Experimentation, effective diffusivity

The parameter / is the characteristic length for a unit cell, E0 is the surface concentration of a carrier protein molecule, and a2, a3, a4, a5 are the reaction rate parameters analogous to that half saturation constants. Table 11.3 displays the experimental effective diffusion coefficients and the volume fraction of intracellular phase A. In the first four sets... 

The experimental effective diffusivity factor should normally approach the value of 2.25 calculated by Kronig and Brink (K5) at low Reynolds numbers. For drops with turbulent circulation described by Handles and Baron s model, R is given (J2) by... 

In practice, lower values than those calculated according to Eq. (30) are found for intermediate Reynolds numbers, only approaching them in the presence of oscillation. A compilation of experimental R factors is available (Cl). The experimental effective diffusivity in gas absorption with internal circulation has been plotted against the axial velocity representing the flow adjacent to the interface (G7), and shows a direct relation between circulation and internal resistance. Similar results are reported (C6) for other gas-liquid and liquid-liquid systems. 

The mass transport influence is easy to diagnose experimentally. One measures the rate at various values of the Thiele modulus the modulus is easily changed by variation of R, the particle size. Cmshing and sieving the particles provide catalyst samples for the experiments. If the rate is independent of the particle size, the effectiveness factor is unity for all of them. If the rate is inversely proportional to particle size, the effectiveness factor is less than unity and

experimental points allow triangulation on the curve of Figure 10 and estimation of Tj and ( ). It is also possible to estimate the effective diffusion coefficient and thereby to estimate Tj and ( ) from a single measurement of the rate (48). 

Diffusion within the largest cavities of a porous medium is assumed to be similar to ordinary or bulk diffusion except that it is hindered by the pore walls (see Eq. 5-236). The tortuosity T that expresses this hindrance has been estimated from geometric arguments. Unfortunately, measured values are often an order of magnitude greater than those estimates. Thus, the effective diffusivity D f (and hence t) is normally determined by comparing a diffusion model to experimental measurements. The normal range of tortuosities for sihca gel, alumina, and other porous solids is 2 < T < 6, but for activated carbon, 5 < T < 65. 

For the effective diffusivity in pores, De = (0/t)D, the void fraction 0 can be measured by a static method to be between 0.2 and 0.7 (Satterfield 1970). The tortuosity factor is more difficult to measure and its value is usually between 3 and 8. Although a preliminary estimate for pore diffusion limitations is always worthwhile, the final check must be made experimentally. Major results of the mathematical treatment involved in pore diffusion limitations with reaction is briefly reviewed next. 

The diffusional transport model for systems in which sorbed molecules can be divided in two populations, one formed by completely immobilized molecules and the other by molecules free to diffuse, has been developed by Vieth and Sladek 33) in a modified form of the Fick s second law. However, if linear isotherms are experimentally found, as in the case of the DGEBA-TETA system in Fig. 4, the diffusion of the penetrant may be described by the classical diffusion law with constant value of the effective diffusion coefficient,... 

It depends only on J sJkj A, which is a dimensionless group known as the Thiele modulus. The Thiele modulus can be measured experimentally by comparing actual rates to intrinsic rates. It can also be predicted from first principles given an estimate of the pore length =2 . Note that the pore radius does not enter the calculations (although the effective diffusivity will be affected by the pore radius when dpore is less than about 100 run). 

In accordance with Pick s Law, diffusive flow always occurs in the direction of decreasing concentration and at a rate, which is proportional to the magnitude of the concentration gradient. Under true conditions of molecular diffusion, the constant of proportionality is equal to the molecular diffusivity of the component i in the system, D, (m /s). For other cases, such as diffusion in porous matrices and for turbulent diffusion applications, an effective diffusivity value is used, which must be determined experimentally. 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

Illustration 12.3 indicates the use of the effective diffusivity approach for estimating catalyst effectiveness factors when this parameter is determined experimentally or may be estimated. 

The measured value of k Sg is 0.716 cm3/(sec-g catalyst) and the ratio of this value to k ltTueSg should be equal to our assumed value for the effectiveness factor, if our assumption was correct. The actual ratio is 0.175, which is at variance with the assumed value. Hence we pick a new value of rj and repeat the procedure until agreement is obtained. This iterative approach produces an effectiveness factor of 0.238, which corresponds to a differs from the experimental value (0.17) and that calculated by the cylindrical pore model (0.61). In the above calculations, an experimental value of eff was not available and this circumstance is largely responsible for the discrepancy. If the combined diffusivity determined in Illustration 12.1 is converted to an effective diffusivity using equation 12.2.9, the value used above corresponds to a tortuosity factor of 2.6. If we had employed Q)c from Illustration 12.1 and a tortuosity factor of unity to calculate eff, we would have determined that rj = 0.65, which is consistent with the value obtained from the straight cylindrical pore model in Illustration 12.2. 

When the hydrogen pressure is 1 atm, and the temperature is 77 °K, the experimentally observed (apparent) rate constant is 0.159 cm3/ sec-g catalyst. Determine the mean pore radius, the effective diffusivity of hydrogen, and the catalyst effectiveness factor. 

Gas diffusion in the nano-porous hydrophobic material under partial pressure gradient and at constant total pressure is theoretically and experimentally investigated. The dusty-gas model is used in which the porous media is presented as a system of hard spherical particles, uniformly distributed in the space. These particles are accepted as gas molecules with infinitely big mass. In the case of gas transport of two-component gas mixture (i = 1,2) the effective diffusion coefficient (Dj)eff of each of the... 

The effective diffusivity De is a characteristic of the particle that must be measured for greatest accuracy. However, in the absence of experimental data, De may be estimated in terms of molecular diffusivity, Dab (for diffusion of A in the binary system A + B), Knudsen diffusivity, DK, particle voidage, p, and a measure of the pore structure called the particle tortuosity, Tp. 

The signal attenuation in the diffusion-weighted spectra does not depend on the binding mode and position but only on the effective diffusion coefficient and on the experimental parameters mentioned above. 

The solute concentration Wg from each run has been expressed as a step function of the distance z, from which (-Wg/Wo) is calculated. The values of all the experimental parameters, D/uL, Dg/uL, Lu, /LD, Cpyti/k and (h f g/yM. ) ( AT) etc. are calculated from the values of the related physical properties in the literature (19). In Figure 3, g is correlated with z/L for all the thirteen runs, for which the values of effective diffusivity Dg in the melted zone have been predicted from the Kraussold correlation (7) using the experimental values of PrGr number and the values of the parameter Pe calculated show that little improvement has been made by using Pe instead of P in the correlation. 

The critical input parameters are then (1) the grain size, which should be known for each case, (2) the Aci temperature which is calculated from thermodynamics, (3) the effective diffusion activation energy, Qea, and (4) the empirical constants aj for each element. Qea and aj were determined by empirically fitting curves derived using Eq. (11.12) to experimentally observed TTT curves, and the final formula for calculating r was given as... 

Laboratory data collected over honeycomb catalyst samples of various lengths and under a variety of experimental conditions were described satisfactorily by the model on a purely predictive basis. Indeed, the effective diffusivities of NO and NH3 were estimated from the pore size distribution measurements and the intrinsic rate parameters were obtained from independent kinetic data collected over the same catalyst ground to very fine particles [27], so that the model did not include any adaptive parameters. 

Very much more is known about the theory of concentration gradients at electrodes than has been mentioned in this brief account. Experimental methods for observing them have also been devised, based on the dependence of refractive index on concentration (the Schlieren method) by means of interferometry (O Brien, 1986). Nevertheless, the basic concept of an effective diffusion-layer thickness, treated here as varying in thickness with fi until the onset of natural convection and as constant with time after convection sets in (though decreasing in value with the degree of disturbance, Table 7.10), is a useful aid to the simple and approximate analysis of many transport-controlled electrodic situations. A few of the uses of the concept of 8 will now be outlined. 

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