Diffusion, coefficients controlled solution

Fig. 7. kf versus linear flow rate calculated for BSA adsorption to fluidized controlled pore glass after Rowe (A, Eq. 19, Ref. 74) and after Fan et al. ( , Eq. 20, Ref. 75). Physical data of adsorbent average particle diameter 200 pm, average particle density 1240 kg/m3, BSA diffusion coefficient in solution 7.3-10 11 m2/s, bed expansion calculated according to Richardson and Zaki, U, and n estimated according to Eqs. (3-5)... 

Water Transport. Two methods of measuring water-vapor transmission rates (WVTR) ate commonly used. The newer method uses a Permatran-W (Modem Controls, Inc.). In this method a film sample is clamped over a saturated salt solution, which generates the desired humidity. Dry air sweeps past the other side of the film and past an infrared detector, which measures the water concentration in the gas. For a caUbrated flow rate of air, the rate of water addition can be calculated from the observed concentration in the sweep gas. From the steady-state rate, the WVTR can be calculated. In principle, the diffusion coefficient could be deterrnined by the method outlined in the previous section. However, only the steady-state region of the response is serviceable. Many different salt solutions can be used to make measurements at selected humidity differences however, in practice,... 

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

Cussler studied diffusion in concentrated associating systems and has shown that, in associating systems, it is the size of diffusing clusters rather than diffusing solutes that controls diffusion. is a reference diffusion coefficient discussed hereafter is the activity of component A and iC is a constant. By assuming that could be predicted by Eq. (5-223) with P = 1, iC was found to be equal to 0.5 based on five binaiy systems and vahdated with a sixth binaiy mixture. The limitations of Eq. (5-225) using and K defined previously have not been explored, so caution is warranted. Gurkan showed that K shoiild actually be closer to 0.3 (rather than 0.5) and discussed the overall results. 

Fig. 2a-c. Kinetic zone diagram for the catalysis at redox modified electrodes a. The kinetic zones are characterized by capital letters R control by rate of mediation reaction, S control by rate of subtrate diffusion, E control by electron diffusion rate, combinations are mixed and borderline cases b. The kinetic parameters on the axes are given in the form of characteristic currents i, current due to exchange reaction, ig current due to electron diffusion, iji current due to substrate diffusion c. The signpost on the left indicates how a position in the diagram will move on changing experimental parameters c% bulk concentration of substrate c, Cq catalyst concentration in the film Dj, Dg diffusion coefficients of substrate and electrons k, rate constant of exchange reaction k distribution coefficient of substrate between film and solution d> film thickness (from ref. 

As reversible ion transfer reactions are diffusion controlled, the mass transport to the interface is given by Fick s second law, which may be directly integrated with the Nernst equation as a boundary condition (see, for instance. Ref. 230 232). A solution for the interfacial concentrations may be obtained, and the maximum forward peak may then be expressed as a function of the interfacial area A, of the potential scan rate v, of the bulk concentration of the ion under study Cj and of its diffusion coefficient D". This leads to the Randles Sevcik equation [233] ... 

The terms in Eq. (6) include the gravitational constant, g, the tube radius, R, the fluid viscosity, p, the solute concentration in the donor phase, C0, and the penetration depth, The density difference between the solution and solvent (ps - p0) is critical to the calculation of a. Thus, this method is dependent upon accurate measurement of density values and close temperature control, particularly when C0 represents a dilute solution. This method has been shown to be sensitive to different diffusion coefficients for various ionic species of citrate and phosphate [5], The variability of this method in terms of the coefficient of variation ranged from 19% for glycine to 2.9% for ortho-aminobenzoic acid. 

Fig. 8. Dependence of (A) corrected diffusion coefficient (D), (B) steady-state fluorescence intensity, and (C) corrected number of particles in the observation volume (N) of Alexa488-coupled IFABP with urea concentration. The diffusion coefficient and number of particles data shown here are corrected for the effect of viscosity and refractive indices of the urea solutions as described in text. For steady-state fluorescence data the protein was excited at 488 nm using a PTI Alphascan fluorometer (Photon Technology International, South Brunswick, New Jersey). Emission spectra at different urea concentrations were recorded between 500 and 600 nm. A baseline control containing only buffer was subtracted from each spectrum. The area of the corrected spectrum was then plotted against denaturant concentrations to obtain the unfolding transition of the protein. Urea data monitored by steady-state fluorescence were fitted to a simple two-state model. Other experimental conditions are the same as in Figure 6.

The Smoluchowski theory for diffusion-controlled reactions, when combined with the Stokes-Einstein equation for the diffusion coefficient, predicts that the rate constant for a diffusion-controlled reaction will be inversely proportional to the solution viscosity.16 Therefore, the literature values for the bimolecular electron transfer reactions (measured for a solution viscosity of r ) were adjusted by multiplying by the factor r 1/r 2 to obtain the adjusted value of the kinetic constant... 

The thermodynamic approach does not make explicit the effects of concentration at the membrane. A good deal of the analysis of concentration polarisation given for ultrafiltration also applies to reverse osmosis. The control of the boundary layer is just as important. The main effects of concentration polarisation in this case are, however, a reduced value of solvent permeation rate as a result of an increased osmotic pressure at the membrane surface given in equation 8.37, and a decrease in solute rejection given in equation 8.38. In many applications it is usual to pretreat feeds in order to remove colloidal material before reverse osmosis. The components which must then be retained by reverse osmosis have higher diffusion coefficients than those encountered in ultrafiltration. Hence, the polarisation modulus given in equation 8.14 is lower, and the concentration of solutes at the membrane seldom results in the formation of a gel. For the case of turbulent flow the Dittus-Boelter correlation may be used, as was the case for ultrafiltration giving a polarisation modulus of ... 

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