Mass transfer velocity dependence

In these models equations of the mass action law for most slow mass transfer processes are replaced by equations in partial derivatives, which include velocity. As the mass transfer velocity depends on water composition and at approach to the saturation it decreases, kinetic mass transfer equations have the nature of recurrent (see equations (2.275) and (2.278)), and that is why modeling of mass transfer kinetics is performed using a two-step method of computation described below. 

When two-phase flow is compared to the single-phase case for the same flow rate of an individual phase, it is an experimental fact that the frictional pressure drop will always be higher for two-phase flow. This higher pressure drop may be caused by the increased velocity of the phases due to the reduction in cross-sectional area available for flow, and also to interactions occurring at the extended gas-liquid interface which exists in most of the possible flow patterns. It is equally true that the heat flux will always be higher for two-phase flow than for the same situation in single-phase flow with the same liquid flow rate. On the other hand, mass transfer will depend upon both the extent of the gas-liquid interface and the relative velocity between the two flowing phases. 

A boundary layer is formed between the two phases (fluid and solid). This is a stagnant film that represents a layer of less movement of the fluid and hence builds up a zone with resistance to mass transfer. The mass transfer coefficient and generally the mass transfer rate depend on the fluid dynamics of the system. Higher fluid velocities significantly reduce the thickness of the film. 

The efficiency of the operation is conditioned by three main factors (1) active enzyme concentration, (2) mass transfer rate, and (3) operational parameters, such as plugging. Mass transfer rate depends in large part upon the linear velocity of the fluid through the bed. Hence, in order to maximize the efficiency, high L/D ratios are required, which may also reduce back mixing. However, the length of the bed is limited by the pressure that immobilized particles may withstand [116]. Flow rate, L/D ratio, pH, and temperature are some of the operational parameters that should be optimized for an efficient operation. In Shukla et al., a fixed bed reactor is used for phenol oxidation by HRP [87]. At least three cycles were needed for 80% phenol removal under the optimal L/D ratio, HRT, temperature, and substrate concentration. 

In all cases, the mass-transfer coefficient depends upon the diffusivity of the transferred material and the thickness of the effective film. The latter is largely determined by the Reynolds number of the moving fluid, that is, its average velocity, density, and viscosity, and some linear dimension of the system. Dimensional analysis gives the following relation ... 

The mass transfer coefficient depends on the flow condition of gas and liquid phases, the interface area is influenced by the geometry of the column internals and local velocity of the two phases. The largest driving force for the mass transfer is the concentration difference when the two phases are uniformly distributed over the entire flow area. This is achieved when a countercurrent flow pattern of the two phases without remixing is reached in a theoretical plate. 

The result obtained from the film theory is that the mass transfer coefficient is directly proportional to the diffusion coefficient. However, the experimental mass transfer data available in the literature [6], for gas-liquid interfaces, indicate that the mass transfer coefficient should rather be proportional with the square root of the diffusion coefficient. Therefore, in many situations the film theory doesn t give a sufficient picture of the mass transfer processes at the interfaces. Furthermore, the mass transfer coefficient dependencies upon variables like fluid viscosity and velocity are not well understood. These dependencies are thus often lumped into the correlations for the film thickness, 1. The film theory is inaccurate for most physical systems, but it is still a simple and useful method that is widely used calculating the interfacial mass transfer fluxes. It is also very useful for analysis of mass transfer with chemical reaction, as the physical mechanisms involved are very complex and the more sophisticated theories do not provide significantly better estimates of the fluxes. Even for the description of many multicomponent systems, the simplicity of the model can be an important advantage. 

The mass-transfer coefficients depend on the mass velocities, which are not constant. The change in liquid rate is very small, and the liquid-film coefficient is based on an average flow rate. The molecular weight of SO2 is 64.1, and the average mass velocity is... 

SCALE-UP. The width of the mass-transfer zone depends on the mass-transfer rate, the flow rate, and the shape of the equilibrium curve. Methods of predicting the concentration profiles and zone width have been published, but lengthy computations are often required, and the results may be inaccurate because of uncertainties in the mass-transfer correlations. Usually adsorbers are scaled up from laboratory tests in a small-diameter bed, and the large unit is designed for the same particle size and superficial velocity. The bed length need not be the same, as shown in the next section. 

In an adiabatic adsorption column the temperature front generally travels at a velocity which is different from the velocity of the primary mass transfer front and, since adsorption equilibrium is temperature dependent, a secondary mass transfer zone is established coincident with the thermal front. In a system with finite heat loss from the column wall one may approach either the isothermal situation with a single mass transfer zone or the adiabatic situation with two mass transfer zones, depending on the relative rates of heat generation and dissipation from the column wall. In the former case the effect of finite heat transfer resistance is to widen the mass transfer zone relative to an isothermal system. 

The external mass transfer resistance depends heavily on fluid flow conditions, such as temperature, pressure, and superficial velocity, in the reactor and the particle size of the catalyst. Varying these parameters can help to reduce the external diffusion restriction, for example, by increasing the velocity of the fluid phase over the particles. 

Removal of impurities in the vapor phase can be achieved by the passage of the vapor through a solid bed of large surface area materials (Ruhl, 1971). Molecular sieves and silica gel are most commonly used for this purpose. No adsorption equilibrium is established, but a mass transfer zone is formed in which the quantity of adsorbed impurity decreases to zero (see Figure 19). The width and the profile of this mass transfer zone depend on the temperature, the flow velocity, the concentration of the components to be adsorbed, and on the kind and granularity of the adsorbing material. 

The Reynolds number includes Voo, the drop velocity relative to its surroundings or slip velocity. If drops move with the surrounding fluid, Voc is negligible, and heat and mass transfer rates depend solely on conduction and diffusion, respectively. If drops are suspended as in fluidization, heat and mass transfer coefficients will increase due to increased slip velocity. 

Eor a linear system f (c) = if, so the wave velocity becomes independent of concentration and, in the absence of dispersive effects such as mass transfer resistance or axial mixing, a concentration perturbation propagates without changing its shape. The propagation velocity is inversely dependent on the adsorption equiUbrium constant. 

Convection heat transfer is dependent largely on the relative velocity between the warm gas and the drying surface. Interest in pulse combustion heat sources anticipates that high frequency reversals of gas flow direction relative to wet material in dispersed-particle dryers can maintain higher gas velocities around the particles for longer periods than possible ia simple cocurrent dryers. This technique is thus expected to enhance heat- and mass-transfer performance. This is apart from the concept that mechanical stresses iaduced ia material by rapid directional reversals of gas flow promote particle deagglomeration, dispersion, and Hquid stream breakup iato fine droplets. Commercial appHcations are needed to confirm the economic value of pulse combustion for drying. 

The general expression given by Eq. (14-8) is more complex than normally is required, but it must be used when the mass-transfer coefficient varies from point to point, as may be the case when the gas is not dilute or when the gas velocity varies as the gas dissolves. The values of yi to be used in Eq. (14-8) depend on the local hquid composition Xi and on the temperature. This dependency is best represented by using the operating and equilibrium lines as discussed later. 

It is now seen that only the resistance to the mass transfer term for the stationary phase is position dependent. All the other terms can be used as developed by Van Deemter, providing the diffusivities are measured at the outlet pressure (atmospheric) and the velocity is that measured at the column exit. 

The relationship between adsorption capacity and surface area under conditions of optimum pore sizes is concentration dependent. It is very important that any evaluation of adsorption capacity be performed under actual concentration conditions. The dimensions and shape of particles affect both the pressure drop through the adsorbent bed and the rate of diffusion into the particles. Pressure drop is lowest when the adsorbent particles are spherical and uniform in size. External mass transfer increases inversely with d (where, d is particle diameter), and the internal adsorption rate varies inversely with d Pressure drop varies with the Reynolds number, and is roughly proportional to the gas velocity through the bed, and inversely proportional to the particle diameter. Assuming all other parameters being constant, adsorbent beds comprised of small particles tend to provide higher adsorption efficiencies, but at the sacrifice of higher pressure drop. This means that sharper and smaller mass-transfer zones will be achieved. 

The air, in passing through the chamber at velocities in the range 1.5 to 3.5 m s" comes into intimate contact with the water, and depending on the conditions required, mass transfer of moisture into the airstream occurs. This transfer produces either addition or removal of moisture hot or chilled water is used in this process. 

Parameters a and b are related to the diffusion coefficient of solutes in the mobile phase, bed porosity, and mass transfer coefficients. They can be determined from the knowledge of two chromatograms obtained at different velocities. If H is unknown, b can be estimated as 3 to 5 times of the mean particle size, where a is highly dependent on the packing and solutes. Then, the parameters can be derived from a single analytical chromatogram. 

Momentum depends upon mass and velocity. The particle approaches the wall with momentum mv and leaves with this same momentum in the opposite direction. The momentum transferred to the wall is, then... 

Hydrodynamics and mass transfer in bubble columns are dependent on the bubble size and the bubble velocity. As the bubble is released from the sparger, it comes into contact with media and microorganisms in the column. In sugar fermentation, glucose is converted to ethanol and carbon dioxide ... 

Most theoretical studies of heat or mass transfer in dispersions have been limited to studies of a single spherical bubble moving steadily under the influence of gravity in a clean system. It is clear, however, that swarms of suspended bubbles, usually entrained by turbulent eddies, have local relative velocities with respect to the continuous phase different from that derived for the case of a steady rise of a single bubble. This is mainly due to the fact that in an ensemble of bubbles the distributions of velocities, temperatures, and concentrations in the vicinity of one bubble are influenced by its neighbors. It is therefore logical to assume that in the case of dispersions the relative velocities and transfer rates depend on quantities characterizing an ensemble of bubbles. For the case of uniformly distributed bubbles, the dispersed-phase volume fraction O, particle-size distribution, and residence-time distribution are such quantities. 

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