Mass transfer interception rates

Thus, at potentials in the range where mass transfer and electrokinetic effects are significant (region 2 of Figure 2.91), a plot of l//c vs. l/w1 2 should be linear, D0 available from the slope and available from the intercept. As a result of the exponential dependence of the rate constant on potential, the actual value of the intercept is strongly dependent on potential. 

Steady-state solutions for diffusion/convection When the air is moving, it becomes more difficult to calculate the interception rate. The mass transfer under these circumstances is generally expressed in dimensionless terms. Adam and Delbriick (1968) were able to generate a formula by making the simplifying assumption that the velocity of the air as it passed around the hair was everywhere constant (U) and very similar to the ambient air flow farther away (U0) ... 

The plot of In ( ) versus T 1 gives a straight line with a slope equal to —E/R and intercept the absolute rate constant kQ as shown in Fig. 7.6. The lowest slope represents reactions controlled by bulk mass transfer, and the largest is... 

Satterfield150 considers a special case of the above equation, in which the gas-phase resistance is neglected (i.e., the second term on the right-hand side of the above equation is zero) and the catalyst effectiveness factor is assumed to be unity. In this case, a series of measurements of AG/R for various catalyst loadings permits a plot of AG/R versus 1/m to be established. The intercept yields the gas-liquid mass-transfer coefficient and the slope yields a combination of the intrinsic rate constant and the liquid- solid mass-transfer coefficient. 

The major difficulty in the analysis of chromatographic data is separating the axial dispersion and mass-transfer contributions since, except for gaseous systems at very low flow rates, the axial dispersion coefficient (Dl) is velocity dependent. For liquid systems Dl varies essentially linearly with velocity so a plot of HETP vs. superficial velocity (ev) should be linear with the mass-transfer resistance directly related to the slope (Fig. 6). For gaseous systems at a high Reynolds number this same plot can be used, but in the low Reynolds number region a plot oiH/v vs. 1 /v may be more convenient since in this region Dl is essentially constant and the intercept thus yields the mass-transfer resistance [43-45]. 

By operating under conditions of high agitation where gas-liquid mass transfer effects are absent, line 1 is produced. The rate constant can be obtained from the slope and estimated value of /cgL- Alternatively, if the rate constant is known, the correct value of ksL for the system at hand can be extracted from the slope. Line 2 is produced at very low agitation, and /tgl may be obtained from the intercept. 

FIG. 22 The dependence of the bulk mass transfer rate on particle radius a for the rotating disk calculated numerically from Eq. (160) (T = 293 K, apparent densities of the particle Ap curve 1, 0.6 kg/dcm curve 2, 0.3 kg/dcm curve 3, 0 curve 4, —0.3 kg/dcm curve 5, —0.6 kg/dcm (the minus sign denotes the gravity force acting opposite the interface). The dashed line denotes the limiting value calculated from Eq. (162) (Stokes law), and the dashed-dotted line represents the results calculated from Eq. (161) (the interception governed flux). (From Ref 37.)... 

The significance of the results presented in Figs. 43 5 is that they allow one to estimate the area over the surface where the flux (transfer rate) become uniform. The mass-transfer-rate constant (reduced flux) measured in this uniformly accessible area and referred to as A is of a particular significance because it can be analyzed theoretically in a much more efficient way than the local flux. Moreover, A can be directly used for determining the significance of various transport mechanisms like diffusion, interception, specific or external force, and so forth. 

With these assumptions in hand, interpretation of real assay data involves plotting a model-derived value for concentration of NAD at the enzyme surface (NAD ). The value for the Mnad+ can be fitted to allow the Lineweaver-Burk plot to intercept the x-axis at a value that yields the value of Km as determined in solution. The value for Vmax is then read as the intercept at the y-axis (Figure 12.2). This approach permits derivation of a Vmax for the electrode that is independent of the effects of mass transfer. If one further assumes that the immobilization process does not affect the turnover rate of the immobilized enzyme (relative to its activity in solution), then this value of Vmax (which represents the total activity of all bound enzyme) can also be used to estimate the amount of immobilized enzyme. This model can be particularly useful when fabricating electrodes using immobilization techniques that entrap a fraction of enzyme from bulk solutions, such as direct physical absorption or co-immobilization within gels. 

To learn that when the rate of electron transfer is slow, a useful approach is to construct Koutecky-Levich plots of (/(current) against l/Tafel plot from these mass-transport-limiting values of the current. 

In this approach, processes represented in Equations 3a and 3e were identified by cmrve fitting a three-parameter equation to the pressure dependence over the range from a few Torr to 2 atm in several systems independently. The composition dependence for the competitive yields in Equations 3b, 3c, and 3d at a fixed pressure of 800 Torr were then determined experimentally by subtracting the noncompetitive contributions from the raw data. Extrapolation of the refined data to pure reactant and infinite dilution with bath gas provides intercepts whose ratio gives the desired relative rate constants. Correction for reduced mass and molecular size then provides relative energy transfer eflBciencies on an equal collision basis. 

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