Continuity equation mass transport

The governing equations for mass flow, energy flow, and contaminant flow in a room will be the continuity equation, Navier-Stokes equations (one in each coordinate direction), the energy equation, and the mass transport equation, respectively. 

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-... 

As with acid-base reactions, the equation of continuity used to describe the mass transport of a solid in a fluid is... 

Continuity equation electrochemical reactor, 30 311 mass transport, 30 312 Continuous-flow stirred-tanlt reactor, 31 189 Continuous reactor, 33 4-5 Continuous stirred-tank reactor, 27 74-77 ControUed-atmosphere studies, choice of materials for construction, 31 188 Conversion theory, 27 50, 51 Coordinatimi number, platinum, 30 265 Coordinative bonding, energy of, 34 158 Coordinative chemisorption on silicon, 34 155-158... 

The overall mass transport coefficient for mass transport from the dispersed organic phase into the continuous aqueous phase was then calculated according to the Calderbank equation (Eq. 4) ... 

Latent heat associated with phase change in two-phase transport has a large impact on the temperature distribution and hence must be included in a nonisothermal model in the two-phase regime. The temperature nonuniformity will in turn affect the saturation pressure, condensation/evaporation rate, and hence the liquid water distribution. Under the local interfacial equilibrium between the two phases, which is an excellent approximation in a PEFG, the mass rate of phase change, ihfg, is readily calculated from the liquid continuity equation, namely... 

To derive the species-continuity equations that follow, it is important to establish some relationships between mass fluxes and species concentration fields. At this point the needed relationships are simply stated in summary form. The details are discussed later in chapters on thermochemical and transport properties. 

When considering the mass continuity of an individual species in a multicomponent mixture, there can be, and typically is, diffusive transport across the control surfaces and the production or destruction of an individual species by volumetric chemical reaction. Despite the fact that individual species may be transported diffusively across a surface, there can be no net mass that is transported across a surface by diffusion alone. Moreover homogeneous chemical reaction cannot alter the net mass in a control volume. For these reasons the overall mass continuity need not consider the individual species. At the conclusion of this section it is shown that that the overall mass continuity equation can be derived by a summation of all the individual species continuity equations. 

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)... 

The objective is to derive a system of equations in general vector form that describes the overall gas-phase mass continuity and the species continuity equations for A and all other species k in the mixture. Assume that there is convective and diffusive transport of the species, but no chemical reaction. 

Beginning with a mass-conservation law, the Reynolds transport theorem, and a differential control volume (Fig. 4.30), derive a steady-state mass-continuity equation for the mean circumferential velocity W in the annular shroud. Remember that the pressure p 6) (and hence the density p(6) and velocity V(6)) are functions of 6 in the annulus. 

Deriving the mass-continuity equation begins with a mass-conservation principle and the Reynolds transport theorem. Unlike the channel with chemically inert walls, when surface chemistry is included the mass-conservation law for the system may have a source term,... 

This result is called the continuity equation for mass [4, 52]. The cause of the flux is called a driving force , which is not used in the Newtonian sense, but instead names any source of perturbation. In the case of mass transport, this cause is typically a gradient (of concentration, electrical potential, or density). 

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 ... 

Because the kinetic and mass-transport phenomena occur in a thin region adjacent to the electrode surface, this area is treated separately from the bulk solution region. Since kinetic effects are manifested within 100 A of the electrode surface, the resulting overpotential is invariably incorporated in the boundary conditions of the problem. Mass transport in the boundary layer is often treated by a separate solution of the convective diffusion equation in this region. Continuity of the current can then be imposed as a matching condition between the boundary layer solution and the solution in the bulk electrolyte. Frequently, Laplace s equation can be used to describe the potential distribution in the bulk electrolyte and provide the basis for determining the current distribution in the bulk electrolyte. 

The principles and basic equations of continuous models have already been introduced in Section 6.2.2. These are based on the well known conservation laws for mass and energy. The diffusion inside the pores is usually described in these models by the Fickian laws or by the theory of multicomponent diffusion (Stefan-Maxwell). However, these approaches basically apply to the mass transport inside the macropores, where the necessary assumption of a continuous fluid phase essentially holds. In contrast, in the microporous case, where the pore size is close to the range of molecular dimensions, only a few molecules will be present within the cross-section of a pore, a fact which poses some doubt on whether the assumption of a continuous phase will be valid. 

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. 

The flow velocities in flame systems are such that transport processes (diffusion and thermal conduction) make appreciable contributions to the overall flows, and must be considered in the analysis of the measured profiles. Indeed, these processes are responsible for the propagation of the flame into the fresh gas supporting it, and the exponential growth zone of the shock tube experiments is replaced by an initial stage of the reaction where active centres are supplied by diffusion from more reacted mixture sightly further downstream. The measured profiles are related to the kinetic reaction rates by means of the continuity equations governing the one-dimensional flowing system. Let Wi represent the concentration (g. cm" ) of any quantity i at distance y and time t, and let F,- represent the overall flux of the quantity (g. cm". sec ). Then continuity considerations require that the sum of the first distance derivative of the flux term and the first time derivative of the concentration term be equal to the mass chemical rate of formation q,- of the quantity, i.e. 

Well-Mixed Cell Model. A conceptually simple approach is based on the representation of the airshed by a three-dimensional array of well-mixed vessels (34, 35, 36). As before, we assume that the airshed has been divided into an array of L cells. Instead of using the array simply as a tool in the finite-difference solution of the continuity equations, let us now assume that each of these cells is actually a well-mixed reactor with inflows and outflows between adjacent cells. If we neglect diffusive transport across the boundaries of the cells and consider only convective transport among cells, a mass balance on species i in cell k is given by... 

Atmospheric GCMs simulate the time evolution of various atmospheric fields (wind speed, temperature, surface pressure, and specific humidity), discretized over the globe, through the integration of the basic physical equations the hydrostatic equation of motion, the thermodynamic equation of state, the mass continuity equation, and a water vapor transport equation. To reproduce the... 

To derive the overall kinetics of a gas/liquid-phase reaction it is required to consider a volume element at the gas/liquid interface and to set up mass balances including the mass transport processes and the catalytic reaction. These balances are either differential in time (batch reactor) or in location (continuous operation). By making suitable assumptions on the hydrodynamics and, hence, the interfacial mass transfer rates, in both phases the concentration of the reactants and products can be calculated by integration of the respective differential equations either as a function of reaction time (batch reactor) or of location (continuously operated reactor). In continuous operation, certain simplifications in setting up the balances are possible if one or all of the phases are well mixed, as in continuously stirred tank reactor, hereby the mathematical treatment is significantly simplified. 

If the flows are unsteady, the terms containing apo can be added on both sides of Eq. (7.10) (refer to Section 6.4). It must be noted that for multiphase flows, the inflow and outflow terms require considerations of interpolations of phase volume fractions in addition to the usual interpolations of velocity and the coefficient of diffusive transport. The source term linearization practices discussed in the previous chapter are also applicable to multiphase flows. It is useful to recognize that special sources for multiphase flows, for example, an interphase mass transfer, is often constituted of terms having similar significance to the a and b terms. Such discretized equations can be formulated for each variable at each computational cell. The issues related to the phase continuity equation, momentum equations and the pressure correction equation are discussed below. 

A mathematical model of a biofilm is more complex than a model of a batch or a continuous culture, mainly because the pertinent mass balance equations contain diffusive mass transport terms. (Recall from Section 2.6.3 that a model of a batch culture contains no mass transport equations at all, while a continuous culture, or chemostat, model contains only simple mass inflow and mass outflow.) The following set of equations characterizes the biofilm of Fig. 3-29 at steady state ... 

The equations (3.109), (3.117) or (3.118) and (3.120) for the velocity, thermal and concentration boundary layers show some noticeable similarities. On the left hand side they contain convective terms , which describe the momentum, heat or mass exchange by convection, whilst on the right hand side a diffusive term for the momentum, heat and mass exchange exists. In addition to this the energy equation for multicomponent mixtures (3.118) and the component continuity equation (3.25) also contain terms for the influence of chemical reactions. The remaining expressions for pressure drop in the momentum equation and mass transport in the energy equation for multicomponent mixtures cannot be compared with each other because they describe two completely different physical phenomena. 

The current at any point in the voltammetry experiment described in Figure 23-5 is determined by a combination of (1) the rate of mass transport of A to the edge of the Nemst diffusion layer by convection and (2) the rate of transport of A from the outer edge of the diffusion layer to the electrode surface. Because the product of the electrolysis P diffuses away from the surface and i.s ultimately swept away by convection, a continuous current is required to maintain the surface concentrations demanded by the Nernst equation. Convection, however, maintains a constant supply of A at the outer edge of the diffusion layer. Thus, a steady-state cuirent results that is determined by the applied potential. 

Darcy s law describes fluid flux in porous media, and must be combined with the continuity equation to develop flow equations. From the flow equations, the spatial and temporal pressure and velocity distributions can be estimated that are needed for the transport equations. The derivation of flow equations starts with the continuity equation, which states that the change in mass or volume within a control volume equals the net flux across the control volume boundary, plus sources and sinks within the control volume. For water within porous media, the continuity equation on a mass basis is ... 

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