Diffusion coefficients resistances

Note that the diffusion coefficient for a polymer through an environment of low molecular weight molecules is typically on the order of magnitude of 10"" m" sec". If the first subscript indicates the diffusing species, and the second the surrounding molecules, and P stands for polymer and S for small molecules, we see that the order of diffusion coefficients is Ds g > Dp g > Dp P sequence which makes sense in terms of relative frictional resistance. 

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. 

A further problem is possible if the reinforcements are very small. Coarsening of the particles or whiskers may occur driven by Ostwald ripening, in which large particles grow through diffusional transport at the expense of smaller ones. This can be minimized by choosing matrices in which the reinforcement elements have very low solid solubilities and diffusion coefficients. Platelets, however, have been shown to be more resistant to coarsening than particles or whiskers. 

Diffusivity and tortuosity affect resistance to diffusion caused by collision with other molecules (bulk diffusion) or by collision with the walls of the pore (Knudsen diffusion). Actual diffusivity in common porous catalysts is intermediate between the two types. Measurements and correlations of diffusivities of both types are Known. Diffusion is expressed per unit cross section and unit thickness of the pellet. Diffusion rate through the pellet then depends on the porosity d and a tortuosity faclor 1 that accounts for increased resistance of crooked and varied-diameter pores. Effective diffusion coefficient is D ff = Empirical porosities range from 0.3 to 0.7, tortuosities from 2 to 7. In the absence of other information, Satterfield Heterogeneous Catalysis in Practice, McGraw-HiU, 1991) recommends taking d = 0.5 and T = 4. In this area, clearly, precision is not a feature. 

A microelectrode has been used by Uchida et al. to study lithium deposition in order to minimize the effect of solution resistance [41], They used a Pt electrode (10-30 jum in diameter) to measure the lithium-ion diffusion coefficient in 1 mol L 1 LiC104/PC electrolyte. The diffusion coefficient was 4.7 x 10-6 cm2 s at 25 °C. 

Archie [23] examined electrical resistivity of various sand formations having pore spaces filled with saline solutions of different salt concentrations. Based upon his own experimental results, he obtained a simple relationship for the conductivity of beds of sand (assuming the sand itself is nonconductive) containing saline solution in terms of the porosity. In terms of diffusion coefficients his expression is... 

There are several correlations for estimating the film mass transfer coefficient, kf, in a batch system. In this work, we estimated kf from the initial concentration decay curve when the diffusion resistance does not prevail [3]. The value of kf obtained firom the initial concentration decay curve is given in Table 2. In this study, the pore diffusion coefficient. Dp, and surface diffusion coefficient, are estimated by pore diffusion model (PDM) and surface diffusion model (SDM) [4], The estimated values of kf. Dp, and A for the phenoxyacetic acids are listed in Table 2. 

Second, hydration of drug molecules may affect their hydrodynamic radii. The diffusion coefficient D is related to the frictional resistance,/, that the diffusing particle experiences in moving through a medium by the equation [51]... 

To summarize, the hydration status of the drug molecule and other components of a pharmaceutical formulation can affect mass transport. Solubility of drug crystals in an aqueous or nonaqueous solvent may depend on the presence or absence of moisture associated with the drug. Hydration may also determine the hydrodynamic radii of molecules. This may affect the frictional resistance and therefore the diffusion coefficient of the drug molecules. Diffusion of drugs in polymeric systems may also be influenced by the percent hydration of the polymers. This is especially tme for hydrogel polymers. Finally, hydration of... 

The Poppe plot is a log-log plot of H/uq = t(JN versus the number of plates with different particle sizes and with lines drawn at constant void time, t(). H is the plate height, Vis the number of plates, and u() is the fluid velocity (assumed equal to the void velocity). The quantity H/u() is called the plate time, which is the time for a theoretical plate to develop and is indicative of the speed of the separation, with units of seconds. In the Poppe plot, a number of parameters including the maximum allowable pressure drop, particle diameter, viscosity, flow resistance, and diffusion coefficient are held constant. 

FIGURE 6.1 A Poppe plot for the required plate number in conventional HPLC. The parameters are taken from Poppe s original paper (Poppe, 1997). The parameters are maximum pressure AP = 4x 107 Pa, viscosity / = 0.001 Pa/s, flow resistance factor

diffusion coefficient D= lx 1CT9 m2/s, and reduced plate height parameters using Knox s plate height model are A — 1, B— 1.5, C = 0.05. 

Resistance to transfer of material by diffusion is caused by collisions with other molecules and with the walls of narrow passages. The corresponding diffusion coefficients are termed molecular diffusivity Dm and Knudsen... 

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