Fluid-phase volume transport

In order to derive a transport equation of the fluid-phase volume fraction af, we will let the internal coordinate be equal to the fluid volume seen by a particle. The fluid-phase volume fraction is then defined by 

The fluid-phase volume-fraction source terms are defined by 

By definition, oTp -i- off = 1 and, in the absence of mass transfer between phases, the right-hand side of Eq. (4.50) is null. In this limit, Eq. (4.56) should reduce to 

A consistent model for the change in fluid volume seen by the particles 

it will be necessary for (Gfi)i to be nonzero in Eq. (4.58) or, in other words, the volume of fluid seen by a particle must change along a particle trajectory. Physically, this change is associated with the presence of other particles (i.e. it cannot be modeled using single-particle physics). A simple consistent model for (Gfi)i can be written as 

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Thus, the sorption of chemicals on the surface of the solid matrix may become important even for substances with medium or even small solid-fluid equilibrium distribution coefficients. For the case of strongly sorbing chemicals only a tiny fraction of the chemical actually remains in the fluid. As diffusion on solids is so small that it usually can be neglected, only the chemical in the fluid phase is available for diffusive transport. Thus, the diffusivity of the total (fluid and sorbed) chemical, the effective diffusivity DieS, may be several orders of magnitude smaller than diffusivity of a nonsorbing chemical. We expect that the fraction which is not directly available for diffusion increases with the chemical s affinity to the sorbed phase. Therefore, the effective diffusivity must be inversely related to the solid-fluid distribution coefficient of the chemical and to the concentration of surface sites per fluid volume. 

Other than the particle dimension d, the porous medium has a system dimension L, which is generally much larger than d. There are cases where L is of the order d such as thin porous layers coated on the heat transfer surfaces. These systems with Lid = 0(1) are treated by the examination of the fluid flow and heat transfer through a small number of particles, a treatment we call direct simulation of the transport. In these treatments, no assumption is made about the existence of the local thermal equilibrium between the finite volumes of the phases. On the other hand, when Lid 1 and when the variation of temperature (or concentration) across d is negligible compared to that across L for both the solid and fluid phases, then we can assume that within a distance d both phases are in thermal equilibrium (local thermal equilibrium). When the solid matrix structure cannot be fully described by the prescription of solid-phase distribution over a distance d, then a representative elementary volume with a linear dimension larger than d is needed. We also have to extend the requirement of a negligible temperature (or concentration) variation to that over the linear dimension of the representa-... 

Transport in the fluid phase inside the packed bed takes place through convection, axial diffusion and flow to or from the zeoHte crystals. A mass balance for a small volume element of the bed results in the following equation for the concentration Cz in the gas phase... 

Here we have emphasized the intrinsic nature of our area-averaged transport equation, and this is especially clear with respect to the last term which represents the rate of reaction per unit volume of the fluid phase. In the study of diffusion and reaction in real porous media (Whitaker, 1986a, 1987), it is traditional to work with the rate of reaction per unit volume of the porous medium. Since the ratio of the fluid volume to the volume of the porous medium is the porosity, i.e. 

One of the most important cases is when there are two (or more) distinct regions within the reactor. This might be a packed bed of porous solids, two fluid phases, partially stagnant regions, or other complicated flows through a vessel that can basically be described by an axial dispersion type model. Transport balances can be made for each phase, per unit reactor volume ... 

At low methanol concentrations the rate drops for all samples. The reason for this is that the polymer loading in the pore volume of the carrier causes a transport limitation for isobutene. So the typical activity pattern for macroporous resins cannot be observed. For the operation of an RD column this is beneficial drastic changes in the rate caused by fluid phase concentration changes do not occur. 

In the above equations, Cpr and Cp< denote heat capacities of the fluid and solid phases, pb is the bed density and hp is the heat transfer coefficient between fluid and particles. Transport of heat through the fluid phase in the axial direction and in the radial direction of the bed by conduction are described by the effective thermal conductivities, ka,i and kas, while in the solid phase thermal conduction can be assumed to be isotropic and the effective thermal conductivity ka can be used to express this effect. Q i represents the heat evolution/absorption by adsorption or desorption on the basis of bed volume. This model neglects the temperature distribution in the radial position of each particle, which may seem contradictory to the case of mass transfer, where intraparticle mass transfer plays a significant role in the overall adsorption rate. Usually in the case of adsorption, the time constant of heat transfer in the particle is smaller than the time constant of intraparticle diffusion, and the temperature in the particle may be assumed to be constant. 

In Chapter 2, rates were derived for reactions confined to an elementary volume of a catalyst particle, uniform in temperature and concentration. In the present chapter the complete particle and its immediate fluid-phase environment will be considered. The conditions in this system are not necessarily uniform, so that the various transport phenomena mentioned in the Introduction to Chapter 2 now have to be accounted for. Accordingly, the particle will be considered either as a pseudocontinuum or as a truly heterogeneous medium. 

It is remarked that in the standard literature on fluid dynamics and transport phenomena three different modeling frameworks, which are named in a physical notation rather than in mathematical terms, have been followed formulating the single phase balance equations [91]. These are (1) The infinitesimal particle approach [2, 3, 67, 91, 145]. In this case a differential cubical fluid particle is considered as it moves through space relative to some fixed coordinate system. By applying the balance principle to this Lagrangian control volume the conservation equations for... 

The porosity or pore water volume fraction of total bed volume e (m m ) is obviously a key independent variable for assessing diffusive transport in porous media. The water that is contained in the bed is called the porewater or interstitial water because it fills the pores or interparticle spaces. It is the key phase for describing chemical mass transport interactions with the overlying water. Hence, all in-bed fluxes of dissolved constituents are transported in this fluid phase. 

Mass transfer through dense polymeric membranes is nowadays accepted to be described by the sorption-diffusion mechanism. According to this, the species being transported dissolve (sorb) in the polymer membrane surface on the higher chemical potential side, diffuse through the polymer free volume in a sorbed phase, and pass into the fluid phase downstream of the membrane (lower chemical potential side). In the case of dense polymeric membranes the polymer is an active participant in both the solution and diffusion processes. However, since in many porous membranes the mass transfer takes place mainly in the pores, the membrane material is not an active participant and only its pore structure is important. ... 

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