Experimentation, effective diffusivity determination

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. 

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. 

These theories fostered a great deal of experimental research to determine the effect of temperature and pressure on the flame velocity and thus to verify which of the theories were correct. In the thermal theory, the higher the ambient temperature, the higher is the final temperature and therefore the faster is the reaction rate and flame velocity. Similarly, in the diffusion theory, the higher the temperature, the greater is the dissociation, the greater is the concentration of radicals to diffuse back, and therefore the faster is the velocity. Consequently, data obtained from temperature and pressure effects did not give conclusive results. 

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... 

In the cases above, a two-parameter model well represents the data. A model with more parameters would be more flexible, but by using a partition constant, K, or a desorption rate constant ka and k, , for the mass-transfer coefficients, the data are well described (see Figs. 3.4-15 and 3.4-13). While K would be a value experimentally determined, kp can be estimated from eqn. (3.4-97) with the external mass-transfer coefficient, km, estimated from the correlation of Stiiber et al. [25] or from that of Tan et al. [27], and the effective diffusivity from the Wakao Smith model [36], Typical values of kp obtained by fitting the data of Tan and Liou are shown in Fig. 3.4-16. As expected, they are below the usual mass-transfer correlations, because internal resistance diminishes the global mass transfer coefficient. These data correspond to the regeneration of spent activated carbon loaded with ethyl acetate, using high-pressure carbon dioxide, published by Tan and Liou [45]. 

The influence of a cut-off relative to the full treatment of electrostatic interactions by Ewald summation on various water parameters has been investigated by Feller et al. [33], These authors performed simulations of pure water and water-DPPC bilayers and also compared the effect of different truncation methods. In the simulations with Ewald summation, the water polarization profiles were in excellent agreement with experimental values from determinations of the hydration force, while they were significantly higher when a cut-off was employed. In addition, the calculated electrostatic potential profile across the bilayer was in much better agreement with experimental values in case of infinite cut-off. However, the values of surface tension and diffusion coefficient of pure water deviated from experiment in the simulations with Ewald summation, pointing out the necessity to reparameterize the water model for use with Ewald summation. 

The literature data on the tortuosity factor r show a large spread, with values ranging from 1.5 to 11. Model predictions lead to values of 1/e s (8), of 2 (parallel-path pore model)(9), of 3 (parallel-cross-linked pore model)(IQ), or 4 as recently calculated by Beeckman and Froment (11) for a random pore model. Therefore, it was decided to determine r experimentally through the measurement of the effective diffusivity by means of a dynamic gas chromatographic technique using a column of 163.5 cm length,... 

Experimentally determined effective transport properties of porous bodies, e.g., effective diffusivity and permeability, can be compared with the respective effective transport properties of reconstructed porous media. Such a comparison was found to be satisfactory in the case of sandstones or other materials with relatively narrow pore size distribution (Bekri et al., 1995 Liang et al., 2000b Yeong and Torquato, 1998b). Critical verification studies of effective transport properties estimated by the concept of reconstructed porous media for porous catalysts with a broad pore size distribution and similar materials are scarce (Mourzenko et al., 2001). Let us employ the sample of the porous... 

Fio. 16. Effective diffusivity of the examined porous sample G1 calculated for (i) Fick s diffusion both in macro- and nano-pores, and (ii) Fick s diffusion in macro-pores only. The experimental value of ij/ = Dcii/D = 0.199 was determined in Grahams diffusion cell. Only grains larger than 10 pm were used in the reconstruction of macro-porous media (from Salejova et al., 2004). 

The extraction of toluene and 1,2 dichlorobenzene from shallow packed beds of porous particles was studied both experimentally and theoretically at various operating conditions. Mathematical extraction models, based on the shrinking core concept, were developed for three different particle geometries. These models contain three adjustable parameters an effective diffusivity, a volumetric fluid-to-particle mass transfer coefficient, and an equilibrium solubility or partition coefficient. K as well as Kq were first determined from initial extraction rates. Then, by fitting experimental extraction data, values of the effective diffusivity were obtained. Model predictions compare well with experimental data and the respective value of the tortuosity factor around 2.5 is in excellent agreement with related literature data. 

The prediction of the parameter values for mass transport through porous materials is too difficult, because we do not know how to take into account the complicated pore geometry as it is in reality. Thus, data for the effective diffusivity or effective molecular and Knudsen diffusivities and the permeability are still more accurately determined experimentally. Experiments of this kind also provide valuable information on the porous structure, such as the average pore size and pore size distribution. 

Several experimental techniques have been developed for the investigation of the mass transport in porous catalysts. Most of them have been employed to determine the effective diffusivities in binary gas mixtures and at isothermal conditions. In some investigations, the experimental data are treated with the more refined dusty gas model (DGM) and its modifications. The diffusion cell and gas chromatographic methods are the most widely used when investigating mass transport in porous catalysts and for the measurement of the effective diffusivities. These methods, with examples of their application in simple situations, are briefly outlined in the following discussion. A review on the methods for experimental evaluation of the effective diffusivity by Haynes [1] and a comprehensive description of the diffusion cell method by Park and Do [2] contain many useful details and additional information. 

The available transport models are not reliable enough for porous material with a complex pore structure and broad pore size distribution. As a result the values of the model par ameters may depend on the operating conditions. Many authors believe that the value of the effective diffusivity D, as determined in a Wicke-Kallenbach steady-state experiment, need not be equal to the value which characterizes the diffusive flux under reaction conditions. It is generally assumed that transient experiments provide more relevant data. One of the arguments is that dead-end pores, which do not influence steady state transport but which contribute under reaction conditions, are accounted for in dynamic experiments. Experimental data confirming or rejecting this opinion are scarce and contradictory [2]. Nevertheless, transient experiments provide important supplementary information and they are definitely required for bidisperse porous material where diffusion in micro- and macropores is described separately with different effective diffusivities. 

A variety of diffusion cells have been developed for transient measurements. The experimental arrangements and the data analysis methods for the determination of effective diffusivities are described in reviews [1,2] and in the papers cited therein. Interesting applications of a diffusion cell with one compartment closed have been described recently for the investigation of the dynamics of ternary gas mixtures [15,16]. In the last papers the DGM and the mean transport pore model have been used to describe the experiments [17]. 

Equation 5.22 shows that Q is determined by an equation describing the steady flow of A through the pellet with a time lag td. To determine td one may plot any quantity proportional to the amount of diffused gas Q, rather than this quantity itself, versus time. The intercept on the t axis will give td and = L2 p/(6td). Note that with this experimental procedure the effective diffusivity can also be found from Equation 5.23, if the steady-state flux is measured. 

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