Fluid side mass transport

For transport of BSA (D = 7.3-10 11 m2/s) in a packed bed of porous adsorbents (dp = 90 pm) at 2 cm/min linear flow rate kf is calculated as 9.4 10 6 m/s when a voidage of the packed bed of 0.4 is assumed. As kf is proportional to Re112 fluid side mass transport efficiency is increased with linear flow rate and under conditions of standard packed bed protein chromatography this is not assumed to be the rate limiting step of the sorption process. 

Summarizing this short discussion it has to be stated, that up to now experiments providing absolute numbers of kf during protein adsorption in fluidized beds are not available, the interpretations are based on correlations derived for small ions. As ion exchange with fluidized resins is performed at much higher Reynolds numbers and mostly is not limited by particle side transport, the validity of the correlations for proteins has to be proven. Nevertheless, the influence of bed expansion at increased linear flow rate cannot be neglected and fluid side mass transport should be considered as a system parameter governing the sorption process in a fluidized bed under certain conditions. 

We note, first, that transport resistance is on the fluid side in most cases the main exception is gas permeation, which, because of the much higher diffusivities involved, is controlled by the membrane resistance. When fluid flow is laminar on the core side, mass transport will be in the entry region... 

The assessment of the role of kf during protein adsorption in a fluidized bed may be performed with the help of a dimensionless transport number. Slater used the correlations provided by Rodrigues to simulate film transport limited adsorption of small ions to fluidized resins [54], In this study dimensionless groups were used to describe the influence of the system parameters particle side transport, liquid dispersion, and fluid side transport. Dispersion was accounted for by the column Peclet number analogous to Bo as introduced above and mass transport from the bulk solution to the resin was summarized in a fluid side transport number NL. 

Sides and Tobias17 observed the coalescence of large bubbles causing fluid motions close to the electrode surface which may be important in the mass transport enhancement due to gas evolution... 

On the other hand, researchers interested in mass transport in water waves have shown that a similar phenomenon occurs for incompressible flow (see [4] for an example). The salient feature is when an obstacle oscillates with high frequency in the quiescent viscous fluid. In this case the steady streaming emerges from both sides of the obstacle along the oscillating direction at small viscosity [5]. Increased importance of the acoustic streaming in small scales has been addressed in [6]. 

Mass transport across isodensity lines should become particularly important when the lubrication approximation breaks down. This should happen near the contact line in the case when two alternative fluid densities near the solid wall are possible. If. say. the boundary densities are Psv 1 and / 5/ = 1 - a, a 1, the three-phase contact line can be viewed as a sharp transition between 0(1) positive and negative values of the nominal thickness h, such that < 1 on either side. This can be treated as a shock of Eq. (91) or (93). The Hugoniot condition, which should ensure zero net flux through the shock, is the equality of chemical potentials on both sides. Unfortunately, this condition cannot be formulated precisely, since the sharp-interface limit of the surface tension term is inapplicable in the shock region. Moreover, our test computations of the profile of the dense layer using Eq. (94) with different boundary conditions imposed on the shock at h = ho showed that the spreading velocity is very sensitive to the conditions on the shock. 

The two-film concept can also be applied to heat transfer operations, as shown in Figure 1.7b. The process is similar to that of mass transport, but differs from it in two important aspects. Eirst, the two fluids (hot and cold) are usually, but not always, separated by a solid partition. This is in contrast to mass transfer operations where direct contact of the phases is the norm. Second, no phase-equilibrium relation needs to be invoked at the interface. Instead, convergence of the two temperature profiles on either side of an interface leads to one and the same temperature at this point. No jiunp-discontinuities in temperature occur at any location along an interface. We... 

The conditions described above ensure that the feed-liquid membrane interface and the liquid membrane-sweep/strip interface will be stable and mass transport will continue. Any loss or change in the liquid membrane is continuously made up by a pressurized membrane liquid reservoir connected to the shell-side membrane fluid. Majumdar et al. (2001) provide a concise description of this technique. The basic technique has been illustrated in Majumdar et al. (1988) and Guha et al. (1992) for gas separation and in Sengupta et al. (1988) for liquid separation. For gas separation, Guha et al. (1992) have demonstrated that feed-gas separation may he implemented with conventional gas permeation, permeate side under vacuum, sweep gas and sweep liquid. In both techniques the overall resistance to transfer maybe described as a sum of five resistances in series ... 

Diffusion with reactive decay and mass transfer to the adjoining fluid phase. This case extends the previous one by including a MTC, k (m/s), on the fluid side to capture this transport resistance. The concentration profile equation for the same idealized initial solid phase conditions is... 

A total of 41 individual transport processes are listed in Table 4.1 as being the most significant ones of concern in this handbook. Seven in the hst are soil-side transport processes and the main focus of Part 2 of this chapter. Several of these processes including diffusion, advection, and fluid-to-solid mass transfer in porous media are also relevant to many other environmental compartments and are covered in more detail in other chapters of this book. Specifically the chapters are mass transport fundamentals from an environmental perspective (Chapter 2) molecular diffusion estimation methods (Chapter 5) advective porewater flux and chemical transport in bed-sediment (Chapter 11) diffusive chemical transport across water and sediment boundary layers (Chapter 12) bioturbation and other sorbed-phase transport processes in surface soils and sediments (Chapter 13), and dispersion and mass transfer in groundwater of the near-surface geologic formations (Chapter 15). 

Chemical transport across the water-sediment interface takes place through numerous chemical, biological, and physical mechanisms (Reible et al., 1991 Thibodeaux and Mackay, 2007). These reflect in large part the characteristics of the particular aquatic system, which include flowing freshwater streams, lakes, estuaries, and the marine, both near shore and beyond. This chapter is focused on so-called diffusive type processes on either side of the sediment-water interface. Specifically, it covers the water-side mass transfer coefficient in the fluid boundary layer above the bed and diffusion within the interparticle pore spaces in the near-surface bed sediment layers. 

Sections 12.2 and 12.3 discuss the mass transport on the water-side of the sediment-water interface and address MTCs influenced by aquatic stream flow and wind-generated convective transport. After a brief description of some key characteristics of this fluid boundary layer, one direct technique of measuring the water-side MTCs are presented. This helps focus the user as to precisely what this subject is about. 

The bed surface, in the context of this section, is considered to be nonmobile porous material made of various size solid particles. The particles typically range in diameter from a few millimeters to less than a micron. The various bed types generally reflect the fluid dynamic nature of the water column above the bed. Table 12.7 contains typical porosity values for sediment. Section 12.2.1 contains an overview description of the types of aquatic streams and currents above the beds. These aquatic systems include rivers, lakes, estuaries, shelf, and marine environments. Unlike the air-water interface, the sediment-water interface has a single fluid (i.e aqueous phase) on either side. Water, the continuous phase, exists from within the column above, through the imaginary interface plane and into the porous bed where it is termed porewater. The interface plane is not a sharp one. It can be considered a thin mixed layer of finite thickness in the context of mass-transfer modeling (DiToro, 2001). Visual and physical examination of thin-sliced (0.1mm) layers of a frozen core sample from a lake sediment bed microcosm showed the presence of a finite flocculent layer positioned between the water side and the particles on the bed surface (Formica et al 1988). Little is known about this layer from a mass-transfer perspective, it will not be considered further. Mass transport in those bed surface layers at and below the first layer of solid particles will be the subject of this section. 

In this Section, it is implicitly assumed that the mass transport resistance at the fluid-membrane interface on either side of the membrane is negligible. Also the following is information that is made available publicly by the membrane manufacturers, when not otherwise noted. As in technical processes, mass transport across semipermeable medical membranes is conveniently related to the concentration and pressme driving forces according to irreversible thermodynamics. Hence, for a two-component mixture the solute and solvent capacity to permeate a semipermeable membrane under an applied pressure and concentration gradient across the membrane can be expressed in terms of the following three parameters Lp, hydraulic permeability Pm, diffusive permeability and a, Staverman reflection coefficient (Kedem and Katchalski, 1958). All of them are more accurately measured experimentally because a limited knowledge of membrane stmcture means that theoretical models provide rather inaccurate predictions. 

---