polymer particles have a high surface/volume ratio and fluxing and gelation with plasticisers is rapid. The use of such polymers was, however, restricted because of the presence of large quantities of soaps and other additives necessary to emulsion polymerisation which adversely affect clarity and electrical insulation properties. 

Since the amount of fissile material in the fuel assemblies is only about 3 percent of the uranium present, it is obvious that there cannot be a large amount of radioactive material in the SNF after fission. The neutron flux produces some newly radioactive material in the form of uranium and plutonium isotopes. The amount of this other newly radioactive material is small compared to the volume of the fuel assembly. These facts prompt some to argue that SNF should be chemically processed and the various components separated into nonradioac-tive material, material that will be radioactive for a long time, and material that could be refabricated into new reactor fuel. Reprocessing the fuel to isolate the plutonium is seen as a reason not to proceed with this technology in the United States. 

Figure 9.8. The flux in the inlet pipe is due solely to convection and has magnitude Qi ai . The flux just inside the reactor at location = 0+ has two components. One component, Qi a(0+), is due to convection. The other component, —DAc[da/dz]Q, is due to difiusion (albeit eddy difiusion) from the relatively high concentrations at the inlet toward the lower concentrations within the reactor. The inflow to the plane at z = 0 must be matched by material leaving the plane at z = 0+ since no reaction occurs in a region that has no volume. Thus,...

---