Particle volume, constant flux

Finally, the diffusion of a chemical may be influenced by another diffusing compound or by the solvent. The latter effect is known as solute-solvent interaction it may become important when solute and solvent form an association that diffuses intact (e.g., by hydration). This may be less relevant for neutral organic compounds, but it plays a central role for diffusing ions. But even for noncharged particles the diffusivities of different chemicals may be coupled. The above example of the glycerol diffusing in water makes this evident in order to keep the volume constant, the diffusive fluxes of water and glycerol must be coupled. 

R bed radius, length, catalyst particle radius, length gas constant, kcal/mol-K rate of reactant consumption, mols/volume slurry-time rate of reaction of A, mols/volume-time flux of A, equation (8-137)... 

The expansion characteristics of homogeneously fluidized beds have been the subject of far more empirical study than theoretical analysis. This could be due to the uncomplicated nature of the experimental procedure, which involves simply the measurement of steady-state bed height Lg as a function of volumetric fluid flux U. The results are usually presented as the relation of U with void fraction e, which, unlike Lg, is independent of the quantity of particles present. The constant particle volume relation. 

The phenomenological constants can be expressed in terms of experimental quantities. The Ly represents the interaction of component i with other particles of component/ In molten salts no component seems particularly suitable for serving as a solvent. The use of a volume fixed frame of reference for defining the fluxes gives a more symmetrical representation. The equations for the phenomenological constants are given by Sund-heim. ... 

The production of species i (number of moles per unit volume and time) is the velocity of reaction,. In the same sense, one understands the molar flux, jh of particles / per unit cross section and unit time. In a linear theory, the rate and the deviation from equilibrium are proportional to each other. The factors of proportionality are called reaction rate constants and transport coefficients respectively. They are state properties and thus depend only on the (local) thermodynamic state variables and not on their derivatives. They can be rationalized by crystal dynamics and atomic kinetics with the help of statistical theories. Irreversible thermodynamics is the theory of the rates of chemical processes in both spatially homogeneous systems (homogeneous reactions) and inhomogeneous systems (transport processes). If transport processes occur in multiphase systems, one is dealing with heterogeneous reactions. Heterogeneous systems stop reacting once one or more of the reactants are consumed and the systems became nonvariant. 

An important property of fluidised beds follows immediately from this simple relation as the bed expands, the product (1-e) L, which represents the total volume of particles per unit cross section, remains unchanged. As the fluid flux is increased, L increases and (1 — s) decreases so as to maintain their product at a constant value. As a consequence, the pressure drop through the fluidised bed remains constant for further increases in fluid velocity, as shown in Figure 9. 

It is possible that the rate-determining process in the kinetics of ion exchange is the film diffusion. Consider a spherical particle encircled by an aqueous solution sphere (see Figure 7.8), in which the zeolite is homoionic at t = 0, the electrolytic solution has a very high volume (i.e., C2A == constant), and the diffusion is stationary and one-dimensional in a direction perpendicular to the zeolite surface [44], Then, under the conditions discussed above, it is possible to calculate the exchange flux of cation B, that is,. 7 , as follows [23] ... 

Now the right-hand side represents an energy flux occurring from the temperature gradient, in the absence of any net particle flow also, at constant volume no work is performed. The resulting Ju thus is a heat flux the proportionality coefficient in (6.6.6) is equivalent to the thermal conductivity, k. This leads to the identification... 

If we assume for a moment that the linear dissolution rates (LDR) are constant as long as the particle survives and that the flux (F) from the surface may be described in terms of moles silica/cm sec, then this surface flux times the molar volume (MV) of the mineral divided by the moles of Si per molecule ( Si) gives the rate at which the radius of a spherical particle or half the length of a cube will disappear by the dissolution process (9). 

The flux is defined up to the constant A. Taking A = (a single particle wavefunction normalized in the volume Q) implies that the relevant observable is 2VJ(r), that is, is the particle flux for a system with a total of N particles with N Sometimes it is convenient to normalize the wavefunction to unit flux, J = 1 by choosing A =... 

Equation (V. 10) is known as Fick s first law of diffusion. In this equation D is the diffusion coefficient in units of mV1. The diffusional flux,/d, represents the amount of substance that crosses a section of unit area, S, per second in the direction normal to that of diffusion. In the above equation dn/dz is the concentration gradient, which for steady-state diffusion is constant in time at any point within the system. The units of j and c should be consistent. We will further express the concentration as either c or n, where c is the number of moles of dispersed particles per unit volume (it is assumed that 1 mole contains 6.02xl023 colloidal particles), and n is the particle number concentration, i.e., the number of particles per unit volume. Consequently, d c and yd are expressed in mol m 2 s 1, and m"2 s"1, respectively. According... 

Another model, known as the Solids Flux Model, was developed by Belfort et al. (1994). This was proposed for sticky particles, which do not backmigrate from the membrane to the bulk solution and cause irreversible fouling. The constant b describes the characteristics of the sublayer and fl>s the solids volume fraction in the feed. 

See, for example, Einstein (1956). The osmotic pressure may be written as p — vkT where V is the number density of particles, k the Boltzmann constant, and T is the absolute temperature. The net osmotic force per unit volume is given by —Vp= —kTVv = /vv where vv is the diffusive flux and / is the friction coefficient that arises from hydrodynamic analysis. The diffusion coefficient must therefore be given by, = kT/f. 

A 10 m long vertical standpipe of inside diameter 0.1 m transports solids at a flux of 100 kg/m s from an upper vessel which is held at a pressure 1.0 bar to a lower vessel held at 1.5 bar. The particle density of the solids is 2500 kg/m and the surface-volume mean particle size is 250 gm. Assuming that the voidage is constant along the standpipe and equal to 0.50, and that the effect of pressure change may be ignored, determine the direction and flow rate of gas passing between the vessels. (Properties of gas in the system density, 1 kg/m viscosity, 2 x 10 Pas.)... 

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