Mass transfer diffusion

This mode is primarily important for mass transfer in a stationary medium such as in a solid and stationary fluid. 

As we remove the separating membrane, mass diffusion takes place in the direction of decrease in concentrations. Hence, species i diffuses from the 

The diffusion rate equation is given by Tick s law of diffusion, which expresses the transfer of a species i in a mixture of i and 

= molar concentration or molar density of species i (kmole/m ) 

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To determine how the height of a theoretical plate can be decreased, it is necessary to understand the experimental factors contributing to the broadening of a solute s chromatographic band. Several theoretical treatments of band broadening have been proposed. We will consider one approach in which the height of a theoretical plate is determined by four contributions multiple paths, longitudinal diffusion, mass transfer in the stationary phase, and mass transfer in the mobile phase. 

Cussler, Diffusion Mass Transfer in Fluid Systems, Cambridge, 1984. 

Controlled by diffusive mass transfer Controlled by chemical factors A major distinction is between reactions that are ... 

Axial diffusion Mass transfer by diffusion along streamlines that occurs at... 

Cussler, E.L. Diffusion. Mass transfer in fluid systems. 2nd edn (Cambridge University Press, Cambridge. 1997). 

Cussler, E. L. (1984). "Diffusion-Mass Transfer in Fluid Systems." Cambridge University Press, Cambridge. 

Cussler, EL, Diffusion, Mass Transfer in Fluid Systems Cambridge University Press Cambridge, UK, 1984. 

In order to verify that the fixed bed and the micro-channel reactor are equivalent concerning chemical conversion, an irreversible first-order reaction A —) B with kinetic constant was considered. For simplicity, the reaction was assumed to occur at the channel surface or at the surface of the catalyst pellets, respectively. Diffusive mass transfer to the surface of the catalyst pellets was described by a correlation given by Villermaux [115]. 

Diffusion Mass transfer driven by a gradient in concentration. 

Osmotic pressure A driving force for convective and diffusive mass transfer that is related to solute concentration. 

EL Cussler. Diffusion Mass Transfer in Fluid Systems. 2nd ed. New York Cambridge Univ Press, 1997, pp 371-389. 

In a typical pulse experiment, a pulse of known size, shape and composition is introduced to a reactor, preferably one with a simple flow pattern, either plug flow or well mixed. The response to the perturbation is then measured behind the reactor. A thermal conductivity detector can be used to compare the shape of the peaks before and after the reactor. This is usually done in the case of non-reacting systems, and moment analysis of the response curve can give information on diffusivities, mass transfer coefficients and adsorption constants. The typical pulse experiment in a reacting system traditionally uses GC analysis by leading the effluent from the reactor directly into a gas chromatographic column. This method yields conversions and selectivities for the total pulse, the time coordinate is lost. 

This simplified description of molecular transfer of hydrogen from the gas phase into the bulk of the liquid phase will be used extensively to describe the coupling of mass transfer with the catalytic reaction. Beside the Henry coefficient (which will be described in Section 45.2.2.2 and is a thermodynamic constant independent of the reactor used), the key parameters governing the mass transfer process are the mass transfer coefficient kL and the specific contact area a. Correlations used for the estimation of these parameters or their product (i.e., the volumetric mass transfer coefficient kLo) will be presented in Section 45.3 on industrial reactors and scale-up issues. Note that the reciprocal of the latter coefficient has a dimension of time and is the characteristic time for the diffusion mass transfer process tdifl-GL=l/kLa (s). 

In general, the intrinsic kinetics, the diffusion, mass transfer and Henry coefficients are either known or can be estimated, while the Hatta number can be determined. This is the first step in assessing the working regime of the reactor. 

We have discussed the positive effects that bacterial displacement and adhesion to substrates exerts on the diffusive mass transfer. Theoretically, direct contact with the substrate could also allow microorganisms to employ other modes of uptake in addition to absorption of water-dissolved molecules. So, which are the physical states for which chemicals can be ingested by microorganisms ... 

Thus, the whole area of fillings for the large lattices considered is concentrated into a narrow range of threshold values of ZB, and ZBpercolation processes, since the sites bound to them are filled through wider bonds. One should expect that nonpercolation bonds have a relatively small effect on diffusion mass transfer that extends from the external surface of a lattice to its bulk. 

The PO mode is a specific elution condition in HPLC enantiomer separation, which has received remarkable popularity especially for macrocyclic antibiotics CSPs and cyclodextrin-based CSPs. It is also applicable and often preferred over RP and NP modes for the separation of chiral acids on the cinchonan carbamate-type CSPs. The beneficial characteristics of the PO mode may arise from (i) the offset of nonspecific hydrophobic interactions, (ii) the faster elution speed, (iii) sometimes enhanced enan-tioselectivities, (iv) favorable peak shapes due to improved diffusive mass transfer in the intraparticulate pores, and last but not least, (v) less stress to the column, which may extend the column lifetime. Hence, it is rational to start separation attempts with such elution conditions. Typical eluents are composed of methanol, acetonitrile (ACN), or methanol-acetonitrile mixtures and to account for the ion-exchange retention mechanism the addition of a competitor acid that acts also as counterion (e.g., 0.5-2% glacial acetic acid or 0.1% formic acid) is required. A good choice for initial tests turned out to be a mobile phase being composed of methanol-glacial acetic acid-ammonium acetate (98 2 0.5 v/v/w). 

In this section, we focus on diffusive mass transfer. The mathematical description of mass transfer is similar to that of heat transfer. Furthermore, heat transfer may also play a role in heterogeneous reactions such as crystal growth and melting. Heat transfer, therefore, will be discussed together with mass transfer and examples may be taken from either mass transfer or heat transfer. 

If the rate is controlled by diffusive mass transfer (Figure 1-1 lb) and if other conditions are kept constant, then (i) the growth (or dissolution) distance is proportional to the square root of time, referred to as the parabolic growth law (an application of the famous square root law for diffusion), (ii) the concentration in the melt is not uniform, (iii) the concentration profile propagates into the melt according to square root of time, and (iv) the interface concentration is near saturation. For the rate to be controlled by diffusion in the fluid, it cannot be stirred. 

The physical transport of mass is essential to many kinetic and d3mamic processes. For example, bubble growth in magma or beer requires mass transfer to bring the gas components to the bubbles radiogenic Ar in a mineral can be lost due to diffusion pollutants in rivers are transported by river flow and diluted by eddy diffusion. Although fluid flow is also important or more important in mass transfer, in this book, we will not deal with fluid flow much because it is the realm of fluid dynamics, not of kinetics. We will focus on diffusive mass transfer, and discuss fluid flow only in relation to diffusion. 

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