The general transport equations

The elemental conservation equation is laid down on the same principles as the continuity equation. The rate of variation of the amount of element i contained in 

This equation holds for any arbitrary volume, and so must be true for the volume element dV. We can therefore drop the integration signs and the common factor dV in order to obtain 

Let us split the flux term pv C into the purely advective term pvC and a diffusive term, hereafter also referred to as T , describing the movement of the species i relative to the center of mass p(v —v)C  

Expanding the derivatives of the left-hand side and the second divergence term on the right-hand side gives 

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The generalized transport equation, equation 17, can be dissected into terms describing bulk flow (term 2), turbulent diffusion (term 3) and other processes, eg, sources or chemical reactions (term 4), each having an impact on the time evolution of the transported property. In many systems, such as urban smog, the processes have very different time scales and can be viewed as being relatively independent over a short time period, allowing the equation to be "spht" into separate operators. This greatly shortens solution times (74). The solution sequence is... 

In this chapter, we shall first make a brief review of the phenomenological aspect of Brownian motion and we shall then show how the general transport equation derived in Section II allows an exact microscopic theory to be developed. 

The motion of a heavy ion in a solvent is, of course, a special case of the general transport equation (112), which for simplicity we shall discuss for a time-independent external field (co = 0) ... 

For simplicity, we shall discuss the case of a time-independent external field the generalized transport equation (111) thus takes the form ... 

In the four preceding sections, we have developed various approximations for the relaxation term in the limiting law for the conductance of electrolytes, starting from the generalized transport equation (111). 

In Section 3.3, the general transport equations for the means, (3.88), and covariances, (3.136), of 0 are derived. These equations contain a number of unclosed terms that must be modeled. For high-Reynolds-number flows, we have seen that simple models are available for the turbulent transport terms (e.g., the gradient-diffusion model for the scalar fluxes). Invoking these models,134 the transport equations become... 

In the case of pure advection (no molecular transport), the diffusion term in the general transport equation (8.2.5) is made equal to zero and time-dependent mass balance is expressed as... 

A conservative property is at steady-state when fluxes, sources, and sinks do not change with time. It is not to be confused with equilibrium which is a state with no flux, no source, and no sink. The general transport equation (8.4.3) of element i at steady-state is... 

Further let s consider the question, which parameters define the value a and, hence, the active time value f. As it is known [5], the relation a/p is connected with exponent p at / in the generalized transport equation as follows ... 

Equation (88) represents the general adsorption/desorption equation in a multi-component solution system which can be used in the general transport equation... 

The simultaneous occurrence of reaction and transport processes can be represented by adding the contributions together and, for the total concentration decrease over time at a given point P(x,y,z) in the media considered by the general transport equation one obtains ... 

For interactions between packaging and product the above descriptions of both material transport processes by diffusion and convection as well as the simultaneous chemical reactions come into consideration. The general transport equation (7-10) is the starting point for solutions of all specific cases occurring in practice. Material loss through poorly sealed regions in the package can be considered as convection currents and/or treated as diffusion in the gas phase. 

Suspended particles arc considered to have finite size thus both the mobility coefficient and diffusion coefficient of the particles depend not only on the size of the particle but also upon the distance between the particle and the collector. A numerical finite-difference technique is used to solve the general transport equation. 

Linear sorption terms have the advantage of simplicity and they provide the possibility to convert them into a retardation factor Rf, so that the general transport equation can be easily expanded by applying the correction term ... 

Convection, diffusion, and dispersion can only describe part of the processes occurring during transport. Only the transport of species that do not react at all with the solid, liquid or gaseous phase (ideal tracers) can be described adequately by the simplified transport equation (Eq. 94). Tritium as well as chloride and bromide can be called ideal tracers in that sense. Their transport can be modeled by the general transport equation as long as no double-porosity aquifers are modeled. Almost all other species in water somehow react with other species or a solid phase. These reactions can be subdivided into the following groups, some of which have already been considered in the previous part of the book. 

Due to the high number of particles in a packed bed, the model for a single particle hes to be simple. Here, an one dimensional approach has been chosen as compromise between accuracy and computing time. The change of a scalar in time within the particle is influenced by diffusion, convection and source terms. Thus, the energy and species distribution over the particle can be described by the general transport equation... 

Field-flow fractionation is, in principle, based on the coupled action of a nonuniform flow velocity profile of a carrier liquid with a nonuniform transverse concentration profile of the analyte caused by an external field applied perpendicularly to the direction of the flow. Based on the magnitude of the acting field, on the properties of the analyte, and, in some cases, on the flow rate of the carrier liquid, different elution modes are observed. They basically differ in the type of the concentration profiles of the analyte. Three types of the concentration profile can be derived by the same procedure from the general transport equation. The differences among them arise from the course and magnitude of the resulting force acting on the analyte (in comparison to the effect of diffusion of the analyte). Based on these concentration profiles, three elution modes are described. 

Generally, the concentration profile of analytes in FFF can be obtained from the solution of the general transport equation. For the sake of simplicity, the concentration profile of the steady-state zone of the analyte along the axis of the applied field is calculated from the one-dimensional transport equation ... 

For the transverse fluctuations with one dynamic variable / = one has the generalized transport equation in the form,... 

The generalized transport equations presented above give a complete picture of the dynamics close to an equilibrium state for arbitrary temporal and spatial scales. 

Many processes in pharmaceutics are related to transport, and the appHcations of the outlined theory are therefore numerous. Notwithstanding their practical importance, the special instances of the general transport equation (11) listed in Table 2 are assumed to be relatively familiar, and will therefore not be discussed further in this chapter. Instead, we focus our attention on applications of hereditary integrals and linear response theory, in particular on dynamic mechanical analysis (DMA) and impedance spectroscopy. 

Chou [23] was the first to derive and publish the generalized transport equation for the Reynolds stresses. The exact transport equation for the Reynolds stresses was established by use of the momentum equation, the continuity equation and a moderate amount of algebra. 

Let -0 be a general instantaneous scalar variable representing quantities like energy, heat, temperature, species mass concentration, etc.. In Cartesian coordinates the general transport equation for tp can be written as... 

The general transport equation for the scalar quantity variance can be formulated, in analogy to the procedure applied for momentum, by subtracting (1.456) from (1.455) to obtain an equation for the turbulent fluctuations (e.g., [153] [167] [154]) ... 

The general transport equation for the specific turbulent fluxes of scalar variables is derived in analogy to the corresponding momentum flux equations, i.e., the Reynolds stress equations. The derivation combines two equations for the fluctuations to produce a flux equation. For the first equation we start with the momentum fluctuation equation (1.389), multiply it by the scalar quantity perturbation ip, and Reynolds average ... 

The moment method can then be employed to derive a generalized equation of change for a mean particle property < ip > in the same manner as described in chap 2 for molecular systems. In particular, the generalized transport equation for < ip r, t) > is derived multiplying (4.1) by a microscopic quantity ip r, c, t) and integrating the resulting relation over the whole velocity space. 

By use of the upwind scheme for the convective terms, the generalized transport equation becomes ... 

Let us now return to the general transport equation (1) for polarized radiation in a plane medium. As a typical transfer problem, we consider the surface Greens function matrix G(t,m 0,), fiQ e[0,l], for a slab with an optical thickness b defined as the solution to the homogeneous transfer equation... 

For the formation and stability of passive films, bulk and surface reactions are important. Ionic reactions in the bulk are called migration, while reactions at the surface are called ion-transfer reactions (ITRs). Both are driven by an electric field d(p/dx, while chemical diffusion plays a role if gradients of the chemical potential exist. The general transport equation is... 

The above relationship allows the simplification of the driving forces in the membrane system. Using the above formulation, substituting the resultant driving forces into the generalized transport equation, Eq. (5.1), and inverting the resultant equations yields the two independent transport equations... 

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