Continuity equations specific forms

Several colloidal systems, that are of practical importance, contain spherically symmetric particles the size of which changes continuously. Polydisperse fluid mixtures can be described by a continuous probability density of one or more particle attributes, such as particle size. Thus, they may be viewed as containing an infinite number of components. It has been several decades since the introduction of polydispersity as a model for molecular mixtures [73], but only recently has it received widespread attention [74-82]. Initially, work was concentrated on nearly monodisperse mixtures and the polydispersity was accounted for by the construction of perturbation expansions with a pure, monodispersive, component as the reference fluid [77,80]. Subsequently, Kofke and Glandt [79] have obtained the equation of state using a theory based on the distinction of particular species in a polydispersive mixture, not by their intermolecular potentials but by a specific form of the distribution of their chemical potentials. Quite recently, Lado [81,82] has generalized the usual OZ equation to the case of a polydispersive mixture. Recently, the latter theory has been also extended to the case of polydisperse quenched-annealed mixtures [83,84]. As this approach has not been reviewed previously, we shall consider it in some detail. 

The device model describes transport in the organic device by the time-dependent continuity equation, with a drift-diffusion form for the current density, coupled to Poisson s equation. To be specific, consider single-carrier structures with holes as the dominant carrier type. In this case,... 

There is a nonzero mass source in the continuity equation, Sm, arising from the summation of all species equations. A general form of this source term is given in Table 1. Use has been made of the assumption that summation of interspecies diffusion within the gas phase is equal to zero. Specifically, one has... 

With some significant simplifying assumptions, the species-continuity equation can be put into a form that is analogous to the thermal-energy equation. Specifically, consider that there is no gas-phase chemistry and that a single species, A, is dilute in an inert carrier gas, B. In this case, considering Eq. 3.128, Eq. 6.24 reduces to... 

Fixed Coordinate Approaches. In the fixed coordinate approach to airshed modeling, the airshed is divided into a three-dimensional grid for the numerical solution of some form of (7), the specific form depending upon the simplifying assumptions made. We classify the general methods for solution of the continuity equations by conventional finite difference methods, particle in cell methods, and variational methods. Finite difference methods and particle in cell methods are discussed here. Variational methods involve assuming the form of the concentration distribution, usually in terms of an expansion of known functions, and evaluating coeflBcients in the expansion. There is currently active interest in the application of these techniques (23) however, they are not yet suflBciently well developed that they may be applied to the solution of three-dimensional time-dependent partial differential equations, such as (7). For this reason we will not discuss these methods here. 

With regard to mass transfer we will restrict ourselves to a binary mixture with components that have approximately the same specific heat capacities, so that the energy equation remains valid in the form given above. In addition the continuity equation for a component holds... 

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. 

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. 

Inspection of the K-H stability condition indicates that the structure of Equation 18 is invariant with the specific modelling of the wall and interfacial shear stresses and evolves essentially from the continuity equations and the left hand side of the momentum equations. On the other hand. Equation 19 for is directly related to the quasi-steady models adopted for the various shear stresses terms (the rhs of the two-fluid momentum Equation 8). In this sense, the form of 18 is general and is affected by the specific modelling of shear stresses only indirectly through the value. Thus, given different correlations for the shear stresses, the general form of 19 provides the corresponding values for... 

The first five terms on the RHS of (4.299) vanish since the specific property tpi equals a constant. The first part of the last term on the RHS of (4.299), vanishes too. Therefore, the complete continuity equations reduce to the form ... 

Quantitative Relationship of Conductivity and Antistatic Action. Assuming that an antistatic finish forms a continuous layer, the conductance it contributes to the fiber is proportional to the volume or weight and specific conductance of the finish. As long as the assumption of continuity is fulfilled it does not matter whether the finish surrounds fine or coarse fibers. Assuming a cylindrical filament of length 1 cm and radius r, denoting the thickness of the finish layer as Ar and the specific conductance of the finish k, the conductance R of the finish layer is given by the equation (84) ... 

Equation 10.1.1 represents a very general formulation of the first law of thermodynamics, which can be readily reduced to a variety of simple forms for specific applications under either steady-state or transient operating conditions. For steady-state applications the time derivative of the system energy is zero. This condition is that of greatest interest in the design of continuous flow reactors. Thus, at steady state,... 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

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