Velocity mass diffusion

Analysis of a method of maximizing the usefiilness of smaH pilot units in achieving similitude is described in Reference 67. The pilot unit should be designed to produce fully developed large bubbles or slugs as rapidly as possible above the inlet. UsuaHy, the basic reaction conditions of feed composition, temperature, pressure, and catalyst activity are kept constant. Constant catalyst activity usuaHy requires use of the same particle size distribution and therefore constant minimum fluidization velocity which is usuaHy much less than the superficial gas velocity. Mass transport from the bubble by diffusion may be less than by convective exchange between the bubble and the surrounding emulsion phase. 

In a boundary layer equation the mass center is considered with the help of the velocity (u, Uy, u ) and therefore a distribution of the velocity of the mass center is desirable. The diffusion velocity and diffusion factor are determined with regard to velocity v, giving a formula for Vax /x, but not for /ax - x useful approach is offered by Eq. (4.268c), using the artificial multiplication factor (v - ax /... 

If the boundary motion is controlled by an independent process, then the boundary motion velocity is independent of diffusion. This can happen if the magma is gradually cooling and crystal growth rate is controlled both by temperature change and mass diffusion. This problem does not have a name. In this case, u depends on time or may be constant. If the dependence of u on time is known, the problem can also be solved. The Stefan problem and the constant-w problem are covered below. 

The mass diffusion velocity of species k relative to the mass average velocity will be denoted V, which is defined by... 

It is possible, and sometimes desirable, to write the mass diffusion velocity V in terms of a mole fraction gradient (rather than the mass fraction gradient in Eq. 12.159). We form a hybrid of Eqs. 12.159 and 12.161 as... 

Writing a generalization of Eq. 12.162, the expression for the mass diffusion velocity of species k in terms of the driving force of species k is... 

No approximation is made when using Eqs. 12.168 and 12.169. Exactly the same mass diffusion velocities and fluxes are obtained in this approach as would be calculated via Eqs. 12.165 and 12.166 (or from the Stefan-Max well approach described next). 

It is sometimes advantageous to write the mass diffusive velocity as a function of the mole-fraction gradient (as was done in Eq. 12.168) rather than the mass-fraction gradient. However, as was seen in Section 12.7.1, a different diffusion coefficient must be employed. Beginning with the expression for the mass diffusion velocity,... 

Table 2.1 presents corresponding well-known empirical force-flux laws that apply under certain conditions. These are Fourier s law of heat flow, a modified version of Fick s law for mass diffusion at constant temperature, and Ohm s law for the electric current density at constant temperature.5 The mobility, Mj, is defined as the velocity of component i induced by a unit force. 

In this example, the equilibrium concentrations maintained at the interface are functions of the interface temperature, which in turn is a function of time. In addition, the velocity of the interface, v (i.e., rate of solidification), depends simultaneously upon the mass diffusion rates and the rates of heat conduction in the two phases, as may be seen by examining the two Stefan conditions that apply at the interface. For the mass flow the condition is... 

Internal diffusional limitations are possible any time that a porous immobilized enzymatic preparation is used. Bernard et al. (1992) studied internal diffusional limitations in the esterification of myristic acid with ethanol, catalyzed by immobilized lipase from Mucor miehei (Lipozyme). No internal mass diffusion would exist if there was no change in the initial velocity of the reaction while the enzyme particle size was changed. Bernard found this was not the case, however, and the initial velocity decreased with increasing particle size. This corresponds to an efficiency of reaction decrease from 0.6 to 0.36 for a particle size increase from 180 pm to 480 pm. Using the Thiele modulus, they also determined that for a reaction efficiency of 90% a particle size of 30 pm would be necessary. While Bernard et al. found that their system was limited by internal diffusion, Steytler et al. (1991) found that when they investigated the effect of different sizes of glass bead, 1 mm and 3 mm, no change in reaction rate was observed. 

Table 1 gives the components present in the crude DDSO and their properties critical pressure (Pc), critical temperature (Tc), critical volume (Vc) and acentric factor (co). These properties were obtained from hypothetical components (a tool of the commercial simulator HYSYS) that are created through the UNIFAC group contribution. The developed DISMOL simulator requires these properties (mean free path enthalpy of vaporization mass diffusivity vapor pressure liquid density heat capacity thermal conductivity viscosity and equipment, process, and system characteristics that are simulation inputs) in calculating other properties of the system, such as evaporation rate, temperature and concentration profiles, residence time, stream compositions, and flow rates (output from the simulation). Furthermore, film thickness and liquid velocity profile on the evaporator are also calculated. 

FIGURE 4 An example of reference velocities. Descriptions of diffusion imply reference to a velocity relative to the system s mass or volume. While the mass often has a nonzero velocity, the volume often shows no velocity. Hence, diffusion is best referred to the volume s average velocity. 

Obtain the Taylor-Prandtl modification of the Reynolds analogy between momentum and heat transfer and give the corresponding analogy for mass transfer. For a particular system a mass transfer coefficient of 8.71 x 10-6 m/s and a heat transfer coefficient of 2730 W/m2K were measured for similar flow conditions. Calculate the ratio of the velocity in the fluid where the laminar sub-layer terminates, to the stream velocity. Molecular diffusivity = 1.5 x 10 9 m2/s. Viscosity = 1 mN s/m2. Density = 1000 kg/m3. Thermal conductivity = 0.48 W/m K. Specific heat capacity = 4.0 kJ/kg K. 

The Lu number is the ratio between the mass diffusion coefficient and the heat diffusion coefficient. It can be interpreted as the ratio between the propagation velocity of the iso-concentration surface and the isothermal surface. In other words, it characterizes the inertia of the temperature field inertia, with respect to the moisture content field (the heat and moisture transfers inertia number). The LUp diffusive filtration number is the ratio between the diffusive filtration field potential (internal pressure field potential) and the temperature field propagation. 

In general, the procedure for designing a bubble column reactor (BCR) (1 ) should start with an exact definition of the requirements, i.e. the required production level, the yields and selectivities. These quantities and the special type of reaction under consideration permits a first choice of the so-called adjustable operational conditions which include phase velocities, temperature, pressure, direction of the flows, i.e. cocurrent or countercurrent operation, etc. In addition, process data are needed. They comprise physical properties of the reaction mixture and its components (densities, viscosities, heat and mass diffusivities, surface tension), phase equilibrium data (above all solubilities) as well as the chemical parameters. The latter are particularly important, as they include all the kinetic and thermodynamic (heat of reaction) information. It is understood that these first level quantities (see Fig. 3) are interrelated in various ways. 

To this point we have limited onr consideration to mass diffitsion in a station aiy medium, and thus the only ntotion involved was the creeping motion of molecules in the direction of decreasing concentration, and there was no motion of the mixture as a whole. Many practical problems, such as the evaporation of water from a lake under the iiifliience of the wind or the mixing of two fluids as they flow in a pipe, involve diffusion in a moving medium where the hoik motion i.s caused by an external force. Mass diffusion in such c.nses is complicated by the fact that chemical species are transported both by diffusion and by the bulk motion of the medium (i.e., convection). The velocities and mass flow rates of species in a moving medium consist of two components one due to molecular diffusion and one due to convection (Fig. 14-29). 

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